[{"data":1,"prerenderedAt":197662},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F0010-biography-of-claude-shannon":3,"i-ci:tag":-1,"i-mingcute:time-line":-1},{"post":4,"nextPost":354,"prevPost":133089},{"id":5,"title":6,"body":7,"description":13,"draft":341,"enableComment":342,"extension":343,"image":330,"important":341,"location":344,"meta":345,"navigation":342,"ogImage":344,"path":346,"seo":347,"stem":348,"summary":349,"tags":350,"time":352,"__hash__":353},"blog\u002Fblog\u002F2026\u002F0010-biography-of-claude-shannon.md","克劳德·香农小传",{"type":8,"value":9,"toc":329},"minimark",[10,14,17,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,76,79,82,85,88,91,94,97,100,103,106,109,112,115,118,121,124,127,130,133,136,139,142,148,151,154,157,160,163,166,169,172,175,179,182,185,188,191,194,197,200,203,206,209,212,215,218,221,226,229,232,236,242,245,248,251,254,257,260,263,266,269,272,275,279,285,288,291,294,297,300,303,306,309,312,315,318,321,324],[11,12,13],"p",{},"美国 20 世纪 50 年代的某个深夜，贝尔实验室的走廊里很安静。一个瘦高的中年人骑着独轮车，从长长的走廊那头慢悠悠地滑过来，手里还抛着几个球。偶尔有加班的同事推开门，看一眼，笑一笑，又把头缩回去继续算他们的方程——大家早就习惯了。",[11,15,16],{},"克劳德·艾尔伍德·香农（Claude Elwood Shannon）最有分量的成就，是用一篇论文给「信息」下了数学定义，由此奠定了整个数字时代的理论根基。可是如果你问他的同事、家人和朋友他是个什么样的人，他们说的多半不是公式，而是独轮车、玩具、小机器和一个永远带着笑意的安静身影。",[18,19,20],"h2",{"id":20},"密歇根小镇的修理少年",[11,22,23],{},"1916 年 4 月 30 日，香农出生在美国密歇根州的佩托斯基，不久后全家搬到北边一个叫盖洛德的小镇，他在那里长大。父亲老克劳德做过商人，也当过遗嘱认证法官；母亲梅布尔是语言教师，后来做了盖洛德中学的校长。姐姐凯瑟琳比他大几岁，日后成了数学教授。家里的气氛是安稳、务实的：父亲有生意头脑，母亲爱读书，孩子们被允许把时间花在自己感兴趣的东西上。",[11,25,26],{},"香农从小就是个「动手的孩子」。他爱做航模，也做过无线电遥控的小船。最著名的一桩少年轶事，是他和一个住在大约半英里外的朋友之间架设的电报线：两个孩子没有现成的电线，就利用沿途的铁丝围栏，把它们接起来当作通信线路，晚上各自守在家里，用莫尔斯电码互相「发报」。对一个中西部小镇的男孩来说，这既是游戏，也是他第一次亲手搭起「通信系统」。",[11,28,29],{},"他还干过一份差事：给西联电报公司送电报。镇上的人看见他骑着车来来往往，也许想不到，这个孩子日后会为「通信」写下一套完整的数学理论。他修收音机的本事在镇上小有名气，据说当地一家百货商店的收音机坏了，也会请他去看看。",[11,31,32],{},"这里得补一点背景：香农的少年时代，正好赶上 1920 年代美国的无线电热。广播电台一座座冒出来，杂志上登着教人自己绕线圈、装矿石收音机的图纸，成千上万个男孩在自家阁楼上折腾「无线电」。香农赶上了这个浪潮的浪尖，而且他不是照着图纸装，是喜欢把东西拆开，弄清里面到底怎么回事。回头看，他一生的路数大概就是这个：感兴趣的不是「怎么用」，而是「它是怎么被装起来的」。",[11,34,35],{},"他的偶像是爱迪生。后来他发现自己和爱迪生居然是远亲，高兴了好一阵子。少年时代他还读到爱伦·坡的小说《金甲虫》，书里主人公靠破译密码找到宝藏，这个故事在他心里种下了对密码的迷恋，多年以后，战时的密码工作和他的信息论，都与这份兴趣有着隐秘的联系。另外，坡还写过一篇《被窃的信》，讲的是「猜对方下一手会怎么出」的心理游戏——二十多年后，香农在一份机器设计的备忘录里，又把这篇小说翻出来引用了一次。",[11,37,38],{},"这个阶段的香农，已经显出日后的底色：不爱张扬，不太合群，但遇到「有意思的东西」就会安静地一头扎进去，一玩就是几个小时。",[18,40,41],{"id":41},"一门选修课和一篇论文",[11,43,44],{},"1932 年，香农进入密歇根大学。他最后同时拿到了电气工程和数学两个学士学位，这个组合在当时并不常见，却像是为他日后的工作预先搭好的脚手架。",[11,46,47],{},"大学里有一件看起来很小的事，影响却极大：他在一门课上接触到了乔治·布尔的逻辑代数——一套只用「真」和「假」、「是」和「非」来做运算的符号系统。大多数同学把它当成一个哲学趣闻，香农却盯着它看了很久，因为他同时懂得电路：开关只有「开」和「关」两种状态，这不正好对应「真」和「假」吗？",[11,49,50],{},"毕业后，他去了麻省理工学院，做万尼瓦尔·布什的助手，管理一台著名的庞然大物——微分分析机。这是一台由轴、齿轮和大量继电器组成的机械模拟计算机，占满一间屋子。香农在它旁边一待就是很长时间，对着一堆纠缠的继电器电路琢磨：能不能找到一套规则，让人不靠反复试错就能设计出正确的开关电路？",[11,52,53],{},"1937 年，他交出了答案——硕士论文《继电器与开关电路的符号分析》。他证明了布尔代数可以用来描述和简化开关电路，也就是说，逻辑可以被「造」出来。",[11,55,56],{},"这篇论文在评价史上的位置，后来被反复提起。历史学家赫尔曼·戈德斯坦 1972 年写下了可能是最实在的一句：它「把数字电路设计从一门手艺变成了一门科学」。数学家所罗门·戈隆布则在 2001 年的悼文里说，这是「有史以来最伟大的硕士论文之一」。流传最广的一版出自哈佛大学的认知心理学家霍华德·加德纳，1985 年他写道，这或许「是这个世纪最重要、也最著名的硕士论文」。这句话后来被到处引用，有时还被人升格成「科学史家的定评」——其实加德纳是心理学家，不是科学史家，这一点值得记一笔。",[11,58,59],{},"还有一个评价来自他的老师本人。布什后来谈到这篇论文时说：「我当时就认为它是一篇经典之作——这个评价我很少给出。」",[11,61,62],{},"比论文本身更能说明问题的，是布什给人写推荐信时的措辞。1938 年，他在一封信里这样描述这个年轻人：「一个相当不合常规的年轻人……他非常害羞内向，极为谦逊，很容易被带偏。」在另一封推荐信里，布什又补了一句：「他腼腆、讨人喜欢，是需要被小心对待的那种人。」一位导师要一边极力推荐、一边提醒别人别把他吓跑，这件事本身就说明了香农是个什么样的人。",[11,64,65],{},"当时的他才二十出头，说起那篇论文却相当轻描淡写，更像是把两样好玩的东西拼到了一起。",[11,67,68],{},"其间他还在贝尔实验室度过了一个夏天，那是他与这家公司缘分的开始。布什后来建议他换个领域，于是他的博士论文竟是关于理论遗传学的，题目叫《理论遗传学的代数学》。为此他去了冷泉港，和遗传学家一起工作。他用数学处理孟德尔遗传，这篇论文并没有像硕士论文那样引起轰动，却再次显示了他的习惯：哪里有「结构」，就想用代数去看一看。",[11,70,71],{},[72,73],"img",{"alt":74,"src":75},"Claude Shannon in 1932","https:\u002F\u002Fimage-assets.dreams.plus\u002F202610162230267.png",[11,77,78],{},"1940 年，他拿到了博士学位。同一年，他与诺玛·莱沃结婚，但这段婚姻很短，两人很快就分开了。香农在感情上一向内敛，很少对外人谈起这些。",[18,80,81],{"id":81},"普林斯顿的拜访与贝尔实验室的夜晚",[11,83,84],{},"博士毕业后，香农去了普林斯顿高等研究院，在那里的几个月里，他见到了爱因斯坦、哥德尔和冯·诺依曼等人。普林斯顿的气氛很学院，学者们常在林荫道上散步、低声争论。香农既敬佩他们，又觉得自己并不真正属于这里：他更愿意在一间堆满零件的屋子里琢磨东西。",[11,86,87],{},"倒是有两位年长的数学家早早看出了他的分量。诺伯特·维纳在 1940 年前后这样评价这个年轻人：「一个才华与智力都极为出众的人……他已经做出了极具原创性的工作，毫无疑问是正在升起的一颗星。」而冯·诺依曼后来跟他的一次闲谈，给这门学科留下了一个沿用至今的词——这件事我们放到 1948 年再说。",[11,89,90],{},"1941 年，他加入了贝尔实验室，那个年代的贝尔实验室是一座奇特的「自由王国」，管理层相信让聪明人自己找事做，终会有收获。香农正好适合这样的环境。战争期间，他参与了防空火力控制系统的研究，也接触了保密通信和密码学方面的工作，其中包括著名的 SIGSALY 加密话音系统的相关研究。",[11,92,93],{},"他在贝尔实验室交到了几个真正投契的朋友，其中两个往后会反复出现。",[11,95,96],{},"一位是约翰·皮尔斯，日后被称作「通信卫星之父」的工程师。皮尔斯差不多是香农在实验室里最好的思想对手，两个人互相扔点子、合写论文、交换成堆的书。皮尔斯留下的说法里有一条后来被引用得最多：香农谈起自己的想法时，常常是从听别人抱怨某个系统开始的——「他漫不经心地听着，然后发明出一个新系统。」皮尔斯三番五次跟他说「这个想法你应该写出来」，香农的回答通常是：「‘应该’是什么意思？」",[11,98,99],{},"另一位是巴尼·奥利弗，后来去了惠普，也是「搜寻地外文明」计划最早的推动者之一。奥利弗与香农、皮尔斯三人合写过那篇《脉码调制的哲学》，还一起申请了专利。奥利弗晚年这样回忆自己在这个圈子里的位置：「我们成了朋友，所以很多理论可以说是我替他接生的。他把想法扔给我……我比谁都更早知道信息论。」",[11,101,102],{},"他的生活节奏在同事眼里有些古怪：不太爱开会，常常关上办公室的门独自思考，下班时间也不固定。他当时住在纽约格林威治村，每天要往返于曼哈顿与实验室之间。据说他爱在深夜工作，走廊里的独轮车也是从这个时期开始出名的。他一边骑，一边思考难题，对他而言，玩耍和思考本是同一件事的两面。",[11,104,105],{},"1943 年初，英国数学家艾伦·图灵访问贝尔实验室。两人在食堂里一起喝茶聊天，由于图灵的密码工作属于机密，他们没法谈具体项目，聊的主要是更宽泛的话题：机器能不能思考？能不能教机器学东西？",[11,107,108],{},"图灵后来在饭桌上向人复述过他印象深刻的一点：「香农不只想往大脑里喂数据，他还想喂文化层面的东西！他想给它放音乐。」（这句话是旁人的转述，见于安德鲁·霍奇斯的图灵传记，不是两人当时的通信。）图灵自己则傲慢得很可爱，他说他并不想造一个「强大的大脑」：「我想要的只是一个平庸的大脑，差不多像美国电话电报公司总裁那样的。」",[11,110,111],{},"有意思的是，两个人对这次交谈的记忆并不一样。近四十年后，香农接受口述史访问时回忆说，他们那次「完全没谈密码」，「谈得更多的是人脑和计算机器」这类问题；至于他自己关于信息的那套想法，图灵当时「并不认为方向是对的」，他「得到的差不多是负面的反馈」。一次被后人追认为「两个天才难得交会」的谈话，在当事人那里，其实还带着一点被质疑的味道。",[11,113,114],{},"这是两个隐居于各自世界里的天才难得的一次交会：一个刚从布莱切利园的机密里走出来，一个正准备写下通信的数学理论，而他们在饭桌上真正关心的，是造一个会想东西的机器。",[11,116,117],{},"在这些年里，香农一边为战争做事，一边在心里慢慢酝酿一个更大的问题：通信的本质到底是什么？在 1945 年的一份内部保密备忘录里，他已经用概率论的语言分析密码系统，战后这份工作以《保密系统的通信理论》为题公开发表。密码学与通信理论，在他脑中是同一枚硬币的两面。",[18,119,120],{"id":120},"一篇论文与一个安静的家",[11,122,123],{},"1948 年，《贝尔系统技术杂志》分两期连载了香农的论文《通信的数学理论》。这篇论文提出了一整套全新的概念：用「比特」衡量信息量；把信息源的不确定性定义为「熵」；证明了在噪声信道中，只要传输速率低于某个「信道容量」，就存在办法让错误率任意小。",[11,125,126],{},"「比特」这个词是同事约翰·图基提出来的，意思是「二进制数字」。图基是普林斯顿的统计学家，战争期间常驻贝尔实验室，据一份 1947 年 1 月的实验室备忘录，这个说法最迟在那时已经定下来了。香农在论文里最早把「bit」用作信息的单位，并且明确注明「此词由 J. W. 图基建议」。据说命名是在实验室的午餐桌上敲定的，有人提议叫「binit」，有人提议叫「bigit」，最后「bit」胜出。一个今天人人都在用的单位，诞生过程就是这么随便。",[11,128,129],{},"关于「熵」这个名字，有一个流传很广的故事：据说冯·诺依曼建议他用这个词，理由是反正没有人真正懂熵，辩论的时候他总能占上风。这个故事细节不一，可信度看起来也难以考证。",[11,131,132],{},"但值得补充的是，这个故事最早的出处其实是香农自己。1971 年，他在《科学美国人》上的一篇文章里回忆道：「我最头疼的是该叫它什么。我想过叫‘信息’，但这个词被用得太滥，于是决定叫‘不确定性’。后来我跟约翰·冯·诺依曼讨论，他有更好的主意。他对我说：‘你就叫它熵吧，有两个理由。第一，你的不确定性函数在统计力学里已经用这个名字了，它本来就有名字。第二，而且更重要：没有人真正知道熵到底是什么，所以辩论的时候你总是占上风。’」——所以这不是后人编的段子，而是他本人的追述；只是「1940 年前后的一次谈话、1971 年才写下来」这条时间线，本身也提醒我们：细节不必太当真。",[11,134,135],{},"论文发表之后的反响，皮尔斯有过一个很好的说法：这篇文章「像一颗炸弹，而且是一颗延时炸弹」。他后来在另一篇回顾里说，这件事「像一道凭空劈下来的闪电」，而且「我不知道还有哪套理论是这样一整套、几乎没什么前身和历史地冒出来的」。皮尔斯也留下了两句让同行记了一辈子的实话：1950 年以前发表的信息论文章，「大多数跟香农的工作没什么关系」；以及「我从没遇到过一个真正弄懂了信息论的物理学家」。",[11,137,138],{},"当时也不是没有冷脸。统计学家约瑟夫·杜布给 1949 年那本书写的书评就不客气，说它更像「有启发性」而非数学性的著作，「而且作者在数学上的意图并不总是体面的」。",[11,140,141],{},"真正把这篇论文推向公众的，是另一个人：洛克菲勒基金会的沃伦·韦弗。韦弗在战时管过科学动员，眼光极好。1949 年，他把自己的通俗介绍与香农的论文合在一起，出了《通信的数学理论》这本书。在给基金会主席的信里，韦弗把香农做的事比作「吉布斯为物理化学做过的事」。而在他自己那篇导言里，他写道：",[143,144,145],"blockquote",{},[11,146,147],{},"「communication（通信）这个词在这里将取一个非常宽泛的含义，包括一个头脑影响另一个头脑的全部过程。这当然不限于书面和口头的言语，还包括音乐、绘画、戏剧、芭蕾，事实上包括人的一切行为。」",[11,149,150],{},"韦弗还给出了那个著名的三层划分——技术的层次（符号传得多准）、语义的层次（符号多准确地传达了意思）、效果的层次（收到的意思多有效地改变了人的行动）。他最后甚至写下这样一句近乎诗的话：在这套理论里，「熵不只说算术的语言，它还说语言的语言」。",[11,152,153],{},"这就是「信息论热」的起点：一篇极其克制的数学论文，配上一篇极其热情的导言。往后几十年里，各种拿「信息」「熵」「冗余」去解释生物、心理、语言、经济、管理的文章层出不穷——而这桩公案，我们后面还会看到香农本人的反应。",[11,155,156],{},"对同行来讲，这篇论文的分量是不需要解释的。多年后，大卫·斯莱皮安在自己编的《信息论关键论文选》前言里写道：「本世纪大概没有哪一件工作，比香农 1948 年那篇文章更深刻地改变了人对通信的理解。」1998 年，在论文发表五十周年的综述里，塞尔希奥·贝尔杜第一次用了那个后来被反复引用的比喻：它是「信息时代的《大宪章》」。顺便说一句，《科学美国人》1990 年那篇专访里其实写的是「数字时代的《大宪章》」——两个版本后来常被混在一起，都算在《科学美国人》头上。",[11,158,159],{},"有意思的是，这样一篇改变世界的论文，作者本人几乎没有因此改变生活。他在发表之后依旧是那个爱摆弄东西、不爱应酬的人。",[11,161,162],{},"1949 年 3 月，他与贝尔实验室里的数值分析员贝蒂（玛丽·伊丽莎白·摩尔）结婚。据说牵线的人是皮尔斯——皮尔斯当时是贝蒂的上司。贝蒂是罗格斯大学的数学毕业生，聪明、幽默、有主见，既是他生活上的伙伴，也是工作上的搭档：他的一些论文手稿甚至是她的字迹抄写的，这让后来的研究者困惑了很久。她会吹钢琴，他会单簧管，两人下班后会找一些二声部的谱子合着弹。两人后来有了三个孩子。",[11,164,165],{},"还有一件圈里人都知道的小事。香农这篇论文出来的时候，诺伯特·维纳其实也在同一片水域里作业——他 1948 年出的《控制论》同样掀起了巨浪。两人的关系始终客气，维纳却在 1946 到 1947 年间私下抱怨过：「我的一个竞争者，贝尔电话公司的香农，要抢在我前面发表文章了。」香农这边对维纳一直很尊敬，说他是「我年轻时候的偶像」；至于两个人气质上的差别，麻省理工的托马斯·凯拉思后来总结得很直白：「维纳喜欢出风头，香农则完全不在乎这些。」",[11,167,168],{},"1949 年到 1950 年，他还发表了关于计算机下棋的论文，提出了让机器评估棋局、搜索走法的基本思路，这些想法后来成了计算机博弈的源头。",[11,170,171],{},"这件事值得多说两句。1950 年 3 月，他在《哲学杂志》上发表了《为计算机下棋编制程序》——这是这个题目上的第一篇技术论文；同年 2 月，《科学美国人》上还有一篇面向大众的《一台会下棋的机器》。文章中他把下棋程序分成两类，一类是硬算到底的「A 型」，一类是只搜有意思的分支的「B 型」，他自己更看好后者；他还估算出棋局的变化量大约是 10 的 120 次方，这个数字后来被叫作「香农数」。更早一点，1949 年他就在家里做出了一台用继电器控制的下棋机器，并且把它演示给国际象棋大师爱德华·拉斯克看过。有意思的是图灵当时也在做类似的事：他和戴维·钱珀瑙恩在 1948 年前后设计过一个下棋程序，因为没机器可用，只能自己拿纸笔一步一步地「手动运行」。图灵 1950 年那篇《计算机器与智能》里，还引用了香农的棋类文章。从香农的 A 型\u002FB 型分类往下，洛斯阿拉莫斯 1956 年的 MANIAC、伯恩斯坦 1958 年在 IBM 704 上的程序、麦卡锡他们在麻省理工写的程序，一直到 1967 年格林布拉特的棋程序，用的都是这套架子。",[11,173,174],{},"香农夫妇很爱玩杂耍、下棋，家里也总有一些稀奇古怪的小发明。对孩子们来说，这位爸爸并不严厉，更像一个爱和他们一起折腾的大朋友。",[18,176,178],{"id":177},"玩具机器与赌场","玩具、机器与赌场",[11,180,181],{},"如果说香农的理论成就是他的「正业」，那么他造的那些机器，就是他生活里最鲜活的部分。",[11,183,184],{},"1950 年，他做了一只名叫「忒修斯」的电动机械鼠，能够在迷宫里摸索前进，一旦走过一遍，它就能记住路线，第二次直接走到出口。名字来自那个用线团走出迷宫的希腊英雄。这只木老鼠本身只是个装着磁铁和几根铜丝触须的壳子，真正的「脑子」藏在地板下面：一整排从电话交换机上拆下来的继电器，驱动着一台带电磁铁的滑车，把老鼠拖着往四个方向走。触须碰到墙，对应的继电器就记下这一格有墙，老鼠拐九十度继续试，直到撞上一块铜做的「奶酪」，蜂鸣器响起来。二十五格的迷宫，大约两个比特记一格，全部记忆加起来也就几十个比特——而它第二次就能一路直着跑到出口。迷宫改了，它还能把学到的忘掉重学。这是最早的「学习机器」之一。香农自己后来开玩笑说：严格讲，「那只老鼠并没有在解迷宫，是迷宫在解那只老鼠。」",[11,186,187],{},"1951 年 3 月，他在梅西基金会的控制论会议上做了题为《一台迷宫求解机的展示》的报告（会议记录到 1952 年才出版，贝尔实验室同年还拍了一部短片，报纸杂志跟着报道了一轮，所以很多书上把年份写成 1952）。据说现场有位神经生理学家拉尔夫·杰拉德，看见老鼠在一个圈里绕来绕去，喊了一声「这是神经症！」，香农于是又给它加了一个「抗神经症回路」。",[11,189,190],{},"他造过一台「读心机器」，能根据人以前猜硬币正反的习惯来预测下一次选择。这件事有个具体的对手。贝尔实验室的同事戴夫·哈格尔巴格先做了一台专门猜硬币的机器，叫 SEER——「序列外推机器人」，机器里面记着八种局面（上一轮自己赢还是输、对手换没换招），每种局面配一个可以上下加减的计数器，靠这点统计去押下一次；据他那篇 1956 年的论文，这台机器跟人对打了九千多局，赢了大约五成三。香农 1953 年 3 月写了一份备忘录，标题就叫《一台读心的（？）机器》，开门见山地说，这是「对哈格尔巴格的机器的一个简化模型」。两人的机器后来被接在一起互相对打，因为没人按按钮，还得再串一台「裁判」机器来产生随机数。谁赢的说法不一：多数说法是香农那台简单些的机器小胜，也有说它「赢得相当轻松」的——但具体比分，恐怕已经没人说得清了。香农在备忘录里还写了一句有趣的话：这类游戏，冯·诺依曼和摩根施特恩用博弈论处理过，爱伦·坡在《被窃的信》里用心理学处理过，而他这台机器「更接近坡的玩法，而不是冯·诺依曼的」。",[11,192,193],{},"他还造过一台只有一个功能的「终极机器」：盒子上有一个开关，你把它拨到「开」，盒盖就会打开，伸出一只机械手，把开关拨回「关」，然后缩回去，盖上盖子。这个点子最初来自他的朋友马文·明斯基——1952 年夏天明斯基在贝尔实验室跟他共事，后来回忆说：「我提议做这个机器，香农喜欢它，就让公司做了一批，送给几位高管。」香农在自己的桌上留了一台。它是被作家阿瑟·克拉克写下来的，克拉克在贝尔实验室看见过它，1958 年这样描述：「它不过是一只小木匣，雪茄盒大小，一面装着开关。你一扳开关，里面就响起一阵愤怒而一本正经的嗡嗡声。盒盖慢慢抬起，一只手从下面伸出来，把开关拨回去，然后缩回盒里。盒盖啪地合上，像棺材合盖一样决绝……一台什么也不做、只把自己关掉的机器，让人有一种说不出的不祥感。」——半个多世纪后，这个点子在网上重新火了一遍，人们叫它「最没用的机器」，做的人多半并不知道它出自明斯基的玩笑和香农的手。",[11,195,196],{},"他还设计了一台只用罗马数字运算的计算机，起名 THROBAC，说的是「节俭的罗马数字回溯式计算机」，名字本身就像个玩笑。这台机器 1953 年 4 月留下了两份技术备忘录，实物今天是麻省理工博物馆的藏品。",[11,198,199],{},"他的「玩具房」是家里最神奇的地方。那里有各式各样的独轮车，其中包括一些特别古怪的设计：有一辆没有座垫，有一辆没有脚蹬，有一辆是双人的，还有一辆的轮毂是偏心的，踩起来人会一上一下地起伏——他说这是头一辆这么做的车，帮他调试的助手查理·曼宁一边担心他的安全，一边为它第一次骑成鼓掌。车库里的独轮车和「大小轮」堆得落了一层灰。有会杂耍的机器人，其中一个被装扮成喜剧演员 W.C. 菲尔兹的形象：那是用儿童建筑玩具搭出来的，用三个钢球，球在一个绷紧的鼓面上弹跳，靠两片摇摆的桨「接一下、抛一下」，出手的误差则用短短的沟槽去纠正——全机没有一个传感器。他还做过三个小丑的「永不掉球」杂耍装置，起因是贝蒂从蛋糕装饰店带回一个四英寸的小丑，「才一块九毛八」。他的得意是：「有史以来最厉害的杂耍家也没法把纪录保持几分钟，可我的小丑整夜都在抛，一次也不掉！」",[11,201,202],{},"据说他还造过会喷火的小号——一吹，喇叭口就喷出火苗；还有火箭动力的飞盘、烧汽油的弹跳杆、一双巨大的泡沫「水上行走鞋」、一根从厨房用缆线连到地下实验室的机械手指（一拉，手指就勾一勾，等于叫人）、一台能设定翻转次数的机械硬币、从门廊下滑到湖边的简易缆车。爱德华·索普第一次走进他家地下室时的感想，被传记作者记了下来：「一个搞小玩意的人的天堂……我这是遇到了终极小玩意爱好者。」",[11,204,205],{},"他对杂耍尤其着迷，甚至总结出了一个「杂耍定理」。它把五个量用一行等式联系起来：设每只手接球到把球抛出的时间为 D，球在空中滞留的时间为 F，一只手空着的比例为 V，那么 (F + D) × 手数 = (V + D) × 球数。这个式子直到今天还在杂耍圈里被引用。他也写过一篇《杂耍的科学》，投给了《科学美国人》，但那份稿子改了又改，最后没发表——他 1987 年接受采访时还说：「我是给《科学美国人》写了一篇，至今没发出来。」文章要等到他去世后，被收进 1993 年的《香农论文集》里才和读者见面。对别人来说，这些都是不务正业；对他来说，这是理解世界的另一种方式。",[11,207,208],{},"他的兴趣还延伸到了钱的世界。1961 年，他与数学家爱德华·索普合作，做了一台可穿戴的轮盘赌预测装置：一个香烟盒大小的模拟计算机，开关藏在鞋里用大脚趾操作，一只脚负责给轮盘计时，另一只负责给球计时，结果则用几个音调通过塞进一只耳朵的小喇叭报出来；连导线都用胶水粘上颜色，尽量跟皮肤和头发混在一起。它预测的不是具体哪一个格子，而是一段扇区。那年 8 月，他们带着它真的去了拉斯维加斯：香农在轮盘旁边计时，索普在赌台另一头下注，两位太太负责望风。机器是灵的，小堆的筹码慢慢变大堆；但麻烦在于那根比头发还细的耳机线老是断，一断就得停半天。有一次同桌一位女士「惊恐地看过来」，索普赶紧走开，才发现「那个小喇叭像一只外星虫子一样从我的耳道里探出头来」。香农自己事后总结得很平：「我们碰到一堆实际问题，实际上从来没赚到钱。」这件事他们守口如瓶到 1966 年才披露；今天它被追认为「第一台可穿戴计算机」，不过这个称呼是后人的追认，不是当时的说法。",[11,210,211],{},"同一件事还有个前传，主角是索普的黑杰克工作。索普那篇论文本来想叫《二十一点的制胜策略》，是香农劝他把标题改得平淡些，最后成了《二十一点的一种有利策略》，1961 年发表在《美国科学院院刊》上——投这篇稿子需要一位院士背书，香农正是那位数学部的院士，据索普回忆，香农在签字前把他「盘问」了一遍才放过他。后来那本《击败庄家》把算牌术传遍了全世界。至于去拉斯维加斯那次，是轮盘赌，不是黑杰克，很多人会把这两件事混在一起。",[11,213,214],{},"他对股票的态度同样如此。香农并不迷恋盯盘，他会买入自己看好的少数几家技术公司并长期持有，比如惠普、摩托罗拉，还有他的老朋友亨利·辛格尔顿掌舵的泰利达因——辛格尔顿从大学时代起就跟他相识，后来还把他请进了董事会。他 1987 年跟采访者算过这笔账：「泰利达因我们拿了二十五年，每年复合增长百分之二十七。」这个数字在当时的圈子里是件稀罕事。至于那个更著名的「年化百分之二十八、跑赢一千多家基金」的说法，出处是庞德斯通 2005 年的书，依据的是一份未曾发表的访谈笔记，听个大概即可，别当成审计过的账目。他自己对这套本事的总结倒是简单：「我们是基本面派，不是技术派。」他对那些画图形的人毫不客气，把「头肩顶」「深 V」之类称作「重要数据的一种非常粗糙的复制品」；至于经济学家说的有效市场，他直说「我根本不信」。他还留下一句在麻省理工流传很广的俏皮话：有人问他为什么不用自己那套选股方法，他说：「算了吧，手续费会把你吃光。」——另有一次他对人说：「我是靠股市挣钱的，不是靠证定理挣钱的。」家里的账目，其实主要是贝蒂在管。女儿佩吉后来说：「这从来都是他们两个人的项目……他们两个都是赌徒。」他们甚至用一台早期的苹果 II 型电脑，每晚算一遍持仓的增长速度。",[11,216,217],{},"同一时期，他还是 1956 年达特茅斯人工智能夏季研讨会的发起者之一，与麦卡锡、明斯基、罗切斯特一起，把「人工智能」这个话题带到了学界的桌面上。有个细节值得记下：他自己并不喜欢「人工智能」这个词，更愿意用老派的「自动机研究」——那年他和麦卡锡合编的那本论文集，名字就叫《自动机研究》。至于那场为期一个多月的讨论会，他本人到底待了多久，史料上并不清楚：会议没有留下正式的出席名单，有回忆说他到场聊过一两天就走了，也有同时代人的笔记把他记成在场者。能确定的是，他不是那个从头住到尾的人。",[11,219,220],{},"然而他又是个清醒的人。1956 年，他在《IRE 信息论汇刊》上写了一篇很短、不到六百词的社论，题目叫《跟风》。这篇文章今天读起来依然刺人：",[143,222,223],{},[11,224,225],{},"「信息论在最近几年变成了一股科学上的跟风热潮。它本来只是通信工程师的一件技术工具，却在通俗媒体和学术媒体上都得到了异乎寻常的关注……它大概已经被吹到了超出它实际成就的高度。」「大自然的秘密很少会一次让出好几条。」「人们会很容易发现，用上几个带劲的词，比如信息、熵、冗余，并不能解决我们所有的问题。」「这个行当无疑是被卖出去的，如果不是被卖过头的话……我们该做的是研究，而不是宣传。几篇第一流的论文，胜过一大批构思拙劣或者半途而废的文章。」",[11,227,228],{},"一个创立了一门学科的人，反而最先劝大家别神化这门学科——这种低调，几乎成了他性格的标签。",[11,230,231],{},"这篇社论不是没人回应，恰恰相反。维纳在三个月后同一份刊物上写了一篇《什么是信息论？》，客气地反驳说，不应该把信息论缩得那么窄；彼得·埃利亚斯则在 1958 年写了一篇讽刺社论，题目叫《两篇著名的论文》，虚构了一篇题为《信息论、光合作用与宗教》的文章来嘲笑滥用，末尾写：「我建议我们别再写这种东西了，把大量人力释放出来，去做那些真正需要研究的、激动人心的重要问题。」它常被后人称作学界最早的一次「索卡尔式恶作剧」的前身。罗伯特·法诺对此的感受很直接：「我不喜欢‘信息论’这个名字。克劳德也不喜欢。因为它听起来像是一门关于‘信息’的理论——可它不是，它是关于信息传输的，不是关于信息的。太多人没弄明白这一点。」贝蒂说得更家常：「那些人把他的话拉来拉去，他有点被惹烦了。他们不明白他想做的是什么。」",[18,233,235],{"id":234},"麻省理工的怪教授","麻省理工的「怪教授」",[11,237,238],{},[72,239],{"alt":240,"src":241},"Shannon SM ’40, PhD ’40, who taught at MIT from 1956 until his retirement in 1978, was known to many as the father of information theory and of modern digital communications.","https:\u002F\u002Fimage-assets.dreams.plus\u002F202610162232493.png",[11,243,244],{},"1956 年 1 月，香农向贝尔实验室请了长假，回到麻省理工做客座教授，头衔是「电气通信访问教授」；1957 年他正式留下，同时挂两个教授头衔——电气工程系的通信科学教授，以及数学系教授。1958 年 10 月，学校宣布他出任新设立的「唐纳科学讲席教授」，这笔讲席由费城的唐纳基金会捐资五十万美元设立，是当时麻省理工收到过的最大一笔讲席捐赠。他和家人搬到了马萨诸塞州的温彻斯特，住进一栋靠近上神秘湖的大房子。房子 1858 年为一位名叫艾伦·德怀特的女士所建，她是托马斯·杰斐逊的曾外孙女，房子的设计据说参考了蒙蒂塞洛。他们后来给它起了个名字：「熵之屋」。",[11,246,247],{},"在学生眼里，香农是个特别的老师。他不爱讲课，更不爱写文章，上课风格随意，有时偏离主题，把话题带到他自己最近在摆弄的玩意上。",[11,249,250],{},"莱昂纳德·克莱因罗克形容他的课「像一顿好饭：你去了，他给你的东西是干净的、直观的」。罗伯特·加拉格记得他讲课时的那种神情：「他不会说‘这一章的要点是什么什么’。他会说：‘昨天晚上我在琢磨这个，想出来一个挺有意思的看法。’说的时候嘴角还带着一点坏笑。」大卫·福尼则是被他的研究讨论班拉进这行的，福尼说，他的教学方法就是讲自己正在做的问题、半成品的结果、猜想；有些问题他有漂亮的结果，有些问题他除了把问题本身说清楚以外毫无进展——「而这对一个正在找论文题目的研究生来说，简直太好了。」赫尔曼·豪斯也记得一件小事：他去旁听香农的讨论班，香农「把我当作和他平等的人对待」，并且对他这个晚辈提了一个关键问题，后来长成了豪斯那本书的第四章。香农的课还有个技术性细节常被提起：他一边回答问题，一边用一只手把一截粉笔抛上抛下，从来没有失手过。有位学生的总结很干脆：「我们都像敬神一样敬香农。」",[11,252,253],{},"做导师这件事，他却躲得很彻底。他一生带出的博士生极少——按后来的统计，只有三个；同一时期，麦卡锡、明斯基那些人各自都带出了一整代人。加拉格的解释很实在：「他的博士生非常少……你要是待在麻省理工，身边站着香农这么个庞然大物，你得有相当大的自我，才敢开口请他带你。」有人劝他多带几个学生，他的回答是：「我当不了导师。我没办法给任何人建议。我不觉得自己有资格去建议别人。」",[11,255,256],{},"他的办公室很乱，邮件和文稿堆得老高，据说他很多信件长期无人回复。这件事有个物证：他给自己存信件的文件夹起了个名字，叫「我拖得太久还没回的信」。记者乔恩·格特纳有一句形容极为传神：「信号可以抵达克劳德·香农。然后就再也传不下去了。」有位 1969 年去麻省理工访问的苏联学者回忆，他在走廊里看见了「香农教授」的门牌，然后过了一年多才意识到，这扇门几乎没开过——香农已经很少来学校了。倒是有位同事借用他那间空掉的办公室时，在里面翻出一张一年前开出、还没兑现的支票。人们还常提到，他在教工俱乐部的午饭桌上算他的股票想法，餐巾纸写满就换一张。",[11,258,259],{},"他对学术的虚名、头衔、荣誉兴趣平平，常常对别人的称赞报以腼腆一笑。他的名头则越来越大：1966 年 12 月，美国国家科学奖章的消息公布，次年 2 月由约翰逊总统在白宫东厅颁发，同一年他还拿到了 IEEE 荣誉奖章。颁奖那天，七岁的女儿佩吉为了穿哪条裙子跟母亲争执了一路，最后是副总统汉弗莱的大笑把她吓到躲在母亲腿后面。贝蒂说：「他是个非常谦逊的人。他得过很多奖，但奖从没让他头脑发热，他也从不谈这些。」他那些荣誉博士的兜帽太多，于是自己做了一个可以旋转的架子挂起来，跟挂领带的架子差不多。",[11,261,262],{},"领奖和演讲则一直是他的软肋。1973 年，第一届「香农讲座」在以色列的阿什凯隆举行，埃尔温·伯利坎普后来回忆：「我从没见过那么严重的怯场。我根本想不到，一个人在朋友面前可以怕成这样。」他需要有人在后台替他做很久的心理建设，还得由一位朋友陪着才肯上台。他后来推掉类似的邀约时，理由是一句玩笑：「退休以后，贝蒂不擦窗户，我不做演讲。」",[11,264,265],{},"这些年里，他的另一桩爱好是音乐：他会吹单簧管，喜欢迪克西兰爵士乐，也常参加小型即兴演奏。楼下客厅里，他吹单簧管、贝蒂弹钢琴的二重奏，是这个家的日常。爵士乐里的即兴与结构，与他的科学趣味可谓同源。",[11,267,268],{},"他爱下棋，但并不在乎输赢；他自己做的下棋机器能赢一般水平的对手，他却更乐意站在旁边看机器怎么「思考」。他骑着独轮车穿过麻省理工的走廊，边走边杂耍，让路过的学生又惊又喜。",[11,270,271],{},"家里那三个孩子，是在一堆零件中间长大的。儿子罗伯特生于 1952 年，后来做计算机，1998 年先于父亲去世；安德鲁生于 1954 年，成了钢琴家和作曲家；女儿佩吉生于 1959 年，后来当了老师。家里那间被报纸叫作「玩具房」的两层附楼，孩子们只叫它「爸爸的房间」。佩吉回忆说，父亲在家工作的时间很多，去学校只是为了上课和见学生，「所以我从小到大觉得他老在身边，跟很多上班的人家不一样」。晚饭后谁洗碗，家里甚至有台会走的发条机器老鼠来决定：把它放在餐桌中间，看它往哪边掉下去，掉到谁那边就谁洗。有一次佩吉把一盒牙签撒在了门廊上，父亲没有生气，而是说：「你知道吗，用这个可以算出圆周率。」——他说的是投针问题。家里饭桌上的话题常常是股票，孩子们很早就被教着读《华尔街日报》。至于辅导作业，他会因为「新数学」的教材顺序不对而发一顿牢骚，然后一路讲到虚数上去，讲到孩子已经完全不指望他把题讲完。",[11,273,274],{},"1978 年，他从教授岗位上退休，此后更多地待在家中，把时间交给他的工坊、他的玩具和他的书。",[18,276,278],{"id":277},"熵之屋里的晚年","「熵之屋」里的晚年",[11,280,281],{},[72,282],{"alt":283,"src":284},"Claude Shannon's electromechanical mouse, which he called Theseus, was one of the earliest attempts to \"teach\" a machine to \"learn\" and one of the first experiments in artificial intelligence.","https:\u002F\u002Fimage-assets.dreams.plus\u002F202610162229686.png",[11,286,287],{},"退休后的香农依然享受生活：造机器，玩杂耍，和贝蒂一起旅行、散步，在家里招待来访的朋友。他不太谈自己的成就，也不爱做回顾。很多年后，来访者发现他的家里很少有显眼的荣誉陈列，反倒到处是工作台、零件和各式玩具。",[11,289,290],{},"他的动手能力没有退化，反而更放得开了。他做了一台解魔方的机器——这是真的，不是传说，那台机器今天也在麻省理工博物馆里；他还为魔方写过一首七十来行、带脚注的打油诗，标明要用《Ta-Ra-Ra Boom-De-Ay》的调子唱。1985 年，他获得了第一届京都奖，去京都领奖时做了一场题为《通信与计算的发展，以及我的爱好》的演讲，那是他晚年最长、也最完整的一次公开表述。他在演讲里说，历史上真正重要的不是帝王将相，而是「达尔文、牛顿、贝多芬这样的思想者和创造者」。",[11,292,293],{},"进入 80 年代，他的记忆开始衰退。1983 年，他去看医生，结论是「很可能处在阿尔茨海默病非常非常早期的阶段」（这是贝蒂的原话）。起初是忘电话号码、忘名字、忘面孔，出门去趟杂货店会忘了怎么回来，字迹也开始发抖。",[11,295,296],{},"1985 年，在英国布莱顿举办的国际信息论研讨会上，出现了一件令人难忘的事。他和贝蒂没有事先打招呼就出现在会场，在几个报告厅里进出，直到有人认出他来，消息才传开。晚宴上主办方把他请上台，他只简短地讲了几句，就被自己的掌声淹没了，最后说了一句：「这——太荒唐了。」然后就没什么可说的了，于是从口袋里掏出三个球，当场抛了起来。事后有人问主持人该怎么形容这件事，得到的回答是：「就像牛顿突然出现在一场物理学会议上。」参会者排起长队请他签名，很多年轻研究者是平生第一次见到这位读了一辈子其论文的人。当时的信息论学会会长也回忆说，那天在场的人里，大部分其实并不认得他，「但有一种嗡嗡声：香农来了，在哪儿，在哪儿？」——那是他最后几次公开露面之一。第二年，学会正式邀请他出席了在密歇根安娜堡举行的研讨会，那时病情已经相当明显：他做了一场十几分钟的晚宴讲话，里面还塞满了趣闻和玩笑，可贝蒂替他挡掉了大部分寒暄。",[11,298,299],{},"此后，病情逐渐加重。女儿佩吉说：「他会到处乱走。我们住在一条繁忙的街边……看着自己的亲人得这种病，是件很可怕的事。」有一阵子，家里参加了当地医院的阿尔茨海默病研究，贝蒂充当对照组。佩吉说：「有些日子他认得人，有些日子不认……有时候还是原来那个爸爸，有时候完全不知道自己在哪儿。」1992 年，佩吉还跟他有过一次关于研究生院的正常交谈；到 1993 年，罗伯特·法诺问他一件从前的事，香农只是回答：「我不记得了。」",[11,301,302],{},"1993 年他摔了一跤，髋骨骨折，康复得很慢。贝蒂本想让他在家里，还在「熵之屋」里布置了一间放得下病床的房间，最后在佩吉的劝说下，把他送进了梅德福的一家护理机构。贝蒂几乎每天去两次，佩吉说她「非常尽心」。贝蒂自己回忆：「中午我过去，护士们会排队坐在一张长凳上等我，因为我一进门，他脸上就会亮起来，咧嘴笑。」他在那里还会把助行架拆开来，琢磨更好的设计；听到音乐就用手指打拍子；护理人员会给他出些简单的算术题；他也会朝着楼梯和门外走。90 年代，他渐渐认不出身边的人。2001 年 2 月 24 日，克劳德·香农在这家护理机构去世，享年 84 岁。他的大脑捐给了阿尔茨海默病研究。",[11,304,305],{},"贝蒂比他多活了十六年，2017 年 5 月去世。他们结婚六十八年，从他还在贝尔实验室算信道容量，一直到「熵之屋」里的人渐渐走完。",[11,307,308],{},"他早年还给自己画过一场想象中葬礼的速写，那张图一点也不悲伤：一支爵士乐队在前面开道，六位独轮车手抬着棺材（图上写着「6 个骑独轮车的人 \u002F 1 位亲人」），后面跟着哭泣的寡妇、八人杂耍队和一台八条腿的杂耍机器，一辆「棋局彩车」上，棋手戴维·利维正在跟电脑对局，再往后是四只按斯金纳的方法训练过的猫、各种小机器、慢跑的人和一支四百一十七人的乐队。一个人能这样安排自己的葬礼，说明他对「正经」和「不正经」的分界，从一开始就没怎么放在心上。",[18,310,311],{"id":311},"一个把工作当成游戏的人",[11,313,314],{},"香农的一生没有戏剧性的冲突，没有激烈的论战，也没有惊人的野心，很多时候他就是在一间堆满零件的屋子里，对着一个有趣的问题发呆。他提出的理论，为今天的手机通信、互联网、数据压缩、纠错码和密码学等整个数字世界打下了地基；而他自己的日子，却过得像一个永远不肯长大的孩子。",[11,316,317],{},"同行对他的评价，几乎都带着一点无从下手的敬意。大卫·斯莱皮安说：「他真的是个天才。我认识的人里，只有他配得上这个词。」又说：「他数学并不算懂得深，但他能现造出他需要的数学。」罗伯特·加拉格的版本是：「他的眼光清澈得惊人。爱因斯坦也是这样。」至于他到底做了什么，加拉格还有一句更朴素的话：「是香农看出，二进制位是一切通信的基本元素。那是他真正的发现，整个通信革命都是从这里长出来的。」托马斯·科弗说：「香农把三个领域炸了出来。」詹姆斯·马西在密码学上补了一句：「毫不夸张地说，香农的文章标志着密码学从一门手艺变成了一门科学。」而马文·明斯基记住的却是另一面：「不管出现什么题目，他都带着兴致扑上去，用某种出人意料的办法去攻它。对他而言，问题看上去越难，找到新东西的机会就越大。」",[11,319,320],{},"他自己对这一切的解释，比任何评论都更实在。1987 年，他对采访者说：「我不认为我做事是为了得奖——虽然另一间屋子里放着两打奖章。我更主要是被好奇心驱动，从来不是为了钱。我只是想知道，东西到底是怎么被装起来的。」「答案找到之后，把它们写下来、发表出来，总是让我很痛苦……我做过很多事，从来没写出来过。太懒了吧。我楼上有一个文件夹，装的全是没写完的论文。」三年后，他又说了一句后来被引用得最多的话：「我一直在追逐自己感兴趣的东西，不太在意它有多少金钱价值，或者对世界有多少价值。我在完全没用的东西上花了很多时间。」",[11,322,323],{},"回望那些日常的小事：少年时在围栏上架起的电报线，独自在走廊里骑独轮车的深夜，家里摆满的各种怪机器，晚年杂耍时那一点羞涩的笑容，你会发现，这些并不是他「正业之外」的点缀，而恰恰是他创造力的来源。他相信好奇心，相信玩耍，也相信，好玩的东西往往藏着深刻的真理。",[143,325,326],{},[11,327,328],{},"卡德班克和斯隆在《自然》上的讣文里写过一句很准的话：香农大概能排进二十世纪科学主要人物名单的前列，「尽管他的名字对公众来说相对陌生」。这大概正是他本人会满意的位置。",{"title":330,"searchDepth":331,"depth":331,"links":332},"",2,[333,334,335,336,337,338,339,340],{"id":20,"depth":331,"text":20},{"id":41,"depth":331,"text":41},{"id":81,"depth":331,"text":81},{"id":120,"depth":331,"text":120},{"id":177,"depth":331,"text":178},{"id":234,"depth":331,"text":235},{"id":277,"depth":331,"text":278},{"id":311,"depth":331,"text":311},false,true,"md",null,{},"\u002Fblog\u002F2026\u002F0010-biography-of-claude-shannon",{"title":6,"description":13},"blog\u002F2026\u002F0010-biography-of-claude-shannon","从密歇根小镇那根架在铁丝围栏上的电报线，到贝尔实验室走廊里的独轮车，再到麻省理工那间堆满零件的办公室。他给「信息」下了数学定义，自己却一辈子只对有意思的东西负责。",[351],"thoughts","2026-10-16T22:03:15+08:00","lG9XuZP_DA5IpAhSLPz6guqBwMoOwfOl7mPBMsPtti8",{"id":355,"title":356,"body":357,"description":133078,"draft":341,"enableComment":342,"extension":343,"image":330,"important":341,"location":344,"meta":133079,"navigation":342,"ogImage":344,"path":133080,"seo":133081,"stem":133082,"summary":133083,"tags":133084,"time":133087,"__hash__":133088},"blog\u002Fblog\u002F2026\u002F0011-about-the-f-divergence.md","f-散度族及其推论",{"type":8,"value":358,"toc":133017},[359,411,525,872,875,879,1099,1263,1339,2658,2663,3503,4170,4607,5064,5550,5554,5876,5919,5923,6059,6237,6294,6837,7009,7494,7707,7711,8762,9097,9195,9199,9440,9713,9749,10219,10374,10903,10906,10910,11012,11316,11321,12286,12954,13766,14609,14741,15670,16221,16519,16860,17394,17700,18316,19450,20270,20813,21159,21412,21650,21681,21777,21781,21813,21816,21819,21836,21839,21843,21846,21934,22170,22710,22875,23419,23422,23712,23997,24558,24561,24859,25554,25918,25922,26084,26561,26833,27229,27940,27943,28380,28383,28820,29089,30053,30306,30947,31185,31353,31540,31950,32456,33174,33900,33956,34201,34462,34879,34883,34885,35052,35121,35761,36156,36548,36557,36903,37518,37522,37524,37656,37724,38252,38259,38632,39005,39958,40807,40811,40813,40994,41310,41849,42011,42419,42482,42959,43577,44489,44680,45106,45110,45194,45535,45981,46357,46566,46838,47024,47562,48103,48529,48532,49387,49476,49745,49749,49752,50321,50788,50998,51126,51667,51880,51889,52391,52859,52863,55282,55286,55581,55585,55841,56508,56512,57471,58162,58664,58668,59023,59349,59454,60187,61060,61064,61885,61920,62171,62337,63383,64036,64042,64308,64806,64999,67189,68203,68513,68788,69300,70988,71878,73648,74411,74414,74825,74828,74831,74835,75141,75479,76245,76248,76609,76641,77130,77193,77988,78366,78414,78418,79300,79606,80748,82442,82446,84265,84271,84415,84817,85361,86488,87429,87435,87903,88588,88995,89068,89107,89505,89834,90086,90832,90897,92178,92823,97329,97336,97340,97343,99256,99515,99684,99688,99692,99695,100098,100304,101094,101220,101732,101736,102152,102410,102870,103576,103580,103641,103773,104160,104508,105738,105742,105928,106283,106286,106764,107154,107666,107949,108370,108630,108634,108637,108971,108975,108979,111939,114998,115001,115005,115058,115305,115518,115644,116005,116280,116657,116778,116783,116787,117824,117827,117831,117834,117893,117896,117900,118090,118094,118097,118626,118981,118985,118988,119560,119680,119684,119732,120135,120138,120142,120207,120211,120214,120568,120571,120575,120656,121010,121286,121444,121681,121947,122270,122642,123052,123927,124459,124729,124732,124736,124740,125091,125096,125835,126288,126626,126628,127229,127391,127397,127743,128259,128265,128839,129081,129085,129678,130069,130489,130837,131310,131314,131572,131887,131993,132527,132530,132534,132537,133014],[11,360,361,362,410],{},"f-散度族是概率分布之间差异的一大类度量，它用一个凸函数 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",[363,444,446,478],{"className":445,"translate":367},[366],[363,447,449],{"className":448},[371],[373,450,451],{"xmlns":375},[377,452,453,475],{},[380,454,455,457,462,466,469,472],{},[383,456,385],{},[458,459,461],"mo",{"stretchy":460},"false","(",[463,464,465],"mn",{},"1",[458,467,468],{"stretchy":460},")",[458,470,471],{},"=",[463,473,474],{},"0",[387,476,477],{"encoding":389},"f(1)=0",[363,479,481,514],{"className":480,"ariaHidden":394},[393],[363,482,484,488,491,495,498,502,507,511],{"className":483},[398],[363,485],{"className":486,"style":487},[402],"height:1em;vertical-align:-0.25em;",[363,489,385],{"className":490,"style":409},[407,408],[363,492,461],{"className":493},[494],"mopen",[363,496,465],{"className":497},[407],[363,499,468],{"className":500},[501],"mclose",[363,503],{"className":504,"style":506},[505],"mspace","margin-right:0.2778em;",[363,508,471],{"className":509},[510],"mrel",[363,512],{"className":513,"style":506},[505],[363,515,517,521],{"className":516},[398],[363,518],{"className":519,"style":520},[402],"height:0.6444em;",[363,522,474],{"className":523},[407],"）定义的散度模板：",[363,526,529],{"className":527,"translate":367},[528],"katex-display",[363,530,532,609],{"className":531,"translate":367},[366],[363,533,535],{"className":534},[371],[373,536,538],{"xmlns":375,"display":537},"block",[377,539,540,606],{},[380,541,542,550,552,555,559,562,564,566,574],{},[543,544,545,548],"msub",{},[383,546,547],{},"D",[383,549,385],{},[458,551,461],{"stretchy":460},[383,553,554],{},"P",[383,556,558],{"mathvariant":557},"normal","∥",[383,560,561],{},"Q",[458,563,468],{"stretchy":460},[458,565,471],{},[543,567,568,572],{},[383,569,571],{"mathvariant":570},"double-struck","E",[383,573,561],{},[380,575,576,579,581,603],{},[458,577,578],{"fence":394},"[",[383,580,385],{},[380,582,583,585,601],{},[458,584,461],{"fence":394},[586,587,588,595],"mfrac",{},[380,589,590,593],{},[383,591,592],{},"d",[383,594,554],{},[380,596,597,599],{},[383,598,592],{},[383,600,561],{},[458,602,468],{"fence":394},[458,604,605],{"fence":394},"]",[387,607,608],{"encoding":389},"D_f(P\\|Q)\n=\\mathbb E_Q\\left[\nf\\left(\\frac{dP}{dQ}\\right)\\right]",[363,610,612,701],{"className":611,"ariaHidden":394},[393],[363,613,615,619,676,679,683,686,689,692,695,698],{"className":614},[398],[363,616],{"className":617,"style":618},[402],"height:1.0361em;vertical-align:-0.2861em;",[363,620,622,626],{"className":621},[407],[363,623,547],{"className":624,"style":625},[407,408],"margin-right:0.0278em;",[363,627,630],{"className":628},[629],"msupsub",[363,631,635,667],{"className":632},[633,634],"vlist-t","vlist-t2",[363,636,639,662],{"className":637},[638],"vlist-r",[363,640,644],{"className":641,"style":643},[642],"vlist","height:0.3361em;",[363,645,647,652],{"style":646},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[363,648],{"className":649,"style":651},[650],"pstrut","height:2.7em;",[363,653,659],{"className":654},[655,656,657,658],"sizing","reset-size6","size3","mtight",[363,660,385],{"className":661,"style":409},[407,408,658],[363,663,666],{"className":664},[665],"vlist-s","​",[363,668,670],{"className":669},[638],[363,671,674],{"className":672,"style":673},[642],"height:0.2861em;",[363,675],{},[363,677,461],{"className":678},[494],[363,680,554],{"className":681,"style":682},[407,408],"margin-right:0.1389em;",[363,684,558],{"className":685},[407],[363,687,561],{"className":688},[407,408],[363,690,468],{"className":691},[501],[363,693],{"className":694,"style":506},[505],[363,696,471],{"className":697},[510],[363,699],{"className":700,"style":506},[505],[363,702,704,708,751,755],{"className":703},[398],[363,705],{"className":706,"style":707},[402],"height:2.4em;vertical-align:-0.95em;",[363,709,711,715],{"className":710},[407],[363,712,571],{"className":713},[407,714],"mathbb",[363,716,718],{"className":717},[629],[363,719,721,743],{"className":720},[633,634],[363,722,724,740],{"className":723},[638],[363,725,728],{"className":726,"style":727},[642],"height:0.3283em;",[363,729,731,734],{"style":730},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[363,732],{"className":733,"style":651},[650],[363,735,737],{"className":736},[655,656,657,658],[363,738,561],{"className":739},[407,408,658],[363,741,666],{"className":742},[665],[363,744,746],{"className":745},[638],[363,747,749],{"className":748,"style":673},[642],[363,750],{},[363,752],{"className":753,"style":754},[505],"margin-right:0.1667em;",[363,756,759,768,771,774,866],{"className":757},[758],"minner",[363,760,764],{"className":761,"style":763},[494,762],"delimcenter","top:0em;",[363,765,578],{"className":766},[767,657],"delimsizing",[363,769,385],{"className":770,"style":409},[407,408],[363,772],{"className":773,"style":754},[505],[363,775,777,783,860],{"className":776},[758],[363,778,780],{"className":779,"style":763},[494,762],[363,781,461],{"className":782},[767,657],[363,784,786,790,857],{"className":785},[407],[363,787],{"className":788},[494,789],"nulldelimiter",[363,791,793],{"className":792},[586],[363,794,796,848],{"className":795},[633,634],[363,797,799,845],{"className":798},[638],[363,800,803,819,830],{"className":801,"style":802},[642],"height:1.3714em;",[363,804,806,810],{"style":805},"top:-2.314em;",[363,807],{"className":808,"style":809},[650],"height:3em;",[363,811,813,816],{"className":812},[407],[363,814,592],{"className":815},[407,408],[363,817,561],{"className":818},[407,408],[363,820,822,825],{"style":821},"top:-3.23em;",[363,823],{"className":824,"style":809},[650],[363,826],{"className":827,"style":829},[828],"frac-line","border-bottom-width:0.04em;",[363,831,833,836],{"style":832},"top:-3.677em;",[363,834],{"className":835,"style":809},[650],[363,837,839,842],{"className":838},[407],[363,840,592],{"className":841},[407,408],[363,843,554],{"className":844,"style":682},[407,408],[363,846,666],{"className":847},[665],[363,849,851],{"className":850},[638],[363,852,855],{"className":853,"style":854},[642],"height:0.8804em;",[363,856],{},[363,858],{"className":859},[501,789],[363,861,863],{"className":862,"style":763},[501,762],[363,864,468],{"className":865},[767,657],[363,867,869],{"className":868,"style":763},[501,762],[363,870,605],{"className":871},[767,657],[11,873,874],{},"它统一了 KL、JS、全变差、卡方、Hellinger 等众多散度，具有非负性、联合凸性、变分对偶表示和数据处理不等式等核心性质，是信息论和生成模型理论中的基础工具。",[18,876,878],{"id":877},"_1-定义","1. 定义",[11,880,881,882,927,928,985,986,1015,1016,1044,1045,1098],{},"设 ",[363,883,885,904],{"className":884,"translate":367},[366],[363,886,888],{"className":887},[371],[373,889,890],{"xmlns":375},[377,891,892,901],{},[380,893,894,896,899],{},[383,895,554],{},[458,897,898],{"separator":394},",",[383,900,561],{},[387,902,903],{"encoding":389},"P,Q",[363,905,907],{"className":906,"ariaHidden":394},[393],[363,908,910,914,917,921,924],{"className":909},[398],[363,911],{"className":912,"style":913},[402],"height:0.8778em;vertical-align:-0.1944em;",[363,915,554],{"className":916,"style":682},[407,408],[363,918,898],{"className":919},[920],"mpunct",[363,922],{"className":923,"style":754},[505],[363,925,561],{"className":926},[407,408]," 是同一个可测空间 ",[363,929,931,956],{"className":930,"translate":367},[366],[363,932,934],{"className":933},[371],[373,935,936],{"xmlns":375},[377,937,938,953],{},[380,939,940,942,946,948,951],{},[458,941,461],{"stretchy":460},[383,943,945],{"mathvariant":944},"script","X",[458,947,898],{"separator":394},[383,949,950],{"mathvariant":944},"A",[458,952,468],{"stretchy":460},[387,954,955],{"encoding":389},"(\\mathcal X,\\mathcal A)",[363,957,959],{"className":958,"ariaHidden":394},[393],[363,960,962,965,968,973,976,979,982],{"className":961},[398],[363,963],{"className":964,"style":487},[402],[363,966,461],{"className":967},[494],[363,969,945],{"className":970,"style":972},[407,971],"mathcal","margin-right:0.1464em;",[363,974,898],{"className":975},[920],[363,977],{"className":978,"style":754},[505],[363,980,950],{"className":981},[407,971],[363,983,468],{"className":984},[501]," 上的两个概率分布，且 ",[363,987,989,1002],{"className":988,"translate":367},[366],[363,990,992],{"className":991},[371],[373,993,994],{"xmlns":375},[377,995,996,1000],{},[380,997,998],{},[383,999,554],{},[387,1001,554],{"encoding":389},[363,1003,1005],{"className":1004,"ariaHidden":394},[393],[363,1006,1008,1012],{"className":1007},[398],[363,1009],{"className":1010,"style":1011},[402],"height:0.6833em;",[363,1013,554],{"className":1014,"style":682},[407,408]," 关于 ",[363,1017,1019,1032],{"className":1018,"translate":367},[366],[363,1020,1022],{"className":1021},[371],[373,1023,1024],{"xmlns":375},[377,1025,1026,1030],{},[380,1027,1028],{},[383,1029,561],{},[387,1031,561],{"encoding":389},[363,1033,1035],{"className":1034,"ariaHidden":394},[393],[363,1036,1038,1041],{"className":1037},[398],[363,1039],{"className":1040,"style":913},[402],[363,1042,561],{"className":1043},[407,408]," 绝对连续（即 ",[363,1046,1048,1067],{"className":1047,"translate":367},[366],[363,1049,1051],{"className":1050},[371],[373,1052,1053],{"xmlns":375},[377,1054,1055,1064],{},[380,1056,1057,1059,1062],{},[383,1058,554],{},[458,1060,1061],{},"≪",[383,1063,561],{},[387,1065,1066],{"encoding":389},"P\\ll Q",[363,1068,1070,1089],{"className":1069,"ariaHidden":394},[393],[363,1071,1073,1077,1080,1083,1086],{"className":1072},[398],[363,1074],{"className":1075,"style":1076},[402],"height:0.7224em;vertical-align:-0.0391em;",[363,1078,554],{"className":1079,"style":682},[407,408],[363,1081],{"className":1082,"style":506},[505],[363,1084,1061],{"className":1085},[510],[363,1087],{"className":1088,"style":506},[505],[363,1090,1092,1095],{"className":1091},[398],[363,1093],{"className":1094,"style":913},[402],[363,1096,561],{"className":1097},[407,408],"），记",[363,1100,1102],{"className":1101,"translate":367},[528],[363,1103,1105,1149],{"className":1104,"translate":367},[366],[363,1106,1108],{"className":1107},[371],[373,1109,1110],{"xmlns":375,"display":537},[377,1111,1112,1146],{},[380,1113,1114,1117,1119,1122,1124,1126,1140,1142,1144],{},[383,1115,1116],{},"t",[458,1118,461],{"stretchy":460},[383,1120,1121],{},"x",[458,1123,468],{"stretchy":460},[458,1125,471],{},[586,1127,1128,1134],{},[380,1129,1130,1132],{},[383,1131,592],{},[383,1133,554],{},[380,1135,1136,1138],{},[383,1137,592],{},[383,1139,561],{},[458,1141,461],{"stretchy":460},[383,1143,1121],{},[458,1145,468],{"stretchy":460},[387,1147,1148],{"encoding":389},"t(x)=\\frac{dP}{dQ}(x)",[363,1150,1152,1179],{"className":1151,"ariaHidden":394},[393],[363,1153,1155,1158,1161,1164,1167,1170,1173,1176],{"className":1154},[398],[363,1156],{"className":1157,"style":487},[402],[363,1159,1116],{"className":1160},[407,408],[363,1162,461],{"className":1163},[494],[363,1165,1121],{"className":1166},[407,408],[363,1168,468],{"className":1169},[501],[363,1171],{"className":1172,"style":506},[505],[363,1174,471],{"className":1175},[510],[363,1177],{"className":1178,"style":506},[505],[363,1180,1182,1186,1254,1257,1260],{"className":1181},[398],[363,1183],{"className":1184,"style":1185},[402],"height:2.2519em;vertical-align:-0.8804em;",[363,1187,1189,1192,1251],{"className":1188},[407],[363,1190],{"className":1191},[494,789],[363,1193,1195],{"className":1194},[586],[363,1196,1198,1243],{"className":1197},[633,634],[363,1199,1201,1240],{"className":1200},[638],[363,1202,1204,1218,1226],{"className":1203,"style":802},[642],[363,1205,1206,1209],{"style":805},[363,1207],{"className":1208,"style":809},[650],[363,1210,1212,1215],{"className":1211},[407],[363,1213,592],{"className":1214},[407,408],[363,1216,561],{"className":1217},[407,408],[363,1219,1220,1223],{"style":821},[363,1221],{"className":1222,"style":809},[650],[363,1224],{"className":1225,"style":829},[828],[363,1227,1228,1231],{"style":832},[363,1229],{"className":1230,"style":809},[650],[363,1232,1234,1237],{"className":1233},[407],[363,1235,592],{"className":1236},[407,408],[363,1238,554],{"className":1239,"style":682},[407,408],[363,1241,666],{"className":1242},[665],[363,1244,1246],{"className":1245},[638],[363,1247,1249],{"className":1248,"style":854},[642],[363,1250],{},[363,1252],{"className":1253},[501,789],[363,1255,461],{"className":1256},[494],[363,1258,1121],{"className":1259},[407,408],[363,1261,468],{"className":1262},[501],[11,1264,1265,1266,1338],{},"为 Radon–Nikodym 导数（在离散情形就是概率质量之比，在密度存在时就是密度之比 ",[363,1267,1269,1301],{"className":1268,"translate":367},[366],[363,1270,1272],{"className":1271},[371],[373,1273,1274],{"xmlns":375},[377,1275,1276,1298],{},[380,1277,1278,1280,1282,1284,1286,1289,1292,1294,1296],{},[383,1279,11],{},[458,1281,461],{"stretchy":460},[383,1283,1121],{},[458,1285,468],{"stretchy":460},[383,1287,1288],{"mathvariant":557},"\u002F",[383,1290,1291],{},"q",[458,1293,461],{"stretchy":460},[383,1295,1121],{},[458,1297,468],{"stretchy":460},[387,1299,1300],{"encoding":389},"p(x)\u002Fq(x)",[363,1302,1304],{"className":1303,"ariaHidden":394},[393],[363,1305,1307,1310,1313,1316,1319,1322,1325,1329,1332,1335],{"className":1306},[398],[363,1308],{"className":1309,"style":487},[402],[363,1311,11],{"className":1312},[407,408],[363,1314,461],{"className":1315},[494],[363,1317,1121],{"className":1318},[407,408],[363,1320,468],{"className":1321},[501],[363,1323,1288],{"className":1324},[407],[363,1326,1291],{"className":1327,"style":1328},[407,408],"margin-right:0.0359em;",[363,1330,461],{"className":1331},[494],[363,1333,1121],{"className":1334},[407,408],[363,1336,468],{"className":1337},[501],"）。",[143,1340,1341,1577,1984,2016,2596],{},[11,1342,1343,1347,1348,1449,1450,1515,1516,1015,1544,1572,1573,1576],{},[1344,1345,1346],"strong",{},"定义 1.1（f-散度）"," 给定一个凸函数 ",[363,1349,1351,1385],{"className":1350,"translate":367},[366],[363,1352,1354],{"className":1353},[371],[373,1355,1356],{"xmlns":375},[377,1357,1358,1382],{},[380,1359,1360,1362,1365,1367,1369,1371,1374,1376,1379],{},[383,1361,385],{},[458,1363,1364],{},":",[458,1366,578],{"stretchy":460},[463,1368,474],{},[458,1370,898],{"separator":394},[383,1372,1373],{"mathvariant":557},"∞",[458,1375,468],{"stretchy":460},[458,1377,1378],{},"→",[383,1380,1381],{"mathvariant":570},"R",[387,1383,1384],{"encoding":389},"f:[0,\\infty)\\to\\mathbb R",[363,1386,1388,1406,1439],{"className":1387,"ariaHidden":394},[393],[363,1389,1391,1394,1397,1400,1403],{"className":1390},[398],[363,1392],{"className":1393,"style":403},[402],[363,1395,385],{"className":1396,"style":409},[407,408],[363,1398],{"className":1399,"style":506},[505],[363,1401,1364],{"className":1402},[510],[363,1404],{"className":1405,"style":506},[505],[363,1407,1409,1412,1415,1418,1421,1424,1427,1430,1433,1436],{"className":1408},[398],[363,1410],{"className":1411,"style":487},[402],[363,1413,578],{"className":1414},[494],[363,1416,474],{"className":1417},[407],[363,1419,898],{"className":1420},[920],[363,1422],{"className":1423,"style":754},[505],[363,1425,1373],{"className":1426},[407],[363,1428,468],{"className":1429},[501],[363,1431],{"className":1432,"style":506},[505],[363,1434,1378],{"className":1435},[510],[363,1437],{"className":1438,"style":506},[505],[363,1440,1442,1446],{"className":1441},[398],[363,1443],{"className":1444,"style":1445},[402],"height:0.6889em;",[363,1447,1381],{"className":1448},[407,714],"，且满足 ",[363,1451,1453,1476],{"className":1452,"translate":367},[366],[363,1454,1456],{"className":1455},[371],[373,1457,1458],{"xmlns":375},[377,1459,1460,1474],{},[380,1461,1462,1464,1466,1468,1470,1472],{},[383,1463,385],{},[458,1465,461],{"stretchy":460},[463,1467,465],{},[458,1469,468],{"stretchy":460},[458,1471,471],{},[463,1473,474],{},[387,1475,477],{"encoding":389},[363,1477,1479,1506],{"className":1478,"ariaHidden":394},[393],[363,1480,1482,1485,1488,1491,1494,1497,1500,1503],{"className":1481},[398],[363,1483],{"className":1484,"style":487},[402],[363,1486,385],{"className":1487,"style":409},[407,408],[363,1489,461],{"className":1490},[494],[363,1492,465],{"className":1493},[407],[363,1495,468],{"className":1496},[501],[363,1498],{"className":1499,"style":506},[505],[363,1501,471],{"className":1502},[510],[363,1504],{"className":1505,"style":506},[505],[363,1507,1509,1512],{"className":1508},[398],[363,1510],{"className":1511,"style":520},[402],[363,1513,474],{"className":1514},[407],"，则 ",[363,1517,1519,1532],{"className":1518,"translate":367},[366],[363,1520,1522],{"className":1521},[371],[373,1523,1524],{"xmlns":375},[377,1525,1526,1530],{},[380,1527,1528],{},[383,1529,554],{},[387,1531,554],{"encoding":389},[363,1533,1535],{"className":1534,"ariaHidden":394},[393],[363,1536,1538,1541],{"className":1537},[398],[363,1539],{"className":1540,"style":1011},[402],[363,1542,554],{"className":1543,"style":682},[407,408],[363,1545,1547,1560],{"className":1546,"translate":367},[366],[363,1548,1550],{"className":1549},[371],[373,1551,1552],{"xmlns":375},[377,1553,1554,1558],{},[380,1555,1556],{},[383,1557,561],{},[387,1559,561],{"encoding":389},[363,1561,1563],{"className":1562,"ariaHidden":394},[393],[363,1564,1566,1569],{"className":1565},[398],[363,1567],{"className":1568,"style":913},[402],[363,1570,561],{"className":1571},[407,408]," 的 ",[1344,1574,1575],{},"f-散度"," 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",[363,1988,1990,2003],{"className":1989,"translate":367},[366],[363,1991,1993],{"className":1992},[371],[373,1994,1995],{"xmlns":375},[377,1996,1997,2001],{},[380,1998,1999],{},[383,2000,561],{},[387,2002,561],{"encoding":389},[363,2004,2006],{"className":2005,"ariaHidden":394},[393],[363,2007,2009,2012],{"className":2008},[398],[363,2010],{"className":2011,"style":913},[402],[363,2013,561],{"className":2014},[407,408]," 的期望形式",[363,2017,2019],{"className":2018,"translate":367},[528],[363,2020,2022,2156],{"className":2021,"translate":367},[366],[363,2023,2025],{"className":2024},[371],[373,2026,2027],{"xmlns":375,"display":537},[377,2028,2029,2153],{},[1591,2030,2031,2103],{"rowspacing":1593,"columnalign":1594,"columnspacing":1595},[1597,2032,2033,2055],{},[1600,2034,2035],{},[1603,2036,2037],{"scriptlevel":474,"displaystyle":394},[380,2038,2039,2045,2047,2049,2051,2053],{},[543,2040,2041,2043],{},[383,2042,547],{},[383,2044,385],{},[458,2046,461],{"stretchy":460},[383,2048,554],{},[383,2050,558],{"mathvariant":557},[383,2052,561],{},[458,2054,468],{"stretchy":460},[1600,2056,2057],{},[1603,2058,2059],{"scriptlevel":474,"displaystyle":394},[380,2060,2061,2063,2065,2071,2073],{},[380,2062],{},[458,2064,471],{},[543,2066,2067,2069],{},[383,2068,571],{"mathvariant":570},[383,2070,561],{},[1645,2072,1652],{},[380,2074,2075,2077,2079,2081,2101],{},[458,2076,578],{"fence":394},[383,2078,385],{},[1645,2080,1652],{},[380,2082,2083,2085,2099],{},[458,2084,461],{"fence":394},[586,2086,2087,2093],{},[380,2088,2089,2091],{},[383,2090,592],{},[383,2092,554],{},[380,2094,2095,2097],{},[383,2096,592],{},[383,2098,561],{},[458,2100,468],{"fence":394},[458,2102,605],{"fence":394},[1597,2104,2105,2111],{},[1600,2106,2107],{},[1603,2108,2109],{"scriptlevel":474,"displaystyle":394},[380,2110],{},[1600,2112,2113],{},[1603,2114,2115],{"scriptlevel":474,"displaystyle":394},[380,2116,2117,2119,2121,2123,2125,2127,2147,2149,2151],{},[380,2118],{},[458,2120,471],{},[458,2122,1635],{},[383,2124,385],{},[1645,2126,1652],{},[380,2128,2129,2131,2145],{},[458,2130,461],{"fence":394},[586,2132,2133,2139],{},[380,2134,2135,2137],{},[383,2136,592],{},[383,2138,554],{},[380,2140,2141,2143],{},[383,2142,592],{},[383,2144,561],{},[458,2146,468],{"fence":394},[383,2148,592],{},[383,2150,561],{},[458,2152,898],{"separator":394},[387,2154,2155],{"encoding":389},"\\begin{aligned}\nD_f(P\\|Q)\n&=\\mathbb E_Q\\!\\left[\nf\\!\\left(\\frac{dP}{dQ}\\right)\\right]\\\\\n&=\\int f\\!\\left(\\frac{dP}{dQ}\\right)dQ,\n\\end{aligned}",[363,2157,2159],{"className":2158,"ariaHidden":394},[393],[363,2160,2162,2166],{"className":2161},[398],[363,2163],{"className":2164,"style":2165},[402],"height:5.1001em;vertical-align:-2.3em;",[363,2167,2169],{"className":2168},[407],[363,2170,2172,2270],{"className":2171},[1591],[363,2173,2175],{"className":2174},[1708],[363,2176,2178,2261],{"className":2177},[633,634],[363,2179,2181,2258],{"className":2180},[638],[363,2182,2185,2249],{"className":2183,"style":2184},[642],"height:2.8em;",[363,2186,2188,2191],{"style":2187},"top:-4.8em;",[363,2189],{"className":2190,"style":1725},[650],[363,2192,2194,2234,2237,2240,2243,2246],{"className":2193},[407],[363,2195,2197,2200],{"className":2196},[407],[363,2198,547],{"className":2199,"style":625},[407,408],[363,2201,2203],{"className":2202},[629],[363,2204,2206,2226],{"className":2205},[633,634],[363,2207,2209,2223],{"className":2208},[638],[363,2210,2212],{"className":2211,"style":643},[642],[363,2213,2214,2217],{"style":646},[363,2215],{"className":2216,"style":651},[650],[363,2218,2220],{"className":2219},[655,656,657,658],[363,2221,385],{"className":2222,"style":409},[407,408,658],[363,2224,666],{"className":2225},[665],[363,2227,2229],{"className":2228},[638],[363,2230,2232],{"className":2231,"style":673},[642],[363,2233],{},[363,2235,461],{"className":2236},[494],[363,2238,554],{"className":2239,"style":682},[407,408],[363,2241,558],{"className":2242},[407],[363,2244,561],{"className":2245},[407,408],[363,2247,468],{"className":2248},[501],[363,2250,2252,2255],{"style":2251},"top:-2.1em;",[363,2253],{"className":2254,"style":1725},[650],[363,2256],{"className":2257},[407],[363,2259,666],{"className":2260},[665],[363,2262,2264],{"className":2263},[638],[363,2265,2268],{"className":2266,"style":2267},[642],"height:2.3em;",[363,2269],{},[363,2271,2273],{"className":2272},[1799],[363,2274,2276,2588],{"className":2275},[633,634],[363,2277,2279,2585],{"className":2278},[638],[363,2280,2282,2455],{"className":2281,"style":2184},[642],[363,2283,2284,2287],{"style":2187},[363,2285],{"className":2286,"style":1725},[650],[363,2288,2290,2293,2296,2299,2302,2342,2345,2348],{"className":2289},[407],[363,2291],{"className":2292},[407],[363,2294],{"className":2295,"style":506},[505],[363,2297,471],{"className":2298},[510],[363,2300],{"className":2301,"style":506},[505],[363,2303,2305,2308],{"className":2304},[407],[363,2306,571],{"className":2307},[407,714],[363,2309,2311],{"className":2310},[629],[363,2312,2314,2334],{"className":2313},[633,634],[363,2315,2317,2331],{"className":2316},[638],[363,2318,2320],{"className":2319,"style":727},[642],[363,2321,2322,2325],{"style":730},[363,2323],{"className":2324,"style":651},[650],[363,2326,2328],{"className":2327},[655,656,657,658],[363,2329,561],{"className":2330},[407,408,658],[363,2332,666],{"className":2333},[665],[363,2335,2337],{"className":2336},[638],[363,2338,2340],{"className":2339,"style":673},[642],[363,2341],{},[363,2343],{"className":2344,"style":1860},[505],[363,2346],{"className":2347,"style":754},[505],[363,2349,2351,2357,2360,2363,2366,2449],{"className":2350},[758],[363,2352,2354],{"className":2353,"style":763},[494,762],[363,2355,578],{"className":2356},[767,657],[363,2358,385],{"className":2359,"style":409},[407,408],[363,2361],{"className":2362,"style":1860},[505],[363,2364],{"className":2365,"style":754},[505],[363,2367,2369,2375,2443],{"className":2368},[758],[363,2370,2372],{"className":2371,"style":763},[494,762],[363,2373,461],{"className":2374},[767,657],[363,2376,2378,2381,2440],{"className":2377},[407],[363,2379],{"className":2380},[494,789],[363,2382,2384],{"className":2383},[586],[363,2385,2387,2432],{"className":2386},[633,634],[363,2388,2390,2429],{"className":2389},[638],[363,2391,2393,2407,2415],{"className":2392,"style":802},[642],[363,2394,2395,2398],{"style":805},[363,2396],{"className":2397,"style":809},[650],[363,2399,2401,2404],{"className":2400},[407],[363,2402,592],{"className":2403},[407,408],[363,2405,561],{"className":2406},[407,408],[363,2408,2409,2412],{"style":821},[363,2410],{"className":2411,"style":809},[650],[363,2413],{"className":2414,"style":829},[828],[363,2416,2417,2420],{"style":832},[363,2418],{"className":2419,"style":809},[650],[363,2421,2423,2426],{"className":2422},[407],[363,2424,592],{"className":2425},[407,408],[363,2427,554],{"className":2428,"style":682},[407,408],[363,2430,666],{"className":2431},[665],[363,2433,2435],{"className":2434},[638],[363,2436,2438],{"className":2437,"style":854},[642],[363,2439],{},[363,2441],{"className":2442},[501,789],[363,2444,2446],{"className":2445,"style":763},[501,762],[363,2447,468],{"className":2448},[767,657],[363,2450,2452],{"className":2451,"style":763},[501,762],[363,2453,605],{"className":2454},[767,657],[363,2456,2457,2460],{"style":2251},[363,2458],{"className":2459,"style":1725},[650],[363,2461,2463,2466,2469,2472,2475,2478,2481,2484,2487,2490,2573,2576,2579,2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0\\,d\\mu\n\\end{aligned}",[363,3660,3662],{"className":3661,"ariaHidden":394},[393],[363,3663,3665,3669],{"className":3664},[398],[363,3666],{"className":3667,"style":3668},[402],"height:5.2839em;vertical-align:-2.392em;",[363,3670,3672],{"className":3671},[407],[363,3673,3675,3773],{"className":3674},[1591],[363,3676,3678],{"className":3677},[1708],[363,3679,3681,3764],{"className":3680},[633,634],[363,3682,3684,3761],{"className":3683},[638],[363,3685,3688,3752],{"className":3686,"style":3687},[642],"height:2.892em;",[363,3689,3691,3694],{"style":3690},"top:-4.892em;",[363,3692],{"className":3693,"style":1725},[650],[363,3695,3697,3737,3740,3743,3746,3749],{"className":3696},[407],[363,3698,3700,3703],{"className":3699},[407],[363,3701,547],{"className":3702,"style":625},[407,408],[363,3704,3706],{"className":3705},[629],[363,3707,3709,3729],{"className":3708},[633,634],[363,3710,3712,3726],{"className":3711},[638],[363,3713,3715],{"className":3714,"style":643},[642],[363,3716,3717,3720],{"style":646},[363,3718],{"className":3719,"style":651},[650],[363,3721,3723],{"className":3722},[655,656,657,658],[363,3724,385],{"className":3725,"style":409},[407,408,658],[363,3727,666],{"className":3728},[665],[363,3730,3732],{"className":3731},[638],[363,3733,3735],{"className":3734,"style":673},[642],[363,3736],{},[363,3738,461],{"className":3739},[494],[363,3741,554],{"className":3742,"style":682},[407,408],[363,3744,558],{"className":3745},[407],[363,3747,561],{"className":3748},[407,408],[363,3750,468],{"className":3751},[501],[363,3753,3755,3758],{"style":3754},"top:-2.145em;",[363,3756],{"className":3757,"style":1725},[650],[363,3759],{"className":3760},[407],[363,3762,666],{"className":3763},[665],[363,3765,3767],{"className":3766},[638],[363,3768,3771],{"className":3769,"style":3770},[642],"height:2.392em;",[363,3772],{},[363,3774,3776],{"className":3775},[1799],[363,3777,3779,4162],{"className":3778},[633,634],[363,3780,3782,4159],{"className":3781},[638],[363,3783,3785,3968],{"className":3784,"style":3687},[642],[363,3786,3787,3790],{"style":3690},[363,3788],{"className":3789,"style":1725},[650],[363,3791,3793,3796,3799,3802,3805,3863,3866,3869,3872,3875,3953,3956,3959,3962,3965],{"className":3792},[407],[363,3794],{"className":3795},[407],[363,3797],{"className":3798,"style":506},[505],[363,3800,471],{"className":3801},[510],[363,3803],{"className":3804,"style":506},[505],[363,3806,3808,3811],{"className":3807},[1832],[363,3809,1635],{"className":3810,"style":1835},[1832,1833,1834],[363,3812,3814],{"className":3813},[629],[363,3815,3817,3854],{"className":3816},[633,634],[363,3818,3820,3851],{"className":3819},[638],[363,3821,3824],{"className":3822,"style":3823},[642],"height:-0.3869em;",[363,3825,3827,3830],{"style":3826},"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;",[363,3828],{"className":3829,"style":651},[650],[363,3831,3833],{"className":3832},[655,656,657,658],[363,3834,3836,3839,3842,3845,3848],{"className":3835},[407,658],[363,3837,3246],{"className":3838},[494,658],[363,3840,1291],{"className":3841,"style":1328},[407,408,658],[363,3843,3251],{"className":3844},[510,658],[363,3846,474],{"className":3847},[407,658],[363,3849,3256],{"className":3850},[501,658],[363,3852,666],{"className":3853},[665],[363,3855,3857],{"className":3856},[638],[363,3858,3861],{"className":3859,"style":3860},[642],"height:1.0869em;",[363,3862],{},[363,3864],{"className":3865,"style":754},[505],[363,3867,385],{"className":3868,"style":409},[407,408],[363,3870],{"className":3871,"style":1860},[505],[363,3873],{"className":3874,"style":754},[505],[363,3876,3878,3884,3947],{"className":3877},[758],[363,3879,3881],{"className":3880,"style":763},[494,762],[363,3882,461],{"className":3883},[767,657],[363,3885,3887,3890,3944],{"className":3886},[407],[363,3888],{"className":3889},[494,789],[363,3891,3893],{"className":3892},[586],[363,3894,3896,3936],{"className":3895},[633,634],[363,3897,3899,3933],{"className":3898},[638],[363,3900,3903,3914,3922],{"className":3901,"style":3902},[642],"height:1.1076em;",[363,3904,3905,3908],{"style":805},[363,3906],{"className":3907,"style":809},[650],[363,3909,3911],{"className":3910},[407],[363,3912,1291],{"className":3913,"style":1328},[407,408],[363,3915,3916,3919],{"style":821},[363,3917],{"className":3918,"style":809},[650],[363,3920],{"className":3921,"style":829},[828],[363,3923,3924,3927],{"style":832},[363,3925],{"className":3926,"style":809},[650],[363,3928,3930],{"className":3929},[407],[363,3931,11],{"className":3932},[407,408],[363,3934,666],{"className":3935},[665],[363,3937,3939],{"className":3938},[638],[363,3940,3942],{"className":3941,"style":854},[642],[363,3943],{},[363,3945],{"className":3946},[501,789],[363,3948,3950],{"className":3949,"style":763},[501,762],[363,3951,468],{"className":3952},[767,657],[363,3954],{"className":3955,"style":754},[505],[363,3957,1291],{"className":3958,"style":1328},[407,408],[363,3960],{"className":3961,"style":754},[505],[363,3963,592],{"className":3964},[407,408],[363,3966,2712],{"className":3967},[407,408],[363,3969,3970,3973],{"style":3754},[363,3971],{"className":3972,"style":1725},[650],[363,3974,3976,3979,3983,3986,3989,3992,4047,4050,4053,4056,4059,4138,4141,4144,4147,4150,4153,4156],{"className":3975},[407],[363,3977],{"className":3978},[407],[363,3980],{"className":3981,"style":3982},[505],"margin-right:2em;",[363,3984],{"className":3985,"style":2935},[505],[363,3987,2818],{"className":3988},[2939],[363,3990],{"className":3991,"style":2935},[505],[363,3993,3995,3998],{"className":3994},[1832],[363,3996,1635],{"className":3997,"style":1835},[1832,1833,1834],[363,3999,4001],{"className":4000},[629],[363,4002,4004,4039],{"className":4003},[633,634],[363,4005,4007,4036],{"className":4006},[638],[363,4008,4010],{"className":4009,"style":3823},[642],[363,4011,4012,4015],{"style":3826},[363,4013],{"className":4014,"style":651},[650],[363,4016,4018],{"className":4017},[655,656,657,658],[363,4019,4021,4024,4027,4030,4033],{"className":4020},[407,658],[363,4022,3246],{"className":4023},[494,658],[363,4025,1291],{"className":4026,"style":1328},[407,408,658],[363,4028,471],{"className":4029},[510,658],[363,4031,474],{"className":4032},[407,658],[363,4034,3256],{"className":4035},[501,658],[363,4037,666],{"className":4038},[665],[363,4040,4042],{"className":4041},[638],[363,4043,4045],{"className":4044,"style":3860},[642],[363,4046],{},[363,4048],{"className":4049,"style":754},[505],[363,4051,385],{"className":4052,"style":409},[407,408],[363,4054],{"className":4055,"style":1860},[505],[363,4057],{"className":4058,"style":754},[505],[363,4060,4062,4069,4132],{"className":4061},[758],[363,4063,4065],{"className":4064,"style":763},[494,762],[363,4066,461],{"className":4067},[767,4068],"size2",[363,4070,4072,4075,4129],{"className":4071},[407],[363,4073],{"className":4074},[494,789],[363,4076,4078],{"className":4077},[586],[363,4079,4081,4120],{"className":4080},[633,634],[363,4082,4084,4117],{"className":4083},[638],[363,4085,4087,4098,4106],{"className":4086,"style":3902},[642],[363,4088,4089,4092],{"style":805},[363,4090],{"className":4091,"style":809},[650],[363,4093,4095],{"className":4094},[407],[363,4096,474],{"className":4097},[407],[363,4099,4100,4103],{"style":821},[363,4101],{"className":4102,"style":809},[650],[363,4104],{"className":4105,"style":829},[828],[363,4107,4108,4111],{"style":832},[363,4109],{"className":4110,"style":809},[650],[363,4112,4114],{"className":4113},[407],[363,4115,11],{"className":4116},[407,408],[363,4118,666],{"className":4119},[665],[363,4121,4123],{"className":4122},[638],[363,4124,4127],{"className":4125,"style":4126},[642],"height:0.686em;",[363,4128],{},[363,4130],{"className":4131},[501,789],[363,4133,4135],{"className":4134,"style":763},[501,762],[363,4136,468],{"className":4137},[767,4068],[363,4139],{"className":4140,"style":2935},[505],[363,4142,3647],{"className":4143},[2939],[363,4145],{"className":4146,"style":2935},[505],[363,4148,474],{"className":4149},[407],[363,4151],{"className":4152,"style":754},[505],[363,4154,592],{"className":4155},[407,408],[363,4157,2712],{"className":4158},[407,408],[363,4160,666],{"className":4161},[665],[363,4163,4165],{"className":4164},[638],[363,4166,4168],{"className":4167,"style":3770},[642],[363,4169],{},[11,4171,4172,4173,4201,4202,4231,4232,4320,4321,4349,4350,4421,4422,4513,4514,4201,4542,4570,4571,4606],{},"第二项正是处理「支撑集不重叠」的地方。若 ",[363,4174,4176,4189],{"className":4175,"translate":367},[366],[363,4177,4179],{"className":4178},[371],[373,4180,4181],{"xmlns":375},[377,4182,4183,4187],{},[380,4184,4185],{},[383,4186,385],{},[387,4188,385],{"encoding":389},[363,4190,4192],{"className":4191,"ariaHidden":394},[393],[363,4193,4195,4198],{"className":4194},[398],[363,4196],{"className":4197,"style":403},[402],[363,4199,385],{"className":4200,"style":409},[407,408]," 在 ",[363,4203,4205,4219],{"className":4204,"translate":367},[366],[363,4206,4208],{"className":4207},[371],[373,4209,4210],{"xmlns":375},[377,4211,4212,4216],{},[380,4213,4214],{},[383,4215,1373],{"mathvariant":557},[387,4217,4218],{"encoding":389},"\\infty",[363,4220,4222],{"className":4221,"ariaHidden":394},[393],[363,4223,4225,4228],{"className":4224},[398],[363,4226],{"className":4227,"style":2693},[402],[363,4229,1373],{"className":4230},[407]," 处有界，可约定 ",[363,4233,4235,4263],{"className":4234,"translate":367},[366],[363,4236,4238],{"className":4237},[371],[373,4239,4240],{"xmlns":375},[377,4241,4242,4260],{},[380,4243,4244,4246,4248,4250,4252,4254,4256,4258],{},[463,4245,474],{},[458,4247,3647],{},[383,4249,385],{},[458,4251,461],{"stretchy":460},[383,4253,1373],{"mathvariant":557},[458,4255,468],{"stretchy":460},[458,4257,471],{},[463,4259,474],{},[387,4261,4262],{"encoding":389},"0\\cdot f(\\infty)=0",[363,4264,4266,4284,4311],{"className":4265,"ariaHidden":394},[393],[363,4267,4269,4272,4275,4278,4281],{"className":4268},[398],[363,4270],{"className":4271,"style":520},[402],[363,4273,474],{"className":4274},[407],[363,4276],{"className":4277,"style":2935},[505],[363,4279,3647],{"className":4280},[2939],[363,4282],{"className":4283,"style":2935},[505],[363,4285,4287,4290,4293,4296,4299,4302,4305,4308],{"className":4286},[398],[363,4288],{"className":4289,"style":487},[402],[363,4291,385],{"className":4292,"style":409},[407,408],[363,4294,461],{"className":4295},[494],[363,4297,1373],{"className":4298},[407],[363,4300,468],{"className":4301},[501],[363,4303],{"className":4304,"style":506},[505],[363,4306,471],{"className":4307},[510],[363,4309],{"className":4310,"style":506},[505],[363,4312,4314,4317],{"className":4313},[398],[363,4315],{"className":4316,"style":520},[402],[363,4318,474],{"className":4319},[407],"；若 ",[363,4322,4324,4337],{"className":4323,"translate":367},[366],[363,4325,4327],{"className":4326},[371],[373,4328,4329],{"xmlns":375},[377,4330,4331,4335],{},[380,4332,4333],{},[383,4334,385],{},[387,4336,385],{"encoding":389},[363,4338,4340],{"className":4339,"ariaHidden":394},[393],[363,4341,4343,4346],{"className":4342},[398],[363,4344],{"className":4345,"style":403},[402],[363,4347,385],{"className":4348,"style":409},[407,408]," 无界，则这一项是否发散决定了 ",[363,4351,4353,4371],{"className":4352,"translate":367},[366],[363,4354,4356],{"className":4355},[371],[373,4357,4358],{"xmlns":375},[377,4359,4360,4368],{},[380,4361,4362],{},[543,4363,4364,4366],{},[383,4365,547],{},[383,4367,385],{},[387,4369,4370],{"encoding":389},"D_f",[363,4372,4374],{"className":4373,"ariaHidden":394},[393],[363,4375,4377,4381],{"className":4376},[398],[363,4378],{"className":4379,"style":4380},[402],"height:0.9694em;vertical-align:-0.2861em;",[363,4382,4384,4387],{"className":4383},[407],[363,4385,547],{"className":4386,"style":625},[407,408],[363,4388,4390],{"className":4389},[629],[363,4391,4393,4413],{"className":4392},[633,634],[363,4394,4396,4410],{"className":4395},[638],[363,4397,4399],{"className":4398,"style":643},[642],[363,4400,4401,4404],{"style":646},[363,4402],{"className":4403,"style":651},[650],[363,4405,4407],{"className":4406},[655,656,657,658],[363,4408,385],{"className":4409,"style":409},[407,408,658],[363,4411,666],{"className":4412},[665],[363,4414,4416],{"className":4415},[638],[363,4417,4419],{"className":4418,"style":673},[642],[363,4420],{}," 是否有限。例如对 KL 的 ",[363,4423,4425,4457],{"className":4424,"translate":367},[366],[363,4426,4428],{"className":4427},[371],[373,4429,4430],{"xmlns":375},[377,4431,4432,4454],{},[380,4433,4434,4436,4438,4440,4442,4444,4446,4449,4452],{},[383,4435,385],{},[458,4437,461],{"stretchy":460},[383,4439,1116],{},[458,4441,468],{"stretchy":460},[458,4443,471],{},[383,4445,1116],{},[383,4447,4448],{},"log",[458,4450,4451],{},"⁡",[383,4453,1116],{},[387,4455,4456],{"encoding":389},"f(t)=t\\log t",[363,4458,4460,4487],{"className":4459,"ariaHidden":394},[393],[363,4461,4463,4466,4469,4472,4475,4478,4481,4484],{"className":4462},[398],[363,4464],{"className":4465,"style":487},[402],[363,4467,385],{"className":4468,"style":409},[407,408],[363,4470,461],{"className":4471},[494],[363,4473,1116],{"className":4474},[407,408],[363,4476,468],{"className":4477},[501],[363,4479],{"className":4480,"style":506},[505],[363,4482,471],{"className":4483},[510],[363,4485],{"className":4486,"style":506},[505],[363,4488,4490,4493,4496,4499,4507,4510],{"className":4489},[398],[363,4491],{"className":4492,"style":403},[402],[363,4494,1116],{"className":4495},[407,408],[363,4497],{"className":4498,"style":754},[505],[363,4500,4502,4503],{"className":4501},[1832],"lo",[363,4504,4506],{"style":4505},"margin-right:0.0139em;","g",[363,4508],{"className":4509,"style":754},[505],[363,4511,1116],{"className":4512},[407,408],"，当 ",[363,4515,4517,4530],{"className":4516,"translate":367},[366],[363,4518,4520],{"className":4519},[371],[373,4521,4522],{"xmlns":375},[377,4523,4524,4528],{},[380,4525,4526],{},[383,4527,554],{},[387,4529,554],{"encoding":389},[363,4531,4533],{"className":4532,"ariaHidden":394},[393],[363,4534,4536,4539],{"className":4535},[398],[363,4537],{"className":4538,"style":1011},[402],[363,4540,554],{"className":4541,"style":682},[407,408],[363,4543,4545,4558],{"className":4544,"translate":367},[366],[363,4546,4548],{"className":4547},[371],[373,4549,4550],{"xmlns":375},[377,4551,4552,4556],{},[380,4553,4554],{},[383,4555,561],{},[387,4557,561],{"encoding":389},[363,4559,4561],{"className":4560,"ariaHidden":394},[393],[363,4562,4564,4567],{"className":4563},[398],[363,4565],{"className":4566,"style":913},[402],[363,4568,561],{"className":4569},[407,408]," 的零测集上放质量时这一项为 ",[363,4572,4574,4590],{"className":4573,"translate":367},[366],[363,4575,4577],{"className":4576},[371],[373,4578,4579],{"xmlns":375},[377,4580,4581,4587],{},[380,4582,4583,4585],{},[458,4584,2818],{},[383,4586,1373],{"mathvariant":557},[387,4588,4589],{"encoding":389},"+\\infty",[363,4591,4593],{"className":4592,"ariaHidden":394},[393],[363,4594,4596,4600,4603],{"className":4595},[398],[363,4597],{"className":4598,"style":4599},[402],"height:0.6667em;vertical-align:-0.0833em;",[363,4601,2818],{"className":4602},[407],[363,4604,1373],{"className":4605},[407],"。因此约定：",[363,4608,4610],{"className":4609,"translate":367},[528],[363,4611,4613,4727],{"className":4612,"translate":367},[366],[363,4614,4616],{"className":4615},[371],[373,4617,4618],{"xmlns":375,"display":537},[377,4619,4620,4724],{},[380,4621,4622,4624,4627,4629,4632,4635,4637,4643,4645,4647,4649,4651,4653,4655],{},[383,4623,554],{},[458,4625,4626],{},"≪̸",[383,4628,561],{},[1645,4630,4631],{}," ",[458,4633,4634],{},"⟹",[1645,4636,4631],{},[543,4638,4639,4641],{},[383,4640,547],{},[383,4642,385],{},[458,4644,461],{"stretchy":460},[383,4646,554],{},[383,4648,558],{"mathvariant":557},[383,4650,561],{},[458,4652,468],{"stretchy":460},[458,4654,471],{},[380,4656,4657,4659],{},[458,4658,3246],{"fence":394},[1591,4660,4664,4699],{"rowspacing":4661,"columnalign":4662,"columnspacing":4663},"0.36em","left left","1em",[1597,4665,4666,4678],{},[1600,4667,4668],{},[1603,4669,4670],{"scriptlevel":474,"displaystyle":460},[380,4671,4672,4674,4676],{},[458,4673,2818],{},[383,4675,1373],{"mathvariant":557},[458,4677,898],{"separator":394},[1600,4679,4680],{},[1603,4681,4682],{"scriptlevel":474,"displaystyle":460},[380,4683,4684,4687,4689,4692,4694,4697],{},[1645,4685,4686],{},"若 ",[383,4688,385],{},[1645,4690,4691],{}," 在 ",[383,4693,1373],{"mathvariant":557},[1645,4695,4696],{}," 处增长",[458,4698,898],{"separator":394},[1597,4700,4701,4712],{},[1600,4702,4703],{},[1603,4704,4705],{"scriptlevel":474,"displaystyle":460},[380,4706,4707,4710],{},[1645,4708,4709],{},"有限极限值",[458,4711,898],{"separator":394},[1600,4713,4714],{},[1603,4715,4716],{"scriptlevel":474,"displaystyle":460},[380,4717,4718,4721],{},[1645,4719,4720],{},"否则",[383,4722,4723],{"mathvariant":557},".",[387,4725,4726],{"encoding":389},"P\\not\\ll Q\\ \\Longrightarrow\\\nD_f(P\\|Q)=\n\\begin{cases}\n+\\infty, & \\text{若 } f \\text{ 在 } \\infty\n\\text{ 处增长},\\\\[2pt]\n\\text{有限极限值}, & \\text{否则}.\n\\end{cases}",[363,4728,4730,4785,4809,4879],{"className":4729,"ariaHidden":394},[393],[363,4731,4733,4736,4739,4742,4775,4779,4782],{"className":4732},[398],[363,4734],{"className":4735,"style":403},[402],[363,4737,554],{"className":4738,"style":682},[407,408],[363,4740],{"className":4741,"style":506},[505],[363,4743,4745],{"className":4744},[510],[363,4746,4749],{"className":4747},[407,4748],"vbox",[363,4750,4753],{"className":4751},[4752],"thinbox",[363,4754,4757,4760,4771],{"className":4755},[4756],"rlap",[363,4758],{"className":4759,"style":403},[402],[363,4761,4764],{"className":4762},[4763],"inner",[363,4765,4767],{"className":4766},[407],[363,4768,4770],{"className":4769},[510],"",[363,4772],{"className":4773},[4774],"fix",[363,4776],{"className":4777},[505,4778],"nobreak",[363,4780,1061],{"className":4781},[510],[363,4783],{"className":4784,"style":506},[505],[363,4786,4788,4791,4794,4797,4800,4803,4806],{"className":4787},[398],[363,4789],{"className":4790,"style":913},[402],[363,4792,561],{"className":4793},[407,408],[363,4795,4631],{"className":4796},[505],[363,4798],{"className":4799,"style":506},[505],[363,4801,4634],{"className":4802},[510],[363,4804,4631],{"className":4805},[505],[363,4807],{"className":4808,"style":506},[505],[363,4810,4812,4815,4855,4858,4861,4864,4867,4870,4873,4876],{"className":4811},[398],[363,4813],{"className":4814,"style":618},[402],[363,4816,4818,4821],{"className":4817},[407],[363,4819,547],{"className":4820,"style":625},[407,408],[363,4822,4824],{"className":4823},[629],[363,4825,4827,4847],{"className":4826},[633,634],[363,4828,4830,4844],{"className":4829},[638],[363,4831,4833],{"className":4832,"style":643},[642],[363,4834,4835,4838],{"style":646},[363,4836],{"className":4837,"style":651},[650],[363,4839,4841],{"className":4840},[655,656,657,658],[363,4842,385],{"className":4843,"style":409},[407,408,658],[363,4845,666],{"className":4846},[665],[363,4848,4850],{"className":4849},[638],[363,4851,4853],{"className":4852,"style":673},[642],[363,4854],{},[363,4856,461],{"className":4857},[494],[363,4859,554],{"className":4860,"style":682},[407,408],[363,4862,558],{"className":4863},[407],[363,4865,561],{"className":4866},[407,408],[363,4868,468],{"className":4869},[501],[363,4871],{"className":4872,"style":506},[505],[363,4874,471],{"className":4875},[510],[363,4877],{"className":4878,"style":506},[505],[363,4880,4882,4886],{"className":4881},[398],[363,4883],{"className":4884,"style":4885},[402],"height:3.08em;vertical-align:-1.29em;",[363,4887,4889,4896,5061],{"className":4888},[758],[363,4890,4892],{"className":4891,"style":763},[494,762],[363,4893,3246],{"className":4894},[767,4895],"size4",[363,4897,4899],{"className":4898},[407],[363,4900,4902,4966,4971],{"className":4901},[1591],[363,4903,4905],{"className":4904},[1799],[363,4906,4908,4957],{"className":4907},[633,634],[363,4909,4911,4954],{"className":4910},[638],[363,4912,4915,4934],{"className":4913,"style":4914},[642],"height:1.79em;",[363,4916,4918,4922],{"style":4917},"top:-3.79em;",[363,4919],{"className":4920,"style":4921},[650],"height:3.008em;",[363,4923,4925,4928,4931],{"className":4924},[407],[363,4926,2818],{"className":4927},[407],[363,4929,1373],{"className":4930},[407],[363,4932,898],{"className":4933},[920],[363,4935,4937,4940],{"style":4936},"top:-2.15em;",[363,4938],{"className":4939,"style":4921},[650],[363,4941,4943,4951],{"className":4942},[407],[363,4944,4947],{"className":4945},[407,4946],"text",[363,4948,4709],{"className":4949},[407,4950],"cjk_fallback",[363,4952,898],{"className":4953},[920],[363,4955,666],{"className":4956},[665],[363,4958,4960],{"className":4959},[638],[363,4961,4964],{"className":4962,"style":4963},[642],"height:1.29em;",[363,4965],{},[363,4967],{"className":4968,"style":4970},[4969],"arraycolsep","width:1em;",[363,4972,4974],{"className":4973},[1799],[363,4975,4977,5053],{"className":4976},[633,634],[363,4978,4980,5050],{"className":4979},[638],[363,4981,4983,5033],{"className":4982,"style":4914},[642],[363,4984,4985,4988],{"style":4917},[363,4986],{"className":4987,"style":4921},[650],[363,4989,4991,5001,5004,5017,5020,5030],{"className":4990},[407],[363,4992,4994,4998],{"className":4993},[407,4946],[363,4995,4997],{"className":4996},[407,4950],"若",[363,4999,4631],{"className":5000},[407],[363,5002,385],{"className":5003,"style":409},[407,408],[363,5005,5007,5010,5014],{"className":5006},[407,4946],[363,5008,4631],{"className":5009},[407],[363,5011,5013],{"className":5012},[407,4950],"在",[363,5015,4631],{"className":5016},[407],[363,5018,1373],{"className":5019},[407],[363,5021,5023,5026],{"className":5022},[407,4946],[363,5024,4631],{"className":5025},[407],[363,5027,5029],{"className":5028},[407,4950],"处增长",[363,5031,898],{"className":5032},[920],[363,5034,5035,5038],{"style":4936},[363,5036],{"className":5037,"style":4921},[650],[363,5039,5041,5047],{"className":5040},[407],[363,5042,5044],{"className":5043},[407,4946],[363,5045,4720],{"class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",[363,5068,5070,5083],{"className":5069,"translate":367},[366],[363,5071,5073],{"className":5072},[371],[373,5074,5075],{"xmlns":375},[377,5076,5077,5081],{},[380,5078,5079],{},[383,5080,385],{},[387,5082,385],{"encoding":389},[363,5084,5086],{"className":5085,"ariaHidden":394},[393],[363,5087,5089,5092],{"className":5088},[398],[363,5090],{"className":5091,"style":403},[402],[363,5093,385],{"className":5094,"style":409},[407,408],[363,5096,5098,5111],{"className":5097,"translate":367},[366],[363,5099,5101],{"className":5100},[371],[373,5102,5103],{"xmlns":375},[377,5104,5105,5109],{},[380,5106,5107],{},[383,5108,1373],{"mathvariant":557},[387,5110,4218],{"encoding":389},[363,5112,5114],{"className":5113,"ariaHidden":394},[393],[363,5115,5117,5120],{"className":5116},[398],[363,5118],{"className":5119,"style":2693},[402],[363,5121,1373],{"className":5122},[407]," 处有界（如 ",[363,5125,5127,5169],{"className":5126,"translate":367},[366],[363,5128,5130],{"className":5129},[371],[373,5131,5132],{"xmlns":375},[377,5133,5134,5166],{},[380,5135,5136,5138,5140,5142,5144,5146,5154,5157,5159,5162,5164],{},[383,5137,385],{},[458,5139,461],{"stretchy":460},[383,5141,1116],{},[458,5143,468],{"stretchy":460},[458,5145,471],{},[1603,5147,5148],{"scriptlevel":474,"displaystyle":460},[586,5149,5150,5152],{},[463,5151,465],{},[463,5153,2811],{},[458,5155,5156],{"stretchy":460},"∣",[383,5158,1116],{},[458,5160,5161],{},"−",[463,5163,465],{},[458,5165,5156],{"stretchy":460},[387,5167,5168],{"encoding":389},"f(t)=\\tfrac12\\lvert t-1\\rvert",[363,5170,5172,5199,5288],{"className":5171,"ariaHidden":394},[393],[363,5173,5175,5178,5181,5184,5187,5190,5193,5196],{"className":5174},[398],[363,5176],{"className":5177,"style":487},[402],[363,5179,385],{"className":5180,"style":409},[407,408],[363,5182,461],{"className":5183},[494],[363,5185,1116],{"className":5186},[407,408],[363,5188,468],{"className":5189},[501],[363,5191],{"className":5192,"style":506},[505],[363,5194,471],{"className":5195},[510],[363,5197],{"className":5198,"style":506},[505],[363,5200,5202,5205,5273,5276,5279,5282,5285],{"className":5201},[398],[363,5203],{"className":5204,"style":2853},[402],[363,5206,5208,5211,5270],{"className":5207},[407],[363,5209],{"className":5210},[494,789],[363,5212,5214],{"className":5213},[586],[363,5215,5217,5262],{"className":5216},[633,634],[363,5218,5220,5259],{"className":5219},[638],[363,5221,5223,5237,5245],{"className":5222,"style":2872},[642],[363,5224,5225,5228],{"style":2875},[363,5226],{"className":5227,"style":809},[650],[363,5229,5231],{"className":5230},[655,656,657,658],[363,5232,5234],{"className":5233},[407,658],[363,5235,2811],{"className":5236},[407,658],[363,5238,5239,5242],{"style":821},[363,5240],{"className":5241,"style":809},[650],[363,5243],{"className":5244,"style":829},[828],[363,5246,5247,5250],{"style":2898},[363,5248],{"className":5249,"style":809},[650],[363,5251,5253],{"className":5252},[655,656,657,658],[363,5254,5256],{"className":5255},[407,658],[363,5257,465],{"className":5258},[407,658],[363,5260,666],{"className":5261},[665],[363,5263,5265],{"className":5264},[638],[363,5266,5268],{"className":5267,"style":2920},[642],[363,5269],{},[363,5271],{"className":5272},[501,789],[363,5274,5156],{"className":5275},[494],[363,5277,1116],{"className":5278},[407,408],[363,5280],{"className":5281,"style":2935},[505],[363,5283,5161],{"className":5284},[2939],[363,5286],{"className":5287,"style":2935},[505],[363,5289,5291,5294,5297],{"className":5290},[398],[363,5292],{"className":5293,"style":487},[402],[363,5295,465],{"className":5296},[407],[363,5298,5156],{"className":5299},[501],"），此时 ",[363,5302,5304,5321],{"className":5303,"translate":367},[366],[363,5305,5307],{"className":5306},[371],[373,5308,5309],{"xmlns":375},[377,5310,5311,5319],{},[380,5312,5313],{},[543,5314,5315,5317],{},[383,5316,547],{},[383,5318,385],{},[387,5320,4370],{"encoding":389},[363,5322,5324],{"className":5323,"ariaHidden":394},[393],[363,5325,5327,5330],{"className":5326},[398],[363,5328],{"className":5329,"style":4380},[402],[363,5331,5333,5336],{"className":5332},[407],[363,5334,547],{"className":5335,"style":625},[407,408],[363,5337,5339],{"className":5338},[629],[363,5340,5342,5362],{"className":5341},[633,634],[363,5343,5345,5359],{"className":5344},[638],[363,5346,5348],{"className":5347,"style":643},[642],[363,5349,5350,5353],{"style":646},[363,5351],{"className":5352,"style":651},[650],[363,5354,5356],{"className":5355},[655,656,657,658],[363,5357,385],{"className":5358,"style":409},[407,408,658],[363,5360,666],{"className":5361},[665],[363,5363,5365],{"className":5364},[638],[363,5366,5368],{"className":5367,"style":673},[642],[363,5369],{}," 取有限极限值。下文如无特别说明，默认 ",[363,5372,5374,5391],{"className":5373,"translate":367},[366],[363,5375,5377],{"className":5376},[371],[373,5378,5379],{"xmlns":375},[377,5380,5381,5389],{},[380,5382,5383,5385,5387],{},[383,5384,554],{},[458,5386,1061],{},[383,5388,561],{},[387,5390,1066],{"encoding":389},[363,5392,5394,5412],{"className":5393,"ariaHidden":394},[393],[363,5395,5397,5400,5403,5406,5409],{"className":5396},[398],[363,5398],{"className":5399,"style":1076},[402],[363,5401,554],{"className":5402,"style":682},[407,408],[363,5404],{"className":5405,"style":506},[505],[363,5407,1061],{"className":5408},[510],[363,5410],{"className":5411,"style":506},[505],[363,5413,5415,5418],{"className":5414},[398],[363,5416],{"className":5417,"style":913},[402],[363,5419,561],{"className":5420},[407,408],"，并把 ",[363,5423,5425,5442],{"className":5424,"translate":367},[366],[363,5426,5428],{"className":5427},[371],[373,5429,5430],{"xmlns":375},[377,5431,5432,5440],{},[380,5433,5434],{},[543,5435,5436,5438],{},[383,5437,547],{},[383,5439,385],{},[387,5441,4370],{"encoding":389},[363,5443,5445],{"className":5444,"ariaHidden":394},[393],[363,5446,5448,5451],{"className":5447},[398],[363,5449],{"className":5450,"style":4380},[402],[363,5452,5454,5457],{"className":5453},[407],[363,5455,547],{"className":5456,"style":625},[407,408],[363,5458,5460],{"className":5459},[629],[363,5461,5463,5483],{"className":5462},[633,634],[363,5464,5466,5480],{"className":5465},[638],[363,5467,5469],{"className":5468,"style":643},[642],[363,5470,5471,5474],{"style":646},[363,5472],{"className":5473,"style":651},[650],[363,5475,5477],{"className":5476},[655,656,657,658],[363,5478,385],{"className":5479,"style":409},[407,408,658],[363,5481,666],{"className":5482},[665],[363,5484,5486],{"className":5485},[638],[363,5487,5489],{"className":5488,"style":673},[642],[363,5490],{}," 看成 ",[363,5493,5495,5519],{"className":5494,"translate":367},[366],[363,5496,5498],{"className":5497},[371],[373,5499,5500],{"xmlns":375},[377,5501,5502,5516],{},[380,5503,5504,5506,5508,5510,5512,5514],{},[458,5505,578],{"stretchy":460},[463,5507,474],{},[458,5509,898],{"separator":394},[458,5511,2818],{},[383,5513,1373],{"mathvariant":557},[458,5515,605],{"stretchy":460},[387,5517,5518],{"encoding":389},"[0,+\\infty]",[363,5520,5522],{"className":5521,"ariaHidden":394},[393],[363,5523,5525,5528,5531,5534,5537,5540,5543,5546],{"className":5524},[398],[363,5526],{"className":5527,"style":487},[402],[363,5529,578],{"className":5530},[494],[363,5532,474],{"className":5533},[407],[363,5535,898],{"className":5536},[920],[363,5538],{"className":5539,"style":754},[505],[363,5541,2818],{"className":5542},[407],[363,5544,1373],{"className":5545},[407],[363,5547,605],{"className":5548},[501]," 中的值。",[2659,5551,5553],{"id":5552},"_12-只依赖似然比","1.2 只依赖似然比",[11,5555,5556,5557,5652,5653,5725,5726,5803,5804,5832,5833,5875],{},"由定义立刻看出：",[363,5558,5560,5588],{"className":5559,"translate":367},[366],[363,5561,5563],{"className":5562},[371],[373,5564,5565],{"xmlns":375},[377,5566,5567,5585],{},[380,5568,5569,5575,5577,5579,5581,5583],{},[543,5570,5571,5573],{},[383,5572,547],{},[383,5574,385],{},[458,5576,461],{"stretchy":460},[383,5578,554],{},[383,5580,558],{"mathvariant":557},[383,5582,561],{},[458,5584,468],{"stretchy":460},[387,5586,5587],{"encoding":389},"D_f(P\\|Q)",[363,5589,5591],{"className":5590,"ariaHidden":394},[393],[363,5592,5594,5597,5637,5640,5643,5646,5649],{"className":5593},[398],[363,5595],{"className":5596,"style":618},[402],[363,5598,5600,5603],{"className":5599},[407],[363,5601,547],{"className":5602,"style":625},[407,408],[363,5604,5606],{"className":5605},[629],[363,5607,5609,5629],{"className":5608},[633,634],[363,5610,5612,5626],{"className":5611},[638],[363,5613,5615],{"className":5614,"style":643},[642],[363,5616,5617,5620],{"style":646},[363,5618],{"className":5619,"style":651},[650],[363,5621,5623],{"className":5622},[655,656,657,658],[363,5624,385],{"className":5625,"style":409},[407,408,658],[363,5627,666],{"className":5628},[665],[363,5630,5632],{"className":5631},[638],[363,5633,5635],{"className":5634,"style":673},[642],[363,5636],{},[363,5638,461],{"className":5639},[494],[363,5641,554],{"className":5642,"style":682},[407,408],[363,5644,558],{"className":5645},[407],[363,5647,561],{"className":5648},[407,408],[363,5650,468],{"className":5651},[501]," 只通过似然比 ",[363,5654,5656,5682],{"className":5655,"translate":367},[366],[363,5657,5659],{"className":5658},[371],[373,5660,5661],{"xmlns":375},[377,5662,5663,5679],{},[380,5664,5665,5667,5669,5671,5673,5675,5677],{},[383,5666,1116],{},[458,5668,471],{},[383,5670,592],{},[383,5672,554],{},[383,5674,1288],{"mathvariant":557},[383,5676,592],{},[383,5678,561],{},[387,5680,5681],{"encoding":389},"t=dP\u002FdQ",[363,5683,5685,5704],{"className":5684,"ariaHidden":394},[393],[363,5686,5688,5692,5695,5698,5701],{"className":5687},[398],[363,5689],{"className":5690,"style":5691},[402],"height:0.6151em;",[363,5693,1116],{"className":5694},[407,408],[363,5696],{"className":5697,"style":506},[505],[363,5699,471],{"className":5700},[510],[363,5702],{"className":5703,"style":506},[505],[363,5705,5707,5710,5713,5716,5719,5722],{"className":5706},[398],[363,5708],{"className":5709,"style":487},[402],[363,5711,592],{"className":5712},[407,408],[363,5714,554],{"className":5715,"style":682},[407,408],[363,5717,1288],{"className":5718},[407],[363,5720,592],{"className":5721},[407,408],[363,5723,561],{"className":5724},[407,408]," 起作用。也就是说，对任意可测映射 ",[363,5727,5729,5753],{"className":5728,"translate":367},[366],[363,5730,5732],{"className":5731},[371],[373,5733,5734],{"xmlns":375},[377,5735,5736,5750],{},[380,5737,5738,5741,5743,5745,5747],{},[383,5739,5740],{},"T",[458,5742,1364],{},[383,5744,945],{"mathvariant":944},[458,5746,1378],{},[383,5748,5749],{"mathvariant":944},"Y",[387,5751,5752],{"encoding":389},"T:\\mathcal X\\to\\mathcal Y",[363,5754,5756,5774,5792],{"className":5755,"ariaHidden":394},[393],[363,5757,5759,5762,5765,5768,5771],{"className":5758},[398],[363,5760],{"className":5761,"style":1011},[402],[363,5763,5740],{"className":5764,"style":682},[407,408],[363,5766],{"className":5767,"style":506},[505],[363,5769,1364],{"className":5770},[510],[363,5772],{"className":5773,"style":506},[505],[363,5775,5777,5780,5783,5786,5789],{"className":5776},[398],[363,5778],{"className":5779,"style":1011},[402],[363,5781,945],{"className":5782,"style":972},[407,971],[363,5784],{"className":5785,"style":506},[505],[363,5787,1378],{"className":5788},[510],[363,5790],{"className":5791,"style":506},[505],[363,5793,5795,5799],{"className":5794},[398],[363,5796],{"className":5797,"style":5798},[402],"height:0.7805em;vertical-align:-0.0972em;",[363,5800,5749],{"className":5801,"style":5802},[407,971],"margin-right:0.0822em;","，它是 ",[363,5805,5807,5820],{"className":5806,"translate":367},[366],[363,5808,5810],{"className":5809},[371],[373,5811,5812],{"xmlns":375},[377,5813,5814,5818],{},[380,5815,5816],{},[383,5817,1116],{},[387,5819,1116],{"encoding":389},[363,5821,5823],{"className":5822,"ariaHidden":394},[393],[363,5824,5826,5829],{"className":5825},[398],[363,5827],{"className":5828,"style":5691},[402],[363,5830,1116],{"className":5831},[407,408]," 的函数，而不是 ",[363,5834,5836,5854],{"className":5835,"translate":367},[366],[363,5837,5839],{"className":5838},[371],[373,5840,5841],{"xmlns":375},[377,5842,5843,5851],{},[380,5844,5845,5847,5849],{},[383,5846,11],{},[458,5848,898],{"separator":394},[383,5850,1291],{},[387,5852,5853],{"encoding":389},"p,q",[363,5855,5857],{"className":5856,"ariaHidden":394},[393],[363,5858,5860,5863,5866,5869,5872],{"className":5859},[398],[363,5861],{"className":5862,"style":2624},[402],[363,5864,11],{"className":5865},[407,408],[363,5867,898],{"className":5868},[920],[363,5870],{"className":5871,"style":754},[505],[363,5873,1291],{"className":5874,"style":1328},[407,408]," 各自的函数。",[11,5877,5878,5879,5918],{},"这一点是 f-散度「缺少几何」的根源，也是它与 Wasserstein 距离分道扬镳的地方（见第 6 节）：只要保持比值 ",[363,5880,5882,5900],{"className":5881,"translate":367},[366],[363,5883,5885],{"className":5884},[371],[373,5886,5887],{"xmlns":375},[377,5888,5889,5897],{},[380,5890,5891,5893,5895],{},[383,5892,11],{},[383,5894,1288],{"mathvariant":557},[383,5896,1291],{},[387,5898,5899],{"encoding":389},"p\u002Fq",[363,5901,5903],{"className":5902,"ariaHidden":394},[393],[363,5904,5906,5909,5912,5915],{"className":5905},[398],[363,5907],{"className":5908,"style":487},[402],[363,5910,11],{"className":5911},[407,408],[363,5913,1288],{"className":5914},[407],[363,5916,1291],{"className":5917,"style":1328},[407,408]," 不变，把两个分布同时「搬到别处」，f-散度的值完全不动。",[2659,5920,5922],{"id":5921},"_13-非单射与仿射自由度","1.3 非单射与仿射自由度",[11,5924,5925,5926,5993,5994,6058],{},"注意 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dQ=1",[363,5952,5954,5984],{"className":5953,"ariaHidden":394},[393],[363,5955,5957,5961,5966,5969,5972,5975,5978,5981],{"className":5956},[398],[363,5958],{"className":5959,"style":5960},[402],"height:1.1111em;vertical-align:-0.3061em;",[363,5962,1635],{"className":5963,"style":5965},[1832,1833,5964],"small-op","margin-right:0.1945em;position:relative;top:-0.0006em;",[363,5967],{"className":5968,"style":754},[505],[363,5970,592],{"className":5971},[407,408],[363,5973,561],{"className":5974},[407,408],[363,5976],{"className":5977,"style":506},[505],[363,5979,471],{"className":5980},[510],[363,5982],{"className":5983,"style":506},[505],[363,5985,5987,5990],{"className":5986},[398],[363,5988],{"className":5989,"style":520},[402],[363,5991,465],{"className":5992},[407],"、",[363,5995,5997,6019],{"className":5996,"translate":367},[366],[363,5998,6000],{"className":5999},[371],[373,6001,6002],{"xmlns":375},[377,6003,6004,6016],{},[380,6005,6006,6008,6010,6012,6014],{},[458,6007,1635],{},[383,6009,592],{},[383,6011,554],{},[458,6013,471],{},[463,6015,465],{},[387,6017,6018],{"encoding":389},"\\int dP=1",[363,6020,6022,6049],{"className":6021,"ariaHidden":394},[393],[363,6023,6025,6028,6031,6034,6037,6040,6043,6046],{"className":6024},[398],[363,6026],{"className":6027,"style":5960},[402],[363,6029,1635],{"className":6030,"style":5965},[1832,1833,5964],[363,6032],{"className":6033,"style":754},[505],[363,6035,592],{"className":6036},[407,408],[363,6038,554],{"className":6039,"style":682},[407,408],[363,6041],{"className":6042,"style":506},[505],[363,6044,471],{"className":6045},[510],[363,6047],{"className":6048,"style":506},[505],[363,6050,6052,6055],{"className":6051},[398],[363,6053],{"className":6054,"style":520},[402],[363,6056,465],{"className":6057},[407],"，因此",[363,6060,6062],{"className":6061,"translate":367},[528],[363,6063,6065,6116],{"className":6064,"translate":367},[366],[363,6066,6068],{"className":6067},[371],[373,6069,6070],{"xmlns":375,"display":537},[377,6071,6072,6113],{},[380,6073,6074,6076,6078,6080,6082,6084,6086,6088,6090,6092,6094,6096,6098,6100,6102,6104,6106,6108,6110],{},[458,6075,1635],{},[458,6077,461],{"stretchy":460},[383,6079,1116],{},[458,6081,5161],{},[463,6083,465],{},[458,6085,468],{"stretchy":460},[1645,6087,1647],{},[383,6089,592],{},[383,6091,561],{},[458,6093,471],{},[458,6095,1635],{},[383,6097,592],{},[383,6099,554],{},[458,6101,5161],{},[458,6103,1635],{},[383,6105,592],{},[383,6107,561],{},[458,6109,471],{},[463,6111,6112],{},"0.",[387,6114,6115],{"encoding":389},"\\int (t-1)\\,dQ=\\int dP-\\int dQ=0 .",[363,6117,6119,6144,6174,6201,6228],{"className":6118,"ariaHidden":394},[393],[363,6120,6122,6126,6129,6132,6135,6138,6141],{"className":6121},[398],[363,6123],{"className":6124,"style":6125},[402],"height:2.2222em;vertical-align:-0.8622em;",[363,6127,1635],{"className":6128,"style":1835},[1832,1833,1834],[363,6130,461],{"className":6131},[494],[363,6133,1116],{"className":6134},[407,408],[363,6136],{"className":6137,"style":2935},[505],[363,6139,5161],{"className":6140},[2939],[363,6142],{"className":6143,"style":2935},[505],[363,6145,6147,6150,6153,6156,6159,6162,6165,6168,6171],{"className":6146},[398],[363,6148],{"className":6149,"style":487},[402],[363,6151,465],{"className":6152},[407],[363,6154,468],{"className":6155},[501],[363,6157],{"className":6158,"style":754},[505],[363,6160,592],{"className":6161},[407,408],[363,6163,561],{"className":6164},[407,408],[363,6166],{"className":6167,"style":506},[505],[363,6169,471],{"className":6170},[510],[363,6172],{"className":6173,"style":506},[505],[363,6175,6177,6180,6183,6186,6189,6192,6195,6198],{"className":6176},[398],[363,6178],{"className":6179,"style":6125},[402],[363,6181,1635],{"className":6182,"style":1835},[1832,1833,1834],[363,6184],{"className":6185,"style":754},[505],[363,6187,592],{"className":6188},[407,408],[363,6190,554],{"className":6191,"style":682},[407,408],[363,6193],{"className":6194,"style":2935},[505],[363,6196,5161],{"className":6197},[2939],[363,6199],{"className":6200,"style":2935},[505],[363,6202,6204,6207,6210,6213,6216,6219,6222,6225],{"className":6203},[398],[363,6205],{"className":6206,"style":6125},[402],[363,6208,1635],{"className":6209,"style":1835},[1832,1833,1834],[363,6211],{"className":6212,"style":754},[505],[363,6214,592],{"className":6215},[407,408],[363,6217,561],{"className":6218},[407,408],[363,6220],{"className":6221,"style":506},[505],[363,6223,471],{"className":6224},[510],[363,6226],{"className":6227,"style":506},[505],[363,6229,6231,6234],{"className":6230},[398],[363,6232],{"className":6233,"style":520},[402],[363,6235,6112],{"className":6236},[407],[11,6238,6239,6240,3093],{},"于是对任意常数 ",[363,6241,6243,6263],{"className":6242,"translate":367},[366],[363,6244,6246],{"className":6245},[371],[373,6247,6248],{"xmlns":375},[377,6249,6250,6260],{},[380,6251,6252,6255,6258],{},[383,6253,6254],{},"c",[458,6256,6257],{},"∈",[383,6259,1381],{"mathvariant":570},[387,6261,6262],{"encoding":389},"c\\in\\mathbb R",[363,6264,6266,6285],{"className":6265,"ariaHidden":394},[393],[363,6267,6269,6273,6276,6279,6282],{"className":6268},[398],[363,6270],{"className":6271,"style":6272},[402],"height:0.5782em;vertical-align:-0.0391em;",[363,6274,6254],{"className":6275},[407,408],[363,6277],{"className":6278,"style":506},[505],[363,6280,6257],{"className":6281},[510],[363,6283],{"className":6284,"style":506},[505],[363,6286,6288,6291],{"className":6287},[398],[363,6289],{"className":6290,"style":1445},[402],[363,6292,1381],{"className":6293},[407,714],[363,6295,6297],{"className":6296,"translate":367},[528],[363,6298,6300,6418],{"className":6299,"translate":367},[366],[363,6301,6303],{"className":6302},[371],[373,6304,6305],{"xmlns":375,"display":537},[377,6306,6307,6415],{},[380,6308,6309,6331,6333,6335,6337,6339,6341,6343,6349,6351,6353,6355,6357,6359,6361,6363,6395,6397,6403,6405,6407,6409,6411,6413],{},[543,6310,6311,6313],{},[383,6312,547],{},[380,6314,6315,6317,6319,6321,6323,6325,6327,6329],{},[383,6316,385],{},[458,6318,2818],{},[383,6320,6254],{},[458,6322,461],{"stretchy":460},[383,6324,1116],{},[458,6326,5161],{},[463,6328,465],{},[458,6330,468],{"stretchy":460},[458,6332,461],{"stretchy":460},[383,6334,554],{},[383,6336,558],{"mathvariant":557},[383,6338,561],{},[458,6340,468],{"stretchy":460},[458,6342,471],{},[543,6344,6345,6347],{},[383,6346,547],{},[383,6348,385],{},[458,6350,461],{"stretchy":460},[383,6352,554],{},[383,6354,558],{"mathvariant":557},[383,6356,561],{},[458,6358,468],{"stretchy":460},[458,6360,2818],{},[383,6362,6254],{},[6364,6365,6366,6389],"munder",{},[6364,6367,6368,6386],{},[380,6369,6370,6372,6374,6376,6378,6380,6382,6384],{},[458,6371,1635],{},[458,6373,461],{"stretchy":460},[383,6375,1116],{},[458,6377,5161],{},[463,6379,465],{},[458,6381,468],{"stretchy":460},[383,6383,592],{},[383,6385,561],{},[458,6387,6388],{"stretchy":394},"⏟",[380,6390,6391,6393],{},[458,6392,471],{},[463,6394,474],{},[458,6396,471],{},[543,6398,6399,6401],{},[383,6400,547],{},[383,6402,385],{},[458,6404,461],{"stretchy":460},[383,6406,554],{},[383,6408,558],{"mathvariant":557},[383,6410,561],{},[458,6412,468],{"stretchy":460},[383,6414,4723],{"mathvariant":557},[387,6416,6417],{"encoding":389},"D_{f+c(t-1)}(P\\|Q)\n=D_f(P\\|Q)+c\\underbrace{\\int(t-1)dQ}_{=0}\n=D_f(P\\|Q).",[363,6419,6421,6519,6589,6773],{"className":6420,"ariaHidden":394},[393],[363,6422,6424,6428,6495,6498,6501,6504,6507,6510,6513,6516],{"className":6423},[398],[363,6425],{"className":6426,"style":6427},[402],"height:1.1052em;vertical-align:-0.3552em;",[363,6429,6431,6434],{"className":6430},[407],[363,6432,547],{"className":6433,"style":625},[407,408],[363,6435,6437],{"className":6436},[629],[363,6438,6440,6486],{"className":6439},[633,634],[363,6441,6443,6483],{"className":6442},[638],[363,6444,6447],{"className":6445,"style":6446},[642],"height:0.3448em;",[363,6448,6450,6453],{"style":6449},"top:-2.5198em;margin-left:-0.0278em;margin-right:0.05em;",[363,6451],{"className":6452,"style":651},[650],[363,6454,6456],{"className":6455},[655,656,657,658],[363,6457,6459,6462,6465,6468,6471,6474,6477,6480],{"className":6458},[407,658],[363,6460,385],{"className":6461,"style":409},[407,408,658],[363,6463,2818],{"className":6464},[2939,658],[363,6466,6254],{"className":6467},[407,408,658],[363,6469,461],{"className":6470},[494,658],[363,6472,1116],{"className":6473},[407,408,658],[363,6475,5161],{"className":6476},[2939,658],[363,6478,465],{"className":6479},[407,658],[363,6481,468],{"className":6482},[501,658],[363,6484,666],{"className":6485},[665],[363,6487,6489],{"className":6488},[638],[363,6490,6493],{"className":6491,"style":6492},[642],"height:0.3552em;",[363,6494],{},[363,6496,461],{"className":6497},[494],[363,6499,554],{"className":6500,"style":682},[407,408],[363,6502,558],{"className":6503},[407],[363,6505,561],{"className":6506},[407,408],[363,6508,468],{"className":6509},[501],[363,6511],{"className":6512,"style":506},[505],[363,6514,471],{"className":6515},[510],[363,6517],{"className":6518,"style":506},[505],[363,6520,6522,6525,6565,6568,6571,6574,6577,6580,6583,6586],{"className":6521},[398],[363,6523],{"className":6524,"style":618},[402],[363,6526,6528,6531],{"className":6527},[407],[363,6529,547],{"className":6530,"style":625},[407,408],[363,6532,6534],{"className":6533},[629],[363,6535,6537,6557],{"className":6536},[633,634],[363,6538,6540,6554],{"className":6539},[638],[363,6541,6543],{"className":6542,"style":643},[642],[363,6544,6545,6548],{"style":646},[363,6546],{"className":6547,"style":651},[650],[363,6549,6551],{"className":6550},[655,656,657,658],[363,6552,385],{"className":6553,"style":409},[407,408,658],[363,6555,666],{"className":6556},[665],[363,6558,6560],{"className":6559},[638],[363,6561,6563],{"className":6562,"style":673},[642],[363,6564],{},[363,6566,461],{"className":6567},[494],[363,6569,554],{"className":6570,"style":682},[407,408],[363,6572,558],{"className":6573},[407],[363,6575,561],{"className":6576},[40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20-1h17z",[363,6703,6704,6707],{"style":6636},[363,6705],{"className":6706,"style":6621},[650],[363,6708,6710,6713,6716,6719,6722,6725,6728,6731,6734,6737],{"className":6709},[407],[363,6711,1635],{"className":6712,"style":1835},[1832,1833,1834],[363,6714,461],{"className":6715},[494],[363,6717,1116],{"className":6718},[407,408],[363,6720],{"className":6721,"style":2935},[505],[363,6723,5161],{"className":6724},[2939],[363,6726],{"className":6727,"style":2935},[505],[363,6729,465],{"className":6730},[407],[363,6732,468],{"className":6733},[501],[363,6735,592],{"className":6736},[407,408],[363,6738,561],{"className":6739},[407,408],[363,6741,666],{"className":6742},[665],[363,6744,6746],{"className":6745},[638],[363,6747,6750],{"className":6748,"style":6749},[642],"height:1.5103em;",[363,6751],{},[363,6753,666],{"className":6754},[665],[363,6756,6758],{"className":6757},[638],[363,6759,6762],{"className":6760,"style":6761},[642],"height:2.1614em;",[363,6763],{},[363,6765],{"className":6766,"style":506},[505],[363,6768,471],{"className":6769},[510],[363,6771],{"className":6772,"style":506},[505],[363,6774,6776,6779,6819,6822,6825,6828,6831,6834],{"className":6775},[398],[363,6777],{"className":6778,"style":618},[402],[363,6780,6782,6785],{"className":6781},[407],[363,6783,547],{"className":6784,"style":625},[407,408],[363,6786,6788],{"className":6787},[629],[363,6789,6791,6811],{"className":6790},[633,634],[363,6792,6794,6808],{"className":6793},[638],[363,6795,6797],{"className":6796,"style":643},[642],[363,6798,6799,6802],{"style":646},[363,6800],{"className":6801,"style":651},[650],[363,6803,6805],{"className":6804},[655,656,657,658],[363,6806,385],{"className":6807,"style":409},[407,408,658],[363,6809,666],{"className":6810},[665],[363,6812,6814],{"className":6813},[638],[363,6815,6817],{"className":6816,"style":673},[642],[363,6818],{},[363,6820,461],{"className":6821},[494],[363,6823,554],{"className":6824,"style":682},[407,408],[363,6826,558],{"className":6827},[407],[363,6829,561],{"className":6830},[407,408],[363,6832,468],{"className":6833},[501],[363,6835,4723],{"className":6836},[407],[11,6838,6839,6842,6843,6912,6913,6941,6942,7008],{},[1344,6840,6841],{},"结论："," ",[363,6844,6846,6863],{"className":6845,"translate":367},[366],[363,6847,6849],{"className":6848},[371],[373,6850,6851],{"xmlns":375},[377,6852,6853,6861],{},[380,6854,6855],{},[543,6856,6857,6859],{},[383,6858,547],{},[383,6860,385],{},[387,6862,4370],{"encoding":389},[363,6864,6866],{"className":6865,"ariaHidden":394},[393],[363,6867,6869,6872],{"className":6868},[398],[363,6870],{"className":6871,"style":4380},[402],[363,6873,6875,6878],{"className":6874},[407],[363,6876,547],{"className":6877,"style":625},[407,408],[363,6879,6881],{"className":6880},[629],[363,6882,6884,6904],{"className":6883},[633,634],[363,6885,6887,6901],{"className":6886},[638],[363,6888,6890],{"className":6889,"style":643},[642],[363,6891,6892,6895],{"style":646},[363,6893],{"className":6894,"style":651},[650],[363,6896,6898],{"className":6897},[655,656,657,658],[363,6899,385],{"className":6900,"style":409},[407,408,658],[363,6902,666],{"className":6903},[665],[363,6905,6907],{"className":6906},[638],[363,6908,6910],{"className":6909,"style":673},[642],[363,6911],{}," 只依赖 ",[363,6914,6916,6929],{"className":6915,"translate":367},[366],[363,6917,6919],{"className":6918},[371],[373,6920,6921],{"xmlns":375},[377,6922,6923,6927],{},[380,6924,6925],{},[383,6926,385],{},[387,6928,385],{"encoding":389},[363,6930,6932],{"className":6931,"ariaHidden":394},[393],[363,6933,6935,6938],{"className":6934},[398],[363,6936],{"className":6937,"style":403},[402],[363,6939,385],{"className":6940,"style":409},[407,408]," 在「模去 ",[363,6943,6945,6969],{"className":6944,"translate":367},[366],[363,6946,6948],{"className":6947},[371],[373,6949,6950],{"xmlns":375},[377,6951,6952,6966],{},[380,6953,6954,6956,6958,6960,6962,6964],{},[383,6955,6254],{},[458,6957,461],{"stretchy":460},[383,6959,1116],{},[458,6961,5161],{},[463,6963,465],{},[458,6965,468],{"stretchy":460},[387,6967,6968],{"encoding":389},"c(t-1)",[363,6970,6972,6996],{"className":6971,"ariaHidden":394},[393],[363,6973,6975,6978,6981,6984,6987,6990,6993],{"className":6974},[398],[363,6976],{"className":6977,"style":487},[402],[363,6979,6254],{"className":6980},[407,408],[363,6982,461],{"className":6983},[494],[363,6985,1116],{"className":6986},[407,408],[363,6988],{"className":6989,"style":2935},[505],[363,6991,5161],{"className":6992},[2939],[363,6994],{"className":6995,"style":2935},[505],[363,6997,6999,7002,7005],{"className":6998},[398],[363,7000],{"className":7001,"style":487},[402],[363,7003,465],{"className":7004},[407],[363,7006,468],{"className":7007},[501],"」之后的等价类。两个常用规范化是：",[7010,7011,7012,7081],"ol",{},[7013,7014,7015,7080],"li",{},[363,7016,7018,7041],{"className":7017,"translate":367},[366],[363,7019,7021],{"className":7020},[371],[373,7022,7023],{"xmlns":375},[377,7024,7025,7039],{},[380,7026,7027,7029,7031,7033,7035,7037],{},[383,7028,385],{},[458,7030,461],{"stretchy":460},[463,7032,465],{},[458,7034,468],{"stretchy":460},[458,7036,471],{},[463,7038,474],{},[387,7040,477],{"encoding":389},[363,7042,7044,7071],{"className":7043,"ariaHidden":394},[393],[363,7045,7047,7050,7053,7056,7059,7062,7065,7068],{"className":7046},[398],[363,7048],{"className":7049,"style":487},[402],[363,7051,385],{"className":7052,"style":409},[407,408],[363,7054,461],{"className":7055},[494],[363,7057,465],{"className":7058},[407],[363,7060,468],{"className":7061},[501],[363,7063],{"className":7064,"style":506},[505],[363,7066,471],{"className":7067},[510],[363,7069],{"className":7070,"style":506},[505],[363,7072,7074,7077],{"className":7073},[398],[363,7075],{"className":7076,"style":520},[402],[363,7078,474],{"className":7079},[407],"（本文采用）；",[7013,7082,7083,7148,7149,7253,7254,3093,7465,7493],{},[363,7084,7086,7109],{"className":7085,"translate":367},[366],[363,7087,7089],{"className":7088},[371],[373,7090,7091],{"xmlns":375},[377,7092,7093,7107],{},[380,7094,7095,7097,7099,7101,7103,7105],{},[383,7096,385],{},[458,7098,461],{"stretchy":460},[463,7100,465],{},[458,7102,468],{"stretchy":460},[458,7104,471],{},[463,7106,474],{},[387,7108,477],{"encoding":389},[363,7110,7112,7139],{"className":7111,"ariaHidden":394},[393],[363,7113,7115,7118,7121,7124,7127,7130,7133,7136],{"className":7114},[398],[363,7116],{"className":7117,"style":487},[402],[363,7119,385],{"className":7120,"style":409},[407,408],[363,7122,461],{"className":7123},[494],[363,7125,465],{"className":7126},[407],[363,7128,468],{"className":7129},[501],[363,7131],{"className":7132,"style":506},[505],[363,7134,471],{"className":7135},[510],[363,7137],{"className":7138,"style":506},[505],[363,7140,7142,7145],{"className":7141},[398],[363,7143],{"className":7144,"style":520},[402],[363,7146,474],{"className":7147},[407]," 且 ",[363,7150,7152,7182],{"className":7151,"translate":367},[366],[363,7153,7155],{"className":7154},[371],[373,7156,7157],{"xmlns":375},[377,7158,7159,7179],{},[380,7160,7161,7169,7171,7173,7175,7177],{},[7162,7163,7164,7166],"msup",{},[383,7165,385],{},[458,7167,7168],{"mathvariant":557,"lspace":1595,"rspace":1595},"′",[458,7170,461],{"stretchy":460},[463,7172,465],{},[458,7174,468],{"stretchy":460},[458,7176,471],{},[463,7178,474],{},[387,7180,7181],{"encoding":389},"f'(1)=0",[363,7183,7185,7244],{"className":7184,"ariaHidden":394},[393],[363,7186,7188,7192,7226,7229,7232,7235,7238,7241],{"className":7187},[398],[363,7189],{"className":7190,"style":7191},[402],"height:1.0019em;vertical-align:-0.25em;",[363,7193,7195,7198],{"className":7194},[407],[363,7196,385],{"className":7197,"style":409},[407,408],[363,7199,7201],{"className":7200},[629],[363,7202,7204],{"className":7203},[633],[363,7205,7207],{"className":7206},[638],[363,7208,7211],{"className":7209,"style":7210},[642],"height:0.7519em;",[363,7212,7214,7217],{"style":7213},"top:-3.063em;margin-right:0.05em;",[363,7215],{"className":7216,"style":651},[650],[363,7218,7220],{"className":7219},[655,656,657,658],[363,7221,7223],{"className":7222},[407,658],[363,7224,7168],{"className":7225},[407,658],[363,7227,461],{"className":7228},[494],[363,7230,465],{"className":7231},[407],[363,7233,468],{"className":7234},[501],[363,7236],{"className":7237,"style":506},[505],[363,7239,471],{"className":7240},[510],[363,7242],{"className":7243,"style":506},[505],[363,7245,7247,7250],{"className":7246},[398],[363,7248],{"className":7249,"style":520},[402],[363,7251,474],{"className":7252},[407],"——此时由凸性 ",[363,7255,7257,7316],{"className":7256,"translate":367},[366],[363,7258,7260],{"className":7259},[371],[373,7261,7262],{"xmlns":375},[377,7263,7264,7313],{},[380,7265,7266,7268,7270,7272,7274,7277,7279,7281,7283,7285,7287,7293,7295,7297,7299,7301,7303,7305,7307,7309,7311],{},[383,7267,385],{},[458,7269,461],{"stretchy":460},[383,7271,1116],{},[458,7273,468],{"stretchy":460},[458,7275,7276],{},"≥",[383,7278,385],{},[458,7280,461],{"stretchy":460},[463,7282,465],{},[458,7284,468],{"stretchy":460},[458,7286,2818],{},[7162,7288,7289,7291],{},[383,7290,385],{},[458,7292,7168],{"mathvariant":557,"lspace":1595,"rspace":1595},[458,7294,461],{"stretchy":460},[463,7296,465],{},[458,7298,468],{"stretchy":460},[458,7300,461],{"stretchy":460},[383,7302,1116],{},[458,7304,5161],{},[463,7306,465],{},[458,7308,468],{"stretchy":460},[458,7310,471],{},[463,7312,474],{},[387,7314,7315],{"encoding":389},"f(t)\\ge f(1)+f'(1)(t-1)=0",[363,7317,7319,7346,7373,7435,7456],{"className":7318,"ariaHidden":394},[393],[363,7320,7322,7325,7328,7331,7334,7337,7340,7343],{"className":7321},[398],[363,7323],{"className":7324,"style":487},[402],[363,7326,385],{"className":7327,"style":409},[407,408],[363,7329,461],{"className":7330},[494],[363,7332,1116],{"className":7333},[407,408],[363,7335,468],{"className":7336},[501],[363,7338],{"className":7339,"style":506},[505],[363,7341,7276],{"className":7342},[510],[363,7344],{"className":7345,"style":506},[505],[363,7347,7349,7352,7355,7358,7361,7364,7367,7370],{"className":7348},[398],[363,7350],{"className":7351,"style":487},[402],[363,7353,385],{"className":7354,"style":409},[407,408],[363,7356,461],{"className":7357},[494],[363,7359,465],{"className":7360},[407],[363,7362,468],{"className":7363},[501],[363,7365],{"className":7366,"style":2935},[505],[363,7368,2818],{"className":7369},[2939],[363,7371],{"className":7372,"style":2935},[505],[363,7374,7376,7379,7411,7414,7417,7420,7423,7426,7429,7432],{"className":7375},[398],[363,7377],{"className":7378,"style":7191},[402],[363,7380,7382,7385],{"className":7381},[407],[363,7383,385],{"className":7384,"style":409},[407,408],[363,7386,7388],{"className":7387},[629],[363,7389,7391],{"className":7390},[633],[363,7392,7394],{"className":7393},[638],[363,7395,7397],{"className":7396,"style":7210},[642],[363,7398,7399,7402],{"style":7213},[363,7400],{"className":7401,"style":651},[650],[363,7403,7405],{"className":7404},[655,656,657,658],[363,7406,7408],{"className":7407},[407,658],[363,7409,7168],{"className":7410},[407,658],[363,7412,461],{"className":7413},[494],[363,7415,465],{"className":7416},[407],[363,7418,468],{"className":7419},[501],[363,7421,461],{"className":7422},[494],[363,7424,1116],{"className":7425},[407,408],[363,7427],{"className":7428,"style":2935},[505],[363,7430,5161],{"className":7431},[2939],[363,7433],{"className":7434,"style":2935},[505],[363,7436,7438,7441,7444,7447,7450,7453],{"className":7437},[398],[363,7439],{"className":7440,"style":487},[402],[363,7442,465],{"className":7443},[407],[363,7445,468],{"className":7446},[501],[363,7448],{"className":7449,"style":506},[505],[363,7451,471],{"className":7452},[510],[363,7454],{"className":7455,"style":506},[505],[363,7457,7459,7462],{"className":7458},[398],[363,7460],{"className":7461,"style":520},[402],[363,7463,474],{"className":7464},[407],[363,7466,7468,7481],{"className":7467,"translate":367},[366],[363,7469,7471],{"className":7470},[371],[373,7472,7473],{"xmlns":375},[377,7474,7475,7479],{},[380,7476,7477],{},[383,7478,385],{},[387,7480,385],{"encoding":389},[363,7482,7484],{"className":7483,"ariaHidden":394},[393],[363,7485,7487,7490],{"className":7486},[398],[363,7488],{"className":7489,"style":403},[402],[363,7491,385],{"className":7492,"style":409},[407,408]," 在整个定义域上非负。",[11,7495,7496,7497,7582,7583,7636,7637,7706],{},"第 3 节里的 KL 的 ",[363,7498,7500,7529],{"className":7499,"translate":367},[366],[363,7501,7503],{"className":7502},[371],[373,7504,7505],{"xmlns":375},[377,7506,7507,7527],{},[380,7508,7509,7511,7513,7515,7517,7519,7521,7523,7525],{},[383,7510,385],{},[458,7512,461],{"stretchy":460},[383,7514,1116],{},[458,7516,468],{"stretchy":460},[458,7518,471],{},[383,7520,1116],{},[383,7522,4448],{},[458,7524,4451],{},[383,7526,1116],{},[387,7528,4456],{"encoding":389},[363,7530,7532,7559],{"className":7531,"ariaHidden":394},[393],[363,7533,7535,7538,7541,7544,7547,7550,7553,7556],{"className":7534},[398],[363,7536],{"className":7537,"style":487},[402],[363,7539,385],{"className":7540,"style":409},[407,408],[363,7542,461],{"className":7543},[494],[363,7545,1116],{"className":7546},[407,408],[363,7548,468],{"className":7549},[501],[363,7551],{"className":7552,"style":506},[505],[363,7554,471],{"className":7555},[510],[363,7557],{"className":7558,"style":506},[505],[363,7560,7562,7565,7568,7571,7576,7579],{"className":7561},[398],[363,7563],{"className":7564,"style":403},[402],[363,7566,1116],{"className":7567},[407,408],[363,7569],{"className":7570,"style":754},[505],[363,7572,4502,7574],{"className":7573},[1832],[363,7575,4506],{"style":4505},[363,7577],{"className":7578,"style":754},[505],[363,7580,1116],{"className":7581},[407,408]," 不满足第 2 条（它在 ",[363,7584,7586,7605],{"className":7585,"translate":367},[366],[363,7587,7589],{"className":7588},[371],[373,7590,7591],{"xmlns":375},[377,7592,7593,7602],{},[380,7594,7595,7597,7600],{},[383,7596,1116],{},[458,7598,7599],{},"\u003C",[463,7601,465],{},[387,7603,7604],{"encoding":389},"t\u003C1",[363,7606,7608,7627],{"className":7607,"ariaHidden":394},[393],[363,7609,7611,7615,7618,7621,7624],{"className":7610},[398],[363,7612],{"className":7613,"style":7614},[402],"height:0.6542em;vertical-align:-0.0391em;",[363,7616,1116],{"className":7617},[407,408],[363,7619],{"className":7620,"style":506},[505],[363,7622,7599],{"className":7623},[510],[363,7625],{"className":7626,"style":506},[505],[363,7628,7630,7633],{"className":7629},[398],[363,7631],{"className":7632,"style":520},[402],[363,7634,465],{"className":7635},[407]," 时取负值），但这无妨：",[363,7638,7640,7657],{"className":7639,"translate":367},[366],[363,7641,7643],{"className":7642},[371],[373,7644,7645],{"xmlns":375},[377,7646,7647,7655],{},[380,7648,7649],{},[543,7650,7651,7653],{},[383,7652,547],{},[383,7654,385],{},[387,7656,4370],{"encoding":389},[363,7658,7660],{"className":7659,"ariaHidden":394},[393],[363,7661,7663,7666],{"className":7662},[398],[363,7664],{"className":7665,"style":4380},[402],[363,7667,7669,7672],{"className":7668},[407],[363,7670,547],{"className":7671,"style":625},[407,408],[363,7673,7675],{"className":7674},[629],[363,7676,7678,7698],{"className":7677},[633,634],[363,7679,7681,7695],{"className":7680},[638],[363,7682,7684],{"className":7683,"style":643},[642],[363,7685,7686,7689],{"style":646},[363,7687],{"className":7688,"style":651},[650],[363,7690,7692],{"className":7691},[655,656,657,658],[363,7693,385],{"className":7694,"style":409},[407,408,658],[363,7696,666],{"className":7697},[665],[363,7699,7701],{"className":7700},[638],[363,7702,7704],{"className":7703,"style":673},[642],[363,7705],{}," 仍然非负（见 4.1），因为「逐点非负」从来不是 f-散度非负性的必要条件。",[2659,7708,7710],{"id":7709},"_14-等价书写形式","1.4 等价书写形式",[363,7712,7714],{"className":7713,"translate":367},[528],[363,7715,7717,7951],{"className":7716,"translate":367},[366],[363,7718,7720],{"className":7719},[371],[373,7721,7722],{"xmlns":375,"display":537},[377,7723,7724,7948],{},[1591,7725,7726,7798,7870],{"rowspacing":1593,"columnalign":1594,"columnspacing":1595},[1597,7727,7728,7750],{},[1600,7729,7730],{},[1603,7731,7732],{"scriptlevel":474,"displaystyle":394},[380,7733,7734,7740,7742,7744,7746,7748],{},[543,7735,7736,7738],{},[383,7737,547],{},[383,7739,385],{},[458,7741,461],{"stretchy":460},[383,7743,554],{},[383,7745,558],{"mathvariant":557},[383,7747,561],{},[458,7749,468],{"stretchy":460},[1600,7751,7752],{},[1603,7753,7754],{"scriptlevel":474,"displaystyle":394},[380,7755,7756,7758,7760,7766,7768],{},[380,7757],{},[458,7759,471],{},[543,7761,7762,7764],{},[383,7763,571],{"mathvariant":570},[383,7765,561],{},[1645,7767,1652],{},[380,7769,7770,7772,7774,7776,7796],{},[458,7771,578],{"fence":394},[383,7773,385],{},[1645,7775,1652],{},[380,7777,7778,7780,7794],{},[458,7779,461],{"fence":394},[586,7781,7782,7788],{},[380,7783,7784,7786],{},[383,7785,592],{},[383,7787,554],{},[380,7789,7790,7792],{},[383,7791,592],{},[383,7793,561],{},[458,7795,468],{"fence":394},[458,7797,605],{"fence":394},[1597,7799,7800,7806],{},[1600,7801,7802],{},[1603,7803,7804],{"scriptlevel":474,"displaystyle":394},[380,7805],{},[1600,7807,7808],{},[1603,7809,7810],{"scriptlevel":474,"displaystyle":394},[380,7811,7812,7814,7816,7822,7824],{},[380,7813],{},[458,7815,471],{},[543,7817,7818,7820],{},[383,7819,571],{"mathvariant":570},[383,7821,554],{},[1645,7823,1652],{},[380,7825,7826,7828,7842,7844,7846,7848,7868],{},[458,7827,578],{"fence":394},[586,7829,7830,7836],{},[380,7831,7832,7834],{},[383,7833,592],{},[383,7835,561],{},[380,7837,7838,7840],{},[383,7839,592],{},[383,7841,554],{},[1645,7843,1647],{},[383,7845,385],{},[1645,7847,1652],{},[380,7849,7850,7852,7866],{},[458,7851,461],{"fence":394},[586,7853,7854,7860],{},[380,7855,7856,7858],{},[383,7857,592],{},[383,7859,554],{},[380,7861,7862,7864],{},[383,7863,592],{},[383,7865,561],{},[458,7867,468],{"fence":394},[458,7869,605],{"fence":394},[1597,7871,7872,7878],{},[1600,7873,7874],{},[1603,7875,7876],{"scriptlevel":474,"displaystyle":394},[380,7877],{},[1600,7879,7880],{},[1603,7881,7882],{"scriptlevel":474,"displaystyle":394},[380,7883,7884,7886,7888,7894,7896,7946],{},[380,7885],{},[458,7887,471],{},[543,7889,7890,7892],{},[383,7891,571],{"mathvariant":570},[383,7893,554],{},[1645,7895,1652],{},[380,7897,7898,7900,7922,7924,7944],{},[458,7899,578],{"fence":394},[7901,7902,7903,7905,7919],"msubsup",{},[383,7904,385],{},[380,7906,7907,7910,7913,7916],{},[383,7908,7909],{},"s",[383,7911,7912],{},"k",[383,7914,7915],{},"e",[383,7917,7918],{},"w",[458,7920,7921],{"lspace":1595,"rspace":1595},"∗",[1645,7923,1652],{},[380,7925,7926,7928,7942],{},[458,7927,461],{"fence":394},[586,7929,7930,7936],{},[380,7931,7932,7934],{},[383,7933,592],{},[383,7935,561],{},[380,7937,7938,7940],{},[383,7939,592],{},[383,7941,554],{},[458,7943,468],{"fence":394},[458,7945,605],{"fence":394},[458,7947,898],{"separator":394},[387,7949,7950],{"encoding":389},"\\begin{aligned}\nD_f(P\\|Q)\n&=\\mathbb E_Q\\!\\left[\nf\\!\\left(\\frac{dP}{dQ}\\right)\\right]\\\\[2pt]\n&=\\mathbb E_P\\!\\left[\\frac{dQ}{dP}\\,\nf\\!\\left(\\frac{dP}{dQ}\\right)\\right]\\\\\n&=\\mathbb 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是后面 4.3 节的「斜共轭」。最后一个等式说明：交换 ",[363,8946,8948,8965],{"className":8947,"translate":367},[366],[363,8949,8951],{"className":8950},[371],[373,8952,8953],{"xmlns":375},[377,8954,8955,8963],{},[380,8956,8957,8959,8961],{},[383,8958,554],{},[458,8960,898],{"separator":394},[383,8962,561],{},[387,8964,903],{"encoding":389},[363,8966,8968],{"className":8967,"ariaHidden":394},[393],[363,8969,8971,8974,8977,8980,8983],{"className":8970},[398],[363,8972],{"className":8973,"style":913},[402],[363,8975,554],{"className":8976,"style":682},[407,408],[363,8978,898],{"className":8979},[920],[363,8981],{"className":8982,"style":754},[505],[363,8984,561],{"className":8985},[407,408]," 只是把 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Q",[363,9224,9226,9245],{"className":9225,"ariaHidden":394},[393],[363,9227,9229,9232,9236,9239,9242],{"className":9228},[398],[363,9230],{"className":9231,"style":1011},[402],[363,9233,945],{"className":9234,"style":9235},[407,408],"margin-right:0.0785em;",[363,9237],{"className":9238,"style":506},[505],[363,9240,9217],{"className":9241},[510],[363,9243],{"className":9244,"style":506},[505],[363,9246,9248,9251],{"className":9247},[398],[363,9249],{"className":9250,"style":913},[402],[363,9252,561],{"className":9253},[407,408],[363,9255,9257,9277],{"className":9256,"translate":367},[366],[363,9258,9260],{"className":9259},[371],[373,9261,9262],{"xmlns":375},[377,9263,9264,9274],{},[380,9265,9266,9268,9270,9272],{},[383,9267,385],{},[458,9269,461],{"stretchy":460},[383,9271,945],{},[458,9273,468],{"stretchy":460},[387,9275,9276],{"encoding":389},"f(X)",[363,9278,9280],{"className":9279,"ariaHidden":394},[393],[363,9281,9283,9286,9289,9292,9295],{"className":9282},[398],[363,9284],{"className":9285,"style":487},[402],[363,9287,385],{"className":9288,"style":409},[407,408],[363,9290,461],{"className":9291},[494],[363,9293,945],{"className":9294,"style":9235},[407,408],[363,9296,468],{"className":9297},[501]," 的期望就是 ",[363,9300,9302,9319],{"className":9301,"translate":367},[366],[363,9303,9305],{"className":9304},[371],[373,9306,9307],{"xmlns":375},[377,9308,9309,9317],{},[380,9310,9311],{},[543,9312,9313,9315],{},[383,9314,547],{},[383,9316,385],{},[387,9318,4370],{"encoding":389},[363,9320,9322],{"className":9321,"ariaHidden":394},[393],[363,9323,9325,9328],{"className":9324},[398],[363,9326],{"className":9327,"style":4380},[402],[363,9329,9331,9334],{"className":9330},[407],[363,9332,547],{"className":9333,"style":625},[407,408],[363,9335,9337],{"className":9336},[629],[363,9338,9340,9360],{"className":9339},[633,634],[363,9341,9343,9357],{"className":9342},[638],[363,9344,9346],{"className":9345,"style":643},[642],[363,9347,9348,9351],{"style":646},[363,9349],{"className":9350,"style":651},[650],[363,9352,9354],{"className":9353},[655,656,657,658],[363,9355,385],{"className":9356,"style":409},[407,408,658],[363,9358,666],{"className":9359},[665],[363,9361,9363],{"className":9362},[638],[363,9364,9366],{"className":9365,"style":673},[642],[363,9367],{},"。因为 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满足",[363,9441,9443],{"className":9442,"translate":367},[528],[363,9444,9446,9508],{"className":9445,"translate":367},[366],[363,9447,9449],{"className":9448},[371],[373,9450,9451],{"xmlns":375,"display":537},[377,9452,9453,9505],{},[380,9454,9455,9461,9463,9465,9467,9469,9471,9485,9487,9489,9491,9493,9495,9497,9499,9501,9503],{},[543,9456,9457,9459],{},[383,9458,571],{"mathvariant":570},[383,9460,561],{},[458,9462,578],{"stretchy":460},[383,9464,1116],{},[458,9466,605],{"stretchy":460},[458,9468,471],{},[458,9470,1635],{},[586,9472,9473,9479],{},[380,9474,9475,9477],{},[383,9476,592],{},[383,9478,554],{},[380,9480,9481,9483],{},[383,9482,592],{},[383,9484,561],{},[1645,9486,1647],{},[383,9488,592],{},[383,9490,561],{},[458,9492,471],{},[458,9494,1635],{},[383,9496,592],{},[383,9498,554],{},[458,9500,471],{},[463,9502,465],{},[458,9504,898],{"separator":394},[387,9506,9507],{"encoding":389},"\\mathbb E_Q[t]=\\int \\frac{dP}{dQ}\\,dQ=\\int 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",[363,9717,9719,9732],{"className":9718,"translate":367},[366],[363,9720,9722],{"className":9721},[371],[373,9723,9724],{"xmlns":375},[377,9725,9726,9730],{},[380,9727,9728],{},[383,9729,385],{},[387,9731,385],{"encoding":389},[363,9733,9735],{"className":9734,"ariaHidden":394},[393],[363,9736,9738,9741],{"className":9737},[398],[363,9739],{"className":9740,"style":403},[402],[363,9742,385],{"className":9743,"style":409},[407,408]," 用 ",[1344,9746,9747],{},"Jensen 不等式","：",[363,9750,9752],{"className":9751,"translate":367},[528],[363,9753,9755,9874],{"className":9754,"translate":367},[366],[363,9756,9758],{"className":9757},[371],[373,9759,9760],{"xmlns":375,"display":537},[377,9761,9762,9871],{},[1591,9763,9764,9841],{"rowspacing":1593,"columnalign":1594,"columnspacing":1595},[1597,9765,9766,9809],{},[1600,9767,9768],{},[1603,9769,9770],{"scriptlevel":474,"displaystyle":394},[380,9771,9772,9778,9780,9782,9784,9786,9788,9790,9796,9799,9801,9803,9805,9807],{},[543,9773,9774,9776],{},[383,9775,547],{},[383,9777,385],{},[458,9779,461],{"stretchy":460},[383,9781,554],{},[383,9783,558],{"mathvariant":557},[383,9785,561],{},[458,9787,468],{"stretchy":460},[458,9789,471],{},[543,9791,9792,9794],{},[383,9793,571],{"mathvariant":570},[383,9795,561],{},[458,9797,578],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},"1.2em",[383,9800,385],{},[458,9802,461],{"stretchy":460},[383,9804,1116],{},[458,9806,468],{"stretchy":460},[458,9808,605],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[1600,9810,9811],{},[1603,9812,9813],{"scriptlevel":474,"displaystyle":394},[380,9814,9815,9817,9819,9821,9823,9825,9827,9833,9835,9837,9839],{},[380,9816],{},[1645,9818,4631],{},[458,9820,7276],{},[1645,9822,4631],{},[383,9824,385],{},[458,9826,461],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[543,9828,9829,9831],{},[383,9830,571],{"mathvariant":570},[383,9832,561],{},[458,9834,578],{"stretchy":460},[383,9836,1116],{},[458,9838,605],{"stretchy":460},[458,9840,468],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[1597,9842,9843,9849],{},[1600,9844,9845],{},[1603,9846,9847],{"scriptlevel":474,"displaystyle":394},[380,9848],{},[1600,9850,9851],{},[1603,9852,9853],{"scriptlevel":474,"displaystyle":394},[380,9854,9855,9857,9859,9861,9863,9865,9867,9869],{},[380,9856],{},[458,9858,471],{},[383,9860,385],{},[458,9862,461],{"stretchy":460},[463,9864,465],{},[458,9866,468],{"stretchy":460},[458,9868,471],{},[463,9870,6112],{},[387,9872,9873],{"encoding":389},"\\begin{aligned}\nD_f(P\\|Q)=\\mathbb E_Q\\big[f(t)\\big]\n&\\ \\ge\\ f\\big(\\mathbb E_Q[t]\\big)\\\\[2pt]\n&=f(1)=0 .\n\\end{aligned}",[363,9875,9877],{"className":9876,"ariaHidden":394},[393],[363,9878,9880,9884],{"className":9879},[398],[363,9881],{"className":9882,"style":9883},[402],"height:2.91em;vertical-align:-1.205em;",[363,9885,9887],{"className":9886},[407],[363,9888,9890,10062],{"className":9889},[1591],[363,9891,9893],{"className":9892},[1708],[363,9894,9896,10053],{"className":9895},[633,634],[363,9897,9899,10050],{"className":9898},[638],[363,9900,9903,10041],{"className":9901,"style":9902},[642],"height:1.705em;",[363,9904,9906,9909],{"style":9905},"top:-3.855em;",[363,9907],{"className":9908,"style":809},[650],[363,9910,9912,9952,9955,9958,9961,9964,9967,9970,9973,9976,10016,10023,10026,10029,10032,10035],{"className":9911},[407],[363,9913,9915,9918],{"className":9914},[407],[363,9916,547],{"className":9917,"style":625},[407,408],[363,9919,9921],{"className":9920},[629],[363,9922,9924,9944],{"className":9923},[633,634],[363,9925,9927,9941],{"className":9926},[638],[363,9928,9930],{"className":9929,"style":643},[642],[363,9931,9932,9935],{"style":646},[363,9933],{"className":9934,"style":651},[650],[363,9936,9938],{"className":9937},[655,656,657,658],[363,9939,385],{"className":9940,"style":409},[407,408,658],[363,9942,666],{"className":9943},[665],[363,9945,9947],{"className":9946},[638],[363,9948,9950],{"className":9949,"style":673},[642],[363,9951],{},[363,9953,461],{"className":9954},[494],[363,9956,554],{"className":9957,"style":682},[407,408],[363,9959,558],{"className":9960},[407],[363,9962,561],{"className":9963},[407,408],[363,9965,468],{"className":9966},[501],[363,9968],{"className":9969,"style":506},[505],[363,9971,471],{"className":9972},[510],[363,9974],{"className":9975,"style":506},[505],[363,9977,9979,9982],{"className":9978},[407],[363,9980,571],{"className":9981},[407,714],[363,9983,9985],{"className":9984},[629],[363,9986,9988,10008],{"className":9987},[633,634],[363,9989,9991,10005],{"className":9990},[638],[363,9992,9994],{"className":9993,"style":727},[642],[363,9995,9996,9999],{"style":730},[363,9997],{"className":9998,"style":651},[650],[363,10000,10002],{"className":10001},[655,656,657,658],[363,10003,561],{"className":10004},[407,408,658],[363,10006,666],{"className":10007},[665],[363,10009,10011],{"className":10010},[638],[363,10012,10014],{"className":10013,"style":673},[642],[363,10015],{},[363,10017,10019],{"className":10018},[407],[363,10020,578],{"className":10021},[767,10022],"size1",[363,10024,385],{"className":10025,"style":409},[407,408],[363,10027,461],{"className":10028},[494],[363,10030,1116],{"className":10031},[407,408],[363,10033,468],{"className":10034},[501],[363,10036,10038],{"className":10037},[407],[363,10039,605],{"className":10040},[767,10022],[363,10042,10044,10047],{"style":10043},"top:-2.155em;",[363,10045],{"className":10046,"style":809},[650],[363,10048],{"className":10049},[407],[363,10051,666],{"className":10052},[665],[363,10054,10056],{"className":10055},[638],[363,10057,10060],{"className":10058,"style":10059},[642],"height:1.205em;",[363,10061],{},[363,10063,10065],{"className":10064},[1799],[363,10066,10068,10211],{"className":10067},[633,634],[363,10069,10071,10208],{"className":10070},[638],[363,10072,10074,10164],{"className":10073,"style":9902},[642],[363,10075,10076,10079],{"style":9905},[363,10077],{"className":10078,"style":809},[650],[363,10080,10082,10085,10088,10091,10094,10097,10100,10103,10109,10149,10152,10155,10158],{"className":10081},[407],[363,10083],{"className":10084},[407],[363,10086,4631],{"className":10087},[505],[363,10089],{"className":10090,"style":506},[505],[363,10092,7276],{"className":10093},[510],[363,10095,4631],{"className":10096},[505],[363,10098],{"className":10099,"style":506},[505],[363,10101,385],{"className":10102,"style":409},[407,408],[363,10104,10106],{"className":10105},[407],[363,10107,461],{"className":10108},[767,10022],[363,10110,10112,10115],{"className":10111},[407],[363,10113,571],{"className":10114},[407,714],[363,10116,10118],{"className":10117},[629],[363,10119,10121,10141],{"className":10120},[633,634],[363,10122,10124,10138],{"className":10123},[638],[363,10125,10127],{"className":10126,"style":727},[642],[363,10128,10129,10132],{"style":730},[363,10130],{"className":10131,"style":651},[650],[363,10133,10135],{"className":10134},[655,656,657,658],[363,10136,561],{"className":10137},[407,408,658],[363,10139,666],{"className":10140},[665],[363,10142,10144],{"className":10143},[638],[363,10145,10147],{"className":10146,"style":673},[642],[363,10148],{},[363,10150,578],{"className":10151},[494],[363,10153,1116],{"className":10154},[407,408],[363,10156,605],{"className":10157},[501],[363,10159,10161],{"className":10160},[407],[363,10162,468],{"className":10163},[767,10022],[363,10165,10166,10169],{"style":10043},[363,10167],{"className":10168,"style":809},[650],[363,10170,10172,10175,10178,10181,10184,10187,10190,10193,10196,10199,10202,10205],{"className":10171},[407],[363,10173],{"className":10174},[407],[363,10176],{"className":10177,"style":506},[505],[363,10179,471],{"className":10180},[510],[363,10182],{"className":10183,"style":506},[505],[363,10185,385],{"className":10186,"style":409},[407,408],[363,10188,461],{"className":10189},[494],[363,10191,465],{"className":10192},[407],[363,10194,468],{"className":10195},[501],[363,10197],{"className":10198,"style":506},[505],[363,10200,471],{"className":10201},[510],[363,10203],{"className":10204,"style":506},[505],[363,10206,6112],{"className":10207},[407],[363,10209,666],{"className":10210},[665],[363,10212,10214],{"className":10213},[638],[363,10215,10217],{"className":10216,"style":10059},[642],[363,10218],{},[11,10220,10221,10222,10250,10251,10279,10280,10373],{},"这就是非负性。",[363,10223,10225,10238],{"className":10224,"translate":367},[366],[363,10226,10228],{"className":10227},[371],[373,10229,10230],{"xmlns":375},[377,10231,10232,10236],{},[380,10233,10234],{},[383,10235,385],{},[387,10237,385],{"encoding":389},[363,10239,10241],{"className":10240,"ariaHidden":394},[393],[363,10242,10244,10247],{"className":10243},[398],[363,10245],{"className":10246,"style":403},[402],[363,10248,385],{"className":10249,"style":409},[407,408]," 的凸性是关键：若 ",[363,10252,10254,10267],{"className":10253,"translate":367},[366],[363,10255,10257],{"className":10256},[371],[373,10258,10259],{"xmlns":375},[377,10260,10261,10265],{},[380,10262,10263],{},[383,10264,385],{},[387,10266,385],{"encoding":389},[363,10268,10270],{"className":10269,"ariaHidden":394},[393],[363,10271,10273,10276],{"className":10272},[398],[363,10274],{"className":10275,"style":403},[402],[363,10277,385],{"className":10278,"style":409},[407,408]," 是凹函数，Jensen 方向反转，",[363,10281,10283,10306],{"className":10282,"translate":367},[366],[363,10284,10286],{"className":10285},[371],[373,10287,10288],{"xmlns":375},[377,10289,10290,10303],{},[380,10291,10292,10298,10301],{},[543,10293,10294,10296],{},[383,10295,547],{},[383,10297,385],{},[458,10299,10300],{},"≤",[463,10302,474],{},[387,10304,10305],{"encoding":389},"D_f\\le 0",[363,10307,10309,10364],{"className":10308,"ariaHidden":394},[393],[363,10310,10312,10315,10355,10358,10361],{"className":10311},[398],[363,10313],{"className":10314,"style":4380},[402],[363,10316,10318,10321],{"className":10317},[407],[363,10319,547],{"className":10320,"style":625},[407,408],[363,10322,10324],{"className":10323},[629],[363,10325,10327,10347],{"className":10326},[633,634],[363,10328,10330,10344],{"className":10329},[638],[363,10331,10333],{"className":10332,"style":643},[642],[363,10334,10335,10338],{"style":646},[363,10336],{"className":10337,"style":651},[650],[363,10339,10341],{"className":10340},[655,656,657,658],[363,10342,385],{"className":10343,"style":409},[407,408,658],[363,10345,666],{"className":10346},[665],[363,10348,10350],{"className":10349},[638],[363,10351,10353],{"className":10352,"style":673},[642],[363,10354],{},[363,10356],{"className":10357,"style":506},[505],[363,10359,10300],{"className":10360},[510],[363,10362],{"className":10363,"style":506},[505],[363,10365,10367,10370],{"className":10366},[398],[363,10368],{"className":10369,"style":520},[402],[363,10371,474],{"className":10372},[407],"，那就不是「散度」了。",[10375,10376,10377,10529],"ul",{},[7013,10378,10379,10410,10411,10528],{},[1344,10380,10381,10409],{},[363,10382,10384,10397],{"className":10383,"translate":367},[366],[363,10385,10387],{"className":10386},[371],[373,10388,10389],{"xmlns":375},[377,10390,10391,10395],{},[380,10392,10393],{},[383,10394,385],{},[387,10396,385],{"encoding":389},[363,10398,10400],{"className":10399,"ariaHidden":394},[393],[363,10401,10403,10406],{"className":10402},[398],[363,10404],{"className":10405,"style":403},[402],[363,10407,385],{"className":10408,"style":409},[407,408]," 凸","：保证 ",[363,10412,10414,10446],{"className":10413,"translate":367},[366],[363,10415,10417],{"className":10416},[371],[373,10418,10419],{"xmlns":375},[377,10420,10421,10443],{},[380,10422,10423,10429,10431,10433,10435,10437,10439,10441],{},[543,10424,10425,10427],{},[383,10426,547],{},[383,10428,385],{},[458,10430,461],{"stretchy":460},[383,10432,554],{},[383,10434,558],{"mathvariant":557},[383,10436,561],{},[458,10438,468],{"stretchy":460},[458,10440,7276],{},[463,10442,474],{},[387,10444,10445],{"encoding":389},"D_f(P\\|Q)\\ge 0",[363,10447,10449,10519],{"className":10448,"ariaHidden":394},[393],[363,10450,10452,10455,10495,10498,10501,10504,10507,10510,10513,10516],{"className":10451},[398],[363,10453],{"className":10454,"style":618},[402],[363,10456,10458,10461],{"className":10457},[407],[363,10459,547],{"className":10460,"style":625},[407,408],[363,10462,10464],{"className":10463},[629],[363,10465,10467,10487],{"className":10466},[633,634],[363,10468,10470,10484],{"className":10469},[638],[363,10471,10473],{"className":10472,"style":643},[642],[363,10474,10475,10478],{"style":646},[363,10476],{"className":10477,"style":651},[650],[363,10479,10481],{"className":10480},[655,656,657,658],[363,10482,385],{"className":10483,"style":409},[407,408,658],[363,10485,666],{"className":10486},[665],[363,10488,10490],{"className":10489},[638],[363,10491,10493],{"className":10492,"style":673},[642],[363,10494],{},[363,10496,461],{"className":10497},[494],[363,10499,554],{"className":10500,"style":682},[407,408],[363,10502,558],{"className":10503},[407],[363,10505,561],{"className":10506},[407,408],[363,10508,468],{"className":10509},[501],[363,10511],{"className":10512,"style":506},[505],[363,10514,7276],{"className":10515},[510],[363,10517],{"className":10518,"style":506},[505],[363,10520,10522,10525],{"className":10521},[398],[363,10523],{"className":10524,"style":520},[402],[363,10526,474],{"className":10527},[407],"。",[7013,10530,10531,10598,10599,10650,10651,10785,10786,10528],{},[1344,10532,10533],{},[363,10534,10536,10559],{"className":10535,"translate":367},[366],[363,10537,10539],{"className":10538},[371],[373,10540,10541],{"xmlns":375},[377,10542,10543,10557],{},[380,10544,10545,10547,10549,10551,10553,10555],{},[383,10546,385],{},[458,10548,461],{"stretchy":460},[463,10550,465],{},[458,10552,468],{"stretchy":460},[458,10554,471],{},[463,10556,474],{},[387,10558,477],{"encoding":389},[363,10560,10562,10589],{"className":10561,"ariaHidden":394},[393],[363,10563,10565,10568,10571,10574,10577,10580,10583,10586],{"className":10564},[398],[363,10566],{"className":10567,"style":487},[402],[363,10569,385],{"className":10570,"style":409},[407,408],[363,10572,461],{"className":10573},[494],[363,10575,465],{"className":10576},[407],[363,10578,468],{"className":10579},[501],[363,10581],{"className":10582,"style":506},[505],[363,10584,471],{"className":10585},[510],[363,10587],{"className":10588,"style":506},[505],[363,10590,10592,10595],{"className":10591},[398],[363,10593],{"className":10594,"style":520},[402],[363,10596,474],{"className":10597},[407],"：保证当 ",[363,10600,10602,10620],{"className":10601,"translate":367},[366],[363,10603,10605],{"className":10604},[371],[373,10606,10607],{"xmlns":375},[377,10608,10609,10617],{},[380,10610,10611,10613,10615],{},[383,10612,554],{},[458,10614,471],{},[383,10616,561],{},[387,10618,10619],{"encoding":389},"P=Q",[363,10621,10623,10641],{"className":10622,"ariaHidden":394},[393],[363,10624,10626,10629,10632,10635,10638],{"className":10625},[398],[363,10627],{"className":10628,"style":1011},[402],[363,10630,554],{"className":10631,"style":682},[407,408],[363,10633],{"className":10634,"style":506},[505],[363,10636,471],{"className":10637},[510],[363,10639],{"className":10640,"style":506},[505],[363,10642,10644,10647],{"className":10643},[398],[363,10645],{"className":10646,"style":913},[402],[363,10648,561],{"className":10649},[407,408]," 时，",[363,10652,10654,10684],{"className":10653,"translate":367},[366],[363,10655,10657],{"className":10656},[371],[373,10658,10659],{"xmlns":375},[377,10660,10661,10681],{},[380,10662,10663,10677,10679],{},[586,10664,10665,10671],{},[380,10666,10667,10669],{},[383,10668,592],{},[383,10670,554],{},[380,10672,10673,10675],{},[383,10674,592],{},[383,10676,561],{},[458,10678,471],{},[463,10680,465],{},[387,10682,10683],{"encoding":389},"\\frac{dP}{dQ}=1",[363,10685,10687,10776],{"className":10686,"ariaHidden":394},[393],[363,10688,10690,10693,10767,10770,10773],{"className":10689},[398],[363,10691],{"className":10692,"style":3016},[402],[363,10694,10696,10699,10764],{"className":10695},[407],[363,10697],{"className":10698},[494,789],[363,10700,10702],{"className":10701},[586],[363,10703,10705,10756],{"className":10704},[633,634],[363,10706,10708,10753],{"className":10707},[638],[363,10709,10711,10728,10736],{"className":10710,"style":3035},[642],[363,10712,10713,10716],{"style":2875},[363,10714],{"className":10715,"style":809},[650],[363,10717,10719],{"className":10718},[655,656,657,658],[363,10720,10722,10725],{"className":10721},[407,658],[363,10723,592],{"className":10724},[407,408,658],[363,10726,561],{"className":10727},[407,408,658],[363,10729,10730,10733],{"style":821},[363,10731],{"className":10732,"style":809},[650],[363,10734],{"className":10735,"style":829},[828],[363,10737,10738,10741],{"style":2898},[363,10739],{"className":10740,"style":809},[650],[363,10742,10744],{"className":10743},[655,656,657,658],[363,10745,10747,10750],{"className":10746},[407,658],[363,10748,592],{"className":10749},[407,408,658],[363,10751,554],{"className":10752,"style":682},[407,408,658],[363,10754,666],{"className":10755},[665],[363,10757,10759],{"className":10758},[638],[363,10760,10762],{"className":10761,"style":3087},[642],[363,10763],{},[363,10765],{"className":10766},[501,789],[363,10768],{"className":10769,"style":506},[505],[363,10771,471],{"className":10772},[510],[363,10774],{"className":10775,"style":506},[505],[363,10777,10779,10782],{"className":10778},[398],[363,10780],{"className":10781,"style":520},[402],[363,10783,465],{"className":10784},[407],"，从而 ",[363,10787,10789,10821],{"className":10788,"translate":367},[366],[363,10790,10792],{"className":10791},[371],[373,10793,10794],{"xmlns":375},[377,10795,10796,10818],{},[380,10797,10798,10804,10806,10808,10810,10812,10814,10816],{},[543,10799,10800,10802],{},[383,10801,547],{},[383,10803,385],{},[458,10805,461],{"stretchy":460},[383,10807,554],{},[383,10809,558],{"mathvariant":557},[383,10811,554],{},[458,10813,468],{"stretchy":460},[458,10815,471],{},[463,10817,474],{},[387,10819,10820],{"encoding":389},"D_f(P\\|P)=0",[363,10822,10824,10894],{"className":10823,"ariaHidden":394},[393],[363,10825,10827,10830,10870,10873,10876,10879,10882,10885,10888,10891],{"className":10826},[398],[363,10828],{"className":10829,"style":618},[402],[363,10831,10833,10836],{"className":10832},[407],[363,10834,547],{"className":10835,"style":625},[407,408],[363,10837,10839],{"className":10838},[629],[363,10840,10842,10862],{"className":10841},[633,634],[363,10843,10845,10859],{"className":10844},[638],[363,10846,10848],{"className":10847,"style":643},[642],[363,10849,10850,10853],{"style":646},[363,10851],{"className":10852,"style":651},[650],[363,10854,10856],{"className":10855},[655,656,657,658],[363,10857,385],{"className":10858,"style":409},[407,408,658],[363,10860,666],{"className":10861},[665],[363,10863,10865],{"className":10864},[638],[363,10866,10868],{"className":10867,"style":673},[642],[363,10869],{},[363,10871,461],{"className":10872},[494],[363,10874,554],{"className":10875,"style":682},[407,408],[363,10877,558],{"className":10878},[407],[363,10880,554],{"className":10881,"style":682},[407,408],[363,10883,468],{"className":10884},[501],[363,10886],{"className":10887,"style":506},[505],[363,10889,471],{"className":10890},[510],[363,10892],{"className":10893,"style":506},[505],[363,10895,10897,10900],{"className":10896},[398],[363,10898],{"className":10899,"style":520},[402],[363,10901,474],{"className":10902},[407],[11,10904,10905],{},"因此 f-散度天然具有非负性和「相同分布散度为零」的性质。",[2659,10907,10909],{"id":10908},"_22-凸性为何必需","2.2 凸性为何必需",[11,10911,10912,10913,10982,10983,11011],{},"有人会问：凸性是不是一个过强的技术条件？答案是否定的。若要求 ",[363,10914,10916,10933],{"className":10915,"translate":367},[366],[363,10917,10919],{"className":10918},[371],[373,10920,10921],{"xmlns":375},[377,10922,10923,10931],{},[380,10924,10925],{},[543,10926,10927,10929],{},[383,10928,547],{},[383,10930,385],{},[387,10932,4370],{"encoding":389},[363,10934,10936],{"className":10935,"ariaHidden":394},[393],[363,10937,10939,10942],{"className":10938},[398],[363,10940],{"className":10941,"style":4380},[402],[363,10943,10945,10948],{"className":10944},[407],[363,10946,547],{"className":10947,"style":625},[407,408],[363,10949,10951],{"className":10950},[629],[363,10952,10954,10974],{"className":10953},[633,634],[363,10955,10957,10971],{"className":10956},[638],[363,10958,10960],{"className":10959,"style":643},[642],[363,10961,10962,10965],{"style":646},[363,10963],{"className":10964,"style":651},[650],[363,10966,10968],{"className":10967},[655,656,657,658],[363,10969,385],{"className":10970,"style":409},[407,408,658],[363,10972,666],{"className":10973},[665],[363,10975,10977],{"className":10976},[638],[363,10978,10980],{"className":10979,"style":673},[642],[363,10981],{}," 对一切「随机变换」（马尔可夫核）单调不增（即第 4.5 节的数据处理不等式），那么 ",[363,10984,10986,10999],{"className":10985,"translate":367},[366],[363,10987,10989],{"className":10988},[371],[373,10990,10991],{"xmlns":375},[377,10992,10993,10997],{},[380,10994,10995],{},[383,10996,385],{},[387,10998,385],{"encoding":389},[363,11000,11002],{"className":11001,"ariaHidden":394},[393],[363,11003,11005,11008],{"className":11004},[398],[363,11006],{"className":11007,"style":403},[402],[363,11009,385],{"className":11010,"style":409},[407,408]," 必须凸。二元分布上就能看到这一点。",[143,11013,11014],{},[11,11015,11016,11019,11020,11215,11216,11257,11258,11286,11287,11315],{},[1344,11017,11018],{},"命题 2.1（Csiszár 刻画）"," 若数据处理不等式 ",[363,11021,11023,11074],{"className":11022,"translate":367},[366],[363,11024,11026],{"className":11025},[371],[373,11027,11028],{"xmlns":375},[377,11029,11030,11071],{},[380,11031,11032,11038,11040,11042,11045,11047,11049,11051,11053,11055,11061,11063,11065,11067,11069],{},[543,11033,11034,11036],{},[383,11035,547],{},[383,11037,385],{},[458,11039,461],{"stretchy":460},[383,11041,554],{},[383,11043,11044],{},"K",[383,11046,558],{"mathvariant":557},[383,11048,561],{},[383,11050,11044],{},[458,11052,468],{"stretchy":460},[458,11054,10300],{},[543,11056,11057,11059],{},[383,11058,547],{},[383,11060,385],{},[458,11062,461],{"stretchy":460},[383,11064,554],{},[383,11066,558],{"mathvariant":557},[383,11068,561],{},[458,11070,468],{"stretchy":460},[387,11072,11073],{"encoding":389},"D_f(PK\\|QK)\\le D_f(P\\|Q)",[363,11075,11077,11154],{"className":11076,"ariaHidden":394},[393],[363,11078,11080,11083,11123,11126,11129,11133,11136,11139,11142,11145,11148,11151],{"className":11079},[398],[363,11081],{"className":11082,"style":618},[402],[363,11084,11086,11089],{"className":11085},[407],[363,11087,547],{"className":11088,"style":625},[407,408],[363,11090,11092],{"className":11091},[629],[363,11093,11095,11115],{"className":11094},[633,634],[363,11096,11098,11112],{"className":11097},[638],[363,11099,11101],{"className":11100,"style":643},[642],[363,11102,11103,11106],{"style":646},[363,11104],{"className":11105,"style":651},[650],[363,11107,11109],{"className":11108},[655,656,657,658],[363,11110,385],{"className":11111,"style":409},[407,408,658],[363,11113,666],{"className":11114},[665],[363,11116,11118],{"className":11117},[638],[363,11119,11121],{"className":11120,"style":673},[642],[363,11122],{},[363,11124,461],{"className":11125},[494],[363,11127,554],{"className":11128,"style":682},[407,408],[363,11130,11044],{"className":11131,"style":11132},[407,408],"margin-right:0.0715em;",[363,11134,558],{"className":11135},[407],[363,11137,561],{"className":11138},[407,408],[363,11140,11044],{"className":11141,"style":11132},[407,408],[363,11143,468],{"className":11144},[501],[363,11146],{"className":11147,"style":506},[505],[363,11149,10300],{"className":11150},[510],[363,11152],{"className":11153,"style":506},[505],[363,11155,11157,11160,11200,11203,11206,11209,11212],{"className":11156},[398],[363,11158],{"className":11159,"style":618},[402],[363,11161,11163,11166],{"className":11162},[407],[363,11164,547],{"className":11165,"style":625},[407,408],[363,11167,11169],{"className":11168},[629],[363,11170,11172,11192],{"className":11171},[633,634],[363,11173,11175,11189],{"className":11174},[638],[363,11176,11178],{"className":11177,"style":643},[642],[363,11179,11180,11183],{"style":646},[363,11181],{"className":11182,"style":651},[650],[363,11184,11186],{"className":11185},[655,656,657,658],[363,11187,385],{"className":11188,"style":409},[407,408,658],[363,11190,666],{"className":11191},[665],[363,11193,11195],{"className":11194},[638],[363,11196,11198],{"className":11197,"style":673},[642],[363,11199],{},[363,11201,461],{"className":11202},[494],[363,11204,554],{"className":11205,"style":682},[407,408],[363,11207,558],{"className":11208},[407],[363,11210,561],{"className":11211},[407,408],[363,11213,468],{"className":11214},[501]," 对一切概率对 ",[363,11217,11219,11236],{"className":11218,"translate":367},[366],[363,11220,11222],{"className":11221},[371],[373,11223,11224],{"xmlns":375},[377,11225,11226,11234],{},[380,11227,11228,11230,11232],{},[383,11229,554],{},[458,11231,898],{"separator":394},[383,11233,561],{},[387,11235,903],{"encoding":389},[363,11237,11239],{"className":11238,"ariaHidden":394},[393],[363,11240,11242,11245,11248,11251,11254],{"className":11241},[398],[363,11243],{"className":11244,"style":913},[402],[363,11246,554],{"className":11247,"style":682},[407,408],[363,11249,898],{"className":11250},[920],[363,11252],{"className":11253,"style":754},[505],[363,11255,561],{"className":11256},[407,408]," 与一切马尔可夫核 ",[363,11259,11261,11274],{"className":11260,"translate":367},[366],[363,11262,11264],{"className":11263},[371],[373,11265,11266],{"xmlns":375},[377,11267,11268,11272],{},[380,11269,11270],{},[383,11271,11044],{},[387,11273,11044],{"encoding":389},[363,11275,11277],{"className":11276,"ariaHidden":394},[393],[363,11278,11280,11283],{"className":11279},[398],[363,11281],{"className":11282,"style":1011},[402],[363,11284,11044],{"className":11285,"style":11132},[407,408]," 成立，则 ",[363,11288,11290,11303],{"className":11289,"translate":367},[366],[363,11291,11293],{"className":11292},[371],[373,11294,11295],{"xmlns":375},[377,11296,11297,11301],{},[380,11298,11299],{},[383,11300,385],{},[387,11302,385],{"encoding":389},[363,11304,11306],{"className":11305,"ariaHidden":394},[393],[363,11307,11309,11312],{"className":11308},[398],[363,11310],{"className":11311,"style":403},[402],[363,11313,385],{"className":11314,"style":409},[407,408]," 凸。",[11,11317,11318],{},[1344,11319,11320],{},"证明（两步）。",[11,11322,11323,11410,11411,2628,11645,11674,11675,11751,11752,5993,11950,12147,12148,12285],{},[1344,11324,11325,11326,6842,11357,11409],{},"第一步：DPI ",[363,11327,11329,11344],{"className":11328,"translate":367},[366],[363,11330,11332],{"className":11331},[371],[373,11333,11334],{"xmlns":375},[377,11335,11336,11341],{},[380,11337,11338],{},[458,11339,11340],{},"⇒",[387,11342,11343],{"encoding":389},"\\Rightarrow",[363,11345,11347],{"className":11346,"ariaHidden":394},[393],[363,11348,11350,11354],{"className":11349},[398],[363,11351],{"className":11352,"style":11353},[402],"height:0.3669em;",[363,11355,11340],{"className":11356},[510],[363,11358,11360,11382],{"className":11359,"translate":367},[366],[363,11361,11363],{"className":11362},[371],[373,11364,11365],{"xmlns":375},[377,11366,11367,11379],{},[380,11368,11369,11371,11373,11375,11377],{},[458,11370,461],{"stretchy":460},[383,11372,554],{},[458,11374,898],{"separator":394},[383,11376,561],{},[458,11378,468],{"stretchy":460},[387,11380,11381],{"encoding":389},"(P,Q)",[363,11383,11385],{"className":11384,"ariaHidden":394},[393],[363,11386,11388,11391,11394,11397,11400,11403,11406],{"className":11387},[398],[363,11389],{"className":11390,"style":487},[402],[363,11392,461],{"className":11393},[494],[363,11395,554],{"className":11396,"style":682},[407,408],[363,11398,898],{"className":11399},[920],[363,11401],{"className":11402,"style":754},[505],[363,11404,561],{"className":11405},[407,408],[363,11407,468],{"className":11408},[501]," 联合凸。"," 设 ",[363,11412,11414,11456],{"className":11413,"translate":367},[366],[363,11415,11417],{"className":11416},[371],[373,11418,11419],{"xmlns":375},[377,11420,11421,11453],{},[380,11422,11423,11429,11431,11437,11439,11445,11447],{},[543,11424,11425,11427],{},[383,11426,554],{},[463,11428,465],{},[458,11430,898],{"separator":394},[543,11432,11433,11435],{},[383,11434,554],{},[463,11436,2811],{},[458,11438,898],{"separator":394},[543,11440,11441,11443],{},[383,11442,561],{},[463,11444,465],{},[458,11446,898],{"separator":394},[543,11448,11449,11451],{},[383,11450,561],{},[463,11452,2811],{},[387,11454,11455],{"encoding":389},"P_1,P_2,Q_1,Q_2",[363,11457,11459],{"className":11458,"ariaHidden":394},[393],[363,11460,11462,11465,11507,11510,11513,11553,11556,11559,11599,11602,11605],{"className":11461},[398],[363,11463],{"className":11464,"style":913},[402],[363,11466,11468,11471],{"className":11467},[407],[363,11469,554],{"className":11470,"style":682},[407,408],[363,11472,11474],{"className":11473},[629],[363,11475,11477,11499],{"className":11476},[633,634],[363,11478,11480,11496],{"className":11479},[638],[363,11481,11484],{"className":11482,"style":11483},[642],"height:0.3011em;",[363,11485,11487,11490],{"style":11486},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[363,11488],{"className":11489,"style":651},[650],[363,11491,11493],{"className":11492},[655,656,657,658],[363,11494,465],{"className":11495},[407,658],[363,11497,666],{"className":11498},[665],[363,11500,11502],{"className":11501},[638],[363,11503,11505],{"className":11504,"style":8316},[642],[363,11506],{},[363,11508,898],{"className":11509},[920],[363,11511],{"className":11512,"style":754},[505],[363,11514,11516,11519],{"className":11515},[407],[363,11517,554],{"className":11518,"style":682},[407,408],[363,11520,11522],{"className":11521},[629],[363,11523,11525,11545],{"className":11524},[633,634],[363,11526,11528,11542],{"className":11527},[638],[363,11529,11531],{"className":11530,"style":11483},[642],[363,11532,11533,11536],{"style":11486},[363,11534],{"className":11535,"style":651},[650],[363,11537,11539],{"className":11538},[655,656,657,658],[363,11540,2811],{"className":11541},[407,658],[363,11543,666],{"className":11544},[665],[363,11546,11548],{"className":11547},[638],[363,11549,11551],{"className":11550,"style":8316},[642],[363,11552],{},[363,11554,898],{"className":11555},[920],[363,11557],{"className":11558,"style":754},[505],[363,11560,11562,11565],{"className":11561},[407],[363,11563,561],{"className":11564},[407,408],[363,11566,11568],{"className":11567},[629],[363,11569,11571,11591],{"className":11570},[633,634],[363,11572,11574,11588],{"className":11573},[638],[363,11575,11577],{"className":11576,"style":11483},[642],[363,11578,11579,11582],{"style":730},[363,11580],{"className":11581,"style":651},[650],[363,11583,11585],{"className":11584},[655,656,657,658],[363,11586,465],{"className":11587},[407,658],[363,11589,666],{"className":11590},[665],[363,11592,11594],{"className":11593},[638],[363,11595,11597],{"className":11596,"style":8316},[642],[363,11598],{},[363,11600,898],{"className":11601},[920],[363,11603],{"className":11604,"style":754},[505],[363,11606,11608,11611],{"className":11607},[407],[363,11609,561],{"className":11610},[407,408],[363,11612,11614],{"className":11613},[629],[363,11615,11617,11637],{"className":11616},[633,634],[363,11618,11620,11634],{"className":11619},[638],[363,11621,11623],{"className":11622,"style":11483},[642],[363,11624,11625,11628],{"style":730},[363,11626],{"className":11627,"style":651},[650],[363,11629,11631],{"className":11630},[655,656,657,658],[363,11632,2811],{"className":11633},[407,658],[363,11635,666],{"className":11636},[665],[363,11638,11640],{"className":11639},[638],[363,11641,11643],{"className":11642,"style":8316},[642],[363,11644],{},[363,11646,11648,11662],{"className":11647,"translate":367},[366],[363,11649,11651],{"className":11650},[371],[373,11652,11653],{"xmlns":375},[377,11654,11655,11659],{},[380,11656,11657],{},[383,11658,945],{"mathvariant":944},[387,11660,11661],{"encoding":389},"\\mathcal X",[363,11663,11665],{"className":11664,"ariaHidden":394},[393],[363,11666,11668,11671],{"className":11667},[398],[363,11669],{"className":11670,"style":1011},[402],[363,11672,945],{"className":11673,"style":972},[407,971]," 上任给的四对分布，",[363,11676,11678,11705],{"className":11677,"translate":367},[366],[363,11679,11681],{"className":11680},[371],[373,11682,11683],{"xmlns":375},[377,11684,11685,11702],{},[380,11686,11687,11690,11692,11694,11696,11698,11700],{},[383,11688,11689],{},"λ",[458,11691,6257],{},[458,11693,461],{"stretchy":460},[463,11695,474],{},[458,11697,898],{"separator":394},[463,11699,465],{},[458,11701,468],{"stretchy":460},[387,11703,11704],{"encoding":389},"\\lambda\\in(0,1)",[363,11706,11708,11727],{"className":11707,"ariaHidden":394},[393],[363,11709,11711,11715,11718,11721,11724],{"className":11710},[398],[363,11712],{"className":11713,"style":11714},[402],"height:0.7335em;vertical-align:-0.0391em;",[363,11716,11689],{"className":11717},[407,408],[363,11719],{"className":11720,"style":506},[505],[363,11722,6257],{"className":11723},[510],[363,11725],{"className":11726,"style":506},[505],[363,11728,11730,11733,11736,11739,11742,11745,11748],{"className":11729},[398],[363,11731],{"className":11732,"style":487},[402],[363,11734,461],{"className":11735},[494],[363,11737,474],{"className":11738},[407],[363,11740,898],{"className":11741},[920],[363,11743],{"className":11744,"style":754},[505],[363,11746,465],{"className":11747},[407],[363,11749,468],{"className":11750},[501],"，记 ",[363,11753,11755,11797],{"className":11754,"translate":367},[366],[363,11756,11758],{"className":11757},[371],[373,11759,11760],{"xmlns":375},[377,11761,11762,11794],{},[380,11763,11764,11766,11768,11770,11776,11778,11780,11782,11784,11786,11788],{},[383,11765,554],{},[458,11767,471],{},[383,11769,11689],{},[543,11771,11772,11774],{},[383,11773,554],{},[463,11775,465],{},[458,11777,2818],{},[458,11779,461],{"stretchy":460},[463,11781,465],{},[458,11783,5161],{},[383,11785,11689],{},[458,11787,468],{"stretchy":460},[543,11789,11790,11792],{},[383,11791,554],{},[463,11793,2811],{},[387,11795,11796],{"encoding":389},"P=\\lambda 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",[363,12149,12151,12188],{"className":12150,"translate":367},[366],[363,12152,12154],{"className":12153},[371],[373,12155,12156],{"xmlns":375},[377,12157,12158,12185],{},[380,12159,12160,12168,12170,12172,12174,12176,12178,12180,12183],{},[12161,12162,12163,12165],"mover",{"accent":394},[383,12164,945],{"mathvariant":944},[458,12166,12167],{},"~",[458,12169,471],{},[458,12171,3246],{"stretchy":460},[463,12173,465],{},[458,12175,898],{"separator":394},[463,12177,2811],{},[458,12179,3256],{"stretchy":460},[458,12181,12182],{},"×",[383,12184,945],{"mathvariant":944},[387,12186,12187],{"encoding":389},"\\tilde{\\mathcal X}=\\{1,2\\}\\times\\mathcal X",[363,12189,12191,12243,12276],{"className":12190,"ariaHidden":394},[393],[363,12192,12194,12198,12234,12237,12240],{"className":12193},[398],[363,12195],{"className":12196,"style":12197},[402],"height:0.9202em;",[363,12199,12202],{"className":12200},[407,12201],"accent",[363,12203,12205],{"className":12204},[633],[363,12206,12208],{"className":12207},[638],[363,12209,12211,12220],{"className":12210,"style":12197},[642],[363,12212,12214,12217],{"style":12213},"top:-3em;",[363,12215],{"className":12216,"style":809},[650],[363,12218,945],{"className":12219,"style":972},[407,971],[363,12221,12223,12226],{"style":12222},"top:-3.6023em;",[363,12224],{"className":12225,"style":809},[650],[363,12227,12231],{"className":12228,"style":12230},[12229],"accent-body","left:-0.1111em;",[363,12232,12167],{"className":12233},[407],[363,12235],{"className":12236,"style":506},[505],[363,12238,471],{"className":12239},[510],[363,12241],{"className":12242,"style":506},[505],[363,12244,12246,12249,12252,12255,12258,12261,12264,12267,12270,12273],{"className":12245},[398],[363,12247],{"className":12248,"style":487},[402],[363,12250,3246],{"className":12251},[494],[363,12253,465],{"className":12254},[407],[363,12256,898],{"className":12257},[920],[363,12259],{"className":12260,"style":754},[505],[363,12262,2811],{"className":12263},[407],[363,12265,3256],{"className":12266},[501],[363,12268],{"className":12269,"style":2935},[505],[363,12271,12182],{"className":12272},[2939],[363,12274],{"className":12275,"style":2935},[505],[363,12277,12279,12282],{"className":12278},[398],[363,12280],{"className":12281,"style":1011},[402],[363,12283,945],{"className":12284,"style":972},[407,971]," 及其上的两个分块分布：",[363,12287,12289],{"className":12288,"translate":367},[528],[363,12290,12292,12427],{"className":12291,"translate":367},[366],[363,12293,12295],{"className":12294},[371],[373,12296,12297],{"xmlns":375,"display":537},[377,12298,12299,12424],{},[380,12300,12301,12307,12309,12311,12314,12316,12318,12320,12322,12324,12330,12332,12338,12340,12342,12344,12346,12348,12354,12356,12358,12360,12362,12364,12366,12368,12370,12376,12378,12384,12386,12388,12390,12392,12394,12400,12402,12404,12406,12408,12414,12416,12418,12420,12422],{},[12161,12302,12303,12305],{"accent":394},[383,12304,554],{},[458,12306,12167],{},[458,12308,461],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[458,12310,461],{"stretchy":460},[383,12312,12313],{},"i",[458,12315,898],{"separator":394},[383,12317,1121],{},[458,12319,468],{"stretchy":460},[458,12321,468],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[458,12323,471],{},[543,12325,12326,12328],{},[383,12327,7918],{},[383,12329,12313],{},[1645,12331,1647],{},[543,12333,12334,12336],{},[383,12335,554],{},[383,12337,12313],{},[458,12339,461],{"stretchy":460},[383,12341,1121],{},[458,12343,468],{"stretchy":460},[458,12345,898],{"separator":394},[505,12347],{"width":3610},[12161,12349,12350,12352],{"accent":394},[383,12351,561],{},[458,12353,12167],{},[458,12355,461],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[458,12357,461],{"stretchy":460},[383,12359,12313],{},[458,12361,898],{"separator":394},[383,12363,1121],{},[458,12365,468],{"stretchy":460},[458,12367,468],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[458,12369,471],{},[543,12371,12372,12374],{},[383,12373,7918],{},[383,12375,12313],{},[1645,12377,1647],{},[543,12379,12380,12382],{},[383,12381,561],{},[383,12383,12313],{},[458,12385,461],{"stretchy":460},[383,12387,1121],{},[458,12389,468],{"stretchy":460},[458,12391,898],{"separator":394},[505,12393],{"width":3610},[543,12395,12396,12398],{},[383,12397,7918],{},[463,12399,465],{},[458,12401,471],{},[383,12403,11689],{},[458,12405,898],{"separator":394},[1645,12407,4631],{},[543,12409,12410,12412],{},[383,12411,7918],{},[463,12413,2811],{},[458,12415,471],{},[463,12417,465],{},[458,12419,5161],{},[383,12421,11689],{},[383,12423,4723],{"mathvariant":557},[387,12425,12426],{"encoding":389},"\\tilde P\\big((i,x)\\big)=w_i\\,P_i(x),\\qquad\n\\tilde Q\\big((i,x)\\big)=w_i\\,Q_i(x),\\qquad\nw_1=\\lambda,\\ w_2=1-\\lambda .",[363,12428,12430,12508,12699,12855,12922,12941],{"className":12429,"ariaHidden":394},[393],[363,12431,12433,12437,12469,12475,12478,12481,12484,12487,12490,12493,12499,12502,12505],{"className":12432},[398],[363,12434],{"className":12435,"style":12436},[402],"height:1.2702em;vertical-align:-0.35em;",[363,12438,12440],{"className":12439},[407,12201],[363,12441,12443],{"className":12442},[633],[363,12444,12446],{"className":12445},[638],[363,12447,12449,12457],{"className":12448,"style":12197},[642],[363,12450,12451,12454],{"style":12213},[363,12452],{"className":12453,"style":809},[650],[363,12455,554],{"className":12456,"style":682},[407,408],[363,12458,12459,12462],{"style":12222},[363,12460],{"className":12461,"style":809},[650],[363,12463,12466],{"className":12464,"style":12465},[12229],"left:-0.1667em;",[363,12467,12167],{"className":12468},[407],[363,12470,12472],{"className":12471},[407],[363,12473,461],{"className":12474},[767,10022],[363,12476,461],{"className":12477},[494],[363,12479,12313],{"className":12480},[407,408],[363,12482,898],{"className":12483},[920],[363,12485],{"className":12486,"style":754},[505],[363,12488,1121],{"className":12489},[407,408],[363,12491,468],{"className":12492},[501],[363,12494,12496],{"className":12495},[407],[363,12497,468],{"className":12498},[767,10022],[363,12500],{"className":12501,"style":506},[505],[363,12503,471],{"className":12504},[510],[363,12506],{"className":12507,"style":506},[505],[363,12509,12511,12514,12556,12559,12599,12602,12605,12608,12611,12614,12617,12660,12666,12669,12672,12675,12678,12681,12684,12690,12693,12696],{"className":12510},[398],[363,12512],{"className":12513,"style":12436},[402],[363,12515,12517,12520],{"className":12516},[407],[363,12518,7918],{"className":12519,"style":8622},[407,408],[363,12521,12523],{"className":12522},[629],[363,12524,12526,12548],{"className":12525},[633,634],[363,12527,12529,12545],{"className":12528},[638],[363,12530,12533],{"className":12531,"style":12532},[642],"height:0.3117em;",[363,12534,12536,12539],{"style":12535},"top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;",[363,12537],{"className":12538,"style":651},[650],[363,12540,12542],{"className":12541},[655,656,657,658],[363,12543,12313],{"className":12544},[407,408,658],[363,12546,666],{"className":12547},[665],[363,12549,12551],{"className":12550},[638],[363,12552,12554],{"className":12553,"style":8316},[642],[363,12555],{},[363,12557],{"className":12558,"style":754},[505],[363,12560,12562,12565],{"className":12561},[407],[363,12563,554],{"className":12564,"style":682},[407,408],[363,12566,12568],{"className":12567},[629],[363,12569,12571,12591],{"className":12570},[633,634],[363,12572,12574,12588],{"className":12573},[638],[363,12575,12577],{"className":12576,"style":12532},[642],[363,12578,12579,12582],{"style":11486},[363,12580],{"className":12581,"style":651},[650],[363,12583,12585],{"className":12584},[655,656,657,658],[363,12586,12313],{"className":12587},[407,408,658],[363,12589,666],{"className":12590},[665],[363,12592,12594],{"className":12593},[638],[363,12595,12597],{"className":12596,"style":8316},[642],[363,12598],{},[363,12600,461],{"className":12601},[494],[363,12603,1121],{"className":12604},[407,408],[363,12606,468],{"className":12607},[501],[363,12609,898],{"className":12610},[920],[363,12612],{"className":12613,"style":3982},[505],[363,12615],{"className":12616,"style":754},[505],[363,12618,12620],{"className":12619},[407,12201],[363,12621,12623,12651],{"className":12622},[633,634],[363,12624,12626,12648],{"className":12625},[638],[363,12627,12629,12637],{"className":12628,"style":12197},[642],[363,12630,12631,12634],{"style":12213},[363,12632],{"className":12633,"style":809},[650],[363,12635,561],{"className":12636},[407,408],[363,12638,12639,12642],{"style":12222},[363,12640],{"className":12641,"style":809},[650],[363,12643,12645],{"className":12644,"style":12465},[12229],[363,12646,12167],{"className":12647},[407],[363,12649,666],{"className":12650},[665],[363,12652,12654],{"className":12653},[638],[363,12655,12658],{"className":12656,"style":12657},[642],"height:0.1944em;",[363,12659],{},[363,12661,12663],{"className":12662},[407],[363,12664,461],{"className":12665},[767,10022],[363,12667,461],{"className":12668},[494],[363,12670,12313],{"className":12671},[407,408],[363,12673,898],{"className":12674},[920],[363,12676],{"className":12677,"style":754},[505],[363,12679,1121],{"className":12680},[407,408],[363,12682,468],{"className":12683},[501],[363,12685,12687],{"className":12686},[407],[363,12688,468],{"className":12689},[767,10022],[363,12691],{"className":12692,"style":506},[505],[363,12694,471],{"className":12695},[510],[363,12697],{"className":12698,"style":506},[505],[363,12700,12702,12705,12745,12748,12788,12791,12794,12797,12800,12803,12806,12846,12849,12852],{"className":12701},[398],[363,12703],{"className":12704,"style":487},[402],[363,12706,12708,12711],{"className":12707},[407],[363,12709,7918],{"className":12710,"style":8622},[407,408],[363,12712,12714],{"className":12713},[629],[363,12715,12717,12737],{"className":12716},[633,634],[363,12718,12720,12734],{"className":12719},[638],[363,12721,12723],{"className":12722,"style":12532},[642],[363,12724,12725,12728],{"style":12535},[363,12726],{"className":12727,"style":651},[650],[363,12729,12731],{"className":12730},[655,656,657,658],[363,12732,12313],{"className":12733},[407,408,658],[363,12735,666],{"className":12736},[665],[363,12738,12740],{"className":12739},[638],[363,12741,12743],{"className":12742,"style":8316},[642],[363,12744],{},[363,12746],{"className":12747,"style":754},[505],[363,12749,12751,12754],{"className":12750},[407],[363,12752,561],{"className":12753},[407,408],[363,12755,12757],{"className":12756},[629],[363,12758,12760,12780],{"className":12759},[633,634],[363,12761,12763,12777],{"className":12762},[638],[363,12764,12766],{"className":12765,"style":12532},[642],[363,12767,12768,12771],{"style":730},[363,12769],{"className":12770,"style":651},[650],[363,12772,12774],{"className":12773},[655,656,657,658],[363,12775,12313],{"className":12776},[407,408,658],[363,12778,666],{"className":12779},[665],[363,12781,12783],{"className":12782},[638],[363,12784,12786],{"className":12785,"style":8316},[642],[363,12787],{},[363,12789,461],{"className":12790},[494],[363,12792,1121],{"className":12793},[407,408],[363,12795,468],{"className":12796},[501],[363,12798,898],{"className":12799},[920],[363,12801],{"className":12802,"style":3982},[505],[363,12804],{"className":12805,"style":754},[505],[363,12807,12809,12812],{"className":12808},[407],[363,12810,7918],{"className":12811,"style":8622},[407,408],[363,12813,12815],{"className":12814},[629],[363,12816,12818,12838],{"className":12817},[633,634],[363,12819,12821,12835],{"className":12820},[638],[363,12822,12824],{"className":12823,"style":11483},[642],[363,12825,12826,12829],{"style":12535},[363,12827],{"className":12828,"style":651},[650],[363,12830,12832],{"className":12831},[655,656,657,658],[363,12833,465],{"className":12834},[407,658],[363,12836,666],{"className":12837},[665],[363,12839,12841],{"className":12840},[638],[363,12842,12844],{"className":12843,"style":8316},[642],[363,12845],{},[363,12847],{"className":12848,"style":506},[505],[363,12850,471],{"className":12851},[510],[363,12853],{"className":12854,"style":506},[505],[363,12856,12858,12861,12864,12867,12870,12873,12913,12916,12919],{"className":12857},[398],[363,12859],{"className":12860,"style":403},[402],[363,12862,11689],{"className":12863},[407,408],[363,12865,898],{"className":12866},[920],[363,12868,4631],{"className":12869},[505],[363,12871],{"className":12872,"style":754},[505],[363,12874,12876,12879],{"className":12875},[407],[363,12877,7918],{"className":12878,"style":8622},[407,408],[363,12880,12882],{"className":12881},[629],[363,12883,12885,12905],{"className":12884},[633,634],[363,12886,12888,12902],{"className":12887},[638],[363,12889,12891],{"className":12890,"style":11483},[642],[363,12892,12893,12896],{"style":12535},[363,12894],{"className":12895,"style":651},[650],[363,12897,12899],{"className":12898},[655,656,657,658],[363,12900,2811],{"className":12901},[407,658],[363,12903,666],{"className":12904},[665],[363,12906,12908],{"className":12907},[638],[363,12909,12911],{"className":12910,"style":8316},[642],[363,12912],{},[363,12914],{"className":12915,"style":506},[505],[363,12917,471],{"className":12918},[510],[363,12920],{"className":12921,"style":506},[505],[363,12923,12925,12929,12932,12935,12938],{"className":12924},[398],[363,12926],{"className":12927,"style":12928},[402],"height:0.7278em;vertical-align:-0.0833em;",[363,12930,465],{"className":12931},[407],[363,12933],{"className":12934,"style":2935},[505],[363,12936,5161],{"className":12937},[2939],[363,12939],{"className":12940,"style":2935},[505],[363,12942,12944,12948,12951],{"className":12943},[398],[363,12945],{"className":12946,"style":12947},[402],"height:0.6944em;",[363,12949,11689],{"className":12950},[407,408],[363,12952,4723],{"className":12953},[407],[11,12955,12956,12957,6842,12960,13061,13062,13090,13091,5993,13179,13279,13280,13693,13694,13765],{},"取",[1344,12958,12959],{},"确定性核",[363,12961,12963,12995],{"className":12962,"translate":367},[366],[363,12964,12966],{"className":12965},[371],[373,12967,12968],{"xmlns":375},[377,12969,12970,12992],{},[380,12971,12972,12974,12976,12978,12980,12982,12984,12986,12988,12990],{},[383,12973,11044],{},[458,12975,1364],{},[1645,12977,1647],{},[458,12979,461],{"stretchy":460},[383,12981,12313],{},[458,12983,898],{"separator":394},[383,12985,1121],{},[458,12987,468],{"stretchy":460},[458,12989,9032],{},[383,12991,1121],{},[387,12993,12994],{"encoding":389},"K:\\,(i,x)\\mapsto x",[363,12996,12998,13019,13052],{"className":12997,"ariaHidden":394},[393],[363,12999,13001,13004,13007,13010,13013,13016],{"className":13000},[398],[363,13002],{"className":13003,"style":1011},[402],[363,13005,11044],{"className":13006,"style":11132},[407,408],[363,13008],{"className":13009,"style":506},[505],[363,13011,1364],{"className":13012},[510],[363,13014],{"className":13015,"style":754},[505],[363,13017],{"className":13018,"style":506},[505],[363,13020,13022,13025,13028,13031,13034,13037,13040,13043,13046,13049],{"className":13021},[398],[363,13023],{"className":13024,"style":487},[402],[363,13026,461],{"className":13027},[494],[363,13029,12313],{"className":13030},[407,408],[363,13032,898],{"className":13033},[920],[363,13035],{"className":13036,"style":754},[505],[363,13038,1121],{"className":13039},[407,408],[363,13041,468],{"className":13042},[501],[363,13044],{"className":13045,"style":506},[505],[363,13047,9032],{"className":13048},[510],[363,13050],{"className":13051,"style":506},[505],[363,13053,13055,13058],{"className":13054},[398],[363,13056],{"className":13057,"style":2693},[402],[363,13059,1121],{"className":13060},[407,408],"（\"擦掉\"分量下标，只保留 ",[363,13063,13065,13078],{"className":13064,"translate":367},[366],[363,13066,13068],{"className":13067},[371],[373,13069,13070],{"xmlns":375},[377,13071,13072,13076],{},[380,13073,13074],{},[383,13075,1121],{},[387,13077,1121],{"encoding":389},[363,13079,13081],{"className":13080,"ariaHidden":394},[393],[363,13082,13084,13087],{"className":13083},[398],[363,13085],{"className":13086,"style":2693},[402],[363,13088,1121],{"className":13089},[407,408],"）。由分块结构立刻算出 ",[363,13092,13094,13118],{"className":13093,"translate":367},[366],[363,13095,13097],{"className":13096},[371],[373,13098,13099],{"xmlns":375},[377,13100,13101,13115],{},[380,13102,13103,13109,13111,13113],{},[12161,13104,13105,13107],{"accent":394},[383,13106,554],{},[458,13108,12167],{},[383,13110,11044],{},[458,13112,471],{},[383,13114,554],{},[387,13116,13117],{"encoding":389},"\\tilde PK=P",[363,13119,13121,13170],{"className":13120,"ariaHidden":394},[393],[363,13122,13124,13127,13158,13161,13164,13167],{"className":13123},[398],[363,13125],{"className":13126,"style":12197},[402],[363,13128,13130],{"className":13129},[407,12201],[363,13131,13133],{"className":13132},[633],[363,13134,13136],{"className":13135},[638],[363,13137,13139,13147],{"className":13138,"style":12197},[642],[363,13140,13141,13144],{"style":12213},[363,13142],{"className":13143,"style":809},[650],[363,13145,554],{"className":13146,"style":682},[407,408],[363,13148,13149,13152],{"style":12222},[363,13150],{"className":13151,"style":809},[650],[363,13153,13155],{"className":13154,"style":12465},[12229],[363,13156,12167],{"className":13157},[407],[363,13159,11044],{"className":13160,"style":11132},[407,408],[363,13162],{"className":13163,"style":506},[505],[363,13165,471],{"className":13166},[510],[363,13168],{"className":13169,"style":506},[505],[363,13171,13173,13176],{"className":13172},[398],[363,13174],{"className":13175,"style":1011},[402],[363,13177,554],{"className":13178,"style":682},[407,408],[363,13180,13182,13206],{"className":13181,"translate":367},[366],[363,13183,13185],{"className":13184},[371],[373,13186,13187],{"xmlns":375},[377,13188,13189,13203],{},[380,13190,13191,13197,13199,13201],{},[12161,13192,13193,13195],{"accent":394},[383,13194,561],{},[458,13196,12167],{},[383,13198,11044],{},[458,13200,471],{},[383,13202,561],{},[387,13204,13205],{"encoding":389},"\\tilde QK=Q",[363,13207,13209,13270],{"className":13208,"ariaHidden":394},[393],[363,13210,13212,13216,13258,13261,13264,13267],{"className":13211},[398],[363,13213],{"className":13214,"style":13215},[402],"height:1.1146em;vertical-align:-0.1944em;",[363,13217,13219],{"className":13218},[407,12201],[363,13220,13222,13250],{"className":13221},[633,634],[363,13223,13225,13247],{"className":13224},[638],[363,13226,13228,13236],{"className":13227,"style":12197},[642],[363,13229,13230,13233],{"style":12213},[363,13231],{"className":13232,"style":809},[650],[363,13234,561],{"className":13235},[407,408],[363,13237,13238,13241],{"style":12222},[363,13239],{"className":13240,"style":809},[650],[363,13242,13244],{"className":13243,"style":12465},[12229],[363,13245,12167],{"className":13246},[407],[363,13248,666],{"className":13249},[665],[363,13251,13253],{"className":13252},[638],[363,13254,13256],{"className":13255,"style":12657},[642],[363,13257],{},[363,13259,11044],{"className":13260,"style":11132},[407,408],[363,13262],{"className":13263,"style":506},[505],[363,13265,471],{"className":13266},[510],[363,13268],{"className":13269,"style":506},[505],[363,13271,13273,13276],{"className":13272},[398],[363,13274],{"className":13275,"style":913},[402],[363,13277,561],{"className":13278},[407,408],"，且似然比是分块恒定的：",[363,13281,13283,13361],{"className":13282,"translate":367},[366],[363,13284,13286],{"className":13285},[371],[373,13287,13288],{"xmlns":375},[377,13289,13290,13358],{},[380,13291,13292,13316,13318,13320,13322,13324,13326,13328,13352,13354,13356],{},[1603,13293,13294],{"scriptlevel":474,"displaystyle":394},[586,13295,13296,13306],{},[380,13297,13298,13300],{},[383,13299,592],{},[12161,13301,13302,13304],{"accent":394},[383,13303,554],{},[458,13305,12167],{},[380,13307,13308,13310],{},[383,13309,592],{},[12161,13311,13312,13314],{"accent":394},[383,13313,561],{},[458,13315,12167],{},[458,13317,461],{"stretchy":460},[383,13319,12313],{},[458,13321,898],{"separator":394},[383,13323,1121],{},[458,13325,468],{"stretchy":460},[458,13327,471],{},[1603,13329,13330],{"scriptlevel":474,"displaystyle":394},[586,13331,13332,13342],{},[380,13333,13334,13336],{},[383,13335,592],{},[543,13337,13338,13340],{},[383,13339,554],{},[383,13341,12313],{},[380,13343,13344,13346],{},[383,13345,592],{},[543,13347,13348,13350],{},[383,13349,561],{},[383,13351,12313],{},[458,13353,461],{"stretchy":460},[383,13355,1121],{},[458,13357,468],{"stretchy":460},[387,13359,13360],{"encoding":389},"\\dfrac{d\\tilde P}{d\\tilde Q}(i,x)=\\dfrac{dP_i}{dQ_i}(x)",[363,13362,13364,13536],{"className":13363,"ariaHidden":394},[393],[363,13365,13367,13371,13509,13512,13515,13518,13521,13524,13527,13530,13533],{"className":13366},[398],[363,13368],{"className":13369,"style":13370},[402],"height:2.6018em;vertical-align:-1.0046em;",[363,13372,13374,13377,13506],{"className":13373},[407],[363,13375],{"className":13376},[494,789],[363,13378,13380],{"className":13379},[586],[363,13381,13383,13497],{"className":13382},[633,634],[363,13384,13386,13494],{"className":13385},[638],[363,13387,13390,13444,13452],{"className":13388,"style":13389},[642],"height:1.5972em;",[363,13391,13393,13396],{"style":13392},"top:-2.1898em;",[363,13394],{"className":13395,"style":809},[650],[363,13397,13399,13402],{"className":13398},[407],[363,13400,592],{"className":13401},[407,408],[363,13403,13405],{"className":13404},[407,12201],[363,13406,13408,13436],{"className":13407},[633,634],[363,13409,13411,13433],{"className":13410},[638],[363,13412,13414,13422],{"className":13413,"style":12197},[642],[363,13415,13416,13419],{"style":12213},[363,13417],{"className":13418,"style":809},[650],[363,13420,561],{"className":13421},[407,408],[363,13423,13424,13427],{"style":12222},[363,13425],{"className":13426,"style":809},[650],[363,13428,13430],{"className":13429,"style":12465},[12229],[363,13431,12167],{"className":13432},[407],[363,13434,666],{"className":13435},[665],[363,13437,13439],{"className":13438},[638],[363,13440,13442],{"className":13441,"style":12657},[642],[363,13443],{},[363,13445,13446,13449],{"style":821},[363,13447],{"className":13448,"style":809},[650],[363,13450],{"className":13451,"style":829},[828],[363,13453,13454,13457],{"style":832},[363,13455],{"className":13456,"style":809},[650],[363,13458,13460,13463],{"className":13459},[407],[363,13461,592],{"className":13462},[407,408],[363,13464,13466],{"className":13465},[407,12201],[363,13467,13469],{"className":13468},[633],[363,13470,13472],{"className":13471},[638],[363,13473,13475,13483],{"className":13474,"style":12197},[642],[363,13476,13477,13480],{"style":12213},[363,13478],{"className":13479,"style":809},[650],[363,13481,554],{"className":13482,"style":682},[407,408],[363,13484,13485,13488],{"style":12222},[363,13486],{"className":13487,"style":809},[650],[363,13489,13491],{"className":13490,"style":12465},[12229],[363,13492,12167],{"className":13493},[407],[363,13495,666],{"className":13496},[665],[363,13498,13500],{"className":13499},[638],[363,13501,13504],{"className":13502,"style":13503},[642],"height:1.0046em;",[363,13505],{},[363,13507],{"className":13508},[501,789],[363,13510,461],{"className":13511},[494],[363,13513,12313],{"className":13514},[407,408],[363,13516,898],{"className":13517},[920],[363,13519],{"className":13520,"style":754},[505],[363,13522,1121],{"className":13523},[407,408],[363,13525,468],{"className":13526},[501],[363,13528],{"className":13529,"style":506},[505],[363,13531,471],{"className":13532},[510],[363,13534],{"className":13535,"style":506},[505],[363,13537,13539,13542,13684,13687,13690],{"className":13538},[398],[363,13540],{"className":13541,"style":1185},[402],[363,13543,13545,13548,13681],{"className":13544},[407],[363,13546],{"className":13547},[494,789],[363,13549,13551],{"className":13550},[586],[363,13552,13554,13673],{"className":13553},[633,634],[363,13555,13557,13670],{"className":13556},[638],[363,13558,13560,13611,13619],{"className":13559,"style":802},[642],[363,13561,13562,13565],{"style":805},[363,13563],{"className":13564,"style":809},[650],[363,13566,13568,13571],{"className":13567},[407],[363,13569,592],{"className":13570},[407,408],[363,13572,13574,13577],{"className":13573},[407],[363,13575,561],{"className":13576},[407,408],[363,13578,13580],{"className":13579},[629],[363,13581,13583,13603],{"className":13582},[633,634],[363,13584,13586,13600],{"className":13585},[638],[363,13587,13589],{"className":13588,"style":12532},[642],[363,13590,13591,13594],{"style":730},[363,13592],{"className":13593,"style":651},[650],[363,13595,13597],{"className":13596},[655,656,657,658],[363,13598,12313],{"className":13599},[407,408,658],[363,13601,666],{"className":13602},[665],[363,13604,13606],{"className":13605},[638],[363,13607,13609],{"className":13608,"style":8316},[642],[363,13610],{},[363,13612,13613,13616],{"style":821},[363,13614],{"className":13615,"style":809},[650],[363,13617],{"className":13618,"style":829},[828],[363,13620,13621,13624],{"style":832},[363,13622],{"className":13623,"style":809},[650],[363,13625,13627,13630],{"className":13626},[407],[363,13628,592],{"className":13629},[407,408],[363,13631,13633,13636],{"className":13632},[407],[363,13634,554],{"className":13635,"style":682},[407,408],[363,13637,13639],{"className":13638},[629],[363,13640,13642,13662],{"className":13641},[633,634],[363,13643,13645,13659],{"className":13644},[638],[363,13646,13648],{"className":13647,"style":12532},[642],[363,13649,13650,13653],{"style":11486},[363,13651],{"className":13652,"style":651},[650],[363,13654,13656],{"className":13655},[655,656,657,658],[363,13657,12313],{"className":13658},[407,408,658],[363,13660,666],{"className":13661},[665],[363,13663,13665],{"className":13664},[638],[363,13666,13668],{"className":13667,"style":8316},[642],[363,13669],{},[363,13671,666],{"className":13672},[665],[363,13674,13676],{"className":13675},[638],[363,13677,13679],{"className":13678,"style":854},[642],[363,13680],{},[363,13682],{"className":13683},[501,789],[363,13685,461],{"className":13686},[494],[363,13688,1121],{"className":13689},[407,408],[363,13691,468],{"className":13692},[501],"（权重 ",[363,13695,13697,13715],{"className":13696,"translate":367},[366],[363,13698,13700],{"className":13699},[371],[373,13701,13702],{"xmlns":375},[377,13703,13704,13712],{},[380,13705,13706],{},[543,13707,13708,13710],{},[383,13709,7918],{},[383,13711,12313],{},[387,13713,13714],{"encoding":389},"w_i",[363,13716,13718],{"className":13717,"ariaHidden":394},[393],[363,13719,13721,13725],{"className":13720},[398],[363,13722],{"className":13723,"style":13724},[402],"height:0.5806em;vertical-align:-0.15em;",[363,13726,13728,13731],{"className":13727},[407],[363,13729,7918],{"className":13730,"style":8622},[407,408],[363,13732,13734],{"className":13733},[629],[363,13735,13737,13757],{"className":13736},[633,634],[363,13738,13740,13754],{"className":13739},[638],[363,13741,13743],{"className":13742,"style":12532},[642],[363,13744,13745,13748],{"style":12535},[363,13746],{"className":13747,"style":651},[650],[363,13749,13751],{"className":13750},[655,656,657,658],[363,13752,12313],{"className":13753},[407,408,658],[363,13755,666],{"className":13756},[665],[363,13758,13760],{"className":13759},[638],[363,13761,13763],{"className":13762,"style":8316},[642],[363,13764],{}," 在分子分母中约去）。代入定义：",[363,13767,13769],{"className":13768,"translate":367},[528],[363,13770,13772,13915],{"className":13771,"translate":367},[366],[363,13773,13775],{"className":13774},[371],[373,13776,13777],{"xmlns":375,"display":537},[377,13778,13779,13912],{},[380,13780,13781,13787,13789,13795,13797,13803,13805,13807,13814,13820,13822,13828,13830,13836,13838,13844,13846,13848,13850,13856,13858,13864,13866,13872,13874,13876,13878,13880,13882,13884,13886,13892,13894,13900,13902,13908,13910],{},[543,13782,13783,13785],{},[383,13784,547],{},[383,13786,385],{},[458,13788,461],{"stretchy":460},[12161,13790,13791,13793],{"accent":394},[383,13792,554],{},[458,13794,12167],{},[383,13796,558],{"mathvariant":557},[12161,13798,13799,13801],{"accent":394},[383,13800,561],{},[458,13802,12167],{},[458,13804,468],{"stretchy":460},[458,13806,471],{},[6364,13808,13809,13812],{},[458,13810,13811],{},"∑",[383,13813,12313],{},[543,13815,13816,13818],{},[383,13817,7918],{},[383,13819,12313],{},[1645,13821,1647],{},[543,13823,13824,13826],{},[383,13825,547],{},[383,13827,385],{},[458,13829,461],{"stretchy":460},[543,13831,13832,13834],{},[383,13833,554],{},[383,13835,12313],{},[383,13837,558],{"mathvariant":557},[543,13839,13840,13842],{},[383,13841,561],{},[383,13843,12313],{},[458,13845,468],{"stretchy":460},[458,13847,471],{},[383,13849,11689],{},[543,13851,13852,13854],{},[383,13853,547],{},[383,13855,385],{},[458,13857,461],{"stretchy":460},[543,13859,13860,13862],{},[383,13861,554],{},[463,13863,465],{},[383,13865,558],{"mathvariant":557},[543,13867,13868,13870],{},[383,13869,561],{},[463,13871,465],{},[458,13873,468],{"stretchy":460},[458,13875,2818],{},[458,13877,461],{"stretchy":460},[463,13879,465],{},[458,13881,5161],{},[383,13883,11689],{},[458,13885,468],{"stretchy":460},[543,13887,13888,13890],{},[383,13889,547],{},[383,13891,385],{},[458,13893,461],{"stretchy":460},[543,13895,13896,13898],{},[383,13897,554],{},[463,13899,2811],{},[383,13901,558],{"mathvariant":557},[543,13903,13904,13906],{},[383,13905,561],{},[463,13907,2811],{},[458,13909,468],{"stretchy":460},[383,13911,4723],{"mathvariant":557},[387,13913,13914],{"encoding":389},"D_f(\\tilde P\\|\\tilde Q)\n=\\sum_i w_i\\,D_f(P_i\\|Q_i)\n=\\lambda D_f(P_1\\|Q_1)+(1-\\lambda)D_f(P_2\\|Q_2).",[363,13916,13918,14056,14297,14444,14465],{"className":13917,"ariaHidden":394},[393],[363,13919,13921,13925,13965,13968,13999,14002,14044,14047,14050,14053],{"className":13920},[398],[363,13922],{"className":13923,"style":13924},[402],"height:1.2063em;vertical-align:-0.2861em;",[363,13926,13928,13931],{"className":13927},[407],[363,13929,547],{"className":13930,"style":625},[407,408],[363,13932,13934],{"className":13933},[629],[363,13935,13937,13957],{"className":13936},[633,634],[363,13938,13940,13954],{"className":13939},[638],[363,13941,13943],{"className":13942,"style":643},[642],[363,13944,13945,13948],{"style":646},[363,13946],{"className":13947,"style":651},[650],[363,13949,13951],{"className":13950},[655,656,657,658],[363,13952,385],{"className":13953,"style":409},[407,408,658],[363,13955,666],{"className":13956},[665],[363,13958,13960],{"className":13959},[638],[363,13961,13963],{"className":13962,"style":673},[642],[363,13964],{},[363,13966,461],{"className":13967},[494],[363,13969,13971],{"className":13970},[407,12201],[363,13972,13974],{"className":13973},[633],[363,13975,13977],{"className":13976},[638],[363,13978,13980,13988],{"className":13979,"style":12197},[642],[363,13981,13982,13985],{"style":12213},[363,13983],{"className":13984,"style":809},[650],[363,13986,554],{"className":13987,"style":682},[407,408],[363,13989,13990,13993],{"style":12222},[363,13991],{"className":13992,"style":809},[650],[363,13994,13996],{"className":13995,"style":12465},[12229],[363,13997,12167],{"className":13998},[407],[363,14000,558],{"className":14001},[407],[363,14003,14005],{"className":14004},[407,12201],[363,14006,14008,14036],{"className":14007},[633,634],[363,14009,14011,14033],{"className":14010},[638],[363,14012,14014,14022],{"className":14013,"style":12197},[642],[363,14015,14016,14019],{"style":12213},[363,14017],{"className":14018,"style":809},[650],[363,14020,561],{"className":14021},[407,408],[363,14023,14024,14027],{"style":12222},[363,14025],{"className":14026,"style":809},[650],[363,14028,14030],{"className":14029,"style":12465},[12229],[363,14031,12167],{"className":14032},[407],[363,14034,666],{"className":14035},[665],[363,14037,14039],{"className":14038},[638],[363,14040,14042],{"className":14041,"style":12657},[642],[363,14043],{},[363,14045,468],{"className":14046},[501],[363,14048],{"className":14049,"style":506},[505],[363,14051,471],{"className":14052},[510],[363,14054],{"className":14055,"style":506},[505],[363,14057,14059,14063,14113,14116,14156,14159,14199,14202,14242,14245,14285,14288,14291,14294],{"className":14058},[398],[363,14060],{"className":14061,"style":14062},[402],"height:2.3277em;vertical-align:-1.2777em;",[363,14064,14067],{"className":14065},[1832,14066],"op-limits",[363,14068,14070,14104],{"className":14069},[633,634],[363,14071,14073,14101],{"className":14072},[638],[363,14074,14077,14090],{"className":14075,"style":14076},[642],"height:1.05em;",[363,14078,14080,14084],{"style":14079},"top:-1.8723em;margin-left:0em;",[363,14081],{"className":14082,"style":14083},[650],"height:3.05em;",[363,14085,14087],{"className":14086},[655,656,657,658],[363,14088,12313],{"className":14089},[407,408,658],[363,14091,14093,14096],{"style":14092},"top:-3.05em;",[363,14094],{"className":14095,"style":14083},[650],[363,14097,14098],{},[363,14099,13811],{"className":14100},[1832,1833,1834],[363,14102,666],{"className":14103},[665],[363,14105,14107],{"className":14106},[638],[363,14108,14111],{"className":14109,"style":14110},[642],"height:1.2777em;",[363,14112],{},[363,14114],{"className":14115,"style":754},[505],[363,14117,14119,14122],{"className":14118},[407],[363,14120,7918],{"className":14121,"style":8622},[407,408],[363,14123,14125],{"className":14124},[629],[363,14126,14128,14148],{"className":14127},[633,634],[363,14129,14131,14145],{"className":14130},[638],[363,14132,14134],{"className":14133,"style":12532},[642],[363,14135,14136,14139],{"style":12535},[363,14137],{"className":14138,"style":651},[650],[363,14140,14142],{"className":14141},[655,656,657,658],[363,14143,12313],{"className":14144},[407,408,658],[363,14146,666],{"className":14147},[665],[363,14149,14151],{"className":14150},[638],[363,14152,14154],{"className":14153,"style":8316},[642],[363,14155],{},[363,14157],{"className":14158,"style":754},[505],[363,14160,14162,14165],{"className":14161},[407],[363,14163,547],{"className":14164,"style":625},[407,408],[363,14166,14168],{"className":14167},[629],[363,14169,14171,14191],{"className":14170},[633,634],[363,14172,14174,14188],{"className":14173},[638],[363,14175,14177],{"className":14176,"style":643},[642],[363,14178,14179,14182],{"style":646},[363,14180],{"className":14181,"style":651},[650],[363,14183,14185],{"className":14184},[655,656,657,658],[363,14186,385],{"className":14187,"style":409},[407,408,658],[363,14189,666],{"className":14190},[665],[363,14192,14194],{"className":14193},[638],[363,14195,14197],{"className":14196,"style":673},[642],[363,14198],{},[363,14200,461],{"className":14201},[494],[363,14203,14205,14208],{"className":14204},[407],[363,14206,554],{"className":14207,"style":682},[407,408],[363,14209,14211],{"className":14210},[629],[363,14212,14214,14234],{"className":14213},[633,634],[363,14215,14217,14231],{"className":14216},[638],[363,14218,14220],{"className":14219,"style":12532},[642],[363,14221,14222,14225],{"style":11486},[363,14223],{"className":14224,"style":651},[650],[363,14226,14228],{"className":14227},[655,656,657,658],[363,14229,12313],{"className":14230},[407,408,658],[363,14232,666],{"className":14233},[665],[363,14235,14237],{"className":14236},[638],[363,14238,14240],{"className":14239,"style":8316},[642],[363,14241],{},[363,14243,558],{"className":14244},[407],[363,14246,14248,14251],{"className":14247},[407],[363,14249,561],{"className":14250},[407,408],[363,14252,14254],{"className":14253},[629],[363,14255,14257,14277],{"className":14256},[633,634],[363,14258,14260,14274],{"className":14259},[638],[363,14261,14263],{"className":14262,"style":12532},[642],[363,14264,14265,14268],{"style":730},[363,14266],{"className":14267,"style":651},[650],[363,14269,14271],{"className":14270},[655,656,657,658],[363,14272,12313],{"className":14273},[407,408,658],[363,14275,666],{"className":14276},[665],[363,14278,14280],{"className":14279},[638],[363,14281,14283],{"className":14282,"style":8316},[642],[363,14284],{},[363,14286,468],{"className":14287},[501],[363,14289],{"className":14290,"style":506},[505],[363,14292,471],{"className":14293},[510],[363,14295],{"className":14296,"style":506},[505],[363,14298,14300,14303,14306,14346,14349,14389,14392,14432,14435,14438,14441],{"className":14299},[398],[363,14301],{"className":14302,"style":618},[402],[363,14304,11689],{"className":14305},[407,408],[363,14307,14309,14312],{"className":14308},[407],[363,14310,547],{"className":14311,"style":625},[407,408],[363,14313,14315],{"className":14314},[629],[363,14316,14318,14338],{"className":14317},[633,634],[363,14319,14321,14335],{"className":14320},[638],[363,14322,14324],{"className":14323,"style":643},[642],[363,14325,14326,14329],{"style":646},[363,14327],{"className":14328,"style":651},[650],[363,14330,14332],{"className":14331},[655,656,657,658],[363,14333,385],{"className":14334,"style":409},[407,408,658],[363,14336,666],{"className":14337},[665],[363,14339,14341],{"className":14340},[638],[363,14342,14344],{"className":14343,"style":673},[642],[363,14345],{},[363,14347,461],{"className":14348},[494],[363,14350,14352,14355],{"className":14351},[407],[363,14353,554],{"className":14354,"style":682},[407,408],[363,14356,14358],{"className":14357},[629],[363,14359,14361,14381],{"className":14360},[633,634],[363,14362,14364,14378],{"className":14363},[638],[363,14365,14367],{"className":14366,"style":11483},[642],[363,14368,14369,14372],{"style":11486},[363,14370],{"className":14371,"style":651},[650],[363,14373,14375],{"className":14374},[655,656,657,658],[363,14376,465],{"className":14377},[407,658],[363,14379,666],{"className":14380},[665],[363,14382,14384],{"className":14383},[638],[363,14385,14387],{"className":14386,"style":8316},[642],[363,14388],{},[363,14390,558],{"className":14391},[407],[363,14393,14395,14398],{"className":14394},[407],[363,14396,561],{"className":14397},[407,408],[363,14399,14401],{"className":14400},[629],[363,14402,14404,14424],{"className":14403},[633,634],[363,14405,14407,14421],{"className":14406},[638],[363,14408,14410],{"className":14409,"style":11483},[642],[363,14411,14412,14415],{"style":730},[363,14413],{"className":14414,"style":651},[650],[363,14416,14418],{"className":14417},[655,656,657,658],[363,14419,465],{"className":14420},[407,658],[363,14422,666],{"className":14423},[665],[363,14425,14427],{"className":14426},[638],[363,14428,14430],{"className":14429,"style":8316},[642],[363,14431],{},[363,14433,468],{"className":14434},[501],[363,14436],{"className":14437,"style":2935},[505],[363,14439,2818],{"className":14440},[2939],[363,14442],{"className":14443,"style":2935},[505],[363,14445,14447,14450,14453,14456,14459,14462],{"className":14446},[398],[363,14448],{"className":14449,"style":487},[402],[363,14451,461],{"className":14452},[494],[363,14454,465],{"className":14455},[407],[363,14457],{"className":14458,"style":2935},[505],[363,14460,5161],{"className":14461},[2939],[363,14463],{"className":14464,"style":2935},[505],[363,14466,14468,14471,14474,14477,14517,14520,14560,14563,14603,14606],{"className":14467},[398],[363,14469],{"className":14470,"style":618},[402],[363,14472,11689],{"className":14473},[407,408],[363,14475,468],{"className":14476},[501],[363,14478,14480,14483],{"className":14479},[407],[363,14481,547],{"className":14482,"style":625},[407,408],[363,14484,14486],{"className":14485},[629],[363,14487,14489,14509],{"className":14488},[633,634],[363,14490,14492,14506],{"className":14491},[638],[363,14493,14495],{"className":14494,"style":643},[642],[363,14496,14497,14500],{"style":646},[363,14498],{"className":14499,"style":651},[650],[363,14501,14503],{"className":14502},[655,656,657,658],[363,14504,385],{"className":14505,"style":409},[407,408,658],[363,14507,666],{"className":14508},[665],[363,14510,14512],{"className":14511},[638],[363,14513,14515],{"className":14514,"style":673},[642],[363,14516],{},[363,14518,461],{"className":14519},[494],[363,14521,14523,14526],{"className":14522},[407],[363,14524,554],{"className":14525,"style":682},[407,408],[363,14527,14529],{"className":14528},[629],[363,14530,14532,14552],{"className":14531},[633,634],[363,14533,14535,14549],{"className":14534},[638],[363,14536,14538],{"className":14537,"style":11483},[642],[363,14539,14540,14543],{"style":11486},[363,14541],{"className":14542,"style":651},[650],[363,14544,14546],{"className":14545},[655,656,657,658],[363,14547,2811],{"className":14548},[407,658],[363,14550,666],{"className":14551},[665],[363,14553,14555],{"className":14554},[638],[363,14556,14558],{"className":14557,"style":8316},[642],[363,14559],{},[363,14561,558],{"className":14562},[407],[363,14564,14566,14569],{"className":14565},[407],[363,14567,561],{"className":14568},[407,408],[363,14570,14572],{"className":14571},[629],[363,14573,14575,14595],{"className":14574},[633,634],[363,14576,14578,14592],{"className":14577},[638],[363,14579,14581],{"className":14580,"style":11483},[642],[363,14582,14583,14586],{"style":730},[363,14584],{"className":14585,"style":651},[650],[363,14587,14589],{"className":14588},[655,656,657,658],[363,14590,2811],{"className":14591},[407,658],[363,14593,666],{"className":14594},[665],[363,14596,14598],{"className":14597},[638],[363,14599,14601],{"className":14600,"style":8316},[642],[363,14602],{},[363,14604,468],{"className":14605},[501],[363,14607,4723],{"className":14608},[407],[11,14610,14611,14612,14740],{},"对 ",[363,14613,14615,14645],{"className":14614,"translate":367},[366],[363,14616,14618],{"className":14617},[371],[373,14619,14620],{"xmlns":375},[377,14621,14622,14642],{},[380,14623,14624,14626,14632,14634,14640],{},[458,14625,461],{"stretchy":460},[12161,14627,14628,14630],{"accent":394},[383,14629,554],{},[458,14631,12167],{},[458,14633,898],{"separator":394},[12161,14635,14636,14638],{"accent":394},[383,14637,561],{},[458,14639,12167],{},[458,14641,468],{"stretchy":460},[387,14643,14644],{"encoding":389},"(\\tilde P,\\tilde Q)",[363,14646,14648],{"className":14647,"ariaHidden":394},[393],[363,14649,14651,14655,14658,14689,14692,14695,14737],{"className":14650},[398],[363,14652],{"className":14653,"style":14654},[402],"height:1.1702em;vertical-align:-0.25em;",[363,14656,461],{"className":14657},[494],[363,14659,14661],{"className":14660},[407,12201],[363,14662,14664],{"className":14663},[633],[363,14665,14667],{"className":14666},[638],[363,14668,14670,14678],{"className":14669,"style":12197},[642],[363,14671,14672,14675],{"style":12213},[363,14673],{"className":14674,"style":809},[650],[363,14676,554],{"className":14677,"style":682},[407,408],[363,14679,14680,14683],{"style":12222},[363,14681],{"className":14682,"style":809},[650],[363,14684,14686],{"className":14685,"style":12465},[12229],[363,14687,12167],{"className":14688},[407],[363,14690,898],{"className":14691},[920],[363,14693],{"className":14694,"style":754},[505],[363,14696,14698],{"className":14697},[407,12201],[363,14699,14701,14729],{"className":14700},[633,634],[363,14702,14704,14726],{"className":14703},[638],[363,14705,14707,14715],{"className":14706,"style":12197},[642],[363,14708,14709,14712],{"style":12213},[363,14710],{"className":14711,"style":809},[650],[363,14713,561],{"className":14714},[407,408],[363,14716,14717,14720],{"style":12222},[363,14718],{"className":14719,"style":809},[650],[363,14721,14723],{"className":14722,"style":12465},[12229],[363,14724,12167],{"className":14725},[407],[363,14727,666],{"className":14728},[665],[363,14730,14732],{"className":14731},[638],[363,14733,14735],{"className":14734,"style":12657},[642],[363,14736],{},[363,14738,468],{"className":14739},[501]," 使用 DPI，即得到联合凸性：",[363,14742,14744],{"className":14743,"translate":367},[528],[363,14745,14747,14927],{"className":14746,"translate":367},[366],[363,14748,14750],{"className":14749},[371],[373,14751,14752],{"xmlns":375,"display":537},[377,14753,14754,14924],{},[1591,14755,14756,14840],{"rowspacing":1593,"columnalign":1594,"columnspacing":1595},[1597,14757,14758,14764],{},[1600,14759,14760],{},[1603,14761,14762],{"scriptlevel":474,"displaystyle":394},[380,14763],{},[1600,14765,14766],{},[1603,14767,14768],{"scriptlevel":474,"displaystyle":394},[380,14769,14770,14772,14778,14780,14782,14788,14790,14792,14794,14796,14798,14800,14806,14808,14810,14812,14814,14820,14822,14824,14826,14828,14830,14832,14838],{},[380,14771],{},[543,14773,14774,14776],{},[383,14775,547],{},[383,14777,385],{},[458,14779,461],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[383,14781,11689],{},[543,14783,14784,14786],{},[383,14785,554],{},[463,14787,465],{},[458,14789,2818],{},[458,14791,461],{"stretchy":460},[463,14793,465],{},[458,14795,5161],{},[383,14797,11689],{},[458,14799,468],{"stretchy":460},[543,14801,14802,14804],{},[383,14803,554],{},[463,14805,2811],{},[1645,14807,1647],{},[458,14809,558],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[1645,14811,1647],{},[383,14813,11689],{},[543,14815,14816,14818],{},[383,14817,561],{},[463,14819,465],{},[458,14821,2818],{},[458,14823,461],{"stretchy":460},[463,14825,465],{},[458,14827,5161],{},[383,14829,11689],{},[458,14831,468],{"stretchy":460},[543,14833,14834,14836],{},[383,14835,561],{},[463,14837,2811],{},[458,14839,468],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[1597,14841,14842,14848],{},[1600,14843,14844],{},[1603,14845,14846],{"scriptlevel":474,"displaystyle":394},[380,14847],{},[1600,14849,14850],{},[1603,14851,14852],{"scriptlevel":474,"displaystyle":394},[380,14853,14854,14856,14858,14860,14862,14868,14870,14876,14878,14884,14886,14888,14890,14892,14894,14896,14898,14904,14906,14912,14914,14920,14922],{},[380,14855],{},[505,14857],{"width":3610},[458,14859,10300],{},[383,14861,11689],{},[543,14863,14864,14866],{},[383,14865,547],{},[383,14867,385],{},[458,14869,461],{"stretchy":460},[543,14871,14872,14874],{},[383,14873,554],{},[463,14875,465],{},[383,14877,558],{"mathvariant":557},[543,14879,14880,14882],{},[383,14881,561],{},[463,14883,465],{},[458,14885,468],{"stretchy":460},[458,14887,2818],{},[458,14889,461],{"stretchy":460},[463,14891,465],{},[458,14893,5161],{},[383,14895,11689],{},[458,14897,468],{"stretchy":460},[543,14899,14900,14902],{},[383,14901,547],{},[383,14903,385],{},[458,14905,461],{"stretchy":460},[543,14907,14908,14910],{},[383,14909,554],{},[463,14911,2811],{},[383,14913,558],{"mathvariant":557},[543,14915,14916,14918],{},[383,14917,561],{},[463,14919,2811],{},[458,14921,468],{"stretchy":460},[383,14923,4723],{"mathvariant":557},[387,14925,14926],{"encoding":389},"\\begin{aligned}\n&D_f\\big(\\lambda P_1+(1-\\lambda)P_2\n\\,\\big\\|\\,\\lambda Q_1+(1-\\lambda)Q_2\\big)\\\\\n&\\qquad\\le \\lambda D_f(P_1\\|Q_1)\n+(1-\\lambda)D_f(P_2\\|Q_2).\n\\end{aligned}",[363,14928,14930],{"className":14929,"ariaHidden":394},[393],[363,14931,14933,14937],{"className":14932},[398],[363,14934],{"className":14935,"style":14936},[402],"height:2.71em;vertical-align:-1.105em;",[363,14938,14940],{"className":14939},[407],[363,14941,14943,14987],{"className":14942},[1591],[363,14944,14946],{"className":14945},[1708],[363,14947,14949,14978],{"className":14948},[633,634],[363,14950,14952,14975],{"className":14951},[638],[363,14953,14956,14966],{"className":14954,"style":14955},[642],"height:1.605em;",[363,14957,14959,14963],{"style":14958},"top:-3.605em;",[363,14960],{"className":14961,"style":14962},[650],"height:2.85em;",[363,14964],{"className":14965},[407],[363,14967,14969,14972],{"style":14968},"top:-2.105em;",[363,14970],{"className":14971,"style":14962},[650],[363,14973],{"className":14974},[407],[363,14976,666],{"className":14977},[665],[363,14979,14981],{"className":14980},[638],[363,14982,14985],{"className":14983,"style":14984},[642],"height:1.105em;",[363,14986],{},[363,14988,14990],{"className":14989},[1799],[363,14991,14993,15662],{"className":14992},[633,634],[363,14994,14996,15659],{"className":14995},[638],[363,14997,14999,15341],{"className":14998,"style":14955},[642],[363,15000,15002,15005],{"style":15001},"top:-3.755em;",[363,15003],{"className":15004,"style":809},[650],[363,15006,15008,15011,15051,15057,15060,15100,15103,15106,15109,15112,15115,15118,15121,15124,15127,15130,15170,15173,15219,15222,15225,15265,15268,15271,15274,15277,15280,15283,15286,15289,15292,15295,15335],{"className":15007},[407],[363,15009],{"className":15010},[407],[363,15012,15014,15017],{"className":15013},[407],[363,15015,547],{"className":15016,"style":625},[407,408],[363,15018,15020],{"className":15019},[629],[363,15021,15023,15043],{"className":15022},[633,634],[363,15024,15026,15040],{"className":15025},[638],[363,15027,15029],{"className":15028,"style":643},[642],[363,15030,15031,15034],{"style":646},[363,15032],{"className":15033,"style":651},[650],[363,15035,15037],{"className":15036},[655,656,657,658],[363,15038,385],{"className":15039,"style":409},[407,408,658],[363,15041,666],{"className":15042},[665],[363,15044,15046],{"className":15045},[638],[363,15047,15049],{"className":15048,"style":673},[642],[363,15050],{},[363,15052,15054],{"className":15053},[407],[363,15055,461],{"className":15056},[767,10022],[363,15058,11689],{"className":15059},[407,408],[363,15061,15063,15066],{"className":15062},[407],[363,15064,554],{"className":15065,"style":682},[407,408],[363,15067,15069],{"className":15068},[629],[363,15070,15072,15092],{"className":15071},[633,634],[363,15073,15075,15089],{"className":15074},[638],[363,15076,15078],{"className":15077,"style":11483},[642],[363,15079,15080,15083],{"style":11486},[363,15081],{"className":15082,"style":651},[650],[363,15084,15086],{"className":15085},[655,656,657,658],[363,15087,465],{"className":15088},[407,658],[363,15090,666],{"className":15091},[665],[363,15093,15095],{"className":15094},[638],[363,15096,15098],{"className":15097,"style":8316},[642],[363,15099],{},[363,15101],{"className":15102,"style":2935},[505],[363,15104,2818],{"className":15105},[2939],[363,15107],{"className":15108,"style":2935},[505],[363,15110,461],{"className":15111},[494],[363,15113,465],{"className":15114},[407],[363,15116],{"className":15117,"style":2935},[505],[363,15119,5161],{"className":15120},[2939],[363,15122],{"className":15123,"style":2935},[505],[363,15125,11689],{"className":15126},[407,408],[363,15128,468],{"className":15129},[501],[363,15131,15133,15136],{"className":15132},[407],[363,15134,554],{"className":15135,"style":682},[407,408],[363,15137,15139],{"className":15138},[629],[363,15140,15142,15162],{"className":15141},[633,634],[363,15143,15145,15159],{"className":15144},[638],[363,15146,15148],{"className":15147,"style":11483},[642],[363,15149,15150,15153],{"style":11486},[363,15151],{"className":15152,"style":651},[650],[363,15154,15156],{"className":15155},[655,656,657,658],[363,15157,2811],{"className":15158},[407,658],[363,15160,666],{"className":15161},[665],[363,15163,15165],{"className":15164},[638],[363,15166,15168],{"className":15167,"style":8316},[642],[363,15169],{},[363,15171],{"className":15172,"style":754},[505],[363,15174,15176],{"className":15175},[407],[363,15177,15180],{"className":15178},[767,15179],"mult",[363,15181,15183,15210],{"className":15182},[633,634],[363,15184,15186,15207],{"className":15185},[638],[363,15187,15190],{"className":15188,"style":15189},[642],"height:0.85em;",[363,15191,15193,15197],{"style":15192},"top:-2.85em;",[363,15194],{"className":15195,"style":15196},[650],"height:3.2em;",[363,15198,15200],{"style":15199},"width:0.556em;height:1.2em;",[6671,15201,15204],{"xmlns":6673,"width":15202,"height":9798,"viewBox":15203},"0.556em","0 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h43z",[363,15208,666],{"className":15209},[665],[363,15211,15213],{"className":15212},[638],[363,15214,15217],{"className":15215,"style":15216},[642],"height:0.35em;",[363,15218],{},[363,15220],{"className":15221,"style":754},[505],[363,15223,11689],{"className":15224},[407,408],[363,15226,15228,15231],{"className":15227},[407],[363,15229,561],{"className":15230},[407,408],[363,15232,15234],{"className":15233},[629],[363,15235,15237,15257],{"className":15236},[633,634],[363,15238,15240,15254],{"className":15239},[638],[363,15241,15243],{"className":15242,"style":11483},[642],[363,15244,15245,15248],{"style":730},[363,15246],{"className":15247,"style":651},[650],[363,15249,15251],{"className":15250},[655,656,657,658],[363,15252,465],{"className":15253},[407,658],[363,15255,666],{"className":15256},[665],[363,15258,15260],{"className":15259},[638],[363,15261,15263],{"className":15262,"style":8316},[642],[363,15264],{},[363,15266],{"className":15267,"style":2935},[505],[363,15269,2818],{"className":15270},[2939],[363,15272],{"className":15273,"style":2935},[505],[363,15275,461],{"className":15276},[494],[363,15278,465],{"className":15279},[407],[363,15281],{"className":15282,"style":2935},[505],[363,15284,5161],{"className":15285},[2939],[363,15287],{"className":15288,"style":2935},[505],[363,15290,11689],{"className":15291},[407,408],[363,15293,468],{"className":15294},[501],[363,15296,15298,15301],{"className":15297},[407],[363,15299,561],{"className":15300},[407,408],[363,15302,15304],{"className":15303},[629],[363,15305,15307,15327],{"className":15306},[633,634],[363,15308,15310,15324],{"className":15309},[638],[363,15311,15313],{"className":15312,"style":11483},[642],[363,15314,15315,15318],{"style":730},[363,15316],{"className":15317,"style":651},[650],[363,15319,15321],{"className":15320},[655,656,657,658],[363,15322,2811],{"className":15323},[407,658],[363,15325,666],{"className":15326},[665],[363,15328,15330],{"className":15329},[638],[363,15331,15333],{"className":15332,"style":8316},[642],[363,15334],{},[363,15336,15338],{"className":15337},[407],[363,15339,468],{"className":15340},[767,10022],[363,15342,15344,15347],{"style":15343},"top:-2.255em;",[363,15345],{"className":15346,"style":809},[650],[363,15348,15350,15353,15356,15359,15362,15365,15368,15408,15411,15451,15454,15494,15497,15500,15503,15506,15509,15512,15515,15518,15521,15524,15527,15567,15570,15610,15613,15653,15656],{"className":15349},[407],[363,15351],{"className":15352},[407],[363,15354],{"className":15355,"style":3982},[505],[363,15357],{"className":15358,"style":506},[505],[363,15360,10300],{"className":15361},[510],[363,15363],{"className":15364,"style":506},[505],[363,15366,11689],{"className":15367},[407,408],[363,15369,15371,15374],{"className":15370},[407],[363,15372,547],{"className":15373,"style":625},[407,408],[363,15375,15377],{"className":15376},[629],[363,15378,15380,15400],{"className":15379},[633,634],[363,15381,15383,15397],{"className":15382},[638],[363,15384,15386],{"className":15385,"style":643},[642],[363,15387,15388,15391],{"style":646},[363,15389],{"className":15390,"style":651},[650],[363,15392,15394],{"className":15393},[655,656,657,658],[363,15395,385],{"className":15396,"style":409},[407,408,658],[363,15398,666],{"className":15399},[665],[363,15401,15403],{"className":15402},[638],[363,15404,15406],{"className":15405,"style":673},[642],[363,15407],{},[363,15409,461],{"className":15410},[494],[363,15412,15414,15417],{"className":15413},[407],[363,15415,554],{"className":15416,"style":682},[407,408],[363,15418,15420],{"className":15419},[629],[363,15421,15423,15443],{"className":15422},[633,634],[363,15424,15426,15440],{"className":15425},[638],[363,15427,15429],{"className":15428,"style":11483},[642],[363,15430,15431,15434],{"style":11486},[363,15432],{"className":15433,"style":651},[650],[363,15435,15437],{"className":15436},[655,656,657,658],[363,15438,465],{"className":15439},[407,658],[363,15441,666],{"className":15442},[665],[363,15444,15446],{"className":15445},[638],[363,15447,15449],{"className":15448,"style":8316},[642],[363,15450],{},[363,15452,558],{"className":15453},[407],[363,15455,15457,15460],{"className":15456},[407],[363,15458,561],{"className":15459},[407,408],[363,15461,15463],{"className":15462},[629],[363,15464,15466,15486],{"className":15465},[633,634],[363,15467,15469,15483],{"className":15468},[638],[363,15470,15472],{"className":15471,"style":11483},[642],[363,15473,15474,15477],{"style":730},[363,15475],{"className":15476,"style":651},[650],[363,15478,15480],{"className":15479},[655,656,657,658],[363,15481,465],{"className":15482},[407,658],[363,15484,666],{"className":15485},[665],[363,15487,15489],{"className":15488},[638],[363,15490,15492],{"className":15491,"style":8316},[642],[363,15493],{},[363,15495,468],{"className":15496},[501],[363,15498],{"className":15499,"style":2935},[505],[363,15501,2818],{"className":15502},[2939],[363,15504],{"className":15505,"style":2935},[505],[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",[363,15676,15678,15691],{"className":15677,"translate":367},[366],[363,15679,15681],{"className":15680},[371],[373,15682,15683],{"xmlns":375},[377,15684,15685,15689],{},[380,15686,15687],{},[458,15688,11340],{},[387,15690,11343],{"encoding":389},[363,15692,15694],{"className":15693,"ariaHidden":394},[393],[363,15695,15697,15700],{"className":15696},[398],[363,15698],{"className":15699,"style":11353},[402],[363,15701,11340],{"className":15702},[510],[363,15704,15706,15719],{"className":15705,"translate":367},[366],[363,15707,15709],{"className":15708},[371],[373,15710,15711],{"xmlns":375},[377,15712,15713,15717],{},[380,15714,15715],{},[383,15716,385],{},[387,15718,385],{"encoding":389},[363,15720,15722],{"className":15721,"ariaHidden":394},[393],[363,15723,15725,15728],{"className":15724},[398],[363,15726],{"className":15727,"style":403},[402],[363,15729,385],{"className":15730,"style":409},[407,408]," 固定 ",[363,15733,15735,15758],{"className":15734,"translate":367},[366],[363,15736,15738],{"className":15737},[371],[373,15739,15740],{"xmlns":375},[377,15741,15742,15755],{},[380,15743,15744,15746,15748,15751,15753],{},[383,15745,7909],{},[458,15747,898],{"separator":394},[383,15749,15750],{},"u",[458,15752,7276],{},[463,15754,474],{},[387,15756,15757],{"encoding":389},"s,u\\ge0",[363,15759,15761,15789],{"className":15760,"ariaHidden":394},[393],[363,15762,15764,15768,15771,15774,15777,15780,15783,15786],{"className":15763},[398],[363,15765],{"className":15766,"style":15767},[402],"height:0.8304em;vertical-align:-0.1944em;",[363,15769,7909],{"className":15770},[407,408],[363,15772,898],{"className":15773},[920],[363,15775],{"className":15776,"style":754},[505],[363,15778,15750],{"className":15779},[407,408],[363,15781],{"className":15782,"style":506},[505],[363,15784,7276],{"className":15785},[510],[363,15787],{"className":15788,"style":506},[505],[363,15790,15792,15795],{"className":15791},[398],[363,15793],{"className":15794,"style":520},[402],[363,15796,474],{"className":15797},[407],[363,15799,15801,15826],{"className":15800,"translate":367},[366],[363,15802,15804],{"className":15803},[371],[373,15805,15806],{"xmlns":375},[377,15807,15808,15824],{},[380,15809,15810,15812,15814,15816,15818,15820,15822],{},[383,15811,11689],{},[458,15813,6257],{},[458,15815,461],{"stretchy":460},[463,15817,474],{},[458,15819,898],{"separator":394},[463,15821,465],{},[458,15823,468],{"stretchy":460},[387,15825,11704],{"encoding":389},[363,15827,15829,15847],{"className":15828,"ariaHidden":394},[393],[363,15830,15832,15835,15838,15841,15844],{"className":15831},[398],[363,15833],{"className":15834,"style":11714},[402],[363,15836,11689],{"className":15837},[407,408],[363,15839],{"className":15840,"style":506},[505],[363,15842,6257],{"className":15843},[510],[363,15845],{"className":15846,"style":506},[505],[363,15848,15850,15853,15856,15859,15862,15865,15868],{"className":15849},[398],[363,15851],{"className":15852,"style":487},[402],[363,15854,461],{"className":15855},[494],[363,15857,474],{"className":15858},[407],[363,15860,898],{"className":15861},[920],[363,15863],{"className":15864,"style":754},[505],[363,15866,465],{"className":15867},[407],[363,15869,468],{"className":15870},[501],"，记目标值 ",[363,15873,15875,15909],{"className":15874,"translate":367},[366],[363,15876,15878],{"className":15877},[371],[373,15879,15880],{"xmlns":375},[377,15881,15882,15906],{},[380,15883,15884,15886,15888,15890,15892,15894,15896,15898,15900,15902,15904],{},[383,15885,7918],{},[458,15887,471],{},[383,15889,11689],{},[383,15891,7909],{},[458,15893,2818],{},[458,15895,461],{"stretchy":460},[463,15897,465],{},[458,15899,5161],{},[383,15901,11689],{},[458,15903,468],{"stretchy":460},[383,15905,15750],{},[387,15907,15908],{"encoding":389},"w=\\lambda s+(1-\\lambda)u",[363,15910,15912,15930,15952,15973],{"className":15911,"ariaHidden":394},[393],[363,15913,15915,15918,15921,15924,15927],{"className":15914},[398],[363,15916],{"className":15917,"style":2693},[402],[363,15919,7918],{"className":15920,"style":8622},[407,408],[363,15922],{"className":15923,"style":506},[505],[363,15925,471],{"className":15926},[510],[363,15928],{"className":15929,"style":506},[505],[363,15931,15933,15937,15940,15943,15946,15949],{"className":15932},[398],[363,15934],{"className":15935,"style":15936},[402],"height:0.7778em;vertical-align:-0.0833em;",[363,15938,11689],{"className":15939},[407,408],[363,15941,7909],{"className":15942},[407,408],[363,15944],{"className":15945,"style":2935},[505],[363,15947,2818],{"className":15948},[2939],[363,15950],{"className":15951,"style":2935},[505],[363,15953,15955,15958,15961,15964,15967,15970],{"className":15954},[398],[363,15956],{"className":15957,"style":487},[402],[363,15959,461],{"className":15960},[494],[363,15962,465],{"className":15963},[407],[363,15965],{"className":15966,"style":2935},[505],[363,15968,5161],{"className":15969},[2939],[363,15971],{"className":15972,"style":2935},[505],[363,15974,15976,15979,15982,15985],{"className":15975},[398],[363,15977],{"className":15978,"style":487},[402],[363,15980,11689],{"className":15981},[407,408],[363,15983,468],{"className":15984},[501],[363,15986,15750],{"className":15987},[407,408],"。在两点空间 ",[363,15990,15992,16014],{"className":15991,"translate":367},[366],[363,15993,15995],{"className":15994},[371],[373,15996,15997],{"xmlns":375},[377,15998,15999,16011],{},[380,16000,16001,16003,16005,16007,16009],{},[458,16002,3246],{"stretchy":460},[463,16004,465],{},[458,16006,898],{"separator":394},[463,16008,2811],{},[458,16010,3256],{"stretchy":460},[387,16012,16013],{"encoding":389},"\\{1,2\\}",[363,16015,16017],{"className":16016,"ariaHidden":394},[393],[363,16018,16020,16023,16026,16029,16032,16035,16038],{"className":16019},[398],[363,16021],{"className":16022,"style":487},[402],[363,16024,3246],{"className":16025},[494],[363,16027,465],{"className":16028},[407],[363,16030,898],{"className":16031},[920],[363,16033],{"className":16034,"style":754},[505],[363,16036,2811],{"className":16037},[407],[363,16039,3256],{"className":16040},[501]," 上取同一个底分布 ",[363,16043,16045,16078],{"className":16044,"translate":367},[366],[363,16046,16048],{"className":16047},[371],[373,16049,16050],{"xmlns":375},[377,16051,16052,16075],{},[380,16053,16054,16056,16058,16060,16063,16065,16067,16069,16071,16073],{},[383,16055,561],{},[458,16057,471],{},[458,16059,461],{"stretchy":460},[383,16061,16062],{},"δ",[458,16064,898],{"separator":394},[1645,16066,1647],{},[463,16068,465],{},[458,16070,5161],{},[383,16072,16062],{},[458,16074,468],{"stretchy":460},[387,16076,16077],{"encoding":389},"Q=(\\delta,\\,1-\\delta)",[363,16079,16081,16099,16133],{"className":16080,"ariaHidden":394},[393],[363,16082,16084,16087,16090,16093,16096],{"className":16083},[398],[363,16085],{"className":16086,"style":913},[402],[363,16088,561],{"className":16089},[407,408],[363,16091],{"className":16092,"style":506},[505],[363,16094,471],{"className":16095},[510],[363,16097],{"className":16098,"style":506},[505],[363,16100,16102,16105,16108,16112,16115,16118,16121,16124,16127,16130],{"className":16101},[398],[363,16103],{"className":16104,"style":487},[402],[363,16106,461],{"className":16107},[494],[363,16109,16062],{"className":16110,"style":16111},[407,408],"margin-right:0.0379em;",[363,16113,898],{"className":16114},[920],[363,16116],{"className":16117,"style":754},[505],[363,16119],{"className":16120,"style":754},[505],[363,16122,465],{"className":16123},[407],[363,16125],{"className":16126,"style":2935},[505],[363,16128,5161],{"className":16129},[2939],[363,16131],{"className":16132,"style":2935},[505],[363,16134,16136,16139,16142],{"className":16135},[398],[363,16137],{"className":16138,"style":487},[402],[363,16140,16062],{"className":16141,"style":16111},[407,408],[363,16143,468],{"className":16144},[501],"（",[363,16147,16149,16175],{"className":16148,"translate":367},[366],[363,16150,16152],{"className":16151},[371],[373,16153,16154],{"xmlns":375},[377,16155,16156,16172],{},[380,16157,16158,16160,16162,16164,16166,16168,16170],{},[383,16159,16062],{},[458,16161,6257],{},[458,16163,461],{"stretchy":460},[463,16165,474],{},[458,16167,898],{"separator":394},[463,16169,465],{},[458,16171,468],{"stretchy":460},[387,16173,16174],{"encoding":389},"\\delta\\in(0,1)",[363,16176,16178,16196],{"className":16177,"ariaHidden":394},[393],[363,16179,16181,16184,16187,16190,16193],{"className":16180},[398],[363,16182],{"className":16183,"style":11714},[402],[363,16185,16062],{"className":16186,"style":16111},[407,408],[363,16188],{"className":16189,"style":506},[505],[363,16191,6257],{"className":16192},[510],[363,16194],{"className":16195,"style":506},[505],[363,16197,16199,16202,16205,16208,16211,16214,16217],{"className":16198},[398],[363,16200],{"className":16201,"style":487},[402],[363,16203,461],{"className":16204},[494],[363,16206,474],{"className":16207},[407],[363,16209,898],{"className":16210},[920],[363,16212],{"className":16213,"style":754},[505],[363,16215,465],{"className":16216},[407],[363,16218,468],{"className":16219},[501]," 充分小），并取似然比在第一坐标上为常数的分布：",[363,16222,16224],{"className":16223,"translate":367},[528],[363,16225,16227,16302],{"className":16226,"translate":367},[366],[363,16228,16230],{"className":16229},[371],[373,16231,16232],{"xmlns":375,"display":537},[377,16233,16234,16299],{},[380,16235,16236,16242,16244,16246,16248,16250,16252,16255,16257,16259,16261,16263,16265,16267,16269,16275,16277,16279,16281,16283,16285,16287,16289,16291,16293,16295,16297],{},[543,16237,16238,16240],{},[383,16239,554],{},[463,16241,465],{},[458,16243,471],{},[458,16245,461],{"stretchy":460},[383,16247,7909],{},[383,16249,16062],{},[458,16251,898],{"separator":394},[1645,16253,16254],{},"  ",[463,16256,465],{},[458,16258,5161],{},[383,16260,7909],{},[383,16262,16062],{},[458,16264,468],{"stretchy":460},[458,16266,898],{"separator":394},[505,16268],{"width":3610},[543,16270,16271,16273],{},[383,16272,554],{},[463,16274,2811],{},[458,16276,471],{},[458,16278,461],{"stretchy":460},[383,16280,15750],{},[383,16282,16062],{},[458,16284,898],{"separator":394},[1645,16286,16254],{},[463,16288,465],{},[458,16290,5161],{},[383,16292,15750],{},[383,16294,16062],{},[458,16296,468],{"stretchy":460},[458,16298,898],{"separator":394},[387,16300,16301],{"encoding":389},"P_1=(s\\delta,\\;1-s\\delta),\\qquad\nP_2=(u\\delta,\\;1-u\\delta),",[363,16303,16305,16361,16395,16465,16501],{"className":16304,"ariaHidden":394},[393],[363,16306,16308,16312,16352,16355,16358],{"className":16307},[398],[363,16309],{"className":16310,"style":16311},[402],"height:0.8333em;vertical-align:-0.15em;",[363,16313,16315,16318],{"className":16314},[407],[363,16316,554],{"className":16317,"style":682},[407,408],[363,16319,16321],{"className":16320},[629],[363,16322,16324,16344],{"className":16323},[633,634],[363,16325,16327,16341],{"className":16326},[638],[363,16328,16330],{"className":16329,"style":11483},[642],[363,16331,16332,16335],{"style":11486},[363,16333],{"className":16334,"style":651},[650],[363,16336,16338],{"className":16337},[655,656,657,658],[363,16339,465],{"className":16340},[407,658],[363,16342,666],{"className":16343},[665],[363,16345,16347],{"className":16346},[638],[363,16348,16350],{"className":16349,"style":8316},[642],[363,16351],{},[363,16353],{"className":16354,"style":506},[505],[363,16356,471],{"className":16357},[510],[363,16359],{"className":16360,"style":506},[505],[363,16362,16364,16367,16370,16374,16377,16380,16383,16386,16389,16392],{"className":16363},[398],[363,16365],{"className":16366,"style":487},[402],[363,16368,461],{"className":16369},[494],[363,16371,16373],{"className":16372,"style":16111},[407,408],"sδ",[363,16375,898],{"className":16376},[920],[363,16378],{"className":16379,"style":506},[505],[363,16381],{"className":16382,"style":754},[505],[363,16384,465],{"className":16385},[407],[363,16387],{"className":16388,"style":2935},[505],[363,16390,5161],{"className":16391},[2939],[363,16393],{"className":16394,"style":2935},[505],[363,16396,16398,16401,16404,16407,16410,16413,16416,16456,16459,16462],{"className":16397},[398],[363,16399],{"className":16400,"style":487},[402],[363,16402,16373],{"className":16403,"style":16111},[407,408],[363,16405,468],{"className":16406},[501],[363,16408,898],{"className":16409},[920],[363,16411],{"className":16412,"style":3982},[505],[363,16414],{"className":16415,"style":754},[505],[363,16417,16419,16422],{"className":16418},[407],[363,16420,554],{"className":16421,"style":682},[407,408],[363,16423,16425],{"className":16424},[629],[363,16426,16428,16448],{"className":16427},[633,634],[363,16429,16431,16445],{"className":16430},[638],[363,16432,16434],{"className":16433,"style":11483},[642],[363,16435,16436,16439],{"style":11486},[363,16437],{"className":16438,"style":651},[650],[363,16440,16442],{"className":16441},[655,656,657,658],[363,16443,2811],{"className":16444},[407,658],[363,16446,666],{"className":16447},[665],[363,16449,16451],{"className":16450},[638],[363,16452,16454],{"className":16453,"style":8316},[642],[363,16455],{},[363,16457],{"className":16458,"style":506},[505],[363,16460,471],{"className":16461},[510],[363,16463],{"className":16464,"style":506},[505],[363,16466,16468,16471,16474,16477,16480,16483,16486,16489,16492,16495,16498],{"className":16467},[398],[363,16469],{"className":16470,"style":487},[402],[363,16472,461],{"className":16473},[494],[363,16475,15750],{"className":16476},[407,408],[363,16478,16062],{"className":16479,"style":16111},[407,408],[363,16481,898],{"className":16482},[920],[363,16484],{"className":16485,"style":506},[505],[363,16487],{"className":16488,"style":754},[505],[363,16490,465],{"className":16491},[407],[363,16493],{"className":16494,"style":2935},[505],[363,16496,5161],{"className":16497},[2939],[363,16499],{"className":16500,"style":2935},[505],[363,16502,16504,16507,16510,16513,16516],{"className":16503},[398],[363,16505],{"className":16506,"style":487},[402],[363,16508,15750],{"className":16509},[407,408],[363,16511,16062],{"className":16512,"style":16111},[407,408],[363,16514,468],{"className":16515},[501],[363,16517,898],{"className":16518},[920],[11,16520,16145,16521,16550,16551,16593,16594,2757,16622,16785,16786,16815,16816,3093],{},[363,16522,16524,16538],{"className":16523,"translate":367},[366],[363,16525,16527],{"className":16526},[371],[373,16528,16529],{"xmlns":375},[377,16530,16531,16535],{},[380,16532,16533],{},[383,16534,16062],{},[387,16536,16537],{"encoding":389},"\\delta",[363,16539,16541],{"className":16540,"ariaHidden":394},[393],[363,16542,16544,16547],{"className":16543},[398],[363,16545],{"className":16546,"style":12947},[402],[363,16548,16062],{"className":16549,"style":16111},[407,408]," 足够小时这都是合法概率分布；",[363,16552,16554,16572],{"className":16553,"translate":367},[366],[363,16555,16557],{"className":16556},[371],[373,16558,16559],{"xmlns":375},[377,16560,16561,16569],{},[380,16562,16563,16565,16567],{},[463,16564,474],{},[458,16566,898],{"separator":394},[463,16568,465],{},[387,16570,16571],{"encoding":389},"0,1",[363,16573,16575],{"className":16574,"ariaHidden":394},[393],[363,16576,16578,16581,16584,16587,16590],{"className":16577},[398],[363,16579],{"className":16580,"style":9706},[402],[363,16582,474],{"className":16583},[407],[363,16585,898],{"className":16586},[920],[363,16588],{"className":16589,"style":754},[505],[363,16591,465],{"className":16592},[407]," 两点上的似然比分别为 ",[363,16595,16597,16610],{"className":16596,"translate":367},[366],[363,16598,16600],{"className":16599},[371],[373,16601,16602],{"xmlns":375},[377,16603,16604,16608],{},[380,16605,16606],{},[383,16607,7909],{},[387,16609,7909],{"encoding":389},[363,16611,16613],{"className":16612,"ariaHidden":394},[393],[363,16614,16616,16619],{"className":16615},[398],[363,16617],{"className":16618,"style":2693},[402],[363,16620,7909],{"className":16621},[407,408],[363,16623,16625,16667],{"className":16624,"translate":367},[366],[363,16626,16628],{"className":16627},[371],[373,16629,16630],{"xmlns":375},[377,16631,16632,16664],{},[380,16633,16634,16636,16638,16640,16642,16644],{},[383,16635,4506],{},[458,16637,461],{"stretchy":460},[383,16639,7909],{},[458,16641,468],{"stretchy":460},[458,16643,471],{},[586,16645,16646,16656],{},[380,16647,16648,16650,16652,16654],{},[463,16649,465],{},[458,16651,5161],{},[383,16653,7909],{},[383,16655,16062],{},[380,16657,16658,16660,16662],{},[463,16659,465],{},[458,16661,5161],{},[383,16663,16062],{},[387,16665,16666],{"encoding":389},"g(s)=\\frac{1-s\\delta}{1-\\delta}",[363,16668,16670,16697],{"className":16669,"ariaHidden":394},[393],[363,16671,16673,16676,16679,16682,16685,16688,16691,16694],{"className":16672},[398],[363,16674],{"className":16675,"style":487},[402],[363,16677,4506],{"className":16678,"style":1328},[407,408],[363,16680,461],{"className":16681},[494],[363,16683,7909],{"className":16684},[407,408],[363,16686,468],{"className":16687},[501],[363,16689],{"className":16690,"style":506},[505],[363,16692,471],{"className":16693},[510],[363,16695],{"className":16696,"style":506},[505],[363,16698,16700,16704],{"className":16699},[398],[363,16701],{"className":16702,"style":16703},[402],"height:1.2834em;vertical-align:-0.4033em;",[363,16705,16707,16710,16782],{"className":16706},[407],[363,16708],{"className":16709},[494,789],[363,16711,16713],{"className":16712},[586],[363,16714,16716,16773],{"className":16715},[633,634],[363,16717,16719,16770],{"className":16718},[638],[363,16720,16722,16742,16750],{"className":16721,"style":3035},[642],[363,16723,16724,16727],{"style":2875},[363,16725],{"className":16726,"style":809},[650],[363,16728,16730],{"className":16729},[655,656,657,658],[363,16731,16733,16736,16739],{"className":16732},[407,658],[363,16734,465],{"className":16735},[407,658],[363,16737,5161],{"className":16738},[2939,658],[363,16740,16062],{"className":16741,"style":16111},[407,408,658],[363,16743,16744,16747],{"style":821},[363,16745],{"className":16746,"style":809},[650],[363,16748],{"className":16749,"style":829},[828],[363,16751,16752,16755],{"style":2898},[363,16753],{"className":16754,"style":809},[650],[363,16756,16758],{"className":16757},[655,656,657,658],[363,16759,16761,16764,16767],{"className":16760},[407,658],[363,16762,465],{"className":16763},[407,658],[363,16765,5161],{"className":16766},[2939,658],[363,16768,16373],{"className":16769,"style":16111},[407,408,658],[363,16771,666],{"className":16772},[665],[363,16774,16776],{"className":16775},[638],[363,16777,16780],{"className":16778,"style":16779},[642],"height:0.4033em;",[363,16781],{},[363,16783],{"className":16784},[501,789],"）。于是一般公式给出：对任何第一坐标似然比为常值 ",[363,16787,16789,16803],{"className":16788,"translate":367},[366],[363,16790,16792],{"className":16791},[371],[373,16793,16794],{"xmlns":375},[377,16795,16796,16801],{},[380,16797,16798],{},[383,16799,16800],{},"r",[387,16802,16800],{"encoding":389},[363,16804,16806],{"className":16805,"ariaHidden":394},[393],[363,16807,16809,16812],{"className":16808},[398],[363,16810],{"className":16811,"style":2693},[402],[363,16813,16800],{"className":16814,"style":625},[407,408]," 的分布 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+(1-\\delta)\\,\nf\\!\\left(\\frac{1-r\\delta}{1-\\delta}\\right).\n\\end{aligned}",[363,16999,17001],{"className":17000,"ariaHidden":394},[393],[363,17002,17004,17008],{"className":17003},[398],[363,17005],{"className":17006,"style":17007},[402],"height:3.91em;vertical-align:-1.705em;",[363,17009,17011],{"className":17010},[407],[363,17012,17014,17166],{"className":17013},[1591],[363,17015,17017],{"className":17016},[1708],[363,17018,17020,17158],{"className":17019},[633,634],[363,17021,17023,17155],{"className":17022},[638],[363,17024,17027,17146],{"className":17025,"style":17026},[642],"height:2.205em;",[363,17028,17030,17033],{"style":17029},"top:-4.805em;",[363,17031],{"className":17032,"style":1725},[650],[363,17034,17036,17076,17082,17085,17088,17091,17094,17097,17134,17137,17140],{"className":17035},[407],[363,17037,17039,17042],{"className":17038},[407],[363,17040,547],{"className":17041,"style":625},[407,408],[363,17043,17045],{"className":17044},[629],[363,17046,17048,17068],{"className":17047},[633,634],[363,17049,17051,17065],{"className":17050},[638],[363,17052,17054],{"className":17053,"style":643},[642],[363,17055,17056,17059],{"style":646},[363,17057],{"className":17058,"style":651},[650],[363,17060,17062],{"className":17061},[655,656,657,658],[363,17063,385],{"className":17064,"style":409},[407,408,658],[363,17066,666],{"className":17067},[665],[363,17069,17071],{"className":17070},[638],[363,17072,17074],{"className":17073,"style":673},[642],[363,17075],{},[363,17077,17079],{"className":17078},[407],[363,17080,461],{"className":17081},[767,10022],[363,17083,554],{"className":17084,"style":682},[407,408],[363,17086,461],{"className":17087},[494],[363,17089,16800],{"className":17090,"style":625},[407,408],[363,17092,468],{"className":17093},[501],[363,17095],{"className":17096,"style":754},[505],[363,17098,17100],{"className":17099},[407],[363,17101,17103],{"className":17102},[767,15179],[363,17104,17106,17126],{"className":17105},[633,634],[363,17107,17109,17123],{"className":17108},[638],[363,17110,17112],{"className":17111,"style":15189},[642],[363,17113,17114,17117],{"style":15192},[363,17115],{"className":17116,"style":15196},[650],[363,17118,17119],{"style":15199},[6671,17120,17121],{"xmlns":6673,"width":15202,"height":9798,"viewBox":15203},[6679,17122],{"d":15206},[363,17124,666],{"className":17125},[665],[363,17127,17129],{"className":17128},[638],[363,17130,17132],{"className":17131,"style":15216},[642],[363,17133],{},[363,17135],{"className":17136,"style":754},[505],[363,17138,561],{"className":17139},[407,408],[363,17141,17143],{"className":17142},[407],[363,17144,468],{"className":17145},[767,10022],[363,17147,17149,17152],{"style":17148},"top:-2.695em;",[363,17150],{"className":17151,"style":1725},[650],[363,17153],{"className":17154},[407],[363,17156,666],{"className":17157},[665],[363,17159,17161],{"className":17160},[638],[363,17162,17164],{"className":17163,"style":9902},[642],[363,17165],{},[363,17167,17169],{"className":17168},[1799],[363,17170,17172,17386],{"className":17171},[633,634],[363,17173,17175,17383],{"className":17174},[638],[363,17176,17178,17216],{"className":17177,"style":17026},[642],[363,17179,17180,17183],{"style":17029},[363,17181],{"className":17182,"style":1725},[650],[363,17184,17186,17189,17192,17195,17198,17201,17204,17207,17210,17213],{"className":17185},[407],[363,17187],{"className":17188},[407],[363,17190],{"className":17191,"style":506},[505],[363,17193,471],{"className":17194},[510],[363,17196],{"className":17197,"style":506},[505],[363,17199,16062],{"className":17200,"style":16111},[407,408],[363,17202],{"className":17203,"style":754},[505],[363,17205,385],{"className":17206,"style":409},[407,408],[363,17208,461],{"className":17209},[494],[363,17211,16800],{"className":17212,"style":625},[407,408],[363,17214,468],{"className":17215},[501],[363,17217,17218,17221],{"style":17148},[363,17219],{"className":17220,"style":1725},[650],[363,17222,17224,17227,17230,17233,17236,17239,17242,17245,17248,17251,17254,17257,17260,17263,17266,17269,17272,17377,17380],{"className":17223},[407],[363,17225],{"className":17226},[407],[363,17228],{"className":17229,"style":3982},[505],[363,17231],{"className":17232,"style":2935},[505],[363,17234,2818],{"className":17235},[2939],[363,17237],{"className":17238,"style":2935},[505],[363,17240,461],{"className":17241},[494],[363,17243,465],{"className":17244},[407],[363,17246],{"className":17247,"style":2935},[505],[363,17249,5161],{"className":17250},[2939],[363,17252],{"className":17253,"style":2935},[505],[363,17255,16062],{"className":17256,"style":16111},[407,408],[363,17258,468],{"className":17259},[501],[363,17261],{"className":17262,"style":754},[505],[363,17264,385],{"className":17265,"style":409},[407,408],[363,17267],{"className":17268,"style":1860},[505],[363,17270],{"className":17271,"style":754},[505],[363,17273,17275,17281,17371],{"className":17274},[758],[363,17276,17278],{"className":17277,"style":763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",[363,17553,17555,17589],{"className":17554,"translate":367},[366],[363,17556,17558],{"className":17557},[371],[373,17559,17560],{"xmlns":375},[377,17561,17562,17586],{},[380,17563,17564,17566,17568,17570,17572,17574,17576,17578,17580,17582,17584],{},[383,17565,11689],{},[383,17567,7909],{},[458,17569,2818],{},[458,17571,461],{"stretchy":460},[463,17573,465],{},[458,17575,5161],{},[383,17577,11689],{},[458,17579,468],{"stretchy":460},[383,17581,15750],{},[458,17583,471],{},[383,17585,7918],{},[387,17587,17588],{"encoding":389},"\\lambda 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下混合，比值线性）代入第一步的联合凸性：",[363,17701,17703],{"className":17702,"translate":367},[528],[363,17704,17706,17903],{"className":17705,"translate":367},[366],[363,17707,17709],{"className":17708},[371],[373,17710,17711],{"xmlns":375,"display":537},[377,17712,17713,17900],{},[1591,17714,17715,17767,17828],{"rowspacing":1593,"columnalign":1594,"columnspacing":1595},[1597,17716,17717,17723],{},[1600,17718,17719],{},[1603,17720,17721],{"scriptlevel":474,"displaystyle":394},[380,17722],{},[1600,17724,17725],{},[1603,17726,17727],{"scriptlevel":474,"displaystyle":394},[380,17728,17729,17731,17733,17735,17737,17739,17741,17743,17745,17747,17749,17751,17753,17755,17757,17759,17761,17763,17765],{},[380,17730],{},[383,17732,16062],{},[383,17734,385],{},[458,17736,461],{"stretchy":460},[383,17738,7918],{},[458,17740,468],{"stretchy":460},[458,17742,2818],{},[458,17744,461],{"stretchy":460},[463,17746,465],{},[458,17748,5161],{},[383,17750,16062],{},[458,17752,468],{"stretchy":460},[383,17754,385],{},[458,17756,461],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[383,17758,4506],{},[458,17760,461],{"stretchy":460},[383,17762,7918],{},[458,17764,468],{"stretchy":460},[458,17766,468],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[1597,17768,17769,17775],{},[1600,17770,17771],{},[1603,17772,17773],{"scriptlevel":474,"displaystyle":394},[380,17774],{},[1600,17776,17777],{},[1603,17778,17779],{"scriptlevel":474,"displaystyle":394},[380,17780,17781,17783,17785,17787,17790,17792,17794,17796,17798,17800,17802,17804,17806,17808,17810,17812,17814,17816,17818,17820,17822,17824,17826],{},[380,17782],{},[458,17784,10300],{},[383,17786,11689],{},[458,17788,578],{"fence":460,"stretchy":394,"minsize":17789,"maxsize":17789},"1.8em",[383,17791,16062],{},[383,17793,385],{},[458,17795,461],{"stretchy":460},[383,17797,7909],{},[458,17799,468],{"stretchy":460},[458,17801,2818],{},[458,17803,461],{"stretchy":460},[463,17805,465],{},[458,17807,5161],{},[383,17809,16062],{},[458,17811,468],{"stretchy":460},[383,17813,385],{},[458,17815,461],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[383,17817,4506],{},[458,17819,461],{"stretchy":460},[383,17821,7909],{},[458,17823,468],{"stretchy":460},[458,17825,468],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[458,17827,605],{"fence":460,"stretchy":394,"minsize":17789,"maxsize":17789},[1597,17829,17830,17836],{},[1600,17831,17832],{},[1603,17833,17834],{"scriptlevel":474,"displaystyle":394},[380,17835],{},[1600,17837,17838],{},[1603,17839,17840],{"scriptlevel":474,"displaystyle":394},[380,17841,17842,17844,17846,17848,17850,17852,17854,17856,17858,17860,17862,17864,17866,17868,17870,17872,17874,17876,17878,17880,17882,17884,17886,17888,17890,17892,17894,17896,17898],{},[380,17843],{},[505,17845],{"width":3610},[458,17847,2818],{},[458,17849,461],{"stretchy":460},[463,17851,465],{},[458,17853,5161],{},[383,17855,11689],{},[458,17857,468],{"stretchy":460},[458,17859,578],{"fence":460,"stretchy":394,"minsize":17789,"maxsize":17789},[383,17861,16062],{},[383,17863,385],{},[458,17865,461],{"stretchy":460},[383,17867,15750],{},[458,17869,468],{"stretchy":460},[458,17871,2818],{},[458,17873,461],{"stretchy":460},[463,17875,465],{},[458,17877,5161],{},[383,17879,16062],{},[458,17881,468],{"stretchy":460},[383,17883,385],{},[458,17885,461],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[383,17887,4506],{},[458,17889,461],{"stretchy":460},[383,17891,15750],{},[458,17893,468],{"stretchy":460},[458,17895,468],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[458,17897,605],{"fence":460,"stretchy":394,"minsize":17789,"maxsize":17789},[458,17899,898],{"separator":394},[387,17901,17902],{"encoding":389},"\\begin{aligned}\n&\\delta f(w)+(1-\\delta)f\\big(g(w)\\big)\\\\\n&\\le \\lambda\\Big[\\delta f(s)\n+(1-\\delta)f\\big(g(s)\\big)\\Big]\\\\\n&\\qquad+(1-\\lambda)\\Big[\\delta f(u)\n+(1-\\delta)f\\big(g(u)\\big)\\Big],\n\\end{aligned}",[363,17904,17906],{"className":17905,"ariaHidden":394},[393],[363,17907,17909,17913],{"className":17908},[398],[363,17910],{"className":17911,"style":17912},[402],"height:5.41em;vertical-align:-2.455em;",[363,17914,17916],{"className":17915},[407],[363,17917,17919,17972],{"className":17918},[1591],[363,17920,17922],{"className":17921},[1708],[363,17923,17925,17963],{"className":17924},[633,634],[363,17926,17928,17960],{"className":17927},[638],[363,17929,17932,17942,17951],{"className":17930,"style":17931},[642],"height:2.955em;",[363,17933,17935,17939],{"style":17934},"top:-5.255em;",[363,17936],{"className":17937,"style":17938},[650],"height:3.15em;",[363,17940],{"className":17941},[407],[363,17943,17945,17948],{"style":17944},"top:-3.445em;",[363,17946],{"className":17947,"style":17938},[650],[363,17949],{"className":17950},[407],[363,17952,17954,17957],{"style":17953},"top:-1.345em;",[363,17955],{"className":17956,"style":17938},[650],[363,17958],{"className":17959},[407],[363,17961,666],{"className":17962},[665],[363,17964,17966],{"className":17965},[638],[363,17967,17970],{"className":17968,"style":17969},[642],"height:2.455em;",[363,17971],{},[363,17973,17975],{"className":17974},[1799],[363,17976,17978,18308],{"className":17977},[633,634],[363,17979,17981,18305],{"className":17980},[638],[363,17982,17984,18067,18174],{"className":17983,"style":17931},[642],[363,17985,17986,17989],{"style":17934},[363,17987],{"className":17988,"style":17938},[650],[363,17990,17992,17995,17998,18001,18004,18007,18010,18013,18016,18019,18022,18025,18028,18031,18034,18037,18040,18043,18049,18052,18055,18058,18061],{"className":17991},[407],[363,17993],{"className":17994},[407],[363,17996,16062],{"className":17997,"style":16111},[407,408],[363,17999,385],{"className":18000,"style":409},[407,408],[363,18002,461],{"className":18003},[494],[363,18005,7918],{"className":18006,"style":8622},[407,408],[363,18008,468],{"className":18009},[501],[363,18011],{"className":18012,"style":2935},[505],[363,18014,2818],{"className":18015},[2939],[363,18017],{"className":18018,"style":2935},[505],[363,18020,461],{"className":18021},[494],[363,18023,465],{"className":18024},[407],[363,18026],{"className":18027,"style":2935},[505],[363,18029,5161],{"className":18030},[2939],[363,18032],{"className":18033,"style":2935},[505],[363,18035,16062],{"className":18036,"style":16111},[407,408],[363,18038,468],{"className":18039},[501],[363,18041,385],{"className":18042,"style":409},[407,408],[363,18044,18046],{"className":18045},[407],[363,18047,461],{"className":18048},[767,10022],[363,18050,4506],{"className":18051,"style":1328},[407,408],[363,18053,461],{"className":18054},[494],[363,18056,7918],{"className":18057,"style":8622},[407,408],[363,18059,468],{"className":18060},[501],[363,18062,18064],{"className":18063},[407],[363,18065,468],{"className":18066},[767,10022],[363,18068,18069,18072],{"style":17944},[363,18070],{"className":18071,"style":17938},[650],[363,18073,18075,18078,18081,18084,18087,18090,18096,18099,18102,18105,18108,18111,18114,18117,18120,18123,18126,18129,18132,18135,18138,18141,18144,18150,18153,18156,18159,18162,18168],{"className":18074},[407],[363,18076],{"className":18077},[407],[363,18079],{"className":18080,"style":506},[505],[363,18082,10300],{"className":18083},[510],[363,18085],{"className":18086,"style":506},[505],[363,18088,11689],{"className":18089},[407,408],[363,18091,18093],{"className":18092},[407],[363,18094,578],{"className":18095},[767,4068],[363,18097,16062],{"className":18098,"style":16111},[407,408],[363,18100,385],{"className":18101,"style":409},[407,408],[363,18103,461],{"className":18104},[494],[363,18106,7909],{"className":18107},[407,408],[363,18109,468],{"className":18110},[501],[363,18112],{"className":18113,"style":2935},[505],[363,18115,2818],{"className":18116},[2939],[363,18118],{"className":18119,"style":2935},[505],[363,18121,461],{"className":18122},[494],[363,18124,465],{"className":18125},[407],[363,18127],{"className":18128,"style":2935},[505],[363,18130,5161],{"className":18131},[2939],[363,18133],{"className":18134,"style":2935},[505],[363,18136,16062],{"className":18137,"style":16111},[407,408],[363,18139,468],{"className":18140},[501],[363,18142,385],{"className":18143,"style":409},[407,408],[363,18145,18147],{"className":18146},[407],[363,18148,461],{"className":18149},[767,10022],[363,18151,4506],{"className":18152,"style":1328},[407,408],[363,18154,461],{"className":18155},[494],[363,18157,7909],{"className":18158},[407,408],[363,18160,468],{"className":18161},[501],[363,18163,18165],{"className":18164},[407],[363,18166,468],{"className":18167},[767,10022],[363,18169,18171],{"className":18170},[407],[363,18172,605],{"className":18173},[767,4068],[363,18175,18176,18179],{"style":17953},[363,18177],{"className":18178,"style":17938},[650],[363,18180,18182,18185,18188,18191,18194,18197,18200,18203,18206,18209,18212,18215,18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},[2939],[363,18469],{"className":18470,"style":2935},[505],[363,18472,16800],{"className":18473,"style":625},[407,408],[363,18475,16062],{"className":18476,"style":16111},[407,408],[363,18478,666],{"className":18479},[665],[363,18481,18483],{"className":18482},[638],[363,18484,18486],{"className":18485,"style":17365},[642],[363,18487],{},[363,18489],{"className":18490},[501,789],"。注意 ",[363,18493,18495,18508],{"className":18494,"translate":367},[366],[363,18496,18498],{"className":18497},[371],[373,18499,18500],{"xmlns":375},[377,18501,18502,18506],{},[380,18503,18504],{},[383,18505,4506],{},[387,18507,4506],{"encoding":389},[363,18509,18511],{"className":18510,"ariaHidden":394},[393],[363,18512,18514,18517],{"className":18513},[398],[363,18515],{"className":18516,"style":2624},[402],[363,18518,4506],{"className":18519,"style":1328},[407,408]," 是仿射函数且 ",[363,18522,18524,18576],{"className":18523,"translate":367},[366],[363,18525,18527],{"className":18526},[371],[373,18528,18529],{"xmlns":375},[377,18530,18531,18573],{},[380,18532,18533,18535,18537,18539,18541,18543,18545,18547,18549,18551,18553,18555,18557,18559,18561,18563,18565,18567,18569,18571],{},[383,18534,4506],{},[458,18536,461],{"stretchy":460},[383,18538,7918],{},[458,18540,468],{"stretchy":460},[458,18542,471],{},[383,18544,11689],{},[383,18546,4506],{},[458,18548,461],{"stretchy":460},[383,18550,7909],{},[458,18552,468],{"stretchy":460},[458,18554,2818],{},[458,18556,461],{"stretchy":460},[463,18558,465],{},[458,18560,5161],{},[383,18562,11689],{},[458,18564,468],{"stretchy":460},[383,18566,4506],{},[458,18568,461],{"stretchy":460},[383,18570,15750],{},[458,18572,468],{"stretchy":460},[387,18574,18575],{"encoding":389},"g(w)=\\lambda g(s)+(1-\\lambda)g(u)",[363,18577,18579,18606,18636,18657],{"className":18578,"ariaHidden":394},[393],[363,18580,18582,18585,18588,18591,18594,18597,18600,18603],{"className":18581},[398],[363,18583],{"className":18584,"style":487},[402],[363,18586,4506],{"className":18587,"style":1328},[407,408],[363,18589,461],{"className":18590},[494],[363,18592,7918],{"className":18593,"style":8622},[407,408],[363,18595,468],{"className":18596},[501],[363,18598],{"className":18599,"style":506},[505],[363,18601,471],{"className":18602},[510],[363,18604],{"className":18605,"style":506},[505],[363,18607,18609,18612,18615,18618,18621,18624,18627,18630,18633],{"className":18608},[398],[363,18610],{"className":18611,"style":487},[402],[363,18613,11689],{"className":18614},[407,408],[363,18616,4506],{"className":18617,"style":1328},[407,408],[363,18619,461],{"className":18620},[494],[363,18622,7909],{"className":18623},[407,408],[363,18625,468],{"className":18626},[501],[363,18628],{"className":18629,"style":2935},[505],[363,18631,2818],{"className":18632},[2939],[363,18634],{"className":18635,"style":2935},[505],[363,18637,18639,18642,18645,18648,18651,18654],{"className":18638},[398],[363,18640],{"className":18641,"style":487},[402],[363,18643,461],{"className":18644},[494],[363,18646,465],{"className":18647},[407],[363,18649],{"className":18650,"style":2935},[505],[363,18652,5161],{"className":18653},[2939],[363,18655],{"className":18656,"style":2935},[505],[363,18658,18660,18663,18666,18669,18672,18675,18678],{"className":18659},[398],[363,18661],{"className":18662,"style":487},[402],[363,18664,11689],{"className":18665},[407,408],[363,18667,468],{"className":18668},[501],[363,18670,4506],{"className":18671,"style":1328},[407,408],[363,18673,461],{"className":18674},[494],[363,18676,15750],{"className":18677},[407,408],[363,18679,468],{"className":18680},[501],"，用 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f(1)+f'(1)(t-1)*f'(1)*",[363,18813,18815,18842,18869,18931,18952],{"className":18814,"ariaHidden":394},[393],[363,18816,18818,18821,18824,18827,18830,18833,18836,18839],{"className":18817},[398],[363,18819],{"className":18820,"style":487},[402],[363,18822,385],{"className":18823,"style":409},[407,408],[363,18825,461],{"className":18826},[494],[363,18828,1116],{"className":18829},[407,408],[363,18831,468],{"className":18832},[501],[363,18834],{"className":18835,"style":506},[505],[363,18837,7276],{"className":18838},[510],[363,18840],{"className":18841,"style":506},[505],[363,18843,18845,18848,18851,18854,18857,18860,18863,18866],{"className":18844},[398],[363,18846],{"className":18847,"style":487},[402],[363,18849,385],{"className":18850,"style":409},[407,408],[363,18852,461],{"className":18853},[494],[363,18855,465],{"className":18856},[407],[363,18858,468],{"className":18859},[501],[363,18861],{"className":18862,"style":2935},[505],[363,18864,2818],{"className":18865},[2939],[363,18867],{"className":18868,"style":2935},[505],[363,18870,18872,18875,18907,18910,18913,18916,18919,18922,18925,18928],{"className":18871},[398],[363,18873],{"className":18874,"style":7191},[402],[363,18876,18878,18881],{"className":18877},[407],[363,18879,385],{"className":18880,"style":409},[407,408],[363,18882,18884],{"className":18883},[629],[363,18885,18887],{"className":18886},[633],[363,18888,18890],{"className":18889},[638],[363,18891,18893],{"className":18892,"style":7210},[642],[363,18894,18895,18898],{"style":7213},[363,18896],{"className":18897,"style":651},[650],[363,18899,18901],{"className":18900},[655,656,657,658],[363,18902,18904],{"className":18903},[407,658],[363,18905,7168],{"className":18906},[407,658],[363,18908,461],{"className":18909},[494],[363,18911,465],{"className":18912},[407],[363,18914,468],{"className":18915},[501],[363,18917,461],{"className":18918},[494],[363,18920,1116],{"className":18921},[407,408],[363,18923],{"className":18924,"style":2935},[505],[363,18926,5161],{"className":18927},[2939],[363,18929],{"className":18930,"style":2935},[505],[363,18932,18934,18937,18940,18943,18946,18949],{"className":18933},[398],[363,18935],{"className":18936,"style":487},[402],[363,18938,465],{"className":18939},[407],[363,18941,468],{"className":18942},[501],[363,18944],{"className":18945,"style":2935},[505],[363,18947,7921],{"className":18948},[2939],[363,18950],{"className":18951,"style":2935},[505],[363,18953,18955,18958,18990,18993,18996,18999],{"className":18954},[398],[363,18956],{"className":18957,"style":7191},[402],[363,18959,18961,18964],{"className":18960},[407],[363,18962,385],{"className":18963,"style":409},[407,408],[363,18965,18967],{"className":18966},[629],[363,18968,18970],{"className":18969},[633],[363,18971,18973],{"className":18972},[638],[363,18974,18976],{"className":18975,"style":7210},[642],[363,18977,18978,18981],{"style":7213},[363,18979],{"className":18980,"style":651},[650],[363,18982,18984],{"className":18983},[655,656,657,658],[363,18985,18987],{"className":18986},[407,658],[363,18988,7168],{"className":18989},[407,658],[363,18991,461],{"className":18992},[494],[363,18994,465],{"className":18995},[407],[363,18997,468],{"className":18998},[501],[363,19000,7921],{"className":19001},[407],"，此处只需次梯度 ",[363,19004,19006,19030],{"className":19005,"translate":367},[366],[363,19007,19009],{"className":19008},[371],[373,19010,19011],{"xmlns":375},[377,19012,19013,19027],{},[380,19014,19015,19021,19023,19025],{},[7162,19016,19017,19019],{},[383,19018,385],{},[458,19020,7168],{"mathvariant":557,"lspace":1595,"rspace":1595},[458,19022,461],{"stretchy":460},[463,19024,465],{},[458,19026,468],{"stretchy":460},[387,19028,19029],{"encoding":389},"f'(1)",[363,19031,19033],{"className":19032,"ariaHidden":394},[393],[363,19034,19036,19039,19071,19074,19077],{"className":19035},[398],[363,19037],{"className":19038,"style":7191},[402],[363,19040,19042,19045],{"className":19041},[407],[363,19043,385],{"className":19044,"style":409},[407,408],[363,19046,19048],{"className":19047},[629],[363,19049,19051],{"className":19050},[633],[363,19052,19054],{"className":19053},[638],[363,19055,19057],{"className":19056,"style":7210},[642],[363,19058,19059,19062],{"style":7213},[363,19060],{"className":19061,"style":651},[650],[363,19063,19065],{"className":19064},[655,656,657,658],[363,19066,19068],{"className":19067},[407,658],[363,19069,7168],{"className":19070},[407,658],[363,19072,461],{"className":19073},[494],[363,19075,465],{"className":19076},[407],[363,19078,468],{"className":19079},[501],"，即 ",[363,19082,19084,19132],{"className":19083,"translate":367},[366],[363,19085,19087],{"className":19086},[371],[373,19088,19089],{"xmlns":375},[377,19090,19091,19129],{},[380,19092,19093,19095,19097,19099,19101,19103,19105,19107,19109,19111,19113,19119,19121,19123,19125,19127],{},[383,19094,385],{},[458,19096,461],{"stretchy":460},[383,19098,1116],{},[458,19100,468],{"stretchy":460},[458,19102,7276],{},[383,19104,385],{},[458,19106,461],{"stretchy":460},[463,19108,465],{},[458,19110,468],{"stretchy":460},[458,19112,2818],{},[543,19114,19115,19117],{},[383,19116,4506],{},[463,19118,465],{},[458,19120,461],{"stretchy":460},[383,19122,1116],{},[458,19124,5161],{},[463,19126,465],{},[458,19128,468],{"stretchy":460},[387,19130,19131],{"encoding":389},"f(t)\\ge f(1)+g_1(t-1)",[363,19133,19135,19162,19189,19251],{"className":19134,"ariaHidden":394},[393],[363,19136,19138,19141,19144,19147,19150,19153,19156,19159],{"className":19137},[398],[363,19139],{"className":19140,"style":487},[402],[363,19142,385],{"className":19143,"style":409},[407,408],[363,19145,461],{"className":19146},[494],[363,19148,1116],{"className":19149},[407,408],[363,19151,468],{"className":19152},[501],[363,19154],{"className":19155,"style":506},[505],[363,19157,7276],{"className":19158},[510],[363,19160],{"className":19161,"style":506},[505],[363,19163,19165,19168,19171,19174,19177,19180,19183,19186],{"className":19164},[398],[363,19166],{"className":19167,"style":487},[402],[363,19169,385],{"className":19170,"style":409},[407,408],[363,19172,461],{"className":19173},[494],[363,19175,465],{"className":19176},[407],[363,19178,468],{"className":19179},[501],[363,19181],{"className":19182,"style":2935},[505],[363,19184,2818],{"className":19185},[2939],[363,19187],{"className":19188,"style":2935},[505],[363,19190,19192,19195,19236,19239,19242,19245,19248],{"className":19191},[398],[363,19193],{"className":19194,"style":487},[402],[363,19196,19198,19201],{"className":19197},[407],[363,19199,4506],{"className":19200,"style":1328},[407,408],[363,19202,19204],{"className":19203},[629],[363,19205,19207,19228],{"className":19206},[633,634],[363,19208,19210,19225],{"className":19209},[638],[363,19211,19213],{"className":19212,"style":11483},[642],[363,19214,19216,19219],{"style":19215},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[363,19217],{"className":19218,"style":651},[650],[363,19220,19222],{"className":19221},[655,656,657,658],[363,19223,465],{"className":19224},[407,658],[363,19226,666],{"className":19227},[665],[363,19229,19231],{"className":19230},[638],[363,19232,19234],{"className":19233,"style":8316},[642],[363,19235],{},[363,19237,461],{"className":19238},[494],[363,19240,1116],{"className":19241},[407,408],[363,19243],{"className":19244,"style":2935},[505],[363,19246,5161],{"className":19247},[2939],[363,19249],{"className":19250,"style":2935},[505],[363,19252,19254,19257,19260],{"className":19253},[398],[363,19255],{"className":19256,"style":487},[402],[363,19258,465],{"className":19259},[407],[363,19261,468],{"className":19262},[501]," 对任一次梯度 ",[363,19265,19267,19285],{"className":19266,"translate":367},[366],[363,19268,19270],{"className":19269},[371],[373,19271,19272],{"xmlns":375},[377,19273,19274,19282],{},[380,19275,19276],{},[543,19277,19278,19280],{},[383,19279,4506],{},[463,19281,465],{},[387,19283,19284],{"encoding":389},"g_1",[363,19286,19288],{"className":19287,"ariaHidden":394},[393],[363,19289,19291,19294],{"className":19290},[398],[363,19292],{"className":19293,"style":2624},[402],[363,19295,19297,19300],{"className":19296},[407],[363,19298,4506],{"className":19299,"style":1328},[407,408],[363,19301,19303],{"className":19302},[629],[363,19304,19306,19326],{"className":19305},[633,634],[363,19307,19309,19323],{"className":19308},[638],[363,19310,19312],{"className":19311,"style":11483},[642],[363,19313,19314,19317],{"style":19215},[363,19315],{"className":19316,"style":651},[650],[363,19318,19320],{"className":19319},[655,656,657,658],[363,19321,465],{"className":19322},[407,658],[363,19324,666],{"className":19325},[665],[363,19327,19329],{"className":19328},[638],[363,19330,19332],{"className":19331,"style":8316},[642],[363,19333],{},"）处理非主项：对凸的 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中关于 ",[363,19422,19424,19437],{"className":19423,"translate":367},[366],[363,19425,19427],{"className":19426},[371],[373,19428,19429],{"xmlns":375},[377,19430,19431,19435],{},[380,19432,19433],{},[383,19434,16800],{},[387,19436,16800],{"encoding":389},[363,19438,19440],{"className":19439,"ariaHidden":394},[393],[363,19441,19443,19446],{"className":19442},[398],[363,19444],{"className":19445,"style":2693},[402],[363,19447,16800],{"className":19448,"style":625},[407,408]," 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项在 ",[363,20417,20419,20432],{"className":20418,"translate":367},[366],[363,20420,20422],{"className":20421},[371],[373,20423,20424],{"xmlns":375},[377,20425,20426,20430],{},[380,20427,20428],{},[383,20429,16800],{},[387,20431,16800],{"encoding":389},[363,20433,20435],{"className":20434,"ariaHidden":394},[393],[363,20436,20438,20441],{"className":20437},[398],[363,20439],{"className":20440,"style":2693},[402],[363,20442,16800],{"className":20443,"style":625},[407,408]," 变化时一致有界，因为 ",[363,20446,20448,20482],{"className":20447,"translate":367},[366],[363,20449,20451],{"className":20450},[371],[373,20452,20453],{"xmlns":375},[377,20454,20455,20479],{},[380,20456,20457,20459,20461,20463,20465,20467,20469,20471,20473,20475,20477],{},[383,20458,4506],{},[458,20460,461],{"stretchy":460},[383,20462,16800],{},[458,20464,468],{"stretchy":460},[458,20466,471],{},[463,20468,465],{},[458,20470,2818],{},[383,20472,19593],{},[458,20474,461],{"stretchy":460},[383,20476,16062],{},[458,20478,468],{"stretchy":460},[387,20480,20481],{"encoding":389},"g(r)=1+O(\\delta)",[363,20483,20485,20512,20530],{"className":20484,"ariaHidden":394},[393],[363,20486,20488,20491,20494,20497,20500,20503,20506,20509],{"className":20487},[398],[363,20489],{"className":20490,"style":487},[402],[363,20492,4506],{"className":20493,"style":1328},[407,408],[363,20495,461],{"className":20496},[494],[363,20498,16800],{"className":20499,"style":625},[407,408],[363,20501,468],{"className":20502},[501],[363,20504],{"className":20505,"style":506},[505],[363,20507,471],{"className":20508},[510],[363,20510],{"className":20511,"style":506},[505],[363,20513,20515,20518,20521,20524,20527],{"className":20514},[398],[363,20516],{"className":20517,"style":12928},[402],[363,20519,465],{"className":20520},[407],[363,20522],{"className":20523,"style":2935},[505],[363,20525,2818],{"className":20526},[2939],[363,20528],{"className":20529,"style":2935},[505],[363,20531,20533,20536,20539,20542,20545],{"className":20532},[398],[363,20534],{"className":20535,"style":487},[402],[363,20537,19593],{"className":20538,"style":625},[407,408],[363,20540,461],{"className":20541},[494],[363,20543,16062],{"className":20544,"style":16111},[407,408],[363,20546,468],{"className":20547},[501],[363,20549,20551,20564],{"className":20550,"translate":367},[366],[363,20552,20554],{"className":20553},[371],[373,20555,20556],{"xmlns":375},[377,20557,20558,20562],{},[380,20559,20560],{},[383,20561,385],{},[387,20563,385],{"encoding":389},[363,20565,20567],{"className":20566,"ariaHidden":394},[393],[363,20568,20570,20573],{"className":20569},[398],[363,20571],{"className":20572,"style":403},[402],[363,20574,385],{"className":20575,"style":409},[407,408]," 连续）。代回不等式：两边的主项（",[363,20578,20580,20602],{"className":20579,"translate":367},[366],[363,20581,20583],{"className":20582},[371],[373,20584,20585],{"xmlns":375},[377,20586,20587,20599],{},[380,20588,20589,20591,20593,20595,20597],{},[383,20590,16062],{},[383,20592,385],{},[458,20594,461],{"stretchy":460},[458,20596,3647],{},[458,20598,468],{"stretchy":460},[387,20600,20601],{"encoding":389},"\\delta f(\\cdot)",[363,20603,20605],{"className":20604,"ariaHidden":394},[393],[363,20606,20608,20611,20614,20617,20620,20623],{"className":20607},[398],[363,20609],{"className":20610,"style":487},[402],[363,20612,16062],{"className":20613,"style":16111},[407,408],[363,20615,385],{"className":20616,"style":409},[407,408],[363,20618,461],{"className":20619},[494],[363,20621,3647],{"className":20622},[407],[363,20624,468],{"className":20625},[501]," 部分）正是 ",[363,20628,20630,20678],{"className":20629,"translate":367},[366],[363,20631,20633],{"className":20632},[371],[373,20634,20635],{"xmlns":375},[377,20636,20637,20675],{},[380,20638,20639,20641,20643,20645,20647,20649,20651,20653,20655,20657,20659,20661,20663,20665,20667,20669,20671,20673],{},[383,20640,16062],{},[458,20642,578],{"stretchy":460},[383,20644,11689],{},[383,20646,385],{},[458,20648,461],{"stretchy":460},[383,20650,7909],{},[458,20652,468],{"stretchy":460},[458,20654,2818],{},[458,20656,461],{"stretchy":460},[463,20658,465],{},[458,20660,5161],{},[383,20662,11689],{},[458,20664,468],{"stretchy":460},[383,20666,385],{},[458,20668,461],{"stretchy":460},[383,20670,15750],{},[458,20672,468],{"stretchy":460},[458,20674,605],{"stretchy":460},[387,20676,20677],{"encoding":389},"\\delta[\\lambda f(s)+(1-\\lambda)f(u)]",[363,20679,20681,20717,20738],{"className":20680,"ariaHidden":394},[393],[363,20682,20684,20687,20690,20693,20696,20699,20702,20705,20708,20711,20714],{"className":20683},[398],[363,20685],{"className":20686,"style":487},[402],[363,20688,16062],{"className":20689,"style":16111},[407,408],[363,20691,578],{"className":20692},[494],[363,20694,11689],{"className":20695},[407,408],[363,20697,385],{"className":20698,"style":409},[407,408],[363,20700,461],{"className":20701},[494],[363,20703,7909],{"className":20704},[407,408],[363,20706,468],{"className":20707},[501],[363,20709],{"className":20710,"style":2935},[505],[363,20712,2818],{"className":20713},[2939],[363,20715],{"className":20716,"style":2935},[505],[363,20718,20720,20723,20726,20729,20732,20735],{"className":20719},[398],[363,20721],{"className":20722,"style":487},[402],[363,20724,461],{"className":20725},[494],[363,20727,465],{"className":20728},[407],[363,20730],{"className":20731,"style":2935},[505],[363,20733,5161],{"className":20734},[2939],[363,20736],{"className":20737,"style":2935},[505],[363,20739,20741,20744,20747,20750,20753,20756,20759],{"className":20740},[398],[363,20742],{"className":20743,"style":487},[402],[363,20745,11689],{"className":20746},[407,408],[363,20748,468],{"className":20749},[501],[363,20751,385],{"className":20752,"style":409},[407,408],[363,20754,461],{"className":20755},[494],[363,20757,15750],{"className":20758},[407,408],[363,20760,20762],{"className":20761},[501],")]",[363,20764,20766,20788],{"className":20765,"translate":367},[366],[363,20767,20769],{"className":20768},[371],[373,20770,20771],{"xmlns":375},[377,20772,20773,20785],{},[380,20774,20775,20777,20779,20781,20783],{},[383,20776,16062],{},[383,20778,385],{},[458,20780,461],{"stretchy":460},[383,20782,7918],{},[458,20784,468],{"stretchy":460},[387,20786,20787],{"encoding":389},"\\delta f(w)",[363,20789,20791],{"className":20790,"ariaHidden":394},[393],[363,20792,20794,20797,20800,20803,20806,20809],{"className":20793},[398],[363,20795],{"className":20796,"style":487},[402],[363,20798,16062],{"className":20799,"style":16111},[407,408],[363,20801,385],{"className":20802,"style":409},[407,408],[363,20804,461],{"className":20805},[494],[363,20807,7918],{"className":20808,"style":8622},[407,408],[363,20810,468],{"className":20811},[501],"；非主项部分",[363,20814,20816],{"className":20815,"translate":367},[528],[363,20817,20819,20905],{"className":20818,"translate":367},[366],[363,20820,20822],{"className":20821},[371],[373,20823,20824],{"xmlns":375,"display":537},[377,20825,20826,20902],{},[380,20827,20828,20830,20832,20834,20836,20838,20840,20846,20848,20850,20852,20854,20856,20858,20860,20862,20864,20866,20868,20870,20872,20874,20876,20878,20880,20882,20884,20886,20888,20890,20892,20894,20896,20898,20900],{},[458,20829,461],{"stretchy":460},[463,20831,465],{},[458,20833,5161],{},[383,20835,16062],{},[458,20837,468],{"stretchy":460},[1645,20839,1647],{},[543,20841,20842,20844],{},[383,20843,4506],{},[463,20845,465],{},[383,20847,16062],{},[458,20849,578],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[383,20851,11689],{},[458,20853,461],{"stretchy":460},[463,20855,465],{},[458,20857,5161],{},[383,20859,7909],{},[458,20861,468],{"stretchy":460},[458,20863,2818],{},[458,20865,461],{"stretchy":460},[463,20867,465],{},[458,20869,5161],{},[383,20871,11689],{},[458,20873,468],{"stretchy":460},[458,20875,461],{"stretchy":460},[463,20877,465],{},[458,20879,5161],{},[383,20881,15750],{},[458,20883,468],{"stretchy":460},[458,20885,5161],{},[458,20887,461],{"stretchy":460},[463,20889,465],{},[458,20891,5161],{},[383,20893,7918],{},[458,20895,468],{"stretchy":460},[458,20897,605],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[458,20899,471],{},[463,20901,474],{},[387,20903,20904],{"encoding":389},"(1-\\delta)\\,g_1\\delta\\big[\\lambda(1-s)\n+(1-\\lambda)(1-u)-(1-w)\\big]=0",[363,20906,20908,20929,21012,21033,21054,21081,21102,21123,21150],{"className":20907,"ariaHidden":394},[393],[363,20909,20911,20914,20917,20920,20923,20926],{"className":20910},[398],[363,20912],{"className":20913,"style":487},[402],[363,20915,461],{"className":20916},[494],[363,20918,465],{"className":20919},[407],[363,20921],{"className":20922,"style":2935},[505],[363,20924,5161],{"className":20925},[2939],[363,20927],{"className":20928,"style":2935},[505],[363,20930,20932,20936,20939,20942,20945,20985,20988,20994,20997,21000,21003,21006,21009],{"className":20931},[398],[363,20933],{"className":20934,"style":20935},[402],"height:1.2em;vertical-align:-0.35em;",[363,20937,16062],{"className":20938,"style":16111},[407,408],[363,20940,468],{"className":20941},[501],[363,20943],{"className":20944,"style":754},[505],[363,20946,20948,20951],{"className":20947},[407],[363,20949,4506],{"className":20950,"style":1328},[407,408],[363,20952,20954],{"className":20953},[629],[363,20955,20957,20977],{"className":20956},[633,634],[363,20958,20960,20974],{"className":20959},[638],[363,20961,20963],{"className":20962,"style":11483},[642],[363,20964,20965,20968],{"style":19215},[363,20966],{"className":20967,"style":651},[650],[363,20969,20971],{"className":20970},[655,656,657,658],[363,20972,465],{"className":20973},[407,658],[363,20975,666],{"className":20976},[665],[363,20978,20980],{"className":20979},[638],[363,20981,20983],{"className":20982,"style":8316},[642],[363,20984],{},[363,20986,16062],{"className":20987,"style":16111},[407,408],[363,20989,20991],{"className":20990},[407],[363,20992,578],{"className":20993},[767,10022],[363,20995,11689],{"className":20996},[407,408],[363,20998,461],{"className":20999},[494],[363,21001,465],{"className":21002},[407],[363,21004],{"className":21005,"style":2935},[505],[363,21007,5161],{"className":21008},[2939],[363,21010],{"className":21011,"style":2935},[505],[363,21013,21015,21018,21021,21024,21027,21030],{"className":21014},[398],[363,21016],{"className":21017,"style":487},[402],[363,21019,7909],{"className":21020},[407,408],[363,21022,468],{"className":21023},[501],[363,21025],{"className":21026,"style":2935},[505],[363,21028,2818],{"className":21029},[2939],[363,21031],{"className":21032,"style":2935},[505],[363,21034,21036,21039,21042,21045,21048,21051],{"className":21035},[398],[363,21037],{"className":21038,"style":487},[402],[363,21040,461],{"className":21041},[494],[363,21043,465],{"className":21044},[407],[363,21046],{"className":21047,"style":2935},[505],[363,21049,5161],{"className":21050},[2939],[363,21052],{"className":21053,"style":2935},[505],[363,21055,21057,21060,21063,21066,21069,21072,21075,21078],{"className":21056},[398],[363,21058],{"className":21059,"style":487},[402],[363,21061,11689],{"className":21062},[407,408],[363,21064,468],{"className":21065},[501],[363,21067,461],{"className":21068},[494],[363,21070,465],{"className":21071},[407],[363,21073],{"className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0，因为 ",[363,21163,21165,21198],{"className":21164,"translate":367},[366],[363,21166,21168],{"className":21167},[371],[373,21169,21170],{"xmlns":375},[377,21171,21172,21196],{},[380,21173,21174,21176,21178,21180,21182,21184,21186,21188,21190,21192,21194],{},[383,21175,7918],{},[458,21177,471],{},[383,21179,11689],{},[383,21181,7909],{},[458,21183,2818],{},[458,21185,461],{"stretchy":460},[463,21187,465],{},[458,21189,5161],{},[383,21191,11689],{},[458,21193,468],{"stretchy":460},[383,21195,15750],{},[387,21197,15908],{"encoding":389},[363,21199,21201,21219,21240,21261],{"className":21200,"ariaHidden":394},[393],[363,21202,21204,21207,21210,21213,21216],{"className":21203},[398],[363,21205],{"className":21206,"style":2693},[402],[363,21208,7918],{"className":21209,"style":8622},[407,408],[363,21211],{"className":21212,"style":506},[505],[363,21214,471],{"className":21215},[510],[363,21217],{"className":21218,"style":506},[505],[363,21220,21222,21225,21228,21231,21234,21237],{"className":21221},[398],[363,21223],{"className":21224,"style":15936},[402],[363,21226,11689],{"className":21227},[407,408],[363,21229,7909],{"className":21230},[407,408],[363,21232],{"className":21233,"style":2935},[505],[363,21235,2818],{"className":21236},[2939],[363,21238],{"className":21239,"style":2935},[505],[363,21241,21243,21246,21249,21252,21255,21258],{"className":21242},[398],[363,21244],{"className":21245,"style":487},[402],[363,21247,461],{"className":21248},[494],[363,21250,465],{"className":21251},[407],[363,21253],{"className":21254,"style":2935},[505],[363,21256,5161],{"className":21257},[2939],[363,21259],{"className":21260,"style":2935},[505],[363,21262,21264,21267,21270,21273],{"className":21263},[398],[363,21265],{"className":21266,"style":487},[402],[363,21268,11689],{"className":21269},[407,408],[363,21271,468],{"className":21272},[501],[363,21274,15750],{"className":21275},[407,408],"）。于是不等式两边同除以 ",[363,21278,21280,21298],{"className":21279,"translate":367},[366],[363,21281,21283],{"className":21282},[371],[373,21284,21285],{"xmlns":375},[377,21286,21287,21295],{},[380,21288,21289,21291,21293],{},[383,21290,16062],{},[458,21292,3251],{},[463,21294,474],{},[387,21296,21297],{"encoding":389},"\\delta>0",[363,21299,21301,21319],{"className":21300,"ariaHidden":394},[393],[363,21302,21304,21307,21310,21313,21316],{"className":21303},[398],[363,21305],{"className":21306,"style":11714},[402],[363,21308,16062],{"className":21309,"style":16111},[407,408],[363,21311],{"className":21312,"style":506},[505],[363,21314,3251],{"className":21315},[510],[363,21317],{"className":21318,"style":506},[505],[363,21320,21322,21325],{"className":21321},[398],[363,21323],{"className":21324,"style":520},[402],[363,21326,474],{"className":21327},[407],"，令 ",[363,21330,21332,21354],{"className":21331,"translate":367},[366],[363,21333,21335],{"className":21334},[371],[373,21336,21337],{"xmlns":375},[377,21338,21339,21351],{},[380,21340,21341,21343,21345],{},[383,21342,16062],{},[458,21344,1378],{},[7162,21346,21347,21349],{},[463,21348,474],{},[458,21350,2818],{},[387,21352,21353],{"encoding":389},"\\delta\\to0^+",[363,21355,21357,21375],{"className":21356,"ariaHidden":394},[393],[363,21358,21360,21363,21366,21369,21372],{"className":21359},[398],[363,21361],{"className":21362,"style":12947},[402],[363,21364,16062],{"className":21365,"style":16111},[407,408],[363,21367],{"className":21368,"style":506},[505],[363,21370,1378],{"className":21371},[510],[363,21373],{"className":21374,"style":506},[505],[363,21376,21378,21382],{"className":21377},[398],[363,21379],{"className":21380,"style":21381},[402],"height:0.7713em;",[363,21383,21385,21388],{"className":21384},[407],[363,21386,474],{"className":21387},[407],[363,21389,21391],{"className":21390},[629],[363,21392,21394],{"className":21393},[633],[363,21395,21397],{"className":21396},[638],[363,21398,21400],{"className":21399,"style":21381},[642],[363,21401,21402,21405],{"style":7213},[363,21403],{"className":21404,"style":651},[650],[363,21406,21408],{"className":21407},[655,656,657,658],[363,21409,2818],{"className":21410},[2939,658],"，得",[363,21413,21415],{"className":21414,"translate":367},[528],[363,21416,21418,21488],{"className":21417,"translate":367},[366],[363,21419,21421],{"className":21420},[371],[373,21422,21423],{"xmlns":375,"display":537},[377,21424,21425,21485],{},[380,21426,21427,21429,21431,21433,21435,21437,21439,21441,21443,21445,21447,21449,21451,21453,21455,21457,21459,21461,21463,21465,21467,21469,21471,21473,21475,21477,21479,21481,21483],{},[383,21428,385],{},[458,21430,461],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[383,21432,11689],{},[383,21434,7909],{},[458,21436,2818],{},[458,21438,461],{"stretchy":460},[463,21440,465],{},[458,21442,5161],{},[383,21444,11689],{},[458,21446,468],{"stretchy":460},[383,21448,15750],{},[458,21450,468],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[458,21452,10300],{},[383,21454,11689],{},[383,21456,385],{},[458,21458,461],{"stretchy":460},[383,21460,7909],{},[458,21462,468],{"stretchy":460},[458,21464,2818],{},[458,21466,461],{"stretchy":460},[463,21468,465],{},[458,21470,5161],{},[383,21472,11689],{},[458,21474,468],{"stretchy":460},[383,21476,385],{},[458,21478,461],{"stretchy":460},[383,21480,15750],{},[458,21482,468],{"stretchy":460},[458,21484,898],{"separator":394},[387,21486,21487],{"encoding":389},"f\\big(\\lambda s+(1-\\lambda)u\\big)\n\\le\\lambda f(s)+(1-\\lambda)f(u),",[363,21489,21491,21521,21542,21572,21602,21623],{"className":21490,"ariaHidden":394},[393],[363,21492,21494,21497,21500,21506,21509,21512,21515,21518],{"className":21493},[398],[363,21495],{"className":21496,"style":20935},[402],[363,21498,385],{"className":21499,"style":409},[407,408],[363,21501,21503],{"className":21502},[407],[363,21504,461],{"className":21505},[767,10022],[363,21507,11689],{"className":21508},[407,408],[363,21510,7909],{"className":21511},[407,408],[363,21513],{"className":21514,"style":2935},[505],[363,21516,2818],{"className":21517},[2939],[363,21519],{"className":21520,"style":2935},[505],[363,21522,21524,21527,21530,21533,21536,21539],{"className":21523},[398],[363,21525],{"className":21526,"style":487},[402],[363,21528,461],{"className":21529},[494],[363,21531,465],{"className":21532},[407],[363,21534],{"className":21535,"style":2935},[505],[363,21537,5161],{"className":21538},[2939],[363,21540],{"className":21541,"style":2935},[505],[363,21543,21545,21548,21551,21554,21557,21563,21566,21569],{"className":21544},[398],[363,21546],{"className":21547,"style":20935},[402],[363,21549,11689],{"className":21550},[407,408],[363,21552,468],{"className":21553},[501],[363,21555,15750],{"className":21556},[407,408],[363,21558,21560],{"className":21559},[407],[363,21561,468],{"className":21562},[767,10022],[363,21564],{"className":21565,"style":506},[505],[363,21567,10300],{"className":21568},[510],[363,21570],{"className":21571,"style":506},[505],[363,21573,21575,21578,21581,21584,21587,21590,21593,21596,21599],{"className":21574},[398],[363,21576],{"className":21577,"style":487},[402],[363,21579,11689],{"className":21580},[407,408],[363,21582,385],{"className":21583,"style":409},[407,408],[363,21585,461],{"className":21586},[494],[363,21588,7909],{"className":21589},[407,408],[363,21591,468],{"className":21592},[501],[363,21594],{"className":21595,"style":2935},[505],[363,21597,2818],{"className":21598},[2939],[363,21600],{"className":21601,"style":2935},[505],[363,21603,21605,21608,21611,21614,21617,21620],{"className":21604},[398],[363,21606],{"className":21607,"style":487},[402],[363,21609,461],{"className":21610},[494],[363,21612,465],{"className":21613},[407],[363,21615],{"className":21616,"style":2935},[505],[363,21618,5161],{"className":21619},[2939],[363,21621],{"className":21622,"style":2935},[505],[363,21624,21626,21629,21632,21635,21638,21641,21644,21647],{"className":21625},[398],[363,21627],{"className":21628,"style":487},[402],[363,21630,11689],{"className":21631},[407,408],[363,21633,468],{"className":21634},[501],[363,21636,385],{"className":21637,"style":409},[407,408],[363,21639,461],{"className":21640},[494],[363,21642,15750],{"className":21643},[407,408],[363,21645,468],{"className":21646},[501],[363,21648,898],{"className":21649},[920],[11,21651,21652,21653,11315],{},"即 ",[363,21654,21656,21669],{"className":21655,"translate":367},[366],[363,21657,21659],{"className":21658},[371],[373,21660,21661],{"xmlns":375},[377,21662,21663,21667],{},[380,21664,21665],{},[383,21666,385],{},[387,21668,385],{"encoding":389},[363,21670,21672],{"className":21671,"ariaHidden":394},[393],[363,21673,21675,21678],{"className":21674},[398],[363,21676],{"className":21677,"style":403},[402],[363,21679,385],{"className":21680,"style":409},[407,408],[11,21682,21683,21684,21687,21688,21716,21717,21747,21748,21776],{},"所以\"凸\"与\"数据处理\"是同一件事的两面。这也是 f-散度族成为\"一切满足数据处理不等式的最一般散度\"的原因：它不是一堆例子的偶然汇总，而是一个有刻画定理支撑的完整族。换句话说，Csiszár–Ali–Silvey 定理给出一个完整的",[1344,21685,21686],{},"刻画","：凸函数 ",[363,21689,21691,21704],{"className":21690,"translate":367},[366],[363,21692,21694],{"className":21693},[371],[373,21695,21696],{"xmlns":375},[377,21697,21698,21702],{},[380,21699,21700],{},[383,21701,385],{},[387,21703,385],{"encoding":389},[363,21705,21707],{"className":21706,"ariaHidden":394},[393],[363,21708,21710,21713],{"className":21709},[398],[363,21711],{"className":21712,"style":403},[402],[363,21714,385],{"className":21715,"style":409},[407,408],"（模去仿射项）",[363,21718,21720,21735],{"className":21719,"translate":367},[366],[363,21721,21723],{"className":21722},[371],[373,21724,21725],{"xmlns":375},[377,21726,21727,21732],{},[380,21728,21729],{},[458,21730,21731],{},"↔",[387,21733,21734],{"encoding":389},"\\leftrightarrow",[363,21736,21738],{"className":21737,"ariaHidden":394},[393],[363,21739,21741,21744],{"className":21740},[398],[363,21742],{"className":21743,"style":11353},[402],[363,21745,21731],{"className":21746},[510]," 联合凸泛函 ",[363,21749,21751,21764],{"className":21750,"translate":367},[366],[363,21752,21754],{"className":21753},[371],[373,21755,21756],{"xmlns":375},[377,21757,21758,21762],{},[380,21759,21760],{},[458,21761,21731],{},[387,21763,21734],{"encoding":389},[363,21765,21767],{"className":21766,"ariaHidden":394},[393],[363,21768,21770,21773],{"className":21769},[398],[363,21771],{"className":21772,"style":11353},[402],[363,21774,21731],{"className":21775},[510]," 满足数据处理不等式的分布差异度量，三者一一对应。",[18,21778,21780],{"id":21779},"_3-常见散度","3. 常见散度",[11,21782,21783,21784,21812],{},"不同的 ",[363,21785,21787,21800],{"className":21786,"translate":367},[366],[363,21788,21790],{"className":21789},[371],[373,21791,21792],{"xmlns":375},[377,21793,21794,21798],{},[380,21795,21796],{},[383,21797,385],{},[387,21799,385],{"encoding":389},[363,21801,21803],{"className":21802,"ariaHidden":394},[393],[363,21804,21806,21809],{"className":21805},[398],[363,21807],{"className":21808,"style":403},[402],[363,21810,385],{"className":21811,"style":409},[407,408]," 对应不同的散度。KL、JS、全变差、卡方、Hellinger、Wasserstein、MMD 等散度，表面上都是「衡量两个分布差多少」，但它们的根本区别可以归结为几个维度：",[11,21814,21815],{},"它们对「差异」的定义方式不同，对「支撑集不重叠」的处理不同，对「几何\u002F度量结构」的利用不同，因而导出的拓扑、梯度行为和优化性质也不同。",[11,21817,21818],{},"可以把不同散度的区别归纳为五个核心问题：",[7010,21820,21821,21824,21827,21830,21833],{},[7013,21822,21823],{},"差异定义在什么量上？是密度比值、密度差、累积分布差，还是搬运质量？",[7013,21825,21826],{},"是否对称？",[7013,21828,21829],{},"是否有界？",[7013,21831,21832],{},"是否利用样本空间的几何结构？",[7013,21834,21835],{},"诱导的拓扑强弱如何？",[11,21837,21838],{},"这五点决定了散度在理论性质和实际优化中的行为差异。下面逐个把常用散度「代入模板」并验算。",[2659,21840,21842],{"id":21841},"_31-kl-散度","3.1 KL 散度",[11,21844,21845],{},"取：",[363,21847,21849],{"className":21848,"translate":367},[528],[363,21850,21852,21881],{"className":21851,"translate":367},[366],[363,21853,21855],{"className":21854},[371],[373,21856,21857],{"xmlns":375,"display":537},[377,21858,21859,21879],{},[380,21860,21861,21863,21865,21867,21869,21871,21873,21875,21877],{},[383,21862,385],{},[458,21864,461],{"stretchy":460},[383,21866,1116],{},[458,21868,468],{"stretchy":460},[458,21870,471],{},[383,21872,1116],{},[383,21874,4448],{},[458,21876,4451],{},[383,21878,1116],{},[387,21880,4456],{"encoding":389},[363,21882,21884,21911],{"className":21883,"ariaHidden":394},[393],[363,21885,21887,21890,21893,21896,21899,21902,21905,21908],{"className":21886},[398],[363,21888],{"className":21889,"style":487},[402],[363,21891,385],{"className":21892,"style":409},[407,408],[363,21894,461],{"className":21895},[494],[363,21897,1116],{"className":21898},[407,408],[363,21900,468],{"className":21901},[501],[363,21903],{"className":21904,"style":506},[505],[363,21906,471],{"className":21907},[510],[363,21909],{"className":21910,"style":506},[505],[363,21912,21914,21917,21920,21923,21928,21931],{"className":21913},[398],[363,21915],{"className":21916,"style":403},[402],[363,21918,1116],{"className":21919},[407,408],[363,21921],{"className":21922,"style":754},[505],[363,21924,4502,21926],{"className":21925},[1832],[363,21927,4506],{"style":4505},[363,21929],{"className":21930,"style":754},[505],[363,21932,1116],{"className":21933},[407,408],[11,21935,21936,21937,21970,21971,3093,22036,22169],{},"（下面 ",[363,21938,21940,21956],{"className":21939,"translate":367},[366],[363,21941,21943],{"className":21942},[371],[373,21944,21945],{"xmlns":375},[377,21946,21947,21953],{},[380,21948,21949,21951],{},[383,21950,4448],{},[458,21952,4451],{},[387,21954,21955],{"encoding":389},"\\log",[363,21957,21959],{"className":21958,"ariaHidden":394},[393],[363,21960,21962,21965],{"className":21961},[398],[363,21963],{"className":21964,"style":403},[402],[363,21966,4502,21968],{"className":21967},[1832],[363,21969,4506],{"style":4505}," 均为自然对数。）则 ",[363,21972,21974,21997],{"className":21973,"translate":367},[366],[363,21975,21977],{"className":21976},[371],[373,21978,21979],{"xmlns":375},[377,21980,21981,21995],{},[380,21982,21983,21985,21987,21989,21991,21993],{},[383,21984,385],{},[458,21986,461],{"stretchy":460},[463,21988,465],{},[458,21990,468],{"stretchy":460},[458,21992,471],{},[463,21994,474],{},[387,21996,477],{"encoding":389},[363,21998,22000,22027],{"className":21999,"ariaHidden":394},[393],[363,22001,22003,22006,22009,22012,22015,22018,22021,22024],{"className":22002},[398],[363,22004],{"className":22005,"style":487},[402],[363,22007,385],{"className":22008,"style":409},[407,408],[363,22010,461],{"className":22011},[494],[363,22013,465],{"className":22014},[407],[363,22016,468],{"className":22017},[501],[363,22019],{"className":22020,"style":506},[505],[363,22022,471],{"className":22023},[510],[363,22025],{"className":22026,"style":506},[505],[363,22028,22030,22033],{"className":22029},[398],[363,22031],{"className":22032,"style":520},[402],[363,22034,474],{"className":22035},[407],[363,22037,22039,22079],{"className":22038,"translate":367},[366],[363,22040,22042],{"className":22041},[371],[373,22043,22044],{"xmlns":375},[377,22045,22046,22076],{},[380,22047,22048,22058,22060,22062,22064,22066,22068,22070,22072,22074],{},[7162,22049,22050,22052],{},[383,22051,385],{},[380,22053,22054,22056],{},[458,22055,7168],{"mathvariant":557},[458,22057,7168],{"mathvariant":557},[458,22059,461],{"stretchy":460},[383,22061,1116],{},[458,22063,468],{"stretchy":460},[458,22065,471],{},[463,22067,465],{},[383,22069,1288],{"mathvariant":557},[383,22071,1116],{},[458,22073,3251],{},[463,22075,474],{},[387,22077,22078],{"encoding":389},"f''(t)=1\u002Ft>0",[363,22080,22082,22139,22160],{"className":22081,"ariaHidden":394},[393],[363,22083,22085,22088,22121,22124,22127,22130,22133,22136],{"className":22084},[398],[363,22086],{"className":22087,"style":7191},[402],[363,22089,22091,22094],{"className":22090},[407],[363,22092,385],{"className":22093,"style":409},[407,408],[363,22095,22097],{"className":22096},[629],[363,22098,22100],{"className":22099},[633],[363,22101,22103],{"className":22102},[638],[363,22104,22106],{"className":22105,"style":7210},[642],[363,22107,22108,22111],{"style":7213},[363,22109],{"className":22110,"style":651},[650],[363,22112,22114],{"className":22113},[655,656,657,658],[363,22115,22117],{"className":22116},[407,658],[363,22118,22120],{"className":22119},[407,658],"′′",[363,22122,461],{"className":22123},[494],[363,22125,1116],{"className":22126},[407,408],[363,22128,468],{"className":22129},[501],[363,22131],{"className":22132,"style":506},[505],[363,22134,471],{"className":22135},[510],[363,22137],{"className":22138,"style":506},[505],[363,22140,22142,22145,22148,22151,22154,22157],{"className":22141},[398],[363,22143],{"className":22144,"style":487},[402],[363,22146,8937],{"className":22147},[407],[363,22149,1116],{"className":22150},[407,408],[363,22152],{"className":22153,"style":506},[505],[363,22155,3251],{"className":22156},[510],[363,22158],{"className":22159,"style":506},[505],[363,22161,22163,22166],{"className":22162},[398],[363,22164],{"className":22165,"style":520},[402],[363,22167,474],{"className":22168},[407]," 凸。代入模板：",[363,22171,22173],{"className":22172,"translate":367},[528],[363,22174,22176,22279],{"className":22175,"translate":367},[366],[363,22177,22179],{"className":22178},[371],[373,22180,22181],{"xmlns":375,"display":537},[377,22182,22183,22276],{},[380,22184,22185,22191,22193,22195,22197,22199,22201,22203,22205,22207,22209,22215,22217,22219,22225,22227,22229,22231,22233,22235,22237,22239,22241,22247,22249,22251,22253,22255,22266,22268,22270,22272,22274],{},[543,22186,22187,22189],{},[383,22188,547],{},[383,22190,385],{},[458,22192,461],{"stretchy":460},[383,22194,554],{},[383,22196,558],{"mathvariant":557},[383,22198,561],{},[458,22200,468],{"stretchy":460},[458,22202,471],{},[458,22204,1635],{},[383,22206,1291],{},[458,22208,3647],{},[586,22210,22211,22213],{},[383,22212,11],{},[383,22214,1291],{},[383,22216,4448],{},[458,22218,4451],{},[586,22220,22221,22223],{},[383,22222,11],{},[383,22224,1291],{},[1645,22226,1647],{},[383,22228,592],{},[383,22230,1121],{},[458,22232,471],{},[458,22234,1635],{},[383,22236,11],{},[383,22238,4448],{},[458,22240,4451],{},[586,22242,22243,22245],{},[383,22244,11],{},[383,22246,1291],{},[1645,22248,1647],{},[383,22250,592],{},[383,22252,1121],{},[458,22254,471],{},[543,22256,22257,22259],{},[383,22258,547],{},[380,22260,22261,22263],{},[383,22262,11044],{},[383,22264,22265],{},"L",[458,22267,461],{"stretchy":460},[383,22269,554],{},[383,22271,558],{"mathvariant":557},[383,22273,561],{},[458,22275,468],{"stretchy":460},[387,22277,22278],{"encoding":389},"D_f(P\\|Q)\n=\\int q\\cdot \\frac{p}{q}\\log\\frac{p}{q}\\,dx\n=\\int p\\log\\frac{p}{q}\\,dx\n=D_{KL}(P\\|Q)",[363,22280,22282,22352,22376,22536,22643],{"className":22281,"ariaHidden":394},[393],[363,22283,22285,22288,22328,22331,22334,22337,22340,22343,22346,22349],{"className":22284},[398],[363,22286],{"className":22287,"style":618},[402],[363,22289,22291,22294],{"className":22290},[407],[363,22292,547],{"className":22293,"style":625},[407,408],[363,22295,22297],{"className":22296},[629],[363,22298,22300,22320],{"className":22299},[633,634],[363,22301,22303,22317],{"className":22302},[638],[363,22304,22306],{"className":22305,"style":643},[642],[363,22307,22308,22311],{"style":646},[363,22309],{"className":22310,"style":651},[650],[363,22312,22314],{"className":22313},[655,656,657,658],[363,22315,385],{"className":22316,"style":409},[407,408,658],[363,22318,666],{"className":22319},[665],[363,22321,22323],{"className":22322},[638],[363,22324,22326],{"className":22325,"style":673},[642],[363,22327],{},[363,22329,461],{"className":22330},[494],[363,22332,554],{"className":22333,"style":682},[407,408],[363,22335,558],{"className":22336},[407],[363,22338,561],{"className":22339},[407,408],[363,22341,468],{"className":22342},[501],[363,22344],{"className":22345,"style":506},[505],[363,22347,471],{"className":22348},[510],[363,22350],{"className":22351,"style":506},[505],[363,22353,22355,22358,22361,22364,22367,22370,22373],{"className":22354},[398],[363,22356],{"className":22357,"style":6125},[402],[363,22359,1635],{"className":22360,"style":1835},[1832,1833,1834],[363,22362],{"className":22363,"style":754},[505],[363,22365,1291],{"className":22366,"style":1328},[407,408],[363,22368],{"className":22369,"style":2935},[505],[363,22371,3647],{"className":22372},[2939],[363,22374],{"className":22375,"style":2935},[505],[363,22377,22379,22383,22445,22448,22453,22456,22518,22521,22524,22527,22530,22533],{"className":22378},[398],[363,22380],{"className":22381,"style":22382},[402],"height:1.988em;vertical-align:-0.8804em;",[363,22384,22386,22389,22442],{"className":22385},[407],[363,22387],{"className":22388},[494,789],[363,22390,22392],{"className":22391},[586],[363,22393,22395,22434],{"className":22394},[633,634],[363,22396,22398,22431],{"className":22397},[638],[363,22399,22401,22412,22420],{"className":22400,"style":3902},[642],[363,22402,22403,22406],{"style":805},[363,22404],{"className":22405,"style":809},[650],[363,22407,22409],{"className":22408},[407],[363,22410,1291],{"className":22411,"style":1328},[407,408],[363,22413,22414,22417],{"style":821},[363,22415],{"className":22416,"style":809},[650],[363,22418],{"className":22419,"style":829},[828],[363,22421,22422,22425],{"style":832},[363,22423],{"className":22424,"style":809},[650],[363,22426,22428],{"className":22427},[407],[363,22429,11],{"className":22430},[407,408],[363,22432,666],{"className":22433},[665],[363,22435,22437],{"className":22436},[638],[363,22438,22440],{"className":22439,"style":854},[642],[363,22441],{},[363,22443],{"className":22444},[501,789],[363,22446],{"className":22447,"style":754},[505],[363,22449,4502,22451],{"className":22450},[1832],[363,22452,4506],{"style":4505},[363,22454],{"className":22455,"style":754},[505],[363,22457,22459,22462,22515],{"className":22458},[407],[363,22460],{"className":22461},[494,789],[363,22463,22465],{"className":22464},[586],[363,22466,22468,22507],{"className":22467},[633,634],[363,22469,22471,22504],{"className":22470},[638],[363,22472,22474,22485,22493],{"className":22473,"style":3902},[642],[363,22475,22476,22479],{"style":805},[363,22477],{"className":22478,"style":809},[650],[363,22480,22482],{"className":22481},[407],[363,22483,1291],{"className":22484,"style":1328},[407,408],[363,22486,22487,22490],{"style":821},[363,22488],{"className":22489,"style":809},[650],[363,22491],{"className":22492,"style":829},[828],[363,22494,22495,22498],{"style":832},[363,22496],{"className":22497,"style":809},[650],[363,22499,22501],{"className":22500},[407],[363,22502,11],{"className":22503},[407,408],[363,22505,666],{"className":22506},[665],[363,22508,22510],{"className":22509},[638],[363,22511,22513],{"className":22512,"style":854},[642],[363,22514],{},[363,22516],{"className":22517},[501,789],[363,22519],{"className":22520,"style":754},[505],[363,22522,592],{"className":22523},[407,408],[363,22525,1121],{"className":22526},[407,408],[363,22528],{"className":22529,"style":506},[505],[363,22531,471],{"className":22532},[510],[363,22534],{"className":22535,"style":506},[505],[363,22537,22539,22543,22546,22549,22552,22555,22560,22563,22625,22628,22631,22634,22637,22640],{"className":22538},[398],[363,22540],{"className":22541,"style":22542},[402],"height:2.2404em;vertical-align:-0.8804em;",[363,22544,1635],{"className":22545,"style":1835},[1832,1833,1834],[363,22547],{"className":22548,"style":754},[505],[363,22550,11],{"className":22551},[407,408],[363,22553],{"className":22554,"style":754},[505],[363,22556,4502,22558],{"className":22557},[1832],[363,22559,4506],{"style":4505}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KL 散度 ",[363,22714,22716,22748],{"className":22715,"translate":367},[366],[363,22717,22719],{"className":22718},[371],[373,22720,22721],{"xmlns":375},[377,22722,22723,22745],{},[380,22724,22725,22735,22737,22739,22741,22743],{},[543,22726,22727,22729],{},[383,22728,547],{},[380,22730,22731,22733],{},[383,22732,11044],{},[383,22734,22265],{},[458,22736,461],{"stretchy":460},[383,22738,561],{},[383,22740,558],{"mathvariant":557},[383,22742,554],{},[458,22744,468],{"stretchy":460},[387,22746,22747],{"encoding":389},"D_{KL}(Q\\|P)",[363,22749,22751],{"className":22750,"ariaHidden":394},[393],[363,22752,22754,22757,22803,22806,22809,22812,22815],{"className":22753},[398],[363,22755],{"className":22756,"style":487},[402],[363,22758,22760,22763],{"className":22759},[407],[363,22761,547],{"className":22762,"style":625},[407,408],[363,22764,22766],{"className":22765},[629],[363,22767,22769,22795],{"className":22768},[633,634],[363,22770,22772,22792],{"className":22771},[638],[363,22773,22775],{"className":22774,"style":727},[642],[363,22776,22777,22780],{"style":646},[363,22778],{"className":22779,"style":651},[650],[363,22781,22783],{"className":22782},[655,656,657,658],[363,22784,22786,22789],{"className":22785},[407,658],[363,22787,11044],{"className":22788,"style":11132},[407,408,658],[363,22790,22265],{"className":22791},[407,408,658],[363,22793,666],{"className":22794},[665],[363,22796,22798],{"className":22797},[638],[363,22799,22801],{"className":22800,"style":8316},[642],[363,22802],{},[363,22804,461],{"className":22805},[494],[363,22807,561],{"className":22808},[407,408],[363,22810,558],{"className":22811},[407],[363,22813,554],{"className":22814,"style":682},[407,408],[363,22816,468],{"className":22817},[501]," 对应斜共轭 ",[363,22820,22822,22848],{"className":22821,"translate":367},[366],[363,22823,22825],{"className":22824},[371],[373,22826,22827],{"xmlns":375},[377,22828,22829,22845],{},[380,22830,22831,22833,22835,22837,22839,22841,22843],{},[383,22832,1116],{},[383,22834,385],{},[458,22836,461],{"stretchy":460},[463,22838,465],{},[383,22840,1288],{"mathvariant":557},[383,22842,1116],{},[458,22844,468],{"stretchy":460},[387,22846,22847],{"encoding":389},"t f(1\u002Ft)",[363,22849,22851],{"className":22850,"ariaHidden":394},[393],[363,22852,22854,22857,22860,22863,22866,22869,22872],{"className":22853},[398],[363,22855],{"className":22856,"style":487},[402],[363,22858,1116],{"className":22859},[407,408],[363,22861,385],{"className":22862,"style":409},[407,408],[363,22864,461],{"className":22865},[494],[363,22867,8937],{"className":22868},[407],[363,22870,1116],{"className":22871},[407,408],[363,22873,468],{"className":22874},[501],[363,22876,22878],{"className":22877,"translate":367},[528],[363,22879,22881,23003],{"className":22880,"translate":367},[366],[363,22882,22884],{"className":22883},[371],[373,22885,22886],{"xmlns":375,"display":537},[377,22887,22888,23000],{},[1591,22889,22890,22924],{"rowspacing":1593,"columnalign":1594,"columnspacing":1595},[1597,22891,22892,22906],{},[1600,22893,22894],{},[1603,22895,22896],{"scriptlevel":474,"displaystyle":394},[380,22897,22898,22900,22902,22904],{},[383,22899,385],{},[458,22901,461],{"stretchy":460},[383,22903,1116],{},[458,22905,468],{"stretchy":460},[1600,22907,22908],{},[1603,22909,22910],{"scriptlevel":474,"displaystyle":394},[380,22911,22912,22914,22916,22918,22920,22922],{},[380,22913],{},[458,22915,471],{},[383,22917,1116],{},[383,22919,4448],{},[458,22921,4451],{},[383,22923,1116],{},[1597,22925,22926,22932],{},[1600,22927,22928],{},[1603,22929,22930],{"scriptlevel":474,"displaystyle":394},[380,22931],{},[1600,22933,22934],{},[1603,22935,22936],{"scriptlevel":474,"displaystyle":394},[380,22937,22938,22940,22942,22944,22960,22962,22964,22966,22968,22970,22972,22978,22980,22982,22988,22990,22992,22994,22996,22998],{},[380,22939],{},[458,22941,4634],{},[1645,22943,4631],{},[7901,22945,22946,22948,22958],{},[383,22947,385],{},[380,22949,22950,22952,22954,22956],{},[383,22951,7909],{},[383,22953,7912],{},[383,22955,7915],{},[383,22957,7918],{},[458,22959,7921],{"lspace":1595,"rspace":1595},[458,22961,461],{"stretchy":460},[383,22963,1116],{},[458,22965,468],{"stretchy":460},[458,22967,471],{},[383,22969,1116],{},[458,22971,3647],{},[586,22973,22974,22976],{},[463,22975,465],{},[383,22977,1116],{},[383,22979,4448],{},[458,22981,4451],{},[586,22983,22984,22986],{},[463,22985,465],{},[383,22987,1116],{},[458,22989,471],{},[458,22991,5161],{},[383,22993,4448],{},[458,22995,4451],{},[383,22997,1116],{},[458,22999,898],{"separator":394},[387,23001,23002],{"encoding":389},"\\begin{aligned}\nf(t)&=t\\log t\\\\[2pt]\n&\\Longrightarrow\\ f^{*}_{skew}(t)\n=t\\cdot\\frac1t\\log\\frac1t=-\\log t,\n\\end{aligned}",[363,23004,23006],{"className":23005,"ariaHidden":394},[393],[363,23007,23009,23013],{"className":23008},[398],[363,23010],{"className":23011,"style":23012},[402],"height:3.7074em;vertical-align:-1.6037em;",[363,23014,23016],{"className":23015},[407],[363,23017,23019,23075],{"className":23018},[1591],[363,23020,23022],{"className":23021},[1708],[363,23023,23025,23066],{"className":23024},[633,634],[363,23026,23028,23063],{"className":23027},[638],[363,23029,23032,23054],{"className":23030,"style":23031},[642],"height:2.1037em;",[363,23033,23035,23039],{"style":23034},"top:-4.5852em;",[363,23036],{"className":23037,"style":23038},[650],"height:3.3214em;",[363,23040,23042,23045,23048,23051],{"className":23041},[407],[363,23043,385],{"className":23044,"style":409},[407,408],[363,23046,461],{"className":23047},[494],[363,23049,1116],{"className":23050},[407,408],[363,23052,468],{"className":23053},[501],[363,23055,23057,23060],{"style":23056},"top:-2.4037em;",[363,23058],{"className":23059,"style":23038},[650],[363,23061],{"className":23062},[407],[363,23064,666],{"className":23065},[665],[363,23067,23069],{"className":23068},[638],[363,23070,23073],{"className":23071,"style":23072},[642],"height:1.6037em;",[363,23074],{},[363,23076,23078],{"className":23077},[1799],[363,23079,23081,23411],{"className":23080},[633,634],[363,23082,23084,23408],{"className":23083},[638],[363,23085,23087,23124],{"className":23086,"style":23031},[642],[363,23088,23089,23092],{"style":23034},[363,23090],{"className":23091,"style":23038},[650],[363,23093,23095,23098,23101,23104,23107,23110,23113,23118,23121],{"className":23094},[407],[363,23096],{"className":23097},[407],[363,23099],{"className":23100,"style":506},[505],[363,23102,471],{"className":23103},[510],[363,23105],{"className":23106,"style":506},[505],[363,23108,1116],{"className":23109},[407,408],[363,23111],{"className":23112,"style":754},[505],[363,23114,4502,23116],{"className":23115},[1832],[363,23117,4506],{"style":4505},[363,23119],{"className":23120,"style":754},[505],[363,23122,1116],{"className":23123},[407,408],[363,23125,23126,23129],{"style":23056},[363,23127],{"className":23128,"style":23038},[650],[363,23130,23132,23135,23138,23141,23144,23147,23213,23216,23219,23222,23225,23228,23231,23234,23237,23240,23243,23306,23309,23314,23317,23379,23382,23385,23388,23391,23394,23399,23402,23405],{"className":23131},[407],[363,23133],{"className":23134},[407],[363,23136],{"className":23137,"style":506},[505],[363,23139,4634],{"className":23140},[510],[363,23142,4631],{"className":23143},[505],[363,23145],{"className":23146,"style":506},[505],[363,23148,23150,23153],{"className":23149},[407],[363,23151,385],{"className":23152,"style":409},[407,408],[363,23154,23156],{"className":23155},[629],[363,23157,23159,23205],{"className":23158},[633,634],[363,23160,23162,23202],{"className":23161},[638],[363,23163,23165,23188],{"className":23164,"style":8596},[642],[363,23166,23167,23170],{"style":8599},[363,23168],{"className":23169,"style":651},[650],[363,23171,23173],{"className":23172},[655,656,657,658],[363,23174,23176,23179,23182,23185],{"className":23175},[407,658],[363,23177,7909],{"className":23178},[407,408,658],[363,23180,7912],{"className":23181,"style":8615},[407,408,658],[363,23183,7915],{"className":23184},[407,408,658],[363,23186,7918],{"className":23187,"style":8622},[407,408,658],[363,23189,23190,23193],{"style":8625},[363,23191],{"className":23192,"style":651},[650],[363,23194,23196],{"className":23195},[655,656,657,658],[363,23197,23199],{"className":23198},[407,658],[363,23200,7921],{"className":23201},[407,658],[363,23203,666],{"className":23204},[665],[363,23206,23208],{"className":23207},[638],[363,23209,23211],{"className":23210,"style":8647},[642],[363,23212],{},[363,23214,461],{"className":23215},[494],[363,23217,1116],{"className":23218},[407,408],[363,23220,468],{"className":23221},[501],[363,23223],{"className":23224,"style":506},[505],[363,23226,471],{"className":23227},[510],[363,23229],{"className":23230,"style":506},[505],[363,23232,1116],{"className":23233},[407,408],[363,23235],{"className":23236,"style":2935},[505],[363,23238,3647],{"className":23239},[2939],[363,23241],{"className":23242,"style":2935},[505],[363,23244,23246,23249,23303],{"className":23245},[407],[363,23247],{"className":23248},[494,789],[363,23250,23252],{"className":23251},[586],[363,23253,23255,23295],{"className":23254},[633,634],[363,23256,23258,23292],{"className":23257},[638],[363,23259,23262,23273,23281],{"className":23260,"style":23261},[642],"height:1.3214em;",[363,23263,23264,23267],{"style":805},[363,23265],{"className":23266,"style":809},[650],[363,23268,23270],{"className":23269},[407],[363,23271,1116],{"className":23272},[407,408],[363,23274,23275,23278],{"style":821},[363,23276],{"className":23277,"style":809},[650],[363,23279],{"className":23280,"style":829},[828],[363,23282,23283,23286],{"style":832},[363,23284],{"className":23285,"style":809},[650],[363,23287,23289],{"className":23288},[407],[363,23290,465],{"className":23291},[407],[363,23293,666],{"className":23294},[665],[363,23296,23298],{"className":23297},[638],[363,23299,23301],{"className":23300,"style":4126},[642],[363,23302],{},[363,23304],{"className":23305},[501,789],[363,23307],{"className":23308,"style":754},[505],[363,23310,4502,23312],{"className":23311},[1832],[363,23313,4506],{"style":4505},[363,23315],{"className":23316,"style":754},[505],[363,23318,23320,23323,23376],{"className":23319},[407],[363,23321],{"className":23322},[494,789],[363,23324,23326],{"className":23325},[586],[363,23327,23329,23368],{"className":23328},[633,634],[363,23330,23332,23365],{"className":23331},[638],[363,23333,23335,23346,23354],{"className":23334,"style":23261},[642],[363,23336,23337,23340],{"style":805},[363,23338],{"className":23339,"style":809},[650],[363,23341,23343],{"className":23342},[407],[363,23344,1116],{"className":23345},[407,408],[363,23347,23348,23351],{"style":821},[363,23349],{"className":23350,"style":809},[650],[363,23352],{"className":23353,"style":829},[828],[363,23355,23356,23359],{"style":832},[363,23357],{"className":23358,"style":809},[650],[363,23360,23362],{"className":23361},[407],[363,23363,465],{"className":23364},[407],[363,23366,666],{"className":23367},[665],[363,23369,23371],{"className":23370},[638],[363,23372,23374],{"className":23373,"style":4126},[642],[363,23375],{},[363,23377],{"className":23378},[501,789],[363,23380],{"className":23381,"style":506},[505],[363,23383,471],{"className":23384},[510],[363,23386],{"className":23387,"style":506},[505],[363,23389,5161],{"className":23390},[407],[363,23392],{"className":23393,"style":754},[505],[363,23395,4502,23397],{"className":23396},[1832],[363,23398,4506],{"style":4505},[363,23400],{"className":23401,"style":754},[505],[363,23403,1116],{"className":23404},[407,408],[363,23406,898],{"className":23407},[920],[363,23409,666],{"className":23410},[665],[363,23412,23414],{"className":23413},[638],[363,23415,23417],{"className":23416,"style":23072},[642],[363,23418],{},[11,23420,23421],{},"所以",[363,23423,23425],{"className":23424,"translate":367},[528],[363,23426,23428,23504],{"className":23427,"translate":367},[366],[363,23429,23431],{"className":23430},[371],[373,23432,23433],{"xmlns":375,"display":537},[377,23434,23435,23501],{},[380,23436,23437,23439,23441,23443,23445,23447,23449,23451,23453,23455,23457,23459,23461,23467,23469,23471,23473,23475,23477,23479,23489,23491,23493,23495,23497,23499],{},[383,23438,385],{},[458,23440,461],{"stretchy":460},[383,23442,1116],{},[458,23444,468],{"stretchy":460},[458,23446,471],{},[458,23448,5161],{},[383,23450,4448],{},[458,23452,4451],{},[383,23454,1116],{},[1645,23456,4631],{},[458,23458,4634],{},[1645,23460,4631],{},[543,23462,23463,23465],{},[383,23464,547],{},[383,23466,385],{},[458,23468,461],{"stretchy":460},[383,23470,554],{},[383,23472,558],{"mathvariant":557},[383,23474,561],{},[458,23476,468],{"stretchy":460},[458,23478,471],{},[543,23480,23481,23483],{},[383,23482,547],{},[380,23484,23485,23487],{},[383,23486,11044],{},[383,23488,22265],{},[458,23490,461],{"stretchy":460},[383,23492,561],{},[383,23494,558],{"mathvariant":557},[383,23496,554],{},[458,23498,468],{"stretchy":460},[383,23500,4723],{"mathvariant":557},[387,23502,23503],{"encoding":389},"f(t)=-\\log t\n\\ \\Longrightarrow\\ D_f(P\\|Q)=D_{KL}(Q\\|P).",[363,23505,23507,23534,23572,23642],{"className":23506,"ariaHidden":394},[393],[363,23508,23510,23513,23516,23519,23522,23525,23528,23531],{"className":23509},[398],[363,23511],{"className":23512,"style":487},[402],[363,23514,385],{"className":23515,"style":409},[407,408],[363,23517,461],{"className":23518},[494],[363,23520,1116],{"className":23521},[407,408],[363,23523,468],{"className":23524},[501],[363,23526],{"className":23527,"style":506},[505],[363,23529,471],{"className":23530},[510],[363,23532],{"className":23533,"style":506},[505],[363,23535,23537,23540,23543,23546,23551,23554,23557,23560,23563,23566,23569],{"className":23536},[398],[363,23538],{"className":23539,"style":403},[402],[363,23541,5161],{"className":23542},[407],[363,23544],{"className":23545,"style":754},[505],[363,23547,4502,23549],{"className":23548},[1832],[363,23550,4506],{"style":4505},[363,23552],{"className":23553,"style":754},[505],[363,23555,1116],{"className":23556},[407,408],[363,23558,4631],{"className":23559},[505],[363,23561],{"className":23562,"style":506},[505],[363,23564,4634],{"className":23565},[510],[363,23567,4631],{"className":23568},[505],[363,23570],{"className":23571,"style":506},[505],[363,23573,23575,23578,23618,23621,23624,23627,23630,23633,23636,23639],{"className":23574},[398],[363,23576],{"className":23577,"style":618},[402],[363,23579,23581,23584],{"className":23580},[407],[363,23582,547],{"className":23583,"style":625},[407,408],[363,23585,23587],{"className":23586},[629],[363,23588,23590,23610],{"className":23589},[633,634],[363,23591,23593,23607],{"className":23592},[638],[363,23594,23596],{"className":23595,"style":643},[642],[363,23597,23598,23601],{"style":646},[363,23599],{"className":23600,"style":651},[650],[363,23602,23604],{"className":23603},[655,656,657,658],[363,23605,385],{"className":23606,"style":409},[407,408,658],[363,23608,666],{"className":23609},[665],[363,23611,23613],{"className":23612},[638],[363,23614,23616],{"className":23615,"style":673},[642],[363,23617],{},[363,23619,461],{"className":23620},[494],[363,23622,554],{"className":23623,"style":682},[407,408],[363,23625,558],{"className":23626},[407],[363,23628,561],{"className":23629},[407,408],[363,23631,468],{"className":23632},[501],[363,23634],{"className":23635,"style":506},[505],[363,23637,471],{"className":23638},[510],[363,23640],{"className":23641,"style":506},[505],[363,23643,23645,23648,23694,23697,23700,23703,23706,23709],{"className":23644},[398],[363,23646],{"className":23647,"style":487},[402],[363,23649,23651,23654],{"className":23650},[407],[363,23652,547],{"className":23653,"style":625},[407,408],[363,23655,23657],{"className":23656},[629],[363,23658,23660,23686],{"className":23659},[633,634],[363,23661,23663,23683],{"className":23662},[638],[363,23664,23666],{"className":23665,"style":727},[642],[363,23667,23668,23671],{"style":646},[363,23669],{"className":23670,"style":651},[650],[363,23672,23674],{"className":23673},[655,656,657,658],[363,23675,23677,23680],{"className":23676},[407,658],[363,23678,11044],{"className":23679,"style":11132},[407,408,658],[363,23681,22265],{"className":23682},[407,408,658],[363,23684,666],{"className":23685},[665],[363,23687,23689],{"className":23688},[638],[363,23690,23692],{"className":23691,"style":8316},[642],[363,23693],{},[363,23695,461],{"className":23696},[494],[363,23698,561],{"className":23699},[407,408],[363,23701,558],{"className":23702},[407],[363,23704,554],{"className":23705,"style":682},[407,408],[363,23707,468],{"className":23708},[501],[363,23710,4723],{"className":23711},[407],[11,23713,23714,23717,23718,23836,23837,23902,23903,23996],{},[1344,23715,23716],{},"对称化：Jeffreys 散度。"," 取 ",[363,23719,23721,23759],{"className":23720,"translate":367},[366],[363,23722,23724],{"className":23723},[371],[373,23725,23726],{"xmlns":375},[377,23727,23728,23756],{},[380,23729,23730,23732,23734,23736,23738,23740,23742,23744,23746,23748,23750,23752,23754],{},[383,23731,385],{},[458,23733,461],{"stretchy":460},[383,23735,1116],{},[458,23737,468],{"stretchy":460},[458,23739,471],{},[458,23741,461],{"stretchy":460},[383,23743,1116],{},[458,23745,5161],{},[463,23747,465],{},[458,23749,468],{"stretchy":460},[383,23751,4448],{},[458,23753,4451],{},[383,23755,1116],{},[387,23757,23758],{"encoding":389},"f(t)=(t-1)\\log t",[363,23760,23762,23789,23810],{"className":23761,"ariaHidden":394},[393],[363,23763,23765,23768,23771,23774,23777,23780,23783,23786],{"className":23764},[398],[363,23766],{"className":23767,"style":487},[402],[363,23769,385],{"className":23770,"style":409},[407,408],[363,23772,461],{"className":23773},[494],[363,23775,1116],{"className":23776},[407,408],[363,23778,468],{"className":23779},[501],[363,23781],{"className":23782,"style":506},[505],[363,23784,471],{"className":23785},[510],[363,23787],{"className":23788,"style":506},[505],[363,23790,23792,23795,23798,23801,23804,23807],{"className":23791},[398],[363,23793],{"className":23794,"style":487},[402],[363,23796,461],{"className":23797},[494],[363,23799,1116],{"className":23800},[407,408],[363,23802],{"className":23803,"style":2935},[505],[363,23805,5161],{"className":23806},[2939],[363,23808],{"className":23809,"style":2935},[505],[363,23811,23813,23816,23819,23822,23825,23830,23833],{"className":23812},[398],[363,23814],{"className":23815,"style":487},[402],[363,23817,465],{"className":23818},[407],[363,23820,468],{"className":23821},[501],[363,23823],{"className":23824,"style":754},[505],[363,23826,4502,23828],{"className":23827},[1832],[363,23829,4506],{"style":4505},[363,23831],{"className":23832,"style":754},[505],[363,23834,1116],{"className":23835},[407,408],"（它满足 ",[363,23838,23840,23863],{"className":23839,"translate":367},[366],[363,23841,23843],{"className":23842},[371],[373,23844,23845],{"xmlns":375},[377,23846,23847,23861],{},[380,23848,23849,23851,23853,23855,23857,23859],{},[383,23850,385],{},[458,23852,461],{"stretchy":460},[463,23854,465],{},[458,23856,468],{"stretchy":460},[458,23858,471],{},[463,23860,474],{},[387,23862,477],{"encoding":389},[363,23864,23866,23893],{"className":23865,"ariaHidden":394},[393],[363,23867,23869,23872,23875,23878,23881,23884,23887,23890],{"className":23868},[398],[363,23870],{"className":23871,"style":487},[402],[363,23873,385],{"className":23874,"style":409},[407,408],[363,23876,461],{"className":23877},[494],[363,23879,465],{"className":23880},[407],[363,23882,468],{"className":23883},[501],[363,23885],{"className":23886,"style":506},[505],[363,23888,471],{"className":23889},[510],[363,23891],{"className":23892,"style":506},[505],[363,23894,23896,23899],{"className":23895},[398],[363,23897],{"className":23898,"style":520},[402],[363,23900,474],{"className":23901},[407],"，且 ",[363,23904,23906,23942],{"className":23905,"translate":367},[366],[363,23907,23909],{"className":23908},[371],[373,23910,23911],{"xmlns":375},[377,23912,23913,23939],{},[380,23914,23915,23917,23919,23921,23923,23925,23927,23929,23931,23933,23935,23937],{},[383,23916,1116],{},[383,23918,385],{},[458,23920,461],{"stretchy":460},[463,23922,465],{},[383,23924,1288],{"mathvariant":557},[383,23926,1116],{},[458,23928,468],{"stretchy":460},[458,23930,471],{},[383,23932,385],{},[458,23934,461],{"stretchy":460},[383,23936,1116],{},[458,23938,468],{"stretchy":460},[387,23940,23941],{"encoding":389},"t f(1\u002Ft)=f(t)",[363,23943,23945,23978],{"className":23944,"ariaHidden":394},[393],[363,23946,23948,23951,23954,23957,23960,23963,23966,23969,23972,23975],{"className":23947},[398],[363,23949],{"className":23950,"style":487},[402],[363,23952,1116],{"className":23953},[407,408],[363,23955,385],{"className":23956,"style":409},[407,408],[363,23958,461],{"className":23959},[494],[363,23961,8937],{"className":23962},[407],[363,23964,1116],{"className":23965},[407,408],[363,23967,468],{"className":23968},[501],[363,23970],{"className":23971,"style":506},[505],[363,23973,471],{"className":23974},[510],[363,23976],{"className":23977,"style":506},[505],[363,23979,23981,23984,23987,23990,23993],{"className":23980},[398],[363,23982],{"className":23983,"style":487},[402],[363,23985,385],{"className":23986,"style":409},[407,408],[363,23988,461],{"className":23989},[494],[363,23991,1116],{"className":23992},[407,408],[363,23994,468],{"className":23995},[501],"），则",[363,23998,24000],{"className":23999,"translate":367},[528],[363,24001,24003,24131],{"className":24002,"translate":367},[366],[363,24004,24006],{"className":24005},[371],[373,24007,24008],{"xmlns":375,"display":537},[377,24009,24010,24128],{},[380,24011,24012,24018,24020,24022,24024,24026,24028,24030,24032,24034,24036,24038,24040,24042,24044,24046,24052,24054,24056,24058,24061,24063,24073,24075,24077,24079,24081,24083,24085,24095,24097,24099,24101,24103,24105,24107,24109,24116,24118,24120,24122,24124,24126],{},[543,24013,24014,24016],{},[383,24015,547],{},[383,24017,385],{},[458,24019,461],{"stretchy":460},[383,24021,554],{},[383,24023,558],{"mathvariant":557},[383,24025,561],{},[458,24027,468],{"stretchy":460},[458,24029,471],{},[458,24031,1635],{},[458,24033,461],{"stretchy":460},[383,24035,11],{},[458,24037,5161],{},[383,24039,1291],{},[458,24041,468],{"stretchy":460},[383,24043,4448],{},[458,24045,4451],{},[586,24047,24048,24050],{},[383,24049,11],{},[383,24051,1291],{},[1645,24053,1647],{},[383,24055,592],{},[383,24057,1121],{},[505,24059],{"linebreak":24060},"newline",[458,24062,471],{},[543,24064,24065,24067],{},[383,24066,547],{},[380,24068,24069,24071],{},[383,24070,11044],{},[383,24072,22265],{},[458,24074,461],{"stretchy":460},[383,24076,554],{},[383,24078,558],{"mathvariant":557},[383,24080,561],{},[458,24082,468],{"stretchy":460},[458,24084,2818],{},[543,24086,24087,24089],{},[383,24088,547],{},[380,24090,24091,24093],{},[383,24092,11044],{},[383,24094,22265],{},[458,24096,461],{"stretchy":460},[383,24098,561],{},[383,24100,558],{"mathvariant":557},[383,24102,554],{},[458,24104,468],{"stretchy":460},[458,24106,471],{},[458,24108,1364],{},[543,24110,24111,24113],{},[383,24112,547],{},[383,24114,24115],{},"J",[458,24117,461],{"stretchy":460},[383,24119,554],{},[458,24121,898],{"separator":394},[383,24123,561],{},[458,24125,468],{"stretchy":460},[458,24127,898],{"separator":394},[387,24129,24130],{"encoding":389},"D_f(P\\|Q)=\\int (p-q)\\log\\frac{p}{q}\\,dx \\\\\n=D_{KL}(P\\|Q)+D_{KL}(Q\\|P)=:D_J(P,Q),",[363,24132,24134,24204,24228,24322,24325,24337,24413,24490],{"className":24133,"ariaHidden":394},[393],[363,24135,24137,24140,24180,24183,24186,24189,24192,24195,24198,24201],{"className":24136},[398],[363,24138],{"className":24139,"style":618},[402],[363,24141,24143,24146],{"className":24142},[407],[363,24144,547],{"className":24145,"style":625},[407,408],[363,24147,24149],{"className":24148},[629],[363,24150,24152,24172],{"className":24151},[633,634],[363,24153,24155,24169],{"className":24154},[638],[363,24156,24158],{"className":24157,"style":643},[642],[363,24159,24160,24163],{"style":646},[363,24161],{"className":24162,"style":651},[650],[363,24164,24166],{"className":24165},[655,656,657,658],[363,24167,385],{"className":24168,"style":409},[407,408,658],[363,24170,666],{"className":24171},[665],[363,24173,24175],{"className":24174},[638],[363,24176,24178],{"className":24177,"style":673},[642],[363,24179],{},[363,24181,461],{"className":24182},[494],[363,24184,554],{"className":24185,"style":682},[407,408],[363,24187,558],{"className":24188},[407],[363,24190,561],{"className":24191},[407,408],[363,24193,468],{"className":24194},[501],[363,24196],{"className":24197,"style":506},[505],[363,24199,471],{"className":24200},[510],[363,24202],{"className":24203,"style":506},[505],[363,24205,24207,24210,24213,24216,24219,24222,24225],{"className":24206},[398],[363,24208],{"className":24209,"style":6125},[402],[363,24211,1635],{"className":24212,"style":1835},[1832,1833,1834],[363,24214,461],{"className":24215},[494],[363,24217,11],{"className":24218},[407,408],[363,24220],{"className":24221,"style":2935},[505],[363,24223,5161],{"className":24224},[2939],[363,24226],{"className":24227,"style":2935},[505],[363,24229,24231,24234,24237,24240,24243,24248,24251,24313,24316,24319],{"className":24230},[398],[363,24232],{"className":24233,"style":22382},[402],[363,24235,1291],{"className":24236,"style":1328},[407,408],[363,24238,468],{"className":24239},[501],[363,24241],{"className":24242,"style":754},[505],[363,24244,4502,24246],{"className":24245},[1832],[363,24247,4506],{"style":4505},[363,24249],{"className":24250,"style":754},[505],[363,24252,24254,24257,24310],{"className":24253},[407],[363,24255],{"className":24256},[494,789],[363,24258,24260],{"className":24259},[586],[363,24261,24263,24302],{"className":24262},[633,634],[363,24264,24266,24299],{"className":24265},[638],[363,24267,24269,24280,24288],{"className":24268,"style":3902},[642],[363,24270,24271,24274],{"style":805},[363,24272],{"className":24273,"style":809},[650],[363,24275,24277],{"className":24276},[407],[363,24278,1291],{"className":24279,"style":1328},[407,408],[363,24281,24282,24285],{"style":821},[363,24283],{"className":24284,"style":809},[650],[363,24286],{"className":24287,"style":829},[828],[363,24289,24290,24293],{"style":832},[363,24291],{"className":24292,"style":809},[650],[363,24294,24296],{"className":24295},[407],[363,24297,11],{"className":24298},[407,408],[363,24300,666],{"className":24301},[665],[363,24303,24305],{"className":24304},[638],[363,24306,24308],{"className":24307,"style":854},[642],[363,24309],{},[363,24311],{"className":24312},[501,789],[363,24314],{"className":24315,"style":754},[505],[363,24317,592],{"className":24318},[407,408],[363,24320,1121],{"className":24321},[407,408],[363,24323],{"className":24324},[505,24060],[363,24326,24328,24331,24334],{"className":24327},[398],[363,24329],{"className":24330,"style":11353},[402],[363,24332,471],{"className":24333},[510],[363,24335],{"className":24336,"style":506},[505],[363,24338,24340,24343,24389,24392,24395,24398,24401,24404,24407,24410],{"className":24339},[398],[363,24341],{"className":24342,"style":487},[402],[363,24344,24346,24349],{"className":24345},[407],[363,24347,547],{"className":24348,"style":625},[407,408],[363,24350,24352],{"className":24351},[629],[363,24353,24355,24381],{"className":24354},[633,634],[363,24356,24358,24378],{"className":24357},[638],[363,24359,24361],{"className":24360,"style":727},[642],[363,24362,24363,24366],{"style":646},[363,24364],{"className":24365,"style":651},[650],[363,24367,24369],{"className":24368},[655,656,657,658],[363,24370,24372,24375],{"className":24371},[407,658],[363,24373,11044],{"className":24374,"style":11132},[407,408,658],[363,24376,22265],{"className":24377},[407,408,658],[363,24379,666],{"className":24380},[665],[363,24382,24384],{"className":24383},[638],[363,24385,24387],{"className":24386,"style":8316},[642],[363,24388],{},[363,24390,461],{"className":24391},[494],[363,24393,554],{"className":24394,"style":682},[407,408],[363,24396,558],{"className":24397},[407],[363,24399,561],{"className":24400},[407,408],[363,24402,468],{"className":24403},[501],[363,24405],{"className":24406,"style":2935},[505],[363,24408,2818],{"className":24409},[2939],[363,24411],{"className":24412,"style":2935},[505],[363,24414,24416,24419,24465,24468,24471,24474,24477,24480,24483,24487],{"className":24415},[398],[363,24417],{"className":24418,"style":487},[402],[363,24420,24422,24425],{"className":24421},[407],[363,24423,547],{"className":24424,"style":625},[407,408],[363,24426,24428],{"className":24427},[629],[363,24429,24431,24457],{"className":24430},[633,634],[363,24432,24434,24454],{"className":24433},[638],[363,24435,24437],{"className":24436,"style":727},[642],[363,24438,24439,24442],{"style":646},[363,24440],{"className":24441,"style":651},[650],[363,24443,24445],{"className":24444},[655,656,657,658],[363,24446,24448,24451],{"className":24447},[407,658],[363,24449,11044],{"className":24450,"style":11132},[407,408,658],[363,24452,22265],{"className":24453},[407,408,658],[363,24455,666],{"className":24456},[665],[363,24458,24460],{"className":24459},[638],[363,24461,24463],{"className":24462,"style":8316},[642],[363,24464],{},[363,24466,461],{"className":24467},[494],[363,24469,561],{"className":24470},[407,408],[363,24472,558],{"className":24473},[407],[363,24475,554],{"className":24476,"style":682},[407,408],[363,24478,468],{"className":24479},[501],[363,24481],{"className":24482,"style":506},[505],[363,24484,24486],{"className":24485},[510],"=:",[363,24488],{"className":24489,"style":506},[505],[363,24491,24493,24496,24537,24540,24543,24546,24549,24552,24555],{"className":24492},[398],[363,24494],{"className":24495,"style":487},[402],[363,24497,24499,24502],{"className":24498},[407],[363,24500,547],{"className":24501,"style":625},[407,408],[363,24503,24505],{"className":24504},[629],[363,24506,24508,24529],{"className":24507},[633,634],[363,24509,24511,24526],{"className":24510},[638],[363,24512,24514],{"className":24513,"style":727},[642],[363,24515,24516,24519],{"style":646},[363,24517],{"className":24518,"style":651},[650],[363,24520,24522],{"className":24521},[655,656,657,658],[363,24523,24115],{"className":24524,"style":24525},[407,408,658],"margin-right:0.0962em;",[363,24527,666],{"className":24528},[665],[363,24530,24532],{"className":24531},[638],[363,24533,24535],{"className":24534,"style":8316},[642],[363,24536],{},[363,24538,461],{"className":24539},[494],[363,24541,554],{"className":24542,"style":682},[407,408],[363,24544,898],{"className":24545},[920],[363,24547],{"className":24548,"style":754},[505],[363,24550,561],{"className":24551},[407,408],[363,24553,468],{"className":24554},[501],[363,24556,898],{"className":24557},[920],[11,24559,24560],{},"这是 KL 的两向之和，也是一个人为构造的对称 f-散度。",[11,24562,24563,24566,24567,24570,24571,24599,24600,24628,24629,24657,24658,24690,24691,24796,24797,24825,24826,24854,24855,24858],{},[1344,24564,24565],{},"意义。"," KL 是最基础的散度，等价于信息论中的",[1344,24568,24569],{},"相对熵","：若真实分布是 ",[363,24572,24574,24587],{"className":24573,"translate":367},[366],[363,24575,24577],{"className":24576},[371],[373,24578,24579],{"xmlns":375},[377,24580,24581,24585],{},[380,24582,24583],{},[383,24584,554],{},[387,24586,554],{"encoding":389},[363,24588,24590],{"className":24589,"ariaHidden":394},[393],[363,24591,24593,24596],{"className":24592},[398],[363,24594],{"className":24595,"style":1011},[402],[363,24597,554],{"className":24598,"style":682},[407,408],"、而用 ",[363,24601,24603,24616],{"className":24602,"translate":367},[366],[363,24604,24606],{"className":24605},[371],[373,24607,24608],{"xmlns":375},[377,24609,24610,24614],{},[380,24611,24612],{},[383,24613,561],{},[387,24615,561],{"encoding":389},[363,24617,24619],{"className":24618,"ariaHidden":394},[393],[363,24620,24622,24625],{"className":24621},[398],[363,24623],{"className":24624,"style":913},[402],[363,24626,561],{"className":24627},[407,408]," 去编码来自 ",[363,24630,24632,24645],{"className":24631,"translate":367},[366],[363,24633,24635],{"className":24634},[371],[373,24636,24637],{"xmlns":375},[377,24638,24639,24643],{},[380,24640,24641],{},[383,24642,554],{},[387,24644,554],{"encoding":389},[363,24646,24648],{"className":24647,"ariaHidden":394},[393],[363,24649,24651,24654],{"className":24650},[398],[363,24652],{"className":24653,"style":1011},[402],[363,24655,554],{"className":24656,"style":682},[407,408]," 的样本，那么 ",[363,24659,24661,24676],{"className":24660,"translate":367},[366],[363,24662,24664],{"className":24663},[371],[373,24665,24666],{"xmlns":375},[377,24667,24668,24674],{},[380,24669,24670,24672],{},[383,24671,4448],{},[458,24673,4451],{},[387,24675,21955],{"encoding":389},[363,24677,24679],{"className":24678,"ariaHidden":394},[393],[363,24680,24682,24685],{"className":24681},[398],[363,24683],{"className":24684,"style":403},[402],[363,24686,4502,24688],{"className":24687},[1832],[363,24689,4506],{"style":4505}," 以 2 为底时 ",[363,24692,24694,24726],{"className":24693,"translate":367},[366],[363,24695,24697],{"className":24696},[371],[373,24698,24699],{"xmlns":375},[377,24700,24701,24723],{},[380,24702,24703,24713,24715,24717,24719,24721],{},[543,24704,24705,24707],{},[383,24706,547],{},[380,24708,24709,24711],{},[383,24710,11044],{},[383,24712,22265],{},[458,24714,461],{"stretchy":460},[383,24716,554],{},[383,24718,558],{"mathvariant":557},[383,24720,561],{},[458,24722,468],{"stretchy":460},[387,24724,24725],{"encoding":389},"D_{KL}(P\\|Q)",[363,24727,24729],{"className":24728,"ariaHidden":394},[393],[363,24730,24732,24735,24781,24784,24787,24790,24793],{"className":24731},[398],[363,24733],{"className":24734,"style":487},[402],[363,24736,24738,24741],{"className":24737},[407],[363,24739,547],{"className":24740,"style":625},[407,408],[363,24742,24744],{"className":24743},[629],[363,24745,24747,24773],{"className":24746},[633,634],[363,24748,24750,24770],{"className":24749},[638],[363,24751,24753],{"className":24752,"style":727},[642],[363,24754,24755,24758],{"style":646},[363,24756],{"className":24757,"style":651},[650],[363,24759,24761],{"className":24760},[655,656,657,658],[363,24762,24764,24767],{"className":24763},[407,658],[363,24765,11044],{"className":24766,"style":11132},[407,408,658],[363,24768,22265],{"className":24769},[407,408,658],[363,24771,666],{"className":24772},[665],[363,24774,24776],{"className":24775},[638],[363,24777,24779],{"className":24778,"style":8316},[642],[363,24780],{},[363,24782,461],{"className":24783},[494],[363,24785,554],{"className":24786,"style":682},[407,408],[363,24788,558],{"className":24789},[407],[363,24791,561],{"className":24792},[407,408],[363,24794,468],{"className":24795},[501]," 正是用 ",[363,24798,24800,24813],{"className":24799,"translate":367},[366],[363,24801,24803],{"className":24802},[371],[373,24804,24805],{"xmlns":375},[377,24806,24807,24811],{},[380,24808,24809],{},[383,24810,561],{},[387,24812,561],{"encoding":389},[363,24814,24816],{"className":24815,"ariaHidden":394},[393],[363,24817,24819,24822],{"className":24818},[398],[363,24820],{"className":24821,"style":913},[402],[363,24823,561],{"className":24824},[407,408]," 编码比用 ",[363,24827,24829,24842],{"className":24828,"translate":367},[366],[363,24830,24832],{"className":24831},[371],[373,24833,24834],{"xmlns":375},[377,24835,24836,24840],{},[380,24837,24838],{},[383,24839,554],{},[387,24841,554],{"encoding":389},[363,24843,24845],{"className":24844,"ariaHidden":394},[393],[363,24846,24848,24851],{"className":24847},[398],[363,24849],{"className":24850,"style":1011},[402],[363,24852,554],{"className":24853,"style":682},[407,408]," 编码",[1344,24856,24857],{},"多付出的比特数","，即",[363,24860,24862],{"className":24861,"translate":367},[528],[363,24863,24865,25007],{"className":24864,"translate":367},[366],[363,24866,24868],{"className":24867},[371],[373,24869,24870],{"xmlns":375,"display":537},[377,24871,24872,25004],{},[1591,24873,24874,24947],{"rowspacing":1593,"columnalign":1594,"columnspacing":1595},[1597,24875,24876,24902],{},[1600,24877,24878],{},[1603,24879,24880],{"scriptlevel":474,"displaystyle":394},[380,24881,24882,24892,24894,24896,24898,24900],{},[543,24883,24884,24886],{},[383,24885,547],{},[380,24887,24888,24890],{},[383,24889,11044],{},[383,24891,22265],{},[458,24893,461],{"stretchy":460},[383,24895,554],{},[383,24897,558],{"mathvariant":557},[383,24899,561],{},[458,24901,468],{"stretchy":460},[1600,24903,24904],{},[1603,24905,24906],{"scriptlevel":474,"displaystyle":394},[380,24907,24908,24910,24912],{},[380,24909],{},[458,24911,471],{},[6364,24913,24914,24944],{},[6364,24915,24916,24942],{},[380,24917,24918,24924,24926,24928,24930,24932,24934,24936,24938,24940],{},[543,24919,24920,24922],{},[383,24921,571],{"mathvariant":570},[383,24923,554],{},[458,24925,578],{"stretchy":460},[458,24927,5161],{},[383,24929,4448],{},[458,24931,4451],{},[383,24933,1291],{},[458,24935,461],{"stretchy":460},[383,24937,945],{},[458,24939,468],{"stretchy":460},[458,24941,605],{"stretchy":460},[458,24943,6388],{"stretchy":394},[1645,24945,24946],{},"交叉熵",[1597,24948,24949,24955],{},[1600,24950,24951],{},[1603,24952,24953],{"scriptlevel":474,"displaystyle":394},[380,24954],{},[1600,24956,24957],{},[1603,24958,24959],{"scriptlevel":474,"displaystyle":394},[380,24960,24961,24963,24965,24967,25002],{},[380,24962],{},[505,24964],{"width":3610},[458,24966,5161],{},[6364,24968,24969,24999],{},[6364,24970,24971,24997],{},[380,24972,24973,24979,24981,24983,24985,24987,24989,24991,24993,24995],{},[543,24974,24975,24977],{},[383,24976,571],{"mathvariant":570},[383,24978,554],{},[458,24980,578],{"stretchy":460},[458,24982,5161],{},[383,24984,4448],{},[458,24986,4451],{},[383,24988,11],{},[458,24990,461],{"stretchy":460},[383,24992,945],{},[458,24994,468],{"stretchy":460},[458,24996,605],{"stretchy":460},[458,24998,6388],{"stretchy":394},[1645,25000,25001],{},"熵",[383,25003,4723],{"mathvariant":557},[387,25005,25006],{"encoding":389},"\\begin{aligned}\nD_{KL}(P\\|Q)\n&=\\underbrace{\\mathbb E_P[-\\log q(X)]}\n_{\\text{交叉熵}}\\\\[2pt]\n&\\qquad-\\underbrace{\\mathbb E_P[-\\log p(X)]}\n_{\\text{熵}} .\n\\end{aligned}",[363,25008,25010],{"className":25009,"ariaHidden":394},[393],[363,25011,25013,25017],{"className":25012},[398],[363,25014],{"className":25015,"style":25016},[402],"height:5.1327em;vertical-align:-2.3163em;",[363,25018,25020],{"className":25019},[407],[363,25021,25023,25127],{"className":25022},[1591],[363,25024,25026],{"className":25025},[1708],[363,25027,25029,25118],{"className":25028},[633,634],[363,25030,25032,25115],{"className":25031},[638],[363,25033,25036,25106],{"className":25034,"style":25035},[642],"height:2.8163em;",[363,25037,25039,25042],{"style":25038},"top:-4.9763em;",[363,25040],{"className":25041,"style":809},[650],[363,25043,25045,25091,25094,25097,25100,25103],{"className":25044},[407],[363,25046,25048,25051],{"className":25047},[407],[363,25049,547],{"className":25050,"style":625},[407,408],[363,25052,25054],{"className":25053},[629],[363,25055,25057,25083],{"className":25056},[633,634],[363,25058,25060,25080],{"className":25059},[638],[363,25061,25063],{"className":25062,"style":727},[642],[363,25064,25065,25068],{"style":646},[363,25066],{"className":25067,"style":651},[650],[363,25069,25071],{"className":25070},[655,656,657,658],[363,25072,25074,25077],{"className":25073},[407,658],[363,25075,11044],{"className":25076,"style":11132},[407,408,658],[363,25078,22265],{"className":25079},[407,408,658],[363,25081,666],{"className":25082},[665],[363,25084,25086],{"className":25085},[638],[363,25087,25089],{"className":25088,"style":8316},[642],[363,25090],{},[363,25092,461],{"className":25093},[494],[363,25095,554],{"className":25096,"style":682},[407,408],[363,25098,558],{"className":25099},[407],[363,25101,561],{"className":25102},[407,408],[363,25104,468],{"className":25105},[501],[363,25107,25109,25112],{"style":25108},"top:-2.26em;",[363,25110],{"className":25111,"style":809},[650],[363,25113],{"className":25114},[407],[363,25116,666],{"className":25117},[665],[363,25119,25121],{"className":25120},[638],[363,25122,25125],{"className":25123,"style":25124},[642],"height:2.3163em;",[363,25126],{},[363,25128,25130],{"className":25129},[1799],[363,25131,25133,25546],{"className":25132},[633,634],[363,25134,25136,25543],{"className":25135},[638],[363,25137,25139,25339],{"className":25138,"style":25035},[642],[363,25140,25141,25144],{"style":25038},[363,25142],{"className":25143,"style":809},[650],[363,25145,25147,25150,25153,25156,25159],{"className":25146},[407],[363,25148],{"className":25149},[407],[363,25151],{"className":25152,"style":506},[505],[363,25154,471],{"className":25155},[510],[363,25157],{"className":25158,"style":506},[505],[363,25160,25162],{"className":25161},[758,6364],[363,25163,25165,25330],{"className":25164},[633,634],[363,25166,25168,25327],{"className":25167},[638],[363,25169,25172,25190],{"className":25170,"style":25171},[642],"height:0.75em;",[363,25173,25175,25178],{"style":25174},"top:-1.4237em;",[363,25176],{"className":25177,"style":809},[650],[363,25179,25181],{"className":25180},[655,656,657,658],[363,25182,25184],{"className":25183},[407,658],[363,25185,25187],{"className":25186},[407,4946,658],[363,25188,24946],{"className":25189},[407,4950,658],[363,25191,25192,25195],{"style":12213},[363,25193],{"className":25194,"style":809},[650],[363,25196,25198],{"className":25197},[758,6364],[363,25199,25201,25318],{"className":25200},[633,634],[363,25202,25204,25315],{"className":25203},[638],[363,25205,25207,25238],{"className":25206,"style":25171},[642],[363,25208,25211,25214],{"className":25209,"style":25210},[6655],"top:-2.102em;",[363,25212],{"className":25213,"style":809},[650],[363,25215,25217,25224,25231],{"className":25216,"style":6664},[6663],[363,25218,25220],{"className":25219,"style":6669},[6668],[6671,25221,25222],{"xmlns":6673,"width":6674,"height":6675,"viewBox":6676,"preserveAspectRatio":6677},[6679,25223],{"d":6681},[363,25225,25227],{"className":25226,"style":6669},[6685],[6671,25228,25229],{"xmlns":6673,"width":6674,"height":6675,"viewBox":6676,"preserveAspectRatio":6688},[6679,25230],{"d":6691},[363,25232,25234],{"className":25233,"style":6669},[6695],[6671,25235,25236],{"xmlns":6673,"width":6674,"height":6675,"viewBox":6676,"preserveAspectRatio":6698},[6679,25237],{"d":6701},[363,25239,25240,25243],{"style":12213},[363,25241],{"className":25242,"style":809},[650],[363,25244,25246,25286,25289,25292,25295,25300,25303,25306,25309,25312],{"className":25245},[407],[363,25247,25249,25252],{"className":25248},[407],[363,25250,571],{"className":25251},[407,714],[363,25253,25255],{"className":25254},[629],[363,25256,25258,25278],{"className":25257},[633,634],[363,25259,25261,25275],{"className":25260},[638],[363,25262,25264],{"className":25263,"style":727},[642],[363,25265,25266,25269],{"style":730},[363,25267],{"className":25268,"style":651},[650],[363,25270,25272],{"className":25271},[655,656,657,658],[363,25273,554],{"className":25274,"style":682},[407,408,658],[363,25276,666],{"className":25277},[665],[363,25279,25281],{"className":25280},[638],[363,25282,25284],{"className":25283,"style":8316},[642],[363,25285],{},[363,25287,578],{"className":25288},[494],[363,25290,5161],{"className":25291},[407],[363,25293],{"className":25294,"style":754},[505],[363,25296,4502,25298],{"className":25297},[1832],[363,25299,4506],{"style":4505},[363,25301],{"className":25302,"style":754},[505],[363,25304,1291],{"className":25305,"style":1328},[407,408],[363,25307,461],{"className":25308},[494],[363,25310,945],{"className":25311,"style":9235},[407,408],[363,25313,20762],{"className":25314},[501],[363,25316,666],{"className":25317},[665],[363,25319,25321],{"className":25320},[638],[363,25322,25325],{"className":25323,"style":25324},[642],"height:0.898em;",[363,25326],{},[363,25328,666],{"className":25329},[665],[363,25331,25333],{"className":25332},[638],[363,25334,25337],{"className":25335,"style":25336},[642],"height:1.5763em;",[363,25338],{},[363,25340,25341,25344],{"style":25108},[363,25342],{"className":25343,"style":809},[650],[363,25345,25347,25350,25353,25356,25359,25362,25537,25540],{"className":25346},[407],[363,25348],{"className":25349},[407],[363,25351],{"className":25352,"style":3982},[505],[363,25354],{"className":25355,"style":2935},[505],[363,25357,5161],{"className":25358},[2939],[363,25360],{"className":25361,"style":2935},[505],[363,25363,25365],{"className":25364},[758,6364],[363,25366,25368,25529],{"className":25367},[633,634],[363,25369,25371,25526],{"className":25370},[638],[363,25372,25374,25391],{"className":25373,"style":25171},[642],[363,25375,25376,25379],{"style":25174},[363,25377],{"className":25378,"style":809},[650],[363,25380,25382],{"className":25381},[655,656,657,658],[363,25383,25385],{"className":25384},[407,658],[363,25386,25388],{"className":25387},[407,4946,658],[363,25389,25001],{"className":25390},[407,4950,658],[363,25392,25393,25396],{"style":12213},[363,25394],{"className":25395,"style":809},[650],[363,25397,25399],{"className":25398},[758,6364],[363,25400,25402,25518],{"className":25401},[633,634],[363,25403,25405,25515],{"className":25404},[638],[363,25406,25408,25438],{"className":25407,"style":25171},[642],[363,25409,25411,25414],{"className":25410,"style":25210},[6655],[363,25412],{"className":25413,"style":809},[650],[363,25415,25417,25424,25431],{"className":25416,"style":6664},[6663],[363,25418,25420],{"className":25419,"style":6669},[6668],[6671,25421,25422],{"xmlns":6673,"width":6674,"height":6675,"viewBox":6676,"preserveAspectRatio":6677},[6679,25423],{"d":6681},[363,25425,25427],{"className":25426,"style":6669},[6685],[6671,25428,25429],{"xmlns":6673,"width":6674,"height":6675,"viewBox":6676,"preserveAspectRatio":6688},[6679,25430],{"d":6691},[363,25432,25434],{"className":25433,"style":6669},[6695],[6671,25435,25436],{"xmlns":6673,"width":6674,"height":6675,"viewBox":6676,"preserveAspectRatio":6698},[6679,25437],{"d":6701},[363,25439,25440,25443],{"style":12213},[363,25441],{"className":25442,"style":809},[650],[363,25444,25446,25486,25489,25492,25495,25500,25503,25506,25509,25512],{"className":25445},[407],[363,25447,25449,25452],{"className":25448},[407],[363,25450,571],{"className":25451},[407,714],[363,25453,25455],{"className":25454},[629],[363,25456,25458,25478],{"className":25457},[633,634],[363,25459,25461,25475],{"className":25460},[638],[363,25462,25464],{"className":25463,"style":727},[642],[363,25465,25466,25469],{"style":730},[363,25467],{"className":25468,"style":651},[650],[363,25470,25472],{"className":25471},[655,656,657,658],[363,25473,554],{"className":25474,"style":682},[407,408,658],[363,25476,666],{"className":25477},[665],[363,25479,25481],{"className":25480},[638],[363,25482,25484],{"className":25483,"style":8316},[642],[363,25485],{},[363,25487,578],{"className":25488},[494],[363,25490,5161],{"className":25491},[407],[363,25493],{"className":25494,"style":754},[505],[363,25496,4502,25498],{"className":25497},[1832],[363,25499,4506],{"style":4505},[363,25501],{"className":25502,"style":754},[505],[363,25504,11],{"className":25505},[407,408],[363,25507,461],{"className":25508},[494],[363,25510,945],{"className":25511,"style":9235},[407,408],[363,25513,20762],{"className":25514},[501],[363,25516,666],{"className":25517},[665],[363,25519,25521],{"className":25520},[638],[363,25522,25524],{"className":25523,"style":25324},[642],[363,25525],{},[363,25527,666],{"className":25528},[665],[363,25530,25532],{"className":25531},[638],[363,25533,25535],{"className":25534,"style":25336},[642],[363,25536],{},[363,25538],{"className":25539,"style":754},[505],[363,25541,4723],{"className":25542},[407],[363,25544,666],{"className":25545},[665],[363,25547,25549],{"className":25548},[638],[363,25550,25552],{"className":25551,"style":25124},[642],[363,25553],{},[11,25555,25556,25557,25661,25662,4201,25690,25718,25719,25722,25723,25751,25752,25856,25857,4201,25885,25913,25914,25917],{},"在统计中它是最大似然估计的对数似然比的期望，决定了 Stein 引理里第二类错误的最优指数，也是 AIC\u002FBIC 这类模型选择的基石；在机器学习中它是最大似然训练、变分推断与知识蒸馏的目标。它的不对称性带来两种相反的行为：",[363,25558,25560,25591],{"className":25559,"translate":367},[366],[363,25561,25563],{"className":25562},[371],[373,25564,25565],{"xmlns":375},[377,25566,25567,25589],{},[380,25568,25569,25579,25581,25583,25585,25587],{},[543,25570,25571,25573],{},[383,25572,547],{},[380,25574,25575,25577],{},[383,25576,11044],{},[383,25578,22265],{},[458,25580,461],{"stretchy":460},[383,25582,554],{},[383,25584,558],{"mathvariant":557},[383,25586,561],{},[458,25588,468],{"stretchy":460},[387,25590,24725],{"encoding":389},[363,25592,25594],{"className":25593,"ariaHidden":394},[393],[363,25595,25597,25600,25646,25649,25652,25655,25658],{"className":25596},[398],[363,25598],{"className":25599,"style":487},[402],[363,25601,25603,25606],{"className":25602},[407],[363,25604,547],{"className":25605,"style":625},[407,408],[363,25607,25609],{"className":25608},[629],[363,25610,25612,25638],{"className":25611},[633,634],[363,25613,25615,25635],{"className":25614},[638],[363,25616,25618],{"className":25617,"style":727},[642],[363,25619,25620,25623],{"style":646},[363,25621],{"className":25622,"style":651},[650],[363,25624,25626],{"className":25625},[655,656,657,658],[363,25627,25629,25632],{"className":25628},[407,658],[363,25630,11044],{"className":25631,"style":11132},[407,408,658],[363,25633,22265],{"className":25634},[407,408,658],[363,25636,666],{"className":25637},[665],[363,25639,25641],{"className":25640},[638],[363,25642,25644],{"className":25643,"style":8316},[642],[363,25645],{},[363,25647,461],{"className":25648},[494],[363,25650,554],{"className":25651,"style":682},[407,408],[363,25653,558],{"className":25654},[407],[363,25656,561],{"className":25657},[407,408],[363,25659,468],{"className":25660},[501]," 要求 ",[363,25663,25665,25678],{"className":25664,"translate":367},[366],[363,25666,25668],{"className":25667},[371],[373,25669,25670],{"xmlns":375},[377,25671,25672,25676],{},[380,25673,25674],{},[383,25675,561],{},[387,25677,561],{"encoding":389},[363,25679,25681],{"className":25680,"ariaHidden":394},[393],[363,25682,25684,25687],{"className":25683},[398],[363,25685],{"className":25686,"style":913},[402],[363,25688,561],{"className":25689},[407,408],[363,25691,25693,25706],{"className":25692,"translate":367},[366],[363,25694,25696],{"className":25695},[371],[373,25697,25698],{"xmlns":375},[377,25699,25700,25704],{},[380,25701,25702],{},[383,25703,554],{},[387,25705,554],{"encoding":389},[363,25707,25709],{"className":25708,"ariaHidden":394},[393],[363,25710,25712,25715],{"className":25711},[398],[363,25713],{"className":25714,"style":1011},[402],[363,25716,554],{"className":25717,"style":682},[407,408]," 有质量处不为零（",[1344,25720,25721],{},"零回避","，mass-covering，倾向覆盖 ",[363,25724,25726,25739],{"className":25725,"translate":367},[366],[363,25727,25729],{"className":25728},[371],[373,25730,25731],{"xmlns":375},[377,25732,25733,25737],{},[380,25734,25735],{},[383,25736,554],{},[387,25738,554],{"encoding":389},[363,25740,25742],{"className":25741,"ariaHidden":394},[393],[363,25743,25745,25748],{"className":25744},[398],[363,25746],{"className":25747,"style":1011},[402],[363,25749,554],{"className":25750,"style":682},[407,408]," 的每一个模式），而 ",[363,25753,25755,25786],{"className":25754,"translate":367},[366],[363,25756,25758],{"className":25757},[371],[373,25759,25760],{"xmlns":375},[377,25761,25762,25784],{},[380,25763,25764,25774,25776,25778,25780,25782],{},[543,25765,25766,25768],{},[383,25767,547],{},[380,25769,25770,25772],{},[383,25771,11044],{},[383,25773,22265],{},[458,25775,461],{"stretchy":460},[383,25777,561],{},[383,25779,558],{"mathvariant":557},[383,25781,554],{},[458,25783,468],{"stretchy":460},[387,25785,22747],{"encoding":389},[363,25787,25789],{"className":25788,"ariaHidden":394},[393],[363,25790,25792,25795,25841,25844,25847,25850,25853],{"className":25791},[398],[363,25793],{"className":25794,"style":487},[402],[363,25796,25798,25801],{"className":25797},[407],[363,25799,547],{"className":25800,"style":625},[407,408],[363,25802,25804],{"className":25803},[629],[363,25805,25807,25833],{"className":25806},[633,634],[363,25808,25810,25830],{"className":25809},[638],[363,25811,25813],{"className":25812,"style":727},[642],[363,25814,25815,25818],{"style":646},[363,25816],{"className":25817,"style":651},[650],[363,25819,25821],{"className":25820},[655,656,657,658],[363,25822,25824,25827],{"className":25823},[407,658],[363,25825,11044],{"className":25826,"style":11132},[407,408,658],[363,25828,22265],{"className":25829},[407,408,658],[363,25831,666],{"className":25832},[665],[363,25834,25836],{"className":25835},[638],[363,25837,25839],{"className":25838,"style":8316},[642],[363,25840],{},[363,25842,461],{"className":25843},[494],[363,25845,561],{"className":25846},[407,408],[363,25848,558],{"className":25849},[407],[363,25851,554],{"className":25852,"style":682},[407,408],[363,25854,468],{"className":25855},[501]," 会把 ",[363,25858,25860,25873],{"className":25859,"translate":367},[366],[363,25861,25863],{"className":25862},[371],[373,25864,25865],{"xmlns":375},[377,25866,25867,25871],{},[380,25868,25869],{},[383,25870,561],{},[387,25872,561],{"encoding":389},[363,25874,25876],{"className":25875,"ariaHidden":394},[393],[363,25877,25879,25882],{"className":25878},[398],[363,25880],{"className":25881,"style":913},[402],[363,25883,561],{"className":25884},[407,408],[363,25886,25888,25901],{"className":25887,"translate":367},[366],[363,25889,25891],{"className":25890},[371],[373,25892,25893],{"xmlns":375},[377,25894,25895,25899],{},[380,25896,25897],{},[383,25898,554],{},[387,25900,554],{"encoding":389},[363,25902,25904],{"className":25903,"ariaHidden":394},[393],[363,25905,25907,25910],{"className":25906},[398],[363,25908],{"className":25909,"style":1011},[402],[363,25911,554],{"className":25912,"style":682},[407,408]," 为零处压到零（",[1344,25915,25916],{},"零强制","，mode-seeking，倾向锁定单一模式）。GAN 的模式崩塌与变分推断的均值场近似，本质都由这一不对称性决定。",[2659,25919,25921],{"id":25920},"_32-js-散度","3.2 JS 散度",[11,25923,25924,25925,9748],{},"先回顾 JS 散度的定义，令 ",[363,25926,25928,25961],{"className":25927,"translate":367},[366],[363,25929,25931],{"className":25930},[371],[373,25932,25933],{"xmlns":375},[377,25934,25935,25958],{},[380,25936,25937,25940,25942,25948,25950,25952,25954,25956],{},[383,25938,25939],{},"M",[458,25941,471],{},[586,25943,25944,25946],{},[463,25945,465],{},[463,25947,2811],{},[458,25949,461],{"stretchy":460},[383,25951,554],{},[458,25953,2818],{},[383,25955,561],{},[458,25957,468],{"stretchy":460},[387,25959,25960],{"encoding":389},"M=\\frac12(P+Q)",[363,25962,25964,25983,26072],{"className":25963,"ariaHidden":394},[393],[363,25965,25967,25970,25974,25977,25980],{"className":25966},[398],[363,25968],{"className":25969,"style":1011},[402],[363,25971,25939],{"className":25972,"style":25973},[407,408],"margin-right:0.109em;",[363,25975],{"className":25976,"style":506},[505],[363,25978,471],{"className":25979},[510],[363,25981],{"className":25982,"style":506},[505],[363,25984,25986,25989,26057,26060,26063,26066,26069],{"className":25985},[398],[363,25987],{"className":25988,"style":2853},[402],[363,25990,25992,25995,26054],{"className":25991},[407],[363,25993],{"className":25994},[494,789],[363,25996,25998],{"className":25997},[586],[363,25999,26001,26046],{"className":26000},[633,634],[363,26002,26004,26043],{"className":26003},[638],[363,26005,26007,26021,26029],{"className":26006,"style":2872},[642],[363,26008,26009,26012],{"style":2875},[363,26010],{"className":26011,"style":809},[650],[363,26013,26015],{"className":26014},[655,656,657,658],[363,26016,26018],{"className":26017},[407,658],[363,26019,2811],{"className":26020},[407,658],[363,26022,26023,26026],{"style":821},[363,26024],{"className":26025,"style":809},[650],[363,26027],{"className":26028,"style":829},[828],[363,26030,26031,26034],{"style":2898},[363,26032],{"className":26033,"style":809},[650],[363,26035,26037],{"className":26036},[655,656,657,658],[363,26038,26040],{"className":26039},[407,658],[363,26041,465],{"className":26042},[407,658],[363,26044,666],{"className":26045},[665],[363,26047,26049],{"className":26048},[638],[363,26050,26052],{"className":26051,"style":2920},[642],[363,26053],{},[363,26055],{"className":26056},[501,789],[363,26058,461],{"className":26059},[494],[363,26061,554],{"className":26062,"style":682},[407,408],[363,26064],{"className":26065,"style":2935},[505],[363,26067,2818],{"className":26068},[2939],[363,26070],{"className":26071,"style":2935},[505],[363,26073,26075,26078,26081],{"className":26074},[398],[363,26076],{"className":26077,"style":487},[402],[363,26079,561],{"className":26080},[407,408],[363,26082,468],{"className":26083},[501],[363,26085,26087],{"className":26086,"translate":367},[528],[363,26088,26090,26189],{"className":26089,"translate":367},[366],[363,26091,26093],{"className":26092},[371],[373,26094,26095],{"xmlns":375,"display":537},[377,26096,26097,26186],{},[380,26098,26099,26110,26112,26114,26116,26118,26120,26122,26128,26138,26140,26152,26154,26160,26170,26172,26184],{},[543,26100,26101,26103],{},[383,26102,547],{},[380,26104,26105,26107],{},[383,26106,24115],{},[383,26108,26109],{},"S",[458,26111,461],{"stretchy":460},[383,26113,554],{},[383,26115,558],{"mathvariant":557},[383,26117,561],{},[458,26119,468],{"stretchy":460},[458,26121,471],{},[586,26123,26124,26126],{},[463,26125,465],{},[463,26127,2811],{},[543,26129,26130,26132],{},[383,26131,547],{},[380,26133,26134,26136],{},[383,26135,11044],{},[383,26137,22265],{},[1645,26139,1652],{},[380,26141,26142,26144,26146,26148,26150],{},[458,26143,461],{"fence":394},[383,26145,554],{},[383,26147,558],{"mathvariant":557},[383,26149,25939],{},[458,26151,468],{"fence":394},[458,26153,2818],{},[586,26155,26156,26158],{},[463,26157,465],{},[463,26159,2811],{},[543,26161,26162,26164],{},[383,26163,547],{},[380,26165,26166,26168],{},[383,26167,11044],{},[383,26169,22265],{},[1645,26171,1652],{},[380,26173,26174,26176,26178,26180,26182],{},[458,26175,461],{"fence":394},[383,26177,561],{},[383,26179,558],{"mathvariant":557},[383,26181,25939],{},[458,26183,468],{"fence":394},[383,26185,4723],{"mathvariant":557},[387,26187,26188],{"encoding":389},"D_{JS}(P\\|Q)=\\frac12 D_{KL}\\!\\left(P\\|M\\right)\n+\\frac12 D_{KL}\\!\\left(Q\\|M\\right).",[363,26190,26192,26269,26417],{"className":26191,"ariaHidden":394},[393],[363,26193,26195,26198,26245,26248,26251,26254,26257,26260,26263,26266],{"className":26194},[398],[363,26196],{"className":26197,"style":487},[402],[363,26199,26201,26204],{"className":26200},[407],[363,26202,547],{"className":26203,"style":625},[407,408],[363,26205,26207],{"className":26206},[629],[363,26208,26210,26237],{"className":26209},[633,634],[363,26211,26213,26234],{"className":26212},[638],[363,26214,26216],{"className":26215,"style":727},[642],[363,26217,26218,26221],{"style":646},[363,26219],{"className":26220,"style":651},[650],[363,26222,26224],{"className":26223},[655,656,657,658],[363,26225,26227,26230],{"className":26226},[407,658],[363,26228,24115],{"className":26229,"style":24525},[407,408,658],[363,26231,26109],{"className":26232,"style":26233},[407,408,658],"margin-right:0.0576em;",[363,26235,666],{"className":26236},[665],[363,26238,26240],{"className":26239},[638],[363,26241,26243],{"className":26242,"style":8316},[642],[363,26244],{},[363,26246,461],{"className":26247},[494],[363,26249,554],{"className":26250,"style":682},[407,408],[363,26252,558],{"className":26253},[407],[363,26255,561],{"className":26256},[407,408],[363,26258,468],{"className":26259},[501],[363,26261],{"className":26262,"style":506},[505],[363,26264,471],{"className":26265},[510],[363,26267],{"className":26268,"style":506},[505],[363,26270,26272,26276,26338,26384,26387,26390,26408,26411,26414],{"className":26271},[398],[363,26273],{"className":26274,"style":26275},[402],"height:2.0074em;vertical-align:-0.686em;",[363,26277,26279,26282,26335],{"className":26278},[407],[363,26280],{"className":26281},[494,789],[363,26283,26285],{"className":26284},[586],[363,26286,26288,26327],{"className":26287},[633,634],[363,26289,26291,26324],{"className":26290},[638],[363,26292,26294,26305,26313],{"className":26293,"style":23261},[642],[363,26295,26296,26299],{"style":805},[363,26297],{"className":26298,"style":809},[650],[363,26300,26302],{"className":26301},[407],[363,26303,2811],{"className":26304},[407],[363,26306,26307,26310],{"style":821},[363,26308],{"className":26309,"style":809},[650],[363,26311],{"className":26312,"style":829},[828],[363,26314,26315,26318],{"style":832},[363,26316],{"className":26317,"style":809},[650],[363,26319,26321],{"className":26320},[407],[363,26322,465],{"className":26323},[407],[363,26325,666],{"className":26326},[665],[363,26328,26330],{"className":26329},[638],[363,26331,26333],{"className":26332,"style":4126},[642],[363,26334],{},[363,26336],{"className":26337},[501,789],[363,26339,26341,26344],{"className":26340},[407],[363,26342,547],{"className":26343,"style":625},[407,408],[363,26345,26347],{"className":26346},[629],[363,26348,26350,26376],{"className":26349},[633,634],[363,26351,26353,26373],{"className":26352},[638],[363,26354,26356],{"className":26355,"style":727},[642],[363,26357,26358,26361],{"style":646},[363,26359],{"className":26360,"style":651},[650],[363,26362,26364],{"className":26363},[655,656,657,658],[363,26365,26367,26370],{"className":26366},[407,658],[363,26368,11044],{"className":26369,"style":11132},[407,408,658],[363,26371,22265],{"className":26372},[407,408,658],[363,26374,666],{"className":26375},[665],[363,26377,26379],{"className":26378},[638],[363,26380,26382],{"className":26381,"style":8316},[642],[363,26383],{},[363,26385],{"className":26386,"style":1860},[505],[363,26388],{"className":26389,"style":754},[505],[363,26391,26393,26396,26399,26402,26405],{"className":26392},[758],[363,26394,461],{"className":26395,"style":763},[494,762],[363,26397,554],{"className":26398,"style":682},[407,408],[363,26400,558],{"className":26401},[407],[363,26403,25939],{"className":26404,"style":25973},[407,408],[363,26406,468],{"className":26407,"style":763},[501,762],[363,26409],{"className":26410,"style":2935},[505],[363,26412,2818],{"className":26413},[2939],[363,26415],{"className":26416,"style":2935},[505],[363,26418,26420,26423,26485,26531,26534,26537,26555,26558],{"className":26419},[398],[363,26421],{"className":26422,"style":26275},[402],[363,26424,26426,26429,26482],{"className":26425},[407],[363,26427],{"className":26428},[494,789],[363,26430,26432],{"className":26431},[586],[363,26433,26435,26474],{"className":26434},[633,634],[363,26436,26438,26471],{"className":26437},[638],[363,26439,26441,26452,26460],{"className":26440,"style":23261},[642],[363,26442,26443,26446],{"style":805},[363,26444],{"className":26445,"style":809},[650],[363,26447,26449],{"className":26448},[407],[363,26450,2811],{"className":26451},[407],[363,26453,26454,26457],{"style":821},[363,26455],{"className":26456,"style":809},[650],[363,26458],{"className":26459,"style":829},[828],[363,26461,26462,26465],{"style":832},[363,26463],{"className":26464,"style":809},[650],[363,26466,26468],{"className":26467},[407],[363,26469,465],{"className":26470},[407],[363,26472,666],{"className":26473},[665],[363,26475,26477],{"className":26476},[638],[363,26478,26480],{"className":26479,"style":4126},[642],[363,26481],{},[363,26483],{"className":26484},[501,789],[363,26486,26488,26491],{"className":26487},[407],[363,26489,547],{"className":26490,"style":625},[407,408],[363,26492,26494],{"className":26493},[629],[363,26495,26497,26523],{"className":26496},[633,634],[363,26498,26500,26520],{"className":26499},[638],[363,26501,26503],{"className":26502,"style":727},[642],[363,26504,26505,26508],{"style":646},[363,26506],{"className":26507,"style":651},[650],[363,26509,26511],{"className":26510},[655,656,657,658],[363,26512,26514,26517],{"className":26513},[407,658],[363,26515,11044],{"className":26516,"style":11132},[407,408,658],[363,26518,22265],{"className":26519},[407,408,658],[363,26521,666],{"className":26522},[665],[363,26524,26526],{"className":26525},[638],[363,26527,26529],{"className":26528,"style":8316},[642],[363,26530],{},[363,26532],{"className":26533,"style":1860},[505],[363,26535],{"className":26536,"style":754},[505],[363,26538,26540,26543,26546,26549,26552],{"className":26539},[758],[363,26541,461],{"className":26542,"style":763},[494,762],[363,26544,561],{"className":26545},[407,408],[363,26547,558],{"className":26548},[407],[363,26550,25939],{"className":26551,"style":25973},[407,408],[363,26553,468],{"className":26554,"style":763},[501,762],[363,26556],{"className":26557,"style":754},[505],[363,26559,4723],{"className":26560},[407],[11,26562,26563,26566,26567,26595,26596,26651,26652,26680,26681,26832],{},[1344,26564,26565],{},"它是单个 f-散度。"," 这一点很容易被误解。我们直接构造 ",[363,26568,26570,26583],{"className":26569,"translate":367},[366],[363,26571,26573],{"className":26572},[371],[373,26574,26575],{"xmlns":375},[377,26576,26577,26581],{},[380,26578,26579],{},[383,26580,385],{},[387,26582,385],{"encoding":389},[363,26584,26586],{"className":26585,"ariaHidden":394},[393],[363,26587,26589,26592],{"className":26588},[398],[363,26590],{"className":26591,"style":403},[402],[363,26593,385],{"className":26594,"style":409},[407,408],"：代入 ",[363,26597,26599,26619],{"className":26598,"translate":367},[366],[363,26600,26602],{"className":26601},[371],[373,26603,26604],{"xmlns":375},[377,26605,26606,26616],{},[380,26607,26608,26610,26612,26614],{},[383,26609,11],{},[458,26611,471],{},[383,26613,1116],{},[383,26615,1291],{},[387,26617,26618],{"encoding":389},"p=tq",[363,26620,26622,26640],{"className":26621,"ariaHidden":394},[393],[363,26623,26625,26628,26631,26634,26637],{"className":26624},[398],[363,26626],{"className":26627,"style":2624},[402],[363,26629,11],{"className":26630},[407,408],[363,26632],{"className":26633,"style":506},[505],[363,26635,471],{"className":26636},[510],[363,26638],{"className":26639,"style":506},[505],[363,26641,26643,26647],{"className":26642},[398],[363,26644],{"className":26645,"style":26646},[402],"height:0.8095em;vertical-align:-0.1944em;",[363,26648,26650],{"className":26649,"style":1328},[407,408],"tq","，于是 ",[363,26653,26655,26668],{"className":26654,"translate":367},[366],[363,26656,26658],{"className":26657},[371],[373,26659,26660],{"xmlns":375},[377,26661,26662,26666],{},[380,26663,26664],{},[383,26665,25939],{},[387,26667,25939],{"encoding":389},[363,26669,26671],{"className":26670,"ariaHidden":394},[393],[363,26672,26674,26677],{"className":26673},[398],[363,26675],{"className":26676,"style":1011},[402],[363,26678,25939],{"className":26679,"style":25973},[407,408]," 的密度为 ",[363,26682,26684,26719],{"className":26683,"translate":367},[366],[363,26685,26687],{"className":26686},[371],[373,26688,26689],{"xmlns":375},[377,26690,26691,26716],{},[380,26692,26693,26696,26698],{},[383,26694,26695],{},"m",[458,26697,471],{},[586,26699,26700,26714],{},[380,26701,26702,26704,26706,26708,26710,26712],{},[458,26703,461],{"stretchy":460},[383,26705,1116],{},[458,26707,2818],{},[463,26709,465],{},[458,26711,468],{"stretchy":460},[383,26713,1291],{},[463,26715,2811],{},[387,26717,26718],{"encoding":389},"m=\\frac{(t+1)q}{2}",[363,26720,26722,26740],{"className":26721,"ariaHidden":394},[393],[363,26723,26725,26728,26731,26734,26737],{"className":26724},[398],[363,26726],{"className":26727,"style":2693},[402],[363,26729,26695],{"className":26730},[407,408],[363,26732],{"className":26733,"style":506},[505],[363,26735,471],{"className":26736},[510],[363,26738],{"className":26739,"style":506},[505],[363,26741,26743,26747],{"className":26742},[398],[363,26744],{"className":26745,"style":26746},[402],"height:1.355em;vertical-align:-0.345em;",[363,26748,26750,26753,26829],{"className":26749},[407],[363,26751],{"className":26752},[494,789],[363,26754,26756],{"className":26755},[586],[363,26757,26759,26821],{"className":26758},[633,634],[363,26760,26762,26818],{"className":26761},[638],[363,26763,26766,26780,26788],{"className":26764,"style":26765},[642],"height:1.01em;",[363,26767,26768,26771],{"style":2875},[363,26769],{"className":26770,"style":809},[650],[363,26772,26774],{"className":26773},[655,656,657,658],[363,26775,26777],{"className":26776},[407,658],[363,26778,2811],{"className":26779},[407,658],[363,26781,26782,26785],{"style":821},[363,26783],{"className":26784,"style":809},[650],[363,26786],{"className":26787,"style":829},[828],[363,26789,26791,26794],{"style":26790},"top:-3.485em;",[363,26792],{"className":26793,"style":809},[650],[363,26795,26797],{"className":26796},[655,656,657,658],[363,26798,26800,26803,26806,26809,26812,26815],{"className":26799},[407,658],[363,26801,461],{"className":26802},[494,658],[363,26804,1116],{"className":26805},[407,408,658],[363,26807,2818],{"className":26808},[2939,658],[363,26810,465],{"className":26811},[407,658],[363,26813,468],{"className":26814},[501,658],[363,26816,1291],{"className":26817,"style":1328},[407,408,658],[363,26819,666],{"className":26820},[665],[363,26822,26824],{"className":26823},[638],[363,26825,26827],{"className":26826,"style":2920},[642],[363,26828],{},[363,26830],{"className":26831},[501,789],"，逐项算：",[363,26834,26836],{"className":26835,"translate":367},[528],[363,26837,26839,26901],{"className":26838,"translate":367},[366],[363,26840,26842],{"className":26841},[371],[373,26843,26844],{"xmlns":375,"display":537},[377,26845,26846,26898],{},[380,26847,26848,26854,26856,26872,26874,26876,26882,26884,26896],{},[586,26849,26850,26852],{},[383,26851,11],{},[383,26853,26695],{},[458,26855,471],{},[586,26857,26858,26864],{},[380,26859,26860,26862],{},[463,26861,2811],{},[383,26863,1116],{},[380,26865,26866,26868,26870],{},[383,26867,1116],{},[458,26869,2818],{},[463,26871,465],{},[458,26873,898],{"separator":394},[505,26875],{"width":3610},[586,26877,26878,26880],{},[383,26879,1291],{},[383,26881,26695],{},[458,26883,471],{},[586,26885,26886,26888],{},[463,26887,2811],{},[380,26889,26890,26892,26894],{},[383,26891,1116],{},[458,26893,2818],{},[463,26895,465],{},[458,26897,898],{"separator":394},[387,26899,26900],{"encoding":389},"\\frac{p}{m}=\\frac{2t}{t+1},\\qquad\n\\frac{q}{m}=\\frac{2}{t+1},",[363,26902,26904,26982,27146],{"className":26903,"ariaHidden":394},[393],[363,26905,26907,26911,26973,26976,26979],{"className":26906},[398],[363,26908],{"className":26909,"style":26910},[402],"height:1.7936em;vertical-align:-0.686em;",[363,26912,26914,26917,26970],{"className":26913},[407],[363,26915],{"className":26916},[494,789],[363,26918,26920],{"className":26919},[586],[363,26921,26923,26962],{"className":26922},[633,634],[363,26924,26926,26959],{"className":26925},[638],[363,26927,26929,26940,26948],{"className":26928,"style":3902},[642],[363,26930,26931,26934],{"style":805},[363,26932],{"className":26933,"style":809},[650],[363,26935,26937],{"className":26936},[407],[363,26938,26695],{"className":26939},[407,408],[363,26941,26942,26945],{"style":821},[363,26943],{"className":26944,"style":809},[650],[363,26946],{"className":26947,"style":829},[828],[363,26949,26950,26953],{"style":832},[363,26951],{"className":26952,"style":809},[650],[363,26954,26956],{"className":26955},[407],[363,26957,11],{"className":26958},[407,408],[363,26960,666],{"className":26961},[665],[363,26963,26965],{"className":26964},[638],[363,26966,26968],{"className":26967,"style":4126},[642],[363,26969],{},[363,26971],{"className":26972},[501,789],[363,26974],{"className":26975,"style":506},[505],[363,26977,471],{"className":26978},[510],[363,26980],{"className":26981,"style":506},[505],[363,26983,26985,26989,27066,27069,27072,27075,27137,27140,27143],{"className":26984},[398],[363,26986],{"className":26987,"style":26988},[402],"height:2.0908em;vertical-align:-0.7693em;",[363,26990,26992,26995,27063],{"className":26991},[407],[363,26993],{"className":26994},[494,789],[363,26996,26998],{"className":26997},[586],[363,26999,27001,27055],{"className":27000},[633,634],[363,27002,27004,27052],{"className":27003},[638],[363,27005,27007,27030,27038],{"className":27006,"style":23261},[642],[363,27008,27009,27012],{"style":805},[363,27010],{"className":27011,"style":809},[650],[363,27013,27015,27018,27021,27024,27027],{"className":27014},[407],[363,27016,1116],{"className":27017},[407,408],[363,27019],{"className":27020,"style":2935},[505],[363,27022,2818],{"className":27023},[2939],[363,27025],{"className":27026,"style":2935},[505],[363,27028,465],{"className":27029},[407],[363,27031,27032,27035],{"style":821},[363,27033],{"className":27034,"style":809},[650],[363,27036],{"className":27037,"style":829},[828],[363,27039,27040,27043],{"style":832},[363,27041],{"className":27042,"style":809},[650],[363,27044,27046,27049],{"className":27045},[407],[363,27047,2811],{"className":27048},[407],[363,27050,1116],{"className":27051},[407,408],[363,27053,666],{"className":27054},[665],[363,27056,27058],{"className":27057},[638],[363,27059,27061],{"className":27060,"style":17365},[642],[363,27062],{},[363,27064],{"className":27065},[501,789],[363,27067,898],{"className":27068},[920],[363,27070],{"className":27071,"style":3982},[505],[363,27073],{"className":27074,"style":754},[505],[363,27076,27078,27081,27134],{"className":27077},[407],[363,27079],{"className":27080},[494,789],[363,27082,27084],{"className":27083},[586],[363,27085,27087,27126],{"className":27086},[633,634],[363,27088,27090,27123],{"className":27089},[638],[363,27091,27093,27104,27112],{"className":27092,"style":3902},[642],[363,27094,27095,27098],{"style":805},[363,27096],{"className":27097,"style":809},[650],[363,27099,27101],{"className":27100},[407],[363,27102,26695],{"className":27103},[407,408],[363,27105,27106,27109],{"style":821},[363,27107],{"className":27108,"style":809},[650],[363,27110],{"className":27111,"style":829},[828],[363,27113,27114,27117],{"style":832},[363,27115],{"className":27116,"style":809},[650],[363,27118,27120],{"className":27119},[407],[363,27121,1291],{"className":27122,"style":1328},[407,408],[363,27124,666],{"className":27125},[665],[363,27127,27129],{"className":27128},[638],[363,27130,27132],{"className":27131,"style":4126},[642],[363,27133],{},[363,27135],{"className":27136},[501,789],[363,27138],{"className":27139,"style":506},[505],[363,27141,471],{"className":27142},[510],[363,27144],{"className":27145,"style":506},[505],[363,27147,27149,27152,27226],{"className":27148},[398],[363,27150],{"className":27151,"style":26988},[402],[363,27153,27155,27158,27223],{"className":27154},[407],[363,27156],{"className":27157},[494,789],[363,27159,27161],{"className":27160},[586],[363,27162,27164,27215],{"className":27163},[633,634],[363,27165,27167,27212],{"className":27166},[638],[363,27168,27170,27193,27201],{"className":27169,"style":23261},[642],[363,27171,27172,27175],{"style":805},[363,27173],{"className":27174,"style":809},[650],[363,27176,27178,27181,27184,27187,27190],{"className":27177},[407],[363,27179,1116],{"className":27180},[407,408],[363,27182],{"className":27183,"style":2935},[505],[363,27185,2818],{"className":27186},[2939],[363,27188],{"className":27189,"style":2935},[505],[363,27191,465],{"className":27192},[407],[363,27194,27195,27198],{"style":821},[363,27196],{"className":27197,"style":809},[650],[363,27199],{"className":27200,"style":829},[828],[363,27202,27203,27206],{"style":832},[363,27204],{"className":27205,"style":809},[650],[363,27207,27209],{"className":27208},[407],[363,27210,2811],{"className":27211},[407],[363,27213,666],{"className":27214},[665],[363,27216,27218],{"className":27217},[638],[363,27219,27221],{"className":27220,"style":17365},[642],[363,27222],{},[363,27224],{"className":27225},[501,789],[363,27227,898],{"className":27228},[920],[363,27230,27232],{"className":27231,"translate":367},[528],[363,27233,27235,27375],{"className":27234,"translate":367},[366],[363,27236,27238],{"className":27237},[371],[373,27239,27240],{"xmlns":375,"display":537},[377,27241,27242,27372],{},[1591,27243,27244,27324],{"rowspacing":1593,"columnalign":1594,"columnspacing":1595},[1597,27245,27246,27272],{},[1600,27247,27248],{},[1603,27249,27250],{"scriptlevel":474,"displaystyle":394},[380,27251,27252,27262,27264,27266,27268,27270],{},[543,27253,27254,27256],{},[383,27255,547],{},[380,27257,27258,27260],{},[383,27259,24115],{},[383,27261,26109],{},[458,27263,461],{"stretchy":460},[383,27265,554],{},[383,27267,558],{"mathvariant":557},[383,27269,561],{},[458,27271,468],{"stretchy":460},[1600,27273,27274],{},[1603,27275,27276],{"scriptlevel":474,"displaystyle":394},[380,27277,27278,27280,27282,27288,27290,27292,27294,27296,27302,27304,27310,27312,27314,27316,27318],{},[380,27279],{},[458,27281,471],{},[586,27283,27284,27286],{},[463,27285,465],{},[463,27287,2811],{},[458,27289,1635],{},[383,27291,11],{},[383,27293,4448],{},[458,27295,4451],{},[586,27297,27298,27300],{},[383,27299,11],{},[383,27301,26695],{},[458,27303,2818],{},[586,27305,27306,27308],{},[463,27307,465],{},[463,27309,2811],{},[458,27311,1635],{},[383,27313,1291],{},[383,27315,4448],{},[458,27317,4451],{},[586,27319,27320,27322],{},[383,27321,1291],{},[383,27323,26695],{},[1597,27325,27326,27332],{},[1600,27327,27328],{},[1603,27329,27330],{"scriptlevel":474,"displaystyle":394},[380,27331],{},[1600,27333,27334],{},[1603,27335,27336],{"scriptlevel":474,"displaystyle":394},[380,27337,27338,27340,27342,27344,27346,27348,27358,27360,27362,27364,27366,27368,27370],{},[380,27339],{},[458,27341,471],{},[458,27343,1635],{},[383,27345,1291],{},[1645,27347,1647],{},[543,27349,27350,27352],{},[383,27351,385],{},[380,27353,27354,27356],{},[383,27355,24115],{},[383,27357,26109],{},[458,27359,461],{"stretchy":460},[383,27361,1116],{},[458,27363,468],{"stretchy":460},[1645,27365,1647],{},[383,27367,592],{},[383,27369,1121],{},[458,27371,898],{"separator":394},[387,27373,27374],{"encoding":389},"\\begin{aligned}\nD_{JS}(P\\|Q)\n&=\\frac12\\int 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f_{JS}(1\u002Ft)=f_{JS}(t)",[363,30115,30117,30193],{"className":30116,"ariaHidden":394},[393],[363,30118,30120,30123,30126,30172,30175,30178,30181,30184,30187,30190],{"className":30119},[398],[363,30121],{"className":30122,"style":487},[402],[363,30124,1116],{"className":30125},[407,408],[363,30127,30129,30132],{"className":30128},[407],[363,30130,385],{"className":30131,"style":409},[407,408],[363,30133,30135],{"className":30134},[629],[363,30136,30138,30164],{"className":30137},[633,634],[363,30139,30141,30161],{"className":30140},[638],[363,30142,30144],{"className":30143,"style":727},[642],[363,30145,30146,30149],{"style":27881},[363,30147],{"className":30148,"style":651},[650],[363,30150,30152],{"className":30151},[655,656,657,658],[363,30153,30155,30158],{"className":30154},[407,658],[363,30156,24115],{"className":30157,"style":24525},[407,408,658],[363,30159,26109],{"className":30160,"style":26233},[407,408,658],[363,30162,666],{"className":30163},[665],[363,30165,30167],{"className":30166},[638],[363,30168,30170],{"className":30169,"style":8316},[642],[363,30171],{},[363,30173,461],{"className":30174},[494],[363,30176,8937],{"className":30177},[407],[363,30179,1116],{"className":30180},[407,408],[363,30182,468],{"className":30183},[501],[363,30185],{"className":30186,"style":506},[505],[363,30188,471],{"className":30189},[510],[363,30191],{"className":30192,"style":506},[505],[363,30194,30196,30199,30245,30248,30251],{"className":30195},[398],[363,30197],{"className":30198,"style":487},[402],[363,30200,30202,30205],{"className":30201},[407],[363,30203,385],{"className":30204,"style":409},[407,408],[363,30206,30208],{"className":30207},[629],[363,30209,30211,30237],{"className":30210},[633,634],[363,30212,30214,30234],{"className":30213},[638],[363,30215,30217],{"className":30216,"style":727},[642],[363,30218,30219,30222],{"style":27881},[363,30220],{"className":30221,"style":651},[650],[363,30223,30225],{"className":30224},[655,656,657,658],[363,30226,30228,30231],{"className":30227},[407,658],[363,30229,24115],{"className":30230,"style":24525},[407,408,658],[363,30232,26109],{"className":30233,"style":26233},[407,408,658],[363,30235,666],{"className":30236},[665],[363,30238,30240],{"className":30239},[638],[363,30241,30243],{"className":30242,"style":8316},[642],[363,30244],{},[363,30246,461],{"className":30247},[494],[363,30249,1116],{"className":30250},[407,408],[363,30252,468],{"className":30253},[501],"。因此当 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时",[363,30307,30309],{"className":30308,"translate":367},[528],[363,30310,30312,30440],{"className":30311,"translate":367},[366],[363,30313,30315],{"className":30314},[371],[373,30316,30317],{"xmlns":375,"display":537},[377,30318,30319,30437],{},[380,30320,30321,30331,30333,30335,30337,30339,30341,30343,30357,30359,30361,30363,30365,30367,30369,30375,30381,30383,30385,30387,30389,30391,30393,30399,30405,30407,30409,30411,30413,30415,30417,30419,30421,30423,30425,30427,30429,30431,30433,30435],{},[543,30322,30323,30325],{},[383,30324,547],{},[380,30326,30327,30329],{},[383,30328,24115],{},[383,30330,26109],{},[458,30332,461],{"stretchy":460},[383,30334,554],{},[383,30336,558],{"mathvariant":557},[383,30338,561],{},[458,30340,468],{"stretchy":460},[458,30342,471],{},[543,30344,30345,30347],{},[383,30346,547],{},[543,30348,30349,30351],{},[383,30350,385],{},[380,30352,30353,30355],{},[383,30354,24115],{},[383,30356,26109],{},[458,30358,461],{"stretchy":460},[383,30360,554],{},[383,30362,558],{"mathvariant":557},[383,30364,561],{},[458,30366,468],{"stretchy":460},[458,30368,471],{},[586,30370,30371,30373],{},[463,30372,465],{},[463,30374,2811],{},[543,30376,30377,30379],{},[383,30378,547],{},[383,30380,385],{},[458,30382,461],{"stretchy":460},[383,30384,554],{},[383,30386,558],{"mathvariant":557},[383,30388,25939],{},[458,30390,468],{"stretchy":460},[458,30392,2818],{},[586,30394,30395,30397],{},[463,30396,465],{},[463,30398,2811],{},[543,30400,30401,30403],{},[383,30402,547],{},[383,30404,385],{},[458,30406,461],{"stretchy":460},[383,30408,561],{},[383,30410,558],{"mathvariant":557},[383,30412,25939],{},[458,30414,468],{"stretchy":460},[458,30416,898],{"separator":394},[505,30418],{"width":3610},[458,30420,461],{"stretchy":460},[383,30422,385],{},[458,30424,471],{},[383,30426,1116],{},[383,30428,4448],{},[458,30430,4451],{},[383,30432,1116],{},[458,30434,468],{"stretchy":460},[458,30436,898],{"separator":394},[387,30438,30439],{"encoding":389},"D_{JS}(P\\|Q)=D_{f_{JS}}(P\\|Q)\n=\\frac12D_f(P\\|M)\n+\\frac12D_f(Q\\|M),\\qquad(f=t\\log t),",[363,30441,30443,30519,30639,30771,30918],{"className":30442,"ariaHidden":394},[393],[363,30444,30446,30449,30495,30498,30501,30504,30507,30510,30513,30516],{"className":30445},[398],[363,30447],{"className":30448,"style":487},[402],[363,30450,30452,30455],{"className":30451},[407],[363,30453,547],{"className":30454,"style":625},[407,408],[363,30456,30458],{"className":30457},[629],[363,30459,30461,30487],{"className":30460},[633,634],[363,30462,30464,30484],{"className":30463},[638],[363,30465,30467],{"className":30466,"style":727},[642],[363,30468,30469,30472],{"style":646},[363,30470],{"className":30471,"style":651},[650],[363,30473,30475],{"className":30474},[655,656,657,658],[363,30476,30478,30481],{"className":30477},[407,658],[363,30479,24115],{"className":30480,"style":24525},[407,408,658],[363,30482,26109],{"className":30483,"style":26233},[407,408,658],[363,30485,666],{"className":30486},[665],[363,30488,30490],{"className":30489},[638],[363,30491,30493],{"className":30492,"style":8316},[642],[363,30494],{},[363,30496,461],{"className":30497},[494],[363,30499,554],{"className":30500,"style":682},[407,408],[363,30502,558],{"className":30503},[407],[363,30505,561],{"className":30506},[407,408],[363,30508,468],{"className":30509},[501],[363,30511],{"className":30512,"style":506},[505],[363,30514,471],{"className":30515},[510],[363,30517],{"className":30518,"style":506},[505],[363,30520,30522,30525,30615,30618,30621,30624,30627,30630,30633,30636],{"className":30521},[398],[363,30523],{"className":30524,"style":618},[402],[363,30526,30528,30531],{"className":30527},[407],[363,30529,547],{"className":30530,"style":625},[407,408],[363,30532,30534],{"className":30533},[629],[363,30535,30537,30607],{"className":30536},[633,634],[363,30538,30540,30604],{"className":30539},[638],[363,30541,30543],{"className":30542,"style":643},[642],[363,30544,30545,30548],{"style":646},[363,30546],{"className":30547,"style":651},[650],[363,30549,30551],{"className":30550},[655,656,657,658],[363,30552,30554],{"className":30553},[407,658],[363,30555,30557,30560],{"className":30556},[407,658],[363,30558,385],{"className":30559,"style":409},[407,408,658],[363,30561,30563],{"className":30562},[629],[363,30564,30566,30595],{"className":30565},[633,634],[363,30567,30569,30592],{"className":30568},[638],[363,30570,30572],{"className":30571,"style":6446},[642],[363,30573,30575,30579],{"style":30574},"top:-2.3567em;margin-left:-0.1076em;margin-right:0.0714em;",[363,30576],{"className":30577,"style":30578},[650],"height:2.5em;",[363,30580,30583],{"className":30581},[655,30582,10022,658],"reset-size3",[363,30584,30586,30589],{"className":30585},[407,658],[363,30587,24115],{"className":30588,"style":24525},[407,408,658],[363,30590,26109],{"className":30591,"style":26233},[407,408,658],[363,30593,666],{"className":30594},[665],[363,30596,30598],{"className":30597},[638],[363,30599,30602],{"className":30600,"style":30601},[642],"height:0.1433em;",[363,30603],{},[363,30605,666],{"className":30606},[665],[363,30608,30610],{"className":30609},[638],[363,30611,30613],{"className":30612,"style":673},[642],[363,30614],{},[363,30616,461],{"className":30617},[494],[363,30619,554],{"className":30620,"style":682},[407,408],[363,30622,558],{"className":30623},[407],[363,30625,561],{"className":30626},[407,408],[363,30628,468],{"className":30629},[501],[363,30631],{"className":30632,"style":506},[505],[363,30634,471],{"className":30635},[510],[363,30637],{"className":30638,"style":506},[505],[363,30640,30642,30645,30707,30747,30750,30753,30756,30759,30762,30765,30768],{"className":30641},[398],[363,30643],{"className":30644,"style":26275},[402],[363,30646,30648,30651,30704],{"className":30647},[407],[363,30649],{"className":30650},[494,789],[363,30652,30654],{"className":30653},[586],[363,30655,30657,30696],{"className":30656},[633,634],[363,30658,30660,30693],{"className":30659},[638],[363,30661,30663,30674,30682],{"className":30662,"style":23261},[642],[363,30664,30665,30668],{"style":805},[363,30666],{"className":30667,"style":809},[650],[363,30669,30671],{"className":30670},[407],[363,30672,2811],{"className":30673},[407],[363,30675,30676,30679],{"style":821},[363,30677],{"className":30678,"style":809},[650],[363,30680],{"className":30681,"style":829},[828],[363,30683,30684,30687],{"style":832},[363,30685],{"className":30686,"style":809},[650],[363,30688,30690],{"className":30689},[407],[363,30691,465],{"className":30692},[407],[363,30694,666],{"className":30695},[665],[363,30697,30699],{"className":30698},[638],[363,30700,30702],{"className":30701,"style":4126},[642],[363,30703],{},[363,30705],{"className":30706},[501,789],[363,30708,30710,30713],{"className":30709},[407],[363,30711,547],{"className":30712,"style":625},[407,408],[363,30714,30716],{"className":30715},[629],[363,30717,30719,30739],{"className":30718},[633,634],[363,30720,30722,30736],{"className":30721},[638],[363,30723,30725],{"className":30724,"style":643},[642],[363,30726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既是「两个 f-散度（相对 ",[363,30951,30953,30966],{"className":30952,"translate":367},[366],[363,30954,30956],{"className":30955},[371],[373,30957,30958],{"xmlns":375},[377,30959,30960,30964],{},[380,30961,30962],{},[383,30963,25939],{},[387,30965,25939],{"encoding":389},[363,30967,30969],{"className":30968,"ariaHidden":394},[393],[363,30970,30972,30975],{"className":30971},[398],[363,30973],{"className":30974,"style":1011},[402],[363,30976,25939],{"className":30977,"style":25973},[407,408],"）的平均」，也是「一个对称 f-散度」。原文「不是单一 ",[363,30980,30982,30999],{"className":30981,"translate":367},[366],[363,30983,30985],{"className":30984},[371],[373,30986,30987],{"xmlns":375},[377,30988,30989,30997],{},[380,30990,30991],{},[543,30992,30993,30995],{},[383,30994,547],{},[383,30996,385],{},[387,30998,4370],{"encoding":389},[363,31000,31002],{"className":31001,"ariaHidden":394},[393],[363,31003,31005,31008],{"className":31004},[398],[363,31006],{"className":31007,"style":4380},[402],[363,31009,31011,31014],{"className":31010},[407],[363,31012,547],{"className":31013,"style":625},[407,408],[363,31015,31017],{"className":31016},[629],[363,31018,31020,31040],{"className":31019},[633,634],[363,31021,31023,31037],{"className":31022},[638],[363,31024,31026],{"className":31025,"style":643},[642],[363,31027,31028,31031],{"style":646},[363,31029],{"className":31030,"style":651},[650],[363,31032,31034],{"className":31033},[655,656,657,658],[363,31035,385],{"className":31036,"style":409},[407,408,658],[363,31038,666],{"className":31039},[665],[363,31041,31043],{"className":31042},[638],[363,31044,31046],{"className":31045,"style":673},[642],[363,31047],{},"」的说法可修正为：",[1344,31050,31051,31052,31102,31103,31183],{},"在 ",[363,31053,31055,31072],{"className":31054,"translate":367},[366],[363,31056,31058],{"className":31057},[371],[373,31059,31060],{"xmlns":375},[377,31061,31062,31070],{},[380,31063,31064,31066,31068],{},[383,31065,554],{},[458,31067,1061],{},[383,31069,561],{},[387,31071,1066],{"encoding":389},[363,31073,31075,31093],{"className":31074,"ariaHidden":394},[393],[363,31076,31078,31081,31084,31087,31090],{"className":31077},[398],[363,31079],{"className":31080,"style":1076},[402],[363,31082,554],{"className":31083,"style":682},[407,408],[363,31085],{"className":31086,"style":506},[505],[363,31088,1061],{"className":31089},[510],[363,31091],{"className":31092,"style":506},[505],[363,31094,31096,31099],{"className":31095},[398],[363,31097],{"className":31098,"style":913},[402],[363,31100,561],{"className":31101},[407,408]," 时它是单一 ",[363,31104,31106,31128],{"className":31105,"translate":367},[366],[363,31107,31109],{"className":31108},[371],[373,31110,31111],{"xmlns":375},[377,31112,31113,31125],{},[380,31114,31115],{},[543,31116,31117,31119],{},[383,31118,385],{},[380,31120,31121,31123],{},[383,31122,24115],{},[383,31124,26109],{},[387,31126,31127],{"encoding":389},"f_{JS}",[363,31129,31131],{"className":31130,"ariaHidden":394},[393],[363,31132,31134,31137],{"className":31133},[398],[363,31135],{"className":31136,"style":403},[402],[363,31138,31140,31143],{"className":31139},[407],[363,31141,385],{"className":31142,"style":409},[407,408],[363,31144,31146],{"className":31145},[629],[363,31147,31149,31175],{"className":31148},[633,634],[363,31150,31152,31172],{"className":31151},[638],[363,31153,31155],{"className":31154,"style":727},[642],[363,31156,31157,31160],{"style":27881},[363,31158],{"className":31159,"style":651},[650],[363,31161,31163],{"className":31162},[655,656,657,658],[363,31164,31166,31169],{"className":31165},[407,658],[363,31167,24115],{"className":31168,"style":24525},[407,408,658],[363,31170,26109],{"className":31171,"style":26233},[407,408,658],[363,31173,666],{"className":31174},[665],[363,31176,31178],{"className":31177},[638],[363,31179,31181],{"className":31180,"style":8316},[642],[363,31182],{}," 的 f-散度","；一般情形则用下面的双变量形式。",[11,31186,31187,31190,31191,31232,31233,31348,31349,31352],{},[1344,31188,31189],{},"一般情形的双变量形式."," 当 ",[363,31192,31194,31211],{"className":31193,"translate":367},[366],[363,31195,31197],{"className":31196},[371],[373,31198,31199],{"xmlns":375},[377,31200,31201,31209],{},[380,31202,31203,31205,31207],{},[383,31204,554],{},[458,31206,898],{"separator":394},[383,31208,561],{},[387,31210,903],{"encoding":389},[363,31212,31214],{"className":31213,"ariaHidden":394},[393],[363,31215,31217,31220,31223,31226,31229],{"className":31216},[398],[363,31218],{"className":31219,"style":913},[402],[363,31221,554],{"className":31222,"style":682},[407,408],[363,31224,898],{"className":31225},[920],[363,31227],{"className":31228,"style":754},[505],[363,31230,561],{"className":31231},[407,408]," 支撑集可能分离时，",[363,31234,31236,31272],{"className":31235,"translate":367},[366],[363,31237,31239],{"className":31238},[371],[373,31240,31241],{"xmlns":375},[377,31242,31243,31269],{},[380,31244,31245,31247,31249,31251,31261,31263,31265,31267],{},[458,31246,461],{"stretchy":460},[383,31248,1116],{},[458,31250,898],{"separator":394},[543,31252,31253,31255],{},[383,31254,385],{},[380,31256,31257,31259],{},[383,31258,24115],{},[383,31260,26109],{},[458,31262,461],{"stretchy":460},[383,31264,1116],{},[458,31266,468],{"stretchy":460},[458,31268,468],{"stretchy":460},[387,31270,31271],{"encoding":389},"(t,f_{JS}(t))",[363,31273,31275],{"className":31274,"ariaHidden":394},[393],[363,31276,31278,31281,31284,31287,31290,31293,31339,31342,31345],{"className":31277},[398],[363,31279],{"className":31280,"style":487},[402],[363,31282,461],{"className":31283},[494],[363,31285,1116],{"className":31286},[407,408],[363,31288,898],{"className":31289},[920],[363,31291],{"className":31292,"style":754},[505],[363,31294,31296,31299],{"className":31295},[407],[363,31297,385],{"className":31298,"style":409},[407,408],[363,31300,31302],{"className":31301},[629],[363,31303,31305,31331],{"className":31304},[633,634],[363,31306,31308,31328],{"className":31307},[638],[363,31309,31311],{"className":31310,"style":727},[642],[363,31312,31313,31316],{"style":27881},[363,31314],{"className":31315,"style":651},[650],[363,31317,31319],{"className":31318},[655,656,657,658],[363,31320,31322,31325],{"className":31321},[407,658],[363,31323,24115],{"className":31324,"style":24525},[407,408,658],[363,31326,26109],{"className":31327,"style":26233},[407,408,658],[363,31329,666],{"className":31330},[665],[363,31332,31334],{"className":31333},[638],[363,31335,31337],{"className":31336,"style":8316},[642],[363,31338],{},[363,31340,461],{"className":31341},[494],[363,31343,1116],{"className":31344},[407,408],[363,31346,19419],{"className":31347},[501]," 的写法会漏掉奇异项。此时把 JS 写成 Csiszár 的",[1344,31350,31351],{},"双变量（齐次、联合凸）泛函","最稳妥：",[363,31354,31356],{"className":31355,"translate":367},[528],[363,31357,31359,31416],{"className":31358,"translate":367},[366],[363,31360,31362],{"className":31361},[371],[373,31363,31364],{"xmlns":375,"display":537},[377,31365,31366,31413],{},[380,31367,31368,31378,31380,31382,31384,31386,31388,31390,31392,31395,31397,31399,31401,31403,31405,31407,31409,31411],{},[543,31369,31370,31372],{},[383,31371,547],{},[380,31373,31374,31376],{},[383,31375,24115],{},[383,31377,26109],{},[458,31379,461],{"stretchy":460},[383,31381,554],{},[383,31383,558],{"mathvariant":557},[383,31385,561],{},[458,31387,468],{"stretchy":460},[458,31389,471],{},[458,31391,1635],{},[383,31393,31394],{},"φ",[458,31396,461],{"stretchy":460},[383,31398,11],{},[458,31400,898],{"separator":394},[383,31402,1291],{},[458,31404,468],{"stretchy":460},[1645,31406,1647],{},[383,31408,592],{},[383,31410,2712],{},[458,31412,898],{"separator":394},[387,31414,31415],{"encoding":389},"D_{JS}(P\\|Q)=\\int \\varphi(p,q)\\,d\\mu,",[363,31417,31419,31495],{"className":31418,"ariaHidden":394},[393],[363,31420,31422,31425,31471,31474,31477,31480,31483,31486,31489,31492],{"className":31421},[398],[363,31423],{"className":31424,"style":487},[402],[363,31426,31428,31431],{"className":31427},[407],[363,31429,547],{"className":31430,"style":625},[407,408],[363,31432,31434],{"className":31433},[629],[363,31435,31437,31463],{"className":31436},[633,634],[363,31438,31440,31460],{"className":31439},[638],[363,31441,31443],{"className":31442,"style":727},[642],[363,31444,31445,31448],{"style":646},[363,31446],{"className":31447,"style":651},[650],[363,31449,31451],{"className":31450},[655,656,657,658],[363,31452,31454,31457],{"className":31453},[407,658],[363,31455,24115],{"className":31456,"style":24525},[407,408,658],[363,31458,26109],{"className":31459,"style":26233},[407,408,658],[363,31461,666],{"className":31462},[665],[363,31464,31466],{"className":31465},[638],[363,31467,31469],{"className":31468,"style":8316},[642],[363,31470],{},[363,31472,461],{"className":31473},[494],[363,31475,554],{"className":31476,"style":682},[407,408],[363,31478,558],{"className":31479},[407],[363,31481,561],{"className":31482},[407,408],[363,31484,468],{"className":31485},[501],[363,31487],{"className":31488,"style":506},[505],[363,31490,471],{"className":31491},[510],[363,31493],{"className":31494,"style":506},[505],[363,31496,31498,31501,31504,31507,31510,31513,31516,31519,31522,31525,31528,31531,31534,31537],{"className":31497},[398],[363,31499],{"className":31500,"style":6125},[402],[363,31502,1635],{"className":31503,"style":1835},[1832,1833,1834],[363,31505],{"className":31506,"style":754},[505],[363,31508,31394],{"className":31509},[407,408],[363,31511,461],{"className":31512},[494],[363,31514,11],{"className":31515},[407,408],[363,31517,898],{"className":31518},[920],[363,31520],{"className":31521,"style":754},[505],[363,31523,1291],{"className":31524,"style":1328},[407,408],[363,31526,468],{"className":31527},[501],[363,31529],{"className":31530,"style":754},[505],[363,31532,592],{"className":31533},[407,408],[363,31535,2712],{"className":31536},[407,408],[363,31538,898],{"className":31539},[920],[363,31541,31543],{"className":31542,"translate":367},[528],[363,31544,31546,31630],{"className":31545,"translate":367},[366],[363,31547,31549],{"className":31548},[371],[373,31550,31551],{"xmlns":375,"display":537},[377,31552,31553,31627],{},[380,31554,31555,31557,31559,31561,31563,31565,31567,31569,31575,31577,31579,31581,31583,31599,31601,31603,31605,31607,31623,31625],{},[383,31556,31394],{},[458,31558,461],{"stretchy":460},[383,31560,11],{},[458,31562,898],{"separator":394},[383,31564,1291],{},[458,31566,468],{"stretchy":460},[458,31568,471],{},[586,31570,31571,31573],{},[463,31572,465],{},[463,31574,2811],{},[458,31576,578],{"fence":460,"stretchy":394,"minsize":17789,"maxsize":17789},[383,31578,11],{},[383,31580,4448],{},[458,31582,4451],{},[586,31584,31585,31591],{},[380,31586,31587,31589],{},[463,31588,2811],{},[383,31590,11],{},[380,31592,31593,31595,31597],{},[383,31594,11],{},[458,31596,2818],{},[383,31598,1291],{},[458,31600,2818],{},[383,31602,1291],{},[383,31604,4448],{},[458,31606,4451],{},[586,31608,31609,31615],{},[380,31610,31611,31613],{},[463,31612,2811],{},[383,31614,1291],{},[380,31616,31617,31619,31621],{},[383,31618,11],{},[458,31620,2818],{},[383,31622,1291],{},[458,31624,605],{"fence":460,"stretchy":394,"minsize":17789,"maxsize":17789},[458,31626,898],{"separator":394},[387,31628,31629],{"encoding":389},"\\varphi(p,q)=\\frac12\\Big[\np\\log\\frac{2p}{p+q}+q\\log\\frac{2q}{p+q}\\Big],",[363,31631,31633,31669,31844],{"className":31632,"ariaHidden":394},[393],[363,31634,31636,31639,31642,31645,31648,31651,31654,31657,31660,31663,31666],{"className":31635},[398],[363,31637],{"className":31638,"style":487},[402],[363,31640,31394],{"className":31641},[407,408],[363,31643,461],{"className":31644},[494],[363,31646,11],{"className":31647},[407,408],[363,31649,898],{"className":31650},[920],[363,31652],{"className":31653,"style":754},[505],[363,31655,1291],{"className":31656,"style":1328},[407,408],[363,31658,468],{"className":31659},[501],[363,31661],{"className":31662,"style":506},[505],[363,31664,471],{"className":31665},[510],[363,31667],{"className":31668,"style":506},[505],[363,31670,31672,31676,31738,31744,31747,31750,31755,31758,31835,31838,31841],{"className":31671},[398],[363,31673],{"className":31674,"style":31675},[402],"height:2.2019em;vertical-align:-0.8804em;",[363,31677,31679,31682,31735],{"className":31678},[407],[363,31680],{"className":31681},[494,789],[363,31683,31685],{"className":31684},[586],[363,31686,31688,31727],{"className":31687},[633,634],[363,31689,31691,31724],{"className":31690},[638],[363,31692,31694,31705,31713],{"className":31693,"style":23261},[642],[363,31695,31696,31699],{"style":805},[363,31697],{"className":31698,"style":809},[650],[363,31700,31702],{"className":31701},[407],[363,31703,2811],{"className":31704},[407],[363,31706,31707,31710],{"style":821},[363,31708],{"className":31709,"style":809},[650],[363,31711],{"className":31712,"style":829},[828],[363,31714,31715,31718],{"style":832},[363,31716],{"className":31717,"style":809},[650],[363,31719,31721],{"className":31720},[407],[363,31722,465],{"className":31723},[407],[363,31725,666],{"className":31726},[665],[363,31728,31730],{"className":31729},[638],[363,31731,31733],{"className":31732,"style":4126},[642],[363,31734],{},[363,31736],{"className":31737},[501,789],[363,31739,31741],{"className":31740},[407],[363,31742,578],{"className":31743},[767,4068],[363,31745,11],{"className":31746},[407,408],[363,31748],{"className":31749,"style":754},[505],[363,31751,4502,31753],{"className":31752},[1832],[363,31754,4506],{"style":4505},[363,31756],{"className":31757,"style":754},[505],[363,31759,31761,31764,31832],{"className":31760},[407],[363,31762],{"className":31763},[494,789],[363,31765,31767],{"className":31766},[586],[363,31768,31770,31824],{"className":31769},[633,634],[363,31771,31773,31821],{"className":31772},[638],[363,31774,31776,31799,31807],{"className":31775,"style":23261},[642],[363,31777,31778,31781],{"style":805},[363,31779],{"className":31780,"style":809},[650],[363,31782,31784,31787,31790,31793,31796],{"className":31783},[407],[363,31785,11],{"className":31786},[407,408],[363,31788],{"className":31789,"style":2935},[505],[363,31791,2818],{"className":31792},[2939],[363,31794],{"className":31795,"style":2935},[505],[363,31797,1291],{"className":31798,"style":1328},[407,408],[363,31800,31801,31804],{"style":821},[363,31802],{"className":31803,"style":809},[650],[363,31805],{"className":31806,"style":829},[828],[363,31808,31809,31812],{"style":832},[363,31810],{"className":31811,"style":809},[650],[363,31813,31815,31818],{"className":31814},[407],[363,31816,2811],{"className":31817},[407],[363,31819,11],{"className":31820},[407,408],[363,31822,666],{"className":31823},[665],[363,31825,31827],{"className":31826},[638],[363,31828,31830],{"className":31829,"style":854},[642],[363,31831],{},[363,31833],{"className":31834},[501,789],[363,31836],{"className":31837,"style":2935},[505],[363,31839,2818],{"className":31840},[2939],[363,31842],{"className":31843,"style":2935},[505],[363,31845,31847,31850,31853,31856,31861,31864,31941,31947],{"className":31846},[398],[363,31848],{"className":31849,"style":31675},[402],[363,31851,1291],{"className":31852,"style":1328},[407,408],[363,31854],{"className":31855,"style":754},[505],[363,31857,4502,31859],{"className":31858},[1832],[363,31860,4506],{"style":4505},[363,31862],{"className":31863,"style":754},[505],[363,31865,31867,31870,31938],{"className":31866},[407],[363,31868],{"className":31869},[494,789],[363,31871,31873],{"className":31872},[586],[363,31874,31876,31930],{"className":31875},[633,634],[363,31877,31879,31927],{"className":31878},[638],[363,31880,31882,31905,31913],{"className":31881,"style":23261},[642],[363,31883,31884,31887],{"style":805},[363,31885],{"className":31886,"style":809},[650],[363,31888,31890,31893,31896,31899,31902],{"className":31889},[407],[363,31891,11],{"className":31892},[407,408],[363,31894],{"className":31895,"style":2935},[505],[363,31897,2818],{"className":31898},[2939],[363,31900],{"className":31901,"style":2935},[505],[363,31903,1291],{"className":31904,"style":1328},[407,408],[363,31906,31907,31910],{"style":821},[363,31908],{"className":31909,"style":809},[650],[363,31911],{"className":31912,"style":829},[828],[363,31914,31915,31918],{"style":832},[363,31916],{"className":31917,"style":809},[650],[363,31919,31921,31924],{"className":31920},[407],[363,31922,2811],{"className":31923},[407],[363,31925,1291],{"className":31926,"style":1328},[407,408],[363,31928,666],{"className":31929},[665],[363,31931,31933],{"className":31932},[638],[363,31934,31936],{"className":31935,"style":854},[642],[363,31937],{},[363,31939],{"className":31940},[501,789],[363,31942,31944],{"className":31943},[407],[363,31945,605],{"className":31946},[767,4068],[363,31948,898],{"className":31949},[920],[11,31951,31952,1015,31981,32033,32034,32086,32087,32244,32245,32331,32332,32455],{},[363,31953,31955,31969],{"className":31954,"translate":367},[366],[363,31956,31958],{"className":31957},[371],[373,31959,31960],{"xmlns":375},[377,31961,31962,31966],{},[380,31963,31964],{},[383,31965,31394],{},[387,31967,31968],{"encoding":389},"\\varphi",[363,31970,31972],{"className":31971,"ariaHidden":394},[393],[363,31973,31975,31978],{"className":31974},[398],[363,31976],{"className":31977,"style":2624},[402],[363,31979,31394],{"className":31980},[407,408],[363,31982,31984,32006],{"className":31983,"translate":367},[366],[363,31985,31987],{"className":31986},[371],[373,31988,31989],{"xmlns":375},[377,31990,31991,32003],{},[380,31992,31993,31995,31997,31999,32001],{},[458,31994,461],{"stretchy":460},[383,31996,11],{},[458,31998,898],{"separator":394},[383,32000,1291],{},[458,32002,468],{"stretchy":460},[387,32004,32005],{"encoding":389},"(p,q)",[363,32007,32009],{"className":32008,"ariaHidden":394},[393],[363,32010,32012,32015,32018,32021,32024,32027,32030],{"className":32011},[398],[363,32013],{"className":32014,"style":487},[402],[363,32016,461],{"className":32017},[494],[363,32019,11],{"className":32020},[407,408],[363,32022,898],{"className":32023},[920],[363,32025],{"className":32026,"style":754},[505],[363,32028,1291],{"className":32029,"style":1328},[407,408],[363,32031,468],{"className":32032},[501]," 联合凸且一次齐次。在 ",[363,32035,32037,32055],{"className":32036,"translate":367},[366],[363,32038,32040],{"className":32039},[371],[373,32041,32042],{"xmlns":375},[377,32043,32044,32052],{},[380,32045,32046,32048,32050],{},[383,32047,1291],{},[458,32049,3251],{},[463,32051,474],{},[387,32053,32054],{"encoding":389},"q>0",[363,32056,32058,32077],{"className":32057,"ariaHidden":394},[393],[363,32059,32061,32065,32068,32071,32074],{"className":32060},[398],[363,32062],{"className":32063,"style":32064},[402],"height:0.7335em;vertical-align:-0.1944em;",[363,32066,1291],{"className":32067,"style":1328},[407,408],[363,32069],{"className":32070,"style":506},[505],[363,32072,3251],{"className":32073},[510],[363,32075],{"className":32076,"style":506},[505],[363,32078,32080,32083],{"className":32079},[398],[363,32081],{"className":32082,"style":520},[402],[363,32084,474],{"className":32085},[407]," 处 ",[363,32088,32090,32138],{"className":32089,"translate":367},[366],[363,32091,32093],{"className":32092},[371],[373,32094,32095],{"xmlns":375},[377,32096,32097,32135],{},[380,32098,32099,32101,32103,32105,32107,32109,32111,32113,32115,32117,32119,32129,32131,32133],{},[383,32100,31394],{},[458,32102,461],{"stretchy":460},[383,32104,1116],{},[383,32106,1291],{},[458,32108,898],{"separator":394},[383,32110,1291],{},[458,32112,468],{"stretchy":460},[458,32114,471],{},[383,32116,1291],{},[1645,32118,1647],{},[543,32120,32121,32123],{},[383,32122,385],{},[380,32124,32125,32127],{},[383,32126,24115],{},[383,32128,26109],{},[458,32130,461],{"stretchy":460},[383,32132,1116],{},[458,32134,468],{"stretchy":460},[387,32136,32137],{"encoding":389},"\\varphi(tq,q)=q\\,f_{JS}(t)",[363,32139,32141,32177],{"className":32140,"ariaHidden":394},[393],[363,32142,32144,32147,32150,32153,32156,32159,32162,32165,32168,32171,32174],{"className":32143},[398],[363,32145],{"className":32146,"style":487},[402],[363,32148,31394],{"className":32149},[407,408],[363,32151,461],{"className":32152},[494],[363,32154,26650],{"className":32155,"style":1328},[407,408],[363,32157,898],{"className":32158},[920],[363,32160],{"className":32161,"style":754},[505],[363,32163,1291],{"className":32164,"style":1328},[407,408],[363,32166,468],{"className":32167},[501],[363,32169],{"className":32170,"style":506},[505],[363,32172,471],{"className":32173},[510],[363,32175],{"className":32176,"style":506},[505],[363,32178,32180,32183,32186,32189,32235,32238,32241],{"className":32179},[398],[363,32181],{"className":32182,"style":487},[402],[363,32184,1291],{"className":32185,"style":1328},[407,408],[363,32187],{"className":32188,"style":754},[505],[363,32190,32192,32195],{"className":32191},[407],[363,32193,385],{"className":32194,"style":409},[407,408],[363,32196,32198],{"className":32197},[629],[363,32199,32201,32227],{"className":32200},[633,634],[363,32202,32204,32224],{"className":32203},[638],[363,32205,32207],{"className":32206,"style":727},[642],[363,32208,32209,32212],{"style":27881},[363,32210],{"className":32211,"style":651},[650],[363,32213,32215],{"className":32214},[655,656,657,658],[363,32216,32218,32221],{"className":32217},[407,658],[363,32219,24115],{"className":32220,"style":24525},[407,408,658],[363,32222,26109],{"className":32223,"style":26233},[407,408,658],[363,32225,666],{"className":32226},[665],[363,32228,32230],{"className":32229},[638],[363,32231,32233],{"className":32232,"style":8316},[642],[363,32234],{},[363,32236,461],{"className":32237},[494],[363,32239,1116],{"className":32240},[407,408],[363,32242,468],{"className":32243},[501],"，两者一致；在 ",[363,32246,32248,32274],{"className":32247,"translate":367},[366],[363,32249,32251],{"className":32250},[371],[373,32252,32253],{"xmlns":375},[377,32254,32255,32271],{},[380,32256,32257,32259,32261,32263,32265,32267,32269],{},[383,32258,1291],{},[458,32260,471],{},[463,32262,474],{},[458,32264,898],{"separator":394},[383,32266,11],{},[458,32268,3251],{},[463,32270,474],{},[387,32272,32273],{"encoding":389},"q=0,p>0",[363,32275,32277,32295,32322],{"className":32276,"ariaHidden":394},[393],[363,32278,32280,32283,32286,32289,32292],{"className":32279},[398],[363,32281],{"className":32282,"style":2624},[402],[363,32284,1291],{"className":32285,"style":1328},[407,408],[363,32287],{"className":32288,"style":506},[505],[363,32290,471],{"className":32291},[510],[363,32293],{"className":32294,"style":506},[505],[363,32296,32298,32301,32304,32307,32310,32313,32316,32319],{"className":32297},[398],[363,32299],{"className":32300,"style":9706},[402],[363,32302,474],{"className":32303},[407],[363,32305,898],{"className":32306},[920],[363,32308],{"className":32309,"style":754},[505],[363,32311,11],{"className":32312},[407,408],[363,32314],{"className":32315,"style":506},[505],[363,32317,3251],{"className":32318},[510],[363,32320],{"className":32321,"style":506},[505],[363,32323,32325,32328],{"className":32324},[398],[363,32326],{"className":32327,"style":520},[402],[363,32329,474],{"className":32330},[407]," 处奇异项 ",[363,32333,32335,32361],{"className":32334,"translate":367},[366],[363,32336,32338],{"className":32337},[371],[373,32339,32340],{"xmlns":375},[377,32341,32342,32358],{},[380,32343,32344,32350,32352,32354,32356],{},[586,32345,32346,32348],{},[463,32347,465],{},[463,32349,2811],{},[383,32351,11],{},[383,32353,4448],{},[458,32355,4451],{},[463,32357,2811],{},[387,32359,32360],{"encoding":389},"\\frac12 p\\log 2",[363,32362,32364],{"className":32363,"ariaHidden":394},[393],[363,32365,32367,32370,32438,32441,32444,32449,32452],{"className":32366},[398],[363,32368],{"className":32369,"style":2853},[402],[363,32371,32373,32376,32435],{"className":32372},[407],[363,32374],{"className":32375},[494,789],[363,32377,32379],{"className":32378},[586],[363,32380,32382,32427],{"className":32381},[633,634],[363,32383,32385,32424],{"className":32384},[638],[363,32386,32388,32402,32410],{"className":32387,"style":2872},[642],[363,32389,32390,32393],{"style":2875},[363,32391],{"className":32392,"style":809},[650],[363,32394,32396],{"className":32395},[655,656,657,658],[363,32397,32399],{"className":32398},[407,658],[363,32400,2811],{"className":32401},[407,658],[363,32403,32404,32407],{"style":821},[363,32405],{"className":32406,"style":809},[650],[363,32408],{"className":32409,"style":829},[828],[363,32411,32412,32415],{"style":2898},[363,32413],{"className":32414,"style":809},[650],[363,32416,32418],{"className":32417},[655,656,657,658],[363,32419,32421],{"className":32420},[407,658],[363,32422,465],{"className":32423},[407,658],[363,32425,666],{"className":32426},[665],[363,32428,32430],{"className":32429},[638],[363,32431,32433],{"className":32432,"style":2920},[642],[363,32434],{},[363,32436],{"className":32437},[501,789],[363,32439,11],{"className":32440},[407,408],[363,32442],{"className":32443,"style":754},[505],[363,32445,4502,32447],{"className":32446},[1832],[363,32448,4506],{"style":4505},[363,32450],{"className":32451,"style":754},[505],[363,32453,2811],{"className":32454},[407]," 会被保留下来（这正是 1.1 节讨论的「支撑不重叠」约定）。",[11,32457,32458,32461,32462,32541,32542,16145,32545,32596,32597,32830,32831,32859,32860,32888,32889,33077,33078,33173],{},[1344,32459,32460],{},"有界性。"," 注意 ",[363,32463,32465,32486],{"className":32464,"translate":367},[366],[363,32466,32468],{"className":32467},[371],[373,32469,32470],{"xmlns":375},[377,32471,32472,32484],{},[380,32473,32474],{},[543,32475,32476,32478],{},[383,32477,385],{},[380,32479,32480,32482],{},[383,32481,24115],{},[383,32483,26109],{},[387,32485,31127],{"encoding":389},[363,32487,32489],{"className":32488,"ariaHidden":394},[393],[363,32490,32492,32495],{"className":32491},[398],[363,32493],{"className":32494,"style":403},[402],[363,32496,32498,32501],{"className":32497},[407],[363,32499,385],{"className":32500,"style":409},[407,408],[363,32502,32504],{"className":32503},[629],[363,32505,32507,32533],{"className":32506},[633,634],[363,32508,32510,32530],{"className":32509},[638],[363,32511,32513],{"className":32512,"style":727},[642],[363,32514,32515,32518],{"style":27881},[363,32516],{"className":32517,"style":651},[650],[363,32519,32521],{"className":32520},[655,656,657,658],[363,32522,32524,32527],{"className":32523},[407,658],[363,32525,24115],{"className":32526,"style":24525},[407,408,658],[363,32528,26109],{"className":32529,"style":26233},[407,408,658],[363,32531,666],{"className":32532},[665],[363,32534,32536],{"className":32535},[638],[363,32537,32539],{"className":32538,"style":8316},[642],[363,32540],{}," 逐点",[1344,32543,32544],{},"无界",[363,32546,32548,32566],{"className":32547,"translate":367},[366],[363,32549,32551],{"className":32550},[371],[373,32552,32553],{"xmlns":375},[377,32554,32555,32563],{},[380,32556,32557,32559,32561],{},[383,32558,1116],{},[458,32560,1378],{},[383,32562,1373],{"mathvariant":557},[387,32564,32565],{"encoding":389},"t\\to\\infty",[363,32567,32569,32587],{"className":32568,"ariaHidden":394},[393],[363,32570,32572,32575,32578,32581,32584],{"className":32571},[398],[363,32573],{"className":32574,"style":5691},[402],[363,32576,1116],{"className":32577},[407,408],[363,32579],{"className":32580,"style":506},[505],[363,32582,1378],{"className":32583},[510],[363,32585],{"className":32586,"style":506},[505],[363,32588,32590,32593],{"className":32589},[398],[363,32591],{"className":32592,"style":2693},[402],[363,32594,1373],{"className":32595},[407]," 时 ",[363,32598,32600,32648],{"className":32599,"translate":367},[366],[363,32601,32603],{"className":32602},[371],[373,32604,32605],{"xmlns":375},[377,32606,32607,32645],{},[380,32608,32609,32619,32621,32623,32625,32627,32633,32635,32637,32639,32641,32643],{},[543,32610,32611,32613],{},[383,32612,385],{},[380,32614,32615,32617],{},[383,32616,24115],{},[383,32618,26109],{},[458,32620,461],{"stretchy":460},[383,32622,1116],{},[458,32624,468],{"stretchy":460},[458,32626,9217],{},[586,32628,32629,32631],{},[463,32630,465],{},[463,32632,2811],{},[383,32634,1116],{},[383,32636,4448],{},[458,32638,4451],{},[463,32640,2811],{},[458,32642,1378],{},[383,32644,1373],{"mathvariant":557},[387,32646,32647],{"encoding":389},"f_{JS}(t)\\sim\\frac12 t\\log 2\\to\\infty",[363,32649,32651,32721,32821],{"className":32650,"ariaHidden":394},[393],[363,32652,32654,32657,32703,32706,32709,32712,32715,32718],{"className":32653},[398],[363,32655],{"className":32656,"style":487},[402],[363,32658,32660,32663],{"className":32659},[407],[363,32661,385],{"className":32662,"style":409},[407,408],[363,32664,32666],{"className":32665},[629],[363,32667,32669,32695],{"className":32668},[633,634],[363,32670,32672,32692],{"className":32671},[638],[363,32673,32675],{"className":32674,"style":727},[642],[363,32676,32677,32680],{"style":27881},[363,32678],{"className":32679,"style":651},[650],[363,32681,32683],{"className":32682},[655,656,657,658],[363,32684,32686,32689],{"className":32685},[407,658],[363,32687,24115],{"className":32688,"style":24525},[407,408,658],[363,32690,26109],{"className":32691,"style":26233},[407,408,658],[363,32693,666],{"className":32694},[665],[363,32696,32698],{"className":32697},[638],[363,32699,32701],{"className":32700,"style":8316},[642],[363,32702],{},[363,32704,461],{"className":32705},[494],[363,32707,1116],{"className":32708},[407,408],[363,32710,468],{"className":32711},[501],[363,32713],{"className":32714,"style":506},[505],[363,32716,9217],{"className":32717},[510],[363,32719],{"className":32720,"style":506},[505],[363,32722,32724,32727,32795,32798,32801,32806,32809,32812,32815,32818],{"className":32723},[398],[363,32725],{"className":32726,"style":2853},[402],[363,32728,32730,32733,32792],{"className":32729},[407],[363,32731],{"className":32732},[494,789],[363,32734,32736],{"className":32735},[586],[363,32737,32739,32784],{"className":32738},[633,634],[363,32740,32742,32781],{"className":32741},[638],[363,32743,32745,32759,32767],{"className":32744,"style":2872},[642],[363,32746,32747,32750],{"style":2875},[363,32748],{"className":32749,"style":809},[650],[363,32751,32753],{"className":32752},[655,656,657,658],[363,32754,32756],{"className":32755},[407,658],[363,32757,2811],{"className":32758},[407,658],[363,32760,32761,32764],{"style":821},[363,32762],{"className":32763,"style":809},[650],[363,32765],{"className":32766,"style":829},[828],[363,32768,32769,32772],{"style":2898},[363,32770],{"className":32771,"style":809},[650],[363,32773,32775],{"className":32774},[655,656,657,658],[363,32776,32778],{"className":32777},[407,658],[363,32779,465],{"className":32780},[407,658],[363,32782,666],{"className":32783},[665],[363,32785,32787],{"className":32786},[638],[363,32788,32790],{"className":32789,"style":2920},[642],[363,32791],{},[363,32793],{"className":32794},[501,789],[363,32796,1116],{"className":32797},[407,408],[363,32799],{"className":32800,"style":754},[505],[363,32802,4502,32804],{"className":32803},[1832],[363,32805,4506],{"style":4505},[363,32807],{"className":32808,"style":754},[505],[363,32810,2811],{"className":32811},[407],[363,32813],{"className":32814,"style":506},[505],[363,32816,1378],{"className":32817},[510],[363,32819],{"className":32820,"style":506},[505],[363,32822,32824,32827],{"className":32823},[398],[363,32825],{"className":32826,"style":2693},[402],[363,32828,1373],{"className":32829},[407],"），但积分有界，因为 ",[363,32832,32834,32847],{"className":32833,"translate":367},[366],[363,32835,32837],{"className":32836},[371],[373,32838,32839],{"xmlns":375},[377,32840,32841,32845],{},[380,32842,32843],{},[383,32844,1116],{},[387,32846,1116],{"encoding":389},[363,32848,32850],{"className":32849,"ariaHidden":394},[393],[363,32851,32853,32856],{"className":32852},[398],[363,32854],{"className":32855,"style":5691},[402],[363,32857,1116],{"className":32858},[407,408]," 大的地方 ",[363,32861,32863,32876],{"className":32862,"translate":367},[366],[363,32864,32866],{"className":32865},[371],[373,32867,32868],{"xmlns":375},[377,32869,32870,32874],{},[380,32871,32872],{},[383,32873,1291],{},[387,32875,1291],{"encoding":389},[363,32877,32879],{"className":32878,"ariaHidden":394},[393],[363,32880,32882,32885],{"className":32881},[398],[363,32883],{"className":32884,"style":2624},[402],[363,32886,1291],{"className":32887,"style":1328},[407,408]," 小、权重趋于 0。严格地用两 KL 形式：",[363,32890,32892,32950],{"className":32891,"translate":367},[366],[363,32893,32895],{"className":32894},[371],[373,32896,32897],{"xmlns":375},[377,32898,32899,32947],{},[380,32900,32901,32903,32905,32907,32909,32911,32913,32915,32917,32919,32921,32923,32929,32931,32933,32935,32937,32939,32941,32943,32945],{},[463,32902,2811],{},[383,32904,11],{},[383,32906,1288],{"mathvariant":557},[458,32908,461],{"stretchy":460},[383,32910,11],{},[458,32912,2818],{},[383,32914,1291],{},[458,32916,468],{"stretchy":460},[458,32918,10300],{},[463,32920,2811],{},[458,32922,11340],{},[380,32924,32925,32927],{},[383,32926,11044],{"mathvariant":557},[383,32928,22265],{"mathvariant":557},[458,32930,461],{"stretchy":460},[383,32932,554],{},[383,32934,558],{"mathvariant":557},[383,32936,25939],{},[458,32938,468],{"stretchy":460},[458,32940,10300],{},[383,32942,4448],{},[458,32944,4451],{},[463,32946,2811],{},[387,32948,32949],{"encoding":389},"2p\u002F(p+q)\\le 2\\Rightarrow \\mathrm{KL}(P\\|M)\\le\\log2",[363,32951,32953,32983,33004,33022,33060],{"className":32952,"ariaHidden":394},[393],[363,32954,32956,32959,32962,32965,32968,32971,32974,32977,32980],{"className":32955},[398],[363,32957],{"className":32958,"style":487},[402],[363,32960,2811],{"className":32961},[407],[363,32963,11],{"className":32964},[407,408],[363,32966,1288],{"className":32967},[407],[363,32969,461],{"className":32970},[494],[363,32972,11],{"className":32973},[407,408],[363,32975],{"className":32976,"style":2935},[505],[363,32978,2818],{"className":32979},[2939],[363,32981],{"className":32982,"style":2935},[505],[363,32984,32986,32989,32992,32995,32998,33001],{"className":32985},[398],[363,32987],{"className":32988,"style":487},[402],[363,32990,1291],{"className":32991,"style":1328},[407,408],[363,32993,468],{"className":32994},[501],[363,32996],{"className":32997,"style":506},[505],[363,32999,10300],{"className":33000},[510],[363,33002],{"className":33003,"style":506},[505],[363,33005,33007,33010,33013,33016,33019],{"className":33006},[398],[363,33008],{"className":33009,"style":520},[402],[363,33011,2811],{"className":33012},[407],[363,33014],{"className":33015,"style":506},[505],[363,33017,11340],{"className":33018},[510],[363,33020],{"className":33021,"style":506},[505],[363,33023,33025,33028,33036,33039,33042,33045,33048,33051,33054,33057],{"className":33024},[398],[363,33026],{"className":33027,"style":487},[402],[363,33029,33031],{"className":33030},[407],[363,33032,33035],{"className":33033},[407,33034],"mathrm","KL",[363,33037,461],{"className":33038},[494],[363,33040,554],{"className":33041,"style":682},[407,408],[363,33043,558],{"className":33044},[407],[363,33046,25939],{"className":33047,"style":25973},[407,408],[363,33049,468],{"className":33050},[501],[363,33052],{"className":33053,"style":506},[505],[363,33055,10300],{"className":33056},[510],[363,33058],{"className":33059,"style":506},[505],[363,33061,33063,33066,33071,33074],{"className":33062},[398],[363,33064],{"className":33065,"style":403},[402],[363,33067,4502,33069],{"className":33068},[1832],[363,33070,4506],{"style":4505},[363,33072],{"className":33073,"style":754},[505],[363,33075,2811],{"className":33076},[407],"，同理 ",[363,33079,33081,33117],{"className":33080,"translate":367},[366],[363,33082,33084],{"className":33083},[371],[373,33085,33086],{"xmlns":375},[377,33087,33088,33114],{},[380,33089,33090,33096,33098,33100,33102,33104,33106,33108,33110,33112],{},[380,33091,33092,33094],{},[383,33093,11044],{"mathvariant":557},[383,33095,22265],{"mathvariant":557},[458,33097,461],{"stretchy":460},[383,33099,561],{},[383,33101,558],{"mathvariant":557},[383,33103,25939],{},[458,33105,468],{"stretchy":460},[458,33107,10300],{},[383,33109,4448],{},[458,33111,4451],{},[463,33113,2811],{},[387,33115,33116],{"encoding":389},"\\mathrm{KL}(Q\\|M)\\le\\log2",[363,33118,33120,33156],{"className":33119,"ariaHidden":394},[393],[363,33121,33123,33126,33132,33135,33138,33141,33144,33147,33150,33153],{"className":33122},[398],[363,33124],{"className":33125,"style":487},[402],[363,33127,33129],{"className":33128},[407],[363,33130,33035],{"className":33131},[407,33034],[363,33133,461],{"className":33134},[494],[363,33136,561],{"className":33137},[407,408],[363,33139,558],{"className":33140},[407],[363,33142,25939],{"className":33143,"style":25973},[407,408],[363,33145,468],{"className":33146},[501],[363,33148],{"className":33149,"style":506},[505],[363,33151,10300],{"className":33152},[510],[363,33154],{"className":33155,"style":506},[505],[363,33157,33159,33162,33167,33170],{"className":33158},[398],[363,33160],{"className":33161,"style":403},[402],[363,33163,4502,33165],{"className":33164},[1832],[363,33166,4506],{"style":4505},[363,33168],{"className":33169,"style":754},[505],[363,33171,2811],{"className":33172},[407],"，故",[363,33175,33177],{"className":33176,"translate":367},[528],[363,33178,33180,33338],{"className":33179,"translate":367},[366],[363,33181,33183],{"className":33182},[371],[373,33184,33185],{"xmlns":375,"display":537},[377,33186,33187,33335],{},[1591,33188,33189,33277],{"rowspacing":1593,"columnalign":1594,"columnspacing":1595},[1597,33190,33191,33221],{},[1600,33192,33193],{},[1603,33194,33195],{"scriptlevel":474,"displaystyle":394},[380,33196,33197,33199,33201,33211,33213,33215,33217,33219],{},[463,33198,474],{},[458,33200,10300],{},[543,33202,33203,33205],{},[383,33204,547],{},[380,33206,33207,33209],{},[383,33208,24115],{},[383,33210,26109],{},[458,33212,461],{"stretchy":460},[383,33214,554],{},[383,33216,558],{"mathvariant":557},[383,33218,561],{},[458,33220,468],{"stretchy":460},[1600,33222,33223],{},[1603,33224,33225],{"scriptlevel":474,"displaystyle":394},[380,33226,33227,33229,33231,33237,33243,33245,33247,33249,33251,33253,33255,33261,33267,33269,33271,33273,33275],{},[380,33228],{},[458,33230,471],{},[586,33232,33233,33235],{},[463,33234,465],{},[463,33236,2811],{},[380,33238,33239,33241],{},[383,33240,11044],{"mathvariant":557},[383,33242,22265],{"mathvariant":557},[458,33244,461],{"stretchy":460},[383,33246,554],{},[383,33248,558],{"mathvariant":557},[383,33250,25939],{},[458,33252,468],{"stretchy":460},[458,33254,2818],{},[586,33256,33257,33259],{},[463,33258,465],{},[463,33260,2811],{},[380,33262,33263,33265],{},[383,33264,11044],{"mathvariant":557},[383,33266,22265],{"mathvariant":557},[458,33268,461],{"stretchy":460},[383,33270,561],{},[383,33272,558],{"mathvariant":557},[383,33274,25939],{},[458,33276,468],{"stretchy":460},[1597,33278,33279,33285],{},[1600,33280,33281],{},[1603,33282,33283],{"scriptlevel":474,"displaystyle":394},[380,33284],{},[1600,33286,33287],{},[1603,33288,33289],{"scriptlevel":474,"displaystyle":394},[380,33290,33291,33293,33295,33301,33303,33305,33307,33309,33315,33317,33319,33321,33323,33325,33327,33329,33332],{},[380,33292],{},[458,33294,10300],{},[586,33296,33297,33299],{},[463,33298,465],{},[463,33300,2811],{},[383,33302,4448],{},[458,33304,4451],{},[463,33306,2811],{},[458,33308,2818],{},[586,33310,33311,33313],{},[463,33312,465],{},[463,33314,2811],{},[383,33316,4448],{},[458,33318,4451],{},[463,33320,2811],{},[458,33322,471],{},[383,33324,4448],{},[458,33326,4451],{},[463,33328,2811],{},[458,33330,33331],{},"≈",[463,33333,33334],{},"0.693.",[387,33336,33337],{"encoding":389},"\\begin{aligned}\n0\\le D_{JS}(P\\|Q)\n&=\\frac12\\mathrm{KL}(P\\|M)\n+\\frac12\\mathrm{KL}(Q\\|M)\\\\[2pt]\n&\\le \\frac12\\log 2+\\frac12\\log 2\n=\\log 2\\approx 0.693 .\n\\end{aligned}",[363,33339,33341],{"className":33340,"ariaHidden":394},[393],[363,33342,33344,33348],{"className":33343},[398],[363,33345],{"className":33346,"style":33347},[402],"height:4.3149em;vertical-align:-1.9074em;",[363,33349,33351],{"className":33350},[407],[363,33352,33354,33469],{"className":33353},[1591],[363,33355,33357],{"className":33356},[1708],[363,33358,33360,33460],{"className":33359},[633,634],[363,33361,33363,33457],{"className":33362},[638],[363,33364,33367,33449],{"className":33365,"style":33366},[642],"height:2.4074em;",[363,33368,33370,33373],{"style":33369},"top:-4.4074em;",[363,33371],{"className":33372,"style":23038},[650],[363,33374,33376,33379,33382,33385,33388,33434,33437,33440,33443,33446],{"className":33375},[407],[363,33377,474],{"className":33378},[407],[363,33380],{"className":33381,"style":506},[505],[363,33383,10300],{"className":33384},[510],[363,33386],{"className":33387,"style":506},[505],[363,33389,33391,33394],{"className":33390},[407],[363,33392,547],{"className":33393,"style":625},[407,408],[363,33395,33397],{"className":33396},[629],[363,33398,33400,33426],{"className":33399},[633,634],[363,33401,33403,33423],{"className":33402},[638],[363,33404,33406],{"className":33405,"style":727},[642],[363,33407,33408,33411],{"style":646},[363,33409],{"className":33410,"style":651},[650],[363,33412,33414],{"className":33413},[655,656,657,658],[363,33415,33417,33420],{"className":33416},[407,658],[363,33418,24115],{"className":33419,"style":24525},[407,408,658],[363,33421,26109],{"className":33422,"style":26233},[407,408,658],[363,33424,666],{"className":33425},[665],[363,33427,33429],{"className":33428},[638],[363,33430,33432],{"className":33431,"style":8316},[642],[363,33433],{},[363,33435,461],{"className":33436},[494],[363,33438,554],{"className":33439,"style":682},[407,408],[363,33441,558],{"className":33442},[407],[363,33444,561],{"className":33445},[407,408],[363,33447,468],{"className":33448},[501],[363,33450,33451,33454],{"style":2251},[363,33452],{"className":33453,"style":23038},[650],[363,33455],{"className":33456},[407],[363,33458,666],{"className":33459},[665],[363,33461,33463],{"className":33462},[638],[363,33464,33467],{"className":33465,"style":33466},[642],"height:1.9074em;",[363,33468],{},[363,33470,33472],{"className":33471},[1799],[363,33473,33475,33892],{"className":33474},[633,634],[363,33476,33478,33889],{"className":33477},[638],[363,33479,33481,33676],{"className":33480,"style":33366},[642],[363,33482,33483,33486],{"style":33369},[363,33484],{"className":33485,"style":23038},[650],[363,33487,33489,33492,33495,33498,33501,33563,33569,33572,33575,33578,33581,33584,33587,33590,33593,33655,33661,33664,33667,33670,33673],{"className":33488},[407],[363,33490],{"className":33491},[407],[363,33493],{"className":33494,"style":506},[505],[363,33496,471],{"className":33497},[510],[363,33499],{"className":33500,"style":506},[505],[363,33502,33504,33507,33560],{"className":33503},[407],[363,33505],{"className":33506},[494,789],[363,33508,33510],{"className":33509},[586],[363,33511,33513,33552],{"className":33512},[633,634],[363,33514,33516,33549],{"className":33515},[638],[363,33517,33519,33530,33538],{"className":33518,"style":23261},[642],[363,33520,33521,33524],{"style":805},[363,33522],{"className":33523,"style":809},[650],[363,33525,33527],{"className":33526},[407],[363,33528,2811],{"className":33529},[407],[363,33531,33532,33535],{"style":821},[363,33533],{"className":33534,"style":809},[650],[363,33536],{"className":33537,"style":829},[828],[363,33539,33540,33543],{"style":832},[363,33541],{"className":33542,"style":809},[650],[363,33544,33546],{"className":33545},[407],[363,33547,465],{"className":33548},[407],[363,33550,666],{"className":33551},[665],[363,33553,33555],{"className":33554},[638],[363,33556,33558],{"className":33557,"style":4126},[642],[363,33559],{},[363,33561],{"className":33562},[501,789],[363,33564,33566],{"className":33565},[407],[363,33567,33035],{"className":33568},[407,33034],[363,33570,461],{"className":33571},[494],[363,33573,554],{"className":33574,"style":682},[407,408],[363,33576,558],{"className":33577},[407],[363,33579,25939],{"className":33580,"style":25973},[407,408],[363,33582,468],{"className":33583},[501],[363,33585],{"className":33586,"style":2935},[505],[363,33588,2818],{"className":33589},[2939],[363,33591],{"className":33592,"style":2935},[505],[363,33594,33596,33599,33652],{"className":33595},[407],[363,33597],{"className":33598},[494,789],[363,33600,33602],{"className":33601},[586],[363,33603,33605,33644],{"className":33604},[633,634],[363,33606,33608,33641],{"className":33607},[638],[363,33609,33611,33622,33630],{"className":33610,"style":23261},[642],[363,33612,33613,33616],{"style":805},[363,33614],{"className":33615,"style":809},[650],[363,33617,33619],{"className":33618},[407],[363,33620,2811],{"className":33621},[407],[363,33623,33624,33627],{"style":821},[363,33625],{"className":33626,"style":809},[650],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",[363,33904,33906,33925],{"className":33905,"translate":367},[366],[363,33907,33909],{"className":33908},[371],[373,33910,33911],{"xmlns":375},[377,33912,33913,33922],{},[380,33914,33915,33917,33920],{},[383,33916,554],{},[458,33918,33919],{},"⊥",[383,33921,561],{},[387,33923,33924],{"encoding":389},"P\\perp Q",[363,33926,33928,33946],{"className":33927,"ariaHidden":394},[393],[363,33929,33931,33934,33937,33940,33943],{"className":33930},[398],[363,33932],{"className":33933,"style":12947},[402],[363,33935,554],{"className":33936,"style":682},[407,408],[363,33938],{"className":33939,"style":506},[505],[363,33941,33919],{"className":33942},[510],[363,33944],{"className":33945,"style":506},[505],[363,33947,33949,33952],{"className":33948},[398],[363,33950],{"className":33951,"style":913},[402],[363,33953,561],{"className":33954},[407,408],"（支撑集不重叠）。有界是 JS 相对 KL 的最大优点（梯度消失则是它最大的缺点，二者互为因果）。",[11,33957,33958,33960,33961,33990,33991,34024,34025,34053,34054,34082,34083,34111,34112,34140,34141,2757,34169,34197,34198,9748],{},[1344,33959,24565],{}," JS 有一个很直观的信息论解释：先掷一枚公平硬币 ",[363,33962,33964,33978],{"className":33963,"translate":367},[366],[363,33965,33967],{"className":33966},[371],[373,33968,33969],{"xmlns":375},[377,33970,33971,33976],{},[380,33972,33973],{},[383,33974,33975],{},"Z",[387,33977,33975],{"encoding":389},[363,33979,33981],{"className":33980,"ariaHidden":394},[393],[363,33982,33984,33987],{"className":33983},[398],[363,33985],{"className":33986,"style":1011},[402],[363,33988,33975],{"className":33989,"style":11132},[407,408],"（正反面各 ",[363,33992,33994,34012],{"className":33993,"translate":367},[366],[363,33995,33997],{"className":33996},[371],[373,33998,33999],{"xmlns":375},[377,34000,34001,34009],{},[380,34002,34003,34005,34007],{},[463,34004,465],{},[383,34006,1288],{"mathvariant":557},[463,34008,2811],{},[387,34010,34011],{"encoding":389},"1\u002F2",[363,34013,34015],{"className":34014,"ariaHidden":394},[393],[363,34016,34018,34021],{"className":34017},[398],[363,34019],{"className":34020,"style":487},[402],[363,34022,34011],{"className":34023},[407],"），正面时从 ",[363,34026,34028,34041],{"className":34027,"translate":367},[366],[363,34029,34031],{"className":34030},[371],[373,34032,34033],{"xmlns":375},[377,34034,34035,34039],{},[380,34036,34037],{},[383,34038,554],{},[387,34040,554],{"encoding":389},[363,34042,34044],{"className":34043,"ariaHidden":394},[393],[363,34045,34047,34050],{"className":34046},[398],[363,34048],{"className":34049,"style":1011},[402],[363,34051,554],{"className":34052,"style":682},[407,408]," 取样、反面时从 ",[363,34055,34057,34070],{"className":34056,"translate":367},[366],[363,34058,34060],{"className":34059},[371],[373,34061,34062],{"xmlns":375},[377,34063,34064,34068],{},[380,34065,34066],{},[383,34067,561],{},[387,34069,561],{"encoding":389},[363,34071,34073],{"className":34072,"ariaHidden":394},[393],[363,34074,34076,34079],{"className":34075},[398],[363,34077],{"className":34078,"style":913},[402],[363,34080,561],{"className":34081},[407,408]," 取样，得到观测 ",[363,34084,34086,34099],{"className":34085,"translate":367},[366],[363,34087,34089],{"className":34088},[371],[373,34090,34091],{"xmlns":375},[377,34092,34093,34097],{},[380,34094,34095],{},[383,34096,945],{},[387,34098,945],{"encoding":389},[363,34100,34102],{"className":34101,"ariaHidden":394},[393],[363,34103,34105,34108],{"className":34104},[398],[363,34106],{"className":34107,"style":1011},[402],[363,34109,945],{"className":34110,"style":9235},[407,408],"；则 JS 正是「看一眼 ",[363,34113,34115,34128],{"className":34114,"translate":367},[366],[363,34116,34118],{"className":34117},[371],[373,34119,34120],{"xmlns":375},[377,34121,34122,34126],{},[380,34123,34124],{},[383,34125,945],{},[387,34127,945],{"encoding":389},[363,34129,34131],{"className":34130,"ariaHidden":394},[393],[363,34132,34134,34137],{"className":34133},[398],[363,34135],{"className":34136,"style":1011},[402],[363,34138,945],{"className":34139,"style":9235},[407,408]," 能否判断硬币正反」所蕴含的信息量，即 ",[363,34142,34144,34157],{"className":34143,"translate":367},[366],[363,34145,34147],{"className":34146},[371],[373,34148,34149],{"xmlns":375},[377,34150,34151,34155],{},[380,34152,34153],{},[383,34154,33975],{},[387,34156,33975],{"encoding":389},[363,34158,34160],{"className":34159,"ariaHidden":394},[393],[363,34161,34163,34166],{"className":34162},[398],[363,34164],{"className":34165,"style":1011},[402],[363,34167,33975],{"className":34168,"style":11132},[407,408],[363,34170,34172,34185],{"className":34171,"translate":367},[366],[363,34173,34175],{"className":34174},[371],[373,34176,34177],{"xmlns":375},[377,34178,34179,34183],{},[380,34180,34181],{},[383,34182,945],{},[387,34184,945],{"encoding":389},[363,34186,34188],{"className":34187,"ariaHidden":394},[393],[363,34189,34191,34194],{"className":34190},[398],[363,34192],{"className":34193,"style":1011},[402],[363,34195,945],{"className":34196,"style":9235},[407,408]," 的",[1344,34199,34200],{},"互信息",[363,34202,34204],{"className":34203,"translate":367},[528],[363,34205,34207,34286],{"className":34206,"translate":367},[366],[363,34208,34210],{"className":34209},[371],[373,34211,34212],{"xmlns":375,"display":537},[377,34213,34214,34283],{},[380,34215,34216,34226,34228,34230,34232,34234,34236,34238,34241,34243,34245,34248,34250,34252,34254,34257,34259,34261,34263,34265,34267,34269,34271,34273,34275,34277,34279,34281],{},[543,34217,34218,34220],{},[383,34219,547],{},[380,34221,34222,34224],{},[383,34223,24115],{},[383,34225,26109],{},[458,34227,461],{"stretchy":460},[383,34229,554],{},[383,34231,558],{"mathvariant":557},[383,34233,561],{},[458,34235,468],{"stretchy":460},[458,34237,471],{},[383,34239,34240],{},"I",[458,34242,461],{"stretchy":460},[383,34244,33975],{},[458,34246,34247],{"separator":394},";",[383,34249,945],{},[458,34251,468],{"stretchy":460},[458,34253,471],{},[383,34255,34256],{},"H",[458,34258,461],{"stretchy":460},[383,34260,945],{},[458,34262,468],{"stretchy":460},[458,34264,5161],{},[383,34266,34256],{},[458,34268,461],{"stretchy":460},[383,34270,945],{},[1645,34272,1647],{},[383,34274,5156],{"mathvariant":557},[1645,34276,1647],{},[383,34278,33975],{},[458,34280,468],{"stretchy":460},[383,34282,4723],{"mathvariant":557},[387,34284,34285],{"encoding":389},"D_{JS}(P\\|Q)=I(Z;X)=H(X)-H(X\\,|\\,Z).",[363,34287,34289,34365,34401,34429],{"className":34288,"ariaHidden":394},[393],[363,34290,34292,34295,34341,34344,34347,34350,34353,34356,34359,34362],{"className":34291},[398],[363,34293],{"className":34294,"style":487},[402],[363,34296,34298,34301],{"className":34297},[407],[363,34299,547],{"className":34300,"style":625},[407,408],[363,34302,34304],{"className":34303},[629],[363,34305,34307,34333],{"className":34306},[633,634],[363,34308,34310,34330],{"className":34309},[638],[363,34311,34313],{"className":34312,"style":727},[642],[363,34314,34315,34318],{"style":646},[363,34316],{"className":34317,"style":651},[650],[363,34319,34321],{"className":34320},[655,656,657,658],[363,34322,34324,34327],{"className":34323},[407,658],[363,34325,24115],{"className":34326,"style":24525},[407,408,658],[363,34328,26109],{"className":34329,"style":26233},[407,408,658],[363,34331,666],{"className":34332},[665],[363,34334,34336],{"className":34335},[638],[363,34337,34339],{"className":34338,"style":8316},[642],[363,34340],{},[363,34342,461],{"className":34343},[494],[363,34345,554],{"className":34346,"style":682},[407,408],[363,34348,558],{"className":34349},[407],[363,34351,561],{"className":34352},[407,408],[363,34354,468],{"className":34355},[501],[363,34357],{"className":34358,"style":506},[505],[363,34360,471],{"className":34361},[510],[363,34363],{"className":34364,"style":506},[505],[363,34366,34368,34371,34374,34377,34380,34383,34386,34389,34392,34395,34398],{"className":34367},[398],[363,34369],{"className":34370,"style":487},[402],[363,34372,34240],{"className":34373,"style":9235},[407,408],[363,34375,461],{"className":34376},[494],[363,34378,33975],{"className":34379,"style":11132},[407,408],[363,34381,34247],{"className":34382},[920],[363,34384],{"className":34385,"style":754},[505],[363,34387,945],{"className":34388,"style":9235},[407,408],[363,34390,468],{"className":34391},[501],[363,34393],{"className":34394,"style":506},[505],[363,34396,471],{"className":34397},[510],[363,34399],{"className":34400,"style":506},[505],[363,34402,34404,34407,34411,34414,34417,34420,34423,34426],{"className":34403},[398],[363,34405],{"className":34406,"style":487},[402],[363,34408,34256],{"className":34409,"style":34410},[407,408],"margin-right:0.0813em;",[363,34412,461],{"className":34413},[494],[363,34415,945],{"className":34416,"style":9235},[407,408],[363,34418,468],{"className":34419},[501],[363,34421],{"className":34422,"style":2935},[505],[363,34424,5161],{"className":34425},[2939],[363,34427],{"className":34428,"style":2935},[505],[363,34430,34432,34435,34438,34441,34444,34447,34450,34453,34456,34459],{"className":34431},[398],[363,34433],{"className":34434,"style":487},[402],[363,34436,34256],{"className":34437,"style":34410},[407,408],[363,34439,461],{"className":34440},[494],[363,34442,945],{"className":34443,"style":9235},[407,408],[363,34445],{"className":34446,"style":754},[505],[363,34448,5156],{"className":34449},[407],[363,34451],{"className":34452,"style":754},[505],[363,34454,33975],{"className":34455,"style":11132},[407,408],[363,34457,468],{"className":34458},[501],[363,34460,4723],{"className":34461},[407],[11,34463,34464,34465,34493,34494,34526,34527,34655,34656,34697,34698,34836,34837,34878],{},"由于 ",[363,34466,34468,34481],{"className":34467,"translate":367},[366],[363,34469,34471],{"className":34470},[371],[373,34472,34473],{"xmlns":375},[377,34474,34475,34479],{},[380,34476,34477],{},[383,34478,33975],{},[387,34480,33975],{"encoding":389},[363,34482,34484],{"className":34483,"ariaHidden":394},[393],[363,34485,34487,34490],{"className":34486},[398],[363,34488],{"className":34489,"style":1011},[402],[363,34491,33975],{"className":34492,"style":11132},[407,408]," 是概率 ",[363,34495,34497,34514],{"className":34496,"translate":367},[366],[363,34498,34500],{"className":34499},[371],[373,34501,34502],{"xmlns":375},[377,34503,34504,34512],{},[380,34505,34506,34508,34510],{},[463,34507,465],{},[383,34509,1288],{"mathvariant":557},[463,34511,2811],{},[387,34513,34011],{"encoding":389},[363,34515,34517],{"className":34516,"ariaHidden":394},[393],[363,34518,34520,34523],{"className":34519},[398],[363,34521],{"className":34522,"style":487},[402],[363,34524,34011],{"className":34525},[407]," 的伯努利变量，",[363,34528,34530,34572],{"className":34529,"translate":367},[366],[363,34531,34533],{"className":34532},[371],[373,34534,34535],{"xmlns":375},[377,34536,34537,34569],{},[380,34538,34539,34541,34543,34545,34547,34549,34551,34553,34555,34557,34559,34561,34563,34565,34567],{},[383,34540,34240],{},[458,34542,461],{"stretchy":460},[383,34544,33975],{},[458,34546,34247],{"separator":394},[383,34548,945],{},[458,34550,468],{"stretchy":460},[458,34552,10300],{},[383,34554,34256],{},[458,34556,461],{"stretchy":460},[383,34558,33975],{},[458,34560,468],{"stretchy":460},[458,34562,471],{},[383,34564,4448],{},[458,34566,4451],{},[463,34568,2811],{},[387,34570,34571],{"encoding":389},"I(Z;X)\\le H(Z)=\\log 2",[363,34573,34575,34611,34638],{"className":34574,"ariaHidden":394},[393],[363,34576,34578,34581,34584,34587,34590,34593,34596,34599,34602,34605,34608],{"className":34577},[398],[363,34579],{"className":34580,"style":487},[402],[363,34582,34240],{"className":34583,"style":9235},[407,408],[363,34585,461],{"className":34586},[494],[363,34588,33975],{"className":34589,"style":11132},[407,408],[363,34591,34247],{"className":34592},[920],[363,34594],{"className":34595,"style":754},[505],[363,34597,945],{"className":34598,"style":9235},[407,408],[363,34600,468],{"className":34601},[501],[363,34603],{"className":34604,"style":506},[505],[363,34606,10300],{"className":34607},[510],[363,34609],{"className":34610,"style":506},[505],[363,34612,34614,34617,34620,34623,34626,34629,34632,34635],{"className":34613},[398],[363,34615],{"className":34616,"style":487},[402],[363,34618,34256],{"className":34619,"style":34410},[407,408],[363,34621,461],{"className":34622},[494],[363,34624,33975],{"className":34625,"style":11132},[407,408],[363,34627,468],{"className":34628},[501],[363,34630],{"className":34631,"style":506},[505],[363,34633,471],{"className":34634},[510],[363,34636],{"className":34637,"style":506},[505],[363,34639,34641,34644,34649,34652],{"className":34640},[398],[363,34642],{"className":34643,"style":403},[402],[363,34645,4502,34647],{"className":34646},[1832],[363,34648,4506],{"style":4505},[363,34650],{"className":34651,"style":754},[505],[363,34653,2811],{"className":34654},[407],"，有界性由此而来，",[363,34657,34659,34677],{"className":34658,"translate":367},[366],[363,34660,34662],{"className":34661},[371],[373,34663,34664],{"xmlns":375},[377,34665,34666,34674],{},[380,34667,34668,34670,34672],{},[383,34669,4448],{},[458,34671,4451],{},[463,34673,2811],{},[387,34675,34676],{"encoding":389},"\\log 2",[363,34678,34680],{"className":34679,"ariaHidden":394},[393],[363,34681,34683,34686,34691,34694],{"className":34682},[398],[363,34684],{"className":34685,"style":403},[402],[363,34687,4502,34689],{"className":34688},[1832],[363,34690,4506],{"style":4505},[363,34692],{"className":34693,"style":754},[505],[363,34695,2811],{"className":34696},[407]," 恰好对应「最多只能获得 1 比特」这一直觉。JS 对称、有界且 ",[363,34699,34701,34726],{"className":34700,"translate":367},[366],[363,34702,34704],{"className":34703},[371],[373,34705,34706],{"xmlns":375},[377,34707,34708,34723],{},[380,34709,34710],{},[34711,34712,34713],"msqrt",{},[543,34714,34715,34717],{},[383,34716,547],{},[380,34718,34719,34721],{},[383,34720,24115],{},[383,34722,26109],{},[387,34724,34725],{"encoding":389},"\\sqrt{D_{JS}}",[363,34727,34729],{"className":34728,"ariaHidden":394},[393],[363,34730,34732,34736],{"className":34731},[398],[363,34733],{"className":34734,"style":34735},[402],"height:1.04em;vertical-align:-0.1883em;",[363,34737,34740],{"className":34738},[407,34739],"sqrt",[363,34741,34743,34827],{"className":34742},[633,634],[363,34744,34746,34824],{"className":34745},[638],[363,34747,34750,34806],{"className":34748,"style":34749},[642],"height:0.8517em;",[363,34751,34753,34756],{"className":34752,"style":12213},[6655],[363,34754],{"className":34755,"style":809},[650],[363,34757,34760],{"className":34758,"style":34759},[407],"padding-left:0.833em;",[363,34761,34763,34766],{"className":34762},[407],[363,34764,547],{"className":34765,"style":625},[407,408],[363,34767,34769],{"className":34768},[629],[363,34770,34772,34798],{"className":34771},[633,634],[363,34773,34775,34795],{"className":34774},[638],[363,34776,34778],{"className":34777,"style":727},[642],[363,34779,34780,34783],{"style":646},[363,34781],{"className":34782,"style":651},[650],[363,34784,34786],{"className":34785},[655,656,657,658],[363,34787,34789,34792],{"className":34788},[407,658],[363,34790,24115],{"className":34791,"style":24525},[407,408,658],[363,34793,26109],{"className":34794,"style":26233},[407,408,658],[363,34796,666],{"className":34797},[665],[363,34799,34801],{"className":34800},[638],[363,34802,34804],{"className":34803,"style":8316},[642],[363,34805],{},[363,34807,34809,34812],{"style":34808},"top:-2.8117em;",[363,34810],{"className":34811,"style":809},[650],[363,34813,34817],{"className":34814,"style":34816},[34815],"hide-tail","min-width:0.853em;height:1.08em;",[6671,34818,34821],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},"1.08em","0 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A}|P(A)-Q(A)|",[363,35841,35843,35922,36008],{"className":35842,"ariaHidden":394},[393],[363,35844,35846,35849,35895,35898,35901,35904,35907,35910,35913,35916,35919],{"className":35845},[398],[363,35847],{"className":35848,"style":487},[402],[363,35850,35852,35855],{"className":35851},[407],[363,35853,547],{"className":35854,"style":625},[407,408],[363,35856,35858],{"className":35857},[629],[363,35859,35861,35887],{"className":35860},[633,634],[363,35862,35864,35884],{"className":35863},[638],[363,35865,35867],{"className":35866,"style":727},[642],[363,35868,35869,35872],{"style":646},[363,35870],{"className":35871,"style":651},[650],[363,35873,35875],{"className":35874},[655,656,657,658],[363,35876,35878,35881],{"className":35877},[407,658],[363,35879,5740],{"className":35880,"style":682},[407,408,658],[363,35882,35222],{"className":35883,"style":2935},[407,408,658],[363,35885,666],{"className":35886},[665],[363,35888,35890],{"className":35889},[638],[363,35891,35893],{"className":35892,"style":8316},[642],[363,35894],{},[363,35896,461],{"className":35897},[494],[363,35899,554],{"className":35900,"style":682},[407,408],[363,35902,898],{"className":35903},[920],[363,35905],{"className":35906,"style":754},[505],[363,35908,561],{"className":35909},[407,408],[363,35911,468],{"className":35912},[501],[363,35914],{"className":35915,"style":506},[505],[363,35917,471],{"className":35918},[510],[363,35920],{"className":35921,"style":506},[505],[363,35923,35925,35929,35981,35984,35987,35990,35993,35996,35999,36002,36005],{"className":35924},[398],[363,35926],{"className":35927,"style":35928},[402],"height:1.0215em;vertical-align:-0.2715em;",[363,35930,35932,35935],{"className":35931},[1832],[363,35933,35804],{"className":35934},[1832],[363,35936,35938],{"className":35937},[629],[363,35939,35941,35972],{"className":35940},[633,634],[363,35942,35944,35969],{"className":35943},[638],[363,35945,35948],{"className":35946,"style":35947},[642],"height:0.2342em;",[363,35949,35951,35954],{"style":35950},"top:-2.4559em;margin-right:0.05em;",[363,35952],{"className":35953,"style":651},[650],[363,35955,35957],{"className":35956},[655,656,657,658],[363,35958,35960,35963,35966],{"className":35959},[407,658],[363,35961,950],{"className":35962},[407,408,658],[363,35964,6257],{"className":35965},[510,658],[363,35967,950],{"className":35968},[407,971,658],[363,35970,666],{"className":35971},[665],[363,35973,35975],{"className":35974},[638],[363,35976,35979],{"className":35977,"style":35978},[642],"height:0.2715em;",[363,35980],{},[363,35982],{"className":35983,"style":754},[505],[363,35985,5156],{"className":35986},[407],[363,35988,554],{"className":35989,"style":682},[407,408],[363,35991,461],{"className":35992},[494],[363,35994,950],{"className":35995},[407,408],[363,35997,468],{"className":35998},[501],[363,36000],{"className":36001,"style":2935},[505],[363,36003,5161],{"className":36004},[2939],[363,36006],{"className":36007,"style":2935},[505],[363,36009,36011,36014,36017,36020,36023,36026],{"className":36010},[398],[363,36012],{"className":36013,"style":487},[402],[363,36015,561],{"className":36016},[407,408],[363,36018,461],{"className":36019},[494],[363,36021,950],{"className":36022},[407,408],[363,36024,468],{"className":36025},[501],[363,36027,5156],{"className":36028},[407],"；两个表达式相等（这是全变差的对偶刻画）。注意 ",[363,36031,36033,36067],{"className":36032,"translate":367},[366],[363,36034,36036],{"className":36035},[371],[373,36037,36038],{"xmlns":375},[377,36039,36040,36064],{},[380,36041,36042,36052,36054,36056,36058,36060,36062],{},[543,36043,36044,36046],{},[383,36045,547],{},[380,36047,36048,36050],{},[383,36049,5740],{},[383,36051,35222],{},[458,36053,6257],{},[458,36055,578],{"stretchy":460},[463,36057,474],{},[458,36059,898],{"separator":394},[463,36061,465],{},[458,36063,605],{"stretchy":460},[387,36065,36066],{"encoding":389},"D_{TV}\\in[0,1]",[363,36068,36070,36131],{"className":36069,"ariaHidden":394},[393],[363,36071,36073,36076,36122,36125,36128],{"className":36072},[398],[363,36074],{"className":36075,"style":16311},[402],[363,36077,36079,36082],{"className":36078},[407],[363,36080,547],{"className":36081,"style":625},[407,408],[363,36083,36085],{"className":36084},[629],[363,36086,36088,36114],{"className":36087},[633,634],[363,36089,36091,36111],{"className":36090},[638],[363,36092,36094],{"className":36093,"style":727},[642],[363,36095,36096,36099],{"style":646},[363,36097],{"className":36098,"style":651},[650],[363,36100,36102],{"className":36101},[655,656,657,658],[363,36103,36105,36108],{"className":36104},[407,658],[363,36106,5740],{"className":36107,"style":682},[407,408,658],[363,36109,35222],{"className":36110,"style":2935},[407,408,658],[363,36112,666],{"className":36113},[665],[363,36115,36117],{"className":36116},[638],[363,36118,36120],{"className":36119,"style":8316},[642],[363,36121],{},[363,36123],{"className":36124,"style":506},[505],[363,36126,6257],{"className":36127},[510],[363,36129],{"className":36130,"style":506},[505],[363,36132,36134,36137,36140,36143,36146,36149,36152],{"className":36133},[398],[363,36135],{"className":36136,"style":487},[402],[363,36138,578],{"className":36139},[494],[363,36141,474],{"className":36142},[407],[363,36144,898],{"className":36145},[920],[363,36147],{"className":36148,"style":754},[505],[363,36150,465],{"className":36151},[407],[363,36153,605],{"className":36154},[501],"，总有界。",[11,36157,36158,9748,36161,36547],{},[1344,36159,36160],{},"斜共轭",[363,36162,36164,36228],{"className":36163,"translate":367},[366],[363,36165,36167],{"className":36166},[371],[373,36168,36169],{"xmlns":375},[377,36170,36171,36225],{},[380,36172,36173,36175,36177,36183,36185,36191,36193,36195,36197,36199,36205,36207,36209,36211,36213,36215,36217,36219,36221,36223],{},[383,36174,1116],{},[458,36176,3647],{},[586,36178,36179,36181],{},[463,36180,465],{},[463,36182,2811],{},[383,36184,5156],{"mathvariant":557},[586,36186,36187,36189],{},[463,36188,465],{},[383,36190,1116],{},[458,36192,5161],{},[463,36194,465],{},[383,36196,5156],{"mathvariant":557},[458,36198,471],{},[586,36200,36201,36203],{},[463,36202,465],{},[463,36204,2811],{},[383,36206,5156],{"mathvariant":557},[463,36208,465],{},[458,36210,5161],{},[383,36212,1116],{},[383,36214,5156],{"mathvariant":557},[458,36216,471],{},[383,36218,385],{},[458,36220,461],{"stretchy":460},[383,36222,1116],{},[458,36224,468],{"stretchy":460},[387,36226,36227],{"encoding":389},"t\\cdot\\frac12|\\frac1t-1|=\\frac12|1-t|=f(t)",[363,36229,36231,36249,36403,36421,36508,36529],{"className":36230,"ariaHidden":394},[393],[363,36232,36234,36237,36240,36243,36246],{"className":36233},[398],[363,36235],{"className":36236,"style":5691},[402],[363,36238,1116],{"className":36239},[407,408],[363,36241],{"className":36242,"style":2935},[505],[363,36244,3647],{"className":36245},[2939],[363,36247],{"className":36248,"style":2935},[505],[363,36250,36252,36255,36323,36326,36394,36397,36400],{"className":36251},[398],[363,36253],{"className":36254,"style":2853},[402],[363,36256,36258,36261,36320],{"className":36257},[407],[363,36259],{"className":36260},[494,789],[363,36262,36264],{"className":36263},[586],[363,36265,36267,36312],{"className":36266},[633,634],[363,36268,36270,36309],{"className":36269},[638],[363,36271,36273,36287,36295],{"className":36272,"style":2872},[642],[363,36274,36275,36278],{"style":2875},[363,36276],{"className":36277,"style":809},[650],[363,36279,36281],{"className":36280},[655,656,657,658],[363,36282,36284],{"className":36283},[407,658],[363,36285,2811],{"className":36286},[407,658],[363,36288,36289,36292],{"style":821},[363,36290],{"className":36291,"style":809},[650],[363,36293],{"className":36294,"style":829},[828],[363,36296,36297,36300],{"style":2898},[363,36298],{"className":36299,"style":809},[650],[363,36301,36303],{"className":36302},[655,656,657,658],[363,36304,36306],{"className":36305},[407,658],[363,36307,465],{"className":36308},[407,658],[363,36310,666],{"className":36311},[665],[363,36313,36315],{"className":36314},[638],[363,36316,36318],{"className":36317,"style":2920},[642],[363,36319],{},[363,36321],{"className":36322},[501,789],[363,36324,5156],{"className":36325},[407],[363,36327,36329,36332,36391],{"className":36328},[407],[363,36330],{"className":36331},[494,789],[363,36333,36335],{"className":36334},[586],[363,36336,36338,36383],{"className":36337},[633,634],[363,36339,36341,36380],{"className":36340},[638],[363,36342,36344,36358,36366],{"className":36343,"style":2872},[642],[363,36345,36346,36349],{"style":2875},[363,36347],{"className":36348,"style":809},[650],[363,36350,36352],{"className":36351},[655,656,657,658],[363,36353,36355],{"className":36354},[407,658],[363,36356,1116],{"className":36357},[407,408,658],[363,36359,36360,36363],{"style":821},[363,36361],{"className":36362,"style":809},[650],[363,36364],{"className":36365,"style":829},[828],[363,36367,36368,36371],{"style":2898},[363,36369],{"className":36370,"style":809},[650],[363,36372,36374],{"className":36373},[655,656,657,658],[363,36375,36377],{"className":36376},[407,658],[363,36378,465],{"className":36379},[407,658],[363,36381,666],{"className":36382},[665],[363,36384,36386],{"className":36385},[638],[363,36387,36389],{"className":36388,"style":2920},[642],[363,36390],{},[363,36392],{"className":36393},[501,789],[363,36395],{"className":36396,"style":2935},[505],[363,36398,5161],{"className":36399},[2939],[363,36401],{"className":36402,"style":2935},[505],[363,36404,36406,36409,36412,36415,36418],{"className":36405},[398],[363,36407],{"className":36408,"style":487},[402],[363,36410,35051],{"className":36411},[407],[363,36413],{"className":36414,"style":506},[505],[363,36416,471],{"className":36417},[510],[363,36419],{"className":36420,"style":506},[505],[363,36422,36424,36427,36495,36499,36502,36505],{"className":36423},[398],[363,36425],{"className":36426,"style":2853},[402],[363,36428,36430,36433,36492],{"className":36429},[407],[363,36431],{"className":36432},[494,789],[363,36434,36436],{"className":36435},[586],[363,36437,36439,36484],{"className":36438},[633,634],[363,36440,36442,36481],{"className":36441},[638],[363,36443,36445,36459,36467],{"className":36444,"style":2872},[642],[363,36446,36447,36450],{"style":2875},[363,36448],{"className":36449,"style":809},[650],[363,36451,36453],{"className":36452},[655,656,657,658],[363,36454,36456],{"className":36455},[407,658],[363,36457,2811],{"className":36458},[407,658],[363,36460,36461,36464],{"style":821},[363,36462],{"className":36463,"style":809},[650],[363,36465],{"className":36466,"style":829},[828],[363,36468,36469,36472],{"style":2898},[363,36470],{"className":36471,"style":809},[650],[363,36473,36475],{"className":36474},[655,656,657,658],[363,36476,36478],{"className":36477},[407,658],[363,36479,465],{"className":36480},[407,658],[363,36482,666],{"className":36483},[665],[363,36485,36487],{"className":36486},[638],[363,36488,36490],{"className":36489,"style":2920},[642],[363,36491],{},[363,36493],{"className":36494},[501,789],[363,36496,36498],{"className":36497},[407],"∣1",[363,36500],{"className":36501,"style":2935},[505],[363,36503,5161],{"className":36504},[2939],[363,36506],{"className":36507,"style":2935},[505],[363,36509,36511,36514,36517,36520,36523,36526],{"className":36510},[398],[363,36512],{"className":36513,"style":487},[402],[363,36515,1116],{"className":36516},[407,408],[363,36518,5156],{"className":36519},[407],[363,36521],{"className":36522,"style":506},[505],[363,36524,471],{"className":36525},[510],[363,36527],{"className":36528,"style":506},[505],[363,36530,36532,36535,36538,36541,36544],{"className":36531},[398],[363,36533],{"className":36534,"style":487},[402],[363,36536,385],{"className":36537,"style":409},[407,408],[363,36539,461],{"className":36540},[494],[363,36542,1116],{"className":36543},[407,408],[363,36545,468],{"className":36546},[501],"，所以 TV 是对称的。",[11,36549,36550,36552,36553,36556],{},[1344,36551,24565],{}," 全变差是概率测度空间上真正的",[1344,36554,36555],{},"度量","（对称、满足三角不等式），它等于两个分布在一件事上能给出的最大概率差：",[363,36558,36560],{"className":36559,"translate":367},[528],[363,36561,36563,36637],{"className":36562,"translate":367},[366],[363,36564,36566],{"className":36565},[371],[373,36567,36568],{"xmlns":375,"display":537},[377,36569,36570,36634],{},[380,36571,36572,36582,36584,36586,36588,36590,36592,36594,36610,36612,36614,36616,36618,36620,36622,36624,36626,36628,36630,36632],{},[543,36573,36574,36576],{},[383,36575,547],{},[380,36577,36578,36580],{},[383,36579,5740],{},[383,36581,35222],{},[458,36583,461],{"stretchy":460},[383,36585,554],{},[458,36587,898],{"separator":394},[383,36589,561],{},[458,36591,468],{"stretchy":460},[458,36593,471],{},[6364,36595,36596,36602],{},[380,36597,36598,36600],{},[383,36599,35804],{},[458,36601,4451],{},[380,36603,36604,36606,36608],{},[383,36605,950],{},[458,36607,6257],{},[383,36609,950],{"mathvariant":944},[458,36611,5156],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[383,36613,554],{},[458,36615,461],{"stretchy":460},[383,36617,950],{},[458,36619,468],{"stretchy":460},[458,36621,5161],{},[383,36623,561],{},[458,36625,461],{"stretchy":460},[383,36627,950],{},[458,36629,468],{"stretchy":460},[458,36631,5156],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[383,36633,4723],{"mathvariant":557},[387,36635,36636],{"encoding":389},"D_{TV}(P,Q)\n=\\sup_{A\\in\\mathcal A}\\big|P(A)-Q(A)\\big| .",[363,36638,36640,36719,36845],{"className":36639,"ariaHidden":394},[393],[363,36641,36643,36646,36692,36695,36698,36701,36704,36707,36710,36713,36716],{"className":36642},[398],[363,36644],{"className":36645,"style":487},[402],[363,36647,36649,36652],{"className":36648},[407],[363,36650,547],{"className":36651,"style":625},[407,408],[363,36653,36655],{"className":36654},[629],[363,36656,36658,36684],{"className":36657},[633,634],[363,36659,36661,36681],{"className":36660},[638],[363,36662,36664],{"className":36663,"style":727},[642],[363,36665,36666,36669],{"style":646},[363,36667],{"className":36668,"style":651},[650],[363,36670,36672],{"className":36671},[655,656,657,658],[363,36673,36675,36678],{"className":36674},[407,658],[363,36676,5740],{"className":36677,"style":682},[407,408,658],[363,36679,35222],{"className":36680,"style":2935},[407,408,658],[363,36682,666],{"className":36683},[665],[363,36685,36687],{"className":36686},[638],[363,36688,36690],{"className":36689,"style":8316},[642],[363,36691],{},[363,36693,461],{"className":36694},[494],[363,36696,554],{"className":36697,"style":682},[407,408],[363,36699,898],{"className":36700},[920],[363,36702],{"className":36703,"style":754},[505],[363,36705,561],{"className":36706},[407,408],[363,36708,468],{"className":36709},[501],[363,36711],{"className":36712,"style":506},[505],[363,36714,471],{"className":36715},[510],[363,36717],{"className":36718,"style":506},[505],[363,36720,36722,36726,36781,36784,36824,36827,36830,36833,36836,36839,36842],{"className":36721},[398],[363,36723],{"className":36724,"style":36725},[402],"height:1.8161em;vertical-align:-0.9661em;",[363,36727,36729],{"className":36728},[1832,14066],[363,36730,36732,36772],{"className":36731},[633,634],[363,36733,36735,36769],{"className":36734},[638],[363,36736,36738,36759],{"className":36737,"style":2693},[642],[363,36739,36741,36744],{"style":36740},"top:-2.1612em;margin-left:0em;",[363,36742],{"className":36743,"style":809},[650],[363,36745,36747],{"className":36746},[655,656,657,658],[363,36748,36750,36753,36756],{"className":36749},[407,658],[363,36751,950],{"className":36752},[407,408,658],[363,36754,6257],{"className":36755},[510,658],[363,36757,950],{"className":36758},[407,971,658],[363,36760,36761,36764],{"style":12213},[363,36762],{"className":36763,"style":809},[650],[363,36765,36766],{},[363,36767,35804],{"className":36768},[1832],[363,36770,666],{"className":36771},[665],[363,36773,36775],{"className":36774},[638],[363,36776,36779],{"className":36777,"style":36778},[642],"height:0.9661em;",[363,36780],{},[363,36782],{"className":36783,"style":754},[505],[363,36785,36787],{"className":36786},[407],[363,36788,36790],{"className":36789},[767,15179],[363,36791,36793,36816],{"className":36792},[633,634],[363,36794,36796,36813],{"className":36795},[638],[363,36797,36799],{"className":36798,"style":15189},[642],[363,36800,36801,36804],{"style":15192},[363,36802],{"className":36803,"style":15196},[650],[363,36805,36807],{"style":36806},"width:0.333em;height:1.2em;",[6671,36808,36810],{"xmlns":6673,"width":35423,"height":9798,"viewBox":36809},"0 0 333 1200",[6679,36811],{"d":36812},"M145 15 v585 v0 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 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",[363,36907,36909,36922],{"className":36908,"translate":367},[366],[363,36910,36912],{"className":36911},[371],[373,36913,36914],{"xmlns":375},[377,36915,36916,36920],{},[380,36917,36918],{},[383,36919,554],{},[387,36921,554],{"encoding":389},[363,36923,36925],{"className":36924,"ariaHidden":394},[393],[363,36926,36928,36931],{"className":36927},[398],[363,36929],{"className":36930,"style":1011},[402],[363,36932,554],{"className":36933,"style":682},[407,408],[363,36935,36937,36950],{"className":36936,"translate":367},[366],[363,36938,36940],{"className":36939},[371],[373,36941,36942],{"xmlns":375},[377,36943,36944,36948],{},[380,36945,36946],{},[383,36947,561],{},[387,36949,561],{"encoding":389},[363,36951,36953],{"className":36952,"ariaHidden":394},[393],[363,36954,36956,36959],{"className":36955},[398],[363,36957],{"className":36958,"style":913},[402],[363,36960,561],{"className":36961},[407,408]," 时，最小平均错误率为 ",[363,36964,36966,37004],{"className":36965,"translate":367},[366],[363,36967,36969],{"className":36968},[371],[373,36970,36971],{"xmlns":375},[377,36972,36973,37001],{},[380,36974,36975,36983,36985,36987,36989,36999],{},[1603,36976,36977],{"scriptlevel":474,"displaystyle":460},[586,36978,36979,36981],{},[463,36980,465],{},[463,36982,2811],{},[458,36984,461],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[463,36986,465],{},[458,36988,5161],{},[543,36990,36991,36993],{},[383,36992,547],{},[380,36994,36995,36997],{},[383,36996,5740],{},[383,36998,35222],{},[458,37000,468],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[387,37002,37003],{"encoding":389},"\\tfrac12\\big(1-D_{TV}\\big)",[363,37005,37007,37099],{"className":37006,"ariaHidden":394},[393],[363,37008,37010,37013,37081,37087,37090,37093,37096],{"className":37009},[398],[363,37011],{"className":37012,"style":20935},[402],[363,37014,37016,37019,37078],{"className":37015},[407],[363,37017],{"className":37018},[494,789],[363,37020,37022],{"className":37021},[586],[363,37023,37025,37070],{"className":37024},[633,634],[363,37026,37028,37067],{"className":37027},[638],[363,37029,37031,37045,37053],{"className":37030,"style":2872},[642],[363,37032,37033,37036],{"style":2875},[363,37034],{"className":37035,"style":809},[650],[363,37037,37039],{"className":37038},[655,656,657,658],[363,37040,37042],{"className":37041},[407,658],[363,37043,2811],{"className":37044},[407,658],[363,37046,37047,37050],{"style":821},[363,37048],{"className":37049,"style":809},[650],[363,37051],{"className":37052,"style":829},[828],[363,37054,37055,37058],{"style":2898},[363,37056],{"className":37057,"style":809},[650],[363,37059,37061],{"className":37060},[655,656,657,658],[363,37062,37064],{"className":37063},[407,658],[363,37065,465],{"className":37066},[407,658],[363,37068,666],{"className":37069},[665],[363,37071,37073],{"className":37072},[638],[363,37074,37076],{"className":37075,"style":2920},[642],[363,37077],{},[363,37079],{"className":37080},[501,789],[363,37082,37084],{"className":37083},[407],[363,37085,461],{"className":37086},[767,10022],[363,37088,465],{"className":37089},[407],[363,37091],{"className":37092,"style":2935},[505],[363,37094,5161],{"className":37095},[2939],[363,37097],{"className":37098,"style":2935},[505],[363,37100,37102,37105,37151],{"className":37101},[398],[363,37103],{"className":37104,"style":20935},[402],[363,37106,37108,37111],{"className":37107},[407],[363,37109,547],{"className":37110,"style":625},[407,408],[363,37112,37114],{"className":37113},[629],[363,37115,37117,37143],{"className":37116},[633,634],[363,37118,37120,37140],{"className":37119},[638],[363,37121,37123],{"className":37122,"style":727},[642],[363,37124,37125,37128],{"style":646},[363,37126],{"className":37127,"style":651},[650],[363,37129,37131],{"className":37130},[655,656,657,658],[363,37132,37134,37137],{"className":37133},[407,658],[363,37135,5740],{"className":37136,"style":682},[407,408,658],[363,37138,35222],{"className":37139,"style":2935},[407,408,658],[363,37141,666],{"className":37142},[665],[363,37144,37146],{"className":37145},[638],[363,37147,37149],{"className":37148,"style":8316},[642],[363,37150],{},[363,37152,37154],{"className":37153},[407],[363,37155,468],{"className":37156},[767,10022],"，因此 TV 直接度量了",[1344,37159,37160],{},"可分性","，「TV 小」就等于「几乎分不开」。TV 是这里所有 f-散度中最「弱」的标尺（总在 ",[363,37163,37165,37187],{"className":37164,"translate":367},[366],[363,37166,37168],{"className":37167},[371],[373,37169,37170],{"xmlns":375},[377,37171,37172,37184],{},[380,37173,37174,37176,37178,37180,37182],{},[458,37175,578],{"stretchy":460},[463,37177,474],{},[458,37179,898],{"separator":394},[463,37181,465],{},[458,37183,605],{"stretchy":460},[387,37185,37186],{"encoding":389},"[0,1]",[363,37188,37190],{"className":37189,"ariaHidden":394},[393],[363,37191,37193,37196,37199,37202,37205,37208,37211],{"className":37192},[398],[363,37194],{"className":37195,"style":487},[402],[363,37197,578],{"className":37198},[494],[363,37200,474],{"className":37201},[407],[363,37203,898],{"className":37204},[920],[363,37206],{"className":37207,"style":754},[505],[363,37209,465],{"className":37210},[407],[363,37212,605],{"className":37213},[501]," 内），但按 Csiszár 定理它与 KL 诱导同一个拓扑，所以常常已经够用；在一维实直线上它退化为 Kolmogorov 距离 ",[363,37216,37218,37283],{"className":37217,"translate":367},[366],[363,37219,37221],{"className":37220},[371],[373,37222,37223],{"xmlns":375},[377,37224,37225,37280],{},[380,37226,37227,37237,37239,37249,37251,37258,37260,37262,37264,37266,37272,37274,37276,37278],{},[543,37228,37229,37231],{},[383,37230,547],{},[380,37232,37233,37235],{},[383,37234,5740],{},[383,37236,35222],{},[458,37238,471],{},[543,37240,37241,37247],{},[380,37242,37243,37245],{},[383,37244,35804],{},[458,37246,4451],{},[383,37248,1121],{},[383,37250,5156],{"mathvariant":557},[543,37252,37253,37256],{},[383,37254,37255],{},"F",[383,37257,554],{},[458,37259,461],{"stretchy":460},[383,37261,1121],{},[458,37263,468],{"stretchy":460},[458,37265,5161],{},[543,37267,37268,37270],{},[383,37269,37255],{},[383,37271,561],{},[458,37273,461],{"stretchy":460},[383,37275,1121],{},[458,37277,468],{"stretchy":460},[383,37279,5156],{"mathvariant":557},[387,37281,37282],{"encoding":389},"D_{TV}=\\sup_x|F_P(x)-F_Q(x)|",[363,37284,37286,37347,37459],{"className":37285,"ariaHidden":394},[393],[363,37287,37289,37292,37338,37341,37344],{"className":37288},[398],[363,37290],{"className":37291,"style":16311},[402],[363,37293,37295,37298],{"className":37294},[407],[363,37296,547],{"className":37297,"style":625},[407,408],[363,37299,37301],{"className":37300},[629],[363,37302,37304,37330],{"className":37303},[633,634],[363,37305,37307,37327],{"className":37306},[638],[363,37308,37310],{"className":37309,"style":727},[642],[363,37311,37312,37315],{"style":646},[363,37313],{"className":37314,"style":651},[650],[363,37316,37318],{"className":37317},[655,656,657,658],[363,37319,37321,37324],{"className":37320},[407,658],[363,37322,5740],{"className":37323,"style":682},[407,408,658],[363,37325,35222],{"className":37326,"style":2935},[407,408,658],[363,37328,666],{"className":37329},[665],[363,37331,37333],{"className":37332},[638],[363,37334,37336],{"className":37335,"style":8316},[642],[363,37337],{},[363,37339],{"className":37340,"style":506},[505],[363,37342,471],{"className":37343},[510],[363,37345],{"className":37346,"style":506},[505],[363,37348,37350,37353,37395,37398,37401,37441,37444,37447,37450,37453,37456],{"className":37349},[398],[363,37351],{"className":37352,"style":487},[402],[363,37354,37356,37359],{"className":37355},[1832],[363,37357,35804],{"className":37358},[1832],[363,37360,37362],{"className":37361},[629],[363,37363,37365,37386],{"className":37364},[633,634],[363,37366,37368,37383],{"className":37367},[638],[363,37369,37372],{"className":37370,"style":37371},[642],"height:0.0573em;",[363,37373,37374,37377],{"style":35950},[363,37375],{"className":37376,"style":651},[650],[363,37378,37380],{"className":37379},[655,656,657,658],[363,37381,1121],{"className":37382},[407,408,658],[363,37384,666],{"className":37385},[665],[363,37387,37389],{"className":37388},[638],[363,37390,37393],{"className":37391,"style":37392},[642],"height:0.2441em;",[363,37394],{},[363,37396],{"className":37397,"style":754},[505],[363,37399,5156],{"className":37400},[407],[363,37402,37404,37407],{"className":37403},[407],[363,37405,37255],{"className":37406,"style":682},[407,408],[363,37408,37410],{"className":37409},[629],[363,37411,37413,37433],{"className":37412},[633,634],[363,37414,37416,37430],{"className":37415},[638],[363,37417,37419],{"className":37418,"style":727},[642],[363,37420,37421,37424],{"style":11486},[363,37422],{"className":37423,"style":651},[650],[363,37425,37427],{"className":37426},[655,656,657,658],[363,37428,554],{"className":37429,"style":682},[407,408,658],[363,37431,666],{"className":37432},[665],[363,37434,37436],{"className":37435},[638],[363,37437,37439],{"className":37438,"style":8316},[642],[363,37440],{},[363,37442,461],{"className":37443},[494],[363,37445,1121],{"className":37446},[407,408],[363,37448,468],{"className":37449},[501],[363,37451],{"className":37452,"style":2935},[505],[363,37454,5161],{"className":37455},[2939],[363,37457],{"className":37458,"style":2935},[505],[363,37460,37462,37465,37505,37508,37511,37514],{"className":37461},[398],[363,37463],{"className":37464,"style":618},[402],[363,37466,37468,37471],{"className":37467},[407],[363,37469,37255],{"className":37470,"style":682},[407,408],[363,37472,37474],{"className":37473},[629],[363,37475,37477,37497],{"className":37476},[633,634],[363,37478,37480,37494],{"className":37479},[638],[363,37481,37483],{"className":37482,"style":727},[642],[363,37484,37485,37488],{"style":11486},[363,37486],{"className":37487,"style":651},[650],[363,37489,37491],{"className":37490},[655,656,657,658],[363,37492,561],{"className":37493},[407,408,658],[363,37495,666],{"className":37496},[665],[363,37498,37500],{"className":37499},[638],[363,37501,37503],{"className":37502,"style":673},[642],[363,37504],{},[363,37506,461],{"className":37507},[494],[363,37509,1121],{"className":37510},[407,408],[363,37512,468],{"className":37513},[501],[363,37515,5156],{"className":37516},[407],"。马尔可夫链的混合时间、耦合构造与统计距离（total variation distance）都以它为核心。",[2659,37519,37521],{"id":37520},"_34-卡方散度","3.4 卡方散度",[11,37523,21845],{},[363,37525,37527],{"className":37526,"translate":367},[528],[363,37528,37530,37566],{"className":37529,"translate":367},[366],[363,37531,37533],{"className":37532},[371],[373,37534,37535],{"xmlns":375,"display":537},[377,37536,37537,37563],{},[380,37538,37539,37541,37543,37545,37547,37549,37551,37553,37555,37557],{},[383,37540,385],{},[458,37542,461],{"stretchy":460},[383,37544,1116],{},[458,37546,468],{"stretchy":460},[458,37548,471],{},[458,37550,461],{"stretchy":460},[383,37552,1116],{},[458,37554,5161],{},[463,37556,465],{},[7162,37558,37559,37561],{},[458,37560,468],{"stretchy":460},[463,37562,2811],{},[387,37564,37565],{"encoding":389},"f(t)=(t-1)^2",[363,37567,37569,37596,37617],{"className":37568,"ariaHidden":394},[393],[363,37570,37572,37575,37578,37581,37584,37587,37590,37593],{"className":37571},[398],[363,37573],{"className":37574,"style":487},[402],[363,37576,385],{"className":37577,"style":409},[407,408],[363,37579,461],{"className":37580},[494],[363,37582,1116],{"className":37583},[407,408],[363,37585,468],{"className":37586},[501],[363,37588],{"className":37589,"style":506},[505],[363,37591,471],{"className":37592},[510],[363,37594],{"className":37595,"style":506},[505],[363,37597,37599,37602,37605,37608,37611,37614],{"className":37598},[398],[363,37600],{"className":37601,"style":487},[402],[363,37603,461],{"className":37604},[494],[363,37606,1116],{"className":37607},[407,408],[363,37609],{"className":37610,"style":2935},[505],[363,37612,5161],{"className":37613},[2939],[363,37615],{"className":37616,"style":2935},[505],[363,37618,37620,37624,37627],{"className":37619},[398],[363,37621],{"className":37622,"style":37623},[402],"height:1.1141em;vertical-align:-0.25em;",[363,37625,465],{"className":37626},[407],[363,37628,37630,37633],{"className":37629},[501],[363,37631,468],{"className":37632},[501],[363,37634,37636],{"className":37635},[629],[363,37637,37639],{"className":37638},[633],[363,37640,37642],{"className":37641},[638],[363,37643,37645],{"className":37644,"style":20107},[642],[363,37646,37647,37650],{"style":8625},[363,37648],{"className":37649,"style":651},[650],[363,37651,37653],{"className":37652},[655,656,657,658],[363,37654,2811],{"className":37655},[407,658],[11,37657,35054,37658,37723],{},[363,37659,37661,37684],{"className":37660,"translate":367},[366],[363,37662,37664],{"className":37663},[371],[373,37665,37666],{"xmlns":375},[377,37667,37668,37682],{},[380,37669,37670,37672,37674,37676,37678,37680],{},[383,37671,385],{},[458,37673,461],{"stretchy":460},[463,37675,465],{},[458,37677,468],{"stretchy":460},[458,37679,471],{},[463,37681,474],{},[387,37683,477],{"encoding":389},[363,37685,37687,37714],{"className":37686,"ariaHidden":394},[393],[363,37688,37690,37693,37696,37699,37702,37705,37708,37711],{"className":37689},[398],[363,37691],{"className":37692,"style":487},[402],[363,37694,385],{"className":37695,"style":409},[407,408],[363,37697,461],{"className":37698},[494],[363,37700,465],{"className":37701},[407],[363,37703,468],{"className":37704},[501],[363,37706],{"className":37707,"style":506},[505],[363,37709,471],{"className":37710},[510],[363,37712],{"className":37713,"style":506},[505],[363,37715,37717,37720],{"className":37716},[398],[363,37718],{"className":37719,"style":520},[402],[363,37721,474],{"className":37722},[407],"、凸，代入：",[363,37725,37727],{"className":37726,"translate":367},[528],[363,37728,37730,37837],{"className":37729,"translate":367},[366],[363,37731,37733],{"className":37732},[371],[373,37734,37735],{"xmlns":375,"display":537},[377,37736,37737,37834],{},[380,37738,37739,37745,37747,37749,37751,37753,37755,37757,37759,37761,37781,37783,37785,37787,37789,37809,37811,37813,37815,37817,37824,37826,37828,37830,37832],{},[543,37740,37741,37743],{},[383,37742,547],{},[383,37744,385],{},[458,37746,461],{"stretchy":460},[383,37748,554],{},[383,37750,558],{"mathvariant":557},[383,37752,561],{},[458,37754,468],{"stretchy":460},[458,37756,471],{},[458,37758,1635],{},[383,37760,1291],{},[7162,37762,37763,37779],{},[380,37764,37765,37767,37773,37775,37777],{},[458,37766,461],{"fence":394},[586,37768,37769,37771],{},[383,37770,11],{},[383,37772,1291],{},[458,37774,5161],{},[463,37776,465],{},[458,37778,468],{"fence":394},[463,37780,2811],{},[383,37782,592],{},[383,37784,1121],{},[458,37786,471],{},[458,37788,1635],{},[586,37790,37791,37807],{},[380,37792,37793,37795,37797,37799,37801],{},[458,37794,461],{"stretchy":460},[383,37796,11],{},[458,37798,5161],{},[383,37800,1291],{},[7162,37802,37803,37805],{},[458,37804,468],{"stretchy":460},[463,37806,2811],{},[383,37808,1291],{},[1645,37810,1647],{},[383,37812,592],{},[383,37814,1121],{},[458,37816,471],{},[7162,37818,37819,37822],{},[383,37820,37821],{},"χ",[463,37823,2811],{},[458,37825,461],{"stretchy":460},[383,37827,554],{},[383,37829,558],{"mathvariant":557},[383,37831,561],{},[458,37833,468],{"stretchy":460},[387,37835,37836],{"encoding":389},"D_f(P\\|Q)=\\int q\\left(\\frac{p}{q}-1\\right)^2dx\n=\\int \\frac{(p-q)^2}{q}\\,dx\n=\\chi^2(P\\|Q)",[363,37838,37840,37910,38064,38202],{"className":37839,"ariaHidden":394},[393],[363,37841,37843,37846,37886,37889,37892,37895,37898,37901,37904,37907],{"className":37842},[398],[363,37844],{"className":37845,"style":618},[402],[363,37847,37849,37852],{"className":37848},[407],[363,37850,547],{"className":37851,"style":625},[407,408],[363,37853,37855],{"className":37854},[629],[363,37856,37858,37878],{"className":37857},[633,634],[363,37859,37861,37875],{"className":37860},[638],[363,37862,37864],{"className":37863,"style":643},[642],[363,37865,37866,37869],{"style":646},[363,37867],{"className":37868,"style":651},[650],[363,37870,37872],{"className":37871},[655,656,657,658],[363,37873,385],{"className":37874,"style":409},[407,408,658],[363,37876,666],{"className":37877},[665],[363,37879,37881],{"className":37880},[638],[363,37882,37884],{"className":37883,"style":673},[642],[363,37885],{},[363,37887,461],{"className":37888},[494],[363,37890,554],{"className":37891,"style":682},[407,408],[363,37893,558],{"className":37894},[407],[363,37896,561],{"className":37897},[407,408],[363,37899,468],{"className":37900},[501],[363,37902],{"className":37903,"style":506},[505],[363,37905,471],{"className":37906},[510],[363,37908],{"className":37909,"style":506},[505],[363,37911,37913,37917,37920,37923,37926,37929,38046,38049,38052,38055,38058,38061],{"className":37912},[398],[363,37914],{"className":37915,"style":37916},[402],"height:2.604em;vertical-align:-0.95em;",[363,37918,1635],{"className":37919,"style":1835},[1832,1833,1834],[363,37921],{"className":37922,"style":754},[505],[363,37924,1291],{"className":37925,"style":1328},[407,408],[363,37927],{"className":37928,"style":754},[505],[363,37930,37932,38021],{"className":37931},[758],[363,37933,37935,37941,38003,38006,38009,38012,38015],{"className":37934},[758],[363,37936,37938],{"className":37937,"style":763},[494,762],[363,37939,461],{"className":37940},[767,657],[363,37942,37944,37947,38000],{"className":37943},[407],[363,37945],{"className":37946},[494,789],[363,37948,37950],{"className":37949},[586],[363,37951,37953,37992],{"className":37952},[633,634],[363,37954,37956,37989],{"className":37955},[638],[363,37957,37959,37970,37978],{"className":37958,"style":3902},[642],[363,37960,37961,37964],{"style":805},[363,37962],{"className":37963,"style":809},[650],[363,37965,37967],{"className":37966},[407],[363,37968,1291],{"className":37969,"style":1328},[407,408],[363,37971,37972,37975],{"style":821},[363,37973],{"className":37974,"style":809},[650],[363,37976],{"className":37977,"style":829},[828],[363,37979,37980,37983],{"style":832},[363,37981],{"className":37982,"style":809},[650],[363,37984,37986],{"className":37985},[407],[363,37987,11],{"className":37988},[407,408],[363,37990,666],{"className":37991},[665],[363,37993,37995],{"className":37994},[638],[363,37996,37998],{"className":37997,"style":854},[642],[363,37999],{},[363,38001],{"className":38002},[501,789],[363,38004],{"className":38005,"style":2935},[505],[363,38007,5161],{"className":38008},[2939],[363,38010],{"className":38011,"style":2935},[505],[363,38013,465],{"className":38014},[407],[363,38016,38018],{"className":38017,"style":763},[501,762],[363,38019,468],{"className":38020},[767,657],[363,38022,38024],{"className":38023},[629],[363,38025,38027],{"className":38026},[633],[363,38028,38030],{"className":38029},[638],[363,38031,38034],{"className":38032,"style":38033},[642],"height:1.654em;",[363,38035,38037,38040],{"style":38036},"top:-3.9029em;margin-right:0.05em;",[363,38038],{"className":38039,"style":651},[650],[363,38041,38043],{"className":38042},[655,656,657,658],[363,38044,2811],{"className":38045},[407,658],[363,38047],{"className":38048,"style":754},[505],[363,38050,592],{"className":38051},[407,408],[363,38053,1121],{"className":38054},[407,408],[363,38056],{"className":38057,"style":506},[505],[363,38059,471],{"className":38060},[510],[363,38062],{"className":38063,"style":506},[505],[363,38065,38067,38071,38074,38077,38184,38187,38190,38193,38196,38199],{"className":38066},[398],[363,38068],{"className":38069,"style":38070},[402],"height:2.3715em;vertical-align:-0.8804em;",[363,38072,1635],{"className":38073,"style":1835},[1832,1833,1834],[363,38075],{"className":38076,"style":754},[505],[363,38078,38080,38083,38181],{"className":38079},[407],[363,38081],{"className":38082},[494,789],[363,38084,38086],{"className":38085},[586],[363,38087,38089,38173],{"className":38088},[633,634],[363,38090,38092,38170],{"className":38091},[638],[363,38093,38096,38107,38115],{"className":38094,"style":38095},[642],"height:1.4911em;",[363,38097,38098,38101],{"style":805},[363,38099],{"className":38100,"style":809},[650],[363,38102,38104],{"className":38103},[407],[363,38105,1291],{"className":38106,"style":1328},[407,408],[363,38108,38109,38112],{"style":821},[363,38110],{"className":38111,"style":809},[650],[363,38113],{"className":38114,"style":829},[828],[363,38116,38117,38120],{"style":832},[363,38118],{"className":38119,"style":809},[650],[363,38121,38123,38126,38129,38132,38135,38138,38141],{"className":38122},[407],[363,38124,461],{"className":38125},[494],[363,38127,11],{"className":38128},[407,408],[363,38130],{"className":38131,"style":2935},[505],[363,38133,5161],{"className":38134},[2939],[363,38136],{"className":38137,"style":2935},[505],[363,38139,1291],{"className":38140,"style":1328},[407,408],[363,38142,38144,38147],{"className":38143},[501],[363,38145,468],{"className":38146},[501],[363,38148,38150],{"className":38149},[629],[363,38151,38153],{"className":38152},[633],[363,38154,38156],{"className":38155},[638],[363,38157,38159],{"className":38158,"style":20400},[642],[363,38160,38161,38164],{"style":7213},[363,38162],{"className":38163,"style":651},[650],[363,38165,38167],{"className":38166},[655,656,657,658],[363,38168,2811],{"className":38169},[407,658],[363,38171,666],{"className":38172},[665],[363,38174,38176],{"className":38175},[638],[363,38177,38179],{"className":38178,"style":854},[642],[363,38180],{},[363,38182],{"className":38183},[501,789],[363,38185],{"className":38186,"style":754},[505],[363,38188,592],{"className":38189},[407,408],[363,38191,1121],{"className":38192},[407,408],[363,38194],{"className":38195,"style":506},[505],[363,38197,471],{"className":38198},[510],[363,38200],{"className":38201,"style":506},[505],[363,38203,38205,38208,38237,38240,38243,38246,38249],{"className":38204},[398],[363,38206],{"className":38207,"style":37623},[402],[363,38209,38211,38214],{"className":38210},[407],[363,38212,37821],{"className":38213},[407,408],[363,38215,38217],{"className":38216},[629],[363,38218,38220],{"className":38219},[633],[363,38221,38223],{"className":38222},[638],[363,38224,38226],{"className":38225,"style":20107},[642],[363,38227,38228,38231],{"style":8625},[363,38229],{"className":38230,"style":651},[650],[363,38232,38234],{"className":38233},[655,656,657,658],[363,38235,2811],{"className":38236},[407,658],[363,38238,461],{"className":38239},[494],[363,38241,554],{"className":38242,"style":682},[407,408],[363,38244,558],{"className":38245},[407],[363,38247,561],{"className":38248},[407,408],[363,38250,468],{"className":38251},[501],[11,38253,38254,38255,38258],{},"这是",[1344,38256,38257],{},"皮尔逊（Pearson）卡方散度","。它非对称、无界。其斜共轭是",[363,38260,38262],{"className":38261,"translate":367},[528],[363,38263,38265,38341],{"className":38264,"translate":367},[366],[363,38266,38268],{"className":38267},[371],[373,38269,38270],{"xmlns":375,"display":537},[377,38271,38272,38338],{},[380,38273,38274,38276,38278,38280,38282,38284,38286,38288,38290,38292,38294,38314,38316,38336],{},[383,38275,1116],{},[1645,38277,1647],{},[383,38279,385],{},[458,38281,461],{"stretchy":460},[463,38283,465],{},[383,38285,1288],{"mathvariant":557},[383,38287,1116],{},[458,38289,468],{"stretchy":460},[458,38291,471],{},[383,38293,1116],{},[7162,38295,38296,38312],{},[380,38297,38298,38300,38306,38308,38310],{},[458,38299,461],{"fence":394},[586,38301,38302,38304],{},[463,38303,465],{},[383,38305,1116],{},[458,38307,5161],{},[463,38309,465],{},[458,38311,468],{"fence":394},[463,38313,2811],{},[458,38315,471],{},[586,38317,38318,38334],{},[380,38319,38320,38322,38324,38326,38328],{},[458,38321,461],{"stretchy":460},[383,38323,1116],{},[458,38325,5161],{},[463,38327,465],{},[7162,38329,38330,38332],{},[458,38331,468],{"stretchy":460},[463,38333,2811],{},[383,38335,1116],{},[458,38337,898],{"separator":394},[387,38339,38340],{"encoding":389},"t\\,f(1\u002Ft)=t\\left(\\frac1t-1\\right)^2\n=\\frac{(t-1)^2}{t},",[363,38342,38344,38380,38516],{"className":38343,"ariaHidden":394},[393],[363,38345,38347,38350,38353,38356,38359,38362,38365,38368,38371,38374,38377],{"className":38346},[398],[363,38348],{"className":38349,"style":487},[402],[363,38351,1116],{"className":38352},[407,408],[363,38354],{"className":38355,"style":754},[505],[363,38357,385],{"className":38358,"style":409},[407,408],[363,38360,461],{"className":38361},[494],[363,38363,8937],{"className":38364},[407],[363,38366,1116],{"className":38367},[407,408],[363,38369,468],{"className":38370},[501],[363,38372],{"className":38373,"style":506},[505],[363,38375,471],{"className":38376},[510],[363,38378],{"className":38379,"style":506},[505],[363,38381,38383,38386,38389,38392,38507,38510,38513],{"className":38382},[398],[363,38384],{"className":38385,"style":37916},[402],[363,38387,1116],{"className":38388},[407,408],[363,38390],{"className":38391,"style":754},[505],[363,38393,38395,38484],{"className":38394},[758],[363,38396,38398,38404,38466,38469,38472,38475,38478],{"className":38397},[758],[363,38399,38401],{"className":38400,"style":763},[494,762],[363,38402,461],{"className":38403},[767,657],[363,38405,38407,38410,38463],{"className":38406},[407],[363,38408],{"className":38409},[494,789],[363,38411,38413],{"className":38412},[586],[363,38414,38416,38455],{"className":38415},[633,634],[363,38417,38419,38452],{"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展开）有 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q)^2\\approx\\frac14\\chi^2",[363,39372,39374,39443,39533],{"className":39373,"ariaHidden":394},[393],[363,39375,39377,39381,39384,39387,39434,39437,39440],{"className":39376},[398],[363,39378],{"className":39379,"style":39380},[402],"height:1.1419em;vertical-align:-0.3369em;",[363,39382,1635],{"className":39383,"style":5965},[1832,1833,5964],[363,39385,461],{"className":39386},[494],[363,39388,39390],{"className":39389},[407,34739],[363,39391,39393,39425],{"className":39392},[633,634],[363,39394,39396,39422],{"className":39395},[638],[363,39397,39400,39409],{"className":39398,"style":39399},[642],"height:0.7031em;",[363,39401,39403,39406],{"className":39402,"style":12213},[6655],[363,39404],{"className":39405,"style":809},[650],[363,39407,11],{"className":39408,"style":34759},[407,408],[363,39410,39412,39415],{"style":39411},"top:-2.6631em;",[363,39413],{"className":39414,"style":809},[650],[363,39416,39418],{"className":39417,"style":34816},[34815],[6671,39419,39420],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},[6679,39421],{"d":34823},[363,39423,666],{"className":39424},[665],[363,39426,39428],{"className":39427},[638],[363,39429,39432],{"className":39430,"style":39431},[642],"height:0.3369em;",[363,39433],{},[363,39435],{"className":39436,"style":2935},[505],[363,39438,5161],{"className":39439},[2939],[363,39441],{"className":39442,"style":2935},[505],[363,39444,39446,39450,39495,39524,39527,39530],{"className":39445},[398],[363,39447],{"className":39448,"style":39449},[402],"height:1.151em;vertical-align:-0.3369em;",[363,39451,39453],{"className":39452},[407,34739],[363,39454,39456,39487],{"className":39455},[633,634],[363,39457,39459,39484],{"className":39458},[638],[363,39460,39462,39472],{"className":39461,"style":39399},[642],[363,39463,39465,39468],{"className":39464,"style":12213},[6655],[363,39466],{"className":39467,"style":809},[650],[363,39469,1291],{"className":39470,"style":39471},[407,408],"margin-right:0.0359em;padding-left:0.833em;",[363,39473,39474,39477],{"style":39411},[363,39475],{"className":39476,"style":809},[650],[363,39478,39480],{"className":39479,"style":34816},[34815],[6671,39481,39482],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},[6679,39483],{"d":34823},[363,39485,666],{"className":39486},[665],[363,39488,39490],{"className":39489},[638],[363,39491,39493],{"className":39492,"style":39431},[642],[363,39494],{},[363,39496,39498,39501],{"className":39497},[501],[363,39499,468],{"className":39500},[501],[363,39502,39504],{"className":39503},[629],[363,39505,39507],{"className":39506},[633],[363,39508,39510],{"className":39509},[638],[363,39511,39513],{"className":39512,"style":20400},[642],[363,39514,39515,39518],{"style":7213},[363,39516],{"className":39517,"style":651},[650],[363,39519,39521],{"className":39520},[655,656,657,658],[363,39522,2811],{"className":39523},[407,658],[363,39525],{"className":39526,"style":506},[505],[363,39528,33331],{"className":39529},[510],[363,39531],{"className":39532,"style":506},[505],[363,39534,39536,39539,39607],{"className":39535},[398],[363,39537],{"className":39538,"style":2853},[402],[363,39540,39542,39545,39604],{"className":39541},[407],[363,39543],{"className":39544},[494,789],[363,39546,39548],{"className":39547},[586],[363,39549,39551,39596],{"className":39550},[633,634],[363,39552,39554,39593],{"className":39553},[638],[363,39555,39557,39571,39579],{"className":39556,"style":2872},[642],[363,39558,39559,39562],{"style":2875},[363,39560],{"className":39561,"style":809},[650],[363,39563,39565],{"className":39564},[655,656,657,658],[363,39566,39568],{"className":39567},[407,658],[363,39569,39361],{"className":39570},[407,658],[363,39572,39573,39576],{"style":821},[363,39574],{"className":39575,"style":809},[650],[363,39577],{"className":39578,"style":829},[828],[363,39580,39581,39584],{"style":2898},[363,39582],{"className":39583,"style":809},[650],[363,39585,39587],{"className":39586},[655,656,657,658],[363,39588,39590],{"className":39589},[407,658],[363,39591,465],{"className":39592},[407,658],[363,39594,666],{"className":39595},[665],[363,39597,39599],{"className":39598},[638],[363,39600,39602],{"className":39601,"style":2920},[642],[363,39603],{},[363,39605],{"className":39606},[501,789],[363,39608,39610,39613],{"className":39609},[407],[363,39611,37821],{"className":39612},[407,408],[363,39614,39616],{"className":39615},[629],[363,39617,39619],{"className":39618},[633],[363,39620,39622],{"className":39621},[638],[363,39623,39625],{"className":39624,"style":20400},[642],[363,39626,39627,39630],{"style":7213},[363,39628],{"className":39629,"style":651},[650],[363,39631,39633],{"className":39632},[655,656,657,658],[363,39634,2811],{"className":39635},[407,658],"、Bhattacharyya 约定的 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4.6 节的局部展开），因此它是所有 f-散度在局部的「公因子」——只是在 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的总体（无限样本）版本：把观测频率看作 ",[363,40297,40299,40312],{"className":40298,"translate":367},[366],[363,40300,40302],{"className":40301},[371],[373,40303,40304],{"xmlns":375},[377,40305,40306,40310],{},[380,40307,40308],{},[383,40309,554],{},[387,40311,554],{"encoding":389},[363,40313,40315],{"className":40314,"ariaHidden":394},[393],[363,40316,40318,40321],{"className":40317},[398],[363,40319],{"className":40320,"style":1011},[402],[363,40322,554],{"className":40323,"style":682},[407,408],"、期望频率看作 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就是「",[363,40439,40441,40454],{"className":40440,"translate":367},[366],[363,40442,40444],{"className":40443},[371],[373,40445,40446],{"xmlns":375},[377,40447,40448,40452],{},[380,40449,40450],{},[383,40451,554],{},[387,40453,554],{"encoding":389},[363,40455,40457],{"className":40456,"ariaHidden":394},[393],[363,40458,40460,40463],{"className":40459},[398],[363,40461],{"className":40462,"style":1011},[402],[363,40464,554],{"className":40465,"style":682},[407,408]," 是否来自 ",[363,40468,40470,40483],{"className":40469,"translate":367},[366],[363,40471,40473],{"className":40472},[371],[373,40474,40475],{"xmlns":375},[377,40476,40477,40481],{},[380,40478,40479],{},[383,40480,561],{},[387,40482,561],{"encoding":389},[363,40484,40486],{"className":40485,"ariaHidden":394},[393],[363,40487,40489,40492],{"className":40488},[398],[363,40490],{"className":40491,"style":913},[402],[363,40493,561],{"className":40494},[407,408],"」的标准化偏差，因此它是独立性检验、拟合优度检验与特征选择的自然尺度。它的分母 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距离还给出了带权核的另一种「局部」相似度。",[2659,40808,40810],{"id":40809},"_35-hellinger-平方","3.5 Hellinger 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凸，代入：",[363,41311,41313],{"className":41312,"translate":367},[528],[363,41314,41316,41406],{"className":41315,"translate":367},[366],[363,41317,41319],{"className":41318},[371],[373,41320,41321],{"xmlns":375,"display":537},[377,41322,41323,41403],{},[380,41324,41325,41331,41333,41335,41337,41339,41341,41343,41345,41347,41369,41371,41373,41375,41377,41379,41383,41385,41389,41395,41397,41399,41401],{},[543,41326,41327,41329],{},[383,41328,547],{},[383,41330,385],{},[458,41332,461],{"stretchy":460},[383,41334,554],{},[383,41336,558],{"mathvariant":557},[383,41338,561],{},[458,41340,468],{"stretchy":460},[458,41342,471],{},[458,41344,1635],{},[383,41346,1291],{},[7162,41348,41349,41367],{},[380,41350,41351,41353,41361,41363,41365],{},[458,41352,461],{"fence":394},[34711,41354,41355],{},[586,41356,41357,41359],{},[383,41358,11],{},[383,41360,1291],{},[458,41362,5161],{},[463,41364,465],{},[458,41366,468],{"fence":394},[463,41368,2811],{},[383,41370,592],{},[383,41372,1121],{},[458,41374,471],{},[458,41376,1635],{},[458,41378,461],{"stretchy":460},[34711,41380,41381],{},[383,41382,11],{},[458,41384,5161],{},[34711,41386,41387],{},[383,41388,1291],{},[7162,41390,41391,41393],{},[458,41392,468],{"stretchy":460},[463,41394,2811],{},[1645,41396,1647],{},[383,41398,592],{},[383,41400,1121],{},[383,41402,4723],{"mathvariant":557},[387,41404,41405],{"encoding":389},"D_f(P\\|Q)\n=\\int q\\left(\\sqrt{\\frac{p}{q}}-1\\right)^2dx\n=\\int (\\sqrt{p}-\\sqrt{q})^2\\,dx 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Bhattacharyya 系数 ",[363,41853,41855,41899],{"className":41854,"translate":367},[366],[363,41856,41858],{"className":41857},[371],[373,41859,41860],{"xmlns":375},[377,41861,41862,41896],{},[380,41863,41864,41866,41868,41870,41872,41874,41876,41878,41880,41882,41890,41892,41894],{},[383,41865,39663],{},[383,41867,39666],{},[458,41869,461],{"stretchy":460},[383,41871,554],{},[458,41873,898],{"separator":394},[383,41875,561],{},[458,41877,468],{"stretchy":460},[458,41879,471],{},[458,41881,1635],{},[34711,41883,41884],{},[380,41885,41886,41888],{},[383,41887,11],{},[383,41889,1291],{},[1645,41891,1647],{},[383,41893,592],{},[383,41895,1121],{},[387,41897,41898],{"encoding":389},"BC(P,Q)=\\int\\sqrt{pq}\\,dx",[363,41900,41902,41941],{"className":41901,"ariaHidden":394},[393],[363,41903,41905,41908,41911,41914,41917,41920,41923,41926,41929,41932,41935,41938],{"className":41904},[398],[363,41906],{"className":41907,"style":487},[402],[363,41909,39663],{"className":41910,"style":39759},[407,408],[363,41912,39666],{"className":41913,"style":11132},[407,408],[363,41915,461],{"className":41916},[494],[363,41918,554],{"className":41919,"style":682},[407,408],[363,41921,898],{"className":41922},[920],[363,41924],{"className":41925,"style":754},[505],[363,41927,561],{"className":41928},[407,408],[363,41930,468],{"className":41931},[501],[363,41933],{"className":41934,"style":506},[505],[363,41936,471],{"className":41937},[510],[363,41939],{"className":41940,"style":506},[505],[363,41942,41944,41947,41950,41953,42001,42004,42007],{"className":41943},[398],[363,41945],{"className":41946,"style":39380},[402],[363,41948,1635],{"className":41949,"style":5965},[1832,1833,5964],[363,41951],{"className":41952,"style":754},[505],[363,41954,41956],{"className":41955},[407,34739],[363,41957,41959,41993],{"className":41958},[633,634],[363,41960,41962,41990],{"className":41961},[638],[363,41963,41965,41978],{"className":41964,"style":39399},[642],[363,41966,41968,41971],{"className":41967,"style":12213},[6655],[363,41969],{"className":41970,"style":809},[650],[363,41972,41974],{"className":41973,"style":34759},[407],[363,41975,41977],{"className":41976,"style":1328},[407,408],"pq",[363,41979,41980,41983],{"style":39411},[363,41981],{"className":41982,"style":809},[650],[363,41984,41986],{"className":41985,"style":34816},[34815],[6671,41987,41988],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},[6679,41989],{"d":34823},[363,41991,666],{"className":41992},[665],[363,41994,41996],{"className":41995},[638],[363,41997,41999],{"className":41998,"style":39431},[642],[363,42000],{},[363,42002],{"className":42003,"style":754},[505],[363,42005,592],{"className":42006},[407,408],[363,42008,1121],{"className":42009},[407,408],"，则",[363,42012,42014],{"className":42013,"translate":367},[528],[363,42015,42017,42093],{"className":42016,"translate":367},[366],[363,42018,42020],{"className":42019},[371],[373,42021,42022],{"xmlns":375,"display":537},[377,42023,42024,42090],{},[380,42025,42026,42028,42030,42034,42036,42040,42046,42048,42050,42052,42054,42056,42058,42060,42062,42064,42072,42074,42076,42078,42080,42082,42084,42086,42088],{},[458,42027,1635],{},[458,42029,461],{"stretchy":460},[34711,42031,42032],{},[383,42033,11],{},[458,42035,5161],{},[34711,42037,42038],{},[383,42039,1291],{},[7162,42041,42042,42044],{},[458,42043,468],{"stretchy":460},[463,42045,2811],{},[458,42047,471],{},[458,42049,1635],{},[383,42051,11],{},[458,42053,2818],{},[458,42055,1635],{},[383,42057,1291],{},[458,42059,5161],{},[463,42061,2811],{},[458,42063,1635],{},[34711,42065,42066],{},[380,42067,42068,42070],{},[383,42069,11],{},[383,42071,1291],{},[458,42073,471],{},[463,42075,2811],{},[458,42077,461],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[463,42079,465],{},[458,42081,5161],{},[383,42083,39663],{},[383,42085,39666],{},[458,42087,468],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[383,42089,4723],{"mathvariant":557},[387,42091,42092],{"encoding":389},"\\int(\\sqrt p-\\sqrt q)^2\n=\\int p+\\int q-2\\int\\sqrt{pq}\n=2\\big(1-BC\\big).",[363,42094,42096,42161,42249,42273,42297,42371,42398],{"className":42095,"ariaHidden":394},[393],[363,42097,42099,42102,42105,42108,42152,42155,42158],{"className":42098},[398],[363,42100],{"className":42101,"style":6125},[402],[363,42103,1635],{"className":42104,"style":1835},[1832,1833,1834],[363,42106,461],{"className":42107},[494],[363,42109,42111],{"className":42110},[407,34739],[363,42112,42114,42144],{"className":42113},[633,634],[363,42115,42117,42141],{"className":42116},[638],[363,42118,42120,42129],{"className":42119,"style":7210},[642],[363,42121,42123,42126],{"className":42122,"style":12213},[6655],[363,42124],{"className":42125,"style":809},[650],[363,42127,11],{"className":42128,"style":34759},[407,408],[363,42130,42131,42134],{"style":41722},[363,42132],{"className":42133,"style":809},[650],[363,42135,42137],{"className":42136,"style":34816},[34815],[6671,42138,42139],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},[6679,42140],{"d":34823},[363,42142,666],{"className":42143},[665],[363,42145,42147],{"className":42146},[638],[363,42148,42150],{"className":42149,"style":41742},[642],[363,42151],{},[363,42153],{"className":42154,"style":2935},[505],[363,42156,5161],{"className":42157},[2939],[363,42159],{"className":42160,"style":2935},[505],[363,42162,42164,42167,42211,42240,42243,42246],{"className":42163},[398],[363,42165],{"className":42166,"style":41760},[402],[363,42168,42170],{"className":42169},[407,34739],[363,42171,42173,42203],{"className":42172},[633,634],[363,42174,42176,42200],{"className":42175},[638],[363,42177,42179,42188],{"className":42178,"style":7210},[642],[363,42180,42182,42185],{"className":42181,"style":12213},[6655],[363,42183],{"className":42184,"style":809},[650],[363,42186,1291],{"className":42187,"style":39471},[407,408],[363,42189,42190,42193],{"style":41722},[363,42191],{"className":42192,"style":809},[650],[363,42194,42196],{"className":42195,"style":34816},[34815],[6671,42197,42198],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},[6679,42199],{"d":34823},[363,42201,666],{"className":42202},[665],[363,42204,42206],{"className":42205},[638],[363,42207,42209],{"className":42208,"style":41742},[642],[363,42210],{},[363,42212,42214,42217],{"className":42213},[501],[363,42215,468],{"className":42216},[501],[363,42218,42220],{"className":42219},[629],[363,42221,42223],{"className":42222},[633],[363,42224,42226],{"className":42225},[638],[363,42227,42229],{"className":42228,"style":20107},[642],[363,42230,42231,42234],{"style":8625},[363,42232],{"className":42233,"style":651},[650],[363,42235,42237],{"className":42236},[655,656,657,658],[363,42238,2811],{"className":42239},[407,658],[363,42241],{"className":42242,"style":506},[505],[363,42244,471],{"className":42245},[510],[363,42247],{"className":42248,"style":506},[505],[363,42250,42252,42255,42258,42261,42264,42267,42270],{"className":42251},[398],[363,42253],{"className":42254,"style":6125},[402],[363,42256,1635],{"className":42257,"style":1835},[1832,1833,1834],[363,42259],{"className":42260,"style":754},[505],[363,42262,11],{"className":42263},[407,408],[363,42265],{"className":42266,"style":2935},[505],[363,42268,2818],{"className":42269},[2939],[363,42271],{"className":42272,"style":2935},[505],[363,42274,42276,42279,42282,42285,42288,42291,42294],{"className":42275},[398],[363,42277],{"className":42278,"style":6125},[402],[363,42280,1635],{"className":42281,"style":1835},[1832,1833,1834],[363,42283],{"className":42284,"style":754},[505],[363,42286,1291],{"className":42287,"style":1328},[407,408],[363,42289],{"className":42290,"style":2935},[505],[363,42292,5161],{"className":42293},[2939],[363,42295],{"className":42296,"style":2935},[505],[363,42298,42300,42303,42306,42309,42312,42315,42362,42365,42368],{"className":42299},[398],[363,42301],{"className":42302,"style":6125},[402],[363,42304,2811],{"className":42305},[407],[363,42307],{"className":42308,"style":754},[505],[363,42310,1635],{"className":42311,"style":1835},[1832,1833,1834],[363,42313],{"className":42314,"style":754},[505],[363,42316,42318],{"className":42317},[407,34739],[363,42319,42321,42354],{"className":42320},[633,634],[363,42322,42324,42351],{"className":42323},[638],[363,42325,42327,42339],{"className":42326,"style":7210},[642],[363,42328,42330,42333],{"className":42329,"style":12213},[6655],[363,42331],{"className":42332,"style":809},[650],[363,42334,42336],{"className":42335,"style":34759},[407],[363,42337,41977],{"className":42338,"style":1328},[407,408],[363,42340,42341,42344],{"style":41722},[363,42342],{"className":42343,"style":809},[650],[363,42345,42347],{"className":42346,"style":34816},[34815],[6671,42348,42349],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},[6679,42350],{"d":34823},[363,42352,666],{"className":42353},[665],[363,42355,42357],{"className":42356},[638],[363,4235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有两种约定：",[363,42483,42485],{"className":42484,"translate":367},[528],[363,42486,42488,42594],{"className":42487,"translate":367},[366],[363,42489,42491],{"className":42490},[371],[373,42492,42493],{"xmlns":375,"display":537},[377,42494,42495,42591],{},[1591,42496,42497,42532],{"rowspacing":1593,"columnalign":1594,"columnspacing":1595},[1597,42498,42499,42512],{},[1600,42500,42501],{},[1603,42502,42503],{"scriptlevel":474,"displaystyle":394},[7901,42504,42505,42507,42510],{},[383,42506,34256],{},[1645,42508,42509],{},"Bhattacharyya",[463,42511,2811],{},[1600,42513,42514],{},[1603,42515,42516],{"scriptlevel":474,"displaystyle":394},[380,42517,42518,42520,42522,42524,42526,42528,42530],{},[380,42519],{},[458,42521,471],{},[463,42523,465],{},[458,42525,5161],{},[383,42527,39663],{},[383,42529,39666],{},[458,42531,898],{"separator":394},[1597,42533,42534,42553],{},[1600,42535,42536],{},[1603,42537,42538],{"scriptlevel":474,"displaystyle":394},[7901,42539,42540,42542,42551],{},[383,42541,34256],{},[380,42543,42544,42547,42549],{},[1645,42545,42546],{},"this",[1645,42548,1647],{},[383,42550,385],{},[463,42552,2811],{},[1600,42554,42555],{},[1603,42556,42557],{"scriptlevel":474,"displaystyle":394},[380,42558,42559,42561,42563,42565,42567,42569,42571,42573,42575,42577,42579,42581,42589],{},[380,42560],{},[458,42562,471],{},[463,42564,2811],{},[458,42566,461],{"stretchy":460},[463,42568,465],{},[458,42570,5161],{},[383,42572,39663],{},[383,42574,39666],{},[458,42576,468],{"stretchy":460},[458,42578,471],{},[463,42580,2811],{},[7901,42582,42583,42585,42587],{},[383,42584,34256],{},[1645,42586,42509],{},[463,42588,2811],{},[383,42590,4723],{"mathvariant":557},[387,42592,42593],{"encoding":389},"\\begin{aligned}\nH^2_{\\text{Bhattacharyya}}&=1-BC,\\\\[2pt]\nH^2_{\\text{this}\\,f}&=2(1-BC)\n=2H^2_{\\text{Bhattacharyya}} 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有界（",[363,43533,43535,43551],{"className":43534,"translate":367},[366],[363,43536,43538],{"className":43537},[371],[373,43539,43540],{"xmlns":375},[377,43541,43542,43548],{},[380,43543,43544,43546],{},[458,43545,10300],{},[463,43547,2811],{},[387,43549,43550],{"encoding":389},"\\le 2",[363,43552,43554,43567],{"className":43553,"ariaHidden":394},[393],[363,43555,43557,43561,43564],{"className":43556},[398],[363,43558],{"className":43559,"style":43560},[402],"height:0.7719em;vertical-align:-0.136em;",[363,43562,10300],{"className":43563},[510],[363,43565],{"className":43566,"style":506},[505],[363,43568,43570,43573],{"className":43569},[398],[363,43571],{"className":43572,"style":520},[402],[363,43574,2811],{"className":43575},[407],"）。对称性可直接验算：",[363,43578,43580],{"className":43579,"translate":367},[528],[363,43581,43583,43759],{"className":43582,"translate":367},[366],[363,43584,43586],{"className":43585},[371],[373,43587,43588],{"xmlns":375,"display":537},[377,43589,43590,43756],{},[1591,43591,43592,43650,43694],{"rowspacing":1593,"columnalign":1594,"columnspacing":1595},[1597,43593,43594,43616],{},[1600,43595,43596],{},[1603,43597,43598],{"scriptlevel":474,"displaystyle":394},[380,43599,43600,43602,43604,43606,43608,43610,43612,43614],{},[383,43601,1116],{},[1645,43603,1647],{},[383,43605,385],{},[458,43607,461],{"stretchy":460},[463,43609,465],{},[383,43611,1288],{"mathvariant":557},[383,43613,1116],{},[458,43615,468],{"stretchy":460},[1600,43617,43618],{},[1603,43619,43620],{"scriptlevel":474,"displaystyle":394},[380,43621,43622,43624,43626,43628],{},[380,43623],{},[458,43625,471],{},[383,43627,1116],{},[7162,43629,43630,43648],{},[380,43631,43632,43634,43642,43644,43646],{},[458,43633,461],{"fence":394},[586,43635,43636,43638],{},[463,43637,465],{},[34711,43639,43640],{},[383,43641,1116],{},[458,43643,5161],{},[463,43645,465],{},[458,43647,468],{"fence":394},[463,43649,2811],{},[1597,43651,43652,43658],{},[1600,43653,43654],{},[1603,43655,43656],{"scriptlevel":474,"displaystyle":394},[380,43657],{},[1600,43659,43660],{},[1603,43661,43662],{"scriptlevel":474,"displaystyle":394},[380,43663,43664,43666,43668,43670,43672,43678,43680,43688,43690,43692],{},[380,43665],{},[458,43667,471],{},[383,43669,1116],{},[458,43671,461],{"fence":460,"stretchy":394,"minsize":17789,"maxsize":17789},[586,43673,43674,43676],{},[463,43675,465],{},[383,43677,1116],{},[458,43679,5161],{},[586,43681,43682,43684],{},[463,43683,2811],{},[34711,43685,43686],{},[383,43687,1116],{},[458,43689,2818],{},[463,43691,465],{},[458,43693,468],{"fence":460,"stretchy":394,"minsize":17789,"maxsize":17789},[1597,43695,43696,43702],{},[1600,43697,43698],{},[1603,43699,43700],{"scriptlevel":474,"displaystyle":394},[380,43701],{},[1600,43703,43704],{},[1603,43705,43706],{"scriptlevel":474,"displaystyle":394},[380,43707,43708,43710,43712,43714,43716,43718,43722,43724,43726,43728,43730,43734,43736,43738,43744,43746,43748,43750,43752,43754],{},[380,43709],{},[458,43711,471],{},[463,43713,465],{},[458,43715,5161],{},[463,43717,2811],{},[34711,43719,43720],{},[383,43721,1116],{},[458,43723,2818],{},[383,43725,1116],{},[458,43727,471],{},[458,43729,461],{"stretchy":460},[34711,43731,43732],{},[383,43733,1116],{},[458,43735,5161],{},[463,43737,465],{},[7162,43739,43740,43742],{},[458,43741,468],{"stretchy":460},[463,43743,2811],{},[458,43745,471],{},[383,43747,385],{},[458,43749,461],{"stretchy":460},[383,43751,1116],{},[458,43753,468],{"stretchy":460},[458,43755,898],{"separator":394},[387,43757,43758],{"encoding":389},"\\begin{aligned}\nt\\,f(1\u002Ft)\n&=t\\left(\\frac{1}{\\sqrt t}-1\\right)^2\\\\\n&=t\\Big(\\frac1t-\\frac{2}{\\sqrt t}+1\\Big)\\\\[2pt]\n&=1-2\\sqrt t+t=(\\sqrt t-1)^2=f(t),\n\\end{aligned}",[363,43760,43762],{"className":43761,"ariaHidden":394},[393],[363,43763,43765,43769],{"className":43764},[398],[363,43766],{"className":43767,"style":43768},[402],"height:6.7569em;vertical-align:-3.1284em;",[363,43770,43772],{"className":43771},[407],[363,43773,43775,43849],{"className":43774},[1591],[363,43776,43778],{"className":43777},[1708],[363,43779,43781,43840],{"className":43780},[633,634],[363,43782,43784,43837],{"className":43783},[638],[363,43785,43788,43819,43828],{"className":43786,"style":43787},[642],"height:3.6284em;",[363,43789,43791,43795],{"style":43790},"top:-5.6284em;",[363,43792],{"className":43793,"style":43794},[650],"height:3.654em;",[363,43796,43798,43801,43804,43807,43810,43813,43816],{"className":43797},[407],[363,43799,1116],{"className":43800},[407,408],[363,43802],{"className":43803,"style":754},[505],[363,43805,385],{"className":43806,"style":409},[407,408],[363,43808,461],{"className":43809},[494],[363,43811,8937],{"className":43812},[407],[363,43814,1116],{"className":43815},[407,408],[363,43817,468],{"className":43818},[501],[363,43820,43822,43825],{"style":43821},"top:-3.057em;",[363,43823],{"className":43824,"style":43794},[650],[363,43826],{"className":43827},[407],[363,43829,43831,43834],{"style":43830},"top:-0.8856em;",[363,43832],{"className":43833,"style":43794},[650],[363,43835],{"className":43836},[407],[363,43838,666],{"className":43839},[665],[363,43841,43843],{"className":43842},[638],[363,43844,43847],{"className":43845,"style":43846},[642],"height:3.1284em;",[363,43848],{},[363,43850,43852],{"className":43851},[1799],[363,43853,43855,44481],{"className":43854},[633,634],[363,43856,43858,44478],{"className":43857},[638],[363,43859,43861,44045,44266],{"className":43860,"style":43787},[642],[363,43862,43863,43866],{"style":43790},[363,43864],{"className":43865,"style":43794},[650],[363,43867,43869,43872,43875,43878,43881,43884,43887],{"className":43868},[407],[363,43870],{"className":43871},[407],[363,43873],{"className":43874,"style":506},[505],[363,43876,471],{"className":43877},[510],[363,43879],{"className":43880,"style":506},[505],[363,43882,1116],{"className":43883},[407,408],[363,43885],{"className":43886,"style":754},[505],[363,43888,43890,44022],{"className":43889},[758],[363,43891,43893,43899,44004,44007,44010,44013,44016],{"className":43892},[758],[363,43894,43896],{"className":43895,"style":763},[494,762],[363,43897,461],{"className":43898},[767,657],[363,43900,43902,43905,44001],{"className":43901},[407],[363,43903],{"className":43904},[494,789],[363,43906,43908],{"className":43907},[586],[363,43909,43911,43992],{"className":43910},[633,634],[363,43912,43914,43989],{"className":43913},[638],[363,43915,43917,43970,43978],{"className":43916,"style":23261},[642],[363,43918,43920,43923],{"style":43919},"top:-2.2175em;",[363,43921],{"className":43922,"style":809},[650],[363,43924,43926],{"className":43925},[407],[363,43927,43929],{"className":43928},[407,34739],[363,43930,43932,43962],{"className":43931},[633,634],[363,43933,43935,43959],{"className":43934},[638],[363,43936,43938,43947],{"className":43937,"style":43055},[642],[363,43939,43941,43944],{"className":43940,"style":12213},[6655],[363,43942],{"className":43943,"style":809},[650],[363,43945,1116],{"className":43946,"style":34759},[407,408],[363,43948,43949,43952],{"style":43067},[363,43950],{"className":43951,"style":809},[650],[363,43953,43955],{"className":43954,"style":34816},[34815],[6671,43956,43957],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},[6679,43958],{"d":34823},[363,43960,666],{"className":43961},[665],[363,43963,43965],{"className":43964},[638],[363,43966,43968],{"className":43967,"style":43087},[642],[363,43969],{},[363,43971,43972,43975],{"style":821},[363,43973],{"className":43974,"style":809},[650],[363,43976],{"className":43977,"style":829},[828],[363,43979,43980,43983],{"style":832},[363,43981],{"className":43982,"style":809},[650],[363,43984,43986],{"className":43985},[407],[363,43987,465],{"className":43988},[407],[363,43990,666],{"className":43991},[665],[363,43993,43995],{"className":43994},[638],[363,43996,43999],{"className":43997,"style":43998},[642],"height:0.93em;",[363,44000],{},[363,44002],{"className":44003},[501,789],[363,44005],{"className":44006,"style":2935},[505],[363,44008,5161],{"className":44009},[2939],[363,44011],{"className":44012,"style":2935},[505],[363,44014,465],{"className":44015},[407],[363,44017,44019],{"className":44018,"style":763},[501,762],[363,44020,468],{"className":44021},[767,657],[363,44023,44025],{"className":44024},[629],[363,44026,44028],{"className":44027},[633],[363,44029,44031],{"className":44030},[638],[363,44032,44034],{"className":44033,"style":38033},[642],[363,44035,44036,44039],{"style":38036},[363,44037],{"className":44038,"style":651},[650],[363,44040,44042],{"className":44041},[655,656,657,658],[363,44043,2811],{"className":44044},[407,658],[363,44046,4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assName":44329,"style":34816},[34815],[6671,44331,44332],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},[6679,44333],{"d":34823},[363,44335,666],{"className":44336},[665],[363,44338,44340],{"className":44339},[638],[363,44341,44343],{"className":44342,"style":40944},[642],[363,44344],{},[363,44346],{"className":44347,"style":2935},[505],[363,44349,2818],{"className":44350},[2939],[363,44352],{"className":44353,"style":2935},[505],[363,44355,1116],{"className":44356},[407,408],[363,44358],{"className":44359,"style":506},[505],[363,44361,471],{"className":44362},[510],[363,44364],{"className":44365,"style":506},[505],[363,44367,461],{"className":44368},[494],[363,44370,44372],{"className":44371},[407,34739],[363,44373,44375,44405],{"className":44374},[633,634],[363,44376,44378,44402],{"className":44377},[638],[363,44379,44381,44390],{"className":44380,"style":40909},[642],[363,44382,44384,44387],{"className":44383,"style":12213},[6655],[363,44385],{"className":44386,"style":809},[650],[363,44388,1116],{"className":44389,"style":34759},[407,408],[363,44391,44392,44395],{"style":40924},[363,44393],{"className":44394,"style":809},[650],[363,44396,44398],{"className":44397,"style":34816},[34815],[6671,44399,44400],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},[6679,44401],{"d":34823},[363,44403,666],{"className":44404},[665],[363,44406,44408],{"className":44407},[638],[363,44409,44411],{"className":44410,"style":40944},[642],[363,44412],{},[363,44414],{"className":44415,"style":2935},[505],[363,44417,5161],{"className":44418},[2939],[363,44420],{"className":44421,"style":2935},[505],[363,44423,465],{"className":44424},[407],[363,44426,44428,44431],{"className":44427},[501],[363,44429,468],{"className":44430},[501],[363,44432,44434],{"className":44433},[629],[363,44435,44437],{"className":44436},[633],[363,44438,44440],{"className":44439},[638],[363,44441,44443],{"className":44442,"style":20107},[642],[363,44444,44445,44448],{"style":8625},[363,44446],{"className":44447,"style":651},[650],[363,44449,44451],{"className":44450},[655,656,657,658],[363,44452,2811],{"className":44453},[407,658],[363,44455],{"className":44456,"style":506},[505],[363,44458,471],{"className":44459},[510],[363,44461],{"className":44462,"style":506},[505],[363,44464,385],{"className":44465,"style":409},[407,408],[363,44467,461],{"className":44468},[494],[363,44470,1116],{"className":44471},[407,408],[363,44473,468],{"className":44474},[501],[363,44476,898],{"className":44477},[920],[363,44479,666],{"className":44480},[665],[363,44482,44484],{"className":44483},[638],[363,44485,44487],{"className":44486,"style":43846},[642],[363,44488],{},[11,44490,21652,44491,44620,44621,44679],{},[363,44492,44494,44526],{"className":44493,"translate":367},[366],[363,44495,44497],{"className":44496},[371],[373,44498,44499],{"xmlns":375},[377,44500,44501,44523],{},[380,44502,44503,44519,44521],{},[7901,44504,44505,44507,44517],{},[383,44506,385],{},[380,44508,44509,44511,44513,44515],{},[383,44510,7909],{},[383,44512,7912],{},[383,44514,7915],{},[383,44516,7918],{},[458,44518,7921],{"lspace":1595,"rspace":1595},[458,44520,471],{},[383,44522,385],{},[387,44524,44525],{"encoding":389},"f^{*}_{skew}=f",[363,44527,44529,44611],{"className":44528,"ariaHidden":394},[393],[363,44530,44532,44536,44602,44605,44608],{"className":44531},[398],[363,44533],{"className":44534,"style":44535},[402],"height:0.9775em;vertical-align:-0.2831em;",[363,44537,44539,44542],{"className":44538},[407],[363,44540,385],{"className":44541,"style":409},[407,408],[363,44543,44545],{"className":44544},[629],[363,44546,44548,44594],{"className":44547},[633,634],[363,44549,44551,44591],{"className":44550},[638],[363,44552,44554,44577],{"className":44553,"style":8847},[642],[363,44555,44556,44559],{"style":8850},[363,44557],{"className":44558,"style":651},[650],[363,44560,44562],{"className":44561},[655,656,657,658],[363,44563,44565,44568,44571,44574],{"className":44564},[407,658],[363,44566,7909],{"className":44567},[407,408,658],[363,44569,7912],{"className":44570,"style":8615},[407,408,658],[363,44572,7915],{"className":44573},[407,408,658],[363,44575,7918],{"className":44576,"style":8622},[407,408,658],[363,44578,44579,44582],{"style":7213},[363,44580],{"className":44581,"style":651},[650],[363,44583,44585],{"className":44584},[655,656,657,658],[363,44586,44588],{"className":44587},[407,658],[363,44589,7921],{"className":44590},[407,658],[363,44592,666],{"className":44593},[665],[363,44595,44597],{"className":44596},[638],[363,44598,44600],{"className":44599,"style":8895},[642],[363,44601],{},[363,44603],{"className":44604,"style":506},[505],[363,44606,471],{"className":44607},[510],[363,44609],{"className":44610,"style":506},[505],[363,44612,44614,44617],{"className":44613},[398],[363,44615],{"className":44616,"style":403},[402],[363,44618,385],{"className":44619,"style":409},[407,408],"，所以 Hellinger",[363,44622,44624,44642],{"className":44623,"translate":367},[366],[363,44625,44627],{"className":44626},[371],[373,44628,44629],{"xmlns":375},[377,44630,44631,44639],{},[380,44632,44633],{},[7162,44634,44635,44637],{},[380,44636],{},[463,44638,2811],{},[387,44640,44641],{"encoding":389},"^2",[363,44643,44645],{"className":44644,"ariaHidden":394},[393],[363,44646,44648,44651],{"className":44647},[398],[363,44649],{"className":44650,"style":20400},[402],[363,44652,44654,44656],{"className":44653},[407],[363,44655],{},[363,44657,44659],{"className":44658},[629],[363,44660,44662],{"className":44661},[633],[363,44663,44665],{"className":44664},[638],[363,44666,44668],{"className":44667,"style":20400},[642],[363,44669,44670,44673],{"style":7213},[363,44671],{"className":44672,"style":651},[650],[363,44674,44676],{"className":44675},[655,656,657,658],[363,44677,2811],{"className":44678},[407,658]," 也满足对称判别准则（4.3 节）。",[11,44681,44682,44684,44685,44799,44800,44803,44804,44860,44861,44917,44918,45076,45077,45105],{},[1344,44683,24565],{}," Hellinger 与 Bhattacharyya 系数 ",[363,44686,44688,44716],{"className":44687,"translate":367},[366],[363,44689,44691],{"className":44690},[371],[373,44692,44693],{"xmlns":375},[377,44694,44695,44713],{},[380,44696,44697,44699,44701,44703,44705],{},[383,44698,39663],{},[383,44700,39666],{},[458,44702,471],{},[458,44704,1635],{},[34711,44706,44707],{},[380,44708,44709,44711],{},[383,44710,11],{},[383,44712,1291],{},[387,44714,44715],{"encoding":389},"BC=\\int\\sqrt{pq}",[363,44717,44719,44740],{"className":44718,"ariaHidden":394},[393],[363,44720,44722,44725,44728,44731,44734,44737],{"className":44721},[398],[363,44723],{"className":44724,"style":1011},[402],[363,44726,39663],{"className":44727,"style":39759},[407,408],[363,44729,39666],{"className":44730,"style":11132},[407,408],[363,44732],{"className":44733,"style":506},[505],[363,44735,471],{"className":44736},[510],[363,44738],{"className":44739,"style":506},[505],[363,44741,44743,44746,44749,44752],{"className":44742},[398],[363,44744],{"className":44745,"style":39380},[402],[363,44747,1635],{"className":44748,"style":5965},[1832,1833,5964],[363,44750],{"className":44751,"style":754},[505],[363,44753,44755],{"className":44754},[407,34739],[363,44756,44758,44791],{"className":44757},[633,634],[363,44759,44761,44788],{"className":44760},[638],[363,44762,44764,44776],{"className":44763,"style":39399},[642],[363,44765,44767,44770],{"className":44766,"style":12213},[6655],[363,44768],{"className":44769,"style":809},[650],[363,44771,44773],{"className":44772,"style":34759},[407],[363,44774,41977],{"className":44775,"style":1328},[407,408],[363,44777,44778,44781],{"style":39411},[363,44779],{"className":44780,"style":809},[650],[363,44782,44784],{"className":44783,"style":34816},[34815],[6671,44785,44786],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},[6679,44787],{"d":34823},[363,44789,666],{"className":44790},[665],[363,44792,44794],{"className":44793},[638],[363,44795,44797],{"className":44796,"style":39431},[642],[363,44798],{}," 把「两个分布的",[1344,44801,44802],{},"重叠程度","」量化：",[363,44805,44807,44827],{"className":44806,"translate":367},[366],[363,44808,44810],{"className":44809},[371],[373,44811,44812],{"xmlns":375},[377,44813,44814,44824],{},[380,44815,44816,44818,44820,44822],{},[383,44817,39663],{},[383,44819,39666],{},[458,44821,471],{},[463,44823,465],{},[387,44825,44826],{"encoding":389},"BC=1",[363,44828,44830,44851],{"className":44829,"ariaHidden":394},[393],[363,44831,44833,44836,44839,44842,44845,44848],{"className":44832},[398],[363,44834],{"className":44835,"style":1011},[402],[363,44837,39663],{"className":44838,"style":39759},[407,408],[363,44840,39666],{"className":44841,"style":11132},[407,408],[363,44843],{"className":44844,"style":506},[505],[363,44846,471],{"className":44847},[510],[363,44849],{"className":44850,"style":506},[505],[363,44852,44854,44857],{"className":44853},[398],[363,44855],{"className":44856,"style":520},[402],[363,44858,465],{"className":44859},[407]," 表示完全重合，",[363,44862,44864,44884],{"className":44863,"translate":367},[366],[363,44865,44867],{"className":44866},[371],[373,44868,44869],{"xmlns":375},[377,44870,44871,44881],{},[380,44872,44873,44875,44877,44879],{},[383,44874,39663],{},[383,44876,39666],{},[458,44878,471],{},[463,44880,474],{},[387,44882,44883],{"encoding":389},"BC=0",[363,44885,44887,44908],{"className":44886,"ariaHidden":394},[393],[363,44888,44890,44893,44896,44899,44902,44905],{"className":44889},[398],[363,44891],{"className":44892,"style":1011},[402],[363,44894,39663],{"className":44895,"style":39759},[407,408],[363,44897,39666],{"className":44898,"style":11132},[407,408],[363,44900],{"className":44901,"style":506},[505],[363,44903,471],{"className":44904},[510],[363,44906],{"className":44907,"style":506},[505],[363,44909,44911,44914],{"className":44910},[398],[363,44912],{"className":44913,"style":520},[402],[363,44915,474],{"className":44916},[407]," 表示支撑分离。由它可造出真正的距离 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",[363,45078,45080,45093],{"className":45079,"translate":367},[366],[363,45081,45083],{"className":45082},[371],[373,45084,45085],{"xmlns":375},[377,45086,45087,45091],{},[380,45088,45089],{},[463,45090,2811],{},[387,45092,2811],{"encoding":389},[363,45094,45096],{"className":45095,"ariaHidden":394},[393],[363,45097,45099,45102],{"className":45098},[398],[363,45100],{"className":45101,"style":520},[402],[363,45103,2811],{"className":45104},[407],"，不会像 KL 那样发散——这让它在数值与统计上格外稳健。Hellinger 距离与 Fisher 信息度量在无穷小尺度上相容（4.6 节），并在非参数估计的 Le Cam 三点法、量子信息中的保真度（fidelity）、以及 Hellinger–Fisher–Rao 几何里扮演中心角色。",[2659,45107,45109],{"id":45108},"_36-α-散度","3.6 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是 Hellinger 型散度。两个端点极限要小心：",[363,46893,46895,46913],{"className":46894,"translate":367},[366],[363,46896,46898],{"className":46897},[371],[373,46899,46900],{"xmlns":375},[377,46901,46902,46910],{},[380,46903,46904],{},[543,46905,46906,46908],{},[383,46907,385],{},[383,46909,45126],{},[387,46911,46912],{"encoding":389},"f_\\alpha",[363,46914,46916],{"className":46915,"ariaHidden":394},[393],[363,46917,46919,46922],{"className":46918},[398],[363,46920],{"className":46921,"style":403},[402],[363,46923,46925,46928],{"className":46924},[407],[363,46926,385],{"className":46927,"style":409},[407,408],[363,46929,46931],{"className":46930},[629],[363,46932,46934,46954],{"className":46933},[633,634],[363,46935,46937,46951],{"className":46936},[638],[363,46938,46940],{"className":46939,"style":45301},[642],[363,46941,46942,46945],{"style":27881},[363,46943],{"className":46944,"style":651},[650],[363,46946,46948],{"className":46947},[655,656,657,658],[363,46949,45126],{"className":46950,"style":45156},[407,408,658],[363,46952,666],{"className":46953},[665],[363,46955,46957],{"className":46956},[638],[363,46958,46960],{"className":46959,"style":8316},[642],[363,46961],{},[363,46963,46965,46986],{"className":46964,"translate":367},[366],[363,46966,46968],{"className":46967},[371],[373,46969,46970],{"xmlns":375},[377,46971,46972,46983],{},[380,46973,46974,46976,46978,46981],{},[383,46975,45126],{},[458,46977,1378],{},[458,46979,46980],{},"±",[463,46982,465],{},[387,46984,46985],{"encoding":389},"\\alpha\\to\\pm1",[363,46987,46989,47007],{"className":46988,"ariaHidden":394},[393],[363,46990,46992,46995,46998,47001,47004],{"className":46991},[398],[363,46993],{"className":46994,"style":2693},[402],[363,46996,45126],{"className":46997,"style":45156},[407,408],[363,46999],{"className":47000,"style":506},[505],[363,47002,1378],{"className":47003},[510],[363,47005],{"className":47006,"style":506},[505],[363,47008,47010,47013,47016],{"className":47009},[398],[363,47011],{"className":47012,"style":12928},[402],[363,47014,46980],{"className":47015},[407],[363,47017,465],{"className":47018},[407]," 时逐点发散，但",[1344,47021,47022],{},"散度值","收敛到 KL：",[363,47025,47027],{"className":47026,"translate":367},[528],[363,47028,47030,47159],{"className":47029,"translate":367},[366],[363,47031,47033],{"className":47032},[371],[373,47034,47035],{"xmlns":375,"display":537},[377,47036,47037,47156],{},[380,47038,47039,47056,47062,47064,47066,47068,47070,47072,47074,47084,47086,47088,47090,47092,47094,47096,47098,47116,47122,47124,47126,47128,47130,47132,47134,47144,47146,47148,47150,47152,47154],{},[6364,47040,47041,47048],{},[380,47042,47043,47046],{},[383,47044,47045],{},"lim",[458,47047,4451],{},[380,47049,47050,47052,47054],{},[383,47051,45126],{},[458,47053,1378],{},[463,47055,465],{},[543,47057,47058,47060],{},[383,47059,547],{},[383,47061,45126],{},[458,47063,461],{"stretchy":460},[383,47065,554],{},[383,47067,558],{"mathvariant":557},[383,47069,561],{},[458,47071,468],{"stretchy":460},[458,47073,471],{},[543,47075,47076,47078],{},[383,47077,547],{},[380,47079,47080,47082],{},[383,47081,11044],{},[383,47083,22265],{},[458,47085,461],{"stretchy":460},[383,47087,554],{},[383,47089,558],{"mathvariant":557},[383,47091,561],{},[458,47093,468],{"stretchy":460},[458,47095,898],{"separator":394},[505,47097],{"width":3610},[6364,47099,47100,47106],{},[380,47101,47102,47104],{},[383,47103,47045],{},[458,47105,4451],{},[380,47107,47108,47110,47112,47114],{},[383,47109,45126],{},[458,47111,1378],{},[458,47113,5161],{},[463,47115,465],{},[543,47117,47118,47120],{},[383,47119,547],{},[383,47121,45126],{},[458,47123,461],{"stretchy":460},[383,47125,554],{},[383,47127,558],{"mathvariant":557},[383,47129,561],{},[458,47131,468],{"stretchy":460},[458,47133,471],{},[543,47135,47136,47138],{},[383,47137,547],{},[380,47139,47140,47142],{},[383,47141,11044],{},[383,47143,22265],{},[458,47145,461],{"stretchy":460},[383,47147,561],{},[383,47149,558],{"mathvariant":557},[383,47151,554],{},[458,47153,468],{"stretchy":460},[383,47155,4723],{"mathvariant":557},[387,47157,47158],{"encoding":389},"\\lim_{\\alpha\\to1}D_\\alpha(P\\|Q)=D_{KL}(P\\|Q),\n\\qquad\n\\lim_{\\alpha\\to-1}D_\\alpha(P\\|Q)=D_{KL}(Q\\|P).",[363,47160,47162,47291,47492],{"className":47161,"ariaHidden":394},[393],[363,47163,47165,47169,47224,47227,47267,47270,47273,47276,47279,47282,47285,47288],{"className":47164},[398],[363,47166],{"className":47167,"style":47168},[402],"height:1.4671em;vertical-align:-0.7171em;",[363,47170,47172],{"className":47171},[1832,14066],[363,47173,47175,47215],{"className":47174},[633,634],[363,47176,47178,47212],{"className":47177},[638],[363,47179,47181,47202],{"className":47180,"style":12947},[642],[363,47182,47184,47187],{"style":47183},"top:-2.3829em;margin-left:0em;",[363,47185],{"className":47186,"style":809},[650],[363,47188,47190],{"className":47189},[655,656,657,658],[363,47191,47193,47196,47199],{"className":47192},[407,658],[363,47194,45126],{"className":47195,"style":45156},[407,408,658],[363,47197,1378],{"className":47198},[510,658],[363,47200,465],{"className":47201},[407,658],[363,47203,47204,47207],{"style":12213},[363,47205],{"className":47206,"style":809},[650],[363,47208,47209],{},[363,47210,47045],{"className":47211},[1832],[363,47213,666],{"className":47214},[665],[363,47216,47218],{"className":47217},[638],[363,47219,47222],{"className":47220,"style":47221},[642],"height:0.7171em;",[363,47223],{},[363,47225],{"className":47226,"style":754},[505],[363,47228,47230,47233],{"className":47229},[407],[363,47231,547],{"className":47232,"style":625},[407,408],[363,47234,47236],{"className":47235},[629],[363,47237,47239,47259],{"className":47238},[633,634],[363,47240,47242,47256],{"className":47241},[638],[363,47243,47245],{"className":47244,"style":45301},[642],[363,47246,47247,47250],{"style":646},[363,47248],{"className":47249,"style":651},[650],[363,47251,47253],{"className":47252},[655,656,657,658],[363,47254,45126],{"className":47255,"style":45156},[407,408,658],[363,47257,666],{"className":47258},[665],[363,47260,47262],{"className":47261},[638],[363,47263,47265],{"className":47264,"style":8316},[642],[363,47266],{},[363,47268,461],{"className":47269},[494],[363,47271,554],{"className":47272,"style":682},[407,408],[363,47274,558],{"className":47275},[407],[363,47277,561],{"className":47278},[407,408],[363,47280,468],{"className":47281},[501],[363,47283],{"className":47284,"style":506},[505],[363,47286,471],{"className":47287},[510],[363,47289],{"className":47290,"style":506},[505],[363,47292,47294,47298,47344,47347,47350,47353,47356,47359,47362,47365,47368,47425,47428,47468,47471,47474,47477,47480,47483,47486,47489],{"className":47293},[398],[363,47295],{"className":47296,"style":47297},[402],"height:1.5254em;vertical-align:-0.7754em;",[363,47299,47301,47304],{"className":47300},[407],[363,47302,547],{"className":47303,"style":625},[407,408],[363,47305,47307],{"className":47306},[629],[363,47308,47310,47336],{"className":47309},[633,634],[363,47311,47313,47333],{"className":47312},[638],[363,47314,47316],{"className":47315,"style":727},[642],[363,47317,47318,47321]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0v40h399900v-40z",[363,49277,49279,49282],{"style":49278},"top:-2.0919em;",[363,49280],{"className":49281,"style":651},[650],[363,49283,49285],{"className":49284},[655,656,657,658,49254],[363,49286,49288,49291,49294],{"className":49287},[407,658],[363,49289,16062],{"className":49290,"style":16111},[407,408,658],[363,49292,1378],{"className":49293},[510,658],[363,49295,474],{"className":49296},[407,658],[363,49298,666],{"className":49299},[665],[363,49301,49303],{"className":49302},[638],[363,49304,49307],{"className":49305,"style":49306},[642],"height:0.6081em;",[363,49308],{},[363,49310],{"className":49311,"style":506},[505],[363,49313,49315,49318],{"className":49314},[407],[363,49316,547],{"className":49317,"style":625},[407,408],[363,49319,49321],{"className":49320},[629],[363,49322,49324,49350],{"className":49323},[633,634],[363,49325,49327,49347],{"className":49326},[638],[363,49328,49330],{"className":49329,"style":727},[642],[363,49331,49332,49335],{"style":646},[363,49333],{"className":49334,"style":651},[650],[363,49336,49338],{"className":49337},[655,656,657,658],[363,49339,49341,49344],{"className":49340},[407,658],[363,49342,11044],{"className":49343,"style":11132},[407,408,658],[363,49345,22265],{"className":49346},[407,408,658],[363,49348,666],{"className":49349},[665],[363,49351,49353],{"className":49352},[638],[363,49354,49356],{"className":49355,"style":8316},[642],[363,49357],{},[363,49359,461],{"className":49360},[494],[363,49362,554],{"className":49363,"style":682},[407,408],[363,49365,558],{"className":49366},[407],[363,49368,561],{"className":49369},[407,408],[363,49371,468],{"className":49372},[501],[363,49374,4723],{"className":49375},[407],[363,49377,666],{"className":49378},[665],[363,49380,49382],{"className":49381},[638],[363,49383,49385],{"className":49384,"style":48844},[642],[363,49386],{},[11,49388,49389,49445,49446,49475],{},[363,49390,49392,49412],{"className":49391,"translate":367},[366],[363,49393,49395],{"className":49394},[371],[373,49396,49397],{"xmlns":375},[377,49398,49399,49409],{},[380,49400,49401,49403,49405,49407],{},[383,49402,45126],{},[458,49404,1378],{},[458,49406,5161],{},[463,49408,465],{},[387,49410,49411],{"encoding":389},"\\alpha\\to-1",[363,49413,49415,49433],{"className":49414,"ariaHidden":394},[393],[363,49416,49418,49421,49424,49427,49430],{"className":49417},[398],[363,49419],{"className":49420,"style":2693},[402],[363,49422,45126],{"className":49423,"style":45156},[407,408],[363,49425],{"className":49426,"style":506},[505],[363,49428,1378],{"className":49429},[510],[363,49431],{"className":49432,"style":506},[505],[363,49434,49436,49439,49442],{"className":49435},[398],[363,49437],{"className":49438,"style":12928},[402],[363,49440,5161],{"className":49441},[407],[363,49443,465],{"className":49444},[407]," 对称地给出反向 KL。所以 α-散度把 KL（两个方向）与 Hellinger 连成一条曲线，",[363,49447,49449,49463],{"className":49448,"translate":367},[366],[363,49450,49452],{"className":49451},[371],[373,49453,49454],{"xmlns":375},[377,49455,49456,49460],{},[380,49457,49458],{},[383,49459,45126],{},[387,49461,49462],{"encoding":389},"\\alpha",[363,49464,49466],{"className":49465,"ariaHidden":394},[393],[363,49467,49469,49472],{"className":49468},[398],[363,49470],{"className":49471,"style":2693},[402],[363,49473,45126],{"className":49474,"style":45156},[407,408]," 是「在对称与不对称之间连续插值」的旋钮。",[11,49477,49478,49480,49481,49509,49510,49544,49545,49573,49574,49624,49625,49628,49629,49657,49658,49686,49687,49715,49716,49744],{},[1344,49479,24565],{}," α-散度的价值在于它把「方向」与「强度」一并参数化：",[363,49482,49484,49497],{"className":49483,"translate":367},[366],[363,49485,49487],{"className":49486},[371],[373,49488,49489],{"xmlns":375},[377,49490,49491,49495],{},[380,49492,49493],{},[383,49494,45126],{},[387,49496,49462],{"encoding":389},[363,49498,49500],{"className":49499,"ariaHidden":394},[393],[363,49501,49503,49506],{"className":49502},[398],[363,49504],{"className":49505,"style":2693},[402],[363,49507,45126],{"className":49508,"style":45156},[407,408]," 从 ",[363,49511,49513,49529],{"className":49512,"translate":367},[366],[363,49514,49516],{"className":49515},[371],[373,49517,49518],{"xmlns":375},[377,49519,49520,49526],{},[380,49521,49522,49524],{},[458,49523,5161],{},[463,49525,465],{},[387,49527,49528],{"encoding":389},"-1",[363,49530,49532],{"className":49531,"ariaHidden":394},[393],[363,49533,49535,49538,49541],{"className":49534},[398],[363,49536],{"className":49537,"style":12928},[402],[363,49539,5161],{"className":49540},[407],[363,49542,465],{"className":49543},[407]," 扫到 ",[363,49546,49548,49561],{"className":49547,"translate":367},[366],[363,49549,49551],{"className":49550},[371],[373,49552,49553],{"xmlns":375},[377,49554,49555,49559],{},[380,49556,49557],{},[463,49558,465],{},[387,49560,465],{"encoding":389},[363,49562,49564],{"className":49563,"ariaHidden":394},[393],[363,49565,49567,49570],{"className":49566},[398],[363,49568],{"className":49569,"style":520},[402],[363,49571,465],{"className":49572},[407]," 时，散度由反向 KL（mode-seeking）连续过渡到正向 KL（mass-covering），中间在 ",[363,49575,49577,49594],{"className":49576,"translate":367},[366],[363,49578,49580],{"className":49579},[371],[373,49581,49582],{"xmlns":375},[377,49583,49584,49592],{},[380,49585,49586,49588,49590],{},[383,49587,45126],{},[458,49589,471],{},[463,49591,474],{},[387,49593,46860],{"encoding":389},[363,49595,49597,49615],{"className":49596,"ariaHidden":394},[393],[363,49598,49600,49603,49606,49609,49612],{"className":49599},[398],[363,49601],{"className":49602,"style":2693},[402],[363,49604,45126],{"className":49605,"style":45156},[407,408],[363,49607],{"className":49608,"style":506},[505],[363,49610,471],{"className":49611},[510],[363,49613],{"className":49614,"style":506},[505],[363,49616,49618,49621],{"className":49617},[398],[363,49619],{"className":49620,"style":520},[402],[363,49622,474],{"className":49623},[407]," 处恰好经过 Hellinger 这个对称点。在 Amari 的",[1344,49626,49627],{},"信息几何","里，α-族是唯一同时拥有 ",[363,49630,49632,49645],{"className":49631,"translate":367},[366],[363,49633,49635],{"className":49634},[371],[373,49636,49637],{"xmlns":375},[377,49638,49639,49643],{},[380,49640,49641],{},[383,49642,45126],{},[387,49644,49462],{"encoding":389},[363,49646,49648],{"className":49647,"ariaHidden":394},[393],[363,49649,49651,49654],{"className":49650},[398],[363,49652],{"className":49653,"style":2693},[402],[363,49655,45126],{"className":49656,"style":45156},[407,408],"-对偶平坦结构的单参数散度族：它诱导的度量是 Fisher 度量，诱导的仿射联络是 ",[363,49659,49661,49674],{"className":49660,"translate":367},[366],[363,49662,49664],{"className":49663},[371],[373,49665,49666],{"xmlns":375},[377,49667,49668,49672],{},[380,49669,49670],{},[383,49671,45126],{},[387,49673,49462],{"encoding":389},[363,49675,49677],{"className":49676,"ariaHidden":394},[393],[363,49678,49680,49683],{"className":49679},[398],[363,49681],{"className":49682,"style":2693},[402],[363,49684,45126],{"className":49685,"style":45156},[407,408],"-联络，这正是它被称为「散度范本」的原因。在机器学习中，",[363,49688,49690,49703],{"className":49689,"translate":367},[366],[363,49691,49693],{"className":49692},[371],[373,49694,49695],{"xmlns":375},[377,49696,49697,49701],{},[380,49698,49699],{},[383,49700,45126],{},[387,49702,49462],{"encoding":389},[363,49704,49706],{"className":49705,"ariaHidden":394},[393],[363,49707,49709,49712],{"className":49708},[398],[363,49710],{"className":49711,"style":2693},[402],[363,49713,45126],{"className":49714,"style":45156},[407,408],"-GAN 与 ",[363,49717,49719,49732],{"className":49718,"translate":367},[366],[363,49720,49722],{"className":49721},[371],[373,49723,49724],{"xmlns":375},[377,49725,49726,49730],{},[380,49727,49728],{},[383,49729,45126],{},[387,49731,49462],{"encoding":389},[363,49733,49735],{"className":49734,"ariaHidden":394},[393],[363,49736,49738,49741],{"className":49737},[398],[363,49739],{"className":49740,"style":2693},[402],[363,49742,45126],{"className":49743,"style":45156},[407,408],"-VI 用一个参数调节「对异常值的鲁棒性」与「模式覆盖」之间的取舍；在信号处理中它用于独立成分分析（ICA）与鲁棒估计。",[2659,49746,49748],{"id":49747},"_37-与-rényi-散度的关系","3.7 与 Rényi 散度的关系",[11,49750,49751],{},"Rényi 散度定义为",[363,49753,49755],{"className":49754,"translate":367},[528],[363,49756,49758,49886],{"className":49757,"translate":367},[366],[363,49759,49761],{"className":49760},[371],[373,49762,49763],{"xmlns":375,"display":537},[377,49764,49765,49883],{},[1591,49766,49767,49847],{"rowspacing":1593,"columnalign":1594,"columnspacing":1595},[1597,49768,49769,49793],{},[1600,49770,49771],{},[1603,49772,49773],{"scriptlevel":474,"displaystyle":394},[380,49774,49775,49783,49785,49787,49789,49791],{},[7901,49776,49777,49779,49781],{},[383,49778,547],{},[383,49780,45126],{},[383,49782,1381],{},[458,49784,461],{"stretchy":460},[383,49786,554],{},[383,49788,558],{"mathvariant":557},[383,49790,561],{},[458,49792,468],{"stretchy":460},[1600,49794,49795],{},[1603,49796,49797],{"scriptlevel":474,"displaystyle":394},[380,49798,49799,49801,49803,49815,49817,49819,49821,49827,49839,49841,49843,49845],{},[380,49800],{},[458,49802,471],{},[586,49804,49805,49807],{},[463,49806,465],{},[380,49808,49809,49811,49813],{},[383,49810,45126],{},[458,49812,5161],{},[463,49814,465],{},[383,49816,4448],{},[458,49818,4451],{},[458,49820,1635],{},[7162,49822,49823,49825],{},[383,49824,11],{},[383,49826,45126],{},[7162,49828,49829,49831],{},[383,49830,1291],{},[380,49832,49833,49835,49837],{},[463,49834,465],{},[458,49836,5161],{},[383,49838,45126],{},[1645,49840,1647],{},[383,49842,592],{},[383,49844,1121],{},[458,49846,898],{"separator":394},[1597,49848,49849,49855],{},[1600,49850,49851],{},[1603,49852,49853],{"scriptlevel":474,"displaystyle":394},[380,49854],{},[1600,49856,49857],{},[1603,49858,49859],{"scriptlevel":474,"displaystyle":394},[380,49860,49861,49863,49865,49867,49869,49871,49873,49875,49877,49880],{},[380,49862],{},[505,49864],{"width":3610},[383,49866,45126],{},[458,49868,3251],{},[463,49870,474],{},[458,49872,898],{"separator":394},[1645,49874,4631],{},[383,49876,45126],{},[458,49878,49879],{"mathvariant":557},"≠",[463,49881,49882],{},"1.",[387,49884,49885],{"encoding":389},"\\begin{aligned}\nD^{R}_{\\alpha}(P\\|Q)\n&=\\frac{1}{\\alpha-1}\n\\log\\int p^{\\alpha}q^{1-\\alpha}\\,dx,\\\\[2pt]\n&\\qquad \\alpha>0,\\ \\alpha\\ne1 .\n\\end{aligned}",[363,49887,49889],{"className":49888,"ariaHidden":394},[393],[363,49890,49892,49896],{"className":49891},[398],[363,49893],{"className":49894,"style":49895},[402],"height:3.7223em;vertical-align:-1.6111em;",[363,49897,49899],{"className":49898},[407],[363,49900,49902,50019],{"className":49901},[1591],[363,49903,49905],{"className":49904},[1708],[363,49906,49908,50010],{"className":49907},[633,634],[363,49909,49911,50007],{"className":49910},[638],[363,49912,49915,49998],{"className":49913,"style":49914},[642],"height:2.1111em;",[363,49916,49918,49921],{"style":49917},"top:-4.1111em;",[363,49919],{"className":49920,"style":6621},[650],[363,49922,49924,49983,49986,49989,49992,49995],{"className":49923},[407],[363,49925,49927,49930],{"className":49926},[407],[363,49928,547],{"className":49929,"style":625},[407,408],[363,49931,49933],{"className":49932},[629],[363,49934,49936,49975],{"className":49935},[633,634],[363,49937,49939,49972],{"className":49938},[638],[363,49940,49942,49957],{"className":49941,"style":38884},[642],[363,49943,49945,49948],{"style":49944},"top:-2.453em;margin-left:-0.0278em;margin-right:0.05em;",[363,49946],{"className":49947,"style":651},[650],[363,49949,49951],{"className":49950},[655,656,657,658],[363,49952,49954],{"className":49953},[407,658],[363,49955,45126],{"className":49956,"style":45156},[407,408,658],[363,49958,49959,49962],{"style":8625},[363,49960],{"className":49961,"style":651},[650],[363,49963,49965],{"className":49964},[655,656,657,658],[363,49966,49968],{"className":49967},[407,658],[363,49969,1381],{"className":49970,"style":49971},[407,408,658],"margin-right:0.0077em;",[363,49973,666],{"className":49974},[665],[363,49976,49978],{"className":49977},[638],[363,49979,49981],{"className":49980,"style":8647},[642],[363,49982],{},[363,49984,461],{"className":49985},[494],[363,49987,554],{"className":49988,"style":682},[407,408],[363,49990,558],{"className":49991},[407],[363,49993,561],{"className":49994},[407,408],[363,49996,468],{"className":49997},[501],[363,49999,50001,50004],{"style":50000},"top:-2.1089em;",[363,50002],{"className":50003,"style":6621},[650],[363,50005],{"className":50006},[407],[363,50008,666],{"className":50009},[665],[363,50011,50013],{"className":50012},[638],[363,50014,50017],{"className":50015,"style":50016},[642],"height:1.6111em;",[363,50018],{},[363,50020,50022],{"className":50021},[1799],[363,50023,50025,50313],{"className":50024},[633,634],[363,50026,50028,50310],{"className":50027},[638],[363,50029,50031,50224],{"className":50030,"style":49914},[642],[363,50032,50033,50036],{"style":49917},[363,50034],{"className":50035,"style":6621},[650],[363,50037,50039,50042,50045,50048,50051,50125,50128,50133,50136,50139,50142,50174,50212,50215,50218,50221],{"className":50038},[407],[363,50040],{"className":50041},[407],[363,50043],{"className":50044,"style":506},[505],[363,50046,471],{"className":50047},[510],[363,50049],{"className":50050,"style":506},[505],[363,50052,50054,50057,50122],{"className":50053},[407],[363,50055],{"className":50056},[494,789],[363,50058,50060],{"className":50059},[586],[363,50061,50063,50114],{"className":50062},[633,634],[363,50064,50066,50111],{"className":50065},[638],[363,50067,50069,50092,50100],{"className":50068,"style":23261},[642],[363,50070,50071,50074],{"style":805},[363,50072],{"className":50073,"style":809},[650],[363,50075,50077,50080,50083,50086,50089],{"className":50076},[407],[363,50078,45126],{"className":50079,"style":45156},[407,408],[363,50081],{"className":50082,"style":2935},[505],[363,50084,5161],{"className":50085},[2939],[363,50087],{"className":50088,"style":2935},[505],[363,50090,465],{"className":50091},[407],[363,50093,50094,50097],{"style":821},[363,50095],{"className":50096,"style":809},[650],[363,50098],{"className":50099,"style":829},[828],[363,50101,50102,50105],{"style":832},[363,50103],{"className":50104,"style":809},[650],[363,50106,50108],{"className":50107},[407],[363,50109,465],{"className":50110},[407],[363,50112,666],{"className":50113},[665],[363,50115,50117],{"className":50116},[638],[363,50118,50120],{"className":50119,"style":17365},[642],[363,50121],{},[363,50123],{"className":50124},[501,789],[363,50126],{"className":50127,"style":754},[505],[363,50129,4502,50131],{"className":50130},[1832],[363,50132,4506],{"style":4505},[363,50134],{"className":50135,"style":754},[505],[363,50137,1635],{"className":50138,"style":1835},[1832,1833,1834],[363,50140],{"className":50141,"style":754},[505],[363,50143,50145,50148],{"className":50144},[407],[363,50146,11],{"className":50147},[407,408],[363,50149,50151],{"className":50150},[629],[363,50152,50154],{"className":50153},[633],[363,50155,50157],{"className":50156},[638],[363,50158,50160],{"className":50159,"style":46283},[642],[363,50161,50162,50165],{"style":8625},[363,50163],{"className":50164,"style":651},[650],[363,50166,50168],{"className":50167},[655,656,657,658],[363,50169,50171],{"className":50170},[407,658],[3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包在积分外面（这正是它满足可乘可加性、但一般不满足数据处理不等式的根源：",[363,50358,50360,50378],{"className":50359,"translate":367},[366],[363,50361,50363],{"className":50362},[371],[373,50364,50365],{"xmlns":375},[377,50366,50367,50375],{},[380,50368,50369,50371,50373],{},[383,50370,45126],{},[458,50372,3251],{},[463,50374,465],{},[387,50376,50377],{"encoding":389},"\\alpha>1",[363,50379,50381,50399],{"className":50380,"ariaHidden":394},[393],[363,50382,50384,50387,50390,50393,50396],{"className":50383},[398],[363,50385],{"className":50386,"style":6272},[402],[363,50388,45126],{"className":50389,"style":45156},[407,408],[363,50391],{"className":50392,"style":506},[505],[363,50394,3251],{"className":50395},[510],[363,50397],{"className":50398,"style":506},[505],[363,50400,50402,50405],{"className":50401},[398],[363,50403],{"className":50404,"style":520},[402],[363,50406,465],{"className":50407},[407]," 的 Rényi 散度可以在随机变换下增大）。它与 α-散度是\"同一积分的两种包装\"：沿用 3.6 节的指数记号 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p^{a}q^{1-a}\\,dx",[363,50636,50638,50694],{"className":50637,"ariaHidden":394},[393],[363,50639,50641,50644,50685,50688,50691],{"className":50640},[398],[363,50642],{"className":50643,"style":16311},[402],[363,50645,50647,50650],{"className":50646},[407],[363,50648,33975],{"className":50649,"style":11132},[407,408],[363,50651,50653],{"className":50652},[629],[363,50654,50656,50677],{"className":50655},[633,634],[363,50657,50659,50674],{"className":50658},[638],[363,50660,50662],{"className":50661,"style":45301},[642],[363,50663,50665,50668],{"style":50664},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[363,50666],{"className":50667,"style":651},[650],[363,50669,50671],{"className":50670},[655,656,657,658],[363,50672,45552],{"className":50673},[407,408,658],[363,50675,666],{"className":50676},[665],[363,50678,50680],{"className":50679},[638],[363,50681,50683],{"className":50682,"style":8316},[642],[363,50684],{},[363,50686],{"className":50687,"style":506},[505],[363,50689,471],{"className":50690},[510],[363,50692],{"className":50693,"style":506},[505],[363,50695,50697,50701,50704,50707,50740,50778,50781,50784],{"className":50696},[398],[363,50698],{"className":50699,"style":50700},[402],"height:1.1202em;vertical-align:-0.3061em;",[363,50702,1635],{"className":50703,"style":5965},[1832,1833,5964],[363,50705],{"className":50706,"style":754},[505],[363,50708,50710,50713],{"className":50709},[407],[363,50711,11],{"className":50712},[407,408],[363,50714,50716],{"className":50715},[629],[363,50717,50719],{"className":50718},[633],[363,50720,50722],{"className":50721},[638],[363,50723,50726],{"className":50724,"style":50725},[642],"height:0.6644em;",[363,50727,50728,50731],{"style":7213},[363,50729],{"className":50730,"style":651},[650],[363,50732,50734],{"className":50733},[655,656,657,658],[363,50735,50737],{"className":50736},[407,658],[363,50738,45552],{"className":50739},[407,408,658],[363,50741,50743,50746],{"className":50742},[407],[363,50744,1291],{"className":50745,"style":1328},[407,408],[363,50747,50749],{"className":50748},[629],[363,50750,50752],{"className":50751},[633],[363,50753,50755],{"className":50754},[638],[363,50756,50758],{"className":50757,"style":20400},[642],[363,50759,50760,50763],{"style":7213},[363,50761],{"className":50762,"style":651},[650],[363,50764,50766],{"className":50765},[655,656,657,658],[363,50767,50769,50772,50775],{"className":50768},[407,658],[363,50770,465],{"className":50771},[407,658],[363,50773,5161],{"className":50774},[2939,658],[363,50776,45552],{"className":50777},[407,408,658],[363,50779],{"className":50780,"style":754},[505],[363,50782,592],{"className":50783},[407,408],[363,50785,1121],{"className":50786},[407,408],"，则 3.6 节给出",[363,50789,50791],{"className":50790,"translate":367},[528],[363,50792,50794,50840],{"className":50793,"translate":367},[366],[363,50795,50797],{"className":50796},[371],[373,50798,50799],{"xmlns":375,"display":537},[377,50800,50801,50837],{},[380,50802,50803,50809,50811,50813,50815,50817,50819,50821,50823,50825,50827,50829,50835],{},[543,50804,50805,50807],{},[383,50806,33975],{},[383,50808,45552],{},[458,50810,471],{},[463,50812,465],{},[458,50814,5161],{},[383,50816,45552],{},[458,50818,461],{"stretchy":460},[463,50820,465],{},[458,50822,5161],{},[383,50824,45552],{},[458,50826,468],{"stretchy":460},[1645,50828,1647],{},[543,50830,50831,50833],{},[383,50832,547],{},[383,50834,45126],{},[458,50836,898],{"separator":394},[387,50838,50839],{"encoding":389},"Z_a=1-a(1-a)\\,D_\\alpha ,",[363,50841,50843,50898,50916,50940],{"className":50842,"ariaHidden":394},[393],[363,50844,50846,50849,50889,50892,50895],{"className":50845},[398],[363,50847],{"className":50848,"style":16311},[402],[363,50850,50852,50855],{"className":50851},[407],[363,50853,33975],{"className":50854,"style":11132},[407,408],[363,50856,50858],{"className":50857},[629],[363,50859,50861,50881],{"className":50860},[633,634],[363,50862,50864,50878],{"className":50863},[638],[363,50865,50867],{"className":50866,"style":45301},[642],[363,50868,50869,50872],{"style":50664},[363,50870],{"className":50871,"style":651},[650],[363,50873,50875],{"className":50874},[655,656,657,658],[363,50876,45552],{"className":50877},[407,408,658],[363,50879,666],{"className":50880},[665],[363,50882,50884],{"className":50883},[638],[363,50885,50887],{"className":50886,"style":8316},[642],[363,50888],{},[363,50890],{"className":50891,"style":506},[505],[363,50893,471],{"className":50894},[510],[363,50896],{"className":50897,"style":506},[505],[363,50899,50901,50904,50907,50910,50913],{"className":50900},[398],[363,50902],{"className":50903,"style":12928},[402],[363,50905,465],{"className":50906},[407],[363,50908],{"className":50909,"style":2935},[505],[363,50911,5161],{"className":50912},[2939],[363,50914],{"className":50915,"style":2935},[505],[363,50917,50919,50922,50925,50928,50931,50934,50937],{"className":50918},[398],[363,50920],{"className":50921,"style":487},[402],[363,50923,45552],{"className":50924},[407,408],[363,50926,461],{"className":50927},[494],[363,50929,465],{"className":50930},[407],[363,50932],{"className":50933,"style":2935},[505],[363,50935,5161],{"className":50936},[2939],[363,50938],{"className":50939,"style":2935},[505],[363,50941,50943,50946,50949,50952,50955,50995],{"className":50942},[398],[363,50944],{"className":50945,"style":487},[402],[363,50947,45552],{"className":50948},[407,408],[363,50950,468],{"className":50951},[501],[363,50953],{"className":50954,"style":754},[505],[363,50956,50958,50961],{"className":50957},[407],[363,50959,547],{"className":50960,"style":625},[407,408],[363,50962,50964],{"className":50963},[629],[363,50965,50967,50987],{"className":50966},[633,634],[363,50968,50970,50984],{"className":50969},[638],[363,50971,50973],{"className":50972,"style":45301},[642],[363,50974,50975,50978],{"style":646},[363,50976],{"className":50977,"style":651},[650],[363,50979,50981],{"className":50980},[655,656,657,658],[363,50982,45126],{"className":50983,"style":45156},[407,408,658],[363,50985,666],{"className":50986},[665],[363,50988,50990],{"className":50989},[638],[363,50991,50993],{"className":50992,"style":8316},[642],[363,50994],{},[363,50996,898],{"className":50997},[920],[11,50999,51000,51001,51004,51005,51033,51034,51096,51097,51125],{},"而把 Rényi 的",[1344,51002,51003],{},"阶数","取为 ",[363,51006,51008,51021],{"className":51007,"translate":367},[366],[363,51009,51011],{"className":51010},[371],[373,51012,51013],{"xmlns":375},[377,51014,51015,51019],{},[380,51016,51017],{},[383,51018,45552],{},[387,51020,45552],{"encoding":389},[363,51022,51024],{"className":51023,"ariaHidden":394},[393],[363,51025,51027,51030],{"className":51026},[398],[363,51028],{"className":51029,"style":2693},[402],[363,51031,45552],{"className":51032},[407,408],"（注意 order ",[363,51035,51037,51051],{"className":51036,"translate":367},[366],[363,51038,51040],{"className":51039},[371],[373,51041,51042],{"xmlns":375},[377,51043,51044,51048],{},[380,51045,51046],{},[458,51047,49879],{"mathvariant":557},[387,51049,51050],{"encoding":389},"\\neq",[363,51052,51054],{"className":51053,"ariaHidden":394},[393],[363,51055,51057,51060],{"className":51056},[398],[363,51058],{"className":51059,"style":403},[402],[363,51061,51063,51090,51093],{"className":51062},[510],[363,51064,51066],{"className":51065},[510],[363,51067,51069],{"className":51068},[407,4748],[363,51070,51072],{"className":51071},[4752],[363,51073,51075,51078,51087],{"className":51074},[4756],[363,51076],{"className":51077,"style":403},[402],[363,51079,51081],{"className":51080},[4763],[363,51082,51084],{"className":51083},[407],[363,51085,4770],{"className":51086},[510],[363,51088],{"className":51089},[4774],[363,51091],{"className":51092},[505,4778],[363,51094,471],{"className":51095},[510]," 3.6 的参数 ",[363,51098,51100,51113],{"className":51099,"translate":367},[366],[363,51101,51103],{"className":51102},[371],[373,51104,51105],{"xmlns":375},[377,51106,51107,51111],{},[380,51108,51109],{},[383,51110,45126],{},[387,51112,49462],{"encoding":389},[363,51114,51116],{"className":51115,"ariaHidden":394},[393],[363,51117,51119,51122],{"className":51118},[398],[363,51120],{"className":51121,"style":2693},[402],[363,51123,45126],{"className":51124,"style":45156},[407,408],"，只是经过线性换算）：",[363,51127,51129],{"className":51128,"translate":367},[528],[363,51130,51132,51236],{"className":51131,"translate":367},[366],[363,51133,51135],{"className":51134},[371],[373,51136,51137],{"xmlns":375,"display":537},[377,51138,51139,51233],{},[380,51140,51141,51149,51151,51153,51155,51157,51159,51161,51173,51175,51177,51183,51185,51187,51199,51201,51203,51205,51207,51209,51211,51213,51215,51217,51219,51221,51223,51229,51231],{},[7901,51142,51143,51145,51147],{},[383,51144,547],{},[383,51146,45552],{},[383,51148,1381],{},[458,51150,461],{"stretchy":460},[383,51152,554],{},[383,51154,558],{"mathvariant":557},[383,51156,561],{},[458,51158,468],{"stretchy":460},[458,51160,471],{},[586,51162,51163,51165],{},[463,51164,465],{},[380,51166,51167,51169,51171],{},[383,51168,45552],{},[458,51170,5161],{},[463,51172,465],{},[383,51174,4448],{},[458,51176,4451],{},[543,51178,51179,51181],{},[383,51180,33975],{},[383,51182,45552],{},[458,51184,471],{},[458,51186,5161],{},[586,51188,51189,51191],{},[463,51190,465],{},[380,51192,51193,51195,51197],{},[463,51194,465],{},[458,51196,5161],{},[383,51198,45552],{},[1645,51200,1647],{},[383,51202,4448],{},[458,51204,4451],{},[458,51206,461],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[463,51208,465],{},[458,51210,5161],{},[383,51212,45552],{},[458,51214,461],{"stretchy":460},[463,51216,465],{},[458,51218,5161],{},[383,51220,45552],{},[458,51222,468],{"stretchy":460},[543,51224,51225,51227],{},[383,51226,547],{},[383,51228,45126],{},[458,51230,468],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[383,51232,4723],{"mathvariant":557},[387,51234,51235],{"encoding":389},"D^{R}_{a}(P\\|Q)=\\frac{1}{a-1}\\log Z_a\n=-\\frac{1}{1-a}\\,\\log\\big(1-a(1-a)D_\\alpha\\big).",[363,51237,51239,51327,51467,51582,51606],{"className":51238,"ariaHidden":394},[393],[363,51240,51242,51246,51303,51306,51309,51312,51315,51318,51321,51324],{"className":51241},[398],[363,51243],{"className":51244,"style":51245},[402],"height:1.1413em;vertical-align:-0.25em;",[363,51247,51249,51252],{"className":51248},[407],[363,51250,547],{"className":51251,"style":625},[407,408],[363,51253,51255],{"className":51254},[629],[363,51256,51258,51295],{"className":51257},[633,634],[363,51259,51261,51292],{"className":51260},[638],[363,51262,51264,51278],{"className":51263,"style":38884},[642],[363,51265,51266,51269],{"style":49944},[363,51267],{"className":51268,"style":651},[650],[363,51270,51272],{"className":51271},[655,656,657,658],[363,51273,51275],{"className":51274},[407,658],[363,51276,45552],{"className":51277},[407,408,658],[363,51279,51280,51283],{"style":8625},[363,51281],{"className":51282,"style":651},[650],[363,51284,51286],{"className":51285},[655,656,657,658],[363,51287,51289],{"className":51288},[407,658],[363,51290,1381],{"className":51291,"style":49971},[407,408,658],[363,51293,666],{"className":51294},[665],[363,51296,51298],{"className":51297},[638],[363,51299,51301],{"className":51300,"style":8647},[642],[363,51302],{},[363,51304,461],{"className":51305},[494],[363,51307,554],{"className":51308,"style":682},[407,408],[363,51310,558],{"className":51311},[407],[363,51313,561],{"className":51314},[407,408],[363,51316,468],{"className":51317},[501],[363,51319],{"className":51320,"style":506},[505],[363,51322,471],{"className":51323},[510],[363,51325],{"className":51326,"style":506},[505],[363,51328,51330,51333,51407,51410,51415,51418,51458,51461,51464],{"className":51329},[398],[363,51331],{"className":51332,"style":26988},[402],[363,51334,51336,51339,51404],{"className":51335},[407],[363,51337],{"className":51338},[494,789],[363,51340,51342],{"className":51341},[586],[363,51343,51345,51396],{"className":51344},[633,634],[363,51346,51348,51393],{"className":51347},[638],[363,51349,51351,51374,51382],{"className":51350,"style":23261},[642],[363,51352,51353,51356],{"style":805},[363,51354],{"className":51355,"style":809},[650],[363,51357,51359,51362,51365,51368,51371],{"className":51358},[407],[363,51360,45552],{"className":51361},[407,408],[363,51363],{"className":51364,"style":2935},[505],[363,51366,5161],{"className":51367},[2939],[363,51369],{"className":51370,"style":2935},[505],[363,51372,465],{"className":51373},[407],[363,51375,51376,51379],{"style":821},[363,51377],{"className":51378,"style":809},[650],[363,51380],{"className":51381,"style":829},[828],[363,51383,51384,51387],{"style":832},[363,51385],{"className":51386,"style":809},[650],[363,51388,51390],{"className":51389},[407],[363,51391,465],{"className":51392},[407],[363,51394,666],{"className":51395},[665],[363,51397,51399],{"className":51398},[638],[363,51400,51402],{"className":51401,"style":17365},[642],[363,51403],{},[363,51405],{"className":51406},[501,789],[363,51408],{"className":51409,"style":754},[505],[363,51411,4502,51413],{"className":51412},[1832],[363,51414,4506],{"style":4505},[363,51416],{"className":51417,"style":754},[505],[363,51419,51421,51424],{"className":51420},[407],[363,51422,33975],{"className":51423,"style":11132},[407,408],[363,51425,51427],{"className":51426},[629],[363,51428,51430,51450],{"className":51429},[633,634],[363,51431,51433,51447],{"className":51432},[638],[363,51434,51436],{"className":51435,"style":45301},[642],[363,51437,51438,51441],{"style":50664},[363,51439],{"className":51440,"style":651},[650],[363,51442,51444],{"className":51443},[655,656,657,658],[363,51445,45552],{"className":51446},[407,408,658],[363,51448,666],{"className":51449},[665],[363,51451,51453],{"className":51452},[638],[363,51454,51456],{"className":51455,"style":8316},[642],[363,51457],{},[363,51459],{"className":51460,"style":506},[505],[363,51462,471],{"className":51463},[510],[363,51465],{"className":51466,"style":506},[505],[363,51468,51470,51473,51476,51550,51553,51556,51561,51564,51570,51573,51576,51579],{"className":51469},[398],[363,51471],{"className":51472,"style":26988},[402],[363,51474,5161],{"className":51475},[407],[363,51477,51479,51482,51547],{"className":51478},[407],[363,51480],{"className":51481},[494,789],[363,51483,51485],{"className":51484},[586],[363,51486,51488,51539],{"className":51487},[633,634],[363,51489,51491,51536],{"className":51490},[638],[363,51492,51494,51517,51525],{"className":51493,"style":23261},[642],[363,51495,51496,51499],{"style":805},[363,51497],{"className":51498,"style":809},[650],[363,51500,51502,51505,51508,51511,51514],{"className":51501},[407],[363,51503,465],{"className":51504},[407],[363,51506],{"className":51507,"style":2935},[505],[363,51509,5161],{"className":51510},[2939],[363,51512],{"className":51513,"style":2935},[505],[363,51515,45552],{"className":51516},[407,408],[363,51518,51519,51522],{"style"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",[363,51671,51673,51699],{"className":51672,"translate":367},[366],[363,51674,51676],{"className":51675},[371],[373,51677,51678],{"xmlns":375},[377,51679,51680,51696],{},[380,51681,51682,51684,51690],{},[458,51683,461],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[586,51685,51686,51688],{},[383,51687,11],{},[383,51689,1291],{},[7162,51691,51692,51694],{},[458,51693,468],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[383,51695,45552],{},[387,51697,51698],{"encoding":389},"\\big(\\frac pq\\big)^a",[363,51700,51702],{"className":51701,"ariaHidden":394},[393],[363,51703,51705,51709,51715,51783],{"className":51704},[398],[363,51706],{"className":51707,"style":51708},[402],"height:1.3854em;vertical-align:-0.4811em;",[363,51710,51712],{"className":51711},[407],[363,51713,461],{"className":51714},[767,10022],[363,51716,51718,51721,51780],{"className":51717},[407],[363,51719],{"className":51720},[494,789],[363,51722,51724],{"className":51723},[586],[363,51725,51727,51772],{"className":51726},[633,634],[363,51728,51730,51769],{"className":51729},[638],[363,51731,51733,51747,51755],{"className":51732,"style":3451},[642],[363,51734,51735,51738],{"style":2875},[363,51736],{"className":51737,"style":809},[650],[363,51739,51741],{"className":51740},[655,656,657,658],[363,51742,51744],{"className":51743},[407,658],[363,51745,1291],{"className":51746,"style":1328},[407,408,658],[363,51748,51749,51752],{"style":821},[363,51750],{"className":51751,"style":809},[650],[363,51753],{"className":51754,"style":829},[828],[363,51756,51757,51760],{"style":3201},[363,51758],{"className":51759,"style":809},[650],[363,51761,51763],{"className":51762},[655,656,657,658],[363,51764,51766],{"className":51765},[407,658],[363,51767,11],{"className":51768},[407,408,658],[363,51770,666],{"className":51771},[665],[363,51773,51775],{"className":51774},[638],[363,51776,51778],{"className":51777,"style":3087},[642],[363,51779],{},[363,51781],{"className":51782},[501,789],[363,51784,51786,51792],{"className":51785},[407],[363,51787,51789],{"className":51788},[407],[363,51790,468],{"className":51791},[767,10022],[363,51793,51795],{"className":51794},[629],[363,51796,51798],{"className":51797},[633],[363,51799,51801],{"className":51800},[638],[363,51802,51805],{"className":51803,"style":51804},[642],"height:0.9043em;",[363,51806,51808,51811],{"style":51807},"top:-3.3029em;margin-right:0.05em;",[363,51809],{"className":51810,"style":651},[650],[363,51812,51814],{"className":51813},[655,656,657,658],[363,51815,45552],{"className":51816},[407,408,658]," 做 ",[363,51819,51821,51834],{"className":51820,"translate":367},[366],[363,51822,51824],{"className":51823},[371],[373,51825,51826],{"xmlns":375},[377,51827,51828,51832],{},[380,51829,51830],{},[383,51831,561],{},[387,51833,561],{"encoding":389},[363,51835,51837],{"className":51836,"ariaHidden":394},[393],[363,51838,51840,51843],{"className":51839},[398],[363,51841],{"className":51842,"style":913},[402],[363,51844,561],{"className":51845},[407,408]," 下的线性平均，Rényi 在外面再套一层对数。所以 f-散度族与 Rényi 族\"同源\"：后者是前者经 ",[363,51848,51850,51865],{"className":51849,"translate":367},[366],[363,51851,51853],{"className":51852},[371],[373,51854,51855],{"xmlns":375},[377,51856,51857,51863],{},[380,51858,51859,51861],{},[383,51860,4448],{},[458,51862,4451],{},[387,51864,21955],{"encoding":389},[363,51866,51868],{"className":51867,"ariaHidden":394},[393],[363,51869,51871,51874],{"className":51870},[398],[363,51872],{"className":51873,"style":403},[402],[363,51875,4502,51877],{"className":51876},[1832],[363,51878,4506],{"style":4505}," 变换后的\"信息量\"版本。",[11,51881,51882,51884,51885,51888],{},[1344,51883,24565],{}," Rényi 散度在",[1344,51886,51887],{},"假设检验","中直接就是误差指数：Chernoff 信息",[363,51890,51892],{"className":51891,"translate":367},[528],[363,51893,51895,52018],{"className":51894,"translate":367},[366],[363,51896,51898],{"className":51897},[371],[373,51899,51900],{"xmlns":375,"display":537},[377,51901,51902,52015],{},[380,51903,51904,51906,51908,51910,51912,51914,51916,51918,51938,51940,51942,51944,51946,51948,51956,51958,51960,51962,51964,51966,51968,51970,51972,51974,51995,51997,52003],{},[383,51905,39666],{},[458,51907,461],{"stretchy":460},[383,51909,554],{},[458,51911,898],{"separator":394},[383,51913,561],{},[458,51915,468],{"stretchy":460},[458,51917,471],{},[6364,51919,51920,51926],{},[380,51921,51922,51924],{},[383,51923,35804],{},[458,51925,4451],{},[380,51927,51928,51930,51932,51934,51936],{},[463,51929,474],{},[458,51931,7599],{},[383,51933,11689],{},[458,51935,7599],{},[463,51937,465],{},[458,51939,461],{"stretchy":460},[463,51941,465],{},[458,51943,5161],{},[383,51945,11689],{},[458,51947,468],{"stretchy":460},[7901,51949,51950,51952,51954],{},[383,51951,547],{},[383,51953,11689],{},[383,51955,1381],{},[458,51957,461],{"stretchy":460},[383,51959,554],{},[383,51961,558],{"mathvariant":557},[383,51963,561],{},[458,51965,468],{"stretchy":460},[458,51967,471],{},[458,51969,5161],{},[383,51971,4448],{},[458,51973,4451],{},[6364,51975,51976,51983],{},[380,51977,51978,51981],{},[383,51979,51980],{},"min",[458,51982,4451],{},[380,51984,51985,51987,51989,51991,51993],{},[463,51986,474],{},[458,51988,10300],{},[383,51990,11689],{},[458,51992,10300],{},[463,51994,465],{},[458,51996,1635],{},[7162,51998,51999,52001],{},[383,52000,11],{},[383,52002,11689],{},[7162,52004,52005,52007],{},[383,52006,1291],{},[380,52008,52009,52011,52013],{},[463,52010,465],{},[458,52012,5161],{},[383,52014,11689],{},[387,52016,52017],{"encoding":389},"C(P,Q)=\\sup_{0\u003C\\lambda\u003C1}\n(1-\\lambda)D^{R}_{\\lambda}(P\\|Q)\n=-\\log\\min_{0\\le\\lambda\\le1}\n\\int p^\\lambda q^{1-\\lambda}",[363,52019,52021,52057,52140,52233],{"className":52020,"ariaHidden":394},[393],[363,52022,52024,52027,52030,52033,52036,52039,52042,52045,52048,52051,52054],{"className":52023},[398],[363,52025],{"className":52026,"style":487},[402],[363,52028,39666],{"className":52029,"style":11132},[407,408],[363,52031,461],{"className":52032},[494],[363,52034,554],{"className":52035,"style":682},[407,408],[363,52037,898],{"className":52038},[920],[363,52040],{"className":52041,"style":754},[505],[363,52043,561],{"className":52044},[407,408],[363,52046,468],{"className":52047},[501],[363,52049],{"className":52050,"style":506},[505],[363,52052,471],{"className":52053},[510],[363,52055],{"className":52056,"style":506},[505],[363,52058,52060,52064,52125,52128,52131,52134,52137],{"className":52059},[398],[363,52061],{"className":52062,"style":52063},[402],"height:1.7239em;vertical-align:-0.9739em;",[363,52065,52067],{"className":52066},[1832,14066],[363,52068,52070,52116],{"className":52069},[633,634],[363,52071,52073,52113],{"className":52072},[638],[363,52074,52076,52103],{"className":52075,"style":2693},[642],[363,52077,52079,52082],{"style":52078},"top:-2.1535em;margin-left:0em;",[363,52080],{"className":52081,"style":809},[650],[363,52083,52085],{"className":52084},[655,656,657,658],[363,52086,52088,52091,52094,52097,52100],{"className":52087},[407,658],[363,52089,474],{"className":52090},[407,658],[363,52092,7599],{"className":52093},[510,658],[363,52095,11689],{"className":52096},[407,408,658],[363,52098,7599],{"className":52099},[510,658],[363,52101,465],{"className":52102},[407,658],[363,52104,52105,52108],{"style":12213},[363,52106],{"className":52107,"style":809},[650],[363,52109,52110],{},[363,52111,35804],{"className":52112},[1832],[363,52114,666],{"className":52115},[665],[363,52117,52119],{"className":52118},[638],[363,52120,52123],{"className":52121,"style":52122},[642],"height:0.9739em;",[363,52124],{},[363,52126,461],{"className":52127},[494],[363,52129,465],{"className":52130},[407],[363,52132],{"className":52133,"style":2935},[505],[363,52135,5161],{"className":52136},[2939],[363,52138],{"className":52139,"style":2935},[505],[363,52141,52143,52146,52149,52152,52209,52212,52215,52218,52221,52224,52227,52230],{"className":52142},[398],[363,52144],{"className":52145,"style":51245},[402],[363,52147,11689],{"className":52148},[407,408],[363,52150,468],{"className":52151},[501],[363,52153,52155,52158],{"className":52154},[407],[363,52156,547],{"className":52157,"style":625},[407,408],[363,52159,52161],{"className":52160},[629],[363,52162,52164,52201],{"className":52163},[633,634],[363,52165,52167,52198],{"className":52166},[638],[363,52168,52170,52184],{"className":52169,"style":38884},[642],[363,52171,52172,52175],{"style":49944},[363,52173],{"className":52174,"style":651},[650],[363,52176,52178],{"className":52177},[655,656,657,658],[363,52179,52181],{"className":52180},[407,658],[363,52182,11689],{"className":52183},[407,408,658],[363,52185,52186,52189],{"style":8625},[363,52187],{"className":52188,"style":651},[650],[363,52190,52192],{"className":52191},[655,656,657,658],[363,52193,52195],{"className":52194},[407,658],[363,52196,1381],{"className":52197,"style":49971},[407,408,658],[363,52199,666],{"className":52200},[665],[363,52202,52204],{"className":52203},[638],[363,52205,52207],{"className":52206,"style":8647},[642],[363,52208],{},[363,52210,461],{"className":52211},[494],[363,52213,554],{"className":52214,"style":682},[407,408],[363,52216,558],{"className":52217},[407],[363,52219,561],{"className":52220},[407,408],[363,52222,468],{"className":52223},[501],[363,52225],{"className":52226,"style":506},[505],[363,52228,471],{"className":52229},[510],[363,52231],{"className":52232,"style":506},[505],[363,52234,52236,52239,52242,52245,52250,52253,52315,52318,52321,52324,52353],{"className":52235},[398],[363,52237],{"className":52238,"style":6125},[402],[363,52240,5161],{"className":52241},[407],[363,52243],{"className":52244,"style":754},[505],[363,52246,4502,52248],{"className":52247},[1832],[363,52249,4506],{"style":4505},[363,52251],{"className":52252,"style":754},[505],[363,52254,52256],{"className":52255},[1832,14066],[363,52257,52259,52306],{"className":52258},[633,634],[363,52260,52262,52303],{"className":52261},[638],[363,52263,52266,52293],{"className":52264,"style":52265},[642],"height:0.6679em;",[363,52267,52269,52272],{"style":52268},"top:-2.3479em;margin-left:0em;",[363,52270],{"className":52271,"style":809},[650],[363,52273,52275],{"className":52274},[655,656,657,658],[363,52276,52278,52281,52284,52287,52290],{"className":52277},[407,658],[363,52279,474],{"className":52280},[407,658],[363,52282,10300],{"className":52283},[510,658],[363,52285,11689],{"className":52286},[407,408,658],[363,52288,10300],{"className":52289},[510,658],[363,52291,465],{"className":52292},[407,658],[363,52294,52295,52298],{"style":12213},[363,52296],{"className":52297,"style":809},[650],[363,52299,52300],{},[363,52301,51980],{"className":52302},[1832],[363,52304,666],{"className":52305},[665],[363,52307,52309],{"className":52308},[638],[363,52310,52313],{"className":52311,"style":52312},[642],"height:0.8473em;",[363,52314],{},[363,52316],{"className":52317,"style":754},[505],[363,52319,1635],{"className":52320,"style":1835},[1832,1833,1834],[363,52322],{"className":52323,"style":754},[505],[363,52325,52327,52330],{"className":52326},[407],[363,52328,11],{"className":52329},[407,408],[363,52331,52333],{"className":52332},[629],[363,52334,52336],{"className":52335},[633],[363,52337,52339],{"className":52338},[638],[363,52340,52342],{"className":52341,"style":48236},[642],[363,52343,52344,52347],{"style":8625},[363,52345],{"className":52346,"style":651},[650],[363,52348,52350],{"className":52349},[655,656,657,658],[363,52351,11689],{"className":52352},[407,408,658],[363,52354,52356,52359],{"className":52355},[407],[363,52357,1291],{"className":52358,"style":1328},[407,408],[363,52360,52362],{"className":52361},[629],[363,52363,52365],{"className":52364},[633],[363,52366,52368],{"className":52367},[638],[363,52369,52371],{"className":52370,"style":48236},[642],[363,52372,52373,52376],{"style":8625},[363,52374],{"className":52375,"style":651},[650],[363,52377,52379],{"className":52378},[655,656,657,658],[363,52380,52382,52385,52388],{"className":52381},[407,658],[363,52383,465],{"className":52384},[407,658],[363,52386,5161],{"className":52387},[2939,658],[363,52389,11689],{"className":52390},[407,408,658],[11,52392,52393,52394,52560,52561,4201,52632,52682,52683,52734,52735,52854,52855,52858],{},"决定对称检验错误概率的衰减速率，Stein 引理给出的单侧指数则是 ",[363,52395,52397,52429],{"className":52396,"translate":367},[366],[363,52398,52400],{"className":52399},[371],[373,52401,52402],{"xmlns":375},[377,52403,52404,52426],{},[380,52405,52406,52414,52416],{},[7901,52407,52408,52410,52412],{},[383,52409,547],{},[463,52411,465],{},[383,52413,1381],{},[458,52415,471],{},[543,52417,52418,52420],{},[383,52419,547],{},[380,52421,52422,52424],{},[383,52423,11044],{},[383,52425,22265],{},[387,52427,52428],{"encoding":389},"D^{R}_{1}=D_{KL}",[363,52430,52432,52508],{"className":52431,"ariaHidden":394},[393],[363,52433,52435,52439,52499,52502,52505],{"className":52434},[398],[363,52436],{"className":52437,"style":52438},[402],"height:1.0894em;vertical-align:-0.2481em;",[363,52440,52442,52445],{"className":52441},[407],[363,52443,547],{"className":52444,"style":625},[407,408],[363,52446,52448],{"className":52447},[629],[363,52449,52451,52490],{"className":52450},[633,634],[363,52452,52454,52487],{"className":52453},[638],[363,52455,52458,52473],{"className":52456,"style":52457},[642],"height:0.8413em;",[363,52459,52461,52464],{"style":52460},"top:-2.4519em;margin-left:-0.0278em;margin-right:0.05em;",[363,52462],{"className":52463,"style":651},[650],[363,52465,52467],{"className":52466},[655,656,657,658],[363,52468,52470],{"className":52469},[407,658],[363,52471,465],{"className":52472},[407,658],[363,52474,52475,52478],{"style":7213},[363,52476],{"className":52477,"style":651},[650],[363,52479,52481],{"className":52480},[655,656,657,658],[363,52482,52484],{"className":52483},[407,658],[363,52485,1381],{"className":52486,"style":49971},[407,408,658],[363,52488,666],{"className":52489},[665],[363,52491,52493],{"className":52492},[638],[363,52494,52497],{"className":52495,"style":52496},[642],"height:0.2481em;",[363,52498],{},[363,52500],{"className":52501,"style":506},[505],[363,52503,471],{"className":52504},[510],[363,52506],{"className":52507,"style":506},[505],[363,52509,52511,52514],{"className":52510},[398],[363,52512],{"className":52513,"style":16311},[402],[363,52515,52517,52520],{"className":52516},[407],[363,52518,547],{"className":52519,"style":625},[407,408],[363,52521,52523],{"className":52522},[629],[363,52524,52526,52552],{"className":52525},[633,634],[363,52527,52529,52549],{"className":52528},[638],[363,52530,52532],{"className":52531,"style":727},[642],[363,52533,52534,52537],{"style":646},[363,52535],{"className":52536,"style":651},[650],[363,52538,52540],{"className":52539},[655,656,657,658],[363,52541,52543,52546],{"className":52542},[407,658],[363,52544,11044],{"className":52545,"style":11132},[407,408,658],[363,52547,22265],{"className":52548},[407,408,658],[363,52550,666],{"className":52551},[665],[363,52553,52555],{"className":52554},[638],[363,52556,52558],{"className":52557,"style":8316},[642],[363,52559],{},"。对应的 Rényi 熵 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时退化为 Shannon 熵、在 ",[363,52684,52686,52704],{"className":52685,"translate":367},[366],[363,52687,52689],{"className":52688},[371],[373,52690,52691],{"xmlns":375},[377,52692,52693,52701],{},[380,52694,52695,52697,52699],{},[383,52696,45126],{},[458,52698,1378],{},[383,52700,1373],{"mathvariant":557},[387,52702,52703],{"encoding":389},"\\alpha\\to\\infty",[363,52705,52707,52725],{"className":52706,"ariaHidden":394},[393],[363,52708,52710,52713,52716,52719,52722],{"className":52709},[398],[363,52711],{"className":52712,"style":2693},[402],[363,52714,45126],{"className":52715,"style":45156},[407,408],[363,52717],{"className":52718,"style":506},[505],[363,52720,1378],{"className":52721},[510],[363,52723],{"className":52724,"style":506},[505],[363,52726,52728,52731],{"className":52727},[398],[363,52729],{"className":52730,"style":2693},[402],[363,52732,1373],{"className":52733},[407]," 时退化为 min-entropy 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行为（不重叠支撑）",[52962,52963,52964,53165,53364,53492,53890,54109,54304,54584,54860],"tbody",{},[52870,52965,52966,53074,53125,53128,53130],{},[52967,52968,52969,52970],"td",{},"KL 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基本性质",[11,55287,55288,55289,16145,55353,55510,55511,10528],{},"下列性质逐个给出推导。约定 ",[363,55290,55292,55314],{"className":55291,"translate":367},[366],[363,55293,55295],{"className":55294},[371],[373,55296,55297],{"xmlns":375},[377,55298,55299,55311],{},[380,55300,55301,55303,55305,55307,55309],{},[383,55302,554],{},[458,55304,898],{"separator":394},[383,55306,561],{},[458,55308,1061],{},[383,55310,2712],{},[387,55312,55313],{"encoding":389},"P,Q\\ll\\mu",[363,55315,55317,55344],{"className":55316,"ariaHidden":394},[393],[363,55318,55320,55323,55326,55329,55332,55335,55338,55341],{"className":55319},[398],[363,55321],{"className":55322,"style":913},[402],[363,55324,554],{"className":55325,"style":682},[407,408],[363,55327,898],{"className":55328},[920],[363,55330],{"className":55331,"style":754},[505],[363,55333,561],{"className":55334},[407,408],[363,55336],{"className":55337,"style":506},[505],[363,55339,1061],{"className":55340},[510],[363,55342],{"className":55343,"style":506},[505],[363,55345,55347,55350],{"className":55346},[398],[363,55348],{"className":55349,"style":2624},[402],[363,55351,2712],{"className":55352},[407,408],[363,55354,55356,55388],{"className":55355,"translate":367},[366],[363,55357,55359],{"className":55358},[371],[373,55360,55361],{"xmlns":375},[377,55362,55363,55385],{},[380,55364,55365,55367,55369,55375,55377,55379,55381,55383],{},[383,55366,2712],{},[458,55368,471],{},[586,55370,55371,55373],{},[463,55372,465],{},[463,55374,2811],{},[458,55376,461],{"stretchy":460},[383,55378,554],{},[458,55380,2818],{},[383,55382,561],{},[458,55384,468],{"stretchy":460},[387,55386,55387],{"encoding":389},"\\mu=\\frac12(P+Q)",[363,55389,55391,55409,55498],{"className":55390,"ariaHidden":394},[393],[363,55392,55394,55397,55400,55403,55406],{"className":55393},[398],[363,55395],{"className":55396,"style":2624},[402],[363,55398,2712],{"className":55399},[407,408],[363,55401],{"className":55402,"style":506},[505],[363,55404,471],{"className":55405},[510],[363,55407],{"className":55408,"style":506},[505],[363,55410,55412,55415,55483,55486,55489,55492,55495],{"className":55411},[398],[363,55413],{"className":55414,"style":2853},[402],[363,55416,55418,55421,55480],{"className":55417},[407],[363,55419],{"className":55420},[494,789],[363,55422,55424],{"className":55423},[586],[363,55425,55427,55472],{"className":55426},[633,634],[363,55428,55430,55469],{"className":55429},[638],[363,55431,55433,55447,55455],{"className":55432,"style":2872},[642],[363,55434,55435,55438],{"style":2875},[363,55436],{"className":55437,"style":809},[650],[363,55439,55441],{"className":55440},[655,656,657,658],[363,55442,55444],{"className":55443},[407,658],[363,55445,2811],{"className":55446},[407,658],[363,55448,55449,55452],{"style":821},[363,55450],{"className":55451,"style":809},[650],[363,55453],{"className":55454,"style":829},[828],[363,55456,55457,55460],{"style":2898},[363,55458],{"className":55459,"style":809},[650],[363,55461,55463],{"className":55462},[655,656,657,658],[363,55464,55466],{"className":55465},[407,658],[363,55467,465],{"className":55468},[407,658],[363,55470,666],{"className":55471},[665],[363,55473,55475],{"className":55474},[638],[363,55476,55478],{"className":55477,"style":2920},[642],[363,55479],{},[363,55481],{"className":55482},[501,789],[363,55484,461],{"className":55485},[494],[363,55487,554],{"className":55488,"style":682},[407,408],[363,55490],{"className":55491,"style":2935},[505],[363,55493,2818],{"className":55494},[2939],[363,55496],{"className":55497,"style":2935},[505],[363,55499,55501,55504,55507],{"className":55500},[398],[363,55502],{"className":55503,"style":487},[402],[363,55505,561],{"className":55506},[407,408],[363,55508,468],{"className":55509},[501]," 或任一控制测度），",[363,55512,55514,55539],{"className":55513,"translate":367},[366],[363,55515,55517],{"className":55516},[371],[373,55518,55519],{"xmlns":375},[377,55520,55521,55537],{},[380,55522,55523,55525,55527,55529,55531,55533,55535],{},[383,55524,1116],{},[458,55526,471],{},[383,55528,592],{},[383,55530,554],{},[383,55532,1288],{"mathvariant":557},[383,55534,592],{},[383,55536,561],{},[387,55538,5681],{"encoding":389},[363,55540,55542,55560],{"className":55541,"ariaHidden":394},[393],[363,55543,55545,55548,55551,55554,55557],{"className":55544},[398],[363,55546],{"className":55547,"style":5691},[402],[363,55549,1116],{"className":55550},[407,408],[363,55552],{"className":55553,"style":506},[505],[363,55555,471],{"className":55556},[510],[363,55558],{"className":55559,"style":506},[505],[363,55561,55563,55566,55569,55572,55575,55578],{"className":55562},[398],[363,55564],{"className":55565,"style":487},[402],[363,55567,592],{"className":55568},[407,408],[363,55570,554],{"className":55571,"style":682},[407,408],[363,55573,1288],{"className":55574},[407],[363,55576,592],{"className":55577},[407,408],[363,55579,561],{"className":55580},[407,408],[2659,55582,55584],{"id":55583},"_41-非负性","4.1 非负性",[143,55586,55587,55634,55759],{},[11,55588,55589,55592,55593,3093],{},[1344,55590,55591],{},"命题 4.1（非负性）"," 对一切概率分布 ",[363,55594,55596,55613],{"className":55595,"translate":367},[366],[363,55597,55599],{"className":55598},[371],[373,55600,55601],{"xmlns":375},[377,55602,55603,55611],{},[380,55604,55605,55607,55609],{},[383,55606,554],{},[458,55608,898],{"separator":394},[383,55610,561],{},[387,55612,903],{"encoding":389},[363,55614,55616],{"className":55615,"ariaHidden":394},[393],[363,55617,55619,55622,55625,55628,55631],{"className":55618},[398],[363,55620],{"className":55621,"style":913},[402],[363,55623,554],{"className":55624,"style":682},[407,408],[363,55626,898],{"className":55627},[920],[363,55629],{"className":55630,"style":754},[505],[363,55632,561],{"className":55633},[407,408],[363,55635,55637],{"className":55636,"translate":367},[528],[363,55638,55640,55674],{"className":55639,"translate":367},[366],[363,55641,55643],{"className":55642},[371],[373,55644,55645],{"xmlns":375,"display":537},[377,55646,55647,55671],{},[380,55648,55649,55655,55657,55659,55661,55663,55665,55667,55669],{},[543,55650,55651,55653],{},[383,55652,547],{},[383,55654,385],{},[458,55656,461],{"stretchy":460},[383,55658,554],{},[383,55660,558],{"mathvariant":557},[383,55662,561],{},[458,55664,468],{"stretchy":460},[458,55666,7276],{},[463,55668,474],{},[458,55670,898],{"separator":394},[387,55672,55673],{"encoding":389},"D_f(P\\|Q)\\ge 0,",[363,55675,55677,55747],{"className":55676,"ariaHidden":394},[393],[363,55678,55680,55683,55723,55726,55729,55732,55735,55738,55741,55744],{"className":55679},[398],[363,55681],{"className":55682,"style":618},[402],[363,55684,55686,55689],{"className":55685},[407],[363,55687,547],{"className":55688,"style":625},[407,408],[363,55690,55692],{"className":55691},[629],[363,55693,55695,55715],{"className":55694},[633,634],[363,55696,55698,55712],{"className":55697},[638],[363,55699,55701],{"className":55700,"style":643},[642],[363,55702,55703,55706],{"style":646},[363,55704],{"className":55705,"style":651},[650],[363,55707,55709],{"className":55708},[655,656,657,658],[363,55710,385],{"className":55711,"style":409},[407,408,658],[363,55713,666],{"className":55714},[665],[363,55716,55718],{"className":55717},[638],[363,55719,55721],{"className":55720,"style":673},[642],[363,55722],{},[363,55724,461],{"className":55725},[494],[363,55727,554],{"className":55728,"style":682},[407,408],[363,55730,558],{"className":55731},[407],[363,55733,561],{"className":55734},[407,408],[363,55736,468],{"className":55737},[501],[363,55739],{"className":55740,"style":506},[505],[363,55742,7276],{"className":55743},[510],[363,55745],{"className":55746,"style":506},[505],[363,55748,55750,55753,55756],{"className":55749},[398],[363,55751],{"className":55752,"style":9706},[402],[363,55754,474],{"className":55755},[407],[363,55757,898],{"className":55758},[920],[11,55760,55761,55762,55790,55791,10528],{},"且当 ",[363,55763,55765,55778],{"className":55764,"translate":367},[366],[363,55766,55768],{"className":55767},[371],[373,55769,55770],{"xmlns":375},[377,55771,55772,55776],{},[380,55773,55774],{},[383,55775,385],{},[387,55777,385],{"encoding":389},[363,55779,55781],{"className":55780,"ariaHidden":394},[393],[363,55782,55784,55787],{"className":55783},[398],[363,55785],{"className":55786,"style":403},[402],[363,55788,385],{"className":55789,"style":409},[407,408]," 严格凸时，等号成立当且仅当 ",[363,55792,55794,55811],{"className":55793,"translate":367},[366],[363,55795,55797],{"className":55796},[371],[373,55798,55799],{"xmlns":375},[377,55800,55801,55809],{},[380,55802,55803,55805,55807],{},[383,55804,554],{},[458,55806,471],{},[383,55808,561],{},[387,55810,10619],{"encoding":389},[363,55812,55814,55832],{"className":55813,"ariaHidden":394},[393],[363,55815,55817,55820,55823,55826,55829],{"className":55816},[398],[363,55818],{"className":55819,"style":1011},[402],[363,55821,554],{"className":55822,"style":682},[407,408],[363,55824],{"className":55825,"style":506},[505],[363,55827,471],{"className":55828},[510],[363,55830],{"className":55831,"style":506},[505],[363,55833,55835,55838],{"className":55834},[398],[363,55836],{"className":55837,"style":913},[402],[363,55839,561],{"className":55840},[407,408],[11,55842,55843,55846,55847,55876,55877,55905,55906,4201,56035,56063,56064,56114,56115,56143,56144,56312,56313,56405,56406,56456,56457,56507],{},[1344,55844,55845],{},"证明。"," Jensen 已给出 ",[363,55848,55850,55864],{"className":55849,"translate":367},[366],[363,55851,55853],{"className":55852},[371],[373,55854,55855],{"xmlns":375},[377,55856,55857,55861],{},[380,55858,55859],{},[458,55860,7276],{},[387,55862,55863],{"encoding":389},"\\ge",[363,55865,55867],{"className":55866,"ariaHidden":394},[393],[363,55868,55870,55873],{"className":55869},[398],[363,55871],{"className":55872,"style":43560},[402],[363,55874,7276],{"className":55875},[510],"（见 2.1）。等号条件：若 ",[363,55878,55880,55893],{"className":55879,"translate":367},[366],[363,55881,55883],{"className":55882},[371],[373,55884,55885],{"xmlns":375},[377,55886,55887,55891],{},[380,55888,55889],{},[383,55890,385],{},[387,55892,385],{"encoding":389},[363,55894,55896],{"className":55895,"ariaHidden":394},[393],[363,55897,55899,55902],{"className":55898},[398],[363,55900],{"className":55901,"style":403},[402],[363,55903,385],{"className":55904,"style":409},[407,408]," 严格凸，Jensen 取等号当且仅当 ",[363,55907,55909,55941],{"className":55908,"translate":367},[366],[363,55910,55912],{"className":55911},[371],[373,55913,55914],{"xmlns":375},[377,55915,55916,55938],{},[380,55917,55918,55920,55922,55928,55930,55932,55934,55936],{},[383,55919,1116],{},[458,55921,471],{},[543,55923,55924,55926],{},[383,55925,571],{"mathvariant":570},[383,55927,561],{},[458,55929,578],{"stretchy":460},[383,55931,1116],{},[458,55933,605],{"stretchy":460},[458,55935,471],{},[463,55937,465],{},[387,55939,55940],{"encoding":389},"t=\\mathbb E_Q[t]=1",[363,55942,55944,55962,56026],{"className":55943,"ariaHidden":394},[393],[363,55945,55947,55950,55953,55956,55959],{"className":55946},[398],[363,55948],{"className":55949,"style":5691},[402],[363,55951,1116],{"className":55952},[407,408],[363,55954],{"className":55955,"style":506},[505],[363,55957,471],{"className":55958},[510],[363,55960],{"className":55961,"style":506},[505],[363,55963,55965,55968,56008,56011,56014,56017,56020,56023],{"className":55964},[398],[363,55966],{"className":55967,"style":618},[402],[363,55969,55971,55974],{"className":55970},[407],[363,55972,571],{"className":55973},[407,714],[363,55975,55977],{"className":55976},[629],[363,55978,55980,56000],{"className":55979},[633,634],[363,55981,55983,55997],{"className":55982},[638],[363,55984,55986],{"className":55985,"style":727},[642],[363,55987,55988,55991],{"style":730},[363,55989],{"className":55990,"style":651},[650],[363,55992,55994],{"className":55993},[655,656,657,658],[363,55995,561],{"className":55996},[407,408,658],[363,55998,666],{"className":55999},[665],[363,56001,56003],{"className":56002},[638],[363,56004,56006],{"className":56005,"style":673},[642],[363,56007],{},[363,56009,578],{"className":56010},[494],[363,56012,1116],{"className":56013},[407,408],[363,56015,605],{"className":56016},[501],[363,56018],{"className":56019,"style":506},[505],[363,56021,471],{"className":56022},[510],[363,56024],{"className":56025,"style":506},[505],[363,56027,56029,56032],{"className":56028},[398],[363,56030],{"className":56031,"style":520},[402],[363,56033,465],{"className":56034},[407],[363,56036,56038,56051],{"className":56037,"translate":367},[366],[363,56039,56041],{"className":56040},[371],[373,56042,56043],{"xmlns":375},[377,56044,56045,56049],{},[380,56046,56047],{},[383,56048,561],{},[387,56050,561],{"encoding":389},[363,56052,56054],{"className":56053,"ariaHidden":394},[393],[363,56055,56057,56060],{"className":56056},[398],[363,56058],{"className":56059,"style":913},[402],[363,56061,561],{"className":56062},[407,408]," 下几乎必然成立，即 ",[363,56065,56067,56084],{"className":56066,"translate":367},[366],[363,56068,56070],{"className":56069},[371],[373,56071,56072],{"xmlns":375},[377,56073,56074,56082],{},[380,56075,56076,56078,56080],{},[383,56077,554],{},[458,56079,471],{},[383,56081,561],{},[387,56083,10619],{"encoding":389},[363,56085,56087,56105],{"className":56086,"ariaHidden":394},[393],[363,56088,56090,56093,56096,56099,56102],{"className":56089},[398],[363,56091],{"className":56092,"style":1011},[402],[363,56094,554],{"className":56095,"style":682},[407,408],[363,56097],{"className":56098,"style":506},[505],[363,56100,471],{"className":56101},[510],[363,56103],{"className":56104,"style":506},[505],[363,56106,56108,56111],{"className":56107},[398],[363,56109],{"className":56110,"style":913},[402],[363,56112,561],{"className":56113},[407,408],"。若 ",[363,56116,56118,56131],{"className":56117,"translate":367},[366],[363,56119,56121],{"className":56120},[371],[373,56122,56123],{"xmlns":375},[377,56124,56125,56129],{},[380,56126,56127],{},[383,56128,385],{},[387,56130,385],{"encoding":389},[363,56132,56134],{"className":56133,"ariaHidden":394},[393],[363,56135,56137,56140],{"className":56136},[398],[363,56138],{"className":56139,"style":403},[402],[363,56141,385],{"className":56142,"style":409},[407,408]," 只是凸而非严格凸（如 ",[363,56145,56147,56184],{"className":56146,"translate":367},[366],[363,56148,56150],{"className":56149},[371],[373,56151,56152],{"xmlns":375},[377,56153,56154,56182],{},[380,56155,56156,56158,56160,56162,56164,56166,56172,56174,56176,56178,56180],{},[383,56157,385],{},[458,56159,461],{"stretchy":460},[383,56161,1116],{},[458,56163,468],{"stretchy":460},[458,56165,471],{},[586,56167,56168,56170],{},[463,56169,465],{},[463,56171,2811],{},[383,56173,5156],{"mathvariant":557},[383,56175,1116],{},[458,56177,5161],{},[463,56179,465],{},[383,56181,5156],{"mathvariant":557},[387,56183,34928],{"encoding":389},[363,56185,56187,56214,56303],{"className":56186,"ariaHidden":394},[393],[363,56188,56190,56193,56196,56199,56202,56205,56208,56211],{"className":56189},[398],[363,56191],{"className":56192,"style":487},[402],[363,56194,385],{"className":56195,"style":409},[407,408],[363,56197,461],{"className":56198},[494],[363,56200,1116],{"className":56201},[407,408],[363,56203,468],{"className":56204},[501],[363,56206],{"className":56207,"style":506},[505],[363,56209,471],{"className":56210},[510],[363,56212],{"className":56213,"style":506},[505],[363,56215,56217,56220,56288,56291,56294,56297,56300],{"className":56216},[398],[363,56218],{"className":56219,"style":2853},[402],[363,56221,56223,56226,56285],{"className":56222},[407],[363,56224],{"className":56225},[494,789],[363,56227,56229],{"className":56228},[586],[363,56230,56232,56277],{"className":56231},[633,634],[363,56233,56235,56274],{"className":56234},[638],[363,56236,56238,56252,56260],{"className":56237,"style":2872},[642],[363,56239,56240,56243],{"style":2875},[363,56241],{"className":56242,"style":809},[650],[363,56244,56246],{"className":56245},[655,656,657,658],[363,56247,56249],{"className":56248},[407,658],[363,56250,2811],{"className":56251},[407,658],[363,56253,56254,56257],{"style":821},[363,56255],{"className":56256,"style":809},[650],[363,56258],{"className":56259,"style":829},[828],[363,56261,56262,56265],{"style":2898},[363,56263],{"className":56264,"style":809},[650],[363,56266,56268],{"className":56267},[655,656,657,658],[363,56269,56271],{"className":56270},[407,658],[363,56272,465],{"className":56273},[407,658],[363,56275,666],{"className":56276},[665],[363,56278,56280],{"className":56279},[638],[363,56281,56283],{"className":56282,"style":2920},[642],[363,56284],{},[363,56286],{"className":56287},[501,789],[363,56289,5156],{"className":56290},[407],[363,56292,1116],{"className":56293},[407,408],[363,56295],{"className":56296,"style":2935},[505],[363,56298,5161],{"className":56299},[2939],[363,56301],{"className":56302,"style":2935},[505],[363,56304,56306,56309],{"className":56305},[398],[363,56307],{"className":56308,"style":487},[402],[363,56310,35051],{"className":56311},[407],"），则只能保证 ",[363,56314,56316,56338],{"className":56315,"translate":367},[366],[363,56317,56319],{"className":56318},[371],[373,56320,56321],{"xmlns":375},[377,56322,56323,56335],{},[380,56324,56325,56331,56333],{},[543,56326,56327,56329],{},[383,56328,547],{},[383,56330,385],{},[458,56332,471],{},[463,56334,474],{},[387,56336,56337],{"encoding":389},"D_f=0",[363,56339,56341,56396],{"className":56340,"ariaHidden":394},[393],[363,56342,56344,56347,56387,56390,56393],{"className":56343},[398],[363,56345],{"className":56346,"style":4380},[402],[363,56348,56350,56353],{"className":56349},[407],[363,56351,547],{"className":56352,"style":625},[407,408],[363,56354,56356],{"className":56355},[629],[363,56357,56359,56379],{"className":56358},[633,634],[363,56360,56362,56376],{"className":56361},[638],[363,56363,56365],{"className":56364,"style":643},[642],[363,56366,56367,56370],{"style":646},[363,56368],{"className":56369,"style":651},[650],[363,56371,56373],{"className":56372},[655,656,657,658],[363,56374,385],{"className":56375,"style":409},[407,408,658],[363,56377,666],{"className":56378},[665],[363,56380,56382],{"className":56381},[638],[363,56383,56385],{"className":56384,"style":673},[642],[363,56386],{},[363,56388],{"className":56389,"style":506},[505],[363,56391,471],{"className":56392},[510],[363,56394],{"className":56395,"style":506},[505],[363,56397,56399,56402],{"className":56398},[398],[363,56400],{"className":56401,"style":520},[402],[363,56403,474],{"className":56404},[407]," 时不必然 ",[363,56407,56409,56426],{"className":56408,"translate":367},[366],[363,56410,56412],{"className":56411},[371],[373,56413,56414],{"xmlns":375},[377,56415,56416,56424],{},[380,56417,56418,56420,56422],{},[383,56419,554],{},[458,56421,471],{},[383,56423,561],{},[387,56425,10619],{"encoding":389},[363,56427,56429,56447],{"className":56428,"ariaHidden":394},[393],[363,56430,56432,56435,56438,56441,56444],{"className":56431},[398],[363,56433],{"className":56434,"style":1011},[402],[363,56436,554],{"className":56437,"style":682},[407,408],[363,56439],{"className":56440,"style":506},[505],[363,56442,471],{"className":56443},[510],[363,56445],{"className":56446,"style":506},[505],[363,56448,56450,56453],{"className":56449},[398],[363,56451],{"className":56452,"style":913},[402],[363,56454,561],{"className":56455},[407,408],"：TV 是反例，",[363,56458,56460,56477],{"className":56459,"translate":367},[366],[363,56461,56463],{"className":56462},[371],[373,56464,56465],{"xmlns":375},[377,56466,56467,56475],{},[380,56468,56469,56471,56473],{},[383,56470,554],{},[458,56472,471],{},[383,56474,561],{},[387,56476,10619],{"encoding":389},[363,56478,56480,56498],{"className":56479,"ariaHidden":394},[393],[363,56481,56483,56486,56489,56492,56495],{"className":56482},[398],[363,56484],{"className":56485,"style":1011},[402],[363,56487,554],{"className":56488,"style":682},[407,408],[363,56490],{"className":56491,"style":506},[505],[363,56493,471],{"className":56494},[510],[363,56496],{"className":56497,"style":506},[505],[363,56499,56501,56504],{"className":56500},[398],[363,56502],{"className":56503,"style":913},[402],[363,56505,561],{"className":56506},[407,408]," 时才是唯一使 TV 为 0 的一对（TV 也对）。",[2659,56509,56511],{"id":56510},"_42-联合凸性","4.2 联合凸性",[143,56513,56514,56986],{},[11,56515,56516,56519,56520,3093,56716,3093,56912,42010],{},[1344,56517,56518],{},"命题 4.2（联合凸性）"," 若 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节）直接看：",[363,57538,57540,57620],{"className":57539,"translate":367},[366],[363,57541,57543],{"className":57542},[371],[373,57544,57545],{"xmlns":375},[377,57546,57547,57617],{},[380,57548,57549,57555,57557,57559,57561,57563,57565,57567,57577,57579,57585,57587,57589,57591,57593,57599,57601,57607,57609,57611,57613,57615],{},[543,57550,57551,57553],{},[383,57552,547],{},[383,57554,385],{},[458,57556,461],{"stretchy":460},[383,57558,554],{},[383,57560,558],{"mathvariant":557},[383,57562,561],{},[458,57564,468],{"stretchy":460},[458,57566,471],{},[543,57568,57569,57575],{},[380,57570,57571,57573],{},[383,57572,35804],{},[458,57574,4451],{},[383,57576,4506],{},[458,57578,3246],{"stretchy":460},[543,57580,57581,57583],{},[383,57582,571],{"mathvariant":570},[383,57584,554],{},[458,57586,578],{"stretchy":460},[383,57588,4506],{},[458,57590,605],{"stretchy":460},[458,57592,5161],{},[543,57594,57595,57597],{},[383,57596,571],{"mathvariant":570},[383,57598,561],{},[458,57600,578],{"stretchy":460},[7162,57602,57603,57605],{},[383,57604,385],{},[458,57606,7921],{},[458,57608,461],{"stretchy":460},[383,57610,4506],{},[458,57612,468],{"stretchy":460},[458,57614,605],{"stretchy":460},[458,57616,3256],{"stretchy":460},[387,57618,57619],{"encoding":389},"D_f(P\\|Q)=\\sup_g\\{\\mathbb E_P[g]-\\mathbb E_Q[f^*(g)]\\}",[363,57621,57623,57693,57802],{"className":57622,"ariaHidden":394},[393],[363,57624,57626,57629,57669,57672,57675,57678,57681,57684,57687,57690],{"className":57625},[398],[363,57627],{"className":57628,"style":618},[402],[363,57630,57632,57635],{"className":57631},[407],[363,57633,547],{"className":57634,"style":625},[407,408],[363,57636,57638],{"className":57637},[629],[363,57639,57641,57661],{"className":57640},[633,634],[363,57642,57644,57658],{"className":57643},[638],[363,57645,57647],{"className":57646,"style":643},[642],[363,57648,57649,57652],{"style":646},[363,57650],{"className":57651,"style":651},[650],[363,57653,57655],{"className":57654},[655,656,657,658],[363,57656,385],{"className":57657,"style":409},[407,408,658],[363,57659,666],{"className":57660},[665],[363,57662,57664],{"className":57663},[638],[363,57665,57667],{"className":57666,"style":673},[642],[363,57668],{},[363,57670,461],{"className":57671},[494],[363,57673,554],{"className":57674,"style":682},[407,408],[363,57676,558],{"className":57677},[407],[363,57679,561],{"className":57680},[407,408],[363,57682,468],{"className":57683},[501],[363,57685],{"className":57686,"style":506},[505],[363,57688,471],{"className":57689},[510],[363,57691],{"className":57692,"style":506},[505],[363,57694,57696,57700,57741,57744,57784,57787,57790,57793,57796,57799],{"className":57695},[398],[363,57697],{"className":57698,"style":57699},[402],"height:1.1302em;vertical-align:-0.3802em;",[363,57701,57703,57706],{"className":57702},[1832],[363,57704,35804],{"className":57705},[1832],[363,57707,57709],{"className":57708},[629],[363,57710,57712,57732],{"className":57711},[633,634],[363,57713,57715,57729],{"className":57714},[638],[363,57716,57718],{"className":57717,"style":37371},[642],[363,57719,57720,57723],{"style":35950},[363,57721],{"className":57722,"style":651},[650],[363,57724,57726],{"className":57725},[655,656,657,658],[363,57727,4506],{"className":57728,"style":1328},[407,408,658],[363,57730,666],{"className":57731},[665],[363,57733,57735],{"className":57734},[638],[363,57736,57739],{"className":57737,"style":57738},[642],"height:0.3802em;",[363,57740],{},[363,57742,3246],{"className":57743},[494],[363,57745,57747,57750],{"className":57746},[407],[363,57748,571],{"className":57749},[407,714],[363,57751,57753],{"className":57752},[629],[363,57754,57756,57776],{"className":57755},[633,634],[363,57757,57759,57773],{"className":57758},[638],[363,57760,57762],{"className":57761,"style":727},[642],[363,57763,57764,57767],{"style":730},[363,57765],{"className":57766,"style":651},[650],[363,57768,57770],{"className":57769},[655,656,657,658],[363,57771,554],{"className":57772,"style":682},[407,408,658],[363,57774,666],{"className":57775},[665],[363,57777,57779],{"className":57778},[638],[363,57780,57782],{"className":57781,"style":8316},[642],[363,57783],{},[363,57785,578],{"className":57786},[494],[363,57788,4506],{"className":57789,"style":1328},[407,408],[363,57791,605],{"className":57792},[501],[363,57794],{"className":57795,"style":2935},[505],[363,57797,5161],{"className":57798},[2939],[363,57800],{"className":57801,"style":2935},[505],[363,57803,57805,57808,57848,57851,57880,57883,57886],{"className":57804},[398],[363,57806],{"className":57807,"style":618},[402],[363,57809,57811,57814],{"className":57810},[407],[363,57812,571],{"className":57813},[407,714],[363,57815,57817],{"className":57816},[629],[363,57818,57820,57840],{"className":57819},[633,634],[363,57821,57823,57837],{"className":57822},[638],[363,57824,57826],{"className":57825,"style":727},[642],[363,57827,57828,57831],{"style":730},[363,57829],{"className":57830,"style":651},[650],[363,57832,57834],{"className":57833},[655,656,657,658],[363,57835,561],{"className":57836},[407,408,658],[363,57838,666],{"className":57839},[665],[363,57841,57843],{"className":57842},[638],[363,57844,57846],{"className":57845,"style":673},[642],[363,57847],{},[363,57849,578],{"className":57850},[494],[363,57852,57854,57857],{"className":57853},[407],[363,57855,385],{"className":57856,"style":409},[407,408],[363,57858,57860],{"className":57859},[629],[363,57861,57863],{"className":57862},[633],[363,57864,57866],{"className":57865},[638],[363,57867,57869],{"className":57868,"style":8847},[642],[363,57870,57871,57874],{"style":7213},[363,57872],{"className":57873,"style":651},[650],[363,57875,57877],{"className":57876},[655,656,657,658],[363,57878,7921],{"className":57879},[2939,658],[363,57881,461],{"className":57882},[494],[363,57884,4506],{"className":57885,"style":1328},[407,408],[363,57887,57889],{"className":57888},[501],")]}","，右端关于 ",[363,57892,57894,57915],{"className":57893,"translate":367},[366],[363,57895,57897],{"className":57896},[371],[373,57898,57899],{"xmlns":375},[377,57900,57901,57913],{},[380,57902,57903,57905,57907,57909,57911],{},[458,57904,461],{"stretchy":460},[383,57906,554],{},[458,57908,898],{"separator":394},[383,57910,561],{},[458,57912,468],{"stretchy":460},[387,57914,11381],{"encoding":389},[363,57916,57918],{"className":57917,"ariaHidden":394},[393],[363,57919,57921,57924,57927,57930,57933,57936,57939],{"className":57920},[398],[363,57922],{"className":57923,"style":487},[402],[363,57925,461],{"className":57926},[494],[363,57928,554],{"className":57929,"style":682},[407,408],[363,57931,898],{"className":57932},[920],[363,57934],{"className":57935,"style":754},[505],[363,57937,561],{"className":57938},[407,408],[363,57940,468],{"className":57941},[501]," 是凸的（",[363,57944,57946,57970],{"className":57945,"translate":367},[366],[363,57947,57949],{"className":57948},[371],[373,57950,57951],{"xmlns":375},[377,57952,57953,57967],{},[380,57954,57955,57961,57963,57965],{},[543,57956,57957,57959],{},[383,57958,571],{"mathvariant":570},[383,57960,554],{},[458,57962,578],{"stretchy":460},[383,57964,4506],{},[458,57966,605],{"stretchy":460},[387,57968,57969],{"encoding":389},"\\mathbb E_P[g]",[363,57971,57973],{"className":57972,"ariaHidden":394},[393],[363,57974,57976,57979,58019,58022,58025],{"className":57975},[398],[363,57977],{"className":57978,"style":487},[402],[363,57980,57982,57985],{"className":57981},[407],[363,57983,571],{"className":57984},[407,714],[363,57986,57988],{"className":57987},[629],[363,57989,57991,58011],{"className":57990},[633,634],[363,57992,57994,58008],{"className":57993},[638],[363,57995,57997],{"className":57996,"style":727},[642],[363,57998,57999,58002],{"style":730},[363,58000],{"className":58001,"style":651},[650],[363,58003,58005],{"className":58004},[655,656,657,658],[363,58006,554],{"className":58007,"style":682},[407,408,658],[363,58009,666],{"className":58010},[665],[363,58012,58014],{"className":58013},[638],[363,58015,58017],{"className":58016,"style":8316},[642],[363,58018],{},[363,58020,578],{"className":58021},[494],[363,58023,4506],{"className":58024,"style":1328},[407,408],[363,58026,605],{"className":58027},[501]," 线性、",[363,58030,58032,58068],{"className":58031,"translate":367},[366],[363,58033,58035],{"className":58034},[371],[373,58036,58037],{"xmlns":375},[377,58038,58039,58065],{},[380,58040,58041,58043,58049,58051,58057,58059,58061,58063],{},[458,58042,5161],{},[543,58044,58045,58047],{},[383,58046,571],{"mathvariant":570},[383,58048,561],{},[458,58050,578],{"stretchy":460},[7162,58052,58053,58055],{},[383,58054,385],{},[458,58056,7921],{},[458,58058,461],{"stretchy":460},[383,58060,4506],{},[458,58062,468],{"stretchy":460},[458,58064,605],{"stretchy":460},[387,58066,58067],{"encoding":389},"-\\mathbb E_Q[f^*(g)]",[363,58069,58071],{"className":58070,"ariaHidden":394},[393],[363,58072,58074,58077,58080,58120,58123,58152,58155,58158],{"className":58073},[398],[363,58075],{"className":58076,"style":618},[402],[363,58078,5161],{"className":58079},[407],[363,58081,58083,58086],{"className":58082},[407],[363,58084,571],{"className":58085},[407,714],[363,58087,58089],{"className":58088},[629],[363,58090,58092,58112],{"className":58091},[633,634],[363,58093,58095,58109],{"className":58094},[638],[363,58096,58098],{"className":58097,"style":727},[642],[363,58099,58100,58103],{"style":730},[363,58101],{"className":58102,"style":651},[650],[363,58104,58106],{"className":58105},[655,656,657,658],[363,58107,561],{"className":58108},[407,408,658],[363,58110,666],{"className":58111},[665],[363,58113,58115],{"className":58114},[638],[363,58116,58118],{"className":58117,"style":673},[642],[363,58119],{},[363,58121,578],{"className":58122},[494],[363,58124,58126,58129],{"className":58125},[407],[363,58127,385],{"className":58128,"style":409},[407,408],[363,58130,58132],{"className":58131},[629],[363,58133,58135],{"className":58134},[633],[363,58136,58138],{"className":58137},[638],[363,58139,58141],{"className":58140,"style":8847},[642],[363,58142,58143,58146],{"style":7213},[363,58144],{"className":58145,"style":651},[650],[363,58147,58149],{"className":58148},[655,656,657,658],[363,58150,7921],{"className":58151},[2939,658],[363,58153,461],{"className":58154},[494],[363,58156,4506],{"className":58157,"style":1328},[407,408],[363,58159,20762],{"className":58160},[501]," 线性），取上确界后仍凸。",[11,58163,58164,58165,3093,58193,58311,58312,3093,58340,11315,58457,9748,58460,58576,58577,58605,58606,58634,58635,58663],{},"等价地，固定 ",[363,58166,58168,58181],{"className":58167,"translate":367},[366],[363,58169,58171],{"className":58170},[371],[373,58172,58173],{"xmlns":375},[377,58174,58175,58179],{},[380,58176,58177],{},[383,58178,561],{},[387,58180,561],{"encoding":389},[363,58182,58184],{"className":58183,"ariaHidden":394},[393],[363,58185,58187,58190],{"className":58186},[398],[363,58188],{"className":58189,"style":913},[402],[363,58191,561],{"className":58192},[407,408],[363,58194,58196,58228],{"className":58195,"translate":367},[366],[363,58197,58199],{"className":58198},[371],[373,58200,58201],{"xmlns":375},[377,58202,58203,58225],{},[380,58204,58205,58207,58209,58215,58217,58219,58221,58223],{},[383,58206,554],{},[458,58208,9032],{},[543,58210,58211,58213],{},[383,58212,547],{},[383,58214,385],{},[458,58216,461],{"stretchy":460},[383,58218,554],{},[383,58220,558],{"mathvariant":557},[383,58222,561],{},[458,58224,468],{"stretchy":460},[387,58226,58227],{"encoding":389},"P\\mapsto D_f(P\\|Q)",[363,58229,58231,58250],{"className":58230,"ariaHidden":394},[393],[363,58232,58234,58238,58241,58244,58247],{"className":58233},[398],[363,58235],{"className":58236,"style":58237},[402],"height:0.6943em;vertical-align:-0.011em;",[363,58239,554],{"className":58240,"style":682},[407,408],[363,58242],{"className":58243,"style":506},[505],[363,58245,9032],{"className":58246},[510],[363,58248],{"className":58249,"style":506},[505],[363,58251,58253,58256,58296,58299,58302,58305,58308],{"className":58252},[398],[363,58254],{"className":58255,"style":618},[402],[363,58257,58259,58262],{"className":58258},[407],[363,58260,547],{"className":58261,"style":625},[407,408],[363,58263,58265],{"className":58264},[629],[363,58266,58268,58288],{"className":58267},[633,634],[363,58269,58271,58285],{"className":58270},[638],[363,58272,58274],{"className":58273,"style":643},[642],[363,58275,58276,58279],{"style":646},[363,58277],{"className":58278,"style":651},[650],[363,58280,58282],{"className":58281},[655,656,657,658],[363,58283,385],{"className":58284,"style":409},[407,408,658],[363,58286,666],{"className":58287},[665],[363,58289,58291],{"className":58290},[638],[363,58292,58294],{"className":58293,"style":673},[642],[363,58295],{},[363,58297,461],{"className":58298},[494],[363,58300,554],{"className":58301,"style":682},[407,408],[363,58303,558],{"className":58304},[407],[363,58306,561],{"className":58307},[407,408],[363,58309,468],{"className":58310},[501]," 凸；固定 ",[363,58313,58315,58328],{"className":58314,"translate":367},[366],[363,58316,58318],{"className":58317},[371],[373,58319,58320],{"xmlns":375},[377,58321,58322,58326],{},[380,58323,58324],{},[383,58325,554],{},[387,58327,554],{"encoding":389},[363,58329,58331],{"className":58330,"ariaHidden":394},[393],[363,58332,58334,58337],{"className":58333},[398],[363,58335],{"className":58336,"style":1011},[402],[363,58338,554],{"className":58339,"style":682},[407,408],[363,58341,58343,58375],{"className":58342,"translate":367},[366],[363,58344,58346],{"className":58345},[371],[373,58347,58348],{"xmlns":375},[377,58349,58350,58372],{},[380,58351,58352,58354,58356,58362,58364,58366,58368,58370],{},[383,58353,561],{},[458,58355,9032],{},[543,58357,58358,58360],{},[383,58359,547],{},[383,58361,385],{},[458,58363,461],{"stretchy":460},[383,58365,554],{},[383,58367,558],{"mathvariant":557},[383,58369,561],{},[458,58371,468],{"stretchy":460},[387,58373,58374],{"encoding":389},"Q\\mapsto D_f(P\\|Q)",[363,58376,58378,58396],{"className":58377,"ariaHidden":394},[393],[363,58379,58381,58384,58387,58390,58393],{"className":58380},[398],[363,58382],{"className":58383,"style":913},[402],[363,58385,561],{"className":58386},[407,408],[363,58388],{"className":58389,"style":506},[505],[363,58391,9032],{"className":58392},[510],[363,58394],{"className":58395,"style":506},[505],[363,58397,58399,58402,58442,58445,58448,58451,58454],{"className":58398},[398],[363,58400],{"className":58401,"style":618},[402],[363,58403,58405,58408],{"className":58404},[407],[363,58406,547],{"className":58407,"style":625},[407,408],[363,58409,58411],{"className":58410},[629],[363,58412,58414,58434],{"className":58413},[633,634],[363,58415,58417,58431],{"className":58416},[638],[363,58418,58420],{"className":58419,"style":643},[642],[363,58421,58422,58425],{"style":646},[363,58423],{"className":58424,"style":651},[650],[363,58426,58428],{"className":58427},[655,656,657,658],[363,58429,385],{"className":58430,"style":409},[407,408,658],[363,58432,666],{"className":58433},[665],[363,58435,58437],{"className":58436},[638],[363,58438,58440],{"className":58439,"style":673},[642],[363,58441],{},[363,58443,461],{"className":58444},[494],[363,58446,554],{"className":58447,"style":682},[407,408],[363,58449,558],{"className":58450},[407],[363,58452,561],{"className":58453},[407,408],[363,58455,468],{"className":58456},[501],[1344,58458,58459],{},"注意",[363,58461,58463,58494],{"className":58462,"translate":367},[366],[363,58464,58466],{"className":58465},[371],[373,58467,58468],{"xmlns":375},[377,58469,58470,58492],{},[380,58471,58472,58474,58476,58482,58484,58486,58488,58490],{},[383,58473,561],{},[458,58475,9032],{},[543,58477,58478,58480],{},[383,58479,547],{},[383,58481,385],{},[458,58483,461],{"stretchy":460},[383,58485,554],{},[383,58487,558],{"mathvariant":557},[383,58489,561],{},[458,58491,468],{"stretchy":460},[387,58493,58374],{"encoding":389},[363,58495,58497,58515],{"className":58496,"ariaHidden":394},[393],[363,58498,58500,58503,58506,58509,58512],{"className":58499},[398],[363,58501],{"className":58502,"style":913},[402],[363,58504,561],{"className":58505},[407,408],[363,58507],{"className":58508,"style":506},[505],[363,58510,9032],{"className":58511},[510],[363,58513],{"className":58514,"style":506},[505],[363,58516,58518,58521,58561,58564,58567,58570,58573],{"className":58517},[398],[363,58519],{"className":58520,"style":618},[402],[363,58522,58524,58527],{"className":58523},[407],[363,58525,547],{"className":58526,"style":625},[407,408],[363,58528,58530],{"className":58529},[629],[363,58531,58533,58553],{"className":58532},[633,634],[363,58534,58536,58550],{"className":58535},[638],[363,58537,58539],{"className":58538,"style":643},[642],[363,58540,58541,58544],{"style":646},[363,58542],{"className":58543,"style":651},[650],[363,58545,58547],{"className":58546},[655,656,657,658],[363,58548,385],{"className":58549,"style":409},[407,408,658],[363,58551,666],{"className":58552},[665],[363,58554,58556],{"className":58555},[638],[363,58557,58559],{"className":58558,"style":673},[642],[363,58560],{},[363,58562,461],{"className":58563},[494],[363,58565,554],{"className":58566,"style":682},[407,408],[363,58568,558],{"className":58569},[407],[363,58571,561],{"className":58572},[407,408],[363,58574,468],{"className":58575},[501]," 凸是「以 ",[363,58578,58580,58593],{"className":58579,"translate":367},[366],[363,58581,58583],{"className":58582},[371],[373,58584,58585],{"xmlns":375},[377,58586,58587,58591],{},[380,58588,58589],{},[383,58590,561],{},[387,58592,561],{"encoding":389},[363,58594,58596],{"className":58595,"ariaHidden":394},[393],[363,58597,58599,58602],{"className":58598},[398],[363,58600],{"className":58601,"style":913},[402],[363,58603,561],{"className":58604},[407,408]," 为参照」的凸性，它与「",[363,58607,58609,58622],{"className":58608,"translate":367},[366],[363,58610,58612],{"className":58611},[371],[373,58613,58614],{"xmlns":375},[377,58615,58616,58620],{},[380,58617,58618],{},[383,58619,561],{},[387,58621,561],{"encoding":389},[363,58623,58625],{"className":58624,"ariaHidden":394},[393],[363,58626,58628,58631],{"className":58627},[398],[363,58629],{"className":58630,"style":913},[402],[363,58632,561],{"className":58633},[407,408]," 出现在分母上」的直觉相反，这个凸性正是变分推断里「对 ",[363,58636,58638,58651],{"className":58637,"translate":367},[366],[363,58639,58641],{"className":58640},[371],[373,58642,58643],{"xmlns":375},[377,58644,58645,58649],{},[380,58646,58647],{},[383,58648,1291],{},[387,58650,1291],{"encoding":389},[363,58652,58654],{"className":58653,"ariaHidden":394},[393],[363,58655,58657,58660],{"className":58656},[398],[363,58658],{"className":58659,"style":2624},[402],[363,58661,1291],{"className":58662,"style":1328},[407,408]," 做平均场优化收敛到全局最优」的根据之一。",[2659,58665,58667],{"id":58666},"_43-斜对称关系","4.3 斜对称关系",[143,58669,58670,58772,58990],{},[11,58671,58672,58675,58676,58771],{},[1344,58673,58674],{},"定义 4.1（斜共轭）"," 对 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t>0",[363,58841,58843,58933,58981],{"className":58842,"ariaHidden":394},[393],[363,58844,58846,58849,58915,58918,58921,58924,58927,58930],{"className":58845},[398],[363,58847],{"className":58848,"style":487},[402],[363,58850,58852,58855],{"className":58851},[407],[363,58853,385],{"className":58854,"style":409},[407,408],[363,58856,58858],{"className":58857},[629],[363,58859,58861,58907],{"className":58860},[633,634],[363,58862,58864,58904],{"className":58863},[638],[363,58865,58867,58890],{"className":58866,"style":8596},[642],[363,58868,58869,58872],{"style":8599},[363,58870],{"className":58871,"style":651},[650],[363,58873,58875],{"className":58874},[655,656,657,658],[363,58876,58878,58881,58884,58887],{"className":58877},[407,658],[363,58879,7909],{"className":58880},[407,408,658],[363,58882,7912],{"className":58883,"style":8615},[407,408,658],[363,58885,7915],{"className":58886},[407,408,658],[363,58888,7918],{"className":58889,"style":8622},[407,408,658],[363,58891,58892,58895],{"style":8625},[363,58893],{"className":58894,"style":651},[650],[363,58896,58898],{"className":58897},[655,656,657,658],[363,58899,58901],{"className":58900},[407,658],[363,58902,7921],{"className":58903},[407,658],[363,58905,666],{"className":58906},[665],[363,58908,58910],{"className":58909},[638],[363,58911,58913],{"className":58912,"style":8647},[642],[363,58914],{},[363,58916,461],{"className":58917},[494],[363,58919,1116],{"className":58920},[407,408],[363,58922,468],{"className":58923},[501],[363,58925],{"className":58926,"style":506},[505],[363,58928,471],{"className":58929},[510],[363,58931],{"className":58932,"style":506},[505],[363,58934,58936,58939,58942,58945,58948,58951,58954,58957,58960,58963,58966,58969,58972,58975,58978],{"className":58935},[398],[363,58937],{"className":58938,"style":487},[402],[363,58940,1116],{"className":58941},[407,408],[363,58943],{"className":58944,"style":754},[505],[363,58946,385],{"className":58947,"style":409},[407,408],[363,58949,461],{"className":58950},[494],[363,58952,8937],{"className":58953},[407],[363,58955,1116],{"className":58956},[407,408],[363,58958,468],{"className":58959},[501],[363,58961,898],{"className":58962},[920],[363,58964],{"className":58965,"style":3982},[505],[363,58967],{"className":58968,"style":754},[505],[363,58970,1116],{"className":58971},[407,408],[363,58973],{"className":58974,"style":506},[505],[363,58976,3251],{"className":58977},[510],[363,58979],{"className":58980,"style":506},[505],[363,58982,58984,58987],{"className":58983},[398],[363,58985],{"className":58986,"style":520},[402],[363,58988,474],{"className":58989},[407],[11,58991,58992,58993,34197,59021,10528],{},"为 ",[363,58994,58996,59009],{"className":58995,"translate":367},[366],[363,58997,58999],{"className":58998},[371],[373,59000,59001],{"xmlns":375},[377,59002,59003,59007],{},[380,59004,59005],{},[383,59006,385],{},[387,59008,385],{"encoding":389},[363,59010,59012],{"className":59011,"ariaHidden":394},[393],[363,59013,59015,59018],{"className":59014},[398],[363,59016],{"className":59017,"style":403},[402],[363,59019,385],{"className":59020,"style":409},[407,408],[1344,59022,36160],{},[143,59024,59025,59072],{},[11,59026,59027,59030,59031,3093],{},[1344,59028,59029],{},"命题 4.3（斜对称）"," 对任意概率分布 ",[363,59032,59034,59051],{"className":59033,"translate":367},[366],[363,59035,59037],{"className":59036},[371],[373,59038,59039],{"xmlns":375},[377,59040,59041,59049],{},[380,59042,59043,59045,59047],{},[383,59044,554],{},[458,59046,898],{"separator":394},[383,59048,561],{},[387,59050,903],{"encoding":389},[363,59052,59054],{"className":59053,"ariaHidden":394},[393],[363,59055,59057,59060,59063,59066,59069],{"className":59056},[398],[363,59058],{"className":59059,"style":913},[402],[363,59061,554],{"className":59062,"style":682},[407,408],[363,59064,898],{"className":59065},[920],[363,59067],{"className":59068,"style":754},[505],[363,59070,561],{"className":59071},[407,408],[363,59073,59075],{"className":59074,"translate":367},[528],[363,59076,59078,59140],{"className":59077,"translate":367},[366],[363,59079,59081],{"className":59080},[371],[373,59082,59083],{"xmlns":375,"display":537},[377,59084,59085,59137],{},[380,59086,59087,59093,59095,59097,59099,59101,59103,59105,59125,59127,59129,59131,59133,59135],{},[543,59088,59089,59091],{},[383,59090,547],{},[383,59092,385],{},[458,59094,461],{"stretchy":460},[383,59096,554],{},[383,59098,558],{"mathvariant":557},[383,59100,561],{},[458,59102,468],{"stretchy":460},[458,59104,471],{},[543,59106,59107,59109],{},[383,59108,547],{},[7901,59110,59111,59113,59123],{},[383,59112,385],{},[380,59114,59115,59117,59119,59121],{},[383,59116,7909],{},[383,59118,7912],{},[383,59120,7915],{},[383,59122,7918],{},[458,59124,7921],{"lspace":1595,"rspace":1595},[458,59126,461],{"stretchy":460},[383,59128,561],{},[383,59130,558],{"mathvariant":557},[383,59132,554],{},[458,59134,468],{"stretchy":460},[383,59136,4723],{"mathvariant":557},[387,59138,59139],{"encoding":389},"D_f(P\\|Q)=D_{f^{*}_{skew}}(Q\\|P).",[363,59141,59143,59213],{"className":59142,"ariaHidden":394},[393],[363,59144,59146,59149,59189,59192,59195,59198,59201,59204,59207,59210],{"className":59145},[398],[363,59147],{"className":59148,"style":618},[402],[363,59150,59152,59155],{"className":59151},[407],[363,59153,547],{"className":59154,"style":625},[407,408],[363,59156,59158],{"className":59157},[629],[363,59159,59161,59181],{"className":59160},[633,634],[363,59162,59164,59178],{"className":59163},[638],[363,59165,59167],{"className":59166,"style":643},[642],[363,59168,59169,59172],{"style":646},[363,59170],{"className":59171,"style":651},[650],[363,59173,59175],{"className":59174},[655,656,657,658],[363,59176,385],{"className":59177,"style":409},[407,408,658],[363,59179,666],{"className":59180},[665],[363,59182,59184],{"className":59183},[638],[363,59185,59187],{"className":59186,"style":673},[642],[363,59188],{},[363,59190,461],{"className":59191},[494],[363,59193,554],{"className":59194,"style":682},[407,408],[363,59196,558],{"className":59197},[407],[363,59199,561],{"className":59200},[407,408],[363,59202,468],{"className":59203},[501],[363,59205],{"className":59206,"style":506},[505],[363,59208,471],{"className":59209},[510],[363,59211],{"className":59212,"style":506},[505],[363,59214,59216,59220,59331,59334,59337,59340,59343,59346],{"className":59215},[398],[363,59217],{"className":59218,"style":59219},[402],"height:1.1431em;vertical-align:-0.3931em;",[363,59221,59223,59226],{"className":59222},[407],[363,59224,547],{"className":59225,"style":625},[407,408],[363,59227,59229],{"className":59228},[629],[363,59230,59232,59322],{"className":59231},[633,634],[363,59233,59235,59319],{"className":59234},[638],[363,59236,59238],{"className":59237,"style":643},[642],[363,59239,59240,59243],{"style":646},[363,59241],{"className":59242,"style":651},[650],[363,59244,59246],{"className":59245},[655,656,657,658],[363,59247,59249],{"className":59248},[407,658],[363,59250,59252,59255],{"className":59251},[407,658],[363,59253,385],{"className":59254,"style":409},[407,408,658],[363,59256,59258],{"className":59257},[629],[363,59259,59261,59310],{"className":59260},[633,634],[363,59262,59264,59307],{"className":59263},[638],[363,59265,59268,59292],{"className":59266,"style":59267},[642],"height:0.6771em;",[363,59269,59271,59274],{"style":59270},"top:-2.1528em;margin-left:-0.1076em;margin-right:0.0714em;",[363,59272],{"className":59273,"style":30578},[650],[363,59275,59277],{"className":59276},[655,30582,10022,658],[363,59278,59280,59283,59286,59289],{"className":59279},[407,658],[363,59281,7909],{"className":59282},[407,408,658],[363,59284,7912],{"className":59285,"style":8615},[407,408,658],[363,59287,7915],{"className":59288},[407,408,658],[363,59290,7918],{"className":59291,"style":8622},[407,408,658],[363,59293,59295,59298],{"style":59294},"top:-2.8448em;margin-right:0.0714em;",[363,59296],{"className":59297,"style":30578},[650],[363,59299,59301],{"className":59300},[655,30582,10022,658],[363,59302,59304],{"className":59303},[407,658],[363,59305,7921],{"className":59306},[407,658],[363,59308,666],{"className":59309},[665],[363,59311,59313],{"className":59312},[638],[363,59314,59317],{"className":59315,"style":59316},[642],"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f(t)\\,dQ\n=\\int f\\!\\left(\\frac1s\\right) s\\,dP\\\\\n&=\\int 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对称（",[363,60263,60265,60311],{"className":60264,"translate":367},[366],[363,60266,60268],{"className":60267},[371],[373,60269,60270],{"xmlns":375},[377,60271,60272,60308],{},[380,60273,60274,60280,60282,60284,60286,60288,60290,60292,60298,60300,60302,60304,60306],{},[543,60275,60276,60278],{},[383,60277,547],{},[383,60279,385],{},[458,60281,461],{"stretchy":460},[383,60283,554],{},[383,60285,558],{"mathvariant":557},[383,60287,561],{},[458,60289,468],{"stretchy":460},[458,60291,471],{},[543,60293,60294,60296],{},[383,60295,547],{},[383,60297,385],{},[458,60299,461],{"stretchy":460},[383,60301,561],{},[383,60303,558],{"mathvariant":557},[383,60305,554],{},[458,60307,468],{"stretchy":460},[387,60309,60310],{"encoding":389},"D_f(P\\|Q)=D_f(Q\\|P)",[363,60312,60314,60384],{"className":60313,"ariaHidden":394},[393],[363,60315,60317,60320,60360,60363,60366,60369,60372,60375,60378,60381],{"className":60316},[398],[363,60318],{"className":60319,"style":618},[402],[363,60321,60323,60326],{"className":60322},[407],[363,60324,547],{"className":60325,"style":625},[407,408],[363,60327,60329],{"className":60328},[629],[363,60330,60332,60352],{"className":60331},[633,634],[363,60333,60335,60349],{"className":60334},[638],[363,60336,60338],{"className":60337,"style":643},[642],[363,60339,60340,60343],{"style":646},[363,60341],{"className":60342,"style":651},[650],[363,60344,60346],{"className":60345},[655,656,657,658],[363,60347,385],{"className":60348,"style":409},[407,408,658],[363,60350,666],{"className":60351},[665],[363,60353,60355],{"className":60354},[638],[363,60356,60358],{"className":60357,"style":673},[642],[363,60359],{},[363,60361,461],{"className":60362},[494],[363,60364,554],{"className":60365,"style":682},[407,408],[363,60367,558],{"className":60368},[407],[363,60370,561],{"className":60371},[407,408],[363,60373,468],{"className":60374},[501],[363,60376],{"className":60377,"style":506},[505],[363,60379,471],{"className":60380},[510],[363,60382],{"className":60383,"style":506},[505],[363,60385,60387,60390,60430,60433,60436,60439,60442],{"className":60386},[398],[363,60388],{"className":60389,"style":618},[402],[363,60391,60393,60396],{"className":60392},[407],[363,60394,547],{"className":60395,"style":625},[407,408],[363,60397,60399],{"className":60398},[629],[363,60400,60402,60422],{"className":60401},[633,634],[363,60403,60405,60419],{"className":60404},[638],[363,60406,60408],{"className":60407,"style":643},[642],[363,60409,60410,60413],{"style":646},[363,60411],{"className":60412,"style":651},[650],[363,60414,60416],{"className":60415},[655,656,657,658],[363,60417,385],{"className":60418,"style":409},[407,408,658],[363,60420,666],{"className":60421},[665],[363,60423,60425],{"className":60424},[638],[363,60426,60428],{"className":60427,"style":673},[642],[363,60429],{},[363,60431,461],{"className":60432},[494],[363,60434,561],{"className":60435},[407,408],[363,60437,558],{"className":60438},[407],[363,60440,554],{"className":60441,"style":682},[407,408],[363,60443,468],{"className":60444},[501]," 对所有 ",[363,60447,60449,60466],{"className":60448,"translate":367},[366],[363,60450,60452],{"className":60451},[371],[373,60453,60454],{"xmlns":375},[377,60455,60456,60464],{},[380,60457,60458,60460,60462],{},[383,60459,554],{},[458,60461,898],{"separator":394},[383,60463,561],{},[387,60465,903],{"encoding":389},[363,60467,60469],{"className":60468,"ariaHidden":394},[393],[363,60470,60472,60475,60478,60481,60484],{"className":60471},[398],[363,60473],{"className":60474,"style":913},[402],[363,60476,554],{"className":60477,"style":682},[407,408],[363,60479,898],{"className":60480},[920],[363,60482],{"className":60483,"style":754},[505],[363,60485,561],{"className":60486},[407,408],"）当且仅当 ",[363,60489,60491,60522],{"className":60490,"translate":367},[366],[363,60492,60494],{"className":60493},[371],[373,60495,60496],{"xmlns":375},[377,60497,60498,60520],{},[380,60499,60500,60516,60518],{},[7901,60501,60502,60504,60514],{},[383,60503,385],{},[380,60505,60506,60508,60510,60512],{},[383,60507,7909],{},[383,60509,7912],{},[383,60511,7915],{},[383,60513,7918],{},[458,60515,7921],{"lspace":1595,"rspace":1595},[458,60517,471],{},[383,60519,385],{},[387,60521,44525],{"encoding":389},[363,60523,60525,60606],{"className":60524,"ariaHidden":394},[393],[363,60526,60528,60531,60597,60600,60603],{"className":60527},[398],[363,60529],{"className":60530,"style":44535},[402],[363,60532,60534,60537],{"className":60533},[407],[363,60535,385],{"className":60536,"style":409},[407,408],[363,60538,60540],{"className":60539},[629],[363,60541,60543,60589],{"className":60542},[633,634],[363,60544,60546,60586],{"className":60545},[638],[363,60547,60549,60572],{"className":60548,"style":8847},[642],[363,60550,60551,60554],{"style":8850},[363,60552],{"className":60553,"style":651},[650],[363,60555,60557],{"className":60556},[655,656,657,658],[363,60558,60560,60563,60566,60569],{"className":60559},[407,658],[363,60561,7909],{"className":60562},[407,408,658],[363,60564,7912],{"className":60565,"style":8615},[407,408,658],[363,60567,7915],{"className":60568},[407,408,658],[363,60570,7918],{"className":60571,"style":8622},[407,408,658],[363,60573,60574,60577],{"style":7213},[363,60575],{"className":60576,"style":651},[650],[363,60578,60580],{"className":60579},[655,656,657,658],[363,60581,60583],{"className":60582},[407,658],[363,60584,7921],{"className":60585},[407,658],[363,60587,666],{"className":60588},[665],[363,60590,60592],{"className":60591},[638],[363,60593,60595],{"className":60594,"style":8895},[642],[363,60596],{},[363,60598],{"className":60599,"style":506},[505],[363,60601,471],{"className":60602},[510],[363,60604],{"className":60605,"style":506},[505],[363,60607,60609,60612],{"className":60608},[398],[363,60610],{"className":60611,"style":403},[402],[363,60613,385],{"className":60614,"style":409},[407,408],"（模去 ",[363,60617,60619,60642],{"className":60618,"translate":367},[366],[363,60620,60622],{"className":60621},[371],[373,60623,60624],{"xmlns":375},[377,60625,60626,60640],{},[380,60627,60628,60630,60632,60634,60636,60638],{},[383,60629,6254],{},[458,60631,461],{"stretchy":460},[383,60633,1116],{},[458,60635,5161],{},[463,60637,465],{},[458,60639,468],{"stretchy":460},[387,60641,6968],{"encoding":389},[363,60643,60645,60669],{"className":60644,"ariaHidden":394},[393],[363,60646,60648,60651,60654,60657,60660,60663,60666],{"className":60647},[398],[363,60649],{"className":60650,"style":487},[402],[363,60652,6254],{"className":60653},[407,408],[363,60655,461],{"className":60656},[494],[363,60658,1116],{"className":60659},[407,408],[363,60661],{"className":60662,"style":2935},[505],[363,60664,5161],{"className":60665},[2939],[363,60667],{"className":60668,"style":2935},[505],[363,60670,60672,60675,60678],{"className":60671},[398],[363,60673],{"className":60674,"style":487},[402],[363,60676,465],{"className":60677},[407],[363,60679,468],{"className":60680},[501],"）。验证 TV：",[363,60683,60685,60739],{"className":60684,"translate":367},[366],[363,60686,60688],{"className":60687},[371],[373,60689,60690],{"xmlns":375},[377,60691,60692,60736],{},[380,60693,60694,60696,60698,60704,60706,60708,60710,60712,60714,60716,60718,60720,60726,60728,60730,60732,60734],{},[383,60695,1116],{},[458,60697,3647],{},[586,60699,60700,60702],{},[463,60701,465],{},[463,60703,2811],{},[383,60705,5156],{"mathvariant":557},[463,60707,465],{},[383,60709,1288],{"mathvariant":557},[383,60711,1116],{},[458,60713,5161],{},[463,60715,465],{},[383,60717,5156],{"mathvariant":557},[458,60719,471],{},[586,60721,60722,60724],{},[463,60723,465],{},[463,60725,2811],{},[383,60727,5156],{"mathvariant":557},[383,60729,1116],{},[458,60731,5161],{},[463,60733,465],{},[383,60735,5156],{"mathvariant":557},[387,60737,60738],{"encoding":389},"t\\cdot\\frac12|1\u002Ft-1|=\\frac12|t-1|",[363,60740,60742,60760,60850,60868,60957],{"className":60741,"ariaHidden":394},[393],[363,60743,60745,60748,60751,60754,60757],{"className":60744},[398],[363,60746],{"className":60747,"style":5691},[402],[363,60749,1116],{"className":60750},[407,408],[363,60752],{"className":60753,"style":2935},[505],[363,60755,3647],{"className":60756},[2939],[363,60758],{"className":60759,"style":2935},[505],[363,60761,60763,60766,60834,60838,60841,60844,60847],{"className":60762},[398],[363,60764],{"className":60765,"style":2853},[402],[363,60767,60769,60772,60831],{"className":60768},[407],[363,60770],{"className":60771},[494,789],[363,60773,60775],{"className":60774},[586],[363,60776,60778,60823],{"className":60777},[633,634],[363,60779,60781,60820],{"className":60780},[638],[363,60782,60784,60798,60806],{"className":60783,"style":2872},[642],[363,60785,60786,60789],{"style":2875},[363,60787],{"className":60788,"style":809},[650],[363,60790,60792],{"className":60791},[655,656,657,658],[363,60793,60795],{"className":60794},[407,658],[363,60796,2811],{"className":60797},[407,658],[363,60799,60800,60803],{"style":821},[363,60801],{"className":60802,"style":809},[650],[363,60804],{"className":60805,"style":829},[828],[363,60807,60808,60811],{"style":2898},[363,60809],{"className":60810,"style":809},[650],[363,60812,60814],{"className":60813},[655,656,657,658],[363,60815,60817],{"className":60816},[407,658],[363,60818,465],{"className":60819},[407,658],[363,60821,666],{"className":60822},[665],[363,60824,60826],{"className":60825},[638],[363,60827,60829],{"className":60828,"style":2920},[642],[363,60830],{},[363,60832],{"className":60833},[501,789],[363,60835,60837],{"className":60836},[407],"∣1\u002F",[363,60839,1116],{"className":60840},[407,408],[363,60842],{"className":60843,"style":2935},[505],[363,60845,5161],{"className":60846},[2939],[363,60848],{"className":60849,"style":2935},[505],[363,60851,60853,60856,60859,60862,60865],{"className":60852},[398],[363,60854],{"className":60855,"style":487},[402],[363,60857,35051],{"className":60858},[407],[363,60860],{"className":60861,"style":506},[505],[363,60863,471],{"className":60864},[510],[363,60866],{"className":60867,"style":506},[505],[363,60869,60871,60874,60942,60945,60948,60951,60954],{"className":60870},[398],[363,60872],{"className":60873,"style":2853},[402],[363,60875,60877,60880,60939],{"className":60876},[407],[363,60878],{"className":60879},[494,789],[363,60881,60883],{"className":60882},[586],[363,60884,60886,60931],{"className":60885},[633,634],[363,60887,60889,60928],{"className":60888},[638],[363,60890,60892,60906,60914],{"className":60891,"style":2872},[642],[363,60893,60894,60897],{"style":2875},[363,60895],{"className":60896,"style":809},[650],[363,60898,60900],{"className":60899},[655,656,657,658],[363,60901,60903],{"className":60902},[407,658],[363,60904,2811],{"className":60905},[407,658],[363,60907,60908,60911],{"style":821},[363,60909],{"className":60910,"style":809},[650],[363,60912],{"className":60913,"style":829},[828],[363,60915,60916,60919],{"style":2898},[363,60917],{"className":60918,"style":809},[650],[363,60920,60922],{"className":60921},[655,656,657,658],[363,60923,60925],{"className":60924},[407,658],[363,60926,465],{"className":60927},[407,658],[363,60929,666],{"className":60930},[665],[363,60932,60934],{"className":60933},[638],[363,60935,60937],{"className":60936,"style":2920},[642],[363,60938],{},[363,60940],{"className":60941},[501,789],[363,60943,5156],{"className":60944},[407],[363,60946,1116],{"className":60947},[407,408],[363,60949],{"className":60950,"style":2935},[505],[363,60952,5161],{"className":60953},[2939],[363,60955],{"className":60956,"style":2935},[505],[363,60958,60960,60963],{"className":60959},[398],[363,60961],{"className":60962,"style":487},[402],[363,60964,35051],{"className":60965},[407],"；JS 的 ",[363,60968,60970,61005],{"className":60969,"translate":367},[366],[363,60971,60973],{"className":60972},[371],[373,60974,60975],{"xmlns":375},[377,60976,60977,61003],{},[380,60978,60979,60981,60983,60985,60987,60989,60991,60993,60995,60997,60999,61001],{},[383,60980,1116],{},[383,60982,385],{},[458,60984,461],{"stretchy":460},[463,60986,465],{},[383,60988,1288],{"mathvariant":557},[383,60990,1116],{},[458,60992,468],{"stretchy":460},[458,60994,471],{},[383,60996,385],{},[458,60998,461],{"stretchy":460},[383,61000,1116],{},[458,61002,468],{"stretchy":460},[387,61004,23941],{"encoding":389},[363,61006,61008,61041],{"className":61007,"ariaHidden":394},[393],[363,61009,61011,61014,61017,61020,61023,61026,61029,61032,61035,61038],{"className":61010},[398],[363,61012],{"className":61013,"style":487},[402],[363,61015,1116],{"className":61016},[407,408],[363,61018,385],{"className":61019,"style":409},[407,408],[363,61021,461],{"className":61022},[494],[363,61024,8937],{"className":61025},[407],[363,61027,1116],{"className":61028},[407,408],[363,61030,468],{"className":61031},[501],[363,61033],{"className":61034,"style":506},[505],[363,61036,471],{"className":61037},[510],[363,61039],{"className":61040,"style":506},[505],[363,61042,61044,61047,61050,61053,61056],{"className":61043},[398],[363,61045],{"className":61046,"style":487},[402],[363,61048,385],{"className":61049,"style":409},[407,408],[363,61051,461],{"className":61052},[494],[363,61054,1116],{"className":61055},[407,408],[363,61057,468],{"className":61058},[501]," 也已验算；KL 显然不满足，故不对称。",[2659,61061,61063],{"id":61062},"_44-变分表示","4.4 变分表示",[143,61065,61066,61160,61386,61389,61765],{},[11,61067,61068,11410,61071,61130,61131,61159],{},[1344,61069,61070],{},"定理 4.1（变分表示 \u002F Fenchel 对偶）",[363,61072,61074,61092],{"className":61073,"translate":367},[366],[363,61075,61077],{"className":61076},[371],[373,61078,61079],{"xmlns":375},[377,61080,61081,61089],{},[380,61082,61083],{},[7162,61084,61085,61087],{},[383,61086,385],{},[458,61088,7921],{},[387,61090,61091],{"encoding":389},"f^*",[363,61093,61095],{"className":61094,"ariaHidden":394},[393],[363,61096,61098,61101],{"className":61097},[398],[363,61099],{"className":61100,"style":403},[402],[363,61102,61104,61107],{"className":61103},[407],[363,61105,385],{"className":61106,"style":409},[407,408],[363,61108,61110],{"className":61109},[629],[363,61111,61113],{"className":61112},[633],[363,61114,61116],{"className":61115},[638],[363,61117,61119],{"className":61118,"style":8847},[642],[363,61120,61121,61124],{"style":7213},[363,61122],{"className":61123,"style":651},[650],[363,61125,61127],{"className":61126},[655,656,657,658],[363,61128,7921],{"className":61129},[2939,658]," 为 ",[363,61132,61134,61147],{"className":61133,"translate":367},[366],[363,61135,61137],{"className":61136},[371],[373,61138,61139],{"xmlns":375},[377,61140,61141,61145],{},[380,61142,61143],{},[383,61144,385],{},[387,61146,385],{"encoding":389},[363,61148,61150],{"className":61149,"ariaHidden":394},[393],[363,61151,61153,61156],{"className":61152},[398],[363,61154],{"className":61155,"style":403},[402],[363,61157,385],{"className":61158,"style":409},[407,408]," 的凸共轭（Fenchel 共轭）",[363,61161,61163],{"className":61162,"translate":367},[528],[363,61164,61166,61228],{"className":61165,"translate":367},[366],[363,61167,61169],{"className":61168},[371],[373,61170,61171],{"xmlns":375,"display":537},[377,61172,61173,61225],{},[380,61174,61175,61181,61183,61185,61187,61189,61205,61207,61209,61211,61213,61215,61217,61219,61221,61223],{},[7162,61176,61177,61179],{},[383,61178,385],{},[458,61180,7921],{},[458,61182,461],{"stretchy":460},[383,61184,40609],{},[458,61186,468],{"stretchy":460},[458,61188,471],{},[6364,61190,61191,61197],{},[380,61192,61193,61195],{},[383,61194,35804],{},[458,61196,4451],{},[380,61198,61199,61201,61203],{},[383,61200,1116],{},[458,61202,7276],{},[463,61204,474],{},[458,61206,3246],{"stretchy":460},[383,61208,40609],{},[383,61210,1116],{},[458,61212,5161],{},[383,61214,385],{},[458,61216,461],{"stretchy":460},[383,61218,1116],{},[458,61220,468],{"stretchy":460},[458,61222,3256],{"stretchy":460},[458,61224,898],{"separator":394},[387,61226,61227],{"encoding":389},"f^*(y)=\\sup_{t\\ge0}\\{yt-f(t)\\},",[363,61229,61231,61284,61364],{"className":61230,"ariaHidden":394},[393],[363,61232,61234,61237,61266,61269,61272,61275,61278,61281],{"className":61233},[398],[363,61235],{"className":61236,"style":487},[402],[363,61238,61240,61243],{"className":61239},[407],[363,61241,385],{"className":61242,"style":409},[407,408],[363,61244,61246],{"className":61245},[629],[363,61247,61249],{"className":61248},[633],[363,61250,61252],{"className":61251},[638],[363,61253,61255],{"className":61254,"style":8596},[642],[363,61256,61257,61260],{"style":8625},[363,61258],{"className":61259,"style":651},[650],[363,61261,61263],{"className":61262},[655,656,657,658],[363,61264,7921],{"className":61265},[2939,658],[363,61267,461],{"className":61268},[494],[363,61270,40609],{"className":61271,"style":1328},[407,408],[363,61273,468],{"className":61274},[501],[363,61276],{"className":61277,"style":506},[505],[363,61279,471],{"className":61280},[510],[363,61282],{"className":61283,"style":506},[505],[363,61285,61287,61291,61346,61349,61352,61355,61358,61361],{"className":61286},[398],[363,61288],{"className":61289,"style":61290},[402],"height:1.7567em;vertical-align:-1.0067em;",[363,61292,61294],{"className":61293},[1832,14066],[363,61295,61297,61337],{"className":61296},[633,634],[363,61298,61300,61334],{"className":61299},[638],[363,61301,61303,61324],{"className":61302,"style":2693},[642],[363,61304,61306,61309],{"style":61305},"top:-2.1885em;margin-left:0em;",[363,61307],{"className":61308,"style":809},[650],[363,61310,61312],{"className":61311},[655,656,657,658],[363,61313,61315,61318,61321],{"className":61314},[407,658],[363,61316,1116],{"className":61317},[407,408,658],[363,61319,7276],{"className":61320},[510,658],[363,61322,474],{"className":61323},[407,658],[363,61325,61326,61329],{"style":12213},[363,61327],{"className":61328,"style":809},[650],[363,61330,61331],{},[363,61332,35804],{"className":61333},[1832],[363,61335,666],{"className":61336},[665],[363,61338,61340],{"className":61339},[638],[363,61341,61344],{"className":61342,"style":61343},[642],"height:1.0067em;",[363,61345],{},[363,61347,3246],{"className":61348},[494],[363,61350,40609],{"className":61351,"style":1328},[407,408],[363,61353,1116],{"className":61354},[407,408],[363,61356],{"className":61357,"style":2935},[505],[363,61359,5161],{"className":61360},[2939],[363,61362],{"className":61363,"style":2935},[505],[363,61365,61367,61370,61373,61376,61379,61383],{"className":61366},[398],[363,61368],{"className":61369,"style":487},[402],[363,61371,385],{"className":61372,"style":409},[407,408],[363,61374,461],{"className":61375},[494],[363,61377,1116],{"className":61378},[407,408],[363,61380,61382],{"className":61381},[501],")}",[363,61384,898],{"className":61385},[920],[11,61387,61388],{},"则",[363,61390,61392],{"className":61391,"translate":367},[528],[363,61393,61395,61479],{"className":61394,"translate":367},[366],[363,61396,61398],{"className":61397},[371],[373,61399,61400],{"xmlns":375,"display":537},[377,61401,61402,61476],{},[380,61403,61404,61410,61412,61414,61416,61418,61420,61422,61432,61474],{},[543,61405,61406,61408],{},[383,61407,547],{},[383,61409,385],{},[458,61411,461],{"stretchy":460},[383,61413,554],{},[383,61415,558],{"mathvariant":557},[383,61417,561],{},[458,61419,468],{"stretchy":460},[458,61421,471],{},[6364,61423,61424,61430],{},[380,61425,61426,61428],{},[383,61427,35804],{},[458,61429,4451],{},[383,61431,4506],{},[380,61433,61434,61436,61442,61444,61446,61448,61450,61456,61458,61464,61466,61468,61470,61472],{},[458,61435,3246],{"fence":394},[543,61437,61438,61440],{},[383,61439,571],{"mathvariant":570},[383,61441,554],{},[458,61443,578],{"stretchy":460},[383,61445,4506],{},[458,61447,605],{"stretchy":460},[458,61449,5161],{},[543,61451,61452,61454],{},[383,61453,571],{"mathvariant":570},[383,61455,561],{},[458,61457,578],{"stretchy":460},[7162,61459,61460,61462],{},[383,61461,385],{},[458,61463,7921],{},[458,61465,461],{"stretchy":460},[383,61467,4506],{},[458,61469,468],{"stretchy":460},[458,61471,605],{"stretchy":460},[458,61473,3256],{"fence":394},[458,61475,898],{"separator":394},[387,61477,61478],{"encoding":389},"D_f(P\\|Q)=\\sup_{g}\n\\left\\{\\mathbb E_P[g]\n-\\mathbb E_Q[f^*(g)]\\right\\},",[363,61480,61482,61552],{"className":61481,"ariaHidden":394},[393],[363,61483,61485,61488,61528,61531,61534,61537,61540,61543,61546,61549],{"className":61484},[398],[363,61486],{"className":61487,"style":618},[402],[363,61489,61491,61494],{"className":61490},[407],[363,61492,547],{"className":61493,"style":625},[407,408],[363,61495,61497],{"className":61496},[629],[363,61498,61500,61520],{"className":61499},[633,634],[363,61501,61503,61517],{"className":61502},[638],[363,61504,61506],{"className":61505,"style":643},[642],[363,61507,61508,61511],{"style":646},[363,61509],{"className":61510,"style":651},[650],[363,61512,61514],{"className":61513},[655,656,657,658],[363,61515,385],{"className":61516,"style":409},[407,408,658],[363,61518,666],{"className":61519},[665],[363,61521,61523],{"className":61522},[638],[363,61524,61526],{"className":61525,"style":673},[642],[363,61527],{},[363,61529,461],{"className":61530},[494],[363,61532,554],{"className":61533,"style":682},[407,408],[363,61535,558],{"className":61536},[407],[363,61538,561],{"className":61539},[407,408],[363,61541,468],{"className":61542},[501],[363,61544],{"className":61545,"style":506},[505],[363,61547,471],{"className":61548},[510],[363,61550],{"className":61551,"style":506},[505],[363,61553,61555,61559,61608,61611,61759,61762],{"className":61554},[398],[363,61556],{"className":61557,"style":61558},[402],"height:1.7805em;vertical-align:-1.0305em;",[363,61560,61562],{"className":61561},[1832,14066],[363,61563,61565,61599],{"className":61564},[633,634],[363,61566,61568,61596],{"className":61567},[638],[363,61569,61571,61586],{"className":61570,"style":2693},[642],[363,61572,61574,61577],{"style":61573},"top:-2.2056em;margin-left:0em;",[363,61575],{"className":61576,"style":809},[650],[363,61578,61580],{"className":61579},[655,656,657,658],[363,61581,61583],{"className":61582},[407,658],[363,61584,4506],{"className":61585,"style":1328},[407,408,658],[363,61587,61588,61591],{"style":12213},[363,61589],{"className":61590,"style":809},[650],[363,61592,61593],{},[363,61594,35804],{"className":61595},[1832],[363,61597,666],{"className":61598},[665],[363,61600,61602],{"className":61601},[638],[363,61603,61606],{"className":61604,"style":61605},[642],"height:1.0305em;",[363,61607],{},[363,61609],{"className":61610,"style":754},[505],[363,61612,61614,61617,61657,61660,61663,61666,61669,61672,61675,61715,61718,61747,61750,61753,61756],{"className":61613},[758],[363,61615,3246],{"className":61616,"style":763},[494,762],[363,61618,61620,61623],{"className":61619},[407],[363,61621,571],{"className":61622},[407,714],[363,61624,61626],{"className":61625},[629],[363,61627,61629,61649],{"className":61628},[633,634],[363,61630,61632,61646],{"className":61631},[638],[363,61633,61635],{"className":61634,"style":727},[642],[363,61636,61637,61640],{"style":730},[363,61638],{"className":61639,"style":651},[650],[363,61641,61643],{"className":61642},[655,656,657,658],[363,61644,554],{"className":61645,"style":682},[407,408,658],[363,61647,666],{"className":61648},[665],[363,61650,61652],{"className":61651},[638],[363,61653,61655],{"className":61654,"style":8316},[642],[363,61656],{},[363,61658,578],{"className":61659},[494],[363,61661,4506],{"className":61662,"style":1328},[407,408],[363,61664,605],{"className":61665},[501],[363,61667],{"className":61668,"style":2935},[505],[363,61670,5161],{"className":61671},[2939],[363,61673],{"className":61674,"style":2935},[505],[363,61676,61678,61681],{"className":61677},[407],[363,61679,571],{"className":61680},[407,714],[363,61682,61684],{"className":61683},[629],[363,61685,61687,61707],{"className":61686},[633,634],[363,61688,61690,61704],{"className":61689},[638],[363,61691,61693],{"className":61692,"style":727},[642],[363,61694,61695,61698],{"style":730},[363,61696],{"className":61697,"style":651},[650],[363,61699,61701],{"className":61700},[655,656,657,658],[363,61702,561],{"className":61703},[407,408,658],[363,61705,666],{"className":61706},[665],[363,61708,61710],{"className":61709},[638],[363,61711,61713],{"className":61712,"style":673},[642],[363,61714],{},[363,61716,578],{"className":61717},[494],[363,61719,61721,61724],{"className":61720},[407],[363,61722,385],{"className":61723,"style":409},[407,408],[363,61725,61727],{"className":61726},[629],[363,61728,61730],{"className":61729},[633],[363,61731,61733],{"className":61732},[638],[363,61734,61736],{"className":61735,"style":8596},[642],[363,61737,61738,61741],{"style":8625},[363,61739],{"className":61740,"style":651},[650],[363,61742,61744],{"className":61743},[655,656,657,658],[363,61745,7921],{"className":61746},[2939,658],[363,61748,461],{"className":61749},[494],[363,61751,4506],{"className":61752,"style":1328},[407,408],[363,61754,20762],{"className":61755},[501],[363,61757,3256],{"className":61758,"style":763},[501,762],[363,61760],{"className":61761,"style":754},[505],[363,61763,898],{"className":61764},[920],[11,61766,61767,61768,10528],{},"其中上确界取遍所有使两个期望都有定义的可测函数 ",[363,61769,61771,61802],{"className":61770,"translate":367},[366],[363,61772,61774],{"className":61773},[371],[373,61775,61776],{"xmlns":375},[377,61777,61778,61799],{},[380,61779,61780,61782,61784,61786,61788,61791,61793],{},[383,61781,4506],{},[458,61783,1364],{},[383,61785,945],{"mathvariant":944},[458,61787,1378],{},[383,61789,61790],{"mathvariant":557},"dom",[458,61792,4451],{},[7162,61794,61795,61797],{},[383,61796,385],{},[458,61798,7921],{},[387,61800,61801],{"encoding":389},"g:\\mathcal X\\to\\operatorname{dom} f^*",[363,61803,61805,61823,61841],{"className":61804,"ariaHidden":394},[393],[363,61806,61808,61811,61814,61817,61820],{"className":61807},[398],[363,61809],{"className":61810,"style":2624},[402],[363,61812,4506],{"className":61813,"style":1328},[407,408],[363,61815],{"className":61816,"style":506},[505],[363,61818,1364],{"className":61819},[510],[363,61821],{"className":61822,"style":506},[505],[363,61824,61826,61829,61832,61835,61838],{"className":61825},[398],[363,61827],{"className":61828,"style":1011},[402],[363,61830,945],{"className":61831,"style":972},[407,971],[363,61833],{"className":61834,"style":506},[505],[363,61836,1378],{"className":61837},[510],[363,61839],{"className":61840,"style":506},[505],[363,61842,61844,61847,61853,61856],{"className":61843},[398],[363,61845],{"className":61846,"style":403},[402],[363,61848,61850],{"className":61849},[1832],[363,61851,61790],{"className":61852},[407,33034],[363,61854],{"className":61855,"style":754},[505],[363,61857,61859,61862],{"className":61858},[407],[363,61860,385],{"className":61861,"style":409},[407,408],[363,61863,61865],{"className":61864},[629],[363,61866,61868],{"className":61867},[633],[363,61869,61871],{"className":61870},[638],[363,61872,61874],{"className":61873,"style":8847},[642],[363,61875,61876,61879],{"style":7213},[363,61877],{"className":61878,"style":651},[650],[363,61880,61882],{"className":61881},[655,656,657,658],[363,61883,7921],{"className":61884},[2939,658],[11,61886,61887,61890,61891,61919],{},[1344,61888,61889],{},"证明（Fenchel 对偶）。"," 由共轭函数的性质（Fenchel–Moreau，",[363,61892,61894,61907],{"className":61893,"translate":367},[366],[363,61895,61897],{"className":61896},[371],[373,61898,61899],{"xmlns":375},[377,61900,61901,61905],{},[380,61902,61903],{},[383,61904,385],{},[387,61906,385],{"encoding":389},[363,61908,61910],{"className":61909,"ariaHidden":394},[393],[363,61911,61913,61916],{"className":61912},[398],[363,61914],{"className":61915,"style":403},[402],[363,61917,385],{"className":61918,"style":409},[407,408]," 凸闭），",[363,61921,61923],{"className":61922,"translate":367},[528],[363,61924,61926,61993],{"className":61925,"translate":367},[366],[363,61927,61929],{"className":61928},[371],[373,61930,61931],{"xmlns":375,"display":537},[377,61932,61933,61990],{},[380,61934,61935,61937,61939,61941,61943,61945,61955,61957,61959,61961,61963,61969,61971,61973,61975,61977,61979,61981,61984,61986,61988],{},[383,61936,385],{},[458,61938,461],{"stretchy":460},[383,61940,1116],{},[458,61942,468],{"stretchy":460},[458,61944,471],{},[6364,61946,61947,61953],{},[380,61948,61949,61951],{},[383,61950,35804],{},[458,61952,4451],{},[383,61954,40609],{},[458,61956,3246],{"stretchy":460},[383,61958,40609],{},[383,61960,1116],{},[458,61962,5161],{},[7162,61964,61965,61967],{},[383,61966,385],{},[458,61968,7921],{},[458,61970,461],{"stretchy":460},[383,61972,40609],{},[458,61974,468],{"stretchy":460},[458,61976,3256],{"stretchy":460},[458,61978,898],{"separator":394},[505,61980],{"width":3610},[383,61982,61983],{"mathvariant":557},"∀",[383,61985,1116],{},[458,61987,7276],{},[463,61989,6112],{},[387,61991,61992],{"encoding":389},"f(t)=\\sup_{y}\\{yt-f^*(y)\\},\n\\qquad \\forall t\\ge0 .",[363,61994,61996,62023,62094,62162],{"className":61995,"ariaHidden":394},[393],[363,61997,61999,62002,62005,62008,62011,62014,62017,62020],{"className":61998},[398],[363,62000],{"className":62001,"style":487},[402],[363,62003,385],{"className":62004,"style":409},[407,408],[363,62006,461],{"className":62007},[494],[363,62009,1116],{"className":62010},[407,408],[363,62012,468],{"className":62013},[501],[363,62015],{"className":62016,"style":506},[505],[363,62018,471],{"className":62019},[510],[363,62021],{"className":62022,"style":506},[505],[363,62024,62026,62029,62076,62079,62082,62085,62088,62091],{"className":62025},[398],[363,62027],{"className":62028,"style":61558},[402],[363,62030,62032],{"className":62031},[1832,14066],[363,62033,62035,62068],{"className":62034},[633,634],[363,62036,62038,62065],{"className":62037},[638],[363,62039,62041,62055],{"className":62040,"style":2693},[642],[363,62042,62043,62046],{"style":61573},[363,62044],{"className":62045,"style":809},[650],[363,62047,62049],{"className":62048},[655,656,657,658],[363,62050,62052],{"className":62051},[407,658],[363,62053,40609],{"className":62054,"style":1328},[407,408,658],[363,62056,62057,62060],{"style":12213},[363,62058],{"className":62059,"style":809},[650],[363,62061,62062],{},[363,62063,35804],{"className":62064},[1832],[363,62066,666],{"className":62067},[665],[363,62069,62071],{"className":62070},[638],[363,62072,62074],{"className":62073,"style":61605},[642],[363,62075],{},[363,62077,3246],{"className":62078},[494],[363,62080,40609],{"className":62081,"style":1328},[407,408],[363,62083,1116],{"className":62084},[407,408],[363,62086],{"className":62087,"style":2935},[505],[363,62089,5161],{"className":62090},[2939],[363,62092],{"className":62093,"style":2935},[505],[363,62095,62097,62100,62129,62132,62135,62138,62141,62144,62147,62150,62153,62156,62159],{"className":62096},[398],[363,62098],{"className":62099,"style":487},[402],[363,62101,62103,62106],{"className":62102},[407],[363,62104,385],{"className":62105,"style":409},[407,408],[363,62107,62109],{"className":62108},[629],[363,62110,62112],{"className":62111},[633],[363,62113,62115],{"className":62114},[638],[363,62116,62118],{"className":62117,"style":8596},[642],[363,62119,62120,62123],{"style":8625},[363,62121],{"className":62122,"style":651},[650],[363,62124,62126],{"className":62125},[655,656,657,658],[363,62127,7921],{"className":62128},[2939,658],[363,62130,461],{"className":62131},[494],[363,62133,40609],{"className":62134,"style":1328},[407,408],[363,62136,61382],{"className":62137},[501],[363,62139,898],{"className":62140},[920],[363,62142],{"className":62143,"style":3982},[505],[363,62145],{"className":62146,"style":754},[505],[363,62148,61983],{"className":62149},[407],[363,62151,1116],{"className":62152},[407,408],[363,62154],{"className":62155,"style":506},[505],[363,62157,7276],{"className":62158},[510],[363,62160],{"className":62161,"style":506},[505],[363,62163,62165,62168],{"className":62164},[398],[363,62166],{"className":62167,"style":520},[402],[363,62169,6112],{"className":62170},[407],[11,62172,17396,62173,62307,62308,62336],{},[363,62174,62176,62206],{"className":62175,"translate":367},[366],[363,62177,62179],{"className":62178},[371],[373,62180,62181],{"xmlns":375},[377,62182,62183,62203],{},[380,62184,62185,62187,62189],{},[383,62186,1116],{},[458,62188,471],{},[586,62190,62191,62197],{},[380,62192,62193,62195],{},[383,62194,592],{},[383,62196,554],{},[380,62198,62199,62201],{},[383,62200,592],{},[383,62202,561],{},[387,62204,62205],{"encoding":389},"t=\\frac{dP}{dQ}",[363,62207,62209,62227],{"className":62208,"ariaHidden":394},[393],[363,62210,62212,62215,62218,62221,62224],{"className":62211},[398],[363,62213],{"className":62214,"style":5691},[402],[363,62216,1116],{"className":62217},[407,408],[363,62219],{"className":62220,"style":506},[505],[363,62222,471],{"className":62223},[510],[363,62225],{"className":62226,"style":506},[505],[363,62228,62230,62233],{"className":62229},[398],[363,62231],{"className":62232,"style":3016},[402],[363,62234,62236,62239,62304],{"className":62235},[407],[363,62237],{"className":62238},[494,789],[363,62240,62242],{"className":62241},[586],[363,62243,62245,62296],{"className":62244},[633,634],[363,62246,62248,62293],{"className":62247},[638],[363,62249,62251,62268,62276],{"className":62250,"style":3035},[642],[363,62252,62253,62256],{"style":2875},[363,62254],{"className":62255,"style":809},[650],[363,62257,62259],{"className":62258},[655,656,657,658],[363,62260,62262,62265],{"className":62261},[407,658],[363,62263,592],{"className":62264},[407,408,658],[363,62266,561],{"className":62267},[407,408,658],[363,62269,62270,62273],{"style":821},[363,62271],{"className":62272,"style":809},[650],[363,62274],{"className":62275,"style":829},[828],[363,62277,62278,62281],{"style":2898},[363,62279],{"className":62280,"style":809},[650],[363,62282,62284],{"className":62283},[655,656,657,658],[363,62285,62287,62290],{"className":62286},[407,658],[363,62288,592],{"className":62289},[407,408,658],[363,62291,554],{"className":62292,"style":682},[407,408,658],[363,62294,666],{"className":62295},[665],[363,62297,62299],{"className":62298},[638],[363,62300,62302],{"className":62301,"style":3087},[642],[363,62303],{},[363,62305],{"className":62306},[501,789]," 代入并对 ",[363,62309,62311,62324],{"className":62310,"translate":367},[366],[363,62312,62314],{"className":62313},[371],[373,62315,62316],{"xmlns":375},[377,62317,62318,62322],{},[380,62319,62320],{},[383,62321,561],{},[387,62323,561],{"encoding":389},[363,62325,62327],{"className":62326,"ariaHidden":394},[393],[363,62328,62330,62333],{"className":62329},[398],[363,62331],{"className":62332,"style":913},[402],[363,62334,561],{"className":62335},[407,408]," 积分：",[363,62338,62340],{"className":62339,"translate":367},[528],[363,62341,62343,62585],{"className":62342,"translate":367},[366],[363,62344,62346],{"className":62345},[371],[373,62347,62348],{"xmlns":375,"display":537},[377,62349,62350,62582],{},[1591,62351,62352,62438,62512],{"rowspacing":1593,"columnalign":1594,"columnspacing":1595},[1597,62353,62354,62376],{},[1600,62355,62356],{},[1603,62357,62358],{"scriptlevel":474,"displaystyle":394},[380,62359,62360,62366,62368,62370,62372,62374],{},[543,62361,62362,62364],{},[383,62363,547],{},[383,62365,385],{},[458,62367,461],{"stretchy":460},[383,62369,554],{},[383,62371,558],{"mathvariant":557},[383,62373,561],{},[458,62375,468],{"stretchy":460},[1600,62377,62378],{},[1603,62379,62380],{"scriptlevel":474,"displaystyle":394},[380,62381,62382,62384,62386,62388,62398,62400,62402,62404,62418,62420,62426,62428,62430,62432,62434,62436],{},[380,62383],{},[458,62385,471],{},[458,62387,1635],{},[6364,62389,62390,62396],{},[380,62391,62392,62394],{},[383,62393,35804],{},[458,62395,4451],{},[383,62397,40609],{},[458,62399,3246],{"fence":460,"stretchy":394,"minsize":17789,"maxsize":17789},[383,62401,40609],{},[1645,62403,1647],{},[586,62405,62406,62412],{},[380,62407,62408,62410],{},[383,62409,592],{},[383,62411,554],{},[380,62413,62414,62416],{},[383,62415,592],{},[383,62417,561],{},[458,62419,5161],{},[7162,62421,62422,62424],{},[383,62423,385],{},[458,62425,7921],{},[458,62427,461],{"stretchy":460},[383,62429,40609],{},[458,62431,468],{"stretchy":460},[458,62433,3256],{"fence":460,"stretchy":394,"minsize":17789,"maxsize":17789},[383,62435,592],{},[383,62437,561],{},[1597,62439,62440,62446],{},[1600,62441,62442],{},[1603,62443,62444],{"scriptlevel":474,"displaystyle":394},[380,62445],{},[1600,62447,62448],{},[1603,62449,62450],{"scriptlevel":474,"displaystyle":394},[380,62451,62452,62454,62456,62458,62460,62470,62472,62474,62476,62478,62492,62494,62500,62502,62504,62506,62508,62510],{},[380,62453],{},[1645,62455,4631],{},[458,62457,7276],{},[1645,62459,4631],{},[6364,62461,62462,62468],{},[380,62463,62464,62466],{},[383,62465,35804],{},[458,62467,4451],{},[383,62469,40609],{},[458,62471,1635],{},[458,62473,3246],{"fence":460,"stretchy":394,"minsize":17789,"maxsize":17789},[383,62475,40609],{},[1645,62477,1647],{},[586,62479,62480,62486],{},[380,62481,62482,62484],{},[383,62483,592],{},[383,62485,554],{},[380,62487,62488,62490],{},[383,62489,592],{},[383,62491,561],{},[458,62493,5161],{},[7162,62495,62496,62498],{},[383,62497,385],{},[458,62499,7921],{},[458,62501,461],{"stretchy":460},[383,62503,40609],{},[458,62505,468],{"stretchy":460},[458,62507,3256],{"fence":460,"stretchy":394,"minsize":17789,"maxsize":17789},[383,62509,592],{},[383,62511,561],{},[1597,62513,62514,62520],{},[1600,62515,62516],{},[1603,62517,62518],{"scriptlevel":474,"displaystyle":394},[380,62519],{},[1600,62521,62522],{},[1603,62523,62524],{"scriptlevel":474,"displaystyle":394},[380,62525,62526,62528,62530,62540,62542,62548,62550,62552,62554,62556,62562,62564,62570,62572,62574,62576,62578,62580],{},[380,62527],{},[458,62529,471],{},[6364,62531,62532,62538],{},[380,62533,62534,62536],{},[383,62535,35804],{},[458,62537,4451],{},[383,62539,40609],{},[458,62541,3246],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[543,62543,62544,62546],{},[383,62545,571],{"mathvariant":570},[383,62547,554],{},[458,62549,578],{"stretchy":460},[383,62551,40609],{},[458,62553,605],{"stretchy":460},[458,62555,5161],{},[543,62557,62558,62560],{},[383,62559,571],{"mathvariant":570},[383,62561,561],{},[458,62563,578],{"stretchy":460},[7162,62565,62566,62568],{},[383,62567,385],{},[458,62569,7921],{},[458,62571,461],{"stretchy":460},[383,62573,40609],{},[458,62575,468],{"stretchy":460},[458,62577,605],{"stretchy":460},[458,62579,3256],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[383,62581,4723],{"mathvariant":557},[387,62583,62584],{"encoding":389},"\\begin{aligned}\nD_f(P\\|Q)\n&=\\int 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E_Q[f^*(y)]\\big\\}.\n\\end{aligned}",[363,62586,62588],{"className":62587,"ariaHidden":394},[393],[363,62589,62591,62595],{"className":62590},[398],[363,62592],{"className":62593,"style":62594},[402],"height:7.2845em;vertical-align:-3.3923em;",[363,62596,62598],{"className":62597},[407],[363,62599,62601,62709],{"className":62600},[1591],[363,62602,62604],{"className":62603},[1708],[363,62605,62607,62700],{"className":62606},[633,634],[363,62608,62610,62697],{"className":62609},[638],[363,62611,62614,62679,62688],{"className":62612,"style":62613},[642],"height:3.8923em;",[363,62615,62617,62621],{"style":62616},"top:-5.8923em;",[363,62618],{"className":62619,"style":62620},[650],"height:3.3714em;",[363,62622,62624,62664,62667,62670,62673,62676],{"className":62623},[407],[363,62625,62627,62630],{"className":62626},[407],[363,62628,547],{"className":62629,"style":625},[407,408],[363,62631,62633],{"className":62632},[629],[363,62634,62636,62656],{"className":62635},[633,634],[363,62637,62639,62653],{"className":62638},[638],[363,62640,62642],{"className":62641,"style":643},[642],[363,62643,62644,62647],{"style":646},[363,62645],{"className":62646,"style":651},[650],[363,62648,62650],{"className":62649},[655,656,657,658],[363,62651,385],{"className":62652,"style":409},[407,408,658],[363,62654,666],{"className":62655},[665],[363,62657,62659],{"className":62658},[638],[363,62660,62662],{"className":62661,"style":673},[642],[363,62663],{},[363,62665,461],{"className":62666},[494],[363,62668,554],{"className":62669,"style":682},[407,408],[363,62671,558],{"className":62672},[407],[363,62674,561],{"className":62675},[407,408],[363,62677,468],{"className":62678},[501],[363,62680,62682,62685],{"style":62681},"top:-3.1903em;",[363,62683],{"className":62684,"style":62620},[650],[363,62686],{"className":62687},[407],[363,62689,62691,62694],{"style":62690},"top:-1.0097em;",[363,62692],{"className":62693,"style":62620},[650],[363,62695],{"className":62696},[407],[363,62698,666],{"className":62699},[665],[363,62701,62703],{"className":62702},[638],[363,62704,62707],{"className":62705,"style":62706},[642],"height:3.3923em;",[363,62708],{},[363,62710,62712],{"className":62711},[1799],[363,62713,62715,63375],{"className":62714},[633,634],[363,62716,62718,63372],{"className":62717},[638],[363,62719,62721,62933,63151],{"className":62720,"style":62613},[642],[363,62722,62723,62726],{"style":62616},[363,62724],{"className":62725,"style":62620},[650],[363,62727,62729,62732,62735,62738,62741,62744,62747,62791,62794,62800,62803,62806,62874,62877,62880,62883,62912,62915,62918,62921,62927,62930],{"className":62728},[407],[363,62730],{"className":62731},[407],[363,62733],{"className":62734,"style":506},[505],[363,62736,471],{"className":62737},[510],[363,62739],{"className":62740,"style":506},[505],[363,62742,1635],{"className":62743,"style":1835},[1832,1833,1834],[363,62745],{"className":62746,"style":754},[505],[363,62748,62750],{"className":62749},[1832,14066],[363,62751,62753,62783],{"className":62752},[633,634],[363,62754,62756,62780],{"className":62755},[638],[363,62757,62759,62770],{"className":62758,"style":2693},[642],[363,62760,62761,62764],{"style":61573},[363,62762],{"className":62763,"style":809},[650],[363,62765,62767],{"className":62766},[655,656,657,658],[363,62768,40609],{"className":62769,"style":1328},[407,408,658],[363,62771,62772,62775],{"style":12213},[363,62773],{"className":62774,"style":809},[650],[363,62776,62777],{},[363,62778,35804],{"className":62779},[1832],[363,62781,666],{"className":62782},[665],[363,62784,62786],{"className":62785},[638],[363,62787,62789],{"className":62788,"style":61605},[642],[363,62790],{},[363,62792],{"className":62793,"style":754},[505],[363,62795,62797],{"className":62796},[407],[363,62798,3246],{"className":62799},[767,4068],[363,62801,40609],{"className":62802,"style":1328},[407,408],[363,62804],{"className":62805,"style":754},[505],[363,62807,62809,62812,62871],{"className":62808},[407],[363,62810],{"className":62811},[494,789],[363,62813,62815],{"className":62814},[586],[363,62816,62818,62863],{"className":62817},[633,634],[363,62819,62821,62860],{"className":62820},[638],[363,62822,62824,62838,62846],{"className":62823,"style":802},[642],[363,62825,62826,62829],{"style":805},[363,62827],{"className":62828,"style":809},[650],[363,62830,62832,62835],{"className":62831},[407],[363,62833,592],{"className":62834},[407,408],[363,62836,561],{"className":62837},[407,408],[363,62839,62840,62843],{"style":821},[363,62841],{"className":62842,"style":809},[650],[363,62844],{"className":62845,"style":829},[828],[363,62847,62848,62851],{"style":832},[363,62849],{"className":62850,"style":809},[650],[363,62852,62854,62857],{"className":62853},[407],[363,62855,592],{"className":62856},[407,408],[363,62858,554],{"className":62859,"style":682},[407,408],[363,62861,666],{"className":62862},[665],[363,62864,62866],{"className":62865},[638],[363,62867,62869],{"className":62868,"style":854},[642],[3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凸，",[363,63445,63447,63501],{"className":63446,"translate":367},[366],[363,63448,63450],{"className":63449},[371],[373,63451,63452],{"xmlns":375},[377,63453,63454,63498],{},[380,63455,63456,63458,63460,63462,63464,63466,63476,63478,63480,63482,63484,63490,63492,63494,63496],{},[383,63457,385],{},[458,63459,461],{"stretchy":460},[383,63461,1116],{},[458,63463,468],{"stretchy":460},[458,63465,471],{},[543,63467,63468,63474],{},[380,63469,63470,63472],{},[383,63471,35804],{},[458,63473,4451],{},[383,63475,40609],{},[458,63477,3246],{"stretchy":460},[383,63479,40609],{},[383,63481,1116],{},[458,63483,5161],{},[7162,63485,63486,63488],{},[383,63487,385],{},[458,63489,7921],{},[458,63491,461],{"stretchy":460},[383,63493,40609],{},[458,63495,468],{"stretchy":460},[458,63497,3256],{"stretchy":460},[387,63499,63500],{"encoding":389},"f(t)=\\sup_y\\{yt-f^*(y)\\}",[363,63502,63504,63531,63595],{"className":63503,"ariaHidden":394},[393],[363,63505,63507,63510,63513,63516,63519,63522,63525,63528],{"className":63506},[398],[363,63508],{"className":63509,"style":487},[402],[363,63511,385],{"className":63512,"style":409},[407,408],[363,63514,461],{"className":63515},[494],[363,63517,1116],{"className":63518},[407,408],[363,63520,468],{"className":63521},[501],[363,63523],{"className":63524,"style":506},[505],[363,63526,471],{"className":63527},[510],[363,63529],{"className":63530,"style":506},[505],[363,63532,63534,63537,63577,63580,63583,63586,63589,63592],{"className":63533},[398],[363,63535],{"className":63536,"style":57699},[402],[363,63538,63540,63543],{"className":63539},[1832],[363,63541,35804],{"className":63542},[1832],[363,63544,63546],{"className":63545},[629],[363,63547,63549,63569],{"className":63548},[633,634],[363,63550,63552,63566],{"className":63551},[638],[363,63553,63555],{"className":63554,"style":37371},[642],[363,63556,63557,63560],{"style":35950},[363,63558],{"className":63559,"style":651},[650],[363,63561,63563],{"className":63562},[655,656,657,658],[363,63564,40609],{"className":63565,"style":1328},[407,408,658],[363,63567,666],{"className":63568},[665],[363,63570,63572],{"className":63571},[638],[363,63573,63575],{"className":63574,"style":57738},[642],[363,63576],{},[363,63578,3246],{"className":63579},[494],[363,63581,40609],{"className":63582,"style":1328},[407,408],[363,63584,1116],{"className":63585},[407,408],[363,63587],{"className":63588,"style":2935},[505],[363,63590,5161],{"className":63591},[2939],[363,63593],{"className":63594,"style":2935},[505],[363,63596,63598,63601,63630,63633,63636],{"className":63597},[398],[363,63599],{"className":63600,"style":487},[402],[363,63602,63604,63607],{"className":63603},[407],[363,63605,385],{"className":63606,"style":409},[407,408],[363,63608,63610],{"className":63609},[629],[363,63611,63613],{"className":63612},[633],[363,63614,63616],{"className":63615},[638],[363,63617,63619],{"className":63618,"style":8847},[642],[363,63620,63621,63624],{"style":7213},[363,63622],{"className":63623,"style":651},[650],[363,63625,63627],{"className":63626},[655,656,657,658],[363,63628,7921],{"className":63629},[2939,658],[363,63631,461],{"className":63632},[494],[363,63634,40609],{"className":63635,"style":1328},[407,408],[363,63637,61382],{"className":63638},[501]," 中的 ",[363,63641,63643,63656],{"className":63642,"translate":367},[366],[363,63644,63646],{"className":63645},[371],[373,63647,63648],{"xmlns":375},[377,63649,63650,63654],{},[380,63651,63652],{},[383,63653,40609],{},[387,63655,40609],{"encoding":389},[363,63657,63659],{"className":63658,"ariaHidden":394},[393],[363,63660,63662,63665],{"className":63661},[398],[363,63663],{"className":63664,"style":2624},[402],[363,63666,40609],{"className":63667,"style":1328},[407,408]," 可取为 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f'(t)",[363,63779,63781,63799],{"className":63780,"ariaHidden":394},[393],[363,63782,63784,63787,63790,63793,63796],{"className":63783},[398],[363,63785],{"className":63786,"style":9059},[402],[363,63788,1116],{"className":63789},[407,408],[363,63791],{"className":63792,"style":506},[505],[363,63794,9032],{"className":63795},[510],[363,63797],{"className":63798,"style":506},[505],[363,63800,63802,63805,63837,63840,63843],{"className":63801},[398],[363,63803],{"className":63804,"style":7191},[402],[363,63806,63808,63811],{"className":63807},[407],[363,63809,385],{"className":63810,"style":409},[407,408],[363,63812,63814],{"className":63813},[629],[363,63815,63817],{"className":63816},[633],[363,63818,63820],{"className":63819},[638],[363,63821,63823],{"className":63822,"style":7210},[642],[363,63824,63825,63828],{"style":7213},[363,63826],{"className":63827,"style":651},[650],[363,63829,63831],{"className":63830},[655,656,657,658],[363,63832,63834],{"className":63833},[407,658],[363,63835,7168],{"className":63836},[407,658],[363,63838,461],{"className":63839},[494],[363,63841,1116],{"className":63842},[407,408],[363,63844,468],{"className":63845},[501]," 单调，于是 ",[363,63848,63850,63896],{"className":63849,"translate":367},[366],[363,63851,63853],{"className":63852},[371],[373,63854,63855],{"xmlns":375},[377,63856,63857,63893],{},[380,63858,63859,63865,63867,63869,63871,63873,63875,63881,63883,63885,63887,63889,63891],{},[7162,63860,63861,63863],{},[383,63862,4506],{},[458,63864,7921],{},[458,63866,461],{"stretchy":460},[383,63868,1121],{},[458,63870,468],{"stretchy":460},[458,63872,1364],{},[458,63874,471],{},[7162,63876,63877,63879],{},[383,63878,385],{},[458,63880,7168],{"mathvariant":557,"lspace":1595,"rspace":1595},[458,63882,461],{"stretchy":460},[383,63884,1116],{},[458,63886,461],{"stretchy":460},[383,63888,1121],{},[458,63890,468],{"stretchy":460},[458,63892,468],{"stretchy":460},[387,63894,63895],{"encoding":389},"g^*(x):=f'(t(x))",[363,63897,63899,63953],{"className":63898,"ariaHidden":394},[393],[363,63900,63902,63905,63934,63937,63940,63943,63946,63950],{"className":63901},[398],[363,63903],{"className":63904,"style":487},[402],[363,63906,63908,63911],{"className":63907},[407],[363,63909,4506],{"className":63910,"style":1328},[407,408],[363,63912,63914],{"className":63913},[629],[363,63915,63917],{"className":63916},[633],[363,63918,63920],{"className":63919},[638],[363,63921,63923],{"className":63922,"style":8847},[642],[363,63924,63925,63928],{"style":7213},[363,63926],{"className":63927,"style":651},[650],[363,63929,63931],{"className":63930},[655,656,657,658],[363,63932,7921],{"className":63933},[2939,658],[363,63935,461],{"className":63936},[494],[363,63938,1121],{"className":63939},[407,408],[363,63941,468],{"className":63942},[501],[363,63944],{"className":63945,"style":506},[505],[363,63947,63949],{"className":63948},[510],":=",[363,63951],{"className":63952,"style":506},[505],[363,63954,63956,63959,63991,63994,63997,64000,64003],{"className":63955},[398],[363,63957],{"className":63958,"style":7191},[402],[363,63960,63962,63965],{"className":63961},[407],[363,63963,385],{"className":63964,"style":409},[407,408],[363,63966,63968],{"className":63967},[629],[363,63969,63971],{"className":63970},[633],[363,63972,63974],{"className":63973},[638],[363,63975,63977],{"className":63976,"style":7210},[642],[363,63978,63979,63982],{"style":7213},[363,63980],{"className":63981,"style":651},[650],[363,63983,63985],{"className":63984},[655,656,657,658],[363,63986,63988],{"className":63987},[407,658],[363,63989,7168],{"className":63990},[407,658],[363,63992,461],{"className":63993},[494],[363,63995,1116],{"className":63996},[407,408],[363,63998,461],{"className":63999},[494],[363,64001,1121],{"className":64002},[407,408],[363,64004,19419],{"className":64005},[501]," 是可测的，并且对每个 ",[363,64008,64010,64023],{"className":64009,"translate":367},[366],[363,64011,64013],{"className":64012},[371],[373,64014,64015],{"xmlns":375},[377,64016,64017,64021],{},[380,64018,64019],{},[383,64020,1121],{},[387,64022,1121],{"encoding":389},[363,64024,64026],{"className":64025,"ariaHidden":394},[393],[363,64027,64029,64032],{"className":64028},[398],[363,64030],{"className":64031,"style":2693},[402],[363,64033,1121],{"className":64034},[407,408]," 达到上确界。于是不等式取等号。",[11,64037,64038,64041],{},[1344,64039,64040],{},"最优解。"," 达到上确界的函数为",[363,64043,64045],{"className":64044,"translate":367},[528],[363,64046,64048,64110],{"className":64047,"translate":367},[366],[363,64049,64051],{"className":64050},[371],[373,64052,64053],{"xmlns":375,"display":537},[377,64054,64055,64107],{},[380,64056,64057,64063,64065,64067,64069,64071,64077,64079,64105],{},[7162,64058,64059,64061],{},[383,64060,4506],{},[458,64062,7921],{},[458,64064,461],{"stretchy":460},[383,64066,1121],{},[458,64068,468],{"stretchy":460},[458,64070,471],{},[7162,64072,64073,64075],{},[383,64074,385],{},[458,64076,7168],{"mathvariant":557,"lspace":1595,"rspace":1595},[1645,64078,1652],{},[380,64080,64081,64083,64097,64099,64101,64103],{},[458,64082,461],{"fence":394},[586,64084,64085,64091],{},[380,64086,64087,64089],{},[383,64088,592],{},[383,64090,554],{},[380,64092,64093,64095],{},[383,64094,592],{},[383,64096,561],{},[458,64098,461],{"stretchy":460},[383,64100,1121],{},[458,64102,468],{"stretchy":460},[458,64104,468],{"fence":394},[458,64106,898],{"separator":394},[387,64108,64109],{"encoding":389},"g^*(x)=f'\\!\\left(\\frac{dP}{dQ}(x)\\right),",[363,64111,64113,64166],{"className":64112,"ariaHidden":394},[393],[363,64114,64116,64119,64148,64151,64154,64157,64160,64163],{"className":64115},[398],[363,64117],{"className":64118,"style":487},[402],[363,64120,64122,64125],{"className":64121},[407],[363,64123,4506],{"className":64124,"style":1328},[407,408],[363,64126,64128],{"className":64127},[629],[363,64129,64131],{"className":64130},[633],[363,64132,64134],{"className":64133},[638],[363,64135,64137],{"className":64136,"style":8596},[642],[363,64138,64139,64142],{"style":8625},[363,64140],{"className":64141,"style":651},[650],[363,64143,64145],{"className":64144},[655,656,657,658],[363,64146,7921],{"className":64147},[2939,658],[363,64149,461],{"className":64150},[494],[363,64152,1121],{"className":64153},[407,408],[363,64155,468],{"className":64156},[501],[363,64158],{"className":64159,"style":506},[505],[363,64161,471],{"className":64162},[510],[363,64164],{"className":64165,"style":506},[505],[363,64167,64169,64172,64204,64207,64210,64302,64305],{"className":64168},[398],[363,64170],{"className":64171,"style":707},[402],[363,64173,64175,64178],{"className":64174},[407],[363,64176,385],{"className":64177,"style":409},[407,408],[363,64179,64181],{"className":64180},[629],[363,64182,64184],{"className":64183},[633],[363,64185,64187],{"className":64186},[638],[363,64188,64190],{"className":64189,"style":29343},[642],[363,64191,64192,64195],{"style":8625},[363,64193],{"className":64194,"style":651},[650],[363,64196,64198],{"className":64197},[655,656,657,658],[363,64199,64201],{"className":64200},[407,658],[363,64202,7168],{"className":64203},[407,658],[363,64205],{"className":64206,"style":1860},[505],[363,64208],{"className":64209,"style":754},[505],[363,64211,64213,64219,64287,64290,64293,64296],{"className":64212},[758],[363,64214,64216],{"className":64215,"style":763},[494,762],[363,64217,461],{"className":64218},[767,657],[363,64220,64222,64225,64284],{"className":64221},[407],[363,64223],{"className":64224},[494,789],[363,64226,64228],{"className":64227},[586],[363,64229,64231,64276],{"className":64230},[633,634],[363,64232,64234,64273],{"className":64233},[638],[363,64235,64237,64251,64259],{"className":64236,"style":802},[642],[363,64238,64239,64242],{"style":805},[363,64240],{"className":64241,"style":809},[650],[363,64243,64245,64248],{"className":64244},[407],[363,64246,592],{"className":64247},[407,408],[363,64249,561],{"className":64250},[407,408],[363,64252,64253,64256],{"style":821},[363,64254],{"className":64255,"style":809},[650],[363,64257],{"className":64258,"style":829},[828],[363,64260,64261,64264],{"style":832},[363,64262],{"className":64263,"style":809},[650],[363,64265,64267,64270],{"className":64266},[407],[363,64268,592],{"className":64269},[407,408],[363,64271,554],{"className":64272,"style":682},[407,408],[363,64274,666],{"className":64275},[665],[363,64277,64279],{"className":64278},[638],[363,64280,64282],{"className":64281,"style":854},[642],[363,64283],{},[363,64285],{"className":64286},[501,789],[363,64288,461],{"className":64289},[494],[363,64291,1121],{"className":64292},[407,408],[363,64294,468],{"className":64295},[501],[363,64297,64299],{"className":64298,"style":763},[501,762],[363,64300,468],{"className":64301},[767,657],[363,64303],{"className":64304,"style":754},[505],[363,64306,898],{"className":64307},[920],[11,64309,64310,64311,4201,64339,64367,64368,64428,64429,64501,64502,45980,64621,64805],{},"条件是 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",[363,64369,64371,64389],{"className":64370,"translate":367},[366],[363,64372,64374],{"className":64373},[371],[373,64375,64376],{"xmlns":375},[377,64377,64378,64386],{},[380,64379,64380],{},[7162,64381,64382,64384],{},[383,64383,4506],{},[458,64385,7921],{},[387,64387,64388],{"encoding":389},"g^*",[363,64390,64392],{"className":64391,"ariaHidden":394},[393],[363,64393,64395,64399],{"className":64394},[398],[363,64396],{"className":64397,"style":64398},[402],"height:0.8831em;vertical-align:-0.1944em;",[363,64400,64402,64405],{"className":64401},[407],[363,64403,4506],{"className":64404,"style":1328},[407,408],[363,64406,64408],{"className":64407},[629],[363,64409,64411],{"className":64410},[633],[363,64412,64414],{"className":64413},[638],[363,64415,64417],{"className":64416,"style":8847},[642],[363,64418,64419,64422],{"style":7213},[363,64420],{"className":64421,"style":651},[650],[363,64423,64425],{"className":64424},[655,656,657,658],[363,64426,7921],{"className":64427},[2939,658]," 落在 ",[363,64430,64432,64454],{"className":64431,"translate":367},[366],[363,64433,64435],{"className":64434},[371],[373,64436,64437],{"xmlns":375},[377,64438,64439,64451],{},[380,64440,64441,64443,64445],{},[383,64442,61790],{"mathvariant":557},[458,64444,4451],{},[7162,64446,64447,64449],{},[383,64448,385],{},[458,64450,7921],{},[387,64452,64453],{"encoding":389},"\\operatorname{dom} f^*",[363,64455,64457],{"className":64456,"ariaHidden":394},[393],[363,64458,64460,64463,64469,64472],{"className":64459},[398],[363,64461],{"className":64462,"style":403},[402],[363,64464,64466],{"className":64465},[1832],[363,64467,61790],{"className":64468},[407,33034],[363,64470],{"className":64471,"style":754},[505],[363,64473,64475,64478],{"className":64474},[407],[363,64476,385],{"className":64477,"style":409},[407,408],[363,64479,64481],{"className":64480},[629],[363,64482,64484],{"className":64483},[633],[363,64485,64487],{"className":64486},[638],[363,64488,64490],{"className":64489,"style":8847},[642],[363,64491,64492,64495],{"style":7213},[363,64493],{"className":64494,"style":651},[650],[363,64496,64498],{"className":64497},[655,656,657,658],[363,64499,7921],{"className":64500},[2939,658]," 内。例如 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log 密度比」是 f-GAN 的最佳策略。",[11,64807,64808,64870,64871,64970,64971,45980],{},[1344,64809,64810,64811,64869],{},"常用 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可加任意常数 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R",[363,65353,65355],{"className":65354,"ariaHidden":394},[393],[363,65356,65358,65361],{"className":65357},[398],[363,65359],{"className":65360,"style":1445},[402],[363,65362,1381],{"className":65363},[407,714],[52967,65365,65366],{},"MINE-f \u002F 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{},[380,67041,67042,67044,67046],{},[463,67043,465],{},[458,67045,5161],{},[383,67047,40609],{},[387,67049,67050],{"encoding":389},"\\frac{y}{1-y}",[363,67052,67054],{"className":67053,"ariaHidden":394},[393],[363,67055,67057,67060],{"className":67056},[398],[363,67058],{"className":67059,"style":3432},[402],[363,67061,67063,67066,67131],{"className":67062},[407],[363,67064],{"className":67065},[494,789],[363,67067,67069],{"className":67068},[586],[363,67070,67072,67123],{"className":67071},[633,634],[363,67073,67075,67120],{"className":67074},[638],[363,67076,67078,67098,67106],{"className":67077,"style":3451},[642],[363,67079,67080,67083],{"style":2875},[363,67081],{"className":67082,"style":809},[650],[363,67084,67086],{"className":67085},[655,656,657,658],[363,67087,67089,67092,67095],{"className":67088},[407,658],[363,67090,465],{"className":67091},[407,658],[363,67093,5161],{"className":67094},[2939,658],[363,67096,40609],{"className":67097,"style":1328},[407,408,658],[363,67099,67100,67103],{"style":821},[363,67101],{"className":67102,"style":809},[650],[363,67104],{"className":67105,"style":829},[828],[363,67107,67108,67111],{"style":3201},[363,67109],{"className":67110,"style":809},[650],[363,67112,67114],{"className":67113},[655,656,657,658],[363,67115,67117],{"className":67116},[407,658],[363,67118,40609],{"className":67119,"style":1328},[407,408,658],[363,67121,666],{"className":67122},[665],[363,67124,67126],{"className":67125},[638],[363,67127,67129],{"className":67128,"style":3087},[642],[363,67130],{},[363,67132],{"className":67133},[501,789],[52967,67135,67136],{},[363,67137,67139,67157],{"className":67138,"translate":367},[366],[363,67140,67142],{"className":67141},[371],[373,67143,67144],{"xmlns":375},[377,67145,67146,67154],{},[380,67147,67148,67150,67152],{},[383,67149,40609],{},[458,67151,7599],{},[463,67153,465],{},[387,67155,67156],{"encoding":389},"y\u003C1",[363,67158,67160,67178],{"className":67159,"ariaHidden":394},[393],[363,67161,67163,67166,67169,67172,67175],{"className":67162},[398],[363,67164],{"className":67165,"style":32064},[402],[363,67167,40609],{"className":67168,"style":1328},[407,408],[363,67170],{"className":67171,"style":506},[505],[363,67173,7599],{"className":67174},[510],[363,67176],{"className":67177,"style":506},[505],[363,67179,67181,67184],{"className":67180},[398],[363,67182],{"className":67183,"style":520},[402],[363,67185,465],{"className":67186},[407],[52967,67188,66544],{},[11,67190,67191,67192,67250,67251,67497,67498,67659,67660,67925,67926,21328,68086,68202],{},"以 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为例验证：",[363,67252,67254,67314],{"className":67253,"translate":367},[366],[363,67255,67257],{"className":67256},[371],[373,67258,67259],{"xmlns":375},[377,67260,67261,67311],{},[380,67262,67263,67269,67271,67273,67275,67277,67287,67289,67291,67293,67295,67297,67299,67301,67303,67309],{},[7162,67264,67265,67267],{},[383,67266,385],{},[458,67268,7921],{},[458,67270,461],{"stretchy":460},[383,67272,40609],{},[458,67274,468],{"stretchy":460},[458,67276,471],{},[543,67278,67279,67285],{},[380,67280,67281,67283],{},[383,67282,35804],{},[458,67284,4451],{},[383,67286,1116],{},[458,67288,3246],{"stretchy":460},[383,67290,1116],{},[383,67292,40609],{},[458,67294,5161],{},[458,67296,461],{"stretchy":460},[383,67298,1116],{},[458,67300,5161],{},[463,67302,465],{},[7162,67304,67305,67307],{},[458,67306,468],{"stretchy":460},[463,67308,2811],{},[458,67310,3256],{"stretchy":460},[387,67312,67313],{"encoding":389},"f^*(y)=\\sup_t\\{ty-(t-1)^2\\}",[363,67315,67317,67370,67435,67456],{"className":67316,"ariaHidden":394},[393],[363,67318,67320,67323,67352,67355,67358,67361,67364,67367],{"className":67319},[398],[363,67321],{"className":67322,"style":487},[402],[363,67324,67326,67329],{"className":67325},[407],[363,67327,385],{"className":67328,"style":409},[407,408],[363,67330,67332],{"className":67331},[629],[363,67333,67335],{"className":67334},[633],[363,67336,67338],{"className":67337},[638],[363,67339,67341],{"className":67340,"style":8847},[642],[363,67342,67343,67346],{"style":7213},[363,67344],{"className":67345,"style":651},[650],[363,67347,67349],{"className":67348},[655,656,657,658],[363,67350,7921],{"className":67351},[2939,658],[363,67353,461],{"className":67354},[494],[363,67356,40609],{"className":67357,"style":1328},[407,408],[363,67359,468],{"className":67360},[501],[363,67362],{"className":67363,"style":506},[505],[363,67365,471],{"className":67366},[510],[363,67368],{"className":67369,"style":506},[505],[363,67371,67373,67376,67417,67420,67423,67426,67429,67432],{"className":67372},[398],[363,67374],{"className":67375,"style":487},[402],[363,67377,67379,67382],{"className":67378},[1832],[363,67380,35804],{"className":67381},[1832],[363,67383,67385],{"className":67384},[629],[363,67386,67388,67409],{"className":67387},[633,634],[363,67389,67391,67406],{"className":67390},[638],[363,67392,67395],{"className":67393,"style":67394},[642],"height:0.1864em;",[363,67396,67397,67400],{"style":35950},[363,67398],{"className":67399,"style":651},[650],[363,67401,67403],{"className":67402},[655,656,657,658],[363,67404,1116],{"className":67405},[407,408,658],[363,67407,666],{"className":67408},[665],[363,67410,67412],{"className":67411},[638],[363,67413,67415],{"className":67414,"style":37392},[642],[363,67416],{},[363,67418,3246],{"className":67419},[494],[363,67421,1116],{"className":67422},[407,408],[363,67424,40609],{"className":67425,"style":1328},[407,408],[363,67427],{"className":67428,"style":2935},[505],[363,67430,5161],{"className":67431},[2939],[363,67433],{"className":67434,"style":2935},[505],[363,67436,67438,67441,67444,67447,67450,67453],{"className":67437},[398],[363,67439],{"className":67440,"style":487},[402],[363,67442,461],{"className":67443},[494],[363,67445,1116],{"className":67446},[407,408],[363,67448],{"className":67449,"style":2935},[505],[363,67451,5161],{"className":67452},[2939],[363,67454],{"className":67455,"style":2935},[505],[363,67457,67459,67462,67465,67494],{"className":67458},[398],[363,67460],{"className":67461,"style":20375},[402],[363,67463,465],{"className":67464},[407],[363,67466,67468,67471],{"className":67467},[501],[363,67469,468],{"className":67470},[501],[363,67472,67474],{"className":67473},[629],[363,67475,67477],{"className":67476},[633],[363,67478,67480],{"className":67479},[638],[363,67481,67483],{"className":67482,"style":20400},[642],[363,67484,67485,67488],{"style":7213},[363,67486],{"className":67487,"style":651},[650],[363,67489,67491],{"className":67490},[655,656,657,658],[363,67492,2811],{"className":67493},[407,658],[363,67495,3256],{"className":67496},[501],"，驻点 ",[363,67499,67501,67545],{"className":67500,"translate":367},[366],[363,67502,67504],{"className":67503},[371],[373,67505,67506],{"xmlns":375},[377,67507,67508,67542],{},[380,67509,67510,67512,67514,67516,67518,67520,67522,67524,67526,67528,67530,67532,67534,67536,67538,67540],{},[383,67511,40609],{},[458,67513,471],{},[463,67515,2811],{},[458,67517,461],{"stretchy":460},[383,67519,1116],{},[458,67521,5161],{},[463,67523,465],{},[458,67525,468],{"stretchy":460},[458,67527,11340],{},[383,67529,1116],{},[458,67531,471],{},[463,67533,465],{},[458,67535,2818],{},[383,67537,40609],{},[383,67539,1288],{"mathvariant":557},[463,67541,2811],{},[387,67543,67544],{"encoding":389},"y=2(t-1)\\Rightarrow t=1+y\u002F2",[363,67546,67548,67566,67590,67611,67629,67647],{"className":67547,"ariaHidden":394},[393],[363,67549,67551,67554,67557,67560,67563],{"className":67550},[398],[363,67552],{"className":67553,"style":2624},[402],[363,67555,40609],{"className":67556,"style":1328},[407,408],[363,67558],{"className":67559,"style":506},[505],[363,67561,471],{"className":67562},[510],[363,67564],{"className":67565,"style":506},[505],[363,67567,67569,67572,67575,67578,67581,67584,67587],{"className":67568},[398],[363,67570],{"className":67571,"style":487},[402],[363,67573,2811],{"className":67574},[407],[363,67576,461],{"className":67577},[494],[363,67579,1116],{"className":67580},[407,408],[363,67582],{"className":67583,"style":2935},[505],[363,67585,5161],{"className":67586},[2939],[363,67588],{"className":67589,"style":2935},[505],[363,67591,67593,67596,67599,67602,67605,67608],{"className":67592},[398],[363,67594],{"className":67595,"style":487},[402],[363,67597,465],{"className":67598},[407],[363,67600,468],{"className":67601},[501],[363,67603],{"className":67604,"style":506},[505],[363,67606,11340],{"className":67607},[510],[363,67609],{"className":67610,"style":506},[505],[363,67612,67614,67617,67620,67623,67626],{"className":67613},[398],[363,67615],{"className":67616,"style":5691},[402],[363,67618,1116],{"className":67619},[407,408],[363,67621],{"className":67622,"style":506},[505],[363,67624,471],{"className":67625},[510],[363,67627],{"className":67628,"style":506},[505],[363,67630,67632,67635,67638,67641,67644],{"className":67631},[398],[363,67633],{"className":67634,"style":12928},[402],[363,67636,465],{"className":67637},[407],[363,67639],{"className":67640,"style":2935},[505],[363,67642,2818],{"className":67643},[2939],[363,67645],{"className":67646,"style":2935},[505],[363,67648,67650,67653,67656],{"className":67649},[398],[363,67651],{"className":67652,"style":487},[402],[363,67654,40609],{"className":67655,"style":1328},[407,408],[363,67657,45522],{"className":67658},[407],"，得 ",[363,67661,67663,67727],{"className":67662,"translate":367},[366],[363,67664,67666],{"className":67665},[371],[373,67667,67668],{"xmlns":375},[377,67669,67670,67724],{},[380,67671,67672,67678,67680,67682,67684,67686,67688,67690,67692,67694,67696,67698,67704,67706,67708,67710,67712,67714,67720,67722],{},[7162,67673,67674,67676],{},[383,67675,385],{},[458,67677,7921],{},[458,67679,471],{},[383,67681,40609],{},[458,67683,461],{"stretchy":460},[463,67685,465],{},[458,67687,2818],{},[383,67689,40609],{},[383,67691,1288],{"mathvariant":557},[463,67693,2811],{},[458,67695,468],{"stretchy":460},[458,67697,5161],{},[7162,67699,67700,67702],{},[383,67701,40609],{},[463,67703,2811],{},[383,67705,1288],{"mathvariant":557},[463,67707,39361],{},[458,67709,471],{},[383,67711,40609],{},[458,67713,2818],{},[7162,67715,67716,67718],{},[383,67717,40609],{},[463,67719,2811],{},[383,67721,1288],{"mathvariant":557},[463,67723,39361],{},[387,67725,67726],{"encoding":389},"f^*=y(1+y\u002F2)-y^2\u002F4=y+y^2\u002F4",[363,67728,67730,67774,67798,67822,67869,67887],{"className":67729,"ariaHidden":394},[393],[363,67731,67733,67736,67765,67768,67771],{"className":67732},[398],[363,67734],{"className":67735,"style":403},[402],[363,67737,67739,67742],{"className":67738},[407],[363,67740,385],{"className":67741,"style":409},[407,408],[363,67743,67745],{"className":67744},[629],[363,67746,67748],{"className":67747},[633],[363,67749,67751],{"className":67750},[638],[363,67752,67754],{"className":67753,"style":8847},[642],[363,67755,67756,67759],{"style":7213},[363,67757],{"className":67758,"style":651},[650],[363,67760,67762],{"className":67761},[655,656,657,658],[363,67763,7921],{"className":67764},[2939,658],[363,67766],{"className":67767,"style":506},[505],[363,67769,471],{"className":67770},[510],[363,67772],{"className":67773,"style":506},[505],[363,67775,67777,67780,67783,67786,67789,67792,67795],{"className":67776},[398],[363,67778],{"className":67779,"style":487},[402],[363,67781,40609],{"className":67782,"style":1328},[407,408],[363,67784,461],{"className":67785},[494],[363,67787,465],{"className":67788},[407],[363,67790],{"className":67791,"style":2935},[505],[363,67793,2818],{"className":67794},[2939],[363,67796],{"className":67797,"style":2935},[505],[363,67799,67801,67804,67807,67810,67813,67816,67819],{"className":67800},[398],[363,67802],{"className":67803,"style":487},[402],[363,67805,40609],{"className":67806,"style":1328},[407,408],[363,67808,45522],{"className":67809},[407],[363,67811,468],{"className":67812},[501],[363,67814],{"className":67815,"style":2935},[505],[363,67817,5161],{"className":67818},[2939],[363,67820],{"className":67821,"style":2935},[505],[363,67823,67825,67828,67857,67860,67863,67866],{"className":67824},[398],[363,67826],{"className":67827,"style":20375},[402],[363,67829,67831,67834],{"className":67830},[407],[363,67832,40609],{"className":67833,"style":1328},[407,408],[363,67835,67837],{"className":67836},[629],[363,67838,67840],{"className":67839},[633],[363,67841,67843],{"className":67842},[638],[363,67844,67846],{"className":67845,"style":20400},[642],[363,67847,67848,67851],{"style":7213},[363,67849],{"className":67850,"style":651},[650],[363,67852,67854],{"className":67853},[655,656,657,658],[363,67855,2811],{"className":67856},[407,658],[363,67858,40744],{"className":67859},[407],[363,67861],{"className":67862,"style":506},[505],[363,67864,471],{"className":67865},[510],[363,67867],{"className":67868,"style":506},[505],[363,67870,67872,67875,67878,67881,67884],{"className":67871},[398],[363,67873],{"className":67874,"style":40693},[402],[363,67876,40609],{"className":67877,"style":1328},[407,408],[363,67879],{"className":67880,"style":2935},[505],[363,67882,2818],{"className":67883},[2939],[363,67885],{"className":67886,"style":2935},[505],[363,67888,67890,67893,67922],{"className":67889},[398],[363,67891],{"className":67892,"style":20375},[402],[363,67894,67896,67899],{"className":67895},[407],[363,67897,40609],{"className":67898,"style":1328},[407,408],[363,67900,67902],{"className":67901},[629],[363,67903,67905],{"className":67904},[633],[363,67906,67908],{"className":67907},[638],[363,67909,67911],{"className":67910,"style":20400},[642],[363,67912,67913,67916],{"style":7213},[363,67914],{"className":67915,"style":651},[650],[363,67917,67919],{"className":67918},[655,656,657,658],[363,67920,2811],{"className":67921},[407,658],[363,67923,40744],{"className":67924},[407],"。以 Hellinger² 为例：",[363,67927,67929,67965],{"className":67928,"translate":367},[366],[363,67930,67932],{"className":67931},[371],[373,67933,67934],{"xmlns":375},[377,67935,67936,67962],{},[380,67937,67938,67940,67942,67944,67946,67948,67950,67952,67954,67958,67960],{},[383,67939,385],{},[458,67941,461],{"stretchy":460},[383,67943,1116],{},[458,67945,468],{"stretchy":460},[458,67947,471],{},[383,67949,1116],{},[458,67951,5161],{},[463,67953,2811],{},[34711,67955,67956],{},[383,67957,1116],{},[458,67959,2818],{},[463,67961,465],{},[387,67963,67964],{"encoding":389},"f(t)=t-2\\sqrt t+1",[363,67966,67968,67995,68014,68077],{"className":67967,"ariaHidden":394},[393],[363,67969,67971,67974,67977,67980,67983,67986,67989,67992],{"className":67970},[398],[363,67972],{"className":67973,"style":487},[402],[363,67975,385],{"className":67976,"style":409},[407,408],[363,67978,461],{"className":67979},[494],[363,67981,1116],{"className":67982},[407,408],[363,67984,468],{"className":67985},[501],[363,67987],{"className":67988,"style":506},[505],[363,67990,471],{"className":67991},[510],[363,67993],{"className":67994,"style":506},[505],[363,67996,67998,68002,68005,68008,68011],{"className":67997},[398],[363,67999],{"className":68000,"style":68001},[402],"height:0.6984em;vertical-align:-0.0833em;",[363,68003,1116],{"className":68004},[407,408],[363,68006],{"className":68007,"style":2935},[505],[363,68009,5161],{"className":68010},[2939],[363,68012],{"className":68013,"style":2935},[505],[363,68015,68017,68021,68024,68068,68071,68074],{"className":68016},[398],[363,68018],{"className":68019,"style":68020},[402],"height:1.04em;vertical-align:-0.1475em;",[363,68022,2811],{"className":68023},[407],[363,68025,68027],{"className":68026},[407,34739],[363,68028,68030,68060],{"className":68029},[633,634],[363,68031,68033,68057],{"className":68032},[638],[363,68034,68036,68045],{"className":68035,"style":43055},[642],[363,68037,68039,68042],{"className":68038,"style":12213},[6655],[363,68040],{"className":68041,"style":809},[650],[363,68043,1116],{"className":68044,"style":34759},[407,408],[363,68046,68047,68050],{"style":43067},[363,68048],{"className":68049,"style":809},[650],[363,68051,68053],{"className":68052,"style":34816},[34815],[6671,68054,68055],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},[6679,68056],{"d":34823},[363,68058,666],{"className":68059},[665],[363,68061,68063],{"className":68062},[638],[363,68064,68066],{"className":68065,"style":43087},[642],[363,68067],{},[363,68069],{"className":68070,"style":2935},[505],[363,68072,2818],{"className":68073},[2939],[363,68075],{"className":68076,"style":2935},[505],[363,68078,68080,68083],{"className":68079},[398],[363,68081],{"className":68082,"style":520},[402],[363,68084,465],{"className":68085},[407],[363,68087,68089,68113],{"className":68088,"translate":367},[366],[363,68090,68092],{"className":68091},[371],[373,68093,68094],{"xmlns":375},[377,68095,68096,68110],{},[380,68097,68098,68100,68102,68106,68108],{},[383,68099,15750],{},[458,68101,471],{},[34711,68103,68104],{},[383,68105,1116],{},[458,68107,7276],{},[463,68109,474],{},[387,68111,68112],{"encoding":389},"u=\\sqrt t\\ge0",[363,68114,68116,68134,68193],{"className":68115,"ariaHidden":394},[393],[363,68117,68119,68122,68125,68128,68131],{"className":68118},[398],[363,68120],{"className":68121,"style":2693},[402],[363,68123,15750],{"className":68124},[407,408],[363,68126],{"className":68127,"style":506},[505],[363,68129,471],{"className":68130},[510],[363,68132],{"className":68133,"style":506},[505],[363,68135,68137,68140,68184,68187,68190],{"className":68136},[398],[363,68138],{"className":68139,"style":68020},[402],[363,68141,68143],{"className":68142},[407,34739],[363,68144,68146,68176],{"className":68145},[633,634],[363,68147,68149,68173],{"className":68148},[638],[363,68150,68152,68161],{"className":68151,"style":43055},[642],[363,68153,68155,68158],{"className":68154,"style":12213},[6655],[363,68156],{"className":68157,"style":809},[650],[363,68159,1116],{"className":68160,"style":34759},[407,408],[363,68162,68163,68166],{"style":43067},[363,68164],{"className":68165,"style":809},[650],[363,68167,68169],{"className":68168,"style":34816},[34815],[6671,68170,68171],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},[6679,68172],{"d":34823},[363,68174,666],{"className":68175},[665],[363,68177,68179],{"className":68178},[638],[363,68180,68182],{"className":68181,"style":43087},[642],[363,68183],{},[363,68185],{"className":68186,"style":506},[505],[363,68188,7276],{"className":68189},[510],[363,68191],{"className":68192,"style":506},[505],[363,68194,68196,68199],{"className":68195},[398],[363,68197],{"className":68198,"style":520},[402],[363,68200,474],{"className":68201},[407],"，把共轭写成有限上确界",[363,68204,68206],{"className":68205,"translate":367},[528],[363,68207,68209,68283],{"className":68208,"translate":367},[366],[363,68210,68212],{"className":68211},[371],[373,68213,68214],{"xmlns":375,"display":537},[377,68215,68216,68280],{},[380,68217,68218,68224,68226,68228,68230,68232,68248,68250,68252,68254,68256,68258,68260,68266,68268,68270,68272,68274,68276,68278],{},[7162,68219,68220,68222],{},[383,68221,385],{},[458,68223,7921],{},[458,68225,461],{"stretchy":460},[383,68227,40609],{},[458,68229,468],{"stretchy":460},[458,68231,471],{},[6364,68233,68234,68240],{},[380,68235,68236,68238],{},[383,68237,35804],{},[458,68239,4451],{},[380,68241,68242,68244,68246],{},[383,68243,15750],{},[458,68245,7276],{},[463,68247,474],{},[458,68249,3246],{"fence":460,"stretchy":394,"minsize":17789,"maxsize":17789},[458,68251,461],{"stretchy":460},[383,68253,40609],{},[458,68255,5161],{},[463,68257,465],{},[458,68259,468],{"stretchy":460},[7162,68261,68262,68264],{},[383,68263,15750],{},[463,68265,2811],{},[458,68267,2818],{},[463,68269,2811],{},[383,68271,15750],{},[458,68273,5161],{},[463,68275,465],{},[458,68277,3256],{"fence":460,"stretchy":394,"minsize":17789,"maxsize":17789},[458,68279,898],{"separator":394},[387,68281,68282],{"encoding":389},"f^*(y)=\\sup_{u\\ge0}\\Big\\{(y-1)u^2+2u-1\\Big\\},",[363,68284,68286,68339,68423,68473,68494],{"className":68285,"ariaHidden":394},[393],[363,68287,68289,68292,68321,68324,68327,68330,68333,68336],{"className":68288},[398],[363,68290],{"className":68291,"style":487},[402],[363,68293,68295,68298],{"className":68294},[407],[363,68296,385],{"className":68297,"style":409},[407,408],[363,68299,68301],{"className":68300},[629],[363,68302,68304],{"className":68303},[633],[363,68305,68307],{"className":68306},[638],[363,68308,68310],{"className":68309,"style":8596},[642],[363,68311,68312,68315],{"style":8625},[363,68313],{"className":68314,"style":651},[650],[363,68316,68318],{"className":68317},[655,656,657,658],[363,68319,7921],{"className":68320},[2939,658],[363,68322,461],{"className":68323},[494],[363,68325,40609],{"className":68326,"style":1328},[407,408],[363,68328,468],{"className":68329},[501],[363,68331],{"className":68332,"style":506},[505],[363,68334,471],{"className":68335},[510],[363,68337],{"className":68338,"style":506},[505],[363,68340,68342,68346,68399,68402,68408,68411,68414,68417,68420],{"className":68341},[398],[363,68343],{"className":68344,"style":68345},[402],"height:2.1567em;vertical-align:-1.0067em;",[363,68347,68349],{"className":68348},[1832,14066],[363,68350,68352,68391],{"className":68351},[633,634],[363,68353,68355,68388],{"className":68354},[638],[363,68356,68358,68378],{"className":68357,"style":2693},[642],[363,68359,68360,68363],{"style":61305},[363,68361],{"className":68362,"style":809},[650],[363,68364,68366],{"className":68365},[655,656,657,658],[363,68367,68369,68372,68375],{"className":68368},[407,658],[363,68370,15750],{"className":68371},[407,408,658],[363,68373,7276],{"className":68374},[510,658],[363,68376,474],{"className":68377},[407,658],[363,68379,68380,68383],{"style":12213},[363,68381],{"className":68382,"style":809},[650],[363,68384,68385],{},[363,68386,35804],{"className":68387},[1832],[363,68389,666],{"className":68390},[665],[363,68392,68394],{"className":68393},[638],[363,68395,68397],{"className":68396,"style":61343},[642],[363,68398],{},[363,68400],{"className":68401,"style":754},[505],[363,68403,68405],{"className":68404},[407],[363,68406,3246],{"className":68407},[767,4068],[363,68409,461],{"className":68410},[494],[363,68412,40609],{"className":68413,"style":1328},[407,408],[363,68415],{"className":68416,"style":2935},[505],[363,68418,5161],{"className":68419},[2939],[363,68421],{"className":68422,"style":2935},[505],[363,68424,68426,68429,68432,68435,68464,68467,68470],{"className":68425},[398],[363,68427],{"className":68428,"style":37623},[402],[363,68430,465],{"className":68431},[407],[363,68433,468],{"className":68434},[501],[363,68436,68438,68441],{"className":68437},[407],[363,68439,15750],{"className":68440},[407,408],[363,68442,68444],{"className":68443},[629],[363,68445,68447],{"className":68446},[633],[363,68448,68450],{"className":68449},[638],[363,68451,68453],{"className":68452,"style":20107},[642],[363,68454,68455,68458],{"style":8625},[363,68456],{"className":68457,"style":651},[650],[363,68459,68461],{"className":68460},[655,656,657,658],[363,68462,2811],{"className":68463},[407,658],[363,68465],{"className":68466,"style":2935},[505],[363,68468,2818],{"className":68469},[2939],[363,68471],{"className":68472,"style":2935},[505],[363,68474,68476,68479,68482,68485,68488,68491],{"className":68475},[398],[363,68477],{"className":68478,"style":12928},[402],[363,68480,2811],{"className":68481},[407],[363,68483,15750],{"className":68484},[407,408],[363,68486],{"className":68487,"style":2935},[505],[363,68489,5161],{"className":68490},[2939],[363,68492],{"className":68493,"style":2935},[505],[363,68495,68497,68501,68504,68510],{"className":68496},[398],[363,68498],{"className":68499,"style":68500},[402],"height:1.8em;vertical-align:-0.65em;",[363,68502,465],{"className":68503},[407],[363,68505,68507],{"className":68506},[407],[363,68508,3256],{"className":68509},[767,4068],[363,68511,898],{"className":68512},[920],[11,68514,68515,68516,68567,68568,68601,68602,68652,68653,68787],{},"当 ",[363,68517,68519,68537],{"className":68518,"translate":367},[366],[363,68520,68522],{"className":68521},[371],[373,68523,68524],{"xmlns":375},[377,68525,68526,68534],{},[380,68527,68528,68530,68532],{},[383,68529,40609],{},[458,68531,7276],{},[463,68533,465],{},[387,68535,68536],{"encoding":389},"y\\ge1",[363,68538,68540,68558],{"className":68539,"ariaHidden":394},[393],[363,68541,68543,68546,68549,68552,68555],{"className":68542},[398],[363,68544],{"className":68545,"style":15767},[402],[363,68547,40609],{"className":68548,"style":1328},[407,408],[363,68550],{"className":68551,"style":506},[505],[363,68553,7276],{"className":68554},[510],[363,68556],{"className":68557,"style":506},[505],[363,68559,68561,68564],{"className":68560},[398],[363,68562],{"className":68563,"style":520},[402],[363,68565,465],{"className":68566},[407]," 时上确界为 ",[363,68569,68571,68586],{"className":68570,"translate":367},[366],[363,68572,68574],{"className":68573},[371],[373,68575,68576],{"xmlns":375},[377,68577,68578,68584],{},[380,68579,68580,68582],{},[458,68581,2818],{},[383,68583,1373],{"mathvariant":557},[387,68585,4589],{"encoding":389},[363,68587,68589],{"className":68588,"ariaHidden":394},[393],[363,68590,68592,68595,68598],{"className":68591},[398],[363,68593],{"className":68594,"style":4599},[402],[363,68596,2818],{"className":68597},[407],[363,68599,1373],{"className":68600},[407],"；当 ",[363,68603,68605,68622],{"className":68604,"translate":367},[366],[363,68606,68608],{"className":68607},[371],[373,68609,68610],{"xmlns":375},[377,68611,68612,68620],{},[380,68613,68614,68616,68618],{},[383,68615,40609],{},[458,68617,7599],{},[463,68619,465],{},[387,68621,67156],{"encoding":389},[363,68623,68625,68643],{"className":68624,"ariaHidden":394},[393],[363,68626,68628,68631,68634,68637,68640],{"className":68627},[398],[363,68629],{"className":68630,"style":32064},[402],[363,68632,40609],{"className":68633,"style":1328},[407,408],[363,68635],{"className":68636,"style":506},[505],[363,68638,7599],{"className":68639},[510],[363,68641],{"className":68642,"style":506},[505],[363,68644,68646,68649],{"className":68645},[398],[363,68647],{"className":68648,"style":520},[402],[363,68650,465],{"className":68651},[407]," 时上式是开口向下的二次函数，最大点 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y\u003C1 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",[363,69993,69995,70029],{"className":69994,"translate":367},[366],[363,69996,69998],{"className":69997},[371],[373,69999,70000],{"xmlns":375},[377,70001,70002,70026],{},[380,70003,70004,70006,70008,70020,70022,70024],{},[383,70005,40609],{},[458,70007,471],{},[7901,70009,70010,70012,70018],{},[383,70011,385],{},[380,70013,70014,70016],{},[383,70015,24115],{},[383,70017,26109],{},[458,70019,7168],{"mathvariant":557,"lspace":1595,"rspace":1595},[458,70021,461],{"stretchy":460},[383,70023,1116],{},[458,70025,468],{"stretchy":460},[387,70027,70028],{"encoding":389},"y=f_{JS}'(t)",[363,70030,70032,70050],{"className":70031,"ariaHidden":394},[393],[363,70033,70035,70038,70041,70044,70047],{"className":70034},[398],[363,70036],{"className":70037,"style":2624},[402],[363,70039,40609],{"className":70040,"style":1328},[407,408],[363,70042],{"className":70043,"style":506},[505],[363,70045,471],{"className":70046},[510],[363,70048],{"className":70049,"style":506},[505],[363,70051,70053,70056,70116,70119,70122],{"className":70052},[398],[363,70054],{"className":70055,"style":69749},[402],[363,70057,70059,70062],{"className":70058},[407],[363,70060,385],{"className":70061,"style":409},[407,408],[363,70063,70065],{"className":70064},[629],[363,70066,70068,70108],{"className":70067},[633,634],[363,70069,70071,70105],{"className":70070},[638],[363,70072,70074,70091],{"className":70073,"style":7210},[642],[363,70075,70076,70079],{"style":69770},[363,70077],{"className":70078,"style":651},[650],[363,70080,70082],{"className":70081},[655,656,657,658],[363,70083,70085,70088],{"className":70084},[407,658],[363,70086,24115],{"className":70087,"style":24525},[407,408,658],[363,70089,26109],{"className":70090,"style":26233},[407,408,658],[363,70092,70093,70096],{"style":7213},[363,70094],{"className":70095,"style":651},[650],[363,70097,70099],{"className":70098},[655,656,657,658],[363,70100,70102],{"className":70101},[407,658],[363,70103,7168],{"className":70104},[407,658],[363,70106,666],{"className":70107},[665],[363,70109,70111],{"className":70110},[638],[363,70112,70114],{"className":70113,"style":38762},[642],[363,70115],{},[363,70117,461],{"className":70118},[494],[363,70120,1116],{"className":70121},[407,408],[363,70123,468],{"className":70124},[501],[363,70126,70128,70168],{"className":70127,"translate":367},[366],[363,70129,70131],{"className":70130},[371],[373,70132,70133],{"xmlns":375},[377,70134,70135,70165],{},[380,70136,70137,70147,70149],{},[7162,70138,70139,70141],{},[383,70140,7915],{},[380,70142,70143,70145],{},[463,70144,2811],{},[383,70146,40609],{},[458,70148,471],{},[586,70150,70151,70157],{},[380,70152,70153,70155],{},[463,70154,2811],{},[383,70156,1116],{},[380,70158,70159,70161,70163],{},[383,70160,1116],{},[458,70162,2818],{},[463,70164,465],{},[387,70166,70167],{"encoding":389},"e^{2y}=\\frac{2t}{t+1}",[363,70169,70171,70221],{"className":70170,"ariaHidden":394},[393],[363,70172,70174,70177,70212,70215,70218],{"className":70173},[398],[363,70175],{"className":70176,"style":20400},[402],[363,70178,70180,70183],{"className":70179},[407],[363,70181,7915],{"className":70182},[407,408],[363,70184,70186],{"className":70185},[629],[363,70187,70189],{"className":70188},[633],[363,70190,70192],{"className":70191},[638],[363,70193,70195],{"className":70194,"style":20400},[642],[363,70196,70197,70200],{"style":7213},[363,70198],{"className":70199,"style":651},[650],[363,70201,70203],{"className":70202},[655,656,657,658],[363,70204,70206,70209],{"className":70205},[407,658],[363,70207,2811],{"className":70208},[407,658],[363,70210,40609],{"className":70211,"style":1328},[407,408,658],[363,70213],{"className":70214,"style":506},[505],[363,70216,471],{"className":70217},[510],[363,70219],{"className":70220,"style":506},[505],[363,70222,70224,70227],{"className":70223},[398],[363,70225],{"className":70226,"style":45777},[402],[363,70228,70230,70233,70301],{"className":70229},[407],[363,70231],{"className":70232},[494,789],[363,70234,70236],{"className":70235},[586],[363,70237,70239,70293],{"className":70238},[633,634],[363,70240,70242,70290],{"className":70241},[638],[363,70243,70245,70265,70273],{"className":70244,"style":2872},[642],[363,70246,70247,70250],{"style":2875},[363,70248],{"className":70249,"style":809},[650],[363,70251,70253],{"className":70252},[655,656,657,658],[363,70254,70256,70259,70262],{"className":70255},[407,658],[363,70257,1116],{"className":70258},[407,408,658],[363,70260,2818],{"className":70261},[2939,658],[363,70263,465],{"className":70264},[407,658],[363,70266,70267,70270],{"style":821},[363,70268],{"className":70269,"style":809},[650],[363,70271],{"className":70272,"style":829},[828],[363,70274,70275,70278],{"style":2898},[363,70276],{"className":70277,"style":809},[650],[363,70279,70281],{"className":70280},[655,656,657,658],[363,70282,70284,70287],{"className":70283},[407,658],[363,70285,2811],{"className":70286},[407,658],[363,70288,1116],{"className":70289},[407,408,658],[363,70291,66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",[363,70306,70308,70354],{"className":70307,"translate":367},[366],[363,70309,70311],{"className":70310},[371],[373,70312,70313],{"xmlns":375},[377,70314,70315,70351],{},[380,70316,70317,70319,70321],{},[383,70318,1116],{},[458,70320,471],{},[1603,70322,70323],{"scriptlevel":474,"displaystyle":394},[586,70324,70325,70335],{},[7162,70326,70327,70329],{},[383,70328,7915],{},[380,70330,70331,70333],{},[463,70332,2811],{},[383,70334,40609],{},[380,70336,70337,70339,70341],{},[463,70338,2811],{},[458,70340,5161],{},[7162,70342,70343,70345],{},[383,70344,7915],{},[380,70346,70347,70349],{},[463,70348,2811],{},[383,70350,40609],{},[387,70352,70353],{"encoding":389},"t=\\dfrac{e^{2y}}{2-e^{2y}}",[363,70355,70357,70375],{"className":70356,"ariaHidden":394},[393],[363,70358,70360,70363,70366,70369,70372],{"className":70359},[398],[363,70361],{"className":70362,"style":5691},[402],[363,70364,1116],{"className":70365},[407,408],[363,70367],{"className":70368,"style":506},[505],[363,70370,471],{"className":70371},[510],[363,70373],{"className":70374,"style":506},[505],[363,70376,70378,70382],{"className":70377},[398],[363,70379],{"className":70380,"style":70381},[402],"height:2.2604em;vertical-align:-0.7693em;",[363,70383,70385,70388,70517],{"className":70384},[407],[363,70386],{"className":70387},[494,789],[363,70389,70391],{"className":70390},[586],[363,70392,70394,70509],{"className":70393},[633,634],[363,70395,70397,70506],{"className":70396},[638],[363,70398,70400,70455,70463],{"className":70399,"style":38095},[642],[363,70401,70402,70405],{"style":805},[363,70403],{"className":70404,"style":809},[650],[363,70406,70408,70411,70414,70417,70420],{"className":70407},[407],[363,70409,2811],{"className":70410},[407],[363,70412],{"className":70413,"style":2935},[505],[363,70415,5161],{"className":70416},[2939],[363,70418],{"className":70419,"style":2935},[505],[363,70421,70423,70426],{"className":70422},[407],[363,70424,7915],{"className":70425},[407,408],[363,70427,70429],{"className":70428},[629],[363,70430,70432],{"className":70431},[633],[363,70433,70435],{"className":70434},[638],[363,70436,70438],{"className":70437,"style":45404},[642],[363,70439,70440,70443],{"style":45407},[363,70441],{"className":70442,"style":651},[650],[363,70444,70446],{"className":70445},[655,656,657,658],[363,70447,70449,70452],{"className":70448},[407,658],[363,70450,2811],{"className":70451},[407,658],[363,70453,40609],{"className":70454,"style":1328},[407,408,658],[363,70456,70457,70460],{"style":821},[363,70458],{"className":70459,"style":809},[650],[363,70461],{"className":70462,"style":829},[828],[363,70464,70465,70468],{"style":832},[363,70466],{"className":70467,"style":809},[650],[363,70469,70471],{"className":70470},[407],[363,70472,70474,70477],{"className":70473},[407],[363,70475,7915],{"className":70476},[407,408],[363,70478,70480],{"className":70479},[629],[363,70481,70483],{"className":70482},[633],[363,70484,70486],{"className":70485},[638],[363,70487,70489],{"className":70488,"style":20400},[642],[363,70490,70491,70494],{"style":7213},[363,70492],{"className":70493,"style":651},[650],[363,70495,70497],{"className":70496},[655,656,657,658],[363,70498,70500,70503],{"className":70499},[407,658],[363,70501,2811],{"className":70502},[407,658],[363,70504,40609],{"className":70505,"style":1328},[407,408,658],[363,70507,666],{"className":70508},[665],[363,70510,70512],{"className":70511},[638],[363,70513,70515],{"className":70514,"style":17365},[642],[363,70516],{},[363,70518],{"className":70519},[501,789],"（这要求 ",[363,70522,70524,70550],{"className":70523,"translate":367},[366],[363,70525,70527],{"className":70526},[371],[373,70528,70529],{"xmlns":375},[377,70530,70531,70547],{},[380,70532,70533,70543,70545],{},[7162,70534,70535,70537],{},[383,70536,7915],{},[380,70538,70539,70541],{},[463,70540,2811],{},[383,70542,40609],{},[458,70544,7599],{},[463,70546,2811],{},[387,70548,70549],{"encoding":389},"e^{2y}\u003C2",[363,70551,70553,70604],{"className":70552,"ariaHidden":394},[393],[363,70554,70556,70560,70595,70598,70601],{"className":70555},[398],[363,70557],{"className":70558,"style":70559},[402],"height:0.8532em;vertical-align:-0.0391em;",[363,70561,70563,70566],{"className":70562},[407],[363,70564,7915],{"className":70565},[407,408],[363,70567,70569],{"className":70568},[629],[363,70570,70572],{"className":70571},[633],[363,70573,70575],{"className":70574},[638],[363,70576,70578],{"className":70577,"style":20400},[642],[363,70579,70580,70583],{"style":7213},[363,70581],{"className":70582,"style":651},[650],[363,70584,70586],{"className":70585},[655,656,657,658],[363,70587,70589,70592],{"className":70588},[407,658],[363,70590,2811],{"className":70591},[407,658],[363,70593,40609],{"className":70594,"style":1328},[407,408,658],[363,70596],{"className":70597,"style":506},[505],[363,70599,7599],{"className":70600},[510],[363,70602],{"className":70603,"style":506},[505],[363,70605,70607,70610],{"className":70606},[398],[363,70608],{"className":70609,"style":520},[402],[363,70611,2811],{"className":70612},[407],[363,70614,70616,70643],{"className":70615,"translate":367},[366],[363,70617,70619],{"className":70618},[371],[373,70620,70621],{"xmlns":375},[377,70622,70623,70641],{},[380,70624,70625,70627,70629],{},[383,70626,40609],{},[458,70628,7599],{},[586,70630,70631,70639],{},[380,70632,70633,70635,70637],{},[383,70634,4448],{},[458,70636,4451],{},[463,70638,2811],{},[463,70640,2811],{},[387,70642,66126],{"encoding":389},[363,70644,70646,70664],{"className":70645,"ariaHidden":394},[393],[363,70647,70649,70652,70655,70658,70661],{"className":70648},[398],[363,70650],{"className":70651,"style":32064},[402],[363,70653,40609],{"className":70654,"style":1328},[407,408],[363,70656],{"className":70657,"style":506},[505],[363,70659,7599],{"className":70660},[510],[363,70662],{"className":70663,"style":506},[505],[363,70665,70667,70670],{"className":70666},[398],[363,70668],{"className":70669,"style":66154},[402],[363,70671,70673,70676,70750],{"className":70672},[407],[363,70674],{"className":70675},[494,789],[363,70677,70679],{"className":70678},[586],[363,70680,70682,70742],{"className":70681},[633,634],[363,70683,70685,70739],{"className":70684},[638],[363,70686,70688,70702,70710],{"className":70687,"style":3173},[642],[363,70689,70690,70693],{"style":2875},[363,70691],{"className":70692,"style":809},[650],[363,70694,70696],{"className":70695},[655,656,657,658],[363,70697,70699],{"className":70698},[407,658],[363,70700,2811],{"className":70701},[407,658],[363,70703,70704,70707],{"style":821},[363,70705],{"className":70706,"style":809},[650],[363,70708],{"className":70709,"style":829},[828],[363,70711,70712,70715],{"style":3201},[363,70713],{"className":70714,"style":809},[650],[363,70716,70718],{"className":70717},[655,656,657,658],[363,70719,70721,70733,70736],{"className":70720},[407,658],[363,70722,70724,70727,70730],{"className":70723},[1832,658],[363,70725,48030],{"className":70726},[658],[363,70728,48034],{"className":70729},[658],[363,70731,4506],{"className":70732,"style":4505},[658],[363,70734],{"className":70735,"style":42738},[505,658],[363,70737,2811],{"className":70738},[407,658],[363,70740,666],{"className":70741},[665],[363,70743,70745],{"className":70744},[638],[363,70746,70748],{"className":70747,"style":2920},[642],[363,70749],{},[363,70751],{"className":70752},[501,789],"）。于是 ",[363,70755,70757,70811],{"className":70756,"translate":367},[366],[363,70758,70760],{"className":70759},[371],[373,70761,70762],{"xmlns":375},[377,70763,70764,70808],{},[380,70765,70766,70768,70770,70782,70784,70786,70788,70790,70792,70794,70796,70806],{},[383,70767,4448],{},[458,70769,4451],{},[586,70771,70772,70780],{},[380,70773,70774,70776,70778],{},[383,70775,1116],{},[458,70777,2818],{},[463,70779,465],{},[463,70781,2811],{},[458,70783,471],{},[458,70785,5161],{},[383,70787,4448],{},[458,70789,4451],{},[458,70791,461],{"stretchy":460},[463,70793,2811],{},[458,70795,5161],{},[7162,70797,70798,70800],{},[383,70799,7915],{},[380,70801,70802,70804],{},[463,70803,2811],{},[383,70805,40609],{},[458,70807,468],{"stretchy":460},[387,70809,70810],{"encoding":389},"\\log\\frac{t+1}{2}=-\\log(2-e^{2y})",[363,70812,70814,70911,70943],{"className":70813,"ariaHidden":394},[393],[363,70815,70817,70820,70825,70828,70902,70905,70908],{"className":70816},[398],[363,70818],{"className":70819,"style":2853},[402],[363,70821,4502,70823],{"className":70822},[1832],[363,70824,4506],{"style":4505},[363,70826],{"className":70827,"style":754},[505],[363,70829,70831,70834,70899],{"className":70830},[407],[363,70832],{"className":70833},[494,789],[363,70835,70837],{"className":70836},[586],[363,70838,70840,70891],{"className":70839},[633,634],[363,70841,70843,70888],{"className":70842},[638],[363,70844,70846,70860,70868],{"className":70845,"style":2872},[642],[363,70847,70848,70851],{"style":2875},[363,70849],{"className":70850,"style":809},[650],[363,70852,70854],{"className":70853},[655,656,657,658],[363,70855,70857],{"className":70856},[407,658],[363,70858,2811],{"className":70859},[407,658],[363,70861,70862,70865],{"style":821},[363,70863],{"className":70864,"style":809},[650],[363,70866],{"className":70867,"style":829},[828],[363,70869,70870,70873],{"style":2898},[363,70871],{"className":70872,"style":809},[650],[363,70874,70876],{"className":70875},[655,656,657,658],[363,70877,70879,70882,70885],{"className":70878},[407,658],[363,70880,1116],{"className":70881},[407,408,658],[363,70883,2818],{"className":70884},[2939,658],[363,70886,465],{"className":70887},[407,658],[363,70889,666],{"className":70890},[665],[363,70892,70894],{"className":70893},[638],[363,70895,70897],{"className":70896,"style":2920},[642],[363,70898],{},[363,70900],{"className":70901},[501,789],[363,70903],{"className":70904,"style":506},[505],[363,70906,471],{"className":70907},[510],[363,70909],{"className":70910,"style":506},[505],[363,70912,70914,70917,70920,70923,70928,70931,70934,70937,70940],{"className":70913},[398],[363,70915],{"className":70916,"style":487},[402],[363,70918,5161],{"className":70919},[407],[363,70921],{"className":70922,"style":754},[505],[363,70924,4502,70926],{"className":70925},[1832],[363,70927,4506],{"style":4505},[363,70929,461],{"className":70930},[494],[363,70932,2811],{"className":70933},[407],[363,70935],{"className":70936,"style":2935},[505],[363,70938,5161],{"className":70939},[2939],[363,70941],{"className":70942,"style":2935},[505],[363,70944,70946,70949,70984],{"className":70945},[398],[363,70947],{"className":70948,"style":20375},[402],[363,70950,70952,70955],{"className":70951},[407],[363,70953,7915],{"className":70954},[407,408],[363,70956,70958],{"className":70957},[629],[363,70959,70961],{"className":70960},[633],[363,70962,70964],{"className":70963},[638],[363,70965,70967],{"className":70966,"style":20400},[642],[363,70968,70969,70972],{"style":7213},[363,70970],{"className":70971,"style":651},[650],[363,70973,70975],{"className":70974},[655,656,657,658],[363,70976,70978,70981],{"className":70977},[407,658],[363,70979,2811],{"className":70980},[407,658],[363,70982,40609],{"className":70983,"style":1328},[407,408,658],[363,70985,468],{"className":70986},[501],"，代入",[363,70989,70991],{"className":70990,"translate":367},[528],[363,70992,70994,71192],{"className":70993,"translate":367},[366],[363,70995,70997],{"className":70996},[371],[373,70998,70999],{"xmlns":375,"display":537},[377,71000,71001,71189],{},[1591,71002,71003,71055,71137],{"rowspacing":1593,"columnalign":1594,"columnspacing":1595},[1597,71004,71005,71023],{},[1600,71006,71007],{},[1603,71008,71009],{"scriptlevel":474,"displaystyle":394},[380,71010,71011,71017,71019,71021],{},[7162,71012,71013,71015],{},[383,71014,385],{},[458,71016,7921],{},[458,71018,461],{"stretchy":460},[383,71020,40609],{},[458,71022,468],{"stretchy":460},[1600,71024,71025],{},[1603,71026,71027],{"scriptlevel":474,"displaystyle":394},[380,71028,71029,71031,71033,71035,71037,71039,71049,71051,71053],{},[380,71030],{},[458,71032,471],{},[383,71034,40609],{},[383,71036,1116],{},[458,71038,5161],{},[543,71040,71041,71043],{},[383,71042,385],{},[380,71044,71045,71047],{},[383,71046,24115],{},[383,71048,26109],{},[458,71050,461],{"stretchy":460},[383,71052,1116],{},[458,71054,468],{"stretchy":460},[1597,71056,71057,71063],{},[1600,71058,71059],{},[1603,71060,71061],{"scriptlevel":4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",[363,72187,72189,72207],{"className":72188,"translate":367},[366],[363,72190,72192],{"className":72191},[371],[373,72193,72194],{"xmlns":375},[377,72195,72196,72204],{},[380,72197,72198,72200,72202],{},[383,72199,1116],{},[458,72201,10300],{},[463,72203,465],{},[387,72205,72206],{"encoding":389},"t\\le1",[363,72208,72210,72228],{"className":72209,"ariaHidden":394},[393],[363,72211,72213,72216,72219,72222,72225],{"className":72212},[398],[363,72214],{"className":72215,"style":43560},[402],[363,72217,1116],{"className":72218},[407,408],[363,72220],{"className":72221,"style":506},[505],[363,72223,10300],{"className":72224},[510],[363,72226],{"className":72227,"style":506},[505],[363,72229,72231,72234],{"className":72230},[398],[363,72232],{"className":72233,"style":520},[402],[363,72235,465],{"className":72236},[407]," 一侧，",[363,72239,72241,72297],{"className":72240,"translate":367},[366],[363,72242,72244],{"className":72243},[371],[373,72245,72246],{"xmlns":375},[377,72247,72248,72294],{},[380,72249,72250,72252,72254,72256,72268,72270,72272,72274,72276,72278,72284,72286,72288],{},[383,72251,40609],{},[383,72253,1116],{},[458,72255,5161],{},[586,72257,72258,72266],{},[380,72259,72260,72262,72264],{},[463,72261,465],{},[458,72263,5161],{},[383,72265,1116],{},[463,72267,2811],{},[458,72269,471],{},[383,72271,1116],{},[458,72273,461],{"stretchy":460},[383,72275,40609],{},[458,72277,2818],{},[586,72279,72280,72282],{},[463,72281,465],{},[463,72283,2811],{},[458,72285,468],{"stretchy":460},[458,72287,5161],{},[586,72289,72290,72292],{},[463,72291,465],{},[463,72293,2811],{},[387,72295,72296],{"encoding":389},"yt-\\frac{1-t}{2}=t(y+\\frac12)-\\frac12",[363,72298,72300,72321,72410,72434,72520],{"className":72299,"ariaHidden":394},[393],[363,72301,72303,72306,72309,72312,72315,72318],{"className":72302},[398],[363,72304],{"className":72305,"style":26646},[402],[363,72307,40609],{"className":72308,"style":1328},[407,408],[363,72310,1116],{"className":72311},[407,408],[363,72313],{"className":72314,"style":2935},[505],[363,72316,5161],{"className":72317},[2939],[363,72319],{"className":72320,"style":2935},[505],[363,72322,72324,72327,72401,72404,72407],{"className":72323},[398],[363,72325],{"className":72326,"style":2853},[402],[363,72328,72330,72333,72398],{"className":72329},[407],[363,72331],{"className":72332},[494,789],[363,72334,72336],{"className":72335},[586],[363,72337,72339,72390],{"className":72338},[633,634],[363,72340,72342,72387],{"className":72341},[638],[363,72343,72345,72359,72367],{"className":72344,"style":2872},[642],[363,72346,72347,72350],{"style":2875},[363,72348],{"className":72349,"style":809},[650],[363,72351,72353],{"className":72352},[655,656,657,658],[363,72354,72356],{"className":72355},[407,658],[363,72357,2811],{"className":72358},[407,658],[363,72360,72361,72364],{"style":821},[363,72362],{"className":72363,"style":809},[650],[363,72365],{"className":72366,"style":829},[828],[363,72368,72369,72372],{"style":2898},[363,72370],{"className":72371,"style":809},[650],[363,72373,72375],{"className":72374},[655,656,657,658],[363,72376,72378,72381,72384],{"className":72377},[407,658],[363,72379,465],{"className":72380},[407,658],[363,72382,5161],{"className":72383},[2939,658],[363,72385,1116],{"className":72386},[407,408,658],[363,72388,666],{"className":72389},[665],[363,72391,72393],{"className":72392},[638],[363,72394,72396],{"className":72395,"style":2920},[642],[363,72397],{},[363,72399],{"className":72400},[501,789],[363,72402],{"className":72403,"style":506},[505],[363,72405,471],{"className":72406},[510],[363,72408],{"className":72409,"style":506},[505],[363,72411,72413,72416,72419,72422,72425,72428,72431],{"className":72412},[398],[363,72414],{"className":72415,"style":487},[402],[363,72417,1116],{"className":72418},[407,408],[363,72420,461],{"className":72421},[494],[363,72423,40609],{"className":72424,"style":1328},[407,408],[363,72426],{"className":72427,"style":2935},[505],[363,72429,2818],{"className":72430},[2939],[363,72432],{"className":72433,"style":2935},[505],[363,72435,72437,72440,72508,72511,72514,72517],{"className":72436},[398],[363,72438],{"className":72439,"style":2853},[402],[363,72441,72443,72446,72505],{"className":72442},[407],[363,72444],{"className":72445},[494,789],[363,72447,72449],{"className":72448},[586],[363,72450,72452,72497],{"className":72451},[633,634],[363,72453,72455,72494],{"className":72454},[638],[363,72456,72458,72472,72480],{"className":72457,"style":2872},[642],[363,72459,72460,72463],{"style":2875},[363,72461],{"className":72462,"style":809},[650],[363,72464,72466],{"className":72465},[655,656,657,658],[363,72467,72469],{"className":72468},[407,658],[363,72470,2811],{"className":72471},[407,658],[363,72473,72474,72477],{"style":821},[363,72475],{"className":72476,"style":809},[650],[363,72478],{"className":72479,"style":829},[828],[363,72481,72482,72485],{"style":2898},[363,72483],{"className":72484,"style":809},[650],[363,72486,72488],{"className":72487},[655,656,657,658],[363,72489,72491],{"className":72490},[407,658],[363,72492,465],{"className":72493},[407,658],[363,72495,666],{"className":72496},[665],[363,72498,72500],{"className":72499},[638],[363,72501,72503],{"className":72502,"style":2920},[642],[363,72504],{},[363,72506],{"className":72507},[501,789],[363,72509,468],{"className":72510},[501],[363,72512],{"className":72513,"style":2935},[505],[363,72515,5161],{"className":72516},[2939],[363,72518],{"className":72519,"style":2935},[505],[363,72521,72523,72526],{"className":72522},[398],[363,72524],{"className":72525,"style":2853},[402],[363,72527,72529,72532,72591],{"className":72528},[407],[363,72530],{"className":72531},[494,789],[363,72533,72535],{"className":72534},[586],[363,72536,72538,72583],{"className":72537},[633,634],[363,72539,72541,725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时两段的最大值都在 ",[363,73134,73136,73154],{"className":73135,"translate":367},[366],[363,73137,73139],{"className":73138},[371],[373,73140,73141],{"xmlns":375},[377,73142,73143,73151],{},[380,73144,73145,73147,73149],{},[383,73146,1116],{},[458,73148,471],{},[463,73150,465],{},[387,73152,73153],{"encoding":389},"t=1",[363,73155,73157,73175],{"className":73156,"ariaHidden":394},[393],[363,73158,73160,73163,73166,73169,73172],{"className":73159},[398],[363,73161],{"className":73162,"style":5691},[402],[363,73164,1116],{"className":73165},[407,408],[363,73167],{"className":73168,"style":506},[505],[363,73170,471],{"className":73171},[510],[363,73173],{"className":73174,"style":506},[505],[363,73176,73178,73181],{"className":73177},[398],[363,73179],{"className":73180,"style":520},[402],[363,73182,465],{"className":73183},[407]," 取到，为 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时第二段严格递增，上确界为 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Kantorovich 型对偶。",[11,74829,74830],{},"这个对偶表示在机器学习中非常有用，例如 f-GAN 就是基于此构造的（第 5 节详述）。",[2659,74832,74834],{"id":74833},"_45-数据处理不等式","4.5 数据处理不等式",[143,74836,74837,74940],{},[11,74838,74839,11410,74842,74870,74871,5993,74905,74939],{},[1344,74840,74841],{},"命题 4.4（数据处理不等式）",[363,74843,74845,74858],{"className":74844,"translate":367},[366],[363,74846,74848],{"className":74847},[371],[373,74849,74850],{"xmlns":375},[377,74851,74852,74856],{},[380,74853,74854],{},[383,74855,11044],{},[387,74857,11044],{"encoding":389},[363,74859,74861],{"className":74860,"ariaHidden":394},[393],[363,74862,74864,74867],{"className":74863},[398],[363,74865],{"className":74866,"style":1011},[402],[363,74868,11044],{"className":74869,"style":11132},[407,408]," 是一个马尔可夫核（随机变换），记 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严格凸，等号成立当且仅当 ",[363,78023,78025,78053],{"className":78024,"translate":367},[366],[363,78026,78028],{"className":78027},[371],[373,78029,78030],{"xmlns":375},[377,78031,78032,78050],{},[380,78033,78034,78040,78042,78044,78046,78048],{},[543,78035,78036,78038],{},[383,78037,571],{"mathvariant":570},[383,78039,561],{},[458,78041,578],{"stretchy":460},[383,78043,1116],{},[383,78045,5156],{"mathvariant":557},[383,78047,5749],{},[458,78049,605],{"stretchy":460},[387,78051,78052],{"encoding":389},"\\mathbb E_Q[t|Y]",[363,78054,78056],{"className":78055,"ariaHidden":394},[393],[363,78057,78059,78062,78102,78105,78108,78111,78114],{"className":78058},[398],[363,78060],{"className":78061,"style":618},[402],[363,78063,78065,78068],{"className":78064},[407],[363,78066,571],{"className":78067},[407,714],[363,78069,78071],{"className":78070},[629],[363,78072,78074,78094],{"className":78073},[633,634],[363,78075,78077,78091],{"className":78076},[638],[363,78078,78080],{"className":78079,"style":727},[642],[363,78081,78082,78085],{"style":730},[363,78083],{"className":78084,"style":651},[650],[363,78086,78088],{"className":78087},[655,656,657,658],[363,78089,561],{"className":78090},[407,408,658],[363,78092,666],{"className":78093},[665],[363,78095,78097],{"className":78096},[638],[363,78098,78100],{"className":78099,"style":673},[642],[363,78101],{},[363,78103,578],{"className":78104},[494],[363,78106,1116],{"className":78107},[407,408],[363,78109,5156],{"className":78110},[407],[363,78112,5749],{"className":78113,"style":2935},[407,408],[363,78115,605],{"className":78116},[501]," 本身是 ",[363,78119,78121,78141],{"className":78120,"translate":367},[366],[363,78122,78124],{"className":78123},[371],[373,78125,78126],{"xmlns":375},[377,78127,78128,78138],{},[380,78129,78130,78132,78134,78136],{},[383,78131,2680],{},[458,78133,461],{"stretchy":460},[383,78135,5749],{},[458,78137,468],{"stretchy":460},[387,78139,78140],{"encoding":389},"\\sigma(Y)",[363,78142,78144],{"className":78143,"ariaHidden":394},[393],[363,78145,78147,78150,78153,78156,78159],{"className":78146},[398],[363,78148],{"className":78149,"style":487},[402],[363,78151,2680],{"className":78152,"style":1328},[407,408],[363,78154,461],{"className":78155},[494],[363,78157,5749],{"className":78158,"style":2935},[407,408],[363,78160,468],{"className":78161},[501],"-可测的（对每个 ",[363,78164,78166,78179],{"className":78165,"translate":367},[366],[363,78167,78169],{"className":78168},[371],[373,78170,78171],{"xmlns":375},[377,78172,78173,78177],{},[380,78174,78175],{},[383,78176,40609],{},[387,78178,40609],{"encoding":389},[363,78180,78182],{"className":78181,"ariaHidden":394},[393],[363,78183,78185,78188],{"className":78184},[398],[363,78186],{"className":78187,"style":2624},[402],[363,78189,40609],{"className":78190,"style":1328},[407,408]," 是常数），即 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可表为 ",[363,78264,78266,78279],{"className":78265,"translate":367},[366],[363,78267,78269],{"className":78268},[371],[373,78270,78271],{"xmlns":375},[377,78272,78273,78277],{},[380,78274,78275],{},[383,78276,5749],{},[387,78278,5749],{"encoding":389},[363,78280,78282],{"className":78281,"ariaHidden":394},[393],[363,78283,78285,78288],{"className":78284},[398],[363,78286],{"className":78287,"style":1011},[402],[363,78289,5749],{"className":78290,"style":2935},[407,408]," 的函数。此时 ",[363,78293,78295,78308],{"className":78294,"translate":367},[366],[363,78296,78298],{"className":78297},[371],[373,78299,78300],{"xmlns":375},[377,78301,78302,78306],{},[380,78303,78304],{},[383,78305,5749],{},[387,78307,5749],{"encoding":389},[363,78309,78311],{"className":78310,"ariaHidden":394},[393],[363,78312,78314,78317],{"className":78313},[398],[363,78315],{"className":78316,"style":1011},[402],[363,78318,5749],{"className":78319,"style":2935},[407,408]," 是关于 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记 ",[363,79306,79308,79348],{"className":79307,"translate":367},[366],[363,79309,79311],{"className":79310},[371],[373,79312,79313],{"xmlns":375},[377,79314,79315,79345],{},[380,79316,79317,79319,79321,79323,79329,79331,79333,79335,79337,79339,79341,79343],{},[383,79318,1116],{},[458,79320,471],{},[383,79322,592],{},[543,79324,79325,79327],{},[383,79326,561],{},[383,79328,39034],{},[383,79330,1288],{"mathvariant":557},[383,79332,592],{},[383,79334,554],{},[458,79336,471],{},[463,79338,465],{},[458,79340,2818],{},[383,79342,39034],{},[383,79344,39037],{},[387,79346,79347],{"encoding":389},"t=dQ_\\varepsilon\u002FdP=1+\\varepsilon 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做 Taylor 展开：",[363,79607,79609],{"className":79608,"translate":367},[528],[363,79610,79612,79840],{"className":79611,"translate":367},[366],[363,79613,79615],{"className":79614},[371],[373,79616,79617],{"xmlns":375,"display":537},[377,79618,79619,79837],{},[1591,79620,79621,79683,79757],{"rowspacing":1593,"columnalign":1594,"columnspacing":1595},[1597,79622,79623,79649],{},[1600,79624,79625],{},[1603,79626,79627],{"scriptlevel":474,"displaystyle":394},[380,79628,79629,79635,79637,79643,79645,79647],{},[543,79630,79631,79633],{},[383,79632,547],{},[383,79634,385],{},[458,79636,461],{"stretchy":460},[543,79638,79639,79641],{},[383,79640,561],{},[383,79642,39034],{},[383,79644,558],{"mathvariant":557},[383,79646,554],{},[458,79648,468],{"stretchy":460},[1600,79650,79651],{},[1603,79652,79653],{"scriptlevel":474,"displaystyle":394},[380,79654,79655,79657,79659,79665,79667,79669,79671,79673,79675,79677,79679,79681],{},[380,79656],{},[458,79658,471],{},[543,79660,79661,79663],{},[383,79662,571],{"mathvariant":570},[383,79664,554],{},[458,79666,578],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[383,79668,385],{},[458,79670,461],{"stretchy":460},[463,79672,465],{},[458,79674,2818],{},[383,79676,39034],{},[383,79678,39037],{},[458,79680,468],{"stretchy":460},[458,79682,605],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[1597,79684,79685,79691],{},[1600,79686,79687],{},[1603,79688,79689],{"scriptlevel":474,"displaystyle":394},[380,79690],{},[1600,79692,79693],{},[1603,79694,79695],{"scriptlevel":474,"displaystyle":394},[380,79696,79697,79699,79701,79719,79721,79723,79729,79731,79733,79735],{},[380,79698],{},[458,79700,471],{},[6364,79702,79703,79717],{},[6364,79704,79705,79715],{},[380,79706,79707,79709,79711,79713],{},[383,79708,385],{},[458,79710,461],{"stretchy":460},[463,79712,465],{},[458,79714,468],{"stretchy":460},[458,79716,6388],{"stretchy":394},[463,79718,474],{},[458,79720,2818],{},[383,79722,39034],{},[7162,79724,79725,79727],{},[383,79726,385],{},[458,79728,7168],{"mathvariant":557,"lspace":1595,"rspace":1595},[458,79730,461],{"stretchy":460},[463,79732,465],{},[458,79734,468],{"stretchy":460},[6364,79736,79737,79755],{},[6364,79738,79739,79753],{},[380,79740,79741,79747,79749,79751],{},[543,79742,79743,79745],{},[383,79744,571],{"mathvariant":570},[383,79746,554],{},[458,79748,578],{"stretchy":460},[383,79750,39037],{},[458,79752,605],{"stretchy":460},[458,79754,6388],{"stretchy":394},[463,79756,474],{},[1597,79758,79759,79765],{},[1600,79760,79761],{},[1603,79762,79763],{"scriptlevel":474,"displaystyle":394},[380,79764],{},[1600,79766,79767],{},[1603,79768,79769],{"scriptlevel":474,"displaystyle":394},[380,79770,79771,79773,79775,79777,79787,79797,79799,79801,79803,79805,79811,79813,79819,79821,79823,79825,79827,79833,79835],{},[380,79772],{},[505,79774],{"width":3610},[458,79776,2818],{},[586,79778,79779,79785],{},[7162,79780,79781,79783],{},[383,79782,39034],{},[463,79784,2811],{},[463,79786,2811],{},[7162,79788,79789,79791],{},[383,79790,385],{},[380,79792,79793,79795],{},[458,79794,7168],{"mathvariant":557},[458,79796,7168],{"mathvariant":557},[458,79798,461],{"stretchy":460},[463,79800,465],{},[458,79802,468],{"stretchy":460},[1645,79804,1647],{},[543,79806,79807,79809],{},[383,79808,571],{"mathvariant":570},[383,79810,554],{},[458,79812,578],{"stretchy":460},[7162,79814,79815,79817],{},[383,79816,39037],{},[463,79818,2811],{},[458,79820,605],{"stretchy":460},[458,79822,2818],{},[383,79824,19593],{},[458,79826,461],{"stretchy":460},[7162,79828,79829,79831],{},[383,79830,39034],{},[463,79832,41107],{},[458,79834,468],{"stretchy":460},[383,79836,4723],{"mathvariant":557},[387,79838,79839],{"encoding":389},"\\begin{aligned}\nD_f(Q_\\varepsilon\\|P)\n&=\\mathbb E_P\\big[f(1+\\varepsilon h)\\big]\\\\[2pt]\n&=\\underbrace{f(1)}_{0}\n+\\varepsilon f'(1)\n\\underbrace{\\mathbb E_P[h]}_{0}\\\\\n&\\qquad+\\frac{\\varepsilon^2}{2}f''(1)\\,\n\\mathbb 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f-散度都只差一个常数 ",[363,80754,80756,80783],{"className":80755,"translate":367},[366],[363,80757,80759],{"className":80758},[371],[373,80760,80761],{"xmlns":375},[377,80762,80763,80781],{},[380,80764,80765,80775,80777,80779],{},[7162,80766,80767,80769],{},[383,80768,385],{},[380,80770,80771,80773],{},[458,80772,7168],{"mathvariant":557},[458,80774,7168],{"mathvariant":557},[458,80776,461],{"stretchy":460},[463,80778,465],{},[458,80780,468],{"stretchy":460},[387,80782,39906],{"encoding":389},[363,80784,80786],{"className":80785,"ariaHidden":394},[393],[363,80787,80789,80792,80824,80827,80830],{"className":80788},[398],[363,80790],{"className":80791,"style":7191},[402],[363,80793,80795,80798],{"className":80794},[407],[363,80796,385],{"className":80797,"style":409},[407,408],[363,80799,80801],{"className":80800},[629],[363,80802,80804],{"className":80803},[633],[363,80805,80807],{"className":80806},[638],[363,80808,80810],{"className":80809,"style":7210},[642],[363,80811,80812,80815],{"style":7213},[363,80813],{"className":80814,"style":651},[650],[363,80816,80818],{"className":80817},[655,656,657,658],[363,80819,80821],{"className":80820},[407,658],[363,80822,22120],{"className":80823},[407,658],[363,80825,461],{"className":80826},[494],[363,80828,465],{"className":80829},[407],[363,80831,468],{"className":80832},[501],"，形状完全一致。",[363,80835,80837,80866],{"className":80836,"translate":367},[366],[363,80838,80840],{"className":80839},[371],[373,80841,80842],{"xmlns":375},[377,80843,80844,80864],{},[380,80845,80846,80848,80850,80852,80854,80856,80858,80860,80862],{},[383,80847,385],{},[458,80849,461],{"stretchy":460},[383,80851,1116],{},[458,80853,468],{"stretchy":460},[458,80855,471],{},[383,80857,1116],{},[383,80859,4448],{},[458,80861,4451],{},[383,80863,1116],{},[387,80865,4456],{"encoding":389},[363,80867,80869,80896],{"className":80868,"ariaHidden":394},[393],[363,80870,80872,80875,80878,80881,80884,80887,80890,80893],{"className":80871},[398],[363,80873],{"className":80874,"style":487},[402],[363,80876,385],{"className":80877,"style":409},[407,408],[363,80879,461],{"className":80880},[494],[363,80882,1116],{"className":80883},[407,408],[363,80885,468],{"className":80886},[501],[363,80888],{"className":80889,"style":506},[505],[363,80891,471],{"className":80892},[510],[363,80894],{"className":80895,"style":506},[505],[363,80897,80899,80902,80905,80908,80913,80916],{"className":80898},[398],[363,80900],{"className":80901,"style":403},[402],[363,80903,1116],{"className":80904},[407,408],[363,80906],{"className":80907,"style":754},[505],[363,80909,4502,80911],{"className":80910},[1832],[363,80912,4506],{"style":4505},[363,80914],{"className":80915,"style":754},[505],[363,80917,1116],{"className":80918},[407,408],[363,80920,80922,80954],{"className":80921,"translate":367},[366],[363,80923,80925],{"className":80924},[371],[373,80926,80927],{"xmlns":375},[377,80928,80929,80951],{},[380,80930,80931,80941,80943,80945,80947,80949],{},[7162,80932,80933,80935],{},[383,80934,385],{},[380,80936,80937,80939],{},[458,80938,7168],{"mathvariant":557},[458,80940,7168],{"mathvariant":557},[458,80942,461],{"stretchy":460},[463,80944,465],{},[458,80946,468],{"stretchy":460},[458,80948,471],{},[463,80950,465],{},[387,80952,80953],{"encoding":389},"f''(1)=1",[363,80955,80957,81013],{"className":80956,"ariaHidden":394},[393],[363,80958,80960,80963,80995,80998,81001,81004,81007,81010],{"className":80959},[398],[363,80961],{"className":80962,"style":7191},[402],[363,80964,80966,80969],{"className":80965},[407],[363,80967,385],{"className":80968,"style":409},[407,408],[363,80970,80972],{"className":80971},[629],[363,80973,80975],{"className":80974},[633],[363,80976,80978],{"className":80977},[638],[363,80979,80981],{"className":80980,"style":7210},[642],[363,80982,80983,80986],{"style":7213},[363,80984],{"className":80985,"style":651},[650],[363,80987,80989],{"className":80988},[655,656,657,658],[363,80990,80992],{"className":80991},[407,658],[363,80993,22120],{"className":80994},[407,658],[363,80996,461],{"className":80997},[494],[363,80999,465],{"className":81000},[407],[363,81002,468],{"className":81003},[501],[363,81005],{"className":81006,"style":506},[505],[363,81008,471],{"className":81009},[510],[363,81011],{"className":81012,"style":506},[505],[363,81014,81016,81019],{"className":81015},[398],[363,81017],{"className":81018,"style":520},[402],[363,81020,465],{"className":81021},[407],"）即得 ",[363,81024,81026,81062],{"className":81025,"translate":367},[366],[363,81027,81029],{"className":81028},[371],[373,81030,81031],{"xmlns":375},[377,81032,81033,81059],{},[380,81034,81035,81045,81047,81053],{},[543,81036,81037,81039],{},[383,81038,547],{},[380,81040,81041,81043],{},[383,81042,11044],{},[383,81044,22265],{},[458,81046,33331],{},[586,81048,81049,81051],{},[463,81050,465],{},[463,81052,2811],{},[7162,81054,81055,81057],{},[383,81056,37821],{},[463,81058,2811],{},[387,81060,81061],{"encoding":389},"D_{KL}\\approx\\frac12\\chi^2",[363,81063,81065,81126],{"className":81064,"ariaHidden":394},[393],[363,81066,81068,81071,81117,81120,81123],{"className":81067},[398],[363,81069],{"className":81070,"style":16311},[402],[363,81072,81074,81077],{"className":81073},[407],[363,81075,547],{"className":81076,"style":625},[407,408],[363,81078,81080],{"className":81079},[629],[363,81081,81083,81109],{"className":81082},[633,634],[363,81084,81086,81106],{"className":81085},[638],[363,81087,81089],{"className":81088,"style":727},[642],[363,81090,81091,81094],{"style":646},[363,81092],{"className":81093,"style":651},[650],[363,81095,81097],{"className":81096},[655,656,657,658],[363,81098,81100,81103],{"className":81099},[407,658],[363,81101,11044],{"className":81102,"style":11132},[407,408,658],[363,81104,22265],{"className":81105},[407,408,658],[363,81107,666],{"className":81108},[665],[363,81110,81112],{"className":81111},[638],[363,81113,81115],{"className":81114,"style":8316},[642],[363,81116],{},[363,81118],{"className":81119,"style":506},[505],[363,81121,33331],{"className":81122},[510],[363,81124],{"className":81125,"style":506},[505],[363,81127,81129,81132,81200],{"className":81128},[398],[363,81130],{"className":81131,"style":2853},[402],[363,81133,81135,81138,81197],{"className":81134},[407],[363,81136],{"className":81137},[494,789],[363,81139,81141],{"className":81140},[586],[363,81142,81144,81189],{"className":81143},[633,634],[363,81145,81147,81186],{"className":81146},[638],[363,81148,81150,81164,81172],{"className":81149,"style":2872},[642],[363,81151,81152,81155],{"style":2875},[363,81153],{"className":81154,"style":809},[650],[363,81156,81158],{"className":81157},[655,656,657,658],[363,81159,81161],{"className":81160},[407,658],[363,81162,2811],{"className":81163},[407,658],[363,81165,81166,81169],{"style":821},[363,81167],{"className":81168,"style":809},[650],[363,81170],{"className":81171,"style":829},[828],[363,81173,81174,81177],{"style":2898},[363,81175],{"className":81176,"style":809},[650],[363,81178,81180],{"className":81179},[655,656,657,658],[363,81181,81183],{"className":81182},[407,658],[363,81184,465],{"className":81185},[407,658],[363,81187,666],{"className":81188},[665],[363,81190,81192],{"className":81191},[638],[363,81193,81195],{"className":81194,"style":2920},[642],[363,81196],{},[363,81198],{"className":81199},[501,789],[363,81201,81203,81206],{"className":81202},[407],[363,81204,37821],{"className":81205},[407,408],[363,81207,81209],{"className":81208},[629],[363,81210,81212],{"className":81211},[633],[363,81213,81215],{"className":81214},[638],[363,81216,81218],{"className":81217,"style":20400},[642],[363,81219,81220,81223],{"style":7213},[363,81221],{"className":81222,"style":651},[650],[363,81224,81226],{"className":81225},[655,656,657,658],[363,81227,2811],{"className":81228},[407,658],"；取 ",[363,81231,81233,81270],{"className":81232,"translate":367},[366],[363,81234,81236],{"className":81235},[371],[373,81237,81238],{"xmlns":375},[377,81239,81240,81268],{},[380,81241,81242,81244,81246,81248,81250,81252,81254,81258,81260,81262],{},[383,81243,385],{},[458,81245,461],{"stretchy":460},[383,81247,1116],{},[458,81249,468],{"stretchy":460},[458,81251,471],{},[458,81253,461],{"stretchy":460},[34711,81255,81256],{},[383,81257,1116],{},[458,81259,5161],{},[463,81261,465],{},[7162,81263,81264,81266],{},[458,81265,468],{"stretchy":460},[463,81267,2811],{},[387,81269,43002],{"encoding":389},[363,81271,81273,81300,81362],{"className":81272,"ariaHidden":394},[393],[363,81274,81276,81279,81282,81285,81288,81291,81294,81297],{"className":81275},[398],[363,81277],{"className":81278,"style":487},[402],[363,81280,385],{"className":81281,"style":409},[407,408],[363,81283,461],{"className":81284},[494],[363,81286,1116],{"className":81287},[407,408],[363,81289,468],{"className":81290},[501],[363,81292],{"className":81293,"style":506},[505],[363,81295,471],{"className":81296},[510],[363,81298],{"className":81299,"style":506},[505],[363,81301,81303,81306,81309,81353,81356,81359],{"className":81302},[398],[363,81304],{"className":81305,"style":43039},[402],[363,81307,461],{"className":81308},[494],[363,81310,81312],{"className":81311},[407,34739],[363,81313,81315,81345],{"className":81314},[633,634],[363,81316,81318,81342],{"className":81317},[638],[363,81319,81321,81330],{"className":81320,"style":43055},[642],[363,81322,81324,81327],{"className":81323,"style":12213},[6655],[363,81325],{"className":81326,"style":809},[650],[363,81328,1116],{"className":81329,"style":34759},[407,408],[363,81331,81332,81335],{"style":43067},[363,81333],{"className":81334,"style":809},[650],[363,81336,81338],{"className":81337,"style":34816},[34815],[6671,81339,81340],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},[6679,81341],{"d":34823},[363,81343,666],{"className":81344},[665],[363,81346,81348],{"className":81347},[638],[363,81349,81351],{"className":81350,"style":43087},[642],[363,81352],{},[363,81354],{"className":81355,"style":2935},[505],[363,81357,5161],{"className":81358},[2939],[363,81360],{"className":81361,"style":2935},[505],[363,81363,81365,81368,81371],{"className":81364},[398],[363,81366],{"className":81367,"style":20375},[402],[363,81369,465],{"className":81370},[407],[363,81372,81374,81377],{"className":81373},[501],[363,81375,468],{"className":81376},[501],[363,81378,81380],{"className":81379},[629],[363,81381,81383],{"className":81382},[633],[363,81384,81386],{"className":81385},[638],[363,81387,81389],{"className":81388,"style":20400},[642],[363,81390,81391,81394],{"style":7213},[363,81392],{"className":81393,"style":651},[650],[363,81395,81397],{"className":81396},[655,656,657,658],[363,81398,2811],{"className":81399},[407,658],[363,81401,81403,81439],{"className":81402,"translate":367},[366],[363,81404,81406],{"className":81405},[371],[373,81407,81408],{"xmlns":375},[377,81409,81410,81436],{},[380,81411,81412,81422,81424,81426,81428,81430],{},[7162,81413,81414,81416],{},[383,81415,385],{},[380,81417,81418,81420],{},[458,81419,7168],{"mathvariant":557},[458,81421,7168],{"mathvariant":557},[458,81423,461],{"stretchy":460},[463,81425,465],{},[458,81427,468],{"stretchy":460},[458,81429,471],{},[586,81431,81432,81434],{},[463,81433,465],{},[463,81435,2811],{},[387,81437,81438],{"encoding":389},"f''(1)=\\frac12",[363,81440,81442,81498],{"className":81441,"ariaHidden":394},[393],[363,81443,81445,81448,81480,81483,81486,81489,81492,81495],{"className":81444},[398],[363,81446],{"className":81447,"style":7191},[402],[363,81449,81451,81454],{"className":81450},[407],[363,81452,385],{"className":81453,"style":409},[407,408],[363,81455,81457],{"className":81456},[629],[363,81458,81460],{"className":81459},[633],[363,81461,81463],{"className":81462},[638],[363,81464,81466],{"className":81465,"style":7210},[642],[363,81467,81468,81471],{"style":7213},[363,81469],{"className":81470,"style":651},[650],[363,81472,81474],{"className":81473},[655,656,657,658],[363,81475,81477],{"className":81476},[407,658],[363,81478,22120],{"className":81479},[407,658],[363,81481,461],{"className":81482},[494],[363,81484,465],{"className":81485},[407],[363,81487,468],{"className":81488},[501],[363,81490],{"className":81491,"style":506},[505],[363,81493,471],{"className":81494},[510],[363,81496],{"className":81497,"style":506},[505],[363,81499,81501,81504],{"className":81500},[398],[363,81502],{"className":81503,"style":2853},[402],[363,81505,81507,81510,81569],{"className":81506},[407],[363,81508],{"className":81509},[494,789],[363,81511,81513],{"className":81512},[586],[363,81514,81516,81561],{"className":81515},[633,634],[363,81517,81519,81558],{"className":81518},[638],[363,81520,81522,81536,81544],{"className":81521,"style":2872},[642],[363,81523,81524,81527],{"style":2875},[363,81525],{"className":81526,"style":809},[650],[363,81528,81530],{"className":81529},[655,656,657,658],[363,81531,81533],{"className":81532},[407,658],[363,81534,2811],{"className":81535},[407,658],[363,81537,81538,81541],{"style":821},[363,81539],{"className":81540,"style":809},[650],[363,81542],{"className":81543,"style":829},[828],[363,81545,81546,81549],{"style":2898},[363,81547],{"className":81548,"style":809},[650],[363,81550,81552],{"className":81551},[655,656,657,658],[363,81553,81555],{"className":81554},[407,658],[363,81556,465],{"className":81557},[407,658],[363,81559,666],{"className":81560},[665],[363,81562,81564],{"className":81563},[638],[363,81565,81567],{"className":81566,"style":2920},[642],[363,81568],{},[363,81570],{"className":81571},[501,789],[363,81573,81575,81620],{"className":81574,"translate":367},[366],[363,81576,81578],{"className":81577},[371],[373,81579,81580],{"xmlns":375},[377,81581,81582,81618],{},[380,81583,81584,81586,81588,81592,81594,81598,81604,81606,81612],{},[458,81585,1635],{},[458,81587,461],{"stretchy":460},[34711,81589,81590],{},[383,81591,11],{},[458,81593,5161],{},[34711,81595,81596],{},[383,81597,1291],{},[7162,81599,81600,81602],{},[458,81601,468],{"stretchy":460},[463,81603,2811],{},[458,81605,33331],{},[586,81607,81608,81610],{},[463,81609,465],{},[463,81611,39361],{},[7162,81613,81614,81616],{},[383,81615,37821],{},[463,81617,2811],{},[387,81619,39370],{"encoding":389},[363,81621,81623,81688,81776],{"className":81622,"ariaHidden":394},[393],[363,81624,81626,81629,81632,81635,81679,81682,81685],{"className":81625},[398],[363,81627],{"className":81628,"style":39380},[402],[363,81630,1635],{"className":81631,"style":5965},[1832,1833,5964],[363,81633,461],{"className":81634},[494],[363,81636,81638],{"className":81637},[407,34739],[363,81639,81641,81671],{"className":81640},[633,634],[363,81642,81644,81668],{"className":81643},[638],[363,81645,81647,81656],{"className":81646,"style":39399},[642],[363,81648,81650,81653],{"className":81649,"style":12213},[6655],[363,81651],{"className":81652,"style":809},[650],[363,81654,11],{"className":81655,"style":34759},[407,408],[363,81657,81658,81661],{"style":39411},[363,81659],{"className":81660,"style":809},[650],[363,81662,81664],{"className":81663,"style":34816},[34815],[6671,81665,81666],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},[6679,81667],{"d":34823},[363,81669,666],{"className":81670},[665],[363,81672,81674],{"className":81673},[638],[363,81675,81677],{"className":81676,"style":39431},[642],[363,81678],{},[363,81680],{"className":81681,"style":2935},[505],[363,81683,5161],{"className":81684},[2939],[363,81686],{"className":81687,"style":2935},[505],[363,81689,81691,81694,81738,81767,81770,81773],{"className":81690},[398],[363,81692],{"className":81693,"style":39449},[402],[363,81695,81697],{"className":81696},[407,34739],[363,81698,81700,81730],{"className":81699},[633,634],[363,81701,81703,81727],{"className":81702},[638],[363,81704,81706,81715],{"className":81705,"style":39399},[642],[363,81707,81709,81712],{"className":81708,"style":12213},[6655],[363,81710],{"className":81711,"style":809},[650],[363,81713,1291],{"className":81714,"style":39471},[407,408],[363,81716,81717,81720],{"style":39411},[363,81718],{"className":81719,"style":809},[650],[363,81721,81723],{"className":81722,"style":34816},[34815],[6671,81724,81725],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},[6679,81726],{"d":34823},[363,81728,666],{"className":81729},[665],[363,81731,81733],{"className":81732},[638],[363,81734,81736],{"className":81735,"style":39431},[642],[363,81737],{},[363,81739,81741,81744],{"className":81740},[501],[363,81742,468],{"className":81743},[501],[363,81745,81747],{"className":81746},[629],[363,81748,81750],{"className":81749},[633],[363,81751,81753],{"className":81752},[638],[363,81754,81756],{"className":81755,"style":20400},[642],[363,81757,81758,81761],{"style":7213},[363,81759],{"className":81760,"style":651},[650],[363,81762,81764],{"className":81763},[655,656,657,658],[363,81765,2811],{"className":81766},[407,658],[363,81768],{"className":81769,"style":506},[505],[363,81771,33331],{"className":81772},[510],[363,81774],{"className":81775,"style":506},[505],[363,81777,81779,81782,81850],{"className":81778},[398],[363,81780],{"className":81781,"style":2853},[402],[363,81783,81785,81788,81847],{"className":81784},[407],[363,81786],{"className":81787},[494,789],[363,81789,81791],{"className":81790},[586],[363,81792,81794,81839],{"className":81793},[633,634],[363,81795,81797,81836],{"className":81796},[638],[363,81798,81800,81814,81822],{"className":81799,"style":2872},[642],[363,81801,81802,81805],{"style":2875},[363,81803],{"className":81804,"style":809},[650],[363,81806,81808],{"className":81807},[655,656,657,658],[363,81809,81811],{"className":81810},[407,658],[363,81812,39361],{"className":81813},[407,658],[363,81815,81816,81819],{"style":821},[363,81817],{"className":81818,"style":809},[650],[363,81820],{"className":81821,"style":829},[828],[363,81823,81824,81827],{"style":2898},[363,81825],{"className":81826,"style":809},[650],[363,81828,81830],{"className":81829},[655,656,657,658],[363,81831,81833],{"className":81832},[407,658],[363,81834,465],{"className":81835},[407,658],[363,81837,666],{"className":81838},[665],[363,81840,81842],{"className":81841},[638],[363,81843,81845],{"className":81844,"style":2920},[642],[363,81846],{},[363,81848],{"className":81849},[501,789],[363,81851,81853,81856],{"className":81852},[407],[363,81854,37821],{"className":81855},[407,408],[363,81857,81859],{"className":81858},[629],[363,81860,81862],{"className":81861},[633],[363,81863,81865],{"className":81864},[638],[363,81866,81868],{"className":81867,"style":20400},[642],[363,81869,81870,81873],{"style":7213},[363,81871],{"className":81872,"style":651},[650],[363,81874,81876],{"className":81875},[655,656,657,658],[363,81877,2811],{"className":81878},[407,658],"，除以 2 或换算成 ",[363,81881,81883,81903],{"className":81882,"translate":367},[366],[363,81884,81886],{"className":81885},[371],[373,81887,81888],{"xmlns":375},[377,81889,81890,81900],{},[380,81891,81892,81894,81896,81898],{},[463,81893,465],{},[458,81895,5161],{},[383,81897,39663],{},[383,81899,39666],{},[387,81901,81902],{"encoding":389},"1-BC",[363,81904,81906,81924],{"className":81905,"ariaHidden":394},[393],[363,81907,81909,81912,81915,81918,81921],{"className":81908},[398],[363,81910],{"className":81911,"style":12928},[402],[363,81913,465],{"className":81914},[407],[363,81916],{"className":81917,"style":2935},[505],[363,81919,5161],{"className":81920},[2939],[363,81922],{"className":81923,"style":2935},[505],[363,81925,81927,81930,81933],{"className":81926},[398],[363,81928],{"className":81929,"style":1011},[402],[363,81931,39663],{"className":81932,"style":39759},[407,408],[363,81934,39666],{"className":81935,"style":11132},[407,408]," 就是 3.4 节引用的 Bhattacharyya 约定 ",[363,81938,81940,81972],{"className":81939,"translate":367},[366],[363,81941,81943],{"className":81942},[371],[373,81944,81945],{"xmlns":375},[377,81946,81947,81969],{},[380,81948,81949,81955,81957,81963],{},[7162,81950,81951,81953],{},[383,81952,34256],{},[463,81954,2811],{},[458,81956,33331],{},[586,81958,81959,81961],{},[463,81960,465],{},[463,81962,39675],{},[7162,81964,81965,81967],{},[383,81966,37821],{},[463,81968,2811],{},[387,81970,81971],{"encoding":389},"H^2\\approx\\frac18\\chi^2",[363,81973,81975,82019],{"className":81974,"ariaHidden":394},[393],[363,81976,81978,81981,82010,82013,82016],{"className":81977},[398],[363,81979],{"className":81980,"style":20400},[402],[363,81982,81984,81987],{"className":81983},[407],[363,81985,34256],{"className":81986,"style":34410},[407,408],[363,81988,81990],{"className":81989},[629],[363,81991,81993],{"className":81992},[633],[363,81994,81996],{"className":81995},[638],[363,81997,81999],{"className":81998,"style":20400},[642],[363,82000,82001,82004],{"style":7213},[363,82002],{"className":82003,"style":651},[650],[363,82005,82007],{"className":82006},[655,656,657,658],[363,82008,2811],{"className":82009},[407,658],[363,82011],{"className":82012,"style":506},[505],[363,82014,33331],{"className":82015},[510],[363,82017],{"className":82018,"style":506},[505],[363,82020,82022,82025,82093],{"className":82021},[398],[363,82023],{"className":82024,"style":2853},[402],[363,82026,82028,82031,82090],{"className":82027},[407],[363,82029],{"className":82030},[494,789],[363,82032,82034],{"className":82033},[586],[363,82035,82037,82082],{"className":82036},[633,634],[363,82038,82040,82079],{"className":82039},[638],[363,82041,82043,82057,82065],{"className":82042,"style":2872},[642],[363,82044,82045,82048],{"style":2875},[363,82046],{"className":82047,"style":809},[650],[363,82049,82051],{"className":82050},[655,656,657,658],[363,82052,82054],{"className":82053},[407,658],[363,82055,39675],{"className":82056},[407,658],[363,82058,82059,82062],{"style":821},[363,82060],{"className":82061,"style":809},[650],[363,82063],{"className":82064,"style":829},[828],[363,82066,82067,82070],{"style":2898},[363,82068],{"className":82069,"style":809},[650],[363,82071,82073],{"className":82072},[655,656,657,658],[363,82074,82076],{"className":82075},[407,658],[363,82077,465],{"className":82078},[407,658],[363,82080,666],{"className":82081},[665],[363,82083,82085],{"className":82084},[638],[363,82086,82088],{"className":82087,"style":2920},[642],[363,82089],{},[363,82091],{"className":82092},[501,789],[363,82094,82096,82099],{"className":82095},[407],[363,82097,37821],{"className":82098},[407,408],[363,82100,82102],{"className":82101},[629],[363,82103,82105],{"className":82104},[633],[363,82106,82108],{"className":82107},[638],[363,82109,82111],{"className":82110,"style":20400},[642],[363,82112,82113,82116],{"style":7213},[363,82114],{"className":82115,"style":651},[650],[363,82117,82119],{"className":82118},[655,656,657,658],[363,82120,2811],{"className":82121},[407,658],"。共同因子 ",[363,82124,82126,82178],{"className":82125,"translate":367},[366],[363,82127,82129],{"className":82128},[371],[373,82130,82131],{"xmlns":375},[377,82132,82133,82175],{},[380,82134,82135,82141,82143,82149,82151,82153,82155,82161,82163,82169,82171,82173],{},[7162,82136,82137,82139],{},[383,82138,39034],{},[463,82140,2811],{},[458,82142,1635],{},[7162,82144,82145,82147],{},[383,82146,39037],{},[463,82148,2811],{},[383,82150,592],{},[383,82152,554],{},[458,82154,471],{},[7162,82156,82157,82159],{},[383,82158,37821],{},[463,82160,2811],{},[458,82162,461],{"stretchy":460},[543,82164,82165,82167],{},[383,82166,561],{},[383,82168,39034],{},[383,82170,558],{"mathvariant":557},[383,82172,554],{},[458,82174,468],{"stretchy":460},[387,82176,82177],{"encoding":389},"\\varepsilon^2\\int h^2 dP=\\chi^2(Q_\\varepsilon\\|P)",[363,82179,82181,82269],{"className":82180,"ariaHidden":394},[393],[363,82182,82184,82187,82216,82219,82222,82225,82254,82257,82260,82263,82266],{"className":82183},[398],[363,82185],{"className":82186,"style":50700},[402],[363,82188,82190,82193],{"className":82189},[407],[363,82191,39034],{"className":82192},[407,408],[363,82194,82196],{"className":82195},[629],[363,82197,82199],{"className":82198},[633],[363,82200,82202],{"className":82201},[638],[363,82203,82205],{"className":82204,"style":20400},[642],[363,82206,82207,82210],{"style":7213},[363,82208],{"className":82209,"style":651},[650],[363,82211,82213],{"className":82212},[655,656,657,658],[363,82214,2811],{"className":82215},[407,658],[363,82217],{"className":82218,"style":754},[505],[363,82220,1635],{"className":82221,"style":5965},[1832,1833,5964],[363,82223],{"className":82224,"style":754},[505],[363,82226,82228,82231],{"className":82227},[407],[363,82229,39037],{"className":82230},[407,408],[363,82232,82234],{"className":82233},[629],[363,82235,82237],{"className":82236},[633],[363,82238,82240],{"className":82239},[638],[363,82241,82243],{"className":82242,"style":20400},[642],[363,82244,82245,82248],{"style":7213},[363,82246],{"className":82247,"style":651},[650],[363,82249,82251],{"className":82250},[655,656,657,658],[363,82252,2811],{"className":82253},[407,658],[363,82255,592],{"className":82256},[407,408],[363,82258,554],{"className":82259,"style":682},[407,408],[363,82261],{"className":82262,"style":506},[505],[363,82264,471],{"className":82265},[510],[363,82267],{"className":82268,"style":506},[505],[363,82270,82272,82275,82304,82307,82347,82350,82353],{"className":82271},[398],[363,82273],{"className":82274,"style":20375},[402],[363,82276,82278,82281],{"className":82277},[407],[363,82279,37821],{"className":82280},[407,408],[363,82282,82284],{"className":82283},[629],[363,82285,82287],{"className":82286},[633],[363,82288,82290],{"className":82289},[638],[363,82291,82293],{"className":82292,"style":20400},[642],[363,82294,82295,82298],{"style":7213},[363,82296],{"className":82297,"style":651},[650],[363,82299,82301],{"className":82300},[655,656,657,658],[363,82302,2811],{"className":82303},[407,658],[363,82305,461],{"className":82306},[494],[363,82308,82310,82313],{"className":82309},[407],[363,82311,561],{"className":82312},[407,408],[363,82314,82316],{"className":82315},[629],[363,82317,82319,82339],{"className":82318},[633,634],[363,82320,82322,82336],{"className":82321},[638],[363,82323,82325],{"className":82324,"style":45301},[642],[363,82326,82327,82330],{"style":730},[363,82328],{"className":82329,"style":651},[650],[363,82331,82333],{"className":82332},[655,656,657,658],[363,82334,39034],{"className":82335},[407,408,658],[363,82337,666],{"className":82338},[665],[363,82340,82342],{"className":82341},[638],[363,82343,82345],{"className":82344,"style":8316},[642],[363,82346],{},[363,82348,558],{"className":82349},[407],[363,82351,554],{"className":82352,"style":682},[407,408],[363,82354,468],{"className":82355},[501]," 就是",[1344,82358,82359],{},"Fisher 信息度量","：它给出统计流形上的黎曼度量，而 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只是「单位选择」。这也解释了为什么 GAN 中「选 JS 还是选 KL」在小偏差区几乎无差别，差别只在远离平衡或支撑不重叠时才显现。",[2659,82443,82445],{"id":82444},"_47-不等式网络","4.7 不等式网络",[143,82447,82448,82644,83150,83611],{},[11,82449,82450,79304,82453,82643],{},[1344,82451,82452],{},"命题 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D_{TV}(P,Q)\\le\n\\sqrt{\\tfrac12 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2 步（二元问题的精确最小值）。"," 固定差 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时值为 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",[363,86858,86860,86887],{"className":86859,"translate":367},[366],[363,86861,86863],{"className":86862},[371],[373,86864,86865],{"xmlns":375},[377,86866,86867,86885],{},[380,86868,86869,86871,86873],{},[383,86870,45126],{},[458,86872,471],{},[586,86874,86875,86883],{},[380,86876,86877,86879,86881],{},[463,86878,465],{},[458,86880,2818],{},[383,86882,16062],{},[463,86884,2811],{},[387,86886,84978],{"encoding":389},[363,86888,86890,86908],{"className":86889,"ariaHidden":394},[393],[363,86891,86893,86896,86899,86902,86905],{"className":86892},[398],[363,86894],{"className":86895,"style":2693},[402],[363,86897,45126],{"className":86898,"style":45156},[407,408],[363,86900],{"className":86901,"style":506},[505],[363,86903,471],{"className":86904},[510],[363,86906],{"className":86907,"style":506},[505],[363,86909,86911,86914],{"className":86910},[398],[363,86912],{"className":86913,"style":85006},[402],[363,86915,86917,86920,86985],{"className":86916},[407],[363,86918],{"className":86919},[494,789],[363,86921,86923],{"className":86922},[586],[363,86924,86926,86977],{"className":86925},[633,634],[363,86927,86929,86974],{"className":86928},[638],[363,86930,86932,86946,86954],{"className":86931,"style":3035},[642],[363,86933,86934,86937],{"style":2875},[363,86935],{"className":86936,"style":809},[650],[363,86938,86940],{"className":86939},[655,656,657,658],[363,86941,86943],{"className":86942},[407,658],[363,86944,2811],{"className":86945},[407,658],[363,86947,86948,86951],{"style":821},[363,86949],{"className":86950,"style":809},[650],[363,86952],{"className":86953,"style":829},[828],[363,86955,86956,86959],{"style":2898},[363,86957],{"className":86958,"style":809},[650],[363,86960,86962],{"className":86961},[655,656,657,658],[363,86963,86965,86968,86971],{"className":86964},[407,658],[363,86966,465],{"className":86967},[407,658],[363,86969,2818],{"className":86970},[2939,658],[363,86972,16062],{"className":86973,"style":16111},[407,408,658],[363,86975,666],{"className":86976},[665],[363,86978,86980],{"className":86979},[638],[363,86981,86983],{"className":86982,"style":2920},[642],[363,86984],{},[363,86986],{"className":86987},[501,789],"，且是",[1344,86990,86991],{},"精确最小值","（Vajda 界）：",[363,86994,86996,87052],{"className":86995,"translate":367},[366],[363,86997,86999],{"className":86998},[371],[373,87000,87001],{"xmlns":375},[377,87002,87003,87049],{},[380,87004,87005,87015,87017,87019,87021,87023,87025,87027,87029,87031,87037,87039,87041,87043,87045,87047],{},[543,87006,87007,87009],{},[383,87008,547],{},[380,87010,87011,87013],{},[383,87012,11044],{},[383,87014,22265],{},[458,87016,461],{"stretchy":460},[383,87018,554],{},[383,87020,558],{"mathvariant":557},[383,87022,561],{},[458,87024,468],{"stretchy":460},[458,87026,7276],{},[383,87028,39037],{},[458,87030,461],{"stretchy":460},[380,87032,87033,87035],{},[383,87034,5740],{"mathvariant":557},[383,87036,35222],{"mathvariant":557},[458,87038,461],{"stretchy":460},[383,87040,554],{},[458,87042,898],{"separator":394},[383,87044,561],{},[458,87046,468],{"stretchy":460},[458,87048,468],{"stretchy":460},[387,87050,87051],{"encoding":389},"D_{KL}(P\\|Q)\\ge h(\\mathrm{TV}(P,Q))",[363,87053,87055,87131],{"className":87054,"ariaHidden":394},[393],[363,87056,87058,87061,87107,87110,87113,87116,87119,87122,87125,87128],{"className":87057},[398],[363,87059],{"className":87060,"style":487},[402],[363,87062,87064,87067],{"className":87063},[407],[363,87065,547],{"className":87066,"style":625},[407,408],[363,87068,87070],{"className":87069},[629],[363,87071,87073,87099],{"className":87072},[633,634],[363,87074,87076,87096],{"className":87075},[638],[363,87077,87079],{"className":87078,"style":727},[642],[363,87080,87081,87084],{"style":646},[363,87082],{"className":87083,"style":651},[650],[363,87085,87087],{"className":87086},[655,656,657,658],[363,87088,87090,87093],{"className":87089},[407,658],[363,87091,11044],{"className":87092,"style":11132},[407,408,658],[363,87094,22265],{"className":87095},[407,408,658],[363,87097,666],{"className":87098},[665],[363,87100,87102],{"className":87101},[638],[363,87103,87105],{"className":87104,"style":8316},[642],[363,87106],{},[363,87108,461],{"className":87109},[494],[363,87111,554],{"className":87112,"style":682},[407,408],[363,87114,558],{"className":87115},[407],[363,87117,561],{"className":87118},[407,408],[363,87120,468],{"className":87121},[501],[363,87123],{"className":87124,"style":506},[505],[363,87126,7276],{"className":87127},[510],[363,87129],{"className":87130,"style":506},[505],[363,87132,87134,87137,87140,87143,87150,87153,87156,87159,87162,87165],{"className":87133},[398],[363,87135],{"className":87136,"style":487},[402],[363,87138,39037],{"className":87139},[407,408],[363,87141,461],{"className":87142},[494],[363,87144,87146],{"className":87145},[407],[363,87147,87149],{"className":87148,"style":4505},[407,33034],"TV",[363,87151,461],{"className":87152},[494],[363,87154,554],{"className":87155,"style":682},[407,408],[363,87157,898],{"className":87158},[920],[363,87160],{"className":87161,"style":754},[505],[363,87163,561],{"className":87164},[407,408],[363,87166,19419],{"className":87167},[501],"，其中 ",[363,87170,87172,87218],{"className":87171,"translate":367},[366],[363,87173,87175],{"className":87174},[371],[373,87176,87177],{"xmlns":375},[377,87178,87179,87215],{},[380,87180,87181,87183,87185,87187,87189,87191,87193,87195,87197],{},[383,87182,39037],{},[458,87184,461],{"stretchy":460},[383,87186,16062],{},[458,87188,468],{"stretchy":460},[458,87190,471],{},[383,87192,16062],{},[383,87194,4448],{},[458,87196,4451],{},[586,87198,87199,87207],{},[380,87200,87201,87203,87205],{},[463,87202,465],{},[458,87204,2818],{},[383,87206,16062],{},[380,87208,87209,87211,87213],{},[463,87210,465],{},[458,87212,5161],{},[383,87214,16062],{},[387,87216,87217],{"encoding":389},"h(\\delta)=\\delta\\log\\frac{1+\\delta}{1-\\delta}",[363,87219,87221,87248],{"className":87220,"ariaHidden":394},[393],[363,87222,87224,87227,87230,87233,87236,87239,87242,87245],{"className":87223},[398],[363,87225],{"className":87226,"style":487},[402],[363,87228,39037],{"className":87229},[407,408],[363,87231,461],{"className":87232},[494],[363,87234,16062],{"className":87235,"style":16111},[407,408],[363,87237,468],{"className":87238},[501],[363,87240],{"className":87241,"style":506},[505],[363,87243,471],{"className":87244},[510],[363,87246],{"className":87247,"style":506},[505],[363,87249,87251,87254,87257,87260,87265,87268],{"className":87250},[398],[363,87252],{"className":87253,"style":16703},[402],[363,87255,16062],{"className":87256,"style":16111},[407,408],[363,87258],{"className":87259,"style":754},[505],[363,87261,4502,87263],{"className":87262},[1832],[363,87264,4506],{"style":4505},[363,87266],{"className":87267,"style":754},[505],[363,87269,87271,87274,87345],{"className":87270},[407],[363,87272],{"className":87273},[494,789],[363,87275,87277],{"className":87276},[586],[363,87278,87280,87337],{"className":87279},[633,634],[363,87281,87283,87334],{"className":87282},[638],[363,87284,87286,87306,87314],{"className":87285,"style":3035},[642],[363,87287,87288,87291],{"style":2875},[363,87289],{"className":87290,"style":809},[650],[363,87292,87294],{"className":87293},[655,656,657,658],[363,87295,87297,87300,87303],{"className":87296},[407,658],[363,87298,465],{"className":87299},[407,658],[363,87301,5161],{"className":87302},[2939,658],[363,87304,16062],{"className":87305,"style":16111},[407,408,658],[363,87307,87308,87311],{"style":821},[363,87309],{"className":87310,"style":809},[650],[363,87312],{"className":87313,"style":829},[828],[363,87315,87316,87319],{"style":2898},[363,87317],{"className":87318,"style":809},[650],[363,87320,87322],{"className":87321},[655,656,657,658],[363,87323,87325,87328,87331],{"className":87324},[407,658],[363,87326,465],{"className":87327},[407,658],[363,87329,2818],{"className":87330},[2939,658],[363,87332,16062],{"className":87333,"style":16111},[407,408,658],[363,87335,666],{"className":87336},[665],[363,87338,87340],{"className":87339},[638],[363,87341,87343],{"className":87342,"style":16779},[642],[363,87344],{},[363,87346],{"className":87347},[501,789],[363,87349,87351,87369],{"className":87350,"translate":367},[366],[363,87352,87354],{"className":87353},[371],[373,87355,87356],{"xmlns":375},[377,87357,87358,87366],{},[380,87359,87360,87362,87364],{},[383,87361,16062],{},[458,87363,1378],{},[463,87365,465],{},[387,87367,87368],{"encoding":389},"\\delta\\to1",[363,87370,87372,87390],{"className":87371,"ariaHidden":394},[393],[363,87373,87375,87378,87381,87384,87387],{"className":87374},[398],[363,87376],{"className":87377,"style":12947},[402],[363,87379,16062],{"className":87380,"style":16111},[407,408],[363,87382],{"className":87383,"style":506},[505],[363,87385,1378],{"className":87386},[510],[363,87388],{"className":87389,"style":506},[505],[363,87391,87393,87396],{"className":87392},[398],[363,87394],{"className":87395,"style":520},[402],[363,87397,465],{"className":87398},[407]," 时发散（与 KL 在支撑分离时为 ",[363,87401,87403,87416],{"className":87402,"translate":367},[366],[363,87404,87406],{"className":87405},[371],[373,87407,87408],{"xmlns":375},[377,87409,87410,87414],{},[380,87411,87412],{},[383,87413,1373],{"mathvariant":557},[387,87415,4218],{"encoding":389},[363,87417,87419],{"className":87418,"ariaHidden":394},[393],[363,87420,87422,87425],{"className":87421},[398],[363,87423],{"className":87424,"style":2693},[402],[363,87426,1373],{"className":87427},[407]," 一致）。",[11,87430,87431,87434],{},[1344,87432,87433],{},"第 3 步（初等放缩）。"," 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f-散度也有平行的\"固定 TV 时最小化 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\\ge\\ D^{R}_{1\u002F2}(P\\|Q)=-2\\log BC 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p-\\sqrt q|\\;|\\sqrt p+\\sqrt q|\\,dx\\\\\n&\\le\\sqrt{\\int(\\sqrt p-\\sqrt q)^2}\n\\cdot\\sqrt{\\int(\\sqrt p+\\sqrt q)^2}\\\\\n&=\\sqrt{2H^2}\\cdot\\sqrt{2(1+BC)}\n=\\sqrt{4H^2(2-H^2)},\n\\end{aligned}",[363,91130,91132],{"className":91131,"ariaHidden":394},[393],[363,91133,91135,91139],{"className":91134},[398],[363,91136],{"className":91137,"style":91138},[402],"height:7.2262em;vertical-align:-3.3631em;",[363,91140,91142],{"className":91141},[407],[363,91143,91145,91210],{"className":91144},[1591],[363,91146,91148],{"className":91147},[1708],[363,91149,91151,91201],{"className":91150},[633,634],[363,91152,91154,91198],{"className":91153},[638],[363,91155,91158,91180,91189],{"className":91156,"style":91157},[642],"height:3.8631em;",[363,91159,91161,91165],{"style":91160},"top:-6.3858em;",[363,91162],{"className":91163,"style":91164},[650],"height:3.8828em;",[363,91166,91168,91171,91174],{"className":91167},[407],[363,91169,2811],{"className":91170},[407],[363,91172],{"className":91173,"style":754},[505],[363,91175,91177],{"className":91176},[407],[363,91178,87149],{"className":91179,"style":4505},[407,33034],[363,91181,91183,91186],{"style":91182},"top:-3.3408em;",[363,91184],{"className":91185,"style":91164},[650],[363,91187],{"className":91188},[407],[363,91190,91192,91195],{"style":91191},"top:-0.8797em;",[363,91193],{"className":91194,"style":91164},[650],[363,91196],{"className":91197},[407],[363,91199,666],{"className":91200},[665],[363,91202,91204],{"className":91203},[638],[363,91205,91208],{"className":91206,"style":91207},[642],"height:3.3631em;",[363,91209],{},[363,91211,91213],{"className":91212},[1799],[363,91214,91216,92170],{"className":91215},[633,634],[363,91217,91219,92167],{"className":91218},[638],[363,91220,91222,91466,91856],{"className":91221,"style":91157},[642],[363,91223,91224,91227],{"style":91160},[363,91225],{"className":91226,"style":91164},[650],[363,91228,91230,91233,91236,91239,91242,91245,91248,91251,91295,91298,91301,91304,91348,91351,91354,91357,91401,91404,91407,91410,91454,91457,91460,91463],{"className":91229},[407],[363,91231],{"className":91232},[407],[363,91234],{"className":91235,"style":506},[505],[363,91237,471],{"className":91238},[510],[363,91240],{"className":91241,"style":506},[505],[363,91243,1635],{"className":91244,"style":1835},[1832,1833,1834],[363,91246],{"className":91247,"style":754},[505],[363,91249,5156],{"className":91250},[407],[363,91252,91254],{"className":91253},[407,34739],[363,91255,91257,91287],{"className":91256},[633,634],[363,91258,91260,91284],{"className":91259},[638],[363,91261,91263,91272],{"className":91262,"style":7210},[642],[363,91264,91266,91269],{"className":91265,"style":12213},[6655],[363,91267],{"className":91268,"style":809},[650],[363,91270,11],{"className":91271,"style":34759},[407,408],[363,91273,91274,91277],{"style":41722},[363,91275],{"className":91276,"style":809},[650],[363,91278,91280],{"className":91279,"style":34816},[34815],[6671,91281,91282],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},[6679,91283],{"d":34823},[363,91285,666],{"className":91286},[665],[363,91288,91290],{"className":91289},[638],[363,91291,91293],{"className":91292,"style":41742},[642],[363,91294],{},[363,91296],{"className":91297,"style":2935},[505],[363,91299,5161],{"className":91300},[2939],[363,91302],{"className":91303,"style":2935},[505],[363,91305,91307],{"className":91306},[407,34739],[363,91308,91310,91340],{"className":91309},[633,634],[363,91311,91313,91337],{"className":91312},[638],[363,91314,91316,91325],{"className":91315,"style":7210},[642],[363,91317,91319,91322],{"className":91318,"style":12213},[6655],[363,91320],{"className":91321,"style":809},[650],[363,91323,1291],{"className":91324,"style":39471},[407,408],[363,91326,91327,91330],{"style":41722},[363,91328],{"className":91329,"style":809},[650],[363,91331,91333],{"className":91332,"style":34816},[34815],[6671,91334,91335],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},[6679,91336],{"d":34823},[363,91338,666],{"className":91339},[665],[363,91341,91343],{"className":91342},[638],[363,91344,91346],{"className":91345,"style":41742},[642],[363,91347],{},[363,91349,5156],{"className":91350},[407],[363,91352],{"className":91353,"style":506},[505],[363,91355,5156],{"className":91356},[407],[363,91358,91360],{"className":91359},[407,34739],[363,91361,91363,91393],{"className":91362},[633,634],[363,91364,91366,91390],{"className":91365},[638],[363,91367,91369,91378],{"className":91368,"style":7210},[642],[363,91370,91372,91375],{"className":91371,"style":12213},[6655],[363,91373],{"className":91374,"style":809},[650],[363,91376,11],{"className":91377,"style":34759},[407,408],[363,91379,91380,91383],{"style":41722},[363,91381],{"className":91382,"style":809},[650],[363,91384,91386],{"className":91385,"style":34816},[34815],[6671,91387,91388],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},[6679,91389],{"d":34823},[363,91391,666],{"className":91392},[665],[363,91394,91396],{"classNa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凸、",[363,92860,92862,92885],{"className":92861,"translate":367},[366],[363,92863,92865],{"className":92864},[371],[373,92866,92867],{"xmlns":375},[377,92868,92869,92883],{},[380,92870,92871,92873,92875,92877,92879,92881],{},[383,92872,385],{},[458,92874,461],{"stretchy":460},[463,92876,465],{},[458,92878,468],{"stretchy":460},[458,92880,471],{},[463,92882,474],{},[387,92884,477],{"encoding":389},[363,92886,92888,92915],{"className":92887,"ariaHidden":394},[393],[363,92889,92891,92894,92897,92900,92903,92906,92909,92912],{"className":92890},[398],[363,92892],{"className":92893,"style":487},[402],[363,92895,385],{"className":92896,"style":409},[407,408],[363,92898,461],{"className":92899},[494],[363,92901,465],{"className":92902},[407],[363,92904,468],{"className":92905},[501],[363,92907],{"className":92908,"style":506},[505],[363,92910,471],{"className":92911},[510],[363,92913],{"className":92914,"style":506},[505],[363,92916,92918,92921],{"className":92917},[398],[363,92919],{"className":92920,"style":520},[402],[363,92922,474],{"className":92923},[407]," 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上连续。",[11,92979,92980,6842,92983,93316,93317,93388,93389,5993,93496,93645,93646,93801,93802,93952,93953,94011,94012,94140,94141,94198,94199,94505],{},[1344,92981,92982],{},"(a)（\"下来\"方向，一般成立）",[363,92984,92986,93056],{"className":92985,"translate":367},[366],[363,92987,92989],{"className":92988},[371],[373,92990,92991],{"xmlns":375},[377,92992,92993,93053],{},[380,92994,92995,93001,93003,93009,93011,93013,93015,93017,93019,93021,93023,93025,93035,93037,93043,93045,93047,93049,93051],{},[543,92996,92997,92999],{},[383,92998,547],{},[383,93000,385],{},[458,93002,461],{"stretchy":460},[543,93004,93005,93007],{},[383,93006,554],{},[383,93008,82683],{},[383,93010,558],{"mathvariant":557},[383,93012,561],{},[458,93014,468],{"stretchy":460},[458,93016,1378],{},[463,93018,474],{},[1645,93020,4631],{},[458,93022,4634],{},[1645,93024,4631],{},[543,93026,93027,93029],{},[383,93028,547],{},[380,93030,93031,93033],{},[383,93032,5740],{},[383,93034,35222],{},[458,93036,461],{"stretchy":460},[543,93038,93039,93041],{},[383,93040,554],{},[383,93042,82683],{},[458,93044,898],{"separator":394},[383,93046,561],{},[458,93048,468],{"stretchy":460},[458,93050,1378],{},[463,93052,474],{},[387,93054,93055],{"encoding":389},"D_f(P_n\\|Q)\\to0\\ \\Longrightarrow\\ D_{TV}(P_n,Q)\\to0",[363,93057,93059,93166,93191,93307],{"className":93058,"ariaHidden":394},[393],[363,93060,93062,93065,93105,93108,93148,93151,93154,93157,93160,93163],{"className":93061},[398],[363,93063],{"className":93064,"style":618},[402],[363,93066,93068,93071],{"className":93067},[407],[363,93069,547],{"className":93070,"style":625},[407,408],[363,93072,93074],{"className":93073},[629],[363,93075,93077,93097],{"className":93076},[633,634],[363,93078,93080,93094],{"className":93079},[638],[363,93081,93083],{"className":93082,"style":643},[642],[363,93084,93085,93088],{"style":646},[363,93086],{"className":93087,"style":651},[650],[363,93089,93091],{"className":93090},[655,656,657,658],[363,93092,385],{"className":93093,"style":409},[407,408,658],[363,93095,666],{"className":93096},[665],[363,93098,93100],{"className":93099},[638],[363,93101,93103],{"className":93102,"style":673},[642],[363,93104],{},[363,93106,461],{"className":93107},[494],[363,93109,93111,93114],{"className":93110},[407],[363,93112,554],{"className":93113,"style":682},[407,408],[363,93115,93117],{"className":93116},[629],[363,93118,93120,93140],{"className":93119},[633,634],[363,93121,93123,93137],{"className":93122},[638],[363,93124,93126],{"className":93125,"style":45301},[642],[363,93127,93128,93131],{"style":11486},[363,93129],{"className":93130,"style":651},[650],[363,93132,93134],{"className":93133},[655,656,657,658],[363,93135,82683],{"className":93136},[407,408,658],[363,93138,666],{"className":93139},[665],[363,93141,93143],{"className":93142},[638],[363,93144,93146],{"className":93145,"style":8316},[642],[363,93147],{},[363,93149,558],{"className":93150},[407],[363,93152,561],{"className":93153},[407,408],[363,93155,468],{"className":93156},[501],[363,93158],{"className":93159,"style":506},[505],[363,93161,1378],{"className":93162},[510],[363,93164],{"className":93165,"style":506},[505],[363,93167,93169,93173,93176,93179,93182,93185,93188],{"className":93168},[398],[363,93170],{"className":93171,"style":93172},[402],"height:0.6684em;vertical-align:-0.024em;",[363,93174,474],{"className":93175},[407],[363,93177,4631],{"className":93178},[505],[363,93180],{"className":93181,"style":506},[505],[363,93183,4634],{"className":93184},[510],[363,93186,4631],{"className":93187},[505],[363,93189],{"className":93190,"style":506},[505],[363,93192,93194,93197,93243,93246,93286,93289,93292,93295,93298,93301,93304],{"className":93193},[398],[363,93195],{"className":93196,"style":487},[402],[363,93198,93200,93203],{"className":93199},[407],[363,93201,547],{"className":93202,"style":625},[407,408],[363,93204,93206],{"className":93205},[629],[363,93207,93209,93235],{"className":93208},[633,634],[363,93210,93212,93232],{"className":93211},[638],[363,93213,93215],{"className":93214,"style":727},[642],[363,93216,93217,93220],{"style":646},[363,93218],{"className":93219,"style":651},[650],[363,93221,93223],{"className":93222},[655,656,657,658],[363,93224,93226,93229],{"className":93225},[407,658],[363,93227,5740],{"className":93228,"style":682},[407,408,658],[363,93230,35222],{"className":93231,"style":2935},[407,408,658],[363,93233,666],{"className":93234},[665],[363,93236,93238],{"className":93237},[638],[363,93239,93241],{"className":93240,"style":8316},[642],[363,93242],{},[363,93244,461],{"className":93245},[494],[363,93247,93249,93252],{"className":93248},[407],[363,93250,554],{"className":93251,"style":682},[407,408],[363,93253,93255],{"className":93254},[629],[363,93256,93258,93278],{"className":93257},[633,634],[363,93259,93261,93275],{"className":93260},[638],[363,93262,93264],{"className":93263,"style":45301},[642],[363,93265,93266,93269],{"style":11486},[363,93267],{"className":93268,"style":651},[650],[363,93270,93272],{"className":93271},[655,656,657,658],[363,93273,82683],{"className":93274},[407,408,658],[363,93276,666],{"className":93277},[665],[363,93279,93281],{"className":93280},[638],[363,93282,93284],{"className":93283,"style":8316},[642],[363,93285],{},[363,93287,898],{"className":93288},[920],[363,93290],{"className":93291,"style":754},[505],[363,93293,561],{"className":93294},[407,408],[363,93296,468],{"className":93297},[501],[363,93299],{"className":93300,"style":506},[505],[363,93302,1378],{"className":93303},[510],[363,93305],{"className":93306,"style":506},[505],[363,93308,93310,93313],{"className":93309},[398],[363,93311],{"className":93312,"style":520},[402],[363,93314,474],{"className":93315},[407],"。每个适度正则的 f-散度都自带一个连续的 TV 下界函数 ",[363,93318,93320,93338],{"className":93319,"translate":367},[366],[363,93321,93323],{"className":93322},[371],[373,93324,93325],{"xmlns":375},[377,93326,93327,93335],{},[380,93328,93329],{},[543,93330,93331,93333],{},[383,93332,4506],{},[383,93334,385],{},[387,93336,93337],{"encoding":389},"g_f",[363,93339,93341],{"className":93340,"ariaHidden":394},[393],[363,93342,93344,93348],{"className":93343},[398],[363,93345],{"className":93346,"style":93347},[402],"height:0.7167em;vertical-align:-0.2861em;",[363,93349,93351,93354],{"className":93350},[407],[363,93352,4506],{"className":93353,"style":1328},[407,408],[363,93355,93357],{"className":93356},[629],[363,93358,93360,93380],{"className":93359},[633,634],[363,93361,93363,93377],{"className":93362},[638],[363,93364,93366],{"className":93365,"style":643},[642],[363,93367,93368,93371],{"style":19215},[363,93369],{"className":93370,"style":651},[650],[363,93372,93374],{"className":93373},[655,656,657,658],[363,93375,385],{"className":93376,"style":409},[407,408,658],[363,93378,666],{"className":93379},[665],[363,93381,93383],{"className":93382},[638],[363,93384,93386],{"className":93385,"style":673},[642],[363,93387],{},"（Vajda \u002F Fedotov–Harremoës），",[363,93390,93392,93420],{"className":93391,"translate":367},[366],[363,93393,93395],{"className":93394},[371],[373,93396,93397],{"xmlns":375},[377,93398,93399,93417],{},[380,93400,93401,93407,93409,93411,93413,93415],{},[543,93402,93403,93405],{},[383,93404,4506],{},[383,93406,385],{},[458,93408,461],{"stretchy":460},[463,93410,474],{},[458,93412,468],{"stretchy":460},[458,93414,471],{},[463,93416,474],{},[387,93418,93419],{"encoding":389},"g_f(0)=0",[363,93421,93423,93487],{"className":93422,"ariaHidden":394},[393],[363,93424,93426,93429,93469,93472,93475,93478,93481,93484],{"className":93425},[398],[363,93427],{"className":93428,"style":618},[402],[363,93430,93432,93435],{"className":93431},[407],[363,93433,4506],{"className":93434,"style":1328},[407,408],[363,93436,93438],{"className":93437},[629],[363,93439,93441,93461],{"className":93440},[633,634],[363,93442,93444,93458],{"className":93443},[638],[363,93445,93447],{"className":93446,"style":643},[642],[363,93448,93449,93452],{"style":19215},[363,93450],{"className":93451,"style":651},[650],[363,93453,93455],{"className":93454},[655,656,657,658],[363,93456,385],{"className":93457,"style":409},[407,408,658],[363,93459,666],{"className":93460},[665],[363,93462,93464],{"className":93463},[638],[363,93465,93467],{"className":93466,"style":673},[642],[363,93468],{},[363,93470,461],{"className":93471},[494],[363,93473,474],{"className":93474},[407],[363,93476,468],{"className":93477},[501],[363,93479],{"className":93480,"style":506},[505],[363,93482,471],{"className":93483},[510],[363,93485],{"className":93486,"style":506},[505],[363,93488,93490,93493],{"className":93489},[398],[363,93491],{"className":93492,"style":520},[402],[363,93494,474],{"className":93495},[407],[363,93497,93499,93539],{"className":93498,"translate":367},[366],[363,93500,93502],{"className":93501},[371],[373,93503,93504],{"xmlns":375},[377,93505,93506,93536],{},[380,93507,93508,93514,93516,93518,93520,93522,93524,93526,93528,93530,93532,93534],{},[543,93509,93510,93512],{},[383,93511,4506],{},[383,93513,385],{},[458,93515,461],{"stretchy":460},[383,93517,1116],{},[458,93519,468],{"stretchy":460},[458,93521,3251],{},[463,93523,474],{},[1645,93525,4631],{},[458,93527,461],{"stretchy":460},[383,93529,1116],{},[458,93531,3251],{},[463,93533,474],{},[458,93535,468],{"stretchy":460},[387,93537,93538],{"encoding":389},"g_f(t)>0\\ (t>0)",[363,93540,93542,93606,93633],{"className":93541,"ariaHidden":394},[393],[363,93543,93545,93548,93588,93591,93594,93597,93600,93603],{"className":93544},[398],[363,93546],{"className":93547,"style":618},[402],[363,93549,93551,93554],{"className":93550},[407],[363,93552,4506],{"className":93553,"style":1328},[407,408],[363,93555,93557],{"className":93556},[629],[363,93558,93560,93580],{"className":93559},[633,634],[363,93561,93563,93577],{"className":93562},[638],[363,93564,93566],{"className":93565,"style":643},[642],[363,93567,93568,93571],{"style":19215},[363,93569],{"className":93570,"style":651},[650],[363,93572,93574],{"className":93573},[655,656,657,658],[363,93575,385],{"className":93576,"style":409},[407,408,658],[363,93578,666],{"className":93579},[665],[363,93581,93583],{"className":93582},[638],[363,93584,93586],{"className":93585,"style":673},[642],[363,93587],{},[363,93589,461],{"className":93590},[494],[363,93592,1116],{"className":93593},[407,408],[363,93595,468],{"className":93596},[501],[363,93598],{"className":93599,"style":506},[505],[363,93601,3251],{"className":93602},[510],[363,93604],{"className":93605,"style":506},[505],[363,93607,93609,93612,93615,93618,93621,93624,93627,93630],{"className":93608},[398],[363,93610],{"className":93611,"style":487},[402],[363,93613,474],{"className":93614},[407],[363,93616,4631],{"className":93617},[505],[363,93619,461],{"className":93620},[494],[363,93622,1116],{"className":93623},[407,408],[363,93625],{"className":93626,"style":506},[505],[363,93628,3251],{"className":93629},[510],[363,93631],{"className":93632,"style":506},[505],[363,93634,93636,93639,93642],{"className":93635},[398],[363,93637],{"className":93638,"style":487},[402],[363,93640,474],{"className":93641},[407],[363,93643,468],{"className":93644},[501],"，使 ",[363,93647,93649,93685],{"className":93648,"translate":367},[366],[363,93650,93652],{"className":93651},[371],[373,93653,93654],{"xmlns":375},[377,93655,93656,93682],{},[380,93657,93658,93664,93666,93672,93674,93680],{},[543,93659,93660,93662],{},[383,93661,547],{},[383,93663,385],{},[458,93665,7276],{},[543,93667,93668,93670],{},[383,93669,4506],{},[383,93671,385],{},[458,93673,461],{"stretchy":460},[380,93675,93676,93678],{},[383,93677,5740],{"mathvariant":557},[383,93679,35222],{"mathvariant":557},[458,93681,468],{"stretchy":460},[387,93683,93684],{"encoding":389},"D_f\\ge g_f(\\mathrm{TV})",[363,93686,93688,93743],{"className":93687,"ariaHidden":394},[393],[363,93689,93691,93694,93734,93737,93740],{"className":93690},[398],[363,93692],{"className":93693,"style":4380},[402],[363,93695,93697,93700],{"className":93696},[407],[363,93698,547],{"className":93699,"style":625},[407,408],[363,93701,93703],{"className":93702},[629],[363,93704,93706,93726],{"className":93705},[633,634],[363,93707,93709,93723],{"className":93708},[638],[363,93710,93712],{"className":93711,"style":643},[642],[363,93713,93714,93717],{"style":646},[363,93715],{"className":93716,"style":651},[650],[363,93718,93720],{"className":93719},[655,656,657,658],[363,93721,385],{"className":93722,"style":409},[407,408,658],[363,93724,666],{"className":93725},[665],[363,93727,93729],{"className":93728},[638],[363,93730,93732],{"className":93731,"style":673},[642],[363,93733],{},[363,93735],{"className":93736,"style":506},[505],[363,93738,7276],{"className":93739},[510],[363,93741],{"className":93742,"style":506},[505],[363,93744,93746,93749,93789,93792,93798],{"className":93745},[398],[363,93747],{"className":93748,"style":618},[402],[363,93750,93752,93755],{"className":93751},[407],[363,93753,4506],{"className":93754,"style":1328},[407,408],[363,93756,93758],{"className":93757},[629],[363,93759,93761,93781],{"className":93760},[633,634],[363,93762,93764,93778],{"className":93763},[638],[363,93765,93767],{"className":93766,"style":643},[642],[363,93768,93769,93772],{"style":19215},[363,93770],{"className":93771,"style":651},[650],[363,93773,93775],{"className":93774},[655,656,657,658],[363,93776,385],{"className":93777,"style":409},[407,408,658],[363,93779,666],{"className":93780},[665],[363,93782,93784],{"className":93783},[638],[363,93785,93787],{"className":93786,"style":673},[642],[363,93788],{},[363,93790,461],{"className":93791},[494],[363,93793,93795],{"className":93794},[407],[363,93796,87149],{"className":93797,"style":4505},[407,33034],[363,93799,468],{"className":93800},[501],"；常用成员的具体形式：KL 给 ",[363,93803,93805,93843],{"className":93804,"translate":367},[366],[363,93806,93808],{"className":93807},[371],[373,93809,93810],{"xmlns":375},[377,93811,93812,93840],{},[380,93813,93814,93816,93818,93828,93830],{},[463,93815,2811],{},[1645,93817,1647],{},[7162,93819,93820,93826],{},[380,93821,93822,93824],{},[383,93823,5740],{"mathvariant":557},[383,93825,35222],{"mathvariant":557},[463,93827,2811],{},[458,93829,10300],{},[543,93831,93832,93834],{},[383,93833,547],{},[380,93835,93836,93838],{},[383,93837,11044],{},[383,93839,22265],{},[387,93841,93842],{"encoding":389},"2\\,\\mathrm{TV}^2\\le 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给 ",[363,94013,94015,94049],{"className":94014,"translate":367},[366],[363,94016,94018],{"className":94017},[371],[373,94019,94020],{"xmlns":375},[377,94021,94022,94046],{},[380,94023,94024,94030,94032,94034,94036],{},[7162,94025,94026,94028],{},[383,94027,37821],{},[463,94029,2811],{},[458,94031,7276],{},[463,94033,39361],{},[1645,94035,1647],{},[7162,94037,94038,94044],{},[380,94039,94040,94042],{},[383,94041,5740],{"mathvariant":557},[383,94043,35222],{"mathvariant":557},[463,94045,2811],{},[387,94047,94048],{"encoding":389},"\\chi^2\\ge4\\,\\mathrm{TV}^2",[363,94050,94052,94096],{"className":94051,"ariaHidden":394},[393],[363,94053,94055,94058,94087,94090,94093],{"className":94054},[398],[363,94056],{"className":94057,"style":40776},[402],[363,94059,94061,94064],{"className":94060},[407],[363,94062,37821],{"className":94063},[407,408],[363,94065,94067],{"className":94066},[629],[363,94068,94070],{"className":94069},[633],[363,94071,94073],{"className":94072},[638],[363,94074,94076],{"className":94075,"style":20400},[642],[363,94077,94078,94081],{"style":7213},[363,94079],{"className":94080,"style":651},[650],[363,94082,94084],{"className":94083},[655,656,657,658],[363,94085,2811],{"className":94086},[407,658],[363,94088],{"className":94089,"style":506},[505],[363,94091,7276],{"className":94092},[510],[363,94094],{"className":94095,"style":506},[505],[363,94097,94099,94102,94105,94108],{"className":94098},[398],[363,94100],{"className":94101,"style":88763},[402],[363,94103,39361],{"className":94104},[407],[363,94106],{"className":94107,"style":754},[505],[363,94109,94111,94117],{"className":94110},[407],[363,94112,94114],{"className":94113},[407],[363,94115,87149],{"className":94116,"style":4505},[407,33034],[363,94118,94120],{"className":94119},[629],[363,94121,94123],{"className":94122},[633],[363,94124,94126],{"className":94125},[638],[363,94127,94129],{"className":94128,"style":88763},[642],[363,94130,94131,94134],{"style":88793},[363,94132],{"className":94133,"style":651},[650],[363,94135,94137],{"className":94136},[655,656,657,658],[363,94138,2811],{"className":94139},[407,658],"，Hellinger",[363,94142,94144,94161],{"className":94143,"translate":367},[366],[363,94145,94147],{"className":94146},[371],[373,94148,94149],{"xmlns":375},[377,94150,94151,94159],{},[380,94152,94153],{},[7162,94154,94155,94157],{},[380,94156],{},[463,94158,2811],{},[387,94160,44641],{"encoding":389},[363,94162,94164],{"className":94163,"ariaHidden":394},[393],[363,94165,94167,94170],{"className":94166},[398],[363,94168],{"className":94169,"style":20400},[402],[363,94171,94173,94175],{"className":94172},[407],[363,94174],{},[363,94176,94178],{"className":94177},[629],[363,94179,94181],{"className":94180},[633],[363,94182,94184],{"className":94183},[638],[363,94185,94187],{"className":94186,"style":20400},[642],[363,94188,94189,94192],{"style":7213},[363,94190],{"className":94191,"style":651},[650],[363,94193,94195],{"className":94194},[655,656,657,658],[363,94196,2811],{"className":94197},[407,658]," 经由 ",[363,94200,94202,94260],{"className":94201,"translate":367},[366],[363,94203,94205],{"className":94204},[371],[373,94206,94207],{"xmlns":375},[377,94208,94209,94257],{},[380,94210,94211,94217,94219,94243,94245],{},[380,94212,94213,94215],{},[383,94214,5740],{"mathvariant":557},[383,94216,35222],{"mathvariant":557},[458,94218,10300],{},[34711,94220,94221],{},[380,94222,94223,94229,94231,94233,94235,94241],{},[7162,94224,94225,94227],{},[383,94226,34256],{},[463,94228,2811],{},[458,94230,461],{"stretchy":460},[463,94232,2811],{},[458,94234,5161],{},[7162,94236,94237,94239],{},[383,94238,34256],{},[463,94240,2811],{},[458,94242,468],{"stretchy":460},[458,94244,10300],{},[34711,94246,94247],{},[380,94248,94249,94251],{},[463,94250,2811],{},[7162,94252,94253,94255],{},[383,94254,34256],{},[463,94256,2811],{},[387,94258,94259],{"encoding":389},"\\mathrm{TV}\\le\\sqrt{H^2(2-H^2)}\\le\\sqrt{2H^2}",[363,94261,94263,94284,94419],{"className":94262,"ariaHidden":394},[393],[363,94264,94266,94269,94275,94278,94281],{"className":94265},[398],[363,94267],{"className":94268,"style":88741},[402],[363,94270,94272],{"className":94271},[407],[363,94273,87149],{"className":94274,"style":4505},[407,33034],[363,94276],{"className":94277,"style":506},[505],[363,94279,10300],{"className":94280},[510],[363,94282],{"className":94283,"style":506},[505],[363,94285,94287,94290,94410,94413,94416],{"className":94286},[398],[363,94288],{"className":94289,"style":92258},[402],[363,94291,94293],{"className":94292},[407,34739],[363,94294,94296,94402],{"className":94295},[633,634],[363,94297,94299,94399],{"className":94298},[638],[363,94300,94302,94387],{"className":94301,"style":92271},[642],[363,94303,94305,94308],{"className":94304,"style":83923},[6655],[363,94306],{"className":94307,"style":15196},[650],[363,94309,94311,94340,94343,94346,94349,94352,94355,94384],{"className":94310,"style":41533},[407],[363,94312,94314,94317],{"className":94313},[407],[363,94315,34256],{"className":94316,"style":34410},[407,408],[363,94318,94320],{"className":94319},[629],[363,94321,94323],{"className":94322},[633],[363,94324,94326],{"className":94325},[638],[363,94327,94329],{"className":94328,"style":45404},[642],[363,94330,94331,94334],{"style":45407},[363,94332],{"className":94333,"style":651},[650],[363,94335,94337],{"className":94336},[655,656,657,658],[363,94338,2811],{"className":94339},[407,658],[363,94341,461],{"className":94342},[494],[363,94344,2811],{"className":94345},[407],[363,94347],{"className":94348,"style":2935},[505],[363,94350,5161],{"className":94351},[2939],[363,94353],{"className":94354,"style":2935},[505],[363,94356,94358,94361],{"className":94357},[407],[363,94359,34256],{"className":94360,"style":34410},[407,408],[363,94362,94364],{"className":94363},[629],[363,94365,94367],{"className":94366},[633],[363,94368,94370],{"className":94369},[638],[363,94371,94373],{"className":94372,"style":45404},[642],[363,94374,94375,94378],{"style":45407},[363,94376],{"className":94377,"style":651},[650],[363,94379,94381],{"className":94380},[655,656,657,658],[363,94382,2811],{"className":94383},[407,658],[363,94385,468],{"className":94386},[501],[363,94388,94389,94392],{"style":92359},[363,94390],{"className":94391,"style":15196},[650],[363,94393,94395],{"className":94394,"style":84015},[34815],[6671,94396,94397],{"xmlns":6673,"width":6674,"height":84018,"viewBox":84019,"preserveAspectRatio":6677},[6679,94398],{"d":84022},[363,94400,666],{"className":94401},[665],[363,94403,94405],{"className":94404},[638],[363,94406,94408],{"className":94407,"style":92379},[642],[363,94409],{},[363,94411],{"className":94412,"style":506},[505],[363,94414,10300],{"className":94415},[510],[363,94417],{"className":94418,"style":506},[505],[363,94420,94422,94426],{"className":94421},[398],[363,94423],{"className":94424,"style":94425},[402],"height:1.04em;vertical-align:-0.0849em;",[363,94427,94429],{"className":94428},[407,34739],[363,94430,94432,94496],{"className":94431},[633,634],[363,94433,94435,94493],{"className":94434},[638],[363,94436,94439,94480],{"className":94437,"style":94438},[642],"height:0.9551em;",[363,94440,94442,94445],{"className":94441,"style":12213},[6655],[363,94443],{"className":94444,"style":809},[650],[363,94446,94448,94451],{"className":94447,"style":34759},[407],[363,94449,2811],{"className":94450},[407],[363,94452,94454,94457],{"className":94453},[407],[363,94455,34256],{"className":94456,"style":34410},[407,408],[363,94458,94460],{"className":94459},[629],[363,94461,94463],{"className":94462},[633],[363,94464,94466],{"className":94465},[638],[363,94467,94469],{"className":94468,"style":45404},[642],[363,94470,94471,94474],{"style":45407},[363,94472],{"className":94473,"style":651},[650],[363,94475,94477],{"className":94476},[655,656,657,658],[363,94478,2811],{"className":94479},[407,658],[363,94481,94483,94486],{"style":94482},"top:-2.9151em;",[363,94484],{"className":94485,"style":809},[650],[363,94487,94489],{"className":94488,"style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\\Longrightarrow\\ 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一般",[1344,94845,94846],{},"不成立","。反例：在 ",[363,94849,94851,94872],{"className":94850,"translate":367},[366],[363,94852,94854],{"className":94853},[371],[373,94855,94856],{"xmlns":375},[377,94857,94858,94870],{},[380,94859,94860,94862,94864,94866,94868],{},[458,94861,578],{"stretchy":460},[463,94863,474],{},[458,94865,898],{"separator":394},[463,94867,465],{},[458,94869,605],{"stretchy":460},[387,94871,37186],{"encoding":389},[363,94873,94875],{"className":94874,"ariaHidden":394},[393],[363,94876,94878,94881,94884,94887,94890,94893,94896],{"className":94877},[398],[363,94879],{"className":94880,"style":487},[402],[363,94882,578],{"className":94883},[494],[363,94885,474],{"className":94886},[407],[363,94888,898],{"className":94889},[920],[363,94891],{"className":94892,"style":754},[505],[363,94894,465],{"className":94895},[407],[363,94897,605],{"className":94898},[501]," 上取 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",[363,95597,95599,95643],{"className":95598,"translate":367},[366],[363,95600,95602],{"className":95601},[371],[373,95603,95604],{"xmlns":375},[377,95605,95606,95640],{},[380,95607,95608,95614,95616,95622,95624,95626,95628,95630,95636,95638],{},[380,95609,95610,95612],{},[383,95611,5740],{"mathvariant":557},[383,95613,35222],{"mathvariant":557},[458,95615,461],{"stretchy":460},[543,95617,95618,95620],{},[383,95619,1291],{},[383,95621,82683],{},[458,95623,898],{"separator":394},[383,95625,11],{},[458,95627,468],{"stretchy":460},[458,95629,33331],{},[543,95631,95632,95634],{},[383,95633,16062],{},[383,95635,82683],{},[458,95637,1378],{},[463,95639,474],{},[387,95641,95642],{"encoding":389},"\\mathrm{TV}(q_n,p)\\approx\\delta_n\\to0",[363,95644,95646,95722,95777],{"className":95645,"ariaHidden":394},[393],[363,95647,95649,95652,95658,95661,95701,95704,95707,95710,95713,95716,95719],{"className":95648},[398],[363,95650],{"className":95651,"style":487},[402],[363,95653,95655],{"className":95654},[407],[363,95656,87149],{"className":95657,"style":4505},[407,33034],[363,95659,461],{"className":95660},[494],[363,95662,95664,95667],{"className":95663},[407],[363,95665,1291],{"className":95666,"style":1328},[407,408],[363,95668,95670],{"className":95669},[629],[363,95671,95673,95693],{"className":95672},[633,634],[363,95674,95676,95690],{"className":95675},[638],[363,95677,95679],{"className":95678,"style":45301},[642],[363,95680,95681,95684],{"style":19215},[363,95682],{"className":95683,"style":651},[650],[363,95685,95687],{"className":95686},[655,656,657,658],[363,95688,82683],{"className":95689},[407,408,658],[363,95691,666],{"className":95692},[665],[363,95694,95696],{"className":95695},[638],[363,95697,95699],{"className":95698,"style":8316},[642],[363,95700],{},[363,95702,898],{"className":95703},[920],[363,95705],{"className":95706,"style":754},[505],[363,95708,11],{"className":95709},[407,408],[363,95711,468],{"className":95712},[501],[363,95714],{"className":95715,"style":506},[505],[363,95717,33331],{"className":95718},[510],[363,95720],{"className":95721,"style":506},[505],[363,95723,95725,95728,95768,95771,95774],{"className":95724},[398],[363,95726],{"className":95727,"style":11824},[402],[363,95729,95731,95734],{"className":95730},[407],[363,95732,16062],{"className":95733,"style":16111},[407,408],[363,95735,95737],{"className":95736},[629],[363,95738,95740,95760],{"className":95739},[633,634],[363,95741,95743,95757],{"className":95742},[638],[363,95744,95746],{"className":95745,"style":45301},[642],[363,95747,95748,95751],{"style":95132},[363,95749],{"className":95750,"style":651},[650],[363,95752,95754],{"className":95753},[655,656,657,658],[363,95755,82683],{"className":95756},[407,408,658],[363,95758,666],{"className":95759},[665],[363,95761,95763],{"className":95762},[638],[363,95764,95766],{"className":95765,"style":8316},[642],[363,95767],{},[363,95769],{"className":95770,"style":506},[505],[363,95772,1378],{"className":95773},[510],[363,95775],{"className":95776,"style":506},[505],[363,95778,95780,95783],{"className":95779},[398],[363,95781],{"className":95782,"style":520},[402],[363,95784,474],{"className":95785},[407],"，但 ",[363,95788,95790,95839],{"className":95789,"translate":367},[366],[363,95791,95793],{"className":95792},[371],[373,95794,95795],{"xmlns":375},[377,95796,95797,95836],{},[380,95798,95799,95805,95807,95813,95815,95817,95819,95822,95828,95830,95832,95834],{},[380,95800,95801,95803],{},[383,95802,11044],{"mathvariant":557},[383,95804,22265],{"mathvariant":557},[458,95806,461],{"stretchy":460},[543,95808,95809,95811],{},[383,95810,1291],{},[383,95812,82683],{},[383,95814,558],{"mathvariant":557},[383,95816,11],{},[458,95818,468],{"stretchy":460},[458,95820,95821],{},"≳",[543,95823,95824,95826],{},[383,95825,16062],{},[383,95827,82683],{},[458,95829,3647],{},[383,95831,82683],{},[458,95833,1378],{},[383,95835,1373],{"mathvariant":557},[387,95837,95838],{"encoding":389},"\\mathrm{KL}(q_n\\|p)\\gtrsim \\delta_n\\cdot n\\to\\infty",[363,95840,95842,95915,95970,95988],{"className":95841,"ariaHidden":394},[393],[363,95843,95845,95848,95854,95857,95897,95900,95903,95906,95909,95912],{"className":95844},[398],[363,95846],{"className":95847,"style":487},[402],[363,95849,95851],{"className":95850},[407],[363,95852,33035],{"className":95853},[407,33034],[363,95855,461],{"className":95856},[494],[363,95858,95860,95863],{"className":95859},[407],[363,95861,1291],{"className":95862,"style":1328},[407,408],[363,95864,95866],{"className":95865},[629],[363,95867,95869,95889],{"className":95868},[633,634],[363,95870,95872,95886],{"className":95871},[638],[363,95873,95875],{"className":95874,"style":45301},[642],[363,95876,95877,95880],{"style":19215},[363,95878],{"className":95879,"style":651},[650],[363,95881,95883],{"className":95882},[655,656,657,658],[363,95884,82683],{"className":95885},[407,408,658],[363,95887,666],{"className":95888},[665],[363,95890,95892],{"className":95891},[638],[363,95893,95895],{"className":95894,"style":8316},[642],[363,95896],{},[363,95898,558],{"className":95899},[407],[363,95901,11],{"className":95902},[407,408],[363,95904,468],{"className":95905},[501],[363,95907],{"className":95908,"style":506},[505],[363,95910,95821],{"className":95911},[510,60175],[363,95913],{"className":95914,"style":506},[505],[363,95916,95918,95921,95961,95964,95967],{"className":95917},[398],[363,95919],{"className":95920,"style":11824},[402],[363,95922,95924,95927],{"className":95923},[407],[363,95925,16062],{"className":95926,"style":16111},[407,408],[363,95928,95930],{"className":95929},[629],[363,95931,95933,95953],{"className":95932},[633,634],[363,95934,95936,95950],{"className":95935},[638],[363,95937,95939],{"className":95938,"style":45301},[642],[363,95940,95941,95944],{"style":95132},[363,95942],{"className":95943,"style":651},[650],[363,95945,95947],{"className":95946},[655,656,657,658],[363,95948,82683],{"className":95949},[407,408,658],[363,95951,666],{"className":95952},[665],[363,95954,95956],{"className":95955},[638],[363,95957,95959],{"className":95958,"style":8316},[642],[363,95960],{},[363,95962],{"className":95963,"style":2935},[505],[363,95965,3647],{"className":95966},[2939],[363,95968],{"className":95969,"style":2935},[505],[363,95971,95973,95976,95979,95982,95985],{"className":95972},[398],[363,95974],{"className":95975,"style":2693},[402],[363,95977,82683],{"className":95978},[407,408],[363,95980],{"className":95981,"style":506},[505],[363,95983,1378],{"className":95984},[510],[363,95986],{"className":95987,"style":506},[505],[363,95989,95991,95994],{"className":95990},[398],[363,95992],{"className":95993,"style":2693},[402],[363,95995,1373],{"className":95996},[407],"。反例的机制很典型：TV 只控制\"总质量差\"，控制不了小测度集合上密度的巨大尖峰，而 KL 对小分母极其敏感——这正是 KL\u002Fχ² \"无界\"分歧的拓扑体现。",[11,95999,96000,96003,96004,96032,96033,60445,96099,96164,96165,23996],{},[1344,96001,96002],{},"(c)（有限字母表\u002F一致比值界下，全部等价）"," 若似然比一致有界（例如：",[363,96005,96007,96020],{"className":96006,"translate":367},[366],[363,96008,96010],{"className":96009},[371],[373,96011,96012],{"xmlns":375},[377,96013,96014,96018],{},[380,96015,96016],{},[383,96017,561],{},[387,96019,561],{"encoding":389},[363,96021,96023],{"className":96022,"ariaHidden":394},[393],[363,96024,96026,96029],{"className":96025},[398],[363,96027],{"className":96028,"style":913},[402],[363,96030,561],{"className":96031},[407,408]," 在有限字母表上且 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的一致 Lipschitz 性给出）。",[11,97330,97331,97332,97335],{},"也就是说，",[1344,97333,97334],{},"\"所有 f-散度诱导同一拓扑\"必须在有限\u002F紧设定或一致比值界下读","：此时它们都等价于全变差拓扑（也叫强拓扑），只是\"度量标尺\"不同；而在一般测度空间上，KL 型拓扑严格强于 TV 拓扑。这一分层对理解第 6 节 f-散度与 Wasserstein 的真正差异很重要：f-散度族的内部差异是\"强度\"差异，而它们与 Wasserstein 的差异是\"性质\"差异。",[2659,97337,97339],{"id":97338},"_48-一般不是度量","4.8 一般不是度量",[11,97341,97342],{},"f-散度一般不对称，也不满足三角不等式，因此通常不是距离度量。但某些特殊形式可以诱导出度量：",[10375,97344,97345,98857],{},[7013,97346,97347,97350,97351,98023,98026,98027,98055,98056,98129,98130,98390,98391,98796,98797,98856],{},[1344,97348,97349],{},"Hellinger 距离","是度量：",[363,97352,97354],{"className":97353,"translate":367},[528],[363,97355,97357,97461],{"className":97356,"translate":367},[366],[363,97358,97360],{"className":97359},[371],[373,97361,97362],{"xmlns":375,"display":537},[377,97363,97364,97458],{},[380,97365,97366,97372,97374,97376,97378,97380,97382,97384,97422,97424,97436,97438,97456],{},[543,97367,97368,97370],{},[383,97369,44932],{},[383,97371,34256],{},[458,97373,461],{"stretchy":460},[383,97375,554],{},[458,97377,898],{"separator":394},[383,97379,561],{},[458,97381,468],{"stretchy":460},[458,97383,471],{},[34711,97385,97386],{},[380,97387,97388,97396,97398,97400,97404,97406,97410,97416,97418,97420],{},[1603,97389,97390],{"scriptlevel":474,"displaystyle":460},[586,97391,97392,97394],{},[463,97393,465],{},[463,97395,2811],{},[458,97397,1635],{},[458,97399,461],{"stretchy":460},[34711,97401,97402],{},[383,97403,11],{},[458,97405,5161],{},[34711,97407,97408],{},[383,97409,1291],{},[7162,97411,97412,97414],{},[458,97413,468],{"stretchy":460},[463,97415,2811],{},[1645,97417,1647],{},[383,97419,592],{},[383,97421,1121],{},[458,97423,471],{},[34711,97425,97426],{},[380,97427,97428,97430,97432,97434],{},[463,97429,465],{},[458,97431,5161],{},[383,97433,39663],{},[383,97435,39666],{},[458,97437,471],{},[34711,97439,97440],{},[380,97441,97442,97444,97446,97448],{},[463,97443,2811],{},[458,97445,5161],{},[463,97447,2811],{},[7901,97449,97450,97452,97454],{},[383,97451,34256],{},[1645,97453,42509],{},[463,97455,2811],{},[383,97457,4723],{"mathvariant":557},[387,97459,97460],{"encoding":389},"\\rho_H(P,Q)\n=\\sqrt{\\tfrac12\\int(\\sqrt p-\\sqrt q)^2\\,dx}\n=\\sqrt{1-BC}\n=\\sqrt{2-2H^2_{\\text{Bhattacharyya}}}.",[363,97462,97464,97537,97809,97890],{"className":97463,"ariaHidden":394},[393],[363,97465,97467,97470,97510,97513,97516,97519,97522,97525,97528,97531,97534],{"className":97466},[398],[363,97468],{"className":97469,"style":487},[402],[363,97471,97473,97476],{"className":97472},[407],[363,97474,44932],{"className":97475},[407,408],[363,97477,97479],{"className":97478},[629],[363,97480,97482,97502],{"className":97481},[633,634],[363,97483,97485,97499],{"className":97484},[638],[363,97486,97488],{"className":97487,"style":727},[642],[363,97489,97490,97493],{"style":730},[363,97491],{"className":97492,"style":651},[650],[363,97494,97496],{"className":97495},[655,656,657,658],[363,97497,34256],{"className":97498,"style":34410},[407,408,658],[363,97500,666],{"className":97501},[665],[363,97503,97505],{"className":97504},[638],[363,97506,97508],{"className":97507,"style":8316},[642],[363,97509],{},[363,97511,461],{"className":97512},[494],[363,97514,554],{"className":97515,"style":682},[407,408],[363,97517,898],{"className":97518},[920],[363,97520],{"className":97521,"style":754},[505],[363,97523,561],{"className":97524},[407,408],[363,97526,468],{"className":97527},[501],[363,97529],{"className":97530,"style":506},[505],[363,97532,471],{"className":97533},[510],[363,97535],{"className":97536,"style":506},[505],[363,97538,97540,97544,97800,97803,97806],{"className":97539},[398],[363,97541],{"className":97542,"style":97543},[402],"height:3.04em;vertical-align:-1.1572em;",[363,97545,97547],{"className":97546},[407,34739],[363,97548,97550,97792],{"className":97549},[633,634],[363,97551,97553,97789],{"className":97552},[638],[363,97554,97556,97777],{"className":97555,"style":91498},[642],[363,97557,97559,97562],{"className":97558,"style":91502},[6655],[363,97560],{"className":97561,"style":91506},[650],[363,97563,97565,97633,97636,97639,97642,97686,97689,97692,97695,97739,97768,97771,97774],{"className":97564,"style":41533},[407],[363,97566,97568,97571,97630],{"className":97567},[407],[363,97569],{"className":97570},[494,789],[363,97572,97574],{"className":97573},[586],[363,97575,97577,97622],{"className":97576},[633,634],[363,97578,97580,97619],{"className":97579},[638],[363,97581,97583,97597,97605],{"className":97582,"style":2872},[642],[363,97584,97585,97588],{"style":2875},[363,97586],{"className":97587,"style":809},[650],[363,97589,97591],{"className":97590},[655,656,657,658],[363,97592,97594],{"className":97593},[407,658],[363,97595,2811],{"className":97596},[407,658],[363,97598,97599,97602],{"style":821},[363,97600],{"className":97601,"style":809},[650],[363,97603],{"className":97604,"style":829},[828],[363,97606,97607,97610],{"style":2898},[363,97608],{"className":97609,"style":809},[650],[363,97611,97613],{"className":97612},[655,656,657,658],[363,97614,97616],{"className":97615},[407,658],[363,97617,465],{"className":97618},[407,658],[363,97620,666],{"className":97621},[665],[363,97623,97625],{"className":97624},[638],[363,97626,97628],{"className":97627,"style":2920},[642],[363,97629],{},[363,97631],{"className":97632},[501,789],[363,97634],{"className":97635,"style":754},[505],[363,97637,1635],{"className":97638,"style":1835},[1832,1833,1834],[363,97640,461],{"className":97641},[494],[363,97643,97645],{"className":97644},[407,34739],[363,97646,97648,97678],{"className":97647},[633,634],[363,97649,97651,97675],{"className":97650},[638],[363,97652,97654,97663],{"className":97653,"style":7210},[642],[363,97655,97657,97660],{"className":97656,"style":12213},[6655],[363,97658],{"className":97659,"style":809},[650],[363,97661,11],{"className":97662,"style":34759},[407,408],[363,97664,97665,97668],{"style":41722},[363,97666],{"className":97667,"style":809},[650],[363,97669,97671],{"className":97670,"style":34816},[34815],[6671,97672,97673],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},[6679,97674],{"d":34823},[363,97676,666],{"className":97677},[665],[363,97679,97681],{"className":97680},[638],[363,97682,97684],{"className":97683,"style":41742},[642],[363,97685],{},[363,97687],{"className":97688,"style":2935},[505],[363,97690,5161],{"className":97691},[2939],[363,97693],{"className":97694,"style":2935},[505],[363,97696,97698],{"className":97697},[407,34739],[363,97699,97701,97731],{"className":97700},[633,634],[363,97702,97704,97728],{"className":97703},[638],[363,97705,97707,97716],{"className":97706,"style":7210},[642],[363,97708,97710,97713],{"className":97709,"style":12213},[6655],[363,97711],{"className":97712,"style":809},[650],[363,97714,1291],{"className":97715,"style":39471},[407,408],[363,97717,97718,97721],{"style":41722},[363,97719],{"className":97720,"style":809},[650],[363,97722,97724],{"className":97723,"style":34816},[34815],[6671,97725,97726],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},[6679,97727],{"d":34823},[363,97729,666],{"className":97730},[66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",[363,98028,98030,98043],{"className":98029,"translate":367},[366],[363,98031,98033],{"className":98032},[371],[373,98034,98035],{"xmlns":375},[377,98036,98037,98041],{},[380,98038,98039],{},[383,98040,554],{},[387,98042,554],{"encoding":389},[363,98044,98046],{"className":98045,"ariaHidden":394},[393],[363,98047,98049,98052],{"className":98048},[398],[363,98050],{"className":98051,"style":1011},[402],[363,98053,554],{"className":98054,"style":682},[407,408]," 映到单位球面上的点 ",[363,98057,98059,98075],{"className":98058,"translate":367},[366],[363,98060,98062],{"className":98061},[371],[373,98063,98064],{"xmlns":375},[377,98065,98066,98072],{},[380,98067,98068],{},[34711,98069,98070],{},[383,98071,11],{},[387,98073,98074],{"encoding":389},"\\sqrt 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p=1",[363,98170,98172,98357,98381],{"className":98171,"ariaHidden":394},[393],[363,98173,98175,98179,98216,98260,98348,98351,98354],{"className":98174},[398],[363,98176],{"className":98177,"style":98178},[402],"height:1.4537em;vertical-align:-0.3997em;",[363,98180,98182],{"className":98181},[407],[363,98183,98185],{"className":98184},[767,15179],[363,98186,98188,98208],{"className":98187},[633,634],[363,98189,98191,98205],{"className":98190},[638],[363,98192,98194],{"className":98193,"style":15189},[642],[363,98195,98196,98199],{"style":15192},[363,98197],{"className":98198,"style":15196},[650],[363,98200,98201],{"style":15199},[6671,98202,98203],{"xmlns":6673,"width":15202,"height":9798,"viewBox":15203},[6679,98204],{"d":15206},[363,98206,666],{"className":98207},[665],[363,98209,98211],{"className":98210},[638],[363,98212,98214],{"className":98213,"style":15216},[642],[363,98215],{},[363,98217,98219],{"className":98218},[407,34739],[363,98220,98222,98252],{"className":98221},[633,634],[363,98223,98225,98249],{"className":98224},[638],[363,98226,98228,98237],{"className":98227,"style":39399},[642],[363,98229,98231,98234],{"className":98230,"style":12213},[6655],[363,98232],{"className":98233,"style":809},[650],[363,98235,11],{"className":98236,"style":34759},[407,408],[363,98238,98239,98242],{"style":39411},[363,98240],{"className":98241,"style":809},[650],[363,98243,98245],{"className":98244,"style":34816},[34815],[6671,98246,98247],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},[6679,98248],{"d":34823},[363,98250,666],{"className":98251},[665],[363,98253,98255],{"className":98254},[638],[363,98256,98258],{"className":98257,"style":39431},[642],[363,98259],{},[363,98261,98263,98300],{"className":98262},[407],[363,98264,98266],{"className":98265},[407],[363,98267,98269],{"className":98268},[767,15179],[363,98270,98272,98292],{"className":98271},[633,634],[363,98273,98275,98289],{"className":98274},[638],[363,98276,98278],{"className":98277,"style":15189},[642],[363,98279,98280,98283],{"style":15192},[363,98281],{"className":98282,"style":15196},[650],[363,98284,98285],{"style":15199},[6671,98286,98287],{"xmlns":6673,"width":15202,"height":9798,"viewBox":15203},[6679,98288],{"d":15206},[363,98290,666],{"className":98291},[665],[363,98293,98295],{"className":98294},[638],[363,98296,98298],{"className":98297,"style":15216},[642],[363,98299],{},[363,98301,98303],{"className":98302},[629],[363,98304,98306,98339],{"className":98305},[633,634],[363,98307,98309,98336],{"className":98308},[638],[363,98310,98313,98325],{"className":98311,"style":98312},[642],"height:1.054em;",[363,98314,98316,98319],{"style":98315},"top:-2.3003em;margin-right:0.05em;",[363,98317],{"className":98318,"style":651},[650],[363,98320,98322],{"className":98321},[655,656,657,658],[363,98323,2811],{"className":98324},[407,658],[363,98326,98327,98330],{"style":51807},[363,98328],{"className":98329,"style":651},[650],[363,98331,98333],{"className":98332},[655,656,657,658],[363,98334,2811],{"className":98335},[407,658],[363,98337,666],{"className":98338},[665],[363,98340,98342],{"className":98341},[638],[363,98343,98346],{"className":98344,"style":98345},[642],"height:0.3997em;",[363,98347],{},[363,98349],{"className":98350,"style":506},[505],[363,98352,471],{"className":98353},[510],[363,98355],{"className":98356,"style":506},[505],[363,98358,98360,98363,98366,98369,98372,98375,98378],{"className":98359},[398],[363,98361],{"className":98362,"style":5960},[402],[363,98364,1635],{"className":98365,"style":5965},[1832,1833,5964],[363,98367],{"className":98368,"style":754},[505],[363,98370,11],{"className":98371},[407,408],[363,98373],{"className":98374,"style":506},[505],[363,98376,471],{"className":98377},[510],[363,98379],{"className":98380,"style":506},[505],[363,98382,98384,98387],{"className":98383},[398],[363,98385],{"className":98386,"style":520},[402],[363,98388,465],{"className":98389},[407],"），则 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就是单位球面上的欧氏距离——度量公理（对称、三角不等式）由 ",[363,98798,98800,98818],{"className":98799,"translate":367},[366],[363,98801,98803],{"className":98802},[371],[373,98804,98805],{"xmlns":375},[377,98806,98807,98815],{},[380,98808,98809],{},[7162,98810,98811,98813],{},[383,98812,22265],{},[463,98814,2811],{},[387,98816,98817],{"encoding":389},"L^2",[363,98819,98821],{"className":98820,"ariaHidden":394},[393],[363,98822,98824,98827],{"className":98823},[398],[363,98825],{"className":98826,"style":20400},[402],[363,98828,98830,98833],{"className":98829},[407],[363,98831,22265],{"className":98832},[407,408],[363,98834,98836],{"className":98835},[629],[363,98837,98839],{"className":98838},[633],[363,98840,98842],{"className":98841},[638],[363,98843,98845],{"className":98844,"style":20400},[642],[363,98846,98847,98850],{"style":7213},[363,98848],{"className":98849,"style":651},[650],[363,98851,98853],{"className":98852},[655,656,657,658],[363,98854,2811],{"className":98855},[407,658]," 距离直接继承，非退化由\"开平方是单射\"保证；",[7013,98858,98859,99175,99176,99255],{},[1344,98860,98861,98999,99000,99174],{},[363,98862,98864,98894],{"className":98863,"translate":367},[366],[363,98865,98867],{"className":98866},[371],[373,98868,98869],{"xmlns":375},[377,98870,98871,98891],{},[380,98872,98873],{},[34711,98874,98875],{},[380,98876,98877,98879,98881],{},[463,98878,2811],{},[1645,98880,1647],{},[543,98882,98883,98885],{},[383,98884,547],{},[380,98886,98887,98889],{},[383,98888,24115],{},[383,98890,26109],{},[387,98892,98893],{"encoding":389},"\\sqrt{2\\,D_{JS}}",[363,98895,98897],{"className":98896,"ariaHidden":394},[393],[363,98898,98900,98903],{"className":98899},[398],[363,98901],{"className":98902,"style":34735},[402],[363,98904,98906],{"className":98905},[407,34739],[363,98907,98909,98991],{"className":98908},[633,634],[363,98910,98912,98988],{"className":98911},[638],[363,98913,98915,98976],{"className":98914,"style":34749},[642],[363,98916,98918,98921],{"className":98917,"style":12213},[6655],[363,98919],{"className":98920,"style":809},[650],[363,98922,98924,98927,98930],{"className":98923,"style":34759},[407],[363,98925,2811],{"className":98926},[407],[363,98928],{"className":98929,"style":754},[505],[363,98931,98933,98936],{"className":98932},[407],[363,98934,547],{"className":98935,"style":625},[407,408],[363,98937,98939],{"className":98938},[629],[363,98940,98942,98968],{"className":98941},[633,634],[363,98943,98945,98965],{"className":98944},[638],[363,98946,98948],{"className":98947,"style":727},[642],[363,98949,98950,98953],{"style":646},[363,98951],{"className":98952,"style":651},[650],[363,98954,98956],{"className":98955},[655,656,657,658],[363,98957,98959,98962],{"className":98958},[407,658],[363,98960,24115],{"className":98961,"style":24525},[407,408,658],[363,98963,26109],{"className":98964,"style":26233},[407,408,658],[363,98966,666],{"className":98967},[665],[363,98969,98971],{"className":98970},[638],[363,98972,98974],{"className":98973,"style":8316},[642],[363,98975],{},[363,98977,98978,98981],{"style":34808},[363,98979],{"className":98980,"style":809},[650],[363,98982,98984],{"className":98983,"style":34816},[34815],[6671,98985,98986],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},[6679,98987],{"d":34823},[363,98989,666],{"className":98990},[665],[363,98992,98994],{"className":98993},[638],[363,98995,98997],{"className":98996,"style":34833},[642],[363,98998],{},"（等价地 ",[363,99001,99003,99031],{"className":99002,"translate":367},[366],[363,99004,99006],{"className":99005},[371],[373,99007,99008],{"xmlns":375},[377,99009,99010,99028],{},[380,99011,99012],{},[34711,99013,99014],{},[543,99015,99016,99018],{},[383,99017,547],{},[543,99019,99020,99022],{},[383,99021,385],{},[380,99023,99024,99026],{},[383,99025,24115],{},[383,99027,26109],{},[387,99029,99030],{"encoding":389},"\\sqrt{D_{f_{JS}}}",[363,99032,99034],{"className":99033,"ariaHidden":394},[393],[363,99035,99037,99041],{"className":99036},[398],[363,99038],{"className":99039,"style":99040},[402],"height:1.24em;vertical-align:-0.3564em;",[363,99042,99044],{"className":99043},[407,34739],[363,99045,99047,99165],{"className":99046},[633,634],[363,99048,99050,99162],{"className":99049},[638],[363,99051,99054,99149],{"className":99052,"style":99053},[642],"height:0.8836em;",[363,99055,99057,99060],{"className":99056,"style":83923},[6655],[363,99058],{"className":99059,"style":15196},[650],[363,99061,99063],{"className":99062,"style":41533},[407],[363,99064,99066,99069],{"className":99065},[407],[363,99067,547],{"className":99068,"style":625},[407,408],[363,99070,99072],{"className":99071},[629],[363,99073,99075,99141],{"className":99074},[633,634],[363,99076,99078,99138],{"className":99077},[638],[363,99079,99081],{"className":99080,"style":643},[642],[363,99082,99083,99086],{"style":646},[363,99084],{"className":99085,"style":651},[650],[363,99087,99089],{"className":99088},[655,656,657,658],[363,99090,99092],{"className":99091},[407,658],[363,99093,99095,99098],{"className":99094},[407,658],[363,99096,385],{"className":99097,"style":409},[407,408,658],[363,99099,99101],{"className":99100},[629],[363,99102,99104,99130],{"className":99103},[633,634],[363,99105,99107,99127],{"className":99106},[638],[363,99108,99110],{"className":99109,"style":6446},[642],[363,99111,99112,99115],{"style":30574},[363,99113],{"className":99114,"style":30578},[650],[363,99116,99118],{"className":99117},[655,30582,10022,658],[363,99119,99121,99124],{"className":99120},[407,658],[363,99122,24115],{"className":99123,"style":24525},[407,408,658],[363,99125,26109],{"className":99126,"style":26233},[407,408,658],[363,99128,666],{"className":99129},[665],[363,99131,99133],{"className":99132},[638],[363,99134,99136],{"className":99135,"style":30601},[642],[363,99137],{},[363,99139,666],{"className":99140},[665],[363,99142,99144],{"className":99143},[638],[363,99145,99147],{"className":99146,"style":673},[642],[363,99148],{},[363,99150,99152,99155],{"style":99151},"top:-2.8436em;",[363,99153],{"className":99154,"style":15196},[650],[363,99156,99158],{"className":99157,"style":84015},[34815],[6671,99159,99160],{"xmlns":6673,"width":6674,"height":84018,"viewBox":84019,"preserveAspectRatio":6677},[6679,99161],{"d":84022},[363,99163,666],{"className":99164},[665],[363,99166,99168],{"className":99167},[638],[363,99169,99172],{"className":99170,"style":99171},[642],"height:0.3564em;",[363,99173],{}," 规范化后）是度量","（Endres–Schindelin 的结论），其证明用到了 ",[363,99177,99179,99200],{"className":99178,"translate":367},[366],[363,99180,99182],{"className":99181},[371],[373,99183,99184],{"xmlns":375},[377,99185,99186,99198],{},[380,99187,99188],{},[543,99189,99190,99192],{},[383,99191,385],{},[380,99193,99194,99196],{},[383,99195,24115],{},[383,99197,26109],{},[387,99199,31127],{"encoding":389},[363,99201,99203],{"className":99202,"ariaHidden":394},[393],[363,99204,99206,99209],{"className":99205},[398],[363,99207],{"className":99208,"style":403},[402],[363,99210,99212,99215],{"className":99211},[407],[363,99213,385],{"className":99214,"style":409},[407,408],[363,99216,99218],{"className":99217},[629],[363,99219,99221,99247],{"className":99220},[633,634],[363,99222,99224,99244],{"className":99223},[638],[363,99225,99227],{"className":99226,"style":727},[642],[363,99228,99229,99232],{"style":27881},[363,99230],{"className":99231,"style":651},[650],[363,99233,99235],{"className":99234},[655,656,657,658],[363,99236,99238,99241],{"className":99237},[407,658],[363,99239,24115],{"className":99240,"style":24525},[407,408,658],[363,99242,26109],{"className":99243,"style":26233},[407,408,658],[363,99245,666],{"className":99246},[665],[363,99248,99250],{"className":99249},[638],[363,99251,99253],{"className":99252,"style":8316},[642],[363,99254],{}," 对应生成元的\"负定型\"性质。",[11,99257,99258,99259,99375,99376,99456,99457,99485,99486,99514],{},"反过来，不对称显然排除度量；即使对称，",[363,99260,99262,99282],{"className":99261,"translate":367},[366],[363,99263,99265],{"className":99264},[371],[373,99266,99267],{"xmlns":375},[377,99268,99269,99279],{},[380,99270,99271],{},[34711,99272,99273],{},[543,99274,99275,99277],{},[383,99276,547],{},[383,99278,385],{},[387,99280,99281],{"encoding":389},"\\sqrt{D_f}",[363,99283,99285],{"className":99284,"ariaHidden":394},[393],[363,99286,99288,99291],{"className":99287},[398],[363,99289],{"className":99290,"style":99040},[402],[363,99292,99294],{"className":99293},[407,34739],[363,99295,99297,99367],{"className":99296},[633,634],[363,99298,99300,99364],{"className":99299},[638],[363,99301,99303,99352],{"className":99302,"style":99053},[642],[363,99304,99306,99309],{"className":99305,"style":83923},[6655],[363,99307],{"className":99308,"style":15196},[650],[363,99310,99312],{"className":99311,"style":41533},[407],[363,99313,99315,99318],{"className":99314},[407],[363,99316,547],{"className":99317,"style":625},[407,408],[363,99319,99321],{"className":99320},[629],[363,99322,99324,99344],{"className":99323},[633,634],[363,99325,99327,99341],{"className":99326},[638],[363,99328,99330],{"className":99329,"style":643},[642],[363,99331,99332,99335],{"style":646},[363,99333],{"className":99334,"style":651},[650],[363,99336,99338],{"className":99337},[655,656,657,658],[363,99339,385],{"className":99340,"style":409},[407,408,658],[363,99342,666],{"className":99343},[665],[363,99345,99347],{"className":99346},[638],[363,99348,99350],{"className":99349,"style":673},[642],[363,99351],{},[363,99353,99354,99357],{"style":99151},[363,99355],{"className":99356,"style":15196},[650],[363,99358,99360],{"className":99359,"style":84015},[34815],[6671,99361,99362],{"xmlns":6673,"width":6674,"height":84018,"viewBox":84019,"preserveAspectRatio":6677},[6679,99363],{"d":84022},[363,99365,666],{"className":99366},[665],[363,99368,99370],{"className":99369},[638],[363,99371,99373],{"className":99372,"style":99171},[642],[363,99374],{}," 也不自动是度量。例如全变差 ",[363,99377,99379,99401],{"className":99378,"translate":367},[366],[363,99380,99382],{"className":99381},[371],[373,99383,99384],{"xmlns":375},[377,99385,99386,99398],{},[380,99387,99388],{},[543,99389,99390,99392],{},[383,99391,547],{},[380,99393,99394,99396],{},[383,99395,5740],{},[383,99397,35222],{},[387,99399,99400],{"encoding":389},"D_{TV}",[363,99402,99404],{"className":99403,"ariaHidden":394},[393],[363,99405,99407,99410],{"className":99406},[398],[363,99408],{"className":99409,"style":16311},[402],[363,99411,99413,99416],{"className":99412},[407],[363,99414,547],{"className":99415,"style":625},[407,408],[363,99417,99419],{"className":99418},[629],[363,99420,99422,99448],{"className":99421},[633,634],[363,99423,99425,99445],{"className":99424},[638],[363,99426,99428],{"className":99427,"style":727},[642],[363,99429,99430,99433],{"style":646},[363,99431],{"className":99432,"style":651},[650],[363,99434,99436],{"className":99435},[655,656,657,658],[363,99437,99439,99442],{"className":99438},[407,658],[363,99440,5740],{"className":99441,"style":682},[407,408,658],[363,99443,35222],{"className":99444,"style":2935},[407,408,658],[363,99446,666],{"className":99447},[665],[363,99449,99451],{"className":99450},[638],[363,99452,99454],{"className":99453,"style":8316},[642],[363,99455],{}," 本身就是度量，但若取某凸积 ",[363,99458,99460,99473],{"className":99459,"translate":367},[366],[363,99461,99463],{"className":99462},[371],[373,99464,99465],{"xmlns":375},[377,99466,99467,99471],{},[380,99468,99469],{},[383,99470,385],{},[387,99472,385],{"encoding":389},[363,99474,99476],{"className":99475,"ariaHidden":394},[393],[363,99477,99479,99482],{"className":99478},[398],[363,99480],{"className":99481,"style":403},[402],[363,99483,385],{"className":99484,"style":409},[407,408]," 的散度再做开方，三角不等式可能失效——这依赖 ",[363,99487,99489,99502],{"className":99488,"translate":367},[366],[363,99490,99492],{"className":99491},[371],[373,99493,99494],{"xmlns":375},[377,99495,99496,99500],{},[380,99497,99498],{},[383,99499,385],{},[387,99501,385],{"encoding":389},[363,99503,99505],{"className":99504,"ariaHidden":394},[393],[363,99506,99508,99511],{"className":99507},[398],[363,99509],{"className":99510,"style":403},[402],[363,99512,385],{"className":99513,"style":409},[407,408]," 是否落在\"负定（negative type）\"函数类中；不是自动成立的。",[11,99516,99517,99520,99521,99523,99524,99574,99575,99603,99604,99683],{},[1344,99518,99519],{},"与度量的对照结论："," f-散度的正确身份是",[1344,99522,52875],{},"（divergence）：非负、在 ",[363,99525,99527,99544],{"className":99526,"translate":367},[366],[363,99528,99530],{"className":99529},[371],[373,99531,99532],{"xmlns":375},[377,99533,99534,99542],{},[380,99535,99536,99538,99540],{},[383,99537,554],{},[458,99539,471],{},[383,99541,561],{},[387,99543,10619],{"encoding":389},[363,99545,99547,99565],{"className":99546,"ariaHidden":394},[393],[363,99548,99550,99553,99556,99559,99562],{"className":99549},[398],[363,99551],{"className":99552,"style":1011},[402],[363,99554,554],{"className":99555,"style":682},[407,408],[363,99557],{"className":99558,"style":506},[505],[363,99560,471],{"className":99561},[510],[363,99563],{"className":99564,"style":506},[505],[363,99566,99568,99571],{"className":99567},[398],[363,99569],{"className":99570,"style":913},[402],[363,99572,561],{"className":99573},[407,408]," 处取零，但不要求对称与三角不等式。当我们需要真距离时（例如做收敛分析、中心化、 ",[363,99576,99578,99591],{"className":99577,"translate":367},[366],[363,99579,99581],{"className":99580},[371],[373,99582,99583],{"xmlns":375},[377,99584,99585,99589],{},[380,99586,99587],{},[383,99588,7912],{},[387,99590,7912],{"encoding":389},[363,99592,99594],{"className":99593,"ariaHidden":394},[393],[363,99595,99597,99600],{"className":99596},[398],[363,99598],{"className":99599,"style":12947},[402],[363,99601,7912],{"className":99602,"style":8615},[407,408],"-中心聚类），就从 f-散度族里挑那些 ",[363,99605,99607,99623],{"className":99606,"translate":367},[366],[363,99608,99610],{"className":99609},[371],[373,99611,99612],{"xmlns":375},[377,99613,99614,99620],{},[380,99615,99616],{},[34711,99617,99618],{},[458,99619,3647],{"lspace":1595,"rspace":1595},[387,99621,99622],{"encoding":389},"\\sqrt{\\cdot}",[363,99624,99626],{"className":99625,"ariaHidden":394},[393],[363,99627,99629,99633],{"className":99628},[398],[363,99630],{"className":99631,"style":99632},[402],"height:1.04em;vertical-align:-0.2328em;",[363,99634,99636],{"className":99635},[407,34739],[363,99637,99639,99674],{"className":99638},[633,634],[363,99640,99642,99671],{"className":99641},[638],[363,99643,99646,99658],{"className":99644,"style":99645},[642],"height:0.8072em;",[363,99647,99649,99652],{"className":99648,"style":12213},[6655],[363,99650],{"className":99651,"style":809},[650],[363,99653,99655],{"className":99654,"style":34759},[407],[363,99656,3647],{"className":99657},[407],[363,99659,99661,99664],{"style":99660},"top:-2.7672em;",[363,99662],{"className":99663,"style":809},[650],[363,99665,99667],{"className":99666,"style":34816},[34815],[6671,99668,99669],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},[6679,99670],{"d":34823},[363,99672,666],{"className":99673},[665],[363,99675,99677],{"className":99676},[638],[363,99678,99681],{"className":99679,"style":99680},[642],"height:0.2328em;",[363,99682],{}," 后成为度量的成员（Hellinger、JS、TV），或干脆换成真正的度量类（第 6 节的 IPM）。",[18,99685,99687],{"id":99686},"_5-变分表示与-f-gan","5. 变分表示与 f-GAN",[2659,99689,99691],{"id":99690},"_51-可训练目标","5.1 可训练目标",[11,99693,99694],{},"4.4 节给出精确对偶：",[363,99696,99698],{"className":99697,"translate":367},[528],[363,99699,99701,99795],{"className":99700,"translate":367},[366],[363,99702,99704],{"className":99703},[371],[373,99705,99706],{"xmlns":375,"display":537},[377,99707,99708,99792],{},[380,99709,99710,99716,99718,99720,99722,99724,99726,99728,99738,99740,99746,99748,99750,99752,99754,99756,99758,99760,99766,99768,99774,99776,99778,99780,99782,99784,99786,99788,99790],{},[543,99711,99712,99714],{},[383,99713,547],{},[383,99715,385],{},[458,99717,461],{"stretchy":460},[383,99719,554],{},[383,99721,558],{"mathvariant":557},[383,99723,561],{},[458,99725,468],{"stretchy":460},[458,99727,471],{},[6364,99729,99730,99736],{},[380,99731,99732,99734],{},[383,99733,35804],{},[458,99735,4451],{},[383,99737,4506],{},[458,99739,3246],{"fence":460,"stretchy":394,"minsize":17789,"maxsize":17789},[543,99741,99742,99744],{},[383,99743,571],{"mathvariant":570},[383,99745,554],{},[458,99747,578],{"stretchy":460},[383,99749,4506],{},[458,99751,461],{"stretchy":460},[383,99753,945],{},[458,99755,468],{"stretchy":460},[458,99757,605],{"stretchy":460},[458,99759,5161],{},[543,99761,99762,99764],{},[383,99763,571],{"mathvariant":570},[383,99765,561],{},[458,99767,578],{"stretchy":460},[7162,99769,99770,99772],{},[383,99771,385],{},[458,99773,7921],{},[458,99775,461],{"stretchy":460},[383,99777,4506],{},[458,99779,461],{"stretchy":460},[383,99781,945],{},[458,99783,468],{"stretchy":460},[458,99785,468],{"stretchy":460},[458,99787,605],{"stretchy":460},[458,99789,3256],{"fence":460,"stretchy":394,"minsize":17789,"maxsize":17789},[383,99791,4723],{"mathvariant":557},[387,99793,99794],{"encoding":389},"D_f(P\\|Q)\n=\\sup_{g}\\Big\\{\\mathbb E_P[g(X)]\n-\\mathbb 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是数据分布、",[363,100130,100132,100145],{"className":100131,"translate":367},[366],[363,100133,100135],{"className":100134},[371],[373,100136,100137],{"xmlns":375},[377,100138,100139,100143],{},[380,100140,100141],{},[383,100142,561],{},[387,100144,561],{"encoding":389},[363,100146,100148],{"className":100147,"ariaHidden":394},[393],[363,100149,100151,100154],{"className":100150},[398],[363,100152],{"className":100153,"style":913},[402],[363,100155,561],{"className":100156},[407,408]," 是生成分布时，右端可直接用两批样本做蒙特卡洛估计，得到 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的一个",[1344,100229,100230],{},"下界","，并用一个网络 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逼近最优解。于是",[363,100305,100307],{"className":100306,"translate":367},[528],[363,100308,100310,100440],{"className":100309,"translate":367},[366],[363,100311,100313],{"className":100312},[371],[373,100314,100315],{"xmlns":375,"display":537},[377,100316,100317,100437],{},[380,100318,100319,100330,100332,100334,100336,100338,100340,100346,100348,100350,100356,100370,100376,100378,100384,100386,100388,100394,100409,100415,100417,100423,100425,100431,100433,100435],{},[543,100320,100321,100328],{},[12161,100322,100323,100325],{"accent":394},[383,100324,547],{},[458,100326,100327],{"stretchy":394},"^",[383,100329,385],{},[458,100331,461],{"stretchy":460},[383,100333,554],{},[383,100335,558],{"mathvariant":557},[383,100337,561],{},[458,100339,34247],{"separator":394},[543,100341,100342,100344],{},[383,100343,4506],{},[383,100345,100250],{},[458,100347,468],{"stretchy":460},[458,100349,471],{},[586,100351,100352,100354],{},[463,100353,465],{},[383,100355,82683],{},[48692,100357,100358,100360,100368],{},[458,100359,13811],{},[380,100361,100362,100364,100366],{},[383,100363,12313],{},[458,100365,471],{},[463,100367,465],{},[383,100369,82683],{},[543,100371,100372,100374],{},[383,100373,4506],{},[383,100375,100250],{},[458,100377,461],{"stretchy":460},[543,100379,100380,100382],{},[383,100381,1121],{},[383,100383,12313],{},[458,100385,468],{"stretchy":460},[458,100387,5161],{},[586,100389,100390,100392],{},[463,100391,465],{},[383,100393,26695],{},[48692,100395,100396,100398,100407],{},[458,100397,13811],{},[380,100399,100400,100403,100405],{},[383,100401,100402],{},"j",[458,100404,471],{},[463,100406,465],{},[383,100408,26695],{},[7162,100410,100411,100413],{},[383,100412,385],{},[458,100414,7921],{},[458,100416,461],{"stretchy":460},[543,100418,100419,100421],{},[383,100420,4506],{},[383,100422,100250],{},[458,100424,461],{"stretchy":460},[543,100426,100427,100429],{},[383,100428,40609],{},[383,100430,100402],{},[458,100432,468],{"stretchy":460},[458,100434,468],{"stretchy":460},[383,100436,4723],{"mathvariant":557},[387,100438,100439],{"encoding":389},"\\widehat 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的最优解为",[363,101816,101818],{"className":101817,"translate":367},[528],[363,101819,101821,101885],{"className":101820,"translate":367},[366],[363,101822,101824],{"className":101823},[371],[373,101825,101826],{"xmlns":375,"display":537},[377,101827,101828,101882],{},[380,101829,101830,101836,101838,101840,101842,101844,101850,101852,101880],{},[7162,101831,101832,101834],{},[383,101833,4506],{},[458,101835,7921],{},[458,101837,461],{"stretchy":460},[383,101839,1121],{},[458,101841,468],{"stretchy":460},[458,101843,471],{},[7162,101845,101846,101848],{},[383,101847,385],{},[458,101849,7168],{"mathvariant":557,"lspace":1595,"rspace":1595},[1645,101851,1652],{},[380,101853,101854,101856,101878],{},[458,101855,461],{"fence":394},[586,101857,101858,101868],{},[380,101859,101860,101862,101864,101866],{},[383,101861,11],{},[458,101863,461],{"stretchy":460},[383,101865,1121],{},[458,101867,468],{"stretchy":460},[380,101869,101870,101872,101874,101876],{},[383,101871,1291],{},[458,101873,461],{"stretchy":460},[383,101875,1121],{},[458,101877,468],{"stretchy":460},[458,101879,468],{"fence":394},[458,101881,898],{"separator":394},[387,101883,101884],{"encoding":389},"g^*(x)=f'\\!\\left(\\frac{p(x)}{q(x)}\\right),",[363,101886,101888,101941],{"className":101887,"ariaHidden":394},[393],[363,101889,101891,101894,101923,101926,101929,101932,101935,101938],{"className":101890},[398],[363,101892],{"className":101893,"style":487},[402],[363,101895,101897,101900],{"className":101896},[407],[363,101898,4506],{"className":101899,"style":1328},[407,408],[363,101901,101903],{"className":101902},[629],[363,101904,101906],{"className":101905},[633],[363,101907,101909],{"className":101908},[638],[363,101910,101912],{"className":101911,"style":8596},[642],[363,101913,101914,101917],{"style":8625},[363,101915],{"className":101916,"style":651},[650],[363,101918,101920],{"className":101919},[655,656,657,658],[363,101921,7921],{"className":101922},[2939,658],[363,101924,461],{"className":101925},[494],[363,101927,1121],{"className":101928},[407,408],[363,101930,468],{"className":101931},[501],[363,101933],{"className":101934,"style":506},[505],[363,101936,471],{"className":101937},[510],[363,101939],{"className":101940,"style":506},[505],[363,101942,101944,101947,101979,101982,101985,102080,102083],{"className":101943},[398],[363,101945],{"className":101946,"style":707},[402],[363,101948,101950,101953],{"className":101949},[407],[363,101951,385],{"className":101952,"style":409},[407,408],[363,101954,101956],{"className":101955},[629],[363,101957,101959],{"className":101958},[633],[363,101960,101962],{"className":101961},[638],[363,101963,101965],{"className":101964,"style":29343},[642],[363,101966,101967,101970],{"style":8625},[363,101968],{"className":101969,"style":651},[650],[363,101971,101973],{"className":101972},[655,656,657,658],[363,101974,101976],{"className":101975},[407,658],[363,101977,7168],{"className":101978},[407,658],[363,101980],{"className":101981,"style":1860},[505],[363,101983],{"className":101984,"style":754},[505],[363,101986,101988,101994,102074],{"className":101987},[758],[363,101989,101991],{"className":101990,"style":763},[494,762],[363,101992,461],{"className":101993},[767,657],[363,101995,101997,102000,102071],{"className":101996},[407],[363,101998],{"className":101999},[494,789],[363,102001,102003],{"className":102002},[586],[363,102004,102006,102063],{"className":102005},[633,634],[363,102007,102009,102060],{"className":102008},[638],[363,102010,102012,102032,102040],{"className":102011,"style":1891},[642],[363,102013,102014,102017],{"style":805},[363,102015],{"className":102016,"style":809},[650],[363,102018,102020,102023,102026,102029],{"className":102019},[407],[363,102021,1291],{"className":102022,"style":1328},[407,408],[363,102024,461],{"className":102025},[494],[363,102027,1121],{"className":102028},[407,408],[363,102030,468],{"className":102031},[501],[363,102033,102034,102037],{"style":821},[363,102035],{"className":102036,"style":809},[650],[363,102038],{"className":102039,"style":829},[828],[363,102041,102042,102045],{"style":832},[363,102043],{"className":102044,"style":809},[650],[363,102046,102048,102051,102054,102057],{"className":102047},[407],[363,102049,11],{"className":102050},[407,408],[363,102052,461],{"className":102053},[494],[363,102055,1121],{"className":102056},[407,408],[363,102058,468],{"className":102059},[501],[363,102061,666],{"className":102062},[665],[363,102064,102066],{"className":102065},[638],[363,102067,102069],{"className":102068,"style":1949},[642],[363,102070],{},[363,102072],{"className":102073},[501,789],[363,102075,102077],{"className":102076,"style":763},[501,762],[363,102078,468],{"className":102079},[767,657],[363,102081],{"className":102082,"style":754},[505],[363,102084,898],{"className":102085},[920],[11,102087,102088,102089,102151],{},"即判别器学到的量是似然比经 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之后的取值。",[11,102153,102154,102156,102157,102201,102202,102365,102366,102409],{},[1344,102155,55845],{}," 目标对 ",[363,102158,102160,102180],{"className":102159,"translate":367},[366],[363,102161,102163],{"className":102162},[371],[373,102164,102165],{"xmlns":375},[377,102166,102167,102177],{},[380,102168,102169,102171,102173,102175],{},[383,102170,4506],{},[458,102172,461],{"stretchy":460},[383,102174,1121],{},[458,102176,468],{"stretchy":460},[387,102178,102179],{"encoding":389},"g(x)",[363,102181,102183],{"className":102182,"ariaHidden":394},[393],[363,102184,102186,102189,102192,102195,102198],{"className":102185},[398],[363,102187],{"className":102188,"style":487},[402],[363,102190,4506],{"className":102191,"style":1328},[407,408],[363,102193,461],{"className":102194},[494],[363,102196,1121],{"className":102197},[407,408],[363,102199,468],{"className":102200},[501]," 的逐点依赖是 ",[363,102203,102205,102261],{"className":102204,"translate":367},[366],[363,102206,102208],{"className":102207},[371],[373,102209,102210],{"xmlns":375},[377,102211,102212,102258],{},[380,102213,102214,102216,102218,102220,102222,102224,102226,102228,102230,102232,102234,102236,102238,102240,102246,102248,102250,102252,102254,102256],{},[383,102215,11],{},[458,102217,461],{"stretchy":460},[383,102219,1121],{},[458,102221,468],{"stretchy":460},[383,102223,4506],{},[458,102225,461],{"stretchy":460},[383,102227,1121],{},[458,102229,468],{"stretchy":460},[458,102231,5161],{},[383,102233,1291],{},[458,102235,461],{"stretchy":460},[383,102237,1121],{},[458,102239,468],{"stretchy":460},[7162,102241,102242,102244],{},[383,102243,385],{},[458,102245,7921],{},[458,102247,461],{"stretchy":460},[383,102249,4506],{},[458,102251,461],{"stretchy":460},[383,102253,1121],{},[458,102255,468],{"stretchy":460},[458,102257,468],{"stretchy":460},[387,102259,102260],{"encoding":389},"p(x)g(x)-q(x)f^*(g(x))",[363,102262,102264,102303],{"className":102263,"ariaHidden":394},[393],[363,102265,102267,102270,102273,102276,102279,102282,102285,102288,102291,102294,102297,102300],{"className":102266},[398],[363,102268],{"className":102269,"style":487},[402],[363,102271,11],{"className":102272},[407,408],[363,102274,461],{"className":102275},[494],[363,102277,1121],{"className":102278},[407,408],[363,102280,468],{"className":102281},[501],[363,102283,4506],{"className":102284,"style":1328},[407,408],[363,102286,461],{"className":102287},[494],[363,102289,1121],{"className":102290},[407,408],[363,102292,468],{"className":102293},[501],[363,102295],{"className":102296,"style":2935},[505],[363,102298,5161],{"className":102299},[2939],[363,102301],{"className":102302,"style":2935},[505],[363,102304,102306,102309,102312,102315,102318,102321,102350,102353,102356,102359,102362],{"className":102305},[398],[363,102307],{"className":102308,"style":487},[402],[363,102310,1291],{"className":102311,"style":1328},[407,408],[363,102313,461],{"className":102314},[494],[363,102316,1121],{"className":102317},[407,408],[363,102319,468],{"className":102320},[501],[363,102322,102324,102327],{"className":102323},[407],[363,102325,385],{"className":102326,"style":409},[407,408],[363,102328,102330],{"className":102329},[629],[363,102331,102333],{"className":102332},[633],[363,102334,102336],{"className":102335},[638],[363,102337,102339],{"className":102338,"style":8847},[642],[363,102340,102341,102344],{"style":7213},[363,102342],{"className":102343,"style":651},[650],[363,102345,102347],{"className":102346},[655,656,657,658],[363,102348,7921],{"className":102349},[2939,658],[363,102351,461],{"className":102352},[494],[363,102354,4506],{"className":102355,"style":1328},[407,408],[363,102357,461],{"className":102358},[494],[363,102360,1121],{"className":102361},[407,408],[363,102363,19419],{"className":102364},[501],"，对 ",[363,102367,102369,102388],{"className":102368,"translate":367},[366],[363,102370,102372],{"className":102371},[371],[373,102373,102374],{"xmlns":375},[377,102375,102376,102386],{},[380,102377,102378,102380,102382,102384],{},[383,102379,4506],{},[458,102381,461],{"stretchy":460},[383,102383,1121],{},[458,102385,468],{"stretchy":460},[387,102387,102179],{"encoding":389},[363,102389,102391],{"className":102390,"ariaHidden":394},[393],[363,102392,102394,102397,102400,102403,102406],{"className":102393},[398],[363,102395],{"className":102396,"style":487},[402],[363,102398,4506],{"className":102399,"style":1328},[407,408],[363,102401,461],{"className":102402},[494],[363,102404,1121],{"className":102405},[407,408],[363,102407,468],{"className":102408},[501]," 求导：",[363,102411,102413],{"className":102412,"translate":367},[528],[363,102414,102416,102536],{"className":102415,"translate":367},[366],[363,102417,102419],{"className":102418},[371],[373,102420,102421],{"xmlns":375,"display":537},[377,102422,102423,102533],{},[380,102424,102425,102427,102429,102431,102433,102435,102437,102439,102441,102443,102445,102455,102457,102459,102461,102463,102465,102467,102469,102471,102473,102475,102477,102487,102489,102495,102497,102499,102501,102503,102505,102527,102529,102531],{},[383,102426,11],{},[458,102428,461],{"stretchy":460},[383,102430,1121],{},[458,102432,468],{"stretchy":460},[458,102434,5161],{},[383,102436,1291],{},[458,102438,461],{"stretchy":460},[383,102440,1121],{},[458,102442,468],{"stretchy":460},[1645,102444,1647],{},[7162,102446,102447,102449],{},[383,102448,385],{},[380,102450,102451,102453],{},[458,102452,7921],{},[458,102454,7168],{"mathvariant":557},[458,102456,461],{"stretchy":460},[383,102458,4506],{},[458,102460,461],{"stretchy":460},[383,102462,1121],{},[458,102464,468],{"stretchy":460},[458,102466,468],{"stretchy":460},[458,102468,471],{},[463,102470,474],{},[1645,102472,4631],{},[458,102474,4634],{},[1645,102476,4631],{},[7162,102478,102479,102481],{},[383,102480,385],{},[380,102482,102483,102485],{},[458,102484,7921],{},[458,102486,7168],{"mathvariant":557},[458,102488,461],{"stretchy":460},[7162,102490,102491,102493],{},[383,102492,4506],{},[458,102494,7921],{},[458,102496,461],{"stretchy":460},[383,102498,1121],{},[458,102500,468],{"stretchy":460},[458,102502,468],{"stretchy":460},[458,102504,471],{},[586,102506,102507,102517],{},[380,102508,102509,102511,102513,102515],{},[383,102510,11],{},[458,102512,461],{"stretchy":460},[383,102514,1121],{},[458,102516,468],{"stretchy":460},[380,102518,102519,102521,102523,102525],{},[383,102520,1291],{},[458,102522,461],{"stretchy":460},[383,102524,1121],{},[458,102526,468],{"stretchy":460},[458,102528,471],{},[383,102530,1116],{},[383,102532,4723],{"mathvariant":557},[387,102534,102535],{"encoding":389},"p(x)-q(x)\\,f^{*\\prime}(g(x))=0\n\\ \\Longrightarrow\\\nf^{*\\prime}(g^*(x))=\\frac{p(x)}{q(x)}=t .",[363,102537,102539,102566,102647,102671,102762,102858],{"className":102538,"ariaHidden":394},[393],[363,102540,102542,102545,102548,102551,102554,102557,102560,102563],{"className":102541},[398],[363,102543],{"className":102544,"style":487},[402],[363,102546,11],{"className":102547},[407,408],[363,102549,461],{"className":102550},[494],[363,102552,1121],{"className":102553},[407,408],[363,102555,468],{"className":102556},[501],[363,102558],{"className":102559,"style":2935},[505],[363,102561,5161],{"className":102562},[2939],[363,102564],{"className":102565,"style":2935},[505],[363,102567,102569,102573,102576,102579,102582,102585,102588,102623,102626,102629,102632,102635,102638,102641,102644],{"className":102568},[398],[363,102570],{"className":102571,"style":102572},[402],"height:1.0519em;vertical-align:-0.25em;",[363,102574,1291],{"className":102575,"style":1328},[407,408],[363,102577,461],{"className":102578},[494],[363,102580,1121],{"className":102581},[407,408],[363,102583,468],{"className":102584},[501],[363,102586],{"className":102587,"style":754},[505],[363,102589,102591,102594],{"className":102590},[407],[363,102592,385],{"className":102593,"style":409},[407,408],[363,102595,102597],{"className":102596},[629],[363,102598,102600],{"className":102599},[633],[363,102601,102603],{"className":102602},[638],[363,102604,102606],{"className":102605,"style":29343},[642],[363,102607,102608,102611],{"style":8625},[363,102609],{"className":102610,"style":651},[650],[363,102612,102614],{"className":102613},[655,656,657,658],[363,102615,102617,102620],{"className":102616},[407,658],[363,102618,7921],{"className":102619},[407,658],[363,102621,7168],{"className":102622},[407,658],[363,102624,461],{"className":102625},[494],[363,102627,4506],{"className":102628,"style":1328},[407,408],[363,102630,461],{"className":102631},[494],[363,102633,1121],{"className":102634},[407,408],[363,102636,19419],{"className":102637},[501],[363,102639],{"className":102640,"style":506},[505],[363,102642,471],{"className":102643},[510],[363,102645],{"className":102646,"style":506},[505],[363,102648,102650,102653,102656,102659,102662,102665,102668],{"className":102649},[398],[363,102651],{"className":102652,"style":93172},[402],[363,102654,474],{"className":102655},[407],[363,102657,4631],{"className":102658},[505],[363,102660],{"className":102661,"style":506},[505],[363,102663,4634],{"className":102664},[510],[363,102666,4631],{"className":102667},[505],[363,102669],{"className":102670,"style":506},[505],[363,102672,102674,102677,102712,102715,102744,102747,102750,102753,102756,102759],{"className":102673},[398],[363,102675],{"className":102676,"style":102572},[402],[363,102678,102680,102683],{"className":102679},[407],[363,102681,385],{"className":102682,"style":409},[407,408],[363,102684,102686],{"className":102685},[629],[363,102687,102689],{"className":102688},[633],[363,102690,102692],{"className":102691},[638],[363,102693,102695],{"className":102694,"style":29343},[642],[363,102696,102697,102700],{"style":8625},[363,102698],{"className":102699,"style":651},[650],[363,102701,102703],{"className":102702},[655,656,657,658],[363,102704,102706,102709],{"className":102705},[407,658],[363,102707,7921],{"className":102708},[407,658],[363,102710,7168],{"className":102711},[407,658],[363,102713,461],{"className":102714},[494],[363,102716,102718,102721],{"className":102717},[407],[363,102719,4506],{"className":102720,"style":1328},[407,408],[363,102722,102724],{"className":102723},[629],[363,102725,102727],{"className":102726},[633],[363,102728,102730],{"className":102729},[638],[363,102731,102733],{"className":102732,"style":8596},[642],[363,102734,102735,102738],{"style":8625},[363,102736],{"className":102737,"style":651},[650],[363,102739,102741],{"className":102740},[655,656,657,658],[363,102742,7921],{"className":102743},[2939,658],[363,102745,461],{"className":102746},[494],[363,102748,1121],{"className":102749},[407,408],[363,102751,19419],{"className":102752},[501],[363,102754],{"className":102755,"style":506},[505],[363,102757,471],{"className":102758},[510],[363,102760],{"className":102761,"style":506},[505],[363,102763,102765,102769,102849,102852,102855],{"className":102764},[398],[363,102766],{"className":102767,"style":102768},[402],"height:2.363em;vertical-align:-0.936em;",[363,102770,102772,102775,102846],{"className":102771},[407],[363,102773],{"className":102774},[494,789],[363,102776,102778],{"className":102777},[586],[363,102779,102781,102838],{"className":102780},[633,634],[363,102782,102784,102835],{"className":102783},[638],[363,102785,102787,102807,102815],{"className":102786,"style":1891},[642],[363,102788,102789,102792],{"style":805},[363,102790],{"className":102791,"style":809},[650],[363,102793,102795,102798,102801,102804],{"className":102794},[407],[363,102796,1291],{"className":102797,"style":1328},[407,408],[363,102799,461],{"className":102800},[494],[363,102802,1121],{"className":102803},[407,408],[363,102805,468],{"className":102806},[501],[363,102808,102809,102812],{"style":821},[363,102810],{"className":102811,"style":809},[650],[363,102813],{"className":102814,"style":829},[828],[363,102816,102817,102820],{"style":832},[363,102818],{"className":102819,"style":809},[650],[363,102821,102823,102826,102829,102832],{"className":102822},[407],[363,102824,11],{"className":102825},[407,408],[363,102827,461],{"className":102828},[494],[363,102830,1121],{"className":102831},[407,408],[363,102833,468],{"className":102834},[501],[363,102836,666],{"className":102837},[665],[363,102839,102841],{"className":102840},[638],[363,102842,102844],{"className":102843,"style":1949},[642],[363,102845],{},[363,102847],{"className":102848},[501,789],[363,102850],{"className":102851,"style":506},[505],[363,102853,471],{"className":102854},[510],[363,102856],{"className":102857,"style":506},[505],[363,102859,102861,102864,102867],{"className":102860},[398],[363,102862],{"className":102863,"style":5691},[402],[363,102865,1116],{"className":102866},[407,408],[363,102868,4723],{"className":102869},[407],[11,102871,34464,102872,103079,103080,103236,103237,103295,103296,103390,103391,103394,103395,103464,103465,103546,103547,103575],{},[363,102873,102875,102921],{"className":102874,"translate":367},[366],[363,102876,102878],{"className":102877},[371],[373,102879,102880],{"xmlns":375},[377,102881,102882,102918],{},[380,102883,102884,102886,102892,102898,102900,102902,102908],{},[458,102885,461],{"stretchy":460},[7162,102887,102888,102890],{},[383,102889,385],{},[458,102891,7921],{},[7162,102893,102894,102896],{},[458,102895,468],{"stretchy":460},[458,102897,7168],{"mathvariant":557,"lspace":1595,"rspace":1595},[458,102899,471],{},[458,102901,461],{"stretchy":460},[7162,102903,102904,102906],{},[383,102905,385],{},[458,102907,7168],{"mathvariant":557,"lspace":1595,"rspace":1595},[7162,102909,102910,102912],{},[458,102911,468],{"stretchy":460},[380,102913,102914,102916],{},[458,102915,5161],{},[463,102917,465],{},[387,102919,102920],{"encoding":389},"(f^*)'=(f')^{-1}",[363,102922,102924,103003],{"className":102923,"ariaHidden":394},[393],[363,102925,102927,102930,102933,102962,102994,102997,103000],{"className":102926},[398],[363,102928],{"className":102929,"style":7191},[402],[363,102931,461],{"className":102932},[494],[363,102934,102936,102939],{"className":102935},[407],[363,102937,385],{"className":102938,"style":409},[407,408],[363,102940,102942],{"className":102941},[629],[363,102943,102945],{"className":102944},[633],[363,102946,102948],{"className":102947},[638],[363,102949,102951],{"className":102950,"style":8847},[642],[363,102952,102953,102956],{"style":7213},[363,102954],{"className":102955,"style":651},[650],[363,102957,102959],{"className":102958},[655,656,657,658],[363,102960,7921],{"className":102961},[2939,658],[363,102963,102965,102968],{"className":102964},[501],[363,102966,468],{"className":102967},[501],[363,102969,102971],{"className":102970},[629],[363,102972,102974],{"className":102973},[633],[363,102975,102977],{"className":102976},[638],[363,102978,102980],{"className":102979,"style":7210},[642],[363,102981,102982,102985],{"style":7213},[363,102983],{"className":102984,"style":651},[650],[363,102986,102988],{"className":102987},[655,656,657,658],[363,102989,102991],{"className":102990},[407,658],[363,102992,7168],{"className":102993},[407,658],[363,102995],{"className":102996,"style":506},[505],[363,102998,471],{"className":102999},[510],[363,103001],{"className":103002,"style":506},[505],[363,103004,103006,103009,103012,103044],{"className":103005},[398],[363,103007],{"className":103008,"style":20375},[402],[363,103010,461],{"className":103011},[494],[363,103013,103015,103018],{"className":103014},[407],[363,103016,385],{"className":103017,"style":409},[407,408],[363,103019,103021],{"className":103020},[629],[363,103022,103024],{"className":103023},[633],[363,103025,103027],{"className":103026},[638],[363,103028,103030],{"className":103029,"style":7210},[642],[363,103031,103032,103035],{"style":7213},[363,103033],{"className":103034,"style":651},[650],[363,103036,103038],{"className":103037},[655,656,657,658],[363,103039,103041],{"className":103040},[407,658],[363,103042,7168],{"className":103043},[407,658],[363,103045,103047,103050],{"className":103046},[501],[363,103048,468],{"className":103049},[501],[363,103051,103053],{"className":103052},[629],[363,103054,103056],{"className":103055},[633],[363,103057,103059],{"className":103058},[638],[363,103060,103062],{"className":103061,"style":20400},[642],[363,103063,103064,103067],{"style":7213},[363,103065],{"className":103066,"style":651},[650],[363,103068,103070],{"className":103069},[655,656,657,658],[363,103071,103073,103076],{"className":103072},[407,658],[363,103074,5161],{"className":103075},[407,658],[363,103077,465],{"className":103078},[407,658],"（Fenchel 的互逆关系），即得 ",[363,103081,103083,103127],{"className":103082,"translate":367},[366],[363,103084,103086],{"className":103085},[371],[373,103087,103088],{"xmlns":375},[377,103089,103090,103124],{},[380,103091,103092,103098,103100,103102,103104,103106,103112,103114,103116,103118,103120,103122],{},[7162,103093,103094,103096],{},[383,103095,4506],{},[458,103097,7921],{},[458,103099,461],{"stretchy":460},[383,103101,1121],{},[458,103103,468],{"stretchy":460},[458,103105,471],{},[7162,103107,103108,103110],{},[383,103109,385],{},[458,103111,7168],{"mathvariant":557,"lspace":1595,"rspace":1595},[458,103113,461],{"stretchy":460},[383,103115,1116],{},[458,103117,461],{"stretchy":460},[383,103119,1121],{},[458,103121,468],{"stretchy":460},[458,103123,468],{"stretchy":460},[387,103125,103126],{"encoding":389},"g^*(x)=f'(t(x))",[363,103128,103130,103183],{"className":103129,"ariaHidden":394},[393],[363,103131,103133,103136,103165,103168,103171,103174,103177,103180],{"className":103132},[398],[363,103134],{"className":103135,"style":487},[402],[363,103137,103139,103142],{"className":103138},[407],[363,103140,4506],{"className":103141,"style":1328},[407,408],[363,103143,103145],{"className":103144},[629],[363,103146,103148],{"className":103147},[633],[363,103149,103151],{"className":103150},[638],[363,103152,103154],{"className":103153,"style":8847},[642],[363,103155,103156,103159],{"style":7213},[363,103157],{"className":103158,"style":651},[650],[363,103160,103162],{"className":103161},[655,656,657,658],[363,103163,7921],{"className":103164},[2939,658],[363,103166,461],{"className":103167},[494],[363,103169,1121],{"className":103170},[407,408],[363,103172,468],{"className":103173},[501],[363,103175],{"className":103176,"style":506},[505],[363,103178,471],{"className":103179},[510],[363,103181],{"className":103182,"style":506},[505],[363,103184,103186,103189,103221,103224,103227,103230,103233],{"className":103185},[398],[363,103187],{"className":103188,"style":7191},[402],[363,103190,103192,103195],{"className":103191},[407],[363,103193,385],{"className":103194,"style":409},[407,408],[363,103196,103198],{"className":103197},[629],[363,103199,103201],{"className":103200},[633],[363,103202,103204],{"className":103203},[638],[363,103205,103207],{"className":103206,"style":7210},[642],[363,103208,103209,103212],{"style":7213},[363,103210],{"className":103211,"style":651},[650],[363,103213,103215],{"className":103214},[655,656,657,658],[363,103216,103218],{"className":103217},[407,658],[363,103219,7168],{"className":103220},[407,658],[363,103222,461],{"className":103223},[494],[363,103225,1116],{"className":103226},[407,408],[363,103228,461],{"className":103229},[494],[363,103231,1121],{"className":103232},[407,408],[363,103234,19419],{"className":103235},[501],"。把 ",[363,103238,103240,103257],{"className":103239,"translate":367},[366],[363,103241,103243],{"className":103242},[371],[373,103244,103245],{"xmlns":375},[377,103246,103247,103255],{},[380,103248,103249],{},[7162,103250,103251,103253],{},[383,103252,4506],{},[458,103254,7921],{},[387,103256,64388],{"encoding":389},[363,103258,103260],{"className":103259,"ariaHidden":394},[393],[363,103261,103263,103266],{"className":103262},[398],[363,103264],{"className":103265,"style":64398},[402],[363,103267,103269,103272],{"className":103268},[407],[363,103270,4506],{"className":103271,"style":1328},[407,408],[363,103273,103275],{"className":103274},[629],[363,103276,103278],{"className":103277},[633],[363,103279,103281],{"className":103280},[638],[363,103282,103284],{"className":103283,"style":8847},[642],[363,103285,103286,103289],{"style":7213},[363,103287],{"className":103288,"style":651},[650],[363,103290,103292],{"className":103291},[655,656,657,658],[363,103293,7921],{"className":103294},[2939,658]," 代回即得 ",[363,103297,103299,103326],{"className":103298,"translate":367},[366],[363,103300,103302],{"className":103301},[371],[373,103303,103304],{"xmlns":375},[377,103305,103306,103324],{},[380,103307,103308,103314,103316,103318,103320,103322],{},[543,103309,103310,103312],{},[383,103311,547],{},[383,103313,385],{},[458,103315,461],{"stretchy":460},[383,103317,554],{},[383,103319,558],{"mathvariant":557},[383,103321,561],{},[458,103323,468],{"stretchy":460},[387,103325,5587],{"encoding":389},[363,103327,103329],{"className":103328,"ariaHidden":394},[393],[363,103330,103332,103335,103375,103378,103381,103384,103387],{"className":103331},[398],[363,103333],{"className":103334,"style":618},[402],[363,103336,103338,103341],{"className":103337},[407],[363,103339,547],{"className":103340,"style":625},[407,408],[363,103342,103344],{"className":103343},[629],[363,103345,103347,103367],{"className":103346},[633,634],[363,103348,103350,103364],{"className":103349},[638],[363,103351,103353],{"className":103352,"style":643},[642],[363,103354,103355,103358],{"style":646},[363,103356],{"className":103357,"style":651},[650],[363,103359,103361],{"className":103360},[655,656,657,658],[363,103362,385],{"className":103363,"style":409},[407,408,658],[363,103365,666],{"className":103366},[665],[363,103368,103370],{"className":103369},[638],[363,103371,103373],{"className":103372,"style":673},[642],[363,103374],{},[363,103376,461],{"className":103377},[494],[363,103379,554],{"className":103380,"style":682},[407,408],[363,103382,558],{"className":103383},[407],[363,103385,561],{"className":103386},[407,408],[363,103388,468],{"className":103389},[501],"（这正是定理 4.1 所给的最优解）。",[1344,103392,103393],{},"实践含义："," 判别器网络 ",[363,103396,103398,103415],{"className":103397,"translate":367},[366],[363,103399,103401],{"className":103400},[371],[373,103402,103403],{"xmlns":375},[377,103404,103405,103413],{},[380,103406,103407],{},[543,103408,103409,103411],{},[383,103410,4506],{},[383,103412,100250],{},[387,103414,100253],{"encoding":389},[363,103416,103418],{"className":103417,"ariaHidden":394},[393],[363,103419,103421,103424],{"className":103420},[398],[363,103422],{"className":103423,"style":2624},[402],[363,103425,103427,103430],{"className":103426},[407],[363,103428,4506],{"className":103429,"style":1328},[407,408],[363,103431,103433],{"className":103432},[629],[363,103434,103436,103456],{"className":103435},[633,634],[363,103437,103439,103453],{"className":103438},[638],[363,103440,103442],{"className":103441,"style":45301},[642],[363,103443,103444,103447],{"style":19215},[363,103445],{"className":103446,"style":651},[650],[363,103448,103450],{"className":103449},[655,656,657,658],[363,103451,100250],{"className":103452,"style":1328},[407,408,658],[363,103454,666],{"className":103455},[665],[363,103457,103459],{"className":103458},[638],[363,103460,103462],{"className":103461,"style":8316},[642],[363,103463],{}," 学的是 ",[363,103466,103468,103493],{"className":103467,"translate":367},[366],[363,103469,103471],{"className":103470},[371],[373,103472,103473],{"xmlns":375},[377,103474,103475,103490],{},[380,103476,103477,103483,103485,103488],{},[7162,103478,103479,103481],{},[383,103480,385],{},[458,103482,7168],{"mathvariant":557,"lspace":1595,"rspace":1595},[458,103484,461],{"stretchy":460},[1645,103486,103487],{},"似然比",[458,103489,468],{"stretchy":460},[387,103491,103492],{"encoding":389},"f'(\\text{似然比})",[363,103494,103496],{"className":103495,"ariaHidden":394},[393],[363,103497,103499,103502,103534,103537,103543],{"className":103498},[398],[363,103500],{"className":103501,"style":7191},[402],[363,103503,103505,103508],{"className":103504},[407],[363,103506,385],{"className":103507,"style":409},[407,408],[363,103509,103511],{"className":103510},[629],[363,103512,103514],{"className":103513},[633],[363,103515,103517],{"className":103516},[638],[363,103518,103520],{"className":103519,"style":7210},[642],[363,103521,103522,103525],{"style":7213},[363,103523],{"className":103524,"style":651},[650],[363,103526,103528],{"className":103527},[655,656,657,658],[363,103529,103531],{"className":103530},[407,658],[363,103532,7168],{"className":103533},[407,658],[363,103535,461],{"className":103536},[494],[363,103538,103540],{"className":103539},[407,4946],[363,103541,103487],{"className":103542},[407,4950],[363,103544,468],{"className":103545},[501],"，而不是似然比本身；不同 ",[363,103548,103550,103563],{"className":103549,"translate":367},[366],[363,103551,103553],{"className":103552},[371],[373,103554,103555],{"xmlns":375},[377,103556,103557,103561],{},[380,103558,103559],{},[383,103560,385],{},[387,103562,385],{"encoding":389},[363,103564,103566],{"className":103565,"ariaHidden":394},[393],[363,103567,103569,103572],{"className":103568},[398],[363,103570],{"className":103571,"style":403},[402],[363,103573,385],{"className":103574,"style":409},[407,408]," 等价于对似然比做了不同的单调「整形」，因此决定了梯度的尺度与饱和方式。",[2659,103577,103579],{"id":103578},"_53-各散度的-gan-目标","5.3 各散度的 GAN 目标",[11,103581,103582,103583,9748],{},"取上表的 ",[363,103584,103586,103603],{"className":103585,"translate":367},[366],[363,103587,103589],{"className":103588},[371],[373,103590,103591],{"xmlns":375},[377,103592,103593,103601],{},[380,103594,103595],{},[7162,103596,103597,103599],{},[383,103598,385],{},[458,103600,7921],{},[387,103602,61091],{"encoding":389},[363,103604,103606],{"className":103605,"ariaHidden":394},[393],[363,103607,103609,103612],{"className":103608},[398],[363,103610],{"className":103611,"style":403},[402],[363,103613,103615,103618],{"className":103614},[407],[363,103616,385],{"className":103617,"style":409},[407,408],[363,103619,103621],{"className":103620},[629],[363,103622,103624],{"className":103623},[633],[363,103625,103627],{"className":103626},[638],[363,103628,103630],{"className":103629,"style":8847},[642],[363,103631,103632,103635],{"style":7213},[363,103633],{"className":103634,"style":651},[650],[363,103636,103638],{"className":103637},[655,656,657,658],[363,103639,7921],{"className":103640},[2939,658],[10375,103642,103643],{},[7013,103644,103645,16145,103647,45980],{},[1344,103646,33035],{},[363,103648,103650,103682],{"className":103649,"translate":367},[366],[363,103651,103653],{"className":103652},[371],[373,103654,103655],{"xmlns":375},[377,103656,103657,103679],{},[380,103658,103659,103665,103667],{},[7162,103660,103661,103663],{},[383,103662,385],{},[458,103664,7921],{},[458,103666,471],{},[7162,103668,103669,103671],{},[383,103670,7915],{},[380,103672,103673,103675,103677],{},[383,103674,40609],{},[458,103676,5161],{},[463,103678,465],{},[387,103680,103681],{"encoding":389},"f^*=e^{y-1}",[363,103683,103685,103729],{"className":103684,"ariaHidden":394},[393],[363,103686,103688,103691,103720,103723,103726],{"className":103687},[398],[363,103689],{"className":103690,"style":403},[402],[363,103692,103694,103697],{"className":103693},[407],[363,103695,385],{"className":103696,"style":409},[407,408],[363,103698,103700],{"className":103699},[629],[363,103701,103703],{"className":103702},[633],[363,103704,103706],{"className":103705},[638],[363,103707,103709],{"className":103708,"style":8847},[642],[363,103710,103711,103714],{"style":7213},[363,103712],{"className":103713,"style":651},[650],[363,103715,103717],{"className":103716},[655,656,657,658],[363,103718,7921],{"className":103719},[2939,658],[363,103721],{"className":103722,"style":506},[505],[363,103724,471],{"className":103725},[510],[363,103727],{"className":103728,"style":506},[505],[363,103730,103732,103735],{"className":103731},[398],[363,103733],{"className":103734,"style":20400},[402],[363,103736,103738,103741],{"className":103737},[407],[363,103739,7915],{"className":103740},[407,408],[363,103742,103744],{"className":103743},[629],[363,103745,103747],{"className":103746},[633],[363,103748,103750],{"className":103749},[638],[363,103751,103753],{"className":103752,"style":20400},[642],[363,103754,103755,103758],{"style":7213},[363,103756],{"className":103757,"style":651},[650],[363,103759,103761],{"className":103760},[655,656,657,658],[363,103762,103764,103767,103770],{"className":103763},[407,658],[363,103765,40609],{"className":103766,"style":1328},[407,408,658],[363,103768,5161],{"className":103769},[2939,658],[363,103771,465],{"className":103772},[407,658],[363,103774,103776],{"className":103775,"translate":367},[528],[363,103777,103779,103865],{"className":103778,"translate":367},[366],[363,103780,103782],{"className":103781},[371],[373,103783,103784],{"xmlns":375,"display":537},[377,103785,103786,103862],{},[380,103787,103788,103798,103800,103802,103804,103806,103808,103810,103820,103822,103828,103830,103832,103834,103836,103842,103844,103856,103858,103860],{},[543,103789,103790,103792],{},[383,103791,547],{},[380,103793,103794,103796],{},[383,103795,11044],{},[383,103797,22265],{},[458,103799,461],{"stretchy":460},[383,103801,554],{},[383,103803,558],{"mathvariant":557},[383,103805,561],{},[458,103807,468],{"stretchy":460},[458,103809,471],{},[6364,103811,103812,103818],{},[380,103813,103814,103816],{},[383,103815,35804],{},[458,103817,4451],{},[383,103819,4506],{},[458,103821,3246],{"fence":460,"stretchy":394,"minsize":17789,"maxsize":17789},[543,103823,103824,103826],{},[383,103825,571],{"mathvariant":570},[383,103827,554],{},[458,103829,578],{"stretchy":460},[383,103831,4506],{},[458,103833,605],{"stretchy":460},[458,103835,5161],{},[543,103837,103838,103840],{},[383,103839,571],{"mathvariant":570},[383,103841,561],{},[458,103843,578],{"stretchy":460},[7162,103845,103846,103848],{},[383,103847,7915],{},[380,103849,103850,103852,103854],{},[383,103851,4506],{},[458,103853,5161],{},[463,103855,465],{},[458,103857,605],{"stretchy":460},[458,103859,3256],{"fence":460,"stretchy":394,"minsize":17789,"maxsize":17789},[383,103861,4723],{"mathvariant":557},[387,103863,103864],{"encoding":389},"D_{KL}(P\\|Q)\n=\\sup_g\\Big\\{\\mathbb E_P[g]\n-\\mathbb E_Q[e^{g-1}]\\Big\\}.",[363,103866,103868,103944,104061],{"className":103867,"ariaHidden":394},[393],[363,103869,103871,103874,103920,103923,103926,103929,103932,103935,103938,103941],{"className":103870},[398],[363,103872],{"className":103873,"style":487},[402],[363,103875,103877,103880],{"className":103876},[407],[363,103878,547],{"className":103879,"style":625},[407,408],[363,103881,103883],{"className":103882},[629],[363,103884,103886,103912],{"className":103885},[633,634],[363,103887,103889,103909],{"className":103888},[638],[363,103890,103892],{"className":103891,"style":727},[642],[363,103893,103894,103897],{"style":646},[363,103895],{"className":103896,"style":651},[650],[363,103898,103900],{"className":103899},[655,656,657,658],[363,103901,103903,103906],{"className":103902},[407,658],[363,103904,11044],{"className":103905,"style":11132},[407,408,658],[363,103907,22265],{"className":103908},[407,408,658],[363,103910,666],{"className":103911},[665],[363,103913,103915],{"className":103914},[638],[363,103916,103918],{"className":103917,"style":8316},[642],[363,103919],{},[363,103921,461],{"className":103922},[494],[363,103924,554],{"className":103925,"style":682},[407,408],[363,103927,558],{"className":103928},[407],[363,103930,561],{"className":103931},[407,408],[363,103933,468],{"className":103934},[501],[363,103936],{"className":103937,"style":506},[505],[363,103939,471],{"className":103940},[510],[363,103942],{"className":103943,"style":506},[505],[363,103945,103947,103950,103994,103997,104003,104043,104046,104049,104052,104055,104058],{"className":103946},[398],[363,103948],{"className":103949,"style":99874},[402],[363,103951,103953],{"className":103952},[1832,14066],[363,103954,103956,103986],{"className":103955},[633,634],[363,103957,103959,103983],{"className":103958},[638],[363,103960,103962,103973],{"className":103961,"style":2693},[642],[363,103963,103964,103967],{"style":61573},[363,103965],{"className":103966,"style":809},[650],[363,103968,103970],{"className":103969},[655,656,657,658],[363,103971,4506],{"className":103972,"style":1328},[407,408,658],[363,103974,103975,103978],{"style":12213},[363,103976],{"className":103977,"style":809},[650],[363,103979,103980],{},[363,103981,35804],{"className":103982},[1832],[363,103984,666],{"className":103985},[665],[363,103987,103989],{"className":103988},[638],[363,103990,103992],{"className":103991,"style":61605},[642],[363,103993],{},[363,103995],{"className":103996,"style":754},[505],[363,103998,104000],{"className":103999},[407],[363,104001,3246],{"className":104002},[767,4068],[363,104004,104006,104009],{"className":104005},[407],[363,104007,571],{"className":104008},[407,714],[363,104010,104012],{"className":104011},[629],[363,104013,104015,104035],{"className":104014},[633,634],[363,104016,104018,104032],{"className":104017},[638],[363,104019,104021],{"className":104020,"style":727},[642],[363,104022,104023,104026],{"style":730},[363,104024],{"className":104025,"style":651},[650],[363,104027,104029],{"className":104028},[655,656,657,658],[363,104030,554],{"className":104031,"style":682},[407,408,658],[363,104033,666],{"className":104034},[665],[363,104036,104038],{"className":104037},[638],[363,104039,104041],{"className":104040,"style":8316},[642],[363,104042],{},[363,104044,578],{"className":104045},[494],[363,104047,4506],{"className":104048,"style":1328},[407,408],[363,104050,605],{"className":104051},[501],[363,104053],{"className":104054,"style":2935},[505],[363,104056,5161],{"className":104057},[2939],[363,104059],{"className":104060,"style":2935},[505],[363,104062,104064,104067,104107,104110,104148,104151,104157],{"className":104063},[398],[363,104065],{"className":104066,"style":68500},[402],[363,104068,104070,104073],{"className":104069},[407],[363,104071,571],{"className":104072},[407,714],[363,104074,104076],{"className":104075},[629],[363,104077,104079,104099],{"className":104078},[633,634],[363,104080,104082,104096],{"className":104081},[638],[363,104083,104085],{"className":104084,"style":727},[642],[363,104086,104087,104090],{"style":730},[363,104088],{"className":104089,"style":651},[650],[363,104091,104093],{"className":104092},[655,656,657,658],[363,104094,561],{"className":104095},[407,408,658],[363,104097,666],{"className":104098},[665],[363,104100,104102],{"className":104101},[638],[363,104103,104105],{"className":104104,"style":673},[642],[363,104106],{},[363,104108,578],{"className":104109},[494],[363,104111,104113,104116],{"className":104112},[407],[363,104114,7915],{"className":104115},[407,408],[363,104117,104119],{"className":104118},[629],[363,104120,104122],{"className":104121},[633],[363,104123,104125],{"className":104124},[638],[363,104126,104128],{"className":104127,"style":20107},[642],[363,104129,104130,104133],{"style":8625},[363,104131],{"className":104132,"style":651},[650],[363,104134,104136],{"className":104135},[655,656,657,658],[363,104137,104139,104142,104145],{"className":104138},[407,658],[363,104140,4506],{"className":104141,"style":1328},[407,408,658],[363,104143,5161],{"className":104144},[2939,658],[363,104146,465],{"className":104147},[407,658],[363,104149,605],{"className":104150},[501],[363,104152,104154],{"className":104153},[407],[363,104155,3256],{"className":104156},[767,4068],[363,104158,4723],{"className":104159},[407],[11,104161,104162,104163,104371,104372,104445,104446,104478,104479,104507],{},"（这是「MINE-f」式下界。经典 Donsker–Varadhan 下界是 ",[363,104164,104166,104212],{"className":104165,"translate":367},[366],[363,104167,104169],{"className":104168},[371],[373,104170,104171],{"xmlns":375},[377,104172,104173,104209],{},[380,104174,104175,104181,104183,104185,104187,104189,104191,104193,104199,104201,104207],{},[543,104176,104177,104179],{},[383,104178,571],{"mathvariant":570},[383,104180,554],{},[458,104182,578],{"stretchy":460},[383,104184,4506],{},[458,104186,605],{"stretchy":460},[458,104188,5161],{},[383,104190,4448],{},[458,104192,4451],{},[543,104194,104195,104197],{},[383,104196,571],{"mathvariant":570},[383,104198,561],{},[458,104200,578],{"stretchy":460},[7162,104202,104203,104205],{},[383,104204,7915],{},[383,104206,4506],{},[458,104208,605],{"stretchy":460},[387,104210,104211],{"encoding":389},"\\mathbb E_P[g]-\\log\\mathbb E_Q[e^{g}]",[363,104213,104215,104279],{"className":104214,"ariaHidden":394},[393],[363,104216,104218,104221,104261,104264,104267,104270,104273,104276],{"className":104217},[398],[363,104219],{"className":104220,"style":487},[402],[363,104222,104224,104227],{"className":104223},[407],[363,104225,571],{"className":104226},[407,714],[363,104228,104230],{"className":104229},[629],[363,104231,104233,104253],{"className":104232},[633,634],[363,104234,104236,104250],{"className":104235},[638],[363,104237,104239],{"className":104238,"style":727},[642],[363,104240,104241,104244],{"style":730},[363,104242],{"className":104243,"style":651},[650],[363,104245,104247],{"className":104246},[655,656,657,658],[363,104248,554],{"className":104249,"style":682},[407,408,658],[363,104251,666],{"className":104252},[665],[363,104254,104256],{"className":104255},[638],[363,104257,104259],{"className":104258,"style":8316},[642],[363,104260],{},[363,104262,578],{"className":104263},[494],[363,104265,4506],{"className":104266,"style":1328},[407,408],[363,104268,605],{"className":104269},[501],[363,104271],{"className":104272,"style":2935},[505],[363,104274,5161],{"className":104275},[2939],[363,104277],{"className":104278,"style":2935},[505],[363,104280,104282,104285,104290,104293,104333,104336,104368],{"className":104281},[398],[363,104283],{"className":104284,"style":618},[402],[363,104286,4502,104288],{"className":104287},[1832],[363,104289,4506],{"style":4505},[363,104291],{"className":104292,"style":754},[505],[363,104294,104296,104299],{"className":104295},[407],[363,104297,571],{"className":104298},[407,714],[363,104300,104302],{"className":104301},[629],[363,104303,104305,104325],{"className":104304},[633,634],[363,104306,104308,104322],{"className":104307},[638],[363,104309,104311],{"className":104310,"style":727},[642],[363,104312,104313,104316],{"style":730},[363,104314],{"className":104315,"style":651},[650],[363,104317,104319],{"className":104318},[655,656,657,658],[363,104320,561],{"className":104321},[407,408,658],[363,104323,666],{"className":104324},[665],[363,104326,104328],{"className":104327},[638],[363,104329,104331],{"className":104330,"style":673},[642],[363,104332],{},[363,104334,578],{"className":104335},[494],[363,104337,104339,104342],{"className":104338},[407],[363,104340,7915],{"className":104341},[407,408],[363,104343,104345],{"className":104344},[629],[363,104346,104348],{"className":104347},[633],[363,104349,104351],{"className":104350},[638],[363,104352,104354],{"className":104353,"style":50725},[642],[363,104355,104356,104359],{"style":7213},[363,104357],{"className":104358,"style":651},[650],[363,104360,104362],{"className":104361},[655,656,657,658],[363,104363,104365],{"className":104364},[407,658],[363,104366,4506],{"className":104367,"style":1328},[407,408,658],[363,104369,605],{"className":104370},[501],"，二者同源但第二项不同：",[363,104373,104375,104398],{"className":104374,"translate":367},[366],[363,104376,104378],{"className":104377},[371],[373,104379,104380],{"xmlns":375},[377,104381,104382,104396],{},[380,104383,104384],{},[7162,104385,104386,104388],{},[383,104387,7915],{},[380,104389,104390,104392,104394],{},[383,104391,40609],{},[458,104393,5161],{},[463,104395,465],{},[387,104397,65285],{"encoding":389},[363,104399,104401],{"className":104400,"ariaHidden":394},[393],[363,104402,104404,104407],{"className":104403},[398],[363,104405],{"className":104406,"style":20400},[402],[363,104408,104410,104413],{"className":104409},[407],[363,104411,7915],{"className":104412},[407,408],[363,104414,104416],{"className":104415},[629],[363,104417,104419],{"className":104418},[633],[363,104420,104422],{"className":104421},[638],[363,104423,104425],{"className":104424,"style":20400},[642],[363,104426,104427,104430],{"style":7213},[363,104428],{"className":104429,"style":651},[650],[363,104431,104433],{"className":104432},[655,656,657,658],[363,104434,104436,104439,104442],{"className":104435},[407,658],[363,104437,40609],{"className":104438,"style":1328},[407,408,658],[363,104440,5161],{"className":104441},[2939,658],[363,104443,465],{"className":104444},[407,658]," 是把 ",[363,104447,104449,104464],{"className":104448,"translate":367},[366],[363,104450,104452],{"className":104451},[371],[373,104453,104454],{"xmlns":375},[377,104455,104456,104462],{},[380,104457,104458,104460],{},[383,104459,4448],{},[458,104461,4451],{},[387,104463,21955],{"encoding":389},[363,104465,104467],{"className":104466,"ariaHidden":394},[393],[363,104468,104470,104473],{"className":104469},[398],[363,104471],{"className":104472,"style":403},[402],[363,104474,4502,104476],{"className":104475},[1832],[363,104477,4506],{"style":4505}," 移到最外层并做 ",[363,104480,104482,104495],{"className":104481,"translate":367},[366],[363,104483,104485],{"className":104484},[371],[373,104486,104487],{"xmlns":375},[377,104488,104489,104493],{},[380,104490,104491],{},[383,104492,385],{},[387,104494,385],{"encoding":389},[363,104496,104498],{"className":104497,"ariaHidden":394},[393],[363,104499,104501,104504],{"className":104500},[398],[363,104502],{"className":104503,"style":403},[402],[363,104505,385],{"className":104506,"style":409},[407,408],"-散度对偶后的结果。）",[10375,104509,104510,104736,105129,105560],{},[7013,104511,104512,104515,104516,49544,104549,104577,104578,104633,104634,104684,104685,104735],{},[1344,104513,104514],{},"反向 KL \u002F α-GAN","：把 α 从 ",[363,104517,104519,104534],{"className":104518,"translate":367},[366],[363,104520,104522],{"className":104521},[371],[373,104523,104524],{"xmlns":375},[377,104525,104526,104532],{},[380,104527,104528,104530],{},[458,104529,5161],{},[463,104531,465],{},[387,104533,49528],{"encoding":389},[363,104535,104537],{"className":104536,"ariaHidden":394},[393],[363,104538,104540,104543,104546],{"className":104539},[398],[363,104541],{"className":104542,"style":12928},[402],[363,104544,5161],{"className":104545},[407],[363,104547,465],{"className":104548},[407],[363,104550,104552,104565],{"className":104551,"translate":367},[366],[363,104553,104555],{"className":104554},[371],[373,104556,104557],{"xmlns":375},[377,104558,104559,104563],{},[380,104560,104561],{},[463,104562,465],{},[387,104564,465],{"encoding":389},[363,104566,104568],{"className":104567,"ariaHidden":394},[393],[363,104569,104571,104574],{"className":104570},[398],[363,104572],{"className":104573,"style":520},[402],[363,104575,465],{"className":104576},[407],"，散度从反向 KL（",[363,104579,104581,104600],{"className":104580,"translate":367},[366],[363,104582,104584],{"className":104583},[371],[373,104585,104586],{"xmlns":375},[377,104587,104588,104598],{},[380,104589,104590,104592,104594,104596],{},[383,104591,45126],{},[458,104593,1378],{},[458,104595,5161],{},[463,104597,465],{},[387,104599,49411],{"encoding":389},[363,104601,104603,104621],{"className":104602,"ariaHidden":394},[393],[363,104604,104606,104609,104612,104615,104618],{"className":104605},[398],[363,104607],{"className":104608,"style":2693},[402],[363,104610,45126],{"className":104611,"style":45156},[407,408],[363,104613],{"className":104614,"style":506},[505],[363,104616,1378],{"className":104617},[510],[363,104619],{"className":104620,"style":506},[505],[363,104622,104624,104627,104630],{"className":104623},[398],[363,104625],{"className":104626,"style":12928},[402],[363,104628,5161],{"className":104629},[407],[363,104631,465],{"className":104632},[407],"，zero-forcing、模式崩塌倾向）经 Hellinger 型对称散度（",[363,104635,104637,104654],{"className":104636,"translate":367},[366],[363,104638,104640],{"className":104639},[371],[373,104641,104642],{"xmlns":375},[377,104643,104644,104652],{},[380,104645,104646,104648,104650],{},[383,104647,45126],{},[458,104649,471],{},[463,104651,474],{},[387,104653,46860],{"encoding":389},[363,104655,104657,104675],{"className":104656,"ariaHidden":394},[393],[363,104658,104660,104663,104666,104669,104672],{"className":104659},[398],[363,104661],{"className":104662,"style":2693},[402],[363,104664,45126],{"className":104665,"style":45156},[407,408],[363,104667],{"className":104668,"style":506},[505],[363,104670,471],{"className":104671},[510],[363,104673],{"className":104674,"style":506},[505],[363,104676,104678,104681],{"className":104677},[398],[363,104679],{"className":104680,"style":520},[402],[363,104682,474],{"className":104683},[407],"）过渡到正向 KL（",[363,104686,104688,104705],{"className":104687,"translate":367},[366],[363,104689,104691],{"className":104690},[371],[373,104692,104693],{"xmlns":375},[377,104694,104695,104703],{},[380,104696,104697,104699,104701],{},[383,104698,45126],{},[458,104700,1378],{},[463,104702,465],{},[387,104704,47587],{"encoding":389},[363,104706,104708,104726],{"className":104707,"ariaHidden":394},[393],[363,104709,104711,104714,104717,104720,104723],{"className":104710},[398],[363,104712],{"className":104713,"style":2693},[402],[363,104715,45126],{"className":104716,"style":45156},[407,408],[363,104718],{"className":104719,"style":506},[505],[363,104721,1378],{"className":104722},[510],[363,104724],{"className":104725,"style":506},[505],[363,104727,104729,104732],{"className":104728},[398],[363,104730],{"className":104731,"style":520},[402],[363,104733,465],{"className":104734},[407],"，zero-avoiding、模式覆盖倾向），可以在\"崩塌—覆盖\"之间连续调节（见 3.6 节的两端极限）。",[7013,104737,104738,9748,104740,3093,104984,105124,105125,105128],{},[1344,104739,53496],{},[363,104741,104743,104793],{"className":104742,"translate":367},[366],[363,104744,104746],{"className":104745},[371],[373,104747,104748],{"xmlns":375},[377,104749,104750,104790],{},[380,104751,104752,104758,104760,104762,104768,104770,104772,104774,104776,104778,104788],{},[7162,104753,104754,104756],{},[383,104755,385],{},[458,104757,7921],{},[458,104759,471],{},[458,104761,5161],{},[586,104763,104764,104766],{},[463,104765,465],{},[463,104767,2811],{},[383,104769,4448],{},[458,104771,4451],{},[458,104773,461],{"stretchy":460},[463,104775,2811],{},[458,104777,5161],{},[7162,104779,104780,104782],{},[383,104781,7915],{},[380,104783,104784,104786],{},[463,104785,2811],{},[383,104787,40609],{},[458,104789,468],{"stretchy":460},[387,104791,104792],{"encoding":389},"f^*=-\\frac12\\log(2-e^{2y})",[363,104794,104796,104840,104940],{"className":104795,"ariaHidden":394},[393],[363,104797,104799,104802,104831,104834,104837],{"className":104798},[398],[363,104800],{"className":104801,"style":403},[402],[363,104803,104805,104808],{"className":104804},[407],[363,104806,385],{"className":104807,"style":409},[407,408],[363,104809,104811],{"className":104810},[629],[363,104812,104814],{"className":104813},[633],[363,104815,104817],{"className":104816},[638],[363,104818,104820],{"className":104819,"style":8847},[642],[363,104821,104822,104825],{"style":7213},[363,104823],{"className":104824,"style":651},[650],[363,104826,104828],{"className":104827},[655,656,657,658],[363,104829,7921],{"className":104830},[2939,658],[363,104832],{"className":104833,"style":506},[505],[363,104835,471],{"className":104836},[510],[363,104838],{"className":104839,"style":506},[505],[363,104841,104843,104846,104849,104917,104920,104925,104928,104931,104934,104937],{"className":104842},[398],[363,104844],{"className":104845,"style":2853},[402],[363,104847,5161],{"className":104848},[407],[363,104850,104852,104855,104914],{"className":104851},[407],[363,104853],{"className":104854},[494,789],[363,104856,104858],{"className":104857},[586],[363,104859,104861,104906],{"className":104860},[633,634],[363,104862,104864,104903],{"className":104863},[638],[363,104865,104867,104881,104889],{"className":104866,"style":2872},[642],[363,104868,104869,104872],{"style":2875},[363,104870],{"className":104871,"style":809},[650],[363,104873,104875],{"className":104874},[655,656,657,658],[363,104876,104878],{"className":104877},[407,658],[363,104879,2811],{"className":104880},[407,658],[363,104882,104883,104886],{"style":821},[363,104884],{"className":104885,"style":809},[650],[363,104887],{"className":104888,"style":829},[828],[363,104890,104891,104894],{"style":2898},[363,104892],{"className":104893,"style":809},[650],[363,104895,104897],{"className":104896},[655,656,657,658],[363,104898,104900],{"className":104899},[407,658],[363,104901,465],{"className":104902},[407,658],[363,104904,666],{"className":104905},[665],[363,104907,104909],{"className":104908},[638],[363,104910,104912],{"className":104911,"style":2920},[642],[363,104913],{},[363,104915],{"className":104916},[501,789],[363,104918],{"className":104919,"style":754},[505],[363,104921,4502,104923],{"className":104922},[1832],[363,104924,4506],{"style":4505},[363,104926,461],{"className":104927},[494],[363,104929,2811],{"className":104930},[407],[363,104932],{"className":104933,"style":2935},[505],[363,104935,5161],{"className":104936},[2939],[363,104938],{"className":104939,"style":2935},[505],[363,104941,104943,104946,104981],{"className":104942},[398],[363,104944],{"className":104945,"style":20375},[402],[363,104947,104949,104952],{"className":104948},[407],[363,104950,7915],{"className":104951},[407,408],[363,104953,104955],{"className":104954},[629],[363,104956,104958],{"className":104957},[633],[363,104959,104961],{"className":104960},[638],[363,104962,104964],{"className":104963,"style":20400},[642],[363,104965,104966,104969],{"style":7213},[363,104967],{"className":104968,"style":651},[650],[363,104970,104972],{"className":104971},[655,656,657,658],[363,104973,104975,104978],{"className":104974},[407,658],[363,104976,2811],{"className":104977},[407,658],[363,104979,40609],{"className":104980,"style":1328},[407,408,658],[363,104982,468],{"className":104983},[501],[363,104985,104987,105014],{"className":104986,"translate":367},[366],[363,104988,104990],{"className":104989},[371],[373,104991,104992],{"xmlns":375},[377,104993,104994,105012],{},[380,104995,104996,104998,105000],{},[383,104997,40609],{},[458,104999,7599],{},[586,105001,105002,105010],{},[380,105003,105004,105006,105008],{},[383,105005,4448],{},[458,105007,4451],{},[463,105009,2811],{},[463,105011,2811],{},[387,105013,66126],{"encoding":389},[363,105015,105017,105035],{"className":105016,"ariaHidden":394},[393],[363,105018,105020,105023,105026,105029,105032],{"className":105019},[398],[363,105021],{"className":105022,"style":32064},[402],[363,105024,40609],{"className":105025,"style":1328},[407,408],[363,105027],{"className":105028,"style":506},[505],[363,105030,7599],{"className":105031},[510],[363,105033],{"className":105034,"style":506},[505],[363,105036,105038,105041],{"className":105037},[398],[363,105039],{"className":105040,"style":66154},[402],[363,105042,105044,105047,105121],{"className":105043},[407],[363,105045],{"className":105046},[494,789],[363,105048,105050],{"className":105049},[586],[363,105051,105053,105113],{"className":105052},[633,634],[363,105054,105056,105110],{"className":105055},[638],[363,105057,105059,105073,105081],{"className":105058,"style":3173},[642],[363,105060,105061,105064],{"style":2875},[363,105062],{"className":105063,"style":809},[650],[363,105065,105067],{"className":105066},[655,656,657,658],[363,105068,105070],{"className":105069},[407,658],[363,105071,2811],{"className":105072},[407,658],[363,105074,105075,105078],{"style":821},[363,105076],{"className":105077,"style":809},[650],[363,105079],{"className":105080,"style":829},[828],[363,105082,105083,105086],{"style":3201},[363,105084],{"className":105085,"style":809},[650],[363,105087,105089],{"className":105088},[655,656,657,658],[363,105090,105092,105104,105107],{"className":105091},[407,658],[363,105093,105095,105098,105101],{"className":105094},[1832,658],[363,105096,48030],{"className":105097},[658],[363,105099,48034],{"className":105100},[658],[363,105102,4506],{"className":105103,"style":4505},[658],[363,105105],{"className":105106,"style":42738},[505,658],[363,105108,2811],{"className":105109},[407,658],[363,105111,666],{"className":105112},[665],[363,105114,105116],{"className":105115},[638],[363,105117,105119],{"className":105118,"style":2920},[642],[363,105120],{},[363,105122],{"className":105123},[501,789],"。代回即",[1344,105126,105127],{},"原始 GAN 的对抗目标","（见 5.4）。",[7013,105130,105131,9748,105192,105332,105333,105555,105556,105559],{},[1344,105132,105133,105191],{},[363,105134,105136,105153],{"className":105135,"translate":367},[366],[363,105137,105139],{"className":105138},[371],[373,105140,105141],{"xmlns":375},[377,105142,105143,105151],{},[380,105144,105145],{},[7162,105146,105147,105149],{},[383,105148,37821],{},[463,105150,2811],{},[387,105152,40766],{"encoding":389},[363,105154,105156],{"className":105155,"ariaHidden":394},[393],[363,105157,105159,105162],{"className":105158},[398],[363,105160],{"className":105161,"style":40776},[402],[363,105163,105165,105168],{"className":105164},[407],[363,105166,37821],{"className":105167},[407,408],[363,105169,105171],{"className":105170},[629],[363,105172,105174],{"className":105173},[633],[363,105175,105177],{"className":105176},[638],[363,105178,105180],{"className":105179,"style":20400},[642],[363,105181,105182,105185],{"style":7213},[363,105183],{"className":105184,"style":651},[650],[363,105186,105188],{"className":105187},[655,656,657,658],[363,105189,2811],{"className":105190},[407,658],"（LSGAN）",[363,105193,105195,105229],{"className":105194,"translate":367},[366],[363,105196,105198],{"className":105197},[371],[373,105199,105200],{"xmlns":375},[377,105201,105202,105226],{},[380,105203,105204,105210,105212,105214,105216,105222,105224],{},[7162,105205,105206,105208],{},[383,105207,385],{},[458,105209,7921],{},[458,105211,471],{},[383,105213,40609],{},[458,105215,2818],{},[7162,105217,105218,105220],{},[383,105219,40609],{},[463,105221,2811],{},[383,105223,1288],{"mathvariant":557},[463,105225,39361],{},[387,105227,105228],{"encoding":389},"f^*=y+y^2\u002F4",[363,105230,105232,105276,105294],{"className":105231,"ariaHidden":394},[393],[363,105233,105235,105238,105267,105270,105273],{"className":105234},[398],[363,105236],{"className":105237,"style":403},[402],[363,105239,105241,105244],{"className":105240},[407],[363,105242,385],{"className":105243,"style":409},[407,408],[363,105245,105247],{"className":105246},[629],[363,105248,105250],{"className":105249},[633],[363,105251,105253],{"className":105252},[638],[363,105254,105256],{"className":105255,"style":8847},[642],[363,105257,105258,105261],{"style":7213},[363,105259],{"className":105260,"style":651},[650],[363,105262,105264],{"className":105263},[655,656,657,658],[363,105265,7921],{"className":105266},[2939,658],[363,105268],{"className":105269,"style":506},[505],[363,105271,471],{"className":105272},[510],[363,105274],{"className":105275,"style":506},[505],[363,105277,105279,105282,105285,105288,105291],{"className":105278},[398],[363,105280],{"className":105281,"style":40693},[402],[363,105283,40609],{"className":105284,"style":1328},[407,408],[363,105286],{"className":105287,"style":2935},[505],[363,105289,2818],{"className":105290},[2939],[363,105292],{"className":105293,"style":2935},[505],[363,105295,105297,105300,105329],{"className":105296},[398],[363,105298],{"className":105299,"style":20375},[402],[363,105301,105303,105306],{"className":105302},[407],[363,105304,40609],{"className":105305,"style":1328},[407,408],[363,105307,105309],{"className":105308},[629],[363,105310,105312],{"className":105311},[633],[363,105313,105315],{"className":105314},[638],[363,105316,105318],{"className":105317,"style":20400},[642],[363,105319,105320,105323],{"style":7213},[363,105321],{"className":105322,"style":651},[650],[363,105324,105326],{"className":105325},[655,656,657,658],[363,105327,2811],{"className":105328},[407,658],[363,105330,40744],{"className":105331},[407],"，目标变为 ",[363,105334,105336,105386],{"className":105335,"translate":367},[366],[363,105337,105339],{"className":105338},[371],[373,105340,105341],{"xmlns":375},[377,105342,105343,105383],{},[380,105344,105345,105351,105353,105355,105357,105359,105365,105367,105369,105371,105377,105379,105381],{},[543,105346,105347,105349],{},[383,105348,571],{"mathvariant":570},[383,105350,554],{},[458,105352,578],{"stretchy":460},[383,105354,4506],{},[458,105356,605],{"stretchy":460},[458,105358,5161],{},[543,105360,105361,105363],{},[383,105362,571],{"mathvariant":570},[383,105364,561],{},[458,105366,578],{"stretchy":460},[383,105368,4506],{},[458,105370,2818],{},[7162,105372,105373,105375],{},[383,105374,4506],{},[463,105376,2811],{},[383,105378,1288],{"mathvariant":557},[463,105380,39361],{},[458,105382,605],{"stretchy":460},[387,105384,105385],{"encoding":389},"\\mathbb E_P[g]-\\mathbb E_Q[g+g^2\u002F4]",[363,105387,105389,105453,105514],{"className":105388,"ariaHidden":394},[393],[363,105390,105392,105395,105435,105438,105441,105444,105447,105450],{"className":105391},[398],[363,105393],{"className":105394,"style":487},[402],[363,105396,105398,105401],{"className":105397},[407],[363,105399,571],{"className":105400},[407,714],[363,105402,105404],{"className":105403},[629],[363,105405,105407,105427],{"className":105406},[633,634],[363,105408,105410,105424],{"className":105409},[638],[363,105411,105413],{"className":105412,"style":727},[642],[363,105414,105415,105418],{"style":730},[363,105416],{"className":105417,"style":651},[650],[363,105419,105421],{"className":105420},[655,656,657,658],[363,105422,554],{"className":105423,"style":682},[407,408,658],[363,105425,666],{"className":105426},[665],[363,105428,105430],{"className":105429},[638],[363,105431,105433],{"className":105432,"style":8316},[642],[363,105434],{},[363,105436,578],{"className":105437},[494],[363,105439,4506],{"className":105440,"style":1328},[407,408],[363,105442,605],{"className":105443},[501],[363,105445],{"className":105446,"style":2935},[505],[363,105448,5161],{"className":105449},[2939],[363,105451],{"className":105452,"style":2935},[505],[363,105454,105456,105459,105499,105502,105505,105508,105511],{"className":105455},[398],[363,105457],{"className":105458,"style":618},[402],[363,105460,105462,105465],{"className":105461},[407],[363,105463,571],{"className":105464},[407,714],[363,105466,105468],{"className":105467},[629],[363,105469,105471,105491],{"className":105470},[633,634],[363,105472,105474,105488],{"className":105473},[638],[363,105475,105477],{"className":105476,"style":727},[642],[363,105478,105479,105482],{"style":730},[363,105480],{"className":105481,"style":651},[650],[363,105483,105485],{"className":105484},[655,656,657,658],[363,105486,561],{"className":105487},[407,408,658],[363,105489,666],{"className":105490},[665],[363,105492,105494],{"className":105493},[638],[363,105495,105497],{"className":105496,"style":673},[642],[363,105498],{},[363,105500,578],{"className":105501},[494],[363,105503,4506],{"className":105504,"style":1328},[407,408],[363,105506],{"className":105507,"style":2935},[505],[363,105509,2818],{"className":105510},[2939],[363,105512],{"className":105513,"style":2935},[505],[363,105515,105517,105520,105549,105552],{"className":105516},[398],[363,105518],{"className":105519,"style":20375},[402],[363,105521,105523,105526],{"className":105522},[407],[363,105524,4506],{"className":105525,"style":1328},[407,408],[363,105527,105529],{"className":105528},[629],[363,105530,105532],{"className":105531},[633],[363,105533,105535],{"className":105534},[638],[363,105536,105538],{"className":105537,"style":20400},[642],[363,105539,105540,105543],{"style":7213},[363,105541],{"className":105542,"style":651},[650],[363,105544,105546],{"className":105545},[655,656,657,658],[363,105547,2811],{"className":105548},[407,658],[363,105550,40744],{"className":105551},[407],[363,105553,605],{"className":105554},[501],"，即对判别器输出做",[1344,105557,105558],{},"最小二乘","回归，梯度不会像 sigmoid 那样饱和。",[7013,105561,105562,9748,105565,3093,105688,10528],{},[1344,105563,105564],{},"Hellinger²",[363,105566,105568,105602],{"className":105567,"translate":367},[366],[363,105569,105571],{"className":105570},[371],[373,105572,105573],{"xmlns":375},[377,105574,105575,105599],{},[380,105576,105577,105583,105585,105587,105589,105591,105593,105595,105597],{},[7162,105578,105579,105581],{},[383,105580,385],{},[458,105582,7921],{},[458,105584,471],{},[383,105586,40609],{},[383,105588,1288],{"mathvariant":557},[458,105590,461],{"stretchy":460},[463,105592,465],{},[458,105594,5161],{},[383,105596,40609],{},[458,105598,468],{"stretchy":460},[387,105600,105601],{"encoding":389},"f^*=y\u002F(1-y)",[363,105603,105605,105649,105676],{"className":105604,"ariaHidden":394},[393],[363,105606,105608,105611,105640,105643,105646],{"className":105607},[398],[363,105609],{"className":105610,"style":403},[402],[363,105612,105614,105617],{"className":105613},[407],[363,105615,385],{"className":105616,"style":409},[407,408],[363,105618,105620],{"className":105619},[629],[363,105621,105623],{"className":105622},[633],[363,105624,105626],{"className":105625},[638],[363,105627,105629],{"className":105628,"style":8847},[642],[363,105630,105631,105634],{"style":7213},[363,105632],{"className":105633,"style":651},[650],[363,105635,105637],{"className":105636},[655,656,657,658],[363,105638,7921],{"className":105639},[2939,658],[363,105641],{"className":105642,"style":506},[505],[363,105644,471],{"className":105645},[510],[363,105647],{"className":105648,"style":506},[505],[363,105650,105652,105655,105658,105661,105664,105667,105670,105673],{"className":105651},[398],[363,105653],{"className":105654,"style":487},[402],[363,105656,40609],{"className":105657,"style":1328},[407,408],[363,105659,1288],{"className":105660},[407],[363,105662,461],{"className":105663},[494],[363,105665,465],{"className":105666},[407],[363,105668],{"className":105669,"style":2935},[505],[363,105671,5161],{"className":105672},[2939],[363,105674],{"className":105675,"style":2935},[505],[363,105677,105679,105682,105685],{"className":105678},[398],[363,105680],{"className":105681,"style":487},[402],[363,105683,40609],{"className":105684,"style":1328},[407,408],[363,105686,468],{"className":105687},[501],[363,105689,105691,105708],{"className":105690,"translate":367},[366],[363,105692,105694],{"className":105693},[371],[373,105695,105696],{"xmlns":375},[377,105697,105698,105706],{},[380,105699,105700,105702,105704],{},[383,105701,40609],{},[458,105703,7599],{},[463,105705,465],{},[387,105707,67156],{"encoding":389},[363,105709,105711,105729],{"className":105710,"ariaHidden":394},[393],[363,105712,105714,105717,105720,105723,105726],{"className":105713},[398],[363,105715],{"className":105716,"style":32064},[402],[363,105718,40609],{"className":105719,"style":1328},[407,408],[363,105721],{"className":105722,"style":506},[505],[363,105724,7599],{"className":105725},[510],[363,105727],{"className":105728,"style":506},[505],[363,105730,105732,105735],{"className":105731},[398],[363,105733],{"className":105734,"style":520},[402],[363,105736,465],{"className":105737},[407],[2659,105739,105741],{"id":105740},"_54-原始-gan-与-js-散度","5.4 原始 GAN 与 JS 散度",[11,105743,105744,105745,105927],{},"原始 GAN 的判别器输出经 sigmoid 得 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JS 散度下的对抗博弈。判别器写成 logit",[363,105929,105931],{"className":105930,"translate":367},[528],[363,105932,105934,106012],{"className":105933,"translate":367},[366],[363,105935,105937],{"className":105936},[371],[373,105938,105939],{"xmlns":375,"display":537},[377,105940,105941,106009],{},[380,105942,105943,105945,105947,105949,105951,105953,105955,105957,105963,105965,105967,105969,105971,105973,105975,105977,105979,105981,105987,105989,105991,105993,105995,105997,105999,106001,106007],{},[383,105944,4506],{},[458,105946,461],{"stretchy":460},[383,105948,1121],{},[458,105950,468],{"stretchy":460},[458,105952,471],{},[383,105954,4448],{},[458,105956,4451],{},[543,105958,105959,105961],{},[383,105960,547],{},[383,105962,7909],{},[458,105964,461],{"stretchy":460},[383,105966,1121],{},[458,105968,468],{"stretchy":460},[458,105970,5161],{},[383,105972,4448],{},[458,105974,4451],{},[458,105976,461],{"stretchy":460},[463,105978,465],{},[458,105980,5161],{},[543,105982,105983,105985],{},[383,105984,547],{},[383,105986,7909],{},[458,105988,461],{"stretchy":460},[383,105990,1121],{},[458,105992,468],{"stretchy":460},[458,105994,468],{"stretchy":460},[458,105996,471],{},[383,105998,4448],{},[458,106000,4451],{},[586,106002,106003,106005],{},[383,106004,11],{},[383,106006,1291],{},[458,106008,898],{"separator":394},[387,106010,106011],{"encoding":389},"g(x)=\\log 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E_P\\Big[\\log\\frac{2p}{p+q}\\Big]\n=\\mathbb E_P[\\log D_s]+\\log2,",[363,108048,108050,108086,108241,108350],{"className":108049,"ariaHidden":394},[393],[363,108051,108053,108056,108062,108065,108068,108071,108074,108077,108080,108083],{"className":108052},[398],[363,108054],{"className":108055,"style":487},[402],[363,108057,108059],{"className":108058},[407],[363,108060,33035],{"className":108061},[407,33034],[363,108063,461],{"className":108064},[494],[363,108066,554],{"className":108067,"style":682},[407,408],[363,108069,558],{"className":108070},[407],[363,108072,25939],{"className":108073,"style":25973},[407,408],[363,108075,468],{"className":108076},[501],[363,108078],{"className":108079,"style":506},[505],[363,108081,471],{"className":108082},[510],[363,108084],{"className":108085,"style":506},[505],[363,108087,108089,108092,108132,108138,108141,108146,108149,108226,108232,108235,108238],{"className":108088},[398],[363,108090],{"className":108091,"style":31675},[402],[363,108093,108095,108098],{"className":108094},[407],[363,108096,571],{"className":108097},[407,714],[363,108099,108101],{"className":108100},[629],[363,108102,108104,108124],{"className":108103},[633,634],[363,108105,108107,108121],{"className":108106},[638],[363,108108,108110],{"className":108109,"style":727},[642],[363,108111,108112,108115],{"style":730},[363,108113],{"className":108114,"style":651},[650],[363,108116,108118],{"className":108117},[655,656,657,658],[363,108119,554],{"className":108120,"style":682},[407,408,658],[363,108122,666],{"className":108123},[665],[363,108125,108127],{"className":108126},[638],[363,108128,108130],{"className":108129,"style":8316},[642],[363,108131],{},[363,108133,108135],{"className":108134},[407],[363,108136,578],{"className":108137},[767,4068],[363,108139],{"className":108140,"style":754},[505],[363,108142,4502,108144],{"className":108143},[1832],[363,108145,4506],{"style":4505},[363,108147],{"className":108148,"style":754},[505],[363,108150,108152,108155,108223],{"className":108151},[407],[363,108153],{"className":108154},[494,789],[363,108156,108158],{"className":108157},[586],[363,108159,108161,108215],{"className":108160},[633,634],[363,108162,108164,108212],{"className":108163},[638],[363,108165,108167,108190,108198],{"className":108166,"style":23261},[642],[363,108168,108169,108172],{"style":805},[363,108170],{"className":108171,"style":809},[650],[363,108173,108175,108178,108181,108184,108187],{"className":108174},[407],[363,108176,11],{"className":108177},[407,408],[363,108179],{"className":108180,"style":2935},[505],[363,108182,2818],{"className":108183},[2939],[363,108185],{"className":108186,"style":2935},[505],[363,108188,1291],{"className":108189,"style":1328},[407,408],[363,108191,108192,108195],{"style":821},[363,108193],{"className":108194,"style":809},[650],[363,108196],{"className":108197,"style":829},[828],[363,108199,108200,108203],{"style":832},[363,108201],{"className":108202,"style":809},[650],[363,108204,108206,108209],{"className":108205},[407],[363,108207,2811],{"className":108208},[407],[363,108210,11],{"className":108211},[407,408],[363,108213,666],{"className":108214},[665],[363,108216,108218],{"className":108217},[638],[363,108219,108221],{"className":108220,"style":854},[642],[363,108222],{},[363,108224],{"className":108225},[501,789],[363,108227,108229],{"className":108228},[407],[363,108230,605],{"className":108231},[767,4068],[363,108233],{"className":108234,"style":506},[505],[363,108236,471],{"className":108237},[510],[363,108239],{"className":108240,"style":506},[505],[363,108242,108244,108247,108287,108290,108295,108298,108338,108341,108344,108347],{"className":108243},[398],[363,108245],{"className":108246,"style":487},[402],[363,108248,108250,108253],{"className":108249},[407],[363,108251,571],{"className":108252},[407,714],[363,108254,108256],{"className":108255},[629],[363,108257,108259,108279],{"className":108258},[633,634],[363,108260,108262,108276],{"className":108261},[638],[363,108263,108265],{"className":108264,"style":727},[642],[363,108266,108267,108270],{"style":730},[363,108268],{"className":108269,"style":651},[650],[363,108271,108273],{"className":108272},[655,656,657,658],[363,108274,554],{"className":108275,"style":682},[407,408,658],[363,108277,666],{"className":108278},[665],[363,108280,108282],{"className":108281},[638],[363,108283,108285],{"className":108284,"style":8316},[642],[363,108286],{},[363,108288,578],{"className":108289},[494],[363,108291,4502,108293],{"className":108292},[1832],[363,108294,4506],{"style":4505},[363,108296],{"className":108297,"style":754},[505],[363,108299,108301,108304],{"className":108300},[407],[363,108302,547],{"className":108303,"style":625},[407,408],[363,108305,108307],{"className":108306},[629],[363,108308,108310,108330],{"className":108309},[633,634],[363,108311,108313,108327],{"className":108312},[638],[363,108314,108316],{"className":108315,"style":45301},[642],[363,108317,108318,108321],{"style":646},[363,108319],{"className":108320,"style":651},[650],[363,108322,108324],{"className":108323},[655,656,657,658],[363,108325,7909],{"className":108326},[407,408,658],[363,108328,666],{"className":108329},[665],[363,108331,108333],{"className":108332},[638],[363,108334,108336],{"className":108335,"style":8316},[642],[363,108337],{},[363,108339,605],{"className":108340},[501],[363,108342],{"className":108343,"style":2935},[505],[363,108345,2818],{"className":108346},[2939],[363,108348],{"className":108349,"style":2935},[505],[363,108351,108353,108356,108361,108364,108367],{"className":108352},[398],[363,108354],{"className":108355,"style":403},[402],[363,108357,4502,108359],{"className":108358},[1832],[363,108360,4506],{"style":4505},[363,108362],{"className":108363,"style":754},[505],[363,108365,2811],{"className":108366},[407],[363,108368,898],{"className":108369},[920],[11,108371,14611,108372,108400,108401,108455,108456,108570,108571,108629],{},[363,108373,108375,108388],{"className":108374,"translate":367},[366],[363,108376,108378],{"className":108377},[371],[373,108379,108380],{"xmlns":375},[377,108381,108382,108386],{},[380,108383,108384],{},[383,108385,561],{},[387,108387,561],{"encoding":389},[363,108389,108391],{"className":108390,"ariaHidden":394},[393],[363,108392,108394,108397],{"className":108393},[398],[363,108395],{"className":108396,"style":913},[402],[363,108398,561],{"className":108399},[407,408]," 同理，两式相加即得。于是「最小化 JS」与「在最优判别器下最大化 GAN 目标」是同一件事，而常数 ",[363,108402,108404,108426],{"className":108403,"translate":367},[366],[363,108405,108407],{"className":108406},[371],[373,108408,108409],{"xmlns":375},[377,108410,108411,108423],{},[380,108412,108413,108415,108417,108419,108421],{},[458,108414,5161],{},[463,108416,2811],{},[383,108418,4448],{},[458,108420,4451],{},[463,108422,2811],{},[387,108424,108425],{"encoding":389},"-2\\log2",[363,108427,108429],{"className":108428,"ariaHidden":394},[393],[363,108430,108432,108435,108438,108441,108444,108449,108452],{"className":108431},[398],[363,108433],{"className":108434,"style":403},[402],[363,108436,5161],{"className":108437},[407],[363,108439,2811],{"className":108440},[407],[363,108442],{"className":108443,"style":754},[505],[363,108445,4502,108447],{"className":108446},[1832],[363,108448,4506],{"style":4505},[363,108450],{"className":108451,"style":754},[505],[363,108453,2811],{"className":108454},[407]," 与生成器无关。这解释了「GAN 训练不收敛」的来源：JS 有界，一旦支撑集几近不重叠，",[363,108457,108459,108489],{"className":108458,"translate":367},[366],[363,108460,108462],{"className":108461},[371],[373,108463,108464],{"xmlns":375},[377,108465,108466,108486],{},[380,108467,108468,108478,108480,108482,108484],{},[543,108469,108470,108472],{},[383,108471,547],{},[380,108473,108474,108476],{},[383,108475,24115],{},[383,108477,26109],{},[458,108479,1378],{},[383,108481,4448],{},[458,108483,4451],{},[463,108485,2811],{},[387,108487,108488],{"encoding":389},"D_{JS}\\to\\log2",[363,108490,108492,108553],{"className":108491,"ariaHidden":394},[393],[363,108493,108495,108498,108544,108547,108550],{"className":108494},[398],[363,108496],{"className":108497,"style":16311},[402],[363,108499,108501,108504],{"className":108500},[407],[363,108502,547],{"className":108503,"style":625},[407,408],[363,108505,108507],{"className":108506},[629],[363,108508,108510,108536],{"className":108509},[633,634],[363,108511,108513,108533],{"className":108512},[638],[363,108514,108516],{"className":108515,"style":727},[642],[363,108517,108518,108521],{"style":646},[363,108519],{"className":108520,"style":651},[650],[363,108522,108524],{"className":108523},[655,656,657,658],[363,108525,108527,108530],{"className":108526},[407,658],[363,108528,24115],{"className":108529,"style":24525},[407,408,658],[363,108531,26109],{"className":108532,"style":26233},[407,408,658],[363,108534,666],{"className":108535},[665],[363,108537,108539],{"className":108538},[638],[363,108540,108542],{"className":108541,"style":8316},[642],[363,108543],{},[363,108545],{"className":108546,"style":506},[505],[363,108548,1378],{"className":108549},[510],[363,108551],{"className":108552,"style":506},[505],[363,108554,108556,108559,108564,108567],{"className":108555},[398],[363,108557],{"className":108558,"style":403},[402],[363,108560,4502,108562],{"className":108561},[1832],[363,108563,4506],{"style":4505},[363,108565],{"className":108566,"style":754},[505],[363,108568,2811],{"className":108569},[407],"，判别器梯度饱和，生成器梯度消失。反过来，换成 LSGAN（",[363,108572,108574,108591],{"className":108573,"translate":367},[366],[363,108575,108577],{"className":108576},[371],[373,108578,108579],{"xmlns":375},[377,108580,108581,108589],{},[380,108582,108583],{},[7162,108584,108585,108587],{},[383,108586,37821],{},[463,108588,2811],{},[387,108590,40766],{"encoding":389},[363,108592,108594],{"className":108593,"ariaHidden":394},[393],[363,108595,108597,108600],{"className":108596},[398],[363,108598],{"className":108599,"style":40776},[402],[363,108601,108603,108606],{"className":108602},[407],[363,108604,37821],{"className":108605},[407,408],[363,108607,108609],{"className":108608},[629],[363,108610,108612],{"className":108611},[633],[363,108613,108615],{"className":108614},[638],[363,108616,108618],{"className":108617,"style":20400},[642],[363,108619,108620,108623],{"style":7213},[363,108621],{"className":108622,"style":651},[650],[363,108624,108626],{"className":108625},[655,656,657,658],[363,108627,2811],{"className":108628},[407,658],"）、Wasserstein（非 f-散度）或 α-GAN 就能改变这一点。",[2659,108631,108633],{"id":108632},"_55-估计性质","5.5 估计性质",[11,108635,108636],{},"用 5.1 的 Monte Carlo 下界做估计时：",[10375,108638,108639,108784,108939],{},[7013,108640,108641,108642,34197,108711,108713,108714,108783],{},"它是 ",[363,108643,108645,108662],{"className":108644,"translate":367},[366],[363,108646,108648],{"className":108647},[371],[373,108649,108650],{"xmlns":375},[377,108651,108652,108660],{},[380,108653,108654],{},[543,108655,108656,108658],{},[383,108657,547],{},[383,108659,385],{},[387,108661,4370],{"encoding":389},[363,108663,108665],{"className":108664,"ariaHidden":394},[393],[363,108666,108668,108671],{"className":108667},[398],[363,108669],{"className":108670,"style":4380},[402],[363,108672,108674,108677],{"className":108673},[407],[363,108675,547],{"className":108676,"style":625},[407,408],[363,108678,108680],{"className":108679},[629],[363,108681,108683,108703],{"className":108682},[633,634],[363,108684,108686,108700],{"className":108685},[638],[363,108687,108689],{"className":108688,"style":643},[642],[363,108690,108691,108694],{"style":646},[363,108692],{"className":108693,"style":651},[650],[363,108695,108697],{"className":108696},[655,656,657,658],[363,108698,385],{"className":108699,"style":409},[407,408,658],[363,108701,666],{"className":108702},[665],[363,108704,108706],{"className":108705},[638],[363,108707,108709],{"className":108708,"style":673},[642],[363,108710],{},[1344,108712,100230],{},"（偏差向下），且偏差来自 ",[363,108715,108717,108734],{"className":108716,"translate":367},[366],[363,108718,108720],{"className":108719},[371],[373,108721,108722],{"xmlns":375},[377,108723,108724,108732],{},[380,108725,108726],{},[543,108727,108728,108730],{},[383,108729,4506],{},[383,108731,100250],{},[387,108733,100253],{"encoding":389},[363,108735,108737],{"className":108736,"ariaHidden":394},[393],[363,108738,108740,108743],{"className":108739},[398],[363,108741],{"className":108742,"style":2624},[402],[363,108744,108746,108749],{"className":108745},[407],[363,108747,4506],{"className":108748,"style":1328},[407,408],[363,108750,108752],{"className":108751},[629],[363,108753,108755,108775],{"className":108754},[633,634],[363,108756,108758,108772],{"className":108757},[638],[363,108759,108761],{"className":108760,"style":45301},[642],[363,108762,108763,108766],{"style":19215},[363,108764],{"className":108765,"style":651},[650],[363,108767,108769],{"className":108768},[655,656,657,658],[363,108770,100250],{"className":108771,"style":1328},[407,408,658],[363,108773,666],{"className":108774},[665],[363,108776,108778],{"className":108777},[638],[363,108779,108781],{"className":108780,"style":8316},[642],[363,108782],{}," 未达到最优 + 样本有限；",[7013,108785,68515,108786,108816,108817,108875,108876,108938],{},[363,108787,108789,108804],{"className":108788,"translate":367},[366],[363,108790,108792],{"className":108791},[371],[373,108793,108794],{"xmlns":375},[377,108795,108796,108802],{},[380,108797,108798,108800],{},[383,108799,35804],{},[458,108801,4451],{},[387,108803,57523],{"encoding":389},[363,108805,108807],{"className":108806,"ariaHidden":394},[393],[363,108808,108810,108813],{"className":108809},[398],[363,108811],{"className":108812,"style":2624},[402],[363,108814,35804],{"className":108815},[1832]," 取到极值时，估计量方差随 ",[363,108818,108820,108837],{"className":108819,"translate":367},[366],[363,108821,108823],{"className":108822},[371],[373,108824,108825],{"xmlns":375},[377,108826,108827,108835],{},[380,108828,108829],{},[7162,108830,108831,108833],{},[383,108832,385],{},[458,108834,7921],{},[387,108836,61091],{"encoding":389},[363,108838,108840],{"className":108839,"ariaHidden":394},[393],[363,108841,108843,108846],{"className":108842},[398],[363,108844],{"className":108845,"style":403},[402],[363,108847,108849,108852],{"className":108848},[407],[363,108850,385],{"className":108851,"style":409},[407,408],[363,108853,108855],{"className":108854},[629],[363,108856,108858],{"className":108857},[633],[363,108859,108861],{"className":108860},[638],[363,108862,108864],{"className":108863,"style":8847},[642],[363,108865,108866,108869],{"style":7213},[363,108867],{"className":108868,"style":651},[650],[363,108870,108872],{"className":108871},[655,656,657,658],[363,108873,7921],{"className":108874},[2939,658]," 增长而增大（如 KL 的 ",[363,108877,108879,108897],{"className":108878,"translate":367},[366],[363,108880,108882],{"className":108881},[371],[373,108883,108884],{"xmlns":375},[377,108885,108886,108894],{},[380,108887,108888],{},[7162,108889,108890,108892],{},[383,108891,7915],{},[383,108893,4506],{},[387,108895,108896],{"encoding":389},"e^{g}",[363,108898,108900],{"className":108899,"ariaHidden":394},[393],[363,108901,108903,108906],{"className":108902},[398],[363,108904],{"className":108905,"style":50725},[402],[363,108907,108909,108912],{"className":108908},[407],[363,108910,7915],{"className":108911},[407,408],[363,108913,108915],{"className":108914},[629],[363,108916,108918],{"className":108917},[633],[363,108919,108921],{"className":108920},[638],[363,108922,108924],{"className":108923,"style":50725},[642],[363,108925,108926,108929],{"style":7213},[363,108927],{"className":108928,"style":651},[650],[363,108930,108932],{"className":108931},[655,656,657,658],[363,108933,108935],{"className":108934},[407,658],[363,108936,4506],{"className":108937,"style":1328},[407,408,658]," 会重尾）；",[7013,108940,108941,108942,108970],{},"一个常用替代是「对称化」或「有界化」",[363,108943,108945,108958],{"className":108944,"translate":367},[366],[363,108946,108948],{"className":108947},[371],[373,108949,108950],{"xmlns":375},[377,108951,108952,108956],{},[380,108953,108954],{},[383,108955,385],{},[387,108957,385],{"encoding":389},[363,108959,108961],{"className":108960,"ariaHidden":394},[393],[363,108962,108964,108967],{"className":108963},[398],[363,108965],{"className":108966,"style":403},[402],[363,108968,385],{"className":108969,"style":409},[407,408],"（如 JS、TV），牺牲敏感性换取稳定梯度。这与第 6 节讨论的「有界\u002F无界」准确性—稳健性权衡是同一件事。",[18,108972,108974],{"id":108973},"_6-与其他散度的比较","6. 与其他散度的比较",[2659,108976,108978],{"id":108977},"_61-汇总","6.1 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",[363,113004,113006,113098],{"className":113005,"translate":367},[366],[363,113007,113009],{"className":113008},[371],[373,113010,113011],{"xmlns":375},[377,113012,113013,113095],{},[380,113014,113015,113021,113023,113025,113027,113029,113031,113033,113035,113037,113039,113041,113047,113049,113051,113061,113063,113065,113067,113077,113079,113081,113083,113085],{},[7162,113016,113017,113019],{},[383,113018,385],{},[458,113020,7921],{},[458,113022,471],{},[383,113024,1116],{},[458,113026,461],{"stretchy":460},[383,113028,40609],{},[458,113030,5161],{},[463,113032,465],{},[458,113034,468],{"stretchy":460},[458,113036,2818],{},[463,113038,2811],{},[458,113040,5161],{},[586,113042,113043,113045],{},[463,113044,465],{},[383,113046,1116],{},[458,113048,471],{},[458,113050,5161],{},[34711,113052,113053],{},[380,113054,113055,113057,113059],{},[463,113056,465],{},[458,113058,5161],{},[383,113060,40609],{},[458,113062,2818],{},[463,113064,2811],{},[458,113066,5161],{},[34711,113068,113069],{},[380,113070,113071,113073,113075],{},[463,113072,465],{},[458,113074,5161],{},[383,113076,40609],{},[458,113078,471],{},[463,113080,2811],{},[458,113082,5161],{},[463,113084,2811],{},[34711,113086,113087],{},[380,113088,113089,113091,113093],{},[463,113090,465],{},[458,113092,5161],{},[383,113094,40609],{},[387,113096,113097],{"encoding":389},"f^*=t(y-1)+2-\\frac1t=-\\sqrt{1-y}+2-\\sqrt{1-y}=2-2\\sqrt{1-y}",[363,113099,113101,113145,113169,113190,113208,113291,113368,113386,113460,113478],{"className":113100,"ariaHidden":394},[393],[363,113102,113104,113107,113136,113139,113142],{"className":113103},[398],[363,113105],{"className":113106,"style":403},[402],[363,113108,113110,113113],{"className":113109},[407],[363,113111,385],{"className":113112,"style":409},[407,408],[363,113114,113116],{"className":113115},[629],[363,113117,113119],{"className":113118},[633],[363,113120,113122],{"className":113121},[638],[363,113123,113125],{"className":113124,"style":8847},[642],[363,113126,113127,113130],{"style":7213},[363,113128],{"className":113129,"style":651},[650],[363,113131,113133],{"className":113132},[655,656,657,658],[363,113134,7921],{"className":113135},[2939,658],[363,113137],{"className":113138,"style":506},[505],[363,113140,471],{"className":113141},[510],[363,113143],{"className":113144,"style":506},[505],[363,113146,113148,113151,113154,113157,113160,113163,113166],{"className":113147},[398],[363,113149],{"className":113150,"style":487},[402],[363,113152,1116],{"className":113153},[407,408],[363,113155,461],{"className":113156},[494],[363,113158,40609],{"className":113159,"style":1328},[407,408],[363,113161],{"className":113162,"style":2935},[505],[363,113164,5161],{"className":113165},[2939],[363,113167],{"className":113168,"style":2935},[505],[363,113170,113172,113175,113178,113181,113184,113187],{"className":113171},[398],[363,113173],{"className":113174,"style":487},[402],[363,113176,465],{"className":113177},[407],[363,113179,468],{"className":113180},[501],[363,113182],{"className":113183,"style":2935},[505],[363,113185,2818],{"className":113186},[2939],[363,113188],{"className":113189,"style":2935},[505],[363,113191,113193,113196,113199,113202,113205],{"className":113192},[398],[363,113194],{"className":113195,"style":12928},[402],[363,113197,2811],{"className":113198},[407],[363,113200],{"className":113201,"style":2935},[505],[363,113203,5161],{"className":113204},[2939],[363,113206],{"className":113207,"style":2935},[505],[363,113209,113211,113214,113282,113285,113288],{"className":113210},[398],[363,113212],{"className":113213,"style":2853},[402],[363,113215,113217,113220,113279],{"className":113216},[407],[363,113218],{"className":113219},[494,789],[363,113221,113223],{"className":113222},[586],[363,113224,113226,113271],{"className":113225},[633,634],[363,113227,113229,113268],{"className":113228},[638],[363,113230,113232,113246,113254],{"className":113231,"style":2872},[642],[363,113233,113234,113237],{"style":2875},[363,113235],{"className":113236,"style":809},[650],[363,113238,113240],{"className":113239},[655,656,657,658],[363,113241,113243],{"className":113242},[407,658],[363,113244,1116],{"className":113245},[407,408,658],[363,113247,113248,113251],{"style":821},[363,113249],{"className":113250,"style":809},[650],[363,113252],{"className":113253,"style":829},[828],[363,113255,113256,113259],{"style":2898},[363,113257],{"className":113258,"style":809},[650],[363,113260,113262],{"className":113261},[655,656,657,658],[363,113263,113265],{"className":113264},[407,658],[363,113266,465],{"className":113267},[407,658],[363,113269,666],{"className":113270},[665],[363,113272,113274],{"className":113273},[638],[363,113275,113277],{"className":113276,"style":2920},[642],[363,113278],{},[363,113280],{"className":113281},[501,789],[363,113283],{"className":113284,"style":506},[505],[363,113286,471],{"className":113287},[510],[363,113289],{"className":113290,"style":506},[505],[363,113292,113294,113297,113300,113359,11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一行取 ",[363,113548,113550,113589],{"className":113549,"translate":367},[366],[363,113551,113553],{"className":113552},[371],[373,113554,113555],{"xmlns":375},[377,113556,113557,113587],{},[380,113558,113559,113561,113563,113575,113577,113579,113581,113583,113585],{},[383,113560,45552],{},[458,113562,471],{},[586,113564,113565,113573],{},[380,113566,113567,113569,113571],{},[463,113568,465],{},[458,113570,2818],{},[383,113572,45126],{},[463,113574,2811],{},[458,113576,6257],{},[458,113578,461],{"stretchy":460},[463,113580,474],{},[458,113582,898],{"separator":394},[463,113584,465],{},[458,113586,468],{"stretchy":460},[387,113588,45581],{"encoding":389},[363,113590,113592,113610,113699],{"className":113591,"ariaHidden":394},[393],[363,113593,113595,113598,113601,113604,113607],{"className":113594},[398],[363,113596],{"className":113597,"style":2693},[402],[363,113599,45552],{"className":113600},[407,408],[363,113602],{"className":113603,"style":506},[505],[363,113605,471],{"className":113606},[510],[363,113608],{"className":113609,"style":506},[505],[363,113611,113613,113616,113690,113693,113696],{"className":113612},[398],[363,113614],{"className":113615,"style":2853},[402],[363,113617,113619,113622,113687],{"className":113618},[407],[363,113620],{"className":113621},[494,789],[363,113623,113625],{"className":113624},[586],[363,113626,113628,113679],{"className":113627},[633,634],[363,113629,113631,113676],{"className":113630},[638],[363,113632,113634,113648,113656],{"className":113633,"style":2872},[642],[363,113635,113636,113639],{"style":2875},[363,113637],{"className":113638,"style":809},[650],[363,113640,113642],{"className":113641},[655,656,657,658],[363,113643,113645],{"className":113644},[407,658],[363,113646,2811],{"className":113647},[407,658],[363,113649,113650,113653],{"style":821},[363,113651],{"className":113652,"style":809},[650],[363,113654],{"className":113655,"style":829},[828],[363,113657,113658,113661],{"style":2898},[363,113659],{"className":113660,"style":809},[650],[363,113662,113664],{"className":113663},[655,656,657,658],[363,113665,113667,113670,113673],{"className":113666},[407,658],[363,113668,465],{"className":113669},[407,658],[363,113671,2818],{"className":113672},[2939,658],[363,113674,45126],{"className":113675,"style":45156},[407,408,658],[363,113677,666],{"className":113678},[665],[363,113680,113682],{"className":113681},[638],[363,113683,113685],{"className":113684,"style":2920},[642],[363,113686],{},[363,113688],{"className":113689},[501,789],[363,113691],{"className":113692,"style":506},[505],[363,113694,6257],{"className":113695},[510],[363,113697],{"className":113698,"style":506},[505],[363,113700,113702,113705,113708,113711,113714,113717,113720],{"className":113701},[398],[363,113703],{"className":113704,"style":487},[402],[363,113706,461],{"className":113707},[494],[363,113709,474],{"className":113710},[407],[363,113712,898],{"className":113713},[920],[363,113715],{"className":113716,"style":754},[505],[363,113718,465],{"className":113719},[407],[363,113721,468],{"className":113722},[501],"，验证：",[363,113725,113727,113803],{"className":113726,"translate":367},[366],[363,113728,113730],{"className":113729},[371],[373,113731,113732],{"xmlns":375},[377,113733,113734,113800],{},[380,113735,113736,113744,113746,113748,113750,113752,113762,113764,113766,113768,113770,113798],{},[7901,113737,113738,113740,113742],{},[383,113739,385],{},[383,113741,45126],{},[458,113743,7921],{},[458,113745,461],{"stretchy":460},[383,113747,40609],{},[458,113749,468],{"stretchy":460},[458,113751,471],{},[543,113753,113754,113760],{},[380,113755,113756,113758],{},[383,113757,35804],{},[458,113759,4451],{},[383,113761,1116],{},[458,113763,3246],{"stretchy":460},[383,113765,1116],{},[383,113767,40609],{},[458,113769,5161],{},[586,113771,113772,113784],{},[380,113773,113774,113776,113778],{},[463,113775,465],{},[458,113777,5161],{},[7162,113779,113780,113782],{},[383,113781,1116],{},[383,113783,45552],{},[380,113785,113786,113788,113790,113792,113794,113796],{},[383,113787,45552],{},[458,113789,461],{"stretchy":460},[463,113791,465],{},[458,113793,5161],{},[383,113795,45552],{},[458,113797,468],{"stretchy":460},[458,113799,3256],{"stretchy":460},[387,113801,113802],{"encoding":389},"f_\\alpha^*(y)=\\sup_t\\{ty-\\frac{1-t^a}{a(1-a)}\\}",[363,113804,113806,113881,113945],{"className":113805,"ariaHidden":394},[393],[363,113807,113809,113812,113863,113866,113869,113872,113875,113878],{"className":113808},[398],[363,113810],{"className":113811,"style":487},[402],[363,113813,113815,113818],{"className":113814},[407],[363,113816,385],{"className":113817,"style":409},[407,408],[363,113819,113821],{"className":113820},[629],[363,113822,113824,113855],{"className":113823},[633,634],[363,113825,113827,113852],{"className":113826},[638],[363,113828,113830,113841],{"className":113829,"style":8847},[642],[363,113831,113832,113835],{"style":8599},[363,113833],{"className":113834,"style":651},[650],[363,113836,113838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666],{"className":114675},[665],[363,114677,114679],{"className":114678},[638],[363,114680,114682],{"className":114681,"style":2920},[642],[363,114683],{},[363,114685],{"className":114686},[501,789],[363,114688,114690,114707],{"className":114689,"translate":367},[366],[363,114691,114693],{"className":114692},[371],[373,114694,114695],{"xmlns":375},[377,114696,114697,114705],{},[380,114698,114699,114701,114703],{},[383,114700,45126],{},[458,114702,471],{},[463,114704,474],{},[387,114706,46860],{"encoding":389},[363,114708,114710,114728],{"className":114709,"ariaHidden":394},[393],[363,114711,114713,114716,114719,114722,114725],{"className":114712},[398],[363,114714],{"className":114715,"style":2693},[402],[363,114717,45126],{"className":114718,"style":45156},[407,408],[363,114720],{"className":114721,"style":506},[505],[363,114723,471],{"className":114724},[510],[363,114726],{"className":114727,"style":506},[505],[363,114729,114731,114734],{"className":114730},[398],[363,114732],{"className":114733,"style":520},[402],[363,114735,474],{"className":114736},[407],[363,114738,114740,114770],{"className":114739,"translate":367},[366],[363,114741,114743],{"className":114742},[371],[373,114744,114745],{"xmlns":375},[377,114746,114747,114767],{},[380,114748,114749,114751,114753,114755,114757,114759,114761,114765],{},[383,114750,385],{},[458,114752,471],{},[463,114754,39361],{},[458,114756,461],{"stretchy":460},[463,114758,465],{},[458,114760,5161],{},[34711,114762,114763],{},[383,114764,1116],{},[458,114766,468],{"stretchy":460},[387,114768,114769],{"encoding":389},"f=4(1-\\sqrt t)",[363,114771,114773,114791,114815],{"className":114772,"ariaHidden":394},[393],[363,114774,114776,114779,114782,114785,114788],{"className":114775},[398],[363,114777],{"className":114778,"style":403},[402],[363,114780,385],{"className":114781,"style":409},[407,408],[363,114783],{"className":114784,"style":506},[505],[363,114786,471],{"className":114787},[510],[363,114789],{"className":114790,"style":506},[505],[363,114792,114794,114797,114800,114803,114806,114809,114812],{"className":114793},[398],[363,114795],{"className":114796,"style":487},[402],[363,114798,39361],{"className":114799},[407],[363,114801,461],{"className":114802},[494],[363,114804,465],{"className":114805},[407],[363,114807],{"className":114808,"style":2935},[505],[363,114810,5161],{"className":114811},[2939],[363,114813],{"className":114814,"style":2935},[505],[363,114816,114818,114821,114865],{"className":114817},[398],[363,114819],{"className":114820,"style":43039},[402],[363,114822,114824],{"className":114823},[407,34739],[363,114825,114827,114857],{"className":114826},[633,634],[363,114828,114830,114854],{"className":114829},[638],[363,114831,114833,114842],{"className":114832,"style":43055},[642],[363,114834,114836,114839],{"className":114835,"style":12213},[6655],[363,114837],{"className":114838,"style":809},[650],[363,114840,1116],{"className":114841,"style":34759},[407,408],[363,114843,114844,114847],{"style":43067},[363,114845],{"className":114846,"style":809},[650],[363,114848,114850],{"className":114849,"style":34816},[34815],[6671,114851,114852],{"xmlns":6673,"width":6674,"height":34819,"viewBox":34820,"preserveAspectRatio":6677},[6679,114853],{"d":34823},[363,114855,666],{"className":114856},[665],[363,114858,114860],{"className":114859},[638],[363,114861,114863],{"className":114862,"style":43087},[642],[363,114864],{},[363,114866,468],{"className":114867},[501],"）时公式给出 ",[363,114870,114872,114898],{"className":114871,"translate":367},[366],[363,114873,114875],{"className":114874},[371],[373,114876,114877],{"xmlns":375},[377,114878,114879,114895],{},[380,114880,114881,114891,114893],{},[586,114882,114883,114885],{},[463,114884,39361],{},[380,114886,114887,114889],{},[458,114888,5161],{},[383,114890,40609],{},[458,114892,5161],{},[463,114894,39361],{},[387,114896,114897],{"encoding":389},"\\frac{4}{-y}-4",[363,114899,114901,114988],{"className":114900,"ariaHidden":394},[393],[363,114902,114904,114908,114979,114982,114985],{"className":114903},[398],[363,114905],{"className":114906,"style":114907},[402],"height:1.3262em;vertical-align:-0.4811em;",[363,114909,114911,114914,114976],{"className":114910},[407],[363,114912],{"className":114913},[494,789],[363,114915,114917],{"className":114916},[586],[363,114918,114920,114968],{"className":114919},[633,634],[363,114921,114923,114965],{"className":114922},[638],[363,114924,114926,114943,114951],{"className":114925,"style":2872},[642],[363,114927,114928,114931],{"style":2875},[363,114929],{"className":114930,"style":809},[650],[363,114932,114934],{"className":114933},[655,656,657,658],[363,114935,114937,114940],{"className":114936},[407,658],[363,114938,5161],{"className":114939},[407,658],[363,114941,40609],{"className":114942,"style":1328},[407,408,658],[363,114944,114945,114948],{"style":821},[363,114946],{"className":114947,"style":809},[650],[363,114949],{"className":114950,"style":829},[828],[363,114952,114953,114956],{"style":2898},[363,114954],{"className":114955,"style":809},[650],[363,114957,114959],{"className":114958},[655,656,657,658],[363,114960,114962],{"className":114961},[407,658],[363,114963,39361],{"className":114964},[407,658],[363,114966,666],{"className":114967},[665],[363,114969,114971],{"className":114970},[638],[363,114972,114974],{"className":114973,"style":3087},[642],[363,114975],{},[363,114977],{"className":114978},[501,789],[363,114980],{"className":114981,"style":2935},[505],[363,114983,5161],{"className":114984},[2939],[363,114986],{"className":114987,"style":2935},[505],[363,114989,114991,114994],{"className":114990},[398],[363,114992],{"className":114993,"style":520},[402],[363,114995,39361],{"className":114996},[407],"，可与直接展开互相验证。）",[11,114999,115000],{},"可见，f-散度族统一了「不对称\u002F对称」「有界\u002F无界」的多种散度。",[2659,115002,115004],{"id":115003},"_62-积分概率度量ipm","6.2 积分概率度量（IPM）",[11,115006,115007,115008,115057],{},"f-散度只依赖似然比 ",[363,115009,115011,115033],{"className":115010,"translate":367},[366],[363,115012,115014],{"className":115013},[371],[373,115015,115016],{"xmlns":375},[377,115017,115018,115030],{},[380,115019,115020,115022,115024,115026,115028],{},[383,115021,592],{},[383,115023,554],{},[383,115025,1288],{"mathvariant":557},[383,115027,592],{},[383,115029,561],{},[387,115031,115032],{"encoding":389},"dP\u002FdQ",[363,115034,115036],{"className":115035,"ariaHidden":394},[393],[363,115037,115039,115042,115045,115048,115051,115054],{"className":115038},[398],[363,115040],{"className":115041,"style":487},[402],[363,115043,592],{"className":115044},[407,408],[363,115046,554],{"className":115047,"style":682},[407,408],[363,115049,1288],{"className":115050},[407],[363,115052,592],{"className":115053},[407,408],[363,115055,561],{"className":115056},[407,408],"，因此把两个分布同时「搬到别处」值不变。这带来一个根本局限：",[363,115059,115061],{"className":115060,"translate":367},[528],[363,115062,115064,115119],{"className":115063,"translate":367},[366],[363,115065,115067],{"className":115066},[371],[373,115068,115069],{"xmlns":375,"display":537},[377,115070,115071,115116],{},[380,115072,115073,115075,115077,115079,115081,115083,115085,115088,115090,115092,115098,115100,115102,115104,115106,115112,115114],{},[1645,115074,4686],{},[383,115076,554],{},[458,115078,33919],{},[383,115080,561],{},[1645,115082,4631],{},[458,115084,461],{"stretchy":460},[1645,115086,115087],{},"支撑集不重叠，如 ",[383,115089,554],{},[458,115091,471],{},[543,115093,115094,115096],{},[383,115095,16062],{},[463,115097,474],{},[458,115099,898],{"separator":394},[1645,115101,4631],{},[383,115103,561],{},[458,115105,471],{},[543,115107,115108,115110],{},[383,115109,16062],{},[463,115111,465],{},[458,115113,468],{"stretchy":460},[458,115115,898],{"separator":394},[387,115117,115118],{"encoding":389},"\\text{若 } P\\perp Q\\\n(\\text{支撑集不重叠，如 } P=\\delta_0,\\ Q=\\delta_1),",[363,115120,115122,115149,115186,115253],{"className":115121,"ariaHidden":394},[393],[363,115123,115125,115128,115137,115140,115143,115146],{"className":115124},[398],[363,115126],{"className":115127,"style":12947},[402],[363,115129,115131,115134],{"className":115130},[407,4946],[363,115132,4997],{"className":115133},[407,4950],[363,115135,4631],{"className":115136},[407],[363,115138,554],{"className":115139,"style":682},[407,408],[363,115141],{"className":115142,"style":506},[505],[363,115144,33919],{"className":115145},[510],[363,115147],{"className":115148,"style":506},[505],[363,115150,115152,115155,115158,115161,115164,115174,115177,115180,115183],{"className":115151},[398],[363,115153],{"className":115154,"style":487},[402],[363,115156,561],{"className":115157},[407,408],[363,115159,4631],{"className":115160},[505],[363,115162,461],{"className":115163},[494],[363,115165,115167,115171],{"className":115166},[407,4946],[363,115168,115170],{"className":115169},[407,4950],"支撑集不重叠，如",[363,115172,4631],{"className":115173},[407],[363,115175,554],{"className":115176,"style":682},[407,408],[363,115178],{"className":115179,"style":506},[505],[363,115181,471],{"className":115182},[510],[363,115184],{"className":115185,"style":506},[505],[363,115187,115189,115192,115232,115235,115238,115241,115244,115247,115250],{"className":115188},[398],[363,115190],{"className":115191,"style":403},[402],[363,115193,115195,115198],{"className":115194},[407],[363,115196,16062],{"className":115197,"style":16111},[407,408],[363,115199,115201],{"className":115200},[629],[363,115202,115204,115224],{"className":115203},[633,634],[363,115205,115207,115221],{"className":115206},[638],[363,115208,115210],{"className":115209,"style":11483},[642],[363,115211,115212,115215],{"style":95132},[363,115213],{"className":115214,"style":651},[650],[363,115216,115218],{"className":115217},[655,656,657,658],[363,115219,474],{"className":115220},[407,658],[363,115222,666],{"className":115223},[665],[363,115225,115227],{"className":115226},[638],[363,115228,115230],{"className":115229,"style":8316},[642],[363,115231],{},[363,115233,898],{"className":115234},[920],[363,115236,4631],{"className":115237},[505],[363,115239],{"className":115240,"style":754},[505],[363,115242,561],{"className":115243},[407,408],[363,115245],{"className":115246,"style":506},[505],[363,115248,471],{"className":115249},[510],[363,115251],{"className":115252,"style":506},[505],[363,115254,115256,115259,115299,115302],{"className":115255},[398],[363,115257],{"className":115258,"style":487},[402],[363,115260,115262,115265],{"className":115261},[407],[363,115263,16062],{"className":115264,"style":16111},[407,408],[363,115266,115268],{"className":115267},[629],[363,115269,115271,115291],{"className":115270},[633,634],[363,115272,115274,115288],{"className":115273},[638],[363,115275,115277],{"className":115276,"style":11483},[642],[363,115278,115279,115282],{"style":95132},[363,115280],{"className":115281,"style":651},[650],[363,115283,115285],{"className":115284},[655,656,657,658],[363,115286,465],{"className":115287},[407,658],[363,115289,666],{"className":115290},[665],[363,115292,115294],{"className":115293},[638],[363,115295,115297],{"className":115296,"style":8316},[642],[363,115298],{},[363,115300,468],{"className":115301},[501],[363,115303,898],{"className":115304},[920],[363,115306,115308],{"className":115307,"translate":367},[528],[363,115309,115311,115361],{"className":115310,"translate":367},[366],[363,115312,115314],{"className":115313},[371],[373,115315,115316],{"xmlns":375,"display":537},[377,115317,115318,115358],{},[380,115319,115320,115330,115332,115334,115336,115338,115340,115342,115352,115354,115356],{},[543,115321,115322,115324],{},[383,115323,547],{},[380,115325,115326,115328],{},[383,115327,24115],{},[383,115329,26109],{},[458,115331,471],{},[383,115333,4448],{},[458,115335,4451],{},[463,115337,2811],{},[458,115339,898],{"separator":394},[505,115341],{"width":4663},[543,115343,115344,115346],{},[383,115345,547],{},[380,115347,115348,115350],{},[383,115349,5740],{},[383,115351,35222],{},[458,115353,471],{},[463,115355,465],{},[458,115357,898],{"separator":394},[387,115359,115360],{"encoding":389},"D_{JS}=\\log2,\\quad D_{TV}=1,",[363,115362,115364,115425,115506],{"className":115363,"ariaHidden":394},[393],[363,115365,115367,115370,115416,115419,115422],{"className":115366},[398],[363,115368],{"className":115369,"style":16311},[402],[363,115371,115373,115376],{"className":115372},[407],[363,115374,547],{"className":115375,"style":625},[407,408],[363,115377,115379],{"className":115378},[629],[363,115380,115382,115408],{"className":115381},[633,634],[363,115383,115385,115405],{"className":115384},[638],[363,115386,115388],{"className":115387,"style":727},[642],[363,115389,115390,115393],{"style":646},[363,115391],{"className":115392,"style":651},[650],[363,115394,115396],{"className":115395},[655,656,657,658],[363,115397,115399,115402],{"className":115398},[407,658],[363,115400,24115],{"className":115401,"style":24525},[407,408,658],[363,115403,26109],{"className":115404,"style":26233},[407,408,658],[363,115406,666],{"className":115407},[665],[363,115409,115411],{"className":115410},[638],[363,115412,115414],{"className":115413,"style":8316},[642],[363,115415],{},[363,115417],{"className":115418,"style":506},[505],[363,115420,471],{"className":115421},[510],[363,115423],{"className":115424,"style":506},[505],[363,115426,115428,115431,115436,115439,115442,115445,115448,115451,115497,115500,115503],{"className":115427},[398],[363,115429],{"className":115430,"style":403},[402],[363,115432,4502,115434],{"className":115433},[1832],[363,115435,4506],{"style":4505},[363,115437],{"className":115438,"style":754},[505],[363,115440,2811],{"className":115441},[407],[363,115443,898],{"className":115444},[920],[363,115446],{"className":115447,"style":88355},[505],[363,115449],{"className":115450,"style":754},[505],[363,115452,115454,115457],{"className":115453},[407],[363,115455,547],{"className":115456,"style":625},[407,408],[363,115458,115460],{"className":115459},[629],[363,115461,115463,115489],{"className":115462},[633,634],[363,115464,115466,115486],{"className":115465},[638],[363,115467,115469],{"className":115468,"style":727},[642],[363,115470,115471,115474],{"style":646},[363,115472],{"className":115473,"style":651},[650],[363,115475,115477],{"className":115476},[655,656,657,658],[363,115478,115480,115483],{"className":115479},[407,658],[363,115481,5740],{"className":115482,"style":682},[407,408,658],[363,115484,35222],{"className":115485,"style":2935},[407,408,658],[363,115487,666],{"className":115488},[665],[363,115490,115492],{"className":115491},[638],[363,115493,115495],{"className":115494,"style":8316},[642],[363,115496],{},[363,115498],{"className":115499,"style":506},[505],[363,115501,471],{"className":115502},[510],[363,115504],{"className":115505,"style":506},[505],[363,115507,115509,115512,115515],{"className":115508},[398],[363,115510],{"className":115511,"style":9706},[402],[363,115513,465],{"className":115514},[407],[363,115516,898],{"className":115517},[920],[11,115519,115520,115521,115524,115525,115589,115590,115640,115641,9748],{},"也就是说，f-散度在支撑分离时",[1344,115522,115523],{},"饱和到常数上界，对生成器参数的微小改动失去梯度","：因为被积分的量 ",[363,115526,115528,115556],{"className":115527,"translate":367},[366],[363,115529,115531],{"className":115530},[371],[373,115532,115533],{"xmlns":375},[377,115534,115535,115553],{},[380,115536,115537,115539,115541,115543,115545,115547,115549,115551],{},[383,115538,1291],{},[1645,115540,1647],{},[383,115542,385],{},[458,115544,461],{"stretchy":460},[383,115546,11],{},[383,115548,1288],{"mathvariant":557},[383,115550,1291],{},[458,115552,468],{"stretchy":460},[387,115554,115555],{"encoding":389},"q\\,f(p\u002Fq)",[363,115557,115559],{"className":115558,"ariaHidden":394},[393],[363,115560,115562,115565,115568,115571,115574,115577,115580,115583,115586],{"className":115561},[398],[363,115563],{"className":115564,"style":487},[402],[363,115566,1291],{"className":115567,"style":1328},[407,408],[363,115569],{"className":115570,"style":754},[505],[363,115572,385],{"className":115573,"style":409},[407,408],[363,115575,461],{"className":115576},[494],[363,115578,11],{"className":115579},[407,408],[363,115581,1288],{"className":115582},[407],[363,115584,1291],{"className":115585,"style":1328},[407,408],[363,115587,468],{"className":115588},[501]," 只在 ",[363,115591,115593,115610],{"className":115592,"translate":367},[366],[363,115594,115596],{"className":115595},[371],[373,115597,115598],{"xmlns":375},[377,115599,115600,115608],{},[380,115601,115602,115604,115606],{},[383,115603,1291],{},[458,115605,3251],{},[463,115607,474],{},[387,115609,32054],{"encoding":389},[363,115611,115613,115631],{"className":115612,"ariaHidden":394},[393],[363,115614,115616,115619,115622,115625,115628],{"className":115615},[398],[363,115617],{"className":115618,"style":32064},[402],[363,115620,1291],{"className":115621,"style":1328},[407,408],[363,115623],{"className":115624,"style":506},[505],[363,115626,3251],{"className":115627},[510],[363,115629],{"className":115630,"style":506},[505],[363,115632,115634,115637],{"className":115633},[398],[363,115635],{"className":115636,"style":520},[402],[363,115638,474],{"className":115639},[407]," 处有定义，当两个支撑几乎不重叠时，极小化散度的博弈只剩\"常数项 − 常数项\"，梯度信息全部丢失（这正是原 GAN 训练崩溃的理论根源）。Wasserstein 距离与 MMD 属于另一族——",[1344,115642,115643],{},"积分概率度量",[363,115645,115647],{"className":115646,"translate":367},[528],[363,115648,115650,115725],{"className":115649,"translate":367},[366],[363,115651,115653],{"className":115652},[371],[373,115654,115655],{"xmlns":375,"display":537},[377,115656,115657,115722],{},[380,115658,115659,115666,115668,115670,115672,115674,115676,115678,115694,115696,115698,115700,115702,115704,115706,115708,115710,115712,115714,115716,115718,115720],{},[543,115660,115661,115664],{},[1645,115662,115663],{},"IPM",[383,115665,37255],{"mathvariant":944},[458,115667,461],{"stretchy":460},[383,115669,554],{},[458,115671,898],{"separator":394},[383,115673,561],{},[458,115675,468],{"stretchy":460},[458,115677,471],{},[6364,115679,115680,115686],{},[380,115681,115682,115684],{},[383,115683,35804],{},[458,115685,4451],{},[380,115687,115688,115690,115692],{},[383,115689,39037],{},[458,115691,6257],{},[383,115693,37255],{"mathvariant":944},[458,115695,5156],{"fence":460,"stretchy":394,"minsize":17789,"maxsize":17789},[458,115697,1635],{},[383,115699,39037],{},[1645,115701,1647],{},[383,115703,592],{},[383,115705,554],{},[458,115707,5161],{},[458,115709,1635],{},[383,115711,39037],{},[1645,115713,1647],{},[383,115715,592],{},[383,115717,561],{},[458,115719,5156],{"fence":460,"stretchy":394,"minsize":17789,"maxsize":17789},[383,115721,4723],{"mathvariant":557},[387,115723,115724],{"encoding":389},"\\text{IPM}_{\\mathcal F}(P,Q)\n=\\sup_{h\\in\\mathcal F}\\Big|\\int h\\,dP\n-\\int h\\,dQ\\Big| .",[363,115726,115728,115808,115941],{"className":115727,"ariaHidden":394},[393],[363,115729,115731,115734,115781,115784,115787,115790,115793,115796,115799,115802,115805],{"className":115730},[398],[363,115732],{"className":115733,"style":487},[402],[363,115735,115737,115743],{"className":115736},[407],[363,115738,115740],{"className":115739},[407,4946],[363,115741,115663],{"className":115742},[407],[363,115744,115746],{"className":115745},[629],[363,115747,115749,115773],{"className":115748},[633,634],[363,115750,115752,115770],{"className":115751},[638],[363,115753,115755],{"className":115754,"style":727},[642],[363,115756,115757,115760],{"style":52818},[363,115758],{"className":115759,"style":651},[650],[363,115761,115763],{"className":115762},[655,656,657,658],[363,115764,115766],{"className":115765},[407,658],[363,115767,37255],{"className":115768,"style":115769},[407,971,658],"margin-right:0.0993em;",[363,115771,666],{"className":115772},[665],[363,115774,115776],{"className":115775},[638],[363,115777,115779],{"className":115778,"style":8316},[642],[363,115780],{},[363,115782,461],{"className":115783},[494],[363,115785,554],{"className":115786,"style":682},[407,408],[363,115788,898],{"className":115789},[920],[363,115791],{"className":115792,"style":754},[505],[363,115794,561],{"className":115795},[407,408],[363,115797,468],{"className":115798},[501],[363,115800],{"className":115801,"style":506},[505],[363,115803,471],{"className":115804},[510],[363,115806],{"className":115807,"style":506},[505],[363,115809,115811,115815,115868,115871,115911,115914,115917,115920,115923,115926,115929,115932,115935,115938],{"className":115810},[398],[363,115812],{"className":115813,"style":115814},[402],"height:2.3339em;vertical-align:-0.9739em;",[363,115816,115818],{"className":115817},[1832,14066],[363,115819,115821,115860],{"className":115820},[633,634],[363,115822,115824,115857],{"className":115823},[638],[363,115825,115827,115847],{"className":115826,"style":2693},[642],[363,115828,115829,115832],{"style":52078},[363,115830],{"className":115831,"style":809},[650],[363,115833,115835],{"className":115834},[655,656,657,658],[363,115836,115838,115841,115844],{"className":115837},[407,658],[363,115839,39037],{"className":115840},[407,408,658],[363,115842,6257],{"className":115843},[510,658],[363,115845,37255],{"className":115846,"style":115769},[407,971,658],[363,115848,115849,115852],{"style":12213},[363,115850],{"className":115851,"style":809},[650],[363,115853,115854],{},[363,115855,35804],{"className":115856},[1832],[363,115858,666],{"className":115859},[665],[363,115861,115863],{"className":115862},[638],[363,115864,115866],{"className":115865,"style":52122},[642],[363,115867],{},[363,115869],{"className":115870,"style":754},[505],[363,115872,115874],{"className":115873},[407],[363,115875,115877],{"className":115876},[767,15179],[363,115878,115880,115903],{"className":115879},[633,634],[363,115881,115883,115900],{"className":115882},[638],[363,115884,115886],{"className":115885,"style":85902},[642],[363,115887,115888,115891],{"style":73907},[363,115889],{"className":115890,"style":82974},[650],[363,115892,115894],{"style":115893},"width:0.333em;height:1.8em;",[6671,115895,115897],{"xmlns":6673,"width":35423,"height":17789,"viewBox":115896},"0 0 333 1800",[6679,115898],{"d":115899},"M145 15 v585 v600 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v-600 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v600 v585 h43z",[363,115901,666],{"className":115902},[665],[363,115904,115906],{"className":115905},[638],[363,115907,115909],{"className":115908,"style":85926},[642],[363,115910],{},[363,115912],{"className":115913,"style":754},[505],[363,115915,1635],{"className":115916,"style":1835},[1832,1833,1834],[363,115918],{"className":115919,"style":754},[505],[363,115921,39037],{"className":115922},[407,408],[363,115924],{"className":115925,"style":754},[505],[363,115927,592],{"className":115928},[407,408],[363,115930,554],{"className":115931,"style":682},[407,408],[363,115933],{"className":115934,"style":2935},[505],[363,115936,5161],{"className":115937},[2939],[363,115939],{"className":115940,"style":2935},[505],[363,115942,115944,115947,115950,115953,115956,115959,115962,115965,116002],{"className":115943},[398],[363,115945],{"className":115946,"style":6125},[402],[363,115948,1635],{"className":115949,"style":1835},[1832,1833,1834],[363,115951],{"className":115952,"style":754},[505],[363,115954,39037],{"className":115955},[407,408],[363,115957],{"className":115958,"style":754},[505],[363,115960,592],{"className":115961},[407,408],[363,115963,561],{"className":115964},[407,408],[363,115966,115968],{"className":115967},[407],[363,115969,115971],{"className":115970},[767,15179],[363,115972,115974,115994],{"className":115973},[633,634],[363,115975,115977,115991],{"className":115976},[638],[363,115978,115980],{"className":115979,"style":85902},[642],[363,115981,115982,115985],{"style":73907},[363,115983],{"className":115984,"style":82974},[650],[363,115986,115987],{"style":115893},[6671,115988,115989],{"xmlns":6673,"width":35423,"height":17789,"viewBox":115896},[6679,115990],{"d":115899},[363,115992,666],{"className":115993},[665],[363,115995,115997],{"className":115996},[638],[363,115998,116000],{"className":115999,"style":85926},[642],[363,116001],{},[363,116003,4723],{"className":116004},[407],[11,116006,116007,116008,116037,116038,116089,116090,116119,116120,116170,116171,116199,116200,116250,116251,116279],{},"它们比较的是「在函数类 ",[363,116009,116011,116025],{"className":116010,"translate":367},[366],[363,116012,116014],{"className":116013},[371],[373,116015,116016],{"xmlns":375},[377,116017,116018,116022],{},[380,116019,116020],{},[383,116021,37255],{"mathvariant":944},[387,116023,116024],{"encoding":389},"\\mathcal F",[363,116026,116028],{"className":116027,"ariaHidden":394},[393],[363,116029,116031,116034],{"className":116030},[398],[363,116032],{"className":116033,"style":1011},[402],[363,116035,37255],{"className":116036,"style":115769},[407,971]," 上积分的差」，而不是似然比，因此利用了样本空间的度量\u002F几何结构。KS 距离（",[363,116039,116041,116059],{"className":116040,"translate":367},[366],[363,116042,116044],{"className":116043},[371],[373,116045,116046],{"xmlns":375},[377,116047,116048,116056],{},[380,116049,116050,116052,116054],{},[383,116051,37255],{"mathvariant":944},[458,116053,471],{},[458,116055,3246],{"stretchy":460},[387,116057,116058],{"encoding":389},"\\mathcal F=\\{",[363,116060,116062,116080],{"className":116061,"ariaHidden":394},[393],[363,116063,116065,116068,116071,116074,116077],{"className":116064},[398],[363,116066],{"className":116067,"style":1011},[402],[363,116069,37255],{"className":116070,"style":115769},[407,971],[363,116072],{"className":116073,"style":506},[505],[363,116075,471],{"className":116076},[510],[363,116078],{"className":116079,"style":506},[505],[363,116081,116083,116086],{"className":116082},[398],[363,116084],{"className":116085,"style":487},[402],[363,116087,3246],{"className":116088},[494],"一维区间指示函数",[363,116091,116093,116107],{"className":116092,"translate":367},[366],[363,116094,116096],{"className":116095},[371],[373,116097,116098],{"xmlns":375},[377,116099,116100,116104],{},[380,116101,116102],{},[458,116103,3256],{"stretchy":460},[387,116105,116106],{"encoding":389},"\\}",[363,116108,116110],{"className":116109,"ariaHidden":394},[393],[363,116111,116113,116116],{"className":116112},[398],[363,116114],{"className":116115,"style":487},[402],[363,116117,3256],{"className":116118},[501],"）、Wasserstein（",[363,116121,116123,116140],{"className":116122,"translate":367},[366],[363,116124,116126],{"className":116125},[371],[373,116127,116128],{"xmlns":375},[377,116129,116130,116138],{},[380,116131,116132,116134,116136],{},[383,116133,37255],{"mathvariant":944},[458,116135,471],{},[458,116137,3246],{"stretchy":460},[387,116139,116058],{"encoding":389},[363,116141,116143,116161],{"className":116142,"ariaHidden":394},[393],[363,116144,116146,116149,116152,116155,116158],{"className":116145},[398],[363,116147],{"className":116148,"style":1011},[402],[363,116150,37255],{"className":116151,"style":115769},[407,971],[363,116153],{"className":116154,"style":506},[505],[363,116156,471],{"className":116157},[510],[363,116159],{"className":116160,"style":506},[505],[363,116162,116164,116167],{"className":116163},[398],[363,116165],{"className":116166,"style":487},[402],[363,116168,3246],{"className":116169},[494],"1-Lipschitz 函数",[363,116172,116174,116187],{"className":116173,"translate":367},[366],[363,116175,116177],{"className":116176},[371],[373,116178,116179],{"xmlns":375},[377,116180,116181,116185],{},[380,116182,116183],{},[458,116184,3256],{"stretchy":460},[387,116186,116106],{"encoding":389},[363,116188,116190],{"className":116189,"ariaHidden":394},[393],[363,116191,116193,116196],{"className":116192},[398],[363,116194],{"className":116195,"style":487},[402],[363,116197,3256],{"className":116198},[501],"）、MMD（",[363,116201,116203,116220],{"className":116202,"translate":367},[366],[363,116204,116206],{"className":116205},[371],[373,116207,116208],{"xmlns":375},[377,116209,116210,116218],{},[380,116211,116212,116214,116216],{},[383,116213,37255],{"mathvariant":944},[458,116215,471],{},[458,116217,3246],{"stretchy":460},[387,116219,116058],{"encoding":389},[363,116221,116223,116241],{"className":116222,"ariaHidden":394},[393],[363,116224,116226,116229,116232,116235,116238],{"className":116225},[398],[363,116227],{"className":116228,"style":1011},[402],[363,116230,37255],{"className":116231,"style":115769},[407,971],[363,116233],{"className":116234,"style":506},[505],[363,116236,471],{"className":116237},[510],[363,116239],{"className":116240,"style":506},[505],[363,116242,116244,116247],{"className":116243},[398],[363,116245],{"className":116246,"style":487},[402],[363,116248,3246],{"className":116249},[494],"RKHS 单位球",[363,116252,116254,116267],{"className":116253,"translate":367},[366],[363,116255,116257],{"className":116256},[371],[373,116258,116259],{"xmlns":375},[377,116260,116261,116265],{},[380,116262,116263],{},[458,116264,3256],{"stretchy":460},[387,116266,116106],{"encoding":389},[363,116268,116270],{"className":116269,"ariaHidden":394},[393],[363,116271,116273,116276],{"className":116272},[398],[363,116274],{"className":116275,"style":487},[402],[363,116277,3256],{"className":116278},[501],"）都是 IPM。上面对 f-散度的变分表示其实是「带惩罚的 IPM」：",[363,116281,116283],{"className":116282,"translate":367},[528],[363,116284,116286,116368],{"className":116285,"translate":367},[366],[363,116287,116289],{"className":116288},[371],[373,116290,116291],{"xmlns":375,"display":537},[377,116292,116293,116365],{},[380,116294,116295,116301,116303,116305,116307,116309,116311,116313,116323,116325,116331,116333,116335,116337,116339,116345,116347,116353,116355,116357,116359,116361,116363],{},[543,116296,116297,116299],{},[383,116298,547],{},[383,116300,385],{},[458,116302,461],{"stretchy":460},[383,116304,554],{},[383,116306,558],{"mathvariant":557},[383,116308,561],{},[458,116310,468],{"stretchy":460},[458,116312,471],{},[6364,116314,116315,116321],{},[380,116316,116317,116319],{},[383,116318,35804],{},[458,116320,4451],{},[383,116322,4506],{},[458,116324,3246],{"fence":460,"stretchy":394,"minsize":17789,"maxsize":17789},[543,116326,116327,116329],{},[383,116328,571],{"mathvariant":570},[383,116330,554],{},[458,116332,578],{"stretchy":460},[383,116334,4506],{},[458,116336,605],{"stretchy":460},[458,116338,5161],{},[543,116340,116341,116343],{},[383,116342,571],{"mathvariant":570},[383,116344,561],{},[458,116346,578],{"stretchy":460},[7162,116348,116349,116351],{},[383,116350,385],{},[458,116352,7921],{},[458,116354,461],{"stretchy":460},[383,116356,4506],{},[458,116358,468],{"stretchy":460},[458,116360,605],{"stretchy":460},[458,116362,3256],{"fence":460,"stretchy":394,"minsize":17789,"maxsize":17789},[458,116364,898],{"separator":394},[387,116366,116367],{"encoding":389},"D_f(P\\|Q)\n=\\sup_{g}\\Big\\{\\mathbb E_P[g]\n-\\mathbb E_Q[f^*(g)]\\Big\\},",[363,116369,116371,116441,116561],{"className":116370,"ariaHidden":394},[393],[363,116372,116374,116377,116417,116420,116423,116426,116429,116432,116435,116438],{"className":116373},[398],[363,116375],{"className":116376,"style":618},[402],[363,116378,116380,116383],{"className":116379},[407],[363,116381,547],{"className":116382,"style":625},[407,408],[363,116384,116386],{"className":116385},[629],[363,116387,116389,116409],{"className":116388},[633,634],[363,116390,116392,116406],{"className":116391},[638],[363,116393,116395],{"className":116394,"style":643},[642],[363,116396,116397,116400],{"style":646},[363,116398],{"className":116399,"style":651},[650],[363,116401,116403],{"className":116402},[655,656,657,658],[363,116404,385],{"className":116405,"style":409},[407,408,658],[363,116407,666],{"className":116408},[665],[363,116410,116412],{"className":116411},[638],[363,116413,116415],{"className":116414,"style":673},[642],[363,116416],{},[363,116418,461],{"className":116419},[494],[363,116421,554],{"className":116422,"style":682},[407,408],[363,116424,558],{"className":116425},[407],[363,116427,561],{"className":116428},[407,408],[363,116430,468],{"className":116431},[501],[363,116433],{"className":116434,"style":506},[505],[363,116436,471],{"className":116437},[510],[363,116439],{"className":116440,"style":506},[505],[363,116442,116444,116447,116494,116497,116503,116543,116546,116549,116552,116555,116558],{"className":116443},[398],[363,116445],{"className":116446,"style":99874},[402],[363,116448,116450],{"className":116449},[1832,14066],[363,116451,116453,116486],{"className":116452},[633,634],[363,116454,116456,116483],{"className":116455},[638],[363,116457,116459,116473],{"className":116458,"style":2693},[642],[363,116460,116461,116464],{"style":61573},[363,116462],{"className":116463,"style":809},[650],[363,116465,116467],{"className":116466},[655,656,657,658],[363,116468,116470],{"className":116469},[407,658],[363,116471,4506],{"className":116472,"style":1328},[407,408,658],[363,116474,116475,116478],{"style":12213},[363,116476],{"className":116477,"style":809},[650],[363,116479,116480],{},[363,116481,35804],{"className":116482},[1832],[363,116484,666],{"className":116485},[665],[363,116487,116489],{"className":116488},[638],[363,116490,116492],{"className":116491,"style":61605},[642],[363,116493],{},[363,116495],{"className":116496,"style":754},[505],[363,116498,116500],{"className":116499},[407],[363,116501,3246],{"className":116502},[767,4068],[363,116504,116506,116509],{"className":116505},[407],[363,116507,571],{"className":116508},[407,714],[363,116510,116512],{"className":116511},[629],[363,116513,116515,116535],{"className":116514},[633,634],[363,116516,116518,116532],{"className":116517},[638],[363,116519,116521],{"className":116520,"style":727},[642],[363,116522,116523,116526],{"style":730},[363,116524],{"className":116525,"style":651},[650],[363,116527,116529],{"className":116528},[655,656,657,658],[363,116530,554],{"className":116531,"style":682},[407,408,658],[363,116533,666],{"className":116534},[665],[363,116536,116538],{"className":116537},[638],[363,116539,116541],{"className":116540,"style":8316},[642],[363,116542],{},[363,116544,578],{"className":116545},[494],[363,116547,4506],{"className":116548,"style":1328},[407,408],[363,116550,605],{"className":116551},[501],[363,116553],{"className":116554,"style":2935},[505],[363,116556,5161],{"className":116557},[2939],[363,116559],{"className":116560,"style":2935},[505],[363,116562,116564,116567,116607,116610,116639,116642,116645,116648,116654],{"className":116563},[398],[363,116565],{"className":116566,"style":68500},[402],[363,116568,116570,116573],{"className":116569},[407],[363,116571,571],{"className":116572},[407,714],[363,116574,116576],{"className":116575},[629],[363,116577,116579,116599],{"className":116578},[633,634],[363,116580,116582,116596],{"className":116581},[638],[363,116583,116585],{"className":116584,"style":727},[642],[363,116586,116587,116590],{"style":730},[363,116588],{"className":116589,"style":651},[650],[363,116591,116593],{"className":116592},[655,656,657,658],[363,116594,561],{"className":116595},[407,408,658],[363,116597,666],{"className":116598},[665],[363,116600,116602],{"className":116601},[638],[363,116603,116605],{"className":116604,"style":673},[642],[363,116606],{},[363,116608,578],{"className":116609},[494],[363,116611,116613,116616],{"className":116612},[407],[363,116614,385],{"className":116615,"style":409},[407,408],[363,116617,116619],{"className":116618},[629],[363,116620,116622],{"className":116621},[633],[363,116623,116625],{"className":116624},[638],[363,116626,116628],{"className":116627,"style":8596},[642],[363,116629,116630,116633],{"style":8625},[363,116631],{"className":116632,"style":651},[650],[363,116634,116636],{"className":116635},[655,656,657,658],[363,116637,7921],{"className":116638},[2939,658],[363,116640,461],{"className":116641},[494],[363,116643,4506],{"c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由核特征映射的条件期望是收缩算子，沿核的具体结构可证其满足数据处理不等式）。",[2659,116784,116786],{"id":116785},"_63-支撑不重叠的极限","6.3 支撑不重叠的极限",[52864,116788,116789,116869],{},[52867,116790,116791],{},[52870,116792,116793,116796,116798,116800,116802,116804,116864,116866],{},[52873,116794,116795],{},"情形",[52873,116797,33035],{},[52873,116799,109250],{},[52873,116801,53496],{},[52873,116803,87149],{},[52873,116805,116806],{},[363,116807,116809,116826],{"className":116808,"translate":367},[366],[363,116810,116812],{"className":116811},[371],[373,116813,116814],{"xmlns":375},[377,116815,116816,116824],{},[380,116817,116818],{},[7162,116819,116820,116822],{},[383,116821,37821],{},[463,116823,2811],{},[387,116825,40766],{"encoding":389},[363,116827,116829],{"className":116828,"ariaHidden":394},[393],[363,116830,116832,116835],{"className":116831},[398],[363,116833],{"className":116834,"style":40776},[402],[363,116836,116838,116841],{"className":116837},[407],[363,116839,37821],{"className":116840},[407,408],[363,116842,116844],{"className":116843},[629],[363,116845,116847],{"className":116846},[633],[363,116848,116850],{"className":116849},[638],[363,116851,116853],{"className":116852,"style":20400},[642],[363,116854,116855,116858],{"style":7213},[363,116856],{"className":116857,"style":651},[650],[363,116859,116861],{"className":116860},[655,656,657,658],[363,116862,2811],{"className":116863},[407,658],[52873,116865,105564],{},[52873,116867,116868],{},"Wasserstein",[52962,116870,116871,117281],{},[52870,116872,116873,117044,117079,117114,117156,117186,117221,117251],{},[52967,116874,116875,117043],{},[363,116876,116878,116912],{"className":116877,"translate":367},[366],[363,116879,116881],{"className":116880},[371],[373,116882,116883],{"xmlns":375},[377,116884,116885,116909],{},[380,116886,116887,116889,116891,116897,116899,116901,116903],{},[383,116888,554],{},[458,116890,471],{},[543,116892,116893,116895],{},[383,116894,16062],{},[463,116896,474],{},[458,116898,898],{"separator":394},[383,116900,561],{},[458,116902,471],{},[543,116904,116905,116907],{},[383,116906,16062],{},[463,116908,465],{},[387,116910,116911],{"encoding":389},"P=\\delta_0,Q=\\delta_1",[363,116913,116915,116933,116997],{"className":116914,"ariaHidden":394},[393],[363,116916,116918,116921,116924,116927,116930],{"className":116917},[398],[363,116919],{"className":116920,"style":1011},[402],[363,116922,554],{"className":116923,"style":682},[407,408],[363,116925],{"className":116926,"style":506},[505],[363,116928,471],{"className":116929},[510],[363,116931],{"className":116932,"style":506},[505],[363,116934,116936,116939,116979,116982,116985,116988,116991,116994],{"className":116935},[398],[363,116937],{"className":116938,"style":403},[402],[363,116940,116942,116945],{"className":116941},[407],[363,116943,16062],{"className":116944,"style":16111},[407,408],[363,116946,116948],{"className":116947},[629],[363,116949,116951,116971],{"className":116950},[633,634],[363,116952,116954,116968],{"className":116953},[638],[363,116955,116957],{"className":116956,"style":11483},[642],[363,116958,116959,116962],{"style":95132},[363,116960],{"className":116961,"style":651},[650],[363,116963,116965],{"className":116964},[655,656,657,658],[363,116966,474],{"className":116967},[407,658],[363,116969,666],{"className":116970},[665],[363,116972,116974],{"className":116973},[638],[363,116975,116977],{"className":116976,"style":8316},[642],[363,116978],{},[363,116980,898],{"className":116981},[920],[363,116983],{"className":116984,"style":754},[505],[363,116986,561],{"className":116987},[407,408],[363,116989],{"className":116990,"style":506},[505],[363,116992,471],{"className":116993},[510],[363,116995],{"className":116996,"style":506},[505],[363,116998,117000,117003],{"className":116999},[398],[363,117001],{"className":117002,"style":11824},[402],[363,117004,117006,117009],{"className":117005},[407],[363,117007,16062],{"className":117008,"style":16111},[407,408],[363,117010,117012],{"className":117011},[629],[363,117013,117015,117035],{"className":117014},[633,634],[363,117016,117018,117032],{"className":117017},[638],[363,117019,117021],{"className":117020,"style":11483},[642],[363,117022,117023,117026],{"style":95132},[363,117024],{"className":117025,"style":651},[650],[363,117027,117029],{"className":117028},[655,656,657,658],[363,117030,465],{"className":117031},[407,658],[363,117033,666],{"className":117034},[665],[363,117036,117038],{"className":117037},[638],[363,117039,117041],{"className":117040,"style":8316},[642],[363,117042],{},"（约）",[52967,117045,117046],{},[363,117047,117049,117064],{"className":117048,"translate":367},[366],[363,117050,117052],{"className":117051},[371],[373,117053,117054],{"xmlns":375},[377,117055,117056,117062],{},[380,117057,117058,117060],{},[458,117059,2818],{},[383,117061,1373],{"mathvariant":557},[387,117063,4589],{"encoding":389},[363,117065,117067],{"className":117066,"ariaHidden":394},[393],[363,117068,117070,117073,117076],{"className":117069},[398],[363,117071],{"className":117072,"style":4599},[402],[363,117074,2818],{"className":117075},[407],[363,117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Wasserstein 随距离线性增长——这正是词嵌入、最优传输、Wasserstein GAN 的动机。",[18,117828,117830],{"id":117829},"_7-机器学习应用","7. 机器学习应用",[11,117832,117833],{},"f-散度族之所以重要，是因为：",[7010,117835,117836,117842,117852,117887],{},[7013,117837,117838,117841],{},[1344,117839,117840],{},"统一框架","：许多生成模型的目标函数可以写成 f-散度。",[7013,117843,117844,117847,117848,117851],{},[1344,117845,117846],{},"变分估计","：通过凸共轭，可以用神经网络估计 f-散度，这就是 ",[1344,117849,117850],{},"f-GAN"," 的核心思想。",[7013,117853,117854,117857,117858,117886],{},[1344,117855,117856],{},"灵活选择","：不同 ",[363,117859,117861,117874],{"className":117860,"translate":367},[366],[363,117862,117864],{"className":117863},[371],[373,117865,117866],{"xmlns":375},[377,117867,117868,117872],{},[380,117869,117870],{},[383,117871,385],{},[387,117873,385],{"encoding":389},[363,117875,117877],{"className":117876,"ariaHidden":394},[393],[363,117878,117880,117883],{"className":117879},[398],[363,117881],{"className":117882,"style":403},[402],[363,117884,385],{"className":117885,"style":409},[407,408]," 对应不同训练性质。例如 JS 有界但可能梯度消失，KL 无界但敏感。",[7013,117888,117889,117892],{},[1344,117890,117891],{},"理论分析","：数据处理不等式、凸性、对偶表示等为收敛性和泛化分析提供工具。",[11,117894,117895],{},"下面把这几条落到具体场景。",[2659,117897,117899],{"id":117898},"_71-f-gan","7.1 f-GAN",[11,117901,117902,117903,118031,118032,118060,118061,118089],{},"见第 5 节。核心是：",[363,117904,117906,117938],{"className":117905,"translate":367},[366],[363,117907,117909],{"className":117908},[371],[373,117910,117911],{"xmlns":375},[377,117912,117913,117935],{},[380,117914,117915,117925],{},[543,117916,117917,117923],{},[380,117918,117919,117921],{},[383,117920,51980],{},[458,117922,4451],{},[383,117924,101148],{},[543,117926,117927,117933],{},[380,117928,117929,117931],{},[383,117930,52759],{},[458,117932,4451],{},[383,117934,100250],{},[387,117936,117937],{"encoding":389},"\\min_\\theta\\max_\\omega",[363,117939,117941],{"className":117940,"ariaHidden":394},[393],[363,117942,117944,117948,117988,117991],{"className":117943},[398],[363,117945],{"className":117946,"style":117947},[402],"height:0.8179em;vertical-align:-0.15em;",[363,117949,117951,117954],{"className":117950},[1832],[363,117952,51980],{"className":117953},[1832],[363,117955,117957],{"className":117956},[629],[363,117958,117960,117980],{"className":117959},[633,634],[363,117961,117963,117977],{"className":117962},[638],[363,117964,117966],{"className":117965,"style":643},[642],[363,117967,117968,117971],{"style":52818},[363,117969],{"className":117970,"style":651},[650],[363,117972,117974],{"className":117973},[655,656,657,658],[363,117975,101148],{"className":117976,"style":625},[407,408,658],[363,117978,666],{"className":117979},[665],[363,117981,117983],{"className":117982},[638],[363,117984,117986],{"className":117985,"style":8316},[642],[363,117987],{},[363,117989],{"className":117990,"style":754},[505],[363,117992,117994,117997],{"className":117993},[1832],[363,117995,52759],{"className":117996},[1832],[363,117998,118000],{"className":117999},[629],[363,118001,118003,118023],{"className":118002},[633,634],[363,118004,118006,118020],{"className":118005},[638],[363,118007,118009],{"className":118008,"style":45301},[642],[363,118010,118011,118014],{"style":52818},[363,118012],{"className":118013,"style":651},[650],[363,118015,118017],{"className":118016},[655,656,657,658],[363,118018,100250],{"className":118019,"style":1328},[407,408,658],[363,118021,666],{"className":118022},[665],[363,118024,118026],{"className":118025},[638],[363,118027,118029],{"className":118028,"style":8316},[642],[363,118030],{}," 的对抗目标 = 对 ",[363,118033,118035,118048],{"className":118034,"translate":367},[366],[363,118036,118038],{"className":118037},[371],[373,118039,118040],{"xmlns":375},[377,118041,118042,118046],{},[380,118043,118044],{},[383,118045,385],{},[387,118047,385],{"encoding":389},[363,118049,118051],{"className":118050,"ariaHidden":394},[393],[363,118052,118054,118057],{"className":118053},[398],[363,118055],{"className":118056,"style":403},[402],[363,118058,385],{"className":118059,"style":409},[407,408]," 做对偶后的 f-散度估计。选择 ",[363,118062,118064,118077],{"className":118063,"translate":367},[366],[363,118065,118067],{"className":118066},[371],[373,118068,118069],{"xmlns":375},[377,118070,118071,118075],{},[380,118072,118073],{},[383,118074,385],{},[387,118076,385],{"encoding":389},[363,118078,118080],{"className":118079,"ariaHidden":394},[393],[363,118081,118083,118086],{"className":118082},[398],[363,118084],{"className":118085,"style":403},[402],[363,118087,385],{"className":118088,"style":409},[407,408]," 就是选择「用哪种散度训练生成器」，这直接决定模式崩塌、梯度消失与训练稳定性的权衡。",[2659,118091,118093],{"id":118092},"_72-变分推断与-elbo","7.2 变分推断与 ELBO",[11,118095,118096],{},"变分推断里最大化证据下界",[363,118098,118100],{"className":118099,"translate":367},[528],[363,118101,118103,118262],{"className":118102,"translate":367},[366],[363,118104,118106],{"className":118105},[371],[373,118107,118108],{"xmlns":375,"display":537},[377,118109,118110,118259],{},[1591,118111,118112,118189],{"rowspacing":1593,"columnalign":1594,"columnspacing":1595},[1597,118113,118114,118128],{},[1600,118115,118116],{},[1603,118117,118118],{"scriptlevel":474,"displaystyle":394},[380,118119,118120,118122,118124,118126],{},[383,118121,22265],{"mathvariant":944},[458,118123,461],{"stretchy":460},[383,118125,1291],{},[458,118127,468],{"stretchy":460},[1600,118129,118130],{},[1603,118131,118132],{"scriptlevel":474,"displaystyle":394},[380,118133,118134,118136,118138,118144,118146,118148,118150,118152,118154,118156,118158,118161,118163,118165,118167,118173,118175,118177,118179,118181,118183,118185,118187],{},[380,118135],{},[458,118137,471],{},[543,118139,118140,118142],{},[383,118141,571],{"mathvariant":570},[383,118143,1291],{},[458,118145,578],{"stretchy":460},[383,118147,4448],{},[458,118149,4451],{},[383,118151,11],{},[458,118153,461],{"stretchy":460},[383,118155,1121],{},[458,118157,898],{"separator":394},[383,118159,118160],{},"z",[458,118162,468],{"stretchy":460},[458,118164,605],{"stretchy":460},[458,118166,5161],{},[543,118168,118169,118171],{},[383,118170,571],{"mathvariant":570},[383,118172,1291],{},[458,118174,578],{"stretchy":460},[383,118176,4448],{},[458,118178,4451],{},[383,118180,1291],{},[458,118182,461],{"stretchy":460},[383,118184,118160],{},[458,118186,468],{"stretchy":460},[458,118188,605],{"stretchy":460},[1597,118190,118191,118197],{},[1600,118192,118193],{},[1603,118194,118195],{"scriptlevel":474,"displaystyle":394},[380,118196],{},[1600,118198,118199],{},[1603,118200,118201],{"scriptlevel":474,"displaystyle":394},[380,118202,118203,118205,118207,118209,118211,118213,118215,118217,118219,118221,118227,118229,118231,118233,118235,118237,118239,118241,118243,118245,118247,118249,118251,118253,118255,118257],{},[380,118204],{},[458,118206,471],{},[383,118208,4448],{},[458,118210,4451],{},[383,118212,11],{},[458,118214,461],{"stretchy":460},[383,118216,1121],{},[458,118218,468],{"stretchy":460},[458,118220,5161],{},[380,118222,118223,118225],{},[383,118224,11044],{"mathvariant":557},[383,118226,22265],{"mathvariant":557},[458,118228,461],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[383,118230,1291],{},[458,118232,461],{"stretchy":460},[383,118234,118160],{},[458,118236,468],{"stretchy":460},[1645,118238,1647],{},[383,118240,558],{"mathvariant":557},[1645,118242,1647],{},[383,118244,11],{},[458,118246,461],{"stretchy":460},[383,118248,118160],{},[383,118250,5156],{"mathvariant":557},[383,118252,1121],{},[458,118254,468],{"stretchy":460},[458,118256,468],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[458,118258,898],{"separator":394},[387,118260,118261],{"encoding":389},"\\begin{aligned}\n\\mathcal L(q)\n&=\\mathbb E_q[\\log p(x,z)]\n-\\mathbb E_q[\\log q(z)]\\\\\n&=\\log p(x)\n-\\mathrm{KL}\\big(q(z)\\,\\|\\,p(z|x)\\big),\n\\end{aligned}",[363,118263,118265],{"className":118264,"ariaHidden":394},[393],[363,118266,118268,118271],{"className":118267},[398],[363,118269],{"className":118270,"style":14936},[402],[363,118272,118274],{"className":118273},[407],[363,118275,118277,118329],{"className":118276},[1591],[363,118278,118280],{"className":118279},[1708],[363,118281,118283,118321],{"className":118282},[633,634],[363,118284,118286,118318],{"className":118285},[638],[363,118287,118289,118310],{"className":118288,"style":14955},[642],[363,118290,118292,118295],{"style":118291},"top:-3.765em;",[363,118293],{"className":118294,"style":809},[650],[363,118296,118298,118301,118304,118307],{"className":118297},[407],[363,118299,22265],{"className":118300},[407,971],[363,118302,461],{"className":118303},[494],[363,118305,1291],{"className":118306,"style":1328},[407,408],[363,118308,468],{"className":118309},[501],[363,118311,118312,118315],{"style":15343},[363,118313],{"className":118314,"style":809},[650],[363,118316],{"className":118317},[407],[363,118319,666],{"className":118320},[665],[363,118322,118324],{"className":118323},[638],[363,118325,118327],{"className":118326,"style":14984},[642],[363,118328],{},[363,118330,118332],{"className":118331},[1799],[363,118333,118335,118618],{"className":118334},[633,634],[363,118336,118338,118615],{"className":118337},[638],[363,118339,118341,118506],{"className":118340,"style":14955},[642],[363,118342,118343,118346],{"style":118291},[363,118344],{"className":118345,"style":809},[650],[363,118347,118349,118352,118355,118358,118361,118401,118404,118409,118412,118415,118418,118421,118424,118427,118431,118434,118437,118440,118443,118483,118486,118491,118494,118497,118500,118503],{"className":118348},[407],[363,118350],{"className":118351},[407],[363,118353],{"className":118354,"style":506},[505],[363,118356,471],{"className":118357},[510],[363,118359],{"className":118360,"style":506},[505],[363,118362,118364,118367],{"className":118363},[407],[363,118365,571],{"className":118366},[407,714],[363,118368,118370],{"className":118369},[629],[363,118371,118373,118393],{"className":118372},[633,634],[363,118374,118376,118390],{"className":118375},[638],[363,118377,118379],{"className":118378,"style":45301},[642],[363,118380,118381,118384],{"style":730},[363,118382],{"className":118383,"style":651},[650],[363,118385,118387],{"className":118386},[655,656,657,658],[363,118388,1291],{"className":118389,"style":1328},[407,408,658],[363,118391,666],{"className":118392},[665],[363,118394,118396],{"className":118395},[638],[363,118397,118399],{"className":118398,"style":673},[642],[363,118400],{},[363,118402,578],{"className":118403},[494],[363,118405,4502,118407],{"className":118406},[1832],[363,118408,4506],{"style":4505},[363,118410],{"className":118411,"style":754},[505],[363,118413,11],{"className":118414},[407,408],[363,118416,461],{"className":118417},[494],[363,118419,1121],{"className":118420},[407,408],[363,118422,898],{"className":118423},[920],[363,118425],{"className":118426,"style":754},[505],[363,118428,118160],{"className":118429,"style":118430},[407,408],"margin-right:0.044em;",[363,118432,20762],{"className":118433},[501],[363,118435],{"className":118436,"style":2935},[505],[363,118438,5161],{"className":118439},[2939],[363,118441],{"className":118442,"style":2935},[505],[363,118444,118446,118449],{"className":118445},[407],[363,118447,571],{"className":118448},[407,714],[363,118450,118452],{"className":118451},[629],[363,118453,118455,118475],{"className":118454},[633,634],[363,118456,118458,118472],{"className":118457},[638],[363,118459,118461],{"className":118460,"style":45301},[642],[363,118462,118463,118466],{"style":730},[363,118464],{"className":118465,"style":651},[650],[363,118467,118469],{"className":118468},[655,656,657,658],[363,118470,1291],{"className":118471,"style":1328},[407,408,658],[363,118473,666],{"className":118474},[665],[363,118476,118478],{"className":118477},[638],[363,118479,118481],{"className":118480,"style":673},[642],[363,118482],{},[363,118484,578],{"className":118485},[494],[363,118487,4502,118489],{"className":118488},[1832],[363,118490,4506],{"style":4505},[363,118492],{"className":118493,"style":754},[505],[363,118495,1291],{"className":118496,"style":1328},[407,408],[363,118498,461],{"className":118499},[494],[363,118501,118160],{"className":118502,"style":118430},[407,408],[363,118504,20762],{"className":118505},[501],[363,118507,118508,118511],{"style":15343},[363,118509],{"className":118510,"style":809},[650],[363,118512,118514,118517,118520,118523,118526,118531,118534,118537,118540,118543,118546,118549,118552,118555,118561,118567,118570,118573,118576,118579,118582,118585,118588,118591,118594,118597,118600,118603,118606,118612],{"className":118513},[407],[363,118515],{"className":118516},[407],[363,118518],{"className":118519,"style":506},[505],[363,118521,471],{"className":118522},[510],[363,118524],{"className":118525,"style":506},[505],[363,118527,4502,118529],{"className":118528},[1832],[363,118530,4506],{"style":4505},[363,118532],{"className":118533,"style":754},[505],[363,118535,11],{"className":118536},[407,408],[363,118538,461],{"className":118539},[494],[363,118541,1121],{"className":118542},[407,408],[363,118544,468],{"className":118545},[501],[363,118547],{"className":118548,"style":2935},[505],[363,118550,5161],{"className":118551},[2939],[363,118553],{"className":118554,"style":2935},[505],[363,118556,118558],{"className":118557},[407],[363,118559,33035],{"className":118560},[407,33034],[363,118562,118564],{"className":118563},[407],[363,118565,461],{"className":118566},[767,10022],[363,118568,1291],{"className":118569,"style":1328},[407,408],[363,118571,461],{"className":118572},[494],[363,118574,118160],{"className":118575,"style":118430},[407,408],[363,118577,468],{"className":118578},[501],[363,118580],{"className":118581,"style":754},[505],[363,118583,558],{"className":118584},[407],[363,118586],{"className":118587,"style":754},[505],[363,118589,11],{"className":118590},[407,408],[363,118592,461],{"className":118593},[494],[363,118595,118160],{"className":118596,"style":118430},[407,408],[363,118598,5156],{"className":118599},[407],[363,118601,1121],{"className":118602},[407,408],[363,118604,468],{"className":118605},[501],[363,118607,118609],{"className":118608},[407],[363,118610,468],{"className":118611},[767,10022],[363,118613,898],{"className":118614},[920],[363,118616,666],{"className":118617},[665],[363,118619,118621],{"className":118620},[638],[363,118622,118624],{"className":118623,"style":14984},[642],[363,118625],{},[11,118627,2598,118628,118689,118690,118761,118762,118863,118864,118892,118893,118951,118952,118980],{},[363,118629,118631,118659],{"className":118630,"translate":367},[366],[363,118632,118634],{"className":118633},[371],[373,118635,118636],{"xmlns":375},[377,118637,118638,118656],{},[380,118639,118640,118646,118648,118650,118652,118654],{},[380,118641,118642,118644],{},[383,118643,11044],{"mathvariant":557},[383,118645,22265],{"mathvariant":557},[458,118647,461],{"stretchy":460},[383,118649,1291],{},[383,118651,558],{"mathvariant":557},[383,118653,11],{},[458,118655,468],{"stretchy":460},[387,118657,118658],{"encoding":389},"\\mathrm{KL}(q\\|p)",[363,118660,118662],{"className":118661,"ariaHidden":394},[393],[363,118663,118665,118668,118674,118677,118680,118683,118686],{"className":118664},[398],[363,118666],{"className":118667,"style":487},[402],[363,118669,118671],{"className":118670},[407],[363,118672,33035],{"className":118673},[407,33034],[363,118675,461],{"className":118676},[494],[363,118678,1291],{"className":118679,"style":1328},[407,408],[363,118681,558],{"className":118682},[407],[363,118684,11],{"className":118685},[407,408],[363,118687,468],{"className":118688},[501]," 是 f-散度（",[363,118691,118693,118717],{"className":118692,"translate":367},[366],[363,118694,118696],{"className":118695},[371],[373,118697,118698],{"xmlns":375},[377,118699,118700,118714],{},[380,118701,118702,118704,118706,118708,118710,118712],{},[383,118703,385],{},[458,118705,471],{},[383,118707,1116],{},[383,118709,4448],{},[458,118711,4451],{},[383,118713,1116],{},[387,118715,118716],{"encoding":389},"f=t\\log t",[363,118718,118720,118738],{"className":118719,"ariaHidden":394},[393],[363,118721,118723,118726,118729,118732,118735],{"className":118722},[398],[363,118724],{"className":118725,"style":403},[402],[363,118727,385],{"className":118728,"style":409},[407,408],[363,118730],{"className":118731,"style":506},[505],[363,118733,471],{"className":118734},[510],[363,118736],{"className":118737,"style":506},[505],[363,118739,118741,118744,118747,118750,118755,118758],{"className":118740},[398],[363,118742],{"className":118743,"style":403},[402],[363,118745,1116],{"className":118746},[407,408],[363,118748],{"className":118749,"style":754},[505],[363,118751,4502,118753],{"className":118752},[1832],[363,118754,4506],{"style":4505},[363,118756],{"className":118757,"style":754},[505],[363,118759,1116],{"className":118760},[407,408],"）。反过来，任何以 ",[363,118763,118765,118797],{"className":118764,"translate":367},[366],[363,118766,118768],{"className":118767},[371],[373,118769,118770],{"xmlns":375},[377,118771,118772,118794],{},[380,118773,118774,118780,118782,118784,118786,118788,118790,118792],{},[543,118775,118776,118778],{},[383,118777,571],{"mathvariant":570},[383,118779,1291],{},[458,118781,578],{"stretchy":460},[383,118783,4448],{},[458,118785,4451],{},[458,118787,461],{"stretchy":460},[458,118789,3647],{},[458,118791,468],{"stretchy":460},[458,118793,605],{"stretchy":460},[387,118795,118796],{"encoding":389},"\\mathbb E_q[\\log(\\cdot)]",[363,118798,118800],{"className":118799,"ariaHidden":394},[393],[363,118801,118803,118806,118846,118849,118854,118857,118860],{"className":118802},[398],[363,118804],{"className":118805,"style":618},[402],[363,118807,118809,118812],{"className":118808},[407],[363,118810,571],{"className":118811},[407,714],[363,118813,118815],{"className":118814},[629],[363,118816,118818,118838],{"className":118817},[633,634],[363,118819,118821,118835],{"className":118820},[638],[363,118822,118824],{"className":118823,"style":45301},[642],[363,118825,118826,118829],{"style":730},[363,118827],{"className":118828,"style":651},[650],[363,118830,118832],{"className":118831},[655,656,657,658],[363,118833,1291],{"className":118834,"style":1328},[407,408,658],[363,118836,666],{"className":118837},[665],[363,118839,118841],{"className":118840},[638],[363,118842,118844],{"className":118843,"style":673},[642],[363,118845],{},[363,118847,578],{"className":118848},[494],[363,118850,4502,118852],{"className":118851},[1832],[363,118853,4506],{"style":4505},[363,118855,461],{"className":118856},[494],[363,118858,3647],{"className":118859},[407],[363,118861,20762],{"className":118862},[501]," 为组成部分的目标都可以用 f-散度的语言重写；",[363,118865,118867,118880],{"className":118866,"translate":367},[366],[363,118868,118870],{"className":118869},[371],[373,118871,118872],{"xmlns":375},[377,118873,118874,118878],{},[380,118875,118876],{},[383,118877,45126],{},[387,118879,49462],{"encoding":389},[363,118881,118883],{"className":118882,"ariaHidden":394},[393],[363,118884,118886,118889],{"className":118885},[398],[363,118887],{"className":118888,"style":2693},[402],[363,118890,45126],{"className":118891,"style":45156},[407,408],"-VI、",[363,118894,118896,118913],{"className":118895,"translate":367},[366],[363,118897,118899],{"className":118898},[371],[373,118900,118901],{"xmlns":375},[377,118902,118903,118911],{},[380,118904,118905],{},[7162,118906,118907,118909],{},[383,118908,37821],{},[463,118910,2811],{},[387,118912,40766],{"encoding":389},[363,118914,118916],{"className":118915,"ariaHidden":394},[393],[363,118917,118919,118922],{"className":118918},[398],[363,118920],{"className":118921,"style":40776},[402],[363,118923,118925,118928],{"className":118924},[407],[363,118926,37821],{"className":118927},[407,408],[363,118929,118931],{"className":118930},[629],[363,118932,118934],{"className":118933},[633],[363,118935,118937],{"className":118936},[638],[363,118938,118940],{"className":118939,"style":20400},[642],[363,118941,118942,118945],{"style":7213},[363,118943],{"className":118944,"style":651},[650],[363,118946,118948],{"className":118947},[655,656,657,658],[363,118949,2811],{"className":118950},[407,658],"-VI 等就换用别的 ",[363,118953,118955,118968],{"className":118954,"translate":367},[366],[363,118956,118958],{"className":118957},[371],[373,118959,118960],{"xmlns":375},[377,118961,118962,118966],{},[380,118963,118964],{},[383,118965,385],{},[387,118967,385],{"encoding":389},[363,118969,118971],{"className":118970,"ariaHidden":394},[393],[363,118972,118974,118977],{"className":118973},[398],[363,118975],{"className":118976,"style":403},[402],[363,118978,385],{"className":118979,"style":409},[407,408],"，在 3.1 节所说的「零回避」（mass-covering）与「零强制」（mode-seeking）两种极端行为之间取舍。",[2659,118982,118984],{"id":118983},"_73-互信息估计","7.3 互信息估计",[11,118986,118987],{},"互信息可写成",[363,118989,118991],{"className":118990,"translate":367},[528],[363,118992,118994,119109],{"className":118993,"translate":367},[366],[363,118995,118997],{"className":118996},[371],[373,118998,118999],{"xmlns":375,"display":537},[377,119000,119001,119106],{},[380,119002,119003,119005,119007,119009,119011,119013,119015,119017,119023,119025,119035,119037,119039,119041,119047,119050,119056,119058,119060,119074,119076,119086,119088,119094,119096,119102,119104],{},[383,119004,34240],{},[458,119006,461],{"stretchy":460},[383,119008,945],{},[458,119010,34247],{"separator":394},[383,119012,5749],{},[458,119014,468],{"stretchy":460},[458,119016,471],{},[380,119018,119019,119021],{},[383,119020,11044],{"mathvariant":557},[383,119022,22265],{"mathvariant":557},[458,119024,461],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[543,119026,119027,119029],{},[383,119028,554],{},[380,119030,119031,119033],{},[383,119032,945],{},[383,119034,5749],{},[1645,119036,1647],{},[383,119038,558],{"mathvariant":557},[1645,119040,1647],{},[543,119042,119043,119045],{},[383,119044,554],{},[383,119046,945],{},[458,119048,119049],{},"⊗",[543,119051,119052,119054],{},[383,119053,554],{},[383,119055,5749],{},[458,119057,468],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[458,119059,471],{},[543,119061,119062,119064],{},[383,119063,547],{},[380,119065,119066,119068,119070,119072],{},[383,119067,1116],{},[383,119069,4448],{},[458,119071,4451],{},[383,119073,1116],{},[458,119075,461],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[543,119077,119078,119080],{},[383,119079,554],{},[380,119081,119082,119084],{},[383,119083,945],{},[383,119085,5749],{},[383,119087,558],{"mathvariant":557},[543,119089,119090,119092],{},[383,119091,554],{},[383,119093,945],{},[458,119095,119049],{},[543,119097,119098,119100],{},[383,119099,554],{},[383,119101,5749],{},[458,119103,468],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[458,119105,898],{"separator":394},[387,119107,119108],{"encoding":389},"I(X;Y)\n=\\mathrm{KL}\\big(P_{XY}\\,\\|\\,P_X\\otimes P_Y\\big)\n=D_{t\\log t}\\big(P_{XY}\\|P_X\\otimes P_Y\\big),",[363,119110,119112,119148,119270,119331,119505],{"className":119111,"ariaHidden":394},[393],[363,119113,119115,119118,119121,119124,119127,119130,119133,119136,119139,119142,119145],{"className":119114},[398],[363,119116],{"className":119117,"style":487},[402],[363,119119,34240],{"className":119120,"style":9235},[407,408],[363,119122,461],{"className":119123},[494],[363,119125,945],{"className":119126,"style":9235},[407,408],[363,119128,34247],{"className":119129},[920],[363,119131],{"className":119132,"style":754},[505],[363,119134,5749],{"className":119135,"style":2935},[407,408],[363,119137,468],{"className":119138},[501],[363,119140],{"className":119141,"style":506},[505],[363,119143,471],{"className":119144},[510],[363,119146],{"className":119147,"style":506},[505],[363,119149,119151,119154,119160,119166,119212,119215,119218,119221,119261,119264,119267],{"className":119150},[398],[363,119152],{"className":119153,"style":20935},[402],[363,119155,119157],{"className":119156},[407],[363,119158,33035],{"className":119159},[407,33034],[363,119161,119163],{"className":119162},[407],[363,119164,461],{"className":119165},[767,10022],[363,119167,119169,119172],{"className":119168},[407],[363,119170,554],{"className":119171,"style":682},[407,408],[363,119173,119175],{"className":119174},[629],[363,119176,119178,119204],{"className":119177},[633,634],[363,119179,119181,119201],{"className":119180},[638],[363,119182,119184],{"className":119183,"style":727},[642],[363,119185,119186,119189],{"style":11486},[363,119187],{"className":119188,"style":651},[650],[363,119190,119192],{"className":119191},[655,656,657,658],[363,119193,119195,119198],{"className":119194},[407,658],[363,119196,945],{"className":119197,"style":9235},[407,408,658],[363,119199,5749],{"className":119200,"style":2935},[407,408,658],[363,119202,666],{"className":119203},[665],[363,119205,119207],{"className":119206},[638],[363,119208,119210],{"className":119209,"style":8316},[642],[363,119211],{},[363,119213],{"className":119214,"style":754},[505],[363,119216,558],{"className":119217},[407],[363,119219],{"className":119220,"style":754},[505],[363,119222,119224,119227],{"className":119223},[407],[363,119225,554],{"className":119226,"style":682},[407,408],[363,119228,119230],{"className":119229},[629],[363,119231,119233,119253],{"className":119232},[633,634],[363,119234,119236,119250],{"className":119235},[638],[363,119237,119239],{"className":119238,"style":727},[642],[363,119240,119241,119244],{"style":11486},[363,119242],{"className":119243,"style":651},[650],[363,119245,119247],{"className":119246},[655,656,657,658],[363,119248,945],{"className":119249,"style":9235},[407,408,658],[363,119251,666],{"className":119252},[665],[363,119254,119256],{"className":119255},[638],[363,119257,119259],{"className":119258,"style":8316},[642],[363,119260],{},[363,119262],{"className":119263,"style":2935},[505],[363,119265,119049],{"className":119266},[2939],[363,119268],{"className":119269,"style":2935},[505],[363,119271,119273,119276,119316,119322,119325,119328],{"className":119272},[398],[363,119274],{"className":119275,"style":20935},[402],[363,119277,119279,119282],{"className":119278},[407],[363,119280,554],{"className":119281,"style":682},[407,408],[363,119283,119285],{"className":119284},[629],[363,119286,119288,119308],{"className":119287},[633,634],[363,119289,119291,119305],{"className":119290},[638],[363,119292,119294],{"className":119293,"style":727},[642],[363,119295,119296,119299],{"style":11486},[363,119297],{"className":119298,"style":651},[650],[363,119300,119302],{"className":119301},[655,656,657,658],[363,119303,5749],{"className":119304,"style":2935},[407,408,658],[363,119306,666],{"className":119307},[665],[363,119309,119311],{"className":119310},[638],[363,119312,119314],{"className":119313,"style":8316},[642],[363,119315],{},[363,119317,119319],{"className":119318},[407],[363,119320,468],{"className":119321},[767,10022],[363,119323],{"className":119324,"style":506},[505],[363,119326,471],{"className":119327},[510],[363,119329],{"className":119330,"style":506},[505],[363,119332,119334,119337,119401,119407,119453,119456,119496,119499,119502],{"className":119333},[398],[363,119335],{"className":119336,"style":20935},[402],[363,119338,119340,119343],{"className":119339},[407],[363,119341,547],{"className":119342,"style":625},[407,408],[363,119344,119346],{"className":119345},[629],[363,119347,119349,119393],{"className":119348},[633,634],[363,119350,119352,119390],{"className":119351},[638],[363,119353,119355],{"className":119354,"style":643},[642],[363,119356,119357,119360],{"style":646},[363,119358],{"className":119359,"style":651},[650],[363,119361,119363],{"className":119362},[655,656,657,658],[363,119364,119366,119369,119372,119384,119387],{"className":119365},[407,658],[363,119367,1116],{"className":119368},[407,408,658],[363,119370],{"className":119371,"style":42738},[505,658],[363,119373,119375,119378,119381],{"className":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f-散度的特例。MINE 用 Donsker–Varadhan 对偶估计它，f-MINE 则对一般 ",[363,119564,119566,119579],{"className":119565,"translate":367},[366],[363,119567,119569],{"className":119568},[371],[373,119570,119571],{"xmlns":375},[377,119572,119573,119577],{},[380,119574,119575],{},[383,119576,385],{},[387,119578,385],{"encoding":389},[363,119580,119582],{"className":119581,"ariaHidden":394},[393],[363,119583,119585,119588],{"className":119584},[398],[363,119586],{"className":119587,"style":403},[402],[363,119589,385],{"className":119590,"style":409},[407,408]," 用 4.4 节的对偶；不同 ",[363,119593,119595,119608],{"className":119594,"translate":367},[366],[363,119596,119598],{"className":119597},[371],[373,119599,119600],{"xmlns":375},[377,119601,119602,119606],{},[380,119603,119604],{},[383,119605,385],{},[387,119607,385],{"encoding":389},[363,119609,119611],{"className":119610,"ariaHidden":394},[393],[363,119612,119614,119617],{"className":119613},[398],[363,119615],{"className":119616,"style":403},[402],[363,119618,385],{"className":119619,"style":409},[407,408],"（KL、JS、",[363,119622,119624,119641],{"className":119623,"translate":367},[366],[363,119625,119627],{"className":119626},[371],[373,119628,119629],{"xmlns":375},[377,119630,119631,119639],{},[380,119632,119633],{},[7162,119634,119635,119637],{},[383,119636,37821],{},[463,119638,2811],{},[387,119640,40766],{"encoding":389},[363,119642,119644],{"className":119643,"ariaHidden":394},[393],[363,119645,119647,119650],{"className":119646},[398],[363,119648],{"className":119649,"style":40776},[402],[363,119651,119653,119656],{"className":119652},[407],[363,119654,37821],{"className":119655},[407,408],[363,119657,119659],{"className":119658},[629],[363,119660,119662],{"className":119661},[633],[363,119663,119665],{"className":119664},[638],[363,119666,119668],{"className":119667,"style":20400},[642],[363,119669,119670,119673],{"style":7213},[363,119671],{"className":119672,"style":651},[650],[363,119674,119676],{"className":119675},[655,656,657,658],[363,119677,2811],{"className":119678},[407,658],"）给出不同的「MI 下界」和不同的方差—偏差折衷。对比式学习的 InfoNCE 也是 KL 下界的一种变体。",[2659,119681,119683],{"id":119682},"_74-表示学习","7.4 表示学习",[11,119685,119686,119687,119731],{},"任何表示 ",[363,119688,119690,119710],{"className":119689,"translate":367},[366],[363,119691,119693],{"className":119692},[371],[373,119694,119695],{"xmlns":375},[377,119696,119697,119707],{},[380,119698,119699,119701,119703,119705],{},[383,119700,5740],{},[458,119702,461],{"stretchy":460},[383,119704,945],{},[458,119706,468],{"stretchy":460},[387,119708,119709],{"encoding":389},"T(X)",[363,119711,119713],{"className":119712,"ariaHidden":394},[393],[363,119714,119716,119719,119722,119725,119728],{"className":119715},[398],[363,119717],{"className":119718,"style":487},[402],[363,119720,5740],{"className":119721,"style":682},[407,408],[363,119723,461],{"className":119724},[494],[363,119726,945],{"className":119727,"style":9235},[407,408],[363,119729,468],{"className":119730},[501]," 都满足（4.5 节）",[363,119733,119735],{"className":119734,"translate":367},[528],[363,119736,119738,119818],{"className":119737,"translate":367},[366],[363,119739,119741],{"className":119740},[371],[373,119742,119743],{"xmlns":375,"display":537},[377,119744,119745,119815],{},[380,119746,119747,119753,119755,119769,119771,119785,119787,119789,119795,119797,119803,119805,119811,119813],{},[543,119748,119749,119751],{},[383,119750,547],{},[383,119752,385],{},[458,119754,461],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[543,119756,119757,119759],{},[383,119758,554],{},[380,119760,119761,119763,119765,119767],{},[383,119762,5740],{},[458,119764,461],{"stretchy":460},[383,119766,945],{},[458,119768,468],{"stretchy":460},[383,119770,558],{"mathvariant":557},[543,119772,119773,119775],{},[383,119774,561],{},[380,119776,119777,119779,119781,119783],{},[383,119778,5740],{},[458,119780,461],{"stretchy":460},[383,119782,945],{},[458,119784,468],{"stretchy":460},[458,119786,468],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[458,119788,10300],{},[543,119790,119791,119793],{},[383,119792,547],{},[383,119794,385],{},[458,119796,461],{"stretchy":460},[543,119798,119799,119801],{},[383,119800,554],{},[383,119802,945],{},[383,119804,558],{"mathvariant":557},[543,119806,119807,119809],{},[383,119808,561],{},[383,119810,945],{},[458,119812,468],{"stretchy":460},[458,119814,898],{"separator":394},[387,119816,119817],{"encoding":389},"D_f\\big(P_{T(X)}\\|Q_{T(X)}\\big)\n\\le D_f(P_X\\|Q_X),",[363,119819,119821,119997],{"className":119820,"ariaHidden":394},[393],[363,119822,119824,119828,119868,119874,119927,119930,119982,119988,119991,119994],{"className":119823},[398],[363,119825],{"className":119826,"style":119827},[402],"height:1.2052em;vertical-align:-0.3552em;",[363,119829,119831,119834],{"className":119830},[407],[363,119832,547],{"className":119833,"style":625},[407,408],[363,119835,119837],{"className":119836},[629],[363,119838,119840,119860],{"className":119839},[633,634],[363,119841,119843,119857],{"className":119842},[638],[363,119844,119846],{"className":119845,"style":643},[642],[363,119847,119848,119851],{"style":646},[363,119849],{"className":119850,"style":651},[650],[363,119852,119854],{"className":119853},[655,656,657,658],[363,119855,385],{"className":119856,"style":409},[407,408,658],[363,119858,666],{"className":119859},[665],[363,119861,119863],{"className":119862},[638],[363,119864,119866],{"className":119865,"style":673},[642],[363,119867],{},[363,119869,119871],{"className":119870},[407],[363,119872,461],{"className":119873},[767,10022],[363,119875,119877,119880],{"className":119876},[407],[363,119878,554],{"className":119879,"style":682},[407,408],[363,119881,119883],{"className":119882},[629],[363,119884,119886,119919],{"className":119885},[633,634],[363,119887,119889,119916],{"className":119888},[638],[363,119890,119892],{"className":119891,"style":6446},[642],[363,119893,119895,119898],{"style":119894},"top:-2.5198em;margin-left:-0.1389em;margin-right:0.05em;",[363,119896],{"className":119897,"style":651},[650],[363,119899,119901],{"className":119900},[655,656,657,658],[363,119902,119904,119907,119910,119913],{"className":119903},[407,658],[363,119905,5740],{"className":119906,"style":682},[407,408,658],[363,119908,461],{"className":119909},[494,658],[363,119911,945],{"className":119912,"style":9235},[407,408,658],[363,119914,468],{"className":119915},[501,658],[363,119917,666],{"className":119918},[665],[363,119920,119922],{"className":119921},[638],[363,119923,119925],{"className":119924,"style":6492},[642],[363,119926],{},[363,119928,558],{"className":119929},[407],[363,119931,119933,119936],{"className":119932},[407],[363,119934,561],{"className":119935},[407,408],[363,119937,119939],{"className":119938},[629],[363,119940,119942,119974],{"className":119941},[633,634],[363,119943,119945,119971],{"className":119944},[638],[363,119946,119948],{"className":119947,"style":6446},[642],[363,119949,119950,119953],{"style":75958},[363,119951],{"className":119952,"style":651},[650],[363,119954,119956],{"className":119955},[655,656,657,658],[363,119957,119959,119962,119965,119968],{"className":119958},[407,658],[363,119960,5740],{"className":119961,"style":682},[407,408,658],[363,119963,461],{"className":119964},[494,658],[363,119966,945],{"className":119967,"style":9235},[407,408,658],[363,119969,468],{"className":119970},[501,658],[363,119972,666],{"className":119973},[665],[363,119975,119977],{"className":119976},[638],[363,119978,119980],{"className":119979,"style":6492},[642],[363,119981],{},[363,119983,119985],{"className":119984},[407],[363,119986,468],{"className":119987},[767,10022],[363,119989],{"className":119990,"style":506},[505],[363,119992,10300],{"className":119993},[510],[363,119995],{"className":119996,"style":506},[505],[363,119998,120000,120003,120043,120046,120086,120089,120129,120132],{"className":119999},[398],[363,120001],{"className":120002,"style":618},[402],[363,120004,120006,120009],{"className":120005},[407],[363,120007,547],{"className":120008,"style":625},[407,408],[363,120010,120012],{"className":120011},[629],[363,120013,120015,120035],{"className":120014},[633,634],[363,120016,120018,120032],{"className":120017},[638],[363,120019,120021],{"className":120020,"style":643},[642],[363,120022,120023,120026],{"style":646},[363,120024],{"className":120025,"style":651},[650],[363,120027,120029],{"className":120028},[655,656,657,658],[363,120030,385],{"className":120031,"style":409},[407,408,658],[363,120033,666],{"className":120034},[665],[363,120036,120038],{"className":120037},[638],[363,120039,120041],{"className":120040,"style":673},[642],[363,120042],{},[363,120044,461],{"className":120045},[494],[363,120047,120049,120052],{"className":120048},[407],[363,120050,554],{"className":120051,"style":682},[407,408],[363,120053,120055],{"className":120054},[629],[363,120056,120058,120078],{"className":120057},[633,634],[363,120059,120061,120075],{"className":120060},[638],[363,120062,120064],{"className":120063,"style":727},[642],[363,120065,120066,120069],{"style":11486},[363,120067],{"className":120068,"style":651},[650],[363,120070,120072],{"className":120071},[655,656,657,658],[363,120073,945],{"className":120074,"style":9235},[407,408,658],[363,120076,666],{"className":120077},[665],[363,120079,120081],{"className":120080},[638],[363,120082,120084],{"className":120083,"style":8316},[642],[363,120085],{},[363,120087,558],{"className":120088},[407],[363,120090,120092,120095],{"className":120091},[407],[363,120093,561],{"className":120094},[407,408],[363,120096,120098],{"className":120097},[629],[363,120099,120101,120121],{"className":120100},[633,634],[363,120102,120104,120118],{"className":120103},[638],[363,120105,120107],{"className":120106,"style":727},[642],[363,120108,120109,120112],{"style":730},[363,120110],{"className":120111,"style":651},[650],[363,120113,120115],{"className":120114},[655,656,657,658],[363,120116,945],{"className":120117,"style":9235},[407,408,658],[363,120119,666],{"className":120120},[665],[363,120122,120124],{"className":120123},[638],[363,120125,120127],{"className":120126,"style":8316},[642],[363,120128],{},[363,120130,468],{"className":120131},[501],[363,120133,898],{"className":120134},[920],[11,120136,120137],{},"即「编码不会创造信息」。这为对比学习、域对齐、特征选择提供了可证伪的约束：任何声称「表示后差异更大」的方法都必然在某个环节违反了 DPI，需要解释为什么。",[2659,120139,120141],{"id":120140},"_75-域适应","7.5 域适应",[11,120143,120144,120145,120175,120176,120206],{},"在源域 ",[363,120146,120148,120162],{"className":120147,"translate":367},[366],[363,120149,120151],{"className":120150},[371],[373,120152,120153],{"xmlns":375},[377,120154,120155,120159],{},[380,120156,120157],{},[383,120158,26109],{"mathvariant":944},[387,120160,120161],{"encoding":389},"\\mathcal S",[363,120163,120165],{"className":120164,"ariaHidden":394},[393],[363,120166,120168,120171],{"className":120167},[398],[363,120169],{"className":120170,"style":1011},[402],[363,120172,26109],{"className":120173,"style":120174},[407,971],"margin-right:0.075em;"," 与目标域 ",[363,120177,120179,120193],{"className":120178,"translate":367},[366],[363,120180,120182],{"className":120181},[371],[373,120183,120184],{"xmlns":375},[377,120185,120186,120190],{},[380,120187,120188],{},[383,120189,5740],{"mathvariant":944},[387,120191,120192],{"encoding":389},"\\mathcal T",[363,120194,120196],{"className":120195,"ariaHidden":394},[393],[363,120197,120199,120202],{"className":120198},[398],[363,120200],{"className":120201,"style":1011},[402],[363,120203,5740],{"className":120204,"style":120205},[407,971],"margin-right:0.2542em;"," 上，衡量分布漂移的现代方法（DANN、ADDA）本质上用 f-散度或 IPM 对齐边缘分布；其中用 JS 的（GAN 式）对齐与用 MMD\u002FWasserstein 的对齐，区别正是 6.2 节的几何与支撑问题。",[2659,120208,120210],{"id":120209},"_76-假设检验","7.6 假设检验",[11,120212,120213],{},"f-散度是假设检验的错误指数：",[10375,120215,120216,120283],{},[7013,120217,120218,120221,120222,20338],{},[1344,120219,120220],{},"Stein 引理","：在固定第一类错误下，第二类错误的最优指数为 ",[363,120223,120225,120253],{"className":120224,"translate":367},[366],[363,120226,120228],{"className":120227},[371],[373,120229,120230],{"xmlns":375},[377,120231,120232,120250],{},[380,120233,120234,120240,120242,120244,120246,120248],{},[380,120235,120236,120238],{},[383,120237,11044],{"mathvariant":557},[383,120239,22265],{"mathvariant":557},[458,120241,461],{"stretchy":460},[383,120243,554],{},[383,120245,558],{"mathvariant":557},[383,120247,561],{},[458,120249,468],{"stretchy":460},[387,120251,120252],{"encoding":389},"\\mathrm{KL}(P\\|Q)",[363,120254,120256],{"className":120255,"ariaHidden":394},[393],[363,120257,120259,120262,120268,120271,120274,120277,120280],{"className":120258},[398],[363,120260],{"className":120261,"style":487},[402],[363,120263,120265],{"className":120264},[407],[363,120266,33035],{"className":120267},[407,33034],[363,120269,461],{"className":120270},[494],[363,120272,554],{"className":120273,"style":682},[407,408],[363,120275,558],{"className":120276},[407],[363,120278,561],{"className":120279},[407,408],[363,120281,468],{"className":120282},[501],[7013,120284,120285,9748,120288,120567],{},[1344,120286,120287],{},"Chernoff 信息",[363,120289,120291,120365],{"className":120290,"translate":367},[366],[363,120292,120294],{"className":120293},[371],[373,120295,120296],{"xmlns":375},[377,120297,120298,120362],{},[380,120299,120300,120302,120304,120306,120308,120310,120312,120314,120334,120336,120338,120340,120342,120344,120352,120354,120356,120358,120360],{},[383,120301,39666],{},[458,120303,461],{"stretchy":460},[383,120305,554],{},[458,120307,898],{"separator":394},[383,120309,561],{},[458,120311,468],{"stretchy":460},[458,120313,471],{},[543,120315,120316,120322],{},[380,120317,120318,120320],{},[383,120319,35804],{},[458,120321,4451],{},[380,120323,120324,120326,120328,120330,120332],{},[463,120325,474],{},[458,120327,7599],{},[383,120329,11689],{},[458,120331,7599],{},[463,120333,465],{},[458,120335,461],{"stretchy":460},[463,120337,465],{},[458,120339,5161],{},[383,120341,11689],{},[458,120343,468],{"stretchy":460},[7901,120345,120346,120348,120350],{},[383,120347,547],{},[383,120349,11689],{},[383,120351,1381],{},[458,120353,461],{"stretchy":460},[383,120355,554],{},[383,120357,558],{"mathvariant":557},[383,120359,561],{},[458,120361,468],{"stretchy":460},[387,120363,120364],{"encoding":389},"C(P,Q)=\\sup_{0\u003C\\lambda\u003C1}(1-\\lambda)D^{R}_{\\lambda}(P\\|Q)",[363,120366,120368,120404,120481],{"className":120367,"ariaHidden":394},[393],[363,120369,120371,120374,120377,120380,120383,120386,120389,120392,120395,120398,120401],{"className":120370},[398],[363,120372],{"className":120373,"style":487},[402],[363,120375,39666],{"className":120376,"style":11132},[407,408],[363,120378,461],{"className":120379},[494],[363,120381,554],{"className":120382,"style":682},[407,408],[363,120384,898],{"className":120385},[920],[363,120387],{"className":120388,"style":754},[505],[363,120390,561],{"className":120391},[407,408],[363,120393,468],{"className":120394},[501],[363,120396],{"className":120397,"style":506},[505],[363,120399,471],{"className":120400},[510],[363,120402],{"className":120403,"style":506},[505],[363,120405,120407,120410,120466,120469,120472,120475,120478],{"className":120406},[398],[363,120408],{"className":120409,"style":35928},[402],[363,120411,120413,120416],{"className":120412},[1832],[363,120414,35804],{"className":120415},[1832],[363,120417,120419],{"className":120418},[629],[363,120420,120422,120458],{"className":120421},[633,634],[363,120423,120425,120455],{"className":120424},[638],[363,120426,120429],{"className":120427,"style":120428},[642],"height:0.242em;",[363,120430,120431,120434],{"style":35950},[363,120432],{"className":120433,"style":651},[650],[363,120435,120437],{"className":120436},[655,656,657,658],[363,120438,120440,120443,120446,120449,120452],{"className":120439},[407,658],[363,120441,474],{"className":120442},[407,658],[363,120444,7599],{"className":120445},[510,658],[363,120447,11689],{"className":120448},[407,408,658],[363,120450,7599],{"className":120451},[510,658],[363,120453,465],{"className":120454},[407,658],[363,120456,666],{"className":120457},[665],[363,120459,120461],{"className":120460},[638],[363,120462,120464],{"className":120463,"style":35978},[642],[363,120465],{},[363,120467,461],{"className":120468},[494],[363,120470,465],{"className":120471},[407],[363,120473],{"className":120474,"style":2935},[505],[363,120476,5161],{"className":120477},[2939],[363,120479],{"className":120480,"style":2935},[505],[363,120482,120484,120488,120491,120494,120552,120555,120558,120561,120564],{"className":120483},[398],[363,120485],{"className":120486,"style":120487},[402],"height:1.1244em;vertical-align:-0.2831em;",[363,120489,11689],{"className":120490},[407,408],[363,120492,468],{"className":120493},[501],[363,120495,120497,120500],{"className":120496},[407],[363,120498,547],{"className":120499,"style":625},[407,408],[363,120501,120503],{"className":120502},[629],[363,120504,120506,120544],{"className":120505},[633,634],[363,120507,120509,120541],{"className":120508},[638],[363,120510,120512,120527],{"className":120511,"style":52457},[642],[363,120513,120515,120518],{"style":120514},"top:-2.4169em;margin-left:-0.0278em;margin-right:0.05em;",[363,120516],{"className":120517,"style":651},[650],[363,120519,120521],{"className":120520},[655,656,657,658],[363,120522,120524],{"className":120523},[407,658],[363,120525,11689],{"className":120526},[407,408,658],[363,120528,120529,120532],{"style":7213},[363,120530],{"className":120531,"style":651},[650],[363,120533,120535],{"className":120534},[655,656,657,658],[363,120536,120538],{"className":120537},[407,658],[363,120539,1381],{"className":120540,"style":49971},[407,408,658],[363,120542,666],{"className":120543},[665],[363,120545,120547],{"className":120546},[638],[363,120548,120550],{"className":120549,"style":8895},[642],[363,120551],{},[363,120553,461],{"className":120554},[494],[363,120556,554],{"className":120557,"style":682},[407,408],[363,120559,558],{"className":120560},[407],[363,120562,561],{"className":120563},[407,408],[363,120565,468],{"className":120566},[501],"，即 Rényi\u002Fα-族中的最优点；对称检验的指数由它决定。",[11,120569,120570],{},"因此 f-散度不只是「训练目标」，也是统计判决的极限性能。",[18,120572,120574],{"id":120573},"_8-差分隐私f-dp","8. 差分隐私（f-DP）",[11,120576,120577,120578,120581,120582,6842,120585,120613,120614,120655],{},"一个较新的应用是把 f-散度的语言搬到隐私保护。差分隐私（DP）要求相邻数据集的输出分布「难以区分」。Dong–Roth–Su 提出 ",[1344,120579,120580],{},"f-差分隐私（f-DP）","：用一个",[1344,120583,120584],{},"权衡函数（tradeoff function）",[363,120586,120588,120601],{"className":120587,"translate":367},[366],[363,120589,120591],{"className":120590},[371],[373,120592,120593],{"xmlns":375},[377,120594,120595,120599],{},[380,120596,120597],{},[383,120598,385],{},[387,120600,385],{"encoding":389},[363,120602,120604],{"className":120603,"ariaHidden":394},[393],[363,120605,120607,120610],{"className":120606},[398],[363,120608],{"className":120609,"style":403},[402],[363,120611,385],{"className":120612,"style":409},[407,408]," 刻画两个输出分布 ",[363,120615,120617,120634],{"className":120616,"translate":367},[366],[363,120618,120620],{"className":120619},[371],[373,120621,120622],{"xmlns":375},[377,120623,120624,120632],{},[380,120625,120626,120628,120630],{},[383,120627,554],{},[458,120629,898],{"separator":394},[383,120631,561],{},[387,120633,903],{"encoding":389},[363,120635,120637],{"className":120636,"ariaHidden":394},[393],[363,120638,120640,120643,120646,120649,120652],{"className":120639},[398],[363,120641],{"className":120642,"style":913},[402],[363,120644,554],{"className":120645,"style":682},[407,408],[363,120647,898],{"className":120648},[920],[363,120650],{"className":120651,"style":754},[505],[363,120653,561],{"className":120654},[407,408]," 上「漏报率—误报率」的权衡曲线：",[363,120657,120659],{"className":120658,"translate":367},[528],[363,120660,120662,120759],{"className":120661,"translate":367},[366],[363,120663,120665],{"className":120664},[371],[373,120666,120667],{"xmlns":375,"display":537},[377,120668,120669,120756],{},[380,120670,120671,120673,120675,120677,120679,120681,120683,120685,120687,120689,120691,120702,120704,120706,120708,120710,120712,120714,120716,120718,120720,120722,120724,120726,120728,120730,120732,120734,120736,120738,120740,120742,120744,120746,120748,120750,120752,120754],{},[383,120672,5740],{},[458,120674,461],{"stretchy":460},[383,120676,554],{},[458,120678,898],{"separator":394},[383,120680,561],{},[458,120682,468],{"stretchy":460},[458,120684,461],{"stretchy":460},[383,120686,45126],{},[458,120688,468],{"stretchy":460},[458,120690,471],{},[6364,120692,120693,120700],{},[380,120694,120695,120698],{},[383,120696,120697],{},"inf",[458,120699,4451],{},[383,120701,950],{},[458,120703,3246],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[383,120705,554],{},[458,120707,461],{"stretchy":460},[383,120709,950],{},[458,120711,468],{"stretchy":460},[1645,120713,4631],{},[458,120715,1364],{},[1645,120717,4631],{},[383,120719,561],{},[458,120721,461],{"stretchy":460},[383,120723,950],{},[458,120725,468],{"stretchy":460},[458,120727,7276],{},[463,120729,465],{},[458,120731,5161],{},[383,120733,45126],{},[458,120735,3256],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[458,120737,898],{"separator":394},[505,120739],{"width":3610},[383,120741,45126],{},[458,120743,6257],{},[458,120745,578],{"stretchy":460},[463,120747,474],{},[458,120749,898],{"separator":394},[463,120751,465],{},[458,120753,605],{"stretchy":460},[383,120755,4723],{"mathvariant":557},[387,120757,120758],{"encoding":389},"T(P,Q)(\\alpha)\n=\\inf_{A}\\big\\{P(A)\\ :\\ Q(A)\\ge1-\\alpha\\big\\},\n\\qquad \\alpha\\in[0,1] .",[363,120760,120762,120807,120902,120929,120947,120983],{"className":120761,"ariaHidden":394},[393],[363,120763,120765,120768,120771,120774,120777,120780,120783,120786,120789,120792,120795,120798,120801,120804],{"className":120764},[398],[363,120766],{"className":120767,"style":487},[402],[363,120769,5740],{"className":120770,"style":682},[407,408],[363,120772,461],{"className":120773},[494],[363,120775,554],{"className":120776,"style":682},[407,408],[363,120778,898],{"className":120779},[920],[363,120781],{"className":120782,"style":754},[505],[363,120784,561],{"className":120785},[407,408],[363,120787,468],{"className":120788},[501],[363,120790,461],{"className":120791},[494],[363,120793,45126],{"className":120794,"style":45156},[407,408],[363,120796,468],{"className":120797},[501],[363,120799],{"className":120800,"style":506},[505],[363,120802,471],{"className":120803},[510],[363,120805],{"className":120806,"style":506},[505],[363,120808,120810,120814,120866,120869,120875,120878,120881,120884,120887,120890,120893,120896,120899],{"className":120809},[398],[363,120811],{"className":120812,"style":120813},[402],"height:1.5943em;vertical-align:-0.7443em;",[363,120815,120817],{"className":120816},[1832,14066],[363,120818,120820,120857],{"className":120819},[633,634],[363,120821,120823,120854],{"className":120822},[638],[363,120824,120826,120841],{"className":120825,"style":12947},[642],[363,120827,120829,120832],{"style":120828},"top:-2.3557em;margin-left:0em;",[363,120830],{"className":120831,"style":809},[650],[363,120833,120835],{"className":120834},[655,656,657,658],[363,120836,120838],{"className":120837},[407,658],[363,120839,950],{"className":120840},[407,408,658],[363,120842,120843,120846],{"style":12213},[363,120844],{"className":120845,"style":809},[650],[363,120847,120848],{},[363,120849,120851,120852],{"className":120850},[1832],"in",[363,120853,385],{"style":95223},[363,120855,666],{"className":120856},[665],[363,120858,120860],{"className":120859},[638],[363,120861,120864],{"className":120862,"style":120863},[642],"height:0.7443em;",[363,120865],{},[363,120867],{"className":120868,"style":754},[505],[363,120870,120872],{"className":120871},[407],[363,120873,3246],{"className":120874},[767,10022],[363,120876,554],{"className":120877,"style":682},[407,408],[363,120879,461],{"className":120880},[494],[363,120882,950],{"className":120883},[407,408],[363,120885,468],{"className":120886},[501],[363,120888,4631],{"className":120889},[505],[363,120891],{"className":120892,"style":506},[505],[363,120894,1364],{"className":120895},[510],[363,120897,4631],{"className":120898},[505],[363,120900],{"className":120901,"style":506},[505],[363,120903,120905,120908,120911,120914,120917,120920,120923,120926],{"className":120904},[398],[363,120906],{"className":120907,"style":487},[402],[363,120909,561],{"className":120910},[407,408],[363,120912,461],{"className":120913},[494],[363,120915,950],{"className":120916},[407,408],[363,120918,468],{"className":120919},[501],[363,120921],{"className":120922,"style":506},[505],[363,120924,7276],{"className":120925},[510],[363,120927],{"className":120928,"style":506},[505],[363,120930,120932,120935,120938,120941,120944],{"className":120931},[398],[363,120933],{"className":120934,"style":12928},[402],[363,120936,465],{"className":120937},[407],[363,120939],{"className":120940,"style":2935},[505],[363,120942,5161],{"className":120943},[2939],[363,120945],{"className":120946,"style":2935},[505],[363,120948,120950,120953,120956,120962,120965,120968,120971,120974,120977,120980],{"className":120949},[398],[363,120951],{"className":120952,"style":20935},[402],[363,120954,45126],{"className":120955,"style":45156},[407,408],[363,120957,120959],{"className":120958},[407],[363,120960,3256],{"className":120961},[767,10022],[363,120963,898],{"className":120964},[920],[363,120966],{"className":120967,"style":3982},[505],[363,120969],{"className":120970,"style":754},[505],[363,120972,45126],{"className":120973,"style":45156},[407,408],[363,120975],{"className":120976,"style":506},[505],[363,120978,6257],{"className":120979},[510],[363,120981],{"className":120982,"style":506},[505],[363,120984,120986,120989,120992,120995,120998,121001,121004,121007],{"className":120985},[398],[363,120987],{"className":120988,"style":487},[402],[363,120990,578],{"className":120991},[494],[363,120993,474],{"className":120994},[407],[363,120996,898],{"className":120997},[920],[363,120999],{"className":121000,"style":754},[505],[363,121002,465],{"className":121003},[407],[363,121005,605],{"className":121006},[501],[363,121008,4723],{"className":121009},[407],[11,121011,121012,121013,78453,121041,121069,121070,121145,121146,121285],{},"机制 ",[363,121014,121016,121029],{"className":121015,"translate":367},[366],[363,121017,121019],{"className":121018},[371],[373,121020,121021],{"xmlns":375},[377,121022,121023,121027],{},[380,121024,121025],{},[383,121026,25939],{},[387,121028,25939],{"encoding":389},[363,121030,121032],{"className":121031,"ariaHidden":394},[393],[363,121033,121035,121038],{"className":121034},[398],[363,121036],{"className":121037,"style":1011},[402],[363,121039,25939],{"className":121040,"style":25973},[407,408],[363,121042,121044,121057],{"className":121043,"translate":367},[366],[363,121045,121047],{"className":121046},[371],[373,121048,121049],{"xmlns":375},[377,121050,121051,121055],{},[380,121052,121053],{},[383,121054,385],{},[387,121056,385],{"encoding":389},[363,121058,121060],{"className":121059,"ariaHidden":394},[393],[363,121061,121063,121066],{"className":121062},[398],[363,121064],{"className":121065,"style":403},[402],[363,121067,385],{"className":121068,"style":409},[407,408],"-DP，若对所有相邻 ",[363,121071,121073,121095],{"className":121072,"translate":367},[366],[363,121074,121076],{"className":121075},[371],[373,121077,121078],{"xmlns":375},[377,121079,121080,121092],{},[380,121081,121082,121084,121086],{},[383,121083,547],{},[458,121085,898],{"separator":394},[7162,121087,121088,121090],{},[383,121089,547],{},[458,121091,7168],{"mathvariant":557,"lspace":1595,"rspace":1595},[387,121093,121094],{"encoding":389},"D,D'",[363,121096,121098],{"className":121097,"ariaHidden":394},[393],[363,121099,121101,121104,121107,121110,121113],{"className":121100},[398],[363,121102],{"className":121103,"style":64544},[402],[363,121105,547],{"className":121106,"style":625},[407,408],[363,121108,898],{"className":121109},[920],[363,121111],{"className":121112,"style":754},[505],[363,121114,121116,121119],{"className":121115},[407],[363,121117,547],{"className":121118,"style":625},[407,408],[363,121120,121122],{"className":121121},[629],[363,121123,121125],{"className":121124},[633],[363,121126,121128],{"className":121127},[638],[363,121129,121131],{"className":121130,"style":7210},[642],[363,121132,121133,121136],{"style":7213},[363,121134],{"className":121135,"style":651},[650],[363,121137,121139],{"className":121138},[655,656,657,658],[363,121140,121142],{"className":121141},[407,658],[363,121143,7168],{"className":121144},[407,658],"，恒有 ",[363,121147,121149,121193],{"className":121148,"translate":367},[366],[363,121150,121152],{"className":121151},[371],[373,121153,121154],{"xmlns":375},[377,121155,121156,121190],{},[380,121157,121158,121160,121162,121164,121166,121168,121170,121172,121174,121176,121182,121184,121186,121188],{},[383,121159,5740],{},[458,121161,461],{"stretchy":460},[383,121163,25939],{},[458,121165,461],{"stretchy":460},[383,121167,547],{},[458,121169,468],{"stretchy":460},[458,121171,898],{"separator":394},[383,121173,25939],{},[458,121175,461],{"stretchy":460},[7162,121177,121178,121180],{},[383,121179,547],{},[458,121181,7168],{"mathvariant":557,"lspace":1595,"rspace":1595},[458,121183,468],{"stretchy":460},[458,121185,468],{"stretchy":460},[458,121187,7276],{},[383,121189,385],{},[387,121191,121192],{"encoding":389},"T(M(D),M(D'))\\ge f",[363,121194,121196,121276],{"className":121195,"ariaHidden":394},[393],[363,121197,121199,121202,121205,121208,121211,121214,121217,121220,121223,121226,121229,121232,121264,121267,121270,121273],{"className":121198},[398],[363,121200],{"className":121201,"style":7191},[402],[363,121203,5740],{"className":121204,"style":682},[407,408],[363,121206,461],{"className":121207},[494],[363,121209,25939],{"className":121210,"style":25973},[407,408],[363,121212,461],{"className":121213},[494],[363,121215,547],{"className":121216,"style":625},[407,408],[363,121218,468],{"className":121219},[501],[363,121221,898],{"className":121222},[920],[363,121224],{"className":121225,"style":754},[505],[363,121227,25939],{"className":121228,"style":25973},[407,408],[363,121230,461],{"className":121231},[494],[363,121233,121235,121238],{"className":121234},[407],[363,121236,547],{"className":121237,"style":625},[407,408],[363,121239,121241],{"className":121240},[629],[363,121242,121244],{"className":121243},[633],[363,121245,121247],{"className":121246},[638],[363,121248,121250],{"className":121249,"style":7210},[642],[363,121251,121252,121255],{"style":7213},[363,121253],{"className":121254,"style":651},[650],[363,121256,121258],{"className":121257},[655,656,657,658],[363,121259,121261],{"className":121260},[407,658],[363,121262,7168],{"className":121263},[407,658],[363,121265,19419],{"className":121266},[501],[363,121268],{"className":121269,"style":506},[505],[363,121271,7276],{"className":121272},[510],[363,121274],{"className":121275,"style":506},[505],[363,121277,121279,121282],{"className":121278},[398],[363,121280],{"className":121281,"style":403},[402],[363,121283,385],{"className":121284,"style":409},[407,408],"。可以证明：",[10375,121287,121288],{},[7013,121289,121290,32461,121293,121365,121366,121414,121415,121443],{},[1344,121291,121292],{},"权衡曲线与 f-散度互相决定。",[363,121294,121296,121326],{"className":121295,"translate":367},[366],[363,121297,121299],{"className":121298},[371],[373,121300,121301],{"xmlns":375},[377,121302,121303,121323],{},[380,121304,121305,121307,121309,121311,121313,121315,121317,121319,121321],{},[383,121306,5740],{},[458,121308,461],{"stretchy":460},[383,121310,554],{},[458,121312,898],{"separator":394},[383,121314,561],{},[458,121316,468],{"stretchy":460},[458,121318,461],{"stretchy":460},[383,121320,45126],{},[458,121322,468],{"stretchy":460},[387,121324,121325],{"encoding":389},"T(P,Q)(\\alpha)",[363,121327,121329],{"className":121328,"ariaHidden":394},[393],[363,121330,121332,121335,121338,121341,121344,121347,121350,121353,121356,121359,121362],{"className":121331},[398],[363,121333],{"className":121334,"style":487},[402],[363,121336,5740],{"className":121337,"style":682},[407,408],[363,121339,461],{"className":121340},[494],[363,121342,554],{"className":121343,"style":682},[407,408],[363,121345,898],{"className":121346},[920],[363,121348],{"className":121349,"style":754},[505],[363,121351,561],{"className":121352},[407,408],[363,121354,468],{"className":121355},[501],[363,121357,461],{"className":121358},[494],[363,121360,45126],{"className":121361,"style":45156},[407,408],[363,121363,468],{"className":121364},[501]," 只依赖似然比 ",[363,121367,121369,121390],{"className":121368,"translate":367},[366],[363,121370,121372],{"className":121371},[371],[373,121373,121374],{"xmlns":375},[377,121375,121376,121388],{},[380,121377,121378,121380,121382,121384,121386],{},[383,121379,592],{},[383,121381,554],{},[383,121383,1288],{"mathvariant":557},[383,121385,592],{},[383,121387,561],{},[387,121389,115032],{"encoding":389},[363,121391,121393],{"className":121392,"ariaHidden":394},[393],[363,121394,121396,121399,121402,121405,121408,121411],{"className":121395},[398],[363,121397],{"className":121398,"style":487},[402],[363,121400,592],{"className":121401},[407,408],[363,121403,554],{"className":121404,"style":682},[407,408],[363,121406,1288],{"className":121407},[407],[363,121409,592],{"className":121410},[407,408],[363,121412,561],{"className":121413},[407,408]," 的分布，因此 ",[363,121416,121418,121431],{"className":121417,"translate":367},[366],[363,121419,121421],{"className":121420},[371],[373,121422,121423],{"xmlns":375},[377,121424,121425,121429],{},[380,121426,121427],{},[383,121428,5740],{},[387,121430,5740],{"encoding":389},[363,121432,121434],{"className":121433,"ariaHidden":394},[393],[363,121435,121437,121440],{"className":121436},[398],[363,121438],{"className":121439,"style":1011},[402],[363,121441,5740],{"className":121442,"style":682},[407,408]," 与\"全体 f-散度\"携带同样的信息。对 hockey-stick 生成元族",[363,121445,121447],{"className":121446,"translate":367},[528],[363,121448,121450,121517],{"className":121449,"translate":367},[366],[363,121451,121453],{"className":121452},[371],[373,121454,121455],{"xmlns":375,"display":537},[377,121456,121457,121514],{},[380,121458,121459,121466,121468,121470,121472,121474,121476,121478,121480,121482,121484,121486,121488,121490,121492,121494,121496,121498,121500,121502,121504,121506,121508,121510,121512],{},[543,121460,121461,121463],{},[383,121462,4506],{},[383,121464,121465],{},"γ",[458,121467,461],{"stretchy":460},[383,121469,1116],{},[458,121471,468],{"stretchy":460},[458,121473,471],{},[383,121475,52759],{},[458,121477,4451],{},[458,121479,3246],{"stretchy":460},[463,121481,474],{},[458,121483,898],{"separator":394},[1645,121485,1647],{},[383,121487,1116],{},[458,121489,5161],{},[383,121491,121465],{},[458,121493,3256],{"stretchy":460},[458,121495,898],{"separator":394},[505,121497],{"width":3610},[383,121499,121465],{},[458,121501,6257],{},[458,121503,578],{"stretchy":460},[463,121505,474],{},[458,121507,898],{"separator":394},[463,121509,465],{},[458,121511,605],{"stretchy":460},[458,121513,898],{"separator":394},[387,121515,121516],{"encoding":389},"g_\\gamma(t)=\\max\\{0,\\,t-\\gamma\\},\n\\qquad \\gamma\\in[0,1],",[363,121518,121520,121585,121621,121654],{"className":121519,"ariaHidden":394},[393],[363,121521,121523,121526,121567,121570,121573,121576,121579,121582],{"className":121522},[398],[363,121524],{"className":121525,"style":618},[402],[363,121527,121529,121532],{"className":121528},[407],[363,121530,4506],{"className":121531,"style":1328},[407,408],[363,121533,121535],{"className":121534},[629],[363,121536,121538,121559],{"className":121537},[633,634],[363,121539,121541,121556],{"className":121540},[638],[363,121542,121544],{"className":121543,"style":45301},[642],[363,121545,121546,121549],{"style":19215},[363,121547],{"className":121548,"style":651},[650],[363,121550,121552],{"className":121551},[655,656,657,658],[363,121553,121465],{"className":121554,"style":121555},[407,408,658],"margin-right:0.0556em;",[363,121557,666],{"className":121558},[665],[363,121560,121562],{"className":121561},[638],[363,121563,121565],{"className":121564,"style":673},[642],[363,121566],{},[363,121568,461],{"className":121569},[494],[363,121571,1116],{"className":121572},[407,408],[363,121574,468],{"className":121575},[501],[363,121577],{"className":121578,"style":506},[505],[363,121580,471],{"className":121581},[510],[363,121583],{"className":121584,"style":506},[505],[363,121586,121588,121591,121594,121597,121600,121603,121606,121609,121612,121615,121618],{"className":121587},[398],[363,121589],{"className":121590,"style":487},[402],[363,121592,52759],{"className":121593},[1832],[363,121595,3246],{"className":121596},[494],[363,121598,474],{"className":121599},[407],[363,121601,898],{"className":121602},[920],[363,121604],{"className":121605,"style":754},[505],[363,121607],{"className":121608,"style":754},[505],[363,121610,1116],{"className":121611},[407,408],[363,121613],{"className":121614,"style":2935},[505],[363,121616,5161],{"className":121617},[2939],[363,121619],{"className":121620,"style":2935},[505],[363,121622,121624,121627,121630,121633,121636,121639,121642,121645,121648,121651],{"className":121623},[398],[363,121625],{"className":121626,"style":487},[402],[363,121628,121465],{"className":121629,"style":121555},[407,408],[363,121631,3256],{"className":121632},[501],[363,121634,898],{"className":121635},[920],[363,121637],{"className":121638,"style":3982},[505],[363,121640],{"className":121641,"style":754},[505],[363,121643,121465],{"className":121644,"style":121555},[407,408],[363,121646],{"className":121647,"style":506},[505],[363,121649,6257],{"className":121650},[510],[363,121652],{"className":121653,"style":506},[505],[363,121655,121657,121660,121663,121666,121669,121672,121675,121678],{"className":121656},[398],[363,121658],{"className":121659,"style":487},[402],[363,121661,578],{"className":121662},[494],[363,121664,474],{"className":121665},[407],[363,121667,898],{"className":121668},[920],[363,121670],{"className":121671,"style":754},[505],[363,121673,465],{"className":121674},[407],[363,121676,605],{"className":121677},[501],[363,121679,898],{"className":121680},[920],[11,121682,121683,121684,121865,121866,9015,121895,121946],{},"（Dong–Roth–Su 的 ",[363,121685,121687,121735],{"className":121686,"translate":367},[366],[363,121688,121690],{"className":121689},[371],[373,121691,121692],{"xmlns":375},[377,121693,121694,121732],{},[380,121695,121696,121702,121704,121706,121708,121710,121712,121714,121716,121718,121720,121722,121724,121726,121728,121730],{},[543,121697,121698,121700],{},[383,121699,39037],{},[383,121701,121465],{},[458,121703,461],{"stretchy":460},[383,121705,1116],{},[458,121707,468],{"stretchy":460},[458,121709,471],{},[383,121711,52759],{},[458,121713,4451],{},[458,121715,3246],{"stretchy":460},[463,121717,474],{},[458,121719,898],{"separator":394},[383,121721,1116],{},[458,121723,5161],{},[463,121725,465],{},[458,121727,2818],{},[383,121729,121465],{},[458,121731,3256],{"stretchy":460},[387,121733,121734],{"encoding":389},"h_\\gamma(t)=\\max\\{0,t-1+\\gamma\\}",[363,121736,121738,121802,121835,121853],{"className":121737,"ariaHidden":394},[393],[363,121739,121741,121744,121784,121787,121790,121793,121796,121799],{"className":121740},[398],[363,121742],{"className":121743,"style":618},[402],[363,121745,121747,121750],{"className":121746},[407],[363,121748,39037],{"className":121749},[407,408],[363,121751,121753],{"className":121752},[629],[363,121754,121756,121776],{"className":121755},[633,634],[363,121757,121759,121773],{"className":121758},[638],[363,121760,121762],{"className":121761,"style":45301},[642],[363,121763,121764,121767],{"style":730},[363,121765],{"className":121766,"style":651},[650],[363,121768,121770],{"className":121769},[655,656,657,658],[363,121771,121465],{"className":121772,"style":121555},[407,408,658],[363,121774,666],{"className":121775},[665],[363,121777,121779],{"className":121778},[638],[363,121780,121782],{"className":121781,"style":673},[642],[363,121783],{},[363,121785,461],{"className":121786},[494],[363,121788,1116],{"className":121789},[407,408],[363,121791,468],{"className":121792},[501],[363,121794],{"className":121795,"style":506},[505],[363,121797,471],{"className":121798},[510],[363,121800],{"className":121801,"style":506},[505],[363,121803,121805,121808,121811,121814,121817,121820,121823,121826,121829,121832],{"className":121804},[398],[363,121806],{"className":121807,"style":487},[402],[363,121809,52759],{"className":121810},[1832],[363,121812,3246],{"className":121813},[494],[363,121815,474],{"className":121816},[407],[363,121818,898],{"className":121819},[920],[363,121821],{"className":121822,"style":754},[505],[363,121824,1116],{"className":121825},[407,408],[363,121827],{"className":121828,"style":2935},[505],[363,121830,5161],{"className":121831},[2939],[363,121833],{"className":121834,"style":2935},[505],[363,121836,121838,121841,121844,121847,121850],{"className":121837},[398],[363,121839],{"className":121840,"style":12928},[402],[363,121842,465],{"className":121843},[407],[363,121845],{"className":121846,"style":2935},[505],[363,121848,2818],{"className":121849},[2939],[363,121851],{"className":121852,"style":2935},[505],[363,121854,121856,121859,121862],{"className":121855},[398],[363,121857],{"className":121858,"style":487},[402],[363,121860,121465],{"className":121861,"style":121555},[407,408],[363,121863,3256],{"className":121864},[501]," 就是把 ",[363,121867,121869,121883],{"className":121868,"translate":367},[366],[363,121870,121872],{"className":121871},[371],[373,121873,121874],{"xmlns":375},[377,121875,121876,121880],{},[380,121877,121878],{},[383,121879,121465],{},[387,121881,121882],{"encoding":389},"\\gamma",[363,121884,121886],{"className":121885,"ariaHidden":394},[393],[363,121887,121889,121892],{"className":121888},[398],[363,121890],{"className":121891,"style":2624},[402],[363,121893,121465],{"className":121894,"style":121555},[407,408],[363,121896,121898,121916],{"className":121897,"translate":367},[366],[363,121899,121901],{"className":121900},[371],[373,121902,121903],{"xmlns":375},[377,121904,121905,121913],{},[380,121906,121907,121909,121911],{},[463,121908,465],{},[458,121910,5161],{},[383,121912,121465],{},[387,121914,121915],{"encoding":389},"1-\\gamma",[363,121917,121919,121937],{"className":121918,"ariaHidden":394},[393],[363,121920,121922,121925,121928,121931,121934],{"className":121921},[398],[363,121923],{"className":121924,"style":12928},[402],[363,121926,465],{"className":121927},[407],[363,121929],{"className":121930,"style":2935},[505],[363,121932,5161],{"className":121933},[2939],[363,121935],{"className":121936,"style":2935},[505],[363,121938,121940,121943],{"className":121939},[398],[363,121941],{"className":121942,"style":2624},[402],[363,121944,121465],{"className":121945,"style":121555},[407,408]," 的同一族），变分表示给出干净的检验集形式：",[363,121948,121950],{"className":121949,"translate":367},[528],[363,121951,121953,122031],{"className":121952,"translate":367},[366],[363,121954,121956],{"className":121955},[371],[373,121957,121958],{"xmlns":375,"display":537},[377,121959,121960,122028],{},[380,121961,121962,121972,121974,121976,121978,121980,121982,121984,122000,122002,122004,122006,122008,122010,122012,122014,122016,122018,122020,122022,122024,122026],{},[543,121963,121964,121966],{},[383,121965,547],{},[543,121967,121968,121970],{},[383,121969,4506],{},[383,121971,121465],{},[458,121973,461],{"stretchy":460},[383,121975,554],{},[383,121977,558],{"mathvariant":557},[383,121979,561],{},[458,121981,468],{"stretchy":460},[458,121983,471],{},[6364,121985,121986,121992],{},[380,121987,121988,121990],{},[383,121989,35804],{},[458,121991,4451],{},[380,121993,121994,121996,121998],{},[383,121995,950],{},[458,121997,6257],{},[383,121999,950],{"mathvariant":944},[458,122001,3246],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[383,122003,554],{},[458,122005,461],{"stretchy":460},[383,122007,950],{},[458,122009,468],{"stretchy":460},[458,122011,5161],{},[383,122013,121465],{},[1645,122015,1647],{},[383,122017,561],{},[458,122019,461],{"stretchy":460},[383,122021,950],{},[458,122023,468],{"stretchy":460},[458,122025,3256],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[383,122027,4723],{"mathvariant":557},[387,122029,122030],{"encoding":389},"D_{g_\\gamma}(P\\|Q)\n=\\sup_{A\\in\\mathcal A}\\big\\{P(A)\n-\\gamma\\,Q(A)\\big\\}.",[363,122032,122034,122148,122237],{"className":122033,"ariaHidden":394},[393],[363,122035,122037,122041,122124,122127,122130,122133,122136,122139,122142,122145],{"className":122036},[398],[363,122038],{"className":122039,"style":122040},[402],"height:1.0973em;vertical-align:-0.3473em;",[363,122042,122044,122047],{"className":122043},[407],[363,122045,547],{"className":122046,"style":625},[407,408],[363,122048,122050],{"className":122049},[629],[363,122051,122053,122115],{"className":122052},[633,634],[363,122054,122056,122112],{"className":122055},[638],[363,122057,122059],{"className":122058,"style":45301},[642],[363,122060,122061,122064],{"style":646},[363,122062],{"className":122063,"style":651},[650],[363,122065,122067],{"className":122066},[655,656,657,658],[363,122068,122070],{"className":122069},[407,658],[363,122071,122073,122076],{"className":122072},[407,658],[363,122074,4506],{"className":122075,"style":1328},[407,408,658],[363,122077,122079],{"className":122078},[629],[363,122080,122082,122103],{"className":122081},[633,634],[363,122083,122085,122100],{"className":122084},[638],[363,122086,122088],{"className":122087,"style":74645},[642],[363,122089,122091,122094],{"style":122090},"top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;",[363,122092],{"className":122093,"style":30578},[650],[363,122095,122097],{"className":122096},[655,30582,10022,658],[363,122098,121465],{"className":122099,"style":121555},[407,408,658],[363,122101,666],{"className":122102},[665],[363,122104,122106],{"className":122105},[638],[363,122107,122110],{"className":122108,"style":122109},[642],"height:0.2819em;",[363,122111],{},[363,122113,666],{"className":122114},[665],[363,122116,122118],{"className":122117},[638],[363,122119,122122],{"className":122120,"style":122121},[642],"height:0.3473em;",[363,122123],{},[363,122125,461],{"className":122126},[494],[363,122128,554],{"className":122129,"style":682},[407,408],[363,122131,558],{"className":122132},[407],[363,122134,561],{"className":122135},[407,408],[363,122137,468],{"className":122138},[501],[363,122140],{"className":122141,"style":506},[505],[363,122143,471],{"className":122144},[510],[363,122146],{"className":122147,"style":506},[505],[363,122149,122151,122154,122207,122210,122216,122219,122222,122225,122228,122231,122234],{"className":122150},[398],[363,122152],{"className":122153,"style":36725},[402],[363,122155,122157],{"className":122156},[1832,14066],[363,122158,122160,122199],{"className":122159},[633,634],[363,122161,122163,122196],{"className":122162},[638],[363,122164,122166,122186],{"className":122165,"style":2693},[642],[363,122167,122168,122171],{"style":36740},[363,122169],{"className":122170,"style":809},[650],[363,122172,122174],{"className":122173},[655,656,657,658],[363,122175,122177,122180,122183],{"className":122176},[407,658],[363,122178,950],{"className":122179},[407,408,658],[363,122181,6257],{"className":122182},[510,658],[363,122184,950],{"className":122185},[407,971,658],[363,122187,122188,122191],{"style":12213},[363,122189],{"className":122190,"style":809},[650],[363,122192,122193],{},[363,122194,35804],{"className":122195},[1832],[363,122197,666],{"className":122198},[665],[363,122200,122202],{"className":122201},[638],[363,122203,122205],{"className":122204,"style":36778},[642],[363,122206],{},[363,122208],{"className":122209,"style":754},[505],[363,122211,122213],{"className":122212},[407],[363,122214,3246],{"className":122215},[767,10022],[363,122217,554],{"className":122218,"style":682},[407,408],[363,122220,461],{"className":122221},[494],[363,122223,950],{"className":122224},[407,408],[363,122226,468],{"className":122227},[501],[363,122229],{"className":122230,"style":2935},[505],[363,122232,5161],{"className":122233},[2939],[363,122235],{"className":122236,"style":2935},[505],[363,122238,122240,122243,122246,122249,122252,122255,122258,122261,122267],{"className":122239},[398],[363,122241],{"className":122242,"style":20935},[402],[363,122244,121465],{"className":122245,"style":121555},[407,408],[363,122247],{"className":122248,"style":754},[505],[363,122250,561],{"className":122251},[407,408],[363,122253,461],{"className":122254},[494],[363,122256,950],{"className":122257},[407,408],[363,122259,468],{"className":122260},[501],[363,122262,122264],{"className":122263},[407],[363,122265,3256],{"className":122266},[767,10022],[363,122268,4723],{"className":122269},[407],[11,122271,122272,122274,122275,122349,122350,2628,122519,122547,122548,122641],{},[1344,122273,107668],{}," 一方面，任意 ",[363,122276,122278,122300],{"className":122277,"translate":367},[366],[363,122279,122281],{"className":122280},[371],[373,122282,122283],{"xmlns":375},[377,122284,122285,122297],{},[380,122286,122287,122289,122291,122293,122295],{},[463,122288,474],{},[458,122290,10300],{},[383,122292,4506],{},[458,122294,10300],{},[463,122296,465],{},[387,122298,122299],{"encoding":389},"0\\le g\\le1",[363,122301,122303,122322,122340],{"className":122302,"ariaHidden":394},[393],[363,122304,122306,122310,122313,122316,122319],{"className":122305},[398],[363,122307],{"className":122308,"style":122309},[402],"height:0.7804em;vertical-align:-0.136em;",[363,122311,474],{"className":122312},[407],[363,122314],{"className":122315,"style":506},[505],[363,122317,10300],{"className":122318},[510],[363,122320],{"className":122321,"style":506},[505],[363,122323,122325,122328,122331,122334,122337],{"className":122324},[398],[363,122326],{"className":122327,"style":15767},[402],[363,122329,4506],{"className":122330,"style":1328},[407,408],[363,122332],{"className":122333,"style":506},[505],[363,122335,10300],{"className":122336},[510],[363,122338],{"className":122339,"style":506},[505],[363,122341,122343,122346],{"className":122342},[398],[363,122344],{"className":122345,"style":520},[402],[363,122347,465],{"className":122348},[407]," 可写为指示函数的线性组合（层叠公式），而 ",[363,122351,122353,122393],{"className":122352,"translate":367},[366],[363,122354,122356],{"className":122355},[371],[373,122357,122358],{"xmlns":375},[377,122359,122360,122390],{},[380,122361,122362,122368,122370,122372,122374,122376,122378,122384,122386,122388],{},[543,122363,122364,122366],{},[383,122365,571],{},[383,122367,554],{},[458,122369,578],{"stretchy":460},[383,122371,4506],{},[458,122373,605],{"stretchy":460},[458,122375,5161],{},[383,122377,121465],{},[543,122379,122380,122382],{},[383,122381,571],{},[383,122383,561],{},[458,122385,578],{"stretchy":460},[383,122387,4506],{},[458,122389,605],{"stretchy":460},[387,122391,122392],{"encoding":389},"E_P[g]-\\gamma E_Q[g]",[363,122394,122396,122461],{"className":122395,"ariaHidden":394},[393],[363,122397,122399,122402,122443,122446,122449,122452,122455,122458],{"className":122398},[398],[363,122400],{"className":122401,"style":487},[402],[363,122403,122405,122408],{"className":122404},[407],[363,122406,571],{"className":122407,"style":26233},[407,408],[363,122409,122411],{"className":122410},[629],[363,122412,122414,122435],{"className":122413},[633,634],[363,122415,122417,122432],{"className":122416},[638],[363,122418,122420],{"className":122419,"style":727},[642],[363,122421,122423,122426],{"style":122422},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[363,122424],{"className":122425,"style":651},[650],[363,122427,122429],{"className":122428},[655,656,657,658],[363,122430,554],{"className":122431,"style":682},[407,408,658],[363,122433,666],{"className":122434},[665],[363,122436,122438],{"className":122437},[638],[363,122439,122441],{"className":122440,"style":8316},[642],[363,122442],{},[363,122444,578],{"className":122445},[494],[363,122447,4506],{"className":122448,"style":1328},[407,408],[363,122450,605],{"className":122451},[501],[363,122453],{"className":122454,"style":2935},[505],[363,122456,5161],{"className":122457},[2939],[363,122459],{"className":122460,"style":2935},[505],[363,122462,122464,122467,122470,122510,122513,122516],{"className":122463},[398],[363,122465],{"className":122466,"style":618},[402],[363,122468,121465],{"className":122469,"style":121555},[407,408],[363,122471,122473,122476],{"className":122472},[407],[363,122474,571],{"className":122475,"style":26233},[407,408],[363,122477,122479],{"className":122478},[629],[363,122480,122482,122502],{"className":122481},[633,634],[363,122483,122485,122499],{"className":122484},[638],[363,122486,122488],{"className":122487,"style":727},[642],[363,122489,122490,122493],{"style":122422},[363,122491],{"className":122492,"style":651},[650],[363,122494,122496],{"className":122495},[655,656,657,658],[363,122497,561],{"className":122498},[407,408,658],[363,122500,666],{"className":122501},[665],[363,122503,122505],{"className":122504},[638],[363,122506,122508],{"className":122507,"style":673},[642],[363,122509],{},[363,122511,578],{"className":122512},[494],[363,122514,4506],{"className":122515,"style":1328},[407,408],[363,122517,605],{"className":122518},[501],[363,122520,122522,122535],{"className":122521,"translate":367},[366],[363,122523,122525],{"className":122524},[371],[373,122526,122527],{"xmlns":375},[377,122528,122529,122533],{},[380,122530,122531],{},[383,122532,4506],{},[387,122534,4506],{"encoding":389},[363,122536,122538],{"className":122537,"ariaHidden":394},[393],[363,122539,122541,122544],{"className":122540},[398],[363,122542],{"className":122543,"style":2624},[402],[363,122545,4506],{"className":122546,"style":1328},[407,408]," 的线性泛函，故其上确界在指示函数 ",[363,122549,122551,122573],{"className":122550,"translate":367},[366],[363,122552,122554],{"className":122553},[371],[373,122555,122556],{"xmlns":375},[377,122557,122558,122570],{},[380,122559,122560,122562,122564],{},[383,122561,4506],{},[458,122563,471],{},[543,122565,122566,122568],{},[463,122567,465],{"mathvariant":75598},[383,122569,950],{},[387,122571,122572],{"encoding":389},"g=\\mathbf1_A",[363,122574,122576,122594],{"className":122575,"ariaHidden":394},[393],[363,122577,122579,122582,122585,122588,122591],{"className":122578},[398],[363,122580],{"className":122581,"style":2624},[402],[363,122583,4506],{"className":122584,"style":1328},[407,408],[363,122586],{"className":122587,"style":506},[505],[363,122589,471],{"className":122590},[510],[363,122592],{"className":122593,"style":506},[505],[363,122595,122597,122601],{"className":122596},[398],[363,122598],{"className":122599,"style":122600},[402],"height:0.7944em;vertical-align:-0.15em;",[363,122602,122604,122607],{"className":122603},[407],[363,122605,465],{"className":122606},[407,75943],[363,122608,122610],{"className":122609},[629],[363,122611,122613,122633],{"className":122612},[633,634],[363,122614,122616,122630],{"className":122615},[638],[363,122617,122619],{"className":122618,"style":727},[642],[363,122620,122621,122624],{"style":730},[363,122622],{"className":122623,"style":651},[650],[363,122625,122627],{"className":122626},[655,656,657,658],[363,122628,950],{"className":122629},[407,408,658],[363,122631,666],{"className":122632},[665],[363,122634,122636],{"className":122635},[638],[363,122637,122639],{"className":122638,"style":8316},[642],[363,122640],{}," 上达到：",[363,122643,122645],{"className":122644,"translate":367},[528],[363,122646,122648,122750],{"className":122647,"translate":367},[366],[363,122649,122651],{"className":122650},[371],[373,122652,122653],{"xmlns":375,"display":537},[377,122654,122655,122747],{},[380,122656,122657,122677,122679,122685,122687,122689,122691,122693,122695,122701,122703,122705,122707,122709,122711,122721,122723,122725,122727,122729,122731,122733,122735,122737,122739,122741,122743,122745],{},[6364,122658,122659,122665],{},[380,122660,122661,122663],{},[383,122662,35804],{},[458,122664,4451],{},[380,122666,122667,122669,122671,122673,122675],{},[463,122668,474],{},[458,122670,10300],{},[383,122672,4506],{},[458,122674,10300],{},[463,122676,465],{},[458,122678,3246],{"stretchy":460},[543,122680,122681,122683],{},[383,122682,571],{},[383,122684,554],{},[458,122686,578],{"stretchy":460},[383,122688,4506],{},[458,122690,605],{"stretchy":460},[458,122692,5161],{},[383,122694,121465],{},[543,122696,122697,122699],{},[383,122698,571],{},[383,122700,561],{},[458,122702,578],{"stretchy":460},[383,122704,4506],{},[458,122706,605],{"stretchy":460},[458,122708,3256],{"stretchy":460},[458,122710,471],{},[6364,122712,122713,122719],{},[380,122714,122715,122717],{},[383,122716,35804],{},[458,122718,4451],{},[383,122720,950],{},[458,122722,3246],{"stretchy":460},[383,122724,554],{},[458,122726,461],{"stretchy":460},[383,122728,950],{},[458,122730,468],{"stretchy":460},[458,122732,5161],{},[383,122734,121465],{},[383,122736,561],{},[458,122738,461],{"stretchy":460},[383,122740,950],{},[458,122742,468],{"stretchy":460},[458,122744,3256],{"stretchy":460},[458,122746,898],{"separator":394},[387,122748,122749],{"encoding":389},"\\sup_{0\\le g\\le1}\\{E_P[g]-\\gamma E_Q[g]\\}\n=\\sup_{A}\\{P(A)-\\gamma Q(A)\\},",[363,122751,122753,122881,122949,123028],{"className":122752,"ariaHidden":394},[393],[363,122754,122756,122760,122820,122823,122863,122866,122869,122872,122875,122878],{"className":122755},[398],[363,122757],{"className":122758,"style":122759},[402],"height:1.7977em;vertical-align:-1.0477em;",[363,122761,122763],{"className":122762},[1832,14066],[363,122764,122766,122811],{"className":122765},[633,634],[363,122767,122769,122808],{"className":122768},[638],[363,122770,122772,122798],{"className":122771,"style":2693},[642],[363,122773,122774,122777],{"style":61305},[363,122775],{"className":122776,"style":809},[650],[363,122778,122780],{"className":122779},[655,656,657,658],[363,122781,122783,122786,122789,122792,122795],{"className":122782},[407,658],[363,122784,474],{"className":122785},[407,658],[363,122787,10300],{"className":122788},[510,658],[363,122790,4506],{"className":122791,"style":1328},[407,408,658],[363,122793,10300],{"className":122794},[510,658],[363,122796,465],{"className":122797},[407,658],[363,122799,122800,122803],{"style":12213},[363,122801],{"className":122802,"style":809},[650],[363,122804,122805],{},[363,122806,35804],{"className":122807},[1832],[363,122809,666],{"className":122810},[665],[363,122812,122814],{"className":122813},[638],[363,122815,122818],{"className":122816,"style":122817},[642],"height:1.0477em;",[363,122819],{},[363,122821,3246],{"className":122822},[494],[363,122824,122826,122829],{"className":122825},[407],[363,122827,571],{"className":122828,"style":26233},[407,408],[363,122830,122832],{"className":122831},[629],[363,122833,122835,122855],{"className":122834},[633,634],[363,122836,122838,122852],{"className":122837},[638],[363,122839,122841],{"className":122840,"style":727},[642],[363,122842,122843,122846],{"style":122422},[363,122844],{"className":122845,"style":651},[650],[363,122847,122849],{"className":122848},[655,656,657,658],[363,122850,554],{"className":122851,"style":682},[407,408,658],[363,122853,666],{"className":122854},[665],[363,122856,122858],{"className":122857},[638],[363,122859,122861],{"className":122860,"style":8316},[642],[363,122862],{},[363,122864,578],{"className":122865},[494],[363,122867,4506],{"className":122868,"style":1328},[407,408],[363,122870,605],{"className":122871},[501],[363,122873],{"className":122874,"style":2935},[505],[363,122876,5161],{"className":122877},[2939],[363,122879],{"className":122880,"style":2935},[505],[363,122882,122884,122887,122890,122930,122933,122936,122940,122943,122946],{"className":122883},[398],[363,122885],{"className":122886,"style":618},[402],[363,122888,121465],{"className":122889,"style":121555},[407,408],[363,122891,122893,122896],{"className":122892},[407],[363,122894,571],{"className":122895,"style":26233},[407,408],[363,122897,122899],{"className":122898},[629],[363,122900,122902,122922],{"className":122901},[633,634],[363,122903,122905,122919],{"className":122904},[638],[363,122906,122908],{"className":122907,"style":727},[642],[363,122909,122910,122913],{"style":122422},[363,122911],{"className":122912,"style":651},[650],[363,122914,122916],{"className":122915},[655,656,657,658],[363,122917,561],{"className":122918},[407,408,658],[363,122920,666],{"className":122921},[665],[363,122923,122925],{"className":122924},[638],[363,122926,122928],{"className":122927,"style":673},[642],[363,122929],{},[363,122931,578],{"className":122932},[494],[363,122934,4506],{"className":122935,"style":1328},[407,408],[363,122937,122939],{"className":122938},[501],"]}",[363,122941],{"className":122942,"style":506},[505],[363,122944,471],{"className":122945},[510],[363,122947],{"className":122948,"style":506},[505],[363,122950,122952,122956,123004,123007,123010,123013,123016,123019,123022,123025],{"className":122951},[398],[363,122953],{"className":122954,"style":122955},[402],"height:1.6888em;vertical-align:-0.9388em;",[363,122957,122959],{"className":122958},[1832,14066],[363,122960,122962,122995],{"className":122961},[633,634],[363,122963,122965,122992],{"className":122964},[638],[363,122966,122968,122982],{"className":122967,"style":2693},[642],[363,122969,122970,122973],{"style":36740},[363,122971],{"className":122972,"style":809},[650],[363,122974,122976],{"className":122975},[655,656,657,658],[363,122977,122979],{"className":122978},[407,658],[363,122980,950],{"className":122981},[407,408,658],[363,122983,122984,122987],{"style":12213},[363,122985],{"className":122986,"style":809},[650],[363,122988,122989],{},[363,122990,35804],{"className":122991},[1832],[363,122993,666],{"className":122994},[665],[363,122996,122998],{"className":122997},[638],[363,122999,123002],{"className":123000,"style":123001},[642],"height:0.9388em;",[363,123003],{},[363,123005,3246],{"className":123006},[494],[363,123008,554],{"className":123009,"style":682},[407,408],[363,123011,461],{"className":123012},[494],[363,123014,950],{"className":123015},[407,408],[363,123017,468],{"className":123018},[501],[363,123020],{"className":123021,"style":2935},[505],[363,123023,5161],{"className":123024},[2939],[363,123026],{"className":123027,"style":2935},[505],[363,123029,123031,123034,123037,123040,123043,123046,123049],{"className":123030},[398],[363,123032],{"className":123033,"style":487},[402],[363,123035,121465],{"className":123036,"style":121555},[407,408],[363,123038,561],{"className":123039},[407,408],[363,123041,461],{"className":123042},[494],[363,123044,950],{"className":123045},[407,408],[363,123047,61382],{"className":123048},[501],[363,123050,898],{"className":123051},[920],[11,123053,123054,123055,123228,123229,19080,123375,123490,123491,123639,123640,123926],{},"且由 ",[363,123056,123058,123100],{"className":123057,"translate":367},[366],[363,123059,123061],{"className":123060},[371],[373,123062,123063],{"xmlns":375},[377,123064,123065,123097],{},[380,123066,123067,123073,123075,123077,123079,123081,123087,123089,123091,123093,123095],{},[543,123068,123069,123071],{},[383,123070,571],{},[383,123072,554],{},[458,123074,578],{"stretchy":460},[383,123076,4506],{},[458,123078,605],{"stretchy":460},[458,123080,471],{},[543,123082,123083,123085],{},[383,123084,571],{},[383,123086,561],{},[458,123088,578],{"stretchy":460},[383,123090,4506],{},[1645,123092,1647],{},[383,123094,1116],{},[458,123096,605],{"stretchy":460},[387,123098,123099],{"encoding":389},"E_P[g]=E_Q[g\\,t]",[363,123101,123103,123167],{"className":123102,"ariaHidden":394},[393],[363,123104,123106,123109,123149,123152,123155,123158,123161,123164],{"className":123105},[398],[363,123107],{"className":123108,"style":487},[402],[363,123110,123112,123115],{"className":123111},[407],[363,123113,571],{"className":123114,"style":26233},[407,408],[363,123116,123118],{"className":123117},[629],[363,123119,123121,123141],{"className":123120},[633,634],[363,123122,123124,123138],{"className":123123},[638],[363,123125,123127],{"className":123126,"style":727},[642],[363,123128,123129,123132],{"style":122422},[363,123130],{"className":123131,"style":651},[650],[363,123133,123135],{"className":123134},[655,656,657,658],[363,123136,554],{"className":123137,"style":682},[407,408,658],[363,123139,666],{"className":123140},[665],[363,123142,123144],{"className":123143},[638],[363,123145,123147],{"className":123146,"style":8316},[642],[363,123148],{},[363,123150,578],{"className":123151},[494],[363,123153,4506],{"className":123154,"style":1328},[407,408],[363,123156,605],{"className":123157},[501],[363,123159],{"className":123160,"style":506},[505],[363,123162,471],{"className":123163},[510],[363,123165],{"className":123166,"style":506},[505],[363,123168,123170,123173,123213,123216,123219,123222,123225],{"className":123169},[398],[363,123171],{"className":123172,"style":618},[402],[363,123174,123176,123179],{"className":123175},[407],[363,123177,571],{"className":123178,"style":26233},[407,408],[363,123180,123182],{"className":123181},[629],[363,123183,123185,123205],{"className":123184},[633,634],[363,123186,123188,123202],{"className":123187},[638],[363,123189,123191],{"className":123190,"style":727},[642],[363,123192,123193,123196],{"style":122422},[363,123194],{"className":123195,"style":651},[650],[363,123197,123199],{"className":123198},[655,656,657,658],[363,123200,561],{"className":123201},[407,408,658],[363,123203,666],{"className":123204},[665],[363,123206,123208],{"className":123207},[638],[363,123209,123211],{"className":123210,"style":673},[642],[363,123212],{},[363,123214,578],{"className":123215},[494],[363,123217,4506],{"className":123218,"style":1328},[407,408],[363,123220],{"className":123221,"style":754},[505],[363,123223,1116],{"className":123224},[407,408],[363,123226,605],{"className":123227},[501],"，左端点态上确界为 ",[363,123230,123232,123268],{"className":123231,"translate":367},[366],[363,123233,123235],{"className":123234},[371],[373,123236,123237],{"xmlns":375},[377,123238,123239,123265],{},[380,123240,123241,123247,123249,123251,123253,123255,123257,123263],{},[543,123242,123243,123245],{},[383,123244,571],{},[383,123246,561],{},[458,123248,578],{"stretchy":460},[458,123250,461],{"stretchy":460},[383,123252,1116],{},[458,123254,5161],{},[383,123256,121465],{},[7162,123258,123259,123261],{},[458,123260,468],{"stretchy":460},[458,123262,2818],{},[458,123264,605],{"stretchy":460},[387,123266,123267],{"encoding":389},"E_Q[(t-\\gamma)^+]",[363,123269,123271,123333],{"className":123270,"ariaHidden":394},[393],[363,123272,123274,123277,123317,123321,123324,123327,123330],{"className":123273},[398],[363,123275],{"className":123276,"style":618},[402],[363,123278,123280,123283],{"className":123279},[407],[363,123281,571],{"className":123282,"style":26233},[407,408],[363,123284,123286],{"className":123285},[629],[363,123287,123289,123309],{"className":123288},[633,634],[363,123290,123292,123306],{"className":123291},[638],[363,123293,123295],{"className":123294,"style":727},[642],[363,123296,123297,123300],{"style":122422},[363,123298],{"className":123299,"style":651},[650],[363,123301,123303],{"className":123302},[655,656,657,658],[363,123304,561],{"className":123305},[407,408,658],[363,123307,666],{"className":123308},[665],[363,123310,123312],{"className":123311},[638],[363,123313,123315],{"className":123314,"style":673},[642],[363,123316],{},[363,123318,123320],{"className":123319},[494],"[(",[363,123322,1116],{"className":123323},[407,408],[363,123325],{"className":123326,"style":2935},[505],[363,123328,5161],{"className":123329},[2939],[363,123331],{"className":123332,"style":2935},[505],[363,123334,123336,123340,123343,123372],{"className":123335},[398],[363,123337],{"className":123338,"style":123339},[402],"height:1.0213em;vertical-align:-0.25em;",[363,123341,121465],{"className":123342,"style":121555},[407,408],[363,123344,123346,123349],{"className":123345},[501],[363,123347,468],{"className":123348},[501],[363,123350,123352],{"className":123351},[629],[363,123353,123355],{"className":123354},[633],[363,123356,123358],{"className":123357},[638],[363,123359,123361],{"className":123360,"style":21381},[642],[363,123362,123363,123366],{"style":7213},[363,123364],{"className":123365,"style":651},[650],[363,123367,123369],{"className":123368},[655,656,657,658],[363,123370,2818],{"className":123371},[2939,658],[363,123373,605],{"className":123374},[501],[363,123376,123378,123400],{"className":123377,"translate":367},[366],[363,123379,123381],{"className":123380},[371],[373,123382,123383],{"xmlns":375},[377,123384,123385,123397],{},[380,123386,123387],{},[543,123388,123389,123391],{},[383,123390,547],{},[543,123392,123393,123395],{},[383,123394,4506],{},[383,123396,121465],{},[387,123398,123399],{"encoding":389},"D_{g_\\gamma}",[363,123401,123403],{"className":123402,"ariaHidden":394},[393],[363,123404,123406,123410],{"className":123405},[398],[363,123407],{"className":123408,"style":123409},[402],"height:1.0307em;vertical-align:-0.3473em;",[363,123411,123413,123416],{"className":123412},[407],[363,123414,547],{"className":123415,"style":625},[407,408],[363,123417,123419],{"className":123418},[629],[363,123420,123422,123482],{"className":123421},[633,634],[363,123423,123425,123479],{"className":123424},[638],[363,123426,123428],{"className":123427,"style":45301},[642],[363,123429,123430,123433],{"style":646},[363,123431],{"className":123432,"style":651},[650],[363,123434,123436],{"className":123435},[655,656,657,658],[363,123437,123439],{"className":123438},[407,658],[363,123440,123442,123445],{"className":123441},[407,658],[363,123443,4506],{"className":123444,"style":1328},[407,408,658],[363,123446,123448],{"className":123447},[629],[363,123449,123451,123471],{"className":123450},[633,634],[363,123452,123454,123468],{"className":123453},[638],[363,123455,123457],{"className":123456,"style":74645},[642],[363,123458,123459,123462],{"style":122090},[363,123460],{"className":123461,"style":30578},[650],[363,123463,123465],{"className":123464},[655,30582,10022,658],[363,123466,121465],{"className":123467,"style":121555},[407,408,658],[363,123469,666],{"className":123470},[665],[363,123472,123474],{"className":123473},[638],[363,123475,123477],{"className":123476,"style":122109},[642],[363,123478],{},[363,123480,666],{"className":123481},[665],[363,123483,123485],{"className":123484},[638],[363,123486,123488],{"className":123487,"style":122121},[642],[363,123489],{},"。另一方面取 ",[363,123492,123494,123530],{"className":123493,"translate":367},[366],[363,123495,123497],{"className":123496},[371],[373,123498,123499],{"xmlns":375},[377,123500,123501,123527],{},[380,123502,123503,123509,123511],{},[7162,123504,123505,123507],{},[383,123506,4506],{},[458,123508,7921],{},[458,123510,471],{},[543,123512,123513,123515],{},[463,123514,465],{"mathvariant":75598},[380,123516,123517,123519,123521,123523,123525],{},[458,123518,3246],{"stretchy":460},[383,123520,1116],{},[458,123522,3251],{},[383,123524,121465],{},[458,123526,3256],{"stretchy":460},[387,123528,123529],{"encoding":389},"g^*=\\mathbf 1_{\\{t>\\gamma\\}}",[363,123531,123533,123577],{"className":123532,"ariaHidden":394},[393],[363,123534,123536,123539,123568,123571,123574],{"className":123535},[398],[363,123537],{"className":123538,"style":64398},[402],[363,123540,123542,123545],{"className":123541},[407],[363,123543,4506],{"className":123544,"style":1328},[407,408],[363,123546,123548],{"className":123547},[629],[363,123549,123551],{"className":123550},[633],[363,123552,123554],{"className":123553},[638],[363,123555,123557],{"className":123556,"style":8847},[642],[363,123558,123559,123562],{"style":7213},[363,123560],{"className":123561,"style":651},[650],[363,123563,123565],{"className":123564},[655,656,657,658],[363,123566,7921],{"className":123567},[2939,658],[363,123569],{"className":123570,"style":506},[505],[363,123572,471],{"className":123573},[510],[363,123575],{"className":123576,"style":506},[505],[363,123578,123580,123584],{"className":123579},[398],[363,123581],{"className":123582,"style":123583},[402],"height:0.9996em;vertical-align:-0.3552em;",[363,123585,123587,123590],{"className":123586},[407],[363,123588,465],{"className":123589},[407,75943],[363,123591,123593],{"className":123592},[629],[363,123594,123596,123631],{"className":123595},[633,634],[363,123597,123599,123628],{"className":123598},[638],[363,123600,123602],{"className":123601,"style":6446},[642],[363,123603,123604,123607],{"style":75958},[363,123605],{"className":123606,"style":651},[650],[363,123608,123610],{"className":123609},[655,656,657,658],[363,123611,123613,123616,123619,123622,123625],{"className":123612},[407,658],[363,123614,3246],{"className":123615},[494,658],[363,123617,1116],{"className":123618},[407,408,658],[363,123620,3251],{"className":123621},[510,658],[363,123623,121465],{"className":123624,"style":121555},[407,408,658],[363,123626,3256],{"className":123627},[501,658],[363,123629,666],{"className":123630},[665],[363,123632,123634],{"className":123633},[638],[363,123635,123637],{"className":123636,"style":6492},[642],[363,123638],{}," 恰有 ",[363,123641,123643,123707],{"className":123642,"translate":367},[366],[363,123644,123646],{"className":123645},[371],[373,123647,123648],{"xmlns":375},[377,123649,123650,123704],{},[380,123651,123652,123658,123660,123666,123668,123670,123672,123674,123676,123678,123680,123686,123688,123690,123692,123694,123696,123702],{},[543,123653,123654,123656],{},[383,123655,571],{},[383,123657,561],{},[458,123659,578],{"stretchy":460},[7162,123661,123662,123664],{},[383,123663,4506],{},[458,123665,7921],{},[458,123667,461],{"stretchy":460},[383,123669,1116],{},[458,123671,5161],{},[383,123673,121465],{},[458,123675,468],{"stretchy":460},[458,123677,605],{"stretchy":460},[458,123679,471],{},[543,123681,123682,123684],{},[383,123683,571],{},[383,123685,561],{},[458,123687,578],{"stretchy":460},[458,123689,461],{"stretchy":460},[383,123691,1116],{},[458,123693,5161],{},[383,123695,121465],{},[7162,123697,123698,123700],{},[458,123699,468],{"stretchy":460},[458,123701,2818],{},[458,123703,605],{"stretchy":460},[387,123705,123706],{"encoding":389},"E_Q[g^*(t-\\gamma)]=E_Q[(t-\\gamma)^+]",[363,123708,123710,123803,123824,123885],{"className":123709,"ariaHidden":394},[393],[363,123711,123713,123716,123756,123759,123788,123791,123794,123797,123800],{"className":123712},[398],[363,123714],{"className":123715,"style":618},[402],[363,123717,123719,123722],{"className":123718},[407],[363,123720,571],{"className":123721,"style":26233},[407,408],[363,123723,123725],{"className":123724},[629],[363,123726,123728,123748],{"className":123727},[633,634],[363,123729,123731,123745],{"className":123730},[638],[363,123732,123734],{"className":123733,"style":727},[642],[363,123735,123736,123739],{"style":122422},[363,123737],{"className":123738,"style":651},[650],[363,123740,123742],{"className":123741},[655,656,657,658],[363,123743,561],{"className":123744},[407,408,658],[363,123746,666],{"className":123747},[665],[363,123749,123751],{"className":123750},[638],[363,123752,123754],{"className":123753,"style":673},[642],[363,123755],{},[363,123757,578],{"className":123758},[494],[363,123760,123762,123765],{"className":123761},[407],[363,123763,4506],{"className":123764,"style":1328},[407,408],[363,123766,123768],{"className":123767},[629],[363,123769,123771],{"className":123770},[633],[363,123772,123774],{"className":123773},[638],[363,123775,123777],{"className":123776,"style":8847},[642],[363,123778,123779,123782],{"style":7213},[363,123780],{"className":123781,"style":651},[650],[363,123783,123785],{"className":123784},[655,656,657,658],[363,123786,7921],{"className":123787},[2939,658],[363,123789,461],{"className":123790},[494],[363,123792,1116],{"className":123793},[407,408],[363,123795],{"className":123796,"style":2935},[505],[363,123798,5161],{"className":123799},[2939],[363,123801],{"className":123802,"style":2935},[505],[363,123804,123806,123809,123812,123815,123818,123821],{"className":123805},[398],[363,123807],{"className":123808,"style":487},[402],[363,123810,121465],{"className":123811,"style":121555},[407,408],[363,123813,20762],{"className":123814},[501],[363,123816],{"className":123817,"style":506},[505],[363,123819,471],{"className":123820},[510],[363,123822],{"className":123823,"style":506},[505],[363,123825,123827,123830,123870,123873,123876,123879,123882],{"className":123826},[398],[363,123828],{"className":123829,"style":618},[402],[363,123831,123833,123836],{"className":123832},[407],[363,123834,571],{"className":123835,"style":26233},[407,408],[363,123837,123839],{"className":123838},[629],[363,123840,123842,123862],{"className":123841},[633,634],[363,123843,123845,123859],{"className":123844},[638],[363,123846,123848],{"className":123847,"style":727},[642],[363,123849,123850,123853],{"style":122422},[363,123851],{"className":123852,"style":651},[650],[363,123854,123856],{"className":123855},[655,656,657,658],[363,123857,561],{"className":123858},[407,408,658],[363,123860,666],{"className":123861},[665],[363,123863,123865],{"className":123864},[638],[363,123866,123868],{"className":123867,"style":673},[642],[363,123869],{},[363,123871,123320],{"className":123872},[494],[363,123874,1116],{"className":123875},[407,408],[363,123877],{"className":123878,"style":2935},[505],[363,123880,5161],{"className":123881},[2939],[363,123883],{"className":123884,"style":2935},[505],[363,123886,123888,123891,123894,123923],{"className":123887},[398],[363,123889],{"className":123890,"style":123339},[402],[363,123892,121465],{"className":123893,"style":121555},[407,408],[363,123895,123897,123900],{"className":123896},[501],[363,123898,468],{"className":123899},[501],[363,123901,123903],{"className":123902},[629],[363,123904,123906],{"className":123905},[633],[363,123907,123909],{"className":123908},[638],[363,123910,123912],{"className":123911,"style":21381},[642],[363,123913,123914,123917],{"style":7213},[363,123915],{"className":123916,"style":651},[650],[363,123918,123920],{"className":123919},[655,656,657,658],[363,123921,2818],{"className":123922},[2939,658],[363,123924,605],{"className":123925},[501],"，故等号可取。",[11,123928,123929,123930,123958,123959,124010,124011,124047,124048,16145,124151,124202,124203,124370,124371,124429,124430,124458],{},"几何解释：",[363,123931,123933,123946],{"className":123932,"translate":367},[366],[363,123934,123936],{"className":123935},[371],[373,123937,123938],{"xmlns":375},[377,123939,123940,123944],{},[380,123941,123942],{},[383,123943,121465],{},[387,123945,121882],{"encoding":389},[363,123947,123949],{"className":123948,"ariaHidden":394},[393],[363,123950,123952,123955],{"className":123951},[398],[363,123953],{"className":123954,"style":2624},[402],[363,123956,121465],{"className":123957,"style":121555},[407,408]," 扫过 ",[363,123960,123962,123983],{"className":123961,"translate":367},[366],[363,123963,123965],{"className":123964},[371],[373,123966,123967],{"xmlns":375},[377,123968,123969,123981],{},[380,123970,123971,123973,123975,123977,123979],{},[458,123972,578],{"stretchy":460},[463,123974,474],{},[458,123976,898],{"separator":394},[463,123978,465],{},[458,123980,605],{"stretchy":460},[387,123982,37186],{"encoding":389},[363,123984,123986],{"className":123985,"ariaHidden":394},[393],[363,123987,123989,123992,123995,123998,124001,124004,124007],{"className":123988},[398],[363,123990],{"className":123991,"style":487},[402],[363,123993,578],{"className":123994},[494],[363,123996,474],{"className":123997},[407],[363,123999,898],{"className":124000},[920],[363,124002],{"className":124003,"style":754},[505],[363,124005,465],{"className":124006},[407],[363,124008,605],{"className":124009},[501]," 时，右端就是权衡曲线（ROC 下边界）各条斜率为 ",[363,124012,124014,124032],{"className":124013,"translate":367},[366],[363,124015,124017],{"className":124016},[371],[373,124018,124019],{"xmlns":375},[377,124020,124021,124029],{},[380,124022,124023,124025,124027],{},[463,124024,465],{},[383,124026,1288],{"mathvariant":557},[383,124028,121465],{},[387,124030,124031],{"encoding":389},"1\u002F\\gamma",[363,124033,124035],{"className":124034,"ariaHidden":394},[393],[363,124036,124038,124041,124044],{"className":124037},[398],[363,124039],{"className":124040,"style":487},[402],[363,124042,8937],{"className":124043},[407],[363,124045,121465],{"className":124046,"style":121555},[407,408]," 的支撑线的截距——",[1344,124049,124050,124051,124079,124080,124150],{},"知道整条曲线 ",[363,124052,124054,124067],{"className":124053,"translate":367},[366],[363,124055,124057],{"className":124056},[371],[373,124058,124059],{"xmlns":375},[377,124060,124061,124065],{},[380,124062,124063],{},[383,124064,5740],{},[387,124066,5740],{"encoding":389},[363,124068,124070],{"className":124069,"ariaHidden":394},[393],[363,124071,124073,124076],{"className":124072},[398],[363,124074],{"className":124075,"style":1011},[402],[363,124077,5740],{"className":124078,"style":682},[407,408]," 就知道全部 ",[363,124081,124083,124101],{"className":124082,"translate":367},[366],[363,124084,124086],{"className":124085},[371],[373,124087,124088],{"xmlns":375},[377,124089,124090,124098],{},[380,124091,124092],{},[543,124093,124094,124096],{},[383,124095,4506],{},[383,124097,121465],{},[387,124099,124100],{"encoding":389},"g_\\gamma",[363,124102,124104],{"className":124103,"ariaHidden":394},[393],[363,124105,124107,124110],{"className":124106},[398],[363,124108],{"className":124109,"style":93347},[402],[363,124111,124113,124116],{"className":124112},[407],[363,124114,4506],{"className":124115,"style":1328},[407,408],[363,124117,124119],{"className":124118},[629],[363,124120,124122,124142],{"className":124121},[633,634],[363,124123,124125,124139],{"className":124124},[638],[363,124126,124128],{"className":124127,"style":45301},[642],[363,124129,124130,124133],{"style":19215},[363,124131],{"className":124132,"style":651},[650],[363,124134,124136],{"className":124135},[655,656,657,658],[363,124137,121465],{"className":124138,"style":121555},[407,408,658],[363,124140,666],{"className":124141},[665],[363,124143,124145],{"className":124144},[638],[363,124146,124148],{"className":124147,"style":673},[642],[363,124149],{},"-散度",[363,124152,124154,124172],{"className":124153,"translate":367},[366],[363,124155,124157],{"className":124156},[371],[373,124158,124159],{"xmlns":375},[377,124160,124161,124169],{},[380,124162,124163,124165,124167],{},[383,124164,121465],{},[458,124166,471],{},[463,124168,465],{},[387,124170,124171],{"encoding":389},"\\gamma=1",[363,124173,124175,124193],{"className":124174,"ariaHidden":394},[393],[363,124176,124178,124181,124184,124187,124190],{"className":124177},[398],[363,124179],{"className":124180,"style":2624},[402],[363,124182,121465],{"className":124183,"style":121555},[407,408],[363,124185],{"className":124186,"style":506},[505],[363,124188,471],{"className":124189},[510],[363,124191],{"className":124192,"style":506},[505],[363,124194,124196,124199],{"className":124195},[398],[363,124197],{"className":124198,"style":520},[402],[363,124200,465],{"className":124201},[407]," 时立刻给出 ",[363,124204,124206,124258],{"className":124205,"translate":367},[366],[363,124207,124209],{"className":124208},[371],[373,124210,124211],{"xmlns":375},[377,124212,124213,124255],{},[380,124214,124215,124221,124223,124233,124235,124237,124239,124241,124243,124245,124247,124249,124251,124253],{},[380,124216,124217,124219],{},[383,124218,5740],{"mathvariant":557},[383,124220,35222],{"mathvariant":557},[458,124222,471],{},[543,124224,124225,124231],{},[380,124226,124227,124229],{},[383,124228,35804],{},[458,124230,4451],{},[383,124232,950],{},[458,124234,3246],{"stretchy":460},[383,124236,554],{},[458,124238,461],{"stretchy":460},[383,124240,950],{},[458,124242,468],{"stretchy":460},[458,124244,5161],{},[383,124246,561],{},[458,124248,461],{"stretchy":460},[383,124250,950],{},[458,124252,468],{"stretchy":460},[458,124254,3256],{"stretchy":460},[387,124256,124257],{"encoding":389},"\\mathrm{TV}=\\sup_A\\{P(A)-Q(A)\\}",[363,124259,124261,124282,124352],{"className":124260,"ariaHidden":394},[393],[363,124262,124264,124267,124273,124276,124279],{"className":124263},[398],[363,124265],{"className":124266,"style":1011},[402],[363,124268,124270],{"className":124269},[407],[363,124271,87149],{"className":124272,"style":4505},[407,33034],[363,124274],{"className":124275,"style":506},[505],[363,124277,471],{"className":124278},[510],[363,124280],{"className":124281,"style":506},[505],[363,124283,124285,124288,124328,124331,124334,124337,124340,124343,124346,124349],{"className":124284},[398],[363,124286],{"className":124287,"style":487},[402],[363,124289,124291,124294],{"className":124290},[1832],[363,124292,35804],{"className":124293},[1832],[363,124295,124297],{"className":124296},[629],[363,124298,124300,124320],{"className":124299},[633,634],[363,124301,124303,124317],{"className":124302},[638],[363,124304,124306],{"className":124305,"style":35947},[642],[363,124307,124308,124311],{"style":35950},[363,124309],{"className":124310,"style":651},[650],[363,124312,124314],{"className":124313},[655,656,657,658],[363,124315,950],{"className":124316},[407,408,658],[363,124318,666],{"className":124319},[665],[363,124321,124323],{"className":124322},[638],[363,124324,124326],{"className":124325,"style":37392},[642],[363,124327],{},[363,124329,3246],{"className":124330},[494],[363,124332,554],{"className":124333,"style":682},[407,408],[363,124335,461],{"className":124336},[494],[363,124338,950],{"className":124339},[407,408],[363,124341,468],{"className":124342},[501],[363,124344],{"className":124345,"style":2935},[505],[363,124347,5161],{"className":124348},[2939],[363,124350],{"className":124351,"style":2935},[505],[363,124353,124355,124358,124361,124364,124367],{"className":124354},[398],[363,124356],{"className":124357,"style":487},[402],[363,124359,561],{"className":124360},[407,408],[363,124362,461],{"className":124363},[494],[363,124365,950],{"className":124366},[407,408],[363,124368,61382],{"className":124369},[501],"）。反过来，知道单个标量散度（如 KL 或 ",[363,124372,124374,124391],{"className":124373,"translate":367},[366],[363,124375,124377],{"className":124376},[371],[373,124378,124379],{"xmlns":375},[377,124380,124381,124389],{},[380,124382,124383],{},[7162,124384,124385,124387],{},[383,124386,37821],{},[463,124388,2811],{},[387,124390,40766],{"encoding":389},[363,124392,124394],{"className":124393,"ariaHidden":394},[393],[363,124395,124397,124400],{"className":124396},[398],[363,124398],{"className":124399,"style":40776},[402],[363,124401,124403,124406],{"className":124402},[407],[363,124404,37821],{"className":124405},[407,408],[363,124407,124409],{"className":124408},[629],[363,124410,124412],{"className":124411},[633],[363,124413,124415],{"className":124414},[638],[363,124416,124418],{"className":124417,"style":20400},[642],[363,124419,124420,124423],{"style":7213},[363,124421],{"className":124422,"style":651},[650],[363,124424,124426],{"className":124425},[655,656,657,658],[363,124427,2811],{"className":124428},[407,658],"）只对 ",[363,124431,124433,124446],{"className":124432,"translate":367},[366],[363,124434,124436],{"className":124435},[371],[373,124437,124438],{"xmlns":375},[377,124439,124440,124444],{},[380,124441,124442],{},[383,124443,5740],{},[387,124445,5740],{"encoding":389},[363,124447,124449],{"className":124448,"ariaHidden":394},[393],[363,124450,124452,124455],{"className":124451},[398],[363,124453],{"className":124454,"style":1011},[402],[363,124456,5740],{"className":124457,"style":682},[407,408]," 施加一个切向约束（4.7 节的不等式网络），信息严格更少；",[10375,124460,124461,124693],{},[7013,124462,124463,124464,20338],{},"在「小隐私预算」下，所有 f-DP 都由**高斯差分隐私（GDP）**统一逼近，其权衡函数 ",[363,124465,124467,124525],{"className":124466,"translate":367},[366],[363,124468,124470],{"className":124469},[371],[373,124471,124472],{"xmlns":375},[377,124473,124474,124522],{},[380,124475,124476,124483,124485,124487,124489,124491,124494,124496,124506,124508,124510,124512,124514,124516,124518,124520],{},[543,124477,124478,124481],{},[383,124479,124480],{},"G",[383,124482,2712],{},[458,124484,461],{"stretchy":460},[383,124486,45126],{},[458,124488,468],{"stretchy":460},[458,124490,471],{},[383,124492,124493],{"mathvariant":557},"Φ",[458,124495,461],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[7162,124497,124498,124500],{},[383,124499,124493],{"mathvariant":557},[380,124501,124502,124504],{},[458,124503,5161],{},[463,124505,465],{},[458,124507,461],{"stretchy":460},[463,124509,465],{},[458,124511,5161],{},[383,124513,45126],{},[458,124515,468],{"stretchy":460},[458,124517,5161],{},[383,124519,2712],{},[458,124521,468],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[387,124523,124524],{"encoding":389},"G_\\mu(\\alpha)=\\Phi\\big(\\Phi^{-1}(1-\\alpha)-\\mu\\big)",[363,124526,124528,124592,124657,124678],{"className":124527,"ariaHidden":394},[393],[363,124529,124531,124534,124574,124577,124580,124583,124586,124589],{"className":124530},[398],[363,124532],{"className":124533,"style":618},[402],[363,124535,124537,124540],{"className":124536},[407],[363,124538,124480],{"className":124539},[407,408],[363,124541,124543],{"className":124542},[629],[363,124544,124546,124566],{"className":124545},[633,634],[363,124547,124549,124563],{"className":124548},[638],[363,124550,124552],{"className":124551,"style":45301},[642],[363,124553,124554,124557],{"style":730},[363,124555],{"className":124556,"style":651},[650],[363,124558,124560],{"className":124559},[655,656,657,658],[363,124561,2712],{"className":124562},[407,408,658],[363,124564,666],{"className":124565},[665],[363,124567,124569],{"className":124568},[638],[363,124570,124572],{"className":124571,"style":673},[642],[363,124573],{},[363,124575,461],{"className":124576},[494],[363,124578,45126],{"className":124579,"style":45156},[407,408],[363,124581,468],{"className":124582},[501],[363,124584],{"className":124585,"style":506},[505],[363,124587,471],{"className":124588},[510],[363,124590],{"className":124591,"style":506},[505],[363,124593,124595,124598,124601,124607,124642,124645,124648,124651,124654],{"className":124594},[398],[363,124596],{"className":124597,"style":20935},[402],[363,124599,124493],{"className":124600},[407],[363,124602,124604],{"className":124603},[407],[363,124605,461],{"className":124606},[767,10022],[363,124608,124610,124613],{"className":124609},[407],[363,124611,124493],{"className":124612},[407],[363,124614,124616],{"className":124615},[629],[363,124617,124619],{"className":124618},[633],[363,124620,124622],{"className":124621},[638],[363,124623,124625],{"className":124624,"style":20400},[642],[363,124626,124627,124630],{"style":7213},[363,124628],{"className":124629,"style":651},[650],[363,124631,124633],{"className":124632},[655,656,657,658],[363,124634,124636,124639],{"className":124635},[407,658],[363,124637,5161],{"className":124638},[407,658],[363,124640,465],{"className":124641},[407,658],[363,124643,461],{"className":124644},[494],[363,124646,465],{"className":124647},[407],[363,124649],{"className":124650,"style":2935},[505],[363,124652,5161],{"className":124653},[2939],[363,124655],{"className":124656,"style":2935},[505],[363,124658,124660,124663,124666,124669,124672,124675],{"className":124659},[398],[363,124661],{"className":124662,"style":487},[402],[363,124664,45126],{"className":124665,"style":45156},[407,408],[363,124667,468],{"className":124668},[501],[363,124670],{"className":124671,"style":2935},[505],[363,124673,5161],{"className":124674},[2939],[363,124676],{"className":124677,"style":2935},[505],[363,124679,124681,124684,124687],{"className":124680},[398],[363,124682],{"className":124683,"style":20935},[402],[363,124685,2712],{"className":124686},[407,408],[363,124688,124690],{"className":124689},[407],[363,124691,468],{"className":124692},[767,10022],[7013,124694,124695,124696,124699,124700,124728],{},"权衡函数族在",[1344,124697,124698],{},"组合","（post-processing、串联）下封闭，并装备了一个与 f-散度\"散度序\"同构的偏序——这与 f-散度族的\"",[363,124701,124703,124716],{"className":124702,"translate":367},[366],[363,124704,124706],{"className":124705},[371],[373,124707,124708],{"xmlns":375},[377,124709,124710,124714],{},[380,124711,124712],{},[458,124713,7276],{},[387,124715,55863],{"encoding":389},[363,124717,124719],{"className":124718,"ariaHidden":394},[393],[363,124720,124722,124725],{"className":124721},[398],[363,124723],{"className":124724,"style":43560},[402],[363,124726,7276],{"className":124727},[510]," 比较\"在结构上完全平行。",[11,124730,124731],{},"这说明 f-散度不仅统一了信息论度量，也统一了现代隐私的定义框架：用一条凸曲线代替一个标量数字。",[18,124733,124735],{"id":124734},"_9-计算实例","9. 计算实例",[2659,124737,124739],{"id":124738},"_91-正态分布","9.1 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计算 ",[363,126211,126213,126237],{"className":126212,"translate":367},[366],[363,126214,126216],{"className":126215},[371],[373,126217,126218],{"xmlns":375},[377,126219,126220,126234],{},[380,126221,126222,126224,126230,126232],{},[458,126223,1635],{},[7162,126225,126226,126228],{},[383,126227,11],{},[463,126229,2811],{},[383,126231,1288],{"mathvariant":557},[383,126233,1291],{},[387,126235,126236],{"encoding":389},"\\int p^2\u002Fq",[363,126238,126240],{"className":126239,"ariaHidden":394},[393],[363,126241,126243,126246,126249,126252,126281,126284],{"className":126242},[398],[363,126244],{"className":126245,"style":50700},[402],[363,126247,1635],{"className":126248,"style":5965},[1832,1833,5964],[363,126250],{"className":126251,"style":754},[505],[363,126253,126255,126258],{"className":126254},[407],[363,126256,11],{"className":126257},[407,408],[363,126259,126261],{"className":126260},[629],[363,126262,126264],{"className":126263},[633],[363,126265,126267],{"className":126266},[638],[363,126268,126270],{"className":126269,"style":20400},[642],[363,126271,126272,126275],{"style":7213},[363,126273],{"className":126274,"style":651},[650],[363,126276,126278],{"className":126277},[655,656,657,658],[363,126279,2811],{"className":126280},[407,658],[363,126282,1288],{"className":126283},[407],[363,126285,1291],{"className":126286,"style":1328},[407,408],"：由配方可得",[363,126289,126291],{"className":126290,"translate":367},[528],[363,126292,126294,126355],{"className":126293,"translate":367},[366],[363,126295,126297],{"className":126296},[371],[373,126298,126299],{"xmlns":375,"display":537},[377,126300,126301,126352],{},[380,126302,126303,126305,126315,126317,126319,126321,126323,126326,126328,126330,126350],{},[458,126304,1635],{},[586,126306,126307,126313],{},[7162,126308,126309,126311],{},[383,126310,11],{},[463,126312,2811],{},[383,126314,1291],{},[1645,126316,1647],{},[383,126318,592],{},[383,126320,1121],{},[458,126322,471],{},[383,126324,126325],{},"exp",[458,126327,4451],{},[1645,126329,1652],{},[380,126331,126332,126334,126348],{},[458,126333,461],{"fence":394},[586,126335,126336,126342],{},[7162,126337,126338,126340],{},[383,126339,126067],{"mathvariant":557},[463,126341,2811],{},[7162,126343,126344,126346],{},[383,126345,2680],{},[463,126347,2811],{},[458,126349,468],{"fence":394},[458,126351,898],{"separator":394},[387,126353,126354],{"encoding":389},"\\int \\frac{p^2}{q}\\,dx\n=\\exp\\!\\left(\\frac{\\Delta^2}{\\sigma^2}\\right),",[363,126356,126358,126476],{"className":126357,"ariaHidden":394},[393],[363,126359,126361,126364,126367,126370,126458,126461,126464,126467,126470,126473],{"className":126360},[398],[363,126362],{"className":126363,"style":38070},[402],[363,126365,1635],{"className":126366,"style":1835},[1832,1833,1834],[363,126368],{"className":126369,"style":754},[505],[363,126371,126373,126376,126455],{"className":126372},[407],[363,126374],{"className":126375},[494,789],[363,126377,126379],{"className":126378},[586],[363,126380,126382,126447],{"className":126381},[633,634],[363,126383,126385,126444],{"className":126384},[638],[363,126386,126388,126399,126407],{"className":126387,"style":38095},[642],[363,126389,126390,126393],{"style":805},[363,126391],{"className":126392,"style":809},[650],[363,126394,126396],{"className":126395},[407],[363,126397,1291],{"className":126398,"style":1328},[407,408],[363,126400,126401,126404],{"style":821},[363,126402],{"className":126403,"style":809},[650],[363,126405],{"className":126406,"style":829},[828],[363,126408,126409,126412],{"style":832},[363,126410],{"className":126411,"style":809},[650],[363,126413,126415],{"className":126414},[407],[363,126416,126418,126421],{"className":126417},[407],[363,126419,11],{"className":126420},[407,408],[363,126422,126424],{"className":126423},[629],[363,126425,126427],{"className":126426},[633],[363,126428,126430],{"className":126429},[638],[363,126431,126433],{"className":126432,"style":20400},[642],[363,126434,126435,126438],{"style":7213},[363,126436],{"className":126437,"style":651},[650],[363,126439,126441],{"className":126440},[655,656,657,658],[363,126442,2811],{"className":126443},[407,658],[363,126445,666],{"className":126446},[665],[363,126448,126450],{"className":126449},[638],[363,126451,126453],{"className":126452,"style":854},[642],[363,126454],{},[363,126456],{"className":126457},[501,789],[363,126459],{"className":126460,"style":754},[505],[363,126462,592],{"className":126463},[407,408],[363,126465,1121],{"className":126466},[407,408],[363,126468],{"className":126469,"style":506},[505],[363,126471,471],{"className":126472},[510],[363,126474],{"className":126475,"style":506},[505],[363,126477,126479,126482,126485,126488,126491,126620,126623],{"className":126478},[398],[363,126480],{"className":126481,"style":87645},[402],[363,126483,126325],{"className":126484},[1832],[363,126486],{"className":126487,"style":1860},[505],[363,126489],{"className":126490,"style":754},[505],[363,126492,126494,126500,126614],{"className":126493},[758],[363,126495,126497],{"className":126496,"style":763},[494,762],[363,126498,461],{"className":126499},[767,657],[363,126501,126503,126506,126611],{"className":126502},[407],[363,126504],{"className":126505},[494,789],[363,126507,126509],{"className":126508},[586],[363,126510,126512,126603],{"className":126511},[633,634],[363,126513,126515,126600],{"className":126514},[638],[363,126516,126518,126555,126563],{"className":126517,"style":38095},[642],[363,126519,126520,126523],{"style":805},[363,126521],{"className":126522,"style":809},[650],[363,126524,126526],{"className":126525},[407],[363,126527,126529,126532],{"className":126528},[407],[363,126530,2680],{"className":126531,"style":1328},[407,408],[363,126533,126535],{"className":126534},[629],[363,126536,126538],{"className":126537},[633],[363,126539,126541],{"className":126540},[638],[363,126542,126544],{"className":126543,"style":45404},[642],[363,126545,126546,126549],{"style":45407},[363,126547],{"className":126548,"style":651},[650],[363,126550,126552],{"className":126551},[655,656,657,658],[363,126553,2811],{"className":126554},[407,658],[363,126556,126557,126560],{"style":821},[363,126558],{"className":126559,"style":809},[650],[363,126561],{"className":126562,"style":829},[828],[363,126564,126565,126568],{"style":832},[363,126566],{"className":126567,"style":809},[650],[363,126569,126571],{"className":126570},[407],[363,126572,126574,126577],{"className":126573},[407],[363,126575,126067],{"className":126576},[407],[363,126578,126580],{"className":126579},[629],[363,126581,126583],{"className":126582},[633],[363,126584,126586],{"className":126585},[638],[363,126587,126589],{"className":126588,"style":20400},[642],[363,126590,126591,126594],{"style":7213},[363,126592],{"className":126593,"style":651},[650],[363,126595,126597],{"className":126596},[655,656,657,658],[363,126598,2811],{"className":126599},[407,658],[363,126601,666],{"className":126602},[665],[363,126604,126606],{"className":126605},[638],[363,126607,126609],{"className":126608,"style":4126},[642],[363,126610],{},[363,126612],{"className":126613},[501,789],[363,126615,126617],{"className":126616,"style":763},[501,762],[363,126618,468],{"className":126619},[767,657],[363,126621],{"className":126622,"style":754},[505],[363,126624,898],{"className":126625},[920],[11,126627,48531],{},[363,126629,126631],{"className":126630,"translate":367},[528],[363,126632,126634,126742],{"className":126633,"translate":367},[366],[363,126635,126637],{"className":126636},[371],[373,126638,126639],{"xmlns":375,"display":537},[377,126640,126641,126739],{},[380,126642,126643,126649,126651,126653,126655,126657,126659,126661,126663,126683,126685,126687,126689,126691,126693,126703,126705,126707,126709,126711,126713,126715,126735,126737],{},[7162,126644,126645,126647],{},[383,126646,37821],{},[463,126648,2811],{},[458,126650,461],{"stretchy":460},[383,126652,554],{},[383,126654,558],{"mathvariant":557},[383,126656,561],{},[458,126658,468],{"stretchy":460},[458,126660,471],{},[458,126662,1635],{},[586,126664,126665,126681],{},[380,126666,126667,126669,126671,126673,126675],{},[458,126668,461],{"stretchy":460},[383,126670,11],{},[458,126672,5161],{},[383,126674,1291],{},[7162,126676,126677,126679],{},[458,126678,468],{"stretchy":460},[463,126680,2811],{},[383,126682,1291],{},[1645,126684,1647],{},[383,126686,592],{},[383,126688,1121],{},[458,126690,471],{},[458,126692,1635],{},[586,126694,126695,126701],{},[7162,126696,126697,126699],{},[383,126698,11],{},[463,126700,2811],{},[383,126702,1291],{},[458,126704,5161],{},[463,126706,465],{},[458,126708,471],{},[383,126710,126325],{},[458,126712,4451],{},[1645,126714,1652],{},[380,126716,126717,126719,126733],{},[458,126718,461],{"fence":394},[586,126720,126721,126727],{},[7162,126722,126723,126725],{},[383,126724,126067],{"mathvariant":557},[463,126726,2811],{},[7162,126728,126729,126731],{},[383,126730,2680],{},[463,126732,2811],{},[458,126734,468],{"fence":394},[458,126736,5161],{},[463,126738,49882],{},[387,126740,126741],{"encoding":389},"\\chi^2(P\\|Q)=\\int\\frac{(p-q)^2}{q}\\,dx\n=\\int\\frac{p^2}{q}-1\n=\\exp\\!\\left(\\frac{\\Delta^2}{\\sigma^2}\\right)\n-1 .",[363,126743,126745,126804,126940,127049,127067,127220],{"className":126744,"ariaHidden":394},[393],[363,126746,126748,126751,126780,126783,126786,126789,126792,126795,126798,126801],{"className":126747},[398],[363,126749],{"className":126750,"style":37623},[402],[363,126752,126754,126757],{"className":126753},[407],[363,126755,37821],{"className":126756},[407,408],[363,126758,126760],{"className":126759},[629],[363,126761,126763],{"className":126762},[633],[363,126764,126766],{"className":126765},[638],[363,126767,126769],{"className":126768,"style":20107},[642],[363,126770,126771,126774],{"style":8625},[363,126772],{"className":126773,"style":651},[650],[363,126775,126777],{"className":126776},[655,656,657,658],[363,126778,2811],{"className":126779},[407,658],[363,126781,461],{"className":126782},[494],[363,126784,554],{"className":126785,"style":682},[407,408],[363,126787,558],{"className":126788},[407],[363,126790,561],{"className":126791},[407,408],[363,126793,468],{"className":126794},[501],[363,126796],{"className":126797,"style":506},[505],[363,126799,471],{"className":126800},[510],[363,126802],{"className":126803,"style":506},[505],[363,126805,126807,126810,126813,126816,126922,126925,126928,126931,126934,126937],{"className":126806},[398],[363,126808],{"className":126809,"style":38070},[402],[363,126811,1635],{"className":126812,"style":1835},[1832,1833,1834],[363,126814],{"className":126815,"style":754},[505],[363,126817,126819,126822,126919],{"className":126818},[407],[363,126820],{"className":126821},[494,789],[363,126823,126825],{"className":126824},[586],[363,126826,126828,126911],{"className":126827},[633,634],[363,126829,126831,126908],{"className":126830},[638],[363,126832,126834,126845,126853],{"className":126833,"style":38095},[642],[363,126835,126836,126839],{"style":805},[363,126837],{"className":126838,"style":809},[650],[363,126840,126842],{"className":126841},[407],[363,126843,1291],{"className":126844,"style":1328},[407,408],[363,126846,126847,126850],{"style":821},[363,126848],{"className":126849,"style":809},[650],[363,126851],{"className":126852,"style":829},[828],[363,126854,126855,126858],{"style":832},[363,126856],{"className":126857,"style":809},[650],[363,126859,126861,126864,126867,126870,126873,126876,126879],{"className":126860},[407],[363,126862,461],{"className":126863},[494],[363,126865,11],{"className":126866},[407,408],[363,126868],{"className":126869,"style":2935},[505],[363,126871,5161],{"className":126872},[2939],[363,126874],{"className":126875,"style":2935},[505],[363,126877,1291],{"className":126878,"style":1328},[407,408],[363,126880,126882,126885],{"className":126881},[501],[363,126883,468],{"className":126884},[501],[363,126886,126888],{"className":126887},[629],[363,126889,126891],{"className":126890},[633],[363,126892,126894],{"className":126893},[638],[363,126895,126897],{"className":126896,"style":20400},[642],[363,126898,126899,126902],{"style":7213},[363,126900],{"className":126901,"style":651},[650],[363,126903,126905],{"className":126904},[655,656,657,658],[363,126906,2811],{"className":126907},[407,658],[363,126909,666],{"className":126910},[665],[363,126912,126914],{"className":126913},[638],[363,126915,126917],{"className":126916,"style":854},[642],[363,126918],{},[363,126920],{"className":126921},[501,789],[363,126923],{"className":126924,"style":754},[505],[363,126926,592],{"className":126927},[407,408],[363,126929,1121],{"className":126930},[407,408],[363,126932],{"className":126933,"style":506},[505],[363,126935,471],{"className":126936},[510],[363,126938],{"className":126939,"style":506},[505],[363,126941,126943,126946,126949,126952,127040,127043,127046],{"className":126942},[398],[363,126944],{"className":126945,"style":38070},[402],[363,126947,1635],{"className":126948,"style":1835},[1832,1833,1834],[363,126950],{"className":126951,"style":754},[505],[363,126953,126955,126958,127037],{"className":126954},[407],[363,126956],{"className":126957},[494,789],[363,126959,126961],{"className":126960},[586],[363,126962,126964,127029],{"className":126963},[633,634],[363,126965,126967,127026],{"className":126966},[638],[363,126968,126970,126981,126989],{"className":126969,"style":38095},[642],[363,126971,126972,126975],{"style":805},[363,126973],{"className":126974,"style":809},[650],[363,126976,126978],{"className":126977},[407],[363,126979,1291],{"className":126980,"style":1328},[407,408],[363,126982,126983,126986],{"style":821},[363,126984],{"className":126985,"style":809},[650],[363,126987],{"className":126988,"style":829},[828],[363,126990,126991,126994],{"style":832},[363,126992],{"className":126993,"style":809},[650],[363,126995,126997],{"className":126996},[407],[363,126998,127000,127003],{"className":126999},[407],[363,127001,11],{"className":127002},[407,408],[363,127004,127006],{"className":127005},[629],[363,127007,127009],{"className":127008},[633],[363,127010,127012],{"className":127011},[638],[363,127013,127015],{"className":127014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也会随 ",[363,127292,127294,127320],{"className":127293,"translate":367},[366],[363,127295,127297],{"className":127296},[371],[373,127298,127299],{"xmlns":375},[377,127300,127301,127317],{},[380,127302,127303,127309,127311],{},[7162,127304,127305,127307],{},[383,127306,126067],{"mathvariant":557},[463,127308,2811],{},[383,127310,1288],{"mathvariant":557},[7162,127312,127313,127315],{},[383,127314,2680],{},[463,127316,2811],{},[387,127318,127319],{"encoding":389},"\\Delta^2\u002F\\sigma^2",[363,127321,127323],{"className":127322,"ariaHidden":394},[393],[363,127324,127326,127329,127358,127361],{"className":127325},[398],[363,127327],{"className":127328,"style":20375},[402],[363,127330,127332,127335],{"className":127331},[407],[363,127333,126067],{"className":127334},[407],[363,127336,127338],{"className":127337},[629],[363,127339,127341],{"className":127340},[633],[363,127342,127344],{"className":127343},[638],[363,127345,127347],{"className":127346,"style":20400},[642],[363,127348,127349,127352],{"style":7213},[363,127350],{"className":127351,"style":651},[650],[363,127353,127355],{"className":127354},[655,656,657,658],[363,127356,2811],{"className":127357},[407,658],[363,127359,1288],{"className":127360},[407],[363,127362,127364,127367],{"className":127363},[407],[363,127365,2680],{"className":127366,"style":1328},[407,408],[363,127368,127370],{"className":127369},[629],[363,127371,127373],{"className":127372},[633],[363,127374,127376],{"className":127375},[638],[363,127377,127379],{"className":127378,"style":20400},[642],[363,127380,127381,127384],{"style":7213},[363,127382],{"className":127383,"style":651},[650],[363,127385,127387],{"className":127386},[655,656,657,658],[363,127388,2811],{"className":127389},[407,658]," 指数爆炸。",[11,127392,127393,127396],{},[1344,127394,127395],{},"Hellinger²（等方差）。"," Bhattacharyya 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由 3.6 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a=\\frac{1+\\alpha}{2},",[363,128380,128382,128437,128544,128756],{"className":128381,"ariaHidden":394},[393],[363,128383,128385,128388,128428,128431,128434],{"className":128384},[398],[363,128386],{"className":128387,"style":16311},[402],[363,128389,128391,128394],{"className":128390},[407],[363,128392,547],{"className":128393,"style":625},[407,408],[363,128395,128397],{"className":128396},[629],[363,128398,128400,128420],{"className":128399},[633,634],[363,128401,128403,128417],{"className":128402},[638],[363,128404,128406],{"className":128405,"style":45301},[642],[363,128407,128408,128411],{"style":646},[363,128409],{"className":128410,"style":651},[650],[363,128412,128414],{"className":128413},[655,656,657,658],[363,128415,45126],{"className":128416,"style":45156},[407,408,658],[363,128418,666],{"className":128419},[665],[363,128421,128423],{"className":128422},[638],[363,128424,128426],{"className":128425,"style":8316},[642],[363,128427],{},[363,128429],{"className":128430,"style":506},[505],[363,128432,471],{"className":128433},[510],[363,128435],{"className":128436,"style":506},[505],[363,128438,128440,128443,128526,128532,128535,128538,128541],{"className":128439},[398],[363,128441],{"className":128442,"style":68952},[402],[363,128444,128446,128449,128523],{"className":128445},[407],[363,128447],{"className":128448},[494,789],[363,128450,128452],{"className":128451},[586],[363,128453,128455,128515],{"className":128454},[633,634],[363,128456,128458,128512],{"className":128457},[638],[363,128459,128461,128493,128501],{"className":128460,"style":23261},[642],[363,128462,128463,128466],{"style":805},[363,128464],{"className":128465,"style":809},[650],[363,128467,128469,128472,128475,128478,128481,128484,128487,128490],{"className":128468},[407],[363,128470,45552],{"className":128471},[407,408],[363,128473,461],{"className":128474},[494],[363,128476,465],{"className":128477},[407],[363,128479],{"className":128480,"style":2935},[505],[363,128482,5161],{"className":128483},[2939],[363,128485],{"className":128486,"style":2935},[505],[363,128488,45552],{"className":128489},[407,408],[363,128491,468],{"className":128492},[501],[363,128494,128495,128498],{"style":821},[363,128496],{"className":128497,"style":809},[650],[363,128499],{"className":128500,"style":829},[828],[363,128502,128503,128506],{"style":832},[363,128504],{"className":128505,"style":809},[650],[363,128507,128509],{"className":128508},[407],[363,128510,465],{"className":128511},[407],[363,128513,666],{"className":128514},[665],[363,128516,128518],{"className":128517},[638],[363,128519,128521],{"className":128520,"style":1949},[642],[363,128522],{},[363,128524],{"className":128525},[501,789],[363,128527,128529],{"className":128528},[407],[363,128530,461],{"className":128531},[767,4068],[363,128533,465],{"className":128534},[407],[363,128536],{"className":128537,"style":2935},[505],[363,128539,5161],{"className":128540},[2939],[363,128542],{"className":128543,"style":2935},[505],[363,128545,128547,128551,128729,128735,128738,128741,128744,128747,128750,128753],{"className":128546},[398],[363,128548],{"className":128549,"style":128550},[402],"height:1.9791em;vertical-align:-0.65em;",[363,128552,128554,128557],{"className":128553},[407],[363,128555,7915],{"className":128556},[407,408],[363,128558,128560],{"className":128559},[629],[363,128561,128563],{"className":128562},[633],[363,128564,128566],{"className":128565},[638],[363,128567,128570],{"className":128568,"style":128569},[642],"height:1.3291em;",[363,128571,128573,128576],{"style":128572},"top:-3.4534em;margin-right:0.05em;",[363,128574],{"className":128575,"style":809},[650],[363,128577,128579],{"className":128578},[655,656,657,658],[363,128580,128582,128585],{"className":128581},[407,658],[363,128583,5161],{"className":128584},[407,658],[363,128586,128588,128591,128726],{"className":128587},[407,658],[363,128589],{"className":128590},[494,789,655,30582,111714],[363,128592,128594],{"className":128593},[586],[363,128595,128597,128717],{"className":128596},[633,634],[363,128598,128600,128714],{"className":128599},[638],[363,128601,128604,128648,128656],{"className":128602,"style":128603},[642],"height:1.251em;",[363,128605,128607,128610],{"style":128606},"top:-2.5062em;",[363,128608],{"className":128609,"style":809},[650],[363,128611,128613],{"className":128612},[655,30582,10022,658],[363,128614,128616,128619],{"className":128615},[407,658],[363,128617,2811],{"className":128618},[407,658],[363,128620,128622,128625],{"className":128621},[407,658],[363,128623,2680],{"className":128624,"style":1328},[407,408,658],[363,128626,128628],{"className":128627},[629],[363,128629,128631],{"className":128630},[633],[363,128632,128634],{"className":128633},[638],[363,128635,128638],{"className":128636,"style":128637},[642],"height:0.9384em;",[363,128639,128641,128645],{"style":128640},"top:-2.9384em;margin-right:0.1em;",[363,128642],{"className":128643,"style":128644},[650],"height:2.6444em;",[363,128646,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4.6 节的局部展开一致（且此时与 ",[363,129053,129055,129068],{"className":129054,"translate":367},[366],[363,129056,129058],{"className":129057},[371],[373,129059,129060],{"xmlns":375},[377,129061,129062,129066],{},[380,129063,129064],{},[383,129065,45126],{},[387,129067,49462],{"encoding":389},[363,129069,129071],{"className":129070,"ariaHidden":394},[393],[363,129072,129074,129077],{"className":129073},[398],[363,129075],{"className":129076,"style":2693},[402],[363,129078,45126],{"className":129079,"style":45156},[407,408]," 无关）。",[2659,129082,129084],{"id":129083},"_92-伯努利分布","9.2 伯努利分布",[11,129086,881,129087,5993,129164,16145,129241,129328,129329,129431,129432,129460,129461,129431,129627,1338],{},[363,129088,129090,129122],{"className":129089,"translate":367},[366],[363,129091,129093],{"className":129092},[371],[373,129094,129095],{"xmlns":375},[377,129096,129097,129119],{},[380,129098,129099,129101,129103,129113,129115,129117],{},[383,129100,554],{},[458,129102,471],{},[380,129104,129105,129107,129109,129111],{},[383,129106,39663],{"mathvariant":557},[383,129108,7915],{"mathvariant":557},[383,129110,16800],{"mathvariant":557},[383,129112,82683],{"mathvariant":557},[458,129114,461],{"stretchy":460},[383,129116,11],{},[458,129118,468],{"stretchy":460},[387,129120,129121],{"encoding":389},"P=\\mathrm{Bern}(p)",[363,129123,129125,129143],{"className":129124,"ariaHidden":394},[393],[363,129126,129128,129131,129134,129137,129140],{"className":129127},[398],[363,129129],{"className":129130,"style":1011},[402],[363,129132,554],{"className":129133,"style":682},[407,408],[363,129135],{"className":129136,"style":506},[505],[363,129138,471],{"className":129139},[510],[363,129141],{"className":129142,"style":506},[505],[363,129144,129146,129149,129155,129158,129161],{"className":129145},[398],[363,129147],{"className":129148,"style":487},[402],[363,129150,129152],{"className":129151},[407],[363,129153,84695],{"className":129154},[407,33034],[363,129156,461],{"className":129157},[494],[363,129159,11],{"className":129160},[407,408],[363,129162,468],{"className":129163},[501],[363,129165,129167,129199],{"className":129166,"translate":367},[366],[363,129168,129170],{"className":129169},[371],[373,129171,129172],{"xmlns":375},[377,129173,129174,129196],{},[380,129175,129176,129178,129180,129190,129192,129194],{},[383,129177,561],{},[458,129179,471],{},[380,129181,129182,129184,129186,129188],{},[383,129183,39663],{"mathvariant":557},[383,129185,7915],{"mathvariant":557},[383,129187,16800],{"mathvariant":557},[383,129189,82683],{"mathvariant":557},[458,129191,461],{"stretchy":460},[383,129193,1291],{},[458,129195,468],{"stretchy":460},[387,129197,129198],{"encoding":389},"Q=\\mathrm{Bern}(q)",[363,129200,129202,129220],{"className":129201,"ariaHidden":394},[393],[363,129203,129205,129208,129211,129214,129217],{"className":129204},[398],[363,129206],{"className":129207,"style":913},[402],[363,129209,561],{"className":129210},[407,408],[363,129212],{"className":129213,"style":506},[505],[363,129215,471],{"className":129216},[510],[363,129218],{"className":129219,"style":506},[505],[363,129221,129223,129226,129232,129235,129238],{"className":129222},[398],[363,129224],{"className":129225,"style":487},[402],[363,129227,129229],{"className":129228},[407],[363,129230,84695],{"className":129231},[407,33034],[363,129233,461],{"className":129234},[494],[363,129236,1291],{"className":129237,"style":1328},[407,408],[363,129239,468],{"className":129240},[501],[363,129242,129244,129274],{"className":129243,"translate":367},[366],[363,129245,129247],{"className":129246},[371],[373,129248,129249],{"xmlns":375},[377,129250,129251,129271],{},[380,129252,129253,129255,129257,129259,129261,129263,129265,129267,129269],{},[383,129254,11],{},[458,129256,898],{"separator":394},[383,129258,1291],{},[458,129260,6257],{},[458,129262,461],{"stretchy":460},[463,129264,474],{},[458,129266,898],{"separator":394},[463,129268,465],{},[458,129270,468],{"stretchy":460},[387,129272,129273],{"encoding":389},"p,q\\in(0,1)",[363,129275,129277,129304],{"className":129276,"ariaHidden":394},[393],[363,129278,129280,129283,129286,129289,129292,129295,129298,129301],{"className":129279},[398],[363,129281],{"className":129282,"style":32064},[402],[363,129284,11],{"className":129285},[407,408],[363,129287,898],{"className":129288},[920],[363,129290],{"className":129291,"style":754},[505],[363,129293,1291],{"className":129294,"style":1328},[407,408],[363,129296],{"className":129297,"style":506},[505],[363,129299,6257],{"className":129300},[510],[363,129302],{"className":129303,"style":506},[505],[363,129305,129307,129310,129313,129316,129319,129322,129325],{"className":129306},[398],[363,129308],{"className":129309,"style":487},[402],[363,129311,461],{"className":129312},[494],[363,129314,474],{"className":129315},[407],[363,129317,898],{"className":129318},[920],[363,129320],{"className":129321,"style":754},[505],[363,129323,465],{"className":129324},[407],[363,129326,468],{"className":129327},[501],"），似然比取值 ",[363,129330,129332,129358],{"className":129331,"translate":367},[366],[363,129333,129335],{"className":129334},[371],[373,129336,129337],{"xmlns":375},[377,129338,129339,129355],{},[380,129340,129341,129347,129349,129351,129353],{},[543,129342,129343,129345],{},[383,129344,1116],{},[463,129346,465],{},[458,129348,471],{},[383,129350,11],{},[383,129352,1288],{"mathvariant":557},[383,129354,1291],{},[387,129356,129357],{"encoding":389},"t_1=p\u002Fq",[363,129359,129361,129416],{"className":129360,"ariaHidden":394},[393],[363,129362,129364,129367,129407,129410,129413],{"className":129363},[398],[363,129365],{"className":129366,"style":96314},[402],[363,129368,129370,129373],{"className":129369},[407],[363,129371,1116],{"className":129372},[407,408],[363,129374,129376],{"className":129375},[629],[363,129377,129379,129399],{"className":129378},[633,634],[363,129380,129382,129396],{"className":129381},[638],[363,129383,129385],{"className":129384,"style":11483},[642],[363,129386,129387,129390],{"style":730},[363,129388],{"className":129389,"style":651},[650],[363,129391,129393],{"className":129392},[655,656,657,658],[363,129394,465],{"className":129395},[407,658],[363,129397,666],{"className":129398},[665],[363,129400,129402],{"className":129401},[638],[363,129403,129405],{"className":129404,"style":8316},[642],[363,129406],{},[363,129408],{"className":129409,"style":506},[505],[363,129411,471],{"className":129412},[510],[363,129414],{"className":129415,"style":506},[505],[363,129417,129419,129422,129425,129428],{"className":129418},[398],[363,129420],{"className":129421,"style":487},[402],[363,129423,11],{"className":129424},[407,408],[363,129426,1288],{"className":129427},[407],[363,129429,1291],{"className":129430,"style":1328},[407,408],"（概率 ",[363,129433,129435,129448],{"className":129434,"translate":367},[366],[363,129436,129438],{"className":129437},[371],[373,129439,129440],{"xmlns":375},[377,129441,129442,129446],{},[380,129443,129444],{},[383,129445,1291],{},[387,129447,1291],{"encoding":389},[363,129449,129451],{"className":129450,"ariaHidden":394},[393],[363,129452,129454,129457],{"className":129453},[398],[363,129455],{"className":129456,"style":2624},[402],[363,129458,1291],{"className":129459,"style":1328},[407,408],"）与 ",[363,129462,129464,129506],{"className":129463,"translate":367},[366],[363,129465,129467],{"className":129466},[371],[373,129468,129469],{"xmlns":375},[377,129470,129471,129503],{},[380,129472,129473,129479,129481,129483,129485,129487,129489,129491,129493,129495,129497,129499,129501],{},[543,129474,129475,129477],{},[383,129476,1116],{},[463,129478,474],{},[458,129480,471],{},[458,129482,461],{"stretchy":460},[463,129484,465],{},[458,129486,5161],{},[383,129488,11],{},[458,129490,468],{"stretchy":460},[383,129492,1288],{"mathvariant":557},[458,129494,461],{"stretchy":460},[463,129496,465],{},[458,129498,5161],{},[383,129500,1291],{},[458,129502,468],{"stretchy":460},[387,129504,129505],{"encoding":389},"t_0=(1-p)\u002F(1-q)",[363,129507,129509,129564,129585,129615],{"className":129508,"ariaHidden":394},[393],[363,129510,129512,129515,129555,129558,129561],{"className":129511},[398],[363,129513],{"className":129514,"style":96314},[402],[363,129516,129518,129521],{"className":129517},[407],[363,129519,1116],{"className":129520},[407,408],[363,129522,129524],{"className":129523},[629],[363,129525,129527,129547],{"className":129526},[633,634],[363,129528,129530,129544],{"className":129529},[638],[363,129531,129533],{"className":129532,"style":11483},[642],[363,129534,129535,129538],{"style":730},[363,129536],{"className":129537,"style":651},[650],[363,129539,129541],{"className":129540},[655,656,657,658],[363,129542,474],{"className":129543},[407,658],[363,129545,666],{"className":129546},[665],[363,129548,129550],{"className":129549},[638],[363,129551,129553],{"className":129552,"style":8316},[642],[363,129554],{},[363,129556],{"className":129557,"style":506},[505],[363,129559,471],{"className":129560},[510],[363,129562],{"className":129563,"style":506},[505],[363,129565,129567,129570,129573,129576,129579,129582],{"className":129566},[398],[363,129568],{"className":129569,"style":487},[402],[363,129571,461],{"className":129572},[494],[363,129574,465],{"className":129575},[407],[363,129577],{"className":129578,"style":2935},[505],[363,129580,5161],{"className":129581},[2939],[363,129583],{"className":129584,"style":2935},[505],[363,129586,129588,129591,129594,129597,129600,129603,129606,129609,129612],{"className":129587},[398],[363,129589],{"className":129590,"style":487},[402],[363,129592,11],{"className":129593},[407,408],[363,129595,468],{"className":129596},[501],[363,129598,1288],{"className":129599},[407],[363,129601,461],{"className":129602},[494],[363,129604,465],{"className":129605},[407],[363,129607],{"className":129608,"style":2935},[505],[363,129610,5161],{"className":129611},[2939],[363,129613],{"className":129614,"style":2935},[505],[363,129616,129618,129621,129624],{"className":129617},[398],[363,129619],{"className":129620,"style":487},[402],[363,129622,1291],{"className":129623,"style":1328},[407,408],[363,129625,468],{"className":129626},[501],[363,129628,129630,129648],{"className":129629,"translate":367},[366],[363,129631,129633],{"className":129632},[371],[373,129634,129635],{"xmlns":375},[377,129636,129637,129645],{},[380,129638,129639,129641,129643],{},[463,129640,465],{},[458,129642,5161],{},[383,129644,1291],{},[387,129646,129647],{"encoding":389},"1-q",[363,129649,129651,129669],{"className":129650,"ariaHidden":394},[393],[363,129652,129654,129657,129660,129663,129666],{"className":129653},[398],[363,129655],{"className":129656,"style":12928},[402],[363,129658,465],{"className":129659},[407],[363,129661],{"className":129662,"style":2935},[505],[363,129664,5161],{"className":129665},[2939],[363,129667],{"className":129668,"style":2935},[505],[363,129670,129672,129675],{"className":129671},[398],[363,129673],{"className":129674,"style":2624},[402],[363,129676,1291],{"className":129677,"style":1328},[407,408],[363,129679,129681],{"className":129680,"translate":367},[528],[363,129682,129684,129766],{"className":129683,"translate":367},[366],[363,129685,129687],{"className":129686},[371],[373,129688,129689],{"xmlns":375,"display":537},[377,129690,129691,129763],{},[380,129692,129693,129703,129705,129707,129709,129711,129713,129715,129717,129719,129721,129727,129729,129731,129733,129735,129737,129739,129741,129743,129761],{},[543,129694,129695,129697],{},[383,129696,547],{},[380,129698,129699,129701],{},[383,129700,11044],{},[383,129702,22265],{},[458,129704,461],{"stretchy":460},[383,129706,554],{},[383,129708,558],{"mathvariant":557},[383,129710,561],{},[458,129712,468],{"stretchy":460},[458,129714,471],{},[383,129716,11],{},[383,129718,4448],{},[458,129720,4451],{},[586,129722,129723,129725],{},[383,129724,11],{},[383,129726,1291],{},[458,129728,2818],{},[458,129730,461],{"stretchy":460},[463,129732,465],{},[458,129734,5161],{},[383,129736,11],{},[458,129738,468],{"stretchy":460},[383,129740,4448],{},[458,129742,4451],{},[586,129744,129745,129753],{},[380,129746,129747,129749,129751],{},[463,129748,465],{},[458,129750,5161],{},[383,129752,11],{},[380,129754,129755,129757,129759],{},[46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",[363,130841,130843,130865],{"className":130842,"translate":367},[366],[363,130844,130846],{"className":130845},[371],[373,130847,130848],{"xmlns":375},[377,130849,130850,130862],{},[380,130851,130852],{},[543,130853,130854,130856],{},[383,130855,547],{},[380,130857,130858,130860],{},[383,130859,11044],{},[383,130861,22265],{},[387,130863,130864],{"encoding":389},"D_{KL}",[363,130866,130868],{"className":130867,"ariaHidden":394},[393],[363,130869,130871,130874],{"className":130870},[398],[363,130872],{"className":130873,"style":16311},[402],[363,130875,130877,130880],{"className":130876},[407],[363,130878,547],{"className":130879,"style":625},[407,408],[363,130881,130883],{"className":130882},[629],[363,130884,130886,130912],{"className":130885},[633,634],[363,130887,130889,130909],{"className":130888},[638],[363,130890,130892],{"className":130891,"style":727},[642],[363,130893,130894,130897],{"style":646},[363,130895],{"className":130896,"style":651},[650],[363,130898,130900],{"className":130899},[655,656,657,658],[363,130901,130903,130906],{"className":130902},[407,658],[363,130904,11044],{"className":130905,"style":11132},[407,408,658],[363,130907,22265],{"className":130908},[407,408,658],[363,130910,666],{"className":130911},[665],[363,130913,130915],{"className":130914},[638],[363,130916,130918],{"className":130917,"style":8316},[642],[363,130919],{},[363,130921,130923,130936],{"className":130922,"translate":367},[366],[363,130924,130926],{"className":130925},[371],[373,130927,130928],{"xmlns":375},[377,130929,130930,130934],{},[380,130931,130932],{},[383,130933,11],{},[387,130935,11],{"encoding":389},[363,130937,130939],{"className":130938,"ariaHidden":394},[393],[363,130940,130942,130945],{"className":130941},[398],[363,130943],{"className":130944,"style":2624},[402],[363,130946,11],{"className":130947},[407,408],[363,130949,130951,130969],{"className":130950,"translate":367},[366],[363,130952,130954],{"className":130953},[371],[373,130955,130956],{"xmlns":375},[377,130957,130958,130966],{},[380,130959,130960,130962,130964],{},[383,130961,11],{},[458,130963,471],{},[383,130965,1291],{},[387,130967,130968],{"encoding":389},"p=q",[363,130970,130972,130990],{"className":130971,"ariaHidden":394},[393],[363,130973,130975,130978,130981,130984,130987],{"className":130974},[398],[363,130976],{"className":130977,"style":2624},[402],[363,130979,11],{"className":130980},[407,408],[363,130982],{"className":130983,"style":506},[505],[363,130985,471],{"className":130986},[510],[363,130988],{"className":130989,"style":506},[505],[363,130991,130993,130996],{"className":130992},[398],[363,130994],{"className":130995,"style":2624},[402],[363,130997,1291],{"className":130998,"style":1328},[407,408]," 处做二阶展开，应回到 ",[363,131001,131003,131063],{"className":131002,"translate":367},[366],[363,131004,131006],{"className":131005},[371],[373,131007,131008],{"xmlns":375},[377,131009,131010,131060],{},[380,131011,131012,131046,131048,131054],{},[586,131013,131014,131030],{},[380,131015,131016,131018,131020,131022,131024],{},[458,131017,461],{"stretchy":460},[383,131019,11],{},[458,131021,5161],{},[383,131023,1291],{},[7162,131025,131026,131028],{},[458,131027,468],{"stretchy":460},[463,131029,2811],{},[380,131031,131032,131034,131036,131038,131040,131042,131044],{},[463,131033,2811],{},[383,131035,1291],{},[458,131037,461],{"stretchy":460},[463,131039,465],{},[458,131041,5161],{},[383,131043,1291],{},[458,131045,468],{"stretchy":460},[458,131047,471],{},[586,131049,131050,131052],{},[463,131051,465],{},[463,131053,2811],{},[7162,131055,131056,131058],{},[383,131057,37821],{},[463,131059,2811],{},[387,131061,131062],{"encoding":389},"\\frac{(p-q)^2}{2q(1-q)}=\\frac12\\chi^2",[363,131064,131066,131206],{"className":131065,"ariaHidden":394},[393],[363,131067,131069,131073,131197,131200,131203],{"className":131068},[398],[363,131070],{"className":131071,"style":131072},[402],"height:1.6289em;vertical-align:-0.52em;",[363,131074,131076,131079,131194],{"className":131075},[407],[363,131077],{"className":131078},[494,789],[363,131080,131082],{"className":131081},[586],[363,131083,131085,131186],{"className":131084},[633,634],[363,131086,131088,131183],{"className":131087},[638],[363,131089,131091,131123,131131],{"className":131090,"style":38820},[642],[363,131092,131093,131096],{"style":2875},[363,131094],{"className":131095,"style":809},[650],[363,131097,131099],{"className":131098},[655,656,657,658],[363,131100,131102,131105,131108,131111,131114,131117,131120],{"className":131101},[407,658],[363,131103,2811],{"className":131104},[407,658],[363,131106,1291],{"className":131107,"style":1328},[407,408,658],[363,131109,461],{"className":131110},[494,658],[363,131112,465],{"className":131113},[407,658],[363,131115,5161],{"className":131116},[2939,658],[363,131118,1291],{"className":131119,"style":1328},[407,408,658],[363,131121,468],{"className":131122},[501,658],[363,131124,131125,131128],{"style":821},[363,131126],{"className":131127,"style":809},[650],[363,131129],{"className":131130,"style":829},[828],[363,131132,131133,131136],{"style":26790},[363,131134],{"className":131135,"style":809},[650],[363,131137,131139],{"className":131138},[655,656,657,658],[363,131140,131142,131145,131148,131151,131154],{"className":131141},[407,658],[363,131143,461],{"className":131144},[494,658],[363,131146,11],{"className":131147},[407,408,658],[363,131149,5161],{"className":131150},[2939,658],[363,131152,1291],{"className":131153,"style":1328},[407,408,658],[363,131155,131157,131160],{"className":131156},[501,658],[363,131158,468],{"className":131159},[501,658],[363,131161,131163],{"className":131162},[629],[363,131164,131166],{"className":131165},[633],[363,131167,131169],{"className":131168},[638],[363,131170,131172],{"className":131171,"style":38884},[642],[363,131173,131174,131177],{"style":38887},[363,131175],{"className":131176,"style":30578},[650],[363,131178,131180],{"className":131179},[655,30582,10022,658],[363,131181,2811],{"className":131182},[407,658],[363,131184,666],{"className":131185},[665],[363,131187,131189],{"className":131188},[638],[363,131190,131192],{"className":131191,"style":45974},[642],[363,131193],{},[363,131195],{"className":131196},[501,789],[363,131198],{"className":131199,"style":506},[505],[363,131201,471],{"className":131202},[510],[363,131204],{"className":131205,"style":506},[505],[363,131207,131209,131212,131280],{"className":131208},[398],[363,131210],{"className":131211,"style":2853},[402],[363,131213,131215,131218,131277],{"className":131214},[407],[363,131216],{"className":131217},[494,789],[363,131219,131221],{"className":131220},[586],[363,131222,131224,131269],{"className":131223},[633,634],[363,131225,131227,131266],{"className":131226},[638],[363,131228,131230,131244,131252],{"className":131229,"style":2872},[642],[363,131231,131232,131235],{"style":2875},[363,131233],{"className":131234,"style":809},[650],[363,131236,131238],{"className":131237},[655,656,657,658],[363,131239,131241],{"className":131240},[407,658],[363,131242,2811],{"className":131243},[407,658],[363,131245,131246,131249],{"style":821},[363,131247],{"className":131248,"style":809},[650],[363,131250],{"className":131251,"style":829},[828],[363,131253,131254,131257],{"style":2898},[363,131255],{"className":131256,"style":809},[650],[363,131258,131260],{"className":131259},[655,656,657,658],[363,131261,131263],{"className":131262},[407,658],[363,131264,465],{"className":131265},[407,658],[363,131267,666],{"className":131268},[665],[363,131270,131272],{"className":131271},[638],[363,131273,131275],{"className":131274,"style":2920},[642],[363,131276],{},[363,131278],{"className":131279},[501,789],[363,131281,131283,131286],{"className":131282},[407],[363,131284,37821],{"className":131285},[407,408],[363,131287,131289],{"className":131288},[629],[363,131290,131292],{"className":131291},[633],[363,131293,131295],{"className":131294},[638],[363,131296,131298],{"className":131297,"style":20400},[642],[363,131299,131300,131303],{"style":7213},[363,131301],{"className":131302,"style":651},[650],[363,131304,131306],{"className":131305},[655,656,657,658],[363,131307,2811],{"className":131308},[407,658],"，与 4.6 节吻合。",[2659,131311,131313],{"id":131312},"_93-用对偶表示反推-kl","9.3 用对偶表示反推 KL",[11,131315,131316,131317,5993,131402,131543,131544,9748],{},"取 ",[363,131318,131320,131349],{"className":131319,"translate":367},[366],[363,131321,131323],{"className":131322},[371],[373,131324,131325],{"xmlns":375},[377,131326,131327,131347],{},[380,131328,131329,131331,131333,131335,131337,131339,131341,131343,131345],{},[383,131330,385],{},[458,131332,461],{"stretchy":460},[383,131334,1116],{},[458,131336,468],{"stretchy":460},[458,131338,471],{},[383,131340,1116],{},[383,131342,4448],{},[458,131344,4451],{},[383,131346,1116],{},[387,131348,4456],{"encoding":389},[363,131350,131352,131379],{"className":131351,"ariaHidden":394},[393],[363,131353,131355,131358,131361,131364,131367,131370,131373,131376],{"className":131354},[398],[363,131356],{"className":131357,"style":487},[402],[363,131359,385],{"className":131360,"style":409},[407,408],[363,131362,461],{"className":131363},[494],[363,131365,1116],{"className":131366},[407,408],[363,131368,468],{"className":131369},[501],[363,131371],{"className":131372,"style":506},[505],[363,131374,471],{"className":131375},[510],[363,131377],{"className":131378,"style":506},[505],[363,131380,131382,131385,131388,131391,131396,131399],{"className":131381},[398],[363,131383],{"className":131384,"style":403},[402],[363,131386,1116],{"className":131387},[407,408],[363,131389],{"className":131390,"style":754},[505],[363,131392,4502,131394],{"className":131393},[1832],[363,131395,4506],{"style":4505},[363,131397],{"className":131398,"style":754},[505],[363,131400,1116],{"className":131401},[407,408],[363,131403,131405,131443],{"className":131404,"translate":367},[366],[363,131406,131408],{"className":131407},[371],[373,131409,131410],{"xmlns":375},[377,131411,131412,131440],{},[380,131413,131414,131420,131422,131424,131426,131428],{},[7162,131415,131416,131418],{},[383,131417,385],{},[458,131419,7921],{},[458,131421,461],{"stretchy":460},[383,131423,40609],{},[458,131425,468],{"stretchy":460},[458,131427,471],{},[7162,131429,131430,131432],{},[383,131431,7915],{},[380,131433,131434,131436,131438],{},[383,131435,40609],{},[458,131437,5161],{},[463,131439,465],{},[387,131441,131442],{"encoding":389},"f^*(y)=e^{y-1}",[363,131444,131446,131499],{"className":131445,"ariaHidden":394},[393],[363,131447,131449,131452,131481,131484,131487,131490,131493,131496],{"className":131448},[398],[363,131450],{"className":131451,"style":487},[402],[363,131453,131455,131458],{"className":131454},[407],[363,131456,385],{"className":131457,"style":409},[407,408],[363,131459,131461],{"className":131460},[629],[363,131462,131464],{"className":131463},[633],[363,131465,131467],{"className":131466},[638],[363,131468,131470],{"className":131469,"style":8847},[642],[363,131471,131472,131475],{"style":7213},[363,131473],{"className":131474,"style":651},[650],[363,131476,131478],{"className":131477},[655,656,657,658],[363,131479,7921],{"className":131480},[2939,658],[363,131482,461],{"className":131483},[494],[363,131485,40609],{"className":131486,"style":1328},[407,408],[363,131488,468],{"className":131489},[501],[363,131491],{"className":131492,"style":506},[505],[363,131494,471],{"className":131495},[510],[363,131497],{"className":131498,"style":506},[505],[363,131500,131502,131505],{"className":131501},[398],[363,131503],{"className":131504,"style":20400},[402],[363,131506,131508,131511],{"className":131507},[407],[363,131509,7915],{"className":131510},[407,408],[363,131512,131514],{"className":131513},[629],[363,131515,131517],{"className":131516},[633],[363,131518,131520],{"className":131519},[638],[363,131521,131523],{"className":131522,"style":20400},[642],[363,131524,131525,131528],{"style":7213},[363,131526],{"className":131527,"style":651},[650],[363,131529,131531],{"className":131530},[655,656,657,658],[363,131532,131534,131537,131540],{"className":131533},[407,658],[363,131535,40609],{"className":131536,"style":1328},[407,408,658],[363,131538,5161],{"className":131539},[2939,658],[363,131541,465],{"className":131542},[407,658],"，则对任意 ",[363,131545,131547,131560],{"className":131546,"translate":367},[366],[363,131548,131550],{"className":131549},[371],[373,131551,131552],{"xmlns":375},[377,131553,131554,131558],{},[380,131555,131556],{},[383,131557,4506],{},[387,131559,4506],{"encoding":389},[363,131561,131563],{"className":131562,"ariaHidden":394},[393],[363,131564,131566,131569],{"className":131565},[398],[363,131567],{"className":131568,"style":2624},[402],[363,131570,4506],{"className":131571,"style":1328},[407,408],[363,131573,131575],{"className":131574,"translate":367},[528],[363,131576,131578,131650],{"className":131577,"translate":367},[366],[363,131579,131581],{"className":131580},[371],[373,131582,131583],{"xmlns":375,"display":537},[377,131584,131585,131647],{},[380,131586,131587,131597,131599,131601,131603,131605,131607,131609,131615,131617,131619,131621,131623,131629,131631,131643,131645],{},[543,131588,131589,131591],{},[383,131590,547],{},[380,131592,131593,131595],{},[383,131594,11044],{},[383,131596,22265],{},[458,131598,461],{"stretchy":460},[383,131600,554],{},[383,131602,558],{"mathvariant":557},[383,131604,561],{},[458,131606,468],{"stretchy":460},[458,131608,7276],{},[543,131610,131611,131613],{},[383,131612,571],{"mathvariant":570},[383,131614,554],{},[458,131616,578],{"stretchy":460},[383,131618,4506],{},[458,131620,605],{"stretchy":460},[458,131622,5161],{},[543,131624,131625,131627],{},[383,131626,571],{"mathvariant":570},[383,131628,561],{},[458,131630,578],{"stretchy":460},[7162,131632,131633,131635],{},[383,131634,7915],{},[380,131636,131637,131639,131641],{},[383,131638,4506],{},[458,131640,5161],{},[463,131642,465],{},[458,131644,605],{"stretchy":460},[458,131646,898],{"separator":394},[387,131648,131649],{"encoding":389},"D_{KL}(P\\|Q)\\ge \\mathbb E_P[g]\n-\\mathbb E_Q[e^{g-1}],",[363,131651,131653,131729,131793],{"className":131652,"ariaHidden":394},[393],[363,131654,131656,131659,131705,131708,131711,131714,131717,131720,131723,131726],{"className":131655},[398],[363,131657],{"className":131658,"style":487},[402],[363,131660,131662,131665],{"className":131661},[407],[363,131663,547],{"className":131664,"style":625},[407,408],[363,131666,131668],{"className":131667},[629],[363,131669,131671,131697],{"className":131670},[633,634],[363,131672,131674,131694],{"className":131673},[638],[363,131675,131677],{"className":131676,"style":727},[642],[363,131678,131679,131682],{"style":646},[363,131680],{"className":131681,"style":651},[650],[363,131683,131685],{"className":131684},[655,656,657,658],[363,131686,131688,131691],{"className":131687},[407,658],[363,131689,11044],{"className":131690,"style":11132},[407,408,658],[363,131692,22265],{"className":131693},[407,408,658],[363,131695,666],{"className":131696},[665],[363,131698,131700],{"className":131699},[638],[363,131701,131703],{"className":131702,"style":8316},[642],[363,131704],{},[363,131706,461],{"className":131707},[494],[363,131709,554],{"className":131710,"style":682},[407,408],[363,131712,558],{"className":131713},[407],[363,131715,561],{"className":131716},[407,408],[363,131718,468],{"className":131719},[501],[363,131721],{"className":131722,"style":506},[505],[363,131724,7276],{"className":131725},[510],[363,131727],{"className":131728,"style":506},[505],[363,131730,131732,131735,131775,131778,131781,131784,131787,131790],{"className":131731},[398],[363,131733],{"className":131734,"style":487},[402],[363,131736,131738,131741],{"className":131737},[407],[363,131739,571],{"className":131740},[407,714],[363,131742,131744],{"className":131743},[629],[363,131745,131747,131767],{"className":131746},[633,634],[363,131748,131750,131764],{"className":131749},[638],[363,131751,131753],{"className":131752,"style":727},[642],[363,131754,131755,131758],{"style":730},[363,131756],{"className":131757,"style":651},[650],[363,131759,131761],{"className":131760},[655,656,657,658],[363,131762,554],{"className":131763,"style":682},[407,408,658],[363,131765,666],{"className":131766},[665],[363,131768,131770],{"className":131769},[638],[363,131771,131773],{"className":131772,"style":8316},[642],[363,131774],{},[363,131776,578],{"className":131777},[494],[363,131779,4506],{"className":131780,"style":1328},[407,408],[363,131782,605],{"className":131783},[501],[363,131785],{"className":131786,"style":2935},[505],[363,131788,5161],{"className":131789},[2939],[363,131791],{"className":131792,"style":2935},[505],[363,131794,131796,131800,131840,131843,131881,131884],{"className":131795},[398],[363,131797],{"className":131798,"style":131799},[402],"height:1.1502em;vertical-align:-0.2861em;",[363,131801,131803,131806],{"className":131802},[407],[363,131804,571],{"className":131805},[407,714],[363,131807,131809],{"className":131808},[629],[363,131810,131812,131832],{"className":131811},[633,634],[363,131813,131815,131829],{"className":131814},[638],[363,131816,131818],{"className":131817,"style":727},[642],[363,131819,131820,131823],{"style":730},[363,131821],{"className":131822,"style":651},[650],[363,131824,131826],{"className":131825},[655,656,657,658],[363,131827,561],{"className":131828},[407,408,658],[363,131830,666],{"className":131831},[665],[363,131833,131835],{"className":131834},[638],[363,131836,131838],{"className":131837,"style":673},[642],[363,131839],{},[363,131841,578],{"className":131842},[494],[363,131844,131846,131849],{"className":131845},[407],[363,131847,7915],{"className":131848},[407,408],[363,131850,131852],{"className":131851},[629],[363,131853,131855],{"className":131854},[633],[363,131856,131858],{"className":131857},[638],[363,131859,131861],{"className":131860,"style":20107},[642],[363,131862,131863,131866],{"style":8625},[363,131864],{"className":131865,"style":651},[650],[363,131867,131869],{"className":131868},[655,656,657,658],[363,131870,131872,131875,131878],{"className":131871},[407,658],[363,131873,4506],{"className":131874,"style":1328},[407,408,658],[363,131876,5161],{"className":131877},[2939,658],[363,131879,465],{"className":131880},[407,658],[363,131882,605],{"className":131883},[501],[363,131885,898],{"className":131886},[920],[11,131888,131889,131890,131992],{},"等号在 ",[363,131891,131893,131927],{"className":131892,"translate":367},[366],[363,131894,131896],{"className":131895},[371],[373,131897,131898],{"xmlns":375},[377,131899,131900,131924],{},[380,131901,131902,131904,131906,131908,131910,131912,131914,131916,131918,131920,131922],{},[383,131903,4506],{},[458,131905,471],{},[383,131907,4448],{},[458,131909,4451],{},[458,131911,461],{"stretchy":460},[383,131913,11],{},[383,131915,1288],{"mathvariant":557},[383,131917,1291],{},[458,131919,468],{"stretchy":460},[458,131921,2818],{},[463,131923,465],{},[387,131925,131926],{"encoding":389},"g=\\log(p\u002Fq)+1",[363,131928,131930,131948,131983],{"className":131929,"ariaHidden":394},[393],[363,131931,131933,131936,131939,131942,131945],{"className":131932},[398],[363,131934],{"className":131935,"style":2624},[402],[363,131937,4506],{"className":131938,"style":1328},[407,408],[363,131940],{"className":131941,"style":506},[505],[363,131943,471],{"className":131944},[510],[363,131946],{"className":131947,"style":506},[505],[363,131949,131951,131954,131959,131962,131965,131968,131971,131974,131977,131980],{"className":131950},[398],[363,131952],{"className":131953,"style":487},[402],[363,131955,4502,131957],{"className":131956},[1832],[363,131958,4506],{"style":4505},[363,131960,461],{"className":131961},[494],[363,131963,11],{"className":131964},[407,408],[363,131966,1288],{"className":131967},[407],[363,131969,1291],{"className":131970,"style":1328},[407,408],[363,131972,468],{"className":131973},[501],[363,131975],{"className":131976,"style":2935},[505],[363,131978,2818],{"className":131979},[2939],[363,131981],{"className":131982,"style":2935},[505],[363,131984,131986,131989],{"className":131985},[398],[363,131987],{"className":131988,"style":520},[402],[363,131990,465],{"className":131991},[407]," 处取到。代回：",[363,131994,131996],{"className":131995,"translate":367},[528],[363,131997,131999,132125],{"className":131998,"translate":367},[366],[363,132000,132002],{"className":132001},[371],[373,132003,132004],{"xmlns":375,"display":537},[377,132005,132006,132122],{},[380,132007,132008,132014,132016,132018,132020,132022,132024,132026,132028,132030,132032,132034,132036,132038,132044,132046,132066,132068,132070,132080,132082,132084,132086,132088,132090,132092,132094,132096,132098,132100,132110,132112,132114,132116,132118,132120],{},[543,132009,132010,132012],{},[383,132011,571],{"mathvariant":570},[383,132013,554],{},[458,132015,578],{"stretchy":460},[383,132017,4448],{},[458,132019,4451],{},[458,132021,461],{"stretchy":460},[383,132023,11],{},[383,132025,1288],{"mathvariant":557},[383,132027,1291],{},[458,132029,468],{"stretchy":460},[458,132031,2818],{},[463,132033,465],{},[458,132035,605],{"stretchy":460},[458,132037,5161],{},[543,132039,132040,132042],{},[383,132041,571],{"mathvariant":570},[383,132043,561],{},[458,132045,578],{"stretchy":460},[7162,132047,132048,132050],{},[383,132049,7915],{},[380,132051,132052,132054,132056,132058,132060,132062,132064],{},[383,132053,4448],{},[458,132055,4451],{},[458,132057,461],{"stretchy":460},[383,132059,11],{},[383,132061,1288],{"mathvariant":557},[383,132063,1291],{},[458,132065,468],{"stretchy":460},[458,132067,605],{"stretchy":460},[458,132069,471],{},[543,132071,132072,132074],{},[383,132073,547],{},[380,132075,132076,132078],{},[383,132077,11044],{},[383,132079,22265],{},[458,132081,461],{"stretchy":460},[383,132083,554],{},[383,132085,558],{"mathvariant":557},[383,132087,561],{},[458,132089,468],{"stretchy":460},[458,132091,2818],{},[463,132093,465],{},[458,132095,5161],{},[463,132097,465],{},[458,132099,471],{},[543,132101,132102,132104],{},[383,132103,547],{},[380,132105,132106,132108],{},[383,132107,11044],{},[383,132109,22265],{},[458,132111,461],{"stretchy":460},[383,132113,554],{},[383,132115,558],{"mathvariant":557},[383,132117,561],{},[458,132119,468],{"stretchy":460},[458,132121,898],{"separator":394},[387,132123,132124],{"encoding":389},"\\mathbb E_P[\\log(p\u002Fq)+1]\n-\\mathbb E_Q[e^{\\log(p\u002Fq)}]\n=D_{KL}(P\\|Q)+1-1=D_{KL}(P\\|Q),",[363,132126,132128,132206,132227,132345,132421,132439,132457],{"className":132127,"ariaHidden":394},[393],[363,132129,132131,132134,132174,132177,132182,132185,132188,132191,132194,132197,132200,132203],{"className":132130},[398],[363,132132],{"className":132133,"style":487},[402],[363,132135,132137,132140],{"className":132136},[407],[363,132138,571],{"className":132139},[407,714],[363,132141,132143],{"className":132142},[629],[363,132144,132146,132166],{"className":132145},[633,634],[363,132147,132149,132163],{"className":132148},[638],[363,132150,132152],{"className":132151,"style":727},[642],[363,132153,132154,132157],{"style":730},[363,132155],{"className":132156,"style":651},[650],[363,132158,132160],{"className":132159},[655,656,657,658],[363,132161,554],{"className":132162,"style":682},[407,408,658],[363,132164,666],{"className":132165},[665],[363,132167,132169],{"className":132168},[638],[363,132170,132172],{"className":132171,"style":8316},[642],[363,132173],{},[363,132175,578],{"className":132176},[494],[363,132178,4502,132180],{"className":132179},[1832],[363,132181,4506],{"style":4505},[363,132183,461],{"className":132184},[494],[363,132186,11],{"className":132187},[407,408],[363,132189,1288],{"className":132190},[407],[363,132192,1291],{"className":132193,"style":1328},[407,408],[363,132195,468],{"className":132196},[501],[363,132198],{"className":132199,"style":2935},[505],[363,132201,2818],{"className":132202},[2939],[363,132204],{"className":132205,"style":2935},[505],[363,132207,132209,132212,132215,132218,132221,132224],{"className":132208},[398],[363,132210],{"className":132211,"style":487},[402],[363,132213,465],{"className":132214},[407],[363,132216,605],{"className":132217},[501],[363,132219],{"className":132220,"style":2935},[505],[363,132222,5161],{"className":132223},[2939],[363,132225],{"className":132226,"style":2935},[505],[363,132228,132230,132234,132274,132277,132333,132336,132339,132342],{"className":132229},[398],[363,132231],{"className":132232,"style":132233},[402],"height:1.2241em;vertical-align:-0.2861em;",[363,132235,132237,132240],{"className":132236},[407],[363,132238,571],{"className":132239},[407,714],[363,132241,132243],{"className":132242},[629],[363,132244,132246,132266],{"className":132245},[633,634],[363,132247,132249,132263],{"className":132248},[638],[363,132250,132252],{"className":132251,"style":727},[642],[363,132253,132254,132257],{"style":730},[363,132255],{"className":132256,"style":651},[650],[363,132258,132260],{"className":132259},[655,656,657,658],[363,132261,561],{"className":132262},[407,408,658],[363,132264,666],{"className":132265},[665],[363,132267,132269],{"className":132268},[638],[363,132270,132272],{"className":132271,"style":673},[642],[363,132273],{},[363,132275,578],{"className":132276},[494],[363,132278,132280,132283],{"className":132279},[407],[363,132281,7915],{"className":132282},[407,408],[363,132284,132286],{"className":132285},[629],[363,132287,132289],{"className":132288},[633],[363,132290,132292],{"className":132291},[638],[363,132293,132295],{"className":132294,"style":45492},[642],[363,132296,132297,132300],{"style":8625},[363,132298],{"className":132299,"style":651},[650],[363,132301,132303],{"className":132302},[655,656,657,658],[363,132304,132306,132318,132321,132324,132327,132330],{"className":132305},[407,658],[363,132307,132309,132312,132315],{"className":132308},[1832,658],[363,132310,48030],{"className":132311},[658],[363,132313,48034],{"className":132314},[658],[363,132316,4506],{"className":132317,"style":4505},[658],[363,132319,461],{"className":132320},[494,658],[363,132322,11],{"className":132323},[407,408,658],[363,132325,1288],{"className":132326},[407,658],[363,132328,1291],{"className":132329,"style":1328},[407,408,658],[363,132331,468],{"className":132332},[501,658],[363,132334,605],{"className":132335},[501],[363,132337],{"className":132338,"style":506},[505],[363,132340,471],{"className":132341},[510],[363,132343],{"className":132344,"style":506},[505],[363,132346,132348,132351,132397,132400,132403,132406,132409,132412,132415,132418],{"className":132347},[398],[363,132349],{"className":132350,"style":487},[402],[363,132352,132354,132357],{"className":132353},[407],[363,132355,547],{"className":132356,"style":625},[407,408],[363,132358,132360],{"className":132359},[629],[363,132361,132363,132389],{"className":132362},[633,634],[363,132364,132366,132386],{"className":132365},[638],[363,132367,132369],{"className":132368,"style":727},[642],[363,132370,132371,132374],{"style":646},[363,132372],{"className":132373,"style":651},[650],[363,132375,132377],{"className":132376},[655,656,657,658],[363,132378,132380,132383],{"className":132379},[407,658],[363,132381,11044],{"className":132382,"style":11132},[407,408,658],[363,132384,22265],{"className":132385},[407,408,658],[363,132387,666],{"className":132388},[665],[363,132390,132392],{"className":132391},[638],[363,132393,132395],{"className":132394,"style":8316},[642],[363,132396],{},[363,132398,461],{"className":132399},[494],[363,132401,554],{"className":132402,"style":682},[407,408],[363,132404,558],{"className":132405},[407],[363,132407,561],{"className":132408},[407,408],[363,132410,468],{"className":132411},[501],[363,132413],{"className":132414,"style":2935},[505],[363,132416,2818],{"className":132417},[2939],[363,132419],{"className":132420,"style":2935},[505],[363,132422,132424,132427,132430,132433,132436],{"className":132423},[398],[363,132425],{"className":132426,"style":12928},[402],[363,132428,465],{"className":132429},[407],[363,132431],{"className":132432,"style":2935},[505],[363,132434,5161],{"className":132435},[2939],[363,132437],{"className":132438,"style":2935},[505],[363,132440,132442,132445,132448,132451,132454],{"className":132441},[398],[363,132443],{"className":132444,"style":520},[402],[363,132446,465],{"className":132447},[407],[363,132449],{"className":132450,"style":506},[505],[363,132452,471],{"className":132453},[510],[363,132455],{"className":132456,"style":506},[505],[363,132458,132460,132463,132509,132512,132515,132518,132521,132524],{"className":132459},[398],[363,132461],{"className":132462,"style":487},[402],[363,132464,132466,132469],{"className":132465},[407],[363,132467,547],{"className":132468,"style":625},[407,408],[363,132470,132472],{"className":132471},[629],[363,132473,132475,132501],{"className":132474},[633,634],[363,132476,132478,132498],{"className":132477},[638],[363,132479,132481],{"className":132480,"style":727},[642],[363,132482,132483,132486],{"style":646},[363,132484],{"className":132485,"style":651},[650],[363,132487,132489],{"className":132488},[655,656,657,658],[363,132490,132492,132495],{"className":132491},[407,658],[363,132493,11044],{"className":132494,"style":11132},[407,408,658],[363,132496,22265],{"className":132497},[407,408,658],[363,132499,666],{"className":132500},[665],[363,132502,132504],{"className":132503},[638],[363,132505,132507],{"className":132506,"style":8316},[642],[363,132508],{},[363,132510,461],{"className":132511},[494],[363,132513,554],{"className":132514,"style":682},[407,408],[363,132516,558],{"className":132517},[407],[363,132519,561],{"className":132520},[407,408],[363,132522,468],{"className":132523},[501],[363,132525,898],{"className":132526},[920],[11,132528,132529],{},"完全自洽。这给出了一个「可验证」的对偶表示练习。",[18,132531,132533],{"id":132532},"_10-结论","10. 结论",[11,132535,132536],{},"f-散度族的故事可以浓缩成三句话：",[7010,132538,132539,132842,133008],{},[7013,132540,132541,9748,132544,3093,132749,92858,132777,10528],{},[1344,132542,132543],{},"一个模板",[363,132545,132547,132603],{"className":132546,"translate":367},[366],[363,132548,132550],{"className":132549},[371],[373,132551,132552],{"xmlns":375},[377,132553,132554,132600],{},[380,132555,132556,132562,132564,132566,132568,132570,132572,132574,132580,132582,132584,132586,132588,132590,132592,132594,132596,132598],{},[543,132557,132558,132560],{},[383,132559,547],{},[383,132561,385],{},[458,132563,461],{"stretchy":460},[383,132565,554],{},[383,132567,558],{"mathvariant":557},[383,132569,561],{},[458,132571,468],{"stretchy":460},[458,132573,471],{},[543,132575,132576,132578],{},[383,132577,571],{"mathvariant":570},[383,132579,561],{},[458,132581,578],{"stretchy":460},[383,132583,385],{},[458,132585,461],{"stretchy":460},[383,132587,592],{},[383,132589,554],{},[383,132591,1288],{"mathvariant":557},[383,132593,592],{},[383,132595,561],{},[458,132597,468],{"stretchy":460},[458,132599,605],{"stretchy":460},[387,132601,132602],{"encoding":389},"D_f(P\\|Q)=\\mathbb E_Q[f(dP\u002FdQ)]",[363,132604,132606,132676],{"className":132605,"ariaHidden":394},[393],[363,132607,132609,132612,132652,132655,132658,132661,132664,132667,132670,132673],{"className":132608},[398],[363,132610],{"className":132611,"style":618},[402],[363,132613,132615,132618],{"className":132614},[407],[363,132616,547],{"className":132617,"style":625},[407,408],[363,132619,132621],{"className":132620},[629],[363,132622,132624,132644],{"className":132623},[633,634],[363,132625,132627,132641],{"className":132626},[638],[363,132628,132630],{"className":132629,"style":643},[642],[363,132631,132632,132635],{"style":646},[363,132633],{"className":132634,"style":651},[650],[363,132636,132638],{"className":132637},[655,656,657,658],[363,132639,385],{"className":132640,"style":409},[407,408,658],[363,132642,666],{"className":132643},[665],[363,132645,132647],{"className":132646},[638],[363,132648,132650],{"className":132649,"style":673},[642],[363,132651],{},[363,132653,461],{"className":132654},[494],[363,132656,554],{"className":132657,"style":682},[407,408],[363,132659,558],{"className":132660},[407],[363,132662,561],{"className":132663},[407,408],[363,132665,468],{"className":132666},[501],[363,132668],{"className":132669,"style":506},[505],[363,132671,471],{"className":132672},[510],[363,132674],{"className":132675,"style":506},[505],[363,132677,132679,132682,132722,132725,132728,132731,132734,132737,132740,132743,132746],{"className":132678},[398],[363,132680],{"className":132681,"style":618},[402],[363,132683,132685,132688],{"className":132684},[407],[363,132686,571],{"className":132687},[407,714],[363,132689,132691],{"className":132690},[629],[363,132692,132694,132714],{"className":132693},[633,634],[363,132695,132697,132711],{"className":132696},[638],[363,132698,132700],{"className":132699,"style":727},[642],[363,132701,132702,132705],{"style":730},[363,132703],{"className":132704,"style":651},[650],[363,132706,132708],{"className":132707},[655,656,657,658],[363,132709,561],{"className":132710},[407,408,658],[363,132712,666],{"className":132713},[665],[363,132715,132717],{"className":132716},[638],[363,132718,132720],{"className":132719,"style":673},[642],[363,132721],{},[363,132723,578],{"className":132724},[494],[363,132726,385],{"className":132727,"style":409},[407,408],[363,132729,461],{"className":132730},[494],[363,132732,592],{"className":132733},[407,408],[363,132735,554],{"className":132736,"style":682},[407,408],[363,132738,1288],{"className":132739},[407],[363,132741,592],{"className":132742},[407,408],[363,132744,561],{"className":132745},[407,408],[363,132747,20762],{"className":132748},[501],[363,132750,132752,132765],{"className":132751,"translate":367},[366],[363,132753,132755],{"className":132754},[371],[373,132756,132757],{"xmlns":375},[377,132758,132759,132763],{},[380,132760,132761],{},[383,132762,385],{},[387,132764,385],{"encoding":389},[363,132766,132768],{"className":132767,"ariaHidden":394},[393],[363,132769,132771,132774],{"className":132770},[398],[363,132772],{"className":132773,"style":403},[402],[363,132775,385],{"className":132776,"style":409},[407,408],[363,132778,132780,132803],{"className":132779,"translate":367},[366],[363,132781,132783],{"className":132782},[371],[373,132784,132785],{"xmlns":375},[377,132786,132787,132801],{},[380,132788,132789,132791,132793,132795,132797,132799],{},[383,132790,385],{},[458,132792,461],{"stretchy":460},[463,132794,465],{},[458,132796,468],{"stretchy":460},[458,132798,471],{},[463,132800,474],{},[387,132802,477],{"encoding":389},[363,132804,132806,132833],{"className":132805,"ariaHidden":394},[393],[363,132807,132809,132812,132815,132818,132821,132824,132827,132830],{"className":132808},[398],[363,132810],{"className":132811,"style":487},[402],[363,132813,385],{"className":132814,"style":409},[407,408],[363,132816,461],{"className":132817},[494],[363,132819,465],{"className":132820},[407],[363,132822,468],{"className":132823},[501],[363,132825],{"className":132826,"style":506},[505],[363,132828,471],{"className":132829},[510],[363,132831],{"className":132832,"style":506},[505],[363,132834,132836,132839],{"className":132835},[398],[363,132837],{"className":132838,"style":520},[402],[363,132840,474],{"className":132841},[407],[7013,132843,132844,9748,132847,132928,132929,133007],{},[1344,132845,132846],{},"两组对偶",[363,132848,132850,132872],{"className":132849,"translate":367},[366],[363,132851,132853],{"className":132852},[371],[373,132854,132855],{"xmlns":375},[377,132856,132857,132869],{},[380,132858,132859,132861,132863],{},[383,132860,385],{},[458,132862,21731],{},[7162,132864,132865,132867],{},[383,132866,385],{},[458,132868,7921],{},[387,132870,132871],{"encoding":389},"f\\leftrightarrow f^*",[363,132873,132875,132893],{"className":132874,"ariaHidden":394},[393],[363,132876,132878,132881,132884,132887,132890],{"className":132877},[398],[363,132879],{"className":132880,"style":403},[402],[363,132882,385],{"className":132883,"style":409},[407,408],[363,132885],{"className":132886,"style":506},[505],[363,132888,21731],{"className":132889},[510],[363,132891],{"className":132892,"style":506},[505],[363,132894,132896,132899],{"className":132895},[398],[363,132897],{"className":132898,"style":403},[402],[363,132900,132902,132905],{"className":132901},[407],[363,132903,385],{"className":132904,"style":409},[407,408],[363,132906,132908],{"className":132907},[629],[363,132909,132911],{"className":132910},[633],[363,132912,132914],{"className":132913},[638],[363,132915,132917],{"className":132916,"style":8847},[642],[363,132918,132919,132922],{"style":7213},[363,132920],{"className":132921,"style":651},[650],[363,132923,132925],{"className":132924},[655,656,657,658],[363,132926,7921],{"className":132927},[2939,658]," 给出变分表示（可用于估计与对抗训练）；",[363,132930,132932,132962],{"className":132931,"translate":367},[366],[363,132933,132935],{"className":132934},[371],[373,132936,132937],{"xmlns":375},[377,132938,132939,132959],{},[380,132940,132941,132943,132945,132947,132949,132951,132953,132955,132957],{},[383,132942,385],{},[458,132944,21731],{},[383,132946,1116],{},[383,132948,385],{},[458,132950,461],{"stretchy":460},[463,132952,465],{},[383,132954,1288],{"mathvariant":557},[383,132956,1116],{},[458,132958,468],{"stretchy":460},[387,132960,132961],{"encoding":389},"f\\leftrightarrow t f(1\u002Ft)",[363,132963,132965,132983],{"className":132964,"ariaHidden":394},[393],[363,132966,132968,132971,132974,132977,132980],{"className":132967},[398],[363,132969],{"className":132970,"style":403},[402],[363,132972,385],{"className":132973,"style":409},[407,408],[363,132975],{"className":132976,"style":506},[505],[363,132978,21731],{"className":132979},[510],[363,132981],{"className":132982,"style":506},[505],[363,132984,132986,132989,132992,132995,132998,133001,133004],{"className":132985},[398],[363,132987],{"className":132988,"style":487},[402],[363,132990,1116],{"className":132991},[407,408],[363,132993,385],{"className":132994,"style":409},[407,408],[363,132996,461],{"className":132997},[494],[363,132999,8937],{"className":133000},[407],[363,133002,1116],{"className":133003},[407,408],[363,133005,468],{"className":133006},[501]," 给出方向对称性（可用于构造对称散度）。",[7013,133009,133010,133013],{},[1344,133011,133012],{},"一个不变量","：数据处理不等式，说明它是「信息量」而非「几何量」。正因如此，它与 Wasserstein\u002FMMD 在支撑分离时行为迥异。",[11,133015,133016],{},"经典文献可参考：Csiszár (1967)、Ali–Silvey (1966) 的原始定义；Liese–Vajda 的专著对 f-散度的对偶、拓扑与充分性做了系统处理；Amari 的信息几何给出 α-族与 Fisher 度量；Nowozin 等 (2016) 的 f-GAN、Belghazi 等 (2018) 的 MINE 把对偶表示变成可训练算法；Dong–Roth–Su (2022) 说明 f-散度的语言如何统一差分隐私。把这几条线索接起来，读者就能在「定义—性质—估计—应用」四个层面上自由穿梭。",{"title":330,"searchDepth":331,"depth":331,"links":133018},[133019,133026,133031,133041,133051,133058,133063,133071,133072,133077],{"id":877,"depth":331,"text":878,"children":133020},[133021,133023,133024,133025],{"id":2661,"depth":133022,"text":2662},3,{"id":5552,"depth":133022,"text":5553},{"id":5921,"depth":133022,"text":5922},{"id":7709,"depth":133022,"text":7710},{"id":9099,"depth":331,"text":133027,"children":133028},"2. 为何 fff 凸且 f(1)=0f(1)=0f(1)=0",[133029,133030],{"id":9197,"depth":133022,"text":9198},{"id":10908,"depth":133022,"text":10909},{"id":21779,"depth":331,"text":21780,"children":133032},[133033,133034,133035,133036,133037,133038,133039,133040],{"id":21841,"depth":133022,"text":21842},{"id":25920,"depth":133022,"text":25921},{"id":34881,"depth":133022,"text":34882},{"id":37520,"depth":133022,"text":37521},{"id":40809,"depth":133022,"text":40810},{"id":45108,"depth":133022,"text":45109},{"id":49747,"depth":133022,"text":49748},{"id":52861,"depth":133022,"text":52862},{"id":55284,"depth":331,"text":55285,"children":133042},[133043,133044,133045,133046,133047,133048,133049,133050],{"id":55583,"depth":133022,"text":55584},{"id":56510,"depth":133022,"text":56511},{"id":58666,"depth":133022,"text":58667},{"id":61062,"depth":133022,"text":61063},{"id":74833,"depth":133022,"text":74834},{"id":78416,"depth":133022,"text":78417},{"id":82444,"depth":133022,"text":82445},{"id":97338,"depth":133022,"text":97339},{"id":99686,"depth":331,"text":99687,"children":133052},[133053,133054,133055,133056,133057],{"id":99690,"depth":133022,"text":99691},{"id":101734,"depth":133022,"text":101735},{"id":103578,"depth":133022,"text":103579},{"id":105740,"depth":133022,"text":105741},{"id":108632,"depth":133022,"text":108633},{"id":108973,"depth":331,"text":108974,"children":133059},[133060,133061,133062],{"id":108977,"depth":133022,"text":108978},{"id":115003,"depth":133022,"text":115004},{"id":116785,"depth":133022,"text":116786},{"id":117829,"depth":331,"text":117830,"children":133064},[133065,133066,133067,133068,133069,133070],{"id":117898,"depth":133022,"text":117899},{"id":118092,"depth":133022,"text":118093},{"id":118983,"depth":133022,"text":118984},{"id":119682,"depth":133022,"text":119683},{"id":120140,"depth":133022,"text":120141},{"id":120209,"depth":133022,"text":120210},{"id":120573,"depth":331,"text":120574},{"id":124734,"depth":331,"text":124735,"children":133073},[133074,133075,133076],{"id":124738,"depth":133022,"text":124739},{"id":129083,"depth":133022,"text":129084},{"id":131312,"depth":133022,"text":131313},{"id":132532,"depth":331,"text":132533},"f-散度族是概率分布之间差异的一大类度量，它用一个凸函数 fff 来统一描述 KL 散度、JS 散度、全变差、卡方散度等许多常见散度。可以把它理解为「散度」的通用模板。",{},"\u002Fblog\u002F2026\u002F0011-about-the-f-divergence",{"title":356,"description":133078},"blog\u002F2026\u002F0011-about-the-f-divergence","f-散度族（f-divergence）用一个凸函数 f 统一刻画两个概率分布之间的差异，KL、JS、全变差、卡方、Hellinger、α-散度都是它的特例。",[133085,133086],"statistics","mathematics","2026-10-17T04:47:08+08:00","2WAe5fQsIoDK6gYqN03WvWAWQ1lJEHGnxsbnKfYymdg",{"id":133090,"title":133091,"body":133092,"description":197652,"draft":341,"enableComment":342,"extension":343,"image":330,"important":341,"location":344,"meta":197653,"navigation":342,"ogImage":344,"path":197654,"seo":197655,"stem":197656,"summary":197657,"tags":197658,"time":197660,"__hash__":197661},"blog\u002Fblog\u002F2026\u002F0007-dirichlet-distribution-and-evidential-deep-learning.md","Dirichlet 分布与证据深度学习",{"type":8,"value":133093,"toc":197612},[133094,133247,133642,133771,133844,133847,133851,133855,134791,135252,136029,136033,137529,137532,138038,138138,138745,139634,140445,140962,142106,142195,142258,142262,143481,144003,144987,145105,145369,145373,145657,146619,146674,147348,147511,147515,147519,148875,151148,151346,151693,152242,152246,152355,152652,153175,153803,154362,154482,154486,155246,155343,156017,156741,157247,157433,157437,158100,158283,158623,158704,158707,159698,161184,161567,161571,162431,162438,163177,164142,164245,164249,164287,164929,164997,165424,166019,166023,166027,166034,167001,167501,167670,167674,167680,168563,169052,169829,169832,169839,169843,170495,170898,171043,171046,171050,171056,171809,172094,172097,172944,172948,173740,173924,173955,174846,175031,175583,175651,175655,175724,175915,176488,178070,179541,179816,180059,180063,180150,180154,180161,180431,180648,180993,180996,181729,182070,182796,183561,183597,183601,183604,184334,185335,185618,185786,185790,185879,185882,186351,186491,188327,189331,189337,189781,190000,190004,190007,190243,190512,190515,190639,190650,190654,190657,192180,192187,192191,192195,192323,192773,192984,193116,193123,193126,193130,193355,193855,193929,194155,194159,194379,194495,194499,194505,194696,194732,194736,194742,195079,195564,195567,195571,196187,196190,196732,197045,197197,197200,197366,197369,197372,197496,197564,197567,197605],[11,133095,881,133096,133125,133126,133217,133218,133246],{},[363,133097,133099,133113],{"className":133098,"translate":367},[366],[363,133100,133102],{"className":133101},[371],[373,133103,133104],{"xmlns":375},[377,133105,133106,133110],{},[380,133107,133108],{},[383,133109,945],{"mathvariant":944},[387,133111,133112],{"encoding":389},"\\mathcal{X}",[363,133114,133116],{"className":133115,"ariaHidden":394},[393],[363,133117,133119,133122],{"className":133118},[398],[363,133120],{"className":133121,"style":1011},[402],[363,133123,945],{"className":133124,"style":972},[407,971]," 为输入空间，",[363,133127,133129,133160],{"className":133128,"translate":367},[366],[363,133130,133132],{"className":133131},[371],[373,133133,133134],{"xmlns":375},[377,133135,133136,133157],{},[380,133137,133138,133140,133142,133144,133146,133148,133151,133153,133155],{},[383,133139,40609],{},[458,133141,6257],{},[458,133143,3246],{"stretchy":460},[463,133145,465],{},[458,133147,898],{"separator":394},[458,133149,133150],{},"…",[458,133152,898],{"separator":394},[383,133154,11044],{},[458,133156,3256],{"stretchy":460},[387,133158,133159],{"encoding":389},"y \\in \\{1, \\dots, K\\}",[363,133161,133163,133181],{"className":133162,"ariaHidden":394},[393],[363,133164,133166,133169,133172,133175,133178],{"className":133165},[398],[363,133167],{"className":133168,"style":32064},[402],[363,133170,40609],{"className":133171,"style":1328},[407,408],[363,133173],{"className":133174,"style":506},[505],[363,133176,6257],{"className":133177},[510],[363,133179],{"className":133180,"style":506},[505],[363,133182,133184,133187,133190,133193,133196,133199,133202,133205,133208,133211,133214],{"className":133183},[398],[363,133185],{"className":133186,"style":487},[402],[363,133188,3246],{"className":133189},[494],[363,133191,465],{"className":133192},[407],[363,133194,898],{"className":133195},[920],[363,133197],{"className":133198,"style":754},[505],[363,133200,133150],{"className":133201},[758],[363,133203],{"className":133204,"style":754},[505],[363,133206,898],{"className":133207},[920],[363,133209],{"className":133210,"style":754},[505],[363,133212,11044],{"className":133213,"style":11132},[407,408],[363,133215,3256],{"className":133216},[501]," 为类别标签。一个 ",[363,133219,133221,133234],{"className":133220,"translate":367},[366],[363,133222,133224],{"className":133223},[371],[373,133225,133226],{"xmlns":375},[377,133227,133228,133232],{},[380,133229,133230],{},[383,133231,11044],{},[387,133233,11044],{"encoding":389},[363,133235,133237],{"className":133236,"ariaHidden":394},[393],[363,133238,133240,133243],{"className":133239},[398],[363,133241],{"className":133242,"style":1011},[402],[363,133244,11044],{"className":133245,"style":11132},[407,408]," 分类器的输出在数学上究竟是什么？若要求它给出「属于各类别的概率」，则输出必须落在",[363,133248,133250],{"className":133249,"translate":367},[528],[363,133251,133253,133337],{"className":133252,"translate":367},[366],[363,133254,133256],{"className":133255},[371],[373,133257,133258],{"xmlns":375,"display":537},[377,133259,133260,133334],{},[380,133261,133262,133274,133276],{},[7162,133263,133264,133266],{},[383,133265,126067],{"mathvariant":557},[380,133267,133268,133270,133272],{},[383,133269,11044],{},[458,133271,5161],{},[463,133273,465],{},[458,133275,471],{},[380,133277,133278,133280,133282,133284,133290,133292,133298,133300,133302,133304,133306,133332],{},[458,133279,3246],{"fence":394},[383,133281,11],{"mathvariant":75598},[458,133283,6257],{},[7162,133285,133286,133288],{},[383,133287,1381],{"mathvariant":570},[383,133289,11044],{},[458,133291,1364],{},[543,133293,133294,133296],{},[383,133295,11],{},[383,133297,7912],{},[458,133299,7276],{},[463,133301,474],{},[458,133303,898],{"separator":394},[1645,133305,4631],{},[1603,133307,133308,133322,133328,133330],{"scriptlevel":474,"displaystyle":460},[7901,133309,133310,133312,133320],{},[458,133311,13811],{},[380,133313,133314,133316,133318],{},[383,133315,7912],{},[458,133317,471],{},[463,133319,465],{},[383,133321,11044],{},[543,133323,133324,133326],{},[383,133325,11],{},[383,133327,7912],{},[458,133329,471],{},[463,133331,465],{},[458,133333,3256],{"fence":394},[387,133335,133336],{"encoding":389},"\\Delta^{K-1} = \\left\\{ \\mathbf{p} \\in \\mathbb{R}^K : p_k \\ge 0,\\ \\textstyle\\sum_{k=1}^K p_k = 1 \\right\\}",[363,133338,133340,133393],{"className":133339,"ariaHidden":394},[393],[363,133341,133343,133346,133384,133387,133390],{"className":133342},[398],[363,133344],{"className":133345,"style":38884},[402],[363,133347,133349,133352],{"className":133348},[407],[363,133350,126067],{"className":133351},[407],[363,133353,133355],{"className":133354},[629],[363,133356,133358],{"className":133357},[633],[363,133359,133361],{"className":133360},[638],[363,133362,133364],{"className":133363,"style":38884},[642],[363,133365,133366,133369],{"style":8625},[363,133367],{"className":133368,"style":651},[650],[363,133370,133372],{"className":133371},[655,656,657,658],[363,133373,133375,133378,133381],{"className":133374},[407,658],[363,133376,11044],{"className":133377,"style":11132},[407,408,658],[363,133379,5161],{"className":133380},[2939,658],[363,133382,465],{"className":133383},[407,658],[363,133385],{"className":133386,"style":506},[505],[363,133388,471],{"className":133389},[510],[363,133391],{"className":133392,"style":506},[505],[363,133394,133396,133399],{"className":133395},[398],[363,133397],{"className":133398,"style":68500},[402],[363,133400,133402,133408,133411,133414,133417,133420,133449,133452,133455,133458,133498,133501,133504,133507,133510,133513,133516,133519,133581,133584,133624,133627,133630,133633,133636],{"className":133401},[758],[363,133403,133405],{"className":133404,"style":763},[494,762],[363,133406,3246],{"className":133407},[767,4068],[363,133409,11],{"className":133410},[407,75943],[363,133412],{"className":133413,"style":506},[505],[363,133415,6257],{"className":133416},[510],[363,133418],{"className":133419,"style":506},[505],[363,133421,133423,133426],{"className":133422},[407],[363,133424,1381],{"className":133425},[407,714],[363,133427,133429],{"className":133428},[629],[363,133430,133432],{"className":133431},[633],[363,133433,133435],{"className":133434},[638],[363,133436,133438],{"className":133437,"style":38884},[642],[363,133439,133440,133443],{"style":8625},[363,133441],{"className":133442,"style":651},[650],[363,133444,133446],{"className":133445},[655,656,657,658],[363,133447,11044],{"className":133448,"style":11132},[407,408,658],[363,133450],{"className":133451,"style":506},[505],[363,133453,1364],{"className":133454},[510],[363,133456],{"className":133457,"style":506},[505],[363,133459,133461,133464],{"className":133460},[407],[363,133462,11],{"className":133463},[407,408],[363,133465,133467],{"className":133466},[629],[363,133468,133470,133490],{"className":133469},[633,634],[363,133471,133473,133487],{"className":133472},[638],[363,133474,133476],{"className":133475,"style":643},[642],[363,133477,133478,133481],{"style":730},[363,133479],{"className":133480,"style":651},[650],[363,133482,133484],{"className":133483},[655,656,657,658],[363,133485,7912],{"className":133486,"style":8615},[407,408,658],[363,133488,666],{"className":133489},[665],[363,133491,133493],{"className":133492},[638],[363,133494,133496],{"className":133495,"style":8316},[642],[363,133497],{},[363,133499],{"className":133500,"style":506},[505],[363,133502,7276],{"className":133503},[510],[363,133505],{"className":133506,"style":506},[505],[363,133508,474],{"className":133509},[407],[363,133511,898],{"className":133512},[920],[363,133514,4631],{"className":133515},[505],[363,133517],{"className":133518,"style":754},[505],[363,133520,133522,133525],{"className":133521},[1832],[363,133523,13811],{"className":133524,"style":40032},[1832,1833,5964],[363,133526,133528],{"className":133527},[629],[363,133529,133531,133573],{"className":133530},[633,634],[363,133532,133534,133570],{"className":133533},[638],[363,133535,133538,133558],{"className":133536,"style":133537},[642],"height:0.9812em;",[363,133539,133540,133543],{"style":40048},[363,133541],{"className":133542,"style":651},[650],[363,133544,133546],{"className":133545},[655,656,657,658],[363,133547,133549,133552,133555],{"className":133548},[407,658],[363,133550,7912],{"className":133551,"style":8615},[407,408,658],[363,133553,471],{"className":133554},[510,658],[363,133556,465],{"className":133557},[407,658],[363,133559,133561,133564],{"style":133560},"top:-3.2029em;margin-right:0.05em;",[363,133562],{"className":133563,"style":651},[650],[363,133565,133567],{"className":133566},[655,656,657,658],[363,133568,11044],{"className":133569,"style":11132},[407,408,658],[363,133571,666],{"className":133572},[665],[363,133574,133576],{"className":133575},[638],[363,133577,133579],{"className":133578,"style":40067},[642],[363,133580],{},[363,133582],{"className":133583,"style":754},[505],[363,133585,133587,133590],{"className":133586},[407],[363,133588,11],{"className":133589},[407,408],[363,133591,133593],{"className":133592},[629],[363,133594,133596,133616],{"className":133595},[633,634],[363,133597,133599,133613],{"className":133598},[638],[363,133600,133602],{"className":133601,"style":643},[642],[363,133603,133604,133607],{"style":730},[363,133605],{"className":133606,"style":651},[650],[363,133608,133610],{"className":133609},[655,656,657,658],[363,133611,7912],{"className":133612,"style":8615},[407,408,658],[363,133614,666],{"className":133615},[665],[363,133617,133619],{"className":133618},[638],[363,133620,133622],{"className":133621,"style":8316},[642],[363,133623],{},[363,133625],{"className":133626,"style":506},[505],[363,133628,471],{"className":133629},[510],[363,133631],{"className":133632,"style":506},[505],[363,133634,465],{"className":133635},[407],[363,133637,133639],{"className":133638,"style":763},[501,762],[363,133640,3256],{"className":133641},[767,4068],[11,133643,133644,133645,133648,133649,133708,133709,133770],{},"之中——",[1344,133646,133647],{},"概率单纯形","上的一个点。这是分类器输出的自然状态空间：它不是 ",[363,133650,133652,133670],{"className":133651,"translate":367},[366],[363,133653,133655],{"className":133654},[371],[373,133656,133657],{"xmlns":375},[377,133658,133659,133667],{},[380,133660,133661],{},[7162,133662,133663,133665],{},[383,133664,1381],{"mathvariant":570},[383,133666,11044],{},[387,133668,133669],{"encoding":389},"\\mathbb{R}^K",[363,133671,133673],{"className":133672,"ariaHidden":394},[393],[363,133674,133676,133679],{"className":133675},[398],[363,133677],{"className":133678,"style":52457},[402],[363,133680,133682,133685],{"className":133681},[407],[363,133683,1381],{"className":133684},[407,714],[363,133686,133688],{"className":133687},[629],[363,133689,133691],{"className":133690},[633],[363,133692,133694],{"className":133693},[638],[363,133695,133697],{"className":133696,"style":52457},[642],[363,133698,133699,133702],{"style":7213},[363,133700],{"className":133701,"style":651},[650],[363,133703,133705],{"className":133704},[655,656,657,658],[363,133706,11044],{"className":133707,"style":11132},[407,408,658],"，而是一个 ",[363,133710,133712,133734],{"className":133711,"translate":367},[366],[363,133713,133715],{"className":133714},[371],[373,133716,133717],{"xmlns":375},[377,133718,133719,133731],{},[380,133720,133721,133723,133725,133727,133729],{},[458,133722,461],{"stretchy":460},[383,133724,11044],{},[458,133726,5161],{},[463,133728,465],{},[458,133730,468],{"stretchy":460},[387,133732,133733],{"encoding":389},"(K-1)",[363,133735,133737,133758],{"className":133736,"ariaHidden":394},[393],[363,133738,133740,133743,133746,133749,133752,133755],{"className":133739},[398],[363,133741],{"className":133742,"style":487},[402],[363,133744,461],{"className":133745},[494],[363,133747,11044],{"className":133748,"style":11132},[407,408],[363,133750],{"className":133751,"style":2935},[505],[363,133753,5161],{"className":133754},[2939],[363,133756],{"className":133757,"style":2935},[505],[363,133759,133761,133764,133767],{"className":133760},[398],[363,133762],{"className":133763,"style":487},[402],[363,133765,465],{"className":133766},[407],[363,133768,468],{"className":133769},[501]," 维的紧凸集。",[11,133772,133773,133774,133777,133778,6842,133781,133811,133812,133840,133841,10528],{},"一旦承认了这一点，一个二阶问题便随即出现：分类器对自身输出的",[1344,133775,133776],{},"把握","有多大？若把输出建模为单纯形上的一个",[1344,133779,133780],{},"确定点",[363,133782,133784,133798],{"className":133783,"translate":367},[366],[363,133785,133787],{"className":133786},[371],[373,133788,133789],{"xmlns":375},[377,133790,133791,133795],{},[380,133792,133793],{},[383,133794,11],{"mathvariant":75598},[387,133796,133797],{"encoding":389},"\\mathbf{p}",[363,133799,133801],{"className":133800,"ariaHidden":394},[393],[363,133802,133804,133808],{"className":133803},[398],[363,133805],{"className":133806,"style":133807},[402],"height:0.6389em;vertical-align:-0.1944em;",[363,133809,11],{"className":133810},[407,75943],"，则这个问题在表达力上是被禁止的：单纯形内部不存在可以指代「无信息」的元素，因为每个点都是一份完整的、和为 ",[363,133813,133815,133828],{"className":133814,"translate":367},[366],[363,133816,133818],{"className":133817},[371],[373,133819,133820],{"xmlns":375},[377,133821,133822,133826],{},[380,133823,133824],{},[463,133825,465],{},[387,133827,465],{"encoding":389},[363,133829,133831],{"className":133830,"ariaHidden":394},[393],[363,133832,133834,133837],{"className":133833},[398],[363,133835],{"className":133836,"style":520},[402],[363,133838,465],{"className":133839},[407]," 的断言。要回答二阶问题，必须把输出从「单纯形上的点」提升为「单纯形上的分布」。承担这一角色的标准对象正是 ",[1344,133842,133843],{},"迪利克雷分布（Dirichlet distribution）",[11,133845,133846],{},"本文的组织如下。第一部分给出迪利克雷分布的形式化定义、生成性刻画与几何图像。第二部分展开其矩结构、重参数化、共轭更新与分裂构造，并澄清「伪计数」这一常见的直观说法在何处成立、在何处失效。第三部分经由主观逻辑（subjective logic）建立迪利克雷分布与神经网络输出之间的双射，由此引入证据深度学习（Evidential Deep Learning, EDL）。第四部分推导 EDL 的损失函数族及其理论缺陷。第五部分讨论其应用形态与边界。第六部分总结。",[18,133848,133850],{"id":133849},"一形式化","一、形式化",[2659,133852,133854],{"id":133853},"_11-概率单纯形","1.1 概率单纯形",[143,133856,133857],{},[11,133858,133859,133862,133863,3093,133914,133966,133967,134494,134495,10528],{},[1344,133860,133861],{},"定义 1.1（概率单纯形）"," 对整数 ",[363,133864,133866,133884],{"className":133865,"translate":367},[366],[363,133867,133869],{"className":133868},[371],[373,133870,133871],{"xmlns":375},[377,133872,133873,133881],{},[380,133874,133875,133877,133879],{},[383,133876,11044],{},[458,133878,7276],{},[463,133880,2811],{},[387,133882,133883],{"encoding":389},"K \\ge 2",[363,133885,133887,133905],{"className":133886,"ariaHidden":394},[393],[363,133888,133890,133893,133896,133899,133902],{"className":133889},[398],[363,133891],{"className":133892,"style":88741},[402],[363,133894,11044],{"className":133895,"style":11132},[407,408],[363,133897],{"className":133898,"style":506},[505],[363,133900,7276],{"className":133901},[510],[363,133903],{"className":133904,"style":506},[505],[363,133906,133908,133911],{"className":133907},[398],[363,133909],{"className":133910,"style":520},[402],[363,133912,2811],{"className":133913},[407],[363,133915,133917,133935],{"className":133916,"translate":367},[366],[363,133918,133920],{"className":133919},[371],[373,133921,133922],{"xmlns":375},[377,133923,133924,133932],{},[380,133925,133926,133928,133930],{},[383,133927,11044],{},[458,133929,5161],{},[463,133931,465],{},[387,133933,133934],{"encoding":389},"K-1",[363,133936,133938,133957],{"className":133937,"ariaHidden":394},[393],[363,133939,133941,133945,133948,133951,133954],{"className":133940},[398],[363,133942],{"className":133943,"style":133944},[402],"height:0.7667em;vertical-align:-0.0833em;",[363,133946,11044],{"className":133947,"style":11132},[407,408],[363,133949],{"className":133950,"style":2935},[505],[363,133952,5161],{"className":133953},[2939],[363,133955],{"className":133956,"style":2935},[505],[363,133958,133960,133963],{"className":133959},[398],[363,133961],{"className":133962,"style":520},[402],[363,133964,465],{"className":133965},[407]," 维标准单纯形定义为\n",[363,133968,133970,134078],{"className":133969,"translate":367},[366],[363,133971,133973],{"className":133972},[371],[373,133974,133975],{"xmlns":375},[377,133976,133977,134075],{},[380,133978,133979,133991,133993],{},[7162,133980,133981,133983],{},[383,133982,126067],{"mathvariant":557},[380,133984,133985,133987,133989],{},[383,133986,11044],{},[458,133988,5161],{},[463,133990,465],{},[458,133992,471],{},[380,133994,133995,133997,133999,134001,134003,134009,134011,134013,134015,134021,134023,134025,134031,134033,134039,134041,134043,134045,134047,134073],{},[458,133996,3246],{"fence":394},[383,133998,1121],{"mathvariant":75598},[458,134000,471],{},[458,134002,461],{"stretchy":460},[543,134004,134005,134007],{},[383,134006,1121],{},[463,134008,465],{},[458,134010,898],{"separator":394},[458,134012,133150],{},[458,134014,898],{"separator":394},[543,134016,134017,134019],{},[383,134018,1121],{},[383,134020,11044],{},[458,134022,468],{"stretchy":460},[458,134024,6257],{},[7162,134026,134027,134029],{},[383,134028,1381],{"mathvariant":570},[383,134030,11044],{},[458,134032,1364],{},[543,134034,134035,134037],{},[383,134036,1121],{},[383,134038,12313],{},[458,134040,7276],{},[463,134042,474],{},[458,134044,898],{"separator":394},[1645,134046,4631],{},[1603,134048,134049,134063,134069,134071],{"scriptlevel":474,"displaystyle":460},[7901,134050,134051,134053,134061],{},[458,134052,13811],{},[380,134054,134055,134057,134059],{},[383,134056,12313],{},[458,134058,471],{},[463,134060,465],{},[383,134062,11044],{},[543,134064,134065,134067],{},[383,134066,1121],{},[383,134068,12313],{},[458,134070,471],{},[463,134072,465],{},[458,134074,3256],{"fence":394},[387,134076,134077],{"encoding":389},"\\Delta^{K-1} = \\left\\{ \\mathbf{x} = (x_1, \\dots, x_K) \\in \\mathbb{R}^K : x_i \\ge 0,\\ \\textstyle\\sum_{i=1}^K x_i = 1 \\right\\}",[363,134079,134081,134134],{"className":134080,"ariaHidden":394},[393],[363,134082,134084,134087,134125,134128,134131],{"className":134083},[398],[363,134085],{"className":134086,"style":52457},[402],[363,134088,134090,134093],{"className":134089},[407],[363,134091,126067],{"className":134092},[407],[363,134094,134096],{"className":134095},[629],[363,134097,134099],{"className":134098},[633],[363,134100,134102],{"className":134101},[638],[363,134103,134105],{"className":134104,"style":52457},[642],[363,134106,134107,134110],{"style":7213},[363,134108],{"className":134109,"style":651},[650],[363,134111,134113],{"className":134112},[655,656,657,658],[363,134114,134116,134119,134122],{"className":134115},[407,658],[363,134117,11044],{"className":134118,"style":11132},[407,408,658],[363,134120,5161],{"className":134121},[2939,658],[363,134123,465],{"className":134124},[407,658],[363,134126],{"className":134127,"style":506},[505],[363,134129,471],{"className":134130},[510],[363,134132],{"className":134133,"style":506},[505],[363,134135,134137,134140],{"className":134136},[398],[363,134138],{"className":134139,"style":68500},[402],[363,134141,134143,134149,134152,134155,134158,134161,134164,134204,134207,134210,134213,134216,134219,134222,134262,134265,134268,134271,134274,134303,134306,134309,134312,134352,134355,134358,134361,134364,134367,134370,134373,134433,134436,134476,134479,134482,134485,134488],{"className":134142},[758],[363,134144,134146],{"className":134145,"style":763},[494,762],[363,134147,3246],{"className":134148},[767,4068],[363,134150,1121],{"className":134151},[407,75943],[363,134153],{"className":134154,"style":506},[505],[363,134156,471],{"className":134157},[510],[363,134159],{"className":134160,"style":506},[505],[363,134162,461],{"className":134163},[494],[363,134165,134167,134170],{"className":134166},[407],[363,134168,1121],{"className":134169},[407,408],[363,134171,134173],{"className":134172},[629],[363,134174,134176,134196],{"className":134175},[633,634],[363,134177,134179,134193],{"className":134178},[638],[363,134180,134182],{"className":134181,"style":11483},[642],[363,134183,134184,134187],{"style":730},[363,134185],{"className":134186,"style":651},[650],[363,134188,134190],{"className":134189},[655,656,657,658],[363,134191,465],{"className":134192},[407,658],[363,134194,666],{"className":134195},[665],[363,134197,134199],{"className":134198},[638],[363,134200,134202],{"className":134201,"style":8316},[642],[363,134203],{},[363,134205,898],{"className":134206},[920],[363,134208],{"className":134209,"style":754},[505],[363,134211,133150],{"className":134212},[758],[363,134214],{"className":134215,"style":754},[505],[363,134217,898],{"className":134218},[920],[363,134220],{"className":134221,"style":754},[505],[363,134223,134225,134228],{"className":134224},[407],[363,134226,1121],{"className":134227},[407,408],[363,134229,134231],{"className":134230},[629],[363,134232,134234,134254],{"className":134233},[633,634],[363,134235,134237,134251],{"className":134236},[638],[363,134238,134240],{"className":134239,"style":727},[642],[363,134241,134242,134245],{"style":730},[363,134243],{"className":134244,"style":651},[650],[363,134246,134248],{"className":134247},[655,656,657,658],[363,134249,11044],{"className":134250,"style":11132},[407,408,658],[363,134252,666],{"className":134253},[665],[363,134255,134257],{"className":134256},[638],[363,134258,134260],{"className":134259,"style":8316},[642],[363,134261],{},[363,134263,468],{"className":134264},[501],[363,134266],{"className":134267,"style":506},[505],[363,134269,6257],{"className":134270},[510],[363,134272],{"className":134273,"style":506},[505],[363,134275,134277,134280],{"className":134276},[407],[363,134278,1381],{"className":134279},[407,714],[363,134281,134283],{"className":134282},[629],[363,134284,134286],{"className":134285},[633],[363,134287,134289],{"className":134288},[638],[363,134290,134292],{"className":134291,"style":52457},[642],[363,134293,134294,134297],{"style":7213},[363,134295],{"className":134296,"style":651},[650],[363,134298,134300],{"className":134299},[655,656,657,658],[363,134301,11044],{"className":134302,"style":11132},[407,408,658],[363,134304],{"className":134305,"style":506},[505],[363,134307,1364],{"className":134308},[510],[363,134310],{"className":134311,"style":506},[505],[363,134313,134315,134318],{"className":134314},[407],[363,134316,1121],{"className":134317},[407,408],[363,134319,134321],{"className":134320},[629],[363,134322,134324,134344],{"className":134323},[633,634],[363,134325,134327,134341],{"className":134326},[638],[363,134328,134330],{"className":134329,"style":12532},[642],[363,134331,134332,134335],{"style":730},[363,134333],{"className":134334,"style":651},[650],[363,134336,134338],{"className":134337},[655,656,657,658],[363,134339,12313],{"className":134340},[407,408,658],[363,134342,666],{"className":134343},[665],[363,134345,134347],{"className":134346},[638],[363,1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",[363,134496,134498,134569],{"className":134497,"translate":367},[366],[363,134499,134501],{"className":134500},[371],[373,134502,134503],{"xmlns":375},[377,134504,134505,134566],{},[380,134506,134507,134510,134512,134524,134526,134528,134530,134532,134544,134546,134552,134554,134556,134558,134560,134562,134564],{},[383,134508,134509],{"mathvariant":557},"int",[458,134511,4451],{},[7162,134513,134514,134516],{},[383,134515,126067],{"mathvariant":557},[380,134517,134518,134520,134522],{},[383,134519,11044],{},[458,134521,5161],{},[463,134523,465],{},[458,134525,471],{},[458,134527,3246],{"stretchy":460},[383,134529,1121],{"mathvariant":75598},[458,134531,6257],{},[7162,134533,134534,134536],{},[383,134535,126067],{"mathvariant":557},[380,134537,134538,134540,134542],{},[383,134539,11044],{},[458,134541,5161],{},[463,134543,465],{},[458,134545,1364],{},[543,134547,134548,134550],{},[383,134549,1121],{},[383,134551,12313],{},[458,134553,3251],{},[463,134555,474],{},[458,134557,898],{"separator":394},[1645,134559,4631],{},[383,134561,61983],{"mathvariant":557},[383,134563,12313],{},[458,134565,3256],{"stretchy":460},[387,134567,134568],{"encoding":389},"\\operatorname{int}\\Delta^{K-1} = \\{\\mathbf{x} \\in \\Delta^{K-1} : x_i > 0,\\ \\forall i\\}",[363,134570,134572,134634,134655,134708,134764],{"className":134571,"ariaHidden":394},[393],[363,134573,134575,134578,134584,134587,134625,134628,134631],{"className":134574},[398],[363,134576],{"className":134577,"style":52457},[402],[363,134579,134581],{"className":134580},[1832],[363,134582,134509],{"className":134583},[407,33034],[363,134585],{"className":134586,"style":754},[505],[363,134588,134590,134593],{"className":134589},[407],[363,134591,126067],{"className":134592},[407],[363,134594,134596],{"className":134595},[629],[363,134597,134599],{"className":134598},[633],[363,134600,134602],{"className":134601},[638],[363,134603,134605],{"className":134604,"style":52457},[642],[363,134606,134607,134610],{"style":7213},[363,134608],{"className":134609,"style":651},[650],[363,134611,134613],{"className":134612},[655,656,657,658],[363,134614,134616,134619,134622],{"className":134615},[407,658],[363,134617,11044],{"className":134618,"style":11132},[407,408,658],[363,134620,5161],{"className":134621},[2939,658],[363,134623,465],{"className":134624},[407,658],[363,134626],{"className":134627,"style":506},[505],[363,134629,471],{"className":134630},[510],[363,134632],{"className":134633,"style":506},[505],[363,134635,134637,134640,134643,134646,134649,134652],{"className":134636},[398],[363,134638],{"className":134639,"style":487},[402],[363,134641,3246],{"className":134642},[494],[363,134644,1121],{"className":134645},[407,75943],[363,134647],{"className":134648,"style":506},[505],[363,134650,6257],{"className":134651},[510],[363,134653],{"className":134654,"style":506},[505],[363,134656,134658,134661,134699,134702,134705],{"className":134657},[398],[363,134659],{"className":134660,"style":52457},[402],[363,134662,134664,134667],{"className":134663},[407],[363,134665,126067],{"className":134666},[407],[363,134668,134670],{"className":134669},[629],[363,134671,134673],{"className":134672},[633],[363,134674,134676],{"className":134675},[638],[363,134677,134679],{"className":134678,"style":52457},[642],[363,134680,134681,134684],{"style":7213},[363,134682],{"className":134683,"style":651},[650],[363,134685,134687],{"className":134686},[655,656,657,658],[363,134688,134690,134693,134696],{"className":134689},[407,658],[363,134691,11044],{"className":134692,"style":11132},[407,408,658],[363,134694,5161],{"className":134695},[2939,658],[363,134697,465],{"className":134698},[407,658],[363,134700],{"className":134701,"style":506},[505],[363,134703,1364],{"className":134704},[510],[363,134706],{"className":134707,"style":506},[505],[363,134709,134711,134715,134755,134758,134761],{"className":134710},[398],[363,134712],{"className":134713,"style":134714},[402],"height:0.6891em;vertical-align:-0.15em;",[363,134716,134718,134721],{"className":134717},[407],[363,134719,1121],{"className":134720},[407,408],[363,134722,134724],{"className":134723},[629],[363,134725,134727,134747],{"className":134726},[633,634],[363,134728,134730,134744],{"className":134729},[638],[363,134731,134733],{"className":134732,"style":12532},[642],[363,134734,134735,134738],{"style":730},[363,134736],{"className":134737,"style":651},[650],[363,134739,134741],{"className":134740},[655,656,657,658],[363,134742,12313],{"className":134743},[407,408,658],[363,134745,666],{"className":134746},[665],[363,134748,134750],{"className":134749},[638],[363,134751,134753],{"className":134752,"style":8316},[642],[363,134754],{},[363,134756],{"className":134757,"style":506},[505],[363,134759,3251],{"className":134760},[510],[363,134762],{"className":134763,"style":506},[505],[363,134765,134767,134770,134773,134776,134779,134782,134785,134788],{"className":134766},[398],[363,134768],{"className":134769,"style":487},[402],[363,134771,474],{"className":134772},[407],[363,134774,898],{"className":134775},[920],[363,134777,4631],{"className":134778},[505],[363,134780],{"className":134781,"style":754},[505],[363,134783,61983],{"className":134784},[407],[363,134786,12313],{"className":134787},[407,408],[363,134789,3256],{"className":134790},[501],[11,134792,134793,2628,134867,134925,134926,135066,135067,135118,135119,135170,135171,135222,135223,135251],{},[363,134794,134796,134820],{"className":134795,"translate":367},[366],[363,134797,134799],{"className":134798},[371],[373,134800,134801],{"xmlns":375},[377,134802,134803,134817],{},[380,134804,134805],{},[7162,134806,134807,134809],{},[383,134808,126067],{"mathvariant":557},[380,134810,134811,134813,134815],{},[383,134812,11044],{},[458,134814,5161],{},[463,134816,465],{},[387,134818,134819],{"encoding":389},"\\Delta^{K-1}",[363,134821,134823],{"className":134822,"ariaHidden":394},[393],[363,134824,134826,134829],{"className":134825},[398],[363,134827],{"className":134828,"style":52457},[402],[363,134830,134832,134835],{"className":134831},[407],[363,134833,126067],{"className":134834},[407],[363,134836,134838],{"className":134837},[629],[363,134839,134841],{"className":134840},[633],[363,134842,134844],{"className":134843},[638],[363,134845,134847],{"className":134846,"style":52457},[642],[363,134848,134849,134852],{"style":7213},[363,134850],{"className":134851,"style":651},[650],[363,134853,134855],{"className":134854},[655,656,657,658],[363,134856,134858,134861,134864],{"className":134857},[407,658],[363,134859,11044],{"className":134860,"style":11132},[407,408,658],[363,134862,5161],{"className":134863},[2939,658],[363,134865,465],{"className":134866},[407,658],[363,134868,134870,134887],{"className":134869,"translate":367},[366],[363,134871,134873],{"className":134872},[371],[373,134874,134875],{"xmlns":375},[377,134876,134877,134885],{},[380,134878,134879],{},[7162,134880,134881,134883],{},[383,134882,1381],{"mathvariant":570},[383,134884,11044],{},[387,134886,133669],{"encoding":389},[363,134888,134890],{"className":134889,"ariaHidden":394},[393],[363,134891,134893,134896],{"className":134892},[398],[363,134894],{"className":134895,"style":52457},[402],[363,134897,134899,134902],{"className":134898},[407],[363,134900,1381],{"className":134901},[407,714],[363,134903,134905],{"className":134904},[629],[363,134906,134908],{"className":134907},[633],[363,134909,134911],{"className":134910},[638],[363,134912,134914],{"className":134913,"style":52457},[642],[363,134915,134916,134919],{"style":7213},[363,134917],{"className":134918,"style":651},[650],[363,134920,134922],{"className":134921},[655,656,657,658],[363,134923,11044],{"className":134924,"style":11132},[407,408,658]," 中的紧凸集，其顶点为坐标向量 ",[363,134927,134929,134959],{"className":134928,"translate":367},[366],[363,134930,134932],{"className":134931},[371],[373,134933,134934],{"xmlns":375},[377,134935,134936,134956],{},[380,134937,134938,134944,134946,134948,134950],{},[543,134939,134940,134942],{},[383,134941,7915],{"mathvariant":75598},[463,134943,465],{},[458,134945,898],{"separator":394},[458,134947,133150],{},[458,134949,898],{"separator":394},[543,134951,134952,134954],{},[383,134953,7915],{"mathvariant":75598},[383,134955,11044],{},[387,134957,134958],{"encoding":389},"\\mathbf{e}_1, \\dots, \\mathbf{e}_K",[363,134960,134962],{"className":134961,"ariaHidden":394},[393],[363,134963,134965,134968,135008,135011,135014,135017,135020,135023,135026],{"className":134964},[398],[363,134966],{"className":134967,"style":133807},[402],[363,134969,134971,134974],{"className":134970},[407],[363,134972,7915],{"className":134973},[407,75943],[363,134975,134977],{"className":134976},[629],[363,134978,134980,135000],{"className":134979},[633,634],[363,134981,134983,134997],{"className":134982},[638],[363,134984,134986],{"className":134985,"style":11483},[642],[363,134987,134988,134991],{"style":730},[363,134989],{"className":134990,"style":651},[650],[363,134992,134994],{"className":134993},[655,656,657,658],[363,134995,465],{"className":134996},[407,658],[363,134998,666],{"className":134999},[665],[363,135001,135003],{"className":135002},[638],[363,135004,135006],{"className":135005,"style":8316},[642],[363,135007],{},[363,135009,898],{"className":135010},[920],[363,135012],{"className":135013,"style":754},[505],[363,135015,133150],{"className":135016},[758],[363,135018],{"className":135019,"style":754},[505],[363,135021,898],{"className":135022},[920],[363,135024],{"className":135025,"style":754},[505],[363,135027,135029,135032],{"className":135028},[407],[363,135030,7915],{"className":135031},[407,75943],[363,135033,135035],{"className":135034},[629],[363,135036,135038,135058],{"className":135037},[633,634],[363,135039,135041,135055],{"className":135040},[638],[363,135042,135044],{"className":135043,"style":727},[642],[363,135045,135046,135049],{"style":730},[363,135047],{"className":135048,"style":651},[650],[363,135050,135052],{"className":135051},[655,656,657,658],[363,135053,11044],{"className":135054,"style":11132},[407,408,658],[363,135056,666],{"className":135057},[665],[363,135059,135061],{"className":135060},[638],[363,135062,135064],{"className":135063,"style":8316},[642],[363,135065],{},"。低维特例给出了全部几何直观：",[363,135068,135070,135088],{"className":135069,"translate":367},[366],[363,135071,135073],{"className":135072},[371],[373,135074,135075],{"xmlns":375},[377,135076,135077,135085],{},[380,135078,135079,135081,135083],{},[383,135080,11044],{},[458,135082,471],{},[463,135084,2811],{},[387,135086,135087],{"encoding":389},"K = 2",[363,135089,135091,135109],{"className":135090,"ariaHidden":394},[393],[363,135092,135094,135097,135100,135103,135106],{"className":135093},[398],[363,135095],{"className":135096,"style":1011},[402],[363,135098,11044],{"className":135099,"style":11132},[407,408],[363,135101],{"className":135102,"style":506},[505],[363,135104,471],{"className":135105},[510],[363,135107],{"className":135108,"style":506},[505],[363,135110,135112,135115],{"className":135111},[398],[363,135113],{"className":135114,"style":520},[402],[363,135116,2811],{"className":135117},[407]," 时它是一维线段，",[363,135120,135122,135140],{"className":135121,"translate":367},[366],[363,135123,135125],{"className":135124},[371],[373,135126,135127],{"xmlns":375},[377,135128,135129,135137],{},[380,135130,135131,135133,135135],{},[383,135132,11044],{},[458,135134,471],{},[463,135136,41107],{},[387,135138,135139],{"encoding":389},"K = 3",[363,135141,135143,135161],{"className":135142,"ariaHidden":394},[393],[363,135144,135146,135149,135152,135155,135158],{"className":135145},[398],[363,135147],{"className":135148,"style":1011},[402],[363,135150,11044],{"className":135151,"style":11132},[407,408],[363,135153],{"className":135154,"style":506},[505],[363,135156,471],{"className":135157},[510],[363,135159],{"className":135160,"style":506},[505],[363,135162,135164,135167],{"className":135163},[398],[363,135165],{"className":135166,"style":520},[402],[363,135168,41107],{"className":135169},[407]," 时是二维三角形，",[363,135172,135174,135192],{"className":135173,"translate":367},[366],[363,135175,135177],{"className":135176},[371],[373,135178,135179],{"xmlns":375},[377,135180,135181,135189],{},[380,135182,135183,135185,135187],{},[383,135184,11044],{},[458,135186,471],{},[463,135188,39361],{},[387,135190,135191],{"encoding":389},"K = 4",[363,135193,135195,135213],{"className":135194,"ariaHidden":394},[393],[363,135196,135198,135201,135204,135207,135210],{"className":135197},[398],[363,135199],{"className":135200,"style":1011},[402],[363,135202,11044],{"className":135203,"style":11132},[407,408],[363,135205],{"className":135206,"style":506},[505],[363,135208,471],{"className":135209},[510],[363,135211],{"className":135212,"style":506},[505],[363,135214,135216,135219],{"className":135215},[398],[363,135217],{"className":135218,"style":520},[402],[363,135220,39361],{"className":135221},[407]," 时是三维四面体。它与 ",[363,135224,135226,135239],{"className":135225,"translate":367},[366],[363,135227,135229],{"className":135228},[371],[373,135230,135231],{"xmlns":375},[377,135232,135233,135237],{},[380,135234,135235],{},[383,135236,11044],{},[387,135238,11044],{"encoding":389},[363,135240,135242],{"className":135241,"ariaHidden":394},[393],[363,135243,135245,135248],{"className":135244},[398],[363,135246],{"className":135247,"style":1011},[402],[363,135249,11044],{"className":135250,"style":11132},[407,408]," 个取值上的离散概率分布全体一一对应，因此「分类器的输出」与「单纯形上的点」是同一件事的两种说法。",[11,135253,135254,135255,135412,135413,135635,135636,135666,135667,135727,135728,135881,135882,135955,135956,136028],{},"本文采用如下测度约定：以 ",[363,135256,135258,135294],{"className":135257,"translate":367},[366],[363,135259,135261],{"className":135260},[371],[373,135262,135263],{"xmlns":375},[377,135264,135265,135291],{},[380,135266,135267,135273,135275,135277,135279],{},[543,135268,135269,135271],{},[383,135270,1121],{},[463,135272,465],{},[458,135274,898],{"separator":394},[458,135276,133150],{},[458,135278,898],{"separator":394},[543,135280,135281,135283],{},[383,135282,1121],{},[380,135284,135285,135287,135289],{},[383,135286,11044],{},[458,135288,5161],{},[463,135290,465],{},[387,135292,135293],{"encoding":389},"x_1, \\dots, x_{K-1}",[363,135295,135297],{"className":135296,"ariaHidden":394},[393],[363,135298,135300,135304,135344,135347,135350,135353,135356,135359,135362],{"className":135299},[398],[363,135301],{"className":135302,"style":135303},[402],"height:0.6389em;vertical-align:-0.2083em;",[363,135305,135307,135310],{"className":135306},[407],[363,135308,1121],{"className":135309},[407,408],[363,135311,135313],{"className":135312},[629],[363,135314,135316,135336],{"className":135315},[633,634],[363,135317,135319,135333],{"className":135318},[638],[363,135320,135322],{"className":135321,"style":11483},[642],[363,135323,135324,135327],{"style":730},[363,135325],{"className":135326,"style":651},[650],[363,135328,135330],{"className":135329},[655,656,657,658],[363,135331,465],{"className":135332},[407,658],[363,135334,666],{"className":135335},[665],[363,135337,135339],{"className":135338},[638],[363,135340,135342],{"className":135341,"style":8316},[642],[363,135343],{},[363,135345,898],{"className":135346},[920],[363,135348],{"className":135349,"style":754},[505],[363,135351,133150],{"className":135352},[758],[363,135354],{"className":135355,"style":754},[505],[363,135357,898],{"className":135358},[920],[363,135360],{"className":135361,"style":754},[505],[363,135363,135365,135368],{"className":135364},[407],[363,135366,1121],{"className":135367},[407,408],[363,135369,135371],{"className":135370},[629],[363,135372,135374,135403],{"className":135373},[633,634],[363,135375,135377,135400],{"className":135376},[638],[363,135378,135380],{"className":135379,"style":727},[642],[363,135381,135382,135385],{"style":730},[363,135383],{"className":135384,"style":651},[650],[363,135386,135388],{"className":135387},[655,656,657,658],[363,135389,135391,135394,135397],{"className":135390},[407,658],[363,135392,11044],{"className":135393,"style":11132},[407,408,658],[363,135395,5161],{"className":135396},[2939,658],[363,135398,465],{"className":135399},[407,658],[363,135401,666],{"className":135402},[665],[363,135404,135406],{"className":135405},[638],[363,135407,135410],{"className":135408,"style":135409},[642],"height:0.2083em;",[363,135411],{}," 为自由坐标、",[363,135414,135416,135458],{"className":135415,"translate":367},[366],[363,135417,135419],{"className":135418},[371],[373,135420,135421],{"xmlns":375},[377,135422,135423,135455],{},[380,135424,135425,135431,135433,135435,135437,135449],{},[543,135426,135427,135429],{},[383,135428,1121],{},[383,135430,11044],{},[458,135432,471],{},[463,135434,465],{},[458,135436,5161],{},[543,135438,135439,135441],{},[458,135440,13811],{},[380,135442,135443,135445,135447],{},[383,135444,12313],{},[458,135446,7599],{},[383,135448,11044],{},[543,135450,135451,135453],{},[383,135452,1121],{},[383,135454,12313],{},[387,135456,135457],{"encoding":389},"x_K = 1 - \\sum_{i\u003CK} x_i",[363,135459,135461,135516,135534],{"className":135460,"ariaHidden":394},[393],[363,135462,135464,135467,135507,135510,135513],{"className":135463},[398],[363,135465],{"className":135466,"style":13724},[402],[363,135468,135470,135473],{"className":135469},[407],[363,135471,1121],{"className":135472},[407,408],[363,135474,135476],{"className":135475},[629],[363,135477,135479,135499],{"className":135478},[633,634],[363,135480,135482,135496],{"className":135481},[638],[363,135483,135485],{"className":135484,"style":727},[642],[363,135486,135487,135490],{"style":730},[363,135488],{"className":135489,"style":651},[650],[363,135491,135493],{"className":135492},[655,656,657,658],[363,135494,11044],{"className":135495,"style":11132},[407,408,658],[363,135497,666],{"className":135498},[665],[363,135500,135502],{"className":135501},[638],[363,135503,135505],{"className":135504,"style":8316},[642],[363,135506],{},[363,135508],{"className":135509,"style":506},[505],[363,135511,471],{"className":135512},[510],[363,135514],{"className":135515,"style":506},[505],[363,135517,135519,135522,135525,135528,135531],{"className":135518},[398],[363,135520],{"className":135521,"style":12928},[402],[363,135523,465],{"className":135524},[407],[363,135526],{"className":135527,"style":2935},[505],[363,135529,5161],{"className":135530},[2939],[363,135532],{"className":135533,"style":2935},[505],[363,135535,135537,135541,135592,135595],{"className":135536},[398],[363,135538],{"className":135539,"style":135540},[402],"height:1.0771em;vertical-align:-0.3271em;",[363,135542,135544,135547],{"className":135543},[1832],[363,135545,13811],{"className":135546,"style":40032},[1832,1833,5964],[363,135548,135550],{"className":135549},[629],[363,135551,135553,135583],{"className":135552},[633,634],[363,135554,135556,135580],{"className":135555},[638],[363,135557,135560],{"className":135558,"style":135559},[642],"height:0.1786em;",[363,135561,135562,135565],{"style":40048},[363,135563],{"className":135564,"style":651},[650],[363,135566,135568],{"className":135567},[655,656,657,658],[363,135569,135571,135574,135577],{"className":135570},[407,658],[363,135572,12313],{"className":135573},[407,408,658],[363,135575,7599],{"className":135576},[510,658],[363,135578,11044],{"className":135579,"style":11132},[407,408,658],[363,135581,666],{"className":135582},[665],[363,135584,135586],{"className":135585},[638],[363,135587,135590],{"className":135588,"style":135589},[642],"height:0.3271em;",[363,135591],{},[363,135593],{"className":135594,"style":754},[505],[363,135596,135598,135601],{"className":135597},[407],[363,135599,1121],{"className":135600},[407,408],[363,135602,135604],{"className":135603},[629],[363,135605,135607,135627],{"className":135606},[633,634],[363,135608,135610,135624],{"className":135609},[638],[363,135611,135613],{"className":135612,"style":12532},[642],[363,135614,135615,135618],{"style":730},[363,135616],{"className":135617,"style":651},[650],[363,135619,135621],{"className":135620},[655,656,657,658],[363,135622,12313],{"className":135623},[407,408,658],[363,135625,666],{"className":135626},[665],[363,135628,135630],{"className":135629},[638],[363,135631,135633],{"className":135632,"style":8316},[642],[363,135634],{},"，并以 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维 Lebesgue 测度 ",[363,135729,135731,135767],{"className":135730,"translate":367},[366],[363,135732,135734],{"className":135733},[371],[373,135735,135736],{"xmlns":375},[377,135737,135738,135764],{},[380,135739,135740,135742,135748,135750,135752],{},[383,135741,592],{"mathvariant":557},[543,135743,135744,135746],{},[383,135745,1121],{},[463,135747,465],{},[458,135749,87511],{},[383,135751,592],{"mathvariant":557},[543,135753,135754,135756],{},[383,135755,1121],{},[380,135757,135758,135760,135762],{},[383,135759,11044],{},[458,135761,5161],{},[463,135763,465],{},[387,135765,135766],{"encoding":389},"\\mathrm{d}x_1 \\cdots \\mathrm{d}x_{K-1}",[363,135768,135770],{"className":135769,"ariaHidden":394},[393],[363,135771,135773,135777,135780,135820,135823,135826,135829,135832],{"className":135772},[398],[363,135774],{"className":135775,"style":135776},[402],"height:0.9028em;vertical-align:-0.2083em;",[363,135778,592],{"className":135779},[407,33034],[363,135781,135783,135786],{"className":135782},[407],[363,135784,1121],{"className":135785},[407,408],[363,135787,135789],{"className":135788},[629],[363,135790,135792,135812],{"className":135791},[633,634],[363,135793,135795,135809],{"className":135794},[638],[363,135796,135798],{"className":135797,"style":11483},[642],[363,135799,135800,135803],{"style":730},[363,135801],{"className":135802,"style":651},[650],[363,135804,135806],{"className":135805},[655,656,657,658],[363,135807,465],{"className":135808},[407,658],[363,135810,666],{"className":135811},[665],[363,135813,135815],{"className":135814},[638],[363,135816,135818],{"className":135817,"style":8316},[642],[363,135819],{},[363,135821],{"className":135822,"style":754},[505],[363,135824,87511],{"className":135825},[758],[363,135827],{"className":135828,"style":754},[505],[363,135830,592],{"className":135831},[407,33034],[363,135833,135835,135838],{"className":135834},[407],[363,135836,1121],{"className":135837},[407,408],[363,135839,135841],{"className":135840},[629],[363,135842,135844,135873],{"className":135843},[633,634],[363,135845,135847,135870],{"className":135846},[638],[363,135848,135850],{"className":135849,"style":727},[642],[363,135851,135852,135855],{"style":730},[363,135853],{"className":135854,"style":651},[650],[363,135856,135858],{"className":135857},[655,656,657,658],[363,135859,135861,135864,135867],{"className":135860},[407,658],[363,135862,11044],{"className":135863,"style":11132},[407,408,658],[363,135865,5161],{"className":135866},[2939,658],[363,135868,465],{"className":135869},[407,658],[363,135871,666],{"className":135872},[665],[363,135874,135876],{"className":135875},[638],[363,135877,135879],{"className":135878,"style":135409},[642],[363,135880],{}," 为参考测度。在该约定下 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的体积为 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形式化定义",[143,136034,136035],{},[11,136036,136037,136040,136041,11751,136311,136522,136523,136551,136552,136639,136640,137028,137029,137057,137058,137093,137094,137172,137173,137176],{},[1344,136038,136039],{},"定义 1.2（Dirichlet 分布）"," 设参数向量 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\\mathbb{R}_{>0}^K",[363,136098,136100,136125,136245],{"className":136099,"ariaHidden":394},[393],[363,136101,136103,136106,136116,136119,136122],{"className":136102},[398],[363,136104],{"className":136105,"style":135662},[402],[363,136107,136109],{"className":136108},[407],[363,136110,136112],{"className":136111},[407],[363,136113,45126],{"className":136114},[407,136115],"boldsymbol",[363,136117],{"className":136118,"style":506},[505],[363,136120,471],{"className":136121},[510],[363,136123],{"className":136124,"style":506},[505],[363,136126,136128,136131,136134,136175,136178,136181,136184,136187,136190,136193,136233,136236,136239,136242],{"className":136127},[398],[363,136129],{"className":136130,"style":487},[402],[363,136132,461],{"className":136133},[494],[363,136135,136137,136140],{"className":136136},[407],[363,136138,45126],{"className":136139,"style":45156},[407,408],[363,136141,136143],{"className":136142},[629],[363,136144,136146,136167],{"className":136145},[633,634],[363,136147,136149,136164],{"className":136148},[638],[363,136150,136152],{"className":136151,"style":11483},[642],[363,136153,136155,136158],{"style":136154},"top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;",[363,136156],{"className":136157,"style":651},[650],[363,136159,136161],{"className":136160},[655,656,657,658],[363,136162,465],{"className":136163},[407,658],[363,136165,666],{"className":136166},[665],[363,136168,136170],{"className":136169},[638],[363,136171,136173],{"className":136172,"style":8316},[642],[363,136174],{},[363,136176,898],{"className":136177},[920],[363,136179],{"className":136180,"style":754},[505],[363,136182,133150],{"className":136183},[758],[363,136185],{"className":136186,"style":754},[505],[363,136188,898],{"className":136189},[920],[363,136191],{"className":136192,"style":754},[505],[363,136194,136196,136199],{"className":136195},[407],[363,136197,45126],{"className":136198,"style":45156},[407,408],[363,136200,136202],{"className":136201},[629],[363,136203,136205,136225],{"className":136204},[633,634],[363,136206,136208,136222],{"className":136207},[638],[363,136209,136211],{"className":136210,"style":727},[642],[363,136212,136213,136216],{"style":136154},[363,136214],{"className":136215,"style":651},[650],[363,136217,136219],{"className":136218},[655,656,657,658],[363,136220,11044],{"className":136221,"style":11132},[407,408,658],[363,136223,666],{"className":136224},[665],[363,136226,136228],{"className":136227},[638],[363,136229,136231],{"className":136230,"style":8316},[642],[363,136232],{},[363,136234,468],{"className":136235},[501],[363,136237],{"className":136238,"style":506},[505],[363,136240,6257],{"className":136241},[510],[363,136243],{"className":136244,"style":506},[505],[363,136246,136248,136252],{"className":136247},[398],[363,136249],{"className":136250,"style":136251},[402],"height:1.1168em;vertical-align:-0.2755em;",[363,136253,136255,136258],{"className":136254},[407],[363,136256,1381],{"className":136257},[407,714],[363,136259,136261],{"className":136260},[629],[363,136262,136264,136302],{"className":136263},[633,634],[363,136265,136267,136299],{"className":136266},[638],[363,136268,136270,136288],{"className":136269,"style":52457},[642],[363,136271,136273,136276],{"style":136272},"top:-2.4519em;margin-left:0em;margin-right:0.05em;",[363,136274],{"className":136275,"style":651},[650],[363,136277,136279],{"className":136278},[655,656,657,658],[363,136280,136282,136285],{"className":136281},[407,658],[363,136283,3251],{"className":136284},[510,658],[363,136286,474],{"className":136287},[407,658],[363,136289,136290,136293],{"style":7213},[363,136291],{"className":136292,"style":651},[650],[363,136294,136296],{"className":136295},[655,656,657,658],[363,136297,11044],{"className":136298,"style":11132},[407,408,658],[363,136300,666],{"className":136301},[665],[363,136303,136305],{"className":136304},[638],[363,136306,136309],{"className":136307,"style":136308},[642],"height:0.2755em;",[363,136310],{},[363,136312,136314,136354],{"className":136313,"translate":367},[366],[363,136315,136317],{"className":136316},[371],[373,136318,136319],{"xmlns":375},[377,136320,136321,136351],{},[380,136322,136323,136329,136331,136345],{},[543,136324,136325,136327],{},[383,136326,45126],{},[463,136328,474],{},[458,136330,471],{},[7901,136332,136333,136335,136343],{},[458,136334,13811],{},[380,136336,136337,136339,136341],{},[383,136338,12313],{},[458,136340,471],{},[463,136342,465],{},[383,136344,11044],{},[543,136346,136347,136349],{},[383,136348,45126],{},[383,136350,12313],{},[387,136352,136353],{"encoding":389},"\\alpha_0 = \\sum_{i=1}^K \\alpha_i",[363,136355,136357,136412],{"className":136356,"ariaHidden":394},[393],[363,136358,136360,136363,136403,136406,136409],{"className":136359},[398],[363,136361],{"className":136362,"style":13724},[402],[363,136364,136366,136369],{"className":136365},[407],[363,136367,45126],{"className":136368,"style":45156},[407,408],[363,136370,136372],{"className":136371},[629],[363,136373,136375,136395],{"className":136374},[633,634],[363,136376,136378,136392],{"className":136377},[638],[363,136379,136381],{"className":136380,"style":11483},[642],[363,136382,136383,136386],{"style":136154},[363,136384],{"className":136385,"style":651},[650],[363,136387,136389],{"className":136388},[655,656,657,658],[363,136390,474],{"className":136391},[407,658],[363,136393,666],{"className":136394},[665],[363,136396,136398],{"className":136397},[638],[363,136399,136401],{"className":136400,"style":8316},[642],[363,136402],{},[363,136404],{"className":136405,"style":506},[505],[363,136407,471],{"className":136408},[510],[363,136410],{"className":136411,"style":506},[505],[363,136413,136415,136419,136479,136482],{"className":136414},[398],[363,136416],{"className":136417,"style":136418},[402],"height:1.2809em;vertical-align:-0.2997em;",[363,136420,136422,136425],{"className":136421},[1832],[363,136423,13811],{"className":136424,"style":40032},[1832,1833,5964],[363,136426,136428],{"className":136427},[629],[363,136429,136431,136471],{"className":136430},[633,634],[363,136432,136434,136468],{"className":136433},[638],[363,136435,136437,136457],{"className":136436,"style":133537},[642],[363,136438,136439,136442],{"style":40048},[363,136440],{"className":136441,"style":651},[650],[363,136443,136445],{"className":136444},[655,656,657,658],[363,136446,136448,136451,136454],{"className":136447},[407,658],[363,136449,12313],{"className":136450},[407,408,658],[363,136452,471],{"className":136453},[510,658],[363,136455,465],{"className":136456},[407,658],[363,136458,136459,136462],{"style":133560},[363,136460],{"className":136461,"style":651},[650],[363,136463,136465],{"className":136464},[655,656,657,658],[363,136466,11044],{"className":136467,"style":11132},[407,408,658],[363,136469,666],{"className":136470},[665],[363,136472,136474],{"className":136473},[638],[363,136475,136477],{"className":136476,"style":40067},[642],[363,136478],{},[363,136480],{"className":136481,"style":754},[505],[363,136483,136485,136488],{"className":136484},[407],[363,136486,45126],{"className":136487,"style":45156},[407,408],[363,136489,136491],{"className":136490},[629],[363,136492,136494,136514],{"className":136493},[633,634],[363,136495,136497,136511],{"className":136496},[638],[363,136498,136500],{"className":136499,"style":12532},[642],[363,136501,136502,136505],{"style":136154},[363,136503],{"className":136504,"style":651},[650],[363,136506,136508],{"className":136507},[655,656,657,658],[363,136509,12313],{"className":136510},[407,408,658],[363,136512,666],{"className":136513},[665],[363,136515,136517],{"className":136516},[638],[363,136518,136520],{"className":136519,"style":8316},[642],[363,136521],{},"。若随机向量 ",[363,136524,136526,136539],{"className":136525,"translate":367},[366],[363,136527,136529],{"className":136528},[371],[373,136530,136531],{"xmlns":375},[377,136532,136533,136537],{},[380,136534,136535],{},[383,136536,1121],{"mathvariant":75598},[387,136538,135652],{"encoding":389},[363,136540,136542],{"className":136541,"ariaHidden":394},[393],[363,136543,136545,136548],{"className":136544},[398],[363,136546],{"className":136547,"style":135662},[402],[363,136549,1121],{"className":136550},[407,75943]," 取值于 ",[363,136553,136555,136583],{"className":136554,"translate":367},[366],[363,136556,136558],{"className":136557},[371],[373,136559,136560],{"xmlns":375},[377,136561,136562,136580],{},[380,136563,136564,136566,136568],{},[383,136565,134509],{"mathvariant":557},[458,136567,4451],{},[7162,136569,136570,136572],{},[383,136571,126067],{"mathvariant":557},[380,136573,136574,136576,136578],{},[383,136575,11044],{},[458,136577,5161],{},[463,136579,465],{},[387,136581,136582],{"encoding":389},"\\operatorname{int}\\Delta^{K-1}",[363,136584,136586],{"className":136585,"ariaHidden":394},[393],[363,136587,136589,136592,136598,136601],{"className":136588},[398],[363,136590],{"className":136591,"style":52457},[402],[363,136593,136595],{"className":136594},[1832],[363,136596,134509],{"className":136597},[407,33034],[363,136599],{"className":136600,"style":754},[505],[363,136602,136604,136607],{"className":136603},[407],[363,136605,126067],{"className":136606},[407],[363,136608,136610],{"className":136609},[629],[363,136611,136613],{"className":136612},[633],[363,136614,136616],{"className":136615},[638],[363,136617,136619],{"className":136618,"style":52457},[642],[363,136620,136621,136624],{"style":7213},[363,136622],{"className":136623,"style":651},[650],[363,136625,136627],{"className":136626},[655,656,657,658],[363,136628,136630,136633,136636],{"className":136629},[407,658],[363,136631,11044],{"className":136632,"style":11132},[407,408,658],[363,136634,5161],{"className":136635},[2939,658],[363,136637,465],{"className":136638},[407,658],"，且关于上述参考测度的密度为\n",[363,136641,136643,136716],{"className":136642,"translate":367},[366],[363,136644,136646],{"className":136645},[371],[373,136647,136648],{"xmlns":375},[377,136649,136650,136713],{},[380,136651,136652,136654,136656,136658,136660,136662,136664,136666,136680,136695],{},[383,136653,385],{},[458,136655,461],{"stretchy":460},[383,136657,1121],{"mathvariant":75598},[458,136659,5156],{},[383,136661,45126],{"mathvariant":136055},[458,136663,468],{"stretchy":460},[458,136665,471],{},[586,136667,136668,136670],{},[463,136669,465],{},[380,136671,136672,136674,136676,136678],{},[383,136673,39663],{},[458,136675,461],{"stretchy":460},[383,136677,45126],{"mathvariant":136055},[458,136679,468],{"stretchy":460},[7901,136681,136682,136685,136693],{},[458,136683,136684],{},"∏",[380,136686,136687,136689,136691],{},[383,136688,12313],{},[458,136690,471],{},[463,136692,465],{},[383,136694,11044],{},[7901,136696,136697,136699,136701],{},[383,136698,1121],{},[383,136700,12313],{},[380,136702,136703,136709,136711],{},[543,136704,136705,136707],{},[383,136706,45126],{},[383,136708,12313],{},[458,136710,5161],{},[463,136712,465],{},[387,136714,136715],{"encoding":389},"f(\\mathbf{x} \\mid \\boldsymbol{\\alpha}) = \\frac{1}{B(\\boldsymbol{\\alpha})} \\prod_{i=1}^K x_i^{\\alpha_i - 1}",[363,136717,136719,136743,136770],{"className":136718,"ariaHidden":394},[393],[363,136720,136722,136725,136728,136731,136734,136737,136740],{"className":136721},[398],[363,136723],{"className":136724,"style":487},[402],[363,136726,385],{"className":136727,"style":409},[407,408],[363,136729,461],{"className":136730},[494],[363,136732,1121],{"className":136733},[407,75943],[363,136735],{"className":136736,"style":506},[505],[363,136738,5156],{"className":136739},[510],[363,136741],{"className":136742,"style":506},[505],[363,136744,136746,136749,136758,136761,136764,136767],{"className":136745},[398],[363,136747],{"className":136748,"style":487},[402],[363,136750,136752],{"className":136751},[407],[363,136753,136755],{"className":136754},[407],[363,136756,45126],{"className":136757},[407,136115],[363,136759,468],{"className":136760},[501],[363,136762],{"className":136763,"style":506},[505],[363,136765,471],{"className":136766},[510],[363,136768],{"className":136769,"style":506},[505],[363,136771,136773,136777,136860,136863,136923,136926],{"className":136772},[398],[363,136774],{"className":136775,"style":136776},[402],"height:1.5012em;vertical-align:-0.52em;",[363,136778,136780,136783,136857],{"className":136779},[407],[363,136781],{"className":136782},[494,789],[363,136784,136786],{"className":136785},[586],[363,136787,136789,136849],{"className":136788},[633,634],[363,136790,136792,136846],{"className":136791},[638],[363,136793,136795,136824,136832],{"className":136794,"style":2872},[642],[363,136796,136797,136800],{"style":2875},[363,136798],{"className":136799,"style":809},[650],[363,136801,136803],{"className":136802},[655,656,657,658],[363,136804,136806,136809,136812,136821],{"className":136805},[407,658],[363,136807,39663],{"className":136808,"style":39759},[407,408,658],[363,136810,461],{"className":136811},[494,658],[363,136813,136815],{"className":136814},[407,658],[363,136816,136818],{"className":136817},[407,658],[363,136819,45126],{"className":136820},[407,136115,658],[363,136822,468],{"className":136823},[501,658],[363,136825,136826,136829],{"style":821},[363,136827],{"className":136828,"style":809},[650],[363,136830],{"className":136831,"style":829},[828],[363,136833,136834,136837],{"style":2898},[363,136835],{"className":136836,"style":809},[650],[363,136838,136840],{"className":136839},[655,656,657,658],[363,136841,136843],{"className":136842},[407,658],[363,136844,465],{"className":136845},[407,658],[363,136847,666],{"className":136848},[665],[363,136850,136852],{"className":136851},[638],[363,136853,136855],{"className":136854,"style":45974},[642],[363,136856],{},[363,136858],{"className":136859},[501,789],[363,136861],{"className":136862,"style":754},[505],[363,136864,136866,136869],{"className":136865},[1832],[363,136867,136684],{"className":136868,"style":40032},[1832,1833,5964],[363,136870,136872],{"className":136871},[629],[363,136873,136875,136915],{"className":136874},[633,634],[363,136876,136878,136912],{"className":136877},[638],[363,136879,136881,136901],{"className":136880,"style":133537},[642],[363,136882,136883,136886],{"style":40048},[363,136884],{"className":136885,"style":651},[650],[363,136887,136889],{"className":136888},[655,656,657,658],[363,136890,136892,136895,136898],{"className":136891},[407,658],[363,136893,12313],{"className":136894},[407,408,658],[363,136896,471],{"className":136897},[510,658],[363,136899,465],{"className":136900},[407,658],[363,136902,136903,136906],{"style":133560},[363,136904],{"className":136905,"style":651},[650],[363,136907,136909],{"className":136908},[655,656,657,658],[363,136910,11044],{"className":136911,"style":11132},[407,408,658],[363,136913,666],{"className":136914},[665],[363,136916,136918],{"className":136917},[638],[363,136919,136921],{"className":136920,"style":40067},[642],[363,136922],{},[363,136924],{"className":136925,"style":754},[505],[363,136927,136929,136932],{"className":136928},[407],[363,136930,1121],{"className":136931},[407,408],[363,136933,136935],{"className":136934},[629],[363,136936,136938,137019],{"className":136937},[633,634],[363,136939,136941,137016],{"className":136940},[638],[363,136942,136945,136957],{"className":136943,"style":136944},[642],"height:0.896em;",[363,136946,136948,136951],{"style":136947},"top:-2.4231em;margin-left:0em;margin-right:0.05em;",[363,136949],{"className":136950,"style":651},[650],[363,136952,136954],{"className":136953},[655,656,657,658],[363,136955,12313],{"className":136956},[407,408,658],[363,136958,136960,136963],{"style":136959},"top:-3.1449em;margin-right:0.05em;",[363,136961],{"className":136962,"style":651},[650],[363,136964,136966],{"className":136965},[655,656,657,658],[363,136967,136969,137010,137013],{"className":136968},[407,658],[363,136970,136972,136975],{"className":136971},[407,658],[363,136973,45126],{"className":136974,"style":45156},[407,408,658],[363,136976,136978],{"className":136977},[629],[363,136979,136981,137002],{"className":136980},[633,634],[363,136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",[363,137030,137032,137045],{"className":137031,"translate":367},[366],[363,137033,137035],{"className":137034},[371],[373,137036,137037],{"xmlns":375},[377,137038,137039,137043],{},[380,137040,137041],{},[383,137042,1121],{"mathvariant":75598},[387,137044,135652],{"encoding":389},[363,137046,137048],{"className":137047,"ariaHidden":394},[393],[363,137049,137051,137054],{"className":137050},[398],[363,137052],{"className":137053,"style":135662},[402],[363,137055,1121],{"className":137056},[407,75943]," 服从参数为 ",[363,137059,137061,137075],{"className":137060,"translate":367},[366],[363,137062,137064],{"className":137063},[371],[373,137065,137066],{"xmlns":375},[377,137067,137068,137072],{},[380,137069,137070],{},[383,137071,45126],{"mathvariant":136055},[387,137073,137074],{"encoding":389},"\\boldsymbol{\\alpha}",[363,137076,137078],{"className":137077,"ariaHidden":394},[393],[363,137079,137081,137084],{"className":137080},[398],[363,137082],{"className":137083,"style":135662},[402],[363,137085,137087],{"className":137086},[407],[363,137088,137090],{"className":137089},[407],[363,137091,45126],{"className":137092},[407,136115]," 的迪利克雷分布，记作 ",[363,137095,137097,137124],{"className":137096,"translate":367},[366],[363,137098,137100],{"className":137099},[371],[373,137101,137102],{"xmlns":375},[377,137103,137104,137121],{},[380,137105,137106,137108,137110,137113,137115,137117,137119],{},[383,137107,1121],{"mathvariant":75598},[458,137109,9217],{},[383,137111,137112],{"mathvariant":557},"Dir",[458,137114,4451],{},[458,137116,461],{"stretchy":460},[383,137118,45126],{"mathvariant":136055},[458,137120,468],{"stretchy":460},[387,137122,137123],{"encoding":389},"\\mathbf{x} \\sim \\operatorname{Dir}(\\boldsymbol{\\alpha})",[363,137125,137127,137145],{"className":137126,"ariaHidden":394},[393],[363,137128,137130,137133,137136,137139,137142],{"className":137129},[398],[363,137131],{"className":137132,"style":135662},[402],[363,137134,1121],{"className":137135},[407,75943],[363,137137],{"className":137138,"style":506},[505],[363,137140,9217],{"className":137141},[510],[363,137143],{"className":137144,"style":506},[505],[363,137146,137148,137151,137157,137160,137169],{"className":137147},[398],[363,137149],{"className":137150,"style":487},[402],[363,137152,137154],{"className":137153},[1832],[363,137155,137112],{"className":137156},[407,33034],[363,137158,461],{"className":137159},[494],[363,137161,137163],{"className":137162},[407],[363,137164,137166],{"className":137165},[407],[363,137167,45126],{"className":137168},[407,136115],[363,137170,468],{"className":137171},[501],"。其中归一化常数为",[1344,137174,137175],{},"多元 Beta 函数",[363,137177,137179,137248],{"className":137178,"translate":367},[366],[363,137180,137182],{"className":137181},[371],[373,137183,137184],{"xmlns":375},[377,137185,137186,137245],{},[380,137187,137188,137190,137192,137194,137196,137198],{},[383,137189,39663],{},[458,137191,461],{"stretchy":460},[383,137193,45126],{"mathvariant":136055},[458,137195,468],{"stretchy":460},[458,137197,471],{},[586,137199,137200,137229],{},[380,137201,137202,137216,137219,137221,137227],{},[7901,137203,137204,137206,137214],{},[458,137205,136684],{},[380,137207,137208,137210,137212],{},[383,137209,12313],{},[458,137211,471],{},[463,137213,465],{},[383,137215,11044],{},[383,137217,137218],{"mathvariant":557},"Γ",[458,137220,461],{"stretchy":460},[543,137222,137223,137225],{},[383,137224,45126],{},[383,137226,12313],{},[458,137228,468],{"stretchy":460},[380,137230,137231,137233],{},[383,137232,137218],{"mathvariant":557},[380,137234,137235,137237,137243],{},[458,137236,461],{"fence":394},[543,137238,137239,137241],{},[383,137240,45126],{},[463,137242,474],{},[458,137244,468],{"fence":394},[387,137246,137247],{"encoding":389},"B(\\boldsymbol{\\alpha}) = \\frac{\\prod_{i=1}^K \\Gamma(\\alpha_i)}{\\Gamma\\left(\\alpha_0\\right)}",[363,137249,137251,137284],{"className":137250,"ariaHidden":394},[393],[363,137252,137254,137257,137260,137263,137272,137275,137278,137281],{"className":137253},[398],[363,137255],{"className":137256,"style":487},[402],[363,137258,39663],{"className":137259,"style":39759},[407,408],[363,137261,461],{"className":137262},[494],[363,137264,137266],{"className":137265},[407],[363,137267,137269],{"className":137268},[407],[363,137270,45126],{"className":137271},[407,136115],[363,137273,468],{"className":137274},[501],[363,137276],{"className":137277,"style":506},[505],[363,137279,471],{"className":137280},[510],[363,137282],{"className":137283,"style":506},[505],[363,137285,137287,137291],{"className":137286},[398],[363,137288],{"className":137289,"style":137290},[402],"height:1.6984em;vertical-align:-0.52em;",[363,137292,137294,137297,137526],{"className":137293},[407],[363,137295],{"className":137296},[494,789],[363,137298,137300],{"className":137299},[586],[363,137301,137303,137518],{"className":137302},[633,634],[363,137304,137306,137515],{"className":137305},[638],[363,137307,137310,137380,137388],{"className":137308,"style":137309},[642],"height:1.1784em;",[363,137311,137312,137315],{"style":2875},[363,137313],{"className":137314,"style":809},[650],[363,137316,137318],{"className":137317},[655,656,657,658],[363,137319,137321,137324],{"className":137320},[407,658],[363,137322,137218],{"className":137323},[407,658],[363,137325,137327,137333,137374],{"className":137326},[758,658],[363,137328,137330],{"className":137329,"style":763},[494,658,762],[363,137331,461],{"className":137332},[658],[363,137334,137336,137339],{"className":137335},[407,658],[363,137337,45126],{"className":137338,"style":45156},[407,408,658],[363,137340,137342],{"className":137341},[629],[363,137343,137345,137366],{"className":137344},[633,634],[363,137346,137348,137363],{"className":137347},[638],[363,137349,137352],{"className":137350,"style":137351},[642],"height:0.3173em;",[363,137353,137354,137357],{"style":136989},[363,137355],{"className":137356,"style":30578},[650],[363,137358,137360],{"className":137359},[655,30582,10022,658],[363,137361,474],{"className":137362},[407,658],[363,137364,666],{"className":137365},[665],[363,137367,137369],{"className":137368},[638],[363,137370,137372],{"className":137371,"style":40142},[642],[363,137373],{},[363,137375,137377],{"className":137376,"style":763},[501,658,762],[363,137378,468],{"className":137379},[658],[363,137381,137382,137385],{"style":821},[363,137383],{"className":137384,"style":809},[650],[363,137386],{"className":137387,"style":829},[828],[363,137389,137391,137394],{"style":137390},"top:-3.535em;",[363,137392],{"className":137393,"style":809},[650],[363,137395,137397],{"className":137396},[655,656,657,658],[363,137398,137400,137463,137466,137469,137472,137512],{"className":137399},[407,658],[363,137401,137403,137406],{"className":137402},[1832,658],[363,137404,136684],{"className":137405,"style":40032},[1832,1833,5964,658],[363,137407,137409],{"className":137408},[629],[363,137410,137412,137454],{"className":137411},[633,634],[363,137413,137415,137451],{"className":137414},[638],[363,137416,137419,137440],{"className":137417,"style":137418},[642],"height:0.9191em;",[363,137420,137422,137425],{"style":137421},"top:-2.1786em;margin-left:0em;margin-right:0.0714em;",[363,137423],{"className":137424,"style":30578},[650],[363,137426,137428],{"className":137427},[655,30582,10022,658],[363,137429,137431,137434,137437],{"className":137430},[407,658],[363,137432,12313],{"className":137433},[407,408,658],[363,137435,471],{"className":137436},[510,658],[363,137438,465],{"className":137439},[407,658],[363,137441,137442,137445],{"style":38887},[363,137443],{"className":137444,"style":30578},[650],[363,137446,137448],{"className":137447},[655,30582,10022,658],[363,137449,11044],{"className":137450,"style":11132},[407,408,658],[363,137452,666],{"className":137453},[665],[363,137455,137457],{"className":137456},[638],[363,137458,137461],{"className":137459,"style":137460},[642],"height:0.3214em;",[363,137462],{},[363,137464],{"className":137465,"style":42738},[505,658],[363,137467,137218],{"className":137468},[407,658],[363,137470,461],{"className":137471},[494,658],[363,137473,137475,137478],{"className":137474},[407,658],[363,137476,45126],{"className":137477,"style":45156},[407,408,658],[363,137479,137481],{"className":137480},[629],[363,137482,137484,137504],{"className":137483},[633,634],[363,137485,137487,137501],{"className":137486},[638],[363,137488,137490],{"className":137489,"style":40120},[642],[363,137491,137492,137495],{"style":136989},[363,137493],{"className":137494,"style":30578},[650],[363,137496,137498],{"className":137497},[655,30582,10022,658],[363,137499,12313],{"className":137500},[407,408,658],[363,137502,666],{"className":137503},[665],[363,137505,137507],{"className":137506},[638],[363,137508,137510],{"className":137509,"style":40142},[642],[363,137511],{},[363,137513,468],{"className":137514},[501,658],[363,137516,666],{"className":137517},[665],[363,137519,137521],{"className":137520},[638],[363,137522,137524],{"className":137523,"style":45974},[642],[363,137525],{},[363,137527],{"className":137528},[501,789],[11,137530,137531],{},"该定义之所以合法，依赖于下述恒等式。它是整个理论的算术基础，值得单独给出证明。",[143,137533,137534],{},[11,137535,137536,137539,137540,137661,137662],{},[1344,137537,137538],{},"命题 1.3（归一化恒等式）"," 对任意 ",[363,137541,137543,137571],{"className":137542,"translate":367},[366],[363,137544,137546],{"className":137545},[371],[373,137547,137548],{"xmlns":375},[377,137549,137550,137568],{},[380,137551,137552,137554,137556],{},[383,137553,45126],{"mathvariant":136055},[458,137555,6257],{},[7901,137557,137558,137560,137566],{},[383,137559,1381],{"mathvariant":570},[380,137561,137562,137564],{},[458,137563,3251],{},[463,137565,474],{},[383,137567,11044],{},[387,137569,137570],{"encoding":389},"\\boldsymbol{\\alpha} \\in \\mathbb{R}_{>0}^K",[363,137572,137574,137598],{"className":137573,"ariaHidden":394},[393],[363,137575,137577,137580,137589,137592,137595],{"className":137576},[398],[363,137578],{"className":137579,"style":6272},[402],[363,137581,137583],{"className":137582},[407],[363,137584,137586],{"className":137585},[407],[363,137587,45126],{"className":137588},[407,136115],[363,137590],{"className":137591,"style":506},[505],[363,137593,6257],{"className":137594},[510],[363,137596],{"className":137597,"style":506},[505],[363,137599,137601,137604],{"className":137600},[398],[363,137602],{"className":137603,"style":136251},[402],[363,137605,137607,137610],{"className":137606},[407],[363,137608,1381],{"className":137609},[407,714],[363,137611,137613],{"className":137612},[629],[363,137614,137616,137653],{"className":137615},[633,634],[363,137617,137619,137650],{"className":137618},[638],[363,137620,137622,137639],{"className":137621,"style":52457},[642],[363,137623,137624,137627],{"style":136272},[363,137625],{"className":137626,"style":651},[650],[363,137628,137630],{"className":137629},[655,656,657,658],[363,137631,137633,137636],{"className":137632},[407,658],[363,137634,3251],{"className":137635},[510,658],[363,137637,474],{"className":137638},[407,658],[363,137640,137641,137644],{"style":7213},[363,137642],{"className":137643,"style":651},[650],[363,137645,137647],{"className":137646},[655,656,657,658],[363,137648,11044],{"className":137649,"style":11132},[407,408,658],[363,137651,666],{"className":137652},[665],[363,137654,137656],{"className":137655},[638],[363,137657,137659],{"className":137658,"style":136308},[642],[363,137660],{},"，\n",[363,137663,137665,137741],{"className":137664,"translate":367},[366],[363,137666,137668],{"className":137667},[371],[373,137669,137670],{"xmlns":375},[377,137671,137672,137738],{},[380,137673,137674,137690,137704,137722,137724,137726,137728,137730,137732,137734,137736],{},[543,137675,137676,137678],{},[458,137677,1635],{},[7162,137679,137680,137682],{},[383,137681,126067],{"mathvariant":557},[380,137683,137684,137686,137688],{},[383,137685,11044],{},[458,137687,5161],{},[463,137689,465],{},[7901,137691,137692,137694,137702],{},[458,137693,136684],{},[380,137695,137696,137698,137700],{},[383,137697,12313],{},[458,137699,471],{},[463,137701,465],{},[383,137703,11044],{},[7901,137705,137706,137708,137710],{},[383,137707,1121],{},[383,137709,12313],{},[380,137711,137712,137718,137720],{},[543,137713,137714,137716],{},[383,137715,45126],{},[383,137717,12313],{},[458,137719,5161],{},[463,137721,465],{},[1645,137723,1647],{},[383,137725,592],{"mathvariant":557},[383,137727,1121],{"mathvariant":75598},[458,137729,471],{},[383,137731,39663],{},[458,137733,461],{"stretchy":460},[383,137735,45126],{"mathvariant":136055},[458,137737,468],{"stretchy":460},[387,137739,137740],{"encoding":389},"\\int_{\\Delta^{K-1}} \\prod_{i=1}^K x_i^{\\alpha_i - 1} \\,\\mathrm{d}\\mathbf{x} = B(\\boldsymbol{\\alpha})",[363,137742,137744,138014],{"className":137743,"ariaHidden":394},[393],[363,137745,137747,137751,137833,137836,137896,137899,137996,137999,138002,138005,138008,138011],{"className":137746},[398],[363,137748],{"className":137749,"style":137750},[402],"height:1.3371em;vertical-align:-0.3558em;",[363,137752,137754,137757],{"className":137753},[1832],[363,137755,1635],{"className":137756,"style":5965},[1832,1833,5964],[363,137758,137760],{"className":137759},[629],[363,137761,137763,137824],{"className":137762},[633,634],[363,137764,137766,137821],{"className":137765},[638],[363,137767,137770],{"className":137768,"style":137769},[642],"height:0.186em;",[363,137771,137773,137776],{"style":137772},"top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;",[363,137774],{"className":137775,"style":651},[650],[363,137777,137779],{"className":137778},[655,656,657,658],[363,137780,137782],{"className":137781},[407,658],[363,137783,137785,137788],{"className":137784},[407,658],[363,137786,126067],{"className":137787},[407,658],[363,137789,137791],{"className":137790},[629],[363,137792,137794],{"className":137793},[633],[363,137795,137797],{"className":137796},[638],[363,137798,137801],{"className":137799,"style":137800},[642],"height:0.7741em;",[363,137802,137803,137806],{"style":45834},[363,137804],{"className":137805,"style":30578},[650],[363,137807,137809],{"className":137808},[655,30582,10022,658],[363,137810,137812,137815,137818],{"className":137811},[407,658],[363,137813,11044],{"className":137814,"style":11132},[407,408,658],[363,137816,5161],{"className":137817},[2939,658],[363,137819,465],{"className":137820},[407,658],[363,137822,666],{"className":137823},[665],[363,137825,137827],{"className":137826},[638],[363,137828,137831],{"className":137829,"style":137830},[642],"height:0.3558em;",[363,137832],{},[363,137834],{"className":137835,"style":754},[505],[363,137837,137839,137842],{"className":137838},[1832],[363,137840,136684],{"className":137841,"style":40032},[1832,1833,5964],[363,137843,137845],{"className":137844},[629],[363,137846,137848,137888],{"className":137847},[633,634],[363,137849,137851,137885],{"className":137850},[638],[363,137852,137854,137874],{"className":137853,"style":133537},[642],[363,137855,137856,137859],{"style":40048},[363,137857],{"className":137858,"style":651},[650],[363,137860,137862],{"className":137861},[655,656,657,658],[363,137863,137865,137868,137871],{"className":137864},[407,658],[363,137866,12313],{"className":137867},[407,408,658],[363,137869,471],{"className":137870},[510,658],[363,137872,465],{"className":137873},[407,658],[363,137875,137876,137879],{"style":133560},[363,137877],{"className":137878,"style":651},[650],[363,137880,137882],{"className":137881},[655,656,657,658],[363,137883,11044],{"className":137884,"style":11132},[407,408,658],[363,137886,666],{"className":137887},[665],[363,137889,137891],{"className":137890},[638],[363,137892,137894],{"className":137893,"style":40067},[642],[363,137895],{},[363,137897],{"className":137898,"style":754},[505],[363,137900,137902,137905],{"className":137901},[407],[363,137903,1121],{"className":137904},[407,408],[363,137906,137908],{"className":137907},[629],[363,137909,137911,137988],{"className":137910},[633,634],[363,137912,137914,137985],{"className":137913},[638],[363,137915,137917,137928],{"className":137916,"style":136944},[642],[363,137918,137919,137922],{"style":136947},[363,137920],{"className":137921,"style":651},[650],[363,137923,137925],{"className":137924},[655,656,657,658],[363,137926,12313],{"className":137927},[407,408,658],[363,137929,137930,137933],{"style":136959},[363,137931],{"className":137932,"style":651},[650],[363,137934,137936],{"className":137935},[655,656,657,658],[363,137937,137939,137979,137982],{"className":137938},[407,658],[363,137940,137942,137945],{"className":137941},[407,658],[363,137943,45126],{"className":137944,"style":45156},[407,408,658],[363,137946,137948],{"className":137947},[629],[363,137949,137951,137971],{"className":137950},[633,634],[363,137952,137954,137968],{"className":137953},[638],[363,137955,137957],{"className":137956,"style":40120},[642],[363,137958,137959,137962],{"style":136989},[363,137960],{"className":137961,"style":30578},[650],[363,137963,137965],{"className":137964},[655,30582,10022,658],[363,137966,12313],{"className":137967},[407,408,658],[363,137969,666],{"className":137970},[665],[363,137972,137974],{"className":137973},[638],[363,137975,137977],{"className":137976,"style":40142},[642],[363,137978],{},[363,137980,5161],{"className":137981},[2939,658],[363,137983,465],{"className":137984},[407,658],[363,137986,666],{"className":137987},[665],[363,137989,137991],{"className":137990},[638],[363,137992,137994],{"className":137993,"style":137025},[642],[363,137995],{},[363,137997],{"className":137998,"style":754},[505],[363,138000,592],{"className":138001},[407,33034],[363,138003,1121],{"className":138004},[407,75943],[363,138006],{"className":138007,"style":506},[505],[363,138009,471],{"className":138010},[510],[363,138012],{"className":138013,"style":506},[505],[363,138015,138017,138020,138023,138026,138035],{"className":138016},[398],[363,138018],{"className":138019,"style":487},[402],[363,138021,39663],{"className":138022,"style":39759},[407,408],[363,138024,461],{"className":138025},[494],[363,138027,138029],{"className":138028},[407],[363,138030,138032],{"className":138031},[407],[363,138033,45126],{"className":138034},[407,136115],[363,138036,468],{"className":138037},[501],[11,138039,138040,138043,138044,138137],{},[1344,138041,138042],{},"证明"," 考虑 ",[363,138045,138047,138071],{"className":138046,"translate":367},[366],[363,138048,138050],{"className":138049},[371],[373,138051,138052],{"xmlns":375},[377,138053,138054,138068],{},[380,138055,138056],{},[7901,138057,138058,138060,138066],{},[383,138059,1381],{"mathvariant":570},[380,138061,138062,138064],{},[458,138063,3251],{},[463,138065,474],{},[383,138067,11044],{},[387,138069,138070],{"encoding":389},"\\mathbb{R}_{>0}^K",[363,138072,138074],{"className":138073,"ariaHidden":394},[393],[363,138075,138077,138080],{"className":138076},[398],[363,138078],{"className":138079,"style":136251},[402],[363,138081,138083,138086],{"className":138082},[407],[363,138084,1381],{"className":138085},[407,714],[363,138087,138089],{"className":138088},[629],[363,138090,138092,138129],{"className":138091},[633,634],[363,138093,138095,138126],{"className":138094},[638],[363,138096,138098,138115],{"className":138097,"style":52457},[642],[363,138099,138100,138103],{"style":136272},[363,138101],{"className":138102,"style":651},[650],[363,138104,138106],{"className":138105},[655,656,657,658],[363,138107,138109,138112],{"className":138108},[407,658],[363,138110,3251],{"className":138111},[510,658],[363,138113,474],{"className":138114},[407,658],[363,138116,138117,138120],{"style":7213},[363,138118],{"className":138119,"style":651},[650],[363,138121,138123],{"className":138122},[655,656,657,658],[363,138124,11044],{"className":138125,"style":11132},[407,408,658],[363,138127,666],{"className":138128},[665],[363,138130,138132],{"className":138131},[638],[363,138133,138135],{"className":138134,"style":136308},[642],[363,138136],{}," 上的 Gamma 积分之积：",[363,138139,138141],{"className":138140,"translate":367},[528],[363,138142,138144,138252],{"className":138143,"translate":367},[366],[363,138145,138147],{"className":138146},[371],[373,138148,138149],{"xmlns":375,"display":537},[377,138150,138151,138249],{},[380,138152,138153,138167,138169,138171,138177,138179,138181,138197,138211,138229,138243,138245,138247],{},[48692,138154,138155,138157,138165],{},[458,138156,136684],{},[380,138158,138159,138161,138163],{},[383,138160,12313],{},[458,138162,471],{},[463,138164,465],{},[383,138166,11044],{},[383,138168,137218],{"mathvariant":557},[458,138170,461],{"stretchy":460},[543,138172,138173,138175],{},[383,138174,45126],{},[383,138176,12313],{},[458,138178,468],{"stretchy":460},[458,138180,471],{},[543,138182,138183,138185],{},[458,138184,1635],{},[7901,138186,138187,138189,138195],{},[383,138188,1381],{"mathvariant":570},[380,138190,138191,138193],{},[458,138192,3251],{},[463,138194,474],{},[383,138196,11044],{},[48692,138198,138199,138201,138209],{},[458,138200,136684],{},[380,138202,138203,138205,138207],{},[383,138204,12313],{},[458,138206,471],{},[463,138208,465],{},[383,138210,11044],{},[7901,138212,138213,138215,138217],{},[383,138214,1116],{},[383,138216,12313],{},[380,138218,138219,138225,138227],{},[543,138220,138221,138223],{},[383,138222,45126],{},[383,138224,12313],{},[458,138226,5161],{},[463,138228,465],{},[7162,138230,138231,138233],{},[383,138232,7915],{},[380,138234,138235,138237],{},[458,138236,5161],{},[543,138238,138239,138241],{},[383,138240,1116],{},[383,138242,12313],{},[1645,138244,1647],{},[383,138246,592],{"mathvariant":557},[383,138248,1116],{"mathvariant":75598},[387,138250,138251],{"encoding":389},"\\prod_{i=1}^K \\Gamma(\\alpha_i) = \\int_{\\mathbb{R}_{>0}^K} \\prod_{i=1}^K t_i^{\\alpha_i - 1} e^{-t_i} 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x_i",[363,138782,138784,138839],{"className":138783,"ariaHidden":394},[393],[363,138785,138787,138790,138830,138833,138836],{"className":138786},[398],[363,138788],{"className":138789,"style":96314},[402],[363,138791,138793,138796],{"className":138792},[407],[363,138794,1116],{"className":138795},[407,408],[363,138797,138799],{"className":138798},[629],[363,138800,138802,138822],{"className":138801},[633,634],[363,138803,138805,138819],{"className":138804},[638],[363,138806,138808],{"className":138807,"style":12532},[642],[363,138809,138810,138813],{"style":730},[363,138811],{"className":138812,"style":651},[650],[363,138814,138816],{"className":138815},[655,656,657,658],[363,138817,12313],{"className":138818},[407,408,658],[363,138820,666],{"className":138821},[665],[363,138823,138825],{"className":138824},[638],[363,138826,138828],{"className":138827,"style":8316},[642],[363,138829],{},[363,138831],{"className":138832,"style":506},[505],[363,138834,471],{"className":138835},[510],[363,138837],{"className":138838,"style":506},[505],[363,138840,138842,138845,138848,138851],{"className":138841},[398],[363,138843],{"className":138844,"style":13724},[402],[363,138846,7909],{"className":138847},[407,408],[363,138849],{"className":138850,"style":754},[505],[363,138852,138854,138857],{"className":138853},[407],[363,138855,1121],{"className":138856},[407,408],[363,138858,138860],{"className":138859},[629],[363,138861,138863,138883],{"className":138862},[633,634],[363,138864,138866,138880],{"className":138865},[638],[363,138867,138869],{"className":138868,"style":12532},[642],[363,138870,138871,138874],{"style":730},[363,138872],{"className":138873,"style":651},[650],[363,138875,138877],{"className":138876},[655,656,657,658],[363,138878,12313],{"className":138879},[407,408,658],[363,138881,666],{"className":138882},[665],[363,138884,138886],{"className":138885},[638],[363,138887,138889],{"className":138888,"style":8316},[642],[363,138890],{},[363,138892,138894,138926],{"className":138893,"translate":367},[366],[363,138895,138897],{"className":138896},[371],[373,138898,138899],{"xmlns":375},[377,138900,138901,138923],{},[380,138902,138903,138905,138907,138913,138919,138921],{},[383,138904,7909],{},[458,138906,471],{},[543,138908,138909,138911],{},[458,138910,13811],{},[383,138912,12313],{},[543,138914,138915,138917],{},[383,138916,1116],{},[383,138918,12313],{},[458,138920,3251],{},[463,138922,474],{},[387,138924,138925],{"encoding":389},"s = \\sum_i t_i > 0",[363,138927,138929,138947,139046],{"className":138928,"ariaHidden":394},[393],[363,138930,138932,138935,138938,138941,138944],{"className":138931},[398],[363,138933],{"className":138934,"style":2693},[402],[363,138936,7909],{"className":138937},[407,408],[363,138939],{"className":138940,"style":506},[505],[363,138942,471],{"className":138943},[510],[363,138945],{"className":138946,"style":506},[505],[363,138948,138950,138954,138994,138997,139037,139040,139043],{"className":138949},[398],[363,138951],{"className":138952,"style":138953},[402],"height:1.0497em;vertical-align:-0.2997em;",[363,138955,138957,138960],{"className":138956},[1832],[363,138958,13811],{"className":138959,"style":40032},[1832,1833,5964],[363,138961,138963],{"className":138962},[629],[363,138964,138966,138986],{"className":138965},[633,634],[363,138967,138969,138983],{"className":138968},[638],[363,138970,138972],{"className":138971,"style":40045},[642],[363,138973,138974,138977],{"style":40048},[363,138975],{"className":138976,"style":651},[650],[363,138978,138980],{"className":138979},[655,656,657,658],[363,138981,12313],{"className":138982},[407,408,658],[363,138984,666],{"className":138985},[665],[363,138987,138989],{"className":138988},[638],[363,138990,138992],{"className":138991,"style":40067},[642],[363,138993],{},[363,138995],{"className":138996,"style":754},[505],[363,138998,139000,139003],{"className":138999},[407],[363,139001,1116],{"className":139002},[407,408],[363,139004,139006],{"className":139005},[629],[363,139007,139009,139029],{"className":139008},[633,634],[363,139010,139012,139026],{"className":139011},[638],[363,139013,139015],{"className":139014,"style":12532},[642],[363,139016,139017,139020],{"style":730},[363,139018],{"className":139019,"style":651},[650],[363,139021,139023],{"className":139022},[655,656,657,658],[363,139024,12313],{"className":139025},[407,408,658],[363,139027,666],{"className":139028},[665],[363,139030,139032],{"className":139031},[638],[363,139033,139035],{"className":139034,"style":8316},[642],[363,139036],{},[363,139038],{"className":139039,"style":506},[505],[363,139041,3251],{"className":139042},[510],[363,139044],{"className":139045,"style":506},[505],[363,139047,139049,139052],{"className":139048},[398],[363,139050],{"className":139051,"style":520},[402],[363,139053,474],{"className":139054},[407],[363,139056,139058,139086],{"className":139057,"translate":367},[366],[363,139059,139061],{"className":139060},[371],[373,139062,139063],{"xmlns":375},[377,139064,139065,139083],{},[380,139066,139067,139069,139071],{},[383,139068,1121],{"mathvariant":75598},[458,139070,6257],{},[7162,139072,139073,139075],{},[383,139074,126067],{"mathvariant":557},[380,139076,139077,139079,139081],{},[383,139078,11044],{},[458,139080,5161],{},[463,139082,465],{},[387,139084,139085],{"encoding":389},"\\mathbf{x} \\in \\Delta^{K-1}",[363,139087,139089,139107],{"className":139088,"ariaHidden":394},[393],[363,139090,139092,139095,139098,139101,139104],{"className":139091},[398],[363,139093],{"className":139094,"style":6272},[402],[363,139096,1121],{"className":139097},[407,75943],[363,139099],{"className":139100,"style":506},[505],[363,139102,6257],{"className":139103},[510],[363,139105],{"className":139106,"style":506},[505],[363,139108,139110,139113],{"className":139109},[398],[363,139111],{"className":139112,"style":52457},[402],[363,139114,139116,139119],{"className":139115},[407],[363,139117,126067],{"className":139118},[407],[363,139120,139122],{"className":139121},[629],[363,139123,139125],{"className":139124},[633],[363,139126,139128],{"className":139127},[638],[363,139129,139131],{"className":139130,"style":52457},[642],[363,139132,139133,139136],{"style":7213},[363,139134],{"className":139135,"style":651},[650],[363,139137,139139],{"className":139138},[655,656,657,658],[363,139140,139142,139145,139148],{"className":139141},[407,658],[363,139143,11044],{"className":139144,"style":11132},[407,408,658],[363,139146,5161],{"className":139147},[2939,658],[363,139149,465],{"className":139150},[407,658],"。该替换的 Jacobi 行列式绝对值为 ",[363,139153,139155,139192],{"className":139154,"translate":367},[366],[363,139156,139158],{"className":139157},[371],[373,139159,139160],{"xmlns":375},[377,139161,139162,139189],{},[380,139163,139164,139166,139169,139171,139173,139175,139177],{},[383,139165,5156],{"mathvariant":557},[383,139167,139168],{},"det",[458,139170,4451],{},[383,139172,24115],{},[383,139174,5156],{"mathvariant":557},[458,139176,471],{},[7162,139178,139179,139181],{},[383,139180,7909],{},[380,139182,139183,139185,139187],{},[383,139184,11044],{},[458,139186,5161],{},[463,139188,465],{},[387,139190,139191],{"encoding":389},"|\\det J| = s^{K-1}",[363,139193,139195,139228],{"className":139194,"ariaHidden":394},[393],[363,139196,139198,139201,139204,139207,139210,139213,139216,139219,139222,139225],{"className":139197},[398],[363,139199],{"className":139200,"style":487},[402],[363,139202,5156],{"className":139203},[407],[363,139205],{"className":139206,"style":754},[505],[363,139208,139168],{"className":139209},[1832],[363,139211],{"className":139212,"style":754},[505],[363,139214,24115],{"className":139215,"style":24525},[407,408],[363,139217,5156],{"className":139218},[407],[363,139220],{"className":139221,"style":506},[505],[363,139223,471],{"className":139224},[510],[363,139226],{"className":139227,"style":506},[505],[363,139229,139231,139234],{"className":139230},[398],[363,139232],{"className":139233,"style":52457},[402],[363,139235,139237,139240],{"className":139236},[407],[363,139238,7909],{"className":139239},[407,408],[363,139241,139243],{"className":139242},[629],[363,139244,139246],{"className":139245},[633],[363,139247,139249],{"className":139248},[638],[363,139250,139252],{"className":139251,"style":52457},[642],[363,139253,139254,139257],{"style":7213},[363,139255],{"className":139256,"style":651},[650],[363,139258,139260],{"className":139259},[655,656,657,658],[363,139261,139263,139266,139269],{"className":139262},[407,658],[363,139264,11044],{"className":139265,"style":11132},[407,408,658],[363,139267,5161],{"className":139268},[2939,658],[363,139270,465],{"className":139271},[407,658],"（可由 ",[363,139274,139276,139293],{"className":139275,"translate":367},[366],[363,139277,139279],{"className":139278},[371],[373,139280,139281],{"xmlns":375},[377,139282,139283,139291],{},[380,139284,139285,139287,139289],{},[383,139286,11044],{},[458,139288,471],{},[463,139290,2811],{},[387,139292,135087],{"encoding":389},[363,139294,139296,139314],{"className":139295,"ariaHidden":394},[393],[363,139297,139299,139302,139305,139308,139311],{"className":139298},[398],[363,139300],{"className":139301,"style":1011},[402],[363,139303,11044],{"className":139304,"style":11132},[407,408],[363,139306],{"className":139307,"style":506},[505],[363,139309,471],{"className":139310},[510],[363,139312],{"className":139313,"style":506},[505],[363,139315,139317,139320],{"className":139316},[398],[363,139318],{"className":139319,"style":520},[402],[363,139321,2811],{"className":139322},[407]," 时的 ",[363,139325,139327,139399],{"className":139326,"translate":367},[366],[363,139328,139330],{"className":139329},[371],[373,139331,139332],{"xmlns":375},[377,139333,139334,139396],{},[380,139335,139336,139338,139340,139390,139392,139394],{},[383,139337,139168],{},[458,139339,4451],{},[380,139341,139342,139344,139388],{},[458,139343,461],{"fence":394},[1591,139345,139348,139366],{"rowspacing":139346,"columnalign":139347,"columnspacing":4663},"0.16em","center center",[1597,139349,139350,139360],{},[1600,139351,139352],{},[1603,139353,139354],{"scriptlevel":474,"displaystyle":460},[543,139355,139356,139358],{},[383,139357,1121],{},[463,139359,465],{},[1600,139361,139362],{},[1603,139363,139364],{"scriptlevel":474,"displaystyle":460},[383,139365,7909],{},[1597,139367,139368,139378],{},[1600,139369,139370],{},[1603,139371,139372],{"scriptlevel":474,"displaystyle":460},[543,139373,139374,139376],{},[383,139375,1121],{},[463,139377,2811],{},[1600,139379,139380],{},[1603,139381,139382],{"scriptlevel":474,"displaystyle":460},[380,139383,139384,139386],{},[458,139385,5161],{},[383,139387,7909],{},[458,139389,468],{"fence":394},[458,139391,471],{},[458,139393,5161],{},[383,139395,7909],{},[387,139397,139398],{"encoding":389},"\\det\\begin{pmatrix} x_1 & s \\\\ x_2 & -s\\end{pmatrix} = -s",[363,139400,139402,139621],{"className":139401,"ariaHidden":394},[393],[363,139403,139405,139408,139411,139414,139612,139615,139618],{"className":139404},[398],[363,139406],{"className":139407,"style":707},[402],[363,139409,139168],{"className":139410},[1832],[363,139412],{"className":139413,"style":754},[505],[363,139415,139417,139423,139606],{"className":139416},[758],[363,139418,139420],{"className":139419,"style":763},[494,762],[363,139421,461],{"className":139422},[767,657],[363,139424,139426],{"className":139425},[407],[363,139427,139429,139551,139555,139558],{"className":139428},[1591],[363,139430,139433],{"className":139431},[139432],"col-align-c",[363,139434,139436,139543],{"className":139435},[633,634],[363,139437,139439,139540],{"className":139438},[638],[363,139440,139442,139491],{"className":139441,"style":1718},[642],[363,139443,139445,139448],{"style":139444},"top:-3.61em;",[363,139446],{"className":139447,"style":809},[650],[363,139449,139451],{"className":139450},[407],[363,139452,139454,139457],{"className":139453},[407],[363,139455,1121],{"className":139456},[407,408],[363,139458,139460],{"className":139459},[629],[363,139461,139463,139483],{"className":139462},[633,634],[363,139464,139466,139480],{"className":139465},[638],[363,139467,139469],{"className":139468,"style":11483},[642],[363,139470,139471,139474],{"style":730},[363,139472],{"className":139473,"style":651},[650],[363,139475,139477],{"className":139476},[655,656,657,658],[363,139478,465],{"className":139479},[407,658],[363,139481,666],{"className":139482},[665],[363,139484,139486],{"className":139485},[638],[363,139487,139489],{"className":139488,"style":8316},[642],[363,139490],{},[363,139492,139494,139497],{"style":139493},"top:-2.41em;",[363,139495],{"className":139496,"style":809},[650],[363,139498,139500],{"className":139499},[407],[363,139501,139503,139506],{"className":139502},[407],[363,139504,1121],{"className":139505},[407,408],[363,139507,139509],{"className":139508},[629],[363,139510,139512,139532],{"className":139511},[633,634],[363,139513,139515,139529],{"className":139514},[638],[363,139516,139518],{"className":139517,"style":11483},[642],[363,139519,139520,139523],{"style":730},[363,139521],{"className":139522,"style":651},[650],[363,139524,139526],{"className":139525},[655,656,657,658],[363,139527,2811],{"className":139528},[407,658],[363,139530,666],{"className":139531},[665],[363,139533,139535],{"className":139534},[638],[363,139536,139538],{"className":139537,"style":8316},[642],[363,139539],{},[363,139541,666],{"className":139542},[665],[363,139544,139546],{"className":139545},[638],[363,139547,139549],{"className":139548,"style":1793},[642],[363,139550],{},[363,139552],{"className":139553,"style":139554},[4969],"width:0.5em;",[363,139556],{"className":139557,"style":139554},[4969],[363,139559,139561],{"className":139560},[139432],[363,139562,139564,139598],{"className":139563},[633,634],[363,139565,139567,139595],{"className":139566},[638],[363,139568,139570,139581],{"className":139569,"style":1718},[642],[363,139571,139572,139575],{"style":139444},[363,139573],{"className":139574,"style":809},[650],[363,139576,139578],{"className":139577},[407],[363,139579,7909],{"className":139580},[407,408],[363,139582,139583,139586],{"style":139493},[363,139584],{"className":139585,"style":809},[650],[363,139587,139589,139592],{"className":139588},[407],[363,139590,5161],{"className":139591},[407],[363,139593,7909],{"className":139594},[407,408],[363,139596,666],{"className":139597},[665],[363,139599,139601],{"className":139600},[638],[363,139602,139604],{"className":139603,"style":1793},[642],[363,139605],{},[363,139607,139609],{"className":139608,"style":763},[501,762],[363,139610,468],{"className":139611},[767,657],[363,139613],{"className":139614,"style":506},[505],[363,139616,471],{"className":139617},[510],[363,139619],{"className":139620,"style":506},[505],[363,139622,139624,139627,139630],{"className":139623},[398],[363,139625],{"className":139626,"style":4599},[402],[363,139628,5161],{"className":139629},[407],[363,139631,7909],{"className":139632},[407,408]," 归纳验证）。代入得",[363,139635,139637],{"className":139636,"translate":367},[528],[363,139638,139640,139794],{"className":139639,"translate":367},[366],[363,139641,139643],{"className":139642},[371],[373,139644,139645],{"xmlns":375,"display":537},[377,139646,139647,139791],{},[380,139648,139649,139663,139665,139667,139673,139675,139677,139685,139687,139689,139705,139717,139731,139733,139735,139741,139757,139779,139781,139783,139785,139787,139789],{},[48692,139650,139651,139653,139661],{},[458,139652,136684],{},[380,139654,139655,139657,139659],{},[383,139656,12313],{},[458,139658,471],{},[463,139660,465],{},[383,139662,11044],{},[383,139664,137218],{"mathvariant":557},[458,139666,461],{"stretchy":460},[543,139668,139669,139671],{},[383,139670,45126],{},[383,139672,12313],{},[458,139674,468],{"stretchy":460},[458,139676,471],{},[7901,139678,139679,139681,139683],{},[458,139680,1635],{},[463,139682,474],{},[383,139684,1373],{"mathvariant":557},[1645,139686,1652],{},[1645,139688,1652],{},[543,139690,139691,139693],{},[458,139692,1635],{},[7162,139694,139695,139697],{},[383,139696,126067],{"mathvariant":557},[380,139698,139699,139701,139703],{},[383,139700,11044],{},[458,139702,5161],{},[463,139704,465],{},[7162,139706,139707,139709],{},[383,139708,7909],{},[380,139710,139711,139713,139715],{},[383,139712,11044],{},[458,139714,5161],{},[463,139716,465],{},[48692,139718,139719,139721,139729],{},[458,139720,136684],{},[380,139722,139723,139725,139727],{},[383,139724,12313],{},[458,139726,471],{},[463,139728,465],{},[383,139730,11044],{},[458,139732,461],{"stretchy":460},[383,139734,7909],{},[543,139736,139737,139739],{},[383,139738,1121],{},[383,139740,12313],{},[7162,139742,139743,139745],{},[458,139744,468],{"stretchy":460},[380,139746,139747,139753,139755],{},[543,139748,139749,139751],{},[383,139750,45126],{},[383,139752,12313],{},[458,139754,5161],{},[463,139756,465],{},[7162,139758,139759,139761],{},[383,139760,7915],{},[380,139762,139763,139765,139767,139773],{},[458,139764,5161],{},[383,139766,7909],{},[6364,139768,139769,139771],{},[458,139770,13811],{},[383,139772,12313],{},[543,139774,139775,139777],{},[383,139776,1121],{},[383,139778,12313],{},[1645,139780,1647],{},[383,139782,592],{"mathvariant":557},[383,139784,1121],{"mathvariant":75598},[1645,139786,1647],{},[383,139788,592],{"mathvariant":557},[383,139790,7909],{},[387,139792,139793],{"encoding":389},"\\prod_{i=1}^K \\Gamma(\\alpha_i) = \\int_0^\\infty \\!\\!\\int_{\\Delta^{K-1}} s^{K-1} \\prod_{i=1}^K (s x_i)^{\\alpha_i - 1} e^{-s \\sum_i x_i} \\,\\mathrm{d}\\mathbf{x} \\,\\mathrm{d}s",[363,139795,139797,139928],{"className":139796,"ariaHidden":394},[393],[363,139798,139800,139803,139867,139870,139873,139876,139916,139919,139922,139925],{"className":139799},[398],[363,139801],{"className":139802,"style":138261},[402],[363,139804,139806],{"className":139805},[1832,14066],[363,139807,139809,139859],{"className":139808},[633,634],[363,139810,139812,139856],{"className":139811},[638],[363,139813,139815,139835,139845],{"className":139814,"style":138274},[642],[363,139816,139817,139820],{"style":14079},[363,139818],{"className":139819,"style":14083},[650],[363,139821,139823],{"className":139822},[655,656,657,658],[363,139824,139826,139829,139832],{"className":139825},[407,658],[363,139827,12313],{"className":139828},[407,408,658],[363,139830,471],{"className":139831},[510,658],[363,139833,465],{"className":139834},[407,658],[363,139836,139837,139840],{"style":14092},[363,139838],{"className":139839,"style":14083},[650],[363,139841,139842],{},[363,139843,136684],{"className":139844},[1832,1833,1834],[363,139846,139847,139850],{"style":100713},[363,139848],{"className":139849,"style":14083},[650],[363,139851,139853],{"className":139852},[655,656,657,658],[363,139854,11044],{"className":139855,"style":11132},[407,408,658],[363,139857,666],{"className":139858},[665],[363,139860,139862],{"className":139861},[638],[363,139863,139865],{"className":139864,"style":14110},[642],[363,139866],{},[363,139868],{"className":139869,"style":754},[505],[363,139871,137218],{"className":139872},[407],[363,139874,461],{"className":139875},[494],[363,139877,139879,139882],{"className":139878},[407],[363,139880,45126],{"className":139881,"style":45156},[407,408],[363,139883,139885],{"className":139884},[629],[363,139886,139888,139908],{"className":139887},[633,634],[363,139889,139891,139905],{"className":139890},[638],[363,139892,139894],{"className":139893,"style":12532},[642],[363,139895,139896,139899],{"style":136154},[363,139897],{"className":139898,"style":651},[650],[363,139900,139902],{"className":139901},[655,656,657,658],[363,139903,12313],{"className":139904},[407,408,658],[363,139906,666],{"className":139907},[665],[363,139909,139911],{"className":139910},[638],[363,139912,139914],{"className":139913,"style":8316},[642],[363,139915],{},[363,139917,468],{"className":139918},[501],[363,139920],{"className":139921,"style":506},[505],[363,139923,471],{"className":139924},[510],[363,139926],{"className":139927,"style":506},[505],[363,139929,139931,139934,139988,139991,139994,139997,140076,140079,140117,140120,140184,140187,140190,140230,140305,140427,140430,140433,140436,140439,140442],{"className":139930},[398],[363,139932],{"className":139933,"style":138261},[402],[363,139935,139937,139940],{"className":139936},[1832],[363,139938,1635],{"className":139939,"style":1835},[1832,1833,1834],[363,139941,139943],{"className":139942},[629],[363,139944,139946,139979],{"className":139945},[633,634],[363,139947,139949,139976],{"className":139948},[638],[363,139950,139953,139964],{"className":139951,"style":139952},[642],"height:1.4143em;",[363,139954,139955,139958],{"style":3826},[363,139956],{"className":139957,"style":651},[650],[363,139959,139961],{"className":139960},[655,656,657,658],[363,139962,474],{"className":139963},[407,658],[363,139965,139967,139970],{"style":139966},"top:-3.8129em;margin-right:0.05em;",[363,139968],{"className":139969,"style":651},[650],[363,139971,139973],{"className":139972},[655,656,657,658],[363,139974,1373],{"className":139975},[407,658],[363,139977,666],{"className":139978},[665],[363,139980,139982],{"className":139981},[638],[363,139983,139986],{"className":139984,"style":139985},[642],"height:0.9119em;",[363,139987],{},[363,139989],{"className":139990,"style":1860},[505],[363,139992],{"className":139993,"style":1860},[505],[363,139995],{"className":139996,"style":754},[505],[363,139998,140000,140003],{"className":139999},[1832],[363,140001,1635],{"className":140002,"style":1835},[1832,1833,1834],[363,140004,140006],{"className":140005},[629],[363,140007,140009,140068],{"className":140008},[633,634],[363,140010,140012,140065],{"className":140011},[638],[363,140013,140016],{"className":140014,"style":140015},[642],"height:-0.3701em;",[363,140017,140018,140021],{"style":3826},[363,140019],{"className":140020,"style":651},[650],[363,140022,140024],{"className":140023},[655,656,657,658],[363,140025,140027],{"className":140026},[407,658],[363,140028,140030,140033],{"className":140029},[407,658],[363,140031,126067],{"className":140032},[407,658],[363,140034,140036],{"className":140035},[629],[363,140037,140039],{"className":140038},[633],[363,140040,140042],{"className":140041},[638],[363,140043,140045],{"className":140044,"style":137800},[642],[363,140046,140047,140050],{"style":45834},[363,140048],{"className":140049,"style":30578},[650],[363,140051,140053],{"className":140052},[655,30582,10022,658],[363,140054,140056,140059,140062],{"className":140055},[407,658],[363,14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,{"className":140336},[407,658],[363,140338,7909],{"className":140339},[407,408,658],[363,140341],{"className":140342,"style":42738},[505,658],[363,140344,140346,140349],{"className":140345},[1832,658],[363,140347,13811],{"className":140348,"style":40032},[1832,1833,5964,658],[363,140350,140352],{"className":140351},[629],[363,140353,140355,140376],{"className":140354},[633,634],[363,140356,140358,140373],{"className":140357},[638],[363,140359,140362],{"className":140360,"style":140361},[642],"height:0.1496em;",[363,140363,140364,140367],{"style":137421},[363,140365],{"className":140366,"style":30578},[650],[363,140368,140370],{"className":140369},[655,30582,10022,658],[363,140371,12313],{"className":140372},[407,408,658],[363,140374,666],{"className":140375},[665],[363,140377,140379],{"className":140378},[638],[363,140380,140382],{"className":140381,"style":137460},[642],[363,140383],{},[363,140385],{"className":140386,"style":42738},[505,658],[363,140388,140390,140393],{"className":140389},[407,658],[363,140391,1121],{"className":140392},[407,408,658],[363,140394,140396],{"className":140395},[629],[363,140397,140399,140419],{"className":140398},[633,634],[363,140400,140402,140416],{"className":140401},[638],[363,140403,140405],{"className":140404,"style":40120},[642],[363,140406,140407,140410],{"style":74648},[363,140408],{"className":140409,"style":30578},[650],[363,140411,140413],{"className":140412},[655,30582,10022,658],[363,140414,12313],{"className":140415},[407,408,658],[363,140417,666],{"className":140418},[665],[363,140420,140422],{"className":140421},[638],[363,140423,140425],{"className":140424,"style":40142},[642],[363,140426],{},[363,140428],{"className":140429,"style":754},[505],[363,140431,592],{"className":140432},[407,33034],[363,140434,1121],{"className":140435},[407,75943],[363,140437],{"className":140438,"style":754},[505],[363,140440,592],{"className":140441},[407,33034],[363,140443,7909],{"className":140444},[407,408],[11,140446,140447,140448,140589,140590,140618,140619,140862,140863,140932,140933,140961],{},"利用约束 ",[363,140449,140451,140479],{"className":140450,"translate":367},[366],[363,140452,140454],{"className":140453},[371],[373,140455,140456],{"xmlns":375},[377,140457,140458,140476],{},[380,140459,140460,140466,140472,140474],{},[543,140461,140462,140464],{},[458,140463,13811],{},[383,140465,12313],{},[543,140467,140468,140470],{},[383,140469,1121],{},[383,140471,12313],{},[458,140473,471],{},[463,140475,465],{},[387,140477,140478],{"encoding":389},"\\sum_i x_i = 1",[363,140480,140482,140580],{"className":140481,"ariaHidden":394},[393],[363,140483,140485,140488,140528,140531,140571,140574,140577],{"className":140484},[398],[363,140486],{"className":140487,"style":138953},[402],[363,140489,140491,140494],{"className":140490},[1832],[363,140492,13811],{"className":140493,"style":40032},[1832,1833,5964],[363,140495,140497],{"className":140496},[629],[363,140498,140500,140520],{"className":140499},[633,634],[363,140501,140503,140517],{"className":140502},[638],[363,140504,140506],{"className":140505,"style":40045},[642],[363,140507,140508,140511],{"style":40048},[363,140509],{"className":140510,"style":651},[650],[363,140512,140514],{"className":140513},[655,656,657,658],[363,140515,12313],{"className":140516},[407,408,658],[363,140518,666],{"className":140519},[665],[363,140521,140523],{"className":140522},[638],[363,140524,140526],{"className":140525,"style":40067},[642],[363,140527],{},[363,140529],{"className":140530,"style":754},[505],[363,140532,140534,140537],{"className":140533},[407],[363,140535,1121],{"className":140536},[407,408],[363,140538,140540],{"className":140539},[629],[363,140541,140543,140563],{"className":140542},[633,634],[363,140544,140546,140560],{"className":140545},[638],[363,140547,140549],{"className":140548,"style":12532},[642],[363,140550,140551,140554],{"style":730},[363,140552],{"className":140553,"style":651},[650],[363,140555,140557],{"className":140556},[655,656,657,658],[363,140558,12313],{"className":140559},[407,408,658],[363,140561,666],{"className":140562},[665],[363,140564,140566],{"className":140565},[638],[363,140567,140569],{"className":140568,"style":8316},[642],[363,140570],{},[363,140572],{"className":140573,"style":506},[505],[363,140575,471],{"className":140576},[510],[363,140578],{"className":140579,"style":506},[505],[363,140581,140583,140586],{"className":140582},[398],[363,140584],{"className":140585,"style":520},[402],[363,140587,465],{"className":140588},[407],"，被积函数中的 ",[363,140591,140593,140606],{"className":140592,"translate":367},[366],[363,140594,140596],{"className":140595},[371],[373,140597,140598],{"xmlns":375},[377,140599,140600,140604],{},[380,140601,140602],{},[383,140603,7909],{},[387,140605,7909],{"encoding":389},[363,140607,140609],{"className":140608,"ariaHidden":394},[393],[363,140610,140612,140615],{"className":140611},[398],[363,140613],{"className":140614,"style":2693},[402],[363,140616,7909],{"className":140617},[407,408]," 因子合并为 ",[363,140620,140622,140676],{"className":140621,"translate":367},[366],[363,140623,140625],{"className":140624},[371],[373,140626,140627],{"xmlns":375},[377,140628,140629,140673],{},[380,140630,140631,140655,140657],{},[7162,140632,140633,140635],{},[383,140634,7909],{},[380,140636,140637,140639,140641,140643,140645,140651,140653],{},[383,140638,11044],{},[458,140640,5161],{},[463,140642,465],{},[458,140644,2818],{},[543,140646,140647,140649],{},[383,140648,45126],{},[463,140650,474],{},[458,140652,5161],{},[383,140654,11044],{},[458,140656,471],{},[7162,140658,140659,140661],{},[383,140660,7909],{},[380,140662,140663,140669,140671],{},[543,140664,140665,140667],{},[383,140666,45126],{},[463,140668,474],{},[458,140670,5161],{},[463,140672,465],{},[387,140674,140675],{"encoding":389},"s^{K - 1 + \\alpha_0 - K} = s^{\\alpha_0 - 1}",[363,140677,140679,140781],{"className":140678,"ariaHidden":394},[393],[363,140680,140682,140685,140772,140775,140778],{"className":140681},[398],[363,140683],{"className":140684,"style":52457},[402],[363,140686,140688,140691],{"className":140687},[407],[363,140689,7909],{"className":140690},[407,408],[363,140692,140694],{"className":140693},[629],[363,140695,140697],{"className":140696},[633],[363,140698,140700],{"className":140699},[638],[363,140701,140703],{"className":140702,"style":52457},[642],[363,140704,140705,140708],{"style":7213},[363,140706],{"className":140707,"style":651},[650],[363,140709,140711],{"className":140710},[655,656,657,658],[363,140712,140714,140717,140720,140723,140726,140766,140769],{"className":140713},[407,658],[363,140715,11044],{"className":140716,"style":11132},[407,408,658],[363,140718,5161],{"className":140719},[2939,658],[363,140721,465],{"className":140722},[407,658],[363,140724,2818],{"className":140725},[2939,658],[363,140727,140729,140732],{"className":140728},[407,658],[363,140730,45126],{"className":140731,"style":45156},[407,408,658],[363,140733,140735],{"className":140734},[629],[363,140736,140738,140758],{"className":140737},[633,634],[363,140739,140741,140755],{"className":140740},[638],[363,140742,140744],{"className":140743,"style":137351},[642],[363,140745,140746,140749],{"style":136989},[363,140747],{"className":140748,"style":30578},[650],[363,140750,140752],{"className":140751},[655,30582,10022,658],[363,140753,474],{"className":140754},[407,658],[363,140756,666],{"className":140757},[665],[363,140759,140761],{"className":140760},[638],[363,140762,140764],{"className":140763,"style":40142},[642],[363,140765],{},[363,140767,5161],{"className":140768},[2939,658],[363,140770,11044],{"className":140771,"style":11132},[407,408,658],[363,140773],{"className":140774,"style":506},[505],[363,140776,471],{"className":140777},[510],[363,140779],{"className":140780,"style":506},[505],[363,140782,140784,140787],{"className":140783},[398],[363,140785],{"className":140786,"style":20400},[402],[363,140788,140790,140793],{"className":140789},[407],[363,140791,7909],{"className":140792},[407,408],[363,140794,140796],{"className":140795},[629],[363,140797,140799],{"className":140798},[633],[363,140800,140802],{"className":140801},[638],[363,140803,140805],{"className":140804,"style":20400},[642],[363,140806,140807,140810],{"style":7213},[363,140808],{"className":140809,"style":651},[650],[363,140811,140813],{"className":140812},[655,656,657,658],[363,140814,140816,140856,140859],{"className":140815},[407,658],[363,140817,140819,140822],{"className":140818},[407,658],[363,140820,45126],{"className":140821,"style":45156},[407,408,658],[363,140823,140825],{"className":140824},[629],[363,140826,140828,140848],{"className":140827},[633,634],[363,140829,140831,140845],{"className":140830},[638],[363,140832,140834],{"className":140833,"style":137351},[642],[363,140835,140836,140839],{"style":136989},[363,140837],{"className":140838,"style":30578},[650],[363,140840,140842],{"className":140841},[655,30582,10022,658],[363,140843,474],{"className":140844},[407,658],[363,140846,666],{"className":140847},[665],[363,140849,140851],{"className":140850},[638],[363,140852,140854],{"className":140853,"style":40142},[642],[363,140855],{},[363,140857,5161],{"className":140858},[2939,658],[363,140860,465],{"className":140861},[407,658],"，指数因子合并为 ",[363,140864,140866,140888],{"className":140865,"translate":367},[366],[363,140867,140869],{"className":140868},[371],[373,140870,140871],{"xmlns":375},[377,140872,140873,140885],{},[380,140874,140875],{},[7162,140876,140877,140879],{},[383,140878,7915],{},[380,140880,140881,140883],{},[458,140882,5161],{},[383,140884,7909],{},[387,140886,140887],{"encoding":389},"e^{-s}",[363,140889,140891],{"className":140890,"ariaHidden":394},[393],[363,140892,140894,140897],{"className":140893},[398],[363,140895],{"className":140896,"style":21381},[402],[363,140898,140900,140903],{"className":140899},[407],[363,140901,7915],{"className":140902},[407,408],[363,140904,140906],{"className":140905},[629],[363,140907,140909],{"className":140908},[633],[363,140910,140912],{"className":140911},[638],[363,140913,140915],{"className":140914,"style":21381},[642],[363,140916,140917,140920],{"style":7213},[363,140918],{"className":140919,"style":651},[650],[363,140921,140923],{"className":140922},[655,656,657,658],[363,140924,140926,140929],{"className":140925},[407,658],[363,140927,5161],{"className":140928},[407,658],[363,140930,7909],{"className":140931},[407,408,658],"，二者均与 ",[363,140934,140936,140949],{"className":140935,"translate":367},[366],[363,140937,140939],{"className":140938},[371],[373,140940,140941],{"xmlns":375},[377,140942,140943,140947],{},[380,140944,140945],{},[383,140946,1121],{"mathvariant":75598},[387,140948,135652],{"encoding":389},[363,140950,140952],{"className":140951,"ariaHidden":394},[393],[363,140953,140955,140958],{"className":140954},[398],[363,140956],{"className":140957,"style":135662},[402],[363,140959,1121],{"className":140960},[407,75943]," 解耦：",[363,140963,140965],{"className":140964,"translate":367},[528],[363,140966,140968,141184],{"className":140967,"translate":367},[366],[363,140969,140971],{"className":140970},[371],[373,140972,140973],{"xmlns":375,"display":537},[377,140974,140975,141181],{},[380,140976,140977,140991,140993,140995,141001,141003,141005,141051,141111,141113,141115,141117,141119,141125,141127,141143,141157,141175,141177,141179],{},[48692,140978,140979,140981,140989],{},[458,140980,136684],{},[380,140982,140983,140985,140987],{},[383,140984,12313],{},[458,140986,471],{},[463,140988,465],{},[383,140990,11044],{},[383,140992,137218],{"mathvariant":557},[458,140994,461],{"stretchy":460},[543,140996,140997,140999],{},[383,140998,45126],{},[383,141000,12313],{},[458,141002,468],{"stretchy":460},[458,141004,471],{},[380,141006,141007,141009,141017,141033,141043,141045,141047,141049],{},[458,141008,461],{"fence":394},[7901,141010,141011,141013,141015],{},[458,141012,1635],{},[463,141014,474],{},[383,141016,1373],{"mathvariant":557},[7162,141018,141019,141021],{},[383,141020,7909],{},[380,141022,141023,141029,141031],{},[543,141024,141025,141027],{},[383,141026,45126],{},[463,141028,474],{},[458,141030,5161],{},[463,141032,465],{},[7162,141034,141035,141037],{},[383,141036,7915],{},[380,141038,141039,141041],{},[458,141040,5161],{},[383,141042,7909],{},[1645,141044,1647],{},[383,141046,592],{"mathvariant":557},[383,141048,7909],{},[458,141050,468],{"fence":394},[380,141052,141053,141055,141071,141085,141103,141105,141107,141109],{},[458,141054,461],{"fence":394},[543,141056,141057,141059],{},[458,141058,1635],{},[7162,141060,141061,141063],{},[383,141062,126067],{"mathvariant":557},[380,141064,141065,141067,141069],{},[383,141066,11044],{},[458,141068,5161],{},[463,141070,465],{},[48692,141072,141073,141075,141083],{},[458,141074,136684],{},[380,141076,141077,141079,141081],{},[383,141078,12313],{},[458,141080,471],{},[463,141082,465],{},[383,141084,11044],{},[7901,141086,141087,141089,141091],{},[383,141088,1121],{},[383,141090,12313],{},[380,141092,141093,141099,141101],{},[543,141094,141095,141097],{},[383,141096,45126],{},[383,141098,12313],{},[458,141100,5161],{},[463,141102,465],{},[1645,141104,1647],{},[383,141106,592],{"mathvariant":557},[383,141108,1121],{"mathvariant":75598},[458,141110,468],{"fence":394},[458,141112,471],{},[505,141114],{"linebreak":24060},[383,141116,137218],{"mathvariant":557},[458,141118,461],{"stretchy":460},[543,141120,141121,141123],{},[383,141122,45126],{},[463,141124,474],{},[458,141126,468],{"stretchy":460},[543,141128,141129,141131],{},[458,141130,1635],{},[7162,141132,141133,141135],{},[383,141134,126067],{"mathvariant":557},[380,141136,141137,141139,141141],{},[383,141138,11044],{},[458,141140,5161],{},[463,141142,465],{},[48692,141144,141145,141147,141155],{},[458,141146,136684],{},[380,141148,141149,141151,141153],{},[383,141150,12313],{},[458,141152,471],{},[463,141154,465],{},[383,141156,11044],{},[7901,141158,141159,141161,141163],{},[383,141160,1121],{},[383,141162,12313],{},[380,141164,141165,141171,141173],{},[543,141166,141167,141169],{},[383,141168,45126],{},[383,141170,12313],{},[458,141172,5161],{},[463,141174,465],{},[1645,141176,1647],{},[383,141178,592],{"mathvariant":557},[383,141180,1121],{"mathvariant":75598},[387,141182,141183],{"encoding":389},"\\prod_{i=1}^K \\Gamma(\\alpha_i) = \\left(\\int_0^\\infty s^{\\alpha_0 - 1} e^{-s} \\,\\mathrm{d}s\\right) \\left(\\int_{\\Delta^{K-1}} \\prod_{i=1}^K x_i^{\\alpha_i - 1} \\,\\mathrm{d}\\mathbf{x}\\right) = \\\\ \\Gamma(\\alpha_0) \\int_{\\Delta^{K-1}} \\prod_{i=1}^K x_i^{\\alpha_i - 1} \\,\\mathrm{d}\\mathbf{x}",[363,141185,141187,141318,141791,141794],{"className":141186,"ariaHidden":394},[393],[363,141188,141190,141193,141257,141260,141263,141266,141306,141309,141312,141315],{"className":141189},[398],[363,141191],{"className":141192,"style":138261},[402],[363,141194,141196],{"className":141195},[1832,14066],[363,141197,141199,141249],{"className":141198},[633,634],[363,141200,141202,141246],{"className":141201},[638],[363,141203,141205,141225,141235],{"className":141204,"style":138274},[642],[363,141206,141207,141210],{"style":14079},[363,141208],{"className":141209,"style":14083},[650],[363,141211,141213],{"className":141212},[655,656,657,658],[363,141214,141216,141219,141222],{"className":141215},[407,658],[363,141217,12313],{"className":141218},[407,408,658],[363,141220,471],{"className":141221},[510,658],[363,141223,465],{"className":141224},[407,658],[363,141226,141227,141230],{"style":14092},[363,141228],{"className":141229,"style":14083},[650],[363,141231,141232],{},[363,141233,136684],{"className":141234},[1832,1833,1834],[363,141236,141237,141240],{"style":100713},[363,141238],{"className":141239,"style":14083},[650],[363,141241,141243],{"className":141242},[655,656,657,658],[363,141244,11044],{"className":141245,"style":11132},[407,408,658],[363,141247,666],{"className":141248},[665],[363,141250,141252],{"className":141251},[638],[363,141253,141255],{"className":141254,"style":14110},[642],[363,141256],{},[363,141258],{"className":141259,"style":754},[505],[363,141261,137218],{"className":141262},[407],[363,141264,461],{"className":141265},[494],[363,141267,141269,141272],{"className":141268},[407],[363,141270,45126],{"className":141271,"style":45156},[407,408],[363,141273,141275],{"className":141274},[629],[363,141276,141278,141298],{"className":141277},[633,634],[363,141279,141281,141295],{"className":141280},[638],[363,141282,141284],{"className":141283,"style":12532},[642],[363,141285,141286,141289],{"style":136154},[363,141287],{"className":141288,"style":651},[650],[363,141290,141292],{"className":141291},[655,656,657,658],[363,141293,12313],{"className":141294},[407,408,658],[363,141296,666],{"className":141297},[665],[363,141299,141301],{"className":141300},[638],[363,141302,141304],{"className":141303,"style":8316},[642],[363,141305],{},[363,141307,468],{"className":141308},[501],[363,141310],{"className":141311,"style":506},[505],[363,141313,471],{"className":141314},[510],[363,141316],{"className":141317,"style":506},[505],[363,141319,141321,141324,141513,141516,141785,141788],{"className":141320},[398],[363,141322],{"className":141323,"style":138261},[402],[363,141325,141327,141333,141384,141387,141462,141498,141501,141504,141507],{"className":141326},[758],[363,141328,141330],{"className":141329,"style":763},[494,762],[363,141331,461],{"className":141332},[767,657],[363,141334,141336,141339],{"className":141335},[1832],[363,141337,1635],{"className":141338,"style":1835},[1832,1833,1834],[363,141340,141342],{"className":141341},[629],[363,141343,141345,141376],{"className":141344},[633,634],[363,141346,141348,141373],{"className":141347},[638],[363,141349,141351,141362],{"className":141350,"style":139952},[642],[363,141352,141353,141356],{"style":3826},[363,141354],{"className":141355,"style":651},[650],[363,141357,141359],{"className":141358},[655,656,657,658],[363,141360,474],{"className":141361},[407,658],[363,141363,141364,141367],{"style":139966},[363,141365],{"className":141366,"style":651},[650],[363,141368,141370],{"className":141369},[655,656,657,658],[363,141371,1373],{"className":141372},[407,658],[363,141374,666],{"className":141375},[665],[363,141377,141379],{"className":141378},[638],[363,141380,141382],{"className":141381,"style":139985},[642],[363,141383],{},[363,141385],{"className":141386,"style":754},[505],[363,141388,141390,141393],{"className":141389},[407],[363,141391,7909],{"className":141392},[407,408],[363,141394,141396],{"className":141395},[629],[363,141397,141399],{"className":141398},[633],[363,141400,141402],{"className":141401},[638],[363,141403,141405],{"className":141404,"style":20107},[642],[363,141406,141407,141410],{"style":8625},[363,141408],{"className":141409,"style":651},[650],[363,141411,141413],{"className":141412},[655,656,657,658],[363,141414,141416,141456,141459],{"className":141415},[407,658],[363,141417,141419,141422],{"className":141418},[407,658],[363,141420,45126],{"className":141421,"style":45156},[407,408,658],[363,141423,141425],{"className":141424},[629],[363,141426,141428,141448],{"className":141427},[633,634],[363,141429,141431,141445],{"className":141430},[638],[363,141432,141434],{"className":141433,"style":137351},[642],[363,141435,141436,141439],{"style":136989},[363,141437],{"className":141438,"style":30578},[650],[363,141440,141442],{"className":141441},[655,30582,10022,658],[363,141443,474],{"className":141444},[407,658],[363,141446,666],{"className":141447},[665],[363,141449,141451],{"className":14145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",[363,142110,142112,142136],{"className":142111,"translate":367},[366],[363,142113,142115],{"className":142114},[371],[373,142116,142117],{"xmlns":375},[377,142118,142119,142133],{},[380,142120,142121,142123,142125,142131],{},[383,142122,137218],{"mathvariant":557},[458,142124,461],{"stretchy":460},[543,142126,142127,142129],{},[383,142128,45126],{},[463,142130,474],{},[458,142132,468],{"stretchy":460},[387,142134,142135],{"encoding":389},"\\Gamma(\\alpha_0)",[363,142137,142139],{"className":142138,"ariaHidden":394},[393],[363,142140,142142,142145,142148,142151,142191],{"className":142141},[398],[363,142143],{"className":142144,"style":487},[402],[363,142146,137218],{"className":142147},[407],[363,142149,461],{"className":142150},[494],[363,142152,142154,142157],{"className":142153},[407],[363,142155,45126],{"className":142156,"style":45156},[407,408],[363,142158,142160],{"className":142159},[629],[363,142161,142163,142183],{"className":142162},[633,634],[363,142164,142166,142180],{"className":142165},[638],[363,142167,142169],{"className":142168,"style":11483},[642],[363,142170,142171,142174],{"style":136154},[363,142172],{"className":142173,"style":651},[650],[363,142175,142177],{"className":142176},[655,656,657,658],[363,142178,474],{"className":142179},[407,658],[363,142181,666],{"className":142182},[665],[363,142184,142186],{"className":142185},[638],[363,142187,142189],{"className":142188,"style":8316},[642],[363,142190],{},[363,142192,468],{"className":142193},[501]," 即得结论。",[11,142196,142197,142198,2757,142226,34197,142254,142257],{},"命题 1.3 的证明过程本身包含了比结论更多的信息：",[363,142199,142201,142214],{"className":142200,"translate":367},[366],[363,142202,142204],{"className":142203},[371],[373,142205,142206],{"xmlns":375},[377,142207,142208,142212],{},[380,142209,142210],{},[383,142211,7909],{},[387,142213,7909],{"encoding":389},[363,142215,142217],{"className":142216,"ariaHidden":394},[393],[363,142218,142220,142223],{"className":142219},[398],[363,142221],{"className":142222,"style":2693},[402],[363,142224,7909],{"className":142225},[407,408],[363,142227,142229,142242],{"className":142228,"translate":367},[366],[363,142230,142232],{"className":142231},[371],[373,142233,142234],{"xmlns":375},[377,142235,142236,142240],{},[380,142237,142238],{},[383,142239,1121],{"mathvariant":75598},[387,142241,135652],{"encoding":389},[363,142243,142245],{"className":142244,"ariaHidden":394},[393],[363,142246,142248,142251],{"className":142247},[398],[363,142249],{"className":142250,"style":135662},[402],[363,142252,1121],{"className":142253},[407,75943],[1344,142255,142256],{},"完全可分离性","。下一小节的生成性刻画正是对这一可分离性的直接读解。",[2659,142259,142261],{"id":142260},"_13-生成性刻画gamma-商表示","1.3 生成性刻画：Gamma 商表示",[143,142263,142264],{},[11,142265,142266,11410,142269,142410,142411,142578,142579,142649,142650,142678,142679,143202,143203,23902,143279,87168,143356,10528],{},[1344,142267,142268],{},"命题 1.4（Gamma 商表示）",[363,142270,142272,142302],{"className":142271,"translate":367},[366],[363,142273,142275],{"className":142274},[371],[373,142276,142277],{"xmlns":375},[377,142278,142279,142299],{},[380,142280,142281,142287,142289,142291,142293],{},[543,142282,142283,142285],{},[383,142284,121465],{},[463,142286,465],{},[458,142288,898],{"separator":394},[458,142290,133150],{},[458,142292,898],{"separator":394},[543,142294,142295,142297],{},[383,142296,121465],{},[383,142298,11044],{},[387,142300,142301],{"encoding":389},"\\gamma_1, \\dots, \\gamma_K",[363,142303,142305],{"className":142304,"ariaHidden":394},[393],[363,142306,142308,142311,142352,142355,142358,142361,142364,142367,142370],{"className":142307},[398],[363,142309],{"className":142310,"style":2624},[402],[363,142312,142314,142317],{"className":142313},[407],[363,142315,121465],{"className":142316,"style":121555},[407,408],[363,142318,142320],{"className":142319},[629],[363,142321,142323,142344],{"className":142322},[633,634],[363,142324,142326,142341],{"className":142325},[638],[363,142327,142329],{"className":142328,"style":11483},[642],[363,142330,142332,142335],{"style":142331},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[363,142333],{"className":142334,"style":651},[650],[363,142336,142338],{"className":142337},[655,656,657,658],[363,142339,465],{"className":142340},[407,658],[363,142342,666],{"className":142343},[665],[363,142345,142347],{"className":142346},[638],[363,142348,142350],{"className":142349,"style":8316},[642],[363,142351],{},[363,142353,898],{"className":142354},[920],[363,142356],{"className":142357,"style":754},[505],[363,142359,133150],{"className":142360},[758],[363,142362],{"className":142363,"style":754},[505],[363,142365,898],{"className":142366},[920],[363,142368],{"className":142369,"style":754},[505],[363,142371,142373,142376],{"className":142372},[407],[363,142374,121465],{"className":142375,"style":121555},[407,408],[363,142377,142379],{"className":142378},[629],[363,142380,142382,142402],{"className":142381},[633,634],[363,142383,142385,142399],{"className":142384},[638],[363,142386,142388],{"className":142387,"style":727},[642],[363,142389,142390,142393],{"style":142331},[363,142391],{"className":142392,"style":651},[650],[363,142394,142396],{"className":142395},[655,656,657,658],[363,142397,11044],{"className":142398,"style":11132},[407,408,658],[363,142400,666],{"className":142401},[665],[363,142403,142405],{"className":142404},[638],[363,142406,142408],{"className":142407,"style":8316},[642],[363,142409],{}," 相互独立且 ",[363,142412,142414,142453],{"className":142413,"translate":367},[366],[363,142415,142417],{"className":142416},[371],[373,142418,142419],{"xmlns":375},[377,142420,142421,142450],{},[380,142422,142423,142429,142431,142434,142436,142438,142444,142446,142448],{},[543,142424,142425,142427],{},[383,142426,121465],{},[383,142428,12313],{},[458,142430,9217],{},[383,142432,142433],{"mathvariant":557},"Gamma",[458,142435,4451],{},[458,142437,461],{"stretchy":460},[543,142439,142440,142442],{},[383,142441,45126],{},[383,142443,12313],{},[458,142445,898],{"separator":394},[463,142447,465],{},[458,142449,468],{"stretchy":460},[387,142451,142452],{"encoding":389},"\\gamma_i \\sim \\operatorname{Gamma}(\\alpha_i, 1)",[363,142454,142456,142511],{"className":142455,"ariaHidden":394},[393],[363,142457,142459,142462,142502,142505,142508],{"className":142458},[398],[363,142460],{"className":142461,"style":2624},[402],[363,142463,142465,142468],{"className":142464},[407],[363,142466,121465],{"className":142467,"style":121555},[407,408],[363,142469,142471],{"className":142470},[629],[363,142472,142474,142494],{"className":142473},[633,634],[363,142475,142477,142491],{"className":142476},[638],[363,142478,142480],{"className":142479,"style":12532},[642],[363,142481,142482,142485],{"style":142331},[363,142483],{"className":142484,"style":651},[650],[363,142486,142488],{"className":142487},[655,656,657,658],[363,142489,12313],{"className":142490},[407,408,658],[363,142492,666],{"className":142493},[665],[363,142495,142497],{"className":142496},[638],[363,142498,142500],{"className":142499,"style":8316},[642],[363,142501],{},[363,142503],{"className":142504,"style":506},[505],[363,142506,9217],{"className":142507},[510],[363,142509],{"className":142510,"style":506},[505],[363,142512,142514,142517,142523,142526,142566,142569,142572,142575],{"className":142513},[398],[363,142515],{"className":142516,"style":487},[402],[363,142518,142520],{"className":142519},[1832],[363,142521,142433],{"className":142522},[407,33034],[363,142524,461],{"className":142525},[494],[363,142527,142529,142532],{"className":142528},[407],[363,142530,45126],{"className":142531,"style":45156},[407,408],[363,142533,142535],{"className":142534},[629],[363,142536,142538,142558],{"className":142537},[633,634],[363,142539,142541,142555],{"className":142540},[638],[363,142542,142544],{"className":142543,"style":12532},[642],[363,142545,142546,142549],{"style":136154},[363,142547],{"className":142548,"style":651},[650],[363,142550,142552],{"className":142551},[655,656,657,658],[363,142553,12313],{"className":142554},[407,408,658],[363,142556,666],{"className":142557},[665],[363,142559,142561],{"className":142560},[638],[363,142562,142564],{"className":142563,"style":8316},[642],[363,142565],{},[363,142567,898],{"className":142568},[920],[363,142570],{"className":142571,"style":754},[505],[363,142573,465],{"className":142574},[407],[363,142576,468],{"className":142577},[501],"（形状 ",[363,142580,142582,142600],{"className":142581,"translate":367},[366],[363,142583,142585],{"className":142584},[371],[373,142586,142587],{"xmlns":375},[377,142588,142589,142597],{},[380,142590,142591],{},[543,142592,142593,142595],{},[383,142594,45126],{},[383,142596,12313],{},[387,142598,142599],{"encoding":389},"\\alpha_i",[363,142601,142603],{"className":142602,"ariaHidden":394},[393],[363,142604,142606,142609],{"className":142605},[398],[363,142607],{"className":142608,"style":13724},[402],[363,142610,142612,142615],{"className":142611},[407],[363,142613,45126],{"className":142614,"style":45156},[407,408],[363,142616,142618],{"className":142617},[629],[363,142619,142621,142641],{"className":142620},[633,634],[363,142622,142624,142638],{"className":142623},[638],[363,142625,142627],{"className":142626,"style":12532},[642],[363,142628,142629,142632],{"style":136154},[363,142630],{"className":142631,"style":651},[650],[363,142633,142635],{"className":142634},[655,656,657,658],[363,142636,12313],{"className":142637},[407,408,658],[363,142639,666],{"className":142640},[665],[363,142642,142644],{"className":142643},[638],[363,142645,142647],{"className":142646,"style":8316},[642],[363,142648],{},"、率 ",[363,142651,142653,142666],{"className":142652,"translate":367},[366],[363,142654,142656],{"className":142655},[371],[373,142657,142658],{"xmlns":375},[377,142659,142660,142664],{},[380,142661,142662],{},[463,142663,465],{},[387,142665,465],{"encoding":389},[363,142667,142669],{"className":142668,"ariaHidden":394},[393],[363,142670,142672,142675],{"className":142671},[398],[363,142673],{"className":142674,"style":520},[402],[363,142676,465],{"className":142677},[407],"）。令\n",[363,142680,142682,142770],{"className":142681,"translate":367},[366],[363,142683,142685],{"className":142684},[371],[373,142686,142687],{"xmlns":375},[377,142688,142689,142767],{},[380,142690,142691,142693,142695,142709,142715,142717,142719,142725,142727,142737,142739,142741,142743,142745,142747,142753,142755,142757,142759,142765],{},[383,142692,7909],{},[458,142694,471],{},[7901,142696,142697,142699,142707],{},[458,142698,13811],{},[380,142700,142701,142703,142705],{},[383,142702,12313],{},[458,142704,471],{},[463,142706,465],{},[383,142708,11044],{},[543,142710,142711,142713],{},[383,142712,121465],{},[383,142714,12313],{},[458,142716,898],{"separator":394},[505,142718],{"width":3610},[543,142720,142721,142723],{},[383,142722,1121],{},[383,142724,12313],{},[458,142726,471],{},[586,142728,142729,142735],{},[543,142730,142731,142733],{},[383,142732,121465],{},[383,142734,12313],{},[383,142736,7909],{},[458,142738,898],{"separator":394},[505,142740],{"width":3610},[383,142742,1121],{"mathvariant":75598},[458,142744,471],{},[458,142746,461],{"stretchy":460},[543,142748,142749,142751],{},[383,142750,1121],{},[463,142752,465],{},[458,142754,898],{"separator":394},[458,142756,133150],{},[458,142758,898],{"separator":394},[543,142760,142761,142763],{},[383,142762,1121],{},[383,142764,11044],{},[458,142766,468],{"stretchy":460},[387,142768,142769],{"encoding":389},"s = \\sum_{i=1}^K \\gamma_i, \\qquad x_i = \\frac{\\gamma_i}{s}, \\qquad \\mathbf{x} = (x_1, \\dots, x_K)",[363,142771,142773,142791,142958,143092],{"className":142772,"ariaHidden":394},[393],[363,142774,142776,142779,142782,142785,142788],{"className":142775},[398],[363,142777],{"className":142778,"style":2693},[402],[363,142780,7909],{"className":142781},[407,408],[363,142783],{"className":142784,"style":506},[505],[363,142786,471],{"className":142787},[510],[363,142789],{"className":142790,"style":506},[505],[363,142792,142794,142797,142857,142860,142900,142903,142906,142909,142949,142952,142955],{"className":142793},[398],[363,142795],{"className":142796,"style":136418},[402],[363,142798,142800,142803],{"className":142799},[1832],[363,142801,13811],{"className":142802,"style":40032},[1832,1833,5964],[363,142804,142806],{"className":142805},[629],[363,142807,142809,142849],{"className":142808},[633,634],[363,142810,142812,142846],{"className":142811},[638],[363,142813,142815,142835],{"className":142814,"style":133537},[642],[363,142816,142817,142820],{"style":40048},[363,142818],{"className":142819,"style":651},[650],[363,142821,142823],{"className":142822},[655,656,657,658],[363,142824,142826,142829,142832],{"className":142825},[407,658],[363,142827,12313],{"className":142828},[407,408,658],[363,142830,471],{"className":142831},[510,658],[363,142833,465],{"className":142834},[407,658],[363,142836,142837,142840],{"style":133560},[363,142838],{"className":142839,"style":651},[650],[363,142841,142843],{"className":142842},[655,656,657,658],[363,142844,11044],{"className":142845,"style":11132},[407,408,658],[363,142847,666],{"className":142848},[665],[363,142850,142852],{"className":142851},[638],[363,142853,142855],{"className":142854,"style":40067},[642],[363,142856],{},[363,142858],{"className":142859,"style":754},[505],[363,142861,142863,142866],{"className":142862},[407],[363,142864,121465],{"className":142865,"style":121555},[407,408],[363,142867,142869],{"className":142868},[629],[363,142870,142872,142892],{"className":142871},[633,634],[363,142873,142875,142889],{"className":142874},[638],[363,142876,142878],{"className":142877,"style":12532},[642],[363,142879,142880,142883],{"style":142331},[363,142881],{"className":142882,"style":651},[650],[363,142884,142886],{"className":142885},[655,656,657,658],[363,142887,12313],{"className":142888},[407,408,658],[363,142890,666],{"className":142891},[665],[363,142893,142895],{"className":142894},[638],[363,142896,142898],{"className":142897,"style":8316},[642],[363,142899],{},[363,142901,898],{"className":142902},[920],[363,142904],{"className":142905,"style":3982},[505],[363,142907],{"className":142908,"style":754},[505],[363,142910,142912,142915],{"className":142911},[407],[363,142913,1121],{"className":142914},[407,408],[363,142916,142918],{"className":142917},[629],[363,142919,142921,142941],{"className":142920},[633,634],[363,142922,142924,142938],{"className":142923},[638],[363,142925,142927],{"className":142926,"style":12532},[642],[363,142928,142929,142932],{"style":730},[363,142930],{"className":142931,"style":651},[650],[363,142933,142935],{"className":142934},[655,656,657,658],[363,142936,12313],{"className":142937},[407,408,658],[363,142939,666],{"className":142940},[665],[363,142942,142944],{"className":142943},[638],[363,142945,142947],{"className":142946,"style":8316},[642],[363,142948],{},[363,142950],{"className":142951,"style":506},[505],[363,142953,471],{"className":142954},[510],[363,142956],{"className":142957,"style":506},[505],[363,142959,142961,142965,143071,143074,143077,143080,143083,143086,143089],{"className":142960},[398],[363,142962],{"className":142963,"style":142964},[402],"height:1.0925em;vertical-align:-0.345em;",[363,142966,142968,142971,143068],{"className":142967},[407],[363,142969],{"className":142970},[494,789],[363,142972,142974],{"className":142973},[586],[363,142975,142977,143060],{"className":142976},[633,634],[363,142978,142980,143057],{"className":142979},[638],[363,142981,142983,142997,143005],{"className":142982,"style":3451},[642],[363,142984,142985,142988],{"style":2875},[363,142986],{"className":142987,"style":809},[650],[363,142989,142991],{"className":142990},[655,656,657,658],[363,142992,142994],{"className":142993},[407,658],[363,142995,7909],{"className":142996},[407,408,658],[363,142998,142999,143002],{"style":821},[363,143000],{"className":143001,"style":809},[650],[363,143003],{"className":143004,"style":829},[828],[363,143006,143007,143010],{"style":3201},[363,143008],{"className":143009,"style":809},[650],[363,143011,143013],{"className":143012},[655,656,657,658],[363,143014,143016],{"className":143015},[407,658],[363,143017,143019,143022],{"className":143018},[407,658],[363,143020,121465],{"className":143021,"style":121555},[407,408,658],[363,143023,143025],{"className":143024},[629],[363,143026,143028,143049],{"className":143027},[633,634],[363,143029,143031,143046],{"className":143030},[638],[363,143032,143034],{"className":143033,"style":40120},[642],[363,143035,143037,143040],{"style":143036},"top:-2.357em;margin-left:-0.0556em;margin-right:0.0714em;",[363,143038],{"className":143039,"style":30578},[650],[363,143041,143043],{"className":143042},[655,30582,10022,658],[363,143044,12313],{"className":143045},[407,408,658],[363,143047,666],{"className":143048},[665],[363,143050,143052],{"className":143051},[638],[363,143053,143055],{"className":143054,"style":40142},[642],[363,143056],{},[363,143058,666],{"className":143059},[665],[363,143061,143063],{"className":143062},[638],[363,143064,143066],{"className":143065,"style":2920},[642],[363,143067],{},[363,143069],{"className":143070},[501,789],[363,143072,898],{"className":143073},[920],[363,143075],{"className":143076,"style":3982},[505],[363,143078],{"className":143079,"style":754},[505],[363,143081,1121],{"className":143082},[407,75943],[363,143084],{"className":143085,"style":506},[505],[363,143087,471],{"className":143088},[510],[363,143090],{"className":143091,"style":506},[505],[363,143093,143095,143098,143101,143141,143144,143147,143150,143153,143156,143159,143199],{"className":143094},[398],[363,143096],{"className":143097,"style":487},[402],[363,143099,461],{"className":143100},[494],[363,143102,143104,143107],{"className":143103},[407],[363,143105,1121],{"className":143106},[407,408],[363,143108,143110],{"className":143109},[629],[363,143111,143113,143133],{"className":143112},[633,634],[363,143114,143116,143130],{"className":143115},[638],[363,143117,143119],{"className":143118,"style":11483},[642],[363,143120,143121,143124],{"style":730},[363,143122],{"className":143123,"style":651},[650],[363,143125,143127],{"className":143126},[655,656,657,658],[363,143128,465],{"className":143129},[407,658],[363,143131,666],{"className":143132},[665],[363,143134,143136],{"className":143135},[638],[363,143137,143139],{"className":143138,"style":8316},[642],[363,143140],{},[363,143142,898],{"className":143143},[920],[363,143145],{"className":143146,"style":754},[505],[363,143148,133150],{"className":143149},[758],[363,143151],{"className":143152,"style":754},[505],[363,143154,898],{"className":143155},[920],[363,143157],{"className":143158,"style":754},[505],[363,143160,143162,143165],{"className":143161},[407],[363,143163,1121],{"className":143164},[407,408],[363,143166,143168],{"className":143167},[629],[363,143169,143171,143191],{"className":143170},[633,634],[363,143172,143174,143188],{"className":143173},[638],[363,143175,143177],{"className":143176,"style":727},[642],[363,143178,143179,143182],{"style":730},[363,143180],{"className":143181,"style":651},[650],[363,143183,143185],{"className":143184},[655,656,657,658],[363,143186,11044],{"className":143187,"style":11132},[407,408,658],[363,143189,666],{"className":143190},[665],[363,143192,143194],{"className":143193},[638],[363,143195,143197],{"className":143196,"style":8316},[642],[363,143198],{},[363,143200,468],{"className":143201},[501],"\n则 ",[363,143204,143206,143231],{"className":143205,"translate":367},[366],[363,143207,143209],{"className":143208},[371],[373,143210,143211],{"xmlns":375},[377,143212,143213,143229],{},[380,143214,143215,143217,143219,143221,143223,143225,143227],{},[383,143216,1121],{"mathvariant":75598},[458,143218,9217],{},[383,143220,137112],{"mathvariant":557},[458,143222,4451],{},[458,143224,461],{"stretchy":460},[383,143226,45126],{"mathvariant":136055},[458,143228,468],{"stretchy":460},[387,143230,137123],{"encoding":389},[363,143232,143234,143252],{"className":143233,"ariaHidden":394},[393],[363,143235,143237,143240,143243,143246,143249],{"className":143236},[398],[363,143238],{"className":143239,"style":135662},[402],[363,143241,1121],{"className":143242},[407,75943],[363,143244],{"className":143245,"style":506},[505],[363,143247,9217],{"className":143248},[510],[363,143250],{"className":143251,"style":506},[505],[363,143253,143255,143258,143264,143267,143276],{"className":143254},[398],[363,143256],{"className":143257,"style":487},[402],[363,143259,143261],{"className":143260},[1832],[363,143262,137112],{"className":143263},[407,33034],[363,143265,461],{"className":143266},[494],[363,143268,143270],{"className":143269},[407],[363,143271,143273],{"className":143272},[407],[363,143274,45126],{"className":143275},[407,136115],[363,143277,468],{"className":143278},[501],[363,143280,143282,143308],{"className":143281,"translate":367},[366],[363,143283,143285],{"className":143284},[371],[373,143286,143287],{"xmlns":375},[377,143288,143289,143305],{},[380,143290,143291,143293,143295,143297,143299,143301,143303],{},[383,143292,1121],{"mathvariant":75598},[458,143294,33919],{},[1645,143296,1652],{},[1645,143298,1652],{},[1645,143300,1652],{},[458,143302,33919],{},[383,143304,7909],{},[387,143306,143307],{"encoding":389},"\\mathbf{x} \\perp\\!\\!\\!\\perp s",[363,143309,143311,143335,143347],{"className":143310,"ariaHidden":394},[393],[363,143312,143314,143317,143320,143323,143326,143329,143332],{"className":143313},[398],[363,143315],{"className":143316,"style":12947},[402],[363,143318,1121],{"className":143319},[407,75943],[363,143321],{"className":143322,"style":506},[505],[363,143324,33919],{"className":143325},[510],[363,143327],{"className":143328,"style":1860},[505],[363,143330],{"className":143331,"style":1860},[505],[363,143333],{"className":143334,"style":1860},[505],[363,143336,143338,143341,143344],{"className":143337},[398],[363,143339],{"className":143340,"style":12947},[402],[363,143342,33919],{"className":143343},[510],[363,143345],{"className":143346,"style":506},[505],[363,143348,143350,143353],{"className":143349},[398],[363,143351],{"className":143352,"style":2693},[402],[363,143354,7909],{"className":143355},[407,408],[363,143357,143359,143393],{"className":143358,"translate":367},[366],[363,143360,143362],{"className":143361},[371],[373,143363,143364],{"xmlns":375},[377,143365,143366,143390],{},[380,143367,143368,143370,143372,143374,143376,143378,143384,143386,143388],{},[383,143369,7909],{},[458,143371,9217],{},[383,143373,142433],{"mathvariant":557},[458,143375,4451],{},[458,143377,461],{"stretchy":460},[543,143379,143380,143382],{},[383,143381,45126],{},[463,143383,474],{},[458,143385,898],{"separator":394},[463,143387,465],{},[458,143389,468],{"stretchy":460},[387,143391,143392],{"encoding":389},"s \\sim \\operatorname{Gamma}(\\alpha_0, 1)",[363,143394,143396,143414],{"className":143395,"ariaHidden":394},[393],[363,143397,143399,143402,143405,143408,143411],{"className":143398},[398],[363,143400],{"className":143401,"style":2693},[402],[363,143403,7909],{"className":143404},[407,408],[363,143406],{"className":143407,"style":506},[505],[363,143409,9217],{"className":143410},[510],[363,143412],{"className":143413,"style":506},[505],[363,143415,143417,143420,143426,143429,143469,143472,143475,143478],{"className":143416},[398],[363,143418],{"className":143419,"style":487},[402],[363,143421,143423],{"className":143422},[1832],[363,143424,142433],{"className":143425},[407,33034],[363,143427,461],{"className":143428},[494],[363,143430,143432,143435],{"className":143431},[407],[363,143433,45126],{"className":143434,"style":45156},[407,408],[363,143436,143438],{"className":143437},[629],[363,143439,143441,143461],{"className":143440},[633,634],[363,143442,143444,143458],{"className":143443},[638],[363,143445,143447],{"className":143446,"style":11483},[642],[363,143448,143449,143452],{"style":136154},[363,143450],{"className":143451,"style":651},[650],[363,143453,143455],{"className":143454},[655,656,657,658],[363,143456,474],{"className":143457},[407,658],[363,143459,666],{"className":143460},[665],[363,143462,143464],{"className":143463},[638],[363,143465,143467],{"className":143466,"style":8316},[642],[363,143468],{},[363,143470,898],{"className":143471},[920],[363,143473],{"className":143474,"style":754},[505],[363,143476,465],{"className":143477},[407],[363,143479,468],{"className":143480},[501],[11,143482,143483,143485,143486,143522,143523,143863,143864,144002],{},[1344,143484,138042],{}," 由独立性，",[363,143487,143489,143503],{"className":143488,"translate":367},[366],[363,143490,143492],{"className":143491},[371],[373,143493,143494],{"xmlns":375},[377,143495,143496,143500],{},[380,143497,143498],{},[383,143499,121465],{"mathvariant":136055},[387,143501,143502],{"encoding":389},"\\boldsymbol{\\gamma}",[363,143504,143506],{"className":143505,"ariaHidden":394},[393],[363,143507,143509,143512],{"className":143508},[398],[363,143510],{"className":143511,"style":133807},[402],[363,143513,143515],{"className":143514},[407],[363,143516,143518],{"className":143517},[407],[363,143519,121465],{"className":143520,"style":143521},[407,136115],"margin-right:0.0639em;"," 的联合密度为 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\\gamma_i^{\\alpha_i - 1} e^{-\\gamma_i} \u002F 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,{},[363,143861,468],{"className":143862},[501],"。施行与命题 1.3 相同的替换 ",[363,143865,143867,143895],{"className":143866,"translate":367},[366],[363,143868,143870],{"className":143869},[371],[373,143871,143872],{"xmlns":375},[377,143873,143874,143892],{},[380,143875,143876,143882,143884,143886],{},[543,143877,143878,143880],{},[383,143879,121465],{},[383,143881,12313],{},[458,143883,471],{},[383,143885,7909],{},[543,143887,143888,143890],{},[383,143889,1121],{},[383,143891,12313],{},[387,143893,143894],{"encoding":389},"\\gamma_i = s x_i",[363,143896,143898,143953],{"className":143897,"ariaHidden":394},[393],[363,143899,143901,143904,143944,143947,143950],{"className":143900},[398],[363,143902],{"className":143903,"style":2624},[402],[363,143905,143907,143910],{"className":143906},[407],[363,143908,121465],{"className":143909,"style":121555},[407,408],[363,143911,143913],{"className":143912},[629],[363,143914,143916,143936],{"className":143915},[633,634],[363,143917,143919,143933],{"className":143918},[638],[363,143920,143922],{"className":143921,"style":12532},[642],[363,143923,143924,143927],{"style":142331},[363,143925],{"className":143926,"style":651},[650],[363,143928,143930],{"className":143929},[655,656,657,658],[363,143931,12313],{"className":143932},[407,408,658],[363,143934,666],{"className":143935},[665],[363,143937,143939],{"className":143938},[638],[363,143940,143942],{"className":143941,"style":8316},[642],[363,143943],{},[363,143945],{"className":143946,"style":506},[505],[363,143948,471],{"className":143949},[510],[363,143951],{"className":143952,"style":506},[505],[363,143954,143956,143959,143962],{"className":143955},[398],[363,143957],{"className":143958,"style":13724},[402],[363,143960,7909],{"className":143961},[407,408],[363,143963,143965,143968],{"className":143964},[407],[363,143966,1121],{"className":143967},[407,408],[363,143969,143971],{"className":143970},[629],[363,143972,143974,143994],{"className":143973},[633,634],[363,143975,143977,143991],{"className":143976},[638],[363,143978,143980],{"className":143979,"style":12532},[642],[363,143981,143982,143985],{"style":730},[363,143983],{"className":143984,"style":651},[650],[363,143986,143988],{"className":143987},[655,656,657,658],[363,143989,12313],{"className":143990},[407,408,658],[363,143992,666],{"className":143993},[665],[363,143995,143997],{"className":143996},[638],[363,143998,144000],{"className":143999,"style":8316},[642],[363,144001],{},"，联合密度变换为",[363,144004,144006],{"className":144005,"translate":367},[528],[363,144007,144009,144197],{"className":144008,"translate":367},[366],[363,144010,144012],{"className":144011},[371],[373,144013,144014],{"xmlns":375,"display":537},[377,144015,144016,144194],{},[380,144017,144018,144020,144022,144024,144026,144028,144030,144032,144076,144078,144154,144156,144158,144160,144162,144164,144166,144172,144174,144176,144178,144180,144182,144184,144186,144188,144190,144192],{},[383,144019,11],{},[458,144021,461],{"stretchy":460},[383,144023,7909],{},[458,144025,898],{"separator":394},[383,144027,1121],{"mathvariant":75598},[458,144029,468],{"stretchy":460},[458,144031,471],{},[586,144033,144034,144062],{},[380,144035,144036,144052],{},[7162,144037,144038,144040],{},[383,144039,7909],{},[380,144041,144042,144048,144050],{},[543,144043,144044,144046],{},[383,144045,45126],{},[463,144047,474],{},[458,144049,5161],{},[463,144051,465],{},[7162,144053,144054,144056],{},[383,144055,7915],{},[380,144057,144058,144060],{},[458,144059,5161],{},[383,144061,7909],{},[380,144063,144064,144066,144068,144074],{},[383,144065,137218],{"mathvariant":557},[458,144067,461],{"stretchy":460},[543,144069,144070,144072],{},[383,144071,45126],{},[463,144073,474],{},[458,144075,468],{"stretchy":460},[458,144077,3647],{},[586,144079,144080,144126],{},[380,144081,144082,144084,144086,144092,144094,144108],{},[383,144083,137218],{"mathvariant":557},[458,144085,461],{"stretchy":460},[543,144087,144088,144090],{},[383,144089,45126],{},[463,144091,474],{},[458,144093,468],{"stretchy":460},[48692,144095,144096,144098,144106],{},[458,144097,136684],{},[380,144099,144100,144102,144104],{},[383,144101,12313],{},[458,144103,471],{},[463,144105,465],{},[383,144107,11044],{},[7901,144109,144110,144112,144114],{},[383,144111,1121],{},[383,144113,12313],{},[380,144115,144116,144122,144124],{},[543,144117,144118,144120],{},[383,144119,45126],{},[383,144121,12313],{},[458,144123,5161],{},[463,144125,465],{},[380,144127,144128,144142,144144,144146,144152],{},[48692,144129,144130,144132,144140],{},[458,144131,136684],{},[380,144133,144134,144136,144138],{},[383,144135,12313],{},[458,144137,471],{},[463,144139,465],{},[383,144141,11044],{},[383,144143,137218],{"mathvariant":557},[458,144145,461],{"stretchy":460},[543,144147,144148,144150],{},[383,144149,45126],{},[383,144151,12313],{},[458,144153,468],{"stretchy":460},[458,144155,471],{},[383,144157,142433],{"mathvariant":557},[458,144159,4451],{},[458,144161,461],{"stretchy":460},[383,144163,7909],{},[458,144165,34247],{"separator":394},[543,144167,144168,144170],{},[383,144169,45126],{},[463,144171,474],{},[458,144173,898],{"separator":394},[463,144175,465],{},[458,144177,468],{"stretchy":460},[458,144179,3647],{},[383,144181,137112],{"mathvariant":557},[458,144183,4451],{},[458,144185,461],{"stretchy":460},[383,144187,1121],{"mathvariant":75598},[458,144189,34247],{"separator":394},[383,144191,45126],{"mathvariant":136055},[458,144193,468],{"stretchy":460},[387,144195,144196],{"encoding":389},"p(s, \\mathbf{x}) = \\frac{s^{\\alpha_0 - 1} e^{-s}}{\\Gamma(\\alpha_0)} \\cdot \\frac{\\Gamma(\\alpha_0) \\prod_{i=1}^K x_i^{\\alpha_i - 1}}{\\prod_{i=1}^K \\Gamma(\\alpha_i)}\n= \\operatorname{Gamma}(s; \\alpha_0, 1) \\cdot \\operatorname{Dir}(\\mathbf{x}; 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的两个因子的乘积，故 ",[363,145049,145051,145064],{"className":145050,"translate":367},[366],[363,145052,145054],{"className":145053},[371],[373,145055,145056],{"xmlns":375},[377,145057,145058,145062],{},[380,145059,145060],{},[383,145061,7909],{},[387,145063,7909],{"encoding":389},[363,145065,145067],{"className":145066,"ariaHidden":394},[393],[363,145068,145070,145073],{"className":145069},[398],[363,145071],{"className":145072,"style":2693},[402],[363,145074,7909],{"className":145075},[407,408],[363,145077,145079,145092],{"className":145078,"translate":367},[366],[363,145080,145082],{"className":145081},[371],[373,145083,145084],{"xmlns":375},[377,145085,145086,145090],{},[380,145087,145088],{},[383,145089,1121],{"mathvariant":75598},[387,145091,135652],{"encoding":389},[363,145093,145095],{"className":145094,"ariaHidden":394},[393],[363,145096,145098,145101],{"className":145097},[398],[363,145099],{"className":145100,"style":135662},[402],[363,145102,1121],{"className":145103},[407,75943]," 独立，且各自的边缘分布如命题所述。",[11,145106,145107,145108,145152,145153,145181,145182,145364,145365,145368],{},"命题 1.4 给出了三条即时的推论。其一，它提供了一个 ",[363,145109,145111,145131],{"className":145110,"translate":367},[366],[363,145112,145114],{"className":145113},[371],[373,145115,145116],{"xmlns":375},[377,145117,145118,145128],{},[380,145119,145120,145122,145124,145126],{},[383,145121,19593],{},[458,145123,461],{"stretchy":460},[383,145125,11044],{},[458,145127,468],{"stretchy":460},[387,145129,145130],{"encoding":389},"O(K)",[363,145132,145134],{"className":145133,"ariaHidden":394},[393],[363,145135,145137,145140,145143,145146,145149],{"className":145136},[398],[363,145138],{"className":145139,"style":487},[402],[363,145141,19593],{"className":145142,"style":625},[407,408],[363,145144,461],{"className":145145},[494],[363,145147,11044],{"className":145148,"style":11132},[407,408],[363,145150,468],{"className":145151},[501]," 的精确采样算法：独立抽取 ",[363,145154,145156,145169],{"className":145155,"translate":367},[366],[363,145157,145159],{"className":145158},[371],[373,145160,145161],{"xmlns":375},[377,145162,145163,145167],{},[380,145164,145165],{},[383,145166,11044],{},[387,145168,11044],{"encoding":389},[363,145170,145172],{"className":145171,"ariaHidden":394},[393],[363,145173,145175,145178],{"className":145174},[398],[363,145176],{"className":145177,"style":1011},[402],[363,145179,11044],{"className":145180,"style":11132},[407,408]," 个 Gamma 变量再归一化，无需拒绝采样或逆变换。其二，它说明参数 ",[363,145183,145185,145217],{"className":145184,"translate":367},[366],[363,145186,145188],{"className":145187},[371],[373,145189,145190],{"xmlns":375},[377,145191,145192,145214],{},[380,145193,145194,145200,145202,145208],{},[543,145195,145196,145198],{},[383,145197,45126],{},[463,145199,474],{},[458,145201,471],{},[543,145203,145204,145206],{},[458,145205,13811],{},[383,145207,12313],{},[543,145209,145210,145212],{},[383,145211,45126],{},[383,145213,12313],{},[387,145215,145216],{"encoding":389},"\\alpha_0 = \\sum_i \\alpha_i",[363,145218,145220,145275],{"className":145219,"ariaHidden":394},[393],[363,145221,145223,145226,145266,145269,145272],{"className":145222},[398],[363,145224],{"className":145225,"style":13724},[402],[363,145227,145229,145232],{"className":145228},[407],[363,145230,45126],{"className":145231,"style":45156},[407,408],[363,145233,145235],{"className":145234},[629],[363,145236,145238,145258],{"className":145237},[633,634],[363,145239,145241,145255],{"className":145240},[638],[363,145242,145244],{"className":145243,"style":11483},[642],[363,145245,145246,145249],{"style":136154},[363,145247],{"className":145248,"style":651},[650],[363,145250,145252],{"className":145251},[655,656,657,658],[363,145253,474],{"className":145254},[407,658],[363,145256,666],{"className":145257},[665],[363,145259,145261],{"className":145260},[638],[363,145262,145264],{"className":145263,"style":8316},[642],[363,145265],{},[363,145267],{"className":145268,"style":506},[505],[363,145270,471],{"className":145271},[510],[363,145273],{"className":145274,"style":506},[505],[363,145276,145278,145281,145321,145324],{"className":145277},[398],[363,145279],{"className":145280,"style":138953},[402],[363,145282,145284,145287],{"className":145283},[1832],[363,145285,13811],{"className":145286,"style":40032},[1832,1833,5964],[363,145288,145290],{"className":145289},[629],[363,145291,145293,145313],{"className":145292},[633,634],[363,145294,145296,145310],{"className":145295},[638],[363,145297,145299],{"className":145298,"style":40045},[642],[363,145300,145301,145304],{"style":40048},[363,145302],{"className":145303,"style":651},[650],[363,145305,145307],{"className":145306},[655,656,657,658],[363,145308,12313],{"className":145309},[407,408,658],[363,145311,666],{"className":145312},[665],[363,145314,145316],{"className":145315},[638],[363,145317,145319],{"className":145318,"style":40067},[642],[363,145320],{},[363,145322],{"className":145323,"style":754},[505],[363,145325,145327,145330],{"className":145326},[407],[363,145328,45126],{"className":145329,"style":45156},[407,408],[363,145331,145333],{"className":145332},[629],[363,145334,145336,145356],{"className":145335},[633,634],[363,145337,145339,145353],{"className":145338},[638],[363,145340,145342],{"className":145341,"style":12532},[642],[363,145343,145344,145347],{"style":136154},[363,145345],{"className":145346,"style":651},[650],[363,145348,145350],{"className":145349},[655,656,657,658],[363,145351,12313],{"className":145352},[407,408,658],[363,145354,666],{"className":145355},[665],[363,145357,145359],{"className":145358},[638],[363,145360,145362],{"className":145361,"style":8316},[642],[363,145363],{}," 在概率意义上是",[1344,145366,145367],{},"先于归一化的总质量","，这一角色将在 2.2 节被重新参数化为一个精度参数。其三，它把迪利克雷分布与 Gamma 分布、进而与 Poisson 过程联系起来，这正是 2.5 节分裂构造的来源。",[2659,145370,145372],{"id":145371},"_14-与-beta-分布多项分布的关系","1.4 与 Beta 分布、多项分布的关系",[11,145374,145375,31190,145378,145428,145429,145585,145586,145656],{},[1344,145376,145377],{},"退化到 Beta 分布。",[363,145379,145381,145398],{"className":145380,"translate":367},[366],[363,145382,145384],{"className":145383},[371],[373,145385,145386],{"xmlns":375},[377,145387,145388,145396],{},[380,145389,145390,145392,145394],{},[383,145391,11044],{},[458,145393,471],{},[463,145395,2811],{},[387,145397,135087],{"encoding":389},[363,145399,145401,145419],{"className":145400,"ariaHidden":394},[393],[363,145402,145404,145407,145410,145413,145416],{"className":145403},[398],[363,145405],{"className":145406,"style":1011},[402],[363,145408,11044],{"className":145409,"style":11132},[407,408],[363,145411],{"className":145412,"style":506},[505],[363,145414,471],{"className":145415},[510],[363,145417],{"className":145418,"style":506},[505],[363,145420,145422,145425],{"className":145421},[398],[363,145423],{"className":145424,"style":520},[402],[363,145426,2811],{"className":145427},[407]," 时，约束 ",[363,145430,145432,145462],{"className":145431,"translate":367},[366],[363,145433,145435],{"className":145434},[371],[373,145436,145437],{"xmlns":375},[377,145438,145439,145459],{},[380,145440,145441,145447,145449,145455,145457],{},[543,145442,145443,145445],{},[383,145444,1121],{},[463,145446,465],{},[458,145448,2818],{},[543,145450,145451,145453],{},[383,145452,1121],{},[463,145454,2811],{},[458,145456,471],{},[463,145458,465],{},[387,145460,145461],{"encoding":389},"x_1 + x_2 = 1",[363,145463,145465,145521,145576],{"className":145464,"ariaHidden":394},[393],[363,145466,145468,145472,145512,145515,145518],{"className":145467},[398],[363,145469],{"className":145470,"style":145471},[402],"height:0.7333em;vertical-align:-0.15em;",[363,145473,145475,145478],{"className":145474},[407],[363,145476,1121],{"className":145477},[407,408],[363,145479,145481],{"className":145480},[629],[363,145482,145484,145504],{"className":145483},[633,634],[363,145485,145487,145501],{"className":145486},[638],[363,145488,145490],{"className":145489,"style":11483},[642],[363,145491,145492,145495],{"style":730},[363,145493],{"className":145494,"style":651},[650],[363,145496,145498],{"className":145497},[655,656,657,658],[363,145499,465],{"className":145500},[407,658],[363,145502,666],{"className":145503},[665],[363,145505,145507],{"className":145506},[638],[363,145508,145510],{"className":145509,"style":8316},[642],[363,145511],{},[363,145513],{"className":145514,"style":2935},[505],[363,145516,2818],{"className":145517},[2939],[363,145519],{"className":145520,"style":2935},[505],[363,145522,145524,145527,145567,145570,145573],{"className":145523},[398],[363,145525],{"className":145526,"style":13724},[402],[363,145528,145530,145533],{"className":145529},[407],[363,145531,1121],{"className":145532},[407,408],[363,145534,145536],{"className":145535},[629],[363,145537,145539,145559],{"className":145538},[633,634],[363,145540,145542,145556],{"className":145541},[638],[363,145543,145545],{"className":145544,"style":11483},[642],[363,145546,145547,145550],{"style":730},[363,145548],{"className":145549,"style":651},[650],[363,145551,145553],{"className":145552},[655,656,657,658],[363,145554,2811],{"className":145555},[407,658],[363,145557,666],{"className":145558},[665],[363,145560,145562],{"className":145561},[638],[363,145563,145565],{"className":145564,"style":8316},[642],[363,145566],{},[363,145568],{"className":145569,"style":506},[505],[363,145571,471],{"className":145572},[510],[363,145574],{"className":145575,"style":506},[505],[363,145577,145579,145582],{"className":145578},[398],[363,145580],{"className":145581,"style":520},[402],[363,145583,465],{"className":145584},[407]," 使分布由单一变量 ",[363,145587,145589,145607],{"className":145588,"translate":367},[366],[363,145590,145592],{"className":145591},[371],[373,145593,145594],{"xmlns":375},[377,145595,145596,145604],{},[380,145597,145598],{},[543,145599,145600,145602],{},[383,145601,1121],{},[463,145603,465],{},[387,145605,145606],{"encoding":389},"x_1",[363,145608,145610],{"className":145609,"ariaHidden":394},[393],[363,145611,145613,145616],{"className":145612},[398],[363,145614],{"className":145615,"style":13724},[402],[363,145617,145619,145622],{"className":145618},[407],[363,145620,1121],{"className":145621},[407,408],[363,145623,145625],{"className":145624},[629],[363,145626,145628,145648],{"className":145627},[633,634],[363,145629,145631,145645],{"className":145630},[638],[363,145632,145634],{"className":145633,"style":11483},[642],[363,145635,145636,145639],{"style":730},[363,145637],{"className":145638,"style":651},[650],[363,145640,145642],{"className":145641},[655,656,657,658],[363,145643,465],{"className":145644},[407,658],[363,145646,666],{"className":145647},[665],[363,145649,145651],{"className":145650},[638],[363,145652,145654],{"className":145653,"style":8316},[642],[363,145655],{}," 完全决定，密度退化为",[363,145658,145660],{"className":145659,"translate":367},[528],[363,145661,145663,145832],{"className":145662,"translate":367},[366],[363,145664,145666],{"className":145665},[371],[373,145667,145668],{"xmlns":375,"display":537},[377,145669,145670,145829],{},[380,145671,145672,145674,145676,145682,145684,145690,145692,145698,145700,145702,145774,145776,145778,145780,145782,145784,145786,145788,145794,145796,145802,145804,145806,145809,145811,145813,145819,145821,145827],{},[383,145673,385],{},[458,145675,461],{"stretchy":460},[543,145677,145678,145680],{},[383,145679,1121],{},[463,145681,465],{},[458,145683,5156],{},[543,145685,145686,145688],{},[383,145687,45126],{},[463,145689,465],{},[458,145691,898],{"separator":394},[543,145693,145694,145696],{},[383,145695,45126],{},[463,145697,2811],{},[458,145699,468],{"stretchy":460},[458,145701,471],{},[586,145703,145704,145752],{},[380,145705,145706,145724,145726,145728,145730,145736],{},[7901,145707,145708,145710,145712],{},[383,145709,1121],{},[463,145711,465],{},[380,145713,145714,145720,145722],{},[543,145715,145716,145718],{},[383,145717,45126],{},[463,145719,465],{},[458,145721,5161],{},[463,145723,465],{},[458,145725,461],{"stretchy":460},[463,145727,465],{},[458,145729,5161],{},[543,145731,145732,145734],{},[383,145733,1121],{},[463,145735,465],{},[7162,145737,145738,145740],{},[458,145739,468],{"stretchy":460},[380,145741,145742,145748,145750],{},[543,145743,145744,145746],{},[383,145745,45126],{},[463,145747,2811],{},[458,145749,5161],{},[463,145751,465],{},[380,145753,145754,145756,145758,145764,145766,145772],{},[383,145755,39663],{},[458,145757,461],{"stretchy":460},[543,145759,145760,145762],{},[383,145761,45126],{},[463,145763,465],{},[458,145765,898],{"separator":394},[543,145767,145768,145770],{},[383,145769,45126],{},[463,145771,2811],{},[458,145773,468],{"stretchy":460},[458,145775,898],{"separator":394},[505,145777],{"width":3610},[1645,145779,88022],{},[505,145781],{"width":3610},[383,145783,137112],{"mathvariant":557},[458,145785,4451],{},[458,145787,461],{"stretchy":460},[543,145789,145790,145792],{},[383,145791,45126],{},[463,145793,465],{},[458,145795,898],{"separator":394},[543,145797,145798,145800],{},[383,145799,45126],{},[463,145801,2811],{},[458,145803,468],{"stretchy":460},[458,145805,94916],{},[383,145807,145808],{"mathvariant":557},"Beta",[458,145810,4451],{},[458,145812,461],{"stretchy":460},[543,145814,145815,145817],{},[383,145816,45126],{},[463,145818,465],{},[458,145820,898],{"separator":394},[543,145822,145823,145825],{},[383,145824,45126],{},[463,145826,2811],{},[458,145828,468],{"stretchy":460},[387,145830,145831],{"encoding":389},"f(x_1 \\mid \\alpha_1, \\alpha_2) = \\frac{x_1^{\\alpha_1 - 1}(1 - x_1)^{\\alpha_2 - 1}}{B(\\alpha_1, \\alpha_2)},\n\\qquad \\text{即} \\qquad \\operatorname{Dir}(\\alpha_1, \\alpha_2) \\equiv \\operatorname{Beta}(\\alpha_1, \\alpha_2)",[363,145833,145835,145896,146000,146515],{"className":145834,"ariaHidden":394},[393],[363,145836,145838,145841,145844,145847,145887,145890,145893],{"className":145837},[398],[363,145839],{"className":145840,"style":487},[402],[363,145842,385],{"className":145843,"style":409},[407,408],[363,145845,461],{"className":145846},[494],[363,145848,145850,145853],{"className":145849},[407],[363,145851,1121],{"className":145852},[407,408],[363,145854,145856],{"className":145855},[629],[363,145857,145859,145879],{"className":145858},[633,634],[363,145860,145862,145876],{"className":145861},[638],[363,145863,145865],{"className":145864,"style":11483},[642],[363,145866,145867,145870],{"style":730},[363,145868],{"className":145869,"style":651},[650],[363,145871,145873],{"className":145872},[655,656,657,658],[363,145874,465],{"className":145875},[407,658],[363,145877,666],{"className":145878},[665],[363,145880,145882],{"className":145881},[638],[363,145883,145885],{"className":145884,"style":8316},[642],[363,145886],{},[363,145888],{"className":145889,"style":506},[505],[363,145891,5156],{"className":145892},[510],[363,145894],{"className":145895,"style":506},[505],[363,145897,145899,145902,145942,145945,145948,145988,145991,145994,145997],{"className":145898},[398],[363,145900],{"className":145901,"style":487},[402],[363,145903,145905,145908],{"className":145904},[407],[363,145906,45126],{"className":145907,"style":45156},[407,408],[363,145909,145911],{"className":145910},[629],[363,145912,145914,145934],{"className":145913},[633,634],[363,145915,145917,145931],{"className":145916},[638],[363,145918,145920],{"className":145919,"style":11483},[642],[363,145921,145922,145925],{"style":136154},[363,145923],{"className":145924,"style":651},[650],[363,145926,145928],{"className":145927},[655,656,657,658],[363,145929,465],{"className":145930},[407,658],[363,145932,666],{"className":145933},[665],[363,145935,145937],{"className":145936},[638],[363,145938,145940],{"className":145939,"style":8316},[642],[363,145941],{},[363,145943,898],{"className":145944},[920],[363,145946],{"className":145947,"style":754},[505],[363,145949,145951,145954],{"className":145950},[407],[363,145952,45126],{"className":145953,"style":45156},[407,408],[363,145955,145957],{"className":145956},[629],[363,145958,145960,145980],{"className":145959},[633,634],[363,145961,145963,145977],{"className":145962},[638],[363,145964,145966],{"className":145965,"style":11483},[642],[363,145967,145968,145971],{"style":136154},[363,145969],{"className":145970,"style":651},[650],[363,145972,145974],{"className":145973},[655,656,657,658],[363,145975,2811],{"className":145976},[407,658],[363,145978,666],{"className":145979},[665],[363,145981,145983],{"className":145982},[638],[363,145984,145986],{"className":145985,"style":8316},[642],[363,145987],{},[363,145989,468],{"className":145990},[501],[363,145992],{"className":145993,"style":506},[505],[363,145995,471],{"className":145996},[510],[363,145998],{"className":145999,"style":506},[505],[363,146001,146003,146007,146387,146390,146393,146396,146402,146405,146408,146414,146417,146457,146460,146463,146503,146506,146509,146512],{"className":146002},[398],[363,146004],{"className":146005,"style":146006},[402],"height:2.509em;vertical-align:-0.936em;",[363,146008,146010,146013,146384],{"className":146009},[407],[363,146011],{"className":146012},[494,789],[363,146014,146016],{"className":146015},[586],[363,146017,146019,146376],{"className":146018},[633,634],[363,146020,146022,146373],{"className":146021},[638],[363,146023,146026,146129,146137],{"className":146024,"style":146025},[642],"height:1.573em;",[363,146027,146028,146031],{"style":805},[363,146029],{"className":146030,"style":809},[650],[363,146032,146034,146037,146040,146080,146083,146086,146126],{"className":146033},[407],[363,146035,39663],{"className":146036,"style":39759},[407,408],[363,146038,461],{"className":146039},[494],[363,146041,146043,146046],{"className":146042},[407],[363,146044,45126],{"className":146045,"style":45156},[407,408],[363,146047,146049],{"className":146048},[629],[363,146050,146052,146072],{"className":146051},[633,634],[363,146053,146055,146069],{"className":146054},[638],[363,146056,146058],{"className":146057,"style":11483},[642],[363,146059,146060,146063],{"style":136154},[363,146061],{"className":146062,"style":651},[650],[363,146064,146066],{"className":146065},[655,656,657,658],[363,146067,465],{"className":146068},[407,658],[363,146070,666],{"className":146071},[665],[363,146073,146075],{"className":146074},[638],[363,146076,146078],{"className":146077,"style":8316},[642],[363,146079],{},[363,146081,898],{"className":146082},[920],[363,146084],{"className":146085,"style":754},[505],[363,146087,146089,146092],{"className":146088},[407],[363,146090,45126],{"className":146091,"style":45156},[407,408],[363,146093,146095],{"className":146094},[629],[363,146096,146098,146118],{"className":146097},[633,634],[363,146099,146101,146115],{"className":146100},[638],[363,146102,146104],{"className":146103,"style":11483},[642],[363,146105,146106,146109],{"style":136154},[363,146107],{"className":146108,"style":651},[650],[363,146110,146112],{"className":146111},[655,656,657,658],[363,146113,2811],{"className":146114},[407,658],[363,146116,666],{"className":146117},[665],[363,146119,146121],{"className":146120},[638],[363,146122,146124],{"className":146123,"style":8316},[642],[363,146125],{},[363,146127,468],{"className":146128},[501],[363,146130,146131,146134],{"style":821},[363,146132],{"className":146133,"style":809},[650],[363,146135],{"className":146136,"style":829},[828],[363,146138,146139,146142],{"style":832},[363,146140],{"className":146141,"style":809},[650],[363,146143,146145,146243,146246,146249,146252,146255,146258,146298],{"className":146144},[407],[363,146146,146148,146151],{"className":146147},[407],[363,146149,1121],{"className":146150},[407,408],[363,146152,146154],{"className":146153},[629],[363,146155,146157,146235],{"className":146156},[633,634],[363,146158,146160,146232],{"className":146159},[638],[363,146161,146163,146175],{"className":146162,"style":136944},[642],[363,146164,146166,146169],{"style":146165},"top:-2.4337em;margin-left:0em;margin-right:0.05em;",[363,146167],{"className":146168,"style":651},[650],[363,146170,146172],{"className":146171},[655,656,657,658],[363,146173,465],{"className":146174},[407,658],[363,146176,146177,146180],{"style":136959},[363,146178],{"className":146179,"style":651},[650],[363,146181,146183],{"className":146182},[655,656,657,658],[363,146184,146186,146226,146229],{"className":146185},[407,658],[363,146187,146189,146192],{"className":146188},[407,658],[363,146190,45126],{"className":146191,"style":45156},[407,408,658],[363,146193,146195],{"className":146194},[629],[363,146196,146198,146218],{"className":146197},[633,634],[363,146199,146201,146215],{"className":146200},[638],[363,146202,146204],{"className":146203,"style":137351},[642],[363,146205,146206,146209],{"style":136989},[363,146207],{"className":146208,"style":30578},[650],[363,146210,146212],{"className":146211},[655,30582,10022,658],[363,146213,465],{"className":146214},[407,658],[363,146216,666],{"className":146217},[665],[363,146219,146221],{"className":146220},[638],[363,146222,146224],{"className":146223,"style":40142},[642],[363,146225],{},[363,146227,5161],{"className":146228},[2939,658],[363,146230,465],{"className":146231},[407,658],[363,146233,666],{"className":146234},[665],[363,146236,146238],{"className":146237},[638],[363,146239,146241],{"className":146240,"style":125540},[642],[363,146242],{},[363,146244,461],{"className":146245},[494],[363,146247,465],{"className":146248},[407],[363,146250],{"className":146251,"style":2935},[505],[363,146253,5161],{"className":146254},[2939],[363,146256],{"className":146257,"style":2935},[505],[363,146259,146261,146264],{"className":146260},[407],[363,146262,1121],{"className":146263},[407,408],[363,146265,146267],{"className":146266},[629],[363,146268,146270,146290],{"className":146269},[633,634],[363,146271,146273,146287],{"className":146272},[638],[363,146274,146276],{"className":146275,"style":11483},[642],[363,146277,146278,146281],{"style":730},[363,146279],{"className":146280,"style":651},[650],[363,146282,146284],{"className":146283},[655,656,657,658],[363,146285,465],{"className":146286},[407,658],[363,146288,666],{"className":146289},[665],[363,146291,146293],{"className":146292},[638],[363,146294,146296],{"className":146295,"style":8316},[642],[363,146297],{},[363,146299,146301,146304],{"className":146300},[501],[363,146302,468],{"className":146303},[501],[363,146305,146307],{"className":146306},[629],[363,146308,146310],{"className":146309},[633],[363,146311,146313],{"className":146312},[638],[363,146314,146316],{"className":146315,"style":20400},[642],[363,146317,146318,146321],{"style":7213},[363,146319],{"className":146320,"style":651},[650],[363,146322,146324],{"className":146323},[655,656,657,658],[363,146325,146327,146367,146370],{"className":146326},[407,658],[363,146328,146330,146333],{"className":146329},[407,658],[363,146331,45126],{"className":146332,"style":45156},[407,408,658],[363,146334,146336],{"className":146335},[629],[363,146337,146339,146359],{"className":146338},[633,634],[363,146340,146342,146356],{"className":146341},[638],[363,146343,146345],{"className":146344,"style":137351},[642],[363,146346,146347,146350],{"style":136989},[363,146348],{"className":146349,"style":30578},[650],[363,146351,146353],{"className":146352},[655,30582,10022,658],[363,146354,2811],{"className":146355},[407,658],[363,146357,666],{"className":146358},[665],[363,146360,146362],{"className":146361},[638],[363,146363,146365],{"className":146364,"style":40142},[642],[363,146366],{},[363,146368,5161],{"className":146369},[2939,658],[363,146371,465],{"className":146372},[407,658],[363,146374,666],{"className":146375},[665],[363,146377,146379],{"className":146378},[63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Beta 分布在维度上的推广。下文所有关于迪利克雷分布的陈述，在 ",[363,146623,146625,146643],{"className":146624,"translate":367},[366],[363,146626,146628],{"className":146627},[371],[373,146629,146630],{"xmlns":375},[377,146631,146632,146640],{},[380,146633,146634,146636,146638],{},[383,146635,11044],{},[458,146637,471],{},[463,146639,2811],{},[387,146641,146642],{"encoding":389},"K=2",[363,146644,146646,146664],{"className":146645,"ariaHidden":394},[393],[363,146647,146649,146652,146655,146658,146661],{"className":146648},[398],[363,146650],{"className":146651,"style":1011},[402],[363,146653,11044],{"className":146654,"style":11132},[407,408],[363,146656],{"className":146657,"style":506},[505],[363,146659,471],{"className":146660},[510],[363,146662],{"className":146663,"style":506},[505],[363,146665,146667,146670],{"className":146666},[398],[363,146668],{"className":146669,"style":520},[402],[363,146671,2811],{"className":146672},[407]," 时都应退化为 Beta 分布的相应陈述，这构成了检验公式正确性的一个廉价判据。",[11,146675,146676,146679,146680,146750,146751,146958,146959,147112,147113,43137,147278,147347],{},[1344,146677,146678],{},"边缘分布仍为 Beta。"," 更一般地，任意单个分量 ",[363,146681,146683,146701],{"className":146682,"translate":367},[366],[363,146684,146686],{"className":146685},[371],[373,146687,146688],{"xmlns":375},[377,146689,146690,146698],{},[380,146691,146692],{},[543,146693,146694,146696],{},[383,146695,1121],{},[383,146697,12313],{},[387,146699,146700],{"encoding":389},"x_i",[363,146702,146704],{"className":146703,"ariaHidden":394},[393],[363,146705,146707,146710],{"className":146706},[398],[363,146708],{"className":146709,"style":13724},[402],[363,146711,146713,146716],{"className":146712},[407],[363,146714,1121],{"className":146715},[407,408],[363,146717,146719],{"className":146718},[629],[363,146720,146722,146742],{"className":146721},[633,634],[363,146723,146725,146739],{"className":146724},[638],[363,146726,146728],{"className":146727,"style":12532},[642],[363,146729,146730,146733],{"style":730},[363,146731],{"className":146732,"style":651},[650],[363,146734,146736],{"className":146735},[655,656,657,658],[363,146737,12313],{"className":146738},[407,408,658],[363,146740,666],{"className":146741},[665],[363,146743,146745],{"className":146744},[638],[363,146746,146748],{"className":146747,"style":8316},[642],[363,146749],{}," 的边缘分布是 ",[363,146752,146754,146796],{"className":146753,"translate":367},[366],[363,146755,146757],{"className":146756},[371],[373,146758,146759],{"xmlns":375},[377,146760,146761,146793],{},[380,146762,146763,146765,146767,146769,146775,146777,146783,146785,146791],{},[383,146764,145808],{"mathvariant":557},[458,146766,4451],{},[458,146768,461],{"stretchy":460},[543,146770,146771,146773],{},[383,146772,45126],{},[383,146774,12313],{},[458,146776,898],{"separator":394},[543,146778,146779,146781],{},[383,146780,45126],{},[463,146782,474],{},[458,146784,5161],{},[543,146786,146787,146789],{},[383,146788,45126],{},[383,146790,12313],{},[458,146792,468],{"stretchy":460},[387,146794,146795],{"encoding":389},"\\operatorname{Beta}(\\alpha_i, \\alpha_0 - \\alpha_i)",[363,146797,146799,146909],{"className":146798,"ariaHidden":394},[393],[363,146800,146802,146805,146811,146814,146854,146857,146860,146900,146903,146906],{"className":146801},[398],[363,146803],{"className":146804,"style":487},[402],[363,146806,146808],{"className":146807},[1832],[363,146809,145808],{"className":146810},[407,33034],[363,146812,461],{"className":146813},[494],[363,146815,146817,146820],{"className":146816},[407],[363,146818,45126],{"className":146819,"style":45156},[407,408],[363,146821,146823],{"className":146822},[629],[363,146824,146826,146846],{"className":146825},[633,634],[363,146827,146829,146843],{"className":146828},[638],[363,146830,146832],{"className":146831,"style":12532},[642],[363,146833,146834,146837],{"style":136154},[363,146835],{"className":146836,"style":651},[650],[363,146838,146840],{"className":146839},[655,656,657,658],[363,146841,12313],{"className":146842},[407,408,658],[363,146844,666],{"className":146845},[665],[363,146847,146849],{"className":146848},[638],[363,146850,146852],{"className":146851,"style":8316},[642],[363,146853],{},[363,146855,898],{"className":146856},[920],[363,146858],{"className":146859,"style":754},[505],[363,146861,146863,146866],{"className":146862},[407],[363,146864,45126],{"className":146865,"style":45156},[407,408],[363,146867,146869],{"className":146868},[629],[363,146870,146872,146892],{"className":146871},[633,634],[363,146873,146875,146889],{"className":146874},[638],[363,146876,146878],{"className":146877,"style":11483},[642],[363,146879,146880,146883],{"style":136154},[363,146881],{"className":146882,"style":651},[650],[363,146884,146886],{"className":146885},[655,656,657,658],[363,146887,474],{"className":146888},[407,658],[363,146890,666],{"className":146891},[665],[363,146893,146895],{"className":146894},[638],[363,146896,146898],{"className":146897,"style":8316},[642],[363,146899],{},[363,146901],{"className":146902,"style":2935},[505],[363,146904,5161],{"className":146905},[2939],[363,146907],{"className":146908,"style":2935},[505],[363,146910,146912,146915,146955],{"className":146911},[398],[363,146913],{"className":146914,"style":487},[402],[363,146916,146918,146921],{"className":146917},[407],[363,146919,45126],{"className":146920,"style":45156},[407,408],[363,146922,146924],{"className":146923},[629],[363,146925,146927,146947],{"className":146926},[633,634],[363,146928,146930,146944],{"className":146929},[638],[363,146931,146933],{"className":146932,"style":12532},[642],[363,146934,146935,146938],{"style":136154},[363,146936],{"className":146937,"style":651},[650],[363,146939,146941],{"className":146940},[655,656,657,658],[363,146942,12313],{"className":146943},[407,408,658],[363,146945,666],{"className":146946},[665],[363,146948,146950],{"className":146949},[638],[363,146951,146953],{"className":146952,"style":8316},[642],[363,146954],{},[363,146956,468],{"className":146957},[501],"。这一点可由命题 1.4 直接看出：",[363,146960,146962,146996],{"className":146961,"translate":367},[366],[363,146963,146965],{"className":146964},[371],[373,146966,146967],{"xmlns":375},[377,146968,146969,146993],{},[380,146970,146971,146977,146979,146985,146987,146989,146991],{},[543,146972,146973,146975],{},[383,146974,1121],{},[383,146976,12313],{},[458,146978,471],{},[543,146980,146981,146983],{},[383,146982,121465],{},[383,146984,12313],{},[383,146986,1288],{"mathvariant":557},[458,146988,461],{"stretchy":460},[383,146990,7909],{},[458,146992,468],{"stretchy":460},[387,146994,146995],{"encoding":389},"x_i = \\gamma_i \u002F (s)",[363,146997,146999,147054],{"className":146998,"ariaHidden":394},[393],[363,147000,147002,147005,147045,147048,147051],{"className":147001},[398],[363,147003],{"className":147004,"style":13724},[402],[363,147006,147008,147011],{"className":147007},[407],[363,147009,1121],{"className":147010},[407,408],[363,147012,147014],{"className":147013},[629],[363,147015,147017,147037],{"className":147016},[633,634],[363,147018,147020,147034],{"className":147019},[638],[363,147021,147023],{"className":147022,"style":12532},[642],[363,147024,147025,147028],{"style":730},[363,147026],{"className":147027,"style":651},[650],[363,147029,147031],{"className":147030},[655,656,657,658],[363,147032,12313],{"className":147033},[407,408,658],[363,147035,666],{"className":147036},[665],[363,147038,147040],{"className":147039},[638],[363,147041,147043],{"className":147042,"style":8316},[642],[363,147044],{},[363,147046],{"className":147047,"style":506},[505],[363,147049,471],{"className":147050},[510],[363,147052],{"className":147053,"style":506},[505],[363,147055,147057,147060,147100,147103,147106,147109],{"className":147056},[398],[363,147058],{"className":147059,"style":487},[402],[363,147061,147063,147066],{"className":147062},[407],[363,147064,121465],{"className":147065,"style":121555},[407,408],[363,147067,147069],{"className":147068},[629],[363,147070,147072,147092],{"className":147071},[633,634],[363,147073,147075,147089],{"className":147074},[638],[363,147076,147078],{"className":147077,"style":12532},[642],[363,147079,147080,147083],{"style":142331},[363,147081],{"className":147082,"style":651},[650],[363,147084,147086],{"className":147085},[655,656,657,658],[363,147087,12313],{"className":147088},[407,408,658],[363,147090,666],{"className":147091},[665],[363,147093,147095],{"className":147094},[638],[363,147096,147098],{"className":147097,"style":8316},[642],[363,147099],{},[363,147101,1288],{"className":147102},[407],[363,147104,461],{"className":147105},[494],[363,147107,7909],{"className":147108},[407,408],[363,147110,468],{"className":147111},[501],"，而 ",[363,147114,147116,147150],{"className":147115,"translate":367},[366],[363,147117,147119],{"className":147118},[371],[373,147120,147121],{"xmlns":375},[377,147122,147123,147147],{},[380,147124,147125,147131,147133,147135,147137,147139,147145],{},[543,147126,147127,147129],{},[383,147128,121465],{},[383,147130,12313],{},[458,147132,33919],{},[458,147134,461],{"stretchy":460},[383,147136,7909],{},[458,147138,5161],{},[543,147140,147141,147143],{},[383,147142,121465],{},[383,147144,12313],{},[458,147146,468],{"stretchy":460},[387,147148,147149],{"encoding":389},"\\gamma_i \\perp (s - \\gamma_i)",[363,147151,147153,147208,147229],{"className":147152,"ariaHidden":394},[393],[363,147154,147156,147159,147199,147202,147205],{"className":147155},[398],[363,147157],{"className":147158,"style":403},[402],[363,147160,147162,147165],{"className":147161},[407],[363,147163,121465],{"className":147164,"style":121555},[407,408],[363,147166,147168],{"className":147167},[629],[363,147169,147171,147191],{"className":147170},[633,634],[363,147172,147174,147188],{"className":147173},[638],[363,147175,147177],{"className":147176,"style":12532},[642],[363,147178,147179,147182],{"style":142331},[363,147180],{"className":147181,"style":651},[650],[363,147183,147185],{"className":147184},[655,656,657,658],[363,147186,12313],{"className":147187},[407,408,658],[363,147189,666],{"className":147190},[665],[363,147192,147194],{"className":147193},[638],[363,147195,147197],{"className":147196,"style":8316},[642],[363,147198],{},[363,147200],{"className":147201,"style":506},[505],[363,147203,33919],{"className":147204},[510],[363,147206],{"className":147207,"style":506},[505],[363,147209,147211,147214,147217,147220,147223,147226],{"className":147210},[398],[363,147212],{"className":147213,"style":487},[402],[363,147215,461],{"className":147216},[494],[363,147218,7909],{"className":147219},[407,408],[363,147221],{"className":147222,"style":2935},[505],[363,147224,5161],{"className":147225},[2939],[363,147227],{"className":147228,"style":2935},[505],[363,147230,147232,147235,147275],{"className":147231},[398],[363,147233],{"className":147234,"style":487},[402],[363,147236,147238,147241],{"className":147237},[407],[363,147239,121465],{"className":147240,"style":121555},[407,408],[363,147242,147244],{"className":147243},[629],[363,147245,147247,147267],{"className":147246},[633,634],[363,147248,147250,147264],{"className":147249},[638],[363,147251,147253],{"className":147252,"style":12532},[642],[363,147254,147255,147258],{"style":142331},[363,147256],{"className":147257,"style":651},[650],[363,147259,147261],{"className":147260},[655,656,657,658],[363,147262,12313],{"className":147263},[407,408,658],[363,147265,666],{"className":147266},[665],[363,147268,147270],{"className":147269},[638],[363,147271,147273],{"className":147272,"style":8316},[642],[363,147274],{},[363,147276,468],{"className":147277},[501],[363,147279,147281,147298],{"className":147280,"translate":367},[366],[363,147282,147284],{"className":147283},[371],[373,147285,147286],{"xmlns":375},[377,147287,147288,147296],{},[380,147289,147290],{},[543,147291,147292,147294],{},[383,147293,1121],{},[383,147295,12313],{},[387,147297,146700],{"encoding":389},[363,147299,147301],{"className":147300,"ariaHidden":394},[393],[363,147302,147304,147307],{"className":147303},[398],[363,147305],{"className":147306,"style":13724},[402],[363,147308,147310,147313],{"className":147309},[407],[363,147311,1121],{"className":147312},[407,408],[363,147314,147316],{"className":147315},[629],[363,147317,147319,147339],{"className":147318},[633,634],[363,147320,147322,147336],{"className":147321},[638],[363,147323,147325],{"className":147324,"style":12532},[642],[363,147326,147327,147330],{"style":730},[363,147328],{"className":147329,"style":651},[650],[363,147331,147333],{"className":147332},[655,656,657,658],[363,147334,12313],{"className":147335},[407,408,658],[363,147337,666],{"className":147338},[665],[363,147340,147342],{"className":147341},[638],[363,147343,147345],{"className":147344,"style":8316},[642],[363,147346],{}," 是相互独立的两个 Gamma 变量之比，其分布为 Beta。",[11,147349,147350,147353,147354,147357,147358,147392,147393,147422,147423,10528,147508],{},[1344,147351,147352],{},"与多项分布的共轭关系。"," 迪利克雷分布之所以在贝叶斯统计中占据中心位置，在于它是多项分布（等价地，分类分布）的",[1344,147355,147356],{},"共轭先验","。其确切含义与算术后果见 2.4 节。此处只指出其直觉：若先验以「伪计数」",[363,147359,147361,147374],{"className":147360,"translate":367},[366],[363,147362,147364],{"className":147363},[371],[373,147365,147366],{"xmlns":375},[377,147367,147368,147372],{},[380,147369,147370],{},[383,147371,45126],{"mathvariant":136055},[387,147373,137074],{"encoding":389},[363,147375,147377],{"className":147376,"ariaHidden":394},[393],[363,147378,147380,147383],{"className":147379},[398],[363,147381],{"className":147382,"style":135662},[402],[363,147384,147386],{"className":147385},[407],[363,147387,147389],{"className":147388},[407],[363,147390,45126],{"className":147391},[407,136115]," 描述对各类别出现次数的初始信念，则观测到计数 ",[363,147394,147396,147410],{"className":147395,"translate":367},[366],[363,147397,147399],{"className":147398},[371],[373,147400,147401],{"xmlns":375},[377,147402,147403,147407],{},[380,147404,147405],{},[383,147406,82683],{"mathvariant":75598},[387,147408,147409],{"encoding":389},"\\mathbf{n}",[363,147411,147413],{"className":147412,"ariaHidden":394},[393],[363,147414,147416,147419],{"className":147415},[398],[363,147417],{"className":147418,"style":135662},[402],[363,147420,82683],{"className":147421},[407,75943]," 后，信念的更新只是 ",[363,147424,147426,147448],{"className":147425,"translate":367},[366],[363,147427,147429],{"className":147428},[371],[373,147430,147431],{"xmlns":375},[377,147432,147433,147445],{},[380,147434,147435,147437,147439,147441,147443],{},[383,147436,45126],{"mathvariant":136055},[458,147438,9032],{},[383,147440,45126],{"mathvariant":136055},[458,147442,2818],{},[383,147444,82683],{"mathvariant":75598},[387,147446,147447],{"encoding":389},"\\boldsymbol{\\alpha} \\mapsto \\boldsymbol{\\alpha} + \\mathbf{n}",[363,147449,147451,147475,147499],{"className":147450,"ariaHidden":394},[393],[363,147452,147454,147457,147466,147469,147472],{"className":147453},[398],[363,147455],{"className":147456,"style":84346},[402],[363,147458,147460],{"className":147459},[407],[363,147461,147463],{"className":147462},[407],[363,147464,45126],{"className":147465},[407,136115],[363,147467],{"className":147468,"style":506},[505],[363,147470,9032],{"className":147471},[510],[363,147473],{"className":147474,"style":506},[505],[363,147476,147478,147481,147490,147493,147496],{"className":147477},[398],[363,147479],{"className":147480,"style":4599},[402],[363,147482,147484],{"className":147483},[407],[363,147485,147487],{"className":147486},[407],[363,147488,45126],{"className":147489},[407,136115],[363,147491],{"className":147492,"style":2935},[505],[363,147494,2818],{"className":147495},[2939],[363,147497],{"className":147498,"style":2935},[505],[363,147500,147502,147505],{"className":147501},[398],[363,147503],{"className":147504,"style":135662},[402],[363,147506,82683],{"className":147507},[407,75943],[1344,147509,147510],{},"贝叶斯更新的全部计算复杂度，在这个共轭对中退化为一次向量加法。",[18,147512,147514],{"id":147513},"二解析性质","二、解析性质",[2659,147516,147518],{"id":147517},"_21-矩结构","2.1 矩结构",[143,147520,147521],{},[11,147522,147523,11410,147526,147602,147603,147646],{},[1344,147524,147525],{},"命题 2.1（期望、方差与协方差）",[363,147527,147529,147554],{"className":147528,"translate":367},[366],[363,147530,147532],{"className":147531},[371],[373,147533,147534],{"xmlns":375},[377,147535,147536,147552],{},[380,147537,147538,147540,147542,147544,147546,147548,147550],{},[383,147539,1121],{"mathvariant":75598},[458,147541,9217],{},[383,147543,137112],{"mathvariant":557},[458,147545,4451],{},[458,147547,461],{"stretchy":460},[383,147549,45126],{"mathvariant":136055},[458,147551,468],{"stretchy":460},[387,147553,137123],{"encoding":389},[363,147555,147557,147575],{"className":147556,"ariaHidden":394},[393],[363,147558,147560,147563,147566,147569,147572],{"className":147559},[398],[363,147561],{"className":147562,"style":135662},[402],[363,147564,1121],{"className":147565},[407,75943],[363,147567],{"className":147568,"style":506},[505],[363,147570,9217],{"className":147571},[510],[363,147573],{"className":147574,"style":506},[505],[363,147576,147578,147581,147587,147590,147599],{"className":147577},[398],[363,147579],{"className":147580,"style":487},[402],[363,147582,147584],{"className":147583},[1832],[363,147585,137112],{"className":147586},[407,33034],[363,147588,461],{"className":147589},[494],[363,147591,147593],{"className":147592},[407],[363,147594,147596],{"className":147595},[407],[363,147597,45126],{"className":147598},[407,136115],[363,147600,468],{"className":147601},[501],"。则对任意 ",[363,147604,147606,147624],{"className":147605,"translate":367},[366],[363,147607,147609],{"className":147608},[371],[373,147610,147611],{"xmlns":375},[377,147612,147613,147621],{},[380,147614,147615,147617,147619],{},[383,147616,12313],{},[458,147618,898],{"separator":394},[383,147620,100402],{},[387,147622,147623],{"encoding":389},"i, j",[363,147625,147627],{"className":147626,"ariaHidden":394},[393],[363,147628,147630,147634,147637,147640,147643],{"className":147629},[398],[363,147631],{"className":147632,"style":147633},[402],"height:0.854em;vertical-align:-0.1944em;",[363,147635,12313],{"className":147636},[407,408],[363,147638,898],{"className":147639},[920],[363,147641],{"className":147642,"style":754},[505],[363,147644,100402],{"className":147645,"style":100930},[407,408],[363,147647,147649,147845],{"className":147648,"translate":367},[366],[363,147650,147652],{"className":147651},[371],[373,147653,147654],{"xmlns":375},[377,147655,147656,147842],{},[380,147657,147658,147660,147662,147668,147670,147672,147686,147688,147690,147693,147695,147697,147703,147705,147707,147759,147761,147763,147766,147768,147770,147776,147778,147784,147786,147788,147830,147832,147834,147836,147838,147840],{},[383,147659,571],{"mathvariant":570},[458,147661,578],{"stretchy":460},[543,147663,147664,147666],{},[383,147665,1121],{},[383,147667,12313],{},[458,147669,605],{"stretchy":460},[458,147671,471],{},[586,147673,147674,147680],{},[543,147675,147676,147678],{},[383,147677,45126],{},[383,147679,12313],{},[543,147681,147682,147684],{},[383,147683,45126],{},[463,147685,474],{},[458,147687,898],{"separator":394},[505,147689],{"width":3610},[383,147691,147692],{"mathvariant":557},"Var",[458,147694,4451],{},[458,147696,461],{"stretchy":460},[543,147698,147699,147701],{},[383,147700,1121],{},[383,147702,12313],{},[458,147704,468],{"stretchy":460},[458,147706,471],{},[586,147708,147709,147735],{},[380,147710,147711,147717,147719,147725,147727,147733],{},[543,147712,147713,147715],{},[383,147714,45126],{},[383,147716,12313],{},[458,147718,461],{"stretchy":460},[543,147720,147721,147723],{},[383,147722,45126],{},[463,147724,474],{},[458,147726,5161],{},[543,147728,147729,147731],{},[383,147730,45126],{},[383,147732,12313],{},[458,147734,468],{"stretchy":460},[380,147736,147737,147745,147747,147753,147755,147757],{},[7901,147738,147739,147741,147743],{},[383,147740,45126],{},[463,147742,474],{},[463,147744,2811],{},[458,147746,461],{"stretchy":460},[543,147748,147749,147751],{},[383,147750,45126],{},[463,147752,474],{},[458,147754,2818],{},[463,147756,465],{},[458,147758,468],{"stretchy":460},[458,147760,898],{"separator":394},[505,147762],{"width":3610},[383,147764,147765],{"mathvariant":557},"Cov",[458,147767,4451],{},[458,147769,461],{"stretchy":460},[543,147771,147772,147774],{},[383,147773,1121],{},[383,147775,12313],{},[458,147777,898],{"separator":394},[543,147779,147780,147782],{},[383,147781,1121],{},[383,147783,100402],{},[458,147785,468],{"stretchy":460},[458,147787,471],{},[586,147789,147790,147806],{},[380,147791,147792,147794,147800],{},[458,147793,5161],{},[543,147795,147796,147798],{},[383,147797,45126],{},[383,147799,12313],{},[543,147801,147802,147804],{},[383,147803,45126],{},[383,147805,100402],{},[380,147807,147808,147816,147818,147824,147826,147828],{},[7901,147809,147810,147812,147814],{},[383,147811,45126],{},[463,147813,474],{},[463,147815,2811],{},[458,147817,461],{"stretchy":460},[543,147819,147820,147822],{},[383,147821,45126],{},[463,147823,474],{},[458,147825,2818],{},[463,147827,465],{},[458,147829,468],{"stretchy":460},[505,147831],{"width":4663},[458,147833,461],{"stretchy":460},[383,147835,12313],{},[458,147837,49879],{"mathvariant":557},[383,147839,100402],{},[458,147841,468],{"stretchy":460},[387,147843,147844],{"encoding":389},"\\mathbb{E}[x_i] = \\frac{\\alpha_i}{\\alpha_0}, \\qquad \\operatorname{Var}(x_i) = \\frac{\\alpha_i(\\alpha_0 - \\alpha_i)}{\\alpha_0^2(\\alpha_0 + 1)}, \\qquad \\operatorname{Cov}(x_i, x_j) = \\frac{-\\alpha_i \\alpha_j}{\\alpha_0^2(\\alpha_0 + 1)} \\quad (i \\ne 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的倒数的期望 ",[363,149613,149615,149659],{"className":149614,"translate":367},[366],[363,149616,149618],{"className":149617},[371],[373,149619,149620],{"xmlns":375},[377,149621,149622,149656],{},[380,149623,149624,149626,149628,149630,149632,149634,149636,149638,149640,149642,149644,149650,149652,149654],{},[383,149625,571],{"mathvariant":570},[458,149627,578],{"stretchy":460},[463,149629,465],{},[383,149631,1288],{"mathvariant":557},[383,149633,7909],{},[458,149635,605],{"stretchy":460},[458,149637,471],{},[463,149639,465],{},[383,149641,1288],{"mathvariant":557},[458,149643,461],{"stretchy":460},[543,149645,149646,149648],{},[383,149647,45126],{},[463,149649,474],{},[458,149651,5161],{},[463,149653,465],{},[458,149655,468],{"stretchy":460},[387,149657,149658],{"encoding":389},"\\mathbb{E}[1\u002Fs] = 1\u002F(\\alpha_0 - 1)",[363,149660,149662,149692,149753],{"className":149661,"ariaHidden":394},[393],[363,149663,149665,149668,149671,149674,149677,149680,149683,149686,149689],{"className":149664},[398],[363,149666],{"className":149667,"style":487},[402],[363,149669,571],{"className":149670},[407,714],[363,149672,578],{"className":149673},[494],[363,149675,8937],{"className":149676},[407],[363,149678,7909],{"className":149679},[407,408],[363,149681,605],{"className":149682},[501],[363,149684],{"className":149685,"style":506},[505],[363,149687,471],{"className":149688},[510],[363,149690],{"className":149691,"style":506},[505],[363,149693,149695,149698,149701,149704,149744,149747,149750],{"className":149694},[398],[363,149696],{"className":149697,"style":487},[402],[363,149699,8937],{"className":149700},[407],[363,149702,461],{"className":149703},[494],[363,149705,149707,149710],{"className":149706},[407],[363,149708,45126],{"className":149709,"style":45156},[407,408],[363,149711,149713],{"className":149712},[629],[363,149714,149716,149736],{"className":149715},[633,634],[363,149717,149719,149733],{"className":149718},[638],[363,149720,149722],{"className":149721,"style":11483},[642],[363,149723,149724,149727],{"style":136154},[363,149725],{"className":149726,"style":651},[650],[363,149728,149730],{"className":149729},[655,656,657,658],[363,149731,474],{"className":149732},[407,658],[363,149734,666],{"className":149735},[665],[363,149737,149739],{"className":149738},[638],[363,149740,149742],{"className":149741,"style":8316},[642],[363,149743],{},[363,149745],{"className":149746,"style":2935},[505],[363,149748,5161],{"className":149749},[2939],[363,149751],{"className":149752,"style":2935},[505],[363,149754,149756,149759,149762],{"className":149755},[398],[363,149757],{"className":149758,"style":487},[402],[363,149760,465],{"className":149761},[407],[363,149763,468],{"className":149764},[501],[363,149766,149768,149798],{"className":149767,"translate":367},[366],[363,149769,149771],{"className":149770},[371],[373,149772,149773],{"xmlns":375},[377,149774,149775,149795],{},[380,149776,149777,149783,149785,149791,149793],{},[543,149778,149779,149781],{},[383,149780,45126],{},[463,149782,474],{},[458,149784,9032],{},[543,149786,149787,149789],{},[383,149788,45126],{},[463,149790,474],{},[458,149792,5161],{},[463,149794,465],{},[387,149796,149797],{"encoding":389},"\\alpha_0 \\mapsto \\alpha_0 - 1",[363,149799,149801,149857,149912],{"className":149800,"ariaHidden":394},[393],[363,149802,149804,149808,149848,149851,149854],{"className":149803},[398],[363,149805],{"className":149806,"style":149807},[402],"height:0.661em;vertical-align:-0.15em;",[363,149809,149811,149814],{"className":149810},[407],[363,149812,45126],{"className":149813,"style":45156},[407,408],[363,149815,149817],{"className":149816},[629],[363,149818,149820,149840],{"className":149819},[633,634],[363,149821,149823,149837],{"className":149822},[638],[363,149824,149826],{"className":149825,"style":11483},[642],[363,149827,149828,149831],{"style":136154},[363,149829],{"className":149830,"style":651},[650],[363,149832,149834],{"className":149833},[655,656,657,658],[363,149835,474],{"className":149836},[407,658],[363,149838,666],{"className":149839},[665],[363,149841,149843],{"className":149842},[638],[363,149844,149846],{"className":149845,"style":8316},[642],[363,149847],{},[363,149849],{"className":149850,"style":506},[505],[363,149852,9032],{"className":149853},[510],[363,149855],{"className":149856,"style":506},[505],[363,149858,149860,149863,149903,149906,149909],{"className":149859},[398],[363,149861],{"className":149862,"style":145471},[402],[363,149864,149866,149869],{"className":149865},[407],[363,149867,45126],{"className":149868,"style":45156},[407,408],[363,149870,149872],{"className":149871},[629],[363,149873,149875,149895],{"className":149874},[633,634],[363,149876,149878,149892],{"className":149877},[638],[363,149879,149881],{"className":149880,"style":11483},[642],[363,149882,149883,149886],{"style":136154},[363,149884],{"className":149885,"style":651},[650],[363,149887,149889],{"className":149888},[655,656,657,658],[363,149890,474],{"className":149891},[407,658],[363,149893,666],{"className":149894},[665],[363,149896,149898],{"className":149897},[638],[363,149899,149901],{"className":149900,"style":8316},[642],[363,149902],{},[363,149904],{"className":149905,"style":2935},[505],[363,149907,5161],{"className":149908},[2939],[363,149910],{"className":149911,"style":2935},[505],[363,149913,149915,149918],{"className":149914},[398],[363,149916],{"className":149917,"style":520},[402],[363,149919,465],{"className":149920},[407]," 的平移技巧，或直接由命题 1.3 的矩恒等式 ",[363,149923,149925,149985],{"className":149924,"translate":367},[366],[363,149926,149928],{"className":149927},[371],[373,149929,149930],{"xmlns":375},[377,149931,149932,149982],{},[380,149933,149934,149936,149938,149944,149956,149958,149960,149962,149964,149966,149968,149970,149972,149974,149976,149978,149980],{},[383,149935,571],{"mathvariant":570},[458,149937,578],{"stretchy":460},[543,149939,149940,149942],{},[458,149941,136684],{},[383,149943,12313],{},[7901,149945,149946,149948,149950],{},[383,149947,1121],{},[383,149949,12313],{},[543,149951,149952,149954],{},[383,149953,84504],{},[383,149955,12313],{},[458,149957,605],{"stretchy":460},[458,149959,471],{},[383,149961,39663],{},[458,149963,461],{"stretchy":460},[383,149965,45126],{"mathvariant":136055},[458,149967,2818],{},[383,149969,84504],{"mathvariant":136055},[458,149971,468],{"stretchy":460},[383,149973,1288],{"mathvariant":557},[383,149975,39663],{},[458,149977,461],{"stretchy":460},[383,149979,45126],{"mathvariant":136055},[458,149981,468],{"stretchy":460},[387,149983,149984],{"encoding":389},"\\mathbb{E}[\\prod_i x_i^{\\beta_i}] = B(\\boldsymbol{\\alpha} + \\boldsymbol{\\beta})\u002FB(\\boldsymbol{\\alpha})",[363,149986,149988,150150,150180],{"className":149987,"ariaHidden":394},[393],[363,149989,149991,149995,149998,150001,150041,150044,150138,150141,150144,150147],{"className":149990},[398],[363,149992],{"className":149993,"style":149994},[402],"height:1.2667em;vertical-align:-0.2997em;",[363,149996,571],{"className":149997},[407,714],[363,149999,578],{"className":150000},[494],[363,150002,150004,150007],{"className":150003},[1832],[363,150005,136684],{"className":150006,"style":40032},[1832,1833,5964],[363,150008,150010],{"className":150009},[629],[363,150011,150013,150033],{"className":150012},[633,634],[363,150014,150016,150030],{"className":150015},[638],[363,150017,150019],{"className":150018,"style":40045},[642],[363,150020,150021,150024],{"style":40048},[363,150022],{"className":150023,"style":651},[650],[363,150025,150027],{"className":150026},[655,656,657,658],[363,150028,12313],{"className":150029},[407,408,658],[363,150031,666],{"className":150032},[665],[363,150034,150036],{"className":150035},[638],[363,150037,150039],{"className":150038,"style":40067},[642],[363,150040],{},[363,150042],{"className":150043,"style":754},[505],[363,150045,150047,150050],{"className":150046},[407],[363,150048,1121],{"className":150049},[407,408],[363,150051,150053],{"className":150052},[629],[363,150054,150056,150130],{"className":150055},[633,634],[363,150057,150059,150127],{"className":150058},[638],[363,150060,150063,150074],{"className":150061,"style":150062},[642],"height:0.967em;",[363,150064,150065,150068],{"style":136947},[363,150066],{"className":150067,"style":651},[650],[363,150069,150071],{"className":150070},[655,656,657,658],[363,150072,12313],{"className":150073},[407,408,658],[363,150075,150077,150080],{"style":150076},"top:-3.1809em;margin-right:0.05em;",[363,150078],{"className":150079,"style":651},[650],[363,150081,150083],{"className":150082},[655,656,657,658],[363,150084,150086],{"className":150085},[407,658],[363,150087,150089,150092],{"className":150088},[407,658],[363,150090,84504],{"className":150091,"style":84726},[407,408,658],[363,150093,150095],{"className":150094},[629],[363,150096,150098,150119],{"className":150097},[633,634],[363,150099,150101,150116],{"className":150100},[638],[363,150102,150104],{"className":150103,"style":40120},[642],[363,150105,150107,150110],{"style":150106},"top:-2.357em;margin-left:-0.0528em;margin-right:0.0714em;",[363,150108],{"className":150109,"style":30578},[650],[363,150111,150113],{"className":150112},[655,30582,10022,658],[363,150114,12313],{"className":150115},[407,408,658],[363,150117,666],{"className":150118},[665],[363,150120,150122],{"className":150121},[638],[363,150123,150125],{"className":150124,"style":40142},[642],[363,150126],{},[363,150128,666],{"className":150129},[665],[363,150131,150133],{"className":150132},[638],[363,150134,150136],{"className":150135,"style":137025},[642],[363,150137],{},[363,150139,605],{"className":150140},[501],[363,150142],{"className":150143,"style":506},[505],[363,150145,471],{"className":150146},[510],[363,150148],{"className":150149,"style":506},[505],[363,150151,150153,150156,150159,150162,150171,150174,150177],{"className":150152},[398],[363,150154],{"className":150155,"style":487},[402],[363,150157,39663],{"className":150158,"style":39759},[407,408],[363,150160,461],{"className":150161},[494],[363,150163,150165],{"className":150164},[407],[363,150166,150168],{"className":150167},[407],[363,150169,45126],{"className":150170},[407,136115],[363,150172],{"className":150173,"style":2935},[505],[363,150175,2818],{"className":150176},[2939],[363,150178],{"className":150179,"style":2935},[505],[363,150181,150183,150186,150196,150199,150202,150205,150208,150217],{"className":150182},[398],[363,150184],{"className":150185,"style":487},[402],[363,150187,150189],{"className":150188},[407],[363,150190,150192],{"className":150191},[407],[363,150193,84504],{"className":150194,"style":150195},[407,136115],"margin-right:0.034em;",[363,150197,468],{"className":150198},[501],[363,150200,1288],{"className":150201},[407],[363,150203,39663],{"className":150204,"style":39759},[407,408],[363,150206,461],{"className":150207},[494],[363,150209,150211],{"className":150210},[407],[363,150212,150214],{"className":150213},[407],[363,150215,45126],{"className":150216},[407,136115],[363,150218,468],{"className":150219},[501],[363,150221,150223,150245],{"className":150222,"translate":367},[366],[363,150224,150226],{"className":150225},[371],[373,150227,150228],{"xmlns":375},[377,150229,150230,150242],{},[380,150231,150232,150238,150240],{},[543,150233,150234,150236],{},[383,150235,84504],{},[383,150237,12313],{},[458,150239,471],{},[463,150241,465],{},[387,150243,150244],{"encoding":389},"\\beta_i = 1",[363,150246,150248,150304],{"className":150247,"ariaHidden":394},[393],[363,150249,150251,150254,150295,150298,150301],{"className":150250},[398],[363,150252],{"className":150253,"style":403},[402],[363,150255,150257,150260],{"className":150256},[407],[363,150258,84504],{"className":150259,"style":84726},[407,408],[363,150261,150263],{"className":150262},[629],[363,150264,150266,150287],{"className":150265},[633,634],[363,150267,150269,150284],{"className":150268},[638],[363,150270,150272],{"className":150271,"style":12532},[642],[363,150273,150275,150278],{"style":150274},"top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;",[363,150276],{"className":150277,"style":651},[650],[363,150279,150281],{"className":150280},[655,656,657,658],[363,150282,12313],{"className":150283},[407,408,658],[363,150285,666],{"className":150286},[665],[363,150288,150290],{"className":150289},[638],[363,150291,150293],{"className":150292,"style":8316},[642],[363,150294],{},[363,150296],{"className":150297,"style":506},[505],[363,150299,471],{"className":150300},[510],[363,150302],{"className":150303,"style":506},[505],[363,150305,150307,150310],{"className":150306},[398],[363,150308],{"className":150309,"style":520},[402],[363,150311,465],{"className":150312},[407]," 得到。二阶矩由 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= \\frac{\\alpha_i(\\alpha_i+1)}{\\alpha_0(\\alpha_0+1)}",[363,150392,150394,150472],{"className":150393,"ariaHidden":394},[393],[363,150395,150397,150401,150404,150407,150460,150463,150466,150469],{"className":150396},[398],[363,150398],{"className":150399,"style":150400},[402],"height:1.0728em;vertical-align:-0.2587em;",[363,150402,571],{"className":150403},[407,714],[363,150405,578],{"className":150406},[494],[363,150408,150410,150413],{"className":150409},[407],[363,150411,1121],{"className":150412},[407,408],[363,150414,150416],{"className":150415},[629],[363,150417,150419,150451],{"className":150418},[633,634],[363,150420,150422,150448],{"className":150421},[638],[363,150423,150425,150437],{"className":150424,"style":20400},[642],[363,150426,150428,150431],{"style":150427},"top:-2.4413em;margin-left:0em;margin-right:0.05em;",[363,150429],{"className":150430,"style":651},[650],[363,150432,150434],{"className":150433},[655,656,657,658],[363,150435,12313],{"className":150436},[407,408,658],[363,150438,150439,150442],{"style":7213},[363,150440],{"className":150441,"style":651},[650],[363,150443,150445],{"className":150444},[655,656,657,658],[363,150446,2811],{"className":150447},[407,658],[363,150449,666],{"className":150450},[665],[363,150452,150454],{"className":150453},[638],[363,150455,150458],{"className":150456,"style":150457},[642],"height:0.2587em;",[363,150459],{},[363,150461,605],{"className":150462},[501],[363,150464],{"className":150465,"style":506},[505],[363,150467,471],{"className":150468},[510],[363,150470],{"className":150471,"style":506},[505],[363,150473,150475,150479],{"className":150474},[398],[363,150476],{"className":150477,"style":150478},[402],"height:1.53em;vertical-align:-0.52em;",[363,150480,150482,150485,150722],{"className":150481},[407],[363,150483],{"className":150484},[494,789],[363,150486,150488],{"className":150487},[586],[363,150489,150491,150714],{"className":150490},[633,634],[363,150492,150494,150711],{"className":150493},[638],[363,150495,150497,150600,150608],{"className":150496,"style":26765},[642],[363,150498,150499,150502],{"style":2875},[363,150500],{"className":150501,"style":809},[650],[363,150503,150505],{"className":150504},[655,656,657,658],[363,150506,150508,150548,150551,150591,150594,150597],{"className":150507},[407,658],[363,150509,150511,150514],{"className":150510},[407,658],[363,150512,45126],{"className":150513,"style":45156},[407,408,658],[363,150515,150517],{"className":150516},[629],[363,150518,150520,150540],{"className":150519},[633,634],[363,150521,150523,150537],{"className":150522},[638],[363,150524,150526],{"className":150525,"style":137351},[642],[363,150527,150528,150531],{"style":136989},[363,150529],{"className":150530,"style":30578},[650],[363,150532,150534],{"className":150533},[655,30582,10022,658],[363,150535,474],{"className":150536},[407,658],[363,150538,666],{"className":150539},[665],[363,150541,150543],{"className":150542},[638],[363,150544,150546],{"className":150545,"style":40142},[642],[363,150547],{},[363,150549,461],{"className":150550},[494,658],[363,150552,150554,150557],{"className":150553},[407,658],[363,150555,45126],{"className":150556,"style":45156},[407,408,658],[363,150558,150560],{"className":150559},[629],[363,150561,150563,150583],{"className":150562},[633,634],[363,150564,150566,150580],{"className":150565},[638],[363,150567,150569],{"className":150568,"style":137351},[642],[363,150570,150571,150574],{"style":136989},[363,150572],{"className":150573,"style":30578},[650],[363,150575,150577],{"className":150576},[655,30582,10022,658],[363,150578,474],{"className":150579},[407,658],[363,150581,666],{"className":150582},[665],[363,150584,150586],{"className":150585},[638],[363,150587,150589],{"className":150588,"style":40142},[642],[363,150590],{},[363,150592,2818],{"className":150593},[2939,658],[363,150595,465],{"className":150596},[407,658],[363,150598,468],{"className":150599},[501,658],[363,150601,150602,150605],{"style":821},[363,150603],{"className":150604,"style":809},[650],[363,150606],{"className":150607,"style":829},[828],[363,150609,150610,150613],{"style":26790},[363,150611],{"className":150612,"style":809},[650],[363,150614,150616],{"className":150615},[655,656,657,658],[363,150617,150619,150659,150662,150702,150705,150708],{"className":150618},[407,658],[363,150620,150622,150625],{"className":150621},[407,658],[363,150623,45126],{"className":150624,"style":45156},[407,408,658],[363,150626,150628],{"className":150627},[629],[363,150629,150631,150651],{"className":150630},[633,634],[363,150632,150634,150648],{"className":150633},[638],[363,150635,150637],{"className":150636,"style":40120},[642],[363,150638,150639,150642],{"style":136989},[363,150640],{"className":150641,"style":30578},[650],[363,150643,150645],{"className":150644},[655,30582,10022,658],[363,150646,12313],{"className":150647},[407,408,658],[363,150649,666],{"className":150650},[665],[363,150652,150654],{"className":150653},[638],[363,1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x_j] = \\frac{\\alpha_i \\alpha_j}{\\alpha_0(\\alpha_0+1)}",[363,150799,150801,150905],{"className":150800,"ariaHidden":394},[393],[363,150802,150804,150807,150810,150813,150853,150893,150896,150899,150902],{"className":150803},[398],[363,150805],{"className":150806,"style":618},[402],[363,150808,571],{"className":150809},[407,714],[363,150811,578],{"className":150812},[494],[363,150814,150816,150819],{"className":150815},[407],[363,150817,1121],{"className":150818},[407,408],[363,150820,150822],{"className":150821},[629],[363,150823,150825,150845],{"className":150824},[633,634],[363,150826,150828,150842],{"className":150827},[638],[363,150829,150831],{"className":150830,"style":12532},[642],[363,150832,150833,150836],{"style":730},[363,150834],{"className":150835,"style":651},[650],[363,150837,150839],{"className":150838},[655,656,657,658],[363,150840,12313],{"className":150841},[407,408,658],[363,150843,666],{"className":150844},[665],[363,150846,150848],{"className":150847},[638],[363,150849,150851],{"className":150850,"style":8316},[642],[363,150852],{},[363,150854,150856,150859],{"className":150855},[407],[363,150857,1121],{"className":150858},[407,408],[363,150860,150862],{"className":150861},[629],[363,150863,150865,150885],{"className":150864},[633,634],[363,150866,150868,150882],{"className":150867},[638],[363,150869,150871],{"className":150870,"style":12532},[642],[363,150872,150873,150876],{"style":730},[363,150874],{"className":150875,"style":651},[650],[363,150877,150879],{"className":150878},[655,656,657,658],[363,150880,100402],{"className":150881,"style":100930},[407,408,658],[363,150883,666],{"className":150884},[665],[363,150886,150888],{"className":150887},[638],[363,150889,150891],{"className":150890,"style":673},[642],[363,150892],{},[363,150894,605],{"className":150895},[501],[363,150897],{"className":150898,"style":506},[505],[363,150900,471],{"className":150901},[510],[363,150903],{"className":150904,"style":506},[505],[363,150906,150908,150912],{"className":150907},[398],[363,150909],{"className":150910,"style":150911},[402],"height:1.3287em;vertical-align:-0.52em;",[363,150913,150915,150918,151144],{"className":150914},[407],[363,150916],{"className":150917},[494,789],[363,150919,150921],{"className":150920},[586],[363,150922,150924,151136],{"className":150923},[633,634],[363,150925,150927,151133],{"className":150926},[638],[363,150928,150931,151034,151042],{"className":150929,"style":150930},[642],"height:0.8087em;",[363,150932,150933,150936],{"style":2875},[363,150934],{"className":150935,"style":809},[650],[363,150937,150939],{"className":150938},[655,656,657,658],[363,150940,150942,150982,150985,151025,151028,151031],{"className":150941},[407,658],[363,150943,150945,150948],{"className":150944},[407,658],[363,150946,45126],{"className":150947,"style":45156},[407,408,658],[363,150949,150951],{"className":150950},[629],[363,150952,150954,150974],{"className":150953},[633,634],[363,150955,150957,150971],{"className":150956},[638],[363,150958,150960],{"className":150959,"style":137351},[642],[363,150961,150962,150965],{"style":136989},[363,150963],{"className":150964,"style":30578},[650],[363,150966,150968],{"className":150967},[655,30582,10022,658],[363,150969,474],{"className":150970},[407,658],[363,150972,666],{"className":150973},[665],[363,150975,150977],{"className":150976},[638],[363,150978,150980],{"className":150979,"style":40142},[642],[363,150981],{},[363,150983,461],{"className":150984},[494,658],[363,150986,150988,150991],{"className":150987},[407,658],[363,150989,45126],{"className":150990,"style":45156},[407,408,658],[363,150992,150994],{"className":150993},[629],[363,150995,150997,151017],{"className":150996},[633,634],[363,150998,151000,151014],{"className":150999},[638],[363,151001,151003],{"className":151002,"style":137351},[642],[363,151004,151005,151008],{"style":136989},[363,151006],{"className":151007,"style":30578},[650],[363,151009,151011],{"className":151010},[655,30582,10022,658],[363,151012,474],{"className":151013},[407,658],[363,151015,666],{"className":151016},[665],[363,151018,151020],{"className":151019},[638],[363,151021,151023],{"className":151022,"style":40142},[642],[363,151024],{},[363,151026,2818],{"className":151027},[2939,658],[363,151029,465],{"className":151030},[407,658],[363,151032,468],{"className":151033},[501,658],[363,151035,151036,151039],{"style":821},[363,151037],{"className":151038,"style":809},[650],[363,151040],{"className":151041,"style":829},[828],[363,151043,151044,151047],{"style":148705},[363,151045],{"className":151046,"style":809},[650],[363,151048,151050],{"className":151049},[655,656,657,658],[363,151051,151053,151093],{"className":151052},[407,658],[363,151054,151056,151059],{"className":151055},[407,658],[363,151057,45126],{"className":151058,"style":45156},[407,408,658],[363,151060,151062],{"className":151061},[629],[363,151063,151065,151085],{"className":151064},[633,634],[363,151066,151068,151082],{"className":151067},[638],[363,151069,1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相减即得。",[11,151149,151150,151151,151345],{},"上述结果可写成更紧凑的矩阵形式。记 ",[363,151152,151154,151198],{"className":151153,"translate":367},[366],[363,151155,151157],{"className":151156},[371],[373,151158,151159],{"xmlns":375},[377,151160,151161,151195],{},[380,151162,151163,151165,151167,151169,151171,151177,151179,151181,151183],{},[383,151164,2712],{"mathvariant":136055},[458,151166,471],{},[383,151168,45126],{"mathvariant":136055},[383,151170,1288],{"mathvariant":557},[543,151172,151173,151175],{},[383,151174,45126],{},[463,151176,474],{},[458,151178,6257],{},[383,151180,134509],{"mathvariant":557},[458,151182,4451],{},[7162,151184,151185,151187],{},[383,151186,126067],{"mathvariant":557},[380,151188,151189,151191,151193],{},[383,151190,11044],{},[458,151192,5161],{},[463,151194,465],{},[387,151196,151197],{"encoding":389},"\\boldsymbol{\\mu} = \\boldsymbol{\\alpha}\u002F\\alpha_0 \\in \\operatorname{int}\\Delta^{K-1}",[363,151199,151201,151225,151292],{"className":151200,"ariaHidden":394},[393],[363,151202,151204,151207,151216,151219,151222],{"className":151203},[398],[363,151205],{"className":151206,"style":133807},[402],[363,151208,151210],{"className":151209},[407],[363,151211,151213],{"className":151212},[407],[363,151214,2712],{"className":151215},[407,136115],[363,151217],{"className":151218,"style":506},[505],[363,151220,471],{"className":151221},[510],[363,151223],{"className":151224,"style":506},[505],[363,151226,151228,151231,151240,151243,151283,151286,151289],{"className":151227},[398],[363,151229],{"className":151230,"style":487},[402],[363,151232,151234],{"className":151233},[407],[363,151235,151237],{"className":151236},[407],[363,151238,45126],{"className":151239},[407,136115],[363,151241,1288],{"className":151242},[407],[363,151244,151246,151249],{"className":151245},[407],[363,151247,45126],{"className":151248,"style":45156},[407,408],[363,151250,151252],{"className":151251},[629],[363,151253,151255,151275],{"className":151254},[633,634],[363,151256,151258,151272],{"className":151257},[638],[363,151259,151261],{"className":151260,"style":11483},[642],[363,151262,151263,151266],{"style":136154},[363,151264],{"className":151265,"style":651},[650],[363,151267,151269],{"className":151268},[655,656,657,658],[363,151270,474],{"className":151271},[407,658],[363,151273,666],{"className":151274},[665],[363,151276,151278],{"className":151277},[638],[363,151279,151281],{"className":151280,"style":8316},[642],[363,151282],{},[363,151284],{"className":151285,"style":506},[505],[363,151287,6257],{"className":151288},[510],[363,151290],{"className":151291,"style":506},[505],[363,151293,151295,151298,151304,151307],{"className":151294},[398],[363,151296],{"className":151297,"style":52457},[402],[363,151299,151301],{"className":151300},[1832],[363,151302,134509],{"className":151303},[407,33034],[363,151305],{"className":151306,"style":754},[505],[363,151308,151310,151313],{"className":151309},[407],[363,151311,126067],{"className":151312},[407],[363,151314,151316],{"className":151315},[629],[363,151317,151319],{"className":151318},[633],[363,151320,151322],{"className":151321},[638],[363,151323,151325],{"className":151324,"style":52457},[642],[363,151326,151327,151330],{"style":7213},[363,151328],{"className":151329,"style":651},[650],[363,151331,151333],{"className":151332},[655,656,657,658],[363,151334,151336,151339,151342],{"className":151335},[407,658],[363,151337,11044],{"className":151338,"style":11132},[407,408,658],[363,151340,5161],{"className":151341},[2939,658],[363,151343,465],{"className":151344},[407,658]," 为均值向量，则",[363,151347,151349],{"className":151348,"translate":367},[528],[363,151350,151352,151425],{"className":151351,"translate":367},[366],[363,151353,151355],{"className":151354},[371],[373,151356,151357],{"xmlns":375,"display":537},[377,151358,151359,151422],{},[380,151360,151361,151364,151366,151368,151370,151372,151374,151376,151378,151394],{},[383,151362,151363],{"mathvariant":75598},"Σ",[458,151365,471],{},[383,151367,147765],{"mathvariant":557},[458,151369,4451],{},[458,151371,461],{"stretchy":460},[383,151373,1121],{"mathvariant":75598},[458,151375,468],{"stretchy":460},[458,151377,471],{},[586,151379,151380,151382],{},[463,151381,465],{},[380,151383,151384,151390,151392],{},[543,151385,151386,151388],{},[383,151387,45126],{},[463,151389,474],{},[458,151391,2818],{},[463,151393,465],{},[380,151395,151396,151398,151401,151403,151405,151407,151409,151411,151413,151420],{},[458,151397,461],{"fence":394},[383,151399,151400],{"mathvariant":557},"diag",[458,151402,4451],{},[458,151404,461],{"stretchy":460},[383,151406,2712],{"mathvariant":136055},[458,151408,468],{"stretchy":460},[458,151410,5161],{},[383,151412,2712],{"mathvariant":136055},[7162,151414,151415,151417],{},[383,151416,2712],{"mathvariant":136055},[383,151418,151419],{"mathvariant":557},"⊤",[458,151421,468],{"fence":394},[387,151423,151424],{"encoding":389},"\\boldsymbol{\\Sigma} = \\operatorname{Cov}(\\mathbf{x}) = \\frac{1}{\\alpha_0 + 1}\\left(\\operatorname{diag}(\\boldsymbol{\\mu}) - \\boldsymbol{\\mu}\\boldsymbol{\\mu}^\\top\\right)",[363,151426,151428,151453,151483],{"className":151427,"ariaHidden":394},[393],[363,151429,151431,151435,151444,151447,151450],{"className":151430},[398],[363,151432],{"className":151433,"style":151434},[402],"height:0.6861em;",[363,151436,151438],{"className":151437},[407],[363,151439,151441],{"className":151440},[407],[363,151442,151363],{"className":151443},[407,75943],[363,151445],{"className":151446,"style":506},[505],[363,151448,471],{"className":151449},[510],[363,151451],{"className":151452,"style":506},[505],[363,151454,151456,151459,151465,151468,151471,151474,151477,151480],{"className":151455},[398],[363,151457],{"className":151458,"style":487},[402],[363,151460,151462],{"className":151461},[1832],[363,151463,147765],{"className":151464,"style":4505},[407,33034],[363,151466,461],{"className":151467},[494],[363,151469,1121],{"className":151470},[407,75943],[363,151472,468],{"className":151473},[501],[363,151475],{"className":151476,"style":506},[505],[363,151478,471],{"className":151479},[510],[363,151481],{"className":151482,"style":506},[505],[363,151484,151486,151490,151601,151604],{"className":151485},[398],[363,151487],{"className":151488,"style":151489},[402],"height:2.1574em;vertical-align:-0.836em;",[363,151491,151493,151496,151598],{"className":151492},[407],[363,151494],{"className":151495},[494,789],[363,151497,151499],{"className":151498},[586],[363,151500,151502,151590],{"className":151501},[633,634],[363,151503,151505,151587],{"className":151504},[638],[363,151506,151508,151568,151576],{"className":151507,"style":23261},[642],[363,151509,151510,151513],{"style":805},[363,151511],{"className":151512,"style":809},[650],[363,151514,151516,151556,151559,151562,151565],{"className":151515},[407],[363,151517,151519,151522],{"className":151518},[407],[363,151520,45126],{"className":151521,"style":45156},[407,408],[363,151523,151525],{"className":151524},[629],[363,151526,151528,151548],{"className":151527},[633,634],[363,151529,151531,151545],{"className":151530},[638],[363,151532,151534],{"className":151533,"style":11483},[642],[363,151535,151536,151539],{"style":136154},[363,151537],{"className":151538,"style":651},[650],[363,151540,151542],{"className":151541},[655,656,657,658],[363,151543,474],{"className":151544},[407,658],[363,151546,666],{"className":151547},[665],[363,151549,151551],{"className":151550},[638],[363,151552,151554],{"className":151553,"style":8316},[642],[363,151555],{},[363,151557],{"className":151558,"style":2935},[505],[363,151560,2818],{"className":151561},[2939],[363,151563],{"className":151564,"style":2935},[505],[363,151566,465],{"className":151567},[407],[363,151569,151570,151573],{"style":821},[363,151571],{"className":151572,"style":809},[650],[363,151574],{"className":151575,"style":829},[828],[363,151577,151578,151581],{"style":832},[363,151579],{"className":151580,"style":809},[650],[363,151582,151584],{"className":151583},[407],[363,151585,465],{"className":151586},[407],[363,151588,666],{"className":151589},[665],[363,151591,151593],{"className":151592},[638],[363,151594,151596],{"className":151595,"style":125439},[642],[363,151597],{},[363,151599],{"className":151600},[501,789],[363,151602],{"className":151603,"style":754},[505],[363,151605,151607,151613,151619,151622,151631,151634,151637,151640,151643,151652,151687],{"className":151606},[758],[363,151608,151610],{"className":151609,"style":763},[494,762],[363,151611,461],{"className":151612},[767,10022],[363,151614,151616],{"className":151615},[1832],[363,151617,151400],{"className":151618,"style":4505},[407,33034],[363,151620,461],{"className":151621},[494],[363,151623,151625],{"className":151624},[407],[363,151626,151628],{"className":151627},[407],[363,151629,2712],{"className":151630},[407,136115],[363,151632,468],{"className":151633},[501],[363,151635],{"className":151636,"style":2935},[505],[363,151638,5161],{"className":151639},[2939],[363,151641],{"className":151642,"style":2935},[505],[363,151644,151646],{"className":151645},[407],[363,151647,151649],{"className":151648},[407],[363,151650,2712],{"className":151651},[407,136115],[363,151653,151655,151664],{"className":151654},[407],[363,151656,151658],{"className":151657},[407],[363,151659,151661],{"className":151660},[407],[363,151662,2712],{"className":151663},[407,136115],[363,151665,151667],{"className":151666},[629],[363,151668,151670],{"className":151669},[633],[363,151671,151673],{"className":151672},[638],[363,151674,151676],{"className":151675,"style":48236},[642],[363,151677,151678,151681],{"style":8625},[363,151679],{"className":151680,"style":651},[650],[363,151682,151684],{"className":151683},[655,656,657,658],[363,151685,151419],{"className":151686},[407,658],[363,151688,151690],{"className":151689,"style":763},[501,762],[363,151691,468],{"className":151692},[767,10022],[11,151694,151695,151696,151746,151747,3093,151844,151879,151880,151909,151910,151970,151971,152212,152213,152241],{},"这一形式揭示了两个结构性事实。第一，协方差矩阵的秩为 ",[363,151697,151699,151716],{"className":151698,"translate":367},[366],[363,151700,151702],{"className":151701},[371],[373,151703,151704],{"xmlns":375},[377,151705,151706,151714],{},[380,151707,151708,151710,151712],{},[383,151709,11044],{},[458,151711,5161],{},[463,151713,465],{},[387,151715,133934],{"encoding":389},[363,151717,151719,151737],{"className":151718,"ariaHidden":394},[393],[363,151720,151722,151725,151728,151731,151734],{"className":151721},[398],[363,151723],{"className":151724,"style":133944},[402],[363,151726,11044],{"className":151727,"style":11132},[407,408],[363,151729],{"className":151730,"style":2935},[505],[363,151732,5161],{"className":151733},[2939],[363,151735],{"className":151736,"style":2935},[505],[363,151738,151740,151743],{"className":151739},[398],[363,151741],{"className":151742,"style":520},[402],[363,151744,465],{"className":151745},[407],"：由于 ",[363,151748,151750,151776],{"className":151749,"translate":367},[366],[363,151751,151753],{"className":151752},[371],[373,151754,151755],{"xmlns":375},[377,151756,151757,151773],{},[380,151758,151759,151765,151767,151769,151771],{},[7162,151760,151761,151763],{},[463,151762,465],{"mathvariant":75598},[383,151764,151419],{"mathvariant":557},[383,151766,151363],{"mathvariant":75598},[463,151768,465],{"mathvariant":75598},[458,151770,471],{},[463,151772,474],{},[387,151774,151775],{"encoding":389},"\\mathbf{1}^\\top \\boldsymbol{\\Sigma} \\mathbf{1} = 0",[363,151777,151779,151835],{"className":151778,"ariaHidden":394},[393],[363,151780,151782,151785,151814,151823,151826,151829,151832],{"className":151781},[398],[363,151783],{"className":151784,"style":47926},[402],[363,151786,151788,151791],{"className":151787},[407],[363,151789,465],{"className":151790},[407,75943],[363,151792,151794],{"className":151793},[629],[363,151795,151797],{"className":151796},[633],[363,151798,151800],{"className":151799},[638],[363,151801,151803],{"className":151802,"style":47926},[642],[363,151804,151805,151808],{"style":7213},[363,151806],{"className":151807,"style":651},[650],[363,151809,151811],{"className":151810},[655,656,657,658],[363,151812,151419],{"className":151813},[407,658],[363,151815,151817],{"className":151816},[407],[363,151818,151820],{"className":151819},[407],[363,151821,151363],{"className":151822},[407,75943],[363,151824,465],{"className":151825},[407,75943],[363,151827],{"className":151828,"style":506},[505],[363,151830,471],{"className":151831},[510],[363,151833],{"className":151834,"style":506},[505],[363,151836,151838,151841],{"className":151837},[398],[363,151839],{"className":151840,"style":520},[402],[363,151842,474],{"className":151843},[407],[363,151845,151847,151861],{"className":151846,"translate":367},[366],[363,151848,151850],{"className":151849},[371],[373,151851,151852],{"xmlns":375},[377,151853,151854,151858],{},[380,151855,151856],{},[383,151857,151363],{"mathvariant":75598},[387,151859,151860],{"encoding":389},"\\boldsymbol{\\Sigma}",[363,151862,151864],{"className":151863,"ariaHidden":394},[393],[363,151865,151867,151870],{"className":151866},[398],[363,151868],{"className":151869,"style":151434},[402],[363,151871,151873],{"className":151872},[407],[363,151874,151876],{"className":151875},[407],[363,151877,151363],{"className":151878},[407,75943]," 必然奇异，其零空间由 ",[363,151881,151883,151897],{"className":151882,"translate":367},[366],[363,151884,151886],{"className":151885},[371],[373,151887,151888],{"xmlns":375},[377,151889,151890,151894],{},[380,151891,151892],{},[463,151893,465],{"mathvariant":75598},[387,151895,151896],{"encoding":389},"\\mathbf{1}",[363,151898,151900],{"className":151899,"ariaHidden":394},[393],[363,151901,151903,151906],{"className":151902},[398],[363,151904],{"className":151905,"style":520},[402],[363,151907,465],{"className":151908},[407,75943]," 张成——这正是「分布在 ",[363,151911,151913,151934],{"className":151912,"translate":367},[366],[363,151914,151916],{"className":151915},[371],[373,151917,151918],{"xmlns":375},[377,151919,151920,151932],{},[380,151921,151922,151924,151926,151928,151930],{},[458,151923,461],{"stretchy":460},[383,151925,11044],{},[458,151927,5161],{},[463,151929,465],{},[458,151931,468],{"stretchy":460},[387,151933,133733],{"encoding":389},[363,151935,151937,151958],{"className":151936,"ariaHidden":394},[393],[363,151938,151940,151943,151946,151949,151952,151955],{"className":151939},[398],[363,151941],{"className":151942,"style":487},[402],[363,151944,461],{"className":151945},[494],[363,151947,11044],{"className":151948,"style":11132},[407,408],[363,151950],{"className":151951,"style":2935},[505],[363,151953,5161],{"className":151954},[2939],[363,151956],{"className":151957,"style":2935},[505],[363,151959,151961,151964,151967],{"className":151960},[398],[363,151962],{"className":151963,"style":487},[402],[363,151965,465],{"className":151966},[407],[363,151968,468],{"className":151969},[501]," 维对象上」的代数痕迹。第二，全部协方差为负：",[363,151972,151974,152024],{"className":151973,"translate":367},[366],[363,151975,151977],{"className":151976},[371],[373,151978,151979],{"xmlns":375},[377,151980,151981,152021],{},[380,151982,151983,151985,151987,151989,151995,151997,152003,152005,152007,152009,152011,152013,152015,152017,152019],{},[383,151984,147765],{"mathvariant":557},[458,151986,4451],{},[458,151988,461],{"stretchy":460},[543,151990,151991,151993],{},[383,151992,1121],{},[383,151994,12313],{},[458,151996,898],{"separator":394},[543,151998,151999,152001],{},[383,152000,1121],{},[383,152002,100402],{},[458,152004,468],{"stretchy":460},[458,152006,7599],{},[463,152008,474],{},[1645,152010,4631],{},[458,152012,461],{"stretchy":460},[383,152014,12313],{},[458,152016,49879],{"mathvariant":557},[383,152018,100402],{},[458,152020,468],{"stretchy":460},[387,152022,152023],{"encoding":389},"\\operatorname{Cov}(x_i, x_j) \u003C 0\\ (i \\ne j)",[363,152025,152027,152140,152200],{"className":152026,"ariaHidden":394},[393],[363,152028,152030,152033,152039,152042,152082,152085,152088,152128,152131,152134,152137],{"className":152029},[398],[363,152031],{"className":152032,"style":618},[402],[363,152034,152036],{"className":152035},[1832],[363,152037,147765],{"className":152038,"style":4505},[407,33034],[363,152040,461],{"className":152041},[494],[363,152043,152045,152048],{"className":152044},[407],[363,152046,1121],{"className":152047},[407,408],[363,152049,152051],{"className":152050},[629],[363,152052,152054,152074],{"className":152053},[633,634],[363,152055,152057,152071],{"className":152056},[638],[363,152058,152060],{"className":152059,"style":12532},[642],[363,152061,152062,152065],{"style":730},[363,152063],{"className":152064,"style":651},[650],[363,152066,152068],{"className":152067},[655,656,657,658],[363,152069,12313],{"className":152070},[407,408,658],[363,152072,666],{"className":152073},[665],[363,152075,152077],{"className":152076},[638],[363,152078,152080],{"className":152079,"style":8316},[642],[363,152081],{},[363,152083,898],{"className":152084},[920],[363,152086],{"className":152087,"style":754},[505],[363,152089,152091,152094],{"className":152090},[407],[363,152092,1121],{"className":152093},[407,408],[363,152095,152097],{"className":152096},[629],[363,152098,152100,152120],{"className":152099},[633,634],[363,152101,152103,152117],{"className":152102},[638],[363,152104,152106],{"className":152105,"style":12532},[642],[363,152107,152108,152111],{"style":730},[363,152109],{"className":152110,"style":651},[650],[363,152112,152114],{"className":152113},[655,656,657,658],[363,152115,100402],{"className":152116,"style":100930},[407,408,658],[363,152118,666],{"className":152119},[665],[363,152121,152123],{"className":152122},[638],[363,152124,152126],{"className":152125,"style":673},[642],[363,152127],{},[363,152129,468],{"className":152130},[501],[363,152132],{"className":152133,"style":506},[505],[363,152135,7599],{"className":152136},[510],[363,152138],{"className":152139,"style":506},[505],[363,152141,152143,152146,152149,152152,152155,152158,152161,152197],{"className":152142},[398],[363,152144],{"className":152145,"style":487},[402],[363,152147,474],{"className":152148},[407],[363,152150,4631],{"className":152151},[505],[363,152153,461],{"className":152154},[494],[363,152156,12313],{"className":152157},[407,408],[363,152159],{"className":152160,"style":506},[505],[363,152162,152164,152191,152194],{"className":152163},[510],[363,152165,152167],{"className":152166},[510],[363,152168,152170],{"className":152169},[407,4748],[363,152171,152173],{"className":152172},[4752],[363,152174,152176,152179,152188],{"className":152175},[4756],[363,152177],{"className":152178,"style":403},[402],[363,152180,152182],{"className":152181},[4763],[363,152183,152185],{"className":152184},[407],[363,152186,4770],{"className":152187},[510],[363,152189],{"className":152190},[4774],[363,152192],{"className":152193},[505,4778],[363,152195,471],{"className":152196},[510],[363,152198],{"className":152199,"style":506},[505],[363,152201,152203,152206,152209],{"className":152202},[398],[363,152204],{"className":152205,"style":487},[402],[363,152207,100402],{"className":152208,"style":100930},[407,408],[363,152210,468],{"className":152211},[501],"，因为分量之间必须通过和为 ",[363,152214,152216,152229],{"className":152215,"translate":367},[366],[363,152217,152219],{"className":152218},[371],[373,152220,152221],{"xmlns":375},[377,152222,152223,152227],{},[380,152224,152225],{},[463,152226,465],{},[387,152228,465],{"encoding":389},[363,152230,152232],{"className":152231,"ariaHidden":394},[393],[363,152233,152235,152238],{"className":152234},[398],[363,152236],{"className":152237,"style":520},[402],[363,152239,465],{"className":152240},[407]," 的约束相互竞争，任何一个分量的增加都以其他分量的期望减少为代价。",[2659,152243,152245],{"id":152244},"_22-均值精度重参数化","2.2 均值—精度重参数化",[11,152247,152248,152249,2757,152284,152354],{},"命题 2.1 中的 ",[363,152250,152252,152266],{"className":152251,"translate":367},[366],[363,152253,152255],{"className":152254},[371],[373,152256,152257],{"xmlns":375},[377,152258,152259,152263],{},[380,152260,152261],{},[383,152262,2712],{"mathvariant":136055},[387,152264,152265],{"encoding":389},"\\boldsymbol{\\mu}",[363,152267,152269],{"className":152268,"ariaHidden":394},[393],[363,152270,152272,152275],{"className":152271},[398],[363,152273],{"className":152274,"style":133807},[402],[363,152276,152278],{"className":152277},[407],[363,152279,152281],{"className":152280},[407],[363,152282,2712],{"className":152283},[407,136115],[363,152285,152287,152305],{"className":152286,"translate":367},[366],[363,152288,152290],{"className":152289},[371],[373,152291,152292],{"xmlns":375},[377,152293,152294,152302],{},[380,152295,152296],{},[543,152297,152298,152300],{},[383,152299,45126],{},[463,152301,474],{},[387,152303,152304],{"encoding":389},"\\alpha_0",[363,152306,152308],{"className":152307,"ariaHidden":394},[393],[363,152309,152311,152314],{"className":152310},[398],[363,152312],{"className":152313,"style":13724},[402],[363,152315,152317,152320],{"className":152316},[407],[363,152318,45126],{"className":152319,"style":45156},[407,408],[363,152321,152323],{"className":152322},[629],[363,152324,152326,152346],{"className":152325},[633,634],[363,152327,152329,152343],{"className":152328},[638],[363,152330,152332],{"className":152331,"style":11483},[642],[363,152333,152334,152337],{"style":136154},[363,152335],{"className":152336,"style":651},[650],[363,152338,152340],{"className":152339},[655,656,657,658],[363,152341,474],{"className":152342},[407,658],[363,152344,666],{"className":152345},[665],[363,152347,152349],{"className":152348},[638],[363,152350,152352],{"className":152351,"style":8316},[642],[363,152353],{}," 是两个正交的自由度：前者决定「概率质量落在何处」，后者决定「分布在该处聚集得多紧」。据此引入重参数化",[363,152356,152358],{"className":152357,"translate":367},[528],[363,152359,152361,152423],{"className":152360,"translate":367},[366],[363,152362,152364],{"className":152363},[371],[373,152365,152366],{"xmlns":375,"display":537},[377,152367,152368,152420],{},[380,152369,152370,152372,152374,152380,152382,152384,152386,152392,152394,152396,152398,152400,152402,152404,152406,152408],{},[383,152371,45126],{"mathvariant":136055},[458,152373,471],{},[543,152375,152376,152378],{},[383,152377,45126],{},[463,152379,474],{},[383,152381,2712],{"mathvariant":136055},[458,152383,898],{"separator":394},[505,152385],{"width":3610},[543,152387,152388,152390],{},[383,152389,45126],{},[463,152391,474],{},[458,152393,3251],{},[463,152395,474],{},[458,152397,898],{"separator":394},[1645,152399,4631],{},[383,152401,2712],{"mathvariant":136055},[458,152403,6257],{},[383,152405,134509],{"mathvariant":557},[458,152407,4451],{},[7162,152409,152410,152412],{},[383,152411,126067],{"mathvariant":557},[380,152413,152414,152416,152418],{},[383,152415,11044],{},[458,152417,5161],{},[463,152419,465],{},[387,152421,152422],{"encoding":389},"\\boldsymbol{\\alpha} = \\alpha_0 \\boldsymbol{\\mu}, \\qquad \\alpha_0 > 0,\\ \\boldsymbol{\\mu} \\in \\operatorname{int}\\Delta^{K-1}",[363,152424,152426,152450,152563,152599],{"className":152425,"ariaHidden":394},[393],[363,152427,152429,152432,152441,152444,152447],{"className":152428},[398],[363,152430],{"className":152431,"style":135662},[402],[363,152433,152435],{"className":152434},[407],[363,152436,152438],{"className":152437},[407],[363,152439,45126],{"className":152440},[407,136115],[363,152442],{"className":152443,"style":506},[505],[363,152445,471],{"className":152446},[510],[363,152448],{"className":152449,"style":506},[505],[363,152451,152453,152456,152496,152505,152508,152511,152514,152554,152557,152560],{"className":152452},[398],[363,152454],{"className":152455,"style":32064},[402],[363,152457,152459,152462],{"className":152458},[407],[363,152460,45126],{"className":152461,"style":45156},[407,408],[363,152463,152465],{"className":152464},[629],[363,152466,152468,152488],{"className":152467},[633,634],[363,152469,152471,152485],{"className":152470},[638],[363,152472,152474],{"className":152473,"style":11483},[642],[363,152475,152476,152479],{"style":136154},[363,152477],{"className":152478,"style":651},[650],[363,152480,152482],{"className":152481},[655,656,657,658],[363,152483,474],{"className":152484},[407,658],[363,152486,666],{"className":152487},[665],[363,152489,152491],{"className":152490},[638],[363,152492,152494],{"className":152493,"style":8316},[642],[363,152495],{},[363,152497,152499],{"className":152498},[407],[363,152500,152502],{"className":152501},[407],[363,152503,2712],{"className":152504},[407,136115],[363,152506,898],{"className":152507},[920],[363,152509],{"className":152510,"style":3982},[505],[363,152512],{"className":152513,"style":754},[505],[363,152515,152517,152520],{"className":152516},[407],[363,152518,45126],{"className":152519,"style":45156},[407,408],[363,152521,152523],{"className":152522},[629],[363,152524,152526,152546],{"className":152525},[633,634],[363,152527,152529,152543],{"className":152528},[638],[363,152530,152532],{"className":152531,"style":11483},[642],[363,152533,152534,152537],{"style":136154},[363,152535],{"className":152536,"style":651},[650],[363,152538,152540],{"className":152539},[655,656,657,658],[363,152541,474],{"className":152542},[407,658],[363,152544,666],{"className":152545},[665],[363,152547,152549],{"className":152548},[638],[363,152550,152552],{"className":152551,"style":8316},[642],[363,152553],{},[363,152555],{"className":152556,"style":506},[505],[363,152558,3251],{"className":152559},[510],[363,152561],{"className":152562,"style":506},[505],[363,152564,152566,152569,152572,152575,152578,152581,152590,152593,152596],{"className":152565},[398],[363,152567],{"className":152568,"style":9706},[402],[363,152570,474],{"className":152571},[407],[363,152573,898],{"className":152574},[920],[363,152576,4631],{"className":152577},[505],[363,152579],{"className":152580,"style":754},[505],[363,152582,152584],{"className":152583},[407],[363,152585,152587],{"className":152586},[407],[363,152588,2712],{"className":152589},[407,136115],[363,152591],{"className":152592,"style":506},[505],[363,152594,6257],{"className":152595},[510],[363,152597],{"className":152598,"style":506},[505],[363,152600,152602,152605,152611,152614],{"className":152601},[398],[363,152603],{"className":152604,"style":38884},[402],[363,152606,152608],{"className":152607},[1832],[363,152609,134509],{"className":152610},[407,33034],[363,152612],{"className":152613,"style":754},[505],[363,152615,152617,152620],{"className":152616},[407],[363,152618,126067],{"className":152619},[407],[363,152621,152623],{"className":152622},[629],[363,152624,152626],{"className":152625},[633],[363,152627,152629],{"className":152628},[638],[363,152630,152632],{"className":152631,"style":38884},[642],[363,152633,152634,152637],{"style":8625},[363,152635],{"className":152636,"style":651},[650],[363,152638,152640],{"className":152639},[655,656,657,658],[363,152641,152643,152646,152649],{"className":152642},[407,658],[363,152644,11044],{"className":152645,"style":11132},[407,408,658],[363,152647,5161],{"className":152648},[2939,658],[363,152650,465],{"className":152651},[407,658],[11,152653,152654,152655,16145,152658,152708,152709,152712,152713,3093,152996,153104,153105,153174],{},"它把参数空间分解为一个",[1344,152656,152657],{},"方向",[363,152659,152661,152678],{"className":152660,"translate":367},[366],[363,152662,152664],{"className":152663},[371],[373,152665,152666],{"xmlns":375},[377,152667,152668,152676],{},[380,152669,152670,152672,152674],{},[383,152671,11044],{},[458,152673,5161],{},[463,152675,465],{},[387,152677,133934],{"encoding":389},[363,152679,152681,152699],{"className":152680,"ariaHidden":394},[393],[363,152682,152684,152687,152690,152693,152696],{"className":152683},[398],[363,152685],{"className":152686,"style":133944},[402],[363,152688,11044],{"className":152689,"style":11132},[407,408],[363,152691],{"className":152692,"style":2935},[505],[363,152694,5161],{"className":152695},[2939],[363,1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维单纯形）与一个",[1344,152710,152711],{},"尺度","（正实数）。在此参数化下，协方差为 ",[363,152714,152716,152768],{"className":152715,"translate":367},[366],[363,152717,152719],{"className":152718},[371],[373,152720,152721],{"xmlns":375},[377,152722,152723,152765],{},[380,152724,152725,152727,152729,152745,152747,152749,152751,152753,152755,152757,152763],{},[383,152726,151363],{"mathvariant":75598},[458,152728,471],{},[586,152730,152731,152733],{},[463,152732,465],{},[380,152734,152735,152741,152743],{},[543,152736,152737,152739],{},[383,152738,45126],{},[463,152740,474],{},[458,152742,2818],{},[463,152744,465],{},[458,152746,461],{"stretchy":460},[383,152748,151400],{"mathvariant":557},[458,152750,4451],{},[383,152752,2712],{"mathvariant":136055},[458,152754,5161],{},[383,152756,2712],{"mathvariant":136055},[7162,152758,152759,152761],{},[383,152760,2712],{"mathvariant":136055},[383,152762,151419],{"mathvariant":557},[458,152764,468],{"stretchy":460},[387,152766,152767],{"encoding":389},"\\boldsymbol{\\Sigma} = \\frac{1}{\\alpha_0+1}(\\operatorname{diag}\\boldsymbol{\\mu} - \\boldsymbol{\\mu}\\boldsymbol{\\mu}^\\top)",[363,152769,152771,152795,152942],{"className":152770,"ariaHidden":394},[393],[363,152772,152774,152777,152786,152789,152792],{"className":152773},[398],[363,152775],{"className":152776,"style":151434},[402],[363,152778,152780],{"className":152779},[407],[363,152781,152783],{"className":152782},[407],[363,152784,151363],{"className":152785},[407,75943],[363,152787],{"className":152788,"style":506},[505],[363,152790,471],{"className":152791},[510],[363,152793],{"className":152794,"style":506},[505],[363,152796,152798,152801,152912,152915,152921,152924,152933,152936,152939],{"className":152797},[398],[363,152799],{"className":152800,"style":149401},[402],[363,152802,152804,152807,152909],{"className":152803},[407],[363,152805],{"className":152806},[494,789],[363,152808,152810],{"className":152809},[586],[363,152811,152813,152901],{"className":152812},[633,634],[363,152814,152816,152898],{"className":152815},[638],[363,152817,152819,152876,152884],{"className":152818,"style":2872},[642],[363,152820,152821,152824],{"style":2875},[363,152822],{"className":152823,"style":809},[650],[363,152825,152827],{"className":152826},[655,656,657,658],[363,152828,152830,152870,152873],{"className":152829},[407,658],[363,152831,152833,152836],{"className":152832},[407,658],[363,152834,45126],{"className":152835,"style":45156},[407,408,658],[363,152837,152839],{"className":152838},[629],[363,152840,152842,152862],{"className":152841},[633,634],[363,152843,152845,152859],{"className":152844},[638],[363,152846,152848],{"className":152847,"style":137351},[642],[363,152849,152850,152853],{"style":136989},[363,152851],{"className":152852,"style":30578},[650],[363,152854,152856],{"className":152855},[655,30582,10022,658],[363,152857,474],{"className":152858},[407,658],[363,152860,666],{"className":152861},[665],[363,152863,152865],{"className":152864},[638],[363,152866,152868],{"className":152867,"style":40142},[642],[363,152869],{},[363,152871,2818],{"className":152872},[2939,658],[363,152874,465],{"className":152875},[407,658],[363,152877,152878,152881],{"style":821},[363,152879],{"className":152880,"style":809},[650],[363,152882],{"className":152883,"style":829},[828],[363,152885,152886,152889],{"style":2898},[363,152887],{"className":152888,"style":809},[650],[363,152890,152892],{"className":152891},[655,656,657,658],[363,152893,152895],{"className":152894},[407,658],[363,152896,465],{"className":152897},[407,658],[363,152899,666],{"className":152900},[665],[363,152902,152904],{"className":152903},[638],[363,152905,152907],{"className":152906,"style":40289},[642],[363,152908],{},[363,152910],{"className":152911},[501,789],[363,152913,461],{"className":152914},[494],[363,152916,152918],{"className":152917},[1832],[363,152919,151400],{"className":152920,"style":4505},[407,33034],[363,152922],{"className":152923,"style":754},[505],[363,152925,152927],{"className":152926},[407],[363,152928,152930],{"className":152929},[407],[363,152931,2712],{"className":152932},[407,136115],[363,152934],{"className":152935,"style":2935},[505],[363,152937,5161],{"className":152938},[2939],[363,152940],{"className":152941,"style":2935},[505],[363,152943,152945,152949,152958,152993],{"className":152944},[398],[363,152946],{"className":152947,"style":152948},[402],"height:1.0991em;vertical-align:-0.25em;",[363,152950,152952],{"className":152951},[407],[363,152953,152955],{"className":152954},[407],[363,152956,2712],{"className":152957},[407,136115],[363,152959,152961,152970],{"className":152960},[407],[363,152962,152964],{"className":152963},[407],[363,152965,152967],{"className":152966},[407],[363,152968,2712],{"className":152969},[407,136115],[363,152971,152973],{"className":152972},[629],[363,152974,152976],{"className":152975},[633],[363,152977,152979],{"className":152978},[638],[363,152980,152982],{"className":152981,"style":47926},[642],[363,152983,152984,152987],{"style":7213},[363,152985],{"className":152986,"style":651},[650],[363,152988,152990],{"className":152989},[655,656,657,658],[363,152991,151419],{"className":152992},[407,658],[363,152994,468],{"className":152995},[501],[1344,152997,152998,152999,153033,153034,153103],{},"与 ",[363,153000,153002,153015],{"className":153001,"translate":367},[366],[363,153003,153005],{"className":153004},[371],[373,153006,153007],{"xmlns":375},[377,153008,153009,153013],{},[380,153010,153011],{},[383,153012,2712],{"mathvariant":136055},[387,153014,152265],{"encoding":389},[363,153016,153018],{"className":153017,"ariaHidden":394},[393],[363,153019,153021,153024],{"className":153020},[398],[363,153022],{"className":153023,"style":133807},[402],[363,153025,153027],{"className":153026},[407],[363,153028,153030],{"className":153029},[407],[363,153031,2712],{"className":153032},[407,136115]," 的取值无关、只随 ",[363,153035,153037,153054],{"className":153036,"translate":367},[366],[363,153038,153040],{"className":153039},[371],[373,153041,153042],{"xmlns":375},[377,153043,153044,153052],{},[380,153045,153046],{},[543,153047,153048,153050],{},[383,153049,45126],{},[463,153051,474],{},[387,153053,152304],{"encoding":389},[363,153055,153057],{"className":153056,"ariaHidden":394},[393],[363,153058,153060,153063],{"className":153059},[398],[363,153061],{"className":153062,"style":13724},[402],[363,153064,153066,153069],{"className":153065},[407],[363,153067,45126],{"className":153068,"style":45156},[407,408],[363,153070,153072],{"className":153071},[629],[363,153073,153075,153095],{"className":153074},[633,634],[363,153076,153078,153092],{"className":153077},[638],[363,153079,153081],{"className":153080,"style":11483},[642],[363,153082,153083,153086],{"style":136154},[363,153084],{"className":153085,"style":651},[650],[363,153087,153089],{"className":153088},[655,656,657,658],[363,153090,474],{"className":153091},[407,658],[363,153093,666],{"className":153094},[665],[363,153096,153098],{"className":153097},[638],[363,153099,153101],{"className":153100,"style":8316},[642],[363,153102],{}," 单调收缩","。因此 ",[363,153106,153108,153125],{"className":153107,"translate":367},[366],[363,153109,153111],{"className":153110},[371],[373,153112,153113],{"xmlns":375},[377,153114,153115,153123],{},[380,153116,153117],{},[543,153118,153119,153121],{},[383,153120,45126],{},[463,153122,474],{},[387,153124,152304],{"encoding":389},[363,153126,153128],{"className":153127,"ariaHidden":394},[393],[363,153129,153131,153134],{"className":153130},[398],[363,153132],{"className":153133,"style":13724},[402],[363,153135,153137,153140],{"className":153136},[407],[363,153138,45126],{"className":153139,"style":45156},[407,408],[363,153141,153143],{"className":153142},[629],[363,153144,153146,153166],{"className":153145},[633,634],[363,153147,153149,153163],{"className":153148},[638],[363,153150,153152],{"className":153151,"style":11483},[642],[363,153153,153154,153157],{"style":136154},[363,153155],{"className":153156,"style":651},[650],[363,153158,153160],{"className":153159},[655,656,657,658],[363,153161,474],{"className":153162},[407,658],[363,153164,666],{"className":153165},[665],[363,153167,153169],{"className":153168},[638],[363,153170,153172],{"className":153171,"style":8316},[642],[363,153173],{}," 是一个纯粹的集中度（或称精度、强度）参数：",[143,153176,153177],{},[11,153178,153179,15731,153182,21328,153297,153492,153493,10650,153585,153686,153687,153721,153722,10528],{},[1344,153180,153181],{},"命题 2.2（集中性极限）",[363,153183,153185,153217],{"className":153184,"translate":367},[366],[363,153186,153188],{"className":153187},[371],[373,153189,153190],{"xmlns":375},[377,153191,153192,153214],{},[380,153193,153194,153196,153198,153200,153202],{},[383,153195,2712],{"mathvariant":136055},[458,153197,6257],{},[383,153199,134509],{"mathvariant":557},[458,153201,4451],{},[7162,153203,153204,153206],{},[383,153205,126067],{"mathvariant":557},[380,153207,153208,153210,153212],{},[383,153209,11044],{},[458,153211,5161],{},[463,153213,465],{},[387,153215,153216],{"encoding":389},"\\boldsymbol{\\mu} \\in \\operatorname{int}\\Delta^{K-1}",[363,153218,153220,153244],{"className":153219,"ariaHidden":394},[393],[363,153221,153223,153226,153235,153238,153241],{"className":153222},[398],[363,153224],{"className":153225,"style":32064},[402],[363,153227,153229],{"className":153228},[407],[363,153230,153232],{"className":153231},[407],[363,153233,2712],{"className":153234},[407,136115],[363,153236],{"className":153237,"style":506},[505],[363,153239,6257],{"className":153240},[510],[363,153242],{"className":153243,"style":506},[505],[363,153245,153247,153250,153256,153259],{"className":153246},[398],[363,153248],{"className":153249,"style":52457},[402],[363,153251,153253],{"className":153252},[1832],[363,153254,134509],{"className":153255},[407,33034],[363,153257],{"className":153258,"style":754},[505],[363,153260,153262,153265],{"className":153261},[407],[363,153263,126067],{"className":153264},[407],[363,153266,153268],{"className":153267},[629],[363,153269,153271],{"className":153270},[633],[363,153272,153274],{"className":153273},[638],[363,153275,153277],{"className":153276,"style":52457},[642],[363,153278,153279,153282],{"style":7213},[363,153280],{"className":153281,"style":651},[650],[363,153283,153285],{"className":153284},[655,656,657,658],[363,153286,153288,153291,153294],{"className":153287},[407,658],[363,153289,11044],{"className":153290,"style":11132},[407,408,658],[363,153292,5161],{"className":153293},[2939,658],[363,153295,465],{"className":153296},[407,658],[363,153298,153300,153338],{"className":153299,"translate":367},[366],[363,153301,153303],{"className":153302},[371],[373,153304,153305],{"xmlns":375},[377,153306,153307,153335],{},[380,153308,153309,153325,153327,153333],{},[7162,153310,153311,153313],{},[383,153312,45126],{"mathvariant":136055},[380,153314,153315,153317,153323],{},[458,153316,461],{"stretchy":460},[543,153318,153319,153321],{},[383,153320,45126],{},[463,153322,474],{},[458,153324,468],{"stretchy":460},[458,153326,471],{},[543,153328,153329,153331],{},[383,153330,45126],{},[463,153332,474],{},[383,153334,2712],{"mathvariant":136055},[387,153336,153337],{"encoding":389},"\\boldsymbol{\\alpha}^{(\\alpha_0)} = \\alpha_0 \\boldsymbol{\\mu}",[363,153339,153341,153437],{"className":153340,"ariaHidden":394},[393],[363,153342,153344,153347,153428,153431,153434],{"className":153343},[398],[363,153345],{"className":153346,"style":55153},[402],[363,153348,153350,153359],{"className":153349},[407],[363,153351,153353],{"className":153352},[407],[363,153354,153356],{"className":153355},[407],[363,153357,45126],{"className":153358},[407,136115],[363,153360,153362],{"className":153361},[629],[363,153363,153365],{"className":153364},[633],[363,153366,153368],{"className":153367},[638],[363,153369,153371],{"className":153370,"style":55153},[642],[363,153372,153373,153376],{"style":7213},[363,153374],{"className":153375,"style":651},[650],[363,153377,153379],{"className":153378},[655,656,657,658],[363,153380,153382,153385,153425],{"className":153381},[407,658],[363,153383,461],{"className":153384},[494,658],[363,153386,153388,153391],{"className":153387},[407,658],[363,153389,45126],{"className":153390,"style":45156},[407,408,658],[363,153392,153394],{"className":153393},[629],[363,153395,153397,153417],{"className":153396},[633,634],[363,153398,153400,153414],{"className":153399},[638],[363,153401,153403],{"className":153402,"style":137351},[642],[363,153404,153405,153408],{"style":136989},[363,153406],{"className":153407,"style":30578},[650],[363,153409,153411],{"className":153410},[655,30582,10022,658],[363,153412,474],{"className":153413},[407,658],[363,153415,666],{"className":153416},[665],[363,153418,153420],{"className":153419},[638],[363,153421,153423],{"className":153422,"style":40142},[642],[363,153424],{},[363,153426,468],{"className":153427},[501,658],[363,153429],{"className":153430,"style":506},[505],[363,153432,471],{"className":153433},[510],[363,153435],{"className":153436,"style":506},[505],[363,153438,153440,153443,153483],{"className":153439},[398],[363,153441],{"className":153442,"style":133807},[402],[363,153444,153446,153449],{"className":153445},[407],[363,153447,45126],{"className":153448,"style":45156},[407,408],[363,153450,153452],{"className":153451},[629],[363,153453,153455,153475],{"className":153454},[633,634],[363,153456,153458,153472],{"className":153457},[638],[363,153459,153461],{"className":153460,"style":11483},[642],[363,153462,153463,153466],{"style":136154},[363,153464],{"className":153465,"style":651},[650],[363,153467,153469],{"className":153468},[655,656,657,658],[363,153470,474],{"className":153471},[407,658],[363,153473,666],{"className":153474},[665],[363,153476,153478],{"className":153477},[638],[363,153479,153481],{"className":153480,"style":8316},[642],[363,153482],{},[363,153484,153486],{"className":153485},[407],[363,153487,153489],{"className":153488},[407],[363,153490,2712],{"className":153491},[407,136115],"。则当 ",[363,153494,153496,153518],{"className":153495,"translate":367},[366],[363,153497,153499],{"className":153498},[371],[373,153500,153501],{"xmlns":375},[377,153502,153503,153515],{},[380,153504,153505,153511,153513],{},[543,153506,153507,153509],{},[383,153508,45126],{},[463,153510,474],{},[458,153512,1378],{},[383,153514,1373],{"mathvariant":557},[387,153516,153517],{"encoding":389},"\\alpha_0 \\to \\infty",[363,153519,153521,153576],{"className":153520,"ariaHidden":394},[393],[363,153522,153524,153527,153567,153570,153573],{"className":153523},[398],[363,153525],{"className":153526,"style":13724},[402],[363,153528,153530,153533],{"className":153529},[407],[363,153531,45126],{"className":153532,"style":45156},[407,408],[363,153534,153536],{"className":153535},[629],[363,153537,153539,153559],{"className":153538},[633,634],[363,153540,153542,153556],{"className":153541},[638],[363,153543,153545],{"className":153544,"style":11483},[642],[363,153546,153547,153550],{"style":136154},[363,153548],{"className":153549,"style":651},[650],[363,153551,153553],{"className":153552},[655,656,657,658],[363,153554,474],{"className":153555},[407,658],[363,153557,666],{"className":153558},[665],[363,153560,153562],{"className":153561},[638],[363,153563,153565],{"className":153564,"style":8316},[642],[363,153566],{},[363,153568],{"className":153569,"style":506},[505],[363,153571,1378],{"className":153572},[510],[363,153574],{"className":153575,"style":506},[505],[363,153577,153579,153582],{"className":153578},[398],[363,153580],{"className":153581,"style":2693},[402],[363,153583,1373],{"className":153584},[407],[363,153586,153588,153616],{"className":153587,"translate":367},[366],[363,153589,153591],{"className":153590},[371],[373,153592,153593],{"xmlns":375},[377,153594,153595,153613],{},[380,153596,153597,153599,153601,153603,153609,153611],{},[383,153598,137112],{"mathvariant":557},[458,153600,4451],{},[458,153602,461],{"stretchy":460},[543,153604,153605,153607],{},[383,153606,45126],{},[463,153608,474],{},[383,153610,2712],{"mathvariant":136055},[458,153612,468],{"stretchy":460},[387,153614,153615],{"encoding":389},"\\operatorname{Dir}(\\alpha_0 \\boldsymbol{\\mu})",[363,153617,153619],{"className":153618,"ariaHidden":394},[393],[363,153620,153622,153625,153631,153634,153674,153683],{"className":153621},[398],[363,153623],{"className":153624,"style":487},[402],[363,153626,153628],{"className":153627},[1832],[363,153629,137112],{"className":153630},[407,33034],[363,153632,461],{"className":153633},[494],[363,153635,153637,153640],{"className":153636},[407],[363,153638,45126],{"className":153639,"style":45156},[407,408],[363,153641,153643],{"className":153642},[629],[363,153644,153646,153666],{"className":153645},[633,634],[363,153647,153649,153663],{"className":153648},[638],[363,153650,153652],{"className":153651,"style":11483},[642],[363,153653,153654,153657],{"style":136154},[363,153655],{"className":153656,"style":651},[650],[363,153658,153660],{"className":153659},[655,656,657,658],[363,153661,474],{"className":153662},[407,658],[363,153664,666],{"className":153665},[665],[363,153667,153669],{"className":153668},[638],[363,153670,153672],{"className":153671,"style":8316},[642],[363,153673],{},[363,153675,153677],{"className":153676},[407],[363,153678,153680],{"className":153679},[407],[363,153681,2712],{"className":153682},[407,136115],[363,153684,468],{"className":153685},[501]," 弱收敛于位于 ",[363,153688,153690,153703],{"className":153689,"translate":367},[366],[363,153691,153693],{"className":153692},[371],[373,153694,153695],{"xmlns":375},[377,153696,153697,153701],{},[380,153698,153699],{},[383,153700,2712],{"mathvariant":136055},[387,153702,152265],{"encoding":389},[363,153704,153706],{"className":153705,"ariaHidden":394},[393],[363,153707,153709,153712],{"className":153708},[398],[363,153710],{"className":153711,"style":133807},[402],[363,153713,153715],{"className":153714},[407],[363,153716,153718],{"className":153717},[407],[363,153719,2712],{"className":153720},[407,136115]," 的退化测度 ",[363,153723,153725,153743],{"className":153724,"translate":367},[366],[363,153726,153728],{"className":153727},[371],[373,153729,153730],{"xmlns":375},[377,153731,153732,153740],{},[380,153733,153734],{},[543,153735,153736,153738],{},[383,153737,16062],{},[383,153739,2712],{"mathvariant":136055},[387,153741,153742],{"encoding":389},"\\delta_{\\boldsymbol{\\mu}}",[363,153744,153746],{"className":153745,"ariaHidden":394},[393],[363,153747,153749,153753],{"className":153748},[398],[363,153750],{"className":153751,"style":153752},[402],"height:0.9805em;vertical-align:-0.2861em;",[363,153754,153756,153759],{"className":153755},[407],[363,153757,16062],{"className":153758,"style":16111},[407,408],[363,153760,153762],{"className":153761},[629],[363,153763,153765,153795],{"className":153764},[633,634],[363,153766,153768,153792],{"className":153767},[638],[363,153769,153772],{"className":153770,"style":153771},[642],"height:0.1611em;",[363,153773,153774,153777],{"style":95132},[363,153775],{"className":153776,"style":651},[650],[363,153778,153780],{"className":153779},[655,656,657,658],[363,153781,153783],{"className":153782},[407,658],[363,153784,153786],{"className":153785},[407,658],[363,153787,153789],{"className":153788},[407,658],[363,153790,2712],{"className":153791},[407,136115,658],[363,153793,666],{"className":153794},[665],[363,153796,153798],{"className":153797},[638],[363,153799,153801],{"className":153800,"style":673},[642],[363,153802],{},[11,153804,153805,153807,153808,7148,153880,154022,154023,154074,154075,154282,154283,10528],{},[1344,153806,138042],{}," 由命题 2.1，",[363,153809,153811,153835],{"className":153810,"translate":367},[366],[363,153812,153814],{"className":153813},[371],[373,153815,153816],{"xmlns":375},[377,153817,153818,153832],{},[380,153819,153820,153822,153824,153826,153828,153830],{},[383,153821,571],{"mathvariant":570},[458,153823,578],{"stretchy":460},[383,153825,1121],{"mathvariant":75598},[458,153827,605],{"stretchy":460},[458,153829,471],{},[383,153831,2712],{"mathvariant":136055},[387,153833,153834],{"encoding":389},"\\mathbb{E}[\\mathbf{x}] = \\boldsymbol{\\mu}",[363,153836,153838,153865],{"className":153837,"ariaHidden":394},[393],[363,153839,153841,153844,153847,153850,153853,153856,153859,153862],{"className":153840},[398],[363,153842],{"className":153843,"style":487},[402],[363,153845,571],{"className":153846},[407,714],[363,153848,578],{"className":153849},[494],[363,153851,1121],{"className":153852},[407,75943],[363,153854,605],{"className":153855},[501],[363,153857],{"className":153858,"style":506},[505],[363,153860,471],{"className":153861},[510],[363,153863],{"className":153864,"style":506},[505],[363,153866,153868,153871],{"className":153867},[398],[363,153869],{"className":153870,"style":133807},[402],[363,153872,153874],{"className":153873},[407],[363,153875,153877],{"className":153876},[407],[363,153878,2712],{"className":153879},[407,136115],[363,153881,153883,153919],{"className":153882,"translate":367},[366],[363,153884,153886],{"className":153885},[371],[373,153887,153888],{"xmlns":375},[377,153889,153890,153916],{},[380,153891,153892,153894,153896,153898,153900,153902,153904,153910,153912,153914],{},[383,153893,151363],{"mathvariant":75598},[458,153895,471],{},[383,153897,19593],{},[458,153899,461],{"stretchy":460},[463,153901,465],{},[383,153903,1288],{"mathvariant":557},[543,153905,153906,153908],{},[383,153907,45126],{},[463,153909,474],{},[458,153911,468],{"stretchy":460},[458,153913,1378],{},[463,153915,474],{"mathvariant":75598},[387,153917,153918],{"encoding":389},"\\boldsymbol{\\Sigma} = O(1\u002F\\alpha_0) \\to \\mathbf{0}",[363,153920,153922,153946,154013],{"className":153921,"ariaHidden":394},[393],[363,153923,153925,153928,153937,153940,153943],{"className":153924},[398],[363,153926],{"className":153927,"style":151434},[402],[363,153929,153931],{"className":153930},[407],[363,153932,153934],{"className":153933},[407],[363,153935,151363],{"className":153936},[407,75943],[363,153938],{"className":153939,"style":506},[505],[363,153941,471],{"className":153942},[510],[363,153944],{"className":153945,"style":506},[505],[363,153947,153949,153952,153955,153958,153961,154001,154004,154007,154010],{"className":153948},[398],[363,153950],{"className":153951,"style":487},[402],[363,153953,19593],{"className":153954,"style":625},[407,408],[363,153956,461],{"className":153957},[494],[363,153959,8937],{"className":153960},[407],[363,153962,153964,153967],{"className":153963},[407],[363,153965,45126],{"className":153966,"style":45156},[407,408],[363,153968,153970],{"className":153969},[629],[363,153971,153973,153993],{"className":153972},[633,634],[363,153974,153976,153990],{"className":153975},[638],[363,153977,153979],{"className":153978,"style":11483},[642],[363,153980,153981,153984],{"style":136154},[363,153982],{"className":153983,"style":651},[650],[363,153985,153987],{"className":153986},[655,656,657,658],[363,153988,474],{"className":153989},[407,658],[363,153991,666],{"className":153992},[665],[363,153994,153996],{"className":153995},[638],[363,153997,153999],{"className":153998,"style":8316},[642],[363,154000],{},[363,154002,468],{"className":154003},[501],[363,154005],{"className":154006,"style":506},[505],[363,154008,1378],{"className":154009},[510],[363,154011],{"className":154012,"style":506},[505],[363,154014,154016,154019],{"className":154015},[398],[363,154017],{"className":154018,"style":520},[402],[363,154020,474],{"className":154021},[407,75943],"。对任意 ",[363,154024,154026,154044],{"className":154025,"translate":367},[366],[363,154027,154029],{"className":154028},[371],[373,154030,154031],{"xmlns":375},[377,154032,154033,154041],{},[380,154034,154035,154037,154039],{},[383,154036,39034],{},[458,154038,3251],{},[463,154040,474],{},[387,154042,154043],{"encoding":389},"\\varepsilon > 0",[363,154045,154047,154065],{"className":154046,"ariaHidden":394},[393],[363,154048,154050,154053,154056,154059,154062],{"className":154049},[398],[363,154051],{"className":154052,"style":6272},[402],[363,154054,39034],{"className":154055},[407,408],[363,154057],{"className":154058,"style":506},[505],[363,154060,3251],{"className":154061},[510],[363,154063],{"className":154064,"style":506},[505],[363,154066,154068,154071],{"className":154067},[398],[363,154069],{"className":154070,"style":520},[402],[363,154072,474],{"className":154073},[407],"，由 Chebyshev 不等式 ",[363,154076,154078,154130],{"className":154077,"translate":367},[366],[363,154079,154081],{"className":154080},[371],[373,154082,154083],{"xmlns":375},[377,154084,154085,154127],{},[380,154086,154087,154089,154091,154093,154095,154097,154099,154101,154103,154105,154107,154109,154111,154113,154115,154117,154123,154125],{},[383,154088,554],{"mathvariant":570},[458,154090,461],{"stretchy":460},[383,154092,558],{"mathvariant":557},[383,154094,1121],{"mathvariant":75598},[458,154096,5161],{},[383,154098,2712],{"mathvariant":136055},[383,154100,558],{"mathvariant":557},[458,154102,3251],{},[383,154104,39034],{},[458,154106,468],{"stretchy":460},[458,154108,10300],{},[383,154110,52870],{"mathvariant":557},[458,154112,4451],{},[383,154114,151363],{"mathvariant":75598},[383,154116,1288],{"mathvariant":557},[7162,154118,154119,154121],{},[383,154120,39034],{},[463,154122,2811],{},[458,154124,1378],{},[463,154126,474],{},[387,154128,154129],{"encoding":389},"\\mathbb{P}(\\|\\mathbf{x} - \\boldsymbol{\\mu}\\| > \\varepsilon) \\le \\operatorname{tr}\\boldsymbol{\\Sigma}\u002F\\varepsilon^2 \\to 0",[363,154131,154133,154160,154187,154208,154273],{"className":154132,"ariaHidden":394},[393],[363,154134,154136,154139,154142,154145,154148,154151,154154,154157],{"className":154135},[398],[363,154137],{"className":154138,"style":487},[402],[363,154140,554],{"className":154141},[407,714],[363,154143,461],{"className":154144},[494],[363,154146,558],{"className":154147},[407],[363,154149,1121],{"className":154150},[407,75943],[363,154152],{"className":154153,"style":2935},[505],[363,154155,5161],{"className":154156},[2939],[363,154158],{"className":154159,"style":2935},[505],[363,154161,154163,154166,154175,154178,154181,154184],{"className":154162},[398],[363,154164],{"className":154165,"style":487},[402],[363,154167,154169],{"className":154168},[407],[363,154170,154172],{"className":154171},[407],[363,154173,2712],{"className":154174},[407,136115],[363,154176,558],{"className":154177},[407],[363,154179],{"className":154180,"style":506},[505],[363,154182,3251],{"className":154183},[510],[363,154185],{"className":154186,"style":506},[505],[363,154188,154190,154193,154196,154199,154202,154205],{"className":154189},[398],[363,154191],{"className":154192,"style":487},[402],[363,154194,39034],{"className":154195},[407,408],[363,154197,468],{"className":154198},[501],[363,154200],{"className":154201,"style":506},[505],[363,154203,10300],{"className":154204},[510],[363,154206],{"className":154207,"style":506},[505],[363,154209,154211,154214,154220,154223,154232,154235,154264,154267,154270],{"className":154210},[398],[363,154212],{"className":154213,"style":20375},[402],[363,154215,154217],{"className":154216},[1832],[363,154218,52870],{"className":154219},[407,33034],[363,154221],{"className":154222,"style":754},[505],[363,154224,154226],{"className":154225},[407],[363,154227,154229],{"className":154228},[407],[363,154230,151363],{"className":154231},[407,75943],[363,154233,1288],{"className":154234},[407],[363,154236,154238,154241],{"className":154237},[407],[363,154239,39034],{"className":154240},[407,408],[363,154242,154244],{"className":154243},[629],[363,154245,154247],{"className":154246},[633],[363,154248,154250],{"className":154249},[638],[363,154251,154253],{"className":154252,"style":20400},[642],[363,154254,154255,154258],{"style":7213},[363,154256],{"className":154257,"style":651},[650],[363,154259,154261],{"className":154260},[655,656,657,658],[363,154262,2811],{"className":154263},[407,658],[363,154265],{"className":154266,"style":506},[505],[363,154268,1378],{"className":154269},[510],[363,154271],{"className":154272,"style":506},[505],[363,154274,154276,154279],{"className":154275},[398],[363,154277],{"className":154278,"style":520},[402],[363,154280,474],{"className":154281},[407],"，故依概率收敛，进而弱收敛于 ",[363,154284,154286,154304],{"className":154285,"translate":367},[366],[363,154287,154289],{"className":154288},[371],[373,154290,154291],{"xmlns":375},[377,154292,154293,154301],{},[380,154294,154295],{},[543,154296,154297,154299],{},[383,154298,16062],{},[383,154300,2712],{"mathvariant":136055},[387,154302,154303],{"encoding":389},"\\delta_{\\boldsymbol\\mu}",[363,154305,154307],{"className":154306,"ariaHidden":394},[393],[363,154308,154310,154313],{"className":154309},[398],[363,154311],{"className":154312,"style":153752},[402],[363,154314,154316,154319],{"className":154315},[407],[363,154317,16062],{"className":154318,"style":16111},[407,408],[363,154320,154322],{"className":154321},[629],[363,154323,154325,154354],{"className":154324},[633,634],[363,154326,154328,154351],{"className":154327},[638],[363,154329,154331],{"className":154330,"style":153771},[642],[363,154332,154333,154336],{"style":95132},[363,154334],{"className":154335,"style":651},[650],[363,154337,154339],{"className":154338},[655,656,657,658],[363,154340,154342],{"className":154341},[407,658],[363,154343,154345],{"className":154344},[407,658],[363,154346,154348],{"className":154347},[407,658],[363,154349,2712],{"className":154350},[407,136115,658],[363,154352,666],{"className":154353},[665],[363,154355,154357],{"className":154356},[638],[363,154358,154360],{"className":154359,"style":673},[642],[363,154361],{},[11,154363,154364,154365,154481],{},"这一命题给出了本文后续所有「不确定性」语汇的几何基础：",[1344,154366,154367,154368,154445,154446,154480],{},"迪利克雷分布的「模糊程度」等于 ",[363,154369,154371,154393],{"className":154370,"translate":367},[366],[363,154372,154374],{"className":154373},[371],[373,154375,154376],{"xmlns":375},[377,154377,154378,154390],{},[380,154379,154380,154382,154384],{},[463,154381,465],{},[383,154383,1288],{"mathvariant":557},[543,154385,154386,154388],{},[383,154387,45126],{},[463,154389,474],{},[387,154391,154392],{"encoding":389},"1\u002F\\alpha_0",[363,154394,154396],{"className":154395,"ariaHidden":394},[393],[363,154397,154399,154402,154405],{"className":154398},[398],[363,154400],{"className":154401,"style":487},[402],[363,154403,8937],{"className":154404},[407],[363,154406,154408,154411],{"className":154407},[407],[363,154409,45126],{"className":154410,"style":45156},[407,408],[363,154412,154414],{"className":154413},[629],[363,154415,154417,154437],{"className":154416},[633,634],[363,154418,154420,154434],{"className":154419},[638],[363,154421,154423],{"className":154422,"style":11483},[642],[363,154424,154425,154428],{"style":136154},[363,154426],{"className":154427,"style":651},[650],[363,154429,154431],{"className":154430},[655,656,657,658],[363,154432,474],{"className":154433},[407,658],[363,154435,666],{"className":154436},[665],[363,154438,154440],{"className":154439},[638],[363,154441,154443],{"className":154442,"style":8316},[642],[363,154444],{}," 的量级，而它的「中心」由 ",[363,154447,154449,154462],{"className":154448,"translate":367},[366],[363,154450,154452],{"className":154451},[371],[373,154453,154454],{"xmlns":375},[377,154455,154456,154460],{},[380,154457,154458],{},[383,154459,2712],{"mathvariant":136055},[387,154461,152265],{"encoding":389},[363,154463,154465],{"className":154464,"ariaHidden":394},[393],[363,154466,154468,154471],{"className":154467},[398],[363,154469],{"className":154470,"style":133807},[402],[363,154472,154474],{"className":154473},[407],[363,154475,154477],{"className":154476},[407],[363,154478,2712],{"className":154479},[407,136115]," 给出。"," 两者在同一族分布中被参数化为彼此独立的量——这正是证据深度学习能够把「预测什么」与「有多确定」分离开来的数学根据。",[2659,154483,154485],{"id":154484},"_23-众数与熵","2.3 众数与熵",[143,154487,154488],{},[11,154489,154490,56519,154493,154585,154586,11286,154615,4201,154687,154760,154761],{},[1344,154491,154492],{},"命题 2.3（众数）",[363,154494,154496,154518],{"className":154495,"translate":367},[366],[363,154497,154499],{"className":154498},[371],[373,154500,154501],{"xmlns":375},[377,154502,154503,154515],{},[380,154504,154505,154511,154513],{},[543,154506,154507,154509],{},[383,154508,45126],{},[383,154510,12313],{},[458,154512,3251],{},[463,154514,465],{},[387,154516,154517],{"encoding":389},"\\alpha_i > 1",[363,154519,154521,154576],{"className":154520,"ariaHidden":394},[393],[363,154522,154524,154527,154567,154570,154573],{"className":154523},[398],[363,154525],{"className":154526,"style":134714},[402],[363,154528,154530,154533],{"className":154529},[407],[363,154531,45126],{"className":154532,"style":45156},[407,408],[363,154534,154536],{"className":154535},[629],[363,154537,154539,154559],{"className":154538},[633,634],[363,154540,154542,154556],{"className":154541},[638],[363,154543,154545],{"className":154544,"style":12532},[642],[363,154546,154547,154550],{"style":136154},[363,154548],{"className":154549,"style":651},[650],[363,154551,154553],{"className":154552},[655,656,657,658],[363,154554,12313],{"className":154555},[407,408,658],[363,154557,666],{"className":154558},[665],[363,154560,154562],{"className":154561},[638],[363,154563,154565],{"className":154564,"style":8316},[642],[363,154566],{},[363,154568],{"className":154569,"style":506},[505],[363,154571,3251],{"className":154572},[510],[363,154574],{"className":154575,"style":506},[505],[363,154577,154579,154582],{"className":154578},[398],[363,154580],{"className":154581,"style":520},[402],[363,154583,465],{"className":154584},[407]," 对一切 ",[363,154587,154589,154602],{"className":154588,"translate":367},[366],[363,154590,154592],{"className":154591},[371],[373,154593,154594],{"xmlns":375},[377,154595,154596,154600],{},[380,154597,154598],{},[383,154599,12313],{},[387,154601,12313],{"encoding":389},[363,154603,154605],{"className":154604,"ariaHidden":394},[393],[363,154606,154608,154612],{"className":154607},[398],[363,154609],{"className":154610,"style":154611},[402],"height:0.6595em;",[363,154613,12313],{"className":154614},[407,408],[363,154616,154618,154642],{"className":154617,"translate":367},[366],[363,154619,154621],{"className":154620},[371],[373,154622,154623],{"xmlns":375},[377,154624,154625,154639],{},[380,154626,154627,154629,154631,154633,154635,154637],{},[383,154628,385],{},[458,154630,461],{"stretchy":460},[383,154632,1121],{"mathvariant":75598},[458,154634,5156],{},[383,154636,45126],{"mathvariant":136055},[458,154638,468],{"stretchy":460},[387,154640,154641],{"encoding":389},"f(\\mathbf{x} \\mid 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的内部有唯一众数\n",[363,154762,154764,154848],{"className":154763,"translate":367},[366],[363,154765,154767],{"className":154766},[371],[373,154768,154769],{"xmlns":375},[377,154770,154771,154845],{},[380,154772,154773,154779,154781],{},[7162,154774,154775,154777],{},[383,154776,1121],{"mathvariant":75598},[458,154778,76343],{},[458,154780,471],{},[380,154782,154783,154785,154811,154813,154815,154817,154843],{},[458,154784,461],{"fence":394},[586,154786,154787,154799],{},[380,154788,154789,154795,154797],{},[543,154790,154791,154793],{},[383,154792,45126],{},[463,154794,465],{},[458,154796,5161],{},[463,154798,465],{},[380,154800,154801,154807,154809],{},[543,154802,154803,154805],{},[383,154804,45126],{},[463,154806,474],{},[458,154808,5161],{},[383,154810,11044],{},[458,154812,898],{"separator":394},[458,154814,133150],{},[458,154816,898],{"separator":394},[586,154818,154819,154831],{},[380,154820,154821,154827,154829],{},[543,154822,154823,154825],{},[383,154824,45126],{},[383,154826,11044],{},[458,154828,5161],{},[463,154830,465],{},[380,154832,154833,154839,154841],{},[543,154834,154835,154837],{},[383,154836,45126],{},[463,154838,474],{},[458,154840,5161],{},[383,154842,11044],{},[458,154844,468],{"fence":394},[387,154846,154847],{"encoding":389},"\\mathbf{x}^\\star = \\left( \\frac{\\alpha_1 - 1}{\\alpha_0 - K}, \\dots, \\frac{\\alpha_K - 1}{\\alpha_0 - K} \\right)",[363,154849,154851,154895],{"className":154850,"ariaHidden":394},[393],[363,154852,154854,154857,154886,154889,154892],{"className":154853},[398],[363,154855],{"className":154856,"style":8847},[402],[363,154858,154860,154863],{"className":154859},[407],[363,154861,1121],{"className":154862},[407,75943],[363,154864,154866],{"className":154865},[629],[363,154867,154869],{"className":154868},[633],[363,154870,154872],{"className":154871},[638],[363,154873,154875],{"className":154874,"style":8847},[642],[363,154876,154877,154880],{"style":7213},[363,154878],{"className":154879,"style":651},[650],[363,154881,154883],{"className":154882},[655,656,657,658],[363,154884,76343],{"className":154885},[2939,658],[363,154887],{"className":154888,"style":506},[505],[363,154890,471],{"className":154891},[510],[363,154893],{"className":154894,"style":506},[505],[363,154896,154898,154901],{"className":154897},[398],[363,154899],{"className":154900,"style":68500},[402],[363,154902,154904,154910,155065,155068,155071,155074,155077,155080,155083,155240],{"className":154903},[758],[363,154905,154907],{"className":154906,"style":763},[494,762],[363,154908,461],{"className":154909},[767,4068],[363,154911,154913,154916,155062],{"className":154912},[407],[363,154914],{"className":154915},[494,789],[363,154917,154919],{"className":154918},[586],[363,154920,154922,155054],{"className":154921},[633,634],[363,154923,154925,155051],{"className":154924},[638],[363,154926,154929,154986,154994],{"className":154927,"style":154928},[642],"height:0.8612em;",[363,154930,154931,154934],{"style":2875},[363,154932],{"className":154933,"style":809},[650],[363,154935,154937],{"className":154936},[655,656,657,658],[363,154938,154940,154980,154983],{"className":154939},[407,658],[363,154941,154943,154946],{"className":154942},[407,658],[363,154944,45126],{"className":154945,"style":45156},[407,408,658],[363,154947,154949],{"className":154948},[629],[363,154950,154952,154972],{"className":154951},[633,634],[363,154953,154955,154969],{"className":154954},[638],[363,154956,154958],{"className":154957,"style":137351},[642],[363,154959,154960,154963],{"style":136989},[363,154961],{"className":154962,"style":30578},[650],[363,154964,154966],{"className":154965},[655,30582,10022,658],[363,154967,474],{"className":154968},[407,658],[363,154970,666],{"className":154971},[665],[363,154973,154975],{"className":154974},[638],[363,154976,154978],{"className":154977,"style":40142},[642],[363,154979],{},[363,154981,5161],{"className":154982},[2939,658],[363,154984,11044],{"className":154985,"style":11132},[407,408,658],[363,154987,154988,154991],{"style":821},[363,154989],{"className":154990,"style":809},[650],[363,154992],{"className":154993,"style":829},[828],[363,154995,154996,154999],{"style":147999},[363,154997],{"className":154998,"style":809},[650],[363,155000,155002],{"className":155001},[655,656,657,658],[363,155003,155005,155045,155048],{"className":155004},[407,658],[363,155006,155008,155011],{"className":155007},[407,658],[363,155009,45126],{"className":155010,"style":45156},[407,408,658],[363,155012,155014],{"className":155013},[629],[363,155015,155017,155037],{"className":155016},[633,634],[363,155018,155020,155034],{"className":155019},[638],[363,155021,155023],{"className":155022,"style":137351},[642],[363,155024,155025,155028],{"style":136989},[363,155026],{"className":155027,"style":30578},[650],[363,155029,155031],{"className":155030},[655,30582,10022,658],[363,155032,465],{"className":155033},[407,658],[363,155035,666],{"className":155036},[665],[363,155038,155040],{"className":155039},[638],[363,155041,155043],{"className":155042,"style":40142},[642],[363,155044],{},[363,155046,5161],{"className":155047},[2939,658],[363,155049,465],{"className":155050},[407,658],[363,155052,666],{"className":155053},[665],[363,155055,155057],{"className":155056},[638],[363,155058,155060],{"className":155059,"style":40289},[642],[363,155061],{},[363,155063],{"className":155064},[501,789],[363,155066,898],{"className":155067},[920],[363,155069],{"className":155070,"style":754},[505],[363,155072,133150],{"className":155073},[758],[363,155075],{"className":155076,"style":754},[505],[363,155078,898],{"className":155079},[920],[363,155081],{"className":155082,"style":754},[505],[363,155084,155086,155089,155237],{"className":155085},[407],[363,155087],{"className":155088},[494,789],[363,155090,155092],{"className":155091},[586],[363,155093,155095,155229],{"className":155094},[633,634],[363,155096,155098,155226],{"className":155097},[638],[363,155099,155102,155159,155167],{"className":155100,"style":155101},[642],"height:0.8614em;",[363,155103,155104,155107],{"style":2875},[363,155105],{"className":155106,"style":809},[650],[363,155108,155110],{"className":155109},[655,656,657,658],[363,155111,155113,155153,155156],{"className":155112},[407,658],[363,155114,155116,155119],{"className":155115},[407,658],[363,155117,45126],{"className":155118,"style":45156},[407,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",[363,155250,155252,155274],{"className":155251,"translate":367},[366],[363,155253,155255],{"className":155254},[371],[373,155256,155257],{"xmlns":375},[377,155258,155259,155271],{},[380,155260,155261,155267,155269],{},[543,155262,155263,155265],{},[383,155264,45126],{},[383,155266,12313],{},[458,155268,10300],{},[463,155270,465],{},[387,155272,155273],{"encoding":389},"\\alpha_i \\le 1",[363,155275,155277,155333],{"className":155276,"ariaHidden":394},[393],[363,155278,155280,155284,155324,155327,155330],{"className":155279},[398],[363,155281],{"className":155282,"style":155283},[402],"height:0.786em;vertical-align:-0.15em;",[363,155285,155287,155290],{"className":155286},[407],[363,155288,45126],{"className":155289,"style":45156},[407,408],[363,155291,155293],{"className":155292},[629],[363,155294,155296,155316],{"className":155295},[633,634],[363,155297,155299,155313],{"className":155298},[638],[363,155300,155302],{"className":155301,"style":12532},[642],[363,155303,155304,155307],{"style":136154},[363,155305],{"className":155306,"style":651},[650],[363,155308,155310],{"className":155309},[655,656,657,658],[363,155311,12313],{"className":155312},[407,408,658],[363,155314,666],{"className":155315},[665],[363,155317,155319],{"className":155318},[638],[363,155320,155322],{"className":155321,"style":8316},[642],[363,155323],{},[363,155325],{"className":155326,"style":506},[505],[363,155328,10300],{"className":155329},[510],[363,155331],{"className":155332,"style":506},[505],[363,155334,155336,155339],{"className":155335},[398],[363,155337],{"className":155338,"style":520},[402],[363,155340,465],{"className":155341},[407]," 时密度在对应边界面上无界，众数退化到边界，此时分布呈现 2.6 节所述的稀疏形态。",[143,155344,155345],{},[11,155346,155347,155350,155351,155427,155428,155924,155925,156016],{},[1344,155348,155349],{},"命题 2.4（微分熵）"," 在 1.1 节的测度约定下，",[363,155352,155354,155379],{"className":155353,"translate":367},[366],[363,155355,155357],{"className":155356},[371],[373,155358,155359],{"xmlns":375},[377,155360,155361,155377],{},[380,155362,155363,155365,155367,155369,155371,155373,155375],{},[383,155364,1121],{"mathvariant":75598},[458,155366,9217],{},[383,155368,137112],{"mathvariant":557},[458,155370,4451],{},[458,155372,461],{"stretchy":460},[383,155374,45126],{"mathvariant":136055},[458,155376,468],{"stretchy":460},[387,155378,137123],{"encoding":389},[363,155380,155382,155400],{"className":155381,"ariaHidden":394},[393],[363,155383,155385,155388,155391,155394,155397],{"className":155384},[398],[363,155386],{"className":155387,"style":135662},[402],[363,155389,1121],{"className":155390},[407,75943],[363,155392],{"className":155393,"style":506},[505],[363,155395,9217],{"className":155396},[510],[363,155398],{"className":155399,"style":506},[505],[363,155401,155403,155406,155412,155415,155424],{"className":155402},[398],[363,155404],{"className":155405,"style":487},[402],[363,155407,155409],{"className":155408},[1832],[363,155410,137112],{"className":155411},[407,33034],[363,155413,461],{"className":155414},[494],[363,155416,155418],{"className":155417},[407],[363,155419,155421],{"className":155420},[407],[363,155422,45126],{"className":155423},[407,136115],[363,155425,468],{"className":155426},[501]," 的微分熵为\n",[363,155429,155431,155540],{"className":155430,"translate":367},[366],[363,155432,155434],{"className":155433},[371],[373,155435,155436],{"xmlns":375},[377,155437,155438,155537],{},[380,155439,155440,155442,155444,155446,155448,155450,155452,155454,155456,155458,155460,155462,155464,155466,155472,155474,155476,155478,155480,155483,155485,155491,155493,155495,155509,155511,155517,155519,155521,155523,155525,155527,155529,155535],{},[383,155441,34256],{},[458,155443,461],{"stretchy":460},[383,155445,1121],{"mathvariant":75598},[458,155447,468],{"stretchy":460},[458,155449,471],{},[383,155451,4448],{},[458,155453,4451],{},[383,155455,39663],{},[458,155457,461],{"stretchy":460},[383,155459,45126],{"mathvariant":136055},[458,155461,468],{"stretchy":460},[458,155463,2818],{},[458,155465,461],{"stretchy":460},[543,155467,155468,155470],{},[383,155469,45126],{},[463,155471,474],{},[458,155473,5161],{},[383,155475,11044],{},[458,155477,468],{"stretchy":460},[1645,155479,1647],{},[383,155481,155482],{},"ψ",[458,155484,461],{"stretchy":460},[543,155486,155487,155489],{},[383,155488,45126],{},[463,155490,474],{},[458,155492,468],{"stretchy":460},[458,155494,5161],{},[7901,155496,155497,155499,155507],{},[458,155498,13811],{},[380,155500,155501,155503,155505],{},[383,155502,12313],{},[458,155504,471],{},[463,155506,465],{},[383,155508,11044],{},[458,155510,461],{"stretchy":460},[543,155512,155513,155515],{},[383,155514,45126],{},[383,155516,12313],{},[458,155518,5161],{},[463,155520,465],{},[458,155522,468],{"stretchy":460},[1645,155524,1647],{},[383,155526,155482],{},[458,155528,461],{"stretchy":460},[543,155530,155531,155533],{},[383,155532,45126],{},[383,155534,12313],{},[458,155536,468],{"stretchy":460},[387,155538,155539],{"encoding":389},"H(\\mathbf{x}) = \\log B(\\boldsymbol{\\alpha}) + (\\alpha_0 - K)\\,\\psi(\\alpha_0) - \\sum_{i=1}^K (\\alpha_i - 1)\\,\\psi(\\alpha_i)",[363,155541,155543,155570,155611,155669,155742,155860],{"className":155542,"ariaHidden":394},[393],[363,155544,155546,155549,155552,155555,155558,155561,155564,155567],{"className":155545},[398],[363,155547],{"className":155548,"style":487},[402],[363,155550,34256],{"className":155551,"style":34410},[407,408],[363,155553,461],{"className":155554},[494],[363,155556,1121],{"className":155557},[407,75943],[363,155559,468],{"className":155560},[501],[363,155562],{"className":155563,"style":506},[505],[363,155565,471],{"className":155566},[510],[363,155568],{"className":155569,"style":506},[505],[363,155571,155573,155576,155581,155584,155587,155590,155599,155602,155605,155608],{"className":155572},[398],[363,155574],{"className":155575,"style":487},[402],[363,155577,4502,155579],{"className":155578},[1832],[363,155580,4506],{"style":4505},[363,155582],{"className":155583,"style":754},[505],[363,155585,39663],{"className":155586,"style":39759},[407,408],[363,155588,461],{"className":155589},[494],[363,155591,155593],{"className":155592},[407],[363,155594,155596],{"className":155595},[407],[363,155597,45126],{"className":155598},[407,136115],[363,155600,468],{"className":155601},[501],[363,155603],{"className":155604,"style":2935},[505],[363,155606,2818],{"className":155607},[2939],[363,155609],{"className":155610,"style":2935},[505],[363,155612,155614,155617,155620,155660,155663,155666],{"className":155613},[398],[363,155615],{"className":155616,"style":487},[402],[363,155618,461],{"className":155619},[494],[363,155621,155623,155626],{"className":155622},[407],[363,155624,45126],{"className":155625,"style":45156},[407,408],[363,155627,155629],{"className":155628},[629],[363,155630,155632,155652],{"className":155631},[633,634],[363,155633,155635,155649],{"className":155634},[638],[363,155636,155638],{"className":155637,"style":11483},[642],[363,155639,155640,155643],{"style":136154},[363,155641],{"className":155642,"style":651},[650],[363,155644,155646],{"className":155645},[655,656,657,658],[363,155647,474],{"className":155648},[407,658],[363,155650,666],{"className":155651},[665],[363,155653,155655],{"className":155654},[638],[363,155656,155658],{"className":155657,"style":8316},[642],[363,155659],{},[363,155661],{"className":155662,"style":2935},[505],[363,155664,5161],{"className":155665},[2939],[363,155667],{"className":155668,"style":2935},[505],[363,155670,155672,155675,155678,155681,155684,155687,155690,155730,155733,155736,155739],{"className":155671},[398],[363,155673],{"className":155674,"style":487},[402],[363,155676,11044],{"className":155677,"style":11132},[407,408],[363,155679,468],{"className":155680},[501],[363,155682],{"className":155683,"style":754},[505],[363,155685,155482],{"className":155686,"style":1328},[407,408],[363,155688,461],{"className":155689},[494],[363,155691,155693,155696],{"className":155692},[407],[363,155694,45126],{"className":155695,"style":45156},[407,408],[363,155697,155699],{"className":155698},[629],[363,155700,155702,155722],{"className":155701},[633,634],[363,155703,155705,155719],{"className":155704},[638],[363,155706,155708],{"className":155707,"style":11483},[642],[363,155709,155710,155713],{"style":136154},[363,155711],{"className":155712,"style":651},[650],[363,155714,155716],{"className":155715},[655,656,657,658],[363,155717,474],{"className":155718},[407,658],[363,155720,666],{"className":155721},[665],[363,155723,155725],{"className":155724},[638],[363,155726,155728],{"className":155727,"style":8316},[642],[363,155729],{},[363,155731,468],{"className":155732},[501],[363,155734],{"className":155735,"style":2935},[505],[363,155737,5161],{"className":155738},[2939],[363,155740],{"className":155741,"style":2935},[505],[363,155743,155745,155748,155808,155811,155851,155854,155857],{"className":155744},[398],[363,155746],{"className":155747,"style":136418},[402],[363,155749,155751,155754],{"className":155750},[1832],[363,155752,13811],{"className":155753,"style":40032},[1832,1833,5964],[363,155755,155757],{"className":155756},[629],[363,155758,155760,155800],{"className":155759},[633,634],[363,155761,155763,155797],{"className":155762},[638],[363,155764,155766,155786],{"className":155765,"style":133537},[642],[363,155767,155768,155771],{"style":40048},[363,155769],{"className":155770,"style":651},[650],[363,155772,155774],{"className":155773},[655,656,657,658],[363,155775,155777,155780,155783],{"className":155776},[407,658],[363,155778,12313],{"className":155779},[407,408,658],[363,155781,471],{"className":155782},[510,658],[363,155784,465],{"className":155785},[407,658],[363,155787,155788,155791],{"style":133560},[363,155789],{"className":155790,"style":651},[650],[363,155792,155794],{"className":155793},[655,656,657,658],[363,155795,11044],{"className":155796,"style":11132},[407,408,658],[363,155798,666],{"className":155799},[665],[363,155801,155803],{"className":155802},[638],[363,155804,155806],{"className":155805,"style":40067},[642],[363,155807],{},[363,155809,461],{"className":155810},[494],[363,155812,155814,155817],{"className":155813},[407],[363,155815,45126],{"className":155816,"style":45156},[407,408],[363,155818,155820],{"className":155819},[629],[363,155821,155823,155843],{"className":155822},[633,634],[363,155824,155826,155840],{"className":155825},[638],[363,155827,155829],{"className":155828,"style":12532},[642],[363,155830,155831,155834],{"style":136154},[363,155832],{"className":155833,"style":651},[650],[363,155835,155837],{"className":155836},[655,656,657,658],[363,155838,12313],{"className":155839},[407,408,658],[363,155841,666],{"className":155842},[665],[363,155844,155846],{"className":155845},[638],[363,155847,155849],{"className":155848,"style":8316},[642],[363,155850],{},[363,155852],{"className":155853,"style":2935},[505],[363,155855,5161],{"className":155856},[2939],[363,155858],{"className":155859,"style":2935},[505],[363,155861,155863,155866,155869,155872,155875,155878,155881,155921],{"className":155862},[398],[363,155864],{"className":155865,"style":487},[402],[363,155867,465],{"className":155868},[407],[363,155870,468],{"className":155871},[501],[363,155873],{"className":155874,"style":754},[505],[363,155876,155482],{"className":155877,"style":1328},[407,408],[363,155879,461],{"className":155880},[494],[363,155882,155884,155887],{"className":155883},[407],[363,155885,45126],{"className":155886,"style":45156},[407,408],[363,155888,155890],{"className":155889},[629],[363,155891,155893,155913],{"className":155892},[633,634],[363,155894,155896,155910],{"className":155895},[638],[363,155897,155899],{"className":155898,"style":12532},[642],[363,155900,155901,155904],{"style":136154},[363,155902],{"className":155903,"style":651},[650],[363,155905,155907],{"className":155906},[655,656,657,658],[363,155908,12313],{"className":155909},[407,408,658],[363,155911,666],{"className":155912},[665],[363,155914,155916],{"className":155915},[638],[363,155917,155919],{"className":155918,"style":8316},[642],[363,155920],{},[363,155922,468],{"className":155923},[501],"\n其中 ",[363,155926,155928,155954],{"className":155927,"translate":367},[366],[363,155929,155931],{"className":155930},[371],[373,155932,155933],{"xmlns":375},[377,155934,155935,155951],{},[380,155936,155937,155939,155941,155947,155949],{},[383,155938,155482],{},[458,155940,471],{},[7162,155942,155943,155945],{},[383,155944,137218],{"mathvariant":557},[458,155946,7168],{"mathvariant":557,"lspace":1595,"rspace":1595},[383,155948,1288],{"mathvariant":557},[383,155950,137218],{"mathvariant":557},[387,155952,155953],{"encoding":389},"\\psi = \\Gamma'\u002F\\Gamma",[363,155955,155957,155975],{"className":155956,"ariaHidden":394},[393],[363,155958,155960,155963,155966,155969,155972],{"className":155959},[398],[363,155961],{"className":155962,"style":403},[402],[363,155964,155482],{"className":155965,"style":1328},[407,408],[363,155967],{"className":155968,"style":506},[505],[363,155970,471],{"className":155971},[510],[363,155973],{"className":155974,"style":506},[505],[363,155976,155978,155981,156013],{"className":155977},[398],[363,155979],{"className":155980,"style":7191},[402],[363,155982,155984,155987],{"className":155983},[407],[363,155985,137218],{"className":155986},[407],[363,155988,155990],{"className":155989},[629],[363,155991,155993],{"className":155992},[633],[363,155994,155996],{"className":155995},[638],[363,155997,155999],{"className":155998,"style":7210},[642],[363,156000,156001,156004],{"style":7213},[363,156002],{"className":156003,"style":651},[650],[363,156005,156007],{"className":156006},[655,656,657,658],[363,156008,156010],{"className":156009},[407,658],[363,156011,7168],{"className":156012},[407,658],[363,156014,143816],{"className":156015},[407]," 为 digamma 函数。",[11,156018,156019,156021,156022,156109,156110,2757,156416,156669,156670,156740],{},[1344,156020,138042],{}," 直接计算 ",[363,156023,156025,156053],{"className":156024,"translate":367},[366],[363,156026,156028],{"className":156027},[371],[373,156029,156030],{"xmlns":375},[377,156031,156032,156050],{},[380,156033,156034,156036,156038,156040,156042,156044,156046,156048],{},[383,156035,34256],{},[458,156037,471],{},[458,156039,5161],{},[458,156041,1635],{},[383,156043,385],{},[383,156045,4448],{},[458,156047,4451],{},[383,156049,385],{},[387,156051,156052],{"encoding":389},"H = -\\int f \\log f",[363,156054,156056,156074],{"className":156055,"ariaHidden":394},[393],[363,156057,156059,156062,156065,156068,156071],{"className":156058},[398],[363,156060],{"className":156061,"style":1011},[402],[363,156063,34256],{"className":156064,"style":34410},[407,408],[363,156066],{"className":156067,"style":506},[505],[363,156069,471],{"className":156070},[510],[363,156072],{"className":156073,"style":506},[505],[363,156075,156077,156080,156083,156086,156089,156092,156095,156098,156103,156106],{"className":156076},[398],[363,156078],{"className":156079,"style":5960},[402],[363,156081,5161],{"className":156082},[407],[363,156084],{"className":156085,"style":754},[505],[363,156087,1635],{"className":156088,"style":5965},[1832,1833,5964],[363,156090],{"className":156091,"style":754},[505],[363,156093,385],{"className":156094,"style":409},[407,408],[363,156096],{"className":156097,"style":754},[505],[363,156099,4502,156101],{"className":156100},[1832],[363,156102,4506],{"style":4505},[363,156104],{"className":156105,"style":754},[505],[363,156107,385],{"className":156108,"style":409},[407,408],"，代入 ",[363,156111,156113,156179],{"className":156112,"translate":367},[366],[363,156114,156116],{"className":156115},[371],[373,156117,156118],{"xmlns":375},[377,156119,156120,156176],{},[380,156121,156122,156124,156126,156128,156130,156132,156134,156136,156138,156140,156142,156144,156146,156152,156154,156160,156162,156164,156166,156168,156170],{},[458,156123,5161],{},[383,156125,4448],{},[458,156127,4451],{},[383,156129,385],{},[458,156131,471],{},[383,156133,4448],{},[458,156135,4451],{},[383,156137,39663],{},[458,156139,461],{"stretchy":460},[383,156141,45126],{"mathvariant":136055},[458,156143,468],{"stretchy":460},[458,156145,5161],{},[543,156147,156148,156150],{},[458,156149,13811],{},[383,156151,12313],{},[458,156153,461],{"stretchy":460},[543,156155,156156,156158],{},[383,156157,45126],{},[383,156159,12313],{},[458,156161,5161],{},[463,156163,465],{},[458,156165,468],{"stretchy":460},[383,156167,4448],{},[458,156169,4451],{},[543,156171,156172,156174],{},[383,156173,1121],{},[383,156175,12313],{},[387,156177,156178],{"encoding":389},"-\\log f = \\log B(\\boldsymbol{\\alpha}) - \\sum_i (\\alpha_i - 1)\\log x_i",[363,156180,156182,156214,156255,156353],{"className":156181,"ariaHidden":394},[393],[363,156183,156185,156188,156191,156194,156199,156202,156205,156208,156211],{"className":156184},[398],[363,156186],{"className":156187,"style":403},[402],[363,156189,5161],{"className":156190},[407],[363,156192],{"className":156193,"style":754},[505],[363,156195,4502,156197],{"className":156196},[1832],[363,156198,4506],{"style":4505},[363,156200],{"className":156201,"style":754},[505],[363,156203,385],{"className":156204,"style":409},[407,408],[363,156206],{"className":156207,"style":506},[505],[363,156209,471],{"className":156210},[510],[363,156212],{"className":156213,"style":506},[505],[363,156215,156217,156220,156225,156228,156231,156234,156243,156246,156249,156252],{"className":156216},[398],[363,156218],{"className":156219,"style":487},[402],[363,156221,4502,156223],{"className":156222},[1832],[363,156224,4506],{"style":4505},[363,156226],{"className":156227,"style":754},[505],[363,156229,39663],{"className":156230,"style":39759},[407,408],[363,156232,461],{"className":156233},[494],[363,156235,156237],{"className":156236},[407],[363,156238,156240],{"className":156239},[407],[363,156241,45126],{"className":156242},[407,136115],[363,156244,468],{"className":156245},[501],[363,156247],{"className":156248,"style":2935},[505],[363,156250,5161],{"className":156251},[2939],[363,156253],{"className":156254,"style":2935},[505],[363,156256,156258,156261,156301,156304,156344,156347,156350],{"className":156257},[398],[363,156259],{"className":156260,"style":138953},[402],[363,156262,156264,156267],{"className":156263},[1832],[363,156265,13811],{"className":156266,"style":40032},[1832,1833,5964],[363,156268,156270],{"className":156269},[629],[363,156271,156273,156293],{"className":156272},[633,634],[363,156274,156276,156290],{"className":156275},[638],[363,156277,156279],{"className":156278,"style":40045},[642],[363,156280,156281,156284],{"style":40048},[363,156282],{"className":156283,"style":651},[650],[363,156285,156287],{"className":156286},[655,656,657,658],[363,156288,12313],{"className":156289},[407,408,658],[363,156291,666],{"className":156292},[665],[363,156294,156296],{"className":156295},[638],[363,156297,156299],{"className":156298,"style":40067},[642],[363,156300],{},[363,156302,461],{"className":156303},[494],[363,156305,156307,156310],{"className":156306},[407],[363,156308,45126],{"className":156309,"style":45156},[407,408],[363,156311,156313],{"className":156312},[629],[363,156314,156316,156336],{"className":156315},[633,634],[363,156317,156319,156333],{"className":156318},[638],[363,156320,156322],{"className":156321,"style":12532},[642],[363,156323,156324,156327],{"style":136154},[363,156325],{"className":156326,"style":651},[650],[363,156328,156330],{"className":156329},[655,656,657,658],[363,156331,12313],{"className":156332},[407,408,658],[363,156334,666],{"className":156335},[665],[363,156337,156339],{"className":156338},[638],[363,156340,156342],{"className":156341,"style":8316},[642],[363,156343],{},[363,156345],{"className":156346,"style":2935},[505],[363,156348,5161],{"className":156349},[2939],[363,156351],{"className":156352,"style":2935},[505],[363,156354,156356,156359,156362,156365,156368,156373,156376],{"className":156355},[398],[363,156357],{"className":156358,"style":487},[402],[363,156360,465],{"className":156361},[407],[363,156363,468],{"className":156364},[501],[363,156366],{"className":156367,"style":754},[505],[363,156369,4502,156371],{"className":156370},[1832],[363,156372,4506],{"style":4505},[363,156374],{"className":156375,"style":754},[505],[363,156377,156379,156382],{"className":156378},[407],[363,156380,1121],{"className":156381},[407,408],[363,156383,156385],{"className":156384},[629],[363,156386,156388,156408],{"className":156387},[633,634],[363,156389,156391,156405],{"className":156390},[638],[363,156392,156394],{"className":156393,"style":12532},[642],[363,156395,156396,156399],{"style":730},[363,156397],{"className":156398,"style":651},[650],[363,156400,156402],{"className":156401},[655,656,657,658],[363,156403,12313],{"className":156404},[407,408,658],[363,156406,666],{"className":156407},[665],[363,156409,156411],{"className":156410},[638],[363,156412,156414],{"className":156413,"style":8316},[642],[363,156415],{},[363,156417,156419,156475],{"className":156418,"translate":367},[366],[363,156420,156422],{"className":156421},[371],[373,156423,156424],{"xmlns":375},[377,156425,156426,156472],{},[380,156427,156428,156430,156432,156434,156436,156442,156444,156446,156448,156450,156456,156458,156460,156462,156464,156470],{},[383,156429,571],{"mathvariant":570},[458,156431,578],{"stretchy":460},[383,156433,4448],{},[458,156435,4451],{},[543,156437,156438,156440],{},[383,156439,1121],{},[383,156441,12313],{},[458,156443,605],{"stretchy":460},[458,156445,471],{},[383,156447,155482],{},[458,156449,461],{"stretchy":460},[543,156451,156452,156454],{},[383,156453,45126],{},[383,156455,12313],{},[458,156457,468],{"stretchy":460},[458,156459,5161],{},[383,156461,155482],{},[458,156463,461],{"stretchy":460},[543,156465,156466,156468],{},[383,156467,45126],{},[463,156469,474],{},[458,156471,468],{"stretchy":460},[387,156473,156474],{"encoding":389},"\\mathbb{E}[\\log x_i] = \\psi(\\alpha_i) - \\psi(\\alpha_0)",[363,156476,156478,156550,156614],{"className":156477,"ariaHidden":394},[393],[363,156479,156481,156484,156487,156490,156495,156498,156538,156541,156544,156547],{"className":156480},[398],[363,156482],{"className":156483,"style":487},[402],[363,156485,571],{"className":156486},[407,714],[363,156488,578],{"className":156489},[494],[363,156491,4502,156493],{"className":156492},[1832],[363,156494,4506],{"style":4505},[363,156496],{"className":156497,"style":754},[505],[363,156499,156501,156504],{"className":156500},[407],[363,156502,1121],{"className":156503},[407,408],[363,156505,156507],{"className":156506},[629],[363,156508,156510,156530],{"className":156509},[633,634],[363,156511,156513,156527],{"className":156512},[638],[363,156514,156516],{"className":156515,"style":12532},[642],[363,156517,156518,156521],{"style":730},[363,156519],{"className":156520,"style":651},[650],[363,156522,156524],{"className":156523},[655,656,657,658],[363,156525,12313],{"className":156526},[407,408,658],[363,156528,666],{"className":156529},[665],[363,156531,156533],{"className":156532},[638],[363,156534,156536],{"className":156535,"style":8316},[642],[363,156537],{},[363,156539,605],{"className":156540},[501],[363,156542],{"className":156543,"style":506},[505],[363,156545,471],{"className":156546},[510],[363,156548],{"className":156549,"style":506},[505],[363,156551,156553,156556,156559,156562,156602,156605,156608,156611],{"className":156552},[398],[363,156554],{"className":156555,"style":487},[402],[363,156557,155482],{"className":156558,"style":1328},[407,408],[363,156560,461],{"className":156561},[494],[363,156563,156565,156568],{"className":156564},[407],[363,156566,45126],{"className":156567,"style":45156},[407,408],[363,156569,156571],{"className":156570},[629],[363,156572,156574,156594],{"className":156573},[633,634],[363,156575,156577,156591],{"className":156576},[638],[363,156578,156580],{"className":156579,"style":12532},[642],[363,156581,156582,156585],{"style":136154},[363,156583],{"className":156584,"style":651},[650],[363,156586,156588],{"className":156587},[655,656,657,658],[363,156589,12313],{"className":156590},[407,408,658],[363,156592,666],{"className":156593},[665],[363,156595,156597],{"className":156596},[638],[363,156598,156600],{"className":156599,"style":8316},[642],[363,156601],{},[363,156603,468],{"className":156604},[501],[363,156606],{"className":156607,"style":2935},[505],[363,156609,5161],{"className":156610},[2939],[363,156612],{"className":156613,"style":2935},[505],[363,156615,156617,156620,156623,156626,156666],{"className":156616},[398],[363,156618],{"className":156619,"style":487},[402],[363,156621,155482],{"className":156622,"style":1328},[407,408],[363,156624,461],{"className":156625},[494],[363,156627,156629,156632],{"className":156628},[407],[363,156630,45126],{"className":156631,"style":45156},[407,408],[363,156633,156635],{"className":156634},[629],[363,156636,156638,156658],{"className":156637},[633,634],[363,156639,156641,156655],{"className":156640},[638],[363,156642,156644],{"className":156643,"style":11483},[642],[363,156645,156646,156649],{"style":136154},[363,156647],{"className":156648,"style":651},[650],[363,156650,156652],{"className":156651},[655,656,657,658],[363,156653,474],{"className":156654},[407,658],[363,156656,666],{"className":156657},[665],[363,156659,156661],{"className":156660},[638],[363,156662,156664],{"className":156663,"style":8316},[642],[363,156665],{},[363,156667,468],{"className":156668},[501],"（由命题 1.3 的矩恒等式对 ",[363,156671,156673,156691],{"className":156672,"translate":367},[366],[363,156674,156676],{"className":156675},[371],[373,156677,156678],{"xmlns":375},[377,156679,156680,156688],{},[380,156681,156682],{},[543,156683,156684,156686],{},[383,156685,84504],{},[383,156687,12313],{},[387,156689,156690],{"encoding":389},"\\beta_i",[363,156692,156694],{"className":156693,"ariaHidden":394},[393],[363,156695,156697,156700],{"className":156696},[398],[363,156698],{"className":156699,"style":403},[402],[363,156701,156703,156706],{"className":156702},[407],[363,156704,84504],{"className":156705,"style":84726},[407,408],[363,156707,156709],{"className":156708},[629],[363,156710,156712,156732],{"className":156711},[633,634],[363,156713,156715,156729],{"className":156714},[638],[363,156716,156718],{"className":156717,"style":12532},[642],[363,156719,156720,156723],{"style":150274},[363,156721],{"className":156722,"style":651},[650],[363,156724,156726],{"className":156725},[655,656,657,658],[363,156727,12313],{"className":156728},[407,408,658],[363,156730,666],{"className":156731},[665],[363,156733,156735],{"className":156734},[638],[363,156736,156738],{"className":156737,"style":8316},[642],[363,156739],{}," 求导得到），整理即得。",[11,156742,156743,156744,156806,156807,4201,156835,156886,156887,157024,157025,157117,157118,157168,157169,157172,157173,157246],{},"在对称情形 ",[363,156745,156747,156767],{"className":156746,"translate":367},[366],[363,156748,156750],{"className":156749},[371],[373,156751,156752],{"xmlns":375},[377,156753,156754,156764],{},[380,156755,156756,156758,156760,156762],{},[383,156757,45126],{"mathvariant":136055},[458,156759,471],{},[383,156761,45126],{},[463,156763,465],{"mathvariant":75598},[387,156765,156766],{"encoding":389},"\\boldsymbol{\\alpha} = \\alpha \\mathbf{1}",[363,156768,156770,156794],{"className":156769,"ariaHidden":394},[393],[363,156771,156773,156776,156785,156788,156791],{"className":156772},[398],[363,156774],{"className":156775,"style":135662},[402],[363,156777,156779],{"className":156778},[407],[363,156780,156782],{"className":156781},[407],[363,156783,45126],{"className":156784},[407,136115],[363,156786],{"className":156787,"style":506},[505],[363,156789,471],{"className":156790},[510],[363,156792],{"className":156793,"style":506},[505],[363,156795,156797,156800,156803],{"className":156796},[398],[363,156798],{"className":156799,"style":520},[402],[363,156801,45126],{"className":156802,"style":45156},[407,408],[363,156804,465],{"className":156805},[407,75943]," 下，",[363,156808,156810,156823],{"className":156809,"translate":367},[366],[363,156811,156813],{"className":156812},[371],[373,156814,156815],{"xmlns":375},[377,156816,156817,156821],{},[380,156818,156819],{},[383,156820,34256],{},[387,156822,34256],{"encoding":389},[363,156824,156826],{"className":156825,"ariaHidden":394},[393],[363,156827,156829,156832],{"className":156828},[398],[363,156830],{"className":156831,"style":1011},[402],[363,156833,34256],{"className":156834,"style":34410},[407,408],[363,156836,156838,156856],{"className":156837,"translate":367},[366],[363,156839,156841],{"className":156840},[371],[373,156842,156843],{"xmlns":375},[377,156844,156845,156853],{},[380,156846,156847,156849,156851],{},[383,156848,45126],{},[458,156850,471],{},[463,156852,465],{},[387,156854,156855],{"encoding":389},"\\alpha = 1",[363,156857,156859,156877],{"className":156858,"ariaHidden":394},[393],[363,156860,156862,156865,156868,156871,156874],{"className":156861},[398],[363,156863],{"className":156864,"style":2693},[402],[363,156866,45126],{"className":156867,"style":45156},[407,408],[363,156869],{"className":156870,"style":506},[505],[363,156872,471],{"className":156873},[510],[363,156875],{"className":156876,"style":506},[505],[363,156878,156880,156883],{"className":156879},[398],[363,156881],{"className":156882,"style":520},[402],[363,156884,465],{"className":156885},[407]," 处取最大值 ",[363,156888,156890,156936],{"className":156889,"translate":367},[366],[363,156891,156893],{"className":156892},[371],[373,156894,156895],{"xmlns":375},[377,156896,156897,156933],{},[380,156898,156899,156901,156903,156905,156907,156909,156911,156913,156915,156917,156919,156921,156923,156925,156927,156929,156931],{},[458,156900,5161],{},[383,156902,4448],{},[458,156904,4451],{},[383,156906,137218],{"mathvariant":557},[458,156908,461],{"stretchy":460},[383,156910,11044],{},[458,156912,468],{"stretchy":460},[458,156914,471],{},[458,156916,5161],{},[383,156918,4448],{},[458,156920,4451],{},[458,156922,461],{"stretchy":460},[383,156924,11044],{},[458,156926,5161],{},[463,156928,465],{},[458,156930,468],{"stretchy":460},[458,156932,135984],{"stretchy":460},[387,156934,156935],{"encoding":389},"-\\log \\Gamma(K) = -\\log (K-1)!",[363,156937,156939,156980,157012],{"className":156938,"ariaHidden":394},[393],[363,156940,156942,156945,156948,156951,156956,156959,156962,156965,156968,156971,156974,156977],{"className":156941},[398],[363,156943],{"className":156944,"style":487},[402],[363,156946,5161],{"className":156947},[407],[363,156949],{"className":156950,"style":754},[505],[363,156952,4502,156954],{"className":156953},[1832],[363,156955,4506],{"style":4505},[363,156957],{"className":156958,"style":754},[505],[363,156960,137218],{"className":156961},[407],[363,156963,461],{"className":156964},[494],[363,156966,11044],{"className":156967,"style":11132},[407,408],[363,156969,468],{"className":156970},[501],[363,156972],{"className":156973,"style":506},[505],[363,156975,471],{"className":156976},[510],[363,156978],{"className":156979,"style":506},[505],[363,156981,156983,156986,156989,156992,156997,157000,157003,157006,157009],{"className":156982},[398],[363,156984],{"className":156985,"style":487},[402],[363,156987,5161],{"className":156988},[407],[363,156990],{"className":156991,"style":754},[505],[363,156993,4502,156995],{"className":156994},[1832],[363,156996,4506],{"style":4505},[363,156998,461],{"className":156999},[494],[363,157001,11044],{"className":157002,"style":11132},[407,408],[363,157004],{"className":157005,"style":2935},[505],[363,157007,5161],{"className":157008},[2939],[363,157010],{"className":157011,"style":2935},[505],[363,157013,157015,157018,157021],{"className":157014},[398],[363,157016],{"className":157017,"style":487},[402],[363,157019,465],{"className":157020},[407],[363,157022,136027],{"className":157023},[501],"。该值为负，并不矛盾：它是相对于体积为 ",[363,157026,157028,157060],{"className":157027,"translate":367},[366],[363,157029,157031],{"className":157030},[371],[373,157032,157033],{"xmlns":375},[377,157034,157035,157057],{},[380,157036,157037,157039,157041,157043,157045,157047,157049,157051,157053,157055],{},[463,157038,465],{},[383,157040,1288],{"mathvariant":557},[458,157042,461],{"stretchy":460},[383,157044,11044],{},[458,157046,5161],{},[463,157048,465],{},[458,157050,468],{"stretchy":460},[458,157052,135984],{"stretchy":460},[458,157054,7599],{},[463,157056,465],{},[387,157058,157059],{"encoding":389},"1\u002F(K-1)! \u003C 1",[363,157061,157063,157087,157108],{"className":157062,"ariaHidden":394},[393],[363,157064,157066,157069,157072,157075,157078,157081,157084],{"className":157065},[398],[363,157067],{"className":157068,"style":487},[402],[363,157070,8937],{"className":157071},[407],[363,157073,461],{"className":157074},[494],[363,157076,11044],{"className":157077,"style":11132},[407,408],[363,157079],{"className":157080,"style":2935},[505],[363,157082,5161],{"className":157083},[2939],[363,157085],{"className":157086,"style":2935},[505],[363,157088,157090,157093,157096,157099,157102,157105],{"className":157089},[398],[363,157091],{"className":157092,"style":487},[402],[363,157094,465],{"className":157095},[407],[363,157097,136027],{"className":157098},[501],[363,157100],{"className":157101,"style":506},[505],[363,157103,7599],{"className":157104},[510],[363,157106],{"className":157107,"style":506},[505],[363,157109,157111,157114],{"className":157110},[398],[363,157112],{"className":157113,"style":520},[402],[363,157115,465],{"className":157116},[407]," 的支撑集而言的。",[363,157119,157121,157138],{"className":157120,"translate":367},[366],[363,157122,157124],{"className":157123},[371],[373,157125,157126],{"xmlns":375},[377,157127,157128,157136],{},[380,157129,157130,157132,157134],{},[383,157131,45126],{},[458,157133,471],{},[463,157135,465],{},[387,157137,156855],{"encoding":389},[363,157139,157141,157159],{"className":157140,"ariaHidden":394},[393],[363,157142,157144,157147,157150,157153,157156],{"className":157143},[398],[363,157145],{"className":157146,"style":2693},[402],[363,157148,45126],{"className":157149,"style":45156},[407,408],[363,157151],{"className":157152,"style":506},[505],[363,157154,471],{"className":157155},[510],[363,157157],{"className":157158,"style":506},[505],[363,157160,157162,157165],{"className":157161},[398],[363,157163],{"className":157164,"style":520},[402],[363,157166,465],{"className":157167},[407]," 对应单纯形上的",[1344,157170,157171],{},"均匀分布","——一个在其支撑集上平坦、却因支撑集体积小于一而具有负微分熵的密度。一般地，由于 ",[363,157174,157176,157199],{"className":157175,"translate":367},[366],[363,157177,157179],{"className":157178},[371],[373,157180,157181],{"xmlns":375},[377,157182,157183,157197],{},[380,157184,157185],{},[7162,157186,157187,157189],{},[383,157188,126067],{"mathvariant":557},[380,157190,157191,157193,157195],{},[383,157192,11044],{},[458,157194,5161],{},[463,157196,465],{},[387,157198,134819],{"encoding":389},[363,157200,157202],{"className":157201,"ariaHidden":394},[393],[363,157203,157205,157208],{"className":157204},[398],[363,157206],{"className":157207,"style":52457},[402],[363,157209,157211,157214],{"className":157210},[407],[363,157212,126067],{"className":157213},[407],[363,157215,157217],{"className":157216},[629],[363,157218,157220],{"className":157219},[633],[363,157221,157223],{"className":157222},[638],[363,157224,157226],{"className":157225,"style":52457},[642],[363,157227,157228,157231],{"style":7213},[363,157229],{"className":157230,"style":651},[650],[363,157232,157234],{"className":157233},[655,656,657,658],[363,157235,157237,157240,157243],{"className":157236},[407,658],[363,157238,11044],{"className":157239,"style":11132},[407,408,658],[363,157241,5161],{"className":157242},[2939,658],[363,157244,465],{"className":157245},[407,658]," 紧且测度有限，均匀分布是最大熵分布，故",[363,157248,157250],{"className":157249,"translate":367},[528],[363,157251,157253,157308],{"className":157252,"translate":367},[366],[363,157254,157256],{"className":157255},[371],[373,157257,157258],{"xmlns":375,"display":537},[377,157259,157260,157305],{},[380,157261,157262,157264,157266,157268,157270,157272,157274,157276,157278,157280,157282,157284,157286,157288,157290,157293,157295,157297,157299,157301,157303],{},[383,157263,34256],{},[458,157265,461],{"stretchy":460},[383,157267,1121],{"mathvariant":75598},[458,157269,468],{"stretchy":460},[458,157271,10300],{},[458,157273,5161],{},[383,157275,4448],{},[458,157277,4451],{},[383,157279,137218],{"mathvariant":557},[458,157281,461],{"stretchy":460},[383,157283,11044],{},[458,157285,468],{"stretchy":460},[458,157287,898],{"separator":394},[505,157289],{"width":3610},[1645,157291,157292],{},"等号成立",[1645,157294,16254],{},[458,157296,96444],{},[1645,157298,16254],{},[383,157300,45126],{"mathvariant":136055},[458,157302,471],{},[463,157304,465],{"mathvariant":75598},[387,157306,157307],{"encoding":389},"H(\\mathbf{x}) \\le -\\log \\Gamma(K), \\qquad \\text{等号成立} \\iff \\boldsymbol{\\alpha} = \\mathbf{1}",[363,157309,157311,157338,157400,157424],{"className":157310,"ariaHidden":394},[393],[363,157312,157314,157317,157320,157323,157326,157329,157332,157335],{"className":157313},[398],[363,157315],{"className":157316,"style":487},[402],[363,157318,34256],{"className":157319,"style":34410},[407,408],[363,157321,461],{"className":157322},[494],[363,157324,1121],{"className":157325},[407,75943],[363,157327,468],{"className":157328},[501],[363,157330],{"className":157331,"style":506},[505],[363,157333,10300],{"className":157334},[510],[363,157336],{"className":157337,"style":506},[505],[363,157339,157341,157344,157347,157350,157355,157358,157361,157364,157367,157370,157373,157376,157379,157385,157388,157391,157394,157397],{"className":157340},[398],[363,157342],{"className":157343,"style":487},[402],[363,157345,5161],{"className":157346},[407],[363,157348],{"className":157349,"style":754},[505],[363,157351,4502,157353],{"className":157352},[1832],[363,157354,4506],{"style":4505},[363,157356],{"className":157357,"style":754},[505],[363,157359,137218],{"className":157360},[407],[363,157362,461],{"className":157363},[494],[363,157365,11044],{"className":157366,"style":11132},[407,408],[363,157368,468],{"className":157369},[501],[363,157371,898],{"className":157372},[920],[363,157374],{"className":157375,"style":3982},[505],[363,157377],{"className":157378,"style":754},[505],[363,157380,157382],{"className":157381},[407,4946],[363,157383,157292],{"className":157384},[407,4950],[363,157386],{"className":157387,"style":506},[505],[363,157389],{"className":157390,"style":506},[505],[363,157392,96444],{"className":157393},[510],[363,157395],{"className":157396,"style":506},[505],[363,157398],{"className":157399,"style":506},[505],[363,157401,157403,157406,157415,157418,157421],{"className":157402},[398],[363,157404],{"className":157405,"style":135662},[402],[363,157407,157409],{"className":157408},[407],[363,157410,157412],{"className":157411},[407],[363,157413,45126],{"className":157414},[407,136115],[363,157416],{"className":157417,"style":506},[505],[363,157419,471],{"className":157420},[510],[363,157422],{"className":157423,"style":506},[505],[363,157425,157427,157430],{"className":157426},[398],[363,157428],{"className":157429,"style":520},[402],[363,157431,465],{"className":157432},[407,75943],[2659,157434,157436],{"id":157435},"_24-共轭性与贝叶斯更新","2.4 共轭性与贝叶斯更新",[143,157438,157439],{},[11,157440,157441,157444,157445,157530,157531,157566,157567,16145,157739,3093,157831,157972,157973],{},[1344,157442,157443],{},"命题 2.5（Dirichlet–Multinomial 共轭）"," 设先验 ",[363,157446,157448,157475],{"className":157447,"translate":367},[366],[363,157449,157451],{"className":157450},[371],[373,157452,157453],{"xmlns":375},[377,157454,157455,157472],{},[380,157456,157457,157460,157462,157464,157466,157468,157470],{},[383,157458,157459],{"mathvariant":136055},"π",[458,157461,9217],{},[383,157463,137112],{"mathvariant":557},[458,157465,4451],{},[458,157467,461],{"stretchy":460},[383,157469,45126],{"mathvariant":136055},[458,157471,468],{"stretchy":460},[387,157473,157474],{"encoding":389},"\\boldsymbol{\\pi} \\sim \\operatorname{Dir}(\\boldsymbol{\\alpha})",[363,157476,157478,157503],{"className":157477,"ariaHidden":394},[393],[363,157479,157481,157484,157494,157497,157500],{"className":157480},[398],[363,157482],{"className":157483,"style":135662},[402],[363,157485,157487],{"className":157486},[407],[363,157488,157490],{"className":157489},[407],[363,157491,157459],{"className":157492,"style":157493},[407,136115],"margin-right:0.037em;",[363,157495],{"className":157496,"style":506},[505],[363,157498,9217],{"className":157499},[510],[363,157501],{"className":157502,"style":506},[505],[363,157504,157506,157509,157515,157518,157527],{"className":157505},[398],[363,157507],{"className":157508,"style":487},[402],[363,157510,157512],{"className":157511},[1832],[363,157513,137112],{"className":157514},[407,33034],[363,157516,461],{"className":157517},[494],[363,157519,157521],{"className":157520},[407],[363,157522,157524],{"className":157523},[407],[363,157525,45126],{"className":157526},[407,136115],[363,157528,468],{"className":157529},[501],"，给定 ",[363,157532,157534,157548],{"className":157533,"translate":367},[366],[363,157535,157537],{"className":157536},[371],[373,157538,157539],{"xmlns":375},[377,157540,157541,157545],{},[380,157542,157543],{},[383,157544,157459],{"mathvariant":136055},[387,157546,157547],{"encoding":389},"\\boldsymbol{\\pi}",[363,157549,157551],{"className":157550,"ariaHidden":394},[393],[363,157552,157554,157557],{"className":157553},[398],[363,157555],{"className":157556,"style":135662},[402],[363,157558,157560],{"className":157559},[407],[363,157561,157563],{"className":157562},[407],[363,157564,157459],{"className":157565,"style":157493},[407,136115]," 观测到计数向量 ",[363,157568,157570,157608],{"className":157569,"translate":367},[366],[363,157571,157573],{"className":157572},[371],[373,157574,157575],{"xmlns":375},[377,157576,157577,157605],{},[380,157578,157579,157581,157583,157585,157591,157593,157595,157597,157603],{},[383,157580,82683],{"mathvariant":75598},[458,157582,471],{},[458,157584,461],{"stretchy":460},[543,157586,157587,157589],{},[383,157588,82683],{},[463,157590,465],{},[458,157592,898],{"separator":394},[458,157594,133150],{},[458,157596,898],{"separator":394},[543,157598,157599,157601],{},[383,157600,82683],{},[383,157602,11044],{},[458,157604,468],{"stretchy":460},[387,157606,157607],{"encoding":389},"\\mathbf{n} = (n_1, \\dots, n_K)",[363,157609,157611,157629],{"className":157610,"ariaHidden":394},[393],[363,157612,157614,157617,157620,157623,157626],{"className":157613},[398],[363,157615],{"className":157616,"style":135662},[402],[363,157618,82683],{"className":157619},[407,75943],[363,157621],{"className":157622,"style":506},[505],[363,157624,471],{"className":157625},[510],[363,157627],{"className":157628,"style":506},[505],[363,157630,157632,157635,157638,157678,157681,157684,157687,157690,157693,157696,157736],{"className":157631},[398],[363,157633],{"className":157634,"style":487},[402],[363,157636,461],{"className":157637},[494],[363,157639,157641,157644],{"className":157640},[407],[363,157642,82683],{"className":157643},[407,408],[363,157645,157647],{"className":157646},[629],[363,157648,157650,157670],{"className":157649},[633,634],[363,157651,157653,157667],{"className":157652},[638],[363,157654,157656],{"className":157655,"style":11483},[642],[363,157657,157658,157661],{"style":730},[363,157659],{"className":157660,"style":651},[650],[363,157662,157664],{"className":157663},[655,656,657,658],[363,157665,465],{"className":157666},[407,658],[363,157668,666],{"className":157669},[665],[363,157671,157673],{"className":157672},[638],[363,157674,157676],{"className":157675,"style":8316},[642],[363,157677],{},[363,157679,898],{"className":157680},[920],[363,157682],{"className":157683,"style":754},[505],[363,157685,133150],{"className":157686},[758],[363,157688],{"className":157689,"style":754},[505],[363,157691,898],{"className":157692},[920],[363,157694],{"className":157695,"style":754},[505],[363,157697,157699,157702],{"className":157698},[407],[363,157700,82683],{"className":157701},[407,408],[363,157703,157705],{"className":157704},[629],[363,157706,157708,157728],{"className":157707},[633,634],[363,157709,157711,157725],{"className":157710},[638],[363,157712,157714],{"className":157713,"style":727},[642],[363,157715,157716,157719],{"style":730},[363,157717],{"className":157718,"style":651},[650],[363,157720,157722],{"className":157721},[655,656,657,658],[363,157723,11044],{"className":157724,"style":11132},[407,408,658],[363,157726,666],{"className":157727},[665],[363,157729,157731],{"className":157730},[638],[363,157732,157734],{"className":157733,"style":8316},[642],[363,157735],{},[363,157737,468],{"className":157738},[501],[363,157740,157742,157764],{"className":157741,"translate":367},[366],[363,157743,157745],{"className":157744},[371],[373,157746,157747],{"xmlns":375},[377,157748,157749,157761],{},[380,157750,157751,157757,157759],{},[543,157752,157753,157755],{},[383,157754,82683],{},[383,157756,12313],{},[458,157758,6257],{},[383,157760,38656],{"mathvariant":570},[387,157762,157763],{"encoding":389},"n_i \\in \\mathbb{N}",[363,157765,157767,157822],{"className":157766,"ariaHidden":394},[393],[363,157768,157770,157773,157813,157816,157819],{"className":157769},[398],[363,157771],{"className":157772,"style":134714},[402],[363,157774,157776,157779],{"className":157775},[407],[363,157777,82683],{"className":157778},[407,408],[363,157780,157782],{"className":157781},[629],[363,157783,157785,157805],{"className":157784},[633,634],[363,157786,157788,157802],{"className":157787},[638],[363,157789,157791],{"className":157790,"style":12532},[642],[363,157792,157793,157796],{"style":730},[363,157794],{"className":157795,"style":651},[650],[363,157797,157799],{"className":157798},[655,656,657,658],[363,157800,12313],{"className":157801},[407,408,658],[363,157803,666],{"className":157804},[665],[363,157806,157808],{"className":157807},[638],[363,157809,157811],{"className":157810,"style":8316},[642],[363,157812],{},[363,157814],{"className":157815,"style":506},[505],[363,157817,6257],{"className":157818},[510],[363,157820],{"className":157821,"style":506},[505],[363,157823,157825,157828],{"className":157824},[398],[363,157826],{"className":157827,"style":1445},[402],[363,157829,38656],{"className":157830},[407,714],[363,157832,157834,157862],{"className":157833,"translate":367},[366],[363,157835,157837],{"className":157836},[371],[373,157838,157839],{"xmlns":375},[377,157840,157841,157859],{},[380,157842,157843,157845,157847,157853],{},[383,157844,38656],{},[458,157846,471],{},[543,157848,157849,157851],{},[458,157850,13811],{},[383,157852,12313],{},[543,157854,157855,157857],{},[383,157856,82683],{},[383,157858,12313],{},[387,157860,157861],{"encoding":389},"N = \\sum_i n_i",[363,157863,157865,157883],{"className":157864,"ariaHidden":394},[393],[363,157866,157868,157871,157874,157877,157880],{"className":157867},[398],[363,157869],{"className":157870,"style":1011},[402],[363,157872,38656],{"className":157873,"style":25973},[407,408],[363,157875],{"className":157876,"style":506},[505],[363,157878,471],{"className":157879},[510],[363,157881],{"className":157882,"style":506},[505],[363,157884,157886,157889,157929,157932],{"className":157885},[398],[363,157887],{"className":157888,"style":138953},[402],[363,157890,157892,157895],{"className":157891},[1832],[363,157893,13811],{"className":157894,"style":40032},[1832,1833,5964],[363,157896,157898],{"className":157897},[629],[363,157899,157901,157921],{"className":157900},[633,634],[363,157902,157904,157918],{"className":157903},[638],[363,157905,157907],{"className":157906,"style":40045},[642],[363,157908,157909,157912],{"style":40048},[363,157910],{"className":157911,"style":651},[650],[363,157913,157915],{"className":157914},[655,656,657,658],[363,157916,12313],{"className":157917},[407,408,658],[363,157919,666],{"className":157920},[665],[363,157922,157924],{"className":157923},[638],[363,157925,157927],{"className":157926,"style":40067},[642],[363,157928],{},[363,157930],{"className":157931,"style":754},[505],[363,157933,157935,157938],{"className":157934},[407],[363,157936,82683],{"className":157937},[407,408],[363,157939,157941],{"className":157940},[629],[363,157942,157944,157964],{"className":157943},[633,634],[363,157945,157947,157961],{"className":157946},[638],[363,157948,157950],{"className":157949,"style":12532},[642],[363,157951,157952,157955],{"style":730},[363,157953],{"className":157954,"style":651},[650],[363,157956,157958],{"className":157957},[655,656,657,658],[363,157959,12313],{"className":157960},[407,408,658],[363,157962,666],{"className":157963},[665],[363,157965,157967],{"className":157966},[638],[363,157968,157970],{"className":157969,"style":8316},[642],[363,157971],{},"），则后验为\n",[363,157974,157976,158010],{"className":157975,"translate":367},[366],[363,157977,157979],{"className":157978},[371],[373,157980,157981],{"xmlns":375},[377,157982,157983,158007],{},[380,157984,157985,157987,157989,157991,157993,157995,157997,157999,158001,158003,158005],{},[383,157986,157459],{"mathvariant":136055},[458,157988,5156],{},[383,157990,82683],{"mathvariant":75598},[458,157992,9217],{},[383,157994,137112],{"mathvariant":557},[458,157996,4451],{},[458,157998,461],{"stretchy":460},[383,158000,45126],{"mathvariant":136055},[458,158002,2818],{},[383,158004,82683],{"mathvariant":75598},[458,158006,468],{"stretchy":460},[387,158008,158009],{"encoding":389},"\\boldsymbol{\\pi} \\mid \\mathbf{n} \\sim \\operatorname{Dir}(\\boldsymbol{\\alpha} + \\mathbf{n})",[363,158011,158013,158037,158055,158088],{"className":158012,"ariaHidden":394},[393],[363,158014,158016,158019,158028,158031,158034],{"className":158015},[398],[363,158017],{"className":158018,"style":487},[402],[363,158020,158022],{"className":158021},[407],[363,158023,158025],{"className":158024},[407],[363,158026,157459],{"className":158027,"style":157493},[407,136115],[363,158029],{"className":158030,"style":506},[505],[363,158032,5156],{"className":158033},[510],[363,158035],{"className":158036,"style":506},[505],[363,158038,158040,158043,158046,158049,158052],{"className":158039},[398],[363,158041],{"className":158042,"style":135662},[402],[363,158044,82683],{"className":158045},[407,75943],[363,158047],{"className":158048,"style":506},[505],[363,158050,9217],{"className":158051},[510],[363,158053],{"className":158054,"style":506},[505],[363,158056,158058,158061,158067,158070,158079,158082,158085],{"className":158057},[398],[363,158059],{"className":158060,"style":487},[402],[363,158062,158064],{"className":158063},[1832],[363,158065,137112],{"className":158066},[407,33034],[363,158068,461],{"className":158069},[494],[363,158071,158073],{"className":158072},[407],[363,158074,158076],{"className":158075},[407],[363,158077,45126],{"className":158078},[407,136115],[363,158080],{"className":158081,"style":2935},[505],[363,158083,2818],{"className":158084},[2939],[363,158086],{"className":158087,"style":2935},[505],[363,158089,158091,158094,158097],{"className":158090},[398],[363,158092],{"className":158093,"style":487},[402],[363,158095,82683],{"className":158096},[407,75943],[363,158098,468],{"className":158099},[501],[11,158101,158102,158104,158105,158282],{},[1344,158103,138042],{}," 多项似然正比于 ",[363,158106,158108,158138],{"className":158107,"translate":367},[366],[363,158109,158111],{"className":158110},[371],[373,158112,158113],{"xmlns":375},[377,158114,158115,158135],{},[380,158116,158117,158123],{},[543,158118,158119,158121],{},[458,158120,136684],{},[383,158122,12313],{},[7901,158124,158125,158127,158129],{},[383,158126,157459],{},[383,158128,12313],{},[543,158130,158131,158133],{},[383,158132,82683],{},[383,158134,12313],{},[387,158136,158137],{"encoding":389},"\\prod_i \\pi_i^{n_i}",[363,158139,158141],{"className":158140,"ariaHidden":394},[393],[363,158142,158144,158147,158187,158190],{"className":158143},[398],[363,158145],{"className":158146,"style":138953},[402],[363,158148,158150,158153],{"className":158149},[1832],[363,158151,136684],{"className":158152,"style":40032},[1832,1833,5964],[363,158154,158156],{"className":158155},[629],[363,158157,158159,158179],{"className":158158},[633,634],[363,158160,158162,158176],{"className":158161},[638],[363,158163,158165],{"className":158164,"style":40045},[642],[363,158166,158167,158170],{"style":40048},[363,158168],{"className":158169,"style":651},[650],[363,158171,158173],{"className":158172},[655,656,657,658],[363,158174,12313],{"className":158175},[407,408,658],[363,158177,666],{"className":158178},[665],[363,158180,158182],{"className":158181},[638],[363,158183,158185],{"className":158184,"style":40067},[642],[363,158186],{},[363,158188],{"className":158189,"style":754},[505],[363,158191,158193,158196],{"className":158192},[407],[363,158194,157459],{"className":158195,"style":1328},[407,408],[363,158197,158199],{"className":158198},[629],[363,158200,158202,158274],{"className":158201},[633,634],[363,158203,158205,158271],{"className":158204},[638],[363,158206,158208,158220],{"className":158207,"style":45831},[642],[363,158209,158211,158214],{"style":158210},"top:-2.4231em;margin-left:-0.0359em;margin-right:0.05em;",[363,158212],{"className":158213,"style":651},[650],[363,158215,158217],{"className":158216},[655,656,657,658],[363,158218,12313],{"className":158219},[407,408,658],[363,158221,158222,158225],{"style":136959},[363,158223],{"className":158224,"style":651},[650],[363,158226,158228],{"className":158227},[655,656,657,658],[363,158229,158231],{"className":158230},[407,658],[363,158232,158234,158237],{"className":158233},[407,658],[363,158235,82683],{"className":158236},[407,408,658],[363,158238,158240],{"className":158239},[629],[363,158241,158243,158263],{"className":158242},[633,634],[363,158244,158246,158260],{"className":158245},[638],[363,158247,158249],{"className":158248,"style":40120},[642],[363,158250,158251,158254],{"style":74648},[363,158252],{"className":158253,"style":30578},[650],[363,158255,158257],{"className":158256},[655,30582,10022,658],[363,158258,12313],{"className":158259},[407,408,658],[363,158261,666],{"className":158262},[665],[363,158264,158266],{"className":158265},[638],[363,158267,158269],{"className":158268,"style":40142},[642],[363,158270],{},[363,158272,666],{"className":158273},[665],[363,158275,158277],{"className":158276},[638],[363,158278,158280],{"className":158279,"style":137025},[642],[363,158281],{},"，与先验密度相乘得",[363,158284,158286],{"className":158285,"translate":367},[528],[363,158287,158289,158356],{"className":158288,"translate":367},[366],[363,158290,158292],{"className":158291},[371],[373,158293,158294],{"xmlns":375,"display":537},[377,158295,158296,158353],{},[380,158297,158298,158300,158302,158304,158306,158308,158310,158313,158327],{},[383,158299,11],{},[458,158301,461],{"stretchy":460},[383,158303,157459],{"mathvariant":136055},[458,158305,5156],{},[383,158307,82683],{"mathvariant":75598},[458,158309,468],{"stretchy":460},[458,158311,158312],{},"∝",[48692,158314,158315,158317,158325],{},[458,158316,136684],{},[380,158318,158319,158321,158323],{},[383,158320,12313],{},[458,158322,471],{},[463,158324,465],{},[383,158326,11044],{},[7901,158328,158329,158331,158333],{},[383,158330,157459],{},[383,158332,12313],{},[380,158334,158335,158341,158343,158349,158351],{},[543,158336,158337,158339],{},[383,158338,45126],{},[383,158340,12313],{},[458,158342,2818],{},[543,158344,158345,158347],{},[383,158346,82683],{},[383,158348,12313],{},[458,158350,5161],{},[463,158352,465],{},[387,158354,158355],{"encoding":389},"p(\\boldsymbol{\\pi} \\mid \\mathbf{n}) \\propto \\prod_{i=1}^K \\pi_i^{\\alpha_i + n_i - 1}",[363,158357,158359,158389,158410],{"className":158358,"ariaHidden":394},[393],[363,158360,158362,158365,158368,158371,158380,158383,158386],{"className":158361},[398],[363,158363],{"className":158364,"style":487},[402],[363,158366,11],{"className":158367},[407,408],[363,158369,461],{"className":158370},[494],[363,158372,158374],{"className":158373},[407],[363,158375,158377],{"className":158376},[407],[363,158378,157459],{"className":158379,"style":157493},[407,136115],[363,158381],{"className":158382,"style":506},[505],[363,158384,5156],{"className":158385},[510],[363,158387],{"className":158388,"style":506},[505],[363,158390,158392,158395,158398,158401,158404,158407],{"className":158391},[398],[363,158393],{"className":158394,"style":487},[402],[363,158396,82683],{"className":158397},[407,75943],[363,158399,468],{"className":158400},[501],[363,158402],{"className":158403,"style":506},[505],[363,158405,158312],{"className":158406},[510],[363,158408],{"className":158409,"style":506},[505],[363,158411,158413,158416,158480,158483],{"className":158412},[398],[363,158414],{"className":158415,"style":138261},[402],[363,158417,158419],{"className":158418},[1832,14066],[363,158420,158422,158472],{"className":158421},[633,634],[363,158423,158425,158469],{"className":158424},[638],[363,158426,158428,158448,158458],{"className":158427,"style":138274},[642],[363,158429,158430,158433],{"style":14079},[363,158431],{"className":158432,"style":14083},[650],[363,158434,158436],{"className":158435},[655,656,657,658],[363,158437,158439,158442,158445],{"className":158438},[407,658],[363,158440,12313],{"className":158441},[407,408,658],[363,158443,471],{"className":158444},[510,658],[363,158446,465],{"className":158447},[407,658],[363,158449,158450,158453],{"style":14092},[363,158451],{"className":158452,"style":14083},[650],[363,158454,158455],{},[363,158456,136684],{"className":158457},[1832,1833,1834],[363,158459,158460,158463],{"style":100713},[363,158461],{"className":158462,"style":14083},[650],[363,158464,158466],{"className":158465},[655,656,657,658],[363,158467,11044],{"className":158468,"style":11132},[407,408,658],[363,158470,666],{"className":158471},[665],[363,158473,158475],{"className":158474},[638],[363,158476,158478],{"className":158477,"style":14110},[642],[363,158479],{},[363,158481],{"className":158482,"style":754},[505],[363,158484,158486,158489],{"className":158485},[407],[363,158487,157459],{"className":158488,"style":1328},[407,408],[363,158490,158492],{"className":158491},[629],[363,158493,158495,158615],{"className":158494},[633,634],[363,158496,158498,158612],{"className":158497},[638],[363,158499,158501,158512],{"className":158500,"style":136944},[642],[363,158502,158503,158506],{"style":158210},[363,158504],{"className":158505,"style":651},[650],[363,158507,158509],{"className":158508},[655,656,657,658],[363,158510,12313],{"className":158511},[407,408,658],[363,158513,158514,158517],{"style":136959},[363,158515],{"className":158516,"style":651},[650],[363,158518,158520],{"className":158519},[655,656,657,658],[363,158521,158523,158563,158566,158606,158609],{"className":158522},[407,658],[363,158524,158526,158529],{"className":158525},[407,658],[363,158527,45126],{"className":158528,"style":45156},[407,408,658],[363,158530,158532],{"className":158531},[629],[363,158533,158535,158555],{"className":158534},[633,634],[363,158536,158538,158552],{"className":158537},[638],[363,158539,158541],{"className":158540,"style":40120},[642],[363,158542,158543,158546],{"style":136989},[363,158544],{"className":158545,"style":30578},[650],[363,158547,158549],{"className":158548},[655,30582,10022,658],[363,158550,12313],{"className":158551},[407,408,658],[363,158553,666],{"className":158554},[665],[363,158556,158558],{"className":158557},[638],[363,158559,158561],{"className":158560,"style":40142},[642],[363,158562],{},[363,158564,2818],{"className":158565},[2939,658],[363,158567,158569,158572],{"className":158568},[407,658],[363,158570,82683],{"className":158571},[407,408,658],[363,158573,158575],{"className":158574},[629],[363,158576,158578,158598],{"className":158577},[633,634],[363,158579,158581,158595],{"className":158580},[638],[363,158582,158584],{"className":158583,"style":40120},[642],[363,158585,158586,158589],{"style":74648},[363,158587],{"className":158588,"style":30578},[650],[363,158590,158592],{"className":158591},[655,30582,10022,658],[363,158593,12313],{"className":158594},[407,408,658],[363,158596,666],{"className":158597},[665],[363,158599,158601],{"className":158600},[638],[363,158602,158604],{"className":158603,"style":40142},[642],[363,158605],{},[363,158607,5161],{"className":158608},[2939,658],[363,158610,465],{"className":158611},[407,658],[363,158613,666],{"className":158614},[665],[363,158616,158618],{"className":158617},[638],[363,158619,158621],{"className":158620,"style":137025},[642],[363,158622],{},[11,158624,158625,158626,158703],{},"该核与 ",[363,158627,158629,158655],{"className":158628,"translate":367},[366],[363,158630,158632],{"className":158631},[371],[373,158633,158634],{"xmlns":375},[377,158635,158636,158652],{},[380,158637,158638,158640,158642,158644,158646,158648,158650],{},[383,158639,137112],{"mathvariant":557},[458,158641,4451],{},[458,158643,461],{"stretchy":460},[383,158645,45126],{"mathvariant":136055},[458,158647,2818],{},[383,158649,82683],{"mathvariant":75598},[458,158651,468],{"stretchy":460},[387,158653,158654],{"encoding":389},"\\operatorname{Dir}(\\boldsymbol{\\alpha} + \\mathbf{n})",[363,158656,158658,158691],{"className":158657,"ariaHidden":394},[393],[363,158659,158661,158664,158670,158673,158682,158685,158688],{"className":158660},[398],[363,158662],{"className":158663,"style":487},[402],[363,158665,158667],{"className":158666},[1832],[363,158668,137112],{"className":158669},[407,33034],[363,158671,461],{"className":158672},[494],[363,158674,158676],{"className":158675},[407],[363,158677,158679],{"className":158678},[407],[363,158680,45126],{"className":158681},[407,136115],[363,158683],{"className":158684,"style":2935},[505],[363,158686,2818],{"className":158687},[2939],[363,158689],{"className":158690,"style":2935},[505],[363,158692,158694,158697,158700],{"className":158693},[398],[363,158695],{"className":158696,"style":487},[402],[363,158698,82683],{"className":158699},[407,75943],[363,158701,468],{"className":158702},[501]," 的核成比例，归一化即得。",[11,158705,158706],{},"由共轭性可立即导出两个推论，它们在应用中的地位不低于命题本身。",[143,158708,158709],{},[11,158710,158711,158714,158715,158743,158744],{},[1344,158712,158713],{},"推论 2.6（Dirichlet–Multinomial 分布）"," 计数向量 ",[363,158716,158718,158731],{"className":158717,"translate":367},[366],[363,158719,158721],{"className":158720},[371],[373,158722,158723],{"xmlns":375},[377,158724,158725,158729],{},[380,158726,158727],{},[383,158728,82683],{"mathvariant":75598},[387,158730,147409],{"encoding":389},[363,158732,158734],{"className":158733,"ariaHidden":394},[393],[363,158735,158737,158740],{"className":158736},[398],[363,158738],{"className":158739,"style":135662},[402],[363,158741,82683],{"className":158742},[407,75943]," 的边缘似然为\n",[363,158745,158747,158913],{"className":158746,"translate":367},[366],[363,158748,158750],{"className":158749},[371],[373,158751,158752],{"xmlns":375},[377,158753,158754,158910],{},[380,158755,158756,158758,158760,158762,158764,158766,158768,158770,158794,158796,158822,158824,158858,158872],{},[383,158757,11],{},[458,158759,461],{"stretchy":460},[383,158761,82683],{"mathvariant":75598},[458,158763,5156],{},[383,158765,45126],{"mathvariant":136055},[458,158767,468],{"stretchy":460},[458,158769,471],{},[586,158771,158772,158778],{},[380,158773,158774,158776],{},[383,158775,38656],{},[458,158777,135984],{"stretchy":460},[380,158779,158780,158786,158792],{},[543,158781,158782,158784],{},[458,158783,136684],{},[383,158785,12313],{},[543,158787,158788,158790],{},[383,158789,82683],{},[383,158791,12313],{},[458,158793,135984],{"stretchy":460},[458,158795,3647],{},[586,158797,158798,158812],{},[380,158799,158800,158802,158804,158806,158808,158810],{},[383,158801,39663],{},[458,158803,461],{"stretchy":460},[383,158805,45126],{"mathvariant":136055},[458,158807,2818],{},[383,158809,82683],{"mathvariant":75598},[458,158811,468],{"stretchy":460},[380,158813,158814,158816,158818,158820],{},[383,158815,39663],{},[458,158817,461],{"stretchy":460},[383,158819,45126],{"mathvariant":136055},[458,158821,468],{"stretchy":460},[458,158823,471],{},[586,158825,158826,158840],{},[380,158827,158828,158830,158832,158838],{},[383,158829,137218],{"mathvariant":557},[458,158831,461],{"stretchy":460},[543,158833,158834,158836],{},[383,158835,45126],{},[463,158837,474],{},[458,158839,468],{"stretchy":460},[380,158841,158842,158844,158846,158852,158854,158856],{},[383,158843,137218],{"mathvariant":557},[458,158845,461],{"stretchy":460},[543,158847,158848,158850],{},[383,158849,45126],{},[463,158851,474],{},[458,158853,2818],{},[383,158855,38656],{},[458,158857,468],{"stretchy":460},[7901,158859,158860,158862,158870],{},[458,158861,136684],{},[380,158863,158864,158866,158868],{},[383,158865,12313],{},[458,158867,471],{},[463,158869,465],{},[383,158871,11044],{},[586,158873,158874,158896],{},[380,158875,158876,158878,158880,158886,158888,158894],{},[383,158877,137218],{"mathvariant":557},[458,158879,461],{"stretchy":460},[543,158881,158882,158884],{},[383,158883,45126],{},[383,158885,12313],{},[458,158887,2818],{},[543,158889,158890,158892],{},[383,158891,82683],{},[383,158893,12313],{},[458,158895,468],{"stretchy":460},[380,158897,158898,158900,158902,158908],{},[383,158899,137218],{"mathvariant":557},[458,158901,461],{"stretchy":460},[543,158903,158904,158906],{},[383,158905,45126],{},[383,158907,12313],{},[458,158909,468],{"stretchy":460},[387,158911,158912],{"encoding":389},"p(\\mathbf{n} \\mid \\boldsymbol{\\alpha}) = \\frac{N!}{\\prod_i n_i!} \\cdot \\frac{B(\\boldsymbol{\\alpha} + \\mathbf{n})}{B(\\boldsymbol{\\alpha})} = \\frac{\\Gamma(\\alpha_0)}{\\Gamma(\\alpha_0 + N)} \\prod_{i=1}^K \\frac{\\Gamma(\\alpha_i + n_i)}{\\Gamma(\\alpha_i)}",[363,158914,158916,158940,158967,159138,159257],{"className":158915,"ariaHidden":394},[393],[363,158917,158919,158922,158925,158928,158931,158934,158937],{"className":158918},[398],[363,158920],{"className":158921,"style":487},[402],[363,158923,11],{"className":158924},[407,408],[363,158926,461],{"className":158927},[494],[363,158929,82683],{"className":158930},[407,75943],[363,158932],{"className":158933,"style":506},[505],[363,158935,5156],{"className":158936},[510],[363,158938],{"className":158939,"style":506},[505],[363,158941,158943,158946,158955,158958,158961,158964],{"className":158942},[398],[363,158944],{"className":158945,"style":487},[402],[363,158947,158949],{"className":158948},[407],[363,158950,158952],{"className":158951},[407],[363,158953,45126],{"className":158954},[407,136115],[363,158956,468],{"className":158957},[501],[363,158959],{"className":158960,"style":506},[505],[363,158962,471],{"className":158963},[510],[363,158965],{"className":158966,"style":506},[505],[363,158968,158970,158974,159129,159132,159135],{"className":158969},[398],[363,158971],{"className":158972,"style":158973},[402],"height:1.4501em;vertical-align:-0.57em;",[363,158975,158977,158980,159126],{"className":158976},[407],[363,158978],{"className":158979},[494,789],[363,158981,158983],{"className":158982},[586],[363,158984,158986,159117],{"className":158985},[633,634],[363,158987,158989,159114],{"className":158988},[638],[363,158990,158992,159089,159097],{"className":158991,"style":3035},[642],[363,158993,158994,158997],{"style":2875},[363,158995],{"className":158996,"style":809},[650],[363,158998,159000],{"className":158999},[655,656,657,658],[363,159001,159003,159043,159046,159086],{"className":159002},[407,658],[363,159004,159006,159009],{"className":159005},[1832,658],[363,159007,136684],{"className":159008,"style":40032},[1832,1833,5964,658],[363,159010,159012],{"className":159011},[629],[363,159013,159015,159035],{"className":159014},[633,634],[363,159016,159018,159032],{"className":159017},[638],[363,159019,159021],{"className":159020,"style":140361},[642],[363,159022,159023,159026],{"style":137421},[363,159024],{"className":159025,"style":30578},[650],[363,159027,159029],{"className":159028},[655,30582,10022,658],[363,159030,12313],{"className":159031},[407,408,658],[363,159033,666],{"className":159034},[665],[363,159036,159038],{"className":159037},[638],[363,159039,159041],{"className":159040,"style":137460},[642],[363,159042],{},[363,159044],{"className":159045,"style":42738},[505,658],[363,159047,159049,159052],{"className":159048},[407,658],[363,159050,82683],{"className":159051},[407,408,658],[363,159053,159055],{"className":159054},[629],[363,159056,159058,159078],{"className":159057},[633,634],[363,159059,159061,159075],{"className":159060},[638],[363,159062,159064],{"className":159063,"style":40120},[642],[363,159065,159066,159069],{"style":74648},[363,159067],{"className":159068,"style":30578},[650],[363,159070,159072],{"className":159071},[655,30582,10022,658],[363,159073,12313],{"className":159074},[407,408,658],[363,159076,666],{"className":159077},[665],[363,159079,159081],{"className":159080},[638],[363,159082,159084],{"className":159083,"style":40142},[642],[363,159085],{},[363,159087,135984],{"className":159088},[501,658],[363,159090,159091,159094],{"style":821},[363,159092],{"className":159093,"style":809},[650],[363,159095],{"className":159096,"style":829},[828],[363,159098,159099,159102],{"style":2898},[363,159100],{"className":159101,"style":809},[650],[363,159103,159105],{"className":159104},[655,656,657,658],[363,159106,159108,159111],{"className":159107},[407,658],[363,159109,38656],{"className":159110,"style":25973},[407,408,658],[363,159112,135984],{"className":159113},[501,658],[363,159115,666],{"className":159116},[665],[363,159118,159120],{"className":159119},[638],[363,159121,159124],{"className":159122,"style":159123},[642],"height:0.57em;",[363,159125],{},[363,159127],{"className":159128},[501,789],[363,159130],{"className":159131,"style":2935},[505],[363,159133,3647],{"className":159134},[2939],[363,159136],{"className":159137,"style":2935},[505],[363,159139,159141,159144,159248,159251,159254],{"className":159140},[398],[363,159142],{"className":159143,"style":150478},[402],[363,159145,159147,159150,159245],{"className":159146},[407],[363,159148],{"className":159149},[494,789],[363,159151,159153],{"className":159152},[586],[363,159154,159156,159237],{"className":159155},[633,634],[363,159157,159159,159234],{"className":159158},[638],[363,159160,159162,159191,159199],{"className":159161,"style":26765},[642],[363,159163,159164,159167],{"style":2875},[363,159165],{"className":159166,"style":809},[650],[363,159168,159170],{"className":159169},[655,656,657,658],[363,159171,159173,159176,159179,159188],{"className":159172},[407,658],[363,159174,39663],{"className":159175,"style":39759},[407,408,658],[363,159177,461],{"className":159178},[494,658],[363,15918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ame":159454,"style":651},[650],[363,159456,159458],{"className":159457},[655,656,657,658],[363,159459,159461,159464,159467],{"className":159460},[407,658],[363,159462,12313],{"className":159463},[407,408,658],[363,159465,471],{"className":159466},[510,658],[363,159468,465],{"className":159469},[407,658],[363,159471,159472,159475],{"style":133560},[363,159473],{"className":159474,"style":651},[650],[363,159476,159478],{"className":159477},[655,656,657,658],[363,159479,11044],{"className":159480,"style":11132},[407,408,658],[363,159482,666],{"className":159483},[665],[363,159485,159487],{"className":159486},[638],[363,159488,159490],{"className":159489,"style":40067},[642],[363,159491],{},[363,159493],{"className":159494,"style":754},[505],[363,159496,159498,159501,159695],{"className":159497},[407],[363,159499],{"className":159500},[494,789],[363,159502,159504],{"className":159503},[586],[363,159505,159507,159687],{"className":159506},[633,634],[363,159508,159510,159684],{"className":159509},[638],[363,159511,159513,159573,159581],{"className":159512,"style":26765},[642],[363,159514,159515,159518],{"style":2875},[363,159516],{"className":159517,"style":809},[650],[363,159519,159521],{"className":159520},[655,656,657,658],[363,159522,159524,159527,159530,159570],{"className":159523},[407,658],[363,159525,137218],{"className":159526},[407,658],[363,159528,461],{"className":159529},[494,658],[363,159531,159533,159536],{"className":159532},[407,658],[363,159534,45126],{"className":159535,"style":45156},[407,408,658],[363,159537,159539],{"className":159538},[629],[363,159540,159542,159562],{"className":159541},[633,634],[363,159543,159545,159559],{"className":159544},[638],[363,159546,159548],{"className":159547,"style":40120},[642],[363,159549,159550,159553],{"style":136989},[363,159551],{"className":159552,"style":30578},[650],[363,159554,159556],{"className":159555},[655,30582,10022,658],[363,159557,12313],{"className":159558},[407,408,658],[363,159560,666],{"className":159561},[665],[363,159563,159565],{"className":159564},[638],[363,159566,159568],{"className":159567,"style":40142},[642],[363,159569],{},[363,159571,468],{"className":159572},[501,658],[363,159574,159575,159578],{"style":821},[363,159576],{"className":159577,"style":809},[650],[363,159579],{"className":159580,"style":829},[828],[363,159582,159583,159586],{"style":26790},[363,159584],{"className":159585,"style":809},[650],[363,159587,159589],{"className":159588},[655,656,657,658],[363,159590,159592,159595,159598,159638,159641,159681],{"className":159591},[407,658],[363,159593,137218],{"className":159594},[407,658],[363,159596,461],{"className":159597},[494,658],[363,159599,159601,159604],{"className":159600},[407,658],[363,159602,45126],{"className":159603,"style":45156},[407,408,658],[363,159605,159607],{"className":159606},[629],[363,159608,159610,159630],{"className":159609},[633,634],[363,159611,159613,159627],{"className":159612},[638],[363,159614,159616],{"className":159615,"style":40120},[642],[363,159617,159618,159621],{"style":136989},[363,159619],{"className":159620,"style":30578},[650],[363,159622,159624],{"className":159623},[655,30582,10022,658],[363,159625,12313],{"className":159626},[407,408,658],[363,159628,666],{"className":159629},[665],[363,159631,159633],{"className":159632},[638],[363,159634,159636],{"className":159635,"style":40142},[642],[363,159637],{},[363,159639,2818],{"className":159640},[2939,658],[363,159642,159644,159647],{"className":159643},[407,658],[363,159645,82683],{"className":159646},[407,408,658],[363,159648,159650],{"className":159649},[629],[363,159651,159653,159673],{"className":159652},[633,634],[363,159654,159656,159670],{"className":159655},[638],[363,159657,159659],{"className":159658,"style":40120},[642],[363,159660,159661,159664],{"style":74648},[363,159662],{"className":159663,"style":30578},[650],[363,159665,159667],{"className":159666},[655,30582,10022,658],[363,159668,12313],{"className":159669},[407,408,658],[363,159671,666],{"className":159672},[665],[363,159674,159676],{"className":159675},[638],[363,159677,159679],{"className":159678,"style":40142},[642],[363,159680],{},[363,159682,468],{"className":159683},[501,658],[363,159685,666],{"className":159686},[665],[363,159688,159690],{"className":159689},[638],[363,159691,159693],{"className":159692,"style":45974},[642],[363,159694],{},[363,159696],{"className":159697},[501,789],[143,159699,159700],{},[11,159701,159702,159705,159706,160084,160085,160192,160193,160196,160197],{},[1344,159703,159704],{},"推论 2.7（后验预测与收缩）"," 对未见样本，\n",[363,159707,159709,159774],{"className":159708,"translate":367},[366],[363,159710,159712],{"className":159711},[371],[373,159713,159714],{"xmlns":375},[377,159715,159716,159771],{},[380,159717,159718,159720,159722,159729,159731,159733,159735,159737,159739,159741],{},[383,159719,554],{"mathvariant":570},[458,159721,461],{"stretchy":460},[543,159723,159724,159726],{},[383,159725,40609],{},[1645,159727,159728],{},"new",[458,159730,471],{},[383,159732,7912],{},[458,159734,5156],{},[383,159736,82683],{"mathvariant":75598},[458,159738,468],{"stretchy":460},[458,159740,471],{},[586,159742,159743,159759],{},[380,159744,159745,159751,159753],{},[543,159746,159747,159749],{},[383,159748,45126],{},[383,159750,7912],{},[458,159752,2818],{},[543,159754,159755,159757],{},[383,159756,82683],{},[383,159758,7912],{},[380,159760,159761,159767,159769],{},[543,159762,159763,159765],{},[383,159764,45126],{},[463,159766,474],{},[458,159768,2818],{},[383,159770,38656],{},[387,159772,159773],{"encoding":389},"\\mathbb{P}(y_{\\text{new}} = k \\mid \\mathbf{n}) = \\frac{\\alpha_k + n_k}{\\alpha_0 + N}",[363,159775,159777,159844,159862,159883],{"className":159776,"ariaHidden":394},[393],[363,159778,159780,159783,159786,159789,159835,159838,159841],{"className":159779},[398],[363,159781],{"className":159782,"style":487},[402],[363,159784,554],{"className":159785},[407,714],[363,159787,461],{"className":159788},[494],[363,159790,159792,159795],{"className":159791},[407],[363,159793,40609],{"className":159794,"style":1328},[407,408],[363,159796,159798],{"className":159797},[629],[363,159799,159801,159827],{"className":159800},[633,634],[363,159802,159804,159824],{"className":159803},[638],[363,159805,159807],{"className":159806,"style":45301},[642],[363,159808,159809,159812],{"style":19215},[363,159810],{"className":159811,"style":651},[650],[363,159813,159815],{"className":159814},[655,656,657,658],[363,159816,159818],{"className":159817},[407,658],[363,159819,159821],{"className":159820},[407,4946,658],[363,159822,159728],{"className":159823},[407,658],[363,159825,666],{"className":159826},[665],[363,159828,159830],{"className":159829},[638],[363,159831,159833],{"className":159832,"style":8316},[642],[363,159834],{},[363,159836],{"className":159837,"style":506},[505],[363,159839,471],{"className":159840},[510],[363,159842],{"className":159843,"style":506},[505],[363,159845,159847,159850,159853,159856,159859],{"className":159846},[398],[363,159848],{"className":159849,"style":487},[402],[363,159851,7912],{"className":159852,"style":8615},[407,408],[363,159854],{"className":159855,"style":506},[505],[363,159857,5156],{"className":159858},[510],[363,159860],{"className":159861,"style":506},[505],[363,159863,159865,159868,159871,159874,159877,159880],{"className":159864},[398],[363,159866],{"className":159867,"style":487},[402],[363,159869,82683],{"className":159870},[407,75943],[363,159872,468],{"className":159873},[501],[363,159875],{"className":159876,"style":506},[505],[363,159878,471],{"className":159879},[510],[363,159881],{"className":159882,"style":506},[505],[363,159884,159886,159890],{"className":159885},[398],[363,159887],{"className":159888,"style":159889},[402],"height:1.2693em;vertical-align:-0.4451em;",[363,159891,159893,159896,160081],{"className":159892},[407],[363,159894],{"className":159895},[494,789],[363,159897,159899],{"className":159898},[586],[363,159900,159902,160073],{"className":159901},[633,634],[363,159903,159905,160070],{"className":159904},[638],[363,159906,159909,159966,159974],{"className":159907,"style":159908},[642],"height:0.8242em;",[363,159910,159911,159914],{"style":2875},[363,159912],{"className":159913,"style":809},[650],[363,159915,159917],{"className":159916},[655,656,657,658],[363,159918,159920,159960,159963],{"className":159919},[407,658],[363,159921,159923,159926],{"className":159922},[407,658],[363,159924,45126],{"className":159925,"style":45156},[407,408,658],[363,159927,159929],{"className":159928},[629],[363,159930,159932,159952],{"className":159931},[633,634],[363,159933,159935,159949],{"className":159934},[638],[363,159936,159938],{"className":159937,"style":137351},[642],[363,159939,159940,159943],{"style":136989},[363,159941],{"className":159942,"style":30578},[650],[363,159944,159946],{"className":159945},[655,30582,10022,658],[363,159947,474],{"className":159948},[407,658],[363,159950,666],{"className":159951},[665],[363,159953,159955],{"className":159954},[638],[363,159956,159958],{"className":159957,"style":40142},[642],[363,159959],{},[363,159961,2818],{"className":159962},[2939,658],[363,159964,38656],{"className":159965,"style":25973},[407,408,658],[363,159967,159968,159971],{"style":821},[363,159969],{"className":159970,"style":809},[650],[363,159972],{"className":159973,"style":829},[828],[363,159975,159977,159980],{"style":159976},"top:-3.4159em;",[363,159978],{"className":159979,"style":809},[650],[363,159981,159983],{"className":159982},[655,656,657,658],[363,159984,159986,160027,160030],{"className":159985},[407,658],[363,159987,159989,159992],{"className":159988},[407,658],[363,159990,45126],{"className":159991,"style":45156},[407,408,658],[363,159993,159995],{"className":159994},[629],[363,159996,159998,160019],{"className":159997},[633,634],[363,159999,160001,160016],{"className":160000},[638],[363,160002,160004],{"className":160003,"style":6446},[642],[363,160005,160007,160010],{"style":160006},"top:-2.3488em;margin-left:-0.0037em;margin-right:0.0714em;",[363,160008],{"className":160009,"style":30578},[650],[363,160011,160013],{"className":160012},[655,30582,10022,658],[363,160014,7912],{"className":160015,"style":8615},[407,408,658],[363,160017,666],{"className":160018},[665],[363,160020,160022],{"className":160021},[638],[363,160023,160025],{"className":160024,"style":101625},[642],[363,160026],{},[363,160028,2818],{"className":160029},[2939,658],[363,160031,160033,160036],{"className":160032},[407,658],[363,160034,82683],{"className":160035},[407,408,658],[363,160037,160039],{"className":160038},[629],[363,160040,160042,160062],{"className":160041},[633,634],[363,160043,160045,160059],{"className":160044},[638],[363,160046,160048],{"className":160047,"style":6446},[642],[363,160049,160050,160053],{"style":101606},[363,160051],{"className":160052,"style":30578},[650],[363,160054,160056],{"className":160055},[655,30582,10022,658],[363,160057,7912],{"className":160058,"style":8615},[407,408,658],[363,160060,666],{"className":160061},[665],[363,160063,160065],{"className":160064},[638],[363,160066,160068],{"className":160067,"style":101625},[642],[363,160069],{},[363,160071,666],{"className":160072},[665],[363,160074,160076],{"className":160075},[638],[363,160077,160079],{"className":160078,"style":40289},[642],[363,160080],{},[363,160082],{"className":160083},[501,789],"\n而 ",[363,160086,160088,160116],{"className":160087,"translate":367},[366],[363,160089,160091],{"className":160090},[371],[373,160092,160093],{"xmlns":375},[377,160094,160095,160113],{},[380,160096,160097,160099,160101,160107,160109,160111],{},[383,160098,571],{"mathvariant":570},[458,160100,578],{"stretchy":460},[543,160102,160103,160105],{},[383,160104,157459],{},[383,160106,7912],{},[458,160108,5156],{},[383,160110,82683],{"mathvariant":75598},[458,160112,605],{"stretchy":460},[387,160114,160115],{"encoding":389},"\\mathbb{E}[\\pi_k \\mid \\mathbf{n}]",[363,160117,160119,160180],{"className":160118,"ariaHidden":394},[393],[363,160120,160122,160125,160128,160131,160171,160174,160177],{"className":160121},[398],[363,160123],{"className":160124,"style":487},[402],[363,160126,571],{"className":160127},[407,714],[363,160129,578],{"className":160130},[494],[363,160132,160134,160137],{"className":160133},[407],[363,160135,157459],{"className":160136,"style":1328},[407,408],[363,160138,160140],{"className":160139},[629],[363,160141,160143,160163],{"className":160142},[633,634],[363,160144,160146,160160],{"className":160145},[638],[363,160147,160149],{"className":160148,"style":643},[642],[363,160150,160151,160154],{"style":19215},[363,160152],{"className":160153,"style":651},[650],[363,160155,160157],{"className":160156},[655,656,657,658],[363,160158,7912],{"className":160159,"style":8615},[407,408,658],[363,160161,666],{"className":160162},[665],[363,160164,160166],{"className":160165},[638],[363,160167,160169],{"className":160168,"style":8316},[642],[363,160170],{},[363,160172],{"className":160173,"style":506},[505],[363,160175,5156],{"className":160176},[510],[363,160178],{"className":160179,"style":506},[505],[363,160181,160183,160186,160189],{"className":160182},[398],[363,160184],{"className":160185,"style":487},[402],[363,160187,82683],{"className":160188},[407,75943],[363,160190,605],{"className":160191},[501]," 是",[1344,160194,160195],{},"先验均值与经验频率的凸组合","：\n",[363,160198,160200,160314],{"className":160199,"translate":367},[366],[363,160201,160203],{"className":160202},[371],[373,160204,160205],{"xmlns":375},[377,160206,160207,160311],{},[380,160208,160209,160211,160213,160219,160221,160223,160225,160227,160256,160258,160272,160274,160299,160301],{},[383,160210,571],{"mathvariant":570},[458,160212,578],{"stretchy":460},[543,160214,160215,160217],{},[383,160216,157459],{},[383,160218,7912],{},[458,160220,5156],{},[383,160222,82683],{"mathvariant":75598},[458,160224,605],{"stretchy":460},[458,160226,471],{},[6364,160228,160229,160253],{},[6364,160230,160231,160251],{},[586,160232,160233,160239],{},[543,160234,160235,160237],{},[383,160236,45126],{},[463,160238,474],{},[380,160240,160241,160247,160249],{},[543,160242,160243,160245],{},[383,160244,45126],{},[463,160246,474],{},[458,160248,2818],{},[383,160250,38656],{},[458,160252,6388],{"stretchy":394},[1645,160254,160255],{},"收缩权重",[458,160257,3647],{},[586,160259,160260,160266],{},[543,160261,160262,160264],{},[383,160263,45126],{},[383,160265,7912],{},[543,160267,160268,160270],{},[383,160269,45126],{},[463,160271,474],{},[458,160273,2818],{},[6364,160275,160276,160296],{},[6364,160277,160278,160294],{},[586,160279,160280,160282],{},[383,160281,38656],{},[380,160283,160284,160290,160292],{},[543,160285,160286,160288],{},[383,160287,45126],{},[463,160289,474],{},[458,160291,2818],{},[383,160293,38656],{},[458,160295,6388],{"stretchy":394},[1645,160297,160298],{},"数据权重",[458,160300,3647],{},[586,160302,160303,160309],{},[543,160304,160305,160307],{},[383,160306,82683],{},[383,160308,7912],{},[383,160310,38656],{},[387,160312,160313],{"encoding":389},"\\mathbb{E}[\\pi_k \\mid \\mathbf{n}] = \\underbrace{\\frac{\\alpha_0}{\\alpha_0 + N}}_{\\text{收缩权重}} \\cdot \\frac{\\alpha_k}{\\alpha_0} + \\underbrace{\\frac{N}{\\alpha_0 + N}}_{\\text{数据权重}} \\cdot \\frac{n_k}{N}",[363,160315,160317,160378,160399,160675,160834,161072],{"className":160316,"ariaHidden":394},[393],[363,160318,160320,160323,160326,160329,160369,160372,160375],{"className":160319},[398],[363,160321],{"className":160322,"style":487},[402],[363,160324,571],{"className":160325},[407,714],[363,160327,578],{"className":160328},[494],[363,160330,160332,160335],{"className":160331},[407],[363,160333,157459],{"className":160334,"style":1328},[407,408],[363,160336,160338],{"className":160337},[629],[363,160339,160341,160361],{"className":160340},[633,634],[363,160342,160344,160358],{"className":160343},[638],[363,160345,160347],{"className":160346,"style":643},[642],[363,160348,160349,160352],{"style":19215},[363,160350],{"className":160351,"style":651},[650],[363,160353,160355],{"className":160354},[655,656,657,658],[363,160356,7912],{"className":160357,"style":8615},[407,408,658],[363,160359,666],{"className":160360},[665],[363,160362,160364],{"className":160363},[638],[363,160365,160367],{"className":160366,"style":8316},[642],[363,160368],{},[363,160370],{"className":160371,"style":506},[505],[363,160373,5156],{"className":160374},[510],[363,160376],{"className":160377,"style":506},[505],[363,160379,160381,160384,160387,160390,160393,160396],{"className":160380},[398],[363,160382],{"className":160383,"style":487},[402],[363,160385,82683],{"className":160386},[407,75943],[363,160388,605],{"className":160389},[501],[363,160391],{"className":160392,"style":506},[505],[363,160394,471],{"className":160395},[510],[363,160397],{"className":160398,"style":506},[505],[363,160400,160402,160406,160666,160669,160672],{"className":160401},[398],[363,160403],{"className":160404,"style":160405},[402],"height:3.2699em;vertical-align:-2.1623em;",[363,160407,160409],{"className":160408},[758,6364],[363,160410,160412,160657],{"className":160411},[633,634],[363,160413,160415,160654],{"className":160414},[638],[363,160416,160418,160437],{"className":160417,"style":3902},[642],[363,160419,160421,160425],{"style":160420},"top:-0.9452em;",[363,160422],{"className":160423,"style":160424},[650],"height:3.1076em;",[363,160426,160428],{"className":160427},[655,656,657,658],[363,160429,160431],{"className":160430},[407,658],[363,160432,160434],{"className":160433},[407,4946,658],[363,160435,160255],{"className":160436},[407,4950,658],[363,160438,160440,160443],{"style":160439},"top:-3.1076em;",[363,160441],{"className":160442,"style":160424},[650],[363,160444,160446],{"className":160445},[758,6364],[363,160447,160449,160645],{"className":160448},[633,634],[363,160450,160452,160642],{"className":160451},[638],[363,160453,160455,160486],{"className":160454,"style":3902},[642],[363,160456,160459,160462],{"className":160457,"style":160458},[6655],"top:-1.6236em;",[363,160460],{"className":160461,"style":160424},[650],[363,160463,160465,160472,160479],{"className":160464,"style":6664},[6663],[363,160466,160468],{"className":160467,"style":6669},[6668],[6671,160469,160470],{"xmlns":6673,"width":6674,"height":6675,"viewBox":6676,"preserveAspectRatio":6677},[6679,160471],{"d":6681},[363,160473,160475],{"className":160474,"style":6669},[6685],[6671,160476,160477],{"xmlns":6673,"width":6674,"height":6675,"viewBox":6676,"preserveAspectRatio":6688},[6679,160478],{"d":6691},[363,160480,160482],{"className":160481,"style":6669},[6695],[6671,160483,160484],{"xmlns":6673,"width":6674,"height":6675,"viewBox":6676,"preserveAspectRatio":6698},[6679,160485],{"d":6701},[363,160487,160488,160491],{"style":160439},[363,160489],{"className":160490,"style":160424},[650],[363,160492,160494],{"className":160493},[407],[363,160495,160497,160500,160639],{"className":160496},[407],[363,160498],{"className":160499},[494,789],[363,160501,160503],{"className":160502},[586],[363,160504,160506,160631],{"className":160505},[633,634],[363,160507,160509,160628],{"className":160508},[638],[363,160510,160512,160572,160580],{"className":160511,"style":3902},[642],[363,160513,160514,160517],{"style":805},[363,160515],{"className":160516,"style":809},[650],[363,160518,160520,160560,160563,160566,160569],{"className":160519},[407],[363,160521,160523,160526],{"className":160522},[407],[363,160524,45126],{"className":160525,"style":45156},[407,408],[363,160527,160529],{"className":160528},[629],[363,160530,160532,160552],{"className":160531},[633,634],[363,160533,160535,160549],{"className":160534},[638],[363,160536,160538],{"className":160537,"style":11483},[642],[363,160539,160540,160543],{"style":136154},[363,160541],{"className":160542,"style":651},[650],[363,160544,160546],{"className":160545},[655,656,657,658],[363,160547,474],{"className":160548},[407,658],[363,160550,666],{"className":160551},[665],[363,160553,160555],{"className":160554},[638],[363,160556,160558],{"className":160557,"style":8316},[642],[363,160559],{},[363,160561],{"className":160562,"style":2935},[505],[363,160564,2818],{"className":160565},[2939],[363,160567],{"className":160568,"style":293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2.7 用一句话概括了加法平滑的全部含义：",[1344,161188,161189,161190,161259,161260,161288],{},"先验强度的倒数 ",[363,161191,161193,161210],{"className":161192,"translate":367},[366],[363,161194,161196],{"className":161195},[371],[373,161197,161198],{"xmlns":375},[377,161199,161200,161208],{},[380,161201,161202],{},[543,161203,161204,161206],{},[383,161205,45126],{},[463,161207,474],{},[387,161209,152304],{"encoding":389},[363,161211,161213],{"className":161212,"ariaHidden":394},[393],[363,161214,161216,161219],{"className":161215},[398],[363,161217],{"className":161218,"style":13724},[402],[363,161220,161222,161225],{"className":161221},[407],[363,161223,45126],{"className":161224,"style":45156},[407,408],[363,161226,161228],{"className":161227},[629],[363,161229,161231,161251],{"className":161230},[633,634],[363,161232,161234,161248],{"className":161233},[638],[363,161235,161237],{"className":161236,"style":11483},[642],[363,161238,161239,161242],{"style":136154},[363,161240],{"className":161241,"style":651},[650],[363,161243,161245],{"className":161244},[655,656,657,658],[363,161246,474],{"className":161247},[407,658],[363,161249,666],{"className":161250},[665],[363,161252,161254],{"className":161253},[638],[363,161255,161257],{"className":161256,"style":8316},[642],[363,161258],{}," 相对于样本量 ",[363,161261,161263,161276],{"className":161262,"translate":367},[366],[363,161264,161266],{"className":161265},[371],[373,161267,161268],{"xmlns":375},[377,161269,161270,161274],{},[380,161271,161272],{},[383,161273,38656],{},[387,161275,38656],{"encoding":389},[363,161277,161279],{"className":161278,"ariaHidden":394},[393],[363,161280,161282,161285],{"className":161281},[398],[363,161283],{"className":161284,"style":1011},[402],[363,161286,38656],{"className":161287,"style":25973},[407,408]," 的大小，就是数据与先验的信任分配比例。",[363,161290,161292,161314],{"className":161291,"translate":367},[366],[363,161293,161295],{"className":161294},[371],[373,161296,161297],{"xmlns":375},[377,161298,161299,161311],{},[380,161300,161301,161307,161309],{},[543,161302,161303,161305],{},[383,161304,45126],{},[383,161306,12313],{},[458,161308,471],{},[463,161310,465],{},[387,161312,161313],{"encoding":389},"\\alpha_i = 1",[363,161315,161317,161372],{"className":161316,"ariaHidden":394},[393],[363,161318,161320,161323,161363,161366,161369],{"className":161319},[398],[363,161321],{"className":161322,"style":13724},[402],[363,161324,161326,161329],{"className":161325},[407],[363,161327,45126],{"className":161328,"style":45156},[407,408],[363,161330,161332],{"className":161331},[629],[363,161333,161335,161355],{"className":161334},[633,634],[363,161336,161338,161352],{"className":161337},[638],[363,161339,161341],{"className":161340,"style":12532},[642],[363,161342,161343,161346],{"style":136154},[363,161344],{"className":161345,"style":651},[650],[363,161347,161349],{"className":161348},[655,656,657,658],[363,161350,12313],{"className":161351},[407,408,658],[363,161353,666],{"className":161354},[665],[363,161356,161358],{"className":161357},[638],[363,161359,161361],{"className":161360,"style":8316},[642],[363,161362],{},[363,161364],{"className":161365,"style":506},[505],[363,161367,471],{"className":161368},[510],[363,161370],{"className":161371,"style":506},[505],[363,161373,161375,161378],{"className":161374},[398],[363,161376],{"className":161377,"style":520},[402],[363,161379,465],{"className":161380},[407]," 得 Laplace 平滑，取 ",[363,161383,161385,161406],{"className":161384,"translate":367},[366],[363,161386,161388],{"className":161387},[371],[373,161389,161390],{"xmlns":375},[377,161391,161392,161404],{},[380,161393,161394,161400,161402],{},[543,161395,161396,161398],{},[383,161397,45126],{},[383,161399,12313],{},[458,161401,3251],{},[463,161403,465],{},[387,161405,154517],{"encoding":389},[363,161407,161409,161464],{"className":161408,"ariaHidden":394},[393],[363,161410,161412,161415,161455,161458,161461],{"className":161411},[398],[363,161413],{"className":161414,"style":134714},[402],[363,161416,161418,161421],{"className":161417},[407],[363,161419,45126],{"className":161420,"style":45156},[407,408],[363,161422,161424],{"className":161423},[629],[363,161425,161427,161447],{"className":161426},[633,634],[363,161428,161430,161444],{"className":161429},[638],[363,161431,161433],{"className":161432,"style":12532},[642],[363,161434,161435,161438],{"style":136154},[363,161436],{"className":161437,"style":651},[650],[363,161439,161441],{"className":161440},[655,656,657,658],[363,161442,12313],{"className":161443},[407,408,658],[363,161445,666],{"className":161446},[665],[363,161448,161450],{"className":161449},[638],[363,161451,161453],{"className":161452,"style":8316},[642],[363,161454],{},[363,161456],{"className":161457,"style":506},[505],[363,161459,3251],{"className":161460},[510],[363,161462],{"className":161463,"style":506},[505],[363,161465,161467,161470],{"className":161466},[398],[363,161468],{"className":161469,"style":520},[402],[363,161471,465],{"className":161472},[407]," 得 Lidstone 平滑，取 ",[363,161475,161477,161499],{"className":161476,"translate":367},[366],[363,161478,161480],{"className":161479},[371],[373,161481,161482],{"xmlns":375},[377,161483,161484,161496],{},[380,161485,161486,161492,161494],{},[543,161487,161488,161490],{},[383,161489,45126],{},[383,161491,12313],{},[458,161493,1378],{},[463,161495,474],{},[387,161497,161498],{"encoding":389},"\\alpha_i \\to 0",[363,161500,161502,161557],{"className":161501,"ariaHidden":394},[393],[363,161503,161505,161508,161548,161551,161554],{"className":161504},[398],[363,161506],{"className":161507,"style":13724},[402],[363,161509,161511,161514],{"className":161510},[407],[363,161512,45126],{"className":161513,"style":45156},[407,408],[363,161515,161517],{"className":161516},[629],[363,161518,161520,161540],{"className":161519},[633,634],[363,161521,161523,161537],{"className":161522},[638],[363,161524,161526],{"className":161525,"style":12532},[642],[363,161527,161528,161531],{"style":136154},[363,161529],{"className":161530,"style":651},[650],[363,161532,161534],{"className":161533},[655,656,657,658],[363,161535,12313],{"className":161536},[407,408,658],[363,161538,666],{"className":161539},[665],[363,161541,161543],{"className":161542},[638],[363,161544,161546],{"className":161545,"style":8316},[642],[363,161547],{},[363,161549],{"className":161550,"style":506},[505],[363,161552,1378],{"className":161553},[510],[363,161555],{"className":161556,"style":506},[505],[363,161558,161560,161563],{"className":161559},[398],[363,161561],{"className":161562,"style":520},[402],[363,161564,474],{"className":161565},[407]," 得最大似然估计——这三个统计实践中彼此孤立出现的技巧，在此被识别为同一族参数的不同取值。",[2659,161568,161570],{"id":161569},"_25-聚合性中立性与分裂构造","2.5 聚合性、中立性与分裂构造",[143,161572,161573],{},[11,161574,161575,11410,161578,3093,161654,2628,161805,161873,161874,1515,162115,87168,162193,10528],{},[1344,161576,161577],{},"命题 2.8（聚合性）",[363,161579,161581,161606],{"className":161580,"translate":367},[366],[363,161582,161584],{"className":161583},[371],[373,161585,161586],{"xmlns":375},[377,161587,161588,161604],{},[380,161589,161590,161592,161594,161596,161598,161600,161602],{},[383,161591,1121],{"mathvariant":75598},[458,161593,9217],{},[383,161595,137112],{"mathvariant":557},[458,161597,4451],{},[458,161599,461],{"stretchy":460},[383,161601,45126],{"mathvariant":136055},[458,161603,468],{"stretchy":460},[387,161605,137123],{"encoding":389},[363,161607,161609,161627],{"className":161608,"ariaHidden":394},[393],[363,161610,161612,161615,161618,161621,161624],{"className":161611},[398],[363,161613],{"className":161614,"style":135662},[402],[363,161616,1121],{"className":161617},[407,75943],[363,161619],{"className":161620,"style":506},[505],[363,161622,9217],{"className":161623},[510],[363,161625],{"className":161626,"style":506},[505],[363,161628,161630,161633,161639,161642,161651],{"className":161629},[398],[363,161631],{"className":161632,"style":487},[402],[363,161634,161636],{"className":161635},[1832],[363,161637,137112],{"className":161638},[407,33034],[363,161640,461],{"className":161641},[494],[363,161643,161645],{"className":161644},[407],[363,161646,161648],{"className":161647},[407],[363,161649,45126],{"className":161650},[407,136115],[363,161652,468],{"className":161653},[501],[363,161655,161657,161691],{"className":161656,"translate":367},[366],[363,161658,161660],{"className":161659},[371],[373,161661,161662],{"xmlns":375},[377,161663,161664,161688],{},[380,161665,161666,161668,161674,161676,161678,161680,161686],{},[458,161667,3246],{"stretchy":460},[543,161669,161670,161672],{},[383,161671,34240],{},[463,161673,465],{},[458,161675,898],{"separator":394},[458,161677,133150],{},[458,161679,898],{"separator":394},[543,161681,161682,161684],{},[383,161683,34240],{},[383,161685,25939],{},[458,161687,3256],{"stretchy":460},[387,161689,161690],{"encoding":389},"\\{I_1, \\dots, I_M\\}",[363,161692,161694],{"className":161693,"ariaHidden":394},[393],[363,161695,161697,161700,161703,161744,161747,161750,161753,161756,161759,161762,161802],{"className":161696},[398],[363,161698],{"className":161699,"style":487},[402],[363,161701,3246],{"className":161702},[494],[363,161704,161706,161709],{"className":161705},[407],[363,161707,34240],{"className":161708,"style":9235},[407,408],[363,161710,161712],{"className":161711},[629],[363,161713,161715,161736],{"className":161714},[633,634],[363,161716,161718,161733],{"className":161717},[638],[363,161719,161721],{"className":161720,"style":11483},[642],[363,161722,161724,161727],{"style":161723},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[363,161725],{"className":161726,"style":651},[650],[363,161728,161730],{"className":161729},[655,656,657,658],[363,161731,465],{"className":161732},[407,658],[363,161734,666],{"className":161735},[665],[363,161737,161739],{"className":161738},[638],[363,161740,161742],{"className":161741,"style":8316},[642],[363,161743],{},[363,161745,898],{"className":161746},[920],[363,161748],{"className":161749,"style":754},[505],[363,161751,133150],{"className":161752},[758],[363,161754],{"className":161755,"style":754},[505],[363,161757,898],{"className":161758},[920],[363,161760],{"className":161761,"style":754},[505],[363,161763,161765,161768],{"className":161764},[407],[363,161766,34240],{"className":161767,"style":9235},[407,408],[363,161769,161771],{"className":161770},[629],[363,161772,161774,161794],{"className":161773},[633,634],[363,161775,161777,161791],{"className":161776},[638],[363,161778,161780],{"className":161779,"style":727},[642],[363,161781,161782,161785],{"style":161723},[363,161783],{"className":161784,"style":651},[650],[363,161786,161788],{"className":161787},[655,656,657,658],[363,161789,25939],{"className":161790,"style":25973},[407,408,658],[363,161792,666],{"className":161793},[665],[363,161795,161797],{"className":161796},[638],[363,161798,161800],{"className":161799,"style":8316},[642],[363,161801],{},[363,161803,3256],{"className":161804},[501],[363,161806,161808,161834],{"className":161807,"translate":367},[366],[363,161809,161811],{"className":161810},[371],[373,161812,161813],{"xmlns":375},[377,161814,161815,161831],{},[380,161816,161817,161819,161821,161823,161825,161827,161829],{},[458,161818,3246],{"stretchy":460},[463,161820,465],{},[458,161822,898],{"separator":394},[458,161824,133150],{},[458,161826,898],{"separator":394},[383,161828,11044],{},[458,161830,3256],{"stretchy":460},[387,161832,161833],{"encoding":389},"\\{1, \\dots, K\\}",[363,161835,161837],{"className":161836,"ariaHidden":394},[393],[363,161838,161840,161843,161846,161849,161852,161855,161858,161861,161864,161867,161870],{"className":161839},[398],[363,161841],{"className":161842,"style":487},[402],[363,161844,3246],{"className":161845},[494],[363,161847,465],{"className":161848},[407],[363,161850,898],{"className":161851},[920],[363,161853],{"className":161854,"style":754},[505],[363,161856,133150],{"className":161857},[758],[363,161859],{"className":161860,"style":754},[505],[363,161862,898],{"className":161863},[920],[363,161865],{"className":161866,"style":754},[505],[363,161868,11044],{"className":161869,"style":11132},[407,408],[363,161871,3256],{"className":161872},[501]," 的一个划分。令 ",[363,161875,161877,161919],{"className":161876,"translate":367},[366],[363,161878,161880],{"className":161879},[371],[373,161881,161882],{"xmlns":375},[377,161883,161884,161916],{},[380,161885,161886,161892,161894,161910],{},[543,161887,161888,161890],{},[383,161889,40609],{},[383,161891,26695],{},[458,161893,471],{},[543,161895,161896,161898],{},[458,161897,13811],{},[380,161899,161900,161902,161904],{},[383,161901,12313],{},[458,161903,6257],{},[543,161905,161906,161908],{},[383,161907,34240],{},[383,161909,26695],{},[543,161911,161912,161914],{},[383,161913,1121],{},[383,161915,12313],{},[387,161917,161918],{"encoding":389},"y_m = \\sum_{i \\in I_m} x_i",[363,161920,161922,161977],{"className":161921,"ariaHidden":394},[393],[363,161923,161925,161928,161968,161971,161974],{"className":161924},[398],[363,161926],{"className":161927,"style":2624},[402],[363,161929,161931,161934],{"className":161930},[407],[363,161932,40609],{"className":161933,"style":1328},[407,408],[363,161935,161937],{"className":161936},[629],[363,161938,161940,161960],{"className":161939},[633,634],[363,161941,161943,161957],{"className":161942},[638],[363,161944,161946],{"className":161945,"style":45301},[642],[363,161947,161948,161951],{"style":19215},[363,161949],{"className":161950,"style":651},[650],[363,161952,161954],{"className":161953},[655,656,657,658],[363,161955,26695],{"className":161956},[407,408,658],[363,161958,666],{"className":161959},[665],[363,161961,161963],{"className":161962},[638],[363,161964,161966],{"className":161965,"style":8316},[642],[363,161967],{},[363,161969],{"className":161970,"style":506},[505],[363,161972,471],{"className":161973},[510],[363,161975],{"className":161976,"style":506},[505],[363,161978,161980,161984,162072,162075],{"className":161979},[398],[363,161981],{"className":161982,"style":161983},[402],"height:1.1498em;vertical-align:-0.3998em;",[363,161985,161987,161990],{"className":161986},[1832],[363,161988,13811],{"className":161989,"style":40032},[1832,1833,5964],[363,161991,161993],{"className":161992},[629],[363,161994,161996,162063],{"className":161995},[633,634],[363,161997,161999,162060],{"className":161998},[638],[363,162000,162002],{"className":162001,"style":135559},[642],[363,162003,162004,162007],{"style":40048},[363,162005],{"className":162006,"style":651},[650],[363,162008,162010],{"className":162009},[655,656,657,658],[363,162011,162013,162016,162019],{"className":162012},[407,658],[363,162014,12313],{"className":162015},[407,408,658],[363,162017,6257],{"className":162018},[510,658],[363,162020,162022,162025],{"className":162021},[407,658],[363,162023,34240],{"className":162024,"style":9235},[407,408,658],[363,162026,162028],{"className":162027},[629],[363,162029,162031,162052],{"className":162030},[633,634],[363,162032,162034,162049],{"className":162033},[638],[363,162035,162037],{"className":162036,"style":74645},[642],[363,162038,162040,162043],{"style":162039},"top:-2.357em;margin-left:-0.0785em;margin-right:0.0714em;",[363,162041],{"className":162042,"style":30578},[650],[363,162044,162046],{"className":162045},[655,30582,10022,658],[363,162047,26695],{"className":162048},[407,408,658],[363,162050,666],{"className":162051},[665],[363,162053,162055],{"className":162054},[638],[363,162056,162058],{"className":162057,"style":40142},[642],[363,162059],{},[363,162061,666],{"className":162062},[665],[363,162064,162066],{"className":162065},[638],[363,162067,162070],{"className":162068,"style":162069},[642],"height:0.3998em;",[363,162071],{},[363,162073],{"className":162074,"style":754},[505],[363,162076,162078,162081],{"className":162077},[407],[363,162079,1121],{"className":162080},[407,408],[363,162082,162084],{"className":162083},[629],[363,162085,162087,162107],{"className":162086},[633,634],[363,162088,162090,162104],{"className":162089},[638],[363,162091,162093],{"className":162092,"style":12532},[642],[363,162094,162095,162098],{"style":730},[363,162096],{"className":162097,"style":651},[650],[363,162099,162101],{"className":162100},[655,656,657,658],[363,162102,12313],{"className":162103},[407,408,658],[363,162105,666],{"className":162106},[665],[363,162108,162110],{"className":162109},[638],[363,162111,162113],{"className":162112,"style":8316},[642],[363,162114],{},[363,162116,162118,162144],{"className":162117,"translate":367},[366],[363,162119,162121],{"className":162120},[371],[373,162122,162123],{"xmlns":375},[377,162124,162125,162141],{},[380,162126,162127,162129,162131,162133,162135,162137,162139],{},[383,162128,40609],{"mathvariant":75598},[458,162130,9217],{},[383,162132,137112],{"mathvariant":557},[458,162134,4451],{},[458,162136,461],{"stretchy":460},[383,162138,84504],{"mathvariant":136055},[458,162140,468],{"stretchy":460},[387,162142,162143],{"encoding":389},"\\mathbf{y} \\sim \\operatorname{Dir}(\\boldsymbol{\\beta})",[363,162145,162147,162166],{"className":162146,"ariaHidden":394},[393],[363,162148,162150,162153,162157,162160,162163],{"className":162149},[398],[363,162151],{"className":162152,"style":133807},[402],[363,162154,40609],{"className":162155,"style":162156},[407,75943],"margin-right:0.016em;",[363,162158],{"className":162159,"style":506},[505],[363,162161,9217],{"className":162162},[510],[363,162164],{"className":162165,"style":506},[505],[363,162167,162169,162172,162178,162181,162190],{"className":162168},[398],[363,162170],{"className":162171,"style":487},[402],[363,162173,162175],{"className":162174},[1832],[363,162176,137112],{"className":162177},[407,33034],[363,162179,461],{"className":162180},[494],[363,162182,162184],{"className":162183},[407],[363,162185,162187],{"className":162186},[407],[363,162188,84504],{"className":162189,"style":150195},[407,136115],[363,162191,468],{"className":162192},[501],[363,162194,162196,162238],{"className":162195,"translate":367},[366],[363,162197,162199],{"className":162198},[371],[373,162200,162201],{"xmlns":375},[377,162202,162203,162235],{},[380,162204,162205,162211,162213,162229],{},[543,162206,162207,162209],{},[383,162208,84504],{},[383,162210,26695],{},[458,162212,471],{},[543,162214,162215,162217],{},[458,162216,13811],{},[380,162218,162219,162221,162223],{},[383,162220,12313],{},[458,162222,6257],{},[543,162224,162225,162227],{},[383,162226,34240],{},[383,162228,26695],{},[543,162230,162231,162233],{},[383,162232,45126],{},[383,162234,12313],{},[387,162236,162237],{"encoding":389},"\\beta_m = \\sum_{i \\in I_m} \\alpha_i",[363,162239,162241,162296],{"className":162240,"ariaHidden":394},[393],[363,162242,162244,162247,162287,162290,162293],{"className":162243},[398],[363,162245],{"className":162246,"style":403},[402],[363,162248,162250,162253],{"className":162249},[407],[363,162251,84504],{"className":162252,"style":84726},[407,408],[363,162254,162256],{"className":162255},[629],[363,162257,162259,162279],{"className":162258},[633,634],[363,162260,162262,162276],{"className":162261},[638],[363,162263,162265],{"className":162264,"style":45301},[642],[363,162266,162267,162270],{"style":150274},[363,162268],{"className":162269,"style":651},[650],[363,162271,162273],{"className":162272},[655,656,657,658],[363,162274,26695],{"className":162275},[407,408,658],[363,162277,666],{"className":162278},[665],[363,162280,162282],{"className":162281},[638],[363,162283,162285],{"className":162284,"style":8316},[642],[363,162286],{},[363,162288],{"className":162289,"style":506},[505],[363,162291,471],{"className":162292},[510],[363,162294],{"className":162295,"style":506},[505],[363,162297,162299,162302,162388,162391],{"className":162298},[398],[363,162300],{"className":162301,"style":161983},[402],[363,162303,162305,162308],{"className":162304},[1832],[363,162306,13811],{"className":162307,"style":40032},[1832,1833,5964],[363,162309,162311],{"className":162310},[629],[363,162312,162314,162380],{"className":162313},[633,634],[363,162315,162317,162377],{"className":162316},[638],[363,162318,162320],{"className":162319,"style":135559},[642],[363,162321,162322,162325],{"style":40048},[363,162323],{"className":162324,"style":651},[650],[363,162326,162328],{"className":162327},[655,656,657,658],[363,162329,162331,162334,162337],{"className":162330},[407,658],[363,162332,12313],{"className":162333},[407,408,658],[363,162335,6257],{"className":162336},[510,658],[363,162338,162340,162343],{"className":162339},[407,658],[363,162341,34240],{"className":162342,"style":9235},[407,408,658],[363,162344,162346],{"className":162345},[629],[363,162347,162349,162369],{"className":162348},[633,634],[363,162350,162352,162366],{"className":162351},[638],[363,162353,162355],{"className":162354,"style":74645},[642],[363,162356,162357,162360],{"style":162039},[363,162358],{"className":162359,"style":30578},[650],[363,162361,162363],{"className":162362},[655,30582,10022,658],[363,162364,26695],{"className":162365},[407,408,658],[363,162367,666],{"className":162368},[665],[363,162370,162372],{"className":162371},[638],[363,162373,162375],{"className":162374,"style":40142},[642],[363,162376],{},[363,162378,666],{"className":162379},[665],[363,162381,162383],{"className":162382},[638],[363,162384,162386],{"className":162385,"style":162069},[642],[363,162387],{},[363,162389],{"className":162390,"style":754},[505],[363,162392,162394,162397],{"className":162393},[407],[363,162395,45126],{"className":162396,"style":45156},[407,408],[363,162398,162400],{"className":162399},[629],[363,162401,162403,162423],{"className":162402},[633,634],[363,162404,162406,162420],{"className":162405},[638],[363,162407,162409],{"className":162408,"style":12532},[642],[363,162410,162411,162414],{"style":136154},[363,162412],{"className":162413,"style":651},[650],[363,162415,162417],{"className":162416},[655,656,657,658],[363,162418,12313],{"className":162419},[407,408,658],[363,162421,666],{"className":162422},[665],[363,162424,162426],{"className":162425},[638],[363,162427,162429],{"className":162428,"style":8316},[642],[363,162430],{},[11,162432,162433,162434,162437],{},"聚合性说明迪利克雷族在",[1344,162435,162436],{},"粗化","（coarsening）下封闭：把若干类别合并，得到的仍是同一族的成员，参数按分量求和。这一性质在层级分类、LDA 中主题的合并、以及 3.5 节中「由多项分布边缘化得到分类预测」的推导中都扮演关键角色。",[11,162439,162440,162441,162444,162445,3093,162473,162543,162544,162798,162799,162905,162906,163176],{},"由聚合性可导出",[1344,162442,162443],{},"中立性（neutrality）","：对任意 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与其余分量的归一化版本 ",[363,162545,162547,162597],{"className":162546,"translate":367},[366],[363,162548,162550],{"className":162549},[371],[373,162551,162552],{"xmlns":375},[377,162553,162554,162594],{},[380,162555,162556],{},[543,162557,162558,162586],{},[380,162559,162560,162562,162568,162570,162572,162574,162576,162582,162584],{},[458,162561,461],{"fence":394},[543,162563,162564,162566],{},[383,162565,1121],{},[383,162567,12313],{},[383,162569,1288],{"mathvariant":557},[458,162571,461],{"stretchy":460},[463,162573,465],{},[458,162575,5161],{},[543,162577,162578,162580],{},[383,162579,1121],{},[383,162581,7912],{},[458,162583,468],{"stretchy":460},[458,162585,468],{"fence":394},[380,162587,162588,162590,162592],{},[383,162589,12313],{},[458,162591,49879],{"mathvariant":557},[383,162593,7912],{},[387,162595,162596],{"encoding":389},"\\left(x_i\u002F(1-x_k)\\right)_{i \\ne k}",[363,162598,162600],{"className":162599,"ariaHidden":394},[393],[363,162601,162603,162607],{"className":162602},[398],[363,162604],{"className":162605,"style":162606},[402],"height:1.1858em;vertical-align:-0.4358em;",[363,162608,162610,162720],{"className":162609},[758],[363,162611,162613,162616,162656,162659,162662,162665,162668,162671,162674,162714,162717],{"className":162612},[758],[363,162614,461],{"className":162615,"style":763},[494,762],[363,162617,162619,162622],{"className":162618},[407],[363,162620,1121],{"className":162621},[407,408],[363,162623,162625],{"className":162624},[629],[363,162626,162628,162648],{"className":162627},[633,634],[363,162629,162631,162645],{"className":162630},[638],[363,162632,162634],{"className":162633,"style":12532},[642],[363,162635,162636,162639],{"style":730},[363,162637],{"className":162638,"style":651},[650],[363,162640,162642],{"className":162641},[655,656,657,658],[363,162643,12313],{"className":162644},[407,408,658],[363,162646,666],{"className":162647},[665],[363,162649,162651],{"className":162650},[638],[363,162652,162654],{"className":162653,"style":8316},[642],[363,162655],{},[363,162657,1288],{"className":162658},[407],[363,162660,461],{"className":162661},[494],[363,162663,465],{"className":162664},[407],[363,162666],{"className":162667,"style":2935},[505],[363,162669,5161],{"className":162670},[2939],[363,162672],{"className":162673,"style":2935},[505],[363,162675,162677,162680],{"className":162676},[407],[363,162678,1121],{"className":162679},[407,408],[363,162681,162683],{"className":162682},[629],[363,162684,162686,162706],{"className":162685},[633,634],[363,162687,162689,162703],{"className":162688},[638],[363,162690,162692],{"className":162691,"style":643},[642],[363,162693,162694,162697],{"style":730},[363,162695],{"className":162696,"style":651},[650],[363,162698,162700],{"className":162699},[655,656,657,658],[363,162701,7912],{"className":162702,"style":8615},[407,408,658],[363,162704,666],{"className":162705},[665],[363,162707,162709],{"className":162708},[638],[363,162710,162712],{"className":162711,"style":8316},[642],[363,162713],{},[363,162715,468],{"className":162716},[501],[363,162718,468],{"className":162719,"style":763},[501,762],[363,162721,162723],{"className":162722},[629],[363,162724,162726,162789],{"className":162725},[633,634],[363,162727,162729,162786],{"className":162728},[638],[363,162730,162732],{"className":162731,"style":67394},[642],[363,162733,162735,162738],{"style":162734},"top:-2.4003em;margin-right:0.05em;",[363,162736],{"className":162737,"style":651},[650],[363,162739,162741],{"className":162740},[655,656,657,658],[363,162742,162744,162747,162783],{"className":162743},[407,658],[363,162745,12313],{"className":162746},[407,408,658],[363,162748,162750,162777,162780],{"className":162749},[510,658],[363,162751,162753],{"className":162752},[510,658],[363,162754,162756],{"className":162755},[407,4748,658],[363,162757,162759],{"className":162758},[4752,658],[363,162760,162762,162765,162774],{"className":162761},[4756,658],[363,162763],{"className":162764,"style":403},[402],[363,162766,162768],{"className":162767},[4763],[363,162769,162771],{"className":162770},[407,658],[363,162772,4770],{"className":162773},[510,658],[363,162775],{"className":162776},[4774],[363,162778],{"className":162779},[505,4778,658],[363,162781,471],{"className":162782},[510,658],[363,162784,7912],{"className":162785,"style":8615},[407,408,658],[363,162787,666],{"className":162788},[665],[363,162790,162792],{"className":162791},[638],[363,162793,162796],{"className":162794,"style":162795},[642],"height:0.4358em;",[363,162797],{}," 相互独立，且后者服从 ",[363,162800,162802,162832],{"className":162801,"translate":367},[366],[363,162803,162805],{"className":162804},[371],[373,162806,162807],{"xmlns":375},[377,162808,162809,162829],{},[380,162810,162811,162813,162815,162817,162827],{},[383,162812,137112],{"mathvariant":557},[458,162814,4451],{},[458,162816,461],{"stretchy":460},[543,162818,162819,162821],{},[383,162820,45126],{"mathvariant":136055},[380,162822,162823,162825],{},[458,162824,5161],{},[383,162826,7912],{},[458,162828,468],{"stretchy":460},[387,162830,162831],{"encoding":389},"\\operatorname{Dir}(\\boldsymbol{\\alpha}_{-k})",[363,162833,162835],{"className":162834,"ariaHidden":394},[393],[363,162836,162838,162841,162847,162850,162902],{"className":162837},[398],[363,162839],{"className":162840,"style":487},[402],[363,162842,162844],{"className":162843},[1832],[363,162845,137112],{"className":162846},[407,33034],[363,162848,461],{"className":162849},[494],[363,162851,162853,162862],{"className":162852},[407],[363,162854,162856],{"className":162855},[407],[363,162857,162859],{"className":162858},[407],[363,162860,45126],{"className":162861},[407,136115],[363,162863,162865],{"className":162864},[629],[363,162866,162868,162894],{"className":162867},[633,634],[363,162869,162871,162891],{"className":162870},[638],[363,162872,162874],{"className":162873,"style":643},[642],[363,162875,162876,162879],{"style":52818},[363,162877],{"className":162878,"style":651},[650],[363,162880,162882],{"className":162881},[655,656,657,658],[363,162883,162885,162888],{"className":162884},[407,658],[363,162886,5161],{"className":162887},[407,658],[363,162889,7912],{"className":162890,"style":8615},[407,408,658],[363,162892,666],{"className":162893},[665],[363,162895,162897],{"className":162896},[638],[363,162898,162900],{"className":162899,"style":135409},[642],[363,162901],{},[363,162903,468],{"className":162904},[501]," 而 ",[363,162907,162909,162959],{"className":162908,"translate":367},[366],[363,162910,162912],{"className":162911},[371],[373,162913,162914],{"xmlns":375},[377,162915,162916,162956],{},[380,162917,162918,162924,162926,162928,162930,162932,162938,162940,162946,162948,162954],{},[543,162919,162920,162922],{},[383,162921,1121],{},[383,162923,7912],{},[458,162925,9217],{},[383,162927,145808],{"mathvariant":557},[458,162929,4451],{},[458,162931,461],{"stretchy":460},[543,162933,162934,162936],{},[383,162935,45126],{},[383,162937,7912],{},[458,162939,898],{"separator":394},[543,162941,162942,162944],{},[383,162943,45126],{},[463,162945,474],{},[458,162947,5161],{},[543,162949,162950,162952],{},[383,162951,45126],{},[383,162953,7912],{},[458,162955,468],{"stretchy":460},[387,162957,162958],{"encoding":389},"x_k \\sim \\operatorname{Beta}(\\alpha_k, \\alpha_0 - \\alpha_k)",[363,162960,162962,163017,163127],{"className":162961,"ariaHidden":394},[393],[363,162963,162965,162968,163008,163011,163014],{"className":162964},[398],[363,162966],{"className":162967,"style":13724},[402],[363,162969,162971,162974],{"className":162970},[407],[363,162972,1121],{"className":162973},[407,408],[363,162975,162977],{"className":162976},[629],[363,162978,162980,163000],{"className":162979},[633,634],[363,162981,162983,162997],{"className":162982},[638],[363,162984,162986],{"className":162985,"style":643},[642],[363,162987,162988,162991],{"style":730},[363,162989],{"className":162990,"style":651},[650],[363,162992,162994],{"className":162993},[655,656,657,658],[363,162995,7912],{"className":162996,"style":8615},[407,408,658],[363,162998,666],{"className":162999},[665],[363,163001,163003],{"className":163002},[638],[363,163004,163006],{"className":163005,"style":8316},[642],[363,163007],{},[363,163009],{"className":163010,"style":506},[505],[363,163012,9217],{"className":163013},[510],[363,163015],{"className":163016,"style":506},[505],[363,163018,163020,163023,163029,163032,163072,163075,163078,163118,163121,163124],{"className":163019},[398],[363,163021],{"className":163022,"style":487},[402],[363,163024,163026],{"className":163025},[1832],[363,163027,145808],{"className":163028},[407,33034],[363,163030,461],{"className":163031},[494],[363,163033,163035,163038],{"className":163034},[407],[363,163036,45126],{"className":163037,"style":45156},[407,408],[363,163039,163041],{"className":163040},[629],[363,163042,163044,163064],{"className":163043},[633,634],[363,163045,163047,163061],{"className":163046},[638],[363,163048,163050],{"className":163049,"style":643},[642],[363,163051,163052,163055],{"style":136154},[363,163053],{"className":163054,"style":651},[650],[363,163056,163058],{"className":163057},[655,656,657,658],[363,163059,7912],{"className":163060,"style":8615},[407,408,658],[363,163062,666],{"className":163063},[665],[363,163065,163067],{"className":163066},[638],[363,163068,163070],{"className":163069,"style":8316},[642],[363,163071],{},[363,163073,898],{"className":163074},[920],[363,163076],{"className":163077,"style":754},[505],[363,163079,163081,163084],{"className":163080},[407],[363,163082,45126],{"className":163083,"style":45156},[407,408],[363,163085,163087],{"className":163086},[629],[363,163088,163090,163110],{"className":163089},[633,634],[363,163091,163093,163107],{"className":163092},[638],[363,163094,163096],{"className":163095,"style":11483},[642],[363,163097,163098,163101],{"style":136154},[363,163099],{"className":163100,"style":651},[650],[363,163102,163104],{"className":163103},[655,656,657,658],[363,163105,474],{"className":163106},[407,658],[363,163108,666],{"className":163109},[665],[363,163111,163113],{"className":163112},[638],[363,163114,163116],{"className":163115,"style":8316},[642],[363,163117],{},[363,163119],{"className":163120,"style":2935},[505],[363,163122,5161],{"className":163123},[2939],[363,163125],{"className":163126,"style":2935},[505],[363,163128,163130,163133,163173],{"className":163129},[398],[363,163131],{"className":163132,"style":487},[402],[363,163134,163136,163139],{"className":163135},[407],[363,163137,45126],{"className":163138,"style":45156},[407,408],[363,163140,163142],{"className":163141},[629],[363,163143,163145,163165],{"className":163144},[633,634],[363,163146,163148,163162],{"className":163147},[638],[363,163149,163151],{"className":163150,"style":643},[642],[363,163152,163153,163156],{"style":136154},[363,163154],{"className":163155,"style":651},[650],[363,163157,163159],{"className":163158},[655,656,657,658],[363,163160,7912],{"className":163161,"style":8615},[407,408,658],[363,163163,666],{"className":163164},[665],[363,163166,163168],{"className":163167},[638],[363,163169,163171],{"className":163170,"style":8316},[642],[363,163172],{},[363,163174,468],{"className":163175},[501],"。反复施用中立性，得到如下构造性表示：",[143,163178,163179],{},[11,163180,163181,11410,163184,163475,163476,143202,163944,10528],{},[1344,163182,163183],{},"命题 2.9（stick-breaking 表示）",[363,163185,163187,163246],{"className":163186,"translate":367},[366],[363,163188,163190],{"className":163189},[371],[373,163191,163192],{"xmlns":375},[377,163193,163194,163243],{},[380,163195,163196,163203,163205,163207,163209,163211],{},[543,163197,163198,163201],{},[383,163199,163200],{},"v",[383,163202,7912],{},[458,163204,9217],{},[383,163206,145808],{"mathvariant":557},[458,163208,4451],{},[1645,163210,1652],{},[380,163212,163213,163215,163221,163223,163235,163241],{},[458,163214,461],{"fence":394},[543,163216,163217,163219],{},[383,163218,45126],{},[383,163220,7912],{},[458,163222,898],{"separator":394},[543,163224,163225,163227],{},[458,163226,13811],{},[380,163228,163229,163231,163233],{},[383,163230,100402],{},[458,163232,3251],{},[383,163234,7912],{},[543,163236,163237,163239],{},[383,163238,45126],{},[383,163240,100402],{},[458,163242,468],{"fence":394},[387,163244,163245],{"encoding":389},"v_k \\sim \\operatorname{Beta}\\!\\left(\\alpha_k, \\sum_{j>k} \\alpha_j\\right)",[363,163247,163249,163304],{"className":163248,"ariaHidden":394},[393],[363,163250,163252,163255,163295,163298,163301],{"className":163251},[398],[363,163253],{"className":163254,"style":13724},[402],[363,163256,163258,163261],{"className":163257},[407],[363,163259,163200],{"className":163260,"style":1328},[407,408],[363,163262,163264],{"className":163263},[629],[363,163265,163267,163287],{"className":163266},[633,634],[363,163268,163270,163284],{"className":163269},[638],[363,163271,163273],{"className":163272,"style":643},[642],[363,163274,163275,163278],{"style":19215},[363,163276],{"className":163277,"style":651},[650],[363,163279,163281],{"className":163280},[655,656,657,658],[363,163282,7912],{"className":163283,"style":8615},[407,408,658],[363,163285,666],{"className":163286},[665],[363,163288,163290],{"className":163289},[638],[363,163291,163293],{"className":163292,"style":8316},[642],[363,163294],{},[363,163296],{"className":163297,"style":506},[505],[363,163299,9217],{"className":163300},[510],[363,163302],{"className":163303,"style":506},[505],[363,163305,163307,163310,163316,163319,163322],{"className":163306},[398],[363,163308],{"className":163309,"style":68500},[402],[363,163311,163313],{"className":163312},[1832],[363,163314,145808],{"className":163315},[407,33034],[363,163317],{"className":163318,"style":1860},[505],[363,163320],{"className":163321,"style":754},[505],[363,163323,163325,163331,163371,163374,163377,163426,163429,163469],{"className":163324},[758],[363,163326,163328],{"className":163327,"style":763},[494,762],[363,163329,461],{"className":163330},[767,4068],[363,163332,163334,163337],{"className":163333},[407],[363,163335,45126],{"className":163336,"style":45156},[407,408],[363,163338,163340],{"className":163339},[629],[363,163341,163343,163363],{"className":163342},[633,634],[363,163344,163346,163360],{"className":163345},[638],[363,163347,163349],{"className":163348,"style":643},[642],[363,163350,163351,163354],{"style":136154},[363,163352],{"className":163353,"style":651},[650],[363,163355,163357],{"className":163356},[655,656,657,658],[363,163358,7912],{"className":163359,"style":8615},[407,408,658],[363,163361,666],{"className":163362},[665],[363,163364,163366],{"className":163365},[638],[363,163367,163369],{"className":163368,"style":8316},[642],[363,163370],{},[363,163372,898],{"className":163373},[920],[363,163375],{"className":163376,"style":754},[505],[363,163378,163380,163383],{"className":163379},[1832],[363,163381,13811],{"className":163382,"style":40032},[1832,1833,5964],[363,163384,163386],{"className":163385},[629],[363,163387,163389,163418],{"className":163388},[633,634],[363,163390,163392,163415],{"className":163391},[638],[363,163393,163395],{"className":163394,"style":67394},[642],[363,163396,163397,163400],{"style":40048},[363,163398],{"className":163399,"style":651},[650],[363,163401,163403],{"className":163402},[655,656,657,658],[363,163404,163406,163409,163412],{"className":163405},[407,658],[363,163407,100402],{"className":163408,"style":100930},[407,408,658],[363,163410,3251],{"className":163411},[510,658],[363,163413,7912],{"className":163414,"style":8615},[407,408,658],[363,163416,666],{"className":163417},[665],[363,163419,163421],{"className":163420},[638],[363,163422,163424],{"className":163423,"style":162795},[642],[363,163425],{},[363,163427],{"className":163428,"style":754},[505],[363,163430,163432,163435],{"className":163431},[407],[363,163433,45126],{"className":163434,"style":45156},[407,408],[363,163436,163438],{"className":163437},[629],[363,163439,163441,163461],{"className":163440},[633,634],[363,163442,163444,163458],{"className":163443},[638],[363,163445,163447],{"className":163446,"style":12532},[642],[363,163448,163449,163452],{"style":136154},[363,163450],{"className":163451,"style":651},[650],[363,163453,163455],{"className":163454},[655,656,657,658],[363,163456,100402],{"className":163457,"style":100930},[407,408,658],[363,163459,666],{"className":163460},[665],[363,163462,163464],{"className":163463},[638],[363,163465,163467],{"className":163466,"style":673},[642],[363,163468],{},[363,163470,163472],{"className":163471,"style":763},[501,762],[363,163473,468],{"className":163474},[767,4068]," 相互独立。令\n",[363,163477,163479,163569],{"className":163478,"translate":367},[366],[363,163480,163482],{"className":163481},[371],[373,163483,163484],{"xmlns":375},[377,163485,163486,163566],{},[380,163487,163488,163494,163496,163502,163504,163506,163512,163514,163520,163532,163534,163536,163538,163544,163546,163548,163550,163552,163554,163556,163558,163560,163562,163564],{},[543,163489,163490,163492],{},[383,163491,1121],{},[463,163493,465],{},[458,163495,471],{},[543,163497,163498,163500],{},[383,163499,163200],{},[463,163501,465],{},[458,163503,898],{"separator":394},[505,163505],{"width":3610},[543,163507,163508,163510],{},[383,163509,1121],{},[383,163511,7912],{},[458,163513,471],{},[543,163515,163516,163518],{},[383,163517,163200],{},[383,163519,7912],{},[543,163521,163522,163524],{},[458,163523,136684],{},[380,163525,163526,163528,163530],{},[383,163527,100402],{},[458,163529,7599],{},[383,163531,7912],{},[458,163533,461],{"stretchy":460},[463,163535,465],{},[458,163537,5161],{},[543,163539,163540,163542],{},[383,163541,163200],{},[383,163543,100402],{},[458,163545,468],{"stretchy":460},[505,163547],{"width":4663},[458,163549,461],{"stretchy":460},[383,163551,7912],{},[458,163553,471],{},[463,163555,2811],{},[458,163557,898],{"separator":394},[458,163559,133150],{},[458,163561,898],{"separator":394},[383,163563,11044],{},[458,163565,468],{"stretchy":460},[387,163567,163568],{"encoding":389},"x_1 = v_1, \\qquad x_k = v_k \\prod_{j \u003C k}(1 - v_j) \\quad (k = 2, \\dots, K)",[363,163570,163572,163627,163731,163844,163911],{"className":163571,"ariaHidden":394},[393],[363,163573,163575,163578,163618,163621,163624],{"className":163574},[398],[363,163576],{"className":163577,"style":13724},[402],[363,163579,163581,163584],{"className":163580},[407],[363,163582,1121],{"className":163583},[407,408],[363,163585,163587],{"className":163586},[629],[363,163588,163590,163610],{"className":163589},[633,634],[363,163591,163593,163607],{"className":163592},[638],[363,163594,163596],{"className":163595,"style":11483},[642],[363,163597,163598,163601],{"style":730},[363,163599],{"className":163600,"style":651},[650],[363,163602,163604],{"className":163603},[655,656,657,658],[363,163605,465],{"className":163606},[407,658],[363,163608,666],{"className":163609},[665],[363,163611,163613],{"className":163612},[638],[363,163614,163616],{"className":163615,"style":8316},[642],[363,163617],{},[363,163619],{"className":163620,"style":506},[505],[363,163622,471],{"className":163623},[510],[363,163625],{"className":163626,"style":506},[505],[363,163628,163630,163633,163673,163676,163679,163682,163722,163725,163728],{"className":163629},[398],[363,163631],{"className":163632,"style":2624},[402],[363,163634,163636,163639],{"className":163635},[407],[363,163637,163200],{"className":163638,"style":1328},[407,408],[363,163640,163642],{"className":163641},[629],[363,163643,163645,163665],{"className":163644},[633,634],[363,163646,163648,163662],{"className":163647},[638],[363,163649,163651],{"className":163650,"style":11483},[642],[363,163652,163653,163656],{"style":19215},[363,163654],{"className":163655,"style":651},[650],[363,163657,163659],{"className":163658},[655,656,657,658],[363,163660,465],{"className":163661},[407,658],[363,163663,666],{"className":163664},[665],[363,163666,163668],{"className":163667},[638],[363,163669,163671],{"className":163670,"style":8316},[642],[363,163672],{},[363,163674,898],{"className":163675},[920],[363,163677],{"className":163678,"style":3982},[505],[363,163680],{"className":163681,"style":754},[505],[363,163683,163685,163688],{"className":163684},[407],[363,163686,1121],{"className":163687},[407,408],[363,163689,163691],{"className":163690},[629],[363,163692,163694,163714],{"className":163693},[633,634],[363,163695,163697,163711],{"className":163696},[638],[363,163698,163700],{"className":163699,"style":643},[642],[363,163701,163702,163705],{"style":730},[363,163703],{"className":163704,"style":651},[650],[363,163706,163708],{"className":163707},[655,656,657,658],[363,163709,7912],{"className":163710,"style":8615},[407,408,658],[363,163712,666],{"className":163713},[665],[363,163715,163717],{"className":163716},[638],[363,163718,163720],{"className":163719,"style":8316},[642],[363,163721],{},[363,163723],{"className":163724,"style":506},[505],[363,163726,471],{"className":163727},[510],[363,163729],{"className":163730,"style":506},[505],[363,163732,163734,163737,163777,163780,163829,163832,163835,163838,163841],{"className":163733},[398],[363,163735],{"className":163736,"style":162606},[402],[363,163738,163740,163743],{"className":163739},[407],[363,163741,163200],{"className":163742,"style":1328},[407,408],[363,163744,163746],{"className":163745},[629],[363,163747,163749,163769],{"className":163748},[633,634],[363,163750,163752,163766],{"className":163751},[638],[363,163753,163755],{"className":163754,"style":643},[642],[363,163756,163757,163760],{"style":19215},[363,163758],{"className":163759,"style":651},[650],[363,163761,163763],{"className":163762},[655,656,657,658],[363,163764,7912],{"className":163765,"style":8615},[407,408,658],[363,163767,666],{"className":163768},[665],[363,163770,163772],{"className":163771},[638],[363,163773,163775],{"className":163774,"style":8316},[642],[363,163776],{},[363,163778],{"className":163779,"style":754},[505],[363,163781,163783,163786],{"className":163782},[1832],[363,163784,136684],{"className":163785,"style":40032},[1832,1833,5964],[363,163787,163789],{"className":163788},[629],[363,163790,163792,163821],{"className":163791},[633,634],[363,163793,163795,163818],{"className":163794},[638],[363,163796,163798],{"className":163797,"style":67394},[642],[363,163799,163800,163803],{"style":40048},[363,163801],{"className":163802,"style":651},[650],[363,163804,163806],{"className":163805},[655,656,657,658],[363,163807,163809,163812,163815],{"className":163808},[407,658],[363,163810,100402],{"className":163811,"style":100930},[407,408,658],[363,163813,7599],{"className":163814},[510,658],[363,163816,7912],{"className":163817,"style":8615},[407,408,658],[363,163819,666],{"className":163820},[665],[363,163822,163824],{"className":163823},[638],[363,163825,163827],{"className":163826,"style":162795},[642],[363,163828],{},[363,163830,461],{"className":163831},[494],[363,163833,465],{"className":163834},[407],[363,163836],{"className":163837,"style":2935},[505],[363,163839,5161],{"className":163840},[2939],[363,163842],{"className":16384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\\dots, x_K) \\sim \\operatorname{Dir}(\\boldsymbol{\\alpha})",[363,163994,163996,164115],{"className":163995,"ariaHidden":394},[393],[363,163997,163999,164002,164005,164045,164048,164051,164054,164057,164060,164063,164103,164106,164109,164112],{"className":163998},[398],[363,164000],{"className":164001,"style":487},[402],[363,164003,461],{"className":164004},[494],[363,164006,164008,164011],{"className":164007},[407],[363,164009,1121],{"className":164010},[407,408],[363,164012,164014],{"className":164013},[629],[363,164015,164017,164037],{"className":164016},[633,634],[363,164018,164020,164034],{"className":164019},[638],[363,164021,164023],{"className":164022,"style":11483},[642],[363,164024,164025,164028],{"style":730},[363,164026],{"className":164027,"style":651},[650],[363,164029,164031],{"className":164030},[655,656,657,658],[363,164032,465],{"className":164033},[407,658],[363,164035,666],{"className":164036},[665],[363,164038,164040],{"className":164039},[638],[363,164041,164043],{"className":164042,"style":8316},[642],[363,164044],{},[363,164046,898],{"className":164047},[920],[363,164049],{"className":164050,"style":754},[505],[363,164052,133150],{"className":164053},[758],[363,164055],{"className":164056,"style":754},[505],[363,164058,898],{"className":164059},[920],[363,164061],{"className":164062,"style":754},[505],[363,164064,164066,164069],{"className":164065},[407],[363,164067,1121],{"className":164068},[407,408],[363,164070,164072],{"className":164071},[629],[363,164073,164075,164095],{"className":164074},[633,634],[363,164076,164078,164092],{"className":164077},[638],[363,164079,164081],{"className":164080,"style":727},[642],[363,164082,164083,164086],{"style":730},[363,164084],{"className":164085,"style":651},[650],[363,164087,164089],{"className":164088},[655,656,657,658],[363,164090,11044],{"className":164091,"style":11132},[407,408,658],[363,164093,666],{"className":164094},[665],[363,164096,164098],{"className":164097},[638],[363,164099,164101],{"className":164100,"style":8316},[642],[363,164102],{},[363,164104,468],{"className":164105},[501],[363,164107],{"className":164108,"style":506},[505],[363,164110,9217],{"className":164111},[510],[363,164113],{"className":164114,"style":506},[505],[363,164116,164118,164121,164127,164130,164139],{"className":164117},[398],[363,164119],{"className":164120,"style":487},[402],[363,164122,164124],{"className":164123},[1832],[363,164125,137112],{"className":164126},[407,33034],[363,164128,461],{"className":164129},[494],[363,164131,164133],{"className":164132},[407],[363,164134,164136],{"className":164135},[407],[363,164137,45126],{"className":164138},[407,136115],[363,164140,468],{"className":164141},[501],[11,164143,164144,164145,164173,164174,164244],{},"命题 2.9 是 Sethuraman 式的「折断木棍」构造在有限维的情形：从长度为 ",[363,164146,164148,164161],{"className":164147,"translate":367},[366],[363,164149,164151],{"className":164150},[371],[373,164152,164153],{"xmlns":375},[377,164154,164155,164159],{},[380,164156,164157],{},[463,164158,465],{},[387,164160,465],{"encoding":389},[363,164162,164164],{"className":164163,"ariaHidden":394},[393],[363,164165,164167,164170],{"className":164166},[398],[363,164168],{"className":164169,"style":520},[402],[363,164171,465],{"className":164172},[407]," 的木棍中依次折断出比例 ",[363,164175,164177,164195],{"className":164176,"translate":367},[366],[363,164178,164180],{"className":164179},[371],[373,164181,164182],{"xmlns":375},[377,164183,164184,164192],{},[380,164185,164186],{},[543,164187,164188,164190],{},[383,164189,163200],{},[383,164191,7912],{},[387,164193,164194],{"encoding":389},"v_k",[363,164196,164198],{"className":164197,"ariaHidden":394},[393],[363,164199,164201,164204],{"className":164200},[398],[363,164202],{"className":164203,"style":13724},[402],[363,164205,164207,164210],{"className":164206},[407],[363,164208,163200],{"className":164209,"style":1328},[407,408],[363,164211,164213],{"className":164212},[629],[363,164214,164216,164236],{"className":164215},[633,634],[363,164217,164219,164233],{"className":164218},[638],[363,164220,164222],{"className":164221,"style":643},[642],[363,164223,164224,164227],{"style":19215},[363,164225],{"className":164226,"style":651},[650],[363,164228,164230],{"className":164229},[655,656,657,658],[363,164231,7912],{"className":164232,"style":8615},[407,408,658],[363,164234,666],{"className":164235},[665],[363,164237,164239],{"className":164238},[638],[363,164240,164242],{"className":164241,"style":8316},[642],[363,164243],{}," 的一段。它在无穷维极限下给出 Dirichlet 过程，因而把本文的有限维话题与贝叶斯非参数方法连接起来。",[2659,164246,164248],{"id":164247},"_26-参数的直观伪计数及其边界","2.6 参数的直观：伪计数及其边界",[11,164250,164251,164252,164286],{},"综合 2.1 至 2.5 节，可以把形状参数 ",[363,164253,164255,164268],{"className":164254,"translate":367},[366],[363,164256,164258],{"className":164257},[371],[373,164259,164260],{"xmlns":375},[377,164261,164262,164266],{},[380,164263,164264],{},[383,164265,45126],{"mathvariant":136055},[387,164267,137074],{"encoding":389},[363,164269,164271],{"className":164270,"ariaHidden":394},[393],[363,164272,164274,164277],{"className":164273},[398],[363,164275],{"className":164276,"style":135662},[402],[363,164278,164280],{"className":164279},[407],[363,164281,164283],{"className":164282},[407],[363,164284,45126],{"className":164285},[407,136115]," 的语义概括为三点：",[7010,164288,164289,164400,164575],{},[7013,164290,164291,164329,164330,164399],{},[1344,164292,164293,164294,164328],{},"在共轭更新中，",[363,164295,164297,164310],{"className":164296,"translate":367},[366],[363,164298,164300],{"className":164299},[371],[373,164301,164302],{"xmlns":375},[377,164303,164304,164308],{},[380,164305,164306],{},[383,164307,45126],{"mathvariant":136055},[387,164309,137074],{"encoding":389},[363,164311,164313],{"className":164312,"ariaHidden":394},[393],[363,164314,164316,164319],{"className":164315},[398],[363,164317],{"className":164318,"style":135662},[402],[363,164320,164322],{"className":164321},[407],[363,164323,164325],{"className":164324},[407],[363,164326,45126],{"className":164327},[407,136115]," 起先验样本计数的角色","（命题 2.5），其总强度 ",[363,164331,164333,164350],{"className":164332,"translate":367},[366],[363,164334,164336],{"className":164335},[371],[373,164337,164338],{"xmlns":375},[377,164339,164340,164348],{},[380,164341,164342],{},[543,164343,164344,164346],{},[383,164345,45126],{},[463,164347,474],{},[387,164349,152304],{"encoding":389},[363,164351,164353],{"className":164352,"ariaHidden":394},[393],[363,164354,164356,164359],{"className":164355},[398],[363,164357],{"className":164358,"style":13724},[402],[363,164360,164362,164365],{"className":164361},[407],[363,164363,45126],{"className":164364,"style":45156},[407,408],[363,164366,164368],{"className":164367},[629],[363,164369,164371,164391],{"className":164370},[633,634],[363,164372,164374,164388],{"className":164373},[638],[363,164375,164377],{"className":164376,"style":11483},[642],[363,164378,164379,164382],{"style":136154},[363,164380],{"className":164381,"style":651},[650],[363,164383,164385],{"className":164384},[655,656,657,658],[363,164386,474],{"className":164387},[407,658],[363,164389,666],{"className":164390},[665],[363,164392,164394],{"className":164393},[638],[363,164395,164397],{"className":164396,"style":8316},[642],[363,164398],{}," 是「先验等价样本量」（推论 2.7）。",[7013,164401,164402,164539,164540,164574],{},[1344,164403,164404,164405,164433,164434,164468,164469,164538],{},"在几何上，",[363,164406,164408,164421],{"className":164407,"translate":367},[366],[363,164409,164411],{"className":164410},[371],[373,164412,164413],{"xmlns":375},[377,164414,164415,164419],{},[380,164416,164417],{},[383,164418,45126],{},[387,164420,49462],{"encoding":389},[363,164422,164424],{"className":164423,"ariaHidden":394},[393],[363,164425,164427,164430],{"className":164426},[398],[363,164428],{"className":164429,"style":2693},[402],[363,164431,45126],{"className":164432,"style":45156},[407,408]," 由方向 ",[363,164435,164437,164450],{"className":164436,"translate":367},[366],[363,164438,164440],{"className":164439},[371],[373,164441,164442],{"xmlns":375},[377,164443,164444,164448],{},[380,164445,164446],{},[383,164447,2712],{"mathvariant":136055},[387,164449,152265],{"encoding":389},[363,164451,164453],{"className":164452,"ariaHidden":394},[393],[363,164454,164456,164459],{"className":164455},[398],[363,164457],{"className":164458,"style":133807},[402],[363,164460,164462],{"className":164461},[407],[363,164463,164465],{"className":164464},[407],[363,164466,2712],{"className":164467},[407,136115]," 与精度 ",[363,164470,164472,164489],{"className":164471,"translate":367},[366],[363,164473,164475],{"className":164474},[371],[373,164476,164477],{"xmlns":375},[377,164478,164479,164487],{},[380,164480,164481],{},[543,164482,164483,164485],{},[383,164484,45126],{},[463,164486,474],{},[387,164488,152304],{"encoding":389},[363,164490,164492],{"className":164491,"ariaHidden":394},[393],[363,164493,164495,164498],{"className":164494},[398],[363,164496],{"className":164497,"style":13724},[402],[363,164499,164501,164504],{"className":164500},[407],[363,164502,45126],{"className":164503,"style":45156},[407,408],[363,164505,164507],{"className":164506},[629],[363,164508,164510,164530],{"className":164509},[633,634],[363,164511,164513,164527],{"className":164512},[638],[363,164514,164516],{"className":164515,"style":11483},[642],[363,164517,164518,164521],{"style":136154},[363,164519],{"className":164520,"style":651},[650],[363,164522,164524],{"className":164523},[655,656,657,658],[363,164525,474],{"className":164526},[407,658],[363,164528,666],{"className":164529},[665],[363,164531,164533],{"className":164532},[638],[363,164534,164536],{"className":164535,"style":8316},[642],[363,164537],{}," 两个正交分量构成","（2.2 节），精度决定围绕 ",[363,164541,164543,164556],{"className":164542,"translate":367},[366],[363,164544,164546],{"className":164545},[371],[373,164547,164548],{"xmlns":375},[377,164549,164550,164554],{},[380,164551,164552],{},[383,164553,2712],{"mathvariant":136055},[387,164555,152265],{"encoding":389},[363,164557,164559],{"className":164558,"ariaHidden":394},[393],[363,164560,164562,164565],{"className":164561},[398],[363,164563],{"className":164564,"style":133807},[402],[363,164566,164568],{"className":164567},[407],[363,164569,164571],{"className":164570},[407],[363,164572,2712],{"className":164573},[407,136115]," 的收缩程度。",[7013,164576,164577,43137,164743,164835,164836,164928],{},[1344,164578,164579,164580,164672,164673,164742],{},"在似然意义上，",[363,164581,164583,164605],{"className":164582,"translate":367},[366],[363,164584,164586],{"className":164585},[371],[373,164587,164588],{"xmlns":375},[377,164589,164590,164602],{},[380,164591,164592,164598,164600],{},[543,164593,164594,164596],{},[383,164595,45126],{},[383,164597,12313],{},[458,164599,5161],{},[463,164601,465],{},[387,164603,164604],{"encoding":389},"\\alpha_i - 1",[363,164606,164608,164663],{"className":164607,"ariaHidden":394},[393],[363,164609,164611,164614,164654,164657,164660],{"className":164610},[398],[363,164612],{"className":164613,"style":145471},[402],[363,164615,164617,164620],{"className":164616},[407],[363,164618,45126],{"className":164619,"style":45156},[407,408],[363,164621,164623],{"className":164622},[629],[363,164624,164626,164646],{"className":164625},[633,634],[363,164627,164629,164643],{"className":164628},[638],[363,164630,164632],{"className":164631,"style":12532},[642],[363,164633,164634,164637],{"style":136154},[363,164635],{"className":164636,"style":651},[650],[363,164638,164640],{"className":164639},[655,656,657,658],[363,164641,12313],{"className":164642},[407,408,658],[363,164644,666],{"className":164645},[665],[363,164647,164649],{"className":164648},[638],[363,164650,164652],{"className":164651,"style":8316},[642],[363,164653],{},[363,164655],{"className":164656,"style":2935},[505],[363,164658,5161],{"className":164659},[2939],[363,164661],{"className":164662,"style":2935},[505],[363,164664,164666,164669],{"className":164665},[398],[363,164667],{"className":164668,"style":520},[402],[363,164670,465],{"className":164671},[407]," 是对数密度中 ",[363,164674,164676,164693],{"className":164675,"translate":367},[366],[363,164677,164679],{"className":164678},[371],[373,164680,164681],{"xmlns":375},[377,164682,164683,164691],{},[380,164684,164685],{},[543,164686,164687,164689],{},[383,164688,1121],{},[383,164690,12313],{},[387,164692,146700],{"encoding":389},[363,164694,164696],{"className":164695,"ariaHidden":394},[393],[363,164697,164699,164702],{"className":164698},[398],[363,164700],{"className":164701,"style":13724},[402],[363,164703,164705,164708],{"className":164704},[407],[363,164706,1121],{"className":164707},[407,408],[363,164709,164711],{"className":164710},[629],[363,164712,164714,164734],{"className":164713},[633,634],[363,164715,164717,164731],{"className":164716},[638],[363,164718,164720],{"className":164719,"style":12532},[642],[363,164721,164722,164725],{"style":730},[363,164723],{"className":164724,"style":651},[650],[363,164726,164728],{"className":164727},[655,656,657,658],[363,164729,12313],{"className":164730},[407,408,658],[363,164732,666],{"className":164733},[665],[363,164735,164737],{"className":164736},[638],[363,164738,164740],{"className":164739,"style":8316},[642],[363,164741],{}," 的指数",[363,164744,164746,164768],{"className":164745,"translate":367},[366],[363,164747,164749],{"className":164748},[371],[373,164750,164751],{"xmlns":375},[377,164752,164753,164765],{},[380,164754,164755,164761,164763],{},[543,164756,164757,164759],{},[383,164758,45126],{},[383,164760,12313],{},[458,164762,7599],{},[463,164764,465],{},[387,164766,164767],{"encoding":389},"\\alpha_i \u003C 1",[363,164769,164771,164826],{"className":164770,"ariaHidden":394},[393],[363,164772,164774,164777,164817,164820,164823],{"className":164773},[398],[363,164775],{"className":164776,"style":134714},[402],[363,164778,164780,164783],{"className":164779},[407],[363,164781,45126],{"className":164782,"style":45156},[407,408],[363,164784,164786],{"className":164785},[629],[363,164787,164789,164809],{"className":164788},[633,634],[363,164790,164792,164806],{"className":164791},[638],[363,164793,164795],{"className":164794,"style":12532},[642],[363,164796,164797,164800],{"style":136154},[363,164798],{"className":164799,"style":651},[650],[363,164801,164803],{"className":164802},[655,656,657,658],[363,164804,12313],{"className":164805},[407,408,658],[363,164807,666],{"className":164808},[665],[363,164810,164812],{"className":164811},[638],[363,164813,164815],{"className":164814,"style":8316},[642],[363,164816],{},[363,164818],{"className":164819,"style":506},[505],[363,164821,7599],{"className":164822},[510],[363,164824],{"className":164825,"style":506},[505],[363,164827,164829,164832],{"className":164828},[398],[363,164830],{"className":164831,"style":520},[402],[363,164833,465],{"className":164834},[407]," 意味着密度在 ",[363,164837,164839,164861],{"className":164838,"translate":367},[366],[363,164840,164842],{"className":164841},[371],[373,164843,164844],{"xmlns":375},[377,164845,164846,164858],{},[380,164847,164848,164854,164856],{},[543,164849,164850,164852],{},[383,164851,1121],{},[383,164853,12313],{},[458,164855,1378],{},[463,164857,474],{},[387,164859,164860],{"encoding":389},"x_i \\to 0",[363,164862,164864,164919],{"className":164863,"ariaHidden":394},[393],[363,164865,164867,164870,164910,164913,164916],{"className":164866},[398],[363,164868],{"className":164869,"style":13724},[402],[363,164871,164873,164876],{"className":164872},[407],[363,164874,1121],{"className":164875},[407,408],[363,164877,164879],{"className":164878},[629],[363,164880,164882,164902],{"className":164881},[633,634],[363,164883,164885,164899],{"className":164884},[638],[363,164886,164888],{"className":164887,"style":12532},[642],[363,164889,164890,164893],{"style":730},[363,164891],{"className":164892,"style":651},[650],[363,164894,164896],{"className":164895},[655,656,657,658],[363,164897,12313],{"className":164898},[407,408,658],[363,164900,666],{"className":164901},[665],[363,164903,164905],{"className":164904},[638],[363,164906,164908],{"className":164907,"style":8316},[642],[363,164909],{},[363,164911],{"className":164912,"style":506},[505],[363,164914,1378],{"className":164915},[510],[363,164917],{"className":164918,"style":506},[505],[363,164920,164922,164925],{"className":164921},[398],[363,164923],{"className":164924,"style":520},[402],[363,164926,474],{"className":164927},[407]," 处发散。",[11,164930,164931,164932,164934,164935,164996],{},"对于",[1344,164933,30058],{},"情形 ",[363,164936,164938,164957],{"className":164937,"translate":367},[366],[363,164939,164941],{"className":164940},[371],[373,164942,164943],{"xmlns":375},[377,164944,164945,164955],{},[380,164946,164947,164949,164951,164953],{},[383,164948,45126],{"mathvariant":136055},[458,164950,471],{},[383,164952,45126],{},[463,164954,465],{"mathvariant":75598},[387,164956,156766],{"encoding":389},[363,164958,164960,164984],{"className":164959,"ariaHidden":394},[393],[363,164961,164963,164966,164975,164978,164981],{"className":164962},[398],[363,164964],{"className":164965,"style":135662},[402],[363,164967,164969],{"className":164968},[407],[363,164970,164972],{"className":164971},[407],[363,164973,45126],{"className":164974},[407,136115],[363,164976],{"className":164977,"style":506},[505],[363,164979,471],{"className":164980},[510],[363,164982],{"className":164983,"style":506},[505],[363,164985,164987,164990,164993],{"className":164986},[398],[363,164988],{"className":164989,"style":520},[402],[363,164991,45126],{"className":164992,"style":45156},[407,408],[363,164994,465],{"className":164995},[407,75943],"，这三重语义给出一条完整的形态谱：",[52864,164998,164999,165016],{},[52867,165000,165001],{},[52870,165002,165003,165007,165010,165013],{},[52873,165004,165006],{"align":165005},"left","参数区间",[52873,165008,165009],{"align":165005},"分布的形态",[52873,165011,165012],{"align":165005},"采样的典型特征",[52873,165014,165015],{"align":165005},"典型用途",[52962,165017,165018,165191,165254],{},[52870,165019,165020,165073,165157,165188],{},[52967,165021,165022],{"align":165005},[363,165023,165025,165043],{"className":165024,"translate":367},[366],[363,165026,165028],{"className":165027},[371],[373,165029,165030],{"xmlns":375},[377,165031,165032,165040],{},[380,165033,165034,165036,165038],{},[383,165035,45126],{},[458,165037,1378],{},[463,165039,474],{},[387,165041,165042],{"encoding":389},"\\alpha \\to 0",[363,165044,165046,165064],{"className":165045,"ariaHidden":394},[393],[363,165047,165049,165052,165055,165058,165061],{"className":165048},[398],[363,165050],{"className":165051,"style":2693},[402],[363,165053,45126],{"className":165054,"style":45156},[407,408],[363,165056],{"className":165057,"style":506},[505],[363,165059,1378],{"className":165060},[510],[363,165062],{"className":165063,"style":506},[505],[363,165065,165067,165070],{"className":165066},[398],[363,165068],{"className":165069,"style":520},[402],[363,165071,474],{"className":165072},[407],[52967,165074,165075,165076,165156],{"align":165005},"质量集中于顶点 ",[363,165077,165079,165101],{"className":165078,"translate":367},[366],[363,165080,165082],{"className":165081},[371],[373,165083,165084],{"xmlns":375},[377,165085,165086,165098],{},[380,165087,165088,165090,165096],{},[458,165089,3246],{"stretchy":460},[543,165091,165092,165094],{},[383,165093,7915],{"mathvariant":75598},[383,165095,12313],{},[458,165097,3256],{"stretchy":460},[387,165099,165100],{"encoding":389},"\\{\\mathbf{e}_i\\}",[363,165102,165104],{"className":165103,"ariaHidden":394},[393],[363,165105,165107,165110,165113,165153],{"className":165106},[398],[363,165108],{"className":165109,"style":487},[402],[363,165111,3246],{"className":165112},[494],[363,165114,165116,165119],{"className":165115},[407],[363,165117,7915],{"className":165118},[407,75943],[363,165120,165122],{"className":165121},[629],[363,165123,165125,165145],{"className":165124},[633,634],[363,165126,165128,165142],{"className":165127},[638],[363,165129,165131],{"className":165130,"style":12532},[642],[363,165132,165133,165136],{"style":730},[363,165134],{"className":165135,"style":651},[650],[363,165137,165139],{"className":165138},[655,656,657,658],[363,165140,12313],{"className":165141},[407,408,658],[363,165143,666],{"className":165144},[665],[363,165146,165148],{"className":165147},[638],[363,165149,165151],{"className":165150,"style":8316},[642],[363,165152],{},[363,165154,3256],{"className":165155},[501]," 邻域",[52967,165158,165159,165160],{"align":165005},"极端稀疏，单个分量接近 ",[363,165161,165163,165176],{"className":165162,"translate":367},[366],[363,165164,165166],{"className":165165},[371],[373,165167,165168],{"xmlns":375},[377,165169,165170,165174],{},[380,165171,165172],{},[463,165173,465],{},[387,165175,465],{"encoding":389},[363,165177,165179],{"className":165178,"ariaHidden":394},[393],[363,165180,165182,165185],{"className":165181},[398],[363,165183],{"className":165184,"style":520},[402],[363,165186,465],{"className":165187},[407],[52967,165189,165190],{"align":165005},"LDA 文档—主题混合（文档只涉少数主题）",[52870,165192,165193,165245,165248,165251],{},[52967,165194,165195],{"align":165005},[363,165196,165198,165215],{"className":165197,"translate":367},[366],[363,165199,165201],{"className":165200},[371],[373,165202,165203],{"xmlns":375},[377,165204,165205,165213],{},[380,165206,165207,165209,165211],{},[383,165208,45126],{},[458,165210,471],{},[463,165212,465],{},[387,165214,156855],{"encoding":389},[363,165216,165218,165236],{"className":165217,"ariaHidden":394},[393],[363,165219,165221,165224,165227,165230,165233],{"className":165220},[398],[363,165222],{"className":165223,"style":2693},[402],[363,165225,45126],{"className":165226,"style":45156},[407,408],[363,165228],{"className":165229,"style":506},[505],[363,165231,471],{"className":165232},[510],[363,165234],{"className":165235,"style":506},[505],[363,165237,165239,165242],{"className":165238},[398],[363,165240],{"className":165241,"style":520},[402],[363,165243,465],{"className":165244},[407],[52967,165246,165247],{"align":165005},"单纯形上的均匀分布",[52967,165249,165250],{"align":165005},"各分量量级相当，无偏好",[52967,165252,165253],{"align":165005},"无信息先验",[52870,165255,165256,165309,165381,165421],{},[52967,165257,165258],{"align":165005},[363,165259,165261,165279],{"className":165260,"translate":367},[366],[363,165262,165264],{"className":165263},[371],[373,165265,165266],{"xmlns":375},[377,165267,165268,165276],{},[380,165269,165270,165272,165274],{},[383,165271,45126],{},[458,165273,3251],{},[463,165275,465],{},[387,165277,165278],{"encoding":389},"\\alpha > 1",[363,165280,165282,165300],{"className":165281,"ariaHidden":394},[393],[363,165283,165285,165288,165291,165294,165297],{"className":165284},[398],[363,165286],{"className":165287,"style":6272},[402],[363,165289,45126],{"className":165290,"style":45156},[407,408],[363,165292],{"className":165293,"style":506},[505],[363,165295,3251],{"className":165296},[510],[363,165298],{"className":165299,"style":506},[505],[363,165301,165303,165306],{"className":165302},[398],[363,165304],{"className":165305,"style":520},[402],[363,165307,465],{"className":165308},[407],[52967,165310,165311,165312,165351,165352,165380],{"align":165005},"集中于中心 ",[363,165313,165315,165333],{"className":165314,"translate":367},[366],[363,165316,165318],{"className":165317},[371],[373,165319,165320],{"xmlns":375},[377,165321,165322,165330],{},[380,165323,165324,165326,165328],{},[463,165325,465],{"mathvariant":75598},[383,165327,1288],{"mathvariant":557},[383,165329,11044],{},[387,165331,165332],{"encoding":389},"\\mathbf{1}\u002FK",[363,165334,165336],{"className":165335,"ariaHidden":394},[393],[363,165337,165339,165342,165345,165348],{"className":165338},[398],[363,165340],{"className":165341,"style":487},[402],[363,165343,465],{"className":165344},[407,75943],[363,165346,1288],{"className":165347},[407],[363,165349,11044],{"className":165350,"style":11132},[407,408],"，随 ",[363,165353,165355,165368],{"className":165354,"translate":367},[366],[363,165356,165358],{"className":165357},[371],[373,165359,165360],{"xmlns":375},[377,165361,165362,165366],{},[380,165363,165364],{},[383,165365,45126],{},[387,165367,49462],{"encoding":389},[363,165369,165371],{"className":165370,"ariaHidden":394},[393],[363,165372,165374,165377],{"className":165373},[398],[363,165375],{"className":165376,"style":2693},[402],[363,165378,45126],{"className":165379,"style":45156},[407,408]," 增大而收缩",[52967,165382,165383,165384,165420],{"align":165005},"各分量接近 ",[363,165385,165387,165405],{"className":165386,"translate":367},[366],[363,165388,165390],{"className":165389},[371],[373,165391,165392],{"xmlns":375},[377,165393,165394,165402],{},[380,165395,165396,165398,165400],{},[463,165397,465],{},[383,165399,1288],{"mathvariant":557},[383,165401,11044],{},[387,165403,165404],{"encoding":389},"1\u002FK",[363,165406,165408],{"className":165407,"ariaHidden":394},[393],[363,165409,165411,165414,165417],{"className":165410},[398],[363,165412],{"className":165413,"style":487},[402],[363,165415,8937],{"className":165416},[407],[363,165418,11044],{"className":165419,"style":11132},[407,408]," 的平滑向量",[52967,165422,165423],{"align":165005},"强正则化、收缩估计",[11,165425,165426,165427,165496,165497,165589,165590,10650,165681,165796,165797,166018],{},"此处必须澄清一个常见但过强的说法：把 ",[363,165428,165430,165447],{"className":165429,"translate":367},[366],[363,165431,165433],{"className":165432},[371],[373,165434,165435],{"xmlns":375},[377,165436,165437,165445],{},[380,165438,165439],{},[543,165440,165441,165443],{},[383,165442,45126],{},[383,165444,12313],{},[387,165446,142599],{"encoding":389},[363,165448,165450],{"className":165449,"ariaHidden":394},[393],[363,165451,165453,165456],{"className":165452},[398],[363,165454],{"className":165455,"style":13724},[402],[363,165457,165459,165462],{"className":165458},[407],[363,165460,45126],{"className":165461,"style":45156},[407,408],[363,165463,165465],{"className":165464},[629],[363,165466,165468,165488],{"className":165467},[633,634],[363,165469,165471,165485],{"className":165470},[638],[363,165472,165474],{"className":165473,"style":12532},[642],[363,165475,165476,165479],{"style":136154},[363,165477],{"className":165478,"style":651},[650],[363,165480,165482],{"className":165481},[655,656,657,658],[363,165483,12313],{"className":165484},[407,408,658],[363,165486,666],{"className":165487},[665],[363,165489,165491],{"className":165490},[638],[363,165492,165494],{"className":165493,"style":8316},[642],[363,165495],{}," 直接称为「观测到的次数」只在 ",[363,165498,165500,165522],{"className":165499,"translate":367},[366],[363,165501,165503],{"className":165502},[371],[373,165504,165505],{"xmlns":375},[377,165506,165507,165519],{},[380,165508,165509,165515,165517],{},[543,165510,165511,165513],{},[383,165512,45126],{},[383,165514,12313],{},[458,165516,7276],{},[463,165518,465],{},[387,165520,165521],{"encoding":389},"\\alpha_i \\ge 1",[363,165523,165525,165580],{"className":165524,"ariaHidden":394},[393],[363,165526,165528,165531,165571,165574,165577],{"className":165527},[398],[363,165529],{"className":165530,"style":155283},[402],[363,165532,165534,165537],{"className":165533},[407],[363,165535,45126],{"className":165536,"style":45156},[407,408],[363,165538,165540],{"className":165539},[629],[363,165541,165543,165563],{"className":165542},[633,634],[363,165544,165546,165560],{"className":165545},[638],[363,165547,165549],{"className":165548,"style":12532},[642],[363,165550,165551,165554],{"style":136154},[363,165552],{"className":165553,"style":651},[650],[363,165555,165557],{"className":165556},[655,656,657,658],[363,165558,12313],{"className":165559},[407,408,658],[363,165561,666],{"className":165562},[665],[363,165564,165566],{"className":165565},[638],[363,165567,165569],{"className":165568,"style":8316},[642],[363,165570],{},[363,165572],{"className":165573,"style":506},[505],[363,165575,7276],{"className":165576},[510],[363,165578],{"className":165579,"style":506},[505],[363,165581,165583,165586],{"className":165582},[398],[363,165584],{"className":165585,"style":520},[402],[363,165587,465],{"className":165588},[407]," 时在字面上成立。当 ",[363,165591,165593,165614],{"className":165592,"translate":367},[366],[363,165594,165596],{"className":165595},[371],[373,165597,165598],{"xmlns":375},[377,165599,165600,165612],{},[380,165601,165602,165608,165610],{},[543,165603,165604,165606],{},[383,165605,45126],{},[383,165607,12313],{},[458,165609,7599],{},[463,165611,465],{},[387,165613,164767],{"encoding":389},[363,165615,165617,165672],{"className":165616,"ariaHidden":394},[393],[363,165618,165620,165623,165663,165666,165669],{"className":165619},[398],[363,165621],{"className":165622,"style":134714},[402],[363,165624,165626,165629],{"className":165625},[407],[363,165627,45126],{"className":165628,"style":45156},[407,408],[363,165630,165632],{"className":165631},[629],[363,165633,165635,165655],{"className":165634},[633,634],[363,165636,165638,165652],{"className":165637},[638],[363,165639,165641],{"className":165640,"style":12532},[642],[363,165642,165643,165646],{"style":136154},[363,165644],{"className":165645,"style":651},[650],[363,165647,165649],{"className":165648},[655,656,657,658],[363,165650,12313],{"className":165651},[407,408,658],[363,165653,666],{"className":165654},[665],[363,165656,165658],{"className":165657},[638],[363,165659,165661],{"className":165660,"style":8316},[642],[363,165662],{},[363,165664],{"className":165665,"style":506},[505],[363,165667,7599],{"className":165668},[510],[363,165670],{"className":165671,"style":506},[505],[363,165673,165675,165678],{"className":165674},[398],[363,165676],{"className":165677,"style":520},[402],[363,165679,465],{"className":165680},[407],[363,165682,165684,165710],{"className":165683,"translate":367},[366],[363,165685,165687],{"className":165686},[371],[373,165688,165689],{"xmlns":375},[377,165690,165691,165707],{},[380,165692,165693,165699,165701,165703,165705],{},[543,165694,165695,165697],{},[383,165696,45126],{},[383,165698,12313],{},[458,165700,5161],{},[463,165702,465],{},[458,165704,7599],{},[463,165706,474],{},[387,165708,165709],{"encoding":389},"\\alpha_i - 1 \u003C 0",[363,165711,165713,165768,165787],{"className":165712,"ariaHidden":394},[393],[363,165714,165716,165719,165759,165762,165765],{"className":165715},[398],[363,165717],{"className":165718,"style":145471},[402],[363,165720,165722,165725],{"className":165721},[407],[363,165723,45126],{"className":165724,"style":45156},[407,408],[363,165726,165728],{"className":165727},[629],[363,165729,165731,165751],{"className":165730},[633,634],[363,165732,165734,165748],{"className":165733},[638],[363,165735,165737],{"className":165736,"style":12532},[642],[363,165738,165739,165742],{"style":136154},[363,165740],{"className":165741,"style":651},[650],[363,165743,165745],{"className":165744},[655,656,657,658],[363,165746,12313],{"className":165747},[407,408,658],[363,165749,666],{"className":165750},[665],[363,165752,165754],{"className":165753},[638],[363,165755,165757],{"className":165756,"style":8316},[642],[363,165758],{},[363,165760],{"className":165761,"style":2935},[505],[363,165763,5161],{"className":165764},[2939],[363,165766],{"className":165767,"style":2935},[505],[363,165769,165771,165775,165778,165781,165784],{"className":165770},[398],[363,165772],{"className":165773,"style":165774},[402],"height:0.6835em;vertical-align:-0.0391em;",[363,165776,465],{"className":165777},[407],[363,165779],{"className":165780,"style":506},[505],[363,165782,7599],{"className":165783},[510],[363,165785],{"className":165786,"style":506},[505],[363,165788,165790,165793],{"className":165789},[398],[363,165791],{"className":165792,"style":520},[402],[363,165794,474],{"className":165795},[407],"，隐含的「先验计数」为负，这在计数语义下不可解释，却对应于密度在边界处的发散——一个在稀疏建模中必不可少的形态。准确的说法是：",[1344,165798,165799,165833,165834,165925,165926,166017],{},[363,165800,165802,165815],{"className":165801,"translate":367},[366],[363,165803,165805],{"className":165804},[371],[373,165806,165807],{"xmlns":375},[377,165808,165809,165813],{},[380,165810,165811],{},[383,165812,45126],{"mathvariant":136055},[387,165814,137074],{"encoding":389},[363,165816,165818],{"className":165817,"ariaHidden":394},[393],[363,165819,165821,165824],{"className":165820},[398],[363,165822],{"className":165823,"style":135662},[402],[363,165825,165827],{"className":165826},[407],[363,165828,165830],{"className":165829},[407],[363,165831,45126],{"className":165832},[407,136115]," 是伪计数概念的连续化，其在 ",[363,165835,165837,165858],{"className":165836,"translate":367},[366],[363,165838,165840],{"className":165839},[371],[373,165841,165842],{"xmlns":375},[377,165843,165844,165856],{},[380,165845,165846,165852,165854],{},[543,165847,165848,165850],{},[383,165849,45126],{},[383,165851,12313],{},[458,165853,7276],{},[463,165855,465],{},[387,165857,165521],{"encoding":389},[363,165859,165861,165916],{"className":165860,"ariaHidden":394},[393],[363,165862,165864,165867,165907,165910,165913],{"className":165863},[398],[363,165865],{"className":165866,"style":155283},[402],[363,165868,165870,165873],{"className":165869},[407],[363,165871,45126],{"className":165872,"style":45156},[407,408],[363,165874,165876],{"className":165875},[629],[363,165877,165879,165899],{"className":165878},[633,634],[363,165880,165882,165896],{"className":165881},[638],[363,165883,165885],{"className":165884,"style":12532},[642],[363,165886,165887,165890],{"style":136154},[363,165888],{"className":165889,"style":651},[650],[363,165891,165893],{"className":165892},[655,656,657,658],[363,165894,12313],{"className":165895},[407,408,658],[363,165897,666],{"className":165898},[665],[363,165900,165902],{"className":165901},[638],[363,165903,165905],{"className":165904,"style":8316},[642],[363,165906],{},[363,165908],{"className":165909,"style":506},[505],[363,165911,7276],{"className":165912},[510],[363,165914],{"className":165915,"style":506},[505],[363,165917,165919,165922],{"className":165918},[398],[363,165920],{"className":165921,"style":520},[402],[363,165923,465],{"className":165924},[407]," 区域退化回字面意义上的计数解释，而在 ",[363,165927,165929,165950],{"className":165928,"translate":367},[366],[363,165930,165932],{"className":165931},[371],[373,165933,165934],{"xmlns":375},[377,165935,165936,165948],{},[380,165937,165938,165944,165946],{},[543,165939,165940,165942],{},[383,165941,45126],{},[383,165943,12313],{},[458,165945,7599],{},[463,165947,465],{},[387,165949,164767],{"encoding":389},[363,165951,165953,166008],{"className":165952,"ariaHidden":394},[393],[363,165954,165956,165959,165999,166002,166005],{"className":165955},[398],[363,165957],{"className":165958,"style":134714},[402],[363,165960,165962,165965],{"className":165961},[407],[363,165963,45126],{"className":165964,"style":45156},[407,408],[363,165966,165968],{"className":165967},[629],[363,165969,165971,165991],{"className":165970},[633,634],[363,165972,165974,165988],{"className":165973},[638],[363,165975,165977],{"className":165976,"style":12532},[642],[363,165978,165979,165982],{"style":136154},[363,165980],{"className":165981,"style":651},[650],[363,165983,165985],{"className":165984},[655,656,657,658],[363,165986,12313],{"className":165987},[407,408,658],[363,165989,666],{"className":165990},[665],[363,165992,165994],{"className":165993},[638],[363,165995,165997],{"className":165996,"style":8316},[642],[363,165998],{},[363,166000],{"className":166001,"style":506},[505],[363,166003,7599],{"className":166004},[510],[363,166006],{"className":166007,"style":506},[505],[363,166009,166011,166014],{"className":166010},[398],[363,166012],{"className":166013,"style":520},[402],[363,166015,465],{"className":166016},[407]," 区域承担的是稀疏性先验的角色。"," 这一区分在 3.4 节将直接决定 EDL 中「证据」一词的适用边界。",[18,166020,166022],{"id":166021},"三主观逻辑到证据深度学习","三、主观逻辑到证据深度学习",[2659,166024,166026],{"id":166025},"_31-主观逻辑的-opinion-表示","3.1 主观逻辑的 opinion 表示",[11,166028,166029,166030,166033],{},"证据深度学习的直接理论前身是 Josang 的",[1344,166031,166032],{},"主观逻辑（subjective logic）","，它为「对一个命题的相信程度」建立了一套带不确定性维度的代数。",[143,166035,166036,166221,166807],{},[11,166037,166038,166041,166042,166132,166133,166220],{},[1344,166039,166040],{},"定义 3.1（opinion）"," 在有限域 ",[363,166043,166045,166075],{"className":166044,"translate":367},[366],[363,166046,166048],{"className":166047},[371],[373,166049,166050],{"xmlns":375},[377,166051,166052,166072],{},[380,166053,166054,166056,166058,166060,166062,166064,166066,166068,166070],{},[383,166055,5749],{"mathvariant":944},[458,166057,471],{},[458,166059,3246],{"stretchy":460},[463,166061,465],{},[458,166063,898],{"separator":394},[458,166065,133150],{},[458,166067,898],{"separator":394},[383,166069,11044],{},[458,166071,3256],{"stretchy":460},[387,166073,166074],{"encoding":389},"\\mathcal{Y} = \\{1, \\dots, K\\}",[363,166076,166078,166096],{"className":166077,"ariaHidden":394},[393],[363,166079,166081,166084,166087,166090,166093],{"className":166080},[398],[363,166082],{"className":166083,"style":5798},[402],[363,166085,5749],{"className":166086,"style":5802},[407,971],[363,166088],{"className":166089,"style":506},[505],[363,166091,471],{"className":166092},[510],[363,166094],{"className":166095,"style":506},[505],[363,166097,166099,166102,166105,166108,166111,166114,166117,166120,166123,166126,166129],{"className":166098},[398],[363,166100],{"className":166101,"style":487},[402],[363,166103,3246],{"className":166104},[494],[363,166106,465],{"className":166107},[407],[363,166109,898],{"className":166110},[920],[363,166112],{"className":166113,"style":754},[505],[363,166115,133150],{"className":166116},[758],[363,166118],{"className":166119,"style":754},[505],[363,166121,898],{"className":166122},[920],[363,166124],{"className":166125,"style":754},[505],[363,166127,11044],{"className":166128,"style":11132},[407,408],[363,166130,3256],{"className":166131},[501]," 上的一个 opinion 是三元组 ",[363,166134,166136,166166],{"className":166135,"translate":367},[366],[363,166137,166139],{"className":166138},[371],[373,166140,166141],{"xmlns":375},[377,166142,166143,166163],{},[380,166144,166145,166147,166149,166151,166153,166155,166157,166159,166161],{},[383,166146,100250],{},[458,166148,471],{},[458,166150,461],{"stretchy":460},[383,166152,83214],{"mathvariant":75598},[458,166154,898],{"separator":394},[383,166156,15750],{},[458,166158,898],{"separator":394},[383,166160,45552],{"mathvariant":75598},[458,166162,468],{"stretchy":460},[387,166164,166165],{"encoding":389},"\\omega = (\\mathbf{b}, u, \\mathbf{a})",[363,166167,166169,166187],{"className":166168,"ariaHidden":394},[393],[363,166170,166172,166175,166178,166181,166184],{"className":166171},[398],[363,166173],{"className":166174,"style":2693},[402],[363,166176,100250],{"className":166177,"style":1328},[407,408],[363,166179],{"className":166180,"style":506},[505],[363,166182,471],{"className":166183},[510],[363,166185],{"className":166186,"style":506},[505],[363,166188,166190,166193,166196,166199,166202,166205,166208,166211,166214,166217],{"className":166189},[398],[363,166191],{"className":166192,"style":487},[402],[363,166194,461],{"className":166195},[494],[363,166197,83214],{"className":166198},[407,75943],[363,166200,898],{"className":166201},[920],[363,166203],{"className":166204,"style":754},[505],[363,166206,15750],{"className":166207},[407,408],[363,166209,898],{"className":166210},[920],[363,166212],{"className":166213,"style":754},[505],[363,166215,45552],{"className":166216},[407,75943],[363,166218,468],{"className":166219},[501],"，其中",[10375,166222,166223,166477,166556],{},[7013,166224,166225,166473,166474,10528],{},[363,166226,166228,166282],{"className":166227,"translate":367},[366],[363,166229,166231],{"className":166230},[371],[373,166232,166233],{"xmlns":375},[377,166234,166235,166279],{},[380,166236,166237,166239,166241,166243,166249,166251,166253,166255,166261,166263,166265,166267,166269,166271,166273],{},[383,166238,83214],{"mathvariant":75598},[458,166240,471],{},[458,166242,461],{"stretchy":460},[543,166244,166245,166247],{},[383,166246,83214],{},[463,166248,465],{},[458,166250,898],{"separator":394},[458,166252,133150],{},[458,166254,898],{"separator":394},[543,166256,166257,166259],{},[383,166258,83214],{},[383,166260,11044],{},[458,166262,468],{"stretchy":460},[458,166264,6257],{},[458,166266,578],{"stretchy":460},[463,166268,474],{},[458,166270,898],{"separator":394},[463,166272,465],{},[7162,166274,166275,166277],{},[458,166276,605],{"stretchy":460},[383,166278,11044],{},[387,166280,166281],{"encoding":389},"\\mathbf{b} = (b_1, \\dots, b_K) \\in [0,1]^K",[363,166283,166285,166303,166422],{"className":166284,"ariaHidden":394},[393],[363,166286,166288,166291,166294,166297,166300],{"className":166287},[398],[363,166289],{"className":166290,"style":12947},[402],[363,166292,83214],{"className":166293},[407,75943],[363,166295],{"className":166296,"style":506},[505],[363,166298,471],{"className":166299},[510],[363,166301],{"className":166302,"style":506},[505],[363,166304,166306,166309,166312,166352,166355,166358,166361,166364,166367,166370,166410,166413,166416,166419],{"className":166305},[398],[363,166307],{"className":166308,"style":487},[402],[363,166310,461],{"className":166311},[494],[363,166313,166315,166318],{"className":166314},[407],[363,166316,83214],{"className":166317},[407,408],[363,166319,166321],{"className":166320},[629],[363,166322,166324,166344],{"className":166323},[633,634],[363,166325,166327,166341],{"className":166326},[638],[363,166328,166330],{"className":166329,"style":11483},[642],[363,166331,166332,166335],{"style":730},[363,166333],{"className":166334,"style":651},[650],[363,166336,166338],{"className":166337},[655,656,657,658],[363,166339,465],{"className":166340},[407,658],[363,166342,666],{"className":166343},[665],[363,166345,166347],{"className":166346},[638],[363,166348,166350],{"className":166349,"style":8316},[642],[363,166351],{},[363,166353,898],{"className":166354},[920],[363,166356],{"className":166357,"style":754},[505],[363,166359,133150],{"className":166360},[758],[363,166362],{"className":166363,"style":754},[505],[363,166365,898],{"className":166366},[920],[363,166368],{"className":166369,"style":754},[505],[363,166371,166373,166376],{"className":166372},[407],[363,166374,83214],{"className":166375},[407,408],[363,166377,166379],{"className":166378},[629],[363,166380,166382,166402],{"className":166381},[633,634],[363,166383,166385,166399],{"className":166384},[638],[363,166386,166388],{"className":166387,"style":727},[642],[363,166389,166390,166393],{"style":730},[363,166391],{"className":166392,"style":651},[650],[363,166394,166396],{"className":166395},[655,656,657,658],[363,166397,11044],{"className":166398,"style":11132},[407,408,658],[363,166400,666],{"className":166401},[665],[363,166403,166405],{"className":166404},[638],[363,166406,166408],{"className":166407,"style":8316},[642],[363,166409],{},[363,166411,468],{"className":166412},[501],[363,166414],{"className":166415,"style":506},[505],[363,166417,6257],{"className":166418},[510],[363,166420],{"className":166421,"style":506},[505],[363,166423,166425,166429,166432,166435,166438,166441,166444],{"className":166424},[398],[363,166426],{"className":166427,"style":166428},[402],"height:1.0913em;vertical-align:-0.25em;",[363,166430,578],{"className":166431},[494],[363,166433,474],{"className":166434},[407],[363,166436,898],{"className":166437},[920],[363,166439],{"className":166440,"style":754},[505],[363,166442,465],{"className":166443},[407],[363,166445,166447,166450],{"className":166446},[501],[363,166448,605],{"className":166449},[501],[363,166451,166453],{"className":166452},[629],[363,166454,166456],{"className":166455},[633],[363,166457,166459],{"className":166458},[638],[363,166460,166462],{"className":166461,"style":52457},[642],[363,166463,166464,166467],{"style":7213},[363,166465],{"className":166466,"style":651},[650],[363,166468,166470],{"className":166469},[655,656,657,658],[363,166471,11044],{"className":166472,"style":11132},[407,408,658]," 为",[1344,166475,166476],{},"信念质量（belief mass）",[7013,166478,166479,166473,166553,10528],{},[363,166480,166482,166508],{"className":166481,"translate":367},[366],[363,166483,166485],{"className":166484},[371],[373,166486,166487],{"xmlns":375},[377,166488,166489,166505],{},[380,166490,166491,166493,166495,166497,166499,166501,166503],{},[383,166492,15750],{},[458,166494,6257],{},[458,166496,578],{"stretchy":460},[463,166498,474],{},[458,166500,898],{"separator":394},[463,166502,465],{},[458,166504,605],{"stretchy":460},[387,166506,166507],{"encoding":389},"u \\in [0,1]",[363,166509,166511,166529],{"className":166510,"ariaHidden":394},[393],[363,166512,166514,166517,166520,166523,166526],{"className":166513},[398],[363,166515],{"className":166516,"style":6272},[402],[363,166518,15750],{"className":166519},[407,408],[363,166521],{"className":166522,"style":506},[505],[363,166524,6257],{"className":166525},[510],[363,166527],{"className":166528,"style":506},[505],[363,166530,166532,166535,166538,166541,166544,166547,166550],{"className":166531},[398],[363,166533],{"className":166534,"style":487},[402],[363,166536,578],{"className":166537},[494],[363,166539,474],{"className":166540},[407],[363,166542,898],{"className":166543},[920],[363,166545],{"className":166546,"style":754},[505],[363,166548,465],{"className":166549},[407],[363,166551,605],{"className":166552},[501],[1344,166554,166555],{},"不确定性质量（uncertainty mass）",[7013,166557,166558,3093,166662,166803,166804,10528],{},[363,166559,166561,166591],{"className":166560,"translate":367},[366],[363,166562,166564],{"className":166563},[371],[373,166565,166566],{"xmlns":375},[377,166567,166568,166588],{},[380,166569,166570,166572,166574,166576,166578,166580,166582],{},[383,166571,45552],{"mathvariant":75598},[458,166573,6257],{},[458,166575,578],{"stretchy":460},[463,166577,474],{},[458,166579,898],{"separator":394},[463,166581,465],{},[7162,166583,166584,166586],{},[458,166585,605],{"stretchy":460},[383,166587,11044],{},[387,166589,166590],{"encoding":389},"\\mathbf{a} \\in [0,1]^K",[363,166592,166594,166612],{"className":166593,"ariaHidden":394},[393],[363,166595,166597,166600,166603,166606,166609],{"className":166596},[398],[363,166598],{"className":166599,"style":6272},[402],[363,166601,45552],{"className":166602},[407,75943],[363,166604],{"className":166605,"style":506},[505],[363,166607,6257],{"className":166608},[510],[363,166610],{"className":166611,"style":506},[505],[363,166613,166615,166618,166621,166624,166627,166630,166633],{"className":166614},[398],[363,166616],{"className":166617,"style":166428},[402],[363,166619,578],{"className":166620},[494],[363,166622,474],{"className":166623},[407],[363,166625,898],{"className":166626},[920],[363,166628],{"className":166629,"style":754},[505],[363,166631,465],{"className":166632},[407],[363,166634,166636,166639],{"className":166635},[501],[363,166637,605],{"className":166638},[501],[363,166640,166642],{"className":166641},[629],[363,166643,166645],{"className":166644},[633],[363,166646,166648],{"className":166647},[638],[363,166649,166651],{"className":166650,"style":52457},[642],[363,166652,166653,166656],{"style":7213},[363,166654],{"className":166655,"style":651},[650],[363,166657,166659],{"className":166658},[655,656,657,658],[363,166660,11044],{"className":166661,"style":11132},[407,408,658],[363,166663,166665,166693],{"className":166664,"translate":367},[366],[363,166666,166668],{"className":166667},[371],[373,166669,166670],{"xmlns":375},[377,166671,166672,166690],{},[380,166673,166674,166680,166686,166688],{},[543,166675,166676,166678],{},[458,166677,13811],{},[383,166679,7912],{},[543,166681,166682,166684],{},[383,166683,45552],{},[383,166685,7912],{},[458,166687,471],{},[463,166689,465],{},[387,166691,166692],{"encoding":389},"\\sum_k a_k = 1",[363,166694,166696,166794],{"className":166695,"ariaHidden":394},[393],[363,166697,166699,166702,166742,166745,166785,166788,166791],{"className":166698},[398],[363,166700],{"className":166701,"style":138953},[402],[363,166703,166705,166708],{"className":166704},[1832],[363,166706,13811],{"className":166707,"style":40032},[1832,1833,5964],[363,166709,166711],{"className":166710},[629],[363,166712,166714,166734],{"className":166713},[633,634],[363,166715,166717,166731],{"className":166716},[638],[363,166718,166720],{"className":166719,"style":67394},[642],[363,166721,166722,166725],{"style":40048},[363,166723],{"className":166724,"style":651},[650],[363,166726,166728],{"className":166727},[655,656,657,658],[363,166729,7912],{"className":166730,"style":8615},[407,408,658],[363,166732,666],{"className":166733},[665],[363,166735,166737],{"className":166736},[638],[363,166738,166740],{"className":166739,"style":40067},[642],[363,166741],{},[363,166743],{"className":166744,"style":754},[505],[363,166746,166748,166751],{"className":166747},[407],[363,166749,45552],{"className":166750},[407,408],[363,166752,166754],{"className":166753},[629],[363,166755,166757,166777],{"className":166756},[633,634],[363,166758,166760,166774],{"className":166759},[638],[363,166761,166763],{"className":166762,"style":643},[642],[363,166764,166765,166768],{"style":730},[363,166766],{"className":166767,"style":651},[650],[363,166769,166771],{"className":166770},[655,656,657,658],[363,166772,7912],{"className":166773,"style":8615},[407,408,658],[363,166775,666],{"className":166776},[665],[363,166778,166780],{"className":166779},[638],[363,166781,166783],{"className":166782,"style":8316},[642],[363,166784],{},[363,166786],{"className":166787,"style":506},[505],[363,166789,471],{"className":166790},[510],[363,166792],{"className":166793,"style":506},[505],[363,166795,166797,166800],{"className":166796},[398],[363,166798],{"className":166799,"style":520},[402],[363,166801,465],{"className":166802},[407],"，为",[1344,166805,166806],{},"基准率（base rate）",[11,166808,166809,166810,10528],{},"并满足单纯形约束 ",[363,166811,166813,166853],{"className":166812,"translate":367},[366],[363,166814,166816],{"className":166815},[371],[373,166817,166818],{"xmlns":375},[377,166819,166820,166850],{},[380,166821,166822,166836,166842,166844,166846,166848],{},[7901,166823,166824,166826,166834],{},[458,166825,13811],{},[380,166827,166828,166830,166832],{},[383,166829,7912],{},[458,166831,471],{},[463,166833,465],{},[383,166835,11044],{},[543,166837,166838,166840],{},[383,166839,83214],{},[383,166841,7912],{},[458,166843,2818],{},[383,166845,15750],{},[458,166847,471],{},[463,166849,465],{},[387,166851,166852],{"encoding":389},"\\sum_{k=1}^K b_k + u = 1",[363,166854,166856,166974,166992],{"className":166855,"ariaHidden":394},[393],[363,166857,166859,166862,166922,166925,166965,166968,166971],{"className":166858},[398],[363,166860],{"className":166861,"style":136418},[402],[363,166863,166865,166868],{"className":166864},[1832],[363,166866,13811],{"className":166867,"style":40032},[1832,1833,5964],[363,166869,166871],{"className":166870},[629],[363,166872,166874,166914],{"className":166873},[633,634],[363,166875,166877,166911],{"className":166876},[638],[363,166878,166880,166900],{"className":166879,"style":133537},[642],[363,166881,166882,166885],{"style":40048},[363,166883],{"className":166884,"style":651},[650],[363,166886,166888],{"className":166887},[655,656,657,658],[363,166889,166891,166894,166897],{"className":166890},[407,658],[363,166892,7912],{"className":166893,"style":8615},[407,408,658],[363,166895,471],{"className":166896},[510,658],[363,166898,465],{"className":166899},[407,658],[363,166901,166902,166905],{"style":133560},[363,166903],{"className":166904,"style":651},[650],[363,166906,166908],{"className":166907},[655,656,657,658],[363,166909,11044],{"className":166910,"style":11132},[407,408,658],[363,166912,666],{"className":166913},[665],[363,166915,166917],{"className":166916},[638],[363,166918,166920],{"className":166919,"style":40067},[642],[363,166921],{},[363,166923],{"className":166924,"style":754},[505],[363,166926,166928,166931],{"className":166927},[407],[363,166929,83214],{"className":166930},[407,408],[363,166932,166934],{"className":166933},[629],[363,166935,166937,166957],{"className":166936},[633,634],[363,166938,166940,166954],{"className":166939},[638],[363,166941,166943],{"className":166942,"style":643},[642],[363,166944,166945,166948],{"style":730},[363,166946],{"className":166947,"style":651},[650],[363,166949,166951],{"className":166950},[655,656,657,658],[363,166952,7912],{"className":166953,"style":8615},[407,408,658],[363,166955,666],{"className":166956},[665],[363,166958,166960],{"className":166959},[638],[363,166961,166963],{"className":166962,"style":8316},[642],[363,166964],{},[363,166966],{"className":166967,"style":2935},[505],[363,166969,2818],{"className":166970},[2939],[363,166972],{"className":166973,"style":2935},[505],[363,166975,166977,166980,166983,166986,166989],{"className":166976},[398],[363,166978],{"className":166979,"style":2693},[402],[363,166981,15750],{"className":166982},[407,408],[363,166984],{"className":166985,"style":506},[505],[363,166987,471],{"className":166988},[510],[363,166990],{"className":166991,"style":506},[505],[363,166993,166995,166998],{"className":166994},[398],[363,166996],{"className":166997,"style":520},[402],[363,166999,465],{"className":167000},[407],[11,167002,167003,34197,167031,167034,167035,167236,167237,10650,167336,10528],{},[363,167004,167006,167019],{"className":167005,"translate":367},[366],[363,167007,167009],{"className":167008},[371],[373,167010,167011],{"xmlns":375},[377,167012,167013,167017],{},[380,167014,167015],{},[383,167016,100250],{},[387,167018,101112],{"encoding":389},[363,167020,167022],{"className":167021,"ariaHidden":394},[393],[363,167023,167025,167028],{"className":167024},[398],[363,167026],{"className":167027,"style":2693},[402],[363,167029,100250],{"className":167030,"style":1328},[407,408],[1344,167032,167033],{},"概率投影（probability projection）"," 定义为 ",[363,167036,167038,167074],{"className":167037,"translate":367},[366],[363,167039,167041],{"className":167040},[371],[373,167042,167043],{"xmlns":375},[377,167044,167045,167071],{},[380,167046,167047,167053,167055,167061,167063,167069],{},[543,167048,167049,167051],{},[383,167050,554],{},[383,167052,7912],{},[458,167054,471],{},[543,167056,167057,167059],{},[383,167058,83214],{},[383,167060,7912],{},[458,167062,2818],{},[543,167064,167065,167067],{},[383,167066,45552],{},[383,167068,7912],{},[383,167070,15750],{},[387,167072,167073],{"encoding":389},"P_k = b_k + a_k u",[363,167075,167077,167132,167187],{"className":167076,"ariaHidden":394},[393],[363,167078,167080,167083,167123,167126,167129],{"className":167079},[398],[363,167081],{"className":167082,"style":16311},[402],[363,167084,167086,167089],{"className":167085},[407],[363,167087,554],{"className":167088,"style":682},[407,408],[363,167090,167092],{"className":167091},[629],[363,167093,167095,167115],{"className":167094},[633,634],[363,167096,167098,167112],{"className":167097},[638],[363,167099,167101],{"className":167100,"style":643},[642],[363,167102,167103,167106],{"style":11486},[363,167104],{"className":167105,"style":651},[650],[363,167107,167109],{"className":167108},[655,656,657,658],[363,167110,7912],{"className":167111,"style":8615},[407,408,658],[363,167113,666],{"className":167114},[665],[363,167116,167118],{"className":167117},[638],[363,167119,167121],{"className":167120,"style":8316},[642],[363,167122],{},[363,167124],{"className":167125,"style":506},[505],[363,167127,471],{"className":167128},[510],[363,167130],{"className":167131,"style":506},[505],[363,167133,167135,167138,167178,167181,167184],{"className":167134},[398],[363,167136],{"className":167137,"style":11824},[402],[363,167139,167141,167144],{"className":167140},[407],[363,167142,83214],{"className":167143},[407,408],[363,167145,167147],{"className":167146},[629],[363,167148,167150,167170],{"className":167149},[633,634],[363,167151,167153,167167],{"className":167152},[638],[363,167154,167156],{"className":167155,"style":643},[642],[363,167157,167158,167161],{"style":730},[363,167159],{"className":167160,"style":651},[650],[363,167162,167164],{"className":167163},[655,656,657,658],[363,167165,7912],{"className":167166,"style":8615},[407,408,658],[363,167168,666],{"className":167169},[665],[363,167171,167173],{"className":167172},[638],[363,167174,167176],{"className":167175,"style":8316},[642],[363,167177],{},[363,167179],{"className":167180,"style":2935},[505],[363,167182,2818],{"className":167183},[2939],[363,167185],{"className":167186,"style":2935},[505],[363,167188,167190,167193,167233],{"className":167189},[398],[363,167191],{"className":167192,"style":13724},[402],[363,167194,167196,167199],{"className":167195},[407],[363,167197,45552],{"className":167198},[407,408],[363,167200,167202],{"className":167201},[629],[363,167203,167205,167225],{"className":167204},[633,634],[363,167206,167208,167222],{"className":167207},[638],[363,167209,167211],{"className":167210,"style":643},[642],[363,167212,167213,167216],{"style":730},[363,167214],{"className":167215,"style":651},[650],[363,167217,167219],{"className":167218},[655,656,657,658],[363,167220,7912],{"className":167221,"style":8615},[407,408,658],[363,167223,666],{"className":167224},[665],[363,167226,167228],{"className":167227},[638],[363,167229,167231],{"className":167230,"style":8316},[642],[363,167232],{},[363,167234,15750],{"className":167235},[407,408],"：在不了解具体类别时退回基准率，在有所了解时按信念质量分配。当基准率取均匀值 ",[363,167238,167240,167266],{"className":167239,"translate":367},[366],[363,167241,167243],{"className":167242},[371],[373,167244,167245],{"xmlns":375},[377,167246,167247,167263],{},[380,167248,167249,167255,167257,167259,167261],{},[543,167250,167251,167253],{},[383,167252,45552],{},[383,167254,7912],{},[458,167256,471],{},[463,167258,465],{},[383,167260,1288],{"mathvariant":557},[383,167262,11044],{},[387,167264,167265],{"encoding":389},"a_k = 1\u002FK",[363,167267,167269,167324],{"className":167268,"ariaHidden":394},[393],[363,167270,167272,167275,167315,167318,167321],{"className":167271},[398],[363,167273],{"className":167274,"style":13724},[402],[363,167276,167278,167281],{"className":167277},[407],[363,167279,45552],{"className":167280},[407,408],[363,167282,167284],{"className":167283},[629],[363,167285,167287,167307],{"className":167286},[633,634],[363,167288,167290,167304],{"className":167289},[638],[363,167291,167293],{"className":167292,"style":643},[642],[363,167294,167295,167298],{"style":730},[363,167296],{"className":167297,"style":651},[650],[363,167299,167301],{"className":167300},[655,656,657,658],[363,167302,7912],{"className":167303,"style":8615},[407,408,658],[363,167305,666],{"className":167306},[665],[363,167308,167310],{"className":167309},[638],[363,167311,167313],{"className":167312,"style":8316},[642],[363,167314],{},[363,167316],{"className":167317,"style":506},[505],[363,167319,471],{"className":167320},[510],[363,167322],{"className":167323,"style":506},[505],[363,167325,167327,167330,167333],{"className":167326},[398],[363,167328],{"className":167329,"style":487},[402],[363,167331,8937],{"className":167332},[407],[363,167334,11044],{"className":167335,"style":11132},[407,408],[363,167337,167339,167373],{"className":167338,"translate":367},[366],[363,167340,167342],{"className":167341},[371],[373,167343,167344],{"xmlns":375},[377,167345,167346,167370],{},[380,167347,167348,167354,167356,167362,167364,167366,167368],{},[543,167349,167350,167352],{},[383,167351,554],{},[383,167353,7912],{},[458,167355,471],{},[543,167357,167358,167360],{},[383,167359,83214],{},[383,167361,7912],{},[458,167363,2818],{},[383,167365,15750],{},[383,167367,1288],{"mathvariant":557},[383,167369,11044],{},[387,167371,167372],{"encoding":389},"P_k = b_k + u\u002FK",[363,167374,167376,167431,167486],{"className":167375,"ariaHidden":394},[393],[363,167377,167379,167382,167422,167425,167428],{"className":167378},[398],[363,167380],{"className":167381,"style":16311},[402],[363,167383,167385,167388],{"className":167384},[407],[363,167386,554],{"className":167387,"style":682},[407,408],[363,167389,167391],{"className":167390},[629],[363,167392,167394,167414],{"className":167393},[633,634],[363,167395,167397,167411],{"className":167396},[638],[363,167398,167400],{"className":167399,"style":643},[642],[363,167401,167402,167405],{"style":11486},[363,167403],{"className":167404,"style":651},[650],[363,167406,167408],{"className":167407},[655,656,657,658],[363,167409,7912],{"className":167410,"style":8615},[407,408,658],[363,167412,666],{"className":167413},[665],[363,167415,167417],{"className":167416},[638],[363,167418,167420],{"className":167419,"style":8316},[642],[363,167421],{},[363,167423],{"className":167424,"style":506},[505],[363,167426,471],{"className":167427},[510],[363,167429],{"className":167430,"style":506},[505],[363,167432,167434,167437,167477,167480,167483],{"className":167433},[398],[363,167435],{"className":167436,"style":11824},[402],[363,167438,167440,167443],{"className":167439},[407],[363,167441,83214],{"className":167442},[407,408],[363,167444,167446],{"className":167445},[629],[363,167447,167449,167469],{"className":167448},[633,634],[363,167450,167452,167466],{"className":167451},[638],[363,167453,167455],{"className":167454,"style":643},[642],[363,167456,167457,167460],{"style":730},[363,167458],{"className":167459,"style":651},[650],[363,167461,167463],{"className":167462},[655,656,657,658],[363,167464,7912],{"className":167465,"style":8615},[407,408,658],[363,167467,666],{"className":167468},[665],[363,167470,167472],{"className":167471},[638],[363,167473,167475],{"className":167474,"style":8316},[642],[363,167476],{},[363,167478],{"className":167479,"style":2935},[505],[363,167481,2818],{"className":167482},[2939],[363,167484],{"className":167485,"style":2935},[505],[363,167487,167489,167492,167495,167498],{"className":167488},[398],[363,167490],{"className":167491,"style":487},[402],[363,167493,15750],{"className":167494},[407,408],[363,167496,1288],{"className":167497},[407],[363,167499,11044],{"className":167500,"style":11132},[407,408],[11,167502,167503,167504,167566,167567,167595,167596,167669],{},"定义 3.1 的关键在于：opinion 所栖居的空间是 ",[363,167505,167507,167525],{"className":167506,"translate":367},[366],[363,167508,167510],{"className":167509},[371],[373,167511,167512],{"xmlns":375},[377,167513,167514,167522],{},[380,167515,167516],{},[7162,167517,167518,167520],{},[383,167519,126067],{"mathvariant":557},[383,167521,11044],{},[387,167523,167524],{"encoding":389},"\\Delta^{K}",[363,167526,167528],{"className":167527,"ariaHidden":394},[393],[363,167529,167531,167534],{"className":167530},[398],[363,167532],{"className":167533,"style":52457},[402],[363,167535,167537,167540],{"className":167536},[407],[363,167538,126067],{"className":167539},[407],[363,167541,167543],{"className":167542},[629],[363,167544,167546],{"className":167545},[633],[363,167547,167549],{"className":167548},[638],[363,167550,167552],{"className":167551,"style":52457},[642],[363,167553,167554,167557],{"style":7213},[363,167555],{"className":167556,"style":651},[650],[363,167558,167560],{"className":167559},[655,656,657,658],[363,167561,167563],{"className":167562},[407,658],[363,167564,11044],{"className":167565,"style":11132},[407,408,658],"——一个 ",[363,167568,167570,167583],{"className":167569,"translate":367},[366],[363,167571,167573],{"className":167572},[371],[373,167574,167575],{"xmlns":375},[377,167576,167577,167581],{},[380,167578,167579],{},[383,167580,11044],{},[387,167582,11044],{"encoding":389},[363,167584,167586],{"className":167585,"ariaHidden":394},[393],[363,167587,167589,167592],{"className":167588},[398],[363,167590],{"className":167591,"style":1011},[402],[363,167593,11044],{"className":167594,"style":11132},[407,408]," 维单纯形，比 ",[363,167597,167599,167622],{"className":167598,"translate":367},[366],[363,167600,167602],{"className":167601},[371],[373,167603,167604],{"xmlns":375},[377,167605,167606,167620],{},[380,167607,167608],{},[7162,167609,167610,167612],{},[383,167611,126067],{"mathvariant":557},[380,167613,167614,167616,167618],{},[383,167615,11044],{},[458,167617,5161],{},[463,167619,465],{},[387,167621,134819],{"encoding":389},[363,167623,167625],{"className":167624,"ariaHidden":394},[393],[363,167626,167628,167631],{"className":167627},[398],[363,167629],{"className":167630,"style":52457},[402],[363,167632,167634,167637],{"className":167633},[407],[363,167635,126067],{"className":167636},[407],[363,167638,167640],{"className":167639},[629],[363,167641,167643],{"className":167642},[633],[363,167644,167646],{"className":167645},[638],[363,167647,167649],{"className":167648,"style":52457},[642],[363,167650,167651,167654],{"style":7213},[363,167652],{"className":167653,"style":651},[650],[363,167655,167657],{"className":167656},[655,656,657,658],[363,167658,167660,167663,167666],{"className":167659},[407,658],[363,167661,11044],{"className":167662,"style":11132},[407,408,658],[363,167664,5161],{"className":167665},[2939,658],[363,167667,465],{"className":167668},[407,658]," 多出一个维度。多出来的这一维正是「不确定性」的自由度，而它恰恰是 softmax 输出所没有的。",[2659,167671,167673],{"id":167672},"_32-dirichlet-与-opinion-的双射","3.2 Dirichlet 与 opinion 的双射",[11,167675,167676,167677,10528],{},"主观逻辑与迪利克雷分布之间不存在类比关系，而是存在严格的",[1344,167678,167679],{},"双射",[143,167681,167682],{},[11,167683,167684,167687,167688,167740,167741,167839,167840],{},[1344,167685,167686],{},"命题 3.2（双射）"," 设先验权重 ",[363,167689,167691,167710],{"className":167690,"translate":367},[366],[363,167692,167694],{"className":167693},[371],[373,167695,167696],{"xmlns":375},[377,167697,167698,167707],{},[380,167699,167700,167703,167705],{},[383,167701,167702],{},"W",[458,167704,471],{},[383,167706,11044],{},[387,167708,167709],{"encoding":389},"W = K",[363,167711,167713,167731],{"className":167712,"ariaHidden":394},[393],[363,167714,167716,167719,167722,167725,167728],{"className":167715},[398],[363,167717],{"className":167718,"style":1011},[402],[363,167720,167702],{"className":167721,"style":682},[407,408],[363,167723],{"className":167724,"style":506},[505],[363,167726,471],{"className":167727},[510],[363,167729],{"className":167730,"style":506},[505],[363,167732,167734,167737],{"className":167733},[398],[363,167735],{"className":167736,"style":1011},[402],[363,167738,11044],{"className":167739,"style":11132},[407,408],"、基准率 ",[363,167742,167744,167769],{"className":167743,"translate":367},[366],[363,167745,167747],{"className":167746},[371],[373,167748,167749],{"xmlns":375},[377,167750,167751,167767],{},[380,167752,167753,167759,167761,167763,167765],{},[543,167754,167755,167757],{},[383,167756,45552],{},[383,167758,7912],{},[458,167760,471],{},[463,167762,465],{},[383,167764,1288],{"mathvariant":557},[383,167766,11044],{},[387,167768,167265],{"encoding":389},[363,167770,167772,167827],{"className":167771,"ariaHidden":394},[393],[363,167773,167775,167778,167818,167821,167824],{"className":167774},[398],[363,167776],{"className":167777,"style":13724},[402],[363,167779,167781,167784],{"className":167780},[407],[363,167782,45552],{"className":167783},[407,408],[363,167785,167787],{"className":167786},[629],[363,167788,167790,167810],{"className":167789},[633,634],[363,167791,167793,167807],{"className":167792},[638],[363,167794,167796],{"className":167795,"style":643},[642],[363,167797,167798,167801],{"style":730},[363,167799],{"className":167800,"style":651},[650],[363,167802,167804],{"className":167803},[655,656,657,658],[363,167805,7912],{"className":167806,"style":8615},[407,408,658],[363,167808,666],{"className":167809},[665],[363,167811,167813],{"className":167812},[638],[363,167814,167816],{"className":167815,"style":8316},[642],[363,167817],{},[363,167819],{"className":167820,"style":506},[505],[363,167822,471],{"className":167823},[510],[363,167825],{"className":167826,"style":506},[505],[363,167828,167830,167833,167836],{"className":167829},[398],[363,167831],{"className":167832,"style":487},[402],[363,167834,8937],{"className":167835},[407],[363,167837,11044],{"className":167838,"style":11132},[407,408],"。则如下两个映射互逆：\n",[363,167841,167843,167945],{"className":167842,"translate":367},[366],[363,167844,167846],{"className":167845},[371],[373,167847,167848],{"xmlns":375},[377,167849,167850,167942],{},[380,167851,167852,167854,167856,167872,167874,167876,167878,167880,167890,167892,167894,167896,167902,167904,167920,167922,167924,167926,167928,167934,167936],{},[383,167853,83214],{"mathvariant":75598},[458,167855,471],{},[586,167857,167858,167866],{},[380,167859,167860,167862,167864],{},[383,167861,45126],{"mathvariant":136055},[458,167863,5161],{},[463,167865,465],{"mathvariant":75598},[543,167867,167868,167870],{},[383,167869,45126],{},[463,167871,474],{},[458,167873,898],{"separator":394},[505,167875],{"width":3610},[383,167877,15750],{},[458,167879,471],{},[586,167881,167882,167884],{},[383,167883,11044],{},[543,167885,167886,167888],{},[383,167887,45126],{},[463,167889,474],{},[505,167891],{"width":3610},[458,167893,96444],{},[505,167895],{"width":3610},[543,167897,167898,167900],{},[383,167899,45126],{},[383,167901,7912],{},[458,167903,471],{},[586,167905,167906,167918],{},[380,167907,167908,167910,167912],{},[383,167909,11044],{},[1645,167911,1647],{},[543,167913,167914,167916],{},[383,167915,83214],{},[383,167917,7912],{},[383,167919,15750],{},[458,167921,2818],{},[463,167923,465],{},[458,167925,898],{"separator":394},[505,167927],{"width":3610},[543,167929,167930,167932],{},[383,167931,45126],{},[463,167933,474],{},[458,167935,471],{},[586,167937,167938,167940],{},[383,167939,11044],{},[383,167941,15750],{},[387,167943,167944],{"encoding":389},"\\mathbf{b} = \\frac{\\boldsymbol{\\alpha} - \\mathbf{1}}{\\alpha_0}, \\qquad u = \\frac{K}{\\alpha_0} \\qquad \\Longleftrightarrow \\qquad \\alpha_k = \\frac{K\\, b_k}{u} + 1, \\qquad \\alpha_0 = \\frac{K}{u}",[363,167946,167948,167966,168110,168238,168293,168421,168488],{"className":167947,"ariaHidden":394},[393],[363,167949,167951,167954,167957,167960,167963],{"className":167950},[398],[363,167952],{"className":167953,"style":12947},[402],[363,167955,83214],{"className":167956},[407,75943],[363,167958],{"className":167959,"style":506},[505],[363,167961,471],{"className":167962},[510],[363,167964],{"className":167965,"style":506},[505],[363,167967,167969,167972,168089,168092,168095,168098,168101,168104,168107],{"className":167968},[398],[363,167970],{"className":167971,"style":149401},[402],[363,167973,167975,167978,168086],{"className":167974},[407],[363,167976],{"className":167977},[494,789],[363,167979,167981],{"className":167980},[586],[363,167982,167984,168078],{"className":167983},[633,634],[363,167985,167987,168075],{"className":167986},[638],[363,167988,167990,168041,168049],{"className":167989,"style":2872},[642],[363,167991,167992,167995],{"style":2875},[363,167993],{"className":167994,"style":809},[650],[363,167996,167998],{"className":167997},[655,656,657,658],[363,167999,168001],{"className":168000},[407,658],[363,168002,168004,168007],{"className":168003},[407,658],[363,168005,45126],{"className":168006,"style":45156},[407,408,658],[363,168008,168010],{"className":168009},[629],[363,168011,168013,168033],{"className":168012},[633,634],[363,168014,168016,168030],{"className":168015},[638],[363,168017,168019],{"className":168018,"style":137351},[642],[363,168020,168021,168024],{"style":136989},[363,168022],{"className":168023,"style":30578},[650],[363,168025,168027],{"className":168026},[655,30582,10022,658],[363,168028,474],{"className":168029},[407,658],[363,168031,666],{"className":168032},[665],[363,168034,168036],{"className":168035},[638],[363,168037,168039],{"className":168038,"style":40142},[642],[363,168040],{},[363,168042,168043,168046],{"style":821},[363,168044],{"className":168045,"style":809},[650],[363,168047],{"className":168048,"style":829},[828],[363,168050,168051,168054],{"style":2898},[363,168052],{"className":168053,"style":809},[650],[363,168055,168057],{"className":168056},[655,656,657,658],[363,168058,168060,168069,168072],{"className":168059},[407,658],[363,168061,168063],{"className":168062},[407,658],[363,168064,168066],{"className":168065},[407,658],[363,168067,45126],{"className":168068},[407,136115,658],[363,168070,5161],{"className":168071},[2939,658],[363,168073,465],{"className":168074},[407,75943,658],[363,168076,666],{"className":168077},[665],[363,168079,168081],{"className":168080},[638],[363,168082,168084],{"className":168083,"style":40289},[642],[363,168085],{},[363,168087],{"className":168088},[501,789],[363,168090,898],{"className":168091},[920],[363,168093],{"className":168094,"style":3982},[505],[363,168096],{"className":168097,"style":754},[505],[363,168099,15750],{"className":168100},[407,408],[363,168102],{"className":168103,"style":506},[505],[363,168105,471],{"className":168106},[510],[363,168108],{"className":168109,"style":506},[505],[363,168111,168113,168117,168223,168226,168229,168232,168235],{"className":168112},[398],[363,168114],{"className":168115,"style":168116},[402],"height:1.3174em;vertical-align:-0.4451em;",[363,168118,168120,168123,168220],{"className":168119},[407],[363,168121],{"className":168122},[494,789],[363,168124,168126],{"className":168125},[586],[363,168127,168129,168212],{"className":168128},[633,634],[363,168130,168132,168209],{"className":168131},[638],[363,168133,168136,168187,168195],{"className":168134,"style":168135},[642],"height:0.8723em;",[363,168137,168138,168141],{"style":2875},[363,168139],{"className":168140,"style":809},[650],[363,168142,168144],{"className":168143},[655,656,657,658],[363,168145,168147],{"className":168146},[407,658],[363,168148,168150,168153],{"className":168149},[407,658],[363,168151,45126],{"className":168152,"style":45156},[407,408,658],[363,168154,168156],{"className":168155},[629],[363,168157,168159,168179],{"className":168158},[633,634],[363,168160,168162,168176],{"className":168161},[638],[363,168163,168165],{"className":168164,"style":137351},[642],[363,168166,168167,168170],{"style":136989},[363,168168],{"className":168169,"style":30578},[650],[363,168171,168173],{"className":168172},[655,30582,10022,658],[363,168174,474],{"className":168175},[407,658],[363,168177,666],{"className":168178},[665],[363,168180,168182],{"className":168181},[638],[363,168183,168185],{"className":168184,"style":40142},[642],[363,168186],{},[363,168188,168189,168192],{"style":821},[363,168190],{"className":168191,"style":809},[650],[363,168193],{"className":168194,"style":829},[828],[363,168196,168197,168200],{"style":2898},[363,168198],{"className":168199,"style":809},[650],[363,168201,168203],{"className":168202},[655,656,657,658],[363,168204,168206],{"className":168205},[407,658],[363,168207,11044],{"className":168208,"style":11132},[407,408,658],[363,168210,666],{"className":168211},[665],[363,1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正向：",[363,168570,168572,168638],{"className":168571,"translate":367},[366],[363,168573,168575],{"className":168574},[371],[373,168576,168577],{"xmlns":375},[377,168578,168579,168635],{},[380,168580,168581,168587,168593,168595,168597,168599,168619,168621,168631,168633],{},[543,168582,168583,168585],{},[458,168584,13811],{},[383,168586,7912],{},[543,168588,168589,168591],{},[383,168590,83214],{},[383,168592,7912],{},[458,168594,2818],{},[383,168596,15750],{},[458,168598,471],{},[586,168600,168601,168613],{},[380,168602,168603,168609,168611],{},[543,168604,168605,168607],{},[383,168606,45126],{},[463,168608,474],{},[458,168610,5161],{},[383,168612,11044],{},[543,168614,168615,168617],{},[383,168616,45126],{},[463,168618,474],{},[458,168620,2818],{},[586,168622,168623,168625],{},[383,168624,11044],{},[543,168626,168627,168629],{},[383,168628,45126],{},[463,168630,474],{},[458,168632,471],{},[463,168634,465],{},[387,168636,168637],{"encoding":389},"\\sum_k b_k + u = \\frac{\\alpha_0 - K}{\\alpha_0} + \\frac{K}{\\alpha_0} = 1",[363,168639,168641,168739,168757,168922,169042],{"className":168640,"ariaHidden":394},[393],[363,168642,168644,168647,168687,168690,168730,168733,168736],{"className":168643},[398],[363,168645],{"className":168646,"style":138953},[402],[363,168648,168650,168653],{"className":168649},[1832],[363,168651,13811],{"className":168652,"style":40032},[1832,1833,5964],[363,168654,168656],{"className":168655},[629],[363,168657,168659,168679],{"className":168658},[633,634],[363,168660,168662,168676],{"className":168661},[638],[363,168663,168665],{"className":168664,"style":67394},[642],[363,168666,168667,168670],{"style":40048},[363,168668],{"className":168669,"style":651},[650],[363,168671,168673],{"className":168672},[655,656,657,658],[363,168674,7912],{"className":168675,"style":8615},[407,408,658],[363,168677,666],{"className":168678},[665],[363,168680,168682],{"className":168681},[638],[363,168683,168685],{"className":168684,"style":40067},[642],[363,168686],{},[363,168688],{"className":168689,"style":754},[505],[363,168691,168693,168696],{"className":168692},[407],[363,168694,83214],{"className":168695},[407,408],[363,168697,168699],{"className":168698},[629],[363,168700,168702,168722],{"className":168701},[633,634],[363,168703,168705,168719],{"className":168704},[638],[363,168706,168708],{"className":168707,"style":643},[642],[363,168709,168710,168713],{"style":730},[363,168711],{"className":168712,"style":651},[650],[363,168714,168716],{"className":168715},[655,656,657,658],[363,168717,7912],{"className":168718,"style":8615},[407,408,658],[363,168720,666],{"className":168721},[665],[363,168723,168725],{"className":168724},[638],[363,168726,168728],{"className":168727,"style":8316},[642],[363,168729],{},[363,168731],{"className":168732,"style":2935},[505],[363,168734,2818],{"className":168735},[2939],[363,168737],{"className":168738,"style":2935},[505],[363,168740,168742,168745,168748,168751,168754],{"className":168741},[398],[363,168743],{"className":168744,"style":2693},[402],[363,168746,15750],{"className":168747},[407,408],[363,168749],{"className":168750,"style":506},[505],[363,168752,471],{"className":168753},[510],[363,168755],{"className":168756,"style":506},[505],[363,168758,168760,168764,168913,168916,168919],{"className":168759},[398],[363,168761],{"className":168762,"style":168763},[402],"height:1.3335em;vertical-align:-0.4451em;",[363,168765,168767,168770,168910],{"className":168766},[407],[363,168768],{"className":168769},[494,789],[363,168771,168773],{"className":168772},[586],[363,168774,168776,168902],{"className":168775},[633,634],[363,168777,168779,168899],{"className":168778},[638],[363,168780,168783,168834,168842],{"className":168781,"style":168782},[642],"height:0.8884em;",[363,168784,168785,168788],{"style":2875},[363,168786],{"className":168787,"style":809},[650],[363,168789,168791],{"className":168790},[655,656,657,658],[363,168792,168794],{"className":168793},[407,658],[363,168795,168797,168800],{"className":168796},[407,658],[363,168798,45126],{"className":168799,"style":45156},[407,408,658],[363,168801,168803],{"className":168802},[629],[363,168804,168806,168826],{"className":168805},[633,634],[363,168807,168809,168823],{"className":168808},[638],[363,168810,168812],{"className":168811,"style":137351},[642],[363,168813,168814,168817],{"style":136989},[363,168815],{"className":168816,"style":30578},[650],[363,168818,168820],{"className":168819},[655,30582,10022,658],[363,168821,474],{"className":168822},[407,658],[363,168824,666],{"className":168825},[665],[363,168827,168829],{"className":168828},[638],[363,168830,168832],{"className":168831,"style":40142},[642],[363,168833],{},[363,168835,168836,168839],{"style":821},[363,168837],{"className":168838,"style":809},[650],[363,168840],{"className":168841,"style":829},[828],[363,168843,168844,168847],{"style":147999},[363,168845],{"className":168846,"style":809},[650],[363,168848,168850],{"className":168849},[655,656,657,658],[363,168851,168853,168893,168896],{"className":168852},[407,658],[363,168854,168856,168859],{"className":168855},[407,658],[363,168857,45126],{"className":168858,"style":45156},[407,408,658],[363,168860,168862],{"className":168861},[629],[363,168863,168865,168885],{"className":168864},[633,634],[363,168866,168868,168882],{"className":168867},[638],[363,168869,168871],{"className":168870,"style":137351},[642],[363,168872,168873,168876],{"style":136989},[363,168874],{"className":168875,"style":30578},[650],[363,168877,168879],{"className":168878},[655,30582,10022,658],[363,168880,474],{"className":168881},[407,658],[363,168883,666],{"className":168884},[665],[363,168886,168888],{"className":168887},[638],[363,168889,168891],{"className":168890,"style":40142},[642],[363,168892],{},[363,168894,5161],{"className":168895},[2939,658],[363,168897,11044],{"className":168898,"style":11132},[407,408,658],[363,168900,666],{"className":168901},[665],[363,168903,168905],{"className":168904},[638],[363,168906,168908],{"className":168907,"style":40289},[642],[363,168909],{},[363,168911],{"className":168912},[501,789],[363,168914],{"className":168915,"style":2935},[505],[363,168917,2818],{"className":168918},[2939],[363,168920],{"className":168921,"style":2935},[505],[363,168923,168925,168928,169033,169036,169039],{"className":168924},[398],[363,168926],{"className":168927,"style":168116},[402],[363,168929,168931,168934,169030],{"className":168930},[407],[363,168932],{"className":168933},[494,789],[363,168935,168937],{"className":168936},[586],[363,168938,168940,169022],{"className":168939},[633,634],[363,168941,168943,169019],{"className":168942},[638],[363,168944,168946,168997,169005],{"className":168945,"style":168135},[642],[363,168947,168948,168951],{"style":2875},[363,168949],{"className":168950,"style":809},[650],[363,168952,168954],{"className":168953},[655,656,657,658],[363,168955,168957],{"className":168956},[407,658],[363,168958,168960,168963],{"className":168959},[407,658],[363,168961,45126],{"className":168962,"style":45156},[407,408,658],[363,168964,168966],{"className":168965},[629],[363,168967,168969,168989],{"className":168968},[633,634],[363,168970,168972,168986],{"className":168971},[638],[363,168973,168975],{"className":168974,"style":137351},[642],[363,168976,168977,168980],{"style":136989},[363,168978],{"className":168979,"style":30578},[650],[363,168981,168983],{"className":168982},[655,30582,10022,658],[363,168984,474],{"className":168985},[407,658],[363,168987,666],{"className":168988},[665],[363,168990,168992],{"className":168991},[638],[363,168993,168995],{"className":168994,"style":40142},[642],[363,168996],{},[363,168998,168999,169002],{"style":821},[363,169000],{"className":169001,"style":809},[650],[363,169003],{"className":169004,"style":829},[828],[363,169006,169007,169010],{"style":2898},[363,169008],{"className":169009,"style":809},[650],[363,169011,169013],{"className":169012},[655,656,657,658],[363,169014,169016],{"className":169015},[407,658],[363,169017,11044],{"className":169018,"style":11132},[407,408,658],[363,169020,666],{"className":169021},[665],[363,169023,169025],{"className":169024},[638],[363,169026,169028],{"className":169027,"style":40289},[642],[363,169029],{},[363,169031],{"className":169032},[501,789],[363,169034],{"className":169035,"style":506},[505],[363,169037,471],{"className":169038},[510],[363,169040],{"className":169041,"style":506},[505],[363,169043,169045,169048],{"className":169044},[398],[363,169046],{"className":169047,"style":520},[402],[363,169049,465],{"className":169050},[407],"，满足定义 3.1 的约束。反向：代入即得恒等。此外，概率投影与迪利克雷期望重合：",[363,169053,169055],{"className":169054,"translate":367},[528],[363,169056,169058,169156],{"className":169057,"translate":367},[366],[363,169059,169061],{"className":169060},[371],[373,169062,169063],{"xmlns":375,"display":537},[377,169064,169065,169153],{},[380,169066,169067,169073,169075,169081,169083,169089,169091,169111,169113,169123,169125,169139,169141,169143,169145,169151],{},[543,169068,169069,169071],{},[383,169070,554],{},[383,169072,7912],{},[458,169074,471],{},[543,169076,169077,169079],{},[383,169078,83214],{},[383,169080,7912],{},[458,169082,2818],{},[586,169084,169085,169087],{},[383,169086,15750],{},[383,169088,11044],{},[458,169090,471],{},[586,169092,169093,169105],{},[380,169094,169095,169101,169103],{},[543,169096,169097,169099],{},[383,169098,45126],{},[383,169100,7912],{},[458,169102,5161],{},[463,169104,465],{},[543,169106,169107,169109],{},[383,169108,45126],{},[463,169110,474],{},[458,169112,2818],{},[586,169114,169115,169117],{},[463,169116,465],{},[543,169118,169119,169121],{},[383,169120,45126],{},[463,169122,474],{},[458,169124,471],{},[586,169126,169127,169133],{},[543,169128,169129,169131],{},[383,169130,45126],{},[383,169132,7912],{},[543,169134,169135,169137],{},[383,169136,45126],{},[463,169138,474],{},[458,169140,471],{},[383,169142,571],{"mathvariant":570},[458,169144,578],{"stretchy":460},[543,169146,169147,169149],{},[383,169148,1121],{},[383,169150,7912],{},[458,169152,605],{"stretchy":460},[387,169154,169155],{"encoding":389},"P_k = b_k + \\frac{u}{K} = \\frac{\\alpha_k - 1}{\\alpha_0} + \\frac{1}{\\alpha_0} = \\frac{\\alpha_k}{\\alpha_0} = \\mathbb{E}[x_k]",[363,169157,169159,169214,169269,169346,169509,169623,169774],{"className":169158,"ariaHidden":394},[393],[363,169160,169162,169165,169205,169208,169211],{"className":169161},[398],[363,169163],{"className":169164,"style":16311},[402],[363,169166,169168,169171],{"className":169167},[407],[363,169169,554],{"className":169170,"style":682},[407,408],[363,169172,169174],{"className":169173},[629],[363,169175,169177,169197],{"className":169176},[633,634],[363,169178,169180,169194],{"className":169179},[638],[363,169181,169183],{"className":169182,"style":643},[642],[363,169184,169185,169188],{"style":11486},[363,169186],{"className":169187,"style":651},[650],[363,169189,169191],{"className":169190},[655,656,657,658],[363,169192,7912],{"className":169193,"style":8615},[407,408,658],[363,169195,666],{"className":169196},[665],[363,169198,169200],{"className":169199},[638],[363,169201,169203],{"className":169202,"style":8316},[642],[363,169204],{},[363,169206],{"className":169207,"style":506},[505],[363,169209,471],{"className":169210},[510],[363,169212],{"className":169213,"style":506},[505],[363,169215,169217,169220,169260,169263,169266],{"className":169216},[398],[363,169218],{"className":169219,"style":11824},[402],[363,169221,169223,169226],{"className":169222},[407],[363,169224,83214],{"className":169225},[407,408],[363,169227,169229],{"className":169228},[629],[363,169230,169232,169252],{"className":169231},[633,634],[363,169233,169235,169249],{"className":169234},[638],[363,169236,169238],{"className":169237,"style":643},[642],[363,169239,169240,169243],{"style":730},[363,169241],{"className":169242,"style":651},[650],[363,169244,169246],{"className":169245},[655,656,657,658],[363,169247,7912],{"className":169248,"style":8615},[407,408,658],[363,169250,666],{"className":169251},[665],[363,169253,169255],{"className":169254},[638],[363,169256,169258],{"className":169257,"style":8316},[642],[363,169259],{},[363,169261],{"className":169262,"style":2935},[505],[363,169264,2818],{"className":169265},[2939],[363,169267],{"className":169268,"style":2935},[505],[363,169270,169272,169275,169337,169340,169343],{"className":169271},[398],[363,169273],{"className":169274,"style":26910},[402],[363,169276,169278,169281,169334],{"className":169277},[407],[363,169279],{"className":169280},[494,789],[363,169282,169284],{"className":169283},[586],[363,169285,169287,169326],{"className":169286},[633,634],[363,169288,169290,169323],{"className":169289},[638],[363,169291,169293,169304,169312],{"className":169292,"style":3902},[642],[363,169294,169295,169298],{"style":805},[363,169296],{"className":169297,"style":809},[650],[363,169299,169301],{"className":169300},[407],[363,169302,11044],{"className":169303,"style":11132},[407,408],[363,169305,169306,169309],{"style":821},[363,169307],{"className":169308,"style":809},[650],[363,169310],{"className":169311,"style":829},[828],[363,169313,169314,169317],{"style":832},[363,169315],{"className":169316,"style":809},[650],[363,169318,169320],{"className":169319},[407],[363,169321,15750],{"className":169322},[407,408],[363,169324,666],{"className":169325},[665],[363,169327,169329],{"className":169328},[638],[363,169330,169332],{"className":169331,"style":4126},[642],[363,169333],{},[363,169335],{"className":169336},[501,789],[363,169338],{"className":169339,"style":506},[505],[363,169341,471],{"className":169342},[510],[363,169344],{"className":169345,"style":506},[505],[363,169347,169349,169352,169500,169503,169506],{"className":169348},[398],[363,169350],{"className":169351,"style":151489},[402],[363,169353,169355,169358,169497],{"className":169354},[407],[363,169356],{"className":169357},[494,789],[363,169359,169361],{"className":169360},[586],[363,169362,169364,169489],{"className":169363},[633,634],[363,169365,169367,169486],{"className":169366},[638],[363,169368,169370,169418,169426],{"className":169369,"style":23261},[642],[363,169371,169372,169375],{"style":805},[363,169373],{"className":169374,"style":809},[650],[363,169376,169378],{"className":169377},[407],[363,169379,169381,169384],{"className":169380},[407],[363,169382,45126],{"className":169383,"style":45156},[407,408],[363,169385,169387],{"className":169386},[629],[363,169388,169390,169410],{"className":169389},[633,634],[363,169391,169393,169407],{"className":169392},[638],[363,169394,169396],{"className":169395,"style":11483},[642],[363,169397,169398,169401],{"style":136154},[363,169399],{"className":169400,"style":651},[650],[363,169402,169404],{"className":169403},[655,656,657,658],[363,169405,474],{"className":169406},[407,658],[363,169408,666],{"className":169409},[665],[363,169411,169413],{"className":169412},[638],[363,169414,169416],{"className":169415,"style":8316},[642],[363,169417],{},[363,169419,169420,169423],{"style":821},[363,169421],{"className":169422,"style":809},[650],[363,169424],{"className":169425,"style":829},[828],[363,169427,169428,169431],{"style":832},[363,169429],{"className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2.1 给出。",[11,169833,169834,169835,169838],{},"命题 3.2 是整篇文章的枢纽：它说明",[1344,169836,169837],{},"在单纯形上输出一个概率分布，与在 opinion 空间中输出一个带不确定性度量的判断，是同一件事。"," 于是，「给网络加一个不确定性维度」这一看似需要新架构的任务，被归结为「让网络的输出参数化一个迪利克雷分布」。证据深度学习正是沿着这条路径展开的。",[2659,169840,169842],{"id":169841},"_33-softmax-的翻译不变性与过度自信","3.3 Softmax 的翻译不变性与过度自信",[11,169844,169845,169846,170000,170001,170117,170118,19080,170190,170494],{},"设神经网络 ",[363,169847,169849,169879],{"className":169848,"translate":367},[366],[363,169850,169852],{"className":169851},[371],[373,169853,169854],{"xmlns":375},[377,169855,169856,169876],{},[380,169857,169858,169864,169866,169868,169870],{},[543,169859,169860,169862],{},[383,169861,385],{},[383,169863,101148],{"mathvariant":136055},[458,169865,1364],{},[383,169867,945],{"mathvariant":944},[458,169869,1378],{},[7162,169871,169872,169874],{},[383,169873,1381],{"mathvariant":570},[383,169875,11044],{},[387,169877,169878],{"encoding":389},"f_{\\boldsymbol{\\theta}}: \\mathcal{X} \\to \\mathbb{R}^K",[363,169880,169882,169947,169965],{"className":169881,"ariaHidden":394},[393],[363,169883,169885,169888,169938,169941,169944],{"className":169884},[398],[363,169886],{"className":169887,"style":403},[402],[363,169889,169891,169894],{"className":169890},[407],[363,169892,385],{"className":169893,"style":409},[407,408],[363,169895,169897],{"className":169896},[629],[363,169898,169900,169930],{"className":169899},[633,634],[363,169901,169903,169927],{"className":169902},[638],[363,169904,169906],{"className":169905,"style":643},[642],[363,169907,169908,169911],{"style":27881},[363,169909],{"className":169910,"style":651},[650],[363,169912,169914],{"className":169913},[655,656,657,658],[363,169915,169917],{"className":169916},[407,658],[363,169918,169920],{"className":169919},[407,658],[363,169921,169923],{"className":169922},[407,658],[363,169924,101148],{"className":169925,"style":169926},[407,136115,658],"margin-right:0.0319em;",[363,169928,666],{"className":169929},[665],[363,169931,169933],{"className":169932},[638],[363,169934,169936],{"className":169935,"style":8316},[642],[363,169937],{},[363,169939],{"className":169940,"style":506},[505],[363,169942,1364],{"className":169943},[510],[363,169945],{"className":169946,"style":506},[505],[363,169948,169950,169953,169956,169959,169962],{"className":169949},[398],[363,169951],{"className":169952,"style":1011},[402],[363,169954,945],{"className":169955,"style":972},[407,971],[363,169957],{"className":169958,"style":506},[505],[363,169960,1378],{"className":169961},[510],[363,169963],{"className":169964,"style":506},[505],[363,169966,169968,169971],{"className":169967},[398],[363,169969],{"className":169970,"style":52457},[402],[363,169972,169974,169977],{"className":169973},[407],[363,169975,1381],{"className":169976},[407,714],[363,169978,169980],{"className":169979},[629],[363,169981,169983],{"className":169982},[633],[363,169984,169986],{"className":169985},[638],[363,169987,169989],{"className":169988,"style":52457},[642],[363,169990,169991,169994],{"style":7213},[363,169992],{"className":169993,"style":651},[650],[363,169995,169997],{"className":169996},[655,656,657,658],[363,169998,11044],{"className":169999,"style":11132},[407,408,658]," 输出 logits ",[363,170002,170004,170032],{"className":170003,"translate":367},[366],[363,170005,170007],{"className":170006},[371],[373,170008,170009],{"xmlns":375},[377,170010,170011,170029],{},[380,170012,170013,170015,170017,170023,170025,170027],{},[383,170014,118160],{"mathvariant":75598},[458,170016,471],{},[543,170018,170019,170021],{},[383,170020,385],{},[383,170022,101148],{"mathvariant":136055},[458,170024,461],{"stretchy":460},[383,170026,1121],{"mathvariant":75598},[458,170028,468],{"stretchy":460},[387,170030,170031],{"encoding":389},"\\mathbf{z} = f_{\\boldsymbol{\\theta}}(\\mathbf{x})",[363,170033,170035,170053],{"className":170034,"ariaHidden":394},[393],[363,170036,170038,170041,170044,170047,170050],{"className":170037},[398],[363,170039],{"className":170040,"style":135662},[402],[363,170042,118160],{"className":170043},[407,75943],[363,170045],{"className":170046,"style":506},[505],[363,170048,471],{"className":170049},[510],[363,170051],{"className":170052,"style":506},[505],[363,170054,170056,170059,170108,170111,170114],{"className":170055},[398],[363,170057],{"className":170058,"style":487},[402],[363,170060,170062,170065],{"className":170061},[407],[363,170063,385],{"className":170064,"style":409},[407,408],[363,170066,170068],{"className":170067},[629],[363,170069,170071,170100],{"className":170070},[633,634],[363,170072,170074,170097],{"className":170073},[638],[363,170075,170077],{"className":170076,"style":643},[642],[363,170078,170079,170082],{"style":27881},[363,170080],{"className":170081,"style":651},[650],[363,170083,170085],{"className":170084},[655,656,657,658],[363,170086,170088],{"className":170087},[407,658],[363,170089,170091],{"className":170090},[407,658],[363,170092,170094],{"className":170093},[407,658],[363,170095,101148],{"className":170096,"style":169926},[407,136115,658],[363,170098,666],{"className":170099},[665],[363,170101,170103],{"className":170102},[638],[363,170104,170106],{"className":170105,"style":8316},[642],[363,170107],{},[363,170109,461],{"className":170110},[494],[363,170112,1121],{"className":170113},[407,75943],[363,170115,468],{"className":170116},[501],"，softmax 给出 ",[363,170119,170121,170148],{"className":170120,"translate":367},[366],[363,170122,170124],{"className":170123},[371],[373,170125,170126],{"xmlns":375},[377,170127,170128,170145],{},[380,170129,170130,170132,170134,170137,170139,170141,170143],{},[383,170131,11],{"mathvariant":75598},[458,170133,471],{},[383,170135,170136],{"mathvariant":557},"softmax",[458,170138,4451],{},[458,170140,461],{"stretchy":460},[383,170142,118160],{"mathvariant":75598},[458,170144,468],{"stretchy":460},[387,170146,170147],{"encoding":389},"\\mathbf{p} = \\operatorname{softmax}(\\mathbf{z})",[363,170149,170151,170169],{"className":170150,"ariaHidden":394},[393],[363,170152,170154,170157,170160,170163,170166],{"className":170153},[398],[363,170155],{"className":170156,"style":133807},[402],[363,170158,11],{"className":170159},[407,75943],[363,170161],{"className":170162,"style":506},[505],[363,170164,471],{"className":170165},[510],[363,170167],{"className":170168,"style":506},[505],[363,170170,170172,170175,170181,170184,170187],{"className":170171},[398],[363,170173],{"className":170174,"style":487},[402],[363,170176,170178],{"className":170177},[1832],[363,170179,170136],{"className":170180},[407,33034],[363,170182,461],{"className":170183},[494],[363,170185,118160],{"className":170186},[407,75943],[363,170188,468],{"className":170189},[501],[363,170191,170193,170241],{"className":170192,"translate":367},[366],[363,170194,170196],{"className":170195},[371],[373,170197,170198],{"xmlns":375},[377,170199,170200,170238],{},[380,170201,170202,170208,170210,170220,170222,170228],{},[543,170203,170204,170206],{},[383,170205,11],{},[383,170207,7912],{},[458,170209,471],{},[7162,170211,170212,170214],{},[383,170213,7915],{},[543,170215,170216,170218],{},[383,170217,118160],{},[383,170219,7912],{},[383,170221,1288],{"mathvariant":557},[543,170223,170224,170226],{},[458,170225,13811],{},[383,170227,100402],{},[7162,170229,170230,170232],{},[383,170231,7915],{},[543,170233,170234,170236],{},[383,170235,118160],{},[383,170237,100402],{},[387,170239,170240],{"encoding":389},"p_k = e^{z_k}\u002F\\sum_j e^{z_j}",[363,170242,170244,170299],{"className":170243,"ariaHidden":394},[393],[363,170245,170247,170250,170290,170293,170296],{"className":170246},[398],[363,170248],{"className":170249,"style":2624},[402],[363,170251,170253,170256],{"className":170252},[407],[363,170254,11],{"className":170255},[407,408],[363,170257,170259],{"className":170258},[629],[363,170260,170262,170282],{"className":170261},[633,634],[363,170263,170265,170279],{"className":170264},[638],[363,170266,170268],{"className":170267,"style":643},[642],[363,170269,170270,170273],{"style":730},[363,170271],{"className":170272,"style":651},[650],[363,170274,170276],{"className":170275},[655,656,657,658],[363,170277,7912],{"className":170278,"style":8615},[407,408,658],[363,170280,666],{"className":170281},[665],[363,170283,170285],{"className":170284},[638],[363,170286,170288],{"className":170287,"style":8316},[642],[363,170289],{},[363,170291],{"className":170292,"style":506},[505],[363,170294,471],{"className":170295},[510],[363,170297],{"className":170298,"style":506},[505],[363,170300,170302,170305,170375,170378,170381,170421,170424],{"className":170301},[398],[363,170303],{"className":170304,"style":162606},[402],[363,170306,170308,170311],{"className":170307},[407],[363,170309,7915],{"className":170310},[407,408],[363,170312,170314],{"className":170313},[629],[363,170315,170317],{"className":170316},[633],[363,170318,170320],{"className":170319},[638],[363,170321,170323],{"className":170322,"style":50725},[642],[363,170324,170325,170328],{"style":7213},[363,170326],{"className":170327,"style":651},[650],[363,170329,170331],{"className":170330},[655,656,657,658],[363,170332,170334],{"className":170333},[407,658],[363,170335,170337,170340],{"className":170336},[407,658],[363,170338,118160],{"className":170339,"style":118430},[407,408,658],[363,170341,170343],{"className":170342},[629],[363,170344,170346,170367],{"className":170345},[633,634],[363,170347,170349,170364],{"className":170348},[638],[363,170350,170352],{"className":170351,"style":6446},[642],[363,170353,170355,170358],{"style":170354},"top:-2.3488em;margin-left:-0.044em;margin-right:0.0714em;",[363,170356],{"className":170357,"style":30578},[650],[363,170359,170361],{"className":170360},[655,30582,10022,658],[363,170362,7912],{"className":170363,"style":8615},[407,408,658],[363,170365,666],{"className":170366},[665],[363,170368,170370],{"className":170369},[638],[363,170371,170373],{"className":170372,"style":101625},[642],[363,170374],{},[363,170376,1288],{"className":170377},[407],[363,170379],{"className":170380,"style":754},[505],[363,170382,170384,170387],{"className":170383},[1832],[363,170385,13811],{"className":170386,"style":40032},[1832,1833,5964],[363,170388,170390],{"className":170389},[629],[363,170391,170393,170413],{"className":170392},[633,634],[363,170394,170396,170410],{"className":170395},[638],[363,170397,170399],{"className":170398,"style":40045},[642],[363,170400,170401,170404],{"style":40048},[363,170402],{"className":170403,"style":651},[650],[363,170405,170407],{"className":170406},[655,656,657,658],[363,170408,100402],{"className":170409,"style":100930},[407,408,658],[363,170411,666],{"className":170412},[665],[363,170414,170416],{"className":170415},[638],[363,170417,170419],{"className":170418,"style":162795},[642],[363,170420],{},[363,170422],{"className":170423,"style":754},[505],[363,170425,170427,170430],{"className":170426},[407],[363,170428,7915],{"className":170429},[407,408],[363,170431,170433],{"className":170432},[629],[363,170434,170436],{"className":170435},[633],[363,170437,170439],{"className":170438},[638],[363,170440,170442],{"className":170441,"style":50725},[642],[363,170443,170444,170447],{"style":7213},[363,170445],{"className":170446,"style":651},[650],[363,170448,170450],{"className":170449},[655,656,657,658],[363,170451,170453],{"className":170452},[407,658],[363,170454,170456,170459],{"className":170455},[407,658],[363,170457,118160],{"className":170458,"style":118430},[407,408,658],[363,170460,170462],{"className":170461},[629],[363,170463,170465,170486],{"className":170464},[633,634],[363,170466,170468,170483],{"className":170467},[638],[363,170469,170471],{"className":170470,"style":40120},[642],[363,170472,170474,170477],{"style":170473},"top:-2.357em;margin-left:-0.044em;margin-right:0.0714em;",[363,170475],{"className":170476,"style":30578},[650],[363,170478,170480],{"className":170479},[655,30582,10022,658],[363,170481,100402],{"className":170482,"style":100930},[407,408,658],[363,170484,666],{"className":170485},[665],[363,170487,170489],{"className":170488},[638],[363,170490,170492],{"className":170491,"style":122109},[642],[363,170493],{},"。这一映射存在两个结构性缺陷。",[11,170496,170497,170500,170501,3093,170529,170647,170648,34197,170729,170732,170733,170784,170785,170858,170859,170897],{},[1344,170498,170499],{},"其一，平移不变性抹去了「无信息」这一状态。"," 对任意常数 ",[363,170502,170504,170517],{"className":170503,"translate":367},[366],[363,170505,170507],{"className":170506},[371],[373,170508,170509],{"xmlns":375},[377,170510,170511,170515],{},[380,170512,170513],{},[383,170514,6254],{},[387,170516,6254],{"encoding":389},[363,170518,170520],{"className":170519,"ariaHidden":394},[393],[363,170521,170523,170526],{"className":170522},[398],[363,170524],{"className":170525,"style":2693},[402],[363,170527,6254],{"className":170528},[407,408],[363,170530,170532,170572],{"className":170531,"translate":367},[366],[363,170533,170535],{"className":170534},[371],[373,170536,170537],{"xmlns":375},[377,170538,170539,170569],{},[380,170540,170541,170543,170545,170547,170549,170551,170553,170555,170557,170559,170561,170563,170565,170567],{},[383,170542,170136],{"mathvariant":557},[458,170544,4451],{},[458,170546,461],{"stretchy":460},[383,170548,118160],{"mathvariant":75598},[458,170550,2818],{},[383,170552,6254],{},[463,170554,465],{"mathvariant":75598},[458,170556,468],{"stretchy":460},[458,170558,471],{},[383,170560,170136],{"mathvariant":557},[458,170562,4451],{},[458,170564,461],{"stretchy":460},[383,170566,118160],{"mathvariant":75598},[458,170568,468],{"stretchy":460},[387,170570,170571],{"encoding":389},"\\operatorname{softmax}(\\mathbf{z} + c\\mathbf{1}) = \\operatorname{softmax}(\\mathbf{z})",[363,170573,170575,170602,170626],{"className":170574,"ariaHidden":394},[393],[363,170576,170578,170581,170587,170590,170593,170596,170599],{"className":170577},[398],[363,170579],{"className":170580,"style":487},[402],[363,170582,170584],{"className":170583},[1832],[363,170585,170136],{"className":170586},[407,33034],[363,170588,461],{"className":170589},[494],[363,170591,118160],{"className":170592},[407,75943],[363,170594],{"className":170595,"style":2935},[505],[363,170597,2818],{"className":170598},[2939],[363,170600],{"className":170601,"style":2935},[505],[363,170603,170605,170608,170611,170614,170617,170620,170623],{"className":170604},[398],[363,170606],{"className":170607,"style":487},[402],[363,170609,6254],{"className":170610},[407,408],[363,170612,465],{"className":170613},[407,75943],[363,170615,468],{"className":170616},[501],[363,170618],{"className":170619,"style":506},[505],[363,170621,471],{"className":170622},[510],[363,170624],{"className":170625,"style":506},[505],[363,170627,170629,170632,170638,170641,170644],{"className":170628},[398],[363,170630],{"className":170631,"style":487},[402],[363,170633,170635],{"className":170634},[1832],[363,170636,170136],{"className":170637},[407,33034],[363,170639,461],{"className":170640},[494],[363,170642,118160],{"className":170643},[407,75943],[363,170645,468],{"className":170646},[501],"。因此 logits 的绝对水平不含任何信息，网络的输出只依赖于 ",[363,170649,170651,170673],{"className":170650,"translate":367},[366],[363,170652,170654],{"className":170653},[371],[373,170655,170656],{"xmlns":375},[377,170657,170658,170670],{},[380,170659,170660,170662,170668],{},[458,170661,3246],{"stretchy":460},[543,170663,170664,170666],{},[383,170665,118160],{},[383,170667,7912],{},[458,170669,3256],{"stretchy":460},[387,170671,170672],{"encoding":389},"\\{z_k\\}",[363,170674,170676],{"className":170675,"ariaHidden":394},[393],[363,170677,170679,170682,170685,170726],{"className":170678},[398],[363,170680],{"className":170681,"style":487},[402],[363,170683,3246],{"className":170684},[494],[363,170686,170688,170691],{"className":170687},[407],[363,170689,118160],{"className":170690,"style":118430},[407,408],[363,170692,170694],{"className":170693},[629],[363,170695,170697,170718],{"className":170696},[633,634],[363,170698,170700,170715],{"className":170699},[638],[363,170701,170703],{"className":170702,"style":643},[642],[363,170704,170706,170709],{"style":170705},"top:-2.55em;margin-left:-0.044em;margin-right:0.05em;",[363,170707],{"className":170708,"style":651},[650],[363,170710,170712],{"className":170711},[655,656,657,658],[363,170713,7912],{"className":170714,"style":8615},[407,408,658],[363,170716,666],{"className":170717},[665],[363,170719,170721],{"className":170720},[638],[363,170722,170724],{"className":170723,"style":8316},[642],[363,170725],{},[363,170727,3256],{"className":170728},[501],[1344,170730,170731],{},"相对差","。后果是：",[363,170734,170736,170754],{"className":170735,"translate":367},[366],[363,170737,170739],{"className":170738},[371],[373,170740,170741],{"xmlns":375},[377,170742,170743,170751],{},[380,170744,170745,170747,170749],{},[383,170746,118160],{"mathvariant":75598},[458,170748,471],{},[463,170750,474],{"mathvariant":75598},[387,170752,170753],{"encoding":389},"\\mathbf{z} = \\mathbf{0}",[363,170755,170757,170775],{"className":170756,"ariaHidden":394},[393],[363,170758,170760,170763,170766,170769,170772],{"className":170759},[398],[363,170761],{"className":170762,"style":135662},[402],[363,170764,118160],{"className":170765},[407,75943],[363,170767],{"className":170768,"style":506},[505],[363,170770,471],{"className":170771},[510],[363,170773],{"className":170774,"style":506},[505],[363,170776,170778,170781],{"className":170777},[398],[363,170779],{"className":170780,"style":520},[402],[363,170782,474],{"className":170783},[407,75943],"（「全零 logits」）与 ",[363,170786,170788,170810],{"className":170787,"translate":367},[366],[363,170789,170791],{"className":170790},[371],[373,170792,170793],{"xmlns":375},[377,170794,170795,170807],{},[380,170796,170797,170799,170801,170803,170805],{},[383,170798,118160],{"mathvariant":75598},[458,170800,471],{},[463,170802,117325],{},[458,170804,3647],{},[463,170806,465],{"mathvariant":75598},[387,170808,170809],{"encoding":389},"\\mathbf{z} = 100 \\cdot \\mathbf{1}",[363,170811,170813,170831,170849],{"className":170812,"ariaHidden":394},[393],[363,170814,170816,170819,170822,170825,170828],{"className":170815},[398],[363,170817],{"className":170818,"style":135662},[402],[363,170820,118160],{"className":170821},[407,75943],[363,170823],{"className":170824,"style":506},[505],[363,170826,471],{"className":170827},[510],[363,170829],{"className":170830,"style":506},[505],[363,170832,170834,170837,170840,170843,170846],{"className":170833},[398],[363,170835],{"className":170836,"style":520},[402],[363,170838,117325],{"className":170839},[407],[363,170841],{"className":170842,"style":2935},[505],[363,170844,3647],{"className":170845},[2939],[363,170847],{"className":170848,"style":2935},[505],[363,170850,170852,170855],{"className":170851},[398],[363,170853],{"className":170854,"style":520},[402],[363,170856,465],{"className":170857},[407,75943]," 不可区分，二者都对应均匀分布 ",[363,170860,170862,170879],{"className":170861,"translate":367},[366],[363,170863,170865],{"className":170864},[371],[373,170866,170867],{"xmlns":375},[377,170868,170869,170877],{},[380,170870,170871,170873,170875],{},[463,170872,465],{"mathvariant":75598},[383,170874,1288],{"mathvariant":557},[383,170876,11044],{},[387,170878,165332],{"encoding":389},[363,170880,170882],{"className":170881,"ariaHidden":394},[393],[363,170883,170885,170888,170891,170894],{"className":170884},[398],[363,170886],{"className":170887,"style":487},[402],[363,170889,465],{"className":170890},[407,75943],[363,170892,1288],{"className":170893},[407],[363,170895,11044],{"className":170896,"style":11132},[407,408],"。既然均匀分布既可由「无任何信号」也可由「极其强烈的信号恰好相互抵消」产生，输出端就不可能把前者作为一个可辨识的状态标记出来。",[11,170899,170900,170903,170904,171035,171036,171039,171040,10528],{},[1344,170901,170902],{},"其二，交叉熵的最优解在可分数据上把 logits 推向发散。"," 在训练集线性可分时，以交叉熵为目标的极大似然估计不存在有限最优解：沿方向 ",[363,170905,170907,170943],{"className":170906,"translate":367},[366],[363,170908,170910],{"className":170909},[371],[373,170911,170912],{"xmlns":375},[377,170913,170914,170940],{},[380,170915,170916,170918,170920,170922,170924,170930,170932,170934,170936,170938],{},[383,170917,118160],{"mathvariant":75598},[458,170919,1378],{},[383,170921,6254],{},[1645,170923,1647],{},[7162,170925,170926,170928],{},[383,170927,118160],{"mathvariant":75598},[458,170929,76343],{},[458,170931,898],{"separator":394},[1645,170933,4631],{},[383,170935,6254],{},[458,170937,1378],{},[383,170939,1373],{"mathvariant":557},[387,170941,170942],{"encoding":389},"\\mathbf{z} \\to c\\,\\mathbf{z}^\\star,\\ c \\to \\infty",[363,170944,170946,170964,171026],{"className":170945,"ariaHidden":394},[393],[363,170947,170949,170952,170955,170958,170961],{"className":170948},[398],[363,170950],{"className":170951,"style":135662},[402],[363,170953,118160],{"className":170954},[407,75943],[363,170956],{"className":170957,"style":506},[505],[363,170959,1378],{"className":170960},[510],[363,170962],{"className":170963,"style":506},[505],[363,170965,170967,170970,170973,170976,171005,171008,171011,171014,171017,171020,171023],{"className":170966},[398],[363,170968],{"className":170969,"style":64398},[402],[363,170971,6254],{"className":170972},[407,408],[363,170974],{"className":170975,"style":754},[505],[363,170977,170979,170982],{"className":170978},[407],[363,170980,118160],{"className":170981},[407,75943],[363,170983,170985],{"className":170984},[629],[363,170986,170988],{"className":170987},[633],[363,170989,170991],{"className":170990},[638],[363,170992,170994],{"className":170993,"style":8847},[642],[363,170995,170996,170999],{"style":7213},[363,170997],{"className":170998,"style":651},[650],[363,171000,171002],{"className":171001},[655,656,657,658],[363,171003,76343],{"className":171004},[2939,658],[363,171006,898],{"className":171007},[920],[363,171009,4631],{"className":171010},[505],[363,171012],{"className":171013,"style":754},[505],[363,171015,6254],{"className":171016},[407,408],[363,171018],{"className":171019,"style":506},[505],[363,171021,1378],{"className":171022},[510],[363,171024],{"className":171025,"style":506},[505],[363,171027,171029,171032],{"className":171028},[398],[363,171030],{"className":171031,"style":2693},[402],[363,171033,1373],{"className":171034},[407]," 可将损失单调压向零。换言之，",[1344,171037,171038],{},"过度自信不是训练不足的症状，而是 MLE 在可分数据上的退化解","——模型越是被「训练充分」，其 softmax 输出越接近 one-hot，而这一行为",[1344,171041,171042],{},"完全不受输入是否为分布内样本的约束",[11,171044,171045],{},"两个缺陷合起来解释了 5.1 节的经验事实：以最大 softmax 概率（MSP）作为分布外检测分数存在系统性偏差，因为 softmax 对任何使 logits 相对间距增大的输入都会返回高置信度的断言，而无从表达「这个输入对我而言是陌生的」。",[2659,171047,171049],{"id":171048},"_34-证据参数化","3.4 证据参数化",[11,171051,171052,171053],{},"证据深度学习对上述缺陷的回应是：",[1344,171054,171055],{},"不再用 softmax 把 logits 压缩为概率，而是用非负激活函数把 logits 提升为证据。",[143,171057,171058],{},[11,171059,171060,11410,171063,171252,171253,171256,171257,171516,171517,171753,171754,10528],{},[1344,171061,171062],{},"定义 3.3（证据网络）",[363,171064,171066,171102],{"className":171065,"translate":367},[366],[363,171067,171069],{"className":171068},[371],[373,171070,171071],{"xmlns":375},[377,171072,171073,171099],{},[380,171074,171075,171081,171083,171085,171087],{},[543,171076,171077,171079],{},[383,171078,4506],{},[383,171080,101148],{"mathvariant":136055},[458,171082,1364],{},[383,171084,945],{"mathvariant":944},[458,171086,1378],{},[7901,171088,171089,171091,171097],{},[383,171090,1381],{"mathvariant":570},[380,171092,171093,171095],{},[458,171094,7276],{},[463,171096,474],{},[383,171098,11044],{},[387,171100,171101],{"encoding":389},"g_{\\boldsymbol{\\theta}}: \\mathcal{X} \\to \\mathbb{R}_{\\ge 0}^K",[363,171103,171105,171169,171187],{"className":171104,"ariaHidden":394},[393],[363,171106,171108,171111,171160,171163,171166],{"className":171107},[398],[363,171109],{"className":171110,"style":2624},[402],[363,171112,171114,171117],{"className":171113},[407],[363,171115,4506],{"className":171116,"style":1328},[407,408],[363,171118,171120],{"className":171119},[629],[363,171121,171123,171152],{"className":171122},[633,634],[363,171124,171126,171149],{"className":171125},[638],[363,171127,171129],{"className":171128,"style":643},[642],[363,171130,171131,171134],{"style":19215},[363,171132],{"className":171133,"style":651},[650],[363,171135,171137],{"className":171136},[655,656,657,658],[363,171138,171140],{"className":171139},[407,658],[363,171141,171143],{"className":171142},[407,658],[363,171144,171146],{"className":171145},[407,658],[363,171147,101148],{"className":171148,"style":169926},[407,136115,658],[363,171150,666],{"className":171151},[665],[363,171153,171155],{"className":171154},[638],[363,171156,171158],{"className":171157,"style":8316},[642],[363,171159],{},[363,171161],{"className":171162,"style":506},[505],[363,171164,1364],{"className":171165},[510],[363,171167],{"className":171168,"style":506},[505],[363,171170,171172,171175,171178,171181,171184],{"className":171171},[398],[363,171173],{"className":171174,"style":1011},[402],[363,171176,945],{"className":171177,"style":972},[407,971],[363,171179],{"className":171180,"style":506},[505],[363,171182,1378],{"className":171183},[510],[363,171185],{"className":171186,"style":506},[505],[363,171188,171190,171194],{"className":171189},[398],[363,171191],{"className":171192,"style":171193},[402],"height:1.1846em;vertical-align:-0.3433em;",[363,171195,171197,171200],{"className":171196},[407],[363,171198,1381],{"className":171199},[407,714],[363,171201,171203],{"className":171202},[629],[363,171204,171206,171243],{"className":171205},[633,634],[363,171207,171209,171240],{"className":171208},[638],[363,171210,171212,171229],{"className":171211,"style":52457},[642],[363,171213,171214,171217],{"style":136272},[363,171215],{"className":171216,"style":651},[650],[363,171218,171220],{"className":171219},[655,656,657,658],[363,171221,171223,171226],{"className":171222},[407,658],[363,171224,7276],{"className":171225},[510,658],[363,171227,474],{"className":171228},[407,658],[363,171230,171231,171234],{"style":7213},[363,171232],{"className":171233,"style":651},[650],[363,171235,171237],{"className":171236},[655,656,657,658],[363,171238,11044],{"className":171239,"style":11132},[407,408,658],[363,171241,666],{"className":171242},[665],[363,171244,171246],{"className":171245},[638],[363,171247,171250],{"className":171248,"style":171249},[642],"height:0.3433em;",[363,171251],{}," 是由主干网络与",[1344,171254,171255],{},"非负激活函数","复合而成的证据函数，输出 ",[363,171258,171260,171312],{"className":171259,"translate":367},[366],[363,171261,171263],{"className":171262},[371],[373,171264,171265],{"xmlns":375},[377,171266,171267,171309],{},[380,171268,171269,171271,171273,171279,171281,171283,171285,171287,171289,171295,171297,171299,171301,171307],{},[383,171270,7915],{"mathvariant":75598},[458,171272,471],{},[543,171274,171275,171277],{},[383,171276,4506],{},[383,171278,101148],{"mathvariant":136055},[458,171280,461],{"stretchy":460},[383,171282,1121],{"mathvariant":75598},[458,171284,468],{"stretchy":460},[458,171286,471],{},[458,171288,461],{"stretchy":460},[543,171290,171291,171293],{},[383,171292,7915],{},[463,171294,465],{},[458,171296,898],{"separator":394},[458,171298,133150],{},[458,171300,898],{"separator":394},[543,171302,171303,171305],{},[383,171304,7915],{},[383,171306,11044],{},[458,171308,468],{"stretchy":460},[387,171310,171311],{"encoding":389},"\\mathbf{e} = g_{\\boldsymbol{\\theta}}(\\mathbf{x}) = (e_1, \\dots, e_K)",[363,171313,171315,171333,171406],{"className":171314,"ariaHidden":394},[393],[363,171316,171318,171321,171324,171327,171330],{"className":171317},[398],[363,171319],{"className":171320,"style":135662},[402],[363,171322,7915],{"className":171323},[407,75943],[363,171325],{"className":171326,"style":506},[505],[363,171328,471],{"className":171329},[510],[363,171331],{"className":171332,"style":506},[505],[363,171334,171336,171339,171388,171391,171394,171397,171400,171403],{"className":171335},[398],[363,171337],{"className":171338,"style":487},[402],[363,171340,171342,171345],{"className":171341},[407],[363,171343,4506],{"className":171344,"style":1328},[407,408],[363,171346,171348],{"className":171347},[629],[363,171349,171351,171380],{"className":171350},[633,634],[363,171352,171354,171377],{"className":171353},[638],[363,171355,171357],{"className":171356,"style":643},[642],[363,171358,171359,171362],{"style":19215},[363,171360],{"className":171361,"style":651},[650],[363,171363,171365],{"className":171364},[655,656,657,658],[363,171366,171368],{"className":171367},[407,658],[363,171369,171371],{"className":171370},[407,658],[363,171372,171374],{"className":171373},[407,658],[363,171375,101148],{"className":171376,"style":169926},[407,136115,658],[363,171378,666],{"className":171379},[665],[363,171381,171383],{"className":171382},[638],[363,171384,171386],{"className":171385,"style":8316},[642],[363,171387],{},[363,171389,461],{"className":171390},[494],[363,171392,1121],{"className":171393},[407,75943],[363,171395,468],{"className":171396},[501],[363,171398],{"className":171399,"style":506},[505],[363,171401,471],{"className":171402},[510],[363,171404],{"className":171405,"style":506},[505],[363,171407,171409,171412,171415,171455,171458,171461,171464,171467,171470,171473,171513],{"className":171408},[398],[363,171410],{"className":171411,"style":487},[402],[363,171413,461],{"className":171414},[494],[363,171416,171418,171421],{"className":171417},[407],[363,171419,7915],{"className":171420},[407,408],[363,171422,171424],{"className":171423},[629],[363,171425,171427,171447],{"className":171426},[633,634],[363,171428,171430,171444],{"className":171429},[638],[363,171431,171433],{"className":171432,"style":11483},[642],[363,171434,171435,171438],{"style":730},[363,171436],{"className":171437,"style":651},[650],[363,171439,171441],{"className":171440},[655,656,657,658],[363,171442,465],{"className":171443},[407,658],[363,171445,666],{"className":171446},[665],[363,171448,171450],{"className":171449},[638],[363,171451,171453],{"className":171452,"style":8316},[642],[363,171454],{},[363,171456,898],{"className":171457},[920],[363,171459],{"className":171460,"style":754},[505],[363,171462,133150],{"className":171463},[758],[363,171465],{"className":171466,"style":754},[505],[363,171468,898],{"className":171469},[920],[363,171471],{"className":171472,"style":754},[505],[363,171474,171476,171479],{"className":171475},[407],[363,171477,7915],{"className":171478},[407,408],[363,171480,171482],{"className":171481},[629],[363,171483,171485,171505],{"className":171484},[633,634],[363,171486,171488,171502],{"className":171487},[638],[363,171489,171491],{"className":171490,"style":727},[642],[363,171492,171493,171496],{"style":730},[363,171494],{"className":171495,"style":651},[650],[363,171497,171499],{"className":171498},[655,656,657,658],[363,171500,11044],{"className":171501,"style":11132},[407,408,658],[363,171503,666],{"className":171504},[665],[363,171506,171508],{"className":171507},[638],[363,171509,171511],{"className":171510,"style":8316},[642],[363,171512],{},[363,171514,468],{"className":171515},[501],"。迪利克雷参数取为\n",[363,171518,171520,171568],{"className":171519,"translate":367},[366],[363,171521,171523],{"className":171522},[371],[373,171524,171525],{"xmlns":375},[377,171526,171527,171565],{},[380,171528,171529,171535,171537,171543,171545,171547,171549,171551,171553,171555,171557,171559,171561,171563],{},[543,171530,171531,171533],{},[383,171532,45126],{},[383,171534,7912],{},[458,171536,471],{},[543,171538,171539,171541],{},[383,171540,7915],{},[383,171542,7912],{},[458,171544,2818],{},[463,171546,465],{},[458,171548,898],{"separator":394},[505,171550],{"width":3610},[1645,171552,88022],{},[505,171554],{"width":3610},[383,171556,45126],{"mathvariant":136055},[458,171558,471],{},[383,171560,7915],{"mathvariant":75598},[458,171562,2818],{},[463,171564,465],{"mathvariant":75598},[387,171566,171567],{"encoding":389},"\\alpha_k = e_k + 1, \\qquad \\text{即} \\qquad \\boldsymbol{\\alpha} = \\mathbf{e} + \\mathbf{1}",[363,171569,171571,171626,171681,171726,171744],{"className":171570,"ariaHidden":394},[393],[363,171572,171574,171577,171617,171620,171623],{"className":171573},[398],[363,171575],{"className":171576,"style":13724},[402],[363,171578,171580,171583],{"className":171579},[407],[363,171581,45126],{"className":171582,"style":45156},[407,408],[363,171584,171586],{"className":171585},[629],[363,171587,171589,171609],{"className":171588},[633,634],[363,171590,171592,171606],{"className":171591},[638],[363,171593,171595],{"className":171594,"style":643},[642],[363,171596,171597,171600],{"style":136154},[363,171598],{"className":171599,"style":651},[650],[363,171601,171603],{"className":171602},[655,656,657,658],[363,171604,7912],{"className":171605,"style":8615},[407,408,658],[363,171607,666],{"className":171608},[665],[363,171610,171612],{"className":171611},[638],[363,171613,171615],{"className":171614,"style":8316},[642],[363,171616],{},[363,171618],{"className":171619,"style":506},[505],[363,171621,471],{"className":171622},[510],[363,171624],{"className":171625,"style":506},[505],[363,171627,171629,171632,171672,171675,171678],{"className":171628},[398],[363,171630],{"className":171631,"style":145471},[402],[363,171633,171635,171638],{"className":171634},[407],[363,171636,7915],{"className":171637},[407,408],[363,171639,171641],{"className":171640},[629],[363,171642,171644,171664],{"className":171643},[633,634],[363,171645,171647,171661],{"className":171646},[638],[363,171648,171650],{"className":171649,"style":643},[642],[363,171651,171652,171655],{"style":730},[363,171653],{"className":171654,"style":651},[650],[363,171656,171658],{"className":171657},[655,656,657,658],[363,171659,7912],{"className":171660,"style":8615},[407,408,658],[363,171662,666],{"className":171663},[665],[363,171665,171667],{"className":171666},[638],[363,171668,171670],{"className":171669,"style":8316},[642],[363,171671],{},[363,171673],{"className":171674,"style":2935},[505],[363,171676,2818],{"className":171677},[2939],[363,171679],{"className":171680,"style":2935},[505],[363,171682,171684,171687,171690,171693,171696,171699,171705,171708,171717,171720,171723],{"className":171683},[398],[363,171685],{"className":171686,"style":913},[402],[363,171688,465],{"className":171689},[407],[363,171691,898],{"className":171692},[920],[363,171694],{"className":171695,"style":3982},[505],[363,171697],{"className":171698,"style":754},[505],[363,171700,171702],{"className":171701},[407,4946],[363,171703,88022],{"className":171704},[407,4950],[363,171706],{"className":171707,"style":3982},[505],[363,171709,171711],{"className":171710},[407],[363,171712,171714],{"className":171713},[407],[363,171715,45126],{"className":171716},[407,136115],[363,171718],{"className":171719,"style":506},[505],[363,171721,471],{"className":171722},[510],[363,171724],{"className":171725,"style":506},[505],[363,171727,171729,171732,171735,171738,171741],{"className":171728},[398],[363,171730],{"className":171731,"style":4599},[402],[363,171733,7915],{"className":171734},[407,75943],[363,171736],{"className":171737,"style":2935},[505],[363,171739,2818],{"className":171740},[2939],[363,171742],{"className":171743,"style":2935},[505],[363,171745,171747,171750],{"className":171746},[398],[363,171748],{"className":171749,"style":520},[402],[363,171751,465],{"className":171752},[407,75943],"\n网络对输入的预测由此被定义为 ",[363,171755,171757,171779],{"className":171756,"translate":367},[366],[363,171758,171760],{"className":171759},[371],[373,171761,171762],{"xmlns":375},[377,171763,171764,171776],{},[380,171765,171766,171768,171770,171772,171774],{},[383,171767,137112],{"mathvariant":557},[458,171769,4451],{},[458,171771,461],{"stretchy":460},[383,171773,45126],{"mathvariant":136055},[458,171775,468],{"stretchy":460},[387,171777,171778],{"encoding":389},"\\operatorname{Dir}(\\boldsymbol{\\alpha})",[363,171780,171782],{"className":171781,"ariaHidden":394},[393],[363,171783,171785,171788,171794,171797,171806],{"className":171784},[398],[363,171786],{"className":171787,"style":487},[402],[363,171789,171791],{"className":171790},[1832],[363,171792,137112],{"className":171793},[407,33034],[363,171795,461],{"className":171796},[494],[363,171798,171800],{"className":171799},[407],[363,171801,171803],{"className":171802},[407],[363,171804,45126],{"className":171805},[407,136115],[363,171807,468],{"className":171808},[501],[11,171810,171811,171812,171891,171892,171993,171994,172064,172065,172093],{},"参数化的语义可直接由命题 3.2 读出：",[363,171813,171815,171837],{"className":171814,"translate":367},[366],[363,171816,171818],{"className":171817},[371],[373,171819,171820],{"xmlns":375},[377,171821,171822,171834],{},[380,171823,171824,171826,171828,171830,171832],{},[383,171825,7915],{"mathvariant":75598},[458,171827,471],{},[383,171829,45126],{"mathvariant":136055},[458,171831,5161],{},[463,171833,465],{"mathvariant":75598},[387,171835,171836],{"encoding":389},"\\mathbf{e} = \\boldsymbol{\\alpha} - \\mathbf{1}",[363,171838,171840,171858,171882],{"className":171839,"ariaHidden":394},[393],[363,171841,171843,171846,171849,171852,171855],{"className":171842},[398],[363,171844],{"className":171845,"style":135662},[402],[363,171847,7915],{"className":171848},[407,75943],[363,171850],{"className":171851,"style":506},[505],[363,171853,471],{"className":171854},[510],[363,171856],{"className":171857,"style":506},[505],[363,171859,171861,171864,171873,171876,171879],{"className":171860},[398],[363,171862],{"className":171863,"style":4599},[402],[363,171865,171867],{"className":171866},[407],[363,171868,171870],{"className":171869},[407],[363,171871,45126],{"className":171872},[407,136115],[363,171874],{"className":171875,"style":2935},[505],[363,171877,5161],{"className":171878},[2939],[363,171880],{"className":171881,"style":2935},[505],[363,171883,171885,171888],{"className":171884},[398],[363,171886],{"className":171887,"style":520},[402],[363,171889,465],{"className":171890},[407,75943]," 正是",[1344,171893,171894,171895,171992],{},"超出均匀先验 ",[363,171896,171898,171922],{"className":171897,"translate":367},[366],[363,171899,171901],{"className":171900},[371],[373,171902,171903],{"xmlns":375},[377,171904,171905,171919],{},[380,171906,171907,171913,171915,171917],{},[543,171908,171909,171911],{},[383,171910,45552],{},[383,171912,7912],{},[383,171914,167702],{},[458,171916,471],{},[463,171918,465],{},[387,171920,171921],{"encoding":389},"a_k W = 1",[363,171923,171925,171983],{"className":171924,"ariaHidden":394},[393],[363,171926,171928,171931,171971,171974,171977,171980],{"className":171927},[398],[363,171929],{"className":171930,"style":16311},[402],[363,171932,171934,171937],{"className":171933},[407],[363,171935,45552],{"className":171936},[407,408],[363,171938,171940],{"className":171939},[629],[363,171941,171943,171963],{"className":171942},[633,634],[363,171944,171946,171960],{"className":171945},[638],[363,171947,171949],{"className":171948,"style":643},[642],[363,171950,171951,171954],{"style":730},[363,171952],{"className":171953,"style":651},[650],[363,171955,171957],{"className":171956},[655,656,657,658],[363,171958,7912],{"className":171959,"style":8615},[407,408,658],[363,171961,666],{"className":171962},[665],[363,171964,171966],{"className":171965},[638],[363,171967,171969],{"className":171968,"style":8316},[642],[363,171970],{},[363,171972,167702],{"className":171973,"style":682},[407,408],[363,171975],{"className":171976,"style":506},[505],[363,171978,471],{"className":171979},[510],[363,171981],{"className":171982,"style":506},[505],[363,171984,171986,171989],{"className":171985},[398],[363,171987],{"className":171988,"style":520},[402],[363,171990,465],{"className":171991},[407]," 的那部分伪计数","，即 2.6 节意义上「可解释为计数」的区域。这就是「证据」一词的来源：",[363,171995,171997,172015],{"className":171996,"translate":367},[366],[363,171998,172000],{"className":171999},[371],[373,172001,172002],{"xmlns":375},[377,172003,172004,172012],{},[380,172005,172006],{},[543,172007,172008,172010],{},[383,172009,7915],{},[383,172011,7912],{},[387,172013,172014],{"encoding":389},"e_k",[363,172016,172018],{"className":172017,"ariaHidden":394},[393],[363,172019,172021,172024],{"className":172020},[398],[363,172022],{"className":172023,"style":13724},[402],[363,172025,172027,172030],{"className":172026},[407],[363,172028,7915],{"className":172029},[407,408],[363,172031,172033],{"className":172032},[629],[363,172034,172036,172056],{"className":172035},[633,634],[363,172037,172039,172053],{"className":172038},[638],[363,172040,172042],{"className":172041,"style":643},[642],[363,172043,172044,172047],{"style":730},[363,172045],{"className":172046,"style":651},[650],[363,172048,172050],{"className":172049},[655,656,657,658],[363,172051,7912],{"className":172052,"style":8615},[407,408,658],[363,172054,666],{"className":172055},[665],[363,172057,172059],{"className":172058},[638],[363,172060,172062],{"className":172061,"style":8316},[642],[363,172063],{}," 是与类别 ",[363,172066,172068,172081],{"className":172067,"translate":367},[366],[363,172069,172071],{"className":172070},[371],[373,172072,172073],{"xmlns":375},[377,172074,172075,172079],{},[380,172076,172077],{},[383,172078,7912],{},[387,172080,7912],{"encoding":389},[363,172082,172084],{"className":172083,"ariaHidden":394},[393],[363,172085,172087,172090],{"className":172086},[398],[363,172088],{"className":172089,"style":12947},[402],[363,172091,7912],{"className":172092,"style":8615},[407,408]," 相关的、由网络从输入中提取出的支持性观测。",[11,172095,172096],{},"非负激活函数的选择影响显著，值得逐一说明：",[10375,172098,172099,172419,172538,172840],{},[7013,172100,172101,172244,172245,172301,172302,172361,172362,172418],{},[363,172102,172104,172147],{"className":172103,"translate":367},[366],[363,172105,172107],{"className":172106},[371],[373,172108,172109],{"xmlns":375},[377,172110,172111,172144],{},[380,172112,172113,172116,172118,172120,172122,172124,172126,172128,172130,172132,172134,172136,172142],{},[383,172114,172115],{"mathvariant":557},"softplus",[458,172117,4451],{},[458,172119,461],{"stretchy":460},[383,172121,118160],{},[458,172123,468],{"stretchy":460},[458,172125,471],{},[383,172127,4448],{},[458,172129,4451],{},[458,172131,461],{"stretchy":460},[463,172133,465],{},[458,172135,2818],{},[7162,172137,172138,172140],{},[383,172139,7915],{},[383,172141,118160],{},[458,172143,468],{"stretchy":460},[387,172145,172146],{"encoding":389},"\\operatorname{softplus}(z) = \\log(1 + e^z)",[363,172148,172150,172180,172206],{"className":172149,"ariaHidden":394},[393],[363,172151,172153,172156,172162,172165,172168,172171,172174,172177],{"className":172152},[398],[363,172154],{"className":172155,"style":487},[402],[363,172157,172159],{"className":172158},[1832],[363,172160,172115],{"className":172161},[407,33034],[363,172163,461],{"className":172164},[494],[363,172166,118160],{"className":172167,"style":118430},[407,408],[363,172169,468],{"className":172170},[501],[363,172172],{"className":172173,"style":506},[505],[363,172175,471],{"className":172176},[510],[363,172178],{"className":172179,"style":506},[505],[363,172181,172183,172186,172191,172194,172197,172200,172203],{"className":172182},[398],[363,172184],{"className":172185,"style":487},[402],[363,172187,4502,172189],{"className":172188},[1832],[363,172190,4506],{"style":4505},[363,172192,461],{"className":172193},[494],[363,172195,465],{"className":172196},[407],[363,172198],{"className":172199,"style":2935},[505],[363,172201,2818],{"className":172202},[2939],[363,172204],{"className":172205,"style":2935},[505],[363,172207,172209,172212,172241],{"className":172208},[398],[363,172210],{"className":172211,"style":487},[402],[363,172213,172215,172218],{"className":172214},[407],[363,172216,7915],{"className":172217},[407,408],[363,172219,172221],{"className":172220},[629],[363,172222,172224],{"className":172223},[633],[363,172225,172227],{"className":172226},[638],[363,172228,172230],{"className":172229,"style":50725},[642],[363,172231,172232,172235],{"style":7213},[363,172233],{"className":172234,"style":651},[650],[363,172236,172238],{"className":172237},[655,656,657,658],[363,172239,118160],{"className":172240,"style":118430},[407,408,658],[363,172242,468],{"className":172243},[501],"：恒正、处处可微、在 ",[363,172246,172248,172268],{"className":172247,"translate":367},[366],[363,172249,172251],{"className":172250},[371],[373,172252,172253],{"xmlns":375},[377,172254,172255,172265],{},[380,172256,172257,172259,172261,172263],{},[383,172258,118160],{},[458,172260,1378],{},[458,172262,5161],{},[383,172264,1373],{"mathvariant":557},[387,172266,172267],{"encoding":389},"z \\to -\\infty",[363,172269,172271,172289],{"className":172270,"ariaHidden":394},[393],[363,172272,172274,172277,172280,172283,172286],{"className":172273},[398],[363,172275],{"className":172276,"style":2693},[402],[363,172278,118160],{"className":172279,"style":118430},[407,408],[363,172281],{"className":172282,"style":506},[505],[363,172284,1378],{"className":172285},[510],[363,172287],{"className":172288,"style":506},[505],[363,172290,172292,172295,172298],{"className":172291},[398],[363,172293],{"className":172294,"style":4599},[402],[363,172296,5161],{"className":172297},[407],[363,172299,1373],{"className":172300},[407]," 时以 ",[363,172303,172305,172323],{"className":172304,"translate":367},[366],[363,172306,172308],{"className":172307},[371],[373,172309,172310],{"xmlns":375},[377,172311,172312,172320],{},[380,172313,172314],{},[7162,172315,172316,172318],{},[383,172317,7915],{},[383,172319,118160],{},[387,172321,172322],{"encoding":389},"e^z",[363,172324,172326],{"className":172325,"ariaHidden":394},[393],[363,172327,172329,172332],{"className":172328},[398],[363,172330],{"className":172331,"style":50725},[402],[363,172333,172335,172338],{"className":172334},[407],[363,172336,7915],{"className":172337},[407,408],[363,172339,172341],{"className":172340},[629],[363,172342,172344],{"className":172343},[633],[363,172345,172347],{"className":172346},[638],[363,172348,172350],{"className":172349,"style":50725},[642],[363,172351,172352,172355],{"style":7213},[363,172353],{"className":172354,"style":651},[650],[363,172356,172358],{"className":172357},[655,656,657,658],[363,172359,118160],{"className":172360,"style":118430},[407,408,658]," 的速度趋零、在 ",[363,172363,172365,172385],{"className":172364,"translate":367},[366],[363,172366,172368],{"className":172367},[371],[373,172369,172370],{"xmlns":375},[377,172371,172372,172382],{},[380,172373,172374,172376,172378,172380],{},[383,172375,118160],{},[458,172377,1378],{},[458,172379,2818],{},[383,172381,1373],{"mathvariant":557},[387,172383,172384],{"encoding":389},"z \\to +\\infty",[363,172386,172388,172406],{"className":172387,"ariaHidden":394},[393],[363,172389,172391,172394,172397,172400,172403],{"className":172390},[398],[363,172392],{"className":172393,"style":2693},[402],[363,172395,118160],{"className":172396,"style":118430},[407,408],[363,172398],{"className":172399,"style":506},[505],[363,172401,1378],{"className":172402},[510],[363,172404],{"className":172405,"style":506},[505],[363,172407,172409,172412,172415],{"className":172408},[398],[363,172410],{"className":172411,"style":4599},[402],[363,172413,2818],{"className":172414},[407],[363,172416,1373],{"className":172417},[407]," 时线性增长。它既提供了精确的零证据状态，又避免了指数爆炸，是实践中默认的选择。",[7013,172420,172421,172467,172468,172537],{},[363,172422,172424,172446],{"className":172423,"translate":367},[366],[363,172425,172427],{"className":172426},[371],[373,172428,172429],{"xmlns":375},[377,172430,172431,172443],{},[380,172432,172433,172435,172437,172439,172441],{},[383,172434,126325],{},[458,172436,4451],{},[458,172438,461],{"stretchy":460},[383,172440,118160],{},[458,172442,468],{"stretchy":460},[387,172444,172445],{"encoding":389},"\\exp(z)",[363,172447,172449],{"className":172448,"ariaHidden":394},[393],[363,172450,172452,172455,172458,172461,172464],{"className":172451},[398],[363,172453],{"className":172454,"style":487},[402],[363,172456,126325],{"className":172457},[1832],[363,172459,461],{"className":172460},[494],[363,172462,118160],{"className":172463,"style":118430},[407,408],[363,172465,468],{"className":172466},[501],"：严格正、光滑，但无上界，使 ",[363,172469,172471,172488],{"className":172470,"translate":367},[366],[363,172472,172474],{"className":172473},[371],[373,172475,172476],{"xmlns":375},[377,172477,172478,172486],{},[380,172479,172480],{},[543,172481,172482,172484],{},[383,172483,45126],{},[463,172485,474],{},[387,172487,152304],{"encoding":389},[363,172489,172491],{"className":172490,"ariaHidden":394},[393],[363,172492,172494,172497],{"className":172493},[398],[363,172495],{"className":172496,"style":13724},[402],[363,172498,172500,172503],{"className":172499},[407],[363,172501,45126],{"className":172502,"style":45156},[407,408],[363,172504,172506],{"className":172505},[629],[363,172507,172509,172529],{"className":172508},[633,634],[363,172510,172512,172526],{"className":172511},[638],[363,172513,172515],{"className":172514,"style":11483},[642],[363,172516,172517,172520],{"style":136154},[363,172518],{"className":172519,"style":651},[650],[363,172521,172523],{"className":172522},[655,656,657,658],[363,172524,474],{"className":172525},[407,658],[363,172527,666],{"className":172528},[665],[363,172530,172532],{"className":172531},[638],[363,172533,172535],{"className":172534,"style":8316},[642],[363,172536],{}," 可任意大，易造成数值溢出与损失对证据的不稳定响应。",[7013,172539,172540,172590,172591,172642,172643,172735,172736,172739,172740,172810,172811,172839],{},[363,172541,172543,172566],{"className":172542,"translate":367},[366],[363,172544,172546],{"className":172545},[371],[373,172547,172548],{"xmlns":375},[377,172549,172550,172563],{},[380,172551,172552,172555,172557,172559,172561],{},[383,172553,172554],{"mathvariant":557},"ReLU",[458,172556,4451],{},[458,172558,461],{"stretchy":460},[383,172560,118160],{},[458,172562,468],{"stretchy":460},[387,172564,172565],{"encoding":389},"\\operatorname{ReLU}(z)",[363,172567,172569],{"className":172568,"ariaHidden":394},[393],[363,172570,172572,172575,172581,172584,172587],{"className":172571},[398],[363,172573],{"className":172574,"style":487},[402],[363,172576,172578],{"className":172577},[1832],[363,172579,172554],{"className":172580},[407,33034],[363,172582,461],{"className":172583},[494],[363,172585,118160],{"className":172586,"style":118430},[407,408],[363,172588,468],{"className":172589},[501],"：计算廉价，但在 ",[363,172592,172594,172612],{"className":172593,"translate":367},[366],[363,172595,172597],{"className":172596},[371],[373,172598,172599],{"xmlns":375},[377,172600,172601,172609],{},[380,172602,172603,172605,172607],{},[383,172604,118160],{},[458,172606,7599],{},[463,172608,474],{},[387,172610,172611],{"encoding":389},"z \u003C 0",[363,172613,172615,172633],{"className":172614,"ariaHidden":394},[393],[363,172616,172618,172621,172624,172627,172630],{"className":172617},[398],[363,172619],{"className":172620,"style":6272},[402],[363,172622,118160],{"className":172623,"style":118430},[407,408],[363,172625],{"className":172626,"style":506},[505],[363,172628,7599],{"className":172629},[510],[363,172631],{"className":172632,"style":506},[505],[363,172634,172636,172639],{"className":172635},[398],[363,172637],{"className":172638,"style":520},[402],[363,172640,474],{"className":172641},[407]," 处梯度恒为零。处于该区域的单元输出证据 ",[363,172644,172646,172668],{"className":172645,"translate":367},[366],[363,172647,172649],{"className":172648},[371],[373,172650,172651],{"xmlns":375},[377,172652,172653,172665],{},[380,172654,172655,172661,172663],{},[543,172656,172657,172659],{},[383,172658,7915],{},[383,172660,7912],{},[458,172662,471],{},[463,172664,474],{},[387,172666,172667],{"encoding":389},"e_k = 0",[363,172669,172671,172726],{"className":172670,"ariaHidden":394},[393],[363,172672,172674,172677,172717,172720,172723],{"className":172673},[398],[363,172675],{"className":172676,"style":13724},[402],[363,172678,172680,172683],{"className":172679},[407],[363,172681,7915],{"className":172682},[407,408],[363,172684,172686],{"className":172685},[629],[363,172687,172689,172709],{"className":172688},[633,634],[363,172690,172692,172706],{"className":172691},[638],[363,172693,172695],{"className":172694,"style":643},[642],[363,172696,172697,172700],{"style":730},[363,172698],{"className":172699,"style":651},[650],[363,172701,172703],{"className":172702},[655,656,657,658],[363,172704,7912],{"className":172705,"style":8615},[407,408,658],[363,172707,666],{"className":172708},[665],[363,172710,172712],{"className":172711},[638],[363,172713,172715],{"className":172714,"style":8316},[642],[363,172716],{},[363,172718],{"className":172719,"style":506},[505],[363,172721,471],{"className":172722},[510],[363,172724],{"className":172725,"style":506},[505],[363,172727,172729,172732],{"className":172728},[398],[363,172730],{"className":172731,"style":520},[402],[363,172733,474],{"className":172734},[407]," 且",[1344,172737,172738],{},"永远无法通过梯度逃离","——",[363,172741,172743,172761],{"className":172742,"translate":367},[366],[363,172744,172746],{"className":172745},[371],[373,172747,172748],{"xmlns":375},[377,172749,172750,172758],{},[380,172751,172752],{},[543,172753,172754,172756],{},[383,172755,45126],{},[383,172757,7912],{},[387,172759,172760],{"encoding":389},"\\alpha_k",[363,172762,172764],{"className":172763,"ariaHidden":394},[393],[363,172765,172767,172770],{"className":172766},[398],[363,172768],{"className":172769,"style":13724},[402],[363,172771,172773,172776],{"className":172772},[407],[363,172774,45126],{"className":172775,"style":45156},[407,408],[363,172777,172779],{"className":172778},[629],[363,172780,172782,172802],{"className":172781},[633,634],[363,172783,172785,172799],{"className":172784},[638],[363,172786,172788],{"className":172787,"style":643},[642],[363,172789,172790,172793],{"style":136154},[363,172791],{"className":172792,"style":651},[650],[363,172794,172796],{"className":172795},[655,656,657,658],[363,172797,7912],{"className":172798,"style":8615},[407,408,658],[363,172800,666],{"className":172801},[665],[363,172803,172805],{"className":172804},[638],[363,172806,172808],{"className":172807,"style":8316},[642],[363,172809],{}," 被锁在 ",[363,172812,172814,172827],{"className":172813,"translate":367},[366],[363,172815,172817],{"className":172816},[371],[373,172818,172819],{"xmlns":375},[377,172820,172821,172825],{},[380,172822,172823],{},[463,172824,465],{},[387,172826,465],{"encoding":389},[363,172828,172830],{"className":172829,"ariaHidden":394},[393],[363,172831,172833,172836],{"className":172832},[398],[363,172834],{"className":172835,"style":520},[402],[363,172837,465],{"className":172838},[407],"，形成死区。这使 ReLU 在证据函数中通常劣于 softplus。",[7013,172841,172842,172914,172915,172943],{},[363,172843,172845,172872],{"className":172844,"translate":367},[366],[363,172846,172848],{"className":172847},[371],[373,172849,172850],{"xmlns":375},[377,172851,172852,172869],{},[380,172853,172854,172857,172859,172861,172863,172865,172867],{},[383,172855,172856],{"mathvariant":557},"ELU",[458,172858,4451],{},[458,172860,461],{"stretchy":460},[383,172862,118160],{},[458,172864,468],{"stretchy":460},[458,172866,2818],{},[463,172868,465],{},[387,172870,172871],{"encoding":389},"\\operatorname{ELU}(z) + 1",[363,172873,172875,172905],{"className":172874,"ariaHidden":394},[393],[363,172876,172878,172881,172887,172890,172893,172896,172899,172902],{"className":172877},[398],[363,172879],{"className":172880,"style":487},[402],[363,172882,172884],{"className":172883},[1832],[363,172885,172856],{"className":172886},[407,33034],[363,172888,461],{"className":172889},[494],[363,172891,118160],{"className":172892,"style":118430},[407,408],[363,172894,468],{"className":172895},[501],[363,172897],{"className":172898,"style":2935},[505],[363,172900,2818],{"className":172901},[2939],[363,172903],{"className":172904,"style":2935},[505],[363,172906,172908,172911],{"className":172907},[398],[363,172909],{"className":172910,"style":520},[402],[363,172912,465],{"className":172913},[407],"：具有负饱和下界 ",[363,172916,172918,172931],{"className":172917,"translate":367},[366],[363,172919,172921],{"className":172920},[371],[373,172922,172923],{"xmlns":375},[377,172924,172925,172929],{},[380,172926,172927],{},[463,172928,474],{},[387,172930,474],{"encoding":389},[363,172932,172934],{"className":172933,"ariaHidden":394},[393],[363,172935,172937,172940],{"className":172936},[398],[363,172938],{"className":172939,"style":520},[402],[363,172941,474],{"className":172942},[407],"，梯度在负半轴非零，是介于二者之间的折中。",[2659,172945,172947],{"id":172946},"_35-预测概率与不确定性的解析形式","3.5 预测概率与不确定性的解析形式",[143,172949,172950],{},[11,172951,172952,172955,172956,173350,173351],{},[1344,172953,172954],{},"命题 3.4（EDL 的预测与不确定性）"," 在定义 3.3 的参数化下，记总强度 ",[363,172957,172959,173029],{"className":172958,"translate":367},[366],[363,172960,172962],{"className":172961},[371],[373,172963,172964],{"xmlns":375},[377,172965,172966,173026],{},[380,172967,172968,172970,172972,172978,172980,172994,173000,173002,173004,173006,173020],{},[383,172969,26109],{},[458,172971,471],{},[543,172973,172974,172976],{},[383,172975,45126],{},[463,172977,474],{},[458,172979,471],{},[7901,172981,172982,172984,172992],{},[458,172983,13811],{},[380,172985,172986,172988,172990],{},[383,172987,7912],{},[458,172989,471],{},[463,172991,465],{},[383,172993,11044],{},[543,172995,172996,172998],{},[383,172997,45126],{},[383,172999,7912],{},[458,173001,471],{},[383,173003,11044],{},[458,173005,2818],{},[7901,173007,173008,173010,173018],{},[458,173009,13811],{},[380,173011,173012,173014,173016],{},[383,173013,7912],{},[458,173015,471],{},[463,173017,465],{},[383,173019,11044],{},[543,173021,173022,173024],{},[383,173023,7915],{},[383,173025,7912],{},[387,173027,173028],{"encoding":389},"S = \\alpha_0 = \\sum_{k=1}^K \\alpha_k = K + \\sum_{k=1}^K e_k",[363,173030,173032,173050,173105,173223,173241],{"className":173031,"ariaHidden":394},[393],[363,173033,173035,173038,173041,173044,173047],{"className":173034},[398],[363,173036],{"className":173037,"style":1011},[402],[363,173039,26109],{"className":173040,"style":26233},[407,408],[363,173042],{"className":173043,"style":506},[505],[363,173045,471],{"className":173046},[510],[363,173048],{"className":173049,"style":506},[505],[363,173051,173053,173056,173096,173099,173102],{"className":173052},[398],[363,173054],{"className":173055,"style":13724},[402],[363,173057,173059,173062],{"className":173058},[407],[363,173060,45126],{"className":173061,"style":45156},[407,408],[363,173063,173065],{"className":173064},[629],[363,173066,173068,173088],{"className":173067},[633,634],[363,173069,173071,173085],{"className":173070},[638],[363,173072,173074],{"className":173073,"style":11483},[642],[363,173075,173076,173079],{"style":136154},[363,173077],{"className":173078,"style":651},[650],[363,173080,173082],{"className":173081},[655,656,657,658],[363,173083,474],{"className":173084},[407,658],[363,173086,666],{"className":173087},[665],[363,173089,173091],{"className":173090},[638],[363,173092,173094],{"className":173093,"style":8316},[642],[363,173095],{},[363,173097],{"className":173098,"style":506},[505],[363,173100,471],{"className":173101},[510],[363,173103],{"className":173104,"style":506},[505],[363,173106,173108,173111,173171,173174,173214,173217,173220],{"className":173107},[398],[363,173109],{"className":173110,"style":136418},[402],[363,173112,173114,173117],{"className":173113},[1832],[363,173115,13811],{"className":173116,"style":40032},[1832,1833,5964],[363,173118,173120],{"className":173119},[629],[363,173121,173123,173163],{"className":173122},[633,634],[363,173124,173126,173160],{"className":173125},[638],[363,173127,173129,173149],{"className":173128,"style":133537},[642],[363,173130,173131,173134],{"style":40048},[363,173132],{"className":173133,"style":651},[650],[363,173135,173137],{"className":173136},[655,656,657,658],[363,173138,173140,173143,173146],{"className":173139},[407,658],[363,173141,7912],{"className":173142,"style":8615},[407,408,658],[363,173144,471],{"className":173145},[510,658],[363,173147,465],{"className":173148},[407,658],[363,173150,173151,173154],{"style":133560},[363,173152],{"className":173153,"style":651},[650],[363,173155,173157],{"className":173156},[655,656,657,658],[363,173158,11044],{"className":173159,"style":11132},[407,408,658],[363,173161,666],{"className":173162},[665],[363,173164,173166],{"className":173165},[638],[363,173167,173169],{"className":173168,"style":40067},[642],[363,173170],{},[363,173172],{"className":173173,"style":754},[505],[363,173175,173177,173180],{"className":173176},[407],[363,173178,45126],{"className":173179,"style":45156},[407,408],[363,173181,173183],{"className":173182},[629],[363,173184,173186,173206],{"className":173185},[633,634],[363,173187,173189,173203],{"className":173188},[638],[363,173190,173192],{"className":173191,"style":643},[642],[363,173193,173194,173197],{"style":136154},[363,173195],{"className":173196,"style":651},[650],[363,173198,173200],{"className":173199},[655,656,657,658],[363,173201,7912],{"className":173202,"style":8615},[407,408,658],[363,173204,666],{"className":173205},[665],[363,173207,173209],{"className":173208},[638],[363,173210,173212],{"className":173211,"style":8316},[642],[363,173213],{},[363,173215],{"className":173216,"style":506},[505],[363,173218,471],{"className":173219},[510],[363,173221],{"className":173222,"style":506},[505],[363,173224,173226,173229,173232,173235,173238],{"className":173225},[398],[363,173227],{"className":173228,"style":133944},[402],[363,173230,11044],{"className":173231,"style":11132},[407,408],[363,173233],{"className":173234,"style":2935},[505],[363,173236,2818],{"className":173237},[2939],[363,173239],{"className":173240,"style":2935},[505],[363,173242,173244,173247,173307,173310],{"className":173243},[398],[363,173245],{"className":173246,"style":136418},[402],[363,173248,173250,173253],{"className":173249},[1832],[363,173251,13811],{"className":173252,"style":40032},[1832,1833,5964],[363,173254,173256],{"className":173255},[629],[363,173257,173259,173299],{"className":173258},[633,634],[363,173260,173262,173296],{"className":173261},[638],[363,173263,173265,173285],{"className":173264,"style":133537},[642],[363,173266,173267,173270],{"style":40048},[363,173268],{"className":173269,"style":651},[650],[363,173271,173273],{"className":173272},[655,656,657,658],[363,173274,173276,173279,173282],{"className":173275},[407,658],[363,173277,7912],{"className":173278,"style":8615},[407,408,658],[363,173280,471],{"className":173281},[510,658],[363,173283,465],{"className":173284},[407,658],[363,173286,173287,173290],{"style":133560},[363,173288],{"className":173289,"style":651},[650],[363,173291,173293],{"className":173292},[655,656,657,658],[363,173294,11044],{"className":173295,"style":11132},[407,408,658],[363,173297,666],{"className":173298},[665],[363,173300,173302],{"className":173301},[638],[363,173303,173305],{"className":173304,"style":40067},[642],[363,173306],{},[363,173308],{"className":173309,"style":754},[505],[363,173311,173313,173316],{"className":173312},[407],[363,173314,7915],{"className":173315},[407,408],[363,173317,173319],{"className":173318},[629],[363,173320,173322,173342],{"className":173321},[633,634],[363,173323,173325,173339],{"className":173324},[638],[363,173326,173328],{"className":173327,"style":643},[642],[363,173329,173330,173333],{"style":730},[363,173331],{"className":173332,"style":651},[650],[363,173334,173336],{"className":173335},[655,656,657,658],[363,173337,7912],{"className":173338,"style":8615},[407,408,658],[363,173340,666],{"className":173341},[665],[363,173343,173345],{"className":173344},[638],[363,173346,173348],{"className":173347,"style":8316},[642],[363,173349],{},"。则\n",[363,173352,173354,173412],{"className":173353,"translate":367},[366],[363,173355,173357],{"className":173356},[371],[373,173358,173359],{"xmlns":375},[377,173360,173361,173409],{},[380,173362,173363,173369,173371,173373,173375,173381,173383,173385,173395,173397,173399,173401,173403],{},[543,173364,173365,173367],{},[383,173366,11],{},[383,173368,7912],{},[458,173370,471],{},[383,173372,571],{"mathvariant":570},[458,173374,578],{"stretchy":460},[543,173376,173377,173379],{},[383,173378,1121],{},[383,173380,7912],{},[458,173382,605],{"stretchy":460},[458,173384,471],{},[586,173386,173387,173393],{},[543,173388,173389,173391],{},[383,173390,45126],{},[383,173392,7912],{},[383,173394,26109],{},[458,173396,898],{"separator":394},[505,173398],{"width":3610},[383,173400,15750],{},[458,173402,471],{},[586,173404,173405,173407],{},[383,173406,11044],{},[383,173408,26109],{},[387,173410,173411],{"encoding":389},"p_k = \\mathbb{E}[x_k] = \\frac{\\alpha_k}{S}, \\qquad u = \\frac{K}{S}",[363,173413,173415,173470,173534,173666],{"className":173414,"ariaHidden":394},[393],[363,173416,173418,173421,173461,173464,173467],{"className":173417},[398],[363,173419],{"className":173420,"style":2624},[402],[363,173422,173424,173427],{"className":173423},[407],[363,173425,11],{"className":173426},[407,408],[363,173428,173430],{"className":173429},[629],[363,173431,173433,173453],{"className":173432},[633,634],[363,173434,173436,173450],{"className":173435},[638],[363,173437,173439],{"className":173438,"style":643},[642],[363,173440,173441,173444],{"style":730},[363,173442],{"className":173443,"style":651},[650],[363,173445,173447],{"className":173446},[655,656,657,658],[363,173448,7912],{"className":173449,"style":8615},[407,408,658],[363,173451,666],{"className":173452},[665],[363,173454,173456],{"className":173455},[638],[363,173457,173459],{"className":173458,"style":8316},[642],[363,173460],{},[363,173462],{"className":173463,"style":506},[505],[363,173465,471],{"className":173466},[510],[363,173468],{"className":173469,"style":506},[505],[363,173471,173473,173476,173479,173482,173522,173525,173528,173531],{"className":173472},[398],[363,173474],{"className":173475,"style":487},[402],[363,173477,571],{"className":173478},[407,714],[363,173480,578],{"className":173481},[494],[363,173483,173485,173488],{"className":173484},[407],[363,173486,1121],{"className":173487},[407,408],[363,173489,173491],{"className":173490},[629],[363,173492,173494,173514],{"className":173493},[633,634],[363,173495,173497,173511],{"className":173496},[638],[363,173498,173500],{"className":173499,"style":643},[642],[363,173501,173502,173505],{"style":730},[363,173503],{"className":173504,"style":651},[650],[363,173506,173508],{"className":173507},[655,656,657,658],[363,173509,7912],{"className":173510,"style":8615},[407,408,658],[363,173512,666],{"className":173513},[665],[363,173515,173517],{"className":173516},[638],[363,173518,173520],{"className":173519,"style":8316},[642],[363,173521],{},[363,173523,605],{"className":173524},[501],[363,173526],{"className":173527,"style":506},[505],[363,173529,471],{"className":173530},[510],[363,173532],{"className":173533,"style":506},[505],[363,173535,173537,173540,173645,173648,173651,173654,173657,173660,173663],{"className":173536},[398],[363,173538],{"className":173539,"style":161078},[402],[363,173541,173543,173546,173642],{"className":173542},[407],[363,173544],{"className":173545},[494,789],[363,173547,173549],{"className":173548},[586],[363,173550,173552,173634],{"className":173551},[633,634],[363,173553,173555,173631],{"className":173554},[638],[363,173556,173558,173572,173580],{"className":173557,"style":160700},[642],[363,173559,173560,173563],{"style":2875},[363,173561],{"className":173562,"style":809},[650],[363,173564,173566],{"className":173565},[655,656,657,658],[363,173567,173569],{"className":173568},[407,658],[363,173570,26109],{"className":173571,"style":26233},[407,408,658],[363,173573,173574,173577],{"style":821},[363,173575],{"className":173576,"style":809},[650],[363,173578],{"className":173579,"style":829},[828],[363,173581,173582,173585],{"style":159976},[363,173583],{"className":173584,"style":809},[650],[363,173586,173588],{"className":173587},[655,656,657,658],[363,173589,173591],{"className":173590},[407,658],[363,173592,173594,173597],{"className":173593},[407,658],[363,173595,45126],{"className":173596,"style":45156},[407,408,658],[363,173598,173600],{"className":173599},[629],[363,173601,173603,173623],{"className":173602},[633,634],[363,173604,173606,173620],{"className":173605},[638],[363,173607,173609],{"className":173608,"style":6446},[642],[363,173610,173611,173614],{"style":160006},[363,173612],{"className":173613,"style":30578},[650],[363,173615,173617],{"className":173616},[655,30582,10022,658],[363,173618,7912],{"className":173619,"style":8615},[407,408,658],[363,173621,666],{"className":173622},[665],[363,173624,173626],{"className":173625},[638],[363,173627,173629],{"className":173628,"style":101625},[642],[363,173630],{},[363,173632,666],{"className":173633},[665],[363,173635,173637],{"className":173636},[638],[363,173638,173640],{"className":173639,"style":2920},[642],[363,173641],{},[363,173643],{"className":173644},[501,789],[363,173646,898],{"className":173647},[920],[363,173649],{"className":173650,"style":3982},[505],[363,173652],{"className":173653,"style":754},[505],[363,173655,15750],{"className":173656},[407,408],[363,173658],{"className":173659,"style":506},[505],[363,173661,471],{"className":173662},[510],[363,173664],{"className":173665,"style":506},[505],[363,173667,173669,173672],{"className":173668},[398],[363,173670],{"className":173671,"style":168494},[402],[363,173673,173675,173678,173737],{"className":173674},[407],[363,173676],{"className":173677},[494,789],[363,173679,173681],{"className":173680},[586],[363,173682,173684,173729],{"className":173683},[633,634],[363,173685,173687,173726],{"className":173686},[638],[363,173688,173690,173704,173712],{"className":173689,"style":168135},[642],[363,173691,173692,173695],{"style":2875},[363,173693],{"className":173694,"style":809},[650],[363,173696,173698],{"className":173697},[655,656,657,658],[363,173699,173701],{"className":173700},[407,658],[363,173702,26109],{"className":173703,"style":26233},[407,408,658],[363,173705,173706,173709],{"style":821},[363,173707],{"className":173708,"style":809},[650],[363,173710],{"className":173711,"style":829},[828],[363,173713,173714,173717],{"style":2898},[363,173715],{"className":173716,"style":809},[650],[363,173718,173720],{"className":173719},[655,656,657,658],[363,173721,173723],{"className":173722},[407,658],[363,173724,11044],{"className":173725,"style":11132},[407,408,658],[363,173727,666],{"className":173728},[665],[363,173730,173732],{"className":173731},[638],[363,173733,173735],{"className":173734,"style":2920},[642],[363,173736],{},[363,173738],{"className":173739},[501,789],[11,173741,173742,173744,173745,173773,173774,173843,173844,10528],{},[1344,173743,138042],{}," 第一式即命题 2.1。第二式即命题 3.2。对 ",[363,173746,173748,173761],{"className":173747,"translate":367},[366],[363,173749,173751],{"className":173750},[371],[373,173752,173753],{"xmlns":375},[377,173754,173755,173759],{},[380,173756,173757],{},[383,173758,15750],{},[387,173760,15750],{"encoding":389},[363,173762,173764],{"className":173763,"ariaHidden":394},[393],[363,173765,173767,173770],{"className":173766},[398],[363,173768],{"className":173769,"style":2693},[402],[363,173771,15750],{"className":173772},[407,408]," 的推导亦可独立地由聚合性（命题 2.8）得到：若把全部类别合并为一个事件，其迪利克雷参数为 ",[363,173775,173777,173794],{"className":173776,"translate":367},[366],[363,173778,173780],{"className":173779},[371],[373,173781,173782],{"xmlns":375},[377,173783,173784,173792],{},[380,173785,173786],{},[543,173787,173788,173790],{},[383,173789,45126],{},[463,173791,474],{},[387,173793,152304],{"encoding":389},[363,173795,173797],{"className":173796,"ariaHidden":394},[393],[363,173798,173800,173803],{"className":173799},[398],[363,173801],{"className":173802,"style":13724},[402],[363,173804,173806,173809],{"className":173805},[407],[363,173807,45126],{"className":173808,"style":45156},[407,408],[363,173810,173812],{"className":173811},[629],[363,173813,173815,173835],{"className":173814},[633,634],[363,173816,173818,173832],{"className":173817},[638],[363,173819,173821],{"className":173820,"style":11483},[642],[363,173822,173823,173826],{"style":136154},[363,173824],{"className":173825,"style":651},[650],[363,173827,173829],{"className":173828},[655,656,657,658],[363,173830,474],{"className":173831},[407,658],[363,173833,666],{"className":173834},[665],[363,173836,173838],{"className":173837},[638],[363,173839,173841],{"className":173840,"style":8316},[642],[363,173842],{},"，不确定性质量即为 ",[363,173845,173847,173869],{"className":173846,"translate":367},[366],[363,173848,173850],{"className":173849},[371],[373,173851,173852],{"xmlns":375},[377,173853,173854,173866],{},[380,173855,173856,173858,173860],{},[383,173857,11044],{},[383,173859,1288],{"mathvariant":557},[543,173861,173862,173864],{},[383,173863,45126],{},[463,173865,474],{},[387,173867,173868],{"encoding":389},"K\u002F\\alpha_0",[363,173870,173872],{"className":173871,"ariaHidden":394},[393],[363,173873,173875,173878,173881,173884],{"className":173874},[398],[363,173876],{"className":173877,"style":487},[402],[363,173879,11044],{"className":173880,"style":11132},[407,408],[363,173882,1288],{"className":173883},[407],[363,173885,173887,173890],{"className":173886},[407],[363,173888,45126],{"className":173889,"style":45156},[407,408],[363,173891,173893],{"className":173892},[629],[363,173894,173896,173916],{"className":173895},[633,634],[363,173897,173899,173913],{"className":173898},[638],[363,173900,173902],{"className":173901,"style":11483},[642],[363,173903,173904,173907],{"style":136154},[363,173905],{"className":173906,"style":651},[650],[363,173908,173910],{"className":173909},[655,656,657,658],[363,173911,474],{"className":173912},[407,658],[363,173914,666],{"className":173915},[665],[363,173917,173919],{"className":173918},[638],[363,173920,173922],{"className":173921,"style":8316},[642],[363,173923],{},[11,173925,31051,173926,173954],{},[363,173927,173929,173942],{"className":173928,"translate":367},[366],[363,173930,173932],{"className":173931},[371],[373,173933,173934],{"xmlns":375},[377,173935,173936,173940],{},[380,173937,173938],{},[383,173939,15750],{},[387,173941,15750],{"encoding":389},[363,173943,173945],{"className":173944,"ariaHidden":394},[393],[363,173946,173948,173951],{"className":173947},[398],[363,173949],{"className":173950,"style":2693},[402],[363,173952,15750],{"className":173953},[407,408]," 的定义下，预测概率具有一个在解释上极有用的分解：",[363,173956,173958],{"className":173957,"translate":367},[528],[363,173959,173961,174075],{"className":173960,"translate":367},[366],[363,173962,173964],{"className":173963},[371],[373,173965,173966],{"xmlns":375,"display":537},[377,173967,173968,174072],{},[380,173969,173970,173976,173978,174003,174005,174020,174022,174024,174026,174028,174030,174032,174034,174060,174062,174064,174066],{},[543,173971,173972,173974],{},[383,173973,11],{},[383,173975,7912],{},[458,173977,471],{},[6364,173979,173980,174000],{},[6364,173981,173982,173998],{},[586,173983,173984,173996],{},[380,173985,173986,173992,173994],{},[543,173987,173988,173990],{},[383,173989,45126],{},[383,173991,7912],{},[458,173993,5161],{},[463,173995,465],{},[383,173997,26109],{},[458,173999,6388],{"stretchy":394},[1645,174001,174002],{},"证据部分",[458,174004,2818],{},[6364,174006,174007,174017],{},[6364,174008,174009,174015],{},[586,174010,174011,174013],{},[463,174012,465],{},[383,174014,26109],{},[458,174016,6388],{"stretchy":394},[1645,174018,174019],{},"先验部分",[458,174021,471],{},[458,174023,461],{"stretchy":460},[463,174025,465],{},[458,174027,5161],{},[383,174029,15750],{},[458,174031,468],{"stretchy":460},[1645,174033,1647],{},[586,174035,174036,174048],{},[380,174037,174038,174044,174046],{},[543,174039,174040,174042],{},[383,174041,45126],{},[383,174043,7912],{},[458,174045,5161],{},[463,174047,465],{},[380,174049,174050,174056,174058],{},[543,174051,174052,174054],{},[383,174053,45126],{},[463,174055,474],{},[458,174057,5161],{},[383,174059,11044],{},[458,174061,2818],{},[383,174063,15750],{},[458,174065,3647],{},[586,174067,174068,174070],{},[463,174069,465],{},[383,174071,11044],{},[387,174073,174074],{"encoding":389},"p_k = \\underbrace{\\frac{\\alpha_k - 1}{S}}_{\\text{证据部分}} + \\underbrace{\\frac{1}{S}}_{\\text{先验部分}} = (1 - u)\\,\\frac{\\alpha_k - 1}{\\alpha_0 - K} + u \\cdot \\frac{1}{K}",[363,174076,174078,174133,174371,174554,174575,174759,174778],{"className":174077,"ariaHidden":394},[393],[363,174079,174081,174084,174124,174127,174130],{"className":174080},[398],[363,174082],{"className":174083,"style":2624},[402],[363,174085,174087,174090],{"className":174086},[407],[363,174088,11],{"className":174089},[407,408],[363,174091,174093],{"className":174092},[629],[363,174094,174096,174116],{"className":174095},[633,634],[363,174097,174099,174113],{"className":174098},[638],[363,174100,174102],{"className":174101,"style":643},[642],[363,174103,174104,174107],{"style":730},[363,174105],{"className":174106,"style":651},[650],[363,174108,174110],{"className":174109},[655,656,657,658],[363,174111,7912],{"className":174112,"style":8615},[407,408,658],[363,174114,666],{"className":174115},[665],[363,174117,174119],{"className":174118},[638],[363,174120,174122],{"className":174121,"style":8316},[642],[363,174123],{},[363,174125],{"className":174126,"style":506},[505],[363,174128,471],{"className":174129},[510],[363,174131],{"className":174132,"style":506},[505],[363,174134,174136,174140,174362,174365,174368],{"className":174135},[398],[363,174137],{"className":174138,"style":174139},[402],"height:3.3338em;vertical-align:-2.0123em;",[363,174141,174143],{"className":174142},[758,6364],[363,174144,174146,174353],{"className":174145},[633,634],[363,174147,174149,174350],{"className":174148},[638],[363,174150,174152,174170],{"className":174151,"style":23261},[642],[363,174153,174155,174158],{"style":174154},"top:-1.3091em;",[363,174156],{"className":174157,"style":23038},[650],[363,174159,174161],{"className":174160},[655,656,657,658],[363,174162,174164],{"className":174163},[407,658],[363,174165,174167],{"className":174166},[407,4946,658],[363,174168,174002],{"className":174169},[407,4950,658],[363,174171,174173,174176],{"style":174172},"top:-3.3214em;",[363,174174],{"className":174175,"style":23038},[650],[363,174177,174179],{"className":174178},[758,6364],[363,174180,174182,174341],{"className":174181},[633,634],[363,174183,174185,174338],{"className":174184},[638],[363,174186,174188,174219],{"className":174187,"style":23261},[642],[363,174189,174192,174195],{"className":174190,"style":174191},[6655],"top:-1.9874em;",[363,174193],{"className":174194,"style":23038},[650],[363,174196,174198,174205,174212],{"className":174197,"style":6664},[6663],[363,174199,174201],{"className":174200,"style":6669},[6668],[6671,174202,174203],{"xmlns":6673,"width":6674,"height":6675,"viewBox":6676,"preserveAspectRatio":6677},[6679,174204],{"d":6681},[363,174206,174208],{"className":174207,"style":6669},[6685],[6671,174209,174210],{"xmlns":6673,"width":6674,"height":6675,"viewBox":6676,"preserveAspectRatio":6688},[6679,174211],{"d":6691},[363,174213,174215],{"className":174214,"style":6669},[6695],[6671,174216,174217],{"xmlns":6673,"width":6674,"height":6675,"viewBox":6676,"preserveAspectRatio":6698},[6679,174218],{"d":6701},[363,174220,174221,174224],{"style":174172},[363,174222],{"className":174223,"style":23038},[650],[363,174225,174227],{"className":174226},[407],[363,174228,174230,174233,174335],{"className":174229},[407],[363,174231],{"className":174232},[494,789],[363,174234,174236],{"className":174235},[586],[363,174237,174239,174327],{"className":174238},[633,634],[363,174240,174242,174324],{"className":174241},[638],[363,174243,174245,174256,174264],{"className":174244,"style":23261},[642],[363,174246,174247,174250],{"style":805},[363,174248],{"className":174249,"style":809},[650],[363,174251,174253],{"className":174252},[407],[363,174254,26109],{"className":174255,"style":26233},[407,408],[363,174257,174258,174261],{"style":821},[363,174259],{"className":174260,"style":809},[650],[363,174262],{"className":174263,"style":829},[828],[363,174265,174266,174269],{"style":832},[363,174267],{"className":174268,"style":809},[650],[363,174270,174272,174312,174315,174318,174321],{"className":174271},[407],[363,174273,174275,174278],{"className":174274},[407],[363,174276,45126],{"className":174277,"style":45156},[407,408],[363,174279,174281],{"className":174280},[629],[363,174282,174284,174304],{"className":174283},[633,634],[363,174285,174287,174301],{"className":174286},[638],[363,174288,174290],{"className":174289,"style":643},[642],[363,174291,174292,174295],{"style":136154},[363,174293],{"className":174294,"style":651},[650],[363,174296,174298],{"className":174297},[655,656,657,658],[363,174299,7912],{"className":174300,"style":8615},[407,408,658],[363,174302,666],{"className":174303},[665],[363,174305,174307],{"className":174306},[638],[363,174308,174310],{"className":174309,"style":8316},[642],[363,174311],{},[363,174313],{"className":174314,"style":2935},[505],[363,174316,5161],{"className":174317},[2939],[363,174319],{"className":174320,"style":2935},[505],[363,174322,465],{"className":174323},[407],[363,174325,666],{"className":174326},[665],[363,174328,174330],{"className":174329},[638],[363,1743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u)",[363,174881,174883,174904],{"className":174882,"ariaHidden":394},[393],[363,174884,174886,174889,174892,174895,174898,174901],{"className":174885},[398],[363,174887],{"className":174888,"style":487},[402],[363,174890,461],{"className":174891},[494],[363,174893,465],{"className":174894},[407],[363,174896],{"className":174897,"style":2935},[505],[363,174899,5161],{"className":174900},[2939],[363,174902],{"className":174903,"style":2935},[505],[363,174905,174907,174910,174913,174916,174919,174922],{"className":174906},[398],[363,174908],{"className":174909,"style":487},[402],[363,174911,15750],{"className":174912},[407,408],[363,174914,898],{"className":174915},[920],[363,174917],{"className":174918,"style":754},[505],[363,174920,15750],{"className":174921},[407,408],[363,174923,468],{"className":174924},[501]," 的凸组合","。当证据充分时（",[363,174928,174930,174948],{"className":174929,"translate":367},[366],[363,174931,174933],{"className":174932},[371],[373,174934,174935],{"xmlns":375},[377,174936,174937,174945],{},[380,174938,174939,174941,174943],{},[383,174940,15750],{},[458,174942,1378],{},[463,174944,474],{},[387,174946,174947],{"encoding":389},"u \\to 0",[363,174949,174951,174969],{"className":174950,"ariaHidden":394},[393],[363,174952,174954,174957,174960,174963,174966],{"className":174953},[398],[363,174955],{"className":174956,"style":2693},[402],[363,174958,15750],{"className":174959},[407,408],[363,174961],{"className":174962,"style":506},[505],[363,174964,1378],{"className":174965},[510],[363,174967],{"className":174968,"style":506},[505],[363,174970,174972,174975],{"className":174971},[398],[363,174973],{"className":174974,"style":520},[402],[363,174976,474],{"className":174977},[407],"），预测趋于证据的方向。当证据消失时（",[363,174980,174982,175000],{"className":174981,"translate":367},[366],[363,174983,174985],{"className":174984},[371],[373,174986,174987],{"xmlns":375},[377,174988,174989,174997],{},[380,174990,174991,174993,174995],{},[383,174992,15750],{},[458,174994,1378],{},[463,174996,465],{},[387,174998,174999],{"encoding":389},"u \\to 1",[363,175001,175003,175021],{"className":175002,"ariaHidden":394},[393],[363,175004,175006,175009,175012,175015,175018],{"className":175005},[398],[363,175007],{"className":175008,"style":2693},[402],[363,175010,15750],{"className":175011},[407,408],[363,175013],{"className":175014,"style":506},[505],[363,175016,1378],{"className":175017},[510],[363,175019],{"className":175020,"style":506},[505],[363,175022,175024,175027],{"className":175023},[398],[363,175025],{"className":175026,"style":520},[402],[363,175028,465],{"className":175029},[407],"），预测退回均匀分布。极限行为因此完全符合直觉：",[363,175032,175034],{"className":175033,"translate":367},[528],[363,175035,175037,175142],{"className":175036,"translate":367},[366],[363,175038,175040],{"className":175039},[371],[373,175041,175042],{"xmlns":375,"display":537},[377,175043,175044,175139],{},[380,175045,175046,175048,175050,175052,175054,175056,175058,175060,175062,175064,175066,175068,175070,175072,175074,175076,175079,175081,175083,175091,175093,175095,175097,175099,175101,175103,175105,175107,175109,175111,175113,175115,175117,175119,175121],{},[383,175047,26109],{},[458,175049,1378],{},[383,175051,11044],{},[1645,175053,4631],{},[458,175055,461],{"stretchy":460},[383,175057,7915],{"mathvariant":75598},[458,175059,1378],{},[463,175061,474],{"mathvariant":75598},[458,175063,468],{"stretchy":460},[458,175065,1364],{},[505,175067],{"width":4663},[383,175069,15750],{},[458,175071,1378],{},[463,175073,465],{},[458,175075,898],{"separator":394},[1645,175077,175078],{},"  ",[383,175080,11],{"mathvariant":75598},[458,175082,1378],{},[1603,175084,175085],{"scriptlevel":474,"displaystyle":460},[586,175086,175087,175089],{},[463,175088,465],{},[383,175090,11044],{},[463,175092,465],{"mathvariant":75598},[505,175094],{"width":3610},[505,175096],{"width":3610},[383,175098,26109],{},[458,175100,1378],{},[383,175102,1373],{"mathvariant":557},[458,175104,1364],{},[505,175106],{"width":4663},[383,175108,15750],{},[458,175110,1378],{},[463,175112,474],{},[458,175114,898],{"separator":394},[1645,175116,175078],{},[383,175118,11],{"mathvariant":75598},[458,175120,1378],{},[586,175122,175123,175125],{},[383,175124,7915],{"mathvariant":75598},[380,175126,175127,175133],{},[6364,175128,175129,175131],{},[458,175130,13811],{},[383,175132,100402],{},[543,175134,175135,175137],{},[383,175136,7915],{},[383,175138,100402],{},[387,175140,175141],{"encoding":389},"S \\to K \\ (\\mathbf{e} \\to \\mathbf{0}): \\quad u \\to 1,\\ \\ \\mathbf{p} \\to \\tfrac{1}{K}\\mathbf{1}\n\\qquad\\qquad\nS \\to \\infty: \\quad u \\to 0,\\ \\ \\mathbf{p} \\to \\frac{\\mathbf{e}}{\\sum_j e_j}",[363,175143,175145,175163,175190,175214,175232,175265,175360,175381,175399,175432],{"className":175144,"ariaHidden":394},[393],[363,175146,175148,175151,175154,175157,175160],{"className":175147},[398],[363,175149],{"className":175150,"style":1011},[402],[363,175152,26109],{"className":175153,"style":26233},[407,408],[363,175155],{"className":175156,"style":506},[505],[363,175158,1378],{"className":175159},[510],[363,175161],{"className":175162,"style":506},[505],[363,175164,175166,175169,175172,175175,175178,175181,175184,175187],{"className":175165},[398],[363,175167],{"className":175168,"style":487},[402],[363,175170,11044],{"className":175171,"style":11132},[407,408],[363,175173,4631],{"className":175174},[505],[363,175176,461],{"className":175177},[494],[363,175179,7915],{"className":175180},[407,75943],[363,175182],{"className":175183,"style":506},[505],[363,175185,1378],{"className":175186},[510],[363,175188],{"className":175189,"style":506},[505],[363,175191,175193,175196,175199,175202,175205,175208,175211],{"className":175192},[398],[363,175194],{"className":175195,"style":487},[402],[363,175197,474],{"className":175198},[407,75943],[363,175200,468],{"className":175201},[501],[363,175203],{"className":175204,"style":506},[505],[363,175206,1364],{"className":175207},[510],[363,175209],{"className":175210,"style":88355},[505],[363,175212],{"className":175213,"style":506},[505],[363,175215,175217,175220,175223,175226,175229],{"className":175216},[398],[363,175218],{"className":175219,"style":2693},[402],[363,175221,15750],{"className":175222},[407,408],[363,175224],{"className":175225,"style":506},[505],[363,175227,1378],{"className":175228},[510],[363,175230],{"className":175231,"style":506},[505],[363,175233,175235,175238,175241,175244,175247,175250,175253,175256,175259,175262],{"className":175234},[398],[363,175236],{"className":175237,"style":9706},[402],[363,175239,465],{"className":175240},[407],[363,175242,898],{"className":175243},[920],[363,175245,4631],{"className":175246},[505],[363,175248,4631],{"className":175249},[505],[363,175251],{"className":175252,"style":754},[505],[363,175254,11],{"className":175255},[407,75943],[363,175257],{"className":175258,"style":506},[505],[363,175260,1378],{"className":175261},[510],[363,175263],{"className":175264,"style":506},[505],[363,175266,175268,175271,175339,175342,175345,175348,175351,175354,175357],{"className":175267},[398],[363,175269],{"className":175270,"style":2853},[402],[363,175272,175274,175277,175336],{"className":175273},[407],[363,175275],{"className":175276},[494,789],[363,175278,175280],{"className":175279},[586],[363,175281,175283,175328],{"className":175282},[633,634],[363,175284,175286,175325],{"className":175285},[638],[363,175287,175289,175303,175311],{"className":175288,"style":2872},[642],[363,175290,175291,175294],{"style":2875},[363,175292],{"className":175293,"style":809},[650],[363,175295,175297],{"className":175296},[655,656,657,658],[363,175298,175300],{"className":175299},[407,658],[363,175301,11044],{"className":175302,"style":11132},[407,408,658],[363,175304,175305,175308],{"style":821},[363,175306],{"className":175307,"style":809},[650],[363,175309],{"className":175310,"style":829},[828],[363,175312,175313,175316],{"style":2898},[363,175314],{"className":175315,"style":809},[650],[363,175317,175319],{"className":175318},[655,656,657,658],[363,175320,175322],{"className":175321},[407,658],[363,175323,465],{"className":175324},[407,658],[363,175326,666],{"className":175327},[665],[363,175329,175331],{"className":175330},[638],[363,175332,175334],{"className":175333,"style":2920},[642],[363,175335],{},[363,175337],{"className":175338},[501,789],[363,175340,465],{"className":175341},[407,75943],[363,175343],{"className":175344,"style":3982},[505],[363,175346],{"className":175347,"style":3982},[505],[363,175349,26109],{"className":175350,"style":26233},[407,408],[363,175352],{"className":175353,"style":506},[505],[363,175355,1378],{"className":175356},[510],[363,175358],{"className":175359,"style":506},[505],[363,175361,175363,175366,175369,175372,175375,175378],{"className":175362},[398],[363,175364],{"className":175365,"style":2693},[402],[363,175367,1373],{"className":175368},[407],[363,175370],{"className":175371,"style":506},[505],[363,175373,1364],{"className":175374},[510],[363,175376],{"className":175377,"style":88355},[505],[363,175379],{"className":175380,"style":506},[505],[363,175382,175384,175387,175390,175393,175396],{"className":175383},[398],[363,175385],{"className":175386,"style":2693},[402],[363,175388,15750],{"className":175389},[407,408],[363,175391],{"className":175392,"style":506},[505],[363,175394,1378],{"className":175395},[510],[363,175397],{"className":175398,"style":506},[505],[363,175400,175402,175405,175408,175411,175414,175417,175420,175423,175426,175429],{"className":175401},[398],[363,175403],{"className":175404,"style":9706},[402],[363,175406,474],{"className":175407},[407],[363,175409,898],{"className":175410},[920],[363,175412,4631],{"className":175413},[505],[363,175415,4631],{"className":175416},[505],[363,175418],{"className":175419,"style":754},[505],[363,175421,11],{"className":175422},[407,75943],[363,175424],{"className":175425,"style":506},[505],[363,175427,1378],{"className":175428},[510],[363,175430],{"className":175431,"style":506},[505],[363,175433,175435,175439],{"className":175434},[398],[363,175436],{"className":175437,"style":175438},[402],"height:2.2433em;vertical-align:-1.1218em;",[363,175440,175442,175445,175580],{"className":175441},[407],[363,175443],{"className":175444},[494,789],[363,175446,175448],{"className":175447},[586],[363,175449,175451,175571],{"className":175450},[633,634],[363,175452,175454,175568],{"className":175453},[638],[363,175455,175458,175549,175557],{"className":175456,"style":175457},[642],"height:1.1214em;",[363,175459,175460,175463],{"style":805},[363,175461],{"className":175462,"style":809},[650],[363,175464,175466,175506,175509],{"className":175465},[407],[363,175467,175469,175472],{"className":175468},[1832],[363,175470,13811],{"className":175471,"style":40032},[1832,1833,5964],[363,175473,175475],{"className":175474},[629],[363,175476,175478,175498],{"className":175477},[633,634],[363,175479,175481,175495],{"className":175480},[638],[363,175482,175484],{"className":175483,"style":40045},[642],[363,175485,175486,175489],{"style":40048},[363,175487],{"className":175488,"style":651},[650],[363,175490,175492],{"className":175491},[655,656,657,658],[363,175493,100402],{"className":175494,"style":100930},[407,408,658],[363,175496,666],{"className":175497},[665],[363,175499,175501],{"className":175500},[638],[363,175502,175504],{"className":175503,"style":162795},[642],[363,175505],{},[363,175507],{"className":175508,"style":754},[505],[363,175510,175512,175515],{"className":175511},[407],[363,175513,7915],{"className":175514},[407,408],[363,175516,175518],{"className":175517},[629],[363,175519,175521,175541],{"className":175520},[633,634],[363,175522,175524,175538],{"className":175523},[638],[363,175525,175527],{"className":175526,"style":12532},[642],[363,175528,175529,175532],{"style":730},[363,175530],{"className":175531,"style":651},[650],[363,175533,175535],{"className":175534},[655,656,657,658],[363,175536,100402],{"className":175537,"style":100930},[407,408,658],[363,175539,666],{"className":175540},[665],[363,175542,175544],{"className":175543},[638],[363,175545,175547],{"className":175546,"style":673},[642],[363,175548],{},[363,175550,175551,175554],{"style":821},[363,175552],{"className":175553,"style":809},[650],[363,175555],{"className":175556,"style":829},[828],[363,175558,175559,175562],{"style":832},[363,175560],{"className":175561,"style":809},[650],[363,175563,175565],{"className":175564},[407],[363,175566,7915],{"className":175567},[407,75943],[363,175569,666],{"className":175570},[665],[363,175572,175574],{"className":175573},[638],[363,175575,175578],{"className":175576,"style":175577},[642],"height:1.1218em;",[363,175579],{},[363,175581],{"className":175582},[501,789],[11,175584,175585,175586,9748,175589,175617,175618,175646,175647,175650],{},"命题 3.4 同时也暴露了该参数化的一个",[1344,175587,175588],{},"结构性局限",[363,175590,175592,175605],{"className":175591,"translate":367},[366],[363,175593,175595],{"className":175594},[371],[373,175596,175597],{"xmlns":375},[377,175598,175599,175603],{},[380,175600,175601],{},[383,175602,15750],{},[387,175604,15750],{"encoding":389},[363,175606,175608],{"className":175607,"ariaHidden":394},[393],[363,175609,175611,175614],{"className":175610},[398],[363,175612],{"className":175613,"style":2693},[402],[363,175615,15750],{"className":175616},[407,408]," 只是总强度 ",[363,175619,175621,175634],{"className":175620,"translate":367},[366],[363,175622,175624],{"className":175623},[371],[373,175625,175626],{"xmlns":375},[377,175627,175628,175632],{},[380,175629,175630],{},[383,175631,26109],{},[387,175633,26109],{"encoding":389},[363,175635,175637],{"className":175636,"ariaHidden":394},[393],[363,175638,175640,175643],{"className":175639},[398],[363,175641],{"className":175642,"style":1011},[402],[363,175644,26109],{"className":175645,"style":26233},[407,408]," 的函数，与证据在各类别之间的",[1344,175648,175649],{},"分布","无关。因此 EDL 的不确定性分数无法区分「少量证据集中于一类」与「大量证据均匀分散」，也无法区分「证据落在正确类别上」与「证据落在错误类别上」。这一局限将在 4.5 节与 5.6 节被重新讨论。",[2659,175652,175654],{"id":175653},"_36-不确定性的分解偶然与认知","3.6 不确定性的分解：偶然与认知",[11,175656,175657,175658,175719,175720,175723],{},"命题 3.4 给出的 ",[363,175659,175661,175683],{"className":175660,"translate":367},[366],[363,175662,175664],{"className":175663},[371],[373,175665,175666],{"xmlns":375},[377,175667,175668,175680],{},[380,175669,175670,175672,175674,175676,175678],{},[383,175671,15750],{},[458,175673,471],{},[383,175675,11044],{},[383,175677,1288],{"mathvariant":557},[383,175679,26109],{},[387,175681,175682],{"encoding":389},"u = K\u002FS",[363,175684,175686,175704],{"className":175685,"ariaHidden":394},[393],[363,175687,175689,175692,175695,175698,175701],{"className":175688},[398],[363,175690],{"className":175691,"style":2693},[402],[363,175693,15750],{"className":175694},[407,408],[363,175696],{"className":175697,"style":506},[505],[363,175699,471],{"className":175700},[510],[363,175702],{"className":175703,"style":506},[505],[363,175705,175707,175710,175713,175716],{"className":175706},[398],[363,175708],{"className":175709,"style":487},[402],[363,175711,11044],{"className":175712,"style":11132},[407,408],[363,175714,1288],{"className":175715},[407],[363,175717,26109],{"className":175718,"style":26233},[407,408]," 是一个",[1344,175721,175722],{},"启发式","的不确定性标量。与之相对，Malinin 与 Gales 在 Prior Networks 的工作中给出了一个有信息论根据的分解，它在迪利克雷族内具有闭式解。",[11,175725,881,175726,7148,175808,175914],{},[363,175727,175729,175754],{"className":175728,"translate":367},[366],[363,175730,175732],{"className":175731},[371],[373,175733,175734],{"xmlns":375},[377,175735,175736,175752],{},[380,175737,175738,175740,175742,175744,175746,175748,175750],{},[383,175739,157459],{"mathvariant":136055},[458,175741,9217],{},[383,175743,137112],{"mathvariant":557},[458,175745,4451],{},[458,175747,461],{"stretchy":460},[383,175749,45126],{"mathvariant":136055},[458,175751,468],{"stretchy":460},[387,175753,157474],{"encoding":389},[363,175755,175757,175781],{"className":175756,"ariaHidden":394},[393],[363,175758,175760,175763,175772,175775,175778],{"className":175759},[398],[363,175761],{"className":175762,"style":135662},[402],[363,175764,175766],{"className":175765},[407],[363,175767,175769],{"className":175768},[407],[363,175770,157459],{"className":175771,"style":157493},[407,136115],[363,175773],{"className":175774,"style":506},[505],[363,175776,9217],{"className":175777},[510],[363,175779],{"className":175780,"style":506},[505],[363,175782,175784,175787,175793,175796,175805],{"className":175783},[398],[363,175785],{"className":175786,"style":487},[402],[363,175788,175790],{"className":175789},[1832],[363,175791,137112],{"className":175792},[407,33034],[363,175794,461],{"className":175795},[494],[363,175797,175799],{"className":175798},[407],[363,175800,175802],{"className":175801},[407],[363,175803,45126],{"className":175804},[407,136115],[363,175806,468],{"className":175807},[501],[363,175809,175811,175842],{"className":175810,"translate":367},[366],[363,175812,175814],{"className":175813},[371],[373,175815,175816],{"xmlns":375},[377,175817,175818,175839],{},[380,175819,175820,175822,175824,175826,175828,175831,175833,175835,175837],{},[383,175821,40609],{},[458,175823,5156],{},[383,175825,157459],{"mathvariant":136055},[458,175827,9217],{},[383,175829,175830],{"mathvariant":557},"Cat",[458,175832,4451],{},[458,175834,461],{"stretchy":460},[383,175836,157459],{"mathvariant":136055},[458,175838,468],{"stretchy":460},[387,175840,175841],{"encoding":389},"y \\mid \\boldsymbol{\\pi} \\sim \\operatorname{Cat}(\\boldsymbol{\\pi})",[363,175843,175845,175863,175887],{"className":175844,"ariaHidden":394},[393],[363,175846,175848,175851,175854,175857,175860],{"className":175847},[398],[363,175849],{"className":175850,"style":487},[402],[363,175852,40609],{"className":175853,"style":1328},[407,408],[363,175855],{"className":175856,"style":506},[505],[363,175858,5156],{"className":175859},[510],[363,175861],{"className":175862,"style":506},[505],[363,175864,175866,175869,175878,175881,175884],{"className":175865},[398],[363,175867],{"className":175868,"style":135662},[402],[363,175870,175872],{"className":175871},[407],[363,175873,175875],{"className":175874},[407],[363,175876,157459],{"className":175877,"style":157493},[407,136115],[363,175879],{"className":175880,"style":506},[505],[363,175882,9217],{"className":175883},[510],[363,175885],{"className":175886,"style":506},[505],[363,175888,175890,175893,175899,175902,175911],{"className":175889},[398],[363,175891],{"className":175892,"style":487},[402],[363,175894,175896],{"className":175895},[1832],[363,175897,175830],{"className":175898},[407,33034],[363,175900,461],{"className":175901},[494],[363,175903,175905],{"className":175904},[407],[363,175906,175908],{"className":175907},[407],[363,175909,157459],{"className":175910,"style":157493},[407,136115],[363,175912,468],{"className":175913},[501],"。考虑三个量：",[7010,175916,175917,176266,176419],{},[7013,175918,175919,175922,175923,87168,176001,10528],{},[1344,175920,175921],{},"总不确定性（total uncertainty）","：预测分布的熵 ",[363,175924,175926,175956],{"className":175925,"translate":367},[366],[363,175927,175929],{"className":175928},[371],[373,175930,175931],{"xmlns":375},[377,175932,175933,175953],{},[380,175934,175935,175937,175939,175941,175943,175945,175947,175949,175951],{},[383,175936,34256],{},[458,175938,578],{"stretchy":460},[383,175940,554],{"mathvariant":570},[458,175942,461],{"stretchy":460},[383,175944,40609],{},[458,175946,5156],{},[383,175948,1121],{"mathvariant":75598},[458,175950,468],{"stretchy":460},[458,175952,605],{"stretchy":460},[387,175954,175955],{"encoding":389},"H[\\mathbb{P}(y \\mid \\mathbf{x})]",[363,175957,175959,175989],{"className":175958,"ariaHidden":394},[393],[363,175960,175962,175965,175968,175971,175974,175977,175980,175983,175986],{"className":175961},[398],[363,175963],{"className":175964,"style":487},[402],[363,175966,34256],{"className":175967,"style":34410},[407,408],[363,175969,578],{"className":175970},[494],[363,175972,554],{"className":175973},[407,714],[363,175975,461],{"className":175976},[494],[363,175978,40609],{"className":175979,"style":1328},[407,408],[363,175981],{"className":175982,"style":506},[505],[363,175984,5156],{"className":175985},[510],[363,175987],{"className":175988,"style":506},[505],[363,175990,175992,175995,175998],{"className":175991},[398],[363,175993],{"className":175994,"style":487},[402],[363,175996,1121],{"className":175997},[407,75943],[363,175999,20762],{"className":176000},[501],[363,176002,176004,176056],{"className":176003,"translate":367},[366],[363,176005,176007],{"className":176006},[371],[373,176008,176009],{"xmlns":375},[377,176010,176011,176053],{},[380,176012,176013,176015,176017,176019,176021,176023,176025,176027,176029,176031,176037,176039,176045,176047],{},[383,176014,554],{"mathvariant":570},[458,176016,461],{"stretchy":460},[383,176018,40609],{},[458,176020,471],{},[383,176022,7912],{},[458,176024,5156],{},[383,176026,1121],{"mathvariant":75598},[458,176028,468],{"stretchy":460},[458,176030,471],{},[543,176032,176033,176035],{},[383,176034,11],{},[383,176036,7912],{},[458,176038,471],{},[543,176040,176041,176043],{},[383,176042,45126],{},[383,176044,7912],{},[383,176046,1288],{"mathvariant":557},[543,176048,176049,176051],{},[383,176050,45126],{},[463,176052,474],{},[387,176054,176055],{"encoding":389},"\\mathbb{P}(y = k \\mid \\mathbf{x}) = p_k = \\alpha_k\u002F\\alpha_0",[363,176057,176059,176083,176101,176122,176177],{"className":176058,"ariaHidden":394},[393],[363,176060,176062,176065,176068,176071,176074,176077,176080],{"className":176061},[398],[363,176063],{"className":176064,"style":487},[402],[363,176066,554],{"className":176067},[407,714],[363,176069,461],{"className":176070},[494],[363,176072,40609],{"className":176073,"style":1328},[407,408],[363,176075],{"className":176076,"style":506},[505],[363,176078,471],{"className":176079},[510],[363,176081],{"className":176082,"style":506},[505],[363,176084,176086,176089,176092,176095,176098],{"className":176085},[398],[363,176087],{"className":176088,"style":487},[402],[363,176090,7912],{"className":176091,"style":8615},[407,408],[363,176093],{"className":176094,"style":506},[505],[363,176096,5156],{"className":176097},[510],[363,176099],{"className":176100,"style":506},[505],[363,176102,176104,176107,176110,176113,176116,176119],{"className":176103},[398],[363,176105],{"className":176106,"style":487},[402],[363,176108,1121],{"className":176109},[407,75943],[363,176111,468],{"className":176112},[501],[363,176114],{"className":176115,"style":506},[505],[363,176117,471],{"className":176118},[510],[363,176120],{"className":176121,"style":506},[505],[363,176123,176125,176128,176168,176171,176174],{"className":176124},[398],[363,176126],{"className":176127,"style":2624},[402],[363,176129,176131,176134],{"className":176130},[407],[363,176132,11],{"className":176133},[407,408],[363,176135,176137],{"className":176136},[629],[363,176138,176140,176160],{"className":176139},[633,634],[363,176141,176143,176157],{"className":176142},[638],[363,176144,176146],{"className":176145,"style":643},[642],[363,176147,176148,176151],{"style":730},[363,176149],{"className":176150,"style":651},[650],[363,176152,176154],{"className":176153},[655,656,657,658],[363,176155,7912],{"className":176156,"style":8615},[407,408,658],[363,176158,666],{"className":176159},[665],[363,176161,176163],{"className":176162},[638],[363,176164,176166],{"className":176165,"style":8316},[642],[363,176167],{},[363,176169],{"className":176170,"style":506},[505],[363,176172,471],{"className":176173},[510],[363,176175],{"className":176176,"style":506},[505],[363,176178,176180,176183,176223,176226],{"className":176179},[398],[363,176181],{"className":176182,"style":487},[402],[363,176184,176186,176189],{"className":176185},[407],[363,176187,45126],{"className":176188,"style":45156},[407,408],[363,176190,176192],{"className":176191},[629],[363,176193,176195,176215],{"className":176194},[633,634],[363,176196,176198,176212],{"className":176197},[638],[363,176199,176201],{"className":176200,"style":643},[642],[363,176202,176203,176206],{"style":136154},[363,176204],{"className":176205,"style":651},[650],[363,176207,176209],{"className":176208},[655,656,657,658],[363,176210,7912],{"className":176211,"style":8615},[407,408,658],[363,176213,666],{"className":176214},[665],[363,176216,176218],{"className":176217},[638],[363,176219,176221],{"className":176220,"style":8316},[642],[363,176222],{},[363,176224,1288],{"className":176225},[407],[363,176227,176229,176232],{"className":176228},[407],[363,176230,45126],{"className":176231,"style":45156},[407,408],[363,176233,176235],{"className":176234},[629],[363,176236,176238,176258],{"className":176237},[633,634],[363,176239,176241,176255],{"className":176240},[638],[363,176242,176244],{"className":176243,"style":11483},[642],[363,176245,176246,176249],{"style":136154},[363,176247],{"className":176248,"style":651},[650],[363,176250,176252],{"className":176251},[655,656,657,658],[363,176253,474],{"className":176254},[407,658],[363,176256,666],{"className":176257},[665],[363,176259,176261],{"className":176260},[638],[363,176262,176264],{"className":176263,"style":8316},[642],[363,176265],{},[7013,176267,176268,176271,176272,176306,176307,10528],{},[1344,176269,176270],{},"偶然不确定性（aleatoric uncertainty）","：给定 ",[363,176273,176275,176288],{"className":176274,"translate":367},[366],[363,176276,176278],{"className":176277},[371],[373,176279,176280],{"xmlns":375},[377,176281,176282,176286],{},[380,176283,176284],{},[383,176285,157459],{"mathvariant":136055},[387,176287,157547],{"encoding":389},[363,176289,176291],{"className":176290,"ariaHidden":394},[393],[363,176292,176294,176297],{"className":176293},[398],[363,176295],{"className":176296,"style":135662},[402],[363,176298,176300],{"className":176299},[407],[363,176301,176303],{"className":176302},[407],[363,176304,157459],{"className":176305,"style":157493},[407,136115]," 后仍存在的熵的期望 ",[363,176308,176310,176340],{"className":176309,"translate":367},[366],[363,176311,176313],{"className":176312},[371],[373,176314,176315],{"xmlns":375},[377,176316,176317,176337],{},[380,176318,176319,176325,176327,176329,176331,176333,176335],{},[543,176320,176321,176323],{},[383,176322,571],{"mathvariant":570},[383,176324,157459],{"mathvariant":136055},[458,176326,578],{"stretchy":460},[383,176328,34256],{},[458,176330,461],{"stretchy":460},[383,176332,157459],{"mathvariant":136055},[458,176334,468],{"stretchy":460},[458,176336,605],{"stretchy":460},[387,176338,176339],{"encoding":389},"\\mathbb{E}_{\\boldsymbol{\\pi}}[H(\\boldsymbol{\\pi})]",[363,176341,176343],{"className":176342,"ariaHidden":394},[393],[363,176344,176346,176349,176398,176401,176404,176407,176416],{"className":176345},[398],[363,176347],{"className":176348,"style":487},[402],[363,176350,176352,176355],{"className":176351},[407],[363,176353,571],{"className":176354},[407,714],[363,176356,176358],{"className":176357},[629],[363,176359,176361,176390],{"className":176360},[633,634],[363,176362,176364,176387],{"className":176363},[638],[363,176365,176367],{"className":176366,"style":153771},[642],[363,176368,176369,176372],{"style":730},[363,176370],{"className":176371,"style":651},[650],[363,176373,176375],{"className":176374},[655,656,657,658],[363,176376,176378],{"className":176377},[407,658],[363,176379,176381],{"className":176380},[407,658],[363,176382,176384],{"className":176383},[407,658],[363,176385,157459],{"className":176386,"style":157493},[407,136115,658],[363,176388,666],{"className":176389},[665],[363,176391,176393],{"className":176392},[638],[363,176394,176396],{"className":176395,"style":8316},[642],[363,176397],{},[363,176399,578],{"className":176400},[494],[363,176402,34256],{"className":176403,"style":34410},[407,408],[363,176405,461],{"className":176406},[494],[363,176408,176410],{"className":176409},[407],[363,176411,176413],{"className":176412},[407],[363,176414,157459],{"className":176415,"style":157493},[407,136115],[363,176417,20762],{"className":176418},[501],[7013,176420,176421,176424,176425,2757,176453,176487],{},[1344,176422,176423],{},"认知不确定性（epistemic uncertainty）","：二者的差，即 ",[363,176426,176428,176441],{"className":176427,"translate":367},[366],[363,176429,176431],{"className":176430},[371],[373,176432,176433],{"xmlns":375},[377,176434,176435,176439],{},[380,176436,176437],{},[383,176438,40609],{},[387,176440,40609],{"encoding":389},[363,176442,176444],{"className":176443,"ariaHidden":394},[393],[363,176445,176447,176450],{"className":176446},[398],[363,176448],{"className":176449,"style":2624},[402],[363,176451,40609],{"className":176452,"style":1328},[407,408],[363,176454,176456,176469],{"className":176455,"translate":367},[366],[363,176457,176459],{"className":176458},[371],[373,176460,176461],{"xmlns":375},[377,176462,176463,176467],{},[380,176464,176465],{},[383,176466,157459],{"mathvariant":136055},[387,176468,157547],{"encoding":389},[363,176470,176472],{"className":176471,"ariaHidden":394},[393],[363,176473,176475,176478],{"className":176474},[398],[363,176476],{"className":176477,"style":135662},[402],[363,176479,176481],{"className":176480},[407],[363,176482,176484],{"className":176483},[407],[363,176485,157459],{"className":176486,"style":157493},[407,136115]," 之间的互信息。",[143,176489,176490],{},[11,176491,176492,79304,176495,3093,176679,173350,176771,177058,177457],{},[1344,176493,176494],{},"命题 3.5（Dirichlet 族内的不确定性分解）",[363,176496,176498,176532],{"className":176497,"translate":367},[366],[363,176499,176501],{"className":176500},[371],[373,176502,176503],{"xmlns":375},[377,176504,176505,176529],{},[380,176506,176507,176513,176515,176521,176523],{},[543,176508,176509,176511],{},[383,176510,11],{},[383,176512,7912],{},[458,176514,471],{},[543,176516,176517,176519],{},[383,176518,45126],{},[383,176520,7912],{},[383,176522,1288],{"mathvariant":557},[543,176524,176525,176527],{},[383,176526,45126],{},[463,176528,474],{},[387,176530,176531],{"encoding":389},"p_k = \\alpha_k\u002F\\alpha_0",[363,176533,176535,176590],{"className":176534,"ariaHidden":394},[393],[363,176536,176538,176541,176581,176584,176587],{"className":176537},[398],[363,176539],{"className":176540,"style":2624},[402],[363,176542,176544,176547],{"className":176543},[407],[363,176545,11],{"className":176546},[407,408],[363,176548,176550],{"className":176549},[629],[363,176551,176553,176573],{"className":176552},[633,634],[363,176554,176556,176570],{"className":176555},[638],[363,176557,176559],{"className":176558,"style":643},[642],[363,176560,176561,176564],{"style":730},[363,176562],{"className":176563,"style":651},[650],[363,176565,176567],{"className":176566},[655,656,657,658],[363,176568,7912],{"className":176569,"style":8615},[407,408,658],[363,176571,666],{"className":176572},[665],[363,176574,176576],{"className":176575},[638],[363,176577,176579],{"className":176578,"style":8316},[642],[363,176580],{},[363,176582],{"className":176583,"style":506},[505],[363,176585,471],{"className":176586},[510],[363,176588],{"className":176589,"style":506},[505],[363,176591,176593,176596,176636,176639],{"className":176592},[398],[363,176594],{"className":176595,"style":487},[402],[363,176597,176599,176602],{"className":176598},[407],[363,176600,45126],{"className":176601,"style":45156},[407,408],[363,176603,176605],{"className":176604},[629],[363,176606,176608,176628],{"className":176607},[633,634],[363,176609,176611,176625],{"className":176610},[638],[363,176612,176614],{"className":176613,"style":643},[642],[363,176615,176616,176619],{"style":136154},[363,176617],{"className":176618,"style":651},[650],[363,176620,176622],{"className":176621},[655,656,657,658],[363,176623,7912],{"className":176624,"style":8615},[407,408,658],[363,176626,666],{"className":176627},[665],[363,176629,176631],{"className":176630},[638],[363,176632,176634],{"className":176633,"style":8316},[642],[363,176635],{},[363,176637,1288],{"className":176638},[407],[363,176640,176642,176645],{"className":176641},[407],[363,176643,45126],{"className":176644,"style":45156},[407,408],[363,176646,176648],{"className":176647},[629],[363,176649,176651,176671],{"className":176650},[633,634],[363,176652,176654,176668],{"className":176653},[638],[363,176655,176657],{"className":176656,"style":11483},[642],[363,176658,176659,176662],{"style":136154},[363,176660],{"className":176661,"style":651},[650],[363,176663,176665],{"className":176664},[655,656,657,658],[363,176666,474],{"className":176667},[407,658],[363,176669,666],{"className":176670},[665],[363,176672,176674],{"className":176673},[638],[363,176675,176677],{"className":176676,"style":8316},[642],[363,176678],{},[363,176680,176682,176704],{"className":176681,"translate":367},[366],[363,176683,176685],{"className":176684},[371],[373,176686,176687],{"xmlns":375},[377,176688,176689,176701],{},[380,176690,176691,176693,176695],{},[383,176692,26109],{},[458,176694,471],{},[543,176696,176697,176699],{},[383,176698,45126],{},[463,176700,474],{},[387,176702,176703],{"encoding":389},"S = \\alpha_0",[363,176705,176707,176725],{"className":176706,"ariaHidden":394},[393],[363,176708,176710,176713,176716,176719,176722],{"className":176709},[398],[363,176711],{"className":176712,"style":1011},[402],[363,176714,26109],{"className":176715,"style":26233},[407,408],[363,176717],{"className":176718,"style":506},[505],[363,176720,471],{"className":176721},[510],[363,176723],{"className":176724,"style":506},[505],[363,176726,176728,176731],{"className":176727},[398],[363,176729],{"className":176730,"style":13724},[402],[363,176732,176734,176737],{"className":176733},[407],[363,176735,45126],{"className":176736,"style":45156},[407,408],[363,176738,176740],{"className":176739},[629],[363,176741,176743,176763],{"className":176742},[633,634],[363,176744,176746,176760],{"className":176745},[638],[363,176747,176749],{"className":176748,"style":11483},[642],[363,176750,176751,176754],{"style":136154},[363,176752],{"className":176753,"style":651},[650],[363,176755,176757],{"className":176756},[655,656,657,658],[363,176758,474],{"className":176759},[407,658],[363,176761,666],{"className":176762},[665],[363,176764,176766],{"className":176765},[638],[363,176767,176769],{"className":176768,"style":8316},[642],[363,176770],{},[363,176772,176774,176838],{"className":176773,"translate":367},[366],[363,176775,176777],{"className":176776},[371],[373,176778,176779],{"xmlns":375},[377,176780,176781,176835],{},[380,176782,176783,176785,176787,176789,176791,176793,176795,176797,176799,176801,176803,176805,176819,176825,176827,176829],{},[383,176784,34256],{},[458,176786,578],{"stretchy":460},[383,176788,554],{"mathvariant":570},[458,176790,461],{"stretchy":460},[383,176792,40609],{},[458,176794,5156],{},[383,176796,1121],{"mathvariant":75598},[458,176798,468],{"stretchy":460},[458,176800,605],{"stretchy":460},[458,176802,471],{},[458,176804,5161],{},[7901,176806,176807,176809,176817],{},[458,176808,13811],{},[380,176810,176811,176813,176815],{},[383,176812,7912],{},[458,176814,471],{},[463,176816,465],{},[383,176818,11044],{},[543,176820,176821,176823],{},[383,176822,11],{},[383,176824,7912],{},[383,176826,4448],{},[458,176828,4451],{},[543,176830,176831,176833],{},[383,176832,11],{},[383,176834,7912],{},[387,176836,176837],{"encoding":389},"H[\\mathbb{P}(y \\mid \\mathbf{x})] = -\\sum_{k=1}^K p_k \\log p_k",[363,176839,176841,176871,176892],{"className":176840,"ariaHidden":394},[393],[363,176842,176844,176847,176850,176853,176856,176859,176862,176865,176868],{"className":176843},[398],[363,176845],{"className":176846,"style":487},[402],[363,176848,34256],{"className":176849,"style":34410},[407,408],[363,176851,578],{"className":176852},[494],[363,176854,554],{"className":176855},[407,714],[363,176857,461],{"className":176858},[494],[363,176860,40609],{"className":176861,"style":1328},[407,408],[363,176863],{"className":176864,"style":506},[505],[363,176866,5156],{"className":176867},[510],[363,176869],{"className":176870,"style":506},[505],[363,176872,176874,176877,176880,176883,176886,176889],{"className":176873},[398],[363,176875],{"className":176876,"style":487},[402],[363,176878,1121],{"className":176879},[407,75943],[363,176881,20762],{"className":176882},[501],[363,176884],{"className":176885,"style":506},[505],[363,176887,471],{"className":176888},[510],[363,176890],{"className":176891,"style":506},[505],[363,176893,176895,176898,176901,176904,176964,176967,177007,177010,177015,177018],{"className":176894},[398],[363,176896],{"className":176897,"style":136418},[402],[363,176899,5161],{"className":176900},[407],[363,176902],{"className":176903,"style":754},[505],[363,176905,176907,176910],{"className":176906},[1832],[363,176908,13811],{"className":176909,"style":40032},[1832,1833,5964],[363,176911,176913],{"className":176912},[629],[363,176914,176916,176956],{"className":176915},[633,634],[363,176917,176919,176953],{"className":176918},[638],[363,176920,176922,176942],{"className":176921,"style":133537},[642],[363,176923,176924,176927],{"style":40048},[363,176925],{"className":176926,"style":651},[650],[363,176928,176930],{"className":176929},[655,656,657,658],[363,176931,176933,176936,176939],{"className":176932},[407,658],[363,176934,7912],{"className":176935,"style":8615},[407,408,658],[363,176937,471],{"className":176938},[510,658],[363,176940,465],{"className":176941},[407,658],[363,176943,176944,176947],{"style":133560},[363,176945],{"className":176946,"style":651},[650],[363,176948,176950],{"className":176949},[655,656,657,658],[363,176951,11044],{"className":176952,"style":11132},[407,408,658],[363,176954,666],{"className":176955},[665],[363,176957,176959],{"className":176958},[638],[363,176960,176962],{"className":176961,"style":40067},[642],[363,176963],{},[363,176965],{"className":176966,"style":754},[505],[363,176968,176970,176973],{"className":176969},[407],[363,176971,11],{"className":176972},[407,408],[363,176974,176976],{"className":176975},[629],[363,176977,176979,176999],{"className":176978},[633,634],[363,176980,176982,176996],{"className":176981},[638],[363,176983,176985],{"className":176984,"style":643},[642],[363,176986,176987,176990],{"style":730},[363,176988],{"className":176989,"style":651},[650],[363,176991,176993],{"className":176992},[655,656,657,658],[363,176994,7912],{"className":176995,"style":8615},[407,408,658],[363,176997,666],{"className":176998},[665],[363,177000,177002],{"className":177001},[638],[363,177003,177005],{"className":177004,"style":8316},[642],[363,177006],{},[363,177008],{"className":177009,"style":754},[505],[363,177011,4502,177013],{"className":177012},[1832],[363,177014,4506],{"style":4505},[363,177016],{"className":177017,"style":754},[505],[363,177019,177021,177024],{"className":177020},[407],[363,177022,11],{"className":177023},[407,408],[363,177025,177027],{"className":177026},[629],[363,177028,177030,177050],{"className":177029},[633,634],[363,177031,177033,177047],{"className":177032},[638],[363,177034,177036],{"className":177035,"style":643},[642],[363,177037,177038,177041],{"style":730},[363,177039],{"className":177040,"style":651},[650],[363,177042,177044],{"className":177043},[655,656,657,658],[363,177045,7912],{"className":177046,"style":8615},[407,408,658],[363,177048,666],{"className":177049},[665],[363,177051,177053],{"className":177052},[638],[363,177054,177056],{"className":177055,"style":8316},[642],[363,177057],{},[363,177059,177061,177145],{"className":177060,"translate":367},[366],[363,177062,177064],{"className":177063},[371],[373,177065,177066],{"xmlns":375},[377,177067,177068,177142],{},[380,177069,177070,177076,177078,177080,177082,177084,177086,177088,177090,177092,177094,177096,177098,177100,177102,177104,177118,177124,177126,177128,177130,177136,177138,177140],{},[543,177071,177072,177074],{},[383,177073,571],{"mathvariant":570},[383,177075,157459],{"mathvariant":136055},[458,177077,578],{"stretchy":460},[383,177079,34256],{},[458,177081,461],{"stretchy":460},[383,177083,157459],{"mathvariant":136055},[458,177085,468],{"stretchy":460},[458,177087,605],{"stretchy":460},[458,177089,471],{},[383,177091,155482],{},[458,177093,461],{"stretchy":460},[383,177095,26109],{},[458,177097,2818],{},[463,177099,465],{},[458,177101,468],{"stretchy":460},[458,177103,5161],{},[7901,177105,177106,177108,177116],{},[458,177107,13811],{},[3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= \\psi(S + 1) - \\sum_{k=1}^K p_k \\,\\psi(\\alpha_k + 1)",[363,177146,177148,177233,177257,177278,177445],{"className":177147,"ariaHidden":394},[393],[363,177149,177151,177154,177203,177206,177209,177212,177221,177224,177227,177230],{"className":177150},[398],[363,177152],{"className":177153,"style":487},[402],[363,177155,177157,177160],{"className":177156},[407],[363,177158,571],{"className":177159},[407,714],[363,177161,177163],{"className":177162},[629],[363,177164,177166,177195],{"className":177165},[633,634],[363,177167,177169,177192],{"className":177168},[638],[363,177170,177172],{"className":177171,"style":153771},[642],[363,177173,177174,177177],{"style":730},[363,177175],{"className":177176,"style":651},[650],[363,177178,177180],{"className":177179},[655,656,657,658],[363,177181,177183],{"className":177182},[407,658],[363,177184,177186],{"className":177185},[407,658],[363,177187,177189],{"className":177188},[407,658],[363,177190,157459],{"className":177191,"style":157493},[407,136115,658],[363,177193,666],{"className":177194},[665],[363,177196,177198],{"className":177197},[638],[363,177199,177201],{"className":177200,"style":8316},[642],[363,177202],{},[363,177204,578],{"className":177205},[494],[363,177207,34256],{"className":177208,"style":34410},[407,408],[363,177210,461],{"className":177211},[494],[363,177213,177215],{"className":177214},[407],[363,177216,177218],{"className":177217},[407],[363,177219,157459],{"className":177220,"style":157493},[407,136115],[363,177222,20762],{"className":177223},[501],[363,177225],{"className":177226,"style":506},[505],[363,177228,471],{"className":177229},[510],[363,177231],{"className":177232,"style":506},[505],[363,177234,177236,177239,177242,177245,177248,177251,177254],{"className":177235},[398],[363,177237],{"className":177238,"style":487},[402],[363,177240,155482],{"className":177241,"style":1328},[407,408],[363,177243,461],{"className":177244},[494],[363,177246,26109],{"className":177247,"style":26233},[407,408],[363,177249],{"className":177250,"style":2935},[505],[363,177252,2818],{"className":177253},[2939],[363,177255],{"className":177256,"style":2935},[505],[363,177258,177260,177263,177266,177269,177272,177275],{"className":177259},[398],[363,177261],{"className":177262,"style":487},[402],[363,177264,465],{"className":177265},[407],[363,177267,468],{"className":177268},[501],[363,177270],{"className":177271,"style":2935},[505],[363,177273,5161],{"className":177274},[2939],[363,177276],{"className":177277,"style":2935},[505],[363,177279,177281,177284,177344,177347,177387,177390,177393,177396,177436,177439,177442],{"className":177280},[398],[363,177282],{"className":177283,"style":136418},[402],[363,177285,177287,177290],{"className":177286},[1832],[363,177288,13811],{"className":177289,"style":40032},[1832,1833,5964],[363,177291,177293],{"className":177292},[629],[363,177294,177296,177336],{"className":177295},[633,634],[363,177297,177299,177333],{"className":177298},[638],[363,177300,177302,177322],{"className":177301,"style":133537},[642],[363,177303,177304,177307],{"style":40048},[363,177305],{"className":177306,"style":651},[650],[363,177308,177310],{"className":177309},[655,656,657,658],[363,177311,177313,177316,177319],{"className":177312},[407,658],[363,177314,7912],{"className":177315,"style":8615},[407,408,658],[363,177317,471],{"className":177318},[510,658],[363,177320,465],{"className":177321},[407,658],[363,177323,177324,177327],{"style":133560},[363,177325],{"className":177326,"style":651},[650],[363,177328,177330],{"className":177329},[655,656,657,658],[363,177331,11044],{"className":177332,"style":11132},[407,408,658],[363,177334,666],{"className":177335},[665],[363,177337,177339],{"className":177338},[638],[363,177340,177342],{"className":177341,"style":40067},[642],[363,177343],{},[363,177345],{"className":177346,"style":754},[505],[363,177348,177350,177353],{"className":177349},[407],[363,177351,11],{"className":177352},[407,408],[363,177354,177356],{"className":177355},[629],[363,177357,177359,177379],{"className":177358},[633,634],[363,177360,177362,177376],{"className":177361},[638],[363,177363,177365],{"className":177364,"style":643},[642],[363,177366,177367,177370],{"style":730},[363,177368],{"className":177369,"style":651},[650],[363,177371,177373],{"className":177372},[655,656,657,658],[363,177374,7912],{"className":177375,"style":8615},[407,408,658],[363,177377,666],{"className":177378},[665],[363,177380,177382],{"className":177381},[638],[363,177383,177385],{"className":177384,"style":8316},[642],[363,177386],{},[363,177388],{"className":177389,"style":754},[505],[363,177391,155482],{"className":177392,"style":1328},[407,408],[363,177394,461],{"className":177395},[494],[363,177397,177399,177402],{"className":177398},[407],[363,177400,45126],{"className":177401,"style":45156},[407,408],[363,177403,177405],{"className":177404},[629],[363,177406,177408,177428],{"className":177407},[633,634],[363,177409,177411,177425],{"className":177410},[638],[363,177412,177414],{"className":17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\\boldsymbol{\\pi}) = H[\\mathbb{P}(y \\mid \\mathbf{x})] - \\mathbb{E}_{\\boldsymbol{\\pi}}[H(\\boldsymbol{\\pi})] = \\sum_{k=1}^K p_k\\big(\\psi(\\alpha_k + 1) - \\log p_k\\big) - \\psi(S + 1)",[363,177593,177595,177637,177667,177688,177773,177944,177965,178034,178058],{"className":177594,"ariaHidden":394},[393],[363,177596,177598,177601,177604,177607,177610,177613,177616,177625,177628,177631,177634],{"className":177597},[398],[363,177599],{"className":177600,"style":487},[402],[363,177602,34240],{"className":177603,"style":9235},[407,408],[363,177605,461],{"className":177606},[494],[363,177608,40609],{"className":177609,"style":1328},[407,408],[363,177611,34247],{"className":177612},[920],[363,177614],{"className":177615,"style":754},[505],[363,177617,177619],{"className":177618},[407],[363,177620,177622],{"className":177621},[407],[363,177623,157459],{"className":177624,"style":157493},[407,136115],[363,177626,468],{"className":177627},[501],[363,177629],{"className":177630,"style":506},[505],[363,177632,471],{"className":177633},[510],[363,177635],{"className":177636,"style":506},[505],[363,177638,177640,177643,177646,177649,177652,177655,177658,177661,177664],{"className":177639},[398],[363,177641],{"className":177642,"style":487},[402],[363,177644,34256],{"className":177645,"style":34410},[407,408],[363,177647,578],{"className":177648},[494],[363,177650,554],{"className":177651},[407,714],[363,177653,461],{"className":177654},[494],[363,177656,40609],{"className":177657,"style":1328},[407,408],[363,177659],{"className":177660,"style":506},[505],[363,177662,5156],{"className":177663},[510],[363,177665],{"className":177666,"style":506},[505],[363,177668,177670,177673,177676,177679,177682,177685],{"className":177669},[398],[363,177671],{"className":177672,"style":487},[402],[363,177674,1121],{"className":177675},[407,75943],[363,177677,20762],{"className":177678},[501],[363,177680],{"className":177681,"style":2935},[505],[363,177683,5161],{"className":177684},[2939],[363,177686],{"className":177687,"style":2935},[505],[363,177689,177691,177694,177743,177746,177749,177752,177761,177764,177767,177770],{"className":177690},[398],[363,177692],{"className":177693,"style":487},[402],[363,177695,177697,177700],{"className":177696},[407],[363,177698,571],{"className":177699},[407,714],[363,177701,177703],{"className":177702},[629],[363,177704,177706,177735],{"className":177705},[633,634],[363,177707,177709,177732],{"className":177708},[638],[363,177710,177712],{"className":177711,"style":153771},[642],[363,177713,177714,177717],{"style":730},[363,177715],{"className":177716,"style":651},[650],[363,177718,177720],{"className":177719},[655,656,657,658],[363,177721,177723],{"className":177722},[407,658],[363,177724,177726],{"className":177725},[407,658],[363,177727,177729],{"className":177728},[407,658],[363,177730,157459],{"className":177731,"style":157493},[407,136115,658],[363,177733,666],{"className":177734},[665],[363,177736,177738],{"className":177737},[638],[363,177739,177741],{"className":177740,"style":8316},[642],[363,177742],{},[363,177744,578],{"className":177745},[494],[363,177747,34256],{"className":177748,"style":34410},[407,408],[363,177750,461],{"className":177751},[494],[363,177753,177755],{"className":177754},[407],[363,177756,177758],{"className":177757},[407],[363,177759,157459],{"className":177760,"style":157493},[407,136115],[363,177762,20762],{"className":177763},[501],[363,177765],{"className":177766,"style":506},[505],[363,177768,471],{"className":177769},[510],[363,177771],{"className":177772,"style":506},[505],[363,177774,177776,177780,177840,177843,177883,177889,177892,177895,177935,177938,177941],{"className":177775},[398],[363,177777],{"className":177778,"style":177779},[402],"height:1.3312em;vertical-align:-0.35em;",[363,177781,177783,177786],{"className":177782},[1832],[363,177784,13811],{"className":177785,"style":40032},[1832,1833,5964],[363,177787,177789],{"className":177788},[629],[363,177790,177792,177832],{"className":177791},[633,634],[363,177793,177795,177829],{"className":177794},[638],[363,177796,177798,177818],{"className":177797,"style":133537},[642],[363,177799,177800,177803],{"style":40048},[363,177801],{"className":177802,"style":651},[650],[363,177804,177806],{"className":177805},[655,656,657,658],[363,177807,177809,177812,177815],{"className":177808},[407,658],[363,177810,7912],{"className":177811,"style":8615},[407,408,658],[363,177813,471],{"className":177814},[510,658],[363,177816,465],{"className":177817},[407,658],[363,177819,177820,177823],{"style":133560},[363,177821],{"className":177822,"style":651},[650],[363,177824,177826],{"className":177825},[655,656,657,658],[363,177827,11044],{"className":177828,"style":11132},[407,408,658],[363,177830,666],{"className":177831},[665],[363,177833,177835],{"className":177834},[638],[363,177836,177838],{"className":177837,"style":40067},[642],[363,177839],{},[363,177841],{"className":177842,"style":754},[505],[363,177844,177846,177849],{"className":177845},[407],[363,177847,11],{"className":177848},[407,408],[363,177850,177852],{"className":177851},[629],[363,177853,177855,177875],{"className":177854},[633,63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关键在于 ",[363,178076,178078,178124],{"className":178077,"translate":367},[366],[363,178079,178081],{"className":178080},[371],[373,178082,178083],{"xmlns":375},[377,178084,178085,178121],{},[380,178086,178087,178089,178091,178093,178095,178097,178099,178105,178111,178113,178115],{},[383,178088,34256],{},[458,178090,461],{"stretchy":460},[383,178092,157459],{"mathvariant":136055},[458,178094,468],{"stretchy":460},[458,178096,471],{},[458,178098,5161],{},[543,178100,178101,178103],{},[458,178102,13811],{},[383,178104,7912],{},[543,178106,178107,178109],{},[383,178108,157459],{},[383,178110,7912],{},[383,178112,4448],{},[458,178114,4451],{},[543,178116,178117,178119],{},[383,178118,157459],{},[383,178120,7912],{},[387,178122,178123],{"encoding":389},"H(\\boldsymbol{\\pi}) = -\\sum_k \\pi_k \\log \\pi_k",[363,178125,178127,178160],{"className":178126,"ariaHidden":394},[393],[363,178128,178130,178133,178136,178139,178148,178151,178154,178157],{"className":178129},[398],[363,178131],{"className":178132,"style":487},[402],[363,178134,34256],{"className":178135,"style":34410},[407,408],[363,178137,461],{"className":178138},[494],[363,178140,178142],{"className":178141},[407],[363,178143,178145],{"className":178144},[407],[363,178146,157459],{"className":178147,"style":157493},[407,136115],[363,178149,468],{"className":178150},[501],[363,178152],{"className":178153,"style":506},[505],[363,178155,471],{"className":178156},[510],[363,178158],{"className":178159,"style":506},[505],[363,178161,178163,178166,178169,178172,178212,178215,178255,178258,178263,178266],{"className":178162},[398],[363,178164],{"className":178165,"style":138953},[402],[363,178167,5161],{"className":178168},[407],[363,178170],{"className":178171,"style":754},[505],[363,178173,178175,178178],{"className":178174},[1832],[363,178176,13811],{"className":178177,"style":40032},[1832,1833,5964],[363,178179,178181],{"className":178180},[629],[363,178182,178184,178204],{"className":178183},[633,634],[363,178185,178187,178201],{"className":178186},[638],[363,178188,178190],{"className":178189,"style":67394},[642],[363,178191,178192,178195],{"style":40048},[363,178193],{"className":178194,"style":651},[650],[363,178196,178198],{"className":178197},[655,656,657,658],[363,178199,7912],{"className":178200,"style":8615},[407,408,658],[363,178202,666],{"className":178203},[665],[363,178205,178207],{"className":178206},[638],[363,178208,178210],{"className":178209,"style":40067},[642],[363,178211],{},[363,178213],{"className":178214,"style":754},[505],[363,178216,178218,178221],{"className":178217},[407],[363,178219,157459],{"className":178220,"style":1328},[407,408],[363,178222,178224],{"className":178223},[629],[363,178225,178227,178247],{"className":178226},[633,634],[363,178228,178230,178244],{"className":178229},[638],[363,178231,178233],{"className":178232,"style":643},[642],[363,178234,178235,178238],{"style":19215},[363,178236],{"className":178237,"style":651},[650],[363,178239,178241],{"className":178240},[655,656,657,658],[363,178242,7912],{"className":178243,"style":8615},[407,408,658],[363,178245,666],{"className":178246},[665],[363,178248,178250],{"className":178249},[638],[363,178251,178253],{"className":178252,"style":8316},[642],[363,178254],{},[363,178256],{"className":178257,"style":754},[505],[363,178259,4502,178261],{"className":178260},[1832],[363,178262,4506],{"style":4505},[363,178264],{"className":178265,"style":754},[505],[363,178267,178269,178272],{"className":178268},[407],[363,178270,157459],{"className":178271,"style":1328},[407,408],[363,178273,178275],{"className":178274},[629],[363,178276,178278,178298],{"className":178277},[633,634],[363,178279,178281,178295],{"className":178280},[638],[363,178282,178284],{"className":178283,"style":643},[642],[363,178285,178286,178289],{"style":19215},[363,178287],{"className":178288,"style":651},[650],[363,178290,178292],{"className":178291},[655,656,657,658],[363,178293,7912],{"className":178294,"style":8615},[407,408,658],[363,178296,666],{"className":178297},[665],[363,178299,178301],{"className":178300},[638],[363,178302,178304],{"className":178303,"style":8316},[642],[363,178305],{},[363,178307,178309,178367],{"className":178308,"translate":367},[366],[363,178310,178312],{"className":178311},[371],[373,178313,178314],{"xmlns":375},[377,178315,178316,178364],{},[380,178317,178318,178320,178322,178324,178326,178328,178330,178332,178334,178336,178342,178344,178346,178352,178354,178356,178362],{},[383,178319,571],{"mathvariant":570},[458,178321,578],{"stretchy":460},[383,178323,34256],{},[458,178325,461],{"stretchy":460},[383,178327,157459],{"mathvariant":136055},[458,178329,468],{"stretchy":460},[458,178331,605],{"stretchy":460},[458,178333,471],{},[458,178335,5161],{},[543,178337,178338,178340],{},[458,178339,13811],{},[383,178341,7912],{},[383,178343,571],{"mathvariant":570},[458,178345,578],{"stretchy":460},[543,178347,178348,178350],{},[383,178349,157459],{},[383,178351,7912],{},[383,178353,4448],{},[458,178355,4451],{},[543,178357,178358,178360],{},[383,178359,157459],{},[383,178361,7912],{},[458,178363,605],{"stretchy":460},[387,178365,178366],{"encoding":389},"\\mathbb{E}[H(\\boldsymbol{\\pi})] = -\\sum_k \\mathbb{E}[\\pi_k \\log \\pi_k]",[363,178368,178370,178409],{"className":178369,"ariaHidden":394},[393],[363,178371,178373,178376,178379,178382,178385,178388,178397,178400,178403,178406],{"className":178372},[398],[363,178374],{"className":178375,"style":487},[402],[363,178377,571],{"className":178378},[407,714],[363,178380,578],{"className":178381},[494],[363,178383,34256],{"className":178384,"style":34410},[407,408],[363,178386,461],{"className":178387},[494],[363,178389,178391],{"className":178390},[407],[363,178392,178394],{"className":178393},[407],[363,178395,157459],{"className":178396,"style":157493},[407,136115],[363,178398,20762],{"className":178399},[501],[363,178401],{"className":178402,"style":506},[505],[363,178404,471],{"className":178405},[510],[363,178407],{"className":178408,"style":506},[505],[363,178410,178412,178415,178418,178421,178461,178464,178467,178470,178510,178513,178518,178521,178561],{"className":178411},[398],[363,178413],{"className":178414,"style":138953},[402],[363,178416,5161],{"className":178417},[407],[363,178419],{"className":178420,"style":754},[505],[363,178422,178424,178427],{"className":178423},[1832],[363,178425,13811],{"className":178426,"style":40032},[1832,1833,5964],[363,178428,178430],{"className":178429},[629],[363,178431,178433,178453],{"className":178432},[633,634],[363,178434,178436,178450],{"className":178435},[638],[363,178437,178439],{"className":178438,"style":67394},[642],[363,178440,178441,178444],{"style":40048},[363,178442],{"className":178443,"style":651},[650],[363,178445,178447],{"className":178446},[655,656,657,658],[363,178448,7912],{"className":178449,"style":8615},[407,408,658],[363,178451,666],{"className":178452},[665],[363,178454,178456],{"className":178455},[638],[363,178457,178459],{"className":178458,"style":40067},[642],[363,178460],{},[363,178462],{"className":178463,"style":754},[505],[363,178465,571],{"className":178466},[407,714],[363,178468,578],{"className":178469},[494],[363,178471,178473,178476],{"className":178472},[407],[363,178474,157459],{"className":178475,"style":1328},[407,408],[363,178477,178479],{"className":178478},[629],[363,178480,178482,178502],{"className":178481},[633,634],[363,178483,178485,178499],{"className":178484},[638],[363,178486,178488],{"className":178487,"style":643},[642],[363,178489,178490,178493],{"style":19215},[363,178491],{"className":178492,"style":651},[650],[363,178494,178496],{"className":178495},[655,656,657,658],[363,178497,7912],{"className":178498,"style":8615},[407,408,658],[363,178500,666],{"className":178501},[665],[363,178503,178505],{"className":178504},[638],[363,178506,178508],{"className":178507,"style":8316},[642],[363,178509],{},[363,178511],{"className":178512,"style":754},[505],[363,178514,4502,178516],{"className":178515},[1832],[363,178517,4506],{"style":4505},[363,178519],{"className":178520,"style":754},[505],[363,178522,178524,178527],{"className":178523},[407],[363,178525,157459],{"className":178526,"style":1328},[407,408],[363,178528,178530],{"className":178529},[629],[363,178531,178533,178553],{"className":178532},[633,634],[363,178534,178536,178550],{"className":178535},[638],[363,178537,178539],{"className":178538,"style":643},[642],[363,178540,178541,178544],{"style":19215},[363,178542],{"className":178543,"style":651},[650],[363,178545,178547],{"className":178546},[655,656,657,658],[363,178548,7912],{"className":178549,"style":8615},[407,408,658],[363,178551,666],{"className":178552},[665],[363,178554,178556],{"className":178555},[638],[363,178557,178559],{"className":178558,"style":8316},[642],[363,178560],{},[363,178562,605],{"className":178563},[501],"。由 1.4 节的边缘 Beta 性，",[363,178566,178568,178614],{"className":178567,"translate":367},[366],[363,178569,178571],{"className":178570},[371],[373,178572,178573],{"xmlns":375},[377,178574,178575,178611],{},[380,178576,178577,178583,178585,178587,178589,178591,178597,178599,178601,178603,178609],{},[543,178578,178579,178581],{},[383,178580,157459],{},[383,178582,7912],{},[458,178584,9217],{},[383,178586,145808],{"mathvariant":557},[458,178588,4451],{},[458,178590,461],{"stretchy":460},[543,178592,178593,178595],{},[383,178594,45126],{},[383,178596,7912],{},[458,178598,898],{"separator":394},[383,178600,26109],{},[458,178602,5161],{},[543,178604,178605,178607],{},[383,178606,45126],{},[383,178608,7912],{},[458,178610,468],{"stretchy":460},[387,178612,178613],{"encoding":389},"\\pi_k \\sim \\operatorname{Beta}(\\alpha_k, S - \\alpha_k)",[363,178615,178617,178672,178745],{"className":178616,"ariaHidden":394},[393],[363,178618,178620,178623,178663,178666,178669],{"className":178619},[398],[363,178621],{"className":178622,"style":13724},[402],[363,178624,178626,178629],{"className":178625},[407],[363,178627,157459],{"className":178628,"style":1328},[407,408],[363,178630,178632],{"className":178631},[629],[363,178633,178635,178655],{"className":178634},[633,634],[363,178636,178638,178652],{"className":178637},[638],[363,178639,178641],{"className":178640,"style":643},[642],[363,178642,178643,178646],{"style":19215},[363,178644],{"className":178645,"style":651},[650],[363,178647,178649],{"className":178648},[655,656,657,658],[363,178650,7912],{"className":178651,"style":8615},[407,408,658],[363,178653,666],{"className":178654},[665],[363,178656,178658],{"className":178657},[638],[363,178659,178661],{"className":178660,"style":8316},[642],[363,178662],{},[363,178664],{"className":178665,"style":506},[505],[363,178667,9217],{"className":178668},[510],[363,178670],{"className":178671,"style":506},[505],[363,178673,178675,178678,178684,178687,178727,178730,178733,178736,178739,178742],{"className":178674},[398],[363,178676],{"className":178677,"style":487},[402],[363,178679,178681],{"className":178680},[1832],[363,178682,145808],{"className":178683},[407,33034],[363,178685,461],{"className":178686},[494],[363,178688,178690,178693],{"className":178689},[407],[363,178691,45126],{"className":178692,"style":45156},[407,408],[363,178694,178696],{"className":178695},[629],[363,178697,178699,178719],{"className":178698},[633,634],[363,178700,178702,178716],{"className":178701},[638],[363,178703,178705],{"className":178704,"style":643},[642],[363,178706,178707,178710],{"style":136154},[363,178708],{"className":178709,"style":651},[650],[363,178711,178713],{"className":178712},[655,656,657,658],[363,178714,7912],{"className":178715,"style":8615},[407,408,658],[363,178717,666],{"className":178718},[665],[363,178720,178722],{"className":178721},[638],[363,178723,178725],{"className":178724,"style":8316},[642],[363,178726],{},[363,178728,898],{"className":178729},[920],[363,178731],{"className":178732,"style":754},[505],[363,178734,26109],{"className":178735,"style":26233},[407,408],[363,178737],{"className":178738,"style":2935},[505],[363,178740,5161],{"className":178741},[2939],[363,178743],{"className":178744,"style":2935},[505],[363,178746,178748,178751,178791],{"className":178747},[398],[363,178749],{"className":178750,"style":487},[402],[363,178752,178754,178757],{"className":178753},[407],[363,178755,45126],{"className":178756,"style":45156},[407,408],[363,178758,178760],{"className":178759},[629],[363,178761,178763,178783],{"className":178762},[633,634],[363,178764,178766,178780],{"className":178765},[638],[363,178767,178769],{"className":178768,"style":643},[642],[363,178770,178771,178774],{"style":136154},[363,178772],{"className":178773,"style":651},[650],[363,178775,178777],{"className":178776},[655,656,657,658],[363,178778,7912],{"className":178779,"style":8615},[407,408,658],[363,178781,666],{"className":178782},[665],[363,178784,178786],{"className":178785},[638],[363,178787,178789],{"className":178788,"style":8316},[642],[363,178790],{},[363,178792,468],{"className":178793},[501],"，而对该 Beta 分布有恒等式 ",[363,178796,178798,178872],{"className":178797,"translate":367},[366],[363,178799,178801],{"className":178800},[371],[373,178802,178803],{"xmlns":375},[377,178804,178805,178869],{},[380,178806,178807,178809,178811,178813,178815,178817,178819,178821,178823,178835,178837,178839,178841,178843,178845,178847,178849,178851,178853,178855,178857,178859,178861,178863,178865,178867],{},[383,178808,571],{"mathvariant":570},[458,178810,578],{"stretchy":460},[383,178812,945],{},[383,178814,4448],{},[458,178816,4451],{},[383,178818,945],{},[458,178820,605],{"stretchy":460},[458,178822,471],{},[586,178824,178825,178827],{},[383,178826,45552],{},[380,178828,178829,178831,178833],{},[383,178830,45552],{},[458,178832,2818],{},[383,178834,83214],{},[458,178836,461],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[383,178838,155482],{},[458,178840,461],{"stretchy":460},[383,178842,45552],{},[458,178844,2818],{},[463,178846,465],{},[458,178848,468],{"stretchy":460},[458,178850,5161],{},[383,178852,155482],{},[458,178854,461],{"stretchy":460},[383,178856,45552],{},[458,178858,2818],{},[383,178860,83214],{},[458,178862,2818],{},[463,178864,465],{},[458,178866,468],{"stretchy":460},[458,178868,468],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[387,178870,178871],{"encoding":389},"\\mathbb{E}[X \\log X] = \\frac{a}{a+b}\\big(\\psi(a+1) - \\psi(a+b+1)\\big)",[363,178873,178875,178916,179020,179041,179065,179083],{"className":178874,"ariaHidden":394},[393],[363,178876,178878,178881,178884,178887,178890,178893,178898,178901,178904,178907,178910,178913],{"className":178877},[398],[363,178879],{"className":178880,"style":487},[402],[363,178882,571],{"className":178883},[407,714],[363,178885,578],{"className":178886},[494],[363,178888,945],{"className":178889,"style":9235},[407,408],[363,178891],{"className":178892,"style":754},[505],[363,178894,4502,178896],{"className":178895},[1832],[363,178897,4506],{"style":4505},[363,178899],{"className":178900,"style":754},[505],[363,178902,945],{"className":178903,"style":9235},[407,408],[363,178905,605],{"className":178906},[501],[363,178908],{"className":178909,"style":506},[505],[363,178911,471],{"className":178912},[510],[363,178914],{"className":178915,"style":506},[505],[363,178917,178919,178922,178996,179002,179005,179008,179011,179014,179017],{"className":178918},[398],[363,178920],{"className":178921,"style":111804},[402],[363,178923,178925,178928,178993],{"className":178924},[407],[363,178926],{"className":178927},[494,789],[363,178929,178931],{"className":178930},[586],[363,178932,178934,178985],{"className":178933},[633,634],[363,178935,178937,178982],{"className":178936},[638],[363,178938,178940,178960,178968],{"className":178939,"style":86079},[642],[363,178941,178942,178945],{"style":2875},[363,178943],{"className":178944,"style":809},[650],[363,178946,178948],{"className":178947},[655,656,657,658],[363,178949,178951,178954,178957],{"className":178950},[407,658],[363,178952,45552],{"className":178953},[407,408,658],[363,178955,2818],{"className":178956},[2939,658],[363,178958,83214],{"className":178959},[407,408,658],[363,178961,178962,178965],{"style":821},[363,178963],{"className":178964,"style":809},[650],[363,178966],{"className":178967,"style":829},[828],[363,178969,178970,178973],{"style":2898},[363,178971],{"className":178972,"style":809},[650],[363,178974,178976],{"className":178975},[655,656,657,658],[363,178977,178979],{"className":178978},[407,658],[363,178980,45552],{"className":178981},[407,408,658],[363,178983,666],{"className":178984},[665],[363,178986,178988],{"className":178987},[638],[363,178989,178991],{"className":178990,"style":16779},[642],[363,178992],{},[363,178994],{"className":178995},[501,789],[363,178997,178999],{"className":178998},[407],[363,179000,461],{"className":179001},[767,10022],[363,179003,155482],{"className":179004,"style":1328},[407,408],[363,179006,461],{"className":179007},[494],[363,179009,45552],{"className":179010},[407,408],[363,179012],{"className":179013,"style":2935},[505],[363,179015,2818],{"className":179016},[2939],[363,179018],{"className":179019,"style":2935},[505],[363,179021,179023,179026,179029,179032,179035,179038],{"className":179022},[398],[363,179024],{"className":179025,"style":487},[402],[363,179027,465],{"className":179028},[407],[363,179030,468],{"className":179031},[501],[363,179033],{"className":179034,"style":2935},[505],[363,179036,5161],{"className":179037},[2939],[363,179039],{"className":179040,"style":2935},[505],[363,179042,179044,179047,179050,179053,179056,179059,179062],{"className":179043},[398],[363,179045],{"className":179046,"style":487},[402],[363,179048,155482],{"className":179049,"style":1328},[407,408],[363,179051,461],{"className":179052},[494],[363,179054,45552],{"className":179055},[407,408],[363,179057],{"className":179058,"style":2935},[505],[363,179060,2818],{"className":179061},[2939],[363,179063],{"className":179064,"style":2935},[505],[363,179066,179068,179071,179074,179077,179080],{"className":179067},[398],[363,179069],{"className":179070,"style":15936},[402],[363,179072,83214],{"className":179073},[407,408],[363,179075],{"className":179076,"style":2935},[505],[363,179078,2818],{"className":179079},[2939],[363,179081],{"className":179082,"style":2935},[505],[363,179084,179086,179089,179092,179095],{"className":179085},[398],[363,179087],{"className":179088,"style":20935},[402],[363,179090,465],{"className":179091},[407],[363,179093,468],{"className":179094},[501],[363,179096,179098],{"className":179097},[407],[363,179099,468],{"className":179100},[767,10022],"（此处 ",[363,179103,179105,179127],{"className":179104,"translate":367},[366],[363,179106,179108],{"className":179107},[371],[373,179109,179110],{"xmlns":375},[377,179111,179112,179124],{},[380,179113,179114,179116,179118],{},[383,179115,45552],{},[458,179117,471],{},[543,179119,179120,179122],{},[383,179121,45126],{},[383,179123,7912],{},[387,179125,179126],{"encoding":389},"a = \\alpha_k",[363,179128,179130,179148],{"className":179129,"ariaHidden":394},[393],[363,179131,179133,179136,179139,179142,179145],{"className":179132},[398],[363,179134],{"className":179135,"style":2693},[402],[363,179137,45552],{"className":179138},[407,408],[363,179140],{"className":179141,"style":506},[505],[363,179143,471],{"className":179144},[510],[363,179146],{"className":179147,"style":506},[505],[363,179149,179151,179154],{"className":179150},[398],[363,179152],{"className":179153,"style":13724},[402],[363,179155,179157,179160],{"className":179156},[407],[363,179158,45126],{"className":179159,"style":45156},[407,408],[363,179161,179163],{"className":179162},[629],[363,179164,179166,179186],{"className":179165},[633,634],[363,179167,179169,179183],{"className":179168},[638],[363,179170,179172],{"className":179171,"style":643},[642],[363,179173,179174,179177],{"style":136154},[363,179175],{"className":179176,"style":651},[650],[363,179178,179180],{"className":179179},[655,656,657,658],[363,179181,7912],{"className":179182,"style":8615},[407,408,658],[363,179184,666],{"className":179185},[665],[363,179187,179189],{"className":179188},[638],[363,179190,179192],{"className":179191,"style":8316},[642],[363,179193],{},[363,179195,179197,179219],{"className":179196,"translate":367},[366],[363,179198,179200],{"className":179199},[371],[373,179201,179202],{"xmlns":375},[377,179203,179204,179216],{},[380,179205,179206,179208,179210,179212,179214],{},[383,179207,45552],{},[458,179209,2818],{},[383,179211,83214],{},[458,179213,471],{},[383,179215,26109],{},[387,179217,179218],{"encoding":389},"a + b = S",[363,179220,179222,179240,179258],{"className":179221,"ariaHidden":394},[393],[363,179223,179225,179228,179231,179234,179237],{"className":179224},[398],[363,179226],{"className":179227,"style":4599},[402],[363,179229,45552],{"className":179230},[407,408],[363,179232],{"className":179233,"style":2935},[505],[363,179235,2818],{"className":179236},[2939],[363,179238],{"className":179239,"style":2935},[505],[363,179241,179243,179246,179249,179252,179255],{"className":179242},[398],[363,179244],{"className":179245,"style":12947},[402],[363,179247,83214],{"className":179248},[407,408],[363,179250],{"className":179251,"style":506},[505],[363,179253,471],{"className":179254},[510],[363,179256],{"className":179257,"style":506},[505],[363,179259,179261,179264],{"className":179260},[398],[363,179262],{"className":179263,"style":1011},[402],[363,179265,26109],{"className":179266,"style":26233},[407,408],"）。代入 ",[363,179269,179271,179305],{"className":179270,"translate":367},[366],[363,179272,179274],{"className":179273},[371],[373,179275,179276],{"xmlns":375},[377,179277,179278,179302],{},[380,179279,179280,179282,179284,179286,179288,179290,179292,179294,179296],{},[383,179281,45552],{},[383,179283,1288],{"mathvariant":557},[458,179285,461],{"stretchy":460},[383,179287,45552],{},[458,179289,2818],{},[383,179291,83214],{},[458,179293,468],{"stretchy":460},[458,179295,471],{},[543,179297,179298,179300],{},[383,179299,11],{},[383,179301,7912],{},[387,179303,179304],{"encoding":389},"a\u002F(a+b) = p_k",[363,179306,179308,179335,179356],{"className":179307,"ariaHidden":394},[393],[363,179309,179311,179314,179317,179320,179323,179326,179329,179332],{"className":179310},[398],[363,179312],{"className":179313,"style":487},[402],[363,179315,45552],{"className":179316},[407,408],[363,179318,1288],{"className":179319},[407],[363,179321,461],{"className":179322},[494],[363,179324,45552],{"className":179325},[407,408],[363,179327],{"className":179328,"style":2935},[505],[363,179330,2818],{"className":179331},[2939],[363,179333],{"className":179334,"style":2935},[505],[363,179336,179338,179341,179344,179347,179350,179353],{"className":179337},[398],[363,179339],{"className":179340,"style":487},[402],[363,179342,83214],{"className":179343},[407,408],[363,179345,468],{"className":179346},[501],[363,179348],{"className":179349,"style":506},[505],[363,179351,471],{"className":179352},[510],[363,179354],{"className":179355,"style":506},[505],[363,179357,179359,179362],{"className":179358},[398],[363,179360],{"className":179361,"style":2624},[402],[363,179363,179365,179368],{"className":179364},[407],[363,179366,11],{"className":179367},[407,408],[363,179369,179371],{"className":179370},[629],[363,179372,179374,179394],{"className":179373},[633,634],[363,179375,179377,179391],{"className":179376},[638],[363,179378,179380],{"className":179379,"style":643},[642],[363,179381,179382,179385],{"style":730},[363,179383],{"className":179384,"style":651},[650],[363,179386,179388],{"className":179387},[655,656,657,658],[363,179389,7912],{"className":179390,"style":8615},[407,408,658],[363,179392,666],{"className":179393},[665],[363,179395,179397],{"className":179396},[638],[363,179398,179400],{"className":179399,"style":8316},[642],[363,179401],{}," 即得第二式。互信息由 ",[363,179404,179406,179450],{"className":179405,"translate":367},[366],[363,179407,179409],{"className":179408},[371],[373,179410,179411],{"xmlns":375},[377,179412,179413,179447],{},[380,179414,179415,179417,179419,179421,179423,179425,179427,179429,179431,179433,179435,179437,179439,179441,179443,179445],{},[383,179416,34240],{},[458,179418,471],{},[383,179420,34256],{},[458,179422,578],{"stretchy":460},[383,179424,571],{"mathvariant":570},[1645,179426,1647],{},[383,179428,157459],{"mathvariant":136055},[458,179430,605],{"stretchy":460},[458,179432,5161],{},[383,179434,571],{"mathvariant":570},[458,179436,578],{"stretchy":460},[383,179438,34256],{},[458,179440,461],{"stretchy":460},[383,179442,157459],{"mathvariant":136055},[458,179444,468],{"stretchy":460},[458,179446,605],{"stretchy":460},[387,179448,179449],{"encoding":389},"I = H[\\mathbb{E}\\,\\boldsymbol{\\pi}] - \\mathbb{E}[H(\\boldsymbol{\\pi})]",[363,179451,179453,179471,179510],{"className":179452,"ariaHidden":394},[393],[363,179454,179456,179459,179462,179465,179468],{"className":179455},[398],[363,179457],{"className":179458,"style":1011},[402],[363,179460,34240],{"className":179461,"style":9235},[407,408],[363,179463],{"className":179464,"style":506},[505],[363,179466,471],{"className":179467},[510],[363,179469],{"className":179470,"style":506},[505],[363,179472,179474,179477,179480,179483,179486,179489,179498,179501,179504,179507],{"className":179473},[398],[363,179475],{"className":179476,"style":487},[402],[363,179478,34256],{"className":179479,"style":34410},[407,408],[363,179481,578],{"className":179482},[494],[363,179484,571],{"className":179485},[407,714],[363,179487],{"className":179488,"style":754},[505],[363,179490,179492],{"className":179491},[407],[363,179493,179495],{"className":179494},[407],[363,179496,157459],{"className":179497,"style":157493},[407,136115],[363,179499,605],{"className":179500},[501],[363,179502],{"className":179503,"style":2935},[505],[363,179505,5161],{"className":179506},[2939],[363,179508],{"className":179509,"style":2935},[505],[363,179511,179513,179516,179519,179522,179525,179528,179537],{"className":179512},[398],[363,179514],{"className":179515,"style":487},[402],[363,179517,571],{"className":179518},[407,714],[363,179520,578],{"className":179521},[494],[363,179523,34256],{"className":179524,"style":34410},[407,408],[363,179526,461],{"className":179527},[494],[363,179529,179531],{"className":179530},[407],[363,179532,179534],{"className":179533},[407],[363,179535,157459],{"className":179536,"style":157493},[407,136115],[363,179538,20762],{"className":179539},[501]," 得到。",[11,179542,179543,179544,179547,179548,4201,179611,179662,179663,179723,179724,179753,179754,179782,179783,179786,179787,179815],{},"这一分解的意义在于：",[1344,179545,179546],{},"它把「数据本身的随机性」与「关于类别分布的认知不足」在数学上分离成了两个可分别计算的量。"," 认知项 ",[363,179549,179551,179575],{"className":179550,"translate":367},[366],[363,179552,179554],{"className":179553},[371],[373,179555,179556],{"xmlns":375},[377,179557,179558,179572],{},[380,179559,179560,179562,179564,179566,179568,179570],{},[383,179561,34240],{},[458,179563,461],{"stretchy":460},[383,179565,40609],{},[458,179567,34247],{"separator":394},[383,179569,157459],{"mathvariant":136055},[458,179571,468],{"stretchy":460},[387,179573,179574],{"encoding":389},"I(y;\\boldsymbol{\\pi})",[363,179576,179578],{"className":179577,"ariaHidden":394},[393],[363,179579,179581,179584,179587,179590,179593,179596,179599,179608],{"className":179580},[398],[363,179582],{"className":179583,"style":487},[402],[363,179585,34240],{"className":179586,"style":9235},[407,408],[363,179588,461],{"className":179589},[494],[363,179591,40609],{"className":179592,"style":1328},[407,408],[363,179594,34247],{"className":179595},[920],[363,179597],{"className":179598,"style":754},[505],[363,179600,179602],{"className":179601},[407],[363,179603,179605],{"className":179604},[407],[363,179606,157459],{"className":179607,"style":157493},[407,136115],[363,179609,468],{"className":179610},[501],[363,179612,179614,179632],{"className":179613,"translate":367},[366],[363,179615,179617],{"className":179616},[371],[373,179618,179619],{"xmlns":375},[377,179620,179621,179629],{},[380,179622,179623,179625,179627],{},[383,179624,26109],{},[458,179626,1378],{},[383,179628,1373],{"mathvariant":557},[387,179630,179631],{"encoding":389},"S \\to \\infty",[363,179633,179635,179653],{"className":179634,"ariaHidden":394},[393],[363,179636,179638,179641,179644,179647,179650],{"className":179637},[398],[363,179639],{"className":179640,"style":1011},[402],[363,179642,26109],{"className":179643,"style":26233},[407,408],[363,179645],{"className":179646,"style":506},[505],[363,179648,1378],{"className":179649},[510],[363,179651],{"className":179652,"style":506},[505],[363,179654,179656,179659],{"className":179655},[398],[363,179657],{"className":179658,"style":2693},[402],[363,179660,1373],{"className":179661},[407],"（证据充分）时趋于零，在证据缺失时增大，与 ",[363,179664,179666,179687],{"className":179665,"translate":367},[366],[363,179667,179669],{"className":179668},[371],[373,179670,179671],{"xmlns":375},[377,179672,179673,179685],{},[380,179674,179675,179677,179679,179681,179683],{},[383,179676,15750],{},[458,179678,471],{},[383,179680,11044],{},[383,179682,1288],{"mathvariant":557},[383,179684,26109],{},[387,179686,175682],{"encoding":389},[363,179688,179690,179708],{"className":179689,"ariaHidden":394},[393],[363,179691,179693,179696,179699,179702,179705],{"className":179692},[398],[363,179694],{"className":179695,"style":2693},[402],[363,179697,15750],{"className":179698},[407,408],[363,179700],{"className":179701,"style":506},[505],[363,179703,471],{"className":179704},[510],[363,179706],{"className":179707,"style":506},[505],[363,179709,179711,179714,179717,179720],{"className":179710},[398],[363,179712],{"className":179713,"style":487},[402],[363,179715,11044],{"className":179716,"style":11132},[407,408],[363,179718,1288],{"className":179719},[407],[363,179721,26109],{"className":179722,"style":26233},[407,408]," 的定性行为一致。但它是通过 ",[363,179725,179727,179741],{"className":179726,"translate":367},[366],[363,179728,179730],{"className":179729},[371],[373,179731,179732],{"xmlns":375},[377,179733,179734,179738],{},[380,179735,179736],{},[383,179737,155482],{},[387,179739,179740],{"encoding":389},"\\psi",[363,179742,179744],{"className":179743,"ariaHidden":394},[393],[363,179745,179747,179750],{"className":179746},[398],[363,179748],{"className":179749,"style":403},[402],[363,179751,155482],{"className":179752,"style":1328},[407,408]," 函数的差构造的，捕捉了 ",[363,179755,179757,179770],{"className":179756,"translate":367},[366],[363,179758,179760],{"className":179759},[371],[373,179761,179762],{"xmlns":375},[377,179763,179764,179768],{},[380,179765,179766],{},[383,179767,15750],{},[387,179769,15750],{"encoding":389},[363,179771,179773],{"className":179772,"ariaHidden":394},[393],[363,179774,179776,179779],{"className":179775},[398],[363,179777],{"className":179778,"style":2693},[402],[363,179780,15750],{"className":179781},[407,408]," 所忽略的",[1344,179784,179785],{},"证据分布","信息，因而是比 ",[363,179788,179790,179803],{"className":179789,"translate":367},[366],[363,179791,179793],{"className":179792},[371],[373,179794,179795],{"xmlns":375},[377,179796,179797,179801],{},[380,179798,179799],{},[383,179800,15750],{},[387,179802,15750],{"encoding":389},[363,179804,179806],{"className":179805,"ariaHidden":394},[393],[363,179807,179809,179812],{"className":179808},[398],[363,179810],{"className":179811,"style":2693},[402],[363,179813,15750],{"className":179814},[407,408]," 更精细的分数。",[11,179817,179818,179819,179853,179854,179888,179889,179983,179984,153104,179987,180049,180050,180054,180055,180058],{},"不过，这一分解的「认知」标签必须谨慎对待。它在形式上是「关于 ",[363,179820,179822,179835],{"className":179821,"translate":367},[366],[363,179823,179825],{"className":179824},[371],[373,179826,179827],{"xmlns":375},[377,179828,179829,179833],{},[380,179830,179831],{},[383,179832,157459],{"mathvariant":136055},[387,179834,157547],{"encoding":389},[363,179836,179838],{"className":179837,"ariaHidden":394},[393],[363,179839,179841,179844],{"className":179840},[398],[363,179842],{"className":179843,"style":135662},[402],[363,179845,179847],{"className":179846},[407],[363,179848,179850],{"className":179849},[407],[363,179851,157459],{"className":179852,"style":157493},[407,136115]," 的不确定性」，而 ",[363,179855,179857,179870],{"className":179856,"translate":367},[366],[363,179858,179860],{"className":179859},[371],[373,179861,179862],{"xmlns":375},[377,179863,179864,179868],{},[380,179865,179866],{},[383,179867,157459],{"mathvariant":136055},[387,179869,157547],{"encoding":389},[363,179871,179873],{"className":179872,"ariaHidden":394},[393],[363,179874,179876,179879],{"className":179875},[398],[363,179877],{"className":179878,"style":135662},[402],[363,179880,179882],{"className":179881},[407],[363,179883,179885],{"className":179884},[407],[363,179886,157459],{"className":179887,"style":157493},[407,136115]," 在 EDL 中是 ",[363,179890,179892,179916],{"className":179891,"translate":367},[366],[363,179893,179895],{"className":179894},[371],[373,179896,179897],{"xmlns":375},[377,179898,179899,179913],{},[380,179900,179901,179907,179909,179911],{},[543,179902,179903,179905],{},[383,179904,4506],{},[383,179906,101148],{"mathvariant":136055},[458,179908,461],{"stretchy":460},[383,179910,1121],{"mathvariant":75598},[458,179912,468],{"stretchy":460},[387,179914,179915],{"encoding":389},"g_{\\boldsymbol{\\theta}}(\\mathbf{x})",[363,179917,179919],{"className":179918,"ariaHidden":394},[393],[363,179920,179922,179925,179974,179977,179980],{"className":179921},[398],[363,179923],{"className":179924,"style":487},[402],[363,179926,179928,179931],{"className":179927},[407],[363,179929,4506],{"className":179930,"style":1328},[407,408],[363,179932,179934],{"className":179933},[629],[363,179935,179937,179966],{"className":179936},[633,634],[363,179938,179940,179963],{"className":179939},[638],[363,179941,179943],{"className":179942,"style":643},[642],[363,179944,179945,179948],{"style":19215},[363,179946],{"className":179947,"style":651},[650],[363,179949,179951],{"className":179950},[655,656,657,658],[363,179952,179954],{"className":179953},[407,658],[363,179955,179957],{"className":179956},[407,658],[363,179958,179960],{"className":179959},[407,658],[363,179961,101148],{"className":179962,"style":169926},[407,136115,658],[363,179964,666],{"className":179965},[665],[363,179967,179969],{"className":179968},[638],[363,179970,179972],{"className":179971,"style":8316},[642],[363,179973],{},[363,179975,461],{"className":179976},[494],[363,179978,1121],{"className":179979},[407,75943],[363,179981,468],{"className":179982},[501]," 的输出，是",[1344,179985,179986],{},"网络参数的确定性函数",[363,179988,179990,180013],{"className":179989,"translate":367},[366],[363,179991,179993],{"className":179992},[371],[373,179994,179995],{"xmlns":375},[377,179996,179997,180011],{},[380,179998,179999,180001,180003,180005,180007,180009],{},[383,180000,34240],{},[458,180002,461],{"stretchy":460},[383,180004,40609],{},[458,180006,34247],{"separator":394},[383,180008,157459],{"mathvariant":136055},[458,180010,468],{"stretchy":460},[387,180012,179574],{"encoding":389},[363,180014,180016],{"className":180015,"ariaHidden":394},[393],[363,180017,180019,180022,180025,180028,180031,180034,180037,180046],{"className":180018},[398],[363,180020],{"className":180021,"style":487},[402],[363,180023,34240],{"className":180024,"style":9235},[407,408],[363,180026,461],{"className":180027},[494],[363,180029,40609],{"className":180030,"style":1328},[407,408],[363,180032,34247],{"className":180033},[920],[363,180035],{"className":180036,"style":754},[505],[363,180038,180040],{"className":180039},[407],[363,180041,180043],{"className":180042},[407],[363,180044,157459],{"className":180045,"style":157493},[407,136115],[363,180047,468],{"className":180048},[501]," 度量的是",[180051,180052,180053],"em",{},"类概率向量","的不确定，而不是",[180051,180056,180057],{},"模型参数","的不确定——后者才是贝叶斯认识论意义上的 epistemic uncertainty。二者的距离，正是 4.5 节全部批评的技术根源。",[18,180060,180062],{"id":180061},"四证据深度学习的损失函数","四、证据深度学习的损失函数",[11,180064,180065,180066,180094,180095,180149],{},"EDL 的训练目标不再是「使 ",[363,180067,180069,180082],{"className":180068,"translate":367},[366],[363,180070,180072],{"className":180071},[371],[373,180073,180074],{"xmlns":375},[377,180075,180076,180080],{},[380,180077,180078],{},[383,180079,11],{"mathvariant":75598},[387,180081,133797],{"encoding":389},[363,180083,180085],{"className":180084,"ariaHidden":394},[393],[363,180086,180088,180091],{"className":180087},[398],[363,180089],{"className":180090,"style":133807},[402],[363,180092,11],{"className":180093},[407,75943]," 逼近 one-hot 标签」，而是「使整族 ",[363,180096,180098,180119],{"className":180097,"translate":367},[366],[363,180099,180101],{"className":180100},[371],[373,180102,180103],{"xmlns":375},[377,180104,180105,180117],{},[380,180106,180107,180109,180111,180113,180115],{},[383,180108,137112],{"mathvariant":557},[458,180110,4451],{},[458,180112,461],{"stretchy":460},[383,180114,45126],{"mathvariant":136055},[458,180116,468],{"stretchy":460},[387,180118,171778],{"encoding":389},[363,180120,180122],{"className":180121,"ariaHidden":394},[393],[363,180123,180125,180128,180134,180137,180146],{"className":180124},[398],[363,180126],{"className":180127,"style":487},[402],[363,180129,180131],{"className":180130},[1832],[363,180132,137112],{"className":180133},[407,33034],[363,180135,461],{"className":180136},[494],[363,180138,180140],{"className":180139},[407],[363,180141,180143],{"className":180142},[407],[363,180144,45126],{"className":180145},[407,136115],[363,180147,468],{"className":180148},[501]," 逼近一个理想的分布」。损失函数因此被组织为两项：一项拟合分类目标，一项约束不确定性的行为。",[2659,180151,180153],{"id":180152},"_41-type-ii-极大似然与对数风险","4.1 Type-II 极大似然与对数风险",[11,180155,180156,180157,180160],{},"最自然的构造是极小化对数损失在迪利克雷先验下的",[1344,180158,180159],{},"贝叶斯风险","（亦称 Type-II 极大似然）：",[363,180162,180164],{"className":180163,"translate":367},[528],[363,180165,180167,180224],{"className":180166,"translate":367},[366],[363,180168,180170],{"className":180169},[371],[373,180171,180172],{"xmlns":375,"display":537},[377,180173,180174,180221],{},[380,180175,180176,180183,180185,180205,180207,180209,180211,180213,180219],{},[7162,180177,180178,180180],{},[383,180179,22265],{"mathvariant":944},[1645,180181,180182],{},"NLL",[458,180184,471],{},[543,180186,180187,180189],{},[383,180188,571],{"mathvariant":570},[380,180190,180191,180193,180195,180197,180199,180201,180203],{},[383,180192,157459],{"mathvariant":136055},[458,180194,9217],{},[383,180196,137112],{"mathvariant":557},[458,180198,4451],{},[458,180200,461],{"stretchy":460},[383,180202,45126],{"mathvariant":136055},[458,180204,468],{"stretchy":460},[458,180206,578],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[458,180208,5161],{},[383,180210,4448],{},[458,180212,4451],{},[543,180214,180215,180217],{},[383,180216,157459],{},[383,180218,40609],{},[458,180220,605],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[387,180222,180223],{"encoding":389},"\\mathcal{L}^{\\text{NLL}} = \\mathbb{E}_{\\boldsymbol{\\pi} \\sim \\operatorname{Dir}(\\boldsymbol{\\alpha})}\\big[-\\log \\pi_y\\big]",[363,180225,180227,180277,180371],{"className":180226,"ariaHidden":394},[393],[363,180228,180230,180233,180268,180271,180274],{"className":180229},[398],[363,180231],{"className":180232,"style":38884},[402],[363,180234,180236,180239],{"className":180235},[407],[363,180237,22265],{"className":180238},[407,971],[363,180240,180242],{"className":180241},[629],[363,180243,180245],{"className":180244},[633],[363,180246,180248],{"className":180247},[638],[363,180249,180251],{"className":180250,"style":38884},[642],[363,180252,180253,180256],{"style":8625},[363,180254],{"className":180255,"style":651},[650],[363,180257,180259],{"className":180258},[655,656,657,658],[363,180260,180262],{"className":180261},[407,658],[363,180263,180265],{"className":180264},[407,4946,658],[363,180266,180182],{"className":180267},[407,658],[363,180269],{"className":180270,"style":506},[505],[363,180272,471],{"className":180273},[510],[363,180275],{"className":180276,"style":506},[505],[363,180278,180280,180283,180356,180362,180365,180368],{"className":180279},[398],[363,180281],{"className":180282,"style":119827},[402],[363,180284,180286,180289],{"className":180285},[407],[363,180287,571],{"className":180288},[407,714],[363,180290,180292],{"className":180291},[629],[363,180293,180295,180348],{"className":180294},[633,634],[363,180296,180298,180345],{"className":180297},[638],[363,180299,180301],{"className":180300,"style":6446},[642],[363,180302,180303,180306],{"style":75958},[363,180304],{"className":180305,"style":651},[650],[363,180307,180309],{"className":180308},[655,656,657,658],[363,180310,180312,180321,180324,180330,180333,180342],{"className":180311},[407,658],[363,180313,180315],{"className":180314},[407,658],[363,180316,180318],{"className":180317},[407,658],[363,180319,157459],{"className":180320,"style":157493},[407,136115,658],[363,180322,9217],{"className":180323},[510,658],[363,180325,180327],{"className":180326},[1832,658],[363,180328,137112],{"className":180329},[407,33034,658],[363,180331,461],{"className":180332},[494,658],[363,180334,180336],{"className":180335},[407,658],[363,180337,180339],{"className":180338},[407,658],[363,180340,45126],{"className":180341},[407,136115,658],[363,180343,468],{"className":180344},[501,658],[363,180346,666],{"className":180347},[665],[363,180349,180351],{"className":180350},[638],[363,180352,180354],{"className":180353,"style":6492},[642],[363,180355],{},[363,180357,180359],{"className":180358},[407],[363,180360,578],{"className":180361},[767,10022],[363,180363],{"className":180364,"style":2935},[505],[363,180366,5161],{"className":180367},[2939],[363,180369],{"className":180370,"style":2935},[505],[363,180372,180374,180377,180382,180385,180425],{"className":180373},[398],[363,180375],{"className":180376,"style":20935},[402],[363,180378,4502,180380],{"className":180379},[1832],[363,180381,4506],{"style":4505},[363,180383],{"className":180384,"style":754},[505],[363,180386,180388,180391],{"className":180387},[407],[363,180389,157459],{"className":180390,"style":1328},[407,408],[363,180392,180394],{"className":180393},[629],[363,180395,180397,180417],{"className":180396},[633,634],[363,180398,180400,180414],{"className":180399},[638],[363,180401,180403],{"className":180402,"style":45301},[642],[363,180404,180405,180408],{"style":19215},[363,180406],{"className":180407,"style":651},[650],[363,180409,180411],{"className":180410},[655,656,657,658],[363,180412,40609],{"className":180413,"style":1328},[407,408,658],[363,180415,666],{"className":180416},[665],[363,180418,180420],{"className":180419},[638],[363,180421,180423],{"className":180422,"style":673},[642],[363,180424],{},[363,180426,180428],{"className":180427},[407],[363,180429,605],{"className":180430},[767,10022],[143,180432,180433],{},[11,180434,180435,180438,180439,180467,180468],{},[1344,180436,180437],{},"命题 4.1（对数风险的闭式）"," 设真实标签为 ",[363,180440,180442,180455],{"className":180441,"translate":367},[366],[363,180443,180445],{"className":180444},[371],[373,180446,180447],{"xmlns":375},[377,180448,180449,180453],{},[380,180450,180451],{},[383,180452,40609],{},[387,180454,40609],{"encoding":389},[363,180456,180458],{"className":180457,"ariaHidden":394},[393],[363,180459,180461,180464],{"className":180460},[398],[363,180462],{"className":180463,"style":2624},[402],[363,180465,40609],{"className":180466,"style":1328},[407,408],"，则\n",[363,180469,180471,180513],{"className":180470,"translate":367},[366],[363,180472,180474],{"className":180473},[371],[373,180475,180476],{"xmlns":375},[377,180477,180478,180510],{},[380,180479,180480,180486,180488,180490,180492,180494,180496,180498,180500,180502,180508],{},[7162,180481,180482,180484],{},[383,180483,22265],{"mathvariant":944},[1645,180485,180182],{},[458,180487,471],{},[383,180489,155482],{},[458,180491,461],{"stretchy":460},[383,180493,26109],{},[458,180495,468],{"stretchy":460},[458,180497,5161],{},[383,180499,155482],{},[458,180501,461],{"stretchy":460},[543,180503,180504,180506],{},[383,180505,45126],{},[383,180507,40609],{},[458,180509,468],{"stretchy":460},[387,180511,180512],{"encoding":389},"\\mathcal{L}^{\\text{NLL}} = \\psi(S) - \\psi(\\alpha_y)",[363,180514,180516,180566,180593],{"className":180515,"ariaHidden":394},[393],[363,180517,180519,180522,180557,180560,180563],{"className":180518},[398],[363,180520],{"className":180521,"style":52457},[402],[363,180523,180525,180528],{"className":180524},[407],[363,180526,22265],{"className":180527},[407,971],[363,180529,180531],{"className":180530},[629],[363,180532,180534],{"className":180533},[633],[363,180535,180537],{"className":180536},[638],[363,180538,180540],{"className":180539,"style":52457},[642],[363,180541,180542,180545],{"style":7213},[363,180543],{"className":180544,"style":651},[650],[363,180546,180548],{"className":180547},[655,656,657,658],[363,180549,180551],{"className":180550},[407,658],[363,180552,180554],{"className":180553},[407,4946,658],[363,180555,180182],{"className":180556},[407,658],[363,180558],{"className":180559,"style":506},[505],[363,180561,471],{"className":180562},[510],[363,180564],{"className":180565,"style":506},[505],[363,180567,180569,180572,180575,180578,180581,180584,180587,180590],{"className":180568},[398],[363,180570],{"className":180571,"style":487},[402],[363,180573,155482],{"className":180574,"style":1328},[407,408],[363,180576,461],{"className":180577},[494],[363,180579,26109],{"className":180580,"style":26233},[407,408],[363,180582,468],{"className":180583},[501],[363,180585],{"className":180586,"style":2935},[505],[363,180588,5161],{"className":180589},[2939],[363,180591],{"className":180592,"style":2935},[505],[363,180594,180596,180599,180602,180605,180645],{"className":180595},[398],[363,180597],{"className":180598,"style":618},[402],[363,180600,155482],{"className":180601,"style":1328},[407,408],[363,180603,461],{"className":180604},[494],[363,180606,180608,180611],{"className":180607},[407],[363,180609,45126],{"className":180610,"style":45156},[407,408],[363,180612,180614],{"className":180613},[629],[363,180615,180617,180637],{"className":180616},[633,634],[363,180618,180620,180634],{"className":180619},[638],[363,180621,180623],{"className":180622,"style":45301},[642],[363,180624,180625,180628],{"style":136154},[363,180626],{"className":180627,"style":651},[650],[363,180629,180631],{"className":180630},[655,656,657,658],[363,180632,40609],{"className":180633,"style":1328},[407,408,658],[363,180635,666],{"className":180636},[665],[363,180638,180640],{"className":180639},[638],[363,180641,180643],{"className":180642,"style":673},[642],[363,180644],{},[363,180646,468],{"className":180647},[501],[11,180649,180650,180652,180653,180906,180907,180958,180959,180992],{},[1344,180651,138042],{}," 由 ",[363,180654,180656,180712],{"className":180655,"translate":367},[366],[363,180657,180659],{"className":180658},[371],[373,180660,180661],{"xmlns":375},[377,180662,180663,180709],{},[380,180664,180665,180667,180669,180671,180673,180679,180681,180683,180685,180687,180693,180695,180697,180699,180701,180707],{},[383,180666,571],{"mathvariant":570},[458,180668,578],{"stretchy":460},[383,180670,4448],{},[458,180672,4451],{},[543,180674,180675,180677],{},[383,180676,1121],{},[383,180678,7912],{},[458,180680,605],{"stretchy":460},[458,180682,471],{},[383,180684,155482],{},[458,180686,461],{"stretchy":460},[543,180688,180689,180691],{},[383,180690,45126],{},[383,180692,7912],{},[458,180694,468],{"stretchy":460},[458,180696,5161],{},[383,180698,155482],{},[458,180700,461],{"stretchy":460},[543,180702,180703,180705],{},[383,180704,45126],{},[463,180706,474],{},[458,180708,468],{"stretchy":460},[387,180710,180711],{"encoding":389},"\\mathbb{E}[\\log x_k] = \\psi(\\alpha_k) - \\psi(\\alpha_0)",[363,180713,180715,180787,180851],{"className":180714,"ariaHidden":394},[393],[363,180716,180718,180721,180724,180727,180732,180735,180775,180778,180781,180784],{"className":180717},[398],[363,180719],{"className":180720,"style":487},[402],[363,180722,571],{"className":180723},[407,714],[363,180725,578],{"className":180726},[494],[363,180728,4502,180730],{"className":180729},[1832],[363,180731,4506],{"style":4505},[363,180733],{"className":180734,"style":754},[505],[363,180736,180738,180741],{"className":180737},[407],[363,180739,1121],{"className":180740},[407,408],[363,180742,180744],{"className":180743},[629],[363,180745,180747,180767],{"className":180746},[633,634],[363,180748,180750,180764],{"className":180749},[638],[363,180751,180753],{"className":180752,"style":643},[642],[363,180754,180755,180758],{"style":730},[363,180756],{"className":180757,"style":651},[650],[363,180759,180761],{"className":180760},[655,656,657,658],[363,180762,7912],{"className":180763,"style":8615},[407,408,658],[363,180765,666],{"className":180766},[665],[363,180768,180770],{"className":180769},[638],[363,180771,180773],{"className":180772,"style":8316},[642],[363,180774],{},[363,180776,605],{"className":180777},[501],[363,180779],{"className":180780,"style":506},[505],[363,180782,471],{"className":180783},[510],[363,180785],{"className":180786,"style":506},[505],[363,180788,180790,180793,180796,180799,180839,180842,180845,180848],{"className":180789},[398],[363,180791],{"className":180792,"style":487},[402],[363,180794,155482],{"className":180795,"style":1328},[407,408],[363,180797,461],{"className":180798},[494],[363,180800,180802,180805],{"className":180801},[407],[363,180803,45126],{"className":180804,"style":45156},[407,408],[363,180806,180808],{"className":180807},[629],[363,180809,180811,180831],{"className":180810},[633,634],[363,180812,180814,180828],{"className":180813},[638],[363,180815,180817],{"className":180816,"style":643},[642],[363,180818,180819,180822],{"style":136154},[363,180820],{"className":180821,"style":651},[650],[363,180823,180825],{"className":180824},[655,656,657,658],[363,180826,7912],{"className":180827,"style":8615},[407,408,658],[363,180829,666],{"className":180830},[665],[363,180832,180834],{"className":180833},[638],[363,180835,180837],{"className":180836,"style":8316},[642],[363,180838],{},[363,180840,468],{"className":180841},[501],[363,180843],{"className":180844,"style":2935},[505],[363,180846,5161],{"className":180847},[2939],[363,180849],{"className":180850,"style":2935},[505],[363,180852,180854,180857,180860,180863,180903],{"className":180853},[398],[363,180855],{"className":180856,"style":487},[402],[363,180858,155482],{"className":180859,"style":1328},[407,408],[363,180861,461],{"className":180862},[494],[363,180864,180866,180869],{"className":180865},[407],[363,180867,45126],{"className":180868,"style":45156},[407,408],[363,180870,180872],{"className":180871},[629],[363,180873,180875,180895],{"className":180874},[633,634],[363,180876,180878,180892],{"className":180877},[638],[363,180879,180881],{"className":180880,"style":11483},[642],[363,180882,180883,180886],{"style":136154},[363,180884],{"className":180885,"style":651},[650],[363,180887,180889],{"className":180888},[655,656,657,658],[363,180890,474],{"className":180891},[407,658],[363,180893,666],{"className":180894},[665],[363,180896,180898],{"className":180897},[638],[363,180899,180901],{"className":180900,"style":8316},[642],[363,180902],{},[363,180904,468],{"className":180905},[501],"（命题 2.4 的证明中已给出），取 ",[363,180908,180910,180928],{"className":180909,"translate":367},[366],[363,180911,180913],{"className":180912},[371],[373,180914,180915],{"xmlns":375},[377,180916,180917,180925],{},[380,180918,180919,180921,180923],{},[383,180920,7912],{},[458,180922,471],{},[383,180924,40609],{},[387,180926,180927],{"encoding":389},"k = y",[363,180929,180931,180949],{"className":180930,"ariaHidden":394},[393],[363,180932,180934,180937,180940,180943,180946],{"className":180933},[398],[363,180935],{"className":180936,"style":12947},[402],[363,180938,7912],{"className":180939,"style":8615},[407,408],[363,180941],{"className":180942,"style":506},[505],[363,180944,471],{"className":180945},[510],[363,180947],{"className":180948,"style":506},[505],[363,180950,180952,180955],{"className":180951},[398],[363,180953],{"className":180954,"style":2624},[402],[363,180956,40609],{"className":180957,"style":1328},[407,408]," 并乘以 ",[363,180960,180962,180977],{"className":180961,"translate":367},[366],[363,180963,180965],{"className":180964},[371],[373,180966,180967],{"xmlns":375},[377,180968,180969,180975],{},[380,180970,180971,180973],{},[458,180972,5161],{},[463,180974,465],{},[387,180976,49528],{"encoding":389},[363,180978,180980],{"className":180979,"ariaHidden":394},[393],[363,180981,180983,180986,180989],{"className":180982},[398],[363,180984],{"className":180985,"style":12928},[402],[363,180987,5161],{"className":180988},[407],[363,180990,465],{"className":180991},[407]," 即得。",[11,180994,180995],{},"该损失的梯度结构值得注意：",[363,180997,180999],{"className":180998,"translate":367},[528],[363,181000,181002,181129],{"className":181001,"translate":367},[366],[363,181003,181005],{"className":181004},[371],[373,181006,181007],{"xmlns":375,"display":537},[377,181008,181009,181126],{},[380,181010,181011,181034,181036,181042,181044,181046,181048,181050,181056,181058,181064,181066,181068,181070,181072,181074,181096,181098,181104,181106,181108,181110,181112,181114,181116,181118,181120,181122,181124],{},[586,181012,181013,181024],{},[380,181014,181015,181018],{},[383,181016,181017],{"mathvariant":557},"∂",[7162,181019,181020,181022],{},[383,181021,22265],{"mathvariant":944},[1645,181023,180182],{},[380,181025,181026,181028],{},[383,181027,181017],{"mathvariant":557},[543,181029,181030,181032],{},[383,181031,45126],{},[383,181033,40609],{},[458,181035,471],{},[7162,181037,181038,181040],{},[383,181039,155482],{},[458,181041,7168],{"mathvariant":557,"lspace":1595,"rspace":1595},[458,181043,461],{"stretchy":460},[383,181045,26109],{},[458,181047,468],{"stretchy":460},[458,181049,5161],{},[7162,181051,181052,181054],{},[383,181053,155482],{},[458,181055,7168],{"mathvariant":557,"lspace":1595,"rspace":1595},[458,181057,461],{"stretchy":460},[543,181059,181060,181062],{},[383,181061,45126],{},[383,181063,40609],{},[458,181065,468],{"stretchy":460},[458,181067,10300],{},[463,181069,474],{},[458,181071,898],{"separator":394},[505,181073],{"width":3610},[586,181075,181076,181086],{},[380,181077,181078,181080],{},[383,181079,181017],{"mathvariant":557},[7162,181081,181082,181084],{},[383,181083,22265],{"mathvariant":944},[1645,181085,180182],{},[380,181087,181088,181090],{},[383,181089,181017],{"mathvariant":557},[543,181091,181092,181094],{},[383,181093,45126],{},[383,181095,100402],{},[458,181097,471],{},[7162,181099,181100,181102],{},[383,181101,155482],{},[458,181103,7168],{"mathvariant":557,"lspace":1595,"rspace":1595},[458,181105,461],{"stretchy":460},[383,181107,26109],{},[458,181109,468],{"stretchy":460},[458,181111,3251],{},[463,181113,474],{},[505,181115],{"width":4663},[458,181117,461],{"stretchy":460},[383,181119,100402],{},[458,181121,49879],{"mathvariant":557},[383,181123,40609],{},[458,181125,468],{"stretchy":460},[387,181127,181128],{"encoding":389},"\\frac{\\partial \\mathcal{L}^{\\text{NLL}}}{\\partial \\alpha_y} = \\psi'(S) - \\psi'(\\alpha_y) \\le 0,\n\\qquad\n\\frac{\\partial \\mathcal{L}^{\\text{NLL}}}{\\partial \\alpha_j} = \\psi'(S) > 0 \\quad (j \\ne y)",[363,181130,181132,181287,181343,181437,181601,181657,181717],{"className":181131,"ariaHidden":394},[393],[363,181133,181135,181139,181278,181281,181284],{"className":181134},[398],[363,181136],{"className":181137,"style":181138},[402],"height:2.4904em;vertical-align:-0.9721em;",[363,181140,181142,181145,181275],{"className":181141},[407],[363,181143],{"className":181144},[494,789],[363,181146,181148],{"className":181147},[586],[363,181149,181151,181266],{"className":181150},[633,634],[363,181152,181154,181263],{"className":181153},[638],[363,181155,181158,181209,181217],{"className":181156,"style":181157},[642],"height:1.5183em;",[363,181159,181160,181163],{"style":805},[363,181161],{"className":181162,"style":809},[650],[363,181164,181166,181169],{"className":181165},[407],[363,181167,181017],{"className":181168,"style":121555},[407],[363,181170,181172,181175],{"className":181171},[407],[363,181173,45126],{"className":181174,"style":45156},[407,408],[363,181176,181178],{"className":181177},[629],[363,181179,181181,181201],{"className":181180},[633,634],[363,181182,181184,181198],{"className":181183},[638],[363,181185,181187],{"className":181186,"style":45301},[642],[363,181188,181189,181192],{"style":136154},[363,181190],{"className":181191,"style":651},[650],[363,181193,181195],{"className":181194},[655,656,657,658],[363,181196,40609],{"className":181197,"style":1328},[407,408,658],[363,181199,666],{"className":181200},[665],[363,181202,181204],{"className":181203},[638],[363,181205,181207],{"className":181206,"style":673},[642],[363,181208],{},[363,181210,181211,181214],{"style":821},[363,181212],{"className":181213,"style":809},[650],[363,181215],{"className":181216,"style":829},[828],[363,181218,181219,181222],{"style":832},[363,181220],{"className":181221,"style":809},[650],[363,181223,181225,181228],{"className":181224},[407],[363,181226,181017],{"className":181227,"style":121555},[407],[363,181229,181231,181234],{"className":181230},[407],[363,181232,22265],{"className":181233},[407,971],[363,181235,181237],{"className":181236},[629],[363,181238,181240],{"className":181239},[633],[363,181241,181243],{"className":181242},[638],[363,181244,181246],{"className":181245,"style":52457},[642],[363,181247,181248,181251],{"style":7213},[363,181249],{"className":181250,"style":651},[650],[363,181252,181254],{"className":181253},[655,656,657,658],[363,181255,181257],{"className":181256},[407,658],[363,181258,181260],{"className":181259},[407,4946,658],[363,181261,180182],{"className":181262},[407,658],[363,181264,666],{"className":181265},[665],[363,181267,181269],{"className":181268},[638],[363,181270,181273],{"className":181271,"style":181272},[642],"height:0.9721em;",[363,181274],{},[363,181276],{"className":181277},[501,789],[363,181279],{"className":181280,"style":506},[505],[363,181282,471],{"className":181283},[510],[363,181285],{"className":181286,"style":506},[505],[363,181288,181290,181293,181325,181328,181331,181334,181337,181340],{"className":181289},[398],[363,181291],{"className":181292,"style":102572},[402],[363,181294,181296,181299],{"className":181295},[407],[363,181297,155482],{"className":181298,"style":1328},[407,408],[363,181300,181302],{"className":181301},[629],[363,181303,181305],{"className":181304},[633],[363,181306,181308],{"className":181307},[638],[363,181309,181311],{"className":181310,"style":29343},[642],[363,181312,181313,181316],{"style":8625},[363,181314],{"className":181315,"style":651},[650],[363,181317,181319],{"className":181318},[655,656,657,658],[363,181320,181322],{"className":181321},[407,658],[363,181323,7168],{"className":181324},[407,658],[363,181326,461],{"className":181327},[494],[363,181329,26109],{"className":181330,"style":26233},[407,408],[363,181332,468],{"className":181333},[501],[363,181335],{"className":181336,"style":2935},[505],[363,181338,5161],{"className":181339},[2939],[363,181341],{"className":181342,"style":2935},[505],[363,181344,181346,181350,181382,181385,181425,181428,181431,181434],{"className":181345},[398],[363,181347],{"className":181348,"style":181349},[402],"height:1.088em;vertical-align:-0.2861em;",[363,181351,181353,181356],{"className":181352},[407],[363,181354,155482],{"className":181355,"style":1328},[407,408],[363,181357,181359],{"className":181358},[629],[363,181360,181362],{"className":181361},[633],[363,181363,181365],{"className":181364},[638],[363,181366,181368],{"className":181367,"style":29343},[642],[363,181369,181370,181373],{"style":8625},[363,181371],{"className":181372,"style":651},[650],[363,181374,181376],{"className":181375},[655,656,657,658],[363,181377,181379],{"className":181378},[407,658],[363,181380,7168],{"className":181381},[407,658],[363,181383,461],{"className":181384},[494],[363,181386,181388,181391],{"className":181387},[407],[363,181389,45126],{"className":181390,"style":45156},[407,408],[363,181392,181394],{"className":181393},[629],[363,181395,181397,181417],{"className":181396},[633,634],[363,181398,181400,181414],{"className":181399},[638],[363,181401,181403],{"className":181402,"style":45301},[642],[363,181404,181405,181408],{"style":136154},[363,181406],{"className":181407,"style":651},[650],[363,181409,181411],{"className":181410},[655,656,657,658],[363,181412,40609],{"className":181413,"style":1328},[407,408,658],[363,181415,666],{"className":181416},[665],[363,181418,181420],{"className":181419},[638],[363,181421,181423],{"className":181422,"style":673},[642],[363,181424],{},[363,181426,468],{"className":181427},[501],[363,181429],{"className":181430,"style":506},[505],[363,181432,10300],{"className":181433},[510],[363,181435],{"className":181436,"style":506},[505],[363,181438,181440,181443,181446,181449,181452,181455,181592,181595,181598],{"className":181439},[398],[363,181441],{"className":181442,"style":181138},[402],[363,181444,474],{"className":181445},[407],[363,181447,898],{"className":181448},[920],[363,181450],{"className":181451,"style":3982},[505],[363,181453],{"className":181454,"style":754},[505],[363,181456,181458,181461,181589],{"className":181457},[407],[363,181459],{"className":181460},[494,789],[363,181462,181464],{"className":181463},[586],[363,181465,181467,181581],{"className":181466},[633,634],[363,181468,181470,181578],{"className":181469},[638],[363,181471,181473,181524,181532],{"className":181472,"style":181157},[642],[363,181474,181475,181478],{"style":805},[363,181476],{"className":181477,"style":809},[650],[363,181479,181481,181484],{"className":181480},[407],[363,181482,181017],{"className":181483,"style":121555},[407],[363,181485,181487,181490],{"className":181486},[407],[363,181488,45126],{"className":181489,"style":45156},[407,408],[363,181491,181493],{"className":181492},[629],[363,181494,181496,181516],{"className":181495},[633,634],[363,181497,181499,181513],{"className":181498},[638],[363,181500,181502],{"className":181501,"style":12532},[642],[363,181503,181504,181507],{"style":136154},[363,181505],{"className":181506,"style":651},[650],[363,181508,181510],{"className":181509},[655,656,657,658],[363,181511,100402],{"className":181512,"style":100930},[407,408,658],[363,181514,666],{"className":181515},[665],[363,181517,181519],{"className":181518},[638],[363,181520,181522],{"className":181521,"style":673},[642],[363,181523],{},[363,181525,181526,181529],{"style":821},[363,181527],{"className":181528,"style":809},[650],[363,181530],{"className":181531,"style":829},[828],[363,181533,181534,181537],{"style":832},[363,181535],{"className":181536,"style":809},[650],[363,181538,181540,181543],{"className":181539},[407],[363,181541,181017],{"className":181542,"style":121555},[407],[363,181544,181546,181549],{"className":181545},[407],[363,181547,22265],{"className":181548},[407,971],[363,181550,181552],{"className":181551},[629],[363,181553,181555],{"className":181554},[633],[363,181556,181558],{"className":181557},[638],[363,181559,181561],{"className":181560,"style":52457},[642],[363,181562,181563,181566],{"style":7213},[363,181564],{"className":181565,"style":651},[650],[363,181567,181569],{"className":181568},[655,656,657,658],[363,181570,181572],{"className":181571},[407,658],[363,181573,181575],{"className":181574},[407,4946,658],[363,181576,180182],{"className":181577},[407,658],[363,181579,666],{"className":181580},[665],[363,181582,181584],{"className":181583},[638],[363,181585,181587],{"className":181586,"style":181272},[642],[363,181588],{},[363,181590],{"className":181591},[501,789],[363,181593],{"className":181594,"style":506},[505],[363,181596,471],{"className":181597},[510],[363,181599],{"className":181600,"style":506},[505],[363,181602,181604,181607,181639,181642,181645,181648,181651,181654],{"className":181603},[398],[363,181605],{"className":181606,"style":102572},[402],[363,181608,181610,181613],{"className":181609},[407],[363,181611,155482],{"className":181612,"style":1328},[407,408],[363,181614,181616],{"className":181615},[629],[363,181617,181619],{"className":181618},[633],[363,181620,181622],{"className":181621},[638],[363,181623,181625],{"className":181624,"style":29343},[642],[363,181626,181627,181630],{"style":8625},[363,181628],{"className":181629,"style":651},[650],[363,181631,181633],{"className":181632},[655,656,657,658],[363,181634,181636],{"className":181635},[407,658],[363,181637,7168],{"className":181638},[407,658],[363,181640,461],{"className":181641},[494],[363,181643,26109],{"className":181644,"style":26233},[407,408],[363,181646,468],{"className":181647},[501],[363,181649],{"className":181650,"style":506},[505],[363,181652,3251],{"className":181653},[510],[363,181655],{"className":181656,"style":506},[505],[363,181658,181660,181663,181666,181669,181672,181675,181678,181714],{"className":181659},[398],[363,181661],{"className":181662,"style":487},[402],[363,181664,474],{"className":181665},[407],[363,181667],{"className":181668,"style":88355},[505],[363,181670,461],{"className":181671},[494],[363,181673,100402],{"className":181674,"style":100930},[407,408],[363,181676],{"className":181677,"style":506},[505],[363,181679,181681,181708,181711],{"className":181680},[510],[363,181682,181684],{"className":181683},[510],[363,181685,181687],{"className":181686},[407,4748],[363,181688,181690],{"className":181689},[4752],[363,181691,181693,181696,181705],{"className":181692},[4756],[363,181694],{"className":181695,"style":403},[402],[363,181697,181699],{"className":181698},[4763],[363,181700,181702],{"className":181701},[407],[363,181703,4770],{"className":181704},[510],[363,181706],{"className":181707},[4774],[363,181709],{"className":181710},[505,4778],[363,181712,471],{"className":181713},[510],[363,181715],{"className":181716,"style":506},[505],[363,181718,181720,181723,181726],{"className":181719},[398],[363,181721],{"className":181722,"style":487},[402],[363,181724,40609],{"className":181725,"style":1328},[407,408],[363,181727,468],{"className":181728},[501],[11,181730,2598,181731,181793,181794,16145,181797,181867,181868,16145,181871,181941,181942,182011,182012,182040,182041,182069],{},[363,181732,181734,181752],{"className":181733,"translate":367},[366],[363,181735,181737],{"className":181736},[371],[373,181738,181739],{"xmlns":375},[377,181740,181741,181749],{},[380,181742,181743],{},[7162,181744,181745,181747],{},[383,181746,155482],{},[458,181748,7168],{"mathvariant":557,"lspace":1595,"rspace":1595},[387,181750,181751],{"encoding":389},"\\psi'",[363,181753,181755],{"className":181754,"ariaHidden":394},[393],[363,181756,181758,181761],{"className":181757},[398],[363,181759],{"className":181760,"style":64544},[402],[363,181762,181764,181767],{"className":181763},[407],[363,181765,155482],{"className":181766,"style":1328},[407,408],[363,181768,181770],{"className":181769},[629],[363,181771,181773],{"className":181772},[633],[363,181774,181776],{"className":181775},[638],[363,181777,181779],{"className":181778,"style":7210},[642],[363,181780,181781,181784],{"style":7213},[363,181782],{"className":181783,"style":651},[650],[363,181785,181787],{"className":181786},[655,656,657,658],[363,181788,181790],{"className":181789},[407,658],[363,181791,7168],{"className":181792},[407,658]," 为 trigamma 函数（恒正、单调递减）。因此梯度沿两个方向作用：",[1344,181795,181796],{},"提升真实类别的证据",[363,181798,181800,181818],{"className":181799,"translate":367},[366],[363,181801,181803],{"className":181802},[371],[373,181804,181805],{"xmlns":375},[377,181806,181807,181815],{},[380,181808,181809],{},[543,181810,181811,181813],{},[383,181812,45126],{},[383,181814,40609],{},[387,181816,181817],{"encoding":389},"\\alpha_y",[363,181819,181821],{"className":181820,"ariaHidden":394},[393],[363,181822,181824,181827],{"className":181823},[398],[363,181825],{"className":181826,"style":93347},[402],[363,181828,181830,181833],{"className":181829},[407],[363,181831,45126],{"className":181832,"style":45156},[407,408],[363,181834,181836],{"className":181835},[629],[363,181837,181839,181859],{"className":181838},[633,634],[363,181840,181842,181856],{"className":181841},[638],[363,181843,181845],{"className":181844,"style":45301},[642],[363,181846,181847,181850],{"style":136154},[363,181848],{"className":181849,"style":651},[650],[363,181851,181853],{"className":181852},[655,656,657,658],[363,181854,40609],{"className":181855,"style":1328},[407,408,658],[363,181857,666],{"className":181858},[665],[363,181860,181862],{"className":181861},[638],[363,181863,181865],{"className":181864,"style":673},[642],[363,181866],{}," 增大），",[1344,181869,181870],{},"压低其余类别的证据",[363,181872,181874,181892],{"className":181873,"translate":367},[366],[363,181875,181877],{"className":181876},[371],[373,181878,181879],{"xmlns":375},[377,181880,181881,181889],{},[380,181882,181883],{},[543,181884,181885,181887],{},[383,181886,45126],{},[383,181888,100402],{},[387,181890,181891],{"encoding":389},"\\alpha_j",[363,181893,181895],{"className":181894,"ariaHidden":394},[393],[363,181896,181898,181901],{"className":181897},[398],[363,181899],{"className":181900,"style":93347},[402],[363,181902,181904,181907],{"className":181903},[407],[363,181905,45126],{"className":181906,"style":45156},[407,408],[363,181908,181910],{"className":181909},[629],[363,181911,181913,181933],{"className":181912},[633,634],[363,181914,181916,181930],{"className":181915},[638],[363,181917,181919],{"className":181918,"style":12532},[642],[363,181920,181921,181924],{"style":136154},[363,181922],{"className":181923,"style":651},[650],[363,181925,181927],{"className":181926},[655,656,657,658],[363,181928,100402],{"className":181929,"style":100930},[407,408,658],[363,181931,666],{"className":181932},[665],[363,181934,181936],{"className":181935},[638],[363,181937,181939],{"className":181938,"style":673},[642],[363,181940],{}," 减小）。第二条作用有一个退化倾向：若标签之外的证据可被任意压低而不受惩罚，网络便有动机把 ",[363,181943,181945,181962],{"className":181944,"translate":367},[366],[363,181946,181948],{"className":181947},[371],[373,181949,181950],{"xmlns":375},[377,181951,181952,181960],{},[380,181953,181954],{},[543,181955,181956,181958],{},[383,181957,45126],{},[383,181959,100402],{},[387,181961,181891],{"encoding":389},[363,181963,181965],{"className":181964,"ariaHidden":394},[393],[363,181966,181968,181971],{"className":181967},[398],[363,181969],{"className":181970,"style":93347},[402],[363,181972,181974,181977],{"className":181973},[407],[363,181975,45126],{"className":181976,"style":45156},[407,408],[363,181978,181980],{"className":181979},[629],[363,181981,181983,182003],{"className":181982},[633,634],[363,181984,181986,182000],{"className":181985},[638],[363,181987,181989],{"className":181988,"style":12532},[642],[363,181990,181991,181994],{"style":136154},[363,181992],{"className":181993,"style":651},[650],[363,181995,181997],{"className":181996},[655,656,657,658],[363,181998,100402],{"className":181999,"style":100930},[407,408,658],[363,182001,666],{"className":182002},[665],[363,182004,182006],{"className":182005},[638],[363,182007,182009],{"className":182008,"style":673},[642],[363,182010],{}," 全部驱向 ",[363,182013,182015,182028],{"className":182014,"translate":367},[366],[363,182016,182018],{"className":182017},[371],[373,182019,182020],{"xmlns":375},[377,182021,182022,182026],{},[380,182023,182024],{},[463,182025,465],{},[387,182027,465],{"encoding":389},[363,182029,182031],{"className":182030,"ariaHidden":394},[393],[363,182032,182034,182037],{"className":182033},[398],[363,182035],{"className":182036,"style":520},[402],[363,182038,465],{"className":182039},[407],"，从而收缩总证据 ",[363,182042,182044,182057],{"className":182043,"translate":367},[366],[363,182045,182047],{"className":182046},[371],[373,182048,182049],{"xmlns":375},[377,182050,182051,182055],{},[380,182052,182053],{},[383,182054,26109],{},[387,182056,26109],{"encoding":389},[363,182058,182060],{"className":182059,"ariaHidden":394},[393],[363,182061,182063,182066],{"className":182062},[398],[363,182064],{"className":182065,"style":1011},[402],[363,182067,26109],{"className":182068,"style":26233},[407,408],"——这正是 4.3 节引入正则项的直接原因。",[143,182071,182072],{},[11,182073,182074,6842,182077,182142,182143],{},[1344,182075,182076],{},"推论 4.2（与插值交叉熵的关系）",[363,182078,182080,182098],{"className":182079,"translate":367},[366],[363,182081,182083],{"className":182082},[371],[373,182084,182085],{"xmlns":375},[377,182086,182087,182095],{},[380,182088,182089],{},[7162,182090,182091,182093],{},[383,182092,22265],{"mathvariant":944},[1645,182094,180182],{},[387,182096,182097],{"encoding":389},"\\mathcal{L}^{\\text{NLL}}",[363,182099,182101],{"className":182100,"ariaHidden":394},[393],[363,182102,182104,182107],{"className":182103},[398],[363,182105],{"className":182106,"style":52457},[402],[363,182108,182110,182113],{"className":182109},[407],[363,182111,22265],{"className":182112},[407,971],[363,182114,182116],{"className":182115},[629],[363,182117,182119],{"className":182118},[633],[363,182120,182122],{"className":182121},[638],[363,182123,182125],{"className":182124,"style":52457},[642],[363,182126,182127,182130],{"style":7213},[363,182128],{"className":182129,"style":651},[650],[363,182131,182133],{"className":182132},[655,656,657,658],[363,182134,182136],{"className":182135},[407,658],[363,182137,182139],{"className":182138},[407,4946,658],[363,182140,180182],{"className":182141},[407,658]," 是 one-hot 标签下插值交叉熵的 Jensen 上界：\n",[363,182144,182146,182250],{"className":182145,"translate":367},[366],[363,182147,182149],{"className":182148},[371],[373,182150,182151],{"xmlns":375},[377,182152,182153,182247],{},[380,182154,182155,182161,182163,182165,182167,182169,182175,182177,182183,182207,182209,182211,182213,182229,182231,182233,182235,182237,182239,182241],{},[7162,182156,182157,182159],{},[383,182158,22265],{"mathvariant":944},[1645,182160,180182],{},[458,182162,471],{},[458,182164,5161],{},[383,182166,4448],{},[458,182168,4451],{},[543,182170,182171,182173],{},[383,182172,11],{},[383,182174,40609],{},[458,182176,2818],{},[586,182178,182179,182181],{},[463,182180,465],{},[463,182182,2811],{},[380,182184,182185,182187,182197,182199,182205],{},[458,182186,461],{"fence":394},[586,182188,182189,182191],{},[463,182190,465],{},[543,182192,182193,182195],{},[383,182194,45126],{},[383,182196,40609],{},[458,182198,5161],{},[586,182200,182201,182203],{},[463,182202,465],{},[383,182204,26109],{},[458,182206,468],{"fence":394},[458,182208,2818],{},[383,182210,19593],{},[1645,182212,1652],{},[380,182214,182215,182217,182227],{},[458,182216,461],{"fence":394},[7162,182218,182219,182221],{},[383,182220,26109],{},[380,182222,182223,182225],{},[458,182224,5161],{},[463,182226,2811],{},[458,182228,468],{"fence":394},[1645,182230,4631],{},[458,182232,7276],{},[1645,182234,4631],{},[458,182236,5161],{},[383,182238,4448],{},[458,182240,4451],{},[543,182242,182243,182245],{},[383,182244,11],{},[383,182246,40609],{},[387,182248,182249],{"encoding":389},"\\mathcal{L}^{\\text{NLL}} = -\\log p_y + \\frac{1}{2}\\left(\\frac{1}{\\alpha_y} - \\frac{1}{S}\\right) + O\\!\\left(S^{-2}\\right) \\ \\ge \\ -\\log p_y",[363,182251,182253,182303,182372,182656,182736],{"className":182252,"ariaHidden":394},[393],[363,182254,182256,182259,182294,182297,182300],{"className":182255},[398],[363,182257],{"className":182258,"style":52457},[402],[363,182260,182262,182265],{"className":182261},[407],[363,182263,22265],{"className":182264},[407,971],[363,182266,182268],{"className":182267},[629],[363,182269,182271],{"className":182270},[633],[363,182272,182274],{"className":182273},[638],[363,182275,182277],{"className":182276,"style":52457},[642],[363,182278,182279,182282],{"style":7213},[363,182280],{"className":182281,"style":651},[650],[363,182283,182285],{"className":182284},[655,656,657,658],[363,182286,182288],{"className":182287},[407,658],[363,182289,182291],{"className":182290},[407,4946,658],[363,182292,180182],{"className":182293},[407,658],[363,182295],{"className":182296,"style":506},[505],[363,182298,471],{"className":182299},[510],[363,182301],{"className":182302,"style":506},[505],[363,182304,182306,182309,182312,182315,182320,182323,182363,182366,182369],{"className":182305},[398],[363,182307],{"className":182308,"style":153752},[402],[363,182310,5161],{"className":182311},[407],[363,182313],{"className":182314,"style":754},[505],[363,182316,4502,182318],{"className":182317},[1832],[363,182319,4506],{"style":4505},[363,182321],{"className":182322,"style":754},[505],[363,182324,182326,182329],{"className":182325},[407],[363,182327,11],{"className":182328},[407,408],[363,182330,182332],{"className":182331},[629],[363,182333,182335,182355],{"className":182334},[633,634],[363,182336,182338,182352],{"className":182337},[638],[363,182339,182341],{"className":182340,"style":45301},[642],[363,182342,182343,182346],{"style":730},[363,182344],{"className":182345,"style":651},[650],[363,182347,182349],{"className":182348},[655,656,657,658],[363,182350,40609],{"className":182351,"style":1328},[407,408,658],[363,182353,666],{"className":182354},[665],[363,182356,182358],{"className":182357},[638],[363,182359,182361],{"className":182360,"style":673},[642],[363,182362],{},[363,182364],{"className":182365,"style":2935},[505],[363,182367,2818],{"className":182368},[2939],[363,182370],{"className":182371,"style":2935},[505],[363,182373,182375,182378,182446,182449,182647,182650,182653],{"className":182374},[398],[363,182376],{"className":182377,"style":68500},[402],[363,182379,182381,182384,182443],{"className":182380},[407],[363,182382],{"className":182383},[494,789],[363,182385,182387],{"className":182386},[586],[363,182388,182390,182435],{"className":182389},[633,634],[363,182391,182393,182432],{"className":182392},[638],[363,182394,182396,182410,182418],{"className":182395,"style":2872},[642],[363,182397,182398,182401],{"style":2875},[363,182399],{"className":182400,"style":809},[650],[363,182402,182404],{"className":182403},[655,656,657,658],[363,182405,182407],{"className":182406},[407,658],[363,182408,2811],{"className":182409},[407,658],[363,182411,182412,182415],{"style":821},[363,182413],{"className":182414,"style":809},[650],[363,182416],{"className":182417,"style":829},[828],[363,182419,182420,182423],{"style":2898},[363,182421],{"className":182422,"style":809},[650],[363,182424,182426],{"className":182425},[655,656,657,658],[363,182427,182429],{"className":182428},[407,658],[363,182430,465],{"className":182431},[407,658],[363,182433,666],{"className":182434},[665],[363,182436,182438],{"className":182437},[638],[363,182439,182441],{"className":182440,"style":2920},[642],[363,182442],{},[363,182444],{"className":182445},[501,789],[363,182447],{"className":182448,"style":754},[505],[363,182450,182452,182458,182564,182567,182570,182573,182641],{"className":182451},[758],[363,182453,182455],{"className":182454,"style":763},[494,762],[363,182456,461],{"className":182457},[767,4068],[363,182459,182461,182464,182561],{"className":182460},[407],[363,182462],{"className":182463},[494,789],[363,182465,182467],{"className":182466},[586],[363,182468,182470,182552],{"className":182469},[633,634],[363,182471,182473,182549],{"className":182472},[638],[363,182474,182476,182527,182535],{"className":182475,"style":2872},[642],[363,182477,182478,182481],{"style":2875},[363,182479],{"className":182480,"style":809},[650],[363,182482,182484],{"className":182483},[655,656,657,658],[363,182485,182487],{"className":182486},[407,658],[363,182488,182490,182493],{"className":182489},[407,658],[363,182491,45126],{"className":182492,"style":45156},[407,408,658],[363,182494,182496],{"className":182495},[629],[363,182497,182499,182519],{"className":182498},[633,634],[363,182500,182502,182516],{"className":182501},[638],[363,182503,182505],{"className":182504,"style":74645},[642],[363,182506,182507,182510],{"style":136989},[363,182508],{"className":182509,"style":30578},[650],[363,182511,182513],{"className":182512},[655,30582,10022,658],[363,182514,40609],{"className":182515,"style":1328},[407,408,658],[363,182517,666],{"className":182518},[665],[36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由 Jensen 不等式 ",[363,182802,182804,182862],{"className":182803,"translate":367},[366],[363,182805,182807],{"className":182806},[371],[373,182808,182809],{"xmlns":375},[377,182810,182811,182859],{},[380,182812,182813,182815,182817,182819,182821,182827,182829,182831,182833,182835,182837,182839,182845,182847,182849,182851,182853],{},[383,182814,571],{"mathvariant":570},[458,182816,578],{"stretchy":460},[383,182818,4448],{},[458,182820,4451],{},[543,182822,182823,182825],{},[383,182824,157459],{},[383,182826,40609],{},[458,182828,605],{"stretchy":460},[458,182830,10300],{},[383,182832,4448],{},[458,182834,4451],{},[383,182836,571],{"mathvariant":570},[458,182838,578],{"stretchy":460},[543,182840,182841,182843],{},[383,182842,157459],{},[383,182844,40609],{},[458,182846,605],{"stretchy":460},[458,182848,471],{},[383,182850,4448],{},[458,182852,4451],{},[543,182854,182855,182857],{},[383,182856,11],{},[383,182858,40609],{},[387,182860,182861],{"encoding":389},"\\mathbb{E}[\\log \\pi_y] \\le \\log \\mathbb{E}[\\pi_y] = \\log p_y",[363,182863,182865,182937,183009],{"className":182864,"ariaHidden":394},[393],[363,182866,182868,182871,182874,182877,182882,182885,182925,182928,182931,182934],{"className":182867},[398],[363,182869],{"className":182870,"style":618},[402],[363,182872,571],{"className":182873},[407,714],[363,182875,578],{"className":182876},[494],[363,182878,4502,182880],{"className":182879},[1832],[363,182881,4506],{"style":4505},[363,182883],{"className":182884,"style":754},[505],[363,182886,182888,182891],{"className":182887},[407],[363,182889,157459],{"className":182890,"style":1328},[407,408],[363,182892,182894],{"className":182893},[629],[363,182895,182897,182917],{"className":182896},[633,634],[363,182898,182900,182914],{"className":182899},[638],[363,182901,182903],{"className":182902,"style":45301},[642],[363,182904,182905,182908],{"style":19215},[363,182906],{"className":182907,"style":651},[650],[363,182909,182911],{"className":182910},[655,656,657,658],[363,182912,40609],{"className":182913,"style":1328},[407,408,658],[363,182915,666],{"className":182916},[665],[363,182918,182920],{"className":182919},[638],[363,182921,182923],{"className":182922,"style":673},[642],[363,182924],{},[363,182926,605],{"className":182927},[501],[363,182929],{"className":182930,"style":506},[505],[363,182932,10300],{"className":182933},[510],[363,182935],{"className":182936,"style":506},[505],[363,182938,182940,182943,182948,182951,182954,182957,182997,183000,183003,183006],{"className":182939},[398],[363,182941],{"className":182942,"style":618},[402],[363,182944,4502,182946],{"className":182945},[1832],[363,182947,4506],{"style":4505},[363,182949],{"className":182950,"style":754},[505],[363,182952,571],{"className":182953},[407,714],[363,182955,578],{"className":182956},[494],[363,182958,182960,182963],{"className":182959},[407],[363,182961,157459],{"className":182962,"style":1328},[407,408],[363,182964,182966],{"className":182965},[629],[363,182967,182969,182989],{"className":182968},[633,634],[363,182970,182972,182986],{"className":182971},[638],[363,182973,182975],{"className":182974,"style":45301},[642],[363,182976,182977,182980],{"style":19215},[363,182978],{"className":182979,"style":651},[650],[363,182981,182983],{"className":182982},[655,656,657,658],[363,182984,40609],{"className":182985,"style":1328},[407,408,658],[363,182987,666],{"className":182988},[665],[363,182990,182992],{"className":182991},[638],[363,182993,182995],{"className":182994,"style":673},[642],[363,182996],{},[363,182998,605],{"className":182999},[501],[363,183001],{"className":183002,"style":506},[505],[363,183004,471],{"className":183005},[510],[363,183007],{"className":183008,"style":506},[505],[363,183010,183012,183015,183020,183023],{"className":183011},[398],[363,183013],{"className":183014,"style":153752},[402],[363,183016,4502,183018],{"className":183017},[1832],[363,183019,4506],{"style":4505},[363,183021],{"className":183022,"style":754},[505],[363,183024,183026,183029],{"className":183025},[407],[363,183027,11],{"className":183028},[407,408],[363,183030,183032],{"className":183031},[629],[363,183033,183035,183055],{"className":183034},[633,634],[363,183036,183038,183052],{"className":183037},[638],[363,183039,183041],{"className":183040,"style":45301},[642],[363,183042,183043,183046],{"style":730},[363,183044],{"className":183045,"style":651},[650],[363,183047,183049],{"className":183048},[655,656,657,658],[363,183050,40609],{"className":183051,"style":1328},[407,408,658],[363,183053,666],{"className":183054},[665],[363,183056,183058],{"className":183057},[638],[363,183059,183061],{"className":183060,"style":673},[642],[363,183062],{},[363,183064,183066,183098],{"className":183065,"translate":367},[366],[363,183067,183069],{"className":183068},[371],[373,183070,183071],{"xmlns":375},[377,183072,183073,183095],{},[380,183074,183075,183081,183083,183085,183087,183089],{},[7162,183076,183077,183079],{},[383,183078,22265],{"mathvariant":944},[1645,183080,180182],{},[458,183082,7276],{},[458,183084,5161],{},[383,183086,4448],{},[458,183088,4451],{},[543,183090,183091,183093],{},[383,183092,11],{},[383,183094,40609],{},[387,183096,183097],{"encoding":389},"\\mathcal{L}^{\\text{NLL}} \\ge -\\log p_y",[363,183099,183101,183152],{"className":183100,"ariaHidden":394},[393],[363,183102,183104,183108,183143,183146,183149],{"className":183103},[398],[363,183105],{"className":183106,"style":183107},[402],"height:0.9773em;vertical-align:-0.136em;",[363,183109,183111,183114],{"className":183110},[407],[363,183112,22265],{"className":183113},[407,971],[363,183115,183117],{"className":183116},[629],[363,183118,183120],{"className":183119},[633],[363,183121,183123],{"className":183122},[638],[363,183124,183126],{"className":183125,"style":52457},[642],[363,183127,183128,183131],{"style":7213},[363,183129],{"className":183130,"style":651},[650],[363,183132,183134],{"className":183133},[655,656,657,658],[363,183135,183137],{"className":183136},[407,658],[363,183138,183140],{"className":183139},[407,4946,658],[363,183141,180182],{"className":183142},[407,658],[363,183144],{"className":183145,"style":506},[505],[363,183147,7276],{"className":183148},[510],[363,183150],{"className":183151,"style":506},[505],[363,183153,183155,183158,183161,183164,183169,183172],{"className":183154},[398],[363,183156],{"className":183157,"style":153752},[402],[363,183159,5161],{"className":183160},[407],[363,183162],{"className":183163,"style":754},[505],[363,183165,4502,183167],{"className":183166},[1832],[363,183168,4506],{"style":4505},[363,183170],{"className":183171,"style":754},[505],[363,183173,183175,183178],{"className":183174},[407],[363,183176,11],{"className":183177},[407,408],[363,183179,183181],{"className":183180},[629],[363,183182,183184,183204],{"className":183183},[633,634],[363,183185,183187,183201],{"className":183186},[638],[363,183188,183190],{"className":183189,"style":45301},[642],[363,183191,183192,183195],{"style":730},[363,183193],{"className":183194,"style":651},[650],[363,183196,183198],{"className":183197},[655,656,657,658],[363,183199,40609],{"className":183200,"style":1328},[407,408,658],[363,183202,666],{"className":183203},[665],[363,183205,183207],{"className":183206},[638],[363,183208,183210],{"className":183209,"style":673},[642],[363,183211],{},"。渐近展开由 ",[363,183214,183216,183274],{"className":183215,"translate":367},[366],[363,183217,183219],{"className":183218},[371],[373,183220,183221],{"xmlns":375},[377,183222,183223,183271],{},[380,183224,183225,183227,183229,183231,183233,183235,183237,183239,183241,183243,183253,183255,183257,183259,183269],{},[383,183226,155482],{},[458,183228,461],{"stretchy":460},[383,183230,45552],{},[458,183232,468],{"stretchy":460},[458,183234,471],{},[383,183236,4448],{},[458,183238,4451],{},[383,183240,45552],{},[458,183242,5161],{},[586,183244,183245,183247],{},[463,183246,465],{},[380,183248,183249,183251],{},[463,183250,2811],{},[383,183252,45552],{},[458,183254,2818],{},[383,183256,19593],{},[458,183258,461],{"stretchy":460},[7162,183260,183261,183263],{},[383,183262,45552],{},[380,183264,183265,183267],{},[458,183266,5161],{},[463,183268,2811],{},[458,183270,468],{"stretchy":460},[387,183272,183273],{"encoding":389},"\\psi(a) = \\log a - \\frac{1}{2a} + O(a^{-2})",[363,183275,183277,183304,183330,183416],{"className":183276,"ariaHidden":394},[393],[363,183278,183280,183283,183286,183289,183292,183295,183298,183301],{"className":183279},[398],[363,183281],{"className":183282,"style":487},[402],[363,183284,155482],{"className":183285,"style":1328},[407,408],[363,183287,461],{"className":183288},[494],[363,183290,45552],{"className":183291},[407,408],[363,183293,468],{"className":183294},[501],[363,183296],{"className":183297,"style":506},[505],[363,183299,471],{"className":183300},[510],[363,183302],{"className":183303,"style":506},[505],[363,183305,183307,183310,183315,183318,183321,183324,183327],{"className":183306},[398],[363,183308],{"className":183309,"style":403},[402],[363,183311,4502,183313],{"className":183312},[1832],[363,183314,4506],{"style":4505},[363,183316],{"className":183317,"style":754},[505],[363,183319,45552],{"className":183320},[407,408],[363,183322],{"className":183323,"style":2935},[505],[363,183325,5161],{"className":183326},[2939],[363,183328],{"className":183329,"style":2935},[505],[363,183331,183333,183336,183407,183410,183413],{"className":183332},[398],[363,183334],{"className":183335,"style":2853},[402],[363,183337,183339,183342,183404],{"className":183338},[407],[363,183340],{"className":183341},[494,789],[363,183343,183345],{"className":183344},[586],[363,183346,183348,183396],{"className":183347},[633,634],[363,183349,183351,183393],{"className":183350},[638],[363,183352,183354,183371,183379],{"className":183353,"style":2872},[642],[363,183355,183356,183359],{"style":2875},[363,183357],{"className":183358,"style":809},[650],[363,183360,183362],{"className":183361},[655,656,657,658],[363,183363,183365,183368],{"className":183364},[407,658],[363,183366,2811],{"className":183367},[407,658],[363,183369,45552],{"className":183370},[407,408,658],[363,183372,183373,183376],{"style":821},[363,183374],{"className":183375,"style":809},[650],[363,183377],{"className":183378,"style":829},[828],[363,183380,183381,183384],{"style":2898},[363,183382],{"className":183383,"style":809},[650],[363,183385,183387],{"className":183386},[655,656,657,658],[363,183388,183390],{"className":183389},[407,658],[363,183391,465],{"className":183392},[407,658],[363,183394,666],{"className":183395},[665],[363,183397,183399],{"className":183398},[638],[363,183400,183402],{"className":183401,"style":2920},[642],[363,183403],{},[363,183405],{"className":183406},[501,789],[363,183408],{"className":183409,"style":2935},[505],[363,183411,2818],{"className":183412},[2939],[363,183414],{"className":183415,"style":2935},[505],[363,183417,183419,183422,183425,183428,183463],{"className":183418},[398],[363,183420],{"className":183421,"style":20375},[402],[363,183423,19593],{"className":183424,"style":625},[407,408],[363,183426,461],{"className":183427},[494],[363,183429,183431,183434],{"className":183430},[407],[363,183432,45552],{"className":183433},[407,408],[363,183435,183437],{"className":183436},[629],[363,183438,183440],{"className":183439},[633],[363,183441,183443],{"className":183442},[638],[363,183444,183446],{"className":183445,"style":20400},[642],[363,183447,183448,183451],{"style":7213},[363,183449],{"className":183450,"style":651},[650],[363,183452,183454],{"className":183453},[655,656,657,658],[363,183455,183457,183460],{"className":183456},[407,658],[363,183458,5161],{"className":183459},[407,658],[363,183461,2811],{"className":183462},[407,658],[363,183464,468],{"className":183465},[501]," 给出，而 ",[363,183468,183470,183492],{"className":183469,"translate":367},[366],[363,183471,183473],{"className":183472},[371],[373,183474,183475],{"xmlns":375},[377,183476,183477,183489],{},[380,183478,183479,183485,183487],{},[543,183480,183481,183483],{},[383,183482,45126],{},[383,183484,40609],{},[458,183486,10300],{},[383,183488,26109],{},[387,183490,183491],{"encoding":389},"\\alpha_y \\le S",[363,183493,183495,183551],{"className":183494,"ariaHidden":394},[393],[363,183496,183498,183502,183542,183545,183548],{"className":183497},[398],[363,183499],{"className":183500,"style":183501},[402],"height:0.9221em;vertical-align:-0.2861em;",[363,183503,183505,183508],{"className":183504},[407],[363,183506,45126],{"className":183507,"style":45156},[407,408],[363,183509,183511],{"className":183510},[629],[363,183512,183514,183534],{"className":183513},[633,634],[363,183515,183517,183531],{"className":183516},[638],[363,183518,183520],{"className":183519,"style":45301},[642],[363,183521,183522,183525],{"style":136154},[363,183523],{"className":183524,"style":651},[650],[363,183526,183528],{"className":183527},[655,656,657,658],[363,183529,40609],{"className":183530,"style":1328},[407,408,658],[363,183532,666],{"className":183533},[665],[363,183535,183537],{"className":183536},[638],[363,183538,183540],{"className":183539,"style":673},[642],[363,183541],{},[363,183543],{"className":183544,"style":506},[505],[363,183546,10300],{"className":183547},[510],[363,183549],{"className":183550,"style":506},[505],[363,183552,183554,183557],{"className":183553},[398],[363,183555],{"className":183556,"style":1011},[402],[363,183558,26109],{"className":183559,"style":26233},[407,408]," 保证修正项非负。",[11,183562,183563,183564,183567,183568,183596],{},"推论 4.2 揭示了一个在文献中常被忽略的事实：",[1344,183565,183566],{},"EDL 的对数损失并非多项—迪利克雷边缘似然的负对数，而是它加上一个显式的、随证据稀缺而增大的修正项。"," 该项在 ",[363,183569,183571,183584],{"className":183570,"translate":367},[366],[363,183572,183574],{"className":183573},[371],[373,183575,183576],{"xmlns":375},[377,183577,183578,183582],{},[380,183579,183580],{},[383,183581,26109],{},[387,183583,26109],{"encoding":389},[363,183585,183587],{"className":183586,"ariaHidden":394},[393],[363,183588,183590,183593],{"className":183589},[398],[363,183591],{"className":183592,"style":1011},[402],[363,183594,26109],{"className":183595,"style":26233},[407,408]," 大时消失，在证据不足时提供额外压力，其作用与下一小节方差惩罚项在平方损失中的地位完全平行。",[2659,183598,183600],{"id":183599},"_42-平方损失的贝叶斯风险","4.2 平方损失的贝叶斯风险",[11,183602,183603],{},"Sensoy 等人的原始工作采用了平方损失而非对数损失，其闭式更为直观。",[143,183605,183606],{},[11,183607,183608,11410,183611,183640,183641],{},[1344,183609,183610],{},"命题 4.3（平方风险的闭式）",[363,183612,183614,183628],{"className":183613,"translate":367},[366],[363,183615,183617],{"className":183616},[371],[373,183618,183619],{"xmlns":375},[377,183620,183621,183625],{},[380,183622,183623],{},[383,183624,40609],{"mathvariant":75598},[387,183626,183627],{"encoding":389},"\\mathbf{y}",[363,183629,183631],{"className":183630,"ariaHidden":394},[393],[363,183632,183634,183637],{"className":183633},[398],[363,183635],{"className":183636,"style":133807},[402],[363,183638,40609],{"className":183639,"style":162156},[407,75943]," 为 one-hot 标签，则\n",[363,183642,183644,183769],{"className":183643,"translate":367},[366],[363,183645,183647],{"className":183646},[371],[373,183648,183649],{"xmlns":375},[377,183650,183651,183766],{},[380,183652,183653,183660,183662,183668,183688,183690,183704],{},[7162,183654,183655,183657],{},[383,183656,22265],{"mathvariant":944},[1645,183658,183659],{},"MSE",[458,183661,471],{},[543,183663,183664,183666],{},[383,183665,571],{"mathvariant":570},[383,183667,157459],{"mathvariant":136055},[380,183669,183670,183672,183674,183676,183678,183680,183686],{},[458,183671,578],{"fence":394},[383,183673,558],{"mathvariant":557},[383,183675,40609],{"mathvariant":75598},[458,183677,5161],{},[383,183679,157459],{"mathvariant":136055},[7162,183681,183682,183684],{},[383,183683,558],{"mathvariant":557},[463,183685,2811],{},[458,183687,605],{"fence":394},[458,183689,471],{},[7901,183691,183692,183694,183702],{},[458,183693,13811],{},[380,183695,183696,183698,183700],{},[383,183697,7912],{},[458,183699,471],{},[463,183701,465],{},[383,183703,11044],{},[380,183705,183706,183708,183710,183716,183718,183724,183730,183732,183764],{},[458,183707,578],{"fence":394},[458,183709,461],{"stretchy":460},[543,183711,183712,183714],{},[383,183713,40609],{},[383,183715,7912],{},[458,183717,5161],{},[543,183719,183720,183722],{},[383,183721,11],{},[383,183723,7912],{},[7162,183725,183726,183728],{},[458,183727,468],{"stretchy":460},[463,183729,2811],{},[458,183731,2818],{},[586,183733,183734,183756],{},[380,183735,183736,183742,183744,183746,183748,183754],{},[543,183737,183738,183740],{},[383,183739,11],{},[383,183741,7912],{},[458,183743,461],{"stretchy":460},[463,183745,465],{},[458,183747,5161],{},[543,183749,183750,183752],{},[383,183751,11],{},[383,183753,7912],{},[458,183755,468],{"stretchy":460},[380,183757,183758,183760,183762],{},[383,183759,26109],{},[458,183761,2818],{},[463,183763,465],{},[458,183765,605],{"fence":394},[387,183767,183768],{"encoding":389},"\\mathcal{L}^{\\text{MSE}} = \\mathbb{E}_{\\boldsymbol{\\pi}}\\left[\\|\\mathbf{y} - \\boldsymbol{\\pi}\\|^2\\right] = \\sum_{k=1}^K \\left[(y_k - p_k)^2 + \\frac{p_k(1 - p_k)}{S + 1}\\right]",[363,183770,183772,183822,183957],{"className":183771,"ariaHidden":394},[393],[363,183773,183775,183778,183813,183816,183819],{"className":183774},[398],[363,183776],{"className":183777,"style":52457},[402],[363,183779,183781,183784],{"className":183780},[407],[363,183782,22265],{"className":183783},[407,971],[363,183785,183787],{"className":183786},[629],[363,183788,183790],{"className":183789},[633],[363,183791,183793],{"className":183792},[638],[363,183794,183796],{"className":183795,"style":52457},[642],[363,183797,183798,183801],{"style":7213},[363,183799],{"className":183800,"style":651},[650],[363,183802,183804],{"className":183803},[655,656,657,658],[363,183805,183807],{"className":183806},[407,658],[363,183808,183810],{"className":183809},[407,4946,658],[363,183811,183659],{"className":183812},[407,658],[363,183814],{"className":183815,"style":506},[505],[363,183817,471],{"className":183818},[510],[363,183820],{"className":183821,"style":506},[505],[363,183823,183825,183828,183877,183880,183948,183951,183954],{"className":183824},[398],[363,183826],{"className":183827,"style":20935},[402],[363,183829,183831,183834],{"className":183830},[407],[363,183832,571],{"className":183833},[407,714],[363,183835,183837],{"className":183836},[629],[363,183838,183840,183869],{"className":183839},[633,634],[363,183841,183843,183866],{"className":183842},[638],[363,183844,183846],{"className":183845,"style":153771},[642],[363,183847,183848,183851],{"style":730},[363,183849],{"className":183850,"style":651},[650],[363,183852,183854],{"className":183853},[655,656,657,658],[363,183855,183857],{"className":183856},[407,658],[363,183858,183860],{"className":183859},[407,658],[363,183861,183863],{"className":183862},[407,658],[363,183864,157459],{"className":183865,"style":157493},[407,136115,658],[363,183867,666],{"className":183868},[665],[363,183870,183872],{"className":183871},[638],[363,183873,183875],{"className":183874,"style":8316},[642],[363,183876],{},[363,183878],{"className":183879,"style":754},[505],[363,183881,183883,183889,183892,183895,183898,183901,183904,183913,183942],{"className":183882},[758],[363,183884,183886],{"className":183885,"style":763},[494,762],[363,183887,578],{"className":183888},[767,10022],[363,183890,558],{"className":183891},[407],[363,183893,40609],{"className":183894,"style":162156},[407,75943],[363,183896],{"className":183897,"style":2935},[505],[363,183899,5161],{"className":183900},[2939],[363,183902],{"className":183903,"style":2935},[505],[363,183905,183907],{"className":183906},[407],[363,183908,183910],{"className":183909},[407],[363,183911,157459],{"className":183912,"style":157493},[407,136115],[363,183914,183916,183919],{"className":183915},[407],[363,183917,558],{"className":183918},[407],[363,183920,183922],{"className":183921},[629],[363,183923,183925],{"className":183924},[633],[363,183926,183928],{"className":183927},[638],[363,183929,183931],{"className":183930,"style":20400},[642],[363,183932,183933,183936],{"style":7213},[363,183934],{"className":183935,"style":651},[650],[363,183937,183939],{"className":183938},[655,656,657,658],[363,183940,2811],{"className":183941},[407,658],[363,183943,183945],{"className":183944,"style":763},[501,762],[363,183946,605],{"className":183947},[767,10022],[363,183949],{"className":183950,"style":506},[505],[363,183952,471],{"className":183953},[510],[363,183955],{"className":183956,"style":506},[505],[363,183958,183960,183963,184023,184026],{"className":183959},[398],[363,183961],{"className":183962,"style":68500},[402],[363,183964,183966,183969],{"className":183965},[1832],[363,183967,13811],{"className":183968,"style":40032},[1832,1833,5964],[363,183970,183972],{"className":183971},[629],[363,183973,183975,184015],{"className":183974},[633,634],[363,183976,183978,184012],{"className":183977},[638],[363,183979,183981,184001],{"className":183980,"style":133537},[642],[363,183982,183983,183986],{"style":40048},[363,183984],{"className":183985,"style":651},[650],[363,183987,183989],{"className":183988},[655,656,657,658],[363,183990,183992,183995,183998],{"className":183991},[407,658],[363,183993,7912],{"className":183994,"style":8615},[407,408,658],[363,183996,471],{"className":183997},[510,658],[363,183999,465],{"className":184000},[407,658],[363,184002,184003,184006],{"style":133560},[363,184004],{"className":184005,"style":651},[650],[363,184007,184009],{"className":184008},[655,656,657,658],[363,184010,11044],{"className":184011,"style":11132},[407,408,658],[363,184013,666],{"className":184014},[665],[363,184016,184018],{"className":184017},[638],[363,184019,184021],{"className":184020,"style":40067},[642],[363,184022],{},[363,184024],{"className":184025,"style":754},[505],[363,184027,184029,184035,184038,184078,184081,184084,184087,184127,184156,184159,184162,184165,184328],{"className":184028},[758],[363,184030,184032],{"className":184031,"style":763},[494,762],[363,184033,578],{"className":184034},[767,4068],[363,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e":184313},[501,658],[363,184315,666],{"className":184316},[665],[363,184318,184320],{"className":184319},[638],[363,184321,184323],{"className":184322,"style":16779},[642],[363,184324],{},[363,184326],{"className":184327},[501,789],[363,184329,184331],{"className":184330,"style":763},[501,762],[363,184332,605],{"className":184333},[767,4068],[11,184335,184336,184338,184339,184773,184774,185334],{},[1344,184337,138042],{}," 逐项展开：",[363,184340,184342,184422],{"className":184341,"translate":367},[366],[363,184343,184345],{"className":184344},[371],[373,184346,184347],{"xmlns":375},[377,184348,184349,184419],{},[380,184350,184351,184353,184355,184357,184363,184365,184371,184377,184379,184381,184383,184389,184391,184397,184403,184405,184407,184409,184411,184417],{},[383,184352,571],{"mathvariant":570},[458,184354,578],{"stretchy":460},[458,184356,461],{"stretchy":460},[543,184358,184359,184361],{},[383,184360,40609],{},[383,184362,7912],{},[458,184364,5161],{},[543,184366,184367,184369],{},[383,184368,157459],{},[383,184370,7912],{},[7162,184372,184373,184375],{},[458,184374,468],{"stretchy":460},[463,184376,2811],{},[458,184378,605],{"stretchy":460},[458,184380,471],{},[458,184382,461],{"stretchy":460},[543,184384,184385,184387],{},[383,184386,40609],{},[383,184388,7912],{},[458,184390,5161],{},[543,184392,184393,184395],{},[383,184394,11],{},[383,184396,7912],{},[7162,184398,184399,184401],{},[458,184400,468],{"stretchy":460},[463,184402,2811],{},[458,184404,2818],{},[383,184406,147692],{"mathvariant":557},[458,184408,4451],{},[458,184410,461],{"stretchy":460},[543,184412,184413,184415],{},[383,184414,157459],{},[383,184416,7912],{},[458,184418,468],{"stretchy":460},[387,184420,184421],{"encoding":389},"\\mathbb{E}[(y_k - \\pi_k)^2] = (y_k - p_k)^2 + \\operatorname{Var}(\\pi_k)",[363,184423,184425,184486,184573,184631,184715],{"className":184424,"ariaHidden":394},[393],[363,184426,184428,184431,184434,184437,184477,184480,184483],{"className":184427},[398],[363,184429],{"className":184430,"style":487},[402],[363,184432,571],{"className":184433},[407,714],[363,184435,123320],{"className":184436},[494],[363,184438,184440,184443],{"className":184439},[407],[363,184441,40609],{"className":184442,"style":1328},[407,408],[363,184444,184446],{"className":184445},[629],[363,184447,184449,184469],{"className":184448},[633,634],[363,184450,184452,184466],{"className":184451},[638],[363,184453,184455],{"className":184454,"style":643},[642],[363,184456,184457,184460],{"style":19215},[363,184458],{"className":184459,"style":651},[650],[363,184461,184463],{"className":184462},[655,656,657,658],[363,184464,7912],{"className":184465,"style":8615},[407,408,658],[363,184467,666],{"className":184468},[665],[363,184470,184472],{"className":184471},[638],[363,184473,184475],{"className":184474,"style":8316},[642],[363,184476],{},[363,184478],{"className":184479,"style":2935},[505],[363,184481,5161],{"className":184482},[2939],[363,184484],{"className":184485,"style":2935},[505],[363,184487,184489,184492,184532,184561,184564,184567,184570],{"className":184488},[398],[363,184490],{"className":184491,"style":20375},[402],[363,184493,184495,184498],{"className":184494},[407],[363,184496,157459],{"className":184497,"style":1328},[407,408],[363,184499,184501],{"className":184500},[629],[363,184502,184504,184524],{"className":184503},[633,634],[363,184505,184507,184521],{"className":184506},[638],[363,184508,184510],{"className":184509,"style":643},[642],[363,184511,184512,184515],{"style":19215},[363,184513],{"className":184514,"style":651},[650],[363,184516,184518],{"className":184517},[655,656,657,658],[363,184519,7912],{"className":184520,"style":8615},[407,408,658],[363,184522,666],{"className":184523},[665],[363,184525,184527],{"className":184526},[638],[363,184528,184530],{"className":184529,"style":8316},[642],[363,184531],{},[363,184533,184535,184538],{"className":184534},[501],[363,184536,468],{"className":184537},[501],[363,184539,184541],{"className":184540},[629],[363,184542,184544],{"className":184543},[633],[363,184545,184547],{"className":184546},[638],[363,184548,184550],{"className":184549,"style":20400},[642],[363,184551,184552,184555],{"style":7213},[363,184553],{"className":184554,"style":651},[650],[363,184556,184558],{"className":184557},[655,656,657,658],[363,184559,2811],{"className":184560},[407,658],[363,184562,605],{"className":184563},[501],[363,184565],{"className":184566,"style":506},[505],[363,184568,471],{"className":184569},[510],[363,184571],{"className":184572,"style":506},[505],[363,184574,184576,184579,184582,184622,184625,184628],{"className":184575},[398],[363,184577],{"className":184578,"style":487},[402],[363,184580,461],{"className":184581},[494],[363,184583,184585,184588],{"className":184584},[407],[363,184586,40609],{"className":184587,"style":1328},[407,408],[363,184589,184591],{"className":184590},[629],[363,184592,184594,184614],{"className":184593},[633,634],[363,184595,184597,184611],{"className":184596},[638],[363,184598,184600],{"className":184599,"style":643},[642],[363,184601,184602,184605],{"style":19215},[363,184603],{"className":184604,"style":651},[650],[363,184606,184608],{"className":184607},[655,656,657,658],[363,184609,7912],{"className":184610,"style":8615},[407,408,658],[363,184612,666],{"className":184613},[665],[363,184615,184617],{"className":184616},[638],[363,184618,184620],{"className":184619,"style":8316},[642],[363,184621],{},[363,184623],{"className":184624,"style":2935},[505],[363,184626,5161],{"className":184627},[2939],[363,184629],{"className":184630,"style":2935},[505],[363,184632,184634,184637,184677,184706,184709,184712],{"className":184633},[398],[363,184635],{"className":184636,"style":20375},[402],[363,184638,184640,184643],{"className":184639},[407],[363,184641,11],{"className":184642},[407,408],[363,184644,184646],{"className":184645},[629],[363,184647,184649,184669],{"className":184648},[633,634],[363,184650,184652,184666],{"className":184651},[638],[363,184653,184655],{"className":184654,"style":643},[642],[363,184656,184657,184660],{"style":730},[363,184658],{"className":184659,"style":651},[650],[363,184661,184663],{"className":184662},[655,656,657,658],[363,184664,7912],{"className":184665,"style":8615},[407,408,658],[363,184667,666],{"className":184668},[665],[363,184670,184672],{"className":184671},[638],[363,184673,184675],{"className":184674,"style":8316},[642],[363,184676],{},[363,184678,184680,184683],{"className":184679},[501],[363,184681,468],{"className":184682},[501],[363,184684,184686],{"className":184685},[629],[363,184687,184689],{"className":184688},[633],[363,184690,184692],{"className":184691},[638],[363,184693,184695],{"className":184694,"style":20400},[642],[363,184696,184697,184700],{"style":7213},[363,184698],{"className":184699,"style":651},[650],[363,184701,184703],{"className":184702},[655,656,657,658],[363,184704,2811],{"className":184705},[407,658],[363,184707],{"className":184708,"style":2935},[505],[363,184710,2818],{"className":184711},[2939],[363,184713],{"className":184714,"style":2935},[505],[363,184716,184718,184721,184727,184730,184770],{"className":184717},[398],[363,184719],{"className":184720,"style":487},[402],[363,184722,184724],{"className":184723},[1832],[363,184725,147692],{"className":184726},[407,33034],[363,184728,461],{"className":184729},[494],[363,184731,184733,184736],{"className":184732},[407],[363,184734,157459],{"className":184735,"style":1328},[407,408],[363,184737,184739],{"className":184738},[629],[363,184740,184742,184762],{"className":184741},[633,634],[363,184743,184745,184759],{"className":184744},[638],[363,184746,184748],{"className":184747,"style":643},[642],[363,184749,184750,184753],{"style":19215},[363,184751],{"className":184752,"style":651},[650],[363,184754,184756],{"className":184755},[655,656,657,658],[363,184757,7912],{"className":184758,"style":8615},[407,408,658],[363,184760,666],{"className":184761},[665],[363,184763,184765],{"className":184764},[638],[363,184766,184768],{"className":184767,"style":8316},[642],[363,184769],{},[363,184771,468],{"className":184772},[501],"。由命题 2.1，",[363,184775,184777,184881],{"className":184776,"translate":367},[366],[363,184778,184780],{"className":184779},[371],[373,184781,184782],{"xmlns":375},[377,184783,184784,184878],{},[380,184785,184786,184788,184790,184792,184798,184800,184802,184844,184846],{},[383,184787,147692],{"mathvariant":557},[458,184789,4451],{},[458,184791,461],{"stretchy":460},[543,184793,184794,184796],{},[383,184795,157459],{},[383,184797,7912],{},[458,184799,468],{"stretchy":460},[458,184801,471],{},[586,184803,184804,184826],{},[380,184805,184806,184812,184814,184816,184818,184824],{},[543,184807,184808,184810],{},[383,184809,45126],{},[383,184811,7912],{},[458,184813,461],{"stretchy":460},[383,184815,26109],{},[458,184817,5161],{},[543,184819,184820,184822],{},[383,184821,45126],{},[383,184823,7912],{},[458,184825,468],{"stretchy":460},[380,184827,184828,184834,184836,184838,184840,184842],{},[7162,184829,184830,184832],{},[383,184831,26109],{},[463,184833,2811],{},[458,184835,461],{"stretchy":460},[383,184837,26109],{},[458,184839,2818],{},[463,184841,465],{},[458,184843,468],{"stretchy":460},[458,184845,471],{},[586,184847,184848,184870],{},[380,184849,184850,184856,184858,184860,184862,184868],{},[543,184851,184852,184854],{},[383,184853,11],{},[383,184855,7912],{},[458,184857,461],{"stretchy":460},[463,184859,465],{},[458,184861,5161],{},[543,184863,184864,184866],{},[383,184865,11],{},[383,184867,7912],{},[458,184869,468],{"stretchy":460},[380,184871,184872,184874,184876],{},[383,184873,26109],{},[458,184875,2818],{},[463,184877,465],{},[387,184879,184880],{"encoding":389},"\\operatorname{Var}(\\pi_k) = \\frac{\\alpha_k(S - \\alpha_k)}{S^2(S+1)} = \\frac{p_k(1-p_k)}{S+1}",[363,184882,184884,184951,185164],{"className":184883,"ariaHidden":394},[393],[363,184885,184887,184890,184896,184899,184939,184942,184945,184948],{"className":184886},[398],[363,184888],{"className":184889,"style":487},[402],[363,184891,184893],{"className":184892},[1832],[363,184894,147692],{"className":184895},[407,33034],[363,184897,461],{"className":184898},[494],[363,184900,184902,184905],{"className":184901},[407],[363,184903,157459],{"className":184904,"style":1328},[407,408],[363,184906,184908],{"className":184907},[629],[363,184909,184911,184931],{"className":184910},[633,634],[363,184912,184914,184928],{"className":184913},[638],[363,184915,184917],{"className":184916,"style":643},[642],[363,184918,184919,184922],{"style":19215},[363,184920],{"className":184921,"style":651},[650],[363,184923,184925],{"className":184924},[655,656,657,658],[363,184926,7912],{"className":184927,"style":8615},[407,408,658],[363,184929,666],{"className":184930},[665],[363,184932,184934],{"className":184933},[638],[363,184935,184937],{"className":184936,"style":8316},[642],[363,184938],{},[363,184940,468],{"className":184941},[501],[363,184943],{"className":184944,"style":506},[505],[363,184946,471],{"className":184947},[510],[363,184949],{"className":184950,"style":506},[505],[363,184952,184954,184957,185155,185158,185161],{"className":184953},[398],[363,184955],{"className":184956,"style":150478},[402],[363,184958,184960,184963,185152],{"className":184959},[407],[363,184961],{"className":184962},[494,789],[363,184964,184966],{"className":184965},[586],[363,184967,184969,185144],{"className":184968},[633,634],[363,184970,184972,185141],{"className":184971},[638],[363,184973,184975,185030,185038],{"className":184974,"style":26765},[642],[363,184976,184977,184980],{"style":2875},[363,184978],{"className":184979,"style":809},[650],[363,184981,184983],{"className":184982},[655,656,657,658],[363,184984,184986,185015,185018,185021,185024,185027],{"className":184985},[407,658],[363,184987,184989,184992],{"className":184988},[407,658],[363,184990,26109],{"className":184991,"style":26233},[407,408,658],[363,184993,184995],{"className":184994},[629],[363,184996,184998],{"className":184997},[633],[363,184999,185001],{"className":185000},[638],[363,185002,185004],{"className":185003,"style":45831},[642],[363,185005,185006,185009],{"style":45834},[363,185007],{"className":185008,"style":30578},[650],[363,185010,185012],{"className":185011},[655,30582,10022,658],[363,185013,2811],{"className":185014},[407,658],[363,185016,461],{"className":185017},[494,658],[363,185019,26109],{"className":185020,"style":26233},[407,408,658],[363,185022,2818],{"className":185023},[2939,658],[363,185025,465],{"className":185026},[407,658],[363,185028,468],{"className":185029},[501,658],[363,185031,185032,185035],{"style":821},[363,185033],{"className":185034,"style":809},[650],[363,185036],{"className":185037,"style":829},[828],[363,185039,185040,185043],{"style":26790},[363,185041],{"className":185042,"style":809},[650],[363,185044,185046],{"className":185045},[655,656,657,658],[363,185047,185049,185089,185092,185095,185098,185138],{"className":185048},[407,658],[363,185050,185052,185055],{"className":185051},[407,658],[363,185053,45126],{"className":185054,"style":45156},[407,408,658],[363,185056,185058],{"className":185057},[629],[363,185059,185061,185081],{"className":185060},[633,634],[363,185062,185064,185078],{"className":185063},[638],[363,185065,185067],{"className":185066,"style":6446},[642],[363,185068,185069,185072],{"style":160006},[363,185070],{"className":185071,"style":30578},[650],[363,185073,185075],{"className":185074},[655,30582,10022,658],[363,185076,7912],{"className":185077,"style":8615},[407,408,658],[363,185079,666],{"className":185080},[665],[363,185082,185084],{"className":185083},[638],[363,185085,185087],{"className":185086,"style":101625},[642],[363,185088],{},[363,185090,461],{"className":185091},[494,658],[363,185093,26109],{"className":185094,"style":26233},[407,408,658],[363,185096,5161],{"className":185097},[2939,658],[363,185099,185101,185104],{"className":185100},[407,658],[363,185102,45126],{"className":185103,"style":45156},[407,408,658],[363,185105,185107],{"className":185106},[629],[363,185108,185110,185130],{"className":185109},[633,634],[363,185111,185113,185127],{"className":185112},[638],[363,185114,185116],{"className":185115,"style":6446},[642],[363,185117,185118,185121],{"style":160006},[363,185119],{"className":185120,"style":30578},[650],[363,185122,185124],{"className":185123},[655,30582,10022,658],[363,185125,7912],{"className":185126,"style":8615},[407,408,658],[363,185128,666],{"className":185129},[665],[363,185131,185133],{"className":185132},[638],[363,185134,185136],{"className":185135,"style":101625},[642],[363,185137],{},[363,185139,468],{"className":185140},[501,658],[363,185142,666],{"className":185143},[665],[363,185145,185147],{"className":185146},[638],[363,185148,185150],{"className":185149,"style":45974},[642],[363,185151],{},[363,185153],{"className":185154},[501,789],[363,185156],{"className":185157,"style":506},[505],[363,185159,471],{"className":185160},[510],[363,185162],{"className":185163,"style":506},[505],[363,185165,185167,185171],{"className":185166},[398],[363,185168],{"className":185169,"style":185170},[402],"height:1.4133em;vertical-align:-0.4033em;",[363,185172,185174,185177,185331],{"className":185173},[407],[363,185175],{"className":185176},[494,789],[363,185178,185180],{"className":185179},[586],[363,185181,185183,185323],{"className":185182},[633,634],[363,185184,185186,185320],{"className":185185},[638],[363,185187,185189,185209,185217],{"className":185188,"style":26765},[642],[363,185190,185191,185194],{"style":2875},[363,185192],{"className":185193,"style":809},[650],[363,185195,185197],{"className":185196},[655,656,657,658],[363,185198,185200,185203,185206],{"className":185199},[407,658],[363,185201,26109],{"className":185202,"style":26233},[407,408,658],[363,185204,2818],{"className":185205},[2939,658],[363,185207,465],{"className":185208},[407,658],[363,185210,185211,185214],{"style":821},[363,185212],{"className":185213,"style":809},[650],[363,185215],{"className":185216,"style":829},[828],[363,185218,185219,185222],{"style":26790},[363,185220],{"className":185221,"style":809},[650],[363,185223,185225],{"className":185224},[655,656,657,658],[363,185226,185228,185268,185271,185274,185277,185317],{"className":185227},[407,658],[363,185229,185231,185234],{"className":185230},[407,658],[363,185232,11],{"className":185233},[407,408,658],[363,185235,185237],{"className":185236},[629],[363,185238,185240,185260],{"className":185239},[633,634],[363,185241,185243,185257],{"className":185242},[638],[363,185244,185246],{"className":185245,"style":6446},[642],[363,185247,185248,185251],{"style":101606},[363,185249],{"className":185250,"style":30578},[650],[363,185252,185254],{"className":185253},[655,30582,10022,658],[363,185255,7912],{"className":185256,"style":8615},[407,408,658],[363,185258,666],{"className":185259},[665],[363,185261,185263],{"className":185262},[638],[363,185264,185266],{"className":185265,"style":101625},[642],[363,185267],{},[363,185269,461],{"className":185270},[494,658],[363,185272,465],{"className":185273},[407,658],[363,185275,5161],{"className":185276},[2939,658],[363,185278,185280,185283],{"className":185279},[407,658],[363,185281,11],{"className":185282},[407,408,658],[363,185284,185286],{"className":185285},[629],[363,185287,185289,185309],{"className":185288},[633,634],[363,185290,185292,185306],{"className":185291},[638],[363,185293,185295],{"className":185294,"style":6446},[642],[363,185296,185297,185300],{"style":101606},[363,185298],{"className":185299,"style":30578},[650],[363,185301,185303],{"className":185302},[655,30582,10022,658],[363,185304,7912],{"className":185305,"style":8615},[407,408,658],[363,185307,666],{"className":185308},[665],[363,185310,185312],{"className":185311},[638],[363,185313,185315],{"className":185314,"style":101625},[642],[363,185316],{},[363,185318,468],{"className":185319},[501,658],[363,185321,666],{"className":185322},[665],[363,185324,185326],{"className":185325},[638],[363,185327,185329],{"className":185328,"style":16779},[642],[363,185330],{},[363,185332],{"className":185333},[501,789],"。求和即得。",[11,185336,185337,185338,185341,185342,185412,185413,185416,185417,185565,185566,185617],{},"该损失的两项具有不同的角色。",[1344,185339,185340],{},"第一项是拟合项","：在 ",[363,185343,185345,185363],{"className":185344,"translate":367},[366],[363,185346,185348],{"className":185347},[371],[373,185349,185350],{"xmlns":375},[377,185351,185352,185360],{},[380,185353,185354],{},[543,185355,185356,185358],{},[383,185357,11],{},[383,185359,7912],{},[387,185361,185362],{"encoding":389},"p_k",[363,185364,185366],{"className":185365,"ariaHidden":394},[393],[363,185367,185369,185372],{"className":185368},[398],[363,185370],{"className":185371,"style":2624},[402],[363,185373,185375,185378],{"className":185374},[407],[363,185376,11],{"className":185377},[407,408],[363,185379,185381],{"className":185380},[629],[363,185382,185384,185404],{"className":185383},[633,634],[363,185385,185387,185401],{"className":185386},[638],[363,185388,185390],{"className":185389,"style":643},[642],[363,185391,185392,185395],{"style":730},[363,185393],{"className":185394,"style":651},[650],[363,185396,185398],{"className":185397},[655,656,657,658],[363,185399,7912],{"className":185400,"style":8615},[407,408,658],[363,185402,666],{"className":185403},[665],[363,185405,185407],{"className":185406},[638],[363,185408,185410],{"className":185409,"style":8316},[642],[363,185411],{}," 退化为确定性预测时，它退化为 one-hot 编码下的均方误差。",[1344,185414,185415],{},"第二项是方差惩罚项","：它正比于每个分量的伯努利方差 ",[363,185418,185420,185452],{"className":185419,"translate":367},[366],[363,185421,185423],{"className":185422},[371],[373,185424,185425],{"xmlns":375},[377,185426,185427,185449],{},[380,185428,185429,185435,185437,185439,185441,185447],{},[543,185430,185431,185433],{},[383,185432,11],{},[383,185434,7912],{},[458,185436,461],{"stretchy":460},[463,185438,465],{},[458,185440,5161],{},[543,185442,185443,185445],{},[383,185444,11],{},[383,185446,7912],{},[458,185448,468],{"stretchy":460},[387,185450,185451],{"encoding":389},"p_k(1-p_k)",[363,185453,185455,185516],{"className":185454,"ariaHidden":394},[393],[363,185456,185458,185461,185501,185504,185507,185510,185513],{"className":185457},[398],[363,185459],{"className":185460,"style":487},[402],[363,185462,185464,185467],{"className":185463},[407],[363,185465,11],{"className":185466},[407,408],[363,185468,185470],{"className":185469},[629],[363,185471,185473,185493],{"className":185472},[633,634],[363,185474,185476,185490],{"className":185475},[638],[363,185477,185479],{"className":185478,"style":643},[642],[363,185480,185481,185484],{"style":730},[363,185482],{"className":185483,"style":651},[650],[363,185485,185487],{"className":185486},[655,656,657,658],[363,185488,7912],{"className":185489,"style":8615},[407,408,658],[363,185491,666],{"className":185492},[665],[363,185494,185496],{"className":185495},[638],[363,185497,185499],{"className":185498,"style":8316},[642],[363,185500],{},[363,185502,461],{"className":185503},[494],[363,185505,465],{"className":185506},[407],[363,185508],{"className":185509,"style":2935},[505],[363,185511,5161],{"className":185512},[2939],[363,185514],{"className":185515,"style":2935},[505],[363,185517,185519,185522,185562],{"className":185518},[398],[363,185520],{"className":185521,"style":487},[402],[363,185523,185525,185528],{"className":185524},[407],[363,185526,11],{"className":185527},[407,408],[363,185529,185531],{"className":185530},[629],[363,185532,185534,185554],{"className":185533},[633,634],[363,185535,185537,185551],{"className":185536},[638],[363,185538,185540],{"className":185539,"style":643},[642],[363,185541,185542,185545],{"style":730},[363,185543],{"className":185544,"style":651},[650],[363,185546,185548],{"className":185547},[655,656,657,658],[363,185549,7912],{"className":185550,"style":8615},[407,408,658],[363,185552,666],{"className":185553},[665],[363,185555,185557],{"className":185556},[638],[363,185558,185560],{"className":185559,"style":8316},[642],[363,185561],{},[363,185563,468],{"className":185564},[501],"，反比于总证据 ",[363,185567,185569,185587],{"className":185568,"translate":367},[366],[363,185570,185572],{"className":185571},[371],[373,185573,185574],{"xmlns":375},[377,185575,185576,185584],{},[380,185577,185578,185580,185582],{},[383,185579,26109],{},[458,185581,2818],{},[463,185583,465],{},[387,185585,185586],{"encoding":389},"S + 1",[363,185588,185590,185608],{"className":185589,"ariaHidden":394},[393],[363,185591,185593,185596,185599,185602,185605],{"className":185592},[398],[363,185594],{"className":185595,"style":133944},[402],[363,185597,26109],{"className":185598,"style":26233},[407,408],[363,185600],{"className":185601,"style":2935},[505],[363,185603,2818],{"className":185604},[2939],[363,185606],{"className":185607,"style":2935},[505],[363,185609,185611,185614],{"className":185610},[398],[363,185612],{"className":185613,"style":520},[402],[363,185615,465],{"className":185616},[407],"。其作用是双重的：",[7010,185619,185620,185656],{},[7013,185621,185622,185623,185651,185652,185655],{},"当证据稀缺（",[363,185624,185626,185639],{"className":185625,"translate":367},[366],[363,185627,185629],{"className":185628},[371],[373,185630,185631],{"xmlns":375},[377,185632,185633,185637],{},[380,185634,185635],{},[383,185636,26109],{},[387,185638,26109],{"encoding":389},[363,185640,185642],{"className":185641,"ariaHidden":394},[393],[363,185643,185645,185648],{"className":185644},[398],[363,185646],{"className":185647,"style":1011},[402],[363,185649,26109],{"className":185650,"style":26233},[407,408]," 小）时，方差项较大，迫使模型",[1344,185653,185654],{},"不得","以低方差的方式给出尖锐预测——这是一种自适应的软惩罚，其强度由数据本身决定，而非人为设定的超参数。",[7013,185657,185658,185659,2757,185687,185756,185757,185785],{},"当证据变化时，",[363,185660,185662,185675],{"className":185661,"translate":367},[366],[363,185663,185665],{"className":185664},[371],[373,185666,185667],{"xmlns":375},[377,185668,185669,185673],{},[380,185670,185671],{},[383,185672,26109],{},[387,185674,26109],{"encoding":389},[363,185676,185678],{"className":185677,"ariaHidden":394},[393],[363,185679,185681,185684],{"className":185680},[398],[363,185682],{"className":185683,"style":1011},[402],[363,185685,26109],{"className":185686,"style":26233},[407,408],[363,185688,185690,185707],{"className":185689,"translate":367},[366],[363,185691,185693],{"className":185692},[371],[373,185694,185695],{"xmlns":375},[377,185696,185697,185705],{},[380,185698,185699],{},[543,185700,185701,185703],{},[383,185702,11],{},[383,185704,7912],{},[387,185706,185362],{"encoding":389},[363,185708,185710],{"className":185709,"ariaHidden":394},[393],[363,185711,185713,185716],{"className":185712},[398],[363,185714],{"className":185715,"style":2624},[402],[363,185717,185719,185722],{"className":185718},[407],[363,185720,11],{"className":185721},[407,408],[363,185723,185725],{"className":185724},[629],[363,185726,185728,185748],{"className":185727},[633,634],[363,185729,185731,185745],{"className":185730},[638],[363,185732,185734],{"className":185733,"style":643},[642],[363,185735,185736,185739],{"style":730},[363,185737],{"className":185738,"style":651},[650],[363,185740,185742],{"className":185741},[655,656,657,658],[363,185743,7912],{"className":185744,"style":8615},[407,408,658],[363,185746,666],{"className":185747},[665],[363,185749,185751],{"className":185750},[638],[363,185752,185754],{"className":185753,"style":8316},[642],[363,185755],{}," 同时进入两项，形成张力：增大 ",[363,185758,185760,185773],{"className":185759,"translate":367},[366],[363,185761,185763],{"className":185762},[371],[373,185764,185765],{"xmlns":375},[377,185766,185767,185771],{},[380,185768,185769],{},[383,185770,26109],{},[387,185772,26109],{"encoding":389},[363,185774,185776],{"className":185775,"ariaHidden":394},[393],[363,185777,185779,185782],{"className":185778},[398],[363,185780],{"className":185781,"style":1011},[402],[363,185783,26109],{"className":185784,"style":26233},[407,408]," 可以压低方差项，却会强化拟合项。因此最优解不会退化为「证据全为零」的平凡解。",[2659,185787,185789],{"id":185788},"_43-证据正则化与退火-kl","4.3 证据正则化与退火 KL",[11,185791,185792,185793,185796,185797,185874,185875,185878],{},"仅有拟合损失仍不足以约束错误预测上的证据。原因在于 4.1 节的梯度分析：拟合项只对",[1344,185794,185795],{},"真实类别","的证据给出直接的提升压力，而对错误类别上的证据，它只通过 ",[363,185798,185800,185824],{"className":185799,"translate":367},[366],[363,185801,185803],{"className":185802},[371],[373,185804,185805],{"xmlns":375},[377,185806,185807,185821],{},[380,185808,185809,185815,185817,185819],{},[7162,185810,185811,185813],{},[383,185812,155482],{},[458,185814,7168],{"mathvariant":557,"lspace":1595,"rspace":1595},[458,185816,461],{"stretchy":460},[383,185818,26109],{},[458,185820,468],{"stretchy":460},[387,185822,185823],{"encoding":389},"\\psi'(S)",[363,185825,185827],{"className":185826,"ariaHidden":394},[393],[363,185828,185830,185833,185865,185868,185871],{"className":185829},[398],[363,185831],{"className":185832,"style":7191},[402],[363,185834,185836,185839],{"className":185835},[407],[363,185837,155482],{"className":185838,"style":1328},[407,408],[363,185840,185842],{"className":185841},[629],[363,185843,185845],{"className":185844},[633],[363,185846,185848],{"className":185847},[638],[363,185849,185851],{"className":185850,"style":7210},[642],[363,185852,185853,185856],{"style":7213},[363,185854],{"className":185855,"style":651},[650],[363,185857,185859],{"className":185858},[655,656,657,658],[363,185860,185862],{"className":185861},[407,658],[363,185863,7168],{"className":185864},[407,658],[363,185866,461],{"className":185867},[494],[363,185869,26109],{"className":185870,"style":26233},[407,408],[363,185872,468],{"className":185873},[501]," 给出一个整体的压制信号，这一信号与证据在各类别间的分配无关。后果是：",[1344,185876,185877],{},"当一个样本被错误分类时，网络可以通过在错误类别上堆积证据来降低损失","，而这恰恰是我们希望惩罚的行为。",[11,185880,185881],{},"正则项的构造基于一个「反事实」的迪利克雷参数：抹去真实类别的证据，保留其余。",[143,185883,185884],{},[11,185885,185886,47619,185889,186064,186065,186093,186094],{},[1344,185887,185888],{},"定义 4.4（反事实参数与证据正则）",[363,185890,185892,185931],{"className":185891,"translate":367},[366],[363,185893,185895],{"className":185894},[371],[373,185896,185897],{"xmlns":375},[377,185898,185899,185928],{},[380,185900,185901,185907,185909,185911,185913,185915,185917,185919,185921,185923,185926],{},[12161,185902,185903,185905],{"accent":394},[383,185904,45126],{"mathvariant":136055},[458,185906,12167],{},[458,185908,471],{},[383,185910,40609],{"mathvariant":75598},[458,185912,2818],{},[458,185914,461],{"stretchy":460},[463,185916,465],{"mathvariant":75598},[458,185918,5161],{},[383,185920,40609],{"mathvariant":75598},[458,185922,468],{"stretchy":460},[458,185924,185925],{},"⊙",[383,185927,45126],{"mathvariant":136055},[387,185929,185930],{"encoding":389},"\\tilde{\\boldsymbol{\\alpha}} = \\mathbf{y} + (\\mathbf{1} - \\mathbf{y}) \\odot \\boldsymbol{\\alpha}",[363,185932,185934,185989,186007,186028,186049],{"className":185933,"ariaHidden":394},[393],[363,185935,185937,185941,185980,185983,185986],{"className":185936},[398],[363,185938],{"className":185939,"style":185940},[402],"height:0.6813em;",[363,185942,185944],{"className":185943},[407,12201],[363,185945,185947],{"className":185946},[633],[363,185948,185950],{"className":185949},[638],[363,185951,185953,185967],{"className":185952,"style":185940},[642],[363,185954,185955,185958],{"style":12213},[363,185956],{"className":185957,"style":809},[650],[363,185959,185961],{"className":185960},[407],[363,185962,185964],{"className":185963},[407],[363,185965,45126],{"className":185966},[407,136115],[363,185968,185970,185973],{"style":185969},"top:-3.3634em;",[363,185971],{"className":185972,"style":809},[650],[363,185974,185977],{"className":185975,"style":185976},[12229],"left:-0.25em;",[363,185978,12167],{"className":185979},[407],[363,185981],{"className":185982,"style":506},[505],[363,185984,471],{"className":185985},[510],[363,185987],{"className":185988,"style":506},[505],[363,185990,185992,185995,185998,186001,186004],{"className":185991},[398],[363,185993],{"className":185994,"style":40693},[402],[363,185996,40609],{"className":185997,"style":162156},[407,75943],[363,185999],{"className":186000,"style":2935},[505],[363,186002,2818],{"className":186003},[2939],[363,186005],{"className":186006,"style":2935},[505],[363,186008,186010,186013,186016,186019,186022,186025],{"className":186009},[398],[363,186011],{"className":186012,"style":487},[402],[363,186014,461],{"className":186015},[494],[363,186017,465],{"className":186018},[407,75943],[363,186020],{"className":186021,"style":2935},[505],[363,186023,5161],{"className":186024},[2939],[363,186026],{"className":186027,"style":2935},[505],[363,186029,186031,186034,186037,186040,186043,186046],{"className":186030},[398],[363,186032],{"className":186033,"style":487},[402],[363,186035,40609],{"className":186036,"style":162156},[407,75943],[363,186038,468],{"className":186039},[501],[363,186041],{"className":186042,"style":2935},[505],[363,186044,185925],{"className":186045},[2939],[363,186047],{"className":186048,"style":2935},[505],[363,186050,186052,186055],{"className":186051},[398],[363,186053],{"className":186054,"style":135662},[402],[363,186056,186058],{"className":186057},[407],[363,186059,186061],{"className":186060},[407],[363,186062,45126],{"className":186063},[407,136115],"，即把真实类别处的参数置为 ",[363,186066,186068,186081],{"className":186067,"translate":367},[366],[363,186069,186071],{"className":186070},[371],[373,186072,186073],{"xmlns":375},[377,186074,186075,186079],{},[380,186076,186077],{},[463,186078,465],{},[387,186080,465],{"encoding":389},[363,186082,186084],{"className":186083,"ariaHidden":394},[393],[363,186085,186087,186090],{"className":186086},[398],[363,186088],{"className":186089,"style":520},[402],[363,186091,465],{"className":186092},[407],"、其余保持不变。证据正则项定义为\n",[363,186095,186097,186158],{"className":186096,"translate":367},[366],[363,186098,186100],{"className":186099},[371],[373,186101,186102],{"xmlns":375},[377,186103,186104,186155],{},[380,186105,186106,186113,186115,186117,186119],{},[7162,186107,186108,186110],{},[383,186109,22265],{"mathvariant":944},[1645,186111,186112],{},"reg",[458,186114,471],{},[383,186116,33035],{"mathvariant":557},[458,186118,4451],{},[380,186120,186121,186123,186125,186127,186129,186135,186137,186139,186141,186143,186145,186147,186149,186151,186153],{},[458,186122,578],{"fence":394},[383,186124,137112],{"mathvariant":557},[458,186126,4451],{},[458,186128,461],{"stretchy":460},[12161,186130,186131,186133],{"accent":394},[383,186132,45126],{"mathvariant":136055},[458,186134,12167],{},[458,186136,468],{"stretchy":460},[1645,186138,1647],{},[458,186140,558],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[1645,186142,1647],{},[383,186144,137112],{"mathvariant":557},[458,186146,4451],{},[458,186148,461],{"stretchy":460},[463,186150,465],{"mathvariant":75598},[458,186152,468],{"stretchy":460},[458,186154,605],{"fence":394},[387,186156,186157],{"encoding":389},"\\mathcal{L}^{\\text{reg}} = \\operatorname{KL}\\left[\\operatorname{Dir}(\\tilde{\\boldsymbol{\\alpha}}) \\,\\big\\|\\, \\operatorname{Dir}(\\mathbf{1})\\right]",[363,186159,186161,186211],{"className":186160,"ariaHidden":394},[393],[363,186162,186164,186167,186202,186205,186208],{"className":186163},[398],[363,186165],{"className":186166,"style":1011},[402],[363,186168,186170,186173],{"className":186169},[407],[363,186171,22265],{"className":186172},[407,971],[363,186174,186176],{"className":186175},[629],[363,186177,186179],{"className":186178},[633],[363,186180,186182],{"className":186181},[638],[363,186183,186185],{"className":186184,"style":50725},[642],[363,186186,186187,186190],{"style":7213},[363,186188],{"className":186189,"style":651},[650],[363,186191,186193],{"className":186192},[655,656,657,658],[363,186194,186196],{"className":186195},[407,658],[363,186197,186199],{"className":186198},[407,4946,658],[363,186200,186112],{"className":186201},[407,658],[363,186203],{"className":186204,"style":506},[505],[363,186206,471],{"className":186207},[510],[363,186209],{"className":186210,"style":506},[505],[363,186212,186214,186217,186223,186226],{"className":186213},[398],[363,186215],{"className":186216,"style":20935},[402],[363,186218,186220],{"className":186219},[1832],[363,186221,33035],{"className":186222},[407,33034],[363,186224],{"className":186225,"style":754},[505],[363,186227,186229,186235,186241,186244,186281,186284,186287,186324,186327,186330,186336,186339,186342,186345],{"className":186228},[758],[363,186230,186232],{"className":186231,"style":763},[494,762],[363,186233,578],{"className":186234},[767,10022],[363,186236,186238],{"className":186237},[1832],[363,186239,137112],{"className":186240},[407,33034],[363,186242,461],{"className":186243},[494],[363,186245,186247],{"className":186246},[407,12201],[363,186248,186250],{"className":186249},[633],[363,186251,186253],{"className":186252},[638],[363,186254,186256,186270],{"className":186255,"style":185940},[642],[363,186257,186258,186261],{"style":12213},[363,186259],{"className":186260,"style":809},[650],[363,186262,186264],{"className":186263},[407],[363,186265,186267],{"className":186266},[407],[363,186268,45126],{"className":186269},[407,136115],[363,186271,186272,186275],{"style":185969},[363,186273],{"className":186274,"style":809},[650],[363,186276,186278],{"className":186277,"style":185976},[12229],[363,186279,12167],{"className":186280},[407],[363,186282,468],{"className":186283},[501],[363,186285],{"className":186286,"style":754},[505],[363,186288,186290],{"className":186289},[407],[363,186291,186293],{"className":186292},[767,15179],[363,186294,186296,186316],{"className":186295},[633,634],[363,186297,186299,186313],{"className":186298},[638],[363,186300,186302],{"className":186301,"style":15189},[642],[363,186303,186304,186307],{"style":15192},[363,186305],{"className":186306,"style":15196},[650],[363,186308,186309],{"style":15199},[6671,186310,186311],{"xmlns":6673,"width":15202,"height":9798,"viewBox":15203},[6679,186312],{"d":15206},[363,186314,666],{"className":186315},[665],[363,186317,186319],{"className":186318},[638],[363,186320,186322],{"className":186321,"style":15216},[642],[363,186323],{},[363,186325],{"className":186326,"style":754},[505],[363,186328],{"className":186329,"style":754},[505],[363,186331,186333],{"className":186332},[1832],[363,186334,137112],{"className":186335},[407,33034],[363,186337,461],{"className":186338},[494],[363,186340,465],{"className":186341},[407,75943],[363,186343,468],{"className":186344},[501],[363,186346,186348],{"className":186347,"style":763},[501,762],[363,186349,605],{"className":186350},[767,10022],[11,186352,5925,186353,186402,186403,186490],{},[363,186354,186356,186378],{"className":186355,"translate":367},[366],[363,186357,186359],{"className":186358},[371],[373,186360,186361],{"xmlns":375},[377,186362,186363,186375],{},[380,186364,186365,186367,186369,186371,186373],{},[383,186366,137112],{"mathvariant":557},[458,186368,4451],{},[458,186370,461],{"stretchy":460},[463,186372,465],{"mathvariant":75598},[458,186374,468],{"stretchy":460},[387,186376,186377],{"encoding":389},"\\operatorname{Dir}(\\mathbf{1})",[363,186379,186381],{"className":186380,"ariaHidden":394},[393],[363,186382,186384,186387,186393,186396,186399],{"className":186383},[398],[363,186385],{"className":186386,"style":487},[402],[363,186388,186390],{"className":186389},[1832],[363,186391,137112],{"className":186392},[407,33034],[363,186394,461],{"className":186395},[494],[363,186397,465],{"className":186398},[407,75943],[363,186400,468],{"className":186401},[501]," 正是无信息均匀先验。该 KL 散度只依赖于非目标类别上的证据：若网络在错误类别上给出大量证据，",[363,186404,186406,186432],{"className":186405,"translate":367},[366],[363,186407,186409],{"className":186408},[371],[373,186410,186411],{"xmlns":375},[377,186412,186413,186429],{},[380,186414,186415,186417,186419,186421,186427],{},[383,186416,137112],{"mathvariant":557},[458,186418,4451],{},[458,186420,461],{"stretchy":460},[12161,186422,186423,186425],{"accent":394},[383,186424,45126],{"mathvariant":136055},[458,186426,12167],{},[458,186428,468],{"stretchy":460},[387,186430,186431],{"encoding":389},"\\operatorname{Dir}(\\tilde{\\boldsymbol{\\alpha}})",[363,186433,186435],{"className":186434,"ariaHidden":394},[393],[363,186436,186438,186441,186447,186450,186487],{"className":186437},[398],[363,186439],{"className":186440,"style":487},[402],[363,186442,186444],{"className":186443},[1832],[363,186445,137112],{"className":186446},[407,33034],[363,186448,461],{"className":186449},[494],[363,186451,186453],{"className":186452},[407,12201],[363,186454,186456],{"className":186455},[633],[363,186457,186459],{"className":186458},[638],[363,186460,186462,186476],{"className":186461,"style":185940},[642],[363,186463,186464,186467],{"style":12213},[363,186465],{"className":186466,"style":809},[650],[363,186468,186470],{"className":186469},[407],[363,186471,186473],{"className":186472},[407],[363,186474,45126],{"className":186475},[407,136115],[363,186477,186478,186481],{"style":185969},[363,186479],{"className":186480,"style":809},[650],[363,186482,186484],{"className":186483,"style":185976},[12229],[363,186485,12167],{"className":186486},[407],[363,186488,468],{"className":186489},[501]," 将显著偏离均匀分布，KL 随之增大。",[143,186492,186493],{},[11,186494,186495,58675,186498,137661,186638,187324,187325,187382,187383,187469,187470],{},[1344,186496,186497],{},"命题 4.5（Dirichlet 的 KL 散度）",[363,186499,186501,186533],{"className":186500,"translate":367},[366],[363,186502,186504],{"className":186503},[371],[373,186505,186506],{"xmlns":375},[377,186507,186508,186530],{},[380,186509,186510,186512,186514,186516,186518],{},[383,186511,45126],{"mathvariant":136055},[458,186513,898],{"separator":394},[383,186515,84504],{"mathvariant":136055},[458,186517,6257],{},[7901,186519,186520,186522,186528],{},[383,186521,1381],{"mathvariant":570},[380,186523,186524,186526],{},[458,186525,3251],{},[463,186527,474],{},[383,186529,11044],{},[387,186531,186532],{"encoding":389},"\\boldsymbol{\\alpha}, \\boldsymbol{\\beta} \\in \\mathbb{R}_{>0}^K",[363,186534,186536,186575],{"className":186535,"ariaHidden":394},[393],[363,186537,186539,186542,186551,186554,186557,186566,186569,186572],{"className":186538},[398],[363,186540],{"className":186541,"style":403},[402],[363,186543,186545],{"className":186544},[407],[363,186546,186548],{"className":186547},[407],[363,186549,45126],{"className":186550},[407,136115],[363,186552,898],{"className":186553},[920],[363,186555],{"className":186556,"style":754},[505],[363,186558,186560],{"className":186559},[407],[363,186561,186563],{"className":186562},[407],[363,186564,84504],{"className":186565,"style":150195},[407,136115],[363,186567],{"className":186568,"style":506},[505],[363,186570,6257],{"className":186571},[510],[363,186573],{"className":186574,"style":506},[505],[363,186576,186578,186581],{"className":186577},[398],[363,186579],{"className":186580,"style":136251},[402],[363,186582,186584,186587],{"className":186583},[407],[363,186585,1381],{"className":186586},[407,714],[363,186588,186590],{"className":186589},[629],[363,186591,186593,186630],{"className":186592},[633,634],[363,186594,186596,186627],{"className":186595},[638],[363,186597,186599,186616],{"className":186598,"style":52457},[642],[363,186600,186601,186604],{"style":136272},[363,186602],{"className":186603,"style":651},[650],[363,186605,186607],{"className":186606},[655,656,657,658],[363,186608,186610,186613],{"className":186609},[407,658],[363,186611,3251],{"className":186612},[510,658],[363,186614,474],{"className":186615},[407,658],[363,186617,186618,186621],{"style":7213},[363,186619],{"className":186620,"style":651},[650],[363,186622,186624],{"className":186623},[655,656,657,658],[363,186625,11044],{"className":186626,"style":11132},[407,408,658],[363,186628,666],{"className":186629},[665],[363,186631,186633],{"className":186632},[638],[363,186634,186636],{"className":186635,"style":136308},[642],[363,186637],{},[363,186639,186641,186781],{"className":186640,"translate":367},[366],[363,186642,186644],{"className":186643},[371],[373,186645,186646],{"xmlns":375},[377,186647,186648,186778],{},[380,186649,186650,186652,186654,186686,186688,186690,186692,186714,186716,186730,186732,186738,186740,186746,186748,186750,186752,186754,186760,186762,186764,186766,186768,186774,186776],{},[383,186651,33035],{"mathvariant":557},[458,186653,4451],{},[380,186655,186656,186658,186660,186662,186664,186666,186668,186670,186672,186674,186676,186678,186680,186682,186684],{},[458,186657,578],{"fence":394},[383,186659,137112],{"mathvariant":557},[458,186661,4451],{},[458,186663,461],{"stretchy":460},[383,186665,45126],{"mathvariant":136055},[458,186667,468],{"stretchy":460},[1645,186669,1647],{},[458,186671,558],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[1645,186673,1647],{},[383,186675,137112],{"mathvariant":557},[458,186677,4451],{},[458,186679,461],{"stretchy":460},[383,186681,84504],{"mathvariant":136055},[458,186683,468],{"stretchy":460},[458,186685,605],{"fence":394},[458,186687,471],{},[383,186689,4448],{},[458,186691,4451],{},[586,186693,186694,186704],{},[380,186695,186696,186698,186700,186702],{},[383,186697,39663],{},[458,186699,461],{"stretchy":460},[383,186701,84504],{"mathvariant":136055},[458,186703,468],{"stretchy":460},[380,186705,186706,186708,186710,186712],{},[383,186707,39663],{},[458,186709,461],{"stretchy":460},[383,186711,45126],{"mathvariant":136055},[458,186713,468],{"stretchy":460},[458,186715,2818],{},[7901,186717,186718,186720,186728],{},[458,186719,13811],{},[380,186721,186722,186724,186726],{},[383,186723,7912],{},[458,186725,471],{},[463,186727,465],{},[383,186729,11044],{},[458,186731,461],{"stretchy":460},[543,186733,186734,186736],{},[383,186735,45126],{},[383,186737,7912],{},[458,186739,5161],{},[543,186741,186742,186744],{},[383,186743,84504],{},[383,186745,7912],{},[458,186747,468],{"stretchy":460},[458,186749,461],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[383,186751,155482],{},[458,186753,461],{"stretchy":460},[543,186755,186756,186758],{},[383,186757,45126],{},[383,186759,7912],{},[458,186761,468],{"stretchy":460},[458,186763,5161],{},[383,186765,155482],{},[458,186767,461],{"stretchy":460},[543,186769,186770,186772],{},[383,186771,45126],{},[463,186773,474],{},[458,186775,468],{"stretchy":460},[458,186777,468],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[387,186779,186780],{"encoding":389},"\\operatorname{KL}\\left[\\operatorname{Dir}(\\boldsymbol{\\alpha}) \\,\\big\\|\\, \\operatorname{Dir}(\\boldsymbol{\\beta})\\right] = \\log\\frac{B(\\boldsymbol{\\beta})}{B(\\boldsymbol{\\alpha})} + \\sum_{k=1}^K (\\alpha_k - \\beta_k)\\big(\\psi(\\alpha_k) - \\psi(\\alpha_0)\\big)",[363,186782,186784,186911,187032,187150,187263],{"className":186783,"ariaHidden":394},[393],[363,186785,186787,186790,186796,186799,186902,186905,186908],{"className":186786},[398],[363,186788],{"className":186789,"style":20935},[402],[363,186791,186793],{"className":186792},[1832],[363,186794,33035],{"className":186795},[407,33034],[363,186797],{"className":186798,"style":754},[505],[363,186800,186802,186808,186814,186817,186826,186829,186832,186869,186872,186875,186881,186884,186893,186896],{"className":186801},[758],[363,186803,186805],{"className":186804,"style":763},[494,762],[363,186806,578],{"className":186807},[767,10022],[363,186809,186811],{"className":186810},[1832],[363,186812,137112],{"className":186813},[407,33034],[363,186815,461],{"className":186816},[494],[363,186818,186820],{"className":186819},[407],[363,186821,186823],{"className":186822},[407],[363,186824,45126],{"className":186825},[407,136115],[363,186827,468],{"className":186828},[501],[363,186830],{"className":186831,"style":754},[505],[363,186833,186835],{"className":186834},[407],[363,186836,186838],{"className":186837},[767,15179],[363,186839,186841,186861],{"className":186840},[633,634],[363,186842,186844,186858],{"className":186843},[638],[363,186845,186847],{"className":186846,"style":15189},[642],[363,186848,186849,186852],{"style":15192},[363,186850],{"className":186851,"style":15196},[650],[363,186853,186854],{"style":15199},[6671,186855,186856],{"xmlns":6673,"width":15202,"height":9798,"viewBox":15203},[6679,186857],{"d":15206},[363,186859,666],{"className":186860},[665],[363,186862,186864],{"className":186863},[638],[363,186865,186867],{"className":186866,"style":15216},[642],[363,186868],{},[363,186870],{"className":186871,"style":754},[505],[363,186873],{"className":186874,"style":754},[505],[363,186876,186878],{"className":186877},[1832],[363,186879,137112],{"className":186880},[407,33034],[363,186882,461],{"className":186883},[494],[363,186885,186887],{"className":186886},[407],[363,186888,186890],{"className":186889},[407],[363,186891,84504],{"className":186892,"style":150195},[407,136115],[363,186894,468],{"className":186895},[501],[363,186897,186899],{"className":186898,"style":763},[501,762],[363,186900,605],{"className":186901},[767,10022],[363,186903],{"className":186904,"style":506},[505],[363,186906,471],{"className":186907},[510],[363,186909],{"className":186910,"style":506},[505],[363,186912,186914,186917,186922,186925,187023,187026,187029],{"className":186913},[398],[363,186915],{"className":186916,"style":150478},[402],[363,186918,4502,186920],{"className":186919},[1832],[363,186921,4506],{"style":4505},[363,186923],{"className":186924,"style":754},[505],[363,186926,186928,186931,187020],{"className":186927},[407],[363,186929],{"className":186930},[494,789],[363,186932,186934],{"className":186933},[586],[363,186935,186937,187012],{"className":186936},[633,634],[363,186938,186940,187009],{"className":186939},[638],[363,186941,186943,186972,186980],{"className":186942,"style":26765},[642],[363,186944,186945,186948],{"style":2875},[363,186946],{"className":186947,"style":809},[650],[363,186949,186951],{"className":186950},[655,656,657,658],[363,186952,186954,186957,186960,186969],{"className":186953},[407,658],[363,186955,39663],{"className":186956,"style":39759},[407,408,658],[363,186958,461],{"className":186959},[494,658],[363,186961,186963],{"className":186962},[407,658],[363,186964,186966],{"className":186965},[407,658],[363,186967,45126],{"className":186968},[407,136115,658],[363,186970,468],{"className":186971},[501,658],[363,186973,186974,186977],{"style":821},[363,186975],{"className":186976,"style":809},[650],[363,186978],{"className":186979,"style":829},[828],[363,186981,186982,186985],{"style":26790},[363,186983],{"className":186984,"style":809},[650],[363,186986,186988],{"className":186987},[655,656,657,658],[363,186989,186991,186994,186997,187006],{"className":186990},[407,658],[363,186992,39663],{"className":186993,"style":39759},[407,408,658],[363,186995,461],{"className":186996},[494,658],[363,186998,187000],{"className":186999},[407,658],[363,187001,187003],{"className":187002},[407,658],[363,187004,84504],{"className":187005,"style":150195},[407,136115,658],[363,187007,468],{"className":187008},[501,658],[363,187010,666],{"className":187011},[665],[363,187013,187015],{"className":187014},[638],[363,187016,187018],{"className":187017,"style":45974},[642],[363,187019],{},[363,187021],{"className":187022},[501,789],[363,187024],{"className":187025,"style":2935},[505],[363,187027,2818],{"className":187028},[2939],[363,187030],{"className":187031,"style":2935},[505],[363,187033,187035,187038,187098,187101,187141,187144,187147],{"className":187034},[398],[363,187036],{"className":187037,"style":136418},[402],[363,187039,187041,187044],{"className":187040},[1832],[363,187042,13811],{"className":187043,"style":40032},[1832,1833,5964],[363,187045,187047],{"className":187046},[629],[363,187048,187050,187090],{"className":187049},[633,634],[363,187051,187053,187087],{"className":187052},[638],[363,187054,187056,187076],{"className":187055,"style":133537},[642],[363,187057,187058,187061],{"style":40048},[363,187059],{"className":187060,"style":651},[650],[363,187062,187064],{"className":187063},[655,656,657,658],[363,187065,187067,187070,187073],{"className":187066},[407,658],[363,187068,7912],{"className":187069,"style":8615},[407,408,658],[363,187071,471],{"className":187072},[510,658],[363,187074,465],{"className":187075},[407,658],[363,187077,187078,187081],{"style":133560},[363,187079],{"className":187080,"style":651},[650],[363,187082,187084],{"className":187083},[655,656,657,658],[363,187085,11044],{"className":187086,"style":11132},[407,408,658],[363,187088,666],{"className":187089},[665],[363,187091,187093],{"className":187092},[638],[363,187094,187096],{"className":187095,"style":40067},[642],[363,187097],{},[363,187099,461],{"className":187100},[494],[363,187102,187104,187107],{"className":187103},[407],[363,187105,45126],{"className":187106,"style":45156},[407,408],[363,187108,187110],{"className":187109},[629],[363,187111,187113,187133],{"className":187112},[633,634],[363,187114,187116,187130],{"className":187115},[638],[363,187117,187119],{"className":187118,"style":643},[642],[363,187120,187121,187124],{"style":136154},[363,187122],{"className":187123,"style":651},[650],[363,187125,187127],{"className":187126},[655,656,657,658],[363,187128,7912],{"className":187129,"style":8615},[407,408,658],[363,187131,666],{"className":187132},[665],[363,187134,187136],{"className":187135},[638],[363,187137,187139],{"className":187138,"style":8316},[642],[363,187140],{},[363,187142],{"className":187143,"style":2935},[505],[363,187145,5161],{"className":187146},[2939],[363,187148],{"className":187149,"style":2935},[505],[363,187151,187153,187156,187196,187199,187205,187208,187211,187251,187254,187257,187260],{"className":187152},[398],[363,187154],{"className":187155,"style":20935},[402],[363,187157,187159,187162],{"className":187158},[407],[363,187160,84504],{"className":187161,"style":84726},[407,408],[363,187163,187165],{"className":187164},[629],[363,187166,187168,187188],{"className":187167},[633,634],[363,187169,187171,187185],{"className":187170},[638],[363,187172,187174],{"className":187173,"style":643},[642],[363,187175,187176,187179],{"style":150274},[363,187177],{"className":187178,"style":651},[650],[363,187180,187182],{"className":187181},[655,656,657,658],[363,187183,7912],{"className":187184,"style":8615},[407,408,658],[363,187186,666],{"className":187187},[665],[363,187189,187191],{"className":187190},[638],[363,187192,187194],{"className":187193,"style":8316},[642],[363,187195],{},[363,187197,468],{"className":187198},[501],[363,187200,187202],{"className":187201},[407],[363,187203,461],{"className":187204},[767,10022],[363,187206,155482],{"className":187207,"style":1328},[407,408],[363,187209,461],{"className":187210},[494],[363,187212,187214,187217],{"className":187213},[407],[363,187215,45126],{"className":187216,"style":45156},[407,408],[363,187218,187220],{"className":187219},[629],[363,187221,187223,187243],{"className":187222},[633,634],[363,187224,187226,187240],{"className":187225},[638],[363,187227,187229],{"className":187228,"style":643},[642],[363,187230,187231,187234],{"style":136154},[363,187232],{"className":187233,"style":651},[650],[363,187235,187237],{"className":187236},[655,656,657,658],[363,187238,7912],{"className":187239,"style":8615},[407,408,658],[363,187241,666],{"className":187242},[665],[363,187244,187246],{"className":187245},[638],[363,187247,187249],{"className":187248,"style":8316},[642],[363,187250],{},[363,187252,468],{"className":187253},[501],[363,187255],{"className":187256,"style":2935},[505],[363,187258,5161],{"className":187259},[2939],[363,187261],{"className":187262,"style":2935},[505],[363,187264,187266,187269,187272,187275,187315,187318],{"className":187265},[398],[363,187267],{"className":187268,"style":20935},[402],[363,187270,155482],{"className":187271,"style":1328},[407,408],[363,187273,461],{"className":187274},[494],[363,187276,187278,187281],{"className":187277},[407],[363,187279,45126],{"className":187280,"style":45156},[407,408],[363,187282,187284],{"className":187283},[629],[363,187285,187287,187307],{"className":187286},[633,634],[363,187288,187290,187304],{"className":187289},[638],[363,187291,187293],{"className":187292,"style":11483},[642],[363,187294,187295,187298],{"style":136154},[363,187296],{"className":187297,"style":651},[650],[363,187299,187301],{"className":187300},[655,656,657,658],[363,187302,474],{"className":187303},[407,658],[363,187305,666],{"className":187306},[665],[363,187308,187310],{"className":187309},[638],[363,187311,187313],{"className":187312,"style":8316},[642],[363,187314],{},[363,187316,468],{"className":187317},[501],[363,187319,187321],{"className":187320},[407],[363,187322,468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",[363,187326,187328,187346],{"className":187327,"translate":367},[366],[363,187329,187331],{"className":187330},[371],[373,187332,187333],{"xmlns":375},[377,187334,187335,187343],{},[380,187336,187337,187339,187341],{},[383,187338,84504],{"mathvariant":136055},[458,187340,471],{},[463,187342,465],{"mathvariant":75598},[387,187344,187345],{"encoding":389},"\\boldsymbol{\\beta} = \\mathbf{1}",[363,187347,187349,187373],{"className":187348,"ariaHidden":394},[393],[363,187350,187352,187355,187364,187367,187370],{"className":187351},[398],[363,187353],{"className":187354,"style":403},[402],[363,187356,187358],{"className":187357},[407],[363,187359,187361],{"className":187360},[407],[363,187362,84504],{"className":187363,"style":150195},[407,136115],[363,187365],{"className":187366,"style":506},[505],[363,187368,471],{"className":187369},[510],[363,187371],{"className":187372,"style":506},[505],[363,187374,187376,187379],{"className":187375},[398],[363,187377],{"className":187378,"style":520},[402],[363,187380,465],{"className":187381},[407,75943],"（此时 ",[363,187384,187386,187420],{"className":187385,"translate":367},[366],[363,187387,187389],{"className":187388},[371],[373,187390,187391],{"xmlns":375},[377,187392,187393,187417],{},[380,187394,187395,187397,187399,187401,187403,187405,187407,187409,187411,187413,187415],{},[383,187396,39663],{},[458,187398,461],{"stretchy":460},[463,187400,465],{"mathvariant":75598},[458,187402,468],{"stretchy":460},[458,187404,471],{},[463,187406,465],{},[383,187408,1288],{"mathvariant":557},[383,187410,137218],{"mathvariant":557},[458,187412,461],{"stretchy":460},[383,187414,11044],{},[458,187416,468],{"stretchy":460},[387,187418,187419],{"encoding":389},"B(\\mathbf{1}) = 1\u002F\\Gamma(K)",[363,187421,187423,187450],{"className":187422,"ariaHidden":394},[393],[363,187424,187426,187429,187432,187435,187438,187441,187444,187447],{"className":187425},[398],[363,187427],{"className":187428,"style":487},[402],[363,187430,39663],{"className":187431,"style":39759},[407,408],[363,187433,461],{"className":187434},[494],[363,187436,465],{"className":187437},[407,75943],[363,187439,468],{"className":187440},[501],[363,187442],{"className":187443,"style":506},[505],[363,187445,471],{"className":187446},[510],[363,187448],{"className":187449,"style":506},[505],[363,187451,187453,187456,187460,187463,187466],{"className":187452},[398],[363,187454],{"className":187455,"style":487},[402],[363,187457,187459],{"className":187458},[407],"1\u002FΓ",[363,187461,461],{"className":187462},[494],[363,187464,11044],{"className":187465,"style":11132},[407,408],[363,187467,468],{"className":187468},[501],"）得\n",[363,187471,187473,187635],{"className":187472,"translate":367},[366],[363,187474,187476],{"className":187475},[371],[373,187477,187478],{"xmlns":375},[377,187479,187480,187632],{},[380,187481,187482,187488,187490,187492,187494,187496,187498,187508,187510,187512,187526,187528,187530,187532,187534,187544,187546,187548,187550,187552,187554,187556,187558,187560,187562,187576,187578,187588,187590,187592,187594,187596,187598,187600,187610,187612,187614,187616,187618,187628,187630],{},[7162,187483,187484,187486],{},[383,187485,22265],{"mathvariant":944},[1645,187487,186112],{},[458,187489,471],{},[383,187491,4448],{},[458,187493,4451],{},[383,187495,137218],{"mathvariant":557},[458,187497,461],{"stretchy":460},[543,187499,187500,187506],{},[12161,187501,187502,187504],{"accent":394},[383,187503,45126],{},[458,187505,12167],{},[463,187507,474],{},[458,187509,468],{"stretchy":460},[458,187511,5161],{},[7901,187513,187514,187516,187524],{},[458,187515,13811],{},[380,187517,187518,187520,187522],{},[383,187519,7912],{},[458,187521,471],{},[463,187523,465],{},[383,187525,11044],{},[383,187527,4448],{},[458,187529,4451],{},[383,187531,137218],{"mathvariant":557},[458,187533,461],{"stretchy":460},[543,187535,187536,187542],{},[12161,187537,187538,187540],{"accent":394},[383,187539,45126],{},[458,187541,12167],{},[383,187543,7912],{},[458,187545,468],{"stretchy":460},[458,187547,5161],{},[383,187549,4448],{},[458,187551,4451],{},[383,187553,137218],{"mathvariant":557},[458,187555,461],{"stretchy":460},[383,187557,11044],{},[458,187559,468],{"stretchy":460},[458,187561,2818],{},[7901,187563,187564,187566,187574],{},[458,187565,13811],{},[380,187567,187568,187570,187572],{},[383,187569,7912],{},[458,187571,471],{},[463,187573,465],{},[383,187575,11044],{},[458,187577,461],{"stretchy":460},[543,187579,187580,187586],{},[12161,187581,187582,187584],{"accent":394},[383,187583,45126],{},[458,187585,12167],{},[383,187587,7912],{},[458,187589,5161],{},[463,187591,465],{},[458,187593,468],{"stretchy":460},[458,187595,461],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[383,187597,155482],{},[458,187599,461],{"stretchy":460},[543,187601,187602,187608],{},[12161,187603,187604,187606],{"accent":394},[383,187605,45126],{},[458,187607,12167],{},[383,187609,7912],{},[458,187611,468],{"stretchy":460},[458,187613,5161],{},[383,187615,155482],{},[458,187617,461],{"stretchy":460},[543,187619,187620,187626],{},[12161,187621,187622,187624],{"accent":394},[383,187623,45126],{},[458,187625,12167],{},[463,187627,474],{},[458,187629,468],{"stretchy":460},[458,187631,468],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[387,187633,187634],{"encoding":389},"\\mathcal{L}^{\\text{reg}} = \\log\\Gamma(\\tilde{\\alpha}_0) - \\sum_{k=1}^K \\log\\Gamma(\\tilde{\\alpha}_k) - \\log\\Gamma(K) + \\sum_{k=1}^K (\\tilde{\\alpha}_k - 1)\\big(\\psi(\\tilde{\\alpha}_k) - \\psi(\\tilde{\\alpha}_0)\\big)",[363,187636,187638,187688,187790,187953,187988,188134,188238],{"className":187637,"ariaHidden":394},[393],[363,187639,187641,187644,187679,187682,187685],{"className":187640},[398],[363,187642],{"className":187643,"style":1011},[402],[363,187645,187647,187650],{"className":187646},[407],[363,187648,22265],{"className":187649},[407,971],[363,187651,187653],{"className":187652},[629],[363,187654,187656],{"className":187655},[633],[363,187657,187659],{"className":187658},[638],[363,187660,187662],{"className":187661,"style":50725},[642],[363,187663,187664,187667],{"style":7213},[363,187665],{"className":187666,"style":651},[650],[363,187668,187670],{"className":187669},[655,656,657,658],[363,187671,187673],{"className":187672},[407,658],[363,187674,187676],{"className":187675},[407,4946,658],[363,187677,186112],{"className":187678},[407,658],[363,187680],{"className":187681,"style":506},[505],[363,187683,471],{"className":187684},[510],[363,187686],{"className":187687,"style":506},[505],[363,187689,187691,187694,187699,187702,187705,187708,187778,187781,187784,187787],{"className":187690},[398],[363,187692],{"className":187693,"style":487},[402],[363,187695,4502,187697],{"className":187696},[1832],[363,187698,4506],{"style":4505},[363,187700],{"className":187701,"style":754},[505],[363,187703,137218],{"className":187704},[407],[363,187706,461],{"className":187707},[494],[363,187709,187711,187744],{"className":187710},[407],[363,187712,187714],{"className":187713},[407,12201],[363,187715,187717],{"className":187716},[633],[363,187718,187720],{"className":187719},[638],[363,187721,187723,187731],{"className":187722,"style":52265},[642],[363,187724,187725,187728],{"style":12213},[363,187726],{"className":187727,"style":809},[650],[363,187729,45126],{"className":187730,"style":45156},[407,408],[363,187732,187734,187737],{"style":187733},"top:-3.35em;",[363,187735],{"className":187736,"style":809},[650],[363,187738,187741],{"className":187739,"style":187740},[12229],"left:-0.2222em;",[363,187742,12167],{"className":187743},[407],[363,187745,187747],{"className":187746},[629],[363,187748,187750,187770],{"className":187749},[633,634],[363,187751,187753,187767],{"className":187752},[638],[363,187754,187756],{"className":187755,"style":11483},[642],[363,187757,187758,187761],{"style":136154},[363,187759],{"className":187760,"style":651},[650],[363,187762,187764],{"className":187763},[655,656,657,658],[363,187765,474],{"className":187766},[407,658],[363,187768,666],{"className":187769},[665],[363,187771,187773],{"className":187772},[638],[363,187774,187776],{"className":187775,"style":8316},[642],[363,187777],{},[363,187779,468],{"className":187780},[501],[363,187782],{"className":187783,"style":2935},[505],[363,187785,5161],{"className":187786},[2939],[363,187788],{"className":187789,"style":2935},[505],[363,187791,187793,187796,187856,187859,187864,187867,187870,187873,187941,187944,187947,187950],{"className":187792},[398],[363,187794],{"className":187795,"style":136418},[402],[363,187797,187799,187802],{"className":187798},[1832],[363,187800,13811],{"className":187801,"style":40032},[1832,1833,5964],[363,187803,187805],{"className":187804},[629],[363,187806,187808,187848],{"className":187807},[633,634],[363,187809,187811,187845],{"className":187810},[638],[363,187812,187814,187834],{"className":187813,"style":133537},[642],[363,187815,187816,187819],{"style":40048},[363,187817],{"className":187818,"style":651},[650],[363,187820,187822],{"className":187821},[655,656,657,658],[363,187823,187825,187828,187831],{"className":187824},[407,658],[363,187826,7912],{"className":187827,"style":8615},[407,408,658],[363,187829,471],{"className":187830},[510,658],[363,187832,465],{"className":187833},[407,658],[363,187835,187836,187839],{"style":133560},[363,187837],{"className":187838,"style":651},[650],[363,187840,187842],{"className":187841},[655,656,657,658],[363,187843,11044],{"className":187844,"style":11132},[407,408,658],[363,187846,666],{"className":187847},[665],[363,187849,187851],{"className":187850},[638],[363,187852,187854],{"className":187853,"style":40067},[642],[363,187855],{},[363,187857],{"className":187858,"style":754},[505],[363,187860,4502,187862],{"className":187861},[1832],[363,187863,4506],{"style":4505},[363,187865],{"className":187866,"style":754},[505],[363,187868,137218],{"className":187869},[407],[363,187871,461],{"className":187872},[494],[363,187874,187876,187907],{"className":187875},[407],[363,187877,187879],{"className":187878},[407,12201],[363,187880,187882],{"className":187881},[633],[363,187883,187885],{"className":187884},[638],[363,187886,187888,187896],{"className":187887,"style":52265},[642],[363,187889,187890,187893],{"style":12213},[363,187891],{"className":187892,"style":809},[650],[363,187894,45126],{"className":187895,"style":45156},[407,408],[363,187897,187898,187901],{"style":187733},[363,187899],{"className":187900,"style":809},[650],[363,187902,18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188175,188178],{"style":12213},[363,188176],{"className":188177,"style":809},[650],[363,188179,45126],{"className":188180,"style":45156},[407,408],[363,188182,188183,188186],{"style":187733},[363,188184],{"className":188185,"style":809},[650],[363,188187,188189],{"className":188188,"style":187740},[12229],[363,188190,12167],{"className":188191},[407],[363,188193,188195],{"className":188194},[629],[363,188196,188198,188218],{"className":188197},[633,634],[363,188199,188201,188215],{"className":188200},[638],[363,188202,188204],{"className":188203,"style":643},[642],[363,188205,188206,188209],{"style":136154},[363,188207],{"className":188208,"style":651},[650],[363,188210,188212],{"className":188211},[655,656,657,658],[363,188213,7912],{"className":188214,"style":8615},[407,408,658],[363,188216,666],{"className":188217},[665],[363,188219,188221],{"className":188220},[638],[363,188222,188224],{"className":188223,"style":8316},[642],[363,188225],{},[363,188227,468],{"className":188228},[501],[363,188230],{"className":188231,"style":2935},[505],[363,188233,5161],{"className":188234},[2939],[363,188236],{"className":188237,"style":2935},[505],[363,188239,188241,188244,188247,188250,188318,188321],{"className":188240},[398],[363,188242],{"className":188243,"style":20935},[402],[363,188245,155482],{"className":188246,"style":1328},[407,408],[363,188248,461],{"className":188249},[494],[363,188251,188253,188284],{"className":188252},[407],[363,188254,188256],{"className":188255},[407,12201],[363,188257,188259],{"className":188258},[633],[363,188260,188262],{"className":188261},[638],[363,188263,188265,188273],{"className":188264,"style":52265},[642],[363,188266,188267,188270],{"style":12213},[363,188268],{"className":188269,"style":809},[650],[363,188271,45126],{"className":188272,"style":45156},[407,408],[363,188274,188275,188278],{"style":187733},[363,188276],{"className":188277,"style":809},[650],[363,188279,188281],{"className":188280,"style":187740},[12229],[363,188282,12167],{"className":188283},[407],[363,188285,188287],{"className":188286},[629],[363,188288,188290,188310],{"className":188289},[633,634],[363,188291,188293,188307],{"className":188292},[638],[363,188294,188296],{"className":188295,"style":11483},[642],[363,188297,188298,188301],{"style":136154},[363,188299],{"className":188300,"style":651},[650],[363,188302,188304],{"className":188303},[655,656,657,658],[363,188305,474],{"className":188306},[407,658],[363,188308,666],{"className":188309},[665],[363,188311,188313],{"className":188312},[638],[363,188314,188316],{"className":188315,"style":8316},[642],[363,188317],{},[363,188319,468],{"className":188320},[501],[363,188322,188324],{"className":188323},[407],[363,188325,468],{"className":188326},[767,10022],[11,188328,188329,188331,188332,188599,188600,2757,188903,189330],{},[1344,188330,138042],{}," 由指数族的一般 KL 公式：",[363,188333,188335,188385],{"className":188334,"translate":367},[366],[363,188336,188338],{"className":188337},[371],[373,188339,188340],{"xmlns":375},[377,188341,188342,188382],{},[380,188343,188344,188346,188348,188350,188356,188358,188360,188362,188368,188370,188372,188374,188380],{},[383,188345,33035],{"mathvariant":557},[458,188347,4451],{},[458,188349,471],{},[543,188351,188352,188354],{},[383,188353,571],{"mathvariant":570},[383,188355,45126],{"mathvariant":136055},[458,188357,578],{"stretchy":460},[383,188359,4448],{},[458,188361,4451],{},[543,188363,188364,188366],{},[383,188365,11],{},[383,188367,45126],{"mathvariant":136055},[458,188369,5161],{},[383,188371,4448],{},[458,188373,4451],{},[543,188375,188376,188378],{},[383,188377,11],{},[383,188379,84504],{"mathvariant":136055},[458,188381,605],{"stretchy":460},[387,188383,188384],{"encoding":389},"\\operatorname{KL} = \\mathbb{E}_{\\boldsymbol{\\alpha}}[\\log p_{\\boldsymbol{\\alpha}} - \\log p_{\\boldsymbol{\\beta}}]",[363,188386,188388,188409,188533],{"className":188387,"ariaHidden":394},[393],[363,188389,188391,188394,188400,188403,188406],{"className":188390},[398],[363,188392],{"className":188393,"style":1011},[402],[363,188395,188397],{"className":188396},[1832],[363,188398,33035],{"className":188399},[407,33034],[363,188401],{"className":188402,"style":506},[505],[363,188404,471],{"className":188405},[510],[363,188407],{"className":188408,"style":506},[505],[363,188410,188412,188415,188464,188467,188472,188475,188524,188527,188530],{"className":188411},[398],[363,188413],{"className":188414,"style":487},[402],[363,188416,188418,188421],{"className":188417},[407],[363,188419,571],{"className":188420},[407,714],[363,188422,188424],{"className":188423},[629],[363,188425,188427,188456],{"className":188426},[633,634],[363,188428,188430,188453],{"className":188429},[638],[363,188431,188433],{"className":188432,"style":153771},[642],[363,188434,188435,188438],{"style":730},[363,188436],{"className":188437,"style":651},[650],[363,188439,188441],{"className":188440},[655,656,657,658],[363,188442,188444],{"className":188443},[407,658],[363,188445,188447],{"className":188446},[407,658],[363,188448,188450],{"className":188449},[407,658],[363,188451,45126],{"className":188452},[407,136115,658],[363,188454,666],{"className":188455},[665],[363,188457,188459],{"className":188458},[638],[363,188460,188462],{"className":188461,"style":8316},[642],[363,188463],{},[363,188465,578],{"className":188466},[494],[363,188468,4502,188470],{"className":188469},[1832],[363,188471,4506],{"style":4505},[363,188473],{"className":188474,"style":754},[505],[363,188476,188478,188481],{"className":188477},[407],[363,188479,11],{"className":188480},[407,408],[363,188482,188484],{"className":188483},[629],[363,188485,188487,188516],{"className":188486},[633,634],[363,188488,188490,188513],{"className":188489},[638],[363,188491,188493],{"className":188492,"style":153771},[642],[363,188494,188495,188498],{"style":730},[363,188496],{"className":188497,"style":651},[650],[363,188499,188501],{"className":188500},[655,656,657,658],[363,188502,188504],{"className":188503},[407,658],[363,188505,188507],{"className":188506},[407,658],[363,188508,188510],{"className":188509},[407,658],[363,188511,45126],{"className":188512},[407,136115,658],[363,188514,666],{"className":188515},[665],[363,188517,188519],{"className":188518},[638],[363,188520,188522],{"className":188521,"style":8316},[642],[363,188523],{},[363,188525],{"className":188526,"style":2935},[505],[363,188528,5161],{"className":188529},[2939],[363,188531],{"className":188532,"style":2935},[505],[363,188534,188536,188539,188544,188547,188596],{"className":188535},[398],[363,188537],{"className":188538,"style":618},[402],[363,188540,4502,188542],{"className":188541},[1832],[363,188543,4506],{"style":4505},[363,188545],{"className":188546,"style":754},[505],[363,188548,188550,188553],{"className":188549},[407],[363,188551,11],{"className":188552},[407,408],[363,188554,188556],{"className":188555},[629],[363,188557,188559,188588],{"className":188558},[633,634],[363,188560,188562,188585],{"className":188561},[638],[363,188563,188565],{"className":188564,"style":643},[642],[363,188566,188567,188570],{"style":730},[363,188568],{"className":188569,"style":651},[650],[363,188571,188573],{"className":188572},[655,656,657,658],[363,188574,188576],{"className":188575},[407,658],[363,188577,188579],{"className":188578},[407,658],[363,188580,188582],{"className":188581},[407,658],[363,188583,84504],{"className":188584,"style":150195},[407,136115,658],[363,188586,666],{"className":188587},[665],[363,188589,188591],{"className":188590},[638],[363,188592,188594],{"className":188593,"style":673},[642],[363,188595],{},[363,188597,605],{"className":188598},[501],"。代入 1.2 节的密度并利用 ",[363,188601,188603,188663],{"className":188602,"translate":367},[366],[363,188604,188606],{"className":188605},[371],[373,188607,188608],{"xmlns":375},[377,188609,188610,188660],{},[380,188611,188612,188618,188620,188622,188624,188630,188632,188634,188636,188638,188644,188646,188648,188650,188652,188658],{},[543,188613,188614,188616],{},[383,188615,571],{"mathvariant":570},[383,188617,45126],{"mathvariant":136055},[458,188619,578],{"stretchy":460},[383,188621,4448],{},[458,188623,4451],{},[543,188625,188626,188628],{},[383,188627,1121],{},[383,188629,7912],{},[458,188631,605],{"stretchy":460},[458,188633,471],{},[383,188635,155482],{},[458,188637,461],{"stretchy":460},[543,188639,188640,188642],{},[383,188641,45126],{},[383,188643,7912],{},[458,188645,468],{"stretchy":460},[458,188647,5161],{},[383,188649,155482],{},[458,188651,461],{"stretchy":460},[543,188653,188654,188656],{},[383,188655,45126],{},[463,188657,474],{},[458,188659,468],{"stretchy":460},[387,188661,188662],{"encoding":389},"\\mathbb{E}_{\\boldsymbol{\\alpha}}[\\log x_k] = \\psi(\\alpha_k) - \\psi(\\alpha_0)",[363,188664,188666,188784,188848],{"className":188665,"ariaHidden":394},[393],[363,188667,188669,188672,188721,188724,188729,188732,188772,188775,188778,188781],{"className":188668},[398],[363,188670],{"className":188671,"style":487},[402],[363,188673,188675,188678],{"className":188674},[407],[363,188676,571],{"className":188677},[407,714],[363,188679,188681],{"className":188680},[629],[363,188682,188684,188713],{"className":188683},[633,634],[363,188685,188687,188710],{"className":188686},[638],[363,188688,188690],{"className":188689,"style":153771},[642],[363,188691,188692,188695],{"style":730},[363,188693],{"className":188694,"style":651},[650],[363,188696,188698],{"className":188697},[655,656,657,658],[363,188699,188701],{"className":188700},[407,658],[363,188702,188704],{"className":188703},[407,658],[363,188705,188707],{"className":188706},[407,658],[363,188708,45126],{"className":188709},[407,136115,658],[363,188711,666],{"className":188712},[665],[363,188714,188716],{"className":188715},[638],[363,188717,188719],{"className":188718,"style":8316},[642],[363,188720],{},[363,188722,578],{"className":188723},[494],[363,188725,4502,188727],{"className":188726},[1832],[363,188728,4506],{"style":4505},[363,188730],{"className":188731,"style":754},[505],[363,188733,188735,188738],{"className":188734},[407],[363,188736,1121],{"className":188737},[407,408],[363,188739,188741],{"className":188740},[629],[363,188742,188744,188764],{"className":188743},[633,634],[363,188745,188747,188761],{"className":188746},[638],[363,188748,188750],{"className":188749,"style":643},[642],[363,188751,188752,188755],{"style":730},[363,188753],{"className":188754,"style":651},[650],[363,188756,188758],{"className":188757},[655,656,657,658],[363,188759,7912],{"className":188760,"style":8615},[407,408,658],[363,188762,666],{"className":188763},[665],[363,188765,188767],{"className":188766},[638],[363,188768,188770],{"className":188769,"style":8316},[642],[363,188771],{},[363,188773,605],{"className":188774},[501],[363,188776],{"className":188777,"style":506},[505],[363,188779,471],{"className":188780},[510],[363,188782],{"className":188783,"style":506},[505],[363,188785,188787,188790,188793,188796,188836,188839,188842,188845],{"className":188786},[398],[363,188788],{"className":188789,"style":487},[402],[363,188791,155482],{"className":188792,"style":1328},[407,408],[363,188794,461],{"className":188795},[494],[363,188797,188799,188802],{"className":188798},[407],[363,188800,45126],{"className":188801,"style":45156},[407,408],[363,188803,188805],{"className":188804},[629],[363,188806,188808,188828],{"className":188807},[633,634],[363,188809,188811,188825],{"className":188810},[638],[363,188812,188814],{"className":188813,"style":643},[642],[363,188815,188816,188819],{"style":136154},[363,188817],{"className":188818,"style":651},[650],[363,188820,188822],{"className":188821},[655,656,657,658],[363,188823,7912],{"className":188824,"style":8615},[407,408,658],[363,188826,666],{"className":188827},[665],[363,188829,188831],{"className":188830},[638],[363,188832,188834],{"className":188833,"style":8316},[642],[363,188835],{},[363,188837,468],{"className":188838},[501],[363,188840],{"className":188841,"style":2935},[505],[363,188843,5161],{"className":188844},[2939],[363,188846],{"className":188847,"style":2935},[505],[363,188849,188851,188854,188857,188860,188900],{"className":188850},[398],[363,188852],{"className":188853,"style":487},[402],[363,188855,155482],{"className":188856,"style":1328},[407,408],[363,188858,461],{"className":188859},[494],[363,188861,188863,188866],{"className":188862},[407],[363,188864,45126],{"className":188865,"style":45156},[407,408],[363,188867,188869],{"className":188868},[629],[363,188870,188872,188892],{"className":188871},[633,634],[363,188873,188875,188889],{"className":188874},[638],[363,188876,188878],{"className":188877,"style":11483},[642],[363,188879,188880,188883],{"style":136154},[363,188881],{"className":188882,"style":651},[650],[363,188884,188886],{"className":188885},[655,656,657,658],[363,188887,474],{"className":188888},[407,658],[363,188890,666],{"className":188891},[665],[363,188893,188895],{"className":188894},[638],[363,188896,188898],{"className":188897,"style":8316},[642],[363,188899],{},[363,188901,468],{"className":188902},[501],[363,188904,188906,188996],{"className":188905,"translate":367},[366],[363,188907,188909],{"className":188908},[371],[373,188910,188911],{"xmlns":375},[377,188912,188913,188993],{},[380,188914,188915,188921,188923,188925,188927,188929,188935,188937,188939,188941,188943,188949,188951,188953,188955,188957,188959,188965,188967,188969,188971,188973,188975,188977,188979,188981,188983,188985,188991],{},[543,188916,188917,188919],{},[458,188918,13811],{},[383,188920,7912],{},[383,188922,571],{"mathvariant":570},[458,188924,578],{"stretchy":460},[383,188926,4448],{},[458,188928,4451],{},[543,188930,188931,188933],{},[383,188932,1121],{},[383,188934,7912],{},[458,188936,605],{"stretchy":460},[458,188938,471],{},[383,188940,155482],{},[458,188942,461],{"stretchy":460},[543,188944,188945,188947],{},[383,188946,45126],{},[463,188948,474],{},[458,188950,468],{"stretchy":460},[458,188952,5161],{},[383,188954,11044],{},[383,188956,155482],{},[458,188958,461],{"stretchy":460},[543,188960,188961,188963],{},[383,188962,45126],{},[463,188964,474],{},[458,188966,468],{"stretchy":460},[458,188968,471],{},[458,188970,5161],{},[458,188972,461],{"stretchy":460},[383,188974,11044],{},[458,188976,5161],{},[463,188978,465],{},[458,188980,468],{"stretchy":460},[383,188982,155482],{},[458,188984,461],{"stretchy":460},[543,188986,188987,188989],{},[383,188988,45126],{},[463,188990,474],{},[458,188992,468],{"stretchy":460},[387,188994,188995],{"encoding":389},"\\sum_k \\mathbb{E}[\\log x_k] = \\psi(\\alpha_0) - K\\psi(\\alpha_0) = -(K-1)\\psi(\\alpha_0)",[363,188997,188999,189114,189178,189245,189269],{"className":188998,"ariaHidden":394},[393],[363,189000,189002,189005,189045,189048,189051,189054,189059,189062,189102,189105,189108,189111],{"className":189001},[398],[363,189003],{"className":189004,"style":138953},[402],[363,189006,189008,189011],{"className":189007},[1832],[363,189009,13811],{"className":189010,"style":40032},[1832,1833,5964],[363,189012,189014],{"className":189013},[629],[363,189015,189017,189037],{"className":189016},[633,634],[363,189018,189020,189034],{"className":189019},[638],[363,189021,189023],{"className":189022,"style":67394},[642],[363,189024,189025,189028],{"style":40048},[363,189026],{"className":189027,"style":651},[650],[363,189029,189031],{"className":189030},[655,656,657,658],[363,189032,7912],{"className":189033,"style":8615},[407,408,658],[363,189035,666],{"className":189036},[665],[363,189038,189040],{"className":189039},[638],[363,189041,189043],{"className":189042,"style":40067},[642],[363,189044],{},[363,189046],{"className":189047,"style":754},[505],[363,189049,571],{"className":189050},[407,714],[363,189052,578],{"className":189053},[494],[363,189055,4502,189057],{"className":189056},[1832],[363,189058,4506],{"style":4505},[363,189060],{"className":189061,"style":754},[505],[363,189063,189065,189068],{"className":189064},[407],[363,189066,1121],{"className":189067},[407,408],[363,189069,189071],{"className":189070},[629],[363,189072,189074,189094],{"className":189073},[633,634],[363,189075,189077,189091],{"className":189076},[638],[363,189078,189080],{"className":189079,"style":643},[642],[363,189081,189082,189085],{"style":730},[363,189083],{"className":189084,"style":651},[650],[363,189086,189088],{"className":189087},[655,656,657,658],[363,189089,7912],{"className":189090,"style":8615},[407,408,658],[363,189092,666],{"className":189093},[665],[363,189095,189097],{"className":189096},[638],[363,189098,189100],{"className":189099,"style":8316},[642],[363,189101],{},[363,189103,605],{"className":189104},[501],[363,189106],{"className":189107,"style":506},[505],[363,189109,471],{"className":189110},[510],[363,189112],{"className":189113,"style":506},[505],[363,189115,189117,189120,189123,189126,189166,189169,189172,189175],{"className":189116},[398],[363,189118],{"className":189119,"style":487},[402],[363,189121,155482],{"className":189122,"style":1328},[407,408],[363,189124,461],{"className":189125},[494],[363,189127,189129,189132],{"className":189128},[407],[363,189130,45126],{"className":189131,"style":45156},[407,408],[363,189133,189135],{"className":189134},[629],[363,189136,189138,189158],{"className":189137},[633,634],[363,189139,189141,189155],{"className":189140},[638],[363,189142,189144],{"className":189143,"style":11483},[642],[363,189145,189146,189149],{"style":136154},[363,189147],{"className":189148,"style":651},[650],[363,189150,189152],{"className":189151},[655,656,657,658],[363,189153,474],{"className":189154},[407,658],[363,189156,666],{"className":189157},[665],[363,189159,189161],{"className":189160},[638],[363,189162,189164],{"className":189163,"style":8316},[642],[363,189165],{},[363,189167,468],{"className":189168},[501],[363,189170],{"className":189171,"style":2935},[505],[363,189173,5161],{"className":189174},[2939],[363,189176],{"className":189177,"style":2935},[505],[363,189179,189181,189184,189187,189190,189193,189233,189236,189239,189242],{"className":189180},[398],[363,189182],{"className":189183,"style":487},[402],[363,189185,11044],{"className":189186,"style":11132},[407,408],[363,189188,155482],{"className":189189,"style":1328},[407,408],[363,189191,461],{"className":189192},[494],[363,189194,189196,189199],{"className":189195},[407],[363,189197,45126],{"className":189198,"style":45156},[407,408],[363,189200,189202],{"className":189201},[629],[363,189203,189205,189225],{"className":189204},[633,634],[363,189206,189208,189222],{"className":189207},[638],[363,189209,189211],{"className":189210,"style":11483},[642],[363,189212,189213,189216],{"style":136154},[363,189214],{"className":189215,"style":651},[650],[363,189217,189219],{"className":189218},[655,656,657,658],[363,189220,474],{"className":189221},[407,658],[363,189223,666],{"className":189224},[665],[363,189226,189228],{"className":189227},[638],[363,189229,189231],{"className":189230,"style":8316},[642],[363,189232],{},[363,189234,468],{"className":189235},[501],[363,189237],{"className":189238,"style":506},[505],[363,189240,471],{"className":189241},[510],[363,189243],{"className":189244,"style":506},[505],[363,189246,189248,189251,189254,189257,189260,189263,189266],{"className":189247},[398],[363,189249],{"className":189250,"style":487},[402],[363,189252,5161],{"className":189253},[407],[363,189255,461],{"className":189256},[494],[363,189258,11044],{"className":189259,"style":11132},[407,408],[363,189261],{"className":189262,"style":2935},[505],[363,189264,5161],{"className":189265},[2939],[363,189267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即可化简。",[11,189332,189333,189334,9748],{},"总损失为两项的加权和，其中权重按训练步",[1344,189335,189336],{},"退火",[363,189338,189340],{"className":189339,"translate":367},[528],[363,189341,189343,189421],{"className":189342,"translate":367},[366],[363,189344,189346],{"className":189345},[371],[373,189347,189348],{"xmlns":375,"display":537},[377,189349,189350,189418],{},[380,189351,189352,189354,189356,189363,189365,189371,189373,189379,189381,189383,189389,189391,189393,189395,189397],{},[383,189353,22265],{"mathvariant":944},[458,189355,471],{},[7162,189357,189358,189360],{},[383,189359,22265],{"mathvariant":944},[1645,189361,189362],{},"fit",[458,189364,2818],{},[543,189366,189367,189369],{},[383,189368,11689],{},[383,189370,1116],{},[1645,189372,1647],{},[7162,189374,189375,189377],{},[383,189376,22265],{"mathvariant":944},[1645,189378,186112],{},[458,189380,898],{"separator":394},[505,189382],{"width":3610},[543,189384,189385,189387],{},[383,189386,11689],{},[383,189388,1116],{},[458,189390,471],{},[383,189392,51980],{},[458,189394,4451],{},[1645,189396,1652],{},[380,189398,189399,189401,189403,189405,189416],{},[458,189400,461],{"fence":394},[463,189402,465],{},[458,189404,898],{"separator":394},[586,189406,189407,189409],{},[383,189408,1116],{},[543,189410,189411,189413],{},[383,189412,5740],{},[1645,189414,189415],{},"anneal",[458,189417,468],{"fence":394},[387,189419,189420],{"encoding":389},"\\mathcal{L} = \\mathcal{L}^{\\text{fit}} + \\lambda_t \\,\\mathcal{L}^{\\text{reg}},\n\\qquad\n\\lambda_t = \\min\\!\\left(1, \\frac{t}{T_{\\text{anneal}}}\\right)",[363,189422,189424,189442,189493,189637],{"className":189423,"ariaHidden":394},[393],[363,189425,189427,189430,189433,189436,189439],{"className":189426},[398],[363,189428],{"className":189429,"style":1011},[402],[363,189431,22265],{"className":189432},[407,971],[363,189434],{"className":189435,"style":506},[505],[363,189437,471],{"className":189438},[510],[363,189440],{"className":189441,"style":506},[505],[363,189443,189445,189449,189484,189487,189490],{"className":189444},[398],[363,189446],{"className":189447,"style":189448},[402],"height:0.9824em;vertical-align:-0.0833em;",[363,189450,189452,189455],{"className":189451},[407],[363,189453,22265],{"className":189454},[407,971],[363,189456,189458],{"className":189457},[629],[363,189459,189461],{"className":189460},[633],[363,189462,189464],{"className":189463},[638],[363,189465,189467],{"className":189466,"style":48236},[642],[363,189468,189469,189472],{"style":8625},[363,189470],{"className":189471,"style":651},[650],[363,189473,189475],{"className":189474},[655,656,657,658],[363,189476,189478],{"className":189477},[407,658],[363,189479,189481],{"className":189480},[407,4946,658],[363,189482,189362],{"className":189483},[407,658],[363,189485],{"className":189486,"style":2935},[505],[363,189488,2818],{"className":189489},[2939],[363,189491],{"className":189492,"style":2935},[505],[363,189494,189496,189500,189541,189544,189579,189582,189585,189588,189628,189631,189634],{"className":189495},[398],[363,189497],{"className":189498,"style":189499},[402],"height:0.9088em;vertical-align:-0.1944em;",[363,189501,189503,189506],{"className":189502},[407],[363,189504,11689],{"className":189505},[407,408],[363,189507,189509],{"className":189508},[629],[363,189510,189512,189533],{"className":189511},[633,634],[363,189513,189515,189530],{"className":189514},[638],[363,189516,189519],{"className":189517,"style":189518},[642],"height:0.2806em;",[363,189520,189521,189524],{"style":730},[363,189522],{"className":189523,"style":651},[650],[363,189525,189527],{"className":189526},[655,656,657,658],[363,189528,1116],{"className":189529},[407,408,658],[363,189531,666],{"className":189532},[665],[363,189534,189536],{"className":189535},[638],[363,189537,189539],{"className":189538,"style":8316},[642],[363,189540],{},[363,189542],{"className":189543,"style":754},[505],[363,189545,189547,189550],{"className":189546},[407],[363,189548,22265],{"className":189549},[407,971],[363,189551,189553],{"className":189552},[629],[363,189554,189556],{"className":189555},[633],[363,189557,189559],{"className":189558},[638],[363,189560,189562],{"className":189561,"style":46283},[642],[363,189563,189564,189567],{"style":8625},[363,189565],{"className":189566,"style":651},[650],[363,189568,189570],{"className":189569},[655,656,657,658],[363,189571,189573],{"className":189572},[407,658],[363,189574,189576],{"className":189575},[407,4946,658],[363,189577,186112],{"className":189578},[407,658],[363,189580,898],{"className":189581},[920],[363,189583],{"className":189584,"style":3982},[505],[363,189586],{"className":189587,"style":754},[505],[363,189589,189591,189594],{"className":189590},[407],[363,189592,11689],{"className":189593},[407,408],[363,189595,189597],{"className":189596},[629],[363,189598,189600,189620],{"className":189599},[633,634],[363,189601,189603,189617],{"className":189602},[638],[363,189604,189606],{"className":189605,"style":189518},[642],[363,189607,189608,189611],{"style":730},[363,189609],{"className":189610,"style":651},[650],[363,189612,189614],{"className":189613},[655,656,657,658],[363,189615,1116],{"className":189616},[407,408,658],[363,189618,666],{"className":189619},[665],[363,189621,189623],{"className":189622},[638],[363,189624,189626],{"className":189625,"style":8316},[642],[363,189627],{},[363,189629],{"className":189630,"style":506},[505],[363,189632,471],{"className":189633},[510],[363,189635],{"className":189636,"style":506},[505],[363,189638,189640,189643,189646,189649,189652],{"className":189639},[398],[363,189641],{"className":189642,"style":707},[402],[363,189644,51980],{"className":189645},[1832],[363,189647],{"className":189648,"style":1860},[505],[363,189650],{"className":189651,"style":754},[505],[363,189653,189655,189661,189664,189667,189670,189775],{"className":189654},[758],[363,189656,189658],{"className":189657,"style":763},[494,762],[363,189659,461],{"className":189660},[767,657],[363,189662,465],{"className":189663},[407],[363,189665,898],{"className":189666},[920],[363,189668],{"className":189669,"style":754},[505],[363,189671,189673,189676,189772],{"className":189672},[407],[363,189674],{"className":189675},[494,789],[363,189677,189679],{"className":189678},[586],[363,189680,189682,189764],{"className":189681},[633,634],[363,189683,189685,189761],{"className":189684},[638],[363,189686,189688,189742,189750],{"className":189687,"style":71438},[642],[363,189689,189690,189693],{"style":805},[363,189691],{"className":189692,"style":809},[650],[363,189694,189696],{"className":189695},[407],[363,189697,189699,189702],{"className":189698},[407],[363,189700,5740],{"className":189701,"style":682},[407,408],[363,189703,189705],{"className":189704},[629],[363,189706,189708,189734],{"className":189707},[633,634],[363,189709,189711,189731],{"className":189710},[638],[363,189712,189714],{"className":189713,"style":643},[642],[363,189715,189716,189719],{"style":11486},[363,189717],{"className":189718,"style":651},[650],[363,189720,189722],{"className":189721},[655,656,657,658],[363,189723,189725],{"className":189724},[407,658],[363,189726,189728],{"className":189727},[407,4946,658],[363,189729,189415],{"className":189730},[407,658],[363,189732,666],{"className":189733},[665],[363,189735,189737],{"className":189736},[638],[363,189738,189740],{"className":189739,"style":8316},[642],[363,189741],{},[363,189743,189744,189747],{"style":821},[363,189745],{"className":189746,"style":809},[650],[363,189748],{"className":189749,"style":829},[828],[363,189751,189752,189755],{"style":832},[363,189753],{"className":189754,"style":809},[650],[363,189756,189758],{"className":189757},[407],[363,189759,1116],{"className":189760},[407,408],[363,189762,666],{"className":189763},[665],[363,189765,189767],{"className":189766},[638],[363,189768,189770],{"className":189769,"style":125439},[642],[363,189771],{},[363,189773],{"className":189774},[501,789],[363,189776,189778],{"className":189777,"style":763},[501,762],[363,189779,468],{"className":189780},[767,657],[11,189782,189783,189784,189876,189877,189969,189970,189999],{},"退火的设计意图是分工时段的划分：训练初期 ",[363,189785,189787,189809],{"className":189786,"translate":367},[366],[363,189788,189790],{"className":189789},[371],[373,189791,189792],{"xmlns":375},[377,189793,189794,189806],{},[380,189795,189796,189802,189804],{},[543,189797,189798,189800],{},[383,189799,11689],{},[383,189801,1116],{},[458,189803,33331],{},[463,189805,474],{},[387,189807,189808],{"encoding":389},"\\lambda_t \\approx 0",[363,189810,189812,189867],{"className":189811,"ariaHidden":394},[393],[363,189813,189815,189818,189858,189861,189864],{"className":189814},[398],[363,189816],{"className":189817,"style":11824},[402],[363,189819,189821,189824],{"className":189820},[407],[363,189822,11689],{"className":189823},[407,408],[363,189825,189827],{"className":189826},[629],[363,189828,189830,189850],{"className":189829},[633,634],[363,189831,189833,189847],{"className":189832},[638],[363,189834,189836],{"className":189835,"style":189518},[642],[363,189837,189838,189841],{"style":730},[363,189839],{"className":189840,"style":651},[650],[363,189842,189844],{"className":189843},[655,656,657,658],[363,189845,1116],{"className":189846},[407,408,658],[363,189848,666],{"className":189849},[665],[363,189851,189853],{"className":189852},[638],[363,189854,189856],{"className":189855,"style":8316},[642],[363,189857],{},[363,189859],{"className":189860,"style":506},[505],[363,189862,33331],{"className":189863},[510],[363,189865],{"className":189866,"style":506},[505],[363,189868,189870,189873],{"className":189869},[398],[363,189871],{"className":189872,"style":520},[402],[363,189874,474],{"className":189875},[407],"，网络可以自由地按拟合损失调整证据的量级，避免在证据尚未成形时被正则项过早地压向零。训练后期 ",[363,189878,189880,189902],{"className":189879,"translate":367},[366],[363,189881,189883],{"className":189882},[371],[373,189884,189885],{"xmlns":375},[377,189886,189887,189899],{},[380,189888,189889,189895,189897],{},[543,189890,189891,189893],{},[383,189892,11689],{},[383,189894,1116],{},[458,189896,1378],{},[463,189898,465],{},[387,189900,189901],{"encoding":389},"\\lambda_t \\to 1",[363,189903,189905,189960],{"className":189904,"ariaHidden":394},[393],[363,189906,189908,189911,189951,189954,189957],{"className":189907},[398],[363,189909],{"className":189910,"style":11824},[402],[363,189912,189914,189917],{"className":189913},[407],[363,189915,11689],{"className":189916},[407,408],[363,189918,189920],{"className":189919},[629],[363,189921,189923,189943],{"className":189922},[633,634],[363,189924,189926,189940],{"className":189925},[638],[363,189927,189929],{"className":189928,"style":189518},[642],[363,189930,189931,189934],{"style":730},[363,189932],{"className":189933,"style":651},[650],[363,189935,189937],{"className":189936},[655,656,657,658],[363,189938,1116],{"className":189939},[407,408,658],[363,189941,666],{"className":189942},[665],[363,189944,189946],{"className":189945},[638],[363,189947,189949],{"className":189948,"style":8316},[642],[363,189950],{},[363,189952],{"className":189953,"style":506},[505],[363,189955,1378],{"className":189956},[510],[363,189958],{"className":189959,"style":506},[505],[363,189961,189963,189966],{"className":189962},[398],[363,189964],{"className":189965,"style":520},[402],[363,189967,465],{"className":189968},[407],"，正则项接管，迫使模型对错误预测给出高不确定性。消融实验一致表明，缺少退火（即使用常数 ",[363,189971,189973,189987],{"className":189972,"translate":367},[366],[363,189974,189976],{"className":189975},[371],[373,189977,189978],{"xmlns":375},[377,189979,189980,189984],{},[380,189981,189982],{},[383,189983,11689],{},[387,189985,189986],{"encoding":389},"\\lambda",[363,189988,189990],{"className":189989,"ariaHidden":394},[393],[363,189991,189993,189996],{"className":189992},[398],[363,189994],{"className":189995,"style":12947},[402],[363,189997,11689],{"className":189998},[407,408],"）会显著损害性能——这与该设计的解释相符，而非仅仅是超参数调优的产物。",[2659,190001,190003],{"id":190002},"_44-统一视角与变体","4.4 统一视角与变体",[11,190005,190006],{},"上述两项可以统一在一个更一般的框架下。所有基于 EDL 的拟合损失都具有形式",[363,190008,190010],{"className":190009,"translate":367},[528],[363,190011,190013,190072],{"className":190012,"translate":367},[366],[363,190014,190016],{"className":190015},[371],[373,190017,190018],{"xmlns":375,"display":537},[377,190019,190020,190069],{},[380,190021,190022,190028,190030,190050],{},[7162,190023,190024,190026],{},[383,190025,22265],{"mathvariant":944},[1645,190027,189362],{},[458,190029,471],{},[543,190031,190032,190034],{},[383,190033,571],{"mathvariant":570},[380,190035,190036,190038,190040,190042,190044,190046,190048],{},[383,190037,157459],{"mathvariant":136055},[458,190039,9217],{},[383,190041,137112],{"mathvariant":557},[458,190043,4451],{},[458,190045,461],{"stretchy":460},[383,190047,45126],{"mathvariant":136055},[458,190049,468],{"stretchy":460},[380,190051,190052,190054,190057,190059,190061,190063,190065,190067],{},[458,190053,578],{"fence":394},[383,190055,190056],{"mathvariant":557},"ℓ",[458,190058,461],{"stretchy":460},[383,190060,40609],{"mathvariant":75598},[458,190062,898],{"separator":394},[383,190064,157459],{"mathvariant":136055},[458,190066,468],{"stretchy":460},[458,190068,605],{"fence":394},[387,190070,190071],{"encoding":389},"\\mathcal{L}^{\\text{fit}} = \\mathbb{E}_{\\boldsymbol{\\pi} \\sim \\operatorname{Dir}(\\boldsymbol{\\alpha})}\\left[\\ell(\\mathbf{y}, \\boldsymbol{\\pi})\\right]",[363,190073,190075,190125],{"className":190074,"ariaHidden":394},[393],[363,190076,190078,190081,190116,190119,190122],{"className":190077},[398],[363,190079],{"className":190080,"style":48236},[402],[363,190082,190084,190087],{"className":190083},[407],[363,190085,22265],{"className":190086},[407,971],[363,190088,190090],{"className":190089},[629],[363,190091,190093],{"className":190092},[633],[363,190094,190096],{"className":190095},[638],[363,190097,190099],{"className":190098,"style":48236},[642],[363,190100,190101,190104],{"style":8625},[363,190102],{"className":190103,"style":651},[650],[363,190105,190107],{"className":190106},[655,656,657,658],[363,190108,190110],{"className":190109},[407,658],[363,190111,190113],{"className":190112},[407,4946,658],[363,190114,189362],{"className":190115},[407,658],[363,190117],{"className":190118,"style":506},[505],[363,190120,471],{"className":190121},[510],[363,190123],{"className":190124,"style":506},[505],[363,190126,190128,190131,190204,190207],{"className":190127},[398],[363,190129],{"className":190130,"style":6427},[402],[363,190132,190134,190137],{"className":190133},[407],[363,190135,571],{"className":190136},[407,714],[363,190138,190140],{"className":190139},[629],[363,190141,190143,190196],{"className":190142},[633,634],[363,190144,190146,190193],{"className":190145},[638],[363,190147,190149],{"className":190148,"style":6446},[642],[363,190150,190151,190154],{"style":75958},[363,190152],{"className":190153,"style":651},[650],[363,190155,190157],{"className":190156},[655,656,657,658],[363,190158,190160,190169,190172,190178,190181,190190],{"className":190159},[407,658],[363,190161,190163],{"className":190162},[407,658],[363,190164,190166],{"className":190165},[407,658],[363,190167,157459],{"className":190168,"style":157493},[407,136115,658],[363,190170,9217],{"className":190171},[510,658],[363,190173,190175],{"className":190174},[1832,658],[363,190176,137112],{"className":190177},[407,33034,658],[363,190179,461],{"className":190180},[494,658],[363,190182,190184],{"className":190183},[407,658],[363,190185,190187],{"className":190186},[407,658],[363,190188,45126],{"className":190189},[407,136115,658],[363,190191,468],{"className":190192},[501,658],[363,190194,666],{"className":190195},[665],[363,190197,190199],{"className":190198},[638],[363,190200,190202],{"className":190201,"style":6492},[642],[363,190203],{},[363,190205],{"className":190206,"style":754},[505],[363,190208,190210,190213,190216,190219,190222,190225,190228,190237,190240],{"className":190209},[758],[363,190211,578],{"className":190212,"style":763},[494,762],[363,190214,190056],{"className":190215},[407],[363,190217,461],{"className":190218},[494],[363,190220,40609],{"className":190221,"style":162156},[407,75943],[363,190223,898],{"className":190224},[920],[363,190226],{"className":190227,"style":754},[505],[363,190229,190231],{"className":190230},[407],[363,190232,190234],{"className":190233},[407],[363,190235,157459],{"className":190236,"style":157493},[407,136115],[363,190238,468],{"className":190239},[501],[363,190241,605],{"className":190242,"style":763},[501,762],[11,190244,88022,190245,190278,190279,190391,190392,190511],{},[1344,190246,190247,190248,190277],{},"对某个一阶损失 ",[363,190249,190251,190265],{"className":190250,"translate":367},[366],[363,190252,190254],{"className":190253},[371],[373,190255,190256],{"xmlns":375},[377,190257,190258,190262],{},[380,190259,190260],{},[383,190261,190056],{"mathvariant":557},[387,190263,190264],{"encoding":389},"\\ell",[363,190266,190268],{"className":190267,"ariaHidden":394},[393],[363,190269,190271,190274],{"className":190270},[398],[363,190272],{"className":190273,"style":12947},[402],[363,190275,190056],{"className":190276},[407]," 的先验期望","。取 ",[363,190280,190282,190310],{"className":190281,"translate":367},[366],[363,190283,190285],{"className":190284},[371],[373,190286,190287],{"xmlns":375},[377,190288,190289,190307],{},[380,190290,190291,190293,190295,190297,190299,190301],{},[383,190292,190056],{"mathvariant":557},[458,190294,471],{},[458,190296,5161],{},[383,190298,4448],{},[458,190300,4451],{},[543,190302,190303,190305],{},[383,190304,157459],{},[383,190306,40609],{},[387,190308,190309],{"encoding":389},"\\ell = -\\log \\pi_y",[363,190311,190313,190331],{"className":190312,"ariaHidden":394},[393],[363,190314,190316,190319,190322,190325,190328],{"className":190315},[398],[363,190317],{"className":190318,"style":12947},[402],[363,190320,190056],{"className":190321},[407],[363,190323],{"className":190324,"style":506},[505],[363,190326,471],{"className":190327},[510],[363,190329],{"className":190330,"style":506},[505],[363,190332,190334,190337,190340,190343,190348,190351],{"className":190333},[398],[363,190335],{"className":190336,"style":153752},[402],[363,190338,5161],{"className":190339},[407],[363,190341],{"className":190342,"style":754},[505],[363,190344,4502,190346],{"className":190345},[1832],[363,190347,4506],{"style":4505},[363,190349],{"className":190350,"style":754},[505],[363,190352,190354,190357],{"className":190353},[407],[363,190355,157459],{"className":190356,"style":1328},[407,408],[363,190358,190360],{"className":190359},[629],[363,190361,190363,190383],{"className":190362},[633,634],[363,190364,190366,190380],{"className":190365},[638],[363,190367,190369],{"className":190368,"style":45301},[642],[363,190370,190371,190374],{"style":19215},[363,190372],{"className":190373,"style":651},[650],[363,190375,190377],{"className":190376},[655,656,657,658],[363,190378,40609],{"className":190379,"style":1328},[407,408,658],[363,190381,666],{"className":190382},[665],[363,190384,190386],{"className":190385},[638],[363,190387,190389],{"className":190388,"style":673},[642],[363,190390],{}," 得命题 4.1 的 digamma 形式。取 ",[363,190393,190395,190425],{"className":190394,"translate":367},[366],[363,190396,190398],{"className":190397},[371],[373,190399,190400],{"xmlns":375},[377,190401,190402,190422],{},[380,190403,190404,190406,190408,190410,190412,190414,190416],{},[383,190405,190056],{"mathvariant":557},[458,190407,471],{},[383,190409,558],{"mathvariant":557},[383,190411,40609],{"mathvariant":75598},[458,190413,5161],{},[383,190415,157459],{"mathvariant":136055},[7162,190417,190418,190420],{},[383,190419,558],{"mathvariant":557},[463,190421,2811],{},[387,190423,190424],{"encoding":389},"\\ell = \\|\\mathbf{y} - \\boldsymbol{\\pi}\\|^2",[363,190426,190428,190446,190467],{"className":190427,"ariaHidden":394},[393],[363,190429,190431,190434,190437,190440,190443],{"className":190430},[398],[363,190432],{"className":190433,"style":12947},[402],[363,190435,190056],{"className":190436},[407],[363,190438],{"className":190439,"style":506},[505],[363,190441,471],{"className":190442},[510],[363,190444],{"className":190445,"style":506},[505],[363,190447,190449,190452,190455,190458,190461,190464],{"className":190448},[398],[363,190450],{"className":190451,"style":487},[402],[363,190453,558],{"className":190454},[407],[363,190456,40609],{"className":190457,"style":162156},[407,75943],[363,190459],{"className":190460,"style":2935},[505],[363,190462,5161],{"className":190463},[2939],[363,190465],{"className":190466,"style":2935},[505],[363,190468,190470,190473,190482],{"className":190469},[398],[363,190471],{"className":190472,"style":20375},[402],[363,190474,190476],{"className":190475},[407],[363,190477,190479],{"className":190478},[407],[363,190480,157459],{"className":190481,"style":157493},[407,136115],[363,190483,190485,190488],{"className":190484},[407],[363,190486,558],{"className":190487},[407],[363,190489,190491],{"className":190490},[629],[363,190492,190494],{"className":190493},[633],[363,190495,190497],{"className":190496},[638],[363,190498,190500],{"className":190499,"style":20400},[642],[363,190501,190502,190505],{"style":7213},[363,190503],{"className":190504,"style":651},[650],[363,190506,190508],{"className":190507},[655,656,657,658],[363,190509,2811],{"className":190510},[407,658]," 得命题 4.3 的平方形式。这一视角把「EDL 的损失设计」转化为「选择哪个一阶损失并在迪利克雷族上求其贝叶斯风险」的标准问题，从而使新损失的设计有章可循。",[11,190513,190514],{},"在此基础上，文献中出现的主要变体可归入三类：",[7010,190516,190517,190623,190633],{},[7013,190518,190519,190522,190523,190593,190594,190622],{},[1344,190520,190521],{},"正则方式不同","。除 4.3 节的 KL 正则外，也有工作直接以「认知不确定性应随证据增加而衰减」作为约束，或对证据施加 ",[363,190524,190526,190544],{"className":190525,"translate":367},[366],[363,190527,190529],{"className":190528},[371],[373,190530,190531],{"xmlns":375},[377,190532,190533,190541],{},[380,190534,190535],{},[543,190536,190537,190539],{},[383,190538,22265],{},[463,190540,2811],{},[387,190542,190543],{"encoding":389},"L_2",[363,190545,190547],{"className":190546,"ariaHidden":394},[393],[363,190548,190550,190553],{"className":190549},[398],[363,190551],{"className":190552,"style":16311},[402],[363,190554,190556,190559],{"className":190555},[407],[363,190557,22265],{"className":190558},[407,408],[363,190560,190562],{"className":190561},[629],[363,190563,190565,190585],{"className":190564},[633,634],[363,190566,190568,190582],{"className":190567},[638],[363,190569,190571],{"className":190570,"style":11483},[642],[363,190572,190573,190576],{"style":730},[363,190574],{"className":190575,"style":651},[650],[363,190577,190579],{"className":190578},[655,656,657,658],[363,190580,2811],{"className":190581},[407,658],[363,190583,666],{"className":190584},[665],[363,190586,190588],{"className":190587},[638],[363,190589,190591],{"className":190590,"style":8316},[642],[363,190592],{}," 惩罚。其共同目标都是阻止 ",[363,190595,190597,190610],{"className":190596,"translate":367},[366],[363,190598,190600],{"className":190599},[371],[373,190601,190602],{"xmlns":375},[377,190603,190604,190608],{},[380,190605,190606],{},[383,190607,26109],{},[387,190609,26109],{"encoding":389},[363,190611,190613],{"className":190612,"ariaHidden":394},[393],[363,190614,190616,190619],{"className":190615},[398],[363,190617],{"className":190618,"style":1011},[402],[363,190620,26109],{"className":190621,"style":26233},[407,408]," 的退化。",[7013,190624,190625,190628,190629,190632],{},[1344,190626,190627],{},"监督方式不同","。Prior Networks 不在期望风险上训练，而是显式地构造一个",[1344,190630,190631],{},"目标迪利克雷分布","（例如：分布内样本用尖锐的 Dirichlet 匹配、分布外样本用平坦的 Dirichlet 匹配），并在分布层面直接计算 KL。这需要分布外样本来定义目标，因此换取的是更强的不确定性控制与更弱的即插即用性。",[7013,190634,190635,190638],{},[1344,190636,190637],{},"回归情形的推广","。把分类的迪利克雷替换为 Normal-Inverse-Gamma 先验，即得 Deep Evidential Regression（5.5 节）。",[11,190640,190641,190642,190645,190646,190649],{},"统一框架也澄清了一个理论定位问题：EDL 与贝叶斯神经网络（BNN）、深度集成（deep ensembles）、MC Dropout 属于",[1344,190643,190644],{},"不同的","不确定性来源。后三者通过对网络参数（或其近似后验）的分布求积分得到预测分布。EDL 则在固定的参数点估计上，对",[1344,190647,190648],{},"输出层的二阶分布","求积分。二者的区分不是工程实现上的，而是概率模型结构上的，这正是下一小节批评的出发点。",[2659,190651,190653],{"id":190652},"_45-使用边界","4.5 使用边界",[11,190655,190656],{},"EDL 在实践中表现良好，但其理论地位常被过度解读。把边界集中列出，与前文的正式推导互为对照：",[7010,190658,190659,191568,191698,191964,192025],{},[7013,190660,190661,190664,190665,190974,190975,191044,191045,10528,191260,191263,191264,191447,191448,2757,191476,191538,191539,191567],{},[1344,190662,190663],{},"它不是贝叶斯不确定性。"," 严格的后验预测分布应由对网络参数的边缘化给出：\n",[363,190666,190668,190753],{"className":190667,"translate":367},[366],[363,190669,190671],{"className":190670},[371],[373,190672,190673],{"xmlns":375},[377,190674,190675,190750],{},[380,190676,190677,190679,190681,190683,190685,190687,190689,190691,190693,190695,190698,190700,190702,190704,190706,190708,190710,190712,190714,190716,190718,190720,190722,190724,190726,190728,190730,190732,190734,190736,190738,190740,190742,190744,190746,190748],{},[383,190678,11],{},[458,190680,461],{"stretchy":460},[383,190682,40609],{},[458,190684,5156],{},[383,190686,1121],{"mathvariant":75598},[458,190688,898],{"separator":394},[383,190690,547],{"mathvariant":944},[458,190692,468],{"stretchy":460},[458,190694,471],{},[458,190696,190697],{},"∬",[383,190699,11],{},[458,190701,461],{"stretchy":460},[383,190703,40609],{},[458,190705,5156],{},[383,190707,157459],{"mathvariant":136055},[458,190709,468],{"stretchy":460},[1645,190711,1647],{},[383,190713,11],{},[458,190715,461],{"stretchy":460},[383,190717,157459],{"mathvariant":136055},[458,190719,5156],{},[383,190721,101148],{"mathvariant":136055},[458,190723,468],{"stretchy":460},[1645,190725,1647],{},[383,190727,11],{},[458,190729,461],{"stretchy":460},[383,190731,101148],{"mathvariant":136055},[458,190733,5156],{},[383,190735,547],{"mathvariant":944},[458,190737,468],{"stretchy":460},[1645,190739,1647],{},[383,190741,592],{"mathvariant":557},[383,190743,157459],{"mathvariant":136055},[1645,190745,1647],{},[383,190747,592],{"mathvariant":557},[383,190749,101148],{"mathvariant":136055},[387,190751,190752],{"encoding":389},"p(y \\mid \\mathbf{x}, \\mathcal{D}) = \\iint p(y \\mid \\boldsymbol{\\pi})\\,p(\\boldsymbol{\\pi} \\mid \\boldsymbol{\\theta})\\,p(\\boldsymbol{\\theta} \\mid \\mathcal{D})\\,\\mathrm{d}\\boldsymbol{\\pi}\\,\\mathrm{d}\\boldsymbol{\\theta}",[363,190754,190756,190780,190810,190842,190887,190932],{"className":190755,"ariaHidden":394},[393],[363,190757,190759,190762,190765,190768,190771,190774,190777],{"className":190758},[398],[363,190760],{"className":190761,"style":487},[402],[363,190763,11],{"className":190764},[407,408],[363,190766,461],{"className":190767},[494],[363,190769,40609],{"className":190770,"style":1328},[407,408],[363,190772],{"className":190773,"style":506},[505],[363,190775,5156],{"className":190776},[510],[363,190778],{"className":190779,"style":506},[505],[363,190781,190783,190786,190789,190792,190795,190798,190801,190804,190807],{"className":190782},[398],[363,190784],{"className":190785,"style":487},[402],[363,190787,1121],{"className":190788},[407,75943],[363,190790,898],{"className":190791},[920],[363,190793],{"className":190794,"style":754},[505],[363,190796,547],{"className":190797,"style":625},[407,971],[363,190799,468],{"className":190800},[501],[363,190802],{"className":190803,"style":506},[505],[363,190805,471],{"className":190806},[510],[363,190808],{"className":190809,"style":506},[505],[363,190811,190813,190817,190821,190824,190827,190830,190833,190836,190839],{"className":190812},[398],[363,190814],{"className":190815,"style":190816},[402],"height:1.111em;vertical-align:-0.306em;",[363,190818,190697],{"className":190819,"style":190820},[1832,1833,5964],"margin-right:0.1945em;position:relative;top:-0.0005em;",[363,190822],{"className":190823,"style":754},[505],[363,190825,11],{"className":190826},[407,408],[363,190828,461],{"className":190829},[494],[363,190831,40609],{"className":190832,"style":1328},[407,408],[363,190834],{"className":190835,"style":506},[505],[363,190837,5156],{"className":190838},[510],[363,190840],{"className":190841,"style":506},[505],[363,190843,190845,190848,190857,190860,190863,190866,190869,190878,190881,190884],{"className":190844},[398],[363,190846],{"className":190847,"style":487},[402],[363,190849,190851],{"className":190850},[407],[363,190852,190854],{"className":190853},[407],[363,190855,157459],{"className":190856,"style":157493},[407,136115],[363,190858,468],{"className":190859},[501],[363,190861],{"className":190862,"style":754},[505],[363,190864,11],{"className":190865},[407,408],[363,190867,461],{"className":190868},[494],[363,190870,190872],{"className":190871},[407],[363,190873,190875],{"className":190874},[407],[363,190876,157459],{"className":190877,"style":157493},[407,136115],[363,190879],{"className":190880,"style":506},[505],[363,190882,5156],{"className":190883},[510],[363,190885],{"className":190886,"style":506},[505],[363,190888,190890,190893,190902,190905,190908,190911,190914,190923,190926,190929],{"className":190889},[398],[363,190891],{"className":190892,"style":487},[402],[363,190894,190896],{"className":190895},[407],[363,190897,190899],{"className":190898},[407],[363,190900,101148],{"className":190901,"style":169926},[407,136115],[363,190903,468],{"className":190904},[501],[363,190906],{"className":190907,"style":754},[505],[363,190909,11],{"className":190910},[407,408],[363,190912,461],{"className":190913},[494],[363,190915,190917],{"className":190916},[407],[363,190918,190920],{"className":190919},[407],[363,190921,101148],{"className":190922,"style":169926},[407,136115],[363,190924],{"className":190925,"style":506},[505],[363,190927,5156],{"className":190928},[510],[363,190930],{"className":190931,"style":506},[505],[363,190933,190935,190938,190941,190944,190947,190950,190959,190962,190965],{"className":190934},[398],[363,190936],{"className":190937,"style":487},[402],[363,190939,547],{"className":190940,"style":625},[407,971],[363,190942,468],{"className":190943},[501],[363,190945],{"className":190946,"style":754},[505],[363,190948,592],{"className":190949},[407,33034],[363,190951,190953],{"className":190952},[407],[363,190954,190956],{"className":190955},[407],[363,190957,157459],{"className":190958,"style":157493},[407,136115],[363,190960],{"className":190961,"style":754},[505],[363,190963,592],{"className":190964},[407,33034],[363,190966,190968],{"className":190967},[407],[363,190969,190971],{"className":190970},[407],[363,190972,101148],{"className":190973,"style":169926},[407,136115],"\n而 EDL 以点估计 ",[363,190976,190978,190996],{"className":190977,"translate":367},[366],[363,190979,190981],{"className":190980},[371],[373,190982,190983],{"xmlns":375},[377,190984,190985,190993],{},[380,190986,190987],{},[12161,190988,190989,190991],{"accent":394},[383,190990,101148],{"mathvariant":136055},[458,190992,100327],{},[387,190994,190995],{"encoding":389},"\\hat{\\boldsymbol{\\theta}}",[363,190997,190999],{"className":190998,"ariaHidden":394},[393],[363,191000,191002,191006],{"className":191001},[398],[363,191003],{"className":191004,"style":191005},[402],"height:0.9579em;",[363,191007,191009],{"className":191008},[407,12201],[363,191010,191012],{"className":191011},[633],[363,191013,191015],{"className":191014},[638],[363,191016,191018,191032],{"className":191017,"style":191005},[642],[363,191019,191020,191023],{"style":12213},[363,191021],{"className":191022,"style":809},[650],[363,191024,191026],{"className":191025},[407],[363,191027,191029],{"className":191028},[407],[363,191030,101148],{"className":191031,"style":169926},[407,136115],[363,191033,191035,191038],{"style":191034},"top:-3.2634em;",[363,191036],{"className":191037,"style":809},[650],[363,191039,191041],{"className":191040,"style":185976},[12229],[363,191042,100327],{"className":191043},[407]," 替换外层积分，取 ",[363,191046,191048,191102],{"className":191047,"translate":367},[366],[363,191049,191051],{"className":191050},[371],[373,191052,191053],{"xmlns":375},[377,191054,191055,191099],{},[380,191056,191057,191059,191061,191063,191065,191067,191069,191071,191073,191075,191077,191079,191081,191083,191085,191087,191089,191095,191097],{},[383,191058,11],{},[458,191060,461],{"stretchy":460},[383,191062,157459],{"mathvariant":136055},[458,191064,5156],{},[383,191066,1121],{"mathvariant":75598},[458,191068,468],{"stretchy":460},[458,191070,471],{},[383,191072,137112],{"mathvariant":557},[458,191074,4451],{},[458,191076,461],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[383,191078,157459],{"mathvariant":136055},[458,191080,34247],{"separator":394},[383,191082,45126],{"mathvariant":136055},[458,191084,461],{"stretchy":460},[383,191086,1121],{"mathvariant":75598},[458,191088,34247],{"separator":394},[12161,191090,191091,191093],{"accent":394},[383,191092,101148],{"mathvariant":136055},[458,191094,100327],{},[458,191096,468],{"stretchy":460},[458,191098,468],{"fence":460,"stretchy":394,"minsize":9798,"maxsize":9798},[387,191100,191101],{"encoding":389},"p(\\boldsymbol{\\pi} \\mid \\mathbf{x}) = \\operatorname{Dir}\\big(\\boldsymbol{\\pi}; \\boldsymbol{\\alpha}(\\mathbf{x}; \\hat{\\boldsymbol{\\theta}})\\big)",[363,191103,191105,191135,191156],{"className":191104,"ariaHidden":394},[393],[363,191106,191108,191111,191114,191117,191126,191129,191132],{"className":191107},[398],[363,191109],{"className":191110,"style":487},[402],[363,191112,11],{"className":191113},[407,408],[363,191115,461],{"className":191116},[494],[363,191118,191120],{"className":191119},[407],[363,191121,191123],{"className":191122},[407],[363,191124,157459],{"className":191125,"style":157493},[407,136115],[363,191127],{"className":191128,"style":506},[505],[363,191130,5156],{"className":191131},[510],[363,191133],{"className":191134,"style":506},[505],[363,191136,191138,191141,191144,191147,191150,191153],{"className":191137},[398],[363,191139],{"className":191140,"style":487},[402],[363,191142,1121],{"className":191143},[407,75943],[363,191145,468],{"className":191146},[501],[363,191148],{"className":191149,"style":506},[505],[363,191151,471],{"className":191152},[510],[363,191154],{"className":191155,"style":506},[505],[363,191157,191159,191163,191169,191172,191178,191187,191190,191193,191202,191205,191208,191211,191214,191251,191254],{"className":191158},[398],[363,191160],{"className":191161,"style":191162},[402],"height:1.3079em;vertical-align:-0.35em;",[363,191164,191166],{"className":191165},[1832],[363,191167,137112],{"className":191168},[407,33034],[363,191170],{"className":191171,"style":754},[505],[363,191173,191175],{"className":191174},[407],[363,191176,461],{"className":191177},[767,10022],[363,191179,191181],{"className":191180},[407],[363,191182,191184],{"className":191183},[407],[363,191185,157459],{"className":191186,"style":157493},[407,136115],[363,191188,34247],{"className":191189},[920],[363,191191],{"className":191192,"style":754},[505],[363,191194,191196],{"className":191195},[407],[363,191197,191199],{"className":191198},[407],[363,191200,45126],{"className":191201},[407,136115],[363,191203,461],{"className":191204},[494],[363,191206,1121],{"className":191207},[407,75943],[363,191209,34247],{"className":191210},[920],[363,191212],{"className":191213,"style":754},[505],[363,191215,191217],{"className":191216},[407,12201],[363,191218,191220],{"className":191219},[633],[363,191221,191223],{"className":191222},[638],[363,191224,191226,191240],{"className":191225,"style":191005},[642],[363,191227,191228,191231],{"style":12213},[363,191229],{"className":191230,"style":809},[650],[363,191232,191234],{"className":191233},[407],[363,191235,191237],{"className":191236},[407],[363,191238,101148],{"className":191239,"style":169926},[407,136115],[363,191241,191242,191245],{"style":191034},[363,191243],{"className":191244,"style":809},[650],[363,191246,191248],{"className":191247,"style":185976},[12229],[363,191249,100327],{"className":191250},[407],[363,191252,468],{"className":191253},[501],[363,191255,191257],{"className":191256},[407],[363,191258,468],{"className":191259},[767,10022],[1344,191261,191262],{},"结果是一个插值（plug-in）估计","，其内层积分给出 ",[363,191265,191267,191300],{"className":191266,"translate":367},[366],[363,191268,191270],{"className":191269},[371],[373,191271,191272],{"xmlns":375},[377,191273,191274,191298],{},[380,191275,191276,191282,191284,191290,191292],{},[543,191277,191278,191280],{},[383,191279,11],{},[383,191281,7912],{},[458,191283,471],{},[543,191285,191286,191288],{},[383,191287,45126],{},[383,191289,7912],{},[383,191291,1288],{"mathvariant":557},[543,191293,191294,191296],{},[383,191295,45126],{},[463,191297,474],{},[387,191299,176531],{"encoding":389},[363,191301,191303,191358],{"className":191302,"ariaHidden":394},[393],[363,191304,191306,191309,191349,191352,191355],{"className":191305},[398],[363,191307],{"className":191308,"style":2624},[402],[363,191310,191312,191315],{"className":191311},[407],[363,191313,11],{"className":191314},[407,408],[363,191316,191318],{"className":191317},[629],[363,191319,191321,191341],{"className":191320},[633,634],[363,191322,191324,191338],{"className":191323},[638],[363,191325,191327],{"className":191326,"style":643},[642],[363,191328,191329,191332],{"style":730},[363,191330],{"className":191331,"style":651},[650],[363,191333,191335],{"className":191334},[655,656,657,658],[363,191336,7912],{"className":191337,"style":8615},[407,408,658],[363,191339,666],{"className":191340},[665],[363,191342,191344],{"className":191343},[638],[363,191345,191347],{"className":191346,"style":8316},[642],[363,191348],{},[363,191350],{"className":191351,"style":506},[505],[363,191353,471],{"className":191354},[510],[363,191356],{"className":191357,"style":506},[505],[363,191359,191361,191364,191404,191407],{"className":191360},[398],[363,191362],{"className":191363,"style":487},[402],[363,191365,191367,191370],{"className":191366},[407],[363,191368,45126],{"className":191369,"style":45156},[407,408],[363,191371,191373],{"className":191372},[629],[363,191374,191376,191396],{"className":191375},[633,634],[363,191377,191379,191393],{"className":191378},[638],[363,191380,191382],{"className":191381,"style":643},[642],[363,191383,191384,191387],{"style":136154},[363,191385],{"className":191386,"style":651},[650],[363,191388,191390],{"className":191389},[655,656,657,658],[363,191391,7912],{"className":191392,"style":8615},[407,408,658],[363,191394,666],{"className":191395},[665],[363,191397,191399],{"className":191398},[638],[363,191400,191402],{"className":191401,"style":8316},[642],[363,191403],{},[363,191405,1288],{"className":191406},[407],[363,191408,191410,191413],{"className":191409},[407],[363,191411,45126],{"className":191412,"style":45156},[407,408],[363,191414,191416],{"className":191415},[629],[363,191417,191419,191439],{"className":191418},[633,634],[363,191420,191422,191436],{"className":191421},[638],[363,191423,191425],{"className":191424,"style":11483},[642],[363,191426,191427,191430],{"style":136154},[363,191428],{"className":191429,"style":651},[650],[363,191431,191433],{"className":191432},[655,656,657,658],[363,191434,474],{"className":191435},[407,658],[363,191437,666],{"className":191438},[665],[363,191440,191442],{"className":191441},[638],[363,191443,191445],{"className":191444,"style":8316},[642],[363,191446],{},"（由聚合性），确实正确。但外层积分被抛弃，",[363,191449,191451,191464],{"className":191450,"translate":367},[366],[363,191452,191454],{"className":191453},[371],[373,191455,191456],{"xmlns":375},[377,191457,191458,191462],{},[380,191459,191460],{},[383,191461,15750],{},[387,191463,15750],{"encoding":389},[363,191465,191467],{"className":191466,"ariaHidden":394},[393],[363,191468,191470,191473],{"className":191469},[398],[363,191471],{"className":191472,"style":2693},[402],[363,191474,15750],{"className":191475},[407,408],[363,191477,191479,191502],{"className":191478,"translate":367},[366],[363,191480,191482],{"className":191481},[371],[373,191483,191484],{"xmlns":375},[377,191485,191486,191500],{},[380,191487,191488,191490,191492,191494,191496,191498],{},[383,191489,34240],{},[458,191491,461],{"stretchy":460},[383,191493,40609],{},[458,191495,34247],{"separator":394},[383,191497,157459],{"mathvariant":136055},[458,191499,468],{"stretchy":460},[387,191501,179574],{"encoding":389},[363,191503,191505],{"className":191504,"ariaHidden":394},[393],[363,191506,191508,191511,191514,191517,191520,191523,191526,191535],{"className":191507},[398],[363,191509],{"className":191510,"style":487},[402],[363,191512,34240],{"className":191513,"style":9235},[407,408],[363,191515,461],{"className":191516},[494],[363,191518,40609],{"className":191519,"style":1328},[407,408],[363,191521,34247],{"className":191522},[920],[363,191524],{"className":191525,"style":754},[505],[363,191527,191529],{"className":191528},[407],[363,191530,191532],{"className":191531},[407],[363,191533,157459],{"className":191534,"style":157493},[407,136115],[363,191536,468],{"className":191537},[501]," 所度量的是「对输出分布的把握」，而非「对模型的把握」。二者在分布外输入上可以严重分歧：一个被错误训练的网络可以以极低的 ",[363,191540,191542,191555],{"className":191541,"translate":367},[366],[363,191543,191545],{"className":191544},[371],[373,191546,191547],{"xmlns":375},[377,191548,191549,191553],{},[380,191550,191551],{},[383,191552,15750],{},[387,191554,15750],{"encoding":389},[363,191556,191558],{"className":191557,"ariaHidden":394},[393],[363,191559,191561,191564],{"className":191560},[398],[363,191562],{"className":191563,"style":2693},[402],[363,191565,15750],{"className":191566},[407,408]," 输出一个错误的类别。",[7013,191569,191570,191573,191574,191634,191635,191697],{},[1344,191571,191572],{},"不确定性分数丢弃了证据的分布信息。"," 如 3.5 节所述，",[363,191575,191577,191598],{"className":191576,"translate":367},[366],[363,191578,191580],{"className":191579},[371],[373,191581,191582],{"xmlns":375},[377,191583,191584,191596],{},[380,191585,191586,191588,191590,191592,191594],{},[383,191587,15750],{},[458,191589,471],{},[383,191591,11044],{},[383,191593,1288],{"mathvariant":557},[383,191595,26109],{},[387,191597,175682],{"encoding":389},[363,191599,191601,191619],{"className":191600,"ariaHidden":394},[393],[363,191602,191604,191607,191610,191613,191616],{"className":191603},[398],[363,191605],{"className":191606,"style":2693},[402],[363,191608,15750],{"className":191609},[407,408],[363,191611],{"className":191612,"style":506},[505],[363,191614,471],{"className":191615},[510],[363,191617],{"className":191618,"style":506},[505],[363,191620,191622,191625,191628,191631],{"className":191621},[398],[363,191623],{"className":191624,"style":487},[402],[363,191626,11044],{"className":191627,"style":11132},[407,408],[363,191629,1288],{"className":191630},[407],[363,191632,26109],{"className":191633,"style":26233},[407,408]," 仅是总证据的函数。它无法表达「证据分散但充分」与「证据集中但贫乏」的区别，也无法区分证据落在何类。",[363,191636,191638,191661],{"className":191637,"translate":367},[366],[363,191639,191641],{"className":191640},[371],[373,191642,191643],{"xmlns":375},[377,191644,191645,191659],{},[380,191646,191647,191649,191651,191653,191655,191657],{},[383,191648,34240],{},[458,191650,461],{"stretchy":460},[383,191652,40609],{},[458,191654,34247],{"separator":394},[383,191656,157459],{"mathvariant":136055},[458,191658,468],{"stretchy":460},[387,191660,179574],{"encoding":389},[363,191662,191664],{"className":191663,"ariaHidden":394},[393],[363,191665,191667,191670,191673,191676,191679,191682,191685,191694],{"className":191666},[398],[363,191668],{"className":191669,"style":487},[402],[363,191671,34240],{"className":191672,"style":9235},[407,408],[363,191674,461],{"className":191675},[494],[363,191677,40609],{"className":191678,"style":1328},[407,408],[363,191680,34247],{"className":191681},[920],[363,191683],{"className":191684,"style":754},[505],[363,191686,191688],{"className":191687},[407],[363,191689,191691],{"className":191690},[407],[363,191692,157459],{"className":191693,"style":157493},[407,136115],[363,191695,468],{"className":191696},[501]," 缓解了这一点，但仍不引入任何关于模型参数的信息。",[7013,191699,191700,191703,191704,191959,191960,191963],{},[1344,191701,191702],{},"拟合损失带有一个与强度耦合的偏置。"," 推论 4.2 表明，对数形式等价于交叉熵加上 ",[363,191705,191707,191749],{"className":191706,"translate":367},[366],[363,191708,191710],{"className":191709},[371],[373,191711,191712],{"xmlns":375},[377,191713,191714,191746],{},[380,191715,191716,191718,191732,191734,191744],{},[458,191717,461],{"stretchy":460},[586,191719,191720,191722],{},[463,191721,465],{},[380,191723,191724,191726],{},[463,191725,2811],{},[543,191727,191728,191730],{},[383,191729,45126],{},[383,191731,40609],{},[458,191733,5161],{},[586,191735,191736,191738],{},[463,191737,465],{},[380,191739,191740,191742],{},[463,191741,2811],{},[383,191743,26109],{},[458,191745,468],{"stretchy":460},[387,191747,191748],{"encoding":389},"(\\frac{1}{2\\alpha_y} - \\frac{1}{2S})",[363,191750,191752,191879],{"className":191751,"ariaHidden":394},[393],[363,191753,191755,191759,191762,191870,191873,191876],{"className":191754},[398],[363,191756],{"className":191757,"style":191758},[402],"height:1.3874em;vertical-align:-0.5423em;",[363,191760,461],{"className":191761},[494],[363,191763,191765,191768,191867],{"className":191764},[407],[363,191766],{"className":191767},[494,789],[363,191769,191771],{"className":191770},[586],[363,191772,191774,191859],{"className":191773},[633,634],[363,191775,191777,191856],{"className":191776},[638],[363,191778,191780,191834,191842],{"className":191779,"style":2872},[642],[363,191781,191782,191785],{"style":2875},[363,191783],{"className":191784,"style":809},[650],[363,191786,191788],{"className":191787},[655,656,657,658],[363,191789,191791,191794],{"className":191790},[407,658],[363,191792,2811],{"className":191793},[407,658],[363,191795,191797,191800],{"className":191796},[407,658],[363,191798,45126],{"className":191799,"style":45156},[407,408,658],[363,191801,191803],{"className":191802},[629],[363,191804,191806,191826],{"className":191805},[633,634],[363,191807,191809,191823],{"className":191808},[638],[363,191810,191812],{"className":191811,"style":74645},[642],[363,191813,191814,191817],{"style":136989},[363,191815],{"className":191816,"style":30578},[650],[363,191818,191820],{"className":191819},[655,30582,10022,658],[363,191821,40609],{"className":191822,"style":1328},[407,408,658],[363,191824,666],{"className":191825},[665],[363,191827,191829],{"className":191828},[638],[363,191830,191832],{"className":191831,"style":122109},[642],[363,191833],{},[363,191835,191836,191839],{"style":821},[363,191837],{"className":191838,"style":809},[650],[363,191840],{"className":191841,"style":829},[828],[363,191843,191844,191847],{"style":2898},[363,191845],{"className":191846,"style":809},[650],[363,191848,191850],{"className":191849},[655,656,657,658],[363,191851,191853],{"className":191852},[407,658],[363,191854,465],{"className":191855},[407,658],[363,191857,666],{"className":191858},[665],[363,191860,191862],{"className":191861},[638],[363,191863,191865],{"className":191864,"style":182558},[642],[363,191866],{},[363,191868],{"className":191869},[501,789],[363,191871],{"className":191872,"style":2935},[505],[363,191874,5161],{"className":191875},[2939],[363,191877],{"className":191878,"style":2935},[505],[363,191880,191882,191885,191956],{"className":191881},[398],[363,191883],{"className":191884,"style":2853},[402],[363,191886,191888,191891,191953],{"className":191887},[407],[363,191889],{"className":191890},[494,789],[363,191892,191894],{"className":191893},[586],[363,191895,191897,191945],{"className":191896},[633,634],[363,191898,191900,191942],{"className":191899},[638],[363,191901,191903,191920,191928],{"className":191902,"style":2872},[642],[363,191904,191905,191908],{"style":2875},[363,191906],{"className":191907,"style":809},[650],[363,191909,191911],{"className":191910},[655,656,657,658],[363,191912,191914,191917],{"className":191913},[407,658],[363,191915,2811],{"className":191916},[407,658],[363,191918,26109],{"className":191919,"style":26233},[407,408,658],[363,191921,191922,191925],{"style":821},[363,191923],{"className":191924,"style":809},[650],[363,191926],{"className":191927,"style":829},[828],[363,191929,191930,191933],{"style":2898},[363,191931],{"className":191932,"style":809},[650],[363,191934,191936],{"className":191935},[655,656,657,658],[363,191937,191939],{"className":191938},[407,658],[363,191940,465],{"className":191941},[407,658],[363,191943,666],{"className":191944},[665],[363,191946,191948],{"className":191947},[638],[363,191949,191951],{"className":191950,"style":2920},[642],[363,191952],{},[363,191954],{"className":191955},[501,789],[363,191957,468],{"className":191958},[501]," 的修正。因此损失的最小值并不位于「插值交叉熵的最优解」，而是位于一个把证据量级也纳入权衡的点上。这不是缺陷本身，但它意味着",[1344,191961,191962],{},"EDL 的优化目标与「拟合条件分布」并非同一个目标","，将其损失解释为似然的做法在术语上应作区分。",[7013,191965,191966,191969,191970,192024],{},[1344,191967,191968],{},"「以损失最小化实现不确定性量化」这一范式受到根本性质疑。"," 有研究指出，当认知不确定性被定义为某个二阶损失（即关于 ",[363,191971,191973,191994],{"className":191972,"translate":367},[366],[363,191974,191976],{"className":191975},[371],[373,191977,191978],{"xmlns":375},[377,191979,191980,191992],{},[380,191981,191982,191984,191986,191988,191990],{},[383,191983,137112],{"mathvariant":557},[458,191985,4451],{},[458,191987,461],{"stretchy":460},[383,191989,45126],{"mathvariant":136055},[458,191991,468],{"stretchy":460},[387,191993,171778],{"encoding":389},[363,191995,191997],{"className":191996,"ariaHidden":394},[393],[363,191998,192000,192003,192009,192012,192021],{"className":191999},[398],[363,192001],{"className":192002,"style":487},[402],[363,192004,192006],{"className":192005},[1832],[363,192007,137112],{"className":192008},[407,33034],[363,192010,461],{"className":192011},[494],[363,192013,192015],{"className":192014},[407],[363,192016,192018],{"className":192017},[407],[363,192019,45126],{"className":192020},[407,136115],[363,192022,468],{"className":192023},[501]," 的泛函）的极小化者时，损失可以在与真实认知状态无关的第二阶分布上取得同一最小值，从而该极小化者一般并不落在具有正确认知解释的分布上。换言之，「损失小」并不蕴含「认知不确定性估计正确」。这一批评与本条前三点在方向上是独立的：即便接受 EDL 的目标函数，其最优解也未必具有所宣称的语义。",[7013,192026,192027,192030,192031,192147,192148,192176,192177,10528],{},[1344,192028,192029],{},"它对主干特征的可分性有隐含依赖。"," 证据由 ",[363,192032,192034,192062],{"className":192033,"translate":367},[366],[363,192035,192037],{"className":192036},[371],[373,192038,192039],{"xmlns":375},[377,192040,192041,192059],{},[380,192042,192043,192045,192047,192053,192055,192057],{},[383,192044,7915],{"mathvariant":75598},[458,192046,471],{},[543,192048,192049,192051],{},[383,192050,4506],{},[383,192052,101148],{"mathvariant":136055},[458,192054,461],{"stretchy":460},[383,192056,1121],{"mathvariant":75598},[458,192058,468],{"stretchy":460},[387,192060,192061],{"encoding":389},"\\mathbf{e} = g_{\\boldsymbol{\\theta}}(\\mathbf{x})",[363,192063,192065,192083],{"className":192064,"ariaHidden":394},[393],[363,192066,192068,192071,192074,192077,192080],{"className":192067},[398],[363,192069],{"className":192070,"style":135662},[402],[363,192072,7915],{"className":192073},[407,75943],[363,192075],{"className":192076,"style":506},[505],[363,192078,471],{"className":192079},[510],[363,192081],{"className":192082,"style":506},[505],[363,192084,192086,192089,192138,192141,192144],{"className":192085},[398],[363,192087],{"className":192088,"style":487},[402],[363,192090,192092,192095],{"className":192091},[407],[363,192093,4506],{"className":192094,"style":1328},[407,408],[363,192096,192098],{"className":192097},[629],[363,192099,192101,192130],{"className":192100},[633,634],[363,192102,192104,192127],{"className":192103},[638],[363,192105,192107],{"className":192106,"style":643},[642],[363,192108,192109,192112],{"style":19215},[363,192110],{"className":192111,"style":651},[650],[363,192113,192115],{"className":192114},[655,656,657,658],[363,192116,192118],{"className":192117},[407,658],[363,192119,192121],{"className":192120},[407,658],[363,192122,192124],{"className":192123},[407,658],[363,192125,101148],{"className":192126,"style":169926},[407,136115,658],[363,192128,666],{"className":192129},[665],[363,192131,192133],{"className":192132},[638],[363,192134,192136],{"className":192135,"style":8316},[642],[363,192137],{},[363,192139,461],{"className":192140},[494],[363,192142,1121],{"className":192143},[407,75943],[363,192145,468],{"className":192146},[501]," 给出，因此 ",[363,192149,192151,192164],{"className":192150,"translate":367},[366],[363,192152,192154],{"className":192153},[371],[373,192155,192156],{"xmlns":375},[377,192157,192158,192162],{},[380,192159,192160],{},[383,192161,1121],{"mathvariant":75598},[387,192163,135652],{"encoding":389},[363,192165,192167],{"className":192166,"ariaHidden":394},[393],[363,192168,192170,192173],{"className":192169},[398],[363,192171],{"className":192172,"style":135662},[402],[363,192174,1121],{"className":192175},[407,75943]," 落在分布外时能否给出低证据，完全取决于特征提取器是否把该区域映射到低响应区。若主干网络对分布外输入仍产生高幅 logits，EDL 的不确定性同样失效——",[1344,192178,192179],{},"它把 OOD 问题从输出层转移到了特征层，而非解决它",[11,192181,192182,192183,192186],{},"综合以上点，恰当的使用立场是：",[1344,192184,192185],{},"把 EDL 视为一种参数高效、结构可解释的不确定性启发式，而非贝叶斯推理的替代品。"," 它在「单次前向代价下同时输出预测与不确定性」这一约束下的性价比是突出的。但凡是论断依赖于「该不确定性具有贝叶斯后验语义」的场合，都应改回到对参数做边缘化的方法，或至少给出与后者的对照实验。",[18,192188,192190],{"id":192189},"五应用","五、应用",[2659,192192,192194],{"id":192193},"_51-分布外检测与开放集识别","5.1 分布外检测与开放集识别",[11,192196,192197,192198,175719,192258,192261,192262,192290,192291,192322],{},"由命题 3.4，",[363,192199,192201,192222],{"className":192200,"translate":367},[366],[363,192202,192204],{"className":192203},[371],[373,192205,192206],{"xmlns":375},[377,192207,192208,192220],{},[380,192209,192210,192212,192214,192216,192218],{},[383,192211,15750],{},[458,192213,471],{},[383,192215,11044],{},[383,192217,1288],{"mathvariant":557},[383,192219,26109],{},[387,192221,175682],{"encoding":389},[363,192223,192225,192243],{"className":192224,"ariaHidden":394},[393],[363,192226,192228,192231,192234,192237,192240],{"className":192227},[398],[363,192229],{"className":192230,"style":2693},[402],[363,192232,15750],{"className":192233},[407,408],[363,192235],{"className":192236,"style":506},[505],[363,192238,471],{"className":192239},[510],[363,192241],{"className":192242,"style":506},[505],[363,192244,192246,192249,192252,192255],{"className":192245},[398],[363,192247],{"className":192248,"style":487},[402],[363,192250,11044],{"className":192251,"style":11132},[407,408],[363,192253,1288],{"className":192254},[407],[363,192256,26109],{"className":192257,"style":26233},[407,408],[1344,192259,192260],{},"单调反比于总证据","的量，因此可以直接充当分布外（OOD）分数：",[363,192263,192265,192278],{"className":192264,"translate":367},[366],[363,192266,192268],{"className":192267},[371],[373,192269,192270],{"xmlns":375},[377,192271,192272,192276],{},[380,192273,192274],{},[383,192275,15750],{},[387,192277,15750],{"encoding":389},[363,192279,192281],{"className":192280,"ariaHidden":394},[393],[363,192282,192284,192287],{"className":192283},[398],[363,192285],{"className":192286,"style":2693},[402],[363,192288,15750],{"className":192289},[407,408]," 高意味着输入未能激活任何类别的证据。以阈值 ",[363,192292,192294,192309],{"className":192293,"translate":367},[366],[363,192295,192297],{"className":192296},[371],[373,192298,192299],{"xmlns":375},[377,192300,192301,192306],{},[380,192302,192303],{},[383,192304,192305],{},"τ",[387,192307,192308],{"encoding":389},"\\tau",[363,192310,192312],{"className":192311,"ariaHidden":394},[393],[363,192313,192315,192318],{"className":192314},[398],[363,192316],{"className":192317,"style":2693},[402],[363,192319,192305],{"className":192320,"style":192321},[407,408],"margin-right:0.1132em;"," 作拒识判决",[363,192324,192326],{"className":192325,"translate":367},[528],[363,192327,192329,192441],{"className":192328,"translate":367},[366],[363,192330,192332],{"className":192331},[371],[373,192333,192334],{"xmlns":375,"display":537},[377,192335,192336,192438],{},[380,192337,192338,192344,192346,192348,192350,192352],{},[12161,192339,192340,192342],{"accent":394},[383,192341,40609],{},[458,192343,100327],{},[458,192345,461],{"stretchy":460},[383,192347,1121],{"mathvariant":75598},[458,192349,468],{"stretchy":460},[458,192351,471],{},[380,192353,192354,192356],{},[458,192355,3246],{"fence":394},[1591,192357,192358,192407],{"rowspacing":4661,"columnalign":4662,"columnspacing":4663},[1597,192359,192360,192389],{},[1600,192361,192362],{},[1603,192363,192364],{"scriptlevel":474,"displaystyle":460},[380,192365,192366,192369,192371,192381,192387],{},[383,192367,192368],{},"arg",[458,192370,4451],{},[543,192372,192373,192379],{},[380,192374,192375,192377],{},[383,192376,52759],{},[458,192378,4451],{},[383,192380,7912],{},[543,192382,192383,192385],{},[383,192384,11],{},[383,192386,7912],{},[458,192388,898],{"separator":394},[1600,192390,192391],{},[1603,192392,192393],{"scriptlevel":474,"displaystyle":460},[380,192394,192395,192397,192399,192401,192403,192405],{},[383,192396,15750],{},[458,192398,461],{"stretchy":460},[383,192400,1121],{"mathvariant":75598},[458,192402,468],{"stretchy":460},[458,192404,10300],{},[383,192406,192305],{},[1597,192408,192409,192420],{},[1600,192410,192411],{},[1603,192412,192413],{"scriptlevel":474,"displaystyle":460},[380,192414,192415,192418],{},[1645,192416,192417],{},"reject",[458,192419,898],{"separator":394},[1600,192421,192422],{},[1603,192423,192424],{"scriptlevel":474,"displaystyle":460},[380,192425,192426,192428,192430,192432,192434,192436],{},[383,192427,15750],{},[458,192429,461],{"stretchy":460},[383,192431,1121],{"mathvariant":75598},[458,192433,468],{"stretchy":460},[458,192435,3251],{},[383,192437,192305],{},[387,192439,192440],{"encoding":389},"\\hat{y}(\\mathbf{x}) = \\begin{cases} \\arg\\max_k p_k, & u(\\mathbf{x}) \\le \\tau \\\\ \\text{reject}, & u(\\mathbf{x}) > \\tau \\end{cases}",[363,192442,192444,192511],{"className":192443,"ariaHidden":394},[393],[363,192445,192447,192450,192493,192496,192499,192502,192505,192508],{"className":192446},[398],[363,192448],{"className":192449,"style":487},[402],[363,192451,192453],{"className":192452},[407,12201],[363,192454,192456,192485],{"className":192455},[633,634],[363,192457,192459,192482],{"className":192458},[638],[363,192460,192462,192470],{"className":192461,"style":12947},[642],[363,192463,192464,192467],{"style":12213},[363,192465],{"className":192466,"style":809},[650],[363,192468,40609],{"className":192469,"style":1328},[407,408],[363,192471,192472,192475],{"style":12213},[363,192473],{"className":192474,"style":809},[650],[363,192476,192479],{"className":192477,"style":192478},[12229],"left:-0.1944em;",[363,192480,100327],{"className":192481},[407],[363,192483,666],{"className":192484},[665],[363,192486,192488],{"className":192487},[638],[363,192489,192491],{"className":192490,"style":12657},[642],[363,192492],{},[363,192494,461],{"className":192495},[494],[363,192497,1121],{"className":192498},[407,75943],[363,192500,468],{"className":192501},[501],[363,192503],{"className":192504,"style":506},[505],[363,192506,471],{"className":192507},[510],[363,192509],{"className":192510,"style":506},[505],[363,192512,192514,192518],{"className":192513},[398],[363,192515],{"className":192516,"style":192517},[402],"height:3em;vertical-align:-1.25em;",[363,192519,192521,192527,192770],{"className":192520},[758],[363,192522,192524],{"className":192523,"style":763},[494,762],[363,192525,3246],{"className":192526},[767,4895],[363,192528,192530],{"className":192529},[407],[363,192531,192533,192680,192683],{"className":192532},[1591],[363,192534,192536],{"className":192535},[1799],[363,192537,192539,192671],{"className":192538},[633,634],[363,192540,192542,192668],{"className":192541},[638],[363,192543,192546,192650],{"className":192544,"style":192545},[642],"height:1.69em;",[363,192547,192549,192552],{"style":192548},"top:-3.69em;",[363,192550],{"className":192551,"style":4921},[650],[363,192553,192555,192561,192564,192604,192607,192647],{"className":192554},[407],[363,192556,192558,192559],{"className":192557},[1832],"ar",[363,192560,4506],{"style":4505},[363,192562],{"className":192563,"style":754},[505],[363,192565,192567,192570],{"className":192566},[1832],[363,192568,52759],{"className":192569},[1832],[363,192571,192573],{"className":192572},[629],[363,192574,192576,192596],{"className":192575},[633,634],[363,192577,192579,192593],{"className":192578},[638],[363,192580,192582],{"className":192581,"style":643},[642],[363,192583,192584,192587],{"style":52818},[363,192585],{"className":192586,"style":651},[650],[363,192588,192590],{"className":192589},[655,656,657,658],[363,192591,7912],{"className":192592,"style":8615},[407,408,658],[363,192594,666],{"className":192595},[665],[363,192597,192599],{"className":192598},[638],[363,192600,192602],{"className":192601,"style":8316},[642],[363,192603],{},[363,192605],{"className":192606,"style":754},[505],[363,192608,192610,192613],{"className":192609},[407],[363,192611,11],{"className":192612},[407,408],[363,192614,192616],{"className":192615},[629],[363,192617,192619,192639],{"className":192618},[633,634],[363,192620,192622,192636],{"className":192621},[638],[363,192623,192625],{"className":192624,"style":643},[642],[363,192626,192627,192630],{"style":730},[363,192628],{"className":192629,"style":651},[650],[363,192631,192633],{"className":192632},[655,656,657,658],[363,192634,7912],{"className":192635,"style":8615},[407,408,658],[363,192637,666],{"className":192638},[665],[363,192640,192642],{"className":192641},[638],[363,192643,192645],{"className":192644,"style":8316},[642],[363,192646],{},[363,192648,898],{"className":192649},[920],[363,192651,192653,192656],{"style":192652},"top:-2.25em;",[363,192654],{"className":192655,"style":4921},[650],[363,192657,192659,192665],{"className":192658},[407],[363,192660,192662],{"className":192661},[407,4946],[363,192663,192417],{"className":192664},[407],[363,192666,898],{"className":192667},[920],[363,192669,666],{"className":192670},[665],[363,192672,192674],{"className":192673},[638],[363,192675,192678],{"className":192676,"style":192677},[642],"height:1.19em;",[363,192679],{},[363,192681],{"className":192682,"style":4970},[4969],[363,192684,192686],{"className":192685},[1799],[363,192687,192689,192762],{"className":192688},[633,634],[363,192690,192692,192759],{"className":192691},[638],[363,192693,192695,192727],{"className":192694,"style":192545},[642],[363,192696,192697,192700],{"style":192548},[363,192698],{"className":192699,"style":4921},[650],[363,192701,192703,192706,192709,192712,192715,192718,192721,192724],{"className":192702},[407],[363,192704,15750],{"className":192705},[407,408],[363,192707,461],{"className":192708},[494],[363,192710,1121],{"className":192711},[407,75943],[363,192713,468],{"className":192714},[501],[363,192716],{"className":192717,"style":506},[505],[363,192719,10300],{"className":192720},[510],[363,192722],{"className":192723,"style":506},[505],[363,192725,192305],{"className":192726,"style":192321},[407,408],[363,192728,192729,192732],{"style":192652},[363,192730],{"className":192731,"style":4921},[650],[363,192733,192735,192738,192741,192744,192747,192750,192753,192756],{"className":192734},[407],[363,192736,15750],{"className":192737},[407,408],[363,192739,461],{"className":192740},[494],[363,192742,1121],{"className":192743},[407,75943],[363,192745,468],{"className":192746},[501],[363,192748],{"className":192749,"style":506},[505],[363,192751,3251],{"className":192752},[510],[363,192754],{"className":192755,"style":506},[505],[363,192757,192305],{"className":192758,"style":192321},[407,408],[363,192760,666],{"className":192761},[665],[363,192763,192765],{"className":192764},[638],[363,192766,192768],{"className":192767,"style":192677},[642],[363,192769],{},[363,192771],{"className":192772},[501,789],[11,192774,192775,192776,192983],{},"即得开放集分类器。与之竞争的分数包括最大 softmax 概率（MSP）、预测熵、ODIN 的温度与输入扰动、Mahalanobis 距离，以及能量分数 ",[363,192777,192779,192824],{"className":192778,"translate":367},[366],[363,192780,192782],{"className":192781},[371],[373,192783,192784],{"xmlns":375},[377,192785,192786,192821],{},[380,192787,192788,192791,192793,192795,192797,192799,192801,192803,192805,192811],{},[383,192789,192790],{"mathvariant":557},"LSE",[458,192792,4451],{},[458,192794,461],{"stretchy":460},[383,192796,118160],{"mathvariant":75598},[458,192798,468],{"stretchy":460},[458,192800,471],{},[383,192802,4448],{},[458,192804,4451],{},[543,192806,192807,192809],{},[458,192808,13811],{},[383,192810,7912],{},[7162,192812,192813,192815],{},[383,192814,7915],{},[543,192816,192817,192819],{},[383,192818,118160],{},[383,192820,7912],{},[387,192822,192823],{"encoding":389},"\\operatorname{LSE}(\\mathbf{z}) = \\log\\sum_k e^{z_k}",[363,192825,192827,192857],{"className":192826,"ariaHidden":394},[393],[363,192828,192830,192833,192839,192842,192845,192848,192851,192854],{"className":192829},[398],[363,192831],{"className":192832,"style":487},[402],[363,192834,192836],{"className":192835},[1832],[363,192837,192790],{"className":192838},[407,33034],[363,192840,461],{"className":192841},[494],[363,192843,118160],{"className":192844},[407,75943],[363,192846,468],{"className":192847},[501],[363,192849],{"className":192850,"style":506},[505],[363,192852,471],{"className":192853},[510],[363,192855],{"className":192856,"style":506},[505],[363,192858,192860,192863,192868,192871,192911,192914],{"className":192859},[398],[363,192861],{"className":192862,"style":138953},[402],[363,192864,4502,192866],{"className":192865},[1832],[363,192867,4506],{"style":4505},[363,192869],{"className":192870,"style":754},[505],[363,192872,192874,192877],{"className":192873},[1832],[363,192875,13811],{"className":192876,"style":40032},[1832,1833,5964],[363,192878,192880],{"className":192879},[629],[363,192881,192883,192903],{"className":192882},[633,634],[363,192884,192886,192900],{"className":192885},[638],[363,192887,192889],{"className":192888,"style":67394},[642],[363,192890,192891,192894],{"style":40048},[363,192892],{"className":192893,"style":651},[650],[363,192895,192897],{"className":192896},[655,656,657,658],[363,192898,7912],{"className":192899,"style":8615},[407,408,658],[363,192901,666],{"className":192902},[665],[363,192904,192906],{"className":192905},[638],[363,192907,192909],{"className":192908,"style":40067},[642],[363,192910],{},[363,192912],{"className":192913,"style":754},[505],[363,192915,192917,192920],{"className":192916},[407],[363,192918,7915],{"className":192919},[407,408],[363,192921,192923],{"className":192922},[629],[363,192924,192926],{"className":192925},[633],[363,192927,192929],{"className":192928},[638],[363,192930,192932],{"className":192931,"style":50725},[642],[363,192933,192934,192937],{"style":7213},[363,192935],{"className":192936,"style":651},[650],[363,192938,192940],{"className":192939},[655,656,657,658],[363,192941,192943],{"className":192942},[407,658],[363,192944,192946,192949],{"className":192945},[407,658],[363,192947,118160],{"className":192948,"style":118430},[407,408,658],[363,192950,192952],{"className":192951},[629],[363,192953,192955,192975],{"className":192954},[633,634],[363,192956,192958,192972],{"className":192957},[638],[363,192959,192961],{"className":192960,"style":6446},[642],[363,192962,192963,192966],{"style":170354},[363,192964],{"className":192965,"style":30578},[650],[363,192967,192969],{"className":192968},[655,30582,10022,658],[363,192970,7912],{"className":192971,"style":8615},[407,408,658],[363,192973,666],{"className":192974},[665],[363,192976,192978],{"className":192977},[638],[363,192979,192981],{"className":192980,"style":101625},[642],[363,192982],{},"。将它们在性质上加以对照：",[52864,192985,192986,193002],{},[52867,192987,192988],{},[52870,192989,192990,192993,192996,192999],{},[52873,192991,192992],{"align":165005},"分数",[52873,192994,192995],{"align":165005},"依赖的结构",[52873,192997,192998],{"align":165005},"是否需要 OOD 数据",[52873,193000,193001],{"align":165005},"不确定性语汇",[52962,193003,193004,193017,193030,193043],{},[52870,193005,193006,193009,193012,193014],{},[52967,193007,193008],{"align":165005},"MSP \u002F 熵",[52967,193010,193011],{"align":165005},"softmax 输出",[52967,193013,53127],{"align":165005},[52967,193015,193016],{"align":165005},"无（受 3.3 节缺陷支配）",[52870,193018,193019,193022,193025,193027],{},[52967,193020,193021],{"align":165005},"Mahalanobis",[52967,193023,193024],{"align":165005},"各特征层的类条件高斯",[52967,193026,53127],{"align":165005},[52967,193028,193029],{"align":165005},"距离，几何语义",[52870,193031,193032,193035,193038,193040],{},[52967,193033,193034],{"align":165005},"能量分数",[52967,193036,193037],{"align":165005},"logits 的对数配分",[52967,193039,53127],{"align":165005},[52967,193041,193042],{"align":165005},"无显式不确定性",[52870,193044,193045,193078,193109,193113],{},[52967,193046,193047],{"align":165005},[1344,193048,193049,193050],{},"EDL 的 ",[363,193051,193053,193066],{"className":193052,"translate":367},[366],[363,193054,193056],{"className":193055},[371],[373,193057,193058],{"xmlns":375},[377,193059,193060,193064],{},[380,193061,193062],{},[383,193063,15750],{},[387,193065,15750],{"encoding":389},[363,193067,193069],{"className":193068,"ariaHidden":394},[393],[363,193070,193072,193075],{"className":193071},[398],[363,193073],{"className":193074,"style":2693},[402],[363,193076,15750],{"className":193077},[407,408],[52967,193079,193080,193081],{"align":165005},"迪利克雷总强度 ",[363,193082,193084,193097],{"className":193083,"translate":367},[366],[363,193085,193087],{"className":193086},[371],[373,193088,193089],{"xmlns":375},[377,193090,193091,193095],{},[380,193092,193093],{},[383,193094,26109],{},[387,193096,26109],{"encoding":389},[363,193098,193100],{"className":193099,"ariaHidden":394},[393],[363,193101,193103,193106],{"className":193102},[398],[363,193104],{"className":193105,"style":1011},[402],[363,193107,26109],{"className":193108,"style":26233},[407,408],[52967,193110,193111],{"align":165005},[1344,193112,53127],{},[52967,193114,193115],{"align":165005},"证据缺失量，可解释",[11,193117,193118,193119,193122],{},"EDL 在此项比较中的结构性优势在于两点。第一，它的不确定性是",[1344,193120,193121],{},"训练目标的一部分","（由 4.3 节的正则项直接优化），而非对训练好的网络做事后加工的产物。第二，它给出的是类概率向量的分布，因此可以在不接触任何 OOD 样本的前提下产生 OOD 分数——这是先验网络之外少有的性质（先验网络需要 OOD 样本以构造目标分布）。",[11,193124,193125],{},"限度同样明确：如 4.5 节第 5 点所述，其成败取决于特征空间的可分性。评价应使用 AUROC、AUPR（In\u002FOut 与 Out\u002FIn 分别报告）、FPR@95TPR 以及开放集风险—覆盖曲线，而非仅报单一的准确率。",[2659,193127,193129],{"id":193128},"_52-校准与选择性预测","5.2 校准与选择性预测",[11,193131,193132,193135,193136,193354],{},[1344,193133,193134],{},"校准","关心的是 ",[363,193137,193139,193183],{"className":193138,"translate":367},[366],[363,193140,193142],{"className":193141},[371],[373,193143,193144],{"xmlns":375},[377,193145,193146,193180],{},[380,193147,193148,193150,193152,193158,193160,193162,193164,193170,193172,193174,193176,193178],{},[383,193149,554],{"mathvariant":570},[458,193151,461],{"stretchy":460},[12161,193153,193154,193156],{"accent":394},[383,193155,40609],{},[458,193157,100327],{},[458,193159,471],{},[383,193161,40609],{},[458,193163,5156],{},[12161,193165,193166,193168],{"accent":394},[383,193167,11],{},[458,193169,100327],{},[458,193171,471],{},[383,193173,1291],{},[458,193175,468],{"stretchy":460},[458,193177,471],{},[383,193179,1291],{},[387,193181,193182],{"encoding":389},"\\mathbb{P}(\\hat{y} = y \\mid \\hat{p} = q) = q",[363,193184,193186,193249,193267,193324,193345],{"className":193185,"ariaHidden":394},[393],[363,193187,193189,193192,193195,193198,193240,193243,193246],{"className":193188},[398],[363,193190],{"className":193191,"style":487},[402],[363,193193,554],{"className":193194},[407,714],[363,193196,461],{"className":193197},[494],[363,193199,193201],{"className":193200},[407,12201],[363,193202,193204,193232],{"className":193203},[633,634],[363,193205,193207,193229],{"className":193206},[638],[363,193208,193210,193218],{"className":193209,"style":12947},[642],[363,193211,193212,193215],{"style":12213},[363,193213],{"className":193214,"style":809},[650],[363,193216,40609],{"className":193217,"style":1328},[407,408],[363,193219,193220,193223],{"style":12213},[363,193221],{"className":193222,"style":809},[650],[363,193224,193226],{"className":193225,"style":192478},[12229],[363,193227,100327],{"className":193228},[407],[363,193230,666],{"className":193231},[665],[363,193233,193235],{"className":193234},[638],[363,193236,193238],{"className":193237,"style":12657},[642],[363,193239],{},[363,193241],{"className":193242,"style":506},[505],[363,193244,471],{"className":193245},[510],[363,193247],{"className":193248,"style":506},[505],[363,193250,193252,193255,193258,193261,193264],{"className":193251},[398],[363,193253],{"className":193254,"style":487},[402],[363,193256,40609],{"className":193257,"style":1328},[407,408],[363,193259],{"className":193260,"style":506},[505],[363,193262,5156],{"className":193263},[510],[363,193265],{"className":193266,"style":506},[505],[363,193268,193270,193273,193315,193318,193321],{"className":193269},[398],[363,193271],{"className":193272,"style":403},[402],[363,193274,193276],{"className":193275},[407,12201],[363,193277,193279,193307],{"className":193278},[633,634],[363,193280,193282,193304],{"className":193281},[638],[363,193283,193285,193293],{"className":193284,"style":12947},[642],[363,193286,193287,193290],{"style":12213},[363,193288],{"className":193289,"style":809},[650],[363,193291,11],{"className":193292},[407,408],[363,193294,193295,193298],{"style":12213},[363,193296],{"className":193297,"style":809},[650],[363,193299,193301],{"className":193300,"style":12465},[12229],[363,193302,100327],{"className":193303},[407],[363,193305,666],{"className":193306},[665],[363,193308,193310],{"className":193309},[638],[363,193311,193313],{"className":193312,"style":12657},[642],[363,193314],{},[363,193316],{"className":193317,"style":506},[505],[363,193319,471],{"className":193320},[510],[363,193322],{"className":193323,"style":506},[505],[363,193325,193327,193330,193333,193336,193339,193342],{"className":193326},[398],[363,193328],{"className":193329,"style":487},[402],[363,193331,1291],{"className":193332,"style":1328},[407,408],[363,193334,468],{"className":193335},[501],[363,193337],{"className":193338,"style":506},[505],[363,193340,471],{"className":193341},[510],[363,193343],{"className":193344,"style":506},[505],[363,193346,193348,193351],{"className":193347},[398],[363,193349],{"className":193350,"style":2624},[402],[363,193352,1291],{"className":193353,"style":1328},[407,408]," 是否成立，常用期望校准误差",[363,193356,193358],{"className":193357,"translate":367},[528],[363,193359,193361,193446],{"className":193360,"translate":367},[366],[363,193362,193364],{"className":193363},[371],[373,193365,193366],{"xmlns":375,"display":537},[377,193367,193368,193443],{},[380,193369,193370,193373,193375,193389,193405,193407,193409,193412,193414,193416,193422,193424,193426,193429,193431,193433,193439,193441],{},[1645,193371,193372],{},"ECE",[458,193374,471],{},[48692,193376,193377,193379,193387],{},[458,193378,13811],{},[380,193380,193381,193383,193385],{},[383,193382,26695],{},[458,193384,471],{},[463,193386,465],{},[383,193388,25939],{},[586,193390,193391,193403],{},[380,193392,193393,193395,193401],{},[383,193394,5156],{"mathvariant":557},[543,193396,193397,193399],{},[383,193398,39663],{"mathvariant":944},[383,193400,26695],{},[383,193402,5156],{"mathvariant":557},[383,193404,82683],{},[1645,193406,1647],{},[458,193408,5156],{"fence":460,"stretchy":394,"minsize":17789,"maxsize":17789},[383,193410,193411],{"mathvariant":557},"acc",[458,193413,4451],{},[458,193415,461],{"stretchy":460},[543,193417,193418,193420],{},[383,193419,39663],{"mathvariant":944},[383,193421,26695],{},[458,193423,468],{"stretchy":460},[458,193425,5161],{},[383,193427,193428],{"mathvariant":557},"conf",[458,193430,4451],{},[458,193432,461],{"stretchy":460},[543,193434,193435,193437],{},[383,193436,39663],{"mathvariant":944},[383,193438,26695],{},[458,193440,468],{"stretchy":460},[458,193442,5156],{"fence":460,"stretchy":394,"minsize":17789,"maxsize":17789},[387,193444,193445],{"encoding":389},"\\text{ECE} = \\sum_{m=1}^{M} \\frac{|\\mathcal{B}_m|}{n}\\,\\Big|\\operatorname{acc}(\\mathcal{B}_m) - \\operatorname{conf}(\\mathcal{B}_m)\\Big|",[363,193447,193449,193470,193760],{"className":193448,"ariaHidden":394},[393],[363,193450,193452,193455,193461,193464,193467],{"className":193451},[398],[363,193453],{"className":193454,"style":1011},[402],[363,193456,193458],{"className":193457},[407,4946],[363,193459,193372],{"className":193460},[407],[363,193462],{"className":193463,"style":506},[505],[363,193465,471],{"className":193466},[510],[363,193468],{"className":193469,"style":506},[505],[363,193471,193473,193477,193546,193549,193656,193659,193696,193699,193705,193708,193748,193751,193754,193757],{"className":193472},[398],[363,193474],{"className":193475,"style":193476},[402],"height:3.0954em;vertical-align:-1.2671em;",[363,193478,193480],{"className":193479},[1832,14066],[363,193481,193483,193537],{"className":193482},[633,634],[363,193484,193486,193534],{"className":193485},[638],[363,193487,193489,193510,193520],{"className":193488,"style":138274},[642],[363,193490,193492,193495],{"style":193491},"top:-1.8829em;margin-left:0em;",[363,193493],{"className":193494,"style":14083},[650],[363,193496,193498],{"className":193497},[655,656,657,658],[363,193499,193501,193504,193507],{"className":193500},[407,658],[363,193502,26695],{"className":193503},[407,408,658],[363,193505,471],{"className":193506},[510,658],[363,193508,465],{"className":193509},[407,658],[363,193511,193512,193515],{"style":14092},[363,193513],{"className":193514,"style":14083},[650],[363,193516,193517],{},[363,193518,13811],{"className":193519},[1832,1833,1834],[363,193521,193522,193525],{"style":100713},[363,193523],{"className":193524,"style":14083},[650],[363,193526,193528],{"className":193527},[655,656,657,658],[363,193529,193531],{"className":193530},[407,658],[363,193532,25939],{"className":193533,"style":25973},[407,408,658],[363,193535,666],{"className":193536},[665],[363,193538,193540],{"className":193539},[638],[363,193541,193544],{"className":193542,"style":193543},[642],"height:1.2671em;",[363,193545],{},[363,193547],{"className":193548,"style":754},[505],[363,193550,193552,193555,193653],{"className":193551},[407],[363,193553],{"className":193554},[494,789],[363,193556,193558],{"className":193557},[586],[363,193559,193561,193645],{"className":193560},[633,634],[363,193562,193564,193642],{"className":193563},[638],[363,193565,193567,193578,193586],{"className":193566,"style":1891},[642],[363,193568,193569,193572],{"style":805},[363,193570],{"className":193571,"style":809},[650],[363,193573,193575],{"className":193574},[407],[363,193576,82683],{"className":193577},[407,408],[363,193579,193580,193583],{"style":821},[363,193581],{"className":193582,"style":809},[650],[363,193584],{"className":193585,"style":829},[828],[363,193587,193588,193591],{"style":832},[363,193589],{"className":193590,"style":809},[650],[363,193592,193594,193597,193639],{"className":193593},[407],[363,193595,5156],{"className":193596},[407],[363,193598,193600,193604],{"className":193599},[407],[363,193601,39663],{"className":193602,"style":193603},[407,971],"margin-right:0.0304em;",[363,193605,193607],{"className":193606},[629],[363,193608,193610,193631],{"className":193609},[633,634],[363,193611,193613,193628],{"className":193612},[638],[363,193614,193616],{"className":193615,"style":45301},[642],[363,193617,193619,193622],{"style":193618},"top:-2.55em;margin-left:-0.0304em;margin-right:0.05em;",[363,193620],{"className":193621,"style":651},[650],[363,193623,193625],{"className":193624},[655,656,657,658],[363,193626,26695],{"className":193627},[407,408,658],[363,193629,666],{"className":193630},[665],[363,193632,193634],{"className":193633},[638],[363,193635,193637],{"className":193636,"style":8316},[642],[363,193638],{},[363,193640,5156],{"className":193641},[407],[363,193643,666],{"className":193644},[665],[363,193646,193648],{"className":193647},[638],[363,193649,193651],{"className":193650,"style":4126},[642],[363,193652],{},[363,193654],{"className":193655},[501,789],[363,193657],{"className":193658,"style":754},[505],[363,193660,193662],{"className":193661},[407],[363,193663,193665],{"className":193664},[767,15179],[363,193666,193668,193688],{"className":193667},[633,634],[363,193669,193671,193685],{"className":193670},[638],[363,193672,193674],{"className":193673,"style":85902},[642],[363,193675,193676,193679],{"style":73907},[363,193677],{"className":193678,"style":82974},[650],[363,193680,193681],{"style":115893},[6671,193682,193683],{"xmlns":6673,"width":35423,"height":17789,"viewBox":115896},[6679,193684],{"d":115899},[363,193686,666],{"className":193687},[665],[363,193689,193691],{"className":193690},[638],[363,193692,193694],{"className":193693,"style":85926},[642],[363,193695],{},[363,193697],{"className":193698,"style":754},[505],[363,193700,193702],{"className":193701},[1832],[363,193703,193411],{"className":193704},[407,33034],[363,193706,461],{"className":193707},[494],[363,193709,193711,193714],{"className":193710},[407],[363,193712,39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",[363,193859,193861,193879],{"className":193860,"translate":367},[366],[363,193862,193864],{"className":193863},[371],[373,193865,193866],{"xmlns":375},[377,193867,193868,193876],{},[380,193869,193870],{},[543,193871,193872,193874],{},[383,193873,39663],{"mathvariant":944},[383,193875,26695],{},[387,193877,193878],{"encoding":389},"\\mathcal{B}_m",[363,193880,193882],{"className":193881,"ariaHidden":394},[393],[363,193883,193885,193888],{"className":193884},[398],[363,193886],{"className":193887,"style":16311},[402],[363,193889,193891,193894],{"className":193890},[407],[363,193892,39663],{"className":193893,"style":193603},[407,971],[363,193895,193897],{"className":193896},[629],[363,193898,193900,193920],{"className":193899},[633,634],[363,193901,193903,193917],{"className":193902},[638],[363,193904,193906],{"className":193905,"style":45301},[642],[363,193907,193908,193911],{"style":193618},[363,193909],{"className":193910,"style":651},[650],[363,193912,193914],{"className":193913},[655,656,657,658],[363,193915,26695],{"className":193916},[407,408,658],[363,193918,666],{"className":193919},[665],[363,193921,193923],{"className":193922},[638],[363,193924,193926],{"className":193925,"style":8316},[642],[363,193927],{}," 是置信度分箱。EDL 的训练目标隐含地抑制了 3.3 节所述的 logits 发散退化，因此在若干基准上比 MSP 更校准。但这不是免费的：如 4.5 节第 7 点，EDL 的一阶校准同样需要验证，不应默认成立。",[11,193930,193931,2628,193934,193962,193963,194029,194030,194062,194063,194091,194092,194154],{},[1344,193932,193933],{},"选择性预测（selective prediction）",[363,193935,193937,193950],{"className":193936,"translate":367},[366],[363,193938,193940],{"className":193939},[371],[373,193941,193942],{"xmlns":375},[377,193943,193944,193948],{},[380,193945,193946],{},[383,193947,15750],{},[387,193949,15750],{"encoding":389},[363,193951,193953],{"className":193952,"ariaHidden":394},[393],[363,193954,193956,193959],{"className":193955},[398],[363,193957],{"className":193958,"style":2693},[402],[363,193960,15750],{"className":193961},[407,408]," 的更自然的落点。经典 Chow 规则指出，在已知真实后验 ",[363,193964,193966,193990],{"className":193965,"translate":367},[366],[363,193967,193969],{"className":193968},[371],[373,193970,193971],{"xmlns":375},[377,193972,193973,193987],{},[380,193974,193975,193977,193979,193981,193983,193985],{},[383,193976,11],{},[458,193978,461],{"stretchy":460},[383,193980,40609],{},[458,193982,5156],{},[383,193984,1121],{"mathvariant":75598},[458,193986,468],{"stretchy":460},[387,193988,193989],{"encoding":389},"p(y \\mid \\mathbf{x})",[363,193991,193993,194017],{"className":193992,"ariaHidden":394},[393],[363,193994,193996,193999,194002,194005,194008,194011,194014],{"className":193995},[398],[363,193997],{"className":193998,"style":487},[402],[363,194000,11],{"className":194001},[407,408],[363,194003,461],{"className":194004},[494],[363,194006,40609],{"className":194007,"style":1328},[407,408],[363,194009],{"className":194010,"style":506},[505],[363,194012,5156],{"className":194013},[510],[363,194015],{"className":194016,"style":506},[505],[363,194018,194020,194023,194026],{"className":194019},[398],[363,194021],{"className":194022,"style":487},[402],[363,194024,1121],{"className":194025},[407,75943],[363,194027,468],{"className":194028},[501]," 时，最优拒识策略是「当最大后验概率低于某阈值时拒识」，其最优性",[1344,194031,194032,194033,194061],{},"依赖于 ",[363,194034,194036,194049],{"className":194035,"translate":367},[366],[363,194037,194039],{"className":194038},[371],[373,194040,194041],{"xmlns":375},[377,194042,194043,194047],{},[380,194044,194045],{},[383,194046,11],{},[387,194048,11],{"encoding":389},[363,194050,194052],{"className":194051,"ariaHidden":394},[393],[363,194053,194055,194058],{"className":194054},[398],[363,194056],{"className":194057,"style":2624},[402],[363,194059,11],{"className":194060},[407,408]," 本身的正确性","。EDL 提供了一条与之互补的路径：直接使用二阶量 ",[363,194064,194066,194079],{"className":194065,"translate":367},[366],[363,194067,194069],{"className":194068},[371],[373,194070,194071],{"xmlns":375},[377,194072,194073,194077],{},[380,194074,194075],{},[383,194076,15750],{},[387,194078,15750],{"encoding":389},[363,194080,194082],{"className":194081,"ariaHidden":394},[393],[363,194083,194085,194088],{"className":194084},[398],[363,194086],{"className":194087,"style":2693},[402],[363,194089,15750],{"className":194090},[407,408]," 或 ",[363,194093,194095,194118],{"className":194094,"translate":367},[366],[363,194096,194098],{"className":194097},[371],[373,194099,194100],{"xmlns":375},[377,194101,194102,194116],{},[380,194103,194104,194106,194108,194110,194112,194114],{},[383,194105,34240],{},[458,194107,461],{"stretchy":460},[383,194109,40609],{},[458,194111,34247],{"separator":394},[383,194113,157459],{"mathvariant":136055},[458,194115,468],{"stretchy":460},[387,194117,179574],{"encoding":389},[363,194119,194121],{"className":194120,"ariaHidden":394},[393],[363,194122,194124,194127,194130,194133,194136,194139,194142,194151],{"className":194123},[398],[363,194125],{"className":194126,"style":487},[402],[363,194128,34240],{"className":194129,"style":9235},[407,408],[363,194131,461],{"className":194132},[494],[363,194134,40609],{"className":194135,"style":1328},[407,408],[363,194137,34247],{"className":194138},[920],[363,194140],{"className":194141,"style":754},[505],[363,194143,194145],{"className":194144},[407],[363,194146,194148],{"className":194147},[407],[363,194149,157459],{"className":194150,"style":157493},[407,136115],[363,194152,468],{"className":194153},[501]," 作为拒识分数，从而绕开对一阶校准的依赖。其代价是引入了一个新的、校准状况未知的量。实践中应同时考察两条路径并报告风险—覆盖曲线下的面积（AURC）。",[2659,194156,194158],{"id":194157},"_53-主动学习与半监督学习","5.3 主动学习与半监督学习",[11,194160,194161,194164,194165,194091,194209,194287,194288,194316,194317,194345,194346,194349,194350,194378],{},[1344,194162,194163],{},"主动学习。"," 采集函数可以直接取 ",[363,194166,194168,194188],{"className":194167,"translate":367},[366],[363,194169,194171],{"className":194170},[371],[373,194172,194173],{"xmlns":375},[377,194174,194175,194185],{},[380,194176,194177,194179,194181,194183],{},[383,194178,15750],{},[458,194180,461],{"stretchy":460},[383,194182,1121],{"mathvariant":75598},[458,194184,468],{"stretchy":460},[387,194186,194187],{"encoding":389},"u(\\mathbf{x})",[363,194189,194191],{"className":194190,"ariaHidden":394},[393],[363,194192,194194,194197,194200,194203,194206],{"className":194193},[398],[363,194195],{"className":194196,"style":487},[402],[363,194198,15750],{"className":194199},[407,408],[363,194201,461],{"className":194202},[494],[363,194204,1121],{"className":194205},[407,75943],[363,194207,468],{"className":194208},[501],[363,194210,194212,194242],{"className":194211,"translate":367},[366],[363,194213,194215],{"className":194214},[371],[373,194216,194217],{"xmlns":375},[377,194218,194219,194239],{},[380,194220,194221,194223,194225,194227,194229,194231,194233,194235,194237],{},[383,194222,34240],{},[458,194224,461],{"stretchy":460},[383,194226,40609],{},[458,194228,34247],{"separator":394},[383,194230,157459],{"mathvariant":136055},[458,194232,468],{"stretchy":460},[458,194234,461],{"stretchy":460},[383,194236,1121],{"mathvariant":75598},[458,194238,468],{"stretchy":460},[387,194240,194241],{"encoding":389},"I(y;\\boldsymbol{\\pi})(\\mathbf{x})",[363,194243,194245],{"className":194244,"ariaHidden":394},[393],[363,194246,194248,194251,194254,194257,194260,194263,194266,194275,194278,194281,194284],{"className":194247},[398],[363,194249],{"className":194250,"style":487},[402],[363,194252,34240],{"className":194253,"style":9235},[407,408],[363,194255,461],{"className":194256},[494],[363,194258,40609],{"className":194259,"style":1328},[407,408],[363,194261,34247],{"className":194262},[920],[363,194264],{"className":194265,"style":754},[505],[363,194267,194269],{"className":194268},[407],[363,194270,194272],{"className":194271},[407],[363,194273,157459],{"className":194274,"style":157493},[407,136115],[363,194276,468],{"className":194277},[501],[363,194279,461],{"className":194280},[494],[363,194282,1121],{"className":194283},[407,75943],[363,194285,468],{"className":194286},[501],"：前者是「总证据最少」，后者是「关于类概率的知识增益最大」。选择 ",[363,194289,194291,194304],{"className":194290,"translate":367},[366],[363,194292,194294],{"className":194293},[371],[373,194295,194296],{"xmlns":375},[377,194297,194298,194302],{},[380,194299,194300],{},[383,194301,34240],{},[387,194303,34240],{"encoding":389},[363,194305,194307],{"className":194306,"ariaHidden":394},[393],[363,194308,194310,194313],{"className":194309},[398],[363,194311],{"className":194312,"style":1011},[402],[363,194314,34240],{"className":194315,"style":9235},[407,408]," 而非 ",[363,194318,194320,194333],{"className":194319,"translate":367},[366],[363,194321,194323],{"className":194322},[371],[373,194324,194325],{"xmlns":375},[377,194326,194327,194331],{},[380,194328,194329],{},[383,194330,15750],{},[387,194332,15750],{"encoding":389},[363,194334,194336],{"className":194335,"ariaHidden":394},[393],[363,194337,194339,194342],{"className":194338},[398],[363,194340],{"className":194341,"style":2693},[402],[363,194343,15750],{"className":194344},[407,408]," 的好处在于它同时考虑证据的量与分布，等价于 BALD 在迪利克雷族内的闭式对应物。EDL 在此场景的最大优势是",[1344,194347,194348],{},"代价","：一次前向即可得到采集分数，无需 MC 采样、无需多次前向、无需集成子模型。其风险同样直接来自 4.5 节第 1 点：若 ",[363,194351,194353,194366],{"className":194352,"translate":367},[366],[363,194354,194356],{"className":194355},[371],[373,194357,194358],{"xmlns":375},[377,194359,194360,194364],{},[380,194361,194362],{},[383,194363,15750],{},[387,194365,15750],{"encoding":389},[363,194367,194369],{"className":194368,"ariaHidden":394},[393],[363,194370,194372,194375],{"className":194371},[398],[363,194373],{"className":194374,"style":2693},[402],[363,194376,15750],{"className":194377},[407,408]," 因特征层的原因而系统性高估某些区域，主动学习会持续采样这些区域，形成采样偏差。缓解办法是在采集函数中叠加批内多样性项。",[11,194380,194381,194384,194385,194413,194414,194465,194466,194494],{},[1344,194382,194383],{},"半监督与伪标签。"," 伪标签流程中，",[363,194386,194388,194401],{"className":194387,"translate":367},[366],[363,194389,194391],{"className":194390},[371],[373,194392,194393],{"xmlns":375},[377,194394,194395,194399],{},[380,194396,194397],{},[383,194398,15750],{},[387,194400,15750],{"encoding":389},[363,194402,194404],{"className":194403,"ariaHidden":394},[393],[363,194405,194407,194410],{"className":194406},[398],[363,194408],{"className":194409,"style":2693},[402],[363,194411,15750],{"className":194412},[407,408]," 提供了比「最大概率」更细的过滤准则：取 ",[363,194415,194417,194435],{"className":194416,"translate":367},[366],[363,194418,194420],{"className":194419},[371],[373,194421,194422],{"xmlns":375},[377,194423,194424,194432],{},[380,194425,194426,194428,194430],{},[383,194427,15750],{},[458,194429,7599],{},[383,194431,192305],{},[387,194433,194434],{"encoding":389},"u \u003C \\tau",[363,194436,194438,194456],{"className":194437,"ariaHidden":394},[393],[363,194439,194441,194444,194447,194450,194453],{"className":194440},[398],[363,194442],{"className":194443,"style":6272},[402],[363,194445,15750],{"className":194446},[407,408],[363,194448],{"className":194449,"style":506},[505],[363,194451,7599],{"className":194452},[510],[363,194454],{"className":194455,"style":506},[505],[363,194457,194459,194462],{"className":194458},[398],[363,194460],{"className":194461,"style":2693},[402],[363,194463,192305],{"className":194464,"style":192321},[407,408]," 的样本赋予伪标签，可以同时排除「概率高但实为分布外」的样本——这正是 MSP 过滤所无法处理的失败模式。反之，当 ",[363,194467,194469,194482],{"className":194468,"translate":367},[366],[363,194470,194472],{"className":194471},[371],[373,194473,194474],{"xmlns":375},[377,194475,194476,194480],{},[380,194477,194478],{},[383,194479,192305],{},[387,194481,192308],{"encoding":389},[363,194483,194485],{"className":194484,"ariaHidden":394},[393],[363,194486,194488,194491],{"className":194487},[398],[363,194489],{"className":194490,"style":2693},[402],[363,194492,192305],{"className":194493,"style":192321},[407,408]," 过紧时，模型会倾向于只在自己已经熟悉的区域采样伪标签，导致训练集分布收缩、对稀有模式进一步失明。该权衡与 5.6 节讨论的长尾问题同源。",[2659,194496,194498],{"id":194497},"_54-安全关键场景","5.4 安全关键场景",[11,194500,194501,194502,9748],{},"在医学影像、自动驾驶与工业质检等场景中，「放弃预测并移交人类」是一个具有明确成本结构的决策，而非纯粹的统计操作。EDL 在这里的价值不是给出更准的标签，而是提供一个",[1344,194503,194504],{},"可嵌入决策规则的二阶量",[10375,194506,194507,194597,194655,194690],{},[7013,194508,194509,194512,194513,194567,194568,194596],{},[1344,194510,194511],{},"分割与体素级不确定性","：对每个体素输出 ",[363,194514,194516,194537],{"className":194515,"translate":367},[366],[363,194517,194519],{"className":194518},[371],[373,194520,194521],{"xmlns":375},[377,194522,194523,194535],{},[380,194524,194525,194527,194529,194531,194533],{},[383,194526,137112],{"mathvariant":557},[458,194528,4451],{},[458,194530,461],{"stretchy":460},[383,194532,45126],{"mathvariant":136055},[458,194534,468],{"stretchy":460},[387,194536,171778],{"encoding":389},[363,194538,194540],{"className":194539,"ariaHidden":394},[393],[363,194541,194543,194546,194552,194555,194564],{"className":194542},[398],[363,194544],{"className":194545,"style":487},[402],[363,194547,194549],{"className":194548},[1832],[363,194550,137112],{"className":194551},[407,33034],[363,194553,461],{"className":194554},[494],[363,194556,194558],{"className":194557},[407],[363,194559,194561],{"className":194560},[407],[363,194562,45126],{"className":194563},[407,136115],[363,194565,468],{"className":194566},[501],"，得到逐像素的 ",[363,194569,194571,194584],{"className":194570,"translate":367},[366],[363,194572,194574],{"className":194573},[371],[373,194575,194576],{"xmlns":375},[377,194577,194578,194582],{},[380,194579,194580],{},[383,194581,15750],{},[387,194583,15750],{"encoding":389},[363,194585,194587],{"className":194586,"ariaHidden":394},[393],[363,194588,194590,194593],{"className":194589},[398],[363,194591],{"className":194592,"style":2693},[402],[363,194594,15750],{"className":194595},[407,408],"，可据此生成「需人工复核区域」的掩码。",[7013,194598,194599,194602,194603,194654],{},[1344,194600,194601],{},"转诊（referral）机制","：将 ",[363,194604,194606,194624],{"className":194605,"translate":367},[366],[363,194607,194609],{"className":194608},[371],[373,194610,194611],{"xmlns":375},[377,194612,194613,194621],{},[380,194614,194615,194617,194619],{},[383,194616,15750],{},[458,194618,3251],{},[383,194620,192305],{},[387,194622,194623],{"encoding":389},"u > \\tau",[363,194625,194627,194645],{"className":194626,"ariaHidden":394},[393],[363,194628,194630,194633,194636,194639,194642],{"className":194629},[398],[363,194631],{"className":194632,"style":6272},[402],[363,194634,15750],{"className":194635},[407,408],[363,194637],{"className":194638,"style":506},[505],[363,194640,3251],{"className":194641},[510],[363,194643],{"className":194644,"style":506},[505],[363,194646,194648,194651],{"className":194647},[398],[363,194649],{"className":194650,"style":2693},[402],[363,194652,192305],{"className":194653,"style":192321},[407,408]," 的样本路由至专家，使自动化系统在其能力边界处显式地退出。",[7013,194656,194657,194660,194661,194689],{},[1344,194658,194659],{},"开放集动作识别","：视频中存在未见动作时，",[363,194662,194664,194677],{"className":194663,"translate":367},[366],[363,194665,194667],{"className":194666},[371],[373,194668,194669],{"xmlns":375},[377,194670,194671,194675],{},[380,194672,194673],{},[383,194674,15750],{},[387,194676,15750],{"encoding":389},[363,194678,194680],{"className":194679,"ariaHidden":394},[393],[363,194681,194683,194686],{"className":194682},[398],[363,194684],{"className":194685,"style":2693},[402],[363,194687,15750],{"className":194688},[407,408]," 提供拒绝信号。",[7013,194691,194692,194695],{},[1344,194693,194694],{},"分子性质预测与虚拟筛选","：在化学空间中标识训练分布之外的候选物，避免对高不确定性的预测投入昂贵的实验验证。",[11,194697,194698,194699,194702,194703,194731],{},"这些应用的共同点是：",[1344,194700,194701],{},"错误的代价不对称，且「不预测」是允许的动作。"," 这也划定了 EDL 不适用的场合——当系统必须对所有输入给出一个标签、且拒识没有对应的动作时，",[363,194704,194706,194719],{"className":194705,"translate":367},[366],[363,194707,194709],{"className":194708},[371],[373,194710,194711],{"xmlns":375},[377,194712,194713,194717],{},[380,194714,194715],{},[383,194716,15750],{},[387,194718,15750],{"encoding":389},[363,194720,194722],{"className":194721,"ariaHidden":394},[393],[363,194723,194725,194728],{"className":194724},[398],[363,194726],{"className":194727,"style":2693},[402],[363,194729,15750],{"className":194730},[407,408]," 至多是一个监控指标。",[2659,194733,194735],{"id":194734},"_55-回归情形的推广","5.5 回归情形的推广",[11,194737,194738,194739,9748],{},"分类的目标是单纯形上的向量，回归的目标是实数。相应地把迪利克雷替换为",[1344,194740,194741],{},"正态—逆 Gamma（Normal-Inverse-Gamma, NIG）先验",[363,194743,194745],{"className":194744,"translate":367},[528],[363,194746,194748,194821],{"className":194747,"translate":367},[366],[363,194749,194751],{"className":194750},[371],[373,194752,194753],{"xmlns":375,"display":537},[377,194754,194755,194818],{},[380,194756,194757,194759,194761,194763,194765,194786,194788,194790,194796,194798,194808,194810,194812,194814,194816],{},[383,194758,2712],{},[458,194760,9217],{},[383,194762,38656],{"mathvariant":944},[1645,194764,1652],{},[380,194766,194767,194769,194771,194773,194784],{},[458,194768,461],{"fence":394},[383,194770,121465],{},[458,194772,898],{"separator":394},[586,194774,194775,194781],{},[7162,194776,194777,194779],{},[383,194778,2680],{},[463,194780,2811],{},[383,194782,194783],{},"υ",[458,194785,468],{"fence":394},[458,194787,898],{"separator":394},[505,194789],{"width":3610},[7162,194791,194792,194794],{},[383,194793,2680],{},[463,194795,2811],{},[458,194797,9217],{},[7162,194799,194800,194802],{},[383,194801,137218],{"mathvariant":557},[380,194803,194804,194806],{},[458,194805,5161],{},[463,194807,465],{},[458,194809,461],{"stretchy":460},[383,194811,45126],{},[458,194813,898],{"separator":394},[383,194815,84504],{},[458,194817,468],{"stretchy":460},[387,194819,194820],{"encoding":389},"\\mu \\sim \\mathcal{N}\\!\\left(\\gamma, \\frac{\\sigma^2}{\\upsilon}\\right),\n\\qquad\n\\sigma^2 \\sim \\Gamma^{-1}(\\alpha, \\beta)",[363,194822,194824,194842,195020],{"className":194823,"ariaHidden":394},[393],[363,194825,194827,194830,194833,194836,194839],{"className":194826},[398],[363,194828],{"className":194829,"style":2624},[402],[363,194831,2712],{"className":194832},[407,408],[363,194834],{"className":194835,"style":506},[505],[363,194837,9217],{"className":194838},[510],[363,194840],{"className":194841,"style":506},[505],[363,194843,194845,194848,194852,194855,194858,194970,194973,194976,194979,194982,195011,195014,195017],{"className":194844},[398],[363,194846],{"className":194847,"style":87645},[402],[363,194849,38656],{"className":194850,"style":194851},[407,971],"margin-right:0.1474em;",[363,194853],{"className":194854,"style":1860},[505],[363,194856],{"className":194857,"style":754},[505],[363,194859,194861,194867,194870,194873,194876,194964],{"className":194860},[758],[363,194862,194864],{"className":194863,"style":763},[494,762],[363,194865,461],{"className":194866},[767,657],[363,194868,121465],{"className":194869,"style":121555},[407,408],[363,194871,898],{"className":194872},[920],[363,194874],{"className":194875,"style":754},[505],[363,194877,194879,194882,194961],{"className":194878},[407],[363,194880],{"className":194881},[494,789],[363,194883,194885],{"className":194884},[586],[363,194886,194888,194953],{"className":194887},[633,634],[363,194889,194891,194950],{"className":194890},[638],[363,194892,194894,194905,194913],{"className":194893,"style":38095},[642],[363,194895,194896,194899],{"style":805},[363,194897],{"className":194898,"style":809},[650],[363,194900,194902],{"className":194901},[407],[363,194903,194783],{"className":194904,"style":1328},[407,408],[363,194906,194907,194910],{"style":821},[363,194908],{"className":194909,"style":809},[650],[363,194911],{"className":194912,"style":829},[828],[363,194914,194915,194918],{"style":832},[363,194916],{"className":194917,"style":809},[650],[363,194919,194921],{"className":194920},[407],[363,194922,194924,194927],{"className":194923},[407],[363,194925,2680],{"className":194926,"style":1328},[407,408],[363,194928,194930],{"className":194929},[629],[363,194931,194933],{"className":194932},[633],[363,194934,194936],{"className":194935},[638],[363,194937,194939],{"className":194938,"style":20400},[642],[363,194940,194941,194944],{"style":7213},[363,194942],{"className":194943,"style":651},[650],[363,194945,194947],{"className":194946},[655,656,657,658],[363,194948,2811],{"className":194949},[407,658],[363,194951,666],{"className":194952},[665],[363,194954,194956],{"className":194955},[638],[363,194957,194959],{"className":194958,"style":4126},[642],[363,194960],{},[363,194962],{"className":194963},[501,789],[363,194965,194967],{"className":194966,"style":763},[501,762],[363,194968,468],{"className":194969},[767,657],[363,194971],{"className":194972,"style":754},[505],[363,194974,898],{"className":194975},[920],[363,194977],{"className":194978,"style":3982},[505],[363,194980],{"className":194981,"style":754},[505],[363,194983,194985,194988],{"className":194984},[407],[363,194986,2680],{"className":194987,"style":1328},[407,408],[363,194989,194991],{"className":194990},[629],[363,194992,194994],{"className":194993},[633],[363,194995,194997],{"className":194996},[638],[363,194998,195000],{"className":194999,"style":20107},[642],[363,195001,195002,195005],{"style":8625},[363,195003],{"className":195004,"style":651},[650],[363,195006,195008],{"className":195007},[655,656,657,658],[363,195009,2811],{"className":195010},[407,658],[363,195012],{"className":195013,"style":506},[505],[363,195015,9217],{"className":195016},[510],[363,195018],{"className":195019,"style":506},[505],[363,195021,195023,195026,195061,195064,195067,195070,195073,195076],{"className":195022},[398],[363,195024],{"className":195025,"style":37623},[402],[363,195027,195029,195032],{"className":195028},[407],[363,195030,137218],{"className":195031},[407],[363,195033,195035],{"className":195034},[629],[363,195036,195038],{"className":195037},[633],[363,195039,195041],{"className":195040},[638],[363,195042,195044],{"className":195043,"style":20107},[642],[363,195045,195046,195049],{"style":8625},[363,195047],{"className":195048,"style":651},[650],[363,195050,195052],{"className":195051},[655,656,657,658],[363,195053,195055,195058],{"className":195054},[407,658],[363,195056,5161],{"className":195057},[407,658],[363,195059,465],{"className":195060},[407,658],[363,195062,461],{"className":195063},[494],[363,195065,45126],{"className":195066,"style":45156},[407,408],[363,195068,898],{"className":195069},[920],[363,195071],{"className":195072,"style":754},[505],[363,195074,84504],{"className":195075,"style":84726},[407,408],[363,195077,468],{"className":195078},[501],[11,195080,195081,195082,195160,195161,195195,195196,195224,195225,195296,195297,195380,195381,195501,195502,195195,195535,195563],{},"网络对每个输入输出四元组 ",[363,195083,195085,195115],{"className":195084,"translate":367},[366],[363,195086,195088],{"className":195087},[371],[373,195089,195090],{"xmlns":375},[377,195091,195092,195112],{},[380,195093,195094,195096,195098,195100,195102,195104,195106,195108,195110],{},[458,195095,461],{"stretchy":460},[383,195097,121465],{},[458,195099,898],{"separator":394},[383,195101,194783],{},[458,195103,898],{"separator":394},[383,195105,45126],{},[458,195107,898],{"separator":394},[383,195109,84504],{},[458,195111,468],{"stretchy":460},[387,195113,195114],{"encoding":389},"(\\gamma, \\upsilon, \\alpha, \\beta)",[363,195116,195118],{"className":195117,"ariaHidden":394},[393],[363,195119,195121,195124,195127,195130,195133,195136,195139,195142,195145,195148,195151,195154,195157],{"className":195120},[398],[363,195122],{"className":195123,"style":487},[402],[363,195125,461],{"className":195126},[494],[363,195128,121465],{"className":195129,"style":121555},[407,408],[363,195131,898],{"className":195132},[920],[363,195134],{"className":195135,"style":754},[505],[363,195137,194783],{"className":195138,"style":1328},[407,408],[363,195140,898],{"className":195141},[920],[363,195143],{"className":195144,"style":754},[505],[363,195146,45126],{"className":195147,"style":45156},[407,408],[363,195149,898],{"className":195150},[920],[363,195152],{"className":195153,"style":754},[505],[363,195155,84504],{"className":195156,"style":84726},[407,408],[363,195158,468],{"className":195159},[501],"，边缘预测为自由度 ",[363,195162,195164,195180],{"className":195163,"translate":367},[366],[363,195165,195167],{"className":195166},[371],[373,195168,195169],{"xmlns":375},[377,195170,195171,195177],{},[380,195172,195173,195175],{},[463,195174,2811],{},[383,195176,45126],{},[387,195178,195179],{"encoding":389},"2\\alpha",[363,195181,195183],{"className":195182,"ariaHidden":394},[393],[363,195184,195186,195189,195192],{"className":195185},[398],[363,195187],{"className":195188,"style":520},[402],[363,195190,2811],{"className":195191},[407],[363,195193,45126],{"className":195194,"style":45156},[407,408]," 的 Student-",[363,195197,195199,195212],{"className":195198,"translate":367},[366],[363,195200,195202],{"className":195201},[371],[373,195203,195204],{"xmlns":375},[377,195205,195206,195210],{},[380,195207,195208],{},[383,195209,1116],{},[387,195211,1116],{"encoding":389},[363,195213,195215],{"className":195214,"ariaHidden":394},[393],[363,195216,195218,195221],{"className":195217},[398],[363,195219],{"className":195220,"style":5691},[402],[363,195222,1116],{"className":195223},[407,408]," 分布。其预测方差可精确分解为偶然部分 ",[363,195226,195228,195254],{"className":195227,"translate":367},[366],[363,195229,195231],{"className":195230},[371],[373,195232,195233],{"xmlns":375},[377,195234,195235,195251],{},[380,195236,195237,195239,195241,195243,195245,195247,195249],{},[383,195238,84504],{},[383,195240,1288],{"mathvariant":557},[458,195242,461],{"stretchy":460},[383,195244,45126],{},[458,195246,5161],{},[463,195248,465],{},[458,195250,468],{"stretchy":460},[387,195252,195253],{"encoding":389},"\\beta\u002F(\\alpha - 1)",[363,195255,195257,195284],{"className":195256,"ariaHidden":394},[393],[363,195258,195260,195263,195266,195269,195272,195275,195278,195281],{"className":195259},[398],[363,195261],{"className":195262,"style":487},[402],[363,195264,84504],{"className":195265,"style":84726},[407,408],[363,195267,1288],{"className":195268},[407],[363,195270,461],{"className":195271},[494],[363,195273,45126],{"className":195274,"style":45156},[407,408],[363,195276],{"className":195277,"style":2935},[505],[363,195279,5161],{"className":195280},[2939],[363,195282],{"className":195283,"style":2935},[505],[363,195285,195287,195290,195293],{"className":195286},[398],[363,195288],{"className":195289,"style":487},[402],[363,195291,465],{"className":195292},[407],[363,195294,468],{"className":195295},[501]," 与认知部分 ",[363,195298,195300,195332],{"className":195299,"translate":367},[366],[363,195301,195303],{"className":195302},[371],[373,195304,195305],{"xmlns":375},[377,195306,195307,195329],{},[380,195308,195309,195311,195313,195315,195317,195319,195321,195323,195325,195327],{},[383,195310,84504],{},[383,195312,1288],{"mathvariant":557},[458,195314,578],{"stretchy":460},[383,195316,194783],{},[458,195318,461],{"stretchy":460},[383,195320,45126],{},[458,195322,5161],{},[463,195324,465],{},[458,195326,468],{"stretchy":460},[458,195328,605],{"stretchy":460},[387,195330,195331],{"encoding":389},"\\beta\u002F[\\upsilon(\\alpha - 1)]",[363,195333,195335,195368],{"className":195334,"ariaHidden":394},[393],[363,195336,195338,195341,195344,195347,195350,195353,195356,195359,195362,195365],{"className":195337},[398],[363,195339],{"className":195340,"style":487},[402],[363,195342,84504],{"className":195343,"style":84726},[407,408],[363,195345,1288],{"className":195346},[407],[363,195348,578],{"className":195349},[494],[363,195351,194783],{"className":195352,"style":1328},[407,408],[363,195354,461],{"className":195355},[494],[363,195357,45126],{"className":195358,"style":45156},[407,408],[363,195360],{"className":195361,"style":2935},[505],[363,195363,5161],{"className":195364},[2939],[363,195366],{"className":195367,"style":2935},[505],[363,195369,195371,195374,195377],{"className":195370},[398],[363,195372],{"className":195373,"style":487},[402],[363,195375,465],{"className":195376},[407],[363,195378,20762],{"className":195379},[501],"，二者之和 ",[363,195382,195384,195426],{"className":195383,"translate":367},[366],[363,195385,195387],{"className":195386},[371],[373,195388,195389],{"xmlns":375},[377,195390,195391,195423],{},[380,195392,195393,195395,195397,195399,195401,195403,195405,195407,195409,195411,195413,195415,195417,195419,195421],{},[383,195394,84504],{},[458,195396,461],{"stretchy":460},[383,195398,194783],{},[458,195400,2818],{},[463,195402,465],{},[458,195404,468],{"stretchy":460},[383,195406,1288],{"mathvariant":557},[458,195408,578],{"stretchy":460},[383,195410,194783],{},[458,195412,461],{"stretchy":460},[383,195414,45126],{},[458,195416,5161],{},[463,195418,465],{},[458,195420,468],{"stretchy":460},[458,195422,605],{"stretchy":460},[387,195424,195425],{"encoding":389},"\\beta(\\upsilon + 1)\u002F[\\upsilon(\\alpha - 1)]",[363,195427,195429,195453,195489],{"className":195428,"ariaHidden":394},[393],[363,195430,195432,195435,195438,195441,195444,195447,195450],{"className":195431},[398],[363,195433],{"className":195434,"style":487},[402],[363,195436,84504],{"className":195437,"style":84726},[407,408],[363,195439,461],{"className":195440},[494],[363,195442,194783],{"className":195443,"style":1328},[407,408],[363,195445],{"className":195446,"style":2935},[505],[363,195448,2818],{"className":195449},[2939],[363,195451],{"className":195452,"style":2935},[505],[363,195454,195456,195459,195462,195465,195468,195471,195474,195477,195480,195483,195486],{"className":195455},[398],[363,195457],{"className":195458,"style":487},[402],[363,195460,465],{"className":195461},[407],[363,195463,468],{"className":195464},[501],[363,195466,1288],{"className":195467},[407],[363,195469,578],{"className":195470},[494],[363,195472,194783],{"className":195473,"style":1328},[407,408],[363,195475,461],{"className":195476},[494],[363,195478,45126],{"className":195479,"style":45156},[407,408],[363,195481],{"className":195482,"style":2935},[505],[363,195484,5161],{"className":195485},[2939],[363,195487],{"className":195488,"style":2935},[505],[363,195490,195492,195495,195498],{"className":195491},[398],[363,195493],{"className":195494,"style":487},[402],[363,195496,465],{"className":195497},[407],[363,195499,20762],{"className":195500},[501]," 即自由度 ",[363,195503,195505,195520],{"className":195504,"translate":367},[366],[363,195506,195508],{"className":195507},[371],[373,195509,195510],{"xmlns":375},[377,195511,195512,195518],{},[380,195513,195514,195516],{},[463,195515,2811],{},[383,195517,45126],{},[387,195519,195179],{"encoding":389},[363,195521,195523],{"className":195522,"ariaHidden":394},[393],[363,195524,195526,195529,195532],{"className":195525},[398],[363,195527],{"className":195528,"style":520},[402],[363,195530,2811],{"className":195531},[407],[363,195533,45126],{"className":195534,"style":45156},[407,408],[363,195536,195538,195551],{"className":195537,"translate":367},[366],[363,195539,195541],{"className":195540},[371],[373,195542,195543],{"xmlns":375},[377,195544,195545,195549],{},[380,195546,195547],{},[383,195548,1116],{},[387,195550,1116],{"encoding":389},[363,195552,195554],{"className":195553,"ariaHidden":394},[393],[363,195555,195557,195560],{"className":195556},[398],[363,195558],{"className":195559,"style":5691},[402],[363,195561,1116],{"className":195562},[407,408]," 分布的方差，形式上与命题 3.5 的分解对应。该推广在分子性质预测、三维重建与不确定性感知的姿态估计中均有应用。",[11,195565,195566],{},"需要指出的是，这一推广的理论状况比分类情形更受争议：有后续研究指出，此类模型所报告的「认知」分量并不随训练数据的增加而按预期衰减，其与偶然分量的分离缺乏理论保证。因此 4.5 节的谨慎立场在回归情形下应进一步加强。",[2659,195568,195570],{"id":195569},"_56-与原型学习超球面几何的融合","5.6 与原型学习、超球面几何的融合",[11,195572,195573,195574,195736,195737,195886,195887,196045,196046,196186],{},"最后讨论一个尚未定型、但在方法上自然的方向：把 EDL 与基于原型的超球面表示学习结合。设主干网络输出归一化特征 ",[363,195575,195577,195615],{"className":195576,"translate":367},[366],[363,195578,195580],{"className":195579},[371],[373,195581,195582],{"xmlns":375},[377,195583,195584,195612],{},[380,195585,195586,195592,195594,195596,195598,195600],{},[543,195587,195588,195590],{},[383,195589,385],{"mathvariant":75598},[383,195591,101148],{"mathvariant":136055},[458,195593,461],{"stretchy":460},[383,195595,1121],{"mathvariant":75598},[458,195597,468],{"stretchy":460},[458,195599,6257],{},[7162,195601,195602,195604],{},[383,195603,26109],{"mathvariant":570},[380,195605,195606,195608,195610],{},[383,195607,592],{},[458,195609,5161],{},[463,195611,465],{},[387,195613,195614],{"encoding":389},"\\mathbf{f}_{\\boldsymbol{\\theta}}(\\mathbf{x}) \\in \\mathbb{S}^{d-1}",[363,195616,195618,195692],{"className":195617,"ariaHidden":394},[393],[363,195619,195621,195624,195674,195677,195680,195683,195686,195689],{"className":195620},[398],[363,195622],{"className":195623,"style":487},[402],[363,195625,195627,195630],{"className":195626},[407],[363,195628,385],{"className":195629,"style":25973},[407,75943],[363,195631,195633],{"className":195632},[629],[363,195634,195636,195666],{"className":195635},[633,634],[363,195637,195639,195663],{"className":195638},[638],[363,195640,195642],{"className":195641,"style":643},[642],[363,195643,195645,195648],{"style":195644},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[363,195646],{"className":195647,"style":651},[650],[363,195649,195651],{"className":195650},[655,656,657,658],[363,195652,195654],{"className":195653},[407,658],[363,195655,195657],{"className":195656},[407,658],[363,195658,195660],{"className":195659},[407,658],[363,195661,101148],{"className":195662,"style":169926},[407,136115,658],[363,195664,666],{"className":195665},[665],[363,195667,195669],{"className":195668},[638],[363,195670,195672],{"className":195671,"style":8316},[642],[363,195673],{},[363,195675,461],{"className":195676},[494],[363,195678,1121],{"className":195679},[407,75943],[363,195681,468],{"className":195682},[501],[363,195684],{"className":195685,"style":506},[505],[363,195687,6257],{"className":195688},[510],[363,195690],{"className":195691,"style":506},[505],[363,195693,195695,195698],{"className":195694},[398],[363,195696],{"className":195697,"style":47926},[402],[363,195699,195701,195704],{"className":195700},[407],[363,195702,26109],{"className":195703},[407,714],[363,195705,195707],{"className":195706},[629],[363,195708,195710],{"className":195709},[633],[363,195711,195713],{"className":195712},[638],[363,195714,195716],{"className":195715,"style":47926},[642],[363,195717,195718,195721],{"style":7213},[363,195719],{"className":195720,"style":651},[650],[363,195722,195724],{"className":195723},[655,656,657,658],[363,195725,195727,195730,195733],{"className":195726},[407,658],[363,195728,592],{"className":195729},[407,408,658],[363,195731,5161],{"className":195732},[2939,658],[363,195734,465],{"className":195735},[407,658],"，各类原型为 ",[363,195738,195740,195777],{"className":195739,"translate":367},[366],[363,195741,195743],{"className":195742},[371],[373,195744,195745],{"xmlns":375},[377,195746,195747,195774],{},[380,195748,195749,195751,195757,195759,195762],{},[458,195750,3246],{"stretchy":460},[543,195752,195753,195755],{},[383,195754,163200],{"mathvariant":75598},[383,195756,7912],{},[458,195758,3256],{"stretchy":460},[458,195760,195761],{},"⊂",[7162,195763,195764,195766],{},[383,195765,26109],{"mathvariant":570},[380,195767,195768,195770,195772],{},[383,195769,592],{},[458,195771,5161],{},[463,195773,465],{},[387,195775,195776],{"encoding":389},"\\{\\mathbf{v}_k\\} \\subset \\mathbb{S}^{d-1}",[363,195778,195780,195842],{"className":195779,"ariaHidden":394},[393],[363,195781,195783,195786,195789,195830,195833,195836,195839],{"className":195782},[398],[363,195784],{"className":195785,"style":487},[402],[363,195787,3246],{"className":195788},[494],[363,195790,195792,195795],{"className":195791},[407],[363,195793,163200],{"className":195794,"style":162156},[407,75943],[363,195796,195798],{"className":195797},[629],[363,195799,195801,195822],{"className":195800},[633,634],[363,195802,195804,195819],{"className":195803},[638],[363,195805,195807],{"className":195806,"style":643},[642],[363,195808,195810,195813],{"style":195809},"top:-2.55em;margin-left:-0.016em;margin-right:0.05em;",[363,195811],{"className":195812,"style":651},[650],[363,195814,195816],{"className":195815},[655,656,657,658],[363,195817,7912],{"className":195818,"style":8615},[407,408,658],[363,195820,666],{"className":195821},[665],[363,195823,195825],{"className":195824},[638],[363,195826,195828],{"className":195827,"style":8316},[642],[363,195829],{},[363,195831,3256],{"className":195832},[501],[363,195834],{"className":195835,"style":506},[505],[363,195837,195761],{"className":195838},[510],[363,195840],{"className":195841,"style":506},[505],[363,195843,195845,195848],{"className":195844},[398],[363,195846],{"className":195847,"style":47926},[402],[363,195849,195851,195854],{"className":195850},[407],[363,195852,26109],{"className":195853},[407,714],[363,195855,195857],{"className":195856},[629],[363,195858,195860],{"className":195859},[633],[363,195861,195863],{"className":195862},[638],[363,195864,195866],{"className":195865,"style":47926},[642],[363,195867,195868,195871],{"style":7213},[363,195869],{"className":195870,"style":651},[650],[363,195872,195874],{"className":195873},[655,656,657,658],[363,195875,195877,195880,195883],{"className":195876},[407,658],[363,195878,592],{"className":195879},[407,408,658],[363,195881,5161],{"className":195882},[2939,658],[363,195884,465],{"className":195885},[407,658],"，相似度 ",[363,195888,195890,195926],{"className":195889,"translate":367},[366],[363,195891,195893],{"className":195892},[371],[373,195894,195895],{"xmlns":375},[377,195896,195897,195923],{},[380,195898,195899,195905,195907,195910,195912,195914,195920],{},[543,195900,195901,195903],{},[383,195902,7909],{},[383,195904,7912],{},[458,195906,471],{},[458,195908,195909],{"stretchy":460},"⟨",[383,195911,385],{"mathvariant":75598},[458,195913,898],{"separator":394},[543,195915,195916,195918],{},[383,195917,163200],{"mathvariant":75598},[383,195919,7912],{},[458,195921,195922],{"stretchy":460},"⟩",[387,195924,195925],{"encoding":389},"s_k = \\langle \\mathbf{f}, \\mathbf{v}_k \\rangle",[363,195927,195929,195984],{"className":195928,"ariaHidden":394},[393],[363,195930,195932,195935,195975,195978,195981],{"className":195931},[398],[363,195933],{"className":195934,"style":13724},[402],[363,195936,195938,195941],{"className":195937},[407],[363,195939,7909],{"className":195940},[407,408],[363,195942,195944],{"className":195943},[629],[363,195945,195947,195967],{"className":195946},[633,634],[363,195948,195950,195964],{"className":195949},[638],[363,195951,195953],{"className":195952,"style":643},[642],[363,195954,195955,195958],{"style":730},[363,195956],{"className":195957,"style":651},[650],[363,195959,195961],{"className":195960},[655,656,657,658],[363,195962,7912],{"className":195963,"style":8615},[407,408,658],[363,195965,666],{"className":195966},[665],[363,195968,195970],{"className":195969},[638],[363,195971,195973],{"className":195972,"style":8316},[642],[363,195974],{},[363,195976],{"className":195977,"style":506},[505],[363,195979,471],{"className":195980},[510],[363,195982],{"className":195983,"style":506},[505],[363,195985,195987,195990,195993,195996,195999,196002,196042],{"className":195986},[398],[363,195988],{"className":195989,"style":487},[402],[363,195991,195909],{"className":195992},[494],[363,195994,385],{"className":195995,"style":25973},[407,75943],[363,195997,898],{"className":195998},[920],[363,196000],{"className":196001,"style":754},[505],[363,196003,196005,196008],{"className":196004},[407],[363,196006,163200],{"className":196007,"style":162156},[407,75943],[363,196009,196011],{"className":196010},[629],[363,196012,196014,196034],{"className":196013},[633,634],[363,196015,196017,196031],{"className":196016},[638],[363,196018,196020],{"className":196019,"style":643},[642],[363,196021,196022,196025],{"style":195809},[363,196023],{"className":196024,"style":651},[650],[363,196026,196028],{"className":196027},[655,656,657,658],[363,196029,7912],{"className":196030,"style":8615},[407,408,658],[363,196032,666],{"className":196033},[665],[363,196035,196037],{"className":196036},[638],[363,196038,196040],{"className":196039,"style":8316},[642],[363,196041],{},[363,196043,195922],{"className":196044},[501],"。传统原型方法以 ",[363,196047,196049,196078],{"className":196048,"translate":367},[366],[363,196050,196052],{"className":196051},[371],[373,196053,196054],{"xmlns":375},[377,196055,196056,196075],{},[380,196057,196058,196064,196066,196069],{},[543,196059,196060,196062],{},[383,196061,118160],{"mathvariant":75598},[383,196063,7912],{},[458,196065,471],{},[383,196067,196068],{},"κ",[543,196070,196071,196073],{},[383,196072,7909],{},[383,196074,7912],{},[387,196076,196077],{"encoding":389},"\\mathbf{z}_k = \\kappa s_k",[363,196079,196081,196137],{"className":196080,"ariaHidden":394},[393],[363,196082,196084,196088,196128,196131,196134],{"className":196083},[398],[363,196085],{"className":196086,"style":196087},[402],"height:0.5944em;vertical-align:-0.15em;",[363,196089,196091,196094],{"className":196090},[407],[363,196092,118160],{"className":196093},[407,75943],[363,196095,196097],{"className":196096},[629],[363,196098,196100,196120],{"className":196099},[633,634],[363,196101,196103,196117],{"className":196102},[638],[363,196104,196106],{"className":196105,"style":643},[642],[363,196107,196108,196111],{"style":730},[363,196109],{"className":196110,"style":651},[650],[363,196112,196114],{"className":196113},[655,656,657,658],[363,196115,7912],{"className":196116,"style":8615},[407,408,658],[363,196118,666],{"className":196119},[665],[363,196121,196123],{"className":196122},[638],[363,196124,196126],{"className":196125,"style":8316},[642],[363,196127],{},[363,196129],{"className":196130,"style":506},[505],[363,196132,471],{"className":196133},[510],[363,196135],{"className":196136,"style":506},[505],[363,196138,196140,196143,196146],{"className":196139},[398],[363,196141],{"className":196142,"style":13724},[402],[363,196144,196068],{"className":196145},[407,408],[363,196147,196149,196152],{"className":196148},[407],[363,196150,7909],{"className":196151},[407,408],[363,196153,196155],{"className":196154},[629],[363,196156,196158,196178],{"className":196157},[633,634],[363,196159,196161,196175],{"className":196160},[638],[363,196162,196164],{"className":196163,"style":643},[642],[363,196165,196166,196169],{"style":730},[363,196167],{"className":196168,"style":651},[650],[363,196170,196172],{"className":196171},[655,656,657,658],[363,196173,7912],{"className":196174,"style":8615},[407,408,658],[363,196176,666],{"className":196177},[665],[363,196179,196181],{"className":196180},[638],[363,196182,196184],{"className":196183,"style":8316},[642],[363,196185],{}," 作为 logits 并配 softmax 交叉熵，由 3.3 节，这仍受平移不变性与 logits 发散两项缺陷的支配。",[11,196188,196189],{},"一个直接的融合方案是把相似度送入证据函数：",[363,196191,196193],{"className":196192,"translate":367},[528],[363,196194,196196,196306],{"className":196195,"translate":367},[366],[363,196197,196199],{"className":196198},[371],[373,196200,196201],{"xmlns":375,"display":537},[377,196202,196203,196303],{},[380,196204,196205,196211,196213,196215,196217,196219,196221,196227,196229,196231,196233,196239,196241,196243,196245,196247,196249,196251,196253,196259,196261,196263,196265,196267,196269,196271,196273,196287,196289,196291,196293,196295,196301],{},[543,196206,196207,196209],{},[383,196208,7915],{},[383,196210,7912],{},[458,196212,471],{},[383,196214,4506],{},[458,196216,461],{"stretchy":460},[383,196218,196068],{},[1645,196220,1647],{},[543,196222,196223,196225],{},[383,196224,7909],{},[383,196226,7912],{},[458,196228,468],{"stretchy":460},[458,196230,898],{"separator":394},[505,196232],{"width":3610},[543,196234,196235,196237],{},[383,196236,45126],{},[383,196238,7912],{},[458,196240,471],{},[463,196242,465],{},[458,196244,2818],{},[383,196246,4506],{},[458,196248,461],{"stretchy":460},[383,196250,196068],{},[1645,196252,1647],{},[543,196254,196255,196257],{},[383,196256,7909],{},[383,196258,7912],{},[458,196260,468],{"stretchy":460},[458,196262,898],{"separator":394},[505,196264],{"width":3610},[383,196266,26109],{},[458,196268,471],{},[383,196270,11044],{},[458,196272,2818],{},[48692,196274,196275,196277,196285],{},[458,196276,13811],{},[380,196278,196279,196281,196283],{},[383,196280,7912],{},[458,196282,471],{},[463,196284,465],{},[383,196286,11044],{},[383,196288,4506],{},[458,196290,461],{"stretchy":460},[383,196292,196068],{},[1645,196294,1647],{},[543,196296,196297,196299],{},[383,196298,7909],{},[383,196300,7912],{},[458,196302,468],{"stretchy":460},[387,196304,196305],{"encoding":389},"e_k = g(\\kappa\\, s_k), \\qquad \\alpha_k = 1 + g(\\kappa\\, s_k), \\qquad S = K + \\sum_{k=1}^K g(\\kappa\\, s_k)",[363,196307,196309,196364,196483,196501,196583,196601],{"className":196308,"ariaHidden":394},[393],[363,196310,196312,196315,196355,196358,196361],{"className":196311},[398],[363,196313],{"className":196314,"style":13724},[402],[363,196316,196318,196321],{"className":196317},[407],[363,196319,7915],{"className":196320},[407,408],[363,196322,196324],{"className":196323},[629],[363,196325,196327,196347],{"className":196326},[633,634],[363,196328,196330,196344],{"className":196329},[638],[363,196331,196333],{"className":196332,"style":643},[642],[363,196334,196335,196338],{"style":730},[363,196336],{"className":196337,"style":651},[650],[363,196339,196341],{"className":196340},[655,656,657,658],[363,196342,7912],{"className":196343,"style":8615},[407,408,658],[363,196345,666],{"className":196346},[665],[363,196348,196350],{"className":196349},[638],[363,196351,196353],{"className":196352,"style":8316},[642],[363,196354],{},[363,196356],{"className":196357,"style":506},[505],[363,196359,471],{"className":196360},[510],[363,196362],{"className":196363,"style":506},[505],[363,196365,196367,196370,196373,196376,196379,196382,196422,196425,196428,196431,196434,196474,196477,196480],{"className":196366},[398],[363,196368],{"className":196369,"style":487},[402],[363,196371,4506],{"className":196372,"style":1328},[407,408],[363,196374,461],{"className":196375},[494],[363,196377,196068],{"className":196378},[407,408],[363,196380],{"className":196381,"style":754},[505],[363,196383,196385,196388],{"className":196384},[407],[363,196386,7909],{"className":196387},[407,408],[363,196389,196391],{"className":196390},[629],[363,196392,196394,196414],{"className":196393},[633,634],[363,196395,196397,196411],{"className":196396},[638],[363,196398,196400],{"className":196399,"style":643},[642],[363,196401,196402,196405],{"style":730},[363,196403],{"className":196404,"style":651},[650],[363,196406,196408],{"className":196407},[655,656,657,658],[363,196409,7912],{"className":196410,"style":8615},[407,408,658],[363,196412,666],{"className":196413},[665],[363,196415,196417],{"className":196416},[638],[363,196418,196420],{"className":196419,"style":8316},[642],[363,196421],{},[363,196423,468],{"className":196424},[501],[363,196426,898],{"className":196427},[920],[363,196429],{"className":196430,"style":3982},[505],[363,196432],{"className":196433,"style":754},[505],[363,196435,196437,196440],{"className":196436},[407],[363,196438,45126],{"className":196439,"style":45156},[407,408],[363,196441,196443],{"className":196442},[629],[363,196444,196446,196466],{"className":196445},[633,634],[363,196447,196449,196463],{"className":196448},[638],[363,196450,196452],{"className":196451,"style":643},[642],[363,196453,196454,196457],{"style":136154},[363,196455],{"className":196456,"style":651},[650],[363,196458,196460],{"className":196459},[655,656,657,658],[363,196461,7912],{"className":196462,"style":8615},[407,408,658],[363,196464,666],{"className":196465},[665],[363,196467,196469],{"className":196468},[638],[363,196470,196472],{"className":196471,"style":8316},[642],[363,196473],{},[363,196475],{"className":196476,"style":506},[505],[363,196478,471],{"className":196479},[510],[363,196481],{"className":196482,"style":506},[505],[363,196484,196486,196489,196492,196495,196498],{"className":196485},[398],[363,196487],{"className":196488,"style":12928},[402],[363,196490,465],{"className":196491},[407],[363,196493],{"className":196494,"style":2935},[505],[363,196496,2818],{"className":196497},[2939],[363,196499],{"className":196500,"style":2935},[505],[363,196502,196504,196507,196510,196513,196516,196519,196559,196562,196565,196568,196571,196574,196577,196580],{"className":196503},[398],[363,196505],{"className":196506,"style":487},[402],[363,196508,4506],{"className":196509,"style":1328},[407,408],[363,196511,461],{"className":196512},[494],[363,196514,196068],{"className":196515},[407,408],[363,196517],{"className":196518,"style":754},[505],[363,196520,196522,196525],{"className":196521},[407],[363,196523,7909],{"className":196524},[407,408],[363,196526,196528],{"className":196527},[629],[363,196529,196531,196551],{"className":196530},[633,634],[363,196532,196534,196548],{"className":196533},[638],[363,196535,196537],{"className":196536,"style":643},[642],[363,196538,196539,196542],{"style":730},[363,196540],{"className":196541,"style":651},[650],[363,196543,196545],{"className":196544},[655,656,657,658],[363,196546,7912],{"className":196547,"style":8615},[407,408,658],[363,196549,666],{"className":196550},[665],[363,196552,196554],{"className":196553},[638],[363,196555,196557],{"className":196556,"style":8316},[642],[363,196558],{},[363,196560,468],{"className":196561},[501],[363,196563,898],{"className":196564},[920],[363,196566],{"className":196567,"style":3982},[505],[363,196569],{"className":196570,"style":754},[505],[363,196572,26109],{"className":196573,"style":26233},[407,408],[363,196575],{"className":196576,"style":506},[505],[363,196578,471],{"className":196579},[510],[363,196581],{"className":196582,"style":506},[505],[363,196584,196586,196589,196592,196595,196598],{"className":196585},[398],[363,196587],{"className":196588,"style":133944},[402],[363,196590,11044],{"className":196591,"style":11132},[407,408],[363,196593],{"className":196594,"style":2935},[505],[363,196596,2818],{"className":196597},[2939],[363,196599],{"className":196600,"style":2935},[505],[363,196602,196604,196608,196674,196677,196680,196683,196686,196689,196729],{"className":196603},[398],[363,196605],{"className":196606,"style":196607},[402],"height:3.1304em;vertical-align:-1.3021em;",[363,196609,196611],{"className":196610},[1832,14066],[363,196612,196614,196665],{"className":196613},[633,634],[363,196615,196617,196662],{"className":196616},[638],[363,196618,196620,196641,196651],{"className":196619,"style":138274},[642],[363,196621,196623,196626],{"style":196622},"top:-1.8479em;margin-left:0em;",[363,196624],{"className":196625,"style":14083},[650],[363,196627,196629],{"className":196628},[655,656,657,658],[363,196630,196632,196635,196638],{"className":196631},[407,658],[363,196633,7912],{"className":196634,"style":8615},[407,408,658],[363,196636,471],{"className":196637},[510,658],[363,196639,465],{"className":196640},[407,658],[363,196642,196643,196646],{"style":14092},[363,196644],{"className":196645,"style":14083},[650],[363,196647,196648],{},[363,196649,13811],{"className":196650},[1832,1833,1834],[363,196652,196653,196656],{"style":100713},[363,196654],{"className":196655,"style":14083},[650],[363,196657,196659],{"className":196658},[655,656,657,658],[363,196660,11044],{"className":196661,"style":11132},[407,408,658],[363,196663,666],{"className":196664},[665],[363,196666,196668],{"className":196667},[638],[363,196669,196672],{"className":196670,"style":196671},[642],"height:1.3021em;",[363,196673],{},[363,196675],{"className":196676,"style":754},[505],[363,196678,4506],{"className":196679,"style":1328},[407,408],[363,196681,461],{"className":196682},[494],[363,196684,196068],{"className":196685},[407,408],[363,196687],{"className":196688,"style":754},[505],[363,196690,196692,196695],{"className":196691},[407],[363,196693,7909],{"className":196694},[407,408],[363,196696,196698],{"className":196697},[629],[363,196699,196701,196721],{"className":196700},[633,634],[363,196702,196704,196718],{"className":196703},[638],[363,196705,196707],{"className":196706,"style":643},[642],[363,196708,196709,196712],{"style":730},[363,196710],{"className":196711,"style":651},[650],[363,196713,196715],{"className":196714},[655,656,657,658],[363,196716,7912],{"className":196717,"style":8615},[407,408,658],[363,196719,666],{"className":196720},[665],[363,196722,196724],{"className":196723},[638],[363,196725,196727],{"className":196726,"style":8316},[642],[363,196728],{},[363,196730,468],{"className":196731},[501],[11,196733,2598,196734,196762,196763,196792,196793,196863,196864,196892,196893,196921,196922,196991,196992,10528,197042],{},[363,196735,196737,196750],{"className":196736,"translate":367},[366],[363,196738,196740],{"className":196739},[371],[373,196741,196742],{"xmlns":375},[377,196743,196744,196748],{},[380,196745,196746],{},[383,196747,4506],{},[387,196749,4506],{"encoding":389},[363,196751,196753],{"className":196752,"ariaHidden":394},[393],[363,196754,196756,196759],{"className":196755},[398],[363,196757],{"className":196758,"style":2624},[402],[363,196760,4506],{"className":196761,"style":1328},[407,408]," 取 3.4 节的非负激活，",[363,196764,196766,196780],{"className":196765,"translate":367},[366],[363,196767,196769],{"className":196768},[371],[373,196770,196771],{"xmlns":375},[377,196772,196773,196777],{},[380,196774,196775],{},[383,196776,196068],{},[387,196778,196779],{"encoding":389},"\\kappa",[363,196781,196783],{"className":196782,"ariaHidden":394},[393],[363,196784,196786,196789],{"className":196785},[398],[363,196787],{"className":196788,"style":2693},[402],[363,196790,196068],{"className":196791},[407,408]," 沿用 vMF 分布的集中参数语义。该构造把几何与不确定性直接挂钩：特征落在某原型附近时，",[363,196794,196796,196814],{"className":196795,"translate":367},[366],[363,196797,196799],{"className":196798},[371],[373,196800,196801],{"xmlns":375},[377,196802,196803,196811],{},[380,196804,196805],{},[543,196806,196807,196809],{},[383,196808,7909],{},[383,196810,7912],{},[387,196812,196813],{"encoding":389},"s_k",[363,196815,196817],{"className":196816,"ariaHidden":394},[393],[363,196818,196820,196823],{"className":196819},[398],[363,196821],{"className":196822,"style":13724},[402],[363,196824,196826,196829],{"className":196825},[407],[363,196827,7909],{"className":196828},[407,408],[363,196830,196832],{"className":196831},[629],[363,196833,196835,196855],{"className":196834},[633,634],[363,196836,196838,196852],{"className":196837},[638],[363,196839,196841],{"className":196840,"style":643},[642],[363,196842,196843,196846],{"style":730},[363,196844],{"className":196845,"style":651},[650],[363,196847,196849],{"className":196848},[655,656,657,658],[363,196850,7912],{"className":196851,"style":8615},[407,408,658],[363,196853,666],{"className":196854},[665],[363,196856,196858],{"className":196857},[638],[363,196859,196861],{"className":196860,"style":8316},[642],[363,196862],{}," 接近 ",[363,196865,196867,196880],{"className":196866,"translate":367},[366],[363,196868,196870],{"className":196869},[371],[373,196871,196872],{"xmlns":375},[377,196873,196874,196878],{},[380,196875,196876],{},[463,196877,465],{},[387,196879,465],{"encoding":389},[363,196881,196883],{"className":196882,"ariaHidden":394},[393],[363,196884,196886,196889],{"className":196885},[398],[363,196887],{"className":196888,"style":520},[402],[363,196890,465],{"className":196891},[407],"、证据充分、",[363,196894,196896,196909],{"className":196895,"translate":367},[366],[363,196897,196899],{"className":196898},[371],[373,196900,196901],{"xmlns":375},[377,196902,196903,196907],{},[380,196904,196905],{},[383,196906,15750],{},[387,196908,15750],{"encoding":389},[363,196910,196912],{"className":196911,"ariaHidden":394},[393],[363,196913,196915,196918],{"className":196914},[398],[363,196916],{"className":196917,"style":2693},[402],[363,196919,15750],{"className":196920},[407,408]," 小。特征落在所有原型之外（其与全部原型的相似度均低或为负）时，各 ",[363,196923,196925,196942],{"className":196924,"translate":367},[366],[363,196926,196928],{"className":196927},[371],[373,196929,196930],{"xmlns":375},[377,196931,196932,196940],{},[380,196933,196934],{},[543,196935,196936,196938],{},[383,196937,7915],{},[383,196939,7912],{},[387,196941,172014],{"encoding":389},[363,196943,196945],{"className":196944,"ariaHidden":394},[393],[363,196946,196948,196951],{"className":196947},[398],[363,196949],{"className":196950,"style":13724},[402],[363,196952,196954,196957],{"className":196953},[407],[363,196955,7915],{"className":196956},[407,408],[363,196958,196960],{"className":196959},[629],[363,196961,196963,196983],{"className":196962},[633,634],[363,196964,196966,196980],{"className":196965},[638],[363,196967,196969],{"className":196968,"style":643},[642],[363,196970,196971,196974],{"style":730},[363,196972],{"className":196973,"style":651},[650],[363,196975,196977],{"className":196976},[655,656,657,658],[363,196978,7912],{"className":196979,"style":8615},[407,408,658],[363,196981,666],{"className":196982},[665],[363,196984,196986],{"className":196985},[638],[363,196987,196989],{"className":196988,"style":8316},[642],[363,196990],{}," 同时趋零、",[363,196993,196995,197012],{"className":196994,"translate":367},[366],[363,196996,196998],{"className":196997},[371],[373,196999,197000],{"xmlns":375},[377,197001,197002,197010],{},[380,197003,197004,197006,197008],{},[383,197005,15750],{},[458,197007,1378],{},[463,197009,465],{},[387,197011,174999],{"encoding":389},[363,197013,197015,197033],{"className":197014,"ariaHidden":394},[393],[363,197016,197018,197021,197024,197027,197030],{"className":197017},[398],[363,197019],{"className":197020,"style":2693},[402],[363,197022,15750],{"className":197023},[407,408],[363,197025],{"className":197026,"style":506},[505],[363,197028,1378],{"className":197029},[510],[363,197031],{"className":197032,"style":506},[505],[363,197034,197036,197039],{"className":197035},[398],[363,197037],{"className":197038,"style":520},[402],[363,197040,465],{"className":197041},[407],[1344,197043,197044],{},"「离原型远」与「证据不足」在此被同一个机制表达。",[11,197046,197047,197048,197051,197052,197055,197056,197128,197129,197157,197158,197161,197162,197196],{},"但这一构造的成立范围必须严格界定。迪利克雷分布定义在",[1344,197049,197050],{},"类概率单纯形","上，原型方法定义在",[1344,197053,197054],{},"球面特征空间","上，二者的「不确定性」不是同一个对象：前者是关于类概率向量的二阶分布，后者是关于特征位置的一阶几何量。",[363,197057,197059,197083],{"className":197058,"translate":367},[366],[363,197060,197062],{"className":197061},[371],[373,197063,197064],{"xmlns":375},[377,197065,197066,197080],{},[380,197067,197068,197070,197072,197074,197076,197078],{},[383,197069,1121],{"mathvariant":75598},[458,197071,9032],{},[383,197073,45126],{"mathvariant":136055},[458,197075,461],{"stretchy":460},[383,197077,1121],{"mathvariant":75598},[458,197079,468],{"stretchy":460},[387,197081,197082],{"encoding":389},"\\mathbf{x} \\mapsto \\boldsymbol{\\alpha}(\\mathbf{x})",[363,197084,197086,197104],{"className":197085,"ariaHidden":394},[393],[363,197087,197089,197092,197095,197098,197101],{"className":197088},[398],[363,197090],{"className":197091,"style":84346},[402],[363,197093,1121],{"className":197094},[407,75943],[363,197096],{"className":197097,"style":506},[505],[363,197099,9032],{"className":197100},[510],[363,197102],{"className":197103,"style":506},[505],[363,197105,197107,197110,197119,197122,197125],{"className":197106},[398],[363,197108],{"className":197109,"style":487},[402],[363,197111,197113],{"className":197112},[407],[363,197114,197116],{"className":197115},[407],[363,197117,45126],{"className":197118},[407,136115],[363,197120,461],{"className":197121},[494],[363,197123,1121],{"className":197124},[407,75943],[363,197126,468],{"className":197127},[501]," 仍然是一个点估计的映射，因此 4.5 节第 1 点的批评在此完全适用——",[363,197130,197132,197145],{"className":197131,"translate":367},[366],[363,197133,197135],{"className":197134},[371],[373,197136,197137],{"xmlns":375},[377,197138,197139,197143],{},[380,197140,197141],{},[383,197142,15750],{},[387,197144,15750],{"encoding":389},[363,197146,197148],{"className":197147,"ariaHidden":394},[393],[363,197149,197151,197154],{"className":197150},[398],[363,197152],{"className":197153,"style":2693},[402],[363,197155,15750],{"className":197156},[407,408]," 度量的是「特征到原型的距离」，而不是「模型对该映射的把握」。要真正结合二者，需要构造（特征空间，类概率空间）的",[1344,197159,197160],{},"联合","模型，例如令 ",[363,197163,197165,197178],{"className":197164,"translate":367},[366],[363,197166,197168],{"className":197167},[371],[373,197169,197170],{"xmlns":375},[377,197171,197172,197176],{},[380,197173,197174],{},[383,197175,45126],{"mathvariant":136055},[387,197177,137074],{"encoding":389},[363,197179,197181],{"className":197180,"ariaHidden":394},[393],[363,197182,197184,197187],{"className":197183},[398],[363,197185],{"className":197186,"style":135662},[402],[363,197188,197190],{"className":197189},[407],[363,197191,197193],{"className":197192},[407],[363,197194,45126],{"className":197195},[407,136115]," 自身携带分布（层级模型），或以 vMF 构造特征的生成式模型、以迪利克雷构造其上的分类头，使两个不确定性来源可以在同一层级结构中分别标识。",[11,197198,197199],{},"即便在当前的简化形式下，该方向仍有两个值得推进的具体问题：",[7010,197201,197202,197266],{},[7013,197203,197204,197207,197208,197236,197237,197265],{},[1344,197205,197206],{},"长尾与稀有类。"," 稀有类的原型天然缺乏证据支撑，",[363,197209,197211,197224],{"className":197210,"translate":367},[366],[363,197212,197214],{"className":197213},[371],[373,197215,197216],{"xmlns":375},[377,197217,197218,197222],{},[380,197219,197220],{},[383,197221,15750],{},[387,197223,15750],{"encoding":389},[363,197225,197227],{"className":197226,"ariaHidden":394},[393],[363,197228,197230,197233],{"className":197229},[398],[363,197231],{"className":197232,"style":2693},[402],[363,197234,15750],{"className":197235},[407,408]," 更高。这既可以被用作保护机制（在损失中对高 ",[363,197238,197240,197253],{"className":197239,"translate":367},[366],[363,197241,197243],{"className":197242},[371],[373,197244,197245],{"xmlns":375},[377,197246,197247,197251],{},[380,197248,197249],{},[383,197250,15750],{},[387,197252,15750],{"encoding":389},[363,197254,197256],{"className":197255,"ariaHidden":394},[393],[363,197257,197259,197262],{"className":197258},[398],[363,197260],{"className":197261,"style":2693},[402],[363,197263,15750],{"className":197264},[407,408]," 样本降低梯度权重），也可能加剧对稀有类的欠拟合（模型学会用「不确定」来逃避稀有类）。二者孰占主导是一个可以实验判定的问题。",[7013,197267,197268,197271,197272,194091,197300,197362,197363],{},[1344,197269,197270],{},"与开集\u002F分布外任务的重叠。"," 在代码漏洞检测这类任务中，异常样本（漏洞）与正常样本在特征空间中高度重叠，其本质是开集问题而非单纯的类别不平衡。",[363,197273,197275,197288],{"className":197274,"translate":367},[366],[363,197276,197278],{"className":197277},[371],[373,197279,197280],{"xmlns":375},[377,197281,197282,197286],{},[380,197283,197284],{},[383,197285,15750],{},[387,197287,15750],{"encoding":389},[363,197289,197291],{"className":197290,"ariaHidden":394},[393],[363,197292,197294,197297],{"className":197293},[398],[363,197295],{"className":197296,"style":2693},[402],[363,197298,15750],{"className":197299},[407,408],[363,197301,197303,197326],{"className":197302,"translate":367},[366],[363,197304,197306],{"className":197305},[371],[373,197307,197308],{"xmlns":375},[377,197309,197310,197324],{},[380,197311,197312,197314,197316,197318,197320,197322],{},[383,197313,34240],{},[458,197315,461],{"stretchy":460},[383,197317,40609],{},[458,197319,34247],{"separator":394},[383,197321,157459],{"mathvariant":136055},[458,197323,468],{"stretchy":460},[387,197325,179574],{"encoding":389},[363,197327,197329],{"className":197328,"ariaHidden":394},[393],[363,197330,197332,197335,197338,197341,197344,197347,197350,197359],{"className":197331},[398],[363,197333],{"className":197334,"style":487},[402],[363,197336,34240],{"className":197337,"style":9235},[407,408],[363,197339,461],{"className":197340},[494],[363,197342,40609],{"className":197343,"style":1328},[407,408],[363,197345,34247],{"className":197346},[920],[363,197348],{"className":197349,"style":754},[505],[363,197351,197353],{"className":197352},[407],[363,197354,197356],{"className":197355},[407],[363,197357,157459],{"className":197358,"style":157493},[407,136115],[363,197360,468],{"className":197361},[501]," 可用作告警与拒识的分数，但必须与 4.5 节第 5 点的前提一并验证：",[1344,197364,197365],{},"只有当特征提取器确实把漏洞样本映射到低证据区时，这一机制才成立，而这正是需要独立验证的假设。",[18,197367,197368],{"id":197368},"结语",[11,197370,197371],{},"回顾全文，迪利克雷分布在本文中依次扮演了三个层次的角色：",[10375,197373,197374,197484,197490],{},[7013,197375,197376,197379,197380,164468,197414,197483],{},[1344,197377,197378],{},"作为单纯形上的分布","，它是「关于概率的概率」这一二阶对象的标准载体，其参数由方向 ",[363,197381,197383,197396],{"className":197382,"translate":367},[366],[363,197384,197386],{"className":197385},[371],[373,197387,197388],{"xmlns":375},[377,197389,197390,197394],{},[380,197391,197392],{},[383,197393,2712],{"mathvariant":136055},[387,197395,152265],{"encoding":389},[363,197397,197399],{"className":197398,"ariaHidden":394},[393],[363,197400,197402,197405],{"className":197401},[398],[363,197403],{"className":197404,"style":133807},[402],[363,197406,197408],{"className":197407},[407],[363,197409,197411],{"className":197410},[407],[363,197412,2712],{"className":197413},[407,136115],[363,197415,197417,197434],{"className":197416,"translate":367},[366],[363,197418,197420],{"className":197419},[371],[373,197421,197422],{"xmlns":375},[377,197423,197424,197432],{},[380,197425,197426],{},[543,197427,197428,197430],{},[383,197429,45126],{},[463,197431,474],{},[387,197433,152304],{"encoding":389},[363,197435,197437],{"className":197436,"ariaHidden":394},[393],[363,197438,197440,197443],{"className":197439},[398],[363,197441],{"className":197442,"style":13724},[402],[363,197444,197446,197449],{"className":197445},[407],[363,197447,45126],{"className":197448,"style":45156},[407,408],[363,197450,197452],{"className":197451},[629],[363,197453,197455,197475],{"className":197454},[633,634],[363,197456,197458,197472],{"className":197457},[638],[363,197459,197461],{"className":197460,"style":11483},[642],[363,197462,197463,197466],{"style":136154},[363,197464],{"className":197465,"style":651},[650],[363,197467,197469],{"className":197468},[655,656,657,658],[363,197470,474],{"className":197471},[407,658],[363,197473,666],{"className":197474},[665],[363,197476,197478],{"className":197477},[638],[363,197479,197481],{"className":197480,"style":8316},[642],[363,197482],{}," 两个正交分量构成（命题 2.2）。",[7013,197485,197486,197489],{},[1344,197487,197488],{},"作为多项分布的共轭先验","，它把贝叶斯更新化简为一次向量加法（命题 2.5），并在聚合性、中立性与 stick-breaking 构造下形成一个结构完整、边界清晰的分布族（命题 2.8、2.9）。",[7013,197491,197492,197495],{},[1344,197493,197494],{},"作为神经网络的输出层","，经由与主观逻辑的严格双射（命题 3.2），它把分类器的输出从「单纯形上的点」提升为「单纯形上的分布」，从而首次让「我不知道」成为一个可表示、可优化的对象。",[11,197497,197498,197499,197502,197503,197563],{},"证据深度学习的核心洞见可以压缩为一句话：",[1344,197500,197501],{},"softmax 的平移不变性丢弃了 logits 的绝对水平，因而无法表示无信息状态。证据函数不是平移不变的，它把这一被丢弃的自由度重新赋值为「证据的量级」。"," 命题 3.4 中 ",[363,197504,197506,197527],{"className":197505,"translate":367},[366],[363,197507,197509],{"className":197508},[371],[373,197510,197511],{"xmlns":375},[377,197512,197513,197525],{},[380,197514,197515,197517,197519,197521,197523],{},[383,197516,15750],{},[458,197518,471],{},[383,197520,11044],{},[383,197522,1288],{"mathvariant":557},[383,197524,26109],{},[387,197526,175682],{"encoding":389},[363,197528,197530,197548],{"className":197529,"ariaHidden":394},[393],[363,197531,197533,197536,197539,197542,197545],{"className":197532},[398],[363,197534],{"className":197535,"style":2693},[402],[363,197537,15750],{"className":197538},[407,408],[363,197540],{"className":197541,"style":506},[505],[363,197543,471],{"className":197544},[510],[363,197546],{"className":197547,"style":506},[505],[363,197549,197551,197554,197557,197560],{"className":197550},[398],[363,197552],{"className":197553,"style":487},[402],[363,197555,11044],{"className":197556,"style":11132},[407,408],[363,197558,1288],{"className":197559},[407],[363,197561,26109],{"className":197562,"style":26233},[407,408]," 的全部行为——证据充分时趋于确定、证据消失时退回均匀——都是这一重新赋值的直接后果。",[11,197565,197566],{},"配合 4.3 节的退火 KL 正则，模型不仅被要求「预测得对」，还被要求「在预测错时承认自己不确定」。",[11,197568,197569,197570,197572,197573,197601,197602],{},"但本文同样花了相当篇幅界定这一范式的边界。EDL 的不确定性是",[1344,197571,175722],{},"的，不是贝叶斯后验不确定性：它以点估计替换了对网络参数的边缘化，因而度量的是对输出分布的把握而非对模型的把握（4.5 节第 1 点）。",[363,197574,197576,197589],{"className":197575,"translate":367},[366],[363,197577,197579],{"className":197578},[371],[373,197580,197581],{"xmlns":375},[377,197582,197583,197587],{},[380,197584,197585],{},[383,197586,15750],{},[387,197588,15750],{"encoding":389},[363,197590,197592],{"className":197591,"ariaHidden":394},[393],[363,197593,197595,197598],{"className":197594},[398],[363,197596],{"className":197597,"style":2693},[402],[363,197599,15750],{"className":197600},[407,408]," 丢弃了证据在类别间的分布信息（第 2 点）。其拟合损失带有与证据量级耦合的偏置，不等同于边缘似然（第 3 点）。并且「以损失最小化实现不确定性量化」这一范式本身受到根本性质疑（第 4 点）。这些限制并不取消该方法的实用价值——在单次前向的代价约束下同时给出预测与不确定性，其性价比是突出的——但它们决定了正确的使用姿态：",[1344,197603,197604],{},"在需要二阶量的场合使用它，在需要后验语义的场合验证它，并在两者都不确定时，回到对参数做边缘化的方法上去。",[11,197606,197607,197608,197611],{},"最后应当记住的是，不确定性量化的可靠程度，与因果推断一样，",[1344,197609,197610],{},"等于你愿意且能够承担的假设的强度","。迪利克雷分布给出的是一个精确而优美的数学结构。把一个精确的结构接到一个近似的映射（神经网络）上，得到的精确性只属于前者。识别出哪一部分是定理、哪一部分是启发式，是使用这套工具时最有价值的一步。",{"title":330,"searchDepth":331,"depth":331,"links":197613},[197614,197620,197628,197636,197643,197651],{"id":133849,"depth":331,"text":133850,"children":197615},[197616,197617,197618,197619],{"id":133853,"depth":133022,"text":133854},{"id":136031,"depth":133022,"text":136032},{"id":142260,"depth":133022,"text":142261},{"id":145371,"depth":133022,"text":145372},{"id":147513,"depth":331,"text":147514,"children":197621},[197622,197623,197624,197625,197626,197627],{"id":147517,"depth":133022,"text":147518},{"id":152244,"depth":133022,"text":152245},{"id":154484,"depth":133022,"text":154485},{"id":157435,"depth":133022,"text":157436},{"id":161569,"depth":133022,"text":161570},{"id":164247,"depth":133022,"text":164248},{"id":166021,"depth":331,"text":166022,"children":197629},[197630,197631,197632,197633,197634,197635],{"id":166025,"depth":133022,"text":166026},{"id":167672,"depth":133022,"text":167673},{"id":169841,"depth":133022,"text":169842},{"id":171048,"depth":133022,"text":171049},{"id":172946,"depth":133022,"text":172947},{"id":175653,"depth":133022,"text":175654},{"id":180061,"depth":331,"text":180062,"children":197637},[197638,197639,197640,197641,197642],{"id":180152,"depth":133022,"text":180153},{"id":183599,"depth":133022,"text":183600},{"id":185788,"depth":133022,"text":185789},{"id":190002,"depth":133022,"text":190003},{"id":190652,"depth":133022,"text":190653},{"id":192189,"depth":331,"text":192190,"children":197644},[197645,197646,197647,197648,197649,197650],{"id":192193,"depth":133022,"text":192194},{"id":193128,"depth":133022,"text":193129},{"id":194157,"depth":133022,"text":194158},{"id":194497,"depth":133022,"text":194498},{"id":194734,"depth":133022,"text":194735},{"id":195569,"depth":133022,"text":195570},{"id":197368,"depth":331,"text":197368},"设 X\\mathcal{X}X 为输入空间，y∈{1,…,K}y \\in \\{1, \\dots, K\\}y∈{1,…,K} 为类别标签。一个 KKK 分类器的输出在数学上究竟是什么？若要求它给出「属于各类别的概率」，则输出必须落在",{},"\u002Fblog\u002F2026\u002F0007-dirichlet-distribution-and-evidential-deep-learning",{"title":133091,"description":197652},"blog\u002F2026\u002F0007-dirichlet-distribution-and-evidential-deep-learning","Dirichlet 是单纯形上的分布、多项分布的共轭先验，也是「关于概率的概率」这一二阶对象的自然载体。本文先给出它在定义、矩结构、共轭更新与分裂构造上的完整形式化，神经网络的输出重解释为证据并参数化迪利克雷分布，推导证据深度学习的对数损失等应用。",[133085,197659],"machine-learning","2026-10-01T18:52:06+08:00","Mepc-PK7MsRVGk2QQbFNDWa8P860uqFfJNNZDD7bKTo",1792183806064]