[{"data":1,"prerenderedAt":121820},["ShallowReactive",2],{"post-\u002Fblog\u002F2025\u002F2025-09-11-representer-theorem":3,"i-ci:tag":33448,"i-mingcute:time-line":33448},{"post":4,"nextPost":33458,"prevPost":61582},{"id":5,"title":6,"body":7,"description":33444,"draft":33445,"enableComment":33446,"extension":33447,"image":33412,"important":33445,"location":33448,"meta":33449,"navigation":33446,"ogImage":33448,"path":33450,"seo":33451,"stem":33452,"summary":33453,"tags":33454,"time":33456,"__hash__":33457},"blog\u002Fblog\u002F2025\u002F2025-09-11-representer-theorem.md","核方法基石：Representer Theorem",{"type":8,"value":9,"toc":33411},"minimark",[10,91,94,99,291,423,601,604,812,845,954,1125,1128,1724,1899,1903,1908,2404,2834,3053,3772,3866,4141,4200,4272,4525,4528,5234,5307,5311,5314,5479,5610,5902,6144,6147,6151,7111,7243,7247,7251,8722,8725,9086,9321,9357,9361,9364,9680,9735,10164,10401,11039,11116,11119,11615,11647,11924,11927,12602,12967,13532,13544,13605,13609,13863,14298,14608,15466,15551,15859,16120,16124,16135,17044,17386,17689,17693,17697,18070,18074,18106,18651,18865,18868,19014,19070,19073,19357,20141,20148,20152,20155,20294,20394,20397,20551,20776,20780,20784,20908,21368,21616,21792,21824,22062,22277,22412,22473,22982,23182,23186,23189,23777,23883,24188,24191,24968,24971,25193,25446,25449,26313,26413,26751,27015,27019,27022,27552,27800,28333,28365,28369,28372,28474,28477,28838,28841,29078,29081,29413,29573,29577,29711,29944,29947,30433,30575,30578,30582,31419,31423,31884,31916,32697,32729,32733,32740,32743,33047,33215,33218,33222,33229,33347],[11,12,13,14,90],"p",{},"核方法最反直觉的一点是：你明明在一个可能是无限维的函数空间里做优化，最后却只需要解一个 ",[15,16,20,49],"span",{"className":17,"translate":19},[18],"katex","no",[15,21,24],{"className":22},[23],"katex-mathml",[25,26,28],"math",{"xmlns":27},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[29,30,31,44],"semantics",{},[32,33,34,38,42],"mrow",{},[35,36,37],"mi",{},"n",[39,40,41],"mo",{},"×",[35,43,37],{},[45,46,48],"annotation",{"encoding":47},"application\u002Fx-tex","n\\times n",[15,50,54,80],{"className":51,"ariaHidden":53},[52],"katex-html","true",[15,55,58,63,68,73,77],{"className":56},[57],"base",[15,59],{"className":60,"style":62},[61],"strut","height:0.6667em;vertical-align:-0.0833em;",[15,64,37],{"className":65},[66,67],"mord","mathnormal",[15,69],{"className":70,"style":72},[71],"mspace","margin-right:0.2222em;",[15,74,41],{"className":75},[76],"mbin",[15,78],{"className":79,"style":72},[71],[15,81,83,87],{"className":82},[57],[15,84],{"className":85,"style":86},[61],"height:0.4306em;",[15,88,37],{"className":89},[66,67]," 的线性系统或二次规划。把这件事说清楚的，就是表示定理。",[11,92,93],{},"本文会把这个结论拆成三件事：RKHS 为什么能定义「范数正则化」；正交分解为什么迫使最优解落在训练点张成的子空间里；以及这个结论的边界——什么样的正则项会让它失效。",[95,96,98],"h2",{"id":97},"一无穷维优化","一、无穷维优化",[11,100,101,102,232,233,261,262,290],{},"线性模型 ",[15,103,105,145],{"className":104,"translate":19},[18],[15,106,108],{"className":107},[23],[25,109,110],{"xmlns":27},[29,111,112,142],{},[32,113,114,117,121,124,127,130,140],{},[35,115,116],{},"f",[39,118,120],{"stretchy":119},"false","(",[35,122,123],{},"x",[39,125,126],{"stretchy":119},")",[39,128,129],{},"=",[131,132,133,136],"msup",{},[35,134,135],{},"w",[35,137,139],{"mathvariant":138},"normal","⊤",[35,141,123],{},[45,143,144],{"encoding":47},"f(x)=w^\\top x",[15,146,148,181],{"className":147,"ariaHidden":53},[52],[15,149,151,155,159,163,166,170,174,178],{"className":150},[57],[15,152],{"className":153,"style":154},[61],"height:1em;vertical-align:-0.25em;",[15,156,116],{"className":157,"style":158},[66,67],"margin-right:0.1076em;",[15,160,120],{"className":161},[162],"mopen",[15,164,123],{"className":165},[66,67],[15,167,126],{"className":168},[169],"mclose",[15,171],{"className":172,"style":173},[71],"margin-right:0.2778em;",[15,175,129],{"className":176},[177],"mrel",[15,179],{"className":180,"style":173},[71],[15,182,184,188,229],{"className":183},[57],[15,185],{"className":186,"style":187},[61],"height:0.8491em;",[15,189,191,195],{"className":190},[66],[15,192,135],{"className":193,"style":194},[66,67],"margin-right:0.0269em;",[15,196,199],{"className":197},[198],"msupsub",[15,200,203],{"className":201},[202],"vlist-t",[15,204,207],{"className":205},[206],"vlist-r",[15,208,211],{"className":209,"style":187},[210],"vlist",[15,212,214,219],{"style":213},"top:-3.063em;margin-right:0.05em;",[15,215],{"className":216,"style":218},[217],"pstrut","height:2.7em;",[15,220,226],{"className":221},[222,223,224,225],"sizing","reset-size6","size3","mtight",[15,227,139],{"className":228},[66,225],[15,230,123],{"className":231},[66,67]," 的表达能力受限于特征 ",[15,234,236,249],{"className":235,"translate":19},[18],[15,237,239],{"className":238},[23],[25,240,241],{"xmlns":27},[29,242,243,247],{},[32,244,245],{},[35,246,123],{},[45,248,123],{"encoding":47},[15,250,252],{"className":251,"ariaHidden":53},[52],[15,253,255,258],{"className":254},[57],[15,256],{"className":257,"style":86},[61],[15,259,123],{"className":260},[66,67]," 的坐标。想拟合非线性关系，一个标准做法是先把 ",[15,263,265,278],{"className":264,"translate":19},[18],[15,266,268],{"className":267},[23],[25,269,270],{"xmlns":27},[29,271,272,276],{},[32,273,274],{},[35,275,123],{},[45,277,123],{"encoding":47},[15,279,281],{"className":280,"ariaHidden":53},[52],[15,282,284,287],{"className":283},[57],[15,285],{"className":286,"style":86},[61],[15,288,123],{"className":289},[66,67]," 映射到一个更大的特征空间：",[15,292,295],{"className":293,"translate":19},[294],"katex-display",[15,296,298,339],{"className":297,"translate":19},[18],[15,299,301],{"className":300},[23],[25,302,304],{"xmlns":27,"display":303},"block",[29,305,306,336],{},[32,307,308,310,312,314,316,318,324,327,329,331,333],{},[35,309,116],{},[39,311,120],{"stretchy":119},[35,313,123],{},[39,315,126],{"stretchy":119},[39,317,129],{},[131,319,320,322],{},[35,321,135],{},[35,323,139],{"mathvariant":138},[35,325,326],{},"φ",[39,328,120],{"stretchy":119},[35,330,123],{},[39,332,126],{"stretchy":119},[39,334,335],{"separator":53},",",[45,337,338],{"encoding":47},"f(x)=w^\\top \\varphi(x),",[15,340,342,369],{"className":341,"ariaHidden":53},[52],[15,343,345,348,351,354,357,360,363,366],{"className":344},[57],[15,346],{"className":347,"style":154},[61],[15,349,116],{"className":350,"style":158},[66,67],[15,352,120],{"className":353},[162],[15,355,123],{"className":356},[66,67],[15,358,126],{"className":359},[169],[15,361],{"className":362,"style":173},[71],[15,364,129],{"className":365},[177],[15,367],{"className":368,"style":173},[71],[15,370,372,376,407,410,413,416,419],{"className":371},[57],[15,373],{"className":374,"style":375},[61],"height:1.1491em;vertical-align:-0.25em;",[15,377,379,382],{"className":378},[66],[15,380,135],{"className":381,"style":194},[66,67],[15,383,385],{"className":384},[198],[15,386,388],{"className":387},[202],[15,389,391],{"className":390},[206],[15,392,395],{"className":393,"style":394},[210],"height:0.8991em;",[15,396,398,401],{"style":397},"top:-3.113em;margin-right:0.05em;",[15,399],{"className":400,"style":218},[217],[15,402,404],{"className":403},[222,223,224,225],[15,405,139],{"className":406},[66,225],[15,408,326],{"className":409},[66,67],[15,411,120],{"className":412},[162],[15,414,123],{"className":415},[66,67],[15,417,126],{"className":418},[169],[15,420,335],{"className":421},[422],"mpunct",[11,424,425,426,541,542,571,572,600],{},"其中 ",[15,427,429,462],{"className":428,"translate":19},[18],[15,430,432],{"className":431},[23],[25,433,434],{"xmlns":27},[29,435,436,459],{},[32,437,438,440,443,447,450],{},[35,439,326],{},[39,441,442],{},":",[35,444,446],{"mathvariant":445},"script","X",[39,448,449],{},"→",[131,451,452,456],{},[35,453,455],{"mathvariant":454},"double-struck","R",[35,457,458],{},"d",[45,460,461],{"encoding":47},"\\varphi:\\mathcal X\\to\\mathbb R^d",[15,463,465,484,505],{"className":464,"ariaHidden":53},[52],[15,466,468,472,475,478,481],{"className":467},[57],[15,469],{"className":470,"style":471},[61],"height:0.625em;vertical-align:-0.1944em;",[15,473,326],{"className":474},[66,67],[15,476],{"className":477,"style":173},[71],[15,479,442],{"className":480},[177],[15,482],{"className":483,"style":173},[71],[15,485,487,491,496,499,502],{"className":486},[57],[15,488],{"className":489,"style":490},[61],"height:0.6833em;",[15,492,446],{"className":493,"style":495},[66,494],"mathcal","margin-right:0.1464em;",[15,497],{"className":498,"style":173},[71],[15,500,449],{"className":501},[177],[15,503],{"className":504,"style":173},[71],[15,506,508,511],{"className":507},[57],[15,509],{"className":510,"style":187},[61],[15,512,514,518],{"className":513},[66],[15,515,455],{"className":516},[66,517],"mathbb",[15,519,521],{"className":520},[198],[15,522,524],{"className":523},[202],[15,525,527],{"className":526},[206],[15,528,530],{"className":529,"style":187},[210],[15,531,532,535],{"style":213},[15,533],{"className":534,"style":218},[217],[15,536,538],{"className":537},[222,223,224,225],[15,539,458],{"className":540},[66,67,225]," 是特征映射。多项式回归、样条基、径向基函数都属于这个套路。问题在于，",[15,543,545,559],{"className":544,"translate":19},[18],[15,546,548],{"className":547},[23],[25,549,550],{"xmlns":27},[29,551,552,556],{},[32,553,554],{},[35,555,326],{},[45,557,558],{"encoding":47},"\\varphi",[15,560,562],{"className":561,"ariaHidden":53},[52],[15,563,565,568],{"className":564},[57],[15,566],{"className":567,"style":471},[61],[15,569,326],{"className":570},[66,67]," 的维数可以很大，甚至是无穷维。例如高斯核对应的是一个无穷维的特征空间；如果显式地写出 ",[15,573,575,588],{"className":574,"translate":19},[18],[15,576,578],{"className":577},[23],[25,579,580],{"xmlns":27},[29,581,582,586],{},[32,583,584],{},[35,585,326],{},[45,587,558],{"encoding":47},[15,589,591],{"className":590,"ariaHidden":53},[52],[15,592,594,597],{"className":593},[57],[15,595],{"className":596,"style":471},[61],[15,598,326],{"className":599},[66,67],"，优化根本无从下手。",[11,602,603],{},"核方法绕开了显式特征：它只通过内积",[15,605,607],{"className":606,"translate":19},[294],[15,608,610,671],{"className":609,"translate":19},[18],[15,611,613],{"className":612},[23],[25,614,615],{"xmlns":27,"display":303},[29,616,617,668],{},[32,618,619,622,624,626,628,636,638,640,643,645,647,649,651,653,655,657,663,665],{},[35,620,621],{},"k",[39,623,120],{"stretchy":119},[35,625,123],{},[39,627,335],{"separator":53},[131,629,630,632],{},[35,631,123],{},[39,633,635],{"mathvariant":138,"lspace":634,"rspace":634},"0em","′",[39,637,126],{"stretchy":119},[39,639,129],{},[39,641,642],{"stretchy":119},"⟨",[35,644,326],{},[39,646,120],{"stretchy":119},[35,648,123],{},[39,650,126],{"stretchy":119},[39,652,335],{"separator":53},[35,654,326],{},[39,656,120],{"stretchy":119},[131,658,659,661],{},[35,660,123],{},[39,662,635],{"mathvariant":138,"lspace":634,"rspace":634},[39,664,126],{"stretchy":119},[39,666,667],{"stretchy":119},"⟩",[45,669,670],{"encoding":47},"k(x,x')=\\langle \\varphi(x),\\varphi(x')\\rangle",[15,672,674,743],{"className":673,"ariaHidden":53},[52],[15,675,677,681,685,688,691,694,698,731,734,737,740],{"className":676},[57],[15,678],{"className":679,"style":680},[61],"height:1.0519em;vertical-align:-0.25em;",[15,682,621],{"className":683,"style":684},[66,67],"margin-right:0.0315em;",[15,686,120],{"className":687},[162],[15,689,123],{"className":690},[66,67],[15,692,335],{"className":693},[422],[15,695],{"className":696,"style":697},[71],"margin-right:0.1667em;",[15,699,701,704],{"className":700},[66],[15,702,123],{"className":703},[66,67],[15,705,707],{"className":706},[198],[15,708,710],{"className":709},[202],[15,711,713],{"className":712},[206],[15,714,717],{"className":715,"style":716},[210],"height:0.8019em;",[15,718,719,722],{"style":397},[15,720],{"className":721,"style":218},[217],[15,723,725],{"className":724},[222,223,224,225],[15,726,728],{"className":727},[66,225],[15,729,635],{"className":730},[66,225],[15,732,126],{"className":733},[169],[15,735],{"className":736,"style":173},[71],[15,738,129],{"className":739},[177],[15,741],{"className":742,"style":173},[71],[15,744,746,749,752,755,758,761,764,767,770,773,776,808],{"className":745},[57],[15,747],{"className":748,"style":680},[61],[15,750,642],{"className":751},[162],[15,753,326],{"className":754},[66,67],[15,756,120],{"className":757},[162],[15,759,123],{"className":760},[66,67],[15,762,126],{"className":763},[169],[15,765,335],{"className":766},[422],[15,768],{"className":769,"style":697},[71],[15,771,326],{"className":772},[66,67],[15,774,120],{"className":775},[162],[15,777,779,782],{"className":778},[66],[15,780,123],{"className":781},[66,67],[15,783,785],{"className":784},[198],[15,786,788],{"className":787},[202],[15,789,791],{"className":790},[206],[15,792,794],{"className":793,"style":716},[210],[15,795,796,799],{"style":397},[15,797],{"className":798,"style":218},[217],[15,800,802],{"className":801},[222,223,224,225],[15,803,805],{"className":804},[66,225],[15,806,635],{"className":807},[66,225],[15,809,811],{"className":810},[169],")⟩",[11,813,814,815,844],{},"来访问特征空间。这个 ",[15,816,818,831],{"className":817,"translate":19},[18],[15,819,821],{"className":820},[23],[25,822,823],{"xmlns":27},[29,824,825,829],{},[32,826,827],{},[35,828,621],{},[45,830,621],{"encoding":47},[15,832,834],{"className":833,"ariaHidden":53},[52],[15,835,837,841],{"className":836},[57],[15,838],{"className":839,"style":840},[61],"height:0.6944em;",[15,842,621],{"className":843,"style":684},[66,67]," 就是核函数。于是，一个在特征空间中的线性模型，可以等价地写成",[15,846,848],{"className":847,"translate":19},[294],[15,849,851,891],{"className":850,"translate":19},[18],[15,852,854],{"className":853},[23],[25,855,856],{"xmlns":27,"display":303},[29,857,858,888],{},[32,859,860,862,864,866,868,870,872,874,876,878,880,882,884,886],{},[35,861,116],{},[39,863,120],{"stretchy":119},[35,865,123],{},[39,867,126],{"stretchy":119},[39,869,129],{},[39,871,642],{"stretchy":119},[35,873,135],{},[39,875,335],{"separator":53},[35,877,326],{},[39,879,120],{"stretchy":119},[35,881,123],{},[39,883,126],{"stretchy":119},[39,885,667],{"stretchy":119},[39,887,335],{"separator":53},[45,889,890],{"encoding":47},"f(x)=\\langle w,\\varphi(x)\\rangle,",[15,892,894,921],{"className":893,"ariaHidden":53},[52],[15,895,897,900,903,906,909,912,915,918],{"className":896},[57],[15,898],{"className":899,"style":154},[61],[15,901,116],{"className":902,"style":158},[66,67],[15,904,120],{"className":905},[162],[15,907,123],{"className":908},[66,67],[15,910,126],{"className":911},[169],[15,913],{"className":914,"style":173},[71],[15,916,129],{"className":917},[177],[15,919],{"className":920,"style":173},[71],[15,922,924,927,930,933,936,939,942,945,948,951],{"className":923},[57],[15,925],{"className":926,"style":154},[61],[15,928,642],{"className":929},[162],[15,931,135],{"className":932,"style":194},[66,67],[15,934,335],{"className":935},[422],[15,937],{"className":938,"style":697},[71],[15,940,326],{"className":941},[66,67],[15,943,120],{"className":944},[162],[15,946,123],{"className":947},[66,67],[15,949,811],{"className":950},[169],[15,952,335],{"className":953},[422],[11,955,956,957,985,986,1014,1015,1043,1044,1124],{},"而 ",[15,958,960,973],{"className":959,"translate":19},[18],[15,961,963],{"className":962},[23],[25,964,965],{"xmlns":27},[29,966,967,971],{},[32,968,969],{},[35,970,135],{},[45,972,135],{"encoding":47},[15,974,976],{"className":975,"ariaHidden":53},[52],[15,977,979,982],{"className":978},[57],[15,980],{"className":981,"style":86},[61],[15,983,135],{"className":984,"style":194},[66,67]," 所在的希尔伯特空间（更准确地说，是 ",[15,987,989,1002],{"className":988,"translate":19},[18],[15,990,992],{"className":991},[23],[25,993,994],{"xmlns":27},[29,995,996,1000],{},[32,997,998],{},[35,999,326],{},[45,1001,558],{"encoding":47},[15,1003,1005],{"className":1004,"ariaHidden":53},[52],[15,1006,1008,1011],{"className":1007},[57],[15,1009],{"className":1010,"style":471},[61],[15,1012,326],{"className":1013},[66,67]," 张成的闭包）就是核 ",[15,1016,1018,1031],{"className":1017,"translate":19},[18],[15,1019,1021],{"className":1020},[23],[25,1022,1023],{"xmlns":27},[29,1024,1025,1029],{},[32,1026,1027],{},[35,1028,621],{},[45,1030,621],{"encoding":47},[15,1032,1034],{"className":1033,"ariaHidden":53},[52],[15,1035,1037,1040],{"className":1036},[57],[15,1038],{"className":1039,"style":840},[61],[15,1041,621],{"className":1042,"style":684},[66,67]," 对应的再生核希尔伯特空间 ",[15,1045,1047,1067],{"className":1046,"translate":19},[18],[15,1048,1050],{"className":1049},[23],[25,1051,1052],{"xmlns":27},[29,1053,1054,1064],{},[32,1055,1056],{},[1057,1058,1059,1062],"msub",{},[35,1060,1061],{"mathvariant":445},"H",[35,1063,621],{},[45,1065,1066],{"encoding":47},"\\mathcal 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empirical risk 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是无穷维的，这看起来是一个无穷维优化问题。表示定理说：它其实是一个 ",[15,1871,1873,1886],{"className":1872,"translate":19},[18],[15,1874,1876],{"className":1875},[23],[25,1877,1878],{"xmlns":27},[29,1879,1880,1884],{},[32,1881,1882],{},[35,1883,37],{},[45,1885,37],{"encoding":47},[15,1887,1889],{"className":1888,"ariaHidden":53},[52],[15,1890,1892,1895],{"className":1891},[57],[15,1893],{"className":1894,"style":86},[61],[15,1896,37],{"className":1897},[66,67]," 维问题。",[95,1900,1902],{"id":1901},"二rkhs-速览","二、RKHS 速览",[1904,1905,1907],"h3",{"id":1906},"_21-正定核与再生性","2.1 正定核与再生性",[11,1909,1910,1911,1940,1941,2038,2039,2043,2044,2072,2073,2239,2240,2403],{},"设 ",[15,1912,1914,1928],{"className":1913,"translate":19},[18],[15,1915,1917],{"className":1916},[23],[25,1918,1919],{"xmlns":27},[29,1920,1921,1925],{},[32,1922,1923],{},[35,1924,446],{"mathvariant":445},[45,1926,1927],{"encoding":47},"\\mathcal 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R",[15,1971,1973,1991,2010,2028],{"className":1972,"ariaHidden":53},[52],[15,1974,1976,1979,1982,1985,1988],{"className":1975},[57],[15,1977],{"className":1978,"style":840},[61],[15,1980,621],{"className":1981,"style":684},[66,67],[15,1983],{"className":1984,"style":173},[71],[15,1986,442],{"className":1987},[177],[15,1989],{"className":1990,"style":173},[71],[15,1992,1994,1998,2001,2004,2007],{"className":1993},[57],[15,1995],{"className":1996,"style":1997},[61],"height:0.7667em;vertical-align:-0.0833em;",[15,1999,446],{"className":2000,"style":495},[66,494],[15,2002],{"className":2003,"style":72},[71],[15,2005,41],{"className":2006},[76],[15,2008],{"className":2009,"style":72},[71],[15,2011,2013,2016,2019,2022,2025],{"className":2012},[57],[15,2014],{"className":2015,"style":490},[61],[15,2017,446],{"className":2018,"style":495},[66,494],[15,2020],{"className":2021,"style":173},[71],[15,2023,449],{"className":2024},[177],[15,2026],{"className":2027,"style":173},[71],[15,2029,2031,2035],{"className":2030},[57],[15,2032],{"className":2033,"style":2034},[61],"height:0.6889em;",[15,2036,455],{"className":2037},[66,517]," 称为",[2040,2041,2042],"strong",{},"正定核","（positive definite kernel），如果对任意 ",[15,2045,2047,2060],{"className":2046,"translate":19},[18],[15,2048,2050],{"className":2049},[23],[25,2051,2052],{"xmlns":27},[29,2053,2054,2058],{},[32,2055,2056],{},[35,2057,37],{},[45,2059,37],{"encoding":47},[15,2061,2063],{"className":2062,"ariaHidden":53},[52],[15,2064,2066,2069],{"className":2065},[57],[15,2067],{"className":2068,"style":86},[61],[15,2070,37],{"className":2071},[66,67],"、任意 ",[15,2074,2076,2111],{"className":2075,"translate":19},[18],[15,2077,2079],{"className":2078},[23],[25,2080,2081],{"xmlns":27},[29,2082,2083,2108],{},[32,2084,2085,2091,2093,2096,2098,2104,2106],{},[1057,2086,2087,2089],{},[35,2088,123],{},[1209,2090,1211],{},[39,2092,335],{"separator":53},[39,2094,2095],{},"…",[39,2097,335],{"separator":53},[1057,2099,2100,2102],{},[35,2101,123],{},[35,2103,37],{},[39,2105,1187],{},[35,2107,446],{"mathvariant":445},[45,2109,2110],{"encoding":47},"x_1,\\dots,x_n\\in\\mathcal X",[15,2112,2114,2230],{"className":2113,"ariaHidden":53},[52],[15,2115,2117,2121,2162,2165,2168,2171,2174,2177,2180,2221,2224,2227],{"className":2116},[57],[15,2118],{"className":2119,"style":2120},[61],"height:0.7335em;vertical-align:-0.1944em;",[15,2122,2124,2127],{"className":2123},[66],[15,2125,123],{"className":2126},[66,67],[15,2128,2130],{"className":2129},[198],[15,2131,2133,2154],{"className":2132},[202,1090],[15,2134,2136,2151],{"className":2135},[206],[15,2137,2140],{"className":2138,"style":2139},[210],"height:0.3011em;",[15,2141,2142,2145],{"style":1584},[15,2143],{"className":2144,"style":218},[217],[15,2146,2148],{"className":2147},[222,223,224,225],[15,2149,1211],{"className":2150},[66,225],[15,2152,1114],{"className":2153},[1113],[15,2155,2157],{"className":2156},[206],[15,2158,2160],{"className":2159,"style":1121},[210],[15,2161],{},[15,2163,335],{"className":2164},[422],[15,2166],{"className":2167,"style":697},[71],[15,2169,2095],{"className":2170},[1413],[15,2172],{"className":2173,"style":697},[71],[15,2175,335],{"className":2176},[422],[15,2178],{"className":2179,"style":697},[71],[15,2181,2183,2186],{"className":2182},[66],[15,2184,123],{"className":2185},[66,67],[15,2187,2189],{"className":2188},[198],[15,2190,2192,2213],{"className":2191},[202,1090],[15,2193,2195,2210],{"className":2194},[206],[15,2196,2199],{"className":2197,"style":2198},[210],"height:0.1514em;",[15,2200,2201,2204],{"style":1584},[15,2202],{"className":2203,"style":218},[217],[15,2205,2207],{"className":2206},[222,223,224,225],[15,2208,37],{"className":2209},[66,67,225],[15,2211,1114],{"className":2212},[1113],[15,2214,2216],{"className":2215},[206],[15,2217,2219],{"className":2218,"style":1121},[210],[15,2220],{},[15,2222],{"className":2223,"style":173},[71],[15,2225,1187],{"className":2226},[177],[15,2228],{"className":2229,"style":173},[71],[15,2231,2233,2236],{"className":2232},[57],[15,2234],{"className":2235,"style":490},[61],[15,2237,446],{"className":2238,"style":495},[66,494]," 和任意 ",[15,2241,2243,2278],{"className":2242,"translate":19},[18],[15,2244,2246],{"className":2245},[23],[25,2247,2248],{"xmlns":27},[29,2249,2250,2275],{},[32,2251,2252,2259,2261,2263,2265,2271,2273],{},[1057,2253,2254,2257],{},[35,2255,2256],{},"c",[1209,2258,1211],{},[39,2260,335],{"separator":53},[39,2262,2095],{},[39,2264,335],{"separator":53},[1057,2266,2267,2269],{},[35,2268,2256],{},[35,2270,37],{},[39,2272,1187],{},[35,2274,455],{"mathvariant":454},[45,2276,2277],{"encoding":47},"c_1,\\dots,c_n\\in\\mathbb R",[15,2279,2281,2394],{"className":2280,"ariaHidden":53},[52],[15,2282,2284,2287,2327,2330,2333,2336,2339,2342,2345,2385,2388,2391],{"className":2283},[57],[15,2285],{"className":2286,"style":2120},[61],[15,2288,2290,2293],{"className":2289},[66],[15,2291,2256],{"className":2292},[66,67],[15,2294,2296],{"className":2295},[198],[15,2297,2299,2319],{"className":2298},[202,1090],[15,2300,2302,2316],{"className":2301},[206],[15,2303,2305],{"className":2304,"style":2139},[210],[15,2306,2307,2310],{"style":1584},[15,2308],{"className":2309,"style":218},[217],[15,2311,2313],{"className":2312},[222,223,224,225],[15,2314,1211],{"className":2315},[66,225],[15,2317,1114],{"className":2318},[1113],[15,2320,2322],{"className":2321},[206],[15,2323,2325],{"className":2324,"style":1121},[210],[15,2326],{},[15,2328,335],{"className":2329},[422],[15,2331],{"className":2332,"style":697},[71],[15,2334,2095],{"className":2335},[1413],[15,2337],{"className":2338,"style":697},[71],[15,2340,335],{"className":2341},[422],[15,2343],{"className":2344,"style":697},[71],[15,2346,2348,2351],{"className":2347},[66],[15,2349,2256],{"className":2350},[66,67],[15,2352,2354],{"className":2353},[198],[15,2355,2357,2377],{"className":2356},[202,1090],[15,2358,2360,2374],{"className":2359},[206],[15,2361,2363],{"className":2362,"style":2198},[210],[15,2364,2365,2368],{"style":1584},[15,2366],{"className":2367,"style":218},[217],[15,2369,2371],{"className":2370},[222,223,224,225],[15,2372,37],{"className":2373},[66,67,225],[15,2375,1114],{"className":2376},[1113],[15,2378,2380],{"className":2379},[206],[15,2381,2383],{"className":2382,"style":1121},[210],[15,2384],{},[15,2386],{"className":2387,"style":173},[71],[15,2389,1187],{"className":2390},[177],[15,2392],{"className":2393,"style":173},[71],[15,2395,2397,2400],{"className":2396},[57],[15,2398],{"className":2399,"style":2034},[61],[15,2401,455],{"className":2402},[66,517],"，都有",[15,2405,2407],{"className":2406,"translate":19},[294],[15,2408,2410,2491],{"className":2409,"translate":19},[18],[15,2411,2413],{"className":2412},[23],[25,2414,2415],{"xmlns":27,"display":303},[29,2416,2417,2488],{},[32,2418,2419,2433,2448,2454,2460,2463,2465,2467,2473,2475,2481,2483,2485],{},[1196,2420,2421,2423,2431],{},[39,2422,1200],{},[32,2424,2425,2427,2429],{},[35,2426,1205],{},[39,2428,129],{},[1209,2430,1211],{},[35,2432,37],{},[1196,2434,2435,2437,2446],{},[39,2436,1200],{},[32,2438,2439,2442,2444],{},[35,2440,2441],{},"j",[39,2443,129],{},[1209,2445,1211],{},[35,2447,37],{},[1057,2449,2450,2452],{},[35,2451,2256],{},[35,2453,1205],{},[1057,2455,2456,2458],{},[35,2457,2256],{},[35,2459,2441],{},[1274,2461,2462],{}," ",[35,2464,621],{},[39,2466,120],{"stretchy":119},[1057,2468,2469,2471],{},[35,2470,123],{},[35,2472,1205],{},[39,2474,335],{"separator":53},[1057,2476,2477,2479],{},[35,2478,123],{},[35,2480,2441],{},[39,2482,126],{"stretchy":119},[39,2484,1801],{},[1209,2486,2487],{},"0.",[45,2489,2490],{"encoding":47},"\\sum_{i=1}^n\\sum_{j=1}^n c_i c_j\\,k(x_i,x_j)\\ge 0.",[15,2492,2494,2825],{"className":2493,"ariaHidden":53},[52],[15,2495,2497,2501,2565,2568,2634,2637,2677,2718,2721,2724,2727,2767,2770,2773,2813,2816,2819,2822],{"className":2496},[57],[15,2498],{"className":2499,"style":2500},[61],"height:3.0652em;vertical-align:-1.4138em;",[15,2502,2504],{"className":2503},[1339,1351],[15,2505,2507,2557],{"className":2506},[202,1090],[15,2508,2510,2554],{"className":2509},[206],[15,2511,2513,2533,2543],{"className":2512,"style":1436},[210],[15,2514,2515,2518],{"style":1439},[15,2516],{"className":2517,"style":1443},[217],[15,2519,2521],{"className":2520},[222,223,224,225],[15,2522,2524,2527,2530],{"className":2523},[66,225],[15,2525,1205],{"className":2526},[66,67,225],[15,2528,129],{"className":2529},[177,225],[15,2531,1211],{"className":2532},[66,225],[15,2534,2535,2538],{"style":1461},[15,2536],{"className":2537,"style":1443},[217],[15,2539,2540],{},[15,2541,1200],{"className":2542},[1339,1470,1471],[15,2544,2545,2548],{"style":1474},[15,2546],{"className":2547,"style":1443},[217],[15,2549,2551],{"className":2550},[222,223,224,225],[15,2552,37],{"className":2553},[66,67,225],[15,2555,1114],{"className":2556},[1113],[15,2558,2560],{"className":2559},[206],[15,2561,2563],{"className":2562,"style":1493},[210],[15,2564],{},[15,2566],{"className":2567,"style":697},[71],[15,2569,2571],{"className":2570},[1339,1351],[15,2572,2574,2625],{"className":2573},[202,1090],[15,2575,2577,2622],{"className":2576},[206],[15,2578,2580,2601,2611],{"className":2579,"style":1436},[210],[15,2581,2582,2585],{"style":1439},[15,2583],{"className":2584,"style":1443},[217],[15,2586,2588],{"className":2587},[222,223,224,225],[15,2589,2591,2595,2598],{"className":2590},[66,225],[15,2592,2441],{"className":2593,"style":2594},[66,67,225],"margin-right:0.0572em;",[15,2596,129],{"className":2597},[177,225],[15,2599,1211],{"className":2600},[66,225],[15,2602,2603,2606],{"style":1461},[15,2604],{"className":2605,"style":1443},[217],[15,2607,2608],{},[15,2609,1200],{"className":2610},[1339,1470,1471],[15,2612,2613,2616],{"style":1474},[15,2614],{"className":2615,"style":1443},[217],[15,2617,2619],{"className":2618},[222,223,224,225],[15,2620,37],{"className":2621},[66,67,225],[15,2623,1114],{"className":2624},[1113],[15,2626,2628],{"className":2627},[206],[15,2629,2632],{"className":2630,"style":2631},[210],"height:1.4138em;",[15,2633],{},[15,2635],{"className":2636,"style":697},[71],[15,2638,2640,2643],{"className":2639},[66],[15,2641,2256],{"className":2642},[66,67],[15,2644,2646],{"className":2645},[198],[15,2647,2649,2669],{"className":2648},[202,1090],[15,2650,2652,2666],{"className":2651},[206],[15,2653,2655],{"className":2654,"style":1528},[210],[15,2656,2657,2660],{"style":1584},[15,2658],{"className":2659,"style":218},[217],[15,2661,2663],{"className":2662},[222,223,224,225],[15,2664,1205],{"className":2665},[66,67,225],[15,2667,1114],{"className":2668},[1113],[15,2670,2672],{"className":2671},[206],[15,2673,2675],{"className":2674,"style":1121},[210],[15,2676],{},[15,2678,2680,2683],{"className":2679},[66],[15,2681,2256],{"className":2682},[66,67],[15,2684,2686],{"className":2685},[198],[15,2687,2689,2709],{"className":2688},[202,1090],[15,2690,2692,2706],{"className":2691},[206],[15,2693,2695],{"className":2694,"style":1528},[210],[15,2696,2697,2700],{"style":1584},[15,2698],{"className":2699,"style":218},[217],[15,2701,2703],{"className":2702},[222,223,224,225],[15,2704,2441],{"className":2705,"style":2594},[66,67,225],[15,2707,1114],{"className":2708},[1113],[15,2710,2712],{"className":2711},[206],[15,2713,2716],{"className":2714,"style":2715},[210],"height:0.2861em;",[15,2717],{},[15,2719],{"className":2720,"style":697},[71],[15,2722,621],{"className":2723,"style":684},[66,67],[15,2725,120],{"className":2726},[162],[15,2728,2730,2733],{"className":2729},[66],[15,2731,123],{"className":2732},[66,67],[15,2734,2736],{"className":2735},[198],[15,2737,2739,2759],{"className":2738},[202,1090],[15,2740,2742,2756],{"className":2741},[206],[15,2743,2745],{"className":2744,"style":1528},[210],[15,2746,2747,2750],{"style":1584},[15,2748],{"className":2749,"style":218},[217],[15,2751,2753],{"className":2752},[222,223,224,225],[15,2754,1205],{"className":2755},[66,67,225],[15,2757,1114],{"className":2758},[1113],[15,2760,2762],{"className":2761},[206],[15,2763,2765],{"className":2764,"style":1121},[210],[15,2766],{},[15,2768,335],{"className":2769},[422],[15,2771],{"className":2772,"style":697},[71],[15,2774,2776,2779],{"className":2775},[66],[15,2777,123],{"className":2778},[66,67],[15,2780,2782],{"className":2781},[198],[15,2783,2785,2805],{"className":2784},[202,1090],[15,2786,2788,2802],{"className":2787},[206],[15,2789,2791],{"className":2790,"style":1528},[210],[15,2792,2793,2796],{"style":1584},[15,2794],{"className":2795,"style":218},[217],[15,2797,2799],{"className":2798},[222,223,224,225],[15,2800,2441],{"className":2801,"style":2594},[66,67,225],[15,2803,1114],{"className":2804},[1113],[15,2806,2808],{"className":2807},[206],[15,2809,2811],{"className":2810,"style":2715},[210],[15,2812],{},[15,2814,126],{"className":2815},[169],[15,2817],{"className":2818,"style":173},[71],[15,2820,1801],{"className":2821},[177],[15,2823],{"className":2824,"style":173},[71],[15,2826,2828,2831],{"className":2827},[57],[15,2829],{"className":2830,"style":1836},[61],[15,2832,2487],{"className":2833},[66],[11,2835,2836,2837,3052],{},"也就是说，核矩阵 ",[15,2838,2840,2885],{"className":2839,"translate":19},[18],[15,2841,2843],{"className":2842},[23],[25,2844,2845],{"xmlns":27},[29,2846,2847,2882],{},[32,2848,2849,2860,2862,2864,2866,2872,2874,2880],{},[1057,2850,2851,2854],{},[35,2852,2853],{},"K",[32,2855,2856,2858],{},[35,2857,1205],{},[35,2859,2441],{},[39,2861,129],{},[35,2863,621],{},[39,2865,120],{"stretchy":119},[1057,2867,2868,2870],{},[35,2869,123],{},[35,2871,1205],{},[39,2873,335],{"separator":53},[1057,2875,2876,2878],{},[35,2877,123],{},[35,2879,2441],{},[39,2881,126],{"stretchy":119},[45,2883,2884],{"encoding":47},"K_{ij}=k(x_i,x_j)",[15,2886,2888,2950],{"className":2887,"ariaHidden":53},[52],[15,2889,2891,2895,2941,2944,2947],{"className":2890},[57],[15,2892],{"className":2893,"style":2894},[61],"height:0.9694em;vertical-align:-0.2861em;",[15,2896,2898,2902],{"className":2897},[66],[15,2899,2853],{"className":2900,"style":2901},[66,67],"margin-right:0.0715em;",[15,2903,2905],{"className":2904},[198],[15,2906,2908,2933],{"className":2907},[202,1090],[15,2909,2911,2930],{"className":2910},[206],[15,2912,2914],{"className":2913,"style":1528},[210],[15,2915,2917,2920],{"style":2916},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[15,2918],{"className":2919,"style":218},[217],[15,2921,2923],{"className":2922},[222,223,224,225],[15,2924,2926],{"className":2925},[66,225],[15,2927,2929],{"className":2928,"style":2594},[66,67,225],"ij",[15,2931,1114],{"className":2932},[1113],[15,2934,2936],{"className":2935},[206],[15,2937,2939],{"className":2938,"style":2715},[210],[15,2940],{},[15,2942],{"className":2943,"style":173},[71],[15,2945,129],{"className":2946},[177],[15,2948],{"className":2949,"style":173},[71],[15,2951,2953,2957,2960,2963,3003,3006,3009,3049],{"className":2952},[57],[15,2954],{"className":2955,"style":2956},[61],"height:1.0361em;vertical-align:-0.2861em;",[15,2958,621],{"className":2959,"style":684},[66,67],[15,2961,120],{"className":2962},[162],[15,2964,2966,2969],{"className":2965},[66],[15,2967,123],{"className":2968},[66,67],[15,2970,2972],{"className":2971},[198],[15,2973,2975,2995],{"className":2974},[202,1090],[15,2976,2978,2992],{"className":2977},[206],[15,2979,2981],{"className":2980,"style":1528},[210],[15,2982,2983,2986],{"style":1584},[15,2984],{"className":2985,"style":218},[217],[15,2987,2989],{"className":2988},[222,223,224,225],[15,2990,1205],{"className":2991},[66,67,225],[15,2993,1114],{"className":2994},[1113],[15,2996,2998],{"className":2997},[206],[15,2999,3001],{"className":3000,"style":1121},[210],[15,3002],{},[15,3004,335],{"className":3005},[422],[15,3007],{"className":3008,"style":697},[71],[15,3010,3012,3015],{"className":3011},[66],[15,3013,123],{"className":3014},[66,67],[15,3016,3018],{"className":3017},[198],[15,3019,3021,3041],{"className":3020},[202,1090],[15,3022,3024,3038],{"className":3023},[206],[15,3025,3027],{"className":3026,"style":1528},[210],[15,3028,3029,3032],{"style":1584},[15,3030],{"className":3031,"style":218},[217],[15,3033,3035],{"className":3034},[222,223,224,225],[15,3036,2441],{"className":3037,"style":2594},[66,67,225],[15,3039,1114],{"className":3040},[1113],[15,3042,3044],{"className":3043},[206],[15,3045,3047],{"className":3046,"style":2715},[210],[15,3048],{},[15,3050,126],{"className":3051},[169]," 总是半正定的。",[3054,3055,3056,3190,3518,3737],"blockquote",{},[11,3057,3058,3061,3062,3090,3091,3160,3161,3189],{},[2040,3059,3060],{},"定理 2.1（Moore–Aronszajn）"," 对任意正定核 ",[15,3063,3065,3078],{"className":3064,"translate":19},[18],[15,3066,3068],{"className":3067},[23],[25,3069,3070],{"xmlns":27},[29,3071,3072,3076],{},[32,3073,3074],{},[35,3075,621],{},[45,3077,621],{"encoding":47},[15,3079,3081],{"className":3080,"ariaHidden":53},[52],[15,3082,3084,3087],{"className":3083},[57],[15,3085],{"className":3086,"style":840},[61],[15,3088,621],{"className":3089,"style":684},[66,67],"，存在唯一的希尔伯特空间 ",[15,3092,3094,3111],{"className":3093,"translate":19},[18],[15,3095,3097],{"className":3096},[23],[25,3098,3099],{"xmlns":27},[29,3100,3101,3109],{},[32,3102,3103],{},[1057,3104,3105,3107],{},[35,3106,1061],{"mathvariant":445},[35,3108,621],{},[45,3110,1066],{"encoding":47},[15,3112,3114],{"className":3113,"ariaHidden":53},[52],[15,3115,3117,3120],{"className":3116},[57],[15,3118],{"className":3119,"style":1076},[61],[15,3121,3123,3126],{"className":3122},[66],[15,3124,1061],{"className":3125,"style":1083},[66,494],[15,3127,3129],{"className":3128},[198],[15,3130,3132,3152],{"className":3131},[202,1090],[15,3133,3135,3149],{"className":3134},[206],[15,3136,3138],{"className":3137,"style":1097},[210],[15,3139,3140,3143],{"style":1100},[15,3141],{"className":3142,"style":218},[217],[15,3144,3146],{"className":3145},[222,223,224,225],[15,3147,621],{"className":3148,"style":684},[66,67,225],[15,3150,1114],{"className":3151},[1113],[15,3153,3155],{"className":3154},[206],[15,3156,3158],{"className":3157,"style":1121},[210],[15,3159],{},"，由 ",[15,3162,3164,3177],{"className":3163,"translate":19},[18],[15,3165,3167],{"className":3166},[23],[25,3168,3169],{"xmlns":27},[29,3170,3171,3175],{},[32,3172,3173],{},[35,3174,446],{"mathvariant":445},[45,3176,1927],{"encoding":47},[15,3178,3180],{"className":3179,"ariaHidden":53},[52],[15,3181,3183,3186],{"className":3182},[57],[15,3184],{"className":3185,"style":490},[61],[15,3187,446],{"className":3188,"style":495},[66,494]," 上的函数组成，使得：",[3191,3192,3193,3372],"ol",{},[3194,3195,3196,3197,3249,3250,3371],"li",{},"对每个 ",[15,3198,3200,3218],{"className":3199,"translate":19},[18],[15,3201,3203],{"className":3202},[23],[25,3204,3205],{"xmlns":27},[29,3206,3207,3215],{},[32,3208,3209,3211,3213],{},[35,3210,123],{},[39,3212,1187],{},[35,3214,446],{"mathvariant":445},[45,3216,3217],{"encoding":47},"x\\in\\mathcal X",[15,3219,3221,3240],{"className":3220,"ariaHidden":53},[52],[15,3222,3224,3228,3231,3234,3237],{"className":3223},[57],[15,3225],{"className":3226,"style":3227},[61],"height:0.5782em;vertical-align:-0.0391em;",[15,3229,123],{"className":3230},[66,67],[15,3232],{"className":3233,"style":173},[71],[15,3235,1187],{"className":3236},[177],[15,3238],{"className":3239,"style":173},[71],[15,3241,3243,3246],{"className":3242},[57],[15,3244],{"className":3245,"style":490},[61],[15,3247,446],{"className":3248,"style":495},[66,494],"，",[15,3251,3253,3286],{"className":3252,"translate":19},[18],[15,3254,3256],{"className":3255},[23],[25,3257,3258],{"xmlns":27},[29,3259,3260,3283],{},[32,3261,3262,3264,3266,3269,3271,3273,3275,3277],{},[35,3263,621],{},[39,3265,120],{"stretchy":119},[39,3267,3268],{},"⋅",[39,3270,335],{"separator":53},[35,3272,123],{},[39,3274,126],{"stretchy":119},[39,3276,1187],{},[1057,3278,3279,3281],{},[35,3280,1061],{"mathvariant":445},[35,3282,621],{},[45,3284,3285],{"encoding":47},"k(\\cdot,x)\\in\\mathcal H_k",[15,3287,3289,3325],{"className":3288,"ariaHidden":53},[52],[15,3290,3292,3295,3298,3301,3304,3307,3310,3313,3316,3319,3322],{"className":3291},[57],[15,3293],{"className":3294,"style":154},[61],[15,3296,621],{"className":3297,"style":684},[66,67],[15,3299,120],{"className":3300},[162],[15,3302,3268],{"className":3303},[66],[15,3305,335],{"className":3306},[422],[15,3308],{"className":3309,"style":697},[71],[15,3311,123],{"className":3312},[66,67],[15,3314,126],{"className":3315},[169],[15,3317],{"className":3318,"style":173},[71],[15,3320,1187],{"className":3321},[177],[15,3323],{"className":3324,"style":173},[71],[15,3326,3328,3331],{"className":3327},[57],[15,3329],{"className":3330,"style":1076},[61],[15,3332,3334,3337],{"className":3333},[66],[15,3335,1061],{"className":3336,"style":1083},[66,494],[15,3338,3340],{"className":3339},[198],[15,3341,3343,3363],{"className":3342},[202,1090],[15,3344,3346,3360],{"className":3345},[206],[15,3347,3349],{"className":3348,"style":1097},[210],[15,3350,3351,3354],{"style":1100},[15,3352],{"className":3353,"style":218},[217],[15,3355,3357],{"className":3356},[222,223,224,225],[15,3358,621],{"className":3359,"style":684},[66,67,225],[15,3361,1114],{"className":3362},[1113],[15,3364,3366],{"className":3365},[206],[15,3367,3369],{"className":3368,"style":1121},[210],[15,3370],{},"；",[3194,3373,3196,3374,3467,3468,3249],{},[15,3375,3377,3399],{"className":3376,"translate":19},[18],[15,3378,3380],{"className":3379},[23],[25,3381,3382],{"xmlns":27},[29,3383,3384,3396],{},[32,3385,3386,3388,3390],{},[35,3387,116],{},[39,3389,1187],{},[1057,3391,3392,3394],{},[35,3393,1061],{"mathvariant":445},[35,3395,621],{},[45,3397,3398],{"encoding":47},"f\\in\\mathcal H_k",[15,3400,3402,3421],{"className":3401,"ariaHidden":53},[52],[15,3403,3405,3409,3412,3415,3418],{"className":3404},[57],[15,3406],{"className":3407,"style":3408},[61],"height:0.8889em;vertical-align:-0.1944em;",[15,3410,116],{"className":3411,"style":158},[66,67],[15,3413],{"className":3414,"style":173},[71],[15,3416,1187],{"className":3417},[177],[15,3419],{"className":3420,"style":173},[71],[15,3422,3424,3427],{"className":3423},[57],[15,3425],{"className":3426,"style":1076},[61],[15,3428,3430,3433],{"className":3429},[66],[15,3431,1061],{"className":3432,"style":1083},[66,494],[15,3434,3436],{"className":3435},[198],[15,3437,3439,3459],{"className":3438},[202,1090],[15,3440,3442,3456],{"className":3441},[206],[15,3443,3445],{"className":3444,"style":1097},[210],[15,3446,3447,3450],{"style":1100},[15,3448],{"className":3449,"style":218},[217],[15,3451,3453],{"className":3452},[222,223,224,225],[15,3454,621],{"className":3455,"style":684},[66,67,225],[15,3457,1114],{"className":3458},[1113],[15,3460,3462],{"className":3461},[206],[15,3463,3465],{"className":3464,"style":1121},[210],[15,3466],{}," 和 ",[15,3469,3471,3488],{"className":3470,"translate":19},[18],[15,3472,3474],{"className":3473},[23],[25,3475,3476],{"xmlns":27},[29,3477,3478,3486],{},[32,3479,3480,3482,3484],{},[35,3481,123],{},[39,3483,1187],{},[35,3485,446],{"mathvariant":445},[45,3487,3217],{"encoding":47},[15,3489,3491,3509],{"className":3490,"ariaHidden":53},[52],[15,3492,3494,3497,3500,3503,3506],{"className":3493},[57],[15,3495],{"className":3496,"style":3227},[61],[15,3498,123],{"className":3499},[66,67],[15,3501],{"className":3502,"style":173},[71],[15,3504,1187],{"className":3505},[177],[15,3507],{"className":3508,"style":173},[71],[15,3510,3512,3515],{"className":3511},[57],[15,3513],{"className":3514,"style":490},[61],[15,3516,446],{"className":3517,"style":495},[66,494],[15,3519,3521],{"className":3520,"translate":19},[294],[15,3522,3524,3577],{"className":3523,"translate":19},[18],[15,3525,3527],{"className":3526},[23],[25,3528,3529],{"xmlns":27,"display":303},[29,3530,3531,3574],{},[32,3532,3533,3535,3537,3539,3541,3543,3545,3547,3549,3551,3561,3563,3565,3567,3569,3571],{},[39,3534,642],{"stretchy":119},[35,3536,116],{},[39,3538,335],{"separator":53},[35,3540,621],{},[39,3542,120],{"stretchy":119},[39,3544,3268],{},[39,3546,335],{"separator":53},[35,3548,123],{},[39,3550,126],{"stretchy":119},[1057,3552,3553,3555],{},[39,3554,667],{"stretchy":119},[1057,3556,3557,3559],{},[35,3558,1061],{"mathvariant":445},[35,3560,621],{},[39,3562,129],{},[35,3564,116],{},[39,3566,120],{"stretchy":119},[35,3568,123],{},[39,3570,126],{"stretchy":119},[35,3572,3573],{"mathvariant":138},".",[45,3575,3576],{"encoding":47},"\\langle f,k(\\cdot,x)\\rangle_{\\mathcal H_k}=f(x).",[15,3578,3580,3716],{"className":3579,"ariaHidden":53},[52],[15,3581,3583,3587,3590,3593,3596,3599,3602,3605,3608,3611,3614,3617,3620,3707,3710,3713],{"className":3582},[57],[15,3584],{"className":3585,"style":3586},[61],"height:1.0059em;vertical-align:-0.2559em;",[15,3588,642],{"className":3589},[162],[15,3591,116],{"className":3592,"style":158},[66,67],[15,3594,335],{"className":3595},[422],[15,3597],{"className":3598,"style":697},[71],[15,3600,621],{"className":3601,"style":684},[66,67],[15,3603,120],{"className":3604},[162],[15,3606,3268],{"className":3607},[66],[15,3609,335],{"className":3610},[422],[15,3612],{"className":3613,"style":697},[71],[15,3615,123],{"className":3616},[66,67],[15,3618,126],{"className":3619},[169],[15,3621,3623,3626],{"className":3622},[169],[15,3624,667],{"className":3625},[169],[15,3627,3629],{"className":3628},[198],[15,3630,3632,3698],{"className":3631},[202,1090],[15,3633,3635,3695],{"className":3634},[206],[15,3636,3639],{"className":3637,"style":3638},[210],"height:0.3283em;",[15,3640,3641,3644],{"style":1584},[15,3642],{"className":3643,"style":218},[217],[15,3645,3647],{"className":3646},[222,223,224,225],[15,3648,3650],{"className":3649},[66,225],[15,3651,3653,3656],{"className":3652},[66,225],[15,3654,1061],{"className":3655,"style":1083},[66,494,225],[15,3657,3659],{"className":3658},[198],[15,3660,3662,3686],{"className":3661},[202,1090],[15,3663,3665,3683],{"className":3664},[206],[15,3666,3669],{"className":3667,"style":3668},[210],"height:0.3448em;",[15,3670,3672,3676],{"style":3671},"top:-2.3488em;margin-left:-0.0097em;margin-right:0.0714em;",[15,3673],{"className":3674,"style":3675},[217],"height:2.5em;",[15,3677,3680],{"className":3678},[222,3679,1508,225],"reset-size3",[15,3681,621],{"className":3682,"style":684},[66,67,225],[15,3684,1114],{"className":3685},[1113],[15,3687,3689],{"className":3688},[206],[15,3690,3693],{"className":3691,"style":3692},[210],"height:0.1512em;",[15,3694],{},[15,3696,1114],{"className":3697},[1113],[15,3699,3701],{"className":3700},[206],[15,3702,3705],{"className":3703,"style":3704},[210],"height:0.2559em;",[15,3706],{},[15,3708],{"className":3709,"style":173},[71],[15,3711,129],{"className":3712},[177],[15,3714],{"className":3715,"style":173},[71],[15,3717,3719,3722,3725,3728,3731,3734],{"className":3718},[57],[15,3720],{"className":3721,"style":154},[61],[15,3723,116],{"className":3724,"style":158},[66,67],[15,3726,120],{"className":3727},[162],[15,3729,123],{"className":3730},[66,67],[15,3732,126],{"className":3733},[169],[15,3735,3573],{"className":3736},[66],[11,3738,3739,3740,3768,3769,1124],{},"这个空间称为 ",[15,3741,3743,3756],{"className":3742,"translate":19},[18],[15,3744,3746],{"className":3745},[23],[25,3747,3748],{"xmlns":27},[29,3749,3750,3754],{},[32,3751,3752],{},[35,3753,621],{},[45,3755,621],{"encoding":47},[15,3757,3759],{"className":3758,"ariaHidden":53},[52],[15,3760,3762,3765],{"className":3761},[57],[15,3763],{"className":3764,"style":840},[61],[15,3766,621],{"className":3767,"style":684},[66,67]," 的",[2040,3770,3771],{},"再生核希尔伯特空间",[11,3773,3774,3775,3778,3779,3807,3808,3865],{},"第二条就是",[2040,3776,3777],{},"再生性","（reproducing property）。它把「函数在 ",[15,3780,3782,3795],{"className":3781,"translate":19},[18],[15,3783,3785],{"className":3784},[23],[25,3786,3787],{"xmlns":27},[29,3788,3789,3793],{},[32,3790,3791],{},[35,3792,123],{},[45,3794,123],{"encoding":47},[15,3796,3798],{"className":3797,"ariaHidden":53},[52],[15,3799,3801,3804],{"className":3800},[57],[15,3802],{"className":3803,"style":86},[61],[15,3805,123],{"className":3806},[66,67]," 点的取值」变成了「与 ",[15,3809,3811,3835],{"className":3810,"translate":19},[18],[15,3812,3814],{"className":3813},[23],[25,3815,3816],{"xmlns":27},[29,3817,3818,3832],{},[32,3819,3820,3822,3824,3826,3828,3830],{},[35,3821,621],{},[39,3823,120],{"stretchy":119},[39,3825,3268],{},[39,3827,335],{"separator":53},[35,3829,123],{},[39,3831,126],{"stretchy":119},[45,3833,3834],{"encoding":47},"k(\\cdot,x)",[15,3836,3838],{"className":3837,"ariaHidden":53},[52],[15,3839,3841,3844,3847,3850,3853,3856,3859,3862],{"className":3840},[57],[15,3842],{"className":3843,"style":154},[61],[15,3845,621],{"className":3846,"style":684},[66,67],[15,3848,120],{"className":3849},[162],[15,3851,3268],{"className":3852},[66],[15,3854,335],{"className":3855},[422],[15,3857],{"className":3858,"style":697},[71],[15,3860,123],{"className":3861},[66,67],[15,3863,126],{"className":3864},[169]," 的内积」。等价地说，取值泛函",[15,3867,3869],{"className":3868,"translate":19},[294],[15,3870,3872,3930],{"className":3871,"translate":19},[18],[15,3873,3875],{"className":3874},[23],[25,3876,3877],{"xmlns":27,"display":303},[29,3878,3879,3927],{},[32,3880,3881,3888,3890,3896,3898,3900,3902,3905,3911,3913,3915,3917,3919,3921,3923,3925],{},[1057,3882,3883,3886],{},[35,3884,3885],{},"δ",[35,3887,123],{},[39,3889,442],{},[1057,3891,3892,3894],{},[35,3893,1061],{"mathvariant":445},[35,3895,621],{},[39,3897,449],{},[35,3899,455],{"mathvariant":454},[39,3901,335],{"separator":53},[71,3903],{"width":3904},"2em",[1057,3906,3907,3909],{},[35,3908,3885],{},[35,3910,123],{},[39,3912,120],{"stretchy":119},[35,3914,116],{},[39,3916,126],{"stretchy":119},[39,3918,129],{},[35,3920,116],{},[39,3922,120],{"stretchy":119},[35,3924,123],{},[39,3926,126],{"stretchy":119},[45,3928,3929],{"encoding":47},"\\delta_x:\\mathcal H_k\\to\\mathbb R,\\qquad \\delta_x(f)=f(x)",[15,3931,3933,3991,4046,4123],{"className":3932,"ariaHidden":53},[52],[15,3934,3936,3940,3982,3985,3988],{"className":3935},[57],[15,3937],{"className":3938,"style":3939},[61],"height:0.8444em;vertical-align:-0.15em;",[15,3941,3943,3947],{"className":3942},[66],[15,3944,3885],{"className":3945,"style":3946},[66,67],"margin-right:0.0379em;",[15,3948,3950],{"className":3949},[198],[15,3951,3953,3974],{"className":3952},[202,1090],[15,3954,3956,3971],{"className":3955},[206],[15,3957,3959],{"className":3958,"style":2198},[210],[15,3960,3962,3965],{"style":3961},"top:-2.55em;margin-left:-0.0379em;margin-right:0.05em;",[15,3963],{"className":3964,"style":218},[217],[15,3966,3968],{"className":3967},[222,223,224,225],[15,3969,123],{"className":3970},[66,67,225],[15,3972,1114],{"className":3973},[1113],[15,3975,3977],{"className":3976},[206],[15,3978,3980],{"className":3979,"style":1121},[210],[15,3981],{},[15,3983],{"className":3984,"style":173},[71],[15,3986,442],{"className":3987},[177],[15,3989],{"className":3990,"style":173},[71],[15,3992,3994,3997,4037,4040,4043],{"className":3993},[57],[15,3995],{"className":3996,"style":1076},[61],[15,3998,4000,4003],{"className":3999},[66],[15,4001,1061],{"className":4002,"style":1083},[66,494],[15,4004,4006],{"className":4005},[198],[15,4007,4009,4029],{"className":4008},[202,1090],[15,4010,4012,4026],{"className":4011},[206],[15,4013,4015],{"className":4014,"style":1097},[210],[15,4016,4017,4020],{"style":1100},[15,4018],{"className":4019,"style":218},[217],[15,4021,4023],{"className":4022},[222,223,224,225],[15,4024,621],{"className":4025,"style":684},[66,67,225],[15,4027,1114],{"className":4028},[1113],[15,4030,4032],{"className":4031},[206],[15,4033,4035],{"className":4034,"style":1121},[210],[15,4036],{},[15,4038],{"className":4039,"style":173},[71],[15,4041,449],{"className":4042},[177],[15,4044],{"className":4045,"style":173},[71],[15,4047,4049,4052,4055,4058,4062,4065,4105,4108,4111,4114,4117,4120],{"className":4048},[57],[15,4050],{"className":4051,"style":154},[61],[15,4053,455],{"className":4054},[66,517],[15,4056,335],{"className":4057},[422],[15,4059],{"className":4060,"style":4061},[71],"margin-right:2em;",[15,4063],{"className":4064,"style":697},[71],[15,4066,4068,4071],{"className":4067},[66],[15,4069,3885],{"className":4070,"style":3946},[66,67],[15,4072,4074],{"className":4073},[198],[15,4075,4077,4097],{"className":4076},[202,1090],[15,4078,4080,4094],{"className":4079},[206],[15,4081,4083],{"className":4082,"style":2198},[210],[15,4084,4085,4088],{"style":3961},[15,4086],{"className":4087,"style":218},[217],[15,4089,4091],{"className":4090},[222,223,224,225],[15,4092,123],{"className":4093},[66,67,225],[15,4095,1114],{"className":4096},[1113],[15,4098,4100],{"className":4099},[206],[15,4101,4103],{"className":4102,"style":1121},[210],[15,4104],{},[15,4106,120],{"className":4107},[162],[15,4109,116],{"className":4110,"style":158},[66,67],[15,4112,126],{"className":4113},[169],[15,4115],{"className":4116,"style":173},[71],[15,4118,129],{"className":4119},[177],[15,4121],{"className":4122,"style":173},[71],[15,4124,4126,4129,4132,4135,4138],{"className":4125},[57],[15,4127],{"className":4128,"style":154},[61],[15,4130,116],{"className":4131,"style":158},[66,67],[15,4133,120],{"className":4134},[162],[15,4136,123],{"className":4137},[66,67],[15,4139,126],{"className":4140},[169],[11,4142,4143,4144,1124],{},"是有界线性泛函，且由 Riesz 表示定理，它对应的表示元正是 ",[15,4145,4147,4170],{"className":4146,"translate":19},[18],[15,4148,4150],{"className":4149},[23],[25,4151,4152],{"xmlns":27},[29,4153,4154,4168],{},[32,4155,4156,4158,4160,4162,4164,4166],{},[35,4157,621],{},[39,4159,120],{"stretchy":119},[39,4161,3268],{},[39,4163,335],{"separator":53},[35,4165,123],{},[39,4167,126],{"stretchy":119},[45,4169,3834],{"encoding":47},[15,4171,4173],{"className":4172,"ariaHidden":53},[52],[15,4174,4176,4179,4182,4185,4188,4191,4194,4197],{"className":4175},[57],[15,4177],{"className":4178,"style":154},[61],[15,4180,621],{"className":4181,"style":684},[66,67],[15,4183,120],{"className":4184},[162],[15,4186,3268],{"className":4187},[66],[15,4189,335],{"className":4190},[422],[15,4192],{"className":4193,"style":697},[71],[15,4195,123],{"className":4196},[66,67],[15,4198,126],{"className":4199},[169],[11,4201,4202,4271],{},[15,4203,4205,4222],{"className":4204,"translate":19},[18],[15,4206,4208],{"className":4207},[23],[25,4209,4210],{"xmlns":27},[29,4211,4212,4220],{},[32,4213,4214],{},[1057,4215,4216,4218],{},[35,4217,1061],{"mathvariant":445},[35,4219,621],{},[45,4221,1066],{"encoding":47},[15,4223,4225],{"className":4224,"ariaHidden":53},[52],[15,4226,4228,4231],{"className":4227},[57],[15,4229],{"className":4230,"style":1076},[61],[15,4232,4234,4237],{"className":4233},[66],[15,4235,1061],{"className":4236,"style":1083},[66,494],[15,4238,4240],{"className":4239},[198],[15,4241,4243,4263],{"className":4242},[202,1090],[15,4244,4246,4260],{"className":4245},[206],[15,4247,4249],{"className":4248,"style":1097},[210],[15,4250,4251,4254],{"style":1100},[15,4252],{"className":4253,"style":218},[217],[15,4255,4257],{"className":4256},[222,223,224,225],[15,4258,621],{"className":4259,"style":684},[66,67,225],[15,4261,1114],{"className":4262},[1113],[15,4264,4266],{"className":4265},[206],[15,4267,4269],{"className":4268,"style":1121},[210],[15,4270],{}," 可以显式地构造为",[15,4273,4275],{"className":4274,"translate":19},[294],[15,4276,4278,4336],{"className":4277,"translate":19},[18],[15,4279,4281],{"className":4280},[23],[25,4282,4283],{"xmlns":27,"display":303},[29,4284,4285,4333],{},[32,4286,4287,4293,4295,4305,4308,4310,4312,4314,4316,4318,4320,4322,4324,4326,4328,4331],{},[1057,4288,4289,4291],{},[35,4290,1061],{"mathvariant":445},[35,4292,621],{},[39,4294,129],{},[4296,4297,4298,4300,4302],"mover",{"accent":53},[35,4299,15],{"mathvariant":138},[39,4301,1170],{},[39,4303,4304],{"stretchy":53},"‾",[39,4306,4307],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},"{",[35,4309,621],{},[39,4311,120],{"stretchy":119},[39,4313,3268],{},[39,4315,335],{"separator":53},[35,4317,123],{},[39,4319,126],{"stretchy":119},[39,4321,442],{},[35,4323,123],{},[39,4325,1187],{},[35,4327,446],{"mathvariant":445},[39,4329,4330],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},"}",[39,4332,335],{"separator":53},[45,4334,4335],{"encoding":47},"\\mathcal H_k=\\overline{\\operatorname{span}}\\bigl\\{k(\\cdot,x):x\\in\\mathcal 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里的函数都「足够好」：它们是由核函数线性组合逼近出来的，而核的选择决定了这个空间的光滑性。",[1904,5308,5310],{"id":5309},"_22-特征映射与核技巧","2.2 特征映射与核技巧",[11,5312,5313],{},"令",[15,5315,5317],{"className":5316,"translate":19},[294],[15,5318,5320,5364],{"className":5319,"translate":19},[18],[15,5321,5323],{"className":5322},[23],[25,5324,5325],{"xmlns":27,"display":303},[29,5326,5327,5361],{},[32,5328,5329,5331,5333,5335,5337,5339,5341,5343,5345,5347,5349,5351,5353,5359],{},[35,5330,326],{},[39,5332,120],{"stretchy":119},[35,5334,123],{},[39,5336,126],{"stretchy":119},[39,5338,129],{},[35,5340,621],{},[39,5342,120],{"stretchy":119},[39,5344,3268],{},[39,5346,335],{"separator":53},[35,5348,123],{},[39,5350,126],{"stretchy":119},[39,5352,1187],{},[1057,5354,5355,5357],{},[35,5356,1061],{"mathvariant":445},[35,5358,621],{},[35,5360,3573],{"mathvariant":138},[45,5362,5363],{"encoding":47},"\\varphi(x)=k(\\cdot,x)\\in\\mathcal H_k.",[15,5365,5367,5394,5430],{"className":5366,"ariaHidden":53},[52],[15,5368,5370,5373,5376,5379,5382,5385,5388,5391],{"className":5369},[57],[15,5371],{"className":5372,"style":154},[61],[15,5374,326],{"className":5375},[66,67],[15,5377,120],{"className":5378},[162],[15,5380,123],{"className":5381},[66,67],[15,5383,126],{"className":5384},[169],[15,5386],{"className":5387,"style":173},[71],[15,5389,129],{"className":5390},[177],[15,5392],{"className":5393,"style":173},[71],[15,5395,5397,5400,5403,5406,5409,5412,5415,5418,5421,5424,5427],{"className":5396},[57],[15,5398],{"className":5399,"style":154},[61],[15,5401,621],{"className":5402,"style":684},[66,67],[15,5404,120],{"className":5405},[162],[15,5407,3268],{"className":5408},[66],[15,5410,335],{"className":5411},[422],[15,5413],{"className":5414,"style":697},[71],[15,5416,123],{"className":5417},[66,67],[15,5419,126],{"className":5420},[169],[15,5422],{"className":5423,"style":173},[71],[15,5425,1187],{"className":5426},[177],[15,5428],{"className":5429,"style":173},[71],[15,5431,5433,5436,5476],{"className":5432},[57],[15,5434],{"className":5435,"style":1076},[61],[15,5437,5439,5442],{"className":5438},[66],[15,5440,1061],{"className":5441,"style":1083},[66,494],[15,5443,5445],{"className":5444},[198],[15,5446,5448,5468],{"className":5447},[202,1090],[15,5449,5451,5465],{"className":5450},[206],[15,5452,5454],{"className":5453,"style":1097},[210],[15,5455,5456,5459],{"style":1100},[15,5457],{"className":5458,"style":218},[217],[15,5460,5462],{"className":5461},[222,223,224,225],[15,5463,621],{"className":5464,"style":684},[66,67,225],[15,5466,1114],{"className":5467},[1113],[15,5469,5471],{"className":5470},[206],[15,5472,5474],{"className":5473,"style":1121},[210],[15,5475],{},[15,5477,3573],{"className":5478},[66],[11,5480,5481,5482,5510,5511,5539,5540,5609],{},"则 ",[15,5483,5485,5498],{"className":5484,"translate":19},[18],[15,5486,5488],{"className":5487},[23],[25,5489,5490],{"xmlns":27},[29,5491,5492,5496],{},[32,5493,5494],{},[35,5495,326],{},[45,5497,558],{"encoding":47},[15,5499,5501],{"className":5500,"ariaHidden":53},[52],[15,5502,5504,5507],{"className":5503},[57],[15,5505],{"className":5506,"style":471},[61],[15,5508,326],{"className":5509},[66,67]," 就是一个从 ",[15,5512,5514,5527],{"className":5513,"translate":19},[18],[15,5515,5517],{"className":5516},[23],[25,5518,5519],{"xmlns":27},[29,5520,5521,5525],{},[32,5522,5523],{},[35,5524,446],{"mathvariant":445},[45,5526,1927],{"encoding":47},[15,5528,5530],{"className":5529,"ariaHidden":53},[52],[15,5531,5533,5536],{"className":5532},[57],[15,5534],{"className":5535,"style":490},[61],[15,5537,446],{"className":5538,"style":495},[66,494]," 到 ",[15,5541,5543,5560],{"className":5542,"translate":19},[18],[15,5544,5546],{"className":5545},[23],[25,5547,5548],{"xmlns":27},[29,5549,5550,5558],{},[32,5551,5552],{},[1057,5553,5554,5556],{},[35,5555,1061],{"mathvariant":445},[35,5557,621],{},[45,5559,1066],{"encoding":47},[15,5561,5563],{"className":5562,"ariaHidden":53},[52],[15,5564,5566,5569],{"className":5565},[57],[15,5567],{"className":5568,"style":1076},[61],[15,5570,5572,5575],{"className":5571},[66],[15,5573,1061],{"className":5574,"style":1083},[66,494],[15,5576,5578],{"className":5577},[198],[15,5579,5581,5601],{"className":5580},[202,1090],[15,5582,5584,5598],{"className":5583},[206],[15,5585,5587],{"className":5586,"style":1097},[210],[15,5588,5589,5592],{"style":1100},[15,5590],{"className":5591,"style":218},[217],[15,5593,5595],{"className":5594},[222,223,224,225],[15,5596,621],{"className":5597,"style":684},[66,67,225],[15,5599,1114],{"className":5600},[1113],[15,5602,5604],{"className":5603},[206],[15,5605,5607],{"className":5606,"style":1121},[210],[15,5608],{}," 的特征映射，并且",[15,5611,5613],{"className":5612,"translate":19},[294],[15,5614,5616,5682],{"className":5615,"translate":19},[18],[15,5617,5619],{"className":5618},[23],[25,5620,5621],{"xmlns":27,"display":303},[29,5622,5623,5679],{},[32,5624,5625,5627,5629,5631,5633,5639,5641,5643,5645,5647,5649,5651,5653,5655,5657,5659,5665,5667,5677],{},[35,5626,621],{},[39,5628,120],{"stretchy":119},[35,5630,123],{},[39,5632,335],{"separator":53},[131,5634,5635,5637],{},[35,5636,123],{},[39,5638,635],{"mathvariant":138,"lspace":634,"rspace":634},[39,5640,126],{"stretchy":119},[39,5642,129],{},[39,5644,642],{"stretchy":119},[35,5646,326],{},[39,5648,120],{"stretchy":119},[35,5650,123],{},[39,5652,126],{"stretchy":119},[39,5654,335],{"separator":53},[35,5656,326],{},[39,5658,120],{"stretchy":119},[131,5660,5661,5663],{},[35,5662,123],{},[39,5664,635],{"mathvariant":138,"lspace":634,"rspace":634},[39,5666,126],{"stretchy":119},[1057,5668,5669,5671],{},[39,5670,667],{"stretchy":119},[1057,5672,5673,5675],{},[35,5674,1061],{"mathvariant":445},[35,5676,621],{},[35,5678,3573],{"mathvariant":138},[45,5680,5681],{"encoding":47},"k(x,x')=\\langle \\varphi(x),\\varphi(x')\\rangle_{\\mathcal H_k}.",[15,5683,5685,5750],{"className":5684,"ariaHidden":53},[52],[15,5686,5688,5691,5694,5697,5700,5703,5706,5738,5741,5744,5747],{"className":5687},[57],[15,5689],{"className":5690,"style":680},[61],[15,5692,621],{"className":5693,"style":684},[66,67],[15,5695,120],{"className":5696},[162],[15,5698,123],{"className":5699},[66,67],[15,5701,335],{"className":5702},[422],[15,5704],{"className":5705,"style":697},[71],[15,5707,5709,5712],{"className":5708},[66],[15,5710,123],{"className":5711},[66,67],[15,5713,5715],{"className":5714},[198],[15,5716,5718],{"className":5717},[202],[15,5719,5721],{"className":5720},[206],[15,5722,5724],{"className":5723,"style":716},[210],[15,5725,5726,5729],{"style":397},[15,5727],{"className":5728,"style":218},[217],[15,5730,5732],{"className":5731},[222,223,224,225],[15,5733,5735],{"className":5734},[66,225],[15,5736,635],{"className":5737},[66,225],[15,5739,126],{"className":5740},[169],[15,5742],{"className":5743,"style":173},[71],[15,5745,129],{"className":5746},[177],[15,5748],{"className":5749,"style":173},[71],[15,5751,5753,5757,5760,5763,5766,5769,5772,5775,5778,5781,5784,5816,5819,5899],{"className":5752},[57],[15,5754],{"className":5755,"style":5756},[61],"height:1.0578em;vertical-align:-0.2559em;",[15,5758,642],{"className":5759},[162],[15,5761,326],{"className":5762},[66,67],[15,5764,120],{"className":5765},[162],[15,5767,123],{"className":5768},[66,67],[15,5770,126],{"className":5771},[169],[15,5773,335],{"className":5774},[422],[15,5776],{"className":5777,"style":697},[71],[15,5779,326],{"className":5780},[66,67],[15,5782,120],{"className":5783},[162],[15,5785,5787,5790],{"className":5786},[66],[15,5788,123],{"className":5789},[66,67],[15,5791,5793],{"className":5792},[198],[15,5794,5796],{"className":5795},[202],[15,5797,5799],{"className":5798},[206],[15,5800,5802],{"className":5801,"style":716},[210],[15,5803,5804,5807],{"style":397},[15,5805],{"className":5806,"style":218},[217],[15,5808,5810],{"className":5809},[222,223,224,225],[15,5811,5813],{"className":5812},[66,225],[15,5814,635],{"className":5815},[66,225],[15,5817,126],{"className":5818},[169],[15,5820,5822,5825],{"className":5821},[169],[15,5823,667],{"className":5824},[169],[15,5826,5828],{"className":5827},[198],[15,5829,5831,5891],{"className":5830},[202,1090],[15,5832,5834,5888],{"className":5833},[206],[15,5835,5837],{"className":5836,"style":3638},[210],[15,5838,5839,5842],{"style":1584},[15,5840],{"className":5841,"style":218},[217],[15,5843,5845],{"className":5844},[222,223,224,225],[15,5846,5848],{"className":5847},[66,225],[15,5849,5851,5854],{"className":5850},[66,225],[15,5852,1061],{"className":5853,"style":1083},[66,494,225],[15,5855,5857],{"className":5856},[198],[15,5858,5860,5880],{"className":5859},[202,1090],[15,5861,5863,5877],{"className":5862},[206],[15,5864,5866],{"className":5865,"style":3668},[210],[15,5867,5868,5871],{"style":3671},[15,5869],{"className":5870,"style":3675},[217],[15,5872,5874],{"className":5873},[222,3679,1508,225],[15,5875,621],{"className":5876,"style":684},[66,67,225],[15,5878,1114],{"className":5879},[1113],[15,5881,5883],{"className":5882},[206],[15,5884,5886],{"className":5885,"style":3692},[210],[15,5887],{},[15,5889,1114],{"className":5890},[1113],[15,5892,5894],{"className":5893},[206],[15,5895,5897],{"className":5896,"style":3704},[210],[15,5898],{},[15,5900,3573],{"className":5901},[66],[11,5903,5904,5905,5908,5909,6023,6024,6114,6115,6143],{},"这就是",[2040,5906,5907],{},"核技巧","（kernel trick）的来源：所有只涉及特征内积的算法，都可以把 ",[15,5910,5912,5950],{"className":5911,"translate":19},[18],[15,5913,5915],{"className":5914},[23],[25,5916,5917],{"xmlns":27},[29,5918,5919,5947],{},[32,5920,5921,5923,5925,5927,5929,5931,5933,5935,5937,5943,5945],{},[39,5922,642],{"stretchy":119},[35,5924,326],{},[39,5926,120],{"stretchy":119},[35,5928,123],{},[39,5930,126],{"stretchy":119},[39,5932,335],{"separator":53},[35,5934,326],{},[39,5936,120],{"stretchy":119},[131,5938,5939,5941],{},[35,5940,123],{},[39,5942,635],{"mathvariant":138,"lspace":634,"rspace":634},[39,5944,126],{"stretchy":119},[39,5946,667],{"stretchy":119},[45,5948,5949],{"encoding":47},"\\langle\\varphi(x),\\varphi(x')\\rangle",[15,5951,5953],{"className":5952,"ariaHidden":53},[52],[15,5954,5956,5960,5963,5966,5969,5972,5975,5978,5981,5984,5987,6020],{"className":5955},[57],[15,5957],{"className":5958,"style":5959},[61],"height:1.0019em;vertical-align:-0.25em;",[15,5961,642],{"className":5962},[162],[15,5964,326],{"className":5965},[66,67],[15,5967,120],{"className":5968},[162],[15,5970,123],{"className":5971},[66,67],[15,5973,126],{"className":5974},[169],[15,5976,335],{"className":5977},[422],[15,5979],{"className":5980,"style":697},[71],[15,5982,326],{"className":5983},[66,67],[15,5985,120],{"className":5986},[162],[15,5988,5990,5993],{"className":5989},[66],[15,5991,123],{"className":5992},[66,67],[15,5994,5996],{"className":5995},[198],[15,5997,5999],{"className":5998},[202],[15,6000,6002],{"className":6001},[206],[15,6003,6006],{"className":6004,"style":6005},[210],"height:0.7519em;",[15,6007,6008,6011],{"style":213},[15,6009],{"className":6010,"style":218},[217],[15,6012,6014],{"className":6013},[222,223,224,225],[15,6015,6017],{"className":6016},[66,225],[15,6018,635],{"className":6019},[66,225],[15,6021,811],{"className":6022},[169]," 替换成 ",[15,6025,6027,6055],{"className":6026,"translate":19},[18],[15,6028,6030],{"className":6029},[23],[25,6031,6032],{"xmlns":27},[29,6033,6034,6052],{},[32,6035,6036,6038,6040,6042,6044,6050],{},[35,6037,621],{},[39,6039,120],{"stretchy":119},[35,6041,123],{},[39,6043,335],{"separator":53},[131,6045,6046,6048],{},[35,6047,123],{},[39,6049,635],{"mathvariant":138,"lspace":634,"rspace":634},[39,6051,126],{"stretchy":119},[45,6053,6054],{"encoding":47},"k(x,x')",[15,6056,6058],{"className":6057,"ariaHidden":53},[52],[15,6059,6061,6064,6067,6070,6073,6076,6079,6111],{"className":6060},[57],[15,6062],{"className":6063,"style":5959},[61],[15,6065,621],{"className":6066,"style":684},[66,67],[15,6068,120],{"className":6069},[162],[15,6071,123],{"className":6072},[66,67],[15,6074,335],{"className":6075},[422],[15,6077],{"className":6078,"style":697},[71],[15,6080,6082,6085],{"className":6081},[66],[15,6083,123],{"className":6084},[66,67],[15,6086,6088],{"className":6087},[198],[15,6089,6091],{"className":6090},[202],[15,6092,6094],{"className":6093},[206],[15,6095,6097],{"className":6096,"style":6005},[210],[15,6098,6099,6102],{"style":213},[15,6100],{"className":6101,"style":218},[217],[15,6103,6105],{"className":6104},[222,223,224,225],[15,6106,6108],{"className":6107},[66,225],[15,6109,635],{"className":6110},[66,225],[15,6112,126],{"className":6113},[169],"，从而在不必显式构造 ",[15,6116,6118,6131],{"className":6117,"translate":19},[18],[15,6119,6121],{"className":6120},[23],[25,6122,6123],{"xmlns":27},[29,6124,6125,6129],{},[32,6126,6127],{},[35,6128,326],{},[45,6130,558],{"encoding":47},[15,6132,6134],{"className":6133,"ariaHidden":53},[52],[15,6135,6137,6140],{"className":6136},[57],[15,6138],{"className":6139,"style":471},[61],[15,6141,326],{"className":6142},[66,67]," 的情况下运行。SVM、核岭回归、核 PCA 都是这样。",[11,6145,6146],{},"需要强调的是，核技巧和表示定理是两件事。核技巧说的是「内积可以隐式计算」；表示定理说的是「最优解具有有限维形式」。前者是计算技巧，后者是结构定理。",[1904,6148,6150],{"id":6149},"_23-常见核与它们的-rkhs","2.3 常见核与它们的 RKHS",[6152,6153,6154,6170],"table",{},[6155,6156,6157],"thead",{},[6158,6159,6160,6164,6167],"tr",{},[6161,6162,6163],"th",{},"核",[6161,6165,6166],{},"形式",[6161,6168,6169],{},"特征维数 \u002F RKHS",[6171,6172,6173,6297,6571,6826,7014],"tbody",{},[6158,6174,6175,6179,6266],{},[6176,6177,6178],"td",{},"线性核",[6176,6180,6181],{},[15,6182,6184,6210],{"className":6183,"translate":19},[18],[15,6185,6187],{"className":6186},[23],[25,6188,6189],{"xmlns":27},[29,6190,6191,6207],{},[32,6192,6193,6195,6197,6199,6205],{},[39,6194,642],{"stretchy":119},[35,6196,123],{},[39,6198,335],{"separator":53},[131,6200,6201,6203],{},[35,6202,123],{},[39,6204,635],{"mathvariant":138,"lspace":634,"rspace":634},[39,6206,667],{"stretchy":119},[45,6208,6209],{"encoding":47},"\\langle x,x'\\rangle",[15,6211,6213],{"className":6212,"ariaHidden":53},[52],[15,6214,6216,6219,6222,6225,6228,6231,6263],{"className":6215},[57],[15,6217],{"className":6218,"style":5959},[61],[15,6220,642],{"className":6221},[162],[15,6223,123],{"className":6224},[66,67],[15,6226,335],{"className":6227},[422],[15,6229],{"className":6230,"style":697},[71],[15,6232,6234,6237],{"className":6233},[66],[15,6235,123],{"className":6236},[66,67],[15,6238,6240],{"className":6239},[198],[15,6241,6243],{"className":6242},[202],[15,6244,6246],{"className":6245},[206],[15,6247,6249],{"className":6248,"style":6005},[210],[15,6250,6251,6254],{"style":213},[15,6252],{"className":6253,"style":218},[217],[15,6255,6257],{"className":6256},[222,223,224,225],[15,6258,6260],{"className":6259},[66,225],[15,6261,635],{"className":6262},[66,225],[15,6264,667],{"className":6265},[169],[6176,6267,6268,6296],{},[15,6269,6271,6284],{"className":6270,"translate":19},[18],[15,6272,6274],{"className":6273},[23],[25,6275,6276],{"xmlns":27},[29,6277,6278,6282],{},[32,6279,6280],{},[35,6281,458],{},[45,6283,458],{"encoding":47},[15,6285,6287],{"className":6286,"ariaHidden":53},[52],[15,6288,6290,6293],{"className":6289},[57],[15,6291],{"className":6292,"style":840},[61],[15,6294,458],{"className":6295},[66,67]," 维",[6158,6298,6299,6302,6450],{},[6176,6300,6301],{},"多项式核",[6176,6303,6304],{},[15,6305,6307,6345],{"className":6306,"translate":19},[18],[15,6308,6310],{"className":6309},[23],[25,6311,6312],{"xmlns":27},[29,6313,6314,6342],{},[32,6315,6316,6318,6320,6322,6324,6330,6332,6334,6336],{},[39,6317,120],{"stretchy":119},[39,6319,642],{"stretchy":119},[35,6321,123],{},[39,6323,335],{"separator":53},[131,6325,6326,6328],{},[35,6327,123],{},[39,6329,635],{"mathvariant":138,"lspace":634,"rspace":634},[39,6331,667],{"stretchy":119},[39,6333,1245],{},[35,6335,2256],{},[131,6337,6338,6340],{},[39,6339,126],{"stretchy":119},[35,6341,11],{},[45,6343,6344],{"encoding":47},"(\\langle 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定理",[3054,7252,7253,7826,8326,8329,8335,8682],{},[11,7254,7255,7258,7259,1755,7287,7315,7316,7584,7585,7727,7728,7825],{},[2040,7256,7257],{},"定理 3.1（表示定理）"," 设 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",[15,7586,7588,7626],{"className":7587,"translate":19},[18],[15,7589,7591],{"className":7590},[23],[25,7592,7593],{"xmlns":27},[29,7594,7595,7623],{},[32,7596,7597,7599,7601,7607,7609,7611,7614,7616,7618,7621],{},[35,7598,1216],{},[39,7600,442],{},[131,7602,7603,7605],{},[35,7604,455],{"mathvariant":454},[35,7606,37],{},[39,7608,449],{},[35,7610,455],{"mathvariant":454},[39,7612,7613],{},"∪",[39,7615,4307],{"stretchy":119},[39,7617,1245],{},[35,7619,7620],{"mathvariant":138},"∞",[39,7622,4330],{"stretchy":119},[45,7624,7625],{"encoding":47},"L:\\mathbb R^n\\to\\mathbb R\\cup\\{+\\infty\\}",[15,7627,7629,7647,7691,7709],{"className":7628,"ariaHidden":53},[52],[15,7630,7632,7635,7638,7641,7644],{"className":7631},[57],[15,7633],{"className":7634,"style":490},[61],[15,7636,1216],{"className":7637},[66,67],[15,7639],{"className":7640,"style":173},[71],[15,7642,442],{"className":7643},[177],[15,7645],{"className":7646,"style":173},[71],[15,7648,7650,7653,7682,7685,7688],{"className":7649},[57],[15,7651],{"className":7652,"style":2034},[61],[15,7654,7656,7659],{"className":7655},[66],[15,7657,455],{"className":7658},[66,517],[15,7660,7662],{"className":7661},[198],[15,7663,7665],{"className":7664},[202],[15,7666,7668],{"className":7667},[206],[15,7669,7671],{"className":7670,"style":6438},[210],[15,7672,7673,7676],{"style":213},[15,7674],{"className":7675,"style":218},[217],[15,7677,7679],{"className":7678},[222,223,224,225],[15,7680,37],{"className":7681},[66,67,225],[15,7683],{"className":7684,"style":173},[71],[15,7686,449],{"className":7687},[177],[15,7689],{"className":7690,"style":173},[71],[15,7692,7694,7697,7700,7703,7706],{"className":7693},[57],[15,7695],{"className":7696,"style":2034},[61],[15,7698,455],{"className":7699},[66,517],[15,7701],{"className":7702,"style":72},[71],[15,7704,7613],{"className":7705},[76],[15,7707],{"className":7708,"style":72},[71],[15,7710,7712,7715,7718,7721,7724],{"className":7711},[57],[15,7713],{"className":7714,"style":154},[61],[15,7716,4307],{"className":7717},[162],[15,7719,1245],{"className":7720},[66],[15,7722,7620],{"className":7723},[66],[15,7725,4330],{"className":7726},[169]," 是任意损失函数，",[15,7729,7731,7762],{"className":7730,"translate":19},[18],[15,7732,7734],{"className":7733},[23],[25,7735,7736],{"xmlns":27},[29,7737,7738,7759],{},[32,7739,7740,7743,7745,7747,7749,7751,7753,7755,7757],{},[35,7741,7742],{"mathvariant":138},"Ω",[39,7744,442],{},[39,7746,1194],{"stretchy":119},[1209,7748,1804],{},[39,7750,335],{"separator":53},[35,7752,7620],{"mathvariant":138},[39,7754,126],{"stretchy":119},[39,7756,449],{},[35,7758,455],{"mathvariant":454},[45,7760,7761],{"encoding":47},"\\Omega:[0,\\infty)\\to\\mathbb R",[15,7763,7765,7783,7816],{"className":7764,"ariaHidden":53},[52],[15,7766,7768,7771,7774,7777,7780],{"className":7767},[57],[15,7769],{"className":7770,"style":490},[61],[15,7772,7742],{"className":7773},[66],[15,7775],{"className":7776,"style":173},[71],[15,7778,442],{"className":7779},[177],[15,7781],{"className":7782,"style":173},[71],[15,7784,7786,7789,7792,7795,7798,7801,7804,7807,7810,7813],{"className":7785},[57],[15,7787],{"className":7788,"style":154},[61],[15,7790,1194],{"className":7791},[162],[15,7793,1804],{"className":7794},[66],[15,7796,335],{"className":7797},[422],[15,7799],{"className":7800,"style":697},[71],[15,7802,7620],{"className":7803},[66],[15,7805,126],{"className":7806},[169],[15,7808],{"className":7809,"style":173},[71],[15,7811,449],{"className":7812},[177],[15,7814],{"className":7815,"style":173},[71],[15,7817,7819,7822],{"className":7818},[57],[15,7820],{"className":7821,"style":2034},[61],[15,7823,455],{"className":7824},[66,517]," 是非减函数。考虑",[15,7827,7829],{"className":7828,"translate":19},[294],[15,7830,7832,7951],{"className":7831,"translate":19},[18],[15,7833,7835],{"className":7834},[23],[25,7836,7837],{"xmlns":27,"display":303},[29,7838,7839,7948],{},[1142,7840,7841],{"width":1144},[1146,7842,7843,7845,7941,7943],{},[1149,7844],{"width":1151},[1149,7846,7847],{},[32,7848,7849,7855,7857,7859,7861,7877,7939],{},[131,7850,7851,7853],{},[35,7852,116],{},[39,7854,1162],{},[39,7856,1187],{},[35,7858,1167],{},[39,7860,1170],{},[1172,7862,7863,7869],{},[32,7864,7865,7867],{},[35,7866,1178],{},[39,7868,1170],{},[32,7870,7871,7873,7875],{},[35,7872,116],{},[39,7874,1187],{},[35,7876,1061],{"mathvariant":445},[32,7878,7879,7881,7883,7885,7887,7889,7895,7897,7899,7901,7903,7905,7907,7913,7915,7917,7919,7921,7923,7925,7927,7935,7937],{},[39,7880,4307],{"fence":53},[35,7882,1216],{},[39,7884,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,7886,116],{},[39,7888,120],{"stretchy":119},[1057,7890,7891,7893],{},[35,7892,123],{},[1209,7894,1211],{},[39,7896,126],{"stretchy":119},[39,7898,335],{"separator":53},[39,7900,2095],{},[39,7902,335],{"separator":53},[35,7904,116],{},[39,7906,120],{"stretchy":119},[1057,7908,7909,7911],{},[35,7910,123],{},[35,7912,37],{},[39,7914,126],{"stretchy":119},[39,7916,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,7918,1245],{},[35,7920,7742],{"mathvariant":138},[39,7922,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,7924,1251],{"mathvariant":138},[35,7926,116],{},[1255,7928,7929,7931,7933],{},[35,7930,1251],{"mathvariant":138},[35,7932,1061],{"mathvariant":445},[1209,7934,1263],{},[39,7936,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,7938,4330],{"fence":53},[35,7940,3573],{"mathvariant":138},[1149,7942],{"width":1151},[1149,7944,7945],{},[1274,7946,7947],{},"(2)",[45,7949,7950],{"encoding":47},"f^*\\in\\arg\\min_{f\\in\\mathcal 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存在一个最优解具有形式",[15,8336,8338],{"className":8337,"translate":19},[294],[15,8339,8341,8424],{"className":8340,"translate":19},[18],[15,8342,8344],{"className":8343},[23],[25,8345,8346],{"xmlns":27,"display":303},[29,8347,8348,8421],{},[1142,8349,8350],{"width":1144},[1146,8351,8352,8354,8414,8416],{},[1149,8353],{"width":1151},[1149,8355,8356],{},[32,8357,8358,8364,8366,8368,8370,8372,8386,8393,8395,8397,8399,8401,8403,8409,8411],{},[131,8359,8360,8362],{},[35,8361,116],{},[39,8363,1162],{},[39,8365,120],{"stretchy":119},[35,8367,123],{},[39,8369,126],{"stretchy":119},[39,8371,129],{},[1196,8373,8374,8376,8384],{},[39,8375,1200],{},[32,8377,8378,8380,8382],{},[35,8379,1205],{},[39,8381,129],{},[1209,8383,1211],{},[35,8385,37],{},[1057,8387,8388,8391],{},[35,8389,8390],{},"α",[35,8392,1205],{},[1274,8394,2462],{},[35,8396,621],{},[39,8398,120],{"stretchy":119},[35,8400,123],{},[39,8402,335],{"separator":53},[1057,8404,8405,8407],{},[35,8406,123],{},[35,8408,1205],{},[39,8410,126],{"stretchy":119},[39,8412,8413],{"separator":53},";",[1149,8415],{"width":1151},[1149,8417,8418],{},[1274,8419,8420],{},"(3)",[45,8422,8423],{"encoding":47},"f^*(x)=\\sum_{i=1}^n\\alpha_i\\,k(x,x_i);\n\\tag{3}",[15,8425,8427,8480,8660],{"className":8426,"ariaHidden":53},[52],[15,8428,8430,8433,8462,8465,8468,8471,8474,8477],{"className":8429},[57],[15,8431],{"className":8432,"style":154},[61],[15,8434,8436,8439],{"className":8435},[66],[15,8437,116],{"className":8438,"style":158},[66,67],[15,8440,8442],{"className":8441},[198],[15,8443,8445],{"className":8444},[202],[15,8446,8448],{"className":8447},[206],[15,8449,8451],{"className":8450,"style":1308},[210],[15,8452,8453,8456],{"style":397},[15,8454],{"className":8455,"style":218},[217],[15,8457,8459],{"className":8458},[222,223,224,225],[15,8460,1162],{"className":8461},[76,225],[15,8463,120],{"className":8464},[162],[15,8466,123],{"className":8467},[66,67],[15,8469,126],{"className":8470},[169],[15,8472],{"className":8473,"style":173},[71],[15,8475,129],{"className":8476},[177],[15,8478],{"className":8479,"style":173},[71],[15,8481,8483,8487,8551,8554,8596,8599,8602,8605,8608,8611,8614,8654,8657],{"className":8482},[57],[15,8484],{"className":8485,"style":8486},[61],"height:2.9291em;vertical-align:-1.2777em;",[15,8488,8490],{"className":8489},[1339,1351],[15,8491,8493,8543],{"className":8492},[202,1090],[15,8494,8496,8540],{"className":8495},[206],[15,8497,8499,8519,8529],{"className":8498,"style":1436},[210],[15,8500,8501,8504],{"style":1439},[15,8502],{"className":8503,"style":1443},[217],[15,8505,8507],{"className":8506},[222,223,224,225],[15,8508,8510,8513,8516],{"className":8509},[66,225],[15,8511,1205],{"className":8512},[66,67,225],[15,8514,129],{"className":8515},[177,225],[15,8517,1211],{"className":8518},[66,225],[15,8520,8521,8524],{"style":1461},[15,8522],{"className":8523,"style":1443},[217],[15,8525,8526],{},[15,8527,1200],{"className":8528},[1339,1470,1471],[15,8530,8531,8534],{"style":1474},[15,8532],{"className":8533,"style":1443},[217],[15,8535,8537],{"className":8536},[222,223,224,225],[15,8538,37],{"className":8539},[66,67,225],[15,8541,1114],{"className":8542},[1113],[15,8544,8546],{"className":8545},[206],[15,8547,8549],{"className":8548,"style":1493},[210],[15,8550],{},[15,8552],{"className":8553,"style":697},[71],[15,8555,8557,8561],{"className":8556},[66],[15,8558,8390],{"className":8559,"style":8560},[66,67],"margin-right:0.0037em;",[15,8562,8564],{"className":8563},[198],[15,8565,8567,8588],{"className":8566},[202,1090],[15,8568,8570,8585],{"className":8569},[206],[15,8571,8573],{"className":8572,"style":1528},[210],[15,8574,8576,8579],{"style":8575},"top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;",[15,8577],{"className":8578,"style":218},[217],[15,8580,8582],{"className":8581},[222,223,224,225],[15,8583,1205],{"className":8584},[66,67,225],[15,8586,1114],{"className":8587},[1113],[15,8589,8591],{"className":8590},[206],[15,8592,8594],{"className":8593,"style":1121},[210],[15,8595],{},[15,8597],{"className":8598,"style":697},[71],[15,8600,621],{"className":8601,"style":684},[66,67],[15,8603,120],{"className":8604},[162],[15,8606,123],{"className":8607},[66,67],[15,8609,335],{"className":8610},[422],[15,8612],{"className":8613,"style":697},[71],[15,8615,8617,8620],{"className":8616},[66],[15,8618,123],{"className":8619},[66,67],[15,8621,8623],{"className":8622},[198],[15,8624,8626,8646],{"className":8625},[202,1090],[15,8627,8629,8643],{"className":8628},[206],[15,8630,8632],{"className":8631,"style":1528},[210],[15,8633,8634,8637],{"style":1584},[15,8635],{"className":8636,"style":218},[217],[15,8638,8640],{"className":8639},[222,223,224,225],[15,8641,1205],{"className":8642},[66,67,225],[15,8644,1114],{"className":8645},[1113],[15,8647,8649],{"className":8648},[206],[15,8650,8652],{"className":8651,"style":1121},[210],[15,8653],{},[15,8655,126],{"className":8656},[169],[15,8658,8413],{"className":8659},[422],[15,8661,8663,8666],{"className":8662},[1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若 ",[15,8689,8691,8705],{"className":8690,"translate":19},[18],[15,8692,8694],{"className":8693},[23],[25,8695,8696],{"xmlns":27},[29,8697,8698,8702],{},[32,8699,8700],{},[35,8701,7742],{"mathvariant":138},[45,8703,8704],{"encoding":47},"\\Omega",[15,8706,8708],{"className":8707,"ariaHidden":53},[52],[15,8709,8711,8714],{"className":8710},[57],[15,8712],{"className":8713,"style":490},[61],[15,8715,7742],{"className":8716},[66]," 严格递增，则",[2040,8719,8720],{},"每个","最优解都具有这个形式。",[11,8723,8724],{},"这里有三个细节值得强调。",[11,8726,8727,8728,8756,8757,8927,8928,8956,8957,8985,8986,9056,9057,9085],{},"第一，",[15,8729,8731,8744],{"className":8730,"translate":19},[18],[15,8732,8734],{"className":8733},[23],[25,8735,8736],{"xmlns":27},[29,8737,8738,8742],{},[32,8739,8740],{},[35,8741,1216],{},[45,8743,1216],{"encoding":47},[15,8745,8747],{"className":8746,"ariaHidden":53},[52],[15,8748,8750,8753],{"className":8749},[57],[15,8751],{"className":8752,"style":490},[61],[15,8754,1216],{"className":8755},[66,67]," 只通过 ",[15,8758,8760,8802],{"className":8759,"translate":19},[18],[15,8761,8763],{"className":8762},[23],[25,8764,8765],{"xmlns":27},[29,8766,8767,8799],{},[32,8768,8769,8771,8773,8779,8781,8783,8785,8787,8789,8791,8797],{},[35,8770,116],{},[39,8772,120],{"stretchy":119},[1057,8774,8775,8777],{},[35,8776,123],{},[1209,8778,1211],{},[39,8780,126],{"stretchy":119},[39,8782,335],{"separator":53},[39,8784,2095],{},[39,8786,335],{"separator":53},[35,8788,116],{},[39,8790,120],{"stretchy":119},[1057,8792,8793,8795],{},[35,8794,123],{},[35,8796,37],{},[39,8798,126],{"stretchy":119},[45,8800,8801],{"encoding":47},"f(x_1),\\dots,f(x_n)",[15,8803,8805],{"className":8804,"ariaHidden":53},[52],[15,8806,8808,8811,8814,8817,8857,8860,8863,8866,8869,8872,8875,8878,8881,8884,8924],{"className":8807},[57],[15,8809],{"className":8810,"style":154},[61],[15,8812,116],{"className":8813,"style":158},[66,67],[15,8815,120],{"className":8816},[162],[15,8818,8820,8823],{"className":8819},[66],[15,8821,123],{"className":8822},[66,67],[15,8824,8826],{"className":8825},[198],[15,8827,8829,8849],{"className":8828},[202,1090],[15,8830,8832,8846],{"className":8831},[206],[15,8833,8835],{"className":8834,"style":2139},[210],[15,8836,8837,8840],{"style":1584},[15,8838],{"className":8839,"style":218},[217],[15,8841,8843],{"className":8842},[222,223,224,225],[15,8844,1211],{"className":8845},[66,225],[15,8847,1114],{"className":8848},[1113],[15,8850,8852],{"className":8851},[206],[15,8853,8855],{"className":8854,"style":1121},[210],[15,8856],{},[15,8858,126],{"className":8859},[169],[15,8861,335],{"className":8862},[422],[15,8864],{"className":8865,"style":697},[71],[15,8867,2095],{"className":8868},[1413],[15,8870],{"className":8871,"style":697},[71],[15,8873,335],{"className":8874},[422],[15,8876],{"className":8877,"style":697},[71],[15,8879,116],{"className":8880,"style":158},[66,67],[15,8882,120],{"className":8883},[162],[15,8885,8887,8890],{"className":8886},[66],[15,8888,123],{"className":8889},[66,67],[15,8891,8893],{"className":8892},[198],[15,8894,8896,8916],{"className":8895},[202,1090],[15,8897,8899,8913],{"className":8898},[206],[15,8900,8902],{"className":8901,"style":2198},[210],[15,8903,8904,8907],{"style":1584},[15,8905],{"className":8906,"style":218},[217],[15,8908,8910],{"className":8909},[222,223,224,225],[15,8911,37],{"className":8912},[66,67,225],[15,8914,1114],{"className":8915},[1113],[15,8917,8919],{"className":8918},[206],[15,8920,8922],{"className":8921,"style":1121},[210],[15,8923],{},[15,8925,126],{"className":8926},[169]," 这 ",[15,8929,8931,8944],{"className":8930,"translate":19},[18],[15,8932,8934],{"className":8933},[23],[25,8935,8936],{"xmlns":27},[29,8937,8938,8942],{},[32,8939,8940],{},[35,8941,37],{},[45,8943,37],{"encoding":47},[15,8945,8947],{"className":8946,"ariaHidden":53},[52],[15,8948,8950,8953],{"className":8949},[57],[15,8951],{"className":8952,"style":86},[61],[15,8954,37],{"className":8955},[66,67]," 个值依赖 ",[15,8958,8960,8973],{"className":8959,"translate":19},[18],[15,8961,8963],{"className":8962},[23],[25,8964,8965],{"xmlns":27},[29,8966,8967,8971],{},[32,8968,8969],{},[35,8970,116],{},[45,8972,116],{"encoding":47},[15,8974,8976],{"className":8975,"ariaHidden":53},[52],[15,8977,8979,8982],{"className":8978},[57],[15,8980],{"className":8981,"style":3408},[61],[15,8983,116],{"className":8984,"style":158},[66,67],"（标签 ",[15,8987,8989,9007],{"className":8988,"translate":19},[18],[15,8990,8992],{"className":8991},[23],[25,8993,8994],{"xmlns":27},[29,8995,8996,9004],{},[32,8997,8998],{},[1057,8999,9000,9002],{},[35,9001,1224],{},[35,9003,1205],{},[45,9005,9006],{"encoding":47},"y_i",[15,9008,9010],{"className":9009,"ariaHidden":53},[52],[15,9011,9013,9016],{"className":9012},[57],[15,9014],{"className":9015,"style":471},[61],[15,9017,9019,9022],{"className":9018},[66],[15,9020,1224],{"className":9021,"style":1515},[66,67],[15,9023,9025],{"className":9024},[198],[15,9026,9028,9048],{"className":9027},[202,1090],[15,9029,9031,9045],{"className":9030},[206],[15,9032,9034],{"className":9033,"style":1528},[210],[15,9035,9036,9039],{"style":1531},[15,9037],{"className":9038,"style":218},[217],[15,9040,9042],{"className":9041},[222,223,224,225],[15,9043,1205],{"className":9044},[66,67,225],[15,9046,1114],{"className":9047},[1113],[15,9049,9051],{"className":9050},[206],[15,9052,9054],{"className":9053,"style":1121},[210],[15,9055],{}," 已吸收进 ",[15,9058,9060,9073],{"className":9059,"translate":19},[18],[15,9061,9063],{"className":9062},[23],[25,9064,9065],{"xmlns":27},[29,9066,9067,9071],{},[32,9068,9069],{},[35,9070,1216],{},[45,9072,1216],{"encoding":47},[15,9074,9076],{"className":9075,"ariaHidden":53},[52],[15,9077,9079,9082],{"className":9078},[57],[15,9080],{"className":9081,"style":490},[61],[15,9083,1216],{"className":9084},[66,67],"）。如果损失还依赖导数、积分或在其他点上的取值，定理的形式会改变。表示定理的标准版本针对的正是这种「点值型」经验风险。",[11,9087,9088,9089,9117,9118,9193,9194,9266,9267,9320],{},"第二，",[15,9090,9092,9105],{"className":9091,"translate":19},[18],[15,9093,9095],{"className":9094},[23],[25,9096,9097],{"xmlns":27},[29,9098,9099,9103],{},[32,9100,9101],{},[35,9102,7742],{"mathvariant":138},[45,9104,8704],{"encoding":47},[15,9106,9108],{"className":9107,"ariaHidden":53},[52],[15,9109,9111,9114],{"className":9110},[57],[15,9112],{"className":9113,"style":490},[61],[15,9115,7742],{"className":9116},[66]," 只要求非减，不要求凸、不要求可微。式 (1) 中的 ",[15,9119,9121,9145],{"className":9120,"translate":19},[18],[15,9122,9124],{"className":9123},[23],[25,9125,9126],{"xmlns":27},[29,9127,9128,9142],{},[32,9129,9130,9132,9134,9136],{},[35,9131,1248],{},[35,9133,1251],{"mathvariant":138},[35,9135,116],{},[131,9137,9138,9140],{},[35,9139,1251],{"mathvariant":138},[1209,9141,1263],{},[45,9143,9144],{"encoding":47},"\\lambda\\|f\\|^2",[15,9146,9148],{"className":9147,"ariaHidden":53},[52],[15,9149,9151,9155,9158,9161,9164],{"className":9150},[57],[15,9152],{"className":9153,"style":9154},[61],"height:1.0641em;vertical-align:-0.25em;",[15,9156,1248],{"className":9157},[66,67],[15,9159,1251],{"className":9160},[66],[15,9162,116],{"className":9163,"style":158},[66,67],[15,9165,9167,9170],{"className":9166},[66],[15,9168,1251],{"className":9169},[66],[15,9171,9173],{"className":9172},[198],[15,9174,9176],{"className":9175},[202],[15,9177,9179],{"className":9178},[206],[15,9180,9182],{"className":9181,"style":6740},[210],[15,9183,9184,9187],{"style":213},[15,9185],{"className":9186,"style":218},[217],[15,9188,9190],{"className":9189},[222,223,224,225],[15,9191,1263],{"className":9192},[66,225]," 对应 ",[15,9195,9197,9224],{"className":9196,"translate":19},[18],[15,9198,9200],{"className":9199},[23],[25,9201,9202],{"xmlns":27},[29,9203,9204,9221],{},[32,9205,9206,9208,9210,9213,9215,9217,9219],{},[35,9207,7742],{"mathvariant":138},[39,9209,120],{"stretchy":119},[35,9211,9212],{},"t",[39,9214,126],{"stretchy":119},[39,9216,129],{},[35,9218,1248],{},[35,9220,9212],{},[45,9222,9223],{"encoding":47},"\\Omega(t)=\\lambda t",[15,9225,9227,9254],{"className":9226,"ariaHidden":53},[52],[15,9228,9230,9233,9236,9239,9242,9245,9248,9251],{"className":9229},[57],[15,9231],{"className":9232,"style":154},[61],[15,9234,7742],{"className":9235},[66],[15,9237,120],{"className":9238},[162],[15,9240,9212],{"className":9241},[66,67],[15,9243,126],{"className":9244},[169],[15,9246],{"className":9247,"style":173},[71],[15,9249,129],{"className":9250},[177],[15,9252],{"className":9253,"style":173},[71],[15,9255,9257,9260,9263],{"className":9256},[57],[15,9258],{"className":9259,"style":840},[61],[15,9261,1248],{"className":9262},[66,67],[15,9264,9212],{"className":9265},[66,67],"（",[15,9268,9270,9289],{"className":9269,"translate":19},[18],[15,9271,9273],{"className":9272},[23],[25,9274,9275],{"xmlns":27},[29,9276,9277,9286],{},[32,9278,9279,9281,9284],{},[35,9280,1248],{},[39,9282,9283],{},">",[1209,9285,1804],{},[45,9287,9288],{"encoding":47},"\\lambda>0",[15,9290,9292,9311],{"className":9291,"ariaHidden":53},[52],[15,9293,9295,9299,9302,9305,9308],{"className":9294},[57],[15,9296],{"className":9297,"style":9298},[61],"height:0.7335em;vertical-align:-0.0391em;",[15,9300,1248],{"className":9301},[66,67],[15,9303],{"className":9304,"style":173},[71],[15,9306,9283],{"className":9307},[177],[15,9309],{"className":9310,"style":173},[71],[15,9312,9314,9317],{"className":9313},[57],[15,9315],{"className":9316,"style":1836},[61],[15,9318,1804],{"className":9319},[66]," 时严格递增），是最常见的特例。",[11,9322,9323,9324,9327,9328,9356],{},"第三，第 1 条是",[2040,9325,9326],{},"存在性","，第 2 条才是「任意最优解」。如果 ",[15,9329,9331,9344],{"className":9330,"translate":19},[18],[15,9332,9334],{"className":9333},[23],[25,9335,9336],{"xmlns":27},[29,9337,9338,9342],{},[32,9339,9340],{},[35,9341,7742],{"mathvariant":138},[45,9343,8704],{"encoding":47},[15,9345,9347],{"className":9346,"ariaHidden":53},[52],[15,9348,9350,9353],{"className":9349},[57],[15,9351],{"className":9352,"style":490},[61],[15,9354,7742],{"className":9355},[66]," 只是非减而非严格递增，可能存在落在子空间之外的最优解；我们会在第四节看到这一点。",[1904,9358,9360],{"id":9359},"_32-证明正交分解","3.2 证明：正交分解",[11,9362,9363],{},"证明的核心是一个正交分解。令",[15,9365,9367],{"className":9366,"translate":19},[294],[15,9368,9370,9443],{"className":9369,"translate":19},[18],[15,9371,9373],{"className":9372},[23],[25,9374,9375],{"xmlns":27,"display":303},[29,9376,9377,9440],{},[32,9378,9379,9385,9387,9389,9391,9393,9395,9397,9399,9401,9407,9409,9411,9413,9415,9417,9419,9421,9423,9429,9431,9433,9436,9438],{},[1057,9380,9381,9383],{},[35,9382,1061],{"mathvariant":445},[1209,9384,1804],{},[39,9386,129],{},[35,9388,15],{"mathvariant":138},[39,9390,1170],{},[39,9392,4307],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,9394,621],{},[39,9396,120],{"stretchy":119},[39,9398,3268],{},[39,9400,335],{"separator":53},[1057,9402,9403,9405],{},[35,9404,123],{},[1209,9406,1211],{},[39,9408,126],{"stretchy":119},[39,9410,335],{"separator":53},[39,9412,2095],{},[39,9414,335],{"separator":53},[35,9416,621],{},[39,9418,120],{"stretchy":119},[39,9420,3268],{},[39,9422,335],{"separator":53},[1057,9424,9425,9427],{},[35,9426,123],{},[35,9428,37],{},[39,9430,126],{"stretchy":119},[39,9432,4330],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,9434,9435],{},"⊆",[35,9437,1061],{"mathvariant":445},[35,9439,3573],{"mathvariant":138},[45,9441,9442],{"encoding":47},"\\mathcal H_0=\\operatorname{span}\\bigl\\{k(\\cdot,x_1),\\dots,k(\\cdot,x_n)\\bigr\\}\\subseteq\\mathcal H.",[15,9444,9446,9501,9668],{"className":9445,"ariaHidden":53},[52],[15,9447,9449,9452,9492,9495,9498],{"className":9448},[57],[15,9450],{"className":9451,"style":1076},[61],[15,9453,9455,9458],{"className":9454},[66],[15,9456,1061],{"className":9457,"style":1083},[66,494],[15,9459,9461],{"className":9460},[198],[15,9462,9464,9484],{"className":9463},[202,1090],[15,9465,9467,9481],{"className":9466},[206],[15,9468,9470],{"className":9469,"style":2139},[210],[15,9471,9472,9475],{"style":1100},[15,9473],{"className":9474,"style":218},[217],[15,9476,9478],{"className":9477},[222,223,224,225],[15,9479,1804],{"className":9480},[66,225],[15,9482,1114],{"className":9483},[1113],[15,9485,9487],{"className":9486},[206],[15,9488,9490],{"className":9489,"style":1121},[210],[15,9491],{},[15,9493],{"className":9494,"style":173},[71],[15,9496,129],{"className":9497},[177],[15,9499],{"className":9500,"style":173},[71],[15,9502,9504,9507,9513,9519,9522,9525,9528,9531,9534,9574,9577,9580,9583,9586,9589,9592,9595,9598,9601,9604,9607,9610,9650,9653,9659,9662,9665],{"className":9503},[57],[15,9505],{"className":9506,"style":4400},[61],[15,9508,9510],{"className":9509},[1339],[15,9511,15],{"className":9512},[66,4429],[15,9514,9516],{"className":9515},[162],[15,9517,4307],{"className":9518},[1422,1508],[15,9520,621],{"className":9521,"style":684},[66,67],[15,9523,120],{"className":9524},[162],[15,9526,3268],{"className":9527},[66],[15,9529,335],{"className":9530},[422],[15,9532],{"className":9533,"style":697},[71],[15,9535,9537,9540],{"className":9536},[66],[15,9538,123],{"className":9539},[66,67],[15,9541,9543],{"className":9542},[198],[15,9544,9546,9566],{"className":9545},[202,1090],[15,9547,9549,9563],{"className":9548},[206],[15,9550,9552],{"className":9551,"style":2139},[210],[15,9553,9554,9557],{"style":1584},[15,9555],{"className":9556,"style":218},[217],[15,9558,9560],{"className":9559},[222,223,224,225],[15,9561,1211],{"className":9562},[66,225],[15,9564,1114],{"className":9565},[1113],[15,9567,9569],{"className":9568},[206],[15,9570,9572],{"className":9571,"style":1121},[210],[15,9573],{},[15,9575,126],{"className":9576},[169],[15,9578,335],{"className":9579},[422],[15,9581],{"className":9582,"style":697},[71],[15,9584,2095],{"className":9585},[1413],[15,9587],{"className":9588,"style":697},[71],[15,9590,335],{"className":9591},[422],[15,9593],{"className":9594,"style":697},[71],[15,9596,621],{"className":9597,"style":684},[66,67],[15,9599,120],{"className":9600},[162],[15,9602,3268],{"className":9603},[66],[15,9605,335],{"className":9606},[422],[15,9608],{"className":9609,"style":697},[71],[15,9611,9613,9616],{"className":9612},[66],[15,9614,123],{"className":9615},[66,67],[15,9617,9619],{"className":9618},[198],[15,9620,9622,9642],{"className":9621},[202,1090],[15,9623,9625,9639],{"className":9624},[206],[15,9626,9628],{"className":9627,"style":2198},[210],[15,9629,9630,9633],{"style":1584},[15,9631],{"className":9632,"style":218},[217],[15,9634,9636],{"className":9635},[222,223,224,225],[15,9637,37],{"className":9638},[66,67,225],[15,9640,1114],{"className":9641},[1113],[15,9643,9645],{"className":9644},[206],[15,9646,9648],{"className":9647,"style":1121},[210],[15,9649],{},[15,9651,126],{"className":9652},[169],[15,9654,9656],{"className":9655},[169],[15,9657,4330],{"className":9658},[1422,1508],[15,9660],{"className":9661,"style":173},[71],[15,9663,9435],{"className":9664},[177],[15,9666],{"className":9667,"style":173},[71],[15,9669,9671,9674,9677],{"className":9670},[57],[15,9672],{"className":9673,"style":490},[61],[15,9675,1061],{"className":9676,"style":1083},[66,494],[15,9678,3573],{"className":9679},[66],[11,9681,9682,9683,9734],{},"对任意 ",[15,9684,9686,9704],{"className":9685,"translate":19},[18],[15,9687,9689],{"className":9688},[23],[25,9690,9691],{"xmlns":27},[29,9692,9693,9701],{},[32,9694,9695,9697,9699],{},[35,9696,116],{},[39,9698,1187],{},[35,9700,1061],{"mathvariant":445},[45,9702,9703],{"encoding":47},"f\\in\\mathcal H",[15,9705,9707,9725],{"className":9706,"ariaHidden":53},[52],[15,9708,9710,9713,9716,9719,9722],{"className":9709},[57],[15,9711],{"className":9712,"style":3408},[61],[15,9714,116],{"className":9715,"style":158},[66,67],[15,9717],{"className":9718,"style":173},[71],[15,9720,1187],{"className":9721},[177],[15,9723],{"className":9724,"style":173},[71],[15,9726,9728,9731],{"className":9727},[57],[15,9729],{"className":9730,"style":490},[61],[15,9732,1061],{"className":9733,"style":1083},[66,494],"，把它分解为",[15,9736,9738],{"className":9737,"translate":19},[294],[15,9739,9741,9813],{"className":9740,"translate":19},[18],[15,9742,9744],{"className":9743},[23],[25,9745,9746],{"xmlns":27,"display":303},[29,9747,9748,9810],{},[32,9749,9750,9752,9754,9760,9762,9769,9771,9773,9779,9781,9787,9789,9792,9798,9800,9808],{},[35,9751,116],{},[39,9753,129],{},[1057,9755,9756,9758],{},[35,9757,116],{},[39,9759,1251],{},[39,9761,1245],{},[1057,9763,9764,9766],{},[35,9765,116],{},[39,9767,9768],{},"⊥",[39,9770,335],{"separator":53},[71,9772],{"width":3904},[1057,9774,9775,9777],{},[35,9776,116],{},[39,9778,1251],{},[39,9780,1187],{},[1057,9782,9783,9785],{},[35,9784,1061],{"mathvariant":445},[1209,9786,1804],{},[39,9788,335],{"separator":53},[71,9790],{"width":9791},"1em",[1057,9793,9794,9796],{},[35,9795,116],{},[39,9797,9768],{},[39,9799,1187],{},[1255,9801,9802,9804,9806],{},[35,9803,1061],{"mathvariant":445},[1209,9805,1804],{},[39,9807,9768],{},[35,9809,3573],{"mathvariant":138},[45,9811,9812],{"encoding":47},"f=f_\\parallel+f_\\perp,\\qquad f_\\parallel\\in\\mathcal H_0,\\quad f_\\perp\\in\\mathcal H_0^\\perp.",[15,9814,9816,9834,9892,9997,10102],{"className":9815,"ariaHidden":53},[52],[15,9817,9819,9822,9825,9828,9831],{"className":9818},[57],[15,9820],{"className":9821,"style":3408},[61],[15,9823,116],{"className":9824,"style":158},[66,67],[15,9826],{"className":9827,"style":173},[71],[15,9829,129],{"className":9830},[177],[15,9832],{"className":9833,"style":173},[71],[15,9835,9837,9841,9883,9886,9889],{"className":9836},[57],[15,9838],{"className":9839,"style":9840},[61],"height:1.0496em;vertical-align:-0.3552em;",[15,9842,9844,9847],{"className":9843},[66],[15,9845,116],{"className":9846,"style":158},[66,67],[15,9848,9850],{"className":9849},[198],[15,9851,9853,9874],{"className":9852},[202,1090],[15,9854,9856,9871],{"className":9855},[206],[15,9857,9859],{"className":9858,"style":3668},[210],[15,9860,9862,9865],{"style":9861},"top:-2.5198em;margin-left:-0.1076em;margin-right:0.05em;",[15,9863],{"className":9864,"style":218},[217],[15,9866,9868],{"className":9867},[222,223,224,225],[15,9869,1251],{"className":9870},[177,225],[15,9872,1114],{"className":9873},[1113],[15,9875,9877],{"className":9876},[206],[15,9878,9881],{"className":9879,"style":9880},[210],"height:0.3552em;",[15,9882],{},[15,9884],{"className":9885,"style":72},[71],[15,9887,1245],{"className":9888},[76],[15,9890],{"className":9891,"style":72},[71],[15,9893,9895,9898,9939,9942,9945,9948,9988,9991,9994],{"className":9894},[57],[15,9896],{"className":9897,"style":9840},[61],[15,9899,9901,9904],{"className":9900},[66],[15,9902,116],{"className":9903,"style":158},[66,67],[15,9905,9907],{"className":9906},[198],[15,9908,9910,9931],{"className":9909},[202,1090],[15,9911,9913,9928],{"className":9912},[206],[15,9914,9916],{"className":9915,"style":1097},[210],[15,9917,9919,9922],{"style":9918},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[15,9920],{"className":9921,"style":218},[217],[15,9923,9925],{"className":9924},[222,223,224,225],[15,9926,9768],{"className":9927},[177,225],[15,9929,1114],{"className":9930},[1113],[15,9932,9934],{"className":9933},[206],[15,9935,9937],{"className":9936,"style":1121},[210],[15,9938],{},[15,9940,335],{"className":9941},[422],[15,9943],{"className":9944,"style":4061},[71],[15,9946],{"className":9947,"style":697},[71],[15,9949,9951,9954],{"className":9950},[66],[15,9952,116],{"className":9953,"style":158},[66,67],[15,9955,9957],{"className":9956},[198],[15,9958,9960,9980],{"className":9959},[202,1090],[15,9961,9963,9977],{"className":9962},[206],[15,9964,9966],{"className":9965,"style":3668},[210],[15,9967,9968,9971],{"style":9861},[15,9969],{"className":9970,"style":218},[217],[15,9972,9974],{"className":9973},[222,223,224,225],[15,9975,1251],{"className":9976},[177,225],[15,9978,1114],{"className":9979},[1113],[15,9981,9983],{"className":9982},[206],[15,9984,9986],{"className":9985,"style":9880},[210],[15,9987],{},[15,9989],{"className":9990,"style":173},[71],[15,9992,1187],{"className":9993},[177],[15,9995],{"className":9996,"style":173},[71],[15,9998,10000,10003,10043,10046,10050,10053,10093,10096,10099],{"className":9999},[57],[15,10001],{"className":10002,"style":3408},[61],[15,10004,10006,10009],{"className":10005},[66],[15,10007,1061],{"className":10008,"style":1083},[66,494],[15,10010,10012],{"className":10011},[198],[15,10013,10015,10035],{"className":10014},[202,1090],[15,10016,10018,10032],{"className":10017},[206],[15,10019,10021],{"className":10020,"style":2139},[210],[15,10022,10023,10026],{"style":1100},[15,10024],{"className":10025,"style":218},[217],[15,10027,10029],{"className":10028},[222,223,224,225],[15,10030,1804],{"className":10031},[66,225],[15,10033,1114],{"className":10034},[1113],[15,10036,10038],{"className":10037},[206],[15,10039,10041],{"className":10040,"style":1121},[210],[15,10042],{},[15,10044,335],{"className":10045},[422],[15,10047],{"className":10048,"style":10049},[71],"margin-right:1em;",[15,10051],{"className":10052,"style":697},[71],[15,10054,10056,10059],{"className":10055},[66],[15,10057,116],{"className":10058,"style":158},[66,67],[15,10060,10062],{"className":10061},[198],[15,10063,10065,10085],{"className":10064},[202,1090],[15,10066,10068,10082],{"className":10067},[206],[15,10069,10071],{"className":10070,"style":1097},[210],[15,10072,10073,10076],{"style":9918},[15,10074],{"className":10075,"style":218},[217],[15,10077,10079],{"className":10078},[222,223,224,225],[15,10080,9768],{"className":10081},[177,225],[15,10083,1114],{"className":10084},[1113],[15,10086,10088],{"className":10087},[206],[15,10089,10091],{"className":10090,"style":1121},[210],[15,10092],{},[15,10094],{"className":10095,"style":173},[71],[15,10097,1187],{"className":10098},[177],[15,10100],{"className":10101,"style":173},[71],[15,10103,10105,10109,10161],{"className":10104},[57],[15,10106],{"className":10107,"style":10108},[61],"height:1.1461em;vertical-align:-0.247em;",[15,10110,10112,10115],{"className":10111},[66],[15,10113,1061],{"className":10114,"style":1083},[66,494],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",[15,10168,10170,10188],{"className":10169,"translate":19},[18],[15,10171,10173],{"className":10172},[23],[25,10174,10175],{"xmlns":27},[29,10176,10177,10185],{},[32,10178,10179],{},[1057,10180,10181,10183],{},[35,10182,123],{},[35,10184,1205],{},[45,10186,10187],{"encoding":47},"x_i",[15,10189,10191],{"className":10190,"ariaHidden":53},[52],[15,10192,10194,10198],{"className":10193},[57],[15,10195],{"className":10196,"style":10197},[61],"height:0.5806em;vertical-align:-0.15em;",[15,10199,10201,10204],{"className":10200},[66],[15,10202,123],{"className":10203},[66,67],[15,10205,10207],{"className":10206},[198],[15,10208,10210,10230],{"className":10209},[202,1090],[15,10211,10213,10227],{"className":10212},[206],[15,10214,10216],{"className":10215,"style":1528},[210],[15,10217,10218,10221],{"style":1584},[15,10219],{"className":10220,"style":218},[217],[15,10222,10224],{"className":10223},[222,223,224,225],[15,10225,1205],{"className":10226},[66,67,225],[15,10228,1114],{"className":10229},[1113],[15,10231,10233],{"className":10232},[206],[15,10234,10236],{"className":10235,"style":1121},[210],[15,10237],{},"，因为 ",[15,10240,10242,10278],{"className":10241,"translate":19},[18],[15,10243,10245],{"className":10244},[23],[25,10246,10247],{"xmlns":27},[29,10248,10249,10275],{},[32,10250,10251,10253,10255,10257,10259,10265,10267,10269],{},[35,10252,621],{},[39,10254,120],{"stretchy":119},[39,10256,3268],{},[39,10258,335],{"separator":53},[1057,10260,10261,10263],{},[35,10262,123],{},[35,10264,1205],{},[39,10266,126],{"stretchy":119},[39,10268,1187],{},[1057,10270,10271,10273],{},[35,10272,1061],{"mathvariant":445},[1209,10274,1804],{},[45,10276,10277],{"encoding":47},"k(\\cdot,x_i)\\in\\mathcal H_0",[15,10279,10281,10354],{"className":10280,"ariaHidden":53},[52],[15,10282,10284,10287,10290,10293,10296,10299,10302,10342,10345,10348,10351],{"className":10283},[57],[15,10285],{"className":10286,"style":154},[61],[15,10288,621],{"className":10289,"style":684},[66,67],[15,10291,120],{"className":10292},[162],[15,10294,3268],{"className":10295},[66],[15,10297,335],{"className":10298},[422],[15,10300],{"className":10301,"style":697},[71],[15,10303,10305,10308],{"className":10304},[66],[15,10306,123],{"className":10307},[66,67],[15,10309,10311],{"className":10310},[198],[15,10312,10314,10334],{"className":10313},[202,1090],[15,10315,10317,10331],{"className":10316},[206],[15,10318,10320],{"className":10319,"style":1528},[210],[15,10321,10322,10325],{"style":1584},[15,10323],{"className":10324,"style":218},[217],[15,10326,10328],{"className":10327},[222,223,224,225],[15,10329,1205],{"className":10330},[66,67,225],[15,10332,1114],{"className":10333},[1113],[15,10335,10337],{"className":10336},[206],[15,10338,10340],{"className":10339,"style":1121},[210],[15,10341],{},[15,10343,126],{"className":10344},[169],[15,10346],{"className":10347,"style":173},[71],[15,10349,1187],{"className":10350},[177],[15,10352],{"className":10353,"style":173},[71],[15,10355,10357,10360],{"className":10356},[57],[15,10358],{"className":10359,"style":1076},[61],[15,10361,10363,10366],{"className":10362},[66],[15,10364,1061],{"className":10365,"style":1083},[66,494],[15,10367,10369],{"className":10368},[198],[15,10370,10372,10392],{"className":10371},[202,1090],[15,10373,10375,10389],{"className":10374},[206],[15,10376,10378],{"className":10377,"style":2139},[210],[15,10379,10380,10383],{"style":1100},[15,10381],{"className":10382,"style":218},[217],[15,10384,10386],{"className":10385},[222,223,224,225],[15,10387,1804],{"className":10388},[66,225],[15,10390,1114],{"className":10391},[1113],[15,10393,10395],{"className":10394},[206],[15,10396,10398],{"className":10397,"style":1121},[210],[15,10399],{},"，所以由再生性",[15,10402,10404],{"className":10403,"translate":19},[294],[15,10405,10407,10540],{"className":10406,"translate":19},[18],[15,10408,10410],{"className":10409},[23],[25,10411,10412],{"xmlns":27,"display":303},[29,10413,10414,10537],{},[32,10415,10416,10418,10420,10426,10428,10430,10432,10434,10436,10438,10440,10442,10444,10450,10452,10454,10457,10459,10461,10467,10469,10471,10473,10475,10477,10483,10485,10487,10489,10491,10497,10499,10501,10503,10505,10507,10513,10515,10517,10519,10525,10527,10533,10535],{},[35,10417,116],{},[39,10419,120],{"stretchy":119},[1057,10421,10422,10424],{},[35,10423,123],{},[35,10425,1205],{},[39,10427,126],{"stretchy":119},[39,10429,129],{},[39,10431,642],{"stretchy":119},[35,10433,116],{},[39,10435,335],{"separator":53},[35,10437,621],{},[39,10439,120],{"stretchy":119},[39,10441,3268],{},[39,10443,335],{"separator":53},[1057,10445,10446,10448],{},[35,10447,123],{},[35,10449,1205],{},[39,10451,126],{"stretchy":119},[39,10453,667],{"stretchy":119},[71,10455],{"linebreak":10456},"newline",[39,10458,129],{},[39,10460,642],{"stretchy":119},[1057,10462,10463,10465],{},[35,10464,116],{},[39,10466,1251],{},[39,10468,335],{"separator":53},[35,10470,621],{},[39,10472,120],{"stretchy":119},[39,10474,3268],{},[39,10476,335],{"separator":53},[1057,10478,10479,10481],{},[35,10480,123],{},[35,10482,1205],{},[39,10484,126],{"stretchy":119},[39,10486,667],{"stretchy":119},[39,10488,1245],{},[39,10490,642],{"stretchy":119},[1057,10492,10493,10495],{},[35,10494,116],{},[39,10496,9768],{},[39,10498,335],{"separator":53},[35,10500,621],{},[39,10502,120],{"stretchy":119},[39,10504,3268],{},[39,10506,335],{"separator":53},[1057,10508,10509,10511],{},[35,10510,123],{},[35,10512,1205],{},[39,10514,126],{"stretchy":119},[39,10516,667],{"stretchy":119},[39,10518,129],{},[1057,10520,10521,10523],{},[35,10522,116],{},[39,10524,1251],{},[39,10526,120],{"stretchy":119},[1057,10528,10529,10531],{},[35,10530,123],{},[35,10532,1205],{},[39,10534,126],{"stretchy":119},[35,10536,3573],{"mathvariant":138},[45,10538,10539],{"encoding":47},"f(x_i)=\\langle f,k(\\cdot,x_i)\\rangle \\\\\n=\\langle f_\\parallel,k(\\cdot,x_i)\\rangle+\\langle f_\\perp,k(\\cdot,x_i)\\rangle\n=f_\\parallel(x_i).",[15,10541,10543,10607,10683,10686,10699,10822,10944],{"className":10542,"ariaHidden":53},[52],[15,10544,10546,10549,10552,10555,10595,10598,10601,10604],{"className":10545},[57],[15,10547],{"className":10548,"style":154},[61],[15,10550,116],{"className":10551,"style":158},[66,67],[15,10553,120],{"className":10554},[162],[15,10556,10558,10561],{"className":10557},[66],[15,10559,123],{"className":10560},[66,67],[15,10562,10564],{"className":10563},[198],[15,10565,10567,10587],{"className":10566},[202,1090],[15,10568,10570,10584],{"className":10569},[206],[15,10571,10573],{"className":10572,"style":1528},[210],[15,10574,10575,10578],{"style":1584},[15,10576],{"className":10577,"style":218},[217],[15,10579,10581],{"className":10580},[222,223,224,225],[15,10582,1205],{"className":10583},[66,67,225],[15,10585,1114],{"className":10586},[1113],[15,10588,10590],{"className":10589},[206],[15,10591,10593],{"className":10592,"style":1121},[210],[15,10594],{},[15,10596,126],{"className":10597},[169],[15,10599],{"className":10600,"style":173},[71],[15,10602,129],{"className":10603},[177],[15,10605],{"className":10606,"style":173},[71],[15,10608,10610,10613,10616,10619,10622,10625,10628,10631,10634,10637,10640,10680],{"className":10609},[57],[15,10611],{"className":10612,"style":154},[61],[15,10614,642],{"className":10615},[162],[15,10617,116],{"className":10618,"style":158},[66,67],[15,10620,335],{"className":10621},[422],[15,10623],{"className":10624,"style":697},[71],[15,10626,621],{"className":10627,"style":684},[66,67],[15,10629,120],{"className":10630},[162],[15,10632,3268],{"className":10633},[66],[15,10635,335],{"className":10636},[422],[15,10638],{"className":10639,"style":697},[71],[15,10641,10643,10646],{"className":10642},[66],[15,10644,123],{"className":10645},[66,67],[15,10647,10649],{"className":10648},[198],[15,10650,10652,10672],{"className":10651},[202,1090],[15,10653,10655,10669],{"className":10654},[206],[15,10656,10658],{"className":10657,"style":1528},[210],[15,10659,10660,10663],{"style":1584},[15,10661],{"className":10662,"style":218},[217],[15,10664,10666],{"className":10665},[222,223,224,225],[15,10667,1205],{"className":10668},[66,67,225],[15,10670,1114],{"className":10671},[1113],[15,10673,10675],{"className":10674},[206],[15,10676,10678],{"className":10677,"style":1121},[210],[15,10679],{},[15,10681,811],{"className":10682},[169],[15,10684],{"className":10685},[71,10456],[15,10687,10689,10693,10696],{"className":10688},[57],[15,10690],{"className":10691,"style":10692},[61],"height:0.3669em;",[15,10694,129],{"className":10695},[177],[15,10697],{"className":10698,"style":173},[71],[15,10700,10702,10706,10709,10749,10752,10755,10758,10761,10764,10767,10770,10810,10813,10816,10819],{"className":10701},[57],[15,10703],{"className":10704,"style":10705},[61],"height:1.1052em;vertical-align:-0.3552em;",[15,10707,642],{"className":10708},[162],[15,10710,10712,10715],{"className":10711},[66],[15,10713,116],{"className":10714,"style":158},[66,67],[15,10716,10718],{"className":10717},[198],[15,10719,10721,10741],{"className":10720},[202,1090],[15,10722,10724,10738],{"className":10723},[206],[15,10725,10727],{"className":10726,"style":3668},[210],[15,10728,10729,10732],{"style":9861},[15,10730],{"className":10731,"style":218},[217],[15,10733,10735],{"className":10734},[222,223,224,225],[15,10736,1251],{"className":10737},[177,225],[15,10739,1114],{"className":10740},[1113],[15,10742,10744],{"className":10743},[206],[15,10745,10747],{"className":10746,"style":9880},[210],[15,10748],{},[15,10750,335],{"className":10751},[422],[15,10753],{"className":10754,"style":697},[71],[15,10756,621],{"className":10757,"style":684},[66,67],[15,10759,120],{"className":10760},[162],[15,10762,3268],{"className":10763},[66],[15,10765,335],{"className":10766},[422],[15,10768],{"className":10769,"style":697},[71],[15,10771,10773,10776],{"className":10772},[66],[15,10774,123],{"className":10775},[66,67],[15,10777,10779],{"className":10778},[198],[15,10780,10782,10802],{"className":10781},[202,1090],[15,10783,10785,10799],{"className":10784},[206],[15,10786,10788],{"className":10787,"style":1528},[210],[15,10789,10790,10793],{"style":1584},[15,10791],{"className":10792,"style":218},[217],[15,10794,10796],{"className":10795},[222,223,224,225],[15,10797,1205],{"className":10798},[66,67,225],[15,10800,1114],{"className":10801},[1113],[15,10803,10805],{"className":10804},[206],[15,10806,10808],{"className":10807,"style":1121},[210],[15,10809],{},[15,10811,811],{"className":10812},[169],[15,10814],{"className":10815,"style":72},[71],[15,10817,1245],{"className":10818},[76],[15,10820],{"className":10821,"style":72},[71],[15,10823,10825,10828,10831,10871,10874,10877,10880,10883,10886,10889,10892,10932,10935,10938,10941],{"className":10824},[57],[15,10826],{"className":10827,"style":154},[61],[15,10829,642],{"className":10830},[162],[15,10832,10834,10837],{"className":10833},[66],[15,10835,116],{"className":10836,"style":158},[66,67],[15,10838,10840],{"className":10839},[198],[15,10841,10843,10863],{"className":10842},[202,1090],[15,10844,10846,10860],{"className":10845},[206],[15,10847,10849],{"className":10848,"style":1097},[210],[15,10850,10851,10854],{"style":9918},[15,10852],{"className":10853,"style":218},[217],[15,10855,10857],{"className":10856},[222,223,224,225],[15,10858,9768],{"className":10859},[177,225],[15,10861,1114],{"className":10862},[1113],[15,10864,10866],{"className":10865},[206],[15,10867,10869],{"className":10868,"style":1121},[210],[15,10870],{},[15,10872,335],{"className":10873},[422],[15,10875],{"className":10876,"style":697},[71],[15,10878,621],{"className":10879,"style":684},[66,67],[15,10881,120],{"className":10882},[162],[15,10884,3268],{"className":10885},[66],[15,10887,335],{"className":10888},[422],[15,10890],{"className":10891,"style":697},[71],[15,10893,10895,10898],{"className":10894},[66],[15,10896,123],{"className":10897},[66,67],[15,10899,10901],{"className":10900},[198],[15,10902,10904,10924],{"className":10903},[202,1090],[15,10905,10907,10921],{"className":10906},[206],[15,10908,10910],{"className":10909,"style":1528},[210],[15,10911,10912,10915],{"style":1584},[15,10913],{"className":10914,"style":218},[217],[15,10916,10918],{"className":10917},[222,223,224,225],[15,10919,1205],{"className":10920},[66,67,225],[15,10922,1114],{"className":10923},[1113],[15,10925,10927],{"className":10926},[206],[15,10928,10930],{"className":10929,"style":1121},[210],[15,10931],{},[15,10933,811],{"className":10934},[169],[15,10936],{"className":10937,"style":173},[71],[15,10939,129],{"className":10940},[177],[15,10942],{"className":10943,"style":173},[71],[15,10945,10947,10950,10990,10993,11033,11036],{"className":10946},[57],[15,10948],{"className":10949,"style":10705},[61],[15,10951,10953,10956],{"className":10952},[66],[15,10954,116],{"className":10955,"style":158},[66,67],[15,10957,10959],{"className":10958},[198],[15,10960,10962,10982],{"className":10961},[202,1090],[15,10963,10965,10979],{"className":10964},[206],[15,10966,10968],{"className":10967,"style":3668},[210],[15,10969,10970,10973],{"style":9861},[15,10971],{"className":10972,"style":218},[217],[15,10974,10976],{"className":10975},[222,223,224,225],[15,10977,1251],{"className":10978},[177,225],[15,10980,1114],{"className":10981},[1113],[15,10983,10985],{"className":10984},[206],[15,10986,10988],{"className":10987,"style":9880},[210],[15,10989],{},[15,10991,120],{"className":10992},[162],[15,10994,10996,10999],{"className":10995},[66],[15,10997,123],{"className":10998},[66,67],[15,11000,11002],{"className":11001},[198],[15,11003,11005,11025],{"className":11004},[202,1090],[15,11006,11008,11022],{"className":11007},[206],[15,11009,11011],{"className":11010,"style":1528},[210],[15,11012,11013,11016],{"style":1584},[15,11014],{"className":11015,"style":218},[217],[15,11017,11019],{"className":11018},[222,223,224,225],[15,11020,1205],{"className":11021},[66,67,225],[15,11023,1114],{"className":11024},[1113],[15,11026,11028],{"className":11027},[206],[15,11029,11031],{"className":11030,"style":1121},[210],[15,11032],{},[15,11034,126],{"className":11035},[169],[15,11037,3573],{"className":11038},[66],[11,11040,11041,11042,11115],{},"也就是说，",[2040,11043,11044,11114],{},[15,11045,11047,11065],{"className":11046,"translate":19},[18],[15,11048,11050],{"className":11049},[23],[25,11051,11052],{"xmlns":27},[29,11053,11054,11062],{},[32,11055,11056],{},[1057,11057,11058,11060],{},[35,11059,116],{},[39,11061,9768],{},[45,11063,11064],{"encoding":47},"f_\\perp",[15,11066,11068],{"className":11067,"ariaHidden":53},[52],[15,11069,11071,11074],{"className":11070},[57],[15,11072],{"className":11073,"style":3408},[61],[15,11075,11077,11080],{"className":11076},[66],[15,11078,116],{"className":11079,"style":158},[66,67],[15,11081,11083],{"className":11082},[198],[15,11084,11086,11106],{"className":11085},[202,1090],[15,11087,11089,11103],{"className":11088},[206],[15,11090,11092],{"className":11091,"style":1097},[210],[15,11093,11094,11097],{"style":9918},[15,11095],{"className":11096,"style":218},[217],[15,11098,11100],{"className":11099},[222,223,224,225],[15,11101,9768],{"className":11102},[177,225],[15,11104,1114],{"className":11105},[1113],[15,11107,11109],{"className":11108},[206],[15,11110,11112],{"className":11111,"style":1121},[210],[15,11113],{}," 在所有训练点上的取值都是零","。它对损失函数没有任何贡献。",[11,11117,11118],{},"再看正则项。由勾股定理，",[15,11120,11122],{"className":11121,"translate":19},[294],[15,11123,11125,11205],{"className":11124,"translate":19},[18],[15,11126,11128],{"className":11127},[23],[25,11129,11130],{"xmlns":27,"display":303},[29,11131,11132,11202],{},[32,11133,11134,11136,11138,11146,11148,11150,11156,11164,11166,11168,11174,11182,11184,11186,11192,11200],{},[35,11135,1251],{"mathvariant":138},[35,11137,116],{},[1255,11139,11140,11142,11144],{},[35,11141,1251],{"mathvariant":138},[35,11143,1061],{"mathvariant":445},[1209,11145,1263],{},[39,11147,129],{},[35,11149,1251],{"mathvariant":138},[1057,11151,11152,11154],{},[35,11153,116],{},[39,11155,1251],{},[1255,11157,11158,11160,11162],{},[35,11159,1251],{"mathvariant":138},[35,11161,1061],{"mathvariant":445},[1209,11163,1263],{},[39,11165,1245],{},[35,11167,1251],{"mathvariant":138},[1057,11169,11170,11172],{},[35,11171,116],{},[39,11173,9768],{},[1255,11175,11176,11178,11180],{},[35,11177,1251],{"mathvariant":138},[35,11179,1061],{"mathvariant":445},[1209,11181,1263],{},[39,11183,1801],{},[35,11185,1251],{"mathvariant":138},[1057,11187,11188,11190],{},[35,11189,116],{},[39,11191,1251],{},[1255,11193,11194,11196,11198],{},[35,11195,1251],{"mathvariant":138},[35,11197,1061],{"mathvariant":445},[1209,11199,1263],{},[35,11201,3573],{"mathvariant":138},[45,11203,11204],{"encoding":47},"\\|f\\|_{\\mathcal H}^2=\\|f_\\parallel\\|_{\\mathcal H}^2+\\|f_\\perp\\|_{\\mathcal H}^2\n\\ge \\|f_\\parallel\\|_{\\mathcal H}^2.",[15,11206,11208,11284,11397,11509],{"className":11207,"ariaHidden":53},[52],[15,11209,11211,11215,11218,11221,11275,11278,11281],{"className":11210},[57],[15,11212],{"className":11213,"style":11214},[61],"height:1.1141em;vertical-align:-0.25em;",[15,11216,1251],{"className":11217},[66],[15,11219,116],{"className":11220,"style":158},[66,67],[15,11222,11224,11227],{"className":11223},[66],[15,11225,1251],{"className":11226},[66],[15,11228,11230],{"className":11229},[198],[15,11231,11233,11267],{"className":11232},[202,1090],[15,11234,11236,11264],{"className":11235},[206],[15,11237,11239,11253],{"className":11238,"style":1650},[210],[15,11240,11241,11244],{"style":1653},[15,11242],{"className":11243,"style":218},[217],[15,11245,11247],{"className":11246},[222,223,224,225],[15,11248,11250],{"className":11249},[66,225],[15,11251,1061],{"className":11252,"style":1083},[66,494,225],[15,11254,11255,11258],{"style":397},[15,11256],{"className":11257,"style":218},[217],[15,11259,11261],{"className":11260},[222,223,224,225],[15,11262,1263],{"className":11263},[66,225],[15,11265,1114],{"className":11266},[1113],[15,11268,11270],{"className":11269},[206],[15,11271,11273],{"className":11272,"style":1686},[210],[15,11274],{},[15,11276],{"className":11277,"style":173},[71],[15,11279,129],{"className":11280},[177],[15,11282],{"className":11283,"style":173},[71],[15,11285,11287,11291,11294,11334,11388,11391,11394],{"className":11286},[57],[15,11288],{"className":11289,"style":11290},[61],"height:1.2193em;vertical-align:-0.3552em;",[15,11292,1251],{"className":11293},[66],[15,11295,11297,11300],{"className":11296},[66],[15,11298,116],{"className":11299,"style":158},[66,67],[15,11301,11303],{"className":11302},[198],[15,11304,11306,11326],{"className":11305},[202,1090],[15,11307,11309,11323],{"className":11308},[206],[15,11310,11312],{"className":11311,"style":3668},[210],[15,11313,11314,11317],{"style":9861},[15,11315],{"className":11316,"style":218},[217],[15,11318,11320],{"className":11319},[222,223,224,225],[15,11321,1251],{"className":11322},[177,225],[15,11324,1114],{"className":11325},[1113],[15,11327,11329],{"className":11328},[206],[15,11330,11332],{"className":11331,"style":9880},[210],[15,11333],{},[15,11335,11337,11340],{"className":11336},[66],[15,11338,1251],{"className":11339},[66],[15,11341,11343],{"className":11342},[198],[15,11344,11346,11380],{"className":11345},[202,1090],[15,11347,11349,11377],{"className":11348},[206],[15,11350,11352,11366],{"className":11351,"style":1650},[210],[15,11353,11354,11357],{"style":1653},[15,11355],{"className":11356,"style":218},[217],[15,11358,11360],{"className":11359},[222,223,224,225],[15,11361,11363],{"className":11362},[66,225],[15,11364,1061],{"className":11365,"style":1083},[66,494,225],[15,11367,11368,11371],{"style":397},[15,11369],{"className":11370,"style":218},[217],[15,11372,11374],{"className":11373},[222,223,224,225],[15,11375,1263],{"className":11376},[66,225],[15,11378,1114],{"className":11379},[1113],[15,11381,11383],{"className":11382},[206],[15,11384,11386],{"className":11385,"style":1686},[210],[15,11387],{},[15,11389],{"className":11390,"style":72},[71],[15,11392,1245],{"className":11393},[76],[15,11395],{"className":11396,"style":72},[71],[15,11398,11400,11403,11406,11446,11500,11503,11506],{"className":11399},[57],[15,11401],{"className":11402,"style":11214},[61],[15,11404,1251],{"className":11405},[66],[15,11407,11409,11412],{"className":11408},[66],[15,11410,116],{"className":11411,"style":158},[66,67],[15,11413,11415],{"className":11414},[198],[15,11416,11418,11438],{"className":11417},[202,1090],[15,11419,11421,11435],{"className":11420},[206],[15,11422,11424],{"className":11423,"style":1097},[210],[15,11425,11426,11429],{"style":9918},[15,11427],{"className":11428,"style":218},[217],[15,11430,11432],{"className":11431},[222,223,224,225],[15,11433,9768],{"className":11434},[177,225],[15,11436,1114],{"className":11437},[1113],[15,11439,11441],{"className":11440},[206],[15,11442,11444],{"className":11443,"style":1121},[210],[15,11445],{},[15,11447,11449,11452],{"className":11448},[66],[15,11450,1251],{"className":11451},[66],[15,11453,11455],{"className":11454},[198],[15,11456,11458,11492],{"className":11457},[202,1090],[15,11459,11461,11489],{"className":11460},[206],[15,11462,11464,11478],{"className":11463,"style":1650},[210],[15,11465,11466,11469],{"style":1653},[15,11467],{"className":11468,"style":218},[217],[15,11470,11472],{"className":11471},[222,223,224,225],[15,11473,11475],{"className":11474},[66,225],[15,11476,1061],{"className":11477,"style":1083},[66,494,225],[15,11479,11480,11483],{"style":397},[15,11481],{"className":11482,"style":218},[217],[15,11484,11486],{"className":11485},[222,223,224,225],[15,11487,1263],{"className":11488},[66,225],[15,11490,1114],{"className":11491},[1113],[15,11493,11495],{"className":11494},[206],[15,11496,11498],{"className":11497,"style":1686},[210],[15,11499],{},[15,11501],{"className":11502,"style":173},[71],[15,11504,1801],{"className":11505},[177],[15,11507],{"className":11508,"style":173},[71],[15,11510,11512,11515,11518,11558,11612],{"className":11511},[57],[15,11513],{"className":11514,"style":11290},[61],[15,11516,1251],{"className":11517},[66],[15,11519,11521,11524],{"className":11520},[66],[15,11522,116],{"className":11523,"style":158},[66,67],[15,11525,11527],{"className":11526},[198],[15,11528,11530,11550],{"className":11529},[202,1090],[15,11531,11533,11547],{"className":11532},[206],[15,11534,11536],{"className":11535,"style":3668},[210],[15,11537,11538,11541],{"style":9861},[15,11539],{"className":11540,"style":218},[217],[15,11542,11544],{"className":11543},[222,223,224,225],[15,11545,1251],{"className":11546},[177,225],[15,11548,1114],{"className":11549},[1113],[15,11551,11553],{"className":11552},[206],[15,11554,11556],{"className":11555,"style":9880},[210],[15,11557],{},[15,11559,11561,11564],{"className":11560},[66],[15,11562,1251],{"className":11563},[66],[15,11565,11567],{"className":11566},[198],[15,11568,11570,11604],{"className":11569},[202,1090],[15,11571,11573,11601],{"className":11572},[206],[15,11574,11576,11590],{"className":11575,"style":1650},[210],[15,11577,11578,11581],{"style":1653},[15,11579],{"className":11580,"style":218},[217],[15,11582,11584],{"className":11583},[222,223,224,225],[15,11585,11587],{"className":11586},[66,225],[15,11588,1061],{"className":11589,"style":1083},[66,494,225],[15,11591,11592,11595],{"style":397},[15,11593],{"className":11594,"style":218},[217],[15,11596,11598],{"className":11597},[222,223,224,225],[15,11599,1263],{"className":11600},[66,225],[15,11602,1114],{"className":11603},[1113],[15,11605,11607],{"className":11606},[206],[15,11608,11610],{"className":11609,"style":1686},[210],[15,11611],{},[15,11613,3573],{"className":11614},[66],[11,11616,11617,11618,11646],{},"因为 ",[15,11619,11621,11634],{"className":11620,"translate":19},[18],[15,11622,11624],{"className":11623},[23],[25,11625,11626],{"xmlns":27},[29,11627,11628,11632],{},[32,11629,11630],{},[35,11631,7742],{"mathvariant":138},[45,11633,8704],{"encoding":47},[15,11635,11637],{"className":11636,"ariaHidden":53},[52],[15,11638,11640,11643],{"className":11639},[57],[15,11641],{"className":11642,"style":490},[61],[15,11644,7742],{"className":11645},[66]," 非减，",[15,11648,11650],{"className":11649,"translate":19},[294],[15,11651,11653,11709],{"className":11652,"translate":19},[18],[15,11654,11656],{"className":11655},[23],[25,11657,11658],{"xmlns":27,"display":303},[29,11659,11660,11706],{},[32,11661,11662,11664,11666,11668,11670,11678,11680,11682,11684,11686,11688,11694,11702,11704],{},[35,11663,7742],{"mathvariant":138},[39,11665,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,11667,1251],{"mathvariant":138},[35,11669,116],{},[1255,11671,11672,11674,11676],{},[35,11673,1251],{"mathvariant":138},[35,11675,1061],{"mathvariant":445},[1209,11677,1263],{},[39,11679,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,11681,1801],{},[35,11683,7742],{"mathvariant":138},[39,11685,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,11687,1251],{"mathvariant":138},[1057,11689,11690,11692],{},[35,11691,116],{},[39,11693,1251],{},[1255,11695,11696,11698,11700],{},[35,11697,1251],{"mathvariant":138},[35,11699,1061],{"mathvariant":445},[1209,11701,1263],{},[39,11703,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,11705,3573],{"mathvariant":138},[45,11707,11708],{"encoding":47},"\\Omega\\bigl(\\|f\\|_{\\mathcal H}^2\\bigr)\\ge  \\Omega\\bigl(\\|f_\\parallel\\|_{\\mathcal H}^2\\bigr).",[15,11710,11712,11803],{"className":11711,"ariaHidden":53},[52],[15,11713,11715,11719,11722,11728,11731,11734,11788,11794,11797,11800],{"className":11714},[57],[15,11716],{"className":11717,"style":11718},[61],"height:1.2141em;vertical-align:-0.35em;",[15,11720,7742],{"className":11721},[66],[15,11723,11725],{"className":11724},[162],[15,11726,120],{"className":11727},[1422,1508],[15,11729,1251],{"className":11730},[66],[15,11732,116],{"className":11733,"style":158},[66,67],[15,11735,11737,11740],{"className":11736},[66],[15,11738,1251],{"className":11739},[66],[15,11741,11743],{"className":11742},[198],[15,11744,11746,11780],{"className":11745},[202,1090],[15,11747,11749,11777],{"className":11748},[206],[15,11750,11752,11766],{"className":11751,"style":1650},[210],[15,11753,11754,11757],{"style":1653},[15,11755],{"className":11756,"style":218},[217],[15,11758,11760],{"className":11759},[222,223,224,225],[15,11761,11763],{"className":11762},[66,225],[15,11764,1061],{"className":11765,"style":1083},[66,494,225],[15,11767,11768,11771],{"style":397},[15,11769],{"className":11770,"style":218},[217],[15,11772,11774],{"className":11773},[222,223,224,225],[15,11775,1263],{"className":11776},[66,225],[15,11778,1114],{"className":11779},[1113],[15,11781,11783],{"className":11782},[206],[15,11784,11786],{"className":11785,"style":1686},[210],[15,11787],{},[15,11789,11791],{"className":11790},[169],[15,11792,126],{"className":11793},[1422,1508],[15,11795],{"className":11796,"style":173},[71],[15,11798,1801],{"className":11799},[177],[15,11801],{"className":11802,"style":173},[71],[15,11804,11806,11809,11812,11818,11821,11861,11915,11921],{"className":11805},[57],[15,11807],{"className":11808,"style":11290},[61],[15,11810,7742],{"className":11811},[66],[15,11813,11815],{"className":11814},[162],[15,11816,120],{"className":11817},[1422,1508],[15,11819,1251],{"className":11820},[66],[15,11822,11824,11827],{"className":11823},[66],[15,11825,116],{"className":11826,"style":158},[66,67],[15,11828,11830],{"className":11829},[198],[15,11831,11833,11853],{"className":11832},[202,1090],[15,11834,11836,11850],{"className":11835},[206],[15,11837,11839],{"className":11838,"style":3668},[210],[15,11840,11841,11844],{"style":9861},[15,11842],{"className":11843,"style":218},[217],[15,11845,11847],{"className":11846},[222,223,224,225],[15,11848,1251],{"className":11849},[177,225],[15,11851,1114],{"className":11852},[1113],[15,11854,11856],{"className":11855},[206],[15,11857,11859],{"className":11858,"style":9880},[210],[15,11860],{},[15,11862,11864,11867],{"className":11863},[66],[15,11865,1251],{"className":11866},[66],[15,11868,11870],{"className":11869},[198],[15,11871,11873,11907],{"className":11872},[202,1090],[15,11874,11876,11904],{"className":11875},[206],[15,11877,11879,11893],{"className":11878,"style":1650},[210],[15,11880,11881,11884],{"style":1653},[15,11882],{"className":11883,"style":218},[217],[15,11885,11887],{"className":11886},[222,223,224,225],[15,11888,11890],{"className":11889},[66,225],[15,11891,1061],{"className":11892,"style":1083},[66,494,225],[15,11894,11895,11898],{"style":397},[15,11896],{"className":11897,"style":218},[217],[15,11899,11901],{"className":11900},[222,223,224,225],[15,11902,1263],{"className":11903},[66,225],[15,11905,1114],{"className":11906},[1113],[15,11908,11910],{"className":11909},[206],[15,11911,11913],{"className":11912,"style":1686},[210],[15,11914],{},[15,11916,11918],{"className":11917},[169],[15,11919,126],{"className":11920},[1422,1508],[15,11922,3573],{"className":11923},[66],[11,11925,11926],{},"把两件事合起来：",[15,11928,11930],{"className":11929,"translate":19},[294],[15,11931,11933,12071],{"className":11932,"translate":19},[18],[15,11934,11936],{"className":11935},[23],[25,11937,11938],{"xmlns":27,"display":303},[29,11939,11940,12068],{},[32,11941,11942,11944,11946,11948,11950,11956,11958,11960,11962,11964,11966,11968,11974,11976,11978,11980,11982,11984,11986,11988,11994,11996,11998,12000,12002,12004,12010,12012,12018,12020,12022,12024,12026,12032,12034,12040,12042,12044,12046,12048,12050,12052,12058,12064,12066],{},[35,11943,1216],{},[39,11945,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,11947,116],{},[39,11949,120],{"stretchy":119},[1057,11951,11952,11954],{},[35,11953,123],{},[1209,11955,1211],{},[39,11957,126],{"stretchy":119},[39,11959,335],{"separator":53},[39,11961,2095],{},[39,11963,335],{"separator":53},[35,11965,116],{},[39,11967,120],{"stretchy":119},[1057,11969,11970,11972],{},[35,11971,123],{},[35,11973,37],{},[39,11975,126],{"stretchy":119},[39,11977,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,11979,1245],{},[35,11981,7742],{"mathvariant":138},[39,11983,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,11985,1251],{"mathvariant":138},[35,11987,116],{},[131,11989,11990,11992],{},[35,11991,1251],{"mathvariant":138},[1209,11993,1263],{},[39,11995,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,11997,1801],{},[71,11999],{"linebreak":10456},[35,12001,1216],{},[39,12003,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[1057,12005,12006,12008],{},[35,12007,116],{},[39,12009,1251],{},[39,12011,120],{"stretchy":119},[1057,12013,12014,12016],{},[35,12015,123],{},[1209,12017,1211],{},[39,12019,126],{"stretchy":119},[39,12021,335],{"separator":53},[39,12023,2095],{},[39,12025,335],{"separator":53},[1057,12027,12028,12030],{},[35,12029,116],{},[39,12031,1251],{},[39,12033,120],{"stretchy":119},[1057,12035,12036,12038],{},[35,12037,123],{},[35,12039,37],{},[39,12041,126],{"stretchy":119},[39,12043,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,12045,1245],{},[35,12047,7742],{"mathvariant":138},[39,12049,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,12051,1251],{"mathvariant":138},[1057,12053,12054,12056],{},[35,12055,116],{},[39,12057,1251],{},[131,12059,12060,12062],{},[35,12061,1251],{"mathvariant":138},[1209,12063,1263],{},[39,12065,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,12067,3573],{"mathvariant":138},[45,12069,12070],{"encoding":47},"L\\bigl(f(x_1),\\dots,f(x_n)\\bigr)+\\Omega\\bigl(\\|f\\|^2\\bigr)\n\\ge \n \\\\ \nL\\bigl(f_\\parallel(x_1),\\dots,f_\\parallel(x_n)\\bigr)+\\Omega\\bigl(\\|f_\\parallel\\|^2\\bigr).",[15,12072,12074,12220,12282,12285,12506],{"className":12073,"ariaHidden":53},[52],[15,12075,12077,12080,12083,12089,12092,12095,12135,12138,12141,12144,12147,12150,12153,12156,12159,12162,12202,12205,12211,12214,12217],{"className":12076},[57],[15,12078],{"className":12079,"style":4400},[61],[15,12081,1216],{"className":12082},[66,67],[15,12084,12086],{"className":12085},[162],[15,12087,120],{"className":12088},[1422,1508],[15,12090,116],{"className":12091,"style":158},[66,67],[15,12093,120],{"className":12094},[162],[15,12096,12098,12101],{"className":12097},[66],[15,12099,123],{"className":12100},[66,67],[15,12102,12104],{"className":12103},[198],[15,12105,12107,12127],{"className":12106},[202,1090],[15,12108,12110,12124],{"className":12109},[206],[15,12111,12113],{"className":12112,"style":2139},[210],[15,12114,12115,12118],{"style":1584},[15,12116],{"className":12117,"style":218},[217],[15,12119,12121],{"className":12120},[222,223,224,225],[15,12122,1211],{"className":12123},[66,225],[15,12125,1114],{"className":12126},[1113],[15,12128,12130],{"className":12129},[206],[15,12131,12133],{"className":12132,"style":1121},[210],[15,12134],{},[15,12136,126],{"className":12137},[169],[15,12139,335],{"className":12140},[422],[15,12142],{"className":12143,"style":697},[71],[15,12145,2095],{"className":12146},[1413],[15,12148],{"className":12149,"style":697},[71],[15,12151,335],{"className":12152},[422],[15,12154],{"className":12155,"style":697},[71],[15,12157,116],{"className":12158,"style":158},[66,67],[15,12160,120],{"className":12161},[162],[15,12163,12165,12168],{"className":12164},[66],[15,12166,123],{"className":12167},[66,67],[15,12169,12171],{"className":12170},[198],[15,12172,12174,12194],{"className":12173},[202,1090],[15,12175,12177,12191],{"className":12176},[206],[15,12178,12180],{"className":12179,"style":2198},[210],[15,12181,12182,12185],{"style":1584},[15,12183],{"className":12184,"style":218},[217],[15,12186,12188],{"className":12187},[222,223,224,225],[15,12189,37],{"className":12190},[66,67,225],[15,12192,1114],{"className":12193},[1113],[15,12195,12197],{"className":12196},[206],[15,12198,12200],{"className":12199,"style":1121},[210],[15,12201],{},[15,12203,126],{"className":12204},[169],[15,12206,12208],{"className":12207},[169],[15,12209,126],{"className":12210},[1422,1508],[15,12212],{"className":12213,"style":72},[71],[15,12215,1245],{"className":12216},[76],[15,12218],{"className":12219,"style":72},[71],[15,12221,12223,12226,12229,12235,12238,12241,12270,12276,12279],{"className":12222},[57],[15,12224],{"className":12225,"style":11718},[61],[15,12227,7742],{"className":12228},[66],[15,12230,12232],{"className":12231},[162],[15,12233,120],{"className":12234},[1422,1508],[15,12236,1251],{"className":12237},[66],[15,12239,116],{"className":12240,"style":158},[66,67],[15,12242,12244,12247],{"className":12243},[66],[15,12245,1251],{"className":12246},[66],[15,12248,12250],{"className":12249},[198],[15,12251,12253],{"className":12252},[202],[15,12254,12256],{"className":12255},[206],[15,12257,12259],{"className":12258,"style":1650},[210],[15,12260,12261,12264],{"style":397},[15,12262],{"className":12263,"style":218},[217],[15,12265,12267],{"className":12266},[222,223,224,225],[15,12268,1263],{"className":12269},[66,225],[15,12271,12273],{"className":12272},[169],[15,12274,126],{"className":12275},[1422,1508],[15,12277],{"className":12278,"style":173},[71],[15,12280,1801],{"className":12281},[177],[15,12283],{"className":12284},[71,10456],[15,12286,12288,12292,12295,12301,12341,12344,12384,12387,12390,12393,12396,12399,12402,12405,12445,12448,12488,12491,12497,12500,12503],{"className":12287},[57],[15,12289],{"className":12290,"style":12291},[61],"height:1.2052em;vertical-align:-0.3552em;",[15,12293,1216],{"className":12294},[66,67],[15,12296,12298],{"className":12297},[162],[15,12299,120],{"className":12300},[1422,1508],[15,12302,12304,12307],{"className":12303},[66],[15,12305,116],{"className":12306,"style":158},[66,67],[15,12308,12310],{"className":12309},[198],[15,12311,12313,12333],{"className":12312},[202,1090],[15,12314,12316,12330],{"className":12315},[206],[15,12317,12319],{"className":12318,"style":3668},[210],[15,12320,12321,12324],{"style":9861},[15,12322],{"className":12323,"style":218},[217],[15,12325,12327],{"className":12326},[222,223,224,225],[15,12328,1251],{"className":12329},[177,225],[15,12331,1114],{"className":12332},[1113],[15,12334,12336],{"className":12335},[206],[15,12337,12339],{"className":12338,"style":9880},[210],[15,12340],{},[15,12342,120],{"className":12343},[162],[15,12345,12347,12350],{"className":12346},[66],[15,12348,123],{"className":12349},[66,67],[15,12351,12353],{"className":12352},[198],[15,12354,12356,12376],{"className":12355},[202,1090],[15,12357,12359,12373],{"className":12358},[206],[15,12360,12362],{"className":12361,"style":2139},[210],[15,12363,12364,12367],{"style":1584},[15,12365],{"className":12366,"style":218},[217],[15,12368,12370],{"className":12369},[222,223,224,225],[15,12371,1211],{"className":12372},[66,225],[15,12374,1114],{"className":12375},[1113],[15,12377,12379],{"className":12378},[206],[15,12380,12382],{"className":12381,"style":1121},[210],[15,12383],{},[15,12385,126],{"className":12386},[169],[15,12388,335],{"className":12389},[422],[15,12391],{"className":12392,"style":697},[71],[15,12394,2095],{"className":12395},[1413],[15,12397],{"className":12398,"style":697},[71],[15,12400,335],{"className":12401},[422],[15,12403],{"className":12404,"style":697},[71],[15,12406,12408,12411],{"className":12407},[66],[15,12409,116],{"className":12410,"style":158},[66,67],[15,12412,12414],{"className":12413},[198],[15,12415,12417,12437],{"className":12416},[202,1090],[15,12418,12420,12434],{"className":12419},[206],[15,12421,12423],{"className":12422,"style":3668},[210],[15,12424,12425,12428],{"style":9861},[15,12426],{"className":12427,"style":218},[217],[15,12429,12431],{"className":12430},[222,223,224,225],[15,12432,1251],{"className":12433},[177,225],[15,12435,1114],{"className":12436},[1113],[15,12438,12440],{"className":12439},[206],[15,12441,12443],{"className":12442,"style":9880},[210],[15,12444],{},[15,12446,120],{"className":12447},[162],[15,12449,12451,12454],{"className":12450},[66],[15,12452,123],{"className":12453},[66,67],[15,12455,12457],{"className":12456},[198],[15,12458,12460,12480],{"className":12459},[202,1090],[15,12461,12463,12477],{"className":12462},[206],[15,12464,12466],{"className":12465,"style":2198},[210],[15,12467,12468,12471],{"style":1584},[15,12469],{"className":12470,"style":218},[217],[15,12472,12474],{"className":12473},[222,223,224,225],[15,12475,37],{"className":12476},[66,67,225],[15,12478,1114],{"className":12479},[1113],[15,12481,12483],{"className":12482},[206],[15,12484,12486],{"className":12485,"style":1121},[210],[15,12487],{},[15,12489,126],{"className":12490},[169],[15,12492,12494],{"className":12493},[169],[15,12495,126],{"className":12496},[1422,1508],[15,12498],{"className":12499,"style":72},[71],[15,12501,1245],{"className":12502},[76],[15,12504],{"className":12505,"style":72},[71],[15,12507,12509,12512,12515,12521,12524,12564,12593,12599],{"className":12508},[57],[15,12510],{"className":12511,"style":11290},[61],[15,12513,7742],{"className":12514},[66],[15,12516,12518],{"className":12517},[162],[15,12519,120],{"className":12520},[1422,1508],[15,12522,1251],{"className":12523},[66],[15,12525,12527,12530],{"className":12526},[66],[15,12528,116],{"className":12529,"style":158},[66,67],[15,12531,12533],{"className":12532},[198],[15,12534,12536,12556],{"className":12535},[202,1090],[15,12537,12539,12553],{"className":12538},[206],[15,12540,12542],{"className":12541,"style":3668},[210],[15,12543,12544,12547],{"style":9861},[15,12545],{"className":12546,"style":218},[217],[15,12548,12550],{"className":12549},[222,223,224,225],[15,12551,1251],{"className":12552},[177,225],[15,12554,1114],{"className":12555},[1113],[15,12557,12559],{"className":12558},[206],[15,12560,12562],{"className":12561,"style":9880},[210],[15,12563],{},[15,12565,12567,12570],{"className":12566},[66],[15,12568,1251],{"className":12569},[66],[15,12571,12573],{"className":12572},[198],[15,12574,12576],{"className":12575},[202],[15,12577,12579],{"className":12578},[206],[15,12580,12582],{"className":12581,"style":1650},[210],[15,12583,12584,12587],{"style":397},[15,12585],{"className":12586,"style":218},[217],[15,12588,12590],{"className":12589},[222,223,224,225],[15,12591,1263],{"className":12592},[66,225],[15,12594,12596],{"className":12595},[169],[15,12597,126],{"className":12598},[1422,1508],[15,12600,3573],{"className":12601},[66],[11,12603,12604,12605,12633,12634,12704,12705,12733,12734,12762,12763,12832,12833,12966],{},"所以，对任意 ",[15,12606,12608,12621],{"className":12607,"translate":19},[18],[15,12609,12611],{"className":12610},[23],[25,12612,12613],{"xmlns":27},[29,12614,12615,12619],{},[32,12616,12617],{},[35,12618,116],{},[45,12620,116],{"encoding":47},[15,12622,12624],{"className":12623,"ariaHidden":53},[52],[15,12625,12627,12630],{"className":12626},[57],[15,12628],{"className":12629,"style":3408},[61],[15,12631,116],{"className":12632,"style":158},[66,67],"，它的投影 ",[15,12635,12637,12655],{"className":12636,"translate":19},[18],[15,12638,12640],{"className":12639},[23],[25,12641,12642],{"xmlns":27},[29,12643,12644,12652],{},[32,12645,12646],{},[1057,12647,12648,12650],{},[35,12649,116],{},[39,12651,1251],{},[45,12653,12654],{"encoding":47},"f_\\parallel",[15,12656,12658],{"className":12657,"ariaHidden":53},[52],[15,12659,12661,12664],{"className":12660},[57],[15,12662],{"className":12663,"style":9840},[61],[15,12665,12667,12670],{"className":12666},[66],[15,12668,116],{"className":12669,"style":158},[66,67],[15,12671,12673],{"className":12672},[198],[15,12674,12676,12696],{"className":12675},[202,1090],[15,12677,12679,12693],{"className":12678},[206],[15,12680,12682],{"className":12681,"style":3668},[210],[15,12683,12684,12687],{"style":9861},[15,12685],{"className":12686,"style":218},[217],[15,12688,12690],{"className":12689},[222,223,224,225],[15,12691,1251],{"className":12692},[177,225],[15,12694,1114],{"className":12695},[1113],[15,12697,12699],{"className":12698},[206],[15,12700,12702],{"className":12701,"style":9880},[210],[15,12703],{}," 的目标值都不比 ",[15,12706,12708,12721],{"className":12707,"translate":19},[18],[15,12709,12711],{"className":12710},[23],[25,12712,12713],{"xmlns":27},[29,12714,12715,12719],{},[32,12716,12717],{},[35,12718,116],{},[45,12720,116],{"encoding":47},[15,12722,12724],{"className":12723,"ariaHidden":53},[52],[15,12725,12727,12730],{"className":12726},[57],[15,12728],{"className":12729,"style":3408},[61],[15,12731,116],{"className":12732,"style":158},[66,67]," 差。若 ",[15,12735,12737,12750],{"className":12736,"translate":19},[18],[15,12738,12740],{"className":12739},[23],[25,12741,12742],{"xmlns":27},[29,12743,12744,12748],{},[32,12745,12746],{},[35,12747,116],{},[45,12749,116],{"encoding":47},[15,12751,12753],{"className":12752,"ariaHidden":53},[52],[15,12754,12756,12759],{"className":12755},[57],[15,12757],{"className":12758,"style":3408},[61],[15,12760,116],{"className":12761,"style":158},[66,67]," 是最优解，则 ",[15,12764,12766,12783],{"className":12765,"translate":19},[18],[15,12767,12769],{"className":12768},[23],[25,12770,12771],{"xmlns":27},[29,12772,12773,12781],{},[32,12774,12775],{},[1057,12776,12777,12779],{},[35,12778,116],{},[39,12780,1251],{},[45,12782,12654],{"encoding":47},[15,12784,12786],{"className":12785,"ariaHidden":53},[52],[15,12787,12789,12792],{"className":12788},[57],[15,12790],{"className":12791,"style":9840},[61],[15,12793,12795,12798],{"className":12794},[66],[15,12796,116],{"className":12797,"style":158},[66,67],[15,12799,12801],{"className":12800},[198],[15,12802,12804,12824],{"className":12803},[202,1090],[15,12805,12807,12821],{"className":12806},[206],[15,12808,12810],{"className":12809,"style":3668},[210],[15,12811,12812,12815],{"style":9861},[15,12813],{"className":12814,"style":218},[217],[15,12816,12818],{"className":12817},[222,223,224,225],[15,12819,1251],{"className":12820},[177,225],[15,12822,1114],{"className":12823},[1113],[15,12825,12827],{"className":12826},[206],[15,12828,12830],{"className":12829,"style":9880},[210],[15,12831],{}," 也是最优解，且 ",[15,12834,12836,12862],{"className":12835,"translate":19},[18],[15,12837,12839],{"className":12838},[23],[25,12840,12841],{"xmlns":27},[29,12842,12843,12859],{},[32,12844,12845,12851,12853],{},[1057,12846,12847,12849],{},[35,12848,116],{},[39,12850,1251],{},[39,12852,1187],{},[1057,12854,12855,12857],{},[35,12856,1061],{"mathvariant":445},[1209,12858,1804],{},[45,12860,12861],{"encoding":47},"f_\\parallel\\in\\mathcal H_0",[15,12863,12865,12920],{"className":12864,"ariaHidden":53},[52],[15,12866,12868,12871,12911,12914,12917],{"className":12867},[57],[15,12869],{"className":12870,"style":9840},[61],[15,12872,12874,12877],{"className":12873},[66],[15,12875,116],{"className":12876,"style":158},[66,67],[15,12878,12880],{"className":12879},[198],[15,12881,12883,12903],{"className":12882},[202,1090],[15,12884,12886,12900],{"className":12885},[206],[15,12887,12889],{"className":12888,"style":3668},[210],[15,12890,12891,12894],{"style":9861},[15,12892],{"className":12893,"style":218},[217],[15,12895,12897],{"className":12896},[222,223,224,225],[15,12898,1251],{"className":12899},[177,225],[15,12901,1114],{"className":12902},[1113],[15,12904,12906],{"className":12905},[206],[15,12907,12909],{"className":12908,"style":9880},[210],[15,12910],{},[15,12912],{"className":12913,"style":173},[71],[15,12915,1187],{"className":12916},[177],[15,12918],{"className":12919,"style":173},[71],[15,12921,12923,12926],{"className":12922},[57],[15,12924],{"className":12925,"style":1076},[61],[15,12927,12929,12932],{"className":12928},[66],[15,12930,1061],{"className":12931,"style":1083},[66,494],[15,12933,12935],{"className":12934},[198],[15,12936,12938,12958],{"className":12937},[202,1090],[15,12939,12941,12955],{"className":12940},[206],[15,12942,12944],{"className":12943,"style":2139},[210],[15,12945,12946,12949],{"style":1100},[15,12947],{"className":12948,"style":218},[217],[15,12950,12952],{"className":12951},[222,223,224,225],[15,12953,1804],{"className":12954},[66,225],[15,12956,1114],{"className":12957},[1113],[15,12959,12961],{"className":12960},[206],[15,12962,12964],{"className":12963,"style":1121},[210],[15,12965],{},"。这就证明了第 1 条。",[11,12968,12969,12970,12998,12999,13132,13133,13306,13307,13335,13336,13428,13429,13499,13500],{},"若 ",[15,12971,12973,12986],{"className":12972,"translate":19},[18],[15,12974,12976],{"className":12975},[23],[25,12977,12978],{"xmlns":27},[29,12979,12980,12984],{},[32,12981,12982],{},[35,12983,7742],{"mathvariant":138},[45,12985,8704],{"encoding":47},[15,12987,12989],{"className":12988,"ariaHidden":53},[52],[15,12990,12992,12995],{"className":12991},[57],[15,12993],{"className":12994,"style":490},[61],[15,12996,7742],{"className":12997},[66]," 严格递增且 ",[15,13000,13002,13025],{"className":13001,"translate":19},[18],[15,13003,13005],{"className":13004},[23],[25,13006,13007],{"xmlns":27},[29,13008,13009,13022],{},[32,13010,13011,13017,13020],{},[1057,13012,13013,13015],{},[35,13014,116],{},[39,13016,9768],{},[39,13018,13019],{"mathvariant":138},"≠",[1209,13021,1804],{},[45,13023,13024],{"encoding":47},"f_\\perp\\neq 0",[15,13026,13028,13123],{"className":13027,"ariaHidden":53},[52],[15,13029,13031,13034,13074,13077,13120],{"className":13030},[57],[15,13032],{"className":13033,"style":3408},[61],[15,13035,13037,13040],{"className":13036},[66],[15,13038,116],{"className":13039,"style":158},[66,67],[15,13041,13043],{"className":13042},[198],[15,13044,13046,13066],{"className":13045},[202,1090],[15,13047,13049,13063],{"className":13048},[206],[15,13050,13052],{"className":13051,"style":1097},[210],[15,13053,13054,13057],{"style":9918},[15,13055],{"className":13056,"style":218},[217],[15,13058,13060],{"className":13059},[222,223,224,225],[15,13061,9768],{"className":13062},[177,225],[15,13064,1114],{"className":13065},[1113],[15,13067,13069],{"className":13068},[206],[15,13070,13072],{"className":13071,"style":1121},[210],[15,13073],{},[15,13075],{"className":13076,"style":173},[71],[15,13078,13080,13113,13117],{"className":13079},[177],[15,13081,13083],{"className":13082},[177],[15,13084,13087],{"className":13085},[66,13086],"vbox",[15,13088,13091],{"className":13089},[13090],"thinbox",[15,13092,13095,13098,13109],{"className":13093},[13094],"rlap",[15,13096],{"className":13097,"style":3408},[61],[15,13099,13102],{"className":13100},[13101],"inner",[15,13103,13105],{"className":13104},[66],[15,13106,13108],{"className":13107},[177],"",[15,13110],{"className":13111},[13112],"fix",[15,13114],{"className":13115},[71,13116],"nobreak",[15,13118,129],{"className":13119},[177],[15,13121],{"className":13122,"style":173},[71],[15,13124,13126,13129],{"className":13125},[57],[15,13127],{"className":13128,"style":1836},[61],[15,13130,1804],{"className":13131},[66],"，则 ",[15,13134,13136,13174],{"className":13135,"translate":19},[18],[15,13137,13139],{"className":13138},[23],[25,13140,13141],{"xmlns":27},[29,13142,13143,13171],{},[32,13144,13145,13147,13149,13155,13157,13159,13165],{},[35,13146,1251],{"mathvariant":138},[35,13148,116],{},[131,13150,13151,13153],{},[35,13152,1251],{"mathvariant":138},[1209,13154,1263],{},[39,13156,9283],{},[35,13158,1251],{"mathvariant":138},[1057,13160,13161,13163],{},[35,13162,116],{},[39,13164,1251],{},[131,13166,13167,13169],{},[35,13168,1251],{"mathvariant":138},[1209,13170,1263],{},[45,13172,13173],{"encoding":47},"\\|f\\|^2>\\|f_\\parallel\\|^2",[15,13175,13177,13227],{"className":13176,"ariaHidden":53},[52],[15,13178,13180,13183,13186,13189,13218,13221,13224],{"className":13179},[57],[15,13181],{"className":13182,"style":9154},[61],[15,13184,1251],{"className":13185},[66],[15,13187,116],{"className":13188,"style":158},[66,67],[15,13190,13192,13195],{"className":13191},[66],[15,13193,1251],{"className":13194},[66],[15,13196,13198],{"className":13197},[198],[15,13199,13201],{"className":13200},[202],[15,13202,13204],{"className":13203},[206],[15,13205,13207],{"className":13206,"style":6740},[210],[15,13208,13209,13212],{"style":213},[15,13210],{"className":13211,"style":218},[217],[15,13213,13215],{"className":13214},[222,223,224,225],[15,13216,1263],{"className":13217},[66,225],[15,13219],{"className":13220,"style":173},[71],[15,13222,9283],{"className":13223},[177],[15,13225],{"className":13226,"style":173},[71],[15,13228,13230,13234,13237,13277],{"className":13229},[57],[15,13231],{"className":13232,"style":13233},[61],"height:1.1693em;vertical-align:-0.3552em;",[15,13235,1251],{"className":13236},[66],[15,13238,13240,13243],{"className":13239},[66],[15,13241,116],{"className":13242,"style":158},[66,67],[15,13244,13246],{"className":13245},[198],[15,13247,13249,13269],{"className":13248},[202,1090],[15,13250,13252,13266],{"className":13251},[206],[15,13253,13255],{"className":13254,"style":3668},[210],[15,13256,13257,13260],{"style":9861},[15,13258],{"className":13259,"style":218},[217],[15,13261,13263],{"className":13262},[222,223,224,225],[15,13264,1251],{"className":13265},[177,225],[15,13267,1114],{"className":13268},[1113],[15,13270,13272],{"className":13271},[206],[15,13273,13275],{"className":13274,"style":9880},[210],[15,13276],{},[15,13278,13280,13283],{"className":13279},[66],[15,13281,1251],{"className":13282},[66],[15,13284,13286],{"className":13285},[198],[15,13287,13289],{"className":13288},[202],[15,13290,13292],{"className":13291},[206],[15,13293,13295],{"className":13294,"style":6740},[210],[15,13296,13297,13300],{"style":213},[15,13298],{"className":13299,"style":218},[217],[15,13301,13303],{"className":13302},[222,223,224,225],[15,13304,1263],{"className":13305},[66,225],"，上面的不等式是严格的，",[15,13308,13310,13323],{"className":13309,"translate":19},[18],[15,13311,13313],{"className":13312},[23],[25,13314,13315],{"xmlns":27},[29,13316,13317,13321],{},[32,13318,13319],{},[35,13320,116],{},[45,13322,116],{"encoding":47},[15,13324,13326],{"className":13325,"ariaHidden":53},[52],[15,13327,13329,13332],{"className":13328},[57],[15,13330],{"className":13331,"style":3408},[61],[15,13333,116],{"className":13334,"style":158},[66,67]," 不可能是最优解。因此每个最优解都必须满足 ",[15,13337,13339,13361],{"className":13338,"translate":19},[18],[15,13340,13342],{"className":13341},[23],[25,13343,13344],{"xmlns":27},[29,13345,13346,13358],{},[32,13347,13348,13354,13356],{},[1057,13349,13350,13352],{},[35,13351,116],{},[39,13353,9768],{},[39,13355,129],{},[1209,13357,1804],{},[45,13359,13360],{"encoding":47},"f_\\perp=0",[15,13362,13364,13419],{"className":13363,"ariaHidden":53},[52],[15,13365,13367,13370,13410,13413,13416],{"className":13366},[57],[15,13368],{"className":13369,"style":3408},[61],[15,13371,13373,13376],{"className":13372},[66],[15,13374,116],{"className":13375,"style":158},[66,67],[15,13377,13379],{"className":13378},[198],[15,13380,13382,13402],{"className":13381},[202,1090],[15,13383,13385,13399],{"className":13384},[206],[15,13386,13388],{"className":13387,"style":1097},[210],[15,13389,13390,13393],{"style":9918},[15,13391],{"className":13392,"style":218},[217],[15,13394,13396],{"className":13395},[222,223,224,225],[15,13397,9768],{"className":13398},[177,225],[15,13400,1114],{"className":13401},[1113],[15,13403,13405],{"className":13404},[206],[15,13406,13408],{"className":13407,"style":1121},[210],[15,13409],{},[15,13411],{"className":13412,"style":173},[71],[15,13414,129],{"className":13415},[177],[15,13417],{"className":13418,"style":173},[71],[15,13420,13422,13425],{"className":13421},[57],[15,13423],{"className":13424,"style":1836},[61],[15,13426,1804],{"className":13427},[66],"，即落在 ",[15,13430,13432,13450],{"className":13431,"translate":19},[18],[15,13433,13435],{"className":13434},[23],[25,13436,13437],{"xmlns":27},[29,13438,13439,13447],{},[32,13440,13441],{},[1057,13442,13443,13445],{},[35,13444,1061],{"mathvariant":445},[1209,13446,1804],{},[45,13448,13449],{"encoding":47},"\\mathcal H_0",[15,13451,13453],{"className":13452,"ariaHidden":53},[52],[15,13454,13456,13459],{"className":13455},[57],[15,13457],{"className":13458,"style":1076},[61],[15,13460,13462,13465],{"className":13461},[66],[15,13463,1061],{"className":13464,"style":1083},[66,494],[15,13466,13468],{"className":13467},[198],[15,13469,13471,13491],{"className":13470},[202,1090],[15,13472,13474,13488],{"className":13473},[206],[15,13475,13477],{"className":13476,"style":2139},[210],[15,13478,13479,13482],{"style":1100},[15,13480],{"className":13481,"style":218},[217],[15,13483,13485],{"className":13484},[222,223,224,225],[15,13486,1804],{"className":13487},[66,225],[15,13489,1114],{"className":13490},[1113],[15,13492,13494],{"className":13493},[206],[15,13495,13497],{"className":13496,"style":1121},[210],[15,13498],{}," 中。第 2 条得证。",[15,13501,13503,13518],{"className":13502,"translate":19},[18],[15,13504,13506],{"className":13505},[23],[25,13507,13508],{"xmlns":27},[29,13509,13510,13515],{},[32,13511,13512],{},[35,13513,13514],{"mathvariant":138},"□",[45,13516,13517],{"encoding":47},"\\square",[15,13519,13521],{"className":13520,"ariaHidden":53},[52],[15,13522,13524,13528],{"className":13523},[57],[15,13525],{"className":13526,"style":13527},[61],"height:0.675em;",[15,13529,13514],{"className":13530},[66,13531],"amsrm",[13533,13534,13535],"div",{},[13536,13537],"img",{"height":13538,"width":13539,"style":13540,"data-d2-theme":13541,"alt":13542,"src":13543},277,760,"max-width:100%;height:auto","default","表示定理的证明结构：正交分解与有限维归约","data:image\u002Fsvg+xml;base64,<svg width="1415" height="277" xmlns="http://www.w3.org/2000/svg" data-d2-version="v0.9.0" viewBox="0 0 1415 277"><svg width="1415" height="277" class="d2-3061015132 d2-svg" viewBox="-1 -1 1415 277"><rect width="1415" height="277" x="-1" y="-1" 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stroke="#0d32b2" d="M413 209h142v66H413z" class="stroke-B1 fill-B6" style="stroke-width:2"/></g><text x="484" y="247.5" fill="#0a0f25" class="text-bold fill-N1" style="text-anchor:middle;font-size:16px">f(xᵢ) = f∥(xᵢ)</text></g><g class="RQ=="><g class="shape"><path fill="#f7f8fe" stroke="#0d32b2" d="M655 37h197v66H655z" class="stroke-B1 fill-B6" style="stroke-width:2"/></g><text x="753.5" y="75.5" fill="#0a0f25" class="text-bold fill-N1" style="text-anchor:middle;font-size:16px">‖f‖² = ‖f∥‖² + ‖f⊥‖²</text></g><g class="Rg=="><g class="shape"><path fill="#f7f8fe" stroke="#0d32b2" d="M682 163h144v66H682z" class="stroke-B1 fill-B6" style="stroke-width:2"/></g><text x="754" y="201.5" fill="#0a0f25" class="text-bold fill-N1" style="text-anchor:middle;font-size:16px">损失只看 f∥</text></g><g class="Rw=="><g class="shape"><path fill="#f7f8fe" stroke="#0d32b2" d="M952 100h200v66H952z" class="stroke-B1 fill-B6" style="stroke-width:2"/></g><text x="1052" y="138.5" fill="#0a0f25" class="text-bold 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k(\\cdot,x_j)",[15,13668,13670,13688],{"className":13669,"ariaHidden":53},[52],[15,13671,13673,13676,13679,13682,13685],{"className":13672},[57],[15,13674],{"className":13675,"style":3408},[61],[15,13677,116],{"className":13678,"style":158},[66,67],[15,13680],{"className":13681,"style":173},[71],[15,13683,129],{"className":13684},[177],[15,13686],{"className":13687,"style":173},[71],[15,13689,13691,13695,13761,13764,13804,13807,13810,13813,13816,13819,13859],{"className":13690},[57],[15,13692],{"className":13693,"style":13694},[61],"height:1.2401em;vertical-align:-0.4358em;",[15,13696,13698,13703],{"className":13697},[1339],[15,13699,1200],{"className":13700,"style":13702},[1339,1470,13701],"small-op","position:relative;top:0em;",[15,13704,13706],{"className":13705},[198],[15,13707,13709,13752],{"className":13708},[202,1090],[15,13710,13712,13749],{"className":13711},[206],[15,13713,13716,13737],{"className":13714,"style":13715},[210],"height:0.8043em;",[15,13717,13719,13722],{"style":13718},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[15,13720],{"className":13721,"style":218},[217],[15,13723,13725],{"className":13724},[222,223,224,225],[15,13726,13728,13731,13734],{"className":13727},[66,225],[15,13729,2441],{"className":13730,"style":2594},[66,67,225],[15,13732,129],{"className":13733},[177,225],[15,13735,1211],{"className":13736},[66,225],[15,13738,13740,13743],{"style":13739},"top:-3.2029em;margin-right:0.05em;",[15,13741],{"className":13742,"style":218},[217],[15,13744,13746],{"className":13745},[222,223,224,225],[15,13747,37],{"className":13748},[66,67,225],[15,13750,1114],{"className":13751},[1113],[15,13753,13755],{"className":13754},[206],[15,13756,13759],{"className":13757,"style":13758},[210],"height:0.4358em;",[15,13760],{},[15,13762],{"className":13763,"style":697},[71],[15,13765,13767,13770],{"className":13766},[66],[15,13768,8390],{"className":13769,"style":8560},[66,67],[15,13771,13773],{"className":13772},[198],[15,13774,13776,13796],{"className":13775},[202,1090],[15,13777,13779,13793],{"className":13778},[206],[15,13780,13782],{"className":13781,"style":1528},[210],[15,13783,13784,13787],{"style":8575},[15,13785],{"className":13786,"style":218},[217],[15,13788,13790],{"className":13789},[222,223,224,225],[15,13791,2441],{"className":13792,"style":2594},[66,67,225],[15,13794,1114],{"className":13795},[1113],[15,13797,13799],{"className":13798},[206],[15,13800,13802],{"className":13801,"style":2715},[210],[15,13803],{},[15,13805,621],{"className":13806,"style":684},[66,67],[15,13808,120],{"className":13809},[162],[15,13811,3268],{"className":13812},[66],[15,13814,335],{"className":13815},[422],[15,13817],{"className":13818,"style":697},[71],[15,13820,13822,13825],{"className":13821},[66],[15,13823,123],{"className":13824},[66,67],[15,13826,13828],{"className":13827},[198],[15,13829,13831,13851],{"className":13830},[202,1090],[15,13832,13834,13848],{"className":13833},[206],[15,13835,13837],{"className":13836,"style":1528},[210],[15,13838,13839,13842],{"style":1584},[15,13840],{"className":13841,"style":218},[217],[15,13843,13845],{"className":13844},[222,223,224,225],[15,13846,2441],{"className":13847,"style":2594},[66,67,225],[15,13849,1114],{"className":13850},[1113],[15,13852,13854],{"className":13853},[206],[15,13855,13857],{"className":13856,"style":2715},[210],[15,13858],{},[15,13860,126],{"className":13861},[169]," 代回目标函数。对每个训练点，",[15,13864,13866],{"className":13865,"translate":19},[294],[15,13867,13869,13953],{"className":13868,"translate":19},[18],[15,13870,13872],{"className":13871},[23],[25,13873,13874],{"xmlns":27,"display":303},[29,13875,13876,13950],{},[32,13877,13878,13880,13882,13888,13890,13892,13906,13912,13914,13916,13918,13924,13926,13932,13934,13936,13938,13940,13942,13948],{},[35,13879,116],{},[39,13881,120],{"stretchy":119},[1057,13883,13884,13886],{},[35,13885,123],{},[35,13887,1205],{},[39,13889,126],{"stretchy":119},[39,13891,129],{},[1196,13893,13894,13896,13904],{},[39,13895,1200],{},[32,13897,13898,13900,13902],{},[35,13899,2441],{},[39,13901,129],{},[1209,13903,1211],{},[35,13905,37],{},[1057,13907,13908,13910],{},[35,13909,8390],{},[35,13911,2441],{},[1274,13913,2462],{},[35,13915,621],{},[39,13917,120],{"stretchy":119},[1057,13919,13920,13922],{},[35,13921,123],{},[35,13923,1205],{},[39,13925,335],{"separator":53},[1057,13927,13928,13930],{},[35,13929,123],{},[35,13931,2441],{},[39,13933,126],{"stretchy":119},[39,13935,129],{},[39,13937,120],{"stretchy":119},[35,13939,2853],{},[35,13941,8390],{},[1057,13943,13944,13946],{},[39,13945,126],{"stretchy":119},[35,13947,1205],{},[39,13949,335],{"separator":53},[45,13951,13952],{"encoding":47},"f(x_i)=\\sum_{j=1}^n\\alpha_j\\,k(x_i,x_j)=(K\\alpha)_i,",[15,13954,13956,14020,14240],{"className":13955,"ariaHidden":53},[52],[15,13957,13959,13962,13965,13968,14008,14011,14014,14017],{"className":13958},[57],[15,13960],{"className":13961,"style":154},[61],[15,13963,116],{"className":13964,"style":158},[66,67],[15,13966,120],{"className":13967},[162],[15,13969,13971,13974],{"className":13970},[66],[15,13972,123],{"className":13973},[66,67],[15,13975,13977],{"className":13976},[198],[15,13978,13980,14000],{"className":13979},[202,1090],[15,13981,13983,13997],{"className":13982},[206],[15,13984,13986],{"className":13985,"style":1528},[210],[15,13987,13988,13991],{"style":1584},[15,13989],{"className":13990,"style":218},[217],[15,13992,13994],{"className":13993},[222,223,224,225],[15,13995,1205],{"className":13996},[66,67,225],[15,13998,1114],{"className":13999},[1113],[15,14001,14003],{"className":14002},[206],[15,14004,14006],{"className":14005,"style":1121},[210],[15,14007],{},[15,14009,126],{"className":14010},[169],[15,14012],{"className":14013,"style":173},[71],[15,14015,129],{"className":14016},[177],[15,14018],{"className":14019,"style":173},[71],[15,14021,14023,14026,14090,14093,14133,14136,14139,14142,14182,14185,14188,14228,14231,14234,14237],{"className":14022},[57],[15,14024],{"className":14025,"style":2500},[61],[15,14027,14029],{"className":14028},[1339,1351],[15,14030,14032,14082],{"className":14031},[202,1090],[15,14033,14035,14079],{"className":14034},[206],[15,14036,14038,14058,14068],{"className":14037,"style":1436},[210],[15,14039,14040,14043],{"style":1439},[15,14041],{"className":14042,"style":1443},[217],[15,14044,14046],{"className":14045},[222,223,224,225],[15,14047,14049,14052,14055],{"className":14048},[66,225],[15,14050,2441],{"className":14051,"style":2594},[66,67,225],[15,14053,129],{"className":14054},[177,225],[15,14056,1211],{"className":14057},[66,225],[15,14059,14060,14063],{"style":1461},[15,14061],{"className":14062,"style":1443},[217],[15,14064,14065],{},[15,14066,1200],{"className":14067},[1339,1470,1471],[15,14069,14070,14073],{"style":1474},[15,14071],{"className":14072,"style":1443},[217],[15,14074,14076],{"className":14075},[222,223,224,225],[15,14077,37],{"className":14078},[66,67,225],[15,14080,1114],{"className":14081},[1113],[15,14083,14085],{"className":14084},[206],[15,14086,14088],{"className":14087,"style":2631},[210],[15,14089],{},[15,14091],{"className":14092,"style":697},[71],[15,14094,14096,14099],{"className":14095},[66],[15,14097,8390],{"className":14098,"style":8560},[66,67],[15,14100,14102],{"className":14101},[198],[15,14103,14105,14125],{"className":14104},[202,1090],[15,14106,14108,14122],{"className":14107},[206],[15,14109,14111],{"className":14110,"style":1528},[210],[15,14112,14113,14116],{"style":8575},[15,14114],{"className":14115,"style":218},[217],[15,14117,14119],{"className":14118},[222,223,224,225],[15,14120,2441],{"className":14121,"style":2594},[66,67,225],[15,14123,1114],{"className":14124},[1113],[15,14126,14128],{"className":14127},[206],[15,14129,14131],{"className":14130,"style":2715},[210],[15,14132],{},[15,14134],{"className":14135,"style":697},[71],[15,14137,621],{"className":14138,"style":684},[66,67],[15,14140,120],{"className":14141},[162],[15,14143,14145,14148],{"className":14144},[66],[15,14146,123],{"className":14147},[66,67],[15,14149,14151],{"className":14150},[198],[15,14152,14154,14174],{"className":14153},[202,1090],[15,14155,14157,14171],{"className":14156},[206],[15,14158,14160],{"className":14159,"style":1528},[210],[15,14161,14162,14165],{"style":1584},[15,14163],{"className":14164,"style":218},[217],[15,14166,14168],{"className":14167},[222,223,224,225],[15,14169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R^{n\\times n}",[15,14332,14334,14353],{"className":14333,"ariaHidden":53},[52],[15,14335,14337,14341,14344,14347,14350],{"className":14336},[57],[15,14338],{"className":14339,"style":14340},[61],"height:0.7224em;vertical-align:-0.0391em;",[15,14342,2853],{"className":14343,"style":2901},[66,67],[15,14345],{"className":14346,"style":173},[71],[15,14348,1187],{"className":14349},[177],[15,14351],{"className":14352,"style":173},[71],[15,14354,14356,14360],{"className":14355},[57],[15,14357],{"className":14358,"style":14359},[61],"height:0.7713em;",[15,14361,14363,14366],{"className":14362},[66],[15,14364,455],{"className":14365},[66,517],[15,14367,14369],{"className":14368},[198],[15,14370,14372],{"className":14371},[202],[15,14373,14375],{"className":14374},[206],[15,14376,14378],{"className":14377,"style":14359},[210],[15,14379,14380,14383],{"style":213},[15,14381],{"className":14382,"style":218},[217],[15,14384,14386],{"className":14385},[222,223,224,225],[15,14387,14389,14392,14395],{"className":14388},[66,225],[15,14390,37],{"className":14391},[66,67,225],[15,14393,41],{"className":14394},[76,225],[15,14396,37],{"className":14397},[66,67,225]," 是核矩阵，",[15,14400,14402,14445],{"className":14401,"translate":19},[18],[15,14403,14405],{"className":14404},[23],[25,14406,14407],{"xmlns":27},[29,14408,14409,14443],{},[32,14410,14411,14421,14423,14425,14427,14433,14435,14441],{},[1057,14412,14413,14415],{},[35,14414,2853],{},[32,14416,14417,14419],{},[35,14418,1205],{},[35,14420,2441],{},[39,14422,129],{},[35,14424,621],{},[39,14426,120],{"stretchy":119},[1057,14428,14429,14431],{},[35,14430,123],{},[35,14432,1205],{},[39,14434,335],{"separator":53},[1057,14436,14437,14439],{},[35,14438,123],{},[35,14440,2441],{},[39,14442,126],{"stretchy":119},[45,14444,2884],{"encoding":47},[15,14446,14448,14506],{"className":14447,"ariaHidden":53},[52],[15,14449,14451,14454,14497,14500,14503],{"className":14450},[57],[15,14452],{"className":14453,"style":2894},[61],[15,14455,14457,14460],{"className":14456},[66],[15,14458,2853],{"className":14459,"style":2901},[66,67],[15,14461,14463],{"className":14462},[198],[15,14464,14466,14489],{"className":14465},[202,1090],[15,14467,14469,14486],{"className":14468},[206],[15,14470,14472],{"className":14471,"style":1528},[210],[15,14473,14474,14477],{"style":2916},[15,14475],{"className":14476,"style":218},[217],[15,14478,14480],{"className":14479},[222,223,224,225],[15,14481,14483],{"className":14482},[66,225],[15,14484,2929],{"className":14485,"style":2594},[66,67,225],[15,14487,1114],{"className":14488},[1113],[15,14490,14492],{"className":14491},[206],[15,14493,14495],{"className":14494,"style":2715},[210],[15,14496],{},[15,14498],{"className":14499,"style":173},[71],[15,14501,129],{"className":14502},[177],[15,14504],{"className":14505,"style":173},[71],[15,14507,14509,14512,14515,14518,14558,14561,14564,14604],{"className":14508},[57],[15,14510],{"className":14511,"style":2956},[61],[15,14513,621],{"className":14514,"style":684},[66,67],[15,14516,120],{"className":14517},[162],[15,14519,14521,14524],{"className":14520},[66],[15,14522,123],{"className":14523},[66,67],[15,14525,14527],{"className":14526},[198],[15,14528,14530,14550],{"className":14529},[202,1090],[15,14531,14533,14547],{"className":14532},[206],[15,14534,14536],{"className":14535,"style":1528},[210],[15,14537,14538,14541],{"style":1584},[15,14539],{"className":14540,"style":218},[217],[15,14542,14544],{"className":14543},[222,223,224,225],[15,14545,1205],{"className":14546},[66,67,225],[15,14548,1114],{"className":14549},[1113],[15,14551,14553],{"className":14552},[206],[15,14554,14556],{"className":14555,"style":1121},[210],[15,14557],{},[15,14559,335],{"className":14560},[422],[15,14562],{"className":14563,"style":697},[71],[15,14565,14567,14570],{"className":14566},[66],[15,14568,123],{"className":14569},[66,67],[15,14571,14573],{"className":14572},[198],[15,14574,14576,14596],{"className":14575},[202,1090],[15,14577,14579,14593],{"className":14578},[206],[15,14580,14582],{"className":14581,"style":1528},[210],[15,14583,14584,14587],{"style":1584},[15,14585],{"className":14586,"style":218},[217],[15,14588,14590],{"className":14589},[222,223,224,225],[15,14591,2441],{"className":14592,"style":2594},[66,67,225],[15,14594,1114],{"className":14595},[1113],[15,14597,14599],{"className":14598},[206],[15,14600,14602],{"className":14601,"style":2715},[210],[15,14603],{},[15,14605,126],{"className":14606},[169],"。正则项则是",[15,14609,14611],{"className":14610,"translate":19},[294],[15,14612,14614,14766],{"className":14613,"translate":19},[18],[15,14615,14617],{"className":14616},[23],[25,14618,14619],{"xmlns":27,"display":303},[29,14620,14621,14763],{},[32,14622,14623,14625,14627,14635,14637,14639,14645,14651,14653,14655,14657,14659,14665,14667,14669,14675,14681,14683,14685,14687,14689,14695,14697,14699,14701,14703,14715,14721,14727,14729,14731,14733,14739,14741,14747,14749,14751,14757,14759,14761],{},[35,14624,1251],{"mathvariant":138},[35,14626,116],{},[1255,14628,14629,14631,14633],{},[35,14630,1251],{"mathvariant":138},[35,14632,1061],{"mathvariant":445},[1209,14634,1263],{},[39,14636,129],{},[39,14638,642],{"fence":53,"stretchy":53,"minsize":4545,"maxsize":4545},[1172,14640,14641,14643],{},[39,14642,1200],{},[35,14644,1205],{},[1057,14646,14647,14649],{},[35,14648,8390],{},[35,14650,1205],{},[35,14652,621],{},[39,14654,120],{"stretchy":119},[39,14656,3268],{},[39,14658,335],{"separator":53},[1057,14660,14661,14663],{},[35,14662,123],{},[35,14664,1205],{},[39,14666,126],{"stretchy":119},[39,14668,335],{"separator":53},[1172,14670,14671,14673],{},[39,14672,1200],{},[35,14674,2441],{},[1057,14676,14677,14679],{},[35,14678,8390],{},[35,14680,2441],{},[35,14682,621],{},[39,14684,120],{"stretchy":119},[39,14686,3268],{},[39,14688,335],{"separator":53},[1057,14690,14691,14693],{},[35,14692,123],{},[35,14694,2441],{},[39,14696,126],{"stretchy":119},[39,14698,667],{"fence":53,"stretchy":53,"minsize":4545,"maxsize":4545},[71,14700],{"linebreak":10456},[39,14702,129],{},[1172,14704,14705,14707],{},[39,14706,1200],{},[32,14708,14709,14711,14713],{},[35,14710,1205],{},[39,14712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H}^2\n=\\Bigl\\langle\\sum_i\\alpha_i k(\\cdot,x_i),\\sum_j\\alpha_j k(\\cdot,x_j)\\Bigr\\rangle \\\\\n=\\sum_{i,j}\\alpha_i\\alpha_j\\,k(x_i,x_j)\n=\\alpha^\\top 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的有限维问题：",[15,15552,15554],{"className":15553,"translate":19},[294],[15,15555,15557,15640],{"className":15556,"translate":19},[18],[15,15558,15560],{"className":15559},[23],[25,15561,15562],{"xmlns":27,"display":303},[29,15563,15564,15637],{},[1142,15565,15566],{"width":1144},[1146,15567,15568,15570,15630,15632],{},[1149,15569],{"width":1151},[1149,15571,15572],{},[32,15573,15574,15594,15628],{},[1172,15575,15576,15582],{},[32,15577,15578,15580],{},[35,15579,1178],{},[39,15581,1170],{},[32,15583,15584,15586,15588],{},[35,15585,8390],{},[39,15587,1187],{},[131,15589,15590,15592],{},[35,15591,455],{"mathvariant":454},[35,15593,37],{},[32,15595,15596,15598,15600,15602,15604,15606,15608,15610,15612,15614,15620,15622,15624,15626],{},[39,15597,4307],{"fence":53},[35,15599,1216],{},[39,15601,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,15603,2853],{},[35,15605,8390],{},[39,15607,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,15609,1245],{},[35,15611,7742],{"mathvariant":138},[39,15613,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[131,15615,15616,15618],{},[35,15617,8390],{},[35,15619,139],{"mathvariant":138},[35,15621,2853],{},[35,15623,8390],{},[39,15625,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,15627,4330],{"fence":53},[35,15629,3573],{"mathvariant":138},[1149,15631],{"width":1151},[1149,15633,15634],{},[1274,15635,15636],{},"(4)",[45,15638,15639],{"encoding":47},"\\min_{\\alpha\\in\\mathbb R^n}\\left\\{L\\bigl(K\\alpha\\bigr)+\\Omega\\bigl(\\alpha^\\top K\\alpha\\bigr)\\right\\}.\n\\tag{4}",[15,15641,15643,15837],{"className":15642,"ariaHidden":53},[52],[15,15644,15646,15650,15733,15736,15831,15834],{"className":15645},[57],[15,15647],{"className":15648,"style":15649},[61],"height:1.6747em;vertical-align:-0.7756em;",[15,15651,15653],{"className":15652},[1339,1351],[15,15654,15656,15724],{"className":15655},[202,1090],[15,15657,15659,15721],{"className":15658},[206],[15,15660,15662,15711],{"className":15661,"style":1361},[210],[15,15663,15665,15668],{"style":15664},"top:-2.3518em;margin-left:0em;",[15,15666],{"className":15667,"style":1368},[217],[15,15669,15671],{"className":15670},[222,223,224,225],[15,15672,15674,15677,15680],{"className":15673},[66,225],[15,15675,8390],{"className":15676,"style":8560},[66,67,225],[15,15678,1187],{"className":15679},[177,225],[15,15681,15683,15686],{"className":15682},[66,225],[15,15684,455],{"className":15685},[66,517,225],[15,15687,15689],{"className":15688},[198],[15,15690,15692],{"className":15691},[202],[15,15693,15695],{"className":15694},[206],[15,15696,15699],{"className":15697,"style":15698},[210],"height:0.5935em;",[15,15700,15702,15705],{"style":15701},"top:-2.786em;margin-right:0.0714em;",[15,15703],{"className":15704,"style":3675},[217],[15,15706,15708],{"className":15707},[222,3679,1508,225],[15,15709,37],{"className":15710},[66,67,225],[15,15712,15713,15716],{"style":1386},[15,15714],{"className":15715,"style":1368},[217],[15,15717,15718],{},[15,15719,1178],{"className":15720},[1339],[15,15722,1114],{"className":15723},[1113],[15,15725,15727],{"className":15726},[206],[15,15728,15731],{"className":15729,"style":15730},[210],"height:0.7756em;",[15,15732],{},[15,15734],{"className":15735,"style":697},[71],[15,15737,15739,15745,15748,15754,15757,15760,15766,15769,15772,15775,15778,15784,15813,15816,15819,15825],{"className":15738},[1413],[15,15740,15742],{"className":15741,"style":1418},[162,1417],[15,15743,4307],{"className":15744},[1422,1508],[15,15746,1216],{"className":15747},[66,67],[15,15749,15751],{"className":15750},[162],[15,15752,120],{"className":15753},[1422,1508],[15,15755,2853],{"className":15756,"style":2901},[66,67],[15,15758,8390],{"className":15759,"style":8560},[66,67],[15,15761,15763],{"className":15762},[169],[15,15764,126],{"className":15765},[1422,1508],[15,15767],{"className":15768,"style":72},[71],[15,15770,1245],{"className":15771},[76],[15,15773],{"className":15774,"style":72},[71],[15,15776,7742],{"className":15777},[66],[15,15779,15781],{"className":15780},[162],[15,15782,120],{"className":15783},[1422,1508],[15,15785,15787,15790],{"className":15786},[66],[15,15788,8390],{"className":15789,"style":8560},[66,67],[15,15791,15793],{"className":15792},[198],[15,15794,15796],{"className":15795},[202],[15,15797,15799],{"className":15798},[206],[15,15800,15802],{"className":15801,"style":394},[210],[15,15803,15804,15807],{"style":397},[15,15805],{"className":15806,"style":218},[217],[15,15808,15810],{"className":15809},[222,223,224,225],[15,15811,139],{"className":15812},[66,225],[15,15814,2853],{"className":15815,"style":2901},[66,67],[15,15817,8390],{"className":15818,"style":8560},[66,67],[15,15820,15822],{"className":15821},[169],[15,15823,126],{"className":15824},[1422,1508],[15,15826,15828],{"className":15827,"style":1418},[169,1417],[15,15829,4330],{"className":15830},[1422,1508],[15,15832],{"className":15833,"style":697},[71],[15,15835,3573],{"className":15836},[66],[15,15838,15840,15843],{"className":15839},[1704],[15,15841],{"className":15842,"style":15649},[61],[15,15844,15846,15849,15856],{"className":15845},[66,1711],[15,15847,120],{"className":15848},[66],[15,15850,15852],{"className":15851},[66],[15,15853,15855],{"className":15854},[66],"4",[15,15857,126],{"className":15858},[66],[11,15860,15861,15862,15890,15891,15919,15920,15948,15949,15977,15978,16031,16032,16060,16061,16090,16091,16119],{},"这就是表示定理的全部实用价值：原本在无穷维空间里找 ",[15,15863,15865,15878],{"className":15864,"translate":19},[18],[15,15866,15868],{"className":15867},[23],[25,15869,15870],{"xmlns":27},[29,15871,15872,15876],{},[32,15873,15874],{},[35,15875,116],{},[45,15877,116],{"encoding":47},[15,15879,15881],{"className":15880,"ariaHidden":53},[52],[15,15882,15884,15887],{"className":15883},[57],[15,15885],{"className":15886,"style":3408},[61],[15,15888,116],{"className":15889,"style":158},[66,67],"，现在只需找 ",[15,15892,15894,15907],{"className":15893,"translate":19},[18],[15,15895,15897],{"className":15896},[23],[25,15898,15899],{"xmlns":27},[29,15900,15901,15905],{},[32,15902,15903],{},[35,15904,37],{},[45,15906,37],{"encoding":47},[15,15908,15910],{"className":15909,"ariaHidden":53},[52],[15,15911,15913,15916],{"className":15912},[57],[15,15914],{"className":15915,"style":86},[61],[15,15917,37],{"className":15918},[66,67]," 个系数。核矩阵 ",[15,15921,15923,15936],{"className":15922,"translate":19},[18],[15,15924,15926],{"className":15925},[23],[25,15927,15928],{"xmlns":27},[29,15929,15930,15934],{},[32,15931,15932],{},[35,15933,2853],{},[45,15935,2853],{"encoding":47},[15,15937,15939],{"className":15938,"ariaHidden":53},[52],[15,15940,15942,15945],{"className":15941},[57],[15,15943],{"className":15944,"style":490},[61],[15,15946,2853],{"className":15947,"style":2901},[66,67]," 是半正定的；若 ",[15,15950,15952,15965],{"className":15951,"translate":19},[18],[15,15953,15955],{"className":15954},[23],[25,15956,15957],{"xmlns":27},[29,15958,15959,15963],{},[32,15960,15961],{},[35,15962,2853],{},[45,15964,2853],{"encoding":47},[15,15966,15968],{"className":15967,"ariaHidden":53},[52],[15,15969,15971,15974],{"className":15970},[57],[15,15972],{"className":15973,"style":490},[61],[15,15975,2853],{"className":15976,"style":2901},[66,67]," 可逆，则 ",[15,15979,15981,16000],{"className":15980,"translate":19},[18],[15,15982,15984],{"className":15983},[23],[25,15985,15986],{"xmlns":27},[29,15987,15988,15997],{},[32,15989,15990,15992,15995],{},[35,15991,8390],{},[39,15993,15994],{},"↦",[35,15996,116],{},[45,15998,15999],{"encoding":47},"\\alpha\\mapsto f",[15,16001,16003,16022],{"className":16002,"ariaHidden":53},[52],[15,16004,16006,16010,16013,16016,16019],{"className":16005},[57],[15,16007],{"className":16008,"style":16009},[61],"height:0.522em;vertical-align:-0.011em;",[15,16011,8390],{"className":16012,"style":8560},[66,67],[15,16014],{"className":16015,"style":173},[71],[15,16017,15994],{"className":16018},[177],[15,16020],{"className":16021,"style":173},[71],[15,16023,16025,16028],{"className":16024},[57],[15,16026],{"className":16027,"style":3408},[61],[15,16029,116],{"className":16030,"style":158},[66,67]," 是一一对应；若 ",[15,16033,16035,16048],{"className":16034,"translate":19},[18],[15,16036,16038],{"className":16037},[23],[25,16039,16040],{"xmlns":27},[29,16041,16042,16046],{},[32,16043,16044],{},[35,16045,2853],{},[45,16047,2853],{"encoding":47},[15,16049,16051],{"className":16050,"ariaHidden":53},[52],[15,16052,16054,16057],{"className":16053},[57],[15,16055],{"className":16056,"style":490},[61],[15,16058,2853],{"className":16059,"style":2901},[66,67]," 奇异，不同的 ",[15,16062,16064,16078],{"className":16063,"translate":19},[18],[15,16065,16067],{"className":16066},[23],[25,16068,16069],{"xmlns":27},[29,16070,16071,16075],{},[32,16072,16073],{},[35,16074,8390],{},[45,16076,16077],{"encoding":47},"\\alpha",[15,16079,16081],{"className":16080,"ariaHidden":53},[52],[15,16082,16084,16087],{"className":16083},[57],[15,16085],{"className":16086,"style":86},[61],[15,16088,8390],{"className":16089,"style":8560},[66,67]," 可能对应同一个函数（相差一个 ",[15,16092,16094,16107],{"className":16093,"translate":19},[18],[15,16095,16097],{"className":16096},[23],[25,16098,16099],{"xmlns":27},[29,16100,16101,16105],{},[32,16102,16103],{},[35,16104,2853],{},[45,16106,2853],{"encoding":47},[15,16108,16110],{"className":16109,"ariaHidden":53},[52],[15,16111,16113,16116],{"className":16112},[57],[15,16114],{"className":16115,"style":490},[61],[15,16117,2853],{"className":16118,"style":2901},[66,67]," 的零空间向量，其范数为零），但函数本身仍然是良定义的。",[1904,16121,16123],{"id":16122},"_34-唯一性系数与偏置项","3.4 唯一性、系数与偏置项",[11,16125,16126,16127,16130,16131,16134],{},"表示定理给出的是",[2040,16128,16129],{},"解的形式","，不是",[2040,16132,16133],{},"解的唯一性","。以下几点容易混淆：",[16136,16137,16138,16566,16631],"ul",{},[3194,16139,16140,8687,16143,16171,16172,16200,16201,16261,16262,16312,16313,16341,16342,16400,16401,16536,16537,16565],{},[2040,16141,16142],{},"函数唯一 vs 系数唯一。",[15,16144,16146,16159],{"className":16145,"translate":19},[18],[15,16147,16149],{"className":16148},[23],[25,16150,16151],{"xmlns":27},[29,16152,16153,16157],{},[32,16154,16155],{},[35,16156,2853],{},[45,16158,2853],{"encoding":47},[15,16160,16162],{"className":16161,"ariaHidden":53},[52],[15,16163,16165,16168],{"className":16164},[57],[15,16166],{"className":16167,"style":490},[61],[15,16169,2853],{"className":16170,"style":2901},[66,67]," 奇异，",[15,16173,16175,16188],{"className":16174,"translate":19},[18],[15,16176,16178],{"className":16177},[23],[25,16179,16180],{"xmlns":27},[29,16181,16182,16186],{},[32,16183,16184],{},[35,16185,8390],{},[45,16187,16077],{"encoding":47},[15,16189,16191],{"className":16190,"ariaHidden":53},[52],[15,16192,16194,16197],{"className":16193},[57],[15,16195],{"className":16196,"style":86},[61],[15,16198,8390],{"className":16199,"style":8560},[66,67]," 不唯一，但 ",[15,16202,16204,16222],{"className":16203,"translate":19},[18],[15,16205,16207],{"className":16206},[23],[25,16208,16209],{"xmlns":27},[29,16210,16211,16219],{},[32,16212,16213],{},[131,16214,16215,16217],{},[35,16216,116],{},[39,16218,1162],{},[45,16220,16221],{"encoding":47},"f^*",[15,16223,16225],{"className":16224,"ariaHidden":53},[52],[15,16226,16228,16231],{"className":16227},[57],[15,16229],{"className":16230,"style":3408},[61],[15,16232,16234,16237],{"className":16233},[66],[15,16235,116],{"className":16236,"style":158},[66,67],[15,16238,16240],{"className":16239},[198],[15,16241,16243],{"className":16242},[202],[15,16244,16246],{"className":16245},[206],[15,16247,16250],{"className":16248,"style":16249},[210],"height:0.6887em;",[15,16251,16252,16255],{"style":213},[15,16253],{"className":16254,"style":218},[217],[15,16256,16258],{"className":16257},[222,223,224,225],[15,16259,1162],{"className":16260},[76,225]," 可能唯一。核岭回归在 ",[15,16263,16265,16282],{"className":16264,"translate":19},[18],[15,16266,16268],{"className":16267},[23],[25,16269,16270],{"xmlns":27},[29,16271,16272,16280],{},[32,16273,16274,16276,16278],{},[35,16275,1248],{},[39,16277,9283],{},[1209,16279,1804],{},[45,16281,9288],{"encoding":47},[15,16283,16285,16303],{"className":16284,"ariaHidden":53},[52],[15,16286,16288,16291,16294,16297,16300],{"className":16287},[57],[15,16289],{"className":16290,"style":9298},[61],[15,16292,1248],{"className":16293},[66,67],[15,16295],{"className":16296,"style":173},[71],[15,16298,9283],{"className":16299},[177],[15,16301],{"className":16302,"style":173},[71],[15,16304,16306,16309],{"className":16305},[57],[15,16307],{"className":16308,"style":1836},[61],[15,16310,1804],{"className":16311},[66]," 时就是如此：目标函数关于 ",[15,16314,16316,16329],{"className":16315,"translate":19},[18],[15,16317,16319],{"className":16318},[23],[25,16320,16321],{"xmlns":27},[29,16322,16323,16327],{},[32,16324,16325],{},[35,16326,116],{},[45,16328,116],{"encoding":47},[15,16330,16332],{"className":16331,"ariaHidden":53},[52],[15,16333,16335,16338],{"className":16334},[57],[15,16336],{"className":16337,"style":3408},[61],[15,16339,116],{"className":16340,"style":158},[66,67]," 严格凸，",[15,16343,16345,16362],{"className":16344,"translate":19},[18],[15,16346,16348],{"className":16347},[23],[25,16349,16350],{"xmlns":27},[29,16351,16352,16360],{},[32,16353,16354],{},[131,16355,16356,16358],{},[35,16357,116],{},[39,16359,1162],{},[45,16361,16221],{"encoding":47},[15,16363,16365],{"className":16364,"ariaHidden":53},[52],[15,16366,16368,16371],{"className":16367},[57],[15,16369],{"className":16370,"style":3408},[61],[15,16372,16374,16377],{"className":16373},[66],[15,16375,116],{"className":16376,"style":158},[66,67],[15,16378,16380],{"className":16379},[198],[15,16381,16383],{"className":16382},[202],[15,16384,16386],{"className":16385},[206],[15,16387,16389],{"className":16388,"style":16249},[210],[15,16390,16391,16394],{"style":213},[15,16392],{"className":16393,"style":218},[217],[15,16395,16397],{"className":16396},[222,223,224,225],[15,16398,1162],{"className":16399},[76,225]," 唯一；而 ",[15,16402,16404,16443],{"className":16403,"translate":19},[18],[15,16405,16407],{"className":16406},[23],[25,16408,16409],{"xmlns":27},[29,16410,16411,16440],{},[32,16412,16413,16415,16417,16419,16421,16423,16425,16428,16438],{},[35,16414,8390],{},[39,16416,129],{},[39,16418,120],{"stretchy":119},[35,16420,2853],{},[39,16422,1245],{},[35,16424,1248],{},[35,16426,16427],{},"I",[131,16429,16430,16432],{},[39,16431,126],{"stretchy":119},[32,16433,16434,16436],{},[39,16435,6602],{},[1209,16437,1211],{},[35,16439,1224],{},[45,16441,16442],{"encoding":47},"\\alpha=(K+\\lambda I)^{-1}y",[15,16444,16446,16464,16485],{"className":16445,"ariaHidden":53},[52],[15,16447,16449,16452,16455,16458,16461],{"className":16448},[57],[15,16450],{"className":16451,"style":86},[61],[15,16453,8390],{"className":16454,"style":8560},[66,67],[15,16456],{"className":16457,"style":173},[71],[15,16459,129],{"className":16460},[177],[15,16462],{"className":16463,"style":173},[71],[15,16465,16467,16470,16473,16476,16479,16482],{"className":16466},[57],[15,16468],{"className":16469,"style":154},[61],[15,16471,120],{"className":16472},[162],[15,16474,2853],{"className":16475,"style":2901},[66,67],[15,16477],{"className":16478,"style":72},[71],[15,16480,1245],{"className":16481},[76],[15,16483],{"className":16484,"style":72},[71],[15,16486,16488,16491,16494,16498,16533],{"className":16487},[57],[15,16489],{"className":16490,"style":9154},[61],[15,16492,1248],{"className":16493},[66,67],[15,16495,16427],{"className":16496,"style":16497},[66,67],"margin-right:0.0785em;",[15,16499,16501,16504],{"className":16500},[169],[15,16502,126],{"className":16503},[169],[15,16505,16507],{"className":16506},[198],[15,16508,16510],{"className":16509},[202],[15,16511,16513],{"className":16512},[206],[15,16514,16516],{"className":16515,"style":6740},[210],[15,16517,16518,16521],{"style":213},[15,16519],{"className":16520,"style":218},[217],[15,16522,16524],{"className":16523},[222,223,224,225],[15,16525,16527,16530],{"className":16526},[66,225],[15,16528,6602],{"className":16529},[66,225],[15,16531,1211],{"className":16532},[66,225],[15,16534,1224],{"className":16535,"style":1515},[66,67]," 只是其中一个系数表示，所有相差 ",[15,16538,16540,16553],{"className":16539,"translate":19},[18],[15,16541,16543],{"className":16542},[23],[25,16544,16545],{"xmlns":27},[29,16546,16547,16551],{},[32,16548,16549],{},[35,16550,2853],{},[45,16552,2853],{"encoding":47},[15,16554,16556],{"className":16555,"ariaHidden":53},[52],[15,16557,16559,16562],{"className":16558},[57],[15,16560],{"className":16561,"style":490},[61],[15,16563,2853],{"className":16564,"style":2901},[66,67]," 零空间的系数给出同一个函数。",[3194,16567,16568,16630],{},[2040,16569,16570,16571,16629],{},"损失不严格凸时，",[15,16572,16574,16591],{"className":16573,"translate":19},[18],[15,16575,16577],{"className":16576},[23],[25,16578,16579],{"xmlns":27},[29,16580,16581,16589],{},[32,16582,16583],{},[131,16584,16585,16587],{},[35,16586,116],{},[39,16588,1162],{},[45,16590,16221],{"encoding":47},[15,16592,16594],{"className":16593,"ariaHidden":53},[52],[15,16595,16597,16600],{"className":16596},[57],[15,16598],{"className":16599,"style":3408},[61],[15,16601,16603,16606],{"className":16602},[66],[15,16604,116],{"className":16605,"style":158},[66,67],[15,16607,16609],{"className":16608},[198],[15,16610,16612],{"className":16611},[202],[15,16613,16615],{"className":16614},[206],[15,16616,16618],{"className":16617,"style":16249},[210],[15,16619,16620,16623],{"style":213},[15,16621],{"className":16622,"style":218},[217],[15,16624,16626],{"className":16625},[222,223,224,225],[15,16627,1162],{"className":16628},[76,225]," 也可能不唯一。"," 例如 hinge loss 在某些数据上会有平坦区域；表示定理仍然成立，但最优解可能是一个集合。",[3194,16632,16633,16636,16637,16703,16704,16732,16733,16761,16762,17010,17011,17039,17040,17043],{},[2040,16634,16635],{},"偏置项要单独处理。"," 如果模型是 ",[15,16638,16640,16664],{"className":16639,"translate":19},[18],[15,16641,16643],{"className":16642},[23],[25,16644,16645],{"xmlns":27},[29,16646,16647,16661],{},[32,16648,16649,16651,16653,16655,16657,16659],{},[35,16650,116],{},[39,16652,120],{"stretchy":119},[35,16654,123],{},[39,16656,126],{"stretchy":119},[39,16658,1245],{},[35,16660,4587],{},[45,16662,16663],{"encoding":47},"f(x)+b",[15,16665,16667,16694],{"className":16666,"ariaHidden":53},[52],[15,16668,16670,16673,16676,16679,16682,16685,16688,16691],{"className":16669},[57],[15,16671],{"className":16672,"style":154},[61],[15,16674,116],{"className":16675,"style":158},[66,67],[15,16677,120],{"className":16678},[162],[15,16680,123],{"className":16681},[66,67],[15,16683,126],{"className":16684},[169],[15,16686],{"className":16687,"style":72},[71],[15,16689,1245],{"className":16690},[76],[15,16692],{"className":16693,"style":72},[71],[15,16695,16697,16700],{"className":16696},[57],[15,16698],{"className":16699,"style":840},[61],[15,16701,4587],{"className":16702},[66,67],"，而 ",[15,16705,16707,16720],{"className":16706,"translate":19},[18],[15,16708,16710],{"className":16709},[23],[25,16711,16712],{"xmlns":27},[29,16713,16714,16718],{},[32,16715,16716],{},[35,16717,4587],{},[45,16719,4587],{"encoding":47},[15,16721,16723],{"className":16722,"ariaHidden":53},[52],[15,16724,16726,16729],{"className":16725},[57],[15,16727],{"className":16728,"style":840},[61],[15,16730,4587],{"className":16731},[66,67]," 不受正则项约束，那么表示定理直接作用于 ",[15,16734,16736,16749],{"className":16735,"translate":19},[18],[15,16737,16739],{"className":16738},[23],[25,16740,16741],{"xmlns":27},[29,16742,16743,16747],{},[32,16744,16745],{},[35,16746,116],{},[45,16748,116],{"encoding":47},[15,16750,16752],{"className":16751,"ariaHidden":53},[52],[15,16753,16755,16758],{"className":16754},[57],[15,16756],{"className":16757,"style":3408},[61],[15,16759,116],{"className":16760,"style":158},[66,67]," 的部分仍然给出 ",[15,16763,16765,16813],{"className":16764,"translate":19},[18],[15,16766,16768],{"className":16767},[23],[25,16769,16770],{"xmlns":27},[29,16771,16772,16810],{},[32,16773,16774,16780,16782,16788,16794,16796,16798,16800,16802,16808],{},[131,16775,16776,16778],{},[35,16777,116],{},[39,16779,1162],{},[39,16781,129],{},[1057,16783,16784,16786],{},[39,16785,1200],{},[35,16787,1205],{},[1057,16789,16790,16792],{},[35,16791,8390],{},[35,16793,1205],{},[35,16795,621],{},[39,16797,120],{"stretchy":119},[39,16799,3268],{},[39,16801,335],{"separator":53},[1057,16803,16804,16806],{},[35,16805,123],{},[35,16807,1205],{},[39,16809,126],{"stretchy":119},[45,16811,16812],{"encoding":47},"f^*=\\sum_i\\alpha_i k(\\cdot,x_i)",[15,16814,16816,16860],{"className":16815,"ariaHidden":53},[52],[15,16817,16819,16822,16851,16854,16857],{"className":16818},[57],[15,16820],{"className":16821,"style":3408},[61],[15,16823,16825,16828],{"className":16824},[66],[15,16826,116],{"className":16827,"style":158},[66,67],[15,16829,16831],{"className":16830},[198],[15,16832,16834],{"className":16833},[202],[15,16835,16837],{"className":16836},[206],[15,16838,16840],{"className":16839,"style":16249},[210],[15,16841,16842,16845],{"style":213},[15,16843],{"className":16844,"style":218},[217],[15,16846,16848],{"className":16847},[222,223,224,225],[15,16849,1162],{"className":16850},[76,225],[15,16852],{"className":16853,"style":173},[71],[15,16855,129],{"className":16856},[177],[15,16858],{"className":16859,"style":173},[71],[15,16861,16863,16867,16909,16912,16952,16955,16958,16961,16964,16967,17007],{"className":16862},[57],[15,16864],{"className":16865,"style":16866},[61],"height:1.0497em;vertical-align:-0.2997em;",[15,16868,16870,16873],{"className":16869},[1339],[15,16871,1200],{"className":16872,"style":13702},[1339,1470,13701],[15,16874,16876],{"className":16875},[198],[15,16877,16879,16900],{"className":16878},[202,1090],[15,16880,16882,16897],{"className":16881},[206],[15,16883,16886],{"className":16884,"style":16885},[210],"height:0.162em;",[15,16887,16888,16891],{"style":13718},[15,16889],{"className":16890,"style":218},[217],[15,16892,16894],{"className":16893},[222,223,224,225],[15,16895,1205],{"className":16896},[66,67,225],[15,16898,1114],{"className":16899},[1113],[15,16901,16903],{"className":16902},[206],[15,16904,16907],{"className":16905,"style":16906},[210],"height:0.2997em;",[15,16908],{},[15,16910],{"className":16911,"style":697},[71],[15,16913,16915,16918],{"className":16914},[66],[15,16916,8390],{"className":16917,"style":8560},[66,67],[15,16919,16921],{"className":16920},[198],[15,16922,16924,16944],{"className":16923},[202,1090],[15,16925,16927,16941],{"className":16926},[206],[15,16928,16930],{"className":16929,"style":1528},[210],[15,16931,16932,16935],{"style":8575},[15,16933],{"className":16934,"style":218},[217],[15,16936,16938],{"className":16937},[222,223,224,225],[15,16939,1205],{"className":16940},[66,67,225],[15,16942,1114],{"className":16943},[1113],[15,16945,16947],{"className":16946},[206],[15,16948,16950],{"className":16949,"style":1121},[210],[15,16951],{},[15,16953,621],{"className":16954,"style":684},[66,67],[15,16956,120],{"className":16957},[162],[15,16959,3268],{"className":16960},[66],[15,16962,335],{"className":16963},[422],[15,16965],{"className":16966,"style":697},[71],[15,16968,16970,16973],{"className":16969},[66],[15,16971,123],{"className":16972},[66,67],[15,16974,16976],{"className":16975},[198],[15,16977,16979,16999],{"className":16978},[202,1090],[15,16980,16982,16996],{"className":16981},[206],[15,16983,16985],{"className":16984,"style":1528},[210],[15,16986,16987,16990],{"style":1584},[15,16988],{"className":16989,"style":218},[217],[15,16991,16993],{"className":16992},[222,223,224,225],[15,16994,1205],{"className":16995},[66,67,225],[15,16997,1114],{"className":16998},[1113],[15,17000,17002],{"className":17001},[206],[15,17003,17005],{"className":17004,"style":1121},[210],[15,17006],{},[15,17008,126],{"className":17009},[169],"，但 ",[15,17012,17014,17027],{"className":17013,"translate":19},[18],[15,17015,17017],{"className":17016},[23],[25,17018,17019],{"xmlns":27},[29,17020,17021,17025],{},[32,17022,17023],{},[35,17024,4587],{},[45,17026,4587],{"encoding":47},[15,17028,17030],{"className":17029,"ariaHidden":53},[52],[15,17031,17033,17036],{"className":17032},[57],[15,17034],{"className":17035,"style":840},[61],[15,17037,4587],{"className":17038},[66,67]," 需要通过有限维优化或 KKT 条件单独求出。更系统的处理是",[2040,17041,17042],{},"半参数表示定理","（semi-parametric representer theorem）：若模型为",[15,17045,17047],{"className":17046,"translate":19},[294],[15,17048,17050,17128],{"className":17049,"translate":19},[18],[15,17051,17053],{"className":17052},[23],[25,17054,17055],{"xmlns":27,"display":303},[29,17056,17057,17125],{},[32,17058,17059,17061,17063,17065,17067,17069,17083,17090,17097,17099,17101,17103,17105,17107,17109,17111,17113,17115,17117,17119,17121,17123],{},[35,17060,116],{},[39,17062,120],{"stretchy":119},[35,17064,123],{},[39,17066,126],{"stretchy":119},[39,17068,129],{},[1196,17070,17071,17073,17081],{},[39,17072,1200],{},[32,17074,17075,17077,17079],{},[35,17076,2441],{},[39,17078,129],{},[1209,17080,1211],{},[35,17082,11],{},[1057,17084,17085,17088],{},[35,17086,17087],{},"β",[35,17089,2441],{},[1057,17091,17092,17095],{},[35,17093,17094],{},"ϕ",[35,17096,2441],{},[39,17098,120],{"stretchy":119},[35,17100,123],{},[39,17102,126],{"stretchy":119},[39,17104,1245],{},[35,17106,1344],{},[39,17108,120],{"stretchy":119},[35,17110,123],{},[39,17112,126],{"stretchy":119},[39,17114,335],{"separator":53},[71,17116],{"width":3904},[35,17118,1344],{},[39,17120,1187],{},[35,17122,1061],{"mathvariant":445},[39,17124,335],{"separator":53},[45,17126,17127],{"encoding":47},"f(x)=\\sum_{j=1}^p\\beta_j\\phi_j(x)+g(x),\\qquad g\\in\\mathcal H,",[15,17129,17131,17158,17334,17373],{"className":17130,"ariaHidden":53},[52],[15,17132,17134,17137,17140,17143,17146,17149,17152,17155],{"className":17133},[57],[15,17135],{"className":17136,"style":154},[61],[15,17138,116],{"className":17139,"style":158},[66,67],[15,17141,120],{"className":17142},[162],[15,17144,123],{"className":17145},[66,67],[15,17147,126],{"className":17148},[169],[15,17150],{"className":17151,"style":173},[71],[15,17153,129],{"className":17154},[177],[15,17156],{"className":17157,"style":173},[71],[15,17159,17161,17165,17231,17234,17276,17316,17319,17322,17325,17328,17331],{"className":17160},[57],[15,17162],{"className":17163,"style":17164},[61],"height:3.1123em;vertical-align:-1.4138em;",[15,17166,17168],{"className":17167},[1339,1351],[15,17169,17171,17223],{"className":17170},[202,1090],[15,17172,17174,17220],{"className":17173},[206],[15,17175,17178,17198,17208],{"className":17176,"style":17177},[210],"height:1.6985em;",[15,17179,17180,17183],{"style":1439},[15,17181],{"className":17182,"style":1443},[217],[15,17184,17186],{"className":17185},[222,223,224,225],[15,17187,17189,17192,17195],{"className":17188},[66,225],[15,17190,2441],{"className":17191,"style":2594},[66,67,225],[15,17193,129],{"className":17194},[177,225],[15,17196,1211],{"className":17197},[66,225],[15,17199,17200,17203],{"style":1461},[15,17201],{"className":17202,"style":1443},[217],[15,17204,17205],{},[15,17206,1200],{"className":17207},[1339,1470,1471],[15,17209,17211,17214],{"style":17210},"top:-4.3471em;margin-left:0em;",[15,17212],{"className":17213,"style":1443},[217],[15,17215,17217],{"className":17216},[222,223,224,225],[15,17218,11],{"className":17219},[66,67,225],[15,17221,1114],{"className":17222},[1113],[15,17224,17226],{"className":17225},[206],[15,17227,17229],{"className":17228,"style":2631},[210],[15,17230],{},[15,17232],{"className":17233,"style":697},[71],[15,17235,17237,17241],{"className":17236},[66],[15,17238,17087],{"className":17239,"style":17240},[66,67],"margin-right:0.0528em;",[15,17242,17244],{"className":17243},[198],[15,17245,17247,17268],{"className":17246},[202,1090],[15,17248,17250,17265],{"className":17249},[206],[15,17251,17253],{"className":17252,"style":1528},[210],[15,17254,17256,17259],{"style":17255},"top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;",[15,17257],{"className":17258,"style":218},[217],[15,17260,17262],{"className":17261},[222,223,224,225],[15,17263,2441],{"className":17264,"style":2594},[66,67,225],[15,17266,1114],{"className":17267},[1113],[15,17269,17271],{"className":17270},[206],[15,17272,17274],{"className":17273,"style":2715},[210],[15,17275],{},[15,17277,17279,17282],{"className":17278},[66],[15,17280,17094],{"className":17281},[66,67],[15,17283,17285],{"className":17284},[198],[15,17286,17288,17308],{"className":17287},[202,1090],[15,17289,17291,17305],{"className":17290},[206],[15,17292,17294],{"className":17293,"style":1528},[210],[15,17295,17296,17299],{"style":1584},[15,17297],{"className":17298,"style":218},[217],[15,17300,17302],{"className":17301},[222,223,224,225],[15,17303,2441],{"className":17304,"style":2594},[66,67,225],[15,17306,1114],{"className":17307},[1113],[15,17309,17311],{"className":17310},[206],[15,17312,17314],{"className":17313,"style":2715},[210],[15,17315],{},[15,17317,120],{"className":17318},[162],[15,17320,123],{"className":17321},[66,67],[15,17323,126],{"className":17324},[169],[15,17326],{"className":17327,"style":72},[71],[15,17329,1245],{"className":17330},[76],[15,17332],{"className":17333,"style":72},[71],[15,17335,17337,17340,17343,17346,17349,17352,17355,17358,17361,17364,17367,17370],{"className":17336},[57],[15,17338],{"className":17339,"style":154},[61],[15,17341,1344],{"className":17342,"style":1515},[66,67],[15,17344,120],{"className":17345},[162],[15,17347,123],{"className":17348},[66,67],[15,17350,126],{"className":17351},[169],[15,17353,335],{"className":17354},[422],[15,17356],{"className":17357,"style":4061},[71],[15,17359],{"className":17360,"style":697},[71],[15,17362,1344],{"className":17363,"style":1515},[66,67],[15,17365],{"className":17366,"style":173},[71],[15,17368,1187],{"className":17369},[177],[15,17371],{"className":17372,"style":173},[71],[15,17374,17376,17380,17383],{"className":17375},[57],[15,17377],{"className":17378,"style":17379},[61],"height:0.8778em;vertical-align:-0.1944em;",[15,17381,1061],{"className":17382,"style":1083},[66,494],[15,17384,335],{"className":17385},[422],[11,17387,425,17388,17529,17530,17629,17630,17658,17659,17688],{},[15,17389,17391,17421],{"className":17390,"translate":19},[18],[15,17392,17394],{"className":17393},[23],[25,17395,17396],{"xmlns":27},[29,17397,17398,17418],{},[32,17399,17400,17406,17408,17410,17412],{},[1057,17401,17402,17404],{},[35,17403,17094],{},[1209,17405,1211],{},[39,17407,335],{"separator":53},[39,17409,2095],{},[39,17411,335],{"separator":53},[1057,17413,17414,17416],{},[35,17415,17094],{},[35,17417,11],{},[45,17419,17420],{"encoding":47},"\\phi_1,\\dots,\\phi_p",[15,17422,17424],{"className":17423,"ariaHidden":53},[52],[15,17425,17427,17431,17471,17474,17477,17480,17483,17486,17489],{"className":17426},[57],[15,17428],{"className":17429,"style":17430},[61],"height:0.9805em;vertical-align:-0.2861em;",[15,17432,17434,17437],{"className":17433},[66],[15,17435,17094],{"className":17436},[66,67],[15,17438,17440],{"className":17439},[198],[15,17441,17443,17463],{"className":17442},[202,1090],[15,17444,17446,17460],{"className":17445},[206],[15,17447,17449],{"className":17448,"style":2139},[210],[15,17450,17451,17454],{"style":1584},[15,17452],{"className":17453,"style":218},[217],[15,17455,17457],{"className":17456},[222,223,224,225],[15,17458,1211],{"className":17459},[66,225],[15,17461,1114],{"className":17462},[1113],[15,17464,17466],{"className":17465},[206],[15,17467,17469],{"className":17468,"style":1121},[210],[15,17470],{},[15,17472,335],{"className":17473},[422],[15,17475],{"className":17476,"style":697},[71],[15,17478,2095],{"className":17479},[1413],[15,17481],{"className":17482,"style":697},[71],[15,17484,335],{"className":17485},[422],[15,17487],{"className":17488,"style":697},[71],[15,17490,17492,17495],{"className":17491},[66],[15,17493,17094],{"className":17494},[66,67],[15,17496,17498],{"className":17497},[198],[15,17499,17501,17521],{"className":17500},[202,1090],[15,17502,17504,17518],{"className":17503},[206],[15,17505,17507],{"className":17506,"style":2198},[210],[15,17508,17509,17512],{"style":1584},[15,17510],{"className":17511,"style":218},[217],[15,17513,17515],{"className":17514},[222,223,224,225],[15,17516,11],{"className":17517},[66,67,225],[15,17519,1114],{"className":17520},[1113],[15,17522,17524],{"className":17523},[206],[15,17525,17527],{"className":17526,"style":2715},[210],[15,17528],{}," 张成一个有限维空间（例如常数、线性函数），正则项只惩罚 ",[15,17531,17533,17557],{"className":17532,"translate":19},[18],[15,17534,17536],{"className":17535},[23],[25,17537,17538],{"xmlns":27},[29,17539,17540,17554],{},[32,17541,17542,17544,17546],{},[35,17543,1251],{"mathvariant":138},[35,17545,1344],{},[1255,17547,17548,17550,17552],{},[35,17549,1251],{"mathvariant":138},[35,17551,1061],{"mathvariant":445},[1209,17553,1263],{},[45,17555,17556],{"encoding":47},"\\|g\\|_{\\mathcal H}^2",[15,17558,17560],{"className":17559,"ariaHidden":53},[52],[15,17561,17563,17567,17570,17573],{"className":17562},[57],[15,17564],{"className":17565,"style":17566},[61],"height:1.0894em;vertical-align:-0.2753em;",[15,17568,1251],{"className":17569},[66],[15,17571,1344],{"className":17572,"style":1515},[66,67],[15,17574,17576,17579],{"className":17575},[66],[15,17577,1251],{"className":17578},[66],[15,17580,17582],{"className":17581},[198],[15,17583,17585,17620],{"className":17584},[202,1090],[15,17586,17588,17617],{"className":17587},[206],[15,17589,17591,17606],{"className":17590,"style":6740},[210],[15,17592,17594,17597],{"style":17593},"top:-2.4247em;margin-left:0em;margin-right:0.05em;",[15,17595],{"className":17596,"style":218},[217],[15,17598,17600],{"className":17599},[222,223,224,225],[15,17601,17603],{"className":17602},[66,225],[15,17604,1061],{"className":17605,"style":1083},[66,494,225],[15,17607,17608,17611],{"style":213},[15,17609],{"className":17610,"style":218},[217],[15,17612,17614],{"className":17613},[222,223,224,225],[15,17615,1263],{"className":17616},[66,225],[15,17618,1114],{"className":17619},[1113],[15,17621,17623],{"className":17622},[206],[15,17624,17627],{"className":17625,"style":17626},[210],"height:0.2753em;",[15,17628],{},"，则最优的 ",[15,17631,17633,17646],{"className":17632,"translate":19},[18],[15,17634,17636],{"className":17635},[23],[25,17637,17638],{"xmlns":27},[29,17639,17640,17644],{},[32,17641,17642],{},[35,17643,1344],{},[45,17645,1344],{"encoding":47},[15,17647,17649],{"className":17648,"ariaHidden":53},[52],[15,17650,17652,17655],{"className":17651},[57],[15,17653],{"className":17654,"style":471},[61],[15,17656,1344],{"className":17657,"style":1515},[66,67]," 仍在训练点的张成空间中，而 ",[15,17660,17662,17676],{"className":17661,"translate":19},[18],[15,17663,17665],{"className":17664},[23],[25,17666,17667],{"xmlns":27},[29,17668,17669,17673],{},[32,17670,17671],{},[35,17672,17087],{},[45,17674,17675],{"encoding":47},"\\beta",[15,17677,17679],{"className":17678,"ariaHidden":53},[52],[15,17680,17682,17685],{"className":17681},[57],[15,17683],{"className":17684,"style":3408},[61],[15,17686,17087],{"className":17687,"style":17240},[66,67]," 通过一个有限维系统求解。SVM 的偏置、平滑样条的多项式部分都属于这种情况。",[95,17690,17692],{"id":17691},"四为什么是范数的非减函数","四、为什么是「范数的非减函数」",[1904,17694,17696],{"id":17695},"_41-充分性","4.1 充分性",[11,17698,17699,17700,17728,17729,17798,17799,17868,17869,3467,18041,18069],{},"第三节的证明已经说明：只要 ",[15,17701,17703,17716],{"className":17702,"translate":19},[18],[15,17704,17706],{"className":17705},[23],[25,17707,17708],{"xmlns":27},[29,17709,17710,17714],{},[32,17711,17712],{},[35,17713,7742],{"mathvariant":138},[45,17715,8704],{"encoding":47},[15,17717,17719],{"className":17718,"ariaHidden":53},[52],[15,17720,17722,17725],{"className":17721},[57],[15,17723],{"className":17724,"style":490},[61],[15,17726,7742],{"className":17727},[66]," 非减，投影 ",[15,17730,17732,17749],{"className":17731,"translate":19},[18],[15,17733,17735],{"className":17734},[23],[25,17736,17737],{"xmlns":27},[29,17738,17739,17747],{},[32,17740,17741],{},[1057,17742,17743,17745],{},[35,17744,116],{},[39,17746,1251],{},[45,17748,12654],{"encoding":47},[15,17750,17752],{"className":17751,"ariaHidden":53},[52],[15,17753,17755,17758],{"className":17754},[57],[15,17756],{"className":17757,"style":9840},[61],[15,17759,17761,17764],{"className":17760},[66],[15,17762,116],{"className":17763,"style":158},[66,67],[15,17765,17767],{"className":17766},[198],[15,17768,17770,17790],{"className":17769},[202,1090],[15,17771,17773,17787],{"className":17772},[206],[15,17774,17776],{"className":17775,"style":3668},[210],[15,17777,17778,17781],{"style":9861},[15,17779],{"className":17780,"style":218},[217],[15,17782,17784],{"className":17783},[222,223,224,225],[15,17785,1251],{"className":17786},[177,225],[15,17788,1114],{"className":17789},[1113],[15,17791,17793],{"className":17792},[206],[15,17794,17796],{"className":17795,"style":9880},[210],[15,17797],{}," 就不会让目标变差，因此总有一个最优解落在 ",[15,17800,17802,17819],{"className":17801,"translate":19},[18],[15,17803,17805],{"className":17804},[23],[25,17806,17807],{"xmlns":27},[29,17808,17809,17817],{},[32,17810,17811],{},[1057,17812,17813,17815],{},[35,17814,1061],{"mathvariant":445},[1209,17816,1804],{},[45,17818,13449],{"encoding":47},[15,17820,17822],{"className":17821,"ariaHidden":53},[52],[15,17823,17825,17828],{"className":17824},[57],[15,17826],{"className":17827,"style":1076},[61],[15,17829,17831,17834],{"className":17830},[66],[15,17832,1061],{"className":17833,"style":1083},[66,494],[15,17835,17837],{"className":17836},[198],[15,17838,17840,17860],{"className":17839},[202,1090],[15,17841,17843,17857],{"className":17842},[206],[15,17844,17846],{"className":17845,"style":2139},[210],[15,17847,17848,17851],{"style":1100},[15,17849],{"className":17850,"style":218},[217],[15,17852,17854],{"className":17853},[222,223,224,225],[15,17855,1804],{"className":17856},[66,225],[15,17858,1114],{"className":17859},[1113],[15,17861,17863],{"className":17862},[206],[15,17864,17866],{"className":17865,"style":1121},[210],[15,17867],{}," 中。这一步只用了 ",[15,17870,17872,17910],{"className":17871,"translate":19},[18],[15,17873,17875],{"className":17874},[23],[25,17876,17877],{"xmlns":27},[29,17878,17879,17907],{},[32,17880,17881,17883,17885,17891,17893,17895,17901],{},[35,17882,1251],{"mathvariant":138},[35,17884,116],{},[131,17886,17887,17889],{},[35,17888,1251],{"mathvariant":138},[1209,17890,1263],{},[39,17892,1801],{},[35,17894,1251],{"mathvariant":138},[1057,17896,17897,17899],{},[35,17898,116],{},[39,17900,1251],{},[131,17902,17903,17905],{},[35,17904,1251],{"mathvariant":138},[1209,17906,1263],{},[45,17908,17909],{"encoding":47},"\\|f\\|^2\\ge\\|f_\\parallel\\|^2",[15,17911,17913,17963],{"className":17912,"ariaHidden":53},[52],[15,17914,17916,17919,17922,17925,17954,17957,17960],{"className":17915},[57],[15,17917],{"className":17918,"style":9154},[61],[15,17920,1251],{"className":17921},[66],[15,17923,116],{"className":17924,"style":158},[66,67],[15,17926,17928,17931],{"className":17927},[66],[15,17929,1251],{"className":17930},[66],[15,17932,17934],{"className":17933},[198],[15,17935,17937],{"className":17936},[202],[15,17938,17940],{"className":17939},[206],[15,17941,17943],{"className":17942,"style":6740},[210],[15,17944,17945,17948],{"style":213},[15,17946],{"className":17947,"style":218},[217],[15,17949,17951],{"className":17950},[222,223,224,225],[15,17952,1263],{"className":17953},[66,225],[15,17955],{"className":17956,"style":173},[71],[15,17958,1801],{"className":17959},[177],[15,17961],{"className":17962,"style":173},[71],[15,17964,17966,17969,17972,18012],{"className":17965},[57],[15,17967],{"className":17968,"style":13233},[61],[15,17970,1251],{"className":17971},[66],[15,17973,17975,17978],{"className":17974},[66],[15,17976,116],{"className":17977,"style":158},[66,67],[15,17979,17981],{"className":17980},[198],[15,17982,17984,18004],{"className":17983},[202,1090],[15,17985,17987,18001],{"className":17986},[206],[15,17988,17990],{"className":17989,"style":3668},[210],[15,17991,17992,17995],{"style":9861},[15,17993],{"className":17994,"style":218},[217],[15,17996,17998],{"className":17997},[222,223,224,225],[15,17999,1251],{"className":18000},[177,225],[15,18002,1114],{"className":18003},[1113],[15,18005,18007],{"className":18006},[206],[15,18008,18010],{"className":18009,"style":9880},[210],[15,18011],{},[15,18013,18015,18018],{"className":18014},[66],[15,18016,1251],{"className":18017},[66],[15,18019,18021],{"className":18020},[198],[15,18022,18024],{"className":18023},[202],[15,18025,18027],{"className":18026},[206],[15,18028,18030],{"className":18029,"style":6740},[210],[15,18031,18032,18035],{"style":213},[15,18033],{"className":18034,"style":218},[217],[15,18036,18038],{"className":18037},[222,223,224,225],[15,18039,1263],{"className":18040},[66,225],[15,18042,18044,18057],{"className":18043,"translate":19},[18],[15,18045,18047],{"className":18046},[23],[25,18048,18049],{"xmlns":27},[29,18050,18051,18055],{},[32,18052,18053],{},[35,18054,7742],{"mathvariant":138},[45,18056,8704],{"encoding":47},[15,18058,18060],{"className":18059,"ariaHidden":53},[52],[15,18061,18063,18066],{"className":18062},[57],[15,18064],{"className":18065,"style":490},[61],[15,18067,7742],{"className":18068},[66]," 的单调性，没有别的东西。",[1904,18071,18073],{"id":18072},"_42-反例非单调正则项","4.2 反例：非单调正则项",[11,18075,18076,18077,18105],{},"反过来，如果 ",[15,18078,18080,18093],{"className":18079,"translate":19},[18],[15,18081,18083],{"className":18082},[23],[25,18084,18085],{"xmlns":27},[29,18086,18087,18091],{},[32,18088,18089],{},[35,18090,7742],{"mathvariant":138},[45,18092,8704],{"encoding":47},[15,18094,18096],{"className":18095,"ariaHidden":53},[52],[15,18097,18099,18102],{"className":18098},[57],[15,18100],{"className":18101,"style":490},[61],[15,18103,7742],{"className":18104},[66]," 不是非减的，表示定理就可能失效。考虑一个最小的例子。",[11,18107,18108,18109,18190,18191,18359,18360,18441,18442,18557,18558,18650],{},"取 ",[15,18110,18112,18134],{"className":18111,"translate":19},[18],[15,18113,18115],{"className":18114},[23],[25,18116,18117],{"xmlns":27},[29,18118,18119,18131],{},[32,18120,18121,18123,18125],{},[35,18122,446],{"mathvariant":445},[39,18124,129],{},[131,18126,18127,18129],{},[35,18128,455],{"mathvariant":454},[1209,18130,1263],{},[45,18132,18133],{"encoding":47},"\\mathcal X=\\mathbb R^2",[15,18135,18137,18155],{"className":18136,"ariaHidden":53},[52],[15,18138,18140,18143,18146,18149,18152],{"className":18139},[57],[15,18141],{"className":18142,"style":490},[61],[15,18144,446],{"className":18145,"style":495},[66,494],[15,18147],{"className":18148,"style":173},[71],[15,18150,129],{"className":18151},[177],[15,18153],{"className":18154,"style":173},[71],[15,18156,18158,18161],{"className":18157},[57],[15,18159],{"className":18160,"style":6740},[61],[15,18162,18164,18167],{"className":18163},[66],[15,18165,455],{"className":18166},[66,517],[15,18168,18170],{"className":18169},[198],[15,18171,18173],{"className":18172},[202],[15,18174,18176],{"className":18175},[206],[15,18177,18179],{"className":18178,"style":6740},[210],[15,18180,18181,18184],{"style":213},[15,18182],{"className":18183,"style":218},[217],[15,18185,18187],{"className":18186},[222,223,224,225],[15,18188,1263],{"className":18189},[66,225],"，线性核 ",[15,18192,18194,18238],{"className":18193,"translate":19},[18],[15,18195,18197],{"className":18196},[23],[25,18198,18199],{"xmlns":27},[29,18200,18201,18235],{},[32,18202,18203,18205,18207,18209,18211,18217,18219,18221,18223,18225,18227,18233],{},[35,18204,621],{},[39,18206,120],{"stretchy":119},[35,18208,123],{},[39,18210,335],{"separator":53},[131,18212,18213,18215],{},[35,18214,123],{},[39,18216,635],{"mathvariant":138,"lspace":634,"rspace":634},[39,18218,126],{"stretchy":119},[39,18220,129],{},[39,18222,642],{"stretchy":119},[35,18224,123],{},[39,18226,335],{"separator":53},[131,18228,18229,18231],{},[35,18230,123],{},[39,18232,635],{"mathvariant":138,"lspace":634,"rspace":634},[39,18234,667],{"stretchy":119},[45,18236,18237],{"encoding":47},"k(x,x')=\\langle x,x'\\rangle",[15,18239,18241,18306],{"className":18240,"ariaHidden":53},[52],[15,18242,18244,18247,18250,18253,18256,18259,18262,18294,18297,18300,18303],{"className":18243},[57],[15,18245],{"className":18246,"style":5959},[61],[15,18248,621],{"className":18249,"style":684},[66,67],[15,18251,120],{"className":18252},[162],[15,18254,123],{"className":18255},[66,67],[15,18257,335],{"className":18258},[422],[15,18260],{"className":18261,"style":697},[71],[15,18263,18265,18268],{"className":18264},[66],[15,18266,123],{"className":18267},[66,67],[15,18269,18271],{"className":18270},[198],[15,18272,18274],{"className":18273},[202],[15,18275,18277],{"className":18276},[206],[15,18278,18280],{"className":18279,"style":6005},[210],[15,18281,18282,18285],{"style":213},[15,18283],{"className":18284,"style":218},[217],[15,18286,18288],{"className":18287},[222,223,224,225],[15,18289,18291],{"className":18290},[66,225],[15,18292,635],{"className":18293},[66,225],[15,18295,126],{"className":18296},[169],[15,18298],{"className":18299,"style":173},[71],[15,18301,129],{"className":18302},[177],[15,18304],{"className":18305,"style":173},[71],[15,18307,18309,18312,18315,18318,18321,18324,18356],{"className":18308},[57],[15,18310],{"className":18311,"style":5959},[61],[15,18313,642],{"className":18314},[162],[15,18316,123],{"className":18317},[66,67],[15,18319,335],{"className":18320},[422],[15,18322],{"className":18323,"style":697},[71],[15,18325,18327,18330],{"className":18326},[66],[15,18328,123],{"className":18329},[66,67],[15,18331,18333],{"className":18332},[198],[15,18334,18336],{"className":18335},[202],[15,18337,18339],{"className":18338},[206],[15,18340,18342],{"className":18341,"style":6005},[210],[15,18343,18344,18347],{"style":213},[15,18345],{"className":18346,"style":218},[217],[15,18348,18350],{"className":18349},[222,223,224,225],[15,18351,18353],{"className":18352},[66,225],[15,18354,635],{"className":18355},[66,225],[15,18357,667],{"className":18358},[169],"，此时 ",[15,18361,18363,18385],{"className":18362,"translate":19},[18],[15,18364,18366],{"className":18365},[23],[25,18367,18368],{"xmlns":27},[29,18369,18370,18382],{},[32,18371,18372,18374,18376],{},[35,18373,1061],{"mathvariant":445},[39,18375,129],{},[131,18377,18378,18380],{},[35,18379,455],{"mathvariant":454},[1209,18381,1263],{},[45,18383,18384],{"encoding":47},"\\mathcal H=\\mathbb R^2",[15,18386,18388,18406],{"className":18387,"ariaHidden":53},[52],[15,18389,18391,18394,18397,18400,18403],{"className":18390},[57],[15,18392],{"className":18393,"style":490},[61],[15,18395,1061],{"className":18396,"style":1083},[66,494],[15,18398],{"className":18399,"style":173},[71],[15,18401,129],{"className":18402},[177],[15,18404],{"className":18405,"style":173},[71],[15,18407,18409,18412],{"className":18408},[57],[15,18410],{"className":18411,"style":6740},[61],[15,18413,18415,18418],{"className":18414},[66],[15,18416,455],{"className":18417},[66,517],[15,18419,18421],{"className":18420},[198],[15,18422,18424],{"className":18423},[202],[15,18425,18427],{"className":18426},[206],[15,18428,18430],{"className":18429,"style":6740},[210],[15,18431,18432,18435],{"style":213},[15,18433],{"className":18434,"style":218},[217],[15,18436,18438],{"className":18437},[222,223,224,225],[15,18439,1263],{"className":18440},[66,225],"。只有一个训练点 ",[15,18443,18445,18475],{"className":18444,"translate":19},[18],[15,18446,18448],{"className":18447},[23],[25,18449,18450],{"xmlns":27},[29,18451,18452,18472],{},[32,18453,18454,18460,18462,18464,18466,18468,18470],{},[1057,18455,18456,18458],{},[35,18457,123],{},[1209,18459,1211],{},[39,18461,129],{},[39,18463,120],{"stretchy":119},[1209,18465,1211],{},[39,18467,335],{"separator":53},[1209,18469,1804],{},[39,18471,126],{"stretchy":119},[45,18473,18474],{"encoding":47},"x_1=(1,0)",[15,18476,18478,18533],{"className":18477,"ariaHidden":53},[52],[15,18479,18481,18484,18524,18527,18530],{"className":18480},[57],[15,18482],{"className":18483,"style":10197},[61],[15,18485,18487,18490],{"className":18486},[66],[15,18488,123],{"className":18489},[66,67],[15,18491,18493],{"className":18492},[198],[15,18494,18496,18516],{"className":18495},[202,1090],[15,18497,18499,18513],{"className":18498},[206],[15,18500,18502],{"className":18501,"style":2139},[210],[15,18503,18504,18507],{"style":1584},[15,18505],{"className":18506,"style":218},[217],[15,18508,18510],{"className":18509},[222,223,224,225],[15,18511,1211],{"className":18512},[66,225],[15,18514,1114],{"className":18515},[1113],[15,18517,18519],{"className":18518},[206],[15,18520,18522],{"className":18521,"style":1121},[210],[15,18523],{},[15,18525],{"className":18526,"style":173},[71],[15,18528,129],{"className":18529},[177],[15,18531],{"className":18532,"style":173},[71],[15,18534,18536,18539,18542,18545,18548,18551,18554],{"className":18535},[57],[15,18537],{"className":18538,"style":154},[61],[15,18540,120],{"className":18541},[162],[15,18543,1211],{"className":18544},[66],[15,18546,335],{"className":18547},[422],[15,18549],{"className":18550,"style":697},[71],[15,18552,1804],{"className":18553},[66],[15,18555,126],{"className":18556},[169],"，标签 ",[15,18559,18561,18583],{"className":18560,"translate":19},[18],[15,18562,18564],{"className":18563},[23],[25,18565,18566],{"xmlns":27},[29,18567,18568,18580],{},[32,18569,18570,18576,18578],{},[1057,18571,18572,18574],{},[35,18573,1224],{},[1209,18575,1211],{},[39,18577,129],{},[1209,18579,1804],{},[45,18581,18582],{"encoding":47},"y_1=0",[15,18584,18586,18641],{"className":18585,"ariaHidden":53},[52],[15,18587,18589,18592,18632,18635,18638],{"className":18588},[57],[15,18590],{"className":18591,"style":471},[61],[15,18593,18595,18598],{"className":18594},[66],[15,18596,1224],{"className":18597,"style":1515},[66,67],[15,18599,18601],{"className":18600},[198],[15,18602,18604,18624],{"className":18603},[202,1090],[15,18605,18607,18621],{"className":18606},[206],[15,18608,18610],{"className":18609,"style":2139},[210],[15,18611,18612,18615],{"style":1531},[15,18613],{"className":18614,"style":218},[217],[15,18616,18618],{"className":18617},[222,223,224,225],[15,18619,1211],{"className":18620},[66,225],[15,18622,1114],{"className":18623},[1113],[15,18625,18627],{"className":18626},[206],[15,18628,18630],{"className":18629,"style":1121},[210],[15,18631],{},[15,18633],{"className":18634,"style":173},[71],[15,18636,129],{"className":18637},[177],[15,18639],{"className":18640,"style":173},[71],[15,18642,18644,18647],{"className":18643},[57],[15,18645],{"className":18646,"style":1836},[61],[15,18648,1804],{"className":18649},[66],"。损失取平方损失",[15,18652,18654],{"className":18653,"translate":19},[294],[15,18655,18657,18707],{"className":18656,"translate":19},[18],[15,18658,18660],{"className":18659},[23],[25,18661,18662],{"xmlns":27,"display":303},[29,18663,18664,18704],{},[32,18665,18666,18668,18670,18672,18674,18680,18682,18684,18686,18688,18690,18696,18702],{},[35,18667,1216],{},[39,18669,120],{"stretchy":119},[35,18671,116],{},[39,18673,120],{"stretchy":119},[1057,18675,18676,18678],{},[35,18677,123],{},[1209,18679,1211],{},[39,18681,126],{"stretchy":119},[39,18683,126],{"stretchy":119},[39,18685,129],{},[35,18687,116],{},[39,18689,120],{"stretchy":119},[1057,18691,18692,18694],{},[35,18693,123],{},[1209,18695,1211],{},[131,18697,18698,18700],{},[39,18699,126],{"stretchy":119},[1209,18701,1263],{},[39,18703,335],{"separator":53},[45,18705,18706],{"encoding":47},"L(f(x_1))=f(x_1)^2,",[15,18708,18710,18781],{"className":18709,"ariaHidden":53},[52],[15,18711,18713,18716,18719,18722,18725,18728,18768,18772,18775,18778],{"className":18712},[57],[15,18714],{"className":18715,"style":154},[61],[15,18717,1216],{"className":18718},[66,67],[15,18720,120],{"className":18721},[162],[15,18723,116],{"className":18724,"style":158},[66,67],[15,18726,120],{"className":18727},[162],[15,18729,18731,18734],{"className":18730},[66],[15,18732,123],{"className":18733},[66,67],[15,18735,18737],{"className":18736},[198],[15,18738,18740,18760],{"className":18739},[202,1090],[15,18741,18743,18757],{"className":18742},[206],[15,18744,18746],{"className":18745,"style":2139},[210],[15,18747,18748,18751],{"style":1584},[15,18749],{"className":18750,"style":218},[217],[15,18752,18754],{"className":18753},[222,223,224,225],[15,18755,1211],{"className":18756},[66,225],[15,18758,1114],{"className":18759},[1113],[15,18761,18763],{"className":18762},[206],[15,18764,18766],{"className":18765,"style":1121},[210],[15,18767],{},[15,18769,18771],{"className":18770},[169],"))",[15,18773],{"className":18774,"style":173},[71],[15,18776,129],{"className":18777},[177],[15,18779],{"className":18780,"style":173},[71],[15,18782,18784,18787,18790,18793,18833,18862],{"className":18783},[57],[15,18785],{"className":18786,"style":11214},[61],[15,18788,116],{"className":18789,"style":158},[66,67],[15,18791,120],{"className":18792},[162],[15,18794,18796,18799],{"className":18795},[66],[15,18797,123],{"className":18798},[66,67],[15,18800,18802],{"className":18801},[198],[15,18803,18805,18825],{"className":18804},[202,1090],[15,18806,18808,18822],{"className":18807},[206],[15,18809,18811],{"className":18810,"style":2139},[210],[15,18812,18813,18816],{"style":1584},[15,18814],{"className":18815,"style":218},[217],[15,18817,18819],{"className":18818},[222,223,224,225],[15,18820,1211],{"className":18821},[66,225],[15,18823,1114],{"className":18824},[1113],[15,18826,18828],{"className":18827},[206],[15,18829,18831],{"className":18830,"style":1121},[210],[15,18832],{},[15,18834,18836,18839],{"className":18835},[169],[15,18837,126],{"className":18838},[169],[15,18840,18842],{"className":18841},[198],[15,18843,18845],{"className":18844},[202],[15,18846,18848],{"className":18847},[206],[15,18849,18851],{"className":18850,"style":1650},[210],[15,18852,18853,18856],{"style":397},[15,18854],{"className":18855,"style":218},[217],[15,18857,18859],{"className":18858},[222,223,224,225],[15,18860,1263],{"className":18861},[66,225],[15,18863,335],{"className":18864},[422],[11,18866,18867],{},"正则项取",[15,18869,18871],{"className":18870,"translate":19},[294],[15,18872,18874,18916],{"className":18873,"translate":19},[18],[15,18875,18877],{"className":18876},[23],[25,18878,18879],{"xmlns":27,"display":303},[29,18880,18881,18913],{},[32,18882,18883,18885,18887,18889,18891,18893,18895,18897,18899,18901,18903,18905,18911],{},[35,18884,7742],{"mathvariant":138},[39,18886,120],{"stretchy":119},[35,18888,116],{},[39,18890,126],{"stretchy":119},[39,18892,129],{},[39,18894,120],{"stretchy":119},[35,18896,1251],{"mathvariant":138},[35,18898,116],{},[35,18900,1251],{"mathvariant":138},[39,18902,6602],{},[1209,18904,1211],{},[131,18906,18907,18909],{},[39,18908,126],{"stretchy":119},[1209,18910,1263],{},[35,18912,3573],{"mathvariant":138},[45,18914,18915],{"encoding":47},"\\Omega(f)=(\\|f\\|-1)^2.",[15,18917,18919,18946,18973],{"className":18918,"ariaHidden":53},[52],[15,18920,18922,18925,18928,18931,18934,18937,18940,18943],{"className":18921},[57],[15,18923],{"className":18924,"style":154},[61],[15,18926,7742],{"className":18927},[66],[15,18929,120],{"className":18930},[162],[15,18932,116],{"className":18933,"style":158},[66,67],[15,18935,126],{"className":18936},[169],[15,18938],{"className":18939,"style":173},[71],[15,18941,129],{"className":18942},[177],[15,18944],{"className":18945,"style":173},[71],[15,18947,18949,18952,18955,18958,18961,18964,18967,18970],{"className":18948},[57],[15,18950],{"className":18951,"style":154},[61],[15,18953,120],{"className":18954},[162],[15,18956,1251],{"className":18957},[66],[15,18959,116],{"className":18960,"style":158},[66,67],[15,18962,1251],{"className":18963},[66],[15,18965],{"className":18966,"style":72},[71],[15,18968,6602],{"className":18969},[76],[15,18971],{"className":18972,"style":72},[71],[15,18974,18976,18979,18982,19011],{"className":18975},[57],[15,18977],{"className":18978,"style":11214},[61],[15,18980,1211],{"className":18981},[66],[15,18983,18985,18988],{"className":18984},[169],[15,18986,126],{"className":18987},[169],[15,18989,18991],{"className":18990},[198],[15,18992,18994],{"className":18993},[202],[15,18995,18997],{"className":18996},[206],[15,18998,19000],{"className":18999,"style":1650},[210],[15,19001,19002,19005],{"style":397},[15,19003],{"className":19004,"style":218},[217],[15,19006,19008],{"className":19007},[222,223,224,225],[15,19009,1263],{"className":19010},[66,225],[15,19012,3573],{"className":19013},[66],[11,19015,19016,19017,19069],{},"这是一个非负的、只依赖范数的正则项，但它在 ",[15,19018,19020,19042],{"className":19019,"translate":19},[18],[15,19021,19023],{"className":19022},[23],[25,19024,19025],{"xmlns":27},[29,19026,19027,19039],{},[32,19028,19029,19031,19033,19035,19037],{},[39,19030,1194],{"stretchy":119},[1209,19032,1804],{},[39,19034,335],{"separator":53},[1209,19036,1211],{},[39,19038,1266],{"stretchy":119},[45,19040,19041],{"encoding":47},"[0,1]",[15,19043,19045],{"className":19044,"ariaHidden":53},[52],[15,19046,19048,19051,19054,19057,19060,19063,19066],{"className":19047},[57],[15,19049],{"className":19050,"style":154},[61],[15,19052,1194],{"className":19053},[162],[15,19055,1804],{"className":19056},[66],[15,19058,335],{"className":19059},[422],[15,19061],{"className":19062,"style":697},[71],[15,19064,1211],{"className":19065},[66],[15,19067,1266],{"className":19068},[169]," 上递减，不是非减函数。",[11,19071,19072],{},"训练点张成的子空间是",[15,19074,19076],{"className":19075,"translate":19},[294],[15,19077,19079,19149],{"className":19078,"translate":19},[18],[15,19080,19082],{"className":19081},[23],[25,19083,19084],{"xmlns":27,"display":303},[29,19085,19086,19146],{},[32,19087,19088,19094,19096,19098,19100,19102,19104,19106,19108,19110,19116,19118,19120,19122,19124,19126,19128,19130,19132,19134,19136,19138,19140,19142,19144],{},[1057,19089,19090,19092],{},[35,19091,1061],{"mathvariant":445},[1209,19093,1804],{},[39,19095,129],{},[35,19097,15],{"mathvariant":138},[39,19099,1170],{},[39,19101,4307],{"stretchy":119},[35,19103,621],{},[39,19105,120],{"stretchy":119},[39,19107,3268],{},[39,19109,335],{"separator":53},[1057,19111,19112,19114],{},[35,19113,123],{},[1209,19115,1211],{},[39,19117,126],{"stretchy":119},[39,19119,4330],{"stretchy":119},[39,19121,129],{},[39,19123,4307],{"stretchy":119},[39,19125,120],{"stretchy":119},[35,19127,4556],{},[39,19129,335],{"separator":53},[1209,19131,1804],{},[39,19133,126],{"stretchy":119},[39,19135,442],{},[35,19137,4556],{},[39,19139,1187],{},[35,19141,455],{"mathvariant":454},[39,19143,4330],{"stretchy":119},[35,19145,3573],{"mathvariant":138},[45,19147,19148],{"encoding":47},"\\mathcal H_0=\\operatorname{span}\\{k(\\cdot,x_1)\\}=\\{(a,0):a\\in\\mathbb R\\}.",[15,19150,19152,19207,19290,19324,19342],{"className":19151,"ariaHidden":53},[52],[15,19153,19155,19158,19198,19201,19204],{"className":19154},[57],[15,19156],{"className":19157,"style":1076},[61],[15,19159,19161,19164],{"className":19160},[66],[15,19162,1061],{"className":19163,"style":1083},[66,494],[15,19165,19167],{"className":19166},[198],[15,19168,19170,19190],{"className":19169},[202,1090],[15,19171,19173,19187],{"className":19172},[206],[15,19174,19176],{"className":19175,"style":2139},[210],[15,19177,19178,19181],{"style":1100},[15,19179],{"className":19180,"style":218},[217],[15,19182,19184],{"className":19183},[222,223,224,225],[15,19185,1804],{"className":19186},[66,225],[15,19188,1114],{"className":19189},[1113],[15,19191,19193],{"className":19192},[206],[15,19194,19196],{"className":19195,"style":1121},[210],[15,19197],{},[15,19199],{"className":19200,"style":173},[71],[15,19202,129],{"className":19203},[177],[15,19205],{"className":19206,"style":173},[71],[15,19208,19210,19213,19219,19222,19225,19228,19231,19234,19237,19277,19281,19284,19287],{"className":19209},[57],[15,19211],{"className":19212,"style":154},[61],[15,19214,19216],{"className":19215},[1339],[15,19217,15],{"className":19218},[66,4429],[15,19220,4307],{"className":19221},[162],[15,19223,621],{"className":19224,"style":684},[66,67],[15,19226,120],{"className":19227},[162],[15,19229,3268],{"className":19230},[66],[15,19232,335],{"className":19233},[422],[15,19235],{"className":19236,"style":697},[71],[15,19238,19240,19243],{"className":19239},[66],[15,19241,123],{"className":19242},[66,67],[15,19244,19246],{"className":19245},[198],[15,19247,19249,19269],{"className":19248},[202,1090],[15,19250,19252,19266],{"className":19251},[206],[15,19253,19255],{"className":19254,"style":2139},[210],[15,19256,19257,19260],{"style":1584},[15,19258],{"className":19259,"style":218},[217],[15,19261,19263],{"className":19262},[222,223,224,225],[15,19264,1211],{"className":19265},[66,225],[15,19267,1114],{"className":19268},[1113],[15,19270,19272],{"className":19271},[206],[15,19273,19275],{"className":19274,"style":1121},[210],[15,19276],{},[15,19278,19280],{"className":19279},[169],")}",[15,19282],{"className":19283,"style":173},[71],[15,19285,129],{"className":19286},[177],[15,19288],{"className":19289,"style":173},[71],[15,19291,19293,19296,19300,19303,19306,19309,19312,19315,19318,19321],{"className":19292},[57],[15,19294],{"className":19295,"style":154},[61],[15,19297,19299],{"className":19298},[162],"{(",[15,19301,4556],{"className":19302},[66,67],[15,19304,335],{"className":19305},[422],[15,19307],{"className":19308,"style":697},[71],[15,19310,1804],{"className":19311},[66],[15,19313,126],{"className":19314},[169],[15,19316],{"className":19317,"style":173},[71],[15,19319,442],{"className":19320},[177],[15,19322],{"className":19323,"style":173},[71],[15,19325,19327,19330,19333,19336,19339],{"className":19326},[57],[15,19328],{"className":19329,"style":3227},[61],[15,19331,4556],{"className":19332},[66,67],[15,19334],{"className":19335,"style":173},[71],[15,19337,1187],{"className":19338},[177],[15,19340],{"className":19341,"style":173},[71],[15,19343,19345,19348,19351,19354],{"className":19344},[57],[15,19346],{"className":19347,"style":154},[61],[15,19349,455],{"className":19350},[66,517],[15,19352,4330],{"className":19353},[169],[15,19355,3573],{"className":19356},[66],[11,19358,19359,19360,19434,19435,19542,19543,17010,19571,19632,19633,19661,19662,19690,19691,19434,19765,19871,19872,16703,19900,19632,19961,19661,19989,20017,20018,20070,20071,20140],{},"在 ",[15,19361,19363,19389],{"className":19362,"translate":19},[18],[15,19364,19366],{"className":19365},[23],[25,19367,19368],{"xmlns":27},[29,19369,19370,19386],{},[32,19371,19372,19374,19376,19378,19380,19382,19384],{},[35,19373,116],{},[39,19375,129],{},[39,19377,120],{"stretchy":119},[1209,19379,1804],{},[39,19381,335],{"separator":53},[1209,19383,1804],{},[39,19385,126],{"stretchy":119},[45,19387,19388],{"encoding":47},"f=(0,0)",[15,19390,19392,19410],{"className":19391,"ariaHidden":53},[52],[15,19393,19395,19398,19401,19404,19407],{"className":19394},[57],[15,19396],{"className":19397,"style":3408},[61],[15,19399,116],{"className":19400,"style":158},[66,67],[15,19402],{"className":19403,"style":173},[71],[15,19405,129],{"className":19406},[177],[15,19408],{"className":19409,"style":173},[71],[15,19411,19413,19416,19419,19422,19425,19428,19431],{"className":19412},[57],[15,19414],{"className":19415,"style":154},[61],[15,19417,120],{"className":19418},[162],[15,19420,1804],{"className":19421},[66],[15,19423,335],{"className":19424},[422],[15,19426],{"className":19427,"style":697},[71],[15,19429,1804],{"className":19430},[66],[15,19432,126],{"className":19433},[169]," 处，",[15,19436,19438,19466],{"className":19437,"translate":19},[18],[15,19439,19441],{"className":19440},[23],[25,19442,19443],{"xmlns":27},[29,19444,19445,19463],{},[32,19446,19447,19449,19451,19457,19459,19461],{},[35,19448,116],{},[39,19450,120],{"stretchy":119},[1057,19452,19453,19455],{},[35,19454,123],{},[1209,19456,1211],{},[39,19458,126],{"stretchy":119},[39,19460,129],{},[1209,19462,1804],{},[45,19464,19465],{"encoding":47},"f(x_1)=0",[15,19467,19469,19533],{"className":19468,"ariaHidden":53},[52],[15,19470,19472,19475,19478,19481,19521,19524,19527,19530],{"className":19471},[57],[15,19473],{"className":19474,"style":154},[61],[15,19476,116],{"className":19477,"style":158},[66,67],[15,19479,120],{"className":19480},[162],[15,19482,19484,19487],{"className":19483},[66],[15,19485,123],{"className":19486},[66,67],[15,19488,19490],{"className":19489},[198],[15,19491,19493,19513],{"className":19492},[202,1090],[15,19494,19496,19510],{"className":19495},[206],[15,19497,19499],{"className":19498,"style":2139},[210],[15,19500,19501,19504],{"style":1584},[15,19502],{"className":19503,"style":218},[217],[15,19505,19507],{"className":19506},[222,223,224,225],[15,19508,1211],{"className":19509},[66,225],[15,19511,1114],{"className":19512},[1113],[15,19514,19516],{"className":19515},[206],[15,19517,19519],{"className":19518,"style":1121},[210],[15,19520],{},[15,19522,126],{"className":19523},[169],[15,19525],{"className":19526,"style":173},[71],[15,19528,129],{"className":19529},[177],[15,19531],{"className":19532,"style":173},[71],[15,19534,19536,19539],{"className":19535},[57],[15,19537],{"className":19538,"style":1836},[61],[15,19540,1804],{"className":19541},[66],"，损失为 ",[15,19544,19546,19559],{"className":19545,"translate":19},[18],[15,19547,19549],{"className":19548},[23],[25,19550,19551],{"xmlns":27},[29,19552,19553,19557],{},[32,19554,19555],{},[1209,19556,1804],{},[45,19558,1804],{"encoding":47},[15,19560,19562],{"className":19561,"ariaHidden":53},[52],[15,19563,19565,19568],{"className":19564},[57],[15,19566],{"className":19567,"style":1836},[61],[15,19569,1804],{"className":19570},[66],[15,19572,19574,19596],{"className":19573,"translate":19},[18],[15,19575,19577],{"className":19576},[23],[25,19578,19579],{"xmlns":27},[29,19580,19581,19593],{},[32,19582,19583,19585,19587,19589,19591],{},[35,19584,1251],{"mathvariant":138},[35,19586,116],{},[35,19588,1251],{"mathvariant":138},[39,19590,129],{},[1209,19592,1804],{},[45,19594,19595],{"encoding":47},"\\|f\\|=0",[15,19597,19599,19623],{"className":19598,"ariaHidden":53},[52],[15,19600,19602,19605,19608,19611,19614,19617,19620],{"className":19601},[57],[15,19603],{"className":19604,"style":154},[61],[15,19606,1251],{"className":19607},[66],[15,19609,116],{"className":19610,"style":158},[66,67],[15,19612,1251],{"className":19613},[66],[15,19615],{"className":19616,"style":173},[71],[15,19618,129],{"className":19619},[177],[15,19621],{"className":19622,"style":173},[71],[15,19624,19626,19629],{"className":19625},[57],[15,19627],{"className":19628,"style":1836},[61],[15,19630,1804],{"className":19631},[66],"，正则项为 ",[15,19634,19636,19649],{"className":19635,"translate":19},[18],[15,19637,19639],{"className":19638},[23],[25,19640,19641],{"xmlns":27},[29,19642,19643,19647],{},[32,19644,19645],{},[1209,19646,1211],{},[45,19648,1211],{"encoding":47},[15,19650,19652],{"className":19651,"ariaHidden":53},[52],[15,19653,19655,19658],{"className":19654},[57],[15,19656],{"className":19657,"style":1836},[61],[15,19659,1211],{"className":19660},[66],"，目标值为 ",[15,19663,19665,19678],{"className":19664,"translate":19},[18],[15,19666,19668],{"className":19667},[23],[25,19669,19670],{"xmlns":27},[29,19671,19672,19676],{},[32,19673,19674],{},[1209,19675,1211],{},[45,19677,1211],{"encoding":47},[15,19679,19681],{"className":19680,"ariaHidden":53},[52],[15,19682,19684,19687],{"className":19683},[57],[15,19685],{"className":19686,"style":1836},[61],[15,19688,1211],{"className":19689},[66],"。在 ",[15,19692,19694,19720],{"className":19693,"translate":19},[18],[15,19695,19697],{"className":19696},[23],[25,19698,19699],{"xmlns":27},[29,19700,19701,19717],{},[32,19702,19703,19705,19707,19709,19711,19713,19715],{},[35,19704,116],{},[39,19706,129],{},[39,19708,120],{"stretchy":119},[1209,19710,1804],{},[39,19712,335],{"separator":53},[1209,19714,1211],{},[39,19716,126],{"stretchy":119},[45,19718,19719],{"encoding":47},"f=(0,1)",[15,19721,19723,19741],{"className":19722,"ariaHidden":53},[52],[15,19724,19726,19729,19732,19735,19738],{"className":19725},[57],[15,19727],{"className":19728,"style":3408},[61],[15,19730,116],{"className":19731,"style":158},[66,67],[15,19733],{"className":19734,"style":173},[71],[15,19736,129],{"className":19737},[177],[15,19739],{"className":19740,"style":173},[71],[15,19742,19744,19747,19750,19753,19756,19759,19762],{"className":19743},[57],[15,19745],{"className":19746,"style":154},[61],[15,19748,120],{"className":19749},[162],[15,19751,1804],{"className":19752},[66],[15,19754,335],{"className":19755},[422],[15,19757],{"className":19758,"style":697},[71],[15,19760,1211],{"className":19761},[66],[15,19763,126],{"className":19764},[169],[15,19766,19768,19795],{"className":19767,"translate":19},[18],[15,19769,19771],{"className":19770},[23],[25,19772,19773],{"xmlns":27},[29,19774,19775,19793],{},[32,19776,19777,19779,19781,19787,19789,19791],{},[35,19778,116],{},[39,19780,120],{"stretchy":119},[1057,19782,19783,19785],{},[35,19784,123],{},[1209,19786,1211],{},[39,19788,126],{"stretchy":119},[39,19790,129],{},[1209,19792,1804],{},[45,19794,19465],{"encoding":47},[15,19796,19798,19862],{"className":19797,"ariaHidden":53},[52],[15,19799,19801,19804,19807,19810,19850,19853,19856,19859],{"className":19800},[57],[15,19802],{"className":19803,"style":154},[61],[15,19805,116],{"className":19806,"style":158},[66,67],[15,19808,120],{"className":19809},[162],[15,19811,19813,19816],{"className":19812},[66],[15,19814,123],{"className":19815},[66,67],[15,19817,19819],{"className":19818},[198],[15,19820,19822,19842],{"className":19821},[202,1090],[15,19823,19825,19839],{"className":19824},[206],[15,19826,19828],{"className":19827,"style":2139},[210],[15,19829,19830,19833],{"style":1584},[15,19831],{"className":19832,"style":218},[217],[15,19834,19836],{"className":19835},[222,223,224,225],[15,19837,1211],{"className":19838},[66,225],[15,19840,1114],{"className":19841},[1113],[15,19843,19845],{"className":19844},[206],[15,19846,19848],{"className":19847,"style":1121},[210],[15,19849],{},[15,19851,126],{"className":19852},[169],[15,19854],{"className":19855,"style":173},[71],[15,19857,129],{"className":19858},[177],[15,19860],{"className":19861,"style":173},[71],[15,19863,19865,19868],{"className":19864},[57],[15,19866],{"className":19867,"style":1836},[61],[15,19869,1804],{"className":19870},[66],"，损失仍为 ",[15,19873,19875,19888],{"className":19874,"translate":19},[18],[15,19876,19878],{"className":19877},[23],[25,19879,19880],{"xmlns":27},[29,19881,19882,19886],{},[32,19883,19884],{},[1209,19885,1804],{},[45,19887,1804],{"encoding":47},[15,19889,19891],{"className":19890,"ariaHidden":53},[52],[15,19892,19894,19897],{"className":19893},[57],[15,19895],{"className":19896,"style":1836},[61],[15,19898,1804],{"className":19899},[66],[15,19901,19903,19925],{"className":19902,"translate":19},[18],[15,19904,19906],{"className":19905},[23],[25,19907,19908],{"xmlns":27},[29,19909,19910,19922],{},[32,19911,19912,19914,19916,19918,19920],{},[35,19913,1251],{"mathvariant":138},[35,19915,116],{},[35,19917,1251],{"mathvariant":138},[39,19919,129],{},[1209,19921,1211],{},[45,19923,19924],{"encoding":47},"\\|f\\|=1",[15,19926,19928,19952],{"className":19927,"ariaHidden":53},[52],[15,19929,19931,19934,19937,19940,19943,19946,19949],{"className":19930},[57],[15,19932],{"className":19933,"style":154},[61],[15,19935,1251],{"className":19936},[66],[15,19938,116],{"className":19939,"style":158},[66,67],[15,19941,1251],{"className":19942},[66],[15,19944],{"className":19945,"style":173},[71],[15,19947,129],{"className":19948},[177],[15,19950],{"className":19951,"style":173},[71],[15,19953,19955,19958],{"className":19954},[57],[15,19956],{"className":19957,"style":1836},[61],[15,19959,1211],{"className":19960},[66],[15,19962,19964,19977],{"className":19963,"translate":19},[18],[15,19965,19967],{"className":19966},[23],[25,19968,19969],{"xmlns":27},[29,19970,19971,19975],{},[32,19972,19973],{},[1209,19974,1804],{},[45,19976,1804],{"encoding":47},[15,19978,19980],{"className":19979,"ariaHidden":53},[52],[15,19981,19983,19986],{"className":19982},[57],[15,19984],{"className":19985,"style":1836},[61],[15,19987,1804],{"className":19988},[66],[15,19990,19992,20005],{"className":19991,"translate":19},[18],[15,19993,19995],{"className":19994},[23],[25,19996,19997],{"xmlns":27},[29,19998,19999,20003],{},[32,20000,20001],{},[1209,20002,1804],{},[45,20004,1804],{"encoding":47},[15,20006,20008],{"className":20007,"ariaHidden":53},[52],[15,20009,20011,20014],{"className":20010},[57],[15,20012],{"className":20013,"style":1836},[61],[15,20015,1804],{"className":20016},[66],"。所以最优解是 ",[15,20019,20021,20043],{"className":20020,"translate":19},[18],[15,20022,20024],{"className":20023},[23],[25,20025,20026],{"xmlns":27},[29,20027,20028,20040],{},[32,20029,20030,20032,20034,20036,20038],{},[39,20031,120],{"stretchy":119},[1209,20033,1804],{},[39,20035,335],{"separator":53},[1209,20037,1211],{},[39,20039,126],{"stretchy":119},[45,20041,20042],{"encoding":47},"(0,1)",[15,20044,20046],{"className":20045,"ariaHidden":53},[52],[15,20047,20049,20052,20055,20058,20061,20064,20067],{"className":20048},[57],[15,20050],{"className":20051,"style":154},[61],[15,20053,120],{"className":20054},[162],[15,20056,1804],{"className":20057},[66],[15,20059,335],{"className":20060},[422],[15,20062],{"className":20063,"style":697},[71],[15,20065,1211],{"className":20066},[66],[15,20068,126],{"className":20069},[169],"，它落在 ",[15,20072,20074,20091],{"className":20073,"translate":19},[18],[15,20075,20077],{"className":20076},[23],[25,20078,20079],{"xmlns":27},[29,20080,20081,20089],{},[32,20082,20083],{},[1057,20084,20085,20087],{},[35,20086,1061],{"mathvariant":445},[1209,20088,1804],{},[45,20090,13449],{"encoding":47},[15,20092,20094],{"className":20093,"ariaHidden":53},[52],[15,20095,20097,20100],{"className":20096},[57],[15,20098],{"className":20099,"style":1076},[61],[15,20101,20103,20106],{"className":20102},[66],[15,20104,1061],{"className":20105,"style":1083},[66,494],[15,20107,20109],{"className":20108},[198],[15,20110,20112,20132],{"className":20111},[202,1090],[15,20113,20115,20129],{"className":20114},[206],[15,20116,20118],{"className":20117,"style":2139},[210],[15,20119,20120,20123],{"style":1100},[15,20121],{"className":20122,"style":218},[217],[15,20124,20126],{"className":20125},[222,223,224,225],[15,20127,1804],{"className":20128},[66,225],[15,20130,1114],{"className":20131},[1113],[15,20133,20135],{"className":20134},[206],[15,20136,20138],{"className":20137,"style":1121},[210],[15,20139],{}," 的正交补里，而不是训练点的张成空间中。",[11,20142,20143,20144,20147],{},"这个反例说明：",[2040,20145,20146],{},"非减性不是证明技巧，而是定理的本质条件。"," 正则项如果「喜欢」某个非零范数，它就会把解推离训练点子空间。",[1904,20149,20151],{"id":20150},"_43-充要条件","4.3 充要条件",[11,20153,20154],{},"上述反例可以推广。Dinuzzo 与 Schölkopf 在 2012 年给出了完整的刻画：在相当一般的条件下，表示定理对所有损失函数和所有数据集成立，当且仅当正则项是范数的非减函数。也就是说，",[15,20156,20158],{"className":20157,"translate":19},[294],[15,20159,20161,20200],{"className":20160,"translate":19},[18],[15,20162,20164],{"className":20163},[23],[25,20165,20166],{"xmlns":27,"display":303},[29,20167,20168,20197],{},[32,20169,20170,20172,20174,20176,20178,20180,20183,20185,20187,20189,20195],{},[35,20171,7742],{"mathvariant":138},[39,20173,120],{"stretchy":119},[35,20175,116],{},[39,20177,126],{"stretchy":119},[39,20179,129],{},[35,20181,20182],{},"h",[39,20184,120],{"stretchy":119},[35,20186,1251],{"mathvariant":138},[35,20188,116],{},[1057,20190,20191,20193],{},[35,20192,1251],{"mathvariant":138},[35,20194,1061],{"mathvariant":445},[39,20196,126],{"stretchy":119},[45,20198,20199],{"encoding":47},"\\Omega(f)=h(\\|f\\|_{\\mathcal H})",[15,20201,20203,20230],{"className":20202,"ariaHidden":53},[52],[15,20204,20206,20209,20212,20215,20218,20221,20224,20227],{"className":20205},[57],[15,20207],{"className":20208,"style":154},[61],[15,20210,7742],{"className":20211},[66],[15,20213,120],{"className":20214},[162],[15,20216,116],{"className":20217,"style":158},[66,67],[15,20219,126],{"className":20220},[169],[15,20222],{"className":20223,"style":173},[71],[15,20225,129],{"className":20226},[177],[15,20228],{"className":20229,"style":173},[71],[15,20231,20233,20236,20239,20242,20245,20248,20291],{"className":20232},[57],[15,20234],{"className":20235,"style":154},[61],[15,20237,20182],{"className":20238},[66,67],[15,20240,120],{"className":20241},[162],[15,20243,1251],{"className":20244},[66],[15,20246,116],{"className":20247,"style":158},[66,67],[15,20249,20251,20254],{"className":20250},[66],[15,20252,1251],{"className":20253},[66],[15,20255,20257],{"className":20256},[198],[15,20258,20260,20283],{"className":20259},[202,1090],[15,20261,20263,20280],{"className":20262},[206],[15,20264,20266],{"className":20265,"style":3638},[210],[15,20267,20268,20271],{"style":1584},[15,20269],{"className":20270,"style":218},[217],[15,20272,20274],{"className":20273},[222,223,224,225],[15,20275,20277],{"className":20276},[66,225],[15,20278,1061],{"className":20279,"style":1083},[66,494,225],[15,20281,1114],{"className":20282},[1113],[15,20284,20286],{"className":20285},[206],[15,20287,20289],{"className":20288,"style":1121},[210],[15,20290],{},[15,20292,126],{"className":20293},[169],[11,20295,20296,20297,20393],{},"对某个非减的 ",[15,20298,20300,20330],{"className":20299,"translate":19},[18],[15,20301,20303],{"className":20302},[23],[25,20304,20305],{"xmlns":27},[29,20306,20307,20327],{},[32,20308,20309,20311,20313,20315,20317,20319,20321,20323,20325],{},[35,20310,20182],{},[39,20312,442],{},[39,20314,1194],{"stretchy":119},[1209,20316,1804],{},[39,20318,335],{"separator":53},[35,20320,7620],{"mathvariant":138},[39,20322,126],{"stretchy":119},[39,20324,449],{},[35,20326,455],{"mathvariant":454},[45,20328,20329],{"encoding":47},"h:[0,\\infty)\\to\\mathbb R",[15,20331,20333,20351,20384],{"className":20332,"ariaHidden":53},[52],[15,20334,20336,20339,20342,20345,20348],{"className":20335},[57],[15,20337],{"className":20338,"style":840},[61],[15,20340,20182],{"className":20341},[66,67],[15,20343],{"className":20344,"style":173},[71],[15,20346,442],{"className":20347},[177],[15,20349],{"className":20350,"style":173},[71],[15,20352,20354,20357,20360,20363,20366,20369,20372,20375,20378,20381],{"className":20353},[57],[15,20355],{"className":20356,"style":154},[61],[15,20358,1194],{"className":20359},[162],[15,20361,1804],{"className":20362},[66],[15,20364,335],{"className":20365},[422],[15,20367],{"className":20368,"style":697},[71],[15,20370,7620],{"className":20371},[66],[15,20373,126],{"className":20374},[169],[15,20376],{"className":20377,"style":173},[71],[15,20379,449],{"className":20380},[177],[15,20382],{"className":20383,"style":173},[71],[15,20385,20387,20390],{"className":20386},[57],[15,20388],{"className":20389,"style":2034},[61],[15,20391,455],{"className":20392},[66,517]," 成立，是表示定理的充要条件。",[11,20395,20396],{},"这里还要区分两个层次：",[16136,20398,20399,20475],{},[3194,20400,20401,20404,20405,20474],{},[2040,20402,20403],{},"非减","：保证存在一个落在 ",[15,20406,20408,20425],{"className":20407,"translate":19},[18],[15,20409,20411],{"className":20410},[23],[25,20412,20413],{"xmlns":27},[29,20414,20415,20423],{},[32,20416,20417],{},[1057,20418,20419,20421],{},[35,20420,1061],{"mathvariant":445},[1209,20422,1804],{},[45,20424,13449],{"encoding":47},[15,20426,20428],{"className":20427,"ariaHidden":53},[52],[15,20429,20431,20434],{"className":20430},[57],[15,20432],{"className":20433,"style":1076},[61],[15,20435,20437,20440],{"className":20436},[66],[15,20438,1061],{"className":20439,"style":1083},[66,494],[15,20441,20443],{"className":20442},[198],[15,20444,20446,20466],{"className":20445},[202,1090],[15,20447,20449,20463],{"className":20448},[206],[15,20450,20452],{"className":20451,"style":2139},[210],[15,20453,20454,20457],{"style":1100},[15,20455],{"className":20456,"style":218},[217],[15,20458,20460],{"className":20459},[222,223,224,225],[15,20461,1804],{"className":20462},[66,225],[15,20464,1114],{"className":20465},[1113],[15,20467,20469],{"className":20468},[206],[15,20470,20472],{"className":20471,"style":1121},[210],[15,20473],{}," 中的最优解；",[3194,20476,20477,20480,20481,20550],{},[2040,20478,20479],{},"严格递增","：保证每个最优解都落在 ",[15,20482,20484,20501],{"className":20483,"translate":19},[18],[15,20485,20487],{"className":20486},[23],[25,20488,20489],{"xmlns":27},[29,20490,20491,20499],{},[32,20492,20493],{},[1057,20494,20495,20497],{},[35,20496,1061],{"mathvariant":445},[1209,20498,1804],{},[45,20500,13449],{"encoding":47},[15,20502,20504],{"className":20503,"ariaHidden":53},[52],[15,20505,20507,20510],{"className":20506},[57],[15,20508],{"className":20509,"style":1076},[61],[15,20511,20513,20516],{"className":20512},[66],[15,20514,1061],{"className":20515,"style":1083},[66,494],[15,20517,20519],{"className":20518},[198],[15,20520,20522,20542],{"className":20521},[202,1090],[15,20523,20525,20539],{"className":20524},[206],[15,20526,20528],{"className":20527,"style":2139},[210],[15,20529,20530,20533],{"style":1100},[15,20531],{"className":20532,"style":218},[217],[15,20534,20536],{"className":20535},[222,223,224,225],[15,20537,1804],{"className":20538},[66,225],[15,20540,1114],{"className":20541},[1113],[15,20543,20545],{"className":20544},[206],[15,20546,20548],{"className":20547,"style":1121},[210],[15,20549],{}," 中。",[11,20552,20553,20554,20606,20607,20635,20636,20705,20706,20775],{},"如果 ",[15,20555,20557,20576],{"className":20556,"translate":19},[18],[15,20558,20560],{"className":20559},[23],[25,20561,20562],{"xmlns":27},[29,20563,20564,20573],{},[32,20565,20566,20568,20571],{},[35,20567,7742],{"mathvariant":138},[39,20569,20570],{},"≡",[1209,20572,1804],{},[45,20574,20575],{"encoding":47},"\\Omega\\equiv 0",[15,20577,20579,20597],{"className":20578,"ariaHidden":53},[52],[15,20580,20582,20585,20588,20591,20594],{"className":20581},[57],[15,20583],{"className":20584,"style":490},[61],[15,20586,7742],{"className":20587},[66],[15,20589],{"className":20590,"style":173},[71],[15,20592,20570],{"className":20593},[177],[15,20595],{"className":20596,"style":173},[71],[15,20598,20600,20603],{"className":20599},[57],[15,20601],{"className":20602,"style":1836},[61],[15,20604,1804],{"className":20605},[66],"，目标函数只依赖 ",[15,20608,20610,20623],{"className":20609,"translate":19},[18],[15,20611,20613],{"className":20612},[23],[25,20614,20615],{"xmlns":27},[29,20616,20617,20621],{},[32,20618,20619],{},[35,20620,116],{},[45,20622,116],{"encoding":47},[15,20624,20626],{"className":20625,"ariaHidden":53},[52],[15,20627,20629,20632],{"className":20628},[57],[15,20630],{"className":20631,"style":3408},[61],[15,20633,116],{"className":20634,"style":158},[66,67]," 在训练点上的取值。任何在这些点上取值相同的函数都是最优解，其中很多并不在 ",[15,20637,20639,20656],{"className":20638,"translate":19},[18],[15,20640,20642],{"className":20641},[23],[25,20643,20644],{"xmlns":27},[29,20645,20646,20654],{},[32,20647,20648],{},[1057,20649,20650,20652],{},[35,20651,1061],{"mathvariant":445},[1209,20653,1804],{},[45,20655,13449],{"encoding":47},[15,20657,20659],{"className":20658,"ariaHidden":53},[52],[15,20660,20662,20665],{"className":20661},[57],[15,20663],{"className":20664,"style":1076},[61],[15,20666,20668,20671],{"className":20667},[66],[15,20669,1061],{"className":20670,"style":1083},[66,494],[15,20672,20674],{"className":20673},[198],[15,20675,20677,20697],{"className":20676},[202,1090],[15,20678,20680,20694],{"className":20679},[206],[15,20681,20683],{"className":20682,"style":2139},[210],[15,20684,20685,20688],{"style":1100},[15,20686],{"className":20687,"style":218},[217],[15,20689,20691],{"className":20690},[222,223,224,225],[15,20692,1804],{"className":20693},[66,225],[15,20695,1114],{"className":20696},[1113],[15,20698,20700],{"className":20699},[206],[15,20701,20703],{"className":20702,"style":1121},[210],[15,20704],{}," 里。表示定理的「存在」版本仍然成立（取 ",[15,20707,20709,20726],{"className":20708,"translate":19},[18],[15,20710,20712],{"className":20711},[23],[25,20713,20714],{"xmlns":27},[29,20715,20716,20724],{},[32,20717,20718],{},[1057,20719,20720,20722],{},[35,20721,116],{},[39,20723,1251],{},[45,20725,12654],{"encoding":47},[15,20727,20729],{"className":20728,"ariaHidden":53},[52],[15,20730,20732,20735],{"className":20731},[57],[15,20733],{"className":20734,"style":9840},[61],[15,20736,20738,20741],{"className":20737},[66],[15,20739,116],{"className":20740,"style":158},[66,67],[15,20742,20744],{"className":20743},[198],[15,20745,20747,20767],{"className":20746},[202,1090],[15,20748,20750,20764],{"className":20749},[206],[15,20751,20753],{"className":20752,"style":3668},[210],[15,20754,20755,20758],{"style":9861},[15,20756],{"className":20757,"style":218},[217],[15,20759,20761],{"className":20760},[222,223,224,225],[15,20762,1251],{"className":20763},[177,225],[15,20765,1114],{"className":20766},[1113],[15,20768,20770],{"className":20769},[206],[15,20771,20773],{"className":20772,"style":9880},[210],[15,20774],{}," 即可），但「任意最优解」的版本不成立。因此，「任意最优解都满足」这一说法需要加上严格递增这个条件。",[95,20777,20779],{"id":20778},"五五个经典例子","五、五个经典例子",[1904,20781,20783],{"id":20782},"_51-核岭回归","5.1 核岭回归",[11,20785,20786,20787,9266,20857,20907],{},"取平方损失和 ",[15,20788,20790,20815],{"className":20789,"translate":19},[18],[15,20791,20793],{"className":20792},[23],[25,20794,20795],{"xmlns":27},[29,20796,20797,20813],{},[32,20798,20799,20801,20803,20805,20807,20809,20811],{},[35,20800,7742],{"mathvariant":138},[39,20802,120],{"stretchy":119},[35,20804,9212],{},[39,20806,126],{"stretchy":119},[39,20808,129],{},[35,20810,1248],{},[35,20812,9212],{},[45,20814,9223],{"encoding":47},[15,20816,20818,20845],{"className":20817,"ariaHidden":53},[52],[15,20819,20821,20824,20827,20830,20833,20836,20839,20842],{"className":20820},[57],[15,20822],{"className":20823,"style":154},[61],[15,20825,7742],{"className":20826},[66],[15,20828,120],{"className":20829},[162],[15,20831,9212],{"className":20832},[66,67],[15,20834,126],{"className":20835},[169],[15,20837],{"className":20838,"style":173},[71],[15,20840,129],{"className":20841},[177],[15,20843],{"className":20844,"style":173},[71],[15,20846,20848,20851,20854],{"className":20847},[57],[15,20849],{"className":20850,"style":840},[61],[15,20852,1248],{"className":20853},[66,67],[15,20855,9212],{"className":20856},[66,67],[15,20858,20860,20877],{"className":20859,"translate":19},[18],[15,20861,20863],{"className":20862},[23],[25,20864,20865],{"xmlns":27},[29,20866,20867,20875],{},[32,20868,20869,20871,20873],{},[35,20870,1248],{},[39,20872,9283],{},[1209,20874,1804],{},[45,20876,9288],{"encoding":47},[15,20878,20880,20898],{"className":20879,"ariaHidden":53},[52],[15,20881,20883,20886,20889,20892,20895],{"className":20882},[57],[15,20884],{"className":20885,"style":9298},[61],[15,20887,1248],{"className":20888},[66,67],[15,20890],{"className":20891,"style":173},[71],[15,20893,9283],{"className":20894},[177],[15,20896],{"className":20897,"style":173},[71],[15,20899,20901,20904],{"className":20900},[57],[15,20902],{"className":20903,"style":1836},[61],[15,20905,1804],{"className":20906},[66],"）：",[15,20909,20911],{"className":20910,"translate":19},[294],[15,20912,20914,21008],{"className":20913,"translate":19},[18],[15,20915,20917],{"className":20916},[23],[25,20918,20919],{"xmlns":27,"display":303},[29,20920,20921,21005],{},[32,20922,20923,20939,21003],{},[1172,20924,20925,20931],{},[32,20926,20927,20929],{},[35,20928,1178],{},[39,20930,1170],{},[32,20932,20933,20935,20937],{},[35,20934,116],{},[39,20936,1187],{},[35,20938,1061],{"mathvariant":445},[32,20940,20941,20943,20957,20959,20965,20967,20969,20971,20977,20979,20985,20987,20989,20991,20993,21001],{},[39,20942,4307],{"fence":53},[1196,20944,20945,20947,20955],{},[39,20946,1200],{},[32,20948,20949,20951,20953],{},[35,20950,1205],{},[39,20952,129],{},[1209,20954,1211],{},[35,20956,37],{},[39,20958,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[1057,20960,20961,20963],{},[35,20962,1224],{},[35,20964,1205],{},[39,20966,6602],{},[35,20968,116],{},[39,20970,120],{"stretchy":119},[1057,20972,20973,20975],{},[35,20974,123],{},[35,20976,1205],{},[39,20978,126],{"stretchy":119},[131,20980,20981,20983],{},[39,20982,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,20984,1263],{},[39,20986,1245],{},[35,20988,1248],{},[35,20990,1251],{"mathvariant":138},[35,20992,116],{},[1255,20994,20995,20997,20999],{},[35,20996,1251],{"mathvariant":138},[35,20998,1061],{"mathvariant":445},[1209,21000,1263],{},[39,21002,4330],{"fence":53},[35,21004,3573],{"mathvariant":138},[45,21006,21007],{"encoding":47},"\\min_{f\\in\\mathcal H}\\left\\{\\sum_{i=1}^n\\bigl(y_i-f(x_i)\\bigr)^2+\\lambda\\|f\\|_{\\mathcal H}^2\\right\\}.",[15,21009,21011],{"className":21010,"ariaHidden":53},[52],[15,21012,21014,21017,21070,21073,21362,21365],{"className":21013},[57],[15,21015],{"className":21016,"style":1335},[61],[15,21018,21020],{"className":21019},[1339,1351],[15,21021,21023,21062],{"className":21022},[202,1090],[15,21024,21026,21059],{"className":21025},[206],[15,21027,21029,21049],{"className":21028,"style":1361},[210],[15,21030,21031,21034],{"style":1364},[15,21032],{"className":21033,"style":1368},[217],[15,21035,21037],{"className":21036},[222,223,224,225],[15,21038,21040,21043,21046],{"className":21039},[66,225],[15,21041,116],{"className":21042,"style":158},[66,67,225],[15,21044,1187],{"className":21045},[177,225],[15,21047,1061],{"className":21048,"style":1083},[66,494,225],[15,21050,21051,21054],{"style":1386},[15,21052],{"className":21053,"style":1368},[217],[15,21055,21056],{},[15,21057,1178],{"className":21058},[1339],[15,21060,1114],{"className":21061},[1113],[15,21063,21065],{"className":21064},[206],[15,21066,21068],{"className":21067,"style":1404},[210],[15,21069],{},[15,21071],{"className":21072,"style":697},[71],[15,21074,21076,21082,21146,21152,21192,21195,21198,21201,21204,21207,21247,21250,21284,21287,21290,21293,21296,21299,21302,21356],{"className":21075},[1413],[15,21077,21079],{"className":21078,"style":1418},[162,1417],[15,21080,4307],{"className":21081},[1422,1423],[15,21083,21085],{"className":21084},[1339,1351],[15,21086,21088,21138],{"className":21087},[202,1090],[15,21089,21091,21135],{"className":21090},[206],[15,21092,21094,21114,21124],{"className":21093,"style":1436},[210],[15,21095,21096,21099],{"style":1439},[15,21097],{"className":21098,"style":1443},[217],[15,21100,21102],{"className":21101},[222,223,224,225],[15,21103,21105,21108,21111],{"className":21104},[66,225],[15,21106,1205],{"className":21107},[66,67,225],[15,21109,129],{"className":21110},[177,225],[15,21112,1211],{"className":21113},[66,225],[15,21115,21116,21119],{"style":1461},[15,21117],{"className":21118,"style":1443},[217],[15,21120,21121],{},[15,21122,1200],{"className":21123},[1339,1470,1471],[15,21125,21126,21129],{"style":1474},[15,21127],{"className":21128,"style":1443},[217],[15,21130,21132],{"className":21131},[222,223,224,225],[15,21133,37],{"className":21134},[66,67,225],[15,21136,1114],{"className":21137},[1113],[15,21139,21141],{"className":21140},[206],[15,21142,21144],{"className":21143,"style":1493},[210],[15,21145],{},[15,21147,21149],{"className":21148},[162],[15,21150,120],{"className":21151},[1422,1508],[15,21153,21155,21158],{"className":21154},[66],[15,21156,1224],{"className":21157,"style":1515},[66,67],[15,21159,21161],{"className":21160},[198],[15,21162,21164,21184],{"className":21163},[202,1090],[15,21165,21167,21181],{"className":21166},[206],[15,21168,21170],{"className":21169,"style":1528},[210],[15,21171,21172,21175],{"style":1531},[15,21173],{"className":21174,"style":218},[217],[15,21176,21178],{"className":21177},[222,223,224,225],[15,21179,1205],{"className":21180},[66,67,225],[15,21182,1114],{"className":21183},[1113],[15,21185,21187],{"className":21186},[206],[15,21188,21190],{"className":21189,"style":1121},[210],[15,21191],{},[15,21193],{"className":21194,"style":72},[71],[15,21196,6602],{"className":21197},[76],[15,21199],{"className":21200,"style":72},[71],[15,21202,116],{"className":21203,"style":158},[66,67],[15,21205,120],{"className":21206},[162],[15,21208,21210,21213],{"className":21209},[66],[15,21211,123],{"className":21212},[66,67],[15,21214,21216],{"className":21215},[198],[15,21217,21219,21239],{"className":21218},[202,1090],[15,21220,21222,21236],{"className":21221},[206],[15,21223,21225],{"className":21224,"style":1528},[210],[15,21226,21227,21230],{"style":1584},[15,21228],{"className":21229,"style":218},[217],[15,21231,21233],{"className":21232},[222,223,224,225],[15,21234,1205],{"className":21235},[66,67,225],[15,21237,1114],{"className":21238},[1113],[15,21240,21242],{"className":21241},[206],[15,21243,21245],{"className":21244,"style":1121},[210],[15,21246],{},[15,21248,126],{"className":21249},[169],[15,21251,21253,21259],{"className":21252},[169],[15,21254,21256],{"className":21255},[169],[15,21257,126],{"className":21258},[1422,1508],[15,21260,21262],{"className":21261},[198],[15,21263,21265],{"className":21264},[202],[15,21266,21268],{"className":21267},[206],[15,21269,21272],{"className":21270,"style":21271},[210],"height:1.054em;",[15,21273,21275,21278],{"style":21274},"top:-3.3029em;margin-right:0.05em;",[15,21276],{"className":21277,"style":218},[217],[15,21279,21281],{"className":21280},[222,223,224,225],[15,21282,1263],{"className":21283},[66,225],[15,21285],{"className":21286,"style":72},[71],[15,21288,1245],{"className":21289},[76],[15,21291],{"className":21292,"style":72},[71],[15,21294,1248],{"className":21295},[66,67],[15,21297,1251],{"className":21298},[66],[15,21300,116],{"className":21301,"style":158},[66,67],[15,21303,21305,21308],{"className":21304},[66],[15,21306,1251],{"className":21307},[66],[15,21309,21311],{"className":21310},[198],[15,21312,21314,21348],{"className":21313},[202,1090],[15,21315,21317,21345],{"className":21316},[206],[15,21318,21320,21334],{"className":21319,"style":1650},[210],[15,21321,21322,21325],{"style":1653},[15,21323],{"className":21324,"style":218},[217],[15,21326,21328],{"className":21327},[222,223,224,225],[15,21329,21331],{"className":21330},[66,225],[15,21332,1061],{"className":21333,"style":1083},[66,494,225],[15,21335,21336,21339],{"style":397},[15,21337],{"className":21338,"style":218},[217],[15,21340,21342],{"className":21341},[222,223,224,225],[15,21343,1263],{"className":21344},[66,225],[15,21346,1114],{"className":21347},[1113],[15,21349,21351],{"className":21350},[206],[15,21352,21354],{"className":21353,"style":1686},[210],[15,21355],{},[15,21357,21359],{"className":21358,"style":1418},[169,1417],[15,21360,4330],{"className":21361},[1422,1423],[15,21363],{"className":21364,"style":697},[71],[15,21366,3573],{"className":21367},[66],[11,21369,21370,21371,21615],{},"由表示定理，",[15,21372,21374,21421],{"className":21373,"translate":19},[18],[15,21375,21377],{"className":21376},[23],[25,21378,21379],{"xmlns":27},[29,21380,21381,21419],{},[32,21382,21383,21389,21391,21397,21403,21405,21407,21409,21411,21417],{},[131,21384,21385,21387],{},[35,21386,116],{},[39,21388,1162],{},[39,21390,129],{},[1057,21392,21393,21395],{},[39,21394,1200],{},[35,21396,1205],{},[1057,21398,21399,21401],{},[35,21400,8390],{},[35,21402,1205],{},[35,21404,621],{},[39,21406,120],{"stretchy":119},[39,21408,3268],{},[39,21410,335],{"separator":53},[1057,21412,21413,21415],{},[35,21414,123],{},[35,21416,1205],{},[39,21418,126],{"stretchy":119},[45,21420,16812],{"encoding":47},[15,21422,21424,21468],{"className":21423,"ariaHidden":53},[52],[15,21425,21427,21430,21459,21462,21465],{"className":21426},[57],[15,21428],{"className":21429,"style":3408},[61],[15,21431,21433,21436],{"className":21432},[66],[15,21434,116],{"className":21435,"style":158},[66,67],[15,21437,21439],{"className":21438},[198],[15,21440,21442],{"className":21441},[202],[15,21443,21445],{"className":21444},[206],[15,21446,21448],{"className":21447,"style":16249},[210],[15,21449,21450,21453],{"style":213},[15,21451],{"className":21452,"style":218},[217],[15,21454,21456],{"className":21455},[222,223,224,225],[15,21457,1162],{"className":21458},[76,225],[15,21460],{"className":21461,"style":173},[71],[15,21463,129],{"className":21464},[177],[15,21466],{"className":21467,"style":173},[71],[15,21469,21471,21474,21514,21517,21557,21560,21563,21566,21569,21572,21612],{"className":21470},[57],[15,21472],{"className":21473,"style":16866},[61],[15,21475,21477,21480],{"className":21476},[1339],[15,21478,1200],{"className":21479,"style":13702},[1339,1470,13701],[15,21481,21483],{"className":21482},[198],[15,21484,21486,21506],{"className":21485},[202,1090],[15,21487,21489,21503],{"className":21488},[206],[15,21490,21492],{"className":21491,"style":16885},[210],[15,21493,21494,21497],{"style":13718},[15,21495],{"className":21496,"style":218},[217],[15,21498,21500],{"className":21499},[222,223,224,225],[15,21501,1205],{"className":21502},[66,67,225],[15,21504,1114],{"className":21505},[1113],[15,21507,21509],{"className":21508},[206],[15,21510,21512],{"className":21511,"style":16906},[210],[15,21513],{},[15,21515],{"className":21516,"style":697},[71],[15,21518,21520,21523],{"className":21519},[66],[15,21521,8390],{"className":21522,"style":8560},[66,67],[15,21524,21526],{"className":21525},[198],[15,21527,21529,21549],{"className":21528},[202,1090],[15,21530,21532,21546],{"className":21531},[206],[15,21533,21535],{"className":21534,"style":1528},[210],[15,21536,21537,21540],{"style":8575},[15,21538],{"className":21539,"style":218},[217],[15,21541,21543],{"className":21542},[222,223,224,225],[15,21544,1205],{"className":21545},[66,67,225],[15,21547,1114],{"className":21548},[1113],[15,21550,21552],{"className":21551},[206],[15,21553,21555],{"className":21554,"style":1121},[210],[15,21556],{},[15,21558,621],{"className":21559,"style":684},[66,67],[15,21561,120],{"className":21562},[162],[15,21564,3268],{"className":21565},[66],[15,21567,335],{"className":21568},[422],[15,21570],{"className":21571,"style":697},[71],[15,21573,21575,21578],{"className":21574},[66],[15,21576,123],{"className":21577},[66,67],[15,21579,21581],{"className":21580},[198],[15,21582,21584,21604],{"className":21583},[202,1090],[15,21585,21587,21601],{"className":21586},[206],[15,21588,21590],{"className":21589,"style":1528},[210],[15,21591,21592,21595],{"style":1584},[15,21593],{"className":21594,"style":218},[217],[15,21596,21598],{"className":21597},[222,223,224,225],[15,21599,1205],{"className":21600},[66,67,225],[15,21602,1114],{"className":21603},[1113],[15,21605,21607],{"className":21606},[206],[15,21608,21610],{"className":21609,"style":1121},[210],[15,21611],{},[15,21613,126],{"className":21614},[169],"。代入式 (4)：",[15,21617,21619],{"className":21618,"translate":19},[294],[15,21620,21622,21668],{"className":21621,"translate":19},[18],[15,21623,21625],{"className":21624},[23],[25,21626,21627],{"xmlns":27,"display":303},[29,21628,21629,21665],{},[32,21630,21631,21633,21635,21637,21639,21641,21647,21649,21651,21653,21659,21661,21663],{},[35,21632,1251],{"mathvariant":138},[35,21634,1224],{},[39,21636,6602],{},[35,21638,2853],{},[35,21640,8390],{},[131,21642,21643,21645],{},[35,21644,1251],{"mathvariant":138},[1209,21646,1263],{},[39,21648,1245],{},[35,21650,1248],{},[1274,21652,2462],{},[131,21654,21655,21657],{},[35,21656,8390],{},[35,21658,139],{"mathvariant":138},[35,21660,2853],{},[35,21662,8390],{},[35,21664,3573],{"mathvariant":138},[45,21666,21667],{"encoding":47},"\\|y-K\\alpha\\|^2+\\lambda\\,\\alpha^\\top K\\alpha.",[15,21669,21671,21692,21742],{"className":21670,"ariaHidden":53},[52],[15,21672,21674,21677,21680,21683,21686,21689],{"className":21673},[57],[15,21675],{"className":21676,"style":154},[61],[15,21678,1251],{"className":21679},[66],[15,21681,1224],{"className":21682,"style":1515},[66,67],[15,21684],{"className":21685,"style":72},[71],[15,21687,6602],{"className":21688},[76],[15,21690],{"className":21691,"style":72},[71],[15,21693,21695,21698,21701,21704,21733,21736,21739],{"className":21694},[57],[15,21696],{"className":21697,"style":11214},[61],[15,21699,2853],{"className":21700,"style":2901},[66,67],[15,21702,8390],{"className":21703,"style":8560},[66,67],[15,21705,21707,21710],{"className":21706},[66],[15,21708,1251],{"className":21709},[66],[15,21711,21713],{"className":21712},[198],[15,21714,21716],{"className":21715},[202],[15,21717,21719],{"className":21718},[206],[15,21720,21722],{"className":21721,"style":1650},[210],[15,21723,21724,21727],{"style":397},[15,21725],{"className":21726,"style":218},[217],[15,21728,21730],{"className":21729},[222,223,224,225],[15,21731,1263],{"className":21732},[66,225],[15,21734],{"className":21735,"style":72},[71],[15,21737,1245],{"className":21738},[76],[15,21740],{"className":21741,"style":72},[71],[15,21743,21745,21748,21751,21754,21783,21786,21789],{"className":21744},[57],[15,21746],{"className":21747,"style":394},[61],[15,21749,1248],{"className":21750},[66,67],[15,21752],{"className":21753,"style":697},[71],[15,21755,21757,21760],{"className":21756},[66],[15,21758,8390],{"className":21759,"style":8560},[66,67],[15,21761,21763],{"className":21762},[198],[15,21764,21766],{"className":21765},[202],[15,21767,21769],{"className":21768},[206],[15,21770,21772],{"className":21771,"style":394},[210],[15,21773,21774,21777],{"style":397},[15,21775],{"className":21776,"style":218},[217],[15,21778,21780],{"className":21779},[222,223,224,225],[15,21781,139],{"className":21782},[66,225],[15,21784,2853],{"className":21785,"style":2901},[66,67],[15,21787,8390],{"className":21788,"style":8560},[66,67],[15,21790,3573],{"className":21791},[66],[11,21793,21794,21795,21823],{},"对 ",[15,21796,21798,21811],{"className":21797,"translate":19},[18],[15,21799,21801],{"className":21800},[23],[25,21802,21803],{"xmlns":27},[29,21804,21805,21809],{},[32,21806,21807],{},[35,21808,8390],{},[45,21810,16077],{"encoding":47},[15,21812,21814],{"className":21813,"ariaHidden":53},[52],[15,21815,21817,21820],{"className":21816},[57],[15,21818],{"className":21819,"style":86},[61],[15,21821,8390],{"className":21822,"style":8560},[66,67]," 求梯度：",[15,21825,21827],{"className":21826,"translate":19},[294],[15,21828,21830,21900],{"className":21829,"translate":19},[18],[15,21831,21833],{"className":21832},[23],[25,21834,21835],{"xmlns":27,"display":303},[29,21836,21837,21897],{},[32,21838,21839,21841,21843,21845,21847,21849,21851,21853,21855,21857,21859,21861,21863,21865,21867,21869,21871,21873,21875,21877,21879,21881,21883,21885,21887,21889,21891,21893,21895],{},[39,21840,6602],{},[1209,21842,1263],{},[35,21844,2853],{},[39,21846,120],{"stretchy":119},[35,21848,1224],{},[39,21850,6602],{},[35,21852,2853],{},[35,21854,8390],{},[39,21856,126],{"stretchy":119},[39,21858,1245],{},[1209,21860,1263],{},[35,21862,1248],{},[35,21864,2853],{},[35,21866,8390],{},[39,21868,129],{},[1209,21870,1263],{},[35,21872,2853],{},[39,21874,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,21876,120],{"stretchy":119},[35,21878,2853],{},[39,21880,1245],{},[35,21882,1248],{},[35,21884,16427],{},[39,21886,126],{"stretchy":119},[35,21888,8390],{},[39,21890,6602],{},[35,21892,1224],{},[39,21894,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,21896,3573],{"mathvariant":138},[45,21898,21899],{"encoding":47},"-2K(y-K\\alpha)+2\\lambda K\\alpha\n=2K\\bigl((K+\\lambda I)\\alpha-y\\bigr).",[15,21901,21903,21933,21957,21984,22017,22044],{"className":21902,"ariaHidden":53},[52],[15,21904,21906,21909,21912,21915,21918,21921,21924,21927,21930],{"className":21905},[57],[15,21907],{"className":21908,"style":154},[61],[15,21910,6602],{"className":21911},[66],[15,21913,1263],{"className":21914},[66],[15,21916,2853],{"className":21917,"style":2901},[66,67],[15,21919,120],{"className":21920},[162],[15,21922,1224],{"className":21923,"style":1515},[66,67],[15,21925],{"className":21926,"style":72},[71],[15,21928,6602],{"className":21929},[76],[15,21931],{"className":21932,"style":72},[71],[15,21934,21936,21939,21942,21945,21948,21951,21954],{"className":21935},[57],[15,21937],{"className":21938,"style":154},[61],[15,21940,2853],{"className":21941,"style":2901},[66,67],[15,21943,8390],{"className":21944,"style":8560},[66,67],[15,21946,126],{"className":21947},[169],[15,21949],{"className":21950,"style":72},[71],[15,21952,1245],{"className":21953},[76],[15,21955],{"className":21956,"style":72},[71],[15,21958,21960,21963,21966,21969,21972,21975,21978,21981],{"className":21959},[57],[15,21961],{"className":21962,"style":840},[61],[15,21964,1263],{"className":21965},[66],[15,21967,1248],{"className":21968},[66,67],[15,21970,2853],{"className":21971,"style":2901},[66,67],[15,21973,8390],{"className":21974,"style":8560},[66,67],[15,21976],{"className":21977,"style":173},[71],[15,21979,129],{"className":21980},[177],[15,21982],{"className":21983,"style":173},[71],[15,21985,21987,21990,21993,21996,22002,22005,22008,22011,22014],{"className":21986},[57],[15,21988],{"className":21989,"style":4400},[61],[15,21991,1263],{"className":21992},[66],[15,21994,2853],{"className":21995,"style":2901},[66,67],[15,21997,21999],{"className":21998},[162],[15,22000,120],{"className":22001},[1422,1508],[15,22003,120],{"className":22004},[162],[15,22006,2853],{"className":22007,"style":2901},[66,67],[15,22009],{"className":22010,"style":72},[71],[15,22012,1245],{"className":22013},[76],[15,22015],{"className":22016,"style":72},[71],[15,22018,22020,22023,22026,22029,22032,22035,22038,22041],{"className":22019},[57],[15,22021],{"className":22022,"style":154},[61],[15,22024,1248],{"className":22025},[66,67],[15,22027,16427],{"className":22028,"style":16497},[66,67],[15,22030,126],{"className":22031},[169],[15,22033,8390],{"className":22034,"style":8560},[66,67],[15,22036],{"className":22037,"style":72},[71],[15,22039,6602],{"className":22040},[76],[15,22042],{"className":22043,"style":72},[71],[15,22045,22047,22050,22053,22059],{"className":22046},[57],[15,22048],{"className":22049,"style":4400},[61],[15,22051,1224],{"className":22052,"style":1515},[66,67],[15,22054,22056],{"className":22055},[169],[15,22057,126],{"className":22058},[1422,1508],[15,22060,3573],{"className":22061},[66],[11,22063,22064,22065,22168,22169,22219,22220,22276],{},"令梯度为零，得到正规方程 ",[15,22066,22068,22102],{"className":22067,"translate":19},[18],[15,22069,22071],{"className":22070},[23],[25,22072,22073],{"xmlns":27},[29,22074,22075,22099],{},[32,22076,22077,22079,22081,22083,22085,22087,22089,22091,22093,22095,22097],{},[35,22078,2853],{},[39,22080,120],{"stretchy":119},[35,22082,2853],{},[39,22084,1245],{},[35,22086,1248],{},[35,22088,16427],{},[39,22090,126],{"stretchy":119},[35,22092,8390],{},[39,22094,129],{},[35,22096,2853],{},[35,22098,1224],{},[45,22100,22101],{"encoding":47},"K(K+\\lambda I)\\alpha=Ky",[15,22103,22105,22129,22156],{"className":22104,"ariaHidden":53},[52],[15,22106,22108,22111,22114,22117,22120,22123,22126],{"className":22107},[57],[15,22109],{"className":22110,"style":154},[61],[15,22112,2853],{"className":22113,"style":2901},[66,67],[15,22115,120],{"className":22116},[162],[15,22118,2853],{"className":22119,"style":2901},[66,67],[15,22121],{"className":22122,"style":72},[71],[15,22124,1245],{"className":22125},[76],[15,22127],{"className":22128,"style":72},[71],[15,22130,22132,22135,22138,22141,22144,22147,22150,22153],{"className":22131},[57],[15,22133],{"className":22134,"style":154},[61],[15,22136,1248],{"className":22137},[66,67],[15,22139,16427],{"className":22140,"style":16497},[66,67],[15,22142,126],{"className":22143},[169],[15,22145,8390],{"className":22146,"style":8560},[66,67],[15,22148],{"className":22149,"style":173},[71],[15,22151,129],{"className":22152},[177],[15,22154],{"className":22155,"style":173},[71],[15,22157,22159,22162,22165],{"className":22158},[57],[15,22160],{"className":22161,"style":17379},[61],[15,22163,2853],{"className":22164,"style":2901},[66,67],[15,22166,1224],{"className":22167,"style":1515},[66,67],"。由于 ",[15,22170,22172,22189],{"className":22171,"translate":19},[18],[15,22173,22175],{"className":22174},[23],[25,22176,22177],{"xmlns":27},[29,22178,22179,22187],{},[32,22180,22181,22183,22185],{},[35,22182,1248],{},[39,22184,9283],{},[1209,22186,1804],{},[45,22188,9288],{"encoding":47},[15,22190,22192,22210],{"className":22191,"ariaHidden":53},[52],[15,22193,22195,22198,22201,22204,22207],{"className":22194},[57],[15,22196],{"className":22197,"style":9298},[61],[15,22199,1248],{"className":22200},[66,67],[15,22202],{"className":22203,"style":173},[71],[15,22205,9283],{"className":22206},[177],[15,22208],{"className":22209,"style":173},[71],[15,22211,22213,22216],{"className":22212},[57],[15,22214],{"className":22215,"style":1836},[61],[15,22217,1804],{"className":22218},[66]," 时 ",[15,22221,22223,22243],{"className":22222,"translate":19},[18],[15,22224,22226],{"className":22225},[23],[25,22227,22228],{"xmlns":27},[29,22229,22230,22240],{},[32,22231,22232,22234,22236,22238],{},[35,22233,2853],{},[39,22235,1245],{},[35,22237,1248],{},[35,22239,16427],{},[45,22241,22242],{"encoding":47},"K+\\lambda I",[15,22244,22246,22264],{"className":22245,"ariaHidden":53},[52],[15,22247,22249,22252,22255,22258,22261],{"className":22248},[57],[15,22250],{"className":22251,"style":1997},[61],[15,22253,2853],{"className":22254,"style":2901},[66,67],[15,22256],{"className":22257,"style":72},[71],[15,22259,1245],{"className":22260},[76],[15,22262],{"className":22263,"style":72},[71],[15,22265,22267,22270,22273],{"className":22266},[57],[15,22268],{"className":22269,"style":840},[61],[15,22271,1248],{"className":22272},[66,67],[15,22274,16427],{"className":22275,"style":16497},[66,67]," 可逆，",[15,22278,22280],{"className":22279,"translate":19},[294],[15,22281,22283,22320],{"className":22282,"translate":19},[18],[15,22284,22286],{"className":22285},[23],[25,22287,22288],{"xmlns":27,"display":303},[29,22289,22290,22318],{},[32,22291,22292,22294,22296,22298,22300,22302,22304,22306,22316],{},[35,22293,8390],{},[39,22295,129],{},[39,22297,120],{"stretchy":119},[35,22299,2853],{},[39,22301,1245],{},[35,22303,1248],{},[35,22305,16427],{},[131,22307,22308,22310],{},[39,22309,126],{"stretchy":119},[32,22311,22312,22314],{},[39,22313,6602],{},[1209,22315,1211],{},[35,22317,1224],{},[45,22319,16442],{"encoding":47},[15,22321,22323,22341,22362],{"className":22322,"ariaHidden":53},[52],[15,22324,22326,22329,22332,22335,22338],{"className":22325},[57],[15,22327],{"className":22328,"style":86},[61],[15,22330,8390],{"className":22331,"style":8560},[66,67],[15,22333],{"className":22334,"style":173},[71],[15,22336,129],{"className":22337},[177],[15,22339],{"className":22340,"style":173},[71],[15,22342,22344,22347,22350,22353,22356,22359],{"className":22343},[57],[15,22345],{"className":22346,"style":154},[61],[15,22348,120],{"className":22349},[162],[15,22351,2853],{"className":22352,"style":2901},[66,67],[15,22354],{"className":22355,"style":72},[71],[15,22357,1245],{"className":22358},[76],[15,22360],{"className":22361,"style":72},[71],[15,22363,22365,22368,22371,22374,22409],{"className":22364},[57],[15,22366],{"className":22367,"style":11214},[61],[15,22369,1248],{"className":22370},[66,67],[15,22372,16427],{"className":22373,"style":16497},[66,67],[15,22375,22377,22380],{"className":22376},[169],[15,22378,126],{"className":22379},[169],[15,22381,22383],{"className":22382},[198],[15,22384,22386],{"className":22385},[202],[15,22387,22389],{"className":22388},[206],[15,22390,22392],{"className":22391,"style":1650},[210],[15,22393,22394,22397],{"style":397},[15,22395],{"className":22396,"style":218},[217],[15,22398,22400],{"className":22399},[222,223,224,225],[15,22401,22403,22406],{"className":22402},[66,225],[15,22404,6602],{"className":22405},[66,225],[15,22407,1211],{"className":22408},[66,225],[15,22410,1224],{"className":22411,"style":1515},[66,67],[11,22413,22414,22415,22443,22444,22472],{},"是一个解（即使 ",[15,22416,22418,22431],{"className":22417,"translate":19},[18],[15,22419,22421],{"className":22420},[23],[25,22422,22423],{"xmlns":27},[29,22424,22425,22429],{},[32,22426,22427],{},[35,22428,2853],{},[45,22430,2853],{"encoding":47},[15,22432,22434],{"className":22433,"ariaHidden":53},[52],[15,22435,22437,22440],{"className":22436},[57],[15,22438],{"className":22439,"style":490},[61],[15,22441,2853],{"className":22442,"style":2901},[66,67]," 奇异，这个 ",[15,22445,22447,22460],{"className":22446,"translate":19},[18],[15,22448,22450],{"className":22449},[23],[25,22451,22452],{"xmlns":27},[29,22453,22454,22458],{},[32,22455,22456],{},[35,22457,8390],{},[45,22459,16077],{"encoding":47},[15,22461,22463],{"className":22462,"ariaHidden":53},[52],[15,22464,22466,22469],{"className":22465},[57],[15,22467],{"className":22468,"style":86},[61],[15,22470,8390],{"className":22471,"style":8560},[66,67]," 也满足正规方程；不同的解给出同一个函数）。预测为",[15,22474,22476],{"className":22475,"translate":19},[294],[15,22477,22479,22601],{"className":22478,"translate":19},[18],[15,22480,22482],{"className":22481},[23],[25,22483,22484],{"xmlns":27,"display":303},[29,22485,22486,22598],{},[32,22487,22488,22494,22496,22498,22500,22502,22504,22506,22508,22514,22516,22518,22520,22522,22524,22534,22536,22538,22540,22542,22544,22546,22548,22550,22552,22554,22556,22558,22560,22566,22568,22570,22572,22574,22576,22578,22580,22582,22588,22590,22596],{},[131,22489,22490,22492],{},[35,22491,116],{},[39,22493,1162],{},[39,22495,120],{"stretchy":119},[35,22497,123],{},[39,22499,126],{"stretchy":119},[39,22501,129],{},[35,22503,621],{},[39,22505,120],{"stretchy":119},[35,22507,123],{},[131,22509,22510,22512],{},[39,22511,126],{"stretchy":119},[35,22513,139],{"mathvariant":138},[39,22515,120],{"stretchy":119},[35,22517,2853],{},[39,22519,1245],{},[35,22521,1248],{},[35,22523,16427],{},[131,22525,22526,22528],{},[39,22527,126],{"stretchy":119},[32,22529,22530,22532],{},[39,22531,6602],{},[1209,22533,1211],{},[35,22535,1224],{},[39,22537,335],{"separator":53},[71,22539],{"linebreak":10456},[35,22541,621],{},[39,22543,120],{"stretchy":119},[35,22545,123],{},[39,22547,126],{"stretchy":119},[39,22549,129],{},[39,22551,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,22553,621],{},[39,22555,120],{"stretchy":119},[35,22557,123],{},[39,22559,335],{"separator":53},[1057,22561,22562,22564],{},[35,22563,123],{},[1209,22565,1211],{},[39,22567,126],{"stretchy":119},[39,22569,335],{"separator":53},[39,22571,2095],{},[39,22573,335],{"separator":53},[35,22575,621],{},[39,22577,120],{"stretchy":119},[35,22579,123],{},[39,22581,335],{"separator":53},[1057,22583,22584,22586],{},[35,22585,123],{},[35,22587,37],{},[39,22589,126],{"stretchy":119},[131,22591,22592,22594],{},[39,22593,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[35,22595,139],{"mathvariant":138},[35,22597,3573],{"mathvariant":138},[45,22599,22600],{"encoding":47},"f^*(x)=k(x)^\\top(K+\\lambda I)^{-1}y,\n\\\\\nk(x)=\\bigl(k(x,x_1),\\dots,k(x,x_n)\\bigr)^\\top.",[15,22602,22604,22657,22716,22769,22772,22799],{"className":22603,"ariaHidden":53},[52],[15,22605,22607,22610,22639,22642,22645,22648,22651,22654],{"className":22606},[57],[15,22608],{"className":22609,"style":154},[61],[15,22611,22613,22616],{"className":22612},[66],[15,22614,116],{"className":22615,"style":158},[66,67],[15,22617,22619],{"className":22618},[198],[15,22620,22622],{"className":22621},[202],[15,22623,22625],{"className":22624},[206],[15,22626,22628],{"className":22627,"style":1308},[210],[15,22629,22630,22633],{"style":397},[15,22631],{"className":22632,"style":218},[217],[15,22634,22636],{"className":22635},[222,223,224,225],[15,22637,1162],{"className":22638},[76,225],[15,22640,120],{"className":22641},[162],[15,22643,123],{"className":22644},[66,67],[15,22646,126],{"className":22647},[169],[15,22649],{"className":22650,"style":173},[71],[15,22652,129],{"className":22653},[177],[15,22655],{"className":22656,"style":173},[71],[15,22658,22660,22663,22666,22669,22672,22701,22704,22707,22710,22713],{"className":22659},[57],[15,22661],{"className":22662,"style":375},[61],[15,22664,621],{"className":22665,"style":684},[66,67],[15,22667,120],{"className":22668},[162],[15,22670,123],{"className":22671},[66,67],[15,22673,22675,22678],{"className":22674},[169],[15,22676,126],{"className":22677},[169],[15,22679,22681],{"className":22680},[198],[15,22682,22684],{"className":22683},[202],[15,22685,22687],{"className":22686},[206],[15,22688,22690],{"className":22689,"style":394},[210],[15,22691,22692,22695],{"style":397},[15,22693],{"className":22694,"style":218},[217],[15,22696,22698],{"className":22697},[222,223,224,225],[15,22699,139],{"className":22700},[66,225],[15,22702,120],{"className":22703},[162],[15,22705,2853],{"className":22706,"style":2901},[66,67],[15,22708],{"className":22709,"style":72},[71],[15,22711,1245],{"className":22712},[76],[15,22714],{"className":22715,"style":72},[71],[15,22717,22719,22722,22725,22728,22763,22766],{"className":22718},[57],[15,22720],{"className":22721,"style":11214},[61],[15,22723,1248],{"className":22724},[66,67],[15,22726,16427],{"className":22727,"style":16497},[66,67],[15,22729,22731,22734],{"className":22730},[169],[15,22732,126],{"className":22733},[169],[15,22735,22737],{"className":22736},[198],[15,22738,22740],{"className":22739},[202],[15,22741,22743],{"className":22742},[206],[15,22744,22746],{"className":22745,"style":1650},[210],[15,22747,22748,22751],{"style":397},[15,22749],{"className":22750,"style":218},[217],[15,22752,22754],{"className":22753},[222,223,224,225],[15,22755,22757,22760],{"className":22756},[66,225],[15,22758,6602],{"className":22759},[66,225],[15,22761,1211],{"className":22762},[66,225],[15,22764,1224],{"className":22765,"style":1515},[66,67],[15,22767,335],{"className":22768},[422],[15,22770],{"className":22771},[71,10456],[15,22773,22775,22778,22781,22784,22787,22790,22793,22796],{"className":22774},[57],[15,22776],{"className":22777,"style":154},[61],[15,22779,621],{"className":22780,"style":684},[66,67],[15,22782,120],{"className":22783},[162],[15,22785,123],{"className":22786},[66,67],[15,22788,126],{"className":22789},[169],[15,22791],{"className":22792,"style":173},[71],[15,22794,129],{"className":22795},[177],[15,22797],{"className":22798,"style":173},[71],[15,22800,22802,22806,22812,22815,22818,22821,22824,22827,22867,22870,22873,22876,22879,22882,22885,22888,22891,22894,22897,22900,22903,22943,22946,22979],{"className":22801},[57],[15,22803],{"className":22804,"style":22805},[61],"height:1.439em;vertical-align:-0.35em;",[15,22807,22809],{"className":22808},[162],[15,22810,120],{"className":22811},[1422,1508],[15,22813,621],{"className":22814,"style":684},[66,67],[15,22816,120],{"className":22817},[162],[15,22819,123],{"className":22820},[66,67],[15,22822,335],{"className":22823},[422],[15,22825],{"className":22826,"style":697},[71],[15,22828,22830,22833],{"className":22829},[66],[15,22831,123],{"className":22832},[66,67],[15,22834,22836],{"className":22835},[198],[15,22837,22839,22859],{"className":22838},[202,1090],[15,22840,22842,22856],{"className":22841},[206],[15,22843,22845],{"className":22844,"style":2139},[210],[15,22846,22847,22850],{"style":1584},[15,22848],{"className":22849,"style":218},[217],[15,22851,22853],{"className":22852},[222,223,224,225],[15,22854,1211],{"className":22855},[66,225],[15,22857,1114],{"className":22858},[1113],[15,22860,22862],{"className":22861},[206],[15,22863,22865],{"className":22864,"style":1121},[210],[15,22866],{},[15,22868,126],{"className":22869},[169],[15,22871,335],{"className":22872},[422],[15,22874],{"className":22875,"style":697},[71],[15,22877,2095],{"className":22878},[1413],[15,22880],{"className":22881,"style":697},[71],[15,22883,335],{"className":22884},[422],[15,22886],{"className":22887,"style":697},[71],[15,22889,621],{"className":22890,"style":684},[66,67],[15,22892,120],{"className":22893},[16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",[15,22986,22988,23013],{"className":22987,"translate":19},[18],[15,22989,22991],{"className":22990},[23],[25,22992,22993],{"xmlns":27},[29,22994,22995,23010],{},[32,22996,22997,23000,23002,23008],{},[35,22998,22999],{},"O",[39,23001,120],{"stretchy":119},[131,23003,23004,23006],{},[35,23005,37],{},[1209,23007,8678],{},[39,23009,126],{"stretchy":119},[45,23011,23012],{"encoding":47},"O(n^3)",[15,23014,23016],{"className":23015,"ariaHidden":53},[52],[15,23017,23019,23022,23026,23029,23058],{"className":23018},[57],[15,23020],{"className":23021,"style":9154},[61],[15,23023,22999],{"className":23024,"style":23025},[66,67],"margin-right:0.0278em;",[15,23027,120],{"className":23028},[162],[15,23030,23032,23035],{"className":23031},[66],[15,23033,37],{"className":23034},[66,67],[15,23036,23038],{"className":23037},[198],[15,23039,23041],{"className":23040},[202],[15,23042,23044],{"className":23043},[206],[15,23045,23047],{"className":23046,"style":6740},[210],[15,23048,23049,23052],{"style":213},[15,23050],{"className":23051,"style":218},[217],[15,23053,23055],{"className":23054},[222,223,224,225],[15,23056,8678],{"className":23057},[66,225],[15,23059,126],{"className":23060},[169]," 时间、",[15,23063,23065,23089],{"className":23064,"translate":19},[18],[15,23066,23068],{"className":23067},[23],[25,23069,23070],{"xmlns":27},[29,23071,23072,23086],{},[32,23073,23074,23076,23078,23084],{},[35,23075,22999],{},[39,23077,120],{"stretchy":119},[131,23079,23080,23082],{},[35,23081,37],{},[1209,23083,1263],{},[39,23085,126],{"stretchy":119},[45,23087,23088],{"encoding":47},"O(n^2)",[15,23090,23092],{"className":23091,"ariaHidden":53},[52],[15,23093,23095,23098,23101,23104,23133],{"className":23094},[57],[15,23096],{"className":23097,"style":9154},[61],[15,23099,22999],{"className":23100,"style":23025},[66,67],[15,23102,120],{"className":23103},[162],[15,23105,23107,23110],{"className":23106},[66],[15,23108,37],{"className":23109},[66,67],[15,23111,23113],{"className":23112},[198],[15,23114,23116],{"className":23115},[202],[15,23117,23119],{"className":23118},[206],[15,23120,23122],{"className":23121,"style":6740},[210],[15,23123,23124,23127],{"style":213},[15,23125],{"className":23126,"style":218},[217],[15,23128,23130],{"className":23129},[222,223,224,225],[15,23131,1263],{"className":23132},[66,225],[15,23134,126],{"className":23135},[169]," 内存；预测一个点需要 ",[15,23138,23140,23160],{"className":23139,"translate":19},[18],[15,23141,23143],{"className":23142},[23],[25,23144,23145],{"xmlns":27},[29,23146,23147,23157],{},[32,23148,23149,23151,23153,23155],{},[35,23150,22999],{},[39,23152,120],{"stretchy":119},[35,23154,37],{},[39,23156,126],{"stretchy":119},[45,23158,23159],{"encoding":47},"O(n)",[15,23161,23163],{"className":23162,"ariaHidden":53},[52],[15,23164,23166,23169,23172,23175,23178],{"className":23165},[57],[15,23167],{"className":23168,"style":154},[61],[15,23170,22999],{"className":23171,"style":23025},[66,67],[15,23173,120],{"className":23174},[162],[15,23176,37],{"className":23177},[66,67],[15,23179,126],{"className":23180},[169]," 次核函数求值。",[1904,23183,23185],{"id":23184},"_52-支持向量机","5.2 支持向量机",[11,23187,23188],{},"软间隔 SVM 在 RKHS 中的 primal 问题是",[15,23190,23192],{"className":23191,"translate":19},[294],[15,23193,23195,23321],{"className":23194,"translate":19},[18],[15,23196,23198],{"className":23197},[23],[25,23199,23200],{"xmlns":27,"display":303},[29,23201,23202,23318],{},[32,23203,23204,23230,23316],{},[1172,23205,23206,23212],{},[32,23207,23208,23210],{},[35,23209,1178],{},[39,23211,1170],{},[32,23213,23214,23216,23218,23220,23222,23224,23226,23228],{},[35,23215,116],{},[39,23217,1187],{},[35,23219,1061],{"mathvariant":445},[39,23221,335],{"separator":53},[1274,23223,2462],{},[35,23225,4587],{},[39,23227,1187],{},[35,23229,455],{"mathvariant":454},[32,23231,23232,23234,23240,23242,23244,23252,23254,23257,23271,23274,23276,23278,23280,23282,23284,23286,23292,23294,23296,23298,23304,23306,23308,23310,23312,23314],{},[39,23233,4307],{"fence":53},[6467,23235,23236,23238],{},[1209,23237,1211],{},[1209,23239,1263],{},[35,23241,1251],{"mathvariant":138},[35,23243,116],{},[1255,23245,23246,23248,23250],{},[35,23247,1251],{"mathvariant":138},[35,23249,1061],{"mathvariant":445},[1209,23251,1263],{},[39,23253,1245],{},[35,23255,23256],{},"C",[1196,23258,23259,23261,23269],{},[39,23260,1200],{},[32,23262,23263,23265,23267],{},[35,23264,1205],{},[39,23266,129],{},[1209,23268,1211],{},[35,23270,37],{},[35,23272,23273],{},"max",[39,23275,1170],{},[39,23277,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,23279,1804],{},[39,23281,335],{"separator":53},[1209,23283,1211],{},[39,23285,6602],{},[1057,23287,23288,23290],{},[35,23289,1224],{},[35,23291,1205],{},[39,23293,120],{"stretchy":119},[35,23295,116],{},[39,23297,120],{"stretchy":119},[1057,23299,23300,23302],{},[35,23301,123],{},[35,23303,1205],{},[39,23305,126],{"stretchy":119},[39,23307,1245],{},[35,23309,4587],{},[39,23311,126],{"stretchy":119},[39,23313,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,23315,4330],{"fence":53},[35,23317,3573],{"mathvariant":138},[45,23319,23320],{"encoding":47},"\\min_{f\\in\\mathcal H,\\,b\\in\\mathbb R}\n\\left\\{\n\\frac12\\|f\\|_{\\mathcal H}^2\n+C\\sum_{i=1}^n\\max\\bigl(0,1-y_i(f(x_i)+b)\\bigr)\n\\right\\}.",[15,23322,23324],{"className":23323,"ariaHidden":53},[52],[15,23325,23327,23330,23399,23402,23771,23774],{"className":23326},[57],[15,23328],{"className":23329,"style":1335},[61],[15,23331,23333],{"className":23332},[1339,1351],[15,23334,23336,23391],{"className":23335},[202,1090],[15,23337,23339,23388],{"className":23338},[206],[15,23340,23342,23378],{"className":23341,"style":1361},[210],[15,23343,23344,23347],{"style":1364},[15,23345],{"className":23346,"style":1368},[217],[15,23348,23350],{"className":23349},[222,223,224,225],[15,23351,23353,23356,23359,23362,23365,23369,23372,23375],{"className":23352},[66,225],[15,23354,116],{"className":23355,"style":158},[66,67,225],[15,23357,1187],{"className":23358},[177,225],[15,23360,1061],{"className":23361,"style":1083},[66,494,225],[15,23363,335],{"className":23364},[422,225],[15,23366],{"className":23367,"style":23368},[71,225],"margin-right:0.1952em;",[15,23370,4587],{"className":23371},[66,67,225],[15,23373,1187],{"className":23374},[177,225],[15,23376,455],{"className":23377},[66,517,225],[15,23379,23380,23383],{"style":1386},[15,23381],{"className":23382,"style":1368},[217],[15,23384,23385],{},[15,23386,1178],{"className":23387},[1339],[15,23389,1114],{"className":23390},[1113],[15,23392,23394],{"className":23393},[206],[15,23395,23397],{"className":23396,"style":1404},[210],[15,23398],{},[15,23400],{"className":23401,"style":697},[71],[15,23403,23405,23411,23480,23483,23486,23540,23543,23546,23549,23552,23555,23619,23622,23625,23631,23634,23637,23640,23643,23646,23649,23652,23692,23695,23698,23701,23741,23744,23747,23750,23753,23756,23759,23765],{"className":23404},[1413],[15,23406,23408],{"className":23407,"style":1418},[162,1417],[15,23409,4307],{"className":23410},[1422,1423],[15,23412,23414,23418,23477],{"className":23413},[66],[15,23415],{"className":23416},[162,23417],"nulldelimiter",[15,23419,23421],{"className":23420},[6467],[15,23422,23424,23468],{"className":23423},[202,1090],[15,23425,23427,23465],{"className":23426},[206],[15,23428,23431,23443,23453],{"className":23429,"style":23430},[210],"height:1.3214em;",[15,23432,23434,23437],{"style":23433},"top:-2.314em;",[15,23435],{"className":23436,"style":1368},[217],[15,23438,23440],{"className":23439},[66],[15,23441,1263],{"className":23442},[66],[15,23444,23446,23449],{"style":23445},"top:-3.23em;",[15,23447],{"className":23448,"style":1368},[217],[15,23450],{"className":23451,"style":4440},[23452],"frac-line",[15,23454,23456,23459],{"style":23455},"top:-3.677em;",[15,23457],{"className":23458,"style":1368},[217],[15,23460,23462],{"className":23461},[66],[15,23463,1211],{"className":23464},[66],[15,23466,1114],{"className":23467},[1113],[15,23469,23471],{"className":23470},[206],[15,23472,23475],{"className":23473,"style":23474},[210],"height:0.686em;",[15,23476],{},[15,23478],{"className":23479},[169,23417],[15,23481,1251],{"className":23482},[66],[15,23484,116],{"className":23485,"style":158},[66,67],[15,23487,23489,23492],{"className":23488},[66],[15,23490,1251],{"className":23491},[66],[15,23493,23495],{"className":23494},[198],[15,23496,23498,23532],{"className":23497},[202,1090],[15,23499,23501,23529],{"className":23500},[206],[15,23502,23504,23518],{"className":23503,"style":1650},[210],[15,23505,23506,23509],{"style":1653},[15,23507],{"className":23508,"style":218},[217],[15,23510,23512],{"className":23511},[222,223,224,225],[15,23513,23515],{"className":23514},[66,225],[15,23516,1061],{"className":23517,"style":1083},[66,494,225],[15,23519,23520,23523],{"style":397},[15,23521],{"className":23522,"style":218},[217],[15,23524,23526],{"className":23525},[222,223,224,225],[15,23527,1263],{"className":23528},[66,225],[15,23530,1114],{"className":23531},[1113],[15,23533,23535],{"className":23534},[206],[15,23536,23538],{"className":23537,"style":1686},[210],[15,23539],{},[15,23541],{"className":23542,"style":72},[71],[15,23544,1245],{"className":23545},[76],[15,23547],{"className":23548,"style":72},[71],[15,23550,23256],{"className":23551,"style":2901},[66,67],[15,23553],{"className":23554,"style":697},[71],[15,23556,23558],{"className":23557},[1339,1351],[15,23559,23561,23611],{"className":23560},[202,1090],[15,23562,23564,23608],{"className":23563},[206],[15,23565,23567,23587,23597],{"className":23566,"style":1436},[210],[15,23568,23569,23572],{"style":1439},[15,23570],{"className":23571,"style":1443},[217],[15,23573,23575],{"className":23574},[222,223,224,225],[15,23576,23578,23581,23584],{"className":23577},[66,225],[15,23579,1205],{"className":23580},[66,67,225],[15,23582,129],{"className":23583},[177,225],[15,23585,1211],{"className":23586},[66,225],[15,23588,23589,23592],{"style":1461},[15,23590],{"className":23591,"style":1443},[217],[15,23593,23594],{},[15,23595,1200],{"className":23596},[1339,1470,1471],[15,23598,23599,23602],{"style":1474},[15,23600],{"className":23601,"style":1443},[217],[15,23603,23605],{"className":23604},[222,223,224,225],[15,23606,37],{"className":23607},[66,67,225],[15,23609,1114],{"className":23610},[1113],[15,23612,23614],{"className":23613},[206],[15,23615,23617],{"className":23616,"style":1493},[210],[15,23618],{},[15,23620],{"className":23621,"style":697},[71],[15,23623,23273],{"className":23624},[1339],[15,23626,23628],{"className":23627},[162],[15,23629,120],{"className":23630},[1422,1508],[15,23632,1804],{"className":23633},[66],[15,23635,335],{"className":23636},[422],[15,23638],{"className":23639,"style":697},[71],[15,23641,1211],{"className":23642},[66],[15,23644],{"className":23645,"style":72},[71],[15,23647,6602],{"className":23648},[76],[15,23650],{"className":23651,"style":72},[71],[15,23653,23655,23658],{"className":23654},[66],[15,23656,1224],{"className":23657,"style":1515},[66,67],[15,23659,23661],{"className":23660},[198],[15,23662,23664,23684],{"className":23663},[202,1090],[15,23665,23667,23681],{"className":23666},[206],[15,23668,23670],{"className":23669,"style":1528},[210],[15,23671,23672,23675],{"style":1531},[15,23673],{"className":23674,"style":218},[217],[15,23676,23678],{"className":23677},[222,223,224,225],[15,23679,1205],{"className":23680},[66,67,225],[15,23682,1114],{"className":23683},[1113],[15,23685,23687],{"className":23686},[206],[15,23688,23690],{"className":23689,"style":1121},[210],[15,23691],{},[15,23693,120],{"className":23694},[162],[15,23696,116],{"className":23697,"style":158},[66,67],[15,23699,120],{"className":23700},[162],[15,23702,23704,23707],{"className":23703},[66],[15,23705,123],{"className":23706},[66,67],[15,23708,23710],{"className":23709},[198],[15,23711,23713,23733],{"className":23712},[202,1090],[15,23714,23716,23730],{"className":23715},[206],[15,23717,23719],{"className":23718,"style":1528},[210],[15,23720,23721,23724],{"style":1584},[15,23722],{"className":23723,"style":218},[217],[15,23725,23727],{"className":23726},[222,223,224,225],[15,23728,1205],{"className":23729},[66,67,225],[15,23731,1114],{"className":23732},[1113],[15,23734,23736],{"className":23735},[206],[15,23737,23739],{"className":23738,"style":1121},[210],[15,23740],{},[15,23742,126],{"className":23743},[169],[15,23745],{"className":23746,"style":72},[71],[15,23748,1245],{"className":23749},[76],[15,23751],{"className":23752,"style":72},[71],[15,23754,4587],{"className":23755},[66,67],[15,23757,126],{"className":23758},[169],[15,23760,23762],{"className":23761},[169],[15,23763,126],{"className":23764},[1422,1508],[15,23766,23768],{"className":23767,"style":1418},[169,1417],[15,23769,4330],{"className":23770},[1422,1423],[15,23772],{"className":23773,"style":697},[71],[15,23775,3573],{"className":23776},[66],[11,23778,23779,23780,23808,23809,23882],{},"固定 ",[15,23781,23783,23796],{"className":23782,"translate":19},[18],[15,23784,23786],{"className":23785},[23],[25,23787,23788],{"xmlns":27},[29,23789,23790,23794],{},[32,23791,23792],{},[35,23793,4587],{},[45,23795,4587],{"encoding":47},[15,23797,23799],{"className":23798,"ariaHidden":53},[52],[15,23800,23802,23805],{"className":23801},[57],[15,23803],{"className":23804,"style":840},[61],[15,23806,4587],{"className":23807},[66,67],"，正则项 ",[15,23810,23812,23840],{"className":23811,"translate":19},[18],[15,23813,23815],{"className":23814},[23],[25,23816,23817],{"xmlns":27},[29,23818,23819,23837],{},[32,23820,23821,23823,23825,23827,23829,23831,23833,23835],{},[35,23822,7742],{"mathvariant":138},[39,23824,120],{"stretchy":119},[35,23826,9212],{},[39,23828,126],{"stretchy":119},[39,23830,129],{},[35,23832,9212],{},[35,23834,6625],{"mathvariant":138},[1209,23836,1263],{},[45,23838,23839],{"encoding":47},"\\Omega(t)=t\u002F2",[15,23841,23843,23870],{"className":23842,"ariaHidden":53},[52],[15,23844,23846,23849,23852,23855,23858,23861,23864,23867],{"className":23845},[57],[15,23847],{"className":23848,"style":154},[61],[15,23850,7742],{"className":23851},[66],[15,23853,120],{"className":23854},[162],[15,23856,9212],{"className":23857},[66,67],[15,23859,126],{"className":23860},[169],[15,23862],{"className":23863,"style":173},[71],[15,23865,129],{"className":23866},[177],[15,23868],{"className":23869,"style":173},[71],[15,23871,23873,23876,23879],{"className":23872},[57],[15,23874],{"className":23875,"style":154},[61],[15,23877,9212],{"className":23878},[66,67],[15,23880,7110],{"className":23881},[66]," 严格递增，损失是 hinge loss。表示定理给出",[15,23884,23886],{"className":23885,"translate":19},[294],[15,23887,23889,23955],{"className":23888,"translate":19},[18],[15,23890,23892],{"className":23891},[23],[25,23893,23894],{"xmlns":27,"display":303},[29,23895,23896,23952],{},[32,23897,23898,23904,23906,23908,23910,23912,23926,23932,23934,23936,23938,23940,23942,23948,23950],{},[131,23899,23900,23902],{},[35,23901,116],{},[39,23903,1162],{},[39,23905,120],{"stretchy":119},[35,23907,123],{},[39,23909,126],{"stretchy":119},[39,23911,129],{},[1196,23913,23914,23916,23924],{},[39,23915,1200],{},[32,23917,23918,23920,23922],{},[35,23919,1205],{},[39,23921,129],{},[1209,23923,1211],{},[35,23925,37],{},[1057,23927,23928,23930],{},[35,23929,17087],{},[35,23931,1205],{},[1274,23933,2462],{},[35,23935,621],{},[39,23937,120],{"stretchy":119},[35,23939,123],{},[39,23941,335],{"separator":53},[1057,23943,23944,23946],{},[35,23945,123],{},[35,23947,1205],{},[39,23949,126],{"stretchy":119},[35,23951,3573],{"mathvariant":138},[45,23953,23954],{"encoding":47},"f^*(x)=\\sum_{i=1}^n\\beta_i\\,k(x,x_i).",[15,23956,23958,24011],{"className":23957,"ariaHidden":53},[52],[15,23959,23961,23964,23993,23996,23999,24002,24005,24008],{"className":23960},[57],[15,23962],{"className":23963,"style":154},[61],[15,23965,23967,23970],{"className":23966},[66],[15,23968,116],{"className":23969,"style":158},[66,67],[15,23971,23973],{"className":23972},[198],[15,23974,23976],{"className":23975},[202],[15,23977,23979],{"className":23978},[206],[15,23980,23982],{"className":23981,"style":1308},[210],[15,23983,23984,23987],{"style":397},[15,23985],{"className":23986,"style":218},[217],[15,23988,23990],{"className":23989},[222,223,224,225],[15,23991,1162],{"className":23992},[76,225],[15,23994,120],{"className":23995},[162],[15,23997,123],{"className":23998},[66,67],[15,24000,126],{"className":24001},[169],[15,24003],{"className":24004,"style":173},[71],[15,24006,129],{"className":24007},[177],[15,24009],{"className":24010,"style":173},[71],[15,24012,24014,24017,24081,24084,24124,24127,24130,24133,24136,24139,24142,24182,24185],{"className":24013},[57],[15,24015],{"className":24016,"style":8486},[61],[15,24018,24020],{"className":24019},[1339,1351],[15,24021,24023,24073],{"className":24022},[202,1090],[15,24024,24026,24070],{"className":24025},[206],[15,24027,24029,24049,24059],{"className":24028,"style":1436},[210],[15,24030,24031,24034],{"style":1439},[15,24032],{"className":24033,"style":1443},[217],[15,24035,24037],{"className":24036},[222,223,224,225],[15,24038,24040,24043,24046],{"className":24039},[66,225],[15,24041,1205],{"className":24042},[66,67,225],[15,24044,129],{"className":24045},[177,225],[15,24047,1211],{"className":24048},[66,225],[15,24050,24051,24054],{"style":1461},[15,24052],{"className":24053,"style":1443},[217],[15,24055,24056],{},[15,24057,1200],{"className":24058},[1339,1470,1471],[15,24060,24061,24064],{"style":1474},[15,24062],{"className":24063,"style":1443},[217],[15,24065,24067],{"className":24066},[222,223,224,225],[15,24068,37],{"className":24069},[66,67,225],[15,24071,1114],{"className":24072},[1113],[15,24074,24076],{"className":24075},[206],[15,24077,24079],{"className":24078,"style":1493},[210],[15,24080],{},[15,24082],{"className":24083,"style":697},[71],[15,24085,24087,24090],{"className":24086},[66],[15,24088,17087],{"className":24089,"style":17240},[66,67],[15,24091,24093],{"className":24092},[198],[15,24094,24096,24116],{"className":24095},[202,1090],[15,24097,24099,24113],{"className":24098},[206],[15,24100,24102],{"className":24101,"style":1528},[210],[15,24103,24104,24107],{"style":17255},[15,24105],{"className":24106,"style":218},[217],[15,24108,24110],{"className":24109},[222,223,224,225],[15,24111,1205],{"className":24112},[66,67,225],[15,24114,1114],{"className":24115},[1113],[15,24117,24119],{"className":24118},[206],[15,24120,24122],{"className":24121,"style":1121},[210],[15,24123],{},[15,24125],{"className":24126,"style":697},[71],[15,24128,621],{"className":24129,"style":684},[66,67],[15,24131,120],{"className":24132},[162],[15,24134,123],{"className":24135},[66,67],[15,24137,335],{"className":24138},[422],[15,24140],{"className":24141,"style":697},[71],[15,24143,24145,24148],{"className":24144},[66],[15,24146,123],{"className":24147},[66,67],[15,24149,24151],{"className":24150},[198],[15,24152,24154,24174],{"className":24153},[202,1090],[15,24155,24157,24171],{"className":24156},[206],[15,24158,24160],{"className":24159,"style":1528},[210],[15,24161,24162,24165],{"style":1584},[15,24163],{"className":24164,"style":218},[217],[15,24166,24168],{"className":24167},[222,223,224,225],[15,24169,1205],{"className":24170},[66,67,225],[15,24172,1114],{"className":24173},[1113],[15,24175,24177],{"className":24176},[206],[15,24178,24180],{"className":24179,"style":1121},[210],[15,24181],{},[15,24183,126],{"className":24184},[169],[15,24186,3573],{"className":24187},[66],[11,24189,24190],{},"代入后得到对偶问题",[15,24192,24194],{"className":24193,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y_i",[15,25303,25305,25360],{"className":25304,"ariaHidden":53},[52],[15,25306,25308,25311,25351,25354,25357],{"className":25307},[57],[15,25309],{"className":25310,"style":3408},[61],[15,25312,25314,25317],{"className":25313},[66],[15,25315,17087],{"className":25316,"style":17240},[66,67],[15,25318,25320],{"className":25319},[198],[15,25321,25323,25343],{"className":25322},[202,1090],[15,25324,25326,25340],{"className":25325},[206],[15,25327,25329],{"className":25328,"style":1528},[210],[15,25330,25331,25334],{"style":17255},[15,25332],{"className":25333,"style":218},[217],[15,25335,25337],{"className":25336},[222,223,224,225],[15,25338,1205],{"className":25339},[66,67,225],[15,25341,1114],{"className":25342},[1113],[15,25344,25346],{"className":25345},[206],[15,25347,25349],{"className":25348,"style":1121},[210],[15,25350],{},[15,25352],{"className":25353,"style":173},[71],[15,25355,129],{"className":25356},[177],[15,25358],{"className":25359,"style":173},[71],[15,25361,25363,25366,25406],{"className":25362},[57],[15,25364],{"className":25365,"style":471},[61],[15,25367,25369,25372],{"className":25368},[66],[15,25370,8390],{"className":25371,"style":8560},[66,67],[15,25373,25375],{"className":25374},[198],[15,25376,25378,25398],{"className":25377},[202,1090],[15,25379,25381,25395],{"className":25380},[206],[15,25382,25384],{"className":25383,"style":1528},[210],[15,25385,25386,25389],{"style":8575},[15,25387],{"className":25388,"style":218},[217],[15,25390,25392],{"className":25391},[222,223,224,225],[15,25393,1205],{"className":25394},[66,67,225],[15,25396,1114],{"className":25397},[1113],[15,25399,25401],{"className":25400},[206],[15,25402,25404],{"className":25403,"style":1121},[210],[15,25405],{},[15,25407,25409,25412],{"className":25408},[66],[15,25410,1224],{"className":25411,"style":1515},[66,67],[15,25413,25415],{"className":25414},[198],[15,25416,25418,25438],{"className":25417},[202,1090],[15,25419,25421,25435],{"className":25420},[206],[15,25422,25424],{"className":25423,"style":1528},[210],[15,25425,25426,25429],{"style":1531},[15,25427],{"className":25428,"style":218},[217],[15,25430,25432],{"className":25431},[222,223,224,225],[15,25433,1205],{"className":25434},[66,67,225],[15,25436,1114],{"className":25437},[1113],[15,25439,25441],{"className":25440},[206],[15,25442,25444],{"className":25443,"style":1121},[210],[15,25445],{},[11,25447,25448],{},"KKT 条件给出：",[16136,25450,25451,25730,26010],{},[3194,25452,12969,25453,13132,25638,3371],{},[15,25454,25456,25498],{"className":25455,"translate":19},[18],[15,25457,25459],{"className":25458},[23],[25,25460,25461],{"xmlns":27},[29,25462,25463,25495],{},[32,25464,25465,25471,25473,25475,25477,25483,25485,25487,25489,25491,25493],{},[1057,25466,25467,25469],{},[35,25468,1224],{},[35,25470,1205],{},[39,25472,120],{"stretchy":119},[35,25474,116],{},[39,25476,120],{"stretchy":119},[1057,25478,25479,25481],{},[35,25480,123],{},[35,25482,1205],{},[39,25484,126],{"stretchy":119},[39,25486,1245],{},[35,25488,4587],{},[39,25490,126],{"stretchy":119},[39,25492,9283],{},[1209,25494,1211],{},[45,25496,25497],{"encoding":47},"y_i(f(x_i)+b)>1",[15,25499,25501,25608,25629],{"className":25500,"ariaHidden":53},[52],[15,25502,25504,25507,25547,25550,25553,25556,25596,25599,25602,25605],{"className":25503},[57],[15,25505],{"className":25506,"style":154},[61],[15,25508,25510,25513],{"className":25509},[66],[15,25511,1224],{"className":25512,"style":1515},[66,67],[15,25514,25516],{"className":25515},[198],[15,25517,25519,25539],{"className":25518},[202,1090],[15,25520,25522,25536],{"className":25521},[206],[15,25523,25525],{"className":25524,"style":1528},[210],[15,25526,25527,25530],{"style":1531},[15,25528],{"className":25529,"style":218},[217],[15,25531,25533],{"className":25532},[222,223,224,225],[15,25534,1205],{"className":25535},[66,67,225],[15,25537,1114],{"className":25538},[1113],[15,25540,25542],{"className":25541},[206],[15,25543,25545],{"className":25544,"style":1121},[210],[15,25546],{},[15,25548,120],{"className":25549},[162],[15,25551,116],{"className":25552,"style":158},[66,67],[15,25554,120],{"className":25555},[162],[15,25557,25559,25562],{"className":25558},[66],[15,25560,123],{"className":25561},[66,67],[15,25563,25565],{"className":25564},[198],[15,25566,25568,25588],{"className":25567},[202,1090],[15,25569,25571,25585],{"className":25570},[206],[15,25572,25574],{"className":25573,"style":1528},[210],[15,25575,25576,25579],{"style":1584},[15,25577],{"className":25578,"style":218},[217],[15,25580,25582],{"className":25581},[222,223,224,225],[15,25583,1205],{"className":25584},[66,67,225],[15,25586,1114],{"className":25587},[1113],[15,25589,25591],{"className":25590},[206],[15,25592,25594],{"className":25593,"style":1121},[210],[15,25595],{},[15,25597,126],{"className":25598},[169],[15,25600],{"className":25601,"style":72},[71],[15,25603,1245],{"className":25604},[76],[15,25606],{"className":25607,"style":72},[71],[15,25609,25611,25614,25617,25620,25623,25626],{"className":25610},[57],[15,25612],{"className":25613,"style":154},[61],[15,25615,4587],{"className":25616},[66,67],[15,25618,126],{"className":25619},[169],[15,25621],{"className":25622,"style":173},[71],[15,25624,9283],{"className":25625},[177],[15,25627],{"className":25628,"style":173},[71],[15,25630,25632,25635],{"className":25631},[57],[15,25633],{"className":25634,"style":1836},[61],[15,25636,1211],{"className":25637},[66],[15,25639,25641,25663],{"className":25640,"translate":19},[18],[15,25642,25644],{"className":25643},[23],[25,25645,25646],{"xmlns":27},[29,25647,25648,25660],{},[32,25649,25650,25656,25658],{},[1057,25651,25652,25654],{},[35,25653,8390],{},[35,25655,1205],{},[39,25657,129],{},[1209,25659,1804],{},[45,25661,25662],{"encoding":47},"\\alpha_i=0",[15,25664,25666,25721],{"className":25665,"ariaHidden":53},[52],[15,25667,25669,25672,25712,25715,25718],{"className":25668},[57],[15,25670],{"className":25671,"style":10197},[61],[15,25673,25675,25678],{"className":25674},[66],[15,25676,8390],{"className":25677,"style":8560},[66,67],[15,25679,25681],{"className":25680},[198],[15,25682,25684,25704],{"className":25683},[202,1090],[15,25685,25687,25701],{"className":25686},[206],[15,25688,25690],{"className":25689,"style":1528},[210],[15,25691,25692,25695],{"style":8575},[15,25693],{"className":25694,"style":218},[217],[15,25696,25698],{"className":25697},[222,223,224,225],[15,25699,1205],{"className":25700},[66,67,225],[15,25702,1114],{"className":25703},[1113],[15,25705,25707],{"className":25706},[206],[15,25708,25710],{"className":25709,"style":1121},[210],[15,25711],{},[15,25713],{"className":25714,"style":173},[71],[15,25716,129],{"className":25717},[177],[15,25719],{"className":25720,"style":173},[71],[15,25722,25724,25727],{"className":25723},[57],[15,25725],{"className":25726,"style":1836},[61],[15,25728,1804],{"className":25729},[66],[3194,25731,12969,25732,13132,25918,3371],{},[15,25733,25735,25778],{"className":25734,"translate":19},[18],[15,25736,25738],{"className":25737},[23],[25,25739,25740],{"xmlns":27},[29,25741,25742,25775],{},[32,25743,25744,25750,25752,25754,25756,25762,25764,25766,25768,25770,25773],{},[1057,25745,25746,25748],{},[35,25747,1224],{},[35,25749,1205],{},[39,25751,120],{"stretchy":119},[35,25753,116],{},[39,25755,120],{"stretchy":119},[1057,25757,25758,25760],{},[35,25759,123],{},[35,25761,1205],{},[39,25763,126]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C",[15,26227,26229,26248,26304],{"className":26228,"ariaHidden":53},[52],[15,26230,26232,26236,26239,26242,26245],{"className":26231},[57],[15,26233],{"className":26234,"style":26235},[61],"height:0.7804em;vertical-align:-0.136em;",[15,26237,1804],{"className":26238},[66],[15,26240],{"className":26241,"style":173},[71],[15,26243,24220],{"className":26244},[177],[15,26246],{"className":26247,"style":173},[71],[15,26249,26251,26255,26295,26298,26301],{"className":26250},[57],[15,26252],{"className":26253,"style":26254},[61],"height:0.786em;vertical-align:-0.15em;",[15,26256,26258,26261],{"className":26257},[66],[15,26259,8390],{"className":26260,"style":8560},[66,67],[15,26262,26264],{"className":26263},[198],[15,26265,26267,26287],{"className":26266},[202,1090],[15,26268,26270,26284],{"className":26269},[206],[15,26271,26273],{"className":26272,"style":1528},[210],[15,26274,26275,26278],{"style":8575},[15,26276],{"className":26277,"style":218},[217],[15,26279,26281],{"className":26280},[222,223,224,225],[15,26282,1205],{"className":26283},[66,67,225],[15,26285,1114],{"className":26286},[1113],[15,26288,26290],{"className":26289},[206],[15,26291,26293],{"className":26292,"style":1121},[210],[15,26294],{},[15,26296],{"className":26297,"style":173},[71],[15,26299,24220],{"className":26300},[177],[15,26302],{"className":26303,"style":173},[71],[15,26305,26307,26310],{"className":26306},[57],[15,26308],{"className":26309,"style":490},[61],[15,26311,23256],{"className":26312,"style":2901},[66,67],[11,26314,26315,26408,26409,26412],{},[15,26316,26318,26340],{"className":26317,"translate":19},[18],[15,26319,26321],{"className":26320},[23],[25,26322,26323],{"xmlns":27},[29,26324,26325,26337],{},[32,26326,26327,26333,26335],{},[1057,26328,26329,26331],{},[35,26330,8390],{},[35,26332,1205],{},[39,26334,9283],{},[1209,26336,1804],{},[45,26338,26339],{"encoding":47},"\\alpha_i>0",[15,26341,26343,26399],{"className":26342,"ariaHidden":53},[52],[15,26344,26346,26350,26390,26393,26396],{"className":26345},[57],[15,26347],{"className":26348,"style":26349},[61],"height:0.6891em;vertical-align:-0.15em;",[15,26351,26353,26356],{"className":26352},[66],[15,26354,8390],{"className":26355,"style":8560},[66,67],[15,26357,26359],{"className":26358},[198],[15,26360,26362,26382],{"className":26361},[202,1090],[15,26363,26365,26379],{"className":26364},[206],[15,26366,26368],{"className":26367,"style":1528},[210],[15,26369,26370,26373],{"style":8575},[15,26371],{"className":26372,"style":218},[217],[15,26374,26376],{"className":26375},[222,223,224,225],[15,26377,1205],{"className":26378},[66,67,225],[15,26380,1114],{"className":26381},[1113],[15,26383,26385],{"className":26384},[206],[15,26386,26388],{"className":26387,"style":1121},[210],[15,26389],{},[15,26391],{"className":26392,"style":173},[71],[15,26394,9283],{"className":26395},[177],[15,26397],{"className":26398,"style":173},[71],[15,26400,26402,26405],{"className":26401},[57],[15,26403],{"className":26404,"style":1836},[61],[15,26406,1804],{"className":26407},[66]," 的点就是",[2040,26410,26411],{},"支持向量","。决策函数为",[15,26414,26416],{"className":26415,"translate":19},[294],[15,26417,26419,26492],{"className":26418,"translate":19},[18],[15,26420,26422],{"className":26421},[23],[25,26423,26424],{"xmlns":27,"display":303},[29,26425,26426,26489],{},[32,26427,26428,26431,26433,26487],{},[35,26429,26430],{"mathvariant":138},"sign",[39,26432,1170],{},[32,26434,26435,26437,26451,26457,26463,26465,26467,26469,26471,26473,26479,26481,26483,26485],{},[39,26436,120],{"fence":53},[1196,26438,26439,26441,26449],{},[39,26440,1200],{},[32,26442,26443,26445,26447],{},[35,26444,1205],{},[39,26446,129],{},[1209,26448,1211],{},[35,26450,37],{},[1057,26452,26453,26455],{},[35,26454,8390],{},[35,26456,1205],{},[1057,26458,26459,26461],{},[35,26460,1224],{},[35,26462,1205],{},[1274,26464,2462],{},[35,26466,621],{},[39,26468,120],{"stretchy":119},[35,26470,123],{},[39,26472,335],{"separator":53},[1057,26474,26475,26477],{},[35,26476,123],{},[35,26478,1205],{},[39,26480,126],{"stretchy":119},[39,26482,1245],{},[35,26484,4587],{},[39,26486,126],{"fence":53},[35,26488,3573],{"mathvariant":138},[45,26490,26491],{"encoding":47},"\\operatorname{sign}\\left(\\sum_{i=1}^n\\alpha_i y_i\\,k(x,x_i)+b\\right).",[15,26493,26495],{"className":26494,"ariaHidden":53},[52],[15,26496,26498,26501,26507,26510,26745,26748],{"className":26497},[57],[15,26499],{"className":26500,"style":1335},[61],[15,26502,26504],{"className":26503},[1339],[15,26505,26430],{"className":26506},[66,4429],[15,26508],{"className":26509,"style":697},[71],[15,26511,26513,26519,26583,26586,26626,26666,26669,26672,26675,26678,26681,26684,26724,26727,26730,26733,26736,26739],{"className":26512},[1413],[15,26514,26516],{"className":26515,"style":1418},[162,1417],[15,26517,120],{"className":26518},[1422,1423],[15,26520,26522],{"className":26521},[1339,1351],[15,26523,26525,26575],{"className":26524},[202,1090],[15,26526,26528,26572],{"className":26527},[206],[15,26529,26531,26551,26561],{"className":26530,"style":1436},[210],[15,26532,26533,26536],{"style":1439},[15,26534],{"className":26535,"style":1443},[217],[15,26537,26539],{"className":26538},[222,223,224,225],[15,26540,26542,26545,26548],{"className":26541},[66,225],[15,26543,1205],{"className":26544},[66,67,225],[15,26546,129],{"className":26547},[177,225],[15,26549,1211],{"className":26550},[66,225],[15,26552,26553,26556],{"style":1461},[15,26554],{"className":26555,"style":1443},[217],[15,26557,26558],{},[15,26559,1200],{"className":26560},[1339,1470,1471],[15,26562,26563,26566],{"style":1474},[15,26564],{"className":26565,"style":1443},[217],[15,26567,26569],{"className":26568},[222,223,224,225],[15,26570,37],{"className":26571},[66,67,225],[15,26573,1114],{"className":26574},[1113],[15,26576,26578],{"className":26577},[206],[15,26579,26581],{"className":26580,"style":1493},[210],[15,26582],{},[15,26584],{"className":26585,"style":697},[71],[15,26587,26589,26592],{"className":26588},[66],[15,26590,8390],{"className":26591,"style":8560},[66,67],[15,26593,26595],{"className":26594},[198],[15,26596,26598,26618],{"className":26597},[202,1090],[15,26599,26601,26615],{"className":26600},[206],[15,26602,26604],{"className":26603,"style":1528},[210],[15,26605,26606,26609],{"style":8575},[15,26607],{"className":26608,"style":218},[217],[15,26610,26612],{"className":26611},[222,223,224,225],[15,26613,1205],{"className":26614},[66,67,225],[15,26616,1114],{"className":26617},[1113],[15,26619,26621],{"className":26620},[206],[15,26622,26624],{"className":26623,"style":1121},[210],[15,26625],{},[15,26627,26629,26632],{"className":26628},[66],[15,26630,1224],{"className":26631,"style":1515},[66,67],[15,26633,26635],{"className":26634},[198],[15,26636,26638,26658],{"className":26637},[202,1090],[15,26639,26641,26655],{"className":26640},[206],[15,26642,26644],{"className":26643,"style":1528},[210],[15,26645,26646,26649],{"style":1531},[15,26647],{"className":26648,"style":218},[217],[15,26650,26652],{"className":26651},[222,223,224,225],[15,26653,1205],{"className":26654},[66,67,225],[15,26656,1114],{"className":26657},[1113],[15,26659,26661],{"className":26660},[206],[15,26662,26664],{"className":26663,"style":1121},[210],[15,26665],{},[15,26667],{"className":26668,"style":697},[71],[15,26670,621],{"className":26671,"style":684},[66,67],[15,26673,120],{"className":26674},[162],[15,26676,123],{"className":26677},[66,67],[15,26679,335],{"className":26680},[422],[15,26682],{"className":26683,"style":697},[71],[15,26685,26687,26690],{"className":26686},[66],[15,26688,123],{"className":26689},[66,67],[15,26691,26693],{"className":26692},[198],[15,26694,26696,26716],{"className":26695},[202,1090],[15,26697,26699,26713],{"className":26698},[206],[15,26700,26702],{"className":26701,"style":1528},[210],[15,26703,26704,26707],{"style":1584},[15,26705],{"className":26706,"style":218},[217],[15,26708,26710],{"className":26709},[222,223,224,225],[15,26711,1205],{"className":26712},[66,67,225],[15,26714,1114],{"className":26715},[1113],[15,26717,26719],{"className":26718},[206],[15,26720,26722],{"className":26721,"style":1121},[210],[15,26723],{},[15,26725,126],{"className":26726},[169],[15,26728],{"className":26729,"style":72},[71],[15,26731,1245],{"className":26732},[76],[15,26734],{"className":26735,"style":72},[71],[15,26737,4587],{"className":26738},[66,67],[15,26740,26742],{"className":26741,"style":1418},[169,1417],[15,26743,126],{"className":26744},[1422,1423],[15,26746],{"className":26747,"style":697},[71],[15,26749,3573],{"className":26750},[66],[11,26752,26753,26754,26757,26758,26816,26817,26845,26846,26915,26916,26985,26986,27014],{},"这里要特别澄清一个常见误解：",[2040,26755,26756],{},"SVM 的稀疏性不是表示定理给的。"," 表示定理只说 ",[15,26759,26761,26778],{"className":26760,"translate":19},[18],[15,26762,26764],{"className":26763},[23],[25,26765,26766],{"xmlns":27},[29,26767,26768,26776],{},[32,26769,26770],{},[131,26771,26772,26774],{},[35,26773,116],{},[39,26775,1162],{},[45,26777,16221],{"encoding":47},[15,26779,26781],{"className":26780,"ariaHidden":53},[52],[15,26782,26784,26787],{"className":26783},[57],[15,26785],{"className":26786,"style":3408},[61],[15,26788,26790,26793],{"className":26789},[66],[15,26791,116],{"className":26792,"style":158},[66,67],[15,26794,26796],{"className":26795},[198],[15,26797,26799],{"className":26798},[202],[15,26800,26802],{"className":26801},[206],[15,26803,26805],{"className":26804,"style":16249},[210],[15,26806,26807,26810],{"style":213},[15,26808],{"className":26809,"style":218},[217],[15,26811,26813],{"className":26812},[222,223,224,225],[15,26814,1162],{"className":26815},[76,225]," 是 ",[15,26818,26820,26833],{"className":26819,"translate":19},[18],[15,26821,26823],{"className":26822},[23],[25,26824,26825],{"xmlns":27},[29,26826,26827,26831],{},[32,26828,26829],{},[35,26830,37],{},[45,26832,37],{"encoding":47},[15,26834,26836],{"className":26835,"ariaHidden":53},[52],[15,26837,26839,26842],{"className":26838},[57],[15,26840],{"className":26841,"style":86},[61],[15,26843,37],{"className":26844},[66,67]," 个核函数的线性组合，",[15,26847,26849,26866],{"className":26848,"translate":19},[18],[15,26850,26852],{"className":26851},[23],[25,26853,26854],{"xmlns":27},[29,26855,26856,26864],{},[32,26857,26858],{},[1057,26859,26860,26862],{},[35,26861,8390],{},[35,26863,1205],{},[45,26865,25216],{"encoding":47},[15,26867,26869],{"className":26868,"ariaHidden":53},[52],[15,26870,26872,26875],{"className":26871},[57],[15,26873],{"className":26874,"style":10197},[61],[15,26876,26878,26881],{"className":26877},[66],[15,26879,8390],{"className":26880,"style":8560},[66,67],[15,26882,26884],{"className":26883},[198],[15,26885,26887,26907],{"className":26886},[202,1090],[15,26888,26890,26904],{"className":26889},[206],[15,26891,26893],{"className":26892,"style":1528},[210],[15,26894,26895,26898],{"style":8575},[15,26896],{"className":26897,"style":218},[217],[15,26899,26901],{"className":26900},[222,223,224,225],[15,26902,1205],{"className":26903},[66,67,225],[15,26905,1114],{"className":26906},[1113],[15,26908,26910],{"className":26909},[206],[15,26911,26913],{"className":26912,"style":1121},[210],[15,26914],{}," 完全可以全部非零。真正让大部分 ",[15,26917,26919,26936],{"className":26918,"translate":19},[18],[15,26920,26922],{"className":26921},[23],[25,26923,26924],{"xmlns":27},[29,26925,26926,26934],{},[32,26927,26928],{},[1057,26929,26930,26932],{},[35,26931,8390],{},[35,26933,1205],{},[45,26935,25216],{"encoding":47},[15,26937,26939],{"className":26938,"ariaHidden":53},[52],[15,26940,26942,26945],{"className":26941},[57],[15,26943],{"className":26944,"style":10197},[61],[15,26946,26948,26951],{"className":26947},[66],[15,26949,8390],{"className":26950,"style":8560},[66,67],[15,26952,26954],{"className":26953},[198],[15,26955,26957,26977],{"className":26956},[202,1090],[15,26958,26960,26974],{"className":26959},[206],[15,26961,26963],{"className":26962,"style":1528},[210],[15,26964,26965,26968],{"style":8575},[15,26966],{"className":26967,"style":218},[217],[15,26969,26971],{"className":26970},[222,223,224,225],[15,26972,1205],{"className":26973},[66,67,225],[15,26975,1114],{"className":26976},[1113],[15,26978,26980],{"className":26979},[206],[15,26981,26983],{"className":26982,"style":1121},[210],[15,26984],{}," 变成零的，是 hinge loss 的 KKT 条件：间隔大于 ",[15,26987,26989,27002],{"className":26988,"translate":19},[18],[15,26990,26992],{"className":26991},[23],[25,26993,26994],{"xmlns":27},[29,26995,26996,27000],{},[32,26997,26998],{},[1209,26999,1211],{},[45,27001,1211],{"encoding":47},[15,27003,27005],{"className":27004,"ariaHidden":53},[52],[15,27006,27008,27011],{"className":27007},[57],[15,27009],{"className":27010,"style":1836},[61],[15,27012,1211],{"className":27013},[66]," 的点对损失没有贡献，对应的乘子为零。",[1904,27016,27018],{"id":27017},"_53-核逻辑回归","5.3 核逻辑回归",[11,27020,27021],{},"取逻辑损失：",[15,27023,27025],{"className":27024,"translate":19},[294],[15,27026,27028,27139],{"className":27027,"translate":19},[18],[15,27029,27031],{"className":27030},[23],[25,27032,27033],{"xmlns":27,"display":303},[29,27034,27035,27136],{},[32,27036,27037,27053,27134],{},[1172,27038,27039,27045],{},[32,27040,27041,27043],{},[35,27042,1178],{},[39,27044,1170],{},[32,27046,27047,27049,27051],{},[35,27048,116],{},[39,27050,1187],{},[35,27052,1061],{"mathvariant":445},[32,27054,27055,27057,27071,27074,27076,27078,27080,27082,27084,27086,27088,27090,27096,27098,27100,27106,27108,27110,27112,27114,27120,27122,27124,27132],{},[39,27056,4307],{"fence":53},[1196,27058,27059,27061,27069],{},[39,27060,1200],{},[32,27062,27063,27065,27067],{},[35,27064,1205],{},[39,27066,129],{},[1209,27068,1211],{},[35,27070,37],{},[35,27072,27073],{},"log",[39,27075,1170],{},[39,27077,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,27079,1211],{},[39,27081,1245],{},[35,27083,6592],{},[39,27085,1170],{},[39,27087,120],{"stretchy":119},[39,27089,6602],{},[1057,27091,27092,27094],{},[35,27093,1224],{},[35,27095,1205],{},[35,27097,116],{},[39,27099,120],{"stretchy":119},[1057,27101,27102,27104],{},[35,27103,123],{},[35,27105,1205],{},[39,27107,126],{"stretchy":119},[39,27109,126],{"stretchy":119},[39,27111,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,27113,1245],{},[6467,27115,27116,27118],{},[35,27117,1248],{},[1209,27119,1263],{},[35,27121,1251],{"mathvariant":138},[35,27123,116],{},[1255,27125,27126,27128,27130],{},[35,27127,1251],{"mathvariant":138},[35,27129,1061],{"mathvariant":445},[1209,27131,1263],{},[39,27133,4330],{"fence":53},[35,27135,3573],{"mathvariant":138},[45,27137,27138],{"encoding":47},"\\min_{f\\in\\mathcal H}\\left\\{\n\\sum_{i=1}^n\\log\\bigl(1+\\exp(-y_i f(x_i))\\bigr)\n+\\frac{\\lambda}{2}\\|f\\|_{\\mathcal H}^2\n\\right\\}.",[15,27140,27142],{"className":27141,"ariaHidden":53},[52],[15,27143,27145,27148,27201,27204,27546,27549],{"className":27144},[57],[15,27146],{"className":27147,"style":1335},[61],[15,27149,27151],{"className":27150},[1339,1351],[15,27152,27154,27193],{"className":27153},[202,1090],[15,27155,27157,27190],{"className":27156},[206],[15,27158,27160,27180],{"className":27159,"style":1361},[210],[15,27161,27162,27165],{"style":1364},[15,27163],{"className":27164,"style":1368},[217],[15,27166,27168],{"className":27167},[222,223,224,225],[15,27169,27171,27174,27177],{"className":27170},[66,225],[15,27172,116],{"className":27173,"style":158},[66,67,225],[15,27175,1187],{"className":27176},[177,225],[15,27178,1061],{"className":27179,"style":1083},[66,494,225],[15,27181,27182,27185],{"style":1386},[15,27183],{"className":27184,"style":1368},[217],[15,27186,27187],{},[15,27188,1178],{"className":27189},[1339],[15,27191,1114],{"className":27192},[1113],[15,27194,27196],{"className":27195},[206],[15,27197,27199],{"className":27198,"style":1404},[210],[15,27200],{},[15,27202],{"className":27203,"style":697},[71],[15,27205,27207,27213,27277,27280,27286,27292,27295,27298,27301,27304,27307,27310,27313,27353,27356,27359,27399,27402,27408,27411,27414,27417,27480,27483,27486,27540],{"className":27206},[1413],[15,27208,27210],{"className":27209,"style":1418},[162,1417],[15,27211,4307],{"className":27212},[1422,1423],[15,27214,27216],{"className":27215},[1339,1351],[15,27217,27219,27269],{"className":27218},[202,1090],[15,27220,27222,27266],{"className":27221},[206],[15,27223,27225,27245,27255],{"className":27224,"style":1436},[210],[15,27226,27227,27230],{"style":1439},[15,27228],{"className":27229,"style":1443},[217],[15,27231,27233],{"className":27232},[222,223,224,225],[15,27234,27236,27239,27242],{"className":27235},[66,225],[15,27237,1205],{"className":27238},[66,67,225],[15,27240,129],{"className":27241},[177,225],[15,27243,1211],{"className":27244},[66,225],[15,27246,27247,27250],{"style":1461},[15,27248],{"className":27249,"style":1443},[217],[15,27251,27252],{},[15,27253,1200],{"className":27254},[1339,1470,1471],[15,27256,27257,27260],{"style":1474},[15,27258],{"className":27259,"style":1443},[217],[15,27261,27263],{"className":27262},[222,223,224,225],[15,27264,37],{"className":27265},[66,67,225],[15,27267,1114],{"className":27268},[1113],[15,27270,27272],{"className":27271},[206],[15,27273,27275],{"className":27274,"style":1493},[210],[15,27276],{},[15,27278],{"className":27279,"style":697},[71],[15,27281,27283,27284],{"className":27282},[1339],"lo",[15,27285,1344],{"style":1343},[15,27287,27289],{"className":27288},[162],[15,27290,120],{"className":27291},[1422,1508],[15,27293,1211],{"className":27294},[66],[15,27296],{"className":27297,"style":72},[71],[15,27299,1245],{"className":27300},[76],[15,27302],{"className":27303,"style":72},[71],[15,27305,6592],{"className":27306},[1339],[15,27308,120],{"className":27309},[162],[15,27311,6602],{"className":27312},[66],[15,27314,27316,27319],{"className":27315},[66],[15,27317,1224],{"className":27318,"style":1515},[66,67],[15,27320,27322],{"className":27321},[198],[15,27323,27325,27345],{"className":27324},[202,1090],[15,27326,27328,27342],{"className":27327},[206],[15,27329,27331],{"className":27330,"style":1528},[210],[15,27332,27333,27336],{"style":1531},[15,27334],{"className":27335,"style":218},[217],[15,27337,27339],{"className":27338},[222,223,224,225],[15,27340,1205],{"className":27341},[66,67,225],[15,27343,1114],{"className":27344},[1113],[15,27346,27348],{"className":27347},[206],[15,27349,27351],{"className":27350,"style":1121},[210],[15,27352],{},[15,27354,116],{"className":27355,"style":158},[66,67],[15,27357,120],{"className":27358},[162],[15,27360,27362,27365],{"className":27361},[66],[15,27363,123],{"className":27364},[66,67],[15,27366,27368],{"className":27367},[198],[15,27369,27371,27391],{"className":27370},[202,1090],[15,27372,27374,27388],{"className":27373},[206],[15,27375,27377],{"className":27376,"style":1528},[210],[15,27378,27379,27382],{"style":1584},[15,27380],{"className":27381,"style":218},[217],[15,27383,27385],{"className":27384},[222,223,224,225],[15,27386,1205],{"className":27387},[66,67,225],[15,27389,1114],{"className":27390},[1113],[15,27392,27394],{"className":27393},[206],[15,27395,27397],{"className":27396,"style":1121},[210],[15,27398],{},[15,27400,18771],{"className":27401},[169],[15,27403,27405],{"className":27404},[169],[15,27406,126],{"className":27407},[1422,1508],[15,27409],{"className":27410,"style":72},[71],[15,27412,1245],{"className":27413},[76],[15,27415],{"className":27416,"style":72},[71],[15,27418,27420,27423,27477],{"className":27419},[66],[15,27421],{"className":27422},[162,23417],[15,27424,27426],{"className":27425},[6467],[15,27427,27429,27469],{"className":27428},[202,1090],[15,27430,27432,27466],{"className":27431},[206],[15,27433,27436,27447,27455],{"className":27434,"style":27435},[210],"height:1.3714em;",[15,27437,27438,27441],{"style":23433},[15,27439],{"className":27440,"style":1368},[217],[15,27442,27444],{"className":27443},[66],[15,27445,1263],{"className":27446},[66],[15,27448,27449,27452],{"style":23445},[15,27450],{"className":27451,"style":1368},[217],[15,27453],{"className":27454,"style":4440},[23452],[15,27456,27457,27460],{"style":23455},[15,27458],{"className":27459,"style":1368},[217],[15,27461,27463],{"className":27462},[66],[15,27464,1248],{"className":27465},[66,67],[15,27467,1114],{"className":27468},[1113],[15,27470,27472],{"className":27471},[206],[15,27473,27475],{"className":27474,"style":23474},[210],[15,27476],{},[15,27478],{"className":27479},[169,23417],[15,27481,1251],{"className":27482},[66],[15,27484,116],{"className":27485,"style":158},[66,67],[15,27487,27489,27492],{"className":27488},[66],[15,27490,1251],{"className":27491},[66],[15,27493,27495],{"className":27494},[198],[15,27496,27498,27532],{"className":27497},[202,1090],[15,27499,27501,27529],{"className":27500},[206],[15,27502,27504,27518],{"className":27503,"style":1650},[210],[15,27505,27506,27509],{"style":1653},[15,27507],{"className":27508,"style":218},[217],[15,27510,27512],{"className":27511},[222,223,224,225],[15,27513,27515],{"className":27514},[66,225],[15,27516,1061],{"className":27517,"style":1083},[66,494,225],[15,27519,27520,27523],{"style":397},[15,27521],{"className":27522,"style":218},[217],[15,27524,27526],{"className":27525},[222,223,224,225],[15,27527,1263],{"className":27528},[66,225],[15,27530,1114],{"className":27531},[1113],[15,27533,27535],{"className":27534},[206],[15,27536,27538],{"className":27537,"style":1686},[210],[15,27539],{},[15,27541,27543],{"className":27542,"style":1418},[169,1417],[15,27544,4330],{"className":27545},[1422,1423],[15,27547],{"className":27548,"style":697},[71],[15,27550,3573],{"className":27551},[66],[11,27553,27554,27555,27799],{},"表示定理同样适用，",[15,27556,27558,27605],{"className":27557,"translate":19},[18],[15,27559,27561],{"className":27560},[23],[25,27562,27563],{"xmlns":27},[29,27564,27565,27603],{},[32,27566,27567,27573,27575,27581,27587,27589,27591,27593,27595,27601],{},[131,27568,27569,27571],{},[35,27570,116],{},[39,27572,1162],{},[39,27574,129],{},[1057,27576,27577,27579],{},[39,27578,1200],{},[35,27580,1205],{},[1057,27582,27583,27585],{},[35,27584,8390],{},[35,27586,1205],{},[35,27588,621],{},[39,27590,120],{"stretchy":119},[39,27592,3268],{},[39,27594,335],{"separator":53},[1057,27596,27597,27599],{},[35,27598,123],{},[35,27600,1205],{},[39,27602,126],{"stretchy":119},[45,27604,16812],{"encoding":47},[15,27606,27608,27652],{"className":27607,"ariaHidden":53},[52],[15,27609,27611,27614,27643,27646,27649],{"className":27610},[57],[15,27612],{"className":27613,"style":3408},[61],[15,27615,27617,27620],{"className":27616},[66],[15,27618,116],{"className":27619,"style":158},[66,67],[15,27621,27623],{"className":27622},[198],[15,27624,27626],{"className":27625},[202],[15,27627,27629],{"className":27628},[206],[15,27630,27632],{"className":27631,"style":16249},[210],[15,27633,27634,27637],{"style":213},[15,27635],{"className":27636,"style":218},[217],[15,27638,27640],{"className":27639},[222,223,224,225],[15,27641,1162],{"className":27642},[76,225],[15,27644],{"className":27645,"style":173},[71],[15,27647,129],{"className":27648},[177],[15,27650],{"className":27651,"style":173},[71],[15,27653,27655,27658,27698,27701,27741,27744,27747,27750,27753,27756,27796],{"className":27654},[57],[15,27656],{"className":27657,"style":16866},[61],[15,27659,27661,27664],{"className":27660},[1339],[15,27662,1200],{"className":27663,"style":13702},[1339,1470,13701],[15,27665,27667],{"className":27666},[198],[15,27668,27670,27690],{"className":27669},[202,1090],[15,27671,27673,27687],{"className":27672},[206],[15,27674,27676],{"className":27675,"style":16885},[210],[15,27677,27678,27681],{"style":13718},[15,27679],{"className":27680,"style":218},[217],[15,27682,27684],{"className":27683},[222,223,224,225],[15,27685,1205],{"className":27686},[66,67,225],[15,27688,1114],{"className":27689},[1113],[15,27691,27693],{"className":27692},[206],[15,27694,27696],{"className":27695,"style":16906},[210],[15,27697],{},[15,27699],{"className":27700,"style":697},[71],[15,27702,27704,27707],{"className":27703},[66],[15,27705,8390],{"className":27706,"style":8560},[66,67],[15,27708,27710],{"className":27709},[198],[15,27711,27713,27733],{"className":27712},[202,1090],[15,27714,27716,27730],{"className":27715},[206],[15,27717,27719],{"className":27718,"style":1528},[210],[15,27720,27721,27724],{"style":8575},[15,27722],{"className":27723,"style":218},[217],[15,27725,27727],{"className":27726},[222,223,224,225],[15,27728,1205],{"className":27729},[66,67,225],[15,27731,1114],{"className":27732},[1113],[15,27734,27736],{"className":27735},[206],[15,27737,27739],{"className":27738,"style":1121},[210],[15,27740],{},[15,27742,621],{"className":27743,"style":684},[66,67],[15,27745,120],{"className":27746},[162],[15,27748,3268],{"className":27749},[66],[15,27751,335],{"className":27752},[422],[15,27754],{"className":27755,"style":697},[71],[15,27757,27759,27762],{"className":27758},[66],[15,27760,123],{"className":27761},[66,67],[15,27763,27765],{"className":27764},[198],[15,27766,27768,27788],{"className":27767},[202,1090],[15,27769,27771,27785],{"className":27770},[206],[15,27772,27774],{"className":27773,"style":1528},[210],[15,27775,27776,27779],{"style":1584},[15,27777],{"className":27778,"style":218},[217],[15,27780,27782],{"className":27781},[222,223,224,225],[15,27783,1205],{"className":27784},[66,67,225],[15,27786,1114],{"className":27787},[1113],[15,27789,27791],{"className":27790},[206],[15,27792,27794],{"className":27793,"style":1121},[210],[15,27795],{},[15,27797,126],{"className":27798},[169],"。有限维问题是",[15,27801,27803],{"className":27802,"translate":19},[294],[15,27804,27806,27918],{"className":27805,"translate":19},[18],[15,27807,27809],{"className":27808},[23],[25,27810,27811],{"xmlns":27,"display":303},[29,27812,27813,27915],{},[32,27814,27815,27835,27913],{},[1172,27816,27817,27823],{},[32,27818,27819,27821],{},[35,27820,1178],{},[39,27822,1170],{},[32,27824,27825,27827,27829],{},[35,27826,8390],{},[39,27828,1187],{},[131,27830,27831,27833],{},[35,27832,455],{"mathvariant":454},[35,27834,37],{},[32,27836,27837,27839,27853,27855,27857,27859,27861,27863,27865,27867,27869,27871,27877,27879,27881,27883,27889,27891,27893,27895,27901,27907,27909,27911],{},[39,27838,4307],{"fence":53},[1196,27840,27841,27843,27851],{},[39,27842,1200],{},[32,27844,27845,27847,27849],{},[35,27846,1205],{},[39,27848,129],{},[1209,27850,1211],{},[35,27852,37],{},[35,27854,27073],{},[39,27856,1170],{},[39,27858,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,27860,1211],{},[39,27862,1245],{},[35,27864,6592],{},[39,27866,1170],{},[39,27868,120],{"stretchy":119},[39,27870,6602],{},[1057,27872,27873,27875],{},[35,27874,1224],{},[35,27876,1205],{},[39,27878,120],{"stretchy":119},[35,27880,2853],{},[35,27882,8390],{},[1057,27884,27885,27887],{},[39,27886,126],{"stretchy":119},[35,27888,1205],{},[39,27890,126],{"stretchy":119},[39,27892,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[39,27894,1245],{},[6467,27896,27897,27899],{},[35,27898,1248],{},[1209,27900,1263],{},[131,27902,27903,27905],{},[35,27904,8390],{},[35,27906,139],{"mathvariant":138},[35,27908,2853],{},[35,27910,8390],{},[39,27912,4330],{"fence":53},[35,27914,3573],{"mathvariant":138},[45,27916,27917],{"encoding":47},"\\min_{\\alpha\\in\\mathbb R^n}\\left\\{\n\\sum_{i=1}^n\\log\\bigl(1+\\exp(-y_i(K\\alpha)_i)\\bigr)\n+\\frac{\\lambda}{2}\\alpha^\\top K\\alpha\n\\right\\}.",[15,27919,27921],{"className":27920,"ariaHidden":53},[52],[15,27922,27924,27927,28006,28009,28327,28330],{"className":27923},[57],[15,27925],{"className":27926,"style":1335},[61],[15,27928,27930],{"className":27929},[1339,1351],[15,27931,27933,27998],{"className":27932},[202,1090],[15,27934,27936,27995],{"className":27935},[206],[15,27937,27939,27985],{"className":27938,"style":1361},[210],[15,27940,27941,27944],{"style":15664},[15,27942],{"className":27943,"style":1368},[217],[15,27945,27947],{"className":27946},[222,223,224,225],[15,27948,27950,27953,27956],{"className":27949},[66,225],[15,27951,8390],{"className":27952,"style":8560},[66,67,225],[15,27954,1187],{"className":27955},[177,225],[15,27957,27959,27962],{"className":27958},[66,225],[15,27960,455],{"className":27961},[66,517,225],[15,27963,27965],{"className":27964},[198],[15,27966,27968],{"className":27967},[202],[15,27969,27971],{"className":27970},[206],[15,27972,27974],{"className":27973,"style":15698},[210],[15,27975,27976,27979],{"style":15701},[15,27977],{"className":27978,"style":3675},[217],[15,27980,27982],{"className":27981},[222,3679,1508,225],[15,27983,37],{"className":27984},[66,67,225],[15,27986,27987,27990],{"style":1386},[15,27988],{"className":27989,"style":1368},[217],[15,27991,27992],{},[15,27993,1178],{"className":27994},[1339],[15,27996,1114],{"className":27997},[1113],[15,27999,28001],{"className":28000},[206],[15,28002,28004],{"className":28003,"style":15730},[210],[15,28005],{},[15,28007],{"className":28008,"style":697},[71],[15,28010,28012,28018,28082,28085,28090,28096,28099,28102,28105,28108,28111,28114,28117,28157,28160,28163,28166,28206,28209,28215,28218,28221,28224,28286,28315,28318,28321],{"className":28011},[1413],[15,28013,28015],{"className":28014,"style":1418},[162,1417],[15,28016,4307],{"className":28017},[1422,1423],[15,28019,28021],{"className":28020},[1339,1351],[15,28022,28024,28074],{"className":28023},[202,1090],[15,28025,28027,28071],{"className":28026},[206],[15,28028,28030,28050,28060],{"className":28029,"style":1436},[210],[15,28031,28032,28035],{"style":1439},[15,28033],{"className":28034,"style":1443},[217],[15,28036,28038],{"className":28037},[222,223,224,225],[15,28039,28041,28044,28047],{"className":28040},[66,225],[15,28042,1205],{"className":28043},[66,67,225],[15,28045,129],{"className":28046},[177,225],[15,28048,1211],{"className":28049},[66,225],[15,28051,28052,28055],{"style":1461},[15,28053],{"className":28054,"style":1443},[217],[15,28056,28057],{},[15,28058,1200],{"className":28059},[1339,1470,1471],[15,28061,28062,28065],{"style":1474},[15,28063],{"className":28064,"style":1443},[217],[15,28066,28068],{"className":28067},[222,223,224,225],[15,28069,37],{"className":28070},[66,67,225],[15,28072,1114],{"className":28073},[1113],[15,28075,28077],{"className":28076},[206],[15,28078,28080],{"className":28079,"style":1493},[210],[15,28081],{},[15,28083],{"className":28084,"style":697},[71],[15,28086,27283,28088],{"className":28087},[1339],[15,28089,1344],{"style":1343},[15,28091,28093],{"className":28092},[162],[15,28094,120],{"className":28095},[1422,1508],[15,28097,1211],{"className":28098},[66],[15,28100],{"className":28101,"style":72},[71],[15,28103,1245],{"className":28104},[76],[15,28106],{"className":28107,"style":72},[71],[15,28109,6592],{"className":28110},[1339],[15,28112,120],{"className":28113},[162],[15,28115,6602],{"className":28116},[66],[15,28118,28120,28123],{"className":28119},[66],[15,28121,1224],{"className":28122,"style":1515},[66,67],[15,28124,28126],{"className":28125},[198],[15,28127,28129,28149],{"className":28128},[202,1090],[15,28130,28132,28146],{"className":28131},[206],[15,28133,28135],{"className":28134,"style":1528},[210],[15,28136,28137,28140],{"style":1531},[15,28138],{"className":28139,"style":218},[217],[15,28141,28143],{"className":28142},[222,223,224,225],[15,28144,1205],{"className":28145},[66,67,225],[15,28147,1114],{"className":28148},[1113],[15,28150,28152],{"className":28151},[206],[15,28153,28155],{"className":28154,"style":1121},[210],[15,28156],{},[15,28158,120],{"className":28159},[162],[15,28161,2853],{"className":28162,"style":2901},[66,67],[15,28164,8390],{"className":28165,"style":8560},[66,67],[15,28167,28169,28172],{"className":28168},[169],[15,28170,126],{"className":28171},[169],[15,28173,28175],{"className":28174},[198],[15,28176,28178,28198],{"className":28177},[202,1090],[15,28179,28181,28195],{"className":28180},[206],[15,28182,28184],{"className":28183,"style":1528},[210],[15,28185,28186,28189],{"style":1584},[15,28187],{"className":28188,"style":218},[217],[15,28190,28192],{"className":28191},[222,223,224,225],[15,28193,1205],{"className":28194},[66,67,225],[15,28196,1114],{"className":28197},[1113],[15,28199,28201],{"className":28200},[206],[15,28202,28204],{"className":28203,"style":1121},[210],[15,28205],{},[15,28207,126],{"className":28208},[169],[15,28210,28212],{"className":28211},[169],[15,28213,126],{"className":28214},[1422,1508],[15,28216],{"className":28217,"style":72},[71],[15,28219,1245],{"className":28220},[76],[15,28222],{"className":28223,"style":72},[71],[15,28225,28227,28230,28283],{"className":28226},[66],[15,28228],{"className":28229},[162,23417],[15,28231,28233],{"className":28232},[6467],[15,28234,28236,28275],{"className":28235},[202,1090],[15,28237,28239,28272],{"className":28238},[206],[15,28240,28242,28253,28261],{"className":28241,"style":27435},[210],[15,28243,28244,28247],{"style":23433},[15,28245],{"className":28246,"style":1368},[217],[15,28248,28250],{"className":28249},[66],[15,28251,1263],{"className":28252},[66],[15,28254,28255,28258],{"style":23445},[15,28256],{"className":28257,"style":1368},[217],[15,28259],{"className":28260,"style":4440},[23452],[15,28262,28263,28266],{"style":23455},[15,28264],{"className":28265,"style":1368},[217],[15,28267,28269],{"className":28268},[66],[15,28270,1248],{"className":28271},[66,67],[15,28273,1114],{"className":28274},[1113],[15,28276,28278],{"className":28277},[206],[15,28279,28281],{"className":28280,"style":23474},[210],[15,28282],{},[15,28284],{"className":28285},[169,23417],[15,28287,28289,28292],{"className":28288},[66],[15,28290,8390],{"className":28291,"style":8560},[66,67],[15,28293,28295],{"className":28294},[198],[15,28296,28298],{"className":28297},[202],[15,28299,28301],{"className":28300},[206],[15,28302,28304],{"className":28303,"style":394},[210],[15,28305,28306,28309],{"style":397},[15,28307],{"className":28308,"style":218},[217],[15,28310,28312],{"className":28311},[222,223,224,225],[15,28313,139],{"className":28314},[66,225],[15,28316,2853],{"className":28317,"style":2901},[66,67],[15,28319,8390],{"className":28320,"style":8560},[66,67],[15,28322,28324],{"className":28323,"style":1418},[169,1417],[15,28325,4330],{"className":28326},[1422,1423],[15,28328],{"className":28329,"style":697},[71],[15,28331,3573],{"className":28332},[66],[11,28334,28335,28336,28364],{},"这是凸的、可微的，但没有闭式解，通常用 Newton 法、IRLS 或梯度类方法求解。与 SVM 不同，逻辑损失不会产生精确的零系数，",[15,28337,28339,28352],{"className":28338,"translate":19},[18],[15,28340,28342],{"className":28341},[23],[25,28343,28344],{"xmlns":27},[29,28345,28346,28350],{},[32,28347,28348],{},[35,28349,8390],{},[45,28351,16077],{"encoding":47},[15,28353,28355],{"className":28354,"ariaHidden":53},[52],[15,28356,28358,28361],{"className":28357},[57],[15,28359],{"className":28360,"style":86},[61],[15,28362,8390],{"className":28363,"style":8560},[66,67]," 一般是稠密的。",[1904,28366,28368],{"id":28367},"_54-高斯过程","5.4 高斯过程",[11,28370,28371],{},"高斯过程（GP）回归从贝叶斯角度给出了同一个解。设先验",[15,28373,28375],{"className":28374,"translate":19},[294],[15,28376,28378,28415],{"className":28377,"translate":19},[18],[15,28379,28381],{"className":28380},[23],[25,28382,28383],{"xmlns":27,"display":303},[29,28384,28385,28412],{},[32,28386,28387,28389,28392,28400,28402,28404,28406,28408,28410],{},[35,28388,116],{},[39,28390,28391],{},"∼",[32,28393,28394,28397],{},[35,28395,28396],{"mathvariant":445},"G",[35,28398,28399],{"mathvariant":445},"P",[39,28401,120],{"stretchy":119},[1209,28403,1804],{},[39,28405,335],{"separator":53},[35,28407,621],{},[39,28409,126],{"stretchy":119},[39,28411,335],{"separator":53},[45,28413,28414],{"encoding":47},"f\\sim\\mathcal{GP}(0,k),",[15,28416,28418,28436],{"className":28417,"ariaHidden":53},[52],[15,28419,28421,28424,28427,28430,28433],{"className":28420},[57],[15,28422],{"className":28423,"style":3408},[61],[15,28425,116],{"className":28426,"style":158},[66,67],[15,28428],{"className":28429,"style":173},[71],[15,28431,28391],{"className":28432},[177],[15,28434],{"className":28435,"style":173},[71],[15,28437,28439,28442,28453,28456,28459,28462,28465,28468,28471],{"className":28438},[57],[15,28440],{"className":28441,"style":154},[61],[15,28443,28445,28449],{"className":28444},[66],[15,28446,28396],{"className":28447,"style":28448},[66,494],"margin-right:0.0593em;",[15,28450,28399],{"className":28451,"style":28452},[66,494],"margin-right:0.0822em;",[15,28454,120],{"className":28455},[162],[15,28457,1804],{"className":28458},[66],[15,28460,335],{"className":28461},[422],[15,28463],{"className":28464,"style":697},[71],[15,28466,621],{"className":28467,"style":684},[66,67],[15,28469,126],{"className":28470},[169],[15,28472,335],{"className":28473},[422],[11,28475,28476],{},"观测模型",[15,28478,28480],{"className":28479,"translate":19},[294],[15,28481,28483,28555],{"className":28482,"translate":19},[18],[15,28484,28486],{"className":28485},[23],[25,28487,28488],{"xmlns":27,"display":303},[29,28489,28490,28552],{},[32,28491,28492,28498,28500,28502,28504,28510,28512,28514,28521,28523,28525,28531,28533,28536,28538,28540,28542,28548,28550],{},[1057,28493,28494,28496],{},[35,28495,1224],{},[35,28497,1205],{},[39,28499,129],{},[35,28501,116],{},[39,28503,120],{"stretchy":119},[1057,28505,28506,28508],{},[35,28507,123],{},[35,28509,1205],{},[39,28511,126],{"stretchy":119},[39,28513,1245],{},[1057,28515,28516,28519],{},[35,28517,28518],{},"ε",[35,28520,1205],{},[39,28522,335],{"separator":53},[71,28524],{"width":3904},[1057,28526,28527,28529],{},[35,28528,28518],{},[35,28530,1205],{},[39,28532,28391],{},[35,28534,28535],{},"N",[39,28537,120],{"stretchy":119},[1209,28539,1804],{},[39,28541,335],{"separator":53},[131,28543,28544,28546],{},[35,28545,6634],{},[1209,28547,1263],{},[39,28549,126],{"stretchy":119},[39,28551,335],{"separator":53},[45,28553,28554],{"encoding":47},"y_i=f(x_i)+\\varepsilon_i,\\qquad\n\\varepsilon_i\\sim N(0,\\sigma^2),",[15,28556,28558,28613,28677,28781],{"className":28557,"ariaHidden":53},[52],[15,28559,28561,28564,28604,28607,28610],{"className":28560},[57],[15,28562],{"className":28563,"style":471},[61],[15,28565,28567,28570],{"className":28566},[66],[15,28568,1224],{"className":28569,"style":1515},[66,67],[15,28571,28573],{"className":28572},[198],[15,28574,28576,28596],{"className":28575},[202,1090],[15,28577,28579,28593],{"className":28578},[206],[15,28580,28582],{"className":28581,"style":1528},[210],[15,28583,28584,28587],{"style":1531},[15,28585],{"className":28586,"style":218},[217],[15,28588,28590],{"className":28589},[222,223,224,225],[15,28591,1205],{"className":28592},[66,67,225],[15,28594,1114],{"className":28595},[1113],[15,28597,28599],{"className":28598},[206],[15,28600,28602],{"className":28601,"style":1121},[210],[15,28603],{},[15,28605],{"className":28606,"style":173},[71],[15,28608,129],{"className":28609},[177],[15,28611],{"className":28612,"style":173},[71],[15,28614,28616,28619,28622,28625,28665,28668,28671,28674],{"className":28615},[57],[15,28617],{"className":28618,"style":154},[61],[15,28620,116],{"className":28621,"style":158},[66,67],[15,28623,120],{"className":28624},[162],[15,28626,28628,28631],{"className":28627},[66],[15,28629,123],{"className":28630},[66,67],[15,28632,28634],{"className":28633},[198],[15,28635,28637,28657],{"className":28636},[202,1090],[15,28638,28640,28654],{"className":28639},[206],[15,28641,28643],{"className":28642,"style":1528},[210],[15,28644,28645,28648],{"style":1584},[15,28646],{"className":28647,"style":218},[217],[15,28649,28651],{"className":28650},[222,223,224,225],[15,28652,1205],{"className":28653},[66,67,225],[15,28655,1114],{"className":28656},[1113],[15,28658,28660],{"className":28659},[206],[15,28661,28663],{"className":28662,"style":1121},[210],[15,28664],{},[15,28666,126],{"className":28667},[169],[15,28669],{"className":28670,"style":72},[71],[15,28672,1245],{"className":28673},[76],[15,28675],{"className":28676,"style":72},[71],[15,28678,28680,28683,28723,28726,28729,28732,28772,28775,28778],{"className":28679},[57],[15,28681],{"className":28682,"style":471},[61],[15,28684,28686,28689],{"className":28685},[66],[15,28687,28518],{"className":28688},[66,67],[15,28690,28692],{"className":28691},[198],[15,28693,28695,28715],{"className":28694},[202,1090],[15,28696,28698,28712],{"className":28697},[206],[15,28699,28701],{"className":28700,"style":1528},[210],[15,28702,28703,28706],{"style":1584},[15,28704],{"className":28705,"style":218},[217],[15,28707,28709],{"className":28708},[222,223,224,225],[15,28710,1205],{"className":28711},[66,67,225],[15,28713,1114],{"className":28714},[1113],[15,28716,28718],{"className":28717},[206],[15,28719,28721],{"className":28720,"style":1121},[210],[15,28722],{},[15,28724,335],{"className":28725},[422],[15,28727],{"className":28728,"style":4061},[71],[15,28730],{"className":28731,"style":697},[71],[15,28733,28735,28738],{"className":28734},[66],[15,28736,28518],{"className":28737},[66,67],[15,28739,28741],{"className":28740},[198],[15,28742,28744,28764],{"className":28743},[202,1090],[15,28745,28747,28761],{"className":28746},[206],[15,28748,28750],{"className":28749,"style":1528},[210],[15,28751,28752,28755],{"style":1584},[15,28753],{"className":28754,"style":218},[217],[15,28756,28758],{"className":28757},[222,223,224,225],[15,28759,1205],{"className":28760},[66,67,225],[15,28762,1114],{"className":28763},[1113],[15,28765,28767],{"className":28766},[206],[15,28768,28770],{"className":28769,"style":1121},[210],[15,28771],{},[15,28773],{"className":28774,"style":173},[71],[15,28776,28391],{"className":28777},[177],[15,28779],{"className":28780,"style":173},[71],[15,28782,28784,28787,28791,28794,28797,28800,28803,28832,28835],{"className":28783},[57],[15,28785],{"className":28786,"style":11214},[61],[15,28788,28535],{"className":28789,"style":28790},[66,67],"margin-right:0.109em;",[15,28792,120],{"className":28793},[162],[15,28795,1804],{"className":28796},[66],[15,28798,335],{"className":28799},[422],[15,28801],{"className":28802,"style":697},[71],[15,28804,28806,28809],{"className":28805},[66],[15,28807,6634],{"className":28808,"style":1515},[66,67],[15,28810,28812],{"className":28811},[198],[15,28813,28815],{"className":28814},[202],[15,28816,28818],{"className":28817},[206],[15,28819,28821],{"className":28820,"style":1650},[210],[15,28822,28823,28826],{"style":397},[15,28824],{"className":28825,"style":218},[217],[15,28827,28829],{"className":28828},[222,223,224,225],[15,28830,1263],{"className":28831},[66,225],[15,28833,126],{"className":28834},[169],[15,28836,335],{"className":28837},[422],[11,28839,28840],{},"则后验均值是",[15,28842,28844],{"className":28843,"translate":19},[294],[15,28845,28847,28910],{"className":28846,"translate":19},[18],[15,28848,28850],{"className":28849},[23],[25,28851,28852],{"xmlns":27,"display":303},[29,28853,28854,28907],{},[32,28855,28856,28859,28861,28863,28865,28867,288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I)^{-1}y,",[15,28911,28913,28940,28999],{"className":28912,"ariaHidden":53},[52],[15,28914,28916,28919,28922,28925,28928,28931,28934,28937],{"className":28915},[57],[15,28917],{"className":28918,"style":154},[61],[15,28920,28858],{"className":28921},[66,67],[15,28923,120],{"className":28924},[162],[15,28926,123],{"className":28927},[66,67],[15,28929,126],{"className":28930},[169],[15,28932],{"className":28933,"style":173},[71],[15,28935,129],{"className":28936},[177],[15,28938],{"className":28939,"style":173},[71],[15,28941,28943,28946,28949,28952,28955,28984,28987,28990,28993,28996],{"className":28942},[57],[15,28944],{"className":28945,"style":375},[61],[15,28947,621],{"className":28948,"style":684},[66,67],[15,28950,120],{"className":28951},[162],[15,28953,123],{"className":28954},[66,67],[15,28956,28958,28961],{"className":28957},[169],[15,28959,126],{"className":28960},[169],[15,28962,28964],{"className":28963},[198],[15,28965,28967],{"className":28966},[202],[15,28968,28970],{"className":28969},[206],[15,28971,28973],{"className":28972,"style":394},[210],[15,28974,28975,28978],{"style":397},[15,28976],{"className":28977,"style":218},[217],[15,28979,28981],{"className":28980},[222,223,224,225],[15,28982,139],{"className":28983},[66,225],[15,28985,120],{"className":28986},[162],[15,28988,2853],{"className":28989,"style":2901},[66,67],[15,28991],{"className":28992,"style":72},[71],[15,28994,1245],{"className":28995},[76],[15,28997],{"className":28998,"style":72},[71],[15,29000,29002,29005,29034,29037,29072,29075],{"className":29001},[57],[15,29003],{"className":29004,"style":11214},[61],[15,29006,29008,29011],{"className":29007},[66],[15,29009,6634],{"className":29010,"style":1515},[66,67],[15,29012,29014],{"className":29013},[198],[15,29015,29017],{"className":29016},[202],[15,29018,29020],{"className":29019},[206],[15,29021,29023],{"className":29022,"style":1650},[210],[15,29024,29025,29028],{"style":397},[15,29026],{"className":29027,"style":218},[217],[15,29029,29031],{"className":29030},[222,223,224,225],[15,29032,1263],{"className":29033},[66,225],[15,29035,16427],{"className":29036,"style":16497},[66,67],[15,29038,29040,29043],{"className":29039},[169],[15,29041,126],{"className":29042},[169],[15,29044,29046],{"className":29045},[198],[15,29047,29049],{"className":29048},[202],[15,29050,29052],{"className":29051},[206],[15,29053,29055],{"className":29054,"style":1650},[210],[15,29056,29057,29060],{"style":397},[15,29058],{"className":29059,"style":218},[217],[15,29061,29063],{"className":29062},[222,223,224,225],[15,29064,29066,29069],{"className":29065},[66,225],[15,29067,6602],{"className":29068},[66,225],[15,29070,1211],{"className":29071},[66,225],[15,29073,1224],{"className":29074,"style":1515},[66,67],[15,29076,335],{"className":29077},[422],[11,29079,29080],{},"后验方差是",[15,29082,29084],{"className":29083,"translate":19},[294],[15,29085,29087,29174],{"className":29086,"translate":19},[18],[15,29088,29090],{"className":29089},[23],[25,29091,29092],{"xmlns":27,"display":303},[29,29093,29094,29171],{},[32,29095,29096,29103,29105,29107,29109,29111,29113,29115,29117,29119,29121,29123,29125,29127,29129,29131,29137,29139,29141,29143,29149,29151,29161,29163,29165,29167,29169],{},[131,29097,29098,29101],{},[35,29099,29100],{},"s",[1209,29102,1263],{},[39,29104,120],{"stretchy":119},[35,29106,123],{},[39,29108,126],{"stretchy":119},[39,29110,129],{},[35,29112,621],{},[39,29114,120],{"stretchy":119},[35,29116,123],{},[39,29118,335],{"separator":53},[35,29120,123],{},[39,29122,126],{"stretchy":119},[39,29124,6602],{},[35,29126,621],{},[39,29128,120],{"stretchy":119},[35,29130,123],{},[131,29132,29133,29135],{},[39,29134,126],{"stretchy":119},[35,29136,139],{"mathvariant":138},[39,29138,120],{"stretchy":119},[35,29140,2853],{},[39,29142,1245],{},[131,29144,29145,29147],{},[35,29146,6634],{},[1209,29148,1263],{},[35,29150,16427],{},[131,29152,29153,29155],{},[39,29154,126],{"stretchy":119},[32,29156,29157,29159],{},[39,29158,6602],{},[1209,29160,1211],{},[35,29162,621],{},[39,29164,120],{"stretchy":119},[35,29166,123],{},[39,29168,126],{"stretchy":119},[35,29170,3573],{"mathvariant":138},[45,29172,29173],{"encoding":47},"s^2(x)=k(x,x)-k(x)^\\top(K+\\sigma^2 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",[15,29417,29419,29441],{"className":29418,"translate":19},[18],[15,29420,29422],{"className":29421},[23],[25,29423,29424],{"xmlns":27},[29,29425,29426,29438],{},[32,29427,29428,29430,29432],{},[35,29429,1248],{},[39,29431,129],{},[131,29433,29434,29436],{},[35,29435,6634],{},[1209,29437,1263],{},[45,29439,29440],{"encoding":47},"\\lambda=\\sigma^2",[15,29442,29444,29462],{"className":29443,"ariaHidden":53},[52],[15,29445,29447,29450,29453,29456,29459],{"className":29446},[57],[15,29448],{"className":29449,"style":840},[61],[15,29451,1248],{"className":29452},[66,67],[15,29454],{"className":29455,"style":173},[71],[15,29457,129],{"className":29458},[177],[15,29460],{"className":29461,"style":173},[71],[15,29463,29465,29468],{"className":29464},[57],[15,29466],{"className":29467,"style":6740},[61],[15,29469,29471,29474],{"className":29470},[66],[15,29472,6634],{"className":29473,"style":1515},[66,67],[15,29475,29477],{"className":29476},[198],[15,29478,29480],{"className":29479},[202],[15,29481,29483],{"className":29482},[206],[15,29484,29486],{"className":29485,"style":6740},[210],[15,29487,29488,29491],{"style":213},[15,29489],{"className":29490,"style":218},[217],[15,29492,29494],{"className":29493},[222,223,224,225],[15,29495,1263],{"className":29496},[66,225]," 的核岭回归解，也就是表示定理给出的形式。区别在于，GP 还给出了不确定度 ",[15,29499,29501,29525],{"className":29500,"translate":19},[18],[15,29502,29504],{"className":29503},[23],[25,29505,29506],{"xmlns":27},[29,29507,29508,29522],{},[32,29509,29510,29516,29518,29520],{},[131,29511,29512,29514],{},[35,29513,29100],{},[1209,29515,1263],{},[39,29517,120],{"stretchy":119},[35,29519,123],{},[39,29521,126],{"stretchy":119},[45,29523,29524],{"encoding":47},"s^2(x)",[15,29526,29528],{"className":29527,"ariaHidden":53},[52],[15,29529,29531,29534,29563,29566,29569],{"className":29530},[57],[15,29532],{"className":29533,"style":9154},[61],[15,29535,29537,29540],{"className":29536},[66],[15,29538,29100],{"className":29539},[66,67],[15,29541,29543],{"className":29542},[198],[15,29544,29546],{"className":29545},[202],[15,29547,29549],{"className":29548},[206],[15,29550,29552],{"className":29551,"style":6740},[210],[15,29553,29554,29557],{"style":213},[15,29555],{"className":29556,"style":218},[217],[15,29558,29560],{"className":29559},[222,223,224,225],[15,29561,1263],{"className":29562},[66,225],[15,29564,120],{"className":29565},[162],[15,29567,123],{"className":29568},[66,67],[15,29570,126],{"className":29571},[169],"；表示定理只描述均值（或点预测）的结构。",[1904,29574,29576],{"id":29575},"_55-平滑样条","5.5 平滑样条",[11,29578,29579,29580,3249,29654,26816,29682,29710],{},"表示定理最早的重要应用之一是平滑样条。取 ",[15,29581,29583,29609],{"className":29582,"translate":19},[18],[15,29584,29586],{"className":29585},[23],[25,29587,29588],{"xmlns":27},[29,29589,29590,29606],{},[32,29591,29592,29594,29596,29598,29600,29602,29604],{},[35,29593,446],{"mathvariant":445},[39,29595,129],{},[39,29597,1194],{"stretchy":119},[1209,29599,1804],{},[39,29601,335],{"separator":53},[1209,29603,1211],{},[39,29605,1266],{"stretchy":119},[45,29607,29608],{"encoding":47},"\\mathcal X=[0,1]",[15,29610,29612,29630],{"className":29611,"ariaHidden":53},[52],[15,29613,29615,29618,29621,29624,29627],{"className":29614},[57],[15,29616],{"className":29617,"style":490},[61],[15,29619,446],{"className":29620,"style":495},[66,494],[15,29622],{"className":29623,"style":173},[71],[15,29625,129],{"className":29626},[177],[15,29628],{"className":29629,"style":173},[71],[15,29631,29633,29636,29639,29642,29645,29648,29651],{"className":29632},[57],[15,29634],{"className":29635,"style":154},[61],[15,29637,1194],{"className":29638},[162],[15,29640,1804],{"className":29641},[66],[15,29643,335],{"className":29644},[422],[15,29646],{"className":29647,"style":697},[71],[15,29649,1211],{"className":29650},[66],[15,29652,1266],{"className":29653},[169],[15,29655,29657,29670],{"className":29656,"translate":19},[18],[15,29658,29660],{"className":29659},[23],[25,29661,29662],{"xmlns":27},[29,29663,29664,29668],{},[32,29665,29666],{},[35,29667,1061],{"mathvariant":445},[45,29669,1742],{"encoding":47},[15,29671,29673],{"className":29672,"ariaHidden":53},[52],[15,29674,29676,29679],{"className":29675},[57],[15,29677],{"className":29678,"style":490},[61],[15,29680,1061],{"className":29681,"style":1083},[66,494],[15,29683,29685,29698],{"className":29684,"translate":19},[18],[15,29686,29688],{"className":29687},[23],[25,29689,29690],{"xmlns":27},[29,29691,29692,29696],{},[32,29693,29694],{},[35,29695,28858],{},[45,29697,28858],{"encoding":47},[15,29699,29701],{"className":29700,"ariaHidden":53},[52],[15,29702,29704,29707],{"className":29703},[57],[15,29705],{"className":29706,"style":86},[61],[15,29708,28858],{"className":29709},[66,67]," 阶 Sobolev 空间，正则项为",[15,29712,29714],{"className":29713,"translate":19},[294],[15,29715,29717,29774],{"className":29716,"translate":19},[18],[15,29718,29720],{"className":29719},[23],[25,29721,29722],{"xmlns":27,"display":303},[29,29723,29724,29771],{},[32,29725,29726,29728,29737,29739,29751,29753,29755,29757,29763,29765,29767,29769],{},[35,29727,1248],{},[1255,29729,29730,29733,29735],{},[39,29731,29732],{},"∫",[1209,29734,1804],{},[1209,29736,1211],{},[39,29738,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[131,29740,29741,29743],{},[35,29742,116],{},[32,29744,29745,29747,29749],{},[39,29746,120],{"stretchy":119},[35,29748,28858],{},[39,29750,126],{"stretchy":119},[39,29752,120],{"stretchy":119},[35,29754,123],{},[39,29756,126],{"stretchy":119},[131,29758,29759,29761],{},[39,29760,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,29762,1263],{},[1274,29764,2462],{},[35,29766,458],{},[35,29768,123],{},[35,29770,3573],{"mathvariant":138},[45,29772,29773],{"encoding":47},"\\lambda\\int_0^1\\bigl(f^{(m)}(x)\\bigr)^2\\,dx.",[15,29775,29777],{"className":29776,"ariaHidden":53},[52],[15,29778,29780,29784,29787,29790,29846,29852,29891,29894,29897,29900,29932,29935,29938,29941],{"className":29779},[57],[15,29781],{"className":29782,"style":29783},[61],"height:2.476em;vertical-align:-0.9119em;",[15,29785,1248],{"className":29786},[66,67],[15,29788],{"className":29789,"style":697},[71],[15,29791,29793,29797],{"className":29792},[1339],[15,29794,29732],{"className":29795,"style":29796},[1339,1470,1471],"margin-right:0.4445em;position:relative;top:-0.0011em;",[15,29798,29800],{"className":29799},[198],[15,29801,29803,29837],{"className":29802},[202,1090],[15,29804,29806,29834],{"className":29805},[206],[15,29807,29810,29822],{"className":29808,"style":29809},[210],"height:1.564em;",[15,29811,29813,29816],{"style":29812},"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;",[15,29814],{"className":29815,"style":218},[217],[15,29817,29819],{"className":29818},[222,223,224,225],[15,29820,1804],{"className":29821},[66,225],[15,29823,29825,29828],{"style":29824},"top:-3.8129em;margin-right:0.05em;",[15,29826],{"className":29827,"style":218},[217],[15,29829,29831],{"className":29830},[222,223,224,225],[15,29832,1211],{"className":29833},[66,225],[15,29835,1114],{"className":29836},[1113],[15,29838,29840],{"className":29839},[206],[15,29841,29844],{"className":29842,"style":29843},[210],"height:0.9119em;",[15,29845],{},[15,29847,29849],{"className":29848},[162],[15,29850,120],{"className":29851},[1422,1508],[15,29853,29855,29858],{"className":29854},[66],[15,29856,116],{"className":29857,"style":158},[66,67],[15,29859,29861],{"className":29860},[198],[15,29862,29864],{"className":29863},[202],[15,29865,29867],{"className":29866},[206],[15,29868,29871],{"className":29869,"style":29870},[210],"height:0.938em;",[15,29872,29873,29876],{"style":397},[15,29874],{"className":29875,"style":218},[217],[15,29877,29879],{"className":29878},[222,223,224,225],[15,29880,29882,29885,29888],{"className":29881},[66,225],[15,29883,120],{"className":29884},[162,225],[15,29886,28858],{"className":29887},[66,67,225],[15,29889,126],{"className":29890},[169,225],[15,29892,120],{"className":29893},[162],[15,29895,123],{"className":29896},[66,67],[15,29898,126],{"className":29899},[169],[15,29901,29903,29909],{"className":29902},[169],[15,29904,29906],{"className":29905},[169],[15,29907,126],{"className":29908},[1422,1508],[15,29910,29912],{"className":29911},[198],[15,29913,29915],{"className":29914},[202],[15,29916,29918],{"className":29917},[206],[15,29919,29921],{"className":29920,"style":21271},[210],[15,29922,29923,29926],{"style":21274},[15,29924],{"className":29925,"style":218},[217],[15,29927,29929],{"className":29928},[222,223,224,225],[15,29930,1263],{"className":29931},[66,225],[15,29933],{"className":29934,"style":697},[71],[15,29936,458],{"className":29937},[66,67],[15,29939,123],{"className":29940},[66,67],[15,29942,3573],{"className":29943},[66],[11,29945,29946],{},"最小化",[15,29948,29950],{"className":29949,"translate":19},[294],[15,29951,29953,30051],{"className":29952,"translate":19},[18],[15,29954,29956],{"className":29955},[23],[25,29957,29958],{"xmlns":27,"display":303},[29,29959,29960,30048],{},[32,29961,29962,29976,29978,29984,29986,29988,29990,29996,29998,30004,30006,30008,30016,30018,30030,30032,30034,30036,30042,30044,30046],{},[1196,29963,29964,29966,29974],{},[39,29965,1200],{},[32,29967,29968,29970,29972],{},[35,29969,1205],{},[39,29971,129],{},[1209,29973,1211],{},[35,29975,37],{},[39,29977,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[1057,29979,29980,29982],{},[35,29981,1224],{},[35,29983,1205],{},[39,29985,6602],{},[35,29987,116],{},[39,29989,120],{"stretchy":119},[1057,29991,29992,29994],{},[35,29993,123],{},[35,29995,1205],{},[39,29997,126],{"stretchy":119},[131,29999,30000,30002],{},[39,30001,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,30003,1263],{},[39,30005,1245],{},[35,30007,1248],{},[1255,30009,30010,30012,30014],{},[39,30011,29732],{},[1209,30013,1804],{},[1209,30015,1211],{},[39,30017,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[131,30019,30020,30022],{},[35,30021,116],{},[32,30023,30024,30026,30028],{},[39,30025,120],{"stretchy":119},[35,30027,28858],{},[39,30029,126],{"stretchy":119},[39,30031,120],{"stretchy":119},[35,30033,123],{},[39,30035,126],{"stretchy":119},[131,30037,30038,30040],{},[39,30039,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,30041,1263],{},[1274,30043,2462],{},[35,30045,458],{},[35,30047,123],{},[45,30049,30050],{"encoding":47},"\\sum_{i=1}^n\\bigl(y_i-f(x_i)\\bigr)^2\n+\\lambda\\int_0^1\\bigl(f^{(m)}(x)\\bigr)^2\\,dx",[15,30052,30054,30179,30276],{"className":30053,"ariaHidden":53},[52],[15,30055,30057,30060,30124,30130,30170,30173,30176],{"className":30056},[57],[15,30058],{"className":30059,"style":8486},[61],[15,30061,30063],{"className":30062},[1339,1351],[15,30064,30066,30116],{"className":30065},[202,1090],[15,30067,30069,30113],{"className":30068},[206],[15,30070,30072,30092,30102],{"className":30071,"style":1436},[210],[15,30073,30074,30077],{"style":1439},[15,30075],{"className":30076,"style":1443},[217],[15,30078,30080],{"className":30079},[222,223,224,225],[15,30081,30083,30086,30089],{"className":30082},[66,225],[15,30084,1205],{"className":30085},[66,67,225],[15,30087,129],{"className":30088},[177,225],[15,30090,1211],{"className":30091},[66,225],[15,30093,30094,30097],{"style":1461},[15,30095],{"className":30096,"style":1443},[217],[15,30098,30099],{},[15,30100,1200],{"className":30101},[1339,1470,1471],[15,30103,30104,30107],{"style":1474},[15,30105],{"className":30106,"style":1443},[217],[15,30108,30110],{"className":30109},[222,223,224,225],[15,30111,37],{"className":30112},[66,67,225],[15,30114,1114],{"className":30115},[1113],[15,30117,30119],{"className":30118},[206],[15,30120,30122],{"className":30121,"style":1493},[210],[15,30123],{},[15,30125,30127],{"className":30126},[162],[15,30128,120],{"className":30129},[1422,1508],[15,30131,30133,30136],{"className":30132},[66],[15,30134,1224],{"className":30135,"style":1515},[66,67],[15,30137,30139],{"className":30138},[198],[15,30140,30142,30162],{"className":30141},[202,1090],[15,30143,30145,30159],{"className":30144},[206],[15,30146,30148],{"className":30147,"style":1528},[210],[15,30149,30150,30153],{"style":1531},[15,30151],{"className":30152,"style":218},[217],[15,30154,30156],{"className":30155},[222,223,224,225],[15,30157,1205],{"className":30158},[66,67,225],[15,30160,1114],{"className":30161},[1113],[15,30163,30165],{"className":30164},[206],[15,30166,30168],{"className":30167,"style":1121},[210],[15,30169],{},[15,30171],{"className":30172,"style":72},[71],[15,30174,6602],{"className":30175},[76],[15,30177],{"className":30178,"style":72},[71],[15,30180,30182,30186,30189,30192,30232,30235,30267,30270,30273],{"className":30181},[57],[15,30183],{"className":30184,"style":30185},[61],"height:1.404em;vertical-align:-0.35em;",[15,30187,116],{"className":30188,"style":158},[66,67],[15,30190,120],{"className":30191},[162],[15,30193,30195,30198],{"className":30194},[66],[15,30196,123],{"className":30197},[66,67],[15,30199,30201],{"className":30200},[198],[15,30202,30204,30224],{"className":30203},[202,1090],[15,30205,30207,30221],{"className":30206},[206],[15,30208,30210],{"className":30209,"style":1528},[210],[15,30211,30212,30215],{"style":1584},[15,30213],{"className":30214,"style":218},[217],[15,30216,30218],{"className":30217},[222,223,224,225],[15,30219,1205],{"className":30220},[66,67,225],[15,30222,1114],{"className":30223},[1113],[15,30225,30227],{"className":30226},[206],[15,30228,30230],{"className":30229,"style":1121},[210],[15,30231],{},[15,30233,126],{"className":30234},[169],[15,30236,30238,30244],{"className":30237},[169],[15,30239,30241],{"className":30240},[169],[15,30242,126],{"className":30243},[1422,1508],[15,30245,30247],{"className":30246},[198],[15,30248,30250],{"className":30249},[202],[15,30251,30253],{"className":30252},[206],[15,30254,30256],{"className":30255,"style":21271},[210],[15,30257,30258,30261],{"style":21274},[15,30259],{"className":30260,"style":218},[217],[15,30262,30264],{"className":30263},[222,223,224,225],[15,30265,1263],{"className":30266},[66,225],[15,30268],{"className":30269,"style":72},[71],[15,30271,1245],{"className":30272},[76],[15,30274],{"className":30275,"style":72},[71],[15,30277,30279,30282,30285,30288,30339,30345,30383,30386,30389,30392,30424,30427,30430],{"className":30278},[57],[15,30280],{"className":30281,"style":29783},[61],[15,30283,1248],{"className":30284},[66,67],[15,30286],{"className":30287,"style":697},[71],[15,30289,30291,30294],{"className":30290},[1339],[15,30292,29732],{"className":30293,"style":29796},[1339,1470,1471],[15,30295,30297],{"className":30296},[198],[15,30298,30300,30331],{"className":30299},[202,1090],[15,30301,30303,30328],{"className":30302},[206],[15,30304,30306,30317],{"className":30305,"style":29809},[210],[15,30307,30308,30311],{"style":29812},[15,30309],{"className":30310,"style":218},[217],[15,30312,30314],{"className":30313},[222,223,224,225],[15,30315,1804],{"className":30316},[66,225],[15,30318,30319,30322],{"style":29824},[15,30320],{"className":30321,"style":218},[217],[15,30323,30325],{"className":30324},[222,223,224,225],[15,30326,1211],{"className":30327},[66,225],[15,30329,1114],{"className":30330},[1113],[15,30332,30334],{"className":30333},[206],[15,30335,30337],{"className":30336,"style":29843},[210],[15,30338],{},[15,30340,30342],{"className":30341},[162],[15,30343,120],{"className":30344},[1422,1508],[15,30346,30348,30351],{"className":30347},[66],[15,30349,116],{"className":30350,"style":158},[66,67],[15,30352,30354],{"className":30353},[198],[15,30355,30357],{"className":30356},[202],[15,30358,30360],{"className":30359},[206],[15,30361,30363],{"className":30362,"style":29870},[210],[15,30364,30365,30368],{"style":397},[15,30366],{"className":30367,"style":218},[217],[15,30369,30371],{"className":30370},[222,223,224,225],[15,30372,30374,30377,30380],{"className":30373},[66,225],[15,30375,120],{"className":30376},[162,225],[15,30378,28858],{"className":30379},[66,67,225],[15,30381,126],{"className":30382},[169,225],[15,30384,120],{"className":30385},[162],[15,30387,123],{"className":30388},[66,67],[15,30390,126],{"className":30391},[169],[15,30393,30395,30401],{"className":30394},[169],[15,30396,30398],{"className":30397},[169],[15,30399,126],{"className":30400},[1422,1508],[15,30402,30404],{"className":30403},[198],[15,30405,30407],{"className":30406},[202],[15,30408,30410],{"className":30409},[206],[15,30411,30413],{"className":30412,"style":21271},[210],[15,30414,30415,30418],{"style":21274},[15,30416],{"className":30417,"style":218},[217],[15,30419,30421],{"className":30420},[222,223,224,225],[15,30422,1263],{"className":30423},[66,225],[15,30425],{"className":30426,"style":697},[71],[15,30428,458],{"className":30429},[66,67],[15,30431,123],{"className":30432},[66,67],[11,30434,30435,30436,30493,30494,30522,30523,30574],{},"的解是一个以训练点为节点的 ",[15,30437,30439,30459],{"className":30438,"translate":19},[18],[15,30440,30442],{"className":30441},[23],[25,30443,30444],{"xmlns":27},[29,30445,30446,30456],{},[32,30447,30448,30450,30452,30454],{},[1209,30449,1263],{},[35,30451,28858],{},[39,30453,6602],{},[1209,30455,1211],{},[45,30457,30458],{"encoding":47},"2m-1",[15,30460,30462,30484],{"className":30461,"ariaHidden":53},[52],[15,30463,30465,30469,30472,30475,30478,30481],{"className":30464},[57],[15,30466],{"className":30467,"style":30468},[61],"height:0.7278em;vertical-align:-0.0833em;",[15,30470,1263],{"className":30471},[66],[15,30473,28858],{"className":30474},[66,67],[15,30476],{"className":30477,"style":72},[71],[15,30479,6602],{"className":30480},[76],[15,30482],{"className":30483,"style":72},[71],[15,30485,30487,30490],{"className":30486},[57],[15,30488],{"className":30489,"style":1836},[61],[15,30491,1211],{"className":30492},[66]," 次自然多项式样条。表示定理给出了「解在训练点核函数张成空间中」的结构；样条的多项式零空间（",[15,30495,30497,30510],{"className":30496,"translate":19},[18],[15,30498,30500],{"className":30499},[23],[25,30501,30502],{"xmlns":27},[29,30503,30504,30508],{},[32,30505,30506],{},[35,30507,28858],{},[45,30509,28858],{"encoding":47},[15,30511,30513],{"className":30512,"ariaHidden":53},[52],[15,30514,30516,30519],{"className":30515},[57],[15,30517],{"className":30518,"style":86},[61],[15,30520,28858],{"className":30521},[66,67]," 维）则由半参数版本处理。",[15,30524,30526,30544],{"className":30525,"translate":19},[18],[15,30527,30529],{"className":30528},[23],[25,30530,30531],{"xmlns":27},[29,30532,30533,30541],{},[32,30534,30535,30537,30539],{},[35,30536,28858],{},[39,30538,129],{},[1209,30540,1263],{},[45,30542,30543],{"encoding":47},"m=2",[15,30545,30547,30565],{"className":30546,"ariaHidden":53},[52],[15,30548,30550,30553,30556,30559,30562],{"className":30549},[57],[15,30551],{"className":30552,"style":86},[61],[15,30554,28858],{"className":30555},[66,67],[15,30557],{"className":30558,"style":173},[71],[15,30560,129],{"className":30561},[177],[15,30563],{"className":30564,"style":173},[71],[15,30566,30568,30571],{"className":30567},[57],[15,30569],{"className":30570,"style":1836},[61],[15,30572,1263],{"className":30573},[66]," 时就是常见的自然三次样条。",[11,30576,30577],{},"这也是 Kimeldorf–Wahba 定理的内容。它说明表示定理并不是核方法时代才出现的技巧，而是从样条估计中生长出来的结构结论。",[1904,30579,30581],{"id":30580},"_56-一张表","5.6 一张表",[6152,30583,30584,30603],{},[6155,30585,30586],{},[6158,30587,30588,30591,30594,30597,30600],{},[6161,30589,30590],{},"方法",[6161,30592,30593],{},"损失",[6161,30595,30596],{},"正则项",[6161,30598,30599],{},"系数",[6161,30601,30602],{},"稀疏性",[6171,30604,30605,30804,30968,31128,31284],{},[6158,30606,30607,30610,30613,30688,30801],{},[6176,30608,30609],{},"核岭回归",[6176,30611,30612],{},"平方",[6176,30614,30615],{},[15,30616,30618,30641],{"className":30617,"translate":19},[18],[15,30619,30621],{"className":30620},[23],[25,30622,30623],{"xmlns":27},[29,30624,30625,30639],{},[32,30626,30627,30629,30631,30633],{},[35,30628,1248],{},[35,30630,1251],{"mathvariant":138},[35,30632,116],{},[131,30634,30635,30637],{},[35,30636,1251],{"mathvariant":138},[1209,30638,1263],{},[45,30640,9144],{"encoding":47},[15,30642,30644],{"className":30643,"ariaHidden":53},[52],[15,30645,30647,30650,30653,30656,30659],{"className":30646},[57],[15,30648],{"className":30649,"style":9154},[61],[15,30651,1248],{"className":30652},[66,67],[15,30654,1251],{"className":30655},[66],[15,30657,116],{"className":30658,"style":158},[66,67],[15,30660,30662,30665],{"className":30661},[66],[15,30663,1251],{"className":30664},[66],[15,30666,30668],{"className":30667},[198],[15,30669,30671],{"className":30670},[202],[15,30672,30674],{"className":30673},[206],[15,30675,30677],{"className":30676,"style":6740},[210],[15,30678,30679,30682],{"style":213},[15,30680],{"className":30681,"style":218},[217],[15,30683,30685],{"className":30684},[222,223,224,225],[15,30686,1263],{"className":30687},[66,225],[6176,30689,30690],{},[15,30691,30693,30727],{"className":30692,"translate":19},[18],[15,30694,30696],{"className":30695},[23],[25,30697,30698],{"xmlns":27},[29,30699,30700,30724],{},[32,30701,30702,30704,30706,30708,30710,30712,30722],{},[39,30703,120],{"stretchy":119},[35,30705,2853],{},[39,30707,1245],{},[35,30709,1248],{},[35,30711,16427],{},[131,30713,30714,30716],{},[39,30715,126],{"stretchy":119},[32,30717,30718,30720],{},[39,30719,6602],{},[1209,30721,1211],{},[35,30723,1224],{},[45,30725,30726],{"encoding":47},"(K+\\lambda I)^{-1}y",[15,30728,30730,30751],{"className":30729,"ariaHidden":53},[52],[15,30731,30733,30736,30739,30742,30745,30748],{"className":30732},[57],[15,30734],{"className":30735,"style":154},[61],[15,30737,120],{"className":30738},[162],[15,30740,2853],{"className":30741,"style":2901},[66,67],[15,30743],{"className":30744,"style":72},[71],[15,30746,1245],{"className":30747},[76],[15,30749],{"className":30750,"style":72},[71],[15,30752,30754,30757,30760,30763,30798],{"className":30753},[57],[15,30755],{"className":30756,"style":9154},[61],[15,30758,1248],{"className":30759},[66,67],[15,30761,16427],{"className":30762,"style":16497},[66,67],[15,30764,30766,30769],{"className":30765},[169],[15,30767,126],{"className":30768},[169],[15,30770,30772],{"className":30771},[198],[15,30773,30775],{"className":30774},[202],[15,30776,30778],{"className":30777},[206],[15,30779,30781],{"className":30780,"style":6740},[210],[15,30782,30783,30786],{"style":213},[15,30784],{"className":30785,"style":218},[217],[15,30787,30789],{"className":30788},[222,223,224,225],[15,30790,30792,30795],{"className":30791},[66,225],[15,30793,6602],{"className":30794},[66,225],[15,30796,1211],{"className":30797},[66,225],[15,30799,1224],{"className":30800,"style":1515},[66,67],[6176,30802,30803],{},"一般稠密",[6158,30805,30806,30809,30812,30962,30965],{},[6176,30807,30808],{},"SVM",[6176,30810,30811],{},"hinge",[6176,30813,30814],{},[15,30815,30817,30845],{"className":30816,"translate":19},[18],[15,30818,30820],{"className":30819},[23],[25,30821,30822],{"xmlns":27},[29,30823,30824,30842],{},[32,30825,30826,30832,30834,30836],{},[6467,30827,30828,30830],{},[1209,30829,1211],{},[1209,30831,1263],{},[35,30833,1251],{"mathvariant":138},[35,30835,116],{},[131,30837,30838,30840],{},[35,30839,1251],{"mathvariant":138},[1209,30841,1263],{},[45,30843,30844],{"encoding":47},"\\frac12\\|f\\|^2",[15,30846,30848],{"className":30847,"ariaHidden":53},[52],[15,30849,30851,30855,30927,30930,30933],{"className":30850},[57],[15,30852],{"className":30853,"style":30854},[61],"height:1.1901em;vertical-align:-0.345em;",[15,30856,30858,30861,30924],{"className":30857},[66],[15,30859],{"className":30860},[162,23417],[15,30862,30864],{"className":30863},[6467],[15,30865,30867,30915],{"className":30866},[202,1090],[15,30868,30870,30912],{"className":30869},[206],[15,30871,30874,30889,30897],{"className":30872,"style":30873},[210],"height:0.8451em;",[15,30875,30877,30880],{"style":30876},"top:-2.655em;",[15,30878],{"className":30879,"style":1368},[217],[15,30881,30883],{"className":30882},[222,223,224,225],[15,30884,30886],{"className":30885},[66,225],[15,30887,1263],{"className":30888},[66,225],[15,30890,30891,30894],{"style":23445},[15,30892],{"className":30893,"style":1368},[217],[15,30895],{"className":30896,"style":4440},[23452],[15,30898,30900,30903],{"style":30899},"top:-3.394em;",[15,30901],{"className":30902,"style":1368},[217],[15,30904,30906],{"className":30905},[222,223,224,225],[15,30907,30909],{"className":30908},[66,225],[15,30910,1211],{"className":30911},[66,225],[15,30913,1114],{"className":30914},[1113],[15,30916,30918],{"className":30917},[206],[15,30919,30922],{"className":30920,"style":30921},[210],"height:0.345em;",[15,30923],{},[15,30925],{"className":30926},[169,23417],[15,30928,1251],{"className":30929},[66],[15,30931,116],{"className":30932,"style":158},[66,67],[15,30934,30936,30939],{"className":30935},[66],[15,30937,1251],{"className":30938},[66],[15,30940,30942],{"className":30941},[198],[15,30943,30945],{"className":30944},[202],[15,30946,30948],{"className":30947},[206],[15,30949,30951],{"className":30950,"style":6740},[210],[15,30952,30953,30956],{"style":213},[15,30954],{"className":30955,"style":218},[217],[15,30957,30959],{"className":30958},[222,223,224,225],[15,30960,1263],{"className":30961},[66,225],[6176,30963,30964],{},"对偶二次规划",[6176,30966,30967],{},"支持向量稀疏",[6158,30969,30970,30973,30976,31123,31126],{},[6176,30971,30972],{},"核逻辑回归",[6176,30974,30975],{},"logistic",[6176,30977,30978],{},[15,30979,30981,31009],{"className":30980,"translate":19},[18],[15,30982,30984],{"className":30983},[23],[25,30985,30986],{"xmlns":27},[29,30987,30988,31006],{},[32,30989,30990,30996,30998,31000],{},[6467,30991,30992,30994],{},[35,30993,1248],{},[1209,30995,1263],{},[35,30997,1251],{"mathvariant":138},[35,30999,116],{},[131,31001,31002,31004],{},[35,31003,1251],{"mathvariant":138},[1209,31005,1263],{},[45,31007,31008],{"encoding":47},"\\frac{\\lambda}{2}\\|f\\|^2",[15,31010,31012],{"className":31011,"ariaHidden":53},[52],[15,31013,31015,31019,31088,31091,31094],{"className":31014},[57],[15,31016],{"className":31017,"style":31018},[61],"height:1.2251em;vertical-align:-0.345em;",[15,31020,31022,31025,31085],{"className":31021},[66],[15,31023],{"className":31024},[162,23417],[15,31026,31028],{"className":31027},[6467],[15,31029,31031,31077],{"className":31030},[202,1090],[15,31032,31034,31074],{"className":31033},[206],[15,31035,31038,31052,31060],{"className":31036,"style":31037},[210],"height:0.8801em;",[15,31039,31040,31043],{"style":30876},[15,31041],{"className":31042,"style":1368},[217],[15,31044,31046],{"className":31045},[222,223,224,225],[15,31047,31049],{"className":31048},[66,225],[15,31050,1263],{"className":31051},[66,225],[15,31053,31054,31057],{"style":23445},[15,31055],{"className":31056,"style":1368},[217],[15,31058],{"className":31059,"style":4440},[23452],[15,31061,31062,31065],{"style":30899},[15,31063],{"className":31064,"style":1368},[217],[15,31066,31068],{"className":31067},[222,223,224,225],[15,31069,31071],{"className":31070},[66,225],[15,31072,1248],{"className":31073},[66,67,225],[15,31075,1114],{"className":31076},[1113],[15,31078,31080],{"className":31079},[206],[15,31081,31083],{"className":31082,"style":30921},[210],[15,31084],{},[15,31086],{"className":31087},[169,23417],[15,31089,1251],{"className":31090},[66],[15,31092,116],{"className":31093,"style":158},[66,67],[15,31095,31097,31100],{"className":31096},[66],[15,31098,1251],{"className":31099},[66],[15,31101,31103],{"className":31102},[198],[15,31104,31106],{"className":31105},[202],[15,31107,31109],{"className":31108},[206],[15,31110,31112],{"className":31111,"style":6740},[210],[15,31113,31114,31117],{"style":213},[15,31115],{"className":31116,"style":218},[217],[15,31118,31120],{"className":31119},[222,223,224,225],[15,31121,1263],{"className":31122},[66,225],[6176,31124,31125],{},"凸优化",[6176,31127,30803],{},[6158,31129,31130,31133,31136,31139,31282],{},[6176,31131,31132],{},"高斯过程",[6176,31134,31135],{},"平方（高斯似然）",[6176,31137,31138],{},"GP 先验",[6176,31140,31141],{},[15,31142,31144,31182],{"className":31143,"translate":19},[18],[15,31145,31147],{"className":31146},[23],[25,31148,31149],{"xmlns":27},[29,31150,31151,31179],{},[32,31152,31153,31155,31157,31159,31165,31167,31177],{},[39,31154,120],{"stretchy":119},[35,31156,2853],{},[39,31158,1245],{},[131,31160,31161,31163],{},[35,31162,6634],{},[1209,31164,1263],{},[35,31166,16427],{},[131,31168,31169,31171],{},[39,31170,126],{"stretchy":119},[32,31172,31173,31175],{},[39,31174,6602],{},[1209,31176,1211],{},[35,31178,1224],{},[45,31180,31181],{"encoding":47},"(K+\\sigma^2 I)^{-1}y",[15,31183,31185,31206],{"className":31184,"ariaHidden":53},[52],[15,31186,31188,31191,31194,31197,31200,31203],{"className":31187},[57],[15,31189],{"className":31190,"style":154},[61],[15,31192,120],{"className":31193},[162],[15,31195,2853],{"className":31196,"style":2901},[66,67],[15,31198],{"className":31199,"style":72},[71],[15,31201,1245],{"className":31202},[76],[15,31204],{"className":31205,"style":72},[71],[15,31207,31209,31212,31241,31244,31279],{"className":31208},[57],[15,31210],{"className":31211,"style":9154},[61],[15,31213,31215,31218],{"className":31214},[66],[15,31216,6634],{"className":31217,"style":1515},[66,67],[15,31219,31221],{"className":31220},[198],[15,31222,31224],{"className":31223},[202],[15,31225,31227],{"className":31226},[206],[15,31228,31230],{"className":31229,"style":6740},[210],[15,31231,31232,31235],{"style":213},[15,31233],{"className":31234,"style":218},[217],[15,31236,31238],{"className":31237},[222,223,224,225],[15,31239,1263],{"className":31240},[66,225],[15,31242,16427],{"className":31243,"style":16497},[66,67],[15,31245,31247,31250],{"className":31246},[169],[15,31248,126],{"className":31249},[169],[15,31251,31253],{"className":31252},[198],[15,31254,31256],{"className":31255},[202],[15,31257,31259],{"className":31258},[206],[15,31260,31262],{"className":31261,"style":6740},[210],[15,31263,31264,31267],{"style":213},[15,31265],{"className":31266,"style":218},[217],[15,31268,31270],{"className":31269},[222,223,224,225],[15,31271,31273,31276],{"className":31272},[66,225],[15,31274,6602],{"className":31275},[66,225],[15,31277,1211],{"className":31278},[66,225],[15,31280,1224],{"className":31281,"style":1515},[66,67],[6176,31283,30803],{},[6158,31285,31286,31289,31291,31414,31417],{},[6176,31287,31288],{},"平滑样条",[6176,31290,30612],{},[6176,31292,31293],{},[15,31294,31296,31330],{"className":31295,"translate":19},[18],[15,31297,31299],{"className":31298},[23],[25,31300,31301],{"xmlns":27},[29,31302,31303,31327],{},[32,31304,31305,31307,31309,31321],{},[39,31306,29732],{},[39,31308,120],{"stretchy":119},[131,31310,31311,31313],{},[35,31312,116],{},[32,31314,31315,31317,31319],{},[39,31316,120],{"stretchy":119},[35,31318,28858],{},[39,31320,126],{"stretchy":119},[131,31322,31323,31325],{},[39,31324,126],{"stretchy":119},[1209,31326,1263],{},[45,31328,31329],{"encoding":47},"\\int(f^{(m)})^2",[15,31331,31333],{"className":31332,"ariaHidden":53},[52],[15,31334,31336,31340,31344,31347,31385],{"className":31335},[57],[15,31337],{"className":31338,"style":31339},[61],"height:1.1941em;vertical-align:-0.3061em;",[15,31341,29732],{"className":31342,"style":31343},[1339,1470,13701],"margin-right:0.1945em;position:relative;top:-0.0006em;",[15,31345,120],{"className":31346},[162],[15,31348,31350,31353],{"className":31349},[66],[15,31351,116],{"className":31352,"style":158},[66,67],[15,31354,31356],{"className":31355},[198],[15,31357,31359],{"className":31358},[202],[15,31360,31362],{"className":31361},[206],[15,31363,31365],{"className":31364,"style":7066},[210],[15,31366,31367,31370],{"style":213},[15,31368],{"className":31369,"style":218},[217],[15,31371,31373],{"className":31372},[222,223,224,225],[15,31374,31376,31379,31382],{"className":31375},[66,225],[15,31377,120],{"className":31378},[162,225],[15,31380,28858],{"className":31381},[66,67,225],[15,31383,126],{"className":31384},[169,225],[15,31386,31388,31391],{"className":31387},[169],[15,31389,126],{"className":31390},[169],[15,31392,31394],{"className":31393},[198],[15,31395,31397],{"className":31396},[202],[15,31398,31400],{"className":31399},[206],[15,31401,31403],{"className":31402,"style":6740},[210],[15,31404,31405,31408],{"style":213},[15,31406],{"className":31407,"style":218},[217],[15,31409,31411],{"className":31410},[222,223,224,225],[15,31412,1263],{"className":31413},[66,225],[6176,31415,31416],{},"样条系数",[6176,31418,30803],{},[95,31420,31422],{"id":31421},"六计算代价与大规模近似","六、计算代价与大规模近似",[11,31424,31425,31426,31454,31455,31483,31484,31512,31513,31586,31587,31660,31661,31704,31705,3249,31803,31883],{},"表示定理把无穷维问题降到了 ",[15,31427,31429,31442],{"className":31428,"translate":19},[18],[15,31430,31432],{"className":31431},[23],[25,31433,31434],{"xmlns":27},[29,31435,31436,31440],{},[32,31437,31438],{},[35,31439,37],{},[45,31441,37],{"encoding":47},[15,31443,31445],{"className":31444,"ariaHidden":53},[52],[15,31446,31448,31451],{"className":31447},[57],[15,31449],{"className":31450,"style":86},[61],[15,31452,37],{"className":31453},[66,67]," 维，但 ",[15,31456,31458,31471],{"className":31457,"translate":19},[18],[15,31459,31461],{"className":31460},[23],[25,31462,31463],{"xmlns":27},[29,31464,31465,31469],{},[32,31466,31467],{},[35,31468,37],{},[45,31470,37],{"encoding":47},[15,31472,31474],{"className":31473,"ariaHidden":53},[52],[15,31475,31477,31480],{"className":31476},[57],[15,31478],{"className":31479,"style":86},[61],[15,31481,37],{"className":31482},[66,67]," 是训练样本数，可能很大。核矩阵 ",[15,31485,31487,31500],{"className":31486,"translate":19},[18],[15,31488,31490],{"className":31489},[23],[25,31491,31492],{"xmlns":27},[29,31493,31494,31498],{},[32,31495,31496],{},[35,31497,2853],{},[45,31499,2853],{"encoding":47},[15,31501,31503],{"className":31502,"ariaHidden":53},[52],[15,31504,31506,31509],{"className":31505},[57],[15,31507],{"className":31508,"style":490},[61],[15,31510,2853],{"className":31511,"style":2901},[66,67]," 需要 ",[15,31514,31516,31539],{"className":31515,"translate":19},[18],[15,31517,31519],{"className":31518},[23],[25,31520,31521],{"xmlns":27},[29,31522,31523,31537],{},[32,31524,31525,31527,31529,31535],{},[35,31526,22999],{},[39,31528,120],{"stretchy":119},[131,31530,31531,31533],{},[35,31532,37],{},[1209,31534,1263],{},[39,31536,126],{"stretchy":119},[45,31538,23088],{"encoding":47},[15,31540,31542],{"className":31541,"ariaHidden":53},[52],[15,31543,31545,31548,31551,31554,31583],{"className":31544},[57],[15,31546],{"className":31547,"style":9154},[61],[15,31549,22999],{"className":31550,"style":23025},[66,67],[15,31552,120],{"className":31553},[162],[15,31555,31557,31560],{"className":31556},[66],[15,31558,37],{"className":31559},[66,67],[15,31561,31563],{"className":31562},[198],[15,31564,31566],{"className":31565},[202],[15,31567,31569],{"className":31568},[206],[15,31570,31572],{"className":31571,"style":6740},[210],[15,31573,31574,31577],{"style":213},[15,31575],{"className":31576,"style":218},[217],[15,31578,31580],{"className":31579},[222,223,224,225],[15,31581,1263],{"className":31582},[66,225],[15,31584,126],{"className":31585},[169]," 内存；核岭回归的 Cholesky 分解需要 ",[15,31588,31590,31613],{"className":31589,"translate":19},[18],[15,31591,31593],{"className":31592},[23],[25,31594,31595],{"xmlns":27},[29,31596,31597,31611],{},[32,31598,31599,31601,31603,31609],{},[35,31600,22999],{},[39,31602,120],{"stretchy":119},[131,31604,31605,31607],{},[35,31606,37],{},[1209,31608,8678],{},[39,31610,126],{"stretchy":119},[45,31612,23012],{"encoding":47},[15,31614,31616],{"className":31615,"ariaHidden":53},[52],[15,31617,31619,31622,31625,31628,31657],{"className":31618},[57],[15,31620],{"className":31621,"style":9154},[61],[15,31623,22999],{"className":31624,"style":23025},[66,67],[15,31626,120],{"className":31627},[162],[15,31629,31631,31634],{"className":31630},[66],[15,31632,37],{"className":31633},[66,67],[15,31635,31637],{"className":31636},[198],[15,31638,31640],{"className":31639},[202],[15,31641,31643],{"className":31642},[206],[15,31644,31646],{"className":31645,"style":6740},[210],[15,31647,31648,31651],{"style":213},[15,31649],{"className":31650,"style":218},[217],[15,31652,31654],{"className":31653},[222,223,224,225],[15,31655,8678],{"className":31656},[66,225],[15,31658,126],{"className":31659},[169]," 时间；SVM 的通用二次规划更慢。预测一个新点需要计算它与所有训练点的核函数值，代价是 ",[15,31662,31664,31683],{"className":31663,"translate":19},[18],[15,31665,31667],{"className":31666},[23],[25,31668,31669],{"xmlns":27},[29,31670,31671,31681],{},[32,31672,31673,31675,31677,31679],{},[35,31674,22999],{},[39,31676,120],{"stretchy":119},[35,31678,37],{},[39,31680,126],{"stretchy":119},[45,31682,23159],{"encoding":47},[15,31684,31686],{"className":31685,"ariaHidden":53},[52],[15,31687,31689,31692,31695,31698,31701],{"className":31688},[57],[15,31690],{"className":31691,"style":154},[61],[15,31693,22999],{"className":31694,"style":23025},[66,67],[15,31696,120],{"className":31697},[162],[15,31699,37],{"className":31700},[66,67],[15,31702,126],{"className":31703},[169],"（SVM 中是 ",[15,31706,31708,31738],{"className":31707,"translate":19},[18],[15,31709,31711],{"className":31710},[23],[25,31712,31713],{"xmlns":27},[29,31714,31715,31735],{},[32,31716,31717,31719,31721,31733],{},[35,31718,22999],{},[39,31720,120],{"stretchy":119},[1057,31722,31723,31725],{},[35,31724,37],{},[32,31726,31727,31730],{},[35,31728,31729],{"mathvariant":138},"S",[35,31731,31732],{"mathvariant":138},"V",[39,31734,126],{"stretchy":119},[45,31736,31737],{"encoding":47},"O(n_{\\mathrm{SV}})",[15,31739,31741],{"className":31740,"ariaHidden":53},[52],[15,31742,31744,31747,31750,31753,31800],{"className":31743},[57],[15,31745],{"className":31746,"style":154},[61],[15,31748,22999],{"className":31749,"style":23025},[66,67],[15,31751,120],{"className":31752},[162],[15,31754,31756,31759],{"className":31755},[66],[15,31757,37],{"className":31758},[66,67],[15,31760,31762],{"className":31761},[198],[15,31763,31765,31792],{"className":31764},[202,1090],[15,31766,31768,31789],{"className":31767},[206],[15,31769,31771],{"className":31770,"style":3638},[210],[15,31772,31773,31776],{"style":1584},[15,31774],{"className":31775,"style":218},[217],[15,31777,31779],{"className":31778},[222,223,224,225],[15,31780,31782],{"className":31781},[66,225],[15,31783,31785],{"className":31784},[66,225],[15,31786,31788],{"className":31787,"style":1343},[66,4429,225],"SV",[15,31790,1114],{"className":31791},[1113],[15,31793,31795],{"className":31794},[206],[15,31796,31798],{"className":31797,"style":1121},[210],[15,31799],{},[15,31801,126],{"className":31802},[169],[15,31804,31806,31828],{"className":31805,"translate":19},[18],[15,31807,31809],{"className":31808},[23],[25,31810,31811],{"xmlns":27},[29,31812,31813,31825],{},[32,31814,31815],{},[1057,31816,31817,31819],{},[35,31818,37],{},[32,31820,31821,31823],{},[35,31822,31729],{"mathvariant":138},[35,31824,31732],{"mathvariant":138},[45,31826,31827],{"encoding":47},"n_{\\mathrm{SV}}",[15,31829,31831],{"className":31830,"ariaHidden":53},[52],[15,31832,31834,31837],{"className":31833},[57],[15,31835],{"className":31836,"style":10197},[61],[15,31838,31840,31843],{"className":31839},[66],[15,31841,37],{"className":31842},[66,67],[15,31844,31846],{"className":31845},[198],[15,31847,31849,31875],{"className":31848},[202,1090],[15,31850,31852,31872],{"className":31851},[206],[15,31853,31855],{"className":31854,"style":3638},[210],[15,31856,31857,31860],{"style":1584},[15,31858],{"className":31859,"style":218},[217],[15,31861,31863],{"className":31862},[222,223,224,225],[15,31864,31866],{"className":31865},[66,225],[15,31867,31869],{"className":31868},[66,225],[15,31870,31788],{"className":31871,"style":1343},[66,4429,225],[15,31873,1114],{"className":31874},[1113],[15,31876,31878],{"className":31877},[206],[15,31879,31881],{"className":31880,"style":1121},[210],[15,31882],{}," 为支持向量数）。",[11,31885,31886,31887,31915],{},"当 ",[15,31888,31890,31903],{"className":31889,"translate":19},[18],[15,31891,31893],{"className":31892},[23],[25,31894,31895],{"xmlns":27},[29,31896,31897,31901],{},[32,31898,31899],{},[35,31900,37],{},[45,31902,37],{"encoding":47},[15,31904,31906],{"className":31905,"ariaHidden":53},[52],[15,31907,31909,31912],{"className":31908},[57],[15,31910],{"className":31911,"style":86},[61],[15,31913,37],{"className":31914},[66,67]," 很大时，常用的近似方法有：",[16136,31917,31918,32438,32634,32691],{},[3194,31919,31920,31923,31924,31976,31977,32233,32234,7183,32307,3371],{},[2040,31921,31922],{},"Nyström 近似","：选 ",[15,31925,31927,31946],{"className":31926,"translate":19},[18],[15,31928,31930],{"className":31929},[23],[25,31931,31932],{"xmlns":27},[29,31933,31934,31943],{},[32,31935,31936,31938,31941],{},[35,31937,28858],{},[39,31939,31940],{},"≪",[35,31942,37],{},[45,31944,31945],{"encoding":47},"m\\ll n",[15,31947,31949,31967],{"className":31948,"ariaHidden":53},[52],[15,31950,31952,31955,31958,31961,31964],{"className":31951},[57],[15,31953],{"className":31954,"style":3227},[61],[15,31956,28858],{"className":31957},[66,67],[15,31959],{"className":31960,"style":173},[71],[15,31962,31940],{"className":31963},[177],[15,31965],{"className":31966,"style":173},[71],[15,31968,31970,31973],{"className":31969},[57],[15,31971],{"className":31972,"style":86},[61],[15,31974,37],{"className":31975},[66,67]," 个「地标点」，用 ",[15,31978,31980,32039],{"className":31979,"translate":19},[18],[15,31981,31983],{"className":31982},[23],[25,31984,31985],{"xmlns":27},[29,31986,31987,32036],{},[32,31988,31989,31991,31994,32006,32024],{},[35,31990,2853],{},[39,31992,31993],{},"≈",[1057,31995,31996,31998],{},[35,31997,2853],{},[32,31999,32000,32002,32004],{},[35,32001,37],{},[39,32003,335],{"separator":53},[35,32005,28858],{},[1255,32007,32008,32010,32018],{},[35,32009,2853],{},[32,32011,32012,32014,32016],{},[35,32013,28858],{},[39,32015,335],{"separator":53},[35,32017,28858],{},[32,32019,32020,32022],{},[39,32021,6602],{},[1209,32023,1211],{},[1057,32025,32026,32028],{},[35,32027,2853],{},[32,32029,32030,32032,32034],{},[35,32031,28858],{},[39,32033,335],{"separator":53},[35,32035,37],{},[45,32037,32038],{"encoding":47},"K\\approx K_{n,m}K_{m,m}^{-1}K_{m,n}",[15,32040,32042,32060],{"className":32041,"ariaHidden":53},[52],[15,32043,32045,32048,32051,32054,32057],{"className":32044},[57],[15,32046],{"className":32047,"style":490},[61],[15,32049,2853],{"className":32050,"style":2901},[66,67],[15,32052],{"className":32053,"style":173},[71],[15,32055,31993],{"className":32056},[177],[15,32058],{"className":32059,"style":173},[71],[15,32061,32063,32067,32116,32184],{"className":32062},[57],[15,32064],{"className":32065,"style":32066},[61],"height:1.1972em;vertical-align:-0.3831em;",[15,32068,32070,32073],{"className":32069},[66],[15,32071,2853],{"className":32072,"style":2901},[66,67],[15,32074,32076],{"className":32075},[198],[15,32077,32079,32108],{"className":32078},[202,1090],[15,32080,32082,32105],{"className":32081},[206],[15,32083,32085],{"className":32084,"style":2198},[210],[15,32086,32087,32090],{"style":2916},[15,32088],{"className":32089,"style":218},[217],[15,32091,32093],{"className":32092},[222,223,224,225],[15,32094,32096,32099,32102],{"className":32095},[66,225],[15,32097,37],{"className":32098},[66,67,225],[15,32100,335],{"className":32101},[422,225],[15,32103,28858],{"className":32104},[66,67,225],[15,32106,1114],{"className":32107},[1113],[15,32109,32111],{"className":32110},[206],[15,32112,32114],{"className":32113,"style":2715},[210],[15,32115],{},[15,32117,32119,32122],{"className":32118},[66],[15,32120,2853],{"className":32121,"style":2901},[66,67],[15,32123,32125],{"className":32124},[198],[15,32126,32128,32175],{"className":32127},[202,1090],[15,32129,32131,32172],{"className":32130},[206],[15,32132,32134,32155],{"className":32133,"style":6740},[210],[15,32135,32137,32140],{"style":32136},"top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;",[15,32138],{"className":32139,"style":218},[217],[15,32141,32143],{"className":32142},[222,223,224,225],[15,32144,32146,32149,32152],{"className":32145},[66,225],[15,32147,28858],{"className":32148},[66,67,225],[15,32150,335],{"className":32151},[422,225],[15,32153,28858],{"className":32154},[66,67,225],[15,32156,32157,32160],{"style":213},[15,32158],{"className":32159,"style":218},[217],[15,32161,32163],{"className":32162},[222,223,224,225],[15,32164,32166,32169],{"className":32165},[66,225],[15,32167,6602],{"className":32168},[66,225],[15,32170,1211],{"className":32171},[66,225],[15,32173,1114],{"className":32174},[1113],[15,32176,32178],{"className":32177},[206],[15,32179,32182],{"className":32180,"style":32181},[210],"height:0.3831em;",[15,32183],{},[15,32185,32187,32190],{"className":32186},[66],[15,32188,2853],{"className":32189,"style":2901},[66,67],[15,32191,32193],{"className":32192},[198],[15,32194,32196,32225],{"className":32195},[202,1090],[15,32197,32199,32222],{"className":32198},[206],[15,32200,32202],{"className":32201,"style":2198},[210],[15,32203,32204,32207],{"style":2916},[15,32205],{"className":32206,"style":218},[217],[15,32208,32210],{"className":32209},[222,223,224,225],[15,32211,32213,32216,32219],{"className":32212},[66,225],[15,32214,28858],{"className":32215},[66,67,225],[15,32217,335],{"className":32218},[422,225],[15,32220,37],{"className":32221},[66,67,225],[15,32223,1114],{"className":32224},[1113],[15,32226,32228],{"className":32227},[206],[15,32229,32231],{"className":32230,"style":2715},[210],[15,32232],{}," 逼近核矩阵，代价从 ",[15,32235,32237,32260],{"className":32236,"translate":19},[18],[15,32238,32240],{"className":32239},[23],[25,32241,32242],{"xmlns":27},[29,32243,32244,32258],{},[32,32245,32246,32248,32250,32256],{},[35,32247,22999],{},[39,32249,120],{"stretchy":119},[131,32251,32252,32254],{},[35,32253,37],{},[1209,32255,8678],{},[39,32257,126],{"stretchy":119},[45,32259,23012],{"encoding":47},[15,32261,32263],{"className":32262,"ariaHidden":53},[52],[15,32264,32266,32269,32272,32275,32304],{"className":32265},[57],[15,32267],{"className":32268,"style":9154},[61],[15,32270,22999],{"className":32271,"style":23025},[66,67],[15,32273,120],{"className":32274},[162],[15,32276,32278,32281],{"className":32277},[66],[15,32279,37],{"className":32280},[66,67],[15,32282,32284],{"className":32283},[198],[15,32285,32287],{"className":32286},[202],[15,32288,32290],{"className":32289},[206],[15,32291,32293],{"className":32292,"style":6740},[210],[15,32294,32295,32298],{"style":213},[15,32296],{"className":32297,"style":218},[217],[15,32299,32301],{"className":32300},[222,223,224,225],[15,32302,8678],{"className":32303},[66,225],[15,32305,126],{"className":32306},[169],[15,32308,32310,32344],{"className":32309,"translate":19},[18],[15,32311,32313],{"className":32312},[23],[25,32314,32315],{"xmlns":27},[29,32316,32317,32341],{},[32,32318,32319,32321,32323,32325,32331,32333,32339],{},[35,32320,22999],{},[39,32322,120],{"stretchy":119},[35,32324,37],{},[131,32326,32327,32329],{},[35,32328,28858],{},[1209,32330,1263],{},[39,32332,1245],{},[131,32334,32335,32337],{},[35,32336,28858],{},[1209,32338,8678],{},[39,32340,126],{"stretchy":119},[45,32342,32343],{"encoding":47},"O(nm^2+m^3)",[15,32345,32347,32400],{"className":32346,"ariaHidden":53},[52],[15,32348,32350,32353,32356,32359,32362,32391,32394,32397],{"className":32349},[57],[15,32351],{"className":32352,"style":9154},[61],[15,32354,22999],{"className":32355,"style":23025},[66,67],[15,32357,120],{"className":32358},[162],[15,32360,37],{"className":32361},[66,67],[15,32363,32365,32368],{"className":32364},[66],[15,32366,28858],{"className":32367},[66,67],[15,32369,32371],{"className":32370},[198],[15,32372,32374],{"className":32373},[202],[15,32375,32377],{"className":32376},[206],[15,32378,32380],{"className":32379,"style":6740},[210],[15,32381,32382,32385],{"style":213},[15,32383],{"className":32384,"style":218},[217],[15,32386,32388],{"className":32387},[222,223,224,225],[15,32389,1263],{"className":32390},[66,225],[15,32392],{"className":32393,"style":72},[71],[15,32395,1245],{"className":32396},[76],[15,32398],{"className":32399,"style":72},[71],[15,32401,32403,32406,32435],{"className":32402},[57],[15,32404],{"className":32405,"style":9154},[61],[15,32407,32409,32412],{"className":32408},[66],[15,32410,28858],{"className":32411},[66,67],[15,32413,32415],{"className":32414},[198],[15,32416,32418],{"className":32417},[202],[15,32419,32421],{"className":32420},[206],[15,32422,32424],{"className":32423,"style":6740},[210],[15,32425,32426,32429],{"style":213},[15,32427],{"className":32428,"style":218},[217],[15,32430,32432],{"className":32431},[222,223,224,225],[15,32433,8678],{"className":32434},[66,225],[15,32436,126],{"className":32437},[169],[3194,32439,32440,32443,32444,32473,32474,32502,32503,3371],{},[2040,32441,32442],{},"随机傅里叶特征","：用 ",[15,32445,32447,32461],{"className":32446,"translate":19},[18],[15,32448,32450],{"className":32449},[23],[25,32451,32452],{"xmlns":27},[29,32453,32454,32459],{},[32,32455,32456],{},[35,32457,32458],{},"D",[45,32460,32458],{"encoding":47},[15,32462,32464],{"className":32463,"ariaHidden":53},[52],[15,32465,32467,32470],{"className":32466},[57],[15,32468],{"className":32469,"style":490},[61],[15,32471,32458],{"className":32472,"style":23025},[66,67]," 个随机特征近似平移不变核，把核方法转成 ",[15,32475,32477,32490],{"className":32476,"translate":19},[18],[15,32478,32480],{"className":32479},[23],[25,32481,32482],{"xmlns":27},[29,32483,32484,32488],{},[32,32485,32486],{},[35,32487,32458],{},[45,32489,32458],{"encoding":47},[15,32491,32493],{"className":32492,"ariaHidden":53},[52],[15,32494,32496,32499],{"className":32495},[57],[15,32497],{"className":32498,"style":490},[61],[15,32500,32458],{"className":32501,"style":23025},[66,67]," 维的线性方法，代价约为 ",[15,32504,32506,32540],{"className":32505,"translate":19},[18],[15,32507,32509],{"className":32508},[23],[25,32510,32511],{"xmlns":27},[29,32512,32513,32537],{},[32,32514,32515,32517,32519,32521,32527,32529,32535],{},[35,32516,22999],{},[39,32518,120],{"stretchy":119},[35,32520,37],{},[131,32522,32523,32525],{},[35,32524,32458],{},[1209,32526,1263],{},[39,32528,1245],{},[131,32530,32531,32533],{},[35,32532,32458],{},[1209,32534,8678],{},[39,32536,126],{"stretchy":119},[45,32538,32539],{"encoding":47},"O(nD^2+D^3)",[15,32541,32543,32596],{"className":32542,"ariaHidden":53},[52],[15,32544,32546,32549,32552,32555,32558,32587,32590,32593],{"className":32545},[57],[15,32547],{"className":32548,"style":9154},[61],[15,32550,22999],{"className":32551,"style":23025},[66,67],[15,32553,120],{"className":32554},[162],[15,32556,37],{"className":32557},[66,67],[15,32559,32561,32564],{"className":32560},[66],[15,32562,32458],{"className":32563,"style":23025},[66,67],[15,32565,32567],{"className":32566},[198],[15,32568,32570],{"className":32569},[202],[15,32571,32573],{"className":32572},[206],[15,32574,32576],{"className":32575,"style":6740},[210],[15,32577,32578,32581],{"style":213},[15,32579],{"className":32580,"style":218},[217],[15,32582,32584],{"className":32583},[222,223,224,225],[15,32585,1263],{"className":32586},[66,225],[15,32588],{"className":32589,"style":72},[71],[15,32591,1245],{"className":32592},[76],[15,32594],{"className":32595,"style":72},[71],[15,32597,32599,32602,32631],{"className":32598},[57],[15,32600],{"className":32601,"style":9154},[61],[15,32603,32605,32608],{"className":32604},[66],[15,32606,32458],{"className":32607,"style":23025},[66,67],[15,32609,32611],{"className":32610},[198],[15,32612,32614],{"className":32613},[202],[15,32615,32617],{"className":32616},[206],[15,32618,32620],{"className":32619,"style":6740},[210],[15,32621,32622,32625],{"style":213},[15,32623],{"className":32624,"style":218},[217],[15,32626,32628],{"className":32627},[222,223,224,225],[15,32629,8678],{"className":32630},[66,225],[15,32632,126],{"className":32633},[169],[3194,32635,32636,32639,32640,32690],{},[2040,32637,32638],{},"原问题方法","：对线性核或可显式低维特征化的核，直接在原空间优化，避免 ",[15,32641,32643,32660],{"className":32642,"translate":19},[18],[15,32644,32646],{"className":32645},[23],[25,32647,32648],{"xmlns":27},[29,32649,32650,32658],{},[32,32651,32652,32654,32656],{},[35,32653,37],{},[39,32655,41],{},[35,32657,37],{},[45,32659,48],{"encoding":47},[15,32661,32663,32681],{"className":32662,"ariaHidden":53},[52],[15,32664,32666,32669,32672,32675,32678],{"className":32665},[57],[15,32667],{"className":32668,"style":62},[61],[15,32670,37],{"className":32671},[66,67],[15,32673],{"className":32674,"style":72},[71],[15,32676,41],{"className":32677},[76],[15,32679],{"className":32680,"style":72},[71],[15,32682,32684,32687],{"className":32683},[57],[15,32685],{"className":32686,"style":86},[61],[15,32688,37],{"className":32689},[66,67]," 矩阵；",[3194,32692,32693,32696],{},[2040,32694,32695],{},"子集选择 \u002F 预算方法","：只保留一部分训练点作为核中心，控制模型规模。",[11,32698,32699,32700,32728],{},"这些方法改变的是「用多少个核函数」或「如何近似核矩阵」，而不是表示定理本身。表示定理保证精确解在 ",[15,32701,32703,32716],{"className":32702,"translate":19},[18],[15,32704,32706],{"className":32705},[23],[25,32707,32708],{"xmlns":27},[29,32709,32710,32714],{},[32,32711,32712],{},[35,32713,37],{},[45,32715,37],{"encoding":47},[15,32717,32719],{"className":32718,"ariaHidden":53},[52],[15,32720,32722,32725],{"className":32721},[57],[15,32723],{"className":32724,"style":86},[61],[15,32726,37],{"className":32727},[66,67]," 维子空间里；近似方法则用低秩矩阵或低维特征来替代这个子空间的计算。",[95,32730,32732],{"id":32731},"七表示定理与深度学习","七、表示定理与深度学习",[11,32734,32735,32736,32739],{},"表示定理并没有随着深度学习的兴起而过时。相反，它在",[2040,32737,32738],{},"神经正切核","（Neural Tangent Kernel, NTK）的框架里重新出现。",[11,32741,32742],{},"考虑一个无限宽的神经网络，用梯度下降在平方损失上训练。在适当的参数化和训练体制下，网络函数会收敛到核回归的解：",[15,32744,32746],{"className":32745,"translate":19},[294],[15,32747,32749,32828],{"className":32748,"translate":19},[18],[15,32750,32752],{"className":32751},[23],[25,32753,32754],{"xmlns":27,"display":303},[29,32755,32756,32825],{},[32,32757,32758,32760,32762,32764,32766,32768,32781,32783,32785,32787,32789,32791,32793,32805,32807,32809,32811,32821,32823],{},[35,32759,116],{},[39,32761,120],{"stretchy":119},[35,32763,123],{},[39,32765,126],{"stretchy":119},[39,32767,129],{},[1057,32769,32770,32772],{},[35,32771,621],{},[32,32773,32774,32776,32779],{},[35,32775,28535],{"mathvariant":138},[35,32777,32778],{"mathvariant":138},"T",[35,32780,2853],{"mathvariant":138},[39,32782,120],{"stretchy":119},[35,32784,123],{},[39,32786,335],{"separator":53},[35,32788,446],{},[39,32790,126],{"stretchy":119},[39,32792,120],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[1057,32794,32795,32797],{},[35,32796,2853],{},[32,32798,32799,32801,32803],{},[35,32800,28535],{"mathvariant":138},[35,32802,32778],{"mathvariant":138},[35,32804,2853],{"mathvariant":138},[39,32806,1245],{},[35,32808,1248],{},[35,32810,16427],{},[131,32812,32813,32815],{},[39,32814,126],{"fence":53,"stretchy":53,"minsize":1219,"maxsize":1219},[32,32816,32817,32819],{},[39,32818,6602],{},[1209,32820,1211],{},[35,32822,1224],{},[39,32824,335],{"separator":53},[45,32826,32827],{"encoding":47},"f(x)=k_{\\mathrm{NTK}}(x,X)\\bigl(K_{\\mathrm{NTK}}+\\lambda I\\bigr)^{-1}y,",[15,32829,32831,32858,32991],{"className":32830,"ariaHidden":53},[52],[15,32832,32834,32837,32840,32843,32846,32849,32852,32855],{"className":32833},[57],[15,32835],{"className":32836,"style":154},[61],[15,32838,116],{"className":32839,"style":158},[66,67],[15,32841,120],{"className":32842},[162],[15,32844,123],{"className":32845},[66,67],[15,32847,126],{"className":32848},[169],[15,32850],{"className":32851,"style":173},[71],[15,32853,129],{"className":32854},[177],[15,32856],{"className":32857,"style":173},[71],[15,32859,32861,32864,32912,32915,32918,32921,32924,32927,32930,32936,32982,32985,32988],{"className":32860},[57],[15,32862],{"className":32863,"style":4400},[61],[15,32865,32867,32870],{"className":32866},[66],[15,32868,621],{"className":32869,"style":684},[66,67],[15,32871,32873],{"className":32872},[198],[15,32874,32876,32904],{"className":32875},[202,1090],[15,32877,32879,32901],{"className":32878},[206],[15,32880,32882],{"className":32881,"style":3638},[210],[15,32883,32885,32888],{"style":32884},"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;",[15,32886],{"className":32887,"style":218},[217],[15,32889,32891],{"className":32890},[222,223,224,225],[15,32892,32894],{"className":32893},[66,225],[15,32895,32897],{"className":32896},[66,225],[15,32898,32900],{"className":32899},[66,4429,225],"NTK",[15,32902,1114],{"className":32903},[1113],[15,32905,32907],{"className":32906},[206],[15,32908,32910],{"className":32909,"style":1121},[210],[15,32911],{},[15,32913,120],{"className":32914},[162],[15,32916,123],{"className":32917},[66,67],[15,32919,335],{"className":32920},[422],[15,32922],{"className":32923,"style":697},[71],[15,32925,446],{"className":32926,"style":16497},[66,67],[15,32928,126],{"className":32929},[169],[15,32931,32933],{"className":32932},[162],[15,32934,120],{"className":32935},[1422,1508],[15,32937,32939,32942],{"className":32938},[66],[15,32940,2853],{"className":32941,"style":2901},[66,67],[15,32943,32945],{"className":32944},[198],[15,32946,32948,32974],{"className":32947},[202,1090],[15,32949,32951,32971],{"className":32950},[206],[15,32952,32954],{"className":32953,"style":3638},[210],[15,32955,32956,32959],{"style":2916},[15,32957],{"className":32958,"style":218},[217],[15,32960,32962],{"className":32961},[222,223,224,225],[15,32963,32965],{"className":32964},[66,225],[15,32966,32968],{"className":32967},[66,225],[15,32969,32900],{"className":32970},[66,4429,225],[15,32972,1114],{"className":32973},[1113],[15,32975,32977],{"className":32976},[206],[15,32978,32980],{"className":32979,"style":1121},[210],[15,32981],{},[15,32983],{"className":32984,"style":72},[71],[15,32986,1245],{"className":32987},[76],[15,32989],{"className":32990,"style":72},[71],[15,32992,32994,32997,33000,33003,33041,33044],{"className":32993},[57],[15,32995],{"className":32996,"style":30185},[61],[15,32998,1248],{"className":32999},[66,67],[15,33001,16427],{"className":33002,"style":16497},[66,67],[15,33004,33006,33012],{"className":33005},[169],[15,33007,33009],{"className":33008},[169],[15,33010,126],{"className":33011},[1422,1508],[15,33013,33015],{"className":33014},[198],[15,33016,33018],{"className":33017},[202],[15,33019,33021],{"className":33020},[206],[15,33022,33024],{"className":33023,"style":21271},[210],[15,33025,33026,33029],{"style":21274},[15,33027],{"className":33028,"style":218},[217],[15,33030,33032],{"className":33031},[222,223,224,225],[15,33033,33035,33038],{"className":33034},[66,225],[15,33036,6602],{"className":33037},[66,225],[15,33039,1211],{"className":33040},[66,225],[15,33042,1224],{"className":33043,"style":1515},[66,67],[15,33045,335],{"className":33046},[422],[11,33048,425,33049,33131,33132,33214],{},[15,33050,33052,33076],{"className":33051,"translate":19},[18],[15,33053,33055],{"className":33054},[23],[25,33056,33057],{"xmlns":27},[29,33058,33059,33073],{},[32,33060,33061],{},[1057,33062,33063,33065],{},[35,33064,621],{},[32,33066,33067,33069,33071],{},[35,33068,28535],{"mathvariant":138},[35,33070,32778],{"mathvariant":138},[35,33072,2853],{"mathvariant":138},[45,33074,33075],{"encoding":47},"k_{\\mathrm{NTK}}",[15,33077,33079],{"className":33078,"ariaHidden":53},[52],[15,33080,33082,33085],{"className":33081},[57],[15,33083],{"className":33084,"style":3939},[61],[15,33086,33088,33091],{"className":33087},[66],[15,33089,621],{"className":33090,"style":684},[66,67],[15,33092,33094],{"className":33093},[198],[15,33095,33097,33123],{"className":33096},[202,1090],[15,33098,33100,33120],{"className":33099},[206],[15,33101,33103],{"className":33102,"style":3638},[210],[15,33104,33105,33108],{"style":32884},[15,33106],{"className":33107,"style":218},[217],[15,33109,33111],{"className":33110},[222,223,224,225],[15,33112,33114],{"className":33113},[66,225],[15,33115,33117],{"className":33116},[66,225],[15,33118,32900],{"className":33119},[66,4429,225],[15,33121,1114],{"className":33122},[1113],[15,33124,33126],{"className":33125},[206],[15,33127,33129],{"className":33128,"style":1121},[210],[15,33130],{}," 是网络对应的神经正切核，",[15,33133,33135,33159],{"className":33134,"translate":19},[18],[15,33136,33138],{"className":33137},[23],[25,33139,33140],{"xmlns":27},[29,33141,33142,33156],{},[32,33143,33144],{},[1057,33145,33146,33148],{},[35,33147,2853],{},[32,33149,33150,33152,33154],{},[35,33151,28535],{"mathvariant":138},[35,33153,32778],{"mathvariant":138},[35,33155,2853],{"mathvariant":138},[45,33157,33158],{"encoding":47},"K_{\\mathrm{NTK}}",[15,33160,33162],{"className":33161,"ariaHidden":53},[52],[15,33163,33165,33168],{"className":33164},[57],[15,33166],{"className":33167,"style":1076},[61],[15,33169,33171,33174],{"className":33170},[66],[15,33172,2853],{"className":33173,"style":2901},[66,67],[15,33175,33177],{"className":33176},[198],[15,33178,33180,33206],{"className":33179},[202,1090],[15,33181,33183,33203],{"className":33182},[206],[15,33184,33186],{"className":33185,"style":3638},[210],[15,33187,33188,33191],{"style":2916},[15,33189],{"className":33190,"style":218},[217],[15,33192,33194],{"className":33193},[222,223,224,225],[15,33195,33197],{"className":33196},[66,225],[15,33198,33200],{"className":33199},[66,225],[15,33201,32900],{"className":33202},[66,4429,225],[15,33204,1114],{"className":33205},[1113],[15,33207,33209],{"className":33208},[206],[15,33210,33212],{"className":33211,"style":1121},[210],[15,33213],{}," 是它在训练点上的核矩阵。这个式子与式 (3) 完全同构：预测值是训练点上核函数值的线性组合。",[11,33216,33217],{},"这解释了为什么在 NTK 机制里，深度网络的预测可以写成「训练目标的核加权和」。但也要注意它的边界：NTK 描述的是无限宽、线性化、平方损失、梯度流（或小学习率）的极限；它并不是对一般深度网络特征学习的完整解释。表示定理在这里的作用，是给这个极限提供一个清晰的结构语言。",[95,33219,33221],{"id":33220},"九结语","九、结语",[11,33223,33224,33225,33228],{},"表示定理的核心是一句几何话：",[2040,33226,33227],{},"损失函数看不见训练点之外的正交分量，而正则项会惩罚它。"," 因此，最优解没有理由保留这个分量；在严格递增的正则项下，它必然落在训练点核函数张成的子空间里。",[11,33230,33231,33232,33346],{},"这个论证只用了两件事：再生性 ",[15,33233,33235,33277],{"className":33234,"translate":19},[18],[15,33236,33238],{"className":33237},[23],[25,33239,33240],{"xmlns":27},[29,33241,33242,33274],{},[32,33243,33244,33246,33248,33250,33252,33254,33256,33258,33260,33262,33264,33266,33268,33270,33272],{},[35,33245,116],{},[39,33247,120],{"stretchy":119},[35,33249,123],{},[39,33251,126],{"stretchy":119},[39,33253,129],{},[39,33255,642],{"stretchy":119},[35,33257,116],{},[39,33259,335],{"separator":53},[35,33261,621],{},[39,33263,120],{"stretchy":119},[39,33265,3268],{},[39,33267,335],{"separator":53},[35,33269,123],{},[39,33271,126],{"stretchy":119},[39,33273,667],{"stretchy":119},[45,33275,33276],{"encoding":47},"f(x)=\\langle f,k(\\cdot,x)\\rangle",[15,33278,33280,33307],{"className":33279,"ariaHidden":53},[52],[15,33281,33283,33286,33289,33292,33295,33298,33301,33304],{"className":33282},[57],[15,33284],{"className":33285,"style":154},[61],[15,33287,116],{"className":33288,"style":158},[66,67],[15,33290,120],{"className":33291},[162],[15,33293,123],{"className":33294},[66,67],[15,33296,126],{"className":33297},[169],[15,33299],{"className":33300,"style":173},[71],[15,33302,129],{"className":33303},[177],[15,33305],{"className":33306,"style":173},[71],[15,33308,33310,33313,33316,33319,33322,33325,33328,33331,33334,33337,33340,33343],{"className":33309},[57],[15,33311],{"className":33312,"style":154},[61],[15,33314,642],{"className":33315},[162],[15,33317,116],{"className":33318,"style":158},[66,67],[15,33320,335],{"className":33321},[422],[15,33323],{"className":33324,"style":697},[71],[15,33326,621],{"className":33327,"style":684},[66,67],[15,33329,120],{"className":33330},[162],[15,33332,3268],{"className":33333},[66],[15,33335,335],{"className":33336},[422],[15,33338],{"className":33339,"style":697},[71],[15,33341,123],{"className":33342},[66,67],[15,33344,811],{"className":33345},[169],"，以及范数的勾股分解。它不需要损失凸、不需要可微、不需要核矩阵可逆。也正因如此，表示定理能同时覆盖核岭回归、SVM、核逻辑回归、高斯过程和平滑样条。",[11,33348,33349,33350],{},"但它的边界同样清楚：正则项必须是范数的非减函数；若只是非减而非严格递增，定理给出的是「存在一个」这样的解，而不是「每一个」解都如此。非单调的正则项可以让解离开训练点子空间，4.2 节的反例说明这不是技术细节，而是定理的本质。",[2040,33351,33352,33353,33381,33382,33410],{},"无穷维并没有消失，它只是被 ",[15,33354,33356,33369],{"className":33355,"translate":19},[18],[15,33357,33359],{"className":33358},[23],[25,33360,33361],{"xmlns":27},[29,33362,33363,33367],{},[32,33364,33365],{},[35,33366,37],{},[45,33368,37],{"encoding":47},[15,33370,33372],{"className":33371,"ariaHidden":53},[52],[15,33373,33375,33378],{"className":33374},[57],[15,33376],{"className":33377,"style":86},[61],[15,33379,37],{"className":33380},[66,67]," 个训练点压成了 ",[15,33383,33385,33398],{"className":33384,"translate":19},[18],[15,33386,33388],{"className":33387},[23],[25,33389,33390],{"xmlns":27},[29,33391,33392,33396],{},[32,33393,33394],{},[35,33395,37],{},[45,33397,37],{"encoding":47},[15,33399,33401],{"className":33400,"ariaHidden":53},[52],[15,33402,33404,33407],{"className":33403},[57],[15,33405],{"className":33406,"style":86},[61],[15,33408,37],{"className":33409},[66,67]," 维。",{"title":33412,"searchDepth":33413,"depth":33413,"links":33414},"",2,[33415,33416,33422,33428,33433,33441,33442,33443],{"id":97,"depth":33413,"text":98},{"id":1901,"depth":33413,"text":1902,"children":33417},[33418,33420,33421],{"id":1906,"depth":33419,"text":1907},3,{"id":5309,"depth":33419,"text":5310},{"id":6149,"depth":33419,"text":6150},{"id":7245,"depth":33413,"text":7246,"children":33423},[33424,33425,33426,33427],{"id":7249,"depth":33419,"text":7250},{"id":9359,"depth":33419,"text":9360},{"id":13607,"depth":33419,"text":13608},{"id":16122,"depth":33419,"text":16123},{"id":17691,"depth":33413,"text":17692,"children":33429},[33430,33431,33432],{"id":17695,"depth":33419,"text":17696},{"id":18072,"depth":33419,"text":18073},{"id":20150,"depth":33419,"text":20151},{"id":20778,"depth":33413,"text":20779,"children":33434},[33435,33436,33437,33438,33439,33440],{"id":20782,"depth":33419,"text":20783},{"id":23184,"depth":33419,"text":23185},{"id":27017,"depth":33419,"text":27018},{"id":28367,"depth":33419,"text":28368},{"id":29575,"depth":33419,"text":29576},{"id":30580,"depth":33419,"text":30581},{"id":31421,"depth":33413,"text":31422},{"id":32731,"depth":33413,"text":32732},{"id":33220,"depth":33413,"text":33221},"核方法最反直觉的一点是：你明明在一个可能是无限维的函数空间里做优化，最后却只需要解一个 n×nn\\times nn×n 的线性系统或二次规划。把这件事说清楚的，就是表示定理。",false,true,"md",null,{},"\u002Fblog\u002F2025\u002F2025-09-11-representer-theorem",{"title":6,"description":33444},"blog\u002F2025\u002F2025-09-11-representer-theorem","表示定理是核方法的一个基石性结论。它指出：对于一个在再生核希尔伯特空间（RKHS）中带有范数正则化的经验风险最小化问题，其最优解一定可以写成核函数在训练样本点上的线性组合。",[33455],"machine-learning","2025-09-11T21:00:12+08:00","wFRdXtfB52qEpuspNQ6_5BNMWT6yuyp37_5ikKAOpvU",{"id":33459,"title":33460,"body":33461,"description":33412,"draft":33445,"enableComment":33446,"extension":33447,"image":33448,"important":33445,"location":33448,"meta":61573,"navigation":33446,"ogImage":33448,"path":61574,"seo":61575,"stem":61576,"summary":61577,"tags":61578,"time":61580,"__hash__":61581},"blog\u002Fblog\u002F2025\u002F2025-09-18-the-stirling-formula.md","斯特林公式",{"type":8,"value":33462,"toc":61569},[33463,33466,33469,33710,33713,34200,34329,35186,35907,35910,36489,36492,37554,37557,37645,39178,39288,40747,40750,40895,40897,41365,41754,42154,42862,42865,43405,43647,44941,45100,45103,45642,45799,46331,46433,47878,47881,48264,48267,48269,48428,48726,48729,49149,49152,49831,49834,50167,50170,51262,51265,51267,51514,51517,51780,51783,51786,52329,52841,52973,52976,53295,53368,54065,54641,54644,55157,55188,55191,55436,55439,55744,55747,56077,56256,56638,56739,57217,57219,57810,57976,57979,58324,58327,58330,58712,58910,59255,59685,59688,60120,60123,60348,60351,60902,60905,61016,61048,61325,61328],[95,33464,33465],{"id":33465},"沃利斯公式",[11,33467,33468],{},"我们先从一个定积分开始：",[15,33470,33472],{"className":33471,"translate":19},[294],[15,33473,33475,33519],{"className":33474,"translate":19},[18],[15,33476,33478],{"className":33477},[23],[25,33479,33480],{"xmlns":27,"display":303},[29,33481,33482,33516],{},[32,33483,33484,33497,33508,33510,33512,33514],{},[1255,33485,33486,33488,33490],{},[39,33487,29732],{},[1209,33489,1804],{},[6467,33491,33492,33495],{},[35,33493,33494],{},"π",[1209,33496,1263],{},[131,33498,33499,33506],{},[32,33500,33501,33504],{},[35,33502,33503],{},"sin",[39,33505,1170],{},[35,33507,37],{},[35,33509,123],{},[1274,33511,2462],{},[35,33513,458],{},[35,33515,123],{},[45,33517,33518],{"encoding":47},"\\int_{0}^{\\frac{\\pi}{2}} \\sin^n x \\, dx",[15,33520,33522],{"className":33521,"ariaHidden":53},[52],[15,33523,33525,33529,33661,33664,33695,33698,33701,33704,33707],{"className":33524},[57],[15,33526],{"className":33527,"style":33528},[61],"height:2.5089em;vertical-align:-0.9119em;",[15,33530,33532,33535],{"className":33531},[1339],[15,33533,29732],{"className":33534,"style":29796},[1339,1470,1471],[15,33536,33538],{"className":33537},[198],[15,33539,33541,33653],{"className":33540},[202,1090],[15,33542,33544,33650],{"className":33543},[206],[15,33545,33548,33563],{"className":33546,"style":33547},[210],"height:1.597em;",[15,33549,33551,33554],{"style":33550},"top:-2.088em;margin-left:-0.4445em;margin-right:0.05em;",[15,33552],{"className":33553,"style":1368},[217],[15,33555,33557],{"className":33556},[222,223,224,225],[15,33558,33560],{"className":33559},[66,225],[15,33561,1804],{"className":33562},[66,225],[15,33564,33566,33569],{"style":33565},"top:-4.1129em;margin-right:0.05em;",[15,33567],{"className":33568,"style":1368},[217],[15,33570,33572],{"className":33571},[222,223,224,225],[15,33573,33575],{"className":33574},[66,225],[15,33576,33578,33582,33647],{"className":33577},[66,225],[15,33579],{"className":33580},[162,23417,222,3679,33581],"size6",[15,33583,33585],{"className":33584},[6467],[15,33586,33588,33638],{"className":33587},[202,1090],[15,33589,33591,33635],{"className":33590},[206],[15,33592,33595,33610,33620],{"className":33593,"style":33594},[210],"height:0.6915em;",[15,33596,33598,33601],{"style":33597},"top:-2.656em;",[15,33599],{"className":33600,"style":1368},[217],[15,33602,33604],{"className":33603},[222,3679,1508,225],[15,33605,33607],{"className":33606},[66,225],[15,33608,1263],{"className":33609},[66,225],[15,33611,33613,33616],{"style":33612},"top:-3.2255em;",[15,33614],{"className":33615,"style":1368},[217],[15,33617],{"className":33618,"style":33619},[23452,225],"border-bottom-width:0.049em;",[15,33621,33623,33626],{"style":33622},"top:-3.384em;",[15,33624],{"className":33625,"style":1368},[217],[15,33627,33629],{"className":33628},[222,3679,1508,225],[15,33630,33632],{"className":33631},[66,225],[15,33633,33494],{"className":33634,"style":1515},[66,67,225],[15,33636,1114],{"className":33637},[1113],[15,33639,33641],{"className":33640},[206],[15,33642,33645],{"className":33643,"style":33644},[210],"height:0.344em;",[15,33646],{},[15,33648],{"className":33649},[169,23417,222,3679,33581],[15,33651,1114],{"className":33652},[1113],[15,33654,33656],{"className":33655},[206],[15,33657,33659],{"className":33658,"style":29843},[210],[15,33660],{},[15,33662],{"className":33663,"style":697},[71],[15,33665,33667,33670],{"className":33666},[1339],[15,33668,33503],{"className":33669},[1339],[15,33671,33673],{"className":33672},[198],[15,33674,33676],{"className":33675},[202],[15,33677,33679],{"className":33678},[206],[15,33680,33683],{"className":33681,"style":33682},[210],"height:0.7222em;",[15,33684,33686,33689],{"style":33685},"top:-3.1208em;margin-right:0.05em;",[15,33687],{"className":33688,"style":218},[217],[15,33690,33692],{"className":33691},[222,223,224,225],[15,33693,37],{"className":33694},[66,67,225],[15,33696],{"className":33697,"style":697},[71],[15,33699,123],{"className":33700},[66,67],[15,33702],{"className":33703,"style":697},[71],[15,33705,458],{"className":33706},[66,67],[15,33708,123],{"className":33709},[66,67],[11,33711,33712],{},"我们可以推导出如下结果：",[15,33714,33716],{"className":33715,"translate":19},[294],[15,33717,33719,33810],{"className":33718,"translate":19},[18],[15,33720,33722],{"className":33721},[23],[25,33723,33724],{"xmlns":27,"display":303},[29,33725,33726,33807],{},[32,33727,33728,33740,33750,33752,33754,33756,33758,33760,33762,33774,33790,33792,33794,33796,33798,33801,33803,33805],{},[1255,33729,33730,33732,33734],{},[39,33731,29732],{},[1209,33733,1804],{},[6467,33735,33736,33738],{},[35,33737,33494],{},[1209,33739,1263],{},[131,33741,33742,33748],{},[32,33743,33744,33746],{},[35,33745,33503],{},[39,33747,1170],{},[35,33749,37],{},[35,33751,123],{},[1274,33753,2462],{},[35,33755,458],{},[35,33757,123],{},[39,33759,129],{},[39,33761,6602],{},[1255,33763,33764,33766,33768],{},[39,33765,29732],{},[1209,33767,1804],{},[6467,33769,33770,33772],{},[35,33771,33494],{},[1209,33773,1263],{},[131,33775,33776,33782],{},[32,33777,33778,33780],{},[35,33779,33503],{},[39,33781,1170],{},[32,33783,33784,33786,33788],{},[35,33785,37],{},[39,33787,6602],{},[1209,33789,1211],{},[35,33791,123],{},[1274,33793,2462],{},[35,33795,458],{},[39,33797,120],{"stretchy":119},[35,33799,33800],{},"cos",[39,33802,1170],{},[35,33804,123],{},[39,33806,126],{"stretchy":119},[45,33808,33809],{"encoding":47},"\\int_{0}^{\\frac{\\pi}{2}} \\sin^n x \\, dx = -\\int_{0}^{\\frac{\\pi}{2}} \\sin^{n-1} x \\, d(\\cos x)",[15,33811,33813,33997],{"className":33812,"ariaHidden":53},[52],[15,33814,33816,33819,33941,33944,33973,33976,33979,33982,33985,33988,33991,33994],{"className":33815},[57],[15,33817],{"className":33818,"style":33528},[61],[15,33820,33822,33825],{"className":33821},[1339],[15,33823,29732],{"className":33824,"style":29796},[1339,1470,1471],[15,33826,33828],{"className":33827},[198],[15,33829,33831,33933],{"className":33830},[202,1090],[15,33832,33834,33930],{"className":33833},[206],[15,33835,33837,33851],{"className":33836,"style":33547},[210],[15,33838,33839,33842],{"style":33550},[15,33840],{"className":33841,"style":1368},[217],[15,33843,33845],{"className":33844},[222,223,224,225],[15,33846,33848],{"className":33847},[66,225],[15,33849,1804],{"className":33850},[66,225],[15,33852,33853,33856],{"style":33565},[15,33854],{"className":33855,"style":1368},[217],[15,33857,33859],{"className":33858},[222,223,224,225],[15,33860,33862],{"className":33861},[66,225],[15,33863,33865,33868,33927],{"className":33864},[66,225],[15,33866],{"className":33867},[162,23417,222,3679,33581],[15,33869,33871],{"className":33870},[6467],[15,33872,33874,33919],{"className":33873},[202,1090],[15,33875,33877,33916],{"className":33876},[206],[15,33878,33880,33894,33902],{"className":33879,"style":33594},[210],[15,33881,33882,33885],{"style":33597},[15,33883],{"className":33884,"style":1368},[217],[15,33886,33888],{"className":33887},[222,3679,1508,225],[15,33889,33891],{"className":33890},[66,225],[15,33892,1263],{"className":33893},[66,225],[15,33895,33896,33899],{"style":33612},[15,33897],{"className":33898,"style":1368},[217],[15,33900],{"className":33901,"style":33619},[23452,225],[15,33903,33904,33907],{"style":33622},[15,33905],{"className":33906,"style":1368},[217],[15,33908,33910],{"className":33909},[222,3679,1508,225],[15,33911,33913],{"className":33912},[66,225],[15,33914,33494],{"className":33915,"style":1515},[66,67,225],[15,33917,1114],{"className":33918},[1113],[15,33920,33922],{"className":33921},[206],[15,33923,33925],{"className":33924,"style":33644},[210],[15,33926],{},[15,33928],{"className":33929},[169,23417,222,3679,33581],[15,33931,1114],{"className":33932},[1113],[15,33934,33936],{"className":33935},[206],[15,33937,33939],{"className":33938,"style":29843},[210],[15,33940],{},[15,33942],{"className":33943,"style":697},[71],[15,33945,33947,33950],{"className":33946},[1339],[15,33948,33503],{"className":33949},[1339],[15,33951,33953],{"className":33952},[198],[15,33954,33956],{"className":33955},[202],[15,33957,33959],{"className":33958},[206],[15,33960,33962],{"className":33961,"style":33682},[210],[15,33963,33964,33967],{"style":33685},[15,33965],{"className":33966,"style":218},[217],[15,33968,33970],{"className":33969},[222,223,224,225],[15,33971,37],{"className":33972},[66,67,225],[15,33974],{"className":33975,"style":697},[71],[15,33977,123],{"className":33978},[66,67],[15,33980],{"className":33981,"style":697},[71],[15,33983,458],{"className":33984},[66,67],[15,33986,123],{"className":33987},[66,67],[15,33989],{"className":33990,"style":173},[71],[15,33992,129],{"className":33993},[177],[15,33995],{"className":33996,"style":173},[71],[15,33998,34000,34003,34006,34009,34131,34134,34173,34176,34179,34182,34185,34188,34191,34194,34197],{"className":33999},[57],[15,34001],{"className":34002,"style":33528},[61],[15,34004,6602],{"className":34005},[66],[15,34007],{"className":34008,"style":697},[71],[15,34010,34012,34015],{"className":34011},[1339],[15,34013,29732],{"className":34014,"style":29796},[1339,1470,1471],[15,34016,34018],{"className":34017},[198],[15,34019,34021,34123],{"className":34020},[202,1090],[15,34022,34024,34120],{"className":34023},[206],[15,34025,34027,34041],{"className":34026,"style":33547},[210],[15,34028,34029,34032],{"style":33550},[15,34030],{"className":34031,"style":1368},[217],[15,34033,34035],{"className":34034},[222,223,224,225],[15,34036,34038],{"className":34037},[66,225],[15,34039,1804],{"className":34040},[66,225],[15,34042,34043,34046],{"style":33565},[15,34044],{"className":34045,"style":1368},[217],[15,34047,34049],{"className":34048},[222,223,224,225],[15,34050,34052],{"className":34051},[66,225],[15,34053,34055,34058,34117],{"className":34054},[66,225],[15,34056],{"className":34057},[162,23417,222,3679,33581],[15,34059,34061],{"className":34060},[6467],[15,34062,34064,34109],{"className":34063},[202,1090],[15,34065,34067,34106],{"className":34066},[206],[15,34068,34070,34084,34092],{"className":34069,"style":33594},[210],[15,34071,34072,34075],{"style":33597},[15,34073],{"className":34074,"style":1368},[217],[15,34076,34078],{"className":34077},[222,3679,1508,225],[15,34079,34081],{"className":34080},[66,225],[15,34082,1263],{"className":34083},[66,225],[15,34085,34086,34089],{"style":33612},[15,34087],{"className":34088,"style":1368},[217],[15,34090],{"className":34091,"style":33619},[23452,225],[15,34093,34094,34097],{"style":33622},[15,34095],{"className":34096,"style":1368},[217],[15,34098,34100],{"className":34099},[222,3679,1508,225],[15,34101,34103],{"className":34102},[66,225],[15,34104,33494],{"className":34105,"style":1515},[66,67,225],[15,34107,1114],{"className":34108},[1113],[15,34110,34112],{"className":34111},[206],[15,34113,34115],{"className":34114,"style":33644},[210],[15,34116],{},[15,34118],{"className":34119},[169,23417,222,3679,33581],[15,34121,1114],{"className":34122},[1113],[15,34124,34126],{"className":34125},[206],[15,34127,34129],{"className":34128,"style":29843},[210],[15,34130],{},[15,34132],{"className":34133,"style":697},[71],[15,34135,34137,34140],{"className":34136},[1339],[15,34138,33503],{"className":34139},[1339],[15,34141,34143],{"className":34142},[198],[15,34144,34146],{"className":34145},[202],[15,34147,34149],{"className":34148},[206],[15,34150,34153],{"className":34151,"style":34152},[210],"height:0.8719em;",[15,34154,34155,34158],{"style":33685},[15,34156],{"className":34157,"style":218},[217],[15,34159,34161],{"className":34160},[222,223,224,225],[15,34162,34164,34167,34170],{"className":34163},[66,225],[15,34165,37],{"className":34166},[66,67,225],[15,34168,6602],{"className":34169},[76,225],[15,34171,1211],{"className":34172},[66,225],[15,34174],{"className":34175,"style":697},[71],[15,34177,123],{"className":34178},[66,67],[15,34180],{"className":34181,"style":697},[71],[15,34183,458],{"className":34184},[66,67],[15,34186,120],{"className":34187},[162],[15,34189,33800],{"className":34190},[1339],[15,34192],{"className":34193,"style":697},[71],[15,34195,123],{"className":34196},[66,67],[15,34198,126],{"className":34199},[169],[11,34201,34202,34203,34328],{},"利用分部积分法，其中 ",[15,34204,34206,34248],{"className":34205,"translate":19},[18],[15,34207,34209],{"className":34208},[23],[25,34210,34211],{"xmlns":27},[29,34212,34213,34245],{},[32,34214,34215,34217,34220,34222,34224,34227,34229,34231,34233,34235,34237,34239,34241,34243],{},[39,34216,29732],{},[35,34218,34219],{},"u",[1274,34221,2462],{},[35,34223,458],{},[35,34225,34226],{},"v",[39,34228,129],{},[35,34230,34219],{},[35,34232,34226],{},[39,34234,6602],{},[39,34236,29732],{},[35,34238,34226],{},[1274,34240,2462],{},[35,34242,458],{},[35,34244,34219],{},[45,34246,34247],{"encoding":47},"\\int u \\, dv = uv - \\int v \\, du",[15,34249,34251,34285,34304],{"className":34250,"ariaHidden":53},[52],[15,34252,34254,34258,34261,34264,34267,34270,34273,34276,34279,34282],{"className":34253},[57],[15,34255],{"className":34256,"style":34257},[61],"height:1.1111em;vertical-align:-0.3061em;",[15,34259,29732],{"className":34260,"style":31343},[1339,1470,13701],[15,34262],{"className":34263,"style":697},[71],[15,34265,34219],{"className":34266},[66,67],[15,34268],{"className":34269,"style":697},[71],[15,34271,458],{"className":34272},[66,67],[15,34274,34226],{"className":34275,"style":1515},[66,67],[15,34277],{"className":34278,"style":173},[71],[15,34280,129],{"className":34281},[177],[15,34283],{"className":34284,"style":173},[71],[15,34286,34288,34291,34295,34298,34301],{"className":34287},[57],[15,34289],{"className":34290,"style":62},[61],[15,34292,34294],{"className":34293,"style":1515},[66,67],"uv",[15,34296],{"className":34297,"style":72},[71],[15,34299,6602],{"className":34300},[76],[15,34302],{"className":34303,"style":72},[71],[15,34305,34307,34310,34313,34316,34319,34322,34325],{"className":34306},[57],[15,34308],{"className":34309,"style":34257},[61],[15,34311,29732],{"className":34312,"style":31343},[1339,1470,13701],[15,34314],{"className":34315,"style":697},[71],[15,34317,34226],{"className":34318,"style":1515},[66,67],[15,34320],{"className":34321,"style":697},[71],[15,34323,458],{"className":34324},[66,67],[15,34326,34219],{"className":34327},[66,67],"，可得：",[15,34330,34332],{"className":34331,"translate":19},[294],[15,34333,34335,34490],{"className":34334,"translate":19},[18],[15,34336,34338],{"className":34337},[23],[25,34339,34340],{"xmlns":27,"display":303},[29,34341,34342,34487],{},[32,34343,34344,34346,34358,34374,34376,34378,34380,34382,34384,34386,34388,34390,34392,34394,34396,34426,34439,34441,34453,34455,34457,34459,34461,34463],{},[39,34345,6602],{},[1255,34347,34348,34350,34352],{},[39,34349,29732],{},[1209,34351,1804],{},[6467,34353,34354,34356],{},[35,34355,33494],{},[1209,34357,1263],{},[131,34359,34360,34366],{},[32,34361,34362,34364],{},[35,34363,33503],{},[39,34365,1170],{},[32,34367,34368,34370,34372],{},[35,34369,37],{},[39,34371,6602],{},[1209,34373,1211],{},[35,34375,123],{},[1274,34377,2462],{},[35,34379,458],{},[39,34381,120],{"stretchy":119},[35,34383,33800],{},[39,34385,1170],{},[35,34387,123],{},[39,34389,126],{"stretchy":119},[71,34391],{"linebreak":10456},[39,34393,129],{},[39,34395,6602],{},[32,34397,34398,34400,34416,34418,34420,34422,34424],{},[39,34399,120],{"fence":53},[131,34401,34402,34408],{},[32,34403,34404,34406],{},[35,34405,33503],{},[39,34407,1170],{},[32,34409,34410,34412,34414],{},[35,34411,37],{},[39,34413,6602],{},[1209,34415,1211],{},[35,34417,123],{},[35,34419,33800],{},[39,34421,1170],{},[35,34423,123],{},[39,34425,126],{"fence":53},[1255,34427,34428,34431,34433],{},[39,34429,34430],{"fence":119,"stretchy":53,"minsize":4545,"maxsize":4545},"∣",[1209,34432,1804],{},[6467,34434,34435,34437],{},[35,34436,33494],{},[1209,34438,1263],{},[39,34440,1245],{},[1255,34442,34443,34445,34447],{},[39,34444,29732],{},[1209,34446,1804],{},[6467,34448,34449,34451],{},[35,34450,33494],{},[1209,34452,1263],{},[35,34454,33800],{},[39,34456,1170],{},[35,34458,123],{},[1274,34460,2462],{},[35,34462,458],{},[32,34464,34465,34467,34483,34485],{},[39,34466,120],{"fence":53},[131,34468,34469,34475],{},[32,34470,34471,34473],{},[35,34472,33503],{},[39,34474,1170],{},[32,34476,34477,34479,34481],{},[35,34478,37],{},[39,34480,6602],{},[1209,34482,1211],{},[35,34484,123],{},[39,34486,126],{"fence":53},[45,34488,34489],{"encoding":47},"-\\int_{0}^{\\frac{\\pi}{2}} \\sin^{n-1} x \\, d(\\cos x) \\\\ = -\\left( \\sin^{n-1} x \\cos x \\right) \\Big|_{0}^{\\frac{\\pi}{2}} + \\int_{0}^{\\frac{\\pi}{2}} \\cos x \\, d\\left(\\sin^{n-1} x 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\\sin^n x \\, dx = (n-1) \\int_{0}^{\\frac{\\pi}{2}} \\left( \\sin^{n-2} x - \\sin^n x \\right) \\, 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\\int_{0}^{\\frac{\\pi}{2}} \\sin^n x \\, dx = (n-1) \\int_{0}^{\\frac{\\pi}{2}} \\sin^{n-2} x \\, dx \\\\ \\implies \\int_{0}^{\\frac{\\pi}{2}} \\sin^n x \\, dx = \\frac{n-1}{n} \\int_{0}^{\\frac{\\pi}{2}} \\sin^{n-2} x \\, dx",[15,36681,36683,36873,36894,37087,37090,37109,37293],{"className":36682,"ariaHidden":53},[52],[15,36684,36686,36689,36692,36695,36817,36820,36849,36852,36855,36858,36861,36864,36867,36870],{"className":36685},[57],[15,36687],{"className":36688,"style":33528},[61],[15,36690,37],{"className":36691},[66,67],[15,36693],{"className":36694,"style":697},[71],[15,36696,36698,36701],{"className":36697},[1339],[15,36699,29732],{"className":36700,"style":29796},[1339,1470,1471],[15,36702,36704],{"className":36703},[198],[15,36705,36707,36809],{"className":36706},[202,1090],[15,36708,36710,36806],{"className":36709},[206],[15,36711,36713,36727],{"className":36712,"style":33547},[210],[15,36714,36715,36718],{"style":33550},[15,36716],{"className":36717,"style":1368},[217],[15,36719,36721],{"className":36720},[222,223,224,225],[15,36722,36724],{"className":36723},[66,225],[15,36725,1804],{"className":36726},[66,225],[15,36728,36729,36732],{"style":33565},[15,36730],{"className":36731,"style":1368},[217],[15,36733,36735],{"className":36734},[222,223,224,225],[15,36736,36738],{"className":36737},[66,225],[15,36739,36741,36744,36803],{"className":36740},[66,225],[15,36742],{"className":36743},[162,23417,222,3679,33581],[15,36745,36747],{"className":36746},[6467],[15,36748,36750,36795],{"className":36749},[202,1090],[15,36751,36753,36792],{"className":36752},[206],[15,36754,36756,36770,36778],{"className":36755,"style":33594},[210],[15,36757,36758,36761],{"style":33597},[15,36759],{"className":36760,"style":1368},[217],[15,36762,36764],{"className":36763},[222,3679,1508,225],[15,36765,36767],{"className":36766},[66,225],[15,36768,1263],{"className":36769},[66,225],[15,36771,36772,36775],{"style":33612},[15,36773],{"className":36774,"style":1368},[217],[15,36776],{"className":36777,"style":33619},[23452,225],[15,36779,36780,36783],{"style":33622},[15,36781],{"className":36782,"style":1368},[217],[15,36784,36786],{"className":36785},[222,3679,1508,225],[15,36787,36789],{"className":36788},[66,225],[15,36790,33494],{"className":36791,"style":1515},[66,67,225],[15,36793,1114],{"className":36794},[1113],[15,36796,36798],{"className":36797},[206],[15,36799,36801],{"className":36800,"style":33644},[210],[15,36802],{},[15,36804],{"className":36805},[169,23417,222,3679,33581],[15,36807,1114],{"className":36808},[1113],[15,36810,36812],{"className":36811},[206],[15,36813,36815],{"className":36814,"style":29843},[210],[15,36816],{},[15,36818],{"className":36819,"style":697},[71],[15,36821,36823,36826],{"className":36822},[1339],[15,36824,33503],{"className":36825},[1339],[15,36827,36829],{"className":36828},[198],[15,36830,36832],{"className":36831},[202],[15,36833,36835],{"className":36834},[206],[15,36836,36838],{"className":36837,"style":33682},[210],[15,36839,36840,36843],{"style":33685},[15,36841],{"className":36842,"style":218},[217],[15,36844,36846],{"className":36845},[222,223,224,225],[15,36847,37],{"className":36848},[66,67,225],[15,36850],{"className":36851,"style":697},[71],[15,36853,123],{"className":36854},[66,67],[15,36856],{"className":36857,"style":697},[71],[15,36859,458],{"className":36860},[66,67],[15,36862,123],{"className":36863},[66,67],[15,36865],{"className":36866,"style":173},[71],[15,36868,129],{"className":36869},[177],[15,36871],{"className":36872,"style":173},[71],[15,36874,36876,36879,36882,36885,36888,36891],{"className":36875},[57],[15,36877],{"className":36878,"style":154},[61],[15,36880,120],{"className":36881},[162],[15,36883,37],{"className":36884},[66,67],[15,36886],{"className":36887,"style":72},[71],[15,36889,6602],{"className":36890},[76],[15,36892],{"className":36893,"style":72},[71],[15,36895,36897,36900,36903,36906,36909,37031,37034,37072,37075,37078,37081,37084],{"className":36896},[57],[15,36898],{"className":36899,"style":33528},[61],[15,36901,1211],{"className":36902},[66],[15,36904,126],{"className":36905},[169],[15,36907],{"className":36908,"style":697},[71],[15,36910,36912,36915],{"className":36911},[1339],[15,36913,29732],{"className":36914,"style":29796},[1339,1470,1471],[15,36916,36918],{"className":36917},[198],[15,36919,36921,37023],{"className":36920},[202,1090],[15,36922,36924,37020],{"className":36923},[206],[15,36925,36927,36941],{"className":36926,"style":33547},[210],[15,36928,36929,36932],{"style":33550},[15,36930],{"className":36931,"style":1368},[217],[15,36933,36935],{"className":36934},[222,223,224,225],[15,36936,36938],{"className":36937},[66,225],[15,36939,1804],{"className":36940},[66,225],[15,36942,36943,36946],{"style":33565},[15,36944],{"className":36945,"style":1368},[217],[15,36947,36949],{"className":36948},[222,223,224,225],[15,36950,36952],{"className":36951},[66,225],[15,36953,36955,36958,37017],{"className":36954},[66,225],[15,36956],{"className":36957},[162,23417,222,3679,33581],[15,36959,36961],{"className":36960},[6467],[15,36962,36964,37009],{"className":36963},[202,1090],[15,36965,36967,37006],{"className":36966},[206],[15,36968,36970,36984,36992],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为偶数时，即 ",[15,37589,37591,37611],{"className":37590,"translate":19},[18],[15,37592,37594],{"className":37593},[23],[25,37595,37596],{"xmlns":27},[29,37597,37598,37608],{},[32,37599,37600,37602,37604,37606],{},[35,37601,37],{},[39,37603,129],{},[1209,37605,1263],{},[35,37607,621],{},[45,37609,37610],{"encoding":47},"n = 2k",[15,37612,37614,37632],{"className":37613,"ariaHidden":53},[52],[15,37615,37617,37620,37623,37626,37629],{"className":37616},[57],[15,37618],{"className":37619,"style":86},[61],[15,37621,37],{"className":37622},[66,67],[15,37624],{"className":37625,"style":173},[71],[15,37627,129],{"className":37628},[177],[15,37630],{"className":37631,"style":173},[71],[15,37633,37635,37638,37641],{"className":37634},[57],[15,37636],{"className":37637,"style":840},[61],[15,37639,1263],{"className":37640},[66],[15,37642,621],{"className":37643,"style":684},[66,67],"：",[15,37646,37648],{"className":37647,"translate":19},[294],[15,37649,37651,37902],{"className":37650,"translate":19},[18],[15,37652,37654],{"className":37653},[23],[25,37655,37656],{"xmlns":27,"display":303},[29,37657,37658,37899],{},[32,37659,37660,37672,37686,37688,37690,37692,37694,37696,37714,37726,37744,37746,37748,37750,37752,37754,37756,37774,37776,37798,37801,37807,37819,37829,37831,37833,37835,37837,37839,37841,37859,37861,37883,37885,37891,37893],{},[1255,37661,37662,37664,37666],{},[39,37663,29732],{},[1209,37665,1804],{},[6467,37667,37668,37670],{},[35,37669,33494],{},[1209,37671,1263],{},[131,37673,37674,37680],{},[32,37675,37676,37678],{},[35,37677,33503],{},[39,37679,1170],{},[32,37681,37682,37684],{},[1209,37683,1263],{},[35,37685,621],{},[35,37687,123],{},[1274,37689,2462],{},[35,37691,458],{},[35,37693,123],{},[39,37695,129],{},[6467,37697,37698,37708],{},[32,37699,37700,37702,37704,37706],{},[1209,37701,1263],{},[35,37703,621],{},[39,37705,6602],{},[1209,37707,1211],{},[32,37709,37710,37712],{},[1209,37711,1263],{},[35,37713,621],{},[1255,37715,37716,37718,37720],{},[39,37717,29732],{},[1209,37719,1804],{},[6467,37721,37722,37724],{},[35,37723,33494],{},[1209,37725,1263],{},[131,37727,37728,37734],{},[32,37729,37730,37732],{},[35,37731,33503],{},[39,37733,1170],{},[32,37735,37736,37738,37740,37742],{},[1209,37737,1263],{},[35,37739,621],{},[39,37741,6602],{},[1209,37743,1263],{},[35,37745,123],{},[1274,37747,2462],{},[35,37749,458],{},[35,37751,123],{},[71,37753],{"linebreak":10456},[39,37755,129],{},[6467,37757,37758,37768],{},[32,37759,37760,37762,37764,37766],{},[1209,37761,1263],{},[35,37763,621],{},[39,37765,6602],{},[1209,37767,1211],{},[32,37769,37770,37772],{},[1209,37771,1263],{},[35,37773,621],{},[39,37775,3268],{},[6467,37777,37778,37788],{},[32,37779,37780,37782,37784,37786],{},[1209,37781,1263],{},[35,37783,621],{},[39,37785,6602],{},[1209,37787,8678],{},[32,37789,37790,37792,37794,37796],{},[1209,37791,1263],{},[35,37793,621],{},[39,37795,6602],{},[1209,37797,1263],{},[39,37799,37800],{},"⋯",[6467,37802,37803,37805],{},[1209,37804,1211],{},[1209,37806,1263],{},[1255,37808,37809,37811,37813],{},[39,37810,29732],{},[1209,37812,1804],{},[6467,37814,37815,37817],{},[35,37816,33494],{},[1209,37818,1263],{},[131,37820,37821,37827],{},[32,37822,37823,37825],{},[35,37824,33503],{},[39,37826,1170],{},[1209,37828,1804],{},[35,37830,123],{},[1274,37832,2462],{},[35,37834,458],{},[35,37836,123],{},[71,37838],{"linebreak":10456},[39,37840,129],{},[6467,37842,37843,37853],{},[32,37844,37845,37847,37849,37851],{},[1209,37846,1263],{},[35,37848,621],{},[39,37850,6602],{},[1209,37852,1211],{},[32,37854,37855,37857],{},[1209,37856,1263],{},[35,37858,621],{},[39,37860,3268],{},[6467,37862,37863,37873],{},[32,37864,37865,37867,37869,37871],{},[1209,37866,1263],{},[35,37868,621],{},[39,37870,6602],{},[1209,37872,8678],{},[32,37874,37875,37877,37879,37881],{},[1209,37876,1263],{},[35,37878,621],{},[39,37880,6602],{},[1209,37882,1263],{},[39,37884,37800],{},[6467,37886,37887,37889],{},[1209,37888,1211],{},[1209,37890,1263],{},[39,37892,3268],{},[6467,37894,37895,37897],{},[35,37896,33494],{},[1209,37898,1263],{},[45,37900,37901],{"encoding":47},"\\int_{0}^{\\frac{\\pi}{2}} \\sin^{2k} x \\, dx = \\frac{2k-1}{2k} \\int_{0}^{\\frac{\\pi}{2}} \\sin^{2k-2} x \\, dx \\\\ = \\frac{2k-1}{2k} \\cdot \\frac{2k-3}{2k-2} \\cdots \\frac{1}{2} \\int_{0}^{\\frac{\\pi}{2}} \\sin^0 x \\, dx \\\\= \\frac{2k-1}{2k} \\cdot \\frac{2k-3}{2k-2} \\cdots \\frac{1}{2} \\cdot 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为奇数时，即 ",[15,39210,39212,39236],{"className":39211,"translate":19},[18],[15,39213,39215],{"className":39214},[23],[25,39216,39217],{"xmlns":27},[29,39218,39219,39233],{},[32,39220,39221,39223,39225,39227,39229,39231],{},[35,39222,37],{},[39,39224,129],{},[1209,39226,1263],{},[35,39228,621],{},[39,39230,1245],{},[1209,39232,1211],{},[45,39234,39235],{"encoding":47},"n = 2k + 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= m (m-2) (m-4) 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上有 ",[15,41904,41906,41978],{"className":41905,"translate":19},[18],[15,41907,41909],{"className":41908},[23],[25,41910,41911],{"xmlns":27},[29,41912,41913,41975],{},[32,41914,41915,41933,41935,41937,41951,41953,41955,41973],{},[131,41916,41917,41923],{},[32,41918,41919,41921],{},[35,41920,33503],{},[39,41922,1170],{},[32,41924,41925,41927,41929,41931],{},[1209,41926,1263],{},[35,41928,621],{},[39,41930,1245],{},[1209,41932,1211],{},[35,41934,123],{},[39,41936,25772],{},[131,41938,41939,41945],{},[32,41940,41941,41943],{},[35,41942,33503],{},[39,41944,1170],{},[32,41946,41947,41949],{},[1209,41948,1263],{},[35,41950,621],{},[35,41952,123],{},[39,41954,25772],{},[131,41956,41957,41963],{},[32,41958,41959,41961],{},[35,41960,33503],{},[39,41962,1170],{},[32,41964,41965,41967,41969,41971],{},[1209,41966,1263],{},[35,41968,621],{},[39,41970,6602],{},[1209,41972,1211],{},[35,41974,123],{},[45,41976,41977],{"encoding":47},"\\sin^{2k+1} x \u003C \\sin^{2k} x \u003C \\sin^{2k-1} x",[15,41979,41981,42044,42100],{"className":41980,"ariaHidden":53},[52],[15,41982,41984,41988,42029,42032,42035,42038,42041],{"className":41983},[57],[15,41985],{"className":41986,"style":41987},[61],"height:0.946em;vertical-align:-0.0391em;",[15,41989,41991,41994],{"className":41990},[1339],[15,41992,33503],{"className":41993},[1339],[15,41995,41997],{"className":41996},[198],[15,41998,42000],{"className":41999},[202],[15,42001,42003],{"className":42002},[206],[15,42004,42006],{"className":42005,"style":38054},[210],[15,42007,42008,42011],{"style":33685},[15,42009],{"className":42010,"style":218},[217],[15,42012,42014],{"className":42013},[222,223,224,225],[15,42015,42017,42020,42023,42026],{"className":42016},[66,225],[15,42018,1263],{"className":42019},[66,225],[15,42021,621],{"className":42022,"style":684},[66,67,225],[15,42024,1245],{"className":42025},[76,225],[15,42027,1211],{"className":42028},[66,225],[15,42030],{"className":42031,"style":697},[71],[15,42033,123],{"className":42034},[66,67],[15,42036],{"className":42037,"style":173},[71],[15,42039,25772],{"className":42040},[177],[15,42042],{"className":42043,"style":173},[71],[15,42045,42047,42050,42085,42088,42091,42094,42097],{"className":42046},[57],[15,42048],{"className":42049,"style":41987},[61],[15,42051,42053,42056],{"className":42052},[1339],[15,42054,33503],{"className":42055},[1339],[15,42057,42059],{"className":42058},[198],[15,42060,42062],{"className":42061},[202],[15,42063,42065],{"className":42064},[206],[15,42066,42068],{"className":42067,"style":38054},[210],[15,42069,42070,42073],{"style":33685},[15,42071],{"className":42072,"style":218},[217],[15,42074,42076],{"className":42075},[222,223,224,225],[15,42077,42079,42082],{"className":42078},[66,225],[15,42080,1263],{"className":42081},[66,225],[15,42083,621],{"className":42084,"style":684},[66,67,225],[15,42086],{"className":42087,"style":697},[71],[15,42089,123],{"className":42090},[66,67],[15,42092],{"className":42093,"style":173},[71],[15,42095,25772],{"className":42096},[177],[15,42098],{"className":42099,"style":173},[71],[15,42101,42103,42106,42147,42150],{"className":42102},[57],[15,42104],{"className":42105,"style":38054},[61],[15,42107,42109,42112],{"className":42108},[1339],[15,42110,33503],{"className":42111},[1339],[15,42113,42115],{"className":42114},[198],[15,42116,42118],{"className":42117},[202],[15,42119,42121],{"className":42120},[206],[15,42122,42124],{"className":42123,"style":38054},[210],[15,42125,42126,42129],{"style":33685},[15,42127],{"className":42128,"style":218},[217],[15,42130,42132],{"className":42131},[222,223,224,225],[15,42133,42135,42138,42141,42144],{"className":42134},[66,225],[15,42136,1263],{"className":42137},[66,225],[15,42139,621],{"className":42140,"style":684},[66,67,225],[15,42142,6602],{"className":42143},[76,225],[15,42145,1211],{"className":42146},[66,225],[15,42148],{"className":42149,"style":697},[71],[15,42151,123],{"className":42152},[66,67],"，因此有：",[15,42155,42157],{"className":42156,"translate":19},[294],[15,42158,42160,42286],{"className":42159,"translate":19},[18],[15,42161,42163],{"className":42162},[23],[25,42164,42165],{"xmlns":27,"display":303},[29,42166,42167,42283],{},[32,42168,42169,42181,42199,42201,42203,42205,42207,42209,42221,42235,42237,42239,42241,42243,42245,42257,42275,42277,42279,42281],{},[1255,42170,42171,42173,42175],{},[39,42172,29732],{},[1209,42174,1804],{},[6467,42176,42177,42179],{},[35,42178,33494],{},[1209,42180,1263],{},[131,42182,42183,42189],{},[32,42184,42185,42187],{},[35,42186,33503],{},[39,42188,1170],{},[32,42190,42191,42193,42195,42197],{},[1209,42192,1263],{},[35,42194,621],{},[39,42196,1245],{},[1209,42198,1211],{},[35,42200,123],{},[1274,42202,2462],{},[35,42204,458],{},[35,42206,123],{},[39,42208,25772],{},[1255,42210,42211,42213,42215],{},[39,42212,29732],{},[1209,42214,1804],{},[6467,42216,42217,42219],{},[35,42218,33494],{},[1209,42220,1263],{},[131,42222,42223,42229],{},[32,42224,42225,42227],{},[35,42226,33503],{},[39,42228,1170],{},[32,42230,42231,42233],{},[1209,42232,1263],{},[35,42234,621],{},[35,42236,123],{},[1274,42238,2462],{},[35,42240,458],{},[35,42242,123],{},[39,42244,25772],{},[1255,42246,42247,42249,42251],{},[39,42248,29732],{},[1209,42250,1804],{},[6467,42252,42253,42255],{},[35,42254,33494],{},[1209,42256,1263],{},[131,42258,42259,42265],{},[32,42260,42261,42263],{},[35,42262,33503],{},[39,42264,1170],{},[32,42266,42267,42269,42271,42273],{},[1209,42268,1263],{},[35,42270,621],{},[39,42272,6602],{},[1209,42274,1211],{},[35,42276,123],{},[1274,42278,2462],{},[35,42280,458],{},[35,42282,123],{},[45,42284,42285],{"encoding":47},"\\int_{0}^{\\frac{\\pi}{2}} \\sin^{2k+1} x \\, dx \u003C \\int_{0}^{\\frac{\\pi}{2}} \\sin^{2k} x \\, dx \u003C \\int_{0}^{\\frac{\\pi}{2}} \\sin^{2k-1} x \\, dx",[15,42287,42289,42485,42675],{"className":42288,"ariaHidden":53},[52],[15,42290,42292,42295,42417,42420,42461,42464,42467,42470,42473,42476,42479,42482],{"className":42291},[57],[15,42293],{"className":42294,"style":33528},[61],[15,42296,42298,42301],{"className":42297},[1339],[15,42299,29732],{"className":42300,"style":29796},[1339,1470,1471],[15,42302,42304],{"className":42303},[198],[15,42305,42307,42409],{"className":42306},[202,1090],[15,42308,42310,42406],{"className":42309},[206],[15,42311,42313,42327],{"className":42312,"style":33547},[210],[15,42314,42315,42318],{"style":33550},[15,42316],{"className":42317,"style":1368},[217],[15,42319,42321],{"className":42320},[222,223,224,225],[15,42322,42324],{"className":42323},[66,225],[15,42325,1804],{"className":42326},[66,225],[15,42328,42329,42332],{"style":33565},[15,42330],{"className":42331,"style":1368},[217],[15,42333,42335],{"className":42334},[222,223,224,225],[15,42336,42338],{"className":42337},[66,225],[15,42339,42341,42344,42403],{"className":42340},[66,225],[15,42342],{"className":42343},[162,23417,222,3679,33581],[15,42345,42347],{"className":42346},[6467],[15,42348,42350,42395],{"className":42349},[202,1090],[15,42351,42353,42392],{"className":42352},[206],[15,42354,42356,42370,42378],{"className":42355,"style":33594},[210],[15,42357,42358,42361],{"style":33597},[15,42359],{"className":42360,"style":1368},[217],[15,42362,42364],{"className":42363},[222,3679,1508,225],[15,42365,42367],{"className":42366},[66,225],[15,42368,1263],{"className":42369},[66,225],[15,42371,42372,42375],{"style":33612},[15,42373],{"className":42374,"style":1368},[217],[15,42376],{"className":42377,"style":33619},[23452,225],[15,42379,42380,42383],{"style":33622},[15,42381],{"className":42382,"style":1368},[217],[15,42384,42386],{"className":42385},[222,3679,1508,225],[15,42387,42389],{"className":42388},[66,225],[15,42390,33494],{"className":42391,"style":1515},[66,67,225],[15,42393,1114],{"className":42394},[1113],[15,42396,42398],{"className":42397},[206],[15,42399,42401],{"className":42400,"style":33644},[210],[15,42402],{},[15,42404],{"className":42405},[169,23417,222,3679,33581],[15,42407,1114],{"className":42408},[1113],[15,42410,42412],{"className":42411},[206],[15,42413,42415],{"className":42414,"style":29843},[210],[15,42416],{},[15,42418],{"className":42419,"style":697},[71],[15,42421,42423,42426],{"className":42422},[1339],[15,42424,33503],{"className":42425},[1339],[15,42427,42429],{"className":42428},[198],[15,42430,42432],{"className":42431},[202],[15,42433,42435],{"className":42434},[206],[15,42436,42438],{"className":42437,"style":38054},[210],[15,42439,42440,42443],{"style":33685},[15,42441],{"className":42442,"style":218},[217],[15,42444,42446],{"className":42445},[222,223,224,225],[15,42447,42449,42452,42455,42458],{"className":42448},[66,225],[15,42450,1263],{"className":42451},[66,225],[15,42453,621],{"className":42454,"style":684},[66,67,225],[15,42456,1245],{"className":42457},[76,225],[15,42459,1211],{"className":42460},[66,225],[15,42462],{"className":42463,"style":697},[71],[15,42465,123],{"className":42466},[66,67],[15,42468],{"className":42469,"style":697},[71],[15,42471,458],{"className":42472},[66,67],[15,42474,123],{"className":42475},[66,67],[15,42477],{"className":42478,"style":173},[71],[15,42480,25772],{"className":42481},[177],[15,42483],{"className":42484,"style":173},[71],[15,42486,42488,42491,42613,42616,42651,42654,42657,42660,42663,42666,42669,42672],{"className":42487},[57],[15,42489],{"className":42490,"style":33528},[61],[15,42492,42494,42497],{"className":42493},[1339],[15,42495,29732],{"className":42496,"style":29796},[1339,1470,1471],[15,42498,42500],{"className":42499},[198],[15,42501,42503,42605],{"className":42502},[202,1090],[15,42504,42506,42602],{"className":42505},[206],[15,42507,42509,42523],{"className":42508,"style":33547},[210],[15,42510,42511,42514],{"style":33550},[15,42512],{"className":42513,"style":1368},[217],[15,42515,42517],{"className":42516},[222,223,224,225],[15,42518,42520],{"className":42519},[66,225],[15,42521,1804],{"className":42522},[66,225],[15,42524,42525,42528],{"style":33565},[15,42526],{"className":42527,"style":1368},[217],[15,42529,42531],{"className":42530},[222,223,224,225],[15,42532,42534],{"className":42533},[66,225],[15,42535,42537,42540,42599],{"className":42536},[66,225],[15,42538],{"className":42539},[162,23417,222,3679,33581],[15,42541,42543],{"className":42542},[6467],[15,42544,42546,42591],{"className":42545},[202,1090],[15,42547,42549,42588],{"className":42548},[206],[15,42550,42552,42566,42574],{"className":42551,"style":33594},[210],[15,42553,42554,42557],{"style":33597},[15,42555],{"className":42556,"style":1368},[217],[15,42558,42560],{"className":42559},[222,3679,1508,225],[15,42561,42563],{"className":42562},[66,225],[15,42564,1263],{"className":42565},[66,225],[15,42567,42568,42571],{"style":33612},[15,42569],{"className":42570,"style":1368},[217],[15,42572],{"className":42573,"style":33619},[23452,225],[15,42575,42576,42579],{"style":33622},[15,42577],{"className":42578,"style":1368},[217],[15,42580,42582],{"className":42581},[222,3679,1508,225],[15,42583,42585],{"className":42584},[66,225],[15,42586,33494],{"className":42587,"style":1515},[66,67,225],[15,42589,1114],{"className":42590},[1113],[15,42592,42594],{"className":42593},[206],[15,42595,42597],{"className":42596,"style":33644},[210],[15,42598],{},[15,42600],{"className":42601},[169,23417,222,3679,33581],[15,42603,1114],{"className":42604},[1113],[15,42606,42608],{"className":42607},[206],[15,42609,42611],{"className":42610,"style":29843},[210],[15,42612],{},[15,42614],{"className":42615,"style":697},[71],[15,42617,42619,42622],{"className":42618},[1339],[15,42620,33503],{"className":42621},[1339],[15,42623,42625],{"className":42624},[198],[15,42626,42628],{"className":42627},[202],[15,42629,42631],{"className":42630},[206],[15,42632,42634],{"className":42633,"style":38054},[210],[15,42635,42636,42639],{"style":33685},[15,42637],{"className":42638,"style":218},[217],[15,42640,42642],{"className":42641},[222,223,224,225],[15,42643,42645,42648],{"className":42644},[66,225],[15,42646,1263],{"className":42647},[66,225],[15,42649,621],{"className":42650,"style":684},[66,67,225],[15,42652],{"className":42653,"style":697},[71],[15,42655,123],{"className":42656},[66,67],[15,42658],{"className":42659,"style":697},[71],[15,42661,458],{"className":42662},[66,67],[15,42664,123],{"className":42665},[66,67],[15,42667],{"className":42668,"style":173},[71],[15,42670,25772],{"className":42671},[177],[15,42673],{"className":42674,"style":173},[71],[15,42676,42678,42681,42803,42806,42847,42850,42853,42856,42859],{"className":42677},[57],[15,42679],{"className":42680,"style":33528},[61],[15,42682,42684,42687],{"className":42683},[1339],[15,42685,29732],{"className":42686,"style":29796},[1339,1470,1471],[15,42688,42690],{"className":42689},[198],[15,42691,42693,42795],{"className":42692},[202,1090],[15,42694,42696,42792],{"className":42695},[206],[15,42697,42699,42713],{"className":42698,"style":33547},[210],[15,42700,42701,42704],{"style":33550},[15,42702],{"className":42703,"style":1368},[217],[15,42705,42707],{"className":42706},[222,223,224,225],[15,42708,42710],{"className":42709},[66,225],[15,42711,1804],{"className":42712},[66,225],[15,42714,42715,42718],{"style":33565},[15,42716],{"className":42717,"style":1368},[217],[15,42719,42721],{"className":42720},[222,223,224,225],[15,42722,42724],{"className":42723},[66,225],[15,42725,42727,42730,42789],{"className":42726},[66,225],[15,42728],{"className":42729},[162,23417,222,3679,33581],[15,42731,42733],{"className":42732},[6467],[15,42734,42736,42781],{"className":42735},[202,1090],[15,42737,42739,42778],{"className":42738},[206],[15,42740,42742,42756,42764],{"className":42741,"style":33594},[210],[15,42743,42744,42747],{"style":33597},[15,42745],{"className":42746,"style":1368},[217],[15,42748,42750],{"className":42749},[222,3679,1508,225],[15,42751,42753],{"className":42752},[66,225],[15,42754,1263],{"className":42755},[66,225],[15,42757,42758,42761],{"style":33612},[15,42759],{"className":42760,"style":1368},[217],[15,42762],{"className":42763,"style":33619},[23452,225],[15,42765,42766,42769],{"style":33622},[15,42767],{"className":42768,"style":1368},[217],[15,42770,42772],{"className":42771},[222,3679,1508,225],[15,42773,42775],{"className":42774},[66,225],[15,42776,33494],{"className":42777,"style":1515},[66,67,225],[15,42779,1114],{"className":42780},[1113],[15,42782,42784],{"className":42783},[206],[15,42785,42787],{"className":42786,"style":33644},[210],[15,42788],{},[15,42790],{"className":42791},[169,23417,222,3679,33581],[15,42793,1114],{"className":42794},[1113],[15,42796,42798],{"className":42797},[206],[15,42799,42801],{"className":42800,"style":29843},[210],[15,42802],{},[15,42804],{"className":42805,"style":697},[71],[15,42807,42809,42812],{"className":42808},[1339],[15,42810,33503],{"className":42811},[1339],[15,42813,42815],{"className":42814},[198],[15,42816,42818],{"className":42817},[202],[15,42819,42821],{"className":42820},[206],[15,42822,42824],{"className":42823,"style":38054},[210],[15,42825,42826,42829],{"style":33685},[15,42827],{"className":42828,"style":218},[217],[15,42830,42832],{"className":42831},[222,223,224,225],[15,42833,42835,42838,42841,42844],{"className":42834},[66,225],[15,42836,1263],{"className":42837},[66,225],[15,42839,621],{"className":42840,"style":684},[66,67,225],[15,42842,6602],{"className":42843},[76,225],[15,42845,1211],{"className":42846},[66,225],[15,42848],{"className":42849,"style":697},[71],[15,42851,123],{"className":42852},[66,67],[15,42854],{"className":42855,"style":697},[71],[15,42857,458],{"className":42858},[66,67],[15,42860,123],{"className":42861},[66,67],[11,42863,42864],{},"代入上面的公式：",[15,42866,42868],{"className":42867,"translate":19},[294],[15,42869,42871,43001],{"className":42870,"translate":19},[18],[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\u003C \\frac{(2k-1)!!}{(2k)!!} \\cdot \\frac{\\pi}{2} \u003C 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上下界之间的差：",[15,43648,43650],{"className":43649,"translate":19},[294],[15,43651,43653,43964],{"className":43652,"translate":19},[18],[15,43654,43656],{"className":43655},[23],[25,43657,43658],{"xmlns":27,"display":303},[29,43659,43660,43961],{},[32,43661,43662,43679,43759,43761,43763,43779,43883,43885,43887,43903,43917,43919,43957,43959],{},[1172,43663,43664,43671],{},[32,43665,43666,43669],{},[35,43667,43668],{},"lim",[39,43670,1170],{},[32,43672,43673,43675,43677],{},[35,43674,621],{},[39,43676,449],{},[35,43678,7620],{"mathvariant":138},[32,43680,43681,43683,43721,43723,43757],{},[39,43682,120],{"fence":53},[6467,43684,43685,43703],{},[32,43686,43687,43689,43691,43693,43695,43697,43699,43701],{},[39,43688,120],{"stretchy":119},[1209,43690,1263],{},[35,43692,621],{},[39,43694,6602],{},[1209,43696,1263],{},[39,43698,126],{"stretchy":119},[39,43700,40769],{"stretchy":119},[39,43702,40769],{"stretchy":119},[32,43704,43705,43707,43709,43711,43713,43715,43717,43719],{},[39,43706,120],{"stretchy":119},[1209,43708,1263],{},[35,43710,621],{},[39,43712,6602],{},[1209,43714,1211],{},[39,43716,126],{"stretchy":119},[39,43718,40769],{"stretchy":119},[39,43720,40769],{"stretchy":119},[39,43722,6602],{},[6467,43724,43725,43739],{},[32,43726,43727,43729,43731,43733,43735,43737],{},[39,43728,120],{"stretchy":119},[1209,43730,1263],{},[35,43732,621],{},[39,43734,126],{"stretchy":119},[39,43736,40769],{"stretchy":119},[39,43738,40769],{"stretchy":119},[32,43740,43741,43743,43745,43747,43749,43751,43753,43755],{},[39,43742,120],{"stretchy":119},[1209,43744,1263],{},[35,43746,621],{},[39,43748,1245],{},[1209,43750,1211],{},[39,43752,126],{"stretchy":119},[39,43754,40769],{"stretchy":119},[39,43756,40769],{"stretchy":119},[39,43758,126],{"fence":53},[71,43760],{"linebreak":10456},[39,43762,129],{},[1172,43764,43765,43771],{},[32,43766,43767,43769],{},[35,43768,43668],{},[39,43770,1170],{},[32,43772,43773,43775,43777],{},[35,43774,621],{},[39,43776,449],{},[35,43778,7620],{"mathvariant":138},[32,43780,43781,43783,43821,43823,43841,43843,43881],{},[39,43782,120],{"fence":53},[6467,43784,43785,43803],{},[32,43786,43787,43789,43791,43793,43795,43797,43799,43801],{},[39,43788,120],{"stretchy":119},[1209,43790,1263],{},[35,43792,621],{},[39,43794,6602],{},[1209,43796,1263],{},[39,43798,126],{"stretchy":119},[39,43800,40769],{"stretchy":119},[39,43802,40769],{"stretchy":119},[32,43804,43805,43807,43809,43811,43813,43815,43817,43819],{},[39,43806,120],{"stretchy":119},[1209,43808,1263],{},[35,43810,621],{},[39,43812,6602],{},[1209,43814,1211],{},[39,43816,126],{"stretchy":119},[39,43818,40769],{"stretchy":119},[39,43820,40769],{"stretchy":119},[39,43822,6602],{},[6467,43824,43825,43831],{},[32,43826,43827,43829],{},[1209,43828,1263],{},[35,43830,621],{},[32,43832,43833,43835,43837,43839],{},[1209,43834,1263],{},[35,43836,621],{},[39,43838,1245],{},[1209,43840,1211],{},[39,43842,3268],{},[6467,43844,43845,43863],{},[32,43846,43847,43849,43851,43853,43855,43857,43859,43861],{},[39,43848,120],{"stretchy":119},[1209,43850,1263],{},[35,43852,621],{},[39,43854,6602],{},[1209,43856,1263],{},[39,43858,126],{"stretchy":119},[39,43860,40769],{"stretchy":119},[39,43862,40769],{"stretchy":119},[32,43864,43865,43867,43869,43871,43873,43875,43877,43879],{},[39,43866,120],{"stretchy":119},[1209,43868,1263],{},[35,43870,621],{},[39,43872,6602],{},[1209,43874,1211],{},[39,43876,126],{"stretchy":119},[39,43878,40769],{"stretchy":119},[39,43880,40769],{"stretchy":119},[39,43882,126],{"fence":53},[71,43884],{"linebreak":10456},[39,43886,129],{},[1172,43888,43889,43895],{},[32,43890,43891,43893],{},[35,43892,43668],{},[39,43894,1170],{},[32,43896,43897,43899,43901],{},[35,43898,621],{},[39,43900,449],{},[35,43902,7620],{"mathvariant":138},[6467,43904,43905,43907],{},[1209,43906,1211],{},[32,43908,43909,43911,43913,43915],{},[1209,43910,1263],{},[35,43912,621],{},[39,43914,1245],{},[1209,43916,1211],{},[39,43918,3268],{},[6467,43920,43921,43939],{},[32,43922,43923,43925,43927,43929,43931,43933,43935,43937],{},[39,43924,120],{"stretchy":119},[1209,43926,1263],{},[35,43928,621],{},[39,43930,6602],{},[1209,43932,1263],{},[39,43934,126],{"stretchy":119},[39,43936,40769],{"stretchy":119},[39,43938,40769],{"stretchy":119},[32,43940,43941,43943,43945,43947,43949,43951,43953,43955],{},[39,43942,120],{"stretchy":119},[1209,43944,1263],{},[35,43946,621],{},[39,43948,6602],{},[1209,43950,1211],{},[39,43952,126],{"stretchy":119},[39,43954,40769],{"stretchy":119},[39,43956,40769],{"stretchy":119},[39,43958,129],{},[1209,43960,1804],{},[45,43962,43963],{"encoding":47},"\\lim_{k \\rightarrow \\infty} \\left( \\frac{(2k-2)!!}{(2k-1)!!} - \\frac{(2k)!!}{(2k+1)!!} \\right) \\\\ = \\lim_{k \\rightarrow \\infty} \\left( \\frac{(2k-2)!!}{(2k-1)!!} - \\frac{2k}{2k+1} \\cdot \\frac{(2k-2)!!}{(2k-1)!!} \\right) \\\\ = \\lim_{k \\rightarrow \\infty} \\frac{1}{2k+1} \\cdot \\frac{(2k-2)!!}{(2k-1)!!} = 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1",[15,44986,44988,45091],{"className":44987,"ariaHidden":53},[52],[15,44989,44991,44995,45082,45085,45088],{"className":44990},[57],[15,44992],{"className":44993,"style":44994},[61],"height:1.2834em;vertical-align:-0.4033em;",[15,44996,44998,45001,45079],{"className":44997},[66],[15,44999],{"className":45000},[162,23417],[15,45002,45004],{"className":45003},[6467],[15,45005,45007,45070],{"className":45006},[202,1090],[15,45008,45010,45067],{"className":45009},[206],[15,45011,45013,45036,45044],{"className":45012,"style":31037},[210],[15,45014,45015,45018],{"style":30876},[15,45016],{"className":45017,"style":1368},[217],[15,45019,45021],{"className":45020},[222,223,224,225],[15,45022,45024,45027,45030,45033],{"className":45023},[66,225],[15,45025,1263],{"className":45026},[66,225],[15,45028,621],{"className":45029,"style":684},[66,67,225],[15,45031,6602],{"className":45032},[76,225],[15,45034,1211],{"className":45035},[66,225],[15,45037,45038,45041],{"style":23445},[15,45039],{"className":45040,"style":1368},[217],[15,45042],{"className":45043,"style":4440},[23452],[15,45045,45046,45049],{"style":30899},[15,45047],{"className":45048,"style":1368},[217],[15,45050,45052],{"className":45051},[222,223,224,225],[15,45053,45055,45058,45061,45064],{"className":45054},[66,225],[15,45056,1263],{"className":45057},[66,225],[15,45059,621],{"className":45060,"style":684},[66,67,225],[15,45062,6602],{"className":45063},[76,225],[15,45065,1263],{"className":45066},[66,225],[15,45068,1114],{"className":45069},[1113],[15,45071,45073],{"className":45072},[206],[15,45074,45077],{"className":45075,"style":45076},[210],"height:0.4033em;",[15,45078],{},[15,45080],{"className":45081},[169,23417],[15,45083],{"className":45084,"style":173},[71],[15,45086,25772],{"className":45087},[177],[15,45089],{"className":45090,"style":173},[71],[15,45092,45094,45097],{"className":45093},[57],[15,45095],{"className":45096,"style":1836},[61],[15,45098,1211],{"className":45099},[66],[11,45101,45102],{},"对于不等式：",[15,45104,45106],{"className":45105,"translate":19},[294],[15,45107,45109,45238],{"className":45108,"translate":19},[18],[15,45110,45112],{"className":45111},[23],[25,45113,45114],{"xmlns":27,"display":303},[29,45115,45116,45236],{},[32,45117,45118,45152,45154,45188,45190,45196,45198],{},[6467,45119,45120,45134],{},[32,45121,45122,45124,45126,45128,45130,45132],{},[39,45123,120],{"stretchy":119},[1209,45125,1263],{},[35,45127,621],{},[39,45129,126],{"stretchy":119},[39,45131,40769],{"stretchy":119},[39,45133,40769],{"stretchy":119},[32,45135,45136,45138,45140,45142,45144,45146,45148,45150],{},[39,45137,120],{"stretchy":119},[1209,45139,1263],{},[35,45141,621],{},[39,45143,1245],{},[1209,45145,1211],{},[39,45147,126],{"stretchy":119},[39,45149,40769],{"stretchy":119},[39,45151,40769],{"stretchy":119},[39,45153,25772],{},[6467,45155,45156,45174],{},[32,45157,45158,45160,45162,45164,45166,45168,45170,45172],{},[39,45159,120],{"stretchy":119},[1209,45161,1263],{},[35,45163,621],{},[39,45165,6602],{},[1209,45167,1211],{},[39,45169,126],{"stretchy":119},[39,45171,40769],{"stretchy":119},[39,45173,40769],{"stretchy":119},[32,45175,45176,45178,45180,45182,45184,45186],{},[39,45177,120],{"stretchy":119},[1209,45179,1263],{},[35,45181,621],{},[39,45183,126],{"stretchy":119},[39,45185,40769],{"stretchy":119},[39,45187,40769],{"stretchy":119},[39,45189,3268],{},[6467,45191,45192,45194],{},[35,45193,33494],{},[1209,45195,1263],{},[39,45197,25772],{},[6467,45199,45200,45218],{},[32,45201,45202,45204,45206,45208,45210,45212,45214,45216],{},[39,45203,120],{"stretchy":119},[1209,45205,1263],{},[35,45207,621],{},[39,45209,6602],{},[1209,45211,1263],{},[39,45213,126],{"stretchy":119},[39,45215,40769],{"stretchy":119},[39,45217,40769],{"stretchy":119},[32,45219,45220,45222,45224,45226,45228,45230,45232,45234],{},[39,45221,120],{"stretchy":119},[1209,45223,1263],{},[35,45225,621],{},[39,45227,6602],{},[1209,45229,1211],{},[39,45231,126],{"stretchy":119},[39,45233,40769],{"stretchy":119},[39,45235,40769],{"stretchy":119},[45,45237,43000],{"encoding":47},[15,45239,45241,45348,45455,45532],{"className":45240,"ariaHidden":53},[52],[15,45242,45244,45247,45339,45342,45345],{"className":45243},[57],[15,45245],{"className":45246,"style":41192},[61],[15,45248,45250,45253,45336],{"className":45249},[66],[15,45251],{"className":45252},[162,23417],[15,45254,45256],{"className":45255},[6467],[15,45257,45259,45328],{"className":45258},[202,1090],[15,45260,45262,45325],{"className":45261},[206],[15,45263,45265,45297,45305],{"className":45264,"style":41211},[210],[15,45266,45267,45270],{"style":23433},[15,45268],{"className":45269,"style":1368},[217],[15,45271,45273,45276,45279,45282,45285,45288,45291,45294],{"className":45272},[66],[15,45274,120],{"className":45275},[162],[15,45277,1263],{"className":45278},[66],[15,45280,621],{"className":45281,"style":684},[66,67],[15,45283],{"className":45284,"style":72},[71],[15,45286,1245],{"className":45287},[76],[15,45289],{"className":45290,"style":72},[71],[15,45292,1211],{"className":45293},[66],[15,45295,41232],{"className":45296},[169],[15,45298,45299,45302],{"style":23445},[15,45300],{"className":45301,"style":1368},[217],[15,45303],{"className":45304,"style":4440},[23452],[15,45306,45307,45310],{"style":23455},[15,45308],{"className":45309,"style":1368},[217],[15,45311,45313,45316,45319,45322],{"className":45312},[66],[15,45314,120],{"className":45315},[162],[15,45317,1263],{"className":45318},[66],[15,45320,621],{"className":45321,"style":684},[66,67],[15,45323,41232],{"className":45324},[169],[15,45326,1114],{"className":45327},[1113],[15,45329,45331],{"className":45330},[206],[15,45332,45334],{"className":45333,"style":41282},[210],[15,45335],{},[15,45337],{"className":45338},[169,23417],[15,45340],{"className":45341,"style":173},[71],[15,45343,25772],{"className":45344},[177],[15,45346],{"className":45347,"style":173},[71],[15,45349,45351,45354,45446,45449,45452],{"className":45350},[57],[15,45352],{"className":45353,"style":41192},[61],[15,45355,45357,45360,45443],{"className":45356},[66],[15,45358],{"className":45359},[162,23417],[15,45361,45363],{"className":45362},[6467],[15,45364,45366,45435],{"className":45365},[202,1090],[15,45367,45369,45432],{"className":45368},[206],[15,45370,45372,45392,45400],{"className":45371,"style":41211},[210],[15,45373,45374,45377],{"style":23433},[15,45375],{"className":45376,"style":1368},[217],[15,45378,45380,45383,45386,45389],{"className":45379},[66],[15,45381,120],{"className":45382},[162],[15,45384,1263],{"className":45385},[66],[15,45387,621],{"className":45388,"style":684},[66,67],[15,45390,41232],{"className":45391},[169],[15,45393,45394,45397],{"style":23445},[15,45395],{"className":45396,"style":1368},[217],[15,45398],{"className":45399,"style":4440},[23452],[15,45401,45402,45405],{"style":23455},[15,45403],{"className":45404,"style":1368},[217],[15,45406,45408,45411,45414,45417,45420,45423,45426,45429],{"className":45407},[66],[15,45409,120],{"className":45410},[162],[15,45412,1263],{"className":45413},[66],[15,45415,621],{"className":45416,"style":684},[66,67],[15,45418],{"className":45419,"style":72},[71],[15,45421,6602],{"className":45422},[76],[15,45424],{"className":45425,"style":72},[71],[15,45427,1211],{"className":45428},[66],[15,45430,41232],{"className":45431},[169],[15,45433,1114],{"className":45434},[1113],[15,45436,45438],{"className":45437},[206],[15,45439,45441],{"className":45440,"style":41282},[210],[15,45442],{},[15,45444],{"className":45445},[169,23417],[15,45447],{"className":45448,"style":72},[71],[15,45450,3268],{"className":45451},[76],[15,45453],{"className":45454,"style":72},[71],[15,45456,45458,45461,45523,45526,45529],{"className":45457},[57],[15,45459],{"className":45460,"style":39114},[61],[15,45462,45464,45467,45520],{"className":45463},[66],[15,45465],{"className":45466},[162,23417],[15,45468,45470],{"className":45469},[6467],[15,45471,45473,45512],{"className":45472},[202,1090],[15,45474,45476,45509],{"className":45475},[206],[15,45477,45479,45490,45498],{"className":45478,"style":39133},[210],[15,45480,45481,45484],{"style":23433},[15,45482],{"className":45483,"style":1368},[217],[15,45485,45487],{"className":45486},[66],[15,45488,1263],{"className":45489},[66],[15,45491,45492,45495],{"style":23445},[15,45493],{"className":45494,"style":1368},[217],[15,45496],{"className":45497,"style":4440},[23452],[15,45499,45500,45503],{"style":23455},[15,45501],{"className":45502,"style":1368},[217],[15,45504,45506],{"className":45505},[66],[15,45507,33494],{"className":45508,"style":1515},[66,67],[15,45510,1114],{"className":45511},[1113],[15,45513,45515],{"className":45514},[206],[15,45516,45518],{"className":45517,"style":23474},[210],[15,45519],{},[15,45521],{"className":45522},[169,23417],[15,45524],{"className":45525,"style":173},[71],[15,45527,25772],{"className":45528},[177],[15,45530],{"className":45531,"style":173},[71],[15,45533,45535,45538],{"className":45534},[57],[15,45536],{"className":45537,"style":41192},[61],[15,45539,45541,45544,45639],{"className":45540},[66],[15,45542],{"className":45543},[162,23417],[15,45545,45547],{"className":45546},[6467],[15,45548,45550,45631],{"className":45549},[202,1090],[15,45551,45553,45628],{"className":45552},[206],[15,45554,45556,45588,45596],{"className":45555,"style":41211},[210],[15,45557,45558,45561],{"style":23433},[15,45559],{"className":45560,"style":1368},[217],[15,45562,45564,45567,45570,45573,45576,45579,45582,45585],{"className":45563},[66],[15,45565,120],{"className":45566},[162],[15,45568,1263],{"className":45569},[66],[15,45571,621],{"className":45572,"style":684},[66,67],[15,45574],{"className":45575,"style":72},[71],[15,45577,6602],{"className":45578},[76],[15,45580],{"className":45581,"style":72},[71],[15,45583,1211],{"className":45584},[66],[15,45586,41232],{"className":45587},[169],[15,45589,45590,45593],{"style":23445},[15,45591],{"className":45592,"style":1368},[217],[15,45594],{"className":45595,"style":4440},[23452],[15,45597,45598,45601],{"style":23455},[15,45599],{"className":45600,"style":1368},[217],[15,45602,45604,45607,45610,45613,45616,45619,45622,45625],{"className":45603},[66],[15,45605,120],{"className":45606},[162],[15,45608,1263],{"className":45609},[66],[15,45611,621],{"className":45612,"style":684},[66,67],[15,45614],{"className":45615,"style":72},[71],[15,45617,6602],{"className":45618},[76],[15,45620],{"className":45621,"style":72},[71],[15,45623,1263],{"className":45624},[66],[15,45626,41232],{"className":45627},[169],[15,45629,1114],{"className":45630},[1113],[15,45632,45634],{"className":45633},[206],[15,45635,45637],{"className":45636,"style":41282},[210],[15,45638],{},[15,45640],{"className":45641},[169,23417],[11,45643,45644,45645,37644],{},"将所有项乘以 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(2k-1)!!} \u003C \\frac{\\pi}{2} \u003C \\frac{(2k-2)!! 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\\rightarrow \\infty} \\left( \\frac{(2k-2)!! 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(2k-1)!!} \\\\= \\lim_{k \\rightarrow \\infty} \\frac{1}{2k} \\cdot \\frac{(2k)!!^2}{(2k+1)!!^2} = 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\\rightarrow \\infty} \\frac{(2k)!!^2}{(2k+1)!! (2k-1)!!} = \\frac{\\pi}{2}",[15,47978,47980,48196],{"className":47979,"ariaHidden":53},[52],[15,47981,47983,47986,48039,48042,48187,48190,48193],{"className":47982},[57],[15,47984],{"className":47985,"style":45944},[61],[15,47987,47989],{"className":47988},[1339,1351],[15,47990,47992,48031],{"className":47991},[202,1090],[15,47993,47995,48028],{"className":47994},[206],[15,47996,47998,48018],{"className":47997,"style":840},[210],[15,47999,48000,48003],{"style":1364},[15,48001],{"className":48002,"style":1368},[217],[15,48004,48006],{"className":48005},[222,223,224,225],[15,48007,48009,48012,48015],{"className":48008},[66,225],[15,48010,621],{"className":48011,"style":684},[66,67,225],[15,48013,449],{"className":48014},[177,225],[15,48016,7620],{"className":48017},[66,225],[15,48019,48020,48023],{"style":1386},[15,48021],{"className":48022,"style":1368},[217],[15,48024,48025],{},[15,48026,43668],{"className":48027},[1339],[15,48029,1114],{"className":48030},[1113],[15,48032,48034],{"className":48033},[206],[15,48035,48037],{"className":48036,"style":44025},[210],[15,48038],{},[15,48040],{"className":48041,"style":697},[71],[15,48043,48045,48048,48184],{"className":48044},[66],[15,48046],{"className":48047},[162,23417],[15,48049,48051],{"className":48050},[6467],[15,48052,48054,48176],{"className":48053},[202,1090],[15,48055,48057,48173],{"className":48056},[206],[15,48058,48060,48116,48124],{"className":48059,"style":45963},[210],[15,48061,48062,48065],{"style":23433},[15,48063],{"className":48064,"style":1368},[217],[15,48066,48068,48071,48074,48077,48080,48083,48086,48089,48092,48095,48098,48101,48104,48107,48110,48113],{"className":48067},[66],[15,48069,120],{"className":48070},[162],[15,48072,1263],{"className":48073},[66],[15,48075,621],{"className":48076,"style":684},[66,67],[15,48078],{"className":48079,"style":72},[71],[15,48081,1245],{"className":48082},[76],[15,48084],{"className":48085,"style":72},[71],[15,48087,1211],{"className":48088},[66],[15,48090,41232],{"className":48091},[169],[15,48093,120],{"className":48094},[162],[15,48096,1263],{"className":48097},[66],[15,48099,621],{"className":48100,"style":684},[66,67],[15,48102],{"className":48103,"style":72},[71],[15,48105,6602],{"className":48106},[76],[15,48108],{"className":48109,"style":72},[71],[15,48111,1211],{"className":48112},[66],[15,48114,41232],{"className":48115},[169],[15,48117,48118,48121],{"style":23445},[15,48119],{"className":48120,"style":1368},[217],[15,48122],{"className":48123,"style":4440},[23452],[15,48125,48126,48129],{"style":23455},[15,48127],{"className":48128,"style":1368},[217],[15,48130,48132,48135,48138,48141,48144],{"className":48131},[66],[15,48133,120],{"className":48134},[162],[15,48136,1263],{"className":48137},[66],[15,48139,621],{"className":48140,"style":684},[66,67],[15,48142,46048],{"className":48143},[169],[15,48145,48147,48150],{"className":48146},[169],[15,48148,40769],{"className":48149},[169],[15,48151,48153],{"className":48152},[198],[15,48154,48156],{"className":48155},[202],[15,48157,48159],{"className":48158},[206],[15,48160,48162],{"className":48161,"style":6740},[210],[15,48163,48164,48167],{"style":213},[15,48165],{"className":48166,"style":218},[217],[15,48168,48170],{"className":48169},[222,223,224,225],[15,48171,1263],{"className":48172},[66,225],[15,48174,1114],{"className":48175},[1113],[15,48177,48179],{"className":48178},[206],[15,48180,48182],{"className":48181,"style":41282},[210],[15,48183],{},[15,48185],{"className":48186},[169,23417],[15,48188],{"className":48189,"style":173},[71],[15,48191,129],{"className":48192},[177],[15,48194],{"className":48195,"style":173},[71],[15,48197,48199,48202],{"className":48198},[57],[15,48200],{"className":48201,"style":39114},[61],[15,48203,48205,48208,48261],{"className":48204},[66],[15,48206],{"className":48207},[162,23417],[15,48209,48211],{"className":48210},[6467],[15,48212,48214,48253],{"className":48213},[202,1090],[15,48215,48217,48250],{"className":48216},[206],[15,48218,48220,48231,48239],{"className":48219,"style":39133},[210],[15,48221,48222,48225],{"style":23433},[15,48223],{"className":48224,"style":1368},[217],[15,48226,48228],{"className":48227},[66],[15,48229,1263],{"className":48230},[66],[15,48232,48233,48236],{"style":23445},[15,48234],{"className":48235,"style":1368},[217],[15,48237],{"className":48238,"style":4440},[23452],[15,48240,48241,48244],{"style":23455},[15,48242],{"className":48243,"style":1368},[217],[15,48245,48247],{"className":48246},[66],[15,48248,33494],{"className":48249,"style":1515},[66,67],[15,48251,1114],{"className":48252},[1113],[15,48254,48256],{"className":48255},[206],[15,48257,48259],{"className":48258,"style":23474},[210],[15,48260],{},[15,48262],{"className":48263},[169,23417],[11,48265,48266],{},"该公式引入了沃利斯的结果，为推导斯特林公式做准备。",[95,48268,33460],{"id":33460},[11,48270,48271,48272,48306,48307,48366,48367,48427],{},"斯特林公式揭示了 ",[15,48273,48275,48291],{"className":48274,"translate":19},[18],[15,48276,48278],{"className":48277},[23],[25,48279,48280],{"xmlns":27},[29,48281,48282,48288],{},[32,48283,48284,48286],{},[35,48285,37],{},[39,48287,40769],{"stretchy":119},[45,48289,48290],{"encoding":47},"n!",[15,48292,48294],{"className":48293,"ariaHidden":53},[52],[15,48295,48297,48300,48303],{"className":48296},[57],[15,48298],{"className":48299,"style":840},[61],[15,48301,37],{"className":48302},[66,67],[15,48304,40769],{"className":48305},[169],"、",[15,48308,48310,48328],{"className":48309,"translate":19},[18],[15,48311,48313],{"className":48312},[23],[25,48314,48315],{"xmlns":27},[29,48316,48317,48325],{},[32,48318,48319],{},[131,48320,48321,48323],{},[35,48322,37],{},[35,48324,37],{},[45,48326,48327],{"encoding":47},"n^n",[15,48329,48331],{"className":48330,"ariaHidden":53},[52],[15,48332,48334,48337],{"className":48333},[57],[15,48335],{"className":48336,"style":6438},[61],[15,48338,48340,48343],{"className":48339},[66],[15,48341,37],{"className":48342},[66,67],[15,48344,48346],{"className":48345},[198],[15,48347,48349],{"className":48348},[202],[15,48350,48352],{"className":48351},[206],[15,48353,48355],{"className":48354,"style":6438},[210],[15,48356,48357,48360],{"style":213},[15,48358],{"className":48359,"style":218},[217],[15,48361,48363],{"className":48362},[222,223,224,225],[15,48364,37],{"className":48365},[66,67,225]," 与 ",[15,48368,48370,48389],{"className":48369,"translate":19},[18],[15,48371,48373],{"className":48372},[23],[25,48374,48375],{"xmlns":27},[29,48376,48377,48386],{},[32,48378,48379],{},[131,48380,48381,48384],{},[35,48382,48383],{},"e",[35,48385,37],{},[45,48387,48388],{"encoding":47},"e^n",[15,48390,48392],{"className":48391,"ariaHidden":53},[52],[15,48393,48395,48398],{"className":48394},[57],[15,48396],{"className":48397,"style":6438},[61],[15,48399,48401,48404],{"className":48400},[66],[15,48402,48383],{"className":48403},[66,67],[15,48405,48407],{"className":48406},[198],[15,48408,48410],{"className":48409},[202],[15,48411,48413],{"className":48412},[206],[15,48414,48416],{"className":48415,"style":6438},[210],[15,48417,48418,48421],{"style":213},[15,48419],{"className":48420,"style":218},[217],[15,48422,48424],{"className":48423},[222,223,224,225],[15,48425,37],{"className":48426},[66,67,225]," 之间的关系。",[3054,48429,48430],{},[11,48431,31886,48432,48460,48461,48494,48495,1124],{},[15,48433,48435,48448],{"className":48434,"translate":19},[18],[15,48436,48438],{"className":48437},[23],[25,48439,48440],{"xmlns":27},[29,48441,48442,48446],{},[32,48443,48444],{},[35,48445,37],{},[45,48447,37],{"encoding":47},[15,48449,48451],{"className":48450,"ariaHidden":53},[52],[15,48452,48454,48457],{"className":48453},[57],[15,48455],{"className":48456,"style":86},[61],[15,48458,37],{"className":48459},[66,67]," 趋近无穷大时，",[15,48462,48464,48479],{"className":48463,"translate":19},[18],[15,48465,48467],{"className":48466},[23],[25,48468,48469],{"xmlns":27},[29,48470,48471,48477],{},[32,48472,48473,48475],{},[35,48474,37],{},[39,48476,40769],{"stretchy":119},[45,48478,48290],{"encoding":47},[15,48480,48482],{"className":48481,"ariaHidden":53},[52],[15,48483,48485,48488,48491],{"className":48484},[57],[15,48486],{"className":48487,"style":840},[61],[15,48489,37],{"className":48490},[66,67],[15,48492,40769],{"className":48493},[169]," 约等于 ",[15,48496,48498,48537],{"className":48497,"translate":19},[18],[15,48499,48501],{"className":48500},[23],[25,48502,48503],{"xmlns":27},[29,48504,48505,48534],{},[32,48506,48507,48518],{},[48508,48509,48510],"msqrt",{},[32,48511,48512,48514,48516],{},[1209,48513,1263],{},[35,48515,33494],{},[35,48517,37],{},[131,48519,48520,48532],{},[32,48521,48522,48524,48530],{},[39,48523,120],{"fence":53},[6467,48525,48526,48528],{},[35,48527,37],{},[35,48529,48383],{},[39,48531,126],{"fence":53},[35,48533,37],{},[45,48535,48536],{"encoding":47},"\\sqrt{2 \\pi n} \\left(\\frac{n}{e}\\right)^n",[15,48538,48540],{"className":48539,"ariaHidden":53},[52],[15,48541,48543,48547,48613,48616],{"className":48542},[57],[15,48544],{"className":48545,"style":48546},[61],"height:1.2572em;vertical-align:-0.35em;",[15,48548,48551],{"className":48549},[66,48550],"sqrt",[15,48552,48554,48604],{"className":48553},[202,1090],[15,48555,48557,48601],{"className":48556},[206],[15,48558,48561,48581],{"className":48559,"style":48560},[210],"height:0.9072em;",[15,48562,48565,48568],{"className":48563,"style":1386},[48564],"svg-align",[15,48566],{"className":48567,"style":1368},[217],[15,48569,48572,48575,48578],{"className":48570,"style":48571},[66],"padding-left:0.833em;",[15,48573,1263],{"className":48574},[66],[15,48576,33494],{"className":48577,"style":1515},[66,67],[15,48579,37],{"className":48580},[66,67],[15,48582,48584,48587],{"style":48583},"top:-2.8672em;",[15,48585],{"className":48586,"style":1368},[217],[15,48588,48592],{"className":48589,"style":48591},[48590],"hide-tail","min-width:0.853em;height:1.08em;",[34828,48593,48598],{"xmlns":34830,"width":48594,"height":48595,"viewBox":48596,"preserveAspectRatio":48597},"400em","1.08em","0 0 400000 1080","xMinYMin slice",[34834,48599],{"d":48600},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[15,48602,1114],{"className":48603},[1113],[15,48605,48607],{"className":48606},[206],[15,48608,48611],{"className":48609,"style":48610},[210],"height:0.1328em;",[15,48612],{},[15,48614],{"className":48615,"style":697},[71],[15,48617,48619,48702],{"className":48618},[1413],[15,48620,48622,48628,48696],{"className":48621},[1413],[15,48623,48625],{"className":48624,"style":1418},[162,1417],[15,48626,120],{"className":48627},[1422,1508],[15,48629,48631,48634,48693],{"className":48630},[66],[15,48632],{"className":48633},[162,23417],[15,48635,48637],{"className":48636},[6467],[15,48638,48640,48685],{"className":48639},[202,1090],[15,48641,48643,48682],{"className":48642},[206],[15,48644,48646,48660,48668],{"className":48645,"style":41848},[210],[15,48647,48648,48651],{"style":30876},[15,48649],{"className":48650,"style":1368},[217],[15,48652,48654],{"className":48653},[222,223,224,225],[15,48655,48657],{"className":48656},[66,225],[15,48658,48383],{"className":48659},[66,67,225],[15,48661,48662,48665],{"style":23445},[15,48663],{"className":48664,"style":1368},[217],[15,48666],{"className":48667,"style":4440},[23452],[15,48669,48670,48673],{"style":30899},[15,48671],{"className":48672,"style":1368},[217],[15,48674,48676],{"className":48675},[222,223,224,225],[15,48677,48679],{"className":48678},[66,225],[15,48680,37],{"className":48681},[66,67,225],[15,48683,1114],{"className":48684},[1113],[15,48686,48688],{"className":48687},[206],[15,48689,48691],{"className":48690,"style":30921},[210],[15,48692],{},[15,48694],{"className":48695},[169,23417],[15,48697,48699],{"className":48698,"style":1418},[169,1417],[15,48700,126],{"className":48701},[1422,1508],[15,48703,48705],{"className":48704},[198],[15,48706,48708],{"className":48707},[202],[15,48709,48711],{"className":48710},[206],[15,48712,48715],{"className":48713,"style":48714},[210],"height:0.9043em;",[15,48716,48717,48720],{"style":21274},[15,48718],{"className":48719,"style":218},[217],[15,48721,48723],{"className":48722},[222,223,224,225],[15,48724,37],{"className":48725},[66,67,225],[11,48727,48728],{},"首先引入一个极限：",[15,48730,48732],{"className":48731,"translate":19},[294],[15,48733,48735,48799],{"className":48734,"translate":19},[18],[15,48736,48738],{"className":48737},[23],[25,48739,48740],{"xmlns":27,"display":303},[29,48741,48742,48796],{},[32,48743,48744,48760,48788,48790],{},[1172,48745,48746,48752],{},[32,48747,48748,48750],{},[35,48749,43668],{},[39,48751,1170],{},[32,48753,48754,48756,48758],{},[35,48755,37],{},[39,48757,449],{},[35,48759,7620],{"mathvariant":138},[131,48761,48762,48782],{},[32,48763,48764,48766,48780],{},[39,48765,120],{"fence":53},[6467,48767,48768,48774],{},[32,48769,48770,48772],{},[35,48771,37],{},[39,48773,40769],{"stretchy":119},[131,48775,48776,48778],{},[35,48777,37],{},[35,48779,37],{},[39,48781,126],{"fence":53},[6467,48783,48784,48786],{},[1209,48785,1211],{},[35,48787,37],{},[39,48789,129],{},[6467,48791,48792,48794],{},[1209,48793,1211],{},[35,48795,48383],{},[45,48797,48798],{"encoding":47},"\\lim_{n \\rightarrow \\infty} \\left( \\frac{n!}{n^n} \\right)^{\\frac{1}{n}} = \\frac{1}{e}",[15,48800,48802,49080],{"className":48801,"ariaHidden":53},[52],[15,48803,48805,48809,48864,48867,49071,49074,49077],{"className":48804},[57],[15,48806],{"className":48807,"style":48808},[61],"height:2.744em;vertical-align:-0.95em;",[15,48810,48812],{"className":48811},[1339,1351],[15,48813,48815,48855],{"className":48814},[202,1090],[15,48816,48818,48852],{"className":48817},[206],[15,48819,48821,48842],{"className":48820,"style":840},[210],[15,48822,48824,48827],{"style":48823},"top:-2.4em;margin-left:0em;",[15,48825],{"className":48826,"style":1368},[217],[15,48828,48830],{"className":48829},[222,223,224,225],[15,48831,48833,48836,48839],{"className":48832},[66,225],[15,48834,37],{"className":48835},[66,67,225],[15,48837,449],{"className":48838},[177,225],[15,48840,7620],{"className":48841},[66,225],[15,48843,48844,48847],{"style":1386},[15,48845],{"className":48846,"style":1368},[217],[15,48848,48849],{},[15,48850,43668],{"className":48851},[1339],[15,48853,1114],{"className":48854},[1113],[15,48856,48858],{"className":48857},[206],[15,48859,48862],{"className":48860,"style":48861},[210],"height:0.7em;",[15,48863],{},[15,48865],{"className":48866,"style":697},[71],[15,48868,48870,48977],{"className":48869},[1413],[15,48871,48873,48879,48971],{"className":48872},[1413],[15,48874,48876],{"className":48875,"style":1418},[162,1417],[15,48877,120],{"className":48878},[1422,224],[15,48880,48882,48885,48968],{"className":48881},[66],[15,48883],{"className":48884},[162,23417],[15,48886,48888],{"className":48887},[6467],[15,48889,48891,48960],{"className":48890},[202,1090],[15,48892,48894,48957],{"className":48893},[206],[15,48895,48897,48935,48943],{"className":48896,"style":27435},[210],[15,48898,48899,48902],{"style":23433},[15,48900],{"className":48901,"style":1368},[217],[15,48903,48905],{"className":48904},[66],[15,48906,48908,48911],{"className":48907},[66],[15,48909,37],{"className":48910},[66,67],[15,48912,48914],{"className":48913},[198],[15,48915,48917],{"className":48916},[202],[15,48918,48920],{"className":48919},[206],[15,48921,48924],{"className":48922,"style":48923},[210],"height:0.5904em;",[15,48925,48926,48929],{"style":46255},[15,48927],{"className":48928,"style":218},[217],[15,48930,48932],{"className":48931},[222,223,224,225],[15,48933,37],{"className":48934},[66,67,225],[15,48936,48937,48940],{"style":23445},[15,48938],{"className":48939,"style":1368},[217],[15,48941],{"className":48942,"style":4440},[23452],[15,48944,48945,48948],{"style":23455},[15,48946],{"className":48947,"style":1368},[217],[15,48949,48951,48954],{"className":48950},[66],[15,48952,37],{"className":48953},[66,67],[15,48955,40769],{"className":48956},[169],[15,48958,1114],{"className":48959},[1113],[15,48961,48963],{"className":48962},[206],[15,48964,48966],{"className":48965,"style":23474},[210],[15,48967],{},[15,48969],{"className":48970},[169,23417],[15,48972,48974],{"className":48973,"style":1418},[169,1417],[15,48975,126],{"className":48976},[1422,224],[15,48978,48980],{"className":48979},[198],[15,48981,48983],{"className":48982},[202],[15,48984,48986],{"className":48985},[206],[15,48987,48990],{"className":48988,"style":48989},[210],"height:1.7939em;",[15,48991,48993,48996],{"style":48992},"top:-4.2029em;margin-right:0.05em;",[15,48994],{"className":48995,"style":1368},[217],[15,48997,48999],{"className":48998},[222,223,224,225],[15,49000,49002],{"className":49001},[66,225],[15,49003,49005,49008,49068],{"className":49004},[66,225],[15,49006],{"className":49007},[162,23417,222,3679,33581],[15,49009,49011],{"className":49010},[6467],[15,49012,49014,49060],{"className":49013},[202,1090],[15,49015,49017,49057],{"className":49016},[206],[15,49018,49021,49035,49043],{"className":49019,"style":49020},[210],"height:0.8443em;",[15,49022,49023,49026],{"style":33597},[15,49024],{"className":49025,"style":1368},[217],[15,49027,49029],{"className":49028},[222,3679,1508,225],[15,49030,49032],{"className":49031},[66,225],[15,49033,37],{"className":49034},[66,67,225],[15,49036,49037,49040],{"style":33612},[15,49038],{"className":49039,"style":1368},[217],[15,49041],{"className":49042,"style":33619},[23452,225],[15,49044,49045,49048],{"style":33622},[15,49046],{"className":49047,"style":1368},[217],[15,49049,49051],{"className":49050},[222,3679,1508,225],[15,49052,49054],{"className":49053},[66,225],[15,49055,1211],{"className":49056},[66,225],[15,49058,1114],{"className":49059},[1113],[15,49061,49063],{"className":49062},[206],[15,49064,49066],{"className":49065,"style":33644},[210],[15,49067],{},[15,49069],{"className":49070},[169,23417,222,3679,33581],[15,49072],{"className":49073,"style":173},[71],[15,49075,129],{"className":49076},[177],[15,49078],{"className":49079,"style":173},[71],[15,49081,49083,49087],{"className":49082},[57],[15,49084],{"className":49085,"style":49086},[61],"height:2.0074em;vertical-align:-0.686em;",[15,49088,49090,49093,49146],{"className":49089},[66],[15,49091],{"className":49092},[162,23417],[15,49094,49096],{"className":49095},[6467],[15,49097,49099,49138],{"className":49098},[202,1090],[15,49100,49102,49135],{"className":49101},[206],[15,49103,49105,49116,49124],{"className":49104,"style":23430},[210],[15,49106,49107,49110],{"style":23433},[15,49108],{"className":49109,"style":1368},[217],[15,49111,49113],{"className":49112},[66],[15,49114,48383],{"className":49115},[66,67],[15,49117,49118,49121],{"style":23445},[15,49119],{"className":49120,"style":1368},[217],[15,49122],{"className":49123,"style":4440},[23452],[15,49125,49126,49129],{"style":23455},[15,49127],{"className":49128,"style":1368},[217],[15,49130,49132],{"className":49131},[66],[15,49133,1211],{"className":49134},[66],[15,49136,1114],{"className":49137},[1113],[15,49139,49141],{"className":49140},[206],[15,49142,49144],{"className":49143,"style":23474},[210],[15,49145],{},[15,49147],{"className":49148},[169,23417],[11,49150,49151],{},"证明：",[15,49153,49155],{"className":49154,"translate":19},[294],[15,49156,49158,49269],{"className":49157,"translate":19},[18],[15,49159,49161],{"className":49160},[23],[25,49162,49163],{"xmlns":27,"display":303},[29,49164,49165,49266],{},[32,49166,49167,49183,49211,49213,49229],{},[1172,49168,49169,49175],{},[32,49170,49171,49173],{},[35,49172,43668],{},[39,49174,1170],{},[32,49176,49177,49179,49181],{},[35,49178,37],{},[39,49180,449],{},[35,49182,7620],{"mathvariant":138},[131,49184,49185,49205],{},[32,49186,49187,49189,49203],{},[39,49188,120],{"fence":53},[6467,49190,49191,49197],{},[32,49192,49193,49195],{},[35,49194,37],{},[39,49196,40769],{"stretchy":119},[131,49198,49199,49201],{},[35,49200,37],{},[35,49202,37],{},[39,49204,126],{"fence":53},[6467,49206,49207,49209],{},[1209,49208,1211],{},[35,49210,37],{},[39,49212,129],{},[1172,49214,49215,49221],{},[32,49216,49217,49219],{},[35,49218,43668],{},[39,49220,1170],{},[32,49222,49223,49225,49227],{},[35,49224,37],{},[39,49226,449],{},[35,49228,7620],{"mathvariant":138},[131,49230,49231,49233],{},[35,49232,48383],{},[32,49234,49235,49241,49244,49246],{},[6467,49236,49237,49239],{},[1209,49238,1211],{},[35,49240,37],{},[35,49242,49243],{},"ln",[39,49245,1170],{},[32,49247,49248,49250,49264],{},[39,49249,120],{"fence":53},[6467,49251,49252,49258],{},[32,49253,49254,49256],{},[35,49255,37],{},[39,49257,40769],{"stretchy":119},[131,49259,49260,49262],{},[35,49261,37],{},[35,49263,37],{},[39,49265,126],{"fence":53},[45,49267,49268],{"encoding":47},"\\lim_{n \\rightarrow \\infty} \\left( \\frac{n!}{n^n} \\right)^{\\frac{1}{n}} = \\lim_{n \\rightarrow \\infty} e^{\\frac{1}{n} \\ln \\left( \\frac{n!}{n^n} \\right)}",[15,49270,49272,49543],{"className":49271,"ariaHidden":53},[52],[15,49273,49275,49278,49331,49334,49534,49537,49540],{"className":49274},[57],[15,49276],{"className":49277,"style":48808},[61],[15,49279,49281],{"className":49280},[1339,1351],[15,49282,49284,49323],{"className":49283},[202,1090],[15,49285,49287,49320],{"className":49286},[206],[15,49288,49290,49310],{"className":49289,"style":840},[210],[15,49291,49292,49295],{"style":48823},[15,49293],{"className":49294,"style":1368},[217],[15,49296,49298],{"className":49297},[222,223,224,225],[15,49299,49301,49304,49307],{"className":49300},[66,225],[15,49302,37],{"className":49303},[66,67,225],[15,49305,449],{"className":49306},[177,225],[15,49308,7620],{"className":49309},[66,225],[15,49311,49312,49315],{"style":1386},[15,49313],{"className":49314,"style":1368},[217],[15,49316,49317],{},[15,49318,43668],{"className":49319},[1339],[15,49321,1114],{"className":49322},[1113],[15,49324,49326],{"className":49325},[206],[15,49327,49329],{"className":49328,"style":48861},[210],[15,49330],{},[15,49332],{"className":49333,"style":697},[71],[15,49335,49337,49443],{"className":49336},[1413],[15,49338,49340,49346,49437],{"className":49339},[1413],[15,49341,49343],{"className":49342,"style":1418},[162,1417],[15,49344,120],{"className":49345},[1422,224],[15,49347,49349,49352,49434],{"className":49348},[66],[15,49350],{"className":49351},[162,23417],[15,49353,49355],{"className":49354},[6467],[15,49356,49358,49426],{"className":49357},[202,1090],[15,49359,49361,49423],{"className":49360},[206],[15,49362,49364,49401,49409],{"className":49363,"style":27435},[210],[15,49365,49366,49369],{"style":23433},[15,49367],{"className":49368,"style":1368},[217],[15,49370,49372],{"className":49371},[66],[15,49373,49375,49378],{"className":49374},[66],[15,49376,37],{"className":49377},[66,67],[15,49379,49381],{"className":49380},[198],[15,49382,49384],{"className":49383},[202],[15,49385,49387],{"className":49386},[206],[15,49388,49390],{"className":49389,"style":48923},[210],[15,49391,49392,49395],{"style":46255},[15,49393],{"className":49394,"style":218},[217],[15,49396,49398],{"className":49397},[222,223,224,225],[15,49399,37],{"className":49400},[66,67,225],[15,49402,49403,49406],{"style":23445},[15,49404],{"className":49405,"style":1368},[217],[15,49407],{"className":49408,"style":4440},[23452],[15,49410,49411,49414],{"style":23455},[15,49412],{"className":49413,"style":1368},[217],[15,49415,49417,49420],{"className":49416},[66],[15,49418,37],{"className":49419},[66,67],[15,49421,40769],{"className":49422},[169],[15,49424,1114],{"className":49425},[1113],[15,49427,49429],{"className":49428},[206],[15,49430,49432],{"className":49431,"style":23474},[210],[15,49433],{},[15,49435],{"className":49436},[169,23417],[15,49438,49440],{"className":49439,"style":1418},[169,1417],[15,49441,126],{"className":49442},[1422,224],[15,49444,49446],{"className":49445},[198],[15,49447,49449],{"className":49448},[202],[15,49450,49452],{"className":49451},[206],[15,49453,49455],{"className":49454,"style":48989},[210],[15,49456,49457,49460],{"style":48992},[15,49458],{"className":49459,"style":1368},[217],[15,49461,49463],{"className":49462},[222,223,224,225],[15,49464,49466],{"className":49465},[66,225],[15,49467,49469,49472,49531],{"className":49468},[66,225],[15,49470],{"className":49471},[162,23417,222,3679,33581],[15,49473,49475],{"className":49474},[6467],[15,49476,49478,49523],{"className":49477},[202,1090],[15,49479,49481,49520],{"className":49480},[206],[15,49482,49484,49498,49506],{"className":49483,"style":49020},[210],[15,49485,49486,49489],{"style":33597},[15,49487],{"className":49488,"style":1368},[217],[15,49490,49492],{"className":49491},[222,3679,1508,225],[15,49493,49495],{"className":49494},[66,225],[15,49496,37],{"className":49497},[66,67,225],[15,49499,49500,49503],{"style":33612},[15,49501],{"className":49502,"style":1368},[217],[15,49504],{"className":49505,"style":33619},[23452,225],[15,49507,49508,49511],{"style":33622},[15,49509],{"className":49510,"style":1368},[217],[15,49512,49514],{"className":49513},[222,3679,1508,225],[15,49515,49517],{"className":49516},[66,225],[15,49518,1211],{"className":49519},[66,225],[15,49521,1114],{"className":49522},[1113],[15,49524,49526],{"className":49525},[206],[15,49527,49529],{"className":49528,"style":33644},[210],[15,49530],{},[15,49532],{"className":49533},[169,23417,222,3679,33581],[15,49535],{"className":49536,"style":173},[71],[15,49538,129],{"className":49539},[177],[15,49541],{"className":49542,"style":173},[71],[15,49544,49546,49550,49603,49606],{"className":49545},[57],[15,49547],{"className":49548,"style":49549},[61],"height:1.8105em;vertical-align:-0.7em;",[15,49551,49553],{"className":49552},[1339,1351],[15,49554,49556,49595],{"className":49555},[202,1090],[15,49557,49559,49592],{"className":49558},[206],[15,49560,49562,49582],{"className":49561,"style":840},[210],[15,49563,49564,49567],{"style":48823},[15,49565],{"className":49566,"style":1368},[217],[15,49568,49570],{"className":49569},[222,223,224,225],[15,49571,49573,49576,49579],{"className":49572},[66,225],[15,49574,37],{"className":49575},[66,67,225],[15,49577,449],{"className":49578},[177,225],[15,49580,7620],{"className":49581},[66,225],[15,49583,49584,49587],{"style":1386},[15,49585],{"className":49586,"style":1368},[217],[15,49588,49589],{},[15,49590,43668],{"className":49591},[1339],[15,49593,1114],{"className":49594},[1113],[15,49596,49598],{"className":49597},[206],[15,49599,49601],{"className":49600,"style":48861},[210],[15,49602],{},[15,49604],{"className":49605,"style":697},[71],[15,49607,49609,49612],{"className":49608},[66],[15,49610,48383],{"className":49611},[66,67],[15,49613,49615],{"className":49614},[198],[15,49616,49618],{"className":49617},[202],[15,49619,49621],{"className":49620},[206],[15,49622,49625],{"className":49623,"style":49624},[210],"height:1.1105em;",[15,49626,49628,49631],{"style":49627},"top:-3.413em;margin-right:0.05em;",[15,49629],{"className":49630,"style":1368},[217],[15,49632,49634],{"className":49633},[222,223,224,225],[15,49635,49637,49705,49708,49718],{"className":49636},[66,225],[15,49638,49640,49643,49702],{"className":49639},[66,225],[15,49641],{"className":49642},[162,23417,222,3679,33581],[15,49644,49646],{"className":49645},[6467],[15,49647,49649,49694],{"className":49648},[202,1090],[15,49650,49652,49691],{"className":49651},[206],[15,49653,49655,49669,49677],{"className":49654,"style":49020},[210],[15,49656,49657,49660],{"style":33597},[15,49658],{"className":49659,"style":1368},[217],[15,49661,49663],{"className":49662},[222,3679,1508,225],[15,49664,49666],{"className":49665},[66,225],[15,49667,37],{"className":49668},[66,67,225],[15,49670,49671,49674],{"style":33612},[15,49672],{"className":49673,"style":1368},[217],[15,49675],{"className":49676,"style":33619},[23452,225],[15,49678,49679,49682],{"style":33622},[15,49680],{"className":49681,"style":1368},[217],[15,49683,49685],{"className":49684},[222,3679,1508,225],[15,49686,49688],{"className":49687},[66,225],[15,49689,1211],{"className":49690},[66,225],[15,49692,1114],{"className":49693},[1113],[15,49695,49697],{"className":49696},[206],[15,49698,49700],{"className":49699,"style":33644},[210],[15,49701],{},[15,49703],{"className":49704},[169,23417,222,3679,33581],[15,49706],{"className":49707,"style":23368},[71,225],[15,49709,49711,49715],{"className":49710},[1339,225],[15,49712,49714],{"className":49713},[225],"l",[15,49716,37],{"className":49717},[225],[15,49719,49721,49728,49825],{"className":49720},[1413,225],[15,49722,49725],{"className":49723,"style":49724},[162,222,3679,33581,225,1417],"top:0.075em;",[15,49726,120],{"className":49727},[225],[15,49729,49731,49734,49822],{"className":49730},[66,225],[15,49732],{"className":49733},[162,23417,222,3679,33581],[15,49735,49737],{"className":49736},[6467],[15,49738,49740,49814],{"className":49739},[202,1090],[15,49741,49743,49811],{"className":49742},[206],[15,49744,49747,49786,49794],{"className":49745,"style":49746},[210],"height:0.88em;",[15,49748,49749,49752],{"style":33597},[15,49750],{"className":49751,"style":1368},[217],[15,49753,49755],{"className":49754},[222,3679,1508,225],[15,49756,49758],{"className":49757},[66,225],[15,49759,49761,49764],{"className":49760},[66,225],[15,49762,37],{"className":49763},[66,67,225],[15,49765,49767],{"className":49766},[198],[15,49768,49770],{"className":49769},[202],[15,49771,49773],{"className":49772},[206],[15,49774,49777],{"className":49775,"style":49776},[210],"height:0.7246em;",[15,49778,49780,49783],{"style":49779},"top:-2.794em;margin-right:0.1em;",[15,49781],{"className":49782,"style":3675},[217],[15,49784,37],{"className":49785},[66,67,225],[15,49787,49788,49791],{"style":33612},[15,49789],{"className":49790,"style":1368},[217],[15,49792],{"className":49793,"style":33619},[23452,225],[15,49795,49796,49799],{"style":33622},[15,49797],{"className":49798,"style":1368},[217],[15,49800,49802],{"className":49801},[222,3679,1508,225],[15,49803,49805,49808],{"className":49804},[66,225],[15,49806,37],{"className":49807},[66,67,225],[15,49809,40769],{"className":49810},[169,225],[15,49812,1114],{"className":49813},[1113],[15,49815,49817],{"className":49816},[206],[15,49818,49820],{"className":49819,"style":33644},[210],[15,49821],{},[15,49823],{"className":49824},[169,23417,222,3679,33581],[15,49826,49828],{"className":49827,"style":49724},[169,222,3679,33581,225,1417],[15,49829,126],{"className":49830},[225],[11,49832,49833],{},"等价于证明：",[15,49835,49837],{"className":49836,"translate":19},[294],[15,49838,49840,49904],{"className":49839,"translate":19},[18],[15,49841,49843],{"className":49842},[23],[25,49844,49845],{"xmlns":27,"display":303},[29,49846,49847,49901],{},[32,49848,49849,49865,49871,49873,49875,49895,49897,49899],{},[1172,49850,49851,49857],{},[32,49852,49853,49855],{},[35,49854,43668],{},[39,49856,1170],{},[32,49858,49859,49861,49863],{},[35,49860,37],{},[39,49862,449],{},[35,49864,7620],{"mathvariant":138},[6467,49866,49867,49869],{},[1209,49868,1211],{},[35,49870,37],{},[35,49872,49243],{},[39,49874,1170],{},[32,49876,49877,49879,49893],{},[39,49878,120],{"fence":53},[6467,49880,49881,49887],{},[32,49882,49883,49885],{},[35,49884,37],{},[39,49886,40769],{"stretchy":119},[131,49888,49889,49891],{},[35,49890,37],{},[35,49892,37],{},[39,49894,126],{"fence":53},[39,49896,129],{},[39,49898,6602],{},[1209,49900,1211],{},[45,49902,49903],{"encoding":47},"\\lim_{n \\rightarrow \\infty} \\frac{1}{n} \\ln \\left( \\frac{n!}{n^n} \\right) = -1",[15,49905,49907,50155],{"className":49906,"ariaHidden":53},[52],[15,49908,49910,49913,49966,49969,50031,50034,50037,50040,50146,50149,50152],{"className":49909},[57],[15,49911],{"className":49912,"style":43973},[61],[15,49914,49916],{"className":49915},[1339,1351],[15,49917,49919,49958],{"className":49918},[202,1090],[15,49920,49922,49955],{"className":49921},[206],[15,49923,49925,49945],{"className":49924,"style":840},[210],[15,49926,49927,49930],{"style":48823},[15,49928],{"className":49929,"style":1368},[217],[15,49931,49933],{"className":49932},[222,223,224,225],[15,49934,49936,49939,49942],{"className":49935},[66,225],[15,49937,37],{"className":49938},[66,67,225],[15,49940,449],{"className":49941},[177,225],[15,49943,7620],{"className":49944},[66,225],[15,49946,49947,49950],{"style":1386},[15,49948],{"className":49949,"style":1368},[217],[15,49951,49952],{},[15,49953,43668],{"className":49954},[1339],[15,49956,1114],{"className":49957},[1113],[15,49959,49961],{"className":49960},[206],[15,49962,49964],{"className":49963,"style":48861},[210],[15,49965],{},[15,49967],{"className":49968,"style":697},[71],[15,49970,49972,49975,50028],{"className":49971},[66],[15,49973],{"className":49974},[162,23417],[15,49976,49978],{"className":49977},[6467],[15,49979,49981,50020],{"className":49980},[202,1090],[15,49982,49984,50017],{"className":49983},[206],[15,49985,49987,49998,50006],{"className":49986,"style":23430},[210],[15,49988,49989,49992],{"style":23433},[15,49990],{"className":49991,"style":1368},[217],[15,49993,49995],{"className":49994},[66],[15,49996,37],{"className":49997},[66,67],[15,49999,50000,50003],{"style":23445},[15,50001],{"className":50002,"style":1368},[217],[15,50004],{"className":50005,"style":4440},[23452],[15,50007,50008,50011],{"style":23455},[15,50009],{"className":50010,"style":1368},[217],[15,50012,50014],{"className":50013},[66],[15,50015,1211],{"className":50016},[66],[15,50018,1114],{"className":50019},[1113],[15,50021,50023],{"className":50022},[206],[15,50024,50026],{"className":50025,"style":23474},[210],[15,50027],{},[15,50029],{"className":50030},[169,23417],[15,50032],{"className":50033,"style":697},[71],[15,50035,49243],{"className":50036},[1339],[15,50038],{"className":50039,"style":697},[71],[15,50041,50043,50049,50140],{"className":50042},[1413],[15,50044,50046],{"className":50045,"style":1418},[162,1417],[15,50047,120],{"className":50048},[1422,224],[15,50050,50052,50055,50137],{"className":50051},[66],[15,50053],{"className":50054},[162,23417],[15,50056,50058],{"className":50057},[6467],[15,50059,50061,50129],{"className":50060},[202,1090],[15,50062,50064,50126],{"className":50063},[206],[15,50065,50067,50104,50112],{"className":50066,"style":27435},[210],[15,50068,50069,50072],{"style":23433},[15,50070],{"className":50071,"style":1368},[217],[15,50073,50075],{"className":50074},[66],[15,50076,50078,50081],{"className":50077},[66],[15,50079,37],{"className":50080},[66,67],[15,50082,50084],{"className":50083},[198],[15,50085,50087],{"className":50086},[202],[15,50088,50090],{"className":50089},[206],[15,50091,50093],{"className":50092,"style":48923},[210],[15,50094,50095,50098],{"style":46255},[15,50096],{"className":50097,"style":218},[217],[15,50099,50101],{"className":50100},[222,223,224,225],[15,50102,37],{"className":50103},[66,67,225],[15,50105,50106,50109],{"style":23445},[15,50107],{"className":50108,"style":1368},[217],[15,50110],{"className":50111,"style":4440},[23452],[15,50113,50114,50117],{"style":23455},[15,50115],{"className":50116,"style":1368},[217],[15,50118,50120,50123],{"className":50119},[66],[15,50121,37],{"className":50122},[66,67],[15,50124,40769],{"className":50125},[169],[15,50127,1114],{"className":50128},[1113],[15,50130,50132],{"className":50131},[206],[15,50133,50135],{"className":50134,"style":23474},[210],[15,50136],{},[15,50138],{"className":50139},[169,23417],[15,50141,50143],{"className":50142,"style":1418},[169,1417],[15,50144,126],{"className":50145},[1422,224],[15,50147],{"className":50148,"style":173},[71],[15,50150,129],{"className":50151},[177],[15,50153],{"className":50154,"style":173},[71],[15,50156,50158,50161,50164],{"className":50157},[57],[15,50159],{"className":50160,"style":30468},[61],[15,50162,6602],{"className":50163},[66],[15,50165,1211],{"className":50166},[66],[11,50168,50169],{},"我们有：",[15,50171,50173],{"className":50172,"translate":19},[294],[15,50174,50176,50360],{"className":50175,"translate":19},[18],[15,50177,50179],{"className":50178},[23],[25,50180,50181],{"xmlns":27,"display":303},[29,50182,50183,50357],{},[32,50184,50185,50201,50207,50209,50211,50231,50233,50235,50251,50257,50301,50303,50305,50313,50315,50317,50319,50321,50323,50325,50327,50329,50331,50333,50335,50337,50339,50341,50343,50351,50353,50355],{},[1172,50186,50187,50193],{},[32,50188,50189,50191],{},[35,50190,43668],{},[39,50192,1170],{},[32,50194,50195,50197,50199],{},[35,50196,37],{},[39,50198,449],{},[35,50200,7620],{"mathvariant":138},[6467,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\\rightarrow \\infty} \\frac{1}{n} \\ln \\left( \\frac{n!}{n^n} \\right) \\\\ = \\lim_{n \\rightarrow \\infty} \\frac{1}{n} \\left( \\ln \\frac{1}{n} + \\ln \\frac{2}{n} + \\cdots + \\ln \\frac{n}{n} \\right) \\\\ = \\int_{0}^{1} \\ln x \\, dx = (x \\ln x - x) \\Big|_{0}^{1} = 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264],{},"证毕。",[11,51266,35909],{},[15,51268,51270],{"className":51269,"translate":19},[294],[15,51271,51273,51311],{"className":51272,"translate":19},[18],[15,51274,51276],{"className":51275},[23],[25,51277,51278],{"xmlns":27,"display":303},[29,51279,51280,51308],{},[32,51281,51282,51296,51298],{},[6467,51283,51284,51290],{},[32,51285,51286,51288],{},[35,51287,37],{},[39,51289,40769],{"stretchy":119},[131,51291,51292,51294],{},[35,51293,37],{},[35,51295,37],{},[39,51297,28391],{},[6467,51299,51300,51302],{},[1209,51301,1211],{},[131,51303,51304,51306],{},[35,51305,48383],{},[35,51307,37],{},[45,51309,51310],{"encoding":47},"\\frac{n!}{n^n} \\sim \\frac{1}{e^n}",[15,51312,51314,51420],{"className":51313,"ariaHidden":53},[52],[15,51315,51317,51320,51411,51414,51417],{"className":51316},[57],[15,51318],{"className":51319,"style":38387},[61],[15,51321,51323,51326,51408],{"className":51322},[66],[15,51324],{"className":51325},[162,23417],[15,51327,51329],{"className":51328},[6467],[15,51330,51332,51400],{"className":51331},[202,1090],[15,51333,51335,51397],{"className":51334},[206],[15,51336,51338,51375,51383],{"className":51337,"style":27435},[210],[15,51339,51340,51343],{"style":23433},[15,51341],{"className":51342,"style":1368},[217],[15,51344,51346],{"className":51345},[66],[15,51347,51349,51352],{"className":51348},[66],[15,51350,37],{"className":51351},[66,67],[15,51353,51355],{"className":51354},[198],[15,51356,51358],{"className":51357},[202],[15,51359,51361],{"className":51360},[206],[15,51362,51364],{"className":51363,"style":48923},[210],[15,51365,51366,51369],{"style":46255},[15,51367],{"className":51368,"style":218},[217],[15,51370,51372],{"className":51371},[222,223,224,225],[15,51373,37],{"className":51374},[66,67,225],[15,51376,51377,51380],{"style":23445},[15,51378],{"className":51379,"style":1368},[217],[15,51381],{"className":51382,"style":4440},[23452],[15,51384,51385,51388],{"style":23455},[15,51386],{"className":51387,"style":1368},[217],[15,51389,51391,51394],{"className":51390},[66],[15,51392,37],{"className":51393},[66,67],[15,51395,40769],{"className":51396},[169],[15,51398,1114],{"className":51399},[1113],[15,51401,51403],{"className":51402},[206],[15,51404,51406],{"className":51405,"style":23474},[210],[15,51407],{},[15,51409],{"className":51410},[169,23417],[15,51412],{"className":51413,"style":173},[71],[15,51415,28391],{"className":51416},[177],[15,51418],{"className":51419,"style":173},[71],[15,51421,51423,51426],{"className":51422},[57],[15,51424],{"className":51425,"style":49086},[61],[15,51427,51429,51432,51511],{"className":51428},[66],[15,51430],{"className":51431},[162,23417],[15,51433,51435],{"className":51434},[6467],[15,51436,51438,51503],{"className":51437},[202,1090],[15,51439,51441,51500],{"className":51440},[206],[15,51442,51444,51481,51489],{"className":51443,"style":23430},[210],[15,51445,51446,51449],{"style":23433},[15,51447],{"className":51448,"style":1368},[217],[15,51450,51452],{"className":51451},[66],[15,51453,51455,51458],{"className":51454},[66],[15,51456,48383],{"className":51457},[66,67],[15,51459,51461],{"className":51460},[198],[15,51462,51464],{"className":51463},[202],[15,51465,51467],{"className":51466},[206],[15,51468,51470],{"className":51469,"style":48923},[210],[15,51471,51472,51475],{"style":46255},[15,51473],{"className":51474,"style":218},[217],[15,51476,51478],{"className":51477},[222,223,224,225],[15,51479,37],{"className":51480},[66,67,225],[15,51482,51483,51486],{"style":23445},[15,51484],{"className":51485,"style":1368},[217],[15,51487],{"className":51488,"style":4440},[23452],[15,51490,51491,51494],{"style":23455},[15,51492],{"className":51493,"style":1368},[217],[15,51495,51497],{"className":51496},[66],[15,51498,1211],{"className":51499},[66],[15,51501,1114],{"className":51502},[1113],[15,51504,51506],{"className":51505},[206],[15,51507,51509],{"className":51508,"style":23474},[210],[15,51510],{},[15,51512],{"className":51513},[169,23417],[11,51515,51516],{},"我们希望验证：",[15,51518,51520],{"className":51519,"translate":19},[294],[15,51521,51523,51575],{"className":51522,"translate":19},[18],[15,51524,51526],{"className":51525},[23],[25,51527,51528],{"xmlns":27,"display":303},[29,51529,51530,51572],{},[32,51531,51532,51548,51562,51568,51570],{},[1172,51533,51534,51540],{},[32,51535,51536,51538],{},[35,51537,43668],{},[39,51539,1170],{},[32,51541,51542,51544,51546],{},[35,51543,37],{},[39,51545,449],{},[35,51547,7620],{"mathvariant":138},[6467,51549,51550,51556],{},[32,51551,51552,51554],{},[35,51553,37],{},[39,51555,40769],{"stretchy":119},[131,51557,51558,51560],{},[35,51559,37],{},[35,51561,37],{},[131,51563,51564,51566],{},[35,51565,48383],{},[35,51567,37],{},[39,51569,129],{},[1209,51571,1211],{},[45,51573,51574],{"encoding":47},"\\lim_{n \\rightarrow \\infty} \\frac{n!}{n^n} e^n = 1",[15,51576,51578,51771],{"className":51577,"ariaHidden":53},[52],[15,51579,51581,51585,51638,51641,51732,51762,51765,51768],{"className":51580},[57],[15,51582],{"className":51583,"style":51584},[61],"height:2.0714em;vertical-align:-0.7em;",[15,51586,51588],{"className":51587},[1339,1351],[15,51589,51591,51630],{"className":51590},[202,1090],[15,51592,51594,51627],{"className":51593},[206],[15,51595,51597,51617],{"className":51596,"style":840},[210],[15,51598,51599,51602],{"style":48823},[15,51600],{"className":51601,"style":1368},[217],[15,51603,51605],{"className":51604},[222,223,224,225],[15,51606,51608,51611,51614],{"className":51607},[66,225],[15,51609,37],{"className":51610},[66,67,225],[15,51612,449],{"className":51613},[177,225],[15,51615,7620],{"className":51616},[66,225],[15,51618,51619,51622],{"style":1386},[15,51620],{"className":51621,"style":1368},[217],[15,51623,51624],{},[15,51625,43668],{"className":51626},[1339],[15,51628,1114],{"className":51629},[1113],[15,51631,51633],{"className":51632},[206],[15,51634,51636],{"className":51635,"style":48861},[210],[15,51637],{},[15,51639],{"className":51640,"style":697},[71],[15,51642,51644,51647,51729],{"className":51643},[66],[15,51645],{"className":51646},[162,23417],[15,51648,51650],{"className":51649},[6467],[15,51651,51653,51721],{"className":51652},[202,1090],[15,51654,51656,51718],{"className":51655},[206],[15,51657,51659,51696,51704],{"className":51658,"style":27435},[210],[15,51660,51661,51664],{"style":23433},[15,51662],{"className":51663,"style":1368},[217],[15,51665,51667],{"className":51666},[66],[15,51668,51670,51673],{"className":51669},[66],[15,51671,37],{"className":51672},[66,67],[15,51674,51676],{"className":51675},[198],[15,51677,51679],{"className":51678},[202],[15,51680,51682],{"className":51681},[206],[15,51683,51685],{"className":51684,"style":48923},[210],[15,51686,51687,51690],{"style":46255},[15,51688],{"className":51689,"style":218},[217],[15,51691,51693],{"className":51692},[222,223,224,225],[15,51694,37],{"className":51695},[66,67,225],[15,51697,51698,51701],{"style":23445},[15,51699],{"className":51700,"style":1368},[217],[15,51702],{"className":51703,"style":4440},[23452],[15,51705,51706,51709],{"style":23455},[15,51707],{"className":51708,"style":1368},[217],[15,51710,51712,51715],{"className":51711},[66],[15,51713,37],{"className":51714},[66,67],[15,51716,40769],{"className":51717},[169],[15,51719,1114],{"className":51720},[1113],[15,51722,51724],{"className":51723},[206],[15,51725,51727],{"className":51726,"style":23474},[210],[15,51728],{},[15,51730],{"className":51731},[169,23417],[15,51733,51735,51738],{"className":51734},[66],[15,51736,48383],{"className":51737},[66,67],[15,51739,51741],{"className":51740},[198],[15,51742,51744],{"className":51743},[202],[15,51745,51747],{"className":51746},[206],[15,51748,51751],{"className":51749,"style":51750},[210],"height:0.7144em;",[15,51752,51753,51756],{"style":397},[15,51754],{"className":51755,"style":218},[217],[15,51757,51759],{"className":51758},[222,223,224,225],[15,51760,37],{"className":51761},[66,67,225],[15,51763],{"className":51764,"style":173},[71],[15,51766,129],{"className":51767},[177],[15,51769],{"className":51770,"style":173},[71],[15,51772,51774,51777],{"className":51773},[57],[15,51775],{"className":51776,"style":1836},[61],[15,51778,1211],{"className":51779},[66],[11,51781,51782],{},"我们试一下。",[11,51784,51785],{},"使用比值判别法：",[15,51787,51789],{"className":51788,"translate":19},[294],[15,51790,51792,51900],{"className":51791,"translate":19},[18],[15,51793,51795],{"className":51794},[23],[25,51796,51797],{"xmlns":27,"display":303},[29,51798,51799,51897],{},[32,51800,51801,51807,51809,51823,51829,51831,51833,51845,51847,51885],{},[1057,51802,51803,51805],{},[35,51804,4556],{},[35,51806,37],{},[39,51808,129],{},[6467,51810,51811,51817],{},[32,51812,51813,51815],{},[35,51814,37],{},[39,51816,40769],{"stretchy":119},[131,51818,51819,51821],{},[35,51820,37],{},[35,51822,37],{},[131,51824,51825,51827],{},[35,51826,48383],{},[35,51828,37],{},[39,51830,335],{"separator":53},[71,51832],{"width":9791},[1057,51834,51835,51837],{},[35,51836,4556],{},[32,51838,51839,51841,51843],{},[35,51840,37],{},[39,51842,1245],{},[1209,51844,1211],{},[39,51846,129],{},[6467,51848,51849,51863],{},[32,51850,51851,51853,51855,51857,51859,51861],{},[39,51852,120],{"stretchy":119},[35,51854,37],{},[39,51856,1245],{},[1209,51858,1211],{},[39,51860,126],{"stretchy":119},[39,51862,40769],{"stretchy":119},[32,51864,51865,51867,51869,51871,51873],{},[39,51866,120],{"stretchy":119},[35,51868,37],{},[39,51870,1245],{},[1209,51872,1211],{},[131,51874,51875,51877],{},[39,51876,126],{"stretchy":119},[32,51878,51879,51881,51883],{},[35,51880,37],{},[39,51882,1245],{},[1209,51884,1211],{},[131,51886,51887,51889],{},[35,51888,48383],{},[32,51890,51891,51893,51895],{},[35,51892,37],{},[39,51894,1245],{},[1209,51896,1211],{},[45,51898,51899],{"encoding":47},"a_n = \\frac{n!}{n^n} e^n, \\quad a_{n+1} = \\frac{(n+1)!}{(n+1)^{n+1}} e^{n+1}",[15,51901,51903,51958,52152],{"className":51902,"ariaHidden":53},[52],[15,51904,51906,51909,51949,51952,51955],{"className":51905},[57],[15,51907],{"className":51908,"style":10197},[61],[15,51910,51912,51915],{"className":51911},[66],[15,51913,4556],{"className":51914},[66,67],[15,51916,51918],{"className":51917},[198],[15,51919,51921,51941],{"className":51920},[202,1090],[15,51922,51924,51938],{"className":51923},[206],[15,51925,51927],{"className":51926,"style":2198},[210],[15,51928,51929,51932],{"style":1584},[15,51930],{"className":51931,"style":218},[217],[15,51933,51935],{"className":51934},[222,223,224,225],[15,51936,37],{"className":51937},[66,67,225],[15,51939,1114],{"className":51940},[1113],[15,51942,51944],{"className":51943},[206],[15,51945,51947],{"className":51946,"style":1121},[210],[15,51948],{},[15,51950],{"className":51951,"style":173},[71],[15,51953,129],{"className":51954},[177],[15,51956],{"className":51957,"style":173},[71],[15,51959,51961,51964,52055,52084,52087,52090,52093,52143,52146,52149],{"className":51960},[57],[15,51962],{"className":51963,"style":38387},[61],[15,51965,51967,51970,52052],{"className":51966},[66],[15,51968],{"className":51969},[162,23417],[15,51971,51973],{"className":51972},[6467],[15,51974,51976,52044],{"className":51975},[202,1090],[15,51977,51979,52041],{"className":51978},[206],[15,51980,51982,52019,52027],{"className":51981,"style":27435},[210],[15,51983,51984,51987],{"style":23433},[15,51985],{"className":51986,"style":1368},[217],[15,51988,51990],{"className":51989},[66],[15,51991,51993,51996],{"className":51992},[66],[15,51994,37],{"className":51995},[66,67],[15,51997,51999],{"className":51998},[198],[15,52000,52002],{"className":52001},[202],[15,52003,52005],{"className":52004},[206],[15,52006,52008],{"className":52007,"style":48923},[210],[15,52009,52010,52013],{"style":46255},[15,52011],{"className":52012,"style":218},[217],[15,52014,52016],{"className":52015},[222,223,224,225],[15,52017,37],{"className":52018},[66,67,225],[15,52020,52021,52024],{"style":23445},[15,52022],{"className":52023,"style":1368},[217],[15,52025],{"className":52026,"style":4440},[23452],[15,52028,52029,52032],{"style":23455},[15,52030],{"className":52031,"style":1368},[217],[15,52033,52035,52038],{"className":52034},[66],[15,52036,37],{"className":52037},[66,67],[15,52039,40769],{"className":52040},[169],[15,52042,1114],{"className":52043},[1113],[15,52045,52047],{"className":52046},[206],[15,52048,52050],{"className":52049,"style":23474},[210],[15,52051],{},[15,52053],{"className":52054},[169,23417],[15,52056,52058,52061],{"className":52057},[66],[15,52059,48383],{"className":52060},[66,67],[15,52062,52064],{"className":52063},[198],[15,52065,52067],{"className":52066},[202],[15,52068,52070],{"className":52069},[206],[15,52071,52073],{"className":52072,"style":51750},[210],[15,52074,52075,52078],{"style":397},[15,52076],{"className":52077,"style":218},[217],[15,52079,52081],{"className":52080},[222,223,224,225],[15,52082,37],{"className":52083},[66,67,225],[15,52085,335],{"className":52086},[422],[15,52088],{"className":52089,"style":10049},[71],[15,52091],{"className":52092,"style":697},[71],[15,52094,52096,52099],{"className":52095},[66],[15,52097,4556],{"className":52098},[66,67],[15,52100,52102],{"className":52101},[198],[15,52103,52105,52134],{"className":52104},[202,1090],[15,52106,52108,52131],{"className":52107},[206],[15,52109,52111],{"className":52110,"style":2139},[210],[15,52112,52113,52116],{"style":1584},[15,52114],{"className":52115,"style":218},[217],[15,52117,52119],{"className":52118},[222,223,224,225],[15,52120,52122,52125,52128],{"className":52121},[66,225],[15,52123,37],{"className":52124},[66,67,225],[15,52126,1245],{"className":52127},[76,225],[15,52129,1211],{"className":52130},[66,225],[15,52132,1114],{"className":52133},[1113],[15,52135,52137],{"className":52136},[206],[15,52138,52141],{"className":52139,"style":52140},[210],"height:0.2083em;",[15,52142],{},[15,52144],{"className":52145,"style":173},[71],[15,52147,129],{"className":52148},[177],[15,52150],{"className":52151,"style":173},[71],[15,52153,52155,52158,52291],{"className":52154},[57],[15,52156],{"className":52157,"style":41192},[61],[15,52159,52161,52164,52288],{"className":52160},[66],[15,52162],{"className":52163},[162,23417],[15,52165,52167],{"className":52166},[6467],[15,52168,52170,52280],{"className":52169},[202,1090],[15,52171,52173,52277],{"className":52172},[206],[15,52174,52176,52240,52248],{"className":52175,"style":41211},[210],[15,52177,52178,52181],{"style":23433},[15,52179],{"className":52180,"style":1368},[217],[15,52182,52184,52187,52190,52193,52196,52199,52202],{"className":52183},[66],[15,52185,120],{"className":52186},[162],[15,52188,37],{"className":52189},[66,67],[15,52191],{"className":52192,"style":72},[71],[15,52194,1245],{"className":52195},[76],[15,52197],{"className":52198,"style":72},[71],[15,52200,1211],{"className":52201},[66],[15,52203,52205,52208],{"className":52204},[169],[15,52206,126],{"className":52207},[169],[15,52209,52211],{"className":52210},[198],[15,52212,52214],{"className":52213},[202],[15,52215,52217],{"className":52216},[206],[15,52218,52220],{"className":52219,"style":46252},[210],[15,52221,52222,52225],{"style":46255},[15,52223],{"className":52224,"style":218},[217],[15,52226,52228],{"className":52227},[222,223,224,225],[15,52229,52231,52234,52237],{"className":52230},[66,225],[15,52232,37],{"className":52233},[66,67,225],[15,52235,1245],{"className":52236},[76,225],[15,52238,1211],{"className":52239},[66,225],[15,52241,52242,52245],{"style":23445},[15,52243],{"className":52244,"style":1368},[217],[15,52246],{"className":52247,"style":4440},[23452],[15,52249,52250,52253],{"style":23455},[15,52251],{"className":52252,"style":1368},[217],[15,52254,52256,52259,52262,52265,52268,52271,52274],{"className":52255},[66],[15,52257,120],{"className":52258},[162],[15,52260,37],{"className":52261},[66,67],[15,52263],{"className":52264,"style":72},[71],[15,52266,1245],{"className":52267},[76],[15,52269],{"className":52270,"style":72},[71],[15,52272,1211],{"className":52273},[66],[15,52275,46048],{"className":52276},[169],[15,52278,1114],{"className":52279},[1113],[15,52281,52283],{"className":52282},[206],[15,52284,52286],{"className":52285,"style":41282},[210],[15,52287],{},[15,52289],{"className":52290},[169,23417],[15,52292,52294,52297],{"className":52293},[66],[15,52295,48383],{"className":52296},[66,67],[15,52298,52300],{"className":52299},[198],[15,52301,52303],{"className":52302},[202],[15,52304,52306],{"className":52305},[206],[15,52307,52309],{"className":52308,"style":1650},[210],[15,52310,52311,52314],{"style":397},[15,52312],{"className":52313,"style":218},[217],[15,52315,52317],{"className":52316},[222,223,224,225],[15,52318,52320,52323,52326],{"className":52319},[66,225],[15,52321,37],{"className":52322},[66,67,225],[15,52324,1245],{"className":52325},[76,225],[15,52327,1211],{"className":52328},[66,225],[15,52330,52332],{"className":52331,"translate":19},[294],[15,52333,52335,52413],{"className":52334,"translate":19},[18],[15,52336,52338],{"className":52337},[23],[25,52339,52340],{"xmlns":27,"display":303},[29,52341,52342,52410],{},[32,52343,52344,52360,52380,52382,52406,52408],{},[1172,52345,52346,52352],{},[32,52347,52348,52350],{},[35,52349,43668],{},[39,52351,1170],{},[32,52353,52354,52356,52358],{},[35,52355,37],{},[39,52357,449],{},[35,52359,7620],{"mathvariant":138},[6467,52361,52362,52374],{},[1057,52363,52364,52366],{},[35,52365,4556],{},[32,52367,52368,52370,52372],{},[35,52369,37],{},[39,52371,1245],{},[1209,52373,1211],{},[1057,52375,52376,52378],{},[35,52377,4556],{},[35,52379,37],{},[39,52381,129],{},[6467,52383,52384,52386],{},[35,52385,48383],{},[131,52387,52388,52404],{},[32,52389,52390,52392,52394,52396,52402],{},[39,52391,120],{"fence":53},[1209,52393,1211],{},[39,52395,1245],{},[6467,52397,52398,52400],{},[1209,52399,1211],{},[35,52401,37],{},[39,52403,126],{"fence":53},[35,52405,37],{},[39,52407,9283],{},[1209,52409,1211],{},[45,52411,52412],{"encoding":47},"\\lim_{n \\rightarrow \\infty} \\frac{a_{n+1}}{a_n} = \\frac{e}{\\left(1 + \\frac{1}{n}\\right)^n} > 1",[15,52414,52416,52634,52832],{"className":52415,"ariaHidden":53},[52],[15,52417,52419,52423,52476,52479,52625,52628,52631],{"className":52418},[57],[15,52420],{"className":52421,"style":52422},[61],"height:1.9436em;vertical-align:-0.836em;",[15,52424,52426],{"className":52425},[1339,1351],[15,52427,52429,52468],{"className":52428},[202,1090],[15,52430,52432,52465],{"className":52431},[206],[15,52433,52435,52455],{"className":52434,"style":840},[210],[15,52436,52437,52440],{"style":48823},[15,52438],{"className":52439,"style":1368},[217],[15,52441,52443],{"className":52442},[222,223,224,225],[15,52444,52446,52449,52452],{"className":52445},[66,225],[15,52447,37],{"className":52448},[66,67,225],[15,52450,449],{"className":52451},[177,225],[15,52453,7620],{"className":52454},[66,225],[15,52456,52457,52460],{"style":1386},[15,52458],{"className":52459,"style":1368},[217],[15,52461,52462],{},[15,52463,43668],{"className":52464},[1339],[15,52466,1114],{"className":52467},[1113],[15,52469,52471],{"className":52470},[206],[15,52472,52474],{"className":52473,"style":48861},[210],[15,52475],{},[15,52477],{"className":52478,"style":697},[71],[15,52480,52482,52485,52622],{"className":52481},[66],[15,52483],{"className":52484},[162,23417],[15,52486,52488],{"className":52487},[6467],[15,52489,52491,52613],{"className":52490},[202,1090],[15,52492,52494,52610],{"className":52493},[206],[15,52495,52497,52545,52553],{"className":52496,"style":39133},[210],[15,52498,52499,52502],{"style":23433},[15,52500],{"className":52501,"style":1368},[217],[15,52503,52505],{"className":52504},[66],[15,52506,52508,52511],{"className":52507},[66],[15,52509,4556],{"className":52510},[66,67],[15,52512,52514],{"className":52513},[198],[15,52515,52517,52537],{"className":52516},[202,1090],[15,52518,52520,52534],{"className":52519},[206],[15,52521,52523],{"className":52522,"style":2198},[210],[15,52524,52525,52528],{"style":1584},[15,52526],{"className":52527,"style":218},[217],[15,52529,52531],{"className":52530},[222,223,224,225],[15,52532,37],{"className":52533},[66,67,225],[15,52535,1114],{"className":52536},[1113],[15,52538,52540],{"className":52539},[206],[15,52541,52543],{"className":52542,"style":1121},[210],[15,52544],{},[15,52546,52547,52550],{"style":23445},[15,52548],{"className":52549,"style":1368},[217],[15,52551],{"className":52552,"style":4440},[23452],[15,52554,52555,52558],{"style":23455},[15,52556],{"className":52557,"style":1368},[217],[15,52559,52561],{"className":52560},[66],[15,52562,52564,52567],{"className":52563},[66],[15,52565,4556],{"className":52566},[66,67],[15,52568,52570],{"className":52569},[198],[15,52571,52573,52602],{"className":52572},[202,1090],[15,52574,52576,52599],{"className":52575},[206],[15,52577,52579],{"className":52578,"style":2139},[210],[15,52580,52581,52584],{"style":1584},[15,52582],{"className":52583,"style":218},[217],[15,52585,52587],{"className":52586},[222,223,224,225],[15,52588,52590,52593,52596],{"className":52589},[66,225],[15,52591,37],{"className":52592},[66,67,225],[15,52594,1245],{"className":52595},[76,225],[15,52597,1211],{"className":52598},[66,225],[15,52600,1114],{"className":52601},[1113],[15,52603,52605],{"className":52604},[206],[15,52606,52608],{"className":52607,"style":52140},[210],[15,52609],{},[15,52611,1114],{"className":52612},[1113],[15,52614,52616],{"className":52615},[206],[15,52617,52620],{"className":52618,"style":52619},[210],"height:0.836em;",[15,52621],{},[15,52623],{"className":52624},[169,23417],[15,52626],{"className":52627,"style":173},[71],[15,52629,129],{"className":52630},[177],[15,52632],{"className":52633,"style":173},[71],[15,52635,52637,52641,52823,52826,52829],{"className":52636},[57],[15,52638],{"className":52639,"style":52640},[61],"height:2.2519em;vertical-align:-1.1443em;",[15,52642,52644,52647,52820],{"className":52643},[66],[15,52645],{"className":52646},[162,23417],[15,52648,52650],{"className":52649},[6467],[15,52651,52653,52811],{"className":52652},[202,1090],[15,52654,52656,52808],{"className":52655},[206],[15,52657,52659,52789,52797],{"className":52658,"style":39133},[210],[15,52660,52662,52665],{"style":52661},"top:-2.2057em;",[15,52663],{"className":52664,"style":1368},[217],[15,52666,52668],{"className":52667},[66],[15,52669,52671,52766],{"className":52670},[1413],[15,52672,52674,52680,52683,52686,52689,52692,52760],{"className":52673},[1413],[15,52675,52677],{"className":52676,"style":1418},[162,1417],[15,52678,120],{"className":52679},[1422,1508],[15,52681,1211],{"className":52682},[66],[15,52684],{"className":52685,"style":72},[71],[15,52687,1245],{"className":52688},[76],[15,52690],{"className":52691,"style":72},[71],[15,52693,52695,52698,52757],{"className":52694},[66],[15,52696],{"className":52697},[162,23417],[15,52699,52701],{"className":52700},[6467],[15,52702,52704,52749],{"className":52703},[202,1090],[15,52705,52707,52746],{"className":52706},[206],[15,52708,52710,52724,52732],{"className":52709,"style":30873},[210],[15,52711,52712,52715],{"style":30876},[15,52713],{"className":52714,"style":1368},[217],[15,52716,52718],{"className":52717},[222,223,224,225],[15,52719,52721],{"className":52720},[66,225],[15,52722,37],{"className":52723},[66,67,225],[15,52725,52726,52729],{"style":23445},[15,52727],{"className":52728,"style":1368},[217],[15,52730],{"className":52731,"style":4440},[23452],[15,52733,52734,52737],{"style":30899},[15,52735],{"className":52736,"style":1368},[217],[15,52738,52740],{"className":52739},[222,223,224,225],[15,52741,52743],{"className":52742},[66,225],[15,52744,1211],{"className":52745},[66,225],[15,52747,1114],{"className":52748},[1113],[15,52750,52752],{"className":52751},[206],[15,52753,52755],{"className":52754,"style":30921},[210],[15,52756],{},[15,52758],{"className":52759},[169,23417],[15,52761,52763],{"className":52762,"style":1418},[169,1417],[15,52764,126],{"className":52765},[1422,1508],[15,52767,52769],{"className":52768},[198],[15,52770,52772],{"className":52771},[202],[15,52773,52775],{"className":52774},[206],[15,52776,52778],{"className":52777,"style":48714},[210],[15,52779,52780,52783],{"style":21274},[15,52781],{"className":52782,"style":218},[217],[15,52784,52786],{"className":52785},[222,223,224,225],[15,52787,37],{"className":52788},[66,67,225],[15,52790,52791,52794],{"style":23445},[15,52792],{"className":52793,"style":1368},[217],[15,52795],{"className":52796,"style":4440},[23452],[15,52798,52799,52802],{"style":23455},[15,52800],{"className":52801,"style":1368},[217],[15,52803,52805],{"className":52804},[66],[15,52806,48383],{"className":52807},[66,67],[15,52809,1114],{"className":52810},[1113],[15,52812,52814],{"className":52813},[206],[15,52815,52818],{"className":52816,"style":52817},[210],"height:1.1443em;",[15,52819],{},[15,52821],{"className":52822},[169,23417],[15,52824],{"className":52825,"style":173},[71],[15,52827,9283],{"className":52828},[177],[15,52830],{"className":52831,"style":173},[71],[15,52833,52835,52838],{"className":52834},[57],[15,52836],{"className":52837,"style":1836},[61],[15,52839,1211],{"className":52840},[66],[11,52842,52843,52844,52872,52873,52901,52902,52972],{},"虽然该表达式的极限是 ",[15,52845,52847,52860],{"className":52846,"translate":19},[18],[15,52848,52850],{"className":52849},[23],[25,52851,52852],{"xmlns":27},[29,52853,52854,52858],{},[32,52855,52856],{},[35,52857,48383],{},[45,52859,48383],{"encoding":47},[15,52861,52863],{"className":52862,"ariaHidden":53},[52],[15,52864,52866,52869],{"className":52865},[57],[15,52867],{"className":52868,"style":86},[61],[15,52870,48383],{"className":52871},[66,67],"，但它总是从下方趋近 ",[15,52874,52876,52889],{"className":52875,"translate":19},[18],[15,52877,52879],{"className":52878},[23],[25,52880,52881],{"xmlns":27},[29,52882,52883,52887],{},[32,52884,52885],{},[35,52886,48383],{},[45,52888,48383],{"encoding":47},[15,52890,52892],{"className":52891,"ariaHidden":53},[52],[15,52893,52895,52898],{"className":52894},[57],[15,52896],{"className":52897,"style":86},[61],[15,52899,48383],{"className":52900},[66,67],"，因此比值仍大于 1，说明 ",[15,52903,52905,52923],{"className":52904,"translate":19},[18],[15,52906,52908],{"className":52907},[23],[25,52909,52910],{"xmlns":27},[29,52911,52912,52920],{},[32,52913,52914],{},[1057,52915,52916,52918],{},[35,52917,4556],{},[35,52919,37],{},[45,52921,52922],{"encoding":47},"a_n",[15,52924,52926],{"className":52925,"ariaHidden":53},[52],[15,52927,52929,52932],{"className":52928},[57],[15,52930],{"className":52931,"style":10197},[61],[15,52933,52935,52938],{"className":52934},[66],[15,52936,4556],{"className":52937},[66,67],[15,52939,52941],{"className":52940},[198],[15,52942,52944,52964],{"className":52943},[202,1090],[15,52945,52947,52961],{"className":52946},[206],[15,52948,52950],{"className":52949,"style":2198},[210],[15,52951,52952,52955],{"style":1584},[15,52953],{"className":52954,"style":218},[217],[15,52956,52958],{"className":52957},[222,223,224,225],[15,52959,37],{"className":52960},[66,67,225],[15,52962,1114],{"className":52963},[1113],[15,52965,52967],{"className":52966},[206],[15,52968,52970],{"className":52969,"style":1121},[210],[15,52971],{}," 是发散的。",[11,52974,52975],{},"数学家们尝试了各种方法，发现：",[15,52977,52979],{"className":52978,"translate":19},[294],[15,52980,52982,53032],{"className":52981,"translate":19},[18],[15,52983,52985],{"className":52984},[23],[25,52986,52987],{"xmlns":27,"display":303},[29,52988,52989,53029],{},[32,52990,52991,52997,52999,53023],{},[1057,52992,52993,52995],{},[35,52994,4556],{},[35,52996,37],{},[39,52998,129],{},[6467,53000,53001,53007],{},[32,53002,53003,53005],{},[35,53004,37],{},[39,53006,40769],{"stretchy":119},[131,53008,53009,53011],{},[35,53010,37],{},[32,53012,53013,53015,53017],{},[35,53014,37],{},[39,53016,1245],{},[6467,53018,53019,53021],{},[1209,53020,1211],{},[1209,53022,1263],{},[131,53024,53025,53027],{},[35,53026,48383],{},[35,53028,37],{},[45,53030,53031],{"encoding":47},"a_n = \\frac{n!}{n^{n + \\frac{1}{2}}} e^n",[15,53033,53035,53090],{"className":53034,"ariaHidden":53},[52],[15,53036,53038,53041,53081,53084,53087],{"className":53037},[57],[15,53039],{"className":53040,"style":10197},[61],[15,53042,53044,53047],{"className":53043},[66],[15,53045,4556],{"className":53046},[66,67],[15,53048,53050],{"className":53049},[198],[15,53051,53053,53073],{"className":53052},[202,1090],[15,53054,53056,53070],{"className":53055},[206],[15,53057,53059],{"className":53058,"style":2198},[210],[15,53060,53061,53064],{"style":1584},[15,53062],{"className":53063,"style":218},[217],[15,53065,53067],{"className":53066},[222,223,224,225],[15,53068,37],{"className":53069},[66,67,225],[15,53071,1114],{"className":53072},[1113],[15,53074,53076],{"className":53075},[206],[15,53077,53079],{"className":53078,"style":1121},[210],[15,53080],{},[15,53082],{"className":53083,"style":173},[71],[15,53085,129],{"className":53086},[177],[15,53088],{"className":53089,"style":173},[71],[15,53091,53093,53097,53266],{"className":53092},[57],[15,53094],{"className":53095,"style":53096},[61],"height:2.201em;vertical-align:-0.8296em;",[15,53098,53100,53103,53263],{"className":53099},[66],[15,53101],{"className":53102},[162,23417],[15,53104,53106],{"className":53105},[6467],[15,53107,53109,53254],{"className":53108},[202,1090],[15,53110,53112,53251],{"className":53111},[206],[15,53113,53115,53229,53237],{"className":53114,"style":27435},[210],[15,53116,53118,53121],{"style":53117},"top:-2.1704em;",[15,53119],{"className":53120,"style":1368},[217],[15,53122,53124],{"className":53123},[66],[15,53125,53127,53130],{"className":53126},[66],[15,53128,37],{"className":53129},[66,67],[15,53131,53133],{"className":53132},[198],[15,53134,53136],{"className":53135},[202],[15,53137,53139],{"className":53138},[206],[15,53140,53143],{"className":53141,"style":53142},[210],"height:0.9396em;",[15,53144,53146,53149],{"style":53145},"top:-3.3486em;margin-right:0.05em;",[15,53147],{"className":53148,"style":1368},[217],[15,53150,53152],{"className":53151},[222,223,224,225],[15,53153,53155,53158,53161],{"className":53154},[66,225],[15,53156,37],{"className":53157},[66,67,225],[15,53159,1245],{"className":53160},[76,225],[15,53162,53164,53167,53226],{"className":53163},[66,225],[15,53165],{"className":53166},[162,23417,222,3679,33581],[15,53168,53170],{"className":53169},[6467],[15,53171,53173,53218],{"className":53172},[202,1090],[15,53174,53176,53215],{"className":53175},[206],[15,53177,53179,53193,53201],{"className":53178,"style":49020},[210],[15,53180,53181,53184],{"style":33597},[15,53182],{"className":53183,"style":1368},[217],[15,53185,53187],{"className":53186},[222,3679,1508,225],[15,53188,53190],{"className":53189},[66,225],[15,53191,1263],{"className":53192},[66,225],[15,53194,53195,53198],{"style":33612},[15,53196],{"className":53197,"style":1368},[217],[15,53199],{"className":53200,"style":33619},[23452,225],[15,53202,53203,53206],{"style":33622},[15,53204],{"className":53205,"style":1368},[217],[15,53207,53209],{"className":53208},[222,3679,1508,225],[15,53210,53212],{"className":53211},[66,225],[15,53213,1211],{"className":53214},[66,225],[15,53216,1114],{"className":53217},[1113],[15,53219,53221],{"className":53220},[206],[15,53222,53224],{"className":53223,"style":33644},[210],[15,53225],{},[15,53227],{"className":53228},[169,23417,222,3679,33581],[15,53230,53231,53234],{"style":23445},[15,53232],{"className":53233,"style":1368},[217],[15,53235],{"className":53236,"style":4440},[23452],[15,53238,53239,53242],{"style":23455},[15,53240],{"className":53241,"style":1368},[217],[15,53243,53245,53248],{"className":53244},[66],[15,53246,37],{"className":53247},[66,67],[15,53249,40769],{"className":53250},[169],[15,53252,1114],{"className":53253},[1113],[15,53255,53257],{"className":53256},[206],[15,53258,53261],{"className":53259,"style":53260},[210],"height:0.8296em;",[15,53262],{},[15,53264],{"className":53265},[169,23417],[15,53267,53269,53272],{"className":53268},[66],[15,53270,48383],{"className":53271},[66,67],[15,53273,53275],{"className":53274},[198],[15,53276,53278],{"className":53277},[202],[15,53279,53281],{"className":53280},[206],[15,53282,53284],{"className":53283,"style":51750},[210],[15,53285,53286,53289],{"style":397},[15,53287],{"className":53288,"style":218},[217],[15,53290,53292],{"className":53291},[222,223,224,225],[15,53293,37],{"className":53294},[66,67,225],[11,53296,53297,53298,53367],{},"为了证明 ",[15,53299,53301,53318],{"className":53300,"translate":19},[18],[15,53302,53304],{"className":53303},[23],[25,53305,53306],{"xmlns":27},[29,53307,53308,53316],{},[32,53309,53310],{},[1057,53311,53312,53314],{},[35,53313,4556],{},[35,53315,37],{},[45,53317,52922],{"encoding":47},[15,53319,53321],{"className":53320,"ariaHidden":53},[52],[15,53322,53324,53327],{"className":53323},[57],[15,53325],{"className":53326,"style":10197},[61],[15,53328,53330,53333],{"className":53329},[66],[15,53331,4556],{"className":53332},[66,67],[15,53334,53336],{"className":53335},[198],[15,53337,53339,53359],{"className":53338},[202,1090],[15,53340,53342,53356],{"className":53341},[206],[15,53343,53345],{"className":53344,"style":2198},[210],[15,53346,53347,53350],{"style":1584},[15,53348],{"className":53349,"style":218},[217],[15,53351,53353],{"className":53352},[222,223,224,225],[15,53354,37],{"className":53355},[66,67,225],[15,53357,1114],{"className":53358},[1113],[15,53360,53362],{"className":53361},[206],[15,53363,53365],{"className":53364,"style":1121},[210],[15,53366],{}," 收敛，再次使用比值判别法：",[15,53369,53371],{"className":53370,"translate":19},[294],[15,53372,53374,53496],{"className":53373,"translate":19},[18],[15,53375,53377],{"className":53376},[23],[25,53378,53379],{"xmlns":27,"display":303},[29,53380,53381,53493],{},[32,53382,53383,53389,53391,53415,53421,53423,53425,53437,53439,53481],{},[1057,53384,53385,53387],{},[35,53386,4556],{},[35,53388,37],{},[39,53390,129],{},[6467,53392,53393,53399],{},[32,53394,53395,53397],{},[35,53396,37],{},[39,53398,40769],{"stretchy":119},[131,53400,53401,53403],{},[35,53402,37],{},[32,53404,53405,53407,53409],{},[35,53406,37],{},[39,53408,1245],{},[6467,53410,53411,53413],{},[1209,53412,1211],{},[1209,53414,1263],{},[131,53416,53417,53419],{},[35,53418,48383],{},[35,53420,37],{},[39,53422,335],{"separator":53},[71,53424],{"width":9791},[1057,53426,53427,53429],{},[35,53428,4556],{},[32,53430,53431,53433,53435],{},[35,53432,37],{},[39,53434,1245],{},[1209,53436,1211],{},[39,53438,129],{},[6467,53440,53441,53455],{},[32,53442,53443,53445,53447,53449,53451,53453],{},[39,53444,120],{"stretchy":119},[35,53446,37],{},[39,53448,1245],{},[1209,53450,1211],{},[39,53452,126],{"stretchy":119},[39,53454,40769],{"stretchy":119},[32,53456,53457,53459,53461,53463,53465],{},[39,53458,120],{"stretchy":119},[35,53460,37],{},[39,53462,1245],{},[1209,53464,1211],{},[131,53466,53467,53469],{},[39,53468,126],{"stretchy":119},[32,53470,53471,53473,53475],{},[35,53472,37],{},[39,53474,1245],{},[6467,53476,53477,53479],{},[1209,53478,8678],{},[1209,53480,1263],{},[131,53482,53483,53485],{},[35,53484,48383],{},[32,53486,53487,53489,53491],{},[35,53488,37],{},[39,53490,1245],{},[1209,53492,1211],{},[45,53494,53495],{"encoding":47},"a_n = \\frac{n!}{n^{n + \\frac{1}{2}}} e^n, \\quad a_{n+1} = \\frac{(n+1)!}{(n+1)^{n + \\frac{3}{2}}} e^{n+1}",[15,53497,53499,53554,53821],{"className":53498,"ariaHidden":53},[52],[15,53500,53502,53505,53545,53548,53551],{"className":53501},[57],[15,53503],{"className":53504,"style":10197},[61],[15,53506,53508,53511],{"className":53507},[66],[15,53509,4556],{"className":53510},[66,67],[15,53512,53514],{"className":53513},[198],[15,53515,53517,53537],{"className":53516},[202,1090],[15,53518,53520,53534],{"className":53519},[206],[15,53521,53523],{"className":53522,"style":2198},[210],[15,53524,53525,53528],{"style":1584},[15,53526],{"className":53527,"style":218},[217],[15,53529,53531],{"className":53530},[222,223,224,225],[15,53532,37],{"className":53533},[66,67,225],[15,53535,1114],{"className":53536},[1113],[15,53538,53540],{"className":53539},[206],[15,53541,53543],{"className":53542,"style":1121},[210],[15,53544],{},[15,53546],{"className":53547,"style":173},[71],[15,53549,129],{"className":53550},[177],[15,53552],{"className":53553,"style":173},[71],[15,53555,53557,53560,53725,53754,53757,53760,53763,53812,53815,53818],{"className":53556},[57],[15,53558],{"className":53559,"style":53096},[61],[15,53561,53563,53566,53722],{"className":53562},[66],[15,53564],{"className":53565},[162,23417],[15,53567,53569],{"className":53568},[6467],[15,53570,53572,53714],{"className":53571},[202,1090],[15,53573,53575,53711],{"className":53574},[206],[15,53576,53578,53689,53697],{"className":53577,"style":27435},[210],[15,53579,53580,53583],{"style":53117},[15,53581],{"className":53582,"style":1368},[217],[15,53584,53586],{"className":53585},[66],[15,53587,53589,53592],{"className":53588},[66],[15,53590,37],{"className":53591},[66,67],[15,53593,53595],{"className":53594},[198],[15,53596,53598],{"className":53597},[202],[15,53599,53601],{"className":53600},[206],[15,53602,53604],{"className":53603,"style":53142},[210],[15,53605,53606,53609],{"style":53145},[15,53607],{"className":53608,"style":1368},[217],[15,53610,53612],{"className":53611},[222,223,224,225],[15,53613,53615,53618,53621],{"className":53614},[66,225],[15,53616,37],{"className":53617},[66,67,225],[15,53619,1245],{"className":53620},[76,225],[15,53622,53624,53627,53686],{"className":53623},[66,225],[15,53625],{"className":53626},[162,23417,222,3679,33581],[15,53628,53630],{"className":53629},[6467],[15,53631,53633,53678],{"className":53632},[202,1090],[15,53634,53636,53675],{"className":53635},[206],[15,53637,53639,53653,53661],{"className":53638,"style":49020},[210],[15,53640,53641,53644],{"style":33597},[15,53642],{"className":53643,"style":1368},[217],[15,53645,53647],{"className":53646},[222,3679,1508,225],[15,53648,53650],{"className":53649},[66,225],[15,53651,1263],{"className":53652},[66,225],[15,53654,53655,53658],{"style":33612},[15,53656],{"className":53657,"style":1368},[217],[15,53659],{"className":53660,"style":33619},[23452,225],[15,53662,53663,53666],{"style":33622},[15,53664],{"className":53665,"style":1368},[217],[15,53667,53669],{"className":53668},[222,3679,1508,225],[15,53670,53672],{"className":53671},[66,225],[15,53673,1211],{"className":53674},[66,225],[15,53676,1114],{"className":53677},[1113],[15,53679,53681],{"className":53680},[206],[15,53682,53684],{"className":53683,"style":33644},[210],[15,53685],{},[15,53687],{"className":53688},[169,23417,222,3679,33581],[15,53690,53691,53694],{"style":23445},[15,53692],{"className":53693,"style":1368},[217],[15,53695],{"className":53696,"style":4440},[23452],[15,53698,53699,53702],{"style":23455},[15,53700],{"className":53701,"style":1368},[217],[15,53703,53705,53708],{"className":53704},[66],[15,53706,37],{"className":53707},[66,67],[15,53709,40769],{"className":53710},[169],[15,53712,1114],{"className":53713},[1113],[15,53715,53717],{"className":53716},[206],[15,53718,53720],{"className":53719,"style":53260},[210],[15,53721],{},[15,53723],{"className":53724},[169,23417],[15,53726,53728,53731],{"className":53727},[66],[15,53729,48383],{"className":53730},[66,67],[15,53732,53734],{"className":53733},[198],[15,53735,53737],{"className":53736},[202],[15,53738,53740],{"className":53739},[206],[15,53741,53743],{"className":53742,"style":51750},[210],[15,53744,53745,53748],{"style":397},[15,53746],{"className":53747,"style":218},[217],[15,53749,53751],{"className":53750},[222,223,224,225],[15,53752,37],{"className":53753},[66,67,225],[15,53755,335],{"className":53756},[422],[15,53758],{"className":53759,"style":10049},[71],[15,53761],{"className":53762,"style":697},[71],[15,53764,53766,53769],{"className":53765},[66],[15,53767,4556],{"className":53768},[66,67],[15,53770,53772],{"className":53771},[198],[15,53773,53775,53804],{"className":53774},[202,1090],[15,53776,53778,53801],{"className":53777},[206],[15,53779,53781],{"className":53780,"style":2139},[210],[15,53782,53783,53786],{"style":1584},[15,53784],{"className":53785,"style":218},[217],[15,53787,53789],{"className":53788},[222,223,224,225],[15,53790,53792,53795,53798],{"className":53791},[66,225],[15,53793,37],{"className":53794},[66,67,2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4103,54104,54106,54108],{},[35,54105,37],{},[39,54107,1245],{},[1209,54109,1211],{},[1057,54111,54112,54114],{},[35,54113,4556],{},[35,54115,37],{},[39,54117,129],{},[6467,54119,54120,54122],{},[35,54121,48383],{},[131,54123,54124,54140],{},[32,54125,54126,54128,54130,54132,54138],{},[39,54127,120],{"fence":53},[1209,54129,1211],{},[39,54131,1245],{},[6467,54133,54134,54136],{},[1209,54135,1211],{},[35,54137,37],{},[39,54139,126],{"fence":53},[32,54141,54142,54144,54146],{},[35,54143,37],{},[39,54145,1245],{},[6467,54147,54148,54150],{},[1209,54149,1211],{},[1209,54151,1263],{},[45,54153,54154],{"encoding":47},"\\lim_{n \\rightarrow \\infty} \\frac{a_{n+1}}{a_n} = \\frac{e}{\\left(1 + \\frac{1}{n}\\right)^{n + \\frac{1}{2}}}",[15,54156,54158,54374],{"className":54157,"ariaHidden":53},[52],[15,54159,54161,54164,54217,54220,54365,54368,54371],{"className":54160},[57],[15,54162],{"className":54163,"style":52422},[61],[15,54165,54167],{"className":54166},[1339,1351],[15,54168,54170,54209],{"className":54169},[202,1090],[15,54171,54173,54206],{"className":54172},[206],[15,54174,54176,54196],{"className":54175,"style":840},[210],[15,54177,54178,54181],{"style":48823},[15,54179],{"className":54180,"style":1368},[217],[15,54182,54184],{"className":54183},[222,223,224,225],[15,54185,54187,54190,54193],{"className":54186},[66,225],[15,54188,37],{"className":54189},[66,67,225],[15,54191,449],{"className":54192},[177,225],[15,54194,7620],{"className":54195},[66,225],[15,54197,54198,54201],{"style":1386},[15,54199],{"className":54200,"style":1368},[217],[15,54202,54203],{},[15,54204,43668],{"className":54205},[1339],[15,54207,1114],{"className":54208},[1113],[15,54210,54212],{"className":54211},[206],[15,54213,54215],{"className":54214,"style":48861},[210],[15,54216],{},[15,54218],{"className":54219,"style":697},[71],[15,54221,54223,54226,54362],{"className":54222},[66],[15,54224],{"className":54225},[162,23417],[15,54227,54229],{"className":54228},[6467],[15,54230,54232,54354],{"className":54231},[202,1090],[15,54233,54235,54351],{"className":54234},[206],[15,54236,54238,54286,54294],{"className":54237,"style":39133},[210],[15,54239,54240,54243],{"style":23433},[15,54241],{"className":54242,"style":1368},[217],[15,54244,54246],{"className":54245},[66],[15,54247,54249,54252],{"className":54248},[66],[15,54250,4556],{"className":54251},[66,67],[15,54253,54255],{"className":54254},[198],[15,54256,54258,54278],{"className":54257},[202,1090],[15,54259,54261,54275],{"className":54260},[206],[15,54262,54264],{"className":54263,"style":2198},[210],[15,54265,54266,54269],{"style":1584},[15,54267],{"className":54268,"style":218},[217],[15,54270,54272],{"className":54271},[222,223,224,225],[15,54273,37],{"className":54274},[66,67,225],[15,54276,1114],{"className":54277},[1113],[15,54279,54281],{"className":54280},[206],[15,54282,54284],{"className":54283,"style":1121},[210],[15,54285],{},[15,54287,54288,54291],{"style":23445},[15,54289],{"className":54290,"style":1368},[217],[15,54292],{"className":54293,"style":4440},[23452],[15,54295,54296,54299],{"style":23455},[15,54297],{"className":54298,"style":1368},[217],[15,54300,54302],{"className":54301},[66],[15,54303,54305,54308],{"className":54304},[66],[15,54306,4556],{"className":54307},[66,67],[15,54309,54311],{"className":54310},[198],[15,54312,54314,54343],{"className":54313},[202,1090],[15,54315,54317,54340],{"className":54316},[206],[15,54318,54320],{"className":54319,"style":2139},[210],[15,54321,54322,54325],{"style":1584},[15,54323],{"className":54324,"style":218},[217],[15,54326,54328],{"className":54327},[222,223,224,225],[15,54329,54331,54334,54337],{"className":54330},[66,225],[15,54332,37],{"className":54333},[66,67,225],[15,54335,1245],{"className":54336},[76,225],[15,54338,1211],{"className":54339},[66,225],[15,54341,1114],{"className":54342},[1113],[15,54344,54346],{"className":54345},[206],[15,54347,54349],{"className":54348,"style":52140},[210],[15,54350],{},[15,54352,1114],{"className":54353},[1113],[15,54355,54357],{"className":54356},[206],[15,54358,54360],{"className":54359,"style":52619},[210],[15,54361],{},[15,54363],{"className":54364},[169,23417],[15,54366],{"className":54367,"style":173},[71],[15,54369,129],{"className":54370},[177],[15,54372],{"className":54373,"style":173},[71],[15,54375,54377,54381],{"className":54376},[57],[15,54378],{"className":54379,"style":54380},[61],"height:2.5415em;vertical-align:-1.4339em;",[15,54382,54384,54387,54638],{"className":54383},[66],[15,54385],{"className":54386},[162,23417],[15,54388,54390],{"className":54389},[6467],[15,54391,54393,54629],{"className":54392},[202,1090],[15,54394,54396,54626],{"className":54395},[206],[15,54397,54399,54605,54614],{"className":54398,"style":39133},[210],[15,54400,54402,54406],{"style":54401},"top:-2.11em;",[15,54403],{"className":54404,"style":54405},[217],"height:3.1939em;",[15,54407,54409],{"className":54408},[66],[15,54410,54412,54507],{"className":54411},[1413],[15,54413,54415,54421,54424,54427,54430,54433,54501],{"className":54414},[1413],[15,54416,54418],{"className":54417,"style":1418},[162,1417],[15,54419,120],{"className":54420},[1422,1508],[15,54422,1211],{"className":54423},[66],[15,54425],{"className":54426,"style":72},[71],[15,54428,1245],{"className":54429},[76],[15,54431],{"className":54432,"style":72},[71],[15,54434,54436,54439,54498],{"className":54435},[66],[15,54437],{"className":54438},[162,23417],[15,54440,54442],{"className":54441},[6467],[15,54443,54445,54490],{"className":54444},[202,1090],[15,54446,54448,54487],{"className":54447},[206],[15,54449,54451,54465,54473],{"className":54450,"style":30873},[210],[15,54452,54453,54456],{"style":30876},[15,54454],{"className":54455,"style":1368},[217],[15,54457,54459],{"className":54458},[222,223,224,225],[15,54460,54462],{"className":54461},[66,225],[15,54463,37],{"className":54464},[66,67,225],[15,54466,54467,54470],{"style":23445},[15,54468],{"className":54469,"style":1368},[217],[15,54471],{"className":54472,"style":4440},[23452],[15,54474,54475,54478],{"style":30899},[15,54476],{"className":54477,"style":1368},[217],[15,54479,54481],{"className":54480},[222,223,224,225],[15,54482,54484],{"className":54483},[66,225],[15,54485,1211],{"className":54486},[66,225],[15,54488,1114],{"className":54489},[1113],[15,54491,54493],{"className":54492},[206],[15,54494,54496],{"className":54495,"style":30921},[210],[15,54497],{},[15,54499],{"className":54500},[169,23417],[15,54502,54504],{"className":54503,"style":1418},[169,1417],[15,54505,126],{"className":54506},[1422,1508],[15,54508,54510],{"className":54509},[198],[15,54511,54513],{"className":54512},[202],[15,54514,54516],{"className":54515},[206],[15,54517,54520],{"className":54518,"style":54519},[210],"height:1.1939em;",[15,54521,54522,54525],{"style":51217},[15,54523],{"className":54524,"style":1368},[217],[15,54526,54528],{"className":54527},[222,223,224,225],[15,54529,54531,54534,54537],{"className":54530},[66,225],[15,54532,37],{"className":54533},[66,67,225],[15,54535,1245],{"className":54536},[76,225],[15,54538,54540,54543,54602],{"className":54539},[66,225],[15,54541],{"className":54542},[162,23417,222,3679,33581],[15,54544,54546],{"className":54545},[6467],[15,54547,54549,54594],{"className":54548},[202,1090],[15,54550,54552,54591],{"className":54551},[206],[15,54553,54555,54569,54577],{"className":54554,"style":49020},[210],[15,54556,54557,54560],{"style":33597},[15,54558],{"className":54559,"style":1368},[217],[15,54561,54563],{"className":54562},[222,3679,1508,225],[15,54564,54566],{"className":54565},[66,225],[15,54567,1263],{"className":54568},[66,225],[15,54570,54571,54574],{"style":33612},[15,54572],{"className":54573,"style":1368},[217],[15,54575],{"className":54576,"style":33619},[23452,225],[15,54578,54579,54582],{"style":33622},[15,54580],{"className":54581,"style":1368},[217],[15,54583,54585],{"className":54584},[222,3679,1508,225],[15,54586,54588],{"className":54587},[66,225],[15,54589,1211],{"className":54590},[66,225],[15,54592,1114],{"className":54593},[1113],[15,54595,54597],{"className":54596},[206],[15,54598,54600],{"className":54599,"style":33644},[210],[15,54601],{},[15,54603],{"className":54604},[169,23417,222,3679,33581],[15,54606,54608,54611],{"style":54607},"top:-3.4239em;",[15,54609],{"className":54610,"style":54405},[217],[15,54612],{"className":54613,"style":4440},[23452],[15,54615,54617,54620],{"style":54616},"top:-3.8709em;",[15,54618],{"className":54619,"style":54405},[217],[15,54621,54623],{"className":54622},[66],[15,54624,48383],{"className":54625},[66,67],[15,54627,1114],{"className":54628},[1113],[15,54630,54632],{"className":54631},[206],[15,54633,54636],{"className":54634,"style":54635},[210],"height:1.4339em;",[15,54637],{},[15,54639],{"className":54640},[169,23417],[11,54642,54643],{},"将其与下式比较：",[15,54645,54647],{"className":54646,"translate":19},[294],[15,54648,54650,54734],{"className":54649,"translate":19},[18],[15,54651,54653],{"className":54652},[23],[25,54654,54655],{"xmlns":27,"display":303},[29,54656,54657,54731],{},[32,54658,54659,54689,54691],{},[131,54660,54661,54677],{},[32,54662,54663,54665,54667,54669,54675],{},[39,54664,120],{"fence":53},[1209,54666,1211],{},[39,54668,1245],{},[6467,54670,54671,54673],{},[1209,54672,1211],{},[35,54674,37],{},[39,54676,126],{"fence":53},[32,54678,54679,54681,54683],{},[35,54680,37],{},[39,54682,1245],{},[6467,54684,54685,54687],{},[1209,54686,1211],{},[1209,54688,1263],{},[39,54690,129],{},[131,54692,54693,54695],{},[35,54694,48383],{},[32,54696,54697,54699,54701,54703,54709,54711,54713,54715],{},[39,54698,120],{"stretchy":119},[35,54700,37],{},[39,54702,1245],{},[6467,54704,54705,54707],{},[1209,54706,1211],{},[1209,54708,1263],{},[39,54710,126],{"stretchy":119},[35,54712,49243],{},[39,54714,1170],{},[32,54716,54717,54719,54721,54723,54729],{},[39,54718,120],{"fence":53},[1209,54720,1211],{},[39,54722,1245],{},[6467,54724,54725,54727],{},[1209,54726,1211],{},[35,54728,37],{},[39,54730,126],{"fence":53},[45,54732,54733],{"encoding":47},"\\left(1 + \\frac{1}{n}\\right)^{n + \\frac{1}{2}} = e^{(n + \\frac{1}{2}) \\ln \\left(1 + \\frac{1}{n}\\right)}",[15,54735,54737,54941],{"className":54736,"ariaHidden":53},[52],[15,54738,54740,54743,54932,54935,54938],{"className":54739},[57],[15,54741],{"className":54742,"style":48808},[61],[15,54744,54746,54835],{"className":54745},[1413],[15,54747,54749,54755,54758,54761,54764,54767,54829],{"className":54748},[1413],[15,54750,54752],{"className":54751,"style":1418},[162,1417],[15,54753,120],{"className":54754},[1422,224],[15,54756,1211],{"className":54757},[66],[15,54759],{"className":54760,"style":72},[71],[15,54762,1245],{"className":54763},[76],[15,54765],{"className":54766,"style":72},[71],[15,54768,54770,54773,54826],{"className":54769},[66],[15,54771],{"className":54772},[162,23417],[15,54774,54776],{"className":54775},[6467],[15,54777,54779,54818],{"className":54778},[202,1090],[15,54780,54782,54815],{"className":54781},[206],[15,54783,54785,54796,54804],{"className":54784,"style":23430},[210],[15,54786,54787,54790],{"style":23433},[15,54788],{"className":54789,"style":1368},[217],[15,54791,54793],{"className":54792},[66],[15,54794,37],{"className":54795},[66,67],[15,54797,54798,54801],{"style":23445},[15,54799],{"className":54800,"style":1368},[217],[15,54802],{"className":54803,"style":4440},[23452],[15,54805,54806,54809],{"style":23455},[15,54807],{"className":54808,"style":1368},[217],[15,54810,54812],{"className":54811},[66],[15,54813,1211],{"className":54814},[66],[15,54816,1114],{"className":54817},[1113],[15,54819,54821],{"className":54820},[206],[15,54822,54824],{"className":54823,"style":23474},[210],[15,54825],{},[15,54827],{"className":54828},[169,23417],[15,54830,54832],{"className":54831,"style":1418},[169,1417],[15,54833,126],{"className":54834},[1422,224],[15,54836,54838],{"className":54837},[198],[15,54839,54841],{"className":54840},[202],[15,54842,54844],{"className":54843},[206],[15,54845,54847],{"className":54846,"style":48989},[210],[15,54848,54849,54852],{"style":48992},[15,54850],{"className":54851,"style":1368},[217],[15,54853,54855],{"className":54854},[222,223,224,225],[15,54856,54858,54861,54864],{"className":54857},[66,225],[15,54859,37],{"className":54860},[66,67,225],[15,54862,1245],{"className":54863},[76,225],[15,54865,54867,54870,54929],{"className":54866},[66,225],[15,54868],{"className":54869},[162,23417,222,3679,33581],[15,54871,54873],{"className":54872},[6467],[15,54874,54876,54921],{"className":54875},[202,1090],[15,54877,54879,54918],{"className":54878},[206],[15,54880,54882,54896,54904],{"className":54881,"style":49020},[210],[15,54883,54884,54887],{"style":33597},[15,54885],{"className":54886,"style":1368},[217],[15,54888,54890],{"className":54889},[222,3679,1508,225],[15,54891,54893],{"className":54892},[66,225],[15,54894,1263],{"className":54895},[66,225],[15,54897,54898,54901],{"style":33612},[15,54899],{"className":54900,"style":1368},[217],[15,54902],{"className":54903,"style":33619},[23452,225],[15,54905,54906,54909],{"style":33622},[15,54907],{"className":54908,"style":1368},[217],[15,54910,54912],{"className":54911},[222,3679,1508,225],[15,54913,54915],{"className":54914},[66,225],[15,54916,1211],{"className":54917},[66,225],[15,54919,1114],{"className":54920},[1113],[15,54922,54924],{"className":54923},[206],[15,54925,54927],{"className":54926,"style":33644},[210],[15,54928],{},[15,54930],{"className":54931},[169,23417,222,3679,33581],[15,54933],{"className":54934,"style":173},[71],[15,54936,129],{"className":54937},[177],[15,54939],{"className":54940,"style":173},[71],[15,54942,54944,54947],{"className":54943},[57],[15,54945],{"className":54946,"style":49624},[61],[15,54948,54950,54953],{"className":54949},[66],[15,54951,48383],{"className":54952},[66,67],[15,54954,54956],{"className":54955},[198],[15,54957,54959],{"className":54958},[202],[15,54960,54962],{"className":54961},[206],[15,54963,54965],{"className":54964,"style":49624},[210],[15,54966,54967,54970],{"style":49627},[15,54968],{"className":54969,"style":1368},[217],[15,54971,54973],{"className":54972},[222,223,224,225],[15,54974,54976,54979,54982,54985,55053,55056,55059,55068],{"className":54975},[66,225],[15,54977,120],{"className":54978},[162,225],[15,54980,37],{"className":54981},[66,67,225],[15,54983,1245],{"className":54984},[76,225],[15,54986,54988,54991,55050],{"className":54987},[66,225],[15,54989],{"className":54990},[162,23417,222,3679,33581],[15,54992,54994],{"className":54993},[6467],[15,54995,54997,55042],{"className":54996},[202,1090],[15,54998,55000,55039],{"className":54999},[206],[15,55001,55003,55017,55025],{"className":55002,"style":49020},[210],[15,55004,55005,55008],{"style":33597},[15,55006],{"className":55007,"style":1368},[217],[15,55009,55011],{"className":55010},[222,3679,1508,225],[15,55012,55014],{"className":55013},[66,225],[15,55015,1263],{"className":55016},[66,225],[15,55018,55019,55022],{"style":33612},[15,55020],{"className":55021,"style":1368},[217],[15,55023],{"className":55024,"style":33619},[23452,225],[15,55026,55027,55030],{"style":33622},[15,55028],{"className":55029,"style":1368},[217],[15,55031,55033],{"className":55032},[222,3679,1508,225],[15,55034,55036],{"className":55035},[66,225],[15,55037,1211],{"className":55038},[66,225],[15,55040,1114],{"className":55041},[1113],[15,55043,55045],{"className":55044},[206],[15,55046,55048],{"className":55047,"style":33644},[210],[15,55049],{},[15,55051],{"className":55052},[169,23417,222,3679,33581],[15,55054,126],{"className":55055},[169,225],[15,55057],{"className":55058,"style":23368},[71,225],[15,55060,55062,55065],{"className":55061},[1339,225],[15,55063,49714],{"className":55064},[225],[15,55066,37],{"className":55067},[225],[15,55069,55071,55077,55080,55083,55151],{"className":55070},[1413,225],[15,55072,55074],{"className":55073,"style":49724},[162,222,3679,33581,225,1417],[15,55075,120],{"className":55076},[225],[15,55078,1211],{"className":55079},[66,225],[15,55081,1245],{"className":55082},[76,225],[15,55084,55086,55089,55148],{"className":55085},[66,225],[15,55087],{"className":55088},[162,23417,222,3679,33581],[15,55090,55092],{"className":55091},[6467],[15,55093,55095,55140],{"className":55094},[202,1090],[15,55096,55098,55137],{"className":55097},[206],[15,55099,55101,55115,55123],{"className":55100,"style":49020},[210],[15,55102,55103,55106],{"style":33597},[15,55104],{"className":55105,"style":1368},[217],[15,55107,55109],{"className":55108},[222,3679,1508,225],[15,55110,55112],{"className":55111},[66,225],[15,55113,37],{"className":55114},[66,67,225],[15,55116,55117,55120],{"style":33612},[15,55118],{"className":55119,"style":1368},[217],[15,55121],{"className":55122,"style":33619},[23452,225],[15,55124,55125,55128],{"style":33622},[15,55126],{"className":55127,"style":1368},[217],[15,55129,55131],{"className":55130},[222,3679,1508,225],[15,55132,55134],{"className":55133},[66,225],[15,55135,1211],{"className":55136},[66,225],[15,55138,1114],{"className":55139},[1113],[15,55141,55143],{"className":55142},[206],[15,55144,55146],{"className":55145,"style":33644},[210],[15,55147],{},[15,55149],{"className":55150},[169,23417,222,3679,33581],[15,55152,55154],{"className":55153,"style":49724},[169,222,3679,33581,225,1417],[15,55155,126],{"className":55156},[225],[11,55158,55159,55160,1124],{},"以及 ",[15,55161,55163,55176],{"className":55162,"translate":19},[18],[15,55164,55166],{"className":55165},[23],[25,55167,55168],{"xmlns":27},[29,55169,55170,55174],{},[32,55171,55172],{},[35,55173,48383],{},[45,55175,48383],{"encoding":47},[15,55177,55179],{"className":55178,"ariaHidden":53},[52],[15,55180,55182,55185],{"className":55181},[57],[15,55183],{"className":55184,"style":86},[61],[15,55186,48383],{"className":55187},[66,67],[11,55189,55190],{},"因此，比较：",[15,55192,55194],{"className":55193,"translate":19},[294],[15,55195,55197,55243],{"className":55196,"translate":19},[18],[15,55198,55200],{"className":55199},[23],[25,55201,55202],{"xmlns":27,"display":303},[29,55203,55204,55240],{},[32,55205,55206,55208,55210,55212,55218,55220,55222,55224],{},[39,55207,120],{"stretchy":119},[35,55209,37],{},[39,55211,1245],{},[6467,55213,55214,55216],{},[1209,55215,1211],{},[1209,55217,1263],{},[39,55219,126],{"stretchy":119},[35,55221,49243],{},[39,55223,1170],{},[32,55225,55226,55228,55230,55232,55238],{},[39,55227,120],{"fence":53},[1209,55229,1211],{},[39,55231,1245],{},[6467,55233,55234,55236],{},[1209,55235,1211],{},[35,55237,37],{},[39,55239,126],{"fence":53},[45,55241,55242],{"encoding":47},"(n + \\frac{1}{2}) \\ln \\left(1 + \\frac{1}{n}\\right)",[15,55244,55246,55267],{"className":55245,"ariaHidden":53},[52],[15,55247,55249,55252,55255,55258,55261,55264],{"className":55248},[57],[15,55250],{"className":55251,"style":154},[61],[15,55253,120],{"className":55254},[162],[15,55256,37],{"className":55257},[66,67],[15,55259],{"className":55260,"style":72},[71],[15,55262,1245],{"className":55263},[76],[15,55265],{"className":55266,"style":72},[71],[15,55268,55270,55273,55335,55338,55341,55344,55347],{"className":55269},[57],[15,55271],{"className":55272,"style":43973},[61],[15,55274,55276,55279,55332],{"className":55275},[66],[15,55277],{"className":55278},[162,23417],[15,55280,55282],{"className":55281},[6467],[15,55283,55285,55324],{"className":55284},[202,1090],[15,55286,55288,55321],{"className":55287},[206],[15,55289,55291,55302,55310],{"className":55290,"style":23430},[210],[15,55292,55293,55296],{"style":23433},[15,55294],{"className":55295,"style":1368},[217],[15,55297,55299],{"className":55298},[66],[15,55300,1263],{"className":55301},[66],[15,55303,55304,55307],{"style":23445},[15,55305],{"className":55306,"style":1368},[217],[15,55308],{"className":55309,"style":4440},[23452],[15,55311,55312,55315],{"style":23455},[15,55313],{"className":55314,"style":1368},[217],[15,55316,55318],{"className":55317},[66],[15,55319,1211],{"className":55320},[66],[15,55322,1114],{"className":55323},[1113],[15,55325,55327],{"className":55326},[206],[15,55328,55330],{"className":55329,"style":23474},[210],[15,55331],{},[15,55333],{"className":55334},[169,23417],[15,55336,126],{"className":55337},[169],[15,55339],{"className":55340,"style":697},[71],[15,55342,49243],{"className":55343},[1339],[15,55345],{"className":55346,"style":697},[71],[15,55348,55350,55356,55359,55362,55365,55368,55430],{"className":55349},[1413],[15,55351,55353],{"className":55352,"style":1418},[162,1417],[15,55354,120],{"className":55355},[1422,224],[15,55357,1211],{"className":55358},[66],[15,55360],{"className":55361,"style":72},[71],[15,55363,1245],{"className":55364},[76],[15,55366],{"className":55367,"style":72},[71],[15,55369,55371,55374,55427],{"className":55370},[66],[15,55372],{"className":55373},[162,23417],[15,55375,55377],{"className":55376},[6467],[15,55378,55380,55419],{"className":55379},[202,1090],[15,55381,55383,55416],{"className":55382},[206],[15,55384,55386,55397,55405],{"className":55385,"style":23430},[210],[15,55387,55388,55391],{"style":23433},[15,55389],{"className":55390,"style":1368},[217],[15,55392,55394],{"className":55393},[66],[15,55395,37],{"className":55396},[66,67],[15,55398,55399,55402],{"style":23445},[15,55400],{"className":55401,"style":1368},[217],[15,55403],{"className":55404,"style":4440},[23452],[15,55406,55407,55410],{"style":23455},[15,55408],{"className":55409,"style":1368},[217],[15,55411,55413],{"className":55412},[66],[15,55414,1211],{"className":55415},[66],[15,55417,1114],{"className":55418},[1113],[15,55420,55422],{"className":55421},[206],[15,55423,55425],{"className":55424,"style":23474},[210],[15,55426],{},[15,55428],{"className":55429},[169,23417],[15,55431,55433],{"className":55432,"style":1418},[169,1417],[15,55434,126],{"className":55435},[1422,224],[11,55437,55438],{},"与 1。",[11,55440,55441,55442,48366,55562,37644],{},"具体来说，比较 ",[15,55443,55445,55473],{"className":55444,"translate":19},[18],[15,55446,55448],{"className":55447},[23],[25,55449,55450],{"xmlns":27},[29,55451,55452,55470],{},[32,55453,55454,55456,55458],{},[35,55455,49243],{},[39,55457,1170],{},[6467,55459,55460,55468],{},[32,55461,55462,55464,55466],{},[35,55463,37],{},[39,55465,1245],{},[1209,55467,1211],{},[35,55469,37],{},[45,55471,55472],{"encoding":47},"\\ln \\frac{n+1}{n}",[15,55474,55476],{"className":55475,"ariaHidden":53},[52],[15,55477,55479,55482,55485,55488],{"className":55478},[57],[15,55480],{"className":55481,"style":30854},[61],[15,55483,49243],{"className":55484},[1339],[15,55486],{"className":55487,"style":697},[71],[15,55489,55491,55494,55559],{"className":55490},[66],[15,55492],{"className":55493},[162,23417],[15,55495,55497],{"className":55496},[6467],[15,55498,55500,55551],{"className":55499},[202,1090],[15,55501,55503,55548],{"className":55502},[206],[15,55504,55506,55520,55528],{"className":55505,"style":30873},[210],[15,55507,55508,55511],{"style":30876},[15,55509],{"className":55510,"style":1368},[217],[15,55512,55514],{"className":55513},[222,223,224,225],[15,55515,55517],{"className":55516},[66,225],[15,55518,37],{"className":55519},[66,67,225],[15,55521,55522,55525],{"style":23445},[15,55523],{"className":55524,"style":1368},[217],[15,55526],{"className":55527,"style":4440},[23452],[15,55529,55530,55533],{"style":30899},[15,55531],{"className":55532,"style":1368},[217],[15,55534,55536],{"className":55535},[222,223,224,225],[15,55537,55539,55542,55545],{"className":55538},[66,225],[15,55540,37],{"className":55541},[66,67,225],[15,55543,1245],{"className":55544},[76,225],[15,55546,1211],{"className":55547},[66,225],[15,55549,1114],{"className":55550},[1113],[15,55552,55554],{"className":55553},[206],[15,55555,55557],{"className":55556,"style":30921},[210],[15,55558],{},[15,55560],{"className":55561},[169,23417],[15,55563,55565,55593],{"className":55564,"translate":19},[18],[15,55566,55568],{"className":55567},[23],[25,55569,55570],{"xmlns":27},[29,55571,55572,55590],{},[32,55573,55574],{},[6467,55575,55576,55578],{},[1209,55577,1211],{},[32,55579,55580,55582,55584],{},[35,55581,37],{},[39,55583,1245],{},[6467,55585,55586,55588],{},[1209,55587,1211],{},[1209,55589,1263],{},[45,55591,55592],{"encoding":47},"\\frac{1}{n + \\frac{1}{2}}",[15,55594,55596],{"className":55595,"ariaHidden":53},[52],[15,55597,55599,55603],{"className":55598},[57],[15,55600],{"className":55601,"style":55602},[61],"height:1.4869em;vertical-align:-0.6418em;",[15,55604,55606,55609,55741],{"className":55605},[66],[15,55607],{"className":55608},[162,23417],[15,55610,55612],{"className":55611},[6467],[15,55613,55615,55732],{"className":55614},[202,1090],[15,55616,55618,55729],{"className":55617},[206],[15,55619,55621,55707,55715],{"className":55620,"style":30873},[210],[15,55622,55624,55627],{"style":55623},"top:-2.599em;",[15,55625],{"className":55626,"style":1368},[217],[15,55628,55630],{"className":55629},[222,223,224,225],[15,55631,55633,55636,55639],{"className":55632},[66,225],[15,55634,37],{"className":55635},[66,67,225],[15,55637,1245],{"className":55638},[76,225],[15,55640,55642,55645,55704],{"className":55641},[66,225],[15,55643],{"className":55644},[162,23417,222,3679,33581],[15,55646,55648],{"className":55647},[6467],[15,55649,55651,55696],{"className":55650},[202,1090],[15,55652,55654,55693],{"className":55653},[206],[15,55655,55657,55671,55679],{"className":55656,"style":49020},[210],[15,55658,55659,55662],{"style":33597},[15,55660],{"className":55661,"style":1368},[217],[15,55663,55665],{"className":55664},[222,3679,1508,225],[15,55666,55668],{"className":55667},[66,225],[15,55669,1263],{"className":55670},[66,225],[15,55672,55673,55676],{"style":33612},[15,55674],{"className":55675,"style":1368},[217],[15,55677],{"className":55678,"style":33619},[23452,225],[15,55680,55681,55684],{"style":33622},[15,55682],{"className":55683,"style":1368},[217],[15,55685,55687],{"className":55686},[222,3679,1508,225],[15,55688,55690],{"className":55689},[66,225],[15,55691,1211],{"className":55692},[66,225],[15,55694,1114],{"className":55695},[1113],[15,55697,55699],{"className":55698},[206],[15,55700,55702],{"className":55701,"style":33644},[210],[15,55703],{},[15,55705],{"className":55706},[169,23417,222,3679,33581],[15,55708,55709,55712],{"style":23445},[15,55710],{"className":55711,"style":1368},[217],[15,55713],{"className":55714,"style":4440},[23452],[15,55716,55717,55720],{"style":30899},[15,55718],{"className":55719,"style":1368},[217],[15,55721,55723],{"className":55722},[222,223,224,225],[15,55724,55726],{"className":55725},[66,225],[15,55727,1211],{"className":55728},[66,225],[15,55730,1114],{"className":55731},[1113],[15,55733,55735],{"className":55734},[206],[15,55736,55739],{"className":55737,"style":55738},[210],"height:0.6418em;",[15,55740],{},[15,55742],{"className":55743},[169,23417],[11,55745,55746],{},"由于：",[15,55748,55750],{"className":55749,"translate":19},[294],[15,55751,55753,55807],{"className":55752,"translate":19},[18],[15,55754,55756],{"className":55755},[23],[25,55757,55758],{"xmlns":27,"display":303},[29,55759,55760,55804],{},[32,55761,55762,55774,55776,55778,55780,55796,55798],{},[6467,55763,55764,55766],{},[1209,55765,1211],{},[32,55767,55768,55770,55772],{},[35,55769,37],{},[39,55771,1245],{},[1209,55773,1211],{},[39,55775,25772],{},[35,55777,49243],{},[39,55779,1170],{},[32,55781,55782,55784,55786,55788,55794],{},[39,55783,120],{"fence":53},[1209,55785,1211],{},[39,55787,1245],{},[6467,55789,55790,55792],{},[1209,55791,1211],{},[35,55793,37],{},[39,55795,126],{"fence":53},[39,55797,25772],{},[6467,55799,55800,55802],{},[1209,55801,1211],{},[35,55803,37],{},[45,55805,55806],{"encoding":47},"\\frac{1}{n + 1} \u003C \\ln \\left(1 + \\frac{1}{n}\\right) \u003C \\frac{1}{n}",[15,55808,55810,55899,56009],{"className":55809,"ariaHidden":53},[52],[15,55811,55813,55816,55890,55893,55896],{"className":55812},[57],[15,55814],{"className":55815,"style":44670},[61],[15,55817,55819,55822,55887],{"className":55818},[66],[15,55820],{"className":55821},[162,23417],[15,55823,55825],{"className":55824},[6467],[15,55826,55828,55879],{"className":55827},[202,1090],[15,55829,55831,55876],{"className":55830},[206],[15,55832,55834,55857,55865],{"className":55833,"style":23430},[210],[15,55835,55836,55839],{"style":23433},[15,55837],{"className":55838,"style":1368},[217],[15,55840,55842,55845,55848,55851,55854],{"className":55841},[66],[15,55843,37],{"className":55844},[66,67],[15,55846],{"className":55847,"style":72},[71],[15,55849,1245],{"className":55850},[76],[15,55852],{"className":55853,"style":72},[71],[15,55855,1211],{"className":55856},[66],[15,55858,55859,55862],{"style":23445},[15,55860],{"className":55861,"style":1368},[217],[15,55863],{"className":55864,"style":4440},[23452],[15,55866,55867,55870],{"style":23455},[15,55868],{"className":55869,"style":1368},[217],[15,55871,55873],{"className":55872},[66],[15,55874,1211],{"className":55875},[66],[15,55877,1114],{"className":55878},[1113],[15,55880,55882],{"className":55881},[206],[15,55883,55885],{"className":55884,"style":38570},[210],[15,55886],{},[15,55888],{"className":55889},[169,23417],[15,55891],{"className":55892,"style":173},[71],[15,55894,25772],{"className":55895},[177],[15,55897],{"className":55898,"style":173},[71],[15,55900,55902,55905,55908,55911,56000,56003,56006],{"className":55901},[57],[15,55903],{"className":55904,"style":43973},[61],[15,55906,49243],{"className":55907},[1339],[15,55909],{"className":55910,"style":697},[71],[15,55912,55914,55920,55923,55926,55929,55932,55994],{"className":55913},[1413],[15,55915,55917],{"className":55916,"style":1418},[162,1417],[15,55918,120],{"className":55919},[1422,224],[15,55921,1211],{"className":55922},[66],[15,55924],{"className":55925,"style":72},[71],[15,55927,1245],{"className":55928},[76],[15,55930],{"className":55931,"style":72},[71],[15,55933,55935,55938,55991],{"className":55934},[66],[15,55936],{"className":55937},[162,23417],[15,55939,55941],{"className":55940},[6467],[15,55942,55944,55983],{"className":55943},[202,1090],[15,55945,55947,55980],{"className":55946},[206],[15,55948,55950,55961,55969],{"className":55949,"style":23430},[210],[15,55951,55952,55955],{"style":23433},[15,55953],{"className":55954,"style":1368},[217],[15,55956,55958],{"className":55957},[66],[15,55959,37],{"className":55960},[66,67],[15,55962,55963,55966],{"style":23445},[15,55964],{"className":55965,"style":1368},[217],[15,55967],{"className":55968,"style":4440},[23452],[15,55970,55971,55974],{"style":23455},[15,55972],{"className":55973,"style":1368},[217],[15,55975,55977],{"className":55976},[66],[15,55978,1211],{"className":55979},[66],[15,55981,1114],{"className":55982},[1113],[15,55984,55986],{"className":55985},[206],[15,55987,55989],{"className":55988,"style":23474},[210],[15,55990],{},[15,55992],{"className":55993},[169,23417],[15,55995,55997],{"className":55996,"style":1418},[169,1417],[15,55998,126],{"className":55999},[1422,224],[15,56001],{"className":56002,"style":173},[71],[15,56004,25772],{"className":56005},[177],[15,56007],{"className":56008,"style":173},[71],[15,56010,56012,56015],{"className":56011},[57],[15,56013],{"className":56014,"style":49086},[61],[15,56016,56018,56021,56074],{"className":56017},[66],[15,56019],{"className":56020},[162,23417],[15,56022,56024],{"className":56023},[6467],[15,56025,56027,56066],{"className":56026},[202,1090],[15,56028,56030,56063],{"className":56029},[206],[15,56031,56033,56044,56052],{"className":56032,"style":23430},[210],[15,56034,56035,56038],{"style":23433},[15,56036],{"className":56037,"style":1368},[217],[15,56039,56041],{"className":56040},[66],[15,56042,37],{"className":56043},[66,67],[15,56045,56046,56049],{"style":23445},[15,56047],{"className":56048,"style":1368},[217],[15,56050],{"className":56051,"style":4440},[23452],[15,56053,56054,56057],{"style":23455},[15,56055],{"className":56056,"style":1368},[217],[15,56058,56060],{"className":56059},[66],[15,56061,1211],{"className":56062},[66],[15,56064,1114],{"className":56065},[1113],[15,56067,56069],{"className":56068},[206],[15,56070,56072],{"className":56071,"style":23474},[210],[15,56073],{},[15,56075],{"className":56076},[169,23417],[11,56078,56079,56080,56170,56171,56255],{},"注意：右侧由不等式 ",[15,56081,56083,56113],{"className":56082,"translate":19},[18],[15,56084,56086],{"className":56085},[23],[25,56087,56088],{"xmlns":27},[29,56089,56090,56110],{},[32,56091,56092,56094,56096,56098,56100,56102,56104,56106,56108],{},[35,56093,49243],{},[39,56095,1170],{},[39,56097,120],{"stretchy":119},[1209,56099,1211],{},[39,56101,1245],{},[35,56103,123],{},[39,56105,126],{"stretchy":119},[39,56107,25772],{},[35,56109,123],{},[45,56111,56112],{"encoding":47},"\\ln(1 + x) \u003C x",[15,56114,56116,56140,56161],{"className":56115,"ariaHidden":53},[52],[15,56117,56119,56122,56125,56128,56131,56134,56137],{"className":56118},[57],[15,56120],{"className":56121,"style":154},[61],[15,56123,49243],{"className":56124},[1339],[15,56126,120],{"className":56127},[162],[15,56129,1211],{"className":56130},[66],[15,56132],{"className":56133,"style":72},[71],[15,56135,1245],{"className":56136},[76],[15,56138],{"className":56139,"style":72},[71],[15,56141,56143,56146,56149,56152,56155,56158],{"className":56142},[57],[15,56144],{"className":56145,"style":154},[61],[15,56147,123],{"className":56148},[66,67],[15,56150,126],{"className":56151},[169],[15,56153],{"className":56154,"style":173},[71],[15,56156,25772],{"className":56157},[177],[15,56159],{"className":56160,"style":173},[71],[15,56162,56164,56167],{"className":56163},[57],[15,56165],{"className":56166,"style":86},[61],[15,56168,123],{"className":56169},[66,67],"（其中 ",[15,56172,56174,56192],{"className":56173,"translate":19},[18],[15,56175,56177],{"className":56176},[23],[25,56178,56179],{"xmlns":27},[29,56180,56181,56189],{},[32,56182,56183,56185,56187],{},[35,56184,123],{},[39,56186,13019],{"mathvariant":138},[1209,56188,1804],{},[45,56190,56191],{"encoding":47},"x \\neq 0",[15,56193,56195,56246],{"className":56194,"ariaHidden":53},[52],[15,56196,56198,56201,56204,56207,56243],{"className":56197},[57],[15,56199],{"className":56200,"style":3408},[61],[15,56202,123],{"className":56203},[66,67],[15,56205],{"className":56206,"style":173},[71],[15,56208,56210,56237,56240],{"className":56209},[177],[15,56211,56213],{"className":56212},[177],[15,56214,56216],{"className":56215},[66,13086],[15,56217,56219],{"className":56218},[13090],[15,56220,56222,56225,56234],{"className":56221},[13094],[15,56223],{"className":56224,"style":3408},[61],[15,56226,56228],{"className":56227},[13101],[15,56229,56231],{"className":56230},[66],[15,56232,13108],{"className":56233},[177],[15,56235],{"className":56236},[13112],[15,56238],{"className":56239},[71,13116],[15,56241,129],{"className":56242},[177],[15,56244],{"className":56245,"style":173},[71],[15,56247,56249,56252],{"className":56248},[57],[15,56250],{"className":56251,"style":1836},[61],[15,56253,1804],{"className":56254},[66],"）得出。",[11,56257,56258,56259,56526,56527,56637],{},"左侧由 ",[15,56260,56262,56314],{"className":56261,"translate":19},[18],[15,56263,56265],{"className":56264},[23],[25,56266,56267],{"xmlns":27},[29,56268,56269,56311],{},[32,56270,56271,56273,56275,56287,56289,56291,56293],{},[1209,56272,1211],{},[39,56274,6602],{},[6467,56276,56277,56279],{},[35,56278,37],{},[32,56280,56281,56283,56285],{},[35,56282,37],{},[39,56284,1245],{},[1209,56286,1211],{},[39,56288,25772],{},[35,56290,49243],{},[39,56292,1170],{},[32,56294,56295,56297,56309],{},[39,56296,120],{"fence":53},[6467,56298,56299,56307],{},[32,56300,56301,56303,56305],{},[35,56302,37],{},[39,56304,1245],{},[1209,56306,1211],{},[35,56308,37],{},[39,56310,126],{"fence":53},[45,56312,56313],{"encoding":47},"1 - \\frac{n}{n+1} \u003C \\ln \\left(\\frac{n+1}{n}\\right)",[15,56315,56317,56335,56425],{"className":56316,"ariaHidden":53},[52],[15,56318,56320,56323,56326,56329,56332],{"className":56319},[57],[15,56321],{"className":56322,"style":30468},[61],[15,56324,1211],{"className":56325},[66],[15,56327],{"className":56328,"style":72},[71],[15,56330,6602],{"className":56331},[76],[15,56333],{"className":56334,"style":72},[71],[15,56336,56338,56342,56416,56419,56422],{"className":56337},[57],[15,56339],{"className":56340,"style":56341},[61],"height:1.0987em;vertical-align:-0.4033em;",[15,56343,56345,56348,56413],{"className":56344},[66],[15,56346],{"className":56347},[162,23417],[15,56349,56351],{"className":56350},[6467],[15,56352,56354,56405],{"className":56353},[202,1090],[15,56355,56357,56402],{"className":56356},[206],[15,56358,56360,56380,56388],{"className":56359,"style":41848},[210],[15,56361,56362,56365],{"style":30876},[15,56363],{"className":56364,"style":1368},[217],[15,56366,56368],{"className":56367},[222,223,224,225],[15,56369,56371,56374,56377],{"className":56370},[66,225],[15,56372,37],{"className":56373},[66,67,225],[15,56375,1245],{"className":56376},[76,225],[15,56378,1211],{"className":56379},[66,225],[15,56381,56382,56385],{"style":23445},[15,56383],{"className":56384,"style":1368},[217],[15,56386],{"className":56387,"style":4440},[23452],[15,56389,56390,56393],{"style":30899},[15,56391],{"className":56392,"style":1368},[217],[15,56394,56396],{"className":56395},[222,223,224,225],[15,56397,56399],{"className":56398},[66,225],[15,56400,37],{"className":56401},[66,67,225],[15,56403,1114],{"className":56404},[1113],[15,56406,56408],{"className":56407},[206],[15,56409,56411],{"className":56410,"style":45076},[210],[15,56412],{},[15,56414],{"className":56415},[169,23417],[15,56417],{"className":56418,"style":173},[71],[15,56420,25772],{"className":56421},[177],[15,56423],{"className":56424,"style":173},[71],[15,56426,56428,56431,56434,56437],{"className":56427},[57],[15,56429],{"className":56430,"style":4400},[61],[15,56432,49243],{"className":56433},[1339],[15,56435],{"className":56436,"style":697},[71],[15,56438,56440,56446,56520],{"className":56439},[1413],[15,56441,56443],{"className":56442,"style":1418},[162,1417],[15,56444,120],{"className":56445},[1422,1508],[15,56447,56449,56452,56517],{"className":56448},[66],[15,56450],{"className":56451},[162,23417],[15,56453,56455],{"className":56454},[6467],[15,56456,56458,56509],{"className":56457},[202,1090],[15,56459,56461,56506],{"className":56460},[206],[15,56462,56464,56478,56486],{"className":56463,"style":30873},[210],[15,56465,56466,56469],{"style":30876},[15,56467],{"className":56468,"style":1368},[217],[15,56470,56472],{"className":56471},[222,223,224,225],[15,56473,56475],{"className":56474},[66,225],[15,56476,37],{"className":56477},[66,67,225],[15,56479,56480,56483],{"style":23445},[15,56481],{"className":56482,"style":1368},[217],[15,56484],{"className":56485,"style":4440},[23452],[15,56487,56488,56491],{"style":30899},[15,56489],{"className":56490,"style":1368},[217],[15,56492,56494],{"className":56493},[222,223,224,225],[15,56495,56497,56500,56503],{"className":56496},[66,225],[15,56498,37],{"className":56499},[66,67,225],[15,56501,1245],{"className":56502},[76,225],[15,56504,1211],{"className":56505},[66,225],[15,56507,1114],{"className":56508},[1113],[15,56510,56512],{"className":56511},[206],[15,56513,56515],{"className":56514,"style":30921},[210],[15,56516],{},[15,56518],{"className":56519},[169,23417],[15,56521,56523],{"className":56522,"style":1418},[169,1417],[15,56524,126],{"className":56525},[1422,1508]," 得出，其中 ",[15,56528,56530,56554],{"className":56529,"translate":19},[18],[15,56531,56533],{"className":56532},[23],[25,56534,56535],{"xmlns":27},[29,56536,56537,56551],{},[32,56538,56539],{},[6467,56540,56541,56543],{},[35,56542,37],{},[32,56544,56545,56547,56549],{},[35,56546,37],{},[39,56548,1245],{},[1209,56550,1211],{},[45,56552,56553],{"encoding":47},"\\frac{n}{n+1}",[15,56555,56557],{"className":56556,"ariaHidden":53},[52],[15,56558,56560,56563],{"className":56559},[57],[15,56561],{"className":56562,"style":56341},[61],[15,56564,56566,56569,56634],{"className":56565},[66],[15,56567],{"className":56568},[162,23417],[15,56570,56572],{"className":56571},[6467],[15,56573,56575,56626],{"className":56574},[202,1090],[15,56576,56578,56623],{"className":56577},[206],[15,56579,56581,56601,56609],{"className":56580,"style":41848},[210],[15,56582,56583,56586],{"style":30876},[15,56584],{"className":56585,"style":1368},[217],[15,56587,56589],{"className":56588},[222,223,224,225],[15,56590,56592,56595,56598],{"className":56591},[66,225],[15,56593,37],{"className":56594},[66,67,225],[15,56596,1245],{"className":56597},[76,225],[15,56599,1211],{"className":56600},[66,225],[15,56602,56603,56606],{"style":23445},[15,56604],{"className":56605,"style":1368},[217],[15,56607],{"className":56608,"style":4440},[23452],[15,56610,56611,56614],{"style":30899},[15,56612],{"className":56613,"style":1368},[217],[15,56615,56617],{"className":56616},[222,223,224,225],[15,56618,56620],{"className":56619},[66,225],[15,56621,37],{"className":56622},[66,67,225],[15,56624,1114],{"className":56625},[1113],[15,56627,56629],{"className":56628},[206],[15,56630,56632],{"className":56631,"style":45076},[210],[15,56633],{},[15,56635],{"className":56636},[169,23417]," 被视为一个整体。这是高中知识。",[11,56639,11617,56640,56738],{},[15,56641,56643,56661],{"className":56642,"translate":19},[18],[15,56644,56646],{"className":56645},[23],[25,56647,56648],{"xmlns":27},[29,56649,56650,56658],{},[32,56651,56652],{},[6467,56653,56654,56656],{},[1209,56655,1211],{},[35,56657,123],{},[45,56659,56660],{"encoding":47},"\\frac{1}{x}",[15,56662,56664],{"className":56663,"ariaHidden":53},[52],[15,56665,56667,56670],{"className":56666},[57],[15,56668],{"className":56669,"style":30854},[61],[15,56671,56673,56676,56735],{"className":56672},[66],[15,56674],{"className":56675},[162,23417],[15,56677,56679],{"className":56678},[6467],[15,56680,56682,56727],{"className":56681},[202,1090],[15,56683,56685,56724],{"className":56684},[206],[15,56686,56688,56702,56710],{"className":56687,"style":30873},[210],[15,56689,56690,56693],{"style":30876},[15,56691],{"className":56692,"style":1368},[217],[15,56694,56696],{"className":56695},[222,223,224,225],[15,56697,56699],{"className":56698},[66,225],[15,56700,123],{"className":56701},[66,67,225],[15,56703,56704,56707],{"style":23445},[15,56705],{"className":56706,"style":1368},[217],[15,56708],{"className":56709,"style":4440},[23452],[15,56711,56712,56715],{"style":30899},[15,56713],{"className":56714,"style":1368},[217],[15,56716,56718],{"className":56717},[222,223,224,225],[15,56719,56721],{"className":56720},[66,225],[15,56722,1211],{"className":56723},[66,225],[15,56725,1114],{"className":56726},[1113],[15,56728,56730],{"className":56729},[206],[15,56731,56733],{"className":56732,"style":30921},[210],[15,56734],{},[15,56736],{"className":56737},[169,23417]," 是凹函数：",[15,56740,56742],{"className":56741,"translate":19},[294],[15,56743,56745,56819],{"className":56744,"translate":19},[18],[15,56746,56748],{"className":56747},[23],[25,56749,56750],{"xmlns":27,"display":303},[29,56751,56752,56816],{},[32,56753,56754,56768,56774,56776,56778,56780,56782,56784,56786,56798,56800],{},[1255,56755,56756,56758,56760],{},[39,56757,29732],{},[35,56759,37],{},[32,56761,56762,56764,56766],{},[35,56763,37],{},[39,56765,1245],{},[1209,56767,1211],{},[6467,56769,56770,56772],{},[1209,56771,1211],{},[35,56773,123],{},[1274,56775,2462],{},[35,56777,458],{},[35,56779,123],{},[39,56781,129],{},[35,56783,49243],{},[39,56785,1170],{},[6467,56787,56788,56796],{},[32,56789,56790,56792,56794],{},[35,56791,37],{},[39,56793,1245],{},[1209,56795,1211],{},[35,56797,37],{},[39,56799,9283],{},[6467,56801,56802,56804],{},[1209,56803,1211],{},[32,56805,56806,56808,56810],{},[35,56807,37],{},[39,56809,1245],{},[6467,56811,56812,56814],{},[1209,56813,1211],{},[1209,56815,1263],{},[45,56817,56818],{"encoding":47},"\\int_{n}^{n+1} \\frac{1}{x} \\, dx = \\ln \\frac{n+1}{n} > \\frac{1}{n + \\frac{1}{2}}",[15,56820,56822,56974,57069],{"className":56821,"ariaHidden":53},[52],[15,56823,56825,56828,56891,56894,56956,56959,56962,56965,56968,56971],{"className":56824},[57],[15,56826],{"className":56827,"style":29783},[61],[15,56829,56831,56834],{"className":56830},[1339],[15,56832,29732],{"className":56833,"style":29796},[1339,1470,1471],[15,56835,56837],{"className":56836},[198],[15,56838,56840,56883],{"className":56839},[202,1090],[15,56841,56843,56880],{"className":56842},[206],[15,56844,56846,56860],{"className":56845,"style":29809},[210],[15,56847,56848,56851],{"style":29812},[15,56849],{"className":56850,"style":218},[217],[15,56852,56854],{"className":56853},[222,223,224,225],[15,56855,56857],{"className":56856},[66,225],[15,56858,37],{"className":56859},[66,67,225],[15,56861,56862,56865],{"style":29824},[15,56863],{"className":56864,"style":218},[217],[15,56866,56868],{"className":56867},[222,223,224,225],[15,56869,56871,56874,56877],{"className":56870},[66,225],[15,56872,37],{"className":56873},[66,67,225],[15,56875,1245],{"className":56876},[76,225],[15,56878,1211],{"className":56879},[66,225],[15,56881,1114],{"className":56882},[1113],[15,56884,56886],{"className":56885},[206],[15,56887,56889],{"className":56888,"style":29843},[210],[15,56890],{},[15,56892],{"className":56893,"style":697},[71],[15,56895,56897,56900,56953],{"className":56896},[66],[15,56898],{"className":56899},[162,23417],[15,56901,56903],{"className":56902},[6467],[15,56904,56906,56945],{"className":56905},[202,1090],[15,56907,56909,56942],{"className":56908},[206],[15,56910,56912,56923,56931],{"className":56911,"style":23430},[210],[15,56913,56914,56917],{"style":23433},[15,56915],{"className":56916,"style":1368},[217],[15,56918,56920],{"className":56919},[66],[15,56921,123],{"className":56922},[66,67],[15,56924,56925,56928],{"style":23445},[15,56926],{"className":56927,"style":1368},[217],[15,56929],{"className":56930,"style":4440},[23452],[15,56932,56933,56936],{"style":23455},[15,56934],{"className":56935,"style":1368},[217],[15,56937,56939],{"className":56938},[66],[15,56940,1211],{"className":56941},[66],[15,56943,1114],{"className":56944},[1113],[15,56946,56948],{"className":56947},[206],[15,56949,56951],{"className":56950,"style":23474},[210],[15,56952],{},[15,56954],{"className":56955},[169,23417],[15,56957],{"className":56958,"style":697},[71],[15,56960,458],{"className":56961},[66,67],[15,56963,123],{"className":56964},[66,67],[15,56966],{"className":56967,"style":173},[71],[15,56969,129],{"className":56970},[177],[15,56972],{"className":56973,"style":173},[71],[15,56975,56977,56980,56983,56986,57060,57063,57066],{"className":56976},[57],[15,56978],{"className":56979,"style":49086},[61],[15,56981,49243],{"className":56982},[1339],[15,56984],{"className":56985,"style":697},[71],[15,56987,56989,56992,57057],{"className":56988},[66],[15,56990],{"className":56991},[162,23417],[15,56993,56995],{"className":56994},[6467],[15,56996,56998,57049],{"className":56997},[202,1090],[15,56999,57001,57046],{"className":57000},[206],[15,57002,57004,57015,57023],{"className":57003,"style":23430},[210],[15,57005,57006,57009],{"style":23433},[15,57007],{"className":57008,"style":1368},[217],[15,57010,57012],{"className":57011},[66],[15,57013,37],{"className":57014},[66,67],[15,57016,57017,57020],{"style":23445},[15,57018],{"className":57019,"style":1368},[217],[15,57021],{"className":57022,"style":4440},[23452],[15,57024,57025,57028],{"style":23455},[15,57026],{"className":57027,"style":1368},[217],[15,57029,57031,57034,57037,57040,57043],{"className":57030},[66],[15,57032,37],{"className":57033},[66,67],[15,57035],{"className":57036,"style":72},[71],[15,57038,1245],{"className":57039},[76],[15,57041],{"className":57042,"style":72},[71],[15,57044,1211],{"className":57045},[66],[15,57047,1114],{"className":57048},[1113],[15,57050,57052],{"className":57051},[206],[15,57053,57055],{"className":57054,"style":23474},[210],[15,57056],{},[15,57058],{"className":57059},[169,23417],[15,57061],{"className":57062,"style":173},[71],[15,57064,9283],{"className":57065},[177],[15,57067],{"className":57068,"style":173},[71],[15,57070,57072,57076],{"className":57071},[57],[15,57073],{"className":57074,"style":57075},[61],"height:2.4015em;vertical-align:-1.0801em;",[15,57077,57079,57082,57214],{"className":57078},[66],[15,57080],{"className":57081},[162,23417],[15,57083,57085],{"className":57084},[6467],[15,57086,57088,57205],{"className":57087},[202,1090],[15,57089,57091,57202],{"className":57090},[206],[15,57092,57094,57183,57191],{"className":57093,"style":23430},[210],[15,57095,57097,57100],{"style":57096},"top:-2.2649em;",[15,57098],{"className":57099,"style":1368},[217],[15,57101,57103,57106,57109,57112,57115],{"className":57102},[66],[15,57104,37],{"className":57105},[66,67],[15,57107],{"className":57108,"style":72},[71],[15,57110,1245],{"className":57111},[76],[15,57113],{"className":57114,"style":72},[71],[15,57116,57118,57121,57180],{"className":57117},[66],[15,57119],{"className":57120},[162,23417],[15,57122,57124],{"className":57123},[6467],[15,57125,57127,57172],{"className":57126},[202,1090],[15,57128,57130,57169],{"className":57129},[206],[15,57131,57133,57147,57155],{"className":57132,"style":30873},[210],[15,57134,57135,57138],{"style":30876},[15,57136],{"className":57137,"style":1368},[217],[15,57139,57141],{"className":57140},[222,223,224,225],[15,57142,57144],{"className":57143},[66,225],[15,57145,1263],{"className":57146},[66,225],[15,57148,57149,57152],{"style":23445},[15,57150],{"className":57151,"style":1368},[217],[15,57153],{"className":57154,"style":4440},[23452],[15,57156,57157,57160],{"style":30899},[15,57158],{"className":57159,"style":1368},[217],[15,57161,57163],{"className":57162},[222,223,224,225],[15,57164,57166],{"className":57165},[66,225],[15,57167,1211],{"className":57168},[66,225],[15,57170,1114],{"className":57171},[1113],[15,57173,57175],{"className":57174},[206],[15,57176,57178],{"className":57177,"style":30921},[210],[15,57179],{},[15,57181],{"className":57182},[169,23417],[15,57184,57185,57188],{"style":23445},[15,57186],{"className":57187,"style":1368},[217],[15,57189],{"className":57190,"style":4440},[23452],[15,57192,57193,57196],{"style":23455},[15,57194],{"className":57195,"style":1368},[217],[15,57197,57199],{"className":57198},[66],[15,57200,1211],{"className":57201},[66],[15,57203,1114],{"className":57204},[1113],[15,57206,57208],{"className":57207},[206],[15,57209,57212],{"className":57210,"style":57211},[210],"height:1.0801em;",[15,57213],{},[15,57215],{"className":57216},[169,23417],[11,57218,35909],{},[15,57220,57222],{"className":57221,"translate":19},[294],[15,57223,57225,57313],{"className":57224,"translate":19},[18],[15,57226,57228],{"className":57227},[23],[25,57229,57230],{"xmlns":27,"display":303},[29,57231,57232,57310],{},[32,57233,57234,57250,57270,57272,57306,57308],{},[1172,57235,57236,57242],{},[32,57237,57238,57240],{},[35,57239,43668],{},[39,57241,1170],{},[32,57243,57244,57246,57248],{},[35,57245,37],{},[39,57247,449],{},[35,57249,7620],{"mathvariant":138},[6467,57251,57252,57264],{},[1057,57253,57254,57256],{},[35,57255,4556],{},[32,57257,57258,57260,57262],{},[35,57259,37],{},[39,57261,1245],{},[1209,57263,1211],{},[1057,57265,57266,57268],{},[35,57267,4556],{},[35,57269,37],{},[39,57271,129],{},[6467,57273,57274,57276],{},[35,57275,48383],{},[131,57277,57278,57294],{},[32,57279,57280,57282,57284,57286,57292],{},[39,57281,120],{"fence":53},[1209,57283,1211],{},[39,57285,1245],{},[6467,57287,57288,57290],{},[1209,57289,1211],{},[35,57291,37],{},[39,57293,126],{"fence":53},[32,57295,57296,57298,57300],{},[35,57297,37],{},[39,57299,1245],{},[6467,57301,57302,57304],{},[1209,57303,1211],{},[1209,57305,1263],{},[39,57307,25772],{},[1209,57309,1211],{},[45,57311,57312],{"encoding":47},"\\lim_{n \\rightarrow \\infty} \\frac{a_{n+1}}{a_n} = \\frac{e}{\\left(1 + \\frac{1}{n}\\right)^{n + \\frac{1}{2}}} \u003C 1",[15,57314,57316,57532,57801],{"className":57315,"ariaHidden":53},[52],[15,57317,57319,57322,57375,57378,57523,57526,57529],{"className":57318},[57],[15,57320],{"className":57321,"style":52422},[61],[15,57323,57325],{"className":57324},[1339,1351],[15,57326,57328,57367],{"className":57327},[202,1090],[15,57329,57331,57364],{"className":57330},[206],[15,57332,57334,57354],{"className":57333,"style":840},[210],[15,57335,57336,57339],{"style":48823},[15,57337],{"className":57338,"style":1368},[217],[15,57340,57342],{"className":57341},[222,223,224,225],[15,57343,57345,57348,57351],{"className":57344},[66,225],[15,57346,37],{"className":57347},[66,67,225],[15,57349,449],{"className":57350},[177,225],[15,57352,7620],{"className":57353},[66,225],[15,57355,57356,57359],{"style":1386},[15,57357],{"className":57358,"style":1368},[217],[15,57360,57361],{},[15,57362,43668],{"className":57363},[1339],[15,57365,1114],{"className":57366},[1113],[15,57368,57370],{"className":57369},[206],[15,57371,57373],{"className":57372,"style":48861},[210],[15,57374],{},[15,57376],{"className":57377,"style":697},[71],[15,57379,57381,57384,57520],{"className":57380},[66],[15,57382],{"className":57383},[162,23417],[15,57385,57387],{"className":57386},[6467],[15,57388,57390,57512],{"className":57389},[202,1090],[15,57391,57393,57509],{"className":57392},[206],[15,57394,57396,57444,57452],{"className":57395,"style":39133},[210],[15,57397,57398,57401],{"style":23433},[15,57399],{"className":57400,"style":1368},[217],[15,57402,57404],{"className":57403},[66],[15,57405,57407,57410],{"className":57406},[66],[15,57408,4556],{"className":57409},[66,67],[15,57411,57413],{"className":57412},[198],[15,57414,57416,57436],{"className":57415},[202,1090],[15,57417,57419,57433],{"className":57418},[206],[15,57420,57422],{"className":57421,"style":2198},[210],[15,57423,57424,57427],{"style":1584},[15,57425],{"className":57426,"style":218},[217],[15,57428,57430],{"className":57429},[222,223,224,225],[15,57431,37],{"className":57432},[66,67,225],[15,57434,1114],{"className":57435},[1113],[15,57437,57439],{"className":57438},[206],[15,57440,57442],{"className":57441,"style":1121},[210],[15,57443],{},[15,57445,57446,57449],{"style":23445},[15,57447],{"className":57448,"style":1368},[217],[15,57450],{"className":57451,"style":4440},[23452],[15,57453,57454,57457],{"style":23455},[15,57455],{"className":57456,"style":1368},[217],[15,57458,57460],{"className":57459},[66],[15,57461,57463,57466],{"className":57462},[66],[15,57464,4556],{"className":57465},[66,67],[15,57467,57469],{"className":57468},[198],[15,57470,57472,57501],{"className":57471},[202,1090],[15,57473,57475,57498],{"className":57474},[206],[15,57476,57478],{"className":57477,"style":2139},[210],[15,57479,57480,57483],{"style":1584},[15,57481],{"className":57482,"style":218},[217],[15,57484,57486],{"className":57485},[222,223,224,225],[15,57487,57489,57492,57495],{"className":57488},[66,225],[15,57490,37],{"className":57491},[66,67,225],[15,57493,1245],{"className":57494},[76,225],[15,57496,1211],{"className":57497},[66,225],[15,57499,1114],{"className":57500},[1113],[15,57502,57504],{"className":57503},[206],[15,57505,57507],{"className":57506,"style":52140},[210],[15,57508],{},[15,57510,1114],{"className":57511},[1113],[15,57513,57515],{"className":57514},[206],[15,57516,57518],{"className":57517,"style":52619},[210],[15,57519],{},[15,57521],{"className":57522},[169,23417],[15,57524],{"className":57525,"style":173},[71],[15,57527,129],{"className":57528},[177],[15,57530],{"className":57531,"style":173},[71],[15,57533,57535,57538,57792,57795,57798],{"className":57534},[57],[15,57536],{"className":57537,"style":54380},[61],[15,57539,57541,57544,57789],{"className":57540},[66],[15,57542],{"className":57543},[162,23417],[15,57545,57547],{"className":57546},[6467],[15,57548,57550,57781],{"className":57549},[202,1090],[15,57551,57553,57778],{"className":57552},[206],[15,57554,57556,57759,57767],{"className":57555,"style":39133},[210],[15,57557,57558,57561],{"style":54401},[15,57559],{"className":57560,"style":54405},[217],[15,57562,57564],{"className":57563},[66],[15,57565,57567,57662],{"className":57566},[1413],[15,57568,57570,57576,57579,57582,57585,57588,57656],{"className":57569},[1413],[15,57571,57573],{"className":57572,"style":1418},[162,1417],[15,57574,120],{"className":57575},[1422,1508],[15,57577,1211],{"className":57578},[66],[15,57580],{"className":57581,"style":72},[71],[15,57583,1245],{"className":57584},[76],[15,57586],{"className":57587,"style":72},[71],[15,57589,57591,57594,57653],{"className":57590},[66],[15,57592],{"className":57593},[162,23417],[15,57595,57597],{"className":57596},[6467],[15,57598,57600,57645],{"className":57599},[202,1090],[15,57601,57603,57642],{"className":57602},[206],[15,57604,57606,57620,57628],{"className":57605,"style":30873},[210],[15,57607,57608,57611],{"style":30876},[15,57609],{"className":57610,"style":1368},[217],[15,57612,57614],{"className":57613},[222,223,224,225],[15,57615,57617],{"className":57616},[66,225],[15,57618,37],{"className":57619},[66,67,225],[15,57621,57622,57625],{"style":23445},[15,57623],{"className":57624,"style":1368},[217],[15,57626],{"className":57627,"style":4440},[23452],[15,57629,57630,57633],{"style":30899},[15,57631],{"className":57632,"style":1368},[217],[15,57634,57636],{"className":57635},[222,223,224,225],[15,57637,57639],{"className":57638},[66,225],[15,57640,1211],{"className":57641},[66,225],[15,57643,1114],{"className":57644},[1113],[15,57646,57648],{"className":57647},[206],[15,57649,57651],{"className":57650,"style":30921},[210],[15,57652],{},[15,57654],{"className":57655},[169,23417],[15,57657,57659],{"className":57658,"style":1418},[169,1417],[15,57660,126],{"className":57661},[1422,1508],[15,57663,57665],{"className":57664},[198],[15,57666,57668],{"className":57667},[202],[15,57669,57671],{"className":57670},[206],[15,57672,57674],{"className":57673,"style":54519},[210],[15,57675,57676,57679],{"style":51217},[15,57677],{"className":57678,"style":1368},[217],[15,57680,57682],{"className":57681},[222,223,224,225],[15,57683,57685,57688,57691],{"className":57684},[66,225],[15,57686,37],{"className":57687},[66,67,225],[15,57689,1245],{"className":57690},[76,225],[15,57692,57694,57697,57756],{"className":57693},[66,225],[15,57695],{"className":57696},[162,23417,222,3679,33581],[15,57698,57700],{"className":57699},[6467],[15,57701,57703,57748],{"className":57702},[202,1090],[15,57704,57706,57745],{"className":57705},[206],[15,57707,57709,57723,57731],{"className":57708,"style":49020},[210],[15,57710,57711,57714],{"style":33597},[15,57712],{"className":57713,"style":1368},[217],[15,57715,57717],{"className":57716},[222,3679,1508,225],[15,57718,57720],{"className":57719},[66,225],[15,57721,1263],{"className":57722},[66,225],[15,57724,57725,57728],{"style":33612},[15,57726],{"className":57727,"style":1368},[217],[15,57729],{"className":57730,"style":33619},[23452,225],[15,57732,57733,57736],{"style":33622},[15,57734],{"className":57735,"style":1368},[217],[15,57737,57739],{"className":57738},[222,3679,1508,225],[15,57740,57742],{"className":57741},[66,225],[15,57743,1211],{"className":57744},[66,225],[15,57746,1114],{"className":57747},[1113],[15,57749,57751],{"className":57750},[206],[15,57752,57754],{"className":57753,"style":33644},[210],[15,57755],{},[15,57757],{"className":57758},[169,23417,222,3679,33581],[15,57760,57761,57764],{"style":54607},[15,57762],{"className":57763,"style":54405},[217],[15,57765],{"className":57766,"style":4440},[23452],[15,57768,57769,57772],{"style":54616},[15,57770],{"className":57771,"style":54405},[217],[15,57773,57775],{"className":57774},[66],[15,57776,48383],{"className":57777},[66,67],[15,57779,1114],{"className":57780},[1113],[15,57782,57784],{"className":57783},[206],[15,57785,57787],{"className":57786,"style":54635},[210],[15,57788],{},[15,57790],{"className":57791},[169,23417],[15,57793],{"className":57794,"style":173},[71],[15,57796,25772],{"className":57797},[177],[15,57799],{"className":57800,"style":173},[71],[15,57802,57804,57807],{"className":57803},[57],[15,57805],{"className":57806,"style":1836},[61],[15,57808,1211],{"className":57809},[66],[11,57811,57812,57813,57882,57883,57975],{},"于是 ",[15,57814,57816,57833],{"className":57815,"translate":19},[18],[15,57817,57819],{"className":57818},[23],[25,57820,57821],{"xmlns":27},[29,57822,57823,57831],{},[32,57824,57825],{},[1057,57826,57827,57829],{},[35,57828,4556],{},[35,57830,37],{},[45,57832,52922],{"encoding":47},[15,57834,57836],{"className":57835,"ariaHidden":53},[52],[15,57837,57839,57842],{"className":57838},[57],[15,57840],{"className":57841,"style":10197},[61],[15,57843,57845,57848],{"className":57844},[66],[15,57846,4556],{"className":57847},[66,67],[15,57849,57851],{"className":57850},[198],[15,57852,57854,57874],{"className":57853},[202,1090],[15,57855,57857,57871],{"className":57856},[206],[15,57858,57860],{"className":57859,"style":2198},[210],[15,57861,57862,57865],{"style":1584},[15,57863],{"className":57864,"style":218},[217],[15,57866,57868],{"className":57867},[222,223,224,225],[15,57869,37],{"className":57870},[66,67,225],[15,57872,1114],{"className":57873},[1113],[15,57875,57877],{"className":57876},[206],[15,57878,57880],{"className":57879,"style":1121},[210],[15,57881],{}," 单调递减且 ",[15,57884,57886,57908],{"className":57885,"translate":19},[18],[15,57887,57889],{"className":57888},[23],[25,57890,57891],{"xmlns":27},[29,57892,57893,57905],{},[32,57894,57895,57901,57903],{},[1057,57896,57897,57899],{},[35,57898,4556],{},[35,57900,37],{},[39,57902,9283],{},[1209,57904,1804],{},[45,57906,57907],{"encoding":47},"a_n > 0",[15,57909,57911,57966],{"className":57910,"ariaHidden":53},[52],[15,57912,57914,57917,57957,57960,57963],{"className":57913},[57],[15,57915],{"className":57916,"style":26349},[61],[15,57918,57920,57923],{"className":57919},[66],[15,57921,4556],{"className":57922},[66,67],[15,57924,57926],{"className":57925},[198],[15,57927,57929,57949],{"className":57928},[202,1090],[15,57930,57932,57946],{"className":57931},[206],[15,57933,57935],{"className":57934,"style":2198},[210],[15,57936,57937,57940],{"style":1584},[15,57938],{"className":57939,"style":218},[217],[15,57941,57943],{"className":57942},[222,223,224,225],[15,57944,37],{"className":57945},[66,67,225],[15,57947,1114],{"className":57948},[1113],[15,57950,57952],{"className":57951},[206],[15,57953,57955],{"className":57954,"style":1121},[210],[15,57956],{},[15,57958],{"className":57959,"style":173},[71],[15,57961,9283],{"className":57962},[177],[15,57964],{"className":57965,"style":173},[71],[15,57967,57969,57972],{"className":57968},[57],[15,57970],{"className":57971,"style":1836},[61],[15,57973,1804],{"className":57974},[66],"，因此它必然收敛，即存在极限。",[11,57977,57978],{},"设：",[15,57980,57982],{"className":57981,"translate":19},[294],[15,57983,57985,58047],{"className":57984,"translate":19},[18],[15,57986,57988],{"className":57987},[23],[25,57989,57990],{"xmlns":27,"display":303},[29,57991,57992,58044],{},[32,57993,57994,58010,58034,58040,58042],{},[1172,57995,57996,58002],{},[32,57997,57998,58000],{},[35,57999,43668],{},[39,58001,1170],{},[32,58003,58004,58006,58008],{},[35,58005,37],{},[39,58007,449],{},[35,58009,7620],{"mathvariant":138},[6467,58011,58012,58018],{},[32,58013,58014,58016],{},[35,58015,37],{},[39,58017,40769],{"stretchy":119},[131,58019,58020,58022],{},[35,58021,37],{},[32,58023,58024,58026,58028],{},[35,58025,37],{},[39,58027,1245],{},[6467,58029,58030,58032],{},[1209,58031,1211],{},[1209,58033,1263],{},[131,58035,58036,58038],{},[35,58037,48383],{},[35,58039,37],{},[39,58041,129],{},[35,58043,4556],{},[45,58045,58046],{"encoding":47},"\\lim_{n \\rightarrow \\infty} \\frac{n!}{n^{n + \\frac{1}{2}}} e^n = a",[15,58048,58050,58315],{"className":58049,"ariaHidden":53},[52],[15,58051,58053,58056,58109,58112,58277,58306,58309,58312],{"className":58052},[57],[15,58054],{"className":58055,"style":53096},[61],[15,58057,58059],{"className":58058},[1339,1351],[15,58060,58062,58101],{"className":58061},[202,1090],[15,58063,58065,58098],{"className":58064},[206],[15,58066,58068,58088],{"className":58067,"style":840},[210],[15,58069,58070,58073],{"style":48823},[15,58071],{"className":58072,"style":1368},[217],[15,58074,58076],{"className":58075},[222,223,224,225],[15,58077,58079,58082,58085],{"className":58078},[66,225],[15,58080,37],{"className":58081},[66,67,225],[15,58083,449],{"className":58084},[177,225],[15,58086,7620],{"className":58087},[66,225],[15,58089,58090,58093],{"style":1386},[15,58091],{"className":58092,"style":1368},[217],[15,58094,58095],{},[15,58096,43668],{"className":58097},[1339],[15,58099,1114],{"className":58100},[1113],[15,58102,58104],{"className":58103},[206],[15,58105,58107],{"className":58106,"style":48861},[210],[15,58108],{},[15,58110],{"className":58111,"style":697},[71],[15,58113,58115,58118,58274],{"className":58114},[66],[15,58116],{"className":58117},[162,23417],[15,58119,58121],{"className":58120},[6467],[15,58122,58124,58266],{"className":58123},[202,1090],[15,58125,58127,58263],{"className":58126},[206],[15,58128,58130,58241,58249],{"className":58129,"style":27435},[210],[15,58131,58132,58135],{"style":53117},[15,58133],{"className":58134,"style":1368},[217],[15,58136,58138],{"className":58137},[66],[15,58139,58141,58144],{"className":58140},[66],[15,58142,37],{"className":58143},[66,67],[15,58145,58147],{"className":58146},[198],[15,58148,58150],{"className":58149},[202],[15,58151,58153],{"className":58152},[206],[15,58154,58156],{"className":58155,"style":53142},[210],[15,58157,58158,58161],{"style":53145},[15,58159],{"className":58160,"style":1368},[217],[15,58162,58164],{"className":58163},[222,223,224,225],[15,58165,58167,58170,58173],{"className":58166},[66,225],[15,58168,37],{"className":58169},[66,67,225],[15,58171,1245],{"className":58172},[76,225],[15,58174,58176,58179,58238],{"className":58175},[66,225],[15,58177],{"className":58178},[162,23417,222,3679,33581],[15,58180,58182],{"className":58181},[6467],[15,58183,58185,58230],{"className":58184},[202,1090],[15,58186,58188,58227],{"className":58187},[206],[15,58189,58191,58205,58213],{"className":58190,"style":49020},[210],[15,58192,58193,58196],{"style":33597},[15,58194],{"className":58195,"style":1368},[217],[15,58197,58199],{"className":58198},[222,3679,1508,225],[15,58200,58202],{"className":58201},[66,225],[15,58203,1263],{"className":58204},[66,225],[15,58206,58207,58210],{"style":33612},[15,58208],{"className":58209,"style":1368},[217],[15,58211],{"className":58212,"style":33619},[23452,225],[15,58214,58215,58218],{"style":33622},[15,58216],{"className":58217,"style":1368},[217],[15,58219,58221],{"className":58220},[222,3679,1508,225],[15,58222,58224],{"className":58223},[66,225],[15,58225,1211],{"className":58226},[66,225],[15,58228,1114],{"className":58229},[1113],[15,58231,58233],{"className":58232},[206],[15,58234,58236],{"className":58235,"style":33644},[210],[15,58237],{},[15,58239],{"className":58240},[169,23417,222,3679,33581],[15,58242,58243,58246],{"style":23445},[15,58244],{"className":58245,"style":1368},[217],[15,58247],{"className":58248,"style":4440},[23452],[15,58250,58251,58254],{"style":23455},[15,58252],{"className":58253,"style":1368},[217],[15,58255,58257,58260],{"className":58256},[66],[15,58258,37],{"className":58259},[66,67],[15,58261,40769],{"className":58262},[169],[15,58264,1114],{"className":58265},[1113],[15,58267,58269],{"className":58268},[206],[15,58270,58272],{"className":58271,"style":53260},[210],[15,58273],{},[15,58275],{"className":58276},[169,23417],[15,58278,58280,58283],{"className":58279},[66],[15,58281,48383],{"className":58282},[66,67],[15,58284,58286],{"className":58285},[198],[15,58287,58289],{"className":58288},[202],[15,58290,58292],{"className":58291},[206],[15,58293,58295],{"className":58294,"style":51750},[210],[15,58296,58297,58300],{"style":397},[15,58298],{"className":58299,"style":218},[217],[15,58301,58303],{"className":58302},[222,223,224,225],[15,58304,37],{"className":58305},[66,67,225],[15,58307],{"className":58308,"style":173},[71],[15,58310,129],{"className":58311},[177],[15,58313],{"className":58314,"style":173},[71],[15,58316,58318,58321],{"className":58317},[57],[15,58319],{"className":58320,"style":86},[61],[15,58322,4556],{"className":58323},[66,67],[11,58325,58326],{},"接下来我们需要求出这个极限。",[11,58328,58329],{},"由前面的沃利斯公式：",[15,58331,58333],{"className":58332,"translate":19},[294],[15,58334,58336,58425],{"className":58335,"translate":19},[18],[15,58337,58339],{"className":58338},[23],[25,58340,58341],{"xmlns":27,"display":303},[29,58342,58343,58423],{},[32,58344,58345,58361,58415,58417],{},[1172,58346,58347,58353],{},[32,58348,58349,58351],{},[35,58350,43668],{},[39,58352,1170],{},[32,58354,58355,58357,58359],{},[35,58356,621],{},[39,58358,449],{},[35,58360,7620],{"mathvariant":138},[6467,58362,58363,58381],{},[32,58364,58365,58367,58369,58371,58373,58375],{},[39,58366,120],{"stretchy":119},[1209,58368,1263],{},[35,58370,621],{},[39,58372,126],{"stretchy":119},[39,58374,40769],{"stretchy":119},[131,58376,58377,58379],{},[39,58378,40769],{"stretchy":119},[1209,58380,1263],{},[32,58382,58383,58385,58387,58389,58391,58393,58395,58397,58399,58401,58403,58405,58407,58409,58411,58413],{},[39,58384,120],{"stretchy":119},[1209,58386,1263],{},[35,58388,621],{},[39,58390,1245],{},[1209,58392,1211],{},[39,58394,126],{"stretchy":119},[39,58396,40769],{"stretchy":119},[39,58398,40769],{"stretchy":119},[39,58400,120],{"stretchy":119},[1209,58402,1263],{},[35,58404,621],{},[39,58406,6602],{},[1209,58408,1211],{},[39,58410,126],{"stretchy":119},[39,58412,40769],{"stretchy":119},[39,58414,40769],{"stretchy":119},[39,58416,129],{},[6467,58418,58419,58421],{},[35,58420,33494],{},[1209,58422,1263],{},[45,58424,47976],{"encoding":47},[15,58426,58428,58644],{"className":58427,"ariaHidden":53},[52],[15,58429,58431,58434,58487,58490,58635,58638,58641],{"className":58430},[57],[15,58432],{"className":58433,"style":45944},[61],[15,58435,58437],{"className":58436},[1339,1351],[15,58438,58440,58479],{"className":58439},[202,1090],[15,58441,58443,58476],{"className":58442},[206],[15,58444,58446,58466],{"className":58445,"style":840},[210],[15,58447,58448,58451],{"style":1364},[15,58449],{"className":58450,"style":1368},[217],[15,58452,58454],{"className":58453},[222,223,224,225],[15,58455,58457,58460,58463],{"className":58456},[66,225],[15,58458,621],{"className":58459,"style":684},[66,67,225],[15,58461,449],{"className":58462},[177,225],[15,58464,7620],{"className":58465},[66,225],[15,58467,58468,58471],{"style":1386},[15,58469],{"className":58470,"style":1368},[217],[15,58472,58473],{},[15,58474,43668],{"className":58475},[1339],[15,58477,1114],{"className":58478},[1113],[15,58480,58482],{"className":58481},[206],[15,58483,58485],{"className":58484,"style":44025},[210],[15,58486],{},[15,58488],{"className":58489,"style":697},[71],[15,58491,58493,58496,58632],{"className":58492},[66],[15,58494],{"className":58495},[162,23417],[15,58497,58499],{"className":58498},[6467],[15,58500,58502,58624],{"className":58501},[202,1090],[15,58503,58505,58621],{"className":58504},[206],[15,58506,58508,58564,58572],{"className":58507,"style":45963},[210],[15,58509,58510,58513],{"style":23433},[15,58511],{"className":58512,"style":1368},[217],[15,58514,58516,58519,58522,58525,58528,58531,58534,58537,58540,58543,58546,58549,58552,58555,58558,58561],{"className":58515},[66],[15,58517,120],{"className":58518},[162],[15,58520,1263],{"className":58521},[66],[15,58523,621],{"className":58524,"style":684},[66,67],[15,58526],{"className":58527,"style":72},[71],[15,58529,1245],{"className":58530},[76],[15,58532],{"className":58533,"style":72},[71],[15,58535,1211],{"className":58536},[66],[15,58538,41232],{"className":58539},[169],[15,58541,120],{"className":58542},[162],[15,58544,1263],{"className":58545},[66],[15,58547,621],{"className":58548,"style":684},[66,67],[15,58550],{"className":58551,"style":72},[71],[15,58553,6602],{"className":58554},[76],[15,58556],{"className":58557,"style":72},[71],[15,58559,1211],{"className":58560},[66],[15,58562,41232],{"className":58563},[169],[15,58565,58566,58569],{"style":23445},[15,58567],{"className":58568,"style":1368},[217],[15,58570],{"className":58571,"style":4440},[23452],[15,58573,58574,58577],{"style":23455},[15,58575],{"className":58576,"style":1368},[217],[15,58578,58580,58583,58586,58589,58592],{"className":58579},[66],[15,58581,120],{"className":58582},[162],[15,58584,1263],{"className":58585},[66],[15,58587,621],{"className":58588,"style":684},[66,67],[15,58590,46048],{"className":58591},[169],[15,58593,58595,58598],{"className":58594},[169],[15,58596,40769],{"className":58597},[169],[15,58599,58601],{"className":58600},[198],[15,58602,58604],{"className":58603},[202],[15,58605,58607],{"className":58606},[206],[15,58608,58610],{"className":58609,"style":6740},[210],[15,58611,58612,58615],{"style":213},[15,58613],{"className":58614,"style":218},[217],[15,58616,58618],{"className":58617},[222,223,224,225],[15,58619,1263],{"className":58620},[66,225],[15,58622,1114],{"className":58623},[1113],[15,58625,58627],{"className":58626},[206],[15,58628,58630],{"className":58629,"style":41282},[210],[15,58631],{},[15,58633],{"className":58634},[169,23417],[15,58636],{"className":58637,"style":173},[71],[15,58639,129],{"className":58640},[177],[15,58642],{"className":58643,"style":173},[71],[15,58645,58647,58650],{"className":58646},[57],[15,58648],{"className":58649,"style":39114},[61],[15,58651,58653,58656,58709],{"className":58652},[66],[15,58654],{"className":58655},[162,23417],[15,58657,58659],{"className":58658},[6467],[15,58660,58662,58701],{"className":58661},[202,1090],[15,58663,58665,58698],{"className":58664},[206],[15,58666,58668,58679,58687],{"className":58667,"style":39133},[210],[15,58669,58670,58673],{"style":23433},[15,58671],{"className":58672,"style":1368},[217],[15,58674,58676],{"className":58675},[66],[15,58677,1263],{"className":58678},[66],[15,58680,58681,58684],{"style":23445},[15,58682],{"className":58683,"style":1368},[217],[15,58685],{"className":58686,"style":4440},[23452],[15,58688,58689,58692],{"style":23455},[15,58690],{"className":58691,"style":1368},[217],[15,58693,58695],{"className":58694},[66],[15,58696,33494],{"className":58697,"style":1515},[66,67],[15,58699,1114],{"className":58700},[1113],[15,58702,58704],{"className":58703},[206],[15,58705,58707],{"className":58706,"style":23474},[210],[15,58708],{},[15,58710],{"className":58711},[169,23417],[15,58713,58715],{"className":58714,"translate":19},[294],[15,58716,58718,58776],{"className":58717,"translate":19},[18],[15,58719,58721],{"className":58720},[23],[25,58722,58723],{"xmlns":27,"display":303},[29,58724,58725,58773],{},[32,58726,58727,58729,58731,58733,58735,58737,58739,58741,58743,58745,58747,58749,58751,58753,58755,58757,58759,58761,58763,58769,58771],{},[39,58728,120],{"stretchy":119},[1209,58730,1263],{},[35,58732,37],{},[39,58734,126],{"stretchy":119},[39,58736,40769],{"stretchy":119},[39,58738,40769],{"stretchy":119},[39,58740,129],{},[1209,58742,1263],{},[35,58744,37],{},[39,58746,120],{"stretchy":119},[1209,58748,1263],{},[35,58750,37],{},[39,58752,6602],{},[1209,58754,1263],{},[39,58756,126],{"stretchy":119},[39,58758,37800],{},[1209,58760,1263],{},[39,58762,129],{},[131,58764,58765,58767],{},[1209,58766,1263],{},[35,58768,37],{},[35,58770,37],{},[39,58772,40769],{"stretchy":119},[45,58774,58775],{"encoding":47},"(2n)!! = 2n (2n-2) \\cdots 2 = 2^n n!",[15,58777,58779,58806,58836,58869],{"className":58778,"ariaHidden":53},[52],[15,58780,58782,58785,58788,58791,58794,58797,58800,58803],{"className":58781},[57],[15,58783],{"className":58784,"style":154},[61],[15,58786,120],{"className":58787},[162],[15,58789,1263],{"className":58790},[66],[15,58792,37],{"className":58793},[66,67],[15,58795,41232],{"className":58796},[169],[15,58798],{"className":58799,"style":173},[71],[15,58801,129],{"className":58802},[177],[15,58804],{"className":58805,"style":173},[71],[15,58807,58809,58812,58815,58818,58821,58824,58827,58830,58833],{"className":58808},[57],[15,58810],{"className":58811,"style":154},[61],[15,58813,1263],{"className":58814},[66],[15,58816,37],{"className":58817},[66,67],[15,58819,120],{"className":58820},[162],[15,58822,1263],{"className":58823},[66],[15,58825,37],{"className":58826},[66,67],[15,58828],{"className":58829,"style":72},[71],[15,58831,6602],{"className":58832},[76],[15,58834],{"className":58835,"style":72},[71],[15,58837,58839,58842,58845,58848,58851,58854,58857,58860,58863,58866],{"className":58838},[57],[15,58840],{"className":58841,"style":154},[61],[15,58843,1263],{"className":58844},[66],[15,58846,126],{"className":58847},[169],[15,58849],{"className":58850,"style":697},[71],[15,58852,37800],{"className":58853},[1413],[15,58855],{"className":58856,"style":697},[71],[15,58858,1263],{"className":58859},[66],[15,58861],{"className":58862,"style":173},[71],[15,58864,129],{"className":58865},[177],[15,58867],{"className":58868,"style":173},[71],[15,58870,58872,58875,58904,58907],{"className":58871},[57],[15,58873],{"className":58874,"style":51750},[61],[15,58876,58878,58881],{"className":58877},[66],[15,58879,1263],{"className":58880},[66],[15,58882,58884],{"className":58883},[198],[15,58885,58887],{"className":58886},[202],[15,58888,58890],{"className":58889},[206],[15,58891,58893],{"className":58892,"style":51750},[210],[15,58894,58895,58898],{"style":397},[15,58896],{"className":58897,"style":218},[217],[15,58899,58901],{"className":58900},[222,223,224,225],[15,58902,37],{"className":58903},[66,67,225],[15,58905,37],{"className":58906},[66,67],[15,58908,40769],{"className":58909},[169],[15,58911,58913],{"className":58912,"translate":19},[294],[15,58914,58916,59002],{"className":58915,"translate":19},[18],[15,58917,58919],{"className":58918},[23],[25,58920,58921],{"xmlns":27,"display":303},[29,58922,58923,58999],{},[32,58924,58925,58927,58929,58931,58933,58935,58937,58939,58941,58943,58971,58973],{},[39,58926,120],{"stretchy":119},[1209,58928,1263],{},[35,58930,37],{},[39,58932,6602],{},[1209,58934,1211],{},[39,58936,126],{"stretchy":119},[39,58938,40769],{"stretchy":119},[39,58940,40769],{"stretchy":119},[39,58942,129],{},[6467,58944,58945,58957],{},[32,58946,58947,58949,58951,58953,58955],{},[39,58948,120],{"stretchy":119},[1209,58950,1263],{},[35,58952,37],{},[39,58954,126],{"stretchy":119},[39,58956,40769],{"stretchy":119},[32,58958,58959,58961,58963,58965,58967,58969],{},[39,58960,120],{"stretchy":119},[1209,58962,1263],{},[35,58964,37],{},[39,58966,126],{"stretchy":119},[39,58968,40769],{"stretchy":119},[39,58970,40769],{"stretchy":119},[39,58972,129],{},[6467,58974,58975,58987],{},[32,58976,58977,58979,58981,58983,58985],{},[39,58978,120],{"stretchy":119},[1209,58980,1263],{},[35,58982,37],{},[39,58984,126],{"stretchy":119},[39,58986,40769],{"stretchy":119},[32,58988,58989,58995,58997],{},[131,58990,58991,58993],{},[1209,58992,1263],{},[35,58994,37],{},[35,58996,37],{},[39,58998,40769],{"stretchy":119},[45,59000,59001],{"encoding":47},"(2n-1)!! = \\frac{(2n)!}{(2n)!!} = \\frac{(2n)!}{2^n n!}",[15,59003,59005,59029,59050,59145],{"className":59004,"ariaHidden":53},[52],[15,59006,59008,59011,59014,59017,59020,59023,59026],{"className":59007},[57],[15,59009],{"className":59010,"style":154},[61],[15,59012,120],{"className":59013},[162],[15,59015,1263],{"className":59016},[66],[15,59018,37],{"className":59019},[66,67],[15,59021],{"className":59022,"style":72},[71],[15,59024,6602],{"className":59025},[76],[15,59027],{"className":59028,"style":72},[71],[15,59030,59032,59035,59038,59041,59044,59047],{"className":59031},[57],[15,59033],{"className":59034,"style":154},[61],[15,59036,1211],{"className":59037},[66],[15,59039,41232],{"className":59040},[169],[15,59042],{"className":59043,"style":173},[71],[15,59045,129],{"className":59046},[177],[15,59048],{"className":59049,"style":173},[71],[15,59051,59053,59056,59136,59139,59142],{"className":59052},[57],[15,59054],{"className":59055,"style":41192},[61],[15,59057,59059,59062,59133],{"className":59058},[66],[15,59060],{"className":59061},[162,23417],[15,59063,59065],{"className":59064},[6467],[15,59066,59068,59125],{"className":59067},[202,1090],[15,59069,59071,59122],{"className":59070},[206],[15,59072,59074,59094,59102],{"className":59073,"style":41211},[210],[15,59075,59076,59079],{"style":23433},[15,59077],{"className":59078,"style":1368},[217],[15,59080,59082,59085,59088,59091],{"className":59081},[66],[15,59083,120],{"className":59084},[162],[15,59086,1263],{"className":59087},[66],[15,59089,37],{"className":59090},[66,67],[15,59092,41232],{"className":59093},[169],[15,59095,59096,59099],{"style":23445},[15,59097],{"className":59098,"style":1368},[217],[15,59100],{"className":59101,"style":4440},[23452],[15,59103,59104,59107],{"style":23455},[15,59105],{"className":59106,"style":1368},[217],[15,59108,59110,59113,59116,59119],{"className":59109},[66],[15,59111,120],{"className":59112},[162],[15,59114,1263],{"className":59115},[66],[15,59117,37],{"className":59118},[66,67],[15,59120,46048],{"className":59121},[169],[15,59123,1114],{"className":59124},[1113],[15,59126,59128],{"className":59127},[206],[15,59129,59131],{"className":59130,"style":41282},[210],[15,59132],{},[15,59134],{"className":59135},[169,23417],[15,59137],{"className":59138,"style":173},[71],[15,59140,129],{"className":59141},[177],[15,59143],{"className":59144,"style":173},[71],[15,59146,59148,59152],{"className":59147},[57],[15,59149],{"className":59150,"style":59151},[61],"height:2.113em;vertical-align:-0.686em;",[15,59153,59155,59158,59252],{"className":59154},[66],[15,59156],{"className":59157},[162,23417],[15,59159,59161],{"className":59160},[6467],[15,59162,59164,59244],{"className":59163},[202,1090],[15,59165,59167,59241],{"className":59166},[206],[15,59168,59170,59213,59221],{"className":59169,"style":41211},[210],[15,59171,59172,59175],{"style":23433},[15,59173],{"className":59174,"style":1368},[217],[15,59176,59178,59207,59210],{"className":59177},[66],[15,59179,59181,59184],{"className":59180},[66],[15,59182,1263],{"className":59183},[66],[15,59185,59187],{"className":59186},[198],[15,59188,59190],{"className":59189},[202],[15,59191,59193],{"className":59192},[206],[15,59194,59196],{"className":59195,"style":48923},[210],[15,59197,59198,59201],{"style":46255},[15,59199],{"className":59200,"style":218},[217],[15,59202,59204],{"className":59203},[222,223,224,225],[15,59205,37],{"className":59206},[66,67,225],[15,59208,37],{"className":59209},[66,67],[15,59211,40769],{"className":59212},[169],[15,59214,59215,59218],{"style":23445},[15,59216],{"className":59217,"style":1368},[217],[15,59219],{"className":59220,"style":4440},[23452],[15,59222,59223,59226],{"style":23455},[15,59224],{"className":59225,"style":1368},[217],[15,59227,59229,59232,59235,59238],{"className":59228},[66],[15,59230,120],{"className":59231},[162],[15,59233,1263],{"className":59234},[66],[15,59236,37],{"className":59237},[66,67],[15,59239,46048],{"className":59240},[169],[15,59242,1114],{"className":59243},[1113],[15,59245,59247],{"className":59246},[206],[15,59248,59250],{"className":59249,"style":23474},[210],[15,59251],{},[15,59253],{"className":59254},[169,23417],[15,59256,59258],{"className":59257,"translate":19},[294],[15,59259,59261,59373],{"className":59260,"translate":19},[18],[15,59262,59264],{"className":59263},[23],[25,59265,59266],{"xmlns":27,"display":303},[29,59267,59268,59370],{},[32,59269,59270,59272,59274,59276,59278,59280,59282,59284,59286,59288,59324,59326],{},[39,59271,120],{"stretchy":119},[1209,59273,1263],{},[35,59275,37],{},[39,59277,1245],{},[1209,59279,1211],{},[39,59281,126],{"stretchy":119},[39,59283,40769],{"stretchy":119},[39,59285,40769],{"stretchy":119},[39,59287,129],{},[6467,59289,59290,59306],{},[32,59291,59292,59294,59296,59298,59300,59302,59304],{},[39,59293,120],{"stretchy":119},[1209,59295,1263],{},[35,59297,37],{},[39,59299,1245],{},[1209,59301,1263],{},[39,59303,126],{"stretchy":119},[39,59305,40769],{"stretchy":119},[32,59307,59308,59310,59312,59314,59316,59318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= \\frac{(2n+2)!}{(2n+2)!!} = \\frac{(2n+2)!}{2^{n+1} (n+1)!}",[15,59374,59376,59400,59421,59540],{"className":59375,"ariaHidden":53},[52],[15,59377,59379,59382,59385,59388,59391,59394,59397],{"className":59378},[57],[15,59380],{"className":59381,"style":154},[61],[15,59383,120],{"className":59384},[162],[15,59386,1263],{"className":59387},[66],[15,59389,37],{"className":59390},[66,67],[15,59392],{"className":59393,"style":72},[71],[15,59395,1245],{"className":59396},[76],[15,59398],{"className":59399,"style":72},[71],[15,59401,59403,59406,59409,59412,59415,59418],{"className":59402},[57],[15,59404],{"className":59405,"style":154},[61],[15,59407,1211],{"className":59408},[66],[15,59410,41232],{"className":59411},[169],[15,59413],{"className":59414,"style":173},[71],[15,59416,129],{"className":59417},[177],[15,59419],{"className":59420,"style":173},[71],[15,59422,59424,59427,59531,59534,59537],{"className":59423},[57],[15,59425],{"className":59426,"style":41192},[61],[15,59428,59430,59433,59528],{"className":59429},[66],[15,59431],{"className":59432},[162,23417],[15,59434,59436],{"className":59435},[6467],[15,59437,59439,59520],{"className":59438},[202,1090],[15,59440,59442,59517],{"className":59441},[206],[15,59443,59445,59477,59485],{"className":59444,"style":41211},[210],[15,59446,59447,59450],{"style":23433},[15,59448],{"className":59449,"style":1368},[217],[15,59451,59453,59456,59459,59462,59465,59468,59471,59474],{"className":59452},[66],[15,59454,120],{"className":59455},[162],[15,59457,1263],{"className":59458},[66],[15,59460,37],{"className":59461},[66,67],[15,59463],{"className":59464,"style":72},[71],[15,59466,1245],{"className":59467},[76],[15,59469],{"className":59470,"style":72},[71],[15,59472,1263],{"className":59473},[66],[15,59475,41232],{"className":59476},[169],[15,59478,59479,59482],{"style":23445},[15,59480],{"className":59481,"style":1368},[217],[15,59483],{"className":59484,"style":4440},[23452],[15,59486,59487,59490],{"style":23455},[15,59488],{"className":59489,"style":1368},[217],[15,59491,59493,59496,59499,59502,59505,59508,59511,59514],{"className":59492},[66],[15,59494,120],{"className":59495},[162],[15,59497,1263],{"className":59498},[66],[15,59500,37],{"className":59501},[66,67],[15,59503],{"className":59504,"style":72},[71],[15,59506,1245],{"className":59507},[76],[15,59509],{"className":59510,"style":72},[71],[15,59512,1263],{"className":59513},[66],[15,59515,46048],{"className":59516},[169],[15,59518,1114],{"className":59519},[1113],[15,59521,59523],{"className":59522},[206],[15,59524,59526],{"className":59525,"style":41282},[210],[15,59527],{},[15,59529],{"className":59530},[169,23417],[15,59532],{"className":59533,"style":173},[71],[15,59535,129],{"className":59536},[177],[15,59538],{"className":59539,"style":173},[71],[15,59541,59543,59546],{"className":59542},[57],[15,59544],{"className":59545,"style":41192},[61],[15,59547,59549,59552,59682],{"className":59548},[66],[15,59550],{"className":59551},[162,23417],[15,59553,59555],{"className":59554},[6467],[15,59556,59558,59674],{"className":59557},[202,1090],[15,59559,59561,59671],{"className":59560},[206],[15,59562,59564,59631,59639],{"className":59563,"style":41211},[210],[15,59565,59566,59569],{"style":23433},[15,59567],{"className":59568,"style":1368},[217],[15,59570,59572,59610,59613,59616,59619,59622,59625,59628],{"className":59571},[66],[15,59573,59575,59578],{"className":59574},[66],[15,59576,1263],{"className":59577},[66],[15,59579,59581],{"className":59580},[198],[15,59582,59584],{"className":59583},[202],[15,59585,59587],{"className":59586},[206],[15,59588,59590],{"className":59589,"style":46252},[210],[15,59591,59592,59595],{"style":46255},[15,59593],{"className":59594,"style":218},[217],[15,59596,59598],{"className":59597},[222,223,224,225],[15,59599,59601,59604,59607],{"className":59600},[66,225],[15,59602,37],{"className":59603},[66,67,225],[15,59605,1245],{"className":59606},[76,225],[15,59608,1211],{"className":59609},[66,225],[15,59611,120],{"className":59612},[162],[15,59614,37],{"className":59615},[66,67],[15,59617],{"className":59618,"style":72},[71],[15,59620,1245],{"className":59621},[76],[15,59623],{"className":59624,"style":72},[71],[15,59626,1211],{"className":59627},[66],[15,59629,46048],{"className":59630},[169],[15,59632,59633,59636],{"style":23445},[15,59634],{"className":59635,"style":1368},[217],[15,59637],{"className":59638,"style":4440},[23452],[15,59640,59641,59644],{"style":23455},[15,59642],{"className":59643,"style":1368},[217],[15,59645,59647,59650,59653,59656,59659,59662,59665,59668],{"className":59646},[66],[15,59648,120],{"className":59649},[162],[15,59651,1263],{"className":59652},[66],[15,59654,37],{"className":59655},[66,67],[15,59657],{"className":59658,"style":72},[71],[15,59660,1245],{"className":59661},[76],[15,59663],{"className":59664,"style":72},[71],[15,59666,1263],{"className":59667},[66],[15,59669,46048],{"className":59670},[169],[15,59672,1114],{"className":59673},[1113],[15,59675,59677],{"className":59676},[206],[15,59678,59680],{"className":59679,"style":41282},[210],[15,59681],{},[15,59683],{"className":59684},[169,23417],[11,59686,59687],{},"于是沃利斯公式变为：",[15,59689,59691],{"className":59690,"translate":19},[294],[15,59692,59694,59791],{"className":59693,"translate":19},[18],[15,59695,59697],{"className":59696},[23],[25,59698,59699],{"xmlns":27,"display":303},[29,59700,59701,59788],{},[32,59702,59703,59719,59780,59782],{},[1172,59704,59705,59711],{},[32,59706,59707,59709],{},[35,59708,43668],{},[39,59710,1170],{},[32,59712,59713,59715,59717],{},[35,59714,37],{},[39,59716,449],{},[35,59718,7620],{"mathvariant":138},[6467,59720,59721,59754],{},[32,59722,59723,59730,59732,59734,59736,59742,59744,59746,59748,59750,59752],{},[131,59724,59725,59728],{},[1209,59726,59727],{},"16",[35,59729,37],{},[39,59731,120],{"stretchy":119},[35,59733,37],{},[39,59735,40769],{"stretchy":119},[131,59737,59738,59740],{},[39,59739,126],{"stretchy":119},[1209,59741,15855],{},[39,59743,120],{"stretchy":119},[1209,59745,1263],{},[35,59747,37],{},[39,59749,1245],{},[1209,59751,1263],{},[39,59753,126],{"stretchy":119},[32,59755,59756,59758,59760,59762,59764,59766,59768,59770,59772,59774,59776,59778],{},[39,59757,120],{"stretchy":119},[1209,59759,1263],{},[35,59761,37],{},[39,59763,126],{"stretchy":119},[39,59765,40769],{"stretchy":119},[39,59767,120],{"stretchy":119},[1209,59769,1263],{},[35,59771,37],{},[39,59773,1245],{},[1209,59775,1263],{},[39,59777,126],{"stretchy":119},[39,59779,40769],{"stretchy":119},[39,59781,129],{},[6467,59783,59784,59786],{},[35,59785,33494],{},[1209,59787,1263],{},[45,59789,59790],{"encoding":47},"\\lim_{n \\rightarrow \\infty} \\frac{16^n (n!)^4 (2n+2)}{(2n)!(2n+2)!} = \\frac{\\pi}{2}",[15,59792,59794,60052],{"className":59793,"ariaHidden":53},[52],[15,59795,59797,59800,59853,59856,60043,60046,60049],{"className":59796},[57],[15,59798],{"className":59799,"style":45944},[61],[15,59801,59803],{"className":59802},[1339,1351],[15,59804,59806,59845],{"className":59805},[202,1090],[15,59807,59809,59842],{"className":59808},[206],[15,59810,59812,59832],{"className":59811,"style":840},[210],[15,59813,59814,59817],{"style":48823},[15,59815],{"className":59816,"style":1368},[217],[15,59818,59820],{"className":59819},[222,223,224,225],[15,59821,59823,59826,59829],{"className":59822},[66,225],[15,59824,37],{"className":59825},[66,67,225],[15,59827,449],{"className":59828},[177,225],[15,59830,7620],{"className":59831},[66,225],[15,59833,59834,59837],{"style":1386},[15,59835],{"className":59836,"style":1368},[217],[15,59838,59839],{},[15,59840,43668],{"className":59841},[1339],[15,59843,1114],{"className":59844},[1113],[15,59846,59848],{"className":59847},[206],[15,59849,59851],{"className":59850,"style":48861},[210],[15,59852],{},[15,59854],{"className":59855,"style":697},[71],[15,59857,59859,59862,60040],{"className":59858},[66],[15,59860],{"className":59861},[162,23417],[15,59863,59865],{"className":59864},[6467],[15,59866,59868,60032],{"className":59867},[202,1090],[15,59869,59871,60029],{"className":59870},[206],[15,59872,59874,59918,59926],{"className":59873,"style":45963},[210],[15,59875,59876,59879],{"style":23433},[15,59877],{"className":59878,"style":1368},[217],[15,59880,59882,59885,59888,59891,59894,59897,59900,59903,59906,59909,59912,59915],{"className":59881},[66],[15,59883,120],{"className":59884},[162],[15,59886,1263],{"className":59887},[66],[15,59889,37],{"className":59890},[66,67],[15,59892,46048],{"className":59893},[169],[15,59895,120],{"className":59896},[162],[15,59898,1263],{"className":59899},[66],[15,59901,37],{"className":59902},[66,67],[15,59904],{"className":59905,"style":72},[71],[15,59907,1245],{"className":59908},[76],[15,59910],{"className":59911,"style":72},[71],[15,59913,1263],{"className":59914},[66],[15,59916,46048],{"className":59917},[169],[15,59919,59920,59923],{"style":23445},[15,59921],{"className":59922,"style":1368},[217],[15,59924],{"className":59925,"style":4440},[23452],[15,59927,59928,59931],{"style":23455},[15,59929],{"className":59930,"style":1368},[217],[15,59932,59934,59937,59967,59970,59973,59976,60005,60008,60011,60014,60017,60020,60023,60026],{"className":59933},[66],[15,59935,1211],{"className":59936},[66],[15,59938,59940,59944],{"className":59939},[66],[15,59941,59943],{"className":59942},[66],"6",[15,59945,59947],{"className":59946},[198],[15,59948,59950],{"className":59949},[202],[15,59951,59953],{"className":59952},[206],[15,59954,59956],{"className":59955,"style":6438},[210],[15,59957,59958,59961],{"style":213},[15,59959],{"className":59960,"style":218},[217],[15,59962,59964],{"className":59963},[222,223,224,225],[15,59965,37],{"className":59966},[66,67,225],[15,59968,120],{"className":59969},[162],[15,59971,37],{"className":59972},[66,67],[15,59974,40769],{"className":59975},[169],[15,59977,59979,59982],{"className":59978},[169],[15,59980,126],{"className":59981},[169],[15,59983,59985],{"className":59984},[198],[15,59986,59988],{"className":59987},[202],[15,59989,59991],{"className":59990},[206],[15,59992,59994],{"className":59993,"style":6740},[210],[15,59995,59996,59999],{"style":213},[15,59997],{"className":59998,"style":218},[217],[15,60000,60002],{"className":60001},[222,223,224,225],[15,60003,15855],{"className":60004},[66,225],[15,60006,120],{"className":60007},[162],[15,60009,1263],{"className":60010},[66],[15,60012,37],{"className":60013},[66,67],[15,60015],{"className":60016,"style":72},[71],[15,60018,1245],{"className":60019},[76],[15,60021],{"className":60022,"style":72},[71],[15,60024,1263],{"className":60025},[66],[15,60027,126],{"className":60028},[169],[15,60030,1114],{"className":60031},[1113],[15,60033,60035],{"className":60034},[206],[15,60036,60038],{"className":60037,"style":41282},[210],[15,60039],{},[15,60041],{"className":60042},[169,23417],[15,60044],{"className":60045,"style":173},[71],[15,60047,129],{"className":60048},[177],[15,60050],{"className":60051,"style":173},[71],[15,60053,60055,60058],{"className":60054},[57],[15,60056],{"className":60057,"style":39114},[61],[15,60059,60061,60064,60117],{"className":60060},[66],[15,60062],{"className":60063},[162,23417],[15,60065,60067],{"className":60066},[6467],[15,60068,60070,60109],{"className":60069},[202,1090],[15,60071,60073,60106],{"className":60072},[206],[15,60074,60076,60087,60095],{"className":60075,"style":39133},[210],[15,60077,60078,60081],{"style":23433},[15,60079],{"className":60080,"style":1368},[217],[15,60082,60084],{"className":60083},[66],[15,60085,1263],{"className":60086},[66],[15,60088,60089,60092],{"style":23445},[15,60090],{"className":60091,"style":1368},[217],[15,60093],{"className":60094,"style":4440},[23452],[15,60096,60097,60100],{"style":23455},[15,60098],{"className":60099,"style":1368},[217],[15,60101,60103],{"className":60102},[66],[15,60104,33494],{"className":60105,"style":1515},[66,67],[15,60107,1114],{"className":60108},[1113],[15,60110,60112],{"className":60111},[206],[15,60113,60115],{"className":60114,"style":23474},[210],[15,60116],{},[15,60118],{"className":60119},[169,23417],[11,60121,60122],{},"利用：",[15,60124,60126],{"className":60125,"translate":19},[294],[15,60127,60129,60175],{"className":60128,"translate":19},[18],[15,60130,60132],{"className":60131},[23],[25,60133,60134],{"xmlns":27,"display":303},[29,60135,60136,60172],{},[32,60137,60138,60140,60142,60144,60146,60156],{},[35,60139,37],{},[39,60141,40769],{"stretchy":119},[39,60143,28391],{},[35,60145,4556],{},[131,60147,60148,60150],{},[35,60149,48383],{},[32,60151,60152,60154],{},[39,60153,6602],{},[35,60155,37],{},[131,60157,60158,60160],{},[35,60159,37],{},[32,60161,60162,60164,60166],{},[35,60163,37],{},[39,60165,1245],{},[6467,60167,60168,60170],{},[1209,60169,1211],{},[1209,60171,1263],{},[45,60173,60174],{"encoding":47},"n! \\sim a e^{-n} n^{n + \\frac{1}{2}}",[15,60176,60178,60199],{"className":60177,"ariaHidden":53},[52],[15,60179,60181,60184,60187,60190,60193,60196],{"className":60180},[57],[15,60182],{"className":60183,"style":840},[61],[15,60185,37],{"className":60186},[66,67],[15,60188,40769],{"className":60189},[169],[15,60191],{"className":60192,"style":173},[71],[15,60194,28391],{"className":60195},[177],[15,60197],{"className":60198,"style":173},[71],[15,60200,60202,60206,60209,60245],{"className":60201},[57],[15,60203],{"className":60204,"style":60205},[61],"height:1.004em;",[15,60207,4556],{"className":60208},[66,67],[15,60210,60212,60215],{"className":60211},[66],[15,60213,48383],{"className":60214},[66,67],[15,60216,60218],{"className":60217},[198],[15,60219,60221],{"className":60220},[202],[15,60222,60224],{"className":60223},[206],[15,60225,60228],{"className":60226,"style":60227},[210],"height:0.8213em;",[15,60229,60230,60233],{"style":397},[15,60231],{"className":60232,"style":218},[217],[15,60234,60236],{"className":60235},[222,223,224,225],[15,60237,60239,60242],{"className":60238},[66,225],[15,60240,6602],{"className":60241},[66,225],[15,60243,37],{"className":60244},[66,67,225],[15,60246,60248,60251],{"className":60247},[66],[15,60249,37],{"className":60250},[66,67],[15,60252,60254],{"className":60253},[198],[15,60255,60257],{"className":60256},[202],[15,60258,60260],{"className":60259},[206],[15,60261,60263],{"className":60262,"style":60205},[210],[15,60264,60265,60268],{"style":49627},[15,60266],{"className":60267,"style":1368},[217],[15,60269,60271],{"className":60270},[222,223,224,225],[15,60272,60274,60277,60280],{"className":60273},[66,225],[15,60275,37],{"className":60276},[66,67,225],[15,60278,1245],{"className":60279},[76,225],[15,60281,60283,60286,60345],{"className":60282},[66,225],[15,60284],{"className":60285},[162,23417,222,3679,33581],[15,60287,60289],{"className":60288},[6467],[15,60290,60292,60337],{"className":60291},[202,1090],[15,60293,60295,60334],{"className":60294},[206],[15,60296,60298,60312,60320],{"className":60297,"style":49020},[210],[15,60299,60300,60303],{"style":33597},[15,60301],{"className":60302,"style":1368},[217],[15,60304,60306],{"className":60305},[222,3679,1508,225],[15,60307,60309],{"className":60308},[66,225],[15,60310,1263],{"className":60311},[66,225],[15,60313,60314,60317],{"style":33612},[15,60315],{"className":60316,"style":1368},[217],[15,60318],{"className":60319,"style":33619},[23452,225],[15,60321,60322,60325],{"style":33622},[15,60323],{"className":60324,"style":1368},[217],[15,60326,60328],{"className":60327},[222,3679,1508,225],[15,60329,60331],{"className":60330},[66,225],[15,60332,1211],{"className":60333},[66,225],[15,60335,1114],{"className":60336},[1113],[15,60338,60340],{"className":60339},[206],[15,60341,60343],{"className":60342,"style":33644},[210],[15,60344],{},[15,60346],{"className":60347},[169,23417,222,3679,33581],[11,60349,60350],{},"代入：",[15,60352,60354],{"className":60353,"translate":19},[294],[15,60355,60357,60447],{"className":60356,"translate":19},[18],[15,60358,60360],{"className":60359},[23],[25,60361,60362],{"xmlns":27,"display":303},[29,60363,60364,60444],{},[32,60365,60366,60382,60436,60438],{},[1172,60367,60368,60374],{},[32,60369,60370,60372],{},[35,60371,43668],{},[39,60373,1170],{},[32,60375,60376,60378,60380],{},[35,60377,37],{},[39,60379,449],{},[35,60381,7620],{"mathvariant":138},[6467,60383,60384,60398],{},[32,60385,60386,60392],{},[131,60387,60388,60390],{},[35,60389,4556],{},[1209,60391,1263],{},[131,60393,60394,60396],{},[35,60395,48383],{},[1209,60397,1263],{},[32,60399,60400,60402],{},[1209,60401,15855],{},[131,60403,60404,60422],{},[32,60405,60406,60408,60420],{},[39,60407,120],{"fence":53},[6467,60409,60410,60418],{},[32,60411,60412,60414,60416],{},[35,60413,37],{},[39,60415,1245],{},[1209,60417,1211],{},[35,60419,37],{},[39,60421,126],{"fence":53},[32,60423,60424,60426,60428,60430],{},[1209,60425,1263],{},[35,60427,37],{},[39,60429,1245],{},[6467,60431,60432,60434],{},[1209,60433,8678],{},[1209,60435,1263],{},[39,60437,129],{},[6467,60439,60440,60442],{},[35,60441,33494],{},[1209,60443,1263],{},[45,60445,60446],{"encoding":47},"\\lim_{n \\rightarrow \\infty} \\frac{a^2 e^2}{4 \\left(\\frac{n+1}{n}\\right)^{2n + \\frac{3}{2}}} = \\frac{\\pi}{2}",[15,60448,60450,60834],{"className":60449,"ariaHidden":53},[52],[15,60451,60453,60457,60510,60513,60825,60828,60831],{"className":60452},[57],[15,60454],{"className":60455,"style":60456},[61],"height:2.925em;vertical-align:-1.4339em;",[15,60458,60460],{"className":60459},[1339,1351],[15,60461,60463,60502],{"className":60462},[202,1090],[15,60464,60466,60499],{"className":60465},[206],[15,60467,60469,60489],{"className":60468,"style":840},[210],[15,60470,60471,60474],{"style":48823},[15,60472],{"className":60473,"style":1368},[217],[15,60475,60477],{"className":60476},[222,223,224,225],[15,60478,60480,60483,60486],{"className":60479},[66,225],[15,60481,37],{"className":60482},[66,67,225],[15,60484,449],{"className":60485},[177,225],[15,60487,7620],{"className":60488},[66,225],[15,60490,60491,60494],{"style":1386},[15,60492],{"className":60493,"style":1368},[217],[15,60495,60496],{},[15,60497,43668],{"className":60498},[1339],[15,60500,1114],{"className":60501},[1113],[15,60503,60505],{"className":60504},[206],[15,60506,60508],{"className":60507,"style":48861},[210],[15,60509],{},[15,60511],{"className":60512,"style":697},[71],[15,60514,60516,60519,60822],{"className":60515},[66],[15,60517],{"className":60518},[162,23417],[15,60520,60522],{"className":60521},[6467],[15,60523,60525,60814],{"className":60524},[202,1090],[15,60526,60528,60811],{"className":60527},[206],[15,60529,60531,60737,60745],{"className":60530,"style":45963},[210],[15,60532,60533,60536],{"style":54401},[15,60534],{"className":60535,"style":54405},[217],[15,60537,60539,60542,60545],{"className":60538},[66],[15,60540,15855],{"className":60541},[66],[15,60543],{"className":60544,"style":697},[71],[15,60546,60548,60637],{"className":60547},[1413],[15,60549,60551,60557,60631],{"className":60550},[1413],[15,60552,60554],{"className":60553,"style":1418},[162,1417],[15,60555,120],{"className":60556},[1422,1508],[15,60558,60560,60563,60628],{"className":60559},[66],[15,60561],{"className":60562},[162,23417],[15,60564,60566],{"className":60565},[6467],[15,60567,60569,60620],{"className":60568},[202,1090],[15,60570,60572,60617],{"className":60571},[206],[15,60573,60575,60589,60597],{"className":60574,"style":30873},[210],[15,60576,60577,60580],{"style":30876},[15,60578],{"className":60579,"style":1368},[217],[15,60581,60583],{"className":60582},[222,223,224,225],[15,60584,60586],{"className":60585},[66,225],[15,60587,37],{"className":60588},[66,67,225],[15,60590,60591,60594],{"style":23445},[15,60592],{"className":60593,"style":1368},[217],[15,60595],{"className":60596,"style":4440},[23452],[15,60598,60599,60602],{"style":30899},[15,60600],{"className":60601,"style":1368},[217],[15,60603,60605],{"className":60604},[222,223,224,225],[15,60606,60608,60611,60614],{"className":60607},[66,225],[15,60609,37],{"className":60610},[66,67,225],[15,60612,1245],{"className":60613},[76,225],[15,60615,1211],{"className":60616},[66,225],[15,60618,1114],{"className":60619},[1113],[15,60621,60623],{"className":60622},[206],[15,60624,60626],{"className":60625,"style":30921},[210],[15,60627],{},[15,60629],{"className":60630},[169,23417],[15,60632,60634],{"className":60633,"style":1418},[169,1417],[15,60635,126],{"className":60636},[1422,1508],[15,60638,60640],{"className":60639},[198],[15,60641,60643],{"className":60642},[202],[15,60644,60646],{"className":60645},[206],[15,60647,60649],{"className":60648,"style":54519},[210],[15,60650,60651,60654],{"style":51217},[15,60652],{"className":60653,"style":1368},[217],[15,60655,60657],{"className":60656},[222,223,224,225],[15,60658,60660,60663,60666,60669],{"className":60659},[66,225],[15,60661,1263],{"className":60662},[66,225],[15,60664,37],{"className":60665},[66,67,225],[15,60667,1245],{"className":60668},[76,225],[15,60670,60672,60675,60734],{"className":60671},[66,225],[15,60673],{"className":60674},[162,23417,222,3679,33581],[15,60676,60678],{"className":60677},[6467],[15,60679,60681,60726],{"className":60680},[202,1090],[15,60682,60684,60723],{"className":60683},[206],[15,60685,60687,60701,60709],{"className":60686,"style":49020},[210],[15,60688,60689,60692],{"style":33597},[15,60690],{"className":60691,"style":1368},[217],[15,60693,60695],{"className":60694},[222,3679,1508,225],[15,60696,60698],{"className":60697},[66,225],[15,60699,1263],{"className":60700},[66,225],[15,60702,60703,60706],{"style":33612},[15,60704],{"className":60705,"style":1368},[217],[15,60707],{"className":60708,"style":33619},[23452,225],[15,60710,60711,60714],{"style":33622},[15,60712],{"className":60713,"style":1368},[217],[15,60715,60717],{"className":60716},[222,3679,1508,225],[15,60718,60720],{"className":60719},[66,225],[15,60721,8678],{"className":60722},[66,225],[15,60724,1114],{"className":60725},[1113],[15,60727,60729],{"className":60728},[206],[15,60730,60732],{"className":60731,"style":33644},[210],[15,60733],{},[15,60735],{"className":60736},[169,23417,222,3679,33581],[15,60738,60739,60742],{"style":54607},[15,60740],{"className":60741,"style":54405},[217],[15,60743],{"className":60744,"style":4440},[23452],[15,60746,60747,60750],{"style":54616},[15,60748],{"className":60749,"style":54405},[217],[15,60751,60753,60782],{"className":60752},[66],[15,60754,60756,60759],{"className":60755},[66],[15,60757,4556],{"className":60758},[66,67],[15,60760,60762],{"className":60761},[198],[15,60763,60765],{"className":60764},[202],[15,60766,60768],{"className":60767},[206],[15,60769,60771],{"className":60770,"style":6740},[210],[15,60772,60773,60776],{"style":213},[15,60774],{"className":60775,"style":218},[217],[15,60777,60779],{"className":60778},[222,223,224,225],[15,60780,1263],{"className":60781},[66,225],[15,60783,60785,60788],{"className":60784},[66],[15,60786,48383],{"className":60787},[66,67],[15,60789,60791],{"className":60790},[198],[15,60792,60794],{"className":60793},[202],[15,60795,60797],{"className":60796},[206],[15,60798,60800],{"className":60799,"style":6740},[210],[15,60801,60802,60805],{"style":213},[15,60803],{"className":60804,"style":218},[217],[15,60806,60808],{"className":60807},[222,223,224,225],[15,60809,1263],{"className":60810},[66,225],[15,60812,1114],{"className":60813},[1113],[15,60815,60817],{"className":60816},[206],[15,60818,60820],{"className":60819,"style":54635},[210],[15,60821],{},[15,60823],{"className":60824},[169,23417],[15,60826],{"className":60827,"style":173},[71],[15,60829,129],{"className":60830},[177],[15,60832],{"className":60833,"style":173},[71],[15,60835,60837,60840],{"className":60836},[57],[15,60838],{"className":60839,"style":39114},[61],[15,60841,60843,60846,60899],{"className":60842},[66],[15,60844],{"className":60845},[162,23417],[15,60847,60849],{"className":60848},[6467],[15,60850,60852,60891],{"className":60851},[202,1090],[15,60853,60855,60888],{"className":60854},[206],[15,60856,60858,60869,60877],{"className":60857,"style":39133},[210],[15,60859,60860,60863],{"style":23433},[15,60861],{"className":60862,"style":1368},[217],[15,60864,60866],{"className":60865},[66],[15,60867,1263],{"className":60868},[66],[15,60870,60871,60874],{"style":23445},[15,60872],{"className":60873,"style":1368},[217],[15,60875],{"className":60876,"style":4440},[23452],[15,60878,60879,60882],{"style":23455},[15,60880],{"className":60881,"style":1368},[217],[15,60883,60885],{"className":60884},[66],[15,60886,33494],{"className":60887,"style":1515},[66,67],[15,60889,1114],{"className":60890},[1113],[15,60892,60894],{"className":60893},[206],[15,60895,60897],{"className":60896,"style":23474},[210],[15,60898],{},[15,60900],{"className":60901},[169,23417],[11,60903,60904],{},"解得：",[15,60906,60908],{"className":60907,"translate":19},[294],[15,60909,60911,60935],{"className":60910,"translate":19},[18],[15,60912,60914],{"className":60913},[23],[25,60915,60916],{"xmlns":27,"display":303},[29,60917,60918,60932],{},[32,60919,60920,60922,60924],{},[35,60921,4556],{},[39,60923,129],{},[48508,60925,60926],{},[32,60927,60928,60930],{},[1209,60929,1263],{},[35,60931,33494],{},[45,60933,60934],{"encoding":47},"a = \\sqrt{2 \\pi}",[15,60936,60938,60956],{"className":60937,"ariaHidden":53},[52],[15,60939,60941,60944,60947,60950,60953],{"className":60940},[57],[15,60942],{"className":60943,"style":86},[61],[15,60945,4556],{"className":60946},[66,67],[15,60948],{"className":60949,"style":173},[71],[15,60951,129],{"className":60952},[177],[15,60954],{"className":60955,"style":173},[71],[15,60957,60959,60963],{"className":60958},[57],[15,60960],{"className":60961,"style":60962},[61],"height:1.04em;vertical-align:-0.0839em;",[15,60964,60966],{"className":60965},[66,48550],[15,60967,60969,61007],{"className":60968},[202,1090],[15,60970,60972,61004],{"className":60971},[206],[15,60973,60976,60991],{"className":60974,"style":60975},[210],"height:0.9561em;",[15,60977,60979,60982],{"className":60978,"style":1386},[48564],[15,60980],{"className":60981,"style":1368},[217],[15,60983,60985,60988],{"className":60984,"style":48571},[66],[15,60986,1263],{"className":60987},[66],[15,60989,33494],{"className":60990,"style":1515},[66,67],[15,60992,60994,60997],{"style":60993},"top:-2.9161em;",[15,60995],{"className":60996,"style":1368},[217],[15,60998,61000],{"className":60999,"style":48591},[48590],[34828,61001,61002],{"xmlns":34830,"width":48594,"height":48595,"viewBox":48596,"preserveAspectRatio":48597},[34834,61003],{"d":48600},[15,61005,1114],{"className":61006},[1113],[15,61008,61010],{"className":61009},[206],[15,61011,61014],{"className":61012,"style":61013},[210],"height:0.0839em;",[15,61015],{},[11,61017,61018,61019,61047],{},"因此，当 ",[15,61020,61022,61035],{"className":61021,"translate":19},[18],[15,61023,61025],{"className":61024},[23],[25,61026,61027],{"xmlns":27},[29,61028,61029,61033],{},[32,61030,61031],{},[35,61032,37],{},[45,61034,37],{"encoding":47},[15,61036,61038],{"className":61037,"ariaHidden":53},[52],[15,61039,61041,61044],{"className":61040},[57],[15,61042],{"className":61043,"style":86},[61],[15,61045,37],{"className":61046},[66,67]," 趋近无穷大时：",[15,61049,61051],{"className":61050,"translate":19},[294],[15,61052,61054,61106],{"className":61053,"translate":19},[18],[15,61055,61057],{"className":61056},[23],[25,61058,61059],{"xmlns":27,"display":303},[29,61060,61061,61103],{},[32,61062,61063,61065,61067,61069,61077,61087],{},[35,61064,37],{},[39,61066,40769],{"stretchy":119},[39,61068,28391],{},[48508,61070,61071],{},[32,61072,61073,61075],{},[1209,61074,1263],{},[35,61076,33494],{},[131,61078,61079,61081],{},[35,61080,48383],{},[32,61082,61083,61085],{},[39,61084,6602],{},[35,61086,37],{},[131,61088,61089,61091],{},[35,61090,37],{},[32,61092,61093,61095,61097],{},[35,61094,37],{},[39,61096,1245],{},[6467,61098,61099,61101],{},[1209,61100,1211],{},[1209,61102,1263],{},[45,61104,61105],{"encoding":47},"n! \\sim \\sqrt{2 \\pi} e^{-n} n^{n + \\frac{1}{2}}",[15,61107,61109,61130],{"className":61108,"ariaHidden":53},[52],[15,61110,61112,61115,61118,61121,61124,61127],{"className":61111},[57],[15,61113],{"className":61114,"style":840},[61],[15,61116,37],{"className":61117},[66,67],[15,61119,40769],{"className":61120},[169],[15,61122],{"className":61123,"style":173},[71],[15,61125,28391],{"className":61126},[177],[15,61128],{"className":61129,"style":173},[71],[15,61131,61133,61137,61187,61222],{"className":61132},[57],[15,61134],{"className":61135,"style":61136},[61],"height:1.0879em;vertical-align:-0.0839em;",[15,61138,61140],{"className":61139},[66,48550],[15,61141,61143,61179],{"className":61142},[202,1090],[15,61144,61146,61176],{"className":61145},[206],[15,61147,61149,61164],{"className":61148,"style":60975},[210],[15,61150,61152,61155],{"className":61151,"style":1386},[48564],[15,61153],{"className":61154,"style":1368},[217],[15,61156,61158,61161],{"className":61157,"style":48571},[66],[15,61159,1263],{"className":61160},[66],[15,61162,33494],{"className":61163,"style":1515},[66,67],[15,61165,61166,61169],{"style":60993},[15,61167],{"className":61168,"style":1368},[217],[15,61170,61172],{"className":61171,"style":48591},[48590],[34828,61173,61174],{"xmlns":34830,"width":48594,"height":48595,"viewBox":48596,"preserveAspectRatio":48597},[34834,61175],{"d":48600},[15,61177,1114],{"className":61178},[1113],[15,61180,61182],{"className":61181},[206],[15,61183,61185],{"className":61184,"style":61013},[210],[15,61186],{},[15,61188,61190,61193],{"className":61189},[66],[15,61191,48383],{"className":61192},[66,67],[15,61194,61196],{"className":61195},[198],[15,61197,61199],{"className":61198},[202],[15,61200,61202],{"className":61201},[206],[15,61203,61205],{"className":61204,"style":60227},[210],[15,61206,61207,61210],{"style":397},[15,61208],{"className":61209,"style":218},[217],[15,61211,61213],{"className":61212},[222,223,224,225],[15,61214,61216,61219],{"className":61215},[66,225],[15,61217,6602],{"className":61218},[66,225],[15,61220,37],{"className":61221},[66,67,225],[15,61223,61225,61228],{"className":61224},[66],[15,61226,37],{"className":61227},[66,67],[15,61229,61231],{"className":61230},[198],[15,61232,61234],{"className":61233},[202],[15,61235,61237],{"className":61236},[206],[15,61238,61240],{"className":61239,"style":60205},[210],[15,61241,61242,61245],{"style":49627},[15,61243],{"className":61244,"style":1368},[217],[15,61246,61248],{"className":61247},[222,223,224,225],[15,61249,61251,61254,61257],{"className":61250},[66,225],[15,61252,37],{"className":61253},[66,67,225],[15,61255,1245],{"className":61256},[76,225],[15,61258,61260,61263,61322],{"className":61259},[66,225],[15,61261],{"className":61262},[162,23417,222,3679,33581],[15,61264,61266],{"className":61265},[6467],[15,61267,61269,61314],{"className":61268},[202,1090],[15,61270,61272,61311],{"className":61271},[206],[15,61273,61275,61289,61297],{"className":61274,"style":49020},[210],[15,61276,61277,61280],{"style":33597},[15,61278],{"className":61279,"style":1368},[217],[15,61281,61283],{"className":61282},[222,3679,1508,225],[15,61284,61286],{"className":61285},[66,225],[15,61287,1263],{"className":61288},[66,225],[15,61290,61291,61294],{"style":33612},[15,61292],{"className":61293,"style":1368},[217],[15,61295],{"className":61296,"style":33619},[23452,225],[15,61298,61299,61302],{"style":33622},[15,61300],{"className":61301,"style":1368},[217],[15,61303,61305],{"className":61304},[222,3679,1508,225],[15,61306,61308],{"className":61307},[66,225],[15,61309,1211],{"className":61310},[66,225],[15,61312,1114],{"className":61313},[1113],[15,61315,61317],{"className":61316},[206],[15,61318,61320],{"className":61319,"style":33644},[210],[15,61321],{},[15,61323],{"className":61324},[169,23417,222,3679,33581],[11,61326,61327],{},"或者等价地：",[15,61329,61331],{"className":61330,"translate":19},[294],[15,61332,61334,61378],{"className":61333,"translate":19},[18],[15,61335,61337],{"className":61336},[23],[25,61338,61339],{"xmlns":27,"display":303},[29,61340,61341,61375],{},[32,61342,61343,61345,61347,61349,61359],{},[35,61344,37],{},[39,61346,40769],{"stretchy":119},[39,61348,28391],{},[48508,61350,61351],{},[32,61352,61353,61355,61357],{},[1209,61354,1263],{},[35,61356,33494],{},[35,61358,37],{},[131,61360,61361,61373],{},[32,61362,61363,61365,61371],{},[39,61364,120],{"fence":53},[6467,61366,61367,61369],{},[35,61368,37],{},[35,61370,48383],{},[39,61372,126],{"fence":53},[35,61374,37],{},[45,61376,61377],{"encoding":47},"n! \\sim \\sqrt{2 \\pi n} \\left(\\frac{n}{e}\\right)^n",[15,61379,61381,61402],{"className":61380,"ariaHidden":53},[52],[15,61382,61384,61387,61390,61393,61396,61399],{"className":61383},[57],[15,61385],{"className":61386,"style":840},[61],[15,61388,37],{"className":61389},[66,67],[15,61391,40769],{"className":61392},[169],[15,61394],{"className":61395,"style":173},[71],[15,61397,28391],{"className":61398},[177],[15,61400],{"className":61401,"style":173},[71],[15,61403,61405,61409,61462,61465],{"className":61404},[57],[15,61406],{"className":61407,"style":61408},[61],"height:1.8903em;vertical-align:-0.686em;",[15,61410,61412],{"className":61411},[66,48550],[15,61413,61415,61454],{"className":61414},[202,1090],[15,61416,61418,61451],{"className":61417},[206],[15,61419,61421,61439],{"className":61420,"style":60975},[210],[15,61422,61424,61427],{"className":61423,"style":1386},[48564],[15,61425],{"className":61426,"style":1368},[217],[15,61428,61430,61433,61436],{"className":61429,"style":48571},[66],[15,61431,1263],{"className":61432},[66],[15,61434,33494],{"className":61435,"style":1515},[66,67],[15,61437,37],{"className":61438},[66,67],[15,61440,61441,61444],{"style":60993},[15,61442],{"className":61443,"style":1368},[217],[15,61445,61447],{"className":61446,"style":48591},[48590],[34828,61448,61449],{"xmlns":34830,"width":48594,"height":48595,"viewBox":48596,"preserveAspectRatio":48597},[34834,61450],{"d":48600},[15,61452,1114],{"className":61453},[1113],[15,61455,61457],{"className":61456},[206],[15,61458,61460],{"className":61459,"style":61013},[210],[15,61461],{},[15,61463],{"className":61464,"style":697},[71],[15,61466,61468,61545],{"className":61467},[1413],[15,61469,61471,61477,61539],{"className":61470},[1413],[15,61472,61474],{"className":61473,"style":1418},[162,1417],[15,61475,120],{"className":61476},[1422,4677],[15,61478,61480,61483,61536],{"className":61479},[66],[15,61481],{"className":61482},[162,23417],[15,61484,61486],{"className":61485},[6467],[15,61487,61489,61528],{"className":61488},[202,1090],[15,61490,61492,61525],{"className":61491},[206],[15,61493,61495,61506,61514],{"className":61494,"style":39133},[210],[15,61496,61497,61500],{"style":23433},[15,61498],{"className":61499,"style":1368},[217],[15,61501,61503],{"className":61502},[66],[15,61504,48383],{"className":61505},[66,67],[15,61507,61508,61511],{"style":23445},[15,61509],{"className":61510,"style":1368},[217],[15,61512],{"className":61513,"style":4440},[23452],[15,61515,61516,61519],{"style":23455},[15,61517],{"className":61518,"style":1368},[217],[15,61520,61522],{"className":61521},[66],[15,61523,37],{"className":61524},[66,67],[15,61526,1114],{"className":61527},[1113],[15,61529,61531],{"className":61530},[206],[15,61532,61534],{"className":61533,"style":23474},[210],[15,61535],{},[15,61537],{"className":61538},[169,23417],[15,61540,61542],{"className":61541,"style":1418},[169,1417],[15,61543,126],{"className":61544},[1422,4677],[15,61546,61548],{"className":61547},[198],[15,61549,61551],{"className":61550},[202],[15,61552,61554],{"className":61553},[206],[15,61555,61558],{"className":61556,"style":61557},[210],"height:1.2043em;",[15,61559,61560,61563],{"style":51217},[15,61561],{"className":61562,"style":218},[217],[15,61564,61566],{"className":61565},[222,223,224,225],[15,61567,37],{"className":61568},[66,67,225],{"title":33412,"searchDepth":33413,"depth":33413,"links":61570},[61571,61572],{"id":33465,"depth":33413,"text":33465},{"id":33460,"depth":33413,"text":33460},{},"\u002Fblog\u002F2025\u002F2025-09-18-the-stirling-formula",{"title":33460,"description":33412},"blog\u002F2025\u002F2025-09-18-the-stirling-formula","斯特林公式揭示了当 n 趋向无穷大时 n!、nⁿ 与 eⁿ 之间的关系。",[61579],"mathematics","2025-09-18T00:00:00+08:00","-Zl1Q4Ie-HEQw2nACoF1zCHXCl2aL6v0T6tWoExy47I",{"id":61583,"title":61584,"body":61585,"description":61589,"draft":33445,"enableComment":33446,"extension":33447,"image":33412,"important":33445,"location":33448,"meta":121811,"navigation":33446,"ogImage":33448,"path":121812,"seo":121813,"stem":121814,"summary":121815,"tags":121816,"time":121818,"__hash__":121819},"blog\u002Fblog\u002F2025\u002F2025-09-02-bei-ye-si-jue-ce.md","贝叶斯决策理论",{"type":8,"value":61586,"toc":121753},[61587,61590,61597,61832,61839,61846,61849,61856,61860,61863,62198,62209,62345,62352,62567,62628,62933,62965,63549,63552,63727,63936,63940,63944,63947,64209,64412,64415,64647,64695,64699,64819,65194,65197,65503,65506,65514,65589,65748,65795,65827,65831,65908,66200,66261,66469,66472,66725,66873,66876,67089,67092,67689,67777,67780,68120,68152,68156,68159,68162,68169,68173,68177,68247,68693,69047,69053,69400,69635,69752,69755,70236,70330,70334,70337,70496,70499,70802,70862,71093,71226,71707,71885,71889,72034,72550,72668,72831,72923,72927,73271,73547,73579,73922,73983,73986,74022,74218,74569,74680,75020,75091,75098,75101,75429,75682,75897,75900,75904,75907,76261,76354,76357,76549,76552,76797,76800,76807,77139,77478,77692,77751,77953,78191,78548,78553,79435,79651,79654,79658,79661,79893,79897,79901,80099,80424,80427,81242,81245,81703,81882,82631,82952,82955,83327,83330,83364,83367,83695,83698,84613,84649,85047,85232,85721,85826,85829,86226,86229,86758,87105,87465,87471,87475,87478,88131,88134,88561,88785,89299,89306,89309,89612,89930,90002,90006,90229,90376,90380,90383,90689,90835,92038,92041,92045,92049,92326,92329,92792,92854,93324,93753,93851,94020,94133,94249,94595,94716,94720,95054,95563,95683,95947,95950,96623,97113,97523,97526,97529,97533,97536,98115,98118,98261,98667,98670,99105,99111,99115,99173,99176,99311,99372,99375,99379,99382,99821,100642,100645,100648,100652,100656,100717,101485,101488,101600,101905,101908,101912,101981,101984,102628,102631,102634,102638,102641,102674,102677,102680,102684,102691,103080,103168,103551,103557,103660,103664,103671,103887,103893,104720,105100,105471,105578,105909,105911,106111,106206,106212,106216,106220,106496,107143,107146,107267,107270,107274,107379,107615,107621,107888,107977,107981,108027,108967,109028,109093,109096,109100,109177,109456,109503,110055,110062,110361,110602,110831,110892,110896,110995,110998,111331,111334,111337,111341,111514,111517,111520,111524,111527,111915,111992,111995,112247,112250,112254,112352,112497,112529,112567,112570,112576,112579,112583,112587,112717,112904,112907,113347,113556,113679,114138,114198,114202,114599,114804,114806,115606,115679,116268,116548,117232,117235,117239,117345,117348,118103,118337,118340,118682,118919,119293,119558,119561,119688,119691,119695,119889,120035,120292,120295,120553,120715,120718,120721,121736,121739,121750],[11,61588,61589],{},"机器学习里，「贝叶斯」这个词至少承担了三项不同的工作，而它们经常被混为一谈。",[11,61591,61592,61593,61596],{},"第一项是",[2040,61594,61595],{},"贝叶斯公式","。它是一条关于条件概率的恒等式，本身没有争议，也不要求任何哲学立场：",[15,61598,61600],{"className":61599,"translate":19},[294],[15,61601,61603,61668],{"className":61602,"translate":19},[18],[15,61604,61606],{"className":61605},[23],[25,61607,61608],{"xmlns":27,"display":303},[29,61609,61610,61665],{},[32,61611,61612,61614,61616,61619,61621,61623,61625,61627,61663],{},[35,61613,11],{},[39,61615,120],{"stretchy":119},[35,61617,61618],{},"θ",[39,61620,34430],{},[35,61622,32458],{},[39,61624,126],{"stretchy":119},[39,61626,129],{},[6467,61628,61629,61653],{},[32,61630,61631,61633,61635,61637,61639,61641,61643,61645,61647,61649,61651],{},[35,61632,11],{},[39,61634,120],{"stretchy":119},[35,61636,32458],{},[39,61638,34430],{},[35,61640,61618],{},[39,61642,126],{"stretchy":119},[1274,61644,2462],{},[35,61646,11],{},[39,61648,120],{"stretchy":119},[35,61650,61618],{},[39,61652,126],{"stretchy":119},[32,61654,61655,61657,61659,61661],{},[35,61656,11],{},[39,61658,120],{"stretchy":119},[35,61660,32458],{},[39,61662,126],{"stretchy":119},[35,61664,3573],{"mathvariant":138},[45,61666,61667],{"encoding":47},"p(\\theta\\mid D)=\\frac{p(D\\mid\\theta)\\,p(\\theta)}{p(D)}.",[15,61669,61671,61695,61716],{"className":61670,"ariaHidden":53},[52],[15,61672,61674,61677,61680,61683,61686,61689,61692],{"className":61673},[57],[15,61675],{"className":61676,"style":154},[61],[15,61678,11],{"className":61679},[66,67],[15,61681,120],{"className":61682},[162],[15,61684,61618],{"className":61685,"style":23025},[66,67],[15,61687],{"className":61688,"style":173},[71],[15,61690,34430],{"className":61691},[177],[15,61693],{"className":61694,"style":173},[71],[15,61696,61698,61701,61704,61707,61710,61713],{"className":61697},[57],[15,61699],{"className":61700,"style":154},[61],[15,61702,32458],{"className":61703,"style":23025},[66,67],[15,61705,126],{"className":61706},[169],[15,61708],{"className":61709,"style":173},[71],[15,61711,129],{"className":61712},[177],[15,61714],{"className":61715,"style":173},[71],[15,61717,61719,61722,61829],{"className":61718},[57],[15,61720],{"className":61721,"style":41192},[61],[15,61723,61725,61728,61826],{"className":61724},[66],[15,61726],{"className":61727},[162,23417],[15,61729,61731],{"className":61730},[6467],[15,61732,61734,61818],{"className":61733},[202,1090],[15,61735,61737,61815],{"className":61736},[206],[15,61738,61740,61760,61768],{"className":61739,"style":41211},[210],[15,61741,61742,61745],{"style":23433},[15,61743],{"className":61744,"style":1368},[217],[15,61746,61748,61751,61754,61757],{"className":61747},[66],[15,61749,11],{"className":61750},[66,67],[15,61752,120],{"className":61753},[162],[15,61755,32458],{"className":61756,"style":23025},[66,67],[15,61758,126],{"className":61759},[169],[15,61761,61762,61765],{"style":23445},[15,61763],{"className":61764,"style":1368},[217],[15,61766],{"className":61767,"style":4440},[23452],[15,61769,61770,61773],{"style":23455},[15,61771],{"className":61772,"style":1368},[217],[15,61774,61776,61779,61782,61785,61788,61791,61794,61797,61800,61803,61806,61809,61812],{"className":61775},[66],[15,61777,11],{"className":61778},[66,67],[15,61780,120],{"className":61781},[162],[15,61783,32458],{"className":61784,"style":23025},[66,67],[15,61786],{"className":61787,"style":173},[71],[15,61789,34430],{"className":61790},[177],[15,61792],{"className":61793,"style":173},[71],[15,61795,61618],{"className":61796,"style":23025},[66,67],[15,61798,126],{"className":61799},[169],[15,61801],{"className":61802,"style":697},[71],[15,61804,11],{"className":61805},[66,67],[15,61807,120],{"className":61808},[162],[15,61810,61618],{"className":61811,"style":23025},[66,67],[15,61813,126],{"className":61814},[169],[15,61816,1114],{"className":61817},[1113],[15,61819,61821],{"className":61820},[206],[15,61822,61824],{"className":61823,"style":41282},[210],[15,61825],{},[15,61827],{"className":61828},[169,23417],[15,61830,3573],{"className":61831},[66],[11,61833,61834,61835,61838],{},"第二项是",[2040,61836,61837],{},"贝叶斯决策","。它问的不是「参数是多少」，而是「在还不确定的情况下，应该采取什么行动」。给定后验分布与一个损失函数，最优行动就是使后验期望损失最小的那个。",[11,61840,61841,61842,61845],{},"第三项是",[2040,61843,61844],{},"贝叶斯学习","。它把模型参数也看成随机变量，用后验分布而不是点估计来描述知识状态；预测时对参数做积分，而不是代入一个「最佳值」。",[11,61847,61848],{},"三者可以独立使用，也可以串成一条链：先验与似然给出后验，后验与损失给出行动，行动的结果又成为新的数据。本文沿着这条链走一遍：先补齐概率论的语言，再建立决策论的骨架，最后看这套骨架如何解释机器学习里的分类器、正则化、错误率、泛化界与不确定性。",[11,61850,61851,61852,61855],{},"这条链的终点是一个看起来过于理想化的量，即",[2040,61853,61854],{},"贝叶斯风险","。它假设真实分布已知，因而在实践中永远算不出来。但正是这个「算不出来的量」充当了学习理论的坐标原点：它告诉我们，在给定问题里错误率的下限在哪里，模型的哪些部分在逼近这个下限，哪些部分只是在拟合噪声。",[95,61857,61859],{"id":61858},"_1-一个决策问题的解剖","1. 一个决策问题的解剖",[11,61861,61862],{},"把所有监督学习问题写成同一个形式，只需要四样东西：",[16136,61864,61865,61925,61982,62040],{},[3194,61866,61867,61870,61871,61924],{},[2040,61868,61869],{},"状态"," ",[15,61872,61874,61893],{"className":61873,"translate":19},[18],[15,61875,61877],{"className":61876},[23],[25,61878,61879],{"xmlns":27},[29,61880,61881,61890],{},[32,61882,61883,61886,61888],{},[35,61884,61885],{},"Y",[39,61887,1187],{},[35,61889,61885],{"mathvariant":445},[45,61891,61892],{"encoding":47},"Y\\in\\mathcal Y",[15,61894,61896,61914],{"className":61895,"ariaHidden":53},[52],[15,61897,61899,61902,61905,61908,61911],{"className":61898},[57],[15,61900],{"className":61901,"style":14340},[61],[15,61903,61885],{"className":61904,"style":72},[66,67],[15,61906],{"className":61907,"style":173},[71],[15,61909,1187],{"className":61910},[177],[15,61912],{"className":61913,"style":173},[71],[15,61915,61917,61921],{"className":61916},[57],[15,61918],{"className":61919,"style":61920},[61],"height:0.7805em;vertical-align:-0.0972em;",[15,61922,61885],{"className":61923,"style":28452},[66,494],"：真实类别、待预测的实数值，或环境里我们关心但看不见的量；",[3194,61926,61927,61870,61930,61981],{},[2040,61928,61929],{},"观测",[15,61931,61933,61951],{"className":61932,"translate":19},[18],[15,61934,61936],{"className":61935},[23],[25,61937,61938],{"xmlns":27},[29,61939,61940,61948],{},[32,61941,61942,61944,61946],{},[35,61943,446],{},[39,61945,1187],{},[35,61947,446],{"mathvariant":445},[45,61949,61950],{"encoding":47},"X\\in\\mathcal X",[15,61952,61954,61972],{"className":61953,"ariaHidden":53},[52],[15,61955,61957,61960,61963,61966,61969],{"className":61956},[57],[15,61958],{"className":61959,"style":14340},[61],[15,61961,446],{"className":61962,"style":16497},[66,67],[15,61964],{"className":61965,"style":173},[71],[15,61967,1187],{"className":61968},[177],[15,61970],{"className":61971,"style":173},[71],[15,61973,61975,61978],{"className":61974},[57],[15,61976],{"className":61977,"style":490},[61],[15,61979,446],{"className":61980,"style":495},[66,494],"：我们拿到的特征；",[3194,61983,61984,61870,61987,62039],{},[2040,61985,61986],{},"行动",[15,61988,61990,62009],{"className":61989,"translate":19},[18],[15,61991,61993],{"className":61992},[23],[25,61994,61995],{"xmlns":27},[29,61996,61997,62006],{},[32,61998,61999,62001,62003],{},[35,62000,4556],{},[39,62002,1187],{},[35,62004,62005],{"mathvariant":445},"A",[45,62007,62008],{"encoding":47},"a\\in\\mathcal A",[15,62010,62012,62030],{"className":62011,"ariaHidden":53},[52],[15,62013,62015,62018,62021,62024,62027],{"className":62014},[57],[15,62016],{"className":62017,"style":3227},[61],[15,62019,4556],{"className":62020},[66,67],[15,62022],{"className":62023,"style":173},[71],[15,62025,1187],{"className":62026},[177],[15,62028],{"className":62029,"style":173},[71],[15,62031,62033,62036],{"className":62032},[57],[15,62034],{"className":62035,"style":490},[61],[15,62037,62005],{"className":62038},[66,494],"：模型输出的预测、决策或控制量；",[3194,62041,62042,61870,62044,62139,62140,62168,62169,62197],{},[2040,62043,30593],{},[15,62045,62047,62073],{"className":62046,"translate":19},[18],[15,62048,62050],{"className":62049},[23],[25,62051,62052],{"xmlns":27},[29,62053,62054,62070],{},[32,62055,62056,62058,62060,62062,62064,62066,62068],{},[35,62057,1216],{},[39,62059,442],{},[35,62061,61885],{"mathvariant":445},[39,62063,41],{},[35,62065,62005],{"mathvariant":445},[39,62067,449],{},[35,62069,455],{"mathvariant":454},[45,62071,62072],{"encoding":47},"L:\\mathcal Y\\times\\mathcal A\\to\\mathbb R",[15,62074,62076,62094,62112,62130],{"className":62075,"ariaHidden":53},[52],[15,62077,62079,62082,62085,62088,62091],{"className":62078},[57],[15,62080],{"className":62081,"style":490},[61],[15,62083,1216],{"className":62084},[66,67],[15,62086],{"className":62087,"style":173},[71],[15,62089,442],{"className":62090},[177],[15,62092],{"className":62093,"style":173},[71],[15,62095,62097,62100,62103,62106,62109],{"className":62096},[57],[15,62098],{"className":62099,"style":61920},[61],[15,62101,61885],{"className":62102,"style":28452},[66,494],[15,62104],{"className":62105,"style":72},[71],[15,62107,41],{"className":62108},[76],[15,62110],{"className":62111,"style":72},[71],[15,62113,62115,62118,62121,62124,62127],{"className":62114},[57],[15,62116],{"className":62117,"style":490},[61],[15,62119,62005],{"className":62120},[66,494],[15,62122],{"className":62123,"style":173},[71],[15,62125,449],{"className":62126},[177],[15,62128],{"className":62129,"style":173},[71],[15,62131,62133,62136],{"className":62132},[57],[15,62134],{"className":62135,"style":2034},[61],[15,62137,455],{"className":62138},[66,517],"：当真实状态为 ",[15,62141,62143,62156],{"className":62142,"translate":19},[18],[15,62144,62146],{"className":62145},[23],[25,62147,62148],{"xmlns":27},[29,62149,62150,62154],{},[32,62151,62152],{},[35,62153,1224],{},[45,62155,1224],{"encoding":47},[15,62157,62159],{"className":62158,"ariaHidden":53},[52],[15,62160,62162,62165],{"className":62161},[57],[15,62163],{"className":62164,"style":471},[61],[15,62166,1224],{"className":62167,"style":1515},[66,67],"、采取行动 ",[15,62170,62172,62185],{"className":62171,"translate":19},[18],[15,62173,62175],{"className":62174},[23],[25,62176,62177],{"xmlns":27},[29,62178,62179,62183],{},[32,62180,62181],{},[35,62182,4556],{},[45,62184,4556],{"encoding":47},[15,62186,62188],{"className":62187,"ariaHidden":53},[52],[15,62189,62191,62194],{"className":62190},[57],[15,62192],{"className":62193,"style":86},[61],[15,62195,4556],{"className":62196},[66,67]," 时付出的代价。",[11,62199,62200,62201,62204,62205,62208],{},"四者合起来称为一个",[2040,62202,62203],{},"决策问题","。一个",[2040,62206,62207],{},"决策规则","（decision rule）是一个映射",[15,62210,62212],{"className":62211,"translate":19},[294],[15,62213,62215,62255],{"className":62214,"translate":19},[18],[15,62216,62218],{"className":62217},[23],[25,62219,62220],{"xmlns":27,"display":303},[29,62221,62222,62252],{},[32,62223,62224,62226,62228,62230,62232,62234,62236,62238,62240,62242,62244,62246,62248,62250],{},[35,62225,3885],{},[39,62227,442],{},[35,62229,446],{"mathvariant":445},[39,62231,449],{},[35,62233,62005],{"mathvariant":445},[39,62235,335],{"separator":53},[71,62237],{"width":3904},[35,62239,123],{},[39,62241,15994],{},[35,62243,3885],{},[39,62245,120],{"stretchy":119},[35,62247,123],{},[39,62249,126],{"stretchy":119},[39,62251,335],{"separator":53},[45,62253,62254],{"encoding":47},"\\delta:\\mathcal X\\to\\mathcal A,\\qquad x\\mapsto \\delta(x),",[15,62256,62258,62276,62294,62324],{"className":62257,"ariaHidden":53},[52],[15,62259,62261,62264,62267,62270,62273],{"className":62260},[57],[15,62262],{"className":62263,"style":840},[61],[15,62265,3885],{"className":62266,"style":3946},[66,67],[15,62268],{"className":62269,"style":173},[71],[15,62271,442],{"className":62272},[177],[15,62274],{"className":62275,"style":173},[71],[15,62277,62279,62282,62285,62288,62291],{"className":62278},[57],[15,62280],{"className":62281,"style":490},[61],[15,62283,446],{"className":62284,"style":495},[66,494],[15,62286],{"className":62287,"style":173},[71],[15,62289,449],{"className":62290},[177],[15,62292],{"className":62293,"style":173},[71],[15,62295,62297,62300,62303,62306,62309,62312,62315,62318,62321],{"className":62296},[57],[15,62298],{"className":62299,"style":17379},[61],[15,62301,62005],{"className":62302},[66,494],[15,62304,335],{"className":62305},[422],[15,62307],{"className":62308,"style":4061},[71],[15,62310],{"className":62311,"style":697},[71],[15,62313,123],{"className":62314},[66,67],[15,62316],{"className":62317,"style":173},[71],[15,62319,15994],{"className":62320},[177],[15,62322],{"className":62323,"style":173},[71],[15,62325,62327,62330,62333,62336,62339,62342],{"className":62326},[57],[15,62328],{"className":62329,"style":154},[61],[15,62331,3885],{"className":62332,"style":3946},[66,67],[15,62334,120],{"className":62335},[162],[15,62337,123],{"className":62338},[66,67],[15,62340,126],{"className":62341},[169],[15,62343,335],{"className":62344},[422],[11,62346,62347,62348,62351],{},"它的",[2040,62349,62350],{},"风险","（risk）定义为",[15,62353,62355],{"className":62354,"translate":19},[294],[15,62356,62358,62425],{"className":62357,"translate":19},[18],[15,62359,62361],{"className":62360},[23],[25,62362,62363],{"xmlns":27,"display":303},[29,62364,62365,62422],{},[32,62366,62367,62369,62371,62373,62375,62377,62398,62400,62402,62404,62406,62408,62410,62412,62414,62416,62418,62420],{},[35,62368,455],{},[39,62370,120],{"stretchy":119},[35,62372,3885],{},[39,62374,126],{"stretchy":119},[39,62376,129],{},[1057,62378,62379,62382],{},[35,62380,62381],{"mathvariant":454},"E",[32,62383,62384,62386,62388,62390,62392,62394,62396],{},[39,62385,120],{"stretchy":119},[35,62387,446],{},[39,62389,335],{"separator":53},[35,62391,61885],{},[39,62393,126],{"stretchy":119},[39,62395,28391],{},[35,62397,28399],{},[39,62399,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,62401,1216],{},[39,62403,120],{"stretchy":119},[35,62405,61885],{},[39,62407,335],{"separator":53},[35,62409,3885],{},[39,62411,120],{"stretchy":119},[35,62413,446],{},[39,62415,126],{"stretchy":119},[39,62417,126],{"stretchy":119},[39,62419,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,62421,3573],{"mathvariant":138},[45,62423,62424],{"encoding":47},"R(\\delta)=\\mathbb E_{(X,Y)\\sim P}\\big[L(Y,\\delta(X))\\big].",[15,62426,62428,62456],{"className":62427,"ariaHidden":53},[52],[15,62429,62431,62434,62438,62441,62444,62447,62450,62453],{"className":62430},[57],[15,62432],{"className":62433,"style":154},[61],[15,62435,455],{"className":62436,"style":62437},[66,67],"margin-right:0.0077em;",[15,62439,120],{"className":62440},[162],[15,62442,3885],{"className":62443,"style":3946},[66,67],[15,62445,126],{"className":62446},[169],[15,62448],{"className":62449,"style":173},[71],[15,62451,129],{"className":62452},[177],[15,62454],{"className":62455,"style":173},[71],[15,62457,62459,62462,62525,62531,62534,62537,62540,62543,62546,62549,62552,62555,62558,62564],{"className":62458},[57],[15,62460],{"className":62461,"style":12291},[61],[15,62463,62465,62468],{"className":62464},[66],[15,62466,62381],{"className":62467},[66,517],[15,62469,62471],{"className":62470},[198],[15,62472,62474,62517],{"className":62473},[202,1090],[15,62475,62477,62514],{"className":62476},[206],[15,62478,62480],{"className":62479,"style":3668},[210],[15,62481,62483,62486],{"style":62482},"top:-2.5198em;margin-left:0em;margin-right:0.05em;",[15,62484],{"className":62485,"style":218},[217],[15,62487,62489],{"className":62488},[222,223,224,225],[15,62490,62492,62495,62498,62501,62504,62507,62510],{"className":62491},[66,225],[15,62493,120],{"className":62494},[162,225],[15,62496,446],{"className":62497,"style":16497},[66,67,225],[15,62499,335],{"className":62500},[422,225],[15,62502,61885],{"className":62503,"style":72},[66,67,225],[15,62505,126],{"className":62506},[169,225],[15,62508,28391],{"className":62509},[177,225],[15,62511,28399],{"className":62512,"style":62513},[66,67,225],"margin-right:0.1389em;",[15,62515,1114],{"className":62516},[1113],[15,62518,62520],{"className":62519},[206],[15,62521,62523],{"className":62522,"style":9880},[210],[15,62524],{},[15,62526,62528],{"className":62527},[66],[15,62529,1194],{"className":62530},[1422,1508],[15,62532,1216],{"className":62533},[66,67],[15,62535,120],{"className":62536},[162],[15,62538,61885],{"className":62539,"style":72},[66,67],[15,62541,335],{"className":62542},[422],[15,62544],{"className":62545,"style":697},[71],[15,62547,3885],{"className":62548,"style":3946},[66,67],[15,62550,120],{"className":62551},[162],[15,62553,446],{"className":62554,"style":16497},[66,67],[15,62556,18771],{"className":62557},[169],[15,62559,62561],{"className":62560},[66],[15,62562,1266],{"className":62563},[1422,1508],[15,62565,3573],{"className":62566},[66],[11,62568,62569,62570,62627],{},"这里期望是对联合分布 ",[15,62571,62573,62597],{"className":62572,"translate":19},[18],[15,62574,62576],{"className":62575},[23],[25,62577,62578],{"xmlns":27},[29,62579,62580,62594],{},[32,62581,62582,62584,62586,62588,62590,62592],{},[35,62583,28399],{},[39,62585,120],{"stretchy":119},[35,62587,446],{},[39,62589,335],{"separator":53},[35,62591,61885],{},[39,62593,126],{"stretchy":119},[45,62595,62596],{"encoding":47},"P(X,Y)",[15,62598,62600],{"className":62599,"ariaHidden":53},[52],[15,62601,62603,62606,62609,62612,62615,62618,62621,62624],{"className":62602},[57],[15,62604],{"className":62605,"style":154},[61],[15,62607,28399],{"className":62608,"style":62513},[66,67],[15,62610,120],{"className":62611},[162],[15,62613,446],{"className":62614,"style":16497},[66,67],[15,62616,335],{"className":62617},[422],[15,62619],{"className":62620,"style":697},[71],[15,62622,61885],{"className":62623,"style":72},[66,67],[15,62625,126],{"className":62626},[169]," 取的。这个分布在实践中未知，于是我们退而求其次：用样本来估计它，或者直接在一个受限的函数类里最小化经验风险。这个「退而求其次」的过程，就是统计学习理论。",[11,62629,62630,62631,62659,62660,62688,62689,62717,62718,62746,62747,62775,62776,62804,62805,62833,62834,1124],{},"同一套语言可以覆盖很多看似不同的任务。医疗诊断中，",[15,62632,62634,62647],{"className":62633,"translate":19},[18],[15,62635,62637],{"className":62636},[23],[25,62638,62639],{"xmlns":27},[29,62640,62641,62645],{},[32,62642,62643],{},[35,62644,61885],{},[45,62646,61885],{"encoding":47},[15,62648,62650],{"className":62649,"ariaHidden":53},[52],[15,62651,62653,62656],{"className":62652},[57],[15,62654],{"className":62655,"style":490},[61],[15,62657,61885],{"className":62658,"style":72},[66,67]," 是是否患病，",[15,62661,62663,62676],{"className":62662,"translate":19},[18],[15,62664,62666],{"className":62665},[23],[25,62667,62668],{"xmlns":27},[29,62669,62670,62674],{},[32,62671,62672],{},[35,62673,4556],{},[45,62675,4556],{"encoding":47},[15,62677,62679],{"className":62678,"ariaHidden":53},[52],[15,62680,62682,62685],{"className":62681},[57],[15,62683],{"className":62684,"style":86},[61],[15,62686,4556],{"className":62687},[66,67]," 是是否治疗，损失把漏诊与误诊的代价写成两个不同的数；垃圾邮件过滤中，",[15,62690,62692,62705],{"className":62691,"translate":19},[18],[15,62693,62695],{"className":62694},[23],[25,62696,62697],{"xmlns":27},[29,62698,62699,62703],{},[32,62700,62701],{},[35,62702,61885],{},[45,62704,61885],{"encoding":47},[15,62706,62708],{"className":62707,"ariaHidden":53},[52],[15,62709,62711,62714],{"className":62710},[57],[15,62712],{"className":62713,"style":490},[61],[15,62715,61885],{"className":62716,"style":72},[66,67]," 是邮件类别，",[15,62719,62721,62734],{"className":62720,"translate":19},[18],[15,62722,62724],{"className":62723},[23],[25,62725,62726],{"xmlns":27},[29,62727,62728,62732],{},[32,62729,62730],{},[35,62731,4556],{},[45,62733,4556],{"encoding":47},[15,62735,62737],{"className":62736,"ariaHidden":53},[52],[15,62738,62740,62743],{"className":62739},[57],[15,62741],{"className":62742,"style":86},[61],[15,62744,4556],{"className":62745},[66,67]," 是放行或拦截；回归中，",[15,62748,62750,62763],{"className":62749,"translate":19},[18],[15,62751,62753],{"className":62752},[23],[25,62754,62755],{"xmlns":27},[29,62756,62757,62761],{},[32,62758,62759],{},[35,62760,4556],{},[45,62762,4556],{"encoding":47},[15,62764,62766],{"className":62765,"ariaHidden":53},[52],[15,62767,62769,62772],{"className":62768},[57],[15,62770],{"className":62771,"style":86},[61],[15,62773,4556],{"className":62774},[66,67]," 是一个实数，损失通常是平方误差；强化学习中，",[15,62777,62779,62792],{"className":62778,"translate":19},[18],[15,62780,62782],{"className":62781},[23],[25,62783,62784],{"xmlns":27},[29,62785,62786,62790],{},[32,62787,62788],{},[35,62789,446],{},[45,62791,446],{"encoding":47},[15,62793,62795],{"className":62794,"ariaHidden":53},[52],[15,62796,62798,62801],{"className":62797},[57],[15,62799],{"className":62800,"style":490},[61],[15,62802,446],{"className":62803,"style":16497},[66,67]," 是历史状态，",[15,62806,62808,62821],{"className":62807,"translate":19},[18],[15,62809,62811],{"className":62810},[23],[25,62812,62813],{"xmlns":27},[29,62814,62815,62819],{},[32,62816,62817],{},[35,62818,4556],{},[45,62820,4556],{"encoding":47},[15,62822,62824],{"className":62823,"ariaHidden":53},[52],[15,62825,62827,62830],{"className":62826},[57],[15,62828],{"className":62829,"style":86},[61],[15,62831,4556],{"className":62832},[66,67]," 是动作，损失是负回报。它们的共同点是：",[2040,62835,62836,62837,62903,62904,62932],{},"不确定性由 ",[15,62838,62840,62864],{"className":62839,"translate":19},[18],[15,62841,62843],{"className":62842},[23],[25,62844,62845],{"xmlns":27},[29,62846,62847,62861],{},[32,62848,62849,62851,62853,62855,62857,62859],{},[35,62850,28399],{},[39,62852,120],{"stretchy":119},[35,62854,61885],{},[39,62856,34430],{},[35,62858,446],{},[39,62860,126],{"stretchy":119},[45,62862,62863],{"encoding":47},"P(Y\\mid X)",[15,62865,62867,62891],{"className":62866,"ariaHidden":53},[52],[15,62868,62870,62873,62876,62879,62882,62885,62888],{"className":62869},[57],[15,62871],{"className":62872,"style":154},[61],[15,62874,28399],{"className":62875,"style":62513},[66,67],[15,62877,120],{"className":62878},[162],[15,62880,61885],{"className":62881,"style":72},[66,67],[15,62883],{"className":62884,"style":173},[71],[15,62886,34430],{"className":62887},[177],[15,62889],{"className":62890,"style":173},[71],[15,62892,62894,62897,62900],{"className":62893},[57],[15,62895],{"className":62896,"style":154},[61],[15,62898,446],{"className":62899,"style":16497},[66,67],[15,62901,126],{"className":62902},[169]," 描述，偏好由 ",[15,62905,62907,62920],{"className":62906,"translate":19},[18],[15,62908,62910],{"className":62909},[23],[25,62911,62912],{"xmlns":27},[29,62913,62914,62918],{},[32,62915,62916],{},[35,62917,1216],{},[45,62919,1216],{"encoding":47},[15,62921,62923],{"className":62922,"ariaHidden":53},[52],[15,62924,62926,62929],{"className":62925},[57],[15,62927],{"className":62928,"style":490},[61],[15,62930,1216],{"className":62931},[66,67]," 描述，决策则是两者的交汇",[11,62934,62935,62936,62964],{},"把联合风险按 ",[15,62937,62939,62952],{"className":62938,"translate":19},[18],[15,62940,62942],{"className":62941},[23],[25,62943,62944],{"xmlns":27},[29,62945,62946,62950],{},[32,62947,62948],{},[35,62949,446],{},[45,62951,446],{"encoding":47},[15,62953,62955],{"className":62954,"ariaHidden":53},[52],[15,62956,62958,62961],{"className":62957},[57],[15,62959],{"className":62960,"style":490},[61],[15,62962,446],{"className":62963,"style":16497},[66,67]," 分解，可以得到一个更有用的形式：",[15,62966,62968],{"className":62967,"translate":19},[294],[15,62969,62971,63131],{"className":62970,"translate":19},[18],[15,62972,62974],{"className":62973},[23],[25,62975,62976],{"xmlns":27,"display":303},[29,62977,62978,63128],{},[1142,62979,62982,63079],{"rowspacing":62980,"columnalign":62981,"columnspacing":634},"0.25em","right left",[1146,62983,62984,62999],{},[1149,62985,62986],{},[62987,62988,62989],"mstyle",{"scriptlevel":1804,"displaystyle":53},[32,62990,62991,62993,62995,62997],{},[35,62992,455],{},[39,62994,120],{"stretchy":119},[35,62996,3885],{},[39,62998,126],{"stretchy":119},[1149,63000,63001],{},[62987,63002,63003],{"scriptlevel":1804,"displaystyle":53},[32,63004,63005,63007,63009,63015,63065,63067,63069,63071,63073,63075,63077],{},[32,63006],{},[39,63008,129],{},[1057,63010,63011,63013],{},[39,63012,29732],{},[35,63014,446],{"mathvariant":445},[32,63016,63017,63019,63025,63027,63029,63031,63033,63035,63037,63039,63041,63043,63045,63047,63049,63051,63053,63055,63057,63059,63061,63063],{},[39,63018,120],{"fence":53},[1057,63020,63021,63023],{},[39,63022,29732],{},[35,63024,61885],{"mathvariant":445},[35,63026,1216],{},[39,63028,120],{"stretchy":119},[35,63030,1224],{},[39,63032,335],{"separator":53},[35,63034,3885],{},[39,63036,120],{"stretchy":119},[35,63038,123],{},[39,63040,126],{"stretchy":119},[39,63042,126],{"stretchy":119},[1274,63044,2462],{},[35,63046,11],{},[39,63048,120],{"stretchy":119},[35,63050,1224],{},[39,63052,34430],{},[35,63054,123],{},[39,63056,126],{"stretchy":119},[1274,63058,2462],{},[35,63060,458],{},[35,63062,1224],{},[39,63064,126],{"fence":53},[35,63066,11],{},[39,63068,120],{"stretchy":119},[35,63070,123],{},[39,63072,126],{"stretchy":119},[1274,63074,2462],{},[35,63076,458],{},[35,63078,123],{},[1146,63080,63081,63087],{},[1149,63082,63083],{},[62987,63084,63085],{"scriptlevel":1804,"displaystyle":53},[32,63086],{},[1149,63088,63089],{},[62987,63090,63091],{"scriptlevel":1804,"displaystyle":53},[32,63092,63093,63095,63097,63103,63105,63108,63110,63112,63114,63116,63118,63120,63122,63124,63126],{},[32,63094],{},[39,63096,129],{},[1057,63098,63099,63101],{},[35,63100,62381],{"mathvariant":454},[35,63102,446],{},[39,63104,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,63106,63107],{},"r",[39,63109,120],{"stretchy":119},[35,63111,3885],{},[39,63113,120],{"stretchy":119},[35,63115,446],{},[39,63117,126],{"stretchy":119},[39,63119,335],{"separator":53},[35,63121,446],{},[39,63123,126],{"stretchy":119},[39,63125,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,63127,335],{"separator":53},[45,63129,63130],{"encoding":47},"\\begin{aligned}\nR(\\delta)\n&=\\int_{\\mathcal X}\\left(\\int_{\\mathcal Y}L(y,\\delta(x))\\,p(y\\mid x)\\,dy\\right)p(x)\\,dx\\\\\n&=\\mathbb E_X\\big[r(\\delta(X),X)\\big],\n\\end{aligned}",[15,63132,63134],{"className":63133,"ariaHidden":53},[52],[15,63135,63137,63141],{"className":63136},[57],[15,63138],{"className":63139,"style":63140},[61],"height:3.94em;vertical-align:-1.72em;",[15,63142,63144],{"className":63143},[66],[15,63145,63147,63204],{"className":63146},[1142],[15,63148,63151],{"className":63149},[63150],"col-align-r",[15,63152,63154,63195],{"className":63153},[202,1090],[15,63155,63157,63192],{"className":63156},[206],[15,63158,63161,63183],{"className":63159,"style":63160},[210],"height:2.22em;",[15,63162,63164,63168],{"style":63163},"top:-4.22em;",[15,63165],{"className":63166,"style":63167},[217],"height:3.45em;",[15,63169,63171,63174,63177,63180],{"className":63170},[66],[15,63172,455],{"className":63173,"style":62437},[66,67],[15,63175,120],{"className":63176},[162],[15,63178,3885],{"className":63179,"style":3946},[66,67],[15,63181,126],{"className":63182},[169],[15,63184,63186,63189],{"style":63185},"top:-2.09em;",[15,63187],{"className":63188,"style":63167},[217],[15,63190],{"className":63191},[66],[15,63193,1114],{"className":63194},[1113],[15,63196,63198],{"className":63197},[206],[15,63199,63202],{"className":63200,"style":63201},[210],"height:1.72em;",[15,63203],{},[15,63205,63208],{"className":63206},[63207],"col-align-l",[15,63209,63211,63541],{"className":63210},[202,1090],[15,63212,63214,63538],{"className":63213},[206],[15,63215,63217,63433],{"className":63216,"style":63160},[210],[15,63218,63219,63222],{"style":63163},[15,63220],{"className":63221,"style":63167},[217],[15,63223,63225,63228,63231,63234,63237,63281,63284,63409,63412,63415,63418,63421,63424,63427,63430],{"className":63224},[66],[15,63226],{"className":63227},[66],[15,63229],{"className":63230,"style":173},[71],[15,63232,129],{"className":63233},[177],[15,63235],{"className":63236,"style":173},[71],[15,63238,63240,63243],{"className":63239},[1339],[15,63241,29732],{"className":63242,"style":29796},[1339,1470,1471],[15,63244,63246],{"className":63245},[198],[15,63247,63249,63273],{"className":63248},[202,1090],[15,63250,63252,63270],{"className":63251},[206],[15,63253,63256],{"className":63254,"style":63255},[210],"height:-0.4336em;",[15,63257,63258,63261],{"style":29812},[15,63259],{"className":63260,"style":218},[217],[15,63262,63264],{"className":63263},[222,223,224,225],[15,63265,63267],{"className":63266},[66,225],[15,63268,446],{"className":63269,"style":495},[66,494,225],[15,63271,1114],{"className":63272},[1113],[15,63274,63276],{"className":63275},[206],[15,63277,63279],{"className":63278,"style":29843},[210],[15,63280],{},[15,63282],{"className":63283,"style":697},[71],[15,63285,63287,63293,63337,63340,63343,63346,63349,63352,63355,63358,63361,63364,63367,63370,63373,63376,63379,63382,63385,63388,63391,63394,63397,63400,63403],{"className":63286},[1413],[15,63288,63290],{"className":63289,"style":1418},[162,1417],[15,63291,120],{"className":63292},[1422,224],[15,63294,63296,63299],{"className":63295},[1339],[15,63297,29732],{"className":63298,"style":29796},[1339,1470,1471],[15,63300,63302],{"className":63301},[198],[15,63303,63305,63328],{"className":63304},[202,1090],[15,63306,63308,63325],{"className":63307},[206],[15,63309,63311],{"className":63310,"style":63255},[210],[15,63312,63313,63316],{"style":29812},[15,63314],{"className":63315,"style":218},[217],[15,63317,63319],{"className":63318},[222,223,224,225],[15,63320,63322],{"className":63321},[66,225],[15,63323,61885],{"className":63324,"style":28452},[66,494,225],[15,63326,1114],{"className":63327},[1113],[15,63329,63331],{"className":63330},[206],[15,63332,63335],{"className":63333,"style":63334},[210],"height:0.98em;",[15,63336],{},[15,63338],{"className":63339,"style":697},[71],[15,63341,1216],{"className":63342},[66,67],[15,63344,120],{"className":63345},[162],[15,63347,1224],{"className":63348,"style":1515},[66,67],[15,63350,335],{"className":63351},[422],[15,63353],{"className":63354,"style":697},[71],[15,63356,3885],{"className":63357,"style":3946},[66,67],[15,63359,120],{"className":63360},[162],[15,63362,123],{"className":63363},[66,67],[15,63365,18771],{"className":63366},[169],[15,63368],{"className":63369,"style":697},[71],[15,63371,11],{"className":63372},[66,67],[15,63374,120],{"className":63375},[162],[15,63377,1224],{"className":63378,"style":1515},[66,67],[15,63380],{"className":63381,"style":173},[71],[15,63383,34430],{"className":63384},[177],[15,63386],{"className":63387,"style":173},[71],[15,63389,123],{"className":63390},[66,67],[15,63392,126],{"className":63393},[169],[15,63395],{"className":63396,"style":697},[71],[15,63398,458],{"className":63399},[66,67],[15,63401,1224],{"className":63402,"style":1515},[66,67],[15,63404,63406],{"className":63405,"style":1418},[169,1417],[15,63407,126],{"className":63408},[1422,224],[15,63410],{"className":63411,"style":697},[71],[15,63413,11],{"className":63414},[66,67],[15,63416,120],{"className":63417},[162],[15,63419,123],{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E\\big[L(Y,a)\\mid X=x\\big]",[15,63611,63613,63649,63694,63712],{"className":63612,"ariaHidden":53},[52],[15,63614,63616,63619,63622,63625,63628,63631,63634,63637,63640,63643,63646],{"className":63615},[57],[15,63617],{"className":63618,"style":154},[61],[15,63620,63107],{"className":63621,"style":23025},[66,67],[15,63623,120],{"className":63624},[162],[15,63626,4556],{"className":63627},[66,67],[15,63629,335],{"className":63630},[422],[15,63632],{"className":63633,"style":697},[71],[15,63635,123],{"className":63636},[66,67],[15,63638,126],{"className":63639},[169],[15,63641],{"className":63642,"style":173},[71],[15,63644,129],{"className":63645},[177],[15,63647],{"className":63648,"style":173},[71],[15,63650,63652,63655,63658,63664,63667,63670,63673,63676,63679,63682,63685,63688,63691],{"className":63651},[57],[15,63653],{"className":63654,"style":4400},[61],[15,63656,62381],{"className":63657},[66,517],[15,63659,63661],{"className":63660},[66],[15,63662,1194],{"className":63663},[1422,1508],[15,63665,1216],{"className":63666},[66,67],[15,63668,120],{"className":63669},[162],[15,63671,61885],{"className":63672,"style":72},[66,67],[15,63674,335],{"className":63675},[422],[15,63677],{"className":63678,"style":697},[71],[15,63680,4556],{"className":63681},[66,67],[15,63683,126],{"className":63684},[169],[15,63686],{"className":63687,"style":173},[71],[15,63689,34430],{"className":63690},[177],[15,63692],{"className":63693,"style":173},[71],[15,63695,63697,63700,63703,63706,63709],{"className":63696},[57],[15,63698],{"className":63699,"style":490},[61],[15,63701,446],{"className":63702,"style":16497},[66,67],[15,63704],{"className":63705,"style":173},[71],[15,63707,129],{"className":63708},[177],[15,63710],{"className":63711,"style":173},[71],[15,63713,63715,63718,63721],{"className":63714},[57],[15,63716],{"className":63717,"style":4400},[61],[15,63719,123],{"className":63720},[66,67],[15,63722,63724],{"className":63723},[66],[15,63725,1266],{"className":63726},[1422,1508],[11,63728,63729,63730,63758,63759,3768,63787,63790,63791,63819,63820,63848,63849,63906,63907,63935],{},"称为在 ",[15,63731,63733,63746],{"className":63732,"translate":19},[18],[15,63734,63736],{"className":63735},[23],[25,63737,63738],{"xmlns":27},[29,63739,63740,63744],{},[32,63741,63742],{},[35,63743,123],{},[45,63745,123],{"encoding":47},[15,63747,63749],{"className":63748,"ariaHidden":53},[52],[15,63750,63752,63755],{"className":63751},[57],[15,63753],{"className":63754,"style":86},[61],[15,63756,123],{"className":63757},[66,67]," 处采取行动 ",[15,63760,63762,63775],{"className":63761,"translate":19},[18],[15,63763,63765],{"className":63764},[23],[25,63766,63767],{"xmlns":27},[29,63768,63769,63773],{},[32,63770,63771],{},[35,63772,4556],{},[45,63774,4556],{"encoding":47},[15,63776,63778],{"className":63777,"ariaHidden":53},[52],[15,63779,63781,63784],{"className":63780},[57],[15,63782],{"className":63783,"style":86},[61],[15,63785,4556],{"className":63786},[66,67],[2040,63788,63789],{},"条件风险","。这个分解是整个理论的第一块基石：联合风险被拆成了「每个观测点上的局部风险」再对 ",[15,63792,63794,63807],{"className":63793,"translate":19},[18],[15,63795,63797],{"className":63796},[23],[25,63798,63799],{"xmlns":27},[29,63800,63801,63805],{},[32,63802,63803],{},[35,63804,446],{},[45,63806,446],{"encoding":47},[15,63808,63810],{"className":63809,"ariaHidden":53},[52],[15,63811,63813,63816],{"className":63812},[57],[15,63814],{"className":63815,"style":490},[61],[15,63817,446],{"className":63818,"style":16497},[66,67]," 平均。于是，只要在每个 ",[15,63821,63823,63836],{"className":63822,"translate":19},[18],[15,63824,63826],{"className":63825},[23],[25,63827,63828],{"xmlns":27},[29,63829,63830,63834],{},[32,63831,63832],{},[35,63833,123],{},[45,63835,123],{"encoding":47},[15,63837,63839],{"className":63838,"ariaHidden":53},[52],[15,63840,63842,63845],{"className":63841},[57],[15,63843],{"className":63844,"style":86},[61],[15,63846,123],{"className":63847},[66,67]," 上把 ",[15,63850,63852,63876],{"className":63851,"translate":19},[18],[15,63853,63855],{"className":63854},[23],[25,63856,63857],{"xmlns":27},[29,63858,63859,63873],{},[32,63860,63861,63863,63865,63867,63869,63871],{},[35,63862,63107],{},[39,63864,120],{"stretchy":119},[35,63866,4556],{},[39,63868,335],{"separator":53},[35,63870,123],{},[39,63872,126],{"stretchy":119},[45,63874,63875],{"encoding":47},"r(a,x)",[15,63877,63879],{"className":63878,"ariaHidden":53},[52],[15,63880,63882,63885,63888,63891,63894,63897,63900,63903],{"className":63881},[57],[15,63883],{"className":63884,"style":154},[61],[15,63886,63107],{"className":63887,"style":23025},[66,67],[15,63889,120],{"className":63890},[162],[15,63892,4556],{"className":63893},[66,67],[15,63895,335],{"className":63896},[422],[15,63898],{"className":63899,"style":697},[71],[15,63901,123],{"className":63902},[66,67],[15,63904,126],{"className":63905},[169]," 压到最低，全局风险就自动最低，不同 ",[15,63908,63910,63923],{"className":63909,"translate":19},[18],[15,63911,63913],{"className":63912},[23],[25,63914,63915],{"xmlns":27},[29,63916,63917,63921],{},[32,63918,63919],{},[35,63920,123],{},[45,63922,123],{"encoding":47},[15,63924,63926],{"className":63925,"ariaHidden":53},[52],[15,63927,63929,63932],{"className":63928},[57],[15,63930],{"className":63931,"style":86},[61],[15,63933,123],{"className":63934},[66,67]," 之间没有耦合。",[95,63937,63939],{"id":63938},"_2-概率的语言","2. 概率的语言",[1904,63941,63943],{"id":63942},"_21-条件概率信息如何改变概率","2.1 条件概率：信息如何改变概率",[11,63945,63946],{},"条件概率的定义是",[15,63948,63950],{"className":63949,"translate":19},[294],[15,63951,63953,64023],{"className":63952,"translate":19},[18],[15,63954,63956],{"className":63955},[23],[25,63957,63958],{"xmlns":27,"display":303},[29,63959,63960,64020],{},[32,63961,63962,63964,63966,63968,63970,63973,63975,63977,64004,64006,64008,64010,64012,64014,64016,64018],{},[35,63963,28399],{},[39,63965,120],{"stretchy":119},[35,63967,62005],{},[39,63969,34430],{},[35,63971,63972],{},"B",[39,63974,126],{"stretchy":119},[39,63976,129],{},[6467,63978,63979,63994],{},[32,63980,63981,63983,63985,63987,63990,63992],{},[35,63982,28399],{},[39,63984,120],{"stretchy":119},[35,63986,62005],{},[39,63988,63989],{},"∩",[35,63991,63972],{},[39,63993,126],{"stretchy":119},[32,63995,63996,63998,64000,64002],{},[35,63997,28399],{},[39,63999,120],{"stretchy":119},[35,64001,63972],{},[39,64003,126],{"stretchy":119},[39,64005,335],{"separator":53},[71,64007],{"width":3904},[35,64009,28399],{},[39,64011,120],{"stretchy":119},[35,64013,63972],{},[39,64015,126],{"stretchy":119},[39,64017,9283],{},[1209,64019,2487],{},[45,64021,64022],{"encoding":47},"P(A\\mid B)=\\frac{P(A\\cap B)}{P(B)},\\qquad P(B)>0.",[15,64024,64026,64050,64072,64200],{"className":64025,"ariaHidden":53},[52],[15,64027,64029,64032,64035,64038,64041,64044,64047],{"className":64028},[57],[15,64030],{"className":64031,"style":154},[61],[15,64033,28399],{"className":64034,"style":62513},[66,67],[15,64036,120],{"className":64037},[162],[15,64039,62005],{"className":64040},[66,67],[15,64042],{"className":64043,"style":173},[71],[15,64045,34430],{"className":64046},[177],[15,64048],{"className":64049,"style":173},[71],[15,64051,64053,64056,64060,64063,64066,64069],{"className":64052},[57],[15,64054],{"className":64055,"style":154},[61],[15,64057,63972],{"className":64058,"style":64059},[66,67],"margin-right:0.0502em;",[15,64061,126],{"className":64062},[169],[15,64064],{"className":64065,"style":173},[71],[15,64067,129],{"className":64068},[177],[15,64070],{"className":64071,"style":173},[71],[15,64073,64075,64078,64170,64173,64176,64179,64182,64185,64188,64191,64194,64197],{"className":64074},[57],[15,64076],{"className":64077,"style":41192},[61],[15,64079,64081,64084,64167],{"className":64080},[66],[15,64082],{"className":64083},[162,23417],[15,64085,64087],{"className":64086},[6467],[15,64088,64090,64159],{"className":64089},[202,1090],[15,64091,64093,64156],{"className":64092},[206],[15,64094,64096,64116,64124],{"className":64095,"style":41211},[210],[15,64097,64098,64101],{"style":23433},[15,64099],{"className":64100,"style":1368},[217],[15,64102,64104,64107,64110,64113],{"className":64103},[66],[15,64105,28399],{"className":64106,"style":62513},[66,67],[15,64108,120],{"className":64109},[162],[15,64111,63972],{"className":64112,"style":64059},[66,67],[15,64114,126],{"className":64115},[169],[15,64117,64118,64121],{"style":23445},[15,64119],{"className":64120,"style":1368},[217],[15,64122],{"className":64123,"style":4440},[23452],[15,64125,64126,64129],{"style":23455},[15,64127],{"className":64128,"style":1368},[217],[15,64130,64132,64135,64138,64141,64144,64147,64150,64153],{"className":64131},[66],[15,64133,28399],{"className":64134,"style":62513},[66,67],[15,64136,120],{"className":64137},[162],[15,64139,62005],{"className":64140},[66,67],[15,64142],{"className":64143,"style":72},[71],[15,64145,63989],{"className":64146},[76],[15,64148],{"className":64149,"style":72},[71],[15,64151,63972],{"className":64152,"style":64059},[66,67],[15,64154,126],{"className":64155},[169],[15,64157,1114],{"className":64158},[1113],[15,64160,64162],{"className":64161},[206],[15,64163,64165],{"className":64164,"style":41282},[210],[15,64166],{},[15,64168],{"className":64169},[169,23417],[15,64171,335],{"className":64172},[422],[15,64174],{"className":64175,"style":4061},[71],[15,64177],{"className":64178,"style":697},[71],[15,64180,28399],{"className":64181,"style":62513},[66,67],[15,64183,120],{"className":64184},[162],[15,64186,63972],{"className":64187,"style":64059},[66,67],[15,64189,126],{"className":64190},[169],[15,64192],{"className":64193,"style":173},[71],[15,64195,9283],{"className":64196},[177],[15,64198],{"className":64199,"style":173},[71],[15,64201,64203,64206],{"className":64202},[57],[15,64204],{"className":64205,"style":1836},[61],[15,64207,2487],{"className":64208},[66],[11,64210,64211,64212,64240,64241,64269,64270,64411],{},"它不是在「重新定义」概率，而是在缩小样本空间：一旦知道 ",[15,64213,64215,64228],{"className":64214,"translate":19},[18],[15,64216,64218],{"className":64217},[23],[25,64219,64220],{"xmlns":27},[29,64221,64222,64226],{},[32,64223,64224],{},[35,64225,63972],{},[45,64227,63972],{"encoding":47},[15,64229,64231],{"className":64230,"ariaHidden":53},[52],[15,64232,64234,64237],{"className":64233},[57],[15,64235],{"className":64236,"style":490},[61],[15,64238,63972],{"className":64239,"style":64059},[66,67]," 发生，所有不含 ",[15,64242,64244,64257],{"className":64243,"translate":19},[18],[15,64245,64247],{"className":64246},[23],[25,64248,64249],{"xmlns":27},[29,64250,64251,64255],{},[32,64252,64253],{},[35,64254,63972],{},[45,64256,63972],{"encoding":47},[15,64258,64260],{"className":64259,"ariaHidden":53},[52],[15,64261,64263,64266],{"className":64262},[57],[15,64264],{"className":64265,"style":490},[61],[15,64267,63972],{"className":64268,"style":64059},[66,67]," 的结果都被排除，原来的概率需要在剩下的世界里重新归一化。连续情形下，条件分布由条件密度 ",[15,64271,64273,64321],{"className":64272,"translate":19},[18],[15,64274,64276],{"className":64275},[23],[25,64277,64278],{"xmlns":27},[29,64279,64280,64318],{},[32,64281,64282,64284,64286,64288,64290,64292,64294,64296,64298,64300,64302,64304,64306,64308,64310,64312,64314,64316],{},[35,64283,11],{},[39,64285,120],{"stretchy":119},[35,64287,1224],{},[39,64289,34430],{},[35,64291,123],{},[39,64293,126],{"stretchy":119},[39,64295,129],{},[35,64297,11],{},[39,64299,120],{"stretchy":119},[35,64301,123],{},[39,64303,335],{"separator":53},[35,64305,1224],{},[39,64307,126],{"stretchy":119},[35,64309,6625],{"mathvariant":138},[35,64311,11],{},[39,64313,120],{"stretchy":119},[35,64315,123],{},[39,64317,126],{"stretchy":119},[45,64319,64320],{"encoding":47},"p(y\\mid x)=p(x,y)\u002Fp(x)",[15,64322,64324,64348,64369],{"className":64323,"ariaHidden":53},[52],[15,64325,64327,64330,64333,64336,64339,64342,64345],{"className":64326},[57],[15,64328],{"className":64329,"style":154},[61],[15,64331,11],{"className":64332},[66,67],[15,64334,120],{"className":64335},[162],[15,64337,1224],{"className":64338,"style":1515},[66,67],[15,64340],{"className":64341,"style":173},[71],[15,64343,34430],{"className":64344},[177],[15,64346],{"className":64347,"style":173},[71],[15,64349,64351,64354,64357,64360,64363,64366],{"className":64350},[57],[15,64352],{"className":64353,"style":154},[61],[15,64355,123],{"className":64356},[66,67],[15,64358,126],{"className":64359},[169],[15,64361],{"className":64362,"style":173},[71],[15,64364,129],{"className":64365},[177],[15,64367],{"className":64368,"style":173},[71],[15,64370,64372,64375,64378,64381,64384,64387,64390,64393,64396,64399,64402,64405,64408],{"className":64371},[57],[15,64373],{"className":64374,"style":154},[61],[15,64376,11],{"className":64377},[66,67],[15,64379,120],{"className":64380},[162],[15,64382,123],{"className":64383},[66,67],[15,64385,335],{"className":64386},[422],[15,64388],{"className":64389,"style":697},[71],[15,64391,1224],{"className":64392,"style":1515},[66,67],[15,64394,126],{"className":64395},[169],[15,64397,6625],{"className":64398},[66],[15,64400,11],{"className":64401},[66,67],[15,64403,120],{"className":64404},[162],[15,64406,123],{"className":64407},[66,67],[15,64409,126],{"className":64410},[169]," 给出；更一般地，它是测度论意义下的正则条件分布，本文不展开技术细节。",[11,64413,64414],{},"由条件概率的定义立刻得到乘法公式",[15,64416,64418],{"className":64417,"translate":19},[294],[15,64419,64421,64491],{"className":64420,"translate":19},[18],[15,64422,64424],{"className":64423},[23],[25,64425,64426],{"xmlns":27,"display":303},[29,64427,64428,64488],{},[32,64429,64430,64432,64434,64436,64438,64440,64442,64444,64446,64448,64450,64452,64454,64456,64458,64460,64462,64464,64466,64468,64470,64472,64474,64476,64478,64480,64482,64484,64486],{},[35,64431,28399],{},[39,64433,120],{"stretchy":119},[35,64435,62005],{},[39,64437,63989],{},[35,64439,63972],{},[39,64441,126],{"stretchy":119},[39,64443,129],{},[35,64445,28399],{},[39,64447,120],{"stretchy":119},[35,64449,62005],{},[39,64451,34430],{},[35,64453,63972],{},[39,64455,126],{"stretchy":119},[35,64457,28399],{},[39,64459,120],{"stretchy":119},[35,64461,63972],{},[39,64463,126],{"stretchy":119},[39,64465,129],{},[35,64467,28399],{},[39,64469,120],{"stretchy":119},[35,64471,63972],{},[39,64473,34430],{},[35,64475,62005],{},[39,64477,126],{"stretchy":119},[35,64479,28399],{},[39,64481,120],{"stretchy":119},[35,64483,62005],{},[39,64485,126],{"stretchy":119},[39,64487,335],{"separator":53},[45,64489,64490],{"encoding":47},"P(A\\cap B)=P(A\\mid B)P(B)=P(B\\mid A)P(A),",[15,64492,64494,64518,64539,64563,64596,64620],{"className":64493,"ariaHidden":53},[52],[15,64495,64497,64500,64503,64506,64509,64512,64515],{"className":64496},[57],[15,64498],{"className":64499,"style":154},[61],[15,64501,28399],{"className":64502,"style":62513},[66,67],[15,64504,120],{"className":64505},[162],[15,64507,62005],{"className":64508},[66,67],[15,64510],{"className":64511,"style":72},[71],[15,64513,63989],{"className":64514},[76],[15,64516],{"className":64517,"style":72},[71],[15,64519,64521,64524,64527,64530,64533,64536],{"className":64520},[57],[15,64522],{"className":64523,"style":154},[61],[15,64525,63972],{"className":64526,"style":64059},[66,67],[15,64528,126],{"className":64529},[169],[15,64531],{"className":64532,"style":173},[71],[15,64534,129],{"className":64535},[177],[15,64537],{"className":64538,"style":173},[71],[15,64540,64542,64545,64548,64551,64554,64557,64560],{"className":64541},[57],[15,64543],{"className":64544,"style":154},[61],[15,64546,28399],{"className":64547,"style":62513},[66,67],[15,64549,120],{"className":64550},[162],[15,64552,62005],{"className":64553},[66,67],[15,64555],{"className":64556,"style":173},[71],[15,64558,34430],{"className":64559},[177],[15,64561],{"className":64562,"style":173},[71],[15,64564,64566,64569,64572,64575,64578,64581,64584,64587,64590,64593],{"className":64565},[57],[15,64567],{"className":64568,"style":154},[61],[15,64570,63972],{"className":64571,"style":64059},[66,67],[15,64573,126],{"className":64574},[169],[15,64576,28399],{"className":64577,"style":62513},[66,67],[15,64579,120],{"className":64580},[162],[15,64582,63972],{"className":64583,"style":64059},[66,67],[15,64585,126],{"className":64586},[169],[15,64588],{"className":64589,"style":173},[71],[15,64591,129],{"className":64592},[177],[15,64594],{"className":64595,"style":173},[71],[15,64597,64599,64602,64605,64608,64611,64614,64617],{"className":64598},[57],[15,64600],{"className":64601,"style":154},[61],[15,64603,28399],{"className":64604,"style":62513},[66,67],[15,64606,120],{"className":64607},[162],[15,64609,63972],{"className":64610,"style":64059},[66,67],[15,64612],{"className":64613,"style":173},[71],[15,64615,34430],{"className":64616},[177],[15,64618],{"className":64619,"style":173},[71],[15,64621,64623,64626,64629,64632,64635,64638,64641,64644],{"className":64622},[57],[15,64624],{"className":64625,"style":154},[61],[15,64627,62005],{"className":64628},[66,67],[15,64630,126],{"className":64631},[169],[15,64633,28399],{"className":64634,"style":62513},[66,67],[15,64636,120],{"className":64637},[162],[15,64639,62005],{"className":64640},[66,67],[15,64642,126],{"className":64643},[169],[15,64645,335],{"className":64646},[422],[11,64648,64649,64650,64694],{},"两边同除以 ",[15,64651,64653,64673],{"className":64652,"translate":19},[18],[15,64654,64656],{"className":64655},[23],[25,64657,64658],{"xmlns":27},[29,64659,64660,64670],{},[32,64661,64662,64664,64666,64668],{},[35,64663,28399],{},[39,64665,120],{"stretchy":119},[35,64667,63972],{},[39,64669,126],{"stretchy":119},[45,64671,64672],{"encoding":47},"P(B)",[15,64674,64676],{"className":64675,"ariaHidden":53},[52],[15,64677,64679,64682,64685,64688,64691],{"className":64678},[57],[15,64680],{"className":64681,"style":154},[61],[15,64683,28399],{"className":64684,"style":62513},[66,67],[15,64686,120],{"className":64687},[162],[15,64689,63972],{"className":64690,"style":64059},[66,67],[15,64692,126],{"className":64693},[169],"，就是贝叶斯公式。",[1904,64696,64698],{"id":64697},"_22-贝叶斯公式","2.2 贝叶斯公式",[11,64700,64701,64702,64730,64731,64759,64760,64788,64789,64818],{},"把 ",[15,64703,64705,64718],{"className":64704,"translate":19},[18],[15,64706,64708],{"className":64707},[23],[25,64709,64710],{"xmlns":27},[29,64711,64712,64716],{},[32,64713,64714],{},[35,64715,63972],{},[45,64717,63972],{"encoding":47},[15,64719,64721],{"className":64720,"ariaHidden":53},[52],[15,64722,64724,64727],{"className":64723},[57],[15,64725],{"className":64726,"style":490},[61],[15,64728,63972],{"className":64729,"style":64059},[66,67]," 换成数据 ",[15,64732,64734,64747],{"className":64733,"translate":19},[18],[15,64735,64737],{"className":64736},[23],[25,64738,64739],{"xmlns":27},[29,64740,64741,64745],{},[32,64742,64743],{},[35,64744,32458],{},[45,64746,32458],{"encoding":47},[15,64748,64750],{"className":64749,"ariaHidden":53},[52],[15,64751,64753,64756],{"className":64752},[57],[15,64754],{"className":64755,"style":490},[61],[15,64757,32458],{"className":64758,"style":23025},[66,67],"，把 ",[15,64761,64763,64776],{"className":64762,"translate":19},[18],[15,64764,64766],{"className":64765},[23],[25,64767,64768],{"xmlns":27},[29,64769,64770,64774],{},[32,64771,64772],{},[35,64773,62005],{},[45,64775,62005],{"encoding":47},[15,64777,64779],{"className":64778,"ariaHidden":53},[52],[15,64780,64782,64785],{"className":64781},[57],[15,64783],{"className":64784,"style":490},[61],[15,64786,62005],{"className":64787},[66,67]," 换成参数或隐变量 ",[15,64790,64792,64806],{"className":64791,"translate":19},[18],[15,64793,64795],{"className":64794},[23],[25,64796,64797],{"xmlns":27},[29,64798,64799,64803],{},[32,64800,64801],{},[35,64802,61618],{},[45,64804,64805],{"encoding":47},"\\theta",[15,64807,64809],{"className":64808,"ariaHidden":53},[52],[15,64810,64812,64815],{"className":64811},[57],[15,64813],{"className":64814,"style":840},[61],[15,64816,61618],{"className":64817,"style":23025},[66,67],"，得到",[15,64820,64822],{"className":64821,"translate":19},[294],[15,64823,64825,64933],{"className":64824,"translate":19},[18],[15,64826,64828],{"className":64827},[23],[25,64829,64830],{"xmlns":27,"display":303},[29,64831,64832,64930],{},[32,64833,64834,64836,64838,64840,64842,64844,64846,64848,64884,64886,64888,64890,64892,64894,64896,64898,64900,64902,64904,64906,64908,64910,64912,64914,64916,64918,64920,64922,64924,64926,64928],{},[35,64835,11],{},[39,64837,120],{"stretchy":119},[35,64839,61618],{},[39,64841,34430],{},[35,64843,32458],{},[39,64845,126],{"stretchy":119},[39,64847,129],{},[6467,64849,64850,64874],{},[32,64851,64852,64854,64856,64858,64860,64862,64864,64866,64868,64870,64872],{},[35,64853,11],{},[39,64855,120],{"stretchy":119},[35,64857,32458],{},[39,64859,34430],{},[35,64861,61618],{},[39,64863,126],{"stretchy":119},[1274,64865,2462],{},[35,64867,11],{},[39,64869,120],{"stretchy":119},[35,64871,61618],{},[39,64873,126],{"stretchy":119},[32,64875,64876,64878,64880,64882],{},[35,64877,11],{},[39,64879,120],{"stretchy":119},[35,64881,32458],{},[39,64883,126],{"stretchy":119},[39,64885,335],{"separator":53},[71,64887],{"width":3904},[35,64889,11],{},[39,64891,120],{"stretchy":119},[35,64893,32458],{},[39,64895,126],{"stretchy":119},[39,64897,129],{},[39,64899,29732],{},[35,64901,11],{},[39,64903,120],{"stretchy":119},[35,64905,32458],{},[39,64907,34430],{},[35,64909,61618],{},[39,64911,126],{"stretchy":119},[1274,64913,2462],{},[35,64915,11],{},[39,64917,120],{"stretchy":119},[35,64919,61618],{},[39,64921,126],{"stretchy":119},[1274,64923,2462],{},[35,64925,458],{},[35,64927,61618],{},[35,64929,3573],{"mathvariant":138},[45,64931,64932],{"encoding":47},"p(\\theta\\mid D)=\\frac{p(D\\mid\\theta)\\,p(\\theta)}{p(D)},\n\\qquad\np(D)=\\int p(D\\mid\\theta)\\,p(\\theta)\\,d\\theta .",[15,64934,64936,64960,64981,65124,65155],{"className":64935,"ariaHidden":53},[52],[15,64937,64939,64942,64945,64948,64951,64954,64957],{"className":64938},[57],[15,64940],{"className":64941,"style":154},[61],[15,64943,11],{"className":64944},[66,67],[15,64946,120],{"className":64947},[162],[15,64949,61618],{"className":64950,"style":23025},[66,67],[15,64952],{"className":64953,"style":173},[71],[15,64955,34430],{"className":64956},[177],[15,64958],{"className":64959,"style":173},[71],[15,64961,64963,64966,64969,64972,64975,64978],{"className":64962},[57],[15,64964],{"className":64965,"style":154},[61],[15,64967,32458],{"className":64968,"style":23025},[66,67],[15,64970,126],{"className":64971},[169],[15,64973],{"className":64974,"style":173},[71],[15,64976,129],{"className":64977},[177],[15,64979],{"className":64980,"style":173},[71],[15,64982,64984,64987,65094,65097,65100,65103,65106,65109,65112,65115,65118,65121],{"className":64983},[57],[15,64985],{"className":64986,"style":41192},[61],[15,64988,64990,64993,65091],{"className":64989},[66],[15,64991],{"className":64992},[162,23417],[15,64994,64996],{"className":64995},[6467],[15,64997,64999,65083],{"className":64998},[202,1090],[15,65000,65002,65080],{"className":65001},[206],[15,65003,65005,65025,65033],{"className":65004,"style":41211},[210],[15,65006,65007,65010],{"style":23433},[15,65008],{"className":65009,"style":1368},[217],[15,65011,65013,65016,65019,65022],{"className":65012},[66],[15,65014,11],{"className":65015},[66,67],[15,65017,120],{"className":65018},[162],[15,65020,32458],{"className":65021,"style":23025},[66,67],[15,65023,126],{"className":65024},[169],[15,65026,65027,65030],{"style":23445},[15,65028],{"className":65029,"style":1368},[217],[15,65031],{"className":65032,"style":4440},[23452],[15,65034,65035,65038],{"style":23455},[15,65036],{"className":65037,"style":1368},[217],[15,65039,65041,65044,65047,65050,65053,65056,65059,65062,65065,65068,65071,65074,65077],{"className":65040},[66],[15,65042,11],{"className":65043},[66,67],[15,65045,120],{"className":65046},[162],[15,65048,32458],{"className":65049,"style":23025},[66,67],[15,65051],{"className":65052,"style":173},[71],[15,65054,34430],{"className":65055},[177],[15,65057],{"className":65058,"style":173},[71],[15,65060,61618],{"className":65061,"style":23025},[66,67],[15,65063,126],{"className":65064},[169],[15,65066],{"className":65067,"style":697},[71],[15,65069,11],{"className":65070},[66,67],[15,65072,120],{"className":65073},[162],[15,65075,61618],{"className":65076,"style":23025},[66,67],[15,65078,126],{"className":65079},[169],[15,65081,1114],{"className":65082},[1113],[15,65084,65086],{"className":65085},[206],[15,65087,65089],{"className":65088,"style":41282},[210],[15,65090],{},[15,65092],{"className":65093},[169,23417],[15,65095,335],{"className":65096},[422],[15,65098],{"className":65099,"style":4061},[71],[15,65101],{"className":65102,"style":697},[71],[15,65104,11],{"className":65105},[66,67],[15,65107,120],{"className":65108},[162],[15,65110,32458],{"className":65111,"style":23025},[66,67],[15,65113,126],{"className":65114},[169],[15,65116],{"className":65117,"style":173},[71],[15,65119,129],{"className":65120},[177],[15,65122],{"className":65123,"style":173},[71],[15,65125,65127,65131,65134,65137,65140,65143,65146,65149,65152],{"className":65126},[57],[15,65128],{"className":65129,"style":65130},[61],"height:2.2222em;vertical-align:-0.8622em;",[15,65132,29732],{"className":65133,"style":29796},[1339,1470,1471],[15,65135],{"className":65136,"style":697},[71],[15,65138,11],{"className":65139},[66,67],[15,65141,120],{"className":65142},[162],[15,65144,32458],{"className":65145,"style":23025},[66,67],[15,65147],{"className":65148,"style":173},[71],[15,65150,34430],{"className":65151},[177],[15,65153],{"className":65154,"style":173},[71],[15,65156,65158,65161,65164,65167,65170,65173,65176,65179,65182,65185,65188,65191],{"className":65157},[57],[15,65159],{"className":65160,"style":154},[61],[15,65162,61618],{"className":65163,"style":23025},[66,67],[15,65165,126],{"className":65166},[169],[15,65168],{"className":65169,"style":697},[71],[15,65171,11],{"className":65172},[66,67],[15,65174,120],{"className":65175},[162],[15,65177,61618],{"className":65178,"style":23025},[66,67],[15,65180,126],{"className":65181},[169],[15,65183],{"className":65184,"style":697},[71],[15,65186,458],{"className":65187},[66,67],[15,65189,61618],{"className":65190,"style":23025},[66,67],[15,65192,3573],{"className":65193},[66],[11,65195,65196],{},"公式的四个部分各有名字：",[6152,65198,65199,65212],{},[6155,65200,65201],{},[6158,65202,65203,65206,65209],{},[6161,65204,65205],{},"名称",[6161,65207,65208],{},"记号",[6161,65210,65211],{},"含义",[6171,65213,65214,65268,65344,65427],{},[6158,65215,65216,65219,65265],{},[6176,65217,65218],{},"先验",[6176,65220,65221],{},[15,65222,65224,65244],{"className":65223,"translate":19},[18],[15,65225,65227],{"className":65226},[23],[25,65228,65229],{"xmlns":27},[29,65230,65231,65241],{},[32,65232,65233,65235,65237,65239],{},[35,65234,11],{},[39,65236,120],{"stretchy":119},[35,65238,61618],{},[39,65240,126],{"stretchy":119},[45,65242,65243],{"encoding":47},"p(\\theta)",[15,65245,65247],{"className":65246,"ariaHidden":53},[52],[15,65248,65250,65253,65256,65259,65262],{"className":65249},[57],[15,65251],{"className":65252,"style":154},[61],[15,65254,11],{"className":65255},[66,67],[15,65257,120],{"className":65258},[162],[15,65260,61618],{"className":65261,"style":23025},[66,67],[15,65263,126],{"className":65264},[169],[6176,65266,65267],{},"看到数据前的信念",[6158,65269,65270,65273,65341],{},[6176,65271,65272],{},"似然",[6176,65274,65275],{},[15,65276,65278,65302],{"className":65277,"translate":19},[18],[15,65279,65281],{"className":65280},[23],[25,65282,65283],{"xmlns":27},[29,65284,65285,65299],{},[32,65286,65287,65289,65291,65293,65295,65297],{},[35,65288,11],{},[39,65290,120],{"stretchy":119},[35,65292,32458],{},[39,65294,34430],{},[35,65296,61618],{},[39,65298,126],{"stretchy":119},[45,65300,65301],{"encoding":47},"p(D\\mid\\theta)",[15,65303,65305,65329],{"className":65304,"ariaHidden":53},[52],[15,65306,65308,65311,65314,65317,65320,65323,65326],{"className":65307},[57],[15,65309],{"className":65310,"style":154},[61],[15,65312,11],{"className":65313},[66,67],[15,65315,120],{"className":65316},[162],[15,65318,32458],{"className":65319,"style":23025},[66,67],[15,65321],{"className":65322,"style":173},[71],[15,65324,34430],{"className":65325},[177],[15,65327],{"className":65328,"style":173},[71],[15,65330,65332,65335,65338],{"className":65331},[57],[15,65333],{"className":65334,"style":154},[61],[15,65336,61618],{"className":65337,"style":23025},[66,67],[15,65339,126],{"className":65340},[169],[6176,65342,65343],{},"给定参数时数据的概率",[6158,65345,65346,65349,65395],{},[6176,65347,65348],{},"证据",[6176,65350,65351],{},[15,65352,65354,65374],{"className":65353,"translate":19},[18],[15,65355,65357],{"className":65356},[23],[25,65358,65359],{"xmlns":27},[29,65360,65361,65371],{},[32,65362,65363,65365,65367,65369],{},[35,65364,11],{},[39,65366,120],{"stretchy":119},[35,65368,32458],{},[39,65370,126],{"stretchy":119},[45,65372,65373],{"encoding":47},"p(D)",[15,65375,65377],{"className":65376,"ariaHidden":53},[52],[15,65378,65380,65383,65386,65389,65392],{"className":65379},[57],[15,65381],{"className":65382,"style":154},[61],[15,65384,11],{"className":65385},[66,67],[15,65387,120],{"className":65388},[162],[15,65390,32458],{"className":65391,"style":23025},[66,67],[15,65393,126],{"className":65394},[169],[6176,65396,65397,65398,65426],{},"与 ",[15,65399,65401,65414],{"className":65400,"translate":19},[18],[15,65402,65404],{"className":65403},[23],[25,65405,65406],{"xmlns":27},[29,65407,65408,65412],{},[32,65409,65410],{},[35,65411,61618],{},[45,65413,64805],{"encoding":47},[15,65415,65417],{"className":65416,"ariaHidden":53},[52],[15,65418,65420,65423],{"className":65419},[57],[15,65421],{"className":65422,"style":840},[61],[15,65424,61618],{"className":65425,"style":23025},[66,67]," 无关的归一化常数，也是模型比较的关键",[6158,65428,65429,65432,65500],{},[6176,65430,65431],{},"后验",[6176,65433,65434],{},[15,65435,65437,65461],{"className":65436,"translate":19},[18],[15,65438,65440],{"className":65439},[23],[25,65441,65442],{"xmlns":27},[29,65443,65444,65458],{},[32,65445,65446,65448,65450,65452,65454,65456],{},[35,65447,11],{},[39,65449,120],{"stretchy":119},[35,65451,61618],{},[39,65453,34430],{},[35,65455,32458],{},[39,65457,126],{"stretchy":119},[45,65459,65460],{"encoding":47},"p(\\theta\\mid D)",[15,65462,65464,65488],{"className":65463,"ariaHidden":53},[52],[15,65465,65467,65470,65473,65476,65479,65482,65485],{"className":65466},[57],[15,65468],{"className":65469,"style":154},[61],[15,65471,11],{"className":65472},[66,67],[15,65474,120],{"className":65475},[162],[15,65477,61618],{"className":65478,"style":23025},[66,67],[15,65480],{"className":65481,"style":173},[71],[15,65483,34430],{"className":65484},[177],[15,65486],{"className":65487,"style":173},[71],[15,65489,65491,65494,65497],{"className":65490},[57],[15,65492],{"className":65493,"style":154},[61],[15,65495,32458],{"className":65496,"style":23025},[66,67],[15,65498,126],{"className":65499},[169],[6176,65501,65502],{},"看到数据后的信念",[11,65504,65505],{},"下面的图把这条链画全。注意「后验」不是终点：它只是决策的输入。",[13533,65507,65508],{},[13536,65509],{"height":65510,"width":65511,"style":13540,"data-d2-theme":13541,"alt":65512,"src":65513},257,720,"贝叶斯决策流程：先验与似然合成后验，后验与损失共同决定最优行动","data:image\u002Fsvg+xml;base64,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\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\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",[11,65515,65516,65517,48366,65560,65588],{},"公式右端的 ",[15,65518,65520,65539],{"className":65519,"translate":19},[18],[15,65521,65523],{"className":65522},[23],[25,65524,65525],{"xmlns":27},[29,65526,65527,65537],{},[32,65528,65529,65531,65533,65535],{},[35,65530,11],{},[39,65532,120],{"stretchy":119},[35,65534,32458],{},[39,65536,126],{"stretchy":119},[45,65538,65373],{"encoding":47},[15,65540,65542],{"className":65541,"ariaHidden":53},[52],[15,65543,65545,65548,65551,65554,65557],{"className":65544},[57],[15,65546],{"className":65547,"style":154},[61],[15,65549,11],{"className":65550},[66,67],[15,65552,120],{"className":65553},[162],[15,65555,32458],{"className":65556,"style":23025},[66,67],[15,65558,126],{"className":65559},[169],[15,65561,65563,65576],{"className":65562,"translate":19},[18],[15,65564,65566],{"className":65565},[23],[25,65567,65568],{"xmlns":27},[29,65569,65570,65574],{},[32,65571,65572],{},[35,65573,61618],{},[45,65575,64805],{"encoding":47},[15,65577,65579],{"className":65578,"ariaHidden":53},[52],[15,65580,65582,65585],{"className":65581},[57],[15,65583],{"className":65584,"style":840},[61],[15,65586,61618],{"className":65587,"style":23025},[66,67]," 无关，所以在求后验时常常被略去，写成",[15,65590,65592],{"className":65591,"translate":19},[294],[15,65593,65595,65646],{"className":65594,"translate":19},[18],[15,65596,65598],{"className":65597},[23],[25,65599,65600],{"xmlns":27,"display":303},[29,65601,65602,65643],{},[32,65603,65604,65606,65608,65610,65612,65614,65616,65619,65621,65623,65625,65627,65629,65631,65633,65635,65637,65639,65641],{},[35,65605,11],{},[39,65607,120],{"stretchy":119},[35,65609,61618],{},[39,65611,34430],{},[35,65613,32458],{},[39,65615,126],{"stretchy":119},[39,65617,65618],{},"∝",[35,65620,11],{},[39,65622,120],{"stretchy":119},[35,65624,32458],{},[39,65626,34430],{},[35,65628,61618],{},[39,65630,126],{"stretchy":119},[1274,65632,2462],{},[35,65634,11],{},[39,65636,120],{"stretchy":119},[35,65638,61618],{},[39,65640,126],{"stretchy":119},[35,65642,3573],{"mathvariant":138},[45,65644,65645],{"encoding":47},"p(\\theta\\mid D)\\propto p(D\\mid\\theta)\\,p(\\theta).",[15,65647,65649,65673,65694,65718],{"className":65648,"ariaHidden":53},[52],[15,65650,65652,65655,65658,65661,65664,65667,65670],{"className":65651},[57],[15,65653],{"className":65654,"style":154},[61],[15,65656,11],{"className":65657},[66,67],[15,65659,120],{"className":65660},[162],[15,65662,61618],{"className":65663,"style":23025},[66,67],[15,65665],{"className":65666,"style":173},[71],[15,65668,34430],{"className":65669},[177],[15,65671],{"className":65672,"style":173},[71],[15,65674,65676,65679,65682,65685,65688,65691],{"className":65675},[57],[15,65677],{"className":65678,"style":154},[61],[15,65680,32458],{"className":65681,"style":23025},[66,67],[15,65683,126],{"className":65684},[169],[15,65686],{"className":65687,"style":173},[71],[15,65689,65618],{"className":65690},[177],[15,65692],{"className":65693,"style":173},[71],[15,65695,65697,65700,65703,65706,65709,65712,65715],{"className":65696},[57],[15,65698],{"className":65699,"style":154},[61],[15,65701,11],{"className":65702},[66,67],[15,65704,120],{"className":65705},[162],[15,65707,32458],{"className":65708,"style":23025},[66,67],[15,65710],{"className":65711,"style":173},[71],[15,65713,34430],{"className":65714},[177],[15,65716],{"className":65717,"style":173},[71],[15,65719,65721,65724,65727,65730,65733,65736,65739,65742,65745],{"className":65720},[57],[15,65722],{"className":65723,"style":154},[61],[15,65725,61618],{"className":65726,"style":23025},[66,67],[15,65728,126],{"className":65729},[169],[15,65731],{"className":65732,"style":697},[71],[15,65734,11],{"className":65735},[66,67],[15,65737,120],{"className":65738},[162],[15,65740,61618],{"className":65741,"style":23025},[66,67],[15,65743,126],{"className":65744},[169],[15,65746,3573],{"className":65747},[66],[11,65749,65750,65751,65794],{},"但这个省略只在「给定模型」内部成立。一旦要比较不同模型，",[15,65752,65754,65773],{"className":65753,"translate":19},[18],[15,65755,65757],{"className":65756},[23],[25,65758,65759],{"xmlns":27},[29,65760,65761,65771],{},[32,65762,65763,65765,65767,65769],{},[35,65764,11],{},[39,65766,120],{"stretchy":119},[35,65768,32458],{},[39,65770,126],{"stretchy":119},[45,65772,65373],{"encoding":47},[15,65774,65776],{"className":65775,"ariaHidden":53},[52],[15,65777,65779,65782,65785,65788,65791],{"className":65778},[57],[15,65780],{"className":65781,"style":154},[61],[15,65783,11],{"className":65784},[66,67],[15,65786,120],{"className":65787},[162],[15,65789,32458],{"className":65790,"style":23025},[66,67],[15,65792,126],{"className":65793},[169]," 就是主角，第 6.5 节会回到这一点。",[11,65796,65797,65798,65826],{},"还要强调一点：贝叶斯公式本身只是概率论的一条定理，接受它不需要接受任何统计哲学。真正带有立场的是下一步——把 ",[15,65799,65801,65814],{"className":65800,"translate":19},[18],[15,65802,65804],{"className":65803},[23],[25,65805,65806],{"xmlns":27},[29,65807,65808,65812],{},[32,65809,65810],{},[35,65811,61618],{},[45,65813,64805],{"encoding":47},[15,65815,65817],{"className":65816,"ariaHidden":53},[52],[15,65818,65820,65823],{"className":65819},[57],[15,65821],{"className":65822,"style":840},[61],[15,65824,61618],{"className":65825,"style":23025},[66,67]," 看成随机变量，并为它指定先验。",[1904,65828,65830],{"id":65829},"_23-betabernoulli一个可以手算的更新","2.3 Beta–Bernoulli：一个可以手算的更新",[11,65832,1910,65833,65907],{},[15,65834,65836,65862],{"className":65835,"translate":19},[18],[15,65837,65839],{"className":65838},[23],[25,65840,65841],{"xmlns":27},[29,65842,65843,65859],{},[32,65844,65845,65847,65849,65851,65853,65855,65857],{},[35,65846,61618],{},[39,65848,1187],{},[39,65850,1194],{"stretchy":119},[1209,65852,1804],{},[39,65854,335],{"separator":53},[1209,65856,1211],{},[39,65858,1266],{"stretchy":119},[45,65860,65861],{"encoding":47},"\\theta\\in[0,1]",[15,65863,65865,65883],{"className":65864,"ariaHidden":53},[52],[15,65866,65868,65871,65874,65877,65880],{"className":65867},[57],[15,65869],{"className":65870,"style":9298},[61],[15,65872,61618],{"className":65873,"style":23025},[66,67],[15,65875],{"className":65876,"style":173},[71],[15,65878,1187],{"className":65879},[177],[15,65881],{"className":65882,"style":173},[71],[15,65884,65886,65889,65892,65895,65898,65901,65904],{"className":65885},[57],[15,65887],{"className":65888,"style":154},[61],[15,65890,1194],{"className":65891},[162],[15,65893,1804],{"className":65894},[66],[15,65896,335],{"className":65897},[422],[15,65899],{"className":65900,"style":697},[71],[15,65902,1211],{"className":65903},[66],[15,65905,1266],{"className":65906},[169]," 是一枚硬币正面向上的概率，先验取 Beta 分布",[15,65909,65911],{"className":65910,"translate":19},[294],[15,65912,65914,65988],{"className":65913,"translate":19},[18],[15,65915,65917],{"className":65916},[23],[25,65918,65919],{"xmlns":27,"display":303},[29,65920,65921,65985],{},[32,65922,65923,65925,65927,65929,65931,65933,65983],{},[35,65924,11],{},[39,65926,120],{"stretchy":119},[35,65928,61618],{},[39,65930,126],{"stretchy":119},[39,65932,129],{},[6467,65934,65935,65969],{},[32,65936,65937,65949,65951,65953,65955,65957],{},[131,65938,65939,65941],{},[35,65940,61618],{},[32,65942,65943,65945,65947],{},[35,65944,8390],{},[39,65946,6602],{},[1209,65948,1211],{},[39,65950,120],{"stretchy":119},[1209,65952,1211],{},[39,65954,6602],{},[35,65956,61618],{},[131,65958,65959,65961],{},[39,65960,126],{"stretchy":119},[32,65962,65963,65965,65967],{},[35,65964,17087],{},[39,65966,6602],{},[1209,65968,1211],{},[32,65970,65971,65973,65975,65977,65979,65981],{},[35,65972,63972],{},[39,65974,120],{"stretchy":119},[35,65976,8390],{},[39,65978,335],{"separator":53},[35,65980,17087],{},[39,65982,126],{"stretchy":119},[35,65984,3573],{"mathvariant":138},[45,65986,65987],{"encoding":47},"p(\\theta)=\\frac{\\theta^{\\alpha-1}(1-\\theta)^{\\beta-1}}{B(\\alpha,\\beta)}.",[15,65989,65991,66018],{"className":65990,"ariaHidden":53},[52],[15,65992,65994,65997,66000,66003,66006,66009,66012,66015],{"className":65993},[57],[15,65995],{"className":65996,"style":154},[61],[15,65998,11],{"className":65999},[66,67],[15,66001,120],{"className":66002},[162],[15,66004,61618],{"className":66005,"style":23025},[66,67],[15,66007,126],{"className":66008},[169],[15,66010],{"className":66011,"style":173},[71],[15,66013,129],{"className":66014},[177],[15,66016],{"className":66017,"style":173},[71],[15,66019,66021,66025,66197],{"className":66020},[57],[15,66022],{"className":66023,"style":66024},[61],"height:2.4621em;vertical-align:-0.936em;",[15,66026,66028,66031,66194],{"className":66027},[66],[15,66029],{"className":66030},[162,23417],[15,66032,66034],{"className":66033},[6467],[15,66035,66037,66186],{"className":66036},[202,1090],[15,66038,66040,66183],{"className":66039},[206],[15,66041,66044,66073,66081],{"className":66042,"style":66043},[210],"height:1.5261em;",[15,66045,66046,66049],{"style":23433},[15,66047],{"className":66048,"style":1368},[217],[15,66050,66052,66055,66058,66061,66064,66067,66070],{"className":66051},[66],[15,66053,63972],{"className":66054,"style":64059},[66,67],[15,66056,120],{"className":66057},[162],[15,66059,8390],{"className":66060,"style":8560},[66,67],[15,66062,335],{"className":66063},[422],[15,66065],{"className":66066,"style":697},[71],[15,66068,17087],{"className":66069,"style":17240},[66,67],[15,66071,126],{"className":66072},[169],[15,66074,66075,66078],{"style":23445},[15,66076],{"className":66077,"style":1368},[217],[15,66079],{"className":66080,"style":4440},[23452],[15,66082,66083,66086],{"style":23455},[15,66084],{"className":66085,"style":1368},[217],[15,66087,66089,66127,66130,66133,66136,66139,66142,66145],{"className":66088},[66],[15,66090,66092,66095],{"className":66091},[66],[15,66093,61618],{"className":66094,"style":23025},[66,67],[15,66096,66098],{"className":66097},[198],[15,66099,66101],{"className":66100},[202],[15,66102,66104],{"className":66103},[206],[15,66105,66107],{"className":66106,"style":6740},[210],[15,66108,66109,66112],{"style":213},[15,66110],{"className":66111,"style":218},[217],[15,66113,66115],{"className":66114},[222,223,224,225],[15,66116,66118,66121,66124],{"className":66117},[66,225],[15,66119,8390],{"className":66120,"style":8560},[66,67,225],[15,66122,6602],{"className":66123},[76,225],[15,66125,1211],{"className":66126},[66,225],[15,66128,120],{"className":66129},[162],[15,66131,1211],{"className":66132},[66],[15,66134],{"className":66135,"style":72},[71],[15,66137,6602],{"className":66138},[76],[15,66140],{"className":66141,"style":72},[71],[15,66143,61618],{"className":66144,"style":23025},[66,67],[15,66146,66148,66151],{"className":66147},[169],[15,66149,126],{"className":66150},[169],[15,66152,66154],{"className":66153},[198],[15,66155,66157],{"className":66156},[202],[15,66158,66160],{"className":66159},[206],[15,66161,66163],{"className":66162,"style":187},[210],[15,66164,66165,66168],{"style":213},[15,66166],{"className":66167,"style":218},[217],[15,66169,66171],{"className":66170},[222,223,224,225],[15,66172,66174,66177,66180],{"className":66173},[66,225],[15,66175,17087],{"className":66176,"style":17240},[66,67,225],[15,66178,6602],{"className":66179},[76,225],[15,66181,1211],{"className":66182},[66,225],[15,66184,1114],{"className":66185},[1113],[15,66187,66189],{"className":66188},[206],[15,66190,66192],{"className":66191,"style":41282},[210],[15,66193],{},[15,66195],{"className":66196},[169,23417],[15,66198,3573],{"className":66199},[66],[11,66201,66202,66203,66231,66232,66260],{},"观测到 ",[15,66204,66206,66219],{"className":66205,"translate":19},[18],[15,66207,66209],{"className":66208},[23],[25,66210,66211],{"xmlns":27},[29,66212,66213,66217],{},[32,66214,66215],{},[35,66216,37],{},[45,66218,37],{"encoding":47},[15,66220,66222],{"className":66221,"ariaHidden":53},[52],[15,66223,66225,66228],{"className":66224},[57],[15,66226],{"className":66227,"style":86},[61],[15,66229,37],{"className":66230},[66,67]," 次投掷中 ",[15,66233,66235,66248],{"className":66234,"translate":19},[18],[15,66236,66238],{"className":66237},[23],[25,66239,66240],{"xmlns":27},[29,66241,66242,66246],{},[32,66243,66244],{},[35,66245,621],{},[45,66247,621],{"encoding":47},[15,66249,66251],{"className":66250,"ariaHidden":53},[52],[15,66252,66254,66257],{"className":66253},[57],[15,66255],{"className":66256,"style":840},[61],[15,66258,621],{"className":66259,"style":684},[66,67]," 次正面，似然为",[15,66262,66264],{"className":66263,"translate":19},[294],[15,66265,66267,66321],{"className":66266,"translate":19},[18],[15,66268,66270],{"className":66269},[23],[25,66271,66272],{"xmlns":27,"display":303},[29,66273,66274,66318],{},[32,66275,66276,66278,66280,66282,66284,66286,66288,66290,66296,66298,66300,66302,66304,66316],{},[35,66277,11],{},[39,66279,120],{"stretchy":119},[35,66281,32458],{},[39,66283,34430],{},[35,66285,61618],{},[39,66287,126],{"stretchy":119},[39,66289,129],{},[131,66291,66292,66294],{},[35,66293,61618],{},[35,66295,621],{},[39,66297,120],{"stretchy":119},[1209,66299,1211],{},[39,66301,6602],{},[35,66303,61618],{},[131,66305,66306,66308],{},[39,66307,126],{"stretchy":119},[32,66309,66310,66312,66314],{},[35,66311,37],{},[39,66313,6602],{},[35,66315,621],{},[35,66317,3573],{"mathvariant":138},[45,66319,66320],{"encoding":47},"p(D\\mid\\theta)=\\theta^k(1-\\theta)^{n-k}.",[15,66322,66324,66348,66369,66419],{"className":66323,"ariaHidden":53},[52],[15,66325,66327,66330,66333,66336,66339,66342,66345],{"className":66326},[57],[15,66328],{"className":66329,"style":154},[61],[15,66331,11],{"className":66332},[66,67],[15,66334,120],{"className":66335},[162],[15,66337,32458],{"className":66338,"style":23025},[66,67],[15,66340],{"className":66341,"style":173},[71],[15,66343,34430],{"className":66344},[177],[15,66346],{"className":66347,"style":173},[71],[15,66349,66351,66354,66357,66360,66363,66366],{"className":66350},[57],[15,66352],{"className":66353,"style":154},[61],[15,66355,61618],{"className":66356,"style":23025},[66,67],[15,66358,126],{"className":66359},[169],[15,66361],{"className":66362,"style":173},[71],[15,66364,129],{"className":66365},[177],[15,66367],{"className":66368,"style":173},[71],[15,66370,66372,66375,66404,66407,66410,66413,66416],{"className":66371},[57],[15,66373],{"className":66374,"style":375},[61],[15,66376,66378,66381],{"className":66377},[66],[15,66379,61618],{"className":66380,"style":23025},[66,67],[15,66382,66384],{"className":66383},[198],[15,66385,66387],{"className":66386},[202],[15,66388,66390],{"className":66389},[206],[15,66391,66393],{"className":66392,"style":394},[210],[15,66394,66395,66398],{"style":397},[15,66396],{"className":66397,"style":218},[217],[15,66399,66401],{"className":66400},[222,223,224,225],[15,66402,621],{"className":66403,"style":684},[66,67,225],[15,66405,120],{"className":66406},[162],[15,66408,1211],{"className":66409},[66],[15,66411],{"className":66412,"style":72},[71],[15,66414,6602],{"className":66415},[76],[15,66417],{"className":66418,"style":72},[71],[15,66420,66422,66425,66428,66466],{"className":66421},[57],[15,66423],{"className":66424,"style":375},[61],[15,66426,61618],{"className":66427,"style":23025},[66,67],[15,66429,66431,66434],{"className":66430},[169],[15,66432,126],{"className":66433},[169],[15,66435,66437],{"className":66436},[198],[15,66438,66440],{"className":66439},[202],[15,66441,66443],{"className":66442},[206],[15,66444,66446],{"className":66445,"style":394},[210],[15,66447,66448,66451],{"style":397},[15,66449],{"className":66450,"style":218},[217],[15,66452,66454],{"className":66453},[222,223,224,225],[15,66455,66457,66460,66463],{"className":66456},[66,225],[15,66458,37],{"className":66459},[66,67,225],[15,66461,6602],{"className":66462},[76,225],[15,66464,621],{"className":66465,"style":684},[66,67,225],[15,66467,3573],{"className":66468},[66],[11,66470,66471],{},"两者相乘，后验的核为",[15,66473,66475],{"className":66474,"translate":19},[294],[15,66476,66478,66550],{"className":66477,"translate":19},[18],[15,66479,66481],{"className":66480},[23],[25,66482,66483],{"xmlns":27,"display":303},[29,66484,66485,66547],{},[32,66486,66487,66489,66491,66493,66495,66497,66499,66501,66517,66519,66521,66523,66525,66545],{},[35,66488,11],{},[39,66490,120],{"stretchy":119},[35,66492,61618],{},[39,66494,34430],{},[35,66496,32458],{},[39,66498,126],{"stretchy":119},[39,66500,65618],{},[131,66502,66503,66505],{},[35,66504,61618],{},[32,66506,66507,66509,66511,66513,66515],{},[35,66508,8390],{},[39,66510,1245],{},[35,66512,621],{},[39,66514,6602],{},[1209,66516,1211],{},[39,66518,120],{"stretchy":119},[1209,66520,1211],{},[39,66522,6602],{},[35,66524,61618],{},[131,66526,66527,66529],{},[39,66528,126],{"stretchy":119},[32,66530,66531,66533,66535,66537,66539,66541,66543],{},[35,66532,17087],{},[39,66534,1245],{},[35,66536,37],{},[39,66538,6602],{},[35,66540,621],{},[39,66542,6602],{},[1209,66544,1211],{},[39,66546,335],{"separator":53},[45,66548,66549],{"encoding":47},"p(\\theta\\mid D)\\propto \\theta^{\\alpha+k-1}(1-\\theta)^{\\beta+n-k-1},",[15,66551,66553,66577,66598,66663],{"className":66552,"ariaHidden":53},[52],[15,66554,66556,66559,66562,66565,66568,66571,66574],{"className":66555},[57],[15,66557],{"className":66558,"style":154},[61],[15,66560,11],{"className":66561},[66,67],[15,66563,120],{"className":66564},[162],[15,66566,61618],{"className":66567,"style":23025},[66,67],[15,66569],{"className":66570,"style":173},[71],[15,66572,34430],{"className":66573},[177],[15,66575],{"className":66576,"style":173},[71],[15,66578,66580,66583,66586,66589,66592,66595],{"className":66579},[57],[15,66581],{"className":66582,"style":154},[61],[15,66584,32458],{"className":66585,"style":23025},[66,67],[15,66587,126],{"className":66588},[169],[15,66590],{"className":66591,"style":173},[71],[15,66593,65618],{"className":66594},[177],[15,66596],{"className":66597,"style":173},[71],[15,66599,66601,66604,66648,66651,66654,66657,66660],{"className":66600},[57],[15,66602],{"className":66603,"style":375},[61],[15,66605,66607,66610],{"className":66606},[66],[15,66608,61618],{"className":66609,"style":23025},[66,67],[15,66611,66613],{"className":66612},[198],[15,66614,66616],{"className":66615},[202],[15,66617,66619],{"className":66618},[206],[15,66620,66622],{"className":66621,"style":394},[210],[15,66623,66624,66627],{"style":397},[15,66625],{"className":66626,"style":218},[217],[15,66628,66630],{"className":66629},[222,223,224,225],[15,66631,66633,66636,66639,66642,66645],{"className":66632},[66,225],[15,66634,8390],{"className":66635,"style":8560},[66,67,225],[15,66637,1245],{"className":66638},[76,225],[15,66640,621],{"className":66641,"style":684},[66,67,225],[15,66643,6602],{"className":66644},[76,225],[15,66646,1211],{"className":66647},[66,225],[15,66649,120],{"className":66650},[162],[15,66652,1211],{"className":66653},[66],[15,66655],{"className":66656,"style":72},[71],[15,66658,6602],{"className":66659},[76],[15,66661],{"className":66662,"style":72},[71],[15,66664,66666,66669,66672,66722],{"className":66665},[57],[15,66667],{"className":66668,"style":375},[61],[15,66670,61618],{"className":66671,"style":23025},[66,67],[15,66673,66675,66678],{"className":66674},[169],[15,66676,126],{"className":66677},[169],[15,66679,66681],{"className":66680},[198],[15,66682,66684],{"className":66683},[202],[15,66685,66687],{"className":66686},[206],[15,66688,66690],{"className":66689,"style":394},[210],[15,66691,66692,66695],{"style":397},[15,66693],{"className":66694,"style":218},[217],[15,66696,66698],{"className":66697},[222,223,224,225],[15,66699,66701,66704,66707,66710,66713,66716,66719],{"className":66700},[66,225],[15,66702,17087],{"className":66703,"style":17240},[66,67,225],[15,66705,1245],{"className":66706},[76,225],[15,66708,37],{"className":66709},[66,67,225],[15,66711,6602],{"className":66712},[76,225],[15,66714,621],{"className":66715,"style":684},[66,67,225],[15,66717,6602],{"className":66718},[76,225],[15,66720,1211],{"className":66721},[66,225],[15,66723,335],{"className":66724},[422],[11,66726,66727,66728,66868,66869,66872],{},"这正是 ",[15,66729,66731,66777],{"className":66730,"translate":19},[18],[15,66732,66734],{"className":66733},[23],[25,66735,66736],{"xmlns":27},[29,66737,66738,66774],{},[32,66739,66740,66750,66752,66754,66756,66758,66760,66762,66764,66766,66768,66770,66772],{},[32,66741,66742,66744,66746,66748],{},[35,66743,63972],{"mathvariant":138},[35,66745,48383],{"mathvariant":138},[35,66747,9212],{"mathvariant":138},[35,66749,4556],{"mathvariant":138},[39,66751,120],{"stretchy":119},[35,66753,8390],{},[39,66755,1245],{},[35,66757,621],{},[39,66759,335],{"separator":53},[1274,66761,24233],{},[35,66763,17087],{},[39,66765,1245],{},[35,66767,37],{},[39,66769,6602],{},[35,66771,621],{},[39,66773,126],{"stretchy":119},[45,66775,66776],{"encoding":47},"\\mathrm{Beta}(\\alpha+k,\\ \\beta+n-k)",[15,66778,66780,66808,66838,66856],{"className":66779,"ariaHidden":53},[52],[15,66781,66783,66786,66793,66796,66799,66802,66805],{"className":66782},[57],[15,66784],{"className":66785,"style":154},[61],[15,66787,66789],{"className":66788},[66],[15,66790,66792],{"className":66791},[66,4429],"Beta",[15,66794,120],{"className":66795},[162],[15,66797,8390],{"className":66798,"style":8560},[66,67],[15,66800],{"className":66801,"style":72},[71],[15,66803,1245],{"className":66804},[76],[15,66806],{"className":66807,"style":72},[71],[15,66809,66811,66814,66817,66820,66823,66826,66829,66832,66835],{"className":66810},[57],[15,66812],{"className":66813,"style":3408},[61],[15,66815,621],{"className":66816,"style":684},[66,67],[15,66818,335],{"className":66819},[422],[15,66821,24233],{"className":66822},[71],[15,66824],{"className":66825,"style":697},[71],[15,66827,17087],{"className":66828,"style":17240},[66,67],[15,66830],{"className":66831,"style":72},[71],[15,66833,1245],{"className":66834},[76],[15,66836],{"className":66837,"style":72},[71],[15,66839,66841,66844,66847,66850,66853],{"className":66840},[57],[15,66842],{"className":66843,"style":62},[61],[15,66845,37],{"className":66846},[66,67],[15,66848],{"className":66849,"style":72},[71],[15,66851,6602],{"className":66852},[76],[15,66854],{"className":66855,"style":72},[71],[15,66857,66859,66862,66865],{"className":66858},[57],[15,66860],{"className":66861,"style":154},[61],[15,66863,621],{"className":66864,"style":684},[66,67],[15,66866,126],{"className":66867},[169],"。归一化常数不用算，因为 Beta 分布的形式已经把它确定了。这就是",[2040,66870,66871],{},"共轭先验","的好处：后验与先验属于同一个分布族，更新只需要改两个参数。",[11,66874,66875],{},"后验均值是",[15,66877,66879],{"className":66878,"translate":19},[294],[15,66880,66882,66932],{"className":66881,"translate":19},[18],[15,66883,66885],{"className":66884},[23],[25,66886,66887],{"xmlns":27,"display":303},[29,66888,66889,66929],{},[32,66890,66891,66893,66895,66897,66899,66901,66903,66905,66927],{},[35,66892,62381],{"mathvariant":454},[39,66894,1194],{"stretchy":119},[35,66896,61618],{},[39,66898,34430],{},[35,66900,32458],{},[39,66902,1266],{"stretchy":119},[39,66904,129],{},[6467,66906,66907,66915],{},[32,66908,66909,66911,66913],{},[35,66910,8390],{},[39,66912,1245],{},[35,66914,621],{},[32,66916,66917,66919,66921,66923,66925],{},[35,66918,8390],{},[39,66920,1245],{},[35,66922,17087],{},[39,66924,1245],{},[35,66926,37],{},[39,66928,335],{"separator":53},[45,66930,66931],{"encoding":47},"\\mathbb E[\\theta\\mid D]=\\frac{\\alpha+k}{\\alpha+\\beta+n},",[15,66933,66935,66959,66980],{"className":66934,"ariaHidden":53},[52],[15,66936,66938,66941,66944,66947,66950,66953,66956],{"className":66937},[57],[15,66939],{"className":66940,"style":154},[61],[15,66942,62381],{"className":66943},[66,517],[15,66945,1194],{"className":66946},[162],[15,66948,61618],{"className":66949,"style":23025},[66,67],[15,66951],{"className":66952,"style":173},[71],[15,66954,34430],{"className":66955},[177],[15,66957],{"className":66958,"style":173},[71],[15,66960,66962,66965,66968,66971,66974,66977],{"className":66961},[57],[15,66963],{"className":66964,"style":154},[61],[15,66966,32458],{"className":66967,"style":23025},[66,67],[15,66969,1266],{"className":66970},[169],[15,66972],{"className":66973,"style":173},[71],[15,66975,129],{"className":66976},[177],[15,66978],{"className":66979,"style":173},[71],[15,66981,66983,66987,67086],{"className":66982},[57],[15,66984],{"className":66985,"style":66986},[61],"height:2.2519em;vertical-align:-0.8804em;",[15,66988,66990,66993,67083],{"className":66989},[66],[15,66991],{"className":66992},[162,23417],[15,66994,66996],{"className":66995},[6467],[15,66997,66999,67074],{"className":66998},[202,1090],[15,67000,67002,67071],{"className":67001},[206],[15,67003,67005,67040,67048],{"className":67004,"style":27435},[210],[15,67006,67007,67010],{"style":23433},[15,67008],{"className":67009,"style":1368},[217],[15,67011,67013,67016,67019,67022,67025,67028,67031,67034,67037],{"className":67012},[66],[15,67014,8390],{"className":67015,"style":8560},[66,67],[15,67017],{"className":67018,"style":72},[71],[15,67020,1245],{"className":67021},[76],[15,67023],{"className":67024,"style":72},[71],[15,67026,17087],{"className":67027,"style":17240},[66,67],[15,67029],{"className":67030,"style":72},[71],[15,67032,1245],{"className":67033},[76],[15,67035],{"className":67036,"style":72},[71],[15,67038,37],{"className":67039},[66,67],[15,67041,67042,67045],{"style":23445},[15,67043],{"className":67044,"style":1368},[217],[15,67046],{"className":67047,"style":4440},[23452],[15,67049,67050,67053],{"style":23455},[15,67051],{"className":67052,"style":1368},[217],[15,67054,67056,67059,67062,67065,67068],{"className":67055},[66],[15,67057,8390],{"className":67058,"style":8560},[66,67],[15,67060],{"className":67061,"style":72},[71],[15,67063,1245],{"className":67064},[76],[15,67066],{"className":67067,"style":72},[71],[15,67069,621],{"className":67070,"style":684},[66,67],[15,67072,1114],{"className":67073},[1113],[15,67075,67077],{"className":67076},[206],[15,67078,67081],{"className":67079,"style":67080},[210],"height:0.8804em;",[15,67082],{},[15,67084],{"className":67085},[169,23417],[15,67087,335],{"className":67088},[422],[11,67090,67091],{},"把它改写成加权平均：",[15,67093,67095],{"className":67094,"translate":19},[294],[15,67096,67098,67198],{"className":67097,"translate":19},[18],[15,67099,67101],{"className":67100},[23],[25,67102,67103],{"xmlns":27,"display":303},[29,67104,67105,67195],{},[32,67106,67107,67129,67131,67153,67155,67167,67169,67185,67187,67193],{},[6467,67108,67109,67117],{},[32,67110,67111,67113,67115],{},[35,67112,8390],{},[39,67114,1245],{},[35,67116,621],{},[32,67118,67119,67121,67123,67125,67127],{},[35,67120,8390],{},[39,67122,1245],{},[35,67124,17087],{},[39,67126,1245],{},[35,67128,37],{},[39,67130,129],{},[6467,67132,67133,67141],{},[32,67134,67135,67137,67139],{},[35,67136,8390],{},[39,67138,1245],{},[35,67140,17087],{},[32,67142,67143,67145,67147,67149,67151],{},[35,67144,8390],{},[39,67146,1245],{},[35,67148,17087],{},[39,67150,1245],{},[35,67152,37],{},[39,67154,3268],{},[6467,67156,67157,67159],{},[35,67158,8390],{},[32,67160,67161,67163,67165],{},[35,67162,8390],{},[39,67164,1245],{},[35,67166,17087],{},[39,67168,1245],{},[6467,67170,67171,67173],{},[35,67172,37],{},[32,67174,67175,67177,67179,67181,67183],{},[35,67176,8390],{},[39,67178,1245],{},[35,67180,17087],{},[39,67182,1245],{},[35,67184,37],{},[39,67186,3268],{},[6467,67188,67189,67191],{},[35,67190,621],{},[35,67192,37],{},[35,67194,3573],{"mathvariant":138},[45,67196,67197],{"encoding":47},"\\frac{\\alpha+k}{\\alpha+\\beta+n}\n=\n\\frac{\\alpha+\\beta}{\\alpha+\\beta+n}\\cdot\\frac{\\alpha}{\\alpha+\\beta}\n+\n\\frac{n}{\\alpha+\\beta+n}\\cdot\\frac{k}{n}.",[15,67199,67201,67314,67427,67517,67618],{"className":67200,"ariaHidden":53},[52],[15,67202,67204,67207,67305,67308,67311],{"className":67203},[57],[15,67205],{"className":67206,"style":66986},[61],[15,67208,67210,67213,67302],{"className":67209},[66],[15,67211],{"className":67212},[162,23417],[15,67214,67216],{"className":67215},[6467],[15,67217,67219,67294],{"className":67218},[202,1090],[15,67220,67222,67291],{"className":67221},[206],[15,67223,67225,67260,67268],{"className":67224,"style":27435},[210],[15,67226,67227,67230],{"style":23433},[15,67228],{"className":67229,"style":1368},[217],[15,67231,67233,67236,67239,67242,67245,67248,67251,67254,67257],{"className":67232},[66],[15,67234,8390],{"className":67235,"style":8560},[66,67],[15,67237],{"className":67238,"style":72},[71],[15,67240,1245],{"className":67241},[76],[15,67243],{"className":67244,"style":72},[71],[15,67246,17087],{"className":67247,"style":17240},[66,67],[15,67249],{"className":67250,"style":72},[71],[15,67252,1245],{"className":67253},[76],[15,67255],{"className":67256,"style":72},[71],[15,67258,37],{"className":67259},[66,67],[15,67261,67262,67265],{"style":23445},[15,67263],{"className":67264,"style":1368},[217],[15,67266],{"className":67267,"style":4440},[23452],[15,67269,67270,67273],{"style":23455},[15,67271],{"className":67272,"style":1368},[217],[15,67274,67276,67279,67282,67285,67288],{"className":67275},[66],[15,67277,8390],{"className":67278,"style":8560},[66,67],[15,67280],{"className":67281,"style":72},[71],[15,67283,1245],{"className":67284},[76],[15,67286],{"className":67287,"style":72},[71],[15,67289,621],{"className":67290,"style":684},[66,67],[15,67292,1114],{"className":67293},[1113],[15,67295,67297],{"className":67296},[206],[15,67298,67300],{"className":67299,"style":67080},[210],[15,67301],{},[15,67303],{"className":67304},[169,23417],[15,67306],{"className":67307,"style":173},[71],[15,67309,129],{"className":67310},[177],[15,67312],{"className":67313,"style":173},[71],[15,67315,67317,67320,67418,67421,67424],{"className":67316},[57],[15,67318],{"className":67319,"style":66986},[61],[15,67321,67323,67326,67415],{"className":67322},[66],[15,67324],{"className":67325},[162,23417],[15,67327,67329],{"className":67328},[6467],[15,67330,67332,67407],{"className":67331},[202,1090],[15,67333,67335,67404],{"className":67334},[206],[15,67336,67338,67373,67381],{"className":67337,"style":27435},[210],[15,67339,67340,67343],{"style":23433},[15,67341],{"className":67342,"style":1368},[217],[15,67344,67346,67349,67352,67355,67358,67361,67364,67367,67370],{"className":67345},[66],[15,67347,8390],{"className":67348,"style":8560},[66,67],[15,67350],{"className":67351,"style":72},[71],[15,67353,1245],{"className":67354},[76],[15,67356],{"className":67357,"style":72},[71],[15,67359,17087],{"className":67360,"style":17240},[66,67],[15,67362],{"className":67363,"style":72},[71],[15,67365,1245],{"className":67366},[76],[15,67368],{"className":67369,"style":72},[71],[15,67371,37],{"className":67372},[66,67],[15,67374,67375,67378],{"style":23445},[15,67376],{"className":67377,"style":1368},[217],[15,67379],{"className":67380,"style":4440},[23452],[15,67382,67383,67386],{"style":23455},[15,67384],{"className":67385,"style":1368},[217],[15,67387,67389,67392,67395,67398,67401],{"className":67388},[66],[15,67390,8390],{"className":67391,"style":8560},[66,67],[15,67393],{"className":67394,"style":72},[71],[15,67396,1245],{"className":67397},[76],[15,67399],{"className":67400,"style":72},[71],[15,67402,17087],{"className":67403,"style":17240},[66,67],[15,67405,1114],{"className":67406},[1113],[15,67408,67410],{"className":67409},[206],[15,67411,67413],{"className":67412,"style":67080},[210],[15,67414],{},[15,67416],{"className":67417},[169,23417],[15,67419],{"className":67420,"style":72},[71],[15,67422,3268],{"className":67423},[76],[15,67425],{"className":67426,"style":72},[71],[15,67428,67430,67434,67508,67511,67514],{"className":67429},[57],[15,67431],{"className":67432,"style":67433},[61],"height:1.988em;vertical-align:-0.8804em;",[15,67435,67437,67440,67505],{"className":67436},[66],[15,67438],{"className":67439},[162,23417],[15,67441,67443],{"className":67442},[6467],[15,67444,67446,67497],{"className":67445},[202,1090],[15,67447,67449,67494],{"className":67448},[206],[15,67450,67452,67475,67483],{"className":67451,"style":39133},[210],[15,67453,67454,67457],{"style":23433},[15,67455],{"className":67456,"style":1368},[217],[15,67458,67460,67463,67466,67469,67472],{"className":67459},[66],[15,67461,8390],{"className":67462,"style":8560},[66,67],[15,67464],{"className":67465,"style":72},[71],[15,67467,1245],{"className":67468},[76],[15,67470],{"className":67471,"style":72},[71],[15,67473,17087],{"className":67474,"style":17240},[66,67],[15,67476,67477,67480],{"style":23445},[15,67478],{"className":67479,"style":1368},[217],[15,67481],{"className":67482,"style":4440},[23452],[15,67484,67485,67488],{"style":23455},[15,67486],{"className":67487,"style":1368},[217],[15,67489,67491],{"className":67490},[66],[15,67492,8390],{"className":67493,"style":8560},[66,67],[15,67495,1114],{"className":67496},[1113],[15,67498,67500],{"className":67499},[206],[15,67501,67503],{"className":67502,"style":67080},[210],[15,67504],{},[15,67506],{"className":67507},[169,23417],[15,67509],{"className":67510,"style":72},[71],[15,67512,1245],{"className":67513},[76],[15,67515],{"className":67516,"style":72},[71],[15,67518,67520,67523,67609,67612,67615],{"className":67519},[57],[15,67521],{"className":67522,"style":67433},[61],[15,67524,67526,67529,67606],{"className":67525},[66],[15,67527],{"className":67528},[162,23417],[15,67530,67532],{"className"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48,67850,67852],{},[35,67845,8390],{},[39,67847,1245],{},[35,67849,17087],{},[39,67851,1245],{},[35,67853,37],{},[35,67855,3573],{"mathvariant":138},[45,67857,67858],{"encoding":47},"p(x_{\\text{new}}=1\\mid D)=\\mathbb E[\\theta\\mid D]=\\frac{\\alpha+k}{\\alpha+\\beta+n}.",[15,67860,67862,67929,67947,67968,67992,68013],{"className":67861,"ariaHidden":53},[52],[15,67863,67865,67868,67871,67874,67920,67923,67926],{"className":67864},[57],[15,67866],{"className":67867,"style":154},[61],[15,67869,11],{"className":67870},[66,67],[15,67872,120],{"className":67873},[162],[15,67875,67877,67880],{"className":67876},[66],[15,67878,123],{"className":67879},[66,67],[15,67881,67883],{"className":67882},[198],[15,67884,67886,67912],{"className":67885},[202,1090],[15,67887,67889,67909],{"className":67888},[206],[15,67890,67892],{"className":67891,"style":2198},[210],[15,67893,67894,67897],{"style":1584},[15,67895],{"className":67896,"style":218},[217],[15,67898,67900],{"className":67899},[222,223,224,225],[15,67901,67903],{"className":67902},[66,225],[15,67904,67906],{"className":67905},[66,1711,225],[15,67907,67805],{"className":67908},[66,225],[15,67910,1114],{"className":67911},[1113],[15,67913,67915],{"className":67914},[206],[15,67916,67918],{"className":67917,"style":1121},[210],[15,67919],{},[15,67921],{"className":67922,"style":173},[71],[15,67924,129],{"className":67925},[177],[15,67927],{"className":67928,"style":173},[71],[15,67930,67932,67935,67938,67941,67944],{"className":67931},[57],[15,67933],{"className":67934,"style":154},[61],[15,67936,1211],{"className":67937},[66],[15,67939],{"className":67940,"style":173},[71],[15,67942,34430],{"className":67943},[177],[15,67945],{"className":67946,"style":173},[71],[15,67948,67950,67953,67956,67959,67962,67965],{"className":67949},[57],[15,67951],{"className":67952,"style":154},[61],[15,67954,32458],{"className":67955,"style":23025},[66,67],[15,67957,126],{"className":67958},[169],[15,67960],{"className":67961,"style":173},[71],[15,67963,129],{"className":67964},[177],[15,67966],{"className":67967,"style":173},[71],[15,67969,67971,67974,67977,67980,67983,67986,67989],{"className":67970},[57],[15,67972],{"className":67973,"style":154},[61],[15,67975,62381],{"className":67976},[66,517],[15,67978,1194],{"className":67979},[162],[15,67981,61618],{"className":67982,"style":23025},[66,67],[15,67984],{"className":67985,"style":173},[71],[15,67987,34430],{"className":67988},[177],[15,67990],{"className":67991,"style":173},[71],[15,67993,67995,67998,68001,68004,68007,68010],{"className":67994},[57],[15,67996],{"className":67997,"style":154},[61],[15,67999,32458],{"className":68000,"style":23025},[66,67],[15,68002,1266],{"className":68003},[169],[15,68005],{"className":68006,"style":173},[71],[15,68008,129],{"className":68009},[177],[15,68011],{"className":68012,"style":173},[71],[15,68014,68016,68019,68117],{"className":68015},[57],[15,68017],{"className":68018,"style":66986},[61],[15,68020,68022,68025,68114],{"className":68021},[66],[15,68023],{"className":68024},[162,23417],[15,68026,68028],{"className":68027},[6467],[15,68029,68031,68106],{"className":68030},[202,1090],[15,68032,68034,68103],{"className":68033},[206],[15,68035,68037,68072,68080],{"className":68036,"style":27435},[210],[15,68038,68039,68042],{"style":23433},[15,68040],{"className":68041,"style":1368},[217],[15,68043,68045,68048,68051,68054,68057,68060,68063,68066,68069],{"className":68044},[66],[15,68046,8390],{"className":68047,"style":8560},[66,67],[15,68049],{"className":68050,"style":72},[71],[15,68052,1245],{"className":68053},[76],[15,68055],{"className":68056,"style":72},[71],[15,68058,17087],{"className":68059,"style":17240},[66,67],[15,68061],{"className":68062,"style":72},[71],[15,68064,1245],{"className":68065},[76],[15,68067],{"className":68068,"style":72},[71],[15,68070,37],{"className":68071},[66,67],[15,68073,68074,68077],{"style":23445},[15,68075],{"className":68076,"style":1368},[217],[15,68078],{"className":68079,"style":4440},[23452],[15,68081,68082,68085],{"style":23455},[15,68083],{"className":68084,"style":1368},[217],[15,68086,68088,68091,68094,68097,68100],{"className":68087},[66],[15,68089,8390],{"className":68090,"style":8560},[66,67],[15,68092],{"className":68093,"style":72},[71],[15,68095,1245],{"className":68096},[76],[15,68098],{"className":68099,"style":72},[71],[15,68101,621],{"className":68102,"style":684},[66,67],[15,68104,1114],{"className":68105},[1113],[15,68107,68109],{"className":68108},[206],[15,68110,68112],{"className":68111,"style":67080},[210],[15,68113],{},[15,68115],{"className":68116},[169,23417],[15,68118,3573],{"className":68119},[66],[11,68121,68122,68123,68151],{},"这里恰好等于后验均值，是因为伯努利概率是 ",[15,68124,68126,68139],{"className":68125,"translate":19},[18],[15,68127,68129],{"className":68128},[23],[25,68130,68131],{"xmlns":27},[29,68132,68133,68137],{},[32,68134,68135],{},[35,68136,61618],{},[45,68138,64805],{"encoding":47},[15,68140,68142],{"className":68141,"ariaHidden":53},[52],[15,68143,68145,68148],{"className":68144},[57],[15,68146],{"className":68147,"style":840},[61],[15,68149,61618],{"className":68150,"style":23025},[66,67]," 的线性函数。对一般模型，后验预测分布并不等于「把后验均值代进去」，这个区别在第 6 节会反复出现。",[1904,68153,68155],{"id":68154},"_24-先验是什么不是什么","2.4 先验是什么，不是什么",[11,68157,68158],{},"先验常被批评为「主观」。这个批评有一半是对的：先验确实是一个假设，而且不同先验会给出不同后验。但「主观」不等于「任意」。先验可以被数据稀释：在正则条件下，后验的渐近行为与先验无关（见 6.4）。先验也可以层级化：给超参数再加一层先验，让数据同时决定参数与超参数。先验还可以被检验：通过后验预测检验、敏感性分析，看结论是否依赖于某个特定选择。",[11,68160,68161],{},"更重要的是，非贝叶斯方法同样有假设。模型类、正则项、损失函数、优化算法，都是假设，只是它们不以「先验」的名字出现。贝叶斯的优点不是没有假设，而是把假设写在了明处，并且提供了一个用概率语言修改假设的机制。",[11,68163,68164,68165,68168],{},"第 7.8 节会给出一个更强的说法：在相当一般的条件下，所有",[2040,68166,68167],{},"可容许","（admissible）的决策规则都可以看成某个先验（或先验序列）下的贝叶斯规则。换句话说，贝叶斯规则不是一个「流派」，而是可容许决策规则的一般形式。这个结论把「为什么要用先验」从个人偏好提升到了决策论的结构性事实。",[95,68170,68172],{"id":68171},"_3-从后验到行动","3. 从后验到行动",[1904,68174,68176],{"id":68175},"_31-条件风险与逐点最优","3.1 条件风险与逐点最优",[11,68178,68179,68180,68246],{},"有了后验 ",[15,68181,68183,68207],{"className":68182,"translate":19},[18],[15,68184,68186],{"className":68185},[23],[25,68187,68188],{"xmlns":27},[29,68189,68190,68204],{},[32,68191,68192,68194,68196,68198,68200,68202],{},[35,68193,11],{},[39,68195,120],{"stretchy":119},[35,68197,1224],{},[39,68199,34430],{},[35,68201,123],{},[39,68203,126],{"stretchy":119},[45,68205,68206],{"encoding":47},"p(y\\mid x)",[15,68208,68210,68234],{"className":68209,"ariaHidden":53},[52],[15,68211,68213,68216,68219,68222,68225,68228,68231],{"className":68212},[57],[15,68214],{"className":68215,"style":154},[61],[15,68217,11],{"className":68218},[66,67],[15,68220,120],{"className":68221},[162],[15,68223,1224],{"className":68224,"style":1515},[66,67],[15,68226],{"className":68227,"style":173},[71],[15,68229,34430],{"className":68230},[177],[15,68232],{"className":68233,"style":173},[71],[15,68235,68237,68240,68243],{"className":68236},[57],[15,68238],{"className":68239,"style":154},[61],[15,68241,123],{"className":68242},[66,67],[15,68244,126],{"className":68245},[169],"，决策问题就变成了一个逐点的优化问题。",[3054,68248,68249,68315,68627],{},[11,68250,68251,68254,68255,68314],{},[2040,68252,68253],{},"定义 3.1（贝叶斯决策规则）"," 若决策规则 ",[15,68256,68258,68276],{"className":68257,"translate":19},[18],[15,68259,68261],{"className":68260},[23],[25,68262,68263],{"xmlns":27},[29,68264,68265,68273],{},[32,68266,68267],{},[131,68268,68269,68271],{},[35,68270,3885],{},[39,68272,1162],{},[45,68274,68275],{"encoding":47},"\\delta^*",[15,68277,68279],{"className":68278,"ariaHidden":53},[52],[15,68280,68282,68285],{"className":68281},[57],[15,68283],{"className":68284,"style":840},[61],[15,68286,68288,68291],{"className":68287},[66],[15,68289,3885],{"className":68290,"style":3946},[66,67],[15,68292,68294],{"className":68293},[198],[15,68295,68297],{"className":68296},[202],[15,68298,68300],{"className":68299},[206],[15,68301,68303],{"className":68302,"style":16249},[210],[15,68304,68305,68308],{"style":213},[15,68306],{"className":68307,"style":218},[217],[15,68309,68311],{"className":68310},[222,223,224,225],[15,68312,1162],{"className":68313},[76,225]," 满足",[15,68316,68318],{"className":68317,"translate":19},[294],[15,68319,68321,68404],{"className":68320,"translate":19},[18],[15,68322,68324],{"className":68323},[23],[25,68325,68326],{"xmlns":27,"display":303},[29,68327,68328,68401],{},[32,68329,68330,68336,68338,68340,68342,68344,68346,68348,68364,68366,68368,68370,68372,68374,68376,68378,68380,68382,68384,68386,68388,68390,68392,68394,68397,68399],{},[131,68331,68332,68334],{},[35,68333,3885],{},[39,68335,1162],{},[39,68337,120],{"stretchy":119},[35,68339,123],{},[39,68341,126],{"stretchy":119},[39,68343,1187],{},[35,68345,1167],{},[39,68347,1170],{},[1172,68349,68350,68356],{},[32,68351,68352,68354],{},[35,68353,1178],{},[39,68355,1170],{},[32,68357,68358,68360,68362],{},[35,68359,4556],{},[39,68361,1187],{},[35,68363,62005],{"mathvariant":445},[1274,68365,24233],{},[35,68367,62381],{"mathvariant":454},[39,68369,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,68371,1216],{},[39,68373,120],{"stretchy":119},[35,68375,61885],{},[39,68377,335],{"separator":53},[35,68379,4556],{},[39,68381,126],{"stretchy":119},[39,68383,34430],{},[35,68385,446],{},[39,68387,129],{},[35,68389,123],{},[39,68391,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[71,68393],{"width":3904},[1274,68395,68396],{},"对几乎所有的 ",[35,68398,123],{},[39,68400,335],{"separator":53},[45,68402,68403],{"encoding":47},"\\delta^*(x)\\in\\arg\\min_{a\\in\\mathcal A}\\ \\mathbb E\\big[L(Y,a)\\mid X=x\\big]\n\\qquad \\text{对几乎所有的 }x,",[15,68405,68407,68460,68574,68592],{"className":68406,"ariaHidden":53},[52],[15,68408,68410,68413,68442,68445,68448,68451,68454,68457],{"className":68409},[57],[15,68411],{"className":68412,"style":154},[61],[15,68414,68416,68419],{"className":68415},[66],[15,68417,3885],{"className":68418,"style":3946},[66,67],[15,68420,68422],{"className":68421},[198],[15,68423,68425],{"className":68424},[202],[15,68426,68428],{"className":68427},[206],[15,68429,68431],{"className":68430,"style":1308},[210],[15,68432,68433,68436],{"style":397},[15,68434],{"className":68435,"style":218},[217],[15,68437,68439],{"className":68438},[222,223,224,225],[15,68440,1162],{"className":68441},[76,225],[15,68443,120],{"className":68444},[162],[15,68446,123],{"className":68447},[66,67],[15,68449,126],{"className":68450},[169],[15,68452],{"className":68453,"style":173},[71],[15,68455,1187],{"className":68456},[177],[15,68458],{"className":68459,"style":173},[71],[15,68461,68463,68467,68472,68475,68529,68532,68535,68538,68544,68547,68550,68553,68556,68559,68562,68565,68568,68571],{"className":68462},[57],[15,68464],{"className":68465,"style":68466},[61],"height:1.6217em;vertical-align:-0.7717em;",[15,68468,1340,68470],{"className":68469},[1339],[15,68471,1344],{"style":1343},[15,68473],{"className":68474,"style":697},[71],[15,68476,68478],{"className":68477},[1339,1351],[15,68479,68481,68520],{"className":68480},[202,1090],[15,68482,68484,68517],{"className":68483},[206],[15,68485,68487,68507],{"className":68486,"style":1361},[210],[15,68488,68489,68492],{"style":24354},[15,68490],{"className":68491,"style":1368},[217],[15,68493,68495],{"className":68494},[222,223,224,225],[15,68496,68498,68501,68504],{"className":68497},[66,225],[15,68499,4556],{"className":68500},[66,67,225],[15,68502,1187],{"className":68503},[177,225],[15,68505,62005],{"className":68506},[66,494,225],[15,68508,68509,68512],{"style":1386},[15,68510],{"className":68511,"style":1368},[217],[15,68513,68514],{},[15,68515,1178],{"className":68516},[1339],[15,68518,1114],{"className":68519},[1113],[15,68521,68523],{"className":68522},[206],[15,68524,68527],{"className":68525,"style":68526},[210],"height:0.7717em;",[15,68528],{},[15,68530,24233],{"className":68531},[71],[15,68533],{"className":68534,"style":697},[71],[15,68536,62381],{"className":68537},[66,517],[15,68539,68541],{"className":68540},[66],[15,68542,1194],{"className":68543},[1422,1508],[15,68545,1216],{"className":68546},[66,67],[15,68548,120],{"className":68549},[162],[15,68551,61885],{"className":68552,"style":72},[66,67],[15,68554,335],{"className":68555},[422],[15,68557],{"className":68558,"style":697},[71],[15,68560,4556],{"className":68561},[66,67],[15,68563,126],{"className":68564},[169],[15,68566],{"className":68567,"style":173},[71],[15,68569,34430],{"className":68570},[177],[15,68572],{"className":68573,"style":173},[71],[15,68575,68577,68580,68583,68586,68589],{"className":68576},[57],[15,68578],{"className":68579,"style":490},[61],[15,68581,446],{"className":68582,"style":16497},[66,67],[15,68584],{"className":68585,"style":173},[71],[15,68587,129],{"className":68588},[177],[15,68590],{"className":68591,"style":173},[71],[15,68593,68595,68598,68601,68607,68610,68621,68624],{"className":68594},[57],[15,68596],{"className":68597,"style":4400},[61],[15,68599,123],{"className":68600},[66,67],[15,68602,68604],{"className":68603},[66],[15,68605,1266],{"className":68606},[1422,1508],[15,68608],{"className":68609,"style":4061},[71],[15,68611,68613,68618],{"className":68612},[66,1711],[15,68614,68617],{"className":68615},[66,68616],"cjk_fallback","对几乎所有的",[15,68619,24233],{"className":68620},[66],[15,68622,123],{"className":68623},[66,67],[15,68625,335],{"className":68626},[422],[11,68628,68629,68630,68688,68689,68692],{},"则称 ",[15,68631,68633,68650],{"className":68632,"translate":19},[18],[15,68634,68636],{"className":68635},[23],[25,68637,68638],{"xmlns":27},[29,68639,68640,68648],{},[32,68641,68642],{},[131,68643,68644,68646],{},[35,68645,3885],{},[39,68647,1162],{},[45,68649,68275],{"encoding":47},[15,68651,68653],{"className":68652,"ariaHidden":53},[52],[15,68654,68656,68659],{"className":68655},[57],[15,68657],{"className":68658,"style":840},[61],[15,68660,68662,68665],{"className":68661},[66],[15,68663,3885],{"className":68664,"style":3946},[66,67],[15,68666,68668],{"className":68667},[198],[15,68669,68671],{"className":68670},[202],[15,68672,68674],{"className":68673},[206],[15,68675,68677],{"className":68676,"style":16249},[210],[15,68678,68679,68682],{"style":213},[15,68680],{"className":68681,"style":218},[217],[15,68683,68685],{"className":68684},[222,223,224,225],[15,68686,1162],{"className":68687},[76,225]," 为",[2040,68690,68691],{},"贝叶斯决策规则","（Bayes decision rule）。",[3054,68694,68695,68730],{},[11,68696,68697,68700,68701,2403],{},[2040,68698,68699],{},"定理 3.1（逐点最优即全局最优）"," 对任意决策规则 ",[15,68702,68704,68718],{"className":68703,"translate":19},[18],[15,68705,68707],{"className":68706},[23],[25,68708,68709],{"xmlns":27},[29,68710,68711,68715],{},[32,68712,68713],{},[35,68714,3885],{},[45,68716,68717],{"encoding":47},"\\delta",[15,68719,68721],{"className":68720,"ariaHidden":53},[52],[15,68722,68724,68727],{"className":68723},[57],[15,68725],{"className":68726,"style":840},[61],[15,68728,3885],{"className":68729,"style":3946},[66,67],[15,68731,68733],{"className":68732,"translate":19},[294],[15,68734,68736,68816],{"className":68735,"translate":19},[18],[15,68737,68739],{"className":68738},[23],[25,68740,68741],{"xmlns":27,"display":303},[29,68742,68743,68813],{},[32,68744,68745,68747,68749,68751,68753,68755,68757,68759,68761,68763,68769,68771,68773,68779,68781,68797,68799,68801,68803,68805,68807,68809,68811],{},[35,68746,455],{},[39,68748,120],{"stretchy":119},[35,68750,3885],{},[39,68752,126],{"stretchy":119},[1274,68754,24233],{},[39,68756,1801],{},[1274,68758,24233],{},[35,68760,455],{},[39,68762,120],{"stretchy":119},[131,68764,68765,68767],{},[35,68766,3885],{},[39,68768,1162],{},[39,68770,126],{"stretchy":119},[39,68772,129],{},[1057,68774,68775,68777],{},[35,68776,62381],{"mathvariant":454},[35,68778,446],{},[39,68780,1194],{"fence":119,"stretchy":53,"minsize":4545,"maxsize":4545},[1172,68782,68783,68789],{},[32,68784,68785,68787],{},[35,68786,1178],{},[39,68788,1170],{},[32,68790,68791,68793,68795],{},[35,68792,4556],{},[39,68794,1187],{},[35,68796,62005],{"mathvariant":445},[35,68798,63107],{},[39,68800,120],{"stretchy":119},[35,68802,4556],{},[39,68804,335],{"separator":53},[35,68806,446],{},[39,68808,126],{"stretchy":119},[39,68810,1266],{"fence":119,"stretchy":53,"minsize":4545,"maxsize":4545},[35,68812,3573],{"mathvariant":138},[45,68814,68815],{"encoding":47},"R(\\delta)\\ \\ge\\ R(\\delta^*)=\\mathbb E_X\\Big[\\min_{a\\in\\mathcal A}r(a,X)\\Big].",[15,68817,68819,68852,68905],{"className":68818,"ariaHidden":53},[52],[15,68820,68822,68825,68828,68831,68834,68837,68840,68843,68846,68849],{"className":68821},[57],[15,68823],{"className":68824,"style":154},[61],[15,68826,455],{"className":68827,"style":62437},[66,67],[15,68829,120],{"className":68830},[162],[15,68832,3885],{"className":68833,"style":3946},[66,67],[15,68835,126],{"className":68836},[169],[15,68838,24233],{"className":68839},[71],[15,68841],{"className":68842,"style":173},[71],[15,68844,1801],{"className":68845},[177],[15,68847,24233],{"className":68848},[71],[15,68850],{"className":68851,"style":173},[71],[15,68853,68855,68858,68861,68864,68893,68896,68899,68902],{"className":68854},[57],[15,68856],{"className":68857,"style":154},[61],[15,68859,455],{"className":68860,"style":62437},[66,67],[15,68862,120],{"className":68863},[162],[15,68865,68867,68870],{"className":68866},[66],[15,68868,3885],{"className":68869,"style":3946},[66,67],[15,68871,68873],{"className":68872},[198],[15,68874,68876],{"className":68875},[202],[15,68877,68879],{"className":68878},[206],[15,68880,68882],{"className":68881,"style":1308},[210],[15,68883,68884,68887],{"style":397},[15,68885],{"className":68886,"style":218},[217],[15,68888,68890],{"className":68889},[222,223,224,225],[15,68891,1162],{"className":68892},[76,225],[15,68894,126],{"className":68895},[169],[15,68897],{"className":68898,"style":173},[71],[15,68900,129],{"className":68901},[177],[15,68903],{"className":68904,"style":173},[71],[15,68906,68908,68912,68952,68958,68961,69014,69017,69020,69023,69026,69029,69032,69035,69038,69044],{"className":68907},[57],[15,68909],{"className":68910,"style":68911},[61],"height:1.9217em;vertical-align:-0.7717em;",[15,68913,68915,68918],{"className":68914},[66],[15,68916,62381],{"className":68917},[66,517],[15,68919,68921],{"className":68920},[198],[15,68922,68924,68944],{"className":68923},[202,1090],[15,68925,68927,68941],{"className":68926},[206],[15,68928,68930],{"className":68929,"style":3638},[210],[15,68931,68932,68935],{"style":1584},[15,68933],{"className":68934,"style":218},[217],[15,68936,68938],{"className":68937},[222,223,224,225],[15,68939,446],{"className":68940,"style":16497},[66,67,225],[15,68942,1114],{"className":68943},[1113],[15,68945,68947],{"className":68946},[206],[15,68948,68950],{"className":68949,"style":1121},[210],[15,68951],{},[15,68953,68955],{"className":68954},[66],[15,68956,1194],{"className":68957},[1422,4677],[15,68959],{"className":68960,"style":697},[71],[15,68962,68964],{"className":68963},[1339,1351],[15,68965,68967,69006],{"className":68966},[202,1090],[15,68968,68970,69003],{"className":68969},[206],[15,68971,68973,68993],{"className":68972,"style":1361},[210],[15,68974,68975,68978],{"style":24354},[15,68976],{"className":68977,"style":1368},[217],[15,68979,68981],{"className":68980},[222,223,224,225],[15,68982,68984,68987,68990],{"className":68983},[66,225],[15,68985,4556],{"className":68986},[66,67,225],[15,68988,1187],{"className":68989},[177,225],[15,68991,62005],{"className":68992},[66,494,225],[15,68994,68995,68998],{"style":1386},[15,68996],{"className":68997,"style":1368},[217],[15,68999,69000],{},[15,69001,1178],{"className":69002},[1339],[15,69004,1114],{"className":69005},[1113],[15,69007,69009],{"className":69008},[206],[15,69010,69012],{"className":69011,"style":68526},[210],[15,69013],{},[15,69015],{"className":69016,"style":697},[71],[15,69018,63107],{"className":69019,"style":23025},[66,67],[15,69021,120],{"className":69022},[162],[15,69024,4556],{"className":69025},[66,67],[15,69027,335],{"className":69028},[422],[15,69030],{"className":69031,"style":697},[71],[15,69033,446],{"className":69034,"style":16497},[66,67],[15,69036,126],{"className":69037},[169],[15,69039,69041],{"className":69040},[66],[15,69042,1266],{"className":69043},[1422,4677],[15,69045,3573],{"className":69046},[66],[11,69048,69049,69052],{},[2040,69050,69051],{},"证明。"," 由塔性质，",[15,69054,69056],{"className":69055,"translate":19},[294],[15,69057,69059,69151],{"className":69058,"translate":19},[18],[15,69060,69062],{"className":69061},[23],[25,69063,69064],{"xmlns":27,"display":303},[29,69065,69066,69148],{},[32,69067,69068,69070,69072,69074,69076,69078,69084,69086,69088,69090,69092,69094,69096,69098,69100,69102,69104,69106,69108,69110,69112,69114,69116,69118,69124,69126,69128,69130,69132,69134,69136,69138,69140,69142,69144,69146],{},[35,69069,455],{},[39,69071,120],{"stretchy":119},[35,69073,3885],{},[39,69075,126],{"stretchy":119},[39,69077,129],{},[1057,69079,69080,69082],{},[35,69081,62381],{"mathvariant":454},[35,69083,446],{},[39,69085,1194],{"fence":119,"stretchy":53,"minsize":4545,"maxsize":4545},[35,69087,62381],{"mathvariant":454},[39,69089,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,69091,1216],{},[39,69093,120],{"stretchy":119},[35,69095,61885],{},[39,69097,335],{"separator":53},[35,69099,3885],{},[39,69101,120],{"stretchy":119},[35,69103,446],{},[39,69105,126],{"stretchy":119},[39,69107,126],{"stretchy":119},[39,69109,34430],{},[35,69111,446],{},[39,69113,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,69115,1266],{"fence":119,"stretchy":53,"minsize":4545,"maxsize":4545},[39,69117,129],{},[1057,69119,69120,69122],{},[35,69121,62381],{"mathvariant":454},[35,69123,446],{},[39,69125,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,69127,63107],{},[39,69129,120],{"stretchy":119},[35,69131,3885],{},[39,69133,120],{"stretchy":119},[35,69135,446],{},[39,69137,126],{"stretchy":119},[39,69139,335],{"separator":53},[35,69141,446],{},[39,69143,126],{"stretchy":119},[39,69145,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,69147,3573],{"mathvariant":138},[45,69149,69150],{"encoding":47},"R(\\delta)=\\mathbb E_X\\Big[\\mathbb E\\big[L(Y,\\delta(X))\\mid X\\big]\\Big]\n=\\mathbb E_X\\big[r(\\delta(X),X)\\big].",[15,69152,69154,69181,69279,69309],{"className":69153,"ariaHidden":53},[52],[15,69155,69157,69160,69163,69166,69169,69172,69175,69178],{"className":69156},[57],[15,69158],{"className":69159,"style":154},[61],[15,69161,455],{"className":69162,"style":62437},[66,67],[15,69164,120],{"className":69165},[162],[15,69167,3885],{"className":69168,"style":3946},[66,67],[15,69170,126],{"className":69171},[169],[15,69173],{"className":69174,"style":173},[71],[15,69176,129],{"className":69177},[177],[15,69179],{"className":69180,"style":173},[71],[15,69182,69184,69188,69228,69234,69237,69243,69246,69249,69252,69255,69258,69261,69264,69267,69270,69273,69276],{"className":69183},[57],[15,69185],{"className":69186,"style":69187},[61],"height:1.8em;vertical-align:-0.65em;",[15,69189,69191,69194],{"className":69190},[66],[15,69192,62381],{"className":69193},[66,517],[15,69195,69197],{"className":69196},[198],[15,69198,69200,69220],{"className":69199},[202,1090],[15,69201,69203,69217],{"className":69202},[206],[15,69204,69206],{"className":69205,"style":3638},[210],[15,69207,69208,69211],{"style":1584},[15,69209],{"className":69210,"style":218},[217],[15,69212,69214],{"className":69213},[222,223,224,225],[15,69215,446],{"className":69216,"style":16497},[66,67,225],[15,69218,1114],{"className":69219},[1113],[15,69221,69223],{"className":69222},[206],[15,69224,69226],{"className":69225,"style":1121},[210],[15,69227],{},[15,69229,69231],{"className":69230},[66],[15,69232,1194],{"className":69233},[1422,4677],[15,69235,62381],{"className":69236},[66,517],[15,69238,69240],{"className":69239},[66],[15,69241,1194],{"className":69242},[1422,1508],[15,69244,1216],{"className":69245},[66,67],[15,69247,120],{"className":69248},[162],[15,69250,61885],{"className":69251,"style":72},[66,67],[15,69253,335],{"className":69254},[422],[15,69256],{"className":69257,"style":697},[71],[15,69259,3885],{"className":69260,"style":3946},[66,67],[15,69262,120],{"className":69263},[162],[15,69265,446],{"className":69266,"style":16497},[66,67],[15,69268,18771],{"className":69269},[169],[15,69271],{"className":69272,"style":173},[71],[15,69274,34430],{"className":69275},[177],[15,69277],{"className":69278,"style":173},[71],[15,69280,69282,69285,69288,69294,69300,69303,69306],{"className":69281},[57],[15,69283],{"className":69284,"style":69187},[61],[15,69286,446],{"className":69287,"style":16497},[66,67],[15,69289,69291],{"className":69290},[66],[15,69292,1266],{"className":69293},[1422,1508],[15,69295,69297],{"className":69296},[66],[15,69298,1266],{"className":69299},[1422,4677],[15,69301],{"className":69302,"style":173},[71],[15,69304,129],{"className":69305},[177],[15,69307],{"className":69308,"style":173},[71],[15,69310,69312,69315,69355,69361,69364,69367,69370,69373,69376,69379,69382,69385,69388,69391,69397],{"className":69311},[57],[15,69313],{"className":69314,"style":4400},[61],[15,69316,69318,69321],{"className":69317},[66],[15,69319,62381],{"className":69320},[66,517],[15,69322,69324],{"className":69323},[198],[15,69325,69327,69347],{"className":69326},[202,1090],[15,69328,69330,69344],{"className":69329},[206],[15,69331,69333],{"className":69332,"style":3638},[210],[15,69334,69335,69338],{"style":1584},[15,69336],{"className":69337,"style":218},[217],[15,69339,69341],{"className":69340},[222,223,224,225],[15,69342,446],{"className":69343,"style":16497},[66,67,225],[15,69345,1114],{"className":69346},[1113],[15,69348,69350],{"className":69349},[206],[15,69351,69353],{"className":69352,"style":1121},[210],[15,69354],{},[15,69356,69358],{"className":69357},[66],[15,69359,1194],{"className":69360},[1422,1508],[15,69362,63107],{"className":69363,"style":23025},[66,67],[15,69365,120],{"className":69366},[162],[15,69368,3885],{"className":69369,"style":3946},[66,67],[15,69371,120],{"className":69372},[162],[15,69374,446],{"className":69375,"style":16497},[66,67],[15,69377,126],{"className":69378},[169],[15,69380,335],{"className":69381},[422],[15,69383],{"className":69384,"style":697},[71],[15,69386,446],{"className":69387,"style":16497},[66,67],[15,69389,126],{"className":69390},[169],[15,69392,69394],{"className":69393},[66],[15,69395,1266],{"className":69396},[1422,1508],[15,69398,3573],{"className":69399},[66],[11,69401,3196,69402,3249,69430,69606,69607],{},[15,69403,69405,69418],{"className":69404,"translate":19},[18],[15,69406,69408],{"className":69407},[23],[25,69409,69410],{"xmlns":27},[29,69411,69412,69416],{},[32,69413,69414],{},[35,69415,123],{},[45,69417,123],{"encoding":47},[15,69419,69421],{"className":69420,"ariaHidden":53},[52],[15,69422,69424,69427],{"className":69423},[57],[15,69425],{"className":69426,"style":86},[61],[15,69428,123],{"className":69429},[66,67],[15,69431,69433,69487],{"className":69432,"translate":19},[18],[15,69434,69436],{"className":69435},[23],[25,69437,69438],{"xmlns":27},[29,69439,69440,69484],{},[32,69441,69442,69444,69446,69448,69450,69452,69454,69456,69458,69460,69462,69472,69474,69476,69478,69480,69482],{},[35,69443,63107],{},[39,69445,120],{"stretchy":119},[35,69447,3885],{},[39,69449,120],{"stretchy":119},[35,69451,123],{},[39,69453,126],{"stretchy":119},[39,69455,335],{"separator":53},[35,69457,123],{},[39,69459,126],{"stretchy":119},[39,69461,1801],{},[1057,69463,69464,69470],{},[32,69465,69466,69468],{},[35,69467,1178],{},[39,69469,1170],{},[35,69471,4556],{},[35,69473,63107],{},[39,69475,120],{"stretchy":119},[35,69477,4556],{},[39,69479,335],{"separator":53},[35,69481,123],{},[39,69483,126],{"stretchy":119},[45,69485,69486],{"encoding":47},"r(\\delta(x),x)\\ge \\min_a r(a,x)",[15,69488,69490,69535],{"className":69489,"ariaHidden":53},[52],[15,69491,69493,69496,69499,69502,69505,69508,69511,69514,69517,69520,69523,69526,69529,69532],{"className":69492},[57],[15,69494],{"className":69495,"style":154},[61],[15,69497,63107],{"className":69498,"style":23025},[66,67],[15,69500,120],{"className":69501},[162],[15,69503,3885],{"className":69504,"style":3946},[66,67],[15,69506,120],{"className":69507},[162],[15,69509,123],{"className":69510},[66,67],[15,69512,126],{"className":69513},[169],[15,69515,335],{"className":69516},[422],[15,69518],{"className":69519,"style":697},[71],[15,69521,123],{"className":69522},[66,67],[15,69524,126],{"className":69525},[169],[15,69527],{"className":69528,"style":173},[71],[15,69530,1801],{"className":69531},[177],[15,69533],{"className":69534,"style":173},[71],[15,69536,69538,69541,69582,69585,69588,69591,69594,69597,69600,69603],{"className":69537},[57],[15,69539],{"className":69540,"style":154},[61],[15,69542,69544,69547],{"className":69543},[1339],[15,69545,1178],{"className":69546},[1339],[15,69548,69550],{"className":69549},[198],[15,69551,69553,69574],{"className":69552},[202,1090],[15,69554,69556,69571],{"className":69555},[206],[15,69557,69559],{"className":69558,"style":2198},[210],[15,69560,69562,69565],{"style":69561},"top:-2.55em;margin-right:0.05em;",[15,69563],{"className":69564,"style":218},[217],[15,69566,69568],{"className":69567},[222,223,224,225],[15,69569,4556],{"className":69570},[66,67,225],[15,69572,1114],{"className":69573},[1113],[15,69575,69577],{"className":69576},[206],[15,69578,69580],{"className":69579,"style":1121},[210],[15,69581],{},[15,69583],{"className":69584,"style":697},[71],[15,69586,63107],{"className":69587,"style":23025},[66,67],[15,69589,120],{"className":69590},[162],[15,69592,4556],{"className":69593},[66,67],[15,69595,335],{"className":69596},[422],[15,69598],{"className":69599,"style":697},[71],[15,69601,123],{"className":69602},[66,67],[15,69604,126],{"className":69605},[169],"，因此逐点取期望即得结论。",[15,69608,69610,69623],{"className":69609,"translate":19},[18],[15,69611,69613],{"className":69612},[23],[25,69614,69615],{"xmlns":27},[29,69616,69617,69621],{},[32,69618,69619],{},[35,69620,13514],{"mathvariant":138},[45,69622,13517],{"encoding":47},[15,69624,69626],{"className":69625,"ariaHidden":53},[52],[15,69627,69629,69632],{"className":69628},[57],[15,69630],{"className":69631,"style":13527},[61],[15,69633,13514],{"className":69634},[66,13531],[11,69636,69637,69638,69669,69670,69706,69707,69751],{},"证明只用了两件事：期望的塔性质，以及「逐点最小 ≤ 先对点取最小再平均」这样一个显而易见的事实。它不要求损失凸、不要求后验有密度、也不要求行动空间是有限的。这种一般性是贝叶斯决策理论最漂亮的地方之一。严格地说，若逐点下确界不可达，把 ",[15,69639,69641,69657],{"className":69640,"translate":19},[18],[15,69642,69644],{"className":69643},[23],[25,69645,69646],{"xmlns":27},[29,69647,69648,69654],{},[32,69649,69650,69652],{},[35,69651,1178],{},[39,69653,1170],{},[45,69655,69656],{"encoding":47},"\\min",[15,69658,69660],{"className":69659,"ariaHidden":53},[52],[15,69661,69663,69666],{"className":69662},[57],[15,69664],{"className":69665,"style":1361},[61],[15,69667,1178],{"className":69668},[1339]," 换成 ",[15,69671,69673,69690],{"className":69672,"translate":19},[18],[15,69674,69676],{"className":69675},[23],[25,69677,69678],{"xmlns":27},[29,69679,69680,69687],{},[32,69681,69682,69685],{},[35,69683,69684],{},"inf",[39,69686,1170],{},[45,69688,69689],{"encoding":47},"\\inf",[15,69691,69693],{"className":69692,"ariaHidden":53},[52],[15,69694,69696,69699],{"className":69695},[57],[15,69697],{"className":69698,"style":840},[61],[15,69700,69702,69703],{"className":69701},[1339],"in",[15,69704,116],{"style":69705},"margin-right:0.0778em;","、把 ",[15,69708,69710,69730],{"className":69709,"translate":19},[18],[15,69711,69713],{"className":69712},[23],[25,69714,69715],{"xmlns":27},[29,69716,69717,69727],{},[32,69718,69719,69721,69723,69725],{},[35,69720,1167],{},[39,69722,1170],{},[35,69724,1178],{},[39,69726,1170],{},[45,69728,69729],{"encoding":47},"\\arg\\min",[15,69731,69733],{"className":69732,"ariaHidden":53},[52],[15,69734,69736,69740,69745,69748],{"className":69735},[57],[15,69737],{"className":69738,"style":69739},[61],"height:0.8623em;vertical-align:-0.1944em;",[15,69741,1340,69743],{"className":69742},[1339],[15,69744,1344],{"style":1343},[15,69746],{"className":69747,"style":697},[71],[15,69749,1178],{"className":69750},[1339]," 换成「取一列逼近的行动」即可，证明不变。",[11,69753,69754],{},"把上式改写成差值，可以得到一个更有解释力的恒等式：",[15,69756,69758],{"className":69757,"translate":19},[294],[15,69759,69761,69860],{"className":69760,"translate":19},[18],[15,69762,69764],{"className":69763},[23],[25,69765,69766],{"xmlns":27,"display":303},[29,69767,69768,69857],{},[32,69769,69770,69772,69774,69776,69778,69780,69786,69788,69794,69796,69853,69855],{},[35,69771,455],{},[39,69773,120],{"stretchy":119},[35,69775,3885],{},[39,69777,126],{"stretchy":119},[39,69779,6602],{},[131,69781,69782,69784],{},[35,69783,455],{},[39,69785,1162],{},[39,69787,129],{},[1057,69789,69790,69792],{},[35,69791,62381],{"mathvariant":454},[35,69793,446],{},[39,69795,1194],{"fence":119,"stretchy":53,"minsize":4545,"maxsize":4545},[1172,69797,69798,69847],{},[1172,69799,69800,69844],{},[32,69801,69802,69804,69806,69808,69810,69812,69814,69816,69818,69820,69822,69832,69834,69836,69838,69840,69842],{},[35,69803,63107],{},[39,69805,120],{"stretchy":119},[35,69807,3885],{},[39,69809,120],{"stretchy":119},[35,69811,446],{},[39,69813,126],{"stretchy":119},[39,69815,335],{"separator":53},[35,69817,446],{},[39,69819,126],{"stretchy":119},[39,69821,6602],{},[1172,69823,69824,69830],{},[32,69825,69826,69828],{},[35,69827,1178],{},[39,69829,1170],{},[35,69831,4556],{},[35,69833,63107],{},[39,69835,120],{"stretchy":119},[35,69837,4556],{},[39,69839,335],{"separator":53},[35,69841,446],{},[39,69843,126],{"stretchy":119},[39,69845,69846],{"stretchy":53},"⏟",[32,69848,69849,69851],{},[39,69850,1801],{},[1209,69852,1804],{},[39,69854,1266],{"fence":119,"stretchy":53,"minsize":4545,"maxsize":4545},[35,69856,3573],{"mathvariant":138},[45,69858,69859],{"encoding":47},"R(\\delta)-R^*\n=\\mathbb E_X\\Big[\\underbrace{r(\\delta(X),X)-\\min_{a}r(a,X)}_{\\ge 0}\\Big].",[15,69861,69863,69890,69934],{"className":69862,"ariaHidden":53},[52],[15,69864,69866,69869,69872,69875,69878,69881,69884,69887],{"className":69865},[57],[15,69867],{"className":69868,"style":154},[61],[15,69870,455],{"className":69871,"style":62437},[66,67],[15,69873,120],{"className":69874},[162],[15,69876,3885],{"className":69877,"style":3946},[66,67],[15,69879,126],{"className":69880},[169],[15,69882],{"className":69883,"style":72},[71],[15,69885,6602],{"className":69886},[76],[15,69888],{"className":69889,"style":72},[71],[15,69891,69893,69896,69925,69928,69931],{"className":69892},[57],[15,69894],{"className":69895,"style":1308},[61],[15,69897,69899,69902],{"className":69898},[66],[15,69900,455],{"className":69901,"style":62437},[66,67],[15,69903,69905],{"className":69904},[198],[15,69906,69908],{"className":69907},[202],[15,69909,69911],{"className":69910},[206],[15,69912,69914],{"className":69913,"style":1308},[210],[15,69915,69916,69919],{"style":397},[15,69917],{"className":69918,"style":218},[217],[15,69920,69922],{"className":69921},[222,223,224,225],[15,69923,1162],{"className":69924},[76,225],[15,69926],{"className":69927,"style":173},[71],[15,69929,129],{"className":69930},[177],[15,69932],{"className":69933,"style":173},[71],[15,69935,69937,69941,69981,69987,69990,70224,70227,70233],{"className":69936},[57],[15,69938],{"className":69939,"style":69940},[61],"height:3.2443em;vertical-align:-2.0943em;",[15,69942,69944,69947],{"className":69943},[66],[15,69945,62381],{"className":69946},[66,517],[15,69948,69950],{"className":69949},[198],[15,69951,69953,69973],{"className":69952},[202,1090],[15,69954,69956,69970],{"className":69955},[206],[15,69957,69959],{"className":69958,"style":3638},[210],[15,69960,69961,69964],{"style":1584},[15,69962],{"className":69963,"style":218},[217],[15,69965,69967],{"className":69966},[222,223,224,225],[15,69968,446],{"className":69969,"style":16497},[66,67,225],[15,69971,1114],{"className":69972},[1113],[15,69974,69976],{"className":69975},[206],[15,69977,69979],{"className":69978,"style":1121},[210],[15,69980],{},[15,69982,69984],{"className":69983},[66],[15,69985,1194],{"className":69986},[1422,4677],[15,69988],{"className":69989,"style":697},[71],[15,69991,69993],{"className":69992},[1413,1172],[15,69994,69996,70215],{"className":69995},[202,1090],[15,69997,69999,70212],{"className":69998},[206],[15,70000,70003,70021],{"className":70001,"style":70002},[210],"height:0.75em;",[15,70004,70006,70009],{"style":70005},"top:-1.0009em;",[15,70007],{"className":70008,"style":1368},[217],[15,70010,70012],{"className":70011},[222,223,224,225],[15,70013,70015,70018],{"className":70014},[66,225],[15,70016,1801],{"className":70017},[177,225],[15,70019,1804],{"className":70020},[66,225],[15,70022,70023,70026],{"style":1386},[15,70024],{"className":70025,"style":1368},[217],[15,70027,70029],{"className":70028},[1413,1172],[15,70030,70032,70203],{"className":70031},[202,1090],[15,70033,70035,70200],{"className":70034},[206],[15,70036,70038,70082],{"className":70037,"style":70002},[210],[15,70039,70042,70045],{"className":70040,"style":70041},[48564],"top:-1.652em;",[15,70043],{"className":70044,"style":1368},[217],[15,70046,70050,70062,70072],{"className":70047,"style":70049},[70048],"stretchy","height:0.548em;min-width:1.6em;",[15,70051,70055],{"className":70052,"style":70054},[70053],"brace-left","height:0.548em;",[34828,70056,70059],{"xmlns":34830,"width":48594,"height":70057,"viewBox":70058,"preserveAspectRatio":48597},"0.548em","0 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20-1h17z",[15,70083,70084,70087],{"style":1386},[15,70085],{"className":70086,"style":1368},[217],[15,70088,70090,70093,70096,70099,70102,70105,70108,70111,70114,70117,70120,70123,70126,70129,70176,70179,70182,70185,70188,70191,70194,70197],{"className":70089},[66],[15,70091,63107],{"className":70092,"style":23025},[66,67],[15,70094,120],{"className":70095},[162],[15,70097,3885],{"className":70098,"style":3946},[66,67],[15,70100,120],{"className":70101},[162],[15,70103,446],{"className":70104,"style":16497},[66,67],[15,70106,126],{"className":70107},[169],[15,70109,335],{"className":70110},[422],[15,70112],{"className":70113,"style":697},[71],[15,70115,446],{"className":70116,"style":16497},[66,67],[15,70118,126],{"className":70119},[169],[15,70121],{"className":70122,"style":72},[71],[15,70124,6602],{"className":70125},[76],[15,70127],{"className":70128,"style":72},[71],[15,70130,70132],{"className":70131},[1339,1351],[15,70133,70135,70168],{"className":70134},[202,1090],[15,70136,70138,70165],{"className":70137},[206],[15,70139,70141,70155],{"className":70140,"style":1361},[210],[15,70142,70143,70146],{"style":48823},[15,70144],{"className":70145,"style":1368},[217],[15,70147,70149],{"className":70148},[222,223,224,225],[15,70150,70152],{"className":70151},[66,225],[15,70153,4556],{"className":70154},[66,67,225],[15,70156,70157,70160],{"style":1386},[15,70158],{"className":70159,"style":1368},[217],[15,70161,70162],{},[15,70163,1178],{"className":70164},[1339],[15,70166,1114],{"className":70167},[1113],[15,70169,70171],{"className":70170},[206],[15,70172,70174],{"className":70173,"style":48861},[210],[15,70175],{},[15,70177],{"className":70178,"style":697},[71],[15,70180,63107],{"className":70181,"style":23025},[66,67],[15,70183,120],{"className":70184},[162],[15,70186,4556],{"className":70187},[66,67],[15,70189,335],{"className":70190},[422],[15,70192],{"className":70193,"style":697},[71],[15,70195,446],{"className":70196,"style":16497},[66,67],[15,70198,126],{"className":70199},[169],[15,70201,1114],{"className":70202},[1113],[15,70204,70206],{"className":70205},[206],[15,70207,70210],{"className":70208,"style":70209},[210],"height:1.348em;",[15,70211],{},[15,70213,1114],{"className":70214},[1113],[15,70216,70218],{"className":70217},[206],[15,70219,70222],{"className":70220,"style":70221},[210],"height:2.0943em;",[15,70223],{},[15,70225],{"className":70226,"style":697},[71],[15,70228,70230],{"className":70229},[66],[15,70231,1266],{"className":70232},[1422,4677],[15,70234,3573],{"className":70235},[66],[11,70237,70238,70239,70267,70268,70271,70272,70300,70301,70329],{},"右边每一项都是非负的，称为在 ",[15,70240,70242,70255],{"className":70241,"translate":19},[18],[15,70243,70245],{"className":70244},[23],[25,70246,70247],{"xmlns":27},[29,70248,70249,70253],{},[32,70250,70251],{},[35,70252,123],{},[45,70254,123],{"encoding":47},[15,70256,70258],{"className":70257,"ariaHidden":53},[52],[15,70259,70261,70264],{"className":70260},[57],[15,70262],{"className":70263,"style":86},[61],[15,70265,123],{"className":70266},[66,67]," 处的",[2040,70269,70270],{},"逐点遗憾","（pointwise regret）。它说明：一个决策规则的次优性完全来自它在某些 ",[15,70273,70275,70288],{"className":70274,"translate":19},[18],[15,70276,70278],{"className":70277},[23],[25,70279,70280],{"xmlns":27},[29,70281,70282,70286],{},[32,70283,70284],{},[35,70285,123],{},[45,70287,123],{"encoding":47},[15,70289,70291],{"className":70290,"ariaHidden":53},[52],[15,70292,70294,70297],{"className":70293},[57],[15,70295],{"className":70296,"style":86},[61],[15,70298,123],{"className":70299},[66,67]," 上没有选到局部最优行动。不同 ",[15,70302,70304,70317],{"className":70303,"translate":19},[18],[15,70305,70307],{"className":70306},[23],[25,70308,70309],{"xmlns":27},[29,70310,70311,70315],{},[32,70312,70313],{},[35,70314,123],{},[45,70316,123],{"encoding":47},[15,70318,70320],{"className":70319,"ariaHidden":53},[52],[15,70321,70323,70326],{"className":70322},[57],[15,70324],{"className":70325,"style":86},[61],[15,70327,123],{"className":70328},[66,67]," 之间的错误不会相互抵消，也不会相互放大。",[1904,70331,70333],{"id":70332},"_32-0-1-损失最大后验","3.2 0-1 损失：最大后验",[11,70335,70336],{},"分类问题最常用的损失是 0-1 损失：",[15,70338,70340],{"className":70339,"translate":19},[294],[15,70341,70343,70384],{"className":70342,"translate":19},[18],[15,70344,70346],{"className":70345},[23],[25,70347,70348],{"xmlns":27,"display":303},[29,70349,70350,70381],{},[32,70351,70352,70354,70356,70358,70360,70362,70364,70366,70369,70371,70373,70375,70377,70379],{},[35,70353,1216],{},[39,70355,120],{"stretchy":119},[35,70357,1224],{},[39,70359,335],{"separator":53},[35,70361,4556],{},[39,70363,126],{"stretchy":119},[39,70365,129],{},[1209,70367,1211],{"mathvariant":70368},"bold",[39,70370,4307],{"stretchy":119},[35,70372,1224],{},[39,70374,13019],{"mathvariant":138},[35,70376,4556],{},[39,70378,4330],{"stretchy":119},[35,70380,3573],{"mathvariant":138},[45,70382,70383],{"encoding":47},"L(y,a)=\\mathbf 1\\{y\\ne a\\}.",[15,70385,70387,70423,70481],{"className":70386,"ariaHidden":53},[52],[15,70388,70390,70393,70396,70399,70402,70405,70408,70411,70414,70417,70420],{"className":70389},[57],[15,70391],{"className":70392,"style":154},[61],[15,70394,1216],{"className":70395},[66,67],[15,70397,120],{"className":70398},[162],[15,70400,1224],{"className":70401,"style":1515},[66,67],[15,70403,335],{"className":70404},[422],[15,70406],{"className":70407,"style":697},[71],[15,70409,4556],{"className":70410},[66,67],[15,70412,126],{"className":70413},[169],[15,70415],{"className":70416,"style":173},[71],[15,70418,129],{"className":70419},[177],[15,70421],{"className":70422,"style":173},[71],[15,70424,70426,70429,70433,70436,70439,70442,70478],{"className":70425},[57],[15,70427],{"className":70428,"style":154},[61],[15,70430,1211],{"className":70431},[66,70432],"mathbf",[15,70434,4307],{"className":70435},[162],[15,70437,1224],{"className":70438,"style":1515},[66,67],[15,70440],{"className":70441,"style":173},[71],[15,70443,70445,70472,70475],{"className":70444},[177],[15,70446,70448],{"className":70447},[177],[15,70449,70451],{"className":70450},[66,13086],[15,70452,70454],{"className":70453},[13090],[15,70455,70457,70460,70469],{"className":70456},[13094],[15,70458],{"className":70459,"style":3408},[61],[15,70461,70463],{"className":70462},[13101],[15,70464,70466],{"className":70465},[66],[15,70467,13108],{"className":70468},[177],[15,70470],{"className":70471},[13112],[15,70473],{"className":70474},[71,13116],[15,70476,129],{"className":70477},[177],[15,70479],{"className":70480,"style":173},[71],[15,70482,70484,70487,70490,70493],{"className":70483},[57],[15,70485],{"className":70486,"style":154},[61],[15,70488,4556],{"className":70489},[66,67],[15,70491,4330],{"className":70492},[169],[15,70494,3573],{"className":70495},[66],[11,70497,70498],{},"此时条件风险为",[15,70500,70502],{"className":70501,"translate":19},[294],[15,70503,70505,70577],{"className":70504,"translate":19},[18],[15,70506,70508],{"className":70507},[23],[25,70509,70510],{"xmlns":27,"display":303},[29,70511,70512,70574],{},[32,70513,70514,70516,70518,70520,70522,70524,70526,70528,70530,70532,70534,70536,70538,70540,70542,70544,70546,70548,70550,70552,70554,70556,70558,70560,70562,70564,70566,70568,70570,70572],{},[35,70515,63107],{},[39,70517,120],{"stretchy":119},[35,70519,4556],{},[39,70521,335],{"separator":53},[35,70523,123],{},[39,70525,126],{"stretchy":119},[39,70527,129],{},[35,70529,62381],{"mathvariant":454},[39,70531,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,70533,1211],{"mathvariant":70368},[39,70535,4307],{"stretchy":119},[35,70537,61885],{},[39,70539,13019],{"mathvariant":138},[35,70541,4556],{},[39,70543,4330],{"stretchy":119},[39,70545,34430],{},[35,70547,446],{},[39,70549,129],{},[35,70551,123],{},[39,70553,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,70555,129],{},[1209,70557,1211],{},[39,70559,6602],{},[35,70561,11],{},[39,70563,120],{"stretchy":119},[35,70565,4556],{},[39,70567,34430],{},[35,70569,123],{},[39,70571,126],{"stretchy":119},[39,70573,335],{"separator":53},[45,70575,70576],{"encoding":47},"r(a,x)=\\mathbb E\\big[\\mathbf 1\\{Y\\ne a\\}\\mid X=x\\big]=1-p(a\\mid x),",[15,70578,70580,70616,70682,70703,70721,70745,70763,70787],{"className":70579,"ariaHidden":53},[52],[15,70581,70583,70586,70589,70592,70595,70598,70601,70604,70607,70610,70613],{"className":70582},[57],[15,70584],{"className":70585,"style":154},[61],[15,70587,63107],{"className":70588,"style":23025},[66,67],[15,70590,120],{"className":70591},[162],[15,70593,4556],{"className":70594},[66,67],[15,70596,335],{"className":70597},[422],[15,70599],{"className":70600,"style":697},[71],[15,70602,123],{"className":70603},[66,67],[15,70605,126],{"className":70606},[169],[15,70608],{"className":70609,"style":173},[71],[15,70611,129],{"className":70612},[177],[15,70614],{"className":70615,"style":173},[71],[15,70617,70619,70622,70625,70631,70634,70637,70640,70643,70679],{"className":70618},[57],[15,70620],{"className":70621,"style":4400},[61],[15,70623,62381],{"className":70624},[66,517],[15,70626,70628],{"className":70627},[66],[15,70629,1194],{"className":70630},[1422,1508],[15,70632,1211],{"className":70633},[66,70432],[15,70635,4307],{"className":70636},[162],[15,70638,61885],{"className":70639,"style":72},[66,67],[15,70641],{"className":70642,"style":173},[71],[15,70644,70646,70673,70676],{"className":70645},[177],[15,70647,70649],{"className":70648},[177],[15,70650,70652],{"className":70651},[66,13086],[15,70653,70655],{"className":70654},[13090],[15,70656,70658,70661,70670],{"className":70657},[13094],[15,70659],{"className":70660,"style":3408},[61],[15,70662,70664],{"className":70663},[13101],[15,70665,70667],{"className":70666},[66],[15,70668,13108],{"className":70669},[177],[15,70671],{"className":70672},[13112],[15,70674],{"className":70675},[71,13116],[15,70677,129],{"className":70678},[177],[15,70680],{"className":70681,"style":173},[71],[15,70683,70685,70688,70691,70694,70697,70700],{"className":70684},[57],[15,70686],{"className":70687,"style":154},[61],[15,70689,4556],{"className":70690},[66,67],[15,70692,4330],{"className":70693},[169],[15,70695],{"className":70696,"style":173},[71],[15,70698,34430],{"className":70699},[177],[15,70701],{"className":70702,"style":173},[71],[15,70704,70706,70709,70712,70715,70718],{"className":70705},[57],[15,70707],{"className":70708,"style":490},[61],[15,70710,446],{"className":70711,"style":16497},[66,67],[15,70713],{"className":70714,"style":173},[71],[15,70716,129],{"className":70717},[177],[15,70719],{"className":70720,"style":173},[71],[15,70722,70724,70727,70730,70736,70739,70742],{"className":70723},[57],[15,70725],{"className":70726,"style":4400},[61],[15,70728,123],{"className":70729},[66,67],[15,70731,70733],{"className":70732},[66],[15,70734,1266],{"className":70735},[1422,1508],[15,70737],{"className":70738,"style":173},[71],[15,70740,129],{"className":70741},[177],[15,70743],{"className":70744,"style":173},[71],[15,70746,70748,70751,70754,70757,70760],{"className":70747},[57],[15,70749],{"className":70750,"style":30468},[61],[15,70752,1211],{"className":70753},[66],[15,70755],{"className":70756,"style":72},[71],[15,70758,6602],{"className":70759},[76],[15,70761],{"className":70762,"style":72},[71],[15,70764,70766,70769,70772,70775,70778,70781,70784],{"className":70765},[57],[15,70767],{"className":70768,"style":154},[61],[15,70770,11],{"className":70771},[66,67],[15,70773,120],{"className":70774},[162],[15,70776,4556],{"className":70777},[66,67],[15,70779],{"className":70780,"style":173},[71],[15,70782,34430],{"className":70783},[177],[15,70785],{"className":70786,"style":173},[71],[15,70788,70790,70793,70796,70799],{"className":70789},[57],[15,70791],{"className":70792,"style":154},[61],[15,70794,123],{"className":70795},[66,67],[15,70797,126],{"className":70798},[169],[15,70800,335],{"className":70801},[422],[11,70803,70804,70805,70861],{},"最小化 ",[15,70806,70808,70831],{"className":70807,"translate":19},[18],[15,70809,70811],{"className":70810},[23],[25,70812,70813],{"xmlns":27},[29,70814,70815,70829],{},[32,70816,70817,70819,70821,70823,70825,70827],{},[35,70818,63107],{},[39,70820,120],{"stretchy":119},[35,70822,4556],{},[39,70824,335],{"separator":53},[35,70826,123],{},[39,70828,126],{"stretchy":119},[45,70830,63875],{"encoding":47},[15,70832,70834],{"className":70833,"ariaHidden":53},[52],[15,70835,70837,70840,70843,70846,70849,70852,70855,70858],{"className":70836},[57],[15,70838],{"className":70839,"style":154},[61],[15,70841,63107],{"className":70842,"style":23025},[66,67],[15,70844,120],{"className":70845},[162],[15,70847,4556],{"className":70848},[66,67],[15,70850,335],{"className":70851},[422],[15,70853],{"className":70854,"style":697},[71],[15,70856,123],{"className":70857},[66,67],[15,70859,126],{"className":70860},[169]," 等价于最大化后验概率：",[15,70863,70865],{"className":70864,"translate":19},[294],[15,70866,70868,70930],{"className":70867,"translate":19},[18],[15,70869,70871],{"className":70870},[23],[25,70872,70873],{"xmlns":27,"display":303},[29,70874,70875,70927],{},[32,70876,70877,70883,70885,70887,70889,70891,70893,70895,70911,70913,70915,70917,70919,70921,70923,70925],{},[131,70878,70879,70881],{},[35,70880,3885],{},[39,70882,1162],{},[39,70884,120],{"stretchy":119},[35,70886,123],{},[39,70888,126],{"stretchy":119},[39,70890,129],{},[35,70892,1167],{},[39,70894,1170],{},[1172,70896,70897,70903],{},[32,70898,70899,70901],{},[35,70900,23273],{},[39,70902,1170],{},[32,70904,70905,70907,70909],{},[35,70906,1224],{},[39,70908,1187],{},[35,70910,61885],{"mathvariant":445},[1274,70912,24233],{},[35,70914,11],{},[39,70916,120],{"stretchy":119},[35,70918,1224],{},[39,70920,34430],{},[35,70922,123],{},[39,70924,126],{"stretchy":119},[35,70926,3573],{"mathvariant":138},[45,70928,70929],{"encoding":47},"\\delta^*(x)=\\arg\\max_{y\\in\\mathcal Y}\\ p(y\\mid x).",[15,70931,70933,70986,71078],{"className":70932,"ariaHidden":53},[52],[15,70934,70936,70939,70968,70971,70974,70977,70980,70983],{"className":70935},[57],[15,70937],{"className":70938,"style":154},[61],[15,70940,70942,70945],{"className":70941},[66],[15,70943,3885],{"className":70944,"style":3946},[66,67],[15,70946,70948],{"className":70947},[198],[15,70949,70951],{"className":70950},[202],[15,70952,70954],{"className":70953},[206],[15,70955,70957],{"className":70956,"style":1308},[210],[15,70958,70959,70962],{"style":397},[15,70960],{"className":70961,"style":218},[217],[15,70963,70965],{"className":70964},[222,223,224,225],[15,70966,1162],{"className":70967},[76,225],[15,70969,120],{"className":70970},[162],[15,70972,123],{"className":70973},[66,67],[15,70975,126],{"className":70976},[169],[15,70978],{"className":70979,"style":173},[71],[15,70981,129],{"className":70982},[177],[15,70984],{"className":70985,"style":173},[71],[15,70987,70989,70993,70998,71001,71054,71057,71060,71063,71066,71069,71072,71075],{"className":70988},[57],[15,70990],{"className":70991,"style":70992},[61],"height:1.6304em;vertical-align:-0.8804em;",[15,70994,1340,70996],{"className":70995},[1339],[15,70997,1344],{"style":1343},[15,70999],{"className":71000,"style":697},[71],[15,71002,71004],{"className":71003},[1339,1351],[15,71005,71007,71046],{"className":71006},[202,1090],[15,71008,71010,71043],{"className":71009},[206],[15,71011,71013,71033],{"className":71012,"style":86},[210],[15,71014,71015,71018],{"style":24354},[15,71016],{"className":71017,"style":1368},[217],[15,71019,71021],{"className":71020},[222,223,224,225],[15,71022,71024,71027,71030],{"className":71023},[66,225],[15,71025,1224],{"className":71026,"style":1515},[66,67,225],[15,71028,1187],{"className":71029},[177,225],[15,71031,61885],{"className":71032,"style":28452},[66,494,225],[15,71034,71035,71038],{"style":1386},[15,71036],{"className":71037,"style":1368},[217],[15,71039,71040],{},[15,71041,23273],{"className":71042},[1339],[15,71044,1114],{"className":71045},[1113],[15,71047,71049],{"className":71048},[206],[15,71050,71052],{"className":71051,"style":67080},[210],[15,71053],{},[15,71055,24233],{"className":71056},[71],[15,71058],{"className":71059,"style":697},[71],[15,71061,11],{"className":71062},[66,67],[15,71064,120],{"className":71065},[162],[15,71067,1224],{"className":71068,"style":1515},[66,67],[15,71070],{"className":71071,"style":173},[71],[15,71073,34430],{"className":71074},[177],[15,71076],{"className":71077,"style":173},[71],[15,71079,71081,71084,71087,71090],{"className":71080},[57],[15,71082],{"className":71083,"style":154},[61],[15,71085,123],{"className":71086},[66,67],[15,71088,126],{"className":71089},[169],[15,71091,3573],{"className":71092},[66],[11,71094,5904,71095,71098,71099,71225],{},[2040,71096,71097],{},"最大后验决策","（maximum a posteriori decision）。二分类时，若 ",[15,71100,71102,71141],{"className":71101,"translate":19},[18],[15,71103,71105],{"className":71104},[23],[25,71106,71107],{"xmlns":27},[29,71108,71109,71138],{},[32,71110,71111,71114,71116,71118,71120,71122,71124,71126,71128,71130,71132,71134,71136],{},[35,71112,71113],{},"η",[39,71115,120],{"stretchy":119},[35,71117,123],{},[39,71119,126],{"stretchy":119},[39,71121,129],{},[35,71123,11],{},[39,71125,120],{"stretchy":119},[35,71127,61885],{},[39,71129,129],{},[1209,71131,1211],{},[39,71133,34430],{},[35,71135,123],{},[39,71137,126],{"stretchy":119},[45,71139,71140],{"encoding":47},"\\eta(x)=p(Y=1\\mid x)",[15,71142,71144,71171,71195,71213],{"className":71143,"ariaHidden":53},[52],[15,71145,71147,71150,71153,71156,71159,71162,71165,71168],{"className":71146},[57],[15,71148],{"className":71149,"style":154},[61],[15,71151,71113],{"className":71152,"style":1515},[66,67],[15,71154,120],{"className":71155},[162],[15,71157,123],{"className":71158},[66,67],[15,71160,126],{"className":71161},[169],[15,71163],{"className":71164,"style":173},[71],[15,71166,129],{"className":71167},[177],[15,71169],{"className":71170,"style":173},[71],[15,71172,71174,71177,71180,71183,71186,71189,71192],{"className":71173},[57],[15,71175],{"className":71176,"style":154},[61],[15,71178,11],{"className":71179},[66,67],[15,71181,120],{"className":71182},[162],[15,71184,61885],{"className":71185,"style":72},[66,67],[15,71187],{"className":71188,"style":173},[71],[15,71190,129],{"className":71191},[177],[15,71193],{"className":71194,"style":173},[71],[15,71196,71198,71201,71204,71207,71210],{"className":71197},[57],[15,71199],{"className":71200,"style":154},[61],[15,71202,1211],{"className":71203},[66],[15,71205],{"className":71206,"style":173},[71],[15,71208,34430],{"className":71209},[177],[15,71211],{"className":71212,"style":173},[71],[15,71214,71216,71219,71222],{"className":71215},[57],[15,71217],{"className":71218,"style":154},[61],[15,71220,123],{"className":71221},[66,67],[15,71223,126],{"className":71224},[169],"，规则简化为",[15,71227,71229],{"className":71228,"translate":19},[294],[15,71230,71232,71342],{"className":71231,"translate":19},[18],[15,71233,71235],{"className":71234},[23],[25,71236,71237],{"xmlns":27,"display":303},[29,71238,71239,71339],{},[32,71240,71241,71247,71249,71251,71253,71255],{},[131,71242,71243,71245],{},[35,71244,3885],{},[39,71246,1162],{},[39,71248,120],{"stretchy":119},[35,71250,123],{},[39,71252,126],{"stretchy":119},[39,71254,129],{},[32,71256,71257,71259],{},[39,71258,4307],{"fence":53},[1142,71260,71263,71301],{"rowspacing":71261,"columnalign":71262,"columnspacing":9791},"0.36em","left left",[1146,71264,71265,71275],{},[1149,71266,71267],{},[62987,71268,71269],{"scriptlevel":1804,"displaystyle":119},[32,71270,71271,71273],{},[1209,71272,1211],{},[39,71274,335],{"separator":53},[1149,71276,71277],{},[62987,71278,71279],{"scriptlevel":1804,"displaystyle":119},[32,71280,71281,71283,71285,71287,71289,71291,71299],{},[35,71282,71113],{},[39,71284,120],{"stretchy":119},[35,71286,123],{},[39,71288,126],{"stretchy":119},[39,71290,1801],{},[62987,71292,71293],{"scriptlevel":1804,"displaystyle":119},[6467,71294,71295,71297],{},[1209,71296,1211],{},[1209,71298,1263],{},[39,71300,335],{"separator":53},[1146,71302,71303,71313],{},[1149,71304,71305],{},[62987,71306,71307],{"scriptlevel":1804,"displaystyle":119},[32,71308,71309,71311],{},[1209,71310,1804],{},[39,71312,335],{"separator":53},[1149,71314,71315],{},[62987,71316,71317],{"scriptlevel":1804,"displaystyle":119},[32,71318,71319,71321,71323,71325,71327,71329,71337],{},[35,71320,71113],{},[39,71322,120],{"stretchy":119},[35,71324,123],{},[39,71326,126],{"stretchy":119},[39,71328,25772],{},[62987,71330,71331],{"scriptlevel":1804,"displaystyle":119},[6467,71332,71333,71335],{},[1209,71334,1211],{},[1209,71336,1263],{},[39,71338,335],{"separator":53},[45,71340,71341],{"encoding":47},"\\delta^*(x)=\n\\begin{cases}\n1, & \\eta(x)\\ge \\tfrac12,\\\\[2pt]\n0, & 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",[15,71711,71713,71749],{"className":71712,"translate":19},[18],[15,71714,71716],{"className":71715},[23],[25,71717,71718],{"xmlns":27},[29,71719,71720,71746],{},[32,71721,71722,71724,71726,71728,71730,71732,71734,71736,71738,71740,71742,71744],{},[39,71723,4307],{"stretchy":119},[35,71725,123],{},[39,71727,442],{},[35,71729,71113],{},[39,71731,120],{"stretchy":119},[35,71733,123],{},[39,71735,126],{"stretchy":119},[39,71737,129],{},[1209,71739,1211],{},[35,71741,6625],{"mathvariant":138},[1209,71743,1263],{},[39,71745,4330],{"stretchy":119},[45,71747,71748],{"encoding":47},"\\{x:\\eta(x)=1\u002F2\\}",[15,71750,71752,71773,71800],{"className":71751,"ariaHidden":53},[52],[15,71753,71755,71758,71761,71764,71767,71770],{"className":71754},[57],[15,71756],{"className":71757,"style":154},[61],[15,71759,4307],{"className":71760},[162],[15,71762,123],{"className":71763},[66,67],[15,71765],{"className":71766,"style":173},[71],[15,71768,442],{"className":71769},[177],[15,71771],{"className":71772,"style":173},[71],[15,71774,71776,71779,71782,71785,71788,71791,71794,71797],{"className":71775},[57],[15,71777],{"className":71778,"style":154},[61],[15,71780,71113],{"className":71781,"style":1515},[66,67],[15,71783,120],{"className":71784},[162],[15,71786,123],{"className":71787},[66,67],[15,71789,126],{"className":71790},[169],[15,71792],{"className":71793,"style":173},[71],[15,71795,129],{"className":71796},[177],[15,71798],{"className":71799,"style":173},[71],[15,71801,71803,71806,71810],{"className":71802},[57],[15,71804],{"className":71805,"style":154},[61],[15,71807,71809],{"className":71808},[66],"1\u002F2",[15,71811,4330],{"className":71812},[169],"。当 ",[15,71815,71817,71845],{"className":71816,"translate":19},[18],[15,71818,71820],{"className":71819},[23],[25,71821,71822],{"xmlns":27},[29,71823,71824,71842],{},[32,71825,71826,71828,71830,71832,71834,71836,71838,71840],{},[35,71827,71113],{},[39,71829,120],{"stretchy":119},[35,71831,123],{},[39,71833,126],{"stretchy":119},[39,71835,129],{},[1209,71837,1211],{},[35,71839,6625],{"mathvariant":138},[1209,71841,1263],{},[45,71843,71844],{"encoding":47},"\\eta(x)=1\u002F2",[15,71846,71848,71875],{"className":71847,"ariaHidden":53},[52],[15,71849,71851,71854,71857,71860,71863,71866,71869,71872],{"className":71850},[57],[15,71852],{"className":71853,"style":154},[61],[15,71855,71113],{"className":71856,"style":1515},[66,67],[15,71858,120],{"className":71859},[162],[15,71861,123],{"className":71862},[66,67],[15,71864,126],{"className":71865},[169],[15,71867],{"className":71868,"style":173},[71],[15,71870,129],{"className":71871},[177],[15,71873],{"className":71874,"style":173},[71],[15,71876,71878,71881],{"className":71877},[57],[15,71879],{"className":71880,"style":154},[61],[15,71882,71809],{"className":71883},[66]," 时两种行动同样好，实践中用固定的打破平局规则即可。",[1904,71886,71888],{"id":71887},"_33-平方损失后验均值","3.3 平方损失：后验均值",[11,71890,71891,71892,72033],{},"回归问题常用平方损失 ",[15,71893,71895,71935],{"className":71894,"translate":19},[18],[15,71896,71898],{"className":71897},[23],[25,71899,71900],{"xmlns":27},[29,71901,71902,71932],{},[32,71903,71904,71906,71908,71910,71912,71914,71916,71918,71920,71922,71924,71926],{},[35,71905,1216],{},[39,71907,120],{"stretchy":119},[35,71909,1224],{},[39,71911,335],{"separator":53},[35,71913,4556],{},[39,71915,126],{"stretchy":119},[39,71917,129],{},[39,71919,120],{"stretchy":119},[35,71921,1224],{},[39,71923,6602],{},[35,71925,4556],{},[131,71927,71928,71930],{},[39,71929,126],{"stretchy":119},[1209,71931,1263],{},[45,71933,71934],{"encoding":47},"L(y,a)=(y-a)^2",[15,71936,71938,71974,71995],{"className":71937,"ariaHidden":53},[52],[15,71939,71941,71944,71947,71950,71953,71956,71959,71962,71965,71968,71971],{"className":71940},[57],[15,71942],{"className":71943,"style":154},[61],[15,71945,1216],{"className":71946},[66,67],[15,71948,120],{"className":71949},[162],[15,71951,1224],{"className":71952,"style":1515},[66,67],[15,71954,335],{"className":71955},[422],[15,71957],{"className":71958,"style":697},[71],[15,71960,4556],{"className":71961},[66,67],[15,71963,126],{"className":71964},[169],[15,71966],{"className":71967,"style":173},[71],[15,71969,129],{"className":71970},[177],[15,71972],{"className":71973,"style":173},[71],[15,71975,71977,71980,71983,71986,71989,71992],{"className":71976},[57],[15,71978],{"className":71979,"style":154},[61],[15,71981,120],{"className":71982},[162],[15,71984,1224],{"className":71985,"style":1515},[66,67],[15,71987],{"className":71988,"style":72},[71],[15,71990,6602],{"className":71991},[76],[15,71993],{"className":71994,"style":72},[71],[15,71996,71998,72001,72004],{"className":71997},[57],[15,71999],{"className":72000,"style":9154},[61],[15,72002,4556],{"className":72003},[66,67],[15,72005,72007,72010],{"className":72006},[169],[15,72008,126],{"className":72009},[169],[15,72011,72013],{"className":72012},[198],[15,72014,72016],{"className":72015},[202],[15,72017,72019],{"className":72018},[206],[15,72020,72022],{"className":72021,"style":6740},[210],[15,72023,72024,72027],{"style":213},[15,72025],{"className":72026,"style":218},[217],[15,72028,72030],{"className":72029},[222,223,224,225],[15,72031,1263],{"className":72032},[66,225],"。此时",[15,72035,72037],{"className":72036,"translate":19},[294],[15,72038,72040,72182],{"className":72039,"translate":19},[18],[15,72041,72043],{"className":72042},[23],[25,72044,72045],{"xmlns":27,"display":303},[29,72046,72047,72179],{},[1142,72048,72049,72107],{"rowspacing":62980,"columnalign":62981,"columnspacing":634},[1146,72050,72051,72069],{},[1149,72052,72053],{},[62987,72054,72055],{"scriptlevel":1804,"displaystyle":53},[32,72056,72057,72059,72061,72063,72065,72067],{},[35,72058,63107],{},[39,72060,120],{"stretchy":119},[35,72062,4556],{},[39,72064,335],{"separator":53},[35,72066,123],{},[39,72068,126],{"stretchy":119},[1149,72070,72071],{},[62987,72072,72073],{"scriptlevel":1804,"displaystyle":53},[32,72074,72075,72077,72079,72081,72083,72085,72087,72089,72091,72097,72099,72101,72103,72105],{},[32,72076],{},[39,72078,129],{},[35,72080,62381],{"mathvariant":454},[39,72082,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,72084,120],{"stretchy":119},[35,72086,61885],{},[39,72088,6602],{},[35,72090,4556],{},[131,72092,72093,72095],{},[39,72094,126],{"stretchy":119},[1209,72096,1263],{},[39,72098,34430],{},[35,72100,446],{},[39,72102,129],{},[35,72104,123],{},[39,72106,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[1146,72108,72109,72115],{},[1149,72110,72111],{},[62987,72112,72113],{"scriptlevel":1804,"displaystyle":53},[32,72114],{},[1149,72116,72117],{},[62987,72118,72119],{"scriptlevel":1804,"displaystyle":53},[32,72120,72121,72123,72125,72133,72135,72137,72139,72141,72143,72145,72147,72149,72151,72153,72155,72157,72159,72161,72163,72165,72167,72169,72171,72177],{},[32,72122],{},[39,72124,129],{},[32,72126,72127,72129,72131],{},[35,72128,31732],{"mathvariant":138},[35,72130,4556],{"mathvariant":138},[35,72132,63107],{"mathvariant":138},[39,72134,120],{"stretchy":119},[35,72136,61885],{},[39,72138,34430],{},[35,72140,446],{},[39,72142,129],{},[35,72144,123],{},[39,72146,126],{"stretchy":119},[39,72148,1245],{},[39,72150,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,72152,4556],{},[39,72154,6602],{},[35,72156,62381],{"mathvariant":454},[39,72158,1194],{"stretchy":119},[35,72160,61885],{},[39,72162,34430],{},[35,72164,446],{},[39,72166,129],{},[35,72168,123],{},[39,72170,1266],{"stretchy":119},[131,72172,72173,72175],{},[39,72174,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,72176,1263],{},[39,72178,335],{"separator":53},[45,72180,72181],{"encoding":47},"\\begin{aligned}\nr(a,x)\n&=\\mathbb E\\big[(Y-a)^2\\mid X=x\\big]\\\\\n&=\\mathrm{Var}(Y\\mid X=x)+\\big(a-\\mathbb E[Y\\mid X=x]\\big)^2,\n\\end{aligned}",[15,72183,72185],{"className":72184,"ariaHidden":53},[52],[15,72186,72188,72192],{"className":72187},[57],[15,72189],{"className":72190,"style":72191},[61],"height:2.9381em;vertical-align:-1.2191em;",[15,72193,72195],{"className":72194},[66],[15,72196,72198,72263],{"className":72197},[1142],[15,72199,72201],{"className":72200},[63150],[15,72202,72204,72254],{"className":72203},[202,1090],[15,72205,72207,72251],{"className":72206},[206],[15,72208,72211,72242],{"className":72209,"style":72210},[210],"height:1.7191em;",[15,72212,72214,72218],{"style":72213},"top:-3.909em;",[15,72215],{"className":72216,"style":72217},[217],"height:3.054em;",[15,72219,72221,72224,72227,72230,72233,72236,72239],{"className":72220},[66],[15,72222,63107],{"className":72223,"style":23025},[66,67],[15,72225,120],{"className":72226},[162],[15,72228,4556],{"className":72229},[66,67],[15,72231,335],{"className":72232},[422],[15,72234],{"className":72235,"style":697},[71],[15,72237,123],{"className":72238},[66,67],[15,72240,126],{"className":72241},[169],[15,72243,72245,72248],{"style":72244},"top:-2.1949em;",[15,72246],{"className":72247,"style":72217},[217],[15,72249],{"className":72250},[66],[15,72252,1114],{"className":72253},[1113],[15,72255,72257],{"className":72256},[206],[15,72258,72261],{"className":72259,"style":72260},[210],"height:1.2191em;",[15,72262],{},[15,72264,72266],{"className":72265},[63207],[15,72267,72269,72542],{"className":72268},[202,1090],[15,72270,72272,72539],{"className":72271},[206],[15,72273,72275,72381],{"className":72274,"style":72210},[210],[15,72276,72277,72280],{"style":72213},[15,72278],{"className":72279,"style":72217},[217],[15,72281,72283,72286,72289,72292,72295,72298,72304,72307,72310,72313,72316,72319,72322,72351,72354,72357,72360,72363,72366,72369,72372,72375],{"className":72282},[66],[15,72284],{"className":72285},[66],[15,72287],{"className":72288,"style":173},[71],[15,72290,129],{"className":72291},[177],[15,72293],{"className":72294,"style":173},[71],[15,72296,62381],{"className":72297},[66,517],[15,72299,72301],{"className":72300},[66],[15,72302,1194],{"className":72303},[1422,1508],[15,72305,120],{"className":72306},[162],[15,72308,61885],{"className":72309,"style":72},[66,67],[15,72311],{"className":72312,"style":72},[71],[15,72314,6602],{"className":72315},[76],[15,72317],{"className":72318,"style":72},[71],[15,72320,4556],{"className":72321},[66,67],[15,72323,72325,72328],{"className":72324},[169],[15,72326,126],{"className":72327},[169],[15,72329,72331],{"className":72330},[198],[15,72332,72334],{"className":72333},[202],[15,72335,72337],{"className":72336},[206],[15,72338,72340],{"className":72339,"style":1650},[210],[15,72341,72342,72345],{"style":397},[15,72343],{"className":72344,"style":218},[217],[15,72346,72348],{"className":72347},[222,223,224,225],[15,72349,1263],{"className":72350},[66,225],[15,72352],{"className":72353,"style":173},[71],[15,72355,34430],{"className":72356},[177],[15,72358],{"className":72359,"style":173},[71],[15,72361,446],{"className":72362,"style":16497},[66,67],[15,72364],{"className":72365,"style":173},[71],[15,72367,129],{"className":72368},[177],[15,72370],{"className":72371,"style":173},[71],[15,72373,123],{"className":72374},[66,67],[15,72376,72378],{"className":72377},[66],[15,72379,1266],{"className":72380},[1422,1508],[15,72382,72383,72386],{"style":72244},[15,72384],{"className":72385,"style":72217},[217],[15,72387,72389,72392,72395,72398,72401,72408,72411,72414,72417,72420,72423,72426,72429,72432,72435,72438,72441,72444,72447,72450,72456,72459,72462,72465,72468,72471,72474,72477,72480,72483,72486,72489,72492,72495,72498,72501,72504,72536],{"className":72388},[66],[15,72390],{"className":72391},[66],[15,72393],{"className":72394,"style":173},[71],[15,72396,129],{"className":72397},[177],[15,72399],{"className":72400,"style":173},[71],[15,72402,72404],{"className":72403},[66],[15,72405,72407],{"className":72406},[66,4429],"Var",[15,72409,120],{"className":72410},[162],[15,72412,61885],{"className":72413,"style":72},[66,67],[15,72415],{"className":72416,"style":173},[71],[15,72418,34430],{"className":72419},[177],[15,72421],{"className":72422,"style":173},[71],[15,72424,446],{"className":72425,"style":16497},[66,67],[15,72427],{"className":72428,"style":173},[71],[15,72430,129],{"className":72431},[177],[15,72433],{"className":72434,"style":173},[71],[15,72436,123],{"className":72437},[66,67],[15,72439,126],{"className":72440},[169],[15,72442],{"className":72443,"style":72},[71],[15,72445,1245],{"className":72446},[76],[15,72448],{"className":72449,"style":72},[71],[15,72451,72453],{"className":72452},[66],[15,72454,120],{"className":72455},[1422,1508],[15,72457,4556],{"className":72458},[66,67],[15,72460],{"className":72461,"style":72},[71],[15,72463,6602],{"className":72464},[76],[15,72466],{"className":72467,"style":72},[71],[15,72469,62381],{"className":72470},[66,517],[15,72472,1194],{"className":72473},[162],[15,72475,61885],{"className":72476,"style":72},[66,67],[15,72478],{"className":72479,"style":173},[71],[15,72481,34430],{"className":72482},[177],[15,72484],{"className":72485,"style":173},[71],[15,72487,446],{"className":72488,"style":16497},[66,67],[15,72490],{"className":72491,"style":173},[71],[15,72493,129],{"className":72494},[177],[15,72496],{"className":72497,"style":173},[71],[15,72499,123],{"className":72500},[66,67],[15,72502,1266],{"className":72503},[169],[15,72505,72507,72513],{"className":72506},[66],[15,72508,72510],{"className":72509},[66],[15,72511,126],{"className":72512},[1422,1508],[15,72514,72516],{"className":72515},[198],[15,72517,72519],{"className":72518},[202],[15,72520,72522],{"className":72521},[206],[15,72523,72525],{"className":72524,"style":21271},[210],[15,72526,72527,72530],{"style":21274},[15,72528],{"className":72529,"style":218},[217],[15,72531,72533],{"className":72532},[222,223,224,225],[15,72534,1263],{"className":72535},[66,225],[15,72537,335],{"className":72538},[422],[15,72540,1114],{"className":72541},[1113],[15,72543,72545],{"className":72544},[206],[15,72546,72548],{"className":72547,"style":72260},[210],[15,72549],{},[11,72551,72552,72553,72663,72664,72667],{},"第二项在 ",[15,72554,72556,72588],{"className":72555,"translate":19},[18],[15,72557,72559],{"className":72558},[23],[25,72560,72561],{"xmlns":27},[29,72562,72563,72585],{},[32,72564,72565,72567,72569,72571,72573,72575,72577,72579,72581,72583],{},[35,72566,4556],{},[39,72568,129],{},[35,72570,62381],{"mathvariant":454},[39,72572,1194],{"stretchy":119},[35,72574,61885],{},[39,72576,34430],{},[35,72578,446],{},[39,72580,129],{},[35,72582,123],{},[39,72584,1266],{"stretchy":119},[45,72586,72587],{"encoding":47},"a=\\mathbb E[Y\\mid X=x]",[15,72589,72591,72609,72633,72651],{"className":72590,"ariaHidden":53},[52],[15,72592,72594,72597,72600,72603,72606],{"className":72593},[57],[15,72595],{"className":72596,"style":86},[61],[15,72598,4556],{"className":72599},[66,67],[15,72601],{"className":72602,"style":173},[71],[15,72604,129],{"className":72605},[177],[15,72607],{"className":72608,"style":173},[71],[15,72610,72612,72615,72618,72621,72624,72627,72630],{"className":72611},[57],[15,72613],{"className":72614,"style":154},[61],[15,72616,62381],{"className":72617},[66,517],[15,72619,1194],{"className":72620},[162],[15,72622,61885],{"className":72623,"style":72},[66,67],[15,72625],{"className":72626,"style":173},[71],[15,72628,34430],{"className":72629},[177],[15,72631],{"className":72632,"style":173},[71],[15,72634,72636,72639,72642,72645,72648],{"className":72635},[57],[15,72637],{"className":72638,"style":490},[61],[15,72640,446],{"className":72641,"style":16497},[66,67],[15,72643],{"className":72644,"style":173},[71],[15,72646,129],{"className":72647},[177],[15,72649],{"className":72650,"style":173},[71],[15,72652,72654,72657,72660],{"className":72653},[57],[15,72655],{"className":72656,"style":154},[61],[15,72658,123],{"className":72659},[66,67],[15,72661,1266],{"className":72662},[169]," 时取到最小值。因此平方损失下的贝叶斯行动是",[2040,72665,72666],{},"后验均值","，也就是回归函数",[15,72669,72671],{"className":72670,"translate":19},[294],[15,72672,72674,72718],{"className":72673,"translate":19},[18],[15,72675,72677],{"className":72676},[23],[25,72678,72679],{"xmlns":27,"display":303},[29,72680,72681,72715],{},[32,72682,72683,72689,72691,72693,72695,72697,72699,72701,72703,72705,72707,72709,72711,72713],{},[131,72684,72685,72687],{},[35,72686,116],{},[39,72688,1162],{},[39,72690,120],{"stretchy":119},[35,72692,123],{},[39,72694,126],{"stretchy":119},[39,72696,129],{},[35,72698,62381],{"mathvariant":454},[39,72700,1194],{"stretchy":119},[35,72702,61885],{},[39,72704,34430],{},[35,72706,446],{},[39,72708,129],{},[35,72710,123],{},[39,72712,1266],{"stretchy":119},[35,72714,3573],{"mathvariant":138},[45,72716,72717],{"encoding":47},"f^*(x)=\\mathbb E[Y\\mid X=x].",[15,72719,72721,72774,72798,72816],{"className":72720,"ariaHidden":53},[52],[15,72722,72724,72727,72756,72759,72762,72765,72768,72771],{"className":72723},[57],[15,72725],{"className":72726,"style":154},[61],[15,72728,72730,72733],{"className":72729},[66],[15,72731,116],{"className":72732,"style":158},[66,67],[15,72734,72736],{"className":72735},[198],[15,72737,72739],{"className":72738},[202],[15,72740,72742],{"className":72741},[206],[15,72743,72745],{"className":72744,"style":1308},[210],[15,72746,72747,72750],{"style":397},[15,72748],{"className":72749,"style":218},[217],[15,72751,72753],{"className":72752},[222,223,224,225],[15,72754,1162],{"className":72755},[76,225],[15,72757,120],{"className":72758},[162],[15,72760,123],{"className":72761},[66,67],[15,72763,126],{"className":72764},[169],[15,72766],{"className":72767,"style":173},[71],[15,72769,129],{"className":72770},[177],[15,72772],{"className":72773,"style":173},[71],[15,72775,72777,72780,72783,72786,72789,72792,72795],{"className":72776},[57],[15,72778],{"className":72779,"style":154},[61],[15,72781,62381],{"className":72782},[66,517],[15,72784,1194],{"className":72785},[162],[15,72787,61885],{"className":72788,"style":72},[66,67],[15,72790],{"className":72791,"style":173},[71],[15,72793,34430],{"className":72794},[177],[15,72796],{"className":72797,"style":173},[71],[15,72799,72801,72804,72807,72810,72813],{"className":72800},[57],[15,72802],{"className":72803,"style":490},[61],[15,72805,446],{"className":72806,"style":16497},[66,67],[15,72808],{"className":72809,"style":173},[71],[15,72811,129],{"className":72812},[177],[15,72814],{"className":72815,"style":173},[71],[15,72817,72819,72822,72825,72828],{"className":72818},[57],[15,72820],{"className":72821,"style":154},[61],[15,72823,123],{"className":72824},[66,67],[15,72826,1266],{"className":72827},[169],[15,72829,3573],{"className":72830},[66],[11,72832,72833,72834,72862,72863,72922],{},"这个结论还可以从另一个角度理解：后验均值是 ",[15,72835,72837,72850],{"className":72836,"translate":19},[18],[15,72838,72840],{"className":72839},[23],[25,72841,72842],{"xmlns":27},[29,72843,72844,72848],{},[32,72845,72846],{},[35,72847,61885],{},[45,72849,61885],{"encoding":47},[15,72851,72853],{"className":72852,"ariaHidden":53},[52],[15,72854,72856,72859],{"className":72855},[57],[15,72857],{"className":72858,"style":490},[61],[15,72860,61885],{"className":72861,"style":72},[66,67]," 在 ",[15,72864,72866,72884],{"className":72865,"translate":19},[18],[15,72867,72869],{"className":72868},[23],[25,72870,72871],{"xmlns":27},[29,72872,72873,72881],{},[32,72874,72875],{},[131,72876,72877,72879],{},[35,72878,1216],{},[1209,72880,1263],{},[45,72882,72883],{"encoding":47},"L^2",[15,72885,72887],{"className":72886,"ariaHidden":53},[52],[15,72888,72890,72893],{"className":72889},[57],[15,72891],{"className":72892,"style":6740},[61],[15,72894,72896,72899],{"className":72895},[66],[15,72897,1216],{"className":72898},[66,67],[15,72900,72902],{"className":72901},[198],[15,72903,72905],{"className":72904},[202],[15,72906,72908],{"className":72907},[206],[15,72909,72911],{"className":72910,"style":6740},[210],[15,72912,72913,72916],{"style":213},[15,72914],{"className":72915,"style":218},[217],[15,72917,72919],{"className":72918},[222,223,224,225],[15,72920,1263],{"className":72921},[66,225]," 意义下最好的预测，任何其他预测都会多出一项方差。",[1904,72924,72926],{"id":72925},"_34-绝对损失后验中位数","3.4 绝对损失：后验中位数",[11,72928,72929,72930,73041,73042,73212,73213,73241,73242,73270],{},"若损失为 ",[15,72931,72933,72969],{"className":72932,"translate":19},[18],[15,72934,72936],{"className":72935},[23],[25,72937,72938],{"xmlns":27},[29,72939,72940,72966],{},[32,72941,72942,72944,72946,72948,72950,72952,72954,72956,72958,72960,72962,72964],{},[35,72943,1216],{},[39,72945,120],{"stretchy":119},[35,72947,1224],{},[39,72949,335],{"separator":53},[35,72951,4556],{},[39,72953,126],{"stretchy":119},[39,72955,129],{},[35,72957,34430],{"mathvariant":138},[35,72959,1224],{},[39,72961,6602],{},[35,72963,4556],{},[35,72965,34430],{"mathvariant":138},[45,72967,72968],{"encoding":47},"L(y,a)=|y-a|",[15,72970,72972,73008,73029],{"className":72971,"ariaHidden":53},[52],[15,72973,72975,72978,72981,72984,72987,72990,72993,72996,72999,73002,73005],{"className":72974},[57],[15,72976],{"className":72977,"style":154},[61],[15,72979,1216],{"className":72980},[66,67],[15,72982,120],{"className":72983},[162],[15,72985,1224],{"className":72986,"style":1515},[66,67],[15,72988,335],{"className":72989},[422],[15,72991],{"className":72992,"style":697},[71],[15,72994,4556],{"className":72995},[66,67],[15,72997,126],{"className":72998},[169],[15,73000],{"className":73001,"style":173},[71],[15,73003,129],{"className":73004},[177],[15,73006],{"className":73007,"style":173},[71],[15,73009,73011,73014,73017,73020,73023,73026],{"className":73010},[57],[15,73012],{"className":73013,"style":154},[61],[15,73015,34430],{"className":73016},[66],[15,73018,1224],{"className":73019,"style":1515},[66,67],[15,73021],{"className":73022,"style":72},[71],[15,73024,6602],{"className":73025},[76],[15,73027],{"className":73028,"style":72},[71],[15,73030,73032,73035,73038],{"className":73031},[57],[15,73033],{"className":73034,"style":154},[61],[15,73036,4556],{"className":73037},[66,67],[15,73039,34430],{"className":73040},[66],"，条件风险为 ",[15,73043,73045,73095],{"className":73044,"translate":19},[18],[15,73046,73048],{"className":73047},[23],[25,73049,73050],{"xmlns":27},[29,73051,73052,73092],{},[32,73053,73054,73056,73058,73060,73062,73064,73066,73068,73070,73072,73074,73076,73078,73080,73082,73084,73086,73088,73090],{},[35,73055,63107],{},[39,73057,120],{"stretchy":119},[35,73059,4556],{},[39,73061,335],{"separator":53},[35,73063,123],{},[39,73065,126],{"stretchy":119},[39,73067,129],{},[35,73069,62381],{"mathvariant":454},[39,73071,1194],{"stretchy":119},[35,73073,34430],{"mathvariant":138},[35,73075,61885],{},[39,73077,6602],{},[35,73079,4556],{},[35,73081,34430],{"mathvariant":138},[39,73083,34430],{},[35,73085,446],{},[39,73087,129],{},[35,73089,123],{},[39,73091,1266],{"stretchy":119},[45,73093,73094],{"encoding":47},"r(a,x)=\\mathbb E[|Y-a|\\mid X=x]",[15,73096,73098,73134,73161,73182,73200],{"className":73097,"ariaHidden":53},[52],[15,73099,73101,73104,73107,73110,73113,73116,73119,73122,73125,73128,73131],{"className":73100},[57],[15,73102],{"className":73103,"style":154},[61],[15,73105,63107],{"className":73106,"style":23025},[66,67],[15,73108,120],{"className":73109},[162],[15,73111,4556],{"className":73112},[66,67],[15,73114,335],{"className":73115},[422],[15,73117],{"className":73118,"style":697},[71],[15,73120,123],{"className":73121},[66,67],[15,73123,126],{"className":73124},[169],[15,73126],{"className":73127,"style":173},[71],[15,73129,129],{"className":73130},[177],[15,73132],{"className":73133,"style":173},[71],[15,73135,73137,73140,73143,73146,73149,73152,73155,73158],{"className":73136},[57],[15,73138],{"className":73139,"style":154},[61],[15,73141,62381],{"className":73142},[66,517],[15,73144,1194],{"className":73145},[162],[15,73147,34430],{"className":73148},[66],[15,73150,61885],{"className":73151,"style":72},[66,67],[15,73153],{"className":73154,"style":72},[71],[15,73156,6602],{"className":73157},[76],[15,73159],{"className":73160,"style":72},[71],[15,73162,73164,73167,73170,73173,73176,73179],{"className":73163},[57],[15,73165],{"className":73166,"style":154},[61],[15,73168,4556],{"className":73169},[66,67],[15,73171,34430],{"className":73172},[66],[15,73174],{"className":73175,"style":173},[71],[15,73177,34430],{"className":73178},[177],[15,73180],{"className":73181,"style":173},[71],[15,73183,73185,73188,73191,73194,73197],{"className":73184},[57],[15,73186],{"className":73187,"style":490},[61],[15,73189,446],{"className":73190,"style":16497},[66,67],[15,73192],{"className":73193,"style":173},[71],[15,73195,129],{"className":73196},[177],[15,73198],{"className":73199,"style":173},[71],[15,73201,73203,73206,73209],{"className":73202},[57],[15,73204],{"className":73205,"style":154},[61],[15,73207,123],{"className":73208},[66,67],[15,73210,1266],{"className":73211},[169],"。对 ",[15,73214,73216,73229],{"className":73215,"translate":19},[18],[15,73217,73219],{"className":73218},[23],[25,73220,73221],{"xmlns":27},[29,73222,73223,73227],{},[32,73224,73225],{},[35,73226,4556],{},[45,73228,4556],{"encoding":47},[15,73230,73232],{"className":73231,"ariaHidden":53},[52],[15,73233,73235,73238],{"className":73234},[57],[15,73236],{"className":73237,"style":86},[61],[15,73239,4556],{"className":73240},[66,67]," 求次梯度，在 ",[15,73243,73245,73258],{"className":73244,"translate":19},[18],[15,73246,73248],{"className":73247},[23],[25,73249,73250],{"xmlns":27},[29,73251,73252,73256],{},[32,73253,73254],{},[35,73255,61885],{},[45,73257,61885],{"encoding":47},[15,73259,73261],{"className":73260,"ariaHidden":53},[52],[15,73262,73264,73267],{"className":73263},[57],[15,73265],{"className":73266,"style":490},[61],[15,73268,61885],{"className":73269,"style":72},[66,67]," 的分布连续时它就是导数：",[15,73272,73274],{"className":73273,"translate":19},[294],[15,73275,73277,73346],{"className":73276,"translate":19},[18],[15,73278,73280],{"className":73279},[23],[25,73281,73282],{"xmlns":27,"display":303},[29,73283,73284,73343],{},[32,73285,73286,73293,73295,73297,73299,73301,73303,73305,73307,73309,73311,73313,73315,73317,73319,73321,73323,73325,73327,73329,73331,73333,73335,73337,73339,73341],{},[1057,73287,73288,73291],{},[35,73289,73290],{"mathvariant":138},"∂",[35,73292,4556],{},[35,73294,63107],{},[39,73296,120],{"stretchy":119},[35,73298,4556],{},[39,73300,335],{"separator":53},[35,73302,123],{},[39,73304,126],{"stretchy":119},[39,73306,129],{},[35,73308,28399],{},[39,73310,120],{"stretchy":119},[35,73312,61885],{},[39,73314,25772],{},[35,73316,4556],{},[39,73318,34430],{},[35,73320,123],{},[39,73322,126],{"stretchy":119},[39,73324,6602],{},[35,73326,28399],{},[39,73328,120],{"stretchy":119},[35,73330,61885],{},[39,73332,9283],{},[35,73334,4556],{},[39,73336,34430],{},[35,73338,123],{},[39,73340,126],{"stretchy":119},[39,73342,335],{"separator":53},[45,73344,73345],{"encoding":47},"\\partial_a r(a,x)=P(Y\u003Ca\\mid x)-P(Y>a\\mid x),",[15,73347,73349,73427,73451,73469,73490,73514,73532],{"className":73348,"ariaHidden":53},[52],[15,73350,73352,73355,73397,73400,73403,73406,73409,73412,73415,73418,73421,73424],{"className":73351},[57],[15,73353],{"className":73354,"style":154},[61],[15,73356,73358,73362],{"className":73357},[66],[15,73359,73290],{"className":73360,"style":73361},[66],"margin-right:0.0556em;",[15,73363,73365],{"className":73364},[198],[15,73366,73368,73389],{"className":73367},[202,1090],[15,73369,73371,73386],{"className":73370},[206],[15,73372,73374],{"className":73373,"style":2198},[210],[15,73375,73377,73380],{"style":73376},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[15,73378],{"className":73379,"style":218},[217],[15,73381,73383],{"className":73382},[222,223,224,225],[15,73384,4556],{"className":73385},[66,67,225],[15,73387,1114],{"className":73388},[1113],[15,73390,73392],{"className":73391},[206],[15,73393,73395],{"className":73394,"style":1121},[210],[15,73396],{},[15,73398,63107],{"className":73399,"style":23025},[66,67],[15,73401,120],{"className":73402},[162],[15,73404,4556],{"className":73405},[66,67],[15,73407,335],{"className":73408},[422],[15,73410],{"className":73411,"style":697},[71],[15,73413,123],{"className":73414},[66,67],[15,73416,126],{"className":73417},[169],[15,73419],{"className":73420,"style":173},[71],[15,73422,129],{"className":73423},[177],[15,73425],{"className":73426,"style":173},[71],[15,73428,73430,73433,73436,73439,73442,73445,73448],{"className":73429},[57],[15,73431],{"className":73432,"style":154},[61],[15,73434,28399],{"className":73435,"style":62513},[66,67],[15,73437,120],{"className":73438},[162],[15,73440,61885],{"className":73441,"style":72},[66,67],[15,73443],{"className":73444,"style":173},[71],[15,73446,25772],{"className":73447},[177],[15,73449],{"className":73450,"style":173},[71],[15,73452,73454,73457,73460,73463,73466],{"className":73453},[57],[15,73455],{"className":73456,"style":154},[61],[15,73458,4556],{"className":73459},[66,67],[15,73461],{"className":73462,"style":173},[71],[15,73464,34430],{"className":73465},[177],[15,73467],{"className":73468,"style":173},[71],[15,73470,73472,73475,73478,73481,73484,73487],{"className":73471},[57],[15,73473],{"className":73474,"style":154},[61],[15,73476,123],{"className":73477},[66,67],[15,73479,126],{"className":73480},[169],[15,73482],{"className":73483,"style":72},[71],[15,73485,6602],{"className":73486},[76],[15,73488],{"className":73489,"style":72},[71],[15,73491,73493,73496,73499,73502,73505,73508,73511],{"className":73492},[57],[15,73494],{"className":73495,"style":154},[61],[15,73497,28399],{"className":73498,"style":62513},[66,67],[15,73500,120],{"className":73501},[162],[15,73503,61885],{"className":73504,"style":72},[66,67],[15,73506],{"className":73507,"style":173},[71],[15,73509,9283],{"className":73510},[177],[15,73512],{"className":73513,"style":173},[71],[15,73515,73517,73520,73523,73526,73529],{"className":73516},[57],[15,73518],{"className":73519,"style":154},[61],[15,73521,4556],{"className":73522},[66,67],[15,73524],{"className":73525,"style":173},[71],[15,73527,34430],{"className":73528},[177],[15,73530],{"className":73531,"style":173},[71],[15,73533,73535,73538,73541,73544],{"className":73534},[57],[15,73536],{"className":73537,"style":154},[61],[15,73539,123],{"className":73540},[66,67],[15,73542,126],{"className":73543},[169],[15,73545,335],{"className":73546},[422],[11,73548,73549,73550,73578],{},"令 ",[15,73551,73553,73566],{"className":73552,"translate":19},[18],[15,73554,73556],{"className":73555},[23],[25,73557,73558],{"xmlns":27},[29,73559,73560,73564],{},[32,73561,73562],{},[1209,73563,1804],{},[45,73565,1804],{"encoding":47},[15,73567,73569],{"className":73568,"ariaHidden":53},[52],[15,73570,73572,73575],{"className":73571},[57],[15,73573],{"className":73574,"style":1836},[61],[15,73576,1804],{"className":73577},[66]," 属于次梯度，等价于",[15,73580,73582],{"className":73581,"translate":19},[294],[15,73583,73585,73651],{"className":73584,"translate":19},[18],[15,73586,73588],{"className":73587},[23],[25,73589,73590],{"xmlns":27,"display":303},[29,73591,73592,73648],{},[32,73593,73594,73596,73598,73600,73602,73604,73606,73608,73610,73612,73618,73620,73622,73624,73626,73628,73630,73632,73634,73636,73638,73640,73646],{},[35,73595,28399],{},[39,73597,120],{"stretchy":119},[35,73599,61885],{},[39,73601,24220],{},[35,73603,4556],{},[39,73605,34430],{},[35,73607,123],{},[39,73609,126],{"stretchy":119},[39,73611,1801],{},[6467,73613,73614,73616],{},[1209,73615,1211],{},[1209,73617,1263],{},[39,73619,335],{"separator":53},[71,73621],{"width":3904},[35,73623,28399],{},[39,73625,120],{"stretchy":119},[35,73627,61885],{},[39,73629,1801],{},[35,73631,4556],{},[39,73633,34430],{},[35,73635,123],{},[39,73637,126],{"stretchy":119},[39,73639,1801],{},[6467,73641,73642,73644],{},[1209,73643,1211],{},[1209,73645,1263],{},[35,73647,3573],{"mathvariant":138},[45,73649,73650],{"encoding":47},"P(Y\\le a\\mid x)\\ge \\frac12,\\qquad P(Y\\ge a\\mid x)\\ge \\frac12 .",[15,73652,73654,73678,73696,73717,73812,73830,73851],{"className":73653,"ariaHidden":53},[52],[15,73655,73657,73660,73663,73666,73669,73672,73675],{"className":73656},[57],[15,73658],{"className":73659,"style":154},[61],[15,73661,28399],{"className":73662,"style":62513},[66,67],[15,73664,120],{"className":73665},[162],[15,73667,61885],{"className":73668,"style":72},[66,67],[15,73670],{"className":73671,"style":173},[71],[15,73673,24220],{"className":73674},[177],[15,73676],{"className":73677,"style":173},[71],[15,73679,73681,73684,73687,73690,73693],{"className":73680},[57],[15,73682],{"className":73683,"style":154},[61],[15,73685,4556],{"className":73686},[66,67],[15,73688],{"className":73689,"style":173},[71],[15,73691,34430],{"className":73692},[177],[15,73694],{"className":73695,"style":173},[71],[15,73697,73699,73702,73705,73708,73711,73714],{"className":73698},[57],[15,73700],{"className":73701,"style":154},[61],[15,73703,123],{"className":73704},[66,67],[15,73706,126],{"className":73707},[169],[15,73709],{"className":73710,"style":173},[71],[15,73712,1801],{"className":73713},[177],[15,73715],{"className":73716,"style":173},[71],[15,73718,73720,73723,73785,73788,73791,73794,73797,73800,73803,73806,73809],{"className":73719},[57],[15,73721],{"className":73722,"style":49086},[61],[15,73724,73726,73729,73782],{"className":73725},[66],[15,73727],{"className":73728},[162,23417],[15,73730,73732],{"className":73731},[6467],[15,73733,73735,73774],{"className":73734},[202,1090],[15,73736,73738,73771],{"className":73737},[206],[15,73739,73741,73752,73760],{"className":73740,"style":23430},[210],[15,73742,73743,73746],{"style":23433},[15,73744],{"className":73745,"style":1368},[217],[15,73747,73749],{"className":73748},[66],[15,73750,1263],{"className":73751},[66],[15,73753,73754,73757],{"style":23445},[15,73755],{"className":73756,"style":1368},[217],[15,73758],{"className":73759,"style":4440},[23452],[15,73761,73762,73765],{"style":23455},[15,73763],{"className":73764,"style":1368},[217],[15,73766,73768],{"className":73767},[66],[15,73769,1211],{"className":73770},[66],[15,73772,1114],{"className":73773},[1113],[15,73775,73777],{"className":73776},[206],[15,73778,73780],{"className":73779,"style":23474},[210],[15,73781],{},[15,73783],{"className":73784},[169,23417],[15,73786,335],{"className":73787},[422],[15,73789],{"className":73790,"style":4061},[71],[15,73792],{"className":73793,"style":697},[71],[15,73795,28399],{"className":73796,"style":62513},[66,67],[15,73798,120],{"className":73799},[162],[15,73801,61885],{"className":73802,"style":72},[66,67],[15,73804],{"className":73805,"style":173},[71],[15,73807,1801],{"className":73808},[177],[15,73810],{"className":73811,"style":173},[71],[15,73813,73815,73818,73821,73824,73827],{"className":73814},[57],[15,73816],{"className":73817,"style":154},[61],[15,73819,4556],{"className":73820},[66,67],[15,73822],{"className":73823,"style":173},[71],[15,73825,34430],{"className":73826},[177],[15,73828],{"className":73829,"style":173},[71],[15,73831,73833,73836,73839,73842,73845,73848],{"className":73832},[57],[15,73834],{"className":73835,"style":154},[61],[15,73837,123],{"className":73838},[66,67],[15,73840,126],{"className":73841},[169],[15,73843],{"className":73844,"style":173},[71],[15,73846,1801],{"className":73847},[177],[15,73849],{"className":73850,"style":173},[71],[15,73852,73854,73857,73919],{"className":73853},[57],[15,73855],{"className":73856,"style":49086},[61],[15,73858,73860,73863,73916],{"className":73859},[66],[15,73861],{"className":73862},[162,23417],[15,73864,73866],{"className":73865},[6467],[15,73867,73869,73908],{"className":73868},[202,1090],[15,73870,73872,73905],{"className":73871},[206],[15,73873,73875,73886,73894],{"className":73874,"style":23430},[210],[15,73876,73877,73880],{"style":23433},[15,73878],{"className":73879,"style":1368},[217],[15,73881,73883],{"className":73882},[66],[15,73884,1263],{"className":73885},[66],[15,73887,73888,73891],{"style":23445},[15,73889],{"className":73890,"style":1368},[217],[15,73892],{"className":73893,"style":4440},[23452],[15,73895,73896,73899],{"style":23455},[15,73897],{"className":73898,"style":1368},[217],[15,73900,73902],{"className":73901},[66],[15,73903,1211],{"className":73904},[66],[15,73906,1114],{"className":73907},[1113],[15,73909,73911],{"className":73910},[206],[15,73912,73914],{"className":73913,"style":23474},[210],[15,73915],{},[15,73917],{"className":73918},[169,23417],[15,73920,3573],{"className":73921},[66],[11,73923,73924,73925,73953,73954,73982],{},"满足这两个不等式的 ",[15,73926,73928,73941],{"className":73927,"translate":19},[18],[15,73929,73931],{"className":73930},[23],[25,73932,73933],{"xmlns":27},[29,73934,73935,73939],{},[32,73936,73937],{},[35,73938,4556],{},[45,73940,4556],{"encoding":47},[15,73942,73944],{"className":73943,"ariaHidden":53},[52],[15,73945,73947,73950],{"className":73946},[57],[15,73948],{"className":73949,"style":86},[61],[15,73951,4556],{"className":73952},[66,67]," 就是后验中位数。直观上，中位数把后验概率一分为二，左右两侧的「质量」相等，因此向左或向右移动 ",[15,73955,73957,73970],{"className":73956,"translate":19},[18],[15,73958,73960],{"className":73959},[23],[25,73961,73962],{"xmlns":27},[29,73963,73964,73968],{},[32,73965,73966],{},[35,73967,4556],{},[45,73969,4556],{"encoding":47},[15,73971,73973],{"className":73972,"ariaHidden":53},[52],[15,73974,73976,73979],{"className":73975},[57],[15,73977],{"className":73978,"style":86},[61],[15,73980,4556],{"className":73981},[66,67]," 都不会降低期望绝对误差。",[11,73984,73985],{},"平方损失对异常值敏感，因为误差被平方；绝对损失只看见误差的大小。这个差异在决策层面的表现，就是均值与中位数的差异。",[1904,73987,73989,73990],{"id":73988},"_35-非对称损失阈值不再在-121212","3.5 非对称损失：阈值不再在 ",[15,73991,73993,74010],{"className":73992,"translate":19},[18],[15,73994,73996],{"className":73995},[23],[25,73997,73998],{"xmlns":27},[29,73999,74000,74008],{},[32,74001,74002,74004,74006],{},[1209,74003,1211],{},[35,74005,6625],{"mathvariant":138},[1209,74007,1263],{},[45,74009,71809],{"encoding":47},[15,74011,74013],{"className":74012,"ariaHidden":53},[52],[15,74014,74016,74019],{"className":74015},[57],[15,74017],{"className":74018,"style":154},[61],[15,74020,71809],{"className":74021},[66],[11,74023,74024,74025,74107,74108,74189,74190,72033],{},"现实中的两类错误很少同等昂贵。设预测为正类的代价为 ",[15,74026,74028,74051],{"className":74027,"translate":19},[18],[15,74029,74031],{"className":74030},[23],[25,74032,74033],{"xmlns":27},[29,74034,74035,74048],{},[32,74036,74037],{},[1057,74038,74039,74041],{},[35,74040,2256],{},[32,74042,74043,74046],{},[35,74044,74045],{"mathvariant":138},"F",[35,74047,28399],{"mathvariant":138},[45,74049,74050],{"encoding":47},"c_{\\mathrm{FP}}",[15,74052,74054],{"className":74053,"ariaHidden":53},[52],[15,74055,74057,74060],{"className":74056},[57],[15,74058],{"className":74059,"style":10197},[61],[15,74061,74063,74066],{"className":74062},[66],[15,74064,2256],{"className":74065},[66,67],[15,74067,74069],{"className":74068},[198],[15,74070,74072,74099],{"className":74071},[202,1090],[15,74073,74075,74096],{"className":74074},[206],[15,74076,74078],{"className":74077,"style":3638},[210],[15,74079,74080,74083],{"style":1584},[15,74081],{"className":74082,"style":218},[217],[15,74084,74086],{"className":74085},[222,223,224,225],[15,74087,74089],{"className":74088},[66,225],[15,74090,74092],{"className":74091},[66,225],[15,74093,74095],{"className":74094},[66,4429,225],"FP",[15,74097,1114],{"className":74098},[1113],[15,74100,74102],{"className":74101},[206],[15,74103,74105],{"className":74104,"style":1121},[210],[15,74106],{},"（假阳性），预测为负类的代价为 ",[15,74109,74111,74133],{"className":74110,"translate":19},[18],[15,74112,74114],{"className":74113},[23],[25,74115,74116],{"xmlns":27},[29,74117,74118,74130],{},[32,74119,74120],{},[1057,74121,74122,74124],{},[35,74123,2256],{},[32,74125,74126,74128],{},[35,74127,74045],{"mathvariant":138},[35,74129,28535],{"mathvariant":138},[45,74131,74132],{"encoding":47},"c_{\\mathrm{FN}}",[15,74134,74136],{"className":74135,"ariaHidden":53},[52],[15,74137,74139,74142],{"className":74138},[57],[15,74140],{"className":74141,"style":10197},[61],[15,74143,74145,74148],{"className":74144},[66],[15,74146,2256],{"className":74147},[66,67],[15,74149,74151],{"className":74150},[198],[15,74152,74154,74181],{"className":74153},[202,1090],[15,74155,74157,74178],{"className":74156},[206],[15,74158,74160],{"className":74159,"style":3638},[210],[15,74161,74162,74165],{"style":1584},[15,74163],{"className":74164,"style":218},[217],[15,74166,74168],{"className":74167},[222,223,224,225],[15,74169,74171],{"className":74170},[66,225],[15,74172,74174],{"className":74173},[66,225],[15,74175,74177],{"className":74176},[66,4429,225],"FN",[15,74179,1114],{"className":74180},[1113],[15,74182,74184],{"className":74183},[206],[15,74185,74187],{"className":74186,"style":1121},[210],[15,74188],{},"（假阴性），正确分类的代价为 ",[15,74191,74193,74206],{"className":74192,"translate":19},[18],[15,74194,74196],{"className":74195},[23],[25,74197,74198],{"xmlns":27},[29,74199,74200,74204],{},[32,74201,74202],{},[1209,74203,1804],{},[45,74205,1804],{"encoding":47},[15,74207,74209],{"className":74208,"ariaHidden":53},[52],[15,74210,74212,74215],{"className":74211},[57],[15,74213],{"className":74214,"style":1836},[61],[15,74216,1804],{"className":74217},[66],[15,74219,74221],{"className":74220,"translate":19},[294],[15,74222,74224,74318],{"className":74223,"translate":19},[18],[15,74225,74227],{"className":74226},[23],[25,74228,74229],{"xmlns":27,"display":303},[29,74230,74231,74315],{},[32,74232,74233,74235,74237,74239,74241,74243,74245,74247,74257,74259,74261,74263,74265,74267,74269,74271,74273,74275,74277,74279,74281,74283,74285,74287,74289,74299,74301,74303,74305,74307,74309,74311,74313],{},[35,74234,63107],{},[39,74236,120],{"stretchy":119},[1209,74238,1211],{},[39,74240,335],{"separator":53},[35,74242,123],{},[39,74244,126],{"stretchy":119},[39,74246,129],{},[1057,74248,74249,74251],{},[35,74250,2256],{},[32,74252,74253,74255],{},[35,74254,74045],{"mathvariant":138},[35,74256,28399],{"mathvariant":138},[1274,74258,2462],{},[35,74260,11],{},[39,74262,120],{"stretchy":119},[1209,74264,1804],{},[39,74266,34430],{},[35,74268,123],{},[39,74270,126],{"stretchy":119},[39,74272,335],{"separator":53},[71,74274],{"width":3904},[35,74276,63107],{},[39,74278,120],{"stretchy":119},[1209,74280,1804],{},[39,74282,335],{"separator":53},[35,74284,123],{},[39,74286,126],{"stretchy":119},[39,74288,129],{},[1057,74290,74291,74293],{},[35,74292,2256],{},[32,74294,74295,74297],{},[35,74296,74045],{"mathvariant":138},[35,74298,28535],{"mathvariant":138},[1274,74300,2462],{},[35,74302,11],{},[39,74304,120],{"stretchy":119},[1209,74306,1211],{},[39,74308,34430],{},[35,74310,123],{},[39,74312,126],{"stretchy":119},[35,74314,3573],{"mathvariant":138},[45,74316,74317],{"encoding":47},"r(1,x)=c_{\\mathrm{FP}}\\,p(0\\mid x),\\qquad\nr(0,x)=c_{\\mathrm{FN}}\\,p(1\\mid x).",[15,74319,74321,74357,74430,74481,74554],{"className":74320,"ariaHidden":53},[52],[15,74322,74324,74327,74330,74333,74336,74339,74342,74345,74348,74351,74354],{"className":74323},[57],[15,74325],{"className":74326,"style":154},[61],[15,74328,63107],{"className":74329,"style":23025},[66,67],[15,74331,120],{"className":74332},[162],[15,74334,1211],{"className":74335},[66],[15,74337,335],{"className":74338},[422],[15,74340],{"className":74341,"style":697},[71],[15,74343,123],{"className":74344},[66,67],[15,74346,126],{"className":74347},[169],[15,74349],{"className":74350,"style":173},[71],[15,74352,129],{"className":74353},[177],[15,74355],{"className":74356,"style":173},[71],[15,74358,74360,74363,74409,74412,74415,74418,74421,74424,74427],{"className":74359},[57],[15,74361],{"className":74362,"style":154},[61],[15,74364,74366,74369],{"className":74365},[66],[15,74367,2256],{"className":74368},[66,67],[15,74370,74372],{"className":74371},[198],[15,74373,74375,74401],{"className":74374},[202,1090],[15,74376,74378,74398],{"className":74377},[206],[15,74379,74381],{"className":74380,"style":3638},[210],[15,74382,74383,74386],{"style":1584},[15,74384],{"className":74385,"style":218},[217],[15,74387,74389],{"className":74388},[222,223,224,225],[15,74390,74392],{"className":74391},[66,225],[15,74393,74395],{"className":74394},[66,225],[15,74396,74095],{"className":74397},[66,4429,225],[15,74399,1114],{"className":74400},[1113],[15,74402,74404],{"className":74403},[206],[15,74405,74407],{"className":74406,"style":1121},[210],[15,74408],{},[15,74410],{"className":74411,"style":697},[71],[15,74413,11],{"className":74414},[66,67],[15,74416,120],{"className":74417},[162],[15,74419,1804],{"className":74420},[66],[15,74422],{"className":74423,"style":173},[71],[15,74425,34430],{"className":74426},[177],[15,74428],{"className":74429,"style":173},[71],[15,74431,74433,74436,74439,74442,74445,74448,74451,74454,74457,74460,74463,74466,74469,74472,74475,74478],{"className":74432},[57],[15,74434],{"className":74435,"style":154},[61],[15,74437,123],{"className":74438},[66,67],[15,74440,126],{"className":74441},[169],[15,74443,335],{"className":74444},[422],[15,74446],{"className":74447,"style":4061},[71],[15,74449],{"className":74450,"style":697},[71],[15,74452,63107],{"className":74453,"style":23025},[66,67],[15,74455,120],{"className":74456},[162],[15,74458,1804],{"className":74459},[66],[15,74461,335],{"className":74462},[422],[15,74464],{"className":74465,"style":697},[71],[15,74467,123],{"className":74468},[66,67],[15,74470,126],{"className":74471},[169],[15,74473],{"className":74474,"style":173},[71],[15,74476,129],{"className":74477},[177],[15,74479],{"className":74480,"style":173},[71],[15,74482,74484,74487,74533,74536,74539,74542,74545,74548,74551],{"className":74483},[57],[15,74485],{"className":74486,"style":154},[61],[15,74488,74490,74493],{"className":74489},[66],[15,74491,2256],{"className":74492},[66,67],[15,74494,74496],{"className":74495},[198],[15,74497,74499,74525],{"className":74498},[202,1090],[15,74500,74502,74522],{"className":74501},[206],[15,74503,74505],{"className":74504,"style":3638},[210],[15,74506,74507,74510],{"style":1584},[15,74508],{"className":74509,"style":218},[217],[15,74511,74513],{"className":74512},[222,223,224,225],[15,74514,74516],{"className":74515},[66,225],[15,74517,74519],{"className":74518},[66,225],[15,74520,74177],{"className":74521},[66,4429,225],[15,74523,1114],{"className":74524},[1113],[15,74526,74528],{"className":74527},[206],[15,74529,74531],{"className":74530,"style":1121},[210],[15,74532],{},[15,74534],{"className":74535,"style":697},[71],[15,74537,11],{"className":74538},[66,67],[15,74540,120],{"className":74541},[162],[15,74543,1211],{"className":74544},[66],[15,74546],{"className":74547,"style":173},[71],[15,74549,34430],{"className":74550},[177],[15,74552],{"className":74553,"style":173},[71],[15,74555,74557,74560,74563,74566],{"className":74556},[57],[15,74558],{"className":74559,"style":154},[61],[15,74561,123],{"className":74562},[66,67],[15,74564,126],{"className":74565},[169],[15,74567,3573],{"className":74568},[66],[11,74570,74571,74572,74679],{},"选择正类当且仅当 ",[15,74573,74575,74613],{"className":74574,"translate":19},[18],[15,74576,74578],{"className":74577},[23],[25,74579,74580],{"xmlns":27},[29,74581,74582,74610],{},[32,74583,74584,74586,74588,74590,74592,74594,74596,74598,74600,74602,74604,74606,74608],{},[35,74585,63107],{},[39,74587,120],{"stretchy":119},[1209,74589,1211],{},[39,74591,335],{"separator":53},[35,74593,123],{},[39,74595,126],{"stretchy":119},[39,74597,24220],{},[35,74599,63107],{},[39,74601,120],{"stretchy":119},[1209,74603,1804],{},[39,74605,335],{"separator":53},[35,74607,123],{},[39,74609,126],{"stretchy":119},[45,74611,74612],{"encoding":47},"r(1,x)\\le r(0,x)",[15,74614,74616,74652],{"className":74615,"ariaHidden":53},[52],[15,74617,74619,74622,74625,74628,74631,74634,74637,74640,74643,74646,74649],{"className":74618},[57],[15,74620],{"className":74621,"style":154},[61],[15,74623,63107],{"className":74624,"style":23025},[66,67],[15,74626,120],{"className":74627},[162],[15,74629,1211],{"className":74630},[66],[15,74632,335],{"className":74633},[422],[15,74635],{"className":74636,"style":697},[71],[15,74638,123],{"className":74639},[66,67],[15,74641,126],{"className":74642},[169],[15,74644],{"className":74645,"style":173},[71],[15,74647,24220],{"className":74648},[177],[15,74650],{"className":74651,"style":173},[71],[15,74653,74655,74658,74661,74664,74667,74670,74673,74676],{"className":74654},[57],[15,74656],{"className":74657,"style":154},[61],[15,74659,63107],{"className":74660,"style":23025},[66,67],[15,74662,120],{"className":74663},[162],[15,74665,1804],{"className":74666},[66],[15,74668,335],{"className":74669},[422],[15,74671],{"className":74672,"style":697},[71],[15,74674,123],{"className":74675},[66,67],[15,74677,126],{"className":74678},[169],"，即",[15,74681,74683],{"className":74682,"translate":19},[294],[15,74684,74686,74754],{"className":74685,"translate":19},[18],[15,74687,74689],{"className":74688},[23],[25,74690,74691],{"xmlns":27,"display":303},[29,74692,74693,74751],{},[32,74694,74695,74697,74699,74701,74703,74705,74707,74709,74711,74713,74749],{},[35,74696,11],{},[39,74698,120],{"stretchy":119},[1209,74700,1211],{},[39,74702,34430],{},[35,74704,123],{},[39,74706,126],{"stretchy":119},[1274,74708,24233],{},[39,74710,1801],{},[1274,74712,24233],{},[6467,74714,74715,74725],{},[1057,74716,74717,74719],{},[35,74718,2256],{},[32,74720,74721,74723],{},[35,74722,74045],{"mathvariant":138},[35,74724,28399],{"mathvariant":138},[32,74726,74727,74737,74739],{},[1057,74728,74729,74731],{},[35,74730,2256],{},[32,74732,74733,74735],{},[35,74734,74045],{"mathvariant":138},[35,74736,28399],{"mathvariant":138},[39,74738,1245],{},[1057,74740,74741,74743],{},[35,74742,2256],{},[32,74744,74745,74747],{},[35,74746,74045],{"mathvariant":138},[35,74748,28535],{"mathvariant":138},[35,74750,3573],{"mathvariant":138},[45,74752,74753],{"encoding":47},"p(1\\mid x)\\ \\ge\\ \\frac{c_{\\mathrm{FP}}}{c_{\\mathrm{FP}}+c_{\\mathrm{FN}}}.",[15,74755,74757,74781,74808],{"className":74756,"ariaHidden":53},[52],[15,74758,74760,74763,74766,74769,74772,74775,74778],{"className":74759},[57],[15,74761],{"className":74762,"style":154},[61],[15,74764,11],{"className":74765},[66,67],[15,74767,120],{"className":74768},[162],[15,74770,1211],{"className":74771},[66],[15,74773],{"className":74774,"style":173},[71],[15,74776,34430],{"className":74777},[177],[15,74779],{"className":74780,"style":173},[71],[15,74782,74784,74787,74790,74793,74796,74799,74802,74805],{"className":74783},[57],[15,74785],{"className":74786,"style":154},[61],[15,74788,123],{"className":74789},[66,67],[15,74791,126],{"className":74792},[169],[15,74794,24233],{"className":74795},[71],[15,74797],{"className":74798,"style":173},[71],[15,74800,1801],{"className":74801},[177],[15,74803,24233],{"className":74804},[71],[15,74806],{"className":74807,"style":173},[71],[15,74809,74811,74814,75017],{"className":74810},[57],[15,74812],{"className":74813,"style":52422},[61],[15,74815,74817,74820,75014],{"className":74816},[66],[15,74818],{"className":74819},[162,23417],[15,74821,74823],{"className":74822},[6467],[15,74824,74826,75006],{"className":74825},[202,1090],[15,74827,74829,75003],{"className":74828},[206],[15,74830,74832,74941,74949],{"className":74831,"style":39133},[210],[15,74833,74834,74837],{"style":23433},[15,74835],{"className":74836,"style":1368},[217],[15,74838,74840,74886,74889,74892,74895],{"className":74839},[66],[15,74841,74843,74846],{"className":74842},[66],[15,74844,2256],{"className":74845},[66,67],[15,74847,74849],{"className":74848},[198],[15,74850,74852,74878],{"className":74851},[202,1090],[15,74853,74855,74875],{"className":74854},[206],[15,74856,74858],{"className":74857,"style":3638},[210],[15,74859,74860,74863],{"style":1584},[15,74861],{"className":74862,"style":218},[217],[15,74864,74866],{"className":74865},[222,223,224,225],[15,74867,74869],{"className":74868},[66,225],[15,74870,74872],{"className":74871},[66,225],[15,74873,74095],{"className":74874},[66,4429,225],[15,74876,1114],{"className":74877},[1113],[15,74879,74881],{"className":74880},[206],[15,74882,74884],{"className":74883,"style":1121},[210],[15,74885],{},[15,74887],{"className":74888,"style":72},[71],[15,74890,1245],{"className":74891},[76],[15,74893],{"className":74894,"style":72},[71],[15,74896,74898,74901],{"className":74897},[66],[15,74899,2256],{"className":74900},[66,67],[15,74902,74904],{"className":74903},[198],[15,74905,74907,74933],{"className":74906},[202,1090],[15,74908,74910,74930],{"className":74909},[206],[15,74911,74913],{"className":74912,"style":3638},[210],[15,74914,74915,74918],{"style":1584},[15,74916],{"className":74917,"style":218},[217],[15,74919,74921],{"className":74920},[222,223,224,225],[15,74922,74924],{"className":74923},[66,225],[15,74925,74927],{"className":74926},[66,225],[15,74928,74177],{"className":74929},[66,4429,225],[15,74931,1114],{"className":74932},[1113],[15,74934,74936],{"className":74935},[206],[15,74937,74939],{"className":74938,"style":1121},[210],[15,74940],{},[15,74942,74943,74946],{"style":23445},[15,74944],{"className":74945,"style":1368},[217],[15,74947],{"className":74948,"style":4440},[23452],[15,74950,74951,74954],{"style":23455},[15,74952],{"className":74953,"style":1368},[217],[15,74955,74957],{"className":74956},[66],[15,74958,74960,74963],{"className":74959},[66],[15,74961,2256],{"className":74962},[66,67],[15,74964,74966],{"className":74965},[198],[15,74967,74969,74995],{"className":74968},[202,1090],[15,74970,74972,74992],{"className":74971},[206],[15,74973,74975],{"className":74974,"style":3638},[210],[15,74976,74977,74980],{"style":1584},[15,74978],{"className":74979,"style":218},[217],[15,74981,74983],{"className":74982},[222,223,224,225],[15,74984,74986],{"className":74985},[66,225],[15,74987,74989],{"className":74988},[66,225],[15,74990,74095],{"className":74991},[66,4429,225],[15,74993,1114],{"className":74994},[1113],[15,74996,74998],{"className":74997},[206],[15,74999,75001],{"className":75000,"style":1121},[210],[15,75002],{},[15,75004,1114],{"className":75005},[1113],[15,75007,75009],{"className":75008},[206],[15,75010,75012],{"className":75011,"style":52619},[210],[15,75013],{},[15,75015],{"className":75016},[169,23417],[15,75018,3573],{"className":75019},[66],[11,75021,75022,75023,75055,75056,75090],{},"阈值不再固定在 ",[15,75024,75026,75043],{"className":75025,"translate":19},[18],[15,75027,75029],{"className":75028},[23],[25,75030,75031],{"xmlns":27},[29,75032,75033,75041],{},[32,75034,75035,75037,75039],{},[1209,75036,1211],{},[35,75038,6625],{"mathvariant":138},[1209,75040,1263],{},[45,75042,71809],{"encoding":47},[15,75044,75046],{"className":75045,"ariaHidden":53},[52],[15,75047,75049,75052],{"className":75048},[57],[15,75050],{"className":75051,"style":154},[61],[15,75053,71809],{"className":75054},[66],"，而是由两类代价的比值决定。若漏诊的代价是误诊的十倍，阈值就降到 ",[15,75057,75059,75078],{"className":75058,"translate":19},[18],[15,75060,75062],{"className":75061},[23],[25,75063,75064],{"xmlns":27},[29,75065,75066,75075],{},[32,75067,75068,75070,75072],{},[1209,75069,1211],{},[35,75071,6625],{"mathvariant":138},[1209,75073,75074],{},"11",[45,75076,75077],{"encoding":47},"1\u002F11",[15,75079,75081],{"className":75080,"ariaHidden":53},[52],[15,75082,75084,75087],{"className":75083},[57],[15,75085],{"className":75086,"style":154},[61],[15,75088,75077],{"className":75089},[66]," 左右，模型会倾向于把更多样本判为正类。",[11,75092,75093,75094,75097],{},"这个式子也说明：类别不平衡本身不是问题，",[2040,75095,75096],{},"损失不对称","才是问题。重采样改变了训练分布，相当于改变了先验；如果最终要解决的是原分布上的决策问题，那么重采样之后还需要把阈值校正回来。",[11,75099,75100],{},"把常见损失与对应的贝叶斯行动放在一起：",[6152,75102,75103,75112],{},[6155,75104,75105],{},[6158,75106,75107,75109],{},[6161,75108,30593],{},[6161,75110,75111],{},"贝叶斯行动",[6171,75113,75114,75122,75129,75137,75369],{},[6158,75115,75116,75119],{},[6176,75117,75118],{},"0-1 损失",[6176,75120,75121],{},"后验众数",[6158,75123,75124,75127],{},[6176,75125,75126],{},"平方损失",[6176,75128,72666],{},[6158,75130,75131,75134],{},[6176,75132,75133],{},"绝对损失",[6176,75135,75136],{},"后验中位数",[6158,75138,75139,75142],{},[6176,75140,75141],{},"非对称 0-1 损失",[6176,75143,75144,75145],{},"阈值 ",[15,75146,75148,75198],{"className":75147,"translate":19},[18],[15,75149,75151],{"className":75150},[23],[25,75152,75153],{"xmlns":27},[29,75154,75155,75195],{},[32,75156,75157,75167,75169,75171,75181,75183,75193],{},[1057,75158,75159,75161],{},[35,75160,2256],{},[32,75162,75163,75165],{},[35,75164,74045],{"mathvariant":138},[35,75166,28399],{"mathvariant":138},[35,75168,6625],{"mathvariant":138},[39,75170,120],{"stretchy":119},[1057,75172,75173,75175],{},[35,75174,2256],{},[32,75176,75177,75179],{},[35,75178,74045],{"mathvariant":138},[35,75180,28399],{"mathvariant":138},[39,75182,1245],{},[1057,75184,75185,75187],{},[35,75186,2256],{},[32,75188,75189,75191],{},[35,75190,74045],{"mathvariant":138},[35,75192,28535],{"mathvariant":138},[39,75194,126],{"stretchy":119},[45,75196,75197],{"encoding":47},"c_{\\mathrm{FP}}\u002F(c_{\\mathrm{FP}}+c_{\\mathrm{FN}})",[15,75199,75201,75314],{"className":75200,"ariaHidden":53},[52],[15,75202,75204,75207,75253,75256,75259,75305,75308,75311],{"className":75203},[57],[15,75205],{"className":75206,"style":154},[61],[15,75208,75210,75213],{"className":75209},[66],[15,75211,2256],{"className":75212},[66,67],[15,75214,75216],{"className":75215},[198],[15,75217,75219,75245],{"className":75218},[202,1090],[15,75220,75222,75242],{"className":75221},[206],[15,75223,75225],{"className":75224,"style":3638},[210],[15,75226,75227,75230],{"style":1584},[15,75228],{"className":75229,"style":218},[217],[15,75231,75233],{"className":75232},[222,223,224,225],[15,75234,75236],{"className":75235},[66,225],[15,75237,75239],{"className":75238},[66,225],[15,75240,74095],{"className":75241},[66,4429,225],[15,75243,1114],{"className":75244},[1113],[15,75246,75248],{"className":75247},[206],[15,75249,75251],{"className":75250,"style":1121},[210],[15,75252],{},[15,75254,6625],{"className":75255},[66],[15,75257,120],{"className":75258},[162],[15,75260,75262,75265],{"className":75261},[66],[15,75263,2256],{"className":75264},[66,67],[15,75266,75268],{"className":75267},[198],[15,75269,75271,75297],{"className":75270},[202,1090],[15,75272,75274,75294],{"className":75273},[206],[15,75275,75277],{"className":75276,"style":3638},[210],[15,75278,75279,75282],{"style":1584},[15,75280],{"className":75281,"style":218},[217],[15,75283,75285],{"className":75284},[222,223,224,225],[15,75286,75288],{"className":75287},[66,225],[15,75289,75291],{"className":75290},[66,225],[15,75292,74095],{"className":75293},[66,4429,225],[15,75295,1114],{"className":75296},[1113],[15,75298,75300],{"className":75299},[206],[15,75301,75303],{"className":75302,"style":1121},[210],[15,75304],{},[15,75306],{"className":75307,"style":72},[71],[15,75309,1245],{"className":75310},[76],[15,75312],{"className":75313,"style":72},[71],[15,75315,75317,75320,75366],{"className":75316},[57],[15,75318],{"className":75319,"style":154},[61],[15,75321,75323,75326],{"className":75322},[66],[15,75324,2256],{"className":75325},[66,67],[15,75327,75329],{"className":75328},[198],[15,75330,75332,75358],{"className":75331},[202,1090],[15,75333,75335,75355],{"className":75334},[206],[15,75336,75338],{"className":75337,"style":3638},[210],[15,75339,75340,75343],{"style":1584},[15,75341],{"className":75342,"style":218},[217],[15,75344,75346],{"className":75345},[222,223,224,225],[15,75347,75349],{"className":75348},[66,225],[15,75350,75352],{"className":75351},[66,225],[15,75353,74177],{"className":75354},[66,4429,225],[15,75356,1114],{"className":75357},[1113],[15,75359,75361],{"className":75360},[206],[15,75362,75364],{"className":75363,"style":1121},[210],[15,75365],{},[15,75367,126],{"className":75368},[169],[6158,75370,75371,75374],{},[6176,75372,75373],{},"带拒绝的 0-1 损失",[6176,75375,75376,75377,75428],{},"最大后验低于 ",[15,75378,75380,75398],{"className":75379,"translate":19},[18],[15,75381,75383],{"className":75382},[23],[25,75384,75385],{"xmlns":27},[29,75386,75387,75395],{},[32,75388,75389,75391,75393],{},[1209,75390,1211],{},[39,75392,6602],{},[35,75394,2256],{},[45,75396,75397],{"encoding":47},"1-c",[15,75399,75401,75419],{"className":75400,"ariaHidden":53},[52],[15,75402,75404,75407,75410,75413,75416],{"className":75403},[57],[15,75405],{"className":75406,"style":30468},[61],[15,75408,1211],{"className":75409},[66],[15,75411],{"className":75412,"style":72},[71],[15,75414,6602],{"className":75415},[76],[15,75417],{"className":75418,"style":72},[71],[15,75420,75422,75425],{"className":75421},[57],[15,75423],{"className":75424,"style":86},[61],[15,75426,2256],{"className":75427},[66,67]," 时拒绝",[11,75430,75431,75432,75435,75436,75510,75511,75652,75653,75681],{},"最后一行是",[2040,75433,75434],{},"拒绝选项","（reject option）：除了预测类别，模型还可以说「我不确定」。若拒绝的代价为 ",[15,75437,75439,75465],{"className":75438,"translate":19},[18],[15,75440,75442],{"className":75441},[23],[25,75443,75444],{"xmlns":27},[29,75445,75446,75462],{},[32,75447,75448,75450,75452,75454,75456,75458,75460],{},[35,75449,2256],{},[39,75451,1187],{},[39,75453,120],{"stretchy":119},[1209,75455,1804],{},[39,75457,335],{"separator":53},[1209,75459,1211],{},[39,75461,126],{"stretchy":119},[45,75463,75464],{"encoding":47},"c\\in(0,1)",[15,75466,75468,75486],{"className":75467,"ariaHidden":53},[52],[15,75469,75471,75474,75477,75480,75483],{"className":75470},[57],[15,75472],{"className":75473,"style":3227},[61],[15,75475,2256],{"className":75476},[66,67],[15,75478],{"className":75479,"style":173},[71],[15,75481,1187],{"className":75482},[177],[15,75484],{"className":75485,"style":173},[71],[15,75487,75489,75492,75495,75498,75501,75504,75507],{"className":75488},[57],[15,75490],{"className":75491,"style":154},[61],[15,75493,120],{"className":75494},[162],[15,75496,1804],{"className":75497},[66],[15,75499,335],{"className":75500},[422],[15,75502],{"className":75503,"style":697},[71],[15,75505,1211],{"className":75506},[66],[15,75508,126],{"className":75509},[169],"，预测的期望代价为 ",[15,75512,75514,75552],{"className":75513,"translate":19},[18],[15,75515,75517],{"className":75516},[23],[25,75518,75519],{"xmlns":27},[29,75520,75521,75549],{},[32,75522,75523,75525,75527,75537,75539,75541,75543,75545,75547],{},[1209,75524,1211],{},[39,75526,6602],{},[1057,75528,75529,75535],{},[32,75530,75531,75533],{},[35,75532,23273],{},[39,75534,1170],{},[35,75536,1224],{},[35,75538,11],{},[39,75540,120],{"stretchy":119},[35,75542,1224],{},[39,75544,34430],{},[35,75546,123],{},[39,75548,126],{"stretchy":119},[45,75550,75551],{"encoding":47},"1-\\max_y p(y\\mid x)",[15,75553,75555,75573,75640],{"className":75554,"ariaHidden":53},[52],[15,75556,75558,75561,75564,75567,75570],{"className":75557},[57],[15,75559],{"className":75560,"style":30468},[61],[15,75562,1211],{"className":75563},[66],[15,75565],{"className":75566,"style":72},[71],[15,75568,6602],{"className":75569},[76],[15,75571],{"className":75572,"style":72},[71],[15,75574,75576,75579,75619,75622,75625,75628,75631,75634,75637],{"className":75575},[57],[15,75577],{"className":75578,"style":2956},[61],[15,75580,75582,75585],{"className":75581},[1339],[15,75583,23273],{"className":75584},[1339],[15,75586,75588],{"className":75587},[198],[15,75589,75591,75611],{"className":75590},[202,1090],[15,75592,75594,75608],{"className":75593},[206],[15,75595,75597],{"className":75596,"style":2198},[210],[15,75598,75599,75602],{"style":69561},[15,75600],{"className":75601,"style":218},[217],[15,75603,75605],{"className":75604},[222,223,224,225],[15,75606,1224],{"className":75607,"style":1515},[66,67,225],[15,75609,1114],{"className":75610},[1113],[15,75612,75614],{"className":75613},[206],[15,75615,75617],{"className":75616,"style":2715},[210],[15,75618],{},[15,75620],{"className":75621,"style":697},[71],[15,75623,11],{"className":75624},[66,67],[15,75626,120],{"className":75627},[162],[15,75629,1224],{"className":75630,"style":1515},[66,67],[15,75632],{"className":75633,"style":173},[71],[15,75635,34430],{"className":75636},[177],[15,75638],{"className":75639,"style":173},[71],[15,75641,75643,75646,75649],{"className":75642},[57],[15,75644],{"className":75645,"style":154},[61],[15,75647,123],{"className":75648},[66,67],[15,75650,126],{"className":75651},[169],"，拒绝的代价为 ",[15,75654,75656,75669],{"className":75655,"translate":19},[18],[15,75657,75659],{"className":75658},[23],[25,75660,75661],{"xmlns":27},[29,75662,75663,75667],{},[32,75664,75665],{},[35,75666,2256],{},[45,75668,2256],{"encoding":47},[15,75670,75672],{"className":75671,"ariaHidden":53},[52],[15,75673,75675,75678],{"className":75674},[57],[15,75676],{"className":75677,"style":86},[61],[15,75679,2256],{"className":75680},[66,67],"，因此最优规则是",[15,75683,75685],{"className":75684,"translate":19},[294],[15,75686,75688,75742],{"className":75687,"translate":19},[18],[15,75689,75691],{"className":75690},[23],[25,75692,75693],{"xmlns":27,"display":303},[29,75694,75695,75739],{},[32,75696,75697,75700,75702,75705,75707,75717,75719,75721,75723,75725,75727,75729,75731,75733,75735,75737],{},[1274,75698,75699],{},"拒绝",[71,75701],{"width":9791},[39,75703,75704],{},"⟺",[71,75706],{"width":9791},[1172,75708,75709,75715],{},[32,75710,75711,75713],{},[35,75712,23273],{},[39,75714,1170],{},[35,75716,1224],{},[35,75718,11],{},[39,75720,120],{"stretchy":119},[35,75722,1224],{},[39,75724,34430],{},[35,75726,123],{},[39,75728,126],{"stretchy":119},[39,75730,25772],{},[1209,75732,1211],{},[39,75734,6602],{},[35,75736,2256],{},[35,75738,3573],{"mathvariant":138},[45,75740,75741],{"encoding":47},"\\text{拒绝}\\quad\\Longleftrightarrow\\quad \\max_y p(y\\mid x)\u003C1-c.",[15,75743,75745,75773,75846,75867,75885],{"className":75744,"ariaHidden":53},[52],[15,75746,75748,75752,75758,75761,75764,75767,75770],{"className":75747},[57],[15,75749],{"className":75750,"style":75751},[61],"height:0.7073em;vertical-align:-0.024em;",[15,75753,75755],{"className":75754},[66,1711],[15,75756,75699],{"className":75757},[66,68616],[15,75759],{"className":75760,"style":10049},[71],[15,75762],{"className":75763,"style":173},[71],[15,75765,75704],{"className":75766},[177],[15,75768],{"className":75769,"style":10049},[71],[15,75771],{"className":75772,"style":173},[71],[15,75774,75776,75780,75825,75828,75831,75834,75837,75840,75843],{"className":75775},[57],[15,75777],{"className":75778,"style":75779},[61],"height:1.5861em;vertical-align:-0.8361em;",[15,75781,75783],{"className":75782},[1339,1351],[15,75784,75786,75816],{"className":75785},[202,1090],[15,75787,75789,75813],{"className":75788},[206],[15,75790,75792,75803],{"className":75791,"style":86},[210],[15,75793,75794,75797],{"style":48823},[15,75795],{"className":75796,"style":1368},[217],[15,75798,75800],{"className":75799},[222,223,224,225],[15,75801,1224],{"className":75802,"style":1515},[66,67,225],[15,75804,75805,75808],{"style":1386},[15,75806],{"className":75807,"style":1368},[217],[15,75809,75810],{},[15,75811,23273],{"className":75812},[1339],[15,75814,1114],{"className":75815},[1113],[15,75817,75819],{"className":75818},[206],[15,75820,75823],{"className":75821,"style":75822},[210],"height:0.8361em;",[15,75824],{},[15,75826],{"className":75827,"style":697},[71],[15,75829,11],{"className":75830},[66,67],[15,75832,120],{"className":75833},[162],[15,75835,1224],{"className":75836,"style":1515},[66,67],[15,75838],{"className":75839,"style":173},[71],[15,75841,34430],{"className":75842},[177],[15,75844],{"className":75845,"style":173},[71],[15,75847,75849,75852,75855,75858,75861,75864],{"className":75848},[57],[15,75850],{"className":75851,"style":154},[61],[15,75853,123],{"className":75854},[66,67],[15,75856,126],{"className":75857},[169],[15,75859],{"className":75860,"style":173},[71],[15,75862,25772],{"className":75863},[177],[15,75865],{"className":75866,"style":173},[71],[15,75868,75870,75873,75876,75879,75882],{"className":75869},[57],[15,75871],{"className":75872,"style":30468},[61],[15,75874,1211],{"className":75875},[66],[15,75877],{"className":75878,"style":72},[71],[15,75880,6602],{"className":75881},[76],[15,75883],{"className":75884,"style":72},[71],[15,75886,75888,75891,75894],{"className":75887},[57],[15,75889],{"className":75890,"style":86},[61],[15,75892,2256],{"className":75893},[66,67],[15,75895,3573],{"className":75896},[66],[11,75898,75899],{},"这就是 Chow 规则。它把「不确定性」直接翻译成了一个可执行的动作，而不是一个只用来画图的分数。",[1904,75901,75903],{"id":75902},"_36-贝叶斯风险","3.6 贝叶斯风险",[11,75905,75906],{},"把定理 3.1 中的最优值单独命名：",[15,75908,75910],{"className":75909,"translate":19},[294],[15,75911,75913,75993],{"className":75912,"translate":19},[18],[15,75914,75916],{"className":75915},[23],[25,75917,75918],{"xmlns":27,"display":303},[29,75919,75920,75990],{},[32,75921,75922,75928,75930,75940,75942,75944,75946,75948,75950,75956,75958,75974,75976,75978,75980,75982,75984,75986,75988],{},[131,75923,75924,75926],{},[35,75925,455],{},[39,75927,1162],{},[39,75929,129],{},[1172,75931,75932,75938],{},[32,75933,75934,75936],{},[35,75935,69684],{},[39,75937,1170],{},[35,75939,3885],{},[35,75941,455],{},[39,75943,120],{"stretchy":119},[35,75945,3885],{},[39,75947,126],{"stretchy":119},[39,75949,129],{},[1057,75951,75952,75954],{},[35,75953,62381],{"mathvariant":454},[35,75955,446],{},[39,75957,1194],{"fence":119,"stretchy":53,"minsize":4545,"maxsize":4545},[1172,75959,75960,75966],{},[32,75961,75962,75964],{},[35,75963,1178],{},[39,75965,1170],{},[32,75967,75968,75970,75972],{},[35,75969,4556],{},[39,75971,1187],{},[35,75973,62005],{"mathvariant":445},[35,75975,63107],{},[39,75977,120],{"stretchy":119},[35,75979,4556],{},[39,75981,335],{"separator":53},[35,75983,446],{},[39,75985,126],{"stretchy":119},[39,75987,1266],{"fence":119,"stretchy":53,"minsize":4545,"maxsize":4545},[35,75989,3573],{"mathvariant":138},[45,75991,75992],{"encoding":47},"R^*=\\inf_{\\delta}R(\\delta)\n=\\mathbb E_X\\Big[\\min_{a\\in\\mathcal A}r(a,X)\\Big].",[15,75994,75996,76040,76120],{"className":75995,"ariaHidden":53},[52],[15,75997,75999,76002,76031,76034,76037],{"className":75998},[57],[15,76000],{"className":76001,"style":1308},[61],[15,76003,76005,76008],{"className":76004},[66],[15,76006,455],{"className":76007,"style":62437},[66,67],[15,76009,76011],{"className":76010},[198],[15,76012,76014],{"className":76013},[202],[15,76015,76017],{"className":76016},[206],[15,76018,76020],{"className":76019,"style":1308},[210],[15,76021,76022,76025],{"style":397},[15,76023],{"className":76024,"style":218},[217],[15,76026,76028],{"className":76027},[222,223,224,225],[15,76029,1162],{"className":76030},[76,225],[15,76032],{"className":76033,"style":173},[71],[15,76035,129],{"className":76036},[177],[15,76038],{"className":76039,"style":173},[71],[15,76041,76043,76047,76096,76099,76102,76105,76108,76111,76114,76117],{"className":76042},[57],[15,76044],{"className":76045,"style":76046},[61],"height:1.5021em;vertical-align:-0.7521em;",[15,76048,76050],{"className":76049},[1339,1351],[15,76051,76053,76088],{"className":76052},[202,1090],[15,76054,76056,76085],{"className":76055},[206],[15,76057,76059,76073],{"className":76058,"style":840},[210],[15,76060,76061,76064],{"style":1364},[15,76062],{"className":76063,"style":1368},[217],[15,76065,76067],{"className":76066},[222,223,224,225],[15,76068,76070],{"className":76069},[66,225],[15,76071,3885],{"className":76072,"style":3946},[66,67,225],[15,76074,76075,76078],{"style":1386},[15,76076],{"className":76077,"style":1368},[217],[15,76079,76080],{},[15,76081,69702,76083],{"className":76082},[1339],[15,76084,116],{"style":69705},[15,76086,1114],{"className":76087},[1113],[15,76089,76091],{"className":76090},[206],[15,76092,76094],{"className":76093,"style":44025},[210],[15,76095],{},[15,76097],{"className":76098,"style":697},[71],[15,76100,455],{"className":76101,"style":62437},[66,67],[15,76103,120],{"className":76104},[162],[15,76106,3885],{"className":76107,"style":3946},[66,67],[15,76109,126],{"className":76110},[169],[15,76112],{"className":76113,"style":173},[71],[15,76115,129],{"className":76116},[177],[15,76118],{"className":76119,"style":173},[71],[15,76121,76123,76126,76166,76172,76175,76228,76231,76234,76237,76240,76243,76246,76249,76252,76258],{"className":76122},[57],[15,76124],{"className":76125,"style":68911},[61],[15,76127,76129,76132],{"className":76128},[66],[15,76130,62381],{"className":76131},[66,517],[15,76133,76135],{"className":76134},[198],[15,76136,76138,76158],{"className":76137},[202,1090],[15,76139,76141,76155],{"className":76140},[206],[15,76142,76144],{"className":76143,"style":3638},[210],[15,76145,76146,76149],{"style":1584},[15,76147],{"className":76148,"style":218},[217],[15,76150,76152],{"className":76151},[222,223,224,225],[15,76153,446],{"className":76154,"style":16497},[66,67,225],[15,76156,1114],{"className":76157},[1113],[15,76159,76161],{"className":76160},[206],[15,76162,76164],{"className":76163,"style":1121},[210],[15,76165],{},[15,76167,76169],{"className":76168},[66],[15,76170,1194],{"className":76171},[1422,4677],[15,76173],{"className":76174,"style":697},[71],[15,76176,76178],{"className":76177},[1339,1351],[15,76179,76181,76220],{"className":76180},[202,1090],[15,76182,76184,76217],{"className":76183},[206],[15,76185,76187,76207],{"className":76186,"style":1361},[210],[15,76188,76189,76192],{"style":24354},[15,76190],{"className":76191,"style":1368},[217],[15,76193,76195],{"className":76194},[222,223,224,225],[15,76196,76198,76201,76204],{"className":76197},[66,225],[15,76199,4556],{"className":76200},[66,67,225],[15,76202,1187],{"className":76203},[177,225],[15,76205,62005],{"className":76206},[66,494,225],[15,76208,76209,76212],{"style":1386},[15,76210],{"className":76211,"style":1368},[217],[15,76213,76214],{},[15,76215,1178],{"className":76216},[1339],[15,76218,1114],{"className":76219},[1113],[15,76221,76223],{"className":76222},[206],[15,76224,76226],{"className":76225,"style":68526},[210],[15,76227],{},[15,76229],{"className":76230,"style":697},[71],[15,76232,63107],{"className":76233,"style":23025},[66,67],[15,76235,120],{"className":76236},[162],[15,76238,4556],{"className":76239},[66,67],[15,76241,335],{"className":76242},[422],[15,76244],{"className":76245,"style":697},[71],[15,76247,446],{"className":76248,"style":16497},[66,67],[15,76250,126],{"className":76251},[169],[15,76253,76255],{"className":76254},[66],[15,76256,1266],{"className":76257},[1422,4677],[15,76259,3573],{"className":76260},[66],[11,76262,76263,2038,76322,76324,76325,76353],{},[15,76264,76266,76284],{"className":76265,"translate":19},[18],[15,76267,76269],{"className":76268},[23],[25,76270,76271],{"xmlns":27},[29,76272,76273,76281],{},[32,76274,76275],{},[131,76276,76277,76279],{},[35,76278,455],{},[39,76280,1162],{},[45,76282,76283],{"encoding":47},"R^*",[15,76285,76287],{"className":76286,"ariaHidden":53},[52],[15,76288,76290,76293],{"className":76289},[57],[15,76291],{"classN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risk），也叫贝叶斯错误率（分类情形）。它是这个问题在信息层面的下限：即使知道真实分布、知道每个 ",[15,76326,76328,76341],{"className":76327,"translate":19},[18],[15,76329,76331],{"className":76330},[23],[25,76332,76333],{"xmlns":27},[29,76334,76335,76339],{},[32,76336,76337],{},[35,76338,123],{},[45,76340,123],{"encoding":47},[15,76342,76344],{"className":76343,"ariaHidden":53},[52],[15,76345,76347,76350],{"className":76346},[57],[15,76348],{"className":76349,"style":86},[61],[15,76351,123],{"className":76352},[66,67]," 上的后验，也只能做到这里。它不是某个算法的性能指标，而是问题本身的属性。",[11,76355,76356],{},"对二分类 0-1 损失，把 3.2 节的规则代回去，得到",[15,76358,76360],{"className":76359,"translate":19},[294],[15,76361,76363,76421],{"className":76362,"translate":19},[18],[15,76364,76366],{"className":76365},[23],[25,76367,76368],{"xmlns":27,"display":303},[29,76369,76370,76418],{},[32,76371,76372,76378,76380,76382,76384,76386,76388,76390,76392,76394,76396,76398,76400,76402,76404,76406,76408,76410,76412,76414,76416],{},[131,76373,76374,76376],{},[35,76375,455],{},[39,76377,1162],{},[39,76379,129],{},[35,76381,62381],{"mathvariant":454},[39,76383,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,76385,1178],{},[39,76387,1170],{},[39,76389,4307],{"stretchy":119},[35,76391,71113],{},[39,76393,120],{"stretchy":119},[35,76395,446],{},[39,76397,126],{"stretchy":119},[39,76399,335],{"separator":53},[1209,76401,1211],{},[39,76403,6602],{},[35,76405,71113],{},[39,76407,120],{"stretchy":119},[35,76409,446],{},[39,76411,126],{"stretchy":119},[39,76413,4330],{"stretchy":119},[39,76415,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,76417,3573],{"mathvariant":138},[45,76419,76420],{"encoding":47},"R^*=\\mathbb E\\big[\\min\\{\\eta(X),1-\\eta(X)\\}\\big].",[15,76422,76424,76468,76522],{"className":76423,"ariaHidden":53},[52],[15,76425,76427,76430,76459,76462,76465],{"className":76426},[57],[15,76428],{"className":76429,"style":1308},[61],[15,76431,76433,76436],{"className":76432},[66],[15,76434,455],{"className":76435,"style":62437},[66,67],[15,76437,76439],{"className":76438},[198],[15,76440,76442],{"className":76441},[202],[15,76443,76445],{"className":76444},[206],[15,76446,76448],{"className":76447,"style":1308},[210],[15,76449,76450,76453],{"style":397},[15,76451],{"className":76452,"style":218},[217],[15,76454,76456],{"className":76455},[222,223,224,225],[15,76457,1162],{"className":76458},[76,225],[15,76460],{"className":76461,"style":173},[71],[15,76463,129],{"className":76464},[177],[15,76466],{"className":76467,"style":173},[71],[15,76469,76471,76474,76477,76483,76486,76489,76492,76495,76498,76501,76504,76507,76510,76513,76516,76519],{"className":76470},[57],[15,76472],{"className":76473,"style":4400},[61],[15,76475,62381],{"className":76476},[66,517],[15,76478,76480],{"className":76479},[66],[15,76481,1194],{"className":76482},[1422,1508],[15,76484],{"className":76485,"style":697},[71],[15,76487,1178],{"className":76488},[1339],[15,76490,4307],{"className":76491},[162],[15,76493,71113],{"className":76494,"style":1515},[66,67],[15,76496,120],{"className":76497},[162],[15,76499,446],{"className":76500,"style":16497},[66,67],[15,76502,126],{"className":76503},[169],[15,76505,335],{"className":76506},[422],[15,76508],{"className":76509,"style":697},[71],[15,76511,1211],{"className":76512},[66],[15,76514],{"className":76515,"style":72},[71],[15,76517,6602],{"className":76518},[76],[15,76520],{"className":76521,"style":72},[71],[15,76523,76525,76528,76531,76534,76537,76540,76546],{"className":76524},[57],[15,76526],{"className":76527,"style":4400},[61],[15,76529,71113],{"className":76530,"style":1515},[66,67],[15,76532,120],{"className":76533},[162],[15,76535,446],{"className":76536,"style":16497},[66,67],[15,76538,19280],{"className":76539},[169],[15,76541,76543],{"className":76542},[66],[15,76544,1266],{"className":76545},[1422,1508],[15,76547,3573],{"className":76548},[66],[11,76550,76551],{},"对多分类，则是",[15,76553,76555],{"className":76554,"translate":19},[294],[15,76556,76558,76618],{"className":76557,"translate":19},[18],[15,76559,76561],{"className":76560},[23],[25,76562,76563],{"xmlns":27,"display":303},[29,76564,76565,76615],{},[32,76566,76567,76573,76575,76577,76579,76581,76583,76599,76601,76603,76605,76607,76609,76611,76613],{},[131,76568,76569,76571],{},[35,76570,455],{},[39,76572,1162],{},[39,76574,129],{},[1209,76576,1211],{},[39,76578,6602],{},[35,76580,62381],{"mathvariant":454},[39,76582,1194],{"fence":119,"stretchy":53,"minsize":4545,"maxsize":4545},[1172,76584,76585,76591],{},[32,76586,76587,76589],{},[35,76588,23273],{},[39,76590,1170],{},[32,76592,76593,76595,76597],{},[35,76594,1224],{},[39,76596,1187],{},[35,76598,61885],{"mathvariant":445},[35,76600,11],{},[39,76602,120],{"stretchy":119},[35,76604,1224],{},[39,76606,34430],{},[35,76608,446],{},[39,76610,126],{"stretchy":119},[39,76612,1266],{"fence":119,"stretchy":53,"minsize":4545,"maxsize":4545},[35,76614,3573],{"mathvariant":138},[45,76616,76617],{"encoding":47},"R^*=1-\\mathbb E\\Big[\\max_{y\\in\\mathcal Y}p(y\\mid X)\\Big].",[15,76619,76621,76665,76683,76776],{"className":76620,"ariaHidden":53},[52],[15,76622,76624,76627,76656,76659,76662],{"className":76623},[57],[15,76625],{"className":76626,"style":1308},[61],[15,76628,76630,76633],{"className":76629},[66],[15,76631,455],{"className":76632,"style":62437},[66,67],[15,76634,76636],{"className":76635},[198],[15,76637,76639],{"className":76638},[202],[15,76640,76642],{"className":76641},[206],[15,76643,76645],{"className":76644,"style":1308},[210],[15,76646,76647,76650],{"style":397},[15,76648],{"className":76649,"style":218},[217],[15,76651,76653],{"className":76652},[222,223,224,225],[15,76654,1162],{"className":76655},[76,225],[15,76657],{"className":76658,"style":173},[71],[15,76660,129],{"className":76661},[177],[15,76663],{"className":76664,"style":173},[71],[15,76666,76668,76671,76674,76677,76680],{"className":76667},[57],[15,76669],{"className":76670,"style":30468},[61],[15,76672,1211],{"className":76673},[66],[15,76675],{"className":76676,"style":72},[71],[15,76678,6602],{"className":76679},[76],[15,76681],{"className":76682,"style":72},[71],[15,76684,76686,76690,76693,76699,76702,76755,76758,76761,76764,76767,76770,76773],{"className":76685},[57],[15,76687],{"className":76688,"style":76689},[61],"height:2.0304em;vertical-align:-0.8804em;",[15,76691,62381],{"className":76692},[66,517],[15,76694,76696],{"className":76695},[66],[15,76697,1194],{"className":76698},[1422,4677],[15,76700],{"className":76701,"style":697},[71],[15,76703,76705],{"className":76704},[1339,1351],[15,76706,76708,76747],{"className":76707},[202,1090],[15,76709,76711,76744],{"className":76710},[206],[15,76712,76714,76734],{"className":76713,"style":86},[210],[15,76715,76716,76719],{"style":24354},[15,76717],{"className":76718,"style":1368},[217],[15,76720,76722],{"className":76721},[222,223,224,225],[15,76723,76725,76728,76731],{"className":76724},[66,225],[15,76726,1224],{"className":76727,"style":1515},[66,67,225],[15,76729,1187],{"className":76730},[177,225],[15,76732,61885],{"className":76733,"style":28452},[66,494,225],[15,76735,76736,76739],{"style":1386},[15,76737],{"className":76738,"style":1368},[217],[15,76740,76741],{},[15,76742,23273],{"className":76743},[1339],[15,76745,1114],{"className":76746},[1113],[15,76748,76750],{"className":76749},[206],[15,76751,76753],{"className":76752,"style":67080},[210],[15,76754],{},[15,76756],{"className":76757,"style":697},[71],[15,76759,11],{"className":76760},[66,67],[15,76762,120],{"className":76763},[162],[15,76765,1224],{"className":76766,"style":1515},[66,67],[15,76768],{"className":76769,"style":173},[71],[15,76771,34430],{"className":76772},[177],[15,76774],{"className":76775,"style":173},[71],[15,76777,76779,76782,76785,76788,76794],{"className":76778},[57],[15,76780],{"className":76781,"style":69187},[61],[15,76783,446],{"className":76784,"style":16497},[66,67],[15,76786,126],{"className":76787},[169],[15,76789,76791],{"className":76790},[66],[15,76792,1266],{"className":76793},[1422,4677],[15,76795,3573],{"className":76796},[66],[11,76798,76799],{},"这两个式子在理论里出现得非常多，因为它们把「分类有多难」压缩成了一个标量。",[11,76801,76802,76803,76806],{},"进一步，二分类的",[2040,76804,76805],{},"超额风险","有一个很干净的形式：",[15,76808,76810],{"className":76809,"translate":19},[294],[15,76811,76813,76899],{"className":76812,"translate":19},[18],[15,76814,76816],{"className":76815},[23],[25,76817,76818],{"xmlns":27,"display":303},[29,76819,76820,76896],{},[32,76821,76822,76824,76826,76828,76830,76832,76838,76840,76842,76844,76846,76848,76850,76852,76854,76856,76858,76860,76862,76864,76866,76868,76870,76872,76874,76876,76878,76884,76886,76888,76890,76892,76894],{},[35,76823,455],{},[39,76825,120],{"stretchy":119},[35,76827,3885],{},[39,76829,126],{"stretchy":119},[39,76831,6602],{},[131,76833,76834,76836],{},[35,76835,455],{},[39,76837,1162],{},[39,76839,129],{},[35,76841,62381],{"mathvariant":454},[39,76843,1194],{"fence":119,"stretchy":53,"minsize":4545,"maxsize":4545},[35,76845,34430],{"mathvariant":138},[1209,76847,1263],{},[35,76849,71113],{},[39,76851,120],{"stretchy":119},[35,76853,446],{},[39,76855,126],{"stretchy":119},[39,76857,6602],{},[1209,76859,1211],{},[35,76861,34430],{"mathvariant":138},[1274,76863,2462],{},[1209,76865,1211],{"mathvariant":70368},[39,76867,4307],{"stretchy":119},[35,76869,3885],{},[39,76871,120],{"stretchy":119},[35,76873,446],{},[39,76875,126],{"stretchy":119},[39,76877,13019],{"mathvariant":138},[131,76879,76880,76882],{},[35,76881,3885],{},[39,76883,1162],{},[39,76885,120],{"stretchy":119},[35,76887,446],{},[39,76889,126],{"stretchy":119},[39,76891,4330],{"stretchy":119},[39,76893,1266],{"fence":119,"stretchy":53,"minsize":4545,"maxsize":4545},[35,76895,3573],{"mathvariant":138},[45,76897,76898],{"encoding":47},"R(\\delta)-R^*\n=\\mathbb E\\Big[|2\\eta(X)-1|\\,\\mathbf 1\\{\\delta(X)\\ne\\delta^*(X)\\}\\Big].",[15,76900,76902,76929,76973,77013,77086],{"className":76901,"ariaHidden":53},[52],[15,76903,76905,76908,76911,76914,76917,76920,76923,76926],{"className":76904},[57],[15,76906],{"className":76907,"style":154},[61],[15,76909,455],{"className":76910,"style":62437},[66,67],[15,76912,120],{"className":76913},[162],[15,76915,3885],{"className":76916,"style":3946},[66,67],[15,76918,126],{"className":76919},[169],[15,76921],{"className":76922,"style":72},[71],[15,76924,6602],{"className":76925},[76],[15,76927],{"className":76928,"style":72},[71],[15,76930,76932,76935,76964,76967,76970],{"className":76931},[57],[15,76933],{"className":76934,"style":1308},[61],[15,76936,76938,76941],{"className":76937},[66],[15,76939,455],{"className":76940,"style":62437},[66,67],[15,76942,76944],{"className":76943},[198],[15,76945,76947],{"className":76946},[202],[15,76948,76950],{"className":76949},[206],[15,76951,76953],{"className":76952,"style":1308},[210],[15,76954,76955,76958],{"style":397},[15,76956],{"className":76957,"style":218},[217],[15,76959,76961],{"className":76960},[222,223,224,225],[15,76962,1162],{"className":76963},[76,225],[15,76965],{"className":76966,"style":173},[71],[15,76968,129],{"className":76969},[177],[15,76971],{"className":76972,"style":173},[71],[15,76974,76976,76979,76982,76988,76992,76995,76998,77001,77004,77007,77010],{"className":76975},[57],[15,76977],{"className":76978,"style":69187},[61],[15,76980,62381],{"className":76981},[66,517],[15,76983,76985],{"className":76984},[66],[15,76986,1194],{"className":76987},[1422,4677],[15,76989,76991],{"className":76990},[66],"∣2",[15,76993,71113],{"className":76994,"style":1515},[66,67],[15,76996,120],{"className":76997},[162],[15,76999,446],{"className":77000,"style":16497},[66,67],[15,77002,126],{"className":77003},[169],[15,77005],{"className":77006,"style":72},[71],[15,77008,6602],{"className":77009},[76],[15,77011],{"className":77012,"style":72},[71],[15,77014,77016,77019,77023,77026,77029,77032,77035,77038,77041,77044,77047,77083],{"className":77015},[57],[15,77017],{"className":77018,"style":154},[61],[15,77020,77022],{"className":77021},[66],"1∣",[15,77024],{"className":77025,"style":697},[71],[15,77027,1211],{"className":77028},[66,70432],[15,77030,4307],{"className":77031},[162],[15,77033,3885],{"className":77034,"style":3946},[66,67],[15,77036,120],{"className":77037},[162],[15,77039,446],{"className":77040,"style":16497},[66,67],[15,77042,126],{"className":77043},[169],[15,77045],{"className":77046,"style":173},[71],[15,77048,77050,77077,77080],{"className":77049},[177],[15,77051,77053],{"className":77052},[177],[15,77054,77056],{"className":77055},[66,13086],[15,77057,77059],{"className":77058},[13090],[15,77060,77062,77065,77074],{"className":77061},[13094],[15,77063],{"className":77064,"style":3408},[61],[15,77066,77068],{"className":77067},[13101],[15,77069,77071],{"className":77070},[66],[15,77072,13108],{"className":77073},[177],[15,77075],{"className":77076},[13112],[15,77078],{"className":77079},[71,13116],[15,77081,129],{"className":77082},[177],[15,77084],{"className":77085,"style":173},[71],[15,77087,77089,77092,77121,77124,77127,77130,77136],{"className":77088},[57],[15,77090],{"className":77091,"style":69187},[61],[15,77093,77095,77098],{"className":77094},[66],[15,77096,3885],{"className":77097,"style":3946},[66,67],[15,77099,77101],{"className":77100},[198],[15,77102,77104],{"className":77103},[202],[15,77105,77107],{"className":77106},[206],[15,77108,77110],{"className":77109,"style":1308},[210],[15,77111,77112,77115],{"style":397},[15,77113],{"className":77114,"style":218},[217],[15,77116,77118],{"className":77117},[222,223,224,225],[15,77119,1162],{"className":77120},[76,225],[15,77122,120],{"className":77123},[162],[15,77125,446],{"className":77126,"style":16497},[66,67],[15,77128,19280],{"className":77129},[169],[15,77131,77133],{"className":77132},[66],[15,77134,1266],{"className":77135},[1422,4677],[15,77137,3573],{"className":77138},[66],[11,77140,77141,77143,77144,77172,77173,77284,77285,77313,77314,77395,77396,77477],{},[2040,77142,69051],{}," 对固定的 ",[15,77145,77147,77160],{"className":77146,"translate":19},[18],[15,77148,77150],{"className":77149},[23],[25,77151,77152],{"xmlns":27},[29,77153,77154,77158],{},[32,77155,77156],{},[35,77157,123],{},[45,77159,123],{"encoding":47},[15,77161,77163],{"className":77162,"ariaHidden":53},[52],[15,77164,77166,77169],{"className":77165},[57],[15,77167],{"className":77168,"style":86},[61],[15,77170,123],{"className":77171},[66,67],"，若 ",[15,77174,77176,77210],{"className":77175,"translate":19},[18],[15,77177,77179],{"className":77178},[23],[25,77180,77181],{"xmlns":27},[29,77182,77183,77207],{},[32,77184,77185,77187,77189,77191,77193,77195,77201,77203,77205],{},[35,77186,3885],{},[39,77188,120],{"stretchy":119},[35,77190,123],{},[39,77192,126],{"stretchy":119},[39,77194,129],{},[131,77196,77197,77199],{},[35,77198,3885],{},[39,77200,1162],{},[39,77202,120],{"stretchy":119},[35,77204,123],{},[39,77206,126],{"stretchy":119},[45,77208,77209],{"encoding":47},"\\delta(x)=\\delta^*(x)",[15,77211,77213,77240],{"className":77212,"ariaHidden":53},[52],[15,77214,77216,77219,77222,77225,77228,77231,77234,77237],{"className":77215},[57],[15,77217],{"className":77218,"style":154},[61],[15,77220,3885],{"className":77221,"style":3946},[66,67],[15,77223,120],{"className":77224},[162],[15,77226,123],{"className":77227},[66,67],[15,77229,126],{"className":77230},[169],[15,77232],{"className":77233,"style":173},[71],[15,77235,129],{"className":77236},[177],[15,77238],{"className":77239,"style":173},[71],[15,77241,77243,77246,77275,77278,77281],{"className":77242},[57],[15,77244],{"className":77245,"style":154},[61],[15,77247,77249,77252],{"className":77248},[66],[15,77250,3885],{"className":77251,"style":3946},[66,67],[15,77253,77255],{"className":77254},[198],[15,77256,77258],{"className":77257},[202],[15,77259,77261],{"className":77260},[206],[15,77262,77264],{"className":77263,"style":16249},[210],[15,77265,77266,77269],{"style":213},[15,77267],{"className":77268,"style":218},[217],[15,77270,77272],{"className":77271},[222,223,224,225],[15,77273,1162],{"className":77274},[76,225],[15,77276,120],{"className":77277},[162],[15,77279,123],{"className":77280},[66,67],[15,77282,126],{"className":77283},[169],"，逐点遗憾为 ",[15,77286,77288,77301],{"className":77287,"translate":19},[18],[15,77289,77291],{"className":77290},[23],[25,77292,77293],{"xmlns":27},[29,77294,77295,77299],{},[32,77296,77297],{},[1209,77298,1804],{},[45,77300,1804],{"encoding":47},[15,77302,77304],{"className":77303,"ariaHidden":53},[52],[15,77305,77307,77310],{"className":77306},[57],[15,77308],{"className":77309,"style":1836},[61],[15,77311,1804],{"className":77312},[66],"；若不等，则所选行动的后验概率是 ",[15,77315,77317,77347],{"className":77316,"translate":19},[18],[15,77318,77320],{"className":77319},[23],[25,77321,77322],{"xmlns":27},[29,77323,77324,77344],{},[32,77325,77326,77328,77330,77332,77334,77336,77338,77340,77342],{},[35,77327,23273],{},[39,77329,1170],{},[39,77331,4307],{"stretchy":119},[35,77333,71113],{},[39,77335,335],{"separator":53},[1209,77337,1211],{},[39,77339,6602],{},[35,77341,71113],{},[39,77343,4330],{"stretchy":119},[45,77345,77346],{"encoding":47},"\\max\\{\\eta,1-\\eta\\}",[15,77348,77350,77383],{"className":77349,"ariaHidden":53},[52],[15,77351,77353,77356,77359,77362,77365,77368,77371,77374,77377,77380],{"className":77352},[57],[15,77354],{"className":77355,"style":154},[61],[15,77357,23273],{"className":77358},[1339],[15,77360,4307],{"className":77361},[162],[15,77363,71113],{"className":77364,"style":1515},[66,67],[15,77366,335],{"className":77367},[422],[15,77369],{"className":77370,"style":697},[71],[15,77372,1211],{"className":77373},[66],[15,77375],{"className":77376,"style":72},[71],[15,77378,6602],{"className":77379},[76],[15,77381],{"className":77382,"style":72},[71],[15,77384,77386,77389,77392],{"className":77385},[57],[15,77387],{"className":77388,"style":154},[61],[15,77390,71113],{"className":77391,"style":1515},[66,67],[15,77393,4330],{"className":77394},[169],"，最优行动的后验概率是 ",[15,77397,77399,77429],{"className":77398,"translate":19},[18],[15,77400,77402],{"className":77401},[23],[25,77403,77404],{"xmlns":27},[29,77405,77406,77426],{},[32,77407,77408,77410,77412,77414,77416,77418,77420,77422,77424],{},[35,77409,1178],{},[39,77411,1170],{},[39,77413,4307],{"stretchy":119},[35,77415,71113],{},[39,77417,335],{"separator":53},[1209,77419,1211],{},[39,77421,6602],{},[35,77423,71113],{},[39,77425,4330],{"stretchy":119},[45,77427,77428],{"encoding":47},"\\min\\{\\eta,1-\\eta\\}",[15,77430,77432,77465],{"className":77431,"ariaHidden":53},[52],[15,77433,77435,77438,77441,77444,77447,77450,77453,77456,77459,77462],{"className":77434},[57],[15,77436],{"className":77437,"style":154},[61],[15,77439,1178],{"className":77440},[1339],[15,77442,4307],{"className":77443},[162],[15,77445,71113],{"className":77446,"style":1515},[66,67],[15,77448,335],{"className":77449},[422],[15,77451],{"className":77452,"style":697},[71],[15,77454,1211],{"className":77455},[66],[15,77457],{"className":77458,"style":72},[71],[15,77460,6602],{"className":77461},[76],[15,77463],{"className":77464,"style":72},[71],[15,77466,77468,77471,77474],{"className":77467},[57],[15,77469],{"className":77470,"style":154},[61],[15,77472,71113],{"className":77473,"style":1515},[66,67],[15,77475,4330],{"className":77476},[169],"，两者之差为",[15,77479,77481],{"className":77480,"translate":19},[294],[15,77482,77484,77550],{"className":77483,"translate":19},[18],[15,77485,77487],{"className":77486},[23],[25,77488,77489],{"xmlns":27,"display":303},[29,77490,77491,77547],{},[32,77492,77493,77495,77497,77499,77501,77503,77505,77507,77509,77511,77513,77515,77517,77519,77521,77523,77525,77527,77529,77531,77533,77535,77537,77539,77541,77543,77545],{},[35,77494,23273],{},[39,77496,1170],{},[39,77498,4307],{"stretchy":119},[35,77500,71113],{},[39,77502,335],{"separator":53},[1209,77504,1211],{},[39,77506,6602],{},[35,77508,71113],{},[39,77510,4330],{"stretchy":119},[39,77512,6602],{},[35,77514,1178],{},[39,77516,1170],{},[39,77518,4307],{"stretchy":119},[35,77520,71113],{},[39,77522,335],{"separator":53},[1209,77524,1211],{},[39,77526,6602],{},[35,77528,71113],{},[39,77530,4330],{"stretchy":119},[39,77532,129],{},[35,77534,34430],{"mathvariant":138},[1209,77536,1263],{},[35,77538,71113],{},[39,77540,6602],{},[1209,77542,1211],{},[35,77544,34430],{"mathvariant":138},[35,77546,3573],{"mathvariant":138},[45,77548,77549],{"encoding":47},"\\max\\{\\eta,1-\\eta\\}-\\min\\{\\eta,1-\\eta\\}=|2\\eta-1|.",[15,77551,77553,77586,77607,77640,77661,77682],{"className":77552,"ariaHidden":53},[52],[15,77554,77556,77559,77562,77565,77568,77571,77574,77577,77580,77583],{"className":77555},[57],[15,77557],{"className":77558,"style":154},[61],[15,77560,23273],{"className":77561},[1339],[15,77563,4307],{"className":77564},[162],[15,77566,71113],{"className":77567,"style":1515},[66,67],[15,77569,335],{"className":77570},[422],[15,77572],{"className":77573,"style":697},[71],[15,77575,1211],{"className":77576},[66],[15,77578],{"className":77579,"style":72},[71],[15,77581,6602],{"className":77582},[76],[15,77584],{"className":77585,"style":72},[71],[15,77587,77589,77592,77595,77598,77601,77604],{"className":77588},[57],[15,77590],{"className":77591,"style":154},[61],[15,77593,71113],{"className":77594,"style":1515},[66,67],[15,77596,4330],{"className":77597},[169],[15,77599],{"className":77600,"style":72},[71],[15,77602,6602],{"className":77603},[76],[15,77605],{"className":77606,"style":72},[71],[15,77608,77610,77613,77616,77619,77622,77625,77628,77631,77634,77637],{"className":77609},[57],[15,77611],{"className":77612,"style":154},[61],[15,77614,1178],{"className":77615},[1339],[15,77617,4307],{"className":77618},[162],[15,77620,71113],{"className":77621,"style":1515},[66,67],[15,77623,335],{"className":77624},[422],[15,77626],{"className":77627,"style":697},[71],[15,77629,1211],{"className":77630},[66],[15,77632],{"className":77633,"style":72},[71],[15,77635,6602],{"className":77636},[76],[15,77638],{"className":77639,"style":72},[71],[15,77641,77643,77646,77649,77652,77655,77658],{"className":77642},[57],[15,77644],{"className":77645,"style":154},[61],[15,77647,71113],{"className":77648,"style":1515},[66,67],[15,77650,4330],{"className":77651},[169],[15,77653],{"className":77654,"style":173},[71],[15,77656,129],{"className":77657},[177],[15,77659],{"className":77660,"style":173},[71],[15,77662,77664,77667,77670,77673,77676,77679],{"className":77663},[57],[15,77665],{"className":77666,"style":154},[61],[15,77668,76991],{"className":77669},[66],[15,77671,71113],{"className":77672,"style":1515},[66,67],[15,77674],{"className":77675,"style":72},[71],[15,77677,6602],{"className":77678},[76],[15,77680],{"className":77681,"style":72},[71],[15,77683,77685,77688],{"className":77684},[57],[15,77686],{"className":77687,"style":154},[61],[15,77689,77691],{"className":77690},[66],"1∣.",[11,77693,21794,77694,77722,77723],{},[15,77695,77697,77710],{"className":77696,"translate":19},[18],[15,77698,77700],{"className":77699},[23],[25,77701,77702],{"xmlns":27},[29,77703,77704,77708],{},[32,77705,77706],{},[35,77707,446],{},[45,77709,446],{"encoding":47},[15,77711,77713],{"className":77712,"ariaHidden":53},[52],[15,77714,77716,77719],{"className":77715},[57],[15,77717],{"className":77718,"style":490},[61],[15,77720,446],{"className":77721,"style":16497},[66,67]," 取期望即得。",[15,77724,77726,77739],{"className":77725,"translate":19},[18],[15,77727,77729],{"className":77728},[23],[25,77730,77731],{"xmlns":27},[29,77732,77733,77737],{},[32,77734,77735],{},[35,77736,13514],{"mathvariant":138},[45,77738,13517],{"encoding":47},[15,77740,77742],{"className":77741,"ariaHidden":53},[52],[15,77743,77745,77748],{"className":77744},[57],[15,77746],{"className":77747,"style":13527},[61],[15,77749,13514],{"className":77750},[66,13531],[11,77752,77753,77754,77829,77830,77862,77863,77891,77892,77920,77921,77949,77950,1124],{},"这个恒等式很有解释力：只有越过决策边界的样本才贡献错误，而且每个样本的权重是 ",[15,77755,77757,77787],{"className":77756,"translate":19},[18],[15,77758,77760],{"className":77759},[23],[25,77761,77762],{"xmlns":27},[29,77763,77764,77784],{},[32,77765,77766,77768,77770,77772,77774,77776,77778,77780,77782],{},[35,77767,34430],{"mathvariant":138},[1209,77769,1263],{},[35,77771,71113],{},[39,77773,120],{"stretchy":119},[35,77775,446],{},[39,77777,126],{"stretchy":119},[39,77779,6602],{},[1209,77781,1211],{},[35,77783,34430],{"mathvariant":138},[45,77785,77786],{"encoding":47},"|2\\eta(X)-1|",[15,77788,77790,77820],{"className":77789,"ariaHidden":53},[52],[15,77791,77793,77796,77799,77802,77805,77808,77811,77814,77817],{"className":77792},[57],[15,77794],{"className":77795,"style":154},[61],[15,77797,76991],{"className":77798},[66],[15,77800,71113],{"className":77801,"style":1515},[66,67],[15,77803,120],{"className":77804},[162],[15,77806,446],{"className":77807,"style":16497},[66,67],[15,77809,126],{"className":77810},[169],[15,77812],{"className":77813,"style":72},[71],[15,77815,6602],{"className":77816},[76],[15,77818],{"className":77819,"style":72},[71],[15,77821,77823,77826],{"className":77822},[57],[15,77824],{"className":77825,"style":154},[61],[15,77827,77022],{"className":77828},[66],"。后验接近 ",[15,77831,77833,77850],{"className":77832,"translate":19},[18],[15,77834,77836],{"className":77835},[23],[25,77837,77838],{"xmlns":27},[29,77839,77840,77848],{},[32,77841,77842,77844,77846],{},[1209,77843,1211],{},[35,77845,6625],{"mathvariant":138},[1209,77847,1263],{},[45,77849,71809],{"encoding":47},[15,77851,77853],{"className":77852,"ariaHidden":53},[52],[15,77854,77856,77859],{"className":77855},[57],[15,77857],{"className":77858,"style":154},[61],[15,77860,71809],{"className":77861},[66]," 的样本位于边界附近，即使分错，代价也小；后验接近 ",[15,77864,77866,77879],{"className":77865,"translate":19},[18],[15,77867,77869],{"className":77868},[23],[25,77870,77871],{"xmlns":27},[29,77872,77873,77877],{},[32,77874,77875],{},[1209,77876,1804],{},[45,77878,1804],{"encoding":47},[15,77880,77882],{"className":77881,"ariaHidden":53},[52],[15,77883,77885,77888],{"className":77884},[57],[15,77886],{"className":77887,"style":1836},[61],[15,77889,1804],{"className":77890},[66]," 或 ",[15,77893,77895,77908],{"className":77894,"translate":19},[18],[15,77896,77898],{"className":77897},[23],[25,77899,77900],{"xmlns":27},[29,77901,77902,77906],{},[32,77903,77904],{},[1209,77905,1211],{},[45,77907,1211],{"encoding":47},[15,77909,77911],{"className":77910,"ariaHidden":53},[52],[15,77912,77914,77917],{"className":77913},[57],[15,77915],{"className":77916,"style":1836},[61],[15,77918,1211],{"className":77919},[66]," 的样本一旦分错，代价就接近 ",[15,77922,77924,77937],{"className":77923,"translate":19},[18],[15,77925,77927],{"className":77926},[23],[25,77928,77929],{"xmlns":27},[29,77930,77931,77935],{},[32,77932,77933],{},[1209,77934,1211],{},[45,77936,1211],{"encoding":47},[15,77938,77940],{"className":77939,"ariaHidden":53},[52],[15,77941,77943,77946],{"className":77942},[57],[15,77944],{"className":77945,"style":1836},[61],[15,77947,1211],{"className":77948},[66],"。换句话说，",[2040,77951,77952],{},"贝叶斯错误率不是由边界的长度决定的，而是由边界附近的后验质量决定的",[11,77954,77955,77956,78111,78112,3249],{},"回归情形也有类似的恒等式。设 ",[15,77957,77959,78001],{"className":77958,"translate":19},[18],[15,77960,77962],{"className":77961},[23],[25,77963,77964],{"xmlns":27},[29,77965,77966,77998],{},[32,77967,77968,77974,77976,77978,77980,77982,77984,77986,77988,77990,77992,77994,77996],{},[131,77969,77970,77972],{},[35,77971,116],{},[39,77973,1162],{},[39,77975,120],{"stretchy":119},[35,77977,123],{},[39,77979,126],{"stretchy":119},[39,77981,129],{},[35,77983,62381],{"mathvariant":454},[39,77985,1194],{"stretchy":119},[35,77987,61885],{},[39,77989,34430],{},[35,77991,446],{},[39,77993,129],{},[35,77995,123],{},[39,77997,1266],{"stretchy":119},[45,77999,78000],{"encoding":47},"f^*(x)=\\mathbb E[Y\\mid X=x]",[15,78002,78004,78057,78081,78099],{"className":78003,"ariaHidden":53},[52],[15,78005,78007,78010,78039,78042,78045,78048,78051,78054],{"className":78006},[57],[15,78008],{"className":78009,"style":154},[61],[15,78011,78013,78016],{"className":78012},[66],[15,78014,116],{"className":78015,"style":158},[66,67],[15,78017,78019],{"className":78018},[198],[15,78020,78022],{"className":78021},[202],[15,78023,78025],{"className":78024},[206],[15,78026,78028],{"className":78027,"style":16249},[210],[15,78029,78030,78033],{"style":213},[15,78031],{"className":78032,"style":218},[217],[15,78034,78036],{"className":78035},[222,223,224,225],[15,78037,1162],{"className":78038},[76,225],[15,78040,120],{"className":78041},[162],[15,78043,123],{"className":78044},[66,67],[15,78046,126],{"className":78047},[169],[15,78049],{"className":78050,"style":173},[71],[15,78052,129],{"className":78053},[177],[15,78055],{"className":78056,"style":173},[71],[15,78058,78060,78063,78066,78069,78072,78075,78078],{"className":78059},[57],[15,78061],{"className":78062,"style":154},[61],[15,78064,62381],{"className":78065},[66,517],[15,78067,1194],{"className":78068},[162],[15,78070,61885],{"className":78071,"style":72},[66,67],[15,78073],{"className":78074,"style":173},[71],[15,78076,34430],{"className":78077},[177],[15,78079],{"className":78080,"style":173},[71],[15,78082,78084,78087,78090,78093,78096],{"className":78083},[57],[15,78085],{"className":78086,"style":490},[61],[15,78088,446],{"className":78089,"style":16497},[66,67],[15,78091],{"className":78092,"style":173},[71],[15,78094,129],{"className":78095},[177],[15,78097],{"className":78098,"style":173},[71],[15,78100,78102,78105,78108],{"className":78101},[57],[15,78103],{"className":78104,"style":154},[61],[15,78106,123],{"className":78107},[66,67],[15,78109,1266],{"className":78110},[169],"，则对任意 ",[15,78113,78115,78134],{"className":78114,"translate":19},[18],[15,78116,78118],{"className":78117},[23],[25,78119,78120],{"xmlns":27},[29,78121,78122,78131],{},[32,78123,78124],{},[4296,78125,78126,78128],{"accent":53},[35,78127,116],{},[39,78129,78130],{},"^",[45,78132,78133],{"encoding":47},"\\hat 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E\\big[(Y-\\hat f(X))^2\\big]-\\mathbb E\\big[(Y-f^*(X))^2\\big]\\\\\n&=\\mathbb E\\big[(\\hat f(X)-f^*(X))^2\\big]\n+2\\,\\mathbb E\\big[(Y-f^*(X))(f^*(X)-\\hat 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Name":79070},[1422,1508],[15,79072,79074,79077],{"style":79073},"top:-2.1421em;",[15,79075],{"className":79076,"style":1368},[217],[15,79078,79080,79083,79086,79089,79092,79095,79101,79104,79146,79149,79152,79155,79158,79161,79164,79193,79196,79199,79202,79231,79237,79240,79243,79246,79249,79252,79255,79261,79264,79267,79270,79273,79276,79305,79308,79311,79314,79317,79346,79349,79352,79355,79358,79361,79364,79406,79409,79412,79415,79421],{"className":79079},[66],[15,79081],{"className":79082},[66],[15,79084],{"className":79085,"style":173},[71],[15,79087,129],{"className":79088},[177],[15,79090],{"className":79091,"style":173},[71],[15,79093,62381],{"className":79094},[66,517],[15,79096,79098],{"className":79097},[66],[15,79099,1194],{"className":79100},[1422,1508],[15,79102,120],{"className":79103},[162],[15,79105,79107],{"className":79106},[66,78147],[15,79108,79110,79138],{"className":79109},[202,1090],[15,79111,79113,79135],{"className":79112},[206],[15,79114,79116,79124],{"className":79115,"style":78157},[210],[15,79117,79118,79121],{"style":1386},[15,79119],{"className":79120,"style":1368},[217],[15,79122,116],{"className":79123,"style":158},[66,67],[15,79125,79126,79129],{"style":78168},[15,79127],{"className":79128,"style":1368},[217],[15,79130,79132],{"className":79131,"style":78176},[78175],[15,79133,78130],{"className":79134},[66],[15,79136,1114],{"className":79137},[1113],[15,79139,79141],{"className":79140},[206],[15,79142,79144],{"className":79143,"style":4450},[210],[15,79145],{},[15,79147,120],{"className":79148},[162],[15,79150,446],{"className":79151,"style":16497},[66,67],[15,79153,126],{"className":79154},[169],[15,79156],{"className":79157,"style":72},[71],[15,79159,6602],{"className":79160},[76],[15,79162],{"className":79163,"style":72},[71],[15,79165,79167,79170],{"className":79166},[66],[15,79168,116],{"className":79169,"style":158},[66,67],[15,79171,79173],{"className":79172},[198],[15,79174,79176],{"className":79175},[202],[15,79177,79179],{"className":79178},[206],[15,79180,79182],{"className":79181,"style":1308},[210],[15,79183,79184,79187],{"style":397},[15,79185],{"className":79186,"style":218},[217],[15,79188,79190],{"className":79189},[222,223,224,225],[15,79191,1162],{"className":79192},[76,225],[15,79194,120],{"className":79195},[162],[15,79197,446],{"className":79198,"style":16497},[66,67],[15,79200,126],{"className":79201},[169],[15,79203,79205,79208],{"className":79204},[169],[15,79206,126],{"className":79207},[169],[15,79209,79211],{"className":79210},[198],[15,79212,79214],{"className":79213},[202],[15,79215,79217],{"className":79216},[206],[15,79218,79220],{"className":79219,"style":1650},[210],[15,79221,79222,79225],{"style":397},[15,79223],{"className":79224,"style":218},[217],[15,79226,79228],{"className":79227},[222,223,224,225],[15,79229,1263],{"className":79230},[66,225],[15,79232,79234],{"className":79233},[66],[15,79235,1266],{"className":79236},[1422,1508],[15,79238],{"className":79239,"style":72},[71],[15,79241,1245],{"className":79242},[76],[15,79244],{"className":79245,"style":72},[71],[15,79247,1263],{"className":79248},[66],[15,79250],{"className":79251,"style":697},[71],[15,79253,62381],{"className":79254},[66,517],[15,79256,79258],{"className":79257},[66],[15,79259,1194],{"className":79260},[1422,1508],[15,79262,120],{"className":79263},[162],[15,79265,61885],{"className":79266,"style":72},[66,67],[15,79268],{"className":79269,"style":72},[71],[15,79271,6602],{"className":79272},[76],[15,79274],{"className":79275,"style":72},[71],[15,79277,79279,79282],{"className":79278},[66],[15,79280,116],{"className":79281,"style":158},[66,67],[15,79283,79285],{"className":79284},[198],[15,79286,79288],{"className":79287},[202],[15,79289,79291],{"className":79290},[206],[15,79292,79294],{"className":79293,"style":1308},[210],[15,79295,79296,79299],{"style":397},[15,79297],{"className":79298,"style":218},[217],[15,79300,79302],{"className":79301},[222,223,224,225],[15,79303,1162],{"className":79304},[76,225],[15,79306,120],{"className":79307},[162],[15,79309,446],{"className":79310,"style":16497},[66,67],[15,79312,18771],{"className":79313},[169],[15,79315,120],{"className":79316},[162],[15,79318,79320,79323],{"className":79319},[66],[15,79321,116],{"className":79322,"style":158},[66,67],[15,79324,79326],{"className":79325},[198],[15,79327,79329],{"className":79328},[202],[15,79330,79332],{"className":79331},[206],[15,79333,79335],{"className":79334,"style":1308},[210],[15,79336,79337,79340],{"style":397},[15,79338],{"className":79339,"style":218},[217],[15,79341,79343],{"className":79342},[222,223,224,225],[15,79344,1162],{"className":79345},[76,225],[15,79347,120],{"className":79348},[162],[15,79350,446],{"className":79351,"style":16497},[66,67],[15,79353,126],{"className":79354},[169],[15,79356],{"className":79357,"style":72},[71],[15,79359,6602],{"className":79360},[76],[15,79362],{"className":79363,"style":72},[71],[15,79365,79367],{"className":79366},[66,78147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",[15,79439,79441,79454],{"className":79440,"translate":19},[18],[15,79442,79444],{"className":79443},[23],[25,79445,79446],{"xmlns":27},[29,79447,79448,79452],{},[32,79449,79450],{},[35,79451,446],{},[45,79453,446],{"encoding":47},[15,79455,79457],{"className":79456,"ariaHidden":53},[52],[15,79458,79460,79463],{"className":79459},[57],[15,79461],{"className":79462,"style":490},[61],[15,79464,446],{"className":79465,"style":16497},[66,67]," 取条件期望后，",[15,79468,79470,79512],{"className":79469,"translate":19},[18],[15,79471,79473],{"className":79472},[23],[25,79474,79475],{"xmlns":27},[29,79476,79477,79509],{},[32,79478,79479,79481,79483,79485,79487,79493,79495,79497,79499,79501,79503,79505,79507],{},[35,79480,62381],{"mathvariant":454},[39,79482,1194],{"stretchy":119},[35,79484,61885],{},[39,79486,6602],{},[131,79488,79489,79491],{},[35,79490,116],{},[39,79492,1162],{},[39,79494,120],{"stretchy":119},[35,79496,446],{},[39,79498,126],{"stretchy":119},[39,79500,34430],{},[35,79502,446],{},[39,79504,1266],{"stretchy":119},[39,79506,129],{},[1209,79508,1804],{},[45,79510,79511],{"encoding":47},"\\mathbb E[Y-f^*(X)\\mid X]=0",[15,79513,79515,79539,79592,79613],{"className":79514,"ariaHidden":53},[52],[15,79516,79518,79521,79524,79527,79530,79533,79536],{"className":79517},[57],[15,79519],{"className":79520,"style":154},[61],[15,79522,62381],{"className":79523},[66,517],[15,79525,1194],{"className":79526},[162],[15,79528,61885],{"className":79529,"style":72},[66,67],[15,79531],{"className":79532,"style":72},[71],[15,79534,6602],{"className":79535},[76],[15,79537],{"className":79538,"style":72},[71],[15,79540,79542,79545,79574,79577,79580,79583,79586,79589],{"className":79541},[57],[15,79543],{"className":79544,"style":154},[61],[15,79546,79548,79551],{"className":79547},[66],[15,79549,116],{"className":79550,"style":158},[66,67],[15,79552,79554],{"className":79553},[198],[15,79555,79557],{"className":79556},[202],[15,79558,79560],{"className":79559},[206],[15,79561,79563],{"className":79562,"style":16249},[210],[15,79564,79565,79568],{"style":213},[15,79566],{"className":79567,"style":218},[217],[15,79569,79571],{"className":79570},[222,223,224,225],[15,79572,1162],{"className":79573},[76,225],[15,79575,120],{"className":79576},[162],[15,79578,446],{"className":79579,"style":16497},[66,67],[15,79581,126],{"className":79582},[169],[15,79584],{"className":79585,"style":173},[71],[15,79587,34430],{"className":79588},[177],[15,79590],{"className":79591,"style":173},[71],[15,79593,79595,79598,79601,79604,79607,79610],{"className":79594},[57],[15,79596],{"className":79597,"style":154},[61],[15,79599,446],{"className":79600,"style":16497},[66,67],[15,79602,1266],{"className":79603},[169],[15,79605],{"className":79606,"style":173},[71],[15,79608,129],{"className":79609},[177],[15,79611],{"className":79612,"style":173},[71],[15,79614,79616,79619],{"className":79615},[57],[15,79617],{"className":79618,"style":1836},[61],[15,79620,1804],{"className":79621},[66],"，因此消失。",[15,79624,79626,79639],{"className":79625,"translate":19},[18],[15,79627,79629],{"className":79628},[23],[25,79630,79631],{"xmlns":27},[29,79632,79633,79637],{},[32,79634,79635],{},[35,79636,13514],{"mathvariant":138},[45,79638,13517],{"encoding":47},[15,79640,79642],{"className":79641,"ariaHidden":53},[52],[15,79643,79645,79648],{"className":79644},[57],[15,79646],{"className":79647,"style":13527},[61],[15,79649,13514],{"className":79650},[66,13531],[11,79652,79653],{},"这两个恒等式把「模型误差」翻译成了「决策遗憾」：分类器的错误只发生在越过边界的点上，回归器的误差则是函数逼近误差的平方。贝叶斯风险就是那个无法被任何模型消除的起点。",[1904,79655,79657],{"id":79656},"_37-map-不是贝叶斯行动","3.7 MAP 不是贝叶斯行动",[11,79659,79660],{},"这是最常见的混淆之一。MAP 是后验的众数，是一个「点」；贝叶斯行动是使后验期望损失最小的动作，依赖损失函数。两者只有在参数空间上取 0-1 损失时才重合。",[11,79662,79663,79664,79730,79731,79796,79797,79825,79826,79892],{},"更麻烦的是，MAP 对重参数化不协变。若 ",[15,79665,79667,79691],{"className":79666,"translate":19},[18],[15,79668,79670],{"className":79669},[23],[25,79671,79672],{"xmlns":27},[29,79673,79674,79688],{},[32,79675,79676,79678,79680,79682,79684,79686],{},[35,79677,17094],{},[39,79679,129],{},[35,79681,20182],{},[39,79683,120],{"stretchy":119},[35,79685,61618],{},[39,79687,126],{"stretchy":119},[45,79689,79690],{"encoding":47},"\\phi=h(\\theta)",[15,79692,79694,79712],{"className":79693,"ariaHidden":53},[52],[15,79695,79697,79700,79703,79706,79709],{"className":79696},[57],[15,79698],{"className":79699,"style":3408},[61],[15,79701,17094],{"className":79702},[66,67],[15,79704],{"className":79705,"style":173},[71],[15,79707,129],{"className":79708},[177],[15,79710],{"className":79711,"style":173},[71],[15,79713,79715,79718,79721,79724,79727],{"className":79714},[57],[15,79716],{"className":79717,"style":154},[61],[15,79719,20182],{"className":79720},[66,67],[15,79722,120],{"className":79723},[162],[15,79725,61618],{"className":79726,"style":23025},[66,67],[15,79728,126],{"className":79729},[169]," 是一个非线性变换，",[15,79732,79734,79757],{"className":79733,"translate":19},[18],[15,79735,79737],{"className":79736},[23],[25,79738,79739],{"xmlns":27},[29,79740,79741,79755],{},[32,79742,79743,79745,79747,79749,79751,79753],{},[35,79744,11],{},[39,79746,120],{"stretchy":119},[35,79748,61618],{},[39,79750,34430],{},[35,79752,32458],{},[39,79754,126],{"stretchy":119},[45,79756,65460],{"encoding":47},[15,79758,79760,79784],{"className":79759,"ariaHidden":53},[52],[15,79761,79763,79766,79769,79772,79775,79778,79781],{"className":79762},[57],[15,79764],{"className":79765,"style":154},[61],[15,79767,11],{"className":79768},[66,67],[15,79770,120],{"className":79771},[162],[15,79773,61618],{"className":79774,"style":23025},[66,67],[15,79776],{"className":79777,"style":173},[71],[15,79779,34430],{"className":79780},[177],[15,79782],{"className":79783,"style":173},[71],[15,79785,79787,79790,79793],{"className":79786},[57],[15,79788],{"className":79789,"style":154},[61],[15,79791,32458],{"className":79792,"style":23025},[66,67],[15,79794,126],{"className":79795},[169]," 的众数经过 ",[15,79798,79800,79813],{"className":79799,"translate":19},[18],[15,79801,79803],{"className":79802},[23],[25,79804,79805],{"xmlns":27},[29,79806,79807,79811],{},[32,79808,79809],{},[35,79810,20182],{},[45,79812,20182],{"encoding":47},[15,79814,79816],{"className":79815,"ariaHidden":53},[52],[15,79817,79819,79822],{"className":79818},[57],[15,79820],{"className":79821,"style":840},[61],[15,79823,20182],{"className":79824},[66,67]," 之后，一般不是 ",[15,79827,79829,79853],{"className":79828,"translate":19},[18],[15,79830,79832],{"className":79831},[23],[25,79833,79834],{"xmlns":27},[29,79835,79836,79850],{},[32,79837,79838,79840,79842,79844,79846,79848],{},[35,79839,11],{},[39,79841,120],{"stretchy":119},[35,79843,17094],{},[39,79845,34430],{},[35,79847,32458],{},[39,79849,126],{"stretchy":119},[45,79851,79852],{"encoding":47},"p(\\phi\\mid D)",[15,79854,79856,79880],{"className":79855,"ariaHidden":53},[52],[15,79857,79859,79862,79865,79868,79871,79874,79877],{"className":79858},[57],[15,79860],{"className":79861,"style":154},[61],[15,79863,11],{"className":79864},[66,67],[15,79866,120],{"className":79867},[162],[15,79869,17094],{"className":79870},[66,67],[15,79872],{"className":79873,"style":173},[71],[15,79875,34430],{"className":79876},[177],[15,79878],{"className":79879,"style":173},[71],[15,79881,79883,79886,79889],{"className":79882},[57],[15,79884],{"className":79885,"style":154},[61],[15,79887,32458],{"className":79888,"style":23025},[66,67],[15,79890,126],{"className":79891},[169]," 的众数，因为密度会多出一个 Jacobian 因子。后验中位数在单调变换下等变，后验均值在仿射变换下等变，而 MAP 连这个性质都没有。这也是为什么在贝叶斯推断里，除非有明确的 0-1 损失，否则「取众数」并不是一个默认正确的选择。",[95,79894,79896],{"id":79895},"_4-贝叶斯错误率","4. 贝叶斯错误率",[1904,79898,79900],{"id":79899},"_41-两类情形全变差与重叠面积","4.1 两类情形：全变差与重叠面积",[11,79902,79903,79904,80028,80029,80098],{},"设两类先验相等，类条件密度为 ",[15,79905,79907,79933],{"className":79906,"translate":19},[18],[15,79908,79910],{"className":79909},[23],[25,79911,79912],{"xmlns":27},[29,79913,79914,79930],{},[32,79915,79916,79922,79924],{},[1057,79917,79918,79920],{},[35,79919,11],{},[1209,79921,1804],{},[39,79923,335],{"separator":53},[1057,79925,79926,79928],{},[35,79927,11],{},[1209,79929,1211],{},[45,79931,79932],{"encoding":47},"p_0,p_1",[15,79934,79936],{"className":79935,"ariaHidden":53},[52],[15,79937,79939,79942,79982,79985,79988],{"className":79938},[57],[15,79940],{"className":79941,"style":471},[61],[15,79943,79945,79948],{"className":79944},[66],[15,79946,11],{"className":79947},[66,67],[15,79949,79951],{"className":79950},[198],[15,79952,79954,79974],{"className":79953},[202,1090],[15,79955,79957,79971],{"className":79956},[206],[15,79958,79960],{"className":79959,"style":2139},[210],[15,79961,79962,79965],{"style":1584},[15,79963],{"className":79964,"style":218},[217],[15,79966,79968],{"className":79967},[222,223,224,225],[15,79969,1804],{"className":79970},[66,225],[15,79972,1114],{"className":79973},[1113],[15,79975,79977],{"className":79976},[206],[15,79978,79980],{"className":79979,"style":1121},[210],[15,79981],{},[15,79983,335],{"className":79984},[422],[15,79986],{"className":79987,"style":697},[71],[15,79989,79991,79994],{"className":79990},[66],[15,79992,11],{"className":79993},[66,67],[15,79995,79997],{"className":79996},[198],[15,79998,80000,80020],{"className":79999},[202,1090],[15,80001,80003,80017],{"className":80002},[206],[15,80004,80006],{"className":80005,"style":2139},[210],[15,80007,80008,80011],{"style":1584},[15,80009],{"className":80010,"style":218},[217],[15,80012,80014],{"className":80013},[222,223,224,225],[15,80015,1211],{"className":80016},[66,225],[15,80018,1114],{"className":80019},[1113],[15,80021,80023],{"className":80022},[206],[15,80024,80026],{"className":80025,"style":1121},[210],[15,80027],{},"。由 3.2 节，贝叶斯规则在 ",[15,80030,80032,80059],{"className":80031,"translate":19},[18],[15,80033,80035],{"className":80034},[23],[25,80036,80037],{"xmlns":27},[29,80038,80039,80057],{},[32,80040,80041,80043,80045,80047,80049,80051,80053,80055],{},[35,80042,71113],{},[39,80044,120],{"stretchy":119},[35,80046,123],{},[39,80048,126],{"stretchy":119},[39,80050,129],{},[1209,80052,1211],{},[35,80054,6625],{"mathvariant":138},[1209,80056,1263],{},[45,80058,71844],{"encoding":47},[15,80060,80062,80089],{"className":80061,"ariaHidden":53},[52],[15,80063,80065,80068,80071,80074,80077,80080,80083,80086],{"className":80064},[57],[15,80066],{"className":80067,"style":154},[61],[15,80069,71113],{"className":80070,"style":1515},[66,67],[15,80072,120],{"className":80073},[162],[15,80075,123],{"className":80076},[66,67],[15,80078,126],{"className":80079},[169],[15,80081],{"className":80082,"style":173},[71],[15,80084,129],{"className":80085},[177],[15,80087],{"className":80088,"style":173},[71],[15,80090,80092,80095],{"className":80091},[57],[15,80093],{"className":80094,"style":154},[61],[15,80096,71809],{"className":80097},[66]," 处切换，其中",[15,80100,80102],{"className":80101,"translate":19},[294],[15,80103,80105,80173],{"className":80104,"translate":19},[18],[15,80106,80108],{"className":80107},[23],[25,80109,80110],{"xmlns":27,"display":303},[29,80111,80112,80170],{},[32,80113,80114,80116,80118,80120,80122,80124,80168],{},[35,80115,71113],{},[39,80117,120],{"stretchy":119},[35,80119,123],{},[39,80121,126],{"stretchy":119},[39,80123,129],{},[6467,80125,80126,80140],{},[32,80127,80128,80134,80136,80138],{},[1057,80129,80130,80132],{},[35,80131,11],{},[1209,80133,1211],{},[39,80135,120],{"stretchy":119},[35,80137,123],{},[39,80139,126],{"stretchy":119},[32,80141,80142,80148,80150,80152,80154,80156,80162,80164,80166],{},[1057,80143,80144,80146],{},[35,80145,11],{},[1209,80147,1804],{},[39,80149,120],{"stretchy":119},[35,80151,123],{},[39,80153,126],{"stretchy":119},[39,80155,1245],{},[1057,80157,80158,80160],{},[35,80159,11],{},[1209,80161,1211],{},[39,80163,120],{"stretchy":119},[35,80165,123],{},[39,80167,126],{"stretchy":119},[35,80169,3573],{"mathvariant":138},[45,80171,80172],{"encoding":47},"\\eta(x)=\\frac{p_1(x)}{p_0(x)+p_1(x)}.",[15,80174,80176,80203],{"className":80175,"ariaHidden":53},[52],[15,80177,80179,80182,80185,80188,80191,80194,80197,80200],{"className":80178},[57],[15,80180],{"className":80181,"style":154},[61],[15,80183,71113],{"className":80184,"style":1515},[66,67],[15,80186,120],{"className":80187},[162],[15,80189,123],{"className":80190},[66,67],[15,80192,126],{"className":80193},[169],[15,80195],{"className":80196,"style":173},[71],[15,80198,129],{"className":80199},[177],[15,80201],{"className":80202,"style":173},[71],[15,80204,80206,80209,80421],{"className":80205},[57],[15,80207],{"className":80208,"style":41192},[61],[15,80210,80212,80215,80418],{"className":80211},[66],[15,80213],{"className":80214},[162,23417],[15,80216,80218],{"className":80217},[6467],[15,80219,80221,80410],{"className":80220},[202,1090],[15,80222,80224,80407],{"className":80223},[206],[15,80225,80227,80342,80350],{"className":80226,"style":41211},[210],[15,80228,80229,80232],{"style":23433},[15,80230],{"className":80231,"style":1368},[217],[15,80233,80235,80275,80278,80281,80284,80287,80290,80293,80333,80336,80339],{"className":80234},[66],[15,80236,80238,80241],{"className":80237},[66],[15,80239,11],{"className":80240},[66,67],[15,80242,80244],{"className":80243},[198],[15,80245,80247,80267],{"className":80246},[202,1090],[15,80248,80250,80264],{"className":80249},[206],[15,80251,80253],{"className":80252,"style":2139},[210],[15,80254,80255,80258],{"style":1584},[15,80256],{"className":80257,"style":218},[217],[15,80259,80261],{"className":80260},[222,223,224,225],[15,80262,1804],{"className":80263},[66,225],[15,80265,1114],{"className":80266},[1113],[15,80268,80270],{"className":80269},[206],[15,80271,80273],{"className":80272,"style":1121},[210],[15,80274],{},[15,80276,120],{"className":80277},[162],[15,80279,123],{"className":80280},[66,67],[15,80282,126],{"className":80283},[169],[15,80285],{"className":80286,"style":72},[71],[15,80288,1245],{"className":80289},[76],[15,80291],{"className":80292,"style":72},[71],[15,80294,80296,80299],{"className":80295},[66],[15,80297,11],{"className":80298},[66,67],[15,80300,80302],{"className":80301},[198],[15,80303,80305,80325],{"className":80304},[202,1090],[15,80306,80308,80322],{"className":80307},[206],[15,80309,80311],{"className":80310,"style":2139},[210],[15,80312,80313,80316],{"style":1584},[15,80314],{"className":80315,"style":218},[217],[15,80317,80319],{"className":80318},[222,223,224,225],[15,80320,1211],{"className":80321},[66,225],[15,80323,1114],{"className":80324},[1113],[15,80326,80328],{"className":80327},[206],[15,80329,80331],{"className":80330,"style":1121},[210],[15,80332],{},[15,80334,120],{"className":80335},[162],[15,80337,123],{"className":80338},[66,67],[15,80340,126],{"className":80341},[169],[15,80343,80344,80347],{"style":23445},[15,80345],{"className":80346,"style":1368},[217],[15,80348],{"className":80349,"style":4440},[23452],[15,80351,80352,80355],{"style":23455},[15,80353],{"className":80354,"style":1368},[217],[15,80356,80358,80398,80401,80404],{"className":80357},[66],[15,80359,80361,80364],{"className":80360},[66],[15,80362,11],{"className":80363},[66,67],[15,80365,80367],{"className":80366},[198],[15,80368,80370,80390],{"className":80369},[202,1090],[15,80371,80373,80387],{"className":80372},[206],[15,80374,80376],{"className":80375,"style":2139},[210],[15,80377,80378,80381],{"style":1584},[15,80379],{"className":80380,"style":218},[217],[15,80382,80384],{"className":80383},[222,223,224,225],[15,80385,1211],{"className":80386},[66,225],[15,80388,1114],{"className":80389},[1113],[15,80391,80393],{"className":80392},[206],[15,80394,80396],{"className":80395,"style":1121},[210],[15,80397],{},[15,80399,120],{"className":80400},[162],[15,80402,123],{"className":80403},[66,67],[15,80405,126],{"className":80406},[169],[15,80408,1114],{"className":80409},[1113],[15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\\min\\left\\{\\tfrac12 p_0(x),\\ \\tfrac12 p_1(x)\\right\\}dx\\\\\n&=\\frac12\\int \\min\\{p_0(x),p_1(x)\\}\\,dx 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是全变差距离。公式的几何意义很直接：两个类条件分布的重叠面积越大，错误率越高；当两个分布完全重合时 ",[15,82737,82739,82767],{"className":82738,"translate":19},[18],[15,82740,82742],{"className":82741},[23],[25,82743,82744],{"xmlns":27},[29,82745,82746,82764],{},[32,82747,82748,82756,82758,82760,82762],{},[1255,82749,82750,82752,82754],{},[35,82751,11],{},[35,82753,48383],{},[39,82755,1162],{},[39,82757,129],{},[1209,82759,1211],{},[35,82761,6625],{"mathvariant":138},[1209,82763,1263],{},[45,82765,82766],{"encoding":47},"p_e^*=1\u002F2",[15,82768,82770,82837],{"className":82769,"ariaHidden":53},[52],[15,82771,82773,82777,82828,82831,82834],{"className":82772},[57],[15,82774],{"className":82775,"style":82776},[61],"height:0.9357em;vertical-align:-0.247em;",[15,82778,82780,82783],{"className":82779},[66],[15,82781,11],{"className":82782},[66,67],[15,82784,82786],{"className":82785},[198],[15,82787,82789,82820],{"className":82788},[202,1090],[15,82790,82792,82817],{"className":82791},[206],[15,82793,82795,82806],{"className":82794,"style":16249},[210],[15,82796,82797,82800],{"style":1653},[15,82798],{"className":82799,"style":218},[217],[15,82801,82803],{"className":82802},[222,223,224,225],[15,82804,48383],{"className":82805},[66,67,225],[15,82807,82808,82811],{"style":213},[15,82809],{"className":82810,"style":218},[217],[15,82812,82814],{"className":82813},[222,223,224,225],[15,82815,1162],{"className":82816},[76,225],[15,82818,1114],{"className":82819},[1113],[15,82821,82823],{"className":82822},[206],[15,82824,82826],{"className":82825,"style":1686},[210],[15,82827],{},[15,82829],{"className":82830,"style":173},[71],[15,82832,129],{"className":82833},[177],[15,82835],{"className":82836,"style":173},[71],[15,82838,82840,82843],{"className":82839},[57],[15,82841],{"className":82842,"style":154},[61],[15,82844,71809],{"className":82845},[66],"，当支撑集不相交时 ",[15,82848,82850,82874],{"className":82849,"translate":19},[18],[15,82851,82853],{"className":82852},[23],[25,82854,82855],{"xmlns":27},[29,82856,82857,82871],{},[32,82858,82859,82867,82869],{},[1255,82860,82861,82863,82865],{},[35,82862,11],{},[35,82864,48383],{},[39,82866,1162],{},[39,82868,129],{},[1209,82870,1804],{},[45,82872,82873],{"encoding":47},"p_e^*=0",[15,82875,82877,82943],{"className":82876,"ariaHidden":53},[52],[15,82878,82880,82883,82934,82937,82940],{"className":82879},[57],[15,82881],{"className":82882,"style":82776},[61],[15,82884,82886,82889],{"className":82885},[66],[15,82887,11],{"className":82888},[66,67],[15,82890,82892],{"className":82891},[198],[15,82893,82895,82926],{"className":82894},[202,1090],[15,82896,82898,82923],{"className":82897},[206],[15,82899,82901,82912],{"className":82900,"style":16249},[210],[15,82902,82903,82906],{"style":1653},[15,82904],{"className":82905,"style":218},[217],[15,82907,82909],{"className":82908},[222,223,224,225],[15,82910,48383],{"className":82911},[66,67,225],[15,82913,82914,82917],{"style":213},[15,82915],{"className":82916,"style":218},[217],[15,82918,82920],{"className":82919},[222,223,224,225],[15,82921,1162],{"className":82922},[76,225],[15,82924,1114],{"className":82925},[1113],[15,82927,82929],{"className":82928},[206],[15,82930,82932],{"className":82931,"style":1686},[210],[15,82933],{},[15,82935],{"className":82936,"style":173},[71],[15,82938,129],{"className":82939},[177],[15,82941],{"className":82942,"style":173},[71],[15,82944,82946,82949],{"className":82945},[57],[15,82947],{"className":82948,"style":1836},[61],[15,82950,1804],{"className":82951},[66],[11,82953,82954],{},"若先验不相等，公式改为",[15,82956,82958],{"className":82957,"translate":19},[294],[15,82959,82961,83041],{"className":82960,"translate":19},[18],[15,82962,82964],{"className":82963},[23],[25,82965,82966],{"xmlns":27,"display":303},[29,82967,82968,83038],{},[32,82969,82970,82978,82980,82982,82984,82986,82988,82994,83000,83002,83004,83006,83008,83010,83016,83022,83024,83026,83028,83030,83032,83034,83036],{},[1255,82971,82972,82974,82976],{},[35,82973,11],{},[35,82975,48383],{},[39,82977,1162],{},[39,82979,129],{},[39,82981,29732],{},[35,82983,1178],{},[39,82985,1170],{},[39,82987,4307],{"stretchy":119},[1057,82989,82990,82992],{},[35,82991,33494],{},[1209,82993,1804],{},[1057,82995,82996,82998],{},[35,82997,11],{},[1209,82999,1804],{},[39,83001,120],{"stretchy":119},[35,83003,123],{},[39,83005,126],{"stretchy":119},[39,83007,335],{"separator":53},[1274,83009,24233],{},[1057,83011,83012,83014],{},[35,83013,33494],{},[1209,83015,1211],{},[1057,83017,83018,83020],{},[35,83019,11],{},[1209,83021,1211],{},[39,83023,120],{"stretchy":119},[35,83025,123],{},[39,83027,126],{"stretchy":119},[39,83029,4330],{"stretchy":119},[1274,83031,2462],{},[35,83033,458],{},[35,83035,123],{},[39,83037,335],{"separator":53},[45,83039,83040],{"encoding":47},"p_e^*=\\int \\min\\{\\pi_0p_0(x),\\ \\pi_1p_1(x)\\}\\,dx,",[15,83042,83044,83110],{"className":83043,"ariaHidden":53},[52],[15,83045,83047,83050,83101,83104,83107],{"className":83046},[57],[15,83048],{"className":83049,"style":82023},[61],[15,83051,83053,83056],{"className":83052},[66],[15,83054,11],{"className":83055},[66,67],[15,83057,83059],{"className":83058},[198],[15,83060,83062,83093],{"className":83061},[202,1090],[15,83063,83065,83090],{"className":83064},[206],[15,83066,83068,83079],{"className":83067,"style":1308},[210],[15,83069,83070,83073],{"style":1653},[15,83071],{"className":83072,"style":218},[217],[15,83074,83076],{"className":83075},[222,223,224,225],[15,83077,48383],{"className":83078},[66,67,225],[15,83080,83081,83084],{"style":397},[15,83082],{"className":83083,"style":218},[217],[15,83085,83087],{"className":83086},[222,223,224,225],[15,83088,1162],{"className":83089},[76,225],[15,83091,1114],{"className":83092},[1113],[15,83094,83096],{"className":83095},[206],[15,83097,83099],{"className":83098,"style":1686},[210],[15,83100],{},[15,83102],{"className":83103,"style":173},[71],[15,83105,129],{"className":83106},[177],[15,83108],{"className":83109,"style":173},[71],[15,83111,83113,83116,83119,83122,83125,83128,83168,83208,83211,83214,83217,83220,83223,83226,83266,83306,83309,83312,83315,83318,83321,83324],{"className":83112},[57],[15,83114],{"className":83115,"style":65130},[61],[15,83117,29732],{"className":83118,"style":29796},[1339,1470,1471],[15,83120],{"className":83121,"style":697},[71],[15,83123,1178],{"className":83124},[1339],[15,83126,4307],{"className":83127},[162],[15,83129,83131,83134],{"className":83130},[66],[15,83132,33494],{"className":83133,"style":1515},[66,67],[15,83135,83137],{"className":83136},[198],[15,83138,83140,83160],{"className":83139},[202,1090],[15,83141,83143,83157],{"className":83142},[206],[15,83144,83146],{"className":83145,"style":2139},[210],[15,83147,83148,83151],{"style":1531},[15,83149],{"className":83150,"style":218},[217],[15,83152,83154],{"className":83153},[222,223,224,225],[15,83155,1804],{"className":83156},[66,225],[15,83158,1114],{"className":83159},[1113],[15,83161,83163],{"className":83162},[206],[15,83164,83166],{"className":83165,"style":1121},[210],[15,83167],{},[15,83169,83171,83174],{"className":83170},[66],[15,83172,11],{"className":83173},[66,67],[15,83175,83177],{"className":83176},[198],[15,83178,83180,83200],{"className":83179},[202,1090],[15,83181,83183,83197],{"className":83182},[206],[15,83184,83186],{"className":83185,"style":2139},[210],[15,83187,83188,83191],{"style":1584},[15,83189],{"className":83190,"style":218},[217],[15,83192,83194],{"className":83193},[222,223,224,225],[15,83195,1804],{"className":83196},[66,225],[15,83198,1114],{"className":83199},[1113],[15,83201,83203],{"className":83202},[206],[15,83204,83206],{"className":83205,"style":1121},[210],[15,83207],{},[15,83209,120],{"className":83210},[162],[15,83212,123],{"className":83213},[66,67],[15,83215,126],{"className":83216},[169],[15,83218,335],{"className":83219},[422],[15,83221,24233],{"className":83222},[71],[15,83224],{"className":83225,"style":697},[71],[15,83227,83229,83232],{"className":83228},[66],[15,83230,33494],{"className":83231,"style":1515},[66,67],[15,83233,83235],{"className":83234},[198],[15,83236,83238,83258],{"className":83237},[202,1090],[15,83239,83241,83255],{"className":83240},[206],[15,83242,83244],{"className":83243,"style":2139},[210],[15,83245,83246,83249],{"style":1531},[15,83247],{"className":83248,"style":218},[217],[15,83250,83252],{"className":83251},[222,223,224,225],[15,83253,1211],{"className":83254},[66,225],[15,83256,1114],{"className":83257},[1113],[15,83259,83261],{"className":83260},[206],[15,83262,83264],{"className":83263,"style":1121},[210],[15,83265],{},[15,83267,83269,83272],{"className":83268},[66],[15,83270,11],{"className":83271},[66,67],[15,83273,83275],{"className":83274},[198],[15,83276,83278,83298],{"className":83277},[202,1090],[15,83279,83281,83295],{"className":83280},[206],[15,83282,83284],{"className":83283,"style":2139},[210],[15,83285,83286,83289],{"style":1584},[15,83287],{"className":83288,"style":218},[217],[15,83290,83292],{"className":83291},[222,223,224,225],[15,83293,1211],{"className":83294},[66,225],[15,83296,1114],{"className":83297},[1113],[15,83299,83301],{"className":83300},[206],[15,83302,83304],{"className":83303,"style":1121},[210],[15,83305],{},[15,83307,120],{"className":83308},[162],[15,83310,123],{"className":83311},[66,67],[15,83313,19280],{"className":83314},[169],[15,83316],{"className":83317,"style":697},[71],[15,83319,458],{"className":83320},[66,67],[15,83322,123],{"className":83323},[66,67],[15,83325,335],{"className":83326},[422],[11,83328,83329],{},"它仍然是一个「重叠面积」，只是两侧的权重不同。先验越偏向某一类，另一类的判定区域就越小，错误率的构成也随之改变。",[1904,83331,83333,83334,83363],{"id":83332},"_42-高斯情形马氏距离与-qqq-函数","4.2 高斯情形：马氏距离与 ",[15,83335,83337,83351],{"className":83336,"translate":19},[18],[15,83338,83340],{"className":83339},[23],[25,83341,83342],{"xmlns":27},[29,83343,83344,83349],{},[32,83345,83346],{},[35,83347,83348],{},"Q",[45,83350,83348],{"encoding":47},[15,83352,83354],{"className":83353,"ariaHidden":53},[52],[15,83355,83357,83360],{"className":83356},[57],[15,83358],{"className":83359,"style":17379},[61],[15,83361,83348],{"className":83362},[66,67]," 函数",[11,83365,83366],{},"设",[15,83368,83370],{"className":83369,"translate":19},[294],[15,83371,83373,83449],{"className":83372,"translate":19},[18],[15,83374,83376],{"className":83375},[23],[25,83377,83378],{"xmlns":27,"display":303},[29,83379,83380,83446],{},[32,83381,83382,83384,83386,83388,83390,83392,83394,83396,83398,83405,83407,83410,83412,83414,83416,83418,83420,83422,83424,83426,83428,83430,83432,83438,83440,83442,83444],{},[35,83383,446],{},[39,83385,34430],{},[35,83387,61885],{},[39,83389,129],{},[1209,83391,1804],{},[39,83393,28391],{},[35,83395,28535],{"mathvariant":445},[39,83397,120],{"stretchy":119},[1057,83399,83400,83403],{},[35,83401,83402],{},"μ",[1209,83404,1804],{},[39,83406,335],{"separator":53},[35,83408,83409],{"mathvariant":138},"Σ",[39,83411,126],{"stretchy":119},[39,83413,335],{"separator":53},[71,83415],{"width":3904},[35,83417,446],{},[39,83419,34430],{},[35,83421,61885],{},[39,83423,129],{},[1209,83425,1211],{},[39,83427,28391],{},[35,83429,28535],{"mathvariant":445},[39,83431,120],{"stretchy":119},[1057,83433,83434,83436],{},[35,83435,83402],{},[1209,83437,1211],{},[39,83439,335],{"separator":53},[35,83441,83409],{"mathvariant":138},[39,83443,126],{"stretchy":119},[39,83445,335],{"separator":53},[45,83447,83448],{"encoding":47},"X\\mid 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",[15,85103,85105,85143],{"className":85104,"translate":19},[18],[15,85106,85108],{"className":85107},[23],[25,85109,85110],{"xmlns":27},[29,85111,85112,85140],{},[32,85113,85114,85116,85118,85124,85126,85128,85130,85132,85138],{},[35,85115,28535],{"mathvariant":445},[39,85117,120],{"stretchy":119},[131,85119,85120,85122],{},[35,85121,84668],{"mathvariant":138},[1209,85123,1263],{},[35,85125,6625],{"mathvariant":138},[1209,85127,1263],{},[39,85129,335],{"separator":53},[1274,85131,24233],{},[131,85133,85134,85136],{},[35,85135,84668],{"mathvariant":138},[1209,85137,1263],{},[39,85139,126],{"stretchy":119},[45,85141,85142],{"encoding":47},"\\mathcal N(\\Delta^2\u002F2,\\ \\Delta^2)",[15,85144,85146],{"className":85145,"ariaHidden":53},[52],[15,85147,85149,85152,85155,85158,85187,85190,85193,85196,85199,85228],{"className":85148},[57],[15,85150],{"className":85151,"style":9154},[61],[15,85153,28535],{"className":85154,"style":83515},[66,494],[15,85156,120],{"className":85157},[162],[15,85159,85161,85164],{"className":85160},[66],[15,85162,84668],{"className":85163},[66],[15,85165,85167],{"className":85166},[198],[15,85168,85170],{"className":85169},[202],[15,85171,85173],{"className":85172},[206],[15,85174,85176],{"className":85175,"style":6740},[210],[15,85177,85178,85181],{"style":213},[15,85179],{"className":85180,"style":218},[217],[15,85182,85184],{"className":85183},[222,223,224,225],[15,85185,1263],{"className":85186},[66,225],[15,85188,7110],{"className":85189},[66],[15,85191,335],{"className":85192},[422],[15,85194,24233],{"className":85195},[71],[15,85197],{"className":85198,"style":697},[71],[15,85200,85202,85205],{"className":85201},[66],[15,85203,84668],{"className":85204},[66],[15,85206,85208],{"className":85207},[198],[15,85209,85211],{"className":85210},[202],[15,85212,85214],{"className":85213},[206],[15,85215,85217],{"className":85216,"style":6740},[210],[15,85218,85219,85222],{"style":213},[15,85220],{"className":85221,"style":218},[217],[15,85223,85225],{"className":85224},[222,223,224,225],[15,85226,1263],{"className":85227},[66,225],[15,85229,126],{"className":85230},[169],"。贝叶斯规则在似然比为零处切换，因此",[15,85233,85235],{"className":85234,"translate":19},[294],[15,85236,85238,85331],{"className":85237,"translate":19},[18],[15,85239,85241],{"className":85240},[23],[25,85242,85243],{"xmlns":27,"display":303},[29,85244,85245,85328],{},[32,85246,85247,85255,85257,85259,85261,85263,85265,85271,85273,85275,85277,85283,85285,85287,85289,85291,85293,85296,85310,85312,85314,85326],{},[1255,85248,85249,85251,85253],{},[35,85250,11],{},[35,85252,48383],{},[39,85254,1162],{},[39,85256,129],{},[35,85258,28399],{},[39,85260,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,85262,28535],{"mathvariant":445},[39,85264,120],{"stretchy":119},[131,85266,85267,85269],{},[35,85268,84668],{"mathvariant":138},[1209,85270,1263],{},[35,85272,6625],{"mathvariant":138},[1209,85274,1263],{},[39,85276,335],{"separator":53},[131,85278,85279,85281],{},[35,85280,84668],{"mathvariant":138},[1209,85282,1263],{},[39,85284,126],{"stretchy":119},[39,85286,25772],{},[1209,85288,1804],{},[39,85290,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,85292,129],{},[35,85294,85295],{"mathvariant":138},"Φ",[32,85297,85298,85300,85302,85308],{},[39,85299,120],{"fence":53},[39,85301,6602],{},[6467,85303,85304,85306],{},[35,85305,84668],{"mathvariant":138},[1209,85307,1263],{},[39,85309,126],{"fence":53},[39,85311,129],{},[35,85313,83348],{},[32,85315,85316,85318,85324],{},[39,85317,120],{"fence":53},[6467,85319,85320,85322],{},[35,85321,84668],{"mathvariant":138},[1209,85323,1263],{},[39,85325,126],{"fence":53},[39,85327,335],{"separator":53},[45,85329,85330],{"encoding":47},"p_e^*=P\\big(\\mathcal N(\\Delta^2\u002F2,\\Delta^2)\u003C0\\big)\n=\\Phi\\left(-\\frac{\\Delta}{2}\\right)\n=Q\\left(\\frac{\\Delta}{2}\\right),",[15,85332,85334,85400,85500,85524,85626],{"className":85333,"ariaHidden":53},[52],[15,85335,85337,85340,85391,85394,85397],{"className":85336},[57],[15,85338],{"className":85339,"style":82023},[61],[15,85341,85343,85346],{"className":85342},[66],[15,85344,11],{"className":85345},[66,67],[15,85347,85349],{"className":85348},[198],[15,85350,85352,85383],{"className":85351},[202,1090],[15,85353,85355,85380],{"className":85354},[206],[15,85356,85358,85369],{"className":85357,"style":1308},[210],[15,85359,85360,85363],{"style":1653},[15,85361],{"className":85362,"style":218},[217],[15,85364,85366],{"className":85365},[222,223,224,225],[15,85367,48383],{"className":85368},[66,67,225],[15,85370,85371,85374],{"style":397},[15,85372],{"className":85373,"style":218},[217],[15,85375,85377],{"className":85376},[222,223,224,225],[15,85378,1162],{"className":85379},[76,225],[15,85381,1114],{"className":85382},[1113],[15,85384,85386],{"className":85385},[206],[15,85387,85389],{"className":85388,"style":1686},[210],[15,85390],{},[15,85392],{"className":85393,"style":173},[71],[15,85395,129],{"className":85396},[177],[15,85398],{"className":85399,"style":173},[71],[15,85401,85403,85406,85409,85415,85418,85421,85450,85453,85456,85459,85488,85491,85494,85497],{"className":85402},[57],[15,85404],{"className":85405,"style":11718},[61],[15,85407,28399],{"className":85408,"style":62513},[66,67],[15,85410,85412],{"className":85411},[66],[15,85413,120],{"className":85414},[1422,1508],[15,85416,28535],{"className":85417,"style":83515},[66,494],[15,85419,120],{"className":85420},[162],[15,85422,85424,85427],{"className":85423},[66],[15,85425,84668],{"className":85426},[66],[15,85428,85430],{"className":85429},[198],[15,85431,85433],{"className":85432},[202],[15,85434,85436],{"className":85435},[206],[15,85437,85439],{"className":85438,"style":1650},[210],[15,85440,85441,85444],{"style":397},[15,85442],{"className":85443,"style":218},[217],[15,85445,85447],{"className":85446},[222,223,224,225],[15,85448,1263],{"className":85449},[66,225],[15,85451,7110],{"className":85452},[66],[15,85454,335],{"className":85455},[422],[15,85457],{"className":85458,"style":697},[71],[15,85460,85462,85465],{"className":85461},[66],[15,85463,84668],{"className":85464},[66],[15,85466,85468],{"className":85467},[198],[15,85469,85471],{"className":85470},[202],[15,85472,85474],{"className":85473},[206],[15,85475,85477],{"className":85476,"style":1650},[210],[15,85478,85479,85482],{"style":397},[15,85480],{"className":85481,"style":218},[217],[15,85483,85485],{"className":85484},[222,223,224,225],[15,85486,1263],{"className":85487},[66,225],[15,85489,126],{"className":85490},[169],[15,85492],{"className":85493,"style":173},[71],[15,85495,25772],{"className":85496},[177],[15,85498],{"className":85499,"style":173},[71],[15,85501,85503,85506,85509,85515,85518,85521],{"className":85502},[57],[15,85504],{"className":85505,"style":4400},[61],[15,85507,1804],{"className":85508},[66],[15,85510,85512],{"className":85511},[66],[15,85513,126],{"className":85514},[1422,1508],[15,85516],{"className":85517,"style":173},[71],[15,85519,129],{"className":85520},[177],[15,85522],{"className":85523,"style":173},[71],[15,85525,85527,85530,85533,85536,85617,85620,85623],{"className":85526},[57],[15,85528],{"className":85529,"style":43973},[61],[15,85531,85295],{"className":85532},[66],[15,85534],{"className":85535,"style":697},[71],[15,85537,85539,85545,85548,85611],{"className":85538},[1413],[15,85540,85542],{"className":85541,"style":1418},[162,1417],[15,85543,120],{"className":85544},[1422,224],[15,85546,6602],{"className":85547},[66],[15,85549,85551,85554,85608],{"className":85550},[66],[15,85552],{"className":85553},[162,23417],[15,85555,85557],{"className":85556},[6467],[15,85558,85560,85600],{"className":85559},[202,1090],[15,85561,85563,85597],{"className":85562},[206],[15,85564,85567,85578,85586],{"className":85565,"style":85566},[210],"height:1.3603em;",[15,85568,85569,85572],{"style":23433},[15,85570],{"className":85571,"style":1368},[217],[15,85573,85575],{"className":85574},[66],[15,85576,1263],{"className":85577},[66],[15,85579,85580,85583],{"style":23445},[15,85581],{"className":85582,"style":1368},[217],[15,85584],{"className":85585,"style":4440},[23452],[15,85587,85588,85591],{"style":23455},[15,85589],{"className":85590,"style":1368},[217],[15,85592,85594],{"className":85593},[66],[15,85595,84668],{"className":85596},[66],[15,85598,1114],{"className":85599},[1113],[15,85601,85603],{"className":85602},[206],[15,85604,85606],{"className":85605,"style":23474},[210],[15,85607],{},[15,85609],{"className":85610},[169,23417],[15,85612,85614],{"className":85613,"style":1418},[169,1417],[15,85615,126],{"className":85616},[1422,224],[15,85618],{"className":85619,"style":173},[71],[15,85621,129],{"className":85622},[177],[15,85624],{"className":85625,"style":173},[71],[15,85627,85629,85632,85635,85638,85715,85718],{"className":85628},[57],[15,85630],{"className":85631,"style":43973},[61],[15,85633,83348],{"className":85634},[66,67],[15,85636],{"className":85637,"style":697},[71],[15,85639,85641,85647,85709],{"className":85640},[1413],[15,85642,85644],{"className":85643,"style":1418},[162,1417],[15,85645,120],{"className":85646},[1422,224],[15,85648,85650,85653,85706],{"className":85649},[66],[15,85651],{"className":85652},[162,23417],[15,85654,85656],{"className":85655},[6467],[15,85657,85659,85698],{"className":85658},[202,1090],[15,85660,85662,85695],{"className":85661},[206],[15,85663,85665,85676,85684],{"className":85664,"style":85566},[210],[15,85666,85667,85670],{"style":23433},[15,85668],{"className":85669,"style":1368},[217],[15,85671,85673],{"className":85672},[66],[15,85674,1263],{"className":85675},[66],[15,85677,85678,85681],{"style":23445},[15,85679],{"className":85680,"style":1368},[217],[15,85682],{"className":85683,"style":4440},[23452],[15,85685,85686,85689],{"style":23455},[15,85687],{"className":85688,"style":1368},[217],[15,85690,85692],{"className":85691},[66],[15,85693,84668],{"className":85694},[66],[15,85696,1114],{"className":85697},[1113],[15,85699,85701],{"className":85700},[206],[15,85702,85704],{"className":85703,"style":23474},[210],[15,85705],{},[15,85707],{"className":85708},[169,23417],[15,85710,85712],{"className":85711,"style":1418},[169,1417],[15,85713,126],{"className":85714},[1422,224],[15,85716],{"className":85717,"style":697},[71],[15,85719,335],{"className":85720},[422],[11,85722,425,85723,85752,85753,1124],{},[15,85724,85726,85740],{"className":85725,"translate":19},[18],[15,85727,85729],{"className":85728},[23],[25,85730,85731],{"xmlns":27},[29,85732,85733,85737],{},[32,85734,85735],{},[35,85736,85295],{"mathvariant":138},[45,85738,85739],{"encoding":47},"\\Phi",[15,85741,85743],{"className":85742,"ariaHidden":53},[52],[15,85744,85746,85749],{"className":85745},[57],[15,85747],{"className":85748,"style":490},[61],[15,85750,85295],{"className":85751},[66]," 是标准正态分布函数，",[15,85754,85756,85778],{"className":85755,"translate":19},[18],[15,85757,85759],{"className":85758},[23],[25,85760,85761],{"xmlns":27},[29,85762,85763,85775],{},[32,85764,85765,85767,85769,85771,85773],{},[35,85766,83348],{},[39,85768,129],{},[1209,85770,1211],{},[39,85772,6602],{},[35,85774,85295],{"mathvariant":138},[45,85776,85777],{"encoding":47},"Q=1-\\Phi",[15,85779,85781,85799,85817],{"className":85780,"ariaHidden":53},[52],[15,85782,85784,85787,85790,85793,85796],{"className":85783},[57],[15,85785],{"className":85786,"style":17379},[61],[15,85788,83348],{"className":85789},[66,67],[15,85791],{"className":85792,"style":173},[71],[15,85794,129],{"className":85795},[177],[15,85797],{"className":85798,"style":173},[71],[15,85800,85802,85805,85808,85811,85814],{"className":85801},[57],[15,85803],{"className":85804,"style":30468},[61],[15,85806,1211],{"className":85807},[66],[15,85809],{"className":85810,"style":72},[71],[15,85812,6602],{"className":85813},[76],[15,85815],{"className":85816,"style":72},[71],[15,85818,85820,85823],{"className":85819},[57],[15,85821],{"className":85822,"style":490},[61],[15,85824,85295],{"className":85825},[66],[11,85827,85828],{},"几个数值可以帮助建立直觉：",[6152,85830,85831,86184],{},[6155,85832,85833],{},[6158,85834,85835,85866,85993],{},[6161,85836,85837],{},[15,85838,85840,85854],{"className":85839,"translate":19},[18],[15,85841,85843],{"className":85842},[23],[25,85844,85845],{"xmlns":27},[29,85846,85847,85851],{},[32,85848,85849],{},[35,85850,84668],{"mathvariant":138},[45,85852,85853],{"encoding":47},"\\Delta",[15,85855,85857],{"className":85856,"ariaHidden":53},[52],[15,85858,85860,85863],{"className":85859},[57],[15,85861],{"className":85862,"style":490},[61],[15,85864,84668],{"className":85865},[66],[6161,85867,85868],{},[15,85869,85871,85905],{"className":85870,"translate":19},[18],[15,85872,85874],{"className":85873},[23],[25,85875,85876],{"xmlns":27},[29,85877,85878,85902],{},[32,85879,85880,85888,85890,85892,85894,85896,85898,85900],{},[1255,85881,85882,85884,85886],{},[35,85883,11],{},[35,85885,48383],{},[39,85887,1162],{},[39,85889,129],{},[35,85891,83348],{},[39,85893,120],{"stretchy":119},[35,85895,84668],{"mathvariant":138},[35,85897,6625],{"mathvariant":138},[1209,85899,1263],{},[39,85901,126],{"stretchy":119},[45,85903,85904],{"encoding":47},"p_e^*=Q(\\Delta\u002F2)",[15,85906,85908,85974],{"className":85907,"ariaHidden":53},[52],[15,85909,85911,85914,85965,85968,85971],{"className":85910},[57],[15,85912],{"className":85913,"style":82776},[61],[15,85915,85917,85920],{"className":85916},[66],[15,85918,11],{"className":85919},[66,67],[15,85921,85923],{"className":85922},[198],[15,85924,85926,85957],{"className":85925},[202,1090],[15,85927,85929,85954],{"className":85928},[206],[15,85930,85932,85943],{"className":85931,"style":16249},[210],[15,85933,85934,85937],{"style":1653},[15,85935],{"className":85936,"style":218},[217],[15,85938,85940],{"className":85939},[222,223,224,225],[15,85941,48383],{"className":85942},[66,67,225],[15,85944,85945,85948],{"style":213},[15,85946],{"className":85947,"style":218},[217],[15,85949,85951],{"className":85950},[222,223,224,225],[15,85952,1162],{"className":85953},[76,225],[15,85955,1114],{"className":85956},[1113],[15,85958,85960],{"className":85959},[206],[15,85961,85963],{"className":85962,"style":1686},[210],[15,85964],{},[15,85966],{"className":85967,"style":173},[71],[15,85969,129],{"className":85970},[177],[15,85972],{"className":85973,"style":173},[71],[15,85975,85977,85980,85983,85986,85990],{"className":85976},[57],[15,85978],{"className":85979,"style":154},[61],[15,85981,83348],{"className":85982},[66,67],[15,85984,120],{"className":85985},[162],[15,85987,85989],{"className":85988},[66],"Δ\u002F2",[15,85991,126],{"className":85992},[169],[6161,85994,85995,85996],{},"Bhattacharyya 上界 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很大时，",[15,87106,87108],{"className":87107,"translate":19},[294],[15,87109,87111,87175],{"className":87110,"translate":19},[18],[15,87112,87114],{"className":87113},[23],[25,87115,87116],{"xmlns":27,"display":303},[29,87117,87118,87172],{},[32,87119,87120,87122,87134,87136,87152,87170],{},[35,87121,83348],{},[32,87123,87124,87126,87132],{},[39,87125,120],{"fence":53},[6467,87127,87128,87130],{},[35,87129,84668],{"mathvariant":138},[1209,87131,1263],{},[39,87133,126],{"fence":53},[39,87135,28391],{},[6467,87137,87138,87140],{},[1209,87139,1263],{},[32,87141,87142,87144],{},[35,87143,84668],{"mathvariant":138},[48508,87145,87146],{},[32,87147,87148,87150],{},[1209,87149,1263],{},[35,87151,33494],{},[131,87153,87154,87156],{},[35,87155,48383],{},[32,87157,87158,87160,87166,87168],{},[39,87159,6602],{},[131,87161,87162,87164],{},[35,87163,84668],{"mathvariant":138},[1209,87165,1263],{},[35,87167,6625],{"mathvariant":138},[1209,87169,86034],{},[39,87171,335],{"separator":53},[45,87173,87174],{"encoding":47},"Q\\left(\\frac{\\Delta}{2}\\right)\\sim 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{},"与上界只差一个多项式因子。指数部分完全一致，这正是 Chernoff 界想要抓住的东西：错误率衰减的",[2040,87469,87470],{},"速率",[1904,87472,87474],{"id":87473},"_43-用散度给错误率定界","4.3 用散度给错误率定界",[11,87476,87477],{},"全变差是精确的，但它通常难以估计。实践中更常用的是 Hellinger 距离与 KL 散度。它们与全变差之间有一条标准的不等式链：",[15,87479,87481],{"className":87480,"translate":19},[294],[15,87482,87484,87604],{"className":87483,"translate":19},[18],[15,87485,87487],{"className":87486},[23],[25,87488,87489],{"xmlns":27,"display":303},[29,87490,87491,87601],{},[32,87492,87493,87495,87501,87503,87509,87519,87521,87523,87525,87531,87533,87539,87541,87543,87545,87551,87553,87559,87561,87567,87569,87571,87573,87575,87581,87583,87589,87591,87597,87599],{},[35,87494,1251],{"mathvariant":138},[1057,87496,87497,87499],{},[35,87498,28399],{},[1209,87500,1804],{},[39,87502,6602],{},[1057,87504,87505,87507],{},[35,87506,28399],{},[1209,87508,1211],{},[1057,87510,87511,87513],{},[35,87512,1251],{"mathvariant":138},[32,87514,87515,87517],{},[35,87516,32778],{"mathvariant":138},[35,87518,31732],{"mathvariant":138},[39,87520,24220],{},[35,87522,1061],{},[39,87524,120],{"stretchy":119},[1057,87526,87527,87529],{},[35,87528,28399],{},[1209,87530,1804],{},[39,87532,335],{"separator":53},[1057,87534,87535,87537],{},[35,87536,28399],{},[1209,87538,1211],{},[39,87540,126],{"stretchy":119},[39,87542,335],{"separator":53},[71,87544],{"width":3904},[131,87546,87547,87549],{},[35,87548,1061],{},[1209,87550,1263],{},[39,87552,120],{"stretchy":119},[1057,87554,87555,87557],{},[35,87556,28399],{},[1209,87558,1804],{},[39,87560,335],{"separator":53},[1057,87562,87563,87565],{},[35,87564,28399],{},[1209,87566,1211],{},[39,87568,126],{"stretchy":119},[39,87570,129],{},[1209,87572,1211],{},[39,87574,6602],{},[32,87576,87577,87579],{},[35,87578,63972],{"mathvariant":138},[35,87580,23256],{"mathvariant":138},[39,87582,120],{"stretchy":119},[1057,87584,87585,87587],{},[35,87586,28399],{},[1209,87588,1804],{},[39,87590,335],{"separator":53},[1057,87592,87593,87595],{},[35,87594,28399],{},[1209,87596,1211],{},[39,87598,126],{"stretchy":119},[39,87600,335],{"separator":53},[45,87602,87603],{"encoding":47},"\\|P_0-P_1\\|_{\\mathrm{TV}}\\le H(P_0,P_1),\n\\qquad\nH^2(P_0,P_1)=1-\\mathrm{BC}(P_0,P_1),",[15,87605,87607,87665,87766,88006,88024],{"className":87606,"ariaHidden":53},[52],[15,87608,87610,87613,87616,87656,87659,87662],{"className":87609},[57],[15,87611],{"className":87612,"style":154},[61],[15,87614,1251],{"className":87615},[66],[15,87617,87619,87622],{"className":87618},[66],[15,87620,28399],{"className":87621,"style":62513},[66,67],[15,87623,87625],{"className":87624},[198],[15,87626,87628,87648],{"className":87627},[202,1090],[15,87629,87631,87645],{"className":87630},[206],[15,87632,87634],{"className":87633,"style":2139},[210],[15,87635,87636,87639],{"style":82499},[15,87637],{"className":87638,"style":218},[217],[15,87640,87642],{"className":87641},[222,223,224,225],[15,87643,1804],{"className":87644},[66,225],[15,87646,1114],{"className":87647},[1113],[15,87649,87651],{"className":87650},[206],[15,87652,87654],{"className":87653,"style":1121},[210],[15,87655],{},[15,87657],{"className":87658,"style":72},[71],[15,87660,6602],{"className":87661},[76],[15,87663],{"className":87664,"style":72},[71],[15,87666,87668,87671,87711,87757,87760,87763],{"className":87667},[57],[15,87669],{"className":87670,"style":154},[61],[15,87672,87674,87677],{"className":87673},[66],[15,87675,28399],{"className":87676,"style":62513},[66,67],[15,87678,87680],{"className":87679},[198],[15,87681,87683,87703],{"className":87682},[202,1090],[15,87684,87686,87700],{"className":87685},[206],[15,87687,87689],{"className":87688,"style":2139},[210],[15,87690,87691,87694],{"style":82499},[15,87692],{"className":87693,"style":218},[217],[15,87695,87697],{"className":87696},[222,223,224,225],[15,87698,1211],{"className":87699},[66,225],[15,87701,1114],{"className":87702},[1113],[15,87704,87706],{"className":87705},[206],[15,87707,87709],{"className":87708,"style":1121},[210],[15,87710],{},[15,87712,87714,87717],{"className":87713},[66],[15,87715,1251],{"className":87716},[66],[15,87718,87720],{"className":87719},[198],[15,87721,87723,87749],{"className":87722},[202,1090],[15,87724,87726,87746],{"className":87725},[206],[15,87727,87729],{"className":87728,"style":3638},[210],[15,87730,87731,87734],{"style":1584},[15,87732],{"className":87733,"style":218},[217],[15,87735,87737],{"className":87736},[222,223,224,225],[15,87738,87740],{"className":87739},[66,225],[15,87741,87743],{"className":87742},[66,225],[15,87744,82610],{"className":87745,"style":1343},[66,4429,225],[15,87747,1114],{"className":87748},[1113],[15,87750,87752],{"className":87751},[206],[15,87753,87755],{"className":87754,"style":1121},[210],[15,87756],{},[15,87758],{"className":87759,"style":173},[71],[15,87761,24220],{"className":87762},[177],[15,87764],{"className":87765,"style":173},[71],[15,87767,87769,87772,87775,87778,87818,87821,87824,87864,87867,87870,87873,87876,87905,87908,87948,87951,87954,87994,87997,88000,88003],{"className":87768},[57],[15,87770],{"className":87771,"style":11214},[61],[15,87773,1061],{"className":87774,"style":7073},[66,67],[15,87776,120],{"className":87777},[162],[15,87779,87781,87784],{"className":87780},[66],[15,87782,28399],{"className":87783,"style":62513},[66,67],[15,87785,87787],{"className":87786},[198],[15,87788,87790,87810],{"className":87789},[202,1090],[15,87791,87793,87807],{"className":87792},[206],[15,87794,87796],{"className":87795,"style":2139},[210],[15,87797,87798,87801],{"style":82499},[15,87799],{"className":87800,"style":218},[217],[15,87802,87804],{"className":87803},[222,223,224,225],[15,87805,1804],{"className":87806},[66,225],[15,87808,1114],{"className":87809},[1113],[15,87811,87813],{"className":87812},[206],[15,87814,87816],{"className":87815,"style":1121},[210],[15,87817],{},[15,87819,335],{"className":87820},[422],[15,87822],{"className":87823,"style":697},[71],[15,87825,87827,87830],{"className":87826},[66],[15,87828,28399],{"className":87829,"style":62513},[66,67],[15,87831,87833],{"className":87832},[198],[15,87834,87836,87856],{"className":87835},[202,1090],[15,87837,87839,87853],{"className":87838},[206],[15,87840,87842],{"className":87841,"style":2139},[210],[15,87843,87844,87847],{"style":82499},[15,87845],{"className":87846,"style":218},[217],[15,87848,87850],{"className":87849},[222,223,224,225],[15,87851,1211],{"className":87852},[66,225],[15,87854,1114],{"className":87855},[1113],[15,87857,87859],{"className":87858},[206],[15,87860,87862],{"className":87861,"style":1121},[210],[15,87863],{},[15,87865,126],{"className":87866},[169],[15,87868,335],{"className":87869},[422],[15,87871],{"className":87872,"style":4061},[71],[15,87874],{"className":87875,"style":697},[71],[15,87877,87879,87882],{"className":87878},[66],[15,87880,1061],{"className":87881,"style":7073},[66,67],[15,87883,87885],{"className":87884},[198],[15,87886,87888],{"className":87887},[202],[15,87889,87891],{"className":87890},[206],[15,87892,87894],{"className":87893,"style":1650},[210],[15,87895,87896,87899],{"style":397},[15,87897],{"className":87898,"style":218},[217],[15,87900,87902],{"className":87901},[222,223,224,225],[15,87903,1263],{"className":87904},[66,225],[15,87906,120],{"className":87907},[162],[15,87909,87911,87914],{"className":87910},[66],[15,87912,28399],{"className":87913,"style":62513},[66,67],[15,87915,87917],{"className":87916},[198],[15,87918,87920,87940],{"className":87919},[202,1090],[15,87921,87923,87937],{"className":87922},[206],[15,87924,87926],{"className":87925,"style":2139},[210],[15,87927,87928,87931],{"style":82499},[15,87929],{"className":87930,"style":218},[217],[15,87932,87934],{"className":87933},[222,223,224,225],[15,87935,1804],{"className":87936},[66,225],[15,87938,1114],{"className":87939},[1113],[15,87941,87943],{"className":87942},[206],[15,87944,87946],{"className":87945,"style":1121},[210],[15,87947],{},[15,87949,335],{"className":87950},[422],[15,87952],{"className":87953,"style":697},[71],[15,87955,87957,87960],{"className":87956},[66],[15,87958,28399],{"className":87959,"style":62513},[66,67],[15,87961,87963],{"className":87962},[198],[15,87964,87966,87986],{"className":87965},[202,1090],[15,87967,87969,87983],{"className":87968},[206],[15,87970,87972],{"className":87971,"style":2139},[210],[15,87973,87974,87977],{"style":82499},[15,87975],{"className":87976,"style":218},[217],[15,87978,87980],{"className":87979},[222,223,224,225],[15,87981,1211],{"className":87982},[66,225],[15,87984,1114],{"className":87985},[1113],[15,87987,87989],{"className":87988},[206],[15,87990,87992],{"className":87991,"style":1121},[210],[15,87993],{},[15,87995,126],{"className":87996},[169],[15,87998],{"className":87999,"style":173},[71],[15,88001,129],{"className":88002},[177],[15,88004],{"className":88005,"style":173},[71],[15,88007,88009,88012,88015,88018,88021],{"className":88008},[57],[15,88010],{"className":88011,"style":30468},[61],[15,88013,1211],{"className":88014},[66],[15,88016],{"className":88017,"style":72},[71],[15,88019,6602],{"className":88020},[76],[15,88022],{"className":88023,"style":72},[71],[15,88025,88027,88030,88036,88039,88079,88082,88085,88125,88128],{"className":88026},[57],[15,88028],{"className":88029,"style":154},[61],[15,88031,88033],{"className":88032},[66],[15,88034,86467],{"className":88035},[66,4429],[15,88037,120],{"className":88038},[162],[15,88040,88042,88045],{"className":88041},[66],[15,88043,28399],{"className":88044,"style":62513},[66,67],[15,88046,88048],{"className":88047},[198],[15,88049,88051,88071],{"className":88050},[202,1090],[15,88052,88054,88068],{"className":88053},[206],[15,88055,88057],{"className":88056,"style":2139},[210],[15,88058,88059,88062],{"style":82499},[15,88060],{"className":88061,"style":218},[217],[15,88063,88065],{"className":88064},[222,223,224,225],[15,88066,1804],{"className":88067},[66,225],[15,88069,1114],{"className":88070},[1113],[15,88072,88074],{"className":88073},[206],[15,88075,88077],{"className":88076,"style":1121},[210],[15,88078],{},[15,88080,335],{"className":88081},[422],[15,88083],{"className":88084,"style":697},[71],[15,88086,88088,88091],{"className":88087},[66],[15,88089,28399],{"className":88090,"style":62513},[66,67],[15,88092,88094],{"className":88093},[198],[15,88095,88097,88117],{"className":88096},[202,1090],[15,88098,88100,88114],{"className":88099},[206],[15,88101,88103],{"className":88102,"style":2139},[210],[15,88104,88105,88108],{"style":82499},[15,88106],{"className":88107,"style":218},[217],[15,88109,88111],{"className":88110},[222,223,224,225],[15,88112,1211],{"className":88113},[66,225],[15,88115,1114],{"className":88116},[1113],[15,88118,88120],{"className":88119},[206],[15,88121,88123],{"className":88122,"style":1121},[210],[15,88124],{},[15,88126,126],{"className":88127},[169],[15,88129,335],{"className":88130},[422],[11,88132,88133],{},"以及",[15,88135,88137],{"className":88136,"translate":19},[294],[15,88138,88140,88214],{"className":88139,"translate":19},[18],[15,88141,88143],{"className":88142},[23],[25,88144,88145],{"xmlns":27,"display":303},[29,88146,88147,88211],{},[32,88148,88149,88155,88157,88163,88165,88171,88173,88175,88181,88191,88193,88199,88201,88207,88209],{},[131,88150,88151,88153],{},[35,88152,1061],{},[1209,88154,1263],{},[39,88156,120],{"stretchy":119},[1057,88158,88159,88161],{},[35,88160,28399],{},[1209,88162,1804],{},[39,88164,335],{"separator":53},[1057,88166,88167,88169],{},[35,88168,28399],{},[1209,88170,1211],{},[39,88172,126],{"stretchy":119},[39,88174,24220],{},[6467,88176,88177,88179],{},[1209,88178,1211],{},[1209,88180,1263],{},[1057,88182,88183,88185],{},[35,88184,32458],{},[32,88186,88187,88189],{},[35,88188,2853],{"mathvariant":138},[35,88190,1216],{"mathvariant":138},[39,88192,120],{"stretchy":119},[1057,88194,88195,88197],{},[35,88196,28399],{},[1209,88198,1804],{},[35,88200,1251],{"mathvariant":138},[1057,88202,88203,88205],{},[35,88204,28399],{},[1209,88206,1211],{},[39,88208,126],{"stretchy":119},[35,88210,3573],{"mathvariant":138},[45,88212,88213],{"encoding":47},"H^2(P_0,P_1)\\le \\frac12 D_{\\mathrm{KL}}(P_0\\|P_1).",[15,88215,88217,88353],{"className":88216,"ariaHidden":53},[52],[15,88218,88220,88223,88252,88255,88295,88298,88301,88341,88344,88347,88350],{"className":88219},[57],[15,88221],{"className":88222,"style":11214},[61],[15,88224,88226,88229],{"className":88225},[66],[15,88227,1061],{"className":88228,"style":7073},[66,67],[15,88230,88232],{"className":88231},[198],[15,88233,88235],{"className":88234},[202],[15,88236,88238],{"className":88237},[206],[15,88239,88241],{"className":88240,"style":1650},[210],[15,88242,88243,88246],{"style":397},[15,88244],{"className":88245,"style":218},[217],[15,88247,88249],{"className":88248},[222,223,224,225],[15,88250,1263],{"className":88251},[66,225],[15,88253,120],{"className":88254},[162],[15,88256,88258,88261],{"className":88257},[66],[15,88259,28399],{"className":88260,"style":62513},[66,67],[15,88262,88264],{"className":88263},[198],[15,88265,88267,88287],{"className":88266},[202,1090],[15,88268,88270,88284],{"className":88269},[206],[15,88271,88273],{"className":88272,"style":2139},[210],[15,88274,88275,88278],{"style":82499},[15,88276],{"className":88277,"style":218},[217],[15,88279,88281],{"className":88280},[222,223,224,225],[15,88282,1804],{"className":88283},[66,225],[15,88285,1114],{"className":88286},[1113],[15,88288,88290],{"className":88289},[206],[15,88291,88293],{"className":88292,"style":1121},[210],[15,88294],{},[15,88296,335],{"className":88297},[422],[15,88299],{"className":88300,"style":697},[71],[15,88302,88304,88307],{"className":88303},[66],[15,88305,28399],{"className":88306,"style":62513},[66,67],[15,88308,88310],{"className":88309},[198],[15,88311,88313,88333],{"className":88312},[202,1090],[15,88314,88316,88330],{"className":88315},[206],[15,88317,88319],{"className":88318,"style":2139},[210],[15,88320,88321,88324],{"style":82499},[15,88322],{"className":88323,"style":218},[217],[15,88325,88327],{"className":88326},[222,223,224,225],[15,88328,1211],{"className":88329},[66,225],[15,88331,1114],{"className":88332},[1113],[15,88334,88336],{"className":88335},[206],[15,88337,88339],{"className":88338,"style":1121},[210],[15,88340],{},[15,88342,126],{"className":88343},[169],[15,88345],{"className":88346,"style":173},[71],[15,88348,24220],{"className":88349},[177],[15,88351],{"className":88352,"style":173},[71],[15,88354,88356,88359,88421,88469,88472,88512,88515,88555,88558],{"className":88355},[57],[15,88357],{"className":88358,"style":49086},[61],[15,88360,88362,88365,88418],{"className":88361},[66],[15,88363],{"className":88364},[162,23417],[15,88366,88368],{"className":88367},[6467],[15,88369,88371,88410],{"className":88370},[202,1090],[15,88372,88374,88407],{"className":88373},[206],[15,88375,88377,88388,88396],{"className":88376,"style":23430},[210],[15,88378,88379,88382],{"style":23433},[15,88380],{"className":88381,"style":1368},[217],[15,88383,88385],{"className":88384},[66],[15,88386,1263],{"className":88387},[66],[15,88389,88390,88393],{"style":23445},[15,88391],{"className":88392,"style":1368},[217],[15,88394],{"className":88395,"style":4440},[23452],[15,88397,88398,88401],{"style":23455},[15,88399],{"className":88400,"style":1368},[217],[15,88402,88404],{"className":88403},[66],[15,88405,1211],{"className":88406},[66],[15,88408,1114],{"className":88409},[1113],[15,88411,88413],{"className":88412},[206],[15,88414,88416],{"className":88415,"style":23474},[210],[15,88417],{},[15,88419],{"className":88420},[169,23417],[15,88422,88424,88427],{"className":88423},[66],[15,88425,32458],{"className":88426,"style":23025},[66,67],[15,88428,88430],{"className":88429},[198],[15,88431,88433,88461],{"className":88432},[202,1090],[15,88434,88436,88458],{"className":88435},[206],[15,88437,88439],{"className":88438,"style":3638},[210],[15,88440,88442,88445],{"style":88441},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[15,88443],{"className":88444,"style":218},[217],[15,88446,88448],{"className":88447},[222,223,224,225],[15,88449,88451],{"className":88450},[66,225],[15,88452,88454],{"className":88453},[66,225],[15,88455,88457],{"className":88456},[66,4429,225],"KL",[15,88459,1114],{"className":88460},[1113],[15,88462,88464],{"className":88463},[206],[15,88465,88467],{"className":88466,"style":1121},[210],[15,88468],{},[15,88470,120],{"className":88471},[162],[15,88473,88475,88478],{"className":88474},[66],[15,88476,28399],{"className":88477,"style":62513},[66,67],[15,88479,88481],{"className":88480},[198],[15,88482,88484,88504],{"className":88483},[202,1090],[15,88485,88487,88501],{"className":88486},[206],[15,88488,88490],{"className":88489,"style":2139},[210],[15,88491,88492,88495],{"style":82499},[15,88493],{"className":88494,"style":218},[217],[15,88496,88498],{"className":88497},[222,223,224,225],[15,88499,1804],{"className":88500},[66,225],[15,88502,1114],{"className":88503},[1113],[15,88505,88507],{"className":88506},[206],[15,88508,88510],{"className":88509,"style":1121},[210],[15,88511],{},[15,88513,1251],{"className":88514},[66],[15,88516,88518,88521],{"className":88517},[66],[15,88519,28399],{"className":88520,"style":62513},[66,67],[15,88522,88524],{"className":88523},[198],[15,88525,88527,88547],{"className":88526},[202,1090],[15,88528,88530,88544],{"className":88529},[206],[15,88531,88533],{"className":88532,"style":2139},[210],[15,88534,88535,88538],{"style":82499},[15,88536],{"className":88537,"style":218},[217],[15,88539,88541],{"className":88540},[222,223,224,225],[15,88542,1211],{"className":88543},[66,225],[15,88545,1114],{"className":88546},[1113],[15,88548,88550],{"className":88549},[206],[15,88551,88553],{"className":88552,"style":1121},[210],[15,88554],{},[15,88556,126],{"className":88557},[169],[15,88559,3573],{"className":88560},[66],[11,88562,88563,88564,88784],{},"把第一条与 ",[15,88565,88567,88611],{"className":88566,"translate":19},[18],[15,88568,88570],{"className":88569},[23],[25,88571,88572],{"xmlns":27},[29,88573,88574,88608],{},[32,88575,88576,88584,88586,88594,88596,88598,88600,88606],{},[1255,88577,88578,88580,88582],{},[35,88579,11],{},[35,88581,48383],{},[39,88583,1162],{},[39,88585,129],{},[62987,88587,88588],{"scriptlevel":1804,"displaystyle":119},[6467,88589,88590,88592],{},[1209,88591,1211],{},[1209,88593,1263],{},[39,88595,120],{"stretchy":119},[1209,88597,1211],{},[39,88599,6602],{},[32,88601,88602,88604],{},[35,88603,32778],{"mathvariant":138},[35,88605,31732],{"mathvariant":138},[39,88607,126],{"stretchy":119},[45,88609,88610],{"encoding":47},"p_e^*=\\tfrac12(1-\\mathrm{TV})",[15,88612,88614,88680,88769],{"className":88613,"ariaHidden":53},[52],[15,88615,88617,88620,88671,88674,88677],{"className":88616},[57],[15,88618],{"className":88619,"style":82776},[61],[15,88621,88623,88626],{"className":88622},[66],[15,88624,11],{"className":88625},[66,67],[15,88627,88629],{"className":88628},[198],[15,88630,88632,88663],{"className":88631},[202,1090],[15,88633,88635,88660],{"className":88634},[206],[15,88636,88638,88649],{"className":88637,"style":16249},[210],[15,88639,88640,88643],{"style":1653},[15,88641],{"className":88642,"style":218},[217],[15,88644,88646],{"className":88645},[222,223,224,225],[15,88647,48383],{"className":88648},[66,67,225],[15,88650,88651,88654],{"style":213},[15,88652],{"className":88653,"style":218},[217],[15,88655,88657],{"className":88656},[222,223,224,225],[15,88658,1162],{"className":88659},[76,225],[15,88661,1114],{"className":88662},[1113],[15,88664,88666],{"className":88665},[206],[15,88667,88669],{"className":88668,"style":1686},[210],[15,88670],{},[15,88672],{"className":88673,"style":173},[71],[15,88675,129],{"className":88676},[177],[15,88678],{"className":88679,"style":173},[71],[15,88681,88683,88686,88754,88757,88760,88763,88766],{"className":88682},[57],[15,88684],{"className":88685,"style":30854},[61],[15,88687,88689,88692,88751],{"className":88688},[66],[15,88690],{"className":88691},[162,23417],[15,88693,88695],{"className":88694},[6467],[15,88696,88698,88743],{"className":88697},[202,1090],[15,88699,88701,88740],{"className":88700},[206],[15,88702,88704,88718,88726],{"className":88703,"style":30873},[210],[15,88705,88706,88709],{"style":30876},[15,88707],{"className":88708,"style":1368},[217],[15,88710,88712],{"className":88711},[222,223,224,225],[15,88713,88715],{"className":88714},[66,225],[15,88716,1263],{"className":88717},[66,225],[15,88719,88720,88723],{"style":23445},[15,88721],{"className":88722,"style":1368},[217],[15,88724],{"className":88725,"style":4440},[23452],[15,88727,88728,88731],{"style":30899},[15,88729],{"className":88730,"style":1368},[217],[15,88732,88734],{"className":88733},[222,223,224,225],[15,88735,88737],{"className":88736},[66,225],[15,88738,1211],{"className":88739},[66,225],[15,88741,1114],{"className":88742},[1113],[15,88744,88746],{"className":88745},[206],[15,88747,88749],{"className":88748,"style":30921},[210],[15,88750],{},[15,88752],{"className":88753},[169,23417],[15,88755,120],{"className":88756},[162],[15,88758,1211],{"className":88759},[66],[15,88761],{"className":88762,"style":72},[71],[15,88764,6602],{"className":88765},[76],[15,88767],{"className":88768,"style":72},[71],[15,88770,88772,88775,88781],{"className":88771},[57],[15,88773],{"className":88774,"style":154},[61],[15,88776,88778],{"className":88777},[66],[15,88779,82610],{"className":88780,"style":1343},[66,4429],[15,88782,126],{"className":88783},[169]," 结合，得到",[15,88786,88788],{"className":88787,"translate":19},[294],[15,88789,88791,88871],{"className":88790,"translate":19},[18],[15,88792,88794],{"className":88793},[23],[25,88795,88796],{"xmlns":27,"display":303},[29,88797,88798,88868],{},[32,88799,88800,88808,88810,88812,88814,88820,88866],{},[1255,88801,88802,88804,88806],{},[35,88803,11],{},[35,88805,48383],{},[39,88807,1162],{},[1274,88809,24233],{},[39,88811,1801],{},[1274,88813,24233],{},[6467,88815,88816,88818],{},[1209,88817,1211],{},[1209,88819,1263],{},[32,88821,88822,88824,88826,88828,88864],{},[39,88823,120],{"fence":53},[1209,88825,1211],{},[39,88827,6602],{},[48508,88829,88830],{},[6467,88831,88832,88862],{},[32,88833,88834,88844,88846,88852,88854,88860],{},[1057,88835,88836,88838],{},[35,88837,32458],{},[32,88839,88840,88842],{},[35,88841,2853],{"mathvariant":138},[35,88843,1216],{"mathvariant":138},[39,88845,120],{"stretchy":119},[1057,88847,88848,88850],{},[35,88849,28399],{},[1209,88851,1804],{},[35,88853,1251],{"mathvariant":138},[1057,88855,88856,88858],{},[35,88857,28399],{},[1209,88859,1211],{},[39,88861,126],{"stretchy":119},[1209,88863,1263],{},[39,88865,126],{"fence":53},[35,88867,3573],{"mathvariant":138},[45,88869,88870],{"encoding":47},"p_e^*\\ \\ge\\ \\frac12\\left(1-\\sqrt{\\frac{D_{\\mathrm{KL}}(P_0\\|P_1)}{2}}\\right).",[15,88872,88874,88946],{"className":88873,"ariaHidden":53},[52],[15,88875,88877,88880,88931,88934,88937,88940,88943],{"className":88876},[57],[15,88878],{"className":88879,"style":82023},[61],[15,88881,88883,88886],{"className":88882},[66],[15,88884,11],{"className":88885},[66,67],[15,88887,88889],{"className":88888},[198],[15,88890,88892,88923],{"className":88891},[202,1090],[15,88893,88895,88920],{"className":88894},[206],[15,88896,88898,88909],{"className":88897,"style":1308},[210],[15,88899,88900,88903],{"style":1653},[15,88901],{"className":88902,"style":218},[217],[15,88904,88906],{"className":88905},[222,223,224,225],[15,88907,48383],{"className":88908},[66,67,225],[15,88910,88911,88914],{"style":397},[15,88912],{"className":88913,"style":218},[217],[15,88915,88917],{"className":88916},[222,223,224,225],[15,88918,1162],{"className":88919},[76,225],[15,88921,1114],{"className":88922},[1113],[15,88924,88926],{"className":88925},[206],[15,88927,88929],{"className":88928,"style":1686},[210],[15,88930],{},[15,88932,24233],{"className":88933},[71],[15,88935],{"className":88936,"style":173},[71],[15,88938,1801],{"className":88939},[177],[15,88941,24233],{"className":88942},[71],[15,88944],{"className":88945,"style":173},[71],[15,88947,88949,88953,89015,89018,89293,89296],{"className":88948},[57],[15,88950],{"className":88951,"style":88952},[61],"height:3em;vertical-align:-1.25em;",[15,88954,88956,88959,89012],{"className":88955},[66],[15,88957],{"className":88958},[162,23417],[15,88960,88962],{"className":88961},[6467],[15,88963,88965,89004],{"className":88964},[202,1090],[15,88966,88968,89001],{"className":88967},[206],[15,88969,88971,88982,88990],{"className":88970,"style":23430},[210],[15,88972,88973,88976],{"style":23433},[15,88974],{"className":88975,"style":1368},[217],[15,88977,88979],{"className":88978},[66],[15,88980,1263],{"className":88981},[66],[15,88983,88984,88987],{"style":23445},[15,88985],{"className":88986,"style":1368},[217],[15,88988],{"className":88989,"style":4440},[23452],[15,88991,88992,88995],{"style":23455},[15,88993],{"className":88994,"style":1368},[217],[15,88996,88998],{"className":88997},[66],[15,88999,1211],{"className":89000},[66],[15,89002,1114],{"className":89003},[1113],[15,89005,89007],{"className":89006},[206],[15,89008,89010],{"className":89009,"style":23474},[210],[15,89011],{},[15,89013],{"className":89014},[169,23417],[15,89016],{"className":89017,"style":697},[71],[15,89019,89021,89027,89030,89033,89036,89039,89287],{"className":89020},[1413],[15,89022,89024],{"className":89023,"style":1418},[162,1417],[15,89025,120],{"className":89026},[1422,1423],[15,89028,1211],{"className":89029},[66],[15,89031],{"className":89032,"style":72},[71],[15,89034,6602],{"className":89035},[76],[15,89037],{"className":89038,"style":72},[71],[15,89040,89042],{"className":89041},[66,48550],[15,89043,89045,89278],{"className":89044},[202,1090],[15,89046,89048,89275],{"className":89047},[206],[15,89049,89052,89258],{"className":89050,"style":89051},[210],"height:1.7044em;",[15,89053,89056,89060],{"className":89054,"style":89055},[48564],"top:-4.4em;",[15,89057],{"className":89058,"style":89059},[217],"height:4.4em;",[15,89061,89064],{"className":89062,"style":89063},[66],"padding-left:1em;",[15,89065,89067,89070,89255],{"className":89066},[66],[15,89068],{"className":89069},[162,23417],[15,89071,89073],{"className":89072},[6467],[15,89074,89076,89247],{"className":89075},[202,1090],[15,89077,89079,89244],{"className":89078},[206],[15,89080,89082,89093,89101],{"className":89081,"style":41211},[210],[15,89083,89084,89087],{"style":23433},[15,89085],{"className":89086,"style":1368},[217],[15,89088,89090],{"className":89089},[66],[15,89091,1263],{"className":89092},[66],[15,89094,89095,89098],{"style":23445},[15,89096],{"className":89097,"style":1368},[217],[15,89099],{"className":89100,"style":4440},[23452],[15,89102,89103,89106],{"style":23455},[15,89104],{"className":89105,"style":1368},[217],[15,89107,89109,89155,89158,89198,89201,89241],{"className":89108},[66],[15,89110,89112,89115],{"className":89111},[66],[15,89113,32458],{"className":89114,"style":23025},[66,67],[15,89116,89118],{"className":89117},[198],[15,89119,89121,89147],{"className":89120},[202,1090],[15,89122,89124,89144],{"className":89123},[206],[15,89125,89127],{"className":89126,"style":3638},[210],[15,89128,89129,89132],{"style":88441},[15,89130],{"className":89131,"style":218},[217],[15,89133,89135],{"className":89134},[222,223,224,225],[15,89136,89138],{"className":89137},[66,225],[15,89139,89141],{"className":89140},[66,225],[15,89142,88457],{"className":89143},[66,4429,225],[15,89145,1114],{"className":89146},[1113],[15,89148,89150],{"className":89149},[206],[15,89151,89153],{"className":89152,"style":1121},[210],[15,89154],{},[15,89156,120],{"className":89157},[162],[15,89159,89161,89164],{"className":89160},[66],[15,89162,28399],{"className":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0 400000 2592",[34834,89273],{"d":89274},"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z",[15,89276,1114],{"className":89277},[1113],[15,89279,89281],{"className":89280},[206],[15,89282,89285],{"className":89283,"style":89284},[210],"height:0.7356em;",[15,89286],{},[15,89288,89290],{"className":89289,"style":1418},[169,1417],[15,89291,126],{"className":89292},[1422,1423],[15,89294],{"className":89295,"style":697},[71],[15,89297,3573],{"className":89298},[66],[11,89300,89301,89302,89305],{},"这是 Le Cam 两点法的核心不等式：如果两个分布很难区分（KL 很小），那么任何分类器的错误率都不可能低。注意方向——",[2040,89303,89304],{},"KL 给的是错误率的下界，不是上界","。KL 很大并不意味着问题容易，只意味着这个下界变得松甚至变得平凡。",[11,89307,89308],{},"上界则需要另一个量：",[15,89310,89312],{"className":89311,"translate":19},[294],[15,89313,89315,89371],{"className":89314,"translate":19},[18],[15,89316,89318],{"className":89317},[23],[25,89319,89320],{"xmlns":27,"display":303},[29,89321,89322,89368],{},[32,89323,89324,89332,89334,89340,89342,89348,89350,89356,89358,89364,89366],{},[1255,89325,89326,89328,89330],{},[35,89327,11],{},[35,89329,48383],{},[39,89331,1162],{},[39,89333,24220],{},[6467,89335,89336,89338],{},[1209,89337,1211],{},[1209,89339,1263],{},[1274,89341,2462],{},[32,89343,89344,89346],{},[35,89345,63972],{"mathvariant":138},[35,89347,23256],{"mathvariant":138},[39,89349,120],{"stretchy":119},[1057,89351,89352,89354],{},[35,89353,28399],{},[1209,89355,1804],{},[39,89357,335],{"separator":53},[1057,89359,89360,89362],{},[35,89361,28399],{},[1209,89363,1211],{},[39,89365,126],{"stretchy":119},[35,89367,3573],{"mathvariant":138},[45,89369,89370],{"encoding":47},"p_e^*\\le \\frac12\\,\\mathrm{BC}(P_0,P_1).",[15,89372,89374,89440],{"className":89373,"ariaHidden":53},[52],[15,89375,89377,89380,89431,89434,89437],{"className":89376},[57],[15,89378],{"className":89379,"style":82023},[61],[15,89381,89383,89386],{"className":89382},[66],[15,89384,11],{"className":89385},[66,67],[15,89387,89389],{"className":89388},[198],[15,89390,89392,89423],{"className":89391},[202,1090],[15,89393,89395,89420],{"className":89394},[206],[15,89396,89398,89409],{"className":89397,"style":1308},[210],[15,89399,89400,89403],{"style":1653},[15,89401],{"className":89402,"style":218},[217],[15,89404,89406],{"className":89405},[222,223,224,225],[15,89407,48383],{"className":89408},[66,67,225],[15,89410,89411,89414],{"style":397},[15,89412],{"className":89413,"style":218},[217],[15,89415,89417],{"className":89416},[222,223,224,225],[15,89418,1162],{"className":89419},[76,225],[15,89421,1114],{"className":89422},[1113],[15,89424,89426],{"className":89425},[206],[15,89427,89429],{"className":89428,"style":1686},[210],[15,89430],{},[15,89432],{"className":89433,"style":173},[71],[15,89435,24220],{"className":89436},[177],[15,89438],{"className":89439,"style":173},[71],[15,89441,89443,89446,89508,89511,89517,89520,89560,89563,89566,89606,89609],{"className":89442},[57],[15,89444],{"className":89445,"style":49086},[61],[15,89447,89449,89452,89505],{"className":89448},[66],[15,89450],{"className":89451},[162,23417],[15,89453,89455],{"className":89454},[6467],[15,89456,89458,89497],{"className":89457},[202,1090],[15,89459,89461,89494],{"className":89460},[206],[15,89462,89464,89475,89483],{"className":89463,"style":23430},[210],[15,89465,89466,89469],{"style":23433},[15,89467],{"className":89468,"style":1368},[217],[15,89470,89472],{"className":89471},[66],[15,89473,1263],{"className":89474},[66],[15,89476,89477,89480],{"style":23445},[15,89478],{"className":89479,"style":1368},[217],[15,89481],{"className":89482,"style":4440},[23452],[15,89484,89485,89488],{"style":23455},[15,89486],{"className":89487,"style":1368},[217],[15,89489,89491],{"className":89490},[66],[15,89492,1211],{"className":89493},[66],[15,89495,1114],{"className":89496},[1113],[15,89498,89500],{"className":89499},[206],[15,89501,89503],{"className":89502,"style":23474},[210],[15,89504],{},[15,89506],{"className":89507},[169,23417],[15,89509],{"className":89510,"style":697},[71],[15,89512,89514],{"className":89513},[66],[15,89515,86467],{"className":89516},[66,4429],[15,89518,120],{"className":89519},[162],[15,89521,89523,89526],{"className":89522},[66],[15,89524,28399],{"className":89525,"style":62513},[66,67],[15,89527,89529],{"className":89528},[198],[15,89530,89532,89552],{"className":89531},[202,1090],[15,89533,89535,89549],{"className":89534},[206],[15,89536,89538],{"className":89537,"style":2139},[210],[15,89539,89540,89543],{"style":82499},[15,89541],{"className":89542,"style":218},[217],[15,89544,89546],{"className":89545},[222,223,224,225],[15,89547,1804],{"className":89548},[66,225],[15,89550,1114],{"className":89551},[1113],[15,89553,89555],{"className":89554},[206],[15,89556,89558],{"className":89557,"style":1121},[210],[15,89559],{},[15,89561,335],{"className":89562},[422],[15,89564],{"className":89565,"style":697},[71],[15,89567,89569,89572],{"className":89568},[66],[15,89570,28399],{"className":89571,"style":62513},[66,67],[15,89573,89575],{"className":89574},[198],[15,89576,89578,89598],{"className":89577},[202,1090],[15,89579,89581,89595],{"className":89580},[206],[15,89582,89584],{"className":89583,"style":2139},[210],[15,89585,89586,89589],{"style":82499},[15,89587],{"className":89588,"style":218},[217],[15,89590,89592],{"className":89591},[222,223,224,225],[15,89593,1211],{"className":89594},[66,225],[15,89596,1114],{"className":89597},[1113],[15,89599,89601],{"className":89600},[206],[15,89602,89604],{"className":89603,"style":1121},[210],[15,89605],{},[15,89607,126],{"className":89608},[169],[15,89610,3573],{"className":89611},[66],[11,89613,89614,89615,89929],{},"这是因为 ",[15,89616,89618,89674],{"className":89617,"translate":19},[18],[15,89619,89621],{"className":89620},[23],[25,89622,89623],{"xmlns":27},[29,89624,89625,89671],{},[32,89626,89627,89629,89631,89633,89635,89641,89643,89649,89651,89653,89655],{},[39,89628,29732],{},[35,89630,1178],{},[39,89632,1170],{},[39,89634,4307],{"stretchy":119},[1057,89636,89637,89639],{},[35,89638,11],{},[1209,89640,1804],{},[39,89642,335],{"separator":53},[1057,89644,89645,89647],{},[35,89646,11],{},[1209,89648,1211],{},[39,89650,4330],{"stretchy":119},[39,89652,24220],{},[39,89654,29732],{},[48508,89656,89657],{},[32,89658,89659,89665],{},[1057,89660,89661,89663],{},[35,89662,11],{},[1209,89664,1804],{},[1057,89666,89667,89669],{},[35,89668,11],{},[1209,89670,1211],{},[45,89672,89673],{"encoding":47},"\\int\\min\\{p_0,p_1\\}\\le\\int\\sqrt{p_0p_1}",[15,89675,89677,89793],{"className":89676,"ariaHidden":53},[52],[15,89678,89680,89683,89686,89689,89692,89695,89735,89738,89741,89781,89784,89787,89790],{"className":89679},[57],[15,89681],{"className":89682,"style":34257},[61],[15,89684,29732],{"className":89685,"style":31343},[1339,1470,13701],[15,89687],{"className":89688,"style":697},[71],[15,89690,1178],{"className":89691},[1339],[15,89693,4307],{"className":89694},[162],[15,89696,89698,89701],{"className":89697},[66],[15,89699,11],{"className":89700},[66,67],[15,89702,89704],{"className":89703},[198],[15,89705,89707,89727],{"className":89706},[202,1090],[15,89708,89710,89724],{"className":89709},[206],[15,89711,89713],{"className":89712,"style":2139},[210],[15,89714,89715,89718],{"style":1584},[15,89716],{"className":89717,"style":218},[217],[15,89719,89721],{"className":89720},[222,223,224,225],[15,89722,1804],{"className":89723},[66,225],[15,89725,1114],{"className":89726},[1113],[15,89728,89730],{"className":89729},[206],[15,89731,89733],{"className":89732,"style":1121},[210],[15,89734],{},[15,89736,335],{"className":89737},[422],[15,89739],{"className":89740,"style":697},[71],[15,89742,89744,89747],{"className":89743},[66],[15,89745,11],{"className":89746},[66,67],[15,89748,89750],{"className":89749},[198],[15,89751,89753,89773],{"className":89752},[202,1090],[15,89754,89756,89770],{"className":89755},[206],[15,89757,89759],{"className":89758,"style":2139},[210],[15,89760,89761,89764],{"style":1584},[15,89762],{"className":89763,"style":218},[217],[15,89765,89767],{"className":89766},[222,223,224,225],[15,89768,1211],{"className":89769},[66,225],[15,89771,1114],{"className":89772},[1113],[15,89774,89776],{"className":89775},[206],[15,89777,89779],{"className":89778,"style":1121},[210],[15,89780],{},[15,89782,4330],{"className":89783},[169],[15,89785],{"className":89786,"style":173},[71],[15,89788,24220],{"className":89789},[177],[15,89791],{"className":89792,"style":173},[71],[15,89794,89796,89799,89802,89805],{"className":89795},[57],[15,89797],{"className":89798,"style":86941},[61],[15,89800,29732],{"className":89801,"style":31343},[1339,1470,13701],[15,89803],{"className":89804,"style":697},[71],[15,89806,89808],{"className":89807},[66,48550],[15,89809,89811,89921],{"className":89810},[202,1090],[15,89812,89814,89918],{"className":89813},[206],[15,89815,89817,89906],{"className":89816,"style":86960},[210],[15,89818,89820,89823],{"className":89819,"style":1386},[48564],[15,89821],{"className":89822,"style":1368},[217],[15,89824,89826,89866],{"className":89825,"style":48571},[66],[15,89827,89829,89832],{"className":89828},[66],[15,89830,11],{"className":89831},[66,67],[15,89833,89835],{"className":89834},[198],[15,89836,89838,89858],{"className":89837},[202,1090],[15,89839,89841,89855],{"className":89840},[206],[15,89842,89844],{"className":89843,"style":2139},[210],[15,89845,89846,89849],{"style":1584},[15,89847],{"className":89848,"style":218},[217],[15,89850,89852],{"className":89851},[222,223,224,225],[15,89853,1804],{"className":89854},[66,225],[15,89856,1114],{"className":89857},[1113],[15,89859,89861],{"className":89860},[206],[15,89862,89864],{"className":89863,"style":1121},[210],[15,89865],{},[15,89867,89869,89872],{"className":89868},[66],[15,89870,11],{"className":89871},[66,67],[15,89873,89875],{"className":89874},[198],[15,89876,89878,89898],{"className":89877},[202,1090],[15,89879,89881,89895],{"className":89880},[206],[15,89882,89884],{"className":89883,"style":2139},[210],[15,89885,89886,89889],{"style":1584},[15,89887],{"className":89888,"style":218},[217],[15,89890,89892],{"className":89891},[222,223,224,225],[15,89893,1211],{"className":89894},[66,225],[15,89896,1114],{"className":89897},[1113],[15,89899,89901],{"className":89900},[206],[15,89902,89904],{"className":89903,"style":1121},[210],[15,89905],{},[15,89907,89908,89911],{"style":87052},[15,89909],{"className":89910,"style":1368},[217],[15,89912,89914],{"className":89913,"style":48591},[48590],[34828,89915,89916],{"xmlns":34830,"width":48594,"height":48595,"viewBox":48596,"preserveAspectRatio":48597},[34834,89917],{"d":48600},[15,89919,1114],{"className":89920},[1113],[15,89922,89924],{"className":89923},[206],[15,89925,89927],{"className":89926,"style":87072},[210],[15,89928],{},"（算术-几何平均不等式）。Bhattacharyya 系数可以直接计算的情形（例如高斯），这个上界就能给出正确的指数衰减率。",[11,89931,89932,89933,89936,89937,89996,89997,90001],{},"把两个方向放在一起，可以看到散度在决策问题里的角色：它们不是「错误率」，而是错误率的",[2040,89934,89935],{},"证书","。全变差是唯一精确的那个，但它通常最难估计；Bhattacharyya 系数给出上界，KL 给出下界，Hellinger 则通过 ",[15,89938,89940,89962],{"className":89939,"translate":19},[18],[15,89941,89943],{"className":89942},[23],[25,89944,89945],{"xmlns":27},[29,89946,89947,89959],{},[32,89948,89949,89955,89957],{},[32,89950,89951,89953],{},[35,89952,32778],{"mathvariant":138},[35,89954,31732],{"mathvariant":138},[39,89956,24220],{},[35,89958,1061],{},[45,89960,89961],{"encoding":47},"\\mathrm{TV}\\le H",[15,89963,89965,89987],{"className":89964,"ariaHidden":53},[52],[15,89966,89968,89972,89978,89981,89984],{"className":89967},[57],[15,89969],{"className":89970,"style":89971},[61],"height:0.8193em;vertical-align:-0.136em;",[15,89973,89975],{"className":89974},[66],[15,89976,82610],{"className":89977,"style":1343},[66,4429],[15,89979],{"className":89980,"style":173},[71],[15,89982,24220],{"className":89983},[177],[15,89985],{"className":89986,"style":173},[71],[15,89988,89990,89993],{"className":89989},[57],[15,89991],{"className":89992,"style":490},[61],[15,89994,1061],{"className":89995,"style":7073},[66,67]," 把两边连起来。",[4556,89998,90000],{"href":89999},"\u002Fblog\u002F2026\u002F0011-about-the-f-divergence","关于 f-散度的那篇文章","，正是把这些证书放在同一个凸函数框架下；而贝叶斯错误率则告诉我们，这些证书最终约束的是什么。",[1904,90003,90005],{"id":90004},"_44-错误率住在哪里","4.4 错误率住在哪里",[11,90007,90008,90009,90153,90154,90228],{},"由 3.6 节的恒等式，只有满足 ",[15,90010,90012,90046],{"className":90011,"translate":19},[18],[15,90013,90015],{"className":90014},[23],[25,90016,90017],{"xmlns":27},[29,90018,90019,90043],{},[32,90020,90021,90023,90025,90027,90029,90031,90037,90039,90041],{},[35,90022,3885],{},[39,90024,120],{"stretchy":119},[35,90026,446],{},[39,90028,126],{"stretchy":119},[39,90030,13019],{"mathvariant":138},[131,90032,90033,90035],{},[35,90034,3885],{},[39,90036,1162],{},[39,90038,120],{"stretchy":119},[35,90040,446],{},[39,90042,126],{"stretchy":119},[45,90044,90045],{"encoding":47},"\\delta(X)\\ne\\delta^*(X)",[15,90047,90049,90109],{"className":90048,"ariaHidden":53},[52],[15,90050,90052,90055,90058,90061,90064,90067,90070,90106],{"className":90051},[57],[15,90053],{"className":90054,"style":154},[61],[15,90056,3885],{"className":90057,"style":3946},[66,67],[15,90059,120],{"className":90060},[162],[15,90062,446],{"className":90063,"style":16497},[66,67],[15,90065,126],{"className":90066},[169],[15,90068],{"className":90069,"style":173},[71],[15,90071,90073,90100,90103],{"className":90072},[177],[15,90074,90076],{"className":90075},[177],[15,90077,90079],{"className":90078},[66,13086],[15,90080,90082],{"className":90081},[13090],[15,90083,90085,90088,90097],{"className":90084},[13094],[15,90086],{"className":90087,"style":3408},[61],[15,90089,90091],{"className":90090},[13101],[15,90092,90094],{"className":90093},[66],[15,90095,13108],{"className":90096},[177],[15,90098],{"className":90099},[13112],[15,90101],{"className":90102},[71,13116],[15,90104,129],{"className":90105},[177],[15,90107],{"className":90108,"style":173},[71],[15,90110,90112,90115,90144,90147,90150],{"className":90111},[57],[15,90113],{"className":90114,"style":154},[61],[15,90116,90118,90121],{"className":90117},[66],[15,90119,3885],{"className":90120,"style":3946},[66,67],[15,90122,90124],{"className":90123},[198],[15,90125,90127],{"className":90126},[202],[15,90128,90130],{"className":90129},[206],[15,90131,90133],{"className":90132,"style":16249},[210],[15,90134,90135,90138],{"style":213},[15,90136],{"className":90137,"style":218},[217],[15,90139,90141],{"className":90140},[222,223,224,225],[15,90142,1162],{"className":90143},[76,225],[15,90145,120],{"className":90146},[162],[15,90148,446],{"className":90149,"style":16497},[66,67],[15,90151,126],{"className":90152},[169]," 的样本才贡献错误，且权重是 ",[15,90155,90157,90186],{"className":90156,"translate":19},[18],[15,90158,90160],{"className":90159},[23],[25,90161,90162],{"xmlns":27},[29,90163,90164,90184],{},[32,90165,90166,90168,90170,90172,90174,90176,90178,90180,90182],{},[35,90167,34430],{"mathvariant":138},[1209,90169,1263],{},[35,90171,71113],{},[39,90173,120],{"stretchy":119},[35,90175,446],{},[39,90177,126],{"stretchy":119},[39,90179,6602],{},[1209,90181,1211],{},[35,90183,34430],{"mathvariant":138},[45,90185,77786],{"encoding":47},[15,90187,90189,90219],{"className":90188,"ariaHidden":53},[52],[15,90190,90192,90195,90198,90201,90204,90207,90210,90213,90216],{"className":90191},[57],[15,90193],{"className":90194,"style":154},[61],[15,90196,76991],{"className":90197},[66],[15,90199,71113],{"className":90200,"style":1515},[66,67],[15,90202,120],{"className":90203},[162],[15,90205,446],{"className":90206,"style":16497},[66,67],[15,90208,126],{"className":90209},[169],[15,90211],{"className":90212,"style":72},[71],[15,90214,6602],{"className":90215},[76],[15,90217],{"className":90218,"style":72},[71],[15,90220,90222,90225],{"className":90221},[57],[15,90223],{"className":90224,"style":154},[61],[15,90226,77022],{"className":90227},[66],"。这意味着贝叶斯错误率不是由决策边界的「长度」或「复杂度」决定的，而是由边界附近的后验质量决定的。",[11,90230,90231,90232,90261,90262,77891,90290,90318,90319,90375],{},"高维空间里，决策边界可以非常复杂，但如果后验在边界两侧都很尖锐（",[15,90233,90235,90249],{"className":90234,"translate":19},[18],[15,90236,90238],{"className":90237},[23],[25,90239,90240],{"xmlns":27},[29,90241,90242,90246],{},[32,90243,90244],{},[35,90245,71113],{},[45,90247,90248],{"encoding":47},"\\eta",[15,90250,90252],{"className":90251,"ariaHidden":53},[52],[15,90253,90255,90258],{"className":90254},[57],[15,90256],{"className":90257,"style":471},[61],[15,90259,71113],{"className":90260,"style":1515},[66,67]," 接近 ",[15,90263,90265,90278],{"className":90264,"translate":19},[18],[15,90266,90268],{"className":90267},[23],[25,90269,90270],{"xmlns":27},[29,90271,90272,90276],{},[32,90273,90274],{},[1209,90275,1804],{},[45,90277,1804],{"encoding":47},[15,90279,90281],{"className":90280,"ariaHidden":53},[52],[15,90282,90284,90287],{"className":90283},[57],[15,90285],{"className":90286,"style":1836},[61],[15,90288,1804],{"className":90289},[66],[15,90291,90293,90306],{"className":90292,"translate":19},[18],[15,90294,90296],{"className":90295},[23],[25,90297,90298],{"xmlns":27},[29,90299,90300,90304],{},[32,90301,90302],{},[1209,90303,1211],{},[45,90305,1211],{"encoding":47},[15,90307,90309],{"className":90308,"ariaHidden":53},[52],[15,90310,90312,90315],{"className":90311},[57],[15,90313],{"className":90314,"style":1836},[61],[15,90316,1211],{"className":90317},[66],"），错误率仍然可以很低。反过来，一个简单的线性边界，如果穿过一大片 ",[15,90320,90322,90344],{"className":90321,"translate":19},[18],[15,90323,90325],{"className":90324},[23],[25,90326,90327],{"xmlns":27},[29,90328,90329,90341],{},[32,90330,90331,90333,90335,90337,90339],{},[35,90332,71113],{},[39,90334,31993],{},[1209,90336,1211],{},[35,90338,6625],{"mathvariant":138},[1209,90340,1263],{},[45,90342,90343],{"encoding":47},"\\eta\\approx 1\u002F2",[15,90345,90347,90366],{"className":90346,"ariaHidden":53},[52],[15,90348,90350,90354,90357,90360,90363],{"className":90349},[57],[15,90351],{"className":90352,"style":90353},[61],"height:0.6776em;vertical-align:-0.1944em;",[15,90355,71113],{"className":90356,"style":1515},[66,67],[15,90358],{"className":90359,"style":173},[71],[15,90361,31993],{"className":90362},[177],[15,90364],{"className":90365,"style":173},[71],[15,90367,90369,90372],{"className":90368},[57],[15,90370],{"className":90371,"style":154},[61],[15,90373,71809],{"className":90374},[66]," 的区域，错误率就会很高。这解释了为什么「边界复杂度」本身不是泛化误差的可靠指标：真正决定错误率的是后验在边界附近的分布，而不是边界的几何形状。",[1904,90377,90379],{"id":90378},"_45-回归的贝叶斯风险","4.5 回归的贝叶斯风险",[11,90381,90382],{},"平方损失下，贝叶斯风险为",[15,90384,90386],{"className":90385,"translate":19},[294],[15,90387,90389,90467],{"className":90388,"translate":19},[18],[15,90390,90392],{"className":90391},[23],[25,90393,90394],{"xmlns":27,"display":303},[29,90395,90396,90464],{},[32,90397,90398,90404,90406,90408,90410,90412,90414,90416,90422,90424,90426,90428,90434,90436,90438,90440,90442,90450,90452,90454,90456,90458,90460,90462],{},[131,90399,90400,90402],{},[35,90401,455],{},[39,90403,1162],{},[39,90405,129],{},[35,90407,62381],{"mathvariant":454},[39,90409,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,90411,120],{"stretchy":119},[35,90413,61885],{},[39,90415,6602],{},[131,90417,90418,90420],{},[35,90419,116],{},[39,90421,1162],{},[39,90423,120],{"stretchy":119},[35,90425,446],{},[39,90427,126],{"stretchy":119},[131,90429,90430,90432],{},[39,90431,126],{"stretchy":119},[1209,90433,1263],{},[39,90435,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,90437,129],{},[35,90439,62381],{"mathvariant":454},[39,90441,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[32,90443,90444,90446,90448],{},[35,90445,31732],{"mathvariant":138},[35,90447,4556],{"mathvariant":138},[35,90449,63107],{"mathvariant":138},[39,90451,120],{"stretchy":119},[35,90453,61885],{},[39,90455,34430],{},[35,90457,446],{},[39,90459,126],{"stretchy":119},[39,90461,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,90463,3573],{"mathvariant":138},[45,90465,90466],{"encoding":47},"R^*= \\mathbb E\\big[(Y-f^*(X))^2\\big]=\\mathbb E\\big[\\mathrm{Var}(Y\\mid X)\\big].",[15,90468,90470,90514,90544,90632,90668],{"className":90469,"ariaHidden":53},[52],[15,90471,90473,90476,90505,90508,90511],{"className":90472},[57],[15,90474],{"className":90475,"style":1308},[61],[15,90477,90479,90482],{"className":90478},[66],[15,90480,455],{"className":90481,"style":62437},[66,67],[15,90483,90485],{"className":90484},[198],[15,90486,90488],{"className":90487},[202],[15,90489,90491],{"className":90490},[206],[15,90492,90494],{"className":90493,"style":1308},[210],[15,90495,90496,90499],{"style":397},[15,90497],{"className":90498,"style":218},[217],[15,90500,90502],{"className":90501},[222,223,224,225],[15,90503,1162],{"className":90504},[76,225],[15,90506],{"className":90507,"style":173},[71],[15,90509,129],{"className":90510},[177],[15,90512],{"className":90513,"style":173},[71],[15,90515,90517,90520,90523,90529,90532,90535,90538,90541],{"className":90516},[57],[15,90518],{"className":90519,"style":4400},[61],[15,90521,62381],{"className":90522},[66,517],[15,90524,90526],{"className":90525},[66],[15,90527,1194],{"className":90528},[1422,1508],[15,90530,120],{"className":90531},[162],[15,90533,61885],{"className":90534,"style":72},[66,67],[15,90536],{"className":90537,"style":72},[71],[15,90539,6602],{"className":90540},[76],[15,90542],{"className":90543,"style":72},[71],[15,90545,90547,90550,90579,90582,90585,90588,90617,90623,90626,90629],{"className":90546},[57],[15,90548],{"className":90549,"style":11718},[61],[15,90551,90553,90556],{"className":90552},[66],[15,90554,116],{"className":90555,"style":158},[66,67],[15,90557,90559],{"className":90558},[198],[15,90560,90562],{"className":90561},[202],[15,90563,90565],{"className":90564},[206],[15,90566,90568],{"className":90567,"style":1308},[210],[15,90569,90570,90573],{"style":397},[15,90571],{"className":90572,"style":218},[217],[15,90574,90576],{"className":90575},[222,223,224,225],[15,90577,1162],{"className":90578},[76,225],[15,90580,120],{"className":90581},[162],[15,90583,446],{"className":90584,"style":16497},[66,67],[15,90586,126],{"className":90587},[169],[15,90589,90591,90594],{"className":90590},[169],[15,90592,126],{"className":90593},[169],[15,90595,90597],{"className":90596},[198],[15,90598,90600],{"className":90599},[202],[15,90601,90603],{"className":90602},[206],[15,90604,90606],{"className":90605,"style":1650},[210],[15,90607,90608,90611],{"style":397},[15,90609],{"className":90610,"style":218},[217],[15,90612,90614],{"className":90613},[222,223,224,225],[15,90615,1263],{"className":90616},[66,225],[15,90618,90620],{"className":90619},[66],[15,90621,1266],{"className":90622},[1422,1508],[15,90624],{"className":90625,"style":173},[71],[15,90627,129],{"className":90628},[177],[15,90630],{"className":90631,"style":173},[71],[15,90633,90635,90638,90641,90647,90653,90656,90659,90662,90665],{"className":90634},[57],[15,90636],{"className":90637,"style":4400},[61],[15,90639,62381],{"className":90640},[66,517],[15,90642,90644],{"className":90643},[66],[15,90645,1194],{"className":90646},[1422,1508],[15,90648,90650],{"className":90649},[66],[15,90651,72407],{"className":90652},[66,4429],[15,90654,120],{"className":90655},[162],[15,90657,61885],{"className":90658,"style":72},[66,67],[15,90660],{"className":90661,"style":173},[71],[15,90663,34430],{"className":90664},[177],[15,90666],{"className":90667,"style":173},[71],[15,90669,90671,90674,90677,90680,90686],{"className":90670},[57],[15,90672],{"className":90673,"style":4400},[61],[15,90675,446],{"className":90676,"style":16497},[66,67],[15,90678,126],{"className":90679},[169],[15,90681,90683],{"className":90682},[66],[15,90684,1266],{"className":90685},[1422,1508],[15,90687,3573],{"className":90688},[66],[11,90690,90691,90692,90720,90721,90834],{},"这是数据生成过程本身的噪声，任何模型都消不掉。对一个在训练集 ",[15,90693,90695,90708],{"className":90694,"translate":19},[18],[15,90696,90698],{"className":90697},[23],[25,90699,90700],{"xmlns":27},[29,90701,90702,90706],{},[32,90703,90704],{},[35,90705,32458],{},[45,90707,32458],{"encoding":47},[15,90709,90711],{"className":90710,"ariaHidden":53},[52],[15,90712,90714,90717],{"className":90713},[57],[15,90715],{"className":90716,"style":490},[61],[15,90718,32458],{"className":90719,"style":23025},[66,67]," 上学到的回归器 ",[15,90722,90724,90746],{"className":90723,"translate":19},[18],[15,90725,90727],{"className":90726},[23],[25,90728,90729],{"xmlns":27},[29,90730,90731,90743],{},[32,90732,90733],{},[1057,90734,90735,90741],{},[4296,90736,90737,90739],{"accent":53},[35,90738,116],{},[39,90740,78130],{},[35,90742,32458],{},[45,90744,90745],{"encoding":47},"\\hat f_D",[15,90747,90749],{"className":90748,"ariaHidden":53},[52],[15,90750,90752,90755],{"className":90751},[57],[15,90753],{"className":90754,"style":78143},[61],[15,90756,90758,90800],{"className":90757},[66],[15,90759,90761],{"className":90760},[66,78147],[15,90762,90764,90792],{"className":90763},[202,1090],[15,90765,90767,90789],{"className":90766},[206],[15,90768,90770,90778],{"className":90769,"style":78157},[210],[15,90771,90772,90775],{"style":1386},[15,90773],{"className":90774,"style":1368},[217],[15,90776,116],{"className":90777,"style":158},[66,67],[15,90779,90780,90783],{"style":78168},[15,90781],{"className":90782,"style":1368},[217],[15,90784,90786],{"className":90785,"style":78176},[78175],[15,90787,78130],{"className":90788},[66],[15,90790,1114],{"className":90791},[1113],[15,90793,90795],{"className":90794},[206],[15,90796,90798],{"className":90797,"style":4450},[210],[15,90799],{},[15,90801,90803],{"className":90802},[198],[15,90804,90806,90826],{"className":90805},[202,1090],[15,90807,90809,90823],{"className":90808},[206],[15,90810,90812],{"className":90811,"style":3638},[210],[15,90813,90814,90817],{"style":9918},[15,90815],{"className":90816,"style":218},[217],[15,90818,90820],{"className":90819},[222,223,224,225],[15,90821,32458],{"className":90822,"style":23025},[66,67,225],[15,90824,1114],{"className":90825},[1113],[15,90827,90829],{"className":90828},[206],[15,90830,90832],{"className":90831,"style":1121},[210],[15,90833],{},"，有标准的偏差-方差分解：",[15,90836,90838],{"className":90837,"translate":19},[294],[15,90839,90841,91019],{"className":90840,"translate":19},[18],[15,90842,90844],{"className":90843},[23],[25,90845,90846],{"xmlns":27,"display":303},[29,90847,90848,91016],{},[32,90849,90850,90856,90858,90860,90862,90864,90874,90876,90878,90880,90886,90888,90890,90892,90908,90910,90969,90971,91014],{},[1057,90851,90852,90854],{},[35,90853,62381],{"mathvariant":454},[35,90855,32458],{},[39,90857,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,90859,120],{"stretchy":119},[35,90861,61885],{},[39,90863,6602],{},[1057,90865,90866,90872],{},[4296,90867,90868,90870],{"accent":53},[35,90869,116],{},[39,90871,78130],{},[35,90873,32458],{},[39,90875,120],{"stretchy":119},[35,90877,446],{},[39,90879,126],{"stretchy":119},[131,90881,90882,90884],{},[39,90883,126],{"stretchy":119},[1209,90885,1263],{},[39,90887,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,90889,129],{},[71,90891],{"linebreak":10456},[1172,90893,90894,90906],{},[1172,90895,90896,90904],{},[1255,90897,90898,90900,90902],{},[35,90899,6634],{},[35,90901,28518],{},[1209,90903,1263],{},[39,90905,69846],{"stretchy":53},[1274,90907,61854],{},[39,90909,1245],{},[1172,90911,90912,90962],{},[1172,90913,90914,90960],{},[32,90915,90916,90918,90924,90926,90928,90930,90932,90938,90948,90950,90952,90954],{},[39,90917,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[131,90919,90920,90922],{},[35,90921,116],{},[39,90923,1162],{},[39,90925,120],{"stretchy":119},[35,90927,446],{},[39,90929,126],{"stretchy":119},[39,90931,6602],{},[1057,90933,90934,90936],{},[35,90935,62381],{"mathvariant":454},[35,90937,32458],{},[1057,90939,90940,90946],{},[4296,90941,90942,90944],{"accent":53},[35,90943,116],{},[39,90945,78130],{},[35,90947,32458],{},[39,90949,120],{"stretchy":119},[35,90951,446],{},[39,90953,126],{"stretchy":119},[131,90955,90956,90958],{},[39,90957,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,90959,1263],{},[39,90961,69846],{"stretchy":53},[131,90963,90964,90967],{},[1274,90965,90966],{},"偏差",[1209,90968,1263],{},[39,90970,1245],{},[1172,90972,90973,91011],{},[1172,90974,90975,91009],{},[32,90976,90977,90989,90991,91001,91003,91005,91007],{},[1057,90978,90979,90987],{},[32,90980,90981,90983,90985],{},[35,90982,31732],{"mathvariant":138},[35,90984,4556],{"mathvariant":138},[35,90986,63107],{"mathvariant":138},[35,90988,32458],{},[39,90990,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[1057,90992,90993,90999],{},[4296,90994,90995,90997],{"accent":53},[35,90996,116],{},[39,90998,78130],{},[35,91000,32458],{},[39,91002,120],{"stretchy":119},[35,91004,446],{},[39,91006,126],{"stretchy":119},[39,91008,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,91010,69846],{"stretchy":53},[1274,91012,91013],{},"方差",[35,91015,3573],{"mathvariant":138},[45,91017,91018],{"encoding":47},"\\mathbb E_D\\big[(Y-\\hat f_D(X))^2\\big]\n= \\\\ \\underbrace{\\sigma_\\varepsilon^2}_{\\text{贝叶斯风险}}\n+\\underbrace{\\big(f^*(X)-\\mathbb E_D\\hat f_D(X)\\big)^2}_{\\text{偏差}^2}\n+\\underbrace{\\mathrm{Var}_D\\big(\\hat 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贝叶斯决策如何长成算法",[1904,92046,92048],{"id":92047},"_51-生成式与判别式","5.1 生成式与判别式",[11,92050,92051,92052,92176,92177,92205,92206,92272,92273,92317,92318,92321,92322,92325],{},"贝叶斯规则只依赖后验概率 ",[15,92053,92055,92092],{"className":92054,"translate":19},[18],[15,92056,92058],{"className":92057},[23],[25,92059,92060],{"xmlns":27},[29,92061,92062,92090],{},[32,92063,92064,92066,92068,92070,92072,92074,92076,92078,92080,92082,92084,92086,92088],{},[35,92065,71113],{},[39,92067,120],{"stretchy":119},[35,92069,123],{},[39,92071,126],{"stretchy":119},[39,92073,129],{},[35,92075,11],{},[39,92077,120],{"stretchy":119},[35,92079,61885],{},[39,92081,129],{},[1209,92083,1211],{},[39,92085,34430],{},[35,92087,123],{},[39,92089,126],{"stretchy":119},[45,92091,71140],{"encoding":47},[15,92093,92095,92122,92146,92164],{"className":92094,"ariaHidden":53},[52],[15,92096,92098,92101,92104,92107,92110,92113,92116,92119],{"className":92097},[57],[15,92099],{"className":92100,"style":154},[61],[15,92102,71113],{"className":92103,"style":1515},[66,67],[15,92105,120],{"className":92106},[162],[15,92108,123],{"className":92109},[66,67],[15,92111,126],{"className":92112},[169],[15,92114],{"className":92115,"style":173},[71],[15,92117,129],{"className":92118},[177],[15,92120],{"className":92121,"style":173},[71],[15,92123,92125,92128,92131,92134,92137,92140,92143],{"className":92124},[57],[15,92126],{"className":92127,"style":154},[61],[15,92129,11],{"className":92130},[66,67],[15,92132,120],{"className":92133},[162],[15,92135,61885],{"className":92136,"style":72},[66,67],[15,92138],{"className":92139,"style":173},[71],[15,92141,129],{"className":92142},[177],[15,92144],{"className":92145,"style":173},[71],[15,92147,92149,92152,92155,92158,92161],{"className":92148},[57],[15,92150],{"className":92151,"style":154},[61],[15,92153,1211],{"className":92154},[66],[15,92156],{"className":92157,"style":173},[71],[15,92159,34430],{"className":92160},[177],[15,92162],{"className":92163,"style":173},[71],[15,92165,92167,92170,92173],{"className":92166},[57],[15,92168],{"className":92169,"style":154},[61],[15,92171,123],{"className":92172},[66,67],[15,92174,126],{"className":92175},[169],"。因此有两条路可以走：直接建模 ",[15,92178,92180,92193],{"className":92179,"translate":19},[18],[15,92181,92183],{"className":92182},[23],[25,92184,92185],{"xmlns":27},[29,92186,92187,92191],{},[32,92188,92189],{},[35,92190,71113],{},[45,92192,90248],{"encoding":47},[15,92194,92196],{"className":92195,"ariaHidden":53},[52],[15,92197,92199,92202],{"className":92198},[57],[15,92200],{"className":92201,"style":471},[61],[15,92203,71113],{"className":92204,"style":1515},[66,67],"，或者分别建模类条件密度 ",[15,92207,92209,92233],{"className":92208,"translate":19},[18],[15,92210,92212],{"className":92211},[23],[25,92213,92214],{"xmlns":27},[29,92215,92216,92230],{},[32,92217,92218,92220,92222,92224,92226,92228],{},[35,92219,11],{},[39,92221,120],{"stretchy":119},[35,92223,123],{},[39,92225,34430],{},[35,92227,1224],{},[39,92229,126],{"stretchy":119},[45,92231,92232],{"encoding":47},"p(x\\mid y)",[15,92234,92236,92260],{"className":92235,"ariaHidden":53},[52],[15,92237,92239,92242,92245,92248,92251,92254,92257],{"className":92238},[57],[15,92240],{"className":92241,"style":154},[61],[15,92243,11],{"className":92244},[66,67],[15,92246,120],{"className":92247},[162],[15,92249,123],{"className":92250},[66,67],[15,92252],{"className":92253,"style":173},[71],[15,92255,34430],{"className":92256},[177],[15,92258],{"className":92259,"style":173},[71],[15,92261,92263,92266,92269],{"className":92262},[57],[15,92264],{"className":92265,"style":154},[61],[15,92267,1224],{"className":92268,"style":1515},[66,67],[15,92270,126],{"className":92271},[169]," 与先验 ",[15,92274,92276,92296],{"className":92275,"translate":19},[18],[15,92277,92279],{"className":92278},[23],[25,92280,92281],{"xmlns":27},[29,92282,92283,92293],{},[32,92284,92285,92287,92289,92291],{},[35,92286,11],{},[39,92288,120],{"stretchy":119},[35,92290,1224],{},[39,92292,126],{"stretchy":119},[45,92294,92295],{"encoding":47},"p(y)",[15,92297,92299],{"className":92298,"ariaHidden":53},[52],[15,92300,92302,92305,92308,92311,92314],{"className":92301},[57],[15,92303],{"className":92304,"style":154},[61],[15,92306,11],{"className":92307},[66,67],[15,92309,120],{"className":92310},[162],[15,92312,1224],{"className":92313,"style":1515},[66,67],[15,92315,126],{"className":92316},[169],"，再用贝叶斯公式合成。前者叫",[2040,92319,92320],{},"判别式","（discriminative），后者叫",[2040,92323,92324],{},"生成式","（generative）。",[11,92327,92328],{},"生成式的合成公式是",[15,92330,92332],{"className":92331,"translate":19},[294],[15,92333,92335,92421],{"className":92334,"translate":19},[18],[15,92336,92338],{"className":92337},[23],[25,92339,92340],{"xmlns":27,"display":303},[29,92341,92342,92418],{},[32,92343,92344,92346,92348,92350,92352,92354,92416],{},[35,92345,71113],{},[39,92347,120],{"stretchy":119},[35,92349,123],{},[39,92351,126],{"stretchy":119},[39,92353,129],{},[6467,92355,92356,92376],{},[32,92357,92358,92364,92370,92372,92374],{},[1057,92359,92360,92362],{},[35,92361,33494],{},[1209,92363,1211],{},[1057,92365,92366,92368],{},[35,92367,11],{},[1209,92369,1211],{},[39,92371,120],{"stretchy":119},[35,92373,123],{},[39,92375,126],{"stretchy":119},[32,92377,92378,92384,92390,92392,92394,92396,92398,92404,92410,92412,92414],{},[1057,92379,92380,92382],{},[35,92381,33494],{},[1209,92383,1804],{},[1057,92385,92386,92388],{},[35,92387,11],{},[1209,92389,1804],{},[39,92391,120],{"stretchy":119},[35,92393,123],{},[39,92395,126],{"stretchy":119},[39,92397,1245],{},[1057,92399,92400,92402],{},[35,92401,33494],{},[1209,92403,1211],{},[1057,92405,92406,92408],{},[35,92407,11],{},[1209,92409,1211],{},[39,92411,120],{"stretchy":119},[35,92413,123],{},[39,92415,126],{"stretchy":119},[35,92417,3573],{"mathvariant":138},[45,92419,92420],{"encoding":47},"\\eta(x)=\\frac{\\pi_1p_1(x)}{\\pi_0p_0(x)+\\pi_1p_1(x)}.",[15,92422,92424,92451],{"className":92423,"ariaHidden":53},[52],[15,92425,92427,92430,92433,92436,92439,92442,92445,92448],{"className":92426},[57],[15,92428],{"className":92429,"style":154},[61],[15,92431,71113],{"className":92432,"style":1515},[66,67],[15,92434,120],{"className":92435},[162],[15,92437,123],{"className":92438},[66,67],[15,92440,126],{"className":92441},[169],[15,92443],{"className":92444,"style":173},[71],[15,92446,129],{"className":92447},[177],[15,92449],{"className":92450,"style":173},[71],[15,92452,92454,92457,92789],{"className":92453},[57],[15,92455],{"className":92456,"style":41192},[61],[15,92458,92460,92463,92786],{"className":92459},[66],[15,92461],{"className":92462},[162,23417],[15,92464,92466],{"className":92465},[6467],[15,92467,92469,92778],{"className":92468},[202,1090],[15,92470,92472,92775],{"className":92471},[206],[15,92473,92475,92670,92678],{"className":92474,"style":41211},[210],[15,92476,92477,92480],{"style":23433},[15,92478],{"className":92479,"style":1368},[217],[15,92481,92483,92523,92563,92566,92569,92572,92575,92578,92581,92621,92661,92664,92667],{"className":92482},[66],[15,92484,92486,92489],{"className":92485},[66],[15,92487,33494],{"className":92488,"style":1515},[66,67],[15,92490,92492],{"className":92491},[198],[15,92493,92495,92515],{"className":92494},[202,1090],[15,92496,92498,92512],{"className":92497},[206],[15,92499,92501],{"className":92500,"style":2139},[210],[15,92502,92503,92506],{"style":1531},[15,92504],{"className":92505,"style":218},[217],[15,92507,92509],{"className":92508},[222,223,224,225],[15,92510,1804],{"className":92511},[66,225],[15,92513,1114],{"className":92514},[1113],[15,92516,92518],{"className":92517},[206],[15,92519,92521],{"className":92520,"style":1121},[210],[15,92522],{},[15,92524,92526,92529],{"className":92525},[66],[15,92527,11],{"className":92528},[66,67],[15,92530,92532],{"className":92531},[198],[15,92533,92535,92555],{"className":92534},[202,1090],[15,92536,92538,92552],{"className":92537},[206],[15,92539,92541],{"className":92540,"style":2139},[210],[15,92542,92543,92546],{"style":1584},[15,92544],{"className":92545,"style":218},[217],[15,92547,92549],{"className":92548},[222,223,224,225],[15,92550,1804],{"className":92551},[66,225],[15,92553,1114],{"className":92554},[1113],[15,92556,92558],{"className":92557},[206],[15,92559,92561],{"className":92560,"style":1121},[210],[15,92562],{},[15,92564,120],{"className":92565},[162],[15,92567,123],{"className":92568},[66,67],[15,92570,126],{"className":92571},[169],[15,92573],{"className":92574,"style":72},[71],[15,92576,1245],{"className":92577},[76],[15,92579],{"className":92580,"style":72},[71],[15,92582,92584,92587],{"className":92583},[66],[15,92585,33494],{"className":92586,"style":1515},[66,67],[15,92588,92590],{"className":92589},[198],[15,92591,92593,92613],{"className":92592},[202,1090],[15,92594,92596,92610],{"className":92595},[206],[15,92597,92599],{"className":92598,"style":2139},[210],[15,92600,92601,92604],{"style":1531},[15,92602],{"className":92603,"style":218},[217],[15,92605,92607],{"className":92606},[222,223,224,225],[15,92608,1211],{"className":92609},[66,225],[15,92611,1114],{"className":92612},[1113],[15,92614,92616],{"className":92615},[206],[15,92617,92619],{"className":92618,"style":1121},[210],[15,92620],{},[15,92622,92624,92627],{"className":92623},[66],[15,92625,11],{"className":92626},[66,67],[15,92628,92630],{"className":92629},[198],[15,92631,92633,92653],{"className":92632},[202,1090],[15,92634,92636,92650],{"className":92635},[206],[15,92637,92639],{"className":92638,"style":2139},[210],[15,92640,92641,92644],{"style":1584},[15,92642],{"className":92643,"style":218},[217],[15,92645,92647],{"className":92646},[222,223,224,225],[15,92648,1211],{"className":92649},[66,225],[15,92651,1114],{"className":92652},[1113],[15,92654,92656],{"className":92655},[206],[15,92657,92659],{"className":92658,"style":1121},[210],[15,92660],{},[15,92662,120],{"className":92663},[162],[15,92665,123],{"className":92666},[66,67],[15,92668,126],{"className":92669},[169],[15,92671,92672,92675],{"style":23445},[15,92673],{"className":92674,"style":1368},[217],[15,92676],{"className":92677,"style":4440},[23452],[15,92679,92680,92683],{"style":23455},[15,92681],{"className":92682,"style":1368},[217],[15,92684,92686,92726,92766,92769,92772],{"className":92685},[66],[15,92687,92689,92692],{"className":92688},[66],[15,92690,33494],{"className":92691,"style":1515},[66,67],[15,92693,92695],{"className":92694},[198],[15,92696,92698,92718],{"className":92697},[202,1090],[15,92699,92701,92715],{"className":92700},[206],[15,92702,92704],{"className":92703,"style":2139},[210],[15,92705,92706,92709],{"style":1531},[15,92707],{"className":92708,"style":218},[217],[15,92710,92712],{"className":92711},[222,223,224,225],[15,92713,1211],{"className":92714},[66,225],[15,92716,1114],{"className":92717},[1113],[15,92719,92721],{"className":92720},[206],[15,92722,92724],{"className":92723,"style":1121},[210],[15,92725],{},[15,92727,92729,92732],{"className":92728},[66],[15,92730,11],{"className":92731},[66,67],[15,92733,92735],{"className":92734},[198],[15,92736,92738,92758],{"className":92737},[202,1090],[15,92739,92741,92755],{"className":92740},[206],[15,92742,92744],{"className":92743,"style":2139},[210],[15,92745,92746,92749],{"style":1584},[15,92747],{"className":92748,"style":218},[217],[15,92750,92752],{"className":92751},[222,223,224,225],[15,92753,1211],{"className":92754},[66,225],[15,92756,1114],{"className":92757},[1113],[15,92759,92761],{"className":92760},[206],[15,92762,92764],{"className":92763,"style":1121},[210],[15,92765],{},[15,92767,120],{"className":92768},[162],[15,92770,123],{"className":92771},[66,67],[15,92773,126],{"className":92774},[169],[15,92776,1114],{"className":92777},[1113],[15,92779,92781],{"className":92780},[206],[15,92782,92784],{"className":92783,"style":41282},[210],[15,92785],{},[15,92787],{"className":92788},[169,23417],[15,92790,3573],{"className":92791},[66],[11,92793,92794,92795,92824,92825,92853],{},"只要两边的密度估计足够准，这个式子就给出贝叶斯后验。线性判别分析（LDA）假设两类共享协方差 ",[15,92796,92798,92812],{"className":92797,"translate":19},[18],[15,92799,92801],{"className":92800},[23],[25,92802,92803],{"xmlns":27},[29,92804,92805,92809],{},[32,92806,92807],{},[35,92808,83409],{"mathvariant":138},[45,92810,92811],{"encoding":47},"\\Sigma",[15,92813,92815],{"className":92814,"ariaHidden":53},[52],[15,92816,92818,92821],{"className":92817},[57],[15,92819],{"className":92820,"style":490},[61],[15,92822,83409],{"className":92823},[66],"，此时对数似然比是 ",[15,92826,92828,92841],{"className":92827,"translate":19},[18],[15,92829,92831],{"className":92830},[23],[25,92832,92833],{"xmlns":27},[29,92834,92835,92839],{},[32,92836,92837],{},[35,92838,123],{},[45,92840,123],{"encoding":47},[15,92842,92844],{"className":92843,"ariaHidden":53},[52],[15,92845,92847,92850],{"className":92846},[57],[15,92848],{"className":92849,"style":86},[61],[15,92851,123],{"className":92852},[66,67]," 的线性函数：",[15,92855,92857],{"className":92856,"translate":19},[294],[15,92858,92860,92949],{"className":92859,"translate":19},[18],[15,92861,92863],{"className":92862},[23],[25,92864,92865],{"xmlns":27,"display":303},[29,92866,92867,92946],{},[32,92868,92869,92871,92873,92903,92905,92907,92913,92915,92921,92927,92937,92939,92941,92944],{},[35,92870,27073],{},[39,92872,1170],{},[6467,92874,92875,92889],{},[32,92876,92877,92883,92885,92887],{},[1057,92878,92879,92881],{},[35,92880,11],{},[1209,92882,1211],{},[39,92884,120],{"stretchy":119},[35,92886,123],{},[39,92888,126],{"stretchy":119},[32,92890,92891,92897,92899,92901],{},[1057,92892,92893,92895],{},[35,92894,11],{},[1209,92896,1804],{},[39,92898,120],{"stretchy":119},[35,92900,123],{},[39,92902,126],{"stretchy":119},[39,92904,129],{},[39,92906,120],{"stretchy":119},[1057,92908,92909,92911],{},[35,92910,83402],{},[1209,92912,1211],{},[39,92914,6602],{},[1057,92916,92917,92919],{},[35,92918,83402],{},[1209,92920,1804],{},[131,92922,92923,92925],{},[39,92924,126],{"stretchy":119},[35,92926,139],{"mathvariant":138},[131,92928,92929,92931],{},[35,92930,83409],{"mathvariant":138},[32,92932,92933,92935],{},[39,92934,6602],{},[1209,92936,1211],{},[35,92938,123],{},[39,92940,1245],{},[1274,92942,92943],{},"常数",[39,92945,335],{"separator":53},[45,92947,92948],{"encoding":47},"\\log\\frac{p_1(x)}{p_0(x)}\n=(\\mu_1-\\mu_0)^\\top\\Sigma^{-1}x+\\text{常数},",[15,92950,92952,93129,93187,93309],{"className":92951,"ariaHidden":53},[52],[15,92953,92955,92958,92963,92966,93120,93123,93126],{"className":92954},[57],[15,92956],{"className":92957,"style":41192},[61],[15,92959,27283,92961],{"className":92960},[1339],[15,92962,1344],{"style":1343},[15,92964],{"className":92965,"style":697},[71],[15,92967,92969,92972,93117],{"className":92968},[66],[15,92970],{"className":92971},[162,23417],[15,92973,92975],{"className":92974},[6467],[15,92976,92978,93109],{"className":92977},[202,1090],[15,92979,92981,93106],{"className":92980},[206],[15,92982,92984,93041,93049],{"className":92983,"style":41211},[210],[15,92985,92986,92989],{"style":23433},[15,92987],{"className":92988,"style":1368},[217],[15,92990,92992,93032,93035,93038],{"className":92991},[66],[15,92993,92995,92998],{"className":92994},[66],[15,92996,11],{"className":92997},[66,67],[15,92999,93001],{"className":93000},[198],[15,93002,93004,93024],{"className":93003},[202,1090],[15,93005,93007,93021],{"className":93006},[206],[15,93008,93010],{"className":93009,"style":2139},[210],[15,93011,93012,93015],{"style":1584},[15,93013],{"className":93014,"style":218},[217],[15,93016,93018],{"className":93017},[222,223,224,225],[15,93019,1804],{"className":93020},[66,225],[15,93022,1114],{"className":93023},[1113],[15,93025,93027],{"className":93026},[206],[15,93028,93030],{"className":93029,"style":1121},[210],[15,93031],{},[15,93033,120],{"className":93034},[162],[15,93036,123],{"className":93037},[66,67],[15,93039,126],{"className":93040},[169],[15,93042,93043,93046],{"style":23445},[15,93044],{"className":93045,"style":1368},[217],[15,93047],{"className":93048,"style":4440},[23452],[15,93050,93051,93054],{"style":23455},[15,93052],{"className":93053,"style":1368},[217],[15,93055,93057,93097,93100,93103],{"className":93056},[66],[15,93058,93060,93063],{"className":93059},[66],[15,93061,11],{"className":93062},[66,67],[15,93064,93066],{"className":93065},[198],[15,93067,93069,93089],{"className":93068},[202,1090],[15,93070,93072,93086],{"className":93071},[206],[15,93073,93075],{"className":93074,"style":2139},[210],[15,93076,93077,93080],{"style":1584},[15,93078],{"className":93079,"style":218},[217],[15,93081,93083],{"className":93082},[222,223,224,225],[15,93084,1211],{"className":93085},[66,225],[15,93087,1114],{"className":93088},[1113],[15,93090,93092],{"className":93091},[206],[15,93093,93095],{"className":93094,"style":1121},[210],[15,93096],{},[15,93098,120],{"className":93099},[162],[15,93101,123],{"className":93102},[66,67],[15,93104,126],{"className":93105},[169],[15,93107,1114],{"className":93108},[1113],[15,93110,93112],{"className":93111},[206],[15,93113,93115],{"className":93114,"style":41282},[210],[15,93116],{},[15,93118],{"className":93119},[169,23417],[15,93121],{"className":93122,"style":173},[71],[15,93124,129],{"className":93125},[177],[15,93127],{"className":93128,"style":173},[71],[15,93130,93132,93135,93138,93178,93181,93184],{"className":93131},[57],[15,93133],{"className":93134,"style":154},[61],[15,93136,120],{"className":93137},[162],[15,93139,93141,93144],{"className":93140},[66],[15,93142,83402],{"className":93143},[66,67],[15,93145,93147],{"className":93146},[198],[15,93148,93150,93170],{"className":93149},[202,1090],[15,93151,93153,93167],{"className":93152},[206],[15,93154,93156],{"className":93155,"style":2139},[210],[15,93157,93158,93161],{"style":1584},[15,93159],{"className":93160,"style":218},[217],[15,93162,93164],{"className":93163},[222,223,224,225],[15,93165,1211],{"className":93166},[66,225],[15,93168,1114],{"className":93169},[1113],[15,93171,93173],{"className":93172},[206],[15,93174,93176],{"className":93175,"style":1121},[210],[15,93177],{},[15,93179],{"className":93180,"style":72},[71],[15,93182,6602],{"className":93183},[76],[15,93185],{"className":93186,"style":72},[71],[15,93188,93190,93193,93233,93262,93297,93300,93303,93306],{"className":93189},[57],[15,93191],{"className":93192,"style":375},[61],[15,93194,93196,93199],{"className":93195},[66],[15,93197,83402],{"className":93198},[66,67],[15,93200,93202],{"className":93201},[198],[15,93203,93205,93225],{"className":93204},[202,1090],[15,93206,93208,93222],{"className":93207},[206],[15,93209,93211],{"className":93210,"style":2139},[210],[15,93212,93213,93216],{"style":1584},[15,93214],{"className":93215,"style":218},[217],[15,93217,93219],{"className":93218},[222,223,224,225],[15,93220,1804],{"className":93221},[66,225],[15,93223,1114],{"className":93224},[1113],[15,93226,93228],{"className":93227},[206],[15,93229,93231],{"className":93230,"style":1121},[210],[15,93232],{},[15,93234,93236,93239],{"className":93235},[169],[15,93237,126],{"className":93238},[169],[15,93240,93242],{"className":93241},[198],[15,93243,93245],{"className":93244},[202],[15,93246,93248],{"className":93247},[206],[15,93249,93251],{"className":93250,"style":394},[210],[15,93252,93253,93256],{"style":397},[15,93254],{"className":93255,"style":218},[217],[15,93257,93259],{"className":93258},[222,223,224,225],[15,93260,139],{"className":93261},[66,225],[15,93263,93265,93268],{"className":93264},[66],[15,93266,83409],{"className":93267},[66],[15,93269,93271],{"className":93270},[198],[15,93272,93274],{"className":93273},[202],[15,93275,93277],{"className":93276},[206],[15,93278,93280],{"className":93279,"style":1650},[210],[15,93281,93282,93285],{"style":397},[15,93283],{"className":93284,"style":218},[217],[15,93286,93288],{"className":93287},[222,223,224,225],[15,93289,93291,93294],{"className":93290},[66,225],[15,93292,6602],{"className":93293},[66,225],[15,93295,1211],{"className":93296},[66,225],[15,93298,123],{"className":93299},[66,67],[15,93301],{"className":93302,"style":72},[71],[15,93304,1245],{"className":93305},[76],[15,93307],{"className":93308,"style":72},[71],[15,93310,93312,93315,93321],{"className":93311},[57],[15,93313],{"className":93314,"style":17379},[61],[15,93316,93318],{"className":93317},[66,1711],[15,93319,92943],{"className":93320},[66,68616],[15,93322,335],{"className":93323},[422],[11,93325,93326,93327,93451,93452,93686,93687,93752],{},"决策边界是一个超平面。二次判别分析（QDA）允许两类有不同的协方差 ",[15,93328,93330,93356],{"className":93329,"translate":19},[18],[15,93331,93333],{"className":93332},[23],[25,93334,93335],{"xmlns":27},[29,93336,93337,93353],{},[32,93338,93339,93345,93347],{},[1057,93340,93341,93343],{},[35,93342,83409],{"mathvariant":138},[1209,93344,1804],{},[39,93346,335],{"separator":53},[1057,93348,93349,93351],{},[35,93350,83409],{"mathvariant":138},[1209,93352,1211],{},[45,93354,93355],{"encoding":47},"\\Sigma_0,\\Sigma_1",[15,93357,93359],{"className":93358,"ariaHidden":53},[52],[15,93360,93362,93365,93405,93408,93411],{"className":93361},[57],[15,93363],{"className":93364,"style":17379},[61],[15,93366,93368,93371],{"className":93367},[66],[15,93369,83409],{"className":93370},[66],[15,93372,93374],{"className":93373},[198],[15,93375,93377,93397],{"className":93376},[202,1090],[15,93378,93380,93394],{"className":93379},[206],[15,93381,93383],{"className":93382,"style":2139},[210],[15,93384,93385,93388],{"style":1584},[15,93386],{"className":93387,"style":218},[217],[15,93389,93391],{"className":93390},[222,223,224,225],[15,93392,1804],{"className":93393},[66,225],[15,93395,1114],{"className":93396},[1113],[15,93398,93400],{"className":93399},[206],[15,93401,93403],{"className":93402,"style":1121},[210],[15,93404],{},[15,93406,335],{"className":93407},[422],[15,93409],{"className":93410,"style":697},[71],[15,93412,93414,93417],{"className":93413},[66],[15,93415,83409],{"className":93416},[66],[15,93418,93420],{"className":93419},[198],[15,93421,93423,93443],{"className":93422},[202,1090],[15,93424,93426,93440],{"className":93425},[206],[15,93427,93429],{"className":93428,"style":2139},[210],[15,93430,93431,93434],{"style":1584},[15,93432],{"className":93433,"style":218},[217],[15,93435,93437],{"className":93436},[222,223,224,225],[15,93438,1211],{"className":93439},[66,225],[15,93441,1114],{"className":93442},[1113],[15,93444,93446],{"className":93445},[206],[15,93447,93449],{"className":93448,"style":1121},[210],[15,93450],{},"，对数似然比出现 ",[15,93453,93455,93505],{"className":93454,"translate":19},[18],[15,93456,93458],{"className":93457},[23],[25,93459,93460],{"xmlns":27},[29,93461,93462,93502],{},[32,93463,93464,93470,93472,93484,93486,93498,93500],{},[131,93465,93466,93468],{},[35,93467,123],{},[35,93469,139],{"mathvariant":138},[39,93471,120],{"stretchy":119},[1255,93473,93474,93476,93478],{},[35,93475,83409],{"mathvariant":138},[1209,93477,1211],{},[32,93479,93480,93482],{},[39,93481,6602],{},[1209,93483,1211],{},[39,93485,6602],{},[1255,93487,93488,93490,93492],{},[35,93489,83409],{"mathvariant":138},[1209,93491,1804],{},[32,93493,93494,93496],{},[39,93495,6602],{},[1209,93497,1211],{},[39,93499,126],{"stretchy":119},[35,93501,123],{},[45,93503,93504],{"encoding":47},"x^\\top(\\Sigma_1^{-1}-\\Sigma_0^{-1})x",[15,93506,93508,93617],{"className":93507,"ariaHidden":53},[52],[15,93509,93511,93515,93544,93547,93608,93611,93614],{"className":93510},[57],[15,93512],{"className":93513,"style":93514},[61],"height:1.1205em;vertical-align:-0.2663em;",[15,93516,93518,93521],{"className":93517},[66],[15,93519,123],{"className":93520},[66,67],[15,93522,93524],{"className":93523},[198],[15,93525,93527],{"className":93526},[202],[15,93528,93530],{"className":93529},[206],[15,93531,93533],{"className":93532,"style":187},[210],[15,93534,93535,93538],{"style":213},[15,93536],{"className":93537,"style":218},[217],[15,93539,93541],{"className":93540},[222,223,224,225],[15,93542,139],{"className":93543},[66,225],[15,93545,120],{"className":93546},[162],[15,93548,93550,93553],{"className":93549},[66],[15,93551,83409],{"className":93552},[66],[15,93554,93556],{"className":93555},[198],[15,93557,93559,93599],{"className":93558},[202,1090],[15,93560,93562,93596],{"className":93561},[206],[15,93563,93566,93578],{"className":93564,"style":93565},[210],"height:0.8542em;",[15,93567,93569,93572],{"style":93568},"top:-2.4337em;margin-left:0em;margin-right:0.05em;",[15,93570],{"className":93571,"style":218},[217],[15,93573,93575],{"className":93574},[222,223,224,225],[15,93576,1211],{"className":93577},[66,225],[15,93579,93581,93584],{"style":93580},"top:-3.1031em;margin-right:0.05em;",[15,93582],{"className":93583,"style":218},[217],[15,93585,93587],{"className":93586},[222,223,224,225],[15,93588,93590,93593],{"className":93589},[66,225],[15,93591,6602],{"className":93592},[66,225],[15,93594,1211],{"className":93595},[66,225],[15,93597,1114],{"className":93598},[1113],[15,93600,93602],{"className":93601},[206],[15,93603,93606],{"className":93604,"style":93605},[210],"height:0.2663em;",[15,93607],{},[15,93609],{"className":93610,"style":72},[71],[15,93612,6602],{"className":93613},[76],[15,93615],{"className":93616,"style":72},[71],[15,93618,93620,93623,93680,93683],{"className":93619},[57],[15,93621],{"className":93622,"style":93514},[61],[15,93624,93626,93629],{"className":93625},[66],[15,93627,83409],{"className":93628},[66],[15,93630,93632],{"className":93631},[198],[15,93633,93635,93672],{"className":93634},[202,1090],[15,93636,93638,93669],{"className":93637},[206],[15,93639,93641,93652],{"className":93640,"style":93565},[210],[15,93642,93643,93646],{"style":93568},[15,93644],{"className":93645,"style":218},[217],[15,93647,93649],{"className":93648},[222,223,224,225],[15,93650,1804],{"className":93651},[66,225],[15,93653,93654,93657],{"style":93580},[15,93655],{"className":93656,"style":218},[217],[15,93658,93660],{"className":93659},[222,223,224,225],[15,93661,93663,93666],{"className":93662},[66,225],[15,93664,6602],{"className":93665},[66,225],[15,93667,1211],{"className":93668},[66,225],[15,93670,1114],{"className":93671},[1113],[15,93673,93675],{"className":93674},[206],[15,93676,93678],{"className":93677,"style":93605},[210],[15,93679],{},[15,93681,126],{"className":93682},[169],[15,93684,123],{"className":93685},[66,67]," 这样的二次项，边界变成二次曲面。朴素贝叶斯再加一层条件独立假设，把 ",[15,93688,93690,93713],{"className":93689,"translate":19},[18],[15,93691,93693],{"className":93692},[23],[25,93694,93695],{"xmlns":27},[29,93696,93697,93711],{},[32,93698,93699,93701,93703,93705,93707,93709],{},[35,93700,11],{},[39,93702,120],{"stretchy":119},[35,93704,123],{},[39,93706,34430],{},[35,93708,1224],{},[39,93710,126],{"stretchy":119},[45,93712,92232],{"encoding":47},[15,93714,93716,93740],{"className":93715,"ariaHidden":53},[52],[15,93717,93719,93722,93725,93728,93731,93734,93737],{"className":93718},[57],[15,93720],{"className":93721,"style":154},[61],[15,93723,11],{"className":93724},[66,67],[15,93726,120],{"className":93727},[162],[15,93729,123],{"className":93730},[66,67],[15,93732],{"className":93733,"style":173},[71],[15,93735,34430],{"className":93736},[177],[15,93738],{"className":93739,"style":173},[71],[15,93741,93743,93746,93749],{"className":93742},[57],[15,93744],{"className":93745,"style":154},[61],[15,93747,1224],{"className":93748,"style":1515},[66,67],[15,93750,126],{"className":93751},[169]," 拆成各个特征的乘积，用很大的偏差换取很小的方差。",[11,93754,93755,93756,93784,93785,93850],{},"两条路的目标是同一个 ",[15,93757,93759,93772],{"className":93758,"translate":19},[18],[15,93760,93762],{"className":93761},[23],[25,93763,93764],{"xmlns":27},[29,93765,93766,93770],{},[32,93767,93768],{},[35,93769,71113],{},[45,93771,90248],{"encoding":47},[15,93773,93775],{"className":93774,"ariaHidden":53},[52],[15,93776,93778,93781],{"className":93777},[57],[15,93779],{"className":93780,"style":471},[61],[15,93782,71113],{"className":93783,"style":1515},[66,67],"，但统计性质不同。生成式利用了 ",[15,93786,93788,93811],{"className":93787,"translate":19},[18],[15,93789,93791],{"className":93790},[23],[25,93792,93793],{"xmlns":27},[29,93794,93795,93809],{},[32,93796,93797,93799,93801,93803,93805,93807],{},[35,93798,11],{},[39,93800,120],{"stretchy":119},[35,93802,123],{},[39,93804,34430],{},[35,93806,1224],{},[39,93808,126],{"stretchy":119},[45,93810,92232],{"encoding":47},[15,93812,93814,93838],{"className":93813,"ariaHidden":53},[52],[15,93815,93817,93820,93823,93826,93829,93832,93835],{"className":93816},[57],[15,93818],{"className":93819,"style":154},[61],[15,93821,11],{"className":93822},[66,67],[15,93824,120],{"className":93825},[162],[15,93827,123],{"className":93828},[66,67],[15,93830],{"className":93831,"style":173},[71],[15,93833,34430],{"className":93834},[177],[15,93836],{"className":93837,"style":173},[71],[15,93839,93841,93844,93847],{"className":93840},[57],[15,93842],{"className":93843,"style":154},[61],[15,93845,1224],{"className":93846,"style":1515},[66,67],[15,93848,126],{"className":93849},[169]," 的结构，在模型正确时样本效率更高；判别式直接优化决策边界，在模型误设时通常更稳健。这不是「谁更好」的问题，而是「你愿意把假设放在哪一侧」的问题。",[11,93852,93853,93854,93857,93858,93886,93887,93960,93961,94019],{},"这里说的是 ",[2040,93855,93856],{},"oracle 贝叶斯规则","：它假设 ",[15,93859,93861,93874],{"className":93860,"translate":19},[18],[15,93862,93864],{"className":93863},[23],[25,93865,93866],{"xmlns":27},[29,93867,93868,93872],{},[32,93869,93870],{},[35,93871,71113],{},[45,93873,90248],{"encoding":47},[15,93875,93877],{"className":93876,"ariaHidden":53},[52],[15,93878,93880,93883],{"className":93879},[57],[15,93881],{"className":93882,"style":471},[61],[15,93884,71113],{"className":93885,"style":1515},[66,67]," 已知。实践中我们把估计的 ",[15,93888,93890,93908],{"className":93889,"translate":19},[18],[15,93891,93893],{"className":93892},[23],[25,93894,93895],{"xmlns":27},[29,93896,93897,93905],{},[32,93898,93899],{},[4296,93900,93901,93903],{"accent":53},[35,93902,71113],{},[39,93904,78130],{},[45,93906,93907],{"encoding":47},"\\hat\\eta",[15,93909,93911],{"className":93910,"ariaHidden":53},[52],[15,93912,93914,93917],{"className":93913},[57],[15,93915],{"className":93916,"style":3408},[61],[15,93918,93920],{"className":93919},[66,78147],[15,93921,93923,93952],{"className":93922},[202,1090],[15,93924,93926,93949],{"className":93925},[206],[15,93927,93929,93937],{"className":93928,"style":840},[210],[15,93930,93931,93934],{"style":1386},[15,93932],{"className":93933,"style":1368},[217],[15,93935,71113],{"className":93936,"style":1515},[66,67],[15,93938,93939,93942],{"style":1386},[15,93940],{"className":93941,"style":1368},[217],[15,93943,93946],{"className":93944,"style":93945},[78175],"left:-0.1944em;",[15,93947,78130],{"className":93948},[66],[15,93950,1114],{"className":93951},[1113],[15,93953,93955],{"className":93954},[206],[15,93956,93958],{"className":93957,"style":4450},[210],[15,93959],{}," 代入，得到的只是 plug-in 分类器；文献里的「贝叶斯分类器」有时指这个 oracle 规则，有时指朴素贝叶斯之类的具体算法，读的时候要分清。plug-in 分类器的风险与 ",[15,93962,93964,93981],{"className":93963,"translate":19},[18],[15,93965,93967],{"className":93966},[23],[25,93968,93969],{"xmlns":27},[29,93970,93971,93979],{},[32,93972,93973],{},[131,93974,93975,93977],{},[35,93976,455],{},[39,93978,1162],{},[45,93980,76283],{"encoding":47},[15,93982,93984],{"className":93983,"ariaHidden":53},[52],[15,93985,93987,93990],{"className":93986},[57],[15,93988],{"className":93989,"style":16249},[61],[15,93991,93993,93996],{"className":93992},[66],[15,93994,455],{"className":93995,"style":62437},[66,67],[15,93997,93999],{"className":93998},[198],[15,94000,94002],{"className":94001},[202],[15,94003,94005],{"className":94004},[206],[15,94006,94008],{"className":94007,"style":16249},[210],[15,94009,94010,94013],{"style":213},[15,94011],{"className":94012,"style":218},[217],[15,94014,94016],{"className":94015},[222,223,224,225],[15,94017,1162],{"className":94018},[76,225]," 之间的差距，正是第 7 节要分解的估计误差。",[11,94021,94022,94023,94067,94068,94100,94101,94129,94130,1124],{},"值得一提的是，朴素贝叶斯的条件独立假设几乎总是错的，但它的分类性能常常不差。原因在于 0-1 损失只关心 ",[15,94024,94026,94046],{"className":94025,"translate":19},[18],[15,94027,94029],{"className":94028},[23],[25,94030,94031],{"xmlns":27},[29,94032,94033,94043],{},[32,94034,94035,94037,94039,94041],{},[35,94036,71113],{},[39,94038,120],{"stretchy":119},[35,94040,123],{},[39,94042,126],{"stretchy":119},[45,94044,94045],{"encoding":47},"\\eta(x)",[15,94047,94049],{"className":94048,"ariaHidden":53},[52],[15,94050,94052,94055,94058,94061,94064],{"className":94051},[57],[15,94053],{"className":94054,"style":154},[61],[15,94056,71113],{"className":94057,"style":1515},[66,67],[15,94059,120],{"className":94060},[162],[15,94062,123],{"className":94063},[66,67],[15,94065,126],{"className":94066},[169]," 是否越过 ",[15,94069,94071,94088],{"className":94070,"translate":19},[18],[15,94072,94074],{"className":94073},[23],[25,94075,94076],{"xmlns":27},[29,94077,94078,94086],{},[32,94079,94080,94082,94084],{},[1209,94081,1211],{},[35,94083,6625],{"mathvariant":138},[1209,94085,1263],{},[45,94087,71809],{"encoding":47},[15,94089,94091],{"className":94090,"ariaHidden":53},[52],[15,94092,94094,94097],{"className":94093},[57],[15,94095],{"className":94096,"style":154},[61],[15,94098,71809],{"className":94099},[66],"，而不要求 ",[15,94102,94104,94117],{"className":94103,"translate":19},[18],[15,94105,94107],{"className":94106},[23],[25,94108,94109],{"xmlns":27},[29,94110,94111,94115],{},[32,94112,94113],{},[35,94114,71113],{},[45,94116,90248],{"encoding":47},[15,94118,94120],{"className":94119,"ariaHidden":53},[52],[15,94121,94123,94126],{"className":94122},[57],[15,94124],{"className":94125,"style":471},[61],[15,94127,71113],{"className":94128,"style":1515},[66,67]," 的数值完全准确；只要假设带来的偏差没有把决策边界推得太远，分类结果就可以是对的。这个现象也提醒我们：",[2040,94131,94132],{},"概率估计的质量与决策的质量不是同一件事",[11,94134,94135,94136,94179,94180,94183,94184,37644],{},"这个区别在损失函数上有一个精确的对应。0-1 损失只关心 ",[15,94137,94139,94158],{"className":94138,"translate":19},[18],[15,94140,94142],{"className":94141},[23],[25,94143,94144],{"xmlns":27},[29,94145,94146,94156],{},[32,94147,94148,94150,94152,94154],{},[35,94149,71113],{},[39,94151,120],{"stretchy":119},[35,94153,123],{},[39,94155,126],{"stretchy":119},[45,94157,94045],{"encoding":47},[15,94159,94161],{"className":94160,"ariaHidden":53},[52],[15,94162,94164,94167,94170,94173,94176],{"className":94163},[57],[15,94165],{"className":94166,"style":154},[61],[15,94168,71113],{"className":94169,"style":1515},[66,67],[15,94171,120],{"className":94172},[162],[15,94174,123],{"className":94175},[66,67],[15,94177,126],{"className":94178},[169]," 是否越过阈值，因此它对概率的微小误差不敏感；交叉熵（对数损失）则是一个",[2040,94181,94182],{},"严格 proper scoring rule","，它的总体最优解恰好是真实的条件分布 ",[15,94185,94187,94210],{"className":94186,"translate":19},[18],[15,94188,94190],{"className":94189},[23],[25,94191,94192],{"xmlns":27},[29,94193,94194,94208],{},[32,94195,94196,94198,94200,94202,94204,94206],{},[35,94197,11],{},[39,94199,120],{"stretchy":119},[35,94201,1224],{},[39,94203,34430],{},[35,94205,123],{},[39,94207,126],{"stretchy":119},[45,94209,68206],{"encoding":47},[15,94211,94213,94237],{"className":94212,"ariaHidden":53},[52],[15,94214,94216,94219,94222,94225,94228,94231,94234],{"className":94215},[57],[15,94217],{"className":94218,"style":154},[61],[15,94220,11],{"className":94221},[66,67],[15,94223,120],{"className":94224},[162],[15,94226,1224],{"className":94227,"style":1515},[66,67],[15,94229],{"className":94230,"style":173},[71],[15,94232,34430],{"className":94233},[177],[15,94235],{"className":94236,"style":173},[71],[15,94238,94240,94243,94246],{"className":94239},[57],[15,94241],{"className":94242,"style":154},[61],[15,94244,123],{"className":94245},[66,67],[15,94247,126],{"className":94248},[169],[15,94250,94252],{"className":94251,"translate":19},[294],[15,94253,94255,94364],{"className":94254,"translate":19},[18],[15,94256,94258],{"className":94257},[23],[25,94259,94260],{"xmlns":27,"display":303},[29,94261,94262,94361],{},[32,94263,94264,94266,94268,94270,94272,94274,94277,94279,94281,94283,94285,94287,94289,94291,94293,94295,94297,94299,94301,94303,94305,94307,94309,94311,94313,94315,94317,94319,94321,94327,94329,94331,94333,94335,94337,94339,94341,94343,94345,94347,94349,94351,94353,94355,94357,94359],{},[35,94265,62381],{"mathvariant":454},[39,94267,1194],{"stretchy":119},[39,94269,6602],{},[35,94271,27073],{},[39,94273,1170],{},[35,94275,94276],{},"q",[39,94278,120],{"stretchy":119},[35,94280,61885],{},[39,94282,34430],{},[35,94284,446],{},[39,94286,126],{"stretchy":119},[39,94288,1266],{"stretchy":119},[39,94290,129],{},[35,94292,62381],{"mathvariant":454},[39,94294,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,94296,1061],{"mathvariant":454},[39,94298,120],{"stretchy":119},[35,94300,11],{},[39,94302,120],{"stretchy":119},[39,94304,3268],{},[39,94306,34430],{},[35,94308,446],{},[39,94310,126],{"stretchy":119},[39,94312,126],{"stretchy":119},[39,94314,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,94316,1245],{},[35,94318,62381],{"mathvariant":454},[39,94320,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[32,94322,94323,94325],{},[35,94324,2853],{"mathvariant":138},[35,94326,1216],{"mathvariant":138},[39,94328,120],{"stretchy":119},[35,94330,11],{},[39,94332,120],{"stretchy":119},[39,94334,3268],{},[39,94336,34430],{},[35,94338,446],{},[39,94340,126],{"stretchy":119},[35,94342,1251],{"mathvariant":138},[35,94344,94276],{},[39,94346,120],{"stretchy":119},[39,94348,3268],{},[39,94350,34430],{},[35,94352,446],{},[39,94354,126],{"stretchy":119},[39,94356,126],{"stretchy":119},[39,94358,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,94360,335],{"separator":53},[45,94362,94363],{"encoding":47},"\\mathbb E[-\\log q(Y\\mid X)]\n=\\mathbb E\\big[\\mathbb H(p(\\cdot\\mid X))\\big]\n+\\mathbb E\\big[\\mathrm{KL}(p(\\cdot\\mid X)\\|q(\\cdot\\mid X))\\big],",[15,94365,94367,94411,94433,94472,94499,94541,94574],{"className":94366,"ariaHidden":53},[52],[15,94368,94370,94373,94376,94379,94382,94385,94390,94393,94396,94399,94402,94405,94408],{"className":94369},[57],[15,94371],{"className":94372,"style":154},[61],[15,94374,62381],{"className":94375},[66,517],[15,94377,1194],{"className":94378},[162],[15,94380,6602],{"className":94381},[66],[15,94383],{"className":94384,"style":697},[71],[15,94386,27283,94388],{"className":94387},[1339],[15,94389,1344],{"style":1343},[15,94391],{"className":94392,"style":697},[71],[15,94394,94276],{"className":94395,"style":1515},[66,67],[15,94397,120],{"className":94398},[162],[15,94400,61885],{"className":94401,"style":72},[66,67],[15,94403],{"className":94404,"style":173},[71],[15,94406,34430],{"className":94407},[177],[15,94409],{"className":94410,"style":173},[71],[15,94412,94414,94417,94420,94424,94427,94430],{"className":94413},[57],[15,94415],{"className":94416,"style":154},[61],[15,94418,446],{"className":94419,"style":16497},[66,67],[15,94421,94423],{"className":94422},[169],")]",[15,94425],{"className":94426,"style":173},[71],[15,94428,129],{"className":94429},[177],[15,94431],{"className":94432,"style":173},[71],[15,94434,94436,94439,94442,94448,94451,94454,94457,94460,94463,94466,94469],{"className":94435},[57],[15,94437],{"className":94438,"style":4400},[61],[15,94440,62381],{"className":94441},[66,517],[15,94443,94445],{"className":94444},[66],[15,94446,1194],{"className":94447},[1422,1508],[15,94449,1061],{"className":94450},[66,517],[15,94452,120],{"className":94453},[162],[15,94455,11],{"className":94456},[66,67],[15,94458,120],{"className":94459},[162],[15,94461,3268],{"className":94462},[66],[15,94464],{"className":94465,"style":173},[71],[15,94467,34430],{"className":94468},[177],[15,94470],{"className":94471,"style":173},[71],[15,94473,94475,94478,94481,94484,94490,94493,94496],{"className":94474},[57],[15,94476],{"className":94477,"style":4400},[61],[15,94479,446],{"className":94480,"style":16497},[66,67],[15,94482,18771],{"className":94483},[169],[15,94485,94487],{"className":94486},[66],[15,94488,1266],{"className":94489},[1422,1508],[15,94491],{"className":94492,"style":72},[71],[15,94494,1245],{"className":94495},[76],[15,94497],{"className":94498,"style":72},[71],[15,94500,94502,94505,94508,94514,94520,94523,94526,94529,94532,94535,94538],{"className":94501},[57],[15,94503],{"className":94504,"style":4400},[61],[15,94506,62381],{"className":94507},[66,517],[15,94509,94511],{"className":94510},[66],[15,94512,1194],{"className":94513},[1422,1508],[15,94515,94517],{"className":94516},[66],[15,94518,88457],{"className":94519},[66,4429],[15,94521,120],{"className":94522},[162],[15,94524,11],{"className":94525},[66,67],[15,94527,120],{"className":94528},[162],[15,94530,3268],{"className":94531},[66],[15,94533],{"className":94534,"style":173},[71],[15,94536,34430],{"className":94537},[177],[15,94539],{"className":94540,"style":173},[71],[15,94542,94544,94547,94550,94553,94556,94559,94562,94565,94568,94571],{"className":94543},[57],[15,94545],{"className":94546,"style":154},[61],[15,94548,446],{"className":94549,"style":16497},[66,67],[15,94551,126],{"className":94552},[169],[15,94554,1251],{"className":94555},[66],[15,94557,94276],{"className":94558,"style":1515},[66,67],[15,94560,120],{"className":94561},[162],[15,94563,3268],{"className":94564},[66],[15,94566],{"className":94567,"style":173},[71],[15,94569,34430],{"className":94570},[177],[15,94572],{"className":94573,"style":173},[71],[15,94575,94577,94580,94583,94586,94592],{"className":94576},[57],[15,94578],{"className":94579,"style":4400},[61],[15,94581,446],{"className":94582,"style":16497},[66,67],[15,94584,18771],{"className":94585},[169],[15,94587,94589],{"className":94588},[66],[15,94590,1266],{"className":94591},[1422,1508],[15,94593,335],{"className":94594},[422],[11,94596,94597,94598,94626,94627,94715],{},"右边第一项与 ",[15,94599,94601,94614],{"className":94600,"translate":19},[18],[15,94602,94604],{"className":94603},[23],[25,94605,94606],{"xmlns":27},[29,94607,94608,94612],{},[32,94609,94610],{},[35,94611,94276],{},[45,94613,94276],{"encoding":47},[15,94615,94617],{"className":94616,"ariaHidden":53},[52],[15,94618,94620,94623],{"className":94619},[57],[15,94621],{"className":94622,"style":471},[61],[15,94624,94276],{"className":94625,"style":1515},[66,67]," 无关，第二项在 ",[15,94628,94630,94658],{"className":94629,"translate":19},[18],[15,94631,94633],{"className":94632},[23],[25,94634,94635],{"xmlns":27},[29,94636,94637,94655],{},[32,94638,94639,94641,94643,94645,94647,94649,94651,94653],{},[35,94640,94276],{},[39,94642,129],{},[35,94644,11],{},[39,94646,120],{"stretchy":119},[39,94648,3268],{},[39,94650,34430],{},[35,94652,446],{},[39,94654,126],{"stretchy":119},[45,94656,94657],{"encoding":47},"q=p(\\cdot\\mid X)",[15,94659,94661,94679,94703],{"className":94660,"ariaHidden":53},[52],[15,94662,94664,94667,94670,94673,94676],{"className":94663},[57],[15,94665],{"className":94666,"style":471},[61],[15,94668,94276],{"className":94669,"style":1515},[66,67],[15,94671],{"className":94672,"style":173},[71],[15,94674,129],{"className":94675},[177],[15,94677],{"className":94678,"style":173},[71],[15,94680,94682,94685,94688,94691,94694,94697,94700],{"className":94681},[57],[15,94683],{"className":94684,"style":154},[61],[15,94686,11],{"className":94687},[66,67],[15,94689,120],{"className":94690},[162],[15,94692,3268],{"className":94693},[66],[15,94695],{"className":94696,"style":173},[71],[15,94698,34430],{"className":94699},[177],[15,94701],{"className":94702,"style":173},[71],[15,94704,94706,94709,94712],{"className":94705},[57],[15,94707],{"className":94708,"style":154},[61],[15,94710,446],{"className":94711,"style":16497},[66,67],[15,94713,126],{"className":94714},[169]," 时为零。所以，在总体意义上，用交叉熵训练模型是在要求它给出校准的概率；用 0-1 损失训练，只要求它给出正确的分类。两者目标不同，不能互相替代。",[1904,94717,94719],{"id":94718},"_52-正则化就是先验","5.2 正则化就是先验",[11,94721,94722,94723,94993,94994,95053],{},"设参数化模型 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x_i,\\theta)",[15,94784,94786,94810,94831,94935],{"className":94785,"ariaHidden":53},[52],[15,94787,94789,94792,94795,94798,94801,94804,94807],{"className":94788},[57],[15,94790],{"className":94791,"style":154},[61],[15,94793,11],{"className":94794},[66,67],[15,94796,120],{"className":94797},[162],[15,94799,32458],{"className":94800,"style":23025},[66,67],[15,94802],{"className":94803,"style":173},[71],[15,94805,34430],{"className":94806},[177],[15,94808],{"className":94809,"style":173},[71],[15,94811,94813,94816,94819,94822,94825,94828],{"className":94812},[57],[15,94814],{"className":94815,"style":154},[61],[15,94817,61618],{"className":94818,"style":23025},[66,67],[15,94820,126],{"className":94821},[169],[15,94823],{"className":94824,"style":173},[71],[15,94826,129],{"className":94827},[177],[15,94829],{"className":94830,"style":173},[71],[15,94832,94834,94837,94877,94880,94883,94886,94926,94929,94932],{"className":94833},[57],[15,94835],{"className":94836,"style":16866},[61],[15,94838,94840,94843],{"className":94839},[1339],[15,94841,94753],{"className":94842,"style":13702},[1339,1470,13701],[15,94844,94846],{"className":94845},[198],[15,94847,94849,94869],{"className":94848},[202,1090],[15,94850,94852,94866],{"className":94851},[206],[15,94853,94855],{"className":94854,"style":16885},[210],[15,94856,94857,94860],{"style":13718},[15,94858],{"className":94859,"style":218},[217],[15,94861,94863],{"className":94862},[222,223,224,225],[15,94864,1205],{"className":94865},[66,67,225],[15,94867,1114],{"className":94868},[1113],[15,94870,94872],{"className":94871},[206],[15,94873,94875],{"className":94874,"style":16906},[210],[15,94876],{},[15,94878],{"className":94879,"style":697},[71],[15,94881,11],{"className":94882},[66,67],[15,94884,120],{"className":94885},[162],[15,94887,94889,94892],{"className":94888},[66],[15,94890,1224],{"className":94891,"style":1515},[66,67],[15,94893,94895],{"className":94894},[198],[15,94896,94898,94918],{"className":94897},[202,1090],[15,94899,94901,94915],{"className":94900},[206],[15,94902,94904],{"className":94903,"style":1528},[210],[15,94905,94906,94909],{"style":1531},[15,94907],{"className":94908,"style":218},[217],[15,94910,94912],{"className":94911},[222,223,224,225],[15,94913,1205],{"className":94914},[66,67,225],[15,94916,1114],{"className":94917},[1113],[15,94919,94921],{"className":94920},[206],[15,94922,94924],{"className":94923,"style":1121},[210],[15,94925],{},[15,94927],{"className":94928,"style":173},[71],[15,94930,34430],{"className":94931},[177],[15,94933],{"className":94934,"style":173},[71],[15,94936,94938,94941,94981,94984,94987,94990],{"className":94937},[57],[15,94939],{"className":94940,"style":154},[61],[15,94942,94944,94947],{"className":94943},[66],[15,94945,123],{"className":94946},[66,67],[15,94948,94950],{"className":94949},[198],[15,94951,94953,94973],{"className":94952},[202,1090],[15,94954,94956,94970],{"className":94955},[206],[15,94957,94959],{"className":94958,"style":1528},[210],[15,94960,94961,94964],{"style":1584},[15,94962],{"className":94963,"style":218},[217],[15,94965,94967],{"className":94966},[222,223,224,225],[15,94968,1205],{"className":94969},[66,67,225],[15,94971,1114],{"className":94972},[1113],[15,94974,94976],{"className":94975},[206],[15,94977,94979],{"className":94978,"style":1121},[210],[15,94980],{},[15,94982,335],{"className":94983},[422],[15,94985],{"className":94986,"style":697},[71],[15,94988,61618],{"className":94989,"style":23025},[66,67],[15,94991,126],{"className":94992},[169],"，噪声为高斯，方差 ",[15,94995,94997,95015],{"className":94996,"translate":19},[18],[15,94998,95000],{"className":94999},[23],[25,95001,95002],{"xmlns":27},[29,95003,95004,95012],{},[32,95005,95006],{},[131,95007,95008,95010],{},[35,95009,6634],{},[1209,95011,1263],{},[45,95013,95014],{"encoding":47},"\\sigma^2",[15,95016,95018],{"className":95017,"ariaHidden":53},[52],[15,95019,95021,95024],{"className":95020},[57],[15,95022],{"className":95023,"style":6740},[61],[15,95025,95027,95030],{"className":95026},[66],[15,95028,6634],{"className":95029,"style":1515},[66,67],[15,95031,95033],{"className":95032},[198],[15,95034,95036],{"className":95035},[202],[15,95037,95039],{"className":95038},[206],[15,95040,95042],{"className":95041,"style":6740},[210],[15,95043,95044,95047],{"style":213},[15,95045],{"className":95046,"style":218},[217],[15,95048,95050],{"className":95049},[222,223,224,225],[15,95051,1263],{"className":95052},[66,225],"。则负对数似然正比于平方误差：",[15,95055,95057],{"className":95056,"translate":19},[294],[15,95058,95060,95150],{"className":95059,"translate":19},[18],[15,95061,95063],{"className":95062},[23],[25,95064,95065],{"xmlns":27,"display":303},[29,95066,95067,95147],{},[32,95068,95069,95071,95073,95075,95077,95079,95081,95083,95085,95087,95089,95103,95109,95111,95117,95119,95125,95127,95133,95135,95141,95143,95145],{},[39,95070,6602],{},[35,95072,27073],{},[39,95074,1170],{},[35,95076,11],{},[39,95078,120],{"stretchy":119},[35,95080,32458],{},[39,95082,34430],{},[35,95084,61618],{},[39,95086,126],{"stretchy":119},[39,95088,129],{},[6467,95090,95091,95093],{},[1209,95092,1211],{},[32,95094,95095,95097],{},[1209,95096,1263],{},[131,95098,95099,95101],{},[35,95100,6634],{},[1209,95102,1263],{},[1172,95104,95105,95107],{},[39,95106,1200],{},[35,95108,1205],{},[39,95110,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[1057,95112,95113,95115],{},[35,95114,1224],{},[35,95116,1205],{},[39,95118,6602],{},[1057,95120,95121,95123],{},[35,95122,116],{},[35,95124,61618],{},[39,95126,120],{"stretchy":119},[1057,95128,95129,95131],{},[35,95130,123],{},[35,95132,1205],{},[39,95134,126],{"stretchy":119},[131,95136,95137,95139],{},[39,95138,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,95140,1263],{},[39,95142,1245],{},[1274,95144,92943],{},[35,95146,3573],{"mathvariant":138},[45,95148,95149],{"encoding":47},"-\\log p(D\\mid\\theta)\n=\\frac{1}{2\\sigma^2}\\sum_i\\big(y_i-f_\\theta(x_i)\\big)^2+\\text{常数}.",[15,95151,95153,95191,95212,95415,95548],{"className":95152,"ariaHidden":53},[52],[15,95154,95156,95159,95162,95165,95170,95173,95176,95179,95182,95185,95188],{"className":95155},[57],[15,95157],{"className":95158,"style":154},[61],[15,95160,6602],{"className":95161},[66],[15,95163],{"className":95164,"style":697},[71],[15,95166,27283,95168],{"className":95167},[1339],[15,95169,1344],{"style":1343},[15,95171],{"className":95172,"style":697},[71],[15,95174,11],{"className":95175},[66,67],[15,95177,120],{"className":95178},[162],[15,95180,32458],{"className":95181,"style":23025},[66,67],[15,95183],{"className":95184,"style":173},[71],[15,95186,34430],{"className":95187},[177],[15,95189],{"className":95190,"style":173},[71],[15,95192,95194,95197,95200,95203,95206,95209],{"className":95193},[57],[15,95195],{"className":95196,"style":154},[61],[15,95198,61618],{"className":95199,"style":23025},[66,67],[15,95201,126],{"className":95202},[169],[15,95204],{"className":95205,"style":173},[71],[15,95207,129],{"className":95208},[177],[15,95210],{"className":95211,"style":173},[71],[15,95213,95215,95219,95310,95313,95357,95360,95366,95406,95409,95412],{"className":95214},[57],[15,95216],{"className":95217,"style":95218},[61],"height:2.5991em;vertical-align:-1.2777em;",[15,95220,95222,95225,95307],{"className":95221},[66],[15,95223],{"className":95224},[162,23417],[15,95226,95228],{"className":95227},[6467],[15,95229,95231,95299],{"className":95230},[202,1090],[15,95232,95234,95296],{"className":95233},[206],[15,95235,95237,95277,95285],{"className":95236,"style":23430},[210],[15,95238,95239,95242],{"style":23433},[15,95240],{"className":95241,"style":1368},[217],[15,95243,95245,95248],{"className":95244},[66],[15,95246,1263],{"className":95247},[66],[15,95249,95251,95254],{"className":95250},[66],[15,95252,6634],{"className":95253,"style":1515},[66,67],[15,95255,95257],{"className":95256},[198],[15,95258,95260],{"className":95259},[202],[15,95261,95263],{"className":95262},[206],[15,95264,95266],{"className":95265,"style":46252},[210],[15,95267,95268,95271],{"style":46255},[15,95269],{"className":95270,"style":218},[217],[15,95272,95274],{"className":95273},[222,223,224,225],[15,95275,1263],{"className":95276},[66,225],[15,95278,95279,95282],{"style":23445},[15,95280],{"className":95281,"style":1368},[217],[15,95283],{"className":95284,"style":4440},[23452],[15,95286,95287,95290],{"style":23455},[15,95288],{"className":95289,"style":1368},[217],[15,95291,95293],{"className":95292},[66],[15,95294,1211],{"className":95295},[66],[15,95297,1114],{"className":95298},[1113],[15,95300,95302],{"className":95301},[206],[15,95303,95305],{"className":95304,"style":23474},[210],[15,95306],{},[15,95308],{"className":95309},[169,23417],[15,95311],{"className":95312,"style":697},[71],[15,95314,95316],{"className":95315},[1339,1351],[15,95317,95319,95349],{"className":95318},[202,1090],[15,95320,95322,95346],{"className":95321},[206],[15,95323,95325,95336],{"className":95324,"style":4690},[210],[15,95326,95327,95330],{"style":1439},[15,95328],{"className":95329,"style":1443},[217],[15,95331,95333],{"className":95332},[222,223,224,225],[15,95334,1205],{"className":95335},[66,67,225],[15,95337,95338,95341],{"style":1461},[15,95339],{"className":95340,"style":1443},[217],[15,95342,95343],{},[15,95344,1200],{"className":95345},[1339,1470,1471],[15,95347,1114],{"className":95348},[1113],[15,95350,95352],{"className":95351},[206],[15,95353,95355],{"className":95354,"style":1493},[210],[15,95356],{},[15,95358],{"className":95359,"style":697},[71],[15,95361,95363],{"className":95362},[66],[15,95364,120],{"className":95365},[1422,1508],[15,95367,95369,95372],{"className":95368},[66],[15,95370,1224],{"className":95371,"style":1515},[66,67],[15,95373,95375],{"className":95374},[198],[15,95376,95378,95398],{"className":95377},[202,1090],[15,95379,95381,95395],{"className":95380},[206],[15,95382,95384],{"className":95383,"style":1528},[210],[15,95385,95386,95389],{"style":1531},[15,95387],{"className":95388,"style":218},[217],[15,95390,95392],{"className":95391},[222,223,224,225],[15,95393,1205],{"className":95394},[66,67,225],[15,95396,1114],{"className":95397},[1113],[15,95399,95401],{"className":95400},[206],[15,95402,95404],{"className":95403,"style":1121},[210],[15,95405],{},[15,95407],{"className":95408,"style":72},[71],[15,95410,6602],{"className":95411},[76],[15,95413],{"className":95414,"style":72},[71],[15,95416,95418,95421,95461,95464,95504,95507,95539,95542,95545],{"className":95417},[57],[15,95419],{"className":95420,"style":30185},[61],[15,95422,95424,95427],{"className":95423},[66],[15,95425,116],{"className":95426,"style":158},[66,67],[15,95428,95430],{"className":95429},[198],[15,95431,95433,95453],{"className":95432},[202,1090],[15,95434,95436,95450],{"className":95435},[206],[15,95437,95439],{"className":95438,"style":1097},[210],[15,95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Laplace 分布 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",[15,97033,97035,97057],{"className":97034,"translate":19},[18],[15,97036,97038],{"className":97037},[23],[25,97039,97040],{"xmlns":27},[29,97041,97042,97054],{},[32,97043,97044,97046,97048],{},[35,97045,1251],{"mathvariant":138},[35,97047,61618],{},[1057,97049,97050,97052],{},[35,97051,1251],{"mathvariant":138},[1209,97053,1211],{},[45,97055,97056],{"encoding":47},"\\|\\theta\\|_1",[15,97058,97060],{"className":97059,"ariaHidden":53},[52],[15,97061,97063,97066,97069,97072],{"className":97062},[57],[15,97064],{"className":97065,"style":154},[61],[15,97067,1251],{"className":97068},[66],[15,97070,61618],{"className":97071,"style":23025},[66,67],[15,97073,97075,97078],{"className":97074},[66],[15,97076,1251],{"className":97077},[66],[15,97079,97081],{"className":97080},[198],[15,97082,97084,97104],{"className":97083},[202,1090],[15,97085,97087,97101],{"className":97086},[206],[15,97088,97090],{"className":97089,"style":2139},[210],[15,97091,97092,97095],{"style":1584},[15,97093],{"className":97094,"style":218},[217],[15,97096,97098],{"className":97097},[222,223,224,225],[15,97099,1211],{"className":97100},[66,225],[15,97102,1114],{"className":97103},[1113],[15,97105,97107],{"className":97106},[206],[15,97108,97110],{"className":97109,"style":1121},[210],[15,97111],{},"，MAP 就变成 Lasso。",[6152,97114,97115,97127],{},[6155,97116,97117],{},[6158,97118,97119,97121,97124],{},[6161,97120,65218],{},[6161,97122,97123],{},"负对数先验",[6161,97125,97126],{},"等价惩罚",[6171,97128,97129,97139,97360,97512],{},[6158,97130,97131,97134,97136],{},[6176,97132,97133],{},"均匀先验",[6176,97135,92943],{},[6176,97137,97138],{},"无（MLE）",[6158,97140,97141,97231,97357],{},[6176,97142,97143,97144],{},"高斯 ",[15,97145,97147,97175],{"className":97146,"translate":19},[18],[15,97148,97150],{"className":97149},[23],[25,97151,97152],{"xmlns":27},[29,97153,97154,97172],{},[32,97155,97156,97158,97160,97162,97164,97170],{},[35,97157,28535],{"mathvariant":445},[39,97159,120],{"stretchy":119},[1209,97161,1804],{},[39,97163,335],{"separator":53},[131,97165,97166,97168],{},[35,97167,95594],{},[1209,97169,1263],{},[39,97171,126],{"stretchy":119},[45,97173,97174],{"encoding":47},"\\mathcal N(0,\\tau^2)",[15,97176,97178],{"className":97177,"ariaHidden":53},[52],[15,97179,97181,97184,97187,97190,97193,97196,97199,97228],{"className":97180},[57],[15,97182],{"className":97183,"style":9154},[61],[15,97185,28535],{"className":97186,"style":83515},[66,494],[15,97188,120],{"className":97189},[162],[15,97191,1804],{"className":97192},[66],[15,97194,335],{"className":97195},[422],[15,97197],{"className":97198,"style":697},[71],[15,97200,97202,97205],{"className":97201},[66],[15,97203,95594],{"className":97204,"style":95652},[66,67],[15,97206,97208],{"className":97207},[198],[15,97209,97211],{"className":97210},[202],[15,97212,97214],{"className":97213},[206],[15,97215,97217],{"className":97216,"style":6740},[210],[15,97218,97219,97222],{"style":213},[15,97220],{"className":97221,"style":218},[217],[15,97223,97225],{"className":97224},[222,223,224,225],[15,97226,1263],{"className":97227},[66,225],[15,97229,126],{"className":97230},[169],[6176,97232,97233],{},[15,97234,97236,97272],{"className":97235,"translate":19},[18],[15,97237,97239],{"className":97238},[23],[25,97240,97241],{"xmlns":27},[29,97242,97243,97269],{},[32,97244,97245,97247,97249,97255,97257,97259,97261,97267],{},[35,97246,1251],{"mathvariant":138},[35,97248,61618],{},[131,97250,97251,97253],{},[35,97252,1251],{"mathvariant":138},[1209,97254,1263],{},[35,97256,6625],{"mathvariant":138},[39,97258,120],{"stretchy":119},[1209,97260,1263],{},[131,97262,97263,97265],{},[35,97264,95594],{},[1209,97266,1263],{},[39,97268,126],{"stretchy":119},[45,97270,97271],{"encoding":47},"\\|\\theta\\|^2\u002F(2\\tau^2)",[15,97273,97275],{"className":97274,"ariaHidden":53},[52],[15,97276,97278,97281,97284,97287,97316,97319,97322,97325,97354],{"className":97277},[57],[15,97279],{"className":97280,"style":9154},[61],[15,97282,1251],{"className":97283},[66],[15,97285,61618],{"className":97286,"style":23025},[66,67],[15,97288,97290,97293],{"className":97289},[66],[15,97291,1251],{"className":97292},[66],[15,97294,97296],{"className":97295},[198],[15,97297,97299],{"className":97298},[202],[15,97300,97302],{"className":97301},[206],[15,97303,97305],{"className":97304,"style":6740},[210],[15,97306,97307,97310],{"style":213},[15,97308],{"className":97309,"style":218},[217],[15,97311,97313],{"className":97312},[222,223,224,225],[15,97314,1263],{"className":97315},[66,225],[15,97317,6625],{"className":97318},[66],[15,97320,120],{"className":97321},[162],[15,97323,1263],{"className":97324},[66],[15,97326,97328,97331],{"className":97327},[66],[15,97329,95594],{"className":97330,"style":95652},[66,67],[15,97332,97334],{"className":97333},[198],[15,97335,97337],{"className":97336},[202],[15,97338,97340],{"className":97339},[206],[15,97341,97343],{"className":97342,"style":6740},[210],[15,97344,97345,97348],{"style":213},[15,97346],{"className":97347,"style":218},[217],[15,97349,97351],{"className":97350},[222,223,224,225],[15,97352,1263],{"className":97353},[66,225],[15,97355,126],{"className":97356},[169],[6176,97358,97359],{},"岭回归、权重衰减",[6158,97361,97362,97417,97509],{},[6176,97363,97364,97365],{},"Laplace ",[15,97366,97368,97390],{"className":97367,"translate":19},[18],[15,97369,97371],{"className":97370},[23],[25,97372,97373],{"xmlns":27},[29,97374,97375,97387],{},[32,97376,97377,97379,97381,97383,97385],{},[39,97378,120],{"stretchy":119},[1209,97380,1804],{},[39,97382,335],{"separator":53},[35,97384,4587],{},[39,97386,126],{"stretchy":119},[45,97388,97389],{"encoding":47},"(0,b)",[15,97391,97393],{"className":97392,"ariaHidden":53},[52],[15,97394,97396,97399,97402,97405,97408,97411,97414],{"className":97395},[57],[15,97397],{"className":97398,"style":154},[61],[15,97400,120],{"className":97401},[162],[15,97403,1804],{"className":97404},[66],[15,97406,335],{"className":97407},[422],[15,97409],{"className":97410,"style":697},[71],[15,97412,4587],{"className":97413},[66,67],[15,97415,126],{"className":97416},[169],[6176,97418,97419],{},[15,97420,97422,97448],{"className":97421,"translate":19},[18],[15,97423,97425],{"className":97424},[23],[25,97426,97427],{"xmlns":27},[29,97428,97429,97445],{},[32,97430,97431,97433,97435,97441,97443],{},[35,97432,1251],{"mathvariant":138},[35,97434,61618],{},[1057,97436,97437,97439],{},[35,97438,1251],{"mathvariant":138},[1209,97440,1211],{},[35,97442,6625],{"mathvariant":138},[35,97444,4587],{},[45,97446,97447],{"encoding":47},"\\|\\theta\\|_1\u002Fb",[15,97449,97451],{"className":97450,"ariaHidden":53},[52],[15,97452,97454,97457,97460,97463,97503,97506],{"className":97453},[57],[15,97455],{"className":97456,"style":154},[61],[15,97458,1251],{"className":97459},[66],[15,97461,61618],{"className":97462,"style":23025},[66,67],[15,97464,97466,97469],{"className":97465},[66],[15,97467,1251],{"className":97468},[66],[15,97470,97472],{"className":97471},[198],[15,97473,97475,97495],{"className":97474},[202,1090],[15,97476,97478,97492],{"className":97477},[206],[15,97479,97481],{"className":97480,"style":2139},[210],[15,97482,97483,97486],{"style":1584},[15,97484],{"className":97485,"style":218},[217],[15,97487,97489],{"className":97488},[222,223,224,225],[15,97490,1211],{"className":97491},[66,225],[15,97493,1114],{"className":97494},[1113],[15,97496,97498],{"className":97497},[206],[15,97499,97501],{"className":97500,"style":1121},[210],[15,97502],{},[15,97504,6625],{"className":97505},[66],[15,97507,4587],{"className":97508},[66,67],[6176,97510,97511],{},"Lasso",[6158,97513,97514,97517,97520],{},[6176,97515,97516],{},"稀疏诱导先验",[6176,97518,97519],{},"非凸惩罚",[6176,97521,97522],{},"特征选择",[11,97524,97525],{},"这张表解释了一个常见的困惑：为什么正则化看起来像是「人为加上去的技巧」，却总是有效。它其实是在补充一个先验：参数不应该太大，或者不应该同时有很多非零分量。正则项不是外来的约束，而是模型假设的一部分，只是以惩罚的形式出现。",[11,97527,97528],{},"需要谨慎的是，深度网络里的权重衰减与高斯先验的对应只在全批量 MAP 的意义上成立。SGD 的噪声、批量大小、参数化方式（例如 BatchNorm 的缩放不变性）都会改变有效的先验。把它当成一个有用的类比，而不是一个严格的等价。",[1904,97530,97532],{"id":97531},"_53-似然比roc-与代价","5.3 似然比、ROC 与代价",[11,97534,97535],{},"二分类的贝叶斯规则可以写成似然比检验：",[15,97537,97539],{"className":97538,"translate":19},[294],[15,97540,97542,97648],{"className":97541,"translate":19},[18],[15,97543,97545],{"className":97544},[23],[25,97546,97547],{"xmlns":27,"display":303},[29,97548,97549,97645],{},[32,97550,97551,97553,97555,97557,97559,97561,97563,97565,97567,97569,97599,97601,97643],{},[35,97552,3885],{},[39,97554,120],{"stretchy":119},[35,97556,123],{},[39,97558,126],{"stretchy":119},[39,97560,129],{},[1209,97562,1211],{},[71,97564],{"width":9791},[39,97566,75704],{},[71,97568],{"width":9791},[6467,97570,97571,97585],{},[32,97572,97573,97579,97581,97583],{},[1057,97574,97575,97577],{},[35,97576,11],{},[1209,97578,1211],{},[39,97580,120],{"stretchy":119},[35,97582,123],{},[39,97584,126],{"stretchy":119},[32,97586,97587,97593,97595,97597],{},[1057,97588,97589,97591],{},[35,97590,11],{},[1209,97592,1804],{},[39,97594,120],{"stretchy":119},[35,97596,123],{},[39,97598,126],{"stretchy":119},[39,97600,1801],{},[6467,97602,97603,97623],{},[32,97604,97605,97611,97613],{},[1057,97606,97607,97609],{},[35,97608,33494],{},[1209,97610,1804],{},[1274,97612,2462],{},[1057,97614,97615,97617],{},[35,97616,2256],{},[32,97618,97619,97621],{},[35,97620,74045],{"mathvariant":138},[35,97622,28399],{"mathvariant":138},[32,97624,97625,97631,97633],{},[1057,97626,97627,97629],{},[35,97628,33494],{},[1209,97630,1211],{},[1274,97632,2462],{},[1057,97634,97635,97637],{},[35,97636,2256],{},[32,97638,97639,97641],{},[35,97640,74045],{"mathvariant":138},[35,97642,28535],{"mathvariant":138},[35,97644,3573],{"mathvariant":138},[45,97646,97647],{"encoding":47},"\\delta(x)=1\n\\quad\\Longleftrightarrow\\quad\n\\frac{p_1(x)}{p_0(x)}\n\\ge\n\\frac{\\pi_0\\,c_{\\mathrm{FP}}}{\\pi_1\\,c_{\\mathrm{FN}}}.",[15,97649,97651,97678,97703,97872],{"className":97650,"ariaHidden":53},[52],[15,97652,97654,97657,97660,97663,97666,97669,97672,97675],{"className":97653},[57],[15,97655],{"className":97656,"style":154},[61],[15,97658,3885],{"className":97659,"style":3946},[66,67],[15,97661,120],{"className":97662},[162],[15,97664,123],{"className":97665},[66,67],[15,97667,126],{"className":97668},[169],[15,97670],{"className":97671,"style":173},[71],[15,97673,129],{"className":97674},[177],[15,97676],{"className":97677,"style":173},[71],[15,97679,97681,97685,97688,97691,97694,97697,97700],{"className":97680},[57],[15,97682],{"className":97683,"style":97684},[61],"height:0.6684em;vertical-align:-0.024em;",[15,97686,1211],{"className":97687},[66],[15,97689],{"className":97690,"style":10049},[71],[15,97692],{"className":97693,"style":173},[71],[15,97695,75704],{"className":97696},[177],[15,97698],{"className":97699,"style":10049},[71],[15,97701],{"className":97702,"style":173},[71],[15,97704,97706,97709,97863,97866,97869],{"className":97705},[57],[15,97707],{"className":97708,"style":41192},[61],[15,97710,97712,97715,97860],{"className":97711},[66],[15,97713],{"className":97714},[162,23417],[15,97716,97718],{"className":97717},[6467],[15,97719,97721,97852],{"className":97720},[202,1090],[15,97722,97724,97849],{"className":97723},[206],[15,97725,97727,97784,97792],{"className":97726,"style":41211},[210],[15,97728,97729,97732],{"style":23433},[15,97730],{"className":97731,"style":1368},[217],[15,97733,97735,97775,97778,97781],{"className":97734},[66],[15,97736,97738,97741],{"className":97737},[66],[15,97739,11],{"className":97740},[66,67],[15,97742,97744],{"className":97743},[198],[15,97745,97747,97767],{"className":97746},[202,1090],[15,97748,97750,97764],{"className":97749},[206],[15,97751,97753],{"className":97752,"style":2139},[210],[15,97754,97755,97758],{"style":1584},[15,97756],{"className":97757,"style":218},[217],[15,97759,97761],{"className":97760},[222,223,224,225],[15,97762,1804],{"className":97763},[66,225],[15,97765,1114],{"className":97766},[1113],[15,97768,97770],{"className":97769},[206],[15,97771,97773],{"className":97772,"style":1121},[210],[15,97774],{},[15,97776,120],{"className":97777},[162],[15,97779,123],{"className":97780},[66,67],[15,97782,126],{"className":97783},[169],[15,97785,97786,97789],{"style":23445},[15,97787],{"className":97788,"style":1368},[217],[15,97790],{"className":97791,"style":4440},[23452],[15,97793,97794,97797],{"style":23455},[15,97795],{"className":97796,"style":1368},[217],[15,97798,97800,97840,97843,97846],{"className":97799},[66],[15,97801,97803,97806],{"className":97802},[66],[15,97804,11],{"className":97805},[66,67],[15,97807,97809],{"className":97808},[198],[15,97810,97812,97832],{"className":97811},[202,1090],[15,97813,97815,97829],{"className":97814},[206],[15,97816,97818],{"className":97817,"style":2139},[210],[15,97819,97820,97823],{"style":1584},[15,97821],{"className":97822,"style":218},[217],[15,97824,97826],{"className":97825},[222,223,224,225],[15,97827,1211],{"className":97828},[66,225],[15,97830,1114],{"className":97831},[1113],[15,97833,97835],{"className":97834},[206],[15,97836,97838],{"className":97837,"style":1121},[210],[15,97839],{},[15,97841,120],{"className":97842},[162],[15,97844,123],{"className":97845},[66,67],[15,97847,126],{"className":97848},[169],[15,97850,1114],{"className":97851},[1113],[15,97853,97855],{"className":97854},[206],[15,97856,97858],{"className":97857,"style":41282},[210],[15,97859],{},[15,97861],{"className":97862},[169,23417],[15,97864],{"className":97865,"style":173},[71],[15,97867,1801],{"className":97868},[177],[15,97870],{"className":97871,"style":173},[71],[15,97873,97875,97878,98112],{"className":97874},[57],[15,97876],{"className":97877,"style":52422},[61],[15,97879,97881,97884,98109],{"className":97880},[66],[15,97882],{"className":97883},[162,23417],[15,97885,97887],{"className":97886},[6467],[15,97888,97890,98101],{"className":97889},[202,1090],[15,97891,97893,98098],{"className":97892},[206],[15,97894,97896,97993,98001],{"className":97895,"style":39133},[210],[15,97897,97898,97901],{"style":23433},[15,97899],{"className":97900,"style":1368},[217],[15,97902,97904,97944,97947],{"className":97903},[66],[15,97905,97907,97910],{"className":97906},[66],[15,97908,33494],{"className":97909,"style":1515},[66,67],[15,97911,97913],{"className":97912},[198],[15,97914,97916,97936],{"className":97915},[202,1090],[15,97917,97919,97933],{"className":97918},[206],[15,97920,97922],{"className":97921,"style":2139},[210],[15,97923,97924,97927],{"style":1531},[15,97925],{"className":97926,"style":218},[217],[15,97928,97930],{"className":97929},[222,223,224,225],[15,97931,1211],{"className":97932},[66,225],[15,97934,1114],{"className":97935},[1113],[15,97937,97939],{"className":97938},[206],[15,97940,97942],{"className":97941,"style":1121},[210],[15,97943],{},[15,97945],{"className":97946,"style":697},[71],[15,97948,97950,97953],{"className":97949},[66],[15,97951,2256],{"className":97952},[66,67],[15,97954,97956],{"className":97955},[198],[15,97957,97959,97985],{"className":97958},[202,1090],[15,97960,97962,97982],{"className":97961},[206],[15,97963,97965],{"className":97964,"style":3638},[210],[15,97966,97967,97970],{"style":1584},[15,97968],{"className":97969,"style":218},[217],[15,97971,97973],{"className":97972},[222,223,224,225],[15,97974,97976],{"className":97975},[66,225],[15,97977,97979],{"className":97978},[66,225],[15,97980,74177],{"className":97981},[66,4429,225],[15,97983,1114],{"className":97984},[1113],[15,97986,97988],{"className":97987},[206],[15,97989,97991],{"className":97990,"style":1121},[210],[15,97992],{},[15,97994,97995,97998],{"style":23445},[15,97996],{"className":97997,"style":1368},[217],[15,97999],{"className":98000,"style":4440},[23452],[15,98002,98003,98006],{"style":23455},[15,98004],{"className":98005,"style":1368},[217],[15,98007,98009,98049,98052],{"className":98008},[66],[15,98010,98012,98015],{"className":98011},[66],[15,98013,33494],{"className":98014,"style":1515},[66,67],[15,98016,98018],{"className":98017},[198],[15,98019,98021,98041],{"className":98020},[202,1090],[15,98022,98024,98038],{"className":98023},[206],[15,98025,98027],{"className":98026,"style":2139},[210],[15,98028,98029,98032],{"style":1531},[15,98030],{"className":98031,"style":218},[217],[15,98033,98035],{"className":98034},[222,223,224,225],[15,98036,1804],{"className":98037},[66,225],[15,98039,1114],{"className":98040},[1113],[15,98042,98044],{"className":98043},[206],[15,98045,98047],{"className":98046,"style":1121},[210],[15,98048],{},[15,98050],{"className":98051,"style":697},[71],[15,98053,98055,98058],{"className":98054},[66],[15,98056,2256],{"className":98057},[66,67],[15,98059,98061],{"className":98060},[198],[15,98062,98064,98090],{"className":98063},[202,1090],[15,98065,98067,98087],{"className":98066},[206],[15,98068,98070],{"className":98069,"style":3638},[210],[15,98071,98072,98075],{"style":1584},[15,98073],{"className":98074,"style":218},[217],[15,98076,98078],{"className":98077},[222,223,224,225],[15,98079,98081],{"className":98080},[66,225],[15,98082,98084],{"className":98083},[66,225],[15,98085,74095],{"className":98086},[66,4429,225],[15,98088,1114],{"className":98089},[1113],[15,98091,98093],{"className":98092},[206],[15,98094,98096],{"className":98095,"style":1121},[210],[15,98097],{},[15,98099,1114],{"className":98100},[1113],[15,98102,98104],{"className":98103},[206],[15,98105,98107],{"className":98106,"style":52619},[210],[15,98108],{},[15,98110],{"className":98111},[169,23417],[15,98113,3573],{"className":98114},[66],[11,98116,98117],{},"左边是数据提供的证据，右边是先验与代价决定的阈值。先验偏向负类，阈值升高；漏诊更贵，阈值降低。这个分解把「数据说什么」与「我们更怕哪种错误」分开了。",[11,98119,98120,98121,98150,98151,98205,98206,98260],{},"把阈值从低到高扫一遍，就得到 ROC 曲线。设阈值为 ",[15,98122,98124,98137],{"className":98123,"translate":19},[18],[15,98125,98127],{"className":98126},[23],[25,98128,98129],{"xmlns":27},[29,98130,98131,98135],{},[32,98132,98133],{},[35,98134,9212],{},[45,98136,9212],{"encoding":47},[15,98138,98140],{"className":98139,"ariaHidden":53},[52],[15,98141,98143,98147],{"className":98142},[57],[15,98144],{"className":98145,"style":98146},[61],"height:0.6151em;",[15,98148,9212],{"className":98149},[66,67],"，记真正率为 ",[15,98152,98154,98180],{"className":98153,"translate":19},[18],[15,98155,98157],{"className":98156},[23],[25,98158,98159],{"xmlns":27},[29,98160,98161,98177],{},[32,98162,98163,98171,98173,98175],{},[32,98164,98165,98167,98169],{},[35,98166,32778],{"mathvariant":138},[35,98168,28399],{"mathvariant":138},[35,98170,455],{"mathvariant":138},[39,98172,120],{"stretchy":119},[35,98174,9212],{},[39,98176,126],{"stretchy":119},[45,98178,98179],{"encoding":47},"\\mathrm{TPR}(t)",[15,98181,98183],{"className":98182,"ariaHidden":53},[52],[15,98184,98186,98189,98196,98199,98202],{"className":98185},[57],[15,98187],{"className":98188,"style":154},[61],[15,98190,98192],{"className":98191},[66],[15,98193,98195],{"className":98194},[66,4429],"TPR",[15,98197,120],{"className":98198},[162],[15,98200,9212],{"className":98201},[66,67],[15,98203,126],{"className":98204},[169],"、假正率为 ",[15,98207,98209,98235],{"className":98208,"translate":19},[18],[15,98210,98212],{"className":98211},[23],[25,98213,98214],{"xmlns":27},[29,98215,98216,98232],{},[32,98217,98218,98226,98228,98230],{},[32,98219,98220,98222,98224],{},[35,98221,74045],{"mathvariant":138},[35,98223,28399],{"mathvariant":138},[35,98225,455],{"mathvariant":138},[39,98227,120],{"stretchy":119},[35,98229,9212],{},[39,98231,126],{"stretchy":119},[45,98233,98234],{"encoding":47},"\\mathrm{FPR}(t)",[15,98236,98238],{"className":98237,"ariaHidden":53},[52],[15,98239,98241,98244,98251,98254,98257],{"className":98240},[57],[15,98242],{"className":98243,"style":154},[61],[15,98245,98247],{"className":98246},[66],[15,98248,98250],{"className":98249},[66,4429],"FPR",[15,98252,120],{"className":98253},[162],[15,98255,9212],{"className":98256},[66,67],[15,98258,126],{"className":98259},[169],"，则期望代价为",[15,98262,98264],{"className":98263,"translate":19},[294],[15,98265,98267,98369],{"className":98266,"translate":19},[18],[15,98268,98270],{"className":98269},[23],[25,98271,98272],{"xmlns":27,"display":303},[29,98273,98274,98366],{},[32,98275,98276,98278,98280,98282,98284,98286,98292,98294,98304,98306,98314,98316,98318,98320,98322,98328,98330,98340,98342,98344,98346,98348,98356,98358,98360,98362,98364],{},[35,98277,23256],{},[39,98279,120],{"stretchy":119},[35,98281,9212],{},[39,98283,126],{"stretchy":119},[39,98285,129],{},[1057,98287,98288,98290],{},[35,98289,33494],{},[1209,98291,1804],{},[1274,98293,2462],{},[1057,98295,98296,98298],{},[35,98297,2256],{},[32,98299,98300,98302],{},[35,98301,74045],{"mathvariant":138},[35,98303,28399],{"mathvariant":138},[1274,98305,2462],{},[32,98307,98308,98310,98312],{},[35,98309,74045],{"mathvariant":138},[35,98311,28399],{"mathvariant":138},[35,98313,455],{"mathvariant":138},[39,98315,120],{"stretchy":119},[35,98317,9212],{},[39,98319,126],{"stretchy":119},[39,98321,1245],{},[1057,98323,98324,98326],{},[35,98325,33494],{},[1209,98327,1211],{},[1274,98329,2462],{},[1057,98331,98332,98334],{},[35,98333,2256],{},[32,98335,98336,98338],{},[35,98337,74045],{"mathvariant":138},[35,98339,28535],{"mathvariant":138},[1274,98341,2462],{},[39,98343,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,98345,1211],{},[39,98347,6602],{},[32,98349,98350,98352,98354],{},[35,98351,32778],{"mathvariant":138},[35,98353,28399],{"mathvariant":138},[35,98355,455],{"mathvariant":138},[39,98357,120],{"stretchy":119},[35,98359,9212],{},[39,98361,126],{"stretchy":119},[39,98363,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,98365,3573],{"mathvariant":138},[45,98367,98368],{"encoding":47},"C(t)=\\pi_0\\,c_{\\mathrm{FP}}\\,\\mathrm{FPR}(t)\n+\\pi_1\\,c_{\\mathrm{FN}}\\,\\big(1-\\mathrm{TPR}(t)\\big).",[15,98370,98372,98399,98521,98637],{"className":98371,"ariaHidden":53},[52],[15,98373,98375,98378,98381,98384,98387,98390,98393,98396],{"className":98374},[57],[15,98376],{"className":98377,"style":154},[61],[15,98379,23256],{"className":98380,"style":2901},[66,67],[15,98382,120],{"className":98383},[162],[15,98385,9212],{"className":98386},[66,67],[15,98388,126],{"className":98389},[169],[15,98391],{"className":98392,"style":173},[71],[15,98394,129],{"className":98395},[177],[15,98397],{"className":98398,"style":173},[71],[15,98400,98402,98405,98445,98448,98494,98497,98503,98506,98509,98512,98515,98518],{"className":98401},[57],[15,98403],{"className":98404,"style":154},[61],[15,98406,98408,98411],{"className":98407},[66],[15,98409,33494],{"className":98410,"style":1515},[66,67],[15,98412,98414],{"className":98413},[198],[15,98415,98417,98437],{"className":98416},[202,1090],[15,98418,98420,98434],{"className":98419},[206],[15,98421,98423],{"className":98422,"style":2139},[210],[15,98424,98425,98428],{"style":1531},[15,98426],{"className":98427,"style":218},[217],[15,98429,98431],{"className":98430},[222,223,224,225],[15,98432,1804],{"className":98433},[66,225],[15,98435,1114],{"className":98436},[1113],[15,98438,98440],{"className":98439},[206],[15,98441,98443],{"className":98442,"style":1121},[210],[15,98444],{},[15,98446],{"className":98447,"style":697},[71],[15,98449,98451,98454],{"className":98450},[66],[15,98452,2256],{"className":98453},[66,67],[15,98455,98457],{"className":98456},[198],[15,98458,98460,98486],{"className":98459},[202,1090],[15,98461,98463,98483],{"className":98462},[206],[15,98464,98466],{"className":98465,"style":3638},[210],[15,98467,98468,98471],{"style":1584},[15,98469],{"className":98470,"style":218},[217],[15,98472,98474],{"className":98473},[222,223,224,225],[15,98475,98477],{"className":98476},[66,225],[15,98478,98480],{"className":98479},[66,225],[15,98481,74095],{"className":98482},[66,4429,225],[15,98484,1114],{"className":98485},[1113],[15,98487,98489],{"className":98488},[206],[15,98490,98492],{"className":98491,"style":1121},[210],[15,98493],{},[15,98495],{"className":98496,"style":697},[71],[15,98498,98500],{"className":98499},[66],[15,98501,98250],{"className":98502},[66,4429],[15,98504,120],{"className":98505},[162],[15,98507,9212],{"className":98508},[66,67],[15,98510,126],{"className":98511},[169],[15,98513],{"className":98514,"style":72},[71],[15,98516,1245],{"className":98517},[76],[15,98519],{"className":98520,"style":72},[71],[15,98522,98524,98527,98567,98570,98616,98619,98625,98628,98631,98634],{"className":98523},[57],[15,98525],{"className":98526,"style":4400},[61],[15,98528,98530,98533],{"className":98529},[66],[15,98531,33494],{"className":98532,"style":1515},[66,67],[15,98534,98536],{"className":98535},[198],[15,98537,98539,98559],{"className":98538},[202,1090],[15,98540,98542,98556],{"className":98541},[206],[15,98543,98545],{"className":98544,"style":2139},[210],[15,98546,98547,98550],{"style":1531},[15,98548],{"className":98549,"style":218},[217],[15,98551,98553],{"className":98552},[222,223,224,225],[15,98554,1211],{"className":98555},[66,225],[15,98557,1114],{"className":98558},[1113],[15,98560,98562],{"className":98561},[206],[15,98563,98565],{"className":98564,"style":1121},[210],[15,98566],{},[15,98568],{"className":98569,"style":697},[71],[15,98571,98573,98576],{"className":98572},[66],[15,98574,2256],{"className":98575},[66,67],[15,98577,98579],{"className":98578},[198],[15,98580,98582,98608],{"className":98581},[202,1090],[15,98583,98585,98605],{"className":98584},[206],[15,98586,98588],{"className":98587,"style":3638},[210],[15,98589,98590,98593],{"style":1584},[15,98591],{"className":98592,"style":218},[217],[15,98594,98596],{"className":98595},[222,223,224,225],[15,98597,98599],{"className":98598},[66,225],[15,98600,98602],{"className":98601},[66,225],[15,98603,74177],{"className":98604},[66,4429,225],[15,98606,1114],{"className":98607},[1113],[15,98609,98611],{"className":98610},[206],[15,98612,98614],{"className":98613,"style":1121},[210],[15,98615],{},[15,98617],{"className":98618,"style":697},[71],[15,98620,98622],{"className":98621},[66],[15,98623,120],{"className":98624},[1422,1508],[15,98626,1211],{"className":98627},[66],[15,98629],{"className":98630,"style":72},[71],[15,98632,6602],{"className":98633},[76],[15,98635],{"className":98636,"style":72},[71],[15,98638,98640,98643,98649,98652,98655,98658,98664],{"className":98639},[57],[15,98641],{"className":98642,"style":4400},[61],[15,98644,98646],{"className":98645},[66],[15,98647,98195],{"className":98648},[66,4429],[15,98650,120],{"className":98651},[162],[15,98653,9212],{"className":98654},[66,67],[15,98656,126],{"className":98657},[169],[15,98659,98661],{"className":98660},[66],[15,98662,126],{"className":98663},[1422,1508],[15,98665,3573],{"className":98666},[66],[11,98668,98669],{},"贝叶斯决策做的不是「选一个看起来最好的点」，而是根据先验与代价，在 ROC 上选一个使这个线性函数最小的点。等代价线的斜率是",[15,98671,98673],{"className":98672,"translate":19},[294],[15,98674,98676,98764],{"className":98675,"translate":19},[18],[15,98677,98679],{"className":98678},[23],[25,98680,98681],{"xmlns":27,"display":303},[29,98682,98683,98761],{},[32,98684,98685,98715,98717,98759],{},[6467,98686,98687,98701],{},[32,98688,98689,98691,98693],{},[35,98690,458],{},[1274,98692,2462],{},[32,98694,98695,98697,98699],{},[35,98696,32778],{"mathvariant":138},[35,98698,28399],{"mathvariant":138},[35,98700,455],{"mathvariant":138},[32,98702,98703,98705,98707],{},[35,98704,458],{},[1274,98706,2462],{},[32,98708,98709,98711,98713],{},[35,98710,74045],{"mathvariant":138},[35,98712,28399],{"mathvariant":138},[35,98714,455],{"mathvariant":138},[39,98716,129],{},[6467,98718,98719,98739],{},[32,98720,98721,98727,98729],{},[1057,98722,98723,98725],{},[35,98724,33494],{},[1209,98726,1804],{},[1274,98728,2462],{},[1057,98730,98731,98733],{},[35,98732,2256],{},[32,98734,98735,98737],{},[35,98736,74045],{"mathvariant":138},[35,98738,28399],{"mathvariant":138},[32,98740,98741,98747,98749],{},[1057,98742,98743,98745],{},[35,98744,33494],{},[1209,98746,1211],{},[1274,98748,2462],{},[1057,98750,98751,98753],{},[35,98752,2256],{},[32,98754,98755,98757],{},[35,98756,74045],{"mathvariant":138},[35,98758,28535],{"mathvariant":138},[35,98760,3573],{"mathvariant":138},[45,98762,98763],{"encoding":47},"\\frac{d\\,\\mathrm{TPR}}{d\\,\\mathrm{FPR}}\n=\\frac{\\pi_0\\,c_{\\mathrm{FP}}}{\\pi_1\\,c_{\\mathrm{FN}}}.",[15,98765,98767,98862],{"className":98766,"ariaHidden":53},[52],[15,98768,98770,98773,98853,98856,98859],{"className":98769},[57],[15,98771],{"className":98772,"style":38387},[61],[15,98774,98776,98779,98850],{"className":98775},[66],[15,98777],{"className":98778},[162,23417],[15,98780,98782],{"className":98781},[6467],[15,98783,98785,98842],{"className":98784},[202,1090],[15,98786,98788,98839],{"className":98787},[206],[15,98789,98791,98811,98819],{"className":98790,"style":27435},[210],[15,98792,98793,98796],{"style":23433},[15,98794],{"className":98795,"style":1368},[217],[15,98797,98799,98802,98805],{"className":98798},[66],[15,98800,458],{"className":98801},[66,67],[15,98803],{"className":98804,"style":697},[71],[15,98806,98808],{"className":98807},[66],[15,98809,98250],{"className":98810},[66,4429],[15,98812,98813,98816],{"style":23445},[15,98814],{"className":98815,"style":1368},[217],[15,98817],{"className":98818,"style":4440},[23452],[15,98820,98821,98824],{"style":23455},[15,98822],{"className":98823,"style":1368},[217],[15,98825,98827,98830,98833],{"className":98826},[66],[15,98828,458],{"className":98829},[66,67],[15,98831],{"className":98832,"style":697},[71],[15,98834,98836],{"className":98835},[66],[15,98837,98195],{"className":98838},[66,4429],[15,98840,1114],{"className":98841},[1113],[15,98843,98845],{"className":98844},[206],[15,98846,98848],{"className":98847,"style":23474},[210],[15,98849],{},[15,98851],{"className":98852},[169,23417],[15,98854],{"className":98855,"style":173},[71],[15,98857,129],{"className":98858},[177],[15,98860],{"className":98861,"style":173},[71],[15,98863,98865,98868,99102],{"className":98864},[57],[15,98866],{"className":98867,"style":52422},[61],[15,98869,98871,98874,99099],{"className":98870},[66],[15,98872],{"className":98873},[162,23417],[15,98875,98877],{"className":98876},[6467],[15,98878,98880,99091],{"className":98879},[202,1090],[15,98881,98883,99088],{"className":98882},[206],[15,98884,98886,98983,98991],{"className":98885,"style":39133},[210],[15,98887,98888,98891],{"style":23433},[15,98889],{"className":98890,"style":1368},[217],[15,98892,98894,98934,98937],{"className":98893},[66],[15,98895,98897,98900],{"className":98896},[66],[15,98898,33494],{"className":98899,"style":1515},[66,67],[15,98901,98903],{"className":98902},[198],[15,98904,98906,98926],{"className":98905},[202,1090],[15,98907,98909,98923],{"className":98908},[206],[15,98910,98912],{"className":98911,"style":2139},[210],[15,98913,98914,98917],{"style":1531},[15,98915],{"className":98916,"style":218},[217],[15,98918,98920],{"className":98919},[222,223,224,225],[15,98921,1211],{"className":98922},[66,225],[15,98924,1114],{"className":98925},[1113],[15,98927,98929],{"className":98928},[206],[15,98930,98932],{"className":98931,"style":1121},[210],[15,98933],{},[15,98935],{"className":98936,"style":697},[71],[15,98938,98940,98943],{"className":98939},[66],[15,98941,2256],{"className":98942},[66,67],[15,98944,98946],{"className":98945},[198],[15,98947,98949,98975],{"className":98948},[202,1090],[15,98950,98952,98972],{"className":98951},[206],[15,98953,98955],{"className":98954,"style":3638},[210],[15,98956,98957,98960],{"style":1584},[15,98958],{"className":98959,"style":218},[217],[15,98961,98963],{"className":98962},[222,223,224,225],[15,98964,98966],{"className":98965},[66,225],[15,98967,98969],{"className":98968},[66,225],[15,98970,74177],{"className":98971},[66,4429,225],[15,98973,1114],{"className":98974},[1113],[15,98976,98978],{"className":98977},[206],[15,98979,98981],{"className":98980,"style":1121},[210],[15,98982],{},[15,98984,98985,98988],{"style":23445},[15,98986],{"className":98987,"style":1368},[217],[15,98989],{"className":98990,"style":4440},[23452],[15,98992,98993,98996],{"style":23455},[15,98994],{"className":98995,"style":1368},[217],[15,98997,98999,99039,99042],{"className":98998},[66],[15,99000,99002,99005],{"className":99001},[66],[15,99003,33494],{"className":99004,"style":1515},[66,67],[15,99006,99008],{"className":99007},[198],[15,99009,99011,99031],{"className":99010},[202,1090],[15,99012,99014,99028],{"className":99013},[206],[15,99015,99017],{"className":99016,"style":2139},[210],[15,99018,99019,99022],{"style":1531},[15,99020],{"className":99021,"style":218},[217],[15,99023,99025],{"className":99024},[222,223,224,225],[15,99026,1804],{"className":99027},[66,225],[15,99029,1114],{"className":99030},[1113],[15,99032,99034],{"className":99033},[206],[15,99035,99037],{"className":99036,"style":1121},[210],[15,99038],{},[15,99040],{"className":99041,"style":697},[71],[15,99043,99045,99048],{"className":99044},[66],[15,99046,2256],{"className":99047},[66,67],[15,99049,99051],{"className":99050},[198],[15,99052,99054,99080],{"className":99053},[202,1090],[15,99055,99057,99077],{"className":99056},[206],[15,99058,99060],{"className":99059,"style":3638},[210],[15,99061,99062,99065],{"style":1584},[15,99063],{"className":99064,"style":218},[217],[15,99066,99068],{"className":99067},[222,223,224,225],[15,99069,99071],{"className":99070},[66,225],[15,99072,99074],{"className":99073},[66,225],[15,99075,74095],{"className":99076},[66,4429,225],[15,99078,1114],{"className":99079},[1113],[15,99081,99083],{"className":99082},[206],[15,99084,99086],{"className":99085,"style":1121},[210],[15,99087],{},[15,99089,1114],{"className":99090},[1113],[15,99092,99094],{"className":99093},[206],[15,99095,99097],{"className":99096,"style":52619},[210],[15,99098],{},[15,99100],{"className":99101},[169,23417],[15,99103,3573],{"className":99104},[66],[11,99106,99107,99108,1124],{},"这也说明 AUC 只是一个与阈值无关的排序指标。两个模型的 AUC 可以一个高一个低，但在某个特定的代价结构下，AUC 低的模型反而更优。",[2040,99109,99110],{},"脱离代价谈「最好的分类器」是没有意义的",[1904,99112,99114],{"id":99113},"_54-拒绝选项与选择性分类","5.4 拒绝选项与选择性分类",[11,99116,99117,99118,99168,99169,99172],{},"3.5 节已经导出了 Chow 规则：当最大后验低于 ",[15,99119,99121,99138],{"className":99120,"translate":19},[18],[15,99122,99124],{"className":99123},[23],[25,99125,99126],{"xmlns":27},[29,99127,99128,99136],{},[32,99129,99130,99132,99134],{},[1209,99131,1211],{},[39,99133,6602],{},[35,99135,2256],{},[45,99137,75397],{"encoding":47},[15,99139,99141,99159],{"className":99140,"ariaHidden":53},[52],[15,99142,99144,99147,99150,99153,99156],{"className":99143},[57],[15,99145],{"className":99146,"style":30468},[61],[15,99148,1211],{"className":99149},[66],[15,99151],{"className":99152,"style":72},[71],[15,99154,6602],{"className":99155},[76],[15,99157],{"className":99158,"style":72},[71],[15,99160,99162,99165],{"className":99161},[57],[15,99163],{"className":99164,"style":86},[61],[15,99166,2256],{"className":99167},[66,67]," 时拒绝。把这句话放到整个数据集上，就得到一条",[2040,99170,99171],{},"风险-覆盖率曲线","：随着拒绝阈值升高，被拒绝的样本增多，覆盖率下降，但在被接受的样本上错误率也下降。",[11,99174,99175],{},"选择性分类的理论目标不是最大化准确率，而是最小化",[15,99177,99179],{"className":99178,"translate":19},[294],[15,99180,99182,99209],{"className":99181,"translate":19},[18],[15,99183,99185],{"className":99184},[23],[25,99186,99187],{"xmlns":27,"display":303},[29,99188,99189,99206],{},[32,99190,99191,99194,99196,99204],{},[1274,99192,99193],{},"选择性风险",[39,99195,129],{},[6467,99197,99198,99201],{},[1274,99199,99200],{},"被接受样本上的错误数",[1274,99202,99203],{},"被接受样本数",[39,99205,335],{"separator":53},[45,99207,99208],{"encoding":47},"\\text{选择性风险}=\\frac{\\text{被接受样本上的错误数}}{\\text{被接受样本数}},",[15,99210,99212,99233],{"className":99211,"ariaHidden":53},[52],[15,99213,99215,99218,99224,99227,99230],{"className":99214},[57],[15,99216],{"className":99217,"style":490},[61],[15,99219,99221],{"className":99220},[66,1711],[15,99222,99193],{"className":99223},[66,68616],[15,99225],{"className":99226,"style":173},[71],[15,99228,129],{"className":99229},[177],[15,99231],{"className":99232,"style":173},[71],[15,99234,99236,99240,99308],{"className":99235},[57],[15,99237],{"className":99238,"style":99239},[61],"height:2.0463em;vertical-align:-0.686em;",[15,99241,99243,99246,99305],{"className":99242},[66],[15,99244],{"className":99245},[162,23417],[15,99247,99249],{"className":99248},[6467],[15,99250,99252,99297],{"className":99251},[202,1090],[15,99253,99255,99294],{"className":99254},[206],[15,99256,99258,99272,99280],{"className":99257,"style":85566},[210],[15,99259,99260,99263],{"style":23433},[15,99261],{"className":99262,"style":1368},[217],[15,99264,99266],{"className":99265},[66],[15,99267,99269],{"className":99268},[66,1711],[15,99270,99203],{"className":99271},[66,68616],[15,99273,99274,99277],{"style":23445},[15,99275],{"className":99276,"style":1368},[217],[15,99278],{"className":99279,"style":4440},[23452],[15,99281,99282,99285],{"style":23455},[15,99283],{"className":99284,"style":1368},[217],[15,99286,99288],{"className":99287},[66],[15,99289,99291],{"className":99290},[66,1711],[15,99292,99200],{"className":99293},[66,68616],[15,99295,1114],{"className":99296},[1113],[15,99298,99300],{"className":99299},[206],[15,99301,99303],{"className":99302,"style":23474},[210],[15,99304],{},[15,99306],{"className":99307},[169,23417],[15,99309,335],{"className":99310},[422],[11,99312,99313,99314,99342,99343,99371],{},"同时满足覆盖率约束。这两个量之间的权衡，仍然由损失函数决定：拒绝的代价 ",[15,99315,99317,99330],{"className":99316,"translate":19},[18],[15,99318,99320],{"className":99319},[23],[25,99321,99322],{"xmlns":27},[29,99323,99324,99328],{},[32,99325,99326],{},[35,99327,2256],{},[45,99329,2256],{"encoding":47},[15,99331,99333],{"className":99332,"ariaHidden":53},[52],[15,99334,99336,99339],{"className":99335},[57],[15,99337],{"className":99338,"style":86},[61],[15,99340,2256],{"className":99341},[66,67]," 越小，模型越倾向于拒绝；",[15,99344,99346,99359],{"className":99345,"translate":19},[18],[15,99347,99349],{"className":99348},[23],[25,99350,99351],{"xmlns":27},[29,99352,99353,99357],{},[32,99354,99355],{},[35,99356,2256],{},[45,99358,2256],{"encoding":47},[15,99360,99362],{"className":99361,"ariaHidden":53},[52],[15,99363,99365,99368],{"className":99364},[57],[15,99366],{"className":99367,"style":86},[61],[15,99369,2256],{"className":99370},[66,67]," 越大，模型越倾向于冒险。",[11,99373,99374],{},"在医疗、自动驾驶、内容审核这些场景里，拒绝选项往往比「提高模型容量」更有效。一个知道自己不知道的模型，比一个在所有样本上都自信地犯错的模型更有用。",[1904,99376,99378],{"id":99377},"_55-类别不平衡改损失还是改数据","5.5 类别不平衡：改损失，还是改数据",[11,99380,99381],{},"类别不平衡经常被当成一个数据处理问题，但在决策论的框架里，它首先是一个先验与损失的问题。",[11,99383,99384,99385,99509,99510,99668,99669,99820],{},"设原分布的类先验为 ",[15,99386,99388,99414],{"className":99387,"translate":19},[18],[15,99389,99391],{"className":99390},[23],[25,99392,99393],{"xmlns":27},[29,99394,99395,99411],{},[32,99396,99397,99403,99405],{},[1057,99398,99399,99401],{},[35,99400,33494],{},[1209,99402,1804],{},[39,99404,335],{"separator":53},[1057,99406,99407,99409],{},[35,99408,33494],{},[1209,99410,1211],{},[45,99412,99413],{"encoding":47},"\\pi_0,\\pi_1",[15,99415,99417],{"className":99416,"ariaHidden":53},[52],[15,99418,99420,99423,99463,99466,99469],{"className":99419},[57],[15,99421],{"className":99422,"style":471},[61],[15,99424,99426,99429],{"className":99425},[66],[15,99427,33494],{"className":99428,"style":1515},[66,67],[15,99430,99432],{"className":99431},[198],[15,99433,99435,99455],{"className":99434},[202,1090],[15,99436,99438,99452],{"className":99437},[206],[15,99439,99441],{"className":99440,"style":2139},[210],[15,99442,99443,99446],{"style":1531},[15,99444],{"className":99445,"style":218},[217],[15,99447,99449],{"className":99448},[222,223,224,225],[15,99450,1804],{"className":99451},[66,225],[15,99453,1114],{"className":99454},[1113],[15,99456,99458],{"className":99457},[206],[15,99459,99461],{"className":99460,"style":1121},[210],[15,99462],{},[15,99464,335],{"className":99465},[422],[15,99467],{"className":99468,"style":697},[71],[15,99470,99472,99475],{"className":99471},[66],[15,99473,33494],{"className":99474,"style":1515},[66,67],[15,99476,99478],{"className":99477},[198],[15,99479,99481,99501],{"className":99480},[202,1090],[15,99482,99484,99498],{"className":99483},[206],[15,99485,99487],{"className":99486,"style":2139},[210],[15,99488,99489,99492],{"style":1531},[15,99490],{"className":99491,"style":218},[217],[15,99493,99495],{"className":99494},[222,223,224,225],[15,99496,1211],{"className":99497},[66,225],[15,99499,1114],{"className":99500},[1113],[15,99502,99504],{"className":99503},[206],[15,99505,99507],{"className":99506,"style":1121},[210],[15,99508],{},"，重采样后的先验为 ",[15,99511,99513,99543],{"className":99512,"translate":19},[18],[15,99514,99516],{"className":99515},[23],[25,99517,99518],{"xmlns":27},[29,99519,99520,99540],{},[32,99521,99522,99530,99532],{},[1255,99523,99524,99526,99528],{},[35,99525,33494],{},[1209,99527,1804],{},[39,99529,635],{"mathvariant":138,"lspace":634,"rspace":634},[39,99531,335],{"separator":53},[1255,99533,99534,99536,99538],{},[35,99535,33494],{},[1209,99537,1211],{},[39,99539,635],{"mathvariant":138,"lspace":634,"rspace":634},[45,99541,99542],{"encoding":47},"\\pi'_0,\\pi'_1",[15,99544,99546],{"className":99545,"ariaHidden":53},[52],[15,99547,99549,99553,99608,99611,99614],{"className":99548},[57],[15,99550],{"className":99551,"style":99552},[61],"height:1em;vertical-align:-0.2481em;",[15,99554,99556,99559],{"className":99555},[66],[15,99557,33494],{"className":99558,"style":1515},[66,67],[15,99560,99562],{"className":99561},[198],[15,99563,99565,99600],{"className":99564},[202,1090],[15,99566,99568,99597],{"className":99567},[206],[15,99569,99571,99583],{"className":99570,"style":6005},[210],[15,99572,99574,99577],{"style":99573},"top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;",[15,99575],{"className":99576,"style":218},[217],[15,99578,99580],{"className":99579},[222,223,224,225],[15,99581,1804],{"className":99582},[66,225],[15,99584,99585,99588],{"style":213},[15,99586],{"className":99587,"style":218},[217],[15,99589,99591],{"className":99590},[222,223,224,225],[15,99592,99594],{"className":99593},[66,225],[15,99595,635],{"className":99596},[66,225],[15,99598,1114],{"className":99599},[1113],[15,99601,99603],{"className":99602},[206],[15,99604,99606],{"className":99605,"style":6773},[210],[15,99607],{},[15,99609,335],{"className":99610},[422],[15,99612],{"className":99613,"style":697},[71],[15,99615,99617,99620],{"className":99616},[66],[15,99618,33494],{"className":99619,"style":1515},[66,67],[15,99621,99623],{"className":99622},[198],[15,99624,99626,99660],{"className":99625},[202,1090],[15,99627,99629,99657],{"className":99628},[206],[15,99630,99632,99643],{"className":99631,"style":6005},[210],[15,99633,99634,99637],{"style":99573},[15,99635],{"className":99636,"style":218},[217],[15,99638,99640],{"className":99639},[222,223,224,225],[15,99641,1211],{"className":99642},[66,225],[15,99644,99645,99648],{"style":213},[15,99646],{"className":99647,"style":218},[217],[15,99649,99651],{"className":99650},[222,223,224,225],[15,99652,99654],{"className":99653},[66,225],[15,99655,635],{"className":99656},[66,225],[15,99658,1114],{"className":99659},[1113],[15,99661,99663],{"className":99662},[206],[15,99664,99666],{"className":99665,"style":6773},[210],[15,99667],{},"。同一个似然比 ",[15,99670,99672,99710],{"className":99671,"translate":19},[18],[15,99673,99675],{"className":99674},[23],[25,99676,99677],{"xmlns":27},[29,99678,99679,99707],{},[32,99680,99681,99687,99689,99691,99693,99695,99701,99703,99705],{},[1057,99682,99683,99685],{},[35,99684,11],{},[1209,99686,1211],{},[39,99688,120],{"stretchy":119},[35,99690,123],{},[39,99692,126],{"stretchy":119},[35,99694,6625],{"mathvariant":138},[1057,99696,99697,99699],{},[35,99698,11],{},[1209,99700,1804],{},[39,99702,120],{"stretchy":119},[35,99704,123],{},[39,99706,126],{"stretchy":119},[45,99708,99709],{"encoding":47},"p_1(x)\u002Fp_0(x)",[15,99711,99713],{"className":99712,"ariaHidden":53},[52],[15,99714,99716,99719,99759,99762,99765,99768,99771,99811,99814,99817],{"className":99715},[57],[15,99717],{"className":99718,"style":154},[61],[15,99720,99722,99725],{"className":99721},[66],[15,99723,11],{"className":99724},[66,67],[15,99726,99728],{"className":99727},[198],[15,99729,99731,99751],{"className":99730},[202,1090],[15,99732,99734,99748],{"className":99733},[206],[15,99735,99737],{"className":99736,"style":2139},[210],[15,99738,99739,99742],{"style":1584},[15,99740],{"className":99741,"style":218},[217],[15,99743,99745],{"className":99744},[222,223,224,225],[15,99746,1211],{"className":99747},[66,225],[15,99749,1114],{"className":99750},[1113],[15,99752,99754],{"className":99753},[206],[15,99755,99757],{"className":99756,"style":1121},[210],[15,99758],{},[15,99760,120],{"className":99761},[162],[15,99763,123],{"className":99764},[66,67],[15,99766,126],{"className":99767},[169],[15,99769,6625],{"className":99770},[66],[15,99772,99774,99777],{"className":99773},[66],[15,99775,11],{"className":99776},[66,67],[15,99778,99780],{"className":99779},[198],[15,99781,99783,99803],{"className":99782},[202,1090],[15,99784,99786,99800],{"className":99785},[206],[15,99787,99789],{"className":99788,"style":2139},[210],[15,99790,99791,99794],{"style":1584},[15,99792],{"className":99793,"style":218},[217],[15,99795,99797],{"className":99796},[222,223,224,225],[15,99798,1804],{"className":99799},[66,225],[15,99801,1114],{"className":99802},[1113],[15,99804,99806],{"className":99805},[206],[15,99807,99809],{"className":99808,"style":1121},[210],[15,99810],{},[15,99812,120],{"className":99813},[162],[15,99815,123],{"className":99816},[66,67],[15,99818,126],{"className":99819},[169]," 在两种分布下给出的后验几率不同：",[15,99822,99824],{"className":99823,"translate":19},[294],[15,99825,99827,99961],{"className":99826,"translate":19},[18],[15,99828,99830],{"className":99829},[23],[25,99831,99832],{"xmlns":27,"display":303},[29,99833,99834,99958],{},[32,99835,99836,99875,99877,99916,99918,99956],{},[6467,99837,99838,99857],{},[32,99839,99840,99847,99849,99851,99853,99855],{},[1057,99841,99842,99844],{},[35,99843,11],{},[1274,99845,99846],{},"orig",[39,99848,120],{"stretchy":119},[1209,99850,1211],{},[39,99852,34430],{},[35,99854,123],{},[39,99856,126],{"stretchy":119},[32,99858,99859,99865,99867,99869,99871,99873],{},[1057,99860,99861,99863],{},[35,99862,11],{},[1274,99864,99846],{},[39,99866,120],{"stretchy":119},[1209,99868,1804],{},[39,99870,34430],{},[35,99872,123],{},[39,99874,126],{"stretchy":119},[39,99876,129],{},[6467,99878,99879,99898],{},[32,99880,99881,99888,99890,99892,99894,99896],{},[1057,99882,99883,99885],{},[35,99884,11],{},[1274,99886,99887],{},"res",[39,99889,120],{"stretchy":119},[1209,99891,1211],{},[39,99893,34430],{},[35,99895,123],{},[39,99897,126],{"stretchy":119},[32,99899,99900,99906,99908,99910,99912,99914],{},[1057,99901,99902,99904],{},[35,99903,11],{},[1274,99905,99887],{},[39,99907,120],{"stretchy":119},[1209,99909,1804],{},[39,99911,34430],{},[35,99913,123],{},[39,99915,126],{"stretchy":119},[39,99917,3268],{},[6467,99919,99920,99938],{},[32,99921,99922,99928,99930],{},[1057,99923,99924,99926],{},[35,99925,33494],{},[1209,99927,1211],{},[1274,99929,2462],{},[1255,99931,99932,99934,99936],{},[35,99933,33494],{},[1209,99935,1804],{},[39,99937,635],{"mathvariant":138,"lspace":634,"rspace":634},[32,99939,99940,99948,99950],{},[1255,99941,99942,99944,99946],{},[35,99943,33494],{},[1209,99945,1211],{},[39,99947,635],{"mathvariant":138,"lspace":634,"rspace":634},[1274,99949,2462],{},[1057,99951,99952,99954],{},[35,99953,33494],{},[1209,99955,1804],{},[35,99957,3573],{"mathvariant":138},[45,99959,99960],{"encoding":47},"\\frac{p_{\\text{orig}}(1\\mid x)}{p_{\\text{orig}}(0\\mid x)}\n=\n\\frac{p_{\\text{res}}(1\\mid x)}{p_{\\text{res}}(0\\mid x)}\n\\cdot\n\\frac{\\pi_1\\,\\pi'_0}{\\pi'_1\\,\\pi_0}.",[15,99962,99964,100172,100377],{"className":99963,"ariaHidden":53},[52],[15,99965,99967,99971,100163,100166,100169],{"className":99966},[57],[15,99968],{"className":99969,"style":99970},[61],"height:2.3991em;vertical-align:-0.9721em;",[15,99972,99974,99977,100160],{"className":99973},[66],[15,99975],{"className":99976},[162,23417],[15,99978,99980],{"className":99979},[6467],[15,99981,99983,100151],{"className":99982},[202,1090],[15,99984,99986,100148],{"className":99985},[206],[15,99987,99989,100065,100073],{"className":99988,"style":41211},[210],[15,99990,99991,99994],{"style":23433},[15,99992],{"className":99993,"style":1368},[217],[15,99995,99997,100044,100047,100050,100053,100056,100059,100062],{"className":99996},[66],[15,99998,100000,100003],{"className":99999},[66],[15,100001,11],{"className":100002},[66,67],[15,100004,100006],{"className":100005},[198],[15,100007,100009,100036],{"className":100008},[202,1090],[15,100010,100012,100033],{"className":100011},[206],[15,100013,100016],{"className":100014,"style":100015},[210],"height:0.3175em;",[15,100017,100018,100021],{"style":1584},[15,100019],{"className":100020,"style":218},[217],[15,100022,100024],{"className":100023},[222,223,224,225],[15,100025,100027],{"className":100026},[66,225],[15,100028,100030],{"className":100029},[66,1711,225],[15,100031,99846],{"className":100032},[66,225],[15,100034,1114],{"className":100035},[1113],[15,100037,100039],{"className":100038},[206],[15,100040,100042],{"className":100041,"style":2715},[210],[15,100043],{},[15,100045,120],{"className":100046},[162],[15,100048,1804],{"className":100049},[66],[15,100051],{"className":100052,"style":173},[71],[15,100054,34430],{"className":100055},[177],[15,100057],{"className":100058,"style":173},[71],[15,100060,123],{"className":100061},[66,67],[15,100063,126],{"className":100064},[169],[15,100066,100067,100070],{"style":23445},[15,100068],{"className":100069,"style":1368},[217],[15,100071],{"className":100072,"style":4440},[23452],[15,100074,100075,100078],{"style":23455},[15,100076],{"className":100077,"style":1368},[217],[15,100079,100081,100127,100130,100133,100136,100139,100142,100145],{"className":100080},[66],[15,100082,100084,100087],{"className":100083},[66],[15,100085,11],{"className":100086},[66,67],[15,100088,100090],{"className":100089},[198],[15,100091,100093,100119],{"className":100092},[202,1090],[15,100094,100096,100116],{"className":100095},[206],[15,100097,100099],{"className":100098,"style":100015},[210],[15,100100,100101,100104],{"style":1584},[15,100102],{"className":100103,"style":218},[217],[15,100105,100107],{"className":100106},[222,223,224,225],[15,100108,100110],{"className":100109},[66,225],[15,100111,100113],{"className":100112},[66,1711,225],[15,100114,99846],{"className":100115},[66,225],[15,100117,1114],{"className":100118},[1113],[15,100120,100122],{"className":100121},[206],[15,100123,100125],{"className":100124,"style":2715},[210],[15,100126],{},[15,100128,120],{"className":100129},[162],[15,100131,1211],{"className":100132},[66],[15,100134],{"className":100135,"style":173},[71],[15,100137,34430],{"className":100138},[177],[15,100140],{"className":100141,"style":173},[71],[15,100143,123],{"className":100144},[66,67],[15,100146,126],{"className":100147},[169],[15,100149,1114],{"className":100150},[1113],[15,100152,100154],{"className":100153},[206],[15,100155,100158],{"className":100156,"style":100157},[210],"height:0.9721em;",[15,100159],{},[15,100161],{"className":100162},[169,23417],[15,100164],{"className":100165,"style":173},[71],[15,100167,129],{"className":100168},[177],[15,100170],{"className":100171,"style":173},[71],[15,100173,100175,100178,100368,100371,100374],{"className":100174},[57],[15,100176],{"className":100177,"style":41192},[61],[15,100179,100181,100184,100365],{"className":100180},[66],[15,100182],{"className":100183},[162,23417],[15,100185,100187],{"className":100186},[6467],[15,100188,100190,100357],{"className":100189},[202,1090],[15,100191,100193,100354],{"className":100192},[206],[15,100194,100196,100271,100279],{"className":100195,"style":41211},[210],[15,100197,100198,100201],{"style":23433},[15,100199],{"className":100200,"style":1368},[217],[15,100202,100204,100250,100253,100256,100259,100262,100265,100268],{"className":100203},[66],[15,100205,100207,100210],{"className":100206},[66],[15,100208,11],{"className":100209},[66,67],[15,100211,100213],{"className":100212},[198],[15,100214,100216,100242],{"className":100215},[202,1090],[15,100217,100219,100239],{"className":100218},[206],[15,100220,100222],{"className":100221,"style":2198},[210],[15,100223,100224,100227],{"style":1584},[15,100225],{"className":100226,"style":218},[217],[15,100228,100230],{"className":100229},[222,223,224,225],[15,100231,100233],{"className":100232},[66,225],[15,100234,100236],{"className":100235},[66,1711,225],[15,100237,99887],{"className":100238},[66,225],[15,100240,1114],{"className":100241},[1113],[15,100243,100245],{"className":100244},[206],[15,100246,100248],{"className":100247,"style":1121},[210],[15,100249],{},[15,100251,120],{"className":100252},[162],[15,100254,1804],{"className":100255},[66],[15,100257],{"className":100258,"style":173},[71],[15,100260,34430],{"className":100261},[177],[15,100263],{"className":100264,"style":173},[71],[15,100266,123],{"className":100267},[66,67],[15,100269,126],{"className":100270},[169],[15,100272,100273,100276],{"style":23445},[15,100274],{"className":100275,"style":1368},[217],[15,100277],{"className":100278,"style":4440},[23452],[15,100280,100281,100284],{"style":23455},[15,100282],{"className":100283,"style":1368},[217],[15,100285,100287,100333,100336,100339,100342,100345,100348,100351],{"className":100286},[66],[15,100288,100290,100293],{"className":100289},[66],[15,100291,11],{"className":100292},[66,67],[15,100294,100296],{"className":100295},[198],[15,100297,100299,100325],{"className":100298},[202,1090],[15,100300,100302,100322],{"className":100301},[206],[15,100303,100305],{"className":100304,"style":2198},[210],[15,100306,100307,100310],{"style":1584},[15,100308],{"className":100309,"style":218},[217],[15,100311,100313],{"className":100312},[222,223,224,225],[15,100314,100316],{"className":100315},[66,225],[15,100317,100319],{"className":100318},[66,1711,225],[15,100320,99887],{"className":100321},[66,225],[15,100323,1114],{"className":100324},[1113],[15,100326,100328],{"className":100327},[206],[15,100329,100331],{"className":100330,"style":1121},[210],[15,100332],{},[15,100334,120],{"className":100335},[162],[15,100337,1211],{"className":100338},[66],[15,100340],{"className":100341,"style":173},[71],[15,100343,34430],{"className":100344},[177],[15,100346],{"className":100347,"style":173},[71],[15,100349,123],{"className":100350},[66,67],[15,100352,126],{"className":100353},[169],[15,100355,1114],{"className":100356},[1113],[15,100358,100360],{"className":100359},[206],[15,100361,100363],{"className":100362,"style":41282},[210],[15,100364],{},[15,100366],{"className":100367},[169,23417],[15,100369],{"className":100370,"style":72},[71],[15,100372,3268],{"className":100373},[76],[15,100375],{"className":100376,"style":72},[71],[15,100378,100380,100384,100639],{"className":100379},[57],[15,100381],{"className":100382,"style":100383},[61],"height:2.3812em;vertical-align:-0.9523em;",[15,100385,100387,100390,100636],{"className":100386},[66],[15,100388],{"className":100389},[162,23417],[15,100391,100393],{"className":100392},[6467],[15,100394,100396,100627],{"className":100395},[202,1090],[15,100397,100399,100624],{"className":100398},[206],[15,100400,100403,100511,100519],{"className":100401,"style":100402},[210],"height:1.4289em;",[15,100404,100405,100408],{"style":23433},[15,100406],{"className":100407,"style":1368},[217],[15,100409,100411,100468,100471],{"className":100410},[66],[15,100412,100414,100417],{"className":100413},[66],[15,100415,33494],{"className":100416,"style":1515},[66,67],[15,100418,100420],{"className":100419},[198],[15,100421,100423,100460],{"className":100422},[202,1090],[15,100424,100426,100457],{"className":100425},[206],[15,100427,100430,100442],{"className":100428,"style":100429},[210],"height:0.7337em;",[15,100431,100433,100436],{"style":100432},"top:-2.4337em;margin-left:-0.0359em;margin-right:0.05em;",[15,100434],{"className":100435,"style":218},[217],[15,100437,100439],{"className":100438},[222,223,224,225],[15,100440,1211],{"className":100441},[66,225],[15,100443,100445,100448],{"style":100444},"top:-3.0448em;margin-right:0.05em;",[15,100446],{"className":100447,"style":218},[217],[15,100449,100451],{"className":100450},[222,223,224,225],[15,100452,100454],{"className":100453},[66,225],[15,100455,635],{"className":100456},[66,225],[15,100458,1114],{"className":100459},[1113],[15,100461,100463],{"className":100462},[206],[15,100464,100466],{"className":100465,"style":93605},[210],[15,100467],{},[15,100469],{"className":100470,"style":697},[71],[15,100472,100474,100477],{"className":100473},[66],[15,100475,33494],{"className":100476,"style":1515},[66,67],[15,100478,100480],{"className":100479},[198],[15,100481,100483,100503],{"className":100482},[202,1090],[15,100484,100486,100500],{"className":100485},[206],[15,100487,100489],{"className":100488,"style":2139},[210],[15,100490,100491,100494],{"style":1531},[15,100492],{"className":100493,"style":218},[217],[15,100495,100497],{"className":100496},[222,223,224,225],[15,100498,1804],{"className":100499},[66,225],[15,100501,1114],{"className":100502},[1113],[15,100504,100506],{"className":100505},[206],[15,100507,100509],{"className":100508,"style":1121},[210],[15,100510],{},[15,100512,100513,100516],{"style":23445},[15,100514],{"className":100515,"style":1368},[217],[15,100517],{"className":100518,"style":4440},[23452],[15,100520,100521,100524],{"style":23455},[15,100522],{"className":100523,"style":1368},[217],[15,100525,100527,100567,100570],{"className":100526},[66],[15,100528,100530,100533],{"className":100529},[66],[15,100531,33494],{"className":100532,"style":1515},[66,67],[15,100534,100536],{"className":100535},[198],[15,100537,100539,100559],{"className":100538},[202,1090],[15,100540,100542,100556],{"className":100541},[206],[15,100543,100545],{"className":100544,"style":2139},[210],[15,100546,100547,100550],{"style":1531},[15,100548],{"className":100549,"style":218},[217],[15,100551,100553],{"className":100552},[222,223,224,225],[15,100554,1211],{"className":100555},[66,225],[15,100557,1114],{"className":100558},[1113],[15,100560,100562],{"className":100561},[206],[15,100563,100565],{"className":100564,"style":1121},[210],[15,100566],{},[15,100568],{"className":100569,"style":697},[71],[15,100571,100573,100576],{"className":100572},[66],[15,100574,33494],{"className":100575,"style":1515},[66,67],[15,100577,100579],{"className":100578},[198],[15,100580,100582,100616],{"className":100581},[202,1090],[15,100583,100585,100613],{"className":100584},[206],[15,100586,100588,100599],{"className":100587,"style":6005},[210],[15,100589,100590,100593],{"style":99573},[15,100591],{"className":100592,"style":218},[217],[15,100594,100596],{"className":100595},[222,223,224,225],[15,100597,1804],{"className":100598},[66,225],[15,100600,100601,100604],{"style":213},[15,100602],{"className":100603,"style":218},[217],[15,100605,100607],{"className":100606},[222,223,224,225],[15,100608,100610],{"className":100609},[66,225],[15,100611,635],{"className":100612},[66,225],[15,100614,1114],{"className":100615},[1113],[15,100617,100619],{"className":100618},[206],[15,100620,100622],{"className":100621,"style":6773},[210],[15,100623],{},[15,100625,1114],{"className":100626},[1113],[15,100628,100630],{"className":100629},[206],[15,100631,100634],{"className":100632,"style":100633},[210],"height:0.9523em;",[15,100635],{},[15,100637],{"className":100638},[169,23417],[15,100640,3573],{"className":100641},[66],[11,100643,100644],{},"如果重采样之后不把这个因子校正回去，决策阈值就偏了。反过来，如果不想重采样，也可以直接在原分布上训练，然后按 3.5 节的公式移动阈值。两条路在理论上等价，差别只在实现难度与估计方差。",[11,100646,100647],{},"真正不能省掉的是损失函数。若两类错误的代价确实不同，那么无论怎么采样，都必须把代价写进决策规则；否则得到的只是一个在改动后的分布上「看起来平衡」的分类器，而不是原问题的最优决策。",[95,100649,100651],{"id":100650},"_6-从参数到分布","6. 从参数到分布",[1904,100653,100655],{"id":100654},"_61-三个预测器","6.1 三个预测器",[11,100657,100658,100659,100687,100688,100716],{},"给定训练集 ",[15,100660,100662,100675],{"className":100661,"translate":19},[18],[15,100663,100665],{"className":100664},[23],[25,100666,100667],{"xmlns":27},[29,100668,100669,100673],{},[32,100670,100671],{},[35,100672,32458],{},[45,100674,32458],{"encoding":47},[15,100676,100678],{"className":100677,"ariaHidden":53},[52],[15,100679,100681,100684],{"className":100680},[57],[15,100682],{"className":100683,"style":490},[61],[15,100685,32458],{"className":100686,"style":23025},[66,67],"，预测 ",[15,100689,100691,100704],{"className":100690,"translate":19},[18],[15,100692,100694],{"className":100693},[23],[25,100695,100696],{"xmlns":27},[29,100697,100698,100702],{},[32,100699,100700],{},[35,100701,61885],{},[45,100703,61885],{"encoding":47},[15,100705,100707],{"className":100706,"ariaHidden":53},[52],[15,100708,100710,100713],{"className":100709},[57],[15,100711],{"className":100712,"style":490},[61],[15,100714,61885],{"className":100715,"style":72},[66,67]," 至少有三层做法：",[3191,100718,100719,100899,101166],{},[3194,100720,100721,100724,100725,100786,100787,100898],{},[2040,100722,100723],{},"代入式","：先估计一个参数 ",[15,100726,100728,100746],{"className":100727,"translate":19},[18],[15,100729,100731],{"className":100730},[23],[25,100732,100733],{"xmlns":27},[29,100734,100735,100743],{},[32,100736,100737],{},[4296,100738,100739,100741],{"accent":53},[35,100740,61618],{},[39,100742,78130],{},[45,100744,100745],{"encoding":47},"\\hat\\theta",[15,100747,100749],{"className":100748,"ariaHidden":53},[52],[15,100750,100752,100755],{"className":100751},[57],[15,100753],{"className":100754,"style":78157},[61],[15,100756,100758],{"className":100757},[66,78147],[15,100759,100761],{"className":100760},[202],[15,100762,100764],{"className":100763},[206],[15,100765,100767,100775],{"className":100766,"style":78157},[210],[15,100768,100769,100772],{"style":1386},[15,100770],{"className":100771,"style":1368},[217],[15,100773,61618],{"className":100774,"style":23025},[66,67],[15,100776,100777,100780],{"style":78168},[15,100778],{"className":100779,"style":1368},[217],[15,100781,100783],{"className":100782,"style":96110},[78175],[15,100784,78130],{"className":100785},[66],"，再用 ",[15,100788,100790,100822],{"className":100789,"translate":19},[18],[15,100791,100793],{"className":100792},[23],[25,100794,100795],{"xmlns":27},[29,100796,100797,100819],{},[32,100798,100799,100801,100803,100805,100807,100809,100811,100817],{},[35,100800,11],{},[39,100802,120],{"stretchy":119},[35,100804,1224],{},[39,100806,34430],{},[35,100808,123],{},[39,100810,335],{"separator":53},[4296,100812,100813,100815],{"accent":53},[35,100814,61618],{},[39,100816,78130],{},[39,100818,126],{"stretchy":119},[45,100820,100821],{"encoding":47},"p(y\\mid x,\\hat\\theta)",[15,100823,100825,100849],{"className":100824,"ariaHidden":53},[52],[15,100826,100828,100831,100834,100837,100840,100843,100846],{"className":100827},[57],[15,100829],{"className":100830,"style":154},[61],[15,100832,11],{"className":100833},[66,67],[15,100835,120],{"className":100836},[162],[15,100838,1224],{"className":100839,"style":1515},[66,67],[15,100841],{"className":100842,"style":173},[71],[15,100844,34430],{"className":100845},[177],[15,100847],{"className":100848,"style":173},[71],[15,100850,100852,100855,100858,100861,100864,100895],{"className":100851},[57],[15,100853],{"className":100854,"style":78282},[61],[15,100856,123],{"className":100857},[66,67],[15,100859,335],{"className":100860},[422],[15,100862],{"className":100863,"style":697},[71],[15,100865,100867],{"className":100866},[66,78147],[15,100868,100870],{"className":100869},[202],[15,100871,100873],{"className":100872},[206],[15,100874,100876,100884],{"className":100875,"style":78157},[210],[15,100877,100878,100881],{"style":1386},[15,100879],{"className":100880,"style":1368},[217],[15,100882,61618],{"className":100883,"style":23025},[66,67],[15,100885,100886,100889],{"style":78168},[15,100887],{"className":100888,"style":1368},[217],[15,100890,100892],{"className":100891,"style":96110},[78175],[15,100893,78130],{"className":100894},[66],[15,100896,126],{"className":100897},[169]," 预测；",[3194,100900,100901,100904,100905],{},[2040,100902,100903],{},"后验预测","：对参数积分，\n",[15,100906,100908],{"className":100907,"translate":19},[294],[15,100909,100911,100982],{"className":100910,"translate":19},[18],[15,100912,100914],{"className":100913},[23],[25,100915,100916],{"xmlns":27,"display":303},[29,100917,100918,100979],{},[32,100919,100920,100927,100929,100931,100933,100935,100937,100939,100941,100943,100945,100947,100949,100951,100953,100955,100957,100959,100961,100963,100965,100967,100969,100971,100973,100975,100977],{},[4296,100921,100922,100924],{"accent":53},[35,100923,11],{},[39,100925,100926],{},"ˉ",[39,100928,120],{"stretchy":119},[35,100930,1224],{},[39,100932,34430],{},[35,100934,123],{},[39,100936,126],{"stretchy":119},[39,100938,129],{},[39,100940,29732],{},[35,100942,11],{},[39,100944,120],{"stretchy":119},[35,100946,1224],{},[39,100948,34430],{},[35,100950,123],{},[39,100952,335],{"separator":53},[35,100954,61618],{},[39,100956,126],{"stretchy":119},[1274,100958,2462],{},[35,100960,11],{},[39,100962,120],{"stretchy":119},[35,100964,61618],{},[39,100966,34430],{},[35,100968,32458],{},[39,100970,126],{"stretchy":119},[1274,100972,2462],{},[35,100974,458],{},[35,100976,61618],{},[39,100978,8413],{"separator":53},[45,100980,100981],{"encoding":47},"\\bar p(y\\mid x)=\\int p(y\\mid x,\\theta)\\,p(\\theta\\mid D)\\,d\\theta;",[15,100983,100985,101049,101070,101100,101142],{"className":100984,"ariaHidden":53},[52],[15,100986,100988,100991,101034,101037,101040,101043,101046],{"className":100987},[57],[15,100989],{"className":100990,"style":154},[61],[15,100992,100994],{"className":100993},[66,78147],[15,100995,100997,101026],{"className":100996},[202,1090],[15,100998,101000,101023],{"className":100999},[206],[15,101001,101004,101012],{"className":101002,"style":101003},[210],"height:0.5678em;",[15,101005,101006,101009],{"style":1386},[15,101007],{"className":101008,"style":1368},[217],[15,101010,11],{"className":101011},[66,67],[15,101013,101014,101017],{"style":1386},[15,101015],{"className":101016,"style":1368},[217],[15,101018,101020],{"className":101019,"style":96110},[78175],[15,101021,100926],{"className":101022},[66],[15,101024,1114],{"className":101025},[1113],[15,101027,101029],{"className":101028},[206],[15,101030,101032],{"className":101031,"style":4450},[210],[15,101033],{},[15,101035,120],{"className":101036},[162],[15,101038,1224],{"className":101039,"style":1515},[66,67],[15,101041],{"className":101042,"style":173},[71],[15,101044,34430],{"className":101045},[177],[15,101047],{"className":101048,"style":173},[71],[15,101050,101052,101055,101058,101061,101064,101067],{"className":101051},[57],[15,101053],{"className":101054,"style":154},[61],[15,101056,123],{"className":101057},[66,67],[15,101059,126],{"className":101060},[169],[15,101062],{"className":101063,"style":173},[71],[15,101065,129],{"className":101066},[177],[15,101068],{"className":101069,"style":173},[71],[15,101071,101073,101076,101079,101082,101085,101088,101091,101094,101097],{"className":101072},[57],[15,101074],{"className":101075,"style":65130},[61],[15,101077,29732],{"className":101078,"style":29796},[1339,1470,1471],[15,101080],{"className":101081,"style":697},[71],[15,101083,11],{"className":101084},[66,67],[15,101086,120],{"className":101087},[162],[15,101089,1224],{"className":101090,"style":1515},[66,67],[15,101092],{"className":101093,"style":173},[71],[15,101095,34430],{"className":101096},[177],[15,101098],{"className":101099,"style":173},[71],[15,101101,101103,101106,101109,101112,101115,101118,101121,101124,101127,101130,101133,101136,101139],{"className":101102},[57],[15,101104],{"className":101105,"style":154},[61],[15,101107,123],{"className":101108},[66,67],[15,101110,335],{"className":101111},[422],[15,101113],{"className":101114,"style":697},[71],[15,101116,61618],{"className":101117,"style":23025},[66,67],[15,101119,126],{"className":101120},[169],[15,101122],{"className":101123,"style":697},[71],[15,101125,11],{"className":101126},[66,67],[15,101128,120],{"className":101129},[162],[15,101131,61618],{"className":101132,"style":23025},[66,67],[15,101134],{"className":101135,"style":173},[71],[15,101137,34430],{"className":101138},[177],[15,101140],{"className":101141,"style":173},[71],[15,101143,101145,101148,101151,101154,101157,101160,101163],{"className":101144},[57],[15,101146],{"className":101147,"style":154},[61],[15,101149,32458],{"className":101150,"style":23025},[66,67],[15,101152,126],{"className":101153},[169],[15,101155],{"className":101156,"style":697},[71],[15,101158,458],{"className":101159},[66,67],[15,101161,61618],{"className":101162,"style":23025},[66,67],[15,101164,8413],{"className":101165},[422],[3194,101167,101168,101170,101171],{},[2040,101169,75111],{},"：在预测分布上再做一次决策，\n",[15,101172,101174],{"className":101173,"translate":19},[294],[15,101175,101177,101257],{"className":101176,"translate":19},[18],[15,101178,101180],{"className":101179},[23],[25,101181,101182],{"xmlns":27,"display":303},[29,101183,101184,101254],{},[32,101185,101186,101192,101194,101196,101198,101200,101202,101204,101214,101216,101218,101220,101222,101224,101226,101228,101230,101236,101238,101240,101242,101244,101246,101248,101250,101252],{},[131,101187,101188,101190],{},[35,101189,4556],{},[39,101191,1162],{},[39,101193,120],{"stretchy":119},[35,101195,123],{},[39,101197,126],{"stretchy":119},[39,101199,129],{},[35,101201,1167],{},[39,101203,1170],{},[1172,101205,101206,101212],{},[32,101207,101208,101210],{},[35,101209,1178],{},[39,101211,1170],{},[35,101213,4556],{},[39,101215,29732],{},[35,101217,1216],{},[39,101219,120],{"stretchy":119},[35,101221,1224],{},[39,101223,335],{"separator":53},[35,101225,4556],{},[39,101227,126],{"stretchy":119},[1274,101229,2462],{},[4296,101231,101232,101234],{"accent":53},[35,101233,11],{},[39,101235,100926],{},[39,101237,120],{"stretchy":119},[35,101239,1224],{},[39,101241,34430],{},[35,101243,123],{},[39,101245,126],{"stretchy":119},[1274,101247,2462],{},[35,101249,458],{},[35,101251,1224],{},[35,101253,3573],{"mathvariant":138},[45,101255,101256],{"encoding":47},"a^*(x)=\\arg\\min_a\\int L(y,a)\\,\\bar p(y\\mid x)\\,dy.",[15,101258,101260,101313,101461],{"className":101259,"ariaHidden":53},[52],[15,101261,101263,101266,101295,101298,101301,101304,101307,101310],{"className":101262},[57],[15,101264],{"className":101265,"style":154},[61],[15,101267,101269,101272],{"className":101268},[66],[15,101270,4556],{"className":101271},[66,67],[15,101273,101275],{"className":101274},[198],[15,101276,101278],{"className":101277},[202],[15,101279,101281],{"className":101280},[206],[15,101282,101284],{"className":101283,"style":1308},[210],[15,101285,101286,101289],{"style":397},[15,101287],{"className":101288,"style":218},[217],[15,101290,101292],{"className":101291},[222,223,224,225],[15,101293,1162],{"className":101294},[76,225],[15,101296,120],{"className":101297},[162],[15,101299,123],{"className":101300},[66,67],[15,101302,126],{"className":101303},[169],[15,101305],{"className":101306,"style":173},[71],[15,101308,129],{"className":101309},[177],[15,101311],{"className":101312,"style":173},[71],[15,101314,101316,101319,101324,101327,101371,101374,101377,101380,101383,101386,101389,101392,101395,101398,101401,101404,101446,101449,101452,101455,101458],{"className":101315},[57],[15,101317],{"className":101318,"style":65130},[61],[15,101320,1340,101322],{"className":101321},[1339],[15,101323,1344],{"style":1343},[15,101325],{"className":101326,"style":697},[71],[15,101328,101330],{"className":101329},[1339,1351],[15,101331,101333,101363],{"className":101332},[202,1090],[15,101334,101336,101360],{"className":101335},[206],[15,101337,101339,101350],{"className":101338,"style":1361},[210],[15,101340,101341,101344],{"style":48823},[15,101342],{"className":101343,"style":1368},[217],[15,101345,101347],{"className":101346},[222,223,224,225],[15,101348,4556],{"className":101349},[66,67,225],[15,101351,101352,101355],{"style":1386},[15,101353],{"className":101354,"style":1368},[217],[15,101356,101357],{},[15,101358,1178],{"className":101359},[1339],[15,101361,1114],{"className":101362},[1113],[15,101364,101366],{"className":101365},[206],[15,101367,101369],{"className":101368,"style":48861},[210],[15,101370],{},[15,101372],{"className":101373,"style":697},[71],[15,101375,29732],{"className":101376,"style":29796},[1339,1470,1471],[15,101378],{"className":101379,"style":697},[71],[15,101381,1216],{"className":101382},[66,67],[15,101384,120],{"className":101385},[162],[15,101387,1224],{"className":101388,"style":1515},[66,67],[15,101390,335],{"className":101391},[422],[15,101393],{"className":101394,"style":697},[71],[15,101396,4556],{"className":101397},[66,67],[15,101399,126],{"className":101400},[169],[15,101402],{"className":101403,"style":697},[71],[15,101405,101407],{"className":101406},[66,78147],[15,101408,101410,101438],{"className":101409},[202,1090],[15,101411,101413,101435],{"className":101412},[206],[15,101414,101416,101424],{"className":101415,"style":101003},[210],[15,101417,101418,101421],{"style":1386},[15,101419],{"className":101420,"style":1368},[217],[15,101422,11],{"className":101423},[66,67],[15,101425,101426,101429],{"style":1386},[15,101427],{"className":101428,"style":1368},[217],[15,101430,101432],{"className":101431,"style":96110},[78175],[15,101433,100926],{"className":101434},[66],[15,101436,1114],{"className":101437},[1113],[15,101439,101441],{"className":101440},[206],[15,101442,101444],{"className":101443,"style":4450},[210],[15,101445],{},[15,101447,120],{"className":101448},[162],[15,101450,1224],{"className":101451,"style":1515},[66,67],[15,101453],{"className":101454,"style":173},[71],[15,101456,34430],{"className":101457},[177],[15,101459],{"className":101460,"style":173},[71],[15,101462,101464,101467,101470,101473,101476,101479,101482],{"className":101463},[57],[15,101465],{"className":101466,"style":154},[61],[15,101468,123],{"className":101469},[66,67],[15,101471,126],{"className":101472},[169],[15,101474],{"className":101475,"style":697},[71],[15,101477,458],{"className":101478},[66,67],[15,101480,1224],{"className":101481,"style":1515},[66,67],[15,101483,3573],{"className":101484},[66],[11,101486,101487],{},"第二层是贝叶斯学习的核心，第三层才是完整的贝叶斯决策。很多讨论停在第二层，把「后验预测」直接当成答案；但在有明确损失的问题里，真正的答案是第三层。",[11,101489,101490,101491,101570,101571,101599],{},"后验预测是一个混合分布，不是「把后验均值代进去」。由于 ",[15,101492,101494,101522],{"className":101493,"translate":19},[18],[15,101495,101497],{"className":101496},[23],[25,101498,101499],{"xmlns":27},[29,101500,101501,101519],{},[32,101502,101503,101505,101507,101509,101511,101513,101515,101517],{},[35,101504,11],{},[39,101506,120],{"stretchy":119},[35,101508,1224],{},[39,101510,34430],{},[35,101512,123],{},[39,101514,335],{"separator":53},[35,101516,61618],{},[39,101518,126],{"stretchy":119},[45,101520,101521],{"encoding":47},"p(y\\mid x,\\theta)",[15,101523,101525,101549],{"className":101524,"ariaHidden":53},[52],[15,101526,101528,101531,101534,101537,101540,101543,101546],{"className":101527},[57],[15,101529],{"className":101530,"style":154},[61],[15,101532,11],{"className":101533},[66,67],[15,101535,120],{"className":101536},[162],[15,101538,1224],{"className":101539,"style":1515},[66,67],[15,101541],{"className":101542,"style":173},[71],[15,101544,34430],{"className":101545},[177],[15,101547],{"className":101548,"style":173},[71],[15,101550,101552,101555,101558,101561,101564,101567],{"className":101551},[57],[15,101553],{"className":101554,"style":154},[61],[15,101556,123],{"className":101557},[66,67],[15,101559,335],{"className":101560},[422],[15,101562],{"className":101563,"style":697},[71],[15,101565,61618],{"className":101566,"style":23025},[66,67],[15,101568,126],{"className":101569},[169]," 通常对 ",[15,101572,101574,101587],{"className":101573,"translate":19},[18],[15,101575,101577],{"className":101576},[23],[25,101578,101579],{"xmlns":27},[29,101580,101581,101585],{},[32,101582,101583],{},[35,101584,61618],{},[45,101586,64805],{"encoding":47},[15,101588,101590],{"className":101589,"ariaHidden":53},[52],[15,101591,101593,101596],{"className":101592},[57],[15,101594],{"className":101595,"style":840},[61],[15,101597,61618],{"className":101598,"style":23025},[66,67]," 非线性，",[15,101601,101603],{"className":101602,"translate":19},[294],[15,101604,101606,101686],{"className":101605,"translate":19},[18],[15,101607,101609],{"className":101608},[23],[25,101610,101611],{"xmlns":27,"display":303},[29,101612,101613,101683],{},[32,101614,101615,101617,101619,101621,101623,101625,101627,101629,101631,101633,101635,101637,101639,101641,101643,101645,101647,101649,101651,101653,101655,101657,101659,101661,101663,101665,101667,101669,101671,101673,101675,101677,101679,101681],{},[39,101616,29732],{},[35,101618,11],{},[39,101620,120],{"stretchy":119},[35,101622,1224],{},[39,101624,34430],{},[35,101626,123],{},[39,101628,335],{"separator":53},[35,101630,61618],{},[39,101632,126],{"stretchy":119},[1274,101634,2462],{},[35,101636,11],{},[39,101638,120],{"stretchy":119},[35,101640,61618],{},[39,101642,34430],{},[35,101644,32458],{},[39,101646,126],{"stretchy":119},[1274,101648,2462],{},[35,101650,458],{},[35,101652,61618],{},[39,101654,13019],{"mathvariant":138},[35,101656,11],{},[39,101658,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,101660,1224],{},[39,101662,34430],{},[35,101664,123],{},[39,101666,335],{"separator":53},[35,101668,62381],{"mathvariant":454},[39,101670,1194],{"stretchy":119},[35,101672,61618],{},[39,101674,34430],{},[35,101676,32458],{},[39,101678,1266],{"stretchy":119},[39,101680,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,101682,3573],{"mathvariant":138},[45,101684,101685],{"encoding":47},"\\int p(y\\mid x,\\theta)\\,p(\\theta\\mid D)\\,d\\theta\n\\ne\np\\big(y\\mid x,\\mathbb E[\\theta\\mid D]\\big).",[15,101687,101689,101719,101761,101824,101851,101884],{"className":101688,"ariaHidden":53},[52],[15,101690,101692,101695,101698,101701,101704,101707,101710,101713,101716],{"className":101691},[57],[15,101693],{"className":101694,"style":65130},[61],[15,101696,29732],{"className":101697,"style":29796},[1339,1470,1471],[15,101699],{"className":101700,"style":697},[71],[15,101702,11],{"className":101703},[66,67],[15,101705,120],{"className":101706},[162],[15,101708,1224],{"className":101709,"style":1515},[66,67],[15,101711],{"className":101712,"style":173},[71],[15,101714,34430],{"className":101715},[177],[15,101717],{"className":101718,"style":173},[71],[15,101720,101722,101725,101728,101731,101734,101737,101740,101743,101746,101749,101752,101755,101758],{"className":101721},[57],[15,101723],{"className":101724,"style":154},[61],[15,101726,123],{"className":101727},[66,67],[15,101729,335],{"className":101730},[422],[15,101732],{"className":101733,"style":697},[71],[15,101735,61618],{"className":101736,"style":23025},[66,67],[15,101738,126],{"className":101739},[169],[15,101741],{"className":101742,"style":697},[71],[15,101744,11],{"className":101745},[66,67],[15,101747,120],{"className":101748},[162],[15,101750,61618],{"className":101751,"style":23025},[66,67],[15,101753],{"className":101754,"style":173},[71],[15,101756,34430],{"className":101757},[177],[15,101759],{"className":101760,"style":173},[71],[15,101762,101764,101767,101770,101773,101776,101779,101782,101785,101821],{"className":101763},[57],[15,101765],{"className":101766,"style":154},[61],[15,101768,32458],{"className":101769,"style":23025},[66,67],[15,101771,126],{"className":101772},[169],[15,101774],{"className":101775,"style":697},[71],[15,101777,458],{"className":101778},[66,67],[15,101780,61618],{"className":101781,"style":23025},[66,67],[15,101783],{"className":101784,"style":173},[71],[15,101786,101788,101815,101818],{"className":101787},[177],[15,101789,101791],{"className":101790},[177],[15,101792,101794],{"className":101793},[66,13086],[15,101795,101797],{"className":101796},[13090],[15,101798,101800,101803,101812],{"className":101799},[13094],[15,101801],{"className":101802,"style":3408},[61],[15,101804,101806],{"className":101805},[13101],[15,101807,101809],{"className":101808},[66],[15,101810,13108],{"className":101811},[177],[15,101813],{"className":101814},[13112],[15,101816],{"className":101817},[71,13116],[15,101819,129],{"className":101820},[177],[15,101822],{"className":101823,"style":173},[71],[15,101825,101827,101830,101833,101839,101842,101845,101848],{"className":101826},[57],[15,101828],{"className":101829,"style":4400},[61],[15,101831,11],{"className":101832},[66,67],[15,101834,101836],{"className":101835},[66],[15,101837,120],{"className":101838},[1422,1508],[15,101840,1224],{"className":101841,"style":1515},[66,67],[15,101843],{"className":101844,"style":173},[71],[15,101846,34430],{"className":101847},[177],[15,101849],{"className":101850,"style":173},[71],[15,101852,101854,101857,101860,101863,101866,101869,101872,101875,101878,101881],{"className":101853},[57],[15,101855],{"className":101856,"style":154},[61],[15,101858,123],{"className":101859},[66,67],[15,101861,335],{"className":101862},[422],[15,101864],{"className":101865,"style":697},[71],[15,101867,62381],{"className":101868},[66,517],[15,101870,1194],{"className":101871},[162],[15,101873,61618],{"className":101874,"style":23025},[66,67],[15,101876],{"className":101877,"style":173},[71],[15,101879,34430],{"className":101880},[177],[15,101882],{"className":101883,"style":173},[71],[15,101885,101887,101890,101893,101896,101902],{"className":101886},[57],[15,101888],{"className":101889,"style":4400},[61],[15,101891,32458],{"className":101892,"style":23025},[66,67],[15,101894,1266],{"className":101895},[169],[15,101897,101899],{"className":101898},[66],[15,101900,126],{"className":101901},[1422,1508],[15,101903,3573],{"className":101904},[66],[11,101906,101907],{},"差别在不确定性大的区域尤其明显。设想后验在两个相距很远的参数值上各有一个峰，代入均值会得到一个位于两峰之间、几乎没有概率质量的预测；后验预测则会把两个峰的概率都保留下来。前者过度自信，后者诚实地表达了「我不知道是哪一个」。",[1904,101909,101911],{"id":101910},"_62-两种不确定性","6.2 两种不确定性",[11,101913,101914,101915,101918,101919,101947,101948,101976,101977,101980],{},"预测的不确定性可以分成两类。",[2040,101916,101917],{},"偶然不确定性","（aleatoric uncertainty）来自数据生成过程本身的噪声：同样的 ",[15,101920,101922,101935],{"className":101921,"translate":19},[18],[15,101923,101925],{"className":101924},[23],[25,101926,101927],{"xmlns":27},[29,101928,101929,101933],{},[32,101930,101931],{},[35,101932,123],{},[45,101934,123],{"encoding":47},[15,101936,101938],{"className":101937,"ariaHidden":53},[52],[15,101939,101941,101944],{"className":101940},[57],[15,101942],{"className":101943,"style":86},[61],[15,101945,123],{"className":101946},[66,67]," 可能对应不同的 ",[15,101949,101951,101964],{"className":101950,"translate":19},[18],[15,101952,101954],{"className":101953},[23],[25,101955,101956],{"xmlns":27},[29,101957,101958,101962],{},[32,101959,101960],{},[35,101961,1224],{},[45,101963,1224],{"encoding":47},[15,101965,101967],{"className":101966,"ariaHidden":53},[52],[15,101968,101970,101973],{"className":101969},[57],[15,101971],{"className":101972,"style":471},[61],[15,101974,1224],{"className":101975,"style":1515},[66,67],"，即使知道真实参数也无法消除。",[2040,101978,101979],{},"认知不确定性","（epistemic uncertainty）来自知识不足：我们不知道哪个参数是对的，但随着数据增多，它会收缩。",[11,101982,101983],{},"对平方损失，这两种不确定性由全方差公式精确分解：",[15,101985,101987],{"className":101986,"translate":19},[294],[15,101988,101990,102128],{"className":101989,"translate":19},[18],[15,101991,101993],{"className":101992},[23],[25,101994,101995],{"xmlns":27,"display":303},[29,101996,101997,102125],{},[32,101998,101999,102007,102009,102011,102013,102015,102017,102019,102021,102023,102072,102074,102123],{},[32,102000,102001,102003,102005],{},[35,102002,31732],{"mathvariant":138},[35,102004,4556],{"mathvariant":138},[35,102006,63107],{"mathvariant":138},[39,102008,120],{"stretchy":119},[35,102010,61885],{},[39,102012,34430],{},[35,102014,123],{},[39,102016,335],{"separator":53},[35,102018,32458],{},[39,102020,126],{"stretchy":119},[39,102022,129],{},[1172,102024,102025,102069],{},[1172,102026,102027,102067],{},[32,102028,102029,102041,102043,102051,102053,102055,102057,102059,102061,102063,102065],{},[1057,102030,102031,102033],{},[35,102032,62381],{"mathvariant":454},[32,102034,102035,102037,102039],{},[35,102036,61618],{},[39,102038,34430],{},[35,102040,32458],{},[39,102042,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[32,102044,102045,102047,102049],{},[35,102046,31732],{"mathvariant":138},[35,102048,4556],{"mathvariant":138},[35,102050,63107],{"mathvariant":138},[39,102052,120],{"stretchy":119},[35,102054,61885],{},[39,102056,34430],{},[35,102058,123],{},[39,102060,335],{"separator":53},[35,102062,61618],{},[39,102064,126],{"stretchy":119},[39,102066,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,102068,69846],{"stretchy":53},[1274,102070,102071],{},"偶然",[39,102073,1245],{},[1172,102075,102076,102120],{},[1172,102077,102078,102118],{},[32,102079,102080,102098,102100,102102,102104,102106,102108,102110,102112,102114,102116],{},[1057,102081,102082,102090],{},[32,102083,102084,102086,102088],{},[35,102085,31732],{"mathvariant":138},[35,102087,4556],{"mathvariant":138},[35,102089,63107],{"mathvariant":138},[32,102091,102092,102094,102096],{},[35,102093,61618],{},[39,102095,34430],{},[35,102097,32458],{},[39,102099,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,102101,62381],{"mathvariant":454},[39,102103,1194],{"stretchy":119},[35,102105,61885],{},[39,102107,34430],{},[35,102109,123],{},[39,102111,335],{"separator":53},[35,102113,61618],{},[39,102115,1266],{"stretchy":119},[39,102117,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,102119,69846],{"stretchy":53},[1274,102121,102122],{},"认知",[35,102124,3573],{"mathvariant":138},[45,102126,102127],{"encoding":47},"\\mathrm{Var}(Y\\mid x,D)\n=\n\\underbrace{\\mathbb E_{\\theta\\mid D}\\big[\\mathrm{Var}(Y\\mid x,\\theta)\\big]}_{\\text{偶然}}\n+\n\\underbrace{\\mathrm{Var}_{\\theta\\mid D}\\big(\\mathbb E[Y\\mid x,\\theta]\\big)}_{\\text{认知}}.",[15,102129,102131,102158,102188,102412],{"className":102130,"ariaHidden":53},[52],[15,102132,102134,102137,102143,102146,102149,102152,102155],{"className":102133},[57],[15,102135],{"className":102136,"style":154},[61],[15,102138,102140],{"className":102139},[66],[15,102141,72407],{"className":102142},[66,4429],[15,102144,120],{"className":102145},[162],[15,102147,61885],{"className":102148,"style":72},[66,67],[15,102150],{"className":102151,"style":173},[71],[15,102153,34430],{"className":102154},[177],[15,102156],{"className":102157,"style":173},[71],[15,102159,102161,102164,102167,102170,102173,102176,102179,102182,102185],{"className":102160},[57],[15,102162],{"className":102163,"style":154},[61],[15,102165,123],{"className":102166},[66,67],[15,102168,335],{"className":102169},[422],[15,102171],{"className":102172,"style":697},[71],[15,102174,32458],{"className":102175,"style":23025},[66,67],[15,102177,126],{"className":102178},[169],[15,102180],{"className":102181,"style":173},[71],[15,102183,129],{"className":102184},[177],[15,102186],{"className":102187,"style":173},[71],[15,102189,102191,102195,102403,102406,102409],{"className":102190},[57],[15,102192],{"className":102193,"style":102194},[61],"height:2.5315em;vertical-align:-1.6815em;",[15,102196,102198],{"className":102197},[1413,1172],[15,102199,102201,102394],{"className":102200},[202,1090],[15,102202,102204,102391],{"className":102203},[206],[15,102205,102208,102226],{"className":102206,"style":102207},[210],"height:0.85em;",[15,102209,102211,102214],{"style":102210},"top:-1.3185em;",[15,102212],{"className":102213,"style":1368},[217],[15,102215,102217],{"className":102216},[222,223,224,225],[15,102218,102220],{"className":102219},[66,225],[15,102221,102223],{"className":102222},[66,1711,225],[15,102224,102071],{"className":102225},[66,68616,225],[15,102227,102228,102231],{"style":1386},[15,102229],{"className":102230,"style":1368},[217],[15,102232,102234],{"className":102233},[1413,1172],[15,102235,102237,102382],{"className":102236},[202,1090],[15,102238,102240,102379],{"className":102239},[206],[15,102241,102243,102274],{"className":102242,"style":102207},[210],[15,102244,102247,102250],{"className":102245,"style":102246},[48564],"top:-1.9968em;",[15,102248],{"className":102249,"style":1368},[217],[15,102251,102253,102260,102267],{"className":102252,"style":70049},[70048],[15,102254,102256],{"className":102255,"style":70054},[70053],[34828,102257,102258],{"xmlns":34830,"width":48594,"height":70057,"viewBox":70058,"preserveAspectRatio":48597},[34834,102259],{"d":70061},[15,102261,102263],{"className":102262,"style":70054},[70065],[34828,102264,102265],{"xmlns":34830,"width":48594,"height":70057,"viewBox":70058,"preserveAspectRatio":70068},[34834,102266],{"d":70071},[15,102268,102270],{"className":102269,"style":70054},[70075],[34828,102271,102272],{"xmlns":34830,"width":48594,"height":70057,"viewBox":70058,"preserveAspectRatio":70078},[34834,102273],{"d":70081},[15,102275,102276,102279],{"style":1386},[15,102277],{"className":102278,"style":1368},[217],[15,102280,102282,102331,102337,102343,102346,102349,102352,102355,102358,102361,102364,102367,102370,102373],{"className":102281},[66],[15,102283,102285,102288],{"className":102284},[66],[15,102286,62381],{"className":102287},[66,517],[15,102289,102291],{"className":102290},[198],[15,102292,102294,102323],{"className":102293},[202,1090],[15,102295,102297,102320],{"className":102296},[206],[15,102298,102300],{"className":102299,"style":3668},[210],[15,102301,102302,102305],{"style":62482},[15,102303],{"className":102304,"style":218},[217],[15,102306,102308],{"className":102307},[222,223,224,225],[15,102309,102311,102314,102317],{"className":102310},[66,225],[15,102312,61618],{"className":102313,"style":23025},[66,67,225],[15,102315,34430],{"className":102316},[177,225],[15,102318,32458],{"className":102319,"style":23025},[66,67,225],[15,102321,1114],{"className":102322},[1113],[15,102324,102326],{"className":102325},[206],[15,102327,102329],{"className":102328,"style":9880},[210],[15,102330],{},[15,102332,102334],{"className":102333},[66],[15,102335,1194],{"className":102336},[1422,1508],[15,102338,102340],{"className":102339},[66],[15,102341,72407],{"className":102342},[66,4429],[15,102344,120],{"className":102345},[162],[15,102347,61885],{"className":102348,"style":72},[66,67],[15,102350],{"className":102351,"style":173},[71],[15,102353,34430],{"className":102354},[177],[15,102356],{"className":102357,"style":173},[71],[15,102359,123],{"className":102360},[66,67],[15,102362,335],{"className":102363},[422],[15,102365],{"className":102366,"style":697},[71],[15,102368,61618],{"className":102369,"style":23025},[66,67],[15,102371,126],{"className":102372},[169],[15,102374,102376],{"className":102375},[66],[15,102377,1266],{"className":102378},[1422,1508],[15,102380,1114],{"className":102381},[1113],[15,102383,102385],{"className":102384},[206],[15,102386,102389],{"className":102387,"style":102388},[210],"height:1.0032em;",[15,102390],{},[15,102392,1114],{"className":102393},[1113],[15,102395,102397],{"className":102396},[206],[15,102398,102401],{"className":102399,"style":102400},[210],"height:1.6815em;",[15,102402],{},[15,102404],{"className":102405,"style":72},[71],[15,102407,1245],{"className":102408},[76],[15,102410],{"className":102411,"style":72},[71],[15,102413,102415,102418,102622,102625],{"className":102414},[57],[15,102416],{"className":102417,"style":102194},[61],[15,102419,102421],{"className":102420},[1413,1172],[15,102422,102424,102614],{"className":102423},[202,1090],[15,102425,102427,102611],{"className":102426},[206],[15,102428,102430,102447],{"className":102429,"style":102207},[210],[15,102431,102432,102435],{"style":102210},[15,102433],{"className":102434,"style":1368},[217],[15,102436,102438],{"className":102437},[222,223,224,225],[15,102439,102441],{"className":102440},[66,225],[15,102442,102444],{"className":102443},[66,1711,225],[15,102445,102122],{"className":102446},[66,68616,225],[15,102448,102449,102452],{"style":1386},[15,102450],{"className":102451,"style":1368},[217],[15,102453,102455],{"className":102454},[1413,1172],[15,102456,102458,102603],{"className":102457},[202,1090],[15,102459,102461,102600],{"className":102460},[206],[15,102462,102464,102494],{"className":102463,"style":102207},[210],[15,102465,102467,102470],{"className":102466,"style":102246},[48564],[15,102468],{"className":102469,"style":1368},[217],[15,102471,102473,102480,102487],{"className":102472,"style":70049},[70048],[15,102474,102476],{"className":102475,"style":70054},[70053],[34828,102477,102478],{"xmlns":34830,"width":48594,"height":70057,"viewBox":70058,"preserveAspectRatio":48597},[34834,102479],{"d":70061},[15,102481,102483],{"className":102482,"style":70054},[70065],[34828,102484,102485],{"xmlns":34830,"width":48594,"height":70057,"viewBox":70058,"preserveAspectRatio":70068},[34834,102486],{"d":70071},[15,102488,102490],{"className":102489,"style":70054},[70075],[34828,102491,102492],{"xmlns":34830,"width":48594,"height":70057,"viewBox":70058,"preserveAspectRatio":70078},[34834,102493],{"d":70081},[15,102495,102496,102499],{"style":1386},[15,102497],{"className":102498,"style":1368},[217],[15,102500,102502,102555,102561,102564,102567,102570,102573,102576,102579,102582,102585,102588,102591,102594],{"className":102501},[66],[15,102503,102505,102511],{"className":102504},[66],[15,102506,102508],{"className":102507},[66],[15,102509,72407],{"className":102510},[66,4429],[15,102512,102514],{"className":102513},[198],[15,102515,102517,102547],{"className":102516},[202,1090],[15,102518,102520,102544],{"className":102519},[206],[15,102521,102523],{"className":102522,"style":3668},[210],[15,102524,102526,102529],{"style":102525},"top:-2.5198em;margin-right:0.05em;",[15,102527],{"className":102528,"style":218},[217],[15,102530,102532],{"className":102531},[222,223,224,225],[15,102533,102535,102538,102541],{"className":102534},[66,225],[15,102536,61618],{"className":102537,"style":23025},[66,67,225],[15,102539,34430],{"className":102540},[177,225],[15,102542,32458],{"className":102543,"style":23025},[66,67,225],[15,102545,1114],{"className":102546},[1113],[15,102548,102550],{"className":102549},[206],[15,102551,102553],{"className":102552,"style":9880},[210],[15,102554],{},[15,102556,102558],{"className":102557},[66],[15,102559,120],{"className":102560},[1422,1508],[15,102562,62381],{"className":102563},[66,517],[15,102565,1194],{"className":102566},[162],[15,102568,61885],{"className":102569,"style":72},[66,67],[15,102571],{"className":102572,"style":173},[71],[15,102574,34430],{"className":102575},[177],[15,102577],{"className":102578,"style":173},[71],[15,102580,123],{"className":102581},[66,67],[15,102583,335],{"className":102584},[422],[15,102586],{"className":102587,"style":697},[71],[15,102589,61618],{"className":102590,"style":23025},[66,67],[15,102592,1266],{"className":102593},[169],[15,102595,102597],{"className":102596},[66],[15,102598,126],{"className":102599},[1422,1508],[15,102601,1114],{"className":102602},[1113],[15,102604,102606],{"className":102605},[206],[15,102607,102609],{"className":102608,"style":102388},[210],[15,102610],{},[15,102612,1114],{"className":102613},[1113],[15,102615,102617],{"className":102616},[206],[15,102618,102620],{"className":102619,"style":102400},[210],[15,102621],{},[15,102623],{"className":102624,"style":697},[71],[15,102626,3573],{"className":102627},[66],[11,102629,102630],{},"第一项是「即使知道模型参数，预测仍然会波动的部分」；第二项是「不同参数给出的预测均值之间的差异」。在数据稀疏的区域，后验宽，第二项大；随着数据增加，后验收缩，第二项趋于零。第一项则不会消失。",[11,102632,102633],{},"这个分解也是主动学习的理论基础：我们想选的样本，是那些能最大程度降低认知不确定性的样本，而不是那些本身就带噪声的样本。",[1904,102635,102637],{"id":102636},"_63-推断本身也是决策","6.3 推断本身也是决策",[11,102639,102640],{},"在参数空间上定义一个损失，报一个点估计，本质上也是一次决策：",[6152,102642,102643,102653],{},[6155,102644,102645],{},[6158,102646,102647,102650],{},[6161,102648,102649],{},"参数空间上的损失",[6161,102651,102652],{},"最优估计",[6171,102654,102655,102661,102667],{},[6158,102656,102657,102659],{},[6176,102658,75126],{},[6176,102660,72666],{},[6158,102662,102663,102665],{},[6176,102664,75133],{},[6176,102666,75136],{},[6158,102668,102669,102671],{},[6176,102670,75118],{},[6176,102672,102673],{},"后验众数（MAP）",[11,102675,102676],{},"所以贝叶斯估计不是与决策论并列的东西，而是决策论在参数空间上的应用。后验均值与 MAP 会打架，是因为它们对应不同的损失，没有谁更「正确」。若你关心的是平方误差，均值是对的；若你关心的是「最可能的那个值」，众数是对的；若你关心的是「误差不超过某个范围」，分位数才是对的。",[11,102678,102679],{},"同样的逻辑适用于区间估计。一个可信区间是一个「集合值」的行动，它的损失函数衡量区间是否覆盖真值、以及区间有多宽。置信区间与可信区间的差异当然有概率哲学的成分；但从决策论看，它们也对应不同的损失：一个关心重复抽样下的覆盖率，一个关心给定数据下的后验覆盖概率。",[1904,102681,102683],{"id":102682},"_64-后验的集中","6.4 后验的集中",[11,102685,102686,102687,102690],{},"在正则条件下，后验会随着数据增多而集中到真实参数附近。",[2040,102688,102689],{},"Bernstein–von Mises 定理","给出了更精确的结论：后验在适当的缩放与中心化之后，趋近于一个正态分布，",[15,102692,102694],{"className":102693,"translate":19},[294],[15,102695,102697,102778],{"className":102696,"translate":19},[18],[15,102698,102700],{"className":102699},[23],[25,102701,102702],{"xmlns":27,"display":303},[29,102703,102704,102775],{},[32,102705,102706,102710,102712,102714,102716,102718,102728,102730,102732,102734,102736,102739,102741,102743,102745,102747,102749,102751,102753,102755,102761,102771,102773],{},[48508,102707,102708],{},[35,102709,37],{},[1274,102711,2462],{},[39,102713,120],{"stretchy":119},[35,102715,61618],{},[39,102717,6602],{},[1057,102719,102720,102726],{},[4296,102721,102722,102724],{"accent":53},[35,102723,61618],{},[39,102725,78130],{},[35,102727,37],{},[39,102729,126],{"stretchy":119},[39,102731,34430],{},[35,102733,32458],{},[1274,102735,24233],{},[39,102737,102738],{},"⇒",[1274,102740,24233],{},[35,102742,28535],{"mathvariant":445},[39,102744,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[1209,102746,1804],{},[39,102748,335],{"separator":53},[1274,102750,24233],{},[35,102752,16427],{},[39,102754,120],{"stretchy":119},[1057,102756,102757,102759],{},[35,102758,61618],{},[1209,102760,1804],{},[131,102762,102763,102765],{},[39,102764,126],{"stretchy":119},[32,102766,102767,102769],{},[39,102768,6602],{},[1209,102770,1211],{},[39,102772,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,102774,335],{"separator":53},[45,102776,102777],{"encoding":47},"\\sqrt n\\,(\\theta-\\hat\\theta_n)\\mid D\n\\ \\Rightarrow\\\n\\mathcal N\\big(0,\\ I(\\theta_0)^{-1}\\big),",[15,102779,102781,102853,102939,102963],{"className":102780,"ariaHidden":53},[52],[15,102782,102784,102788,102835,102838,102841,102844,102847,102850],{"className":102783},[57],[15,102785],{"className":102786,"style":102787},[61],"height:1.0992em;vertical-align:-0.25em;",[15,102789,102791],{"className":102790},[66,48550],[15,102792,102794,102826],{"className":102793},[202,1090],[15,102795,102797,102823],{"className":102796},[206],[15,102798,102801,102810],{"className":102799,"style":102800},[210],"height:0.8492em;",[15,102802,102804,102807],{"className":102803,"style":1386},[48564],[15,102805],{"className":102806,"style":1368},[217],[15,102808,37],{"className":102809,"style":48571},[66,67],[15,102811,102813,102816],{"style":102812},"top:-2.8092em;",[15,102814],{"className":102815,"style":1368},[217],[15,102817,102819],{"className":102818,"style":48591},[48590],[34828,102820,102821],{"xmlns":34830,"width":48594,"height":48595,"viewBox":48596,"preserveAspectRatio":48597},[34834,102822],{"d":48600},[15,102824,1114],{"className":102825},[1113],[15,102827,102829],{"className":102828},[206],[15,102830,102833],{"className":102831,"style":102832},[210],"height:0.1908em;",[15,102834],{},[15,102836],{"className":102837,"style":697},[71],[15,102839,120],{"className":102840},[162],[15,102842,61618],{"className":102843,"style":23025},[66,67],[15,102845],{"className":102846,"style":72},[71],[15,102848,6602],{"className":102849},[76],[15,102851],{"className":102852,"style":72},[71],[15,102854,102856,102859,102927,102930,102933,102936],{"className":102855},[57],[15,102857],{"className":102858,"style":78282},[61],[15,102860,102862,102893],{"className":102861},[66],[15,102863,102865],{"className":102864},[66,78147],[15,102866,102868],{"className":102867},[202],[15,102869,102871],{"className":102870},[206],[15,102872,102874,102882],{"className":102873,"style":78157},[210],[15,102875,102876,102879],{"style":1386},[15,102877],{"className":102878,"style":1368},[217],[15,102880,61618],{"className":102881,"style":23025},[66,67],[15,102883,102884,102887],{"style":78168},[15,102885],{"className":102886,"style":1368},[217],[15,102888,102890],{"className":102889,"style":96110},[78175],[15,102891,78130],{"className":102892},[66],[15,102894,102896],{"className":102895},[198],[15,102897,102899,102919],{"className":102898},[202,1090],[15,102900,102902,102916],{"className":102901},[206],[15,102903,102905],{"className":102904,"style":2198},[210],[15,102906,102907,102910],{"style":88441},[15,102908],{"className":102909,"style":218},[217],[15,102911,102913],{"className":102912},[222,223,224,225],[15,102914,37],{"className":102915},[66,67,225],[15,102917,1114],{"className":102918},[1113],[15,102920,102922],{"className":102921},[206],[15,102923,102925],{"className":102924,"style":1121},[210],[15,102926],{},[15,102928,126],{"className":102929},[169],[15,102931],{"className":102932,"style":173},[71],[15,102934,34430],{"className":102935},[177],[15,102937],{"className":102938,"style":173},[71],[15,102940,102942,102945,102948,102951,102954,102957,102960],{"className":102941},[57],[15,102943],{"className":102944,"style":490},[61],[15,102946,32458],{"className":102947,"style":23025},[66,67],[15,102949,24233],{"className":102950},[71],[15,102952],{"className":102953,"style":173},[71],[15,102955,102738],{"className":102956},[177],[15,102958,24233],{"className":102959},[71],[15,102961],{"className":102962,"style":173},[71],[15,102964,102966,102969,102972,102978,102981,102984,102987,102990,102993,102996,103036,103071,103077],{"className":102965},[57],[15,102967],{"className":102968,"style":11718},[61],[15,102970,28535],{"className":102971,"style":83515},[66,494],[15,102973,102975],{"className":102974},[66],[15,102976,120],{"className":102977},[1422,1508],[15,102979,1804],{"className":102980},[66],[15,102982,335],{"className":102983},[422],[15,102985,24233],{"className":102986},[71],[15,102988],{"className":102989,"style":697},[71],[15,102991,16427],{"className":102992,"style":16497},[66,67],[15,102994,120],{"className":102995},[162],[15,102997,102999,103002],{"className":102998},[66],[15,103000,61618],{"className":103001,"style":23025},[66,67],[15,103003,103005],{"className":103004},[198],[15,103006,103008,103028],{"className":103007},[202,1090],[15,103009,103011,103025],{"className":103010},[206],[15,103012,103014],{"className":103013,"style":2139},[210],[15,103015,103016,103019],{"style":88441},[15,103017],{"className":103018,"style":218},[217],[15,103020,103022],{"className":103021},[222,223,224,225],[15,103023,1804],{"className":103024},[66,225],[15,103026,1114],{"className":103027},[1113],[15,103029,103031],{"className":103030},[206],[15,103032,103034],{"className":103033,"style":1121},[210],[15,103035],{},[15,103037,103039,103042],{"className":103038},[169],[15,103040,126],{"className":103041},[169],[15,103043,103045],{"className":103044},[198],[15,103046,103048],{"className":103047},[202],[15,103049,103051],{"className":103050},[206],[15,103052,103054],{"className":103053,"style":1650},[210],[15,103055,103056,103059],{"style":397},[15,103057],{"className":103058,"style":218},[217],[15,103060,103062],{"className":103061},[222,223,224,225],[15,103063,103065,103068],{"className":103064},[66,225],[15,103066,6602],{"className":103067},[66,225],[15,103069,1211],{"className":103070},[66,225],[15,103072,103074],{"className":103073},[66],[15,103075,126],{"className":103076},[1422,1508],[15,103078,335],{"className":103079},[422],[11,103081,425,103082,103167],{},[15,103083,103085,103109],{"className":103084,"translate":19},[18],[15,103086,103088],{"className":103087},[23],[25,103089,103090],{"xmlns":27},[29,103091,103092,103106],{},[32,103093,103094,103096,103098,103104],{},[35,103095,16427],{},[39,103097,120],{"stretchy":119},[1057,103099,103100,103102],{},[35,103101,61618],{},[1209,103103,1804],{},[39,103105,126],{"stretchy":119},[45,103107,103108],{"encoding":47},"I(\\theta_0)",[15,103110,103112],{"className":103111,"ariaHidden":53},[52],[15,103113,103115,103118,103121,103124,103164],{"className":103114},[57],[15,103116],{"className":103117,"style":154},[61],[15,103119,16427],{"className":103120,"style":16497},[66,67],[15,103122,120],{"className":103123},[162],[15,103125,103127,103130],{"className":103126},[66],[15,103128,61618],{"className":103129,"style":23025},[66,67],[15,103131,103133],{"className":103132},[198],[15,103134,103136,103156],{"className":103135},[202,1090],[15,103137,103139,103153],{"className":103138},[206],[15,103140,103142],{"className":103141,"style":2139},[210],[15,103143,103144,103147],{"style":88441},[15,103145],{"className":103146,"style":218},[217],[15,103148,103150],{"className":103149},[222,223,224,225],[15,103151,1804],{"className":103152},[66,225],[15,103154,1114],{"className":103155},[1113],[15,103157,103159],{"className":103158},[206],[15,103160,103162],{"className":103161,"style":1121},[210],[15,103163],{},[15,103165,126],{"className":103166},[169]," 是 Fisher 信息。等价地，后验近似为",[15,103169,103171],{"className":103170,"translate":19},[294],[15,103172,103174,103250],{"className":103173,"translate":19},[18],[15,103175,103177],{"className":103176},[23],[25,103178,103179],{"xmlns":27,"display":303},[29,103180,103181,103247],{},[32,103182,103183,103185,103187,103189,103191,103193,103195,103197,103199,103245],{},[35,103184,11],{},[39,103186,120],{"stretchy":119},[35,103188,61618],{},[39,103190,34430],{},[35,103192,32458],{},[39,103194,126],{"stretchy":119},[39,103196,31993],{},[35,103198,28535],{"mathvariant":445},[32,103200,103201,103203,103213,103215,103217,103223,103225,103227,103233,103243],{},[39,103202,120],{"fence":53},[1057,103204,103205,103211],{},[4296,103206,103207,103209],{"accent":53},[35,103208,61618],{},[39,103210,78130],{},[35,103212,37],{},[39,103214,335],{"separator":53},[1274,103216,24233],{},[6467,103218,103219,103221],{},[1209,103220,1211],{},[35,103222,37],{},[35,103224,16427],{},[39,103226,120],{"stretchy":119},[1057,103228,103229,103231],{},[35,103230,61618],{},[1209,103232,1804],{},[131,103234,103235,103237],{},[39,103236,126],{"stretchy":119},[32,103238,103239,103241],{},[39,103240,6602],{},[1209,103242,1211],{},[39,103244,126],{"fence":53},[35,103246,3573],{"mathvariant":138},[45,103248,103249],{"encoding":47},"p(\\theta\\mid D)\\approx\n\\mathcal N\\left(\\hat\\theta_n,\\ \\frac{1}{n}I(\\theta_0)^{-1}\\right).",[15,103251,103253,103277,103298],{"className":103252,"ariaHidden":53},[52],[15,103254,103256,103259,103262,103265,103268,103271,103274],{"className":103255},[57],[15,103257],{"className":103258,"style":154},[61],[15,103260,11],{"className":103261},[66,67],[15,103263,120],{"className":103264},[162],[15,103266,61618],{"className":103267,"style":23025},[66,67],[15,103269],{"className":103270,"style":173},[71],[15,103272,34430],{"className":103273},[177],[15,103275],{"className":103276,"style":173},[71],[15,103278,103280,103283,103286,103289,103292,103295],{"className":103279},[57],[15,103281],{"className":103282,"style":154},[61],[15,103284,32458],{"className":103285,"style":23025},[66,67],[15,103287,126],{"className":103288},[169],[15,103290],{"className":103291,"style":173},[71],[15,103293,31993],{"className":103294},[177],[15,103296],{"className":103297,"style":173},[71],[15,103299,103301,103304,103307,103310,103545,103548],{"className":103300},[57],[15,103302],{"className":103303,"style":43973},[61],[15,103305,28535],{"className":103306,"style":83515},[66,494],[15,103308],{"className":103309,"style":697},[71],[15,103311,103313,103319,103387,103390,103393,103396,103458,103461,103464,103504,103539],{"className":103312},[1413],[15,103314,103316],{"className":103315,"style":1418},[162,1417],[15,103317,120],{"className":103318},[1422,224],[15,103320,103322,103353],{"className":103321},[66],[15,103323,103325],{"className":103324},[66,78147],[15,103326,103328],{"className":103327},[202],[15,103329,103331],{"className":103330},[206],[15,103332,103334,103342],{"className":103333,"style":78157},[210],[15,103335,103336,103339],{"style":1386},[15,103337],{"className":103338,"style":1368},[217],[15,103340,61618],{"className":103341,"style":23025},[66,67],[15,103343,103344,103347],{"style":78168},[15,103345],{"className":103346,"style":1368},[217],[15,103348,103350],{"className":103349,"style":96110},[78175],[15,103351,78130],{"className":103352},[66],[15,103354,103356],{"className":103355},[198],[15,103357,103359,103379],{"className":103358},[202,1090],[15,103360,103362,103376],{"className":103361},[206],[15,103363,103365],{"className":103364,"style":2198},[210],[15,103366,103367,103370],{"style":88441},[15,103368],{"className":103369,"style":218},[217],[15,103371,103373],{"className":103372},[222,223,224,225],[15,103374,37],{"className":103375},[66,67,225],[15,103377,1114],{"className":103378},[1113],[15,103380,103382],{"className":103381},[206],[15,103383,103385],{"className":103384,"style":1121},[210],[15,103386],{},[15,103388,335],{"className":103389},[422],[15,103391,24233],{"className":103392},[71],[15,103394],{"className":103395,"style":697},[71],[15,103397,103399,103402,103455],{"className":103398},[66],[15,103400],{"className":103401},[162,23417],[15,103403,103405],{"className":103404},[6467],[15,103406,103408,103447],{"className":103407},[202,1090],[15,103409,103411,103444],{"className":103410},[206],[15,103412,103414,103425,103433],{"className":103413,"style":23430},[210],[15,103415,103416,103419],{"style":23433},[15,103417],{"className":103418,"style":1368},[217],[15,103420,103422],{"className":103421},[66],[15,103423,37],{"className":103424},[66,67],[15,103426,103427,103430],{"style":23445},[15,103428],{"className":103429,"style":1368},[217],[15,103431],{"className":103432,"style":4440},[23452],[15,103434,103435,103438],{"style":23455},[15,103436],{"className":103437,"style":1368},[217],[15,103439,103441],{"className":103440},[66],[15,103442,1211],{"className":103443},[66],[15,103445,1114],{"className":103446},[1113],[15,103448,103450],{"className":103449},[206],[15,103451,103453],{"className":103452,"style":23474},[210],[15,103454],{},[15,103456],{"className":103457},[169,23417],[15,103459,16427],{"className":103460,"style":16497},[66,67],[15,103462,120],{"className":103463},[162],[15,103465,103467,103470],{"className":103466},[66],[15,103468,61618],{"className":103469,"style":23025},[66,67],[15,103471,103473],{"className":103472},[198],[15,103474,103476,103496],{"className":103475},[202,1090],[15,103477,103479,103493],{"className":103478},[206],[15,103480,103482],{"className":103481,"style":2139},[210],[15,103483,103484,103487],{"style":88441},[15,103485],{"className":103486,"style":218},[217],[15,103488,103490],{"className":103489},[222,223,224,225],[15,103491,1804],{"className":103492},[66,225],[15,103494,1114],{"className":103495},[1113],[15,103497,103499],{"className":103498},[206],[15,103500,103502],{"className":103501,"style":1121},[210],[15,103503],{},[15,103505,103507,103510],{"className":103506},[169],[15,103508,126],{"className":103509},[169],[15,103511,103513],{"className":103512},[198],[15,103514,103516],{"className":103515},[202],[15,103517,103519],{"className":103518},[206],[15,103520,103522],{"className":103521,"style":1650},[210],[15,103523,103524,103527],{"style":397},[15,103525],{"className":103526,"style":218},[217],[15,103528,103530],{"className":103529},[222,223,224,225],[15,103531,103533,103536],{"className":103532},[66,225],[15,103534,6602],{"className":103535},[66,225],[15,103537,1211],{"className":103538},[66,225],[15,103540,103542],{"className":103541,"style":1418},[169,1417],[15,103543,126],{"className":103544},[1422,224],[15,103546],{"className":103547,"style":697},[71],[15,103549,3573],{"className":103550},[66],[11,103552,103553,103556],{},[2040,103554,103555],{},"Doob 定理","则从另一个角度保证一致性：对先验几乎所有的参数值，后验都会收敛到真值附近的任意邻域。注意「先验几乎处处」这个限定：先验可能把质量放在一个坏集合上，只是那个集合的概率为零。",[11,103558,103559,103560,103659],{},"这两个定理解释了为什么在大样本下，贝叶斯与频率派的答案往往一致。后验的宽度以 ",[15,103561,103563,103589],{"className":103562,"translate":19},[18],[15,103564,103566],{"className":103565},[23],[25,103567,103568],{"xmlns":27},[29,103569,103570,103586],{},[32,103571,103572,103574,103576,103578,103580,103584],{},[35,103573,22999],{},[39,103575,120],{"stretchy":119},[1209,103577,1211],{},[35,103579,6625],{"mathvariant":138},[48508,103581,103582],{},[35,103583,37],{},[39,103585,126],{"stretchy":119},[45,103587,103588],{"encoding":47},"O(1\u002F\\sqrt n)",[15,103590,103592],{"className":103591,"ariaHidden":53},[52],[15,103593,103595,103599,103602,103605,103609,103656],{"className":103594},[57],[15,103596],{"className":103597,"style":103598},[61],"height:1.0503em;vertical-align:-0.25em;",[15,103600,22999],{"className":103601,"style":23025},[66,67],[15,103603,120],{"className":103604},[162],[15,103606,103608],{"className":103607},[66],"1\u002F",[15,103610,103612],{"className":103611},[66,48550],[15,103613,103615,103647],{"className":103614},[202,1090],[15,103616,103618,103644],{"className":103617},[206],[15,103619,103622,103631],{"className":103620,"style":103621},[210],"height:0.8003em;",[15,103623,103625,103628],{"className":103624,"style":1386},[48564],[15,103626],{"className":103627,"style":1368},[217],[15,103629,37],{"className":103630,"style":48571},[66,67],[15,103632,103634,103637],{"style":103633},"top:-2.7603em;",[15,103635],{"className":103636,"style":1368},[217],[15,103638,103640],{"className":103639,"style":48591},[48590],[34828,103641,103642],{"xmlns":34830,"width":48594,"height":48595,"viewBox":48596,"preserveAspectRatio":48597},[34834,103643],{"d":48600},[15,103645,1114],{"className":103646},[1113],[15,103648,103650],{"className":103649},[206],[15,103651,103654],{"className":103652,"style":103653},[210],"height:0.2397em;",[15,103655],{},[15,103657,126],{"className":103658},[169]," 收缩，先验对形状的影响随之减弱，后验主要由似然决定。但这也划出了边界：当模型奇异（参数不可识别、网络层数过多、对称性导致 Fisher 信息退化）时，Bernstein–von Mises 失效，后验的形状本身携带信息，不能用一个高斯替代。这正是奇异学习理论要处理的情形。",[1904,103661,103663],{"id":103662},"_65-边缘似然与奥卡姆剃刀","6.5 边缘似然与奥卡姆剃刀",[11,103665,103666,103667,103670],{},"贝叶斯模型比较的核心量是",[2040,103668,103669],{},"边缘似然","（marginal likelihood），也就是贝叶斯公式的分母：",[15,103672,103674],{"className":103673,"translate":19},[294],[15,103675,103677,103743],{"className":103676,"translate":19},[18],[15,103678,103680],{"className":103679},[23],[25,103681,103682],{"xmlns":27,"display":303},[29,103683,103684,103740],{},[32,103685,103686,103688,103690,103692,103694,103696,103698,103700,103702,103704,103706,103708,103710,103712,103714,103716,103718,103720,103722,103724,103726,103728,103730,103732,103734,103736,103738],{},[35,103687,11],{},[39,103689,120],{"stretchy":119},[35,103691,32458],{},[39,103693,34430],{},[35,103695,95977],{},[39,103697,126],{"stretchy":119},[39,103699,129],{},[39,103701,29732],{},[35,103703,11],{},[39,103705,120],{"stretchy":119},[35,103707,32458],{},[39,103709,34430],{},[35,103711,61618],{},[39,103713,335],{"separator":53},[35,103715,95977],{},[39,103717,126],{"stretchy":119},[1274,103719,2462],{},[35,103721,11],{},[39,103723,120],{"stretchy":119},[35,103725,61618],{},[39,103727,34430],{},[35,103729,95977],{},[39,103731,126],{"stretchy":119},[1274,103733,2462],{},[35,103735,458],{},[35,103737,61618],{},[35,103739,3573],{"mathvariant":138},[45,103741,103742],{"encoding":47},"p(D\\mid M)=\\int p(D\\mid\\theta,M)\\,p(\\theta\\mid M)\\,d\\theta .",[15,103744,103746,103770,103791,103821,103863],{"className":103745,"ariaHidden":53},[52],[15,103747,103749,103752,103755,103758,103761,103764,103767],{"className":103748},[57],[15,103750],{"className":103751,"style":154},[61],[15,103753,11],{"className":103754},[66,67],[15,103756,120],{"className":103757},[162],[15,103759,32458],{"className":103760,"style":23025},[66,67],[15,103762],{"className":103763,"style":173},[71],[15,103765,34430],{"className":103766},[177],[15,103768],{"className":103769,"style":173},[71],[15,103771,103773,103776,103779,103782,103785,103788],{"className":103772},[57],[15,103774],{"className":103775,"style":154},[61],[15,103777,95977],{"className":103778,"style":28790},[66,67],[15,103780,126],{"className":103781},[169],[15,103783],{"className":103784,"style":173},[71],[15,103786,129],{"className":103787},[177],[15,103789],{"className":103790,"style":173},[71],[15,103792,103794,103797,103800,103803,103806,103809,103812,103815,103818],{"className":103793},[57],[15,103795],{"className":103796,"style":65130},[61],[15,103798,29732],{"className":103799,"style":29796},[1339,1470,1471],[15,103801],{"className":103802,"style":697},[71],[15,103804,11],{"className":103805},[66,67],[15,103807,120],{"className":103808},[162],[15,103810,32458],{"className":103811,"style":23025},[66,67],[15,103813],{"className":103814,"style":173},[71],[15,103816,34430],{"className":103817},[177],[15,103819],{"className":103820,"style":173},[71],[15,103822,103824,103827,103830,103833,103836,103839,103842,103845,103848,103851,103854,103857,103860],{"className":103823},[57],[15,103825],{"className":103826,"style":154},[61],[15,103828,61618],{"className":103829,"style":23025},[66,67],[15,103831,335],{"className":103832},[422],[15,103834],{"className":103835,"style":697},[71],[15,103837,95977],{"className":103838,"style":28790},[66,67],[15,103840,126],{"className":103841},[169],[15,103843],{"className":103844,"style":697},[71],[15,103846,11],{"className":103847},[66,67],[15,103849,120],{"className":103850},[162],[15,103852,61618],{"className":103853,"style":23025},[66,67],[15,103855],{"className":103856,"style":173},[71],[15,103858,34430],{"className":103859},[177],[15,103861],{"className":103862,"style":173},[71],[15,103864,103866,103869,103872,103875,103878,103881,103884],{"className":103865},[57],[15,103867],{"className":103868,"style":154},[61],[15,103870,95977],{"className":103871,"style":28790},[66,67],[15,103873,126],{"className":103874},[169],[15,103876],{"className":103877,"style":697},[71],[15,103879,458],{"className":103880},[66,67],[15,103882,61618],{"className":103883,"style":23025},[66,67],[15,103885,3573],{"className":103886},[66],[11,103888,103889,103890,37644],{},"两个模型的比较用",[2040,103891,103892],{},"贝叶斯因子",[15,103894,103896],{"className":103895,"translate":19},[294],[15,103897,103899,104050],{"className":103898,"translate":19},[18],[15,103900,103902],{"className":103901},[23],[25,103903,103904],{"xmlns":27,"display":303},[29,103905,103906,104047],{},[32,103907,103908,103919,103921,103959,103961,103963,104001,104003,104013,104015,104045],{},[1057,103909,103910,103916],{},[32,103911,103912,103914],{},[35,103913,63972],{"mathvariant":138},[35,103915,74045],{"mathvariant":138},[1209,103917,103918],{},"12",[39,103920,129],{},[6467,103922,103923,103941],{},[32,103924,103925,103927,103929,103931,103933,103939],{},[35,103926,11],{},[39,103928,120],{"stretchy":119},[35,103930,32458],{},[39,103932,34430],{},[1057,103934,103935,103937],{},[35,103936,95977],{},[1209,103938,1211],{},[39,103940,126],{"stretchy":119},[32,103942,103943,103945,103947,103949,103951,103957],{},[35,103944,11],{},[39,103946,120],{"stretchy":119},[35,103948,32458],{},[39,103950,34430],{},[1057,103952,103953,103955],{},[35,103954,95977],{},[1209,103956,1263],{},[39,103958,126],{"stretchy":119},[39,103960,335],{"separator":53},[71,103962],{"width":3904},[6467,103964,103965,103983],{},[32,103966,103967,103969,103971,103977,103979,103981],{},[35,103968,11],{},[39,103970,120],{"stretchy":119},[1057,103972,103973,103975],{},[35,103974,95977],{},[1209,103976,1211],{},[39,103978,34430],{},[35,103980,32458],{},[39,103982,126],{"stretchy":119},[32,103984,103985,103987,103989,103995,103997,103999],{},[35,103986,11],{},[39,103988,120],{"stretchy":119},[1057,103990,103991,103993],{},[35,103992,95977],{},[1209,103994,1263],{},[39,103996,34430],{},[35,103998,32458],{},[39,104000,126],{"stretchy":119},[39,104002,129],{},[1057,104004,104005,104011],{},[32,104006,104007,104009],{},[35,104008,63972],{"mathvariant":138},[35,104010,74045],{"mathvariant":138},[1209,104012,103918],{},[39,104014,3268],{},[6467,104016,104017,104031],{},[32,104018,104019,104021,104023,104029],{},[35,104020,11],{},[39,104022,120],{"stretchy":119},[1057,104024,104025,104027],{},[35,104026,95977],{},[1209,104028,1211],{},[39,104030,126],{"stretchy":119},[32,104032,104033,104035,104037,104043],{},[35,104034,11],{},[39,104036,120],{"stretchy":119},[1057,104038,104039,104041],{},[35,104040,95977],{},[1209,104042,1263],{},[39,104044,126],{"stretchy":119},[35,104046,3573],{"mathvariant":138},[45,104048,104049],{"encoding":47},"\\mathrm{BF}_{12}=\\frac{p(D\\mid M_1)}{p(D\\mid M_2)},\n\\qquad\n\\frac{p(M_1\\mid D)}{p(M_2\\mid D)}\n=\\mathrm{BF}_{12}\\cdot\\frac{p(M_1)}{p(M_2)}.",[15,104051,104053,104115,104496,104557],{"className":104052,"ariaHidden":53},[52],[15,104054,104056,104059,104106,104109,104112],{"className":104055},[57],[15,104057],{"className":104058,"style":1076},[61],[15,104060,104062,104069],{"className":104061},[66],[15,104063,104065],{"className":104064},[66],[15,104066,104068],{"className":104067},[66,4429],"BF",[15,104070,104072],{"className":104071},[198],[15,104073,104075,104098],{"className":104074},[202,1090],[15,104076,104078,104095],{"className":104077},[206],[15,104079,104081],{"className":104080,"style":2139},[210],[15,104082,104083,104086],{"style":69561},[15,104084],{"className":104085,"style":218},[217],[15,104087,104089],{"className":104088},[222,223,224,225],[15,104090,104092],{"className":104091},[66,225],[15,104093,103918],{"className":104094},[66,225],[15,104096,1114],{"className":104097},[1113],[15,104099,104101],{"className":104100},[206],[15,104102,104104],{"className":104103,"style":1121},[210],[15,104105],{},[15,104107],{"className":104108,"style":173},[71],[15,104110,129],{"className":104111},[177],[15,104113],{"className":104114,"style":173},[71],[15,104116,104118,104121,104300,104303,104306,104309,104487,104490,104493],{"className":104117},[57],[15,104119],{"className":104120,"style":41192},[61],[15,104122,104124,104127,104297],{"className":104123},[66],[15,104125],{"className":104126},[162,23417],[15,104128,104130],{"className":104129},[6467],[15,104131,104133,104289],{"className":104132},[202,1090],[15,104134,104136,104286],{"className":104135},[206],[15,104137,104139,104209,104217],{"className":104138,"style":41211},[210],[15,104140,104141,104144],{"style":23433},[15,104142],{"className":104143,"style":1368},[217],[15,104145,104147,104150,104153,104156,104159,104162,104165,104206],{"className":104146},[66],[15,104148,11],{"className":104149},[66,67],[15,104151,120],{"className":104152},[162],[15,104154,32458],{"className":104155,"style":23025},[66,67],[15,104157],{"className":104158,"style":173},[71],[15,104160,34430],{"className":104161},[177],[15,104163],{"className":104164,"style":173},[71],[15,104166,104168,104171],{"className":104167},[66],[15,104169,95977],{"className":104170,"style":28790},[66,67],[15,104172,104174],{"className":104173},[198],[15,104175,104177,104198],{"className":104176},[202,1090],[15,104178,104180,104195],{"className":104179},[206],[15,104181,104183],{"className":104182,"style":2139},[210],[15,104184,104186,104189],{"style":104185},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[15,104187],{"className":104188,"style":218},[217],[15,104190,104192],{"className":104191},[222,223,224,225],[15,104193,1263],{"className":104194},[66,225],[15,104196,1114],{"className":104197},[1113],[15,104199,104201],{"className":104200},[206],[15,104202,104204],{"className":104203,"style":1121},[210],[15,104205],{},[15,104207,126],{"className":104208},[169],[15,104210,104211,104214],{"style":23445},[15,104212],{"className":104213,"style":1368},[217],[15,104215],{"className":104216,"style":4440},[23452],[15,104218,104219,104222],{"style":23455},[15,104220],{"className":104221,"style":1368},[217],[15,104223,104225,104228,104231,104234,104237,104240,104243,104283],{"className":104224},[66],[15,104226,11],{"className":104227},[66,67],[15,104229,120],{"className":104230},[162],[15,104232,32458],{"className":104233,"style":23025},[66,67],[15,104235],{"className":104236,"style":173},[71],[15,104238,34430],{"className":104239},[177],[15,104241],{"className":104242,"style":173},[71],[15,104244,104246,104249],{"className":104245},[66],[15,104247,95977],{"className":104248,"style":28790},[66,67],[15,104250,104252],{"className":104251},[198],[15,104253,104255,104275],{"className":104254},[202,1090],[15,104256,104258,104272],{"className":104257},[206],[15,104259,104261],{"className":104260,"style":2139},[210],[15,104262,104263,104266],{"style":104185},[15,104264],{"className":104265,"style":218},[217],[15,104267,104269],{"className":104268},[222,223,224,225],[15,104270,1211],{"className":104271},[66,225],[15,104273,1114],{"className":104274},[1113],[15,104276,104278],{"className":104277},[206],[15,104279,104281],{"className":104280,"style":1121},[210],[15,104282],{},[15,104284,126],{"className":104285},[169],[15,104287,1114],{"className":104288},[1113],[15,104290,104292],{"className":104291},[206],[15,104293,104295],{"className":104294,"style":41282},[210],[15,104296],{},[15,104298],{"className":104299},[169,23417],[15,104301,335],{"className":104302},[422],[15,104304],{"className":104305,"style":4061},[71],[15,104307],{"className":104308,"style":697},[71],[15,104310,104312,104315,104484],{"className":104311},[66],[15,104313],{"className":104314},[162,23417],[15,104316,104318],{"className":104317},[6467],[15,104319,104321,104476],{"className":104320},[202,1090],[15,104322,104324,104473],{"className":104323},[206],[15,104325,104327,104396,104404],{"className":104326,"style":41211},[21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Laplace 近似可以看出这一点。设 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的主项。参数越多，惩罚越重；但更准确地说，惩罚的不是维度本身，而是参数空间中「能给出不同预测」的有效体积。一个参数如果对预测没有影响，它就不会贡献体积，也就不会被惩罚。",[11,106207,106208,106209,1124],{},"边缘似然也有它自己的问题：它对先验敏感。一个过分分散的先验会把概率质量摊到大量「几乎不预测数据」的区域，导致模型被过度惩罚。这就是 Lindley 悖论的一个侧面：当原假设是一个点、备择假设是一个很宽的分布时，贝叶斯因子可能强烈支持原假设，即使频率派的检验已经拒绝了它。",[2040,106210,106211],{},"贝叶斯方法不会自动给出「正确」的模型选择，它只是把选择的标准写清楚了",[95,106213,106215],{"id":106214},"_7-机器学习理论中的位置","7. 机器学习理论中的位置",[1904,106217,106219],{"id":106218},"_71-风险分解","7.1 风险分解",[11,106221,1910,106222,106250,106251,3249,106433,106495],{},[15,106223,106225,106238],{"className":106224,"translate":19},[18],[15,106226,106228],{"className":106227},[23],[25,106229,106230],{"xmlns":27},[29,106231,106232,106236],{},[32,106233,106234],{},[35,106235,1061],{"mathvariant":445},[45,106237,1742],{"encoding":47},[15,106239,106241],{"className":106240,"ariaHidden":53},[52],[15,106242,106244,106247],{"className":106243},[57],[15,106245],{"className":106246,"style":490},[61],[15,106248,1061],{"className":106249,"style":1083},[66,494]," 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是从数据中学到的模型。则",[15,106497,106499],{"className":106498,"translate":19},[294],[15,106500,106502,106604],{"className":106501,"translate":19},[18],[15,106503,106505],{"className":106504},[23],[25,106506,106507],{"xmlns":27,"display":303},[29,106508,106509,106601],{},[32,106510,106511,106513,106515,106521,106523,106525,106531,106533,106562,106564,106599],{},[35,106512,455],{},[39,106514,120],{"stretchy":119},[4296,106516,106517,106519],{"accent":53},[35,106518,20182],{},[39,106520,78130],{},[39,106522,126],{"stretchy":119},[39,106524,6602],{},[131,106526,106527,106529],{},[35,106528,455],{},[39,106530,1162],{},[39,106532,129],{},[1172,106534,106535,106559],{},[1172,106536,106537,106557],{},[32,106538,106539,106541,106547,106549,106555],{},[39,106540,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[1057,106542,106543,106545],{},[35,106544,455],{},[35,106546,1061],{"mathvariant":445},[39,106548,6602],{},[131,106550,106551,106553],{},[35,106552,455],{},[39,106554,1162],{},[39,106556,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,106558,69846],{"stretchy":53},[1274,106560,106561],{},"近似误差",[39,106563,1245],{},[1172,106565,106566,106596],{},[1172,106567,106568,106594],{},[32,106569,106570,106572,106574,106576,106582,106584,106586,106592],{},[39,106571,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,106573,455],{},[39,106575,120],{"stretchy":119},[4296,106577,106578,106580],{"accent":53},[35,106579,20182],{},[39,106581,78130],{},[39,106583,126],{"stretchy":119},[39,106585,6602],{},[1057,106587,106588,106590],{},[35,106589,455],{},[35,106591,1061],{"mathvariant":445},[39,106593,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,106595,69846],{"stretchy":53},[1274,106597,106598],{},"估计误差",[35,106600,3573],{"mathvariant":138},[45,106602,106603],{"encoding":47},"R(\\hat h)-R^*\n=\n\\underbrace{\\big(R_{\\mathcal H}-R^*\\big)}_{\\text{近似误差}}\n+\n\\underbrace{\\big(R(\\hat h)-R_{\\mathcal H}\\big)}_{\\text{估计误差}}.",[15,106605,106607,106662,106706,106921],{"className":106606,"ariaHidden":53},[52],[15,106608,106610,106613,106616,106619,106650,106653,106656,106659],{"className":106609},[57],[15,106611],{"className":106612,"style":78282},[61],[15,106614,455],{"className":106615,"style":62437},[66,67],[15,106617,120],{"className":106618},[162],[15,106620,106622],{"className":106621},[66,78147],[15,106623,106625],{"className":106624},[202],[15,106626,106628],{"className":106627},[206],[15,106629,106631,106639],{"className":106630,"style":78157},[210],[15,106632,106633,106636],{"style":1386},[15,106634],{"className":106635,"style":1368},[217],[15,106637,20182],{"className":106638},[66,67],[15,106640,106641,106644],{"style":78168},[15,106642],{"className":106643,"style":1368},[217],[15,106645,106647],{"className":106646,"style":106491},[78175],[15,106648,78130],{"className":106649},[66],[15,106651,126],{"className":106652},[169],[15,106654],{"className":106655,"style":72},[71],[15,106657,6602],{"className":106658},[76],[15,106660],{"className":106661,"style":72},[71],[15,106663,106665,106668,106697,106700,106703],{"className":106664},[57],[15,106666],{"className":106667,"style":1308},[61],[15,106669,106671,106674],{"className":106670},[66],[15,106672,455],{"className":106673,"style":62437},[66,67],[15,106675,106677],{"className":106676},[198],[15,106678,106680],{"className":106679},[202],[15,106681,106683],{"className":106682},[206],[15,106684,106686],{"className":106685,"style":1308},[210],[15,106687,106688,106691],{"style":397},[15,106689],{"className":106690,"style":218},[217],[15,106692,106694],{"className":106693},[222,223,224,225],[15,106695,1162],{"className":106696},[76,225],[15,106698],{"className":106699,"style":173},[71],[15,106701,129],{"className":106702},[177],[15,106704],{"className":106705,"style":173},[71],[15,106707,106709,106713,106912,106915,106918],{"className":106708},[57],[15,106710],{"className":106711,"style":106712},[61],"height:2.5263em;vertical-align:-1.6763em;",[15,106714,106716],{"className":106715},[1413,1172],[15,106717,106719,106904],{"className":106718},[202,1090],[15,106720,106722,106901],{"className":106721},[206],[15,106723,106725,106742],{"className":106724,"style":102207},[210],[15,106726,106727,106730],{"style":91794},[15,106728],{"className":106729,"style":1368},[217],[15,106731,106733],{"className":106732},[222,223,224,225],[15,106734,106736],{"className":106735},[66,225],[15,106737,106739],{"className":106738},[66,1711,225],[15,106740,106561],{"className":106741},[66,68616,225],[15,106743,106744,106747],{"style":1386},[15,106745],{"className":106746,"style":1368},[217],[15,106748,106750],{"className":106749},[1413,1172],[15,106751,106753,106893],{"className":106752},[202,1090],[15,106754,106756,106890],{"className":106755},[206],[15,106757,106759,106789],{"className":106758,"style":102207},[210],[15,106760,106762,106765],{"className":106761,"style":91830},[48564],[15,106763],{"className":106764,"style":1368},[217],[15,106766,106768,106775,106782],{"className":106767,"style":70049},[70048],[15,106769,106771],{"className":106770,"style":70054},[70053],[34828,106772,106773],{"xmlns":34830,"width":48594,"height":70057,"viewBox":70058,"preserveAspectRatio":48597},[34834,106774],{"d":70061},[15,106776,106778],{"className":106777,"style":70054},[70065],[34828,106779,106780],{"xmlns":34830,"width":48594,"height":70057,"viewBox":70058,"preserveAspectRatio":70068},[34834,106781],{"d":70071},[15,106783,106785],{"className":106784,"style":70054},[70075],[34828,106786,106787],{"xmlns":34830,"width":48594,"height":70057,"viewBox":70058,"preserveAspectRatio":70078},[34834,106788],{"d":70081},[15,106790,106791,106794],{"style":1386},[15,106792],{"className":106793,"style":1368},[217],[15,106795,106797,106803,106846,106849,106852,106855,106884],{"className":106796},[66],[15,106798,106800],{"className":106799},[66],[15,106801,120],{"className":106802},[1422,1508],[15,106804,106806,106809],{"className":106805},[66],[15,106807,455],{"className":106808,"style":62437},[66,67],[15,106810,106812],{"className":106811},[198],[15,106813,106815,106838],{"className":106814},[202,1090],[15,106816,106818,106835],{"className":106817},[206],[15,106819,106821],{"className":106820,"style":3638},[210],[15,106822,106823,106826],{"style":106327},[15,106824],{"className":106825,"style":218},[217],[15,106827,106829],{"className":106828},[222,223,224,225],[15,106830,106832],{"className":106831},[66,225],[15,106833,1061],{"className":106834,"style":1083},[66,494,225],[15,106836,1114],{"className":106837},[1113],[15,106839,106841],{"className":106840},[206],[15,106842,106844],{"className":106843,"style":1121},[210],[15,106845],{},[15,106847],{"className":106848,"style":72},[71],[15,106850,6602],{"className":106851},[76],[15,106853],{"className":106854,"style":72},[71],[15,106856,106858,106861],{"className":106857},[66],[15,106859,455],{"className":106860,"style":62437},[66,67],[15,106862,106864],{"c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e":92029},[210],[15,107136],{},[15,107138],{"className":107139,"style":697},[71],[15,107141,3573],{"className":107142},[66],[11,107144,107145],{},"第一项由模型类决定：模型类太小，无论如何训练都逼近不了贝叶斯风险。第二项由样本量与模型复杂度决定：模型类太大，有限样本上的经验风险会偏离真实风险。",[11,107147,107148,107149,107207,107208,107266],{},"贝叶斯风险 ",[15,107150,107152,107169],{"className":107151,"translate":19},[18],[15,107153,107155],{"className":107154},[23],[25,107156,107157],{"xmlns":27},[29,107158,107159,107167],{},[32,107160,107161],{},[131,107162,107163,107165],{},[35,107164,455],{},[39,107166,1162],{},[45,107168,76283],{"encoding":47},[15,107170,107172],{"className":107171,"ariaHidden":53},[52],[15,107173,107175,107178],{"className":107174},[57],[15,107176],{"className":107177,"style":16249},[61],[15,107179,107181,107184],{"className":107180},[66],[15,107182,455],{"className":107183,"style":62437},[66,67],[15,107185,107187],{"className":107186},[198],[15,107188,107190],{"className":107189},[202],[15,107191,107193],{"className":107192},[206],[15,107194,107196],{"className":107195,"style":16249},[210],[15,107197,107198,107201],{"style":213},[15,107199],{"className":107200,"style":218},[217],[15,107202,107204],{"className":107203},[222,223,224,225],[15,107205,1162],{"className":107206},[76,225]," 是这两项的共同起点。它既不属于模型，也不属于算法，而是问题本身的下限。学习理论的大部分内容，都可以理解为在给定 ",[15,107209,107211,107228],{"className":107210,"translate":19},[18],[15,107212,107214],{"className":107213},[23],[25,107215,107216],{"xmlns":27},[29,107217,107218,107226],{},[32,107219,107220],{},[131,107221,107222,107224],{},[35,107223,455],{},[39,107225,1162],{},[45,107227,76283],{"encoding":47},[15,107229,107231],{"className":107230,"ariaHidden":53},[52],[15,107232,107234,107237],{"className":107233},[57],[15,107235],{"className":107236,"style":16249},[61],[15,107238,107240,107243],{"className":107239},[66],[15,107241,455],{"className":107242,"style":62437},[66,67],[15,107244,107246],{"className":107245},[198],[15,107247,107249],{"className":107248},[202],[15,107250,107252],{"className":107251},[206],[15,107253,107255],{"className":107254,"style":16249},[210],[15,107256,107257,107260],{"style":213},[15,107258],{"className":107259,"style":218},[217],[15,107261,107263],{"className":107262},[222,223,224,225],[15,107264,1162],{"className":107265},[76,225]," 的情况下，分别研究这两项能压到多低。",[11,107268,107269],{},"这也解释了为什么「更复杂的模型」有时有用、有时有害：它降低近似误差，却可能提高估计误差。深度学习成功的部分原因，是它在大规模数据上把估计误差压得足够低，同时用极宽的模型类把近似误差压得很小。这不是对偏差-方差权衡的否定，而是权衡曲线在数据量增大后的重新定位。",[1904,107271,107273],{"id":107272},"_72-一致性","7.2 一致性",[11,107275,107276,107277,2038,107347,107350,107351,3249],{},"一个决策规则序列 ",[15,107278,107280,107298],{"className":107279,"translate":19},[18],[15,107281,107283],{"className":107282},[23],[25,107284,107285],{"xmlns":27},[29,107286,107287,107295],{},[32,107288,107289],{},[1057,107290,107291,107293],{},[35,107292,3885],{},[35,107294,37],{},[45,107296,107297],{"encoding":47},"\\delta_n",[15,107299,107301],{"className":107300,"ariaHidden":53},[52],[15,107302,107304,107307],{"className":107303},[57],[15,107305],{"className":107306,"style":3939},[61],[15,107308,107310,107313],{"className":107309},[66],[15,107311,3885],{"className":107312,"style":3946},[66,67],[15,107314,107316],{"className":107315},[198],[15,107317,107319,107339],{"className":107318},[202,1090],[15,107320,107322,107336],{"className":107321},[206],[15,107323,107325],{"className":107324,"style":2198},[210],[15,107326,107327,107330],{"style":3961},[15,107328],{"className":107329,"style":218},[217],[15,107331,107333],{"className":107332},[222,223,224,225],[15,107334,37],{"className":107335},[66,67,225],[15,107337,1114],{"className":107338},[1113],[15,107340,107342],{"className":107341},[206],[15,107343,107345],{"className":107344,"style":1121},[210],[15,107346],{},[2040,107348,107349],{},"普遍一致的","（universally consistent），若对任意分布 ",[15,107352,107354,107367],{"className":107353,"translate":19},[18],[15,107355,107357],{"className":107356},[23],[25,107358,107359],{"xmlns":27},[29,107360,107361,107365],{},[32,107362,107363],{},[35,107364,28399],{},[45,107366,28399],{"encoding":47},[15,107368,107370],{"className":107369,"ariaHidden":53},[52],[15,107371,107373,107376],{"className":107372},[57],[15,107374],{"className":107375,"style":490},[61],[15,107377,28399],{"className":107378,"style":62513},[66,67],[15,107380,107382],{"className":107381,"translate":19},[294],[15,107383,107385,107437],{"className":107384,"translate":19},[18],[15,107386,107388],{"className":107387},[23],[25,107389,107390],{"xmlns":27,"display":303},[29,107391,107392,107434],{},[32,107393,107394,107396,107398,107404,107406,107408,107426,107428],{},[35,107395,455],{},[39,107397,120],{"stretchy":119},[1057,107399,107400,107402],{},[35,107401,3885],{},[35,107403,37],{},[39,107405,126],{"stretchy":119},[1274,107407,24233],{},[4296,107409,107410,107413],{},[39,107411,449],{"stretchy":53,"minsize":107412},"3.0em",[107414,107415,107418],"mpadded",{"width":107416,"lspace":107417},"+0.6em","0.3em",[32,107419,107420,107422,107424],{},[1274,107421,24233],{},[35,107423,28399],{},[1274,107425,24233],{},[1274,107427,24233],{},[131,107429,107430,107432],{},[35,107431,455],{},[39,107433,1162],{},[45,107435,107436],{"encoding":47},"R(\\delta_n)\\ \\xrightarrow{\\ P\\ }\\ R^*",[15,107438,107440,107580],{"className":107439,"ariaHidden":53},[52],[15,107441,107443,107447,107450,107453,107493,107496,107499,107502,107574,107577],{"className":107442},[57],[15,107444],{"className":107445,"style":107446},[61],"height:1.3503em;vertical-align:-0.25em;",[15,107448,455],{"className":107449,"style":62437},[66,67],[15,107451,120],{"className":107452},[162],[15,107454,107456,107459],{"className":107455},[66],[15,107457,3885],{"className":107458,"style":3946},[66,67],[15,107460,107462],{"className":107461},[198],[15,107463,107465,107485],{"className":107464},[202,1090],[15,107466,107468,107482],{"className":107467},[206],[15,107469,107471],{"className":107470,"style":2198},[210],[15,107472,107473,107476],{"style":3961},[15,107474],{"className":107475,"style":218},[217],[15,107477,107479],{"className":107478},[222,223,224,225],[15,107480,37],{"className":107481},[66,67,225],[15,107483,1114],{"className":107484},[1113],[15,107486,107488],{"className":107487},[206],[15,107489,107491],{"className":107490,"style":1121},[210],[15,107492],{},[15,107494,126],{"className":107495},[169],[15,107497,24233],{"className":107498},[71],[15,107500],{"className":107501,"style":173},[71],[15,107503,107506],{"className":107504},[177,107505],"x-arrow",[15,107507,107509,107565],{"className":107508},[202,1090],[15,107510,107512,107562],{"className":107511},[206],[15,107513,107516,107544],{"className":107514,"style":107515},[210],"height:1.1003em;",[15,107517,107519,107522],{"style":107518},"top:-3.322em;",[15,107520],{"className":107521,"style":218},[217],[15,107523,107526],{"className":107524},[222,223,224,225,107525],"x-arrow-pad",[15,107527,107529,107535,107538],{"className":107528},[66,225],[15,107530,107532],{"className":107531},[71,225],[15,107533,24233],{"className":107534},[225],[15,107536,28399],{"className":107537,"style":62513},[66,67,225],[15,107539,107541],{"className":107540},[71,225],[15,107542,24233],{"className":107543},[225],[15,107545,107548,107551],{"className":107546,"style":107547},[48564],"top:-2.689em;",[15,107549],{"className":107550,"style":218},[217],[15,107552,107555],{"className":107553,"style":107554},[48590],"height:0.522em;min-width:1.469em;",[34828,107556,107559],{"xmlns":34830,"width":48594,"height":107557,"viewBox":107558,"preserveAspectRatio":70078},"0.522em","0 0 400000 522",[34834,107560],{"d":107561},"M0 241v40h399891c-47.3 35.3-84 78-110 128\n-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20\n 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7\n 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85\n-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5\n-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67\n 151.7 139 205zm0 0v40h399900v-40z",[15,107563,1114],{"className":107564},[1113],[15,107566,107568],{"className":107567},[206],[15,107569,107572],{"className":107570,"style":107571},[210],"height:0.011em;",[15,107573],{},[15,107575,24233],{"className":107576},[71],[15,107578],{"className":107579,"style":173},[71],[15,107581,107583,107586],{"className":107582},[57],[15,107584],{"className":107585,"style":1308},[61],[15,107587,107589,107592],{"className":107588},[66],[15,107590,455],{"className":107591,"style":62437},[66,67],[15,107593,107595],{"className":107594},[198],[15,107596,107598],{"className":107597},[202],[15,107599,107601],{"className":107600},[206],[15,107602,107604],{"className":107603,"style":1308},[210],[15,107605,107606,107609],{"style":397},[15,107607],{"className":107608,"style":218},[217],[15,107610,107612],{"className":107611},[222,223,224,225],[15,107613,1162],{"className":107614},[76,225],[11,107616,107617,107618,107620],{},"依概率成立。注意这里收敛的是",[2040,107619,62350],{},"，不是后验估计。一个模型可以是一致的，但给出的概率完全不校准；反过来，一个校准很好的模型也可能因为模型类太小而无法逼近贝叶斯风险。",[11,107622,107623,107651,107652,107703,107704,13132,107765,107793,107794,107822,107823,107887],{},[15,107624,107626,107639],{"className":107625,"translate":19},[18],[15,107627,107629],{"className":107628},[23],[25,107630,107631],{"xmlns":27},[29,107632,107633,107637],{},[32,107634,107635],{},[35,107636,621],{},[45,107638,621],{"encoding":47},[15,107640,107642],{"className":107641,"ariaHidden":53},[52],[15,107643,107645,107648],{"className":107644},[57],[15,107646],{"className":107647,"style":840},[61],[15,107649,621],{"className":107650,"style":684},[66,67]," 近邻是最经典的例子。Stone 定理说：若 ",[15,107653,107655,107673],{"className":107654,"translate":19},[18],[15,107656,107658],{"className":107657},[23],[25,107659,107660],{"xmlns":27},[29,107661,107662,107670],{},[32,107663,107664,107666,107668],{},[35,107665,621],{},[39,107667,449],{},[35,107669,7620],{"mathvariant":138},[45,107671,107672],{"encoding":47},"k\\to\\infty",[15,107674,107676,107694],{"className":107675,"ariaHidden":53},[52],[15,107677,107679,107682,107685,107688,107691],{"className":107678},[57],[15,107680],{"className":107681,"style":840},[61],[15,107683,621],{"className":107684,"style":684},[66,67],[15,107686],{"className":107687,"style":173},[71],[15,107689,449],{"className":107690},[177],[15,107692],{"className":107693,"style":173},[71],[15,107695,107697,107700],{"className":107696},[57],[15,107698],{"className":107699,"style":86},[61],[15,107701,7620],{"className":107702},[66]," 且 ",[15,107705,107707,107729],{"className":107706,"translate":19},[18],[15,107708,107710],{"className":107709},[23],[25,107711,107712],{"xmlns":27},[29,107713,107714,107726],{},[32,107715,107716,107718,107720,107722,107724],{},[35,107717,621],{},[35,107719,6625],{"mathvariant":138},[35,107721,37],{},[39,107723,449],{},[1209,107725,1804],{},[45,107727,107728],{"encoding":47},"k\u002Fn\\to0",[15,107730,107732,107756],{"className":107731,"ariaHidden":53},[52],[15,107733,107735,107738,107741,107744,107747,107750,107753],{"className":107734},[57],[15,107736],{"className":107737,"style":154},[61],[15,107739,621],{"className":107740,"style":684},[66,67],[15,107742,6625],{"className":107743},[66],[15,107745,37],{"className":107746},[66,67],[15,107748],{"className":107749,"style":173},[71],[15,107751,449],{"className":107752},[177],[15,107754],{"className":107755,"style":173},[71],[15,107757,107759,107762],{"className":107758},[57],[15,107760],{"className":107761,"style":1836},[61],[15,107763,1804],{"className":107764},[66],[15,107766,107768,107781],{"className":107767,"translate":19},[18],[15,107769,107771],{"className":107770},[23],[25,107772,107773],{"xmlns":27},[29,107774,107775,107779],{},[32,107776,107777],{},[35,107778,621],{},[45,107780,621],{"encoding":47},[15,107782,107784],{"className":107783,"ariaHidden":53},[52],[15,107785,107787,107790],{"className":107786},[57],[15,107788],{"className":107789,"style":840},[61],[15,107791,621],{"className":107792,"style":684},[66,67],"-NN 是普遍一致的。",[15,107795,107797,107810],{"className":107796,"translate":19},[18],[15,107798,107800],{"className":107799},[23],[25,107801,107802],{"xmlns":27},[29,107803,107804,107808],{},[32,107805,107806],{},[1209,107807,1211],{},[45,107809,1211],{"encoding":47},[15,107811,107813],{"className":107812,"ariaHidden":53},[52],[15,107814,107816,107819],{"className":107815},[57],[15,107817],{"className":107818,"style":1836},[61],[15,107820,1211],{"className":107821},[66],"-NN 更极端：它的渐近错误率不超过 ",[15,107824,107826,107846],{"className":107825,"translate":19},[18],[15,107827,107829],{"className":107828},[23],[25,107830,107831],{"xmlns":27},[29,107832,107833,107843],{},[32,107834,107835,107837],{},[1209,107836,1263],{},[131,107838,107839,107841],{},[35,107840,455],{},[39,107842,1162],{},[45,107844,107845],{"encoding":47},"2R^*",[15,107847,107849],{"className":107848,"ariaHidden":53},[52],[15,107850,107852,107855,107858],{"className":107851},[57],[15,107853],{"className":107854,"style":16249},[61],[15,107856,1263],{"className":107857},[66],[15,107859,107861,107864],{"className":107860},[66],[15,107862,455],{"className":107863,"style":62437},[66,67],[15,107865,107867],{"className":107866},[198],[15,107868,107870],{"className":107869},[202],[15,107871,107873],{"className":107872},[206],[15,107874,107876],{"className":107875,"style":16249},[210],[15,107877,107878,107881],{"style":213},[15,107879],{"className":107880,"style":218},[217],[15,107882,107884],{"className":107883},[222,223,224,225],[15,107885,1162],{"className":107886},[76,225],"。这个「二倍」的常数很说明问题——即使是最简单的局部方法，也能在渐近意义下接近贝叶斯风险，代价是收敛速度很慢，而且在高维里被维数灾难拖垮。",[11,107889,107890,107891,107976],{},"一致性是一个渐近性质，它不告诉你有限样本下会发生什么。一个普遍一致的估计器可能在 ",[15,107892,107894,107917],{"className":107893,"translate":19},[18],[15,107895,107897],{"className":107896},[23],[25,107898,107899],{"xmlns":27},[29,107900,107901,107914],{},[32,107902,107903,107905,107907],{},[35,107904,37],{},[39,107906,129],{},[131,107908,107909,107912],{},[1209,107910,107911],{},"10",[1209,107913,59943],{},[45,107915,107916],{"encoding":47},"n=10^6",[15,107918,107920,107938],{"className":107919,"ariaHidden":53},[52],[15,107921,107923,107926,107929,107932,107935],{"className":107922},[57],[15,107924],{"className":107925,"style":86},[61],[15,107927,37],{"className":107928},[66,67],[15,107930],{"className":107931,"style":173},[71],[15,107933,129],{"className":107934},[177],[15,107936],{"className":107937,"style":173},[71],[15,107939,107941,107944,107947],{"className":107940},[57],[15,107942],{"className":107943,"style":6740},[61],[15,107945,1211],{"className":107946},[66],[15,107948,107950,107953],{"className":107949},[66],[15,107951,1804],{"className":107952},[66],[15,107954,107956],{"className":107955},[198],[15,107957,107959],{"className":107958},[202],[15,107960,107962],{"className":107961},[206],[15,107963,107965],{"className":107964,"style":6740},[210],[15,107966,107967,107970],{"style":213},[15,107968],{"className":107969,"style":218},[217],[15,107971,107973],{"className":107972},[222,223,224,225],[15,107974,59943],{"className":107975},[66,225]," 时仍然表现很差；一个不一致的估计器可能在某个特定问题上表现得很好。贝叶斯风险给出的是终点，不是路线。",[1904,107978,107980],{"id":107979},"_73-pac-bayes把信念放进泛化界","7.3 PAC-Bayes：把「信念」放进泛化界",[11,107982,107983,107984,108026],{},"PAC-Bayes 把贝叶斯式的分布对象与频率式的泛化保证放在同一个不等式里。这里的 ",[15,107985,107987,108005],{"className":107986,"translate":19},[18],[15,107988,107990],{"className":107989},[23],[25,107991,107992],{"xmlns":27},[29,107993,107994,108002],{},[32,107995,107996,107998,108000],{},[35,107997,28399],{},[39,107999,335],{"separator":53},[35,108001,83348],{},[45,108003,108004],{"encoding":47},"P,Q",[15,108006,108008],{"className":108007,"ariaHidden":53},[52],[15,108009,108011,108014,108017,108020,108023],{"className":108010},[57],[15,108012],{"className":108013,"style":17379},[61],[15,108015,28399],{"className":108016,"style":62513},[66,67],[15,108018,335],{"className":108019},[422],[15,108021],{"className":108022,"style":697},[71],[15,108024,83348],{"className":108025},[66,67]," 是假设空间上的分布，不要与数据分布混淆。",[3054,108028,108029,108272,108861],{},[11,108030,108031,108034,108035,108063,108064,108115,108116,108190,108191,108242,108243,108271],{},[2040,108032,108033],{},"定理 7.1（McAllester 界）"," 设先验 ",[15,108036,108038,108051],{"className":108037,"translate":19},[18],[15,108039,108041],{"className":108040},[23],[25,108042,108043],{"xmlns":27},[29,108044,108045,108049],{},[32,108046,108047],{},[35,108048,28399],{},[45,108050,28399],{"encoding":47},[15,108052,108054],{"className":108053,"ariaHidden":53},[52],[15,108055,108057,108060],{"className":108056},[57],[15,108058],{"className":108059,"style":490},[61],[15,108061,28399],{"className":108062,"style":62513},[66,67]," 在看到数据前固定，损失取值于 ",[15,108065,108067,108088],{"className":108066,"translate":19},[18],[15,108068,108070],{"className":108069},[23],[25,108071,108072],{"xmlns":27},[29,108073,108074,108086],{},[32,108075,108076,108078,108080,108082,108084],{},[39,108077,1194],{"stretchy":119},[1209,108079,1804],{},[39,108081,335],{"separator":53},[1209,108083,1211],{},[39,108085,1266],{"stretchy":119},[45,108087,19041],{"encoding":47},[15,108089,108091],{"className":108090,"ariaHidden":53},[52],[15,108092,108094,108097,108100,108103,108106,108109,108112],{"className":108093},[57],[15,108095],{"className":108096,"style":154},[61],[15,108098,1194],{"className":108099},[162],[15,108101,1804],{"className":108102},[66],[15,108104,335],{"className":108105},[422],[15,108107],{"className":108108,"style":697},[71],[15,108110,1211],{"className":108111},[66],[15,108113,1266],{"className":108114},[169],"。对任意 ",[15,108117,108119,108145],{"className":108118,"translate":19},[18],[15,108120,108122],{"className":108121},[23],[25,108123,108124],{"xmlns":27},[29,108125,108126,108142],{},[32,108127,108128,108130,108132,108134,108136,108138,108140],{},[35,108129,3885],{},[39,108131,1187],{},[39,108133,120],{"stretchy":119},[1209,108135,1804],{},[39,108137,335],{"separator":53},[1209,108139,1211],{},[39,108141,126],{"stretchy":119},[45,108143,108144],{"encoding":47},"\\delta\\in(0,1)",[15,108146,108148,108166],{"className":108147,"ariaHidden":53},[52],[15,108149,108151,108154,108157,108160,108163],{"className":108150},[57],[15,108152],{"className":108153,"style":9298},[61],[15,108155,3885],{"className":108156,"style":3946},[66,67],[15,108158],{"className":108159,"style":173},[71],[15,108161,1187],{"className":108162},[177],[15,108164],{"className":108165,"style":173},[71],[15,108167,108169,108172,108175,108178,108181,108184,108187],{"className":108168},[57],[15,108170],{"className":108171,"style":154},[61],[15,108173,120],{"className":108174},[162],[15,108176,1804],{"className":108177},[66],[15,108179,335],{"className":108180},[422],[15,108182],{"className":108183,"style":697},[71],[15,108185,1211],{"className":108186},[66],[15,108188,126],{"className":108189},[169],"，以至少 ",[15,108192,108194,108212],{"className":108193,"translate":19},[18],[15,108195,108197],{"className":108196},[23],[25,108198,108199],{"xmlns":27},[29,108200,108201,108209],{},[32,108202,108203,108205,108207],{},[1209,108204,1211],{},[39,108206,6602],{},[35,108208,3885],{},[45,108210,108211],{"encoding":47},"1-\\delta",[15,108213,108215,108233],{"className":108214,"ariaHidden":53},[52],[15,108216,108218,108221,108224,108227,108230],{"className":108217},[57],[15,108219],{"className":108220,"style":30468},[61],[15,108222,1211],{"className":108223},[66],[15,108225],{"className":108226,"style":72},[71],[15,108228,6602],{"className":108229},[76],[15,108231],{"className":108232,"style":72},[71],[15,108234,108236,108239],{"className":108235},[57],[15,108237],{"className":108238,"style":840},[61],[15,108240,3885],{"className":108241,"style":3946},[66,67]," 的概率，对所有后验分布 ",[15,108244,108246,108259],{"className":108245,"translate":19},[18],[15,108247,108249],{"className":108248},[23],[25,108250,108251],{"xmlns":27},[29,108252,108253,108257],{},[32,108254,108255],{},[35,108256,83348],{},[45,108258,83348],{"encoding":47},[15,108260,108262],{"className":108261,"ariaHidden":53},[52],[15,108263,108265,108268],{"className":108264},[57],[15,108266],{"className":108267,"style":17379},[61],[15,108269,83348],{"className":108270},[66,67]," 同时成立：",[15,108273,108275],{"className":108274,"translate":19},[294],[15,108276,108278,108400],{"className":108277,"translate":19},[18],[15,108279,108281],{"className":108280},[23],[25,108282,108283],{"xmlns":27,"display":303},[29,108284,108285,108397],{},[32,108286,108287,108299,108301,108303,108305,108307,108309,108311,108313,108325,108327,108337,108339,108341,108343,108345,108347,108395],{},[1057,108288,108289,108291],{},[35,108290,62381],{"mathvariant":454},[32,108292,108293,108295,108297],{},[35,108294,20182],{},[39,108296,28391],{},[35,108298,83348],{},[39,108300,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,108302,455],{},[39,108304,120],{"stretchy":119},[35,108306,20182],{},[39,108308,126],{"stretchy":119},[39,108310,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,108312,24220],{},[1057,108314,108315,108317],{},[35,108316,62381],{"mathvariant":454},[32,108318,108319,108321,108323],{},[35,108320,20182],{},[39,108322,28391],{},[35,108324,83348],{},[39,108326,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[1057,108328,108329,108335],{},[4296,108330,108331,108333],{"accent":53},[35,108332,455],{},[39,108334,78130],{},[35,108336,37],{},[39,108338,120],{"stretchy":119},[35,108340,20182],{},[39,108342,126],{"stretchy":119},[39,108344,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,108346,1245],{},[48508,108348,108349],{},[6467,108350,108351,108389],{},[32,108352,108353,108359,108361,108363,108365,108367,108369,108371,108373,108375,108377,108379,108383,108385,108387],{},[32,108354,108355,108357],{},[35,108356,2853],{"mathvariant":138},[35,108358,1216],{"mathvariant":138},[39,108360,120],{"stretchy":119},[35,108362,83348],{},[35,108364,1251],{"mathvariant":138},[35,108366,28399],{},[39,108368,126],{"stretchy":119},[39,108370,1245],{},[35,108372,27073],{},[39,108374,1170],{},[39,108376,120],{"stretchy":119},[1209,108378,1263],{},[48508,108380,108381],{},[35,108382,37],{},[35,108384,6625],{"mathvariant":138},[35,108386,3885],{},[39,108388,126],{"stretchy":119},[32,108390,108391,108393],{},[1209,108392,1263],{},[35,108394,37],{},[39,108396,335],{"separator":53},[45,108398,108399],{"encoding":47},"\\mathbb E_{h\\sim Q}\\big[R(h)\\big]\n\\le\n\\mathbb E_{h\\sim Q}\\big[\\hat R_n(h)\\big]\n+\n\\sqrt{\\frac{\\mathrm{KL}(Q\\|P)+\\log(2\\sqrt n\u002F\\delta)}{2n}},",[15,108401,108403,108491,108647],{"className":108402,"ariaHidden":53},[52],[15,108404,108406,108409,108458,108464,108467,108470,108473,108476,108482,108485,108488],{"className":108405},[57],[15,108407],{"className":108408,"style":4400},[61],[15,108410,108412,108415],{"className":108411},[66],[15,108413,62381],{"className":108414},[66,517],[15,108416,108418],{"className":108417},[198],[15,108419,108421,108450],{"className":108420},[202,1090],[15,108422,108424,108447],{"className":108423},[206],[15,108425,108427],{"className":108426,"style":1097},[210],[15,108428,108429,108432],{"style":1584},[15,108430],{"className":108431,"style":218},[217],[15,108433,108435],{"className":108434},[222,223,224,225],[15,108436,108438,108441,108444],{"className":108437},[66,225],[15,108439,20182],{"className":108440},[66,67,225],[15,108442,28391],{"className":108443},[177,225],[15,108445,83348],{"className":108446},[66,67,225],[15,108448,1114],{"className":108449},[1113],[15,108451,108453],{"className":108452},[206],[15,108454,108456],{"className":108455,"style":2715},[210],[15,108457],{},[15,108459,108461],{"className":108460},[66],[15,108462,1194],{"className":108463},[1422,1508],[15,108465,455],{"className":108466,"style":62437},[66,67],[15,108468,120],{"className":108469},[162],[15,108471,20182],{"className":108472},[66,67],[15,108474,126],{"className":108475},[169],[15,108477,108479],{"className":108478},[66],[15,108480,1266],{"className":108481},[1422,1508],[15,108483],{"className":108484,"style":173},[71],[15,108486,24220],{"className":108487},[177],[15,108489],{"className":108490,"style":173},[71],[15,108492,108494,108498,108547,108553,108623,108626,108629,108632,108638,108641,108644],{"className":108493},[57],[15,108495],{"className":108496,"style":108497},[61],"height:1.2968em;vertical-align:-0.35em;",[15,108499,108501,108504],{"className":108500},[66],[15,108502,62381],{"className":108503},[66,517],[15,108505,108507],{"className":108506},[198],[15,108508,108510,108539],{"className":108509},[202,1090],[15,108511,108513,108536],{"className":108512},[206],[15,108514,108516],{"className":108515,"style":1097},[210],[15,108517,108518,108521],{"style":1584},[15,108519],{"className":108520,"style":218},[217],[15,108522,108524],{"className":108523},[222,223,224,225],[15,108525,108527,108530,108533],{"className":108526},[66,225],[15,108528,20182],{"className":108529},[66,67,225],[15,108531,28391],{"className":108532},[177,225],[15,108534,83348],{"className":108535},[66,67,225],[15,108537,1114],{"className":108538},[1113],[15,108540,108542],{"className":108541},[206],[15,108543,108545],{"className":108544,"style":2715},[210],[15,108546],{},[15,108548,108550],{"className":108549},[66],[15,108551,1194],{"className":108552},[1422,1508],[15,108554,108556,108589],{"className":108555},[66],[15,108557,108559],{"className":108558},[66,78147],[15,108560,108562],{"className":108561},[202],[15,108563,108565],{"className":108564},[206],[15,108566,108569,108577],{"className":108567,"style":108568},[210],"height:0.9468em;",[15,108570,108571,108574],{"style":1386},[15,108572],{"className":108573,"style":1368},[217],[15,108575,455],{"className":108576,"style":62437},[66,67],[15,108578,108580,108583],{"style":108579},"top:-3.2523em;",[15,108581],{"className":108582,"style":1368},[217],[15,108584,108586],{"className":108585,"style":96110},[78175],[15,108587,78130],{"className":108588},[66],[15,108590,108592],{"className":108591},[198],[15,108593,108595,108615],{"className":108594},[202,1090],[15,108596,108598,108612],{"className":108597},[206],[15,108599,108601],{"className":108600,"style":2198},[210],[15,108602,108603,108606],{"style":106327},[15,108604],{"className":108605,"style":218},[217],[15,108607,108609],{"className":108608},[222,223,224,225],[15,108610,37],{"className":108611},[66,67,225],[15,108613,1114],{"className":108614},[1113],[15,108616,108618],{"className":108617},[206],[15,108619,108621],{"className":108620,"style":1121},[210],[15,108622],{},[15,108624,120],{"className":108625},[162],[15,108627,20182],{"className":108628},[66,67],[15,108630,126],{"className":108631},[169],[15,108633,108635],{"className":108634},[66],[15,108636,1266],{"className":108637},[1422,1508],[15,108639],{"className":108640,"style":72},[71],[15,108642,1245],{"className":108643},[76],[15,108645],{"className":108646,"style":72},[71],[15,108648,108650,108654,108858],{"className":108649},[57],[15,108651],{"className":108652,"style":108653},[61],"height:2.44em;vertical-align:-0.7105em;",[15,108655,108657],{"className":108656},[66,48550],[15,108658,108660,108849],{"className":108659},[202,1090],[15,108661,108663,108846],{"className":108662},[206],[15,108664,108667,108833],{"className":108665,"style":108666},[210],"height:1.7295em;",[15,108668,108670,108673],{"className":108669,"style":89055},[48564],[15,108671],{"className":108672,"style":89059},[217],[15,108674,108676],{"className":108675,"style":89063},[66],[15,108677,108679,108682,108830],{"className":108678},[66],[15,108680],{"className":108681},[162,23417],[15,108683,108685],{"className":108684},[6467],[15,108686,108688,108822],{"className":108687},[202,1090],[15,108689,108691,108819],{"className":108690},[206],[15,108692,108695,108709,108717],{"className":108693,"style":108694},[210],"height:1.4773em;",[15,108696,108697,108700],{"style":23433},[15,108698],{"className":108699,"style":1368},[217],[15,108701,108703,108706],{"className":108702},[66],[15,108704,1263],{"className":108705},[66],[15,108707,37],{"className":108708},[66,67],[15,108710,108711,108714],{"style":23445},[15,108712],{"className":108713,"style":1368},[217],[15,108715],{"className":108716,"style":4440},[23452],[15,108718,108719,108722],{"style":23455},[15,108720],{"className":108721,"style":1368},[217],[15,108723,108725,108731,108734,108737,108740,108743,108746,108749,108752,108755,108760,108763,108766,108810,108813,108816],{"className":108724},[66],[15,108726,108728],{"className":108727},[66],[15,108729,88457],{"className":108730},[66,4429],[15,108732,120],{"className":108733},[162],[15,108735,83348],{"className":108736},[66,67],[15,108738,1251],{"className":108739},[66],[15,108741,28399],{"className":108742,"style":62513},[66,67],[15,108744,126],{"className":108745},[169],[15,108747],{"className":108748,"style":72},[71],[15,108750,1245],{"className":108751},[76],[15,108753],{"className":108754,"style":72},[71],[15,108756,27283,108758],{"className":108757},[1339],[15,108759,1344],{"style":1343},[15,108761,120],{"className":108762},[162],[15,108764,1263],{"className":108765},[66],[15,108767,108769],{"className":108768},[66,48550],[15,108770,108772,108802],{"className":108771},[202,1090],[15,108773,108775,108799],{"className":108774},[206],[15,108776,108778,108787],{"className":108777,"style":103621},[210],[15,108779,108781,108784],{"className":108780,"style":1386},[48564],[15,108782],{"className":108783,"style":1368},[217],[15,108785,37],{"className":108786,"style":48571},[66,67],[15,108788,108789,108792],{"style":103633},[15,108790],{"className":108791,"style":1368},[217],[15,108793,108795],{"className":108794,"style":48591},[48590],[34828,108796,108797],{"xmlns":34830,"width":48594,"height":48595,"viewBox":48596,"preserveAspectRatio":48597},[34834,108798],{"d":48600},[15,108800,1114],{"className":108801},[1113],[15,108803,108805],{"className":108804},[206],[15,108806,108808],{"className":108807,"style":103653},[210],[15,108809],{},[15,108811,6625],{"className":108812},[66],[15,108814,3885],{"className":108815,"style":3946},[66,67],[15,108817,126],{"className":108818},[169],[15,108820,1114],{"className":108821},[1113],[15,108823,108825],{"className":108824},[206],[15,108826,108828],{"className":108827,"style":23474},[210],[15,108829],{},[15,108831],{"className":108832},[169,23417],[15,108834,108836,108839],{"style":108835},"top:-3.6895em;",[15,108837],{"className":108838,"style":89059},[217],[15,108840,108842],{"className":108841,"style":89267},[48590],[34828,108843,108844],{"xmlns":34830,"width":48594,"height":89270,"viewBox":89271,"preserveAspectRatio":48597},[34834,108845],{"d":89274},[15,108847,1114],{"className":108848},[1113],[15,108850,108852],{"className":108851},[206],[15,108853,108856],{"className":108854,"style":108855},[210],"height:0.7105em;",[15,108857],{},[15,108859,335],{"className":108860},[422],[11,108862,425,108863,108966],{},[15,108864,108866,108888],{"className":108865,"translate":19},[18],[15,108867,108869],{"className":108868},[23],[25,108870,108871],{"xmlns":27},[29,108872,108873,108885],{},[32,108874,108875],{},[1057,108876,108877,108883],{},[4296,108878,108879,108881],{"accent":53},[35,108880,455],{},[39,108882,78130],{},[35,108884,37],{},[45,108886,108887],{"encoding":47},"\\hat R_n",[15,108889,108891],{"className":108890,"ariaHidden":53},[52],[15,108892,108894,108898],{"className":108893},[57],[15,108895],{"className":108896,"style":108897},[61],"height:1.0968em;vertical-align:-0.15em;",[15,108899,108901,108932],{"className":108900},[66],[15,108902,108904],{"className":108903},[66,78147],[15,108905,108907],{"className":108906},[202],[15,108908,108910],{"className":108909},[206],[15,108911,108913,108921],{"className":108912,"style":108568},[210],[15,108914,108915,108918],{"style":1386},[15,108916],{"className":108917,"style":1368},[217],[15,108919,455],{"className":108920,"style":62437},[66,67],[15,108922,108923,108926],{"style":108579},[15,108924],{"className":108925,"style":1368},[217],[15,108927,108929],{"className":108928,"style":96110},[78175],[15,108930,78130],{"className":108931},[66],[15,108933,108935],{"className":108934},[198],[15,108936,108938,108958],{"className":108937},[202,1090],[15,108939,108941,108955],{"className":108940},[206],[15,108942,108944],{"className":108943,"style":2198},[210],[15,108945,108946,108949],{"style":106327},[15,108947],{"className":108948,"style":218},[217],[15,108950,108952],{"className":108951},[222,223,224,225],[15,108953,37],{"className":108954},[66,67,225],[15,108956,1114],{"className":108957},[1113],[15,108959,108961],{"className":108960},[206],[15,108962,108964],{"className":108963,"style":1121},[210],[15,108965],{}," 是经验风险。",[11,108968,108969,108970,108998,108999,109027],{},"这个界的美妙之处在于：",[15,108971,108973,108986],{"className":108972,"translate":19},[18],[15,108974,108976],{"className":108975},[23],[25,108977,108978],{"xmlns":27},[29,108979,108980,108984],{},[32,108981,108982],{},[35,108983,28399],{},[45,108985,28399],{"encoding":47},[15,108987,108989],{"className":108988,"ariaHidden":53},[52],[15,108990,108992,108995],{"className":108991},[57],[15,108993],{"className":108994,"style":490},[61],[15,108996,28399],{"className":108997,"style":62513},[66,67]," 必须在看到数据前固定，",[15,109000,109002,109015],{"className":109001,"translate":19},[18],[15,109003,109005],{"className":109004},[23],[25,109006,109007],{"xmlns":27},[29,109008,109009,109013],{},[32,109010,109011],{},[35,109012,83348],{},[45,109014,83348],{"encoding":47},[15,109016,109018],{"className":109017,"ariaHidden":53},[52],[15,109019,109021,109024],{"className":109020},[57],[15,109022],{"className":109023,"style":17379},[61],[15,109025,83348],{"className":109026},[66,67]," 可以是任何依赖于数据的分布。KL 项衡量「为了解释数据，你把信念从先验挪了多远」。先验越集中、后验越贴近先验，界就越紧；后验越远离先验，界就越松。",[11,109029,109030,109031,109034,109035,109063,109064,109092],{},"这正是奥卡姆剃刀的概率版本，只不过它作用在",[2040,109032,109033],{},"分布","而不是单个假设上。一个在大量假设上分散的 ",[15,109036,109038,109051],{"className":109037,"translate":19},[18],[15,109039,109041],{"className":109040},[23],[25,109042,109043],{"xmlns":27},[29,109044,109045,109049],{},[32,109046,109047],{},[35,109048,83348],{},[45,109050,83348],{"encoding":47},[15,109052,109054],{"className":109053,"ariaHidden":53},[52],[15,109055,109057,109060],{"className":109056},[57],[15,109058],{"className":109059,"style":17379},[61],[15,109061,83348],{"className":109062},[66,67]," 会付出很大的 KL，即使它的经验风险很低；一个集中在少数假设上的 ",[15,109065,109067,109080],{"className":109066,"translate":19},[18],[15,109068,109070],{"className":109069},[23],[25,109071,109072],{"xmlns":27},[29,109073,109074,109078],{},[32,109075,109076],{},[35,109077,83348],{},[45,109079,83348],{"encoding":47},[15,109081,109083],{"className":109082,"ariaHidden":53},[52],[15,109084,109086,109089],{"className":109085},[57],[15,109087],{"className":109088,"style":17379},[61],[15,109090,83348],{"className":109091},[66,67]," 则可以用较小的 KL 换取较好的界。",[11,109094,109095],{},"PAC-Bayes 也有它的局限：界在深度网络里常常很松，常数项可能大到没有实际意义。但它提供了一个正确的概念框架：泛化不是「参数少」的同义词，而是「后验没有比先验复杂太多」。后来的 flatness、sharpness、信息论泛化界，都可以在这个框架里找到位置。",[1904,109097,109099],{"id":109098},"_74-变分推断当后验收敛不了","7.4 变分推断：当后验收敛不了",[11,109101,109102,109103,109146,109147,109176],{},"精确后验通常不可行，因为归一化常数 ",[15,109104,109106,109125],{"className":109105,"translate":19},[18],[15,109107,109109],{"className":109108},[23],[25,109110,109111],{"xmlns":27},[29,109112,109113,109123],{},[32,109114,109115,109117,109119,109121],{},[35,109116,11],{},[39,109118,120],{"stretchy":119},[35,109120,32458],{},[39,109122,126],{"stretchy":119},[45,109124,65373],{"encoding":47},[15,109126,109128],{"className":109127,"ariaHidden":53},[52],[15,109129,109131,109134,109137,109140,109143],{"className":109130},[57],[15,109132],{"className":109133,"style":154},[61],[15,109135,11],{"className":109136},[66,67],[15,109138,120],{"className":109139},[162],[15,109141,32458],{"className":109142,"style":23025},[66,67],[15,109144,126],{"className":109145},[169]," 是一个高维积分。变分推断用一个受限制的分布族 ",[15,109148,109150,109164],{"className":109149,"translate":19},[18],[15,109151,109153],{"className":109152},[23],[25,109154,109155],{"xmlns":27},[29,109156,109157,109161],{},[32,109158,109159],{},[35,109160,83348],{"mathvariant":445},[45,109162,109163],{"encoding":47},"\\mathcal Q",[15,109165,109167],{"className":109166,"ariaHidden":53},[52],[15,109168,109170,109173],{"className":109169},[57],[15,109171],{"className":109172,"style":61920},[61],[15,109174,83348],{"className":109175},[66,494]," 去逼近后验，目标是",[15,109178,109180],{"className":109179,"translate":19},[294],[15,109181,109183,109263],{"className":109182,"translate":19},[18],[15,109184,109186],{"className":109185},[23],[25,109187,109188],{"xmlns":27,"display":303},[29,109189,109190,109260],{},[32,109191,109192,109198,109200,109202,109204,109220,109222,109228,109230,109232,109234,109236,109238,109240,109242,109244,109246,109248,109250,109252,109254,109256,109258],{},[131,109193,109194,109196],{},[35,109195,94276],{},[39,109197,1162],{},[39,109199,129],{},[35,109201,1167],{},[39,109203,1170],{},[1172,109205,109206,109212],{},[32,109207,109208,109210],{},[35,109209,1178],{},[39,109211,1170],{},[32,109213,109214,109216,109218],{},[35,109215,94276],{},[39,109217,1187],{},[35,109219,83348],{"mathvariant":445},[1274,109221,24233],{},[32,109223,109224,109226],{},[35,109225,2853],{"mathvariant":138},[35,109227,1216],{"mathvariant":138},[39,109229,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,109231,94276],{},[39,109233,120],{"stretchy":119},[35,109235,61618],{},[39,109237,126],{"stretchy":119},[1274,109239,24233],{},[35,109241,1251],{"mathvariant":138},[1274,109243,24233],{},[35,109245,11],{},[39,109247,120],{"stretchy":119},[35,109249,61618],{},[39,109251,34430],{},[35,109253,32458],{},[39,109255,126],{"stretchy":119},[39,109257,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,109259,3573],{"mathvariant":138},[45,109261,109262],{"encoding":47},"q^*=\\arg\\min_{q\\in\\mathcal Q}\\ \\mathrm{KL}\\big(q(\\theta)\\ \\|\\ p(\\theta\\mid D)\\big).",[15,109264,109266,109310,109435],{"className":109265,"ariaHidden":53},[52],[15,109267,109269,109272,109301,109304,109307],{"className":109268},[57],[15,109270],{"className":109271,"style":1289},[61],[15,109273,109275,109278],{"className":109274},[66],[15,109276,94276],{"className":109277,"style":1515},[66,67],[15,109279,109281],{"className":109280},[198],[15,109282,109284],{"className":109283},[202],[15,109285,109287],{"className":109286},[206],[15,109288,109290],{"className":109289,"style":1308},[210],[15,109291,109292,109295],{"style":397},[15,109293],{"className":109294,"style":218},[217],[15,109296,109298],{"className":109297},[222,223,224,225],[15,109299,1162],{"className":109300},[76,225],[15,109302],{"className":109303,"style":173},[71],[15,109305,129],{"className":109306},[177],[15,109308],{"className":109309,"style":173},[71],[15,109311,109313,109317,109322,109325,109378,109381,109384,109390,109396,109399,109402,109405,109408,109411,109414,109417,109420,109423,109426,109429,109432],{"className":109312},[57],[15,109314],{"className":109315,"style":109316},[61],"height:1.7304em;vertical-align:-0.8804em;",[15,109318,1340,109320],{"className":109319},[1339],[15,109321,1344],{"style":1343},[15,109323],{"className":109324,"style":697},[71],[15,109326,109328],{"className":109327},[1339,1351],[15,109329,109331,109370],{"className":109330},[202,1090],[15,109332,109334,109367],{"className":109333},[206],[15,109335,109337,109357],{"className":109336,"style":1361},[210],[15,109338,109339,109342],{"style":24354},[15,109340],{"className":109341,"style":1368},[217],[15,109343,109345],{"className":109344},[222,223,224,225],[15,109346,109348,109351,109354],{"className":109347},[66,225],[15,109349,94276],{"className":109350,"style":1515},[66,67,225],[15,109352,1187],{"className":109353},[177,225],[15,109355,83348],{"className":109356},[66,494,225],[15,109358,109359,109362],{"style":1386},[15,109360],{"className":109361,"style":1368},[217],[15,109363,109364],{},[15,109365,1178],{"className":109366},[1339],[15,109368,1114],{"className":109369},[1113],[15,109371,109373],{"className":109372},[206],[15,109374,109376],{"className":109375,"style":67080},[210],[15,109377],{},[15,109379,24233],{"className":109380},[71],[15,109382],{"className":109383,"style":697},[71],[15,109385,109387],{"className":109386},[66],[15,109388,88457],{"className":109389},[66,4429],[15,109391,109393],{"className":109392},[66],[15,109394,120],{"className":109395},[1422,1508],[15,109397,94276],{"className":109398,"style":1515},[66,67],[15,109400,120],{"className":109401},[162],[15,109403,61618],{"className":109404,"style":23025},[66,67],[15,109406,126],{"className":109407},[169],[15,109409,24233],{"className":109410},[71],[15,109412,1251],{"className":109413},[66],[15,109415,24233],{"className":109416},[71],[15,109418,11],{"className":109419},[66,67],[15,109421,120],{"className":109422},[162],[15,109424,61618],{"className":109425,"style":23025},[66,67],[15,109427],{"className":109428,"style":173},[71],[15,109430,34430],{"className":109431},[177],[15,109433],{"className":109434,"style":173},[71],[15,109436,109438,109441,109444,109447,109453],{"className":109437},[57],[15,109439],{"className":109440,"style":4400},[61],[15,109442,32458],{"className":109443,"style":23025},[66,67],[15,109445,126],{"className":109446},[169],[15,109448,109450],{"className":109449},[66],[15,109451,126],{"className":109452},[1422,1508],[15,109454,3573],{"className":109455},[66],[11,109457,109458,109459,109502],{},"直接最小化这个 KL 仍然需要 ",[15,109460,109462,109481],{"className":109461,"translate":19},[18],[15,109463,109465],{"className":109464},[23],[25,109466,109467],{"xmlns":27},[29,109468,109469,109479],{},[32,109470,109471,109473,109475,109477],{},[35,109472,11],{},[39,109474,120],{"stretchy":119},[35,109476,32458],{},[39,109478,126],{"stretchy":119},[45,109480,65373],{"encoding":47},[15,109482,109484],{"className":109483,"ariaHidden":53},[52],[15,109485,109487,109490,109493,109496,109499],{"className":109486},[57],[15,109488],{"className":109489,"style":154},[61],[15,109491,11],{"className":109492},[66,67],[15,109494,120],{"className":109495},[162],[15,109497,32458],{"className":109498,"style":23025},[66,67],[15,109500,126],{"className":109501},[169],"，于是利用恒等式",[15,109504,109506],{"className":109505,"translate":19},[294],[15,109507,109509,109643],{"className":109508,"translate":19},[18],[15,109510,109512],{"className":109511},[23],[25,109513,109514],{"xmlns":27,"display":303},[29,109515,109516,109640],{},[32,109517,109518,109520,109522,109524,109526,109528,109530,109532,109600,109602,109608,109610,109612,109614,109616,109618,109620,109622,109624,109626,109628,109630,109632,109634,109636,109638],{},[35,109519,27073],{},[39,109521,1170],{},[35,109523,11],{},[39,109525,120],{"stretchy":119},[35,109527,32458],{},[39,109529,126],{"stretchy":119},[39,109531,129],{},[1172,109533,109534,109582],{},[1172,109535,109536,109580],{},[32,109537,109538,109544],{},[1057,109539,109540,109542],{},[35,109541,62381],{"mathvariant":454},[35,109543,94276],{},[32,109545,109546,109548,109550,109552,109578],{},[39,109547,1194],{"fence":53},[35,109549,27073],{},[39,109551,1170],{},[6467,109553,109554,109568],{},[32,109555,109556,109558,109560,109562,109564,109566],{},[35,109557,11],{},[39,109559,120],{"stretchy":119},[35,109561,32458],{},[39,109563,335],{"separator":53},[35,109565,61618],{},[39,109567,126],{"stretchy":119},[32,109569,109570,109572,109574,109576],{},[35,109571,94276],{},[39,109573,120],{"stretchy":119},[35,109575,61618],{},[39,109577,126],{"stretchy":119},[39,109579,1266],{"fence":53},[39,109581,69846],{"stretchy":53},[32,109583,109584,109594,109596,109598],{},[32,109585,109586,109588,109590,109592],{},[35,109587,62381],{"mathvariant":138},[35,109589,1216],{"mathvariant":138},[35,109591,63972],{"mathvariant":138},[35,109593,22999],{"mathvariant":138},[39,109595,120],{"stretchy":119},[35,109597,94276],{},[39,109599,126],{"stretchy":119},[39,109601,1245],{},[32,109603,109604,109606],{},[35,109605,2853],{"mathvariant":138},[35,109607,1216],{"mathvariant":138},[39,109609,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,109611,94276],{},[39,109613,120],{"stretchy":119},[35,109615,61618],{},[39,109617,126],{"stretchy":119},[1274,109619,24233],{},[35,109621,1251],{"mathvariant":138},[1274,109623,24233],{},[35,109625,11],{},[39,109627,120],{"stretchy":119},[35,109629,61618],{},[39,109631,34430],{},[35,109633,32458],{},[39,109635,126],{"stretchy":119},[39,109637,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,109639,335],{"separator":53},[45,109641,109642],{"encoding":47},"\\log p(D)\n=\n\\underbrace{\\mathbb E_{q}\\left[\\log\\frac{p(D,\\theta)}{q(\\theta)}\\right]}_{\\mathrm{ELBO}(q)}\n+\n\\mathrm{KL}\\big(q(\\theta)\\ \\|\\ p(\\theta\\mid D)\\big),",[15,109644,109646,109681,109977,110034],{"className":109645,"ariaHidden":53},[52],[15,109647,109649,109652,109657,109660,109663,109666,109669,109672,109675,109678],{"className":109648},[57],[15,109650],{"className":109651,"style":154},[61],[15,109653,27283,109655],{"className":109654},[1339],[15,109656,1344],{"style":1343},[15,109658],{"className":109659,"style":697},[71],[15,109661,11],{"className":109662},[66,67],[15,109664,120],{"className":109665},[162],[15,109667,32458],{"className":109668,"style":23025},[66,67],[15,109670,126],{"className":109671},[169],[15,109673],{"className":109674,"style":173},[71],[15,109676,129],{"className":109677},[177],[15,109679],{"className":109680,"style":173},[71],[15,109682,109684,109688,109968,109971,109974],{"className":109683},[57],[15,109685],{"className":109686,"style":109687},[61],"height:3.948em;vertical-align:-2.498em;",[15,109689,109691],{"className":109690},[1413,1172],[15,109692,109694,109959],{"className":109693},[202,1090],[15,109695,109697,109956],{"className":109696},[206],[15,109698,109701,109729],{"className":109699,"style":109700},[210],"height:1.45em;",[15,109702,109704,109707],{"style":109703},"top:-1.127em;",[15,109705],{"className":109706,"style":63167},[217],[15,109708,109710],{"className":109709},[222,223,224,225],[15,109711,109713,109720,109723,109726],{"className":109712},[66,225],[15,109714,109716],{"className":109715},[66,225],[15,109717,109719],{"className":109718},[66,4429,225],"ELBO",[15,109721,120],{"className":109722},[162,225],[15,109724,94276],{"className":109725,"style":1515},[66,67,225],[15,109727,126],{"className":109728},[169,225],[15,109730,109732,109735],{"style":109731},"top:-3.45em;",[15,109733],{"className":109734,"style":63167},[217],[15,109736,109738],{"className":109737},[1413,1172],[15,109739,109741,109947],{"className":109740},[202,1090],[15,109742,109744,109944],{"className":109743},[206],[15,109745,109747,109778],{"className":109746,"style":109700},[210],[15,109748,109751,109754],{"className":109749,"style":109750},[48564],"top:-1.852em;",[15,109752],{"className":109753,"style":63167},[217],[15,109755,109757,109764,109771],{"className":109756,"style":70049},[70048],[15,109758,109760],{"className":109759,"style":70054},[70053],[34828,109761,109762],{"xmlns":34830,"width":48594,"height":70057,"viewBox":70058,"preserveAspectRatio":48597},[34834,109763],{"d":70061},[15,109765,109767],{"className":109766,"style":70054},[70065],[34828,109768,109769],{"xmlns":34830,"width":48594,"height":70057,"viewBox":70058,"preserveAspectRatio":70068},[34834,109770],{"d":70071},[15,109772,109774],{"className":109773,"style":70054},[70075],[34828,109775,109776],{"xmlns":34830,"width":48594,"height":70057,"viewBox":70058,"preserveAspectRatio":70078},[34834,109777],{"d":70081},[15,109779,109780,109783],{"style":109731},[15,109781],{"className":109782,"style":63167},[217],[15,109784,109786,109829,109832],{"className":109785},[66],[15,109787,109789,109792],{"className":109788},[66],[15,109790,62381],{"className":109791},[66,517],[15,109793,109795],{"className":109794},[198],[15,109796,109798,109821],{"className":109797},[202,1090],[15,109799,109801,109818],{"className":109800},[206],[15,109802,109804],{"className":109803,"style":2198},[210],[15,109805,109806,109809],{"style":1584},[15,109807],{"className":109808,"style":218},[217],[15,109810,109812],{"className":109811},[222,223,224,225],[15,109813,109815],{"className":109814},[66,225],[15,109816,94276],{"className":109817,"style":1515},[66,67,225],[15,109819,1114],{"className":109820},[1113],[15,109822,109824],{"className":109823},[206],[15,109825,109827],{"className":109826,"style":2715},[210],[15,109828],{},[15,109830],{"className":109831,"style":697},[71],[15,109833,109835,109841,109846,109849,109938],{"className":109834},[1413],[15,109836,109838],{"className":109837,"style":1418},[162,1417],[15,109839,1194],{"className":109840},[1422,224],[15,109842,27283,109844],{"className":109843},[1339],[15,109845,1344],{"style":1343},[15,109847],{"className":109848,"style":697},[71],[15,109850,109852,109855,109935],{"className":109851},[66],[15,109853],{"className":109854},[162,23417],[15,109856,109858],{"className":109857},[6467],[15,109859,109861,109927],{"className":109860},[202,1090],[15,109862,109864,109924],{"className":109863},[206],[15,109865,109867,109887,109895],{"className":109866,"style":41211},[210],[15,109868,109869,109872],{"style":23433},[15,109870],{"className":109871,"style":1368},[217],[15,109873,109875,109878,109881,109884],{"className":109874},[66],[15,109876,94276],{"className":109877,"style":1515},[66,67],[15,109879,120],{"className":109880},[162],[15,109882,61618],{"className":109883,"style":23025},[66,67],[15,109885,126],{"className":109886},[169],[15,109888,109889,109892],{"style":23445},[15,109890],{"className":109891,"style":1368},[217],[15,109893],{"className":109894,"style":4440},[23452],[15,109896,109897,109900],{"style":23455},[15,109898],{"className":109899,"s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E_q\\big[\\log p(D\\mid\\theta)\\big]\n-\n\\mathrm{KL}\\big(q(\\theta)\\ \\|\\ p(\\theta)\\big).",[15,110161,110163,110193,110274,110301],{"className":110162,"ariaHidden":53},[52],[15,110164,110166,110169,110175,110178,110181,110184,110187,110190],{"className":110165},[57],[15,110167],{"className":110168,"style":154},[61],[15,110170,110172],{"className":110171},[66],[15,110173,109719],{"className":110174},[66,4429],[15,110176,120],{"className":110177},[162],[15,110179,94276],{"className":110180,"style":1515},[66,67],[15,110182,126],{"className":110183},[169],[15,110185],{"className":110186,"style":173},[71],[15,110188,129],{"className":110189},[177],[15,110191],{"className":110192,"style":173},[71],[15,110194,110196,110199,110239,110245,110248,110253,110256,110259,110262,110265,110268,110271],{"className":110195},[57],[15,110197],{"className":110198,"style":4400},[61],[15,110200,110202,110205],{"className":110201},[66],[15,110203,62381],{"className":110204},[66,517],[15,110206,110208],{"className":110207},[198],[15,110209,110211,110231],{"className":110210},[202,1090],[15,110212,110214,110228],{"className":110213},[206],[15,110215,110217],{"className":110216,"style":2198},[210],[15,110218,110219,110222],{"style":1584},[15,110220],{"className":110221,"style":218},[217],[15,110223,110225],{"className":110224},[222,223,224,225],[15,110226,94276],{"className":110227,"style":1515},[66,67,225],[15,110229,1114],{"className":110230},[1113],[15,110232,110234],{"className":110233},[206],[15,110235,110237],{"className":110236,"style":2715},[210],[15,110238],{},[15,110240,110242],{"className":110241},[66],[15,110243,1194],{"className":110244},[1422,1508],[15,110246],{"className":110247,"style":697},[71],[15,110249,27283,110251],{"className":110250},[1339],[15,110252,1344],{"style":1343},[15,110254],{"className":110255,"style":697},[71],[15,110257,11],{"className":110258},[66,67],[15,110260,120],{"className":110261},[162],[15,110263,32458],{"className":110264,"style":23025},[66,67],[15,110266],{"className":110267,"style":173},[71],[15,110269,34430],{"className":110270},[177],[15,110272],{"className":110273,"style":173},[71],[15,110275,110277,110280,110283,110286,110292,110295,110298],{"className":110276},[57],[15,110278],{"className":110279,"style":4400},[61],[15,110281,61618],{"className":110282,"style":23025},[66,67],[15,110284,126],{"className":110285},[169],[15,110287,110289],{"className":110288},[66],[15,110290,1266],{"className":110291},[1422,1508],[15,110293],{"className":110294,"style":72},[71],[15,110296,6602],{"className":110297},[76],[15,110299],{"className":110300,"style":72},[71],[15,110302,110304,110307,110313,110319,110322,110325,110328,110331,110334,110337,110340,110343,110346,110349,110352,110358],{"className":110303},[57],[15,110305],{"className":110306,"style":4400},[61],[15,110308,110310],{"className":110309},[66],[15,110311,88457],{"className":110312},[66,4429],[15,110314,110316],{"className":110315},[66],[15,110317,120],{"className":110318},[1422,1508],[15,110320,94276],{"className":110321,"style":1515},[66,67],[15,110323,120],{"className":110324},[162],[15,110326,61618],{"className":110327,"style":23025},[66,67],[15,110329,126],{"className":110330},[169],[15,110332,24233],{"className":110333},[71],[15,110335,1251],{"className":110336},[66],[15,110338,24233],{"className":110339},[71],[15,110341,11],{"className":110342},[66,67],[15,110344,120],{"className":110345},[162],[15,110347,61618],{"className":110348,"style":23025},[66,67],[15,110350,126],{"className":110351},[169],[15,110353,110355],{"className":110354},[66],[15,110356,126],{"className":110357},[1422,1508],[15,110359,3573],{"className":110360},[66],[11,110362,110363,110364,110392,110393,110421,110422,10238,110517,48366,110573,110601],{},"第一项鼓励 ",[15,110365,110367,110380],{"className":110366,"translate":19},[18],[15,110368,110370],{"className":110369},[23],[25,110371,110372],{"xmlns":27},[29,110373,110374,110378],{},[32,110375,110376],{},[35,110377,94276],{},[45,110379,94276],{"encoding":47},[15,110381,110383],{"className":110382,"ariaHidden":53},[52],[15,110384,110386,110389],{"className":110385},[57],[15,110387],{"className":110388,"style":471},[61],[15,110390,94276],{"className":110391,"style":1515},[66,67]," 解释数据，第二项惩罚 ",[15,110394,110396,110409],{"className":110395,"translate":19},[18],[15,110397,110399],{"className":110398},[23],[25,110400,110401],{"xmlns":27},[29,110402,110403,110407],{},[32,110404,110405],{},[35,110406,94276],{},[45,110408,94276],{"encoding":47},[15,110410,110412],{"className":110411,"ariaHidden":53},[52],[15,110413,110415,110418],{"className":110414},[57],[15,110416],{"className":110417,"style":471},[61],[15,110419,94276],{"className":110420,"style":1515},[66,67]," 偏离先验。最大化 ELBO 等价于最小化 ",[15,110423,110425,110463],{"className":110424,"translate":19},[18],[15,110426,110428],{"className":110427},[23],[25,110429,110430],{"xmlns":27},[29,110431,110432,110460],{},[32,110433,110434,110440,110442,110444,110446,110448,110450,110452,110454,110456,110458],{},[32,110435,110436,110438],{},[35,110437,2853],{"mathvariant":138},[35,110439,1216],{"mathvariant":138},[39,110441,120],{"stretchy":119},[35,110443,94276],{},[35,110445,1251],{"mathvariant":138},[35,110447,11],{},[39,110449,120],{"stretchy":119},[35,110451,61618],{},[39,110453,34430],{},[35,110455,32458],{},[39,110457,126],{"stretchy":119},[39,110459,126],{"stretchy":119},[45,110461,110462],{"encoding":47},"\\mathrm{KL}(q\\|p(\\theta\\mid D))",[15,110464,110466,110505],{"className":110465,"ariaHidden":53},[52],[15,110467,110469,110472,110478,110481,110484,110487,110490,110493,110496,110499,110502],{"className":110468},[57],[15,110470],{"className":110471,"style":154},[61],[15,110473,110475],{"className":110474},[66],[15,110476,88457],{"className":110477},[66,4429],[15,110479,120],{"className":110480},[162],[15,110482,94276],{"className":110483,"style":1515},[66,67],[15,110485,1251],{"className":110486},[66],[15,110488,11],{"className":110489},[66,67],[15,110491,120],{"className":110492},[162],[15,110494,61618],{"className":110495,"style":23025},[66,67],[15,110497],{"className":110498,"style":173},[71],[15,110500,34430],{"className":110501},[177],[15,110503],{"className":110504,"style":173},[71],[15,110506,110508,110511,110514],{"className":110507},[57],[15,110509],{"className":110510,"style":154},[61],[15,110512,32458],{"className":110513,"style":23025},[66,67],[15,110515,18771],{"className":110516},[169],[15,110518,110520,110544],{"className":110519,"translate":19},[18],[15,110521,110523],{"className":110522},[23],[25,110524,110525],{"xmlns":27},[29,110526,110527,110541],{},[32,110528,110529,110531,110533,110535,110537,110539],{},[35,110530,27073],{},[39,110532,1170],{},[35,110534,11],{},[39,110536,120],{"stretchy":119},[35,110538,32458],{},[39,110540,126],{"stretchy":119},[45,110542,110543],{"encoding":47},"\\log p(D)",[15,110545,110547],{"className":110546,"ariaHidden":53},[52],[15,110548,110550,110553,110558,110561,110564,110567,110570],{"className":110549},[57],[15,110551],{"className":110552,"style":154},[61],[15,110554,27283,110556],{"className":110555},[1339],[15,110557,1344],{"style":1343},[15,110559],{"className":110560,"style":697},[71],[15,110562,11],{"className":110563},[66,67],[15,110565,120],{"className":110566},[162],[15,110568,32458],{"className":110569,"style":23025},[66,67],[15,110571,126],{"className":110572},[169],[15,110574,110576,110589],{"className":110575,"translate":19},[18],[15,110577,110579],{"className":110578},[23],[25,110580,110581],{"xmlns":27},[29,110582,110583,110587],{},[32,110584,110585],{},[35,110586,94276],{},[45,110588,94276],{"encoding":47},[15,110590,110592],{"className":110591,"ariaHidden":53},[52],[15,110593,110595,110598],{"className":110594},[57],[15,110596],{"className":110597,"style":471},[61],[15,110599,94276],{"className":110600,"style":1515},[66,67]," 无关。",[11,110603,110604,110605,110666,110667,110670,110671,110699,110700,110666,110761,110764,110765,110830],{},"KL 的方向很重要。",[15,110606,110608,110636],{"className":110607,"translate":19},[18],[15,110609,110611],{"className":110610},[23],[25,110612,110613],{"xmlns":27},[29,110614,110615,110633],{},[32,110616,110617,110623,110625,110627,110629,110631],{},[32,110618,110619,110621],{},[35,110620,2853],{"mathvariant":138},[35,110622,1216],{"mathvariant":138},[39,110624,120],{"stretchy":119},[35,110626,94276],{},[35,110628,1251],{"mathvariant":138},[35,110630,11],{},[39,110632,126],{"stretchy":119},[45,110634,110635],{"encoding":47},"\\mathrm{KL}(q\\|p)",[15,110637,110639],{"className":110638,"ariaHidden":53},[52],[15,110640,110642,110645,110651,110654,110657,110660,110663],{"className":110641},[57],[15,110643],{"className":110644,"style":154},[61],[15,110646,110648],{"className":110647},[66],[15,110649,88457],{"className":110650},[66,4429],[15,110652,120],{"className":110653},[162],[15,110655,94276],{"className":110656,"style":1515},[66,67],[15,110658,1251],{"className":110659},[66],[15,110661,11],{"className":110662},[66,67],[15,110664,126],{"className":110665},[169]," 倾向",[2040,110668,110669],{},"寻模","（mode-seeking）：只要 ",[15,110672,110674,110687],{"className":110673,"translate":19},[18],[15,110675,110677],{"className":110676},[23],[25,110678,110679],{"xmlns":27},[29,110680,110681,110685],{},[32,110682,110683],{},[35,110684,94276],{},[45,110686,94276],{"encoding":47},[15,110688,110690],{"className":110689,"ariaHidden":53},[52],[15,110691,110693,110696],{"className":110692},[57],[15,110694],{"className":110695,"style":471},[61],[15,110697,94276],{"className":110698,"style":1515},[66,67]," 覆盖了后验的一个峰，它就不太在意漏掉其他峰，因此常常低估方差。反方向 ",[15,110701,110703,110731],{"className":110702,"translate":19},[18],[15,110704,110706],{"className":110705},[23],[25,110707,110708],{"xmlns":27},[29,110709,110710,110728],{},[32,110711,110712,110718,110720,110722,110724,110726],{},[32,110713,110714,110716],{},[35,110715,2853],{"mathvariant":138},[35,110717,1216],{"mathvariant":138},[39,110719,120],{"stretchy":119},[35,110721,11],{},[35,110723,1251],{"mathvariant":138},[35,110725,94276],{},[39,110727,126],{"stretchy":119},[45,110729,110730],{"encoding":47},"\\mathrm{KL}(p\\|q)",[15,110732,110734],{"className":110733,"ariaHidden":53},[52],[15,110735,110737,110740,110746,110749,110752,110755,110758],{"className":110736},[57],[15,110738],{"className":110739,"style":154},[61],[15,110741,110743],{"className":110742},[66],[15,110744,88457],{"className":110745},[66,4429],[15,110747,120],{"className":110748},[162],[15,110750,11],{"className":110751},[66,67],[15,110753,1251],{"className":110754},[66],[15,110756,94276],{"className":110757,"style":1515},[66,67],[15,110759,126],{"className":110760},[169],[2040,110762,110763],{},"覆盖质量","（mass-covering），但它的计算需要 ",[15,110766,110768,110791],{"className":110767,"translate":19},[18],[15,110769,110771],{"className":110770},[23],[25,110772,110773],{"xmlns":27},[29,110774,110775,110789],{},[32,110776,110777,110779,110781,110783,110785,110787],{},[35,110778,11],{},[39,110780,120],{"stretchy":119},[35,110782,61618],{},[39,110784,34430],{},[35,110786,32458],{},[39,110788,126],{"stretchy":119},[45,110790,65460],{"encoding":47},[15,110792,110794,110818],{"className":110793,"ariaHidden":53},[52],[15,110795,110797,110800,110803,110806,110809,110812,110815],{"className":110796},[57],[15,110798],{"className":110799,"style":154},[61],[15,110801,11],{"className":110802},[66,67],[15,110804,120],{"className":110805},[162],[15,110807,61618],{"className":110808,"style":23025},[66,67],[15,110810],{"className":110811,"style":173},[71],[15,110813,34430],{"className":110814},[177],[15,110816],{"className":110817,"style":173},[71],[15,110819,110821,110824,110827],{"className":110820},[57],[15,110822],{"className":110823,"style":154},[61],[15,110825,32458],{"className":110826,"style":23025},[66,67],[15,110828,126],{"className":110829},[169]," 的归一化常数。变分推断通常用前者，因为 ELBO 只需要未归一化的联合分布。",[11,110832,110833,110834,110863,110864,1124],{},"从决策论的角度看，变分推断本身也是一个决策问题：在受限的分布族里，选一个使 KL 最小的近似。这里的「损失」是 KL，行动是一个分布，而不是一个点。贝叶斯决策理论的框架在这里再次出现，只是行动空间从 ",[15,110835,110837,110851],{"className":110836,"translate":19},[18],[15,110838,110840],{"className":110839},[23],[25,110841,110842],{"xmlns":27},[29,110843,110844,110848],{},[32,110845,110846],{},[35,110847,62005],{"mathvariant":445},[45,110849,110850],{"encoding":47},"\\mathcal A",[15,110852,110854],{"className":110853,"ariaHidden":53},[52],[15,110855,110857,110860],{"className":110856},[57],[15,110858],{"className":110859,"style":490},[61],[15,110861,62005],{"className":110862},[66,494]," 变成了 ",[15,110865,110867,110880],{"className":110866,"translate":19},[18],[15,110868,110870],{"className":110869},[23],[25,110871,110872],{"xmlns":27},[29,110873,110874,110878],{},[32,110875,110876],{},[35,110877,83348],{"mathvariant":445},[45,110879,109163],{"encoding":47},[15,110881,110883],{"className":110882,"ariaHidden":53},[52],[15,110884,110886,110889],{"className":110885},[57],[15,110887],{"className":110888,"style":61920},[61],[15,110890,83348],{"className":110891},[66,494],[1904,110893,110895],{"id":110894},"_75-贝叶斯神经网络与不确定性","7.5 贝叶斯神经网络与不确定性",[11,110897,110898,110899,110927,110928,110994],{},"把参数换成网络权重 ",[15,110900,110902,110915],{"className":110901,"translate":19},[18],[15,110903,110905],{"className":110904},[23],[25,110906,110907],{"xmlns":27},[29,110908,110909,110913],{},[32,110910,110911],{},[35,110912,135],{},[45,110914,135],{"encoding":47},[15,110916,110918],{"className":110917,"ariaHidden":53},[52],[15,110919,110921,110924],{"className":110920},[57],[15,110922],{"className":110923,"style":86},[61],[15,110925,135],{"className":110926,"style":194},[66,67],"，后验 ",[15,110929,110931,110955],{"className":110930,"translate":19},[18],[15,110932,110934],{"className":110933},[23],[25,110935,110936],{"xmlns":27},[29,110937,110938,110952],{},[32,110939,110940,110942,110944,110946,110948,110950],{},[35,110941,11],{},[39,110943,120],{"stretchy":119},[35,110945,135],{},[39,110947,34430],{},[35,110949,32458],{},[39,110951,126],{"stretchy":119},[45,110953,110954],{"encoding":47},"p(w\\mid D)",[15,110956,110958,110982],{"className":110957,"ariaHidden":53},[52],[15,110959,110961,110964,110967,110970,110973,110976,110979],{"className":110960},[57],[15,110962],{"className":110963,"style":154},[61],[15,110965,11],{"className":110966},[66,67],[15,110968,120],{"className":110969},[162],[15,110971,135],{"className":110972,"style":194},[66,67],[15,110974],{"className":110975,"style":173},[71],[15,110977,34430],{"className":110978},[177],[15,110980],{"className":110981,"style":173},[71],[15,110983,110985,110988,110991],{"className":110984},[57],[15,110986],{"className":110987,"style":154},[61],[15,110989,32458],{"className":110990,"style":23025},[66,67],[15,110992,126],{"className":110993},[169]," 就是贝叶斯神经网络。精确后验不可行，实践中用 MC dropout、Laplace 近似、变分推断或深度集成来近似。",[11,110996,110997],{},"对分类问题，预测熵可以分解为",[15,110999,111001],{"className":111000,"translate":19},[294],[15,111002,111004,111090],{"className":111003,"translate":19},[18],[15,111005,111007],{"className":111006},[23],[25,111008,111009],{"xmlns":27,"display":303},[29,111010,111011,111087],{},[32,111012,111013,111015,111017,111019,111021,111023,111025,111027,111029,111031,111043,111045,111047,111049,111051,111053,111055,111057,111059,111061,111063,111065,111067,111069,111071,111073,111075,111077,111079,111081,111083,111085],{},[35,111014,1061],{"mathvariant":454},[39,111016,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,111018,61885],{},[39,111020,34430],{},[35,111022,123],{},[39,111024,335],{"separator":53},[35,111026,32458],{},[39,111028,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,111030,129],{},[1057,111032,111033,111035],{},[35,111034,62381],{"mathvariant":454},[32,111036,111037,111039,111041],{},[35,111038,135],{},[39,111040,34430],{},[35,111042,32458],{},[39,111044,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,111046,1061],{"mathvariant":454},[39,111048,1194],{"stretchy":119},[35,111050,61885],{},[39,111052,34430],{},[35,111054,123],{},[39,111056,335],{"separator":53},[35,111058,135],{},[39,111060,1266],{"stretchy":119},[39,111062,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,111064,1245],{},[35,111066,16427],{"mathvariant":454},[39,111068,120],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,111070,61885],{},[39,111072,8413],{"separator":53},[35,111074,135],{},[39,111076,34430],{},[35,111078,123],{},[39,111080,335],{"separator":53},[35,111082,32458],{},[39,111084,126],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,111086,3573],{"mathvariant":138},[45,111088,111089],{"encoding":47},"\\mathbb H\\big[Y\\mid x,D\\big]\n=\n\\mathbb E_{w\\mid D}\\big[\\mathbb H[Y\\mid x,w]\\big]\n+\n\\mathbb I\\big(Y;w\\mid x,D\\big).",[15,111091,111093,111120,111153,111232,111268,111304],{"className":111092,"ariaHidden":53},[52],[15,111094,111096,111099,111102,111108,111111,111114,111117],{"className":111095},[57],[15,111097],{"className":111098,"style":4400},[61],[15,111100,1061],{"className":111101},[66,517],[15,111103,111105],{"className":111104},[66],[15,111106,1194],{"className":111107},[1422,1508],[15,111109,61885],{"className":111110,"style":72},[66,67],[15,111112],{"className":111113,"style":173},[71],[15,111115,34430],{"className":111116},[177],[15,111118],{"className":111119,"style":173},[71],[15,111121,111123,111126,111129,111132,111135,111138,111144,111147,111150],{"className":111122},[57],[15,111124],{"className":111125,"style":4400},[61],[15,111127,123],{"className":111128},[66,67],[15,111130,335],{"className":111131},[422],[15,111133],{"className":111134,"style":697},[71],[15,111136,32458],{"className":111137,"style":23025},[66,67],[15,111139,111141],{"className":111140},[66],[15,111142,1266],{"className":111143},[1422,1508],[15,111145],{"className":111146,"style":173},[71],[15,111148,129],{"className":111149},[177],[15,111151],{"className":111152,"style":173},[71],[15,111154,111156,111159,111208,111214,111217,111220,111223,111226,111229],{"className":111155},[57],[15,111157],{"className":111158,"style":12291},[61],[15,111160,111162,111165],{"className":111161},[66],[15,111163,62381],{"className":111164},[66,517],[15,111166,111168],{"className":111167},[198],[15,111169,111171,111200],{"className":111170},[202,1090],[15,111172,111174,111197],{"className":111173},[206],[15,111175,111177],{"className":111176,"style":3668},[210],[15,111178,111179,111182],{"style":62482},[15,111180],{"className":111181,"style":218},[217],[15,111183,111185],{"className":111184},[222,223,224,225],[15,111186,111188,111191,111194],{"className":111187},[66,225],[15,111189,135],{"className":111190,"style":194},[66,67,225],[15,111192,34430],{"className":111193},[177,225],[15,111195,32458],{"className":111196,"style":23025},[66,67,225],[15,111198,1114],{"className":111199},[1113],[15,111201,111203],{"className":111202},[206],[15,111204,111206],{"className":111205,"style":9880},[210],[15,111207],{},[15,111209,111211],{"className":111210},[66],[15,111212,1194],{"className":111213},[1422,1508],[15,111215,1061],{"className":111216},[66,517],[15,111218,1194],{"className":111219},[162],[15,111221,61885],{"className":111222,"style":72},[66,67],[15,111224],{"className":111225,"style":173},[71],[15,111227,34430],{"className":111228},[177],[15,111230],{"className":111231,"style":173},[71],[15,111233,111235,111238,111241,111244,111247,111250,111253,111259,111262,111265],{"className":111234},[57],[15,111236],{"className":111237,"style":4400},[61],[15,111239,123],{"className":111240},[66,67],[15,111242,335],{"className":111243},[422],[15,111245],{"className":111246,"style":697},[71],[15,111248,135],{"className":111249,"style":194},[66,67],[15,111251,1266],{"className":111252},[169],[15,111254,111256],{"className":111255},[66],[15,111257,1266],{"className":111258},[1422,1508],[15,111260],{"className":111261,"style":72},[71],[15,111263,1245],{"className":111264},[76],[15,111266],{"className":111267,"style":72},[71],[15,111269,111271,111274,111277,111283,111286,111289,111292,111295,111298,111301],{"className":111270},[57],[15,111272],{"className":111273,"style":4400},[61],[15,111275,16427],{"className":111276},[66,517],[15,111278,111280],{"className":111279},[66],[15,111281,120],{"className":111282},[1422,1508],[15,111284,61885],{"className":111285,"style":72},[66,67],[15,111287,8413],{"className":111288},[422],[15,111290],{"className":111291,"style":697},[71],[15,111293,135],{"className":111294,"style":194},[66,67],[15,111296],{"className":111297,"style":173},[71],[15,111299,34430],{"className":111300},[177],[15,111302],{"className":111303,"style":173},[71],[15,111305,111307,111310,111313,111316,111319,111322,111328],{"className":111306},[57],[15,111308],{"className":111309,"style":4400},[61],[15,111311,123],{"className":111312},[66,67],[15,111314,335],{"className":111315},[422],[15,111317],{"className":111318,"style":697},[71],[15,111320,32458],{"className":111321,"style":23025},[66,67],[15,111323,111325],{"className":111324},[66],[15,111326,126],{"className":111327},[1422,1508],[15,111329,3573],{"className":111330},[66],[11,111332,111333],{},"第一项是偶然不确定性：即使固定权重，类别仍然有噪声。第二项是认知不确定性：不同权重给出的预测分布之间的差异。BALD 等主动学习准则，就是用它来挑选「最能减少认知不确定性」的样本。",[11,111335,111336],{},"但「贝叶斯」这个标签不保证校准。一个近似的后验可能给出过度自信的预测；深度集成不是严格的贝叶斯方法，却常常校准得更好。判断的标准应该是决策质量与校准，而不是方法的名字。一个模型如果能在高损失的区域拒绝预测，或者把不确定性传递给下游决策，它就已经在做贝叶斯决策了，哪怕它没有显式地计算后验。",[1904,111338,111340],{"id":111339},"_76-序贯决策后验采样与探索","7.6 序贯决策：后验采样与探索",[11,111342,111343,111344,111347,111348,111513],{},"在序贯决策里，后验不只是用来「估计」，还用来「探索」。",[2040,111345,111346],{},"Thompson 采样","的规则很简单：每一步从后验中采样一个环境参数 ",[15,111349,111351,111388],{"className":111350,"translate":19},[18],[15,111352,111354],{"className":111353},[23],[25,111355,111356],{"xmlns":27},[29,111357,111358,111385],{},[32,111359,111360,111367,111369,111371,111373,111375,111377,111383],{},[4296,111361,111362,111364],{"accent":53},[35,111363,61618],{},[39,111365,111366],{},"~",[39,111368,28391],{},[35,111370,11],{},[39,111372,120],{"stretchy":119},[35,111374,61618],{},[39,111376,34430],{},[1057,111378,111379,111381],{},[35,111380,32458],{},[35,111382,9212],{},[39,111384,126],{"stretchy":119},[45,111386,111387],{"encoding":47},"\\tilde\\theta\\sim p(\\theta\\mid D_t)",[15,111389,111391,111439,111463],{"className":111390,"ariaHidden":53},[52],[15,111392,111394,111398,111430,111433,111436],{"className":111393},[57],[15,111395],{"className":111396,"style":111397},[61],"height:0.9313em;",[15,111399,111401],{"className":111400},[66,78147],[15,111402,111404],{"className":111403},[202],[15,111405,111407],{"className":111406},[206],[15,111408,111410,111418],{"className":111409,"style":111397},[210],[15,111411,111412,111415],{"style":1386},[15,111413],{"className":111414,"style":1368},[217],[15,111416,61618],{"className":111417,"style":23025},[66,67],[15,111419,111421,111424],{"style":111420},"top:-3.6134em;",[15,111422],{"className":111423,"style":1368},[217],[15,111425,111427],{"className":111426,"style":96110},[78175],[15,111428,111366],{"className":111429},[66],[15,111431],{"className":111432,"style":173},[71],[15,111434,28391],{"className":111435},[177],[15,111437],{"className":111438,"style":173},[71],[15,111440,111442,111445,111448,111451,111454,111457,111460],{"className":111441},[57],[15,111443],{"className":111444,"style":154},[61],[15,111446,11],{"className":111447},[66,67],[15,111449,120],{"className":111450},[162],[15,111452,61618],{"className":111453,"style":23025},[66,67],[15,111455],{"className":111456,"style":173},[71],[15,111458,34430],{"className":111459},[177],[15,111461],{"className":111462,"style":173},[71],[15,111464,111466,111469,111510],{"className":111465},[57],[15,111467],{"className":111468,"style":154},[61],[15,111470,111472,111475],{"className":111471},[66],[15,111473,32458],{"className":111474,"style":23025},[66,67],[15,111476,111478],{"className":111477},[198],[15,111479,111481,111502],{"className":111480},[202,1090],[15,111482,111484,111499],{"className":111483},[206],[15,111485,111488],{"className":111486,"style":111487},[210],"height:0.2806em;",[15,111489,111490,111493],{"style":88441},[15,111491],{"className":111492,"style":218},[217],[15,111494,111496],{"className":111495},[222,223,224,225],[15,111497,9212],{"className":111498},[66,67,225],[15,111500,1114],{"className":111501},[1113],[15,111503,111505],{"className":111504},[206],[15,111506,111508],{"className":111507,"style":1121},[210],[15,111509],{},[15,111511,126],{"className":111512},[169],"，然后按这个采样出的环境做最优动作。",[11,111515,111516],{},"这个规则自动平衡了利用与探索。不确定性大的动作在采样中更可能显得最优，因此会被更频繁地尝试；随着数据增多，后验收窄，采样出的环境越来越接近真实环境，策略逐渐转向利用。它的贝叶斯遗憾有理论保证，而且实现起来只需要一个后验采样器。",[11,111518,111519],{},"从决策论的角度看，Thompson 采样把「信息」当成了决策的一部分：当前动作的收益不只是它带来的即时回报，还包括它揭示的信息。完整的贝叶斯解决方案是动态规划，在信念空间（belief space）上求解；Thompson 采样是它的一个可计算的近似。",[1904,111521,111523],{"id":111522},"_77-主动学习与贝叶斯优化","7.7 主动学习与贝叶斯优化",[11,111525,111526],{},"主动学习问的是：下一个标签应该买哪一个？从决策论看，这是一个两级决策：先选一个样本，再在拿到标签后做预测。理想的采集函数是「期望后验风险降低」：",[15,111528,111530],{"className":111529,"translate":19},[294],[15,111531,111533,111632],{"className":111532,"translate":19},[18],[15,111534,111536],{"className":111535},[23],[25,111537,111538],{"xmlns":27,"display":303},[29,111539,111540,111629],{},[32,111541,111542,111549,111551,111553,111555,111565,111567,111593,111627],{},[1057,111543,111544,111546],{},[35,111545,123],{},[1274,111547,111548],{},"next",[39,111550,129],{},[35,111552,1167],{},[39,111554,1170],{},[1172,111556,111557,111563],{},[32,111558,111559,111561],{},[35,111560,1178],{},[39,111562,1170],{},[35,111564,123],{},[1274,111566,24233],{},[1057,111568,111569,111571],{},[35,111570,62381],{"mathvariant":454},[32,111572,111573,111575,111577,111579,111581,111583,111585,111587,111589,111591],{},[35,111574,1224],{},[39,111576,28391],{},[35,111578,11],{},[39,111580,120],{"stretchy":119},[35,111582,1224],{},[39,111584,34430],{},[35,111586,123],{},[39,111588,335],{"separator":53},[35,111590,32458],{},[39,111592,126],{"stretchy":119},[32,111594,111595,111597,111603,111605,111607,111609,111611,111613,111615,111617,111619,111621,111623,111625],{},[39,111596,1194],{"fence":53},[131,111598,111599,111601],{},[35,111600,455],{},[39,111602,1162],{},[39,111604,120],{"stretchy":119},[35,111606,32458],{},[39,111608,7613],{},[39,111610,4307],{"stretchy":119},[39,111612,120],{"stretchy":119},[35,111614,123],{},[39,111616,335],{"separator":53},[35,111618,1224],{},[39,111620,126],{"stretchy":119},[39,111622,4330],{"stretchy":119},[39,111624,126],{"stretchy":119},[39,111626,1266],{"fence":53},[39,111628,335],{"separator":53},[45,111630,111631],{"encoding":47},"x_{\\text{next}}\n=\n\\arg\\min_{x}\\ \n\\mathbb E_{y\\sim p(y\\mid x,D)}\n\\left[\nR^*(D\\cup\\{(x,y)\\})\n\\right],",[15,111633,111635,111696],{"className":111634,"ariaHidden":53},[52],[15,111636,111638,111641,111687,111690,111693],{"className":111637},[57],[15,111639],{"className":111640,"style":10197},[61],[15,111642,111644,111647],{"className":111643},[66],[15,111645,123],{"className":111646},[66,67],[15,111648,111650],{"className":111649},[198],[15,111651,111653,111679],{"className":111652},[202,1090],[15,111654,111656,111676],{"className":111655},[206],[15,111657,111659],{"className":111658,"style":111487},[210],[15,111660,111661,111664],{"style":1584},[15,111662],{"className":111663,"style":218},[217],[15,111665,111667],{"className":111666},[222,223,224,225],[15,111668,111670],{"className":111669},[66,225],[15,111671,111673],{"className":111672},[66,1711,225],[15,111674,111548],{"className":111675},[66,225],[15,111677,1114],{"className":111678},[1113],[15,111680,111682],{"className":111681},[206],[15,111683,111685],{"className":111684,"style":1121},[210],[15,111686],{},[15,111688],{"className":111689,"style":173},[71],[15,111691,129],{"className":111692},[177],[15,111694],{"className":111695,"style":173},[71],[15,111697,111699,111703,111708,111711,111758,111761,111764,111834,111837,111909,111912],{"className":111698},[57],[15,111700],{"className":111701,"style":111702},[61],"height:1.45em;vertical-align:-0.7em;",[15,111704,1340,111706],{"className":111705},[1339],[15,111707,1344],{"style":1343},[15,111709],{"className":111710,"style":697},[71],[15,111712,111714],{"className":111713},[1339,1351],[15,111715,111717,111750],{"className":111716},[202,1090],[15,111718,111720,111747],{"className":111719},[206],[15,111721,111723,111737],{"className":111722,"style":1361},[210],[15,111724,111725,111728],{"style":48823},[15,111726],{"className":111727,"style":1368},[217],[15,111729,111731],{"className":111730},[222,223,224,225],[15,111732,111734],{"className":111733},[66,225],[15,111735,123],{"className":111736},[66,67,225],[15,111738,111739,111742],{"style":1386},[15,111740],{"className":111741,"style":1368},[217],[15,111743,111744],{},[15,111745,1178],{"className":111746},[1339],[15,111748,1114],{"className":111749},[1113],[15,111751,111753],{"className":111752},[206],[15,111754,111756],{"className":111755,"style":48861},[210],[15,111757],{},[15,111759,24233],{"className":111760},[71],[15,111762],{"className":111763,"style":697},[71],[15,111765,111767,111770],{"className":111766},[66],[15,111768,62381],{"className":111769},[66,517],[15,111771,111773],{"className":111772},[198],[15,111774,111776,111826],{"className":111775},[202,1090],[15,111777,111779,111823],{"className":111778},[206],[15,111780,111782],{"className":111781,"style":3668},[210],[15,111783,111784,111787],{"style":62482},[15,111785],{"className":111786,"style":218},[217],[15,111788,111790],{"className":111789},[222,223,224,225],[15,111791,111793,111796,111799,111802,111805,111808,111811,111814,111817,111820],{"className":111792},[66,225],[15,111794,1224],{"className":111795,"style":1515},[66,67,225],[15,111797,28391],{"className":111798},[177,225],[15,111800,11],{"className":111801},[66,67,225],[15,111803,120],{"className":111804},[162,225],[15,111806,1224],{"className":111807,"style":1515},[66,67,225],[15,111809,34430],{"className":111810},[177,225],[15,111812,123],{"className":111813},[66,67,225],[15,111815,335],{"className":111816},[422,225],[15,111818,32458],{"className":111819,"style":23025},[66,67,225],[15,111821,126],{"className":111822},[169,225],[15,111824,1114],{"className":111825},[1113],[15,111827,111829],{"className":111828},[206],[15,111830,111832],{"className":111831,"style":9880},[210],[15,111833],{},[15,111835],{"className":111836,"style":697},[71],[15,111838,111840,111843,111872,111875,111878,111881,111884,111887,111890,111893,111896,111899,111902,111906],{"className":111839},[1413],[15,111841,1194],{"className":111842,"style":1418},[162,1417],[15,111844,111846,111849],{"className":111845},[66],[15,111847,455],{"className":111848,"style":62437},[66,67],[15,111850,111852],{"className":111851},[198],[15,111853,111855],{"className":111854},[202],[15,111856,111858],{"className":111857},[206],[15,111859,111861],{"className":111860,"style":1308},[210],[15,111862,111863,111866],{"style":397},[15,111864],{"className":111865,"style":218},[217],[15,111867,111869],{"className":111868},[222,223,224,225],[15,111870,1162],{"className":111871},[76,225],[15,111873,120],{"className":111874},[162],[15,111876,32458],{"className":111877,"style":23025},[66,67],[15,111879],{"className":111880,"style":72},[71],[15,111882,7613],{"className":111883},[76],[15,111885],{"className":111886,"style":72},[71],[15,111888,19299],{"className":111889},[162],[15,111891,123],{"className":111892},[66,67],[15,111894,335],{"className":111895},[422],[15,111897],{"className":111898,"style":697},[71],[15,111900,1224],{"className":111901,"style":1515},[66,67],[15,111903,111905],{"className":111904},[169],")})",[15,111907,1266],{"className":111908,"style":1418},[169,1417],[15,111910],{"className":111911,"style":697},[71],[15,111913,335],{"className":111914},[422],[11,111916,425,111917,111991],{},[15,111918,111920,111944],{"className":111919,"translate":19},[18],[15,111921,111923],{"className":111922},[23],[25,111924,111925],{"xmlns":27},[29,111926,111927,111941],{},[32,111928,111929,111935,111937,111939],{},[131,111930,111931,111933],{},[35,111932,455],{},[39,111934,1162],{},[39,111936,120],{"stretchy":119},[35,111938,32458],{},[39,111940,126],{"stretchy":119},[45,111942,111943],{"encoding":47},"R^*(D)",[15,111945,111947],{"className":111946,"ariaHidden":53},[52],[15,111948,111950,111953,111982,111985,111988],{"className":111949},[57],[15,111951],{"className":111952,"style":154},[61],[15,111954,111956,111959],{"className":111955},[66],[15,111957,455],{"className":111958,"style":62437},[66,67],[15,111960,111962],{"className":111961},[198],[15,111963,111965],{"className":111964},[202],[15,111966,111968],{"className":111967},[206],[15,111969,111971],{"className":111970,"style":16249},[210],[15,111972,111973,111976],{"style":213},[15,111974],{"className":111975,"style":218},[217],[15,111977,111979],{"className":111978},[222,223,224,225],[15,111980,1162],{"className":111981},[76,225],[15,111983,120],{"className":111984},[162],[15,111986,32458],{"className":111987,"style":23025},[66,67],[15,111989,126],{"className":111990},[169]," 是当前数据下的贝叶斯风险。这个量通常算不出来，于是有了各种近似：熵、互信息、BALD、期望模型变化。",[11,111993,111994],{},"贝叶斯优化把目标函数本身当成随机变量，用后验选择下一个评估点。Expected Improvement 就是",[15,111996,111998],{"className":111997,"translate":19},[294],[15,111999,112001,112071],{"className":112000,"translate":19},[18],[15,112002,112004],{"className":112003},[23],[25,112005,112006],{"xmlns":27,"display":303},[29,112007,112008,112068],{},[32,112009,112010,112016,112018,112020,112022,112024,112026,112028,112030,112032,112034,112036,112038,112040,112048,112050,112052,112054,112056,112058,112060,112062,112064,112066],{},[32,112011,112012,112014],{},[35,112013,62381],{"mathvariant":138},[35,112015,16427],{"mathvariant":138},[39,112017,120],{"stretchy":119},[35,112019,123],{},[39,112021,126],{"stretchy":119},[39,112023,129],{},[35,112025,62381],{"mathvariant":454},[39,112027,1194],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[35,112029,23273],{},[39,112031,1170],{},[39,112033,4307],{"stretchy":119},[1209,112035,1804],{},[39,112037,335],{"separator":53},[1274,112039,24233],{},[1057,112041,112042,112044,112046],{},[35,112043,116],{},[35,112045,1178],{},[39,112047,1170],{},[39,112049,6602],{},[35,112051,116],{},[39,112053,120],{"stretchy":119},[35,112055,123],{},[39,112057,126],{"stretchy":119},[39,112059,4330],{"stretchy":119},[39,112061,34430],{},[35,112063,32458],{},[39,112065,1266],{"fence":119,"stretchy":53,"minsize":1219,"maxsize":1219},[39,112067,335],{"separator":53},[45,112069,112070],{"encoding":47},"\\mathrm{EI}(x)=\\mathbb E\\big[\\max\\{0,\\ f_{\\min}-f(x)\\}\\mid D\\big],",[15,112072,112074,112105,112202,112229],{"className":112073,"ariaHidden":53},[52],[15,112075,112077,112080,112087,112090,112093,112096,112099,112102],{"className":112076},[57],[15,112078],{"className":112079,"style":154},[61],[15,112081,112083],{"className":112082},[66],[15,112084,112086],{"className":112085},[66,4429],"EI",[15,112088,120],{"className":112089},[162],[15,112091,123],{"className":112092},[66,67],[15,112094,126],{"className":112095},[169],[15,112097],{"className":112098,"style":173},[71],[15,112100,129],{"className":112101},[177],[15,112103],{"className":112104,"style":173},[71],[15,112106,112108,112111,112114,112120,112123,112126,112129,112132,112135,112138,112141,112193,112196,112199],{"className":112107},[57],[15,112109],{"className":112110,"style":4400},[61],[15,112112,62381],{"className":112113},[66,517],[15,112115,112117],{"className":112116},[66],[15,112118,1194],{"className":112119},[1422,1508],[15,112121],{"className":112122,"style":697},[71],[15,112124,23273],{"className":112125},[1339],[15,112127,4307],{"className":112128},[162],[15,112130,1804],{"className":112131},[66],[15,112133,335],{"className":112134},[422],[15,112136,24233],{"className":112137},[71],[15,112139],{"className":112140,"style":697},[71],[15,112142,112144,112147],{"className":112143},[66],[15,112145,116],{"className":112146,"style":158},[66,67],[15,112148,112150],{"className":112149},[198],[15,112151,112153,112185],{"className":112152},[202,1090],[15,112154,112156,112182],{"className":112155},[206],[15,112157,112159],{"className":112158,"style":100015},[210],[15,112160,112161,112164],{"style":9918},[15,112162],{"className":112163,"style":218},[217],[15,112165,112167],{"className":112166},[222,223,224,225],[15,112168,112170],{"className":112169},[66,225],[15,112171,112173,112176,112179],{"className":112172},[1339,225],[15,112174,28858],{"className":112175},[225],[15,112177,1205],{"className":112178},[225],[15,112180,37],{"className":112181},[225],[15,112183,1114],{"className":112184},[1113],[15,112186,112188],{"className":112187},[206],[15,112189,112191],{"className":112190,"style":1121},[210],[15,112192],{},[15,112194],{"className":112195,"style":72},[71],[15,112197,6602],{"className":112198},[76],[15,112200],{"className":112201,"style":72},[71],[15,112203,112205,112208,112211,112214,112217,112220,112223,112226],{"className":112204},[57],[15,112206],{"className":112207,"style":154},[61],[15,112209,116],{"className":112210,"style":158},[66,67],[15,112212,120],{"className":112213},[162],[15,112215,123],{"className":112216},[66,67],[15,112218,19280],{"className":112219},[169],[15,112221],{"className":112222,"style":173},[71],[15,112224,34430],{"className":112225},[177],[15,112227],{"className":112228,"style":173},[71],[15,112230,112232,112235,112238,112244],{"className":112231},[57],[15,112233],{"className":112234,"style":4400},[61],[15,112236,32458],{"className":112237,"style":23025},[66,67],[15,112239,112241],{"className":112240},[66],[15,112242,1266],{"className":112243},[1422,1508],[15,112245,335],{"className":112246},[422],[11,112248,112249],{},"它把「可能变好多少」与「变好的概率」乘在一起。探索与利用的权衡，在这里被写成了一个一维积分的最大化问题。",[1904,112251,112253],{"id":112252},"_78-可容许性与完整类定理","7.8 可容许性与完整类定理",[11,112255,112256,112257,2038,112285,112288,112289,112351],{},"最后回到 2.4 节留下的那个说法。一个决策规则 ",[15,112258,112260,112273],{"className":112259,"translate":19},[18],[15,112261,112263],{"className":112262},[23],[25,112264,112265],{"xmlns":27},[29,112266,112267,112271],{},[32,112268,112269],{},[35,112270,3885],{},[45,112272,68717],{"encoding":47},[15,112274,112276],{"className":112275,"ariaHidden":53},[52],[15,112277,112279,112282],{"className":112278},[57],[15,112280],{"className":112281,"style":840},[61],[15,112283,3885],{"className":112284,"style":3946},[66,67],[2040,112286,112287],{},"可容许的","（admissible），若不存在另一个规则 ",[15,112290,112292,112310],{"className":112291,"translate":19},[18],[15,112293,112295],{"className":112294},[23],[25,112296,112297],{"xmlns":27},[29,112298,112299,112307],{},[32,112300,112301],{},[131,112302,112303,112305],{},[35,112304,3885],{},[39,112306,635],{"mathvariant":138,"lspace":634,"rspace":634},[45,112308,112309],{"encoding":47},"\\delta'",[15,112311,112313],{"className":112312,"ariaHidden":53},[52],[15,112314,112316,112319],{"className":112315},[57],[15,112317],{"className":112318,"style":6005},[61],[15,112320,112322,112325],{"className":112321},[66],[15,112323,3885],{"className":112324,"style":3946},[66,67],[15,112326,112328],{"className":112327},[198],[15,112329,112331],{"className":112330},[202],[15,112332,112334],{"className":112333},[206],[15,112335,112337],{"className":112336,"style":6005},[210],[15,112338,112339,112342],{"style":213},[15,112340],{"className":112341,"style":218},[217],[15,112343,112345],{"className":112344},[222,223,224,225],[15,112346,112348],{"className":112347},[66,225],[15,112349,635],{"className":112350},[66,225],"，使得",[15,112353,112355],{"className":112354,"translate":19},[294],[15,112356,112358,112401],{"className":112357,"translate":19},[18],[15,112359,112361],{"className":112360},[23],[25,112362,112363],{"xmlns":27,"display":303},[29,112364,112365,112398],{},[32,112366,112367,112369,112371,112377,112379,112381,112383,112385,112387,112389,112391,112394,112396],{},[35,112368,455],{},[39,112370,120],{"stretchy":119},[131,112372,112373,112375],{},[35,112374,3885],{},[39,112376,635],{"mathvariant":138,"lspace":634,"rspace":634},[39,112378,126],{"stretchy":119},[39,112380,24220],{},[35,112382,455],{},[39,112384,120],{"stretchy":119},[35,112386,3885],{},[39,112388,126],{"stretchy":119},[71,112390],{"width":9791},[1274,112392,112393],{},"对所有分布 ",[35,112395,28399],{},[39,112397,335],{"separator":53},[45,112399,112400],{"encoding":47},"R(\\delta')\\le R(\\delta)\\quad\\text{对所有分布 }P,",[15,112402,112404,112460],{"className":112403,"ariaHidden":53},[52],[15,112405,112407,112410,112413,112416,112448,112451,112454,112457],{"className":112406},[57],[15,112408],{"className":112409,"style":680},[61],[15,112411,455],{"className":112412,"style":62437},[66,67],[15,112414,120],{"className":112415},[162],[15,112417,112419,112422],{"className":112418},[66],[15,112420,3885],{"className":112421,"style":3946},[66,67],[15,112423,112425],{"className":112424},[198],[15,112426,112428],{"className":112427},[202],[15,112429,112431],{"className":112430},[206],[15,112432,112434],{"className":112433,"style":716},[210],[15,112435,112436,112439],{"style":397},[15,112437],{"className":112438,"style":218},[217],[15,112440,112442],{"className":112441},[222,223,224,225],[15,112443,112445],{"className":112444},[66,225],[15,112446,635],{"className":112447},[66,225],[15,112449,126],{"className":112450},[169],[15,112452],{"className":112453,"style":173},[71],[15,112455,24220],{"className":112456},[177],[15,112458],{"className":112459,"style":173},[71],[15,112461,112463,112466,112469,112472,112475,112478,112481,112491,112494],{"className":112462},[57],[15,112464],{"className":112465,"style":154},[61],[15,112467,455],{"className":112468,"style":62437},[66,67],[15,112470,120],{"className":112471},[162],[15,112473,3885],{"className":112474,"style":3946},[66,67],[15,112476,126],{"className":112477},[169],[15,112479],{"className":112480,"style":10049},[71],[15,112482,112484,112488],{"className":112483},[66,1711],[15,112485,112487],{"className":112486},[66,68616],"对所有分布",[15,112489,24233],{"className":112490},[66],[15,112492,28399],{"className":112493,"style":62513},[66,67],[15,112495,335],{"className":112496},[422],[11,112498,112499,112500,112528],{},"且至少对一个 ",[15,112501,112503,112516],{"className":112502,"translate":19},[18],[15,112504,112506],{"className":112505},[23],[25,112507,112508],{"xmlns":27},[29,112509,112510,112514],{},[32,112511,112512],{},[35,112513,28399],{},[45,112515,28399],{"encoding":47},[15,112517,112519],{"className":112518,"ariaHidden":53},[52],[15,112520,112522,112525],{"className":112521},[57],[15,112523],{"className":112524,"style":490},[61],[15,112526,28399],{"className":112527,"style":62513},[66,67]," 严格小于。换句话说，可容许就是「没有被另一个规则全面支配」。一个处处比别的规则差的决策规则，理应被淘汰；可容许性把这个直觉形式化了。",[3054,112530,112531],{},[11,112532,112533,112536,112537,112566],{},[2040,112534,112535],{},"定理 7.2（Wald 完整类定理）"," 在损失函数凸、行动空间紧、实验族满足适当正则条件（例如被某个 ",[15,112538,112540,112554],{"className":112539,"translate":19},[18],[15,112541,112543],{"className":112542},[23],[25,112544,112545],{"xmlns":27},[29,112546,112547,112551],{},[32,112548,112549],{},[35,112550,6634],{},[45,112552,112553],{"encoding":47},"\\sigma",[15,112555,112557],{"className":112556,"ariaHidden":53},[52],[15,112558,112560,112563],{"className":112559},[57],[15,112561],{"className":112562,"style":86},[61],[15,112564,6634],{"className":112565,"style":1515},[66,67],"-有限测度支配）时，每一个可容许的决策规则都是某个先验（或先验序列的极限）下的贝叶斯规则。",[11,112568,112569],{},"这个定理的意义在于：在适当条件下，它把「频率派的极大极小规则」「看起来没有先验的规则」也纳入了贝叶斯框架。先验不再是一个可选的信仰，而是可容许性这个要求所对应的对偶变量。你可以拒绝某个具体的先验，但很难拒绝「存在某个先验」这件事。",[11,112571,112572,112573,1124],{},"定理的条件不能省。损失不凸、行动空间不紧，或者问题是非参数的，结论可能失效；此时需要广义先验、极限贝叶斯规则，或者放弃完整类定理的结论。这些技术细节不改变核心信息：",[2040,112574,112575],{},"贝叶斯规则不是许多决策规则中的一种，而是所有不被支配的决策规则的一般形式",[11,112577,112578],{},"从实践的角度看，这个定理不告诉你该选哪个先验，只告诉你：如果你想要一个不被全面支配的规则，你总可以把它写成某个先验下的贝叶斯规则。先验的选择仍然是建模的一部分，但「要不要用先验」已经不再是一个真正的选项。",[95,112580,112582],{"id":112581},"_8-几个可以手算的例子","8. 几个可以手算的例子",[1904,112584,112586],{"id":112585},"_81-betabernoulli-的完整更新","8.1 Beta–Bernoulli 的完整更新",[11,112588,112589,112590,112658,112659,66231,112687,112716],{},"取先验 ",[15,112591,112593,112625],{"className":112592,"translate":19},[18],[15,112594,112596],{"className":112595},[23],[25,112597,112598],{"xmlns":27},[29,112599,112600,112622],{},[32,112601,112602,112612,112614,112616,112618,112620],{},[32,112603,112604,112606,112608,112610],{},[35,112605,63972],{"mathvariant":138},[35,112607,48383],{"mathvariant":138},[35,112609,9212],{"mathvariant":138},[35,112611,4556],{"mathvariant":138},[39,112613,120],{"stretchy":119},[1209,112615,8678],{},[39,112617,335],{"separator":53},[1209,112619,8678],{},[39,112621,126],{"stretchy":119},[45,112623,112624],{"encoding":47},"\\mathrm{Beta}(3,3)",[15,112626,112628],{"className":112627,"ariaHidden":53},[52],[15,112629,112631,112634,112640,112643,112646,112649,112652,112655],{"className":112630},[57],[15,112632],{"className":112633,"style":154},[61],[15,112635,112637],{"className":112636},[66],[15,112638,66792],{"className":112639},[66,4429],[15,112641,120],{"className":112642},[162],[15,112644,8678],{"className":112645},[66],[15,112647,335],{"className":112648},[422],[15,112650],{"className":112651,"style":697},[71],[15,112653,8678],{"className":112654},[66],[15,112656,126],{"className":112657},[169],"，观测到 ",[15,112660,112662,112675],{"className":112661,"translate":19},[18],[15,112663,112665],{"className":112664},[23],[25,112666,112667],{"xmlns":27},[29,112668,112669,112673],{},[32,112670,112671],{},[1209,112672,107911],{},[45,112674,107911],{"encoding":47},[15,112676,112678],{"className":112677,"ariaHidden":53},[52],[15,112679,112681,112684],{"className":112680},[57],[15,112682],{"className":112683,"style":1836},[61],[15,112685,107911],{"className":112686},[66],[15,112688,112690,112704],{"className":112689,"translate":19},[18],[15,112691,112693],{"className":112692},[23],[25,112694,112695],{"xmlns":27},[29,112696,112697,112702],{},[32,112698,112699],{},[1209,112700,112701],{},"7",[45,112703,112701],{"encoding":47},[15,112705,112707],{"className":112706,"ariaHidden":53},[52],[15,112708,112710,112713],{"className":112709},[57],[15,112711],{"className":112712,"style":1836},[61],[15,112714,112701],{"className":112715},[66]," 次正面。后验为",[15,112718,112720],{"className":112719,"translate":19},[294],[15,112721,112723,112789],{"className":112722,"translate":19},[18],[15,112724,112726],{"className":112725},[23],[25,112727,112728],{"xmlns":27,"display":303},[29,112729,112730,112786],{},[32,112731,112732,112742,112744,112746,112748,112750,112752,112754,112756,112758,112760,112762,112764,112774,112776,112778,112780,112782,112784],{},[32,112733,112734,112736,112738,112740],{},[35,112735,63972],{"mathvariant":138},[35,112737,48383],{"mathvariant":138},[35,112739,9212],{"mathvariant":138},[35,112741,4556],{"mathvariant":138},[39,112743,120],{"stretchy":119},[1209,112745,8678],{},[39,112747,1245],{},[1209,112749,112701],{},[39,112751,335],{"separator":53},[1274,112753,24233],{},[1209,112755,8678],{},[39,112757,1245],{},[1209,112759,8678],{},[39,112761,126],{"stretchy":119},[39,112763,129],{},[32,112765,112766,112768,112770,112772],{},[35,112767,63972],{"mathvariant":138},[35,112769,48383],{"mathvariant":138},[35,112771,9212],{"mathvariant":138},[35,112773,4556],{"mathvariant":138},[39,112775,120],{"stretchy":119},[1209,112777,107911],{},[39,112779,335],{"separator":53},[1209,112781,59943],{},[39,112783,126],{"stretchy":119},[35,112785,3573],{"mathvariant":138},[45,112787,112788],{"encoding":47},"\\mathrm{Beta}(3+7,\\ 3+3)=\\mathrm{Beta}(10,6).",[15,112790,112792,112819,112850,112871],{"className":112791,"ariaHidden":53},[52],[15,112793,112795,112798,112804,112807,112810,112813,112816],{"className":112794},[57],[15,112796],{"className":112797,"style":154},[61],[15,112799,112801],{"className":112800},[66],[15,112802,66792],{"className":112803},[66,4429],[15,112805,120],{"className":112806},[162],[15,112808,8678],{"className":112809},[66],[15,112811],{"className":112812,"style":72},[71],[15,112814,1245],{"className":112815},[76],[15,112817],{"className":112818,"style":72},[71],[15,112820,112822,112826,112829,112832,112835,112838,112841,112844,112847],{"className":112821},[57],[15,112823],{"className":112824,"style":112825},[61],"height:0.8389em;vertical-align:-0.1944em;",[15,112827,112701],{"className":112828},[66],[15,112830,335],{"className":112831},[422],[15,112833,24233],{"className":112834},[71],[15,112836],{"className":112837,"style":697},[71],[15,112839,8678],{"className":112840},[66],[15,112842],{"className":112843,"style":72},[71],[15,112845,1245],{"className":112846},[76],[15,112848],{"className":112849,"style":72},[71],[15,112851,112853,112856,112859,112862,112865,112868],{"className":112852},[57],[15,112854],{"className":112855,"style":154},[61],[15,112857,8678],{"className":112858},[66],[15,112860,126],{"className":112861},[169],[15,112863],{"className":112864,"style":173},[71],[15,112866,129],{"className":112867},[177],[15,112869],{"className":112870,"style":173},[71],[15,112872,112874,112877,112883,112886,112889,112892,112895,112898,112901],{"className":112873},[57],[15,112875],{"className":112876,"style":154},[61],[15,112878,112880],{"className":112879},[66],[15,112881,66792],{"className":112882},[66,4429],[15,112884,120],{"className":112885},[162],[15,112887,107911],{"className":112888},[66],[15,112890,335],{"className":112891},[422],[15,112893],{"className":112894,"style":697},[71],[15,112896,59943],{"className":112897},[66],[15,112899,126],{"className":112900},[169],[15,112902,3573],{"className":112903},[66],[11,112905,112906],{},"把几个估计量放在一起：",[6152,112908,112909,112919],{},[6155,112910,112911],{},[6158,112912,112913,112916],{},[6161,112914,112915],{},"量",[6161,112917,112918],{},"值",[6171,112920,112921,112985,113049,113112,113177],{},[6158,112922,112923,112926],{},[6176,112924,112925],{},"先验均值",[6176,112927,112928],{},[15,112929,112931,112954],{"className":112930,"translate":19},[18],[15,112932,112934],{"className":112933},[23],[25,112935,112936],{"xmlns":27},[29,112937,112938,112951],{},[32,112939,112940,112942,112944,112946,112948],{},[1209,112941,8678],{},[35,112943,6625],{"mathvariant":138},[1209,112945,59943],{},[39,112947,129],{},[1209,112949,112950],{},"0.5",[45,112952,112953],{"encoding":47},"3\u002F6=0.5",[15,112955,112957,112976],{"className":112956,"ariaHidden":53},[52],[15,112958,112960,112963,112967,112970,112973],{"className":112959},[57],[15,112961],{"className":112962,"style":154},[61],[15,112964,112966],{"className":112965},[66],"3\u002F6",[15,112968],{"className":112969,"style":173},[71],[15,112971,129],{"className":112972},[177],[15,112974],{"className":112975,"style":173},[71],[15,112977,112979,112982],{"className":112978},[57],[15,112980],{"className":112981,"style":1836},[61],[15,112983,112950],{"className":112984},[66],[6158,112986,112987,112990],{},[6176,112988,112989],{},"最大似然估计",[6176,112991,112992],{},[15,112993,112995,113018],{"className":112994,"translate":19},[18],[15,112996,112998],{"className":112997},[23],[25,112999,113000],{"xmlns":27},[29,113001,113002,113015],{},[32,113003,113004,113006,113008,113010,113012],{},[1209,113005,112701],{},[35,113007,6625],{"mathvariant":138},[1209,113009,107911],{},[39,113011,129],{},[1209,113013,113014],{},"0.7",[45,113016,113017],{"encoding":47},"7\u002F10=0.7",[15,113019,113021,113040],{"className":113020,"ariaHidden":53},[52],[15,113022,113024,113027,113031,113034,113037],{"className":113023},[57],[15,113025],{"className":113026,"style":154},[61],[15,113028,113030],{"className":113029},[66],"7\u002F10",[15,113032],{"className":113033,"style":173},[71],[15,113035,129],{"className":113036},[177],[15,113038],{"className":113039,"style":173},[71],[15,113041,113043,113046],{"className":113042},[57],[15,113044],{"className":113045,"style":1836},[61],[15,113047,113014],{"className":113048},[66],[6158,113050,113051,113053],{},[6176,113052,72666],{},[6176,113054,113055],{},[15,113056,113058,113081],{"className":113057,"translate":19},[18],[15,113059,113061],{"className":113060},[23],[25,113062,113063],{"xmlns":27},[29,113064,113065,113078],{},[32,113066,113067,113069,113071,113073,113075],{},[1209,113068,107911],{},[35,113070,6625],{"mathvariant":138},[1209,113072,59727],{},[39,113074,129],{},[1209,113076,113077],{},"0.625",[45,113079,113080],{"encoding":47},"10\u002F16=0.625",[15,113082,113084,113103],{"className":113083,"ariaHidden":53},[52],[15,113085,113087,113090,113094,113097,113100],{"className":113086},[57],[15,113088],{"className":113089,"style":154},[61],[15,113091,113093],{"className":113092},[66],"10\u002F16",[15,113095],{"className":113096,"style":173},[71],[15,113098,129],{"className":113099},[177],[15,113101],{"className":113102,"style":173},[71],[15,113104,113106,113109],{"className":113105},[57],[15,113107],{"className":113108,"style":1836},[61],[15,113110,113077],{"className":113111},[66],[6158,113113,113114,113116],{},[6176,113115,75121],{},[6176,113117,113118],{},[15,113119,113121,113146],{"className":113120,"translate":19},[18],[15,113122,113124],{"className":113123},[23],[25,113125,113126],{"xmlns":27},[29,113127,113128,113143],{},[32,113129,113130,113133,113135,113138,113140],{},[1209,113131,113132],{},"9",[35,113134,6625],{"mathvariant":138},[1209,113136,113137],{},"14",[39,113139,31993],{},[1209,113141,113142],{},"0.643",[45,113144,113145],{"encoding":47},"9\u002F14\\approx0.643",[15,113147,113149,113168],{"className":113148,"ariaHidden":53},[52],[15,113150,113152,113155,113159,113162,113165],{"className":113151},[57],[15,113153],{"className":113154,"style":154},[61],[15,113156,113158],{"className":113157},[66],"9\u002F14",[15,113160],{"className":113161,"style":173},[71],[15,113163,31993],{"className":113164},[177],[15,113166],{"className":113167,"style":173},[71],[15,113169,113171,113174],{"className":113170},[57],[15,113172],{"className":113173,"style":1836},[61],[15,113175,113142],{"className":113176},[66],[6158,113178,113179,113317],{},[6176,113180,113181,113182],{},"后验预测 ",[15,113183,113185,113217],{"className":113184,"translate":19},[18],[15,113186,113188],{"className":113187},[23],[25,113189,113190],{"xmlns":27},[29,113191,113192,113214],{},[32,113193,113194,113196,113198,113204,113206,113208,113210,113212],{},[35,113195,11],{},[39,113197,120],{"stretchy":119},[1057,113199,113200,113202],{},[35,113201,123],{},[1274,113203,67805],{},[39,113205,129],{},[1209,113207,1211],{},[39,113209,34430],{},[35,113211,32458],{},[39,113213,126],{"stretchy":119},[45,113215,113216],{"encoding":47},"p(x_{\\text{new}}=1\\mid D)",[15,113218,113220,113287,113305],{"className":113219,"ariaHidden":53},[52],[15,113221,113223,113226,113229,113232,113278,113281,113284],{"className":113222},[57],[15,113224],{"className":113225,"style":154},[61],[15,113227,11],{"className":113228},[66,67],[15,113230,120],{"className":113231},[162],[15,113233,113235,113238],{"className":113234},[66],[15,113236,123],{"className":113237},[66,67],[15,113239,113241],{"className":113240},[198],[15,113242,113244,113270],{"className":113243},[202,1090],[15,113245,113247,113267],{"className":113246},[206],[15,113248,113250],{"className":113249,"style":2198},[210],[15,113251,113252,113255],{"style":1584},[15,113253],{"className":113254,"style":218},[217],[15,113256,113258],{"className":113257},[222,223,224,225],[15,113259,113261],{"className":113260},[66,225],[15,113262,113264],{"className":113263},[66,1711,225],[15,113265,67805],{"className":113266},[66,225],[15,113268,1114],{"className":113269},[1113],[15,113271,113273],{"className":113272},[206],[15,113274,113276],{"className":113275,"style":1121},[210],[15,113277],{},[15,113279],{"className":113280,"style":173},[71],[15,113282,129],{"className":113283},[177],[15,113285],{"className":113286,"style":173},[71],[15,113288,113290,113293,113296,113299,113302],{"className":113289},[57],[15,113291],{"className":113292,"style":154},[61],[15,113294,1211],{"className":113295},[66],[15,113297],{"className":113298,"style":173},[71],[15,113300,34430],{"className":113301},[177],[15,113303],{"className":113304,"style":173},[71],[15,113306,113308,113311,113314],{"className":113307},[57],[15,113309],{"className":113310,"style":154},[61],[15,113312,32458],{"className":113313,"style":23025},[66,67],[15,113315,126],{"className":113316},[169],[6176,113318,113319],{},[15,113320,113322,113335],{"className":113321,"translate":19},[18],[15,113323,113325],{"className":113324},[23],[25,113326,113327],{"xmlns":27},[29,113328,113329,113333],{},[32,113330,113331],{},[1209,113332,113077],{},[45,113334,113077],{"encoding":47},[15,113336,113338],{"className":113337,"ariaHidden":53},[52],[15,113339,113341,113344],{"className":113340},[57],[15,113342],{"className":113343,"style":1836},[61],[15,113345,113077],{"className":113346},[66],[11,113348,113349,113350,113378,113379,113407,113408,113437,113438,113467,113468,113527,113528,1124],{},"后验均值被先验从 ",[15,113351,113353,113366],{"className":113352,"translate":19},[18],[15,113354,113356],{"className":113355},[23],[25,113357,113358],{"xmlns":27},[29,113359,113360,113364],{},[32,113361,113362],{},[1209,113363,113014],{},[45,113365,113014],{"encoding":47},[15,113367,113369],{"className":113368,"ariaHidden":53},[52],[15,113370,113372,113375],{"className":113371},[57],[15,113373],{"className":113374,"style":1836},[61],[15,113376,113014],{"className":113377},[66]," 拉回到 ",[15,113380,113382,113395],{"className":113381,"translate":19},[18],[15,113383,113385],{"className":113384},[23],[25,113386,113387],{"xmlns":27},[29,113388,113389,113393],{},[32,113390,113391],{},[1209,113392,113077],{},[45,113394,113077],{"encoding":47},[15,113396,113398],{"className":113397,"ariaHidden":53},[52],[15,113399,113401,113404],{"className":113400},[57],[15,113402],{"className":113403,"style":1836},[61],[15,113405,113077],{"className":113406},[66],"，这就是收缩。数据越多，拉力越弱：若观测到 ",[15,113409,113411,113425],{"className":113410,"translate":19},[18],[15,113412,113414],{"className":113413},[23],[25,113415,113416],{"xmlns":27},[29,113417,113418,113423],{},[32,113419,113420],{},[1209,113421,113422],{},"100",[45,113424,113422],{"encoding":47},[15,113426,113428],{"className":113427,"ariaHidden":53},[52],[15,113429,113431,113434],{"className":113430},[57],[15,113432],{"className":113433,"style":1836},[61],[15,113435,113422],{"className":113436},[66]," 次中 ",[15,113439,113441,113455],{"className":113440,"translate":19},[18],[15,113442,113444],{"className":113443},[23],[25,113445,113446],{"xmlns":27},[29,113447,113448,113453],{},[32,113449,113450],{},[1209,113451,113452],{},"70",[45,113454,113452],{"encoding":47},[15,113456,113458],{"className":113457,"ariaHidden":53},[52],[15,113459,113461,113464],{"className":113460},[57],[15,113462],{"className":113463,"style":1836},[61],[15,113465,113452],{"className":113466},[66]," 次正面，后验均值变为 ",[15,113469,113471,113496],{"className":113470,"translate":19},[18],[15,113472,113474],{"className":113473},[23],[25,113475,113476],{"xmlns":27},[29,113477,113478,113493],{},[32,113479,113480,113483,113485,113488,113490],{},[1209,113481,113482],{},"73",[35,113484,6625],{"mathvariant":138},[1209,113486,113487],{},"106",[39,113489,31993],{},[1209,113491,113492],{},"0.689",[45,113494,113495],{"encoding":47},"73\u002F106\\approx0.689",[15,113497,113499,113518],{"className":113498,"ariaHidden":53},[52],[15,113500,113502,113505,113509,113512,113515],{"className":113501},[57],[15,113503],{"className":113504,"style":154},[61],[15,113506,113508],{"className":113507},[66],"73\u002F106",[15,113510],{"className":113511,"style":173},[71],[15,113513,31993],{"className":113514},[177],[15,113516],{"className":113517,"style":173},[71],[15,113519,113521,113524],{"className":113520},[57],[15,113522],{"className":113523,"style":1836},[61],[15,113525,113492],{"className":113526},[66],"，已经非常接近 ",[15,113529,113531,113544],{"className":113530,"translate":19},[18],[15,113532,113534],{"className":113533},[23],[25,113535,113536],{"xmlns":27},[29,113537,113538,113542],{},[32,113539,113540],{},[1209,113541,113014],{},[45,113543,113014],{"encoding":47},[15,113545,113547],{"className":113546,"ariaHidden":53},[52],[15,113548,113550,113553],{"className":113549},[57],[15,113551],{"className":113552,"style":1836},[61],[15,113554,113014],{"className":113555},[66],[11,113557,113558,113559,113591,113592,113620,113621,113649,113650,113678],{},"如果决策是「猜下一次是否为正面」，且两类错误的代价相同，那么只要后验预测大于 ",[15,113560,113562,113579],{"className":113561,"translate":19},[18],[15,113563,113565],{"className":113564},[23],[25,113566,113567],{"xmlns":27},[29,113568,113569,113577],{},[32,113570,113571,113573,113575],{},[1209,113572,1211],{},[35,113574,6625],{"mathvariant":138},[1209,113576,1263],{},[45,113578,71809],{"encoding":47},[15,113580,113582],{"className":113581,"ariaHidden":53},[52],[15,113583,113585,113588],{"className":113584},[57],[15,113586],{"className":113587,"style":154},[61],[15,113589,71809],{"className":113590},[66]," 就猜正面，",[15,113593,113595,113608],{"className":113594,"translate":19},[18],[15,113596,113598],{"className":113597},[23],[25,113599,113600],{"xmlns":27},[29,113601,113602,113606],{},[32,113603,113604],{},[1209,113605,113077],{},[45,113607,113077],{"encoding":47},[15,113609,113611],{"className":113610,"ariaHidden":53},[52],[15,113612,113614,113617],{"className":113613},[57],[15,113615],{"className":113616,"style":1836},[61],[15,113618,113077],{"className":113619},[66]," 满足条件。若两类错误的代价不同：把反面误判为正面的代价为 ",[15,113622,113624,113637],{"className":113623,"translate":19},[18],[15,113625,113627],{"className":113626},[23],[25,113628,113629],{"xmlns":27},[29,113630,113631,113635],{},[32,113632,113633],{},[1209,113634,1263],{},[45,113636,1263],{"encoding":47},[15,113638,113640],{"className":113639,"ariaHidden":53},[52],[15,113641,113643,113646],{"className":113642},[57],[15,113644],{"className":113645,"style":1836},[61],[15,113647,1263],{"className":113648},[66],"，把正面误判为反面的代价为 ",[15,113651,113653,113666],{"className":113652,"translate":19},[18],[15,113654,113656],{"className":113655},[23],[25,113657,113658],{"xmlns":27},[29,113659,113660,113664],{},[32,113661,113662],{},[1209,113663,1211],{},[45,113665,1211],{"encoding":47},[15,113667,113669],{"className":113668,"ariaHidden":53},[52],[15,113670,113672,113675],{"className":113671},[57],[15,113673],{"className":113674,"style":1836},[61],[15,113676,1211],{"className":113677},[66],"，则阈值变为",[15,113680,113682],{"className":113681,"translate":19},[294],[15,113683,113685,113757],{"className":113684,"translate":19},[18],[15,113686,113688],{"className":113687},[23],[25,113689,113690],{"xmlns":27,"display":303},[29,113691,113692,113754],{},[32,113693,113694,113730,113732,113744,113746,113752],{},[6467,113695,113696,113706],{},[1057,113697,113698,113700],{},[35,113699,2256],{},[32,113701,113702,113704],{},[35,113703,74045],{"mathvariant":138},[35,113705,28399],{"mathvariant":138},[32,113707,113708,113718,113720],{},[1057,113709,113710,113712],{},[35,113711,2256],{},[32,113713,113714,113716],{},[35,113715,74045],{"mathvariant":138},[35,113717,28399],{"mathvariant":138},[39,113719,1245],{},[1057,113721,113722,113724],{},[35,113723,2256],{},[32,113725,113726,113728],{},[35,113727,74045],{"mathvariant":138},[35,113729,28535],{"mathvariant":138},[39,113731,129],{},[6467,113733,113734,113736],{},[1209,113735,1263],{},[32,113737,113738,113740,113742],{},[1209,113739,1263],{},[39,113741,1245],{},[1209,113743,1211],{},[39,113745,129],{},[6467,113747,113748,113750],{},[1209,113749,1263],{},[1209,113751,8678],{},[39,113753,335],{"separator":53},[45,113755,113756],{"encoding":47},"\\frac{c_{\\mathrm{FP}}}{c_{\\mathrm{FP}}+c_{\\mathrm{FN}}}\n=\\frac{2}{2+1}=\\frac23,",[15,113758,113760,113978,114067],{"className":113759,"ariaHidden":53},[52],[15,113761,113763,113766,113969,113972,113975],{"className":113762},[57],[15,113764],{"className":113765,"style":52422},[61],[15,113767,113769,113772,113966],{"className":113768},[66],[15,113770],{"className":113771},[162,23417],[15,113773,113775],{"className":113774},[6467],[15,113776,113778,113958],{"className":113777},[202,1090],[15,113779,113781,113955],{"className":113780},[206],[15,113782,113784,113893,113901],{"className":113783,"style":39133},[210],[15,113785,113786,113789],{"style":23433},[15,113787],{"className":113788,"style":1368},[217],[15,113790,113792,113838,113841,113844,113847],{"className":113791},[66],[15,113793,113795,113798],{"className":113794},[66],[15,113796,2256],{"className":113797},[66,67],[15,113799,113801],{"className":113800},[198],[15,113802,113804,113830],{"className":113803},[202,1090],[15,113805,113807,113827],{"className":113806},[206],[15,113808,113810],{"className":113809,"style":3638},[210],[15,113811,113812,113815],{"style":1584},[15,113813],{"className":113814,"style":218},[217],[15,113816,113818],{"className":113817},[222,223,224,225],[15,113819,113821],{"className":113820},[66,225],[15,113822,113824],{"className":113823},[66,225],[15,113825,74095],{"className":113826},[66,4429,225],[15,113828,1114],{"className":113829},[1113],[15,113831,113833],{"className":113832},[206],[15,113834,113836],{"className":113835,"style":1121},[210],[15,113837],{},[15,113839],{"className":113840,"style":72},[71],[15,113842,1245],{"className":113843},[76],[15,113845],{"className":113846,"style":72},[71],[15,113848,113850,113853],{"className":113849},[66],[15,113851,2256],{"className":113852},[66,67],[15,113854,113856],{"className":113855},[198],[15,113857,113859,113885],{"className":113858},[202,1090],[15,113860,113862,113882],{"className":113861},[206],[15,113863,113865],{"className":113864,"style":3638},[210],[15,113866,113867,113870],{"style":1584},[15,113868],{"className":113869,"style":218},[217],[15,113871,113873],{"className":113872},[222,223,224,225],[15,113874,113876],{"className":113875},[66,225],[15,113877,113879],{"className":113878},[66,225],[15,113880,74177],{"className":113881},[66,4429,225],[15,113883,1114],{"className":113884},[1113],[15,113886,113888],{"className":113887},[206],[15,113889,113891],{"className":113890,"style":1121},[210],[15,113892],{},[15,113894,113895,113898],{"style":23445},[15,113896],{"className":113897,"style":1368},[217],[15,113899],{"className":113900,"style":4440},[23452],[15,113902,113903,113906],{"style":23455},[15,113904],{"className":113905,"style":1368},[217],[15,113907,113909],{"className":113908},[66],[15,113910,113912,113915],{"className":113911},[66],[15,113913,2256],{"className":113914},[66,67],[15,113916,113918],{"className":113917},[198],[15,113919,113921,113947],{"className":113920},[202,1090],[15,113922,113924,113944],{"className":113923},[206],[15,113925,113927],{"className":113926,"style":3638},[210],[15,113928,113929,113932],{"style":1584},[15,113930],{"className":113931,"style":218},[217],[15,113933,113935],{"className":113934},[222,223,224,225],[15,113936,113938],{"className":113937},[66,225],[15,113939,113941],{"className":113940},[66,225],[15,113942,74095],{"className":113943},[66,4429,225],[15,113945,1114],{"className":113946},[1113],[15,113948,113950],{"className":113949},[206],[15,113951,113953],{"className":113952,"style":1121},[210],[15,113954],{},[15,113956,1114],{"className":113957},[1113],[15,113959,113961],{"className":113960},[206],[15,113962,113964],{"className":113963,"style":52619},[210],[15,113965],{},[15,113967],{"className":113968},[169,23417],[15,113970],{"className":113971,"style":173},[71],[15,113973,129],{"className":113974},[177],[15,113976],{"className":113977,"style":173},[71],[15,113979,113981,113984,114058,114061,114064],{"className":113980},[57],[15,113982],{"className":113983,"style":44670},[61],[15,113985,113987,113990,114055],{"className":113986},[66],[15,113988],{"className":113989},[162,23417],[15,113991,113993],{"className":113992},[6467],[15,113994,113996,114047],{"className":113995},[202,1090],[15,113997,113999,114044],{"className":113998},[206],[15,114000,114002,114025,114033],{"className":114001,"style":23430},[210],[15,114003,114004,114007],{"style":23433},[15,114005],{"className":114006,"style":1368},[217],[15,114008,114010,114013,114016,114019,114022],{"className":114009},[66],[15,114011,1263],{"className":114012},[66],[15,114014],{"className":114015,"style":72},[71],[15,114017,1245],{"className":114018},[76],[15,114020],{"className":114021,"style":72},[71],[15,114023,1211],{"className":114024},[66],[15,114026,114027,114030],{"style":23445},[15,114028],{"className":114029,"style":1368},[217],[15,114031],{"className":114032,"style":4440},[23452],[15,114034,114035,114038],{"style":23455},[15,114036],{"className":114037,"style":1368},[217],[15,114039,114041],{"className":114040},[66],[15,114042,1263],{"className":114043},[66],[15,114045,1114],{"className":114046},[1113],[15,114048,114050],{"className":114049},[206],[15,114051,114053],{"className":114052,"style":38570},[210],[15,114054],{},[15,114056],{"className":114057},[169,23417],[15,114059],{"className":114060,"style":173},[71],[15,114062,129],{"className":114063},[177],[15,114065],{"className":114066,"style":173},[71],[15,114068,114070,114073,114135],{"className":114069},[57],[15,114071],{"className":114072,"style":49086},[61],[15,114074,114076,114079,114132],{"className":114075},[66],[15,114077],{"className":114078},[162,23417],[15,114080,114082],{"className":114081},[6467],[15,114083,114085,114124],{"className":114084},[202,1090],[15,114086,114088,114121],{"className":114087},[206],[15,114089,114091,114102,114110],{"className":114090,"style":23430},[210],[15,114092,114093,114096],{"style":23433},[15,114094],{"className":114095,"style":1368},[217],[15,114097,114099],{"className":114098},[66],[15,114100,8678],{"className":114101},[66],[15,114103,114104,114107],{"style":23445},[15,114105],{"className":114106,"style":1368},[217],[15,114108],{"className":114109,"style":4440},[23452],[15,114111,114112,114115],{"style":23455},[15,114113],{"className":114114,"style":1368},[217],[15,114116,114118],{"className":114117},[66],[15,114119,1263],{"className":114120},[66],[15,114122,1114],{"className":114123},[1113],[15,114125,114127],{"className":114126},[206],[15,114128,114130],{"className":114129,"style":23474},[210],[15,114131],{},[15,114133],{"className":114134},[169,23417],[15,114136,335],{"className":114137},[422],[11,114139,956,114140,114197],{},[15,114141,114143,114165],{"className":114142,"translate":19},[18],[15,114144,114146],{"className":114145},[23],[25,114147,114148],{"xmlns":27},[29,114149,114150,114162],{},[32,114151,114152,114154,114156,114158,114160],{},[1209,114153,113077],{},[39,114155,25772],{},[1209,114157,1263],{},[35,114159,6625],{"mathvariant":138},[1209,114161,8678],{},[45,114163,114164],{"encoding":47},"0.625\u003C2\u002F3",[15,114166,114168,114187],{"className":114167,"ariaHidden":53},[52],[15,114169,114171,114175,114178,114181,114184],{"className":114170},[57],[15,114172],{"className":114173,"style":114174},[61],"height:0.6835em;vertical-align:-0.0391em;",[15,114176,113077],{"className":114177},[66],[15,114179],{"className":114180,"style":173},[71],[15,114182,25772],{"className":114183},[177],[15,114185],{"className":114186,"style":173},[71],[15,114188,114190,114193],{"className":114189},[57],[15,114191],{"className":114192,"style":154},[61],[15,114194,114196],{"className":114195},[66],"2\u002F3","，最优行动变成猜反面。同一个后验，换一个损失，结论就变了。这正是贝叶斯决策理论与「最大后验」的分水岭：后验只有一个，行动却取决于代价。",[1904,114199,114201],{"id":114200},"_82-高斯先验与收缩","8.2 高斯先验与收缩",[11,114203,114204,114205,114355,114356,114506,114507,114569,114570,114598],{},"设观测模型 ",[15,114206,114208,114244],{"className":114207,"translate":19},[18],[15,114209,114211],{"className":114210},[23],[25,114212,114213],{"xmlns":27},[29,114214,114215,114241],{},[32,114216,114217,114223,114225,114227,114229,114231,114233,114239],{},[1057,114218,114219,114221],{},[35,114220,1224],{},[35,114222,1205],{},[39,114224,28391],{},[35,114226,28535],{"mathvariant":445},[39,114228,120],{"stretchy":119},[35,114230,61618],{},[39,114232,335],{"separator":53},[131,114234,114235,114237],{},[35,114236,6634],{},[1209,114238,1263],{},[39,114240,126],{"stretchy":119},[45,114242,114243],{"encoding":47},"y_i\\sim\\mathcal N(\\theta,\\sigma^2)",[15,114245,114247,114302],{"className":114246,"ariaHidden":53},[52],[15,114248,114250,114253,114293,114296,114299],{"className":114249},[57],[15,114251],{"className":114252,"style":471},[61],[15,114254,114256,114259],{"className":114255},[66],[15,114257,1224],{"className":114258,"style":1515},[66,67],[15,114260,114262],{"className":114261},[198],[15,114263,114265,114285],{"className":114264},[202,1090],[15,114266,114268,114282],{"className":114267},[206],[15,114269,114271],{"className":114270,"style":1528},[210],[15,114272,114273,114276],{"style":1531},[15,114274],{"className":114275,"style":218},[217],[15,114277,114279],{"className":114278},[222,223,224,225],[15,114280,1205],{"className":114281},[66,67,225],[15,114283,1114],{"className":114284},[1113],[15,114286,114288],{"className":114287},[206],[15,114289,114291],{"className":114290,"style":1121},[210],[15,114292],{},[15,114294],{"className":114295,"style":173},[71],[15,114297,28391],{"className":114298},[177],[15,114300],{"className":114301,"style":173},[71],[15,114303,114305,114308,114311,114314,114317,114320,114323,114352],{"className":114304},[57],[15,114306],{"className":114307,"style":9154},[61],[15,114309,28535],{"className":114310,"style":83515},[66,494],[15,114312,120],{"className":114313},[162],[15,114315,61618],{"className":114316,"style":23025},[66,67],[15,114318,335],{"className":114319},[422],[15,114321],{"className":114322,"style":697},[71],[15,114324,114326,114329],{"className":114325},[66],[15,114327,6634],{"className":114328,"style":1515},[66,67],[15,114330,114332],{"className":114331},[198],[15,114333,114335],{"className":114334},[202],[15,114336,114338],{"className":114337},[206],[15,114339,114341],{"className":114340,"style":6740},[210],[15,114342,114343,114346],{"style":213},[15,114344],{"className":114345,"style":218},[217],[15,114347,114349],{"className":114348},[222,223,224,225],[15,114350,1263],{"className":114351},[66,225],[15,114353,126],{"className":114354},[169],"，先验 ",[15,114357,114359,114395],{"className":114358,"translate":19},[18],[15,114360,114362],{"className":114361},[23],[25,114363,114364],{"xmlns":27},[29,114365,114366,114392],{},[32,114367,114368,114370,114372,114374,114376,114382,114384,114390],{},[35,114369,61618],{},[39,114371,28391],{},[35,114373,28535],{"mathvariant":445},[39,114375,120],{"stretchy":119},[1057,114377,114378,114380],{},[35,114379,83402],{},[1209,114381,1804],{},[39,114383,335],{"separator":53},[131,114385,114386,114388],{},[35,114387,95594],{},[1209,114389,1263],{},[39,114391,126],{"stretchy":119},[45,114393,114394],{"encoding":47},"\\theta\\sim\\mathcal N(\\mu_0,\\tau^2)",[15,114396,114398,114416],{"className":114397,"ariaHidden":53},[52],[15,114399,114401,114404,114407,114410,114413],{"className":114400},[57],[15,114402],{"className":114403,"style":840},[61],[15,114405,61618],{"className":114406,"style":23025},[66,67],[15,114408],{"className":114409,"style":173},[71],[15,114411,28391],{"className":114412},[177],[15,114414],{"className":114415,"style":173},[71],[15,114417,114419,114422,114425,114428,114468,114471,114474,114503],{"className":114418},[57],[15,114420],{"className":114421,"style":9154},[61],[15,114423,28535],{"className":114424,"style":83515},[66,494],[15,114426,120],{"className":114427},[162],[15,114429,114431,114434],{"className":114430},[66],[15,114432,83402],{"className":114433},[66,67],[15,114435,114437],{"className":114436},[198],[15,114438,114440,114460],{"className":114439},[202,1090],[15,114441,114443,114457],{"className":114442},[206],[15,114444,114446],{"className":114445,"style":2139},[210],[15,114447,114448,114451],{"style":1584},[15,114449],{"className":114450,"style":218},[217],[15,114452,114454],{"className":114453},[222,223,224,225],[15,114455,1804],{"className":114456},[66,225],[15,114458,1114],{"className":114459},[1113],[15,114461,114463],{"className":114462},[206],[15,114464,114466],{"className":114465,"style":1121},[210],[15,114467],{},[15,114469,335],{"className":114470},[422],[15,114472],{"className":114473,"style":697},[71],[15,114475,114477,114480],{"className":114476},[66],[15,114478,95594],{"className":114479,"style":95652},[66,67],[15,114481,114483],{"className":114482},[198],[15,114484,114486],{"className":114485},[202],[15,114487,114489],{"className":114488},[206],[15,114490,114492],{"className":114491,"style":6740},[210],[15,114493,114494,114497],{"style":213},[15,114495],{"className":114496,"style":218},[217],[15,114498,114500],{"className":114499},[222,223,224,225],[15,114501,1263],{"className":114502},[66,225],[15,114504,126],{"className":114505},[169],"，样本均值为 ",[15,114508,114510,114528],{"className":114509,"translate":19},[18],[15,114511,114513],{"className":114512},[23],[25,114514,114515],{"xmlns":27},[29,114516,114517,114525],{},[32,114518,114519],{},[4296,114520,114521,114523],{"accent":53},[35,114522,123],{},[39,114524,100926],{},[45,114526,114527],{"encoding":47},"\\bar x",[15,114529,114531],{"className":114530,"ariaHidden":53},[52],[15,114532,114534,114537],{"className":114533},[57],[15,114535],{"className":114536,"style":101003},[61],[15,114538,114540],{"className":114539},[66,78147],[15,114541,114543],{"className":114542},[202],[15,114544,114546],{"className":114545},[206],[15,114547,114549,114557],{"className":114548,"style":101003},[210],[15,114550,114551,114554],{"style":1386},[15,114552],{"className":114553,"style":1368},[217],[15,114555,123],{"className":114556},[66,67],[15,114558,114559,114562],{"style":1386},[15,114560],{"className":114561,"style":1368},[217],[15,114563,114566],{"className":114564,"style":114565},[78175],"left:-0.2222em;",[15,114567,100926],{"className":114568},[66],"，样本量为 ",[15,114571,114573,114586],{"className":114572,"translate":19},[18],[15,114574,114576],{"className":114575},[23],[25,114577,114578],{"xmlns":27},[29,114579,114580,114584],{},[32,114581,114582],{},[35,114583,37],{},[45,114585,37],{"encoding":47},[15,114587,114589],{"className":114588,"ariaHidden":53},[52],[15,114590,114592,114595],{"className":114591},[57],[15,114593],{"className":114594,"style":86},[61],[15,114596,37],{"className":114597},[66,67],"。后验仍是高斯：",[15,114600,114602],{"className":114601,"translate":19},[294],[15,114603,114605,114649],{"className":114604,"translate":19},[18],[15,114606,114608],{"className":114607},[23],[25,114609,114610],{"xmlns":27,"display":303},[29,114611,114612,114646],{},[32,114613,114614,114616,114618,114620,114622,114624,114626,114632,114634,114642,114644],{},[35,114615,61618],{},[39,114617,34430],{},[35,114619,32458],{},[39,114621,28391],{},[35,114623,28535],{"mathvariant":445},[39,114625,120],{"stretchy":119},[1057,114627,114628,114630],{},[35,114629,83402],{},[35,114631,37],{},[39,114633,335],{"separator":53},[1255,114635,114636,114638,114640],{},[35,114637,95594],{},[35,114639,37],{},[1209,114641,1263],{},[39,114643,126],{"stretchy":119},[39,114645,335],{"separator":53},[45,114647,114648],{"encoding":47},"\\theta\\mid 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x\u002F\\sigma^2}{1\u002F\\tau^2+n\u002F\\sigma^2},\n\\qquad\n\\frac{1}{\\tau_n^2}=\\frac{1}{\\tau^2}+\\frac{n}{\\sigma^2}.",[15,114935,114937,114992,115406,115509],{"className":114936,"ariaHidden":53},[52],[15,114938,114940,114943,114983,114986,114989],{"className":114939},[57],[15,114941],{"className":114942,"style":471},[61],[15,114944,114946,114949],{"className":114945},[66],[15,114947,83402],{"className":114948},[66,67],[15,114950,114952],{"className":114951},[198],[15,114953,114955,114975],{"className":114954},[202,1090],[15,114956,114958,114972],{"className":114957},[206],[15,114959,114961],{"className":114960,"style":2198},[210],[15,114962,114963,114966],{"style":1584},[15,114964],{"className":114965,"style":218},[217],[15,114967,114969],{"className":114968},[222,223,224,225],[15,114970,37],{"className":114971},[66,67,225],[15,114973,1114],{"className":114974},[1113],[15,114976,114978],{"className":114977},[206],[15,114979,114981],{"className":114980,"style":1121},[210],[15,114982],{},[15,114984],{"className":114985,"style":173},[71],[15,114987,129],{"className":114988},[177],[15,114990],{"className":114991,"style":173},[71],[15,114993,114995,114998,115277,115280,115283,115286,115397,115400,115403],{"className":114994},[57],[15,114996],{"className":114997,"style":45944},[61],[15,114999,115001,115004,115274],{"className":115000},[66],[15,115002],{"className":115003},[162,23417],[15,115005,115007],{"className":115006},[6467],[15,115008,115010,115266],{"className":115009},[202,1090],[15,115011,115013,115263],{"className":115012},[206],[15,115014,115016,115100,115108],{"className":115015,"style":45963},[210],[15,115017,115018,115021],{"style":23433},[15,115019],{"className":115020,"style":1368},[217],[15,115022,115024,115027,115056,115059,115062,115065,115068,115071],{"className":115023},[66],[15,115025,103608],{"className":115026},[66],[15,115028,115030,115033],{"className":115029},[66],[15,115031,95594],{"className":115032,"style":95652},[66,67],[15,115034,115036],{"className":115035},[198],[15,115037,115039],{"className":115038},[202],[15,115040,115042],{"className":115041},[206],[15,115043,115045],{"className":115044,"style":46252},[210],[15,115046,115047,115050],{"style":46255},[15,115048],{"className":115049,"style":218},[217],[15,115051,115053],{"className":115052},[222,223,224,225],[15,115054,1263],{"className":115055},[66,225],[15,115057],{"className":115058,"style":72},[71],[15,115060,1245],{"className":115061},[76],[15,115063],{"className":115064,"style":72},[71],[15,115066,37],{"className":115067},[66,67],[15,115069,6625],{"className":115070},[66],[15,115072,115074,115077],{"className":115073},[66],[15,115075,6634],{"className":115076,"style":1515},[66,67],[15,115078,115080],{"className":115079},[198],[15,115081,115083],{"className":115082},[202],[15,115084,115086],{"className":115085},[206],[15,115087,115089],{"className":115088,"style":46252},[210],[15,115090,115091,115094],{"style":46255},[15,115092],{"className":115093,"style":218},[217],[15,115095,115097],{"className":115096},[222,223,224,225],[15,115098,1263],{"className":115099},[66,225],[15,115101,115102,115105],{"style":23445},[15,115103],{"className":115104,"style":1368},[217],[15,115106],{"className":115107,"style":4440},[23452],[15,115109,115110,115113],{"style":23455},[15,115111],{"className":115112,"style":1368},[217],[15,115114,115116,115156,115159,115188,115191,115194,115197,115200,115231,115234],{"className":115115},[66],[15,115117,115119,115122],{"className":115118},[66],[15,115120,83402],{"className":115121},[66,67],[15,115123,115125],{"className":115124},[198],[15,115126,115128,115148],{"className":115127},[202,1090],[15,115129,115131,115145],{"className":115130},[206],[15,115132,115134],{"className":115133,"style":2139},[210],[15,115135,115136,115139],{"style":1584},[15,115137],{"className":115138,"style":218},[217],[15,115140,115142],{"className":115141},[222,223,224,225],[15,115143,1804],{"className":115144},[66,225],[15,115146,1114],{"className":115147},[1113],[15,115149,115151],{"className":115150},[206],[15,115152,115154],{"className":115153,"style":1121},[210],[15,115155],{},[15,115157,6625],{"className":115158},[66],[15,115160,115162,115165],{"className":115161},[66],[15,115163,95594],{"className":115164,"style":95652},[66,67],[15,115166,115168],{"className":115167},[198],[15,115169,115171],{"className":115170},[202],[15,115172,115174],{"className":115173},[206],[15,115175,115177],{"className":115176,"style":6740},[210],[15,115178,115179,115182],{"style":213},[15,115180],{"className":115181,"style":218},[217],[15,115183,115185],{"className":115184},[222,223,224,225],[15,115186,1263],{"className":115187},[66,225],[15,115189],{"className":115190,"style":72},[71],[15,115192,1245],{"className":115193},[76],[15,115195],{"className":115196,"style":72},[71],[15,115198,37],{"className":115199},[66,67],[15,115201,115203],{"className":115202},[66,78147],[15,115204,115206],{"className":115205},[202],[15,115207,115209],{"clas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写成加权平均：",[15,115680,115682],{"className":115681,"translate":19},[294],[15,115683,115685,115787],{"className":115684,"translate":19},[18],[15,115686,115688],{"className":115687},[23],[25,115689,115690],{"xmlns":27,"display":303},[29,115691,115692,115784],{},[32,115693,115694,115700,115702,115736,115738,115744,115746,115774,115776,115782],{},[1057,115695,115696,115698],{},[35,115697,83402],{},[35,115699,37],{},[39,115701,129],{},[6467,115703,115704,115716],{},[32,115705,115706,115712,115714],{},[131,115707,115708,115710],{},[35,115709,6634],{},[1209,115711,1263],{},[35,115713,6625],{"mathvariant":138},[35,115715,37],{},[32,115717,115718,115724,115726,115732,115734],{},[131,115719,115720,115722],{},[35,115721,95594],{},[1209,115723,1263],{},[39,115725,1245],{},[131,115727,115728,115730],{},[35,115729,6634],{},[1209,115731,1263],{},[35,115733,6625],{"mathvariant":138},[35,115735,37],{},[1274,115737,2462],{},[1057,115739,115740,115742],{},[35,115741,83402],{},[1209,115743,1804],{},[39,115745,1245],{},[6467,115747,115748,115754],{},[131,115749,115750,115752],{},[35,115751,95594],{},[1209,115753,1263],{},[32,115755,115756,115762,115764,115770,115772],{},[131,115757,115758,115760],{},[35,115759,95594],{},[1209,115761,1263],{},[39,115763,1245],{},[131,115765,115766,115768],{},[35,115767,6634],{},[1209,115769,1263],{},[35,115771,6625],{"mathvariant":138},[35,115773,37],{},[1274,115775,2462],{},[4296,115777,115778,115780],{"accent":53},[35,115779,123],{},[39,115781,100926],{},[35,115783,3573],{"mathvariant":138},[45,115785,115786],{"encoding":47},"\\mu_n\n=\n\\frac{\\sigma^2\u002Fn}{\\tau^2+\\sigma^2\u002Fn}\\,\\mu_0\n+\n\\frac{\\tau^2}{\\tau^2+\\sigma^2\u002Fn}\\,\\bar x 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x=1",[15,116491,116493,116539],{"className":116492,"ariaHidden":53},[52],[15,116494,116496,116499,116530,116533,116536],{"className":116495},[57],[15,116497],{"className":116498,"style":101003},[61],[15,116500,116502],{"className":116501},[66,78147],[15,116503,116505],{"className":116504},[202],[15,116506,116508],{"className":116507},[206],[15,116509,116511,116519],{"className":116510,"style":101003},[210],[15,116512,116513,116516],{"style":1386},[15,116514],{"className":116515,"style":1368},[217],[15,116517,123],{"className":116518},[66,67],[15,116520,116521,116524],{"style":1386},[15,116522],{"className":116523,"style":1368},[217],[15,116525,116527],{"className":116526,"style":114565},[78175],[15,116528,100926],{"className":116529},[66],[15,116531],{"className":116532,"style":173},[71],[15,116534,129],{"className":116535},[177],[15,116537],{"className":116538,"style":173},[71],[15,116540,116542,116545],{"className":116541},[57],[15,116543],{"className":116544,"style":1836},[61],[15,116546,1211],{"className":116547},[66],[6152,116549,116550,116743],{},[6155,116551,116552],{},[6158,116553,116554,116584,116656],{},[6161,116555,116556],{},[15,116557,116559,116572],{"className":116558,"translate":19},[18],[15,116560,116562],{"className":116561},[23],[25,116563,116564],{"xmlns":27},[29,116565,116566,116570],{},[32,116567,116568],{},[35,116569,37],{},[45,116571,37],{"encoding":47},[15,116573,116575],{"className":116574,"ariaHidden":53},[52],[15,116576,116578,116581],{"className":116577},[57],[15,116579],{"className":116580,"style":86},[61],[15,116582,37],{"className":116583},[66,67],[6161,116585,116586,116587],{},"后验均值 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",[15,116660,116662,116682],{"className":116661,"translate":19},[18],[15,116663,116665],{"className":116664},[23],[25,116666,116667],{"xmlns":27},[29,116668,116669,116679],{},[32,116670,116671],{},[1255,116672,116673,116675,116677],{},[35,116674,95594],{},[35,116676,37],{},[1209,116678,1263],{},[45,116680,116681],{"encoding":47},"\\tau_n^2",[15,116683,116685],{"className":116684,"ariaHidden":53},[52],[15,116686,116688,116692],{"className":116687},[57],[15,116689],{"className":116690,"style":116691},[61],"height:1.0611em;vertical-align:-0.247em;",[15,116693,116695,116698],{"className":116694},[66],[15,116696,95594],{"className":116697,"style":95652},[66,67],[15,116699,116701],{"className":116700},[198],[15,116702,116704,116735],{"className":116703},[202,1090],[15,116705,116707,116732],{"className":116706},[206],[15,116708,116710,116721],{"className":116709,"style":6740},[210],[15,116711,116712,116715],{"style":114766},[15,116713],{"className":116714,"style":218},[217],[15,116716,116718],{"className":116717},[222,223,224,225],[15,116719,37],{"className":116720},[66,67,225],[15,116722,116723,116726],{"style":213},[15,116724],{"className":116725,"style":218},[217],[15,116727,116729],{"className":116728},[222,223,224,225],[15,116730,1263],{"className":116731},[66,225],[15,116733,1114],{"className":116734},[1113],[15,116736,116738],{"className":116737},[206],[15,116739,116741],{"className":116740,"style":1686},[210],[15,116742],{},[6171,116744,116745,116862,116985,117109],{},[6158,116746,116747,116749,116806],{},[6176,116748,1211],{},[6176,116750,116751],{},[15,116752,116754,116776],{"className":116753,"translate":19},[18],[15,116755,116757],{"className":116756},[23],[25,116758,116759],{"xmlns":27},[29,116760,116761,116773],{},[32,116762,116763,116765,116767,116769,116771],{},[1209,116764,1211],{},[35,116766,6625],{"mathvariant":138},[1209,116768,1263],{},[39,116770,129],{},[1209,116772,112950],{},[45,116774,116775],{"encoding":47},"1\u002F2=0.5",[15,116777,116779,116797],{"className":116778,"ariaHidden":53},[52],[15,116780,116782,116785,116788,116791,116794],{"className":116781},[57],[15,116783],{"className":116784,"style":154},[61],[15,116786,71809],{"className":116787},[66],[15,116789],{"className":116790,"style":173},[71],[15,116792,129],{"className":116793},[177],[15,116795],{"className":116796,"style":173},[71],[15,116798,116800,116803],{"className":116799},[57],[15,116801],{"className":116802,"style":1836},[61],[15,116804,112950],{"className":116805},[66],[6176,116807,116808],{},[15,116809,116811,116832],{"className":116810,"translate":19},[18],[15,116812,116814],{"className":116813},[23],[25,116815,116816],{"xmlns":27},[29,116817,116818,116830],{},[32,116819,116820,116822,116824,116826,116828],{},[1209,116821,1211],{},[35,116823,6625],{"mathvariant":138},[1209,116825,1263],{},[39,116827,129],{},[1209,116829,112950],{},[45,116831,116775],{"encoding":47},[15,116833,116835,116853],{"className":116834,"ariaHidden":53},[52],[15,116836,116838,116841,116844,116847,116850],{"className":116837},[57],[15,116839],{"className":116840,"style":154},[61],[15,116842,71809],{"className":116843},[66],[15,116845],{"className":116846,"style":173},[71],[15,116848,129],{"className":116849},[177],[15,116851],{"className":116852,"style":173},[71],[15,116854,116856,116859],{"className":116855},[57],[15,116857],{"className":116858,"style":1836},[61],[15,116860,112950],{"className":116861},[66],[6158,116863,116864,116866,116926],{},[6176,116865,15855],{},[6176,116867,116868],{},[15,116869,116871,116895],{"className":116870,"translate":19},[18],[15,116872,116874],{"className":116873},[23],[25,116875,116876],{"xmlns":27},[29,116877,116878,116892],{},[32,116879,116880,116882,116884,116887,116889],{},[1209,116881,15855],{},[35,116883,6625],{"mathvariant":138},[1209,116885,116886],{},"5",[39,116888,129],{},[1209,116890,116891],{},"0.8",[45,116893,116894],{"encoding":47},"4\u002F5=0.8",[15,116896,116898,116917],{"className":116897,"ariaHidden":53},[52],[15,116899,116901,116904,116908,116911,116914],{"className":116900},[57],[15,116902],{"className":116903,"style":154},[61],[15,116905,116907],{"className":116906},[66],"4\u002F5",[15,116909],{"className":116910,"style":173},[71],[15,116912,129],{"className":116913},[177],[15,116915],{"className":116916,"style":173},[71],[15,116918,116920,116923],{"className":116919},[57],[15,116921],{"className":116922,"style":1836},[61],[15,116924,116891],{"className":116925},[66],[6176,116927,116928],{},[15,116929,116931,116954],{"className":116930,"translate":19},[18],[15,116932,116934],{"className":116933},[23],[25,116935,116936],{"xmlns":27},[29,116937,116938,116951],{},[32,116939,116940,116942,116944,116946,116948],{},[1209,116941,1211],{},[35,116943,6625],{"mathvariant":138},[1209,116945,116886],{},[39,116947,129],{},[1209,116949,116950],{},"0.2",[45,116952,116953],{"encoding":47},"1\u002F5=0.2",[15,116955,116957,116976],{"className":116956,"ariaHidden":53},[52],[15,116958,116960,116963,116967,116970,116973],{"className":116959},[57],[15,116961],{"className":116962,"style":154},[61],[15,116964,116966],{"className":116965},[66],"1\u002F5",[15,116968],{"className":116969,"style":173},[71],[15,116971,129],{"className":116972},[177],[15,116974],{"className":116975,"style":173},[71],[15,116977,116979,116982],{"className":116978},[57],[15,116980],{"className":116981,"style":1836},[61],[15,116983,116950],{"className":116984},[66],[6158,116986,116987,116990,117050],{},[6176,116988,116989],{},"25",[6176,116991,116992],{},[15,116993,116995,117019],{"className":116994,"translate":19},[18],[15,116996,116998],{"className":116997},[23],[25,116999,117000],{"xmlns":27},[29,117001,117002,117016],{},[32,117003,117004,117006,117008,117011,117013],{},[1209,117005,116989],{},[35,117007,6625],{"mathvariant":138},[1209,117009,117010],{},"26",[39,117012,31993],{},[1209,117014,117015],{},"0.962",[45,117017,117018],{"encoding":47},"25\u002F26\\approx0.962",[15,117020,117022,117041],{"className":117021,"ariaHidden":53},[52],[15,117023,117025,117028,117032,117035,117038],{"className":117024},[57],[15,117026],{"className":117027,"style":154},[61],[15,117029,117031],{"className":117030},[66],"25\u002F26",[15,117033],{"className":117034,"style":173},[71],[15,117036,31993],{"className":117037},[177],[15,117039],{"className":117040,"style":173},[71],[15,117042,117044,117047],{"className":117043},[57],[15,117045],{"className":117046,"style":1836},[61],[15,117048,117015],{"className":117049},[66],[6176,117051,117052],{},[15,117053,117055,117078],{"className":117054,"translate":19},[18],[15,117056,117058],{"className":117057},[23],[25,117059,117060],{"xmlns":27},[29,117061,117062,117075],{},[32,117063,117064,117066,117068,117070,117072],{},[1209,117065,1211],{},[35,117067,6625],{"mathvariant":138},[1209,117069,117010],{},[39,117071,31993],{},[1209,117073,117074],{},"0.038",[45,117076,117077],{"encoding":47},"1\u002F26\\approx0.038",[15,117079,117081,117100],{"className":117080,"ariaHidden":53},[52],[15,117082,117084,117087,117091,117094,117097],{"className":117083},[57],[15,117085],{"className":117086,"style":154},[61],[15,117088,117090],{"className":117089},[66],"1\u002F26",[15,117092],{"className":117093,"style":173},[71],[15,117095,31993],{"className":117096},[177],[15,117098],{"className":117099,"style":173},[71],[15,117101,117103,117106],{"className":117102},[57],[15,117104],{"className":117105,"style":1836},[61],[15,117107,117074],{"className":117108},[66],[6158,117110,117111,117113,117173],{},[6176,117112,113422],{},[6176,117114,117115],{},[15,117116,117118,117142],{"className":117117,"translate":19},[18],[15,117119,117121],{"className":117120},[23],[25,117122,117123],{"xmlns":27},[29,117124,117125,117139],{},[32,117126,117127,117129,117131,117134,117136],{},[1209,117128,113422],{},[35,117130,6625],{"mathvariant":138},[1209,117132,117133],{},"101",[39,117135,31993],{},[1209,117137,117138],{},"0.990",[45,117140,117141],{"encoding":47},"100\u002F101\\approx0.990",[15,117143,117145,117164],{"className":117144,"ariaHidden":53},[52],[15,117146,117148,117151,117155,117158,117161],{"className":117147},[57],[15,117149],{"className":117150,"style":154},[61],[15,117152,117154],{"className":117153},[66],"100\u002F101",[15,117156],{"className":117157,"style":173},[71],[15,117159,31993],{"className":117160},[177],[15,117162],{"className":117163,"style":173},[71],[15,117165,117167,117170],{"className":117166},[57],[15,117168],{"className":117169,"style":1836},[61],[15,117171,117138],{"className":117172},[66],[6176,117174,117175],{},[15,117176,117178,117201],{"className":117177,"translate":19},[18],[15,117179,117181],{"className":117180},[23],[25,117182,117183],{"xmlns":27},[29,117184,117185,117198],{},[32,117186,117187,117189,117191,117193,117195],{},[1209,117188,1211],{},[35,117190,6625],{"mathvariant":138},[1209,117192,117133],{},[39,117194,31993],{},[1209,117196,117197],{},"0.0099",[45,117199,117200],{"encoding":47},"1\u002F101\\approx0.0099",[15,117202,117204,117223],{"className":117203,"ariaHidden":53},[52],[15,117205,117207,117210,117214,117217,117220],{"className":117206},[57],[15,117208],{"className":117209,"style":154},[61],[15,117211,117213],{"className":117212},[66],"1\u002F101",[15,117215],{"className":117216,"style":173},[71],[15,117218,31993],{"className":117219},[177],[15,117221],{"className":117222,"style":173},[71],[15,117224,117226,117229],{"className":117225},[57],[15,117227],{"className":117228,"style":1836},[61],[15,117230,117197],{"className":117231},[66],[11,117233,117234],{},"样本越多，后验越靠近样本均值，方差越小。先验的作用不是「永远拉住估计」，而是在小样本时提供一个稳定的起点，随着证据积累自动退场。",[1904,117236,117238],{"id":117237},"_83-医疗筛查先验似然与代价","8.3 医疗筛查：先验、似然与代价",[11,117240,117241,117242,117276,117277,117310,117311,117344],{},"设某疾病的患病率为 ",[15,117243,117245,117262],{"className":117244,"translate":19},[18],[15,117246,117248],{"className":117247},[23],[25,117249,117250],{"xmlns":27},[29,117251,117252,117259],{},[32,117253,117254,117256],{},[1209,117255,1211],{},[35,117257,117258],{"mathvariant":138},"%",[45,117260,117261],{"encoding":47},"1\\%",[15,117263,117265],{"className":117264,"ariaHidden":53},[52],[15,117266,117268,117272],{"className":117267},[57],[15,117269],{"className":117270,"style":117271},[61],"height:0.8056em;vertical-align:-0.0556em;",[15,117273,117275],{"className":117274},[66],"1%","，检测的灵敏度为 ",[15,117278,117280,117297],{"className":117279,"translate":19},[18],[15,117281,117283],{"className":117282},[23],[25,117284,117285],{"xmlns":27},[29,117286,117287,117294],{},[32,117288,117289,117292],{},[1209,117290,117291],{},"99",[35,117293,117258],{"mathvariant":138},[45,117295,117296],{"encoding":47},"99\\%",[15,117298,117300],{"className":117299,"ariaHidden":53},[52],[15,117301,117303,117306],{"className":117302},[57],[15,117304],{"className":117305,"style":117271},[61],[15,117307,117309],{"className":117308},[66],"99%","，特异度为 ",[15,117312,117314,117331],{"className":117313,"translate":19},[18],[15,117315,117317],{"className":117316},[23],[25,117318,117319],{"xmlns":27},[29,117320,117321,117328],{},[32,117322,117323,117326],{},[1209,117324,117325],{},"95",[35,117327,117258],{"mathvariant":138},[45,117329,117330],{"encoding":47},"95\\%",[15,117332,117334],{"className":117333,"ariaHidden":53},[52],[15,117335,117337,117340],{"className":117336},[57],[15,117338],{"className":117339,"style":117271},[61],[15,117341,117343],{"className":117342},[66],"95%","。一个人检测阳性，患病概率是多少？",[11,117346,117347],{},"用贝叶斯公式：",[15,117349,117351],{"className":117350,"translate":19},[294],[15,117352,117354,117552],{"className":117353,"translate":19},[18],[15,117355,117357],{"className":117356},[23],[25,117358,117359],{"xmlns":27,"display":303},[29,117360,117361,117549],{},[1142,117362,117363,117466,117513],{"rowspacing":62980,"columnalign":62981,"columnspacing":634},[1146,117364,117365,117383],{},[1149,117366,117367],{},[62987,117368,117369],{"scriptlevel":1804,"displaystyle":53},[32,117370,117371,117373,117375,117377,117379,117381],{},[35,117372,28399],{},[39,117374,120],{"stretchy":119},[35,117376,32458],{},[39,117378,34430],{},[39,117380,1245],{},[39,117382,126],{"stretchy":119},[1149,117384,117385],{},[62987,117386,117387],{"scriptlevel":1804,"displaystyle":53},[32,117388,117389,117391,117393],{},[32,117390],{},[39,117392,129],{},[6467,117394,117395,117417],{},[32,117396,117397,117399,117401,117403,117405,117407,117409,117411,117413,117415],{},[35,117398,28399],{},[39,117400,120],{"stretchy":119},[39,117402,1245],{},[39,117404,34430],{},[35,117406,32458],{},[39,117408,126],{"stretchy":119},[35,117410,28399],{},[39,117412,120],{"stretchy":119},[35,117414,32458],{},[39,117416,126],{"stretchy":119},[32,117418,117419,117421,117423,117425,117427,117429,117431,117433,117435,117437,117439,117441,117443,117445,117447,117449,117452,117454,117456,117458,117460,117462,117464],{},[35,117420,28399],{},[39,117422,120],{"stretchy":119},[39,117424,1245],{},[39,117426,34430],{},[35,117428,32458],{},[39,117430,126],{"stretchy":119},[35,117432,28399],{},[39,117434,120],{"stretchy":119},[35,117436,32458],{},[39,117438,126],{"stretchy":119},[39,117440,1245],{},[35,117442,28399],{},[39,117444,120],{"stretchy":119},[39,117446,1245],{},[39,117448,34430],{},[35,117450,117451],{"mathvariant":138},"¬",[35,117453,32458],{},[39,117455,126],{"stretchy":119},[35,117457,28399],{},[39,117459,120],{"stretchy":119},[35,117461,117451],{"mathvariant":138},[35,117463,32458],{},[39,117465,126],{"stretchy":119},[1146,117467,117468,117474],{},[1149,117469,117470],{},[62987,117471,117472],{"scriptlevel":1804,"displaystyle":53},[32,117473],{},[1149,117475,117476],{},[62987,117477,117478],{"scriptlevel":1804,"displaystyle":53},[32,117479,117480,117482,117484],{},[32,117481],{},[39,117483,129],{},[6467,117485,117486,117496],{},[32,117487,117488,117491,117493],{},[1209,117489,117490],{},"0.99",[39,117492,41],{},[1209,117494,117495],{},"0.01",[32,117497,117498,117500,117502,117504,117506,117509,117511],{},[1209,117499,117490],{},[39,117501,41],{},[1209,117503,117495],{},[39,117505,1245],{},[1209,117507,117508],{},"0.05",[39,117510,41],{},[1209,117512,117490],{},[1146,117514,117515,117521],{},[1149,117516,117517],{},[62987,117518,117519],{"scriptlevel":1804,"displaystyle":53},[32,117520],{},[1149,117522,117523],{},[62987,117524,117525],{"scriptlevel":1804,"displaystyle":53},[32,117526,117527,117529,117531,117544,117546],{},[32,117528],{},[39,117530,129],{},[6467,117532,117533,117535],{},[1209,117534,117197],{},[32,117536,117537,117539,117541],{},[1209,117538,117197],{},[39,117540,1245],{},[1209,117542,117543],{},"0.0495",[39,117545,31993],{},[1209,117547,117548],{},"0.167.",[45,117550,117551],{"encoding":47},"\\begin{aligned}\nP(D\\mid +)\n&=\\frac{P(+\\mid D)P(D)}{P(+\\mid D)P(D)+P(+\\mid \\neg D)P(\\neg D)}\\\\\n&=\\frac{0.99\\times0.01}{0.99\\times0.01+0.05\\times0.99}\\\\\n&=\\frac{0.0099}{0.0099+0.0495}\n\\approx0.167 .\n\\end{aligned}",[15,117553,117555],{"className":117554,"ariaHidden":53},[52],[15,117556,117558,117562],{"className":117557},[57],[15,117559],{"className":117560,"style":117561},[61],"height:7.1445em;vertical-align:-3.3223em;",[15,117563,117565],{"className":117564},[66],[15,117566,117568,117645],{"className":117567},[1142],[15,117569,117571],{"className":117570},[63150],[15,117572,117574,117636],{"className":117573},[202,1090],[15,117575,117577,117633],{"className":117576},[206],[15,117578,117581,117615,117624],{"className":117579,"style":117580},[210],"height:3.8223em;",[15,117582,117584,117588],{"style":117583},"top:-5.8223em;",[15,117585],{"className":117586,"style":117587},[217],"height:3.427em;",[15,117589,117591,117594,117597,117600,117603,117606,117609,117612],{"className":117590},[66],[15,117592,28399],{"className":117593,"style":62513},[66,67],[15,117595,120],{"className":117596},[162],[15,117598,32458],{"className":117599,"style":23025},[66,67],[15,117601],{"className":117602,"style":173},[71],[15,117604,34430],{"className":117605},[177],[15,117607],{"className":117608,"style":173},[71],[15,117610,1245],{"className":117611},[66],[15,117613,126],{"className":117614},[169],[15,117616,117618,117621],{"style":117617},"top:-3.2648em;",[15,117619],{"className":117620,"style":117587},[217],[15,117622],{"className":117623},[66],[15,117625,117627,117630],{"style":117626},"top:-0.8741em;",[15,117628],{"className":117629,"style":117587},[217],[15,117631],{"className":117632},[66],[15,117634,1114],{"className":117635},[1113],[15,117637,117639],{"className":117638},[206],[15,117640,117643],{"className":117641,"style":117642},[210],"height:3.3223em;",[15,117644],{},[15,117646,117648],{"className":117647},[63207],[15,117649,117651,118095],{"className":117650},[202,1090],[15,117652,117654,118092],{"className":117653},[206],[15,117655,117657,117856,117986],{"className":117656,"style":117580},[210],[15,117658,117659,117662],{"style":117583},[15,117660],{"className":117661,"style":117587},[217],[15,117663,117665,117668,117671,117674,117677],{"className":117664},[66],[15,117666],{"className":117667},[66],[15,117669],{"className":117670,"style":173},[71],[15,117672,129],{"className":117673},[177],[15,117675],{"className":117676,"style":173},[71],[15,117678,117680,117683,117853],{"className":117679},[66],[15,117681],{"className":117682},[162,23417],[15,117684,117686],{"className":117685},[6467],[15,117687,117689,117845],{"className":117688},[202,1090],[15,117690,117692,117842],{"className":117691},[206],[15,117693,117695,117790,117798],{"className":117694,"style":41211},[210],[15,117696,117697,117700],{"style":23433},[15,117698],{"className":117699,"style":1368},[217],[15,117701,117703,117706,117709,117712,117715,117718,117721,117724,117727,117730,117733,117736,117739,117742,117745,117748,117751,117754,117757,117760,117763,117766,117769,117772,117775,117778,117781,117784,117787],{"className":117702},[66],[15,117704,28399],{"className":117705,"style":62513},[66,67],[15,117707,120],{"className":117708},[162],[15,117710,1245],{"className":117711},[66],[15,117713],{"className":117714,"style":173},[71],[15,117716,34430],{"className":117717},[177],[15,117719],{"className":117720,"style":173},[71],[15,117722,32458],{"className":117723,"style":23025},[66,67],[15,117725,126],{"className":117726},[169],[15,117728,28399],{"className":117729,"style":62513},[66,67],[15,117731,120],{"className":117732},[162],[15,117734,32458],{"className":117735,"style":23025},[66,67],[15,117737,126],{"className":117738},[169],[15,117740],{"className":117741,"style":72},[71],[15,117743,1245],{"className":117744},[76],[15,117746],{"className":117747,"style":72},[71],[15,117749,28399],{"className":117750,"style":62513},[66,67],[15,117752,120],{"className":117753},[162],[15,117755,1245],{"className":117756},[66],[15,117758],{"className":117759,"style":173},[71],[15,117761,34430],{"className":117762},[177],[15,117764],{"className":117765,"style":173},[71],[15,117767,117451],{"className":117768},[66],[15,117770,32458],{"className":117771,"style":23025},[66,67],[15,117773,126],{"className":117774},[169],[15,117776,28399],{"className":117777,"style":62513},[66,67],[15,117779,120],{"className":117780},[162],[15,117782,117451],{"className":117783},[66],[15,117785,32458],{"className":117786,"style":23025},[66,67],[15,117788,126],{"className":117789},[169],[15,117791,117792,117795],{"style":23445},[15,117793],{"className":117794,"style":1368},[217],[15,117796],{"className":117797,"style":4440},[23452],[15,117799,117800,117803],{"style":23455},[15,117801],{"className":117802,"style":1368},[217],[15,117804,117806,117809,117812,117815,117818,117821,117824,117827,117830,117833,117836,117839],{"className":117805},[66],[15,117807,28399],{"className":117808,"style":62513},[66,67],[15,117810,120],{"className":117811},[162],[15,117813,1245],{"className":117814},[66],[15,117816],{"className":117817,"style":173},[71],[15,117819,34430],{"className":117820},[177],[15,11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",[15,118107,118109,118126],{"className":118108,"translate":19},[18],[15,118110,118112],{"className":118111},[23],[25,118113,118114],{"xmlns":27},[29,118115,118116,118123],{},[32,118117,118118,118121],{},[1209,118119,118120],{},"16.7",[35,118122,117258],{"mathvariant":138},[45,118124,118125],{"encoding":47},"16.7\\%",[15,118127,118129],{"className":118128,"ariaHidden":53},[52],[15,118130,118132,118135],{"className":118131},[57],[15,118133],{"className":118134,"style":117271},[61],[15,118136,118138],{"className":118137},[66],"16.7%","，因为健康人占了绝大多数，",[15,118141,118143,118159],{"className":118142,"translate":19},[18],[15,118144,118146],{"className":118145},[23],[25,118147,118148],{"xmlns":27},[29,118149,118150,118156],{},[32,118151,118152,118154],{},[1209,118153,116886],{},[35,118155,117258],{"mathvariant":138},[45,118157,118158],{"encoding":47},"5\\%",[15,118160,118162],{"className":118161,"ariaHidden":53},[52],[15,118163,118165,118168],{"className":118164},[57],[15,118166],{"className":118167,"style":117271},[61],[15,118169,118171],{"className":118170},[66],"5%"," 的假阳性率乘以 ",[15,118174,118176,118191],{"className":118175,"translate":19},[18],[15,118177,118179],{"className":118178},[23],[25,118180,118181],{"xmlns":27},[29,118182,118183,118189],{},[32,118184,118185,118187],{},[1209,118186,117291],{},[35,118188,117258],{"mathvariant":138},[45,118190,117296],{"encoding":47},[15,118192,118194],{"className":118193,"ariaHidden":53},[52],[15,118195,118197,118200],{"className":118196},[57],[15,118198],{"className":118199,"style":117271},[61],[15,118201,117309],{"className":118202},[66]," 的基础人数，产生了一个比真阳性更大的绝对数量。这就是基础率谬误：把 ",[15,118205,118207,118231],{"className":118206,"translate":19},[18],[15,118208,118210],{"className":118209},[23],[25,118211,118212],{"xmlns":27},[29,118213,118214,118228],{},[32,118215,118216,118218,118220,118222,118224,118226],{},[35,118217,28399],{},[39,118219,120],{"stretchy":119},[39,118221,1245],{},[39,118223,34430],{},[35,118225,32458],{},[39,118227,126],{"stretchy":119},[45,118229,118230],{"encoding":47},"P(+\\mid D)",[15,118232,118234,118258],{"className":118233,"ariaHidden":53},[52],[15,118235,118237,118240,118243,118246,118249,118252,118255],{"className":118236},[57],[15,118238],{"className":118239,"style":154},[61],[15,118241,28399],{"className":118242,"style":62513},[66,67],[15,118244,120],{"className":118245},[162],[15,118247,1245],{"className":118248},[66],[15,118250],{"className":118251,"style":173},[71],[15,118253,34430],{"className":118254},[177],[15,118256],{"className":118257,"style":173},[71],[15,118259,118261,118264,118267],{"className":118260},[57],[15,118262],{"className":118263,"style":154},[61],[15,118265,32458],{"className":118266,"style":23025},[66,67],[15,118268,126],{"className":118269},[169],[15,118271,118273,118297],{"className":118272,"translate":19},[18],[15,118274,118276],{"className":118275},[23],[25,118277,118278],{"xmlns":27},[29,118279,118280,118294],{},[32,118281,118282,118284,118286,118288,118290,118292],{},[35,118283,28399],{},[39,118285,120],{"stretchy":119},[35,118287,32458],{},[39,118289,34430],{},[39,118291,1245],{},[39,118293,126],{"stretchy":119},[45,118295,118296],{"encoding":47},"P(D\\mid +)",[15,118298,118300,118324],{"className":118299,"ariaHidden":53},[52],[15,118301,118303,118306,118309,118312,118315,118318,118321],{"className":118302},[57],[15,118304],{"className":118305,"style":154},[61],[15,118307,28399],{"className":118308,"style":62513},[66,67],[15,118310,120],{"className":118311},[162],[15,118313,32458],{"className":118314,"style":23025},[66,67],[15,118316],{"className":118317,"style":173},[71],[15,118319,34430],{"className":118320},[177],[15,118322],{"className":118323,"style":173},[71],[15,118325,118327,118330,118333],{"className":118326},[57],[15,118328],{"className":118329,"style":154},[61],[15,118331,1245],{"className":118332},[66],[15,118334,126],{"className":118335},[169]," 混为一谈。",[11,118338,118339],{},"患病率变化时，同一个检测给出的后验完全不同：",[6152,118341,118342,118416],{},[6155,118343,118344],{},[6158,118345,118346,118349],{},[6161,118347,118348],{},"患病率",[6161,118350,118351],{},[15,118352,118354,118377],{"className":118353,"translate":19},[18],[15,118355,118357],{"className":118356},[23],[25,118358,118359],{"xmlns":27},[29,118360,118361,118375],{},[32,118362,118363,118365,118367,118369,118371,118373],{},[35,118364,28399],{},[39,118366,120],{"stretchy":119},[35,118368,32458],{},[39,118370,34430],{},[39,118372,1245],{},[39,118374,126],{"stretchy":119},[45,118376,118296],{"encoding":47},[15,118378,118380,118404],{"className":118379,"ariaHidden":53},[52],[15,118381,118383,118386,118389,118392,118395,118398,118401],{"className":118382},[57],[15,118384],{"className":118385,"style":154},[61],[15,118387,28399],{"className":118388,"style":62513},[66,67],[15,118390,120],{"className":118391},[162],[15,118393,32458],{"className":118394,"style":23025},[66,67],[15,118396],{"className":118397,"style":173},[71],[15,118399,34430],{"className":118400},[177],[15,118402],{"className":118403,"style":173},[71],[15,118405,118407,118410,118413],{"className":118406},[57],[15,118408],{"className":118409,"style":154},[61],[15,118411,1245],{"className":118412},[66],[15,118414,126],{"className":118415},[169],[6171,118417,118418,118485,118550,118615],{},[6158,118419,118420,118454],{},[6176,118421,118422],{},[15,118423,118425,118441],{"className":118424,"translate":19},[18],[15,118426,118428],{"className":118427},[23],[25,118429,118430],{"xmlns":27},[29,118431,118432,118438],{},[32,118433,118434,118436],{},[1209,118435,112950],{},[35,118437,117258],{"mathvariant":138},[45,118439,118440],{"encoding":47},"0.5\\%",[15,118442,118444],{"className":118443,"ariaHidden":53},[52],[15,118445,118447,118450],{"className":118446},[57],[15,118448],{"className":118449,"style":117271},[61],[15,118451,118453],{"className":118452},[66],"0.5%",[6176,118455,118456],{},[15,118457,118459,118473],{"className":118458,"translate":19},[18],[15,118460,118462],{"className":118461},[23],[25,118463,118464],{"xmlns":27},[29,118465,118466,118471],{},[32,118467,118468],{},[1209,118469,118470],{},"0.090",[45,118472,118470],{"encoding":47},[15,118474,118476],{"className":118475,"ariaHidden":53},[52],[15,118477,118479,118482],{"className":118478},[57],[15,118480],{"className":118481,"style":1836},[61],[15,118483,118470],{"className":118484},[66],[6158,118486,118487,118519],{},[6176,118488,118489],{},[15,118490,118492,118507],{"className":118491,"translate":19},[18],[15,118493,118495],{"className":118494},[23],[25,118496,118497],{"xmlns":27},[29,118498,118499,118505],{},[32,118500,118501,118503],{},[1209,118502,1211],{},[35,118504,117258],{"mathvariant":138},[45,118506,117261],{"encoding":47},[15,118508,118510],{"className":118509,"ariaHidden":53},[52],[15,118511,118513,118516],{"className":118512},[57],[15,118514],{"className":118515,"style":117271},[61],[15,118517,117275],{"className":118518},[66],[6176,118520,118521],{},[15,118522,118524,118538],{"className":118523,"translate":19},[18],[15,118525,118527],{"className":118526},[23],[25,118528,118529],{"xmlns":27},[29,118530,118531,118536],{},[32,118532,118533],{},[1209,118534,118535],{},"0.167",[45,118537,118535],{"encoding":47},[15,118539,118541],{"className":118540,"ariaHidden":53},[52],[15,118542,118544,118547],{"className":118543},[57],[15,118545],{"className":118546,"style":1836},[61],[15,118548,118535],{"className":118549},[66],[6158,118551,118552,118584],{},[6176,118553,118554],{},[15,118555,118557,118572],{"className":118556,"translate":19},[18],[15,118558,118560],{"className":118559},[23],[25,118561,118562],{"xmlns":27},[29,118563,118564,118570],{},[32,118565,118566,118568],{},[1209,118567,116886],{},[35,118569,117258],{"mathvariant":138},[45,118571,118158],{"encoding":47},[15,118573,118575],{"className":118574,"ariaHidden":53},[52],[15,118576,118578,118581],{"className":118577},[57],[15,118579],{"className":118580,"style":117271},[61],[15,118582,118171],{"className":118583},[66],[6176,118585,118586],{},[15,118587,118589,118603],{"className":118588,"translate":19},[18],[15,118590,118592],{"className":118591},[23],[25,118593,118594],{"xmlns":27},[29,118595,118596,118601],{},[32,118597,118598],{},[1209,118599,118600],{},"0.510",[45,118602,118600],{"encoding":47},[15,118604,118606],{"className":118605,"ariaHidden":53},[52],[15,118607,118609,118612],{"className":118608},[57],[15,118610],{"className":118611,"style":1836},[61],[15,118613,118600],{"className":118614},[66],[6158,118616,118617,118651],{},[6176,118618,118619],{},[15,118620,118622,118638],{"className":118621,"translate":19},[18],[15,118623,118625],{"className":118624},[23],[25,118626,118627],{"xmlns":27},[29,118628,118629,118635],{},[32,118630,118631,118633],{},[1209,118632,107911],{},[35,118634,117258],{"mathvariant":138},[45,118636,118637],{"encoding":47},"10\\%",[15,118639,118641],{"className":118640,"ariaHidden":53},[52],[15,118642,118644,118647],{"className":118643},[57],[15,118645],{"className":118646,"style":117271},[61],[15,118648,118650],{"className":118649},[66],"10%",[6176,118652,118653],{},[15,118654,118656,118670],{"className":118655,"translate":19},[18],[15,118657,118659],{"className":118658},[23],[25,118660,118661],{"xmlns":27},[29,118662,118663,118668],{},[32,118664,118665],{},[1209,118666,118667],{},"0.688",[45,118669,118667],{"encoding":47},[15,118671,118673],{"className":118672,"ariaHidden":53},[52],[15,118674,118676,118679],{"className":118675},[57],[15,118677],{"className":118678,"style":1836},[61],[15,118680,118667],{"className":118681},[66],[11,118683,118684,118685,118713,118714,48306,118816,118918],{},"现在加入代价。设漏诊的代价是误诊的 ",[15,118686,118688,118701],{"className":118687,"translate":19},[18],[15,118689,118691],{"className":118690},[23],[25,118692,118693],{"xmlns":27},[29,118694,118695,118699],{},[32,118696,118697],{},[1209,118698,107911],{},[45,118700,107911],{"encoding":47},[15,118702,118704],{"className":118703,"ariaHidden":53},[52],[15,118705,118707,118710],{"className":118706},[57],[15,118708],{"className":118709,"style":1836},[61],[15,118711,107911],{"className":118712},[66]," 倍，即 ",[15,118715,118717,118743],{"className":118716,"translate":19},[18],[15,118718,118720],{"className":118719},[23],[25,118721,118722],{"xmlns":27},[29,118723,118724,118740],{},[32,118725,118726,118736,118738],{},[1057,118727,118728,118730],{},[35,118729,2256],{},[32,118731,118732,118734],{},[35,118733,74045],{"mathvariant":138},[35,118735,28535],{"mathvariant":138},[39,118737,129],{},[1209,118739,107911],{},[45,118741,118742],{"encoding":47},"c_{\\mathrm{FN}}=10",[15,118744,118746,118807],{"className":118745,"ariaHidden":53},[52],[15,118747,118749,118752,118798,118801,118804],{"className":118748},[57],[15,118750],{"className":118751,"style":10197},[61],[15,118753,118755,118758],{"className":118754},[66],[15,118756,2256],{"className":118757},[66,67],[15,118759,118761],{"className":118760},[198],[15,118762,118764,118790],{"className":118763},[202,1090],[15,118765,118767,118787],{"className":118766},[206],[15,118768,118770],{"className":118769,"style":3638},[210],[15,118771,118772,118775],{"style":1584},[15,118773],{"className":118774,"style":218},[217],[15,118776,118778],{"className":118777},[222,223,224,225],[15,118779,118781],{"className":118780},[66,225],[15,118782,118784],{"className":118783},[66,225],[15,118785,74177],{"className":118786},[66,4429,225],[15,118788,1114],{"className":118789},[1113],[15,118791,118793],{"className":118792},[206],[15,118794,118796],{"className":118795,"style":1121},[210],[15,118797],{},[15,118799],{"className":118800,"style":173},[71],[15,118802,129],{"className":118803},[177],[15,118805],{"className":118806,"style":173},[71],[15,118808,118810,118813],{"className":118809},[57],[15,118811],{"className":118812,"style":1836},[61],[15,118814,107911],{"className":118815},[66],[15,118817,118819,118845],{"className":118818,"translate":19},[18],[15,118820,118822],{"className":118821},[23],[25,118823,118824],{"xmlns":27},[29,118825,118826,118842],{},[32,118827,118828,118838,118840],{},[1057,118829,118830,118832],{},[35,118831,2256],{},[32,118833,118834,118836],{},[35,118835,74045],{"mathvariant":138},[35,118837,28399],{"mathvariant":138},[39,118839,129],{},[1209,118841,1211],{},[45,118843,118844],{"encoding":47},"c_{\\mathrm{FP}}=1",[15,118846,118848,118909],{"className":118847,"ariaHidden":53},[52],[15,118849,118851,118854,118900,118903,118906],{"className":118850},[57],[15,118852],{"className":118853,"style":10197},[61],[15,118855,118857,118860],{"className":118856},[66],[15,118858,2256],{"className":118859},[66,67],[15,118861,118863],{"className":118862},[198],[15,118864,118866,118892],{"className":118865},[202,1090],[15,118867,118869,118889],{"className":118868},[206],[15,118870,118872],{"className":118871,"style":3638},[210],[15,118873,118874,118877],{"style":1584},[15,118875],{"className":118876,"style":218},[217],[15,118878,118880],{"className":118879},[222,223,224,225],[15,118881,118883],{"className":118882},[66,225],[15,118884,118886],{"className":118885},[66,225],[15,118887,74095],{"className":118888},[66,4429,225],[15,118890,1114],{"className":118891},[1113],[15,118893,118895],{"className":118894},[206],[15,118896,118898],{"className":118897,"style":1121},[210],[15,118899],{},[15,118901],{"className":118902,"style":173},[71],[15,118904,129],{"className":118905},[177],[15,118907],{"className":118908,"style":173},[71],[15,118910,118912,118915],{"className":118911},[57],[15,118913],{"className":118914,"style":1836},[61],[15,118916,1211],{"className":118917},[66],"，则治疗阈值是",[15,118920,118922],{"className":118921,"translate":19},[294],[15,118923,118925,118986],{"className":118924,"translate":19},[18],[15,118926,118928],{"className":118927},[23],[25,118929,118930],{"xmlns":27,"display":303},[29,118931,118932,118983],{},[32,118933,118934,118970,118972,118978,118980],{},[6467,118935,118936,118946],{},[1057,118937,118938,118940],{},[35,118939,2256],{},[32,118941,118942,118944],{},[35,118943,74045],{"mathvariant":138},[35,118945,28399],{"mathvariant":138},[32,118947,118948,118958,118960],{},[1057,118949,118950,118952],{},[35,118951,2256],{},[32,118953,118954,118956],{},[35,118955,74045],{"mathvariant":138},[35,118957,28399],{"mathvariant":138},[39,118959,1245],{},[1057,118961,118962,118964],{},[35,118963,2256],{},[32,118965,118966,118968],{},[35,118967,74045],{"mathvariant":138},[35,118969,28535],{"mathvariant":138},[39,118971,129],{},[6467,118973,118974,118976],{},[1209,118975,1211],{},[1209,118977,75074],{},[39,118979,31993],{},[1209,118981,118982],{},"0.091.",[45,118984,118985],{"encoding":47},"\\frac{c_{\\mathrm{FP}}}{c_{\\mathrm{FP}}+c_{\\mathrm{FN}}}\n=\\frac{1}{11}\\approx0.091.",[15,118987,118989,119207,119284],{"className":118988,"ariaHidden":53},[52],[15,118990,118992,118995,119198,119201,119204],{"className":118991},[57],[15,118993],{"className":118994,"style":52422},[61],[15,118996,118998,119001,119195],{"className":118997},[66],[15,118999],{"className":119000},[162,23417],[15,119002,119004],{"className":119003},[6467],[15,119005,119007,119187],{"className":119006},[202,109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",[15,119297,119299,119314],{"className":119298,"translate":19},[18],[15,119300,119302],{"className":119301},[23],[25,119303,119304],{"xmlns":27},[29,119305,119306,119312],{},[32,119307,119308,119310],{},[1209,119309,1211],{},[35,119311,117258],{"mathvariant":138},[45,119313,117261],{"encoding":47},[15,119315,119317],{"className":119316,"ariaHidden":53},[52],[15,119318,119320,119323],{"className":119319},[57],[15,119321],{"className":119322,"style":117271},[61],[15,119324,117275],{"className":119325},[66]," 时，",[15,119328,119330,119363],{"className":119329,"translate":19},[18],[15,119331,119333],{"className":119332},[23],[25,119334,119335],{"xmlns":27},[29,119336,119337,119360],{},[32,119338,119339,119341,119343,119345,119347,119349,119351,119353,119355,119357],{},[35,119340,28399],{},[39,119342,120],{"stretchy":119},[35,119344,32458],{},[39,119346,34430],{},[39,119348,1245],{},[39,119350,126],{"stretchy":119},[39,119352,31993],{},[1209,119354,118535],{},[39,119356,9283],{},[1209,119358,119359],{},"0.091",[45,119361,119362],{"encoding":47},"P(D\\mid +)\\approx0.167>0.091",[15,119364,119366,119390,119411,119429],{"className":119365,"ariaHidden":53},[52],[15,119367,119369,119372,119375,119378,119381,119384,119387],{"className":119368},[57],[15,119370],{"className":119371,"style":154},[61],[15,119373,28399],{"className":119374,"style":62513},[66,67],[15,119376,120],{"className":119377},[162],[15,119379,32458],{"className":119380,"style":23025},[66,67],[15,119382],{"className":119383,"style":173},[71],[15,119385,34430],{"className":119386},[177],[15,119388],{"className":119389,"style":173},[71],[15,119391,119393,119396,119399,119402,119405,119408],{"className":119392},[57],[15,119394],{"className":119395,"style":154},[61],[15,119397,1245],{"className":119398},[66],[15,119400,126],{"className":119401},[169],[15,119403],{"className":119404,"style":173},[71],[15,119406,31993],{"className":119407},[177],[15,119409],{"className":119410,"style":173},[71],[15,119412,119414,119417,119420,119423,119426],{"className":119413},[57],[15,119415],{"className":119416,"style":114174},[61],[15,119418,118535],{"className":119419},[66],[15,119421],{"className":119422,"style":173},[71],[15,119424,9283],{"className":119425},[177],[15,119427],{"className":119428,"style":173},[71],[15,119430,119432,119435],{"className":119431},[57],[15,119433],{"className":119434,"style":1836},[61],[15,119436,119359],{"className":119437},[66],"，应当治疗；在患病率 ",[15,119440,119442,119457],{"className":119441,"translate":19},[18],[15,119443,119445],{"className":119444},[23],[25,119446,119447],{"xmlns":27},[29,119448,119449,119455],{},[32,119450,119451,119453],{},[1209,119452,112950],{},[35,119454,117258],{"mathvariant":138},[45,119456,118440],{"encoding":47},[15,119458,119460],{"className":119459,"ariaHidden":53},[52],[15,119461,119463,119466],{"className":119462},[57],[15,119464],{"className":119465,"style":117271},[61],[15,119467,118453],{"className":119468},[66],[15,119470,119472,119500],{"className":119471,"translate":19},[18],[15,119473,119475],{"className":119474},[23],[25,119476,119477],{"xmlns":27},[29,119478,119479,119497],{},[32,119480,119481,119483,119485,119487,119489,119491,119493,119495],{},[35,119482,28399],{},[39,119484,120],{"stretchy":119},[35,119486,32458],{},[39,119488,34430],{},[39,119490,1245],{},[39,119492,126],{"stretchy":119},[39,119494,31993],{},[1209,119496,118470],{},[45,119498,119499],{"encoding":47},"P(D\\mid +)\\approx0.090",[15,119501,119503,119527,119548],{"className":119502,"ariaHidden":53},[52],[15,119504,119506,119509,119512,119515,119518,119521,119524],{"className":119505},[57],[15,119507],{"className":119508,"style":154},[61],[15,119510,28399],{"className":119511,"style":62513},[66,67],[15,119513,120],{"className":119514},[162],[15,119516,32458],{"className":119517,"style":23025},[66,67],[15,119519],{"className":119520,"style":173},[71],[15,119522,34430],{"className":119523},[177],[15,119525],{"className":119526,"style":173},[71],[15,119528,119530,119533,119536,119539,119542,119545],{"className":119529},[57],[15,119531],{"className":119532,"style":154},[61],[15,119534,1245],{"className":119535},[66],[15,119537,126],{"className":119538},[169],[15,119540],{"className":119541,"style":173},[71],[15,119543,31993],{"className":119544},[177],[15,119546],{"className":119547,"style":173},[71],[15,119549,119551,119554],{"className":119550},[57],[15,119552],{"className":119553,"style":1836},[61],[15,119555,118470],{"className":119556},[66],"，已经落到阈值以下，治疗与不治疗的期望代价几乎相同。同一个阳性结果，在不同的先验与代价下导向不同的行动，这不是检测本身变了，而是决策问题变了。",[11,119559,119560],{},"这个例子也说明了为什么「准确率」不是一个好的汇报指标。检测的准确率是",[15,119562,119564],{"className":119563,"translate":19},[294],[15,119565,119567,119601],{"className":119566,"translate":19},[18],[15,119568,119570],{"className":119569},[23],[25,119571,119572],{"xmlns":27,"display":303},[29,119573,119574,119598],{},[32,119575,119576,119578,119580,119582,119584,119587,119589,119591,119593,119596],{},[1209,119577,117490],{},[39,119579,41],{},[1209,119581,117495],{},[39,119583,1245],{},[1209,119585,119586],{},"0.95",[39,119588,41],{},[1209,119590,117490],{},[39,119592,31993],{},[1209,119594,119595],{},"0.9504",[39,119597,335],{"separator":53},[45,119599,119600],{"encoding":47},"0.99\\times0.01+0.95\\times0.99\\approx0.9504,",[15,119602,119604,119622,119640,119658,119676],{"className":119603,"ariaHidden":53},[52],[15,119605,119607,119610,119613,119616,119619],{"className":119606},[57],[15,119608],{"className":119609,"style":30468},[61],[15,119611,117490],{"className":119612},[66],[15,119614],{"className":119615,"style":72},[71],[15,119617,41],{"className":119618},[76],[15,119620],{"className":119621,"style":72},[71],[15,119623,119625,119628,119631,119634,119637],{"className":119624},[57],[15,119626],{"className":119627,"style":30468},[61],[15,119629,117495],{"className":119630},[66],[15,119632],{"className":119633,"style":72},[71],[15,119635,1245],{"className":119636},[76],[15,119638],{"className":119639,"style":72},[71],[15,119641,119643,119646,119649,119652,119655],{"className":119642},[57],[15,119644],{"className":119645,"style":30468},[61],[15,119647,119586],{"className":119648},[66],[15,119650],{"className":119651,"style":72},[71],[15,119653,41],{"className":119654},[76],[15,119656],{"className":119657,"style":72},[71],[15,119659,119661,119664,119667,119670,119673],{"className":119660},[57],[15,119662],{"className":119663,"style":1836},[61],[15,119665,117490],{"className":119666},[66],[15,119668],{"className":119669,"style":173},[71],[15,119671,31993],{"className":119672},[177],[15,119674],{"className":119675,"style":173},[71],[15,119677,119679,119682,119685],{"className":119678},[57],[15,119680],{"className":119681,"style":112825},[61],[15,119683,119595],{"className":119684},[66],[15,119686,335],{"className":119687},[422],[11,119689,119690],{},"看起来很高，但它几乎完全由健康人贡献；在真正需要决策的那一小部分人身上，后验概率才是关键。",[1904,119692,119694],{"id":119693},"_84-高斯两类问题","8.4 高斯两类问题",[11,119696,119697,119698,119779,119780,48366,119808,119836,119837,119888],{},"设两类共享方差 ",[15,119699,119701,119723],{"className":119700,"translate":19},[18],[15,119702,119704],{"className":119703},[23],[25,119705,119706],{"xmlns":27},[29,119707,119708,119720],{},[32,119709,119710,119716,119718],{},[131,119711,119712,119714],{},[35,119713,6634],{},[1209,119715,1263],{},[39,119717,129],{},[1209,119719,1211],{},[45,119721,119722],{"encoding":47},"\\sigma^2=1",[15,119724,119726,119770],{"className":119725,"ariaHidden":53},[52],[15,119727,119729,119732,119761,119764,119767],{"className":119728},[57],[15,119730],{"className":119731,"style":6740},[61],[15,119733,119735,119738],{"className":119734},[66],[15,119736,6634],{"className":119737,"style":1515},[66,67],[15,119739,119741],{"className":119740},[198],[15,119742,119744],{"className":119743},[202],[15,119745,119747],{"className":119746},[206],[15,119748,119750],{"className":119749,"style":6740},[210],[15,119751,119752,119755],{"style":213},[15,119753],{"className":119754,"style":218},[217],[15,119756,119758],{"className":119757},[222,223,224,225],[15,119759,1263],{"className":119760},[66,225],[15,119762],{"className":119763,"style":173},[71],[15,119765,129],{"className":119766},[177],[15,119768],{"className":119769,"style":173},[71],[15,119771,119773,119776],{"className":119772},[57],[15,119774],{"className":119775,"style":1836},[61],[15,119777,1211],{"className":119778},[66],"，均值分别为 ",[15,119781,119783,119796],{"className":119782,"translate":19},[18],[15,119784,119786],{"className":119785},[23],[25,119787,119788],{"xmlns":27},[29,119789,119790,119794],{},[32,119791,119792],{},[1209,119793,1804],{},[45,119795,1804],{"encoding":47},[15,119797,119799],{"className":119798,"ariaHidden":53},[52],[15,119800,119802,119805],{"className":119801},[57],[15,119803],{"className":119804,"style":1836},[61],[15,119806,1804],{"className":119807},[66],[15,119809,119811,119824],{"className":119810,"translate":19},[18],[15,119812,119814],{"className":119813},[23],[25,119815,119816],{"xmlns":27},[29,119817,119818,119822],{},[32,119819,119820],{},[1209,119821,1263],{},[45,119823,1263],{"encoding":47},[15,119825,119827],{"className":119826,"ariaHidden":53},[52],[15,119828,119830,119833],{"className":119829},[57],[15,119831],{"className":119832,"style":1836},[61],[15,119834,1263],{"className":119835},[66],"，先验相等。则 ",[15,119838,119840,119858],{"className":119839,"translate":19},[18],[15,119841,119843],{"className":119842},[23],[25,119844,119845],{"xmlns":27},[29,119846,119847,119855],{},[32,119848,119849,119851,119853],{},[35,119850,84668],{"mathvariant":138},[39,119852,129],{},[1209,119854,1263],{},[45,119856,119857],{"encoding":47},"\\Delta=2",[15,119859,119861,119879],{"className":119860,"ariaHidden":53},[52],[15,119862,119864,119867,119870,119873,119876],{"className":119863},[57],[15,119865],{"className":119866,"style":490},[61],[15,119868,84668],{"className":119869},[66],[15,119871],{"className":119872,"style":173},[71],[15,119874,129],{"className":119875},[177],[15,119877],{"className":119878,"style":173},[71],[15,119880,119882,119885],{"className":119881},[57],[15,119883],{"className":119884,"style":1836},[61],[15,119886,1263],{"className":119887},[66],"，贝叶斯错误率为",[15,119890,119892],{"className":119891,"translate":19},[294],[15,119893,119895,119930],{"className":119894,"translate":19},[18],[15,119896,119898],{"className":119897},[23],[25,119899,119900],{"xmlns":27,"display":303},[29,119901,119902,119927],{},[32,119903,119904,119912,119914,119916,119918,119920,119922,119924],{},[1255,119905,119906,119908,119910],{},[35,119907,11],{},[35,119909,48383],{},[39,119911,1162],{},[39,119913,129],{},[35,119915,83348],{},[39,119917,120],{"stretchy":119},[1209,119919,1211],{},[39,119921,126],{"stretchy":119},[39,119923,31993],{},[1209,119925,119926],{},"0.1587.",[45,119928,119929],{"encoding":47},"p_e^*=Q(1)\\approx0.1587.",[15,119931,119933,119999,120026],{"className":119932,"ariaHidden":53},[52],[15,119934,119936,119939,119990,119993,119996],{"className":119935},[57],[15,119937],{"className":119938,"style":82023},[61],[15,119940,119942,119945],{"className":119941},[66],[15,119943,11],{"className":119944},[66,67],[15,119946,119948],{"className":119947},[198],[15,119949,119951,119982],{"className":119950},[202,1090],[15,119952,119954,119979],{"className":119953},[206],[15,119955,119957,119968],{"className":119956,"style":1308},[210],[15,119958,119959,119962],{"style":1653},[15,119960],{"className":119961,"style":218},[217],[15,119963,119965],{"className":119964},[222,223,224,225],[15,119966,48383],{"className":119967},[66,67,225],[15,119969,119970,119973],{"style":397},[15,119971],{"className":119972,"style":218},[217],[15,119974,119976],{"className":119975},[222,223,224,225],[15,119977,1162],{"className":119978},[76,225],[15,119980,1114],{"className":119981},[1113],[15,119983,119985],{"className":119984},[206],[15,119986,119988],{"className":119987,"style":1686},[210],[15,119989],{},[15,119991],{"className":119992,"style":173},[71],[15,119994,129],{"className":119995},[177],[15,119997],{"className":119998,"style":173},[71],[15,120000,120002,120005,120008,120011,120014,120017,120020,120023],{"className":120001},[57],[15,120003],{"className":120004,"style":154},[61],[15,120006,83348],{"className":120007},[66,67],[15,120009,120],{"className":120010},[162],[15,120012,1211],{"className":120013},[66],[15,120015,126],{"className":120016},[169],[15,120018],{"className":120019,"style":173},[71],[15,120021,31993],{"className":120022},[177],[15,120024],{"className":120025,"style":173},[71],[15,120027,120029,120032],{"className":120028},[57],[15,120030],{"className":120031,"style":1836},[61],[15,120033,119926],{"className":120034},[66],[11,120036,120037,120038,3249,120066,120117,120118,120184,120185,120291],{},"若把均值差扩大到 ",[15,120039,120041,120054],{"className":120040,"translate":19},[18],[15,120042,120044],{"className":120043},[23],[25,120045,120046],{"xmlns":27},[29,120047,120048,120052],{},[32,120049,120050],{},[1209,120051,15855],{},[45,120053,15855],{"encoding":47},[15,120055,120057],{"className":120056,"ariaHidden":53},[52],[15,120058,120060,120063],{"className":120059},[57],[15,120061],{"className":120062,"style":1836},[61],[15,120064,15855],{"className":120065},[66],[15,120067,120069,120087],{"className":120068,"translate":19},[18],[15,120070,120072],{"className":120071},[23],[25,120073,120074],{"xmlns":27},[29,120075,120076,120084],{},[32,120077,120078,120080,120082],{},[35,120079,84668],{"mathvariant":138},[39,120081,129],{},[1209,120083,15855],{},[45,120085,120086],{"encoding":47},"\\Delta=4",[15,120088,120090,120108],{"className":120089,"ariaHidden":53},[52],[15,120091,120093,120096,120099,120102,120105],{"className":120092},[57],[15,120094],{"className":120095,"style":490},[61],[15,120097,84668],{"className":120098},[66],[15,120100],{"className":120101,"style":173},[71],[15,120103,129],{"className":120104},[177],[15,120106],{"className":120107,"style":173},[71],[15,120109,120111,120114],{"className":120110},[57],[15,120112],{"className":120113,"style":1836},[61],[15,120115,15855],{"className":120116},[66],"，错误率降为 ",[15,120119,120121,120145],{"className":120120,"translate":19},[18],[15,120122,120124],{"className":120123},[23],[25,120125,120126],{"xmlns":27},[29,120127,120128,120142],{},[32,120129,120130,120132,120134,120136,120138,120140],{},[35,120131,83348],{},[39,120133,120],{"stretchy":119},[1209,120135,1263],{},[39,120137,126],{"stretchy":119},[39,120139,31993],{},[1209,120141,86222],{},[45,120143,120144],{"encoding":47},"Q(2)\\approx0.0228",[15,120146,120148,120175],{"className":120147,"ariaHidden":53},[52],[15,120149,120151,120154,120157,120160,120163,120166,120169,120172],{"className":120150},[57],[15,120152],{"className":120153,"style":154},[61],[15,120155,83348],{"className":120156},[66,67],[15,120158,120],{"className":120159},[162],[15,120161,1263],{"className":120162},[66],[15,120164,126],{"className":120165},[169],[15,120167],{"className":120168,"style":173},[71],[15,120170,31993],{"className":120171},[177],[15,120173],{"className":120174,"style":173},[71],[15,120176,120178,120181],{"className":120177},[57],[15,120179],{"className":120180,"style":1836},[61],[15,120182,86222],{"className":120183},[66],"。错误率随马氏距离以 ",[15,120186,120188,120218],{"className":120187,"translate":19},[18],[15,120189,120191],{"className":120190},[23],[25,120192,120193],{"xmlns":27},[29,120194,120195,120215],{},[32,120196,120197],{},[131,120198,120199,120201],{},[35,120200,48383],{},[32,120202,120203,120205,120211,120213],{},[39,120204,6602],{},[131,120206,120207,120209],{},[35,120208,84668],{"mathvariant":138},[1209,120210,1263],{},[35,120212,6625],{"mathvariant":138},[1209,120214,86034],{},[45,120216,120217],{"encoding":47},"e^{-\\Delta^2\u002F8}",[15,120219,120221],{"className":120220,"ariaHidden":53},[52],[15,120222,120224,120227],{"className":120223},[57],[15,120225],{"className":120226,"style":86134},[61],[15,120228,120230,120233],{"className":120229},[66],[15,120231,48383],{"className":120232},[66,67],[15,120234,120236],{"className":120235},[198],[15,120237,120239],{"className":120238},[202],[15,120240,120242],{"className":120241},[206],[15,120243,120245],{"className":120244,"style":86134},[210],[15,120246,120247,120250],{"style":213},[15,120248],{"className":120249,"style":218},[217],[15,120251,120253],{"className":120252},[222,223,224,225],[15,120254,120256,120259,120288],{"className":120255},[66,225],[15,120257,6602],{"className":120258},[66,225],[15,120260,120262,120265],{"className":120261},[66,225],[15,120263,84668],{"className":120264},[66,225],[15,120266,120268],{"className":120267},[198],[15,120269,120271],{"className":120270},[202],[15,120272,120274],{"className":120273},[206],[15,120275,120277],{"className":120276,"style":86167},[210],[15,120278,120279,120282],{"style":86170},[15,120280],{"className":120281,"style":3675},[217],[15,120283,120285],{"className":120284},[222,3679,1508,225],[15,120286,1263],{"className":120287},[66,225],[15,120289,86183],{"className":120290},[66,225]," 的速率衰减，但永远不为零，因为两个高斯的支撑集在整个实轴上重叠。",[11,120293,120294],{},"Bhattacharyya 上界给出",[15,120296,120298],{"className":120297,"translate":19},[294],[15,120299,120301,120349],{"className":120300,"translate":19},[18],[15,120302,120304],{"className":120303},[23],[25,120305,120306],{"xmlns":27,"display":303},[29,120307,120308,120346],{},[32,120309,120310,120318,120320,120326,120344],{},[1255,120311,120312,120314,120316],{},[35,120313,11],{},[35,120315,48383],{},[39,120317,1162],{},[39,120319,24220],{},[6467,120321,120322,120324],{},[1209,120323,1211],{},[1209,120325,1263],{},[131,120327,120328,120330],{},[35,120329,48383],{},[32,120331,120332,120334,120340,120342],{},[39,120333,6602],{},[131,120335,120336,120338],{},[35,120337,84668],{"mathvariant":138},[1209,120339,1263],{},[35,120341,6625],{"mathvariant":138},[1209,120343,86034],{},[39,120345,335],{"separator":53},[45,120347,120348],{"encoding":47},"p_e^*\\le\\frac12e^{-\\Delta^2\u002F8},",[15,120350,120352,120418],{"className":120351,"ariaHidden":53},[52],[15,120353,120355,120358,120409,120412,120415],{"className":120354},[57],[15,120356],{"className":120357,"style":82023},[61],[15,120359,120361,120364],{"className":120360},[66],[15,120362,11],{"className":120363},[66,67],[15,120365,120367],{"className":120366},[198],[15,120368,120370,120401],{"className":120369},[202,1090],[15,120371,120373,120398],{"className":120372},[206],[15,120374,120376,120387],{"className":120375,"style":1308},[210],[15,120377,120378,120381],{"style":1653},[15,120379],{"className":120380,"style":218},[217],[15,120382,120384],{"className":120383},[222,223,224,225],[15,120385,48383],{"className":120386},[66,67,225],[15,120388,120389,120392],{"style":397},[15,120390],{"className":120391,"style":218},[217],[15,120393,120395],{"className":120394},[222,223,224,225],[15,120396,1162],{"className":120397},[76,225],[15,120399,1114],{"className":120400},[1113],[15,120402,120404],{"className":120403},[206],[15,120405,120407],{"className":120406,"style":1686},[210],[15,120408],{},[15,120410],{"className":120411,"style":173},[71],[15,120413,24220],{"className":120414},[177],[15,120416],{"className":120417,"style":173},[71],[15,120419,120421,120424,120486,120550],{"className":120420},[57],[15,120422],{"className":120423,"style":49086},[61],[15,120425,120427,120430,120483],{"className":120426},[66],[15,120428],{"className":120429},[162,23417],[15,120431,120433],{"className":120432},[6467],[15,120434,120436,120475],{"className":120435},[202,1090],[15,120437,120439,120472],{"className":120438},[206],[15,120440,120442,120453,120461],{"className":120441,"style":23430},[210],[15,120443,120444,120447],{"style":23433},[15,120445],{"className":120446,"style":1368},[217],[15,120448,120450],{"className":120449},[66],[15,120451,1263],{"className":120452},[66],[15,120454,120455,120458],{"style":23445},[15,120456],{"className":120457,"style":1368},[217],[15,120459],{"className":120460,"style":4440},[23452],[15,120462,120463,120466],{"style":23455},[15,120464],{"className":120465,"style":1368},[217],[15,120467,120469],{"className":120468},[66],[15,120470,1211],{"className":120471},[66],[15,120473,1114],{"className":120474},[1113],[15,120476,120478],{"className":120477},[206],[15,120479,120481],{"className":120480,"style":23474},[210],[15,120482],{},[15,120484],{"className":120485},[169,23417],[15,120487,120489,120492],{"className":120488},[66],[15,120490,48383],{"className":120491},[66,67],[15,120493,120495],{"className":120494},[198],[15,120496,120498],{"className":120497},[202],[15,120499,120501],{"className":120500},[206],[15,120502,120504],{"className":120503,"style":87415},[210],[15,120505,120506,120509],{"style":397},[15,120507],{"className":120508,"style":218},[217],[15,120510,120512],{"className":120511},[222,223,224,225],[15,120513,120515,120518,120547],{"className":120514},[66,225],[15,120516,6602],{"className":120517},[66,225],[15,120519,120521,120524],{"className":120520},[66,225],[15,120522,84668],{"className":120523},[66,225],[15,120525,120527],{"className":120526},[198],[15,120528,120530],{"className":120529},[202],[15,120531,120533],{"className":120532},[206],[15,120534,120536],{"className":120535,"style":86167},[210],[15,120537,120538,120541],{"style":86170},[15,120539],{"className":120540,"style":3675},[217],[15,120542,120544],{"className":120543},[222,3679,1508,225],[15,120545,1263],{"className":120546},[66,225],[15,120548,86183],{"className":120549},[66,225],[15,120551,335],{"className":120552},[422],[11,120554,19359,120555,120605,120606,120635,120636,120605,120686,120714],{},[15,120556,120558,120575],{"className":120557,"translate":19},[18],[15,120559,120561],{"className":120560},[23],[25,120562,120563],{"xmlns":27},[29,120564,120565,120573],{},[32,120566,120567,120569,120571],{},[35,120568,84668],{"mathvariant":138},[39,120570,129],{},[1209,120572,1263],{},[45,120574,119857],{"encoding":47},[15,120576,120578,120596],{"className":120577,"ariaHidden":53},[52],[15,120579,120581,120584,120587,120590,120593],{"className":120580},[57],[15,120582],{"className":120583,"style":490},[61],[15,120585,84668],{"className":120586},[66],[15,120588],{"className":120589,"style":173},[71],[15,120591,129],{"className":120592},[177],[15,120594],{"className":120595,"style":173},[71],[15,120597,120599,120602],{"className":120598},[57],[15,120600],{"className":120601,"style":1836},[61],[15,120603,1263],{"className":120604},[66]," 时为 ",[15,120607,120609,120623],{"className":120608,"translate":19},[18],[15,120610,120612],{"className":120611},[23],[25,120613,120614],{"xmlns":27},[29,120615,120616,120621],{},[32,120617,120618],{},[1209,120619,120620],{},"0.303",[45,120622,120620],{"encoding":47},[15,120624,120626],{"className":120625,"ariaHidden":53},[52],[15,120627,120629,120632],{"className":120628},[57],[15,120630],{"className":120631,"style":1836},[61],[15,120633,120620],{"className":120634},[66],"，在 ",[15,120637,120639,120656],{"className":120638,"translate":19},[18],[15,120640,120642],{"className":120641},[23],[25,120643,120644],{"xmlns":27},[29,120645,120646,120654],{},[32,120647,120648,120650,120652],{},[35,120649,84668],{"mathvariant":138},[39,120651,129],{},[1209,120653,15855],{},[45,120655,120086],{"encoding":47},[15,120657,120659,120677],{"className":120658,"ariaHidden":53},[52],[15,120660,120662,120665,120668,120671,120674],{"className":120661},[57],[15,120663],{"className":120664,"style":490},[61],[15,120666,84668],{"className":120667},[66],[15,120669],{"className":120670,"style":173},[71],[15,120672,129],{"className":120673},[177],[15,120675],{"className":120676,"style":173},[71],[15,120678,120680,120683],{"className":120679},[57],[15,120681],{"className":120682,"style":1836},[61],[15,120684,15855],{"className":120685},[66],[15,120687,120689,120702],{"className":120688,"translate":19},[18],[15,120690,120692],{"className":120691},[23],[25,120693,120694],{"xmlns":27},[29,120695,120696,120700],{},[32,120697,120698],{},[1209,120699,86225],{},[45,120701,86225],{"encoding":47},[15,120703,120705],{"className":120704,"ariaHidden":53},[52],[15,120706,120708,120711],{"className":120707},[57],[15,120709],{"className":120710,"style":1836},[61],[15,120712,86225],{"className":120713},[66],"。上界不紧，但它的指数部分与真实错误率一致；这正是我们在 4.2 节强调的「速率」而非「常数」。",[95,120716,120717],{"id":120717},"结语",[11,120719,120720],{},"把全文的线索压缩成一行：",[15,120722,120724],{"className":120723,"translate":19},[294],[15,120725,120727,120893],{"className":120726,"translate":19},[18],[15,120728,120730],{"className":120729},[23],[25,120731,120732],{"xmlns":27,"display":303},[29,120733,120734,120890],{},[32,120735,120736,120754,120756,120778,120780,120782,120785,120787,120809,120811,120813,120815,120817,120865,120867,120869,120871,120873,120888],{},[1172,120737,120738,120752],{},[1172,120739,120740,120750],{},[32,120741,120742,120744,120746,120748],{},[35,120743,11],{},[39,120745,120],{"stretchy":119},[35,120747,61618],{},[39,120749,126],{"stretchy":119},[39,120751,69846],{"stretchy":53},[1274,120753,65218],{},[39,120755,41],{},[1172,120757,120758,120776],{},[1172,120759,120760,120774],{},[32,120761,120762,120764,120766,120768,120770,120772],{},[35,120763,11],{},[39,120765,120],{"stretchy":119},[35,120767,32458],{},[39,120769,34430],{},[35,120771,61618],{},[39,120773,126],{"stretchy":119},[39,120775,69846],{"stretchy":53},[1274,120777,65272],{},[1274,120779,24233],{},[71,120781],{"linebreak":10456},[39,120783,120784],{},"⟶",[1274,120786,24233],{},[1172,120788,120789,120807],{},[1172,120790,120791,120805],{},[32,120792,120793,120795,120797,120799,120801,120803],{},[35,120794,11],{},[39,120796,120],{"stretchy":119},[35,120798,61618],{},[39,120800,34430],{},[35,120802,32458],{},[39,120804,126],{"stretchy":119},[39,120806,69846],{"stretchy":53},[1274,120808,65431],{},[1274,120810,24233],{},[71,120812],{"linebreak":10456},[39,120814,120784],{},[1274,120816,24233],{},[1172,120818,120819,120863],{},[1172,120820,120821,120861],{},[32,120822,120823,120825,120827,120837,120839,120841,120843,120845,120847,120849,120851,120853,120855,120857,120859],{},[35,120824,1167],{},[39,120826,1170],{},[1172,120828,120829,120835],{},[32,120830,120831,120833],{},[35,120832,1178],{},[39,120834,1170],{},[35,120836,4556],{},[1274,120838,24233],{},[35,120840,62381],{"mathvariant":454},[39,120842,1194],{"stretchy":119},[35,120844,1216],{},[39,120846,120],{"stretchy":119},[35,120848,61885],{},[39,120850,335],{"separator":53},[35,120852,4556],{},[39,120854,126],{"stretchy":119},[39,120856,34430],{},[35,120858,32458],{},[39,120860,1266],{"stretchy":119},[39,120862,69846],{"stretchy":53},[1274,120864,61986],{},[1274,120866,24233],{},[71,120868],{"linebreak":10456},[39,120870,120784],{},[1274,120872,24233],{},[1172,120874,120875,120885],{},[1172,120876,120877,120883],{},[131,120878,120879,120881],{},[35,120880,455],{},[39,120882,1162],{},[39,120884,69846],{"stretchy":53},[1274,120886,120887],{},"下限",[35,120889,3573],{"mathvariant":138},[45,120891,120892],{"encoding":47},"\\underbrace{p(\\theta)}_{\\text{先验}}\n\\times\n\\underbrace{p(D\\mid\\theta)}_{\\text{似然}}\n\\ \\\\ \\longrightarrow\\\n\\underbrace{p(\\theta\\mid D)}_{\\text{后验}}\n\\ \\\\ \\longrightarrow\\\n\\underbrace{\\arg\\min_a\\ \\mathbb E[L(Y,a)\\mid D]}_{\\text{行动}}\n\\ \\\\ \\longrightarrow\\\n\\underbrace{R^*}_{\\text{下限}}.",[15,120894,120896,121034,121173,121176,121191,121330,121333,121348,121566,121569,121584],{"className":120895,"ariaHidden":53},[52],[15,120897,120899,120903,121025,121028,121031],{"className":120898},[57],[15,120900],{"className":120901,"style":120902},[61],"height:2.3263em;vertical-align:-1.5763em;",[15,120904,120906],{"className":120905},[1413,1172],[15,120907,120909,121016],{"className":120908},[202,1090],[15,120910,120912,121013],{"className":120911},[206],[15,120913,120915,120933],{"className":120914,"style":70002},[210],[15,120916,120918,120921],{"style":120917},"top:-1.4237em;",[15,120919],{"className":120920,"style":1368},[217],[15,120922,120924],{"className":120923},[222,223,224,225],[15,120925,120927],{"className":120926},[66,225],[15,120928,120930],{"className":120929},[66,1711,225],[15,120931,65218],{"className":120932},[66,68616,225],[15,120934,120935,120938],{"style":1386},[15,120936],{"className":120937,"style":1368},[217],[15,120939,120941],{"className":120940},[1413,1172],[15,120942,120944,121004],{"className":120943},[202,1090],[15,120945,120947,121001],{"className":120946},[206],[15,120948,120950,120981],{"className":120949,"style":70002},[210],[15,120951,120954,120957],{"className":120952,"style":120953},[48564],"top:-2.102em;",[15,120955],{"className":120956,"style":1368},[217],[15,120958,120960,120967,120974],{"className":120959,"style":70049},[70048],[15,120961,120963],{"className":120962,"style":70054},[70053],[34828,120964,120965],{"xmlns":34830,"width":48594,"height":70057,"viewBox":70058,"preserveAspectRatio":48597},[34834,120966],{"d":70061},[15,120968,120970],{"className":120969,"style":70054},[70065],[34828,120971,120972],{"xmlns":34830,"width":48594,"height":70057,"viewBox":70058,"preserveAspectRatio":70068},[34834,120973],{"d":70071},[15,120975,120977],{"className":120976,"style":70054},[70075],[34828,120978,120979],{"xmlns":34830,"width":48594,"height":70057,"viewBox":70058,"preserveAspectRatio":70078},[34834,120980],{"d":70081},[15,120982,120983,120986],{"style":1386},[15,120984],{"className":120985,"style":1368},[217],[15,120987,120989,120992,120995,120998],{"className":120988},[66],[15,120990,11],{"className":120991},[66,67],[15,120993,120],{"className":120994},[162],[15,120996,61618],{"className":120997,"style":23025},[66,67],[15,120999,126],{"className":121000},[169],[15,121002,1114],{"className":121003},[1113],[15,121005,121007],{"className":121006},[206],[15,121008,121011],{"className":121009,"style":121010},[210],"height:0.898em;",[15,121012],{},[15,121014,1114],{"className":121015},[1113],[15,121017,121019],{"className":121018},[206],[15,121020,121023],{"className":121021,"style":121022},[210],"height:1.5763em;",[15,121024],{},[15,121026],{"className":121027,"style":72},[71],[15,121029,41],{"className":121030},[76],[15,121032],{"className":121033,"style":72},[71],[15,121035,121037,121040,121170],{"className":121036},[57],[15,121038],{"className":121039,"style":120902},[61],[15,121041,121043],{"className":121042},[1413,1172],[15,121044,121046,121162],{"className":121045},[202,1090],[15,121047,121049,121159],{"className":121048},[206],[15,121050,121052,121069],{"className":121051,"style":70002},[210],[15,121053,121054,121057],{"style":120917},[15,121055],{"className":121056,"style":1368},[217],[15,121058,121060],{"className":121059},[222,223,224,225],[15,121061,121063],{"className":121062},[66,225],[15,121064,121066],{"className":121065},[66,1711,225],[15,121067,65272],{"className":121068},[66,68616,225],[15,121070,121071,121074],{"style":1386},[15,121072],{"className":121073,"style":1368},[217],[15,121075,121077],{"className":121076},[1413,1172],[15,121078,121080,121151],{"className":121079},[202,1090],[15,121081,121083,121148],{"className":121082},[206],[15,121084,121086,121116],{"className":121085,"style":70002},[210],[15,121087,121089,121092],{"className":121088,"style":120953},[48564],[15,121090],{"className":121091,"style":1368},[217],[15,121093,121095,121102,121109],{"className":121094,"style":70049},[70048],[15,121096,121098],{"className":121097,"style":70054},[70053],[34828,121099,121100],{"xmlns":34830,"width":48594,"height":70057,"viewBox":70058,"preserveAspectRatio":48597},[34834,121101],{"d":70061},[15,121103,121105],{"className":121104,"style":70054},[70065],[34828,121106,121107],{"xmlns":34830,"width":48594,"height":70057,"viewBox":70058,"preserveAspectRatio":70068},[34834,121108],{"d":70071},[15,121110,121112],{"className":121111,"style":70054},[70075],[34828,121113,121114],{"xmlns":34830,"width":48594,"height":70057,"viewBox":70058,"preserveAspectRatio":70078},[34834,121115],{"d":70081},[15,121117,121118,121121],{"style":1386},[15,121119],{"className":121120,"style":1368},[217],[15,121122,121124,121127,121130,121133,121136,121139,121142,121145],{"className":121123},[66],[15,121125,11],{"className":121126},[66,67],[15,121128,120],{"className":121129},[162],[15,121131,32458],{"className":121132,"style"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