[{"data":1,"prerenderedAt":109783},["ShallowReactive",2],{"post-\u002Fblog\u002F2025\u002F2025-09-27-fano-bu-deng-shi-yu-le-cam-fang-fa":3,"i-ci:tag":45589,"i-mingcute:time-line":45589},{"post":4,"nextPost":45600,"prevPost":81708},{"id":5,"title":6,"body":7,"description":13,"draft":45586,"enableComment":45587,"extension":45588,"image":45552,"important":45586,"location":45589,"meta":45590,"navigation":45587,"ogImage":45589,"path":45591,"seo":45592,"stem":45593,"summary":45594,"tags":45595,"time":45598,"__hash__":45599},"blog\u002Fblog\u002F2025\u002F2025-09-27-fano-bu-deng-shi-yu-le-cam-fang-fa.md","Fano 不等式与 Le Cam 方法",{"type":8,"value":9,"toc":45551},"minimark",[10,14,634,637,642,869,872,1289,1296,1303,1306,1310,1315,1540,1845,2209,2956,3052,3441,3506,3891,3895,3959,4302,4305,4490,4497,4501,4742,4981,5144,5147,5151,5155,5158,6378,6381,6585,6804,7270,7273,7937,7940,8860,8945,9062,9905,10111,10364,11102,11105,11738,12517,12521,12524,12879,12882,13319,13322,14069,14373,14412,14416,14643,15322,15663,16245,16323,16327,16359,16706,17376,17466,17470,17474,18289,18292,18708,18830,19166,19169,19543,19547,19550,21518,21521,21788,22028,22616,22732,22988,23224,23714,23746,24288,24841,25216,25219,25880,25883,26576,26782,26786,26793,27297,27391,27788,28092,28096,28100,28132,29922,30131,30617,30967,31186,31189,31697,31885,32340,32343,32347,32495,32836,33281,33284,34078,34114,34118,34195,34198,34543,34693,34761,35045,35427,35811,35814,36190,36331,36967,37073,37105,37217,37219,37742,38058,38063,38237,38244,38557,38818,39014,39316,39708,40041,40747,41077,41459,41600,41603,42556,42668,43002,43128,43424,43427,43707,43711,44757,44760,44764,44950,44953,45090,45093,45287,45290,45358,45448,45451],[11,12,13],"p",{},"统计学的下界证明有一个几乎统一的套路：先把估计问题换成一个检验问题，再证明这个检验问题很难。",[11,15,16,17,268,269,299,300,339,340,375,376,411,412,509,510,604,605,633],{},"设 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N(\\theta,1)",[18,91,94,236],{"className":92,"ariaHidden":51},[93],"katex-html",[18,95,98,103,162,166,171,175,178,181,184,225,229,233],{"className":96},[97],"base",[18,99],{"className":100,"style":102},[101],"strut","height:0.8778em;vertical-align:-0.1944em;",[18,104,107,112],{"className":105},[106],"mord",[18,108,43],{"className":109,"style":111},[106,110],"mathnormal","margin-right:0.0785em;",[18,113,116],{"className":114},[115],"msupsub",[18,117,121,153],{"className":118},[119,120],"vlist-t","vlist-t2",[18,122,125,148],{"className":123},[124],"vlist-r",[18,126,130],{"className":127,"style":129},[128],"vlist","height:0.3011em;",[18,131,133,138],{"style":132},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[18,134],{"className":135,"style":137},[136],"pstrut","height:2.7em;",[18,139,145],{"className":140},[141,142,143,144],"sizing","reset-size6","size3","mtight",[18,146,47],{"className":147},[106,144],[18,149,152],{"className":150},[151],"vlist-s","​",[18,154,156],{"className":155},[124],[18,157,160],{"className":158,"style":159},[128],"height:0.15em;",[18,161],{},[18,163,52],{"className":164},[165],"mpunct",[18,167],{"className":168,"style":170},[169],"mspace","margin-right:0.1667em;",[18,172,55],{"className":173},[174],"minner",[18,176],{"className":177,"style":170},[169],[18,179,52],{"className":180},[165],[18,182],{"className":183,"style":170},[169],[18,185,187,190],{"className":186},[106],[18,188,43],{"className":189,"style":111},[106,110],[18,191,193],{"className":192},[115],[18,194,196,217],{"className":195},[119,120],[18,197,199,214],{"className":198},[124],[18,200,203],{"className":201,"style":202},[128],"height:0.1514em;",[18,204,205,208],{"style":132},[18,206],{"className":207,"style":137},[136],[18,209,211],{"className":210},[141,142,143,144],[18,212,64],{"className":213},[106,110,144],[18,215,152],{"className":216},[151],[18,218,220],{"className":219},[124],[18,221,223],{"className":222,"style":159},[128],[18,224],{},[18,226],{"className":227,"style":228},[169],"margin-right:0.2778em;",[18,230,67],{"className":231},[232],"mrel",[18,234],{"className":235,"style":228},[169],[18,237,239,243,247,251,255,258,261,264],{"className":238},[97],[18,240],{"className":241,"style":242},[101],"height:1em;vertical-align:-0.25em;",[18,244,70],{"className":245,"style":246},[106,110],"margin-right:0.109em;",[18,248,74],{"className":249},[250],"mopen",[18,252,77],{"className":253,"style":254},[106,110],"margin-right:0.0278em;",[18,256,52],{"className":257},[165],[18,259],{"className":260,"style":170},[169],[18,262,47],{"className":263},[106],[18,265,84],{"className":266},[267],"mclose","，要估计 ",[18,270,272,286],{"className":271,"translate":22},[21],[18,273,275],{"className":274},[26],[28,276,277],{"xmlns":30},[32,278,279,283],{},[35,280,281],{},[41,282,77],{},[86,284,285],{"encoding":88},"\\theta",[18,287,289],{"className":288,"ariaHidden":51},[93],[18,290,292,296],{"className":291},[97],[18,293],{"className":294,"style":295},[101],"height:0.6944em;",[18,297,77],{"className":298,"style":254},[106,110],"。样本均值给出风险 ",[18,301,303,323],{"className":302,"translate":22},[21],[18,304,306],{"className":305},[26],[28,307,308],{"xmlns":30},[32,309,310,320],{},[35,311,312,314,318],{},[45,313,47],{},[41,315,317],{"mathvariant":316},"normal","\u002F",[41,319,64],{},[86,321,322],{"encoding":88},"1\u002Fn",[18,324,326],{"className":325,"ariaHidden":51},[93],[18,327,329,332,336],{"className":328},[97],[18,330],{"className":331,"style":242},[101],[18,333,335],{"className":334},[106],"1\u002F",[18,337,64],{"className":338},[106,110],"。这个 ",[18,341,343,360],{"className":342,"translate":22},[21],[18,344,346],{"className":345},[26],[28,347,348],{"xmlns":30},[32,349,350,358],{},[35,351,352,354,356],{},[45,353,47],{},[41,355,317],{"mathvariant":316},[41,357,64],{},[86,359,322],{"encoding":88},[18,361,363],{"className":362,"ariaHidden":51},[93],[18,364,366,369,372],{"className":365},[97],[18,367],{"className":368,"style":242},[101],[18,370,335],{"className":371},[106],[18,373,64],{"className":374},[106,110]," 是不是最优的？也就是说，是否存在某个估计量，能让最坏情况下的风险比 ",[18,377,379,396],{"className":378,"translate":22},[21],[18,380,382],{"className":381},[26],[28,383,384],{"xmlns":30},[32,385,386,394],{},[35,387,388,390,392],{},[45,389,47],{},[41,391,317],{"mathvariant":316},[41,393,64],{},[86,395,322],{"encoding":88},[18,397,399],{"className":398,"ariaHidden":51},[93],[18,400,402,405,408],{"className":401},[97],[18,403],{"className":404,"style":242},[101],[18,406,335],{"className":407},[106],[18,409,64],{"className":410},[106,110]," 更小？Le Cam 的两点方法给出的回答是：不存在。它的论证只用了两个候选参数 ",[18,413,415,439],{"className":414,"translate":22},[21],[18,416,418],{"className":417},[26],[28,419,420],{"xmlns":30},[32,421,422,436],{},[35,423,424,431,434],{},[38,425,426,428],{},[41,427,77],{},[45,429,430],{},"0",[49,432,433],{},"=",[45,435,430],{},[86,437,438],{"encoding":88},"\\theta_0=0",[18,440,442,499],{"className":441,"ariaHidden":51},[93],[18,443,445,449,490,493,496],{"className":444},[97],[18,446],{"className":447,"style":448},[101],"height:0.8444em;vertical-align:-0.15em;",[18,450,452,455],{"className":451},[106],[18,453,77],{"className":454,"style":254},[106,110],[18,456,458],{"className":457},[115],[18,459,461,482],{"className":460},[119,120],[18,462,464,479],{"className":463},[124],[18,465,467],{"className":466,"style":129},[128],[18,468,470,473],{"style":469},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[18,471],{"className":472,"style":137},[136],[18,474,476],{"className":475},[141,142,143,144],[18,477,430],{"className":478},[106,144],[18,480,152],{"className":481},[151],[18,483,485],{"className":484},[124],[18,486,488],{"className":487,"style":159},[128],[18,489],{},[18,491],{"className":492,"style":228},[169],[18,494,433],{"className":495},[232],[18,497],{"className":498,"style":228},[169],[18,500,502,506],{"className":501},[97],[18,503],{"className":504,"style":505},[101],"height:0.6444em;",[18,507,430],{"className":508},[106]," 与 ",[18,511,513,536],{"className":512,"translate":22},[21],[18,514,516],{"className":515},[26],[28,517,518],{"xmlns":30},[32,519,520,533],{},[35,521,522,528,530],{},[38,523,524,526],{},[41,525,77],{},[45,527,47],{},[49,529,433],{},[41,531,532],{"mathvariant":316},"Δ",[86,534,535],{"encoding":88},"\\theta_1=\\Delta",[18,537,539,594],{"className":538,"ariaHidden":51},[93],[18,540,542,545,585,588,591],{"className":541},[97],[18,543],{"className":544,"style":448},[101],[18,546,548,551],{"className":547},[106],[18,549,77],{"className":550,"style":254},[106,110],[18,552,554],{"className":553},[115],[18,555,557,577],{"className":556},[119,120],[18,558,560,574],{"className":559},[124],[18,561,563],{"className":562,"style":129},[128],[18,564,565,568],{"style":469},[18,566],{"className":567,"style":137},[136],[18,569,571],{"className":570},[141,142,143,144],[18,572,47],{"className":573},[106,144],[18,575,152],{"className":576},[151],[18,578,580],{"className":579},[124],[18,581,583],{"className":582,"style":159},[128],[18,584],{},[18,586],{"className":587,"style":228},[169],[18,589,433],{"className":590},[232],[18,592],{"className":593,"style":228},[169],[18,595,597,601],{"className":596},[97],[18,598],{"className":599,"style":600},[101],"height:0.6833em;",[18,602,532],{"className":603},[106],"——如果连这两个点都分不清，那就更不可能把 ",[18,606,608,621],{"className":607,"translate":22},[21],[18,609,611],{"className":610},[26],[28,612,613],{"xmlns":30},[32,614,615,619],{},[35,616,617],{},[41,618,77],{},[86,620,285],{"encoding":88},[18,622,624],{"className":623,"ariaHidden":51},[93],[18,625,627,630],{"className":626},[97],[18,628],{"className":629,"style":295},[101],[18,631,77],{"className":632,"style":254},[106,110]," 估准。",[11,635,636],{},"本文介绍这套语言里最常用的三个工具：Le Cam 的两点方法、Fano 不等式，以及 Assouad 的超立方体引理。三者的共同点是：先选一个有限的「困难子问题」，把估计下界转化为检验下界，再用总变差、KL 散度或互信息来度量「有多难分」。",[638,639,641],"h2",{"id":640},"一极小极大下界在说什么","一、极小极大：下界在说什么",[11,643,644,645,715,716,720,721,749,750,810,811,839,840,868],{},"一个估计量 ",[18,646,648,668],{"className":647,"translate":22},[21],[18,649,651],{"className":650},[26],[28,652,653],{"xmlns":30},[32,654,655,665],{},[35,656,657],{},[658,659,660,662],"mover",{"accent":51},[41,661,77],{},[49,663,664],{},"^",[86,666,667],{"encoding":88},"\\hat\\theta",[18,669,671],{"className":670,"ariaHidden":51},[93],[18,672,674,678],{"className":673},[97],[18,675],{"className":676,"style":677},[101],"height:0.9579em;",[18,679,682],{"className":680},[106,681],"accent",[18,683,685],{"className":684},[119],[18,686,688],{"className":687},[124],[18,689,691,701],{"className":690,"style":677},[128],[18,692,694,698],{"style":693},"top:-3em;",[18,695],{"className":696,"style":697},[136],"height:3em;",[18,699,77],{"className":700,"style":254},[106,110],[18,702,704,707],{"style":703},"top:-3.2634em;",[18,705],{"className":706,"style":697},[136],[18,708,712],{"className":709,"style":711},[710],"accent-body","left:-0.1667em;",[18,713,664],{"className":714},[106]," 的优劣，通常用它的",[717,718,719],"strong",{},"风险","衡量：在真实参数为 ",[18,722,724,737],{"className":723,"translate":22},[21],[18,725,727],{"className":726},[26],[28,728,729],{"xmlns":30},[32,730,731,735],{},[35,732,733],{},[41,734,77],{},[86,736,285],{"encoding":88},[18,738,740],{"className":739,"ariaHidden":51},[93],[18,741,743,746],{"className":742},[97],[18,744],{"className":745,"style":295},[101],[18,747,77],{"className":748,"style":254},[106,110]," 时，",[18,751,753,770],{"className":752,"translate":22},[21],[18,754,756],{"className":755},[26],[28,757,758],{"xmlns":30},[32,759,760,768],{},[35,761,762],{},[658,763,764,766],{"accent":51},[41,765,77],{},[49,767,664],{},[86,769,667],{"encoding":88},[18,771,773],{"className":772,"ariaHidden":51},[93],[18,774,776,779],{"className":775},[97],[18,777],{"className":778,"style":677},[101],[18,780,782],{"className":781},[106,681],[18,783,785],{"className":784},[119],[18,786,788],{"className":787},[124],[18,789,791,799],{"className":790,"style":677},[128],[18,792,793,796],{"style":693},[18,794],{"className":795,"style":697},[136],[18,797,77],{"className":798,"style":254},[106,110],[18,800,801,804],{"style":703},[18,802],{"className":803,"style":697},[136],[18,805,807],{"className":806,"style":711},[710],[18,808,664],{"className":809},[106]," 的平均损失。但真实参数是未知的，所以我们不能只关心某一个 ",[18,812,814,827],{"className":813,"translate":22},[21],[18,815,817],{"className":816},[26],[28,818,819],{"xmlns":30},[32,820,821,825],{},[35,822,823],{},[41,824,77],{},[86,826,285],{"encoding":88},[18,828,830],{"className":829,"ariaHidden":51},[93],[18,831,833,836],{"className":832},[97],[18,834],{"className":835,"style":295},[101],[18,837,77],{"className":838,"style":254},[106,110]," 下的风险。极小极大（minimax）的立场是：先让大自然挑一个最不利的 ",[18,841,843,856],{"className":842,"translate":22},[21],[18,844,846],{"className":845},[26],[28,847,848],{"xmlns":30},[32,849,850,854],{},[35,851,852],{},[41,853,77],{},[86,855,285],{"encoding":88},[18,857,859],{"className":858,"ariaHidden":51},[93],[18,860,862,865],{"className":861},[97],[18,863],{"className":864,"style":295},[101],[18,866,77],{"className":867,"style":254},[106,110],"，再让统计学家挑估计量，看统计学家能保证什么。",[11,870,871],{},"用博弈的话说，极小极大风险是",[18,873,876],{"className":874,"translate":22},[875],"katex-display",[18,877,879,972],{"className":878,"translate":22},[21],[18,880,882],{"className":881},[26],[28,883,885],{"xmlns":30,"display":884},"block",[32,886,887,969],{},[35,888,889,906,925,929,942,946,949,951,953,955,961,963,966],{},[890,891,892,900],"munder",{},[35,893,894,897],{},[41,895,896],{},"inf",[49,898,899],{},"⁡",[658,901,902,904],{"accent":51},[41,903,77],{},[49,905,664],{},[890,907,908,915],{},[35,909,910,913],{},[41,911,912],{},"sup",[49,914,899],{},[35,916,917,919,922],{},[41,918,77],{},[49,920,921],{},"∈",[41,923,924],{"mathvariant":316},"Θ",[926,927,928],"mtext",{}," ",[38,930,931,935],{},[41,932,934],{"mathvariant":933},"double-struck","E",[38,936,937,940],{},[41,938,939],{},"P",[41,941,77],{},[49,943,945],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},"1.2em","[",[41,947,948],{"mathvariant":316},"ℓ",[49,950,74],{"stretchy":73},[41,952,77],{},[49,954,52],{"separator":51},[658,956,957,959],{"accent":51},[41,958,77],{},[49,960,664],{},[49,962,84],{"stretchy":73},[49,964,965],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},"]",[41,967,968],{"mathvariant":316},".",[86,970,971],{"encoding":88},"\\inf_{\\hat\\theta}\\sup_{\\theta\\in\\Theta}\\ \\mathbb E_{P_\\theta}\\bigl[\\ell(\\theta,\\hat\\theta)\\bigr].",[18,973,975],{"className":974,"ariaHidden":51},[93],[18,976,978,982,1068,1071,1127,1130,1133,1224,1231,1234,1237,1240,1243,1246,1277,1280,1286],{"className":977},[97],[18,979],{"className":980,"style":981},[101],"height:1.9318em;vertical-align:-0.9739em;",[18,983,987],{"className":984},[985,986],"mop","op-limits",[18,988,990,1059],{"className":989},[119,120],[18,991,993,1056],{"className":992},[124],[18,994,996,1041],{"className":995,"style":295},[128],[18,997,999,1002],{"style":998},"top:-2.1635em;margin-left:0em;",[18,1000],{"className":1001,"style":697},[136],[18,1003,1005],{"className":1004},[141,142,143,144],[18,1006,1008],{"className":1007},[106,144],[18,1009,1011],{"className":1010},[106,681,144],[18,1012,1014],{"className":1013},[119],[18,1015,1017],{"className":1016},[124],[18,1018,1020,1029],{"className":1019,"style":677},[128],[18,1021,1023,1026],{"style":1022},"top:-2.7em;",[18,1024],{"className":1025,"style":137},[136],[18,1027,77],{"className":1028,"style":254},[106,110,144],[18,1030,1032,1035],{"style":1031},"top:-2.9634em;",[18,1033],{"className":1034,"style":137},[136],[18,1036,1038],{"className":1037,"style":711},[710],[18,1039,664],{"className":1040},[106,144],[18,1042,1043,1046],{"style":693},[18,1044],{"className":1045,"style":697},[136],[18,1047,1048],{},[18,1049,1051,1052],{"className":1050},[985],"in",[18,1053,1055],{"style":1054},"margin-right:0.0778em;","f",[18,1057,152],{"className":1058},[151],[18,1060,1062],{"className":1061},[124],[18,1063,1066],{"className":1064,"style":1065},[128],"height:0.9365em;",[18,1067],{},[18,1069],{"className":1070,"style":170},[169],[18,1072,1074],{"className":1073},[985,986],[18,1075,1077,1118],{"className":1076},[119,120],[18,1078,1080,1115],{"className":1079},[124],[18,1081,1084,1105],{"className":1082,"style":1083},[128],"height:0.4306em;",[18,1085,1087,1090],{"style":1086},"top:-2.1535em;margin-left:0em;",[18,1088],{"className":1089,"style":697},[136],[18,1091,1093],{"className":1092},[141,142,143,144],[18,1094,1096,1099,1102],{"className":1095},[106,144],[18,1097,77],{"className":1098,"style":254},[106,110,144],[18,1100,921],{"className":1101},[232,144],[18,1103,924],{"className":1104},[106,144],[18,1106,1107,1110],{"style":693},[18,1108],{"className":1109,"style":697},[136],[18,1111,1112],{},[18,1113,912],{"className":1114},[985],[18,1116,152],{"className":1117},[151],[18,1119,1121],{"className":1120},[124],[18,1122,1125],{"className":1123,"style":1124},[128],"height:0.9739em;",[18,1126],{},[18,1128,928],{"className":1129},[169],[18,1131],{"className":1132,"style":170},[169],[18,1134,1136,1140],{"className":1135},[106],[18,1137,934],{"className":1138},[106,1139],"mathbb",[18,1141,1143],{"className":1142},[115],[18,1144,1146,1215],{"className":1145},[119,120],[18,1147,1149,1212],{"className":1148},[124],[18,1150,1153],{"className":1151,"style":1152},[128],"height:0.3283em;",[18,1154,1156,1159],{"style":1155},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[18,1157],{"className":1158,"style":137},[136],[18,1160,1162],{"className":1161},[141,142,143,144],[18,1163,1165],{"className":1164},[106,144],[18,1166,1168,1172],{"className":1167},[106,144],[18,1169,939],{"className":1170,"style":1171},[106,110,144],"margin-right:0.1389em;",[18,1173,1175],{"className":1174},[115],[18,1176,1178,1203],{"className":1177},[119,120],[18,1179,1181,1200],{"className":1180},[124],[18,1182,1185],{"className":1183,"style":1184},[128],"height:0.3448em;",[18,1186,1188,1192],{"style":1187},"top:-2.3488em;margin-left:-0.1389em;margin-right:0.0714em;",[18,1189],{"className":1190,"style":1191},[136],"height:2.5em;",[18,1193,1197],{"className":1194},[141,1195,1196,144],"reset-size3","size1",[18,1198,77],{"className":1199,"style":254},[106,110,144],[18,1201,152],{"className":1202},[151],[18,1204,1206],{"className":1205},[124],[18,1207,1210],{"className":1208,"style":1209},[128],"height:0.1512em;",[18,1211],{},[18,1213,152],{"className":1214},[151],[18,1216,1218],{"className":1217},[124],[18,1219,1222],{"className":1220,"style":1221},[128],"height:0.2559em;",[18,1223],{},[18,1225,1227],{"className":1226},[250],[18,1228,945],{"className":1229},[1230,1196],"delimsizing",[18,1232,948],{"className":1233},[106],[18,1235,74],{"className":1236},[250],[18,1238,77],{"className":1239,"style":254},[106,110],[18,1241,52],{"className":1242},[165],[18,1244],{"className":1245,"style":170},[169],[18,1247,1249],{"className":1248},[106,681],[18,1250,1252],{"className":1251},[119],[18,1253,1255],{"className":1254},[124],[18,1256,1258,1266],{"className":1257,"style":677},[128],[18,1259,1260,1263],{"style":693},[18,1261],{"className":1262,"style":697},[136],[18,1264,77],{"className":1265,"style":254},[106,110],[18,1267,1268,1271],{"style":703},[18,1269],{"className":1270,"style":697},[136],[18,1272,1274],{"className":1273,"style":711},[710],[18,1275,664],{"className":1276},[106],[18,1278,84],{"className":1279},[267],[18,1281,1283],{"className":1282},[267],[18,1284,965],{"className":1285},[1230,1196],[18,1287,968],{"className":1288},[106],[11,1290,1291,1292,1295],{},"这个量回答的是：",[717,1293,1294],{},"在所有可能的参数中，最坏情况下的风险最小能压到多少。"," 一个上界对应一个具体的估计量；一个下界则对应一句否定：无论你多聪明，都不可能比这个数更好。",[11,1297,1298,1299,1302],{},"下界为什么难？因为你要同时对",[717,1300,1301],{},"所有","估计量说话。逐一检查是不可能的。所以下界证明总是走同一条路：把参数空间缩小到一个有限的、精心挑选的子集，使得在这个子集上「区分参数」本身就已经很难；如果区分都做不到，估计就更做不到。",[11,1304,1305],{},"这就是「归约到检验」。接下来的三种方法，区别只在于子集有多大、用什么距离度量「难分」。",[638,1307,1309],{"id":1308},"二决策论框架","二、决策论框架",[1311,1312,1314],"h3",{"id":1313},"_21-模型损失与风险","2.1 模型、损失与风险",[11,1316,16,1317,1447,1448,1480,1481,1510,1511,1539],{},[18,1318,1320,1353],{"className":1319,"translate":22},[21],[18,1321,1323],{"className":1322},[26],[28,1324,1325],{"xmlns":30},[32,1326,1327,1350],{},[35,1328,1329,1332,1338,1341,1343,1345,1347],{},[49,1330,1331],{"stretchy":73},"{",[38,1333,1334,1336],{},[41,1335,939],{},[41,1337,77],{},[49,1339,1340],{},":",[41,1342,77],{},[49,1344,921],{},[41,1346,924],{"mathvariant":316},[49,1348,1349],{"stretchy":73},"}",[86,1351,1352],{"encoding":88},"\\{P_\\theta:\\theta\\in\\Theta\\}",[18,1354,1356,1416,1435],{"className":1355,"ariaHidden":51},[93],[18,1357,1359,1362,1365,1407,1410,1413],{"className":1358},[97],[18,1360],{"className":1361,"style":242},[101],[18,1363,1331],{"className":1364},[250],[18,1366,1368,1371],{"className":1367},[106],[18,1369,939],{"className":1370,"style":1171},[106,110],[18,1372,1374],{"className":1373},[115],[18,1375,1377,1399],{"className":1376},[119,120],[18,1378,1380,1396],{"className":1379},[124],[18,1381,1384],{"className":1382,"style":1383},[128],"height:0.3361em;",[18,1385,1387,1390],{"style":1386},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[18,1388],{"className":1389,"style":137},[136],[18,1391,1393],{"className":1392},[141,142,143,144],[18,1394,77],{"className":1395,"style":254},[106,110,144],[18,1397,152],{"className":1398},[151],[18,1400,1402],{"className":1401},[124],[18,1403,1405],{"className":1404,"style":159},[128],[18,1406],{},[18,1408],{"className":1409,"style":228},[169],[18,1411,1340],{"className":1412},[232],[18,1414],{"className":1415,"style":228},[169],[18,1417,1419,1423,1426,1429,1432],{"className":1418},[97],[18,1420],{"className":1421,"style":1422},[101],"height:0.7335em;vertical-align:-0.0391em;",[18,1424,77],{"className":1425,"style":254},[106,110],[18,1427],{"className":1428,"style":228},[169],[18,1430,921],{"className":1431},[232],[18,1433],{"className":1434,"style":228},[169],[18,1436,1438,1441,1444],{"className":1437},[97],[18,1439],{"className":1440,"style":242},[101],[18,1442,924],{"className":1443},[106],[18,1445,1349],{"className":1446},[267]," 是样本空间 ",[18,1449,1451,1466],{"className":1450,"translate":22},[21],[18,1452,1454],{"className":1453},[26],[28,1455,1456],{"xmlns":30},[32,1457,1458,1463],{},[35,1459,1460],{},[41,1461,43],{"mathvariant":1462},"script",[86,1464,1465],{"encoding":88},"\\mathcal X",[18,1467,1469],{"className":1468,"ariaHidden":51},[93],[18,1470,1472,1475],{"className":1471},[97],[18,1473],{"className":1474,"style":600},[101],[18,1476,43],{"className":1477,"style":1479},[106,1478],"mathcal","margin-right:0.1464em;"," 上的一族分布，参数空间 ",[18,1482,1484,1498],{"className":1483,"translate":22},[21],[18,1485,1487],{"className":1486},[26],[28,1488,1489],{"xmlns":30},[32,1490,1491,1495],{},[35,1492,1493],{},[41,1494,924],{"mathvariant":316},[86,1496,1497],{"encoding":88},"\\Theta",[18,1499,1501],{"className":1500,"ariaHidden":51},[93],[18,1502,1504,1507],{"className":1503},[97],[18,1505],{"className":1506,"style":600},[101],[18,1508,924],{"className":1509},[106]," 可以是任意集合。我们观测 ",[18,1512,1514,1527],{"className":1513,"translate":22},[21],[18,1515,1517],{"className":1516},[26],[28,1518,1519],{"xmlns":30},[32,1520,1521,1525],{},[35,1522,1523],{},[41,1524,64],{},[86,1526,64],{"encoding":88},[18,1528,1530],{"className":1529,"ariaHidden":51},[93],[18,1531,1533,1536],{"className":1532},[97],[18,1534],{"className":1535,"style":1083},[101],[18,1537,64],{"className":1538},[106,110]," 个独立同分布的样本",[18,1541,1543],{"className":1542,"translate":22},[875],[18,1544,1546,1607],{"className":1545,"translate":22},[21],[18,1547,1549],{"className":1548},[26],[28,1550,1551],{"xmlns":30,"display":884},[32,1552,1553,1604],{},[35,1554,1555,1562,1564,1566,1572,1574,1576,1578,1584,1586,1588,1602],{},[1556,1557,1558,1560],"msup",{},[41,1559,43],{},[41,1561,64],{},[49,1563,433],{},[49,1565,74],{"stretchy":73},[38,1567,1568,1570],{},[41,1569,43],{},[45,1571,47],{},[49,1573,52],{"separator":51},[49,1575,55],{},[49,1577,52],{"separator":51},[38,1579,1580,1582],{},[41,1581,43],{},[41,1583,64],{},[49,1585,84],{"stretchy":73},[49,1587,67],{},[1589,1590,1591,1593,1595],"msubsup",{},[41,1592,939],{},[41,1594,77],{},[35,1596,1597,1600],{},[49,1598,1599],{},"⊗",[41,1601,64],{},[49,1603,52],{"separator":51},[86,1605,1606],{"encoding":88},"X^n=(X_1,\\dots,X_n)\\sim P_\\theta^{\\otimes n},",[18,1608,1610,1656,1775],{"className":1609,"ariaHidden":51},[93],[18,1611,1613,1617,1647,1650,1653],{"className":1612},[97],[18,1614],{"className":1615,"style":1616},[101],"height:0.7144em;",[18,1618,1620,1623],{"className":1619},[106],[18,1621,43],{"className":1622,"style":111},[106,110],[18,1624,1626],{"className":1625},[115],[18,1627,1629],{"className":1628},[119],[18,1630,1632],{"className":1631},[124],[18,1633,1635],{"className":1634,"style":1616},[128],[18,1636,1638,1641],{"style":1637},"top:-3.113em;margin-right:0.05em;",[18,1639],{"className":1640,"style":137},[136],[18,1642,1644],{"className":1643},[141,142,143,144],[18,1645,64],{"className":1646},[106,110,144],[18,1648],{"className":1649,"style":228},[169],[18,1651,433],{"className":1652},[232],[18,1654],{"className":1655,"style":228},[169],[18,1657,1659,1662,1665,1705,1708,1711,1714,1717,1720,1723,1763,1766,1769,1772],{"className":1658},[97],[18,1660],{"className":1661,"style":242},[101],[18,1663,74],{"className":1664},[250],[18,1666,1668,1671],{"className":1667},[106],[18,1669,43],{"className":1670,"style":111},[106,110],[18,1672,1674],{"className":1673},[115],[18,1675,1677,1697],{"className":1676},[119,120],[18,1678,1680,1694],{"className":1679},[124],[18,1681,1683],{"className":1682,"style":129},[128],[18,1684,1685,1688],{"style":132},[18,1686],{"className":1687,"style":137},[136],[18,1689,1691],{"className":1690},[141,142,143,144],[18,1692,47],{"className":1693},[106,144],[18,1695,152],{"className":1696},[151],[18,1698,1700],{"className":1699},[124],[18,1701,1703],{"className":1702,"style":159},[128],[18,1704],{},[18,1706,52],{"className":1707},[165],[18,1709],{"className":1710,"style":170},[169],[18,1712,55],{"className":1713},[174],[18,1715],{"className":1716,"style":170},[169],[18,1718,52],{"className":1719},[165],[18,1721],{"className":1722,"style":170},[169],[18,1724,1726,1729],{"className":1725},[106],[18,1727,43],{"className":1728,"style":111},[106,110],[18,1730,1732],{"className":1731},[115],[18,1733,1735,1755],{"className":1734},[119,120],[18,1736,1738,1752],{"className":1737},[124],[18,1739,1741],{"className":1740,"style":202},[128],[18,1742,1743,1746],{"style":132},[18,1744],{"className":1745,"style":137},[136],[18,1747,1749],{"className":1748},[141,142,143,144],[18,1750,64],{"className":1751},[106,110,144],[18,1753,152],{"className":1754},[151],[18,1756,1758],{"className":1757},[124],[18,1759,1761],{"className":1760,"style":159},[128],[18,1762],{},[18,1764,84],{"className":1765},[267],[18,1767],{"className":1768,"style":228},[169],[18,1770,67],{"className":1771},[232],[18,1773],{"className":1774,"style":228},[169],[18,1776,1778,1782,1842],{"className":1777},[97],[18,1779],{"className":1780,"style":1781},[101],"height:1.1128em;vertical-align:-0.2914em;",[18,1783,1785,1788],{"className":1784},[106],[18,1786,939],{"className":1787,"style":1171},[106,110],[18,1789,1791],{"className":1790},[115],[18,1792,1794,1833],{"className":1793},[119,120],[18,1795,1797,1830],{"className":1796},[124],[18,1798,1801,1813],{"className":1799,"style":1800},[128],"height:0.8213em;",[18,1802,1804,1807],{"style":1803},"top:-2.4086em;margin-left:-0.1389em;margin-right:0.05em;",[18,1805],{"className":1806,"style":137},[136],[18,1808,1810],{"className":1809},[141,142,143,144],[18,1811,77],{"className":1812,"style":254},[106,110,144],[18,1814,1815,1818],{"style":1637},[18,1816],{"className":1817,"style":137},[136],[18,1819,1821],{"className":1820},[141,142,143,144],[18,1822,1824,1827],{"className":1823},[106,144],[18,1825,1599],{"className":1826},[106,144],[18,1828,64],{"className":1829},[106,110,144],[18,1831,152],{"className":1832},[151],[18,1834,1836],{"className":1835},[124],[18,1837,1840],{"className":1838,"style":1839},[128],"height:0.2914em;",[18,1841],{},[18,1843,52],{"className":1844},[165],[11,1846,1847,1848,1851,1852,1990,1991,1994,1995,2118,2119,2179,2180,2208],{},"一个",[717,1849,1850],{},"估计量","是任意可测映射 ",[18,1853,1855,1886],{"className":1854,"translate":22},[21],[18,1856,1858],{"className":1857},[26],[28,1859,1860],{"xmlns":30},[32,1861,1862,1883],{},[35,1863,1864,1870,1872,1878,1881],{},[658,1865,1866,1868],{"accent":51},[41,1867,77],{},[49,1869,664],{},[49,1871,1340],{},[1556,1873,1874,1876],{},[41,1875,43],{"mathvariant":1462},[41,1877,64],{},[49,1879,1880],{},"→",[41,1882,924],{"mathvariant":316},[86,1884,1885],{"encoding":88},"\\hat\\theta:\\mathcal X^n\\to\\Theta",[18,1887,1889,1935,1981],{"className":1888,"ariaHidden":51},[93],[18,1890,1892,1895,1926,1929,1932],{"className":1891},[97],[18,1893],{"className":1894,"style":677},[101],[18,1896,1898],{"className":1897},[106,681],[18,1899,1901],{"className":1900},[119],[18,1902,1904],{"className":1903},[124],[18,1905,1907,1915],{"className":1906,"style":677},[128],[18,1908,1909,1912],{"style":693},[18,1910],{"className":1911,"style":697},[136],[18,1913,77],{"className":1914,"style":254},[106,110],[18,1916,1917,1920],{"style":703},[18,1918],{"className":1919,"style":697},[136],[18,1921,1923],{"className":1922,"style":711},[710],[18,1924,664],{"className":1925},[106],[18,1927],{"className":1928,"style":228},[169],[18,1930,1340],{"className":1931},[232],[18,1933],{"className":1934,"style":228},[169],[18,1936,1938,1941,1972,1975,1978],{"className":1937},[97],[18,1939],{"className":1940,"style":600},[101],[18,1942,1944,1947],{"className":1943},[106],[18,1945,43],{"className":1946,"style":1479},[106,1478],[18,1948,1950],{"className":1949},[115],[18,1951,1953],{"className":1952},[119],[18,1954,1956],{"className":1955},[124],[18,1957,1960],{"className":1958,"style":1959},[128],"height:0.6644em;",[18,1961,1963,1966],{"style":1962},"top:-3.063em;margin-right:0.05em;",[18,1964],{"className":1965,"style":137},[136],[18,1967,1969],{"className":1968},[141,142,143,144],[18,1970,64],{"className":1971},[106,110,144],[18,1973],{"className":1974,"style":228},[169],[18,1976,1880],{"className":1977},[232],[18,1979],{"className":1980,"style":228},[169],[18,1982,1984,1987],{"className":1983},[97],[18,1985],{"className":1986,"style":600},[101],[18,1988,924],{"className":1989},[106],"。",[717,1992,1993],{},"损失函数"," ",[18,1996,1998,2034],{"className":1997,"translate":22},[21],[18,1999,2001],{"className":2000},[26],[28,2002,2003],{"xmlns":30},[32,2004,2005,2031],{},[35,2006,2007,2009,2011,2013,2016,2018,2020,2022,2024,2026,2029],{},[41,2008,948],{"mathvariant":316},[49,2010,1340],{},[41,2012,924],{"mathvariant":316},[49,2014,2015],{},"×",[41,2017,924],{"mathvariant":316},[49,2019,1880],{},[49,2021,945],{"stretchy":73},[45,2023,430],{},[49,2025,52],{"separator":51},[41,2027,2028],{"mathvariant":316},"∞",[49,2030,84],{"stretchy":73},[86,2032,2033],{"encoding":88},"\\ell:\\Theta\\times\\Theta\\to[0,\\infty)",[18,2035,2037,2055,2076,2094],{"className":2036,"ariaHidden":51},[93],[18,2038,2040,2043,2046,2049,2052],{"className":2039},[97],[18,2041],{"className":2042,"style":295},[101],[18,2044,948],{"className":2045},[106],[18,2047],{"className":2048,"style":228},[169],[18,2050,1340],{"className":2051},[232],[18,2053],{"className":2054,"style":228},[169],[18,2056,2058,2062,2065,2069,2073],{"className":2057},[97],[18,2059],{"className":2060,"style":2061},[101],"height:0.7667em;vertical-align:-0.0833em;",[18,2063,924],{"className":2064},[106],[18,2066],{"className":2067,"style":2068},[169],"margin-right:0.2222em;",[18,2070,2015],{"className":2071},[2072],"mbin",[18,2074],{"className":2075,"style":2068},[169],[18,2077,2079,2082,2085,2088,2091],{"className":2078},[97],[18,2080],{"className":2081,"style":600},[101],[18,2083,924],{"className":2084},[106],[18,2086],{"className":2087,"style":228},[169],[18,2089,1880],{"className":2090},[232],[18,2092],{"className":2093,"style":228},[169],[18,2095,2097,2100,2103,2106,2109,2112,2115],{"className":2096},[97],[18,2098],{"className":2099,"style":242},[101],[18,2101,945],{"className":2102},[250],[18,2104,430],{"className":2105},[106],[18,2107,52],{"className":2108},[165],[18,2110],{"className":2111,"style":170},[169],[18,2113,2028],{"className":2114},[106],[18,2116,84],{"className":2117},[267]," 度量估计值 ",[18,2120,2122,2139],{"className":2121,"translate":22},[21],[18,2123,2125],{"className":2124},[26],[28,2126,2127],{"xmlns":30},[32,2128,2129,2137],{},[35,2130,2131],{},[658,2132,2133,2135],{"accent":51},[41,2134,77],{},[49,2136,664],{},[86,2138,667],{"encoding":88},[18,2140,2142],{"className":2141,"ariaHidden":51},[93],[18,2143,2145,2148],{"className":2144},[97],[18,2146],{"className":2147,"style":677},[101],[18,2149,2151],{"className":2150},[106,681],[18,2152,2154],{"className":2153},[119],[18,2155,2157],{"className":2156},[124],[18,2158,2160,2168],{"className":2159,"style":677},[128],[18,2161,2162,2165],{"style":693},[18,2163],{"className":2164,"style":697},[136],[18,2166,77],{"className":2167,"style":254},[106,110],[18,2169,2170,2173],{"style":703},[18,2171],{"className":2172,"style":697},[136],[18,2174,2176],{"className":2175,"style":711},[710],[18,2177,664],{"className":2178},[106]," 与真值 ",[18,2181,2183,2196],{"className":2182,"translate":22},[21],[18,2184,2186],{"className":2185},[26],[28,2187,2188],{"xmlns":30},[32,2189,2190,2194],{},[35,2191,2192],{},[41,2193,77],{},[86,2195,285],{"encoding":88},[18,2197,2199],{"className":2198,"ariaHidden":51},[93],[18,2200,2202,2205],{"className":2201},[97],[18,2203],{"className":2204,"style":295},[101],[18,2206,77],{"className":2207,"style":254},[106,110]," 的差距，常用的有：",[2210,2211,2212,2453,2672],"ul",{},[2213,2214,2215,2216,2452],"li",{},"平方损失 ",[18,2217,2219,2272],{"className":2218,"translate":22},[21],[18,2220,2222],{"className":2221},[26],[28,2223,2224],{"xmlns":30},[32,2225,2226,2269],{},[35,2227,2228,2230,2232,2234,2236,2242,2244,2246,2249,2251,2254,2260],{},[41,2229,948],{"mathvariant":316},[49,2231,74],{"stretchy":73},[41,2233,77],{},[49,2235,52],{"separator":51},[658,2237,2238,2240],{"accent":51},[41,2239,77],{},[49,2241,664],{},[49,2243,84],{"stretchy":73},[49,2245,433],{},[49,2247,2248],{"stretchy":73},"∥",[41,2250,77],{},[49,2252,2253],{},"−",[658,2255,2256,2258],{"accent":51},[41,2257,77],{},[49,2259,664],{},[1589,2261,2262,2264,2267],{},[49,2263,2248],{"stretchy":73},[45,2265,2266],{},"2",[45,2268,2266],{},[86,2270,2271],{"encoding":88},"\\ell(\\theta,\\hat\\theta)=\\lVert\\theta-\\hat\\theta\\rVert_2^2",[18,2273,2275,2340,2361],{"className":2274,"ariaHidden":51},[93],[18,2276,2278,2282,2285,2288,2291,2294,2297,2328,2331,2334,2337],{"className":2277},[97],[18,2279],{"className":2280,"style":2281},[101],"height:1.2079em;vertical-align:-0.25em;",[18,2283,948],{"className":2284},[106],[18,2286,74],{"className":2287},[250],[18,2289,77],{"className":2290,"style":254},[106,110],[18,2292,52],{"className":2293},[165],[18,2295],{"className":2296,"style":170},[169],[18,2298,2300],{"className":2299},[106,681],[18,2301,2303],{"className":2302},[119],[18,2304,2306],{"className":2305},[124],[18,2307,2309,2317],{"className":2308,"style":677},[128],[18,2310,2311,2314],{"style":693},[18,2312],{"className":2313,"style":697},[136],[18,2315,77],{"className":2316,"style":254},[106,110],[18,2318,2319,2322],{"style":703},[18,2320],{"className":2321,"style":697},[136],[18,2323,2325],{"className":2324,"style":711},[710],[18,2326,664],{"className":2327},[106],[18,2329,84],{"className":2330},[267],[18,2332],{"className":2333,"style":228},[169],[18,2335,433],{"className":2336},[232],[18,2338],{"className":2339,"style":228},[169],[18,2341,2343,2346,2349,2352,2355,2358],{"className":2342},[97],[18,2344],{"className":2345,"style":242},[101],[18,2347,2248],{"className":2348},[250],[18,2350,77],{"className":2351,"style":254},[106,110],[18,2353],{"className":2354,"style":2068},[169],[18,2356,2253],{"className":2357},[2072],[18,2359],{"className":2360,"style":2068},[169],[18,2362,2364,2367,2398],{"className":2363},[97],[18,2365],{"className":2366,"style":2281},[101],[18,2368,2370],{"className":2369},[106,681],[18,2371,2373],{"className":2372},[119],[18,2374,2376],{"className":2375},[124],[18,2377,2379,2387],{"className":2378,"style":677},[128],[18,2380,2381,2384],{"style":693},[18,2382],{"className":2383,"style":697},[136],[18,2385,77],{"className":2386,"style":254},[106,110],[18,2388,2389,2392],{"style":703},[18,2390],{"className":2391,"style":697},[136],[18,2393,2395],{"className":2394,"style":711},[710],[18,2396,664],{"className":2397},[106],[18,2399,2401,2404],{"className":2400},[267],[18,2402,2248],{"className":2403},[267],[18,2405,2407],{"className":2406},[115],[18,2408,2410,2443],{"className":2409},[119,120],[18,2411,2413,2440],{"className":2412},[124],[18,2414,2417,2429],{"className":2415,"style":2416},[128],"height:0.8141em;",[18,2418,2420,2423],{"style":2419},"top:-2.4519em;margin-left:0em;margin-right:0.05em;",[18,2421],{"className":2422,"style":137},[136],[18,2424,2426],{"className":2425},[141,142,143,144],[18,2427,2266],{"className":2428},[106,144],[18,2430,2431,2434],{"style":1962},[18,2432],{"className":2433,"style":137},[136],[18,2435,2437],{"className":2436},[141,142,143,144],[18,2438,2266],{"className":2439},[106,144],[18,2441,152],{"className":2442},[151],[18,2444,2446],{"className":2445},[124],[18,2447,2450],{"className":2448,"style":2449},[128],"height:0.2481em;",[18,2451],{},"；",[2213,2454,2455,2456,2452],{},"绝对损失 ",[18,2457,2459,2507],{"className":2458,"translate":22},[21],[18,2460,2462],{"className":2461},[26],[28,2463,2464],{"xmlns":30},[32,2465,2466,2504],{},[35,2467,2468,2470,2472,2474,2476,2482,2484,2486,2488,2490,2492,2498],{},[41,2469,948],{"mathvariant":316},[49,2471,74],{"stretchy":73},[41,2473,77],{},[49,2475,52],{"separator":51},[658,2477,2478,2480],{"accent":51},[41,2479,77],{},[49,2481,664],{},[49,2483,84],{"stretchy":73},[49,2485,433],{},[49,2487,2248],{"stretchy":73},[41,2489,77],{},[49,2491,2253],{},[658,2493,2494,2496],{"accent":51},[41,2495,77],{},[49,2497,664],{},[38,2499,2500,2502],{},[49,2501,2248],{"stretchy":73},[45,2503,47],{},[86,2505,2506],{"encoding":88},"\\ell(\\theta,\\hat\\theta)=\\lVert\\theta-\\hat\\theta\\rVert_1",[18,2508,2510,2574,2595],{"className":2509,"ariaHidden":51},[93],[18,2511,2513,2516,2519,2522,2525,2528,2531,2562,2565,2568,2571],{"className":2512},[97],[18,2514],{"className":2515,"style":2281},[101],[18,2517,948],{"className":2518},[106],[18,2520,74],{"className":2521},[250],[18,2523,77],{"className":2524,"style":254},[106,110],[18,2526,52],{"className":2527},[165],[18,2529],{"className":2530,"style":170},[169],[18,2532,2534],{"className":2533},[106,681],[18,2535,2537],{"className":2536},[119],[18,2538,2540],{"className":2539},[124],[18,2541,2543,2551],{"className":2542,"style":677},[128],[18,2544,2545,2548],{"style":693},[18,2546],{"className":2547,"style":697},[136],[18,2549,77],{"className":2550,"style":254},[106,110],[18,2552,2553,2556],{"style":703},[18,2554],{"className":2555,"style":697},[136],[18,2557,2559],{"className":2558,"style":711},[710],[18,2560,664],{"className":2561},[106],[18,2563,84],{"className":2564},[267],[18,2566],{"className":2567,"style":228},[169],[18,2569,433],{"className":2570},[232],[18,2572],{"className":2573,"style":228},[169],[18,2575,2577,2580,2583,2586,2589,2592],{"className":2576},[97],[18,2578],{"className":2579,"style":242},[101],[18,2581,2248],{"className":2582},[250],[18,2584,77],{"className":2585,"style":254},[106,110],[18,2587],{"className":2588,"style":2068},[169],[18,2590,2253],{"className":2591},[2072],[18,2593],{"className":2594,"style":2068},[169],[18,2596,2598,2601,2632],{"className":2597},[97],[18,2599],{"className":2600,"style":2281},[101],[18,2602,2604],{"className":2603},[106,681],[18,2605,2607],{"className":2606},[119],[18,2608,2610],{"className":2609},[124],[18,2611,2613,2621],{"className":2612,"style":677},[128],[18,2614,2615,2618],{"style":693},[18,2616],{"className":2617,"style":697},[136],[18,2619,77],{"className":2620,"style":254},[106,110],[18,2622,2623,2626],{"style":703},[18,2624],{"className":2625,"style":697},[136],[18,2627,2629],{"className":2628,"style":711},[710],[18,2630,664],{"className":2631},[106],[18,2633,2635,2638],{"className":2634},[267],[18,2636,2248],{"className":2637},[267],[18,2639,2641],{"className":2640},[115],[18,2642,2644,2664],{"className":2643},[119,120],[18,2645,2647,2661],{"className":2646},[124],[18,2648,2650],{"className":2649,"style":129},[128],[18,2651,2652,2655],{"style":1155},[18,2653],{"className":2654,"style":137},[136],[18,2656,2658],{"className":2657},[141,142,143,144],[18,2659,47],{"className":2660},[106,144],[18,2662,152],{"className":2663},[151],[18,2665,2667],{"className":2666},[124],[18,2668,2670],{"className":2669,"style":159},[128],[18,2671],{},[2213,2673,2674,2702,2703,2731,2732,1990],{},[18,2675,2677,2690],{"className":2676,"translate":22},[21],[18,2678,2680],{"className":2679},[26],[28,2681,2682],{"xmlns":30},[32,2683,2684,2688],{},[35,2685,2686],{},[45,2687,430],{},[86,2689,430],{"encoding":88},[18,2691,2693],{"className":2692,"ariaHidden":51},[93],[18,2694,2696,2699],{"className":2695},[97],[18,2697],{"className":2698,"style":505},[101],[18,2700,430],{"className":2701},[106],"-",[18,2704,2706,2719],{"className":2705,"translate":22},[21],[18,2707,2709],{"className":2708},[26],[28,2710,2711],{"xmlns":30},[32,2712,2713,2717],{},[35,2714,2715],{},[45,2716,47],{},[86,2718,47],{"encoding":88},[18,2720,2722],{"className":2721,"ariaHidden":51},[93],[18,2723,2725,2728],{"className":2724},[97],[18,2726],{"className":2727,"style":505},[101],[18,2729,47],{"className":2730},[106]," 损失 ",[18,2733,2735,2783],{"className":2734,"translate":22},[21],[18,2736,2738],{"className":2737},[26],[28,2739,2740],{"xmlns":30},[32,2741,2742,2780],{},[35,2743,2744,2746,2748,2750,2752,2758,2760,2762,2765,2767,2769,2772,2778],{},[41,2745,948],{"mathvariant":316},[49,2747,74],{"stretchy":73},[41,2749,77],{},[49,2751,52],{"separator":51},[658,2753,2754,2756],{"accent":51},[41,2755,77],{},[49,2757,664],{},[49,2759,84],{"stretchy":73},[49,2761,433],{},[45,2763,47],{"mathvariant":2764},"bold",[49,2766,945],{"stretchy":73},[41,2768,77],{},[49,2770,2771],{"mathvariant":316},"≠",[658,2773,2774,2776],{"accent":51},[41,2775,77],{},[49,2777,664],{},[49,2779,965],{"stretchy":73},[86,2781,2782],{"encoding":88},"\\ell(\\theta,\\hat\\theta)=\\mathbf 1[\\theta\\neq\\hat\\theta]",[18,2784,2786,2850,2916],{"className":2785,"ariaHidden":51},[93],[18,2787,2789,2792,2795,2798,2801,2804,2807,2838,2841,2844,2847],{"className":2788},[97],[18,2790],{"className":2791,"style":2281},[101],[18,2793,948],{"className":2794},[106],[18,2796,74],{"className":2797},[250],[18,2799,77],{"className":2800,"style":254},[106,110],[18,2802,52],{"className":2803},[165],[18,2805],{"className":2806,"style":170},[169],[18,2808,2810],{"className":2809},[106,681],[18,2811,2813],{"className":2812},[119],[18,2814,2816],{"className":2815},[124],[18,2817,2819,2827],{"className":2818,"style":677},[128],[18,2820,2821,2824],{"style":693},[18,2822],{"className":2823,"style":697},[136],[18,2825,77],{"className":2826,"style":254},[106,110],[18,2828,2829,2832],{"style":703},[18,2830],{"className":2831,"style":697},[136],[18,2833,2835],{"className":2834,"style":711},[710],[18,2836,664],{"className":2837},[106],[18,2839,84],{"className":2840},[267],[18,2842],{"className":2843,"style":228},[169],[18,2845,433],{"className":2846},[232],[18,2848],{"className":2849,"style":228},[169],[18,2851,2853,2856,2860,2863,2866,2869,2913],{"className":2852},[97],[18,2854],{"className":2855,"style":242},[101],[18,2857,47],{"className":2858},[106,2859],"mathbf",[18,2861,945],{"className":2862},[250],[18,2864,77],{"className":2865,"style":254},[106,110],[18,2867],{"className":2868,"style":228},[169],[18,2870,2872,2906,2910],{"className":2871},[232],[18,2873,2875],{"className":2874},[232],[18,2876,2879],{"className":2877},[106,2878],"vbox",[18,2880,2883],{"className":2881},[2882],"thinbox",[18,2884,2887,2891,2902],{"className":2885},[2886],"rlap",[18,2888],{"className":2889,"style":2890},[101],"height:0.8889em;vertical-align:-0.1944em;",[18,2892,2895],{"className":2893},[2894],"inner",[18,2896,2898],{"className":2897},[106],[18,2899,2901],{"className":2900},[232],"",[18,2903],{"className":2904},[2905],"fix",[18,2907],{"className":2908},[169,2909],"nobreak",[18,2911,433],{"className":2912},[232],[18,2914],{"className":2915,"style":228},[169],[18,2917,2919,2922,2953],{"className":2918},[97],[18,2920],{"className":2921,"style":2281},[101],[18,2923,2925],{"className":2924},[106,681],[18,2926,2928],{"className":2927},[119],[18,2929,2931],{"className":2930},[124],[18,2932,2934,2942],{"className":2933,"style":677},[128],[18,2935,2936,2939],{"style":693},[18,2937],{"className":2938,"style":697},[136],[18,2940,77],{"className":2941,"style":254},[106,110],[18,2943,2944,2947],{"style":703},[18,2945],{"className":2946,"style":697},[136],[18,2948,2950],{"className":2949,"style":711},[710],[18,2951,664],{"className":2952},[106],[18,2954,965],{"className":2955},[267],[11,2957,2958,2959,3019,3020,3048,3049,3051],{},"估计量 ",[18,2960,2962,2979],{"className":2961,"translate":22},[21],[18,2963,2965],{"className":2964},[26],[28,2966,2967],{"xmlns":30},[32,2968,2969,2977],{},[35,2970,2971],{},[658,2972,2973,2975],{"accent":51},[41,2974,77],{},[49,2976,664],{},[86,2978,667],{"encoding":88},[18,2980,2982],{"className":2981,"ariaHidden":51},[93],[18,2983,2985,2988],{"className":2984},[97],[18,2986],{"className":2987,"style":677},[101],[18,2989,2991],{"className":2990},[106,681],[18,2992,2994],{"className":2993},[119],[18,2995,2997],{"className":2996},[124],[18,2998,3000,3008],{"className":2999,"style":677},[128],[18,3001,3002,3005],{"style":693},[18,3003],{"className":3004,"style":697},[136],[18,3006,77],{"className":3007,"style":254},[106,110],[18,3009,3010,3013],{"style":703},[18,3011],{"className":3012,"style":697},[136],[18,3014,3016],{"className":3015,"style":711},[710],[18,3017,664],{"className":3018},[106]," 在 ",[18,3021,3023,3036],{"className":3022,"translate":22},[21],[18,3024,3026],{"className":3025},[26],[28,3027,3028],{"xmlns":30},[32,3029,3030,3034],{},[35,3031,3032],{},[41,3033,77],{},[86,3035,285],{"encoding":88},[18,3037,3039],{"className":3038,"ariaHidden":51},[93],[18,3040,3042,3045],{"className":3041},[97],[18,3043],{"className":3044,"style":295},[101],[18,3046,77],{"className":3047,"style":254},[106,110]," 处的",[717,3050,719],{},"是",[18,3053,3055],{"className":3054,"translate":22},[875],[18,3056,3058,3137],{"className":3057,"translate":22},[21],[18,3059,3061],{"className":3060},[26],[28,3062,3063],{"xmlns":30,"display":884},[32,3064,3065,3134],{},[35,3066,3067,3074,3076,3082,3084,3086,3102,3104,3106,3108,3110,3112,3118,3120,3126,3128,3130,3132],{},[38,3068,3069,3072],{},[41,3070,3071],{},"R",[41,3073,77],{},[49,3075,74],{"stretchy":73},[658,3077,3078,3080],{"accent":51},[41,3079,77],{},[49,3081,664],{},[49,3083,84],{"stretchy":73},[49,3085,433],{},[38,3087,3088,3090],{},[41,3089,934],{"mathvariant":933},[1589,3091,3092,3094,3096],{},[41,3093,939],{},[41,3095,77],{},[35,3097,3098,3100],{},[49,3099,1599],{},[41,3101,64],{},[49,3103,945],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[41,3105,948],{"mathvariant":316},[49,3107,74],{"stretchy":73},[41,3109,77],{},[49,3111,52],{"separator":51},[658,3113,3114,3116],{"accent":51},[41,3115,77],{},[49,3117,664],{},[49,3119,74],{"stretchy":73},[1556,3121,3122,3124],{},[41,3123,43],{},[41,3125,64],{},[49,3127,84],{"stretchy":73},[49,3129,84],{"stretchy":73},[49,3131,965],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[41,3133,968],{"mathvariant":316},[86,3135,3136],{"encoding":88},"R_\\theta(\\hat\\theta)=\\mathbb E_{P_\\theta^{\\otimes n}}\\bigl[\\ell(\\theta,\\hat\\theta(X^n))\\bigr].",[18,3138,3140,3234],{"className":3139,"ariaHidden":51},[93],[18,3141,3143,3146,3188,3191,3222,3225,3228,3231],{"className":3142},[97],[18,3144],{"className":3145,"style":2281},[101],[18,3147,3149,3153],{"className":3148},[106],[18,3150,3071],{"className":3151,"style":3152},[106,110],"margin-right:0.0077em;",[18,3154,3156],{"className":3155},[115],[18,3157,3159,3180],{"className":3158},[119,120],[18,3160,3162,3177],{"className":3161},[124],[18,3163,3165],{"className":3164,"style":1383},[128],[18,3166,3168,3171],{"style":3167},"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;",[18,3169],{"className":3170,"style":137},[136],[18,3172,3174],{"className":3173},[141,142,143,144],[18,3175,77],{"className":3176,"style":254},[106,110,144],[18,3178,152],{"className":3179},[151],[18,3181,3183],{"className":3182},[124],[18,3184,3186],{"className":3185,"style":159},[128],[18,3187],{},[18,3189,74],{"className":3190},[250],[18,3192,3194],{"className":3193},[106,681],[18,3195,3197],{"className":3196},[119],[18,3198,3200],{"className":3199},[124],[18,3201,3203,3211],{"className":3202,"style":677},[128],[18,3204,3205,3208],{"style":693},[18,3206],{"className":3207,"style":697},[136],[18,3209,77],{"className":3210,"style":254},[106,110],[18,3212,3213,3216],{"style":703},[18,3214],{"className":3215,"style":697},[136],[18,3217,3219],{"className":3218,"style":711},[710],[18,3220,664],{"className":3221},[106],[18,3223,84],{"className":3224},[267],[18,3226],{"className":3227,"style":228},[169],[18,3229,433],{"className":3230},[232],[18,3232],{"className":3233,"style":228},[169],[18,3235,3237,3241,3344,3350,3353,3356,3359,3362,3365,3396,3399,3428,3432,3438],{"className":3236},[97],[18,3238],{"className":3239,"style":3240},[101],"height:1.4308em;vertical-align:-0.4729em;",[18,3242,3244,3247],{"className":3243},[106],[18,3245,934],{"className":3246},[106,1139],[18,3248,3250],{"className":3249},[115],[18,3251,3253,3335],{"className":3252},[119,120],[18,3254,3256,3332],{"className":3255},[124],[18,3257,3259],{"className":3258,"style":1184},[128],[18,3260,3262,3265],{"style":3261},"top:-2.4701em;margin-left:0em;margin-right:0.05em;",[18,3263],{"className":3264,"style":137},[136],[18,3266,3268],{"className":3267},[141,142,143,144],[18,3269,3271],{"className":3270},[106,144],[18,3272,3274,3277],{"className":3273},[106,144],[18,3275,939],{"className":3276,"style":1171},[106,110,144],[18,3278,3280],{"className":3279},[115],[18,3281,3283,3323],{"className":3282},[119,120],[18,3284,3286,3320],{"className":3285},[124],[18,3287,3290,3302],{"className":3288,"style":3289},[128],"height:0.821em;",[18,3291,3293,3296],{"style":3292},"top:-2.1528em;margin-left:-0.1389em;margin-right:0.0714em;",[18,3294],{"className":3295,"style":1191},[136],[18,3297,3299],{"className":3298},[141,1195,1196,144],[18,3300,77],{"className":3301,"style":254},[106,110,144],[18,3303,3305,3308],{"style":3304},"top:-2.9043em;margin-right:0.0714em;",[18,3306],{"className":3307,"style":1191},[136],[18,3309,3311],{"className":3310},[141,1195,1196,144],[18,3312,3314,3317],{"className":3313},[106,144],[18,3315,1599],{"className":3316},[106,144],[18,3318,64],{"className":3319},[106,110,144],[18,3321,152],{"className":3322},[151],[18,3324,3326],{"className":3325},[124],[18,3327,3330],{"className":3328,"style":3329},[128],"height:0.3472em;",[18,3331],{},[18,3333,152],{"className":3334},[151],[18,3336,3338],{"className":3337},[124],[18,3339,3342],{"className":3340,"style":3341},[128],"height:0.4729em;",[18,3343],{},[18,3345,3347],{"className":3346},[250],[18,3348,945],{"className":3349},[1230,1196],[18,3351,948],{"className":3352},[106],[18,3354,74],{"className":3355},[250],[18,3357,77],{"className":3358,"style":254},[106,110],[18,3360,52],{"className":3361},[165],[18,3363],{"className":3364,"style":170},[169],[18,3366,3368],{"className":3367},[106,681],[18,3369,3371],{"className":3370},[119],[18,3372,3374],{"className":3373},[124],[18,3375,3377,3385],{"className":3376,"style":677},[128],[18,3378,3379,3382],{"style":693},[18,3380],{"className":3381,"style":697},[136],[18,3383,77],{"className":3384,"style":254},[106,110],[18,3386,3387,3390],{"style":703},[18,3388],{"className":3389,"style":697},[136],[18,3391,3393],{"className":3392,"style":711},[710],[18,3394,664],{"className":3395},[106],[18,3397,74],{"className":3398},[250],[18,3400,3402,3405],{"className":3401},[106],[18,3403,43],{"className":3404,"style":111},[106,110],[18,3406,3408],{"className":3407},[115],[18,3409,3411],{"className":3410},[119],[18,3412,3414],{"className":3413},[124],[18,3415,3417],{"className":3416,"style":1616},[128],[18,3418,3419,3422],{"style":1637},[18,3420],{"className":3421,"style":137},[136],[18,3423,3425],{"className":3424},[141,142,143,144],[18,3426,64],{"className":3427},[106,110,144],[18,3429,3431],{"className":3430},[267],"))",[18,3433,3435],{"className":3434},[267],[18,3436,965],{"className":3437},[1230,1196],[18,3439,968],{"className":3440},[106],[11,3442,3443,3444,3447,3448,3476,3477,3505],{},"注意这里的期望是对",[717,3445,3446],{},"数据","取的，",[18,3449,3451,3464],{"className":3450,"translate":22},[21],[18,3452,3454],{"className":3453},[26],[28,3455,3456],{"xmlns":30},[32,3457,3458,3462],{},[35,3459,3460],{},[41,3461,77],{},[86,3463,285],{"encoding":88},[18,3465,3467],{"className":3466,"ariaHidden":51},[93],[18,3468,3470,3473],{"className":3469},[97],[18,3471],{"className":3472,"style":295},[101],[18,3474,77],{"className":3475,"style":254},[106,110]," 固定。极小极大风险则再对 ",[18,3478,3480,3493],{"className":3479,"translate":22},[21],[18,3481,3483],{"className":3482},[26],[28,3484,3485],{"xmlns":30},[32,3486,3487,3491],{},[35,3488,3489],{},[41,3490,77],{},[86,3492,285],{"encoding":88},[18,3494,3496],{"className":3495,"ariaHidden":51},[93],[18,3497,3499,3502],{"className":3498},[97],[18,3500],{"className":3501,"style":295},[101],[18,3503,77],{"className":3504,"style":254},[106,110]," 取最坏情况：",[18,3507,3509],{"className":3508,"translate":22},[875],[18,3510,3512,3592],{"className":3511,"translate":22},[21],[18,3513,3515],{"className":3514},[26],[28,3516,3517],{"xmlns":30,"display":884},[32,3518,3519,3589],{},[35,3520,3521,3529,3531,3533,3535,3537,3539,3541,3555,3571,3577,3579,3585,3587],{},[38,3522,3523,3527],{},[41,3524,3526],{"mathvariant":3525},"fraktur","M",[41,3528,64],{},[49,3530,74],{"stretchy":73},[41,3532,924],{"mathvariant":316},[49,3534,52],{"separator":51},[41,3536,948],{"mathvariant":316},[49,3538,84],{"stretchy":73},[49,3540,433],{},[890,3542,3543,3549],{},[35,3544,3545,3547],{},[41,3546,896],{},[49,3548,899],{},[658,3550,3551,3553],{"accent":51},[41,3552,77],{},[49,3554,664],{},[890,3556,3557,3563],{},[35,3558,3559,3561],{},[41,3560,912],{},[49,3562,899],{},[35,3564,3565,3567,3569],{},[41,3566,77],{},[49,3568,921],{},[41,3570,924],{"mathvariant":316},[38,3572,3573,3575],{},[41,3574,3071],{},[41,3576,77],{},[49,3578,74],{"stretchy":73},[658,3580,3581,3583],{"accent":51},[41,3582,77],{},[49,3584,664],{},[49,3586,84],{"stretchy":73},[41,3588,968],{"mathvariant":316},[86,3590,3591],{"encoding":88},"\\mathfrak M_n(\\Theta,\\ell)=\\inf_{\\hat\\theta}\\sup_{\\theta\\in\\Theta}R_\\theta(\\hat\\theta).",[18,3593,3595,3669],{"className":3594,"ariaHidden":51},[93],[18,3596,3598,3601,3642,3645,3648,3651,3654,3657,3660,3663,3666],{"className":3597},[97],[18,3599],{"className":3600,"style":242},[101],[18,3602,3604,3608],{"className":3603},[106],[18,3605,3526],{"className":3606},[106,3607],"mathfrak",[18,3609,3611],{"className":3610},[115],[18,3612,3614,3634],{"className":3613},[119,120],[18,3615,3617,3631],{"className":3616},[124],[18,3618,3620],{"className":3619,"style":202},[128],[18,3621,3622,3625],{"style":1155},[18,3623],{"className":3624,"style":137},[136],[18,3626,3628],{"className":3627},[141,142,143,144],[18,3629,64],{"className":3630},[106,110,144],[18,3632,152],{"className":3633},[151],[18,3635,3637],{"className":3636},[124],[18,3638,3640],{"className":3639,"style":159},[128],[18,3641],{},[18,3643,74],{"className":3644},[250],[18,3646,924],{"className":3647},[106],[18,3649,52],{"className":3650},[165],[18,3652],{"className":3653,"style":170},[169],[18,3655,948],{"className":3656},[106],[18,3658,84],{"className":3659},[267],[18,3661],{"className":3662,"style":228},[169],[18,3664,433],{"className":3665},[232],[18,3667],{"className":3668,"style":228},[169],[18,3670,3672,3675,3752,3755,3808,3811,3851,3854,3885,3888],{"className":3671},[97],[18,3673],{"className":3674,"style":981},[101],[18,3676,3678],{"className":3677},[985,986],[18,3679,3681,3744],{"className":3680},[119,120],[18,3682,3684,3741],{"className":3683},[124],[18,3685,3687,3729],{"className":3686,"style":295},[128],[18,3688,3689,3692],{"style":998},[18,3690],{"className":3691,"style":697},[136],[18,3693,3695],{"className":3694},[141,142,143,144],[18,3696,3698],{"className":3697},[106,144],[18,3699,3701],{"className":3700},[106,681,144],[18,3702,3704],{"className":3703},[119],[18,3705,3707],{"className":3706},[124],[18,3708,3710,3718],{"className":3709,"style":677},[128],[18,3711,3712,3715],{"style":1022},[18,3713],{"className":3714,"style":137},[136],[18,3716,77],{"className":3717,"style":254},[106,110,144],[18,3719,3720,3723],{"style":1031},[18,3721],{"className":3722,"style":137},[136],[18,3724,3726],{"className":3725,"style":711},[710],[18,3727,664],{"className":3728},[106,144],[18,3730,3731,3734],{"style":693},[18,3732],{"className":3733,"style":697},[136],[18,3735,3736],{},[18,3737,1051,3739],{"className":3738},[985],[18,3740,1055],{"style":1054},[18,3742,152],{"className":3743},[151],[18,3745,3747],{"className":3746},[124],[18,3748,3750],{"className":3749,"style":1065},[128],[18,3751],{},[18,3753],{"className":3754,"style":170},[169],[18,3756,3758],{"className":3757},[985,986],[18,3759,3761,3800],{"className":3760},[119,120],[18,3762,3764,3797],{"className":3763},[124],[18,3765,3767,3787],{"className":3766,"style":1083},[128],[18,3768,3769,3772],{"style":1086},[18,3770],{"className":3771,"style":697},[136],[18,3773,3775],{"className":3774},[141,142,143,144],[18,3776,3778,3781,3784],{"className":3777},[106,144],[18,3779,77],{"className":3780,"style":254},[106,110,144],[18,3782,921],{"className":3783},[232,144],[18,3785,924],{"className":3786},[106,144],[18,3788,3789,3792],{"style":693},[18,3790],{"className":3791,"style":697},[136],[18,3793,3794],{},[18,3795,912],{"className":3796},[985],[18,3798,152],{"className":3799},[151],[18,3801,3803],{"className":3802},[124],[18,3804,3806],{"className":3805,"style":1124},[128],[18,3807],{},[18,3809],{"className":3810,"style":170},[169],[18,3812,3814,3817],{"className":3813},[106],[18,3815,3071],{"className":3816,"style":3152},[106,110],[18,3818,3820],{"className":3819},[115],[18,3821,3823,3843],{"className":3822},[119,120],[18,3824,3826,3840],{"className":3825},[124],[18,3827,3829],{"className":3828,"style":1383},[128],[18,3830,3831,3834],{"style":3167},[18,3832],{"className":3833,"style":137},[136],[18,3835,3837],{"className":3836},[141,142,143,144],[18,3838,77],{"className":3839,"style":254},[106,110,144],[18,3841,152],{"className":3842},[151],[18,3844,3846],{"className":3845},[124],[18,3847,3849],{"className":3848,"style":159},[128],[18,3850],{},[18,3852,74],{"className":3853},[250],[18,3855,3857],{"className":3856},[106,681],[18,3858,3860],{"className":3859},[119],[18,3861,3863],{"className":3862},[124],[18,3864,3866,3874],{"className":3865,"style":677},[128],[18,3867,3868,3871],{"style":693},[18,3869],{"className":3870,"style":697},[136],[18,3872,77],{"className":3873,"style":254},[106,110],[18,3875,3876,3879],{"style":703},[18,3877],{"className":3878,"style":697},[136],[18,3880,3882],{"className":3881,"style":711},[710],[18,3883,664],{"className":3884},[106],[18,3886,84],{"className":3887},[267],[18,3889,968],{"className":3890},[106],[1311,3892,3894],{"id":3893},"_22-一个先验视角","2.2 一个先验视角",[11,3896,3897,3898,3926,3927,3958],{},"极小极大风险有一个贝叶斯下界。对任意 ",[18,3899,3901,3914],{"className":3900,"translate":22},[21],[18,3902,3904],{"className":3903},[26],[28,3905,3906],{"xmlns":30},[32,3907,3908,3912],{},[35,3909,3910],{},[41,3911,924],{"mathvariant":316},[86,3913,1497],{"encoding":88},[18,3915,3917],{"className":3916,"ariaHidden":51},[93],[18,3918,3920,3923],{"className":3919},[97],[18,3921],{"className":3922,"style":600},[101],[18,3924,924],{"className":3925},[106]," 上的先验 ",[18,3928,3930,3945],{"className":3929,"translate":22},[21],[18,3931,3933],{"className":3932},[26],[28,3934,3935],{"xmlns":30},[32,3936,3937,3942],{},[35,3938,3939],{},[41,3940,3941],{},"π",[86,3943,3944],{"encoding":88},"\\pi",[18,3946,3948],{"className":3947,"ariaHidden":51},[93],[18,3949,3951,3954],{"className":3950},[97],[18,3952],{"className":3953,"style":1083},[101],[18,3955,3941],{"className":3956,"style":3957},[106,110],"margin-right:0.0359em;","，记贝叶斯风险为",[18,3960,3962],{"className":3961,"translate":22},[875],[18,3963,3965,4041],{"className":3964,"translate":22},[21],[18,3966,3968],{"className":3967},[26],[28,3969,3970],{"xmlns":30,"display":884},[32,3971,3972,4038],{},[35,3973,3974,3981,3983,3985,3987,3989,4003,4006,4012,4014,4020,4022,4025,4027,4029,4032,4034,4036],{},[38,3975,3976,3979],{},[41,3977,3978],{"mathvariant":3525},"B",[41,3980,64],{},[49,3982,74],{"stretchy":73},[41,3984,3941],{},[49,3986,84],{"stretchy":73},[49,3988,433],{},[890,3990,3991,3997],{},[35,3992,3993,3995],{},[41,3994,896],{},[49,3996,899],{},[658,3998,3999,4001],{"accent":51},[41,4000,77],{},[49,4002,664],{},[49,4004,4005],{},"∫",[38,4007,4008,4010],{},[41,4009,3071],{},[41,4011,77],{},[49,4013,74],{"stretchy":73},[658,4015,4016,4018],{"accent":51},[41,4017,77],{},[49,4019,664],{},[49,4021,84],{"stretchy":73},[926,4023,4024],{}," ",[41,4026,3941],{},[49,4028,74],{"stretchy":73},[41,4030,4031],{},"d",[41,4033,77],{},[49,4035,84],{"stretchy":73},[41,4037,968],{"mathvariant":316},[86,4039,4040],{"encoding":88},"\\mathfrak B_n(\\pi)=\\inf_{\\hat\\theta}\\int R_\\theta(\\hat\\theta)\\,\\pi(d\\theta).",[18,4042,4044,4108],{"className":4043,"ariaHidden":51},[93],[18,4045,4047,4050,4090,4093,4096,4099,4102,4105],{"className":4046},[97],[18,4048],{"className":4049,"style":242},[101],[18,4051,4053,4056],{"className":4052},[106],[18,4054,3978],{"className":4055},[106,3607],[18,4057,4059],{"className":4058},[115],[18,4060,4062,4082],{"className":4061},[119,120],[18,4063,4065,4079],{"className":4064},[124],[18,4066,4068],{"className":4067,"style":202},[128],[18,4069,4070,4073],{"style":1155},[18,4071],{"className":4072,"style":137},[136],[18,4074,4076],{"className":4075},[141,142,143,144],[18,4077,64],{"className":4078},[106,110,144],[18,4080,152],{"className":4081},[151],[18,4083,4085],{"className":4084},[124],[18,4086,4088],{"className":4087,"style":159},[128],[18,4089],{},[18,4091,74],{"className":4092},[250],[18,4094,3941],{"className":4095,"style":3957},[106,110],[18,4097,84],{"className":4098},[267],[18,4100],{"className":4101,"style":228},[169],[18,4103,433],{"className":4104},[232],[18,4106],{"className":4107,"style":228},[169],[18,4109,4111,4115,4192,4195,4201,4204,4244,4247,4278,4281,4284,4287,4290,4293,4296,4299],{"className":4110},[97],[18,4112],{"className":4113,"style":4114},[101],"height:2.2965em;vertical-align:-0.9365em;",[18,4116,4118],{"className":4117},[985,986],[18,4119,4121,4184],{"className":4120},[119,120],[18,4122,4124,4181],{"className":4123},[124],[18,4125,4127,4169],{"className":4126,"style":295},[128],[18,4128,4129,4132],{"style":998},[18,4130],{"className":4131,"style":697},[136],[18,4133,4135],{"className":4134},[141,142,143,144],[18,4136,4138],{"className":4137},[106,144],[18,4139,4141],{"className":4140},[106,681,144],[18,4142,4144],{"className":4143},[119],[18,4145,4147],{"className":4146},[124],[18,4148,4150,4158],{"className":4149,"style":677},[128],[18,4151,4152,4155],{"style":1022},[18,4153],{"className":4154,"style":137},[136],[18,4156,77],{"className":4157,"style":254},[106,110,144],[18,4159,4160,4163],{"style":1031},[18,4161],{"className":4162,"style":137},[136],[18,4164,4166],{"className":4165,"style":711},[710],[18,4167,664],{"className":4168},[106,144],[18,4170,4171,4174],{"style":693},[18,4172],{"className":4173,"style":697},[136],[18,4175,4176],{},[18,4177,1051,4179],{"className":4178},[985],[18,4180,1055],{"style":1054},[18,4182,152],{"className":4183},[151],[18,4185,4187],{"className":4186},[124],[18,4188,4190],{"className":4189,"style":1065},[128],[18,4191],{},[18,4193],{"className":4194,"style":170},[169],[18,4196,4005],{"className":4197,"style":4200},[985,4198,4199],"op-symbol","large-op","margin-right:0.4445em;position:relative;top:-0.0011em;",[18,4202],{"className":4203,"style":170},[169],[18,4205,4207,4210],{"className":4206},[106],[18,4208,3071],{"className":4209,"style":3152},[106,110],[18,4211,4213],{"className":4212},[115],[18,4214,4216,4236],{"className":4215},[119,120],[18,4217,4219,4233],{"className":4218},[124],[18,4220,4222],{"className":4221,"style":1383},[128],[18,4223,4224,4227],{"style":3167},[18,4225],{"className":4226,"style":137},[136],[18,4228,4230],{"className":4229},[141,142,143,144],[18,4231,77],{"className":4232,"style":254},[106,110,144],[18,4234,152],{"className":4235},[151],[18,4237,4239],{"className":4238},[124],[18,4240,4242],{"className":4241,"style":159},[128],[18,4243],{},[18,4245,74],{"className":4246},[250],[18,4248,4250],{"className":4249},[106,681],[18,4251,4253],{"className":4252},[119],[18,4254,4256],{"className":4255},[124],[18,4257,4259,4267],{"className":4258,"style":677},[128],[18,4260,4261,4264],{"style":693},[18,4262],{"className":4263,"style":697},[136],[18,4265,77],{"className":4266,"style":254},[106,110],[18,4268,4269,4272],{"style":703},[18,4270],{"className":4271,"style":697},[136],[18,4273,4275],{"className":4274,"style":711},[710],[18,4276,664],{"className":4277},[106],[18,4279,84],{"className":4280},[267],[18,4282],{"className":4283,"style":170},[169],[18,4285,3941],{"className":4286,"style":3957},[106,110],[18,4288,74],{"className":4289},[250],[18,4291,4031],{"className":4292},[106,110],[18,4294,77],{"className":4295,"style":254},[106,110],[18,4297,84],{"className":4298},[267],[18,4300,968],{"className":4301},[106],[11,4303,4304],{},"由于「积分 ≤ 最大值」，",[18,4306,4308],{"className":4307,"translate":22},[875],[18,4309,4311,4356],{"className":4310,"translate":22},[21],[18,4312,4314],{"className":4313},[26],[28,4315,4316],{"xmlns":30,"display":884},[32,4317,4318,4353],{},[35,4319,4320,4326,4328,4330,4332,4335,4341,4343,4345,4347,4349,4351],{},[38,4321,4322,4324],{},[41,4323,3978],{"mathvariant":3525},[41,4325,64],{},[49,4327,74],{"stretchy":73},[41,4329,3941],{},[49,4331,84],{"stretchy":73},[49,4333,4334],{},"≤",[38,4336,4337,4339],{},[41,4338,3526],{"mathvariant":3525},[41,4340,64],{},[49,4342,74],{"stretchy":73},[41,4344,924],{"mathvariant":316},[49,4346,52],{"separator":51},[41,4348,948],{"mathvariant":316},[49,4350,84],{"stretchy":73},[41,4352,968],{"mathvariant":316},[86,4354,4355],{"encoding":88},"\\mathfrak B_n(\\pi)\\le \\mathfrak M_n(\\Theta,\\ell).",[18,4357,4359,4423],{"className":4358,"ariaHidden":51},[93],[18,4360,4362,4365,4405,4408,4411,4414,4417,4420],{"className":4361},[97],[18,4363],{"className":4364,"style":242},[101],[18,4366,4368,4371],{"className":4367},[106],[18,4369,3978],{"className":4370},[106,3607],[18,4372,4374],{"className":4373},[115],[18,4375,4377,4397],{"className":4376},[119,120],[18,4378,4380,4394],{"className":4379},[124],[18,4381,4383],{"className":4382,"style":202},[128],[18,4384,4385,4388],{"style":1155},[18,4386],{"className":4387,"style":137},[136],[18,4389,4391],{"className":4390},[141,142,143,144],[18,4392,64],{"className":4393},[106,110,144],[18,4395,152],{"className":4396},[151],[18,4398,4400],{"className":4399},[124],[18,4401,4403],{"className":4402,"style":159},[128],[18,4404],{},[18,4406,74],{"className":4407},[250],[18,4409,3941],{"className":4410,"style":3957},[106,110],[18,4412,84],{"className":4413},[267],[18,4415],{"className":4416,"style":228},[169],[18,4418,4334],{"className":4419},[232],[18,4421],{"className":4422,"style":228},[169],[18,4424,4426,4429,4469,4472,4475,4478,4481,4484,4487],{"className":4425},[97],[18,4427],{"className":4428,"style":242},[101],[18,4430,4432,4435],{"className":4431},[106],[18,4433,3526],{"className":4434},[106,3607],[18,4436,4438],{"className":4437},[115],[18,4439,4441,4461],{"className":4440},[119,120],[18,4442,4444,4458],{"className":4443},[124],[18,4445,4447],{"className":4446,"style":202},[128],[18,4448,4449,4452],{"style":1155},[18,4450],{"className":4451,"style":137},[136],[18,4453,4455],{"className":4454},[141,142,143,144],[18,4456,64],{"className":4457},[106,110,144],[18,4459,152],{"className":4460},[151],[18,4462,4464],{"className":4463},[124],[18,4465,4467],{"className":4466,"style":159},[128],[18,4468],{},[18,4470,74],{"className":4471},[250],[18,4473,924],{"className":4474},[106],[18,4476,52],{"className":4477},[165],[18,4479],{"className":4480,"style":170},[169],[18,4482,948],{"className":4483},[106],[18,4485,84],{"className":4486},[267],[18,4488,968],{"className":4489},[106],[11,4491,4492,4493,4496],{},"所以，",[717,4494,4495],{},"任何一个先验下的贝叶斯风险，都是极小极大风险的下界","。这个观察非常有用：下界证明往往先选一个先验（通常是有限集上的均匀分布），再算贝叶斯风险。Le Cam 与 Fano 本质上都是在做这件事，只是把贝叶斯风险进一步转化成了检验误差。",[1311,4498,4500],{"id":4499},"_23-归约到有限子问题","2.3 归约到有限子问题",[11,4502,4503,4504,4741],{},"对任意有限子集 ",[18,4505,4507,4554],{"className":4506,"translate":22},[21],[18,4508,4510],{"className":4509},[26],[28,4511,4512],{"xmlns":30},[32,4513,4514,4551],{},[35,4515,4516,4522,4524,4526,4532,4534,4536,4538,4544,4546,4549],{},[38,4517,4518,4520],{},[41,4519,924],{"mathvariant":316},[45,4521,430],{},[49,4523,433],{},[49,4525,1331],{"stretchy":73},[38,4527,4528,4530],{},[41,4529,77],{},[45,4531,47],{},[49,4533,52],{"separator":51},[49,4535,55],{},[49,4537,52],{"separator":51},[38,4539,4540,4542],{},[41,4541,77],{},[41,4543,3526],{},[49,4545,1349],{"stretchy":73},[49,4547,4548],{},"⊆",[41,4550,924],{"mathvariant":316},[86,4552,4553],{"encoding":88},"\\Theta_0=\\{\\theta_1,\\dots,\\theta_M\\}\\subseteq\\Theta",[18,4555,4557,4613,4732],{"className":4556,"ariaHidden":51},[93],[18,4558,4560,4564,4604,4607,4610],{"className":4559},[97],[18,4561],{"className":4562,"style":4563},[101],"height:0.8333em;vertical-align:-0.15em;",[18,4565,4567,4570],{"className":4566},[106],[18,4568,924],{"className":4569},[106],[18,4571,4573],{"className":4572},[115],[18,4574,4576,4596],{"className":4575},[119,120],[18,4577,4579,4593],{"className":4578},[124],[18,4580,4582],{"className":4581,"style":129},[128],[18,4583,4584,4587],{"style":1155},[18,4585],{"className":4586,"style":137},[136],[18,4588,4590],{"className":4589},[141,142,143,144],[18,4591,430],{"className":4592},[106,144],[18,4594,152],{"className":4595},[151],[18,4597,4599],{"className":4598},[124],[18,4600,4602],{"className":4601,"style":159},[128],[18,4603],{},[18,4605],{"className":4606,"style":228},[169],[18,4608,433],{"className":4609},[232],[18,4611],{"className":4612,"style":228},[169],[18,4614,4616,4619,4622,4662,4665,4668,4671,4674,4677,4680,4720,4723,4726,4729],{"className":4615},[97],[18,4617],{"className":4618,"style":242},[101],[18,4620,1331],{"className":4621},[250],[18,4623,4625,4628],{"className":4624},[106],[18,4626,77],{"className":4627,"style":254},[106,110],[18,4629,4631],{"className":4630},[115],[18,4632,4634,4654],{"className":4633},[119,120],[18,4635,4637,4651],{"className":4636},[124],[18,4638,4640],{"className":4639,"style":129},[128],[18,4641,4642,4645],{"style":469},[18,4643],{"className":4644,"style":137},[136],[18,4646,4648],{"className":4647},[141,142,143,144],[18,4649,47],{"className":4650},[106,144],[18,4652,152],{"className":4653},[151],[18,4655,4657],{"className":4656},[124],[18,4658,4660],{"className":4659,"style":159},[128],[18,4661],{},[18,4663,52],{"className":4664},[165],[18,4666],{"className":4667,"style":170},[169],[18,4669,55],{"className":4670},[174],[18,4672],{"className":4673,"style":170},[169],[18,4675,52],{"className":4676},[165],[18,4678],{"className":4679,"style":170},[169],[18,4681,4683,4686],{"className":4682},[106],[18,4684,77],{"className":4685,"style":254},[106,110],[18,4687,4689],{"className":4688},[115],[18,4690,4692,4712],{"className":4691},[119,120],[18,4693,4695,4709],{"className":4694},[124],[18,4696,4698],{"className":4697,"style":1152},[128],[18,4699,4700,4703],{"style":469},[18,4701],{"className":4702,"style":137},[136],[18,4704,4706],{"className":4705},[141,142,143,144],[18,4707,3526],{"className":4708,"style":246},[106,110,144],[18,4710,152],{"className":4711},[151],[18,4713,4715],{"className":4714},[124],[18,4716,4718],{"className":4717,"style":159},[128],[18,4719],{},[18,4721,1349],{"className":4722},[267],[18,4724],{"className":4725,"style":228},[169],[18,4727,4548],{"className":4728},[232],[18,4730],{"className":4731,"style":228},[169],[18,4733,4735,4738],{"className":4734},[97],[18,4736],{"className":4737,"style":600},[101],[18,4739,924],{"className":4740},[106],"，",[18,4743,4745],{"className":4744,"translate":22},[875],[18,4746,4748,4801],{"className":4747,"translate":22},[21],[18,4749,4751],{"className":4750},[26],[28,4752,4753],{"xmlns":30,"display":884},[32,4754,4755,4798],{},[35,4756,4757,4763,4765,4767,4769,4771,4773,4776,4782,4784,4790,4792,4794,4796],{},[38,4758,4759,4761],{},[41,4760,3526],{"mathvariant":3525},[41,4762,64],{},[49,4764,74],{"stretchy":73},[41,4766,924],{"mathvariant":316},[49,4768,52],{"separator":51},[41,4770,948],{"mathvariant":316},[49,4772,84],{"stretchy":73},[49,4774,4775],{},"≥",[38,4777,4778,4780],{},[41,4779,3526],{"mathvariant":3525},[41,4781,64],{},[49,4783,74],{"stretchy":73},[38,4785,4786,4788],{},[41,4787,924],{"mathvariant":316},[45,4789,430],{},[49,4791,52],{"separator":51},[41,4793,948],{"mathvariant":316},[49,4795,84],{"stretchy":73},[41,4797,968],{"mathvariant":316},[86,4799,4800],{"encoding":88},"\\mathfrak M_n(\\Theta,\\ell)\\ge \\mathfrak M_n(\\Theta_0,\\ell).",[18,4802,4804,4877],{"className":4803,"ariaHidden":51},[93],[18,4805,4807,4810,4850,4853,4856,4859,4862,4865,4868,4871,4874],{"className":4806},[97],[18,4808],{"className":4809,"style":242},[101],[18,4811,4813,4816],{"className":4812},[106],[18,4814,3526],{"className":4815},[106,3607],[18,4817,4819],{"className":4818},[115],[18,4820,4822,4842],{"className":4821},[119,120],[18,4823,4825,4839],{"className":4824},[124],[18,4826,4828],{"className":4827,"style":202},[128],[18,4829,4830,4833],{"style":1155},[18,4831],{"className":4832,"style":137},[136],[18,4834,4836],{"className":4835},[141,142,143,144],[18,4837,64],{"className":4838},[106,110,144],[18,4840,152],{"className":4841},[151],[18,4843,4845],{"className":4844},[124],[18,4846,4848],{"className":4847,"style":159},[128],[18,4849],{},[18,4851,74],{"className":4852},[250],[18,4854,924],{"className":4855},[106],[18,4857,52],{"className":4858},[165],[18,4860],{"className":4861,"style":170},[169],[18,4863,948],{"className":4864},[106],[18,4866,84],{"className":4867},[267],[18,4869],{"className":4870,"style":228},[169],[18,4872,4775],{"className":4873},[232],[18,4875],{"className":4876,"style":228},[169],[18,4878,4880,4883,4923,4926,4966,4969,4972,4975,4978],{"className":4879},[97],[18,4881],{"className":4882,"style":242},[101],[18,4884,4886,4889],{"className":4885},[106],[18,4887,3526],{"className":4888},[106,3607],[18,4890,4892],{"className":4891},[115],[18,4893,4895,4915],{"className":4894},[119,120],[18,4896,4898,4912],{"className":4897},[124],[18,4899,4901],{"className":4900,"style":202},[128],[18,4902,4903,4906],{"style":1155},[18,4904],{"className":4905,"style":137},[136],[18,4907,4909],{"className":4908},[141,142,143,144],[18,4910,64],{"className":4911},[106,110,144],[18,4913,152],{"className":4914},[151],[18,4916,4918],{"className":4917},[124],[18,4919,4921],{"className":4920,"style":159},[128],[18,4922],{},[18,4924,74],{"className":4925},[250],[18,4927,4929,4932],{"className":4928},[106],[18,4930,924],{"className":4931},[106],[18,4933,4935],{"className":4934},[115],[18,4936,4938,4958],{"className":4937},[119,120],[18,4939,4941,4955],{"className":4940},[124],[18,4942,4944],{"className":4943,"style":129},[128],[18,4945,4946,4949],{"style":1155},[18,4947],{"className":4948,"style":137},[136],[18,4950,4952],{"className":4951},[141,142,143,144],[18,4953,430],{"className":4954},[106,144],[18,4956,152],{"className":4957},[151],[18,4959,4961],{"className":4960},[124],[18,4962,4964],{"className":4963,"style":159},[128],[18,4965],{},[18,4967,52],{"className":4968},[165],[18,4970],{"className":4971,"style":170},[169],[18,4973,948],{"className":4974},[106],[18,4976,84],{"className":4977},[267],[18,4979,968],{"className":4980},[106],[11,4982,4983,4984,5012,5013,5085,5086,5114,5115,5143],{},"所以我们只需要在一个有限问题上证明下界。有限问题可以看成一场 ",[18,4985,4987,5000],{"className":4986,"translate":22},[21],[18,4988,4990],{"className":4989},[26],[28,4991,4992],{"xmlns":30},[32,4993,4994,4998],{},[35,4995,4996],{},[41,4997,3526],{},[86,4999,3526],{"encoding":88},[18,5001,5003],{"className":5002,"ariaHidden":51},[93],[18,5004,5006,5009],{"className":5005},[97],[18,5007],{"className":5008,"style":600},[101],[18,5010,3526],{"className":5011,"style":246},[106,110]," 元检验：自然以某个先验选出 ",[18,5014,5016,5035],{"className":5015,"translate":22},[21],[18,5017,5019],{"className":5018},[26],[28,5020,5021],{"xmlns":30},[32,5022,5023,5032],{},[35,5024,5025],{},[38,5026,5027,5029],{},[41,5028,77],{},[41,5030,5031],{},"K",[86,5033,5034],{"encoding":88},"\\theta_K",[18,5036,5038],{"className":5037,"ariaHidden":51},[93],[18,5039,5041,5044],{"className":5040},[97],[18,5042],{"className":5043,"style":448},[101],[18,5045,5047,5050],{"className":5046},[106],[18,5048,77],{"className":5049,"style":254},[106,110],[18,5051,5053],{"className":5052},[115],[18,5054,5056,5077],{"className":5055},[119,120],[18,5057,5059,5074],{"className":5058},[124],[18,5060,5062],{"className":5061,"style":1152},[128],[18,5063,5064,5067],{"style":469},[18,5065],{"className":5066,"style":137},[136],[18,5068,5070],{"className":5069},[141,142,143,144],[18,5071,5031],{"className":5072,"style":5073},[106,110,144],"margin-right:0.0715em;",[18,5075,152],{"className":5076},[151],[18,5078,5080],{"className":5079},[124],[18,5081,5083],{"className":5082,"style":159},[128],[18,5084],{},"，统计学家看到数据后要猜出 ",[18,5087,5089,5102],{"className":5088,"translate":22},[21],[18,5090,5092],{"className":5091},[26],[28,5093,5094],{"xmlns":30},[32,5095,5096,5100],{},[35,5097,5098],{},[41,5099,5031],{},[86,5101,5031],{"encoding":88},[18,5103,5105],{"className":5104,"ariaHidden":51},[93],[18,5106,5108,5111],{"className":5107},[97],[18,5109],{"className":5110,"style":600},[101],[18,5112,5031],{"className":5113,"style":5073},[106,110],"。如果连「猜出是哪一个」都做不到，那么「把 ",[18,5116,5118,5131],{"className":5117,"translate":22},[21],[18,5119,5121],{"className":5120},[26],[28,5122,5123],{"xmlns":30},[32,5124,5125,5129],{},[35,5126,5127],{},[41,5128,77],{},[86,5130,285],{"encoding":88},[18,5132,5134],{"className":5133,"ariaHidden":51},[93],[18,5135,5137,5140],{"className":5136},[97],[18,5138],{"className":5139,"style":295},[101],[18,5141,77],{"className":5142,"style":254},[106,110]," 估到很小的损失」当然也做不到。",[11,5145,5146],{},"把这句话写成不等式，就是 Le Cam 与 Fano 的工具。",[638,5148,5150],{"id":5149},"三le-cam-两点方法","三、Le Cam 两点方法",[1311,5152,5154],{"id":5153},"_31-定理与证明","3.1 定理与证明",[11,5156,5157],{},"先看最简单的情形：只有两个候选参数。",[5159,5160,5161,5572],"blockquote",{},[11,5162,5163,5166,5167,4741,5313,5481,5482,5511,5512,4741],{},[717,5164,5165],{},"定理 3.1（Le Cam 两点方法）"," 设 ",[18,5168,5170,5200],{"className":5169,"translate":22},[21],[18,5171,5173],{"className":5172},[26],[28,5174,5175],{"xmlns":30},[32,5176,5177,5197],{},[35,5178,5179,5185,5187,5193,5195],{},[38,5180,5181,5183],{},[41,5182,77],{},[45,5184,430],{},[49,5186,52],{"separator":51},[38,5188,5189,5191],{},[41,5190,77],{},[45,5192,47],{},[49,5194,921],{},[41,5196,924],{"mathvariant":316},[86,5198,5199],{"encoding":88},"\\theta_0,\\theta_1\\in\\Theta",[18,5201,5203,5304],{"className":5202,"ariaHidden":51},[93],[18,5204,5206,5209,5249,5252,5255,5295,5298,5301],{"className":5205},[97],[18,5207],{"className":5208,"style":2890},[101],[18,5210,5212,5215],{"className":5211},[106],[18,5213,77],{"className":5214,"style":254},[106,110],[18,5216,5218],{"className":5217},[115],[18,5219,5221,5241],{"className":5220},[119,120],[18,5222,5224,5238],{"className":5223},[124],[18,5225,5227],{"className":5226,"style":129},[128],[18,5228,5229,5232],{"style":469},[18,5230],{"className":5231,"style":137},[136],[18,5233,5235],{"className":5234},[141,142,143,144],[18,5236,430],{"className":5237},[106,144],[18,5239,152],{"className":5240},[151],[18,5242,5244],{"className":5243},[124],[18,5245,5247],{"className":5246,"style":159},[128],[18,5248],{},[18,5250,52],{"className":5251},[165],[18,5253],{"className":5254,"style":170},[169],[18,5256,5258,5261],{"className":5257},[106],[18,5259,77],{"className":5260,"style":254},[106,110],[18,5262,5264],{"className":5263},[115],[18,5265,5267,5287],{"className":5266},[119,120],[18,5268,5270,5284],{"className":5269},[124],[18,5271,5273],{"className":5272,"style":129},[128],[18,5274,5275,5278],{"style":469},[18,5276],{"className":5277,"style":137},[136],[18,5279,5281],{"className":5280},[141,142,143,144],[18,5282,47],{"className":5283},[106,144],[18,5285,152],{"className":5286},[151],[18,5288,5290],{"className":5289},[124],[18,5291,5293],{"className":5292,"style":159},[128],[18,5294],{},[18,5296],{"className":5297,"style":228},[169],[18,5299,921],{"className":5300},[232],[18,5302],{"className":5303,"style":228},[169],[18,5305,5307,5310],{"className":5306},[97],[18,5308],{"className":5309,"style":600},[101],[18,5311,924],{"className":5312},[106],[18,5314,5316,5355],{"className":5315,"translate":22},[21],[18,5317,5319],{"className":5318},[26],[28,5320,5321],{"xmlns":30},[32,5322,5323,5352],{},[35,5324,5325,5327,5329,5335,5337,5343,5345,5347,5349],{},[41,5326,948],{"mathvariant":316},[49,5328,74],{"stretchy":73},[38,5330,5331,5333],{},[41,5332,77],{},[45,5334,430],{},[49,5336,52],{"separator":51},[38,5338,5339,5341],{},[41,5340,77],{},[45,5342,47],{},[49,5344,84],{"stretchy":73},[49,5346,4775],{},[45,5348,2266],{},[41,5350,5351],{},"δ",[86,5353,5354],{"encoding":88},"\\ell(\\theta_0,\\theta_1)\\ge 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E_{P_0}\\ell(\\theta_0,\\hat\\theta)+\\mathbb E_{P_1}\\ell(\\theta_1,\\hat\\theta)\n\\ge \\\\ \\delta\\bigl[1+P_0(\\psi=1)-P_1(\\psi=1)\\bigr]\n\\ge \\\\ \\delta\\bigl(1-\\lVert 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me":8430},[165],[18,8432],{"className":8433,"style":170},[169],[18,8435,8437],{"className":8436},[106,681],[18,8438,8440],{"className":8439},[119],[18,8441,8443],{"className":8442},[124],[18,8444,8446,8454],{"className":8445,"style":677},[128],[18,8447,8448,8451],{"style":693},[18,8449],{"className":8450,"style":697},[136],[18,8452,77],{"className":8453,"style":254},[106,110],[18,8455,8456,8459],{"style":703},[18,8457],{"className":8458,"style":697},[136],[18,8460,8462],{"className":8461,"style":711},[710],[18,8463,664],{"className":8464},[106],[18,8466,84],{"className":8467},[267],[18,8469],{"className":8470,"style":228},[169],[18,8472,4775],{"className":8473},[232],[18,8475],{"className":8476},[169,8022],[18,8478,8480,8483,8486,8492,8495,8498,8501],{"className":8479},[97],[18,8481],{"className":8482,"style":6101},[101],[18,8484,5351],{"className":8485,"style":5480},[106,110],[18,8487,8489],{"className":8488},[250],[18,8490,945],{"className":8491},[1230,1196],[18,8493,47],{"className":8494},[106],[18,8496],{"className":8497,"style":2068},[169],[18,8499,7987],{"className":8500},[2072],[18,8502],{"className":8503,"style":2068},[169],[18,8505,8507,8510,8550,8553,8556,8559,8562],{"className":8506},[97],[18,8508],{"className":8509,"style":242},[101],[18,8511,8513,8516],{"className":8512},[106],[18,8514,939],{"className":8515,"style":1171},[106,110],[18,8517,8519],{"className":8518},[115],[18,8520,8522,8542],{"className":8521},[119,120],[18,8523,8525,8539],{"className":8524},[124],[18,8526,8528],{"className":8527,"style":129},[128],[18,8529,8530,8533],{"style":1386},[18,8531],{"className":8532,"style":137},[136],[18,8534,8536],{"className":8535},[141,142,143,144],[18,8537,430],{"className":8538},[106,144],[18,8540,152],{"className":8541},[151],[18,8543,8545],{"className":8544},[124],[18,8546,8548],{"className":8547,"style":159},[128],[18,8549],{},[18,8551,74],{"className":8552},[250],[18,8554,6398],{"className":8555,"style":3957},[106,110],[18,8557],{"className":8558,"style":228},[169],[18,8560,433],{"className":8561},[232],[18,8563],{"className":8564,"style":228},[169],[18,8566,8568,8571,8574,8577,8580,8583],{"className":8567},[97],[18,8569],{"className":8570,"style":242},[101],[18,8572,47],{"className":8573},[106],[18,8575,84],{"className":8576},[267],[18,8578],{"className":8579,"style":2068},[169],[18,8581,2253],{"className":8582},[2072],[18,8584],{"className":8585,"style":2068},[169],[18,8587,8589,8592,8632,8635,8638,8641,8644],{"className":8588},[97],[18,8590],{"className":8591,"style":242},[101],[18,8593,8595,8598],{"className":8594},[106],[18,8596,939],{"className":8597,"style":1171},[106,110],[18,8599,8601],{"className":8600},[115],[18,8602,8604,8624],{"className":8603},[119,120],[18,8605,8607,8621],{"className":8606},[124],[18,8608,8610],{"className":8609,"style":129},[128],[18,8611,8612,8615],{"style":1386},[18,8613],{"className":8614,"style":137},[136],[18,8616,8618],{"className":8617},[141,142,143,144],[18,8619,47],{"className":8620},[106,144],[18,8622,152],{"className":8623},[151],[18,8625,8627],{"className":8626},[124],[18,8628,8630],{"className":8629,"style":159},[128],[18,8631],{},[18,8633,74],{"className":8634},[250],[18,8636,6398],{"className":8637,"style":3957},[106,110],[18,8639],{"className":8640,"style":228},[169],[18,8642,433],{"className":8643},[232],[18,8645],{"className":8646,"style":228},[169],[18,8648,8650,8653,8656,8659,8665,8668],{"className":8649},[97],[18,8651],{"className":8652,"style":6101},[101],[18,8654,47],{"className":8655},[106],[18,8657,84],{"className":8658},[267],[18,8660,8662],{"className":8661},[267],[18,8663,965],{"className":8664},[1230,1196],[18,8666],{"className":8667,"style":228},[169],[18,8669,4775],{"className":8670},[232],[18,8672],{"className":8673},[169,8022],[18,8675,8677,8680,8683,8689,8692,8695,8698],{"className":8676},[97],[18,8678],{"className":8679,"style":6101},[101],[18,8681,5351],{"className":8682,"style":5480},[106,110],[18,8684,8686],{"className":8685},[250],[18,8687,74],{"className":8688},[1230,1196],[18,8690,47],{"className":8691},[106],[18,8693],{"className":8694,"style":2068},[169],[18,8696,2253],{"className":8697},[2072],[18,8699],{"className":8700,"style":2068},[169],[18,8702,8704,8707,8710,8750,8753,8756],{"className":8703},[97],[18,8705],{"className":8706,"style":242},[101],[18,8708,2248],{"className":8709},[250],[18,8711,8713,8716],{"className":8712},[106],[18,8714,939],{"className":8715,"style":1171},[106,110],[18,8717,8719],{"className":8718},[115],[18,8720,8722,8742],{"className":8721},[119,120],[18,8723,8725,8739],{"className":8724},[124],[18,8726,8728],{"className":8727,"style":129},[128],[18,8729,8730,8733],{"style":1386},[18,8731],{"className":8732,"style":137},[136],[18,8734,8736],{"className":8735},[141,142,143,144],[18,8737,430],{"className":8738},[106,144],[18,8740,152],{"className":8741},[151],[18,8743,8745],{"className":8744},[124],[18,8746,8748],{"className":8747,"style":159},[128],[18,8749],{},[18,8751],{"className":8752,"st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",[18,8864,8866,8898],{"className":8865,"translate":22},[21],[18,8867,8869],{"className":8868},[26],[28,8870,8871],{"xmlns":30},[32,8872,8873,8895],{},[35,8874,8875,8877,8879,8881,8883,8889,8891,8893],{},[41,8876,5351],{},[49,8878,74],{"stretchy":73},[45,8880,47],{},[49,8882,2253],{},[35,8884,8885,8887],{},[41,8886,5703],{"mathvariant":316},[41,8888,5706],{"mathvariant":316},[49,8890,84],{"stretchy":73},[41,8892,317],{"mathvariant":316},[45,8894,2266],{},[86,8896,8897],{"encoding":88},"\\delta(1-\\mathrm{TV})\u002F2",[18,8899,8901,8925],{"className":8900,"ariaHidden":51},[93],[18,8902,8904,8907,8910,8913,8916,8919,8922],{"className":8903},[97],[18,8905],{"className":8906,"style":242},[101],[18,8908,5351],{"className":8909,"style":5480},[106,110],[18,8911,74],{"className":8912},[250],[18,8914,47],{"className":8915},[106],[18,8917],{"className":8918,"style":2068},[169],[18,8920,2253],{"className":8921},[2072],[18,8923],{"className":8924,"style":2068},[169],[18,8926,8928,8931,8937,8940],{"className":8927},[97],[18,8929],{"className":8930,"style":242},[101],[18,8932,8934],{"className":8933},[106],[18,8935,6356],{"className":8936,"style":6355},[106,6354],[18,8938,84],{"className":8939},[267],[18,8941,8943],{"className":8942},[106],"\u002F2","。证毕。",[11,8946,8947,8948,2702,8976,9004,9005,9061],{},"如果取 ",[18,8949,8951,8964],{"className":8950,"translate":22},[21],[18,8952,8954],{"className":8953},[26],[28,8955,8956],{"xmlns":30},[32,8957,8958,8962],{},[35,8959,8960],{},[45,8961,430],{},[86,8963,430],{"encoding":88},[18,8965,8967],{"className":8966,"ariaHidden":51},[93],[18,8968,8970,8973],{"className":8969},[97],[18,8971],{"className":8972,"style":505},[101],[18,8974,430],{"className":8975},[106],[18,8977,8979,8992],{"className":8978,"translate":22},[21],[18,8980,8982],{"className":8981},[26],[28,8983,8984],{"xmlns":30},[32,8985,8986,8990],{},[35,8987,8988],{},[45,8989,47],{},[86,8991,47],{"encoding":88},[18,8993,8995],{"className":8994,"ariaHidden":51},[93],[18,8996,8998,9001],{"className":8997},[97],[18,8999],{"className":9000,"style":505},[101],[18,9002,47],{"className":9003},[106]," 损失、",[18,9006,9008,9030],{"className":9007,"translate":22},[21],[18,9009,9011],{"className":9010},[26],[28,9012,9013],{"xmlns":30},[32,9014,9015,9027],{},[35,9016,9017,9019,9021,9023,9025],{},[41,9018,5351],{},[49,9020,433],{},[45,9022,47],{},[41,9024,317],{"mathvariant":316},[45,9026,2266],{},[86,9028,9029],{"encoding":88},"\\delta=1\u002F2",[18,9031,9033,9051],{"className":9032,"ariaHidden":51},[93],[18,9034,9036,9039,9042,9045,9048],{"className":9035},[97],[18,9037],{"className":9038,"style":295},[101],[18,9040,5351],{"className":9041,"style":5480},[106,110],[18,9043],{"className":9044,"style":228},[169],[18,9046,433],{"className":9047},[232],[18,9049],{"className":9050,"style":228},[169],[18,9052,9054,9057],{"className":9053},[97],[18,9055],{"className":9056,"style":242},[101],[18,9058,9060],{"className":9059},[106],"1\u002F2","，就得到最常见的检验形式：",[18,9063,9065],{"className":9064,"translate":22},[875],[18,9066,9068,9202],{"className":9067,"translate":22},[21],[18,9069,9071],{"className":9070},[26],[28,9072,9073],{"xmlns":30,"display":884},[32,9074,9075,9199],{},[35,9076,9077,9091,9111,9113,9125,9127,9133,9135,9141,9143,9145,9151,9153,9155,9157,9159,9171,9173,9185,9195,9197],{},[890,9078,9079,9085],{},[35,9080,9081,9083],{},[41,9082,896],{},[49,9084,899],{},[658,9086,9087,9089],{"accent":51},[41,9088,77],{},[49,9090,664],{},[890,9092,9093,9099],{},[35,9094,9095,9097],{},[41,9096,5593],{},[49,9098,899],{},[35,9100,9101,9103,9105,9107,9109],{},[41,9102,5600],{},[49,9104,433],{},[45,9106,430],{},[49,9108,52],{"separator":51},[45,9110,47],{},[926,9112,928],{},[1589,9114,9115,9117,9119],{},[41,9116,939],{},[41,9118,5600],{},[35,9120,9121,9123],{},[49,9122,1599],{},[41,9124,64],{},[49,9126,74],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[658,9128,9129,9131],{"accent":51},[41,9130,77],{},[49,9132,664],{},[49,9134,2771],{"mathvariant":316},[38,9136,9137,9139],{},[41,9138,77],{},[41,9140,5600],{},[49,9142,84],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,9144,4775],{},[5654,9146,9147,9149],{},[45,9148,47],{},[45,9150,2266],{},[49,9152,74],{"fence":51,"stretchy":51,"minsize":5662,"maxsize":5662},[45,9154,47],{},[49,9156,2253],{},[49,9158,2248],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[1589,9160,9161,9163,9165],{},[41,9162,939],{},[45,9164,430],{},[35,9166,9167,9169],{},[49,9168,1599],{},[41,9170,64],{},[49,9172,2253],{},[1589,9174,9175,9177,9179],{},[41,9176,939],{},[45,9178,47],{},[35,9180,9181,9183],{},[49,9182,1599],{},[41,9184,64],{},[38,9186,9187,9189],{},[49,9188,2248],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[35,9190,9191,9193],{},[41,9192,5703],{"mathvariant":316},[41,9194,5706],{"mathvariant":316},[49,9196,84],{"fence":51,"stretchy":51,"minsize":5662,"maxsize":5662},[41,9198,968],{"mathvariant":316},[86,9200,9201],{"encoding":88},"\\inf_{\\hat\\theta}\\max_{i=0,1}\\ P_i^{\\otimes n}\\bigl(\\hat\\theta\\neq\\theta_i\\bigr)\n\\ge\n\\frac12\\Bigl(1-\\bigl\\lVert P_0^{\\otimes n}-P_1^{\\otimes n}\\bigr\\rVert_{\\mathrm{TV}}\\Bigr).",[18,9203,9205,9495,9556,9644,9753],{"className":9204,"ariaHidden":51},[93],[18,9206,9208,9212,9289,9292,9351,9354,9357,9416,9422,9453,9456,9492],{"className":9207},[97],[18,9209],{"className":9210,"style":9211},[101],"height:1.8944em;vertical-align:-0.9365em;",[18,9213,9215],{"className":9214},[985,986],[18,9216,9218,9281],{"className":9217},[119,120],[18,9219,9221,9278],{"className":9220},[124],[18,9222,9224,9266],{"className":9223,"style":295},[128],[18,9225,9226,9229],{"style":998},[18,9227],{"className":9228,"style":697},[136],[18,9230,9232],{"className":9231},[141,142,143,144],[18,9233,9235],{"className":9234},[106,144],[18,9236,9238],{"className":9237},[106,681,144],[18,9239,9241],{"className":9240},[119],[18,9242,9244],{"className":9243},[124],[18,9245,9247,9255],{"className":9246,"style":677},[128],[18,9248,9249,9252],{"style":1022},[18,9250],{"className":9251,"style":137},[136],[18,9253,77],{"className":9254,"style":254},[106,110,144],[18,9256,9257,9260],{"style":1031},[18,9258],{"className":9259,"style":137},[136],[18,9261,9263],{"className":9262,"style":711},[710],[18,9264,664],{"className":9265},[106,144],[18,9267,9268,9271],{"style":693},[18,9269],{"className":9270,"style":697},[136],[18,9272,9273],{},[18,9274,1051,9276],{"className":9275},[985],[18,9277,1055],{"style":1054},[18,9279,152],{"className":9280},[151],[18,9282,9284],{"className":9283},[124],[18,9285,9287],{"className":9286,"style":1065},[128],[18,9288],{},[18,9290],{"className":9291,"style":170},[169],[18,9293,9295],{"className":9294},[985,986],[18,9296,9298,9343],{"className":9297},[119,120],[18,9299,9301,9340],{"className":9300},[124],[18,9302,9304,9330],{"className":9303,"style":1083},[128],[18,9305,9306,9309],{"style":5738},[18,9307],{"className":9308,"style":697},[136],[18,9310,9312],{"className":9311},[141,142,143,144],[18,9313,9315,9318,9321,9324,9327],{"className":9314},[106,144],[18,9316,5600],{"className":9317},[106,110,144],[18,9319,433],{"className":9320},[232,144],[18,9322,430],{"className":9323},[106,144],[18,9325,52],{"className":9326},[165,144],[18,9328,47],{"className":9329},[106,144],[18,9331,9332,9335],{"style":693},[18,9333],{"className":9334,"style":697},[136],[18,9336,9337],{},[18,9338,5593],{"className":9339},[985],[18,9341,152],{"className":9342},[151],[18,9344,9346],{"className":9345},[124],[18,9347,9349],{"className":9348,"style":5782},[128],[18,9350],{},[18,9352,928],{"className":9353},[169],[18,9355],{"className":9356,"style":170},[169],[18,9358,9360,9363],{"className":9359},[106],[18,9361,939],{"className":9362,"style":1171},[106,110],[18,9364,9366],{"className":9365},[115],[18,9367,9369,9407],{"className":9368},[119,120],[18,9370,9372,9404],{"className":9371},[124],[18,9373,9375,9387],{"className":9374,"style":1800},[128],[18,9376,9378,9381],{"style":9377},"top:-2.433em;margin-left:-0.1389em;margin-right:0.05em;",[18,9379],{"className":9380,"style":137},[136],[18,9382,9384],{"className":9383},[141,142,143,144],[18,9385,5600],{"className":9386},[106,110,144],[18,9388,9389,9392],{"style":1637},[18,9390],{"className":9391,"style":137},[136],[18,9393,9395],{"className":9394},[141,142,143,144],[18,9396,9398,9401],{"className":9397},[106,144],[18,9399,1599],{"className":9400},[106,144],[18,9402,64],{"className":9403},[106,110,144],[18,9405,152],{"className":9406},[151],[18,9408,9410],{"className":9409},[124],[18,9411,9414],{"className":9412,"style":9413},[128],"height:0.267em;",[18,9415],{},[18,9417,9419],{"className":9418},[250],[18,9420,74],{"className":9421},[1230,1196],[18,9423,9425],{"className":9424},[106,681],[18,9426,9428],{"className":9427},[119],[18,9429,9431],{"className":9430},[124],[18,9432,9434,9442],{"className":9433,"style":677},[128],[18,9435,9436,9439],{"style":693},[18,9437],{"className":9438,"style":697},[136],[18,9440,77],{"className":9441,"style":254},[106,110],[18,9443,9444,9447],{"style":703},[18,9445],{"className":9446,"style":697},[136],[18,9448,9450],{"className":9449,"style":711},[710],[18,9451,664],{"className":9452},[106],[18,9454],{"className":9455,"style":228},[169],[18,9457,9459,9486,9489],{"className":9458},[232],[18,9460,9462],{"className":9461},[232],[18,9463,9465],{"className":9464},[106,2878],[18,9466,9468],{"className":9467},[2882],[18,9469,9471,9474,9483],{"className":9470},[2886],[18,9472],{"className":9473,"style":2890},[101],[18,9475,9477],{"className":9476},[2894],[18,9478,9480],{"className":9479},[106],[18,9481,2901],{"className":9482},[232],[18,9484],{"className":9485},[2905],[18,9487],{"className":9488},[169,2909],[18,9490,433],{"className":9491},[232],[18,9493],{"className":9494,"style":228},[169],[18,9496,9498,9501,9541,9547,9550,9553],{"className":9497},[97],[18,9499],{"className":9500,"style":6101},[101],[18,9502,9504,9507],{"className":9503},[106],[18,9505,77],{"className":9506,"style":254},[106,110],[18,9508,9510],{"className":9509},[115],[18,9511,9513,9533],{"className":9512},[119,120],[18,9514,9516,9530],{"className":9515},[124],[18,9517,9519],{"className":9518,"style":5921},[128],[18,9520,9521,9524],{"style":469},[18,9522],{"className":9523,"style":137},[136],[18,9525,9527],{"className":9526},[141,142,143,144],[18,9528,5600],{"className":9529},[106,110,144],[18,9531,152],{"className":9532},[151],[18,9534,9536],{"className":9535},[124],[18,9537,9539],{"className":9538,"style":159},[128],[18,9540],{},[18,9542,9544],{"className":9543},[267],[18,9545,84],{"className":9546},[1230,1196],[18,9548],{"className":9549,"style":228},[169],[18,9551,4775],{"className":9552},[232],[18,9554],{"className":9555,"style":228},[169],[18,9557,9559,9563,9626,9632,9635,9638,9641],{"className":9558},[97],[18,9560],{"className":9561,"style":9562},[101],"height:2.0074em;vertical-align:-0.686em;",[18,9564,9566,9569,9623],{"className":9565},[106],[18,9567],{"className":9568},[250,6012],[18,9570,9572],{"className":9571},[5654],[18,9573,9575,9615],{"className":9574},[119,120],[18,9576,9578,9612],{"className":9577},[124],[18,9579,9582,9593,9601],{"className":9580,"style":9581},[128],"height:1.3214em;",[18,9583,9584,9587],{"style":6028},[18,9585],{"className":9586,"style":697},[136],[18,9588,9590],{"className":9589},[106],[18,9591,2266],{"className":9592},[106],[18,9594,9595,9598],{"style":6040},[18,9596],{"className":9597,"style":697},[136],[18,9599],{"className":9600,"style":6048},[6047],[18,9602,9603,9606],{"style":6051},[18,9604],{"className":9605,"style":697},[136],[18,9607,9609],{"className":9608},[106],[18,9610,47],{"className":9611},[106],[18,9613,152],{"className":9614},[151],[18,9616,9618],{"className":9617},[124],[18,9619,9621],{"className":9620,"style":6070},[128],[18,9622],{},[18,9624],{"className":9625},[267,6012],[18,9627,9629],{"className":9628},[250],[18,9630,74],{"className":9631},[1230,6082],[18,9633,47],{"className":9634},[106],[18,9636],{"className":9637,"style":2068},[169],[18,9639,2253],{"className":9640},[2072],[18,9642],{"className":9643,"style":2068},[169],[18,9645,9647,9650,9687,9744,9747,9750],{"className":9646},[97],[18,9648],{"className":9649,"style":6101},[101],[18,9651,9653],{"className":9652},[250],[18,9654,9656],{"className":9655},[1230,6108],[18,9657,9659,9679],{"className":9658},[119,120],[18,9660,9662,9676],{"className":9661},[124],[18,9663,9665],{"className":9664,"style":6118},[128],[18,9666,9667,9670],{"style":6121},[18,9668],{"className":9669,"style":6125},[136],[18,9671,9672],{"style":6128},[6130,9673,9674],{"xmlns":6132,"width":6133,"height":944,"viewBox":6134},[6136,9675],{"d":6138},[18,9677,152],{"className":9678},[151],[18,9680,9682],{"className":9681},[124],[18,9683,9685],{"className":9684,"style":6148},[128],[18,9686],{},[18,9688,9690,9693],{"className":9689},[106],[18,9691,939],{"className":9692,"style":1171},[106,110],[18,9694,9696],{"className":9695},[115],[18,9697,9699,9736],{"className":9698},[119,120],[18,9700,9702,9733],{"className":9701},[124],[18,9703,9705,9716],{"className":9704,"style":1800},[128],[18,9706,9707,9710],{"style":6171},[18,9708],{"className":9709,"style":137},[136],[18,9711,9713],{"className":9712},[141,142,143,144],[18,9714,430],{"className":9715},[106,144],[18,9717,9718,9721],{"style":1637},[18,9719],{"className":9720,"style":137},[136],[18,9722,9724],{"className":9723},[141,142,143,144],[18,9725,9727,9730],{"className":9726},[106,144],[18,9728,1599],{"className":9729},[106,144],[18,9731,64],{"className":9732},[106,110,144],[18,9734,152],{"className":9735},[151],[18,9737,9739],{"className":9738},[124],[18,9740,9742],{"className":9741,"style":6207},[128],[18,9743],{},[18,9745],{"className":9746,"style":2068},[169],[18,9748,2253],{"className":9749},[2072],[18,9751],{"className":9752,"style":2068},[169],[18,9754,9756,9759,9816,9896,9902],{"className":9755},[97],[18,9757],{"className":9758,"style":6225},[101],[18,9760,9762,9765],{"className":9761},[106],[18,9763,939],{"className":9764,"style":1171},[106,110],[18,9766,9768],{"className":9767},[115],[18,9769,9771,9808],{"className":9770},[119,120],[18,9772,9774,9805],{"className":9773},[124],[18,9775,9777,9788],{"className":9776,"style":1800},[128],[18,9778,9779,9782],{"style":6171},[18,9780],{"className":9781,"style":137},[136],[18,9783,9785],{"className":9784},[141,142,143,144],[18,9786,47],{"className":9787},[106,144],[18,9789,9790,9793],{"style":1637},[18,9791],{"className":9792,"style":137},[136],[18,9794,9796],{"className":9795},[141,142,143,144],[18,9797,9799,9802],{"className":9798},[106,144],[18,9800,1599],{"className":9801},[106,144],[18,9803,64],{"className":9804},[106,110,144],[18,9806,152],{"className":9807},[151],[18,9809,9811],{"className":9810},[124],[18,9812,9814],{"className":9813,"style":6207},[128],[18,9815],{},[18,9817,9819,9856],{"className":9818},[267],[18,9820,9822],{"className":9821},[267],[18,9823,9825],{"className":9824},[1230,6108],[18,9826,9828,9848],{"className":9827},[119,120],[18,9829,9831,9845],{"className":9830},[124],[18,9832,9834],{"className":9833,"style":6118},[128],[18,9835,9836,9839],{"style":6121},[18,9837],{"className":9838,"style":6125},[136],[18,9840,9841],{"style":6128},[6130,9842,9843],{"xmlns":6132,"width":6133,"height":944,"viewBox":6134},[6136,9844],{"d":6138},[18,9846,152],{"className":9847},[151],[18,9849,9851],{"className":9850},[124],[18,9852,9854],{"className":9853,"style":6148},[128],[18,9855],{},[18,9857,9859],{"className":9858},[115],[18,9860,9862,9888],{"className":9861},[119,120],[18,9863,9865,9885],{"className":9864},[124],[18,9866,9868],{"className":9867,"style":6335},[128],[18,9869,9870,9873],{"style":6338},[18,9871],{"className":9872,"style":137},[136],[18,9874,9876],{"className":9875},[141,142,143,144],[18,9877,9879],{"className":9878},[106,144],[18,9880,9882],{"className":9881},[106,144],[18,9883,6356],{"className":9884,"style":6355},[106,6354,144],[18,9886,152],{"className":9887},[151],[18,9889,9891],{"className":9890},[124],[18,9892,9894],{"className":9893,"style":6366},[128],[18,9895],{},[18,9897,9899],{"className":9898},[267],[18,9900,84],{"className":9901},[1230,6082],[18,9903,968],{"className":9904},[106],[11,9906,9907,9908,9943,9944,10077,10078,10110],{},"这个式子的意思是：",[717,9909,9910,9911,1990],{},"两个分布的总变差越小，任何检验的错误率就越接近 ",[18,9912,9914,9931],{"className":9913,"translate":22},[21],[18,9915,9917],{"className":9916},[26],[28,9918,9919],{"xmlns":30},[32,9920,9921,9929],{},[35,9922,9923,9925,9927],{},[45,9924,47],{},[41,9926,317],{"mathvariant":316},[45,9928,2266],{},[86,9930,9060],{"encoding":88},[18,9932,9934],{"className":9933,"ariaHidden":51},[93],[18,9935,9937,9940],{"className":9936},[97],[18,9938],{"className":9939,"style":242},[101],[18,9941,9060],{"className":9942},[106]," 当 ",[18,9945,9947,9973],{"className":9946,"translate":22},[21],[18,9948,9950],{"className":9949},[26],[28,9951,9952],{"xmlns":30},[32,9953,9954,9970],{},[35,9955,9956,9962,9964],{},[38,9957,9958,9960],{},[41,9959,939],{},[45,9961,430],{},[49,9963,433],{},[38,9965,9966,9968],{},[41,9967,939],{},[45,9969,47],{},[86,9971,9972],{"encoding":88},"P_0=P_1",[18,9974,9976,10031],{"className":9975,"ariaHidden":51},[93],[18,9977,9979,9982,10022,10025,10028],{"className":9978},[97],[18,9980],{"className":9981,"style":4563},[101],[18,9983,9985,9988],{"className":9984},[106],[18,9986,939],{"className":9987,"style":1171},[106,110],[18,9989,9991],{"className":9990},[115],[18,9992,9994,10014],{"className":9993},[119,120],[18,9995,9997,10011],{"className":9996},[124],[18,9998,10000],{"className":9999,"style":129},[128],[18,10001,10002,10005],{"style":1386},[18,10003],{"className":10004,"style":137},[136],[18,10006,10008],{"className":10007},[141,142,143,144],[18,10009,430],{"className":10010},[106,144],[18,10012,152],{"className":10013},[151],[18,10015,10017],{"className":10016},[124],[18,10018,10020],{"className":10019,"style":159},[128],[18,10021],{},[18,10023],{"className":10024,"style":228},[169],[18,10026,433],{"className":10027},[232],[18,10029],{"className":10030,"style":228},[169],[18,10032,10034,10037],{"className":10033},[97],[18,10035],{"className":10036,"style":4563},[101],[18,10038,10040,10043],{"className":10039},[106],[18,10041,939],{"className":10042,"style":1171},[106,110],[18,10044,10046],{"className":10045},[115],[18,10047,10049,10069],{"className":10048},[119,120],[18,10050,10052,10066],{"className":10051},[124],[18,10053,10055],{"className":10054,"style":129},[128],[18,10056,10057,10060],{"style":1386},[18,10058],{"className":10059,"style":137},[136],[18,10061,10063],{"className":10062},[141,142,143,144],[18,10064,47],{"className":10065},[106,144],[18,10067,152],{"className":10068},[151],[18,10070,10072],{"className":10071},[124],[18,10073,10075],{"className":10074,"style":159},[128],[18,10076],{}," 时，错误率至少是 ",[18,10079,10081,10098],{"className":10080,"translate":22},[21],[18,10082,10084],{"className":10083},[26],[28,10085,10086],{"xmlns":30},[32,10087,10088,10096],{},[35,10089,10090,10092,10094],{},[45,10091,47],{},[41,10093,317],{"mathvariant":316},[45,10095,2266],{},[86,10097,9060],{"encoding":88},[18,10099,10101],{"className":10100,"ariaHidden":51},[93],[18,10102,10104,10107],{"className":10103},[97],[18,10105],{"className":10106,"style":242},[101],[18,10108,9060],{"className":10109},[106],"，这当然——数据里根本没有信息。",[11,10112,10113,10114,10142,10143,10363],{},"定理 3.1 要求 ",[18,10115,10117,10130],{"className":10116,"translate":22},[21],[18,10118,10120],{"className":10119},[26],[28,10121,10122],{"xmlns":30},[32,10123,10124,10128],{},[35,10125,10126],{},[41,10127,948],{"mathvariant":316},[86,10129,5498],{"encoding":88},[18,10131,10133],{"className":10132,"ariaHidden":51},[93],[18,10134,10136,10139],{"className":10135},[97],[18,10137],{"className":10138,"style":295},[101],[18,10140,948],{"className":10141},[106]," 满足三角不等式，平方损失并不满足。对平方损失，有一个常数稍有不同的版本。设 ",[18,10144,10146,10186],{"className":10145,"translate":22},[21],[18,10147,10149],{"className":10148},[26],[28,10150,10151],{"xmlns":30},[32,10152,10153,10183],{},[35,10154,10155,10157,10159,10161,10167,10169,10175],{},[41,10156,532],{"mathvariant":316},[49,10158,433],{},[49,10160,2248],{"stretchy":73},[38,10162,10163,10165],{},[41,10164,77],{},[45,10166,430],{},[49,10168,2253],{},[38,10170,10171,10173],{},[41,10172,77],{},[45,10174,47],{},[1589,10176,10177,10179,10181],{},[49,10178,2248],{"stretchy":73},[45,10180,2266],{},[45,10182,2266],{},[86,10184,10185],{"encoding":88},"\\Delta=\\lVert\\theta_0-\\theta_1\\rVert_2^2",[18,10187,10189,10207,10265],{"className":10188,"ariaHidden":51},[93],[18,10190,10192,10195,10198,10201,10204],{"className":10191},[97],[18,10193],{"className":10194,"style":600},[101],[18,10196,532],{"className":10197},[106],[18,10199],{"className":10200,"style":228},[169],[18,10202,433],{"className":10203},[232],[18,10205],{"className":10206,"style":228},[169],[18,10208,10210,10213,10216,10256,10259,10262],{"className":10209},[97],[18,10211],{"className":10212,"style":242},[101],[18,10214,2248],{"className":10215},[250],[18,10217,10219,10222],{"className":10218},[106],[18,10220,77],{"className":10221,"style":254},[106,110],[18,10223,10225],{"className":10224},[115],[18,10226,10228,10248],{"className":10227},[119,120],[18,10229,10231,10245],{"className":10230},[124],[18,10232,10234],{"className":10233,"style":129},[128],[18,10235,10236,10239],{"style":469},[18,10237],{"className":10238,"style":137},[136],[18,10240,10242],{"className":10241},[141,142,143,144],[18,10243,430],{"className":10244},[106,144],[18,10246,152],{"className":10247},[151],[18,10249,10251],{"className":10250},[124],[18,10252,10254],{"className":10253,"style":159},[128],[18,10255],{},[18,10257],{"className":10258,"style":2068},[169],[18,10260,2253],{"className":10261},[2072],[18,10263],{"className":10264,"style":2068},[169],[18,10266,10268,10272,10312],{"className":10267},[97],[18,10269],{"className":10270,"style":10271},[101],"height:1.0641em;vertical-align:-0.25em;",[18,10273,10275,10278],{"className":10274},[106],[18,10276,77],{"className":10277,"style":254},[106,110],[18,10279,10281],{"className":10280},[115],[18,10282,10284,10304],{"className":10283},[119,120],[18,10285,10287,10301],{"className":10286},[124],[18,10288,10290],{"className":10289,"style":129},[128],[18,10291,10292,10295],{"style":469},[18,10293],{"className":10294,"style":137},[136],[18,10296,10298],{"className":10297},[141,142,143,144],[18,10299,47],{"className":10300},[106,144],[18,10302,152],{"className":10303},[151],[18,10305,10307],{"className":10306},[124],[18,10308,10310],{"className":10309,"style":159},[128],[18,10311],{},[18,10313,10315,10318],{"className":10314},[267],[18,10316,2248],{"className":10317},[267],[18,10319,10321],{"className":10320},[115],[18,10322,10324,10355],{"className":10323},[119,120],[18,10325,10327,10352],{"className":10326},[124],[18,10328,10330,10341],{"className":10329,"style":2416},[128],[18,10331,10332,10335],{"style":2419},[18,10333],{"className":10334,"style":137},[136],[18,10336,10338],{"className":10337},[141,142,143,144],[18,10339,2266],{"className":10340},[106,144],[18,10342,10343,10346],{"style":1962},[18,10344],{"className":10345,"style":137},[136],[18,10347,10349],{"className":10348},[141,142,143,144],[18,10350,2266],{"className":10351},[106,144],[18,10353,152],{"className":10354},[151],[18,10356,10358],{"className":10357},[124],[18,10359,10361],{"className":10360,"style":2449},[128],[18,10362],{},"，则",[18,10365,10367],{"className":10366,"translate":22},[875],[18,10368,10370,10499],{"className":10369,"translate":22},[21],[18,10371,10373],{"className":10372},[26],[28,10374,10375],{"xmlns":30,"display":884},[32,10376,10377,10496],{},[35,10378,10379,10399,10415,10417,10423,10425,10431,10439,10441,10448,10450,10452,10454,10456,10468,10470,10482,10492,10494],{},[890,10380,10381,10387],{},[35,10382,10383,10385],{},[41,10384,5593],{},[49,10386,899],{},[35,10388,10389,10391,10393,10395,10397],{},[41,10390,5600],{},[49,10392,433],{},[45,10394,430],{},[49,10396,52],{"separator":51},[45,10398,47],{},[38,10400,10401,10403],{},[41,10402,934],{"mathvariant":933},[1589,10404,10405,10407,10409],{},[41,10406,939],{},[41,10408,5600],{},[35,10410,10411,10413],{},[49,10412,1599],{},[41,10414,64],{},[49,10416,2248],{"stretchy":73},[38,10418,10419,10421],{},[41,10420,77],{},[41,10422,5600],{},[49,10424,2253],{},[658,10426,10427,10429],{"accent":51},[41,10428,77],{},[49,10430,664],{},[1589,10432,10433,10435,10437],{},[49,10434,2248],{"stretchy":73},[45,10436,2266],{},[45,10438,2266],{},[49,10440,4775],{},[5654,10442,10443,10445],{},[41,10444,532],{"mathvariant":316},[45,10446,10447],{},"4",[49,10449,74],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[45,10451,47],{},[49,10453,2253],{},[49,10455,2248],{"stretchy":73},[1589,10457,10458,10460,10462],{},[41,10459,939],{},[45,10461,430],{},[35,10463,10464,10466],{},[49,10465,15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n}\\rVert_{\\mathrm{TV}}\\bigr).",[18,10500,10502,10720,10820,10908,10984],{"className":10501,"ariaHidden":51},[93],[18,10503,10505,10509,10568,10571,10668,10671,10711,10714,10717],{"className":10504},[97],[18,10506],{"className":10507,"style":10508},[101],"height:1.6138em;vertical-align:-0.8638em;",[18,10510,10512],{"className":10511},[985,986],[18,10513,10515,10560],{"className":10514},[119,120],[18,10516,10518,10557],{"className":10517},[124],[18,10519,10521,10547],{"className":10520,"style":1083},[128],[18,10522,10523,10526],{"style":5738},[18,10524],{"className":10525,"style":697},[136],[18,10527,10529],{"className":10528},[141,142,143,144],[18,10530,10532,10535,10538,10541,10544],{"className":10531},[106,144],[18,10533,5600],{"className":10534},[106,110,144],[18,10536,433],{"className":10537},[232,144],[18,10539,430],{"className":10540},[106,144],[18,10542,52],{"className":10543},[165,144],[18,10545,47],{"className":10546},[106,144],[18,10548,10549,10552],{"style":693},[18,10550],{"className":10551,"style":697},[136],[18,10553,10554],{},[18,10555,5593],{"className":10556},[985],[18,10558,152],{"className":10559},[151],[18,10561,10563],{"className":10562},[124],[18,10564,10566],{"className":10565,"style":5782},[128],[18,10567],{},[18,10569],{"className":10570,"style":170},[169],[18,10572,10574,10577],{"className":10573},[106],[18,10575,934],{"className":10576},[106,1139],[18,10578,10580],{"className":10579},[115],[18,10581,10583,10660],{"className":10582},[119,120],[18,10584,10586,10657],{"className":10585},[124],[18,10587,10589],{"className":10588,"style":1184},[128],[18,10590,10591,10594],{"style":3261},[18,10592],{"className":10593,"style":137},[136],[18,10595,10597],{"className":10596},[141,142,143,144],[18,10598,10600],{"className":10599},[106,144],[18,10601,10603,10606],{"className":10602},[106,144],[18,10604,939],{"className":10605,"style":1171},[106,110,144],[18,10607,10609],{"className":10608},[115],[18,10610,10612,10649],{"className":10611},[119,120],[18,10613,10615,10646],{"className":10614},[124],[18,10616,10618,10629],{"className":10617,"style":3289},[128],[18,10619,10620,10623],{"style":5840},[18,10621],{"className":10622,"style":1191},[136],[18,10624,10626],{"className":10625},[141,1195,1196,144],[18,10627,5600],{"className":10628},[106,110,144],[18,10630,10631,10634],{"style":3304},[18,10632],{"className":10633,"style":1191},[136],[18,10635,10637],{"className":10636},[141,1195,1196,144],[18,10638,10640,10643],{"className":10639},[106,144],[18,10641,1599],{"className":10642},[106,144],[18,10644,64],{"className":10645},[106,110,144],[18,10647,152],{"className":10648},[151],[18,10650,10652],{"className":10651},[124],[18,10653,10655],{"className":10654,"style":5876},[128],[18,10656],{},[18,10658,152],{"className":10659},[151],[18,10661,10663],{"className":10662},[124],[18,10664,10666],{"className":10665,"style":5888},[128],[18,10667],{},[18,10669,2248],{"className":10670},[250],[18,10672,10674,10677],{"className":10673},[106],[18,10675,77],{"className":10676,"style":254},[106,110],[18,10678,10680],{"className":10679},[115],[18,10681,10683,10703],{"className":10682},[119,120],[18,10684,10686,10700],{"className":10685},[124],[18,10687,10689],{"className":10688,"style":5921},[128],[18,10690,10691,10694],{"style":469},[18,10692],{"className":10693,"style":137},[136],[18,10695,10697],{"className":10696},[141,142,143,144],[18,10698,5600],{"className":10699},[106,110,144],[18,10701,152],{"className":10702},[151],[18,10704,10706],{"className":10705},[124],[18,10707,10709],{"className":10708,"style":159},[128],[18,10710],{},[18,10712],{"className":10713,"style":2068},[169],[18,10715,2253],{"className":10716},[2072],[18,10718],{"className":10719,"style":2068},[169],[18,10721,10723,10726,10757,10811,10814,10817],{"className":10722},[97],[18,10724],{"className":10725,"style":2281},[101],[18,10727,10729],{"className":10728},[106,681],[18,10730,10732],{"className":10731},[119],[18,10733,10735],{"className":10734},[124],[18,10736,10738,10746],{"className":10737,"style":677},[128],[18,10739,10740,10743],{"style":693},[18,10741],{"className":10742,"style":697},[136],[18,10744,77],{"className":10745,"style":254},[106,110],[18,10747,10748,10751],{"style":703},[18,10749],{"className":10750,"style":697},[136],[18,10752,10754],{"className":10753,"style":711},[710],[18,10755,664],{"className":10756},[106],[18,10758,10760,10763],{"className":10759},[267],[18,10761,2248],{"className":10762},[267],[18,10764,10766],{"className":10765},[115],[18,10767,10769,10802],{"className":10768},[119,120],[18,10770,10772,10799],{"className":10771},[124],[18,10773,10776,10788],{"className":10774,"style":10775},[128],"height:0.8641em;",[18,10777,10779,10782],{"style":10778},"top:-2.453em;margin-left:0em;margin-right:0.05em;",[18,10780],{"className":10781,"style":137},[136],[18,10783,10785],{"className":10784},[141,142,143,144],[18,10786,2266],{"className":10787},[106,144],[18,10789,10790,10793],{"style":1637},[18,10791],{"className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\\\\\n\\pi(X^n)=\\Pr(\\theta=\\theta_0\\mid X^n).",[18,11240,11242,11361,11454,11475,11562,11565,11618,11642,11697],{"className":11241,"ariaHidden":51},[93],[18,11243,11245,11248,11317,11320,11349,11352,11355,11358],{"className":11244},[97],[18,11246],{"className":11247,"style":2281},[101],[18,11249,11251,11282],{"className":11250},[106],[18,11252,11254],{"className":11253},[106,681],[18,11255,11257],{"className":11256},[119],[18,11258,11260],{"className":11259},[124],[18,11261,11263,11271],{"className":11262,"style":677},[128],[18,11264,11265,11268],{"style":693},[18,11266],{"className":11267,"style":697},[136],[18,11269,77],{"className":11270,"style":254},[106,110],[18,11272,11273,11276],{"style":703},[18,11274],{"className":11275,"style":697},[136],[18,11277,11279],{"className":11278,"style":711},[710],[18,11280,664],{"className":11281},[106],[18,11283,11285],{"className":11284},[115],[18,11286,11288,11309],{"className":11287},[119,120],[18,11289,11291,11306],{"className":11290},[124],[18,11292,11294],{"className":11293,"style":1152},[128],[18,11295,11296,11299],{"style":469},[18,11297],{"className":11298,"style":137},[136],[18,11300,11302],{"className":11301},[141,142,143,144],[18,11303,3978],{"className":11304,"style":11305},[106,110,144],"margin-right:0.0502em;",[18,11307,152],{"className":11308},[151],[18,11310,11312],{"className":11311},[124],[18,11313,11315],{"className":11314,"style":159},[128],[18,11316],{},[18,11318,74],{"className":11319},[250],[18,11321,11323,11326],{"className":11322},[106],[18,11324,43],{"className":11325,"style":111},[106,110],[18,11327,11329],{"className":11328},[115],[18,11330,11332],{"className":11331},[119],[18,11333,11335],{"className":11334},[124],[18,11336,11338],{"className":11337,"style":1616},[128],[18,11339,11340,11343],{"style":1637},[18,11341],{"className":11342,"style":137},[136],[18,11344,11346],{"className":11345},[141,142,143,144],[18,11347,64],{"className":11348},[106,110,144],[18,11350,84],{"className":11351},[267],[18,11353],{"className":11354,"style":228},[169],[18,11356,433],{"className":11357},[232],[18,11359],{"className":11360,"style":228},[169],[18,11362,11364,11367,11370,11373,11402,11405,11445,11448,11451],{"className":11363},[97],[18,11365],{"className":11366,"style":242},[101],[18,11368,3941],{"className":11369,"style":3957},[106,110],[18,11371,74],{"className":11372},[250],[18,11374,11376,11379],{"className":11375},[106],[18,11377,43],{"className":11378,"style":111},[106,110],[18,11380,11382],{"className":11381},[115],[18,11383,11385],{"className":11384},[119],[18,11386,11388],{"className":11387},[124],[18,11389,11391],{"className":11390,"style":1616},[128],[18,11392,11393,11396],{"style":1637},[18,11394],{"className":11395,"style":137},[136],[18,11397,11399],{"className":11398},[141,142,143,144],[18,11400,64],{"className":11401},[106,110,144],[18,11403,84],{"className":11404},[267],[18,11406,11408,11411],{"className":11407},[106],[18,11409,77],{"className":11410,"style":254},[106,110],[18,11412,11414],{"className":11413},[115],[18,11415,11417,11437],{"className":11416},[119,120],[18,11418,11420,11434],{"className":11419},[124],[18,11421,11423],{"className":11422,"style":129},[128],[18,11424,11425,11428],{"style":469},[18,11426],{"className":11427,"style":137},[136],[18,11429,11431],{"className":11430},[141,142,143,144],[18,11432,430],{"className":11433},[106,144],[18,11435,152],{"className":11436},[151],[18,11438,11440],{"className":11439},[124],[18,11441,11443],{"className":11442,"style":159},[128],[18,11444],{},[18,11446],{"className":11447,"style":2068},[169],[18,11449,7987],{"className":11450},[2072],[18,11452],{"className":11453,"style":2068},[169],[18,11455,11457,11460,11463,11466,11469,11472],{"className":11456},[97],[18,11458],{"className":11459,"style":242},[101],[18,11461,74],{"className":11462},[250],[18,11464,47],{"className":11465},[106],[18,11467],{"className":11468,"style":2068},[169],[18,11470,2253],{"className":11471},[2072],[18,11473],{"className":11474,"style":2068},[169],[18,11476,11478,11481,11484,11487,11516,11519,11559],{"className":11477},[97],[18,11479],{"className":11480,"style":242},[101],[18,11482,3941],{"className":11483,"style":3957},[106,110],[18,11485,74],{"className":11486},[250],[18,11488,11490,11493],{"className":11489},[106],[18,11491,43],{"className":11492,"style":111},[106,110],[18,11494,11496],{"className":11495},[115],[18,11497,11499],{"className":11498},[119],[18,11500,11502],{"className":11501},[124],[18,11503,11505],{"className":11504,"style":1616},[128],[18,11506,11507,11510],{"style":1637},[18,11508],{"className":11509,"style":137},[136],[18,11511,11513],{"className":11512},[141,142,143,144],[18,11514,64],{"className":11515},[106,110,144],[18,11517,3431],{"className":11518},[267],[18,11520,11522,11525],{"className":11521},[106],[18,11523,77],{"className":11524,"style":254},[106,110],[18,11526,11528],{"className":11527},[115],[18,11529,11531,11551],{"className":11530},[119,120],[18,11532,11534,11548],{"className":11533},[124],[18,11535,11537]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",[18,11742,11744,11798],{"className":11743,"translate":22},[21],[18,11745,11747],{"className":11746},[26],[28,11748,11749],{"xmlns":30},[32,11750,11751,11795],{},[35,11752,11753,11755,11757,11759,11761,11763,11765,11771,11773,11775,11777,11779,11781,11783,11789,11791,11793],{},[41,11754,532],{"mathvariant":316},[926,11756,4024],{},[41,11758,934],{"mathvariant":933},[49,11760,945],{"stretchy":73},[41,11762,3941],{},[49,11764,74],{"stretchy":73},[1556,11766,11767,11769],{},[41,11768,43],{},[41,11770,64],{},[49,11772,84],{"stretchy":73},[49,11774,74],{"stretchy":73},[45,11776,47],{},[49,11778,2253],{},[41,11780,3941],{},[49,11782,74],{"stretchy":73},[1556,11784,11785,11787],{},[41,11786,43],{},[41,11788,64],{},[49,11790,84],{"stretchy":73},[49,11792,84],{"stretchy":73},[49,11794,965],{"stretchy":73},[86,11796,11797],{"encoding":88},"\\Delta\\,\\mathbb E[\\pi(X^n)(1-\\pi(X^n))]",[18,11799,11801,11872],{"className":11800,"ariaHidden":51},[93],[18,11802,11804,11807,11810,11813,11816,11819,11822,11825,11854,11857,11860,11863,11866,11869],{"className":11803},[97],[18,11805],{"className":11806,"style":242},[101],[18,11808,532],{"className":11809},[106],[18,11811],{"className":11812,"style":170},[169],[18,11814,934],{"className":11815},[106,1139],[18,11817,945],{"className":11818},[250],[18,11820,3941],{"className":11821,"style":3957},[106,110],[18,11823,74],{"className":11824},[250],[18,11826,11828,11831],{"className":11827},[106],[18,11829,43],{"className":11830,"style":111},[106,110],[18,11832,11834],{"className":11833},[115],[18,11835,11837],{"className":11836},[119],[18,11838,11840],{"className":11839},[124],[18,11841,11843],{"className":11842,"style":1959},[128],[18,11844,11845,11848],{"style":1962},[18,11846],{"className":11847,"style":137},[136],[18,11849,11851],{"className":11850},[141,142,143,144],[18,11852,64],{"className":11853},[106,110,144],[18,11855,84],{"className":11856},[267],[18,11858,74],{"className":11859},[250],[18,11861,47],{"className":11862},[106],[18,11864],{"className":11865,"style":2068},[169],[18,11867,2253],{"className":11868},[2072],[18,11870],{"className":11871,"style":2068},[169],[18,11873,11875,11878,11881,11884,11913],{"className":11874},[97],[18,11876],{"className":11877,"style":242},[101],[18,11879,3941],{"className":11880,"style":3957},[106,110],[18,11882,74],{"className":11883},[250],[18,11885,11887,11890],{"className":11886},[106],[18,11888,43],{"className":11889,"style":111},[106,110],[18,11891,11893],{"className":11892},[115],[18,11894,11896],{"className":11895},[119],[18,11897,11899],{"className":11898},[124],[18,11900,11902],{"className":11901,"style":1959},[128],[18,11903,11904,11907],{"style":1962},[18,11905],{"className":11906,"style":137},[136],[18,11908,11910],{"className":11909},[141,142,143,144],[18,11911,64],{"className":11912},[106,110,144],[18,11914,11916],{"className":11915},[267],"))]","。由 ",[18,11919,11921,11972],{"className":11920,"translate":22},[21],[18,11922,11924],{"className":11923},[26],[28,11925,11926],{"xmlns":30},[32,11927,11928,11969],{},[35,11929,11930,11932,11934,11936,11938,11940,11942,11944,11950,11953,11955,11957,11959,11961,11963,11965,11967],{},[41,11931,3941],{},[49,11933,74],{"stretchy":73},[45,11935,47],{},[49,11937,2253],{},[41,11939,3941],{},[49,11941,84],{"stretchy":73},[49,11943,4775],{},[5654,11945,11946,11948],{},[45,11947,47],{},[45,11949,2266],{},[41,11951,11952],{},"min",[49,11954,899],{},[49,11956,74],{"stretchy":73},[41,11958,3941],{},[49,11960,52],{"separator":51},[45,11962,47],{},[49,11964,2253],{},[41,11966,3941],{},[49,11968,84],{"stretchy":73},[86,11970,11971],{"encoding":88},"\\pi(1-\\pi)\\ge\\frac12\\min(\\pi,1-\\pi)",[18,11973,11975,11999,12020,12129],{"className":11974,"ariaHidden":51},[93],[18,11976,11978,11981,11984,11987,11990,11993,11996],{"className":11977},[97],[18,11979],{"className":11980,"style":242},[101],[18,11982,3941],{"className":11983,"style":3957},[106,110],[18,11985,74],{"className":11986},[250],[18,11988,47],{"className":11989},[106],[18,11991],{"className":11992,"style":2068},[169],[18,11994,2253],{"className":11995},[2072],[18,11997],{"className":11998,"style":2068},[169],[18,12000,12002,12005,12008,12011,12014,12017],{"className":12001},[97],[18,12003],{"className":12004,"style":242},[101],[18,12006,3941],{"className":12007,"style":3957},[106,110],[18,12009,84],{"className":12010},[267],[18,12012],{"className":12013,"style":228},[169],[18,12015,4775],{"className":12016},[232],[18,12018],{"className":12019,"style":228},[169],[18,12021,12023,12027,12099,12102,12105,12108,12111,12114,12117,12120,12123,12126],{"className":12022},[97],[18,12024],{"className":12025,"style":12026},[101],"height:1.1901em;vertical-align:-0.345em;",[18,12028,12030,12033,12096],{"className":12029},[106],[18,12031],{"className":12032},[250,6012],[18,12034,12036],{"className":12035},[5654],[18,12037,12039,12087],{"className":12038},[119,120],[18,12040,12042,12084],{"className":12041},[124],[18,12043,12046,12061,12069],{"className":12044,"style":12045},[128],"height:0.8451em;",[18,12047,12049,12052],{"style":12048},"top:-2.655em;",[18,12050],{"className":12051,"style":697},[136],[18,12053,12055],{"className":12054},[141,142,143,144],[18,12056,12058],{"className":12057},[106,144],[18,12059,2266],{"className":12060},[106,144],[18,12062,12063,12066],{"style":6040},[18,12064],{"className":12065,"style":697},[136],[18,12067],{"className":12068,"style":6048},[6047],[18,12070,12072,12075],{"style":12071},"top:-3.394em;",[18,12073],{"className":12074,"style":697},[136],[18,12076,12078],{"className":12077},[141,142,143,144],[18,12079,12081],{"className":12080},[106,144],[18,12082,47],{"className":12083},[106,144],[18,12085,152],{"className":12086},[151],[18,12088,12090],{"className":12089},[124],[18,12091,12094],{"className":12092,"style":12093},[128],"height:0.345em;",[18,12095],{},[18,12097],{"className":12098},[267,6012],[18,12100],{"className":12101,"style":170},[169],[18,12103,11952],{"className":12104},[985],[18,12106,74],{"className":12107},[250],[18,12109,3941],{"className":12110,"style":3957},[106,110],[18,12112,52],{"className":12113},[165],[18,12115],{"className":12116,"style":170},[169],[18,12118,47],{"className":12119},[106],[18,12121],{"className":12122,"style":2068},[169],[18,12124,2253],{"className":12125},[2072],[18,12127],{"className":12128,"style":2068},[169],[18,12130,12132,12135,12138],{"className":12131},[97],[18,12133],{"className":12134,"style":242},[101],[18,12136,3941],{"className":12137,"style":3957},[106,110],[18,12139,84],{"className":12140},[267]," 与均匀先验下贝叶斯检验误差的下界 ",[18,12143,12145,12203],{"className":12144,"translate":22},[21],[18,12146,12148],{"className":12147},[26],[28,12149,12150],{"xmlns":30},[32,12151,12152,12200],{},[35,12153,12154,12156,12158,12160,12162,12164,12166,12168,12170,12172,12174,12176,12178,12180,12186,12188,12190,12192,12198],{},[41,12155,934],{"mathvariant":933},[49,12157,945],{"stretchy":73},[41,12159,11952],{},[49,12161,899],{},[49,12163,74],{"stretchy":73},[41,12165,3941],{},[49,12167,52],{"separator":51},[45,12169,47],{},[49,12171,2253],{},[41,12173,3941],{},[49,12175,84],{"stretchy":73},[49,12177,965],{"stretchy":73},[49,12179,4775],{},[5654,12181,12182,12184],{},[45,12183,47],{},[45,12185,2266],{},[49,12187,74],{"stretchy":73},[45,12189,47],{},[49,12191,2253],{},[35,12193,12194,12196],{},[41,12195,5703],{"mathvariant":316},[41,12197,5706],{"mathvariant":316},[49,12199,84],{"stretchy":73},[86,12201,12202],{"encoding":88},"\\mathbb E[\\min(\\pi,1-\\pi)]\\ge\\frac12(1-\\mathrm{TV})",[18,12204,12206,12245,12267,12356],{"className":12205,"ariaHidden":51},[93],[18,12207,12209,12212,12215,12218,12221,12224,12227,12230,12233,12236,12239,12242],{"className":12208},[97],[18,12210],{"className":12211,"style":242},[101],[18,12213,934],{"className":12214},[106,1139],[18,12216,945],{"className":12217},[250],[18,12219,11952],{"className":12220},[985],[18,12222,74],{"className":12223},[250],[18,12225,3941],{"className":12226,"style":3957},[106,110],[18,12228,52],{"className":12229},[165],[18,12231],{"className":12232,"style":170},[169],[18,12234,47],{"className":12235},[106],[18,12237],{"className":12238,"style":2068},[169],[18,12240,2253],{"className":12241},[2072],[18,12243],{"className":12244,"style":2068},[169],[18,12246,12248,12251,12254,12258,12261,12264],{"className":12247},[97],[18,12249],{"className":12250,"style":242},[101],[18,12252,3941],{"className":12253,"style":3957},[106,110],[18,12255,12257],{"className":12256},[267],")]",[18,12259],{"className":12260,"style":228},[169],[18,12262,4775],{"className":12263},[232],[18,12265],{"className":12266,"style":228},[169],[18,12268,12270,12273,12341,12344,12347,12350,12353],{"className":12269},[97],[18,12271],{"className":12272,"style":12026},[101],[18,12274,12276,12279,12338],{"className":12275},[106],[18,12277],{"className":12278},[250,6012],[18,12280,12282],{"className":12281},[5654],[18,12283,12285,12330],{"className":12284},[119,120],[18,12286,12288,12327],{"className":12287},[124],[18,12289,12291,12305,12313],{"className":12290,"style":12045},[128],[18,12292,12293,12296],{"style":12048},[18,12294],{"className":12295,"style":697},[136],[18,12297,12299],{"className":12298},[141,142,143,144],[18,12300,12302],{"className":12301},[106,144],[18,12303,2266],{"className":12304},[106,144],[18,12306,12307,12310],{"style":6040},[18,12308],{"className":12309,"style":697},[136],[18,12311],{"className":12312,"style":6048},[6047],[18,12314,12315,12318],{"style":12071},[18,12316],{"className":12317,"style":697},[136],[18,12319,12321],{"className":12320},[141,142,143,144],[18,12322,12324],{"className":12323},[106,144],[18,12325,47],{"className":12326},[106,144],[18,12328,152],{"className":12329},[151],[18,12331,12333],{"className":12332},[124],[18,12334,12336],{"className":12335,"style":12093},[128],[18,12337],{},[18,12339],{"className":12340},[267,6012],[18,12342,74],{"className":12343},[250],[18,12345,47],{"className":12346},[106],[18,12348],{"className":12349,"style":2068},[169],[18,12351,2253],{"className":12352},[2072],[18,12354],{"className":12355,"style":2068},[169],[18,12357,12359,12362,12368],{"className":12358},[97],[18,12360],{"className":12361,"style":242},[101],[18,12363,12365],{"className":12364},[106],[18,12366,6356],{"className":12367,"style":6355},[106,6354],[18,12369,84],{"className":12370},[267],"，风险至少是 ",[18,12373,12375,12407],{"className":12374,"translate":22},[21],[18,12376,12378],{"className":12377},[26],[28,12379,12380],{"xmlns":30},[32,12381,12382,12404],{},[35,12383,12384,12390,12392,12394,12396,12402],{},[5654,12385,12386,12388],{},[41,12387,532],{"mathvariant":316},[45,12389,10447],{},[49,12391,74],{"stretchy":73},[45,12393,47],{},[49,12395,2253],{},[35,12397,12398,12400],{},[41,12399,5703],{"mathvariant":316},[41,12401,5706],{"mathvariant":316},[49,12403,84],{"stretchy":73},[86,12405,12406],{"encoding":88},"\\frac{\\Delta}{4}(1-\\mathrm{TV})",[18,12408,12410,12501],{"className":12409,"ariaHidden":51},[93],[18,12411,12413,12417,12486,12489,12492,12495,12498],{"className":12412},[97],[18,12414],{"className":12415,"style":12416},[101],"height:1.2173em;vertical-align:-0.345em;",[18,12418,12420,12423,12483],{"className":12419},[106],[18,12421],{"className":12422},[250,6012],[18,12424,12426],{"className":12425},[5654],[18,12427,12429,12475],{"className":12428},[119,120],[18,12430,12432,12472],{"className":12431},[124],[18,12433,12436,12450,12458],{"className":12434,"style":12435},[128],"height:0.8723em;",[18,12437,12438,12441],{"style":12048},[18,12439],{"className":12440,"style":697},[136],[18,12442,12444],{"className":12443},[141,142,143,144],[18,12445,12447],{"className":12446},[106,144],[18,12448,10447],{"className":12449},[106,144],[18,12451,12452,12455],{"style":6040},[18,12453],{"className":12454,"style":697},[136],[18,12456],{"className":12457,"style":6048},[6047],[18,12459,12460,12463],{"style":12071},[18,12461],{"className":12462,"style":697},[136],[18,12464,12466],{"className":12465},[141,142,143,144],[18,12467,12469],{"className":12468},[106,144],[18,12470,532],{"className":12471},[106,144],[18,12473,152],{"className":12474},[151],[18,12476,12478],{"className":12477},[124],[18,12479,12481],{"className":12480,"style":12093},[128],[18,12482],{},[18,12484],{"className":12485},[267,6012],[18,12487,74],{"className":12488},[250],[18,12490,47],{"className":12491},[106],[18,12493],{"className":12494,"style":2068},[169],[18,12496,2253],{"className":12497},[2072],[18,12499],{"className":12500,"style":2068},[169],[18,12502,12504,12507,12513],{"className":12503},[97],[18,12505],{"className":12506,"style":242},[101],[18,12508,12510],{"className":12509},[106],[18,12511,6356],{"className":12512,"style":6355},[106,6354],[18,12514,84],{"className":12515},[267],"。极小极大风险不小于贝叶斯风险，得证。",[1311,12518,12520],{"id":12519},"_32-从-tv-到-kl","3.2 从 TV 到 KL",[11,12522,12523],{},"总变差通常不好直接算，尤其对乘积分布。Pinsker 不等式把它换成 KL 散度：",[18,12525,12527],{"className":12526,"translate":22},[875],[18,12528,12530,12598],{"className":12529,"translate":22},[21],[18,12531,12533],{"className":12532},[26],[28,12534,12535],{"xmlns":30,"display":884},[32,12536,12537,12595],{},[35,12538,12539,12541,12543,12545,12548,12558,12560,12593],{},[49,12540,2248],{"stretchy":73},[41,12542,939],{},[49,12544,2253],{},[41,12546,12547],{},"Q",[38,12549,12550,12552],{},[49,12551,2248],{"stretchy":73},[35,12553,12554,12556],{},[41,12555,5703],{"mathvariant":316},[41,12557,5706],{"mathvariant":316},[49,12559,4334],{},[12561,12562,12563],"msqrt",{},[35,12564,12565,12571,12583,12585,12587,12589,12591],{},[5654,12566,12567,12569],{},[45,12568,47],{},[45,12570,2266],{},[38,12572,12573,12576],{},[41,12574,12575],{},"D",[35,12577,12578,12580],{},[41,12579,5031],{"mathvariant":316},[41,12581,12582],{"mathvariant":316},"L",[49,12584,74],{"stretchy":73},[41,12586,939],{},[41,12588,2248],{"mathvariant":316},[41,12590,12547],{},[49,12592,84],{"stretchy":73},[41,12594,968],{"mathvariant":316},[86,12596,12597],{"encoding":88},"\\lVert P-Q\\rVert_{\\mathrm{TV}}\\le\\sqrt{\\frac12 D_{\\mathrm{KL}}(P\\Vert Q)}.",[18,12599,12601,12622,12686],{"className":12600,"ariaHidden":51},[93],[18,12602,12604,12607,12610,12613,12616,12619],{"className":12603},[97],[18,12605],{"className":12606,"style":242},[101],[18,12608,2248],{"className":12609},[250],[18,12611,939],{"className":12612,"style":1171},[106,110],[18,12614],{"className":12615,"style":2068},[169],[18,12617,2253],{"className":12618},[2072],[18,12620],{"className":12621,"style":2068},[169],[18,12623,12625,12628,12631,12677,12680,12683],{"className":12624},[97],[18,12626],{"className":12627,"style":242},[101],[18,12629,12547],{"className":12630},[106,110],[18,12632,12634,12637],{"className":12633},[267],[18,12635,2248],{"className":12636},[267],[18,12638,12640],{"className":12639},[115],[18,12641,12643,12669],{"className":12642},[119,120],[18,12644,12646,12666],{"className":12645},[124],[18,12647,12649],{"className":12648,"style":1152},[128],[18,12650,12651,12654],{"style":1155},[18,12652],{"className":12653,"style":137},[136],[18,12655,12657],{"className":12656},[141,142,143,144],[18,12658,12660],{"className":12659},[106,144],[18,12661,12663],{"className":12662},[106,144],[18,12664,6356],{"className":12665,"style":6355},[106,6354,144],[18,12667,152],{"className":12668},[151],[18,12670,12672],{"className":12671},[124],[18,12673,12675],{"className":12674,"style":159},[128],[18,12676],{},[18,12678],{"className":12679,"style":228},[169],[18,12681,4334],{"className":12682},[232],[18,12684],{"className":12685,"style":228},[169],[18,12687,12689,12693,12876],{"className":12688},[97],[18,12690],{"className":12691,"style":12692},[101],"height:2.44em;vertical-align:-0.7884em;",[18,12694,12697],{"className":12695},[106,12696],"sqrt",[18,12698,12700,12867],{"className":12699},[119,120],[18,12701,12703,12864],{"className":12702},[124],[18,12704,12707,12844],{"className":12705,"style":12706},[128],"height:1.6516em;",[18,12708,12712,12716],{"className":12709,"style":12711},[12710],"svg-align","top:-4.4em;",[18,12713],{"className":12714,"style":12715},[136],"height:4.4em;",[18,12717,12720,12782,12829,12832,12835,12838,12841],{"className":12718,"style":12719},[106],"padding-left:1em;",[18,12721,12723,12726,12779],{"className":12722},[106],[18,12724],{"className":12725},[250,6012],[18,12727,12729],{"className":12728},[5654],[18,12730,12732,12771],{"className":12731},[119,120],[18,12733,12735,12768],{"className":12734},[124],[18,12736,12738,12749,12757],{"className":12737,"style":9581},[128],[18,12739,12740,12743],{"style":6028},[18,12741],{"className":12742,"style":697},[136],[18,12744,12746],{"className":12745},[106],[18,12747,2266],{"className":12748},[106],[18,12750,12751,12754],{"style":6040},[18,12752],{"className":12753,"style":697},[136],[18,12755],{"className":12756,"style":6048},[6047],[18,12758,12759,12762],{"style":6051},[18,12760],{"className":12761,"style":697},[136],[18,12763,12765],{"className":12764},[106],[18,12766,47],{"className":12767},[106],[18,12769,152],{"className":12770},[151],[18,12772,12774],{"className":12773},[124],[18,12775,12777],{"className":12776,"style":6070},[128],[18,12778],{},[18,12780],{"className":12781},[267,6012],[18,12783,12785,12788],{"className":12784},[106],[18,12786,12575],{"className":12787,"style":254},[106,110],[18,12789,12791],{"className":12790},[115],[18,12792,12794,12821],{"className":12793},[119,120],[18,12795,12797,12818],{"className":12796},[124],[18,12798,12800],{"className":12799,"style":1152},[128],[18,12801,12802,12805],{"style":469},[18,12803],{"className":12804,"style":137},[136],[18,12806,12808],{"className":12807},[141,142,143,144],[18,12809,12811],{"className":12810},[106,144],[18,12812,12814],{"className":12813},[106,144],[18,12815,12817],{"className":12816},[106,6354,144],"KL",[18,12819,152],{"className":12820},[151],[18,12822,12824],{"className":12823},[124],[18,12825,12827],{"className":12826,"style":159},[128],[18,12828],{},[18,12830,74],{"className":12831},[250],[18,12833,939],{"className":12834,"style":1171},[106,110],[18,12836,2248],{"className":12837},[106],[18,12839,12547],{"className":12840},[106,110],[18,12842,84],{"className":12843},[267],[18,12845,12847,12850],{"style":12846},"top:-3.6116em;",[18,12848],{"className":12849,"style":12715},[136],[18,12851,12855],{"className":12852,"style":12854},[12853],"hide-tail","min-width:1.02em;height:2.48em;",[6130,12856,12861],{"xmlns":6132,"width":12857,"height":12858,"viewBox":12859,"preserveAspectRatio":12860},"400em","2.48em","0 0 400000 2592","xMinYMin slice",[6136,12862],{"d":12863},"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",[18,12865,152],{"className":12866},[151],[18,12868,12870],{"className":12869},[124],[18,12871,12874],{"className":12872,"style":12873},[128],"height:0.7884em;",[18,12875],{},[18,12877,968],{"className":12878},[106],[11,12880,12881],{},"而 KL 对乘积分布是可加的：",[18,12883,12885],{"className":12884,"translate":22},[875],[18,12886,12888,12976],{"className":12887,"translate":22},[21],[18,12889,12891],{"className":12890},[26],[28,12892,12893],{"xmlns":30,"display":884},[32,12894,12895,12973],{},[35,12896,12897,12907,12909,12921,12923,12935,12937,12939,12941,12943,12953,12955,12961,12963,12969,12971],{},[38,12898,12899,12901],{},[41,12900,12575],{},[35,12902,12903,12905],{},[41,12904,5031],{"mathvariant":316},[41,12906,12582],{"mathvariant":316},[49,12908,74],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[1589,12910,12911,12913,12915],{},[41,12912,939],{},[45,12914,430],{},[35,12916,12917,12919],{},[49,12918,1599],{},[41,12920,64],{},[41,12922,2248],{"mathvariant":316},[1589,12924,12925,12927,12929],{},[41,12926,939],{},[45,12928,47],{},[35,12930,12931,12933],{},[49,12932,1599],{},[41,12934,64],{},[49,12936,84],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,12938,433],{},[41,12940,64],{},[926,12942,4024],{},[38,12944,12945,12947],{},[41,12946,12575],{},[35,12948,12949,12951],{},[41,12950,5031],{"mathvariant":316},[41,12952,12582],{"mathvariant":316},[49,12954,74],{"stretchy":73},[38,12956,12957,12959],{},[41,12958,939],{},[45,12960,430],{},[41,12962,2248],{"mathvariant":316},[38,12964,12965,12967],{},[41,12966,939],{},[45,12968,47],{},[49,12970,84],{"stretchy":73},[41,12972,968],{"mathvariant":316},[86,12974,12975],{"encoding":88},"D_{\\mathrm{KL}}\\bigl(P_0^{\\otimes n}\\Vert P_1^{\\otimes n}\\bigr)=n\\,D_{\\mathrm{KL}}(P_0\\Vert 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3.1：",[18,13323,13325],{"className":13324,"translate":22},[875],[18,13326,13328,13454],{"className":13327,"translate":22},[21],[18,13329,13331],{"className":13330},[26],[28,13332,13333],{"xmlns":30,"display":884},[32,13334,13335,13451],{},[35,13336,13337,13357,13373,13375,13377,13383,13385,13391,13393,13395,13401,13449],{},[890,13338,13339,13345],{},[35,13340,13341,13343],{},[41,13342,5593],{},[49,13344,899],{},[35,13346,13347,13349,13351,13353,13355],{},[41,13348,5600],{},[49,13350,433],{},[45,13352,430],{},[49,13354,52],{"separator":51},[45,13356,47],{},[38,13358,13359,13361],{},[41,13360,934],{"mathvariant":933},[1589,13362,13363,13365,13367],{},[41,13364,939],{},[41,13366,5600],{},[35,13368,13369,13371],{},[49,13370,1599],{},[41,13372,64],{},[41,13374,948],{"mathvariant":316},[49,13376,74],{"stretchy":73},[38,13378,13379,13381],{},[41,13380,77],{},[41,13382,5600],{},[49,13384,52],{"separator":51},[658,13386,13387,13389],{"accent":51},[41,13388,77],{},[49,13390,664],{},[49,13392,84],{"stretchy":73},[49,13394,4775],{},[5654,13396,13397,13399],{},[41,13398,5351],{},[45,13400,2266],{},[35,13402,13403,13405,13407,13409,13447],{},[49,13404,74],{"fence":51},[45,13406,47],{},[49,13408,2253],{},[12561,13410,13411],{},[35,13412,13413,13419,13429,13431,13437,13439,13445],{},[5654,13414,13415,13417],{},[41,13416,64],{},[45,13418,2266],{},[38,13420,13421,13423],{},[41,13422,12575],{},[35,13424,13425,13427],{},[41,13426,5031],{"mathvariant":316},[41,13428,12582],{"mathvariant":316},[49,13430,74],{"stretchy":73},[38,13432,13433,13435],{},[41,13434,939],{},[45,13436,430],{},[41,13438,2248],{"mathvariant":316},[38,13440,13441,13443],{},[41,13442,939],{},[45,13444,47],{},[49,13446,84],{"stretchy":73},[49,13448,84],{"fence":51},[41,13450,968],{"mathvariant":316},[86,13452,13453],{"encoding":88},"\\max_{i=0,1}\\mathbb 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P_1)}\\right).",[18,13455,13457,13717],{"className":13456,"ariaHidden":51},[93],[18,13458,13460,13463,13522,13525,13622,13625,13628,13668,13671,13674,13705,13708,13711,13714],{"className":13459},[97],[18,13461],{"className":13462,"style":5723},[101],[18,13464,13466],{"className":13465},[985,986],[18,13467,13469,13514],{"className":13468},[119,120],[18,13470,13472,13511],{"className":13471},[124],[18,13473,13475,13501],{"className":13474,"style":1083},[128],[18,13476,13477,13480],{"style":5738},[18,13478],{"className":13479,"style":697},[136],[18,13481,13483],{"className":13482},[141,142,143,144],[18,13484,13486,13489,13492,13495,13498],{"className":13485},[106,144],[18,13487,5600],{"className":13488},[106,110,144],[18,13490,433],{"className":13491},[232,144],[18,13493,430],{"className":13494},[106,144],[18,13496,52],{"className":13497},[165,144],[18,13499,47],{"className":13500},[106,144],[18,13502,13503,13506],{"style":693},[18,13504],{"className":13505,"style":697},[136],[18,13507,13508],{},[18,13509,5593],{"className":13510},[985],[18,13512,152],{"className":13513},[151],[18,13515,13517],{"className":13516},[124],[18,13518,13520],{"className":13519,"style":5782},[128],[18,13521],{},[18,13523],{"className":13524,"style":170},[169],[18,13526,13528,13531],{"className":13527},[106],[18,13529,934],{"className":13530},[106,1139],[18,13532,13534],{"className":13533},[115],[18,13535,13537,13614],{"className":13536},[119,120],[18,13538,13540,13611],{"className":13539},[124],[18,13541,13543],{"className":13542,"style":1184},[128],[18,13544,13545,13548],{"style":3261},[18,13546],{"className":13547,"style":137},[136],[18,13549,13551],{"className":13550},[141,142,143,144],[18,13552,13554],{"className":13553},[106,144],[18,13555,13557,13560],{"className":13556},[106,144],[18,13558,939],{"className":13559,"style":1171},[106,110,144],[18,13561,13563],{"className":13562},[115],[18,13564,13566,13603],{"className":13565},[119,120],[18,13567,13569,13600],{"className":13568},[124],[18,13570,13572,13583],{"className":13571,"style":3289},[128],[18,13573,13574,13577],{"style":5840},[18,13575],{"className":13576,"style":1191},[136],[18,13578,13580],{"className":13579},[141,1195,1196,144],[18,13581,5600],{"className":13582},[106,110,144],[18,13584,13585,13588],{"style":3304},[18,13586],{"className":13587,"style":1191},[136],[18,13589,13591],{"className":13590},[141,1195,1196,144],[18,13592,13594,13597],{"className":13593},[106,144],[18,13595,1599],{"className":13596},[106,144],[18,13598,64],{"className":13599},[106,110,144],[18,13601,152],{"className":13602},[151],[18,13604,13606],{"className":13605},[124],[18,13607,13609],{"className":13608,"style":5876},[128],[18,13610],{},[18,13612,152],{"className":13613},[151],[18,13615,13617],{"className":13616},[124],[18,13618,13620],{"className":13619,"style":5888},[128],[18,13621],{},[18,13623,948],{"className":13624},[106],[18,13626,74],{"className":13627},[250],[18,13629,13631,13634],{"className":13630},[106],[18,13632,77],{"className":13633,"style":254},[106,110],[18,13635,13637],{"className":13636},[115],[18,13638,13640,13660],{"className":13639},[119,120],[18,13641,13643,13657],{"className":13642},[124],[18,13644,13646],{"className":13645,"style":5921},[128],[18,13647,13648,13651],{"style":469},[18,13649],{"className":13650,"style":137},[136],[18,13652,13654],{"className":13653},[141,142,143,144],[18,13655,5600],{"className":13656},[106,110,144],[18,13658,152],{"className":13659},[151],[18,13661,13663],{"className":13662},[124],[18,13664,13666],{"className":13665,"style":159},[128],[18,13667],{},[18,13669,52],{"className":13670},[165],[18,13672],{"className":13673,"style":170},[169],[18,13675,13677],{"className":13676},[106,681],[18,13678,13680],{"className":13679},[119],[18,13681,13683],{"className":13682},[124],[18,13684,13686,13694],{"className":13685,"style":677},[128],[18,13687,13688,13691],{"style":693},[18,13689],{"className":13690,"style":697},[136],[18,13692,77],{"className":13693,"style":254},[106,110],[18,13695,13696,13699],{"style":703},[18,13697],{"className":13698,"style":697},[136],[18,13700,13702],{"className":13701,"style":711},[710],[18,13703,664],{"className":13704},[106],[18,13706,84],{"className":13707},[267],[18,13709],{"className":13710,"style":228},[169],[18,13712,4775],{"className":13713},[232],[18,13715],{"className":13716,"style":228},[169],[18,13718,13720,13724,13786,13789,14063,14066],{"className":13719},[97],[18,13721],{"className":13722,"style":13723},[101],"height:2.4947em;vertical-align:-0.95em;",[18,13725,13727,13730,13783],{"className":13726},[106],[18,13728],{"className":13729},[250,6012],[18,13731,13733],{"className":13732},[5654],[18,13734,13736,13775],{"className":13735},[119,120],[18,13737,13739,13772],{"className":13738},[124],[18,13740,13742,13753,13761],{"className":13741,"style":6025},[128],[18,13743,13744,13747],{"style":6028},[18,13745],{"className":13746,"style":697},[136],[18,13748,13750],{"className":13749},[106],[18,13751,2266],{"className":13752},[106],[18,13754,13755,13758],{"style":6040},[18,13756],{"className":13757,"style":697},[136],[18,13759],{"className":13760,"style":6048},[6047],[18,13762,13763,13766],{"style":6051},[18,13764],{"className":13765,"style":697},[136],[18,13767,13769],{"className":13768},[106],[18,13770,5351],{"className":13771,"style":5480},[106,110],[18,13773,152],{"className":13774},[151],[18,13776,13778],{"className":13777},[124],[18,13779,13781],{"className":13780,"style":6070},[128],[18,13782],{},[18,13784],{"className":13785},[267,6012],[18,13787],{"className":13788,"style":170},[169],[18,13790,13792,13800,13803,13806,13809,13812,14057],{"className":13791},[174],[18,13793,13797],{"className":13794,"style":13796},[250,13795],"delimcenter","top:0em;",[18,13798,74],{"className":13799},[1230,143],[18,13801,47],{"className":13802},[106],[18,13804],{"className":13805,"style":2068},[169],[18,13807,2253],{"className":13808},[2072],[18,13810],{"className":13811,"style":2068},[169],[18,13813,13815],{"className":13814},[106,12696],[18,13816,13818,14048],{"className":13817},[119,120],[18,13819,13821,14045],{"className":13820},[124],[18,13822,13825,14032],{"className":13823,"style":13824},[128],"height:1.5447em;",[18,13826,13828,13831],{"className":13827,"style":12711},[12710],[18,13829],{"className":13830,"style":12715},[136],[18,13832,13834,13897,13943,13946,13986,13989,14029],{"className":13833,"style":12719},[106],[18,13835,13837,13840,13894],{"className":13836},[106],[18,13838],{"className":13839},[250,6012],[18,13841,13843],{"className":13842},[5654],[18,13844,13846,13886],{"className":13845},[119,120],[18,13847,13849,13883],{"className":13848},[124],[18,13850,13853,13864,13872],{"className":13851,"style":13852},[128],"height:1.1076em;",[18,13854,13855,13858],{"style":6028},[18,13856],{"className":13857,"style":697},[136],[18,13859,13861],{"className":13860},[106],[18,13862,2266],{"className":13863},[106],[18,13865,13866,13869],{"style":6040},[18,13867],{"className":13868,"style":697},[136],[18,13870],{"className":13871,"style":6048},[6047],[18,13873,13874,13877],{"style":6051},[18,13875],{"className":13876,"style":697},[136],[18,13878,13880],{"className":13879},[106],[18,13881,64],{"className":13882},[106,110],[18,13884,152],{"className":13885},[151],[18,13887,13889],{"className":13888},[124],[18,13890,13892],{"className":13891,"style":6070},[128],[18,13893],{},[18,13895],{"className":13896},[267,6012],[18,13898,13900,13903],{"className":13899},[106],[18,13901,12575],{"className":13902,"style":254},[106,110],[18,13904,13906],{"className":13905},[115],[18,13907,13909,13935],{"className":13908},[119,120],[18,13910,13912,13932],{"className":13911},[124],[18,13913,13915],{"className":13914,"style":1152},[128],[18,13916,13917,13920],{"style":469},[18,13918],{"className":13919,"style":137},[136],[18,13921,13923],{"className":13922},[141,142,143,144],[18,13924,13926],{"className":13925},[106,144],[18,13927,13929],{"className":13928},[106,144],[18,13930,12817],{"className":13931},[106,6354,144],[18,13933,152],{"className":13934},[151],[18,13936,13938],{"className":13937},[124],[18,13939,13941],{"className":13940,"style":159},[128],[18,13942],{},[18,13944,74],{"className":13945},[250],[18,13947,13949,13952],{"className":13948},[106],[18,13950,939],{"className":13951,"style":1171},[106,110],[18,13953,13955],{"className":13954},[115],[18,13956,13958,13978],{"className":13957},[119,120],[18,13959,13961,13975],{"className":13960},[124],[18,13962,13964],{"className":13963,"style":129},[128],[18,13965,13966,13969],{"style":1386},[18,13967],{"className":13968,"style":137},[136],[18,13970,13972],{"className":13971},[141,142,143,144],[18,13973,430],{"className":13974},[106,144],[18,13976,152],{"className":13977},[151],[18,13979,13981],{"className":13980},[124],[18,13982,13984],{"className":13983,"style":159},[128],[18,13985],{},[18,13987,2248],{"className":13988},[106],[18,13990,13992,13995],{"className":13991},[106],[18,13993,939],{"className":13994,"style":1171},[106,110],[18,13996,13998],{"className":13997},[115],[18,13999,14001,14021],{"className":14000},[119,120],[18,14002,14004,14018],{"className":14003},[124],[18,14005,14007],{"className":14006,"style":129},[128],[18,14008,14009,14012],{"style":1386},[18,14010],{"className":14011,"style":137},[136],[18,14013,14015],{"className":14014},[141,142,143,144],[18,14016,47],{"className":14017},[106,144],[18,14019,152],{"className":14020},[151],[18,14022,14024],{"className":14023},[124],[18,14025,14027],{"className":14026,"style":159},[128],[18,14028],{},[18,14030,84],{"className":14031},[267],[18,14033,14035,14038],{"style":14034},"top:-3.5047em;",[18,14036],{"className":14037,"style":12715},[136],[18,14039,14041],{"className":14040,"style":12854},[12853],[6130,14042,14043],{"xmlns":6132,"width":12857,"height":12858,"viewBox":12859,"preserveAspectRatio":12860},[6136,14044],{"d":12863},[18,14046,152],{"className":14047},[151],[18,14049,14051],{"className":14050},[124],[18,14052,14055],{"className":14053,"style":14054},[128],"height:0.8953em;",[18,14056],{},[18,14058,14060],{"className":14059,"style":13796},[267,13795],[18,14061,84],{"className":14062},[1230,143],[18,14064],{"className":14065,"style":170},[169],[18,14067,968],{"className":14068},[106],[11,14070,14071,14072,1994,14075,14104,14105,1994,14108,14193,14194,14372],{},"这个式子把下界拆成了两个因子：",[717,14073,14074],{},"损失间隔",[18,14076,14078,14092],{"className":14077,"translate":22},[21],[18,14079,14081],{"className":14080},[26],[28,14082,14083],{"xmlns":30},[32,14084,14085,14089],{},[35,14086,14087],{},[41,14088,5351],{},[86,14090,14091],{"encoding":88},"\\delta",[18,14093,14095],{"className":14094,"ariaHidden":51},[93],[18,14096,14098,14101],{"className":14097},[97],[18,14099],{"className":14100,"style":295},[101],[18,14102,5351],{"className":14103,"style":5480},[106,110],"，和",[717,14106,14107],{},"信息量",[18,14109,14111,14135],{"className":14110,"translate":22},[21],[18,14112,14114],{"className":14113},[26],[28,14115,14116],{"xmlns":30},[32,14117,14118,14132],{},[35,14119,14120,14122],{},[41,14121,64],{},[38,14123,14124,14126],{},[41,14125,12575],{},[35,14127,14128,14130],{},[41,14129,5031],{"mathvariant":316},[41,14131,12582],{"mathvariant":316},[86,14133,14134],{"encoding":88},"nD_{\\mathrm{KL}}",[18,14136,14138],{"className":14137,"ariaHidden":51},[93],[18,14139,14141,14144,14147],{"className":14140},[97],[18,14142],{"className":14143,"style":4563},[101],[18,14145,64],{"className":14146},[106,110],[18,14148,14150,14153],{"className":14149},[106],[18,14151,12575],{"className":14152,"style":254},[106,110],[18,14154,14156],{"className":14155},[115],[18,14157,14159,14185],{"className":14158},[119,120],[18,14160,14162,14182],{"className":14161},[124],[18,14163,14165],{"className":14164,"style":1152},[128],[18,14166,14167,14170],{"style":469},[18,14168],{"className":14169,"style":137},[136],[18,14171,14173],{"className":14172},[141,142,143,144],[18,14174,14176],{"className":14175},[106,144],[18,14177,14179],{"className":14178},[106,144],[18,14180,12817],{"className":14181},[106,6354,144],[18,14183,152],{"className":14184},[151],[18,14186,14188],{"className":14187},[124],[18,14189,14191],{"className":14190,"style":159},[128],[18,14192],{},"。要让下界为正，必须 ",[18,14195,14197,14229],{"className":14196,"translate":22},[21],[18,14198,14200],{"className":14199},[26],[28,14201,14202],{"xmlns":30},[32,14203,14204,14226],{},[35,14205,14206,14212,14222,14224],{},[5654,14207,14208,14210],{},[41,14209,64],{},[45,14211,2266],{},[38,14213,14214,14216],{},[41,14215,12575],{},[35,14217,14218,14220],{},[41,14219,5031],{"mathvariant":316},[41,14221,12582],{"mathvariant":316},[49,14223,6684],{},[45,14225,47],{},[86,14227,14228],{"encoding":88},"\\frac n2D_{\\mathrm{KL}}\u003C1",[18,14230,14232,14363],{"className":14231,"ariaHidden":51},[93],[18,14233,14235,14239,14308,14354,14357,14360],{"className":14234},[97],[18,14236],{"className":14237,"style":14238},[101],"height:1.0404em;vertical-align:-0.345em;",[18,14240,14242,14245,14305],{"className":14241},[106],[18,14243],{"className":14244},[250,6012],[18,14246,14248],{"className":14247},[5654],[18,14249,14251,14297],{"className":14250},[119,120],[18,14252,14254,14294],{"className":14253},[124],[18,14255,14258,14272,14280],{"className":14256,"style":14257},[128],"height:0.6954em;",[18,14259,14260,14263],{"style":12048},[18,14261],{"className":14262,"style":697},[136],[18,14264,14266],{"className":14265},[141,142,143,144],[18,14267,14269],{"className":14268},[106,144],[18,14270,2266],{"className":14271},[106,144],[18,14273,14274,14277],{"style":6040},[18,14275],{"className":14276,"style":697},[136],[18,14278],{"className":14279,"style":6048},[6047],[18,14281,14282,14285],{"style":12071},[18,14283],{"className":14284,"style":697},[136],[18,14286,14288],{"className":14287},[141,142,143,144],[18,14289,14291],{"className":14290},[106,144],[18,14292,64],{"className":14293},[106,110,144],[18,14295,152],{"className":14296},[151],[18,14298,14300],{"className":14299},[124],[18,14301,14303],{"className":14302,"style":12093},[128],[18,14304],{},[18,14306],{"className":14307},[267,6012],[18,14309,14311,14314],{"className":14310},[106],[18,14312,12575],{"className":14313,"style":254},[106,110],[18,14315,14317],{"className":14316},[115],[18,14318,14320,14346],{"className":14319},[119,120],[18,14321,14323,14343],{"className":14322},[124],[18,14324,14326],{"className":14325,"style":1152},[128],[18,14327,14328,14331],{"style":469},[18,14329],{"className":14330,"style":137},[136],[18,14332,14334],{"className":14333},[141,142,143,144],[18,14335,14337],{"className":14336},[106,144],[18,14338,14340],{"className":14339},[106,144],[18,14341,12817],{"className":14342},[106,6354,144],[18,14344,152],{"className":14345},[151],[18,14347,14349],{"className":14348},[124],[18,14350,14352],{"className":14351,"style":159},[128],[18,14353],{},[18,14355],{"className":14356,"style":228},[169],[18,14358,6684],{"className":14359},[232],[18,14361],{"className":14362,"style":228},[169],[18,14364,14366,14369],{"className":14365},[97],[18,14367],{"className":14368,"style":505},[101],[18,14370,47],{"className":14371},[106],"；换句话说，如果两个分布在这个样本量下已经能被可靠地区分，两点方法就给不出非平凡的下界了。",[11,14374,14375,14376,14411],{},"这也解释了为什么下界里的样本量总以「",[18,14377,14379,14395],{"className":14378,"translate":22},[21],[18,14380,14382],{"className":14381},[26],[28,14383,14384],{"xmlns":30},[32,14385,14386,14392],{},[35,14387,14388,14390],{},[41,14389,64],{},[49,14391,2015],{},[86,14393,14394],{"encoding":88},"n\\times",[18,14396,14398],{"className":14397,"ariaHidden":51},[93],[18,14399,14401,14405,14408],{"className":14400},[97],[18,14402],{"className":14403,"style":14404},[101],"height:0.6667em;vertical-align:-0.0833em;",[18,14406,64],{"className":14407},[106,110],[18,14409,2015],{"className":14410},[106]," 单样本信息」的形式出现：独立样本的信息是线性累加的。",[1311,14413,14415],{"id":14414},"_33-一个例子bernoulli-均值","3.3 一个例子：Bernoulli 均值",[11,14417,16,14418,4741,14537,14642],{},[18,14419,14421,14457],{"className":14420,"translate":22},[21],[18,14422,14424],{"className":14423},[26],[28,14425,14426],{"xmlns":30},[32,14427,14428,14454],{},[35,14429,14430,14436,14438,14448,14450,14452],{},[38,14431,14432,14434],{},[41,14433,43],{},[41,14435,5600],{},[49,14437,67],{},[35,14439,14440,14442,14445],{},[41,14441,3978],{"mathvariant":316},[41,14443,14444],{"mathvariant":316},"e",[41,14446,14447],{"mathvariant":316},"r",[49,14449,74],{"stretchy":73},[41,14451,11],{},[49,14453,84],{"stretchy":73},[86,14455,14456],{"encoding":88},"X_i\\sim\\mathrm{Ber}(p)",[18,14458,14460,14515],{"className":14459,"ariaHidden":51},[93],[18,14461,14463,14466,14506,14509,14512],{"className":14462},[97],[18,14464],{"className":14465,"style":4563},[101],[18,14467,14469,14472],{"className":14468},[106],[18,14470,43],{"className":14471,"style":111},[106,110],[18,14473,14475],{"className":14474},[115],[18,14476,14478,14498],{"className":14477},[119,120],[18,14479,14481,14495],{"className":14480},[124],[18,14482,14484],{"className":14483,"style":5921},[128],[18,14485,14486,14489],{"style":132},[18,14487],{"className":14488,"style":137},[136],[18,14490,14492],{"className":14491},[141,142,143,144],[18,14493,5600],{"className":14494},[106,110,144],[18,14496,152],{"className":14497},[151],[18,14499,14501],{"className":14500},[124],[18,14502,14504],{"className":14503,"style":159},[128],[18,14505],{},[18,14507],{"className":14508,"style":228},[169],[18,14510,67],{"className":14511},[232],[18,14513],{"className":14514,"style":228},[169],[18,14516,14518,14521,14528,14531,14534],{"className":14517},[97],[18,14519],{"className":14520,"style":242},[101],[18,14522,14524],{"className":14523},[106],[18,14525,14527],{"className":14526},[106,6354],"Ber",[18,14529,74],{"className":14530},[250],[18,14532,11],{"className":14533},[106,110],[18,14535,84],{"className":14536},[267],[18,14538,14540,14578],{"className":14539,"translate":22},[21],[18,14541,14543],{"className":14542},[26],[28,14544,14545],{"xmlns":30},[32,14546,14547,14575],{},[35,14548,14549,14551,14553,14555,14557,14559,14561,14563,14565,14567,14569,14571,14573],{},[41,14550,11],{},[49,14552,921],{},[49,14554,1331],{"stretchy":73},[45,14556,47],{},[41,14558,317],{"mathvariant":316},[45,14560,2266],{},[49,14562,52],{"separator":51},[45,14564,47],{},[41,14566,317],{"mathvariant":316},[45,14568,2266],{},[49,14570,7987],{},[41,14572,532],{"mathvariant":316},[49,14574,1349],{"stretchy":73},[86,14576,14577],{"encoding":88},"p\\in\\{1\u002F2,1\u002F2+\\Delta\\}",[18,14579,14581,14600,14630],{"className":14580,"ariaHidden":51},[93],[18,14582,14584,14588,14591,14594,14597],{"className":14583},[97],[18,14585],{"className":14586,"style":14587},[101],"height:0.7335em;vertical-align:-0.1944em;",[18,14589,11],{"className":14590},[106,110],[18,14592],{"className":14593,"style":228},[169],[18,14595,921],{"className":14596},[232],[18,14598],{"className":14599,"style":228},[169],[18,14601,14603,14606,14609,14612,14615,14618,14621,14624,14627],{"className":14602},[97],[18,14604],{"className":14605,"style":242},[101],[18,14607,1331],{"className":14608},[250],[18,14610,9060],{"className":14611},[106],[18,14613,52],{"className":14614},[165],[18,14616],{"className":14617,"style":170},[169],[18,14619,9060],{"className":14620},[106],[18,14622],{"className":14623,"style":2068},[169],[18,14625,7987],{"className":14626},[2072],[18,14628],{"className":14629,"style":2068},[169],[18,14631,14633,14636,14639],{"className":14632},[97],[18,14634],{"className":14635,"style":242},[101],[18,14637,532],{"className":14638},[106],[18,14640,1349],{"className":14641},[267],"，损失为平方损失。两个分布的单样本 KL 是",[18,14644,14646],{"className":14645,"translate":22},[875],[18,14647,14649,14780],{"className":14648,"translate":22},[21],[18,14650,14652],{"className":14651},[26],[28,14653,14654],{"xmlns":30,"display":884},[32,14655,14656,14777],{},[35,14657,14658,14668,14670,14678,14680,14689,14691,14693,14701,14703,14711,14713,14715,14717,14719,14721,14723,14725,14731,14734,14736,14738,14740,14742,14744,14750,14752,14755,14757,14763,14765,14767,14769,14771,14773,14775],{},[38,14659,14660,14662],{},[41,14661,12575],{},[35,14663,14664,14666],{},[41,14665,5031],{"mathvariant":316},[41,14667,12582],{"mathvariant":316},[49,14669,74],{"fence":51,"stretchy":51,"minsize":5662,"maxsize":5662},[35,14671,14672,14674,14676],{},[41,14673,3978],{"mathvariant":316},[41,14675,14444],{"mathvariant":316},[41,14677,14447],{"mathvariant":316},[49,14679,74],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[14681,14682,14683],"mstyle",{"scriptlevel":430,"displaystyle":73},[5654,14684,14685,14687],{},[45,14686,47],{},[45,14688,2266],{},[49,14690,84],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,14692,2248],{"fence":73,"stretchy":51,"minsize":5662,"maxsize":5662},[35,14694,14695,14697,14699],{},[41,14696,3978],{"mathvariant":316},[41,14698,14444],{"mathvariant":316},[41,14700,14447],{"mathvariant":316},[49,14702,74],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[14681,14704,14705],{"scriptlevel":430,"displaystyle":73},[5654,14706,14707,14709],{},[45,14708,47],{},[45,14710,2266],{},[49,14712,7987],{},[41,14714,532],{"mathvariant":316},[49,14716,84],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,14718,84],{"fence":51,"stretchy":51,"minsize":5662,"maxsize":5662},[169,14720],{"linebreak":8022},[49,14722,433],{},[49,14724,2253],{},[5654,14726,14727,14729],{},[45,14728,47],{},[45,14730,2266],{},[41,14732,14733],{},"log",[49,14735,899],{},[49,14737,74],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[45,14739,47],{},[49,14741,2253],{},[45,14743,10447],{},[1556,14745,14746,14748],{},[41,14747,532],{"mathvariant":316},[45,14749,2266],{},[49,14751,84],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,14753,14754],{},"≈",[45,14756,2266],{},[1556,14758,14759,14761],{},[41,14760,532],{"mathvariant":316},[45,14762,2266],{},[169,14764],{"width":7344},[49,14766,74],{"stretchy":73},[41,14768,532],{"mathvariant":316},[49,14770,1880],{},[45,14772,430],{},[49,14774,84],{"stretchy":73},[41,14776,968],{"mathvariant":316},[86,14778,14779],{"encoding":88},"D_{\\mathrm{KL}}\\Bigl(\\mathrm{Ber}\\bigl(\\tfrac12\\bigr)\\Big\\Vert\\mathrm{Ber}\\bigl(\\tfrac12+\\Delta\\bigr)\\Bigr) \\\\\n=-\\frac12\\log\\bigl(1-4\\Delta^2\\bigr)\n\\approx 2\\Delta^2\n\\qquad(\\Delta\\to0).",[18,14781,14783,15060,15081,15084,15097,15196,15250,15307],{"className":14782,"ariaHidden":51},[93],[18,14784,14786,14789,14835,14841,14847,14853,14921,14927,14971,14977,14983,15051,15054,15057],{"className":14785},[97],[18,14787],{"className":14788,"style":6225},[101],[18,14790,14792,14795],{"className":14791},[106],[18,14793,12575],{"className":14794,"style":254},[106,110],[18,14796,14798],{"className":14797},[115],[18,14799,14801,14827],{"className":14800},[119,120],[18,14802,14804,14824],{"className":14803},[124],[18,14805,14807],{"className":14806,"style":1152},[128],[18,14808,14809,14812],{"style":469},[18,14810],{"className":14811,"style":137},[136],[18,14813,14815],{"className":14814},[141,142,143,144],[18,14816,14818],{"className":14817},[106,144],[18,14819,14821],{"className":14820},[106,144],[18,14822,12817],{"className":14823},[106,6354,144],[18,14825,152],{"className":14826},[151],[18,14828,14830],{"className":14829},[124],[18,14831,14833],{"className":14832,"style":159},[128],[18,14834],{},[18,14836,14838],{"className":14837},[250],[18,14839,74],{"className":14840},[1230,6082],[18,14842,14844],{"className":14843},[106],[18,14845,14527],{"className":14846},[106,6354],[18,14848,14850],{"className":14849},[250],[18,14851,74],{"className":14852},[1230,1196],[18,14854,14856,14859,14918],{"className":14855},[106],[18,14857],{"className":14858},[250,6012],[18,14860,14862],{"className":14861},[5654],[18,14863,14865,14910],{"className":14864},[119,120],[18,14866,14868,14907],{"className":14867},[124],[18,14869,14871,14885,14893],{"className":14870,"style":12045},[128],[18,14872,14873,14876],{"style":12048},[18,14874],{"className":14875,"style":697},[136],[18,14877,14879],{"className":14878},[141,142,143,144],[18,14880,14882],{"className":14881},[106,144],[18,14883,2266],{"className":14884},[106,144],[18,14886,14887,14890],{"style":6040},[18,14888],{"className":14889,"style":697},[136],[18,14891],{"className":14892,"style":6048},[6047],[18,14894,14895,14898],{"style":12071},[18,14896],{"className":14897,"style":697},[136],[18,14899,14901],{"className":14900},[141,142,143,144],[18,14902,14904],{"className":14903},[106,144],[18,14905,47],{"className":14906},[106,144],[18,14908,152],{"className":14909},[151],[18,14911,14913],{"className":14912},[124],[18,14914,14916],{"className":14915,"style":12093},[128],[18,14917],{},[18,14919],{"className":14920},[267,6012],[18,14922,14924],{"className":14923},[267],[18,14925,84],{"className":14926},[1230,1196],[18,14928,14930],{"className":14929},[106],[18,14931,14933],{"className":14932},[1230,6108],[18,14934,14936,14962],{"className":14935},[119,120],[18,14937,14939,14959],{"className":14938},[124],[18,14940,14943],{"className":14941,"style":14942},[128],"height:1.15em;",[18,14944,14946,14950],{"style":14945},"top:-3.15em;",[18,14947],{"className":14948,"style":14949},[136],"height:3.8em;",[18,14951,14953],{"style":14952},"width:0.556em;height:1.8em;",[6130,14954,14956],{"xmlns":6132,"width":6133,"height":5662,"viewBox":14955},"0 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E\\bigl[(p_i-\\hat p)^2\\bigr]\n\\ge\n\\frac{\\Delta^2}{4}\\left(1-\\frac12\\right)\n=\\frac{1}{32n}.",[18,15763,15765,15881,15974,16171],{"className":15764,"ariaHidden":51},[93],[18,15766,15768,15772,15817,15820,15823,15829,15832,15872,15875,15878],{"className":15767},[97],[18,15769],{"className":15770,"style":15771},[101],"height:1.5777em;vertical-align:-0.7277em;",[18,15773,15775],{"className":15774},[985,986],[18,15776,15778,15808],{"className":15777},[119,120],[18,15779,15781,15805],{"className":15780},[124],[18,15782,15784,15795],{"className":15783,"style":1083},[128],[18,15785,15786,15789],{"style":5738},[18,15787],{"className":15788,"style":697},[136],[18,15790,15792],{"className":15791},[141,142,143,144],[18,15793,5600],{"className":15794},[106,110,144],[18,15796,15797,15800],{"style":693},[18,15798],{"className":15799,"style":697},[136],[18,15801,15802],{},[18,15803,5593],{"className":15804},[985],[18,15806,152],{"className":15807},[151],[18,15809,15811],{"className":15810},[124],[18,15812,15815],{"className":15813,"style":15814},[128],"height:0.7277em;",[18,15816],{},[18,15818],{"className":15819,"style":170},[169],[18,15821,934],{"className":15822},[106,1139],[18,15824,15826],{"className":15825},[250],[18,15827,945],{"className":15828},[1230,1196],[18,15830,74],{"className":15831},[250],[18,15833,15835,15838],{"className":15834},[106],[18,15836,11],{"className":15837},[106,110],[18,15839,15841],{"className":15840},[115],[18,15842,15844,15864],{"className":15843},[119,120],[18,15845,15847,15861],{"className":15846},[124],[18,15848,15850],{"className":15849,"style":5921},[128],[18,15851,15852,15855],{"style":1155},[18,15853],{"className":15854,"style":137},[136],[18,15856,15858],{"className":15857},[141,142,143,144],[18,15859,5600],{"className":15860},[106,110,144],[18,15862,152],{"className":15863},[151],[18,15865,15867],{"className":15866},[124],[18,15868,15870],{"className":15869,"style":159},[128],[18,15871],{},[18,15873],{"className":15874,"style":2068},[169],[18,15876,2253],{"className":15877},[2072],[18,15879],{"className":15880,"style":2068},[169],[18,15882,15884,15887,15930,15959,15965,15968,15971],{"className":15883},[97],[18,15885],{"className":15886,"style":15202},[101],[18,15888,15890],{"className":15889},[106,681],[18,15891,15893,15921],{"className":15892},[119,120],[18,15894,15896,15918],{"className":15895},[124],[18,15897,15899,15907],{"className":15898,"style":295},[128],[18,15900,15901,15904],{"style":693},[18,15902],{"className":15903,"style":697},[136],[18,15905,11],{"className":15906},[106,110],[18,15908,15909,15912],{"style":693},[18,15910],{"className":15911,"style":697},[136],[18,15913,15915],{"className":15914,"style":711},[710],[18,15916,664],{"className":15917},[106],[18,15919,152],{"className":15920},[151],[18,15922,15924],{"className":15923},[124],[18,15925,15928],{"className":15926,"style":15927},[128],"height:0.1944em;",[18,15929],{},[18,15931,15933,15936],{"className":15932},[267],[18,15934,84],{"className":15935},[267],[18,15937,15939],{"className":15938},[115],[18,15940,15942],{"className":15941},[119],[18,15943,15945],{"className":15944},[124],[18,15946,15948],{"className":15947,"style":10775},[128],[18,15949,15950,15953],{"style":1637},[18,15951],{"className":15952,"style":137},[136],[18,15954,15956],{"className":15955},[141,142,143,144],[18,15957,2266],{"className":15958},[106,144],[18,15960,15962],{"className":15961},[267],[18,15963,965],{"className":15964},[1230,1196],[18,15966],{"className":15967,"style":228},[169],[18,15969,4775],{"className":15970},[232],[18,15972],{"className":15973,"style":228},[169],[18,15975,15977,15981,16070,16073,16162,16165,16168],{"className":15976},[97],[18,15978],{"className":15979,"style":15980},[101],"height:2.4411em;vertical-align:-0.95em;",[18,15982,15984,15987,16067],{"className":15983},[106],[18,15985],{"className":15986},[250,6012],[18,15988,15990],{"className":15989},[5654],[18,15991,15993,16059],{"className":15992},[119,120],[18,15994,15996,16056],{"className":15995},[124],[18,15997,16000,16011,16019],{"className":15998,"style":15999},[128],"height:1.4911em;",[18,16001,16002,16005],{"style":6028},[18,16003],{"className":16004,"style":697},[136],[18,16006,16008],{"className":16007},[106],[18,16009,10447],{"className":16010},[106],[18,16012,16013,16016],{"style":6040},[18,16014],{"className":16015,"style":697},[136],[18,16017],{"className":16018,"style":6048},[6047],[18,16020,16021,16024],{"style":6051},[18,16022],{"className":16023,"style":697},[136],[18,16025,16027],{"className":16026},[106],[18,16028,16030,16033],{"className":16029},[106],[18,16031,532],{"className":16032},[106],[18,16034,16036],{"className":16035},[115],[18,16037,16039],{"className":16038},[119],[18,16040,16042],{"className":16041},[124],[18,16043,16045],{"className":16044,"style":2416},[128],[18,16046,16047,16050],{"style":1962},[18,16048],{"className":16049,"style":137},[136],[18,16051,16053],{"className":16052},[141,142,143,144],[18,16054,2266],{"className":16055},[106,144],[18,16057,152],{"className":16058},[151],[18,16060,16062],{"className":16061},[124],[18,16063,16065],{"className":16064,"style":6070},[128],[18,16066],{},[18,16068],{"className":16069},[267,6012],[18,16071],{"className":16072,"style":170},[169],[18,16074,16076,16082,16085,16088,16091,16094,16156],{"className":16075},[174],[18,16077,16079],{"className":16078,"style":13796},[250,13795],[18,16080,74],{"className":16081},[1230,143],[18,16083,47],{"className":16084},[106],[18,16086],{"className":16087,"style":2068},[169],[18,16089,2253],{"className":16090},[2072],[18,16092],{"className":16093,"style":2068},[169],[18,16095,16097,16100,16153],{"className":16096},[106],[18,16098],{"className":16099},[250,6012],[18,16101,16103],{"className":16102},[5654],[18,16104,16106,16145],{"className":16105},[119,120],[18,16107,16109,16142],{"className":16108},[124],[18,16110,16112,16123,16131],{"className":16111,"style":9581},[128],[18,16113,16114,16117],{"style":6028},[18,16115],{"className":16116,"style":697},[136],[18,16118,16120],{"className":16119},[106],[18,16121,2266],{"className":16122},[106],[18,16124,16125,16128],{"style":6040},[18,16126],{"className":16127,"style":697},[136],[18,16129],{"className":16130,"style":6048},[6047],[18,16132,16133,16136],{"style":6051},[18,16134],{"className":16135,"style":697},[136],[18,16137,16139],{"className":16138},[106],[18,16140,47],{"className":16141},[106],[18,16143,152],{"className":16144},[151],[18,16146,16148],{"className":16147},[124],[18,16149,16151],{"className":16150,"style":6070},[128],[18,16152],{},[18,16154],{"className":16155},[267,6012],[18,16157,16159],{"className":16158,"style":13796},[267,13795],[18,16160,84],{"className":16161},[1230,143],[18,16163],{"className":16164,"style":228},[169],[18,16166,433],{"className":16167},[232],[18,16169],{"className":16170,"style":228},[169],[18,16172,16174,16177,16242],{"className":16173},[97],[18,16175],{"className":16176,"style":9562},[101],[18,16178,16180,16183,16239],{"className":16179},[106],[18,16181],{"className":16182},[250,6012],[18,16184,16186],{"className":16185},[5654],[18,16187,16189,16231],{"className":16188},[119,120],[18,16190,16192,16228],{"className":16191},[124],[18,16193,16195,16209,16217],{"className":16194,"style":9581},[128],[18,16196,16197,16200],{"style":6028},[18,16198],{"className":16199,"style":697},[136],[18,16201,16203,16206],{"className":16202},[106],[18,16204,15754],{"className":16205},[106],[18,16207,64],{"className":16208},[106,110],[18,16210,16211,16214],{"style":6040},[18,16212],{"className":16213,"style":697},[136],[18,16215],{"className":16216,"style":6048},[6047],[18,16218,16219,16222],{"style":6051},[18,16220],{"className":16221,"style":697},[136],[18,16223,16225],{"className":16224},[106],[18,16226,47],{"className":16227},[106],[18,16229,152],{"className":16230},[151],[18,16232,16234],{"className":16233},[124],[18,16235,16237],{"className":16236,"style":6070},[128],[18,16238],{},[18,16240],{"className":16241},[267,6012],[18,16243,968],{"className":16244},[106],[11,16246,16247,16248,16283,16284],{},"所以 Bernoulli 均值的极小极大速率是 ",[18,16249,16251,16268],{"className":16250,"translate":22},[21],[18,16252,16254],{"className":16253},[26],[28,16255,16256],{"xmlns":30},[32,16257,16258,16266],{},[35,16259,16260,16262,16264],{},[45,16261,47],{},[41,16263,317],{"mathvariant":316},[41,16265,64],{},[86,16267,322],{"encoding":88},[18,16269,16271],{"className":16270,"ariaHidden":51},[93],[18,16272,16274,16277,16280],{"className":16273},[97],[18,16275],{"className":16276,"style":242},[101],[18,16278,335],{"className":16279},[106],[18,16281,64],{"className":16282},[106,110],"。这个结论不稀奇，但推导方式值得记住：",[717,16285,16286,16287,16322],{},"我们只用了两个点，就把「",[18,16288,16290,16307],{"className":16289,"translate":22},[21],[18,16291,16293],{"className":16292},[26],[28,16294,16295],{"xmlns":30},[32,16296,16297,16305],{},[35,16298,16299,16301,16303],{},[45,16300,47],{},[41,16302,317],{"mathvariant":316},[41,16304,64],{},[86,16306,322],{"encoding":88},[18,16308,16310],{"className":16309,"ariaHidden":51},[93],[18,16311,16313,16316,16319],{"className":16312},[97],[18,16314],{"className":16315,"style":242},[101],[18,16317,335],{"className":16318},[106],[18,16320,64],{"className":16321},[106,110]," 不可能被突破」这件事证明完了。",[1311,16324,16326],{"id":16325},"_34-两点方法的局限","3.4 两点方法的局限",[11,16328,16329,16330,16358],{},"两点方法简单、常数清楚，但它只能看见一个方向。考虑 ",[18,16331,16333,16346],{"className":16332,"translate":22},[21],[18,16334,16336],{"className":16335},[26],[28,16337,16338],{"xmlns":30},[32,16339,16340,16344],{},[35,16341,16342],{},[41,16343,4031],{},[86,16345,4031],{"encoding":88},[18,16347,16349],{"className":16348,"ariaHidden":51},[93],[18,16350,16352,16355],{"className":16351},[97],[18,16353],{"className":16354,"style":295},[101],[18,16356,4031],{"className":16357},[106,110]," 维高斯均值：",[18,16360,16362],{"className":16361,"translate":22},[875],[18,16363,16365,16437],{"className":16364,"translate":22},[21],[18,16366,16368],{"className":16367},[26],[28,16369,16370],{"xmlns":30,"display":884},[32,16371,16372,16434],{},[35,16373,16374,16380,16382,16384,16386,16392,16394,16396,16398,16400,16402,16409,16416,16418,16420,16422,16424,16426,16432],{},[38,16375,16376,16378],{},[41,16377,43],{},[45,16379,47],{},[49,16381,52],{"separator":51},[49,16383,55],{},[49,16385,52],{"separator":51},[38,16387,16388,16390],{},[41,16389,43],{},[41,16391,64],{},[49,16393,67],{},[41,16395,70],{},[49,16397,74],{"stretchy":73},[41,16399,77],{},[49,16401,52],{"separator":51},[1556,16403,16404,16407],{},[41,16405,16406],{},"σ",[45,16408,2266],{},[38,16410,16411,16414],{},[41,16412,16413],{},"I",[41,16415,4031],{},[49,16417,84],{"stretchy":73},[49,16419,52],{"separator":51},[169,16421],{"width":7344},[41,16423,77],{},[49,16425,921],{},[1556,16427,16428,16430],{},[41,16429,3071],{"mathvariant":933},[41,16431,4031],{},[41,16433,968],{"mathvariant":316},[86,16435,16436],{"encoding":88},"X_1,\\dots,X_n\\sim N(\\theta,\\sigma^2 I_d),\\qquad \\theta\\in\\mathbb R^d.",[18,16438,16440,16553,16667],{"className":16439,"ariaHidden":51},[93],[18,16441,16443,16446,16486,16489,16492,16495,16498,16501,16504,16544,16547,16550],{"className":16442},[97],[18,16444],{"className":16445,"style":102},[101],[18,16447,16449,16452],{"className":16448},[106],[18,16450,43],{"className":16451,"style":111},[106,110],[18,16453,16455],{"className":16454},[115],[18,16456,16458,16478],{"className":16457},[119,120],[18,16459,16461,16475],{"className":16460},[124],[18,16462,16464],{"className":16463,"style":129},[128],[18,16465,16466,16469],{"style":132},[18,16467],{"className":16468,"style":137},[136],[18,16470,16472],{"className":16471},[141,142,143,144],[18,16473,47],{"className":16474},[106,144],[18,16476,152],{"className":16477},[151],[18,16479,16481],{"className":16480},[124],[18,16482,16484],{"className":16483,"style":159},[128],[18,16485],{},[18,16487,52],{"className":16488},[165],[18,16490],{"className":16491,"style":170},[169],[18,16493,55],{"className":16494},[174],[18,16496],{"className":16497,"style":170},[169],[18,16499,52],{"className":16500},[165],[18,16502],{"className":16503,"style":170},[169],[18,16505,16507,16510],{"className":16506},[106],[18,16508,43],{"className":16509,"style":111},[106,110],[18,16511,16513],{"className":16512},[115],[18,16514,16516,16536],{"className":16515},[119,120],[18,16517,16519,16533],{"className":16518},[124],[18,16520,16522],{"className":16521,"style":202},[128],[18,16523,16524,16527],{"style":132},[18,16525],{"className":16526,"style":137},[136],[18,16528,16530],{"className":16529},[141,142,143,144],[18,16531,64],{"className":16532},[106,110,144],[18,16534,152],{"className":16535},[151],[18,16537,16539],{"className":16538},[124],[18,16540,16542],{"className":16541,"style":159},[128],[18,16543],{},[18,16545],{"className":16546,"style":228},[169],[18,16548,67],{"className":16549},[232],[18,16551],{"className":16552,"style":228},[169],[18,16554,16556,16559,16562,16565,16568,16571,16574,16603,16643,16646,16649,16652,16655,16658,16661,16664],{"className":16555},[97],[18,16557],{"className":16558,"style":15256},[101],[18,16560,70],{"className":16561,"style":246},[106,110],[18,16563,74],{"className":16564},[250],[18,16566,77],{"className":16567,"style":254},[106,110],[18,16569,52],{"className":16570},[165],[18,16572],{"className":16573,"style":170},[169],[18,16575,16577,16580],{"className":16576},[106],[18,16578,16406],{"className":16579,"style":3957},[106,110],[18,16581,16583],{"className":16582},[115],[18,16584,16586],{"className":16585},[119],[18,16587,16589],{"className":16588},[124],[18,16590,16592],{"className":16591,"style":10775},[128],[18,16593,16594,16597],{"style":1637},[18,16595],{"className":16596,"style":137},[136],[18,16598,16600],{"className":16599},[141,142,143,144],[18,16601,2266],{"className":16602},[106,144],[18,16604,16606,16609],{"className":16605},[106],[18,16607,16413],{"className":16608,"style":111},[106,110],[18,16610,16612],{"className":16611},[115],[18,16613,16615,16635],{"className":16614},[119,120],[18,16616,16618,16632],{"className":16617},[124],[18,16619,16621],{"className":16620,"style":1383},[128],[18,16622,16623,16626],{"style":132},[18,16624],{"className":16625,"style":137},[136],[18,16627,16629],{"className":16628},[141,142,143,144],[18,16630,4031],{"className":16631},[106,110,144],[18,16633,152],{"className":16634},[151],[18,16636,16638],{"className":16637},[124],[18,16639,16641],{"className":16640,"style":159},[128],[18,16642],{},[18,16644,84],{"className":16645},[267],[18,16647,52],{"className":16648},[165],[18,16650],{"className":16651,"style":7676},[169],[18,16653],{"className":16654,"style":170},[169],[18,16656,77],{"className":16657,"style":254},[106,110],[18,16659],{"className":16660,"style":228},[169],[18,16662,921],{"className":16663},[232],[18,16665],{"className":16666,"style":228},[169],[18,16668,16670,16674,16703],{"className":16669},[97],[18,16671],{"className":16672,"style":16673},[101],"height:0.8991em;",[18,16675,16677,16680],{"className":16676},[106],[18,16678,3071],{"className":16679},[106,1139],[18,16681,16683],{"className":16682},[115],[18,16684,16686],{"className":16685},[119],[18,16687,16689],{"className":16688},[124],[18,16690,16692],{"className":16691,"style":16673},[128],[18,16693,16694,16697],{"style":1637},[18,16695],{"className":16696,"style":137},[136],[18,16698,16700],{"className":16699},[141,142,143,144],[18,16701,4031],{"className":16702},[106,110,144],[18,16704,968],{"className":16705},[106],[11,16707,16708,16709,16800,16801,16941,16942,17061,17062,17129,17130,17236,17237,17272,17273,17347,17348,1990],{},"如果只取 ",[18,16710,16712,16733],{"className":16711,"translate":22},[21],[18,16713,16715],{"className":16714},[26],[28,16716,16717],{"xmlns":30},[32,16718,16719,16731],{},[35,16720,16721,16727,16729],{},[38,16722,16723,16725],{},[41,16724,77],{},[45,16726,430],{},[49,16728,433],{},[45,16730,430],{},[86,16732,438],{"encoding":88},[18,16734,16736,16791],{"className":16735,"ariaHidden":51},[93],[18,16737,16739,16742,16782,16785,16788],{"className":16738},[97],[18,16740],{"className":16741,"style":448},[101],[18,16743,16745,16748],{"className":16744},[106],[18,16746,77],{"className":16747,"style":254},[106,110],[18,16749,16751],{"className":16750},[115],[18,16752,16754,16774],{"className":16753},[119,120],[18,16755,16757,16771],{"className":16756},[124],[18,16758,16760],{"className":16759,"style":129},[128],[18,16761,16762,16765],{"style":469},[18,16763],{"className":16764,"style":137},[136],[18,16766,16768],{"className":16767},[141,142,143,144],[18,16769,430],{"className":16770},[106,144],[18,16772,152],{"className":16773},[151],[18,16775,16777],{"className":16776},[124],[18,16778,16780],{"className":16779,"style":159},[128],[18,16781],{},[18,16783],{"className":16784,"style":228},[169],[18,16786,433],{"className":16787},[232],[18,16789],{"className":16790,"style":228},[169],[18,16792,16794,16797],{"className":16793},[97],[18,16795],{"className":16796,"style":505},[101],[18,16798,430],{"className":16799},[106],"、",[18,16802,16804,16833],{"className":16803,"translate":22},[21],[18,16805,16807],{"className":16806},[26],[28,16808,16809],{"xmlns":30},[32,16810,16811,16830],{},[35,16812,16813,16819,16821,16824],{},[38,16814,16815,16817],{},[41,16816,77],{},[45,16818,47],{},[49,16820,433],{},[41,16822,16823],{},"a",[38,16825,16826,16828],{},[41,16827,14444],{},[45,16829,47],{},[86,16831,16832],{"encoding":88},"\\theta_1=a e_1",[18,16834,16836,16891],{"className":16835,"ariaHidden":51},[93],[18,16837,16839,16842,16882,16885,16888],{"className":16838},[97],[18,16840],{"className":16841,"style":448},[101],[18,16843,16845,16848],{"className":16844},[106],[18,16846,77],{"className":16847,"style":254},[106,110],[18,16849,16851],{"className":16850},[115],[18,16852,16854,16874],{"className":16853},[119,120],[18,16855,16857,16871],{"className":16856},[124],[18,16858,16860],{"className":16859,"style":129},[128],[18,16861,16862,16865],{"style":469},[18,16863],{"className":16864,"style":137},[136],[18,16866,16868],{"className":16867},[141,142,143,144],[18,16869,47],{"className":16870},[106,144],[18,16872,152],{"className":16873},[151],[18,16875,16877],{"className":16876},[124],[18,16878,16880],{"className":16879,"style":159},[128],[18,16881],{},[18,16883],{"className":16884,"style":228},[169],[18,16886,433],{"className":16887},[232],[18,16889],{"className":16890,"style":228},[169],[18,16892,16894,16898,16901],{"className":16893},[97],[18,16895],{"className":16896,"style":16897},[101],"height:0.5806em;vertical-align:-0.15em;",[18,16899,16823],{"className":16900},[106,110],[18,16902,16904,16907],{"className":16903},[106],[18,16905,14444],{"className":16906},[106,110],[18,16908,16910],{"className":16909},[115],[18,16911,16913,16933],{"className":16912},[119,120],[18,16914,16916,16930],{"className":16915},[124],[18,16917,16919],{"className":16918,"style":129},[128],[18,16920,16921,16924],{"style":1155},[18,16922],{"className":16923,"style":137},[136],[18,16925,16927],{"className":16926},[141,142,143,144],[18,16928,47],{"className":16929},[106,144],[18,16931,152],{"className":16932},[151],[18,16934,16936],{"className":16935},[124],[18,16937,16939],{"className":16938,"style":159},[128],[18,16940],{},"，那么 KL 是 ",[18,16943,16945,16979],{"className":16944,"translate":22},[21],[18,16946,16948],{"className":16947},[26],[28,16949,16950],{"xmlns":30},[32,16951,16952,16976],{},[35,16953,16954,16956,16962,16964,16966,16968,16974],{},[41,16955,64],{},[1556,16957,16958,16960],{},[41,16959,16823],{},[45,16961,2266],{},[41,16963,317],{"mathvariant":316},[49,16965,74],{"stretchy":73},[45,16967,2266],{},[1556,16969,16970,16972],{},[41,16971,16406],{},[45,16973,2266],{},[49,16975,84],{"stretchy":73},[86,16977,16978],{"encoding":88},"n a^2\u002F(2\\sigma^2)",[18,16980,16982],{"className":16981,"ariaHidden":51},[93],[18,16983,16985,16988,16991,17020,17023,17026,17029,17058],{"className":16984},[97],[18,16986],{"className":16987,"style":10271},[101],[18,16989,64],{"className":16990},[106,110],[18,16992,16994,16997],{"className":16993},[106],[18,16995,16823],{"className":16996},[106,110],[18,16998,17000],{"className":16999},[115],[18,17001,17003],{"className":17002},[119],[18,17004,17006],{"className":17005},[124],[18,17007,17009],{"className":17008,"style":2416},[128],[18,17010,17011,17014],{"style":1962},[18,17012],{"className":17013,"style":137},[136],[18,17015,17017],{"className":17016},[141,142,143,144],[18,17018,2266],{"className":17019},[106,144],[18,17021,317],{"className":17022},[106],[18,17024,74],{"className":17025},[250],[18,17027,2266],{"className":17028},[106],[18,17030,17032,17035],{"className":17031},[106],[18,17033,16406],{"className":17034,"style":3957},[106,110],[18,17036,17038],{"className":17037},[115],[18,17039,17041],{"className":17040},[119],[18,17042,17044],{"className":17043},[124],[18,17045,17047],{"className":17046,"style":2416},[128],[18,17048,17049,17052],{"style":1962},[18,17050],{"className":17051,"style":137},[136],[18,17053,17055],{"className":17054},[141,142,143,144],[18,17056,2266],{"className":17057},[106,144],[18,17059,84],{"className":17060},[267],"，Le Cam 给出的下界是 ",[18,17063,17065,17087],{"className":17064,"translate":22},[21],[18,17066,17068],{"className":17067},[26],[28,17069,17070],{"xmlns":30},[32,17071,17072,17084],{},[35,17073,17074,17080,17082],{},[1556,17075,17076,17078],{},[41,17077,16823],{},[45,17079,2266],{},[41,17081,317],{"mathvariant":316},[45,17083,10447],{},[86,17085,17086],{"encoding":88},"a^2\u002F4",[18,17088,17090],{"className":17089,"ariaHidden":51},[93],[18,17091,17093,17096,17125],{"className":17092},[97],[18,17094],{"className":17095,"style":10271},[101],[18,17097,17099,17102],{"className":17098},[106],[18,17100,16823],{"className":17101},[106,110],[18,17103,17105],{"className":17104},[115],[18,17106,17108],{"className":17107},[119],[18,17109,17111],{"className":17110},[124],[18,17112,17114],{"className":17113,"style":2416},[128],[18,17115,17116,17119],{"style":1962},[18,17117],{"className":17118,"style":137},[136],[18,17120,17122],{"className":17121},[141,142,143,144],[18,17123,2266],{"className":17124},[106,144],[18,17126,17128],{"className":17127},[106],"\u002F4"," 的量级，取 ",[18,17131,17133,17158],{"className":17132,"translate":22},[21],[18,17134,17136],{"className":17135},[26],[28,17137,17138],{"xmlns":30},[32,17139,17140,17155],{},[35,17141,17142,17144,17147,17149,17151],{},[41,17143,16823],{},[49,17145,17146],{},"≍",[41,17148,16406],{},[41,17150,317],{"mathvariant":316},[12561,17152,17153],{},[41,17154,64],{},[86,17156,17157],{"encoding":88},"a\\asymp\\sigma\u002F\\sqrt n",[18,17159,17161,17180],{"className":17160,"ariaHidden":51},[93],[18,17162,17164,17168,17171,17174,17177],{"className":17163},[97],[18,17165],{"className":17166,"style":17167},[101],"height:0.4637em;",[18,17169,16823],{"className":17170},[106,110],[18,17172],{"className":17173,"style":228},[169],[18,17175,17146],{"className":17176},[232],[18,17178],{"className":17179,"style":228},[169],[18,17181,17183,17186,17189,17192],{"className":17182},[97],[18,17184],{"className":17185,"style":15501},[101],[18,17187,16406],{"className":17188,"style":3957},[106,110],[18,17190,317],{"className":17191},[106],[18,17193,17195],{"className":17194},[106,12696],[18,17196,17198,17228],{"className":17197},[119,120],[18,17199,17201,17225],{"className":17200},[124],[18,17202,17204,17213],{"className":17203,"style":15400},[128],[18,17205,17207,17210],{"className":17206,"style":693},[12710],[18,17208],{"className":17209,"style":697},[136],[18,17211,64],{"className":17212,"style":15410},[106,110],[18,17214,17215,17218],{"style":15413},[18,17216],{"className":17217,"style":697},[136],[18,17219,17221],{"className":17220,"style":15420},[12853],[6130,17222,17223],{"xmlns":6132,"width":12857,"height":15423,"viewBox":15424,"preserveAspectRatio":12860},[6136,17224],{"d":15427},[18,17226,152],{"className":17227},[151],[18,17229,17231],{"className":17230},[124],[18,17232,17234],{"className":17233,"style":15437},[128],[18,17235],{}," 后只有 ",[18,17238,17240,17257],{"className":17239,"translate":22},[21],[18,17241,17243],{"className":17242},[26],[28,17244,17245],{"xmlns":30},[32,17246,17247,17255],{},[35,17248,17249,17251,17253],{},[45,17250,47],{},[41,17252,317],{"mathvariant":316},[41,17254,64],{},[86,17256,322],{"encoding":88},[18,17258,17260],{"className":17259,"ariaHidden":51},[93],[18,17261,17263,17266,17269],{"className":17262},[97],[18,17264],{"className":17265,"style":242},[101],[18,17267,335],{"className":17268},[106],[18,17270,64],{"className":17271},[106,110],"。但真实的极小极大风险是 ",[18,17274,17276,17300],{"className":17275,"translate":22},[21],[18,17277,17279],{"className":17278},[26],[28,17280,17281],{"xmlns":30},[32,17282,17283,17297],{},[35,17284,17285,17291,17293,17295],{},[1556,17286,17287,17289],{},[41,17288,16406],{},[45,17290,2266],{},[41,17292,4031],{},[41,17294,317],{"mathvariant":316},[41,17296,64],{},[86,17298,17299],{"encoding":88},"\\sigma^2d\u002Fn",[18,17301,17303],{"className":17302,"ariaHidden":51},[93],[18,17304,17306,17309,17338,17341,17344],{"className":17305},[97],[18,17307],{"className":17308,"style":10271},[101],[18,17310,17312,17315],{"className":17311},[106],[18,17313,16406],{"className":17314,"style":3957},[106,110],[18,17316,17318],{"className":17317},[115],[18,17319,17321],{"className":17320},[119],[18,17322,17324],{"className":17323},[124],[18,17325,17327],{"className":17326,"style":2416},[128],[18,17328,17329,17332],{"style":1962},[18,17330],{"className":17331,"style":137},[136],[18,17333,17335],{"className":17334},[141,142,143,144],[18,17336,2266],{"className":17337},[106,144],[18,17339,4031],{"className":17340},[106,110],[18,17342,317],{"className":17343},[106],[18,17345,64],{"className":17346},[106,110],"——差了整整一个 ",[18,17349,17351,17364],{"className":17350,"translate":22},[21],[18,17352,17354],{"className":17353},[26],[28,17355,17356],{"xmlns":30},[32,17357,17358,17362],{},[35,17359,17360],{},[41,17361,4031],{},[86,17363,4031],{"encoding":88},[18,17365,17367],{"className":17366,"ariaHidden":51},[93],[18,17368,17370,17373],{"className":17369},[97],[18,17371],{"className":17372,"style":295},[101],[18,17374,4031],{"className":17375},[106,110],[11,17377,17378,17379,17407,17408,17436,17437,17465],{},"原因很直接：两点方法只探测了一个坐标方向，而 ",[18,17380,17382,17395],{"className":17381,"translate":22},[21],[18,17383,17385],{"className":17384},[26],[28,17386,17387],{"xmlns":30},[32,17388,17389,17393],{},[35,17390,17391],{},[41,17392,4031],{},[86,17394,4031],{"encoding":88},[18,17396,17398],{"className":17397,"ariaHidden":51},[93],[18,17399,17401,17404],{"className":17400},[97],[18,17402],{"className":17403,"style":295},[101],[18,17405,4031],{"className":17406},[106,110]," 维问题的困难来自 ",[18,17409,17411,17424],{"className":17410,"translate":22},[21],[18,17412,17414],{"className":17413},[26],[28,17415,17416],{"xmlns":30},[32,17417,17418,17422],{},[35,17419,17420],{},[41,17421,4031],{},[86,17423,4031],{"encoding":88},[18,17425,17427],{"className":17426,"ariaHidden":51},[93],[18,17428,17430,17433],{"className":17429},[97],[18,17431],{"className":17432,"style":295},[101],[18,17434,4031],{"className":17435},[106,110]," 个方向同时存在的不确定性。要捕捉这一点，需要让假设集合随着 ",[18,17438,17440,17453],{"className":17439,"translate":22},[21],[18,17441,17443],{"className":17442},[26],[28,17444,17445],{"xmlns":30},[32,17446,17447,17451],{},[35,17448,17449],{},[41,17450,4031],{},[86,17452,4031],{"encoding":88},[18,17454,17456],{"className":17455,"ariaHidden":51},[93],[18,17457,17459,17462],{"className":17458},[97],[18,17460],{"className":17461,"style":295},[101],[18,17463,4031],{"className":17464},[106,110]," 指数增长。这就是 Fano 不等式和 Assouad 引理的用武之地。",[638,17467,17469],{"id":17468},"四fano-不等式","四、Fano 不等式",[1311,17471,17473],{"id":17472},"_41-信息论形式","4.1 信息论形式",[5159,17475,17476,17966,18228],{},[11,17477,17478,5166,17481,3019,17509,17577,17578,17606,17607,4741,17635,17769,17770,10363],{},[717,17479,17480],{},"定理 4.1（Fano 不等式）",[18,17482,17484,17497],{"className":17483,"translate":22},[21],[18,17485,17487],{"className":17486},[26],[28,17488,17489],{"xmlns":30},[32,17490,17491,17495],{},[35,17492,17493],{},[41,17494,5031],{},[86,17496,5031],{"encoding":88},[18,17498,17500],{"className":17499,"ariaHidden":51},[93],[18,17501,17503,17506],{"className":17502},[97],[18,17504],{"className":17505,"style":600},[101],[18,17507,5031],{"className":17508,"style":5073},[106,110],[18,17510,17512,17538],{"className":17511,"translate":22},[21],[18,17513,17515],{"className":17514},[26],[28,17516,17517],{"xmlns":30},[32,17518,17519,17535],{},[35,17520,17521,17523,17525,17527,17529,17531,17533],{},[49,17522,1331],{"stretchy":73},[45,17524,47],{},[49,17526,52],{"separator":51},[49,17528,55],{},[49,17530,52],{"separator":51},[41,17532,3526],{},[49,17534,1349],{"stretchy":73},[86,17536,17537],{"encoding":88},"\\{1,\\dots,M\\}",[18,17539,17541],{"className":17540,"ariaHidden":51},[93],[18,17542,17544,17547,17550,17553,17556,17559,17562,17565,17568,17571,17574],{"className":17543},[97],[18,17545],{"className":17546,"style":242},[101],[18,17548,1331],{"className":17549},[250],[18,17551,47],{"className":17552},[106],[18,17554,52],{"className":17555},[165],[18,17557],{"className":17558,"style":170},[169],[18,17560,55],{"className":17561},[174],[18,17563],{"className":17564,"style":170},[169],[18,17566,52],{"className":17567},[165],[18,17569],{"className":17570,"style":170},[169],[18,17572,3526],{"className":17573,"style":246},[106,110],[18,17575,1349],{"className":17576},[267]," 上均匀分布，",[18,17579,17581,17594],{"className":17580,"translate":22},[21],[18,17582,17584],{"className":17583},[26],[28,17585,17586],{"xmlns":30},[32,17587,17588,17592],{},[35,17589,17590],{},[41,17591,43],{},[86,17593,43],{"encoding":88},[18,17595,17597],{"className":17596,"ariaHidden":51},[93],[18,17598,17600,17603],{"className":17599},[97],[18,17601],{"className":17602,"style":600},[101],[18,17604,43],{"className":17605,"style":111},[106,110]," 的分布依赖于 ",[18,17608,17610,17623],{"className":17609,"translate":22},[21],[18,17611,17613],{"className":17612},[26],[28,17614,17615],{"xmlns":30},[32,17616,17617,17621],{},[35,17618,17619],{},[41,17620,5031],{},[86,17622,5031],{"encoding":88},[18,17624,17626],{"className":17625,"ariaHidden":51},[93],[18,17627,17629,17632],{"className":17628},[97],[18,17630],{"className":17631,"style":600},[101],[18,17633,5031],{"className":17634,"style":5073},[106,110],[18,17636,17638,17670],{"className":17637,"translate":22},[21],[18,17639,17641],{"className":17640},[26],[28,17642,17643],{"xmlns":30},[32,17644,17645,17667],{},[35,17646,17647,17653,17655,17661,17663,17665],{},[658,17648,17649,17651],{"accent":51},[41,17650,5031],{},[49,17652,664],{},[49,17654,433],{},[658,17656,17657,17659],{"accent":51},[41,17658,5031],{},[49,17660,664],{},[49,17662,74],{"stretchy":73},[41,17664,43],{},[49,17666,84],{"stretchy":73},[86,17668,17669],{"encoding":88},"\\hat K=\\hat K(X)",[18,17671,17673,17722],{"className":17672,"ariaHidden":51},[93],[18,17674,17676,17680,17713,17716,17719],{"className":17675},[97],[18,17677],{"className":17678,"style":17679},[101],"height:0.9468em;",[18,17681,17683],{"className":17682},[106,681],[18,17684,17686],{"className":17685},[119],[18,17687,17689],{"className":17688},[124],[18,17690,17692,17700],{"className":17691,"style":17679},[128],[18,17693,17694,17697],{"style":693},[18,17695],{"className":17696,"style":697},[136],[18,17698,5031],{"className":17699,"style":5073},[106,110],[18,17701,17703,17706],{"style":17702},"top:-3.2523em;",[18,17704],{"className":17705,"style":697},[136],[18,17707,17710],{"className":17708,"style":17709},[710],"left:-0.1944em;",[18,17711,664],{"className":17712},[106],[18,17714],{"className":17715,"style":228},[169],[18,17717,433],{"className":17718},[232],[18,17720],{"className":17721,"style":228},[169],[18,17723,17725,17729,17760,17763,17766],{"className":17724},[97],[18,17726],{"className":17727,"style":17728},[101],"height:1.1968em;vertical-align:-0.25em;",[18,17730,17732],{"className":17731},[106,681],[18,17733,17735],{"className":17734},[119],[18,17736,17738],{"className":17737},[124],[18,17739,17741,17749],{"className":17740,"style":17679},[128],[18,17742,17743,17746],{"style":693},[18,17744],{"className":17745,"style":697},[136],[18,17747,5031],{"className":17748,"style":5073},[106,110],[18,17750,17751,17754],{"style":17702},[18,17752],{"className":17753,"style":697},[136],[18,17755,17757],{"className":17756,"style":17709},[710],[18,17758,664],{"className":17759},[106],[18,17761,74],{"className":17762},[250],[18,17764,43],{"className":17765,"style":111},[106,110],[18,17767,84],{"className":17768},[267]," 是任意估计。记 ",[18,17771,17773,17811],{"className":17772,"translate":22},[21],[18,17774,17776],{"className":17775},[26],[28,17777,17778],{"xmlns":30},[32,17779,17780,17808],{},[35,17781,17782,17788,17790,17792,17794,17796,17802,17804,17806],{},[38,17783,17784,17786],{},[41,17785,939],{},[41,17787,14444],{},[49,17789,433],{},[41,17791,11208],{},[49,17793,899],{},[49,17795,74],{"stretchy":73},[658,17797,17798,17800],{"accent":51},[41,17799,5031],{},[49,17801,664],{},[49,17803,2771],{"mathvariant":316},[41,17805,5031],{},[49,17807,84],{"stretchy":73},[86,17809,17810],{"encoding":88},"P_e=\\Pr(\\hat K\\neq K)",[18,17812,17814,17869,17954],{"className":17813,"ariaHidden":51},[93],[18,17815,17817,17820,17860,17863,17866],{"className":17816},[97],[18,17818],{"className":17819,"style":4563},[101],[18,17821,17823,17826],{"className":17822},[106],[18,17824,939],{"className":17825,"style":1171},[106,110],[18,17827,17829],{"className":17828},[115],[18,17830,17832,17852],{"className":17831},[119,120],[18,17833,17835,17849],{"className":17834},[124],[18,17836,17838],{"className":17837,"style":202},[128],[18,17839,17840,17843],{"style":1386},[18,17841],{"className":17842,"style":137},[136],[18,17844,17846],{"className":17845},[141,142,143,144],[18,17847,14444],{"className":17848},[106,110,144],[18,17850,152],{"className":17851},[151],[18,17853,17855],{"className":17854},[124],[18,17856,17858],{"className":17857,"style":159},[128],[18,17859],{},[18,17861],{"className":17862,"style":228},[169],[18,17864,433],{"className":17865},[232],[18,17867],{"className":17868,"style":228},[169],[18,17870,17872,17875,17878,17881,17912,17915,17951],{"className":17871},[97],[18,17873],{"className":17874,"style":17728},[101],[18,17876,11208],{"className":17877},[985],[18,17879,74],{"className":17880},[250],[18,17882,17884],{"className":17883},[106,681],[18,17885,17887],{"className":17886},[119],[18,17888,17890],{"className":17889},[124],[18,17891,17893,17901],{"className":17892,"style":17679},[128],[18,17894,17895,17898],{"style":693},[18,17896],{"className":17897,"style":697},[136],[18,17899,5031],{"className":17900,"style":5073},[106,110],[18,17902,17903,17906],{"style":17702},[18,17904],{"className":17905,"style":697},[136],[18,17907,17909],{"className":17908,"style":17709},[710],[18,17910,664],{"className":17911},[106],[18,17913],{"className":17914,"style":228},[169],[18,17916,17918,17945,17948],{"className":17917},[232],[18,17919,17921],{"className":17920},[232],[18,17922,17924],{"className":17923},[106,2878],[18,17925,17927],{"className":17926},[2882],[18,17928,17930,17933,17942],{"className":17929},[2886],[18,17931],{"className":17932,"style":2890},[101],[18,17934,17936],{"className":17935},[2894],[18,17937,17939],{"className":17938},[106],[18,17940,2901],{"className":17941},[232],[18,17943],{"className":17944},[2905],[18,17946],{"className":17947},[169,2909],[18,17949,433],{"className":17950},[232],[18,17952],{"className":17953,"style":228},[169],[18,17955,17957,17960,17963],{"className":17956},[97],[18,17958],{"className":17959,"style":242},[101],[18,17961,5031],{"className":17962,"style":5073},[106,110],[18,17964,84],{"className":17965},[267],[18,17967,17969],{"className":17968,"translate":22},[875],[18,17970,17972,18031],{"className":17971,"translate":22},[21],[18,17973,17975],{"className":17974},[26],[28,17976,17977],{"xmlns":30,"display":884},[32,17978,17979,18028],{},[35,17980,17981,17987,17989,17991,17993,18026],{},[38,17982,17983,17985],{},[41,17984,939],{},[41,17986,14444],{},[49,17988,4775],{},[45,17990,47],{},[49,17992,2253],{},[5654,17994,17995,18018],{},[35,17996,17997,17999,18001,18003,18006,18008,18010,18012,18014,18016],{},[41,17998,16413],{},[49,18000,74],{"stretchy":73},[41,18002,5031],{},[49,18004,18005],{"separator":51},";",[41,18007,43],{},[49,18009,84],{"stretchy":73},[49,18011,7987],{},[41,18013,14733],{},[49,18015,899],{},[45,18017,2266],{},[35,18019,18020,18022,18024],{},[41,18021,14733],{},[49,18023,899],{},[41,18025,3526],{},[49,18027,52],{"separator":51},[86,18029,18030],{"encoding":88},"P_e\\ge 1-\\frac{I(K;X)+\\log 2}{\\log M},",[18,18032,18034,18089,18108],{"className":18033,"ariaHidden":51},[93],[18,18035,18037,18040,18080,18083,18086],{"className":18036},[97],[18,18038],{"className":18039,"style":4563},[101],[18,18041,18043,18046],{"className":18042},[106],[18,18044,939],{"className":18045,"style":1171},[106,110],[18,18047,18049],{"className":18048},[115],[18,18050,18052,18072],{"className":18051},[119,120],[18,18053,18055,18069],{"className":18054},[124],[18,18056,18058],{"className":18057,"style":202},[128],[18,18059,18060,18063],{"style":1386},[18,18061],{"className":18062,"style":137},[136],[18,18064,18066],{"className":18065},[141,142,143,144],[18,18067,14444],{"className":18068},[106,110,144],[18,18070,152],{"className":18071},[151],[18,18073,18075],{"className":18074},[124],[18,18076,18078],{"className":18077,"style":159},[128],[18,18079],{},[18,18081],{"className":18082,"style":228},[169],[18,18084,4775],{"className":18085},[232],[18,18087],{"className":18088,"style":228},[169],[18,18090,18092,18096,18099,18102,18105],{"className":18091},[97],[18,18093],{"className":18094,"style":18095},[101],"height:0.7278em;vertical-align:-0.0833em;",[18,18097,47],{"className":18098},[106],[18,18100],{"className":18101,"style":2068},[169],[18,18103,2253],{"className":18104},[2072],[18,18106],{"className":18107,"style":2068},[169],[18,18109,18111,18115,18225],{"className":18110},[97],[18,18112],{"className":18113,"style":18114},[101],"height:2.3074em;vertical-align:-0.8804em;",[18,18116,18118,18121,18222],{"className":18117},[106],[18,18119],{"className":18120},[250,6012],[18,18122,18124],{"className":18123},[5654],[18,18125,18127,18213],{"className":18126},[119,120],[18,18128,18130,18210],{"className":18129},[124],[18,18131,18134,18153,18161],{"className":18132,"style":18133},[128],"height:1.427em;",[18,18135,18136,18139],{"style":6028},[18,18137],{"className":18138,"style":697},[136],[18,18140,18142,18147,18150],{"className":18141},[106],[18,18143,15174,18145],{"className":18144},[985],[18,18146,15177],{"style":6355},[18,18148],{"className":18149,"style":170},[169],[18,18151,3526],{"className":18152,"style":246},[106,110],[18,18154,18155,18158],{"style":6040},[18,18156],{"className":18157,"style":697},[136],[18,18159],{"className":18160,"style":6048},[6047],[18,18162,18163,18166],{"style":6051},[18,18164],{"className":18165,"style":697},[136],[18,18167,18169,18172,18175,18178,18181,18184,18187,18190,18193,18196,18199,18204,18207],{"className":18168},[106],[18,18170,16413],{"className":18171,"style":111},[106,110],[18,18173,74],{"className":18174},[250],[18,18176,5031],{"className":18177,"style":5073},[106,110],[18,18179,18005],{"className":18180},[165],[18,18182],{"className":18183,"style":170},[169],[18,18185,43],{"className":18186,"style":111},[106,110],[18,18188,84],{"className":18189},[267],[18,18191],{"className":18192,"style":2068},[169],[18,18194,7987],{"className":18195},[2072],[18,18197],{"className":18198,"style":2068},[169],[18,18200,15174,18202],{"className":18201},[985],[18,18203,15177],{"style":6355},[18,18205],{"className":18206,"style":170},[169],[18,18208,2266],{"className":18209},[106],[18,18211,152],{"className":18212},[151],[18,18214,18216],{"className":18215},[124],[18,18217,18220],{"className":18218,"style":18219},[128],"height:0.8804em;",[18,18221],{},[18,18223],{"className":18224},[267,6012],[18,18226,52],{"className":18227},[165],[11,18229,18230,18231,18288],{},"其中 ",[18,18232,18234,18258],{"className":18233,"translate":22},[21],[18,18235,18237],{"className":18236},[26],[28,18238,18239],{"xmlns":30},[32,18240,18241,18255],{},[35,18242,18243,18245,18247,18249,18251,18253],{},[41,18244,16413],{},[49,18246,74],{"stretchy":73},[41,18248,5031],{},[49,18250,18005],{"separator":51},[41,18252,43],{},[49,18254,84],{"stretchy":73},[86,18256,18257],{"encoding":88},"I(K;X)",[18,18259,18261],{"className":18260,"ariaHidden":51},[93],[18,18262,18264,18267,18270,18273,18276,18279,18282,18285],{"className":18263},[97],[18,18265],{"className":18266,"style":242},[101],[18,18268,16413],{"className":18269,"style":111},[106,110],[18,18271,74],{"className":18272},[250],[18,18274,5031],{"className":18275,"style":5073},[106,110],[18,18277,18005],{"className":18278},[165],[18,18280],{"className":18281,"style":170},[169],[18,18283,43],{"className":18284,"style":111},[106,110],[18,18286,84],{"className":18287},[267]," 是互信息。",[11,18290,18291],{},"证明分两步。第一步是条件熵的界：",[18,18293,18295],{"className":18294,"translate":22},[875],[18,18296,18298,18388],{"className":18297,"translate":22},[21],[18,18299,18301],{"className":18300},[26],[28,18302,18303],{"xmlns":30,"display":884},[32,18304,18305,18385],{},[35,18306,18307,18310,18312,18314,18316,18322,18324,18326,18329,18331,18337,18339,18341,18347,18349,18351,18353,18355,18357,18359,18361,18363,18365,18367,18369,18371,18377,18379,18381,18383],{},[41,18308,18309],{},"H",[49,18311,74],{"stretchy":73},[41,18313,5031],{},[49,18315,11225],{},[658,18317,18318,18320],{"accent":51},[41,18319,5031],{},[49,18321,664],{},[49,18323,84],{"stretchy":73},[49,18325,4334],{},[41,18327,18328],{},"h",[49,18330,74],{"stretchy":73},[38,18332,18333,18335],{},[41,18334,939],{},[41,18336,14444],{},[49,18338,84],{"stretchy":73},[49,18340,7987],{},[38,18342,18343,18345],{},[41,18344,939],{},[41,18346,14444],{},[41,18348,14733],{},[49,18350,899],{},[49,18352,74],{"stretchy":73},[41,18354,3526],{},[49,18356,2253],{},[45,18358,47],{},[49,18360,84],{"stretchy":73},[49,18362,4334],{},[41,18364,14733],{},[49,18366,899],{},[45,18368,2266],{},[49,18370,7987],{},[38,18372,18373,18375],{},[41,18374,939],{},[41,18376,14444],{},[41,18378,14733],{},[49,18380,899],{},[41,18382,3526],{},[49,18384,52],{"separator":51},[86,18386,18387],{"encoding":88},"H(K\\mid \\hat K)\\le h(P_e)+P_e\\log(M-1)\\le \\log 2+P_e\\log M,",[18,18389,18391,18416,18465,18529,18598,18619,18645],{"className":18390,"ariaHidden":51},[93],[18,18392,18394,18397,18401,18404,18407,18410,18413],{"className":18393},[97],[18,18395],{"className":18396,"style":242},[101],[18,18398,18309],{"className":18399,"style":18400},[106,110],"margin-right:0.0813em;",[18,18402,74],{"className":18403},[250],[18,18405,5031],{"className":18406,"style":5073},[106,110],[18,18408],{"className":18409,"style":228},[169],[18,18411,11225],{"className":18412},[232],[18,18414],{"className":18415,"style":228},[169],[18,18417,18419,18422,18453,18456,18459,18462],{"className":18418},[97],[18,18420],{"className":18421,"style":17728},[101],[18,18423,18425],{"className":18424},[106,681],[18,18426,18428],{"className":18427},[119],[18,18429,18431],{"className":18430},[124],[18,18432,18434,18442],{"className":18433,"style":17679},[128],[18,18435,18436,18439],{"style":693},[18,18437],{"className":18438,"style":697},[136],[18,18440,5031],{"className":18441,"style":5073},[106,110],[18,18443,18444,18447],{"style":17702},[18,18445],{"className":18446,"style":697},[136],[18,18448,18450],{"className":18449,"style":17709},[710],[18,18451,664],{"className":18452},[106],[18,18454,84],{"className":18455},[267],[18,18457],{"className":18458,"style":228},[169],[18,18460,4334],{"className":18461},[232],[18,18463],{"className":18464,"style":228},[169],[18,18466,18468,18471,18474,18477,18517,18520,18523,18526],{"className":18467},[97],[18,18469],{"className":18470,"style":242},[101],[18,18472,18328],{"className":18473},[106,110],[18,18475,74],{"className":18476},[250],[18,18478,18480,18483],{"className":18479},[106],[18,18481,939],{"className":18482,"style":1171},[106,110],[18,18484,18486],{"className":18485},[115],[18,18487,18489,18509],{"className":18488},[119,120],[18,18490,18492,18506],{"className":18491},[124],[18,18493,18495],{"className":18494,"style":202},[128],[18,18496,18497,18500],{"style":1386},[18,18498],{"className":18499,"style":137},[136],[18,18501,18503],{"className":18502},[141,142,143,144],[18,18504,14444],{"className":18505},[106,110,144],[18,18507,152],{"className":18508},[151],[18,18510,18512],{"className":18511},[124],[18,18513,18515],{"className":18514,"style":159},[128],[18,18516],{},[18,18518,84],{"className":18519},[267],[18,18521],{"className":18522,"style":2068},[169],[18,18524,7987],{"className":18525},[2072],[18,18527],{"className":18528,"style":2068},[169],[18,18530,18532,18535,18575,18578,18583,18586,18589,18592,18595],{"className":18531},[97],[18,18533],{"className":18534,"style":242},[101],[18,18536,18538,18541],{"className":18537},[106],[18,18539,939],{"className":18540,"style":1171},[106,110],[18,18542,18544],{"className":18543},[115],[18,18545,18547,18567],{"className":18546},[119,120],[18,18548,18550,18564],{"className":18549},[124],[18,18551,18553],{"className":18552,"style":202},[128],[18,18554,18555,18558],{"style":1386},[18,18556],{"className":18557,"style":137},[136],[18,18559,18561],{"className":18560},[141,142,143,144],[18,18562,14444],{"className":18563},[106,110,144],[18,18565,152],{"className":18566},[151],[18,18568,18570],{"className":18569},[124],[18,18571,18573],{"className":18572,"style":159},[128],[18,18574],{},[18,18576],{"className":18577,"style":170},[169],[18,18579,15174,18581],{"className":18580},[985],[18,18582,15177],{"style":6355},[18,18584,74],{"className":18585},[250],[18,18587,3526],{"className":18588,"style":246},[106,110],[18,18590],{"className":18591,"style":2068},[169],[18,18593,2253],{"className":18594},[2072],[18,18596],{"className":18597,"style":2068},[169],[18,18599,18601,18604,18607,18610,18613,18616],{"className":18600},[97],[18,18602],{"className":18603,"style":242},[101],[18,18605,47],{"className":18606},[106],[18,18608,84],{"className":18609},[267],[18,18611],{"className":18612,"style":228},[169],[18,18614,4334],{"className":18615},[232],[18,18617],{"className":18618,"style":228},[169],[18,18620,18622,18625,18630,18633,18636,18639,18642],{"className":18621},[97],[18,18623],{"className":18624,"style":2890},[101],[18,18626,15174,18628],{"className":18627},[985],[18,18629,15177],{"style":6355},[18,18631],{"className":18632,"style":170},[169],[18,18634,2266],{"className":18635},[106],[18,18637],{"className":18638,"style":2068},[169],[18,18640,7987],{"className":18641},[2072],[18,18643],{"className":18644,"style":2068},[169],[18,18646,18648,18651,18691,18694,18699,18702,18705],{"className":18647},[97],[18,18649],{"className":18650,"style":2890},[101],[18,18652,18654,18657],{"className":18653},[106],[18,18655,939],{"className":18656,"style":1171},[106,110],[18,18658,18660],{"className":18659},[115],[18,18661,18663,18683],{"className":18662},[119,120],[18,18664,18666,18680],{"className":18665},[124],[18,18667,18669],{"className":18668,"style":202},[128],[18,18670,18671,18674],{"style":1386},[18,18672],{"className":18673,"style":137},[136],[18,18675,18677],{"className":18676},[141,142,143,144],[18,18678,14444],{"className":18679},[106,110,144],[18,18681,152],{"className":18682},[151],[18,18684,18686],{"className":18685},[124],[18,18687,18689],{"className":18688,"style":159},[128],[18,18690],{},[18,18692],{"className":18693,"style":170},[169],[18,18695,15174,18697],{"className":18696},[985],[18,18698,15177],{"style":6355},[18,18700],{"className":18701,"style":170},[169],[18,18703,3526],{"className":18704,"style":246},[106,110],[18,18706,52],{"className":18707},[165],[11,18709,18230,18710,18829],{},[18,18711,18713,18745],{"className":18712,"translate":22},[21],[18,18714,18716],{"className":18715},[26],[28,18717,18718],{"xmlns":30},[32,18719,18720,18742],{},[35,18721,18722,18724,18726,18732,18734,18736,18738,18740],{},[41,18723,18328],{},[49,18725,74],{"stretchy":73},[38,18727,18728,18730],{},[41,18729,939],{},[41,18731,14444],{},[49,18733,84],{"stretchy":73},[49,18735,4334],{},[41,18737,14733],{},[49,18739,899],{},[45,18741,2266],{},[86,18743,18744],{"encoding":88},"h(P_e)\\le\\log 2",[18,18746,18748,18812],{"className":18747,"ariaHidden":51},[93],[18,18749,18751,18754,18757,18760,18800,18803,18806,18809],{"className":18750},[97],[18,18752],{"className":18753,"style":242},[101],[18,18755,18328],{"className":18756},[106,110],[18,18758,74],{"className":18759},[250],[18,18761,18763,18766],{"className":18762},[106],[18,18764,939],{"className":18765,"style":1171},[106,110],[18,18767,18769],{"className":18768},[115],[18,18770,18772,18792],{"className":18771},[119,120],[18,18773,18775,18789],{"className":18774},[124],[18,18776,18778],{"className":18777,"style":202},[128],[18,18779,18780,18783],{"style":1386},[18,18781],{"className":18782,"style":137},[136],[18,18784,18786],{"className":18785},[141,142,143,144],[18,18787,14444],{"className":18788},[106,110,144],[18,18790,152],{"className":18791},[151],[18,18793,18795],{"className":18794},[124],[18,18796,18798],{"className":18797,"style":159},[128],[18,18799],{},[18,18801,84],{"className":18802},[267],[18,18804],{"className":18805,"style":228},[169],[18,18807,4334],{"className":18808},[232],[18,18810],{"className":18811,"style":228},[169],[18,18813,18815,18818,18823,18826],{"className":18814},[97],[18,18816],{"className":18817,"style":2890},[101],[18,18819,15174,18821],{"className":18820},[985],[18,18822,15177],{"style":6355},[18,18824],{"className":18825,"style":170},[169],[18,18827,2266],{"className":18828},[106]," 是二元熵的最大值。第二步是数据处理不等式：",[18,18831,18833],{"className":18832,"translate":22},[875],[18,18834,18836,18926],{"className":18835,"translate":22},[21],[18,18837,18839],{"className":18838},[26],[28,18840,18841],{"xmlns":30,"display":884},[32,18842,18843,18923],{},[35,18844,18845,18847,18849,18851,18853,18859,18861,18863,18865,18867,18869,18871,18873,18875,18877,18879,18881,18883,18885,18887,18889,18891,18893,18895,18897,18899,18901,18903,18905,18907,18909,18911,18913,18915,18917,18919,18921],{},[41,18846,18309],{},[49,18848,74],{"stretchy":73},[41,18850,5031],{},[49,18852,11225],{},[658,18854,18855,18857],{"accent":51},[41,18856,5031],{},[49,18858,664],{},[49,18860,84],{"stretchy":73},[49,18862,4775],{},[41,18864,18309],{},[49,18866,74],{"stretchy":73},[41,18868,5031],{},[49,18870,11225],{},[41,18872,43],{},[49,18874,84],{"stretchy":73},[49,18876,433],{},[41,18878,18309],{},[49,18880,74],{"stretchy":73},[41,18882,5031],{},[49,18884,84],{"stretchy":73},[49,18886,2253],{},[41,18888,16413],{},[49,18890,74],{"stretchy":73},[41,18892,5031],{},[49,18894,18005],{"separator":51},[41,18896,43],{},[49,18898,84],{"stretchy":73},[49,18900,433],{},[41,18902,14733],{},[49,18904,899],{},[41,18906,3526],{},[49,18908,2253],{},[41,18910,16413],{},[49,18912,74],{"stretchy":73},[41,18914,5031],{},[49,18916,18005],{"separator":51},[41,18918,43],{},[49,18920,84],{"stretchy":73},[41,18922,968],{"mathvariant":316},[86,18924,18925],{"encoding":88},"H(K\\mid \\hat K)\\ge H(K\\mid X)=H(K)-I(K;X)=\\log M-I(K;X).",[18,18927,18929,18953,19002,19026,19047,19074,19110,19136],{"className":18928,"ariaHidden":51},[93],[18,18930,18932,18935,18938,18941,18944,18947,18950],{"className":18931},[97],[18,18933],{"className":18934,"style":242},[101],[18,18936,18309],{"className":18937,"style":18400},[106,110],[18,18939,74],{"className":18940},[250],[18,18942,5031],{"className":18943,"style":5073},[106,110],[18,18945],{"className":18946,"style":228},[169],[18,18948,11225],{"className":18949},[232],[18,18951],{"className":18952,"style":228},[169],[18,18954,18956,18959,18990,18993,18996,18999],{"className":18955},[97],[18,18957],{"className":18958,"style":17728},[101],[18,18960,18962],{"className":18961},[106,681],[18,18963,18965],{"className":18964},[119],[18,18966,18968],{"className":18967},[124],[18,18969,18971,18979],{"className":18970,"style":17679},[128],[18,18972,18973,18976],{"style":693},[18,18974],{"className":18975,"style":697},[136],[18,18977,5031],{"className":18978,"style":5073},[106,110],[18,18980,18981,18984],{"style":17702},[18,18982],{"className":18983,"style":697},[136],[18,18985,18987],{"className":18986,"style":17709},[710],[18,18988,664],{"className":18989},[106],[18,18991,84],{"className":18992},[267],[18,18994],{"className":18995,"style":228},[169],[18,18997,4775],{"className":18998},[232],[18,19000],{"className":19001,"style":228},[169],[18,19003,19005,19008,19011,19014,19017,19020,19023],{"className":19004},[97],[18,19006],{"className":19007,"style":242},[101],[18,19009,18309],{"className":19010,"style":18400},[106,110],[18,19012,74],{"className":19013},[250],[18,19015,5031],{"className":19016,"style":5073},[106,110],[18,19018],{"className":19019,"style":228},[169],[18,19021,11225],{"className":19022},[232],[18,19024],{"className":19025,"style":228},[169],[18,19027,19029,19032,19035,19038,19041,19044],{"className":19028},[97],[18,19030],{"className":19031,"style":242},[101],[18,19033,43],{"className":19034,"style":111},[106,110],[18,19036,84],{"className":19037},[267],[18,19039],{"className":19040,"style":228},[169],[18,19042,433],{"className":19043},[232],[18,19045],{"className":19046,"style":228},[169],[18,19048,19050,19053,19056,19059,19062,19065,19068,19071],{"className":19049},[97],[18,19051],{"className":19052,"style":242},[101],[18,19054,18309],{"className":19055,"style":18400},[106,110],[18,19057,74],{"className":19058},[250],[18,19060,5031],{"className":19061,"style":5073},[106,110],[18,19063,84],{"className":19064},[267],[18,19066],{"className":19067,"style":2068},[169],[18,19069,2253],{"className":19070},[2072],[18,19072],{"className":19073,"style":2068},[169],[18,19075,19077,19080,19083,19086,19089,19092,19095,19098,19101,19104,19107],{"className":19076},[97],[18,19078],{"className":19079,"style":242},[101],[18,19081,16413],{"className":19082,"style":111},[106,110],[18,19084,74],{"className":19085},[250],[18,19087,5031],{"className":19088,"style":5073},[106,110],[18,19090,18005],{"className":19091},[165],[18,19093],{"className":19094,"style":170},[169],[18,19096,43],{"className":19097,"style":111},[106,110],[18,19099,84],{"className":19100},[267],[18,19102],{"className":19103,"style":228},[169],[18,19105,433],{"className":19106},[232],[18,19108],{"className":19109,"style":228},[169],[18,19111,19113,19116,19121,19124,19127,19130,19133],{"className":19112},[97],[18,19114],{"className":19115,"style":2890},[101],[18,19117,15174,19119],{"className":19118},[985],[18,19120,15177],{"style":6355},[18,19122],{"className":19123,"style":170},[169],[18,19125,3526],{"className":19126,"style":246},[106,110],[18,19128],{"className":19129,"style":2068},[169],[18,19131,2253],{"className":19132},[2072],[18,19134],{"className":19135,"style":2068},[169],[18,19137,19139,19142,19145,19148,19151,19154,19157,19160,19163],{"className":19138},[97],[18,19140],{"className":19141,"style":242},[101],[18,19143,16413],{"className":19144,"style":111},[106,110],[18,19146,74],{"className":19147},[250],[18,19149,5031],{"className":19150,"style":5073},[106,110],[18,19152,18005],{"className":19153},[165],[18,19155],{"className":19156,"style":170},[169],[18,19158,43],{"className":19159,"style":111},[106,110],[18,19161,84],{"className":19162},[267],[18,19164,968],{"className":19165},[106],[11,19167,19168],{},"两式合并，整理即得。",[11,19170,19171,19172,19430,19431,19472,19473,19513,19514,19542],{},"这个不等式有一个很直观的读法：",[717,19173,19174,19175,19203,19204,19232,19233,19284,19285,19371,19372,19400,19401,19429],{},"如果 ",[18,19176,19178,19191],{"className":19177,"translate":22},[21],[18,19179,19181],{"className":19180},[26],[28,19182,19183],{"xmlns":30},[32,19184,19185,19189],{},[35,19186,19187],{},[41,19188,43],{},[86,19190,43],{"encoding":88},[18,19192,19194],{"className":19193,"ariaHidden":51},[93],[18,19195,19197,19200],{"className":19196},[97],[18,19198],{"className":19199,"style":600},[101],[18,19201,43],{"className":19202,"style":111},[106,110]," 几乎不包含关于 ",[18,19205,19207,19220],{"className":19206,"translate":22},[21],[18,19208,19210],{"className":19209},[26],[28,19211,19212],{"xmlns":30},[32,19213,19214,19218],{},[35,19215,19216],{},[41,19217,5031],{},[86,19219,5031],{"encoding":88},[18,19221,19223],{"className":19222,"ariaHidden":51},[93],[18,19224,19226,19229],{"className":19225},[97],[18,19227],{"className":19228,"style":600},[101],[18,19230,5031],{"className":19231,"style":5073},[106,110]," 的信息（",[18,19234,19236,19254],{"className":19235,"translate":22},[21],[18,19237,19239],{"className":19238},[26],[28,19240,19241],{"xmlns":30},[32,19242,19243,19251],{},[35,19244,19245,19247,19249],{},[41,19246,16413],{},[49,19248,14754],{},[45,19250,430],{},[86,19252,19253],{"encoding":88},"I\\approx0",[18,19255,19257,19275],{"className":19256,"ariaHidden":51},[93],[18,19258,19260,19263,19266,19269,19272],{"className":19259},[97],[18,19261],{"className":19262,"style":600},[101],[18,19264,16413],{"className":19265,"style":111},[106,110],[18,19267],{"className":19268,"style":228},[169],[18,19270,14754],{"className":19271},[232],[18,19273],{"className":19274,"style":228},[169],[18,19276,19278,19281],{"className":19277},[97],[18,19279],{"className":19280,"style":505},[101],[18,19282,430],{"className":19283},[106],"），那么任何猜测的错误率至少是 ",[18,19286,19288,19318],{"className":19287,"translate":22},[21],[18,19289,19291],{"className":19290},[26],[28,19292,19293],{"xmlns":30},[32,19294,19295,19315],{},[35,19296,19297,19299,19301,19303,19305,19307,19309,19311,19313],{},[45,19298,47],{},[49,19300,2253],{},[41,19302,14733],{},[49,19304,899],{},[45,19306,2266],{},[41,19308,317],{"mathvariant":316},[41,19310,14733],{},[49,19312,899],{},[41,19314,3526],{},[86,19316,19317],{"encoding":88},"1-\\log 2\u002F\\log M",[18,19319,19321,19339],{"className":19320,"ariaHidden":51},[93],[18,19322,19324,19327,19330,19333,19336],{"className":19323},[97],[18,19325],{"className":19326,"style":18095},[101],[18,19328,47],{"className":19329},[106],[18,19331],{"className":19332,"style":2068},[169],[18,19334,2253],{"className":19335},[2072],[18,19337],{"className":19338,"style":2068},[169],[18,19340,19342,19345,19350,19353,19357,19360,19365,19368],{"className":19341},[97],[18,19343],{"className":19344,"style":242},[101],[18,19346,15174,19348],{"className":19347},[985],[18,19349,15177],{"style":6355},[18,19351],{"className":19352,"style":170},[169],[18,19354,19356],{"className":19355},[106],"2\u002F",[18,19358],{"className":19359,"style":170},[169],[18,19361,15174,19363],{"className":19362},[985],[18,19364,15177],{"style":6355},[18,19366],{"className":19367,"style":170},[169],[18,19369,3526],{"className":19370,"style":246},[106,110],"，当 ",[18,19373,19375,19388],{"className":19374,"translate":22},[21],[18,19376,19378],{"className":19377},[26],[28,19379,19380],{"xmlns":30},[32,19381,19382,19386],{},[35,19383,19384],{},[41,19385,3526],{},[86,19387,3526],{"encoding":88},[18,19389,19391],{"className":19390,"ariaHidden":51},[93],[18,19392,19394,19397],{"className":19393},[97],[18,19395],{"className":19396,"style":600},[101],[18,19398,3526],{"className":19399,"style":246},[106,110]," 很大时接近 ",[18,19402,19404,19417],{"className":19403,"translate":22},[21],[18,19405,19407],{"className":19406},[26],[28,19408,19409],{"xmlns":30},[32,19410,19411,19415],{},[35,19412,19413],{},[45,19414,47],{},[86,19416,47],{"encoding":88},[18,19418,19420],{"className":19419,"ariaHidden":51},[93],[18,19421,19423,19426],{"className":19422},[97],[18,19424],{"className":19425,"style":505},[101],[18,19427,47],{"className":19428},[106],"，也就是接近「瞎猜」。"," 反过来，要让错误率小，互信息必须达到 ",[18,19432,19434,19452],{"className":19433,"translate":22},[21],[18,19435,19437],{"className":19436},[26],[28,19438,19439],{"xmlns":30},[32,19440,19441,19449],{},[35,19442,19443,19445,19447],{},[41,19444,14733],{},[49,19446,899],{},[41,19448,3526],{},[86,19450,19451],{"encoding":88},"\\log M",[18,19453,19455],{"className":19454,"ariaHidden":51},[93],[18,19456,19458,19461,19466,19469],{"className":19457},[97],[18,19459],{"className":19460,"style":2890},[101],[18,19462,15174,19464],{"className":19463},[985],[18,19465,15177],{"style":6355},[18,19467],{"className":19468,"style":170},[169],[18,19470,3526],{"className":19471,"style":246},[106,110]," 的量级——你需要至少 ",[18,19474,19476,19493],{"className":19475,"translate":22},[21],[18,19477,19479],{"className":19478},[26],[28,19480,19481],{"xmlns":30},[32,19482,19483,19491],{},[35,19484,19485,19487,19489],{},[41,19486,14733],{},[49,19488,899],{},[41,19490,3526],{},[86,19492,19451],{"encoding":88},[18,19494,19496],{"className":19495,"ariaHidden":51},[93],[18,19497,19499,19502,19507,19510],{"className":19498},[97],[18,19500],{"className":19501,"style":2890},[101],[18,19503,15174,19505],{"className":19504},[985],[18,19506,15177],{"style":6355},[18,19508],{"className":19509,"style":170},[169],[18,19511,3526],{"className":19512,"style":246},[106,110]," 奈特的信息才能区分 ",[18,19515,19517,19530],{"className":19516,"translate":22},[21],[18,19518,19520],{"className":19519},[26],[28,19521,19522],{"xmlns":30},[32,19523,19524,19528],{},[35,19525,19526],{},[41,19527,3526],{},[86,19529,3526],{"encoding":88},[18,19531,19533],{"className":19532,"ariaHidden":51},[93],[18,19534,19536,19539],{"className":19535},[97],[18,19537],{"className":19538,"style":600},[101],[18,19540,3526],{"className":19541,"style":246},[106,110]," 个等可能的选择。",[1311,19544,19546],{"id":19545},"_42-从检验到估计","4.2 从检验到估计",[11,19548,19549],{},"把 Fano 用在极小极大下界上，需要一个把「损失」翻译成「猜错」的分离条件。",[5159,19551,19552,19778,20031,20343,20955,21039],{},[11,19553,19554,5166,19557,19585,19586,19614,19615,19777],{},[717,19555,19556],{},"定理 4.2（Fano 方法，度量形式）",[18,19558,19560,19573],{"className":19559,"translate":22},[21],[18,19561,19563],{"className":19562},[26],[28,19564,19565],{"xmlns":30},[32,19566,19567,19571],{},[35,19568,19569],{},[41,19570,4031],{},[86,19572,4031],{"encoding":88},[18,19574,19576],{"className":19575,"ariaHidden":51},[93],[18,19577,19579,19582],{"className":19578},[97],[18,19580],{"className":19581,"style":295},[101],[18,19583,4031],{"className":19584},[106,110]," 是 ",[18,19587,19589,19602],{"className":19588,"translate":22},[21],[18,19590,19592],{"className":19591},[26],[28,19593,19594],{"xmlns":30},[32,19595,19596,19600],{},[35,19597,19598],{},[41,19599,924],{"mathvariant":316},[86,19601,1497],{"encoding":88},[18,19603,19605],{"className":19604,"ariaHidden":51},[93],[18,19606,19608,19611],{"className":19607},[97],[18,19609],{"className":19610,"style":600},[101],[18,19612,924],{"className":19613},[106]," 上的度量，",[18,19616,19618,19652],{"className":19617,"translate":22},[21],[18,19619,19621],{"className":19620},[26],[28,19622,19623],{"xmlns":30},[32,19624,19625,19649],{},[35,19626,19627,19633,19635,19637,19639,19645,19647],{},[38,19628,19629,19631],{},[41,19630,77],{},[45,19632,47],{},[49,19634,52],{"separator":51},[49,19636,55],{},[49,19638,52],{"separator":51},[38,19640,19641,19643],{},[41,19642,77],{},[41,19644,3526],{},[49,19646,921],{},[41,19648,924],{"mathvariant":316},[86,19650,19651],{"encoding":88},"\\theta_1,\\dots,\\theta_M\\in\\Theta",[18,19653,19655,19768],{"className":19654,"ariaHidden":51},[93],[18,19656,19658,19661,19701,19704,19707,19710,19713,19716,19719,19759,19762,19765],{"className":19657},[97],[18,19659],{"className":19660,"style":2890},[101],[18,19662,19664,19667],{"className":19663},[106],[18,19665,77],{"className":19666,"style":254},[106,110],[18,19668,19670],{"className":19669},[115],[18,19671,19673,19693],{"className":19672},[119,120],[18,19674,19676,19690],{"className":19675},[124],[18,19677,19679],{"className":19678,"style":129},[128],[18,19680,19681,19684],{"style":469},[18,19682],{"className":19683,"style":137},[136],[18,19685,19687],{"className":19686},[141,142,143,144],[18,19688,47],{"className":19689},[106,144],[18,19691,152],{"className":19692},[151],[18,19694,19696],{"className":19695},[124],[18,19697,19699],{"className":19698,"style":159},[128],[18,19700],{},[18,19702,52],{"className":19703},[165],[18,19705],{"className":19706,"style":170},[169],[18,19708,55],{"className":19709},[174],[18,19711],{"className":19712,"style":170},[169],[18,19714,52],{"className":19715},[165],[18,19717],{"className":19718,"style":170},[169],[18,19720,19722,19725],{"className":19721},[106],[18,19723,77],{"className":19724,"style":254},[106,110],[18,19726,19728],{"className":19727},[115],[18,19729,19731,19751],{"className":19730},[119,120],[18,19732,19734,19748],{"className":19733},[124],[18,19735,19737],{"className":19736,"style":1152},[128],[18,19738,19739,19742],{"style":469},[18,19740],{"className":19741,"style":137},[136],[18,19743,19745],{"className":19744},[141,142,143,144],[18,19746,3526],{"className":19747,"style":246},[106,110,144],[18,19749,152],{"className":19750},[151],[18,19752,19754],{"className":19753},[124],[18,19755,19757],{"className":19756,"style":159},[128],[18,19758],{},[18,19760],{"className":19761,"style":228},[169],[18,19763,921],{"className":19764},[232],[18,19766],{"className":19767,"style":228},[169],[18,19769,19771,19774],{"className":19770},[97],[18,19772],{"className":19773,"style":600},[101],[18,19775,924],{"className":19776},[106]," 满足",[18,19779,19781],{"className":19780,"translate":22},[875],[18,19782,19784,19837],{"className":19783,"translate":22},[21],[18,19785,19787],{"className":19786},[26],[28,19788,19789],{"xmlns":30,"display":884},[32,19790,19791,19834],{},[35,19792,19793,19795,19797,19803,19805,19812,19814,19816,19818,19820,19822,19824,19826,19828,19830,19832],{},[41,19794,4031],{},[49,19796,74],{"stretchy":73},[38,19798,19799,19801],{},[41,19800,77],{},[41,19802,5600],{},[49,19804,52],{"separator":51},[38,19806,19807,19809],{},[41,19808,77],{},[41,19810,19811],{},"j",[49,19813,84],{"stretchy":73},[49,19815,4775],{},[45,19817,2266],{},[41,19819,5351],{},[169,19821],{"width":7344},[49,19823,74],{"stretchy":73},[41,19825,5600],{},[49,19827,2771],{"mathvariant":316},[41,19829,19811],{},[49,19831,84],{"stretchy":73},[41,19833,968],{"mathvariant":316},[86,19835,19836],{"encoding":88},"d(\\theta_i,\\theta_j)\\ge 2\\delta\\qquad(i\\neq 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d(\\theta_K,\\theta_j)-d(\\theta_K,\\hat\\theta)>2\\delta-\\delta=\\delta>d(\\theta_K,\\hat\\theta),",[18,22149,22151,22253,22363,22464,22485,22503,22521],{"className":22150,"ariaHidden":51},[93],[18,22152,22154,22158,22161,22164,22195,22198,22201,22241,22244,22247,22250],{"className":22153},[97],[18,22155],{"className":22156,"style":22157},[101],"height:1.244em;vertical-align:-0.2861em;",[18,22159,4031],{"className":22160},[106,110],[18,22162,74],{"className":22163},[250],[18,22165,22167],{"className":22166},[106,681],[18,22168,22170],{"className":22169},[119],[18,22171,22173],{"className":22172},[124],[18,22174,22176,22184],{"className":22175,"style":677},[128],[18,22177,22178,22181],{"style":693},[18,22179],{"className":22180,"style":697},[136],[18,22182,77],{"className":22183,"style":254},[106,110],[18,22185,22186,22189],{"style":703},[18,22187],{"className":22188,"style":697},[136],[18,22190,22192],{"className":22191,"style":711},[710],[18,22193,664],{"className":22194},[106],[18,22196,52],{"className":22197},[165],[18,22199],{"className":22200,"style":170},[169],[18,22202,22204,22207],{"className":22203},[106],[18,22205,77],{"className":22206,"style":254},[106,110],[18,22208,22210],{"className":22209},[115],[18,22211,22213,22233],{"className":22212},[119,120],[18,22214,22216,22230],{"className":22215},[124],[18,22217,22219],{"className":22218,"style":5921},[128],[18,22220,22221,22224],{"style":469},[18,22222],{"className":22223,"style":137},[136],[18,22225,22227],{"className":22226},[141,142,143,144],[18,22228,19811],{"className":22229,"style":19928},[106,110,144],[18,22231,152],{"className":22232},[151],[18,22234,22236],{"className":22235},[124],[18,22237,22239],{"className":22238,"style":19938},[128],[18,22240],{},[18,22242,84],{"className":22243},[267],[18,22245],{"className":22246,"style":228},[169],[18,22248,4775],{"className":22249},[232],[18,22251],{"className":22252,"style":228},[169],[18,22254,22256,22259,22262,22265,22305,22308,22311,22351,22354,22357,22360],{"className":22255},[97],[18,22257],{"className":22258,"style":19846},[101],[18,22260,4031],{"className":22261},[106,110],[18,22263,74],{"className":22264},[250],[18,22266,22268,22271],{"className":22267},[106],[18,22269,77],{"className":22270,"style":254},[106,110],[18,22272,22274],{"className":22273},[115],[18,22275,22277,22297],{"className":22276},[119,120],[18,22278,22280,22294],{"className":22279},[124],[18,22281,22283],{"className":22282,"style":1152},[128],[18,22284,22285,22288],{"style":469},[18,22286],{"className":22287,"style":137},[136],[18,22289,22291],{"className":22290},[141,142,143,144],[18,22292,5031],{"className":22293,"style":5073},[106,110,144],[18,22295,152],{"className":22296},[151],[18,22298,22300],{"className":22299},[124],[18,22301,22303],{"className":22302,"style":159},[128],[18,22304],{},[18,22306,52],{"className":22307},[165],[18,22309],{"className":22310,"style":170},[169],[18,22312,22314,22317],{"className":22313},[106],[18,22315,77],{"className":22316,"style":254},[106,110],[18,22318,22320],{"className":22319},[115],[18,22321,22323,22343],{"className":22322},[119,120],[18,22324,22326,22340],{"className":22325},[124],[18,22327,22329],{"className":22328,"style":5921},[128],[18,22330,22331,22334],{"style":469},[18,22332],{"className":22333,"style":137},[136],[18,22335,22337],{"className":22336},[141,142,143,144],[18,22338,19811],{"className":22339,"style":19928},[106,110,144],[18,22341,152],{"className":22342},[151],[18,22344,22346],{"className":22345},[124],[18,22347,22349],{"className":22348,"style":19938},[128],[18,22350],{},[18,22352,84],{"className":22353},[267],[18,22355],{"className":22356,"style":2068},[169],[18,22358,2253],{"className":22359},[2072],[18,22361],{"className":22362,"style":2068},[169],[18,22364,22366,22369,22372,22375,22415,22418,22421,22452,22455,22458,22461],{"className":22365},[97],[18,22367],{"className":22368,"style":2281},[101],[18,22370,4031],{"className":22371},[106,110],[18,22373,74],{"className":22374},[250],[18,22376,22378,22381],{"className":22377},[106],[18,22379,77],{"className":22380,"style":254},[106,110],[18,22382,22384],{"className":22383},[115],[18,22385,22387,22407],{"className":22386},[119,120],[18,22388,22390,22404],{"className":22389},[124],[18,22391,22393],{"className":22392,"style":1152},[128],[18,22394,22395,22398],{"style":469},[18,22396],{"className":22397,"style":137},[136],[18,22399,22401],{"className":22400},[141,142,143,144],[18,22402,5031],{"className":22403,"style":5073},[106,110,144],[18,22405,152],{"className":22406},[151],[18,22408,22410],{"className":22409},[124],[18,22411,22413],{"className":22412,"style":159},[128],[18,22414],{},[18,22416,52],{"className":22417},[165],[18,22419],{"className":22420,"style":170},[169],[18,22422,22424],{"className":22423},[106,681],[18,22425,22427],{"className":22426},[119],[18,22428,22430],{"className":22429},[124],[18,22431,22433,22441],{"className":22432,"style":677},[128],[18,22434,22435,22438],{"style":693},[18,22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",[18,22620,22622,22635],{"className":22621,"translate":22},[21],[18,22623,22625],{"className":22624},[26],[28,22626,22627],{"xmlns":30},[32,22628,22629,22633],{},[35,22630,22631],{},[41,22632,5031],{},[86,22634,5031],{"encoding":88},[18,22636,22638],{"className":22637,"ariaHidden":51},[93],[18,22639,22641,22644],{"className":22640},[97],[18,22642],{"className":22643,"style":600},[101],[18,22645,5031],{"className":22646,"style":5073},[106,110]," 是唯一的最小值点，",[18,22649,22651,22673],{"className":22650,"translate":22},[21],[18,22652,22654],{"className":22653},[26],[28,22655,22656],{"xmlns":30},[32,22657,22658,22670],{},[35,22659,22660,22666,22668],{},[658,22661,22662,22664],{"accent":51},[41,22663,5031],{},[49,22665,664],{},[49,22667,433],{},[41,22669,5031],{},[86,22671,22672],{"encoding":88},"\\hat K=K",[18,22674,22676,22722],{"className":22675,"ariaHidden":51},[93],[18,22677,22679,22682,22713,22716,22719],{"className":22678},[97],[18,22680],{"className":22681,"style":17679},[101],[18,22683,22685],{"className":22684},[106,681],[18,22686,22688],{"className":22687},[119],[18,22689,22691],{"className":22690},[124],[18,22692,22694,22702],{"className":22693,"style":17679},[128],[18,22695,22696,22699],{"style":693},[18,22697],{"className":22698,"style":697},[136],[18,22700,5031],{"className":22701,"style":5073},[106,110],[18,22703,22704,22707],{"style":17702},[18,22705],{"className":22706,"style":697},[136],[18,22708,22710],{"className":22709,"style":17709},[710],[18,22711,664],{"className":22712},[106],[18,22714],{"className":22715,"style":228},[169],[18,22717,433],{"className":22718},[232],[18,22720],{"className":22721,"style":228},[169],[18,22723,22725,22728],{"className":22724},[97],[18,22726],{"className":22727,"style":600},[101],[18,22729,5031],{"className":22730,"style":5073},[106,110],"。因此",[18,22733,22735],{"className":22734,"translate":22},[875],[18,22736,22738,22796],{"className":22737,"translate":22},[21],[18,22739,22741],{"className":22740},[26],[28,22742,22743],{"xmlns":30,"display":884},[32,22744,22745,22793],{},[35,22746,22747,22749,22751,22753,22759,22761,22767,22769,22771,22773,22775,22777,22779,22785,22787,22789,22791],{},[49,22748,1331],{"stretchy":73},[41,22750,4031],{},[49,22752,74],{"stretchy":73},[38,22754,22755,22757],{},[41,22756,77],{},[41,22758,5031],{},[49,22760,52],{"separator":51},[658,22762,22763,22765],{"accent":51},[41,22764,77],{},[49,22766,664],{},[49,22768,84],{"stretchy":73},[49,22770,6684],{},[41,22772,5351],{},[49,22774,1349],{"stretchy":73},[49,22776,4548],{},[49,22778,1331],{"stretchy":73},[658,22780,22781,22783],{"accent":51},[41,22782,5031],{},[49,22784,664],{},[49,22786,433],{},[41,22788,5031],{},[49,22790,1349],{"stretchy":73},[49,22792,52],{"separator":51},[86,22794,22795],{"encoding":88},"\\{d(\\theta_K,\\hat\\theta)\u003C\\delta\\}\\subseteq\\{\\hat K=K\\},",[18,22797,22799,22903,22924,22973],{"className":22798,"ariaHidden":51},[93],[18,22800,22802,22805,22808,22811,22814,22854,22857,22860,22891,22894,22897,22900],{"className":22801},[97],[18,22803],{"className":22804,"style":2281},[101],[18,22806,1331],{"className":22807},[250],[18,22809,4031],{"className":22810},[106,110],[18,22812,74],{"className":22813},[250],[18,22815,22817,22820],{"className":22816},[106],[18,22818,77],{"className":22819,"style":254},[106,110],[18,22821,22823],{"className":22822},[115],[18,22824,22826,22846],{"className":22825},[119,120],[18,22827,22829,22843],{"className":22828},[124],[18,22830,22832],{"className":22831,"style":1152},[128],[18,22833,22834,22837],{"style":469},[18,22835],{"className":22836,"style":137},[136],[18,22838,22840],{"className":22839},[141,142,143,144],[18,22841,5031],{"className":22842,"style":5073},[106,110,144],[18,22844,152],{"className":22845},[151],[18,22847,22849],{"className":22848},[124],[18,22850,22852],{"className":22851,"style":159},[128],[18,22853],{},[18,22855,52],{"className":22856},[165],[18,22858],{"className":22859,"style":170},[169],[18,22861,22863],{"className":22862},[106,681],[18,22864,22866],{"className":22865},[119],[18,22867,22869],{"className":22868},[124],[18,22870,22872,22880],{"className":22871,"style":677},[128],[18,22873,22874,22877],{"style":693},[18,22875],{"className":22876,"style":697},[136],[18,22878,77],{"className":22879,"style":254},[106,110],[18,22881,22882,22885],{"style":703},[18,22883],{"className":22884,"style":697},[136],[18,22886,22888],{"className":22887,"style":711},[710],[18,22889,664],{"className":22890},[106],[18,22892,84],{"className":22893},[267],[18,22895],{"className":22896,"style":228},[169],[18,22898,6684],{"className":22899},[232],[18,22901],{"className":22902,"style":228},[169],[18,22904,22906,22909,22912,22915,22918,22921],{"className":22905},[97],[18,22907],{"className":22908,"style":242},[101],[18,22910,5351],{"className":22911,"style":5480},[106,110],[18,22913,1349],{"className":22914},[267],[18,22916],{"className":22917,"style":228},[169],[18,22919,4548],{"className":22920},[232],[18,22922],{"className":22923,"style":228},[169],[18,22925,22927,22930,22933,22964,22967,22970],{"className":22926},[97],[18,22928],{"className":22929,"style":17728},[101],[18,22931,1331],{"className":22932},[250],[18,22934,22936],{"className":22935},[106,681],[18,22937,22939],{"className":22938},[119],[18,22940,22942],{"className":22941},[124],[18,22943,22945,22953],{"className":22944,"style":17679},[128],[18,22946,22947,22950],{"style":693},[18,22948],{"className":22949,"style":697},[136],[18,22951,5031],{"className":22952,"style":5073},[106,110],[18,22954,22955,22958],{"style":17702},[18,22956],{"className":22957,"style":697},[136],[18,22959,22961],{"className":22960,"style":17709},[710],[18,22962,664],{"className":22963},[106],[18,22965],{"className":22966,"style":228},[169],[18,22968,433],{"className":22969},[232],[18,22971],{"className":22972,"style":228},[169],[18,22974,22976,22979,22982,22985],{"className":22975},[97],[18,22977],{"className":22978,"style":242},[101],[18,22980,5031],{"className":22981,"style":5073},[106,110],[18,22983,1349],{"className":22984},[267],[18,22986,52],{"className":22987},[165],[11,22989,22990,22991,23223],{},"从而 ",[18,22992,22994,23046],{"className":22993,"translate":22},[21],[18,22995,22997],{"className":22996},[26],[28,22998,22999],{"xmlns":30},[32,23000,23001,23043],{},[35,23002,23003,23009,23011,23013,23015,23017,23019,23021,23027,23029,23035,23037,23039,23041],{},[38,23004,23005,23007],{},[41,23006,939],{},[41,23008,14444],{},[49,23010,4334],{},[41,23012,11208],{},[49,23014,899],{},[49,23016,74],{"stretchy":73},[41,23018,4031],{},[49,23020,74],{"stretchy":73},[38,23022,23023,23025],{},[41,23024,77],{},[41,23026,5031],{},[49,23028,52],{"separator":51},[658,23030,23031,23033],{"accent":51},[41,23032,77],{},[49,23034,664],{},[49,23036,84],{"stretchy":73},[49,23038,4775],{},[41,23040,5351],{},[49,23042,84],{"stretchy":73},[86,23044,23045],{"encoding":88},"P_e\\le 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E\\bigl[d(\\theta_K,\\hat\\theta)\\bigr]\n\\ge \\delta\\,\\Pr\\bigl(d\\ge\\delta\\bigr)\n\\ge \\delta\\,P_e\n\\ge \\delta\\left(1-\\frac{I+\\log 2}{\\log M}\\right).",[18,23339,23341,23458,23494,23518,23579],{"className":23340,"ariaHidden":51},[93],[18,23342,23344,23348,23351,23357,23360,23363,23403,23406,23409,23440,23443,23449,23452,23455],{"className":23343},[97],[18,23345],{"className":23346,"style":23347},[101],"height:1.3079em;vertical-align:-0.35em;",[18,23349,934],{"className":23350},[106,1139],[18,23352,23354],{"className":23353},[250],[18,23355,945],{"className":23356},[1230,1196],[18,23358,4031],{"className":23359},[106,110],[18,23361,74],{"className":23362},[250],[18,23364,23366,23369],{"className":23365},[106],[18,23367,77],{"className":23368,"style":254},[106,110],[18,23370,23372],{"className":23371},[115],[18,23373,23375,23395],{"className":23374},[119,120],[18,23376,23378,23392],{"className":23377},[124],[18,23379,23381],{"className":23380,"style":1152},[128],[18,23382,23383,23386],{"style":469},[18,23384],{"className":23385,"style":137},[136],[18,23387,23389],{"className":23388},[141,142,143,144],[18,23390,5031],{"className":23391,"style":5073},[106,110,144],[18,23393,152],{"className":23394},[151],[18,23396,23398],{"className":23397},[124],[18,23399,23401],{"className":23400,"style":159},[128],[18,23402],{},[18,23404,52],{"className":23405},[165],[18,23407],{"className":23408,"style":170},[169],[18,23410,23412],{"className":23411},[106,681],[18,23413,23415],{"className":23414},[119],[18,23416,23418],{"className":23417},[124],[18,23419,23421,23429],{"className":23420,"style":677},[128],[18,23422,23423,23426],{"style":693},[18,23424],{"className":23425,"style":697},[136],[18,23427,77],{"className":23428,"style":254},[106,110],[18,23430,23431,23434],{"style":703},[18,23432],{"className":23433,"style":697},[136],[18,23435,23437],{"className":23436,"style":711},[710],[18,23438,664],{"className":23439},[106],[18,23441,84],{"className":23442},[267],[18,23444,23446],{"className":23445},[267],[18,23447,965],{"className":23448},[1230,1196],[18,23450],{"className":23451,"style":228},[169],[18,23453,4775],{"className":23454},[232],[18,23456],{"className":23457,"style":228},[169],[18,23459,23461,23464,23467,23470,23473,23476,23482,23485,23488,23491],{"className":23460},[97],[18,23462],{"className":23463,"style":6101},[101],[18,23465,5351],{"className":23466,"style":5480},[106,110],[18,23468],{"className":23469,"style":170},[169],[18,23471],{"className":23472,"style":170},[169],[18,23474,11208],{"className":23475},[985],[18,23477,23479],{"className":23478},[250],[18,23480,74],{"className":23481},[1230,1196],[18,23483,4031],{"className":23484},[106,110],[18,23486],{"className":23487,"style":228},[169],[18,23489,4775],{"className":23490},[232],[18,23492],{"className":23493,"style":228},[169],[18,23495,23497,23500,23503,23509,23512,23515],{"className":23496},[97],[18,23498],{"className":23499,"style":6101},[101],[18,23501,5351],{"className":23502,"style":5480},[106,110],[18,23504,23506],{"className":23505},[267],[18,23507,84],{"className":23508},[1230,1196],[18,23510],{"className":23511,"style":228},[169],[18,23513,4775],{"className":23514},[232],[18,23516],{"className":23517,"style":228},[169],[18,23519,23521,23524,23527,23530,23570,23573,23576],{"className":23520},[97],[18,23522],{"className":23523,"style":448},[101],[18,23525,5351],{"className":23526,"style":5480},[106,110],[18,23528],{"className":23529,"style":170},[169],[18,23531,23533,23536],{"className":23532},[106],[18,23534,939],{"className":23535,"style":1171},[106,110],[18,23537,23539],{"className":23538},[115],[18,23540,23542,23562],{"className":23541},[119,120],[18,23543,23545,23559],{"className":23544},[124],[18,23546,23548],{"className":23547,"style":202},[128],[18,23549,23550,23553],{"style":1386},[18,23551],{"className":23552,"style":137},[136],[18,23554,23556],{"className":23555},[141,142,143,144],[18,23557,14444],{"className":23558},[106,110,144],[18,23560,152],{"className":23561},[151],[18,23563,23565],{"className":23564},[124],[18,23566,23568],{"className":23567,"style":159},[128],[18,23569],{},[18,23571],{"className":23572,"style":228},[169],[18,23574,4775],{"className":23575},[232],[18,23577],{"className":23578,"style":228},[169],[18,23580,23582,23585,23588,23591,23708,23711],{"className":23581},[97],[18,23583],{"className":23584,"style":20807},[101],[18,23586,5351],{"className":23587,"style":5480},[106,110],[18,23589],{"className":23590,"style":170},[169],[18,23592,23594,23600,23603,23606,23609,23612,23702],{"className":23593},[174],[18,23595,23597],{"className":23596,"style":13796},[250,13795],[18,23598,74],{"className":23599},[1230,143],[18,23601,47],{"className":23602},[106],[18,23604],{"className":23605,"style":2068},[169],[18,23607,2253],{"className":23608},[2072],[18,23610],{"className":23611,"style":2068},[169],[18,23613,23615,23618,23699],{"className":23614},[106],[18,23616],{"className":23617},[250,6012],[18,23619,23621],{"className":23620},[5654],[18,23622,23624,23691],{"className":23623},[119,120],[18,23625,23627,23688],{"className":23626},[124],[18,23628,23630,23649,23657],{"className":23629,"style":6025},[128],[18,23631,23632,23635],{"style":6028},[18,23633],{"className":23634,"style":697},[136],[18,23636,23638,23643,23646],{"className":23637},[106],[18,23639,15174,23641],{"className":23640},[985],[18,23642,15177],{"style":6355},[18,23644],{"className":23645,"style":170},[169],[18,23647,3526],{"className":23648,"style":246},[106,110],[18,23650,23651,23654],{"style":6040},[18,23652],{"className":23653,"style":697},[136],[18,23655],{"className":23656,"style":6048},[6047],[18,23658,23659,23662],{"style":6051},[18,23660],{"className":23661,"style":697},[136],[18,23663,23665,23668,23671,23674,23677,23682,23685],{"className":23664},[106],[18,23666,16413],{"className":23667,"style":111},[106,110],[18,23669],{"className":23670,"style":2068},[169],[18,23672,7987],{"className":23673},[2072],[18,23675],{"className":23676,"style":2068},[169],[18,23678,15174,23680],{"className":23679},[985],[18,23681,15177],{"style":6355},[18,23683],{"className":23684,"style":170},[169],[18,23686,2266],{"className":23687},[106],[18,23689,152],{"className":23690},[151],[18,23692,23694],{"className":23693},[124],[18,23695,23697],{"className":23696,"style":18219},[128],[18,23698],{},[18,23700],{"className":23701},[267,6012],[18,23703,23705],{"className":23704,"style":13796},[267,13795],[18,23706,84],{"className":23707},[1230,143],[18,23709],{"className":23710,"style":170},[169],[18,23712,968],{"className":23713},[106],[11,23715,23716,23717,23745],{},"左边正是 ",[18,23718,23720,23733],{"className":23719,"translate":22},[21],[18,23721,23723],{"className":23722},[26],[28,23724,23725],{"xmlns":30},[32,23726,23727,23731],{},[35,23728,23729],{},[41,23730,3526],{},[86,23732,3526],{"encoding":88},[18,23734,23736],{"className":23735,"ariaHidden":51},[93],[18,23737,23739,23742],{"className":23738},[97],[18,23740],{"className":23741,"style":600},[101],[18,23743,3526],{"className":23744,"style":246},[106,110]," 个风险的平均，平均不超过最大，定理得证。",[11,23747,23748,23749,23830,23831,23859,23860,23939,23940,23969,23970,24181,24182,24287],{},"对平方损失 ",[18,23750,23752,23774],{"className":23751,"translate":22},[21],[18,23753,23755],{"className":23754},[26],[28,23756,23757],{"xmlns":30},[32,23758,23759,23771],{},[35,23760,23761,23763,23765],{},[41,23762,948],{"mathvariant":316},[49,23764,433],{},[1556,23766,23767,23769],{},[41,23768,4031],{},[45,23770,2266],{},[86,23772,23773],{"encoding":88},"\\ell=d^2",[18,23775,23777,23795],{"className":23776,"ariaHidden":51},[93],[18,23778,23780,23783,23786,23789,23792],{"className":23779},[97],[18,23781],{"className":23782,"style":295},[101],[18,23784,948],{"className":23785},[106],[18,23787],{"className":23788,"style":228},[169],[18,23790,433],{"className":23791},[232],[18,23793],{"className":23794,"style":228},[169],[18,23796,23798,23801],{"className":23797},[97],[18,23799],{"className":23800,"style":2416},[101],[18,23802,23804,23807],{"className":23803},[106],[18,23805,4031],{"className":23806},[106,110],[18,23808,23810],{"className":23809},[115],[18,23811,23813],{"className":23812},[119],[18,23814,23816],{"className":23815},[124],[18,23817,23819],{"className":23818,"style":2416},[128],[18,23820,23821,23824],{"style":1962},[18,23822],{"className":23823,"style":137},[136],[18,23825,23827],{"className":23826},[141,142,143,144],[18,23828,2266],{"className":23829},[106,144],"，同样的解码论证把 ",[18,23832,23834,23847],{"className":23833,"translate":22},[21],[18,23835,23837],{"className":23836},[26],[28,23838,23839],{"xmlns":30},[32,23840,23841,23845],{},[35,23842,23843],{},[41,23844,4031],{},[86,23846,4031],{"encoding":88},[18,23848,23850],{"className":23849,"ariaHidden":51},[93],[18,23851,23853,23856],{"className":23852},[97],[18,23854],{"className":23855,"style":295},[101],[18,23857,4031],{"className":23858},[106,110]," 换成 ",[18,23861,23863,23879],{"className":23862,"translate":22},[21],[18,23864,23866],{"className":23865},[26],[28,23867,23868],{"xmlns":30},[32,23869,23870,23876],{},[35,23871,23872],{},[12561,23873,23874],{},[41,23875,948],{"mathvariant":316},[86,23877,23878],{"encoding":88},"\\sqrt{\\ell}",[18,23880,23882],{"className":23881,"ariaHidden":51},[93],[18,23883,23885,23889],{"className":23884},[97],[18,23886],{"className":23887,"style":23888},[101],"height:1.04em;vertical-align:-0.1078em;",[18,23890,23892],{"className":23891},[106,12696],[18,23893,23895,23930],{"className":23894},[119,120],[18,23896,23898,23927],{"className":23897},[124],[18,23899,23902,23914],{"className":23900,"style":23901},[128],"height:0.9322em;",[18,23903,23905,23908],{"className":23904,"style":693},[12710],[18,23906],{"className":23907,"style":697},[136],[18,23909,23911],{"className":23910,"style":15410},[106],[18,23912,948],{"className":23913},[106],[18,23915,23917,23920],{"style":23916},"top:-2.8922em;",[18,23918],{"className":23919,"style":697},[136],[18,23921,23923],{"className":23922,"style":15420},[12853],[6130,23924,23925],{"xmlns":6132,"width":12857,"height":15423,"viewBox":15424,"preserveAspectRatio":12860},[6136,23926],{"d":15427},[18,23928,152],{"className":23929},[151],[18,23931,23933],{"className":23932},[124],[18,23934,23937],{"className":23935,"style":23936},[128],"height:0.1078em;",[18,23938],{},"。若平方间隔为 ",[18,23941,23943,23957],{"className":23942,"translate":22},[21],[18,23944,23946],{"className":23945},[26],[28,23947,23948],{"xmlns":30},[32,23949,23950,23954],{},[35,23951,23952],{},[41,23953,532],{"mathvariant":316},[86,23955,23956],{"encoding":88},"\\Delta",[18,23958,23960],{"className":23959,"ariaHidden":51},[93],[18,23961,23963,23966],{"className":23962},[97],[18,23964],{"className":23965,"style":600},[101],[18,23967,532],{"className":23968},[106],"，即 ",[18,23971,23973,24011],{"className":23972,"translate":22},[21],[18,23974,23976],{"className":23975},[26],[28,23977,23978],{"xmlns":30},[32,23979,23980,24008],{},[35,23981,23982,23984,23986,23992,23994,24000,24002,24004],{},[41,23983,4031],{},[49,23985,74],{"stretchy":73},[38,23987,23988,23990],{},[41,23989,77],{},[41,23991,5600],{},[49,23993,52],{"separator":51},[38,23995,23996,23998],{},[41,23997,77],{},[41,23999,19811],{},[49,24001,84],{"stretchy":73},[49,24003,4775],{},[12561,24005,24006],{},[41,24007,532],{"mathvariant":316},[86,24009,24010],{"encoding":88},"d(\\theta_i,\\theta_j)\\ge\\sqrt{\\Delta}",[18,24012,24014,24124],{"className":24013,"ariaHidden":51},[93],[18,24015,24017,24020,24023,24026,24066,24069,24072,24112,24115,24118,24121],{"className":24016},[97],[18,24018],{"className":24019,"style":19846},[101],[18,24021,4031],{"className":24022},[106,110],[18,24024,74],{"className":24025},[250],[18,24027,24029,24032],{"className":24028},[106],[18,24030,77],{"className":24031,"style":254},[106,110],[18,24033,24035],{"className":24034},[115],[18,24036,24038,24058],{"className":24037},[119,120],[18,24039,24041,24055],{"className":24040},[124],[18,24042,24044],{"className":24043,"style":5921},[128],[18,24045,24046,24049],{"style":469},[18,24047],{"className":24048,"style":137},[136],[18,24050,24052],{"className":24051},[141,142,143,144],[18,24053,5600],{"className":24054},[106,110,144],[18,24056,152],{"className":24057},[151],[18,24059,24061],{"className":24060},[124],[18,24062,24064],{"className":24063,"style":159},[128],[18,24065],{},[18,24067,52],{"className":24068},[165],[18,24070],{"className":24071,"style":170},[169],[18,24073,24075,24078],{"className":24074},[106],[18,24076,77],{"className":24077,"style":254},[106,110],[18,24079,24081],{"className":24080},[115],[18,24082,24084,24104],{"className":24083},[119,120],[18,24085,24087,24101],{"className":24086},[124],[18,24088,24090],{"className":24089,"style":5921},[128],[18,24091,24092,24095],{"style":469},[18,24093],{"className":24094,"style":137},[136],[18,24096,24098],{"className":24097},[141,142,143,144],[18,24099,19811],{"className":24100,"style":19928},[106,110,144],[18,24102,152],{"className":24103},[151],[18,24105,24107],{"className":24106},[124],[18,24108,24110],{"className":24109,"style":19938},[128],[18,24111],{},[18,24113,84],{"className":24114},[267],[18,24116],{"className":24117,"style":228},[169],[18,24119,4775],{"className":24120},[232],[18,24122],{"className":24123,"style":228},[169],[18,24125,24127,24131],{"className":24126},[97],[18,24128],{"className":24129,"style":24130},[101],"height:1.04em;vertical-align:-0.1133em;",[18,24132,24134],{"className":24133},[106,12696],[18,24135,24137,24172],{"className":24136},[119,120],[18,24138,24140,24169],{"className":24139},[124],[18,24141,24144,24156],{"className":24142,"style":24143},[128],"height:0.9267em;",[18,24145,24147,24150],{"className":24146,"style":693},[12710],[18,24148],{"className":24149,"style":697},[136],[18,24151,24153],{"className":24152,"style":15410},[106],[18,24154,532],{"className":24155},[106],[18,24157,24159,24162],{"style":24158},"top:-2.8867em;",[18,24160],{"className":24161,"style":697},[136],[18,24163,24165],{"className":24164,"style":15420},[12853],[6130,24166,24167],{"xmlns":6132,"width":12857,"height":15423,"viewBox":15424,"preserveAspectRatio":12860},[6136,24168],{"d":15427},[18,24170,152],{"className":24171},[151],[18,24173,24175],{"className":24174},[124],[18,24176,24179],{"className":24177,"style":24178},[128],"height:0.1133em;",[18,24180],{},"，则取 ",[18,24183,24185,24209],{"className":24184,"translate":22},[21],[18,24186,24188],{"className":24187},[26],[28,24189,24190],{"xmlns":30},[32,24191,24192,24206],{},[35,24193,24194,24196,24198,24202,24204],{},[41,24195,5351],{},[49,24197,433],{},[12561,24199,24200],{},[41,24201,532],{"mathvariant":316},[41,24203,317],{"mathvariant":316},[45,24205,2266],{},[86,24207,24208],{"encoding":88},"\\delta=\\sqrt{\\Delta}\u002F2",[18,24210,24212,24230],{"className":24211,"ariaHidden":51},[93],[18,24213,24215,24218,24221,24224,24227],{"className":24214},[97],[18,24216],{"className":24217,"style":295},[101],[18,24219,5351],{"className":24220,"style":5480},[106,110],[18,24222],{"className":24223,"style":228},[169],[18,24225,433],{"className":24226},[232],[18,24228],{"className":24229,"style":228},[169],[18,24231,24233,24237,24284],{"className":24232},[97],[18,24234],{"className":24235,"style":24236},[101],"height:1.1767em;vertical-align:-0.25em;",[18,24238,24240],{"className":24239},[106,12696],[18,24241,24243,24276],{"className":24242},[119,120],[18,24244,24246,24273],{"className":24245},[124],[18,24247,24249,24261],{"className":24248,"style":24143},[128],[18,24250,24252,24255],{"className":24251,"style":693},[12710],[18,24253],{"className":24254,"style":697},[136],[18,24256,24258],{"className":24257,"style":15410},[106],[18,24259,532],{"className":24260},[106],[18,24262,24263,24266],{"style":24158},[18,24264],{"className":24265,"style":697},[136],[18,24267,24269],{"className":24268,"style":15420},[12853],[6130,24270,24271],{"xmlns":6132,"width":12857,"height":15423,"viewBox":15424,"preserveAspectRatio":12860},[6136,24272],{"d":15427},[18,24274,152],{"className":24275},[151],[18,24277,24279],{"className":24278},[124],[18,24280,24282],{"className":24281,"style":24178},[128],[18,24283],{},[18,24285,8943],{"className":24286},[106],"，得到",[18,24289,24291],{"className":24290,"translate":22},[875],[18,24292,24294,24396],{"className":24293,"translate":22},[21],[18,24295,24297],{"className":24296},[26],[28,24298,24299],{"xmlns":30,"display":884},[32,24300,24301,24393],{},[35,24302,24303,24309,24315,24317,24319,24321,24323,24329,24331,24337,24339,24341,24343,24349,24391],{},[5654,24304,24305,24307],{},[45,24306,47],{},[41,24308,3526],{},[890,24310,24311,24313],{},[49,24312,20369],{},[41,24314,5600],{},[41,24316,934],{"mathvariant":933},[49,24318,945],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[41,24320,948],{"mathvariant":316},[49,24322,74],{"stretchy":73},[38,24324,24325,24327],{},[41,24326,77],{},[41,24328,5600],{},[49,24330,52],{"separator":51},[658,24332,24333,24335],{"accent":51},[41,24334,77],{},[49,24336,664],{},[49,24338,84],{"stretchy":73},[49,24340,965],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,24342,4775],{},[5654,24344,24345,24347],{},[41,24346,532],{"mathvariant":316},[45,24348,10447],{},[35,24350,24351,24353,24355,24357,24389],{},[49,24352,74],{"fence":51},[45,24354,47],{},[49,24356,2253],{},[5654,24358,24359,24381],{},[35,24360,24361,24363,24365,24367,24369,24371,24373,24375,24377,24379],{},[41,24362,16413],{},[49,24364,74],{"stretchy":73},[41,24366,5031],{},[49,24368,18005],{"separator":51},[41,24370,43],{},[49,24372,84],{"stretchy":73},[49,24374,7987],{},[41,24376,14733],{},[49,24378,899],{},[45,24380,2266],{},[35,24382,24383,24385,24387],{},[41,24384,14733],{},[49,24386,899],{},[41,24388,3526],{},[49,24390,84],{"fence":51},[41,24392,968],{"mathvariant":316},[86,24394,24395],{"encoding":88},"\\frac1M\\sum_i\\mathbb E\\bigl[\\ell(\\theta_i,\\hat\\theta)\\bigr]\n\\ge\n\\frac{\\Delta}{4}\\left(1-\\frac{I(K;X)+\\log 2}{\\log M}\\right).",[18,24397,24399,24629],{"className":24398,"ariaHidden":51},[93],[18,24400,24402,24406,24468,24471,24516,24519,24522,24528,24531,24534,24574,24577,24580,24611,24614,24620,24623,24626],{"className":24401},[97],[18,24403],{"className":24404,"style":24405},[101],"height:2.5991em;vertical-align:-1.2777em;",[18,24407,24409,24412,24465],{"className":24408},[106],[18,24410],{"className":24411},[250,6012],[18,24413,24415],{"className":24414},[5654],[18,24416,24418,24457],{"className":24417},[119,120],[18,24419,24421,24454],{"className":24420},[124],[18,24422,24424,24435,24443],{"className":24423,"style":9581},[128],[18,24425,24426,24429],{"style":6028},[18,24427],{"className":24428,"style":697},[136],[18,24430,24432],{"className":24431},[106],[18,24433,3526],{"className":24434,"style":246},[106,110],[18,24436,24437,24440],{"style":6040},[18,24438],{"className":24439,"style":697},[136],[18,24441],{"className":24442,"style":6048},[6047],[18,24444,24445,24448],{"style":6051},[18,24446],{"className":24447,"style":697},[136],[18,24449,24451],{"className":24450},[106],[18,24452,47],{"className":24453},[106],[18,24455,152],{"className":24456},[151],[18,24458,24460],{"className":24459},[124],[18,24461,24463],{"className":24462,"style":6070},[128],[18,24464],{},[18,24466],{"className":24467},[267,6012],[18,24469],{"className":24470,"style":170},[169],[18,24472,24474],{"className":24473},[985,986],[18,24475,24477,24508],{"className":24476},[119,120],[18,24478,24480,24505],{"className":24479},[124],[18,24481,24484,24495],{"className":24482,"style":24483},[128],"height:1.05em;",[18,24485,24486,24489],{"style":20555},[18,24487],{"className":24488,"style":20559},[136],[18,24490,24492],{"className":24491},[141,142,143,144],[18,24493,5600],{"className":24494},[106,110,144],[18,24496,24497,24500],{"style":20577},[18,24498],{"className":24499,"style":20559},[136],[18,24501,24502],{},[18,24503,20369],{"className":24504},[985,4198,4199],[18,24506,152],{"className":24507},[151],[18,24509,24511],{"className":24510},[124],[18,24512,24514],{"className":24513,"style":20607},[128],[18,24515],{},[18,24517],{"className":24518,"style":170},[169],[18,24520,934],{"className":24521},[106,1139],[18,24523,24525],{"className":24524},[250],[18,24526,945],{"className":24527},[1230,1196],[18,24529,948],{"className":24530},[106],[18,24532,74],{"className":24533},[250],[18,24535,24537,24540],{"className":24536},[106],[18,24538,77],{"className":24539,"style":254},[106,110],[18,24541,24543],{"className":24542},[115],[18,24544,24546,24566],{"className":24545},[119,120],[18,24547,24549,24563],{"className":24548},[124],[18,24550,24552],{"className":24551,"style":5921},[128],[18,24553,24554,24557],{"style":469},[18,24555],{"className":24556,"style":137},[136],[18,24558,24560],{"className":24559},[141,142,143,144],[18,24561,5600],{"className":24562},[106,110,144],[18,24564,152],{"className":24565},[151],[18,24567,24569],{"className":24568},[124],[18,24570,24572],{"className":24571,"style":159},[128],[18,24573],{},[18,24575,52],{"className":24576},[165],[18,24578],{"className":24579,"style":170},[169],[18,24581,24583],{"className":24582},[106,681],[18,24584,24586],{"className":24585},[119],[18,24587,24589],{"className":24588},[124],[18,24590,24592,24600],{"className":24591,"style":677},[128],[18,24593,24594,24597],{"style":693},[18,24595],{"className":24596,"style":697},[136],[18,24598,77],{"className":24599,"style":254},[106,110],[18,24601,24602,24605],{"style":703},[18,24603],{"className":24604,"style":697},[136],[18,24606,24608],{"className":24607,"style":711},[710],[18,24609,664],{"className":24610},[106],[18,24612,84],{"className":24613},[267],[18,24615,24617],{"className":24616},[267],[18,24618,965],{"className":24619},[1230,1196],[18,24621],{"className":24622,"style":228},[169],[18,24624,4775],{"className":24625},[232],[18,24627],{"className":24628,"style":228},[169],[18,24630,24632,24635,24697,24700,24835,24838],{"className":24631},[97],[18,24633],{"className":24634,"style":20807},[101],[18,24636,24638,24641,24694],{"className":24637},[106],[18,24639],{"className":24640},[250,6012],[18,24642,24644],{"className":24643},[5654],[18,24645,24647,24686],{"className":24646},[119,120],[18,24648,24650,24683],{"className":24649},[124],[18,24651,24653,24664,24672],{"className":24652,"style":10845},[128],[18,24654,24655,24658],{"style":6028},[18,24656],{"className":24657,"style":697},[136],[18,24659,24661],{"className":24660},[106],[18,24662,10447],{"className":24663},[106],[18,24665,24666,24669],{"style":6040},[18,24667],{"className":24668,"style":697},[136],[18,24670],{"className":24671,"style":6048},[6047],[18,24673,24674,24677],{"style":6051},[18,24675],{"className":24676,"style":697},[136],[18,24678,24680],{"className":24679},[106],[18,24681,532],{"className":24682},[106],[18,24684,152],{"className":24685},[151],[18,24687,24689],{"className":24688},[124],[18,24690,24692],{"className":24691,"style":6070},[128],[18,24693],{},[18,24695],{"className":24696},[267,6012],[18,24698],{"className":24699,"style":170},[169],[18,24701,24703,24709,24712,24715,24718,24721,24829],{"className":24702},[174],[18,24704,24706],{"className":24705,"style":13796},[250,13795],[18,24707,74],{"className":24708},[1230,143],[18,24710,47],{"className":24711},[106],[18,24713],{"className":24714,"style":2068},[169],[18,24716,2253],{"className":24717},[2072],[18,24719],{"className":24720,"style":2068},[169],[18,24722,24724,24727,24826],{"className":24723},[106],[18,24725],{"className":24726},[250,6012],[18,24728,24730],{"className":24729},[5654],[18,24731,24733,24818],{"className":24732},[119,120],[18,24734,24736,24815],{"className":24735},[124],[18,24737,24739,24758,24766],{"className":24738,"style":18133},[128],[18,24740,24741,24744],{"style":6028},[18,24742],{"className":24743,"style":697},[136],[18,24745,24747,24752,24755],{"className":24746},[106],[18,24748,15174,24750],{"className":24749},[985],[18,24751,15177],{"style":6355},[18,24753],{"className":24754,"style":170},[169],[18,24756,3526],{"className":24757,"style":246},[106,110],[18,24759,24760,24763],{"style":6040},[18,24761],{"className":24762,"style":697},[136],[18,24764],{"className":24765,"style":6048},[6047],[18,24767,24768,24771],{"style":6051},[18,24769],{"className":24770,"style":697},[136],[18,24772,24774,24777,24780,24783,24786,24789,24792,24795,24798,24801,24804,24809,24812],{"className":24773},[106],[18,24775,16413],{"className":24776,"style":111},[106,110],[18,24778,74],{"className":24779},[250],[18,24781,5031],{"className":24782,"style":5073},[106,110],[18,24784,18005],{"className":24785},[165],[18,24787],{"className":24788,"style":170},[169],[18,24790,43],{"className":24791,"style":111},[106,110],[18,24793,84],{"className":24794},[267],[18,24796],{"className":24797,"style":2068},[169],[18,24799,7987],{"className":24800},[2072],[18,24802],{"className":24803,"style":2068},[169],[18,24805,15174,24807],{"className":24806},[985],[18,24808,15177],{"style":6355},[18,24810],{"className":24811,"style":170},[169],[18,24813,2266],{"className":24814},[106],[18,24816,152],{"className":24817},[151],[18,24819,24821],{"className":24820},[124],[18,24822,24824],{"className":24823,"style":18219},[128],[18,24825],{},[18,24827],{"className":24828},[267,6012],[18,24830,24832],{"className":24831,"style":13796},[267,13795],[18,24833,84],{"className":24834},[1230,143],[18,24836],{"className":24837,"style":170},[169],[18,24839,968],{"className":24840},[106],[11,24842,24843,24844,24905,24906,1990,25178,25215],{},"这里用了 ",[18,24845,24847,24869],{"className":24846,"translate":22},[21],[18,24848,24850],{"className":24849},[26],[28,24851,24852],{"xmlns":30},[32,24853,24854,24866],{},[35,24855,24856,24858,24860,24862,24864],{},[49,24857,1331],{"stretchy":73},[41,24859,4031],{},[49,24861,6684],{},[41,24863,5351],{},[49,24865,1349],{"stretchy":73},[86,24867,24868],{"encoding":88},"\\{d\u003C\\delta\\}",[18,24870,24872,24893],{"className":24871,"ariaHidden":51},[93],[18,24873,24875,24878,24881,24884,24887,24890],{"className":24874},[97],[18,24876],{"className":24877,"style":242},[101],[18,24879,1331],{"className":24880},[250],[18,24882,4031],{"className":24883},[106,110],[18,24885],{"className":24886,"style":228},[169],[18,24888,6684],{"className":24889},[232],[18,24891],{"className":24892,"style":228},[169],[18,24894,24896,24899,24902],{"className":24895},[97],[18,24897],{"className":24898,"style":242},[101],[18,24900,5351],{"className":24901,"style":5480},[106,110],[18,24903,1349],{"className":24904},[267]," 蕴含解码正确，因此 ",[18,24907,24909,24969],{"className":24908,"translate":22},[21],[18,24910,24912],{"className":24911},[26],[28,24913,24914],{"xmlns":30},[32,24915,24916,24966],{},[35,24917,24918,24920,24922,24928,24930,24932,24938,24940,24942,24944,24946,24948,24950,24952,24954,24960],{},[41,24919,934],{"mathvariant":933},[49,24921,945],{"stretchy":73},[1556,24923,24924,24926],{},[41,24925,4031],{},[45,24927,2266],{},[49,24929,965],{"stretchy":73},[49,24931,4775],{},[1556,24933,24934,24936],{},[41,24935,5351],{},[45,24937,2266],{},[41,24939,11208],{},[49,24941,899],{},[49,24943,74],{"stretchy":73},[41,24945,4031],{},[49,24947,4775],{},[41,24949,5351],{},[49,24951,84],{"stretchy":73},[49,24953,4775],{},[1556,24955,24956,24958],{},[41,24957,5351],{},[45,24959,2266],{},[38,24961,24962,24964],{},[41,24963,939],{},[41,24965,14444],{},[86,24967,24968],{"encoding":88},"\\mathbb E[d^2]\\ge\\delta^2\\Pr(d\\ge\\delta)\\ge\\delta^2P_e",[18,24970,24972,25025,25081,25102],{"className":24971,"ariaHidden":51},[93],[18,24973,24975,24978,24981,24984,25013,25016,25019,25022],{"className":24974},[97],[18,24976],{"className":24977,"style":10271},[101],[18,24979,934],{"className":24980},[106,1139],[18,24982,945],{"className":24983},[250],[18,24985,24987,24990],{"className":24986},[106],[18,24988,4031],{"className":24989},[106,110],[18,24991,24993],{"className":24992},[115],[18,24994,24996],{"className":24995},[119],[18,24997,24999],{"className":24998},[124],[18,25000,25002],{"className":25001,"style":2416},[128],[18,25003,25004,25007],{"style":1962},[18,25005],{"className":25006,"style":137},[136],[18,25008,25010],{"className":25009},[141,142,143,144],[18,25011,2266],{"className":25012},[106,144],[18,25014,965],{"className":25015},[267],[18,25017],{"className":25018,"style":228},[169],[18,25020,4775],{"className":25021},[232],[18,25023],{"className":25024,"style":228},[169],[18,25026,25028,25031,25060,25063,25066,25069,25072,25075,25078],{"className":25027},[97],[18,25029],{"className":25030,"style":10271},[101],[18,25032,25034,25037],{"className":25033},[106],[18,25035,5351],{"className":25036,"style":5480},[106,110],[18,25038,25040],{"className":25039},[115],[18,25041,25043],{"className":25042},[119],[18,25044,25046],{"className":25045},[124],[18,25047,25049],{"className":25048,"style":2416},[128],[18,25050,25051,25054],{"style":1962},[18,25052],{"className":25053,"style":137},[136],[18,25055,25057],{"className":25056},[141,142,143,144],[18,25058,2266],{"className":25059},[106,144],[18,25061],{"className":25062,"style":170},[169],[18,25064,11208],{"className":25065},[985],[18,25067,74],{"className":25068},[250],[18,25070,4031],{"className":25071},[106,110],[18,25073],{"className":25074,"style":228},[169],[18,25076,4775],{"className":25077},[232],[18,25079],{"className":25080,"style":228},[169],[18,25082,25084,25087,25090,25093,25096,25099],{"className":25083},[97],[18,25085],{"className":25086,"style":242},[101],[18,25088,5351],{"className":25089,"style":5480},[106,110],[18,25091,84],{"className":25092},[267],[18,25094],{"className":25095,"style":228},[169],[18,25097,4775],{"className":25098},[232],[18,25100],{"className":25101,"style":228},[169],[18,25103,25105,25109,25138],{"className":25104},[97],[18,25106],{"className":25107,"style":25108},[101],"height:0.9641em;vertical-align:-0.15em;",[18,25110,25112,25115],{"className":25111},[106],[18,25113,5351],{"className":25114,"style":5480},[106,110],[18,25116,25118],{"className":25117},[115],[18,25119,25121],{"className":25120},[119],[18,25122,25124],{"className":25123},[124],[18,25125,25127],{"className":25126,"style":2416},[128],[18,25128,25129,25132],{"style":1962},[18,25130],{"className":25131,"style":137},[136],[18,25133,25135],{"className":25134},[141,142,143,144],[18,25136,2266],{"className":25137},[106,144],[18,25139,25141,25144],{"className":25140},[106],[18,25142,939],{"className":25143,"style":1171},[106,110],[18,25145,25147],{"className":25146},[115],[18,25148,25150,25170],{"className":25149},[119,120],[18,25151,25153,25167],{"className":25152},[124],[18,25154,25156],{"className":25155,"style":202},[128],[18,25157,25158,25161],{"style":1386},[18,25159],{"className":25160,"style":137},[136],[18,25162,25164],{"className":25163},[141,142,143,144],[18,25165,14444],{"className":25166},[106,110,144],[18,25168,152],{"className":25169},[151],[18,25171,25173],{"className":25172},[124],[18,25174,25176],{"className":25175,"style":159},[128],[18,25177],{},[717,25179,25180,25181],{},"平方损失的常数是 ",[18,25182,25184,25202],{"className":25183,"translate":22},[21],[18,25185,25187],{"className":25186},[26],[28,25188,25189],{"xmlns":30},[32,25190,25191,25199],{},[35,25192,25193,25195,25197],{},[41,25194,532],{"mathvariant":316},[41,25196,317],{"mathvariant":316},[45,25198,10447],{},[86,25200,25201],{"encoding":88},"\\Delta\u002F4",[18,25203,25205],{"className":25204,"ariaHidden":51},[93],[18,25206,25208,25211],{"className":25207},[97],[18,25209],{"className":25210,"style":242},[101],[18,25212,25214],{"className":25213},[106],"Δ\u002F4","，与 Le Cam 的平方损失版本一致。本文后面的例子只关心速率，常数不影响结论。",[11,25217,25218],{},"剩下的任务是把互信息控制住。由定义，",[18,25220,25222],{"className":25221,"translate":22},[875],[18,25223,25225,25340],{"className":25224,"translate":22},[21],[18,25226,25228],{"className":25227},[26],[28,25229,25230],{"xmlns":30,"display":884},[32,25231,25232,25337],{},[35,25233,25234,25236,25238,25240,25242,25244,25246,25248,25254,25268,25278,25280,25286,25288,25295,25297,25299,25301,25307,25309,25315,25329,25335],{},[41,25235,16413],{},[49,25237,74],{"stretchy":73},[41,25239,5031],{},[49,25241,18005],{"separator":51},[41,25243,43],{},[49,25245,84],{"stretchy":73},[49,25247,433],{},[5654,25249,25250,25252],{},[45,25251,47],{},[41,25253,3526],{},[20365,25255,25256,25258,25266],{},[49,25257,20369],{},[35,25259,25260,25262,25264],{},[41,25261,5600],{},[49,25263,433],{},[45,25265,47],{},[41,25267,3526],{},[38,25269,25270,25272],{},[41,25271,12575],{},[35,25273,25274,25276],{},[41,25275,5031],{"mathvariant":316},[41,25277,12582],{"mathvariant":316},[49,25279,74],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[38,25281,25282,25284],{},[41,25283,939],{},[41,25285,5600],{},[41,25287,2248],{"mathvariant":316},[658,25289,25290,25292],{"accent":51},[41,25291,939],{},[49,25293,25294],{},"ˉ",[49,25296,84],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,25298,52],{"separator":51},[169,25300],{"width":7344},[658,25302,25303,25305],{"accent":51},[41,25304,939],{},[49,25306,25294],{},[49,25308,433],{},[5654,25310,25311,25313],{},[45,25312,47],{},[41,25314,3526],{},[20365,25316,25317,25319,25327],{},[49,25318,20369],{},[35,25320,25321,25323,25325],{},[41,25322,5600],{},[49,25324,433],{},[45,25326,47],{},[41,25328,3526],{},[38,25330,25331,25333],{},[41,25332,939],{},[41,25334,5600],{},[41,25336,968],{"mathvariant":316},[86,25338,25339],{"encoding":88},"I(K;X)=\\frac1M\\sum_{i=1}^M D_{\\mathrm{KL}}\\bigl(P_i\\Vert \\bar P\\bigr),\n\\qquad \\bar P=\\frac1M\\sum_{i=1}^M P_i.",[18,25341,25343,25379,25699],{"className":25342,"ariaHidden":51},[93],[18,25344,25346,25349,25352,25355,25358,25361,25364,25367,25370,25373,25376],{"className":25345},[97],[18,25347],{"className":25348,"style":242},[101],[18,25350,16413],{"className":25351,"style":111},[106,110],[18,25353,74],{"className":25354},[250],[18,25356,5031],{"className":25357,"style":5073},[106,110],[18,25359,18005],{"className":25360},[165],[18,25362],{"className":25363,"style":170},[169],[18,25365,43],{"className":25366,"style":111},[106,110],[18,25368,84],{"className":25369},[267],[18,25371],{"className":25372,"style":228},[169],[18,25374,433],{"className":25375},[232],[18,25377],{"className":25378,"style":228},[169],[18,25380,25382,25385,25447,25450,25514,25517,25563,25569,25609,25612,25644,25650,25653,25656,25659,25690,25693,25696],{"className":25381},[97],[18,25383],{"className":25384,"style":20474},[101],[18,25386,25388,25391,25444],{"className":25387},[106],[18,25389],{"className":25390},[250,6012],[18,25392,25394],{"className":25393},[5654],[18,25395,25397,25436],{"className":25396},[119,120],[18,25398,25400,25433],{"className":25399},[124],[18,25401,25403,25414,25422],{"className":25402,"style":9581},[128],[18,25404,25405,25408],{"style":6028},[18,25406],{"className":25407,"style":697},[136],[18,25409,25411],{"className":25410},[106],[18,25412,3526],{"className":25413,"style":246},[106,110],[18,25415,25416,25419],{"style":6040},[18,25417],{"className":25418,"style":697},[136],[18,25420],{"className":25421,"style":6048},[6047],[18,25423,25424,25427],{"style":6051},[18,25425],{"className":25426,"style":697},[136],[18,25428,25430],{"className":25429},[106],[18,25431,47],{"className":25432},[106],[18,25434,152],{"className":25435},[151],[18,25437,25439],{"className":25438},[124],[18,25440,25442],{"className":25441,"style":6070},[128],[18,25443],{},[18,25445],{"className":25446},[267,6012],[18,25448],{"className":25449,"style":170},[169],[18,25451,25453],{"className":25452},[985,986],[18,25454,25456,25506],{"className":25455},[119,120],[18,25457,25459,25503],{"className":25458},[124],[18,25460,25462,25482,25492],{"className":25461,"style":20552},[128],[18,25463,25464,25467],{"style":20555},[18,25465],{"className":25466,"style":20559},[136],[18,25468,25470],{"className":25469},[141,142,143,144],[18,25471,25473,25476,25479],{"className":25472},[106,144],[18,25474,5600],{"className":25475},[106,110,144],[18,25477,433],{"className":25478},[232,144],[18,25480,47],{"className":25481},[106,144],[18,25483,25484,25487],{"style":20577},[18,25485],{"className":25486,"style":20559},[136],[18,25488,25489],{},[18,25490,20369],{"className":25491},[985,4198,4199],[18,25493,25494,25497],{"style":20588},[18,25495],{"className":25496,"style":20559},[136],[18,25498,25500],{"className":25499},[141,142,143,144],[18,25501,3526],{"className":25502,"style":246},[106,110,144],[18,25504,152],{"className":25505},[151],[18,25507,25509],{"className":25508},[124],[18,25510,25512],{"className":25511,"style":20607},[128],[18,25513],{},[18,25515],{"className":25516,"style":170},[169],[18,25518,25520,25523],{"className":25519},[106],[18,25521,12575],{"className":25522,"style":254},[106,110],[18,25524,25526],{"className":25525},[115],[18,25527,25529,25555],{"className":25528},[119,120],[18,25530,25532,25552],{"className":25531},[124],[18,25533,25535],{"className":25534,"style":1152},[128],[18,25536,25537,25540],{"style":469},[18,25538],{"className":25539,"style":137},[136],[18,25541,25543],{"className":25542},[141,142,143,144],[18,25544,25546],{"className":25545},[106,144],[18,25547,25549],{"className":25548},[106,144],[18,25550,12817],{"className":25551},[106,6354,144],[18,25553,152],{"className":25554},[151],[18,25556,25558],{"className":25557},[124],[18,25559,25561],{"className":25560,"style":159},[128],[18,25562],{},[18,25564,25566],{"className":25565},[250],[18,25567,74],{"className":25568},[1230,1196],[18,25570,25572,25575],{"className":25571},[106],[18,25573,939],{"className":25574,"style":1171},[106,110],[18,25576,25578],{"className":25577},[115],[18,25579,25581,25601],{"className":25580},[119,120],[18,25582,25584,25598],{"className":25583},[124],[18,25585,25587],{"className":25586,"style":5921},[128],[18,25588,25589,25592],{"style":1386},[18,25590],{"className":25591,"style":137},[136],[18,25593,25595],{"className":25594},[141,142,143,144],[18,25596,5600],{"className":25597},[106,110,144],[18,25599,152],{"className":25600},[151],[18,25602,25604],{"className":25603},[124],[18,25605,25607],{"className":25606,"style":159},[128],[18,25608],{},[18,25610,2248],{"className":25611},[106],[18,25613,25615],{"className":25614},[106,681],[18,25616,25618],{"className":25617},[119],[18,25619,25621],{"className":25620},[124],[18,25622,25625,25633],{"className":25623,"style":25624},[128],"height:0.8201em;",[18,25626,25627,25630],{"style":693},[18,25628],{"className":25629,"style":697},[136],[18,25631,939],{"className":25632,"style":1171},[106,110],[18,25634,2563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KL 对第二个变量的凸性，",[18,25884,25886],{"className":25885,"translate":22},[875],[18,25887,25889,26019],{"className":25888,"translate":22},[21],[18,25890,25892],{"className":25891},[26],[28,25893,25894],{"xmlns":30,"display":884},[32,25895,25896,26016],{},[35,25897,25898,25900,25902,25904,25906,25908,25910,25912,25922,25940,25950,25952,25958,25960,25966,25968,25970,25986,25996,25998,26004,26006,26012,26014],{},[41,25899,16413],{},[49,25901,74],{"stretchy":73},[41,25903,5031],{},[49,25905,18005],{"separator":51},[41,25907,43],{},[49,25909,84],{"stretchy":73},[49,25911,4334],{},[5654,25913,25914,25916],{},[45,25915,47],{},[1556,25917,25918,25920],{},[41,25919,3526],{},[45,25921,2266],{},[20365,25923,25924,25926,25938],{},[49,25925,20369],{},[35,25927,25928,25930,25932,25934,25936],{},[41,25929,5600],{},[49,25931,52],{"separator":51},[41,25933,19811],{},[49,25935,433],{},[45,25937,47],{},[41,25939,3526],{},[38,25941,25942,25944],{},[41,25943,12575],{},[35,25945,25946,25948],{},[41,25947,5031],{"mathvariant":316},[41,25949,12582],{"mathvariant":316},[49,25951,74],{"stretchy":73},[38,25953,25954,25956],{},[41,25955,939],{},[41,25957,5600],{},[41,25959,2248],{"mathvariant":316},[38,25961,25962,25964],{},[41,25963,939],{},[41,25965,19811],{},[49,25967,84],{"stretchy":73},[49,25969,4334],{},[890,25971,25972,25978],{},[35,25973,25974,25976],{},[41,25975,5593],{},[49,25977,899],{},[35,25979,25980,25982,25984],{},[41,25981,5600],{},[49,25983,52],{"separator":51},[41,25985,19811],{},[38,25987,25988,25990],{},[41,25989,12575],{},[35,25991,25992,25994],{},[41,25993,5031],{"mathvariant":316},[41,25995,12582],{"mathvariant":316},[49,25997,74],{"stretchy":73},[38,25999,26000,26002],{},[41,26001,939],{},[41,26003,5600],{},[41,26005,2248],{"mathvariant":316},[38,26007,26008,26010],{},[41,26009,939],{},[41,26011,19811],{},[49,26013,84],{"stretchy":73},[41,26015,968],{"mathvariant":316},[86,26017,26018],{"encoding":88},"I(K;X)\\le \\frac1{M^2}\\sum_{i,j=1}^M D_{\\mathrm{KL}}(P_i\\Vert P_j)\n\\le \\max_{i,j}D_{\\mathrm{KL}}(P_i\\Vert P_j).",[18,26020,26022,26058,26376],{"className":26021,"ariaHidden":51},[93],[18,26023,26025,26028,26031,26034,26037,26040,26043,26046,26049,26052,26055],{"className":26024},[97],[18,26026],{"className":26027,"style":242},[101],[18,26029,16413],{"className":26030,"style":111},[106,110],[18,26032,74],{"className":26033},[250],[18,26035,5031],{"className":26036,"style":5073},[106,110],[18,26038,18005],{"className":26039},[165],[18,26041],{"className":26042,"style":170},[169],[18,26044,43],{"className":26045,"style":111},[106,110],[18,26047,84],{"className":26048},[267],[18,26050],{"className":26051,"style":228},[169],[18,26053,4334],{"className":26054},[232],[18,26056],{"className":26057,"style":228},[169],[18,26059,26061,26065,26155,26158,26229,26232,26278,26281,26321,26324,26364,26367,26370,26373],{"className":26060},[97],[18,26062],{"className":26063,"style":26064},[101],"height:3.2421em;vertical-align:-1.4138em;",[18,26066,26068,26071,26152],{"className":26067},[106],[18,26069],{"className":26070},[250,6012],[18,26072,26074],{"className":26073},[5654],[18,26075,26077,26144],{"className":26076},[119,120],[18,26078,26080,26141],{"className":26079},[124],[18,26081,26083,26122,26130],{"className":26082,"style":9581},[128],[18,26084,26085,26088],{"style":6028},[18,26086],{"className":26087,"style":697},[136],[18,26089,26091],{"className":26090},[106],[18,26092,26094,26097],{"className":26093},[106],[18,26095,3526],{"className":26096,"style":246},[106,110],[18,26098,26100],{"className":26099},[115],[18,26101,26103],{"className":26102},[119],[18,26104,26106],{"className":26105},[124],[18,26107,26110],{"className":26108,"style":26109},[128],"height:0.7401em;",[18,26111,26113,26116],{"style":26112},"top:-2.989em;margin-right:0.05em;",[18,26114],{"className":26115,"style":137},[136],[18,26117,26119],{"className":26118},[141,142,143,144],[18,26120,2266],{"className":26121},[106,144],[18,26123,26124,26127],{"style":6040},[18,26125],{"className":26126,"style":697},[136],[18,26128],{"className":26129,"style":6048},[6047],[18,26131,26132,26135],{"style":6051},[18,26133],{"className":26134,"style":697},[136],[18,26136,26138],{"className":26137},[106],[18,26139,47],{"className":26140},[106],[18,26142,152],{"className":26143},[151],[18,26145,26147],{"className":26146},[124],[18,26148,26150],{"className":26149,"style":6070},[128],[18,26151],{},[18,26153],{"className":26154},[267,6012],[18,26156],{"className":26157,"style":170},[169],[18,26159,26161],{"className":26160},[985,986],[18,26162,26164,26220],{"className":26163},[119,120],[18,26165,26167,26217],{"className":26166},[124],[18,26168,26170,26196,26206],{"className":26169,"style":20552},[128],[18,26171,26172,26175],{"style":20555},[18,26173],{"className":26174,"style":20559},[136],[18,26176,26178],{"className":26177},[141,142,143,144],[18,26179,26181,26184,26187,26190,26193],{"className":26180},[106,144],[18,26182,5600],{"className":26183},[106,110,144],[18,26185,52],{"className":26186},[165,144],[18,26188,19811],{"className":26189,"style":19928},[106,110,144],[18,26191,433],{"className":26192},[232,144],[18,26194,47],{"className":26195},[106,144],[18,26197,26198,26201],{"style":20577},[18,26199],{"className":26200,"style":20559},[136],[18,26202,26203],{},[18,26204,20369],{"className":26205},[985,4198,4199],[18,26207,26208,26211],{"style":20588},[18,26209],{"className":26210,"style":20559},[136],[18,26212,26214],{"className":26213},[141,142,143,144],[18,26215,3526],{"className":26216,"style":246},[106,110,144],[18,26218,152],{"className":26219},[151],[18,26221,26223],{"className":26222},[124],[18,26224,26227],{"className":26225,"style":26226},[128],"height:1.4138em;",[18,26228],{},[18,26230],{"className":26231,"style":170},[169],[18,26233,26235,26238],{"className":26234},[106],[18,26236,12575],{"className":26237,"style":254},[106,110],[18,26239,26241],{"className":26240},[115],[18,26242,26244,26270],{"className":26243},[119,120],[18,26245,26247,26267],{"className":26246},[124],[18,26248,26250],{"className":26249,"style":1152},[128],[18,26251,26252,26255],{"style":469},[18,26253],{"className":26254,"style":137},[136],[18,26256,26258],{"className":26257},[141,142,143,144],[18,26259,26261],{"className":26260},[106,144],[18,26262,26264],{"className":26263},[106,144],[18,26265,12817],{"className":26266},[106,6354,144],[18,26268,152],{"className":26269},[151],[18,26271,26273],{"className":26272},[124],[18,26274,26276],{"className":26275,"style":159},[128],[18,26277],{},[18,26279,74],{"className":26280},[250],[18,26282,26284,26287],{"className":26283},[106],[18,26285,939],{"className":26286,"style":1171},[106,110],[18,26288,26290],{"className":26289},[115],[18,26291,26293,26313],{"className":26292},[119,120],[18,26294,26296,26310],{"className":26295},[124],[18,26297,26299],{"className":26298,"style":5921},[128],[18,26300,26301,26304],{"style":1386},[18,26302],{"className":26303,"style":137},[136],[18,26305,26307],{"className":26306},[141,142,143,144],[18,26308,5600],{"className":26309},[106,110,144],[18,26311,152],{"className":26312},[151],[18,26314,26316],{"className":26315},[124],[18,26317,26319],{"className":26318,"style":159},[128],[18,26320],{},[18,26322,2248],{"className":26323},[106],[18,26325,26327,26330],{"className":26326},[106],[18,26328,939],{"className":26329,"style":1171},[106,110],[18,26331,26333],{"className":26332},[115],[18,26334,26336,26356],{"className":26335},[119,120],[18,26337,26339,26353],{"className":26338},[124],[18,26340,26342],{"className":26341,"style":5921},[128],[18,26343,26344,26347],{"style":1386},[18,26345],{"className":26346,"style":137},[136],[18,26348,26350],{"className":26349},[141,142,143,144],[18,26351,19811],{"className":26352,"style":19928},[106,110,144],[18,26354,152],{"className":26355},[151],[18,26357,26359],{"className":26358},[124],[18,26360,26362],{"className":26361,"style":19938},[128],[18,26363],{},[18,26365,84],{"className":26366},[267],[18,26368],{"className":26369,"style":228},[169],[18,26371,4334],{"className":26372},[232],[18,26374],{"className":26375,"style":228},[169],[18,26377,26379,26382,26435,26438,26484,26487,26527,26530,26570,26573],{"className":26378},[97],[18,26380],{"className":26381,"style":10508},[101],[18,26383,26385],{"className":26384},[985,986],[18,26386,26388,26427],{"className":26387},[119,120],[18,26389,26391,26424],{"className":26390},[124],[18,26392,26394,26414],{"className":26393,"style":1083},[128],[18,26395,26396,26399],{"style":5738},[18,26397],{"className":26398,"style":697},[136],[18,26400,26402],{"className":26401},[141,142,143,144],[18,26403,26405,26408,26411],{"className":26404},[106,144],[18,26406,5600],{"className":26407},[106,110,144],[18,26409,52],{"className":26410},[165,144],[18,26412,19811],{"className":26413,"style":19928},[106,110,144],[18,26415,26416,26419],{"style":693},[18,26417],{"className":26418,"style":697},[136],[18,26420,26421],{},[18,26422,5593],{"className":26423},[985],[18,26425,152],{"className":26426},[151],[18,26428,26430],{"className":26429},[124],[18,26431,26433],{"className":26432,"style":5782},[128],[18,26434],{},[18,26436],{"className":26437,"style":170},[169],[18,26439,26441,26444],{"className":26440},[106],[18,26442,12575],{"className":26443,"style":254},[106,110],[18,26445,26447],{"className":26446},[115],[18,26448,26450,26476],{"className":26449},[119,120],[18,26451,26453,26473],{"className":26452},[124],[18,26454,26456],{"className":26455,"style":1152},[128],[18,26457,26458,26461],{"style":469},[18,26459],{"className":26460,"style":137},[136],[18,26462,26464],{"className":26463},[141,142,143,144],[18,26465,26467],{"className":26466},[106,144],[18,26468,26470],{"className":26469},[106,144],[18,26471,12817],{"className":26472},[106,6354,144],[18,26474,152],{"className":26475},[151],[18,26477,26479],{"className":26478},[124],[18,26480,26482],{"className":26481,"style":159},[128],[18,26483],{},[18,26485,74],{"className":26486},[250],[18,26488,26490,26493],{"className":26489},[106],[18,26491,939],{"className":26492,"style":1171},[106,110],[18,26494,26496],{"className":26495},[115],[18,26497,26499,26519],{"className":26498},[119,120],[18,26500,26502,26516],{"className":26501},[124],[18,26503,26505],{"className":26504,"style":5921},[128],[18,26506,26507,26510],{"style":1386},[18,26508],{"className":26509,"style":137},[136],[18,26511,26513],{"className":26512},[141,142,143,144],[18,26514,5600],{"className":26515},[106,110,144],[18,26517,152],{"className":26518},[151],[18,26520,26522],{"className":26521},[124],[18,26523,26525],{"className":26524,"style":159},[128],[18,26526],{},[18,26528,2248],{"className":26529},[106],[18,26531,26533,26536],{"className":26532},[106],[18,26534,939],{"className":26535,"style":1171},[106,110],[18,26537,26539],{"className":26538},[115],[18,26540,26542,26562],{"className":26541},[119,120],[18,26543,26545,26559],{"className":26544},[124],[18,26546,26548],{"className":26547,"style":5921},[128],[18,26549,26550,26553],{"style":1386},[18,26551],{"className":26552,"style":137},[136],[18,26554,26556],{"className":26555},[141,142,143,144],[18,26557,19811],{"className":26558,"style":19928},[106,110,144],[18,26560,152],{"className":26561},[151],[18,26563,26565],{"className":26564},[124],[18,26566,26568],{"className":26567,"style":19938},[128],[18,26569],{},[18,26571,84],{"className":26572},[267],[18,26574,968],{"className":26575},[106],[11,26577,26578,26579,26649,26650,26678,26679,26707,26708,509,26748,26781],{},"如果每个 ",[18,26580,26582,26600],{"className":26581,"translate":22},[21],[18,26583,26585],{"className":26584},[26],[28,26586,26587],{"xmlns":30},[32,26588,26589,26597],{},[35,26590,26591],{},[38,26592,26593,26595],{},[41,26594,939],{},[41,26596,5600],{},[86,26598,26599],{"encoding":88},"P_i",[18,26601,26603],{"className":26602,"ariaHidden":51},[93],[18,26604,26606,26609],{"className":26605},[97],[18,26607],{"className":26608,"style":4563},[101],[18,26610,26612,26615],{"className":26611},[106],[18,26613,939],{"className":26614,"style":1171},[106,110],[18,26616,26618],{"className":26617},[115],[18,26619,26621,26641],{"className":26620},[119,120],[18,26622,26624,26638],{"className":26623},[124],[18,26625,26627],{"className":26626,"style":5921},[128],[18,26628,26629,26632],{"style":1386},[18,26630],{"className":26631,"style":137},[136],[18,26633,26635],{"className":26634},[141,142,143,144],[18,26636,5600],{"className":26637},[106,110,144],[18,26639,152],{"className":26640},[151],[18,26642,26644],{"className":26643},[124],[18,26645,26647],{"className":26646,"style":159},[128],[18,26648],{}," 都是 ",[18,26651,26653,26666],{"className":26652,"translate":22},[21],[18,26654,26656],{"className":26655},[26],[28,26657,26658],{"xmlns":30},[32,26659,26660,26664],{},[35,26661,26662],{},[41,26663,64],{},[86,26665,64],{"encoding":88},[18,26667,26669],{"className":26668,"ariaHidden":51},[93],[18,26670,26672,26675],{"className":26671},[97],[18,26673],{"className":26674,"style":1083},[101],[18,26676,64],{"className":26677},[106,110]," 个独立样本的分布，那么成对 KL 是单样本 KL 的 ",[18,26680,26682,26695],{"className":26681,"translate":22},[21],[18,26683,26685],{"className":26684},[26],[28,26686,26687],{"xmlns":30},[32,26688,26689,26693],{},[35,26690,26691],{},[41,26692,64],{},[86,26694,64],{"encoding":88},[18,26696,26698],{"className":26697,"ariaHidden":51},[93],[18,26699,26701,26704],{"className":26700},[97],[18,26702],{"className":26703,"style":1083},[101],[18,26705,64],{"className":26706},[106,110]," 倍。这样，Fano 下界就变成了「",[18,26709,26711,26728],{"className":26710,"translate":22},[21],[18,26712,26714],{"className":26713},[26],[28,26715,26716],{"xmlns":30},[32,26717,26718,26726],{},[35,26719,26720,26722,26724],{},[41,26721,14733],{},[49,26723,899],{},[41,26725,3526],{},[86,26727,19451],{"encoding":88},[18,26729,26731],{"className":26730,"ariaHidden":51},[93],[18,26732,26734,26737,26742,26745],{"className":26733},[97],[18,26735],{"className":26736,"style":2890},[101],[18,26738,15174,26740],{"className":26739},[985],[18,26741,15177],{"style":6355},[18,26743],{"className":26744,"style":170},[169],[18,26746,3526],{"className":26747,"style":246},[106,110],[18,26749,26751,26766],{"className":26750,"translate":22},[21],[18,26752,26754],{"className":26753},[26],[28,26755,26756],{"xmlns":30},[32,26757,26758,26764],{},[35,26759,26760,26762],{},[41,26761,64],{},[49,26763,2015],{},[86,26765,14394],{"encoding":88},[18,26767,26769],{"className":26768,"ariaHidden":51},[93],[18,26770,26772,26775,26778],{"className":26771},[97],[18,26773],{"className":26774,"style":14404},[101],[18,26776,64],{"className":26777},[106,110],[18,26779,2015],{"className":26780},[106],"KL 的竞争」。",[1311,26783,26785],{"id":26784},"_43-广义-fano-与局部版本","4.3 广义 Fano 与局部版本",[11,26787,26788,26789,26792],{},"定理 4.2 常被称为",[717,26790,26791],{},"广义 Fano 不等式","。它的使用可以总结成一句口诀：",[18,26794,26796],{"className":26795,"translate":22},[875],[18,26797,26799,26908],{"className":26798,"translate":22},[21],[18,26800,26802],{"className":26801},[26],[28,26803,26804],{"xmlns":30,"display":884},[32,26805,26806,26905],{},[35,26807,26808],{},[26809,26810,26812],"menclose",{"notation":26811},"box",[14681,26813,26814],{"scriptlevel":430,"displaystyle":73},[14681,26815,26816],{"scriptlevel":430,"displaystyle":73},[14681,26817,26818],{"scriptlevel":430,"displaystyle":51},[35,26819,26820,26823,26825,26827,26829,26831,26834,26836,26838,26841,26857,26867,26869,26875,26877,26883,26885,26887,26890,26896,26899,26901,26903],{},[926,26821,26822],{}," 只要 ",[41,26824,14733],{},[49,26826,899],{},[41,26828,3526],{},[926,26830,928],{},[49,26832,26833],{},"≫",[926,26835,928],{},[41,26837,64],{},[49,26839,26840],{},"⋅",[890,26842,26843,26849],{},[35,26844,26845,26847],{},[41,26846,5593],{},[49,26848,899],{},[35,26850,26851,26853,26855],{},[41,26852,5600],{},[49,26854,52],{"separator":51},[41,26856,19811],{},[38,26858,26859,26861],{},[41,26860,12575],{},[35,26862,26863,26865],{},[41,26864,5031],{"mathvariant":316},[41,26866,12582],{"mathvariant":316},[49,26868,74],{"stretchy":73},[38,26870,26871,26873],{},[41,26872,939],{},[41,26874,5600],{},[41,26876,2248],{"mathvariant":316},[38,26878,26879,26881],{},[41,26880,939],{},[41,26882,19811],{},[49,26884,84],{"stretchy":73},[49,26886,52],{"separator":51},[926,26888,26889],{}," 就有 ",[38,26891,26892,26894],{},[41,26893,3526],{"mathvariant":3525},[41,26895,64],{},[49,26897,26898],{},"≳",[41,26900,5351],{},[41,26902,968],{"mathvariant":316},[926,26904,928],{},[86,26906,26907],{"encoding":88},"\\boxed{\\ \\text{只要}\\ \\log M\\ \\gg\\ n\\cdot \\max_{i,j}D_{\\mathrm{KL}}(P_i\\Vert P_j),\\ \\text{就有}\\ \\mathfrak M_n\\gtrsim\\delta.\\ }",[18,26909,26911],{"className":26910,"ariaHidden":51},[93],[18,26912,26914,26918],{"className":26913},[97],[18,26915],{"className":26916,"style":26917},[101],"height:2.2938em;vertical-align:-1.2038em;",[18,26919,26921],{"className":26920},[106],[18,26922,26924,27288],{"className":26923},[119,120],[18,26925,26927,27285],{"className":26926},[124],[18,26928,26931,27273],{"className":26929,"style":26930},[128],"height:1.09em;",[18,26932,26934,26938],{"style":26933},"top:-4.2938em;",[18,26935],{"className":26936,"style":26937},[136],"height:4.2938em;",[18,26939,26942],{"className":26940},[26941],"boxpad",[18,26943,26945],{"className":26944},[106],[18,26946,26948,26951,26960,26963,26966,26971,26974,26977,26980,26983,26986,26989,26992,26995,26998,27001,27004,27057,27060,27106,27109,27149,27152,27192,27195,27198,27201,27204,27211,27214,27254,27257,27261,27264,27267,27270],{"className":26947},[106],[18,26949,928],{"className":26950},[169],[18,26952,26955],{"className":26953},[106,26954],"text",[18,26956,26959],{"className":26957},[106,26958],"cjk_fallback","只要",[18,26961,928],{"className":26962},[169],[18,26964],{"className":26965,"style":170},[169],[18,26967,15174,26969],{"className":26968},[985],[18,26970,15177],{"style":6355},[18,26972],{"className":26973,"style":170},[169],[18,26975,3526],{"className":26976,"style":246},[106,110],[18,26978,928],{"className":26979},[169],[18,26981],{"className":26982,"style":228},[169],[18,26984,26833],{"className":26985},[232],[18,26987,928],{"className":26988},[169],[18,26990],{"className":26991,"style":228},[169],[18,26993,64],{"className":26994},[106,110],[18,26996],{"className":26997,"style":2068},[169],[18,26999,26840],{"className":27000},[2072],[18,27002],{"className":27003,"style":2068},[169],[18,27005,27007],{"className":27006},[985,986],[18,27008,27010,27049],{"className":27009},[119,120],[18,27011,27013,27046],{"className":27012},[124],[18,27014,27016,27036],{"className":27015,"style":1083},[128],[18,27017,27018,27021],{"style":5738},[18,27019],{"className":27020,"style":697},[136],[18,27022,27024],{"className":27023},[141,142,143,144],[18,27025,27027,27030,27033],{"className":27026},[106,144],[18,27028,5600],{"className":27029},[106,110,144],[18,27031,52],{"className":27032},[165,144],[18,27034,19811],{"className":27035,"style":19928},[106,110,144],[18,27037,27038,27041],{"style":693},[18,27039],{"className":27040,"style":697},[136],[18,27042,27043],{},[18,27044,5593],{"className":27045},[985],[18,27047,152],{"className":27048},[151],[18,27050,27052],{"className":27051},[124],[18,27053,27055],{"className":27054,"style":5782},[128],[18,27056],{},[18,27058],{"className":27059,"style":170},[169],[18,27061,27063,27066],{"className":27062},[106],[18,27064,12575],{"className":27065,"style":254},[106,110],[18,27067,27069],{"className":27068},[115],[18,27070,27072,27098],{"className":27071},[119,120],[18,27073,27075,27095],{"className":27074},[124],[18,27076,27078],{"className":27077,"style":1152},[128],[18,27079,27080,27083],{"style":469},[18,27081],{"className":27082,"style":137},[136],[18,27084,27086],{"className":27085},[141,142,143,144],[18,27087,27089],{"className":27088},[106,144],[18,27090,27092],{"className":27091},[106,144],[18,27093,12817],{"className":27094},[106,6354,144],[18,27096,152],{"className":27097},[151],[18,27099,27101],{"className":27100},[124],[18,27102,27104],{"className":27103,"style":159},[128],[18,27105],{},[18,27107,74],{"className":27108},[250],[18,27110,27112,27115],{"className":27111},[106],[18,27113,939],{"className":27114,"style":1171},[106,110],[18,27116,27118],{"className":27117},[115],[18,27119,27121,27141],{"className":27120},[119,120],[18,27122,27124,27138],{"className":27123},[124],[18,27125,27127],{"className":27126,"style":5921},[128],[18,27128,27129,27132],{"style":1386},[18,27130],{"className":27131,"style":137},[136],[18,27133,27135],{"className":27134},[141,142,143,144],[18,27136,5600],{"className":27137},[106,110,144],[18,27139,152],{"className":27140},[151],[18,27142,27144],{"className":27143},[124],[18,27145,27147],{"className":27146,"style":159},[128],[18,27148],{},[18,27150,2248],{"className":27151},[106],[18,27153,27155,27158],{"className":27154},[106],[18,27156,939],{"className":27157,"style":1171},[106,110],[18,27159,27161],{"className":27160},[115],[18,27162,27164,27184],{"className":27163},[119,120],[18,27165,27167,27181],{"className":27166},[124],[18,27168,27170],{"className":27169,"style":5921},[128],[18,27171,27172,27175],{"style":1386},[18,27173],{"className":27174,"style":137},[136],[18,27176,27178],{"className":27177},[141,142,143,144],[18,27179,19811],{"className":27180,"style":19928},[106,110,144],[18,27182,152],{"className":27183},[151],[18,27185,27187],{"className":27186},[124],[18,27188,27190],{"className":27189,"style":19938},[128],[18,27191],{},[18,27193,84],{"className":27194},[267],[18,27196,52],{"className":27197},[165],[18,27199,928],{"className":27200},[169],[18,27202],{"className":27203,"style":170},[169],[18,27205,27207],{"className":27206},[106,26954],[18,27208,27210],{"className":27209},[106,26958],"就有",[18,27212,928],{"className":27213},[169],[18,27215,27217,27220],{"className":27216},[106],[18,27218,3526],{"className":27219},[106,3607],[18,27221,27223],{"className":27222},[115],[18,27224,27226,27246],{"className":27225},[119,120],[18,27227,27229,27243],{"className":27228},[124],[18,27230,27232],{"className":27231,"style":202},[128],[18,27233,27234,27237],{"style":1155},[18,27235],{"className":27236,"style":137},[136],[18,27238,27240],{"className":27239},[141,142,143,144],[18,27241,64],{"className":27242},[106,110,144],[18,27244,152],{"className":27245},[151],[18,27247,27249],{"className":27248},[124],[18,27250,27252],{"className":27251,"style":159},[128],[18,27253],{},[18,27255],{"className":27256,"style":228},[169],[18,27258,26898],{"className":27259},[232,27260],"amsrm",[18,27262],{"className":27263,"style":228},[169],[18,27265,5351],{"className":27266,"style":5480},[106,110],[18,27268,968],{"className":27269},[106],[18,27271,928],{"className":27272},[169],[18,27274,27276,27279],{"style":27275},"top:-3.09em;",[18,27277],{"className":27278,"style":26937},[136],[18,27280],{"className":27281,"style":27284},[27282,27283],"stretchy","fbox","height:2.2938em;border-style:solid;border-width:0.04em;",[18,27286,152],{"className":27287},[151],[18,27289,27291],{"className":27290},[124],[18,27292,27295],{"className":27293,"style":27294},[128],"height:1.2038em;",[18,27296],{},[11,27298,27299,27300,27328,27329,27357,27358,27361,27362,27390],{},"这里 ",[18,27301,27303,27316],{"className":27302,"translate":22},[21],[18,27304,27306],{"className":27305},[26],[28,27307,27308],{"xmlns":30},[32,27309,27310,27314],{},[35,27311,27312],{},[41,27313,3526],{},[86,27315,3526],{"encoding":88},[18,27317,27319],{"className":27318,"ariaHidden":51},[93],[18,27320,27322,27325],{"className":27321},[97],[18,27323],{"className":27324,"style":600},[101],[18,27326,3526],{"className":27327,"style":246},[106,110]," 是「用 ",[18,27330,27332,27345],{"className":27331,"translate":22},[21],[18,27333,27335],{"className":27334},[26],[28,27336,27337],{"xmlns":30},[32,27338,27339,27343],{},[35,27340,27341],{},[41,27342,5351],{},[86,27344,14091],{"encoding":88},[18,27346,27348],{"className":27347,"ariaHidden":51},[93],[18,27349,27351,27354],{"className":27350},[97],[18,27352],{"className":27353,"style":295},[101],[18,27355,5351],{"className":27356,"style":5480},[106,110]," 分开的假设个数」，也就是损失空间里的",[717,27359,27360],{},"打包数","（packing number）。于是下界问题变成了一个几何问题：在参数空间里能放多少个半径为 ",[18,27363,27365,27378],{"className":27364,"translate":22},[21],[18,27366,27368],{"className":27367},[26],[28,27369,27370],{"xmlns":30},[32,27371,27372,27376],{},[35,27373,27374],{},[41,27375,5351],{},[86,27377,14091],{"encoding":88},[18,27379,27381],{"className":27380,"ariaHidden":51},[93],[18,27382,27384,27387],{"className":27383},[97],[18,27385],{"className":27386,"style":295},[101],[18,27388,5351],{"className":27389,"style":5480},[106,110]," 的小球，同时又让它们在 KL 意义下足够接近？",[11,27392,27393,27394,27753,27754,27787],{},"Yang 与 Barron 的版本直接用 ",[18,27395,27397,27463],{"className":27396,"translate":22},[21],[18,27398,27400],{"className":27399},[26],[28,27401,27402],{"xmlns":30},[32,27403,27404,27460],{},[35,27405,27406,27408,27410,27412,27414,27416,27418,27420,27426,27432,27442,27444,27450,27452,27458],{},[41,27407,16413],{},[49,27409,74],{"stretchy":73},[41,27411,5031],{},[49,27413,18005],{"separator":51},[41,27415,43],{},[49,27417,84],{"stretchy":73},[49,27419,433],{},[5654,27421,27422,27424],{},[45,27423,47],{},[41,27425,3526],{},[38,27427,27428,27430],{},[49,27429,20369],{},[41,27431,5600],{},[38,27433,27434,27436],{},[41,27435,12575],{},[35,27437,27438,27440],{},[41,27439,5031],{"mathvariant":316},[41,27441,12582],{"mathvariant":316},[49,27443,74],{"stretchy":73},[38,27445,27446,27448],{},[41,27447,939],{},[41,27449,5600],{},[41,27451,2248],{"mathvariant":316},[658,27453,27454,27456],{"accent":51},[41,27455,939],{},[49,27457,25294],{},[49,27459,84],{"stretchy":73},[86,27461,27462],{"encoding":88},"I(K;X)=\\frac1M\\sum_i D_{\\mathrm{KL}}(P_i\\Vert\\bar P)",[18,27464,27466,27502],{"className":27465,"ariaHidden":51},[93],[18,27467,27469,27472,27475,27478,27481,27484,27487,27490,27493,27496,27499],{"className":27468},[97],[18,27470],{"className":27471,"style":242},[101],[18,27473,16413],{"className":27474,"style":111},[106,110],[18,27476,74],{"className":27477},[250],[18,27479,5031],{"className":27480,"style":5073},[106,110],[18,27482,18005],{"className":27483},[165],[18,27485],{"className":27486,"style":170},[169],[18,27488,43],{"className":27489,"style":111},[106,110],[18,27491,84],{"className":27492},[267],[18,27494],{"className":27495,"style":228},[169],[18,27497,433],{"className":27498},[232],[18,27500],{"className":27501,"style":228},[169],[18,27503,27505,27508,27576,27579,27624,27627,27673,27676,27716,27719,27750],{"className":27504},[97],[18,27506],{"className":27507,"style":12026},[101],[18,27509,27511,27514,27573],{"className":27510},[106],[18,27512],{"className":27513},[250,6012],[18,27515,27517],{"className":27516},[5654],[18,27518,27520,27565],{"className":27519},[119,120],[18,27521,27523,27562],{"className":27522},[124],[18,27524,27526,27540,27548],{"className":27525,"style":12045},[128],[18,27527,27528,27531],{"style":12048},[18,27529],{"className":27530,"style":697},[136],[18,27532,27534],{"className":27533},[141,142,143,144],[18,27535,27537],{"className":27536},[106,144],[18,27538,3526],{"className":27539,"style":246},[106,110,144],[18,27541,27542,27545],{"style":6040},[18,27543],{"className":27544,"style":697},[136],[18,27546],{"className":27547,"style":6048},[6047],[18,27549,27550,27553],{"style":12071},[18,27551],{"className":27552,"style":697},[136],[18,27554,27556],{"className":27555},[141,142,143,144],[18,27557,27559],{"className":27558},[106,144],[18,27560,47],{"className":27561},[106,144],[18,27563,152],{"className":27564},[151],[18,27566,27568],{"className":27567},[124],[18,27569,27571],{"className":27570,"style":12093},[128],[18,27572],{},[18,27574],{"className":27575},[267,6012],[18,27577],{"className":27578,"style":170},[169],[18,27580,27582,27587],{"className":27581},[985],[18,27583,20369],{"className":27584,"style":27586},[985,4198,27585],"small-op","position:relative;top:0em;",[18,27588,27590],{"className":27589},[115],[18,27591,27593,27615],{"className":27592},[119,120],[18,27594,27596,27612],{"className":27595},[124],[18,27597,27600],{"className":27598,"style":27599},[128],"height:0.162em;",[18,27601,27603,27606],{"style":27602},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[18,27604],{"className":27605,"style":137},[136],[18,27607,27609],{"className":27608},[141,142,143,144],[18,27610,5600],{"className":27611},[106,110,144],[18,27613,152],{"className":27614},[151],[18,27616,27618],{"className":27617},[124],[18,27619,27622],{"className":27620,"style":27621},[128],"height:0.2997em;",[18,27623],{},[18,27625],{"className":27626,"style":170},[169],[18,27628,27630,27633],{"className":27629},[106],[18,27631,12575],{"className":27632,"style":254},[106,110],[18,27634,27636],{"className":27635},[115],[18,27637,27639,27665],{"className":27638},[119,120],[18,27640,27642,27662],{"className":27641},[124],[18,27643,27645],{"className":27644,"style":1152},[128],[18,27646,27647,27650],{"style":469},[18,27648],{"className":27649,"style":137},[136],[18,27651,27653],{"className":27652},[141,142,143,144],[18,27654,27656],{"className":27655},[106,144],[18,27657,27659],{"className":27658},[106,144],[18,27660,12817],{"className":27661},[106,6354,144],[18,27663,152],{"className":27664},[151],[18,27666,27668],{"className":27667},[124],[18,27669,27671],{"className":27670,"style":159},[128],[18,27672],{},[18,27674,74],{"className":27675},[250],[18,27677,27679,27682],{"className":27678},[106],[18,27680,939],{"className":27681,"style":1171},[106,110],[18,27683,27685],{"className":27684},[115],[18,27686,27688,27708],{"className":27687},[119,120],[18,27689,27691,27705],{"className":27690},[124],[18,27692,27694],{"className":27693,"style":5921},[128],[18,27695,27696,27699],{"style":1386},[18,27697],{"className":27698,"style":137},[136],[18,27700,27702],{"className":27701},[141,142,143,144],[18,27703,5600],{"className":27704},[106,110,144],[18,27706,152],{"className":27707},[151],[18,27709,27711],{"className":27710},[124],[18,27712,27714],{"className":27713,"style":159},[128],[18,27715],{},[18,27717,2248],{"className":27718},[106],[18,27720,27722],{"className":27721},[106,681],[18,27723,27725],{"className":27724},[119],[18,27726,27728],{"className":27727},[124],[18,27729,27731,27739],{"className":27730,"style":25624},[128],[18,27732,27733,27736],{"style":693},[18,27734],{"className":27735,"style":697},[136],[18,27737,939],{"className":27738,"style":1171},[106,110],[18,27740,27741,27744],{"style":17702},[18,27742],{"className":27743,"style":697},[136],[18,27745,27747],{"className":27746,"style":711},[710],[18,27748,25294],{"className":27749},[106],[18,27751,84],{"className":27752},[267],"，而不是把它放缩成最大成对 KL。这在局部问题里更精确，常常能去掉多余的 ",[18,27755,27757,27773],{"className":27756,"translate":22},[21],[18,27758,27760],{"className":27759},[26],[28,27761,27762],{"xmlns":30},[32,27763,27764,27770],{},[35,27765,27766,27768],{},[41,27767,14733],{},[49,27769,899],{},[86,27771,27772],{"encoding":88},"\\log",[18,27774,27776],{"className":27775,"ariaHidden":51},[93],[18,27777,27779,27782],{"className":27778},[97],[18,27780],{"className":27781,"style":2890},[101],[18,27783,15174,27785],{"className":27784},[985],[18,27786,15177],{"style":6355}," 因子。其思想是：不要要求所有假设两两都接近，只要它们整体上离它们的「重心」接近就够了。",[11,27789,27790,27791,27794,27795,27865,27866,27894,27895,28005,28006,28034,28035,28063,28064,1990],{},"Fano 还有一个",[717,27792,27793],{},"局部版本","：不要求 ",[18,27796,27798,27816],{"className":27797,"translate":22},[21],[18,27799,27801],{"className":27800},[26],[28,27802,27803],{"xmlns":30},[32,27804,27805,27813],{},[35,27806,27807],{},[38,27808,27809,27811],{},[41,27810,77],{},[41,27812,5600],{},[86,27814,27815],{"encoding":88},"\\theta_i",[18,27817,27819],{"className":27818,"ariaHidden":51},[93],[18,27820,27822,27825],{"className":27821},[97],[18,27823],{"className":27824,"style":448},[101],[18,27826,27828,27831],{"className":27827},[106],[18,27829,77],{"className":27830,"style":254},[106,110],[18,27832,27834],{"className":27833},[115],[18,27835,27837,27857],{"className":27836},[119,120],[18,27838,27840,27854],{"className":27839},[124],[18,27841,27843],{"className":27842,"style":5921},[128],[18,27844,27845,27848],{"style":469},[18,27846],{"className":27847,"style":137},[136],[18,27849,27851],{"className":27850},[141,142,143,144],[18,27852,5600],{"className":27853},[106,110,144],[18,27855,152],{"className":27856},[151],[18,27858,27860],{"className":27859},[124],[18,27861,27863],{"className":27862,"style":159},[128],[18,27864],{}," 覆盖整个参数空间，只要求它们落在一个半径 ",[18,27867,27869,27882],{"className":27868,"translate":22},[21],[18,27870,27872],{"className":27871},[26],[28,27873,27874],{"xmlns":30},[32,27875,27876,27880],{},[35,27877,27878],{},[41,27879,14447],{},[86,27881,14447],{"encoding":88},[18,27883,27885],{"className":27884,"ariaHidden":51},[93],[18,27886,27888,27891],{"className":27887},[97],[18,27889],{"className":27890,"style":1083},[101],[18,27892,14447],{"className":27893,"style":254},[106,110]," 的小球里。非参数速率 ",[18,27896,27898,27936],{"className":27897,"translate":22},[21],[18,27899,27901],{"className":27900},[26],[28,27902,27903],{"xmlns":30},[32,27904,27905,27933],{},[35,27906,27907],{},[1556,27908,27909,27911],{},[41,27910,64],{},[35,27912,27913,27915,27917,27919,27921,27923,27925,27927,27929,27931],{},[49,27914,2253],{},[45,27916,2266],{},[41,27918,20986],{},[41,27920,317],{"mathvariant":316},[49,27922,74],{"stretchy":73},[45,27924,2266],{},[41,27926,20986],{},[49,27928,7987],{},[41,27930,4031],{},[49,27932,84],{"stretchy":73},[86,27934,27935],{"encoding":88},"n^{-2\\beta\u002F(2\\beta+d)}",[18,27937,27939],{"className":27938,"ariaHidden":51},[93],[18,27940,27942,27946],{"className":27941},[97],[18,27943],{"className":27944,"style":27945},[101],"height:0.888em;",[18,27947,27949,27952],{"className":27948},[106],[18,27950,64],{"className":27951},[106,110],[18,27953,27955],{"className":27954},[115],[18,27956,27958],{"className":27957},[119],[18,27959,27961],{"className":27960},[124],[18,27962,27964],{"className":27963,"style":27945},[128],[18,27965,27966,27969],{"style":1962},[18,27967],{"className":27968,"style":137},[136],[18,27970,27972],{"className":27971},[141,142,143,144],[18,27973,27975,27978,27981,27984,27987,27990,27993,27996,27999,28002],{"className":27974},[106,144],[18,27976,2253],{"className":27977},[106,144],[18,27979,2266],{"className":27980},[106,144],[18,27982,20986],{"className":27983,"style":21038},[106,110,144],[18,27985,317],{"className":27986},[106,144],[18,27988,74],{"className":27989},[250,144],[18,27991,2266],{"className":27992},[106,144],[18,27994,20986],{"className":27995,"style":21038},[106,110,144],[18,27997,7987],{"className":27998},[2072,144],[18,28000,4031],{"className":28001},[106,110,144],[18,28003,84],{"className":28004},[267,144]," 正是这样来的——局部的打包数随 ",[18,28007,28009,28022],{"className":28008,"translate":22},[21],[18,28010,28012],{"className":28011},[26],[28,28013,28014],{"xmlns":30},[32,28015,28016,28020],{},[35,28017,28018],{},[41,28019,14447],{},[86,28021,14447],{"encoding":88},[18,28023,28025],{"className":28024,"ariaHidden":51},[93],[18,28026,28028,28031],{"className":28027},[97],[18,28029],{"className":28030,"style":1083},[101],[18,28032,14447],{"className":28033,"style":254},[106,110]," 增长，而 KL 随 ",[18,28036,28038,28051],{"className":28037,"translate":22},[21],[18,28039,28041],{"className":28040},[26],[28,28042,28043],{"xmlns":30},[32,28044,28045,28049],{},[35,28046,28047],{},[41,28048,14447],{},[86,28050,14447],{"encoding":88},[18,28052,28054],{"className":28053,"ariaHidden":51},[93],[18,28055,28057,28060],{"className":28056},[97],[18,28058],{"className":28059,"style":1083},[101],[18,28061,14447],{"className":28062,"style":254},[106,110]," 增长，两者平衡给出最优的 ",[18,28065,28067,28080],{"className":28066,"translate":22},[21],[18,28068,28070],{"className":28069},[26],[28,28071,28072],{"xmlns":30},[32,28073,28074,28078],{},[35,28075,28076],{},[41,28077,14447],{},[86,28079,14447],{"encoding":88},[18,28081,28083],{"className":28082,"ariaHidden":51},[93],[18,28084,28086,28089],{"className":28085},[97],[18,28087],{"className":28088,"style":1083},[101],[18,28090,14447],{"className":28091,"style":254},[106,110],[638,28093,28095],{"id":28094},"五assouad-引理","五、Assouad 引理",[1311,28097,28099],{"id":28098},"_51-超立方体","5.1 超立方体",[11,28101,28102,28103,28131],{},"Fano 处理的是「",[18,28104,28106,28119],{"className":28105,"translate":22},[21],[18,28107,28109],{"className":28108},[26],[28,28110,28111],{"xmlns":30},[32,28112,28113,28117],{},[35,28114,28115],{},[41,28116,3526],{},[86,28118,3526],{"encoding":88},[18,28120,28122],{"className":28121,"ariaHidden":51},[93],[18,28123,28125,28128],{"className":28124},[97],[18,28126],{"className":28127,"style":600},[101],[18,28129,3526],{"className":28130,"style":246},[106,110]," 个任意假设」，Assouad 处理的是一种特殊但极有用的结构：超立方体。",[5159,28133,28134,28345,29114,29117],{},[11,28135,28136,5166,28139,4741,28244,28314,28315,28344],{},[717,28137,28138],{},"定理 5.1（Assouad 引理，平方损失）",[18,28140,28142,28173],{"className":28141,"translate":22},[21],[18,28143,28145],{"className":28144},[26],[28,28146,28147],{"xmlns":30},[32,28148,28149,28170],{},[35,28150,28151,28153,28155,28157,28159,28161,28163],{},[41,28152,924],{"mathvariant":316},[49,28154,433],{},[49,28156,1331],{"stretchy":73},[45,28158,430],{},[49,28160,52],{"separator":51},[45,28162,47],{},[1556,28164,28165,28167],{},[49,28166,1349],{"stretchy":73},[41,28168,28169],{},"m",[86,28171,28172],{"encoding":88},"\\Theta=\\{0,1\\}^m",[18,28174,28176,28194],{"className":28175,"ariaHidden":51},[93],[18,28177,28179,28182,28185,28188,28191],{"className":28178},[97],[18,28180],{"className":28181,"style":600},[101],[18,28183,924],{"className":28184},[106],[18,28186],{"className":28187,"style":228},[169],[18,28189,433],{"className":28190},[232],[18,28192],{"className":28193,"style":228},[169],[18,28195,28197,28200,28203,28206,28209,28212,28215],{"className":28196},[97],[18,28198],{"className":28199,"style":242},[101],[18,28201,1331],{"className":28202},[250],[18,28204,430],{"className":28205},[106],[18,28207,52],{"className":28208},[165],[18,28210],{"className":28211,"style":170},[169],[18,28213,47],{"className":28214},[106],[18,28216,28218,28221],{"className":28217},[267],[18,28219,1349],{"className":28220},[267],[18,28222,28224],{"className":28223},[115],[18,28225,28227],{"className":28226},[119],[18,28228,28230],{"className":28229},[124],[18,28231,28233],{"className":28232,"style":1959},[128],[18,28234,28235,28238],{"style":1962},[18,28236],{"className":28237,"style":137},[136],[18,28239,28241],{"className":28240},[141,142,143,144],[18,28242,28169],{"className":28243},[106,110,144],[18,28245,28247,28265],{"className":28246,"translate":22},[21],[18,28248,28250],{"className":28249},[26],[28,28251,28252],{"xmlns":30},[32,28253,28254,28262],{},[35,28255,28256],{},[38,28257,28258,28260],{},[41,28259,939],{},[41,28261,77],{},[86,28263,28264],{"encoding":88},"P_\\theta",[18,28266,28268],{"className":28267,"ariaHidden":51},[93],[18,28269,28271,28274],{"className":28270},[97],[18,28272],{"className":28273,"style":4563},[101],[18,28275,28277,28280],{"className":28276},[106],[18,28278,939],{"className":28279,"style":1171},[106,110],[18,28281,28283],{"className":28282},[115],[18,28284,28286,28306],{"className":28285},[119,120],[18,28287,28289,28303],{"className":28288},[124],[18,28290,28292],{"className":28291,"style":1383},[128],[18,28293,28294,28297],{"style":1386},[18,28295],{"className":28296,"style":137},[136],[18,28298,28300],{"className":28299},[141,142,143,144],[18,28301,77],{"className":28302,"style":254},[106,110,144],[18,28304,152],{"className":28305},[151],[18,28307,28309],{"className":28308},[124],[18,28310,28312],{"className":28311,"style":159},[128],[18,28313],{}," 是观测分布，损失为平方欧氏距离。对每个坐标 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X]",[18,30000,30002,30094,30118],{"className":30001,"ariaHidden":51},[93],[18,30003,30005,30008,30076,30079,30082,30085,30088,30091],{"className":30004},[97],[18,30006],{"className":30007,"style":2281},[101],[18,30009,30011,30042],{"className":30010},[106],[18,30012,30014],{"className":30013},[106,681],[18,30015,30017],{"className":30016},[119],[18,30018,30020],{"className":30019},[124],[18,30021,30023,30031],{"className":30022,"style":677},[128],[18,30024,30025,30028],{"style":693},[18,30026],{"className":30027,"style":697},[136],[18,30029,77],{"className":30030,"style":254},[106,110],[18,30032,30033,30036],{"style":703},[18,30034],{"className":30035,"style":697},[136],[18,30037,30039],{"className":30038,"style":711},[710],[18,30040,664],{"className":30041},[106],[18,30043,30045],{"className":30044},[115],[18,30046,30048,30068],{"className":30047},[119,120],[18,30049,30051,30065],{"className":30050},[124],[18,30052,30054],{"className":30053,"style":1152},[128],[18,30055,30056,30059],{"style":469},[18,30057],{"className":30058,"style":137},[136],[18,30060,30062],{"className":30061},[141,142,143,144],[18,30063,3978],{"className":30064,"style":11305},[106,110,144],[18,30066,152],{"className":30067},[151],[18,30069,30071],{"className":30070},[124],[18,30072,30074],{"className":30073,"style":159},[128],[18,30075],{},[18,30077,74],{"className":30078},[250],[18,30080,43],{"className":30081,"style":111},[106,110],[18,30083,84],{"className":30084},[267],[18,30086],{"className":30087,"style":228},[169],[18,30089,433],{"className":30090},[232],[18,30092],{"className":30093,"style":228},[169],[18,30095,30097,30100,30103,30106,30109,30112,30115],{"className":30096},[97],[18,30098],{"className":30099,"style":242},[101],[18,30101,934],{"className":30102},[106,1139],[18,30104,945],{"className":30105},[250],[18,30107,77],{"className":30108,"style":254},[106,110],[18,30110],{"className":30111,"style":228},[169],[18,30113,11225],{"className":30114},[232],[18,30116],{"className":30117,"style":228},[169],[18,30119,30121,30124,30127],{"className":30120},[97],[18,30122],{"className":30123,"style":242},[101],[18,30125,43],{"className":30126,"style":111},[106,110],[18,30128,965],{"className":30129},[267],"，其贝叶斯风险为",[18,30132,30134],{"className":30133,"translate":22},[875],[18,30135,30137,30243],{"className":30136,"translate":22},[21],[18,30138,30140],{"className":30139},[26],[28,30141,30142],{"xmlns":30,"display":884},[32,30143,30144,30240],{},[35,30145,30146,30160,30162,30164,30170,30172,30174,30176,30178,30180,30182,30188,30190,30192,30194,30196,30198,30200,30202,30208,30210,30212,30214,30216,30218,30220,30222,30228,30230,30232,30234,30236,30238],{},[20365,30147,30148,30150,30158],{},[49,30149,20369],{},[35,30151,30152,30154,30156],{},[41,30153,19811],{},[49,30155,433],{},[45,30157,47],{},[41,30159,28169],{},[41,30161,934],{"mathvariant":933},[49,30163,945],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[38,30165,30166,30168],{},[41,30167,3941],{},[41,30169,19811],{},[49,30171,74],{"stretchy":73},[41,30173,43],{},[49,30175,84],{"stretchy":73},[49,30177,74],{"stretchy":73},[45,30179,47],{},[49,30181,2253],{},[38,30183,30184,30186],{},[41,30185,3941],{},[41,30187,19811],{},[49,30189,74],{"stretchy":73},[41,30191,43],{},[49,30193,84],{"stretchy":73},[49,30195,84],{"stretchy":73},[49,30197,965],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,30199,52],{"separator":51},[169,30201],{"width":7344},[38,30203,30204,30206],{},[41,30205,3941],{},[41,30207,19811],{},[49,30209,74],{"stretchy":73},[41,30211,43],{},[49,30213,84],{"stretchy":73},[49,30215,433],{},[41,30217,11208],{},[49,30219,899],{},[49,30221,74],{"stretchy":73},[38,30223,30224,30226],{},[41,30225,77],{},[41,30227,19811],{},[49,30229,433],{},[45,30231,47],{},[49,30233,11225],{},[41,30235,43],{},[49,30237,84],{"stretchy":73},[41,30239,968],{"mathvariant":316},[86,30241,30242],{"encoding":88},"\\sum_{j=1}^m\\mathbb E\\bigl[\\pi_j(X)(1-\\pi_j(X))\\bigr],\n\\qquad\n\\pi_j(X)=\\Pr(\\theta_j=1\\mid X).",[18,30244,30246,30395,30523,30584,30602],{"className":30245,"ariaHidden":51},[93],[18,30247,30249,30253,30318,30321,30324,30330,30371,30374,30377,30380,30383,30386,30389,30392],{"className":30248},[97],[18,30250],{"className":30251,"style":30252},[101],"height:3.0652em;vertical-align:-1.4138em;",[18,30254,30256],{"className":30255},[985,986],[18,30257,30259,30310],{"className":30258},[119,120],[18,30260,30262,30307],{"className":30261},[124],[18,30263,30266,30286,30296],{"className":30264,"style":30265},[128],"height:1.6514em;",[18,30267,30268,30271],{"style":20555},[18,30269],{"className":30270,"style":20559},[136],[18,30272,30274],{"className":30273},[141,142,143,144],[18,30275,30277,30280,30283],{"className":30276},[106,144],[18,30278,19811],{"className":30279,"style":19928},[106,110,144],[18,30281,433],{"className":30282},[232,144],[18,30284,47],{"className":30285},[106,144],[18,30287,30288,30291],{"style":20577},[18,30289],{"className":30290,"style":20559},[136],[18,30292,30293],{},[18,30294,20369],{"className":30295},[985,4198,4199],[18,30297,30298,30301],{"style":20588},[18,30299],{"className":30300,"style":20559},[136],[18,30302,30304],{"className":30303},[141,142,143,144],[18,30305,28169],{"className":30306},[106,110,144],[18,30308,152],{"className":30309},[151],[18,30311,30313],{"className":30312},[124],[18,30314,30316],{"className":30315,"style":26226},[128],[18,30317],{},[18,30319],{"className":30320,"style":170},[169],[18,30322,934],{"className":30323},[106,1139],[18,30325,30327],{"className":30326},[250],[18,30328,945],{"className":30329},[1230,1196],[18,30331,30333,30336],{"className":30332},[106],[18,30334,3941],{"className":30335,"style":3957},[106,110],[18,30337,30339],{"className":30338},[115],[18,30340,30342,30363],{"className":30341},[119,120],[18,30343,30345,30360],{"className":30344},[124],[18,30346,30348],{"className":30347,"style":5921},[128],[18,30349,30351,30354],{"style":30350},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[18,30352],{"className":30353,"style":137},[136],[18,30355,30357],{"className":30356},[141,142,143,144],[18,30358,19811],{"className":30359,"style":19928},[106,110,144],[18,30361,152],{"className":30362},[151],[18,30364,30366],{"className":30365},[124],[18,30367,30369],{"className":30368,"style":19938},[128],[18,30370],{},[18,30372,74],{"className":30373},[250],[18,30375,43],{"className":30376,"style":111},[106,110],[18,30378,84],{"className":30379},[267],[18,30381,74],{"className":30382},[250],[18,30384,47],{"className":30385},[106],[18,30387],{"className":30388,"style":2068},[169],[18,30390,2253],{"className":30391},[2072],[18,30393],{"className":30394,"style":2068},[169],[18,30396,30398,30401,30441,30444,30447,30450,30456,30459,30462,30465,30505,30508,30511,30514,30517,30520],{"className":30397},[97],[18,30399],{"className":30400,"style":6101},[101],[18,30402,30404,30407],{"className":30403},[106],[18,30405,3941],{"className":30406,"style":3957},[106,110],[18,30408,30410],{"className":30409},[115],[18,30411,30413,30433],{"className":30412},[119,120],[18,30414,30416,30430],{"className":30415},[124],[18,30417,30419],{"className":30418,"style":5921},[128],[18,30420,30421,30424],{"style":30350},[18,30422],{"className":30423,"style":137},[136],[18,30425,30427],{"className":30426},[141,142,143,144],[18,30428,19811],{"className":30429,"style":19928},[106,110,144],[18,30431,152],{"className":30432},[151],[18,30434,30436],{"className":30435},[124],[18,30437,30439],{"className":30438,"style":19938},[128],[18,30440],{},[18,30442,74],{"className":30443},[250],[18,30445,43],{"className":30446,"style":111},[106,110],[18,30448,3431],{"className":30449},[267],[18,30451,30453],{"className":30452},[267],[18,30454,965],{"className":30455},[1230,1196],[18,30457,52],{"className":30458},[165],[18,30460],{"className":30461,"style":7676},[169],[18,30463],{"className":30464,"style":170},[169],[18,30466,30468,30471],{"className":30467},[106],[18,30469,3941],{"className":30470,"style":3957},[106,110],[18,30472,30474],{"className":30473},[115],[18,30475,30477,30497],{"className":30476},[119,120],[18,30478,30480,30494],{"className":30479},[124],[18,30481,30483],{"className":30482,"style":5921},[128],[18,30484,30485,30488],{"style":30350},[18,30486],{"className":30487,"style":137},[136],[18,30489,30491],{"className":30490},[141,142,143,144],[18,30492,19811],{"className":30493,"style":19928},[106,110,144],[18,30495,152],{"className":30496},[151],[18,30498,30500],{"className":30499},[124],[18,30501,30503],{"className":30502,"style":19938},[128],[18,30504],{},[18,30506,74],{"className":30507},[250],[18,30509,43],{"className":30510,"style":111},[106,110],[18,30512,84],{"className":30513},[267],[18,30515],{"className":30516,"style":228},[169],[18,30518,433],{"className":30519},[232],[18,30521],{"className":30522,"style":228},[169],[18,30524,30526,30529,30532,30535,30575,30578,30581],{"className":30525},[97],[18,30527],{"className":30528,"style":19846},[101],[18,30530,11208],{"className":30531},[985],[18,30533,74],{"className":30534},[250],[18,30536,30538,30541],{"className":30537},[106],[18,30539,77],{"className":30540,"style":254},[106,110],[18,30542,30544],{"className":30543},[115],[18,30545,30547,30567],{"className":30546},[119,120],[18,30548,30550,30564],{"className":30549},[124],[18,30551,30553],{"className":30552,"style":5921},[128],[18,30554,30555,30558],{"style":469},[18,30556],{"className":30557,"style":137},[136],[18,30559,30561],{"className":30560},[141,142,143,144],[18,30562,19811],{"className":30563,"style":19928},[106,110,144],[18,30565,152],{"className":30566},[151],[18,30568,30570],{"className":30569},[124],[18,30571,30573],{"className":30572,"style":19938},[128],[18,30574],{},[18,30576],{"className":30577,"style":228},[169],[18,30579,433],{"className":30580},[232],[18,30582],{"className":30583,"style":228},[169],[18,30585,30587,30590,30593,30596,30599],{"className":30586},[97],[18,30588],{"className":30589,"style":242},[101],[18,30591,47],{"className":30592},[106],[18,30594],{"className":30595,"style":228},[169],[18,30597,11225],{"className":30598},[232],[18,30600],{"className":30601,"style":228},[169],[18,30603,30605,30608,30611,30614],{"className":30604},[97],[18,30606],{"className":30607,"style":242},[101],[18,30609,43],{"className":30610,"style":111},[106,110],[18,30612,84],{"className":30613},[267],[18,30615,968],{"className":30616},[106],[11,30618,30619,30620,30648,30649,30720,30721,509,30806,30891,30892,4741],{},"固定 ",[18,30621,30623,30636],{"className":30622,"translate":22},[21],[18,30624,30626],{"className":30625},[26],[28,30627,30628],{"xmlns":30},[32,30629,30630,30634],{},[35,30631,30632],{},[41,30633,19811],{},[86,30635,19811],{"encoding":88},[18,30637,30639],{"className":30638,"ariaHidden":51},[93],[18,30640,30642,30645],{"className":30641},[97],[18,30643],{"className":30644,"style":28340},[101],[18,30646,19811],{"className":30647,"style":19928},[106,110],"。边际后验 ",[18,30650,30652,30670],{"className":30651,"translate":22},[21],[18,30653,30655],{"className":30654},[26],[28,30656,30657],{"xmlns":30},[32,30658,30659,30667],{},[35,30660,30661],{},[38,30662,30663,30665],{},[41,30664,3941],{},[41,30666,19811],{},[86,30668,30669],{"encoding":88},"\\pi_j",[18,30671,30673],{"className":30672,"ariaHidden":51},[93],[18,30674,30676,30680],{"className":30675},[97],[18,30677],{"className":30678,"style":30679},[101],"height:0.7167em;vertical-align:-0.2861em;",[18,30681,30683,30686],{"className":30682},[106],[18,30684,3941],{"className":30685,"style":3957},[106,110],[18,30687,30689],{"className":30688},[115],[18,30690,30692,30712],{"className":30691},[119,120],[18,30693,30695,30709],{"className":30694},[124],[18,30696,30698],{"className":30697,"style":5921},[128],[18,30699,30700,30703],{"style":30350},[18,30701],{"className":30702,"style":137},[136],[18,30704,30706],{"className":30705},[141,142,143,144],[18,30707,19811],{"className":30708,"style":19928},[106,110,144],[18,30710,152],{"className":30711},[151],[18,30713,30715],{"className":30714},[124],[18,30716,30718],{"className":30717,"style":19938},[128],[18,30719],{}," 正是 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在均匀先验下的后验概率。对任意 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3.1 的检验形式。于是每个坐标贡献至少 ",[18,31701,31703,31739],{"className":31702,"translate":22},[21],[18,31704,31706],{"className":31705},[26],[28,31707,31708],{"xmlns":30},[32,31709,31710,31736],{},[35,31711,31712,31718,31720,31722,31724,31734],{},[5654,31713,31714,31716],{},[45,31715,47],{},[45,31717,10447],{},[49,31719,74],{"stretchy":73},[45,31721,47],{},[49,31723,2253],{},[38,31725,31726,31732],{},[35,31727,31728,31730],{},[41,31729,5703],{"mathvariant":316},[41,31731,5706],{"mathvariant":316},[41,31733,19811],{},[49,31735,84],{"stretchy":73},[86,31737,31738],{"encoding":88},"\\frac14(1-\\mathrm{TV}_j)",[18,31740,31742,31831],{"className":31741,"ariaHidden":51},[93],[18,31743,31745,31748,31816,31819,31822,31825,31828],{"className":31744},[97],[18,31746],{"className":31747,"style":12026},[101],[18,31749,31751,31754,31813],{"className":31750},[106],[18,31752],{"className":31753},[250,6012],[18,31755,31757],{"className":31756},[5654],[18,31758,31760,31805],{"className":31759},[119,120],[18,31761,31763,31802],{"className":31762},[124],[18,31764,31766,31780,31788],{"className":31765,"style":12045},[128],[18,31767,31768,31771],{"style":12048},[18,31769],{"className":31770,"style":697},[136],[18,31772,31774],{"className":31773},[141,142,143,144],[18,31775,31777],{"className":31776},[106,144],[18,31778,10447],{"className":31779},[106,144],[18,31781,31782,31785],{"style":6040},[18,31783],{"className":31784,"style":697},[136],[18,31786],{"className":31787,"style":6048},[6047],[18,31789,31790,31793],{"style":12071},[18,31791],{"className":31792,"style":697},[136],[18,31794,31796],{"className":31795},[141,142,143,144],[18,31797,31799],{"className":31798},[106,144],[18,31800,47],{"className":31801},[106,144],[18,31803,152],{"className":31804},[151],[18,31806,31808],{"className":31807},[124],[18,31809,31811],{"className":31810,"style":12093},[128],[18,31812],{},[18,31814],{"className":31815},[267,6012],[18,31817,74],{"className":31818},[250],[18,31820,47],{"className":31821},[106],[18,31823],{"className":31824,"style":2068},[169],[18,31826,2253],{"className":31827},[2072],[18,31829],{"className":31830,"style":2068},[169],[18,31832,31834,31837,31881],{"className":31833},[97],[18,31835],{"className":31836,"style":19846},[101],[18,31838,31840,31846],{"className":31839},[106],[18,31841,31843],{"className":31842},[106],[18,31844,6356],{"className":31845,"style":6355},[106,6354],[18,31847,31849],{"className":31848},[115],[18,31850,31852,31873],{"className":31851},[119,120],[18,31853,31855,31870],{"className":31854},[124],[18,31856,31858],{"className":31857,"style":5921},[128],[18,31859,31861,31864],{"style":31860},"top:-2.55em;margin-right:0.05em;",[18,31862],{"className":31863,"style":137},[136],[18,31865,31867],{"className":31866},[141,142,143,144],[18,31868,19811],{"className":31869,"style":19928},[106,110,144],[18,31871,152],{"className":31872},[151],[18,31874,31876],{"className":31875},[124],[18,31877,31879],{"className":31878,"style":19938},[128],[18,31880],{},[18,31882,84],{"className":31883},[267],"，求和得",[18,31886,31888],{"className":31887,"translate":22},[875],[18,31889,31891,31977],{"className":31890,"translate":22},[21],[18,31892,31894],{"className":31893},[26],[28,31895,31896],{"xmlns":30,"display":884},[32,31897,31898,31974],{},[35,31899,31900,31906,31908,31914,31972],{},[38,31901,31902,31904],{},[41,31903,3978],{"mathvariant":3525},[41,31905,64],{},[49,31907,4775],{},[5654,31909,31910,31912],{},[41,31911,28169],{},[45,31913,10447],{},[35,31915,31916,31918,31920,31922,31932,31934,31946,31948,31960,31970],{},[49,31917,74],{"fence":51},[45,31919,47],{},[49,31921,2253],{},[890,31923,31924,31930],{},[35,31925,31926,31928],{},[41,31927,5593],{},[49,31929,899],{},[41,31931,19811],{},[49,31933,2248],{"stretchy":73},[38,31935,31936,31938],{},[41,31937,12547],{},[35,31939,31940,31942,31944],{},[41,31941,19811],{},[49,31943,52],{"separator":51},[45,31945,430],{},[49,31947,2253],{},[38,31949,31950,31952],{},[41,31951,12547],{},[35,31953,31954,31956,31958],{},[41,31955,19811],{},[49,31957,52],{"separator":51},[45,31959,47],{},[38,31961,31962,31964],{},[49,31963,2248],{"stretchy":73},[35,31965,31966,31968],{},[41,31967,5703],{"mathvariant":316},[41,31969,5706],{"mathvariant":316},[49,31971,84],{"fence":51},[41,31973,968],{"mathvariant":316},[86,31975,31976],{"encoding":88},"\\mathfrak B_n\\ge\\frac m4\\left(1-\\max_j\\lVert Q_{j,0}-Q_{j,1}\\rVert_{\\mathrm{TV}}\\right).",[18,31978,31980,32036],{"className":31979,"ariaHidden":51},[93],[18,31981,31983,31987,32027,32030,32033],{"className":31982},[97],[18,31984],{"className":31985,"style":31986},[101],"height:0.8414em;vertical-align:-0.15em;",[18,31988,31990,31993],{"className":31989},[106],[18,31991,3978],{"className":31992},[106,3607],[18,31994,31996],{"className":31995},[115],[18,31997,31999,32019],{"className":31998},[119,120],[18,32000,32002,32016],{"className":32001},[124],[18,32003,32005],{"className":32004,"style":202},[128],[18,32006,32007,32010],{"style":1155},[18,32008],{"className":32009,"style":137},[136],[18,32011,32013],{"className":32012},[141,142,143,144],[18,32014,64],{"className":32015},[106,110,144],[18,32017,152],{"className":32018},[151],[18,32020,32022],{"className":32021},[124],[18,32023,32025],{"className":32024,"style":159},[128],[18,32026],{},[18,32028],{"className":32029,"style":228},[169],[18,32031,4775],{"className":32032},[232],[18,32034],{"className":32035,"style":228},[169],[18,32037,32039,32042,32104,32107,32334,32337],{"className":32038},[97],[18,32040],{"className":32041,"style":20807},[101],[18,32043,32045,32048,32101],{"className":32044},[106],[18,32046],{"className":32047},[250,6012],[18,32049,32051],{"className":32050},[5654],[18,32052,32054,32093],{"className":32053},[119,120],[18,32055,32057,32090],{"className":32056},[124],[18,32058,32060,32071,32079],{"className":32059,"style":13852},[128],[18,32061,32062,32065],{"style":6028},[18,32063],{"className":32064,"style":697},[136],[18,32066,32068],{"className":32067},[106],[18,32069,10447],{"className":32070},[106],[18,32072,32073,32076],{"style":6040},[18,32074],{"className":32075,"style":697},[136],[18,32077],{"className":32078,"style":6048},[6047],[18,32080,32081,32084],{"style":6051},[18,32082],{"className":32083,"style":697},[136],[18,32085,32087],{"className":32086},[106],[18,32088,28169],{"className":32089},[106,110],[18,32091,152],{"className":32092},[151],[18,32094,32096],{"className":32095},[124],[18,32097,32099],{"className":32098,"style":6070},[128],[18,32100],{},[18,32102],{"className":32103},[267,6012],[18,32105],{"className":32106,"style":170},[169],[18,32108,32110,32116,32119,32122,32125,32128,32172,32175,32224,32227,32230,32233,32282,32328],{"className":32109},[174],[18,32111,32113],{"className":32112,"style":13796},[250,13795],[18,32114,74],{"className":32115},[1230,143],[18,32117,47],{"className":32118},[106],[18,32120],{"className":32121,"style":2068},[169],[18,32123,2253],{"className":32124},[2072],[18,32126],{"className":32127,"style":2068},[169],[18,32129,32131],{"className":32130},[985,986],[18,32132,32134,32164],{"className":32133},[119,120],[18,32135,32137,32161],{"className":32136},[124],[18,32138,32140,32151],{"className":32139,"style":1083},[128],[18,32141,32142,32145],{"style":5738},[18,32143],{"className":32144,"style":697},[136],[18,32146,32148],{"className":32147},[141,142,143,144],[18,32149,19811],{"className":32150,"style":19928},[106,110,144],[18,32152,32153,32156],{"style":693},[18,32154],{"className":32155,"style":697},[136],[18,32157,32158],{},[18,32159,5593],{"className":32160},[985],[18,32162,152],{"className":32163},[151],[18,32165,32167],{"className":32166},[124],[18,32168,32170],{"className":32169,"style":5782},[128],[18,32171],{},[18,32173,2248],{"className":32174},[250],[18,32176,32178,32181],{"className":32177},[106],[18,32179,12547],{"className":32180},[106,110],[18,32182,32184],{"className":32183},[115],[18,32185,32187,32216],{"className":32186},[119,120],[18,32188,32190,32213],{"className":32189},[124],[18,32191,32193],{"className":32192,"style":5921},[128],[18,32194,32195,32198],{"style":1155},[18,32196],{"className":32197,"style":137},[136],[18,32199,32201],{"className":32200},[141,142,143,144],[18,32202,32204,32207,32210],{"className":32203},[106,144],[18,32205,19811],{"className":32206,"style":19928},[106,110,144],[18,32208,52],{"className":32209},[165,144],[18,32211,430],{"className":32212},[106,144],[18,32214,152],{"className":32215},[151],[18,32217,32219],{"className":32218},[124],[18,32220,32222],{"className":32221,"style":19938},[128],[18,32223],{},[18,32225],{"className":32226,"style":2068},[169],[18,32228,2253],{"className":32229},[2072],[18,32231],{"className":32232,"style":2068},[169],[18,32234,32236,32239],{"className":32235},[106],[18,32237,12547],{"className":32238},[106,110],[18,32240,32242],{"className":32241},[115],[18,32243,32245,32274],{"className":32244},[119,120],[18,32246,32248,32271],{"className":32247},[124],[18,32249,32251],{"className":32250,"style":5921},[128],[18,32252,32253,32256],{"style":1155},[18,32254],{"className":32255,"style":137},[136],[18,32257,32259],{"className":32258},[141,142,143,144],[18,32260,32262,32265,32268],{"className":32261},[106,144],[18,32263,19811],{"className":32264,"style":19928},[106,110,144],[18,32266,52],{"className":32267},[165,144],[18,32269,47],{"className":32270},[106,144],[18,322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逐坐标下界",[11,32348,32349,32350,32378,32379,32407,32408,32436,32437,32465,32466,32494],{},"Assouad 的威力在于它把 ",[18,32351,32353,32366],{"className":32352,"translate":22},[21],[18,32354,32356],{"className":32355},[26],[28,32357,32358],{"xmlns":30},[32,32359,32360,32364],{},[35,32361,32362],{},[41,32363,28169],{},[86,32365,28169],{"encoding":88},[18,32367,32369],{"className":32368,"ariaHidden":51},[93],[18,32370,32372,32375],{"className":32371},[97],[18,32373],{"className":32374,"style":1083},[101],[18,32376,28169],{"className":32377},[106,110]," 维问题拆成了 ",[18,32380,32382,32395],{"className":32381,"translate":22},[21],[18,32383,32385],{"className":32384},[26],[28,32386,32387],{"xmlns":30},[32,32388,32389,32393],{},[35,32390,32391],{},[41,32392,28169],{},[86,32394,28169],{"encoding":88},[18,32396,32398],{"className":32397,"ariaHidden":51},[93],[18,32399,32401,32404],{"className":32400},[97],[18,32402],{"className":32403,"style":1083},[101],[18,32405,28169],{"className":32406},[106,110]," 个一维问题。只要每个坐标上的两个混合分布足够接近，",[18,32409,32411,32424],{"className":32410,"translate":22},[21],[18,32412,32414],{"className":32413},[26],[28,32415,32416],{"xmlns":30},[32,32417,32418,32422],{},[35,32419,32420],{},[41,32421,28169],{},[86,32423,28169],{"encoding":88},[18,32425,32427],{"className":32426,"ariaHidden":51},[93],[18,32428,32430,32433],{"className":32429},[97],[18,32431],{"className":32432,"style":1083},[101],[18,32434,28169],{"className":32435},[106,110]," 个坐标就会贡献 ",[18,32438,32440,32453],{"className":32439,"translate":22},[21],[18,32441,32443],{"className":32442},[26],[28,32444,32445],{"xmlns":30},[32,32446,32447,32451],{},[35,32448,32449],{},[41,32450,28169],{},[86,32452,28169],{"encoding":88},[18,32454,32456],{"className":32455,"ariaHidden":51},[93],[18,32457,32459,32462],{"className":32458},[97],[18,32460],{"className":32461,"style":1083},[101],[18,32463,28169],{"className":32464},[106,110]," 倍的下界——这正是高维速率里 ",[18,32467,32469,32482],{"className":32468,"translate":22},[21],[18,32470,32472],{"className":32471},[26],[28,32473,32474],{"xmlns":30},[32,32475,32476,32480],{},[35,32477,32478],{},[41,32479,4031],{},[86,32481,4031],{"encoding":88},[18,32483,32485],{"className":32484,"ariaHidden":51},[93],[18,32486,32488,32491],{"className":32487},[97],[18,32489],{"className":32490,"style":295},[101],[18,32492,4031],{"className":32493},[106,110]," 的来源。",[11,32496,32497,32498,32603,32604,32696,32697,32835],{},"对 ",[18,32499,32501,32532],{"className":32500,"translate":22},[21],[18,32502,32504],{"className":32503},[26],[28,32505,32506],{"xmlns":30},[32,32507,32508,32529],{},[35,32509,32510,32512,32514,32516,32518,32520,32523],{},[41,32511,924],{"mathvariant":316},[49,32513,433],{},[49,32515,1331],{"stretchy":73},[45,32517,430],{},[49,32519,52],{"separator":51},[41,32521,32522],{},"ε",[1556,32524,32525,32527],{},[49,32526,1349],{"stretchy":73},[41,32528,28169],{},[86,32530,32531],{"encoding":88},"\\Theta=\\{0,\\varepsilon\\}^m",[18,32533,32535,32553],{"className":32534,"ariaHidden":51},[93],[18,32536,32538,32541,32544,32547,32550],{"className":32537},[97],[18,32539],{"className":32540,"style":600},[101],[18,32542,924],{"className":32543},[106],[18,32545],{"className":32546,"style":228},[169],[18,32548,433],{"className":32549},[232],[18,32551],{"className":32552,"style":228},[169],[18,32554,32556,32559,32562,32565,32568,32571,32574],{"className":32555},[97],[18,32557],{"className":32558,"style":242},[101],[18,32560,1331],{"className":32561},[250],[18,32563,430],{"className":32564},[106],[18,32566,52],{"className":32567},[165],[18,32569],{"className":32570,"style":170},[169],[18,32572,32522],{"className":32573},[106,110],[18,32575,32577,32580],{"className":32576},[267],[18,32578,1349],{"className":32579},[267],[18,32581,32583],{"className":32582},[115],[18,32584,32586],{"className":32585},[119],[18,32587,32589],{"className":32588},[124],[18,32590,32592],{"className":32591,"style":1959},[128],[18,32593,32594,32597],{"style":1962},[18,32595],{"className":32596,"style":137},[136],[18,32598,32600],{"className":32599},[141,142,143,144],[18,32601,28169],{"className":32602},[106,110,144],"，令 ",[18,32605,32607,32633],{"className":32606,"translate":22},[21],[18,32608,32610],{"className":32609},[26],[28,32611,32612],{"xmlns":30},[32,32613,32614,32630],{},[35,32615,32616,32618,32620,32622],{},[41,32617,77],{},[49,32619,433],{},[41,32621,32522],{},[1556,32623,32624,32626],{},[41,32625,77],{},[49,32627,32629],{"mathvariant":316,"lspace":32628,"rspace":32628},"0em","′",[86,32631,32632],{"encoding":88},"\\theta=\\varepsilon\\theta'",[18,32634,32636,32654],{"className":32635,"ariaHidden":51},[93],[18,32637,32639,32642,32645,32648,32651],{"className":32638},[97],[18,32640],{"className":32641,"style":295},[101],[18,32643,77],{"className":32644,"style":254},[106,110],[18,32646],{"className":32647,"style":228},[169],[18,32649,433],{"className":32650},[232],[18,32652],{"className":32653,"style":228},[169],[18,32655,32657,32661,32664],{"className":32656},[97],[18,32658],{"className":32659,"style":32660},[101],"height:0.7519em;",[18,32662,32522],{"className":32663},[106,110],[18,32665,32667,32670],{"className":32666},[106],[18,32668,77],{"className":32669,"style":254},[106,110],[18,32671,32673],{"className":32672},[115],[18,32674,32676],{"className":32675},[119],[18,32677,32679],{"className":32678},[124],[18,32680,32682],{"className":32681,"style":32660},[128],[18,32683,32684,32687],{"style":1962},[18,32685],{"className":32686,"style":137},[136],[18,32688,32690],{"className":32689},[141,142,143,144],[18,32691,32693],{"className":32692},[106,144],[18,32694,32629],{"className":32695},[106,144],"，其中 ",[18,32698,32700,32734],{"className":32699,"translate":22},[21],[18,32701,32703],{"className":32702},[26],[28,32704,32705],{"xmlns":30},[32,32706,32707,32731],{},[35,32708,32709,32715,32717,32719,32721,32723,32725],{},[1556,32710,32711,32713],{},[41,32712,77],{},[49,32714,32629],{"mathvariant":316,"lspace":32628,"rspace":32628},[49,32716,921],{},[49,32718,1331],{"stretchy":73},[45,32720,430],{},[49,32722,52],{"separator":51},[45,32724,47],{},[1556,32726,32727,32729],{},[49,32728,1349],{"stretchy":73},[41,32730,28169],{},[86,32732,32733],{"encoding":88},"\\theta'\\in\\{0,1\\}^m",[18,32735,32737,32785],{"className":32736,"ariaHidden":51},[93],[18,32738,32740,32744,32776,32779,32782],{"className":32739},[97],[18,32741],{"className":32742,"style":32743},[101],"height:0.791em;vertical-align:-0.0391em;",[18,32745,32747,32750],{"className":32746},[106],[18,32748,77],{"className":32749,"style":254},[106,110],[18,32751,32753],{"className":32752},[115],[18,32754,32756],{"className":32755},[119],[18,32757,32759],{"className":32758},[124],[18,32760,32762],{"className":32761,"style":32660},[128],[18,32763,32764,32767],{"style":1962},[18,32765],{"className":32766,"style":137},[136],[18,32768,32770],{"className":32769},[141,142,143,144],[18,32771,32773],{"className":32772},[106,144],[18,32774,32629],{"className":32775},[106,144],[18,32777],{"className":32778,"style":228},[169],[18,32780,921],{"className":32781},[232],[18,32783],{"className":32784,"style":228},[169],[18,32786,32788,32791,32794,32797,32800,32803,32806],{"className":32787},[97],[18,32789],{"className":32790,"style":242},[101],[18,32792,1331],{"className":32793},[250],[18,32795,430],{"className":32796},[106],[18,32798,52],{"className":32799},[165],[18,32801],{"className":32802,"style":170},[169],[18,32804,47],{"className":32805},[106],[18,32807,32809,32812],{"className":32808},[267],[18,32810,1349],{"className":32811},[267],[18,32813,32815],{"className":32814},[115],[18,32816,32818],{"className":32817},[119],[18,32819,32821],{"className":32820},[124],[18,32822,32824],{"className":32823,"style":1959},[128],[18,32825,32826,32829],{"style":1962},[18,32827],{"className":32828,"style":137},[136],[18,32830,32832],{"className":32831},[141,142,143,144],[18,32833,28169],{"className":32834},[106,110,144],"。平方损失满足",[18,32837,32839],{"className":32838,"translate":22},[875],[18,32840,32842,32912],{"className":32841,"translate":22},[21],[18,32843,32845],{"className":32844},[26],[28,32846,32847],{"xmlns":30,"display":884},[32,32848,32849,32909],{},[35,32850,32851,32853,32859,32861,32863,32871,32873,32879,32881,32887,32889,32891,32893,32899,32907],{},[49,32852,2248],{"stretchy":73},[658,32854,32855,32857],{"accent":51},[41,32856,77],{},[49,32858,664],{},[49,32860,2253],{},[41,32862,77],{},[1589,32864,32865,32867,32869],{},[49,32866,2248],{"stretchy":73},[45,32868,2266],{},[45,32870,2266],{},[49,32872,433],{},[1556,32874,32875,32877],{},[41,32876,32522],{},[45,32878,2266],{},[49,32880,2248],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[658,32882,32883,32885],{"accent":51},[41,32884,77],{},[49,32886,664],{},[41,32888,317],{"mathvariant":316},[41,32890,32522],{},[49,32892,2253],{},[1556,32894,32895,32897],{},[41,32896,77],{},[49,32898,32629],{"mathvariant":316,"lspace":32628,"rspace":32628},[1589,32900,32901,32903,32905],{},[49,32902,2248],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[45,32904,2266],{},[45,32906,2266],{},[49,32908,52],{"separator":51},[86,32910,32911],{"encoding":88},"\\lVert\\hat\\theta-\\theta\\rVert_2^2\n=\\varepsilon^2\\bigl\\lVert\\hat\\theta\u002F\\varepsilon-\\theta'\\bigr\\rVert_2^2,",[18,32913,32915,32964,33033,33151],{"className":32914,"ariaHidden":51},[93],[18,32916,32918,32921,32924,32955,32958,32961],{"className":32917},[97],[18,32919],{"className":32920,"style":2281},[101],[18,32922,2248],{"className":32923},[250],[18,32925,32927],{"className":32926},[106,681],[18,32928,32930],{"className":32929},[119],[18,32931,32933],{"className":32932},[124],[18,32934,32936,32944],{"className":32935,"style":677},[128],[18,32937,32938,32941],{"style":693},[18,32939],{"className":32940,"style":697},[136],[18,32942,77],{"className":32943,"style":254},[106,110],[18,32945,32946,32949],{"style":703},[18,32947],{"className":32948,"style":697},[136],[18,32950,32952],{"className":32951,"style":711},[710],[18,32953,664],{"className":32954},[106],[18,32956],{"className":32957,"style":2068},[169],[18,32959,2253],{"className":32960},[2072],[18,32962],{"className":32963,"style":2068},[169],[18,32965,32967,32970,32973,33024,33027,33030],{"className":32966},[97],[18,32968],{"className":32969,"style":15256},[101],[18,32971,77],{"className":32972,"style":254},[106,110],[18,32974,32976,32979],{"className":32975},[267],[18,32977,2248],{"className":32978},[267],[18,32980,32982],{"className":32981},[115],[18,32983,32985,33016],{"className":32984},[119,120],[18,32986,32988,33013],{"className":32987},[124],[18,32989,32991,33002],{"className":32990,"style":10775},[128],[18,32992,32993,32996],{"style":10778},[18,32994],{"className":32995,"style":137},[136],[18,32997,32999],{"className":32998},[141,142,143,144],[18,33000,2266],{"className":33001},[106,144],[18,33003,33004,33007],{"style":1637},[18,33005],{"className":33006,"style":137},[136],[18,33008,33010],{"className":33009},[141,142,143,144],[18,33011,2266],{"className":33012},[106,144],[18,33014,152],{"className":33015},[151],[18,33017,33019],{"className":33018},[124],[18,33020,33022],{"className":33021,"style":10808},[128],[18,33023],{},[18,33025],{"className":33026,"style":228},[169],[18,33028,433],{"className":33029},[232],[18,33031],{"className":33032,"style":228},[169],[18,33034,33036,33039,33068,33105,33136,33139,33142,33145,33148],{"className":33035},[97],[18,33037],{"className":33038,"style":23347},[101],[18,33040,33042,33045],{"className":33041},[106],[18,33043,32522],{"className":33044},[106,110],[18,33046,33048],{"className":33047},[115],[18,33049,33051],{"className":33050},[119],[18,33052,33054],{"className":33053},[124],[18,33055,33057],{"className":33056,"style":10775},[128],[18,33058,33059,33062],{"style":1637},[18,33060],{"className":33061,"style":137},[136],[18,33063,33065],{"className":33064},[141,142,143,144],[18,33066,2266],{"className":33067},[106,144],[18,33069,33071],{"className":33070},[250],[18,33072,33074],{"className":33073},[1230,6108],[18,33075,33077,33097],{"className":33076},[119,120],[18,33078,33080,33094],{"className":33079},[124],[18,33081,33083],{"className":33082,"style":6118},[128],[18,33084,33085,33088],{"style":6121},[18,33086],{"className":33087,"style":6125},[136],[18,33089,33090],{"style":6128},[6130,33091,33092],{"xmlns":6132,"width":6133,"height":944,"viewBox":6134},[6136,33093],{"d":6138},[18,33095,152],{"className":33096},[151],[18,33098,33100],{"className":33099},[124],[18,33101,33103],{"className":33102,"style":6148},[128],[18,33104],{},[18,33106,33108],{"className":33107},[106,681],[18,33109,33111],{"className":33110},[119],[18,33112,33114],{"className":33113},[124],[18,33115,33117,33125],{"className":33116,"style":677},[128],[18,33118,33119,33122],{"style":693},[18,33120],{"className":33121,"style":697},[136],[18,33123,77],{"className":33124,"style":254},[106,110],[18,33126,33127,33130],{"style":703},[18,33128],{"className":33129,"style":697},[136],[18,33131,33133],{"className":33132,"style":711},[710],[18,33134,664],{"className":33135},[106],[18,33137,317],{"className":33138},[106],[18,33140,32522],{"className":33141},[106,110],[18,33143],{"className":33144,"style":2068},[169],[18,33146,2253],{"className":33147},[2072],[18,33149],{"className":33150,"style":2068},[169],[18,33152,33154,33158,33191,33278],{"className":33153},[97],[18,33155],{"className":33156,"style":33157},[101],"height:1.4537em;vertical-align:-0.3997em;",[18,33159,33161,33164],{"className":33160},[106],[18,33162,77],{"className":33163,"style":254},[106,110],[18,33165,33167],{"className":33166},[115],[18,33168,33170],{"className":33169},[119],[18,33171,33173],{"className":33172},[124],[18,33174,33177],{"className":33175,"style":33176},[128],"height:0.8019em;",[18,33178,33179,33182],{"style":1637},[18,33180],{"className":33181,"style":137},[136],[18,33183,33185],{"className":33184},[141,142,143,144],[18,33186,33188],{"className":33187},[106,144],[18,33189,32629],{"className":33190},[106,144],[18,33192,33194,33231],{"className":33193},[267],[18,33195,33197],{"className":33196},[267],[18,33198,33200],{"className":33199},[1230,6108],[18,33201,33203,33223],{"className":33202},[119,120],[18,33204,33206,33220],{"className":33205},[124],[18,33207,33209],{"className":33208,"style":6118},[128],[18,33210,33211,33214],{"style":6121},[18,33212],{"className":33213,"style":6125},[136],[18,33215,33216],{"style":6128},[6130,33217,33218],{"xmlns":6132,"width":6133,"height":944,"viewBox":6134},[6136,33219],{"d":6138},[18,33221,152],{"className":33222},[151],[18,33224,33226],{"className":33225},[124],[18,33227,33229],{"className":33228,"style":6148},[128],[18,33230],{},[18,33232,33234],{"className":33233},[115],[18,33235,33237,33270],{"className":33236},[119,120],[18,33238,33240,33267],{"className":33239},[124],[18,33241,33244,33255],{"className":33242,"style":33243},[128],"height:1.054em;",[18,33245,33246,33249],{"style":6338},[18,33247],{"className":33248,"style":137},[136],[18,33250,33252],{"className":33251},[141,142,143,144],[18,33253,2266],{"className":33254},[106,144],[18,33256,33258,33261],{"style":33257},"top:-3.3029em;margin-right:0.05em;",[18,33259],{"className":33260,"style":137},[136],[18,33262,33264],{"className":33263},[141,142,143,144],[18,33265,2266],{"className":33266},[106,144],[18,33268,152],{"className":33269},[151],[18,33271,33273],{"className":33272},[124],[18,33274,33276],{"className":33275,"style":6366},[128],[18,33277],{},[18,33279,52],{"className":33280},[165],[11,33282,33283],{},"所以定理 5.1 直接给出",[18,33285,33287],{"className":33286,"translate":22},[875],[18,33288,33290,33442],{"className":33289,"translate":22},[21],[18,33291,33293],{"className":33292},[26],[28,33294,33295],{"xmlns":30,"display":884},[32,33296,33297,33439],{},[35,33298,33299,33313,33341,33343,33345,33351,33353,33355,33363,33365,33379,33437],{},[890,33300,33301,33307],{},[35,33302,33303,33305],{},[41,33304,896],{},[49,33306,899],{},[658,33308,33309,33311],{"accent":51},[41,33310,77],{},[49,33312,664],{},[890,33314,33315,33321],{},[35,33316,33317,33319],{},[41,33318,5593],{},[49,33320,899],{},[35,33322,33323,33325,33327,33329,33331,33333,33335],{},[41,33324,77],{},[49,33326,921],{},[49,33328,1331],{"stretchy":73},[45,33330,430],{},[49,33332,52],{"separator":51},[41,33334,32522],{},[1556,33336,33337,33339],{},[49,33338,1349],{"stretchy":73},[41,33340,28169],{},[41,33342,934],{"mathvariant":933},[49,33344,2248],{"stretchy":73},[658,33346,33347,33349],{"accent":51},[41,33348,77],{},[49,33350,664],{},[49,33352,2253],{},[41,33354,77],{},[1589,33356,33357,33359,33361],{},[49,33358,2248],{"stretchy":73},[45,33360,2266],{},[45,33362,2266],{},[49,33364,4775],{},[5654,33366,33367,33377],{},[35,33368,33369,33375],{},[1556,33370,33371,33373],{},[41,33372,32522],{},[45,33374,2266],{},[41,33376,28169],{},[45,33378,10447],{},[35,33380,33381,33383,33385,33387,33397,33399,33411,33413,33425,33435],{},[49,33382,74],{"fence":51},[45,33384,47],{},[49,33386,2253],{},[890,33388,33389,33395],{},[35,33390,33391,33393],{},[41,33392,5593],{},[49,33394,899],{},[41,33396,19811],{},[49,33398,2248],{"stretchy":73},[38,33400,33401,33403],{},[41,33402,12547],{},[35,33404,33405,33407,33409],{},[41,33406,19811],{},[49,33408,52],{"separator":51},[45,33410,430],{},[49,33412,2253],{},[38,33414,33415,33417],{},[41,33416,12547],{},[35,33418,33419,33421,33423],{},[41,33420,19811],{},[49,33422,52],{"separator":51},[45,33424,47],{},[38,33426,33427,33429],{},[49,33428,2248],{"stretchy":73},[35,33430,33431,33433],{},[41,33432,5703],{"mathvariant":316},[41,33434,5706],{"mathvariant":316},[49,33436,84],{"fence":51},[41,33438,968],{"mathvariant":316},[86,33440,33441],{"encoding":88},"\\inf_{\\hat\\theta}\\max_{\\theta\\in\\{0,\\varepsilon\\}^m}\\mathbb E\\lVert\\hat\\theta-\\theta\\rVert_2^2\n\\ge\n\\frac{\\varepsilon^2m}{4}\\left(1-\\max_j\\lVert 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高斯均值：",[18,34123,34125,34148],{"className":34124,"translate":22},[21],[18,34126,34128],{"className":34127},[26],[28,34129,34130],{"xmlns":30},[32,34131,34132,34146],{},[35,34133,34134,34140,34142,34144],{},[1556,34135,34136,34138],{},[41,34137,16406],{},[45,34139,2266],{},[41,34141,4031],{},[41,34143,317],{"mathvariant":316},[41,34145,64],{},[86,34147,17299],{"encoding":88},[18,34149,34151],{"className":34150,"ariaHidden":51},[93],[18,34152,34154,34157,34186,34189,34192],{"className":34153},[97],[18,34155],{"className":34156,"style":10271},[101],[18,34158,34160,34163],{"className":34159},[106],[18,34161,16406],{"className":34162,"style":3957},[106,110],[18,34164,34166],{"className":34165},[115],[18,34167,34169],{"className":34168},[119],[18,34170,34172],{"className":34171},[124],[18,34173,34175],{"className":34174,"style":2416},[128],[18,34176,34177,34180],{"style":1962},[18,34178],{"className":34179,"style":137},[136],[18,34181,34183],{"className":34182},[141,142,143,144],[18,34184,2266],{"className":34185},[106,144],[18,34187,4031],{"className":34188},[106,110],[18,34190,317],{"className":34191},[106],[18,34193,64],{"className":34194},[106,110],[11,34196,34197],{},"模型：",[18,34199,34201],{"className":34200,"translate":22},[875],[18,34202,34204,34274],{"className":34203,"translate":22},[21],[18,34205,34207],{"className":34206},[26],[28,34208,34209],{"xmlns":30,"display":884},[32,34210,34211,34271],{},[35,34212,34213,34219,34221,34223,34225,34231,34233,34235,34237,34239,34241,34247,34253,34255,34257,34259,34261,34263,34269],{},[38,34214,34215,34217],{},[41,34216,43],{},[45,34218,47],{},[49,34220,52],{"separator":51},[49,34222,55],{},[49,34224,52],{"separator":51},[38,34226,34227,34229],{},[41,34228,43],{},[41,34230,64],{},[49,34232,67],{},[41,34234,70],{},[49,34236,74],{"stretchy":73},[41,34238,77],{},[49,34240,52],{"separator":51},[1556,34242,34243,34245],{},[41,34244,16406],{},[45,34246,2266],{},[38,34248,34249,34251],{},[41,34250,16413],{},[41,34252,4031],{},[49,34254,84],{"stretchy":73},[49,34256,52],{"separator":51},[169,34258],{"width":7344},[41,34260,77],{},[49,34262,921],{},[1556,34264,34265,34267],{},[41,34266,3071],{"mathvariant":933},[41,34268,4031],{},[49,34270,52],{"separator":51},[86,34272,34273],{"encoding":88},"X_1,\\dots,X_n\\sim N(\\theta,\\sigma^2 I_d),\\qquad \\theta\\in\\mathbb R^d,",[18,34275,34277,34390,34504],{"className":34276,"ariaHidden":51},[93],[18,34278,34280,34283,34323,34326,34329,34332,34335,34338,34341,34381,34384,34387],{"className":34279},[97],[18,34281],{"className":34282,"style":102},[101],[18,34284,34286,34289],{"className":34285},[106],[18,34287,43],{"className":34288,"style":111},[106,110],[18,34290,34292],{"className":34291},[115],[18,34293,34295,34315],{"className":34294},[119,120],[18,34296,34298,34312],{"className":34297},[124],[18,34299,34301],{"className":34300,"style":129},[128],[18,34302,34303,34306],{"style":132},[18,34304],{"className":34305,"style":137},[136],[18,34307,34309],{"className":34308},[141,142,143,144],[18,34310,47],{"className":34311},[106,144],[18,34313,152],{"className":34314},[151],[18,34316,34318],{"className":34317},[124],[18,34319,34321],{"className":34320,"style":159},[128],[18,34322],{},[18,34324,52],{"className":34325},[165],[18,34327],{"className":34328,"style":170},[169],[18,34330,55],{"className":34331},[174],[18,34333],{"className":34334,"style":170},[169],[18,34336,52],{"className":34337},[165],[18,34339],{"className":34340,"style":170},[169],[18,34342,34344,34347],{"className":34343},[106],[18,34345,43],{"className":34346,"style":111},[106,110],[18,34348,34350],{"className":34349},[115],[18,34351,34353,34373],{"className":34352},[119,120],[18,34354,34356,34370],{"className":34355},[124],[18,34357,34359],{"className":34358,"style":202},[128],[18,34360,34361,34364],{"style":132},[18,34362],{"className":34363,"style":137},[136],[18,34365,34367],{"className":34366},[141,142,143,144],[18,34368,64],{"className":34369},[106,110,144],[18,34371,152],{"className":34372},[151],[18,34374,34376],{"className":34375},[124],[18,34377,34379],{"className":34378,"style":159},[128],[18,34380],{},[18,34382],{"className":34383,"style":228},[169],[18,34385,67],{"className":34386},[232],[18,34388],{"className":34389,"style":228},[169],[18,34391,34393,34396,34399,34402,34405,34408,34411,34440,34480,34483,34486,34489,34492,34495,34498,34501],{"className":34392},[97],[18,34394],{"className":34395,"style":15256},[101],[18,34397,70],{"className":34398,"style":246},[106,110],[18,34400,74],{"className":34401},[250],[18,34403,77],{"className":34404,"style":254},[106,110],[18,34406,52],{"className":34407},[165],[18,34409],{"className":34410,"style":170},[169],[18,34412,34414,34417],{"className":34413},[106],[18,34415,16406],{"className":34416,"style":3957},[106,110],[18,34418,34420],{"className":34419},[115],[18,34421,34423],{"className":34422},[119],[18,34424,34426],{"className":34425},[124],[18,34427,34429],{"className":34428,"style":10775},[128],[18,34430,34431,34434],{"style":1637},[18,34432],{"className":34433,"style":137},[136],[18,34435,34437],{"className":34436},[141,142,143,144],[18,34438,2266],{"className":34439},[106,144],[18,34441,34443,34446],{"className":34442},[106],[18,34444,16413],{"className":34445,"style":111},[106,110],[18,34447,34449],{"className":34448},[115],[18,34450,34452,34472],{"className":34451},[119,120],[18,34453,34455,34469],{"className":34454},[124],[18,34456,34458],{"className":34457,"style":1383},[128],[18,34459,34460,34463],{"style":132},[18,34461],{"className":34462,"style":137},[136],[18,34464,34466],{"className":34465},[141,142,143,144],[18,34467,4031],{"className":34468},[106,110,144],[18,34470,152],{"className":34471},[151],[18,34473,34475],{"className":34474},[124],[18,34476,34478],{"className":34477,"style":159},[128],[18,34479],{},[18,34481,84],{"className":34482},[267],[18,34484,52],{"className":34485},[165],[18,34487],{"className":34488,"style":7676},[169],[18,34490],{"className":34491,"style":170},[169],[18,34493,77],{"className":34494,"style":254},[106,110],[18,34496],{"className":34497,"style":228},[169],[18,34499,921],{"className":34500},[232],[18,34502],{"className":34503,"style":228},[169],[18,34505,34507,34511,34540],{"className":34506},[97],[18,34508],{"className":34509,"style":34510},[101],"height:1.0935em;vertical-align:-0.1944em;",[18,34512,34514,34517],{"className":34513},[106],[18,34515,3071],{"className":34516},[106,1139],[18,34518,34520],{"className":34519},[115],[18,34521,34523],{"className":34522},[119],[18,34524,34526],{"className":34525},[124],[18,34527,34529],{"className":34528,"style":16673},[128],[18,34530,34531,34534],{"style":1637},[18,34532],{"className":34533,"style":137},[136],[18,34535,34537],{"className":34536},[141,142,143,144],[18,34538,4031],{"className":34539},[106,110,144],[18,34541,52],{"className":34542},[165],[11,34544,34545,34546,1990],{},"损失为 ",[18,34547,34549,34581],{"className":34548,"translate":22},[21],[18,34550,34552],{"className":34551},[26],[28,34553,34554],{"xmlns":30},[32,34555,34556,34578],{},[35,34557,34558,34560,34566,34568,34570],{},[49,34559,2248],{"stretchy":73},[658,34561,34562,34564],{"accent":51},[41,34563,77],{},[49,34565,664],{},[49,34567,2253],{},[41,34569,77],{},[1589,34571,34572,34574,34576],{},[49,34573,2248],{"stretchy":73},[45,34575,2266],{},[45,34577,2266],{},[86,34579,34580],{"encoding":88},"\\lVert\\hat\\theta-\\theta\\rVert_2^2",[18,34582,34584,34633],{"className":34583,"ariaHidden":51},[93],[18,34585,34587,34590,34593,34624,34627,34630],{"className":34586},[97],[18,34588],{"className":34589,"style":2281},[101],[18,34591,2248],{"className":34592},[250],[18,34594,34596],{"className":34595},[106,681],[18,34597,34599],{"className":34598},[119],[18,34600,34602],{"className":34601},[124],[18,34603,34605,34613],{"className":34604,"style":677},[128],[18,34606,34607,34610],{"style":693},[18,34608],{"className":34609,"style":697},[136],[18,34611,77],{"className":34612,"style":254},[106,110],[18,34614,34615,34618],{"style":703},[18,34616],{"className":34617,"style":697},[136],[18,34619,34621],{"className":34620,"style":711},[710],[18,34622,664],{"className":34623},[106],[18,34625],{"className":34626,"style":2068},[169],[18,34628,2253],{"className":34629},[2072],[18,34631],{"className":34632,"style":2068},[169],[18,34634,34636,34639,34642],{"className":34635},[97],[18,34637],{"className":34638,"style":10271},[101],[18,34640,77],{"className":34641,"style":254},[106,110],[18,34643,34645,34648],{"className":34644},[267],[18,34646,2248],{"className":34647},[267],[18,34649,34651],{"className":34650},[115],[18,34652,34654,34685],{"className":34653},[119,120],[18,34655,34657,34682],{"className":34656},[124],[18,34658,34660,34671],{"className":34659,"style":2416},[128],[18,34661,34662,34665],{"style":2419},[18,34663],{"className":34664,"style":137},[136],[18,34666,34668],{"className":34667},[141,142,143,144],[18,34669,2266],{"className":34670},[106,144],[18,34672,34673,34676],{"style":1962},[18,34674],{"className":34675,"style":137},[136],[18,34677,34679],{"className":34678},[141,142,143,144],[18,34680,2266],{"className":34681},[106,144],[18,34683,152],{"className":34684},[151],[18,34686,34688],{"className":34687},[124],[18,34689,34691],{"className":34690,"style":2449},[128],[18,34692],{},[11,34694,34695,34698,34699,34760],{},[717,34696,34697],{},"上界。"," 样本均值 ",[18,34700,34702,34720],{"className":34701,"translate":22},[21],[18,34703,34705],{"className":34704},[26],[28,34706,34707],{"xmlns":30},[32,34708,34709,34717],{},[35,34710,34711],{},[658,34712,34713,34715],{"accent":51},[41,34714,43],{},[49,34716,25294],{},[86,34718,34719],{"encoding":88},"\\bar X",[18,34721,34723],{"className":34722,"ariaHidden":51},[93],[18,34724,34726,34729],{"className":34725},[97],[18,34727],{"className":34728,"style":25624},[101],[18,34730,34732],{"className":34731},[106,681],[18,34733,34735],{"className":34734},[119],[18,34736,34738],{"className":34737},[124],[18,34739,34741,34749],{"className":34740,"style":25624},[128],[18,34742,34743,34746],{"style":693},[18,34744],{"className":34745,"style":697},[136],[18,34747,43],{"className":34748,"style":111},[106,110],[18,34750,34751,34754],{"style":17702},[18,34752],{"className":34753,"style":697},[136],[18,34755,34757],{"className":34756,"style":711},[710],[18,34758,25294],{"className":34759},[106]," 的风险是",[18,34762,34764],{"className":34763,"translate":22},[875],[18,34765,34767,34819],{"className":34766,"translate":22},[21],[18,34768,34770],{"className":34769},[26],[28,34771,34772],{"xmlns":30,"display":884},[32,34773,34774,34816],{},[35,34775,34776,34778,34780,34786,34788,34790,34798,34800,34814],{},[41,34777,934],{"mathvariant":933},[49,34779,2248],{"stretchy":73},[658,34781,34782,34784],{"accent":51},[41,34783,43],{},[49,34785,25294],{},[49,34787,2253],{},[41,34789,77],{},[1589,34791,34792,34794,34796],{},[49,34793,2248],{"stretchy":73},[45,34795,2266],{},[45,34797,2266],{},[49,34799,433],{},[5654,34801,34802,34812],{},[35,34803,34804,34810],{},[1556,34805,34806,34808],{},[41,34807,16406],{},[45,34809,2266],{},[41,34811,4031],{},[41,34813,64],{},[41,34815,968],{"mathvariant":316},[86,34817,34818],{"encoding":88},"\\mathbb E\\lVert\\bar X-\\theta\\rVert_2^2=\\frac{\\sigma^2d}{n}.",[18,34820,34822,34875,34944],{"className":34821,"ariaHidden":51},[93],[18,34823,34825,34829,34832,34835,34866,34869,34872],{"className":34824},[97],[18,34826],{"className":34827,"style":34828},[101],"height:1.0701em;vertical-align:-0.25em;",[18,34830,934],{"className":34831},[106,1139],[18,34833,2248],{"className":34834},[250],[18,34836,34838],{"className":34837},[106,681],[18,34839,34841],{"className":34840},[119],[18,34842,34844],{"className":34843},[124],[18,34845,34847,34855],{"className":34846,"style":25624},[128],[18,34848,34849,34852],{"style":693},[18,34850],{"className":34851,"style":697},[136],[18,34853,43],{"className":34854,"style":111},[106,110],[18,34856,34857,34860],{"style":17702},[18,34858],{"className":34859,"style":697},[136],[18,34861,34863],{"className":34862,"style":711},[710],[18,34864,25294],{"className":34865},[106],[18,34867],{"className":34868,"style":2068},[169],[18,34870,2253],{"className":34871},[2072],[18,34873],{"className":34874,"style":2068},[169],[18,34876,34878,34881,34884,34935,34938,34941],{"className":34877},[97],[18,34879],{"className":34880,"style":15256},[101],[18,34882,77],{"className":34883,"style":254},[106,110],[18,34885,34887,34890],{"className":34886},[267],[18,34888,2248],{"className":34889},[267],[18,34891,34893],{"className":34892},[115],[18,34894,34896,34927],{"className":34895},[119,120],[18,34897,34899,34924],{"className":34898},[124],[18,34900,34902,34913],{"className":34901,"style":10775},[128],[18,34903,34904,34907],{"style":10778},[18,34905],{"className":34906,"style":137},[136],[18,34908,34910],{"className":34909},[141,142,143,144],[18,34911,2266],{"className":34912},[106,144],[18,34914,34915,34918],{"style":1637},[18,34916],{"className":34917,"style":137},[136],[18,34919,34921],{"className":34920},[141,142,143,144],[18,34922,2266],{"className":34923},[106,144],[18,34925,152],{"className":34926},[151],[18,34928,34930],{"className":34929},[124],[18,34931,34933],{"className":34932,"style":10808},[128],[18,34934],{},[18,34936],{"className":34937,"style":228},[169],[18,34939,433],{"className":34940},[232],[18,34942],{"className":34943,"style":228},[169],[18,34945,34947,34951,35042],{"className":34946},[97],[18,34948],{"className":34949,"style":34950},[101],"height:2.1771em;vertical-align:-0.686em;",[18,34952,34954,34957,35039],{"className":34953},[106],[18,34955],{"className":34956},[250,6012],[18,34958,34960],{"className":34959},[5654],[18,34961,34963,35031],{"className":34962},[119,120],[18,34964,34966,35028],{"className":34965},[124],[18,34967,34969,34980,34988],{"className":34968,"style":15999},[128],[18,34970,34971,34974],{"style":6028},[18,34972],{"className":34973,"style":697},[136],[18,34975,34977],{"className":34976},[106],[18,34978,64],{"className":34979},[106,110],[18,34981,34982,34985],{"style":6040},[18,34983],{"className":34984,"style":697},[136],[18,34986],{"className":34987,"style":6048},[6047],[18,34989,34990,34993],{"style":6051},[18,34991],{"className":34992,"style":697},[136],[18,34994,34996,35025],{"className":34995},[106],[18,34997,34999,35002],{"className":34998},[106],[18,35000,16406],{"className":35001,"style":3957},[106,110],[18,35003,35005],{"className":35004},[115],[18,35006,35008],{"className":35007},[119],[18,35009,35011],{"className":35010},[124],[18,35012,35014],{"className":35013,"style":2416},[128],[18,35015,35016,35019],{"style":1962},[18,35017],{"className":35018,"style":137},[136],[18,35020,35022],{"className":35021},[141,142,143,144],[18,35023,2266],{"className":35024},[106,144],[18,35026,4031],{"className":35027},[106,110],[18,35029,152],{"className":35030},[151],[18,35032,35034],{"className":35033},[124],[18,35035,35037],{"className":35036,"style":6070},[128],[18,35038],{},[18,35040],{"className":35041},[267,6012],[18,35043,968],{"className":35044},[106],[11,35046,35047,35050,35051,35198,35199,35227,35228,509,35312,35396,35397,35426],{},[717,35048,35049],{},"下界。"," 把参数限制到超立方体 ",[18,35052,35054,35088],{"className":35053,"translate":22},[21],[18,35055,35057],{"className":35056},[26],[28,35058,35059],{"xmlns":30},[32,35060,35061,35085],{},[35,35062,35063,35069,35071,35073,35075,35077,35079],{},[38,35064,35065,35067],{},[41,35066,924],{"mathvariant":316},[45,35068,430],{},[49,35070,433],{},[49,35072,1331],{"stretchy":73},[45,35074,430],{},[49,35076,52],{"separator":51},[41,35078,32522],{},[1556,35080,35081,35083],{},[49,35082,1349],{"stretchy":73},[41,35084,4031],{},[86,35086,35087],{"encoding":88},"\\Theta_0=\\{0,\\varepsilon\\}^d",[18,35089,35091,35146],{"className":35090,"ariaHidden":51},[93],[18,35092,35094,35097,35137,35140,35143],{"className":35093},[97],[18,35095],{"className":35096,"style":4563},[101],[18,35098,35100,35103],{"className":35099},[106],[18,35101,924],{"className":35102},[106],[18,35104,35106],{"className":35105},[115],[18,35107,35109,35129],{"className":35108},[119,120],[18,35110,35112,35126],{"className":35111},[124],[18,35113,35115],{"className":35114,"style":129},[128],[18,35116,35117,35120],{"style":1155},[18,35118],{"className":35119,"style":137},[136],[18,35121,35123],{"className":35122},[141,142,143,144],[18,35124,430],{"className":35125},[106,144],[18,35127,152],{"className":35128},[151],[18,35130,35132],{"className":35131},[124],[18,35133,35135],{"className":35134,"style":159},[128],[18,35136],{},[18,35138],{"className":35139,"style":228},[169],[18,35141,433],{"className":35142},[232],[18,35144],{"className":35145,"style":228},[169],[18,35147,35149,35153,35156,35159,35162,35165,35168],{"className":35148},[97],[18,35150],{"className":35151,"style":35152},[101],"height:1.0991em;vertical-align:-0.25em;",[18,35154,1331],{"className":35155},[250],[18,35157,430],{"className":35158},[106],[18,35160,52],{"className":35161},[165],[18,35163],{"className":35164,"style":170},[169],[18,35166,32522],{"className":35167},[106,110],[18,35169,35171,35174],{"className":35170},[267],[18,35172,1349],{"className":35173},[267],[18,35175,35177],{"className":35176},[115],[18,35178,35180],{"className":35179},[119],[18,35181,35183],{"className":35182},[124],[18,35184,35187],{"className":35185,"style":35186},[128],"height:0.8491em;",[18,35188,35189,35192],{"style":1962},[18,35190],{"className":35191,"style":137},[136],[18,35193,35195],{"className":35194},[141,142,143,144],[18,35196,4031],{"className":35197},[106,110,144],"。对坐标 ",[18,35200,35202,35215],{"className":35201,"translate":22},[21],[18,35203,35205],{"className":35204},[26],[28,35206,35207],{"xmlns":30},[32,35208,35209,35213],{},[35,35210,35211],{},[41,35212,19811],{},[86,35214,19811],{"encoding":88},[18,35216,35218],{"className":35217,"ariaHidden":51},[93],[18,35219,35221,35224],{"className":35220},[97],[18,35222],{"className":35223,"style":28340},[101],[18,35225,19811],{"className":35226,"style":19928},[106,110],"，混合分布 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只在这一维的均值上相差 ",[18,35398,35400,35414],{"className":35399,"translate":22},[21],[18,35401,35403],{"className":35402},[26],[28,35404,35405],{"xmlns":30},[32,35406,35407,35411],{},[35,35408,35409],{},[41,35410,32522],{},[86,35412,35413],{"encoding":88},"\\varepsilon",[18,35415,35417],{"className":35416,"ariaHidden":51},[93],[18,35418,35420,35423],{"className":35419},[97],[18,35421],{"className":35422,"style":1083},[101],[18,35424,32522],{"className":35425},[106,110],"。由 KL 的联合凸性，",[18,35428,35430],{"className":35429,"translate":22},[875],[18,35431,35433,35511],{"className":35432,"translate":22},[21],[18,35434,35436],{"className":35435},[26],[28,35437,35438],{"xmlns":30,"display":884},[32,35439,35440,35508],{},[35,35441,35442,35452,35454,35466,35468,35480,35482,35484,35506],{},[38,35443,35444,35446],{},[41,35445,12575],{},[35,35447,35448,35450],{},[41,35449,5031],{"mathvariant":316},[41,35451,12582],{"mathvariant":316},[49,35453,74],{"stretchy":73},[38,35455,35456,35458],{},[41,35457,12547],{},[35,35459,35460,35462,35464],{},[41,35461,19811],{},[49,35463,52],{"separator":51},[45,35465,430],{},[41,35467,2248],{"mathvariant":316},[38,35469,35470,35472],{},[41,35471,12547],{},[35,35473,35474,35476,35478],{},[41,35475,19811],{},[49,35477,52],{"separator":51},[45,35479,47],{},[49,35481,84],{"stretchy":73},[49,35483,4334],{},[5654,35485,35486,35496],{},[35,35487,35488,35490],{},[41,35489,64],{},[1556,35491,35492,35494],{},[41,35493,32522],{},[45,35495,2266],{},[35,35497,35498,35500],{},[45,35499,2266],{},[1556,35501,35502,35504],{},[41,35503,16406],{},[45,35505,2266],{},[41,35507,968],{"mathvariant":316},[86,35509,35510],{"encoding":88},"D_{\\mathrm{KL}}(Q_{j,0}\\Vert Q_{j,1})\n\\le \\frac{n\\varepsilon^2}{2\\sigma^2}.",[18,35512,35514,35682],{"className":35513,"ariaHidden":51},[93],[18,35515,35517,35520,35566,35569,35618,35621,35670,35673,35676,35679],{"className":35516},[97],[18,35518],{"className":35519,"style":19846},[101],[18,35521,35523,35526],{"className":35522},[106],[18,35524,12575],{"className":35525,"style":254},[106,110],[18,35527,35529],{"className":35528},[115],[18,35530,35532,35558],{"className":35531},[119,120],[18,35533,35535,35555],{"className":35534},[124],[18,35536,35538],{"className":35537,"style":1152},[128],[18,35539,35540,35543],{"style":469},[18,35541],{"className":35542,"style":137},[136],[18,35544,35546],{"className":35545},[141,142,143,144],[18,35547,35549],{"className":35548},[106,144],[18,35550,35552],{"className":35551},[106,144],[18,35553,12817],{"className":35554},[106,6354,144],[18,35556,152],{"className":35557},[151],[18,35559,35561],{"className":35560},[124],[18,35562,35564],{"className":35563,"style":159},[128],[18,35565],{},[18,35567,74],{"className":35568},[250],[18,35570,35572,35575],{"className":35571},[106],[18,35573,12547],{"className":35574},[106,110],[18,35576,35578],{"className":35577},[115],[18,35579,35581,35610],{"className":35580},[119,120],[18,35582,35584,35607],{"className":35583},[124],[18,35585,35587],{"className":35586,"style":5921},[128],[18,35588,35589,35592],{"style":1155},[18,35590],{"className":35591,"style":137},[136],[18,35593,35595],{"className":35594},[141,142,143,144],[18,35596,35598,35601,35604],{"className":35597},[106,144],[18,35599,19811],{"className":35600,"style":19928},[106,110,144],[18,35602,52],{"className":35603},[165,144],[18,35605,430],{"className":35606},[106,144],[18,35608,152],{"className":35609},[151],[18,35611,35613],{"className":35612},[124],[18,35614,35616],{"className":35615,"style":19938},[128],[18,35617],{},[18,35619,2248],{"className":35620},[106],[18,35622,35624,35627],{"className":35623},[106],[18,35625,12547],{"className":35626},[106,110],[18,35628,35630],{"className":35629},[115],[18,35631,35633,35662],{"className":35632},[119,120],[18,35634,35636,35659],{"className":35635},[124],[18,35637,35639],{"className":35638,"style":5921},[128],[18,35640,35641,35644],{"style":1155},[18,35642],{"className":35643,"style":137},[136],[18,35645,35647],{"className":35646},[141,142,143,144],[18,35648,35650,35653,35656],{"className":35649},[106,144],[18,35651,19811],{"className":35652,"style":19928},[106,110,144],[18,35654,52],{"className":35655},[165,144],[18,35657,47],{"className":35658},[106,144],[18,35660,152],{"className":35661},[151],[18,35663,35665],{"className":35664},[124],[18,35666,35668],{"className":35667,"style":19938},[128],[18,35669],{},[18,35671,84],{"className":35672},[267],[18,35674],{"className":35675,"style":228},[169],[18,35677,4334],{"className":35678},[232],[18,35680],{"className":35681,"style":228},[169],[18,35683,35685,35688,35808],{"className":35684},[97],[18,35686],{"className":35687,"style":34950},[101],[18,35689,35691,35694,35805],{"className":35690},[106],[18,35692],{"className":35693},[250,6012],[18,35695,35697],{"className":35696},[5654],[18,35698,35700,35797],{"className":35699},[119,120],[18,35701,35703,35794],{"className":35702},[124],[18,35704,35706,35746,35754],{"className":35705,"style":15999},[128],[18,35707,35708,35711],{"style":6028},[18,35709],{"className":35710,"style":697},[136],[18,35712,35714,35717],{"className":35713},[106],[18,35715,2266],{"className":35716},[106],[18,35718,35720,35723],{"className":35719},[106],[18,35721,16406],{"className":35722,"style":3957},[106,110],[18,35724,35726],{"className":35725},[115],[18,35727,35729],{"className":35728},[119],[18,35730,35732],{"className":35731},[124],[18,35733,35735],{"className":35734,"style":26109},[128],[18,35736,35737,35740],{"style":26112},[18,35738],{"className":35739,"style":137},[136],[18,35741,35743],{"className":35742},[141,142,143,144],[18,35744,2266],{"className":35745},[106,144],[18,35747,35748,35751],{"style":6040},[18,35749],{"className":35750,"style":697},[136],[18,35752],{"className":35753,"style":6048},[6047],[18,35755,35756,35759],{"style":6051},[18,35757],{"className":35758,"style":697},[136],[18,35760,35762,35765],{"className":35761},[106],[18,35763,64],{"className":35764},[106,110],[18,35766,35768,35771],{"className":35767},[106],[18,35769,32522],{"className":35770},[106,110],[18,35772,35774],{"className":35773},[115],[18,35775,35777],{"className":35776},[119],[18,35778,35780],{"className":35779},[124],[18,35781,35783],{"className":35782,"style":2416},[128],[18,35784,35785,35788],{"style":1962},[18,35786],{"className":35787,"style":137},[136],[18,35789,35791],{"className":35790},[141,142,143,144],[18,35792,2266],{"className":35793},[106,144],[18,35795,152],{"className":35796},[151],[18,35798,35800],{"className":35799},[124],[18,35801,35803],{"className":35802,"style":6070},[128],[18,35804],{},[18,35806],{"className":35807},[267,6012],[18,35809,968],{"className":35810},[106],[11,35812,35813],{},"Pinsker 给出",[18,35815,35817],{"className":35816,"translate":22},[875],[18,35818,35820,35890],{"className":35819,"translate":22},[21],[18,35821,35823],{"className":35822},[26],[28,35824,35825],{"xmlns":30,"display":884},[32,35826,35827,35887],{},[35,35828,35829,35831,35843,35845,35857,35867,35869,35885],{},[49,35830,2248],{"stretchy":73},[38,35832,35833,35835],{},[41,35834,12547],{},[35,35836,35837,35839,35841],{},[41,35838,19811],{},[49,35840,52],{"separator":51},[45,35842,430],{},[49,35844,2253],{},[38,35846,35847,35849],{},[41,35848,12547],{},[35,35850,35851,35853,35855],{},[41,35852,19811],{},[49,35854,52],{"separator":51},[45,35856,47],{},[38,35858,35859,35861],{},[49,35860,2248],{"stretchy":73},[35,35862,35863,35865],{},[41,35864,5703],{"mathvariant":316},[41,35866,5706],{"mathvariant":316},[49,35868,4334],{},[5654,35870,35871,35879],{},[35,35872,35873,35875],{},[41,35874,32522],{},[12561,35876,35877],{},[41,35878,64],{},[35,35880,35881,35883],{},[45,35882,2266],{},[41,35884,16406],{},[41,35886,968],{"mathvariant":316},[86,35888,35889],{"encoding":88},"\\lVert Q_{j,0}-Q_{j,1}\\rVert_{\\mathrm{TV}}\n\\le \\frac{\\varepsilon\\sqrt n}{2\\sigma}.",[18,35891,35893,35960,36070],{"className":35892,"ariaHidden":51},[93],[18,35894,35896,35899,35902,35951,35954,35957],{"className":35895},[97],[18,35897],{"className":35898,"style":19846},[101],[18,35900,2248],{"className":35901},[250],[18,35903,35905,35908],{"className":35904},[106],[18,35906,12547],{"className":35907},[106,110],[18,35909,35911],{"className":35910},[115],[18,35912,35914,35943],{"className":35913},[119,120],[18,35915,35917,35940],{"className":35916},[124],[18,35918,35920],{"className":35919,"style":5921},[128],[18,35921,35922,35925],{"style":1155},[18,35923],{"className":35924,"style":137},[136],[18,35926,35928],{"className":35927},[141,142,143,144],[18,35929,35931,35934,35937],{"className":35930},[106,144],[18,35932,19811],{"className":35933,"style":19928},[106,110,144],[18,35935,52],{"className":35936},[165,144],[18,35938,430],{"className":35939},[106,144],[18,35941,152],{"className":35942},[151],[18,35944,35946],{"className":35945},[124],[18,35947,35949],{"className":35948,"style":19938},[128],[18,35950],{},[18,35952],{"className":35953,"style":2068},[169],[18,35955,2253],{"className":35956},[2072],[18,35958],{"className":35959,"style":2068},[169],[18,35961,35963,35966,36015,36061,36064,36067],{"className":35962},[97],[18,35964],{"className":35965,"style":19846},[101],[18,35967,35969,35972],{"className":35968},[106],[18,35970,12547],{"className":35971},[106,110],[18,35973,35975],{"className":35974},[115],[18,35976,35978,36007],{"className":35977},[119,120],[18,35979,35981,36004],{"className":35980},[124],[18,35982,35984],{"className":35983,"style":5921},[128],[18,35985,35986,35989],{"style":1155},[18,35987],{"className":35988,"style":137},[136],[18,35990,35992],{"className":35991},[141,142,143,144],[18,35993,35995,35998,36001],{"className":35994},[106,144],[18,35996,19811],{"className":35997,"style":19928},[106,110,144],[18,35999,52],{"className":36000},[165,144],[18,36002,47],{"className":36003},[106,144],[18,36005,152],{"className":36006},[151],[18,36008,36010],{"className":36009},[124],[18,36011,36013],{"className":36012,"style":19938},[128],[18,36014],{},[18,36016,36018,36021],{"className":36017},[267],[18,36019,2248],{"className":36020},[267],[18,36022,36024],{"className":36023},[115],[18,36025,36027,36053],{"className":36026},[119,120],[18,36028,36030,36050],{"className":36029},[124],[18,36031,36033],{"className":36032,"style":1152},[128],[18,36034,36035,36038],{"style":1155},[18,36036],{"className":36037,"style":137},[136],[18,36039,36041],{"className":36040},[141,142,143,144],[18,36042,36044],{"className":36043},[106,144],[18,36045,36047],{"className":36046},[106,144],[18,36048,6356],{"className":36049,"style":6355},[106,6354,144],[18,36051,152],{"className":36052},[151],[18,36054,36056],{"className":36055},[124],[18,36057,36059],{"className":36058,"style":159},[128],[18,36060],{},[18,36062],{"className":36063,"style":228},[169],[18,36065,4334],{"className":36066},[232],[18,36068],{"className":36069,"style":228},[169],[18,36071,36073,36077,36187],{"className":36072},[97],[18,36074],{"className":36075,"style":36076},[101],"height:2.1633em;vertical-align:-0.686em;",[18,36078,36080,36083,36184],{"className":36079},[106],[18,36081],{"className":36082},[250,6012],[18,36084,36086],{"className":36085},[5654],[18,36087,36089,36176],{"className":36088},[119,120],[18,36090,36092,36173],{"className":36091},[124],[18,36093,36096,36110,36118],{"className":36094,"style":36095},[128],"height:1.4773em;",[18,36097,36098,36101],{"style":6028},[18,36099],{"className":36100,"style":697},[136],[18,36102,36104,36107],{"className":36103},[106],[18,36105,2266],{"className":36106},[106],[18,36108,16406],{"className":36109,"style":3957},[106,110],[18,36111,36112,36115],{"style":6040},[18,36113],{"className":36114,"style":697},[136],[18,36116],{"className":36117,"style":6048},[6047],[18,36119,36120,36123],{"style":6051},[18,36121],{"className":36122,"style":697},[136],[18,36124,36126,36129],{"className":36125},[106],[18,36127,32522],{"className":36128},[106,110],[18,36130,36132],{"className":36131},[106,12696],[18,36133,36135,36165],{"className":36134},[119,120],[18,36136,36138,36162],{"className":36137},[124],[18,36139,36141,36150],{"className":36140,"style":15400},[128],[18,36142,36144,36147],{"className":36143,"style":693},[12710],[18,36145],{"className":36146,"style":697},[136],[18,36148,64],{"className":36149,"style":15410},[106,110],[18,36151,36152,36155],{"style":15413},[18,36153],{"className":36154,"style":697},[136],[18,36156,36158],{"className":36157,"style":15420},[12853],[6130,36159,36160],{"xmlns":6132,"width":12857,"height":15423,"viewBox":15424,"preserveAspectRatio":12860},[6136,36161],{"d":15427},[18,36163,152],{"className":36164},[151],[18,36166,36168],{"className":36167},[124],[18,36169,36171],{"className":36170,"style":15437},[128],[18,36172],{},[18,36174,152],{"className":36175},[151],[18,36177,36179],{"className":36178},[124],[18,36180,36182],{"className":36181,"style":6070},[128],[18,36183],{},[18,36185],{"className":36186},[267,6012],[18,36188,968],{"className":36189},[106],[11,36191,36192,36193,36297,36298,36330],{},"取 ",[18,36194,36196,36220],{"className":36195,"translate":22},[21],[18,36197,36199],{"className":36198},[26],[28,36200,36201],{"xmlns":30},[32,36202,36203,36217],{},[35,36204,36205,36207,36209,36211,36213],{},[41,36206,32522],{},[49,36208,433],{},[41,36210,16406],{},[41,36212,317],{"mathvariant":316},[12561,36214,36215],{},[41,36216,64],{},[86,36218,36219],{"encoding":88},"\\varepsilon=\\sigma\u002F\\sqrt n",[18,36221,36223,36241],{"className":36222,"ariaHidden":51},[93],[18,36224,36226,36229,36232,36235,36238],{"className":36225},[97],[18,36227],{"className":36228,"style":1083},[101],[18,36230,32522],{"className":36231},[106,110],[18,36233],{"className":36234,"style":228},[169],[18,36236,433],{"className":36237},[232],[18,36239],{"className":36240,"style":228},[169],[18,36242,36244,36247,36250,36253],{"className":36243},[97],[18,36245],{"className":36246,"style":15501},[101],[18,36248,16406],{"className":36249,"style":3957},[106,110],[18,36251,317],{"className":36252},[106],[18,36254,36256],{"className":36255},[106,12696],[18,36257,36259,36289],{"className":36258},[119,120],[18,36260,36262,36286],{"className":36261},[124],[18,36263,36265,36274],{"className":36264,"style":15400},[128],[18,36266,36268,36271],{"className":36267,"style":693},[12710],[18,36269],{"className":36270,"style":697},[136],[18,36272,64],{"className":36273,"style":15410},[106,110],[18,36275,36276,36279],{"style":15413},[18,36277],{"className":36278,"style":697},[136],[18,36280,36282],{"className":36281,"style":15420},[12853],[6130,36283,36284],{"xmlns":6132,"width":12857,"height":15423,"viewBox":15424,"preserveAspectRatio":12860},[6136,36285],{"d":15427},[18,36287,152],{"className":36288},[151],[18,36290,36292],{"className":36291},[124],[18,36293,36295],{"className":36294,"style":15437},[128],[18,36296],{},"，则 TV 至多 ",[18,36299,36301,36318],{"className":36300,"translate":22},[21],[18,36302,36304],{"className":36303},[26],[28,36305,36306],{"xmlns":30},[32,36307,36308,36316],{},[35,36309,36310,36312,36314],{},[45,36311,47],{},[41,36313,317],{"mathvariant":316},[45,36315,2266],{},[86,36317,9060],{"encoding":88},[18,36319,36321],{"className":36320,"ariaHidden":51},[93],[18,36322,36324,36327],{"className":36323},[97],[18,36325],{"className":36326,"style":242},[101],[18,36328,9060],{"className":36329},[106],"。Assouad：",[18,36332,36334],{"className":36333,"translate":22},[875],[18,36335,36337,36446],{"className":36336,"translate":22},[21],[18,36338,36340],{"className":36339},[26],[28,36341,36342],{"xmlns":30,"display":884},[32,36343,36344,36443],{},[35,36345,36346,36366,36368,36370,36376,36378,36380,36388,36390,36404,36420,36422,36441],{},[890,36347,36348,36354],{},[35,36349,36350,36352],{},[41,36351,5593],{},[49,36353,899],{},[35,36355,36356,36358,36360],{},[41,36357,77],{},[49,36359,921],{},[38,36361,36362,36364],{},[41,36363,924],{"mathvariant":316},[45,36365,430],{},[41,36367,934],{"mathvariant":933},[49,36369,2248],{"stretchy":73},[658,36371,36372,36374],{"accent":51},[41,36373,77],{},[49,36375,664],{},[49,36377,2253],{},[41,36379,77],{},[1589,36381,36382,36384,36386],{},[49,36383,2248],{"stretchy":73},[45,36385,2266],{},[45,36387,2266],{},[49,36389,4775],{},[5654,36391,36392,36402],{},[35,36393,36394,36400],{},[1556,36395,36396,36398],{},[41,36397,32522],{},[45,36399,2266],{},[41,36401,4031],{},[45,36403,10447],{},[35,36405,36406,36408,36410,36412,36418],{},[49,36407,74],{"fence":51},[45,36409,47],{},[49,36411,2253],{},[5654,36413,36414,36416],{},[45,36415,47],{},[45,36417,2266],{},[49,36419,84],{"fence":51},[49,36421,433],{},[5654,36423,36424,36434],{},[35,36425,36426,36432],{},[1556,36427,36428,36430],{},[41,36429,16406],{},[45,36431,2266],{},[41,36433,4031],{},[35,36435,36436,36439],{},[45,36437,36438],{},"8",[41,36440,64],{},[41,36442,968],{"mathvariant":316},[86,36444,36445],{"encoding":88},"\\max_{\\theta\\in\\Theta_0}\\mathbb E\\lVert\\hat\\theta-\\theta\\rVert_2^2\n\\ge\n\\frac{\\varepsilon^2d}{4}\\left(1-\\frac12\\right)\n=\\frac{\\sigma^2d}{8n}.",[18,36447,36449,36597,36666,36864],{"className":36448,"ariaHidden":51},[93],[18,36450,36452,36456,36548,36551,36554,36557,36588,36591,36594],{"className":36451},[97],[18,36453],{"className":36454,"style":36455},[101],"height:1.8101em;vertical-align:-0.8522em;",[18,36457,36459],{"className":36458},[985,986],[18,36460,36462,36539],{"className":36461},[119,120],[18,36463,36465,36536],{"className":36464},[124],[18,36466,36468,36526],{"className":36467,"style":1083},[128],[18,36469,36470,36473],{"style":29370},[18,36471],{"className":36472,"style":697},[136],[18,36474,36476],{"className":36475},[141,142,143,144],[18,36477,36479,36482,36485],{"className":36478},[106,144],[18,36480,77],{"className":36481,"style":254},[106,110,144],[18,36483,921],{"className":36484},[232,144],[18,36486,36488,36491],{"className":36487},[106,144],[18,36489,924],{"className":36490},[106,144],[18,36492,36494],{"className":36493},[115],[18,36495,36497,36518],{"className":36496},[119,120],[18,36498,36500,36515],{"className":36499},[124],[18,36501,36503],{"className":36502,"style":7459},[128],[18,36504,36506,36509],{"style":36505},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[18,36507],{"className":36508,"style":1191},[136],[18,36510,36512],{"className":36511},[141,1195,1196,144],[18,36513,430],{"className":36514},[106,144],[18,36516,152],{"className":36517},[151],[18,36519,36521],{"className":36520},[124],[18,36522,36524],{"className":36523,"style":7481},[128],[18,36525],{},[18,36527,36528,36531],{"style":693},[18,36529],{"className":36530,"style":697},[136],[18,36532,36533],{},[18,36534,5593],{"className":36535},[985],[18,36537,152],{"className":36538},[151],[18,36540,36542],{"className":36541},[124],[18,36543,36546],{"className":36544,"style":36545},[128],"height:0.8522em;",[18,36547],{},[18,36549],{"className":36550,"style":170},[169],[18,36552,934],{"className":36553},[106,1139],[18,36555,2248],{"className":36556},[250],[18,36558,36560],{"className":36559},[106,681],[18,36561,36563],{"className":36562},[119],[18,36564,36566],{"className":36565},[124],[18,36567,36569,36577],{"className":36568,"style":677},[128],[18,36570,36571,36574],{"style":693},[18,36572],{"className":36573,"style":697},[136],[18,36575,77],{"className":36576,"style":254},[106,110],[18,36578,36579,36582],{"style":703},[18,36580],{"className":36581,"style":697},[136],[18,36583,36585],{"className":36584,"style":711},[710],[18,36586,664],{"className":36587},[106],[18,36589],{"className":36590,"style":2068},[169],[18,36592,2253],{"className":36593},[2072],[18,36595],{"className":36596,"style":2068},[169],[18,36598,36600,36603,36606,36657,36660,36663],{"className":36599},[97],[18,36601],{"className":36602,"style":15256},[101],[18,36604,77],{"className":36605,"style":254},[106,110],[18,36607,36609,36612],{"className":36608},[267],[18,36610,2248],{"className":36611},[267],[18,36613,36615],{"className":36614},[115],[18,36616,36618,36649],{"className":36617},[119,120],[18,36619,36621,36646],{"className":36620},[124],[18,36622,36624,36635],{"className":36623,"style":10775},[128],[18,36625,36626,36629],{"style":10778},[18,36627],{"className":36628,"style":137},[136],[18,36630,36632],{"className":36631},[141,142,143,144],[18,36633,2266],{"className":36634},[106,144],[18,36636,36637,36640],{"style":1637},[18,36638],{"className":36639,"style":137},[136],[18,36641,36643],{"className":36642},[141,142,143,144],[18,36644,2266],{"className":36645},[106,144],[18,36647,152],{"className":36648},[151],[18,36650,36652],{"className":36651},[124],[18,36653,36655],{"className":36654,"style":10808},[128],[18,36656],{},[18,36658],{"className":36659,"style":228},[169],[18,36661,4775],{"className":36662},[232],[18,36664],{"className":36665,"style":228},[169],[18,36667,36669,36672,36763,36766,36855,36858,36861],{"className":36668},[97],[18,36670],{"className":36671,"style":15980},[101],[18,36673,36675,36678,36760],{"className":36674},[106],[18,36676],{"className":36677},[250,6012],[18,36679,36681],{"className":36680},[5654],[18,36682,36684,36752],{"className":36683},[119,120],[18,36685,36687,36749],{"className":36686},[124],[18,36688,36690,36701,36709],{"className":36689,"style":15999},[128],[18,36691,36692,36695],{"style":6028},[18,36693],{"className":36694,"style":697},[136],[18,36696,36698],{"className":36697},[106],[18,36699,10447],{"className":36700},[106],[18,36702,36703,36706],{"style":6040},[18,36704],{"className":36705,"style":697},[136],[18,36707],{"className":36708,"style":6048},[6047],[18,36710,36711,36714],{"style":6051},[18,36712],{"className":36713,"style":697},[136],[18,36715,36717,36746],{"className":36716},[106],[18,36718,36720,36723],{"className":36719},[106],[18,36721,32522],{"className":36722},[106,110],[18,36724,36726],{"className":36725},[115],[18,36727,36729],{"className":36728},[119],[18,36730,36732],{"className":36731},[124],[18,36733,36735],{"className":36734,"style":2416},[128],[18,36736,36737,36740],{"style":1962},[18,36738],{"className":36739,"style":137},[136],[18,36741,36743],{"className":36742},[141,142,143,144],[18,36744,2266],{"className":36745},[106,144],[18,36747,4031],{"className":36748},[106,110],[18,36750,152],{"className":36751},[151],[18,36753,36755],{"className":36754},[124],[18,36756,36758],{"className":36757,"style":6070},[128],[18,36759],{},[18,36761],{"className":36762},[267,6012],[18,36764],{"className":36765,"style":170},[169],[18,36767,36769,36775,36778,36781,36784,36787,36849],{"className":36768},[174],[18,36770,36772],{"className":36771,"style":13796},[250,13795],[18,36773,74],{"className":36774},[1230,143],[18,36776,47],{"className":36777},[106],[18,36779],{"className":36780,"style":2068},[169],[18,36782,2253],{"className":36783},[2072],[18,36785],{"className":36786,"style":2068},[169],[18,36788,36790,36793,36846],{"className":36789},[106],[18,36791],{"className":36792},[250,6012],[18,36794,36796],{"className":36795},[5654],[18,36797,36799,36838],{"className":36798},[119,120],[18,36800,36802,36835],{"className":36801},[124],[18,36803,36805,36816,36824],{"className":36804,"style":9581},[128],[18,36806,36807,36810],{"style":6028},[18,36808],{"className":36809,"style":697},[136],[18,36811,36813],{"className":36812},[106],[18,36814,2266],{"className":36815},[106],[18,36817,36818,36821],{"style":6040},[18,36819],{"className":36820,"style":697},[136],[18,36822],{"className":36823,"style":6048},[6047],[18,36825,36826,36829],{"style":6051},[18,36827],{"className":36828,"style":697},[136],[18,36830,36832],{"className":36831},[106],[18,36833,47],{"className":36834},[106],[18,36836,152],{"className":36837},[151],[18,36839,36841],{"className":36840},[124],[18,36842,36844],{"className":36843,"style":6070},[128],[18,36845],{},[18,36847],{"className":36848},[267,6012],[18,36850,36852],{"className":36851,"style":13796},[267,13795],[18,36853,84],{"className":36854},[1230,143],[18,36856],{"className":36857,"style":228},[169],[18,36859,433],{"className":36860},[232],[18,36862],{"className":36863,"style":228},[169],[18,36865,36867,36870,36964],{"className":36866},[97],[18,36868],{"className":36869,"style":34950},[101],[18,36871,36873,36876,36961],{"className":36872},[106],[18,36874],{"className":36875},[250,6012],[18,36877,36879],{"className":36878},[5654],[18,36880,36882,36953],{"className":36881},[119,120],[18,36883,36885,36950],{"className":36884},[124],[18,36886,36888,36902,36910],{"className":36887,"style":15999},[128],[18,36889,36890,36893],{"style":6028},[18,36891],{"className":36892,"style":697},[136],[18,36894,36896,36899],{"className":36895},[106],[18,36897,36438],{"className":36898},[106],[18,36900,64],{"className":36901},[106,110],[18,36903,36904,36907],{"style":6040},[18,36905],{"className":36906,"style":697},[136],[18,36908],{"className":36909,"style":6048},[6047],[18,36911,36912,36915],{"style":6051},[18,36913],{"className":36914,"style":697},[136],[18,36916,36918,36947],{"className":36917},[106],[18,36919,36921,36924],{"className":36920},[106],[18,36922,16406],{"className":36923,"style":3957},[106,110],[18,36925,36927],{"className":36926},[115],[18,36928,36930],{"className":36929},[119],[18,36931,36933],{"className":36932},[124],[18,36934,36936],{"className":36935,"style":2416},[128],[18,36937,36938,36941],{"style":1962},[18,36939],{"className":36940,"style":137},[136],[18,36942,36944],{"className":36943},[141,142,143,144],[18,36945,2266],{"className":36946},[106,144],[18,36948,4031],{"className":36949},[106,110],[18,36951,152],{"className":36952},[151],[18,36954,36956],{"className":36955},[124],[18,36957,36959],{"className":36958,"style":6070},[128],[18,36960],{},[18,36962],{"className":36963},[267,6012],[18,36965,968],{"className":36966},[106],[11,36968,36969,36970,36998,36999,37072],{},"上下界只差常数 ",[18,36971,36973,36986],{"className":36972,"translate":22},[21],[18,36974,36976],{"className":36975},[26],[28,36977,36978],{"xmlns":30},[32,36979,36980,36984],{},[35,36981,36982],{},[45,36983,36438],{},[86,36985,36438],{"encoding":88},[18,36987,36989],{"className":36988,"ariaHidden":51},[93],[18,36990,36992,36995],{"className":36991},[97],[18,36993],{"className":36994,"style":505},[101],[18,36996,36438],{"className":36997},[106],"，速率 ",[18,37000,37002,37025],{"className":37001,"translate":22},[21],[18,37003,37005],{"className":37004},[26],[28,37006,37007],{"xmlns":30},[32,37008,37009,37023],{},[35,37010,37011,37017,37019,37021],{},[1556,37012,37013,37015],{},[41,37014,16406],{},[45,37016,2266],{},[41,37018,4031],{},[41,37020,317],{"mathvariant":316},[41,37022,64],{},[86,37024,17299],{"encoding":88},[18,37026,37028],{"className":37027,"ariaHidden":51},[93],[18,37029,37031,37034,37063,37066,37069],{"className":37030},[97],[18,37032],{"className":37033,"style":10271},[101],[18,37035,37037,37040],{"className":37036},[106],[18,37038,16406],{"className":37039,"style":3957},[106,110],[18,37041,37043],{"className":37042},[115],[18,37044,37046],{"className":37045},[119],[18,37047,37049],{"className":37048},[124],[18,37050,37052],{"className":37051,"style":2416},[128],[18,37053,37054,37057],{"style":1962},[18,37055],{"className":37056,"style":137},[136],[18,37058,37060],{"className":37059},[141,142,143,144],[18,37061,2266],{"className":37062},[106,144],[18,37064,4031],{"className":37065},[106,110],[18,37067,317],{"className":37068},[106],[18,37070,64],{"className":37071},[106,110]," 被夹住了。",[11,37074,37075,37076,37104],{},"这个例子也说明了 Fano 与 Assouad 的分工：Fano 需要构造一个 ",[18,37077,37079,37092],{"className":37078,"translate":22},[21],[18,37080,37082],{"className":37081},[26],[28,37083,37084],{"xmlns":30},[32,37085,37086,37090],{},[35,37087,37088],{},[41,37089,5351],{},[86,37091,14091],{"encoding":88},[18,37093,37095],{"className":37094,"ariaHidden":51},[93],[18,37096,37098,37101],{"className":37097},[97],[18,37099],{"className":37100,"style":295},[101],[18,37102,5351],{"className":37103,"style":5480},[106,110],"-分离的打包，Assouad 则直接利用乘积结构，把打包问题化成了逐坐标的 TV 计算。",[1311,37106,37108,37109],{"id":37107},"_62-非参数回归n2β2βdn-2beta2betadn2β2βd","6.2 非参数回归：",[18,37110,37112,37149],{"className":37111,"translate":22},[21],[18,37113,37115],{"className":37114},[26],[28,37116,37117],{"xmlns":30},[32,37118,37119,37147],{},[35,37120,37121],{},[1556,37122,37123,37125],{},[41,37124,64],{},[35,37126,37127,37129,37131,37133,37135,37137,37139,37141,37143,37145],{},[49,37128,2253],{},[45,37130,2266],{},[41,37132,20986],{},[41,37134,317],{"mathvariant":316},[49,37136,74],{"stretchy":73},[45,37138,2266],{},[41,37140,20986],{},[49,37142,7987],{},[41,37144,4031],{},[49,37146,84],{"stretchy":73},[86,37148,27935],{"encoding":88},[18,37150,37152],{"className":37151,"ariaHidden":51},[93],[18,37153,37155,37158],{"className":37154},[97],[18,37156],{"className":37157,"style":27945},[101],[18,37159,37161,37164],{"className":37160},[106],[18,37162,64],{"className":37163},[106,110],[18,37165,37167],{"className":37166},[115],[18,37168,37170],{"className":37169},[119],[18,37171,37173],{"className":37172},[124],[18,37174,37176],{"className":37175,"style":27945},[128],[18,37177,37178,37181],{"style":1962},[18,37179],{"className":37180,"style":137},[136],[18,37182,37184],{"className":37183},[141,142,143,144],[18,37185,37187,37190,37193,37196,37199,37202,37205,37208,37211,37214],{"className":37186},[106,144],[18,37188,2253],{"className":37189},[106,144],[18,37191,2266],{"className":37192},[106,144],[18,37194,20986],{"className":37195,"style":21038},[106,110,144],[18,37197,317],{"className":37198},[106,144],[18,37200,74],{"className":37201},[250,144],[18,37203,2266],{"className":37204},[106,144],[18,37206,20986],{"className":37207,"style":21038},[106,110,144],[18,37209,7987],{"className":37210},[2072,144],[18,37212,4031],{"className":37213},[106,110,144],[18,37215,84],{"className":37216},[267,144],[11,37218,34197],{},[18,37220,37222],{"className":37221,"translate":22},[875],[18,37223,37225,37337],{"className":37224,"translate":22},[21],[18,37226,37228],{"className":37227},[26],[28,37229,37230],{"xmlns":30,"display":884},[32,37231,37232,37334],{},[35,37233,37234,37241,37243,37245,37247,37253,37255,37257,37264,37266,37268,37274,37276,37286,37288,37290,37292,37294,37296,37302,37304,37306,37308,37314,37316,37318,37320,37322,37324,37330,37332],{},[38,37235,37236,37239],{},[41,37237,37238],{},"Y",[41,37240,5600],{},[49,37242,433],{},[41,37244,1055],{},[49,37246,74],{"stretchy":73},[38,37248,37249,37251],{},[41,37250,43],{},[41,37252,5600],{},[49,37254,84],{"stretchy":73},[49,37256,7987],{},[38,37258,37259,37262],{},[41,37260,37261],{},"ξ",[41,37263,5600],{},[49,37265,52],{"separator":51},[169,37267],{"width":7344},[38,37269,37270,37272],{},[41,37271,43],{},[41,37273,5600],{},[49,37275,67],{},[35,37277,37278,37280,37282,37284],{},[41,37279,20054],{"mathvariant":316},[41,37281,64],{"mathvariant":316},[41,37283,5600],{"mathvariant":316},[41,37285,1055],{"mathvariant":316},[49,37287,74],{"stretchy":73},[49,37289,945],{"stretchy":73},[45,37291,430],{},[49,37293,52],{"separator":51},[45,37295,47],{},[1556,37297,37298,37300],{},[49,37299,965],{"stretchy":73},[41,37301,4031],{},[49,37303,84],{"stretchy":73},[49,37305,52],{"separator":51},[169,37307],{"width":7344},[38,37309,37310,37312],{},[41,37311,37261],{},[41,37313,5600],{},[49,37315,67],{},[41,37317,70],{},[49,37319,74],{"stretchy":73},[45,37321,430],{},[49,37323,52],{"separator":51},[1556,37325,37326,37328],{},[41,37327,16406],{},[45,37329,2266],{},[49,37331,84],{"stretchy":73},[49,37333,52],{"separator":51},[86,37335,37336],{"encoding":88},"Y_i=f(X_i)+\\xi_i,\\qquad X_i\\sim\\mathrm{Unif}([0,1]^d),\\qquad \\xi_i\\sim N(0,\\sigma^2),",[18,37338,37340,37396,37461,37567,37686],{"className":37339,"ariaHidden":51},[93],[18,37341,37343,37346,37387,37390,37393],{"className":37342},[97],[18,37344],{"className":37345,"style":4563},[101],[18,37347,37349,37352],{"className":37348},[106],[18,37350,37238],{"className":37351,"style":2068},[106,110],[18,37353,37355],{"className":37354},[115],[18,37356,37358,37379],{"className":37357},[119,120],[18,37359,37361,37376],{"className":37360},[124],[18,37362,37364],{"className":37363,"style":5921},[128],[18,37365,37367,37370],{"style":37366},"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;",[18,37368],{"className":37369,"style":137},[136],[18,37371,37373],{"className":37372},[141,142,143,144],[18,37374,5600],{"className":37375},[106,110,144],[18,37377,152],{"className":37378},[151],[18,37380,37382],{"className":37381},[124],[18,37383,37385],{"className":37384,"style":159},[128],[18,37386],{},[18,37388],{"className":37389,"style":228},[169],[18,37391,433],{"className":37392},[232],[18,37394],{"className":37395,"style":228},[169],[18,37397,37399,37402,37406,37409,37449,37452,37455,37458],{"className":37398},[97],[18,37400],{"className":37401,"style":242},[101],[18,37403,1055],{"className":37404,"style":37405},[106,110],"margin-right:0.1076em;",[18,37407,74],{"className":37408},[250],[18,37410,37412,37415],{"className":37411},[106],[18,37413,43],{"className":37414,"style":111},[106,110],[18,37416,37418],{"className":37417},[115],[18,37419,37421,37441],{"className":37420},[119,120],[18,37422,37424,37438],{"className":37423},[124],[18,37425,37427],{"className":37426,"style":5921},[128],[18,37428,37429,37432],{"style":132},[18,37430],{"className":37431,"style":137},[136],[18,37433,37435],{"className":37434},[141,142,143,144],[18,37436,5600],{"className":37437},[106,110,144],[18,37439,152],{"className":37440},[151],[18,37442,37444],{"className":37443},[124],[18,37445,37447],{"className":37446,"style":159},[128],[18,37448],{},[18,37450,84],{"className":37451},[267],[18,37453],{"className":37454,"style":2068},[169],[18,37456,7987],{"className":37457},[2072],[18,37459],{"className":37460,"style":2068},[169],[18,37462,37464,37467,37509,37512,37515,37518,37558,37561,37564],{"className":37463},[97],[18,37465],{"className":37466,"style":2890},[101],[18,37468,37470,37474],{"className":37469},[106],[18,37471,37261],{"className":37472,"style":37473},[106,110],"margin-right:0.046em;",[18,37475,37477],{"className":37476},[115],[18,37478,37480,37501],{"className":37479},[119,120],[18,37481,37483,37498],{"className":37482},[124],[18,37484,37486],{"className":37485,"style":5921},[128],[18,37487,37489,37492],{"style":37488},"top:-2.55em;margin-left:-0.046em;margin-right:0.05em;",[18,37490],{"className":37491,"style":137},[136],[18,37493,37495],{"className":37494},[141,142,143,144],[18,37496,5600],{"className":37497},[106,110,144],[18,37499,152],{"className":37500},[151],[18,37502,37504],{"className":37503},[124],[18,37505,37507],{"className":37506,"style":159},[128],[18,37508],{},[18,37510,52],{"className":37511},[165],[18,37513],{"className":37514,"style":7676},[169],[18,37516],{"className":37517,"style":170},[169],[18,37519,37521,37524],{"className":37520},[106],[18,37522,43],{"className":37523,"style":111},[106,110],[18,37525,37527],{"className":37526},[115],[18,37528,37530,37550],{"className":37529},[119,120],[18,37531,37533,37547],{"className":37532},[124],[18,37534,37536],{"className":37535,"style":5921},[128],[18,37537,37538,37541],{"style":132},[18,37539],{"className":37540,"style":137},[136],[18,37542,37544],{"className":37543},[141,142,143,144],[18,37545,5600],{"className":37546},[106,110,144],[18,37548,152],{"className":37549},[151],[18,37551,37553],{"className":37552},[124],[18,37554,37556],{"className":37555,"style":159},[128],[18,37557],{},[18,37559],{"className":37560,"style":228},[169],[18,37562,67],{"className":37563},[232],[18,37565],{"className":37566,"style":228},[169],[18,37568,37570,37574,37580,37584,37587,37590,37593,37596,37625,37628,37631,37634,37637,37677,37680,37683],{"className":37569},[97],[18,37571],{"className":37572,"style":37573},[101],"height:1.1491em;vertical-align:-0.25em;",[18,37575,37577],{"className":37576},[106],[18,37578,20111],{"className":37579,"style":1054},[106,6354],[18,37581,37583],{"className":37582},[250],"([",[18,37585,430],{"className":37586},[106],[18,37588,52],{"className":37589},[165],[18,37591],{"className":37592,"style":170},[169],[18,37594,47],{"className":37595},[106],[18,37597,37599,37602],{"className":37598},[267],[18,37600,965],{"className":37601},[267],[18,37603,37605],{"className":37604},[115],[18,37606,37608],{"className":37607},[119],[18,37609,37611],{"className":37610},[124],[18,37612,37614],{"className":37613,"style":16673},[128],[18,37615,37616,37619],{"style":1637},[18,37617],{"className":37618,"style":137},[136],[18,37620,37622],{"className":37621},[141,142,143,144],[18,37623,4031],{"className":37624},[106,110,144],[18,37626,84],{"classNa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属于 ",[18,37774,37776,37790],{"className":37775,"translate":22},[21],[18,37777,37779],{"className":37778},[26],[28,37780,37781],{"xmlns":30},[32,37782,37783,37787],{},[35,37784,37785],{},[41,37786,20986],{},[86,37788,37789],{"encoding":88},"\\beta",[18,37791,37793],{"className":37792,"ariaHidden":51},[93],[18,37794,37796,37799],{"className":37795},[97],[18,37797],{"className":37798,"style":2890},[101],[18,37800,20986],{"className":37801,"style":21038},[106,110],"-Hölder 球 ",[18,37804,37806,37831],{"className":37805,"translate":22},[21],[18,37807,37809],{"className":37808},[26],[28,37810,37811],{"xmlns":30},[32,37812,37813,37828],{},[35,37814,37815,37818,37820,37822,37824,37826],{},[41,37816,37817],{"mathvariant":316},"Σ",[49,37819,74],{"stretchy":73},[41,37821,20986],{},[49,37823,52],{"separator":51},[41,37825,12582],{},[49,37827,84],{"stretchy":73},[86,37829,37830],{"encoding":88},"\\Sigma(\\beta,L)",[18,37832,37834],{"className":37833,"ariaHidden":51},[93],[18,37835,37837,37840,37843,37846,37849,37852,37855,37858],{"className":37836},[97],[18,37838],{"className":37839,"style":242},[101],[18,37841,37817],{"className":37842},[106],[18,37844,74],{"className":37845},[250],[18,37847,20986],{"className":37848,"style":21038},[106,110],[18,37850,52],{"className":37851},[165],[18,37853],{"className":37854,"style":170},[169],[18,37856,12582],{"className":37857},[106,110],[18,37859,84],{"className":37860},[267],"，损失为 ",[18,37863,37865,37901],{"className":37864,"translate":22},[21],[18,37866,37868],{"className":37867},[26],[28,37869,37870],{"xmlns":30},[32,37871,37872,37898],{},[35,37873,37874,37876,37882,37884,37886],{},[49,37875,2248],{"stretchy":73},[658,37877,37878,37880],{"accent":51},[41,37879,1055],{},[49,37881,664],{},[49,37883,2253],{},[41,37885,1055],{},[1589,37887,37888,37890,37896],{},[49,37889,2248],{"stretchy":73},[1556,37891,37892,37894],{},[41,37893,12582],{},[45,37895,2266],{},[45,37897,2266],{},[86,37899,37900],{"encoding":88},"\\lVert\\hat f-f\\rVert_{L^2}^2",[18,37902,37904,37965],{"className":37903,"ariaHidden":51},[93],[18,37905,37907,37910,37913,37956,37959,37962],{"className":37906},[97],[18,37908],{"className":37909,"style":2281},[101],[18,37911,2248],{"className":37912},[250],[18,37914,37916],{"className":37915},[106,681],[18,37917,37919,37948],{"className":37918},[119,120],[18,37920,37922,37945],{"className":37921},[124],[18,37923,37925,37933],{"className":37924,"style":677},[128],[18,37926,37927,37930],{"style":693},[18,37928],{"className":37929,"style":697},[136],[18,37931,1055],{"className":37932,"style":37405},[106,110],[18,37934,37935,37938],{"style":703},[18,37936],{"className":37937,"style":697},[136],[18,37939,37942],{"className":37940,"style":37941},[710],"left:-0.0833em;",[18,37943,664],{"className":37944},[106],[18,37946,152],{"className":37947},[151],[18,37949,37951],{"className":37950},[124],[18,37952,37954],{"className":37953,"style":15927},[128],[18,37955],{},[18,37957],{"className":37958,"style":2068},[169],[18,37960,2253],{"className":37961},[2072],[18,37963],{"className":37964,"style":2068},[169],[18,37966,37968,37972,37975],{"className":37967},[97],[18,37969],{"className":37970,"style":37971},[101],"height:1.1335em;vertical-align:-0.3194em;",[18,37973,1055],{"className":37974,"style":37405},[106,110],[18,37976,37978,37981],{"className":37977},[267],[18,37979,2248],{"className":37980},[267],[18,37982,37984],{"className":37983},[115],[18,37985,37987,38049],{"className":37986},[119,120],[18,37988,37990,38046],{"className":37989},[124],[18,37991,37993,38035],{"className":37992,"style":2416},[128],[18,37994,37996,37999],{"style":37995},"top:-2.3806em;margin-left:0em;margin-right:0.05em;",[18,37997],{"className":37998,"style":137},[136],[18,38000,38002],{"className":38001},[141,142,143,144],[18,38003,38005],{"className":38004},[106,144],[18,38006,38008,38011],{"className":38007},[106,144],[18,38009,12582],{"className":38010},[106,110,144],[18,38012,38014],{"className":38013},[115],[18,38015,38017],{"className":38016},[119],[18,38018,38020],{"className":38019},[124],[18,38021,38024],{"className":38022,"style":38023},[128],"height:0.7463em;",[18,38025,38026,38029],{"style":33595},[18,38027],{"className":38028,"style":1191},[136],[18,38030,38032],{"className":38031},[141,1195,1196,144],[18,38033,2266],{"className":38034},[106,144],[18,38036,38037,38040],{"style":1962},[18,38038],{"className":38039,"style":137},[136],[18,38041,38043],{"className":38042},[141,142,143,144],[18,38044,2266],{"className":38045},[106,144],[18,38047,152],{"className":38048},[151],[18,38050,38052],{"className":38051},[124],[18,38053,38056],{"className":38054,"style":38055},[128],"height:0.3194em;",[18,38057],{},[11,38059,38060,38062],{},[717,38061,34697],{}," 核估计或局部多项式估计可以达到",[18,38064,38066],{"className":38065,"translate":22},[875],[18,38067,38069,38107],{"className":38068,"translate":22},[21],[18,38070,38072],{"className":38071},[26],[28,38073,38074],{"xmlns":30,"display":884},[32,38075,38076,38104],{},[35,38077,38078],{},[1556,38079,38080,38082],{},[41,38081,64],{},[35,38083,38084,38086],{},[49,38085,2253],{},[5654,38087,38088,38094],{},[35,38089,38090,38092],{},[45,38091,2266],{},[41,38093,20986],{},[35,38095,38096,38098,38100,38102],{},[45,38097,2266],{},[41,38099,20986],{},[49,38101,7987],{},[41,38103,4031],{},[86,38105,38106],{"encoding":88},"n^{-\\frac{2\\beta}{2\\beta+d}}",[18,38108,38110],{"className":38109,"ariaHidden":51},[93],[18,38111,38113,38117],{"className":38112},[97],[18,38114],{"className":38115,"style":38116},[101],"height:1.1167em;",[18,38118,38120,38123],{"className":38119},[106],[18,38121,64],{"className":38122},[106,110],[18,38124,38126],{"className":38125},[115],[18,38127,38129],{"className":38128},[119],[18,38130,38132],{"className":38131},[124],[18,38133,38135],{"className":38134,"style":38116},[128],[18,38136,38138,38141],{"style":38137},"top:-3.4458em;margin-right:0.05em;",[18,38139],{"className":38140,"style":697},[136],[18,38142,38144],{"className":38143},[141,142,143,144],[18,38145,38147,38150],{"className":38146},[106,144],[18,38148,2253],{"className":38149},[106,144],[18,38151,38153,38157,38234],{"className":38152},[106,144],[18,38154],{"className":38155},[250,6012,141,1195,38156],"size6",[18,38158,38160],{"className":38159},[5654],[18,38161,38163,38225],{"className":38162},[119,120],[18,38164,38166,38222],{"className":38165},[124],[18,38167,38170,38194,38204],{"className":38168,"style":38169},[128],"height:0.9584em;",[18,38171,38173,38176],{"style":38172},"top:-2.656em;",[18,38174],{"className":38175,"style":697},[136],[18,38177,38179],{"className":38178},[141,1195,1196,144],[18,38180,38182,38185,38188,38191],{"className":38181},[106,144],[18,38183,2266],{"className":38184},[106,144],[18,38186,20986],{"className":38187,"style":21038},[106,110,144],[18,38189,7987],{"className":38190},[2072,144],[18,38192,4031],{"className":38193},[106,110,144],[18,38195,38197,38200],{"style":38196},"top:-3.2255em;",[18,38198],{"className":38199,"style":697},[136],[18,38201],{"className":38202,"style":38203},[6047,144],"border-bottom-width:0.049em;",[18,38205,38207,38210],{"style":38206},"top:-3.4624em;",[18,38208],{"className":38209,"style":697},[136],[18,38211,38213],{"className":38212},[141,1195,1196,144],[18,38214,38216,38219],{"className":38215},[106,144],[18,38217,2266],{"className":38218},[106,144],[18,38220,20986],{"className":38221,"style":21038},[106,110,144],[18,38223,152],{"className":38224},[151],[18,38226,38228],{"className":38227},[124],[18,38229,38232],{"className":38230,"style":38231},[128],"height:0.4829em;",[18,38233],{},[18,38235],{"className":38236},[267,6012,141,1195,38156],[11,38238,38239,38240,38243],{},"（见 Tsybakov 第 1 章）。",[717,38241,38242],{},"下界","用 Fano。",[11,38245,36192,38246,38320,38321,38403,38404,38495,38496,38524,38525,38556],{},[18,38247,38249,38275],{"className":38248,"translate":22},[21],[18,38250,38252],{"className":38251},[26],[28,38253,38254],{"xmlns":30},[32,38255,38256,38272],{},[35,38257,38258,38260,38262,38264,38266,38268,38270],{},[41,38259,18328],{},[49,38261,921],{},[49,38263,74],{"stretchy":73},[45,38265,430],{},[49,38267,52],{"separator":51},[45,38269,47],{},[49,38271,84],{"stretchy":73},[86,38273,38274],{"encoding":88},"h\\in(0,1)",[18,38276,38278,38296],{"className":38277,"ariaHidden":51},[93],[18,38279,38281,38284,38287,38290,38293],{"className":38280},[97],[18,38282],{"className":38283,"style":1422},[101],[18,38285,18328],{"className":38286},[106,110],[18,38288],{"className":38289,"style":228},[169],[18,38291,921],{"className":38292},[232],[18,38294],{"className":38295,"style":228},[169],[18,38297,38299,38302,38305,38308,38311,38314,38317],{"className":38298},[97],[18,38300],{"className":38301,"style":242},[101],[18,38303,74],{"className":38304},[250],[18,38306,430],{"className":38307},[106],[18,38309,52],{"className":38310},[165],[18,38312],{"className":38313,"style":170},[169],[18,38315,47],{"className":38316},[106],[18,38318,84],{"className":38319},[267],"，把 ",[18,38322,38324,38350],{"className":38323,"translate":22},[21],[18,38325,38327],{"className":38326},[26],[28,38328,38329],{"xmlns":30},[32,38330,38331,38347],{},[35,38332,38333,38335,38337,38339,38341],{},[49,38334,945],{"stretchy":73},[45,38336,430],{},[49,38338,52],{"separator":51},[45,38340,47],{},[1556,38342,38343,38345],{},[49,38344,965],{"stretchy":73},[41,38346,4031],{},[86,38348,38349],{"encoding":88},"[0,1]^d",[18,38351,38353],{"className":38352,"ariaHidden":51},[93],[18,38354,38356,38359,38362,38365,38368,38371,38374],{"className":38355},[97],[18,38357],{"className":38358,"style":35152},[101],[18,38360,945],{"className":38361},[250],[18,38363,430],{"className":38364},[106],[18,38366,52],{"className":38367},[165],[18,38369],{"className":38370,"style":170},[169],[18,38372,47],{"className":38373},[106],[18,38375,38377,38380],{"className":38376},[267],[18,38378,965],{"className":38379},[267],[18,38381,38383],{"className":38382},[115],[18,38384,38386],{"className":38385},[119],[18,38387,38389],{"className":38388},[124],[18,38390,38392],{"className":38391,"style":35186},[128],[18,38393,38394,38397],{"style":1962},[18,38395],{"className":38396,"style":137},[136],[18,38398,38400],{"className":38399},[141,142,143,144],[18,38401,4031],{"className":38402},[106,110,144]," 分成 ",[18,38405,38407,38433],{"className":38406,"translate":22},[21],[18,38408,38410],{"className":38409},[26],[28,38411,38412],{"xmlns":30},[32,38413,38414,38430],{},[35,38415,38416,38418,38420],{},[41,38417,28169],{},[49,38419,433],{},[1556,38421,38422,38424],{},[41,38423,18328],{},[35,38425,38426,38428],{},[49,38427,2253],{},[41,38429,4031],{},[86,38431,38432],{"encoding":88},"m=h^{-d}",[18,38434,38436,38454],{"className":38435,"ariaHidden":51},[93],[18,38437,38439,38442,38445,38448,38451],{"className":38438},[97],[18,38440],{"className":38441,"style":1083},[101],[18,38443,28169],{"className":38444},[106,110],[18,38446],{"className":38447,"style":228},[169],[18,38449,433],{"className":38450},[232],[18,38452],{"className":38453,"style":228},[169],[18,38455,38457,38460],{"className":38456},[97],[18,38458],{"className":38459,"style":35186},[101],[18,38461,38463,38466],{"className":38462},[106],[18,38464,18328],{"className":38465},[106,110],[18,38467,38469],{"className":38468},[115],[18,38470,38472],{"className":38471},[119],[18,38473,38475],{"className":38474},[124],[18,38476,38478],{"className":38477,"style":35186},[128],[18,38479,38480,38483],{"style":1962},[18,38481],{"className":38482,"style":137},[136],[18,38484,38486],{"className":38485},[141,142,143,144],[18,38487,38489,38492],{"className":38488},[106,144],[18,38490,2253],{"className":38491},[106,144],[18,38493,4031],{"className":38494},[106,110,144]," 个边长为 ",[18,38497,38499,38512],{"className":38498,"translate":22},[21],[18,38500,38502],{"className":38501},[26],[28,38503,38504],{"xmlns":30},[32,38505,38506,38510],{},[35,38507,38508],{},[41,38509,18328],{},[86,38511,18328],{"encoding":88},[18,38513,38515],{"className":38514,"ariaHidden":51},[93],[18,38516,38518,38521],{"className":38517},[97],[18,38519],{"className":38520,"style":295},[101],[18,38522,18328],{"className":38523},[106,110]," 的小方块。取一个光滑的鼓包函数 ",[18,38526,38528,38543],{"className":38527,"translate":22},[21],[18,38529,38531],{"className":38530},[26],[28,38532,38533],{"xmlns":30},[32,38534,38535,38540],{},[35,38536,38537],{},[41,38538,38539],{},"φ",[86,38541,38542],{"encoding":88},"\\varphi",[18,38544,38546],{"className":38545,"ariaHidden":51},[93],[18,38547,38549,38553],{"className":38548},[97],[18,38550],{"className":38551,"style":38552},[101],"height:0.625em;vertical-align:-0.1944em;",[18,38554,38539],{"className":38555},[106,110],"，支撑在单位方块内，令",[18,38558,38560],{"className":38559,"translate":22},[875],[18,38561,38563,38614],{"className":38562,"translate":22},[21],[18,38564,38566],{"className":38565},[26],[28,38567,38568],{"xmlns":30,"display":884},[32,38569,38570,38611],{},[35,38571,38572,38578,38580,38583,38585,38587,38589,38591,38607,38609],{},[38,38573,38574,38576],{},[41,38575,38539],{},[41,38577,19811],{},[49,38579,74],{"stretchy":73},[41,38581,38582],{},"x",[49,38584,84],{"stretchy":73},[49,38586,433],{},[41,38588,38539],{},[49,38590,74],{"fence":51,"stretchy":51,"minsize":5662,"maxsize":5662},[5654,38592,38593,38605],{},[35,38594,38595,38597,38599],{},[41,38596,38582],{},[49,38598,2253],{},[38,38600,38601,38603],{},[41,38602,38582],{},[41,38604,19811],{},[41,38606,18328],{},[49,38608,84],{"fence":51,"stretchy":51,"minsize":5662,"maxsize":5662},[49,38610,52],{"separator":51},[86,38612,38613],{"encoding":88},"\\varphi_j(x)=\\varphi\\Bigl(\\frac{x-x_j}{h}\\Bigr),",[18,38615,38617,38681],{"className":38616,"ariaHidden":51},[93],[18,38618,38620,38623,38663,38666,38669,38672,38675,38678],{"className":38619},[97],[18,38621],{"className":38622,"style":19846},[101],[18,38624,38626,38629],{"className":38625},[106],[18,38627,38539],{"className":38628},[106,110],[18,38630,38632],{"className":38631},[115],[18,38633,38635,38655],{"className":38634},[119,120],[18,38636,38638,38652],{"className":38637},[124],[18,38639,38641],{"className":38640,"style":5921},[128],[18,38642,38643,38646],{"style":1155},[18,38644],{"className":38645,"style":137},[136],[18,38647,38649],{"className":38648},[141,142,143,144],[18,38650,19811],{"className":38651,"style":19928},[106,110,144],[18,38653,152],{"className":38654},[151],[18,38656,38658],{"className":38657},[124],[18,38659,38661],{"className":38660,"style":19938},[128],[18,38662],{},[18,38664,74],{"className":38665},[250],[18,38667,38582],{"className":38668},[106,110],[18,38670,84],{"className":38671},[267],[18,38673],{"className":38674,"style":228},[169],[18,38676,433],{"className":38677},[232],[18,38679],{"className":38680,"style":228},[169],[18,38682,38684,38688,38691,38697,38809,38815],{"className":38683},[97],[18,38685],{"className":38686,"style":38687},[101],"height:1.9463em;vertical-align:-0.686em;",[18,38689,38539],{"className":38690},[106,110],[18,38692,38694],{"className":38693},[250],[18,38695,74],{"className":38696},[1230,6082],[18,38698,38700,38703,38806],{"className":38699},[106],[18,38701],{"className":38702},[250,6012],[18,38704,38706],{"className":38705},[5654],[18,38707,38709,38798],{"className":38708},[119,120],[18,38710,38712,38795],{"className":38711},[124],[18,38713,38716,38727,38735],{"className":38714,"style":38715},[128],"height:1.2603em;",[18,38717,38718,38721],{"style":6028},[18,38719],{"className":38720,"style":697},[136],[18,38722,38724],{"className":38723},[106],[18,38725,18328],{"className":38726},[106,110],[18,38728,38729,38732],{"style":6040},[18,38730],{"className":38731,"style":697},[136],[18,38733],{"className":38734,"style":6048},[6047],[18,38736,38737,38740],{"style":6051},[18,38738],{"className":38739,"style":697},[136],[18,38741,38743,38746,38749,38752,38755],{"className":38742},[106],[18,38744,38582],{"className":38745},[106,110],[18,38747],{"className":38748,"style":2068},[169],[18,38750,2253],{"className":38751},[2072],[18,38753],{"className":38754,"style":2068},[169],[18,38756,38758,38761],{"className":38757},[106],[18,38759,38582],{"className":38760},[106,110],[18,38762,38764],{"className":38763},[115],[18,38765,38767,38787],{"className":38766},[119,120],[18,38768,38770,38784],{"className":38769},[124],[18,38771,38773],{"className":38772,"style":5921},[128],[18,38774,38775,38778],{"style":1155},[18,38776],{"className":38777,"style":137},[136],[18,38779,38781],{"className":38780},[141,142,143,144],[18,38782,19811],{"className":38783,"style":19928},[106,110,144],[18,38785,152],{"className":38786},[151],[18,38788,38790],{"className":38789},[124],[18,38791,38793],{"className":38792,"style":19938},[128],[18,38794],{},[18,38796,152],{"className":38797},[151],[18,38799,38801],{"className":38800},[124],[18,38802,38804],{"className":38803,"style":6070},[128],[18,38805],{},[18,38807],{"className":38808},[267,6012],[18,38810,38812],{"className":38811},[267],[18,38813,84],{"className":38814},[1230,6082],[18,38816,52],{"className":38817},[165],[11,38819,18230,38820,38890,38891,38919,38920,39013],{},[18,38821,38823,38841],{"className":38822,"translate":22},[21],[18,38824,38826],{"className":38825},[26],[28,38827,38828],{"xmlns":30},[32,38829,38830,38838],{},[35,38831,38832],{},[38,38833,38834,38836],{},[41,38835,38582],{},[41,38837,19811],{},[86,38839,38840],{"encoding":88},"x_j",[18,38842,38844],{"className":38843,"ariaHidden":51},[93],[18,38845,38847,38850],{"className":38846},[97],[18,38848],{"className":38849,"style":30679},[101],[18,38851,38853,38856],{"className":38852},[106],[18,38854,38582],{"className":38855},[106,110],[18,38857,38859],{"className":38858},[115],[18,38860,38862,38882],{"className":38861},[119,120],[18,38863,38865,38879],{"className":38864},[124],[18,38866,38868],{"className":38867,"style":5921},[128],[18,38869,38870,38873],{"style":1155},[18,38871],{"className":38872,"style":137},[136],[18,38874,38876],{"className":38875},[141,142,143,144],[18,38877,19811],{"className":38878,"style":19928},[106,110,144],[18,38880,152],{"className":38881},[151],[18,38883,38885],{"className":38884},[124],[18,38886,38888],{"className":38887,"style":19938},[128],[18,38889],{}," 是第 ",[18,38892,38894,38907],{"className":38893,"translate":22},[21],[18,38895,38897],{"className":38896},[26],[28,38898,38899],{"xmlns":30},[32,38900,38901,38905],{},[35,38902,38903],{},[41,38904,19811],{},[86,38906,19811],{"encoding":88},[18,38908,38910],{"className":38909,"ariaHidden":51},[93],[18,38911,38913,38916],{"className":38912},[97],[18,38914],{"className":38915,"style":28340},[101],[18,38917,19811],{"className":38918,"style":19928},[106,110]," 个方块的中心。对任意子集 ",[18,38921,38923,38954],{"className":38922,"translate":22},[21],[18,38924,38926],{"className":38925},[26],[28,38927,38928],{"xmlns":30},[32,38929,38930,38951],{},[35,38931,38932,38935,38937,38939,38941,38943,38945,38947,38949],{},[41,38933,38934],{},"S",[49,38936,4548],{},[49,38938,1331],{"stretchy":73},[45,38940,47],{},[49,38942,52],{"separator":51},[49,38944,55],{},[49,38946,52],{"separator":51},[41,38948,28169],{},[49,38950,1349],{"stretchy":73},[86,38952,38953],{"encoding":88},"S\\subseteq\\{1,\\dots,m\\}",[18,38955,38957,38977],{"className":38956,"ariaHidden":51},[93],[18,38958,38960,38964,38968,38971,38974],{"className":38959},[97],[18,38961],{"className":38962,"style":38963},[101],"height:0.8193em;vertical-align:-0.136em;",[18,38965,38934],{"className":38966,"style":38967},[106,110],"margin-right:0.0576em;",[18,38969],{"className":38970,"style":228},[169],[18,38972,4548],{"className":38973},[232],[18,38975],{"className":38976,"style":228},[169],[18,38978,38980,38983,38986,38989,38992,38995,38998,39001,39004,39007,39010],{"className":38979},[97],[18,38981],{"className":38982,"style":242},[101],[18,38984,1331],{"className":38985},[250],[18,38987,47],{"className":38988},[106],[18,38990,52],{"className":38991},[165],[18,38993],{"className":38994,"style":170},[169],[18,38996,55],{"className":38997},[174],[18,38999],{"className":39000,"style":170},[169],[18,39002,52],{"className":39003},[165],[18,39005],{"className":39006,"style":170},[169],[18,39008,28169],{"className":39009},[106,110],[18,39011,1349],{"className":39012},[267],"，定义",[18,39015,39017],{"className":39016,"translate":22},[875],[18,39018,39020,39081],{"className":39019,"translate":22},[21],[18,39021,39023],{"className":39022},[26],[28,39024,39025],{"xmlns":30,"display":884},[32,39026,39027,39078],{},[35,39028,39029,39035,39037,39039,39051,39057,39059,39061,39063,39065,39068,39070,39076],{},[38,39030,39031,39033],{},[41,39032,1055],{},[41,39034,38934],{},[49,39036,433],{},[41,39038,16823],{},[890,39040,39041,39043],{},[49,39042,20369],{},[35,39044,39045,39047,39049],{},[41,39046,19811],{},[49,39048,921],{},[41,39050,38934],{},[38,39052,39053,39055],{},[41,39054,38539],{},[41,39056,19811],{},[49,39058,52],{"separator":51},[169,39060],{"width":7344},[41,39062,16823],{},[49,39064,433],{},[41,39066,39067],{},"c",[926,39069,4024],{},[1556,39071,39072,39074],{},[41,39073,18328],{},[41,39075,20986],{},[41,39077,968],{"mathvariant":316},[86,39079,39080],{"encoding":88},"f_S=a\\sum_{j\\in S}\\varphi_j,\\qquad a=c\\,h^\\beta.",[18,39082,39084,39140,39272],{"className":39083,"ariaHidden":51},[93],[18,39085,39087,39090,39131,39134,39137],{"className":39086},[97],[18,39088],{"className":39089,"style":2890},[101],[18,39091,39093,39096],{"className":39092},[106],[18,39094,1055],{"className":39095,"style":37405},[106,110],[18,39097,39099],{"className":39098},[115],[18,39100,39102,39123],{"className":39101},[119,120],[18,39103,39105,39120],{"className":39104},[124],[18,39106,39108],{"className":39107,"style":1152},[128],[18,39109,39111,39114],{"style":39110},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[18,39112],{"className":39113,"style":137},[136],[18,39115,39117],{"className":39116},[141,142,143,144],[18,39118,38934],{"className":39119,"style":38967},[106,110,144],[18,39121,152],{"className":39122},[151],[18,39124,39126],{"className":39125},[124],[18,39127,39129],{"className":39128,"style":159},[128],[18,39130],{},[18,39132],{"className":39133,"style":228},[169],[18,39135,433],{"className":39136},[232],[18,39138],{"className":39139,"style":228},[169],[18,39141,39143,39147,39150,39153,39208,39211,39251,39254,39257,39260,39263,39266,39269],{"className":39142},[97],[18,39144],{"className":39145,"style":39146},[101],"height:2.4804em;vertical-align:-1.4304em;",[18,39148,16823],{"className":39149},[106,110],[18,39151],{"className":39152,"style":170},[169],[18,39154,39156],{"className":39155},[985,986],[18,39157,39159,39199],{"className":39158},[119,120],[18,39160,39162,39196],{"className":39161},[124],[18,39163,39165,39186],{"className":39164,"style":24483},[128],[18,39166,39168,39171],{"style":39167},"top:-1.8557em;margin-left:0em;",[18,39169],{"className":39170,"style":20559},[136],[18,39172,39174],{"className":39173},[141,142,143,144],[18,39175,39177,39180,39183],{"className":39176},[106,144],[18,39178,19811],{"className":39179,"style":19928},[106,110,144],[18,39181,921],{"className":39182},[232,144],[18,39184,38934],{"className":39185,"style":38967},[106,110,144],[18,39187,39188,39191],{"style":20577},[18,39189],{"className":39190,"style":20559},[136],[18,39192,39193],{},[18,39194,20369],{"className":39195},[985,4198,4199],[18,39197,152],{"className":39198},[151],[18,39200,39202],{"className":39201},[124],[18,39203,39206],{"className":39204,"style":39205},[128],"height:1.4304em;",[18,39207],{},[18,39209],{"className":39210,"style":170},[169],[18,39212,39214,39217],{"className":39213},[106],[18,39215,38539],{"className":39216},[106,110],[18,39218,39220],{"className":39219},[115],[18,39221,39223,39243],{"className":39222},[119,120],[18,39224,39226,39240],{"className":39225},[124],[18,39227,39229],{"className":39228,"style":5921},[128],[18,39230,39231,39234],{"style":1155},[18,39232],{"className":39233,"style":137},[136],[18,39235,39237],{"className":39236},[141,142,143,144],[18,39238,19811],{"className":39239,"style":19928},[106,110,144],[18,39241,152],{"className":39242},[151],[18,39244,39246],{"className":39245},[124],[18,39247,39249],{"className":39248,"style":19938},[128],[18,39250],{},[18,39252,52],{"className":39253},[165],[18,39255],{"className":39256,"style":7676},[169],[18,39258],{"className":39259,"style":170},[169],[18,39261,16823],{"className":39262},[106,110],[18,39264],{"className":39265,"style":228},[169],[18,39267,433],{"className":39268},[232],[18,39270],{"className":39271,"style":228},[169],[18,39273,39275,39278,39281,39284,39313],{"className":39274},[97],[18,39276],{"className":39277,"style":16673},[101],[18,39279,39067],{"className":39280},[106,110],[18,39282],{"className":39283,"style":170},[169],[18,39285,39287,39290],{"className":39286},[106],[18,39288,18328],{"className":39289},[106,110],[18,39291,39293],{"className":39292},[115],[18,39294,39296],{"className":39295},[119],[18,39297,39299],{"className":39298},[124],[18,39300,39302],{"className":39301,"style":16673},[128],[18,39303,39304,39307],{"style":1637},[18,39305],{"className":39306,"style":137},[136],[18,39308,39310],{"className":39309},[141,142,143,144],[18,39311,20986],{"className":39312,"style":21038},[106,110,144],[18,39314,968],{"className":39315},[106],[11,39317,39318,39319,39347,39348,39468,39469,39528,39529,39557,39558,39586,39587,1990],{},"当 ",[18,39320,39322,39335],{"className":39321,"translate":22},[21],[18,39323,39325],{"className":39324},[26],[28,39326,39327],{"xmlns":30},[32,39328,39329,39333],{},[35,39330,39331],{},[41,39332,39067],{},[86,39334,39067],{"encoding":88},[18,39336,39338],{"className":39337,"ariaHidden":51},[93],[18,39339,39341,39344],{"className":39340},[97],[18,39342],{"className":39343,"style":1083},[101],[18,39345,39067],{"className":39346},[106,110]," 足够小时，",[18,39349,39351,39383],{"className":39350,"translate":22},[21],[18,39352,39354],{"className":39353},[26],[28,39355,39356],{"xmlns":30},[32,39357,39358,39380],{},[35,39359,39360,39366,39368,39370,39372,39374,39376,39378],{},[38,39361,39362,39364],{},[41,39363,1055],{},[41,39365,38934],{},[49,39367,921],{},[41,39369,37817],{"mathvariant":316},[49,39371,74],{"stretchy":73},[41,39373,20986],{},[49,39375,52],{"separator":51},[41,39377,12582],{},[49,39379,84],{"stretchy":73},[86,39381,39382],{"encoding":88},"f_S\\in\\Sigma(\\beta,L)",[18,39384,39386,39441],{"className":39385,"ariaHidden":51},[93],[18,39387,39389,39392,39432,39435,39438],{"className":39388},[97],[18,39390],{"className":39391,"style":2890},[101],[18,39393,39395,39398],{"className":39394},[106],[18,39396,1055],{"className":39397,"style":37405},[106,110],[18,39399,39401],{"className":39400},[115],[18,39402,39404,39424],{"className":39403},[119,120],[18,39405,39407,39421],{"className":39406},[124],[18,39408,39410],{"className":39409,"style":1152},[128],[18,39411,39412,39415],{"style":39110},[18,39413],{"className":39414,"style":137},[136],[18,39416,39418],{"className":39417},[141,142,143,144],[18,39419,38934],{"className":39420,"style":38967},[106,110,144],[18,39422,152],{"className":39423},[151],[18,39425,39427],{"className":39426},[124],[18,39428,39430],{"className":39429,"style":159},[128],[18,39431],{},[18,39433],{"className":39434,"style":228},[169],[18,39436,921],{"className":39437},[232],[18,39439],{"className":39440,"style":228},[169],[18,39442,39444,39447,39450,39453,39456,39459,39462,39465],{"className":39443},[97],[18,39445],{"className":39446,"style":242},[101],[18,39448,37817],{"className":39449},[106],[18,39451,74],{"className":39452},[250],[18,39454,20986],{"className":39455,"style":21038},[106,110],[18,39457,52],{"className":39458},[165],[18,39460],{"className":39461,"style":170},[169],[18,39463,12582],{"className":39464},[106,110],[18,39466,84],{"className":39467},[267],"：鼓包的高度是 ",[18,39470,39472,39490],{"className":39471,"translate":22},[21],[18,39473,39475],{"className":39474},[26],[28,39476,39477],{"xmlns":30},[32,39478,39479,39487],{},[35,39480,39481],{},[1556,39482,39483,39485],{},[41,39484,18328],{},[41,39486,20986],{},[86,39488,39489],{"encoding":88},"h^\\beta",[18,39491,39493],{"className":39492,"ariaHidden":51},[93],[18,39494,39496,39499],{"className":39495},[97],[18,39497],{"className":39498,"style":35186},[101],[18,39500,39502,39505],{"className":39501},[106],[18,39503,18328],{"className":39504},[106,110],[18,39506,39508],{"className":39507},[115],[18,39509,39511],{"className":39510},[119],[18,39512,39514],{"className":39513},[124],[18,39515,39517],{"className":39516,"style":35186},[128],[18,39518,39519,39522],{"style":1962},[18,39520],{"className":39521,"style":137},[136],[18,39523,39525],{"className":39524},[141,142,143,144],[18,39526,20986],{"className":39527,"style":21038},[106,110,144],"、宽度是 ",[18,39530,39532,39545],{"className":39531,"translate":22},[21],[18,39533,39535],{"className":39534},[26],[28,39536,39537],{"xmlns":30},[32,39538,39539,39543],{},[35,39540,39541],{},[41,39542,18328],{},[86,39544,18328],{"encoding":88},[18,39546,39548],{"className":39547,"ariaHidden":51},[93],[18,39549,39551,39554],{"className":39550},[97],[18,39552],{"className":39553,"style":295},[101],[18,39555,18328],{"className":39556},[106,110],"，所以 ",[18,39559,39561,39574],{"className":39560,"translate":22},[21],[18,39562,39564],{"className":39563},[26],[28,39565,39566],{"xmlns":30},[32,39567,39568,39572],{},[35,39569,39570],{},[41,39571,20986],{},[86,39573,37789],{"encoding":88},[18,39575,39577],{"className":39576,"ariaHidden":51},[93],[18,39578,39580,39583],{"className":39579},[97],[18,39581],{"className":39582,"style":2890},[101],[18,39584,20986],{"className":39585,"style":21038},[106,110]," 阶导数的量级是 ",[18,39588,39590,39620],{"className":39589,"translate":22},[21],[18,39591,39593],{"className":39592},[26],[28,39594,39595],{"xmlns":30},[32,39596,39597,39617],{},[35,39598,39599,39605,39607,39613,39615],{},[1556,39600,39601,39603],{},[41,39602,18328],{},[41,39604,20986],{},[41,39606,317],{"mathvariant":316},[1556,39608,39609,39611],{},[41,39610,18328],{},[41,39612,20986],{},[49,39614,433],{},[45,39616,47],{},[86,39618,39619],{"encoding":88},"h^\\beta\u002Fh^\\beta=1",[18,39621,39623,39699],{"className":39622,"ariaHidden":51},[93],[18,39624,39626,39629,39658,39661,39690,39693,39696],{"className":39625},[97],[18,39627],{"className":39628,"style":35152},[101],[18,39630,39632,39635],{"className":39631},[106],[18,39633,18328],{"className":39634},[106,110],[18,39636,39638],{"className":39637},[115],[18,39639,39641],{"className":39640},[119],[18,39642,39644],{"className":39643},[124],[18,39645,39647],{"className":39646,"style":35186},[128],[18,39648,39649,39652],{"style":1962},[18,39650],{"className":39651,"style":137},[136],[18,39653,39655],{"className":39654},[141,142,143,144],[18,39656,20986],{"className":39657,"style":21038},[106,110,144],[18,39659,317],{"className":39660},[106],[18,39662,39664,39667],{"className":39663},[106],[18,39665,18328],{"className":39666},[106,110],[18,39668,39670],{"className":39669},[115],[18,39671,39673],{"className":39672},[119],[18,39674,39676],{"className":39675},[124],[18,39677,39679],{"className":39678,"style":35186},[128],[18,39680,39681,39684],{"style":1962},[18,39682],{"className":39683,"style":137},[136],[18,39685,39687],{"className":39686},[141,142,143,144],[18,39688,20986],{"className":39689,"style":21038},[106,110,144],[18,39691],{"className":39692,"style":228},[169],[18,39694,433],{"className":39695},[232],[18,39697],{"className":39698,"style":228},[169],[18,39700,39702,39705],{"className":39701},[97],[18,39703],{"className":39704,"style":505},[101],[18,39706,47],{"className":39707},[106],[11,39709,39710,39711,39872,39873,39932,39933,10363],{},"两个函数 ",[18,39712,39714,39744],{"className":39713,"translate":22},[21],[18,39715,39717],{"className":39716},[26],[28,39718,39719],{"xmlns":30},[32,39720,39721,39741],{},[35,39722,39723,39729,39731],{},[38,39724,39725,39727],{},[41,39726,1055],{},[41,39728,38934],{},[49,39730,52],{"separator":51},[38,39732,39733,39735],{},[41,39734,1055],{},[1556,39736,39737,39739],{},[41,39738,38934],{},[49,39740,32629],{"mathvariant":316,"lspace":32628,"rspace":32628},[86,39742,39743],{"encoding":88},"f_S,f_{S'}",[18,39745,39747],{"className":39746,"ariaHidden":51},[93],[18,39748,39750,39753,39793,39796,39799],{"className":39749},[97],[18,39751],{"className":39752,"style":2890},[101],[18,39754,39756,39759],{"className":39755},[106],[18,39757,1055],{"className":39758,"style":37405},[106,110],[18,39760,39762],{"className":39761},[115],[18,39763,39765,39785],{"className":39764},[119,120],[18,39766,39768,39782],{"className":39767},[124],[18,39769,39771],{"className":39770,"style":1152},[128],[18,39772,39773,39776],{"style":39110},[18,39774],{"className":39775,"style":137},[136],[18,39777,39779],{"className":39778},[141,142,143,144],[18,39780,38934],{"className":39781,"style":38967},[106,110,144],[18,39783,152],{"className":39784},[151],[18,39786,39788],{"className":39787},[124],[18,39789,39791],{"className":39790,"style":159},[128],[18,39792],{},[18,39794,52],{"className":39795},[165],[18,39797],{"className":39798,"style":170},[169],[18,39800,39802,39805],{"className":39801},[106],[18,39803,1055],{"className":39804,"style":37405},[106,110],[18,39806,39808],{"className":39807},[115],[18,39809,39811,39864],{"className":39810},[119,120],[18,39812,39814,39861],{"className":39813},[124],[18,39815,39817],{"className":39816,"style":1152},[128],[18,39818,39819,39822],{"style":39110},[18,39820],{"className":39821,"style":137},[136],[18,39823,39825],{"className":39824},[141,142,143,144],[18,39826,39828],{"className":39827},[106,144],[18,39829,39831,39834],{"className":39830},[106,144],[18,39832,38934],{"className":39833,"style":38967},[106,110,144],[18,39835,39837],{"className":39836},[115],[18,39838,39840],{"className":39839},[119],[18,39841,39843],{"className":39842},[124],[18,39844,39847],{"className":39845,"style":39846},[128],"height:0.6828em;",[18,39848,39849,39852],{"style":33595},[18,39850],{"className":39851,"style":1191},[136],[18,39853,39855],{"className":39854},[141,1195,1196,144],[18,39856,39858],{"className":39857},[106,144],[18,39859,32629],{"className":39860},[106,144],[18,39862,152],{"className":39863},[151],[18,39865,39867],{"className":39866},[124],[18,39868,39870],{"className":39869,"style":159},[128],[18,39871],{}," 的 ",[18,39874,39876,39894],{"className":39875,"translate":22},[21],[18,39877,39879],{"className":39878},[26],[28,39880,39881],{"xmlns":30},[32,39882,39883,39891],{},[35,39884,39885],{},[1556,39886,39887,39889],{},[41,39888,12582],{},[45,39890,2266],{},[86,39892,39893],{"encoding":88},"L^2",[18,39895,39897],{"className":39896,"ariaHidden":51},[93],[18,39898,39900,39903],{"className":39899},[97],[18,39901],{"className":39902,"style":2416},[101],[18,39904,39906,39909],{"className":39905},[106],[18,39907,12582],{"className":39908},[106,110],[18,39910,39912],{"className":39911},[115],[18,39913,39915],{"className":39914},[119],[18,39916,39918],{"className":39917},[124],[18,39919,39921],{"className":39920,"style":2416},[128],[18,39922,39923,39926],{"style":1962},[18,39924],{"className":39925,"style":137},[136],[18,39927,39929],{"className":39928},[141,142,143,144],[18,39930,2266],{"className":39931},[106,144]," 距离只取决于它们不同的那些鼓包。若 ",[18,39934,39936,39968],{"className":39935,"translate":22},[21],[18,39937,39939],{"className":39938},[26],[28,39940,39941],{"xmlns":30},[32,39942,39943,39965],{},[35,39944,39945,39947,39949,39952,39958,39960,39962],{},[41,39946,11225],{"mathvariant":316},[41,39948,38934],{},[41,39950,39951],{"mathvariant":316},"△",[1556,39953,39954,39956],{},[41,39955,38934],{},[49,39957,32629],{"mathvariant":316,"lspace":32628,"rspace":32628},[41,39959,11225],{"mathvariant":316},[49,39961,433],{},[41,39963,39964],{},"k",[86,39966,39967],{"encoding":88},"|S\\triangle 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个码字（",[18,40947,40949,40971],{"className":40948,"translate":22},[21],[18,40950,40952],{"className":40951},[26],[28,40953,40954],{"xmlns":30},[32,40955,40956,40968],{},[35,40957,40958,40964,40966],{},[38,40959,40960,40962],{},[41,40961,39067],{},[45,40963,430],{},[49,40965,6885],{},[45,40967,430],{},[86,40969,40970],{"encoding":88},"c_0>0",[18,40972,40974,41030],{"className":40973,"ariaHidden":51},[93],[18,40975,40977,40981,41021,41024,41027],{"className":40976},[97],[18,40978],{"className":40979,"style":40980},[101],"height:0.6891em;vertical-align:-0.15em;",[18,40982,40984,40987],{"className":40983},[106],[18,40985,39067],{"className":40986},[106,110],[18,40988,40990],{"className":40989},[115],[18,40991,40993,41013],{"className":40992},[119,120],[18,40994,40996,41010],{"className":40995},[124],[18,40997,40999],{"className":40998,"style":129},[128],[18,41000,41001,41004],{"style":1155},[18,41002],{"className":41003,"style":137},[136],[18,41005,41007],{"className":41006},[141,142,143,144],[18,41008,430],{"className":41009},[106,144],[18,41011,152],{"className":41012},[151],[18,41014,41016],{"className":41015},[124],[18,41017,41019],{"className":41018,"style":159},[128],[18,41020],{},[18,41022],{"className":41023,"style":228},[169],[18,41025,6885],{"className":41026},[232],[18,41028],{"className":41029,"style":228},[169],[18,41031,41033,41036],{"className":41032},[97],[18,41034],{"className":41035,"style":505},[101],[18,41037,430],{"className":41038},[106]," 是常数），任意两个码字至少在 ",[18,41041,41043,41061],{"className":41042,"translate":22},[21],[18,41044,41046],{"className":41045},[26],[28,41047,41048],{"xmlns":30},[32,41049,41050,41058],{},[35,41051,41052,41054,41056],{},[41,41053,28169],{},[41,41055,317],{"mathvariant":316},[45,41057,10447],{},[86,41059,41060],{"encoding":88},"m\u002F4",[18,41062,41064],{"className":41063,"ariaHidden":51},[93],[18,41065,41067,41070,41073],{"className":41066},[97],[18,41068],{"className":41069,"style":242},[101],[18,41071,28169],{"className":41072},[106,110],[18,41074,17128],{"className":41075},[106]," 个位置不同。于是",[18,41078,41080],{"className":41079,"translate":22},[875],[18,41081,41083,41147],{"className":41082,"translate":22},[21],[18,41084,41086],{"className":41085},[26],[28,41087,41088],{"xmlns":30,"display":884},[32,41089,41090,41144],{},[35,41091,41092,41094,41100,41102,41112,41124,41126,41132,41142],{},[49,41093,2248],{"stretchy":73},[38,41095,41096,41098],{},[41,41097,1055],{},[41,41099,38934],{},[49,41101,2253],{},[38,41103,41104,41106],{},[41,41105,1055],{},[1556,41107,41108,41110],{},[41,41109,38934],{},[49,41111,32629],{"mathvariant":316,"lspace":32628,"rspace":32628},[1589,41113,41114,41116,41122],{},[49,41115,2248],{"stretchy":73},[1556,41117,41118,41120],{},[41,41119,12582],{},[45,41121,2266],{},[45,41123,2266],{},[49,41125,4775],{},[38,41127,41128,41130],{},[41,41129,39067],{},[45,41131,47],{},[1556,41133,41134,41136],{},[41,41135,18328],{},[35,41137,41138,41140],{},[45,41139,2266],{},[41,41141,20986],{},[49,41143,52],{"separator":51},[86,41145,41146],{"encoding":88},"\\lVert f_S-f_{S'}\\rVert_{L^2}^2\n\\ge c_1h^{2\\beta},",[18,41148,41150,41208,41375],{"className":41149,"ariaHidden":51},[93],[18,41151,41153,41156,41159,41199,41202,41205],{"className":41152},[97],[18,41154],{"className":41155,"style":242},[101],[18,41157,2248],{"className":41158},[250],[18,41160,41162,41165],{"className":41161},[106],[18,41163,1055],{"className":41164,"style":37405},[106,110],[18,41166,41168],{"className":41167},[115],[18,41169,41171,41191],{"className":41170},[119,120],[18,41172,41174,41188],{"className":41173},[124],[18,41175,41177],{"className":41176,"style":1152},[128],[18,41178,41179,41182],{"style":39110},[18,41180],{"className":41181,"style":137},[136],[18,41183,41185],{"className":41184},[141,142,143,144],[18,41186,38934],{"className":41187,"style":38967},[106,110,144],[18,41189,152],{"className":41190},[151],[18,41192,41194],{"className":41193},[124],[18,41195,41197],{"className":41196,"style":159},[128],[18,41198],{},[18,41200],{"className":41201,"style":2068},[169],[18,41203,2253],{"className":41204},[2072],[18,41206],{"className":41207,"style":2068},[169],[18,41209,41211,41214,41286,41366,41369,41372],{"className":41210},[97],[18,41212],{"className":41213,"style":40240},[101],[18,41215,41217,41220],{"className":41216},[106],[18,41218,1055],{"className":41219,"style":37405},[106,110],[18,41221,41223],{"className":41222},[115],[18,41224,41226,41278],{"className":41225},[119,120],[18,41227,41229,41275],{"className":41228},[124],[18,41230,41232],{"className":41231,"style":1152},[128],[18,41233,41234,41237],{"style":39110},[18,41235],{"className":41236,"style":137},[136],[18,41238,41240],{"className":41239},[141,142,143,144],[18,41241,41243],{"className":41242},[106,144],[18,41244,41246,41249],{"className":41245},[106,144],[18,41247,38934],{"className":41248,"style":38967},[106,110,144],[18,41250,41252],{"className":41251},[115],[18,41253,41255],{"className":41254},[119],[18,41256,41258],{"className":41257},[124],[18,41259,41261],{"className":41260,"style":39846},[128],[18,41262,41263,41266],{"style":33595},[18,41264],{"className":41265,"style":1191},[136],[18,41267,41269],{"className":41268},[141,1195,1196,144],[18,41270,41272],{"className":41271},[106,144],[18,41273,32629],{"className":41274},[106,144],[18,41276,152],{"className":41277},[151],[18,41279,41281],{"className":41280},[124],[18,41282,41284],{"className":41283,"style":159},[128],[18,41285],{},[18,41287,41289,41292],{"className":41288},[267],[18,41290,2248],{"className":41291},[267],[18,41293,41295],{"className":41294},[115],[18,41296,41298,41358],{"className":41297},[119,120],[18,41299,41301,41355],{"className":41300},[124],[18,41302,41304,41344],{"className":41303,"style":10775},[128],[18,41305,41306,41309],{"style":40333},[18,41307],{"className":41308,"style":137},[136],[18,41310,41312],{"className":41311},[141,142,143,144],[18,41313,41315],{"className":41314},[106,144],[18,41316,41318,41321],{"className":41317},[106,144],[18,41319,12582],{"className":41320},[106,110,144],[18,41322,41324],{"className":41323},[115],[18,41325,41327],{"className":41326},[119],[18,41328,41330],{"className":41329},[124],[18,41331,41333],{"className":41332,"style":38023},[128],[18,41334,41335,41338],{"style":33595},[18,41336],{"className":41337,"style":1191},[136],[18,41339,41341],{"className":41340},[141,1195,1196,144],[18,41342,2266],{"className":41343},[106,144],[18,41345,41346,41349],{"style":1637},[18,41347],{"className":41348,"style":137},[136],[18,41350,41352],{"className":41351},[141,142,143,144],[18,41353,2266],{"className":41354},[106,144],[18,41356,152],{"className":41357},[151],[18,41359,41361],{"className":41360},[124],[18,41362,41364],{"className":41363,"style":40392},[128],[18,41365],{},[18,41367],{"className":41368,"style":228},[169],[18,41370,4775],{"className":41371},[232],[18,41373],{"className":41374,"style":228},[169],[18,41376,41378,41381,41421,41456],{"className":41377},[97],[18,41379],{"className":41380,"style":34510},[101],[18,41382,41384,41387],{"className":41383},[106],[18,41385,39067],{"className":41386},[106,110],[18,41388,41390],{"className":41389},[115],[18,41391,41393,41413],{"className":41392},[119,120],[18,41394,41396,41410],{"className":41395},[124],[18,41397,41399],{"className":41398,"style":129},[128],[18,41400,41401,41404],{"style":1155},[18,41402],{"className":41403,"style":137},[136],[18,41405,41407],{"className":41406},[141,142,143,144],[18,41408,47],{"className":41409},[106,144],[18,41411,152],{"className":41412},[151],[18,41414,41416],{"className":41415},[124],[18,41417,41419],{"className":41418,"style":159},[128],[18,41420],{},[18,41422,41424,41427],{"className":41423},[106],[18,41425,18328],{"className":41426},[106,110],[18,41428,41430],{"className":41429},[115],[18,41431,41433],{"className":41432},[119],[18,41434,41436],{"className":41435},[124],[18,41437,41439],{"className":41438,"style":16673},[128],[18,41440,41441,41444],{"style":1637},[18,41442],{"className":41443,"style":137},[136],[18,41445,41447],{"className":41446},[141,142,143,144],[18,41448,41450,41453],{"className":41449},[106,144],[18,41451,2266],{"className":41452},[106,144],[18,41454,20986],{"className":41455,"style":21038},[106,110,144],[18,41457,52],{"className":41458},[165],[11,41460,41461,41462,1990],{},"即平方间隔 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KL 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h^{2\\beta}",[18,42595,42597,42623],{"className":42596,"ariaHidden":51},[93],[18,42598,42600,42603,42608,42611,42614,42617,42620],{"className":42599},[97],[18,42601],{"className":42602,"style":2890},[101],[18,42604,15174,42606],{"className":42605},[985],[18,42607,15177],{"style":6355},[18,42609],{"className":42610,"style":170},[169],[18,42612,3526],{"className":42613,"style":246},[106,110],[18,42615],{"className":42616,"style":228},[169],[18,42618,26833],{"className":42619},[232],[18,42621],{"className":42622,"style":228},[169],[18,42624,42626,42629,42632],{"className":42625},[97],[18,42627],{"className":42628,"style":35186},[101],[18,42630,64],{"className":42631},[106,110],[18,42633,42635,42638],{"className":42634},[106],[18,42636,18328],{"className":42637},[106,110],[18,42639,42641],{"className":42640},[115],[18,42642,42644],{"className":42643},[119],[18,42645,42647],{"className":42646},[124],[18,42648,42650],{"className":42649,"style":35186},[128],[18,42651,42652,42655],{"style":1962},[18,42653],{"className":42654,"style":137},[136],[18,42656,42658],{"className":42657},[141,142,143,144],[18,42659,42661,42664],{"className":42660},[106,144],[18,42662,2266],{"className":42663},[106,144],[18,42665,20986],{"className":42666,"style":21038},[106,110,144],"，即",[18,42669,42671],{"className":42670,"translate":22},[875],[18,42672,42674,42747],{"className":42673,"translate":22},[21],[18,42675,42677],{"className":42676},[26],[28,42678,42679],{"xmlns":30,"display":884},[32,42680,42681,42744],{},[35,42682,42683,42693,42695,42697,42707,42710,42713,42715,42717,42720,42742],{},[1556,42684,42685,42687],{},[41,42686,18328],{},[35,42688,42689,42691],{},[49,42690,2253],{},[41,42692,4031],{},[49,42694,26898],{},[41,42696,64],{},[1556,42698,42699,42701],{},[41,42700,18328],{},[35,42702,42703,42705],{},[45,42704,2266],{},[41,42706,20986],{},[169,42708],{"width":42709},"1em",[49,42711,42712],{},"⟺",[169,42714],{"width":42709},[41,42716,18328],{},[49,42718,42719],{},"≲",[1556,42721,42722,42724],{},[41,42723,64],{},[35,42725,42726,42728],{},[49,42727,2253],{},[5654,42729,42730,42732],{},[45,42731,47],{},[35,42733,42734,42736,42738,42740],{},[45,42735,2266],{},[41,42737,20986],{},[49,42739,7987],{},[41,42741,4031],{},[41,42743,968],{"mathvariant":316},[86,42745,42746],{"encoding":88},"h^{-d}\\gtrsim n h^{2\\beta}\n\\quad\\Longleftrightarrow\\quad\nh\\lesssim n^{-\\frac{1}{2\\beta+d}}.",[18,42748,42750,42801,42862,42881],{"className":42749,"ariaHidden":51},[93],[18,42751,42753,42757,42792,42795,42798],{"className":42752},[97],[18,42754],{"className":42755,"style":42756},[101],"height:1.1287em;vertical-align:-0.2296em;",[18,42758,42760,42763],{"className":42759},[106],[18,42761,18328],{"className":42762},[106,110],[18,42764,42766],{"className":42765},[115],[18,42767,42769],{"className":42768},[119],[18,42770,42772],{"className":42771},[124],[18,42773,42775],{"className":42774,"style":16673},[128],[18,42776,42777,42780],{"style":1637},[18,42778],{"className":42779,"style":137},[136],[18,42781,42783],{"className":42782},[141,142,143,144],[18,42784,42786,42789],{"className":42785},[106,144],[18,42787,2253],{"className":42788},[106,144],[18,42790,4031],{"className":42791},[106,110,144],[18,42793],{"className":42794,"style":228},[169],[18,42796,26898],{"className":42797},[232,27260],[18,42799],{"className":42800,"style":228},[169],[18,42802,42804,42808,42811,42846,42850,42853,42856,42859],{"className":42803},[97],[18,42805],{"className":42806,"style":42807},[101],"height:0.9231em;vertical-align:-0.024em;",[18,42809,64],{"className":42810},[106,110],[18,42812,42814,42817],{"className":42813},[106],[18,42815,18328],{"className":42816},[106,110],[18,42818,42820],{"className":42819},[115],[18,42821,42823],{"className":42822},[119],[18,42824,42826],{"className":42825},[124],[18,42827,42829],{"className":42828,"style":16673},[128],[18,42830,42831,42834],{"style":1637},[18,42832],{"className":42833,"style":137},[136],[18,42835,42837],{"className":42836},[141,142,143,144],[18,42838,42840,42843],{"className":42839},[106,144],[18,42841,2266],{"className":42842},[106,144],[18,42844,20986],{"className":42845,"style":21038},[106,110,144],[18,42847],{"className":42848,"style":42849},[169],"margin-right:1em;",[18,42851],{"className":42852,"style":228},[169],[18,42854,42712],{"className":42855},[232],[18,42857],{"className":42858,"style":42849},[169],[18,42860],{"className":42861,"style":228},[169],[18,42863,42865,42869,42872,42875,42878],{"className":42864},[97],[18,42866],{"className":42867,"style":42868},[101],"height:0.9592em;vertical-align:-0.2296em;",[18,42870,18328],{"className":42871},[106,110],[18,42873],{"className":42874,"style":228},[169],[18,42876,42719],{"className":42877},[232,27260],[18,42879],{"className":42880,"style":228},[169],[18,42882,42884,42888,42999],{"className":42883},[97],[18,42885],{"className":42886,"style":42887},[101],"height:1.0368em;",[18,42889,42891,42894],{"className":42890},[106],[18,42892,64],{"className":42893},[106,110],[18,42895,42897],{"className":42896},[115],[18,42898,42900],{"className":42899},[119],[18,42901,42903],{"className":42902},[124],[18,42904,42906],{"className":42905,"style":42887},[128],[18,42907,42908,42911],{"style":38137},[18,42909],{"className":42910,"style":697},[136],[18,42912,42914],{"className":42913},[141,142,143,144],[18,42915,42917,42920],{"className":42916},[106,144],[18,42918,2253],{"className":42919},[106,144],[18,42921,42923,42926,42996],{"className":42922},[106,144],[18,42924],{"className":42925},[250,6012,141,1195,38156],[18,42927,42929],{"className":42928},[5654],[18,42930,42932,42988],{"className":42931},[119,120],[18,42933,42935,42985],{"className":42934},[124],[18,42936,42939,42962,42970],{"className":42937,"style":42938},[128],"height:0.8443em;",[18,42940,42941,42944],{"style":38172},[18,42942],{"className":42943,"style":697},[136],[18,42945,42947],{"className":42946},[141,1195,1196,144],[18,42948,42950,42953,42956,42959],{"className":42949},[106,144],[18,42951,2266],{"className":42952},[106,144],[18,42954,20986],{"className":42955,"style":21038},[106,110,144],[18,42957,7987],{"className":42958},[2072,144],[18,42960,4031],{"className":42961},[106,110,144],[18,42963,42964,42967],{"style":38196},[18,42965],{"className":42966,"style":697},[136],[18,42968],{"className":42969,"style":38203},[6047,144],[18,42971,42973,42976],{"style":42972},"top:-3.384em;",[18,42974],{"className":42975,"style":697},[136],[18,42977,42979],{"className":42978},[141,1195,1196,144],[18,42980,42982],{"className":42981},[106,144],[18,42983,47],{"className":42984},[106,144],[18,42986,152],{"className":42987},[151],[18,42989,42991],{"className":42990},[124],[18,42992,42994],{"className":42993,"style":38231},[128],[18,42995],{},[18,42997],{"className":42998},[267,6012,141,1195,38156],[18,43000,968],{"className":43001},[106],[11,43003,36192,43004,43127],{},[18,43005,43007,43047],{"className":43006,"translate":22},[21],[18,43008,43010],{"className":43009},[26],[28,43011,43012],{"xmlns":30},[32,43013,43014,43044],{},[35,43015,43016,43018,43020],{},[41,43017,18328],{},[49,43019,17146],{},[1556,43021,43022,43024],{},[41,43023,64],{},[35,43025,43026,43028,43030,43032,43034,43036,43038,43040,43042],{},[49,43027,2253],{},[45,43029,47],{},[41,43031,317],{"mathvariant":316},[49,43033,74],{"stretchy":73},[45,43035,2266],{},[41,43037,20986],{},[49,43039,7987],{},[41,43041,4031],{},[49,43043,84],{"stretchy":73},[86,43045,43046],{"encoding":88},"h\\asymp n^{-1\u002F(2\\beta+d)}",[18,43048,43050,43068],{"className":43049,"ariaHidden":51},[93],[18,43051,43053,43056,43059,43062,43065],{"className":43052},[97],[18,43054],{"className":43055,"style":295},[101],[18,43057,18328],{"className":43058},[106,110],[18,43060],{"className":43061,"style":228},[169],[18,43063,17146],{"className":43064},[232],[18,43066],{"className":43067,"style":228},[169],[18,43069,43071,43074],{"className":43070},[97],[18,43072],{"className":43073,"style":27945},[101],[18,43075,43077,43080],{"className":43076},[106],[18,43078,64],{"className":43079},[106,110],[18,43081,43083],{"className":43082},[115],[18,43084,43086],{"className":43085},[119],[18,43087,43089],{"className":43088},[124],[18,43090,43092],{"className":43091,"style":27945},[128],[18,43093,43094,43097],{"style":1962},[18,43095],{"className":43096,"style":137},[136],[18,43098,43100],{"className":43099},[141,142,143,144],[18,43101,43103,43106,43109,43112,43115,43118,43121,43124],{"className":43102},[106,144],[18,43104,2253],{"className":43105},[106,144],[18,43107,335],{"className":43108},[106,144],[18,43110,74],{"className":43111},[250,144],[18,43113,2266],{"className":43114},[106,144],[18,43116,20986],{"className":43117,"style":21038},[106,110,144],[18,43119,7987],{"className":43120},[2072,144],[18,43122,4031],{"className":43123},[106,110,144],[18,43125,84],{"className":43126},[267,144],"，下界就是",[18,43129,43131],{"className":43130,"translate":22},[875],[18,43132,43134,43194],{"className":43133,"translate":22},[21],[18,43135,43137],{"className":43136},[26],[28,43138,43139],{"xmlns":30,"display":884},[32,43140,43141,43191],{},[35,43142,43143,43149,43151,43161,43163,43189],{},[38,43144,43145,43147],{},[41,43146,3526],{"mathvariant":3525},[41,43148,64],{},[49,43150,26898],{},[1556,43152,43153,43155],{},[41,43154,18328],{},[35,43156,43157,43159],{},[45,43158,2266],{},[41,43160,20986],{},[49,43162,17146],{},[1556,43164,43165,43167],{},[41,43166,64],{},[35,43168,43169,43171],{},[49,43170,2253],{},[5654,43172,43173,43179],{},[35,43174,43175,43177],{},[45,43176,2266],{},[41,43178,20986],{},[35,43180,43181,43183,43185,43187],{},[45,43182,2266],{},[41,43184,20986],{},[49,43186,7987],{},[41,43188,4031],{},[49,43190,52],{"separator":51},[86,43192,43193],{"encoding":88},"\\mathfrak M_n\\gtrsim h^{2\\beta}\n\\asymp n^{-\\frac{2\\beta}{2\\beta+d}},",[18,43195,43197,43252,43302],{"className":43196,"ariaHidden":51},[93],[18,43198,43200,43203,43243,43246,43249],{"className":43199},[97],[18,43201],{"className":43202,"style":42868},[101],[18,43204,43206,43209],{"className":43205},[106],[18,43207,3526],{"className":43208},[106,3607],[18,43210,43212],{"className":43211},[115],[18,43213,43215,43235],{"className":43214},[119,120],[18,43216,43218,43232],{"className":43217},[124],[18,43219,43221],{"className":43220,"style":202},[128],[18,43222,43223,43226],{"style":1155},[18,43224],{"className":43225,"style":137},[136],[18,43227,43229],{"className":43228},[141,142,143,144],[18,43230,64],{"className":43231},[106,110,144],[18,43233,152],{"className":43234},[151],[18,43236,43238],{"className":43237},[124],[18,43239,43241],{"className":43240,"style":159},[128],[18,43242],{},[18,43244],{"className":43245,"style":228},[169],[18,43247,26898],{"className":43248},[232,27260],[18,43250],{"className":43251,"style":228},[169],[18,43253,43255,43258,43293,43296,43299],{"className":43254},[97],[18,43256],{"className":43257,"style":16673},[101],[18,43259,43261,43264],{"className":43260},[106],[18,43262,18328],{"className":43263},[106,110],[18,43265,43267],{"className":43266},[115],[18,43268,43270],{"className":43269},[119],[18,43271,43273],{"className":43272},[124],[18,43274,43276],{"className":43275,"style":16673},[128],[18,43277,43278,43281],{"style":1637},[18,43279],{"className":43280,"style":137},[136],[18,43282,43284],{"className":43283},[141,142,143,144],[18,43285,43287,43290],{"className":43286},[106,144],[18,43288,2266],{"className":43289},[106,144],[18,43291,20986],{"className":43292,"style":21038},[106,110,144],[18,43294],{"className":43295,"style":228},[169],[18,43297,17146],{"className":43298},[232],[18,43300],{"className":43301,"style":228},[169],[18,43303,43305,43309,43421],{"className":43304},[97],[18,43306],{"className":43307,"style":43308},[101],"height:1.3111em;vertical-align:-0.1944em;",[18,43310,43312,43315],{"className":43311},[106],[18,43313,64],{"className":43314},[106,110],[18,43316,43318],{"className":43317},[115],[18,43319,43321],{"className":43320},[119],[18,43322,43324],{"className":43323},[124],[18,43325,43327],{"className":43326,"style":38116},[128],[18,43328,43329,43332],{"style":38137},[18,43330],{"className":43331,"style":697},[136],[18,43333,43335],{"className":43334},[141,142,143,144],[18,43336,43338,43341],{"className":43337},[106,144],[18,43339,2253],{"className":43340},[106,144],[18,43342,43344,43347,43418],{"className":43343},[106,144],[18,43345],{"className":43346},[250,6012,141,1195,38156],[18,43348,43350],{"className":43349},[5654],[18,43351,43353,43410],{"className":43352},[119,120],[18,43354,43356,43407],{"className":43355},[124],[18,43357,43359,43382,43390],{"className":43358,"style":38169},[128],[18,43360,43361,43364],{"style":38172},[18,43362],{"className":43363,"style":697},[136],[18,43365,43367],{"className":43366},[141,1195,1196,144],[18,43368,43370,43373,43376,43379],{"className":43369},[106,144],[18,43371,2266],{"className":43372},[106,144],[18,43374,20986],{"className":43375,"style":21038},[106,110,144],[18,43377,7987],{"className":43378},[2072,144],[18,43380,4031],{"className":43381},[106,110,144],[18,43383,43384,43387],{"style":38196},[18,43385],{"className":43386,"style":697},[136],[18,43388],{"className":43389,"style":38203},[6047,144],[18,43391,43392,43395],{"style":38206},[18,43393],{"className":43394,"style":697},[136],[18,43396,43398],{"className":43397},[141,1195,1196,144],[18,43399,43401,43404],{"className":43400},[106,144],[18,43402,2266],{"className":43403},[106,144],[18,43405,20986],{"className":43406,"style":21038},[106,110,144],[18,43408,152],{"className":43409},[151],[18,43411,43413],{"className":43412},[124],[18,43414,43416],{"className":43415,"style":38231},[128],[18,43417],{},[18,43419],{"className":43420},[267,6012,141,1195,38156],[18,43422,52],{"className":43423},[165],[11,43425,43426],{},"与上界同阶。证毕。",[11,43428,43429,43430,43677,43678,43706],{},"这个构造值得反复看：",[717,43431,43432,43433,43461,43462,43531,43532,43606,43607,43676],{},"鼓包的宽度 ",[18,43434,43436,43449],{"className":43435,"translate":22},[21],[18,43437,43439],{"className":43438},[26],[28,43440,43441],{"xmlns":30},[32,43442,43443,43447],{},[35,43444,43445],{},[41,43446,18328],{},[86,43448,18328],{"encoding":88},[18,43450,43452],{"className":43451,"ariaHidden":51},[93],[18,43453,43455,43458],{"className":43454},[97],[18,43456],{"className":43457,"style":295},[101],[18,43459,18328],{"className":43460},[106,110]," 控制两件事——损失间隔（",[18,43463,43465,43487],{"className":43464,"translate":22},[21],[18,43466,43468],{"className":43467},[26],[28,43469,43470],{"xmlns":30},[32,43471,43472,43484],{},[35,43473,43474],{},[1556,43475,43476,43478],{},[41,43477,18328],{},[35,43479,43480,43482],{},[45,43481,2266],{},[41,43483,20986],{},[86,43485,43486],{"encoding":88},"h^{2\\beta}",[18,43488,43490],{"className":43489,"ariaHidden":51},[93],[18,43491,43493,43496],{"className":43492},[97],[18,43494],{"className":43495,"style":35186},[101],[18,43497,43499,43502],{"className":43498},[106],[18,43500,18328],{"className":43501},[106,110],[18,43503,43505],{"className":43504},[115],[18,43506,43508],{"className":43507},[119],[18,43509,43511],{"className":43510},[124],[18,43512,43514],{"className":43513,"style":35186},[128],[18,43515,43516,43519],{"style":1962},[18,43517],{"className":43518,"style":137},[136],[18,43520,43522],{"className":43521},[141,142,143,144],[18,43523,43525,43528],{"className":43524},[106,144],[18,43526,2266],{"className":43527},[106,144],[18,43529,20986],{"className":43530,"style":21038},[106,110,144],"）和 KL（",[18,43533,43535,43559],{"className":43534,"translate":22},[21],[18,43536,43538],{"className":43537},[26],[28,43539,43540],{"xmlns":30},[32,43541,43542,43556],{},[35,43543,43544,43546],{},[41,43545,64],{},[1556,43547,43548,43550],{},[41,43549,18328],{},[35,43551,43552,43554],{},[45,43553,2266],{},[41,43555,20986],{},[86,43557,43558],{"encoding":88},"n h^{2\\beta}",[18,43560,43562],{"className":43561,"ariaHidden":51},[93],[18,43563,43565,43568,43571],{"className":43564},[97],[18,43566],{"className":43567,"style":35186},[101],[18,43569,64],{"className":43570},[106,110],[18,43572,43574,43577],{"className":43573},[106],[18,43575,18328],{"className":43576},[106,110],[18,43578,43580],{"className":43579},[115],[18,43581,43583],{"className":43582},[119],[18,43584,43586],{"className":43585},[124],[18,43587,43589],{"className":43588,"style":35186},[128],[18,43590,43591,43594],{"style":1962},[18,43592],{"className":43593,"style":137},[136],[18,43595,43597],{"className":43596},[141,142,143,144],[18,43598,43600,43603],{"className":43599},[106,144],[18,43601,2266],{"className":43602},[106,144],[18,43604,20986],{"className":43605,"style":21038},[106,110,144],"），而鼓包的个数 ",[18,43608,43610,43632],{"className":43609,"translate":22},[21],[18,43611,43613],{"className":43612},[26],[28,43614,43615],{"xmlns":30},[32,43616,43617,43629],{},[35,43618,43619],{},[1556,43620,43621,43623],{},[41,43622,18328],{},[35,43624,43625,43627],{},[49,43626,2253],{},[41,43628,4031],{},[86,43630,43631],{"encoding":88},"h^{-d}",[18,43633,43635],{"className":43634,"ariaHidden":51},[93],[18,43636,43638,43641],{"className":43637},[97],[18,43639],{"className":43640,"style":35186},[101],[18,43642,43644,43647],{"className":43643},[106],[18,43645,18328],{"className":43646},[106,110],[18,43648,43650],{"className":43649},[115],[18,43651,43653],{"className":43652},[119],[18,43654,43656],{"className":43655},[124],[18,43657,43659],{"className":43658,"style":35186},[128],[18,43660,43661,43664],{"style":1962},[18,43662],{"className":43663,"style":137},[136],[18,43665,43667],{"className":43666},[141,142,143,144],[18,43668,43670,43673],{"className":43669},[106,144],[18,43671,2253],{"className":43672},[106,144],[18,43674,4031],{"className":43675},[106,110,144]," 控制假设的个数。"," 三者平衡，就得到非参数速率。维度 ",[18,43679,43681,43694],{"className":43680,"translate":22},[21],[18,43682,43684],{"className":43683},[26],[28,43685,43686],{"xmlns":30},[32,43687,43688,43692],{},[35,43689,43690],{},[41,43691,4031],{},[86,43693,4031],{"encoding":88},[18,43695,43697],{"className":43696,"ariaHidden":51},[93],[18,43698,43700,43703],{"className":43699},[97],[18,43701],{"className":43702,"style":295},[101],[18,43704,4031],{"className":43705},[106,110]," 出现在指数上，正是因为它决定了「同样宽度下能放多少个鼓包」。",[1311,43708,43710],{"id":43709},"_63-一些经典速率","6.3 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维高斯均值",[43736,43768,43769],{},[18,43770,43772,43798],{"className":43771,"translate":22},[21],[18,43773,43775],{"className":43774},[26],[28,43776,43777],{"xmlns":30},[32,43778,43779,43795],{},[35,43780,43781,43783,43785,43791,43793],{},[49,43782,2248],{"stretchy":73},[41,43784,77],{},[38,43786,43787,43789],{},[49,43788,2248],{"stretchy":73},[45,43790,2266],{},[49,43792,4334],{},[41,43794,3071],{},[86,43796,43797],{"encoding":88},"\\lVert\\theta\\rVert_2\\le 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d",[18,44364,44366,44427,44445],{"className":44365,"ariaHidden":51},[93],[18,44367,44369,44372,44375,44378,44418,44421,44424],{"className":44368},[97],[18,44370],{"className":44371,"style":242},[101],[18,44373,2248],{"className":44374},[250],[18,44376,77],{"className":44377,"style":254},[106,110],[18,44379,44381,44384],{"className":44380},[267],[18,44382,2248],{"className":44383},[267],[18,44385,44387],{"className":44386},[115],[18,44388,44390,44410],{"className":44389},[119,120],[18,44391,44393,44407],{"className":44392},[124],[18,44394,44396],{"className":44395,"style":129},[128],[18,44397,44398,44401],{"style":1155},[18,44399],{"className":44400,"style":137},[136],[18,44402,44404],{"className":44403},[141,142,143,144],[18,44405,430],{"className":44406},[106,144],[18,44408,152],{"className":44409},[151],[18,44411,44413],{"className":44412},[124],[18,44414,44416],{"className":44415,"style":159},[128],[18,44417],{},[18,44419],{"className":44420,"style":228},[169],[18,44422,4334],{"className":44423},[232],[18,44425],{"className":44426,"style":228},[169],[18,44428,44430,44433,44436,44439,44442],{"className":44429},[97],[18,44431],{"className":44432,"style":30930},[101],[18,44434,44311],{"className":44435},[106,110],[18,44437],{"className":44438,"style":228},[169],[18,44440,44357],{"className":44441},[232],[18,44443],{"className":44444,"style":228},[169],[18,44446,44448,44451],{"className":44447},[97],[18,44449],{"className":44450,"style":295},[101],[18,44452,4031],{"className":44453},[106,110],[43736,44455,44456],{},[18,44457,44459,44499],{"className":44458,"translate":22},[21],[18,44460,44462],{"className":44461},[26],[28,44463,44464],{"xmlns":30},[32,44465,44466,44496],{},[35,44467,44468,44484,44486,44488],{},[14681,44469,44470],{"scriptlevel":430,"displaystyle":51},[5654,44471,44472,44482],{},[35,44473,44474,44480],{},[1556,44475,44476,44478],{},[41,44477,16406],{},[45,44479,2266],{},[41,44481,44311],{},[41,44483,64],{},[41,44485,14733],{},[49,44487,899],{},[14681,44489,44490],{"scriptlevel":430,"displaystyle":51},[5654,44491,44492,44494],{},[41,44493,4031],{},[41,44495,44311],{},[86,44497,44498],{"encoding":88},"\\dfrac{\\sigma^2s}{n}\\log\\dfrac ds",[18,44500,44502],{"className":44501,"ariaHidden":51},[93],[18,44503,44505,44508,44599,44602,44607,44610],{"className":44504},[97],[18,44506],{"className":44507,"style":34950},[101],[18,44509,44511,44514,44596],{"className":44510},[106],[18,44512],{"className":44513},[250,6012],[18,44515,44517],{"className":44516},[5654],[18,44518,44520,44588],{"className":44519},[119,120],[18,44521,44523,44585],{"className":44522},[124],[18,44524,44526,44537,44545],{"className":44525,"style":15999},[128],[18,44527,44528,44531],{"style":6028},[18,44529],{"className":44530,"style":697},[136],[18,44532,44534],{"className":44533},[106],[18,44535,64],{"className":44536},[106,110],[18,44538,44539,44542],{"style":6040},[18,44540],{"className":44541,"style":697},[136],[18,44543],{"className":44544,"style":6048},[6047],[18,44546,44547,44550],{"style":6051},[18,44548],{"className":44549,"style":697},[136],[18,44551,44553,44582],{"className":44552},[106],[18,44554,44556,44559],{"className":44555},[106],[18,44557,16406],{"className":44558,"style":3957},[106,110],[18,44560,44562],{"className":44561},[115],[18,44563,44565],{"className":44564},[119],[18,44566,44568],{"className":44567},[124],[18,44569,44571],{"className":44570,"style":2416},[128],[18,44572,44573,44576],{"style":1962},[18,44574],{"className":44575,"style":137},[136],[18,44577,44579],{"className":44578},[141,142,143,144],[18,44580,2266],{"className":44581},[106,144],[18,44583,44311],{"className":44584},[106,110],[18,44586,152],{"className":44587},[151],[18,44589,44591],{"className":44590},[124],[18,44592,44594],{"className":44593,"style":6070},[128],[18,44595],{},[18,44597],{"className":44598},[267,6012],[18,44600],{"className":44601,"style":170},[169],[18,44603,15174,44605],{"className":44604},[985],[18,44606,15177],{"style":6355},[18,44608],{"className":44609,"style":170},[169],[18,44611,44613,44616,44669],{"className":44612},[106],[18,44614],{"className":44615},[250,6012],[18,44617,44619],{"className":44618},[5654],[18,44620,44622,44661],{"className":44621},[119,120],[18,44623,44625,44658],{"className":44624},[124],[18,44626,44628,44639,44647],{"className":44627,"style":6025},[128],[18,44629,44630,44633],{"style":6028},[18,44631],{"className":44632,"style":697},[136],[18,44634,44636],{"className":44635},[106],[18,44637,44311],{"className":44638},[106,110],[18,44640,44641,44644],{"style":6040},[18,44642],{"className":44643,"style":697},[136],[18,44645],{"className":44646,"style":6048},[6047],[18,44648,44649,44652],{"style":6051},[18,44650],{"className":44651,"style":697},[136],[18,44653,44655],{"className":44654},[106],[18,44656,4031],{"className":44657},[106,110],[18,44659,152],{"className":44660},[151],[18,44662,44664],{"className":44663},[124],[18,44665,44667],{"className":44666,"style":6070},[128],[18,44668],{},[18,44670],{"className":44671},[267,6012],[43718,44673,44674,44677,44680],{},[43736,44675,44676],{},"一维单调回归",[43736,44678,44679],{},"单调、有界",[43736,44681,44682],{},[18,44683,44685,44712],{"className":44684,"translate":22},[21],[18,44686,44688],{"className":44687},[26],[28,44689,44690],{"xmlns":30},[32,44691,44692,44709],{},[35,44693,44694],{},[1556,44695,44696,44698],{},[41,44697,64],{},[35,44699,44700,44702,44704,44706],{},[49,44701,2253],{},[45,44703,2266],{},[41,44705,317],{"mathvariant":316},[45,44707,44708],{},"3",[86,44710,44711],{"encoding":88},"n^{-2\u002F3}",[18,44713,44715],{"className":44714,"ariaHidden":51},[93],[18,44716,44718,44721],{"className":44717},[97],[18,44719],{"className":44720,"style":27945},[101],[18,44722,44724,44727],{"className":44723},[106],[18,44725,64],{"className":44726},[106,110],[18,44728,44730],{"className":44729},[115],[18,44731,44733],{"className":44732},[119],[18,44734,44736],{"className":44735},[124],[18,44737,44739],{"className":44738,"style":27945},[128],[18,44740,44741,44744],{"style":1962},[18,44742],{"className":44743,"style":137},[136],[18,44745,44747],{"className":44746},[141,142,143,144],[18,44748,44750,44753],{"className":44749},[106,144],[18,44751,2253],{"className":44752},[106,144],[18,44754,44756],{"className":44755},[106,144],"2\u002F3",[11,44758,44759],{},"这张表里的每一个速率，都可以用本文的三种方法之一推出来。稀疏均值的下界通常用 Fano 加一个随机支撑的打包；单调回归则用 Le Cam 加一个局部的「台阶」构造。",[638,44761,44763],{"id":44762},"七三种方法的比较","七、三种方法的比较",[43712,44765,44766,44785],{},[43715,44767,44768],{},[43718,44769,44770,44773,44776,44779,44782],{},[43721,44771,44772],{},"方法",[43721,44774,44775],{},"假设个数",[43721,44777,44778],{},"距离",[43721,44780,44781],{},"优势",[43721,44783,44784],{},"局限",[43731,44786,44787,44831,44875],{},[43718,44788,44789,44792,44822,44825,44828],{},[43736,44790,44791],{},"Le Cam",[43736,44793,44794],{},[18,44795,44797,44810],{"className":44796,"translate":22},[21],[18,44798,44800],{"className":44799},[26],[28,44801,44802],{"xmlns":30},[32,44803,44804,44808],{},[35,44805,44806],{},[45,44807,2266],{},[86,44809,2266],{"encoding":88},[18,44811,44813],{"className":44812,"ariaHidden":51},[93],[18,44814,44816,44819],{"className":44815},[97],[18,44817],{"className":44818,"style":505},[101],[18,44820,2266],{"className":44821},[106],[43736,44823,44824],{},"TV \u002F KL",[43736,44826,44827],{},"简单，常数清楚",[43736,44829,44830],{},"只探测一个方向，高维问题会松",[43718,44832,44833,44836,44866,44869,44872],{},[43736,44834,44835],{},"Fano",[43736,44837,44838],{},[18,44839,44841,44854],{"className":44840,"translate":22},[21],[18,44842,44844],{"className":44843},[26],[28,44845,44846],{"xmlns":30},[32,44847,44848,44852],{},[35,44849,44850],{},[41,44851,3526],{},[86,44853,3526],{"encoding":88},[18,44855,44857],{"className":44856,"ariaHidden":51},[93],[18,44858,44860,44863],{"className":44859},[97],[18,44861],{"className":44862,"style":600},[101],[18,44864,3526],{"className":44865,"style":246},[106,110],[43736,44867,44868],{},"互信息 \u002F KL",[43736,44870,44871],{},"通用，适合高维与非参数",[43736,44873,44874],{},"需要构造打包，常数较松",[43718,44876,44877,44880,44941,44944,44947],{},[43736,44878,44879],{},"Assouad",[43736,44881,44882],{},[18,44883,44885,44903],{"className":44884,"translate":22},[21],[18,44886,44888],{"className":44887},[26],[28,44889,44890],{"xmlns":30},[32,44891,44892,44900],{},[35,44893,44894],{},[1556,44895,44896,44898],{},[45,44897,2266],{},[41,44899,28169],{},[86,44901,44902],{"encoding":88},"2^m",[18,44904,44906],{"className":44905,"ariaHidden":51},[93],[18,44907,44909,44912],{"className":44908},[97],[18,44910],{"className":44911,"style":1959},[101],[18,44913,44915,44918],{"className":44914},[106],[18,44916,2266],{"className":44917},[106],[18,44919,44921],{"className":44920},[115],[18,44922,44924],{"className":44923},[119],[18,44925,44927],{"className":44926},[124],[18,44928,44930],{"className":44929,"style":1959},[128],[18,44931,44932,44935],{"style":1962},[18,44933],{"className":44934,"style":137},[136],[18,44936,44938],{"className":44937},[141,142,143,144],[18,44939,28169],{"className":44940},[106,110,144],[43736,44942,44943],{},"逐坐标 TV",[43736,44945,44946],{},"乘积结构下最锋利",[43736,44948,44949],{},"只适用于超立方体结构",[11,44951,44952],{},"三者的关系可以这样理解：",[2210,44954,44955,44961,44996],{},[2213,44956,44957,44960],{},[717,44958,44959],{},"Le Cam 是 Fano 的两点版本。"," 当只有两个假设时，Fano 退化为两点检验；Le Cam 用 TV 给出了更干净的常数。",[2213,44962,44963,44966,44967,44995],{},[717,44964,44965],{},"Fano 是 Le Cam 的多假设推广。"," 它把「两个点分不清」推广为「",[18,44968,44970,44983],{"className":44969,"translate":22},[21],[18,44971,44973],{"className":44972},[26],[28,44974,44975],{"xmlns":30},[32,44976,44977,44981],{},[35,44978,44979],{},[41,44980,3526],{},[86,44982,3526],{"encoding":88},[18,44984,44986],{"className":44985,"ariaHidden":51},[93],[18,44987,44989,44992],{"className":44988},[97],[18,44990],{"className":44991,"style":600},[101],[18,44993,3526],{"className":44994,"style":246},[106,110]," 个点平均分不清」。",[2213,44997,44998,45001,45002,45060,45061,45089],{},[717,44999,45000],{},"Assouad 是 Fano 在乘积结构上的精细化。"," 它不直接对 ",[18,45003,45005,45022],{"className":45004,"translate":22},[21],[18,45006,45008],{"className":45007},[26],[28,45009,45010],{"xmlns":30},[32,45011,45012,45020],{},[35,45013,45014],{},[1556,45015,45016,45018],{},[45,45017,2266],{},[41,45019,28169],{},[86,45021,44902],{"encoding":88},[18,45023,45025],{"className":45024,"ariaHidden":51},[93],[18,45026,45028,45031],{"className":45027},[97],[18,45029],{"className":45030,"style":1959},[101],[18,45032,45034,45037],{"className":45033},[106],[18,45035,2266],{"className":45036},[106],[18,45038,45040],{"className":45039},[115],[18,45041,45043],{"className":45042},[119],[18,45044,45046],{"className":45045},[124],[18,45047,45049],{"className":45048,"style":1959},[128],[18,45050,45051,45054],{"style":1962},[18,45052],{"className":45053,"style":137},[136],[18,45055,45057],{"className":45056},[141,142,143,144],[18,45058,28169],{"className":45059},[106,110,144]," 个假设做 Fano，而是逐坐标做 Le Cam，再把 ",[18,45062,45064,45077],{"className":45063,"translate":22},[21],[18,45065,45067],{"className":45066},[26],[28,45068,45069],{"xmlns":30},[32,45070,45071,45075],{},[35,45072,45073],{},[41,45074,28169],{},[86,45076,28169],{"encoding":88},[18,45078,45080],{"className":45079,"ariaHidden":51},[93],[18,45081,45083,45086],{"className":45082},[97],[18,45084],{"className":45085,"style":1083},[101],[18,45087,28169],{"className":45088},[106,110]," 个下界加起来。",[11,45091,45092],{},"实际使用时，选择哪种方法取决于参数空间的结构：",[2210,45094,45095,45098,45142],{},[2213,45096,45097],{},"一维、局部、两样本问题：Le Cam 通常足够，而且常数最干净。",[2213,45099,45100,45101,45141],{},"高维、非参数、需要 ",[18,45102,45104,45121],{"className":45103,"translate":22},[21],[18,45105,45107],{"className":45106},[26],[28,45108,45109],{"xmlns":30},[32,45110,45111,45119],{},[35,45112,45113,45115,45117],{},[41,45114,14733],{},[49,45116,899],{},[41,45118,3526],{},[86,45120,19451],{"encoding":88},[18,45122,45124],{"className":45123,"ariaHidden":51},[93],[18,45125,45127,45130,45135,45138],{"className":45126},[97],[18,45128],{"className":45129,"style":2890},[101],[18,45131,15174,45133],{"className":45132},[985],[18,45134,15177],{"style":6355},[18,45136],{"className":45137,"style":170},[169],[18,45139,3526],{"className":45140,"style":246},[106,110]," 的地方：Fano 是标准工具。",[2213,45143,45144,45145,45215,45216,45286],{},"乘积结构、",[18,45146,45148,45166],{"className":45147,"translate":22},[21],[18,45149,45151],{"className":45150},[26],[28,45152,45153],{"xmlns":30},[32,45154,45155,45163],{},[35,45156,45157],{},[38,45158,45159,45161],{},[41,45160,948],{"mathvariant":316},[45,45162,2266],{},[86,45164,45165],{"encoding":88},"\\ell_2",[18,45167,45169],{"className":45168,"ariaHidden":51},[93],[18,45170,45172,45175],{"className":45171},[97],[18,45173],{"className":45174,"style":448},[101],[18,45176,45178,45181],{"className":45177},[106],[18,45179,948],{"className":45180},[106],[18,45182,45184],{"className":45183},[115],[18,45185,45187,45207],{"className":45186},[119,120],[18,45188,45190,45204],{"className":45189},[124],[18,45191,45193],{"className":45192,"style":129},[128],[18,45194,45195,45198],{"style":1155},[18,45196],{"className":45197,"style":137},[136],[18,45199,45201],{"className":45200},[141,142,143,144],[18,45202,2266],{"className":45203},[106,144],[18,45205,152],{"className":45206},[151],[18,45208,45210],{"className":45209},[124],[18,45211,45213],{"className":45212,"style":159},[128],[18,45214],{}," 或 ",[18,45217,45219,45237],{"className":45218,"translate":22},[21],[18,45220,45222],{"className":45221},[26],[28,45223,45224],{"xmlns":30},[32,45225,45226,45234],{},[35,45227,45228],{},[38,45229,45230,45232],{},[41,45231,948],{"mathvariant":316},[41,45233,2028],{"mathvariant":316},[86,45235,45236],{"encoding":88},"\\ell_\\infty",[18,45238,45240],{"className":45239,"ariaHidden":51},[93],[18,45241,45243,45246],{"className":45242},[97],[18,45244],{"className":45245,"style":448},[101],[18,45247,45249,45252],{"className":45248},[106],[18,45250,948],{"className":45251},[106],[18,45253,45255],{"className":45254},[115],[18,45256,45258,45278],{"className":45257},[119,120],[18,45259,45261,45275],{"className":45260},[124],[18,45262,45264],{"className":45263,"style":202},[128],[18,45265,45266,45269],{"style":1155},[18,45267],{"className":45268,"style":137},[136],[18,45270,45272],{"className":45271},[141,142,143,144],[18,45273,2028],{"className":45274},[106,144],[18,45276,152],{"className":45277},[151],[18,45279,45281],{"className":45280},[124],[18,45282,45284],{"className":45283,"style":159},[128],[18,45285],{}," 速率：Assouad 最方便。",[638,45288,45289],{"id":45289},"结语",[11,45291,45292,45293,45328,45329,45357],{},"回到开头的问题：为什么 ",[18,45294,45296,45313],{"className":45295,"translate":22},[21],[18,45297,45299],{"className":45298},[26],[28,45300,45301],{"xmlns":30},[32,45302,45303,45311],{},[35,45304,45305,45307,45309],{},[45,45306,47],{},[41,45308,317],{"mathvariant":316},[41,45310,64],{},[86,45312,322],{"encoding":88},[18,45314,45316],{"className":45315,"ariaHidden":51},[93],[18,45317,45319,45322,45325],{"className":45318},[97],[18,45320],{"className":45321,"style":242},[101],[18,45323,335],{"className":45324},[106],[18,45326,64],{"className":45327},[106,110]," 不能再小？因为两个足够接近的分布，在 ",[18,45330,45332,45345],{"className":45331,"translate":22},[21],[18,45333,45335],{"className":45334},[26],[28,45336,45337],{"xmlns":30},[32,45338,45339,45343],{},[35,45340,45341],{},[41,45342,64],{},[86,45344,64],{"encoding":88},[18,45346,45348],{"className":45347,"ariaHidden":51},[93],[18,45349,45351,45354],{"className":45350},[97],[18,45352],{"className":45353,"style":1083},[101],[18,45355,64],{"className":45356},[106,110]," 个样本下仍然难以区分；而任何估计器都必须先区分它们。这个论证只需要两个点，所以叫两点方法。",[11,45359,45360,45361,45389,45390,45418,45419,45447],{},"但当困难来自许多方向时，两个点就不够了。Fano 把「两个点」换成「",[18,45362,45364,45377],{"className":45363,"translate":22},[21],[18,45365,45367],{"className":45366},[26],[28,45368,45369],{"xmlns":30},[32,45370,45371,45375],{},[35,45372,45373],{},[41,45374,3526],{},[86,45376,3526],{"encoding":88},[18,45378,45380],{"className":45379,"ariaHidden":51},[93],[18,45381,45383,45386],{"className":45382},[97],[18,45384],{"className":45385,"style":600},[101],[18,45387,3526],{"className":45388,"style":246},[106,110]," 个点」，用互信息度量整体的不可区分性；Assouad 则把 ",[18,45391,45393,45406],{"className":45392,"translate":22},[21],[18,45394,45396],{"className":45395},[26],[28,45397,45398],{"xmlns":30},[32,45399,45400,45404],{},[35,45401,45402],{},[41,45403,28169],{},[86,45405,28169],{"encoding":88},[18,45407,45409],{"className":45408,"ariaHidden":51},[93],[18,45410,45412,45415],{"className":45411},[97],[18,45413],{"className":45414,"style":1083},[101],[18,45416,28169],{"className":45417},[106,110]," 维问题拆成 ",[18,45420,45422,45435],{"className":45421,"translate":22},[21],[18,45423,45425],{"className":45424},[26],[28,45426,45427],{"xmlns":30},[32,45428,45429,45433],{},[35,45430,45431],{},[41,45432,28169],{},[86,45434,28169],{"encoding":88},[18,45436,45438],{"className":45437,"ariaHidden":51},[93],[18,45439,45441,45444],{"className":45440},[97],[18,45442],{"className":45443,"style":1083},[101],[18,45445,28169],{"className":45446},[106,110]," 个一维问题，逐坐标地累加下界。",[11,45449,45450],{},"如果只记三句话：",[45452,45453,45454,45457,45548],"ol",{},[2213,45455,45456],{},"下界证明的第一步永远是归约：把估计问题换成一个有限检验问题。",[2213,45458,45459,45460,45488,45489,45547],{},"Le Cam 用两个点，Fano 用 ",[18,45461,45463,45476],{"className":45462,"translate":22},[21],[18,45464,45466],{"className":45465},[26],[28,45467,45468],{"xmlns":30},[32,45469,45470,45474],{},[35,45471,45472],{},[41,45473,3526],{},[86,45475,3526],{"encoding":88},[18,45477,45479],{"className":45478,"ariaHidden":51},[93],[18,45480,45482,45485],{"className":45481},[97],[18,45483],{"className":45484,"style":600},[101],[18,45486,3526],{"className":45487,"style":246},[106,110]," 个点，Assouad 用 ",[18,45490,45492,45509],{"className":45491,"translate":22},[21],[18,45493,45495],{"className":45494},[26],[28,45496,45497],{"xmlns":30},[32,45498,45499,45507],{},[35,45500,45501],{},[1556,45502,45503,45505],{},[45,45504,2266],{},[41,45506,28169],{},[86,45508,44902],{"encoding":88},[18,45510,45512],{"className":45511,"ariaHidden":51},[93],[18,45513,45515,45518],{"className":45514},[97],[18,45516],{"className":45517,"style":1959},[101],[18,45519,45521,45524],{"className":45520},[106],[18,45522,2266],{"className":45523},[106],[18,45525,45527],{"className":45526},[115],[18,45528,45530],{"className":45529},[119],[18,45531,45533],{"className":45532},[124],[18,45534,45536],{"className":45535,"style":1959},[128],[18,45537,45538,45541],{"style":1962},[18,45539],{"className":45540,"style":137},[136],[18,45542,45544],{"className":45543},[141,142,143,144],[18,45545,28169],{"className":45546},[106,110,144]," 个点；工具在变，思想都是「构造一个在损失上离得远、在信息上离得近的假设集合」。",[2213,45549,45550],{},"信息论下界告诉你统计上不可能；它不告诉你计算上不可能。深度学习时代真正难的，往往是后一个问题。",{"title":45552,"searchDepth":45553,"depth":45553,"links":45554},"",2,[45555,45556,45562,45568,45573,45577,45584,45585],{"id":640,"depth":45553,"text":641},{"id":1308,"depth":45553,"text":1309,"children":45557},[45558,45560,45561],{"id":1313,"depth":45559,"text":1314},3,{"id":3893,"depth":45559,"text":3894},{"id":4499,"depth":45559,"text":4500},{"id":5149,"depth":45553,"text":5150,"children":45563},[45564,45565,45566,45567],{"id":5153,"depth":45559,"text":5154},{"id":12519,"depth":45559,"text":12520},{"id":14414,"depth":45559,"text":14415},{"id":16325,"depth":45559,"text":16326},{"id":17468,"depth":45553,"text":17469,"children":45569},[45570,45571,45572],{"id":17472,"depth":45559,"text":17473},{"id":19545,"depth":45559,"text":19546},{"id":26784,"depth":45559,"text":26785},{"id":28094,"depth":45553,"text":28095,"children":45574},[45575,45576],{"id":28098,"depth":45559,"text":28099},{"id":32345,"depth":45559,"text":32346},{"id":34116,"depth":45553,"text":34117,"children":45578},[45579,45581,45583],{"id":34120,"depth":45559,"text":45580},"6.1 高斯均值：σ2d\u002Fn\\sigma^2d\u002Fnσ2d\u002Fn",{"id":37107,"depth":45559,"text":45582},"6.2 非参数回归：n−2β\u002F(2β+d)n^{-2\\beta\u002F(2\\beta+d)}n−2β\u002F(2β+d)",{"id":43709,"depth":45559,"text":43710},{"id":44762,"depth":45553,"text":44763},{"id":45289,"depth":45553,"text":45289},false,true,"md",null,{},"\u002Fblog\u002F2025\u002F2025-09-27-fano-bu-deng-shi-yu-le-cam-fang-fa",{"title":6,"description":13},"blog\u002F2025\u002F2025-09-27-fano-bu-deng-shi-yu-le-cam-fang-fa","极小极大下界的三种基本工具：Le Cam 两点方法、Fano 不等式与 Assouad 引理。本文从决策论框架出发，给出完整证明与常数，并用高斯均值、非参数回归等例子展示如何把信息论不等式变成速率下界。",[45596,45597],"mathematics","machine-learning","2025-09-27T16:28:39+08:00","PCaKGvsKVNVeAyaMYmmlfttL41cAUYqcbixx-fGLAmI",{"id":45601,"title":45602,"body":45603,"description":45607,"draft":45586,"enableComment":45587,"extension":45588,"image":45552,"important":45586,"location":45589,"meta":81700,"navigation":45587,"ogImage":45589,"path":81701,"seo":81702,"stem":81703,"summary":81704,"tags":81705,"time":81706,"__hash__":81707},"blog\u002Fblog\u002F2025\u002F2025-10-05-vapnik-chervonenkis-dimension.md","Vapnik-Chervonenkis Dimension 简论",{"type":8,"value":45604,"toc":81668},[45605,45608,45814,46074,46077,46080,46084,46088,46416,46767,46798,46838,47548,47555,47796,47800,48016,50049,50179,50211,50215,50350,50736,50772,50964,50996,51000,51004,51263,51651,51953,52159,52603,52760,52763,52767,52829,53049,53274,53811,53982,54043,54232,55223,55730,55813,56042,56398,56671,56913,57031,57147,57435,57765,58060,58443,58679,59026,59029,59095,59213,59529,59744,60053,60182,60551,60612,61090,61424,62168,62663,63167,63170,63230,63362,63365,63368,64219,64890,64894,64900,64910,64974,64981,64985,64989,64992,65343,65592,66176,66180,66580,67043,67248,67479,67843,68799,68802,69465,69468,69472,69526,69863,69927,71277,71653,71660,71664,71668,71675,71798,72567,72570,72752,73986,74270,74273,74582,74703,74707,74760,75171,75380,75559,75764,75861,75865,76070,76076,76080,76112,76340,76343,76541,76577,76580,76584,76587,76751,76813,76819,76890,77085,77092,77104,77108,77112,77304,78362,78485,78608,78612,78615,78618,78972,79012,79015,79019,79022,79027,79059,79331,79547,79579,79665,80260,80321,80444,80450,80486,80491,80580,80583,80586,80589,80594,80597,80661,80667,80670,80674,80681,80684,80747,80751,80820,80823,81653,81660,81662],[11,45606,45607],{},"先看一个只有 1 个参数的分类器：",[18,45609,45611],{"className":45610,"translate":22},[875],[18,45612,45614,45674],{"className":45613,"translate":22},[21],[18,45615,45617],{"className":45616},[26],[28,45618,45619],{"xmlns":30,"display":884},[32,45620,45621,45671],{},[35,45622,45623,45629,45631,45633,45635,45637,45640,45642,45644,45647,45649,45651,45653,45655,45657,45659,45661,45663,45665,45667,45669],{},[38,45624,45625,45627],{},[41,45626,18328],{},[41,45628,77],{},[49,45630,74],{"stretchy":73},[41,45632,38582],{},[49,45634,84],{"stretchy":73},[49,45636,433],{},[41,45638,45639],{"mathvariant":316},"sign",[49,45641,899],{},[49,45643,74],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[41,45645,45646],{},"sin",[49,45648,899],{},[49,45650,74],{"stretchy":73},[41,45652,77],{},[41,45654,38582],{},[49,45656,84],{"stretchy":73},[49,45658,84],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,45660,52],{"separator":51},[169,45662],{"width":7344},[41,45664,77],{},[49,45666,921],{},[41,45668,3071],{"mathvariant":933},[41,45670,968],{"mathvariant":316},[86,45672,45673],{"encoding":88},"h_\\theta(x)=\\operatorname{sign}\\bigl(\\sin(\\theta x)\\bigr),\\qquad \\theta\\in\\mathbb R.",[18,45675,45677,45741,45801],{"className":45676,"ariaHidden":51},[93],[18,45678,45680,45683,45723,45726,45729,45732,45735,45738],{"className":45679},[97],[18,45681],{"className":45682,"style":242},[101],[18,45684,45686,45689],{"className":45685},[106],[18,45687,18328],{"className":45688},[106,110],[18,45690,45692],{"className":45691},[115],[18,45693,45695,45715],{"className":45694},[119,120],[18,45696,45698,45712],{"className":45697},[124],[18,45699,45701],{"className":45700,"style":1383},[128],[18,45702,45703,45706],{"style":1155},[18,45704],{"className":45705,"style":137},[136],[18,45707,45709],{"className":45708},[141,142,143,144],[18,45710,77],{"className":45711,"style":254},[106,110,144],[18,45713,152],{"className":45714},[151],[18,45716,45718],{"className":45717},[124],[18,45719,45721],{"className":45720,"style":159},[128],[18,45722],{},[18,45724,74],{"className":45725},[250],[18,45727,38582],{"className":45728},[106,110],[18,45730,84],{"className":45731},[267],[18,45733],{"className":45734,"style":228},[169],[18,45736,433],{"className":45737},[232],[18,45739],{"className":45740,"style":228},[169],[18,45742,45744,45747,45753,45759,45762,45765,45768,45771,45774,45780,45783,45786,45789,45792,45795,45798],{"className":45743},[97],[18,45745],{"className":45746,"style":6101},[101],[18,45748,45750],{"className":45749},[985],[18,45751,45639],{"className":45752},[106,6354],[18,45754,45756],{"className":45755},[250],[18,45757,74],{"className":45758},[1230,1196],[18,45760,45646],{"className":45761},[985],[18,45763,74],{"className":45764},[250],[18,45766,77],{"className":45767,"style":254},[106,110],[18,45769,38582],{"className":45770},[106,110],[18,45772,84],{"className":45773},[267],[18,45775,45777],{"className":45776},[267],[18,45778,84],{"className":45779},[1230,1196],[18,45781,52],{"className":45782},[165],[18,45784],{"className":45785,"style":7676},[169],[18,45787],{"className":45788,"style":170},[169],[18,45790,77],{"className":45791,"style":254},[106,110],[18,45793],{"className":45794,"style":228},[169],[18,45796,921],{"className":45797},[232],[18,45799],{"className":45800,"style":228},[169],[18,45802,45804,45808,45811],{"className":45803},[97],[18,45805],{"className":45806,"style":45807},[101],"height:0.6889em;",[18,45809,3071],{"className":45810},[106,1139],[18,45812,968],{"className":45813},[106],[11,45815,45816,45817,46010,46011,46039,46040,46073],{},"它只有一个参数。但它的 VC 维是无穷大：对任意多的点，只要把这些点取在 ",[18,45818,45820,45870],{"className":45819,"translate":22},[21],[18,45821,45823],{"className":45822},[26],[28,45824,45825],{"xmlns":30},[32,45826,45827,45867],{},[35,45828,45829,45839,45841,45851,45853,45855,45857],{},[1556,45830,45831,45833],{},[45,45832,2266],{},[35,45834,45835,45837],{},[49,45836,2253],{},[45,45838,47],{},[49,45840,52],{"separator":51},[1556,45842,45843,45845],{},[45,45844,2266],{},[35,45846,45847,45849],{},[49,45848,2253],{},[45,45850,2266],{},[49,45852,52],{"separator":51},[49,45854,55],{},[49,45856,52],{"separator":51},[1556,45858,45859,45861],{},[45,45860,2266],{},[35,45862,45863,45865],{},[49,45864,2253],{},[41,45866,28169],{},[86,45868,45869],{"encoding":88},"2^{-1},2^{-2},\\dots,2^{-m}",[18,45871,45873],{"className":45872,"ariaHidden":51},[93],[18,45874,45876,45880,45915,45918,45921,45956,45959,45962,45965,45968,45971,45974],{"className":45875},[97],[18,45877],{"className":45878,"style":45879},[101],"height:1.0085em;vertical-align:-0.1944em;",[18,45881,45883,45886],{"className":45882},[106],[18,45884,2266],{"className":45885},[106],[18,45887,45889],{"className":45888},[115],[18,45890,45892],{"className":45891},[119],[18,45893,45895],{"className":45894},[124],[18,45896,45898],{"className":45897,"style":2416},[128],[18,45899,45900,45903],{"style":1962},[18,45901],{"className":45902,"style":137},[136],[18,45904,45906],{"className":45905},[141,142,143,144],[18,45907,45909,45912],{"className":45908},[106,144],[18,45910,2253],{"className":45911},[106,144],[18,45913,47],{"className":45914},[106,144],[18,45916,52],{"className":45917},[165],[18,45919],{"className":45920,"style":170},[169],[18,45922,45924,45927],{"className":45923},[106],[18,45925,2266],{"className":45926},[106],[18,45928,45930],{"className":45929},[115],[18,45931,45933],{"className":45932},[119],[18,45934,45936],{"className":45935},[124],[18,45937,45939],{"className":45938,"style":2416},[128],[18,45940,45941,45944],{"style":1962},[18,45942],{"className":45943,"style":137},[136],[18,45945,45947],{"className":45946},[141,142,143,144],[18,45948,45950,45953],{"className":45949},[106,144],[18,45951,2253],{"className":45952},[106,144],[18,45954,2266],{"className":45955},[106,144],[18,45957,52],{"className":45958},[165],[18,45960],{"className":45961,"style":170},[169],[18,45963,55],{"className":45964},[174],[18,45966],{"className":45967,"style":170},[169],[18,45969,52],{"className":45970},[165],[18,45972],{"className":45973,"style":170},[169],[18,45975,45977,45980],{"className":45976},[106],[18,45978,2266],{"className":45979},[106],[18,45981,45983],{"className":45982},[115],[18,45984,45986],{"className":45985},[119],[18,45987,45989],{"className":45988},[124],[18,45990,45993],{"className":45991,"style":45992},[128],"height:0.7713em;",[18,45994,45995,45998],{"style":1962},[18,45996],{"className":45997,"style":137},[136],[18,45999,46001],{"className":46000},[141,142,143,144],[18,46002,46004,46007],{"className":46003},[106,144],[18,46005,2253],{"className":46006},[106,144],[18,46008,28169],{"className":46009},[106,110,144]," 上，就总存在一个 ",[18,46012,46014,46027],{"className":46013,"translate":22},[21],[18,46015,46017],{"className":46016},[26],[28,46018,46019],{"xmlns":30},[32,46020,46021,46025],{},[35,46022,46023],{},[41,46024,77],{},[86,46026,285],{"encoding":88},[18,46028,46030],{"className":46029,"ariaHidden":51},[93],[18,46031,46033,46036],{"className":46032},[97],[18,46034],{"className":46035,"style":295},[101],[18,46037,77],{"className":46038,"style":254},[106,110],"，把任意指定的 ",[18,46041,46043,46061],{"className":46042,"translate":22},[21],[18,46044,46046],{"className":46045},[26],[28,46047,46048],{"xmlns":30},[32,46049,46050,46058],{},[35,46051,46052,46054,46056],{},[45,46053,430],{},[41,46055,317],{"mathvariant":316},[45,46057,47],{},[86,46059,46060],{"encoding":88},"0\u002F1",[18,46062,46064],{"className":46063,"ariaHidden":51},[93],[18,46065,46067,46070],{"className":46066},[97],[18,46068],{"className":46069,"style":242},[101],[18,46071,46060],{"className":46072},[106]," 标注实现出来。反过来，一个有 2500 万参数的 ResNet-50，VC 维虽然有限，却大到让所有经典泛化界都变成一句废话——界大于 1，等于什么都没说。",[11,46075,46076],{},"这两个事实放在一起，说明「参数个数」和「容量」是两件不同的事。VC 维（Vapnik–Chervonenkis dimension）就是 Vapnik 与 Chervonenkis 在 1971 年为了把后者说清楚而发明的量：它不数参数，不假设分布，只看一个假设类作为「函数集合」的组合结构有多丰富。",[11,46078,46079],{},"本文的路线是：第一至四节把定义、Sauer–Shelah 引理、一致收敛定理这条逻辑链完整搭起来；第五至七节讨论它在深度学习里的处境——为什么界会空、空界是否意味着理论失效、今天用什么替代它；第八节清理一批常见误解。",[638,46081,46083],{"id":46082},"一学习问题","一、学习问题",[1311,46085,46087],{"id":46086},"_11-设定","1.1 设定",[11,46089,46090,46091,46119,46120,46196,46197,46200,46201,46297,46298,1994,46301,46331,46332,46385,46386,46415],{},"设输入空间为 ",[18,46092,46094,46107],{"className":46093,"translate":22},[21],[18,46095,46097],{"className":46096},[26],[28,46098,46099],{"xmlns":30},[32,46100,46101,46105],{},[35,46102,46103],{},[41,46104,43],{"mathvariant":1462},[86,46106,1465],{"encoding":88},[18,46108,46110],{"className":46109,"ariaHidden":51},[93],[18,46111,46113,46116],{"className":46112},[97],[18,46114],{"className":46115,"style":600},[101],[18,46117,43],{"className":46118,"style":1479},[106,1478],"，标签空间为 ",[18,46121,46123,46149],{"className":46122,"translate":22},[21],[18,46124,46126],{"className":46125},[26],[28,46127,46128],{"xmlns":30},[32,46129,46130,46146],{},[35,46131,46132,46134,46136,46138,46140,46142,46144],{},[41,46133,37238],{"mathvariant":1462},[49,46135,433],{},[49,46137,1331],{"stretchy":73},[45,46139,430],{},[49,46141,52],{"separator":51},[45,46143,47],{},[49,46145,1349],{"stretchy":73},[86,46147,46148],{"encoding":88},"\\mathcal Y=\\{0,1\\}",[18,46150,46152,46172],{"className":46151,"ariaHidden":51},[93],[18,46153,46155,46159,46163,46166,46169],{"className":46154},[97],[18,46156],{"className":46157,"style":46158},[101],"height:0.7805em;vertical-align:-0.0972em;",[18,46160,37238],{"className":46161,"style":46162},[106,1478],"margin-right:0.0822em;",[18,46164],{"className":46165,"style":228},[169],[18,46167,433],{"className":46168},[232],[18,46170],{"className":46171,"style":228},[169],[18,46173,46175,46178,46181,46184,46187,46190,46193],{"className":46174},[97],[18,46176],{"className":46177,"style":242},[101],[18,46179,1331],{"className":46180},[250],[18,46182,430],{"className":46183},[106],[18,46185,52],{"className":46186},[165],[18,46188],{"className":46189,"style":170},[169],[18,46191,47],{"className":46192},[106],[18,46194,1349],{"className":46195},[267],"。一个",[717,46198,46199],{},"假设","是函数 ",[18,46202,46204,46234],{"className":46203,"translate":22},[21],[18,46205,46207],{"className":46206},[26],[28,46208,46209],{"xmlns":30},[32,46210,46211,46231],{},[35,46212,46213,46215,46217,46219,46221,46223,46225,46227,46229],{},[41,46214,18328],{},[49,46216,1340],{},[41,46218,43],{"mathvariant":1462},[49,46220,1880],{},[49,46222,1331],{"stretchy":73},[45,46224,430],{},[49,46226,52],{"separator":51},[45,46228,47],{},[49,46230,1349],{"stretchy":73},[86,46232,46233],{"encoding":88},"h:\\mathcal X\\to\\{0,1\\}",[18,46235,46237,46255,46273],{"className":46236,"ariaHidden":51},[93],[18,46238,46240,46243,46246,46249,46252],{"className":46239},[97],[18,46241],{"className":46242,"style":295},[101],[18,46244,18328],{"className":46245},[106,110],[18,46247],{"className":46248,"style":228},[169],[18,46250,1340],{"className":46251},[232],[18,46253],{"className":46254,"style":228},[169],[18,46256,46258,46261,46264,46267,46270],{"className":46257},[97],[18,46259],{"className":46260,"style":600},[101],[18,46262,43],{"className":46263,"style":1479},[106,1478],[18,46265],{"className":46266,"style":228},[169],[18,46268,1880],{"className":46269},[232],[18,46271],{"className":46272,"style":228},[169],[18,46274,46276,46279,46282,46285,46288,46291,46294],{"className":46275},[97],[18,46277],{"className":46278,"style":242},[101],[18,46280,1331],{"className":46281},[250],[18,46283,430],{"className":46284},[106],[18,46286,52],{"className":46287},[165],[18,46289],{"className":46290,"style":170},[169],[18,46292,47],{"className":46293},[106],[18,46295,1349],{"className":46296},[267],"，一个",[717,46299,46300],{},"假设类",[18,46302,46304,46318],{"className":46303,"translate":22},[21],[18,46305,46307],{"className":46306},[26],[28,46308,46309],{"xmlns":30},[32,46310,46311,46315],{},[35,46312,46313],{},[41,46314,18309],{"mathvariant":1462},[86,46316,46317],{"encoding":88},"\\mathcal 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H}L_S(h).",[18,47616,47618,47673],{"className":47617,"ariaHidden":51},[93],[18,47619,47621,47624,47664,47667,47670],{"className":47620},[97],[18,47622],{"className":47623,"style":448},[101],[18,47625,47627,47630],{"className":47626},[106],[18,47628,18328],{"className":47629},[106,110],[18,47631,47633],{"className":47632},[115],[18,47634,47636,47656],{"className":47635},[119,120],[18,47637,47639,47653],{"className":47638},[124],[18,47640,47642],{"className":47641,"style":1152},[128],[18,47643,47644,47647],{"style":1155},[18,47645],{"className":47646,"style":137},[136],[18,47648,47650],{"className":47649},[141,142,143,144],[18,47651,38934],{"className":47652,"style":38967},[106,110,144],[18,47654,152],{"className":47655},[151],[18,47657,47659],{"className":47658},[124],[18,47660,47662],{"className":47661,"style":159},[128],[18,47663],{},[18,47665],{"className":47666,"style":228},[169],[18,47668,921],{"className":47669},[232],[18,47671],{"className":47672,"style":228},[169],[18,47674,47676,47680,47685,47688,47741,47744,47784,47787,47790,47793],{"className":47675},[97],[18,47677],{"className":47678,"style":47679},[101],"height:1.5295em;vertical-align:-0.7795em;",[18,47681,21642,47683],{"className":47682},[985],[18,47684,15177],{"style":6355},[18,47686],{"className":47687,"style":170},[169],[18,47689,47691],{"className":47690},[985,986],[18,47692,47694,47733],{"className":47693},[119,120],[18,47695,47697,47730],{"className":47696},[124],[18,47698,47700,47720],{"className":47699,"style":21660},[128],[18,47701,47702,47705],{"style":29370},[18,47703],{"className":47704,"style":697},[136],[18,47706,47708],{"className":47707},[141,142,143,144],[18,47709,47711,47714,47717],{"className":47710},[106,144],[18,47712,18328],{"className":47713},[106,110,144],[18,47715,921],{"className":47716},[232,144],[18,47718,18309],{"className":47719,"style":46330},[106,1478,144],[18,47721,47722,47725],{"style":693},[18,47723],{"className":47724,"style":697},[136],[18,47726,47727],{},[18,47728,11952],{"className":47729},[985],[18,47731,152],{"className":47732},[151],[18,47734,47736],{"className":47735},[124],[18,47737,47739],{"className":47738,"style":29408},[128],[18,47740],{},[18,47742],{"className":47743,"style":170},[169],[18,47745,47747,47750],{"className":47746},[106],[18,47748,12582],{"className":47749},[106,110],[18,47751,47753],{"className":47752},[115],[18,47754,47756,47776],{"className":47755},[119,120],[18,47757,47759,47773],{"className":47758},[124],[18,47760,47762],{"className":47761,"style":1152},[128],[18,47763,47764,47767],{"style":1155},[18,47765],{"className":47766,"style":137},[136],[18,47768,47770],{"className":47769},[141,142,143,144],[18,47771,38934],{"className":47772,"style":38967},[106,110,144],[18,47774,152],{"className":47775},[151],[18,47777,47779],{"className":47778},[124],[18,47780,47782],{"className":47781,"style":159},[128],[18,47783],{},[18,47785,74],{"className":47786},[250],[18,47788,18328],{"className":47789},[106,110],[18,47791,84],{"className":47792},[267],[18,47794,968],{"className":47795},[106],[1311,47797,47799],{"id":47798},"_12-误差分解","1.2 误差分解",[11,47801,47802,47803,38320,47886,48015],{},"对任意 ",[18,47804,47806,47829],{"className":47805,"translate":22},[21],[18,47807,47809],{"className":47808},[26],[28,47810,47811],{"xmlns":30},[32,47812,47813,47826],{},[35,47814,47815,47822,47824],{},[1556,47816,47817,47819],{},[41,47818,18328],{},[49,47820,47821],{},"∗",[49,47823,921],{},[41,47825,18309],{"mathvariant":1462},[86,47827,47828],{"encoding":88},"h^*\\in\\mathcal H",[18,47830,47832,47877],{"className":47831,"ariaHidden":51},[93],[18,47833,47835,47838,47868,47871,47874],{"className":47834},[97],[18,47836],{"className":47837,"style":1422},[101],[18,47839,47841,47844],{"className":47840},[106],[18,47842,18328],{"className":47843},[106,110],[18,47845,47847],{"className":47846},[115],[18,47848,47850],{"className":47849},[119],[18,47851,47853],{"className":47852},[124],[18,47854,47857],{"className":47855,"style":47856},[128],"height:0.6887em;",[18,47858,47859,47862],{"style":1962},[18,47860],{"className":47861,"style":137},[136],[18,47863,47865],{"className":47864},[141,142,143,144],[18,47866,47821],{"className":47867},[2072,144],[18,47869],{"className":47870,"style":228},[169],[18,47872,921],{"className":47873},[232],[18,47875],{"className":47876,"style":228},[169],[18,47878,47880,47883],{"className":47879},[97],[18,47881],{"className":47882,"style":600},[101],[18,47884,18309],{"className":47885,"style":46330},[106,1478],[18,47887,47889,47917],{"className":47888,"translate":22},[21],[18,47890,47892],{"className":47891},[26],[28,47893,47894],{"xmlns":30},[32,47895,47896,47914],{},[35,47897,47898,47904,47906,47912],{},[38,47899,47900,47902],{},[41,47901,12582],{},[41,47903,12575],{"mathvariant":1462},[49,47905,74],{"stretchy":73},[38,47907,47908,47910],{},[41,47909,18328],{},[41,47911,38934],{},[49,47913,84],{"stretchy":73},[86,47915,47916],{"encoding":88},"L_{\\mathcal D}(h_S)",[18,47918,47920],{"className":47919,"ariaHidden":51},[93],[18,47921,47923,47926,47969,47972,48012],{"className":47922},[97],[18,47924],{"className":47925,"style":242},[101],[18,47927,47929,47932],{"className":47928},[106],[18,47930,12582],{"className":47931},[106,110],[18,47933,47935],{"className":47934},[115],[18,47936,47938,47961],{"className":47937},[119,120],[18,47939,47941,47958],{"className":47940},[124],[18,47942,47944],{"className":47943,"style":1152},[128],[18,47945,47946,47949],{"style":1155},[18,47947],{"className":47948,"style":137},[136],[18,47950,47952],{"className":47951},[141,142,143,144],[18,47953,47955],{"className":47954},[106,144],[18,47956,12575],{"className":47957,"style":254},[106,1478,144],[18,47959,152],{"className":47960},[151],[18,47962,47964],{"className":47963},[124],[18,47965,47967],{"className":47966,"style":159},[128],[18,47968],{},[18,47970,74],{"className":47971},[250],[18,47973,47975,47978],{"className":47974},[106],[18,47976,18328],{"className":47977},[106,110],[18,47979,47981],{"className":47980},[115],[18,47982,47984,48004],{"className":47983},[119,120],[18,47985,47987,48001],{"className":47986},[124],[18,47988,47990],{"className":47989,"style":1152},[128],[18,47991,47992,47995],{"style":1155},[18,47993],{"className":47994,"style":137},[136],[18,47996,47998],{"className":47997},[141,142,143,144],[18,47999,38934],{"className":48000,"style":38967},[106,110,144],[18,48002,152],{"className":48003},[151],[18,48005,48007],{"className":48006},[124],[18,48008,48010],{"className":48009,"style":159},[128],[18,48011],{},[18,48013,84],{"className":48014},[267]," 拆开：",[18,48017,48019],{"className":48018,"translate":22},[875],[18,48020,48022,48403],{"className":48021,"translate":22},[21],[18,48023,48025],{"className":48024},[26],[28,48026,48027],{"xmlns":30,"display":884},[32,48028,48029,48400],{},[48030,48031,48034,48126,48200,48314],"mtable",{"rowspacing":48032,"columnalign":48033,"columnspacing":32628},"0.25em","right left",[48035,48036,48037,48060],"mtr",{},[48038,48039,48040],"mtd",{},[14681,48041,48042],{"scriptlevel":430,"displaystyle":51},[35,48043,48044,48050,48052,48058],{},[38,48045,48046,48048],{},[41,48047,12582],{},[41,48049,12575],{"mathvariant":1462},[49,48051,74],{"stretchy":73},[38,48053,48054,48056],{},[41,48055,18328],{},[41,48057,38934],{},[49,48059,84],{"stretchy":73},[48038,48061,48062],{},[14681,48063,48064],{"scriptlevel":430,"displaystyle":51},[35,48065,48066,48068,48070,48072,48078,48080,48086,48088,48090,48096,48098,48104,48106,48108,48110,48116,48118,48124],{},[35,48067],{},[49,48069,433],{},[49,48071,945],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[38,48073,48074,48076],{},[41,48075,12582],{},[41,48077,12575],{"mathvariant":1462},[49,48079,74],{"stretchy":73},[38,48081,48082,48084],{},[41,48083,18328],{},[41,48085,38934],{},[49,48087,84],{"stretchy":73},[49,48089,2253],{},[38,48091,48092,48094],{},[41,48093,12582],{},[41,48095,38934],{},[49,48097,74],{"stretchy":73},[38,48099,48100,48102],{},[41,48101,18328],{},[41,48103,38934],{},[49,48105,84],{"stretchy":73},[49,48107,965],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,48109,7987],{},[38,48111,48112,48114],{},[41,48113,12582],{},[41,48115,38934],{},[49,48117,74],{"stretchy":73},[38,48119,48120,48122],{},[41,48121,18328],{},[41,48123,38934],{},[49,48125,84],{"stretchy":73},[48035,48127,48128,48134],{},[48038,48129,48130],{},[14681,48131,48132],{"scriptlevel":430,"displaystyle":51},[35,48133],{},[48038,48135,48136],{},[14681,48137,48138],{"scriptlevel":430,"displaystyle":51},[35,48139,48140,48142,48144,48146,48152,48154,48160,48162,48164,48170,48172,48178,48180,48182,48184,48190,48192,48198],{},[35,48141],{},[49,48143,4334],{},[49,48145,945],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[38,48147,48148,48150],{},[41,48149,12582],{},[41,48151,12575],{"mathvariant":1462},[49,48153,74],{"stretchy":73},[38,48155,48156,48158],{},[41,48157,18328],{},[41,48159,38934],{},[49,48161,84],{"stretchy":73},[49,48163,2253],{},[38,48165,48166,48168],{},[41,48167,12582],{},[41,48169,38934],{},[49,48171,74],{"stretchy":73},[38,48173,48174,48176],{},[41,48175,18328],{},[41,48177,38934],{},[49,48179,84],{"stretchy":73},[49,48181,965],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,48183,7987],{},[38,48185,48186,48188],{},[41,48187,12582],{},[41,48189,38934],{},[49,48191,74],{"stretchy":73},[1556,48193,48194,48196],{},[41,48195,18328],{},[49,48197,47821],{},[49,48199,84],{"stretchy":73},[48035,48201,48202,48208],{},[48038,48203,48204],{},[14681,48205,48206],{"scriptlevel":430,"displaystyle":51},[35,48207],{},[48038,48209,48210],{},[14681,48211,48212],{"scriptlevel":430,"displaystyle":51},[35,48213,48214,48216,48218,48220,48226,48228,48234,48236,48238,48244,48246,48252,48254,48256,48258,48260,48266,48268,48274,48276,48278,48284,48286,48292,48294,48296,48298,48304,48306,48312],{},[35,48215],{},[49,48217,433],{},[49,48219,945],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[38,48221,48222,48224],{},[41,48223,12582],{},[41,48225,12575],{"mathvariant":1462},[49,48227,74],{"stretchy":73},[38,48229,48230,48232],{},[41,48231,18328],{},[41,48233,38934],{},[49,48235,84],{"stretchy":73},[49,48237,2253],{},[38,48239,48240,48242],{},[41,48241,12582],{},[41,48243,38934],{},[49,48245,74],{"stretchy":73},[38,48247,48248,48250],{},[41,48249,18328],{},[41,48251,38934],{},[49,48253,84],{"stretchy":73},[49,48255,965],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,48257,7987],{},[49,48259,945],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[38,48261,48262,48264],{},[41,48263,12582],{},[41,48265,38934],{},[49,48267,74],{"stretchy":73},[1556,48269,48270,48272],{},[41,48271,18328],{},[49,48273,47821],{},[49,48275,84],{"stretchy":73},[49,48277,2253],{},[38,48279,48280,48282],{},[41,48281,12582],{},[41,48283,12575],{"mathvariant":1462},[49,48285,74],{"stretchy":73},[1556,48287,48288,48290],{},[41,48289,18328],{},[49,48291,47821],{},[49,48293,84],{"stretchy":73},[49,48295,965],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,48297,7987],{},[38,48299,48300,48302],{},[41,48301,12582],{},[41,48303,12575],{"mathvariant":1462},[49,48305,74],{"stretchy":73},[1556,48307,48308,48310],{},[41,48309,18328],{},[49,48311,47821],{},[49,48313,84],{"stretchy":73},[48035,48315,48316,48322],{},[48038,48317,48318],{},[14681,48319,48320],{"scriptlevel":430,"displaystyle":51},[35,48321],{},[48038,48323,48324],{},[14681,48325,48326],{"scriptlevel":430,"displaystyle":51},[35,48327,48328,48330,48332,48338,48340,48346,48348,48350,48352,48368,48370,48376,48378,48380,48382,48384,48390,48392,48394,48396,48398],{},[35,48329],{},[49,48331,4334],{},[38,48333,48334,48336],{},[41,48335,12582],{},[41,48337,12575],{"mathvariant":1462},[49,48339,74],{"stretchy":73},[1556,48341,48342,48344],{},[41,48343,18328],{},[49,48345,47821],{},[49,48347,84],{"stretchy":73},[49,48349,7987],{},[45,48351,2266],{},[890,48353,48354,48360],{},[35,48355,48356,48358],{},[41,48357,912],{},[49,48359,899],{},[35,48361,48362,48364,48366],{},[41,48363,18328],{},[49,48365,921],{},[41,48367,18309],{"mathvariant":1462},[49,48369,11225],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[38,48371,48372,48374],{},[41,48373,12582],{},[41,48375,12575],{"mathvariant":1462},[49,48377,74],{"stretchy":73},[41,48379,18328],{},[49,48381,84],{"stretchy":73},[49,48383,2253],{},[38,48385,48386,48388],{},[41,48387,12582],{},[41,48389,38934],{},[49,48391,74],{"stretchy":73},[41,48393,18328],{},[49,48395,84],{"stretchy":73},[49,48397,11225],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[41,48399,968],{"mathvariant":316},[86,48401,48402],{"encoding":88},"\\begin{aligned}\nL_{\\mathcal D}(h_S)\n&=\\bigl[L_{\\mathcal D}(h_S)-L_S(h_S)\\bigr]+L_S(h_S)\\\\\n&\\le \\bigl[L_{\\mathcal D}(h_S)-L_S(h_S)\\bigr]+L_S(h^*)\\\\\n&=\\bigl[L_{\\mathcal D}(h_S)-L_S(h_S)\\bigr]+\\bigl[L_S(h^*)-L_{\\mathcal D}(h^*)\\bigr]+L_{\\mathcal D}(h^*)\\\\\n&\\le L_{\\mathcal D}(h^*)+2\\sup_{h\\in\\mathcal H}\\bigl|L_{\\mathcal 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是",[717,50172,50173],{},"近似误差","：假设类里最好的那个假设有多好。第二项是",[717,50176,50177],{},"估计误差","：经验风险对真实风险的均匀偏差。",[11,50180,50181,50182,50210],{},"这个分解是整篇文章的地图。VC 理论只关心第二项。它不问你的网络能不能逼近目标函数——那是逼近论的问题；它只问：当你在 ",[18,50183,50185,50198],{"className":50184,"translate":22},[21],[18,50186,50188],{"className":50187},[26],[28,50189,50190],{"xmlns":30},[32,50191,50192,50196],{},[35,50193,50194],{},[41,50195,18309],{"mathvariant":1462},[86,50197,46317],{"encoding":88},[18,50199,50201],{"className":50200,"ariaHidden":51},[93],[18,50202,50204,50207],{"className":50203},[97],[18,50205],{"className":50206,"style":600},[101],[18,50208,18309],{"className":50209,"style":46330},[106,1478]," 里挑一个经验风险最小的假设时，经验风险与真实风险的差距能被控制到什么程度。",[1311,50212,50214],{"id":50213},"_13-一致收敛","1.3 一致收敛",[11,50216,50217,50218,50246,50247,19371,50298,50349],{},"如果对任意分布 ",[18,50219,50221,50234],{"className":50220,"translate":22},[21],[18,50222,50224],{"className":50223},[26],[28,50225,50226],{"xmlns":30},[32,50227,50228,50232],{},[35,50229,50230],{},[41,50231,12575],{"mathvariant":1462},[86,50233,46402],{"encoding":88},[18,50235,50237],{"className":50236,"ariaHidden":51},[93],[18,50238,50240,50243],{"className":50239},[97],[18,50241],{"className":50242,"style":600},[101],[18,50244,12575],{"className":50245,"style":254},[106,1478]," 和任意 ",[18,50248,50250,50268],{"className":50249,"translate":22},[21],[18,50251,50253],{"className":50252},[26],[28,50254,50255],{"xmlns":30},[32,50256,50257,50265],{},[35,50258,50259,50261,50263],{},[41,50260,32522],{},[49,50262,6885],{},[45,50264,430],{},[86,50266,50267],{"encoding":88},"\\varepsilon>0",[18,50269,50271,50289],{"className":50270,"ariaHidden":51},[93],[18,50272,50274,50277,50280,50283,50286],{"className":50273},[97],[18,50275],{"className":50276,"style":30930},[101],[18,50278,32522],{"className":50279},[106,110],[18,50281],{"className":50282,"style":228},[169],[18,50284,6885],{"className":50285},[232],[18,50287],{"className":50288,"style":228},[169],[18,50290,50292,50295],{"className":50291},[97],[18,50293],{"className":50294,"style":505},[101],[18,50296,430],{"className":50297},[106],[18,50299,50301,50319],{"className":50300,"translate":22},[21],[18,50302,50304],{"className":50303},[26],[28,50305,50306],{"xmlns":30},[32,50307,50308,50316],{},[35,50309,50310,50312,50314],{},[41,50311,28169],{},[49,50313,1880],{},[41,50315,2028],{"mathvariant":316},[86,50317,50318],{"encoding":88},"m\\to\\infty",[18,50320,50322,50340],{"className":50321,"ariaHidden":51},[93],[18,50323,50325,50328,50331,50334,50337],{"className":50324},[97],[18,50326],{"className":50327,"style":1083},[101],[18,50329,28169],{"className":50330},[106,110],[18,50332],{"className":50333,"style":228},[169],[18,50335,1880],{"className":50336},[232],[18,50338],{"className":50339,"style":228},[169],[18,50341,50343,50346],{"className":50342},[97],[18,50344],{"className":50345,"style":1083},[101],[18,50347,2028],{"className":50348},[106]," 时",[18,50351,50353],{"className":50352,"translate":22},[875],[18,50354,50356,50434],{"className":50355,"translate":22},[21],[18,50357,50359],{"className":50358},[26],[28,50360,50361],{"xmlns":30,"display":884},[32,50362,50363,50431],{},[35,50364,50365,50367,50369,50425,50427,50429],{},[41,50366,11208],{},[49,50368,899],{},[35,50370,50371,50373,50389,50391,50397,50399,50401,50403,50405,50411,50413,50415,50417,50419,50421,50423],{},[49,50372,945],{"fence":51},[890,50374,50375,50381],{},[35,50376,50377,50379],{},[41,50378,912],{},[49,50380,899],{},[35,50382,50383,50385,50387],{},[41,50384,18328],{},[49,50386,921],{},[41,50388,18309],{"mathvariant":1462},[49,50390,11225],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[38,50392,50393,50395],{},[41,50394,12582],{},[41,50396,12575],{"mathvariant":1462},[49,50398,74],{"stretchy":73},[41,50400,18328],{},[49,50402,84],{"stretchy":73},[49,50404,2253],{},[38,50406,50407,50409],{},[41,50408,12582],{},[41,50410,38934],{},[49,50412,74],{"stretchy":73},[41,50414,18328],{},[49,50416,84],{"stretchy":73},[49,50418,11225],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,50420,6885],{},[41,50422,32522],{},[49,50424,965],{"fence":51},[49,50426,1880],{},[45,50428,430],{},[49,50430,52],{"separator":51},[86,50432,50433],{"encoding":88},"\\Pr\\left[\\sup_{h\\in\\mathcal H}\\bigl|L_{\\mathcal D}(h)-L_S(h)\\bigr|>\\varepsilon\\right]\\to 0,",[18,50435,50437,50723],{"className":50436,"ariaHidden":51},[93],[18,50438,50440,50444,50447,50450,50714,50717,50720],{"className":50439},[97],[18,50441],{"className":50442,"style":50443},[101],"height:2.4239em;vertical-align:-0.9739em;",[18,50445,11208],{"className":50446},[985],[18,50448],{"className":50449,"style":170},[169],[18,50451,50453,50459,50512,50549,50592,50595,50598,50601,50604,50607,50610,50650,50653,50656,50659,50696,50699,50702,50705,50708],{"className":50452},[174],[18,50454,50456],{"className":50455,"style":13796},[250,13795],[18,50457,945],{"className":50458},[1230,143],[18,50460,50462],{"className":50461},[985,986],[18,50463,50465,50504],{"className":50464},[119,120],[18,50466,50468,50501],{"className":50467},[124],[18,50469,50471,50491],{"className":50470,"style":1083},[128],[18,50472,50473,50476],{"style":1086},[18,50474],{"className":50475,"style":697},[136],[18,50477,50479],{"className":50478},[141,142,143,144],[18,50480,50482,50485,50488],{"className":50481},[106,144],[18,50483,18328],{"className":50484},[106,110,144],[18,50486,921],{"className":50487},[232,144],[18,50489,18309],{"className":50490,"style":46330},[106,1478,144],[18,50492,50493,50496],{"style":693},[18,50494],{"className":50495,"style":697},[136],[18,50497,50498],{},[18,50499,912],{"className":50500},[985],[18,50502,152],{"className":50503},[151],[18,50505,50507],{"className":50506},[124],[18,50508,50510],{"className":50509,"style":1124},[128],[18,50511],{},[18,50513,50515],{"className":50514},[250],[18,50516,50518],{"className":50517},[1230,6108],[18,50519,50521,50541],{"className":50520},[119,120],[18,50522,50524,50538],{"className":50523},[124],[18,50525,50527],{"className":50526,"style":6118},[128],[18,50528,50529,50532],{"style":6121},[18,50530],{"className":50531,"style":6125},[136],[18,50533,50534],{"style":49869},[6130,50535,50536],{"xmlns":6132,"width":49872,"height":944,"viewBox":49873},[6136,50537],{"d":49876},[18,50539,152],{"className":50540},[151],[18,50542,50544],{"className":50543},[124],[18,50545,50547],{"className":50546,"style":6148},[128],[18,50548],{},[18,50550,50552,50555],{"className":50551},[106],[18,50553,12582],{"className":50554},[106,110],[18,50556,50558],{"className":50557},[115],[18,50559,50561,50584],{"className":50560},[119,120],[18,50562,50564,50581],{"className":50563},[124],[18,50565,50567],{"className":50566,"style":1152},[128],[18,50568,50569,50572],{"style":1155},[18,50570],{"className":50571,"style":137},[136],[18,50573,50575],{"className":50574},[141,142,143,144],[18,50576,50578],{"className":50577},[106,144],[18,50579,12575],{"className":50580,"style":254},[106,1478,144],[18,50582,152],{"className":50583},[151],[18,50585,50587],{"className":50586},[124],[18,50588,50590],{"className":50589,"style":159},[128],[18,50591],{},[18,50593,74],{"className":50594},[250],[18,50596,18328],{"className":50597},[106,110],[18,50599,84],{"className":50600},[267],[18,50602],{"className":50603,"style":2068},[169],[18,50605,2253],{"className":50606},[2072],[18,50608],{"className":50609,"style":2068},[169],[18,50611,50613,50616],{"className":50612},[106],[18,50614,12582],{"className":50615},[106,110],[18,50617,50619],{"className":50618},[115],[18,50620,50622,50642],{"className":50621},[119,120],[18,50623,50625,50639],{"className":50624},[124],[18,50626,50628],{"className":50627,"style":1152},[128],[18,50629,50630,50633],{"style":1155},[18,50631],{"className":50632,"style":137},[136],[18,50634,50636],{"className":50635},[141,142,143,144],[18,50637,38934],{"className":50638,"style":38967},[106,110,144],[18,50640,152],{"className":50641},[151],[18,50643,50645],{"className":50644},[124],[18,50646,50648],{"className":50647,"style":159},[128],[18,50649],{},[18,50651,74],{"className":50652},[250],[18,50654,18328],{"className":50655},[106,110],[18,50657,84],{"className":50658},[267],[18,50660,50662],{"className":50661},[267],[18,50663,50665],{"className":50664},[1230,6108],[18,50666,50668,50688],{"className":50667},[119,120],[18,50669,50671,50685],{"className":50670},[124],[18,50672,50674],{"className":50673,"style":6118},[128],[18,50675,50676,50679],{"style":6121},[18,50677],{"className":50678,"style":6125},[136],[18,50680,50681],{"style":49869},[6130,50682,50683],{"xmlns":6132,"width":49872,"height":944,"viewBox":49873},[6136,50684],{"d":49876},[18,50686,152],{"className":50687},[151],[18,50689,50691],{"className":50690},[124],[18,50692,50694],{"className":50693,"style":6148},[128],[18,50695],{},[18,50697],{"className":50698,"style":228},[169],[18,50700,6885],{"className":50701},[232],[18,50703],{"className":50704,"style":228},[169],[18,50706,32522],{"className":50707},[106,110],[18,50709,50711],{"className":50710,"style":13796},[267,13795],[18,50712,965],{"className":50713},[1230,143],[18,50715],{"className":50716,"style":228},[169],[18,50718,1880],{"className":50719},[232],[18,50721],{"className":50722,"style":228},[169],[18,50724,50726,50730,50733],{"className":50725},[97],[18,50727],{"className":50728,"style":50729},[101],"height:0.8389em;vertical-align:-0.1944em;",[18,50731,430],{"className":50732},[106],[18,50734,52],{"className":50735},[165],[11,50737,50738,50739,50767,50768,50771],{},"就说 ",[18,50740,50742,50755],{"className":50741,"translate":22},[21],[18,50743,50745],{"className":50744},[26],[28,50746,50747],{"xmlns":30},[32,50748,50749,50753],{},[35,50750,50751],{},[41,50752,18309],{"mathvariant":1462},[86,50754,46317],{"encoding":88},[18,50756,50758],{"className":50757,"ariaHidden":51},[93],[18,50759,50761,50764],{"className":50760},[97],[18,50762],{"className":50763,"style":600},[101],[18,50765,18309],{"className":50766,"style":46330},[106,1478]," 具有",[717,50769,50770],{},"一致收敛性","（uniform convergence）。",[11,50773,50774,50805,50806,50834,50835,50863,50864,50934,50935,50963],{},[18,50775,50777,50793],{"className":50776,"translate":22},[21],[18,50778,50780],{"className":50779},[26],[28,50781,50782],{"xmlns":30},[32,50783,50784,50790],{},[35,50785,50786,50788],{},[41,50787,912],{},[49,50789,899],{},[86,50791,50792],{"encoding":88},"\\sup",[18,50794,50796],{"className":50795,"ariaHidden":51},[93],[18,50797,50799,50802],{"className":50798},[97],[18,50800],{"className":50801,"style":38552},[101],[18,50803,912],{"className":50804},[985]," 在概率里面，这个顺序很关键：不是「对每个 ",[18,50807,50809,50822],{"className":50808,"translate":22},[21],[18,50810,50812],{"className":50811},[26],[28,50813,50814],{"xmlns":30},[32,50815,50816,50820],{},[35,50817,50818],{},[41,50819,18328],{},[86,50821,18328],{"encoding":88},[18,50823,50825],{"className":50824,"ariaHidden":51},[93],[18,50826,50828,50831],{"className":50827},[97],[18,50829],{"className":50830,"style":295},[101],[18,50832,18328],{"className":50833},[106,110]," 分别收敛」，而是「所有 ",[18,50836,50838,50851],{"className":50837,"translate":22},[21],[18,50839,50841],{"className":50840},[26],[28,50842,50843],{"xmlns":30},[32,50844,50845,50849],{},[35,50846,50847],{},[41,50848,18328],{},[86,50850,18328],{"encoding":88},[18,50852,50854],{"className":50853,"ariaHidden":51},[93],[18,50855,50857,50860],{"className":50856},[97],[18,50858],{"className":50859,"style":295},[101],[18,50861,18328],{"className":50862},[106,110]," 同时收敛」。前者由大数定律自动成立，后者才是学习所需要的——因为 ",[18,50865,50867,50885],{"className":50866,"translate":22},[21],[18,50868,50870],{"className":50869},[26],[28,50871,50872],{"xmlns":30},[32,50873,50874,50882],{},[35,50875,50876],{},[38,50877,50878,50880],{},[41,50879,18328],{},[41,50881,38934],{},[86,50883,50884],{"encoding":88},"h_S",[18,50886,50888],{"className":50887,"ariaHidden":51},[93],[18,50889,50891,50894],{"className":50890},[97],[18,50892],{"className":50893,"style":448},[101],[18,50895,50897,50900],{"className":50896},[106],[18,50898,18328],{"className":50899},[106,110],[18,50901,50903],{"className":50902},[115],[18,50904,50906,50926],{"className":50905},[119,120],[18,50907,50909,50923],{"className":50908},[124],[18,50910,50912],{"className":50911,"style":1152},[128],[18,50913,50914,50917],{"style":1155},[18,50915],{"className":50916,"style":137},[136],[18,50918,50920],{"className":50919},[141,142,143,144],[18,50921,38934],{"className":50922,"style":38967},[106,110,144],[18,50924,152],{"className":50925},[151],[18,50927,50929],{"className":50928},[124],[18,50930,50932],{"className":50931,"style":159},[128],[18,50933],{}," 本身依赖数据，你不能先固定 ",[18,50936,50938,50951],{"className":50937,"translate":22},[21],[18,50939,50941],{"className":50940},[26],[28,50942,50943],{"xmlns":30},[32,50944,50945,50949],{},[35,50946,50947],{},[41,50948,18328],{},[86,50950,18328],{"encoding":88},[18,50952,50954],{"className":50953,"ariaHidden":51},[93],[18,50955,50957,50960],{"className":50956},[97],[18,50958],{"className":50959,"style":295},[101],[18,50961,18328],{"className":50962},[106,110]," 再用大数定律。",[11,50965,50966,50967,50995],{},"于是问题变成：什么样的 ",[18,50968,50970,50983],{"className":50969,"translate":22},[21],[18,50971,50973],{"className":50972},[26],[28,50974,50975],{"xmlns":30},[32,50976,50977,50981],{},[35,50978,50979],{},[41,50980,18309],{"mathvariant":1462},[86,50982,46317],{"encoding":88},[18,50984,50986],{"className":50985,"ariaHidden":51},[93],[18,50987,50989,50992],{"className":50988},[97],[18,50990],{"className":50991,"style":600},[101],[18,50993,18309],{"className":50994,"style":46330},[106,1478]," 具有一致收敛性？答案由 VC 维给出。",[638,50997,50999],{"id":50998},"二打散与-vc-维","二、打散与 VC 维",[1311,51001,51003],{"id":51002},"_21-二分","2.1 二分",[11,51005,51006,51007,1990,51202,3019,51230,51258,51259,51262],{},"固定一个有限的点集 ",[18,51008,51010,51053],{"className":51009,"translate":22},[21],[18,51011,51013],{"className":51012},[26],[28,51014,51015],{"xmlns":30},[32,51016,51017,51050],{},[35,51018,51019,51022,51024,51026,51032,51034,51036,51038,51044,51046,51048],{},[41,51020,51021],{},"C",[49,51023,433],{},[49,51025,1331],{"stretchy":73},[38,51027,51028,51030],{},[41,51029,39067],{},[45,51031,47],{},[49,51033,52],{"separator":51},[49,51035,55],{},[49,51037,52],{"separator":51},[38,51039,51040,51042],{},[41,51041,39067],{},[41,51043,28169],{},[49,51045,1349],{"stretchy":73},[49,51047,4548],{},[41,51049,43],{"mathvariant":1462},[86,51051,51052],{"encoding":88},"C=\\{c_1,\\dots,c_m\\}\\subseteq\\mathcal X",[18,51054,51056,51074,51193],{"className":51055,"ariaHidden":51},[93],[18,51057,51059,51062,51065,51068,51071],{"className":51058},[97],[18,51060],{"className":51061,"style":600},[101],[18,51063,51021],{"className":51064,"style":5073},[106,110],[18,51066],{"className":51067,"style":228},[169],[18,51069,433],{"className":51070},[232],[18,51072],{"className":51073,"style":228},[169],[18,51075,51077,51080,51083,51123,51126,51129,51132,51135,51138,51141,51181,51184,51187,51190],{"className":51076},[97],[18,51078],{"className":51079,"style":242},[101],[18,51081,1331],{"className":51082},[250],[18,51084,51086,51089],{"className":51085},[106],[18,51087,39067],{"className":51088},[106,110],[18,51090,51092],{"className":51091},[115],[18,51093,51095,51115],{"className":51094},[119,120],[18,51096,51098,51112],{"className":51097},[124],[18,51099,51101],{"className":51100,"style":129},[128],[18,51102,51103,51106],{"style":1155},[18,51104],{"className":51105,"style":137},[136],[18,51107,51109],{"className":51108},[141,142,143,144],[18,51110,47],{"className":51111},[106,144],[18,51113,152],{"className":51114},[151],[18,51116,51118],{"className":51117},[124],[18,51119,51121],{"className":51120,"style":159},[128],[18,51122],{},[18,51124,52],{"className":51125},[165],[18,51127],{"className":51128,"style":170},[169],[18,51130,55],{"className":51131},[174],[18,51133],{"className":51134,"style":170},[169],[18,51136,52],{"className":51137},[165],[18,51139],{"className":51140,"style":170},[169],[18,51142,51144,51147],{"className":51143},[106],[18,51145,39067],{"className":51146},[106,110],[18,51148,51150],{"className":51149},[115],[18,51151,51153,51173],{"className":51152},[119,120],[18,51154,51156,51170],{"className":51155},[124],[18,51157,51159],{"className":51158,"style":202},[128],[18,51160,51161,51164],{"style":1155},[18,51162],{"className":51163,"style":137},[136],[18,51165,51167],{"className":51166},[141,142,143,144],[18,51168,28169],{"className":51169},[106,110,144],[18,51171,152],{"className":51172},[151],[18,51174,51176],{"className":51175},[124],[18,51177,51179],{"className":51178,"style":159},[128],[18,51180],{},[18,51182,1349],{"className":51183},[267],[18,51185],{"className":51186,"style":228},[169],[18,51188,4548],{"className":51189},[232],[18,51191],{"className":51192,"style":228},[169],[18,51194,51196,51199],{"className":51195},[97],[18,51197],{"className":51198,"style":600},[101],[18,51200,43],{"className":51201,"style":1479},[106,1478],[18,51203,51205,51218],{"className":51204,"translate":22},[21],[18,51206,51208],{"className":51207},[26],[28,51209,51210],{"xmlns":30},[32,51211,51212,51216],{},[35,51213,51214],{},[41,51215,18309],{"mathvariant":1462},[86,51217,46317],{"encoding":88},[18,51219,51221],{"className":51220,"ariaHidden":51},[93],[18,51222,51224,51227],{"className":51223},[97],[18,51225],{"className":51226,"style":600},[101],[18,51228,18309],{"className":51229,"style":46330},[106,1478],[18,51231,51233,51246],{"className":51232,"translate":22},[21],[18,51234,51236],{"className":51235},[26],[28,51237,51238],{"xmlns":30},[32,51239,51240,51244],{},[35,51241,51242],{},[41,51243,51021],{},[86,51245,51021],{"encoding":88},[18,51247,51249],{"className":51248,"ariaHidden":51},[93],[18,51250,51252,51255],{"className":51251},[97],[18,51253],{"className":51254,"style":600},[101],[18,51256,51021],{"className":51257,"style":5073},[106,110]," 上的",[717,51260,51261],{},"限制","（restriction）是",[18,51264,51266],{"className":51265,"translate":22},[875],[18,51267,51269,51355],{"className":51268,"translate":22},[21],[18,51270,51272],{"className":51271},[26],[28,51273,51274],{"xmlns":30,"display":884},[32,51275,51276,51352],{},[35,51277,51278,51280,51286,51288,51290,51292,51294,51296,51302,51304,51306,51308,51310,51312,51314,51320,51322,51324,51326,51328,51330,51332,51334,51336,51338,51340,51342,51344,51350],{},[41,51279,18309],{"mathvariant":1462},[38,51281,51282,51284],{},[41,51283,11225],{"mathvariant":316},[41,51285,51021],{},[49,51287,433],{},[49,51289,1331],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,51291,74],{"stretchy":73},[41,51293,18328],{},[49,51295,74],{"stretchy":73},[38,51297,51298,51300],{},[41,51299,39067],{},[45,51301,47],{},[49,51303,84],{"stretchy":73},[49,51305,52],{"separator":51},[49,51307,55],{},[49,51309,52],{"separator":51},[41,51311,18328],{},[49,51313,74],{"stretchy":73},[38,51315,51316,51318],{},[41,51317,39067],{},[41,51319,28169],{},[49,51321,84],{"stretchy":73},[49,51323,84],{"stretchy":73},[49,51325,1340],{},[41,51327,18328],{},[49,51329,921],{},[41,51331,18309],{"mathvariant":1462},[49,51333,1349],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,51335,4548],{},[49,51337,1331],{"stretchy":73},[45,51339,430],{},[49,51341,52],{"separator":51},[45,51343,47],{},[1556,51345,51346,51348],{},[49,51347,1349],{"stretchy":73},[41,51349,28169],{},[41,51351,968],{"mathvariant":316},[86,51353,51354],{"encoding":88},"\\mathcal H|_C=\\bigl\\{(h(c_1),\\dots,h(c_m)):h\\in\\mathcal H\\bigr\\}\\subseteq\\{0,1\\}^m.",[18,51356,51358,51416,51556,51574,51598],{"className":51357,"ariaHidden":51},[93],[18,51359,51361,51364,51367,51407,51410,51413],{"className":51360},[97],[18,51362],{"className":51363,"style":242},[101],[18,51365,18309],{"className":51366,"style":46330},[106,1478],[18,51368,51370,51373],{"className":51369},[106],[18,51371,11225],{"className":51372},[106],[18,51374,51376],{"className":51375},[115],[18,51377,51379,51399],{"className":51378},[119,120],[18,51380,51382,51396],{"className":51381},[124],[18,51383,51385],{"className":51384,"style":1152},[128],[18,51386,51387,51390],{"style":1155},[18,51388],{"className":51389,"style":137},[136],[18,51391,51393],{"className":51392},[141,142,143,144],[18,51394,51021],{"className":51395,"style":5073},[106,110,144],[18,51397,152],{"className":51398},[151],[18,51400,51402],{"className":51401},[124],[18,51403,51405],{"className":51404,"style":159},[128],[18,51406],{},[18,51408],{"className":51409,"style":228},[169],[18,51411,433],{"className":51412},[232],[18,51414],{"className":51415,"style":228},[169],[18,51417,51419,51422,51428,51431,51434,51437,51477,51480,51483,51486,51489,51492,51495,51498,51501,51504,51544,51547,51550,51553],{"className":51418},[97],[18,51420],{"className":51421,"style":6101},[101],[18,51423,51425],{"className":51424},[250],[18,51426,1331],{"className":51427},[1230,1196],[18,51429,74],{"className":51430},[250],[18,51432,18328],{"className":51433},[106,110],[18,51435,74],{"className":51436},[250],[18,51438,51440,51443],{"className":51439},[106],[18,51441,39067],{"className":51442},[106,110],[18,51444,51446],{"className":51445},[115],[18,51447,51449,51469],{"className":51448},[119,120],[18,51450,51452,51466],{"className":51451},[124],[18,51453,51455],{"className":51454,"style":129},[128],[18,51456,51457,51460],{"style":1155},[18,51458],{"className":51459,"style":137},[136],[18,51461,51463],{"className":51462},[141,142,143,144],[18,51464,47],{"className":51465},[106,144],[18,51467,152],{"className":51468},[151],[18,51470,51472],{"className":51471},[124],[18,51473,51475],{"className":51474,"style":159},[128],[18,51476],{},[18,51478,84],{"className":51479},[267],[18,51481,52],{"className":51482},[165],[18,51484],{"className":51485,"style":170},[169],[18,51487,55],{"className":51488},[174],[18,51490],{"className":51491,"style":170},[169],[18,51493,52],{"className":51494},[165],[18,51496],{"className":51497,"style":170},[169],[18,51499,18328],{"className":51500},[106,110],[18,51502,74],{"className":51503},[250],[18,51505,51507,51510],{"className":51506},[106],[18,51508,39067],{"className":51509},[106,110],[18,51511,51513],{"className":51512},[115],[18,51514,51516,51536],{"className":51515},[119,120],[18,51517,51519,51533],{"className":51518},[124],[18,51520,51522],{"className":51521,"style":202},[128],[18,51523,51524,51527],{"style":1155},[18,51525],{"className":51526,"style":137},[136],[18,51528,51530],{"className":51529},[141,142,143,144],[18,51531,28169],{"className":51532},[106,110,144],[18,51534,152],{"className":51535},[151],[18,51537,51539],{"className":51538},[124],[18,51540,51542],{"className":51541,"style":159},[128],[18,51543],{},[18,51545,3431],{"className":51546},[267],[18,51548],{"className":51549,"style":228},[169],[18,51551,1340],{"className":51552},[232],[18,51554],{"className":51555,"style":228},[169],[18,51557,51559,51562,51565,51568,51571],{"className":51558},[97],[18,51560],{"className":51561,"style":1422},[101],[18,51563,18328],{"className":51564},[106,110],[18,51566],{"className":51567,"style":228},[169],[18,51569,921],{"className":51570},[232],[18,51572],{"className":51573,"style":228},[169],[18,51575,51577,51580,51583,51589,51592,51595],{"className":51576},[97],[18,51578],{"className":51579,"style":6101},[101],[18,51581,18309],{"className":51582,"style":46330},[106,1478],[18,51584,51586],{"className":51585},[267],[18,51587,1349],{"className":51588},[1230,1196],[18,51590],{"className":51591,"style":228},[169],[18,51593,4548],{"className":51594},[232],[18,51596],{"className":51597,"style":228},[169],[18,51599,51601,51604,51607,51610,51613,51616,51619,51648],{"className":51600},[97],[18,51602],{"className":51603,"style":242},[101],[18,51605,1331],{"className":51606},[250],[18,51608,430],{"className":51609},[106],[18,51611,52],{"className":51612},[165],[18,51614],{"className":51615,"style":170},[169],[18,51617,47],{"className":51618},[106],[18,51620,51622,51625],{"className":51621},[267],[18,51623,1349],{"className":51624},[267],[18,51626,51628],{"className":51627},[115],[18,51629,51631],{"className":51630},[119],[18,51632,51634],{"className":51633},[124],[18,51635,51637],{"className":51636,"style":1616},[128],[18,51638,51639,51642],{"style":1637},[18,51640],{"className":51641,"style":137},[136],[18,51643,51645],{"className":51644},[141,142,143,144],[18,51646,28169],{"className":51647},[106,110,144],[18,51649,968],{"className":51650},[106],[11,51652,51653,51654,51657,51658,51743,51744,51802,51803,51861,51862,51890,51891,51923,51924,51952],{},"这个集合里的每个元素称为一个",[717,51655,51656],{},"二分","（dichotomy）。",[18,51659,51661,51685],{"className":51660,"translate":22},[21],[18,51662,51664],{"className":51663},[26],[28,51665,51666],{"xmlns":30},[32,51667,51668,51682],{},[35,51669,51670,51672,51674,51680],{},[41,51671,11225],{"mathvariant":316},[41,51673,18309],{"mathvariant":1462},[38,51675,51676,51678],{},[41,51677,11225],{"mathvariant":316},[41,51679,51021],{},[41,51681,11225],{"mathvariant":316},[86,51683,51684],{"encoding":88},"|\\mathcal H|_C|",[18,51686,51688],{"className":51687,"ariaHidden":51},[93],[18,51689,51691,51694,51697,51700,51740],{"className":51690},[97],[18,51692],{"className":51693,"style":242},[101],[18,51695,11225],{"className":51696},[106],[18,51698,18309],{"className":51699,"style":46330},[106,1478],[18,51701,51703,51706],{"className":51702},[106],[18,51704,11225],{"className":51705},[106],[18,51707,51709],{"className":51708},[115],[18,51710,51712,51732],{"className":51711},[119,120],[18,51713,51715,51729],{"className":51714},[124],[18,51716,51718],{"className":51717,"style":1152},[128],[18,51719,51720,51723],{"style":1155},[18,51721],{"className":51722,"style":137},[136],[18,51724,51726],{"className":51725},[141,142,143,144],[18,51727,51021],{"className":51728,"style":5073},[106,110,144],[18,51730,152],{"className":51731},[151],[18,51733,51735],{"className":51734},[124],[18,51736,51738],{"className":51737,"style":159},[128],[18,51739],{},[18,51741,11225],{"className":51742},[106]," 最多是 ",[18,51745,51747,51764],{"className":51746,"translate":22},[21],[18,51748,51750],{"className":51749},[26],[28,51751,51752],{"xmlns":30},[32,51753,51754,51762],{},[35,51755,51756],{},[1556,51757,51758,51760],{},[45,51759,2266],{},[41,51761,28169],{},[86,51763,44902],{"encoding":88},[18,51765,51767],{"className":51766,"ariaHidden":51},[93],[18,51768,51770,51773],{"className":51769},[97],[18,51771],{"className":51772,"style":1959},[101],[18,51774,51776,51779],{"className":51775},[106],[18,51777,2266],{"className":51778},[106],[18,51780,51782],{"className":51781},[115],[18,51783,51785],{"className":51784},[119],[18,51786,51788],{"className":51787},[124],[18,51789,51791],{"className":51790,"style":1959},[128],[18,51792,51793,51796],{"style":1962},[18,51794],{"className":51795,"style":137},[136],[18,51797,51799],{"className":51798},[141,142,143,144],[18,51800,28169],{"className":51801},[106,110,144],"。如果取到 ",[18,51804,51806,51823],{"className":51805,"translate":22},[21],[18,51807,51809],{"className":51808},[26],[28,51810,51811],{"xmlns":30},[32,51812,51813,51821],{},[35,51814,51815],{},[1556,51816,51817,51819],{},[45,51818,2266],{},[41,51820,28169],{},[86,51822,44902],{"encoding":88},[18,51824,51826],{"className":51825,"ariaHidden":51},[93],[18,51827,51829,51832],{"className":51828},[97],[18,51830],{"className":51831,"style":1959},[101],[18,51833,51835,51838],{"className":51834},[106],[18,51836,2266],{"className":51837},[106],[18,51839,51841],{"className":51840},[115],[18,51842,51844],{"className":51843},[119],[18,51845,51847],{"className":51846},[124],[18,51848,51850],{"className":51849,"style":1959},[128],[18,51851,51852,51855],{"style":1962},[18,51853],{"className":51854,"style":137},[136],[18,51856,51858],{"className":51857},[141,142,143,144],[18,51859,28169],{"className":51860},[106,110,144],"，说明对这 ",[18,51863,51865,51878],{"className":51864,"translate":22},[21],[18,51866,51868],{"className":51867},[26],[28,51869,51870],{"xmlns":30},[32,51871,51872,51876],{},[35,51873,51874],{},[41,51875,28169],{},[86,51877,28169],{"encoding":88},[18,51879,51881],{"className":51880,"ariaHidden":51},[93],[18,51882,51884,51887],{"className":51883},[97],[18,51885],{"className":51886,"style":1083},[101],[18,51888,28169],{"className":51889},[106,110]," 个点的每一种 ",[18,51892,51894,51911],{"className":51893,"translate":22},[21],[18,51895,51897],{"className":51896},[26],[28,51898,51899],{"xmlns":30},[32,51900,51901,51909],{},[35,51902,51903,51905,51907],{},[45,51904,430],{},[41,51906,317],{"mathvariant":316},[45,51908,47],{},[86,51910,46060],{"encoding":88},[18,51912,51914],{"className":51913,"ariaHidden":51},[93],[18,51915,51917,51920],{"className":51916},[97],[18,51918],{"className":51919,"style":242},[101],[18,51921,46060],{"className":51922},[106]," 标注，",[18,51925,51927,51940],{"className":51926,"translate":22},[21],[18,51928,51930],{"className":51929},[26],[28,51931,51932],{"xmlns":30},[32,51933,51934,51938],{},[35,51935,51936],{},[41,51937,18309],{"mathvariant":1462},[86,51939,46317],{"encoding":88},[18,51941,51943],{"className":51942,"ariaHidden":51},[93],[18,51944,51946,51949],{"className":51945},[97],[18,51947],{"className":51948,"style":600},[101],[18,51950,18309],{"className":51951,"style":46330},[106,1478]," 里都有一个假设恰好实现它。",[5159,51954,51955],{},[11,51956,51957,51960,51961,52098,52099,1994,52127,52130,52131,1990],{},[717,51958,51959],{},"定义 2.1（打散）"," 若 ",[18,51962,51964,51996],{"className":51963,"translate":22},[21],[18,51965,51967],{"className":51966},[26],[28,51968,51969],{"xmlns":30},[32,51970,51971,51993],{},[35,51972,51973,51975,51977,51983,51985,51987],{},[41,51974,11225],{"mathvariant":316},[41,51976,18309],{"mathvariant":1462},[38,51978,51979,51981],{},[41,51980,11225],{"mathvariant":316},[41,51982,51021],{},[41,51984,11225],{"mathvariant":316},[49,51986,433],{},[1556,51988,51989,51991],{},[45,51990,2266],{},[41,51992,28169],{},[86,51994,51995],{"encoding":88},"|\\mathcal H|_C|=2^m",[18,51997,51999,52063],{"className":51998,"ariaHidden":51},[93],[18,52000,52002,52005,52008,52011,52051,52054,52057,52060],{"className":52001},[97],[18,52003],{"className":52004,"style":242},[101],[18,52006,11225],{"className":52007},[106],[18,52009,18309],{"className":52010,"style":46330},[106,1478],[18,52012,52014,52017],{"className":52013},[106],[18,52015,11225],{"className":52016},[106],[18,52018,52020],{"className":52019},[115],[18,52021,52023,52043],{"className":52022},[119,120],[18,52024,52026,52040],{"className":52025},[124],[18,52027,52029],{"className":52028,"style":1152},[128],[18,52030,52031,52034],{"style":1155},[18,52032],{"className":52033,"style":137},[136],[18,52035,52037],{"className":52036},[141,142,143,144],[18,52038,51021],{"className":52039,"style":5073},[106,110,144],[18,52041,152],{"className":52042},[151],[18,52044,52046],{"className":52045},[124],[18,52047,52049],{"className":52048,"style":159},[128],[18,52050],{},[18,52052,11225],{"className":52053},[106],[18,52055],{"className":52056,"style":228},[169],[18,52058,433],{"className":52059},[232],[18,52061],{"className":52062,"style":228},[169],[18,52064,52066,52069],{"className":52065},[97],[18,52067],{"className":52068,"style":1959},[101],[18,52070,52072,52075],{"className":52071},[106],[18,52073,2266],{"className":52074},[106],[18,52076,52078],{"className":52077},[115],[18,52079,52081],{"className":52080},[119],[18,52082,52084],{"className":52083},[124],[18,52085,52087],{"className":52086,"style":1959},[128],[18,52088,52089,52092],{"style":1962},[18,52090],{"className":52091,"style":137},[136],[18,52093,52095],{"className":52094},[141,142,143,144],[18,52096,28169],{"className":52097},[106,110,144],"，称 ",[18,52100,52102,52115],{"className":52101,"translate":22},[21],[18,52103,52105],{"className":52104},[26],[28,52106,52107],{"xmlns":30},[32,52108,52109,52113],{},[35,52110,52111],{},[41,52112,18309],{"mathvariant":1462},[86,52114,46317],{"encoding":88},[18,52116,52118],{"className":52117,"ariaHidden":51},[93],[18,52119,52121,52124],{"className":52120},[97],[18,52122],{"className":52123,"style":600},[101],[18,52125,18309],{"className":52126,"style":46330},[106,1478],[717,52128,52129],{},"打散","（shatter）了 ",[18,52132,52134,52147],{"className":52133,"translate":22},[21],[18,52135,52137],{"className":52136},[26],[28,52138,52139],{"xmlns":30},[32,52140,52141,52145],{},[35,52142,52143],{},[41,52144,51021],{},[86,52146,51021],{"encoding":88},[18,52148,52150],{"className":52149,"ariaHidden":51},[93],[18,52151,52153,52156],{"className":52152},[97],[18,52154],{"className":52155,"style":600},[101],[18,52157,51021],{"className":52158,"style":5073},[106,110],[5159,52160,52161,52199,52529],{},[11,52162,52163,52166,52167,39872,52195,52198],{},[717,52164,52165],{},"定义 2.2（VC 维）"," 假设类 ",[18,52168,52170,52183],{"className":52169,"translate":22},[21],[18,52171,52173],{"className":52172},[26],[28,52174,52175],{"xmlns":30},[32,52176,52177,52181],{},[35,52178,52179],{},[41,52180,18309],{"mathvariant":1462},[86,52182,46317],{"encoding":88},[18,52184,52186],{"className":52185,"ariaHidden":51},[93],[18,52187,52189,52192],{"className":52188},[97],[18,52190],{"className":52191,"style":600},[101],[18,52193,18309],{"className":52194,"style":46330},[106,1478],[717,52196,52197],{},"VC 维","是它能打散的最大有限集的大小：",[18,52200,52202],{"className":52201,"translate":22},[875],[18,52203,52205,52291],{"className":52204,"translate":22},[21],[18,52206,52208],{"className":52207},[26],[28,52209,52210],{"xmlns":30,"display":884},[32,52211,52212,52288],{},[35,52213,52214,52217,52219,52221,52223,52225,52227,52229,52231,52233,52235,52237,52240,52242,52244,52246,52248,52250,52252,52254,52256,52258,52260,52262,52264,52266,52268,52274,52276,52278,52284,52286],{},[41,52215,52216],{"mathvariant":316},"VCdim",[49,52218,899],{},[49,52220,74],{"stretchy":73},[41,52222,18309],{"mathvariant":1462},[49,52224,84],{"stretchy":73},[49,52226,433],{},[41,52228,912],{},[49,52230,899],{},[49,52232,1331],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[41,52234,28169],{},[49,52236,1340],{},[41,52238,52239],{"mathvariant":316},"∃",[41,52241,51021],{},[49,52243,4548],{},[41,52245,43],{"mathvariant":1462},[49,52247,52],{"separator":51},[926,52249,928],{},[41,52251,11225],{"mathvariant":316},[41,52253,51021],{},[41,52255,11225],{"mathvariant":316},[49,52257,433],{},[41,52259,28169],{},[49,52261,52],{"separator":51},[926,52263,928],{},[49,52265,11225],{"stretchy":73},[41,52267,18309],{"mathvariant":1462},[38,52269,52270,52272],{},[41,52271,11225],{"mathvariant":316},[41,52273,51021],{},[49,52275,11225],{"stretchy":73},[49,52277,433],{},[1556,52279,52280,52282],{},[45,52281,2266],{},[41,52283,28169],{},[49,52285,1349],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[41,52287,968],{"mathvariant":316},[86,52289,52290],{"encoding":88},"\\operatorname{VCdim}(\\mathcal H)=\\sup\\bigl\\{m:\\exists C\\subseteq\\mathcal X,\\ |C|=m,\\ \\lvert\\mathcal H|_C\\rvert=2^m\\bigr\\}.",[18,52292,52294,52324,52351,52373,52409,52485],{"className":52293,"ariaHidden":51},[93],[18,52295,52297,52300,52306,52309,52312,52315,52318,52321],{"className":52296},[97],[18,52298],{"className":52299,"style":242},[101],[18,52301,52303],{"className":52302},[985],[18,52304,52216],{"className":52305},[106,6354],[18,52307,74],{"className":52308},[250],[18,52310,18309],{"className":52311,"style":46330},[106,1478],[18,52313,84],{"className":52314},[267],[18,52316],{"className":52317,"style":228},[169],[18,52319,433],{"className":52320},[232],[18,52322],{"className":52323,"style":228},[169],[18,52325,52327,52330,52333,52339,52342,52345,52348],{"className":52326},[97],[18,52328],{"className":52329,"style":6101},[101],[18,52331,912],{"className":52332},[985],[18,52334,52336],{"className":52335},[250],[18,52337,1331],{"className":52338},[1230,1196],[18,52340,28169],{"className":52341},[106,110],[18,52343],{"className":52344,"style":228},[169],[18,52346,1340],{"className":52347},[232],[18,52349],{"className":52350,"style":228},[169],[18,52352,52354,52358,52361,52364,52367,52370],{"className":52353},[97],[18,52355],{"className":52356,"style":52357},[101],"height:0.8304em;vertical-align:-0.136em;",[18,52359,52239],{"className":52360},[106],[18,52362,51021],{"className":52363,"style":5073},[106,110],[18,52365],{"className":52366,"style":228},[169],[18,52368,4548],{"className":52369},[232],[18,52371],{"className":52372,"style":228},[169],[18,52374,52376,52379,52382,52385,52388,52391,52394,52397,52400,52403,52406],{"className":52375},[97],[18,52377],{"className":52378,"style":242},[101],[18,52380,43],{"className":52381,"style":1479},[106,1478],[18,52383,52],{"className":52384},[165],[18,52386,928],{"className":52387},[169],[18,52389],{"className":52390,"style":170},[169],[18,52392,11225],{"className":52393},[106],[18,52395,51021],{"className":52396,"style":5073},[106,110],[18,52398,11225],{"className":52399},[106],[18,52401],{"className":52402,"style":228},[169],[18,52404,433],{"className":52405},[232],[18,52407],{"className":52408,"style":228},[169],[18,52410,52412,52415,52418,52421,52424,52427,52430,52433,52473,52476,52479,52482],{"className":52411},[97],[18,52413],{"className":52414,"style":242},[101],[18,52416,28169],{"className":52417},[106,110],[18,52419,52],{"className":52420},[165],[18,52422,928],{"className":52423},[169],[18,52425],{"className":52426,"style":170},[169],[18,52428,11225],{"className":52429},[250],[18,52431,18309],{"className":52432,"style":46330},[106,1478],[18,52434,52436,52439],{"className":52435},[106],[18,52437,11225],{"className":52438},[106],[18,52440,52442],{"className":52441},[115],[18,52443,52445,52465],{"className":52444},[119,120],[18,52446,52448,52462],{"className":52447},[124],[18,52449,52451],{"className":52450,"style":1152},[128],[18,52452,52453,52456],{"style":1155},[18,52454],{"className":52455,"style":137},[136],[18,52457,52459],{"className":52458},[141,142,143,144],[18,52460,51021],{"className":52461,"style":5073},[106,110,144],[18,52463,152],{"className":52464},[151],[18,52466,52468],{"className":52467},[124],[18,52469,52471],{"className":52470,"style":159},[128],[18,52472],{},[18,52474,11225],{"className":52475},[267],[18,52477],{"className":52478,"style":228},[169],[18,52480,433],{"className":52481},[232],[18,52483],{"className":52484,"style":228},[169],[18,52486,52488,52491,52520,52526],{"className":52487},[97],[18,52489],{"className":52490,"style":6101},[101],[18,52492,52494,52497],{"className":52493},[106],[18,52495,2266],{"className":52496},[106],[18,52498,52500],{"className":52499},[115],[18,52501,52503],{"className":52502},[119],[18,52504,52506],{"className":52505},[124],[18,52507,52509],{"className":52508,"style":1616},[128],[18,52510,52511,52514],{"style":1637},[18,52512],{"className":52513,"style":137},[136],[18,52515,52517],{"className":52516},[141,142,143,144],[18,52518,28169],{"className":52519},[106,110,144],[18,52521,52523],{"className":52522},[267],[18,52524,1349],{"className":52525},[1230,1196],[18,52527,968],{"className":52528},[106],[11,52530,52531,52532,1990],{},"若右侧无界，则记 ",[18,52533,52535,52561],{"className":52534,"translate":22},[21],[18,52536,52538],{"className":52537},[26],[28,52539,52540],{"xmlns":30},[32,52541,52542,52558],{},[35,52543,52544,52546,52548,52550,52552,52554,52556],{},[41,52545,52216],{"mathvariant":316},[49,52547,899],{},[49,52549,74],{"stretchy":73},[41,52551,18309],{"mathvariant":1462},[49,52553,84],{"stretchy":73},[49,52555,433],{},[41,52557,2028],{"mathvariant":316},[86,52559,52560],{"encoding":88},"\\operatorname{VCdim}(\\mathcal H)=\\infty",[18,52562,52564,52594],{"className":52563,"ariaHidden":51},[93],[18,52565,52567,52570,52576,52579,52582,52585,52588,52591],{"className":52566},[97],[18,52568],{"className":52569,"style":242},[101],[18,52571,52573],{"className":52572},[985],[18,52574,52216],{"className":52575},[106,6354],[18,52577,74],{"className":52578},[250],[18,52580,18309],{"className":52581,"style":46330},[106,1478],[18,52583,84],{"className":52584},[267],[18,52586],{"className":52587,"style":228},[169],[18,52589,433],{"className":52590},[232],[18,52592],{"className":52593,"style":228},[169],[18,52595,52597,52600],{"className":52596},[97],[18,52598],{"className":52599,"style":1083},[101],[18,52601,2028],{"className":52602},[106],[11,52604,52605,52606,52609,52610,52638,52639,1994,52641,52699,52700,1994,52702,52730,52731,52759],{},"请把定义里的量词读三遍：",[717,52607,52608],{},"存在","一个点集 ",[18,52611,52613,52626],{"className":52612,"translate":22},[21],[18,52614,52616],{"className":52615},[26],[28,52617,52618],{"xmlns":30},[32,52619,52620,52624],{},[35,52621,52622],{},[41,52623,51021],{},[86,52625,51021],{"encoding":88},[18,52627,52629],{"className":52628,"ariaHidden":51},[93],[18,52630,52632,52635],{"className":52631},[97],[18,52633],{"className":52634,"style":600},[101],[18,52636,51021],{"className":52637,"style":5073},[106,110],"，使得对",[717,52640,1301],{},[18,52642,52644,52661],{"className":52643,"translate":22},[21],[18,52645,52647],{"className":52646},[26],[28,52648,52649],{"xmlns":30},[32,52650,52651,52659],{},[35,52652,52653],{},[1556,52654,52655,52657],{},[45,52656,2266],{},[41,52658,28169],{},[86,52660,44902],{"encoding":88},[18,52662,52664],{"className":52663,"ariaHidden":51},[93],[18,52665,52667,52670],{"className":52666},[97],[18,52668],{"className":52669,"style":1959},[101],[18,52671,52673,52676],{"className":52672},[106],[18,52674,2266],{"className":52675},[106],[18,52677,52679],{"className":52678},[115],[18,52680,52682],{"className":52681},[119],[18,52683,52685],{"className":52684},[124],[18,52686,52688],{"className":52687,"style":1959},[128],[18,52689,52690,52693],{"style":1962},[18,52691],{"className":52692,"style":137},[136],[18,52694,52696],{"className":52695},[141,142,143,144],[18,52697,28169],{"className":52698},[106,110,144]," 种标注，都",[717,52701,52608],{},[18,52703,52705,52718],{"className":52704,"translate":22},[21],[18,52706,52708],{"className":52707},[26],[28,52709,52710],{"xmlns":30},[32,52711,52712,52716],{},[35,52713,52714],{},[41,52715,18328],{},[86,52717,18328],{"encoding":88},[18,52719,52721],{"className":52720,"ariaHidden":51},[93],[18,52722,52724,52727],{"className":52723},[97],[18,52725],{"className":52726,"style":295},[101],[18,52728,18328],{"className":52729},[106,110],"。不是「对所有 ",[18,52732,52734,52747],{"className":52733,"translate":22},[21],[18,52735,52737],{"className":52736},[26],[28,52738,52739],{"xmlns":30},[32,52740,52741,52745],{},[35,52742,52743],{},[41,52744,51021],{},[86,52746,51021],{"encoding":88},[18,52748,52750],{"className":52749,"ariaHidden":51},[93],[18,52751,52753,52756],{"className":52752},[97],[18,52754],{"className":52755,"style":600},[101],[18,52757,51021],{"className":52758,"style":5073},[106,110],"」，也不是「对某一种标注」。",[11,52761,52762],{},"这个「存在」是 VC 维最容易被误读的地方，也是它作为最坏情况度量的根源：它问的是假设类在某个精心挑选的点集上最多能变出多少花样，而不是它在典型数据上会怎么做。",[1311,52764,52766],{"id":52765},"_22-几个例子","2.2 几个例子",[11,52768,52769],{},[717,52770,52771,52772,52828],{},"例 1（半直线，",[18,52773,52775,52795],{"className":52774,"translate":22},[21],[18,52776,52778],{"className":52777},[26],[28,52779,52780],{"xmlns":30},[32,52781,52782,52792],{},[35,52783,52784,52786,52788,52790],{},[41,52785,52216],{"mathvariant":316},[49,52787,899],{},[49,52789,433],{},[45,52791,47],{},[86,52793,52794],{"encoding":88},"\\operatorname{VCdim}=1",[18,52796,52798,52819],{"className":52797,"ariaHidden":51},[93],[18,52799,52801,52804,52810,52813,52816],{"className":52800},[97],[18,52802],{"className":52803,"style":295},[101],[18,52805,52807],{"className":52806},[985],[18,52808,52216],{"className":52809},[106,6354],[18,52811],{"className":52812,"style":228},[169],[18,52814,433],{"className":52815},[232],[18,52817],{"className":52818,"style":228},[169],[18,52820,52822,52825],{"className":52821},[97],[18,52823],{"className":52824,"style":505},[101],[18,52826,47],{"className":52827},[106],"）",[11,52830,36192,52831,4741,52998,1990],{},[18,52832,52834,52880],{"className":52833,"translate":22},[21],[18,52835,52837],{"className":52836},[26],[28,52838,52839],{"xmlns":30},[32,52840,52841,52877],{},[35,52842,52843,52845,52847,52849,52851,52854,52856,52858,52860,52862,52865,52867,52869,52871,52873,52875],{},[41,52844,18309],{"mathvariant":1462},[49,52846,433],{},[49,52848,1331],{"stretchy":73},[41,52850,38582],{},[49,52852,52853],{},"↦",[45,52855,47],{"mathvariant":2764},[49,52857,945],{"stretchy":73},[41,52859,38582],{},[49,52861,4775],{},[41,52863,52864],{},"t",[49,52866,965],{"stretchy":73},[49,52868,1340],{},[41,52870,52864],{},[49,52872,921],{},[41,52874,3071],{"mathvariant":933},[49,52876,1349],{"stretchy":73},[86,52878,52879],{"encoding":88},"\\mathcal H=\\{x\\mapsto\\mathbf 1[x\\ge t]:t\\in\\mathbb R\\}",[18,52881,52883,52901,52922,52946,52967,52986],{"className":52882,"ariaHidden":51},[93],[18,52884,52886,52889,52892,52895,52898],{"className":52885},[97],[18,52887],{"className":52888,"style":600},[101],[18,52890,18309],{"className":52891,"style":46330},[106,1478],[18,52893],{"className":52894,"style":228},[169],[18,52896,433],{"className":52897},[232],[18,52899],{"className":52900,"style":228},[169],[18,52902,52904,52907,52910,52913,52916,52919],{"className":52903},[97],[18,52905],{"className":52906,"style":242},[101],[18,52908,1331],{"className":52909},[250],[18,52911,38582],{"className":52912},[106,110],[18,52914],{"className":52915,"style":228},[169],[18,52917,52853],{"className":52918},[232],[18,52920],{"className":52921,"style":228},[169],[18,52923,52925,52928,52931,52934,52937,52940,52943],{"className":52924},[97],[18,52926],{"className":52927,"style":242},[101],[18,52929,47],{"className":52930},[106,2859],[18,52932,945],{"className":52933},[250],[18,52935,38582],{"className":52936},[106,110],[18,52938],{"className":52939,"style":228},[169],[18,52941,4775],{"className":52942},[232],[18,52944],{"className":52945,"style":228},[169],[18,52947,52949,52952,52955,52958,52961,52964],{"className":52948},[97],[18,52950],{"className":52951,"style":242},[101],[18,52953,52864],{"className":52954},[106,110],[18,52956,965],{"className":52957},[267],[18,52959],{"className":52960,"style":228},[169],[18,52962,1340],{"className":52963},[232],[18,52965],{"className":52966,"style":228},[169],[18,52968,52970,52974,52977,52980,52983],{"className":52969},[97],[18,52971],{"className":52972,"style":52973},[101],"height:0.6542em;vertical-align:-0.0391em;",[18,52975,52864],{"className":52976},[106,110],[18,52978],{"className":52979,"style":228},[169],[18,52981,921],{"className":52982},[232],[18,52984],{"className":52985,"style":228},[169],[18,52987,52989,52992,52995],{"className":52988},[97],[18,52990],{"className":52991,"style":242},[101],[18,52993,3071],{"className":52994},[106,1139],[18,52996,1349],{"className":52997},[267],[18,52999,53001,53019],{"className":53000,"translate":22},[21],[18,53002,53004],{"className":53003},[26],[28,53005,53006],{"xmlns":30},[32,53007,53008,53016],{},[35,53009,53010,53012,53014],{},[41,53011,43],{"mathvariant":1462},[49,53013,433],{},[41,53015,3071],{"mathvariant":933},[86,53017,53018],{"encoding":88},"\\mathcal X=\\mathbb R",[18,53020,53022,53040],{"className":53021,"ariaHidden":51},[93],[18,53023,53025,53028,53031,53034,53037],{"className":53024},[97],[18,53026],{"className":53027,"style":600},[101],[18,53029,43],{"className":53030,"style":1479},[106,1478],[18,53032],{"className":53033,"style":228},[169],[18,53035,433],{"className":53036},[232],[18,53038],{"className":53039,"style":228},[169],[18,53041,53043,53046],{"className":53042},[97],[18,53044],{"className":53045,"style":45807},[101],[18,53047,3071],{"className":53048},[106,1139],[11,53050,53051,53052,53113,53114,53166,53167,4741,53195,53166,53246,1990],{},"一个点 ",[18,53053,53055,53077],{"className":53054,"translate":22},[21],[18,53056,53058],{"className":53057},[26],[28,53059,53060],{"xmlns":30},[32,53061,53062,53074],{},[35,53063,53064,53066,53068,53070,53072],{},[41,53065,51021],{},[49,53067,433],{},[49,53069,1331],{"stretchy":73},[45,53071,430],{},[49,53073,1349],{"stretchy":73},[86,53075,53076],{"encoding":88},"C=\\{0\\}",[18,53078,53080,53098],{"className":53079,"ariaHidden":51},[93],[18,53081,53083,53086,53089,53092,53095],{"className":53082},[97],[18,53084],{"className":53085,"style":600},[101],[18,53087,51021],{"className":53088,"style":5073},[106,110],[18,53090],{"className":53091,"style":228},[169],[18,53093,433],{"className":53094},[232],[18,53096],{"className":53097,"style":228},[169],[18,53099,53101,53104,53107,53110],{"className":53100},[97],[18,53102],{"className":53103,"style":242},[101],[18,53105,1331],{"className":53106},[250],[18,53108,430],{"className":53109},[106],[18,53111,1349],{"className":53112},[267]," 显然可以被打散：",[18,53115,53117,53135],{"className":53116,"translate":22},[21],[18,53118,53120],{"className":53119},[26],[28,53121,53122],{"xmlns":30},[32,53123,53124,53132],{},[35,53125,53126,53128,53130],{},[41,53127,52864],{},[49,53129,4334],{},[45,53131,430],{},[86,53133,53134],{"encoding":88},"t\\le 0",[18,53136,53138,53157],{"className":53137,"ariaHidden":51},[93],[18,53139,53141,53145,53148,53151,53154],{"className":53140},[97],[18,53142],{"className":53143,"style":53144},[101],"height:0.7719em;vertical-align:-0.136em;",[18,53146,52864],{"className":53147},[106,110],[18,53149],{"className":53150,"style":228},[169],[18,53152,4334],{"className":53153},[232],[18,53155],{"className":53156,"style":228},[169],[18,53158,53160,53163],{"className":53159},[97],[18,53161],{"className":53162,"style":505},[101],[18,53164,430],{"className":53165},[106]," 输出 ",[18,53168,53170,53183],{"className":53169,"translate":22},[21],[18,53171,53173],{"className":53172},[26],[28,53174,53175],{"xmlns":30},[32,53176,53177,53181],{},[35,53178,53179],{},[45,53180,47],{},[86,53182,47],{"encoding":88},[18,53184,53186],{"className":53185,"ariaHidden":51},[93],[18,53187,53189,53192],{"className":53188},[97],[18,53190],{"className":53191,"style":505},[101],[18,53193,47],{"className":53194},[106],[18,53196,53198,53216],{"className":53197,"translate":22},[21],[18,53199,53201],{"className":53200},[26],[28,53202,53203],{"xmlns":30},[32,53204,53205,53213],{},[35,53206,53207,53209,53211],{},[41,53208,52864],{},[49,53210,6885],{},[45,53212,430],{},[86,53214,53215],{"encoding":88},"t>0",[18,53217,53219,53237],{"className":53218,"ariaHidden":51},[93],[18,53220,53222,53225,53228,53231,53234],{"className":53221},[97],[18,53223],{"className":53224,"style":52973},[101],[18,53226,52864],{"className":53227},[106,110],[18,53229],{"className":53230,"style":228},[169],[18,53232,6885],{"className":53233},[232],[18,53235],{"className":53236,"style":228},[169],[18,53238,53240,53243],{"className":53239},[97],[18,53241],{"className":53242,"style":505},[101],[18,53244,430],{"className":53245},[106],[18,53247,53249,53262],{"className":53248,"translate":22},[21],[18,53250,53252],{"className":53251},[26],[28,53253,53254],{"xmlns":30},[32,53255,53256,53260],{},[35,53257,53258],{},[45,53259,430],{},[86,53261,430],{"encoding":88},[18,53263,53265],{"className":53264,"ariaHidden":51},[93],[18,53266,53268,53271],{"className":53267},[97],[18,53269],{"className":53270,"style":505},[101],[18,53272,430],{"className":53273},[106],[11,53275,53276,53277,53410,53411,53463,53464,53556,53557,53649,53650,53782,53783,1990],{},"两个点 ",[18,53278,53280,53306],{"className":53279,"translate":22},[21],[18,53281,53283],{"className":53282},[26],[28,53284,53285],{"xmlns":30},[32,53286,53287,53303],{},[35,53288,53289,53295,53297],{},[38,53290,53291,53293],{},[41,53292,38582],{},[45,53294,47],{},[49,53296,6684],{},[38,53298,53299,53301],{},[41,53300,38582],{},[45,53302,2266],{},[86,53304,53305],{"encoding":88},"x_1\u003Cx_2",[18,53307,53309,53364],{"className":53308,"ariaHidden":51},[93],[18,53310,53312,53315,53355,53358,53361],{"className":53311},[97],[18,53313],{"className":53314,"style":40980},[101],[18,53316,53318,53321],{"className":53317},[106],[18,53319,38582],{"className":53320},[106,110],[18,53322,53324],{"className":53323},[115],[18,53325,53327,53347],{"className":53326},[119,120],[18,53328,53330,53344],{"className":53329},[124],[18,53331,53333],{"className":53332,"style":129},[128],[18,53334,53335,53338],{"style":1155},[18,53336],{"className":53337,"style":137},[136],[18,53339,53341],{"className":53340},[141,142,143,144],[18,53342,47],{"className":53343},[106,144],[18,53345,152],{"className":53346},[151],[18,53348,53350],{"className":53349},[124],[18,53351,53353],{"className":53352,"style":159},[128],[18,53354],{},[18,53356],{"className":53357,"style":228},[169],[18,53359,6684],{"className":53360},[232],[18,53362],{"className":53363,"style":228},[169],[18,53365,53367,53370],{"className":53366},[97],[18,53368],{"className":53369,"style":16897},[101],[18,53371,53373,53376],{"className":53372},[106],[18,53374,38582],{"className":53375},[106,110],[18,53377,53379],{"className":53378},[115],[18,53380,53382,53402],{"className":53381},[119,120],[18,53383,53385,53399],{"className":53384},[124],[18,53386,53388],{"className":53387,"style":129},[128],[18,53389,53390,53393],{"style":1155},[18,53391],{"className":53392,"style":137},[136],[18,53394,53396],{"className":53395},[141,142,143,144],[18,53397,2266],{"className":53398},[106,144],[18,53400,152],{"className":53401},[151],[18,53403,53405],{"className":53404},[124],[18,53406,53408],{"className":53407,"style":159},[128],[18,53409],{}," 则不行：标注 ",[18,53412,53414,53436],{"className":53413,"translate":22},[21],[18,53415,53417],{"className":53416},[26],[28,53418,53419],{"xmlns":30},[32,53420,53421,53433],{},[35,53422,53423,53425,53427,53429,53431],{},[49,53424,74],{"stretchy":73},[45,53426,47],{},[49,53428,52],{"separator":51},[45,53430,430],{},[49,53432,84],{"stretchy":73},[86,53434,53435],{"encoding":88},"(1,0)",[18,53437,53439],{"className":53438,"ariaHidden":51},[93],[18,53440,53442,53445,53448,53451,53454,53457,53460],{"className":53441},[97],[18,53443],{"className":53444,"style":242},[101],[18,53446,74],{"className":53447},[250],[18,53449,47],{"className":53450},[106],[18,53452,52],{"className":53453},[165],[18,53455],{"className":53456,"style":170},[169],[18,53458,430],{"className":53459},[106],[18,53461,84],{"className":53462},[267]," 要求 ",[18,53465,53467,53489],{"className":53466,"translate":22},[21],[18,53468,53470],{"className":53469},[26],[28,53471,53472],{"xmlns":30},[32,53473,53474,53486],{},[35,53475,53476,53478,53480],{},[41,53477,52864],{},[49,53479,4334],{},[38,53481,53482,53484],{},[41,53483,38582],{},[45,53485,47],{},[86,53487,53488],{"encoding":88},"t\\le x_1",[18,53490,53492,53510],{"className":53491,"ariaHidden":51},[93],[18,53493,53495,53498,53501,53504,53507],{"className":53494},[97],[18,53496],{"className":53497,"style":53144},[101],[18,53499,52864],{"className":53500},[106,110],[18,53502],{"className":53503,"style":228},[169],[18,53505,4334],{"className":53506},[232],[18,53508],{"className":53509,"style":228},[169],[18,53511,53513,53516],{"className":53512},[97],[18,53514],{"className":53515,"style":16897},[101],[18,53517,53519,53522],{"className":53518},[106],[18,53520,38582],{"className":53521},[106,110],[18,53523,53525],{"className":53524},[115],[18,53526,53528,53548],{"className":53527},[119,120],[18,53529,53531,53545],{"className":53530},[124],[18,53532,53534],{"className":53533,"style":129},[128],[18,53535,53536,53539],{"style":1155},[18,53537],{"className":53538,"style":137},[136],[18,53540,53542],{"className":53541},[141,142,143,144],[18,53543,47],{"className":53544},[106,144],[18,53546,152],{"className":53547},[151],[18,53549,53551],{"className":53550},[124],[18,53552,53554],{"className":53553,"style":159},[128],[18,53555],{}," 且 ",[18,53558,53560,53582],{"className":53559,"translate":22},[21],[18,53561,53563],{"className":53562},[26],[28,53564,53565],{"xmlns":30},[32,53566,53567,53579],{},[35,53568,53569,53571,53573],{},[41,53570,52864],{},[49,53572,6885],{},[38,53574,53575,53577],{},[41,53576,38582],{},[45,53578,2266],{},[86,53580,53581],{"encoding":88},"t>x_2",[18,53583,53585,53603],{"className":53584,"ariaHidden":51},[93],[18,53586,53588,53591,53594,53597,53600],{"className":53587},[97],[18,53589],{"className":53590,"style":52973},[101],[18,53592,52864],{"className":53593},[106,110],[18,53595],{"className":53596,"style":228},[169],[18,53598,6885],{"className":53599},[232],[18,53601],{"className":53602,"style":228},[169],[18,53604,53606,53609],{"className":53605},[97],[18,53607],{"className":53608,"style":16897},[101],[18,53610,53612,53615],{"className":53611},[106],[18,53613,38582],{"className":53614},[106,110],[18,53616,53618],{"className":53617},[115],[18,53619,53621,53641],{"className":53620},[119,120],[18,53622,53624,53638],{"className":53623},[124],[18,53625,53627],{"className":53626,"style":129},[128],[18,53628,53629,53632],{"style":1155},[18,53630],{"className":53631,"style":137},[136],[18,53633,53635],{"className":53634},[141,142,143,144],[18,53636,2266],{"className":53637},[106,144],[18,53639,152],{"className":53640},[151],[18,53642,53644],{"className":53643},[124],[18,53645,53647],{"className":53646,"style":159},[128],[18,53648],{},"，与 ",[18,53651,53653,53678],{"className":53652,"translate":22},[21],[18,53654,53656],{"className":53655},[26],[28,53657,53658],{"xmlns":30},[32,53659,53660,53676],{},[35,53661,53662,53668,53670],{},[38,53663,53664,53666],{},[41,53665,38582],{},[45,53667,47],{},[49,53669,6684],{},[38,53671,53672,53674],{},[41,53673,38582],{},[45,53675,2266],{},[86,53677,53305],{"encoding":88},[18,53679,53681,53736],{"className":53680,"ariaHidden":51},[93],[18,53682,53684,53687,53727,53730,53733],{"className":53683},[97],[18,53685],{"className":53686,"style":40980},[101],[18,53688,53690,53693],{"className":53689},[106],[18,53691,38582],{"className":53692},[106,110],[18,53694,53696],{"className":53695},[115],[18,53697,53699,53719],{"className":53698},[119,120],[18,53700,53702,53716],{"className":53701},[124],[18,53703,53705],{"className":53704,"style":129},[128],[18,53706,53707,53710],{"style":1155},[18,53708],{"className":53709,"style":137},[136],[18,53711,53713],{"className":53712},[141,142,143,144],[18,53714,47],{"className":53715},[106,144],[18,53717,152],{"className":53718},[151],[18,53720,53722],{"className":53721},[124],[18,53723,53725],{"className":53724,"style":159},[128],[18,53726],{},[18,53728],{"className":53729,"style":228},[169],[18,53731,6684],{"className":53732},[232],[18,53734],{"className":53735,"style":228},[169],[18,53737,53739,53742],{"className":53738},[97],[18,53740],{"className":53741,"style":16897},[101],[18,53743,53745,53748],{"className":53744},[106],[18,53746,38582],{"className":53747},[106,110],[18,53749,53751],{"className":53750},[115],[18,53752,53754,53774],{"className":53753},[119,120],[18,53755,53757,53771],{"className":53756},[124],[18,53758,53760],{"className":53759,"style":129},[128],[18,53761,53762,53765],{"style":1155},[18,53763],{"className":53764,"style":137},[136],[18,53766,53768],{"className":53767},[141,142,143,144],[18,53769,2266],{"className":53770},[106,144],[18,53772,152],{"className":53773},[151],[18,53775,53777],{"className":53776},[124],[18,53778,53780],{"className":53779,"style":159},[128],[18,53781],{}," 矛盾。所以 VC 维是 ",[18,53784,53786,53799],{"className":53785,"translate":22},[21],[18,53787,53789],{"className":53788},[26],[28,53790,53791],{"xmlns":30},[32,53792,53793,53797],{},[35,53794,53795],{},[45,53796,47],{},[86,53798,47],{"encoding":88},[18,53800,53802],{"className":53801,"ariaHidden":51},[93],[18,53803,53805,53808],{"className":53804},[97],[18,53806],{"className":53807,"style":505},[101],[18,53809,47],{"className":53810},[106],[11,53812,53813,53814,53842,53843,53871,53872,53923,53924,1990],{},"这个例子说明，VC 维和「能拟合多少点」不是一回事：半直线可以把任意多个点拟合成「前面全是 ",[18,53815,53817,53830],{"className":53816,"translate":22},[21],[18,53818,53820],{"className":53819},[26],[28,53821,53822],{"xmlns":30},[32,53823,53824,53828],{},[35,53825,53826],{},[45,53827,430],{},[86,53829,430],{"encoding":88},[18,53831,53833],{"className":53832,"ariaHidden":51},[93],[18,53834,53836,53839],{"className":53835},[97],[18,53837],{"className":53838,"style":505},[101],[18,53840,430],{"className":53841},[106],"、后面全是 ",[18,53844,53846,53859],{"className":53845,"translate":22},[21],[18,53847,53849],{"className":53848},[26],[28,53850,53851],{"xmlns":30},[32,53852,53853,53857],{},[35,53854,53855],{},[45,53856,47],{},[86,53858,47],{"encoding":88},[18,53860,53862],{"className":53861,"ariaHidden":51},[93],[18,53863,53865,53868],{"className":53864},[97],[18,53866],{"className":53867,"style":505},[101],[18,53869,47],{"className":53870},[106],"」的形状，但它只能实现 ",[18,53873,53875,53893],{"className":53874,"translate":22},[21],[18,53876,53878],{"className":53877},[26],[28,53879,53880],{"xmlns":30},[32,53881,53882,53890],{},[35,53883,53884,53886,53888],{},[41,53885,28169],{},[49,53887,7987],{},[45,53889,47],{},[86,53891,53892],{"encoding":88},"m+1",[18,53894,53896,53914],{"className":53895,"ariaHidden":51},[93],[18,53897,53899,53902,53905,53908,53911],{"className":53898},[97],[18,53900],{"className":53901,"style":14404},[101],[18,53903,28169],{"className":53904},[106,110],[18,53906],{"className":53907,"style":2068},[169],[18,53909,7987],{"className":53910},[2072],[18,53912],{"className":53913,"style":2068},[169],[18,53915,53917,53920],{"className":53916},[97],[18,53918],{"className":53919,"style":505},[101],[18,53921,47],{"className":53922},[106]," 种二分，远少于 ",[18,53925,53927,53944],{"className":53926,"translate":22},[21],[18,53928,53930],{"className":53929},[26],[28,53931,53932],{"xmlns":30},[32,53933,53934,53942],{},[35,53935,53936],{},[1556,53937,53938,53940],{},[45,53939,2266],{},[41,53941,28169],{},[86,53943,44902],{"encoding":88},[18,53945,53947],{"className":53946,"ariaHidden":51},[93],[18,53948,53950,53953],{"className":53949},[97],[18,53951],{"className":53952,"style":1959},[101],[18,53954,53956,53959],{"className":53955},[106],[18,53957,2266],{"className":53958},[106],[18,53960,53962],{"className":53961},[115],[18,53963,53965],{"className":53964},[119],[18,53966,53968],{"className":53967},[124],[18,53969,53971],{"className":53970,"style":1959},[128],[18,53972,53973,53976],{"style":1962},[18,53974],{"className":53975,"style":137},[136],[18,53977,53979],{"className":53978},[141,142,143,144],[18,53980,28169],{"className":53981},[106,110,144],[11,53983,53984],{},[717,53985,53986,53987,52828],{},"例 2（区间，",[18,53988,53990,54010],{"className":53989,"translate":22},[21],[18,53991,53993],{"className":53992},[26],[28,53994,53995],{"xmlns":30},[32,53996,53997,54007],{},[35,53998,53999,54001,54003,54005],{},[41,54000,52216],{"mathvariant":316},[49,54002,899],{},[49,54004,433],{},[45,54006,2266],{},[86,54008,54009],{"encoding":88},"\\operatorname{VCdim}=2",[18,54011,54013,54034],{"className":54012,"ariaHidden":51},[93],[18,54014,54016,54019,54025,54028,54031],{"className":54015},[97],[18,54017],{"className":54018,"style":295},[101],[18,54020,54022],{"className":54021},[985],[18,54023,52216],{"className":54024},[106,6354],[18,54026],{"className":54027,"style":228},[169],[18,54029,433],{"className":54030},[232],[18,54032],{"className":54033,"style":228},[169],[18,54035,54037,54040],{"className":54036},[97],[18,54038],{"className":54039,"style":505},[101],[18,54041,2266],{"className":54042},[106],[11,54044,36192,54045,1990],{},[18,54046,54048,54097],{"className":54047,"translate":22},[21],[18,54049,54051],{"className":54050},[26],[28,54052,54053],{"xmlns":30},[32,54054,54055,54094],{},[35,54056,54057,54059,54061,54063,54065,54067,54069,54071,54073,54075,54077,54079,54082,54084,54086,54088,54090,54092],{},[41,54058,18309],{"mathvariant":1462},[49,54060,433],{},[49,54062,1331],{"stretchy":73},[41,54064,38582],{},[49,54066,52853],{},[45,54068,47],{"mathvariant":2764},[49,54070,945],{"stretchy":73},[41,54072,16823],{},[49,54074,4334],{},[41,54076,38582],{},[49,54078,4334],{},[41,54080,54081],{},"b",[49,54083,965],{"stretchy":73},[49,54085,1340],{},[41,54087,16823],{},[49,54089,4334],{},[41,54091,54081],{},[49,54093,1349],{"stretchy":73},[86,54095,54096],{"encoding":88},"\\mathcal H=\\{x\\mapsto\\mathbf 1[a\\le x\\le b]:a\\le b\\}",[18,54098,54100,54118,54139,54163,54181,54202,54220],{"className":54099,"ariaHidden":51},[93],[18,54101,54103,54106,54109,54112,54115],{"className":54102},[97],[18,54104],{"className":54105,"style":600},[101],[18,54107,18309],{"className":54108,"style":46330},[106,1478],[18,54110],{"className":54111,"style":228},[169],[18,54113,433],{"className":54114},[232],[18,54116],{"className":54117,"style":228},[169],[18,54119,54121,54124,54127,54130,54133,54136],{"className":54120},[97],[18,54122],{"className":54123,"style":242},[101],[18,54125,1331],{"className":54126},[250],[18,54128,38582],{"className":54129},[106,110],[18,54131],{"className":54132,"style":228},[169],[18,54134,52853],{"className":54135},[232],[18,54137],{"className":54138,"style":228},[169],[18,54140,54142,54145,54148,54151,54154,54157,54160],{"className":54141},[97],[18,54143],{"className":54144,"style":242},[101],[18,54146,47],{"className":54147},[106,2859],[18,54149,945],{"className":54150},[250],[18,54152,16823],{"className":54153},[106,110],[18,54155],{"className":54156,"style":228},[169],[18,54158,4334],{"className":54159},[232],[18,54161],{"className":54162,"style":228},[169],[18,54164,54166,54169,54172,54175,54178],{"className":54165},[97],[18,54167],{"className":54168,"style":53144},[101],[18,54170,38582],{"className":54171},[106,110],[18,54173],{"className":54174,"style":228},[169],[18,54176,4334],{"className":54177},[232],[18,54179],{"className":54180,"style":228},[169],[18,54182,54184,54187,54190,54193,54196,54199],{"className":54183},[97],[18,54185],{"className":54186,"style":242},[101],[18,54188,54081],{"className":54189},[106,110],[18,54191,965],{"className":54192},[267],[18,54194],{"className":54195,"style":228},[169],[18,54197,1340],{"className":54198},[232],[18,54200],{"className":54201,"style":228},[169],[18,54203,54205,54208,54211,54214,54217],{"className":54204},[97],[18,54206],{"className":54207,"style":53144},[101],[18,54209,16823],{"className":54210},[106,110],[18,54212],{"className":54213,"style":228},[169],[18,54215,4334],{"className":54216},[232],[18,54218],{"className":54219,"style":228},[169],[18,54221,54223,54226,54229],{"className":54222},[97],[18,54224],{"className":54225,"style":242},[101],[18,54227,54081],{"className":54228},[106,110],[18,54230,1349],{"className":54231},[267],[11,54233,53276,54234,54366,54367,54419,54420,2452,54593,54419,54644,2452,54881,54419,54933,2452,55170,55222],{},[18,54235,54237,54262],{"className":54236,"translate":22},[21],[18,54238,54240],{"className":54239},[26],[28,54241,54242],{"xmlns":30},[32,54243,54244,54260],{},[35,54245,54246,54252,54254],{},[38,54247,54248,54250],{},[41,54249,38582],{},[45,54251,47],{},[49,54253,6684],{},[38,54255,54256,54258],{},[41,54257,38582],{},[45,54259,2266],{},[86,54261,53305],{"encoding":88},[18,54263,54265,54320],{"className":54264,"ariaHidden":51},[93],[18,54266,54268,54271,54311,54314,54317],{"className":54267},[97],[18,54269],{"className":54270,"style":40980},[101],[18,54272,54274,54277],{"className":54273},[106],[18,54275,38582],{"className":54276},[106,110],[18,54278,54280],{"className":54279},[115],[18,54281,54283,54303],{"className":54282},[119,120],[18,54284,54286,54300],{"className":54285},[124],[18,54287,54289],{"className":54288,"style":129},[128],[18,54290,54291,54294],{"style":1155},[18,54292],{"className":54293,"style":137},[136],[18,54295,54297],{"className":54296},[141,142,143,144],[18,54298,47],{"className":54299},[106,144],[18,54301,152],{"className":54302},[151],[18,54304,54306],{"className":54305},[124],[18,54307,54309],{"className":54308,"style":159},[128],[18,54310],{},[18,54312],{"className":54313,"style":228},[169],[18,54315,6684],{"className":54316},[232],[18,54318],{"className":54319,"style":228},[169],[18,54321,54323,54326],{"className":54322},[97],[18,54324],{"className":54325,"style":16897},[101],[18,54327,54329,54332],{"className":54328},[106],[18,54330,38582],{"className":54331},[106,110],[18,54333,54335],{"className":54334},[115],[18,54336,54338,54358],{"className":54337},[119,120],[18,54339,54341,54355],{"className":54340},[124],[18,54342,54344],{"className":54343,"style":129},[128],[18,54345,54346,54349],{"style":1155},[18,54347],{"className":54348,"style":137},[136],[18,54350,54352],{"className":54351},[141,142,143,144],[18,54353,2266],{"className":54354},[106,144],[18,54356,152],{"className":54357},[151],[18,54359,54361],{"className":54360},[124],[18,54362,54364],{"className":54363,"style":159},[128],[18,54365],{}," 可以被打散：",[18,54368,54370,54392],{"className":54369,"translate":22},[21],[18,54371,54373],{"className":54372},[26],[28,54374,54375],{"xmlns":30},[32,54376,54377,54389],{},[35,54378,54379,54381,54383,54385,54387],{},[49,54380,74],{"stretchy":73},[45,54382,47],{},[49,54384,52],{"separator":51},[45,54386,47],{},[49,54388,84],{"stretchy":73},[86,54390,54391],{"encoding":88},"(1,1)",[18,54393,54395],{"className":54394,"ariaHidden":51},[93],[18,54396,54398,54401,54404,54407,54410,54413,54416],{"className":54397},[97],[18,54399],{"className":54400,"style":242},[101],[18,54402,74],{"className":54403},[250],[18,54405,47],{"className":54406},[106],[18,54408,52],{"className":54409},[165],[18,54411],{"className":54412,"style":170},[169],[18,54414,47],{"className":54415},[106],[18,54417,84],{"className":54418},[267]," 取 ",[18,54421,54423,54459],{"className":54422,"translate":22},[21],[18,54424,54426],{"className":54425},[26],[28,54427,54428],{"xmlns":30},[32,54429,54430,54456],{},[35,54431,54432,54434,54436,54442,54444,54446,54448,54450],{},[41,54433,16823],{},[49,54435,4334],{},[38,54437,54438,54440],{},[41,54439,38582],{},[45,54441,47],{},[49,54443,52],{"separator":51},[926,54445,928],{},[41,54447,54081],{},[49,54449,4775],{},[38,54451,54452,54454],{},[41,54453,38582],{},[45,54455,2266],{},[86,54457,54458],{"encoding":88},"a\\le x_1,\\ b\\ge x_2",[18,54460,54462,54480,54547],{"className":54461,"ariaHidden":51},[93],[18,54463,54465,54468,54471,54474,54477],{"className":54464},[97],[18,54466],{"className":54467,"style":53144},[101],[18,54469,16823],{"className":54470},[106,110],[18,54472],{"className":54473,"style":228},[169],[18,54475,4334],{"className":54476},[232],[18,54478],{"className":54479,"style":228},[169],[18,54481,54483,54486,54526,54529,54532,54535,54538,54541,54544],{"className":54482},[97],[18,54484],{"className":54485,"style":2890},[101],[18,54487,54489,54492],{"className":54488},[106],[18,54490,38582],{"className":54491},[106,110],[18,54493,54495],{"className":54494},[115],[18,54496,54498,54518],{"className":54497},[119,120],[18,54499,54501,54515],{"className":54500},[124],[18,54502,54504],{"className":54503,"style":129},[128],[18,54505,54506,54509],{"style":1155},[18,54507],{"className":54508,"style":137},[136],[18,54510,54512],{"className":54511},[141,142,143,144],[18,54513,47],{"className":54514},[106,144],[18,54516,152],{"className":54517},[151],[18,54519,54521],{"className":54520},[124],[18,54522,54524],{"className":54523,"style":159},[128],[18,54525],{},[18,54527,52],{"className":54528},[165],[18,54530,928],{"className":54531},[169],[18,54533],{"className":54534,"style":170},[169],[18,54536,54081],{"className":54537},[106,110],[18,54539],{"className":54540,"style":228},[169],[18,54542,4775],{"className":54543},[232],[18,54545],{"className":54546,"style":228},[169],[18,54548,54550,54553],{"className":54549},[97],[18,54551],{"className":54552,"style":16897},[101],[18,54554,54556,54559],{"className":54555},[106],[18,54557,38582],{"className":54558},[106,110],[18,54560,54562],{"className":54561},[115],[18,54563,54565,54585],{"className":54564},[119,120],[18,54566,54568,54582],{"className":54567},[124],[18,54569,54571],{"className":54570,"style":129},[128],[18,54572,54573,54576],{"style":1155},[18,54574],{"className":54575,"style":137},[136],[18,54577,54579],{"className":54578},[141,142,143,144],[18,54580,2266],{"className":54581},[106,144],[18,54583,152],{"className":54584},[151],[18,54586,54588],{"className":54587},[124],[18,54589,54591],{"className":54590,"style":159},[128],[18,54592],{},[18,54594,54596,54617],{"className":54595,"translate":22},[21],[18,54597,54599],{"className":54598},[26],[28,54600,54601],{"xmlns":30},[32,54602,54603,54615],{},[35,54604,54605,54607,54609,54611,54613],{},[49,54606,74],{"stretchy":73},[45,54608,47],{},[49,54610,52],{"separator":51},[45,54612,430],{},[49,54614,84],{"stretchy":73},[86,54616,53435],{"encoding":88},[18,54618,54620],{"className":54619,"ariaHidden":51},[93],[18,54621,54623,54626,54629,54632,54635,54638,54641],{"className":54622},[97],[18,54624],{"className":54625,"style":242},[101],[18,54627,74],{"className":54628},[250],[18,54630,47],{"className":54631},[106],[18,54633,52],{"className":54634},[165],[18,54636],{"className":54637,"style":170},[169],[18,54639,430],{"className":54640},[106],[18,54642,84],{"className":54643},[267],[18,54645,54647,54695],{"className":54646,"translate":22},[21],[18,54648,54650],{"className":54649},[26],[28,54651,54652],{"xmlns":30},[32,54653,54654,54692],{},[35,54655,54656,54658,54660,54666,54668,54670,54672,54674,54676,54682,54684,54690],{},[41,54657,16823],{},[49,54659,4334],{},[38,54661,54662,54664],{},[41,54663,38582],{},[45,54665,47],{},[49,54667,52],{"separator":51},[926,54669,928],{},[41,54671,54081],{},[49,54673,921],{},[49,54675,945],{"stretchy":73},[38,54677,54678,54680],{},[41,54679,38582],{},[45,54681,47],{},[49,54683,52],{"separator":51},[38,54685,54686,54688],{},[41,54687,38582],{},[45,54689,2266],{},[49,54691,84],{"stretchy":73},[86,54693,54694],{"encoding":88},"a\\le x_1,\\ b\\in[x_1,x_2)",[18,54696,54698,54716,54783],{"className":54697,"ariaHidden":51},[93],[18,54699,54701,54704,54707,54710,54713],{"className":54700},[97],[18,54702],{"className":54703,"style":53144},[101],[18,54705,16823],{"className":54706},[106,110],[18,54708],{"className":54709,"style":228},[169],[18,54711,4334],{"className":54712},[232],[18,54714],{"className":54715,"style":228},[169],[18,54717,54719,54722,54762,54765,54768,54771,54774,54777,54780],{"className":54718},[97],[18,54720],{"className":54721,"style":2890},[101],[18,54723,54725,54728],{"className":54724},[106],[18,54726,38582],{"className":54727},[106,110],[18,54729,54731],{"className":54730},[115],[18,54732,54734,54754],{"className":54733},[119,120],[18,54735,54737,54751],{"className":54736},[124],[18,54738,54740],{"className":54739,"style":129},[128],[18,54741,54742,54745],{"style":1155},[18,54743],{"className":54744,"style":137},[136],[18,54746,54748],{"className":54747},[141,142,143,144],[18,54749,47],{"className":54750},[106,144],[18,54752,152],{"className":54753},[151],[18,54755,54757],{"className":54756},[124],[18,54758,54760],{"className":54759,"style":159},[128],[18,54761],{},[18,54763,52],{"className":54764},[165],[18,54766,928],{"className":54767},[169],[18,54769],{"className":54770,"style":170},[169],[18,54772,54081],{"className":54773},[106,110],[18,54775],{"className":54776,"style":228},[169],[18,54778,921],{"className":54779},[232],[18,54781],{"className":54782,"style":228},[169],[18,54784,54786,54789,54792,54832,54835,54838,54878],{"className":54785},[97],[18,54787],{"className":54788,"style":242},[101],[18,54790,945],{"className":54791},[250],[18,54793,54795,54798],{"className":54794},[106],[18,54796,38582],{"className":54797},[106,110],[18,54799,54801],{"className":54800},[115],[18,54802,54804,54824],{"className":54803},[119,120],[18,54805,54807,54821],{"className":54806},[124],[18,54808,54810],{"className":54809,"style":129},[128],[18,54811,54812,54815],{"style":1155},[18,54813],{"className":54814,"style":137},[136],[18,54816,54818],{"className":54817},[141,142,143,144],[18,54819,47],{"className":54820},[106,144],[18,54822,152],{"className":54823},[151],[18,54825,54827],{"className":54826},[124],[18,54828,54830],{"className":54829,"style":159},[128],[18,54831],{},[18,54833,52],{"className":54834},[165],[18,54836],{"className":54837,"style":170},[169],[18,54839,54841,54844],{"className":54840},[106],[18,54842,38582],{"className":54843},[106,110],[18,54845,54847],{"className":54846},[115],[18,54848,54850,54870],{"className":54849},[119,120],[18,54851,54853,54867],{"className":54852},[124],[18,54854,54856],{"className":54855,"style":129},[128],[18,54857,54858,54861],{"style":1155},[18,54859],{"className":54860,"style":137},[136],[18,54862,54864],{"className":54863},[141,142,143,144],[18,54865,2266],{"className":54866},[106,144],[18,54868,152],{"className":54869},[151],[18,54871,54873],{"className":54872},[124],[18,54874,54876],{"className":54875,"style":159},[128],[18,54877],{},[18,54879,84],{"className":54880},[267],[18,54882,54884,54906],{"className":54883,"translate":22},[21],[18,54885,54887],{"className":54886},[26],[28,54888,54889],{"xmlns":30},[32,54890,54891,54903],{},[35,54892,54893,54895,54897,54899,54901],{},[49,54894,74],{"stretchy":73},[45,54896,430],{},[49,54898,52],{"separator":51},[45,54900,47],{},[49,54902,84],{"stretchy":73},[86,54904,54905],{"encoding":88},"(0,1)",[18,54907,54909],{"className":54908,"ariaHidden":51},[93],[18,54910,54912,54915,54918,54921,54924,54927,54930],{"className":54911},[97],[18,54913],{"className":54914,"style":242},[101],[18,54916,74],{"className":54917},[250],[18,54919,430],{"className":54920},[106],[18,54922,52],{"className":54923},[165],[18,54925],{"className":54926,"style":170},[169],[18,54928,47],{"className":54929},[106],[18,54931,84],{"className":54932},[267],[18,54934,54936,54984],{"className":54935,"translate":22},[21],[18,54937,54939],{"className":54938},[26],[28,54940,54941],{"xmlns":30},[32,54942,54943,54981],{},[35,54944,54945,54947,54949,54951,54957,54959,54965,54967,54969,54971,54973,54975],{},[41,54946,16823],{},[49,54948,921],{},[49,54950,74],{"stretchy":73},[38,54952,54953,54955],{},[41,54954,38582],{},[45,54956,47],{},[49,54958,52],{"separator":51},[38,54960,54961,54963],{},[41,54962,38582],{},[45,54964,2266],{},[49,54966,965],{"stretchy":73},[49,54968,52],{"separator":51},[926,54970,928],{},[41,54972,54081],{},[49,54974,4775],{},[38,54976,54977,54979],{},[41,54978,38582],{},[45,54980,2266],{},[86,54982,54983],{"encoding":88},"a\\in(x_1,x_2],\\ b\\ge x_2",[18,54985,54987,55005,55124],{"className":54986,"ariaHidden":51},[93],[18,54988,54990,54993,54996,54999,55002],{"className":54989},[97],[18,54991],{"className":54992,"style":30930},[101],[18,54994,16823],{"className":54995},[106,110],[18,54997],{"className":54998,"style":228},[169],[18,55000,921],{"className":55001},[232],[18,55003],{"className":55004,"style":228},[169],[18,55006,55008,55011,55014,55054,55057,55060,55100,55103,55106,55109,55112,55115,55118,55121],{"className":55007},[97],[18,55009],{"className":55010,"style":242},[101],[18,55012,74],{"className":55013},[250],[18,55015,55017,55020],{"className":55016},[106],[18,55018,38582],{"className":55019},[106,110],[18,55021,55023],{"className":55022},[115],[18,55024,55026,55046],{"className":55025},[119,120],[18,55027,55029,55043],{"className":55028},[124],[18,55030,55032],{"className":55031,"style":129},[128],[18,55033,55034,55037],{"style":1155},[18,55035],{"className":55036,"style":137},[136],[18,55038,55040],{"className":55039},[141,142,143,144],[18,55041,47],{"className":55042},[106,144],[18,55044,152],{"className":55045},[151],[18,55047,55049],{"className":55048},[124],[18,55050,55052],{"className":55051,"style":159},[128],[18,55053],{},[18,55055,52],{"className":55056},[165],[18,55058],{"className":55059,"style":170},[169],[18,55061,55063,55066],{"className":55062},[106],[18,55064,38582],{"className":55065},[106,110],[18,55067,55069],{"className":55068},[115],[18,55070,55072,55092],{"className":55071},[119,120],[18,55073,55075,55089],{"className":55074},[124],[18,55076,55078],{"className":55077,"style":129},[128],[18,55079,55080,55083],{"style":1155},[18,55081],{"className":55082,"style":137},[136],[18,55084,55086],{"className":55085},[141,142,143,144],[18,55087,2266],{"className":55088},[106,144],[18,55090,152],{"className":55091},[151],[18,55093,55095],{"className":55094},[124],[18,55096,55098],{"className":55097,"style":159},[128],[18,55099],{},[18,55101,965],{"className":55102},[267],[18,55104,52],{"className":55105},[165],[18,55107,928],{"className":55108},[169],[18,55110],{"className":55111,"style":170},[169],[18,55113,54081],{"className":55114},[106,110],[18,55116],{"className":55117,"style":228},[169],[18,55119,4775],{"className":55120},[232],[18,55122],{"className":55123,"style":228},[169],[18,55125,55127,55130],{"className":55126},[97],[18,55128],{"className":55129,"style":16897},[101],[18,55131,55133,55136],{"className":55132},[106],[18,55134,38582],{"className":55135},[106,110],[18,55137,55139],{"className":55138},[115],[18,55140,55142,55162],{"className":55141},[119,120],[18,55143,55145,55159],{"className":55144},[124],[18,55146,55148],{"className":55147,"style":129},[128],[18,55149,55150,55153],{"style":1155},[18,55151],{"className":55152,"style":137},[136],[18,55154,55156],{"className":55155},[141,142,143,144],[18,55157,2266],{"className":55158},[106,144],[18,55160,152],{"className":55161},[151],[18,55163,55165],{"className":55164},[124],[18,55166,55168],{"className":55167,"style":159},[128],[18,55169],{},[18,55171,55173,55195],{"className":55172,"translate":22},[21],[18,55174,55176],{"className":55175},[26],[28,55177,55178],{"xmlns":30},[32,55179,55180,55192],{},[35,55181,55182,55184,55186,55188,55190],{},[49,55183,74],{"stretchy":73},[45,55185,430],{},[49,55187,52],{"separator":51},[45,55189,430],{},[49,55191,84],{"stretchy":73},[86,55193,55194],{"encoding":88},"(0,0)",[18,55196,55198],{"className":55197,"ariaHidden":51},[93],[18,55199,55201,55204,55207,55210,55213,55216,55219],{"className":55200},[97],[18,55202],{"className":55203,"style":242},[101],[18,55205,74],{"className":55206},[250],[18,55208,430],{"className":55209},[106],[18,55211,52],{"className":55212},[165],[18,55214],{"className":55215,"style":170},[169],[18,55217,430],{"className":55218},[106],[18,55220,84],{"className":55221},[267]," 取一个避开两点的区间。",[11,55224,55225,55226,55422,55423,55488,55489,509,55559,55629,55630,55700,55701,55729],{},"三个点 ",[18,55227,55229,55263],{"className":55228,"translate":22},[21],[18,55230,55232],{"className":55231},[26],[28,55233,55234],{"xmlns":30},[32,55235,55236,55260],{},[35,55237,55238,55244,55246,55252,55254],{},[38,55239,55240,55242],{},[41,55241,38582],{},[45,55243,47],{},[49,55245,6684],{},[38,55247,55248,55250],{},[41,55249,38582],{},[45,55251,2266],{},[49,55253,6684],{},[38,55255,55256,55258],{},[41,55257,38582],{},[45,55259,44708],{},[86,55261,55262],{"encoding":88},"x_1\u003Cx_2\u003Cx_3",[18,55264,55266,55321,55376],{"className":55265,"ariaHidden":51},[93],[18,55267,55269,55272,55312,55315,55318],{"className":55268},[97],[18,55270],{"className":55271,"style":40980},[101],[18,55273,55275,55278],{"className":55274},[106],[18,55276,38582],{"className":55277},[106,110],[18,55279,55281],{"className":55280},[115],[18,55282,55284,55304],{"className":55283},[119,120],[18,55285,55287,55301],{"className":55286},[124],[18,55288,55290],{"className":55289,"style":129},[128],[18,55291,55292,55295],{"style":1155},[18,55293],{"className":55294,"style":137},[136],[18,55296,55298],{"className":55297},[141,142,143,144],[18,55299,47],{"className":55300},[106,144],[18,55302,152],{"className":55303},[151],[18,55305,55307],{"className":55306},[124],[18,55308,55310],{"className":55309,"style":159},[128],[18,55311],{},[18,55313],{"className":55314,"style":228},[169],[18,55316,6684],{"className":55317},[232],[18,55319],{"className":55320,"style":228},[169],[18,55322,55324,55327,55367,55370,55373],{"className":55323},[97],[18,55325],{"className":55326,"style":40980},[101],[18,55328,55330,55333],{"className":55329},[106],[18,55331,38582],{"className":55332},[106,110],[18,55334,55336],{"className":55335},[115],[18,55337,55339,55359],{"className":55338},[119,120],[18,55340,55342,55356],{"className":55341},[124],[18,55343,55345],{"className":55344,"style":129},[128],[18,55346,55347,55350],{"style":1155},[18,55348],{"className":55349,"style":137},[136],[18,55351,55353],{"className":55352},[141,142,143,144],[18,55354,2266],{"className":55355},[106,144],[18,55357,152],{"className":55358},[151],[18,55360,55362],{"className":55361},[124],[18,55363,55365],{"className":55364,"style":159},[128],[18,55366],{},[18,55368],{"className":55369,"style":228},[169],[18,55371,6684],{"className":55372},[232],[18,55374],{"className":55375,"style":228},[169],[18,55377,55379,55382],{"className":55378},[97],[18,55380],{"className":55381,"style":16897},[101],[18,55383,55385,55388],{"className":55384},[106],[18,55386,38582],{"className":55387},[106,110],[18,55389,55391],{"className":55390},[115],[18,55392,55394,55414],{"className":55393},[119,120],[18,55395,55397,55411],{"className":55396},[124],[18,55398,55400],{"className":55399,"style":129},[128],[18,55401,55402,55405],{"style":1155},[18,55403],{"className":55404,"style":137},[136],[18,55406,55408],{"className":55407},[141,142,143,144],[18,55409,44708],{"className":55410},[106,144],[18,55412,152],{"className":55413},[151],[18,55415,55417],{"className":55416},[124],[18,55418,55420],{"className":55419,"style":159},[128],[18,55421],{}," 不行：标注 ",[18,55424,55426,55452],{"className":55425,"translate":22},[21],[18,55427,55429],{"className":55428},[26],[28,55430,55431],{"xmlns":30},[32,55432,55433,55449],{},[35,55434,55435,55437,55439,55441,55443,55445,55447],{},[49,55436,74],{"stretchy":73},[45,55438,47],{},[49,55440,52],{"separator":51},[45,55442,430],{},[49,55444,52],{"separator":51},[45,55446,47],{},[49,55448,84],{"stretchy":73},[86,55450,55451],{"encoding":88},"(1,0,1)",[18,55453,55455],{"className":55454,"ariaHidden":51},[93],[18,55456,55458,55461,55464,55467,55470,55473,55476,55479,55482,55485],{"className":55457},[97],[18,55459],{"className":55460,"style":242},[101],[18,55462,74],{"className":55463},[250],[18,55465,47],{"className":55466},[106],[18,55468,52],{"className":55469},[165],[18,55471],{"className":55472,"style":170},[169],[18,55474,430],{"className":55475},[106],[18,55477,52],{"className":55478},[165],[18,55480],{"className":55481,"style":170},[169],[18,55483,47],{"className":55484},[106],[18,55486,84],{"className":55487},[267]," 要求区间同时包含 ",[18,55490,55492,55510],{"className":55491,"translate":22},[21],[18,55493,55495],{"className":55494},[26],[28,55496,55497],{"xmlns":30},[32,55498,55499,55507],{},[35,55500,55501],{},[38,55502,55503,55505],{},[41,55504,38582],{},[45,55506,47],{},[86,55508,55509],{"encoding":88},"x_1",[18,55511,55513],{"className":55512,"ariaHidden":51},[93],[18,55514,55516,55519],{"className":55515},[97],[18,55517],{"className":55518,"style":16897},[101],[18,55520,55522,55525],{"className":55521},[106],[18,55523,38582],{"className":55524},[106,110],[18,55526,55528],{"className":55527},[115],[18,55529,55531,55551],{"className":55530},[119,120],[18,55532,55534,55548],{"className":55533},[124],[18,55535,55537],{"className":55536,"style":129},[128],[18,55538,55539,55542],{"style":1155},[18,55540],{"className":55541,"style":137},[136],[18,55543,55545],{"className":55544},[141,142,143,144],[18,55546,47],{"className":55547},[106,144],[18,55549,152],{"className":55550},[151],[18,55552,55554],{"className":55553},[124],[18,55555,55557],{"className":55556,"style":159},[128],[18,55558],{},[18,55560,55562,55580],{"className":55561,"translate":22},[21],[18,55563,55565],{"className":55564},[26],[28,55566,55567],{"xmlns":30},[32,55568,55569,55577],{},[35,55570,55571],{},[38,55572,55573,55575],{},[41,55574,38582],{},[45,55576,44708],{},[86,55578,55579],{"encoding":88},"x_3",[18,55581,55583],{"className":55582,"ariaHidden":51},[93],[18,55584,55586,55589],{"className":55585},[97],[18,55587],{"className":55588,"style":16897},[101],[18,55590,55592,55595],{"className":55591},[106],[18,55593,38582],{"className":55594},[106,110],[18,55596,55598],{"className":55597},[115],[18,55599,55601,55621],{"className":55600},[119,120],[18,55602,55604,55618],{"className":55603},[124],[18,55605,55607],{"className":55606,"style":129},[128],[18,55608,55609,55612],{"style":1155},[18,55610],{"className":55611,"style":137},[136],[18,55613,55615],{"className":55614},[141,142,143,144],[18,55616,44708],{"className":55617},[106,144],[18,55619,152],{"className":55620},[151],[18,55622,55624],{"className":55623},[124],[18,55625,55627],{"className":55626,"style":159},[128],[18,55628],{},"，而区间是凸的，必然包含中间的 ",[18,55631,55633,55651],{"className":55632,"translate":22},[21],[18,55634,55636],{"className":55635},[26],[28,55637,55638],{"xmlns":30},[32,55639,55640,55648],{},[35,55641,55642],{},[38,55643,55644,55646],{},[41,55645,38582],{},[45,55647,2266],{},[86,55649,55650],{"encoding":88},"x_2",[18,55652,55654],{"className":55653,"ariaHidden":51},[93],[18,55655,55657,55660],{"className":55656},[97],[18,55658],{"className":55659,"style":16897},[101],[18,55661,55663,55666],{"className":55662},[106],[18,55664,38582],{"className":55665},[106,110],[18,55667,55669],{"className":55668},[115],[18,55670,55672,55692],{"className":55671},[119,120],[18,55673,55675,55689],{"className":55674},[124],[18,55676,55678],{"className":55677,"style":129},[128],[18,55679,55680,55683],{"style":1155},[18,55681],{"className":55682,"style":137},[136],[18,55684,55686],{"className":55685},[141,142,143,144],[18,55687,2266],{"className":55688},[106,144],[18,55690,152],{"className":55691},[151],[18,55693,55695],{"className":55694},[124],[18,55696,55698],{"className":55697,"style":159},[128],[18,55699],{},"，与中间的 ",[18,55702,55704,55717],{"className":55703,"translate":22},[21],[18,55705,55707],{"className":55706},[26],[28,55708,55709],{"xmlns":30},[32,55710,55711,55715],{},[35,55712,55713],{},[45,55714,430],{},[86,55716,430],{"encoding":88},[18,55718,55720],{"className":55719,"ariaHidden":51},[93],[18,55721,55723,55726],{"className":55722},[97],[18,55724],{"className":55725,"style":505},[101],[18,55727,430],{"className":55728},[106]," 矛盾。",[11,55731,55732],{},[717,55733,55734,55735,52828],{},"例 3（仿射超平面，",[18,55736,55738,55762],{"className":55737,"translate":22},[21],[18,55739,55741],{"className":55740},[26],[28,55742,55743],{"xmlns":30},[32,55744,55745,55759],{},[35,55746,55747,55749,55751,55753,55755,55757],{},[41,55748,52216],{"mathvariant":316},[49,55750,899],{},[49,55752,433],{},[41,55754,4031],{},[49,55756,7987],{},[45,55758,47],{},[86,55760,55761],{"encoding":88},"\\operatorname{VCdim}=d+1",[18,55763,55765,55786,55804],{"className":55764,"ariaHidden":51},[93],[18,55766,55768,55771,55777,55780,55783],{"className":55767},[97],[18,55769],{"className":55770,"style":295},[101],[18,55772,55774],{"className":55773},[985],[18,55775,52216],{"className":55776},[106,6354],[18,55778],{"className":55779,"style":228},[169],[18,55781,433],{"className":55782},[232],[18,55784],{"className":55785,"style":228},[169],[18,55787,55789,55792,55795,55798,55801],{"className":55788},[97],[18,55790],{"className":55791,"style":7224},[101],[18,55793,4031],{"className":55794},[106,110],[18,55796],{"className":55797,"style":2068},[169],[18,55799,7987],{"className":55800},[2072],[18,55802],{"className":55803,"style":2068},[169],[18,55805,55807,55810],{"className":55806},[97],[18,55808],{"className":55809,"style":505},[101],[18,55811,47],{"className":55812},[106],[11,55814,36192,55815,4741,55896,1990],{},[18,55816,55818,55840],{"className":55817,"translate":22},[21],[18,55819,55821],{"className":55820},[26],[28,55822,55823],{"xmlns":30},[32,55824,55825,55837],{},[35,55826,55827,55829,55831],{},[41,55828,43],{"mathvariant":1462},[49,55830,433],{},[1556,55832,55833,55835],{},[41,55834,3071],{"mathvariant":933},[41,55836,4031],{},[86,55838,55839],{"encoding":88},"\\mathcal X=\\mathbb R^d",[18,55841,55843,55861],{"className":55842,"ariaHidden":51},[93],[18,55844,55846,55849,55852,55855,55858],{"className":55845},[97],[18,55847],{"className":55848,"style":600},[101],[18,55850,43],{"className":55851,"style":1479},[106,1478],[18,55853],{"className":55854,"style":228},[169],[18,55856,433],{"className":55857},[232],[18,55859],{"className":55860,"style":228},[169],[18,55862,55864,55867],{"className":55863},[97],[18,55865],{"className":55866,"style":35186},[101],[18,55868,55870,55873],{"className":55869},[106],[18,55871,3071],{"className":55872},[106,1139],[18,55874,55876],{"className":55875},[115],[18,55877,55879],{"className":55878},[119],[18,55880,55882],{"className":55881},[124],[18,55883,55885],{"className":55884,"style":35186},[128],[18,55886,55887,55890],{"style":1962},[18,55888],{"className":55889,"style":137},[136],[18,55891,55893],{"className":55892},[141,142,143,144],[18,55894,4031],{"className":55895},[106,110,144],[18,55897,55899,55942],{"className":55898,"translate":22},[21],[18,55900,55902],{"className":55901},[26],[28,55903,55904],{"xmlns":30},[32,55905,55906,55939],{},[35,55907,55908,55910,55912,55914,55916,55918,55920,55922,55924,55927,55929,55931,55933,55935,55937],{},[41,55909,18309],{"mathvariant":1462},[49,55911,433],{},[49,55913,1331],{"stretchy":73},[41,55915,38582],{},[49,55917,52853],{},[41,55919,45639],{"mathvariant":316},[49,55921,899],{},[49,55923,74],{"stretchy":73},[41,55925,55926],{},"w",[49,55928,26840],{},[41,55930,38582],{},[49,55932,7987],{},[41,55934,54081],{},[49,55936,84],{"stretchy":73},[49,55938,1349],{"stretchy":73},[86,55940,55941],{"encoding":88},"\\mathcal H=\\{x\\mapsto\\operatorname{sign}(w\\cdot x+b)\\}",[18,55943,55945,55963,55984,56012,56030],{"className":55944,"ariaHidden":51},[93],[18,55946,55948,55951,55954,55957,55960],{"className":55947},[97],[18,55949],{"className":55950,"style":600},[101],[18,55952,18309],{"className":55953,"style":46330},[106,1478],[18,55955],{"className":55956,"style":228},[169],[18,55958,433],{"className":55959},[232],[18,55961],{"className":55962,"style":228},[169],[18,55964,55966,55969,55972,55975,55978,55981],{"className":55965},[97],[18,55967],{"className":55968,"style":242},[101],[18,55970,1331],{"className":55971},[250],[18,55973,38582],{"className":55974},[106,110],[18,55976],{"className":55977,"style":228},[169],[18,55979,52853],{"className":55980},[232],[18,55982],{"className":55983,"style":228},[169],[18,55985,55987,55990,55996,55999,56003,56006,56009],{"className":55986},[97],[18,55988],{"className":55989,"style":242},[101],[18,55991,55993],{"className":55992},[985],[18,55994,45639],{"className":55995},[106,6354],[18,55997,74],{"className":55998},[250],[18,56000,55926],{"className":56001,"style":56002},[106,110],"margin-right:0.0269em;",[18,56004],{"className":56005,"style":2068},[169],[18,56007,26840],{"className":56008},[2072],[18,56010],{"className":56011,"style":2068},[169],[18,56013,56015,56018,56021,56024,56027],{"className":56014},[97],[18,56016],{"className":56017,"style":14404},[101],[18,56019,38582],{"className":56020},[106,110],[18,56022],{"className":56023,"style":2068},[169],[18,56025,7987],{"className":56026},[2072],[18,56028],{"className":56029,"style":2068},[169],[18,56031,56033,56036,56039],{"className":56032},[97],[18,56034],{"className":56035,"style":242},[101],[18,56037,54081],{"className":56038},[106,110],[18,56040,46722],{"className":56041},[267],[11,56043,56044,56046,56047,56232,56233,56284,56285,56397],{},[717,56045,38242],{},"：取 ",[18,56048,56050,56092],{"className":56049,"translate":22},[21],[18,56051,56053],{"className":56052},[26],[28,56054,56055],{"xmlns":30},[32,56056,56057,56089],{},[35,56058,56059,56061,56063,56065,56067,56069,56075,56077,56079,56081,56087],{},[41,56060,51021],{},[49,56062,433],{},[49,56064,1331],{"stretchy":73},[45,56066,430],{},[49,56068,52],{"separator":51},[38,56070,56071,56073],{},[41,56072,14444],{},[45,56074,47],{},[49,56076,52],{"separator":51},[49,56078,55],{},[49,56080,52],{"separator":51},[38,56082,56083,56085],{},[41,56084,14444],{},[41,56086,4031],{},[49,56088,1349],{"stretchy":73},[86,56090,56091],{"encoding":88},"C=\\{0,e_1,\\dots,e_d\\}",[18,56093,56095,56113],{"className":56094,"ariaHidden":51},[93],[18,56096,56098,56101,56104,56107,56110],{"className":56097},[97],[18,56099],{"className":56100,"style":600},[101],[18,56102,51021],{"className":56103,"style":5073},[106,110],[18,56105],{"className":56106,"style":228},[169],[18,56108,433],{"className":56109},[232],[18,56111],{"className":56112,"style":228},[169],[18,56114,56116,56119,56122,56125,56128,56131,56171,56174,56177,56180,56183,56186,56189,56229],{"className":56115},[97],[18,56117],{"className":56118,"style":242},[101],[18,56120,1331],{"className":56121},[250],[18,56123,430],{"className":56124},[106],[18,56126,52],{"className":56127},[165],[18,56129],{"className":56130,"style":170},[169],[18,56132,56134,56137],{"className":56133},[106],[18,56135,14444],{"className":56136},[106,110],[18,56138,56140],{"className":56139},[115],[18,56141,56143,56163],{"className":56142},[119,120],[18,56144,56146,56160],{"className":56145},[124],[18,56147,56149],{"className":56148,"style":129},[128],[18,56150,56151,56154],{"style":1155},[18,56152],{"className":56153,"style":137},[136],[18,56155,56157],{"className":56156},[141,142,143,144],[18,56158,47],{"className":56159},[106,144],[18,56161,152],{"className":56162},[151],[18,56164,56166],{"className":56165},[124],[18,56167,56169],{"className":56168,"style":159},[128],[18,56170],{},[18,56172,52],{"className":56173},[165],[18,56175],{"className":56176,"style":170},[169],[18,56178,55],{"className":56179},[174],[18,56181],{"className":56182,"style":170},[169],[18,56184,52],{"className":56185},[165],[18,56187],{"className":56188,"style":170},[169],[18,56190,56192,56195],{"className":56191},[106],[18,56193,14444],{"className":56194},[106,110],[18,56196,56198],{"className":56197},[115],[18,56199,56201,56221],{"className":56200},[119,120],[18,56202,56204,56218],{"className":56203},[124],[18,56205,56207],{"className":56206,"style":1383},[128],[18,56208,56209,56212],{"style":1155},[18,56210],{"className":56211,"style":137},[136],[18,56213,56215],{"className":56214},[141,142,143,144],[18,56216,4031],{"className":56217},[106,110,144],[18,56219,152],{"className":56220},[151],[18,56222,56224],{"className":56223},[124],[18,56225,56227],{"className":56226,"style":159},[128],[18,56228],{},[18,56230,1349],{"className":56231},[267],"，这是 ",[18,56234,56236,56254],{"className":56235,"translate":22},[21],[18,56237,56239],{"className":56238},[26],[28,56240,56241],{"xmlns":30},[32,56242,56243,56251],{},[35,56244,56245,56247,56249],{},[41,56246,4031],{},[49,56248,7987],{},[45,56250,47],{},[86,56252,56253],{"encoding":88},"d+1",[18,56255,56257,56275],{"className":56256,"ariaHidden":51},[93],[18,56258,56260,56263,56266,56269,56272],{"className":56259},[97],[18,56261],{"className":56262,"style":7224},[101],[18,56264,4031],{"className":56265},[106,110],[18,56267],{"className":56268,"style":2068},[169],[18,56270,7987],{"className":56271},[2072],[18,56273],{"className":56274,"style":2068},[169],[18,56276,56278,56281],{"className":56277},[97],[18,56279],{"className":56280,"style":505},[101],[18,56282,47],{"className":56283},[106]," 个仿射无关的点。对任意标注 ",[18,56286,56288,56323],{"className":56287,"translate":22},[21],[18,56289,56291],{"className":56290},[26],[28,56292,56293],{"xmlns":30},[32,56294,56295,56320],{},[35,56296,56297,56299,56301,56303,56306,56308],{},[41,56298,46352],{},[49,56300,921],{},[49,56302,1331],{"stretchy":73},[49,56304,56305],{},"±",[45,56307,47],{},[1556,56309,56310,56312],{},[49,56311,1349],{"stretchy":73},[35,56313,56314,56316,56318],{},[41,56315,4031],{},[49,56317,7987],{},[45,56319,47],{},[86,56321,56322],{"encoding":88},"y\\in\\{\\pm1\\}^{d+1}",[18,56324,56326,56344],{"className":56325,"ariaHidden":51},[93],[18,56327,56329,56332,56335,56338,56341],{"className":56328},[97],[18,56330],{"className":56331,"style":14587},[101],[18,56333,46352],{"className":56334,"style":3957},[106,110],[18,56336],{"className":56337,"style":228},[169],[18,56339,921],{"className":56340},[232],[18,56342],{"className":56343,"style":228},[169],[18,56345,56347,56350,56353,56356,56359],{"className":56346},[97],[18,56348],{"className":56349,"style":35152},[101],[18,56351,1331],{"className":56352},[250],[18,56354,56305],{"className":56355},[106],[18,56357,47],{"className":56358},[106],[18,56360,56362,56365],{"className":56361},[267],[18,56363,1349],{"className":56364},[267],[18,56366,56368],{"className":56367},[115],[18,56369,56371],{"className":56370},[119],[18,56372,56374],{"className":56373},[124],[18,56375,56377],{"className":56376,"style":35186},[128],[18,56378,56379,56382],{"style":1962},[18,56380],{"className":56381,"style":137},[136],[18,56383,56385],{"className":56384},[141,142,143,144],[18,56386,56388,56391,56394],{"className":56387},[106,144],[18,56389,4031],{"className":56390},[106,110,144],[18,56392,7987],{"className":56393},[2072,144],[18,56395,47],{"className":56396},[106,144],"，方程组",[18,56399,56401],{"className":56400,"translate":22},[875],[18,56402,56404,56460],{"className":56403,"translate":22},[21],[18,56405,56407],{"className":56406},[26],[28,56408,56409],{"xmlns":30,"display":884},[32,56410,56411,56457],{},[35,56412,56413,56415,56417,56423,56425,56427,56429,56435,56437,56439,56441,56443,56445,56447,56449,56451,56453,56455],{},[41,56414,55926],{},[49,56416,26840],{},[38,56418,56419,56421],{},[41,56420,38582],{},[41,56422,5600],{},[49,56424,7987],{},[41,56426,54081],{},[49,56428,433],{},[38,56430,56431,56433],{},[41,56432,46352],{},[41,56434,5600],{},[49,56436,52],{"separator":51},[169,56438],{"width":7344},[41,56440,5600],{},[49,56442,433],{},[45,56444,47],{},[49,56446,52],{"separator":51},[49,56448,55],{},[49,56450,52],{"separator":51},[41,56452,4031],{},[49,56454,7987],{},[45,56456,47],{},[86,56458,56459],{"encoding":88},"w\\cdot x_i+b=y_i,\\qquad i=1,\\dots,d+1",[18,56461,56463,56482,56538,56556,56623,56662],{"className":56462,"ariaHidden":51},[93],[18,56464,56466,56470,56473,56476,56479],{"className":56465},[97],[18,56467],{"className":56468,"style":56469},[101],"height:0.4445em;",[18,56471,55926],{"className":56472,"style":56002},[106,110],[18,56474],{"className":56475,"style":2068},[169],[18,56477,26840],{"className":56478},[2072],[18,56480],{"className":56481,"style":2068},[169],[18,56483,56485,56489,56529,56532,56535],{"className":56484},[97],[18,56486],{"className":56487,"style":56488},[101],"height:0.7333em;vertical-align:-0.15em;",[18,56490,56492,56495],{"className":56491},[106],[18,56493,38582],{"className":56494},[106,110],[18,56496,56498],{"className":56497},[115],[18,56499,56501,56521],{"className":56500},[119,120],[18,56502,56504,56518],{"className":56503},[124],[18,56505,56507],{"className":56506,"style":5921},[128],[18,56508,56509,56512],{"style":1155},[18,56510],{"className":56511,"style":137},[136],[18,56513,56515],{"className":56514},[141,142,143,144],[18,56516,5600],{"className":56517},[106,110,144],[18,56519,152],{"className":56520},[151],[18,56522,56524],{"className":56523},[124],[18,56525,56527],{"className":56526,"style":159},[128],[18,56528],{},[18,56530],{"className":56531,"style":2068},[169],[18,56533,7987],{"className":56534},[2072],[18,56536],{"className":56537,"style":2068},[169],[18,56539,56541,56544,56547,56550,56553],{"className":56540},[97],[18,56542],{"className":56543,"style":295},[101],[18,56545,54081],{"className":56546},[106,110],[18,56548],{"className":56549,"style":228},[169],[18,56551,433],{"className":56552},[232],[18,56554],{"className":56555,"style":228},[169],[18,56557,56559,56562,56602,56605,56608,56611,56614,56617,56620],{"className":56558},[97],[18,56560],{"className":56561,"style":28340},[101],[18,56563,56565,56568],{"className":56564},[106],[18,56566,46352],{"className":56567,"style":3957},[106,110],[18,56569,56571],{"className":56570},[115],[18,56572,56574,56594],{"className":56573},[119,120],[18,56575,56577,56591],{"className":56576},[124],[18,56578,56580],{"className":56579,"style":5921},[128],[18,56581,56582,56585],{"style":30350},[18,56583],{"className":56584,"style":137},[136],[18,56586,56588],{"className":56587},[141,142,143,144],[18,56589,5600],{"className":56590},[106,110,144],[18,56592,152],{"className":56593},[151],[18,56595,56597],{"className":56596},[124],[18,56598,56600],{"className":56599,"style":159},[128],[18,56601],{},[18,56603,52],{"className":56604},[165],[18,56606],{"className":56607,"style":7676},[169],[18,56609],{"className":56610,"style":170},[169],[18,56612,5600],{"className":56613},[106,110],[18,56615],{"className":56616,"style":228},[169],[18,56618,433],{"className":56619},[232],[18,56621],{"className":56622,"style":228},[169],[18,56624,56626,56629,56632,56635,56638,56641,56644,56647,56650,56653,56656,56659],{"className":56625},[97],[18,56627],{"className":56628,"style":2890},[101],[18,56630,47],{"className":56631},[106],[18,56633,52],{"className":56634},[165],[18,56636],{"className":56637,"style":170},[169],[18,56639,55],{"className":56640},[174],[18,56642],{"className":56643,"style":170},[169],[18,56645,52],{"className":56646},[165],[18,56648],{"className":56649,"style":170},[169],[18,56651,4031],{"className":56652},[106,110],[18,56654],{"className":56655,"style":2068},[169],[18,56657,7987],{"className":56658},[2072],[18,56660],{"className":56661,"style":2068},[169],[18,56663,56665,56668],{"className":56664},[97],[18,56666],{"className":56667,"style":505},[101],[18,56669,47],{"className":56670},[106],[11,56672,56673,56674,56767,56768,56883,56884,56912],{},"有唯一解，因为以 ",[18,56675,56677,56703],{"className":56676,"translate":22},[21],[18,56678,56680],{"className":56679},[26],[28,56681,56682],{"xmlns":30},[32,56683,56684,56700],{},[35,56685,56686,56688,56694,56696,56698],{},[49,56687,74],{"stretchy":73},[38,56689,56690,56692],{},[41,56691,38582],{},[41,56693,5600],{},[49,56695,52],{"separator":51},[45,56697,47],{},[49,56699,84],{"stretchy":73},[86,56701,56702],{"encoding":88},"(x_i,1)",[18,56704,56706],{"className":56705,"ariaHidden":51},[93],[18,56707,56709,56712,56715,56755,56758,56761,56764],{"className":56708},[97],[18,56710],{"className":56711,"style":242},[101],[18,56713,74],{"className":56714},[250],[18,56716,56718,56721],{"className":56717},[106],[18,56719,38582],{"className":56720},[106,110],[18,56722,56724],{"className":56723},[115],[18,56725,56727,56747],{"className":56726},[119,120],[18,56728,56730,56744],{"className":56729},[124],[18,56731,56733],{"className":56732,"style":5921},[128],[18,56734,56735,56738],{"style":1155},[18,56736],{"className":56737,"style":137},[136],[18,56739,56741],{"className":56740},[141,142,143,144],[18,56742,5600],{"className":56743},[106,110,144],[18,56745,152],{"className":56746},[151],[18,56748,56750],{"className":56749},[124],[18,56751,56753],{"className":56752,"style":159},[128],[18,56754],{},[18,56756,52],{"className":56757},[165],[18,56759],{"className":56760,"style":170},[169],[18,56762,47],{"className":56763},[106],[18,56765,84],{"className":56766},[267]," 为列的 ",[18,56769,56771,56805],{"className":56770,"translate":22},[21],[18,56772,56774],{"className":56773},[26],[28,56775,56776],{"xmlns":30},[32,56777,56778,56802],{},[35,56779,56780,56782,56784,56786,56788,56790,56792,56794,56796,56798,56800],{},[49,56781,74],{"stretchy":73},[41,56783,4031],{},[49,56785,7987],{},[45,56787,47],{},[49,56789,84],{"stretchy":73},[49,56791,2015],{},[49,56793,74],{"stretchy":73},[41,56795,4031],{},[49,56797,7987],{},[45,56799,47],{},[49,56801,84],{"stretchy":73},[86,56803,56804],{"encoding":88},"(d+1)\\times(d+1)",[18,56806,56808,56829,56850,56871],{"className":56807,"ariaHidden":51},[93],[18,56809,56811,56814,56817,56820,56823,56826],{"className":56810},[97],[18,56812],{"className":56813,"style":242},[101],[18,56815,74],{"className":56816},[250],[18,56818,4031],{"className":56819},[106,110],[18,56821],{"className":56822,"style":2068},[169],[18,56824,7987],{"className":56825},[2072],[18,56827],{"className":56828,"style":2068},[169],[18,56830,56832,56835,56838,56841,56844,56847],{"className":56831},[97],[18,56833],{"className":56834,"style":242},[101],[18,56836,47],{"className":56837},[106],[18,56839,84],{"className":56840},[267],[18,56842],{"className":56843,"style":2068},[169],[18,56845,2015],{"className":56846},[2072],[18,56848],{"className":56849,"style":2068},[169],[18,56851,56853,56856,56859,56862,56865,56868],{"className":56852},[97],[18,56854],{"className":56855,"style":242},[101],[18,56857,74],{"className":56858},[250],[18,56860,4031],{"className":56861},[106,110],[18,56863],{"className":56864,"style":2068},[169],[18,56866,7987],{"className":56867},[2072],[18,56869],{"className":56870,"style":2068},[169],[18,56872,56874,56877,56880],{"className":56873},[97],[18,56875],{"className":56876,"style":242},[101],[18,56878,47],{"className":56879},[106],[18,56881,84],{"className":56882},[267]," 矩阵可逆。于是每种标注都能被某个仿射函数实现，",[18,56885,56887,56900],{"className":56886,"translate":22},[21],[18,56888,56890],{"className":56889},[26],[28,56891,56892],{"xmlns":30},[32,56893,56894,56898],{},[35,56895,56896],{},[41,56897,51021],{},[86,56899,51021],{"encoding":88},[18,56901,56903],{"className":56902,"ariaHidden":51},[93],[18,56904,56906,56909],{"className":56905},[97],[18,56907],{"className":56908,"style":600},[101],[18,56910,51021],{"className":56911,"style":5073},[106,110]," 被打散。",[11,56914,56915,56918,56919,56978,56979,57030],{},[717,56916,56917],{},"上界","：",[18,56920,56922,56940],{"className":56921,"translate":22},[21],[18,56923,56925],{"className":56924},[26],[28,56926,56927],{"xmlns":30},[32,56928,56929,56937],{},[35,56930,56931],{},[1556,56932,56933,56935],{},[41,56934,3071],{"mathvariant":933},[41,56936,4031],{},[86,56938,56939],{"encoding":88},"\\mathbb R^d",[18,56941,56943],{"className":56942,"ariaHidden":51},[93],[18,56944,56946,56949],{"className":56945},[97],[18,56947],{"className":56948,"style":35186},[101],[18,56950,56952,56955],{"className":56951},[106],[18,56953,3071],{"className":56954},[106,1139],[18,56956,56958],{"className":56957},[115],[18,56959,56961],{"className":56960},[119],[18,56962,56964],{"className":56963},[124],[18,56965,56967],{"className":56966,"style":35186},[128],[18,56968,56969,56972],{"style":1962},[18,56970],{"className":56971,"style":137},[136],[18,56973,56975],{"className":56974},[141,142,143,144],[18,56976,4031],{"className":56977},[106,110,144]," 中任意 ",[18,56980,56982,57000],{"className":56981,"translate":22},[21],[18,56983,56985],{"className":56984},[26],[28,56986,56987],{"xmlns":30},[32,56988,56989,56997],{},[35,56990,56991,56993,56995],{},[41,56992,4031],{},[49,56994,7987],{},[45,56996,2266],{},[86,56998,56999],{"encoding":88},"d+2",[18,57001,57003,57021],{"className":57002,"ariaHidden":51},[93],[18,57004,57006,57009,57012,57015,57018],{"className":57005},[97],[18,57007],{"className":57008,"style":7224},[101],[18,57010,4031],{"className":57011},[106,110],[18,57013],{"className":57014,"style":2068},[169],[18,57016,7987],{"className":57017},[2072],[18,57019],{"className":57020,"style":2068},[169],[18,57022,57024,57027],{"className":57023},[97],[18,57025],{"className":57026,"style":505},[101],[18,57028,2266],{"className":57029},[106]," 个点都仿射相关，因此可以用 Radon 定理处理。",[5159,57032,57033],{},[11,57034,57035,1994,57038,56978,57096,57146],{},[717,57036,57037],{},"定理（Radon，1921）",[18,57039,57041,57058],{"className":57040,"translate":22},[21],[18,57042,57044],{"className":57043},[26],[28,57045,57046],{"xmlns":30},[32,57047,57048,57056],{},[35,57049,57050],{},[1556,57051,57052,57054],{},[41,57053,3071],{"mathvariant":933},[41,57055,4031],{},[86,57057,56939],{"encoding":88},[18,57059,57061],{"className":57060,"ariaHidden":51},[93],[18,57062,57064,57067],{"className":57063},[97],[18,57065],{"className":57066,"style":35186},[101],[18,57068,57070,57073],{"className":57069},[106],[18,57071,3071],{"className":57072},[106,1139],[18,57074,57076],{"className":57075},[115],[18,57077,57079],{"className":57078},[119],[18,57080,57082],{"className":57081},[124],[18,57083,57085],{"className":57084,"style":35186},[128],[18,57086,57087,57090],{"style":1962},[18,57088],{"className":57089,"style":137},[136],[18,57091,57093],{"className":57092},[141,142,143,144],[18,57094,4031],{"className":57095},[106,110,144],[18,57097,57099,57116],{"className":57098,"translate":22},[21],[18,57100,57102],{"className":57101},[26],[28,57103,57104],{"xmlns":30},[32,57105,57106,57114],{},[35,57107,57108,57110,57112],{},[41,57109,4031],{},[49,57111,7987],{},[45,57113,2266],{},[86,57115,56999],{"encoding":88},[18,57117,57119,57137],{"className":57118,"ariaHidden":51},[93],[18,57120,57122,57125,57128,57131,57134],{"className":57121},[97],[18,57123],{"className":57124,"style":7224},[101],[18,57126,4031],{"className":57127},[106,110],[18,57129],{"className":57130,"style":2068},[169],[18,57132,7987],{"className":57133},[2072],[18,57135],{"className":57136,"style":2068},[169],[18,57138,57140,57143],{"className":57139},[97],[18,57141],{"className":57142,"style":505},[101],[18,57144,2266],{"className":57145},[106]," 个点都可以分成两个不相交的子集，使二者的凸包相交。",[11,57148,57149,57150,4741,57310,57360,57361,57434],{},"证明很短。把点写成齐次坐标 ",[18,57151,57153,57193],{"className":57152,"translate":22},[21],[18,57154,57156],{"className":57155},[26],[28,57157,57158],{"xmlns":30},[32,57159,57160,57190],{},[35,57161,57162,57164,57170,57172,57174,57176,57178],{},[49,57163,74],{"stretchy":73},[38,57165,57166,57168],{},[41,57167,38582],{},[41,57169,5600],{},[49,57171,52],{"separator":51},[45,57173,47],{},[49,57175,84],{"stretchy":73},[49,57177,921],{},[1556,57179,57180,57182],{},[41,57181,3071],{"mathvariant":933},[35,57183,57184,57186,57188],{},[41,57185,4031],{},[49,57187,7987],{},[45,57189,47],{},[86,57191,57192],{"encoding":88},"(x_i,1)\\in\\mathbb R^{d+1}",[18,57194,57196,57266],{"className":57195,"ariaHidden":51},[93],[18,57197,57199,57202,57205,57245,57248,57251,57254,57257,57260,57263],{"className":57198},[97],[18,57200],{"className":57201,"style":242},[101],[18,57203,74],{"className":57204},[250],[18,57206,57208,57211],{"className":57207},[106],[18,57209,38582],{"className":57210},[106,110],[18,57212,57214],{"className":57213},[115],[18,57215,57217,57237],{"className":57216},[119,120],[18,57218,57220,57234],{"className":57219},[124],[18,57221,57223],{"className":57222,"style":5921},[128],[18,57224,57225,57228],{"style":1155},[18,57226],{"className":57227,"style":137},[136],[18,57229,57231],{"className":57230},[141,142,143,144],[18,57232,5600],{"className":57233},[106,110,144],[18,57235,152],{"className":57236},[151],[18,57238,57240],{"className":57239},[124],[18,57241,57243],{"className":57242,"style":159},[128],[18,57244],{},[18,57246,52],{"className":57247},[165],[18,57249],{"className":57250,"style":170},[169],[18,57252,47],{"className":57253},[106],[18,57255,84],{"className":57256},[267],[18,57258],{"className":57259,"style":228},[169],[18,57261,921],{"className":57262},[232],[18,57264],{"className":57265,"style":228},[169],[18,57267,57269,57272],{"className":57268},[97],[18,57270],{"className":57271,"style":35186},[101],[18,57273,57275,57278],{"className":57274},[106],[18,57276,3071],{"className":57277},[106,1139],[18,57279,57281],{"className":57280},[115],[18,57282,57284],{"className":57283},[119],[18,57285,57287],{"className":57286},[124],[18,57288,57290],{"className":57289,"style":35186},[128],[18,57291,57292,57295],{"style":1962},[18,57293],{"className":57294,"style":137},[136],[18,57296,57298],{"className":57297},[141,142,143,144],[18,57299,57301,57304,57307],{"className":57300},[106,144],[18,57302,4031],{"className":57303},[106,110,144],[18,57305,7987],{"className":57306},[2072,144],[18,57308,47],{"className":57309},[106,144],[18,57311,57313,57330],{"className":57312,"translate":22},[21],[18,57314,57316],{"className":57315},[26],[28,57317,57318],{"xmlns":30},[32,57319,57320,57328],{},[35,57321,57322,57324,57326],{},[41,57323,4031],{},[49,57325,7987],{},[45,57327,2266],{},[86,57329,56999],{"encoding":88},[18,57331,57333,57351],{"className":57332,"ariaHidden":51},[93],[18,57334,57336,57339,57342,57345,57348],{"className":57335},[97],[18,57337],{"className":57338,"style":7224},[101],[18,57340,4031],{"className":57341},[106,110],[18,57343],{"className":57344,"style":2068},[169],[18,57346,7987],{"className":57347},[2072],[18,57349],{"className":57350,"style":2068},[169],[18,57352,57354,57357],{"className":57353},[97],[18,57355],{"className":57356,"style":505},[101],[18,57358,2266],{"className":57359},[106]," 个向量线性相关，故存在不全为零的系数 ",[18,57362,57364,57383],{"className":57363,"translate":22},[21],[18,57365,57367],{"className":57366},[26],[28,57368,57369],{"xmlns":30},[32,57370,57371,57380],{},[35,57372,57373],{},[38,57374,57375,57378],{},[41,57376,57377],{},"α",[41,57379,5600],{},[86,57381,57382],{"encoding":88},"\\alpha_i",[18,57384,57386],{"className":57385,"ariaHidden":51},[93],[18,57387,57389,57392],{"className":57388},[97],[18,57390],{"className":57391,"style":16897},[101],[18,57393,57395,57399],{"className":57394},[106],[18,57396,57377],{"className":57397,"style":57398},[106,110],"margin-right:0.0037em;",[18,57400,57402],{"className":57401},[115],[18,57403,57405,57426],{"className":57404},[119,120],[18,57406,57408,57423],{"className":57407},[124],[18,57409,57411],{"className":57410,"style":5921},[128],[18,57412,57414,57417],{"style":57413},"top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;",[18,57415],{"className":57416,"style":137},[136],[18,57418,57420],{"className":57419},[141,142,143,144],[18,57421,5600],{"className":57422},[106,110,144],[18,57424,152],{"className":57425},[151],[18,57427,57429],{"className":57428},[124],[18,57430,57432],{"className":57431,"style":159},[128],[18,57433],{},"，使",[18,57436,57438],{"className":57437,"translate":22},[875],[18,57439,57441,57496],{"className":57440,"translate":22},[21],[18,57442,57444],{"className":57443},[26],[28,57445,57446],{"xmlns":30,"display":884},[32,57447,57448,57493],{},[35,57449,57450,57456,57462,57468,57470,57472,57474,57476,57482,57488,57490],{},[890,57451,57452,57454],{},[49,57453,20369],{},[41,57455,5600],{},[38,57457,57458,57460],{},[41,57459,57377],{},[41,57461,5600],{},[38,57463,57464,57466],{},[41,57465,38582],{},[41,57467,5600],{},[49,57469,433],{},[45,57471,430],{},[49,57473,52],{"separator":51},[169,57475],{"width":7344},[890,57477,57478,57480],{},[49,57479,20369],{},[41,57481,5600],{},[38,57483,57484,57486],{},[41,57485,57377],{},[41,57487,5600],{},[49,57489,433],{},[45,57491,57492],{},"0.",[86,57494,57495],{"encoding":88},"\\sum_i \\alpha_i x_i=0,\\qquad \\sum_i\\alpha_i=0.",[18,57497,57499,57642,57756],{"className":57498,"ariaHidden":51},[93],[18,57500,57502,57506,57550,57553,57593,57633,57636,57639],{"className":57501},[97],[18,57503],{"className":57504,"style":57505},[101],"height:2.3277em;vertical-align:-1.2777em;",[18,57507,57509],{"className":57508},[985,986],[18,57510,57512,57542],{"className":57511},[119,120],[18,57513,57515,57539],{"className":57514},[124],[18,57516,57518,57529],{"className":57517,"style":24483},[128],[18,57519,57520,57523],{"style":20555},[18,57521],{"className":57522,"style":20559},[136],[18,57524,57526],{"className":57525},[141,142,143,144],[18,57527,5600],{"className":57528},[106,110,144],[18,57530,57531,57534],{"style":20577},[18,57532],{"className":57533,"style":20559},[136],[18,57535,57536],{},[18,57537,20369],{"className":57538},[985,4198,4199],[18,57540,152],{"className":57541},[151],[18,57543,57545],{"className":57544},[124],[18,57546,57548],{"className":57547,"style":20607},[128],[18,57549],{},[18,57551],{"className":57552,"style":170},[169],[18,57554,57556,57559],{"className":57555},[106],[18,57557,57377],{"className":57558,"style":57398},[106,110],[18,57560,57562],{"className":57561},[115],[18,57563,57565,57585],{"className":57564},[119,120],[18,57566,57568,57582],{"className":57567},[124],[18,57569,57571],{"className":57570,"style":5921},[128],[18,57572,57573,57576],{"style":57413},[18,57574],{"className":57575,"style":137},[136],[18,57577,57579],{"className":57578},[141,142,143,144],[18,57580,5600],{"className":57581},[106,110,144],[18,57583,152],{"className":57584},[151],[18,57586,57588],{"className":57587},[124],[18,57589,57591],{"className":57590,"style":159},[128],[18,57592],{},[18,57594,57596,57599],{"className":57595},[106],[18,57597,38582],{"className":57598},[106,110],[18,57600,57602],{"className":57601},[115],[18,57603,57605,57625],{"className":57604},[119,120],[18,57606,57608,57622],{"className":57607},[124],[18,57609,57611],{"className":57610,"style":5921},[128],[18,57612,57613,57616],{"style":1155},[18,57614],{"className":57615,"style":137},[136],[18,57617,57619],{"className":57618},[141,142,143,144],[18,57620,5600],{"className":57621},[106,110,144],[18,57623,152],{"className":57624},[151],[18,57626,57628],{"className":57627},[124],[18,57629,57631],{"className":57630,"style":159},[128],[18,57632],{},[18,57634],{"className":57635,"style":228},[169],[18,57637,433],{"className":57638},[232],[18,57640],{"className":57641,"style":228},[169],[18,57643,57645,57648,57651,57654,57657,57660,57704,57707,57747,57750,57753],{"className":57644},[97],[18,57646],{"className":57647,"style":57505},[101],[18,57649,430],{"className":57650},[106],[18,57652,52],{"className":57653},[165],[18,57655],{"className":57656,"style":7676},[169],[18,57658],{"className":57659,"style":170},[169],[18,57661,57663],{"className":57662},[985,986],[18,57664,57666,57696],{"className":57665},[119,120],[18,57667,57669,57693],{"className":57668},[124],[18,57670,57672,57683],{"className":57671,"style":24483},[128],[18,57673,57674,57677],{"style":20555},[18,57675],{"className":57676,"style":20559},[136],[18,57678,57680],{"className":57679},[141,142,143,144],[18,57681,5600],{"className":57682},[106,110,144],[18,57684,57685,57688],{"style":20577},[18,57686],{"className":57687,"style":20559},[136],[18,57689,57690],{},[18,57691,20369],{"className":57692},[985,4198,4199],[18,57694,152],{"className":57695},[151],[18,57697,57699],{"className":57698},[124],[18,57700,57702],{"className":57701,"style":20607},[128],[18,57703],{},[18,57705],{"className":57706,"style":170},[169],[18,57708,57710,57713],{"className":57709},[106],[18,57711,57377],{"className":57712,"style":57398},[106,110],[18,57714,57716],{"className":57715},[115],[18,57717,57719,57739],{"className":57718},[119,120],[18,57720,57722,57736],{"className":57721},[124],[18,57723,57725],{"className":57724,"style":5921},[128],[18,57726,57727,57730],{"style":57413},[18,57728],{"className":57729,"style":137},[136],[18,57731,57733],{"className":57732},[141,142,143,144],[18,57734,5600],{"className":57735},[106,110,144],[18,57737,152],{"className":57738},[151],[18,57740,57742],{"className":57741},[124],[18,57743,57745],{"className":57744,"style":159},[128],[18,57746],{},[18,57748],{"className":57749,"style":228},[169],[18,57751,433],{"className":57752},[232],[18,57754],{"className":57755,"style":228},[169],[18,57757,57759,57762],{"className":57758},[97],[18,57760],{"className":57761,"style":505},[101],[18,57763,57492],{"className":57764},[106],[11,57766,20033,57767,4741,57913,58059],{},[18,57768,57770,57804],{"className":57769,"translate":22},[21],[18,57771,57773],{"className":57772},[26],[28,57774,57775],{"xmlns":30},[32,57776,57777,57801],{},[35,57778,57779,57781,57783,57785,57787,57789,57795,57797,57799],{},[41,57780,939],{},[49,57782,433],{},[49,57784,1331],{"stretchy":73},[41,57786,5600],{},[49,57788,1340],{},[38,57790,57791,57793],{},[41,57792,57377],{},[41,57794,5600],{},[49,57796,6885],{},[45,57798,430],{},[49,57800,1349],{"stretchy":73},[86,57802,57803],{"encoding":88},"P=\\{i:\\alpha_i>0\\}",[18,57805,57807,57825,57846,57901],{"className":57806,"ariaHidden":51},[93],[18,57808,57810,57813,57816,57819,57822],{"className":57809},[97],[18,57811],{"className":57812,"style":600},[101],[18,57814,939],{"className":57815,"style":1171},[106,110],[18,57817],{"className":57818,"style":228},[169],[18,57820,433],{"className":57821},[232],[18,57823],{"className":57824,"style":228},[169],[18,57826,57828,57831,57834,57837,57840,57843],{"className":57827},[97],[18,57829],{"className":57830,"style":242},[101],[18,57832,1331],{"className":57833},[250],[18,57835,5600],{"className":57836},[106,110],[18,57838],{"className":57839,"style":228},[169],[18,57841,1340],{"className":57842},[232],[18,57844],{"className":57845,"style":228},[169],[18,57847,57849,57852,57892,57895,57898],{"className":57848},[97],[18,57850],{"className":57851,"style":40980},[101],[18,57853,57855,57858],{"className":57854},[106],[18,57856,57377],{"className":57857,"style":57398},[106,110],[18,57859,57861],{"className":57860},[115],[18,57862,57864,57884],{"className":57863},[119,120],[18,57865,57867,57881],{"className":57866},[124],[18,57868,57870],{"className":57869,"style":5921},[128],[18,57871,57872,57875],{"style":57413},[18,57873],{"className":57874,"style":137},[136],[18,57876,57878],{"className":57877},[141,142,143,144],[18,57879,5600],{"className":57880},[106,110,144],[18,57882,152],{"className":57883},[151],[18,57885,57887],{"className":57886},[124],[18,57888,57890],{"className":57889,"style":159},[128],[18,57891],{},[18,57893],{"className":57894,"style":228},[169],[18,57896,6885],{"className":57897},[232],[18,57899],{"className":57900,"style":228},[169],[18,57902,57904,57907,57910],{"className":57903},[97],[18,57905],{"className":57906,"style":242},[101],[18,57908,430],{"className":57909},[106],[18,57911,1349],{"className":57912},[267],[18,57914,57916,57950],{"className":57915,"translate":22},[21],[18,57917,57919],{"className":57918},[26],[28,57920,57921],{"xmlns":30},[32,57922,57923,57947],{},[35,57924,57925,57927,57929,57931,57933,57935,57941,57943,57945],{},[41,57926,70],{},[49,57928,433],{},[49,57930,1331],{"stretchy":73},[41,57932,5600],{},[49,57934,1340],{},[38,57936,57937,57939],{},[41,57938,57377],{},[41,57940,5600],{},[49,57942,6684],{},[45,57944,430],{},[49,57946,1349],{"stretchy":73},[86,57948,57949],{"encoding":88},"N=\\{i:\\alpha_i\u003C0\\}",[18,57951,57953,57971,57992,58047],{"className":57952,"ariaHidden":51},[93],[18,57954,57956,57959,57962,57965,57968],{"className":57955},[97],[18,57957],{"className":57958,"style":600},[101],[18,57960,70],{"className":57961,"style":246},[106,110],[18,57963],{"className":57964,"style":228},[169],[18,57966,433],{"className":57967},[232],[18,57969],{"className":57970,"style":228},[169],[18,57972,57974,57977,57980,57983,57986,57989],{"className":57973},[97],[18,57975],{"className":57976,"style":242},[101],[18,57978,1331],{"className":57979},[250],[18,57981,5600],{"className":57982},[106,110],[18,57984],{"className":57985,"style":228},[169],[18,57987,1340],{"className":57988},[232],[18,57990],{"className":57991,"style":228},[169],[18,57993,57995,57998,58038,58041,58044],{"className":57994},[97],[18,57996],{"className":57997,"style":40980},[101],[18,57999,58001,58004],{"className":58000},[106],[18,58002,57377],{"className":58003,"style":57398},[106,110],[18,58005,58007],{"className":58006},[115],[18,58008,58010,58030],{"className":58009},[119,120],[18,58011,58013,58027],{"className":58012},[124],[18,58014,58016],{"className":58015,"style":5921},[128],[18,58017,58018,58021],{"style":57413},[18,58019],{"className":58020,"style":137},[136],[18,58022,58024],{"className":58023},[141,142,143,144],[18,58025,5600],{"className":58026},[106,110,144],[18,58028,152],{"className":58029},[151],[18,58031,58033],{"className":58032},[124],[18,58034,58036],{"className":58035,"style":159},[128],[18,58037],{},[18,58039],{"className":58040,"style":228},[169],[18,58042,6684],{"className":58043},[232],[18,58045],{"className":58046,"style":228},[169],[18,58048,58050,58053,58056],{"className":58049},[97],[18,58051],{"className":58052,"style":242},[101],[18,58054,430],{"className":58055},[106],[18,58057,1349],{"className":58058},[267],"。两者都非空，且",[18,58061,58063],{"className":58062,"translate":22},[875],[18,58064,58066,58136],{"className":58065,"translate":22},[21],[18,58067,58069],{"className":58068},[26],[28,58070,58071],{"xmlns":30,"display":884},[32,58072,58073,58133],{},[35,58074,58075,58087,58093,58099,58101,58113,58115,58117,58123,58125,58131],{},[890,58076,58077,58079],{},[49,58078,20369],{},[35,58080,58081,58083,58085],{},[41,58082,5600],{},[49,58084,921],{},[41,58086,939],{},[38,58088,58089,58091],{},[41,58090,57377],{},[41,58092,5600],{},[38,58094,58095,58097],{},[41,58096,38582],{},[41,58098,5600],{},[49,58100,433],{},[890,58102,58103,58105],{},[49,58104,20369],{},[35,58106,58107,58109,58111],{},[41,58108,5600],{},[49,58110,921],{},[41,58112,70],{},[49,58114,74],{"stretchy":73},[49,58116,2253],{},[38,58118,58119,58121],{},[41,58120,57377],{},[41,58122,5600],{},[49,58124,84],{"stretchy":73},[38,58126,58127,58129],{},[41,58128,38582],{},[41,58130,5600],{},[41,58132,968],{"mathvariant":316},[86,58134,58135],{"encoding":88},"\\sum_{i\\in P}\\alpha_i x_i=\\sum_{i\\in N}(-\\alpha_i)x_i.",[18,58137,58139,58292],{"className":58138,"ariaHidden":51},[93],[18,58140,58142,58146,58200,58203,58243,58283,58286,58289],{"className":58141},[97],[18,58143],{"className":58144,"style":58145},[101],"height:2.3717em;vertical-align:-1.3217em;",[18,58147,58149],{"className":58148},[985,986],[18,58150,58152,58191],{"className":58151},[119,120],[18,58153,58155,58188],{"className":58154},[124],[18,58156,58158,58178],{"className":58157,"style":24483},[128],[18,58159,58160,58163],{"style":39167},[18,58161],{"className":58162,"style":20559},[136],[18,58164,58166],{"className":58165},[141,142,143,144],[18,58167,58169,58172,58175],{"className":58168},[106,144],[18,58170,5600],{"className":58171},[106,110,144],[18,58173,921],{"className":58174},[232,144],[18,58176,939],{"className":58177,"style":1171},[106,110,144],[18,58179,58180,58183],{"style":20577},[18,58181],{"className":58182,"style":20559},[136],[18,58184,58185],{},[18,58186,20369],{"className":58187},[985,4198,4199],[18,58189,152],{"className":58190},[151],[18,58192,58194],{"className":58193},[124],[18,58195,58198],{"className":58196,"style":58197},[128],"height:1.3217em;",[18,58199],{},[18,58201],{"className":58202,"style":170},[169],[18,58204,58206,58209],{"className":58205},[106],[18,58207,57377],{"className":58208,"style":57398},[106,110],[18,58210,58212],{"className":58211},[115],[18,58213,58215,58235],{"className":58214},[119,120],[18,58216,58218,58232],{"className":58217},[124],[18,58219,58221],{"className":58220,"style":5921},[128],[18,58222,58223,58226],{"style":57413},[18,58224],{"className":58225,"style":137},[136],[18,58227,58229],{"className":58228},[141,142,143,144],[18,58230,5600],{"className":58231},[106,110,144],[18,58233,152],{"className":58234},[151],[18,58236,58238],{"className":58237},[124],[18,58239,58241],{"className":58240,"style":159},[128],[18,58242],{},[18,58244,58246,58249],{"className":58245},[106],[18,58247,38582],{"className":58248},[106,110],[18,58250,58252],{"className":58251},[115],[18,58253,58255,58275],{"className":58254},[119,120],[18,58256,58258,58272],{"className":58257},[124],[18,58259,58261],{"className":58260,"style":5921},[128],[18,58262,58263,58266],{"style":1155},[18,58264],{"className":58265,"style":137},[136],[18,58267,58269],{"className":58268},[141,142,143,144],[18,58270,5600],{"className":58271},[106,110,144],[18,58273,152],{"className":58274},[151],[18,58276,58278],{"className":58277},[124],[18,58279,58281],{"className":58280,"style":159},[128],[18,58282],{},[18,58284],{"className":58285,"style":228},[169],[18,58287,433],{"className":58288},[232],[18,58290],{"className":58291,"style":228},[169],[18,58293,58295,58298,58351,58354,58357,58397,58400,58440],{"className":58294},[97],[18,58296],{"className":58297,"style":58145},[101],[18,58299,58301],{"className":58300},[985,986],[18,58302,58304,58343],{"className":58303},[119,120],[18,58305,58307,58340],{"className":58306},[124],[18,58308,58310,58330],{"className":58309,"style":24483},[128],[18,58311,58312,58315],{"style":39167},[18,58313],{"className":58314,"style":20559},[136],[18,58316,58318],{"className":58317},[141,142,143,144],[18,58319,58321,58324,58327],{"className":58320},[106,144],[18,58322,5600],{"className":58323},[106,110,144],[18,58325,921],{"className":58326},[232,144],[18,58328,70],{"className":58329,"style":246},[106,110,144],[18,58331,58332,58335],{"style":20577},[18,58333],{"className":58334,"style":20559},[136],[18,58336,58337],{},[18,58338,20369],{"className":58339},[985,4198,4199],[18,58341,152],{"className":58342},[151],[18,58344,58346],{"className":58345},[124],[18,58347,58349],{"className":58348,"style":58197},[128],[18,58350],{},[18,58352,74],{"className":58353},[250],[18,58355,2253],{"className":58356},[106],[18,58358,58360,58363],{"className":58359},[106],[18,58361,57377],{"className":58362,"style":57398},[106,110],[18,58364,58366],{"className":58365},[115],[18,58367,58369,58389],{"className":58368},[119,120],[18,58370,58372,58386],{"className":58371},[124],[18,58373,58375],{"className":58374,"style":5921},[128],[18,58376,58377,58380],{"style":57413},[18,58378],{"className":58379,"style":137},[136],[18,58381,58383],{"className":58382},[141,142,143,144],[18,58384,5600],{"className":58385},[106,110,144],[18,58387,152],{"className":58388},[151],[18,58390,58392],{"className":58391},[124],[18,58393,58395],{"className":58394,"style":159},[128],[18,58396],{},[18,58398,84],{"className":58399},[267],[18,58401,58403,58406],{"className":58402},[106],[18,58404,38582],{"className":58405},[106,110],[18,58407,58409],{"className":58408},[115],[18,58410,58412,58432],{"className":58411},[119,120],[18,58413,58415,58429],{"className":58414},[124],[18,58416,58418],{"className":58417,"style":5921},[128],[18,58419,58420,58423],{"style":1155},[18,58421],{"className":58422,"style":137},[136],[18,58424,58426],{"className":58425},[141,142,143,144],[18,58427,5600],{"className":58428},[106,110,144],[18,58430,152],{"className":58431},[151],[18,58433,58435],{"className":58434},[124],[18,58436,58438],{"className":58437,"style":159},[128],[18,58439],{},[18,58441,968],{"className":58442},[106],[11,58444,58445,58446,58499,58500,58528,58529,1990],{},"两边系数之和相同，记为 ",[18,58447,58449,58468],{"className":58448,"translate":22},[21],[18,58450,58452],{"className":58451},[26],[28,58453,58454],{"xmlns":30},[32,58455,58456,58465],{},[35,58457,58458,58461,58463],{},[41,58459,58460],{},"A",[49,58462,6885],{},[45,58464,430],{},[86,58466,58467],{"encoding":88},"A>0",[18,58469,58471,58490],{"className":58470,"ariaHidden":51},[93],[18,58472,58474,58478,58481,58484,58487],{"className":58473},[97],[18,58475],{"className":58476,"style":58477},[101],"height:0.7224em;vertical-align:-0.0391em;",[18,58479,58460],{"className":58480},[106,110],[18,58482],{"className":58483,"style":228},[169],[18,58485,6885],{"className":58486},[232],[18,58488],{"className":58489,"style":228},[169],[18,58491,58493,58496],{"className":58492},[97],[18,58494],{"className":58495,"style":505},[101],[18,58497,430],{"className":58498},[106],"。除以 ",[18,58501,58503,58516],{"className":58502,"translate":22},[21],[18,58504,58506],{"className":58505},[26],[28,58507,58508],{"xmlns":30},[32,58509,58510,58514],{},[35,58511,58512],{},[41,58513,58460],{},[86,58515,58460],{"encoding":88},[18,58517,58519],{"className":58518,"ariaHidden":51},[93],[18,58520,58522,58525],{"className":58521},[97],[18,58523],{"className":58524,"style":600},[101],[18,58526,58460],{"className":58527},[106,110],"，得到同一点的两个凸组合表示，于是 ",[18,58530,58532,58573],{"className":58531,"translate":22},[21],[18,58533,58535],{"className":58534},[26],[28,58536,58537],{"xmlns":30},[32,58538,58539,58570],{},[35,58540,58541,58544,58546,58548,58550,58552,58555,58557,58559,58561,58563,58565,58567],{},[41,58542,58543],{"mathvariant":316},"conv",[49,58545,899],{},[49,58547,74],{"stretchy":73},[41,58549,939],{},[49,58551,84],{"stretchy":73},[49,58553,58554],{},"∩",[41,58556,58543],{"mathvariant":316},[49,58558,899],{},[49,58560,74],{"stretchy":73},[41,58562,70],{},[49,58564,84],{"stretchy":73},[49,58566,2771],{"mathvariant":316},[41,58568,58569],{"mathvariant":316},"∅",[86,58571,58572],{"encoding":88},"\\operatorname{conv}(P)\\cap\\operatorname{conv}(N)\\neq\\varnothing",[18,58574,58576,58606,58669],{"className":58575,"ariaHidden":51},[93],[18,58577,58579,58582,58588,58591,58594,58597,58600,58603],{"className":58578},[97],[18,58580],{"className":58581,"style":242},[101],[18,58583,58585],{"className":58584},[985],[18,58586,58543],{"className":58587,"style":6355},[106,6354],[18,58589,74],{"className":58590},[250],[18,58592,939],{"className":58593,"style":1171},[106,110],[18,58595,84],{"className":58596},[267],[18,58598],{"className":58599,"style":2068},[169],[18,58601,58554],{"className":58602},[2072],[18,58604],{"className":58605,"style":2068},[169],[18,58607,58609,58612,58618,58621,58624,58627,58630,58666],{"className":58608},[97],[18,58610],{"className":58611,"style":242},[101],[18,58613,58615],{"className":58614},[985],[18,58616,58543],{"className":58617,"style":6355},[106,6354],[18,58619,74],{"className":58620},[250],[18,58622,70],{"className":58623,"style":246},[106,110],[18,58625,84],{"className":58626},[267],[18,58628],{"className":58629,"style":228},[169],[18,58631,58633,58660,58663],{"className":58632},[232],[18,58634,58636],{"className":58635},[232],[18,58637,58639],{"className":58638},[106,2878],[18,58640,58642],{"className":58641},[2882],[18,58643,58645,58648,58657],{"className":58644},[2886],[18,58646],{"className":58647,"style":2890},[101],[18,58649,58651],{"className":58650},[2894],[18,58652,58654],{"className":58653},[106],[18,58655,2901],{"className":58656},[232],[18,58658],{"className":58659},[2905],[18,58661],{"className":58662},[169,2909],[18,58664,433],{"className":58665},[232],[18,58667],{"className":58668,"style":228},[169],[18,58670,58672,58676],{"className":58671},[97],[18,58673],{"className":58674,"style":58675},[101],"height:0.6633em;vertical-align:-0.0817em;",[18,58677,58569],{"className":58678},[106,27260],[11,58680,58681,58682,58710,58711,58739,58740,58768,58769,58818,58819,58847,58848,58897,58898,58948,58949,1990],{},"现在取标注「",[18,58683,58685,58698],{"className":58684,"translate":22},[21],[18,58686,58688],{"className":58687},[26],[28,58689,58690],{"xmlns":30},[32,58691,58692,58696],{},[35,58693,58694],{},[41,58695,939],{},[86,58697,939],{"encoding":88},[18,58699,58701],{"className":58700,"ariaHidden":51},[93],[18,58702,58704,58707],{"className":58703},[97],[18,58705],{"className":58706,"style":600},[101],[18,58708,939],{"className":58709,"style":1171},[106,110]," 为正、",[18,58712,58714,58727],{"className":58713,"translate":22},[21],[18,58715,58717],{"className":58716},[26],[28,58718,58719],{"xmlns":30},[32,58720,58721,58725],{},[35,58722,58723],{},[41,58724,70],{},[86,58726,70],{"encoding":88},[18,58728,58730],{"className":58729,"ariaHidden":51},[93],[18,58731,58733,58736],{"className":58732},[97],[18,58734],{"className":58735,"style":600},[101],[18,58737,70],{"className":58738,"style":246},[106,110]," 为负」（系数为零的点任意归入一侧，它们不影响下面的凸组合）。任何包含 ",[18,58741,58743,58756],{"className":58742,"translate":22},[21],[18,58744,58746],{"className":58745},[26],[28,58747,58748],{"xmlns":30},[32,58749,58750,58754],{},[35,58751,58752],{},[41,58753,939],{},[86,58755,939],{"encoding":88},[18,58757,58759],{"className":58758,"ariaHidden":51},[93],[18,58760,58762,58765],{"className":58761},[97],[18,58763],{"className":58764,"style":600},[101],[18,58766,939],{"className":58767,"style":1171},[106,110]," 的半空间都包含 ",[18,58770,58772,58794],{"className":58771,"translate":22},[21],[18,58773,58775],{"className":58774},[26],[28,58776,58777],{"xmlns":30},[32,58778,58779,58791],{},[35,58780,58781,58783,58785,58787,58789],{},[41,58782,58543],{"mathvariant":316},[49,58784,899],{},[49,58786,74],{"stretchy":73},[41,58788,939],{},[49,58790,84],{"stretchy":73},[86,58792,58793],{"encoding":88},"\\operatorname{conv}(P)",[18,58795,58797],{"className":58796,"ariaHidden":51},[93],[18,58798,58800,58803,58809,58812,58815],{"className":58799},[97],[18,58801],{"className":58802,"style":242},[101],[18,58804,58806],{"className":58805},[985],[18,58807,58543],{"className":58808,"style":6355},[106,6354],[18,58810,74],{"className":58811},[250],[18,58813,939],{"className":58814,"style":1171},[106,110],[18,58816,84],{"className":58817},[267],"；任何不包含 ",[18,58820,58822,58835],{"className":58821,"translate":22},[21],[18,58823,58825],{"className":58824},[26],[28,58826,58827],{"xmlns":30},[32,58828,58829,58833],{},[35,58830,58831],{},[41,58832,70],{},[86,58834,70],{"encoding":88},[18,58836,58838],{"className":58837,"ariaHidden":51},[93],[18,58839,58841,58844],{"className":58840},[97],[18,58842],{"className":58843,"style":600},[101],[18,58845,70],{"className":58846,"style":246},[106,110]," 的半空间，其补集是凸的，因而也不包含 ",[18,58849,58851,58873],{"className":58850,"translate":22},[21],[18,58852,58854],{"className":58853},[26],[28,58855,58856],{"xmlns":30},[32,58857,58858,58870],{},[35,58859,58860,58862,58864,58866,58868],{},[41,58861,58543],{"mathvariant":316},[49,58863,899],{},[49,58865,74],{"stretchy":73},[41,58867,70],{},[49,58869,84],{"stretchy":73},[86,58871,58872],{"encoding":88},"\\operatorname{conv}(N)",[18,58874,58876],{"className":58875,"ariaHidden":51},[93],[18,58877,58879,58882,58888,58891,58894],{"className":58878},[97],[18,58880],{"className":58881,"style":242},[101],[18,58883,58885],{"className":58884},[985],[18,58886,58543],{"className":58887,"style":6355},[106,6354],[18,58889,74],{"className":58890},[250],[18,58892,70],{"className":58893,"style":246},[106,110],[18,58895,84],{"className":58896},[267],"。两者相交，矛盾。所以 ",[18,58899,58901,58918],{"className":58900,"translate":22},[21],[18,58902,58904],{"className":58903},[26],[28,58905,58906],{"xmlns":30},[32,58907,58908,58916],{},[35,58909,58910,58912,58914],{},[41,58911,4031],{},[49,58913,7987],{},[45,58915,2266],{},[86,58917,56999],{"encoding":88},[18,58919,58921,58939],{"className":58920,"ariaHidden":51},[93],[18,58922,58924,58927,58930,58933,58936],{"className":58923},[97],[18,58925],{"className":58926,"style":7224},[101],[18,58928,4031],{"className":58929},[106,110],[18,58931],{"className":58932,"style":2068},[169],[18,58934,7987],{"className":58935},[2072],[18,58937],{"className":58938,"style":2068},[169],[18,58940,58942,58945],{"className":58941},[97],[18,58943],{"className":58944,"style":505},[101],[18,58946,2266],{"className":58947},[106]," 个点无法被打散，",[18,58950,58952,58975],{"className":58951,"translate":22},[21],[18,58953,58955],{"className":58954},[26],[28,58956,58957],{"xmlns":30},[32,58958,58959,58973],{},[35,58960,58961,58963,58965,58967,58969,58971],{},[41,58962,52216],{"mathvariant":316},[49,58964,899],{},[49,58966,433],{},[41,58968,4031],{},[49,58970,7987],{},[45,58972,47],{},[86,58974,55761],{"encoding":88},[18,58976,58978,58999,59017],{"className":58977,"ariaHidden":51},[93],[18,58979,58981,58984,58990,58993,58996],{"className":58980},[97],[18,58982],{"className":58983,"style":295},[101],[18,58985,58987],{"className":58986},[985],[18,58988,52216],{"className":58989},[106,6354],[18,58991],{"className":58992,"style":228},[169],[18,58994,433],{"className":58995},[232],[18,58997],{"className":58998,"style":228},[169],[18,59000,59002,59005,59008,59011,59014],{"className":59001},[97],[18,59003],{"className":59004,"style":7224},[101],[18,59006,4031],{"className":59007},[106,110],[18,59009],{"className":59010,"style":2068},[169],[18,59012,7987],{"className":59013},[2072],[18,59015],{"className":59016,"style":2068},[169],[18,59018,59020,59023],{"className":59019},[97],[18,59021],{"className":59022,"style":505},[101],[18,59024,47],{"className":59025},[106],[11,59027,59028],{},"这个证明值得记住：它把「线性分类器的容量」归结为「仿射无关点的最大数目」，而后者正是线性代数意义上的维数加一。VC 维里的「维」字，在这里有了具体的所指。",[11,59030,59031],{},[717,59032,59033,59034,52828],{},"例 4（轴对齐矩形，",[18,59035,59037,59059],{"className":59036,"translate":22},[21],[18,59038,59040],{"className":59039},[26],[28,59041,59042],{"xmlns":30},[32,59043,59044,59056],{},[35,59045,59046,59048,59050,59052,59054],{},[41,59047,52216],{"mathvariant":316},[49,59049,899],{},[49,59051,433],{},[45,59053,2266],{},[41,59055,4031],{},[86,59057,59058],{"encoding":88},"\\operatorname{VCdim}=2d",[18,59060,59062,59083],{"className":59061,"ariaHidden":51},[93],[18,59063,59065,59068,59074,59077,59080],{"className":59064},[97],[18,59066],{"className":59067,"style":295},[101],[18,59069,59071],{"className":59070},[985],[18,59072,52216],{"className":59073},[106,6354],[18,59075],{"className":59076,"style":228},[169],[18,59078,433],{"className":59079},[232],[18,59081],{"className":59082,"style":228},[169],[18,59084,59086,59089,59092],{"className":59085},[97],[18,59087],{"className":59088,"style":295},[101],[18,59090,2266],{"className":59091},[106],[18,59093,4031],{"className":59094},[106,110],[11,59096,59097,19585,59125,59183,59184,59212],{},[18,59098,59100,59113],{"className":59099,"translate":22},[21],[18,59101,59103],{"className":59102},[26],[28,59104,59105],{"xmlns":30},[32,59106,59107,59111],{},[35,59108,59109],{},[41,59110,18309],{"mathvariant":1462},[86,59112,46317],{"encoding":88},[18,59114,59116],{"className":59115,"ariaHidden":51},[93],[18,59117,59119,59122],{"className":59118},[97],[18,59120],{"className":59121,"style":600},[101],[18,59123,18309],{"className":59124,"style":46330},[106,1478],[18,59126,59128,59145],{"className":59127,"translate":22},[21],[18,59129,59131],{"className":59130},[26],[28,59132,59133],{"xmlns":30},[32,59134,59135,59143],{},[35,59136,59137],{},[1556,59138,59139,59141],{},[41,59140,3071],{"mathvariant":933},[41,59142,4031],{},[86,59144,56939],{"encoding":88},[18,59146,59148],{"className":59147,"ariaHidden":51},[93],[18,59149,59151,59154],{"className":59150},[97],[18,59152],{"className":59153,"style":35186},[101],[18,59155,59157,59160],{"className":59156},[106],[18,59158,3071],{"className":59159},[106,1139],[18,59161,59163],{"className":59162},[115],[18,59164,59166],{"className":59165},[119],[18,59167,59169],{"className":59168},[124],[18,59170,59172],{"className":59171,"style":35186},[128],[18,59173,59174,59177],{"style":1962},[18,59175],{"className":59176,"style":137},[136],[18,59178,59180],{"className":59179},[141,142,143,144],[18,59181,4031],{"className":59182},[106,110,144]," 中所有轴对齐矩形（即 ",[18,59185,59187,59200],{"className":59186,"translate":22},[21],[18,59188,59190],{"className":59189},[26],[28,59191,59192],{"xmlns":30},[32,59193,59194,59198],{},[35,59195,59196],{},[41,59197,4031],{},[86,59199,4031],{"encoding":88},[18,59201,59203],{"className":59202,"ariaHidden":51},[93],[18,59204,59206,59209],{"className":59205},[97],[18,59207],{"className":59208,"style":295},[101],[18,59210,4031],{"className":59211},[106,110]," 个区间的乘积）的指示函数。",[11,59214,59215,59216,59272,59273,59301,59302,59330,59331,59401,59402,59430,59431,59500,59501,1990],{},"上界：任取 ",[18,59217,59219,59239],{"className":59218,"translate":22},[21],[18,59220,59222],{"className":59221},[26],[28,59223,59224],{"xmlns":30},[32,59225,59226,59236],{},[35,59227,59228,59230,59232,59234],{},[45,59229,2266],{},[41,59231,4031],{},[49,59233,7987],{},[45,59235,47],{},[86,59237,59238],{"encoding":88},"2d+1",[18,59240,59242,59263],{"className":59241,"ariaHidden":51},[93],[18,59243,59245,59248,59251,59254,59257,59260],{"className":59244},[97],[18,59246],{"className":59247,"style":7224},[101],[18,59249,2266],{"className":59250},[106],[18,59252,4031],{"className":59253},[106,110],[18,59255],{"className":59256,"style":2068},[169],[18,59258,7987],{"className":59259},[2072],[18,59261],{"className":59262,"style":2068},[169],[18,59264,59266,59269],{"className":59265},[97],[18,59267],{"className":59268,"style":505},[101],[18,59270,47],{"className":59271},[106]," 个点。若存在一点 ",[18,59274,59276,59289],{"className":59275,"translate":22},[21],[18,59277,59279],{"className":59278},[26],[28,59280,59281],{"xmlns":30},[32,59282,59283,59287],{},[35,59284,59285],{},[41,59286,11],{},[86,59288,11],{"encoding":88},[18,59290,59292],{"className":59291,"ariaHidden":51},[93],[18,59293,59295,59298],{"className":59294},[97],[18,59296],{"className":59297,"style":38552},[101],[18,59299,11],{"className":59300},[106,110],"，使它在每个坐标上都不是严格极值——即对每个坐标 ",[18,59303,59305,59318],{"className":59304,"translate":22},[21],[18,59306,59308],{"className":59307},[26],[28,59309,59310],{"xmlns":30},[32,59311,59312,59316],{},[35,59313,59314],{},[41,59315,19811],{},[86,59317,19811],{"encoding":88},[18,59319,59321],{"className":59320,"ariaHidden":51},[93],[18,59322,59324,59327],{"className":59323},[97],[18,59325],{"className":59326,"style":28340},[101],[18,59328,19811],{"className":59329,"style":19928},[106,110],"，都有别的点在 ",[18,59332,59334,59352],{"className":59333,"translate":22},[21],[18,59335,59337],{"className":59336},[26],[28,59338,59339],{"xmlns":30},[32,59340,59341,59349],{},[35,59342,59343],{},[38,59344,59345,59347],{},[41,59346,11],{},[41,59348,19811],{},[86,59350,59351],{"encoding":88},"p_j",[18,59353,59355],{"className":59354,"ariaHidden":51},[93],[18,59356,59358,59361],{"className":59357},[97],[18,59359],{"className":59360,"style":30679},[101],[18,59362,59364,59367],{"className":59363},[106],[18,59365,11],{"className":59366},[106,110],[18,59368,59370],{"className":59369},[115],[18,59371,59373,59393],{"className":59372},[119,120],[18,59374,59376,59390],{"className":59375},[124],[18,59377,59379],{"className":59378,"style":5921},[128],[18,59380,59381,59384],{"style":1155},[18,59382],{"className":59383,"style":137},[136],[18,59385,59387],{"className":59386},[141,142,143,144],[18,59388,19811],{"className":59389,"style":19928},[106,110,144],[18,59391,152],{"className":59392},[151],[18,59394,59396],{"className":59395},[124],[18,59397,59399],{"className":59398,"style":19938},[128],[18,59400],{}," 的左边和右边——那么标注「",[18,59403,59405,59418],{"className":59404,"translate":22},[21],[18,59406,59408],{"className":59407},[26],[28,59409,59410],{"xmlns":30},[32,59411,59412,59416],{},[35,59413,59414],{},[41,59415,11],{},[86,59417,11],{"encoding":88},[18,59419,59421],{"className":59420,"ariaHidden":51},[93],[18,59422,59424,59427],{"className":59423},[97],[18,59425],{"className":59426,"style":38552},[101],[18,59428,11],{"className":59429},[106,110]," 为负、其余为正」无法实现：任何包含其余所有点的矩形，在每个坐标上的区间都必须覆盖 ",[18,59432,59434,59451],{"className":59433,"translate":22},[21],[18,59435,59437],{"className":59436},[26],[28,59438,59439],{"xmlns":30},[32,59440,59441,59449],{},[35,59442,59443],{},[38,59444,59445,59447],{},[41,59446,11],{},[41,59448,19811],{},[86,59450,59351],{"encoding":88},[18,59452,59454],{"className":59453,"ariaHidden":51},[93],[18,59455,59457,59460],{"className":59456},[97],[18,59458],{"className":59459,"style":30679},[101],[18,59461,59463,59466],{"className":59462},[106],[18,59464,11],{"className":59465},[106,110],[18,59467,59469],{"className":59468},[115],[18,59470,59472,59492],{"className":59471},[119,120],[18,59473,59475,59489],{"className":59474},[124],[18,59476,59478],{"className":59477,"style":5921},[128],[18,59479,59480,59483],{"style":1155},[18,59481],{"className":59482,"style":137},[136],[18,59484,59486],{"className":59485},[141,142,143,144],[18,59487,19811],{"className":59488,"style":19928},[106,110,144],[18,59490,152],{"className":59491},[151],[18,59493,59495],{"className":59494},[124],[18,59496,59498],{"className":59497,"style":19938},[128],[18,59499],{}," 两侧的值，因而必然包含 ",[18,59502,59504,59517],{"className":59503,"translate":22},[21],[18,59505,59507],{"className":59506},[26],[28,59508,59509],{"xmlns":30},[32,59510,59511,59515],{},[35,59512,59513],{},[41,59514,11],{},[86,59516,11],{"encoding":88},[18,59518,59520],{"className":59519,"ariaHidden":51},[93],[18,59521,59523,59526],{"className":59522},[97],[18,59524],{"className":59525,"style":38552},[101],[18,59527,11],{"className":59528},[106,110],[11,59530,59531,59532,59560,59561,59589,59590,59618,59619,59653,59654,59709,59710,59743],{},"这样的 ",[18,59533,59535,59548],{"className":59534,"translate":22},[21],[18,59536,59538],{"className":59537},[26],[28,59539,59540],{"xmlns":30},[32,59541,59542,59546],{},[35,59543,59544],{},[41,59545,11],{},[86,59547,11],{"encoding":88},[18,59549,59551],{"className":59550,"ariaHidden":51},[93],[18,59552,59554,59557],{"className":59553},[97],[18,59555],{"className":59556,"style":38552},[101],[18,59558,11],{"className":59559},[106,110]," 一定存在。否则每个点都在某个坐标上严格小于所有其他点、或严格大于所有其他点。把「坐标 ",[18,59562,59564,59577],{"className":59563,"translate":22},[21],[18,59565,59567],{"className":59566},[26],[28,59568,59569],{"xmlns":30},[32,59570,59571,59575],{},[35,59572,59573],{},[41,59574,19811],{},[86,59576,19811],{"encoding":88},[18,59578,59580],{"className":59579,"ariaHidden":51},[93],[18,59581,59583,59586],{"className":59582},[97],[18,59584],{"className":59585,"style":28340},[101],[18,59587,19811],{"className":59588,"style":19928},[106,110]," 的最小值」和「坐标 ",[18,59591,59593,59606],{"className":59592,"translate":22},[21],[18,59594,59596],{"className":59595},[26],[28,59597,59598],{"xmlns":30},[32,59599,59600,59604],{},[35,59601,59602],{},[41,59603,19811],{},[86,59605,19811],{"encoding":88},[18,59607,59609],{"className":59608,"ariaHidden":51},[93],[18,59610,59612,59615],{"className":59611},[97],[18,59613],{"className":59614,"style":28340},[101],[18,59616,19811],{"className":59617,"style":19928},[106,110]," 的最大值」看作 ",[18,59620,59622,59638],{"className":59621,"translate":22},[21],[18,59623,59625],{"className":59624},[26],[28,59626,59627],{"xmlns":30},[32,59628,59629,59635],{},[35,59630,59631,59633],{},[45,59632,2266],{},[41,59634,4031],{},[86,59636,59637],{"encoding":88},"2d",[18,59639,59641],{"className":59640,"ariaHidden":51},[93],[18,59642,59644,59647,59650],{"className":59643},[97],[18,59645],{"className":59646,"style":295},[101],[18,59648,2266],{"className":59649},[106],[18,59651,4031],{"className":59652},[106,110]," 个位置，每个点占据其中一个；若两个点占据同一个位置，立刻矛盾（不可能同时严格小于对方）。",[18,59655,59657,59676],{"className":59656,"translate":22},[21],[18,59658,59660],{"className":59659},[26],[28,59661,59662],{"xmlns":30},[32,59663,59664,59674],{},[35,59665,59666,59668,59670,59672],{},[45,59667,2266],{},[41,59669,4031],{},[49,59671,7987],{},[45,59673,47],{},[86,59675,59238],{"encoding":88},[18,59677,59679,59700],{"className":59678,"ariaHidden":51},[93],[18,59680,59682,59685,59688,59691,59694,59697],{"className":59681},[97],[18,59683],{"className":59684,"style":7224},[101],[18,59686,2266],{"className":59687},[106],[18,59689,4031],{"className":59690},[106,110],[18,59692],{"className":59693,"style":2068},[169],[18,59695,7987],{"className":59696},[2072],[18,59698],{"className":59699,"style":2068},[169],[18,59701,59703,59706],{"className":59702},[97],[18,59704],{"className":59705,"style":505},[101],[18,59707,47],{"className":59708},[106]," 个点放进 ",[18,59711,59713,59728],{"className":59712,"translate":22},[21],[18,59714,59716],{"className":59715},[26],[28,59717,59718],{"xmlns":30},[32,59719,59720,59726],{},[35,59721,59722,59724],{},[45,59723,2266],{},[41,59725,4031],{},[86,59727,59637],{"encoding":88},[18,59729,59731],{"className":59730,"ariaHidden":51},[93],[18,59732,59734,59737,59740],{"className":59733},[97],[18,59735],{"className":59736,"style":295},[101],[18,59738,2266],{"className":59739},[106],[18,59741,4031],{"className":59742},[106,110]," 个位置，必有一对冲突。所以上界成立。",[11,59745,59746,59747,59780,59781,749,59832,59992,59993,1990],{},"下界是初等的但需要一点构造：确实存在 ",[18,59748,59750,59765],{"className":59749,"translate":22},[21],[18,59751,59753],{"className":59752},[26],[28,59754,59755],{"xmlns":30},[32,59756,59757,59763],{},[35,59758,59759,59761],{},[45,59760,2266],{},[41,59762,4031],{},[86,59764,59637],{"encoding":88},[18,59766,59768],{"className":59767,"ariaHidden":51},[93],[18,59769,59771,59774,59777],{"className":59770},[97],[18,59772],{"className":59773,"style":295},[101],[18,59775,2266],{"className":59776},[106],[18,59778,4031],{"className":59779},[106,110]," 个点可以被矩形打散（",[18,59782,59784,59802],{"className":59783,"translate":22},[21],[18,59785,59787],{"className":59786},[26],[28,59788,59789],{"xmlns":30},[32,59790,59791,59799],{},[35,59792,59793,59795,59797],{},[41,59794,4031],{},[49,59796,433],{},[45,59798,2266],{},[86,59800,59801],{"encoding":88},"d=2",[18,59803,59805,59823],{"className":59804,"ariaHidden":51},[93],[18,59806,59808,59811,59814,59817,59820],{"className":59807},[97],[18,59809],{"className":59810,"style":295},[101],[18,59812,4031],{"className":59813},[106,110],[18,59815],{"className":59816,"style":228},[169],[18,59818,433],{"className":59819},[232],[18,59821],{"className":59822,"style":228},[169],[18,59824,59826,59829],{"className":59825},[97],[18,59827],{"className":59828,"style":505},[101],[18,59830,2266],{"className":59831},[106],[18,59833,59835,59893],{"className":59834,"translate":22},[21],[18,59836,59838],{"className":59837},[26],[28,59839,59840],{"xmlns":30},[32,59841,59842,59890],{},[35,59843,59844,59846,59848,59850,59852,59854,59856,59858,59860,59862,59864,59866,59868,59870,59872,59874,59876,59878,59880,59882,59884,59886,59888],{},[49,59845,74],{"stretchy":73},[45,59847,47],{},[49,59849,52],{"separator":51},[45,59851,2266],{},[49,59853,84],{"stretchy":73},[49,59855,52],{"separator":51},[49,59857,74],{"stretchy":73},[45,59859,2266],{},[49,59861,52],{"separator":51},[45,59863,10447],{},[49,59865,84],{"stretchy":73},[49,59867,52],{"separator":51},[49,59869,74],{"stretchy":73},[45,59871,44708],{},[49,59873,52],{"separator":51},[45,59875,47],{},[49,59877,84],{"stretchy":73},[49,59879,52],{"separator":51},[49,59881,74],{"stretchy":73},[45,59883,10447],{},[49,59885,52],{"separator":51},[45,59887,44708],{},[49,59889,84],{"stretchy":73},[86,59891,59892],{"encoding":88},"(1,2),(2,4),(3,1),(4,3)",[18,59894,59896],{"className":59895,"ariaHidden":51},[93],[18,59897,59899,59902,59905,59908,59911,59914,59917,59920,59923,59926,59929,59932,59935,59938,59941,59944,59947,59950,59953,59956,59959,59962,59965,59968,59971,59974,59977,59980,59983,59986,59989],{"className":59898},[97],[18,59900],{"className":59901,"style":242},[101],[18,59903,74],{"className":59904},[250],[18,59906,47],{"className":59907},[106],[18,59909,52],{"className":59910},[165],[18,59912],{"className":59913,"style":170},[169],[18,59915,2266],{"className":59916},[106],[18,59918,84],{"className":59919},[267],[18,59921,52],{"className":59922},[165],[18,59924],{"className":59925,"style":170},[169],[18,59927,74],{"className":59928},[250],[18,59930,2266],{"className":59931},[106],[18,59933,52],{"className":59934},[165],[18,59936],{"className":59937,"style":170},[169],[18,59939,10447],{"className":59940},[106],[18,59942,84],{"className":59943},[267],[18,59945,52],{"className":59946},[165],[18,59948],{"className":59949,"style":170},[169],[18,59951,74],{"className":59952},[250],[18,59954,44708],{"className":59955},[106],[18,59957,52],{"className":59958},[165],[18,59960],{"className":59961,"style":170},[169],[18,59963,47],{"className":59964},[106],[18,59966,84],{"className":59967},[267],[18,59969,52],{"className":59970},[165],[18,59972],{"className":59973,"style":170},[169],[18,59975,74],{"className":59976},[250],[18,59978,10447],{"className":59979},[106],[18,59981,52],{"className":59982},[165],[18,59984],{"className":59985,"style":170},[169],[18,59987,44708],{"className":59988},[106],[18,59990,84],{"className":59991},[267]," 这四个点就可以；注意不能取正方形的四个角）。于是 ",[18,59994,59996,60017],{"className":59995,"translate":22},[21],[18,59997,59999],{"className":59998},[26],[28,60000,60001],{"xmlns":30},[32,60002,60003,60015],{},[35,60004,60005,60007,60009,60011,60013],{},[41,60006,52216],{"mathvariant":316},[49,60008,899],{},[49,60010,433],{},[45,60012,2266],{},[41,60014,4031],{},[86,60016,59058],{"encoding":88},[18,60018,60020,60041],{"className":60019,"ariaHidden":51},[93],[18,60021,60023,60026,60032,60035,60038],{"className":60022},[97],[18,60024],{"className":60025,"style":295},[101],[18,60027,60029],{"className":60028},[985],[18,60030,52216],{"className":60031},[106,6354],[18,60033],{"className":60034,"style":228},[169],[18,60036,433],{"className":60037},[232],[18,60039],{"className":60040,"style":228},[169],[18,60042,60044,60047,60050],{"className":60043},[97],[18,60045],{"className":60046,"style":295},[101],[18,60048,2266],{"className":60049},[106],[18,60051,4031],{"className":60052},[106,110],[11,60054,60055],{},[717,60056,60057,60058,52828],{},"例 5（有限类，",[18,60059,60061,60095],{"className":60060,"translate":22},[21],[18,60062,60064],{"className":60063},[26],[28,60065,60066],{"xmlns":30},[32,60067,60068,60092],{},[35,60069,60070,60072,60074,60076,60086,60088,60090],{},[41,60071,52216],{"mathvariant":316},[49,60073,899],{},[49,60075,4334],{},[38,60077,60078,60084],{},[35,60079,60080,60082],{},[41,60081,14733],{},[49,60083,899],{},[45,60085,2266],{},[41,60087,11225],{"mathvariant":316},[41,60089,18309],{"mathvariant":1462},[41,60091,11225],{"mathvariant":316},[86,60093,60094],{"encoding":88},"\\operatorname{VCdim}\\le\\log_2|\\mathcal H|",[18,60096,60098,60119],{"className":60097,"ariaHidden":51},[93],[18,60099,60101,60104,60110,60113,60116],{"className":60100},[97],[18,60102],{"className":60103,"style":52357},[101],[18,60105,60107],{"className":60106},[985],[18,60108,52216],{"className":60109},[106,6354],[18,60111],{"className":60112,"style":228},[169],[18,60114,4334],{"className":60115},[232],[18,60117],{"className":60118,"style":228},[169],[18,60120,60122,60125,60170,60173,60176,60179],{"className":60121},[97],[18,60123],{"className":60124,"style":242},[101],[18,60126,60128,60133],{"className":60127},[985],[18,60129,15174,60131],{"className":60130},[985],[18,60132,15177],{"style":6355},[18,60134,60136],{"className":60135},[115],[18,60137,60139,60161],{"className":60138},[119,120],[18,60140,60142,60158],{"className":60141},[124],[18,60143,60146],{"className":60144,"style":60145},[128],"height:0.207em;",[18,60147,60149,60152],{"style":60148},"top:-2.4559em;margin-right:0.05em;",[18,60150],{"className":60151,"style":137},[136],[18,60153,60155],{"className":60154},[141,142,143,144],[18,60156,2266],{"className":60157},[106,144],[18,60159,152],{"className":60160},[151],[18,60162,60164],{"className":60163},[124],[18,60165,60168],{"className":60166,"style":60167},[128],"height:0.2441em;",[18,60169],{},[18,60171],{"className":60172,"style":170},[169],[18,60174,11225],{"className":60175},[106],[18,60177,18309],{"className":60178,"style":46330},[106,1478],[18,60180,11225],{"className":60181},[106],[11,60183,21790,60184,60212,60213,15558,60241,39557,60425,1990],{},[18,60185,60187,60200],{"className":60186,"translate":22},[21],[18,60188,60190],{"className":60189},[26],[28,60191,60192],{"xmlns":30},[32,60193,60194,60198],{},[35,60195,60196],{},[41,60197,18309],{"mathvariant":1462},[86,60199,46317],{"encoding":88},[18,60201,60203],{"className":60202,"ariaHidden":51},[93],[18,60204,60206,60209],{"className":60205},[97],[18,60207],{"className":60208,"style":600},[101],[18,60210,18309],{"className":60211,"style":46330},[106,1478]," 有限且能打散 ",[18,60214,60216,60229],{"className":60215,"translate":22},[21],[18,60217,60219],{"className":60218},[26],[28,60220,60221],{"xmlns":30},[32,60222,60223,60227],{},[35,60224,60225],{},[41,60226,51021],{},[86,60228,51021],{"encoding":88},[18,60230,60232],{"className":60231,"ariaHidden":51},[93],[18,60233,60235,60238],{"className":60234},[97],[18,60236],{"className":60237,"style":600},[101],[18,60239,51021],{"className":60240,"style":5073},[106,110],[18,60242,60244,60290],{"className":60243,"translate":22},[21],[18,60245,60247],{"className":60246},[26],[28,60248,60249],{"xmlns":30},[32,60250,60251,60287],{},[35,60252,60253,60265,60267,60269,60271,60277,60279,60281,60283,60285],{},[1556,60254,60255,60257],{},[45,60256,2266],{},[35,60258,60259,60261,60263],{},[41,60260,11225],{"mathvariant":316},[41,60262,51021],{},[41,60264,11225],{"mathvariant":316},[49,60266,433],{},[41,60268,11225],{"mathvariant":316},[41,60270,18309],{"mathvariant":1462},[38,60272,60273,60275],{},[41,60274,11225],{"mathvariant":316},[41,60276,51021],{},[41,60278,11225],{"mathvariant":316},[49,60280,4334],{},[41,60282,11225],{"mathvariant":316},[41,60284,18309],{"mathvariant":1462},[41,60286,11225],{"mathvariant":316},[86,60288,60289],{"encoding":88},"2^{|C|}=|\\mathcal H|_C|\\le|\\mathcal H|",[18,60291,60293,60346,60410],{"className":60292,"ariaHidden":51},[93],[18,60294,60296,60299,60337,60340,60343],{"className":60295},[97],[18,60297],{"className":60298,"style":27945},[101],[18,60300,60302,60305],{"className":60301},[106],[18,60303,2266],{"className":60304},[106],[18,60306,60308],{"className":60307},[115],[18,60309,60311],{"className":60310},[119],[18,60312,60314],{"className":60313},[124],[18,60315,60317],{"className":60316,"style":27945},[128],[18,60318,60319,60322],{"style":1962},[18,60320],{"className":60321,"style":137},[136],[18,60323,60325],{"className":60324},[141,142,143,144],[18,60326,60328,60331,60334],{"className":60327},[106,144],[18,60329,11225],{"className":60330},[106,144],[18,60332,51021],{"className":60333,"style":5073},[106,110,144],[18,60335,11225],{"className":60336},[106,144],[18,60338],{"className":60339,"style":228},[169],[18,60341,433],{"className":60342},[232],[18,60344],{"className":60345,"style":228},[169],[18,60347,60349,60352,60355,60358,60398,60401,60404,60407],{"className":60348},[97],[18,60350],{"className":60351,"style":242},[101],[18,60353,11225],{"className":60354},[106],[18,60356,18309],{"className":60357,"style":46330},[106,1478],[18,60359,60361,60364],{"className":60360},[106],[18,60362,11225],{"className":60363},[106],[18,60365,60367],{"className":60366},[115],[18,60368,60370,60390],{"className":60369},[119,120],[18,60371,60373,60387],{"className":60372},[124],[18,60374,60376],{"className":60375,"style":1152},[128],[18,60377,60378,60381],{"style":1155},[18,60379],{"className":60380,"style":137},[136],[18,60382,60384],{"className":60383},[141,142,143,144],[18,60385,51021],{"className":60386,"style":5073},[106,110,144],[18,60388,152],{"className":60389},[151],[18,60391,60393],{"className":60392},[124],[18,60394,60396],{"className":60395,"style":159},[128],[18,60397],{},[18,60399,11225],{"className":60400},[106],[18,60402],{"className":60403,"style":228},[169],[18,60405,4334],{"className":60406},[232],[18,60408],{"className":60409,"style":228},[169],[18,60411,60413,60416,60419,60422],{"className":60412},[97],[18,60414],{"className":60415,"style":242},[101],[18,60417,11225],{"className":60418},[106],[18,60420,18309],{"className":60421,"style":46330},[106,1478],[18,60423,11225],{"className":60424},[106],[18,60426,60428,60464],{"className":60427,"translate":22},[21],[18,60429,60431],{"className":60430},[26],[28,60432,60433],{"xmlns":30},[32,60434,60435,60461],{},[35,60436,60437,60439,60441,60443,60445,60455,60457,60459],{},[41,60438,11225],{"mathvariant":316},[41,60440,51021],{},[41,60442,11225],{"mathvariant":316},[49,60444,4334],{},[38,60446,60447,60453],{},[35,60448,60449,60451],{},[41,60450,14733],{},[49,60452,899],{},[45,60454,2266],{},[41,60456,11225],{"mathvariant":316},[41,60458,18309],{"mathvariant":1462},[41,60460,11225],{"mathvariant":316},[86,60462,60463],{"encoding":88},"|C|\\le\\log_2|\\mathcal H|",[18,60465,60467,60491],{"className":60466,"ariaHidden":51},[93],[18,60468,60470,60473,60476,60479,60482,60485,60488],{"className":60469},[97],[18,60471],{"className":60472,"style":242},[101],[18,60474,11225],{"className":60475},[106],[18,60477,51021],{"className":60478,"style":5073},[106,110],[18,60480,11225],{"className":60481},[106],[18,60483],{"className":60484,"style":228},[169],[18,60486,4334],{"className":60487},[232],[18,60489],{"className":60490,"style":228},[169],[18,60492,60494,60497,60539,60542,60545,60548],{"className":60493},[97],[18,60495],{"className":60496,"style":242},[101],[18,60498,60500,60505],{"className":60499},[985],[18,60501,15174,60503],{"className":60502},[985],[18,60504,15177],{"style":6355},[18,60506,60508],{"className":60507},[115],[18,60509,60511,60531],{"className":60510},[119,120],[18,60512,60514,60528],{"className":60513},[124],[18,60515,60517],{"className":60516,"style":60145},[128],[18,60518,60519,60522],{"style":60148},[18,60520],{"className":60521,"style":137},[136],[18,60523,60525],{"className":60524},[141,142,143,144],[18,60526,2266],{"className":60527},[106,144],[18,60529,152],{"className":60530},[151],[18,60532,60534],{"className":60533},[124],[18,60535,60537],{"className":60536,"style":60167},[128],[18,60538],{},[18,60540],{"className":60541,"style":170},[169],[18,60543,11225],{"className":60544},[106],[18,60546,18309],{"className":60547,"style":46330},[106,1478],[18,60549,11225],{"className":60550},[106],[11,60552,60553],{},[717,60554,60555,60556,52828],{},"例 6（正弦类，",[18,60557,60559,60579],{"className":60558,"translate":22},[21],[18,60560,60562],{"className":60561},[26],[28,60563,60564],{"xmlns":30},[32,60565,60566,60576],{},[35,60567,60568,60570,60572,60574],{},[41,60569,52216],{"mathvariant":316},[49,60571,899],{},[49,60573,433],{},[41,60575,2028],{"mathvariant":316},[86,60577,60578],{"encoding":88},"\\operatorname{VCdim}=\\infty",[18,60580,60582,60603],{"className":60581,"ariaHidden":51},[93],[18,60583,60585,60588,60594,60597,60600],{"className":60584},[97],[18,60586],{"className":60587,"style":295},[101],[18,60589,60591],{"className":60590},[985],[18,60592,52216],{"className":60593},[106,6354],[18,60595],{"className":60596,"style":228},[169],[18,60598,433],{"className":60599},[232],[18,60601],{"className":60602,"style":228},[169],[18,60604,60606,60609],{"className":60605},[97],[18,60607],{"className":60608,"style":1083},[101],[18,60610,2028],{"className":60611},[106],[11,60613,36192,60614,4741,60780,60830,60831,60859,60860,60993,60994,28344],{},[18,60615,60617,60669],{"className":60616,"translate":22},[21],[18,60618,60620],{"className":60619},[26],[28,60621,60622],{"xmlns":30},[32,60623,60624,60666],{},[35,60625,60626,60628,60630,60632,60634,60636,60638,60640,60642,60644,60646,60648,60650,60652,60654,60656,60658,60660,60662,60664],{},[41,60627,18309],{"mathvariant":1462},[49,60629,433],{},[49,60631,1331],{"stretchy":73},[41,60633,38582],{},[49,60635,52853],{},[41,60637,45639],{"mathvariant":316},[49,60639,899],{},[49,60641,74],{"stretchy":73},[41,60643,45646],{},[49,60645,899],{},[49,60647,74],{"stretchy":73},[41,60649,77],{},[41,60651,38582],{},[49,60653,84],{"stretchy":73},[49,60655,84],{"stretchy":73},[49,60657,1340],{},[41,60659,77],{},[49,60661,6885],{},[45,60663,430],{},[49,60665,1349],{"stretchy":73},[86,60667,60668],{"encoding":88},"\\mathcal H=\\{x\\mapsto\\operatorname{sign}(\\sin(\\theta 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",[18,60832,60834,60847],{"className":60833,"translate":22},[21],[18,60835,60837],{"className":60836},[26],[28,60838,60839],{"xmlns":30},[32,60840,60841,60845],{},[35,60842,60843],{},[41,60844,28169],{},[86,60846,28169],{"encoding":88},[18,60848,60850],{"className":60849,"ariaHidden":51},[93],[18,60851,60853,60856],{"className":60852},[97],[18,60854],{"className":60855,"style":1083},[101],[18,60857,28169],{"className":60858},[106,110],"，取 ",[18,60861,60863,60893],{"className":60862,"translate":22},[21],[18,60864,60866],{"className":60865},[26],[28,60867,60868],{"xmlns":30},[32,60869,60870,60890],{},[35,60871,60872,60878,60880],{},[38,60873,60874,60876],{},[41,60875,38582],{},[41,60877,5600],{},[49,60879,433],{},[1556,60881,60882,60884],{},[45,60883,2266],{},[35,60885,60886,60888],{},[49,60887,2253],{},[41,60889,5600],{},[86,60891,60892],{"encoding":88},"x_i=2^{-i}",[18,60894,60896,60951],{"className":60895,"ariaHidden":51},[93],[18,60897,60899,60902,60942,60945,60948],{"className":60898},[97],[18,60900],{"className":60901,"style":16897},[101],[18,60903,60905,60908],{"className":60904},[106],[18,60906,38582],{"className":60907},[106,110],[18,60909,60911],{"className":60910},[115],[18,60912,60914,60934],{"className":60913},[119,120],[18,60915,60917,60931],{"className":60916},[124],[18,60918,60920],{"className":60919,"style":5921},[128],[18,60921,60922,60925],{"style":1155},[18,60923],{"className":60924,"style":137},[136],[18,60926,60928],{"className":60927},[141,142,143,144],[18,60929,5600],{"className":60930},[106,110,144],[18,60932,152],{"className":60933},[151],[18,60935,60937],{"className":60936},[124],[18,60938,60940],{"className":60939,"style":159},[128],[18,60941],{},[18,60943],{"className":60944,"style":228},[169],[18,60946,433],{"className":60947},[232],[18,60949],{"className":60950,"style":228},[169],[18,60952,60954,60958],{"className":60953},[97],[18,60955],{"className":60956,"style":60957},[101],"height:0.8247em;",[18,60959,60961,60964],{"className":60960},[106],[18,60962,2266],{"className":60963},[106],[18,60965,60967],{"className":60966},[115],[18,60968,60970],{"className":60969},[119],[18,60971,60973],{"className":60972},[124],[18,60974,60976],{"className":60975,"style":60957},[128],[18,60977,60978,60981],{"style":1962},[18,60979],{"className":60980,"style":137},[136],[18,60982,60984],{"className":60983},[141,142,143,144],[18,60985,60987,60990],{"className":60986},[106,144],[18,60988,2253],{"className":60989},[106,144],[18,60991,5600],{"className":60992},[106,110,144],"，给定任意标注 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取模。",[18,61568,61570,61588],{"className":61569,"translate":22},[21],[18,61571,61573],{"className":61572},[26],[28,61574,61575],{"xmlns":30},[32,61576,61577,61585],{},[35,61578,61579,61581,61583],{},[41,61580,19811],{},[49,61582,6885],{},[41,61584,5600],{},[86,61586,61587],{"encoding":88},"j>i",[18,61589,61591,61609],{"className":61590,"ariaHidden":51},[93],[18,61592,61594,61597,61600,61603,61606],{"className":61593},[97],[18,61595],{"className":61596,"style":28340},[101],[18,61598,19811],{"className":61599,"style":19928},[106,110],[18,61601],{"className":61602,"style":228},[169],[18,61604,6885],{"className":61605},[232],[18,61607],{"className":61608,"style":228},[169],[18,61610,61612,61615],{"className":61611},[97],[18,61613],{"className":61614,"style":61452},[101],[18,61616,5600],{"className":61617},[106,110]," 的项贡献 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的整数倍，可以丢掉；",[18,61654,61656,61674],{"className":61655,"translate":22},[21],[18,61657,61659],{"className":61658},[26],[28,61660,61661],{"xmlns":30},[32,61662,61663,61671],{},[35,61664,61665,61667,61669],{},[41,61666,19811],{},[49,61668,433],{},[41,61670,5600],{},[86,61672,61673],{"encoding":88},"j=i",[18,61675,61677,61695],{"className":61676,"ariaHidden":51},[93],[18,61678,61680,61683,61686,61689,61692],{"className":61679},[97],[18,61681],{"className":61682,"style":28340},[101],[18,61684,19811],{"className":61685,"style":19928},[106,110],[18,61687],{"className":61688,"style":228},[169],[18,61690,433],{"className":61691},[232],[18,61693],{"className":61694,"style":228},[169],[18,61696,61698,61701],{"className":61697},[97],[18,61699],{"className":61700,"style":61452},[101],[18,61702,5600],{"className":61703},[106,110],[18,61705,61707,61720],{"className":61706,"translate":22},[21],[18,61708,61710],{"className":61709},[26],[28,61711,61712],{"xmlns":30},[32,61713,61714,61718],{},[35,61715,61716],{},[41,61717,3941],{},[86,61719,3944],{"encoding":88},[18,61721,61723],{"className":61722,"ariaHidden":51},[93],[18,61724,61726,61729],{"className":61725},[97],[18,61727],{"className":61728,"style":1083},[101],[18,61730,3941],{"className":61731,"style":3957},[106,110],"（当 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的项合并为 ",[18,61884,61886,61914],{"className":61885,"translate":22},[21],[18,61887,61889],{"className":61888},[26],[28,61890,61891],{"xmlns":30},[32,61892,61893,61911],{},[35,61894,61895,61897,61903,61905],{},[41,61896,3941],{},[38,61898,61899,61901],{},[41,61900,39964],{},[41,61902,5600],{},[41,61904,317],{"mathvariant":316},[1556,61906,61907,61909],{},[45,61908,2266],{},[41,61910,5600],{},[86,61912,61913],{"encoding":88},"\\pi 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k_i\\le 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x_i\\equiv \\pi\\left(2^{-i-1}+\\mathbf 1[y_i=-1]+\\frac{k_i}{2^i}\\right)\\pmod{2\\pi}.",[18,62282,62284,62342,62628],{"className":62283,"ariaHidden":51},[93],[18,62285,62287,62290,62293,62333,62336,62339],{"className":62286},[97],[18,62288],{"className":62289,"style":448},[101],[18,62291,77],{"className":62292,"style":254},[106,110],[18,62294,62296,62299],{"className":62295},[106],[18,62297,38582],{"className":62298},[106,110],[18,62300,62302],{"className":62301},[115],[18,62303,62305,62325],{"className":62304},[119,120],[18,62306,62308,62322],{"className":62307},[124],[18,62309,62311],{"className":62310,"style":5921},[128],[18,62312,62313,62316],{"style":1155},[18,62314],{"className":62315,"style":137},[136],[18,62317,62319],{"className":62318},[141,142,143,144],[18,62320,5600],{"className":62321},[106,110,144],[18,62323,152],{"className":62324},[151],[18,62326,62328],{"className":62327},[124],[18,62329,62331],{"className":62330,"style":159},[128],[18,62332],{},[18,62334],{"className":62335,"style":228},[169],[18,62337,62193],{"className":62338},[232],[18,62340],{"className":62341,"style":228},[169],[18,62343,62345,62348,62351,62354,62618,62622,62625],{"className":62344},[97],[18,62346],{"className":62347,"style":20807},[101],[18,62349,3941],{"className":62350,"style":3957},[106,110],[18,62352],{"className":62353,"style":170},[169],[18,62355,62357,62363,62404,62407,62410,62413,62416,62419,62459,62462,62465,62468,62471,62474,62477,62480,62483,62486,62612],{"className":62356},[174],[18,62358,62360],{"className":62359,"style":13796},[250,13795],[18,62361,74],{"className":62362},[1230,143],[18,62364,62366,62369],{"className":62365},[106],[18,62367,2266],{"className":62368},[106],[18,62370,62372],{"className":62371},[115],[18,62373,62375],{"className":62374},[119],[18,62376,62378],{"className":62377},[124],[18,62379,62381],{"className":62380,"style":61400},[128],[18,62382,62383,62386],{"style":1637},[18,62384],{"className":62385,"style":137},[136],[18,62387,62389],{"className":62388},[141,142,143,144],[18,62390,62392,62395,62398,62401],{"className":62391},[106,144],[18,62393,2253],{"className":62394},[106,144],[18,62396,5600],{"className":62397},[106,110,144],[18,62399,2253],{"className":62400},[2072,144],[18,62402,47],{"className":62403},[106,144],[18,62405],{"className":62406,"style":2068},[169],[18,62408,7987],{"className":62409},[2072],[18,62411],{"className":62412,"style":2068},[169],[18,62414,47],{"className":62415},[106,2859],[18,62417,945],{"className":62418},[250],[18,62420,62422,62425],{"className":62421},[106],[18,62423,46352],{"className":62424,"style":3957},[106,110],[18,62426,62428],{"className":62427},[115],[18,62429,62431,62451],{"className":62430},[119,120],[18,62432,62434,62448],{"className":62433},[124],[18,62435,62437],{"className":62436,"style":5921},[128],[18,62438,62439,62442],{"style":30350},[18,62440],{"className":62441,"style":137},[136],[18,62443,62445],{"className":62444},[141,142,143,144],[18,62446,5600],{"className":62447},[106,110,144],[18,62449,152],{"className":62450},[151],[18,62452,62454],{"className":62453},[124],[18,62455,62457],{"className":62456,"style":159},[128],[18,62458],{},[18,62460],{"className":62461,"style":228},[169],[18,62463,433],{"className":62464},[232],[18,62466],{"className":62467,"style":228},[169],[18,62469,2253],{"className":62470},[106],[18,62472,47],{"className":62473},[106],[18,62475,965],{"className":62476},[267],[18,62478],{"className":62479,"style":2068},[169],[18,62481,7987],{"className":62482},[2072],[18,62484],{"className":62485,"style":2068},[169],[18,62487,62489,62492,62609],{"className":62488},[106],[18,62490],{"className":62491},[250,6012],[18,62493,62495],{"className":62494},[5654],[18,62496,62498,62601],{"className":62497},[119,120],[18,62499,62501,62598],{"className":62500},[124],[18,62502,62504,62542,62550],{"className":62503,"style":6025},[128],[18,62505,62506,62509],{"style":6028},[18,62507],{"className":62508,"style":697},[136],[18,62510,62512],{"className":62511},[106],[18,62513,62515,62518],{"className":62514},[106],[18,62516,2266],{"className":62517},[106],[18,62519,62521],{"className":62520},[115],[18,62522,62524],{"className":62523},[119],[18,62525,62527],{"className":62526},[124],[18,62528,62531],{"className":62529,"style":62530},[128],"height:0.7507em;",[18,62532,62533,62536],{"style":26112},[18,62534],{"className":62535,"style":137},[136],[18,62537,62539],{"className":62538},[141,142,143,144],[18,62540,5600],{"className":62541},[106,110,144],[18,62543,62544,62547],{"style":6040},[18,62545],{"className":62546,"style":697},[136],[18,62548],{"className":62549,"style":6048},[6047],[18,62551,62552,62555],{"style":6051},[18,62553],{"className":62554,"style":697},[136],[18,62556,62558],{"className":62557},[106],[18,62559,62561,62564],{"className":62560},[106],[18,62562,39964],{"className":62563,"style":40040},[106,110],[18,62565,62567],{"className":62566},[115],[18,62568,62570,62590],{"className":62569},[119,120],[18,62571,62573,62587],{"className":62572},[124],[18,62574,62576],{"className":62575,"style":5921},[128],[18,62577,62578,62581],{"style":61947},[18,62579],{"className":62580,"style":137},[136],[18,62582,62584],{"className":62583},[141,142,143,144],[18,62585,5600],{"className":62586},[106,110,144],[18,62588,152],{"className":62589},[151],[18,62591,62593],{"className":62592},[124],[18,62594,62596],{"className":62595,"style":159},[128],[18,62597],{},[18,62599,152],{"className":62600},[151],[18,62602,62604],{"className":62603},[124],[18,62605,62607],{"className":62606,"style":6070},[128],[18,62608],{},[18,62610],{"className":62611},[267,6012],[18,62613,62615],{"className":62614,"style":13796},[267,13795],[18,62616,84],{"className":62617},[1230,143],[18,62619],{"className":62620},[169,62621],"allowbreak",[18,62623],{"className":62624,"style":42849},[169],[18,62626],{"className":62627,"style":170},[169],[18,62629,62631,62634,62637,62647,62651,62654,62657,62660],{"className":62630},[97],[18,62632],{"className":62633,"style":242},[101],[18,62635,74],{"className":62636},[250],[18,62638,62640],{"className":62639},[106],[18,62641,62643],{"className":62642},[106],[18,62644,62646],{"className":62645},[106,6354],"mod",[18,62648],{"className":62649,"style":62650},[169],"margin-right:0.3333em;",[18,62652,2266],{"className":62653},[106],[18,62655,3941],{"className":62656,"style":3957},[106,110],[18,62658,84],{"className":62659},[267],[18,62661,968],{"className":62662},[106],[11,62664,39318,62665,62762,62763,62814,62815,62911,62912,62964,62965,63137,63138,63166],{},[18,62666,62668,62692],{"className":62667,"translate":22},[21],[18,62669,62671],{"className":62670},[26],[28,62672,62673],{"xmlns":30},[32,62674,62675,62689],{},[35,62676,62677,62683,62685,62687],{},[38,62678,62679,62681],{},[41,62680,46352],{},[41,62682,5600],{},[49,62684,433],{},[49,62686,7987],{},[45,62688,47],{},[86,62690,62691],{"encoding":88},"y_i=+1",[18,62693,62695,62750],{"className":62694,"ariaHidden":51},[93],[18,62696,62698,62701,62741,62744,62747],{"className":62697},[97],[18,62699],{"className":62700,"style":38552},[101],[18,62702,62704,62707],{"className":62703},[106],[18,62705,46352],{"className":62706,"style":3957},[106,110],[18,62708,62710],{"className":62709},[115],[18,62711,62713,62733],{"className":62712},[119,120],[18,62714,62716,62730],{"className":62715},[124],[18,62717,62719],{"className":62718,"style":5921},[128],[18,62720,62721,62724],{"style":30350},[18,62722],{"className":62723,"style":137},[136],[18,62725,62727],{"className":62726},[141,142,143,144],[18,62728,5600],{"className":62729},[106,110,144],[18,62731,152],{"className":62732},[151],[18,62734,62736],{"className":62735},[124],[18,62737,62739],{"className":62738,"style":159},[128],[18,62740],{},[18,62742],{"className":62743,"style":228},[169],[18,62745,433],{"className":62746},[232],[18,62748],{"className":62749,"style":228},[169],[18,62751,62753,62756,62759],{"className":62752},[97],[18,62754],{"className":62755,"style":18095},[101],[18,62757,7987],{"className":62758},[106],[18,62760,47],{"className":62761},[106]," 时，括号内的值落在 ",[18,62764,62766,62787],{"className":62765,"translate":22},[21],[18,62767,62769],{"className":62768},[26],[28,62770,62771],{"xmlns":30},[32,62772,62773,62785],{},[35,62774,62775,62777,62779,62781,62783],{},[49,62776,74],{"stretchy":73},[45,62778,430],{},[49,62780,52],{"separator":51},[45,62782,47],{},[49,62784,84],{"stretchy":73},[86,62786,54905],{"encoding":88},[18,62788,62790],{"className":62789,"ariaHidden":51},[93],[18,62791,62793,62796,62799,62802,62805,62808,62811],{"className":62792},[97],[18,62794],{"className":62795,"style":242},[101],[18,62797,74],{"className":62798},[250],[18,62800,430],{"className":62801},[106],[18,62803,52],{"className":62804},[165],[18,62806],{"className":62807,"style":170},[169],[18,62809,47],{"className":62810},[106],[18,62812,84],{"className":62813},[267],"；当 ",[18,62816,62818,62841],{"className":62817,"translate":22},[21],[18,62819,62821],{"className":62820},[26],[28,62822,62823],{"xmlns":30},[32,62824,62825,62839],{},[35,62826,62827,62833,62835,62837],{},[38,62828,62829,62831],{},[41,62830,46352],{},[41,62832,5600],{},[49,62834,433],{},[49,62836,2253],{},[45,62838,47],{},[86,62840,61759],{"encoding":88},[18,62842,62844,62899],{"className":62843,"ariaHidden":51},[93],[18,62845,62847,62850,62890,62893,62896],{"className":62846},[97],[18,62848],{"className":62849,"style":38552},[101],[18,62851,62853,62856],{"className":62852},[106],[18,62854,46352],{"className":62855,"style":3957},[106,110],[18,62857,62859],{"className":62858},[115],[18,62860,62862,62882],{"className":62861},[119,120],[18,62863,62865,62879],{"className":62864},[124],[18,62866,62868],{"className":62867,"style":5921},[128],[18,62869,62870,62873],{"style":30350},[18,62871],{"className":62872,"style":137},[136],[18,62874,62876],{"className":62875},[141,142,143,144],[18,62877,5600],{"className":62878},[106,110,144],[18,62880,152],{"className":62881},[151],[18,62883,62885],{"className":62884},[124],[18,62886,62888],{"className":62887,"style":159},[128],[18,62889],{},[18,62891],{"className":62892,"style":228},[169],[18,62894,433],{"className":62895},[232],[18,62897],{"className":62898,"style":228},[169],[18,62900,62902,62905,62908],{"className":62901},[97],[18,62903],{"className":62904,"style":18095},[101],[18,62906,2253],{"className":62907},[106],[18,62909,47],{"className":62910},[106]," 时落在 ",[18,62913,62915,62937],{"className":62914,"translate":22},[21],[18,62916,62918],{"className":62917},[26],[28,62919,62920],{"xmlns":30},[32,62921,62922,62934],{},[35,62923,62924,62926,62928,62930,62932],{},[49,62925,74],{"stretchy":73},[45,62927,47],{},[49,62929,52],{"separator":51},[45,62931,2266],{},[49,62933,84],{"stretchy":73},[86,62935,62936],{"encoding":88},"(1,2)",[18,62938,62940],{"className":62939,"ariaHidden":51},[93],[18,62941,62943,62946,62949,62952,62955,62958,62961],{"className":62942},[97],[18,62944],{"className":62945,"style":242},[101],[18,62947,74],{"className":62948},[250],[18,62950,47],{"className":62951},[106],[18,62953,52],{"className":62954},[165],[18,62956],{"className":62957,"style":170},[169],[18,62959,2266],{"className":62960},[106],[18,62962,84],{"className":62963},[267],"。所以 ",[18,62966,62968,63012],{"className":62967,"translate":22},[21],[18,62969,62971],{"className":62970},[26],[28,62972,62973],{"xmlns":30},[32,62974,62975,63009],{},[35,62976,62977,62979,62981,62983,62985,62987,62989,62991,62997,62999,63001,63003],{},[41,62978,45639],{"mathvariant":316},[49,62980,899],{},[49,62982,74],{"stretchy":73},[41,62984,45646],{},[49,62986,899],{},[49,62988,74],{"stretchy":73},[41,62990,77],{},[38,62992,62993,62995],{},[41,62994,38582],{},[41,62996,5600],{},[49,62998,84],{"stretchy":73},[49,63000,84],{"stretchy":73},[49,63002,433],{},[38,63004,63005,63007],{},[41,63006,46352],{},[41,63008,5600],{},[86,63010,63011],{"encoding":88},"\\operatorname{sign}(\\sin(\\theta x_i))=y_i",[18,63013,63015,63091],{"className":63014,"ariaHidden":51},[93],[18,63016,63018,63021,63027,63030,63033,63036,63039,63079,63082,63085,63088],{"className":63017},[97],[18,63019],{"className":63020,"style":242},[101],[18,63022,63024],{"className":63023},[985],[18,63025,45639],{"className":63026},[106,6354],[18,63028,74],{"className":63029},[250],[18,63031,45646],{"className":63032},[985],[18,63034,74],{"className":63035},[250],[18,63037,77],{"className":63038,"style":254},[106,110],[18,63040,63042,63045],{"className":63041},[106],[18,63043,38582],{"className":63044},[106,110],[18,63046,63048],{"className":63047},[115],[18,63049,63051,63071],{"className":63050},[119,120],[18,63052,63054,63068],{"className":63053},[124],[18,63055,63057],{"className":63056,"style":5921},[128],[18,63058,63059,63062],{"style":1155},[18,63060],{"className":63061,"style":137},[136],[18,63063,63065],{"className":63064},[141,142,143,144],[18,63066,5600],{"className":63067},[106,110,144],[18,63069,152],{"className":63070},[151],[18,63072,63074],{"className":63073},[124],[18,63075,63077],{"className":63076,"style":159},[128],[18,63078],{},[18,63080,3431],{"className":63081},[267],[18,63083],{"className":63084,"style":228},[169],[18,63086,433],{"className":63087},[232],[18,63089],{"className":63090,"style":228},[169],[18,63092,63094,63097],{"className":63093},[97],[18,63095],{"className":63096,"style":38552},[101],[18,63098,63100,63103],{"className":63099},[106],[18,63101,46352],{"className":63102,"style":3957},[106,110],[18,63104,63106],{"className":63105},[115],[18,63107,63109,63129],{"className":63108},[119,120],[18,63110,63112,63126],{"className":63111},[124],[18,63113,63115],{"className":63114,"style":5921},[128],[18,63116,63117,63120],{"style":30350},[18,63118],{"className":63119,"style":137},[136],[18,63121,63123],{"className":63122},[141,142,143,144],[18,63124,5600],{"className":63125},[106,110,144],[18,63127,152],{"className":63128},[151],[18,63130,63132],{"className":63131},[124],[18,63133,63135],{"className":63134,"style":159},[128],[18,63136],{},"。任意 ",[18,63139,63141,63154],{"className":63140,"translate":22},[21],[18,63142,63144],{"className":63143},[26],[28,63145,63146],{"xmlns":30},[32,63147,63148,63152],{},[35,63149,63150],{},[41,63151,28169],{},[86,63153,28169],{"encoding":88},[18,63155,63157],{"className":63156,"ariaHidden":51},[93],[18,63158,63160,63163],{"className":63159},[97],[18,63161],{"className":63162,"style":1083},[101],[18,63164,28169],{"className":63165},[106,110]," 个点都能被打散，VC 维无穷。",[11,63168,63169],{},"一个参数，无穷容量。这就是「参数量不等于容量」最干净的反例。",[11,63171,63172],{},[717,63173,63174,63175,52828],{},"例 7（最近邻，",[18,63176,63178,63197],{"className":63177,"translate":22},[21],[18,63179,63181],{"className":63180},[26],[28,63182,63183],{"xmlns":30},[32,63184,63185,63195],{},[35,63186,63187,63189,63191,63193],{},[41,63188,52216],{"mathvariant":316},[49,63190,899],{},[49,63192,433],{},[41,63194,2028],{"mathvariant":316},[86,63196,60578],{"encoding":88},[18,63198,63200,63221],{"className":63199,"ariaHidden":51},[93],[18,63201,63203,63206,63212,63215,63218],{"className":63202},[97],[18,63204],{"className":63205,"style":295},[101],[18,63207,63209],{"className":63208},[985],[18,63210,52216],{"className":63211},[106,6354],[18,63213],{"className":63214,"style":228},[169],[18,63216,433],{"className":63217},[232],[18,63219],{"className":63220,"style":228},[169],[18,63222,63224,63227],{"className":63223},[97],[18,63225],{"className":63226,"style":1083},[101],[18,63228,2028],{"className":63229},[106],[11,63231,63232,63233,63261,63262,63290,63291,63361],{},"1-NN 没有参数：给定训练集，它把测试点判为最近训练点的标签。给定任意有限点集 ",[18,63234,63236,63249],{"className":63235,"translate":22},[21],[18,63237,63239],{"className":63238},[26],[28,63240,63241],{"xmlns":30},[32,63242,63243,63247],{},[35,63244,63245],{},[41,63246,51021],{},[86,63248,51021],{"encoding":88},[18,63250,63252],{"className":63251,"ariaHidden":51},[93],[18,63253,63255,63258],{"className":63254},[97],[18,63256],{"className":63257,"style":600},[101],[18,63259,51021],{"className":63260,"style":5073},[106,110],"（点互不相同、没有距离并列）和任意标注，直接令训练集就是 ",[18,63263,63265,63278],{"className":63264,"translate":22},[21],[18,63266,63268],{"className":63267},[26],[28,63269,63270],{"xmlns":30},[32,63271,63272,63276],{},[35,63273,63274],{},[41,63275,51021],{},[86,63277,51021],{"encoding":88},[18,63279,63281],{"className":63280,"ariaHidden":51},[93],[18,63282,63284,63287],{"className":63283},[97],[18,63285],{"className":63286,"style":600},[101],[18,63288,51021],{"className":63289,"style":5073},[106,110]," 本身（标签按标注给）；对每个 ",[18,63292,63294,63312],{"className":63293,"translate":22},[21],[18,63295,63297],{"className":63296},[26],[28,63298,63299],{"xmlns":30},[32,63300,63301,63309],{},[35,63302,63303],{},[38,63304,63305,63307],{},[41,63306,38582],{},[41,63308,5600],{},[86,63310,63311],{"encoding":88},"x_i",[18,63313,63315],{"className":63314,"ariaHidden":51},[93],[18,63316,63318,63321],{"className":63317},[97],[18,63319],{"className":63320,"style":16897},[101],[18,63322,63324,63327],{"className":63323},[106],[18,63325,38582],{"className":63326},[106,110],[18,63328,63330],{"className":63329},[115],[18,63331,63333,63353],{"className":63332},[119,120],[18,63334,63336,63350],{"className":63335},[124],[18,63337,63339],{"className":63338,"style":5921},[128],[18,63340,63341,63344],{"style":1155},[18,63342],{"className":63343,"style":137},[136],[18,63345,63347],{"className":63346},[141,142,143,144],[18,63348,5600],{"className":63349},[106,110,144],[18,63351,152],{"className":63352},[151],[18,63354,63356],{"className":63355},[124],[18,63357,63359],{"className":63358,"style":159},[128],[18,63360],{},"，最近邻是它自己，预测即标注。所以任意有限集都被打散，VC 维无穷。",[11,63363,63364],{},"这个例子说明，VC 维无穷并不妨碍一个算法在真实数据上表现良好；它只说明不存在一个对所有分布一致的泛化保证。",[11,63366,63367],{},"把上面几个例子汇总：",[43712,63369,63370,63381],{},[43715,63371,63372],{},[43718,63373,63374,63376,63379],{},[43721,63375,46300],{},[43721,63377,63378],{},"输入空间",[43721,63380,52197],{},[43731,63382,63383,63449,63514,63631,63731,63848,63991,64124],{},[43718,63384,63385,63388,63419],{},[43736,63386,63387],{},"半直线",[43736,63389,63390],{},[18,63391,63393,63407],{"className":63392,"translate":22},[21],[18,63394,63396],{"className":63395},[26],[28,63397,63398],{"xmlns":30},[32,63399,63400,63404],{},[35,63401,63402],{},[41,63403,3071],{"mathvariant":933},[86,63405,63406],{"encoding":88},"\\mathbb R",[18,63408,63410],{"className":63409,"ariaHidden":51},[93],[18,63411,63413,63416],{"className":63412},[97],[18,63414],{"className":63415,"style":45807},[101],[18,63417,3071],{"className":63418},[106,1139],[43736,63420,63421],{},[18,63422,63424,63437],{"className":63423,"translate":22},[21],[18,63425,63427],{"className":63426},[26],[28,63428,63429],{"xmlns":30},[32,63430,63431,63435],{},[35,63432,63433],{},[45,63434,47],{},[86,63436,47],{"encoding":88},[18,63438,63440],{"className":63439,"ariaHidden":51},[93],[18,63441,63443,63446],{"className":63442},[97],[18,63444],{"className":63445,"style":505},[101],[18,63447,47],{"className":63448},[106],[43718,63450,63451,63454,63484],{},[43736,63452,63453],{},"区间",[43736,63455,63456],{},[18,63457,63459,63472],{"className":63458,"translate":22},[21],[18,63460,63462],{"className":63461},[26],[28,63463,63464],{"xmlns":30},[32,63465,63466,63470],{},[35,63467,63468],{},[41,63469,3071],{"mathvariant":933},[86,63471,63406],{"encoding":88},[18,63473,63475],{"className":63474,"ariaHidden":51},[93],[18,63476,63478,63481],{"className":63477},[97],[18,63479],{"className":63480,"style":45807},[101],[18,63482,3071],{"className":63483},[106,1139],[43736,63485,63486],{},[18,63487,63489,63502],{"className":63488,"translate":22},[21],[18,63490,63492],{"className":63491},[26],[28,63493,63494],{"xmlns":30},[32,63495,63496,63500],{},[35,63497,63498],{},[45,63499,2266],{},[86,63501,2266],{"encoding":88},[18,63503,63505],{"className":63504,"ariaHidden":51},[93],[18,63506,63508,63511],{"className":63507},[97],[18,63509],{"className":63510,"style":505},[101],[18,63512,2266],{"className":63513},[106],[43718,63515,63516,63519,63579],{},[43736,63517,63518],{},"仿射超平面",[43736,63520,63521],{},[18,63522,63524,63541],{"className":63523,"translate":22},[21],[18,63525,63527],{"className":63526},[26],[28,63528,63529],{"xmlns":30},[32,63530,63531,63539],{},[35,63532,63533],{},[1556,63534,63535,63537],{},[41,63536,3071],{"mathvariant":933},[41,63538,4031],{},[86,63540,56939],{"encoding":88},[18,63542,63544],{"className":63543,"ariaHidden":51},[93],[18,63545,63547,63550],{"className":63546},[97],[18,63548],{"className":63549,"style":35186},[101],[18,63551,63553,63556],{"className":63552},[106],[18,63554,3071],{"className":63555},[106,1139],[18,63557,63559],{"className":63558},[115],[18,63560,63562],{"className":63561},[119],[18,63563,63565],{"className":63564},[124],[18,63566,63568],{"className":63567,"style":35186},[128],[18,63569,63570,63573],{"style":1962},[18,63571],{"className":63572,"style":137},[136],[18,63574,63576],{"className":63575},[141,142,143,144],[18,63577,4031],{"className":63578},[106,110,144],[43736,63580,63581],{},[18,63582,63584,63601],{"className":63583,"translate":22},[21],[18,63585,63587],{"className":63586},[26],[28,63588,63589],{"xmlns":30},[32,63590,63591,63599],{},[35,63592,63593,63595,63597],{},[41,63594,4031],{},[49,63596,7987],{},[45,63598,47],{},[86,63600,56253],{"encoding":88},[18,63602,63604,63622],{"className":63603,"ariaHidden":51},[93],[18,63605,63607,63610,63613,63616,63619],{"className":63606},[97],[18,63608],{"className":63609,"style":7224},[101],[18,63611,4031],{"className":63612},[106,110],[18,63614],{"className":63615,"style":2068},[169],[18,63617,7987],{"className":63618},[2072],[18,63620],{"className":63621,"style":2068},[169],[18,63623,63625,63628],{"className":63624},[97],[18,63626],{"className":63627,"style":505},[101],[18,63629,47],{"className":63630},[106],[43718,63632,63633,63636,63696],{},[43736,63634,63635],{},"轴对齐矩形",[43736,63637,63638],{},[18,63639,63641,63658],{"className":63640,"translate":22},[21],[18,63642,63644],{"className":63643},[26],[28,63645,63646],{"xmlns":30},[32,63647,63648,63656],{},[35,63649,63650],{},[1556,63651,63652,63654],{},[41,63653,3071],{"mathvariant":933},[41,63655,4031],{},[86,63657,56939],{"encoding":88},[18,63659,63661],{"className":63660,"ariaHidden":51},[93],[18,63662,63664,63667],{"className":63663},[97],[18,63665],{"className":63666,"style":35186},[101],[18,63668,63670,63673],{"className":63669},[106],[18,63671,3071],{"className":63672},[106,1139],[18,63674,63676],{"className":63675},[115],[18,63677,63679],{"className":63678},[119],[18,63680,63682],{"className":63681},[124],[18,63683,63685],{"className":63684,"style":35186},[128],[18,63686,63687,63690],{"style":1962},[18,63688],{"className":63689,"style":137},[136],[18,63691,63693],{"className":63692},[141,142,143,144],[18,63694,4031],{"className":63695},[106,110,144],[43736,63697,63698],{},[18,63699,63701,63716],{"className":63700,"translate":22},[21],[18,63702,63704],{"className":63703},[26],[28,63705,63706],{"xmlns":30},[32,63707,63708,63714],{},[35,63709,63710,63712],{},[45,63711,2266],{},[41,63713,4031],{},[86,63715,59637],{"encoding":88},[18,63717,63719],{"className":63718,"ariaHidden":51},[93],[18,63720,63722,63725,63728],{"className":63721},[97],[18,63723],{"className":63724,"style":295},[101],[18,63726,2266],{"className":63727},[106],[18,63729,4031],{"className":63730},[106,110],[43718,63732,63733,63736,63796],{},[43736,63734,63735],{},"球",[43736,63737,63738],{},[18,63739,63741,63758],{"className":63740,"translate":22},[21],[18,63742,63744],{"className":63743},[26],[28,63745,63746],{"xmlns":30},[32,63747,63748,63756],{},[35,63749,63750],{},[1556,63751,63752,63754],{},[41,63753,3071],{"mathvariant":933},[41,63755,4031],{},[86,63757,56939],{"encoding":88},[18,63759,63761],{"className":63760,"ariaHidden":51},[93],[18,63762,63764,63767],{"className":63763},[97],[18,63765],{"className":63766,"style":35186},[101],[18,63768,63770,63773],{"className":63769},[106],[18,63771,3071],{"className":63772},[106,1139],[18,63774,63776],{"className":63775},[115],[18,63777,63779],{"className":63778},[119],[18,63780,63782],{"className":63781},[124],[18,63783,63785],{"className":63784,"style":35186},[128],[18,63786,63787,63790],{"style":1962},[18,63788],{"className":63789,"style":137},[136],[18,63791,63793],{"className":63792},[141,142,143,144],[18,63794,4031],{"className":63795},[106,110,144],[43736,63797,63798],{},[18,63799,63801,63818],{"className":63800,"translate":22},[21],[18,63802,63804],{"className":63803},[26],[28,63805,63806],{"xmlns":30},[32,63807,63808,63816],{},[35,63809,63810,63812,63814],{},[41,63811,4031],{},[49,63813,7987],{},[45,63815,47],{},[86,63817,56253],{"encoding":88},[18,63819,63821,63839],{"className":63820,"ariaHidden":51},[93],[18,63822,63824,63827,63830,63833,63836],{"className":63823},[97],[18,63825],{"className":63826,"style":7224},[101],[18,63828,4031],{"className":63829},[106,110],[18,63831],{"className":63832,"style":2068},[169],[18,63834,7987],{"className":63835},[2072],[18,63837],{"className":63838,"style":2068},[169],[18,63840,63842,63845],{"className":63841},[97],[18,63843],{"className":63844,"style":505},[101],[18,63846,47],{"className":63847},[106],[43718,63849,63850,63881,63884],{},[43736,63851,63852,63853],{},"有限类 ",[18,63854,63856,63869],{"className":63855,"translate":22},[21],[18,63857,63859],{"className":63858},[26],[28,63860,63861],{"xmlns":30},[32,63862,63863,63867],{},[35,63864,63865],{},[41,63866,18309],{"mathvariant":1462},[86,63868,46317],{"encoding":88},[18,63870,63872],{"className":63871,"ariaHidden":51},[93],[18,63873,63875,63878],{"className":63874},[97],[18,63876],{"className":63877,"style":600},[101],[18,63879,18309],{"className":63880,"style":46330},[106,1478],[43736,63882,63883],{},"任意",[43736,63885,63886],{},[18,63887,63889,63919],{"className":63888,"translate":22},[21],[18,63890,63892],{"className":63891},[26],[28,63893,63894],{"xmlns":30},[32,63895,63896,63916],{},[35,63897,63898,63900,63910,63912,63914],{},[49,63899,4334],{},[38,63901,63902,63908],{},[35,63903,63904,63906],{},[41,63905,14733],{},[49,63907,899],{},[45,63909,2266],{},[49,63911,11225],{"stretchy":73},[41,63913,18309],{"mathvariant":1462},[49,63915,11225],{"stretchy":73},[86,63917,63918],{"encoding":88},"\\le\\log_2\\lvert\\mathcal H\\rvert",[18,63920,63922,63934],{"className":63921,"ariaHidden":51},[93],[18,63923,63925,63928,63931],{"className":63924},[97],[18,63926],{"className":63927,"style":53144},[101],[18,63929,4334],{"className":63930},[232],[18,63932],{"className":63933,"style":228},[169],[18,63935,63937,63940,63982,63985,63988],{"className":63936},[97],[18,63938],{"className":63939,"style":242},[101],[18,63941,63943,63948],{"className":63942},[985],[18,63944,15174,63946],{"className":63945},[985],[18,63947,15177],{"style":6355},[18,63949,63951],{"className":63950},[115],[18,63952,63954,63974],{"className":63953},[119,120],[18,63955,63957,63971],{"className":63956},[124],[18,63958,63960],{"className":63959,"style":60145},[128],[18,63961,63962,63965],{"style":60148},[18,63963],{"className":63964,"style":137},[136],[18,63966,63968],{"className":63967},[141,142,143,144],[18,63969,2266],{"className":63970},[106,144],[18,63972,152],{"className":63973},[151],[18,63975,63977],{"className":63976},[124],[18,63978,63980],{"className":63979,"style":60167},[128],[18,63981],{},[18,63983,11225],{"className":63984},[250],[18,63986,18309],{"className":63987,"style":46330},[106,1478],[18,63989,11225],{"className":63990},[267],[43718,63992,63993,64063,64093],{},[43736,63994,63995],{},[18,63996,63998,64030],{"className":63997,"translate":22},[21],[18,63999,64001],{"className":64000},[26],[28,64002,64003],{"xmlns":30},[32,64004,64005,64027],{},[35,64006,64007,64009,64011,64013,64015,64017,64019,64021,64023,64025],{},[41,64008,45639],{"mathvariant":316},[49,64010,899],{},[49,64012,74],{"stretchy":73},[41,64014,45646],{},[49,64016,899],{},[49,64018,74],{"stretchy":73},[41,64020,77],{},[41,64022,38582],{},[49,64024,84],{"stretchy":73},[49,64026,84],{"stretchy":73},[86,64028,64029],{"encoding":88},"\\operatorname{sign}(\\sin(\\theta x))",[18,64031,64033],{"className":64032,"ariaHidden":51},[93],[18,64034,64036,64039,64045,64048,64051,64054,64057,64060],{"className":64035},[97],[18,64037],{"className":64038,"style":242},[101],[18,64040,64042],{"className":64041},[985],[18,64043,45639],{"className":64044},[106,6354],[18,64046,74],{"className":64047},[250],[18,64049,45646],{"className":64050},[985],[18,64052,74],{"className":64053},[250],[18,64055,77],{"className":64056,"style":254},[106,110],[18,64058,38582],{"className":64059},[106,110],[18,64061,3431],{"className":64062},[267],[43736,64064,64065],{},[18,64066,64068,64081],{"className":64067,"translate":22},[21],[18,64069,64071],{"className":64070},[26],[28,64072,64073],{"xmlns":30},[32,64074,64075,64079],{},[35,64076,64077],{},[41,64078,3071],{"mathvariant":933},[86,64080,63406],{"encoding":88},[18,64082,64084],{"className":64083,"ariaHidden":51},[93],[18,64085,64087,64090],{"className":64086},[97],[18,64088],{"className":64089,"style":45807},[101],[18,64091,3071],{"className":64092},[106,1139],[43736,64094,64095],{},[18,64096,64098,64112],{"className":64097,"translate":22},[21],[18,64099,64101],{"className":64100},[26],[28,64102,64103],{"xmlns":30},[32,64104,64105,64109],{},[35,64106,64107],{},[41,64108,2028],{"mathvariant":316},[86,64110,64111],{"encoding":88},"\\infty",[18,64113,64115],{"className":64114,"ariaHidden":51},[93],[18,64116,64118,64121],{"className":64117},[97],[18,64119],{"className":64120,"style":1083},[101],[18,64122,2028],{"className":64123},[106],[43718,64125,64126,64129,64189],{},[43736,64127,64128],{},"1-NN",[43736,64130,64131],{},[18,64132,64134,64151],{"className":64133,"translate":22},[21],[18,64135,64137],{"className":64136},[26],[28,64138,64139],{"xmlns":30},[32,64140,64141,64149],{},[35,64142,64143],{},[1556,64144,64145,64147],{},[41,64146,3071],{"mathvariant":933},[41,64148,4031],{},[86,64150,56939],{"encoding":88},[18,64152,64154],{"className":64153,"ariaHidden":51},[93],[18,64155,64157,64160],{"className":64156},[97],[18,64158],{"className":64159,"style":35186},[101],[18,64161,64163,64166],{"className":64162},[106],[18,64164,3071],{"className":64165},[106,1139],[18,64167,64169],{"className":64168},[115],[18,64170,64172],{"className":64171},[119],[18,64173,64175],{"className":64174},[124],[18,64176,64178],{"className":64177,"style":35186},[128],[18,64179,64180,64183],{"style":1962},[18,64181],{"className":64182,"style":137},[136],[18,64184,64186],{"className":64185},[141,142,143,144],[18,64187,4031],{"className":64188},[106,110,144],[43736,64190,64191],{},[18,64192,64194,64207],{"className":64193,"translate":22},[21],[18,64195,64197],{"className":64196},[26],[28,64198,64199],{"xmlns":30},[32,64200,64201,64205],{},[35,64202,64203],{},[41,64204,2028],{"mathvariant":316},[86,64206,64111],{"encoding":88},[18,64208,64210],{"className":64209,"ariaHidden":51},[93],[18,64211,64213,64216],{"className":64212},[97],[18,64214],{"className":64215,"style":1083},[101],[18,64217,2028],{"className":64218},[106],[5159,64220,64221],{},[11,64222,64223,64226,64227,64399,64400,64496,64497,64549,64550,64742,64743,64771,64772,64831,64832,64860,64861,64889],{},[717,64224,64225],{},"注（怎么算）"," 对线性分类器，固定点集 ",[18,64228,64230,64268],{"className":64229,"translate":22},[21],[18,64231,64233],{"className":64232},[26],[28,64234,64235],{"xmlns":30},[32,64236,64237,64265],{},[35,64238,64239,64241,64243,64245,64251,64253,64255,64257,64263],{},[41,64240,51021],{},[49,64242,433],{},[49,64244,1331],{"stretchy":73},[38,64246,64247,64249],{},[41,64248,38582],{},[45,64250,47],{},[49,64252,52],{"separator":51},[49,64254,55],{},[49,64256,52],{"separator":51},[38,64258,64259,64261],{},[41,64260,38582],{},[41,64262,39964],{},[49,64264,1349],{"stretchy":73},[86,64266,64267],{"encoding":88},"C=\\{x_1,\\dots,x_k\\}",[18,64269,64271,64289],{"className":64270,"ariaHidden":51},[93],[18,64272,64274,64277,64280,64283,64286],{"className":64273},[97],[18,64275],{"className":64276,"style":600},[101],[18,64278,51021],{"className":64279,"style":5073},[106,110],[18,64281],{"className":64282,"style":228},[169],[18,64284,433],{"className":64285},[232],[18,64287],{"className":64288,"style":228},[169],[18,64290,64292,64295,64298,64338,64341,64344,64347,64350,64353,64356,64396],{"className":64291},[97],[18,64293],{"className":64294,"style":242},[101],[18,64296,1331],{"className":64297},[250],[18,64299,64301,64304],{"className":64300},[106],[18,64302,38582],{"className":64303},[106,110],[18,64305,64307],{"className":64306},[115],[18,64308,64310,64330],{"className":64309},[119,120],[18,64311,64313,64327],{"className":64312},[124],[18,64314,64316],{"className":64315,"style":129},[128],[18,64317,64318,64321],{"style":1155},[18,64319],{"className":64320,"style":137},[136],[18,64322,64324],{"className":64323},[141,142,143,144],[18,64325,47],{"className":64326},[106,144],[18,64328,152],{"className":64329},[151],[18,64331,64333],{"className":64332},[124],[18,64334,64336],{"className":64335,"style":159},[128],[18,64337],{},[18,64339,52],{"className":64340},[165],[18,64342],{"className":64343,"style":170},[169],[18,64345,55],{"className":64346},[174],[18,64348],{"className":64349,"style":170},[169],[18,64351,52],{"className":64352},[165],[18,64354],{"className":64355,"style":170},[169],[18,64357,64359,64362],{"className":64358},[106],[18,64360,38582],{"className":64361},[106,110],[18,64363,64365],{"className":64364},[115],[18,64366,64368,64388],{"className":64367},[119,120],[18,64369,64371,64385],{"className":64370},[124],[18,64372,64374],{"className":64373,"style":1383},[128],[18,64375,64376,64379],{"style":1155},[18,64377],{"className":64378,"style":137},[136],[18,64380,64382],{"className":64381},[141,142,143,144],[18,64383,39964],{"className":64384,"style":40040},[106,110,144],[18,64386,152],{"className":64387},[151],[18,64389,64391],{"className":64390},[124],[18,64392,64394],{"className":64393,"style":159},[128],[18,64395],{},[18,64397,1349],{"className":64398},[267],"，对每个标注 ",[18,64401,64403,64431],{"className":64402,"translate":22},[21],[18,64404,64406],{"className":64405},[26],[28,64407,64408],{"xmlns":30},[32,64409,64410,64428],{},[35,64411,64412,64414,64416,64418,64420,64422],{},[41,64413,46352],{},[49,64415,921],{},[49,64417,1331],{"stretchy":73},[49,64419,56305],{},[45,64421,47],{},[1556,64423,64424,64426],{},[49,64425,1349],{"stretchy":73},[41,64427,39964],{},[86,64429,64430],{"encoding":88},"y\\in\\{\\pm1\\}^k",[18,64432,64434,64452],{"className":64433,"ariaHidden":51},[93],[18,64435,64437,64440,64443,64446,64449],{"className":64436},[97],[18,64438],{"className":64439,"style":14587},[101],[18,64441,46352],{"className":64442,"style":3957},[106,110],[18,64444],{"className":64445,"style":228},[169],[18,64447,921],{"className":64448},[232],[18,64450],{"className":64451,"style":228},[169],[18,64453,64455,64458,64461,64464,64467],{"className":64454},[97],[18,64456],{"className":64457,"style":35152},[101],[18,64459,1331],{"className":64460},[250],[18,64462,56305],{"className":64463},[106],[18,64465,47],{"className":64466},[106],[18,64468,64470,64473],{"className":64469},[267],[18,64471,1349],{"className":64472},[267],[18,64474,64476],{"className":64475},[115],[18,64477,64479],{"className":64478},[119],[18,64480,64482],{"className":64481},[124],[18,64483,64485],{"className":64484,"style":35186},[128],[18,64486,64487,64490],{"style":1962},[18,64488],{"className":64489,"style":137},[136],[18,64491,64493],{"className":64492},[141,142,143,144],[18,64494,39964],{"className":64495,"style":40040},[106,110,144]," 解可行性问题：是否存在 ",[18,64498,64500,64522],{"className":64499,"translate":22},[21],[18,64501,64503],{"className":64502},[26],[28,64504,64505],{"xmlns":30},[32,64506,64507,64519],{},[35,64508,64509,64511,64513,64515,64517],{},[49,64510,74],{"stretchy":73},[41,64512,55926],{},[49,64514,52],{"separator":51},[41,64516,54081],{},[49,64518,84],{"stretchy":73},[86,64520,64521],{"encoding":88},"(w,b)",[18,64523,64525],{"className":64524,"ariaHidden":51},[93],[18,64526,64528,64531,64534,64537,64540,64543,64546],{"className":64527},[97],[18,64529],{"className":64530,"style":242},[101],[18,64532,74],{"className":64533},[250],[18,64535,55926],{"className":64536,"style":56002},[106,110],[18,64538,52],{"className":64539},[165],[18,64541],{"className":64542,"style":170},[169],[18,64544,54081],{"className":64545},[106,110],[18,64547,84],{"className":64548},[267]," 使 ",[18,64551,64553,64593],{"className":64552,"translate":22},[21],[18,64554,64556],{"className":64555},[26],[28,64557,64558],{"xmlns":30},[32,64559,64560,64590],{},[35,64561,64562,64568,64570,64572,64574,64580,64582,64584,64586,64588],{},[38,64563,64564,64566],{},[41,64565,46352],{},[41,64567,5600],{},[49,64569,74],{"stretchy":73},[41,64571,55926],{},[49,64573,26840],{},[38,64575,64576,64578],{},[41,64577,38582],{},[41,64579,5600],{},[49,64581,7987],{},[41,64583,54081],{},[49,64585,84],{"stretchy":73},[49,64587,4775],{},[45,64589,47],{},[86,64591,64592],{"encoding":88},"y_i(w\\cdot x_i+b)\\ge 1",[18,64594,64596,64657,64712,64733],{"className":64595,"ariaHidden":51},[93],[18,64597,64599,64602,64642,64645,64648,64651,64654],{"className":64598},[97],[18,64600],{"className":64601,"style":242},[101],[18,64603,64605,64608],{"className":64604},[106],[18,64606,46352],{"className":64607,"style":3957},[106,110],[18,64609,64611],{"className":64610},[115],[18,64612,64614,64634],{"className":64613},[119,120],[18,64615,64617,64631],{"className":64616},[124],[18,64618,64620],{"className":64619,"style":5921},[128],[18,64621,64622,64625],{"style":30350},[18,64623],{"className":64624,"style":137},[136],[18,64626,64628],{"className":64627},[141,142,143,144],[18,64629,5600],{"className":64630},[106,110,144],[18,64632,152],{"className":64633},[151],[18,64635,64637],{"className":64636},[124],[18,64638,64640],{"className":64639,"style":159},[128],[18,64641],{},[18,64643,74],{"className":64644},[250],[18,64646,55926],{"className":64647,"style":56002},[106,110],[18,64649],{"className":64650,"style":2068},[169],[18,64652,26840],{"className":64653},[2072],[18,64655],{"className":64656,"style":2068},[169],[18,64658,64660,64663,64703,64706,64709],{"className":64659},[97],[18,64661],{"className":64662,"style":56488},[101],[18,64664,64666,64669],{"className":64665},[106],[18,64667,38582],{"className":64668},[106,110],[18,64670,64672],{"className":64671},[115],[18,64673,64675,64695],{"className":64674},[119,120],[18,64676,64678,64692],{"className":64677},[124],[18,64679,64681],{"className":64680,"style":5921},[128],[18,64682,64683,64686],{"style":1155},[18,64684],{"className":64685,"style":137},[136],[18,64687,64689],{"className":64688},[141,142,143,144],[18,64690,5600],{"className":64691},[106,110,144],[18,64693,152],{"className":64694},[151],[18,64696,64698],{"className":64697},[124],[18,64699,64701],{"className":64700,"style":159},[128],[18,64702],{},[18,64704],{"className":64705,"style":2068},[169],[18,64707,7987],{"className":64708},[2072],[18,64710],{"className":64711,"style":2068},[169],[18,64713,64715,64718,64721,64724,64727,64730],{"className":64714},[97],[18,64716],{"className":64717,"style":242},[101],[18,64719,54081],{"className":64720},[106,110],[18,64722,84],{"className":64723},[267],[18,64725],{"className":64726,"style":228},[169],[18,64728,4775],{"className":64729},[232],[18,64731],{"className":64732,"style":228},[169],[18,64734,64736,64739],{"className":64735},[97],[18,64737],{"className":64738,"style":505},[101],[18,64740,47],{"className":64741},[106]," 对所有 ",[18,64744,64746,64759],{"className":64745,"translate":22},[21],[18,64747,64749],{"className":64748},[26],[28,64750,64751],{"xmlns":30},[32,64752,64753,64757],{},[35,64754,64755],{},[41,64756,5600],{},[86,64758,5600],{"encoding":88},[18,64760,64762],{"className":64761,"ariaHidden":51},[93],[18,64763,64765,64768],{"className":64764},[97],[18,64766],{"className":64767,"style":61452},[101],[18,64769,5600],{"className":64770},[106,110]," 成立。这是一个线性规划。若 ",[18,64773,64775,64793],{"className":64774,"translate":22},[21],[18,64776,64778],{"className":64777},[26],[28,64779,64780],{"xmlns":30},[32,64781,64782,64790],{},[35,64783,64784],{},[1556,64785,64786,64788],{},[45,64787,2266],{},[41,64789,39964],{},[86,64791,64792],{"encoding":88},"2^k",[18,64794,64796],{"className":64795,"ariaHidden":51},[93],[18,64797,64799,64802],{"className":64798},[97],[18,64800],{"className":64801,"style":35186},[101],[18,64803,64805,64808],{"className":64804},[106],[18,64806,2266],{"className":64807},[106],[18,64809,64811],{"className":64810},[115],[18,64812,64814],{"className":64813},[119],[18,64815,64817],{"className":64816},[124],[18,64818,64820],{"className":64819,"style":35186},[128],[18,64821,64822,64825],{"style":1962},[18,64823],{"className":64824,"style":137},[136],[18,64826,64828],{"className":64827},[141,142,143,144],[18,64829,39964],{"className":64830,"style":40040},[106,110,144]," 个标注全部可行，",[18,64833,64835,64848],{"className":64834,"translate":22},[21],[18,64836,64838],{"className":64837},[26],[28,64839,64840],{"xmlns":30},[32,64841,64842,64846],{},[35,64843,64844],{},[41,64845,51021],{},[86,64847,51021],{"encoding":88},[18,64849,64851],{"className":64850,"ariaHidden":51},[93],[18,64852,64854,64857],{"className":64853},[97],[18,64855],{"className":64856,"style":600},[101],[18,64858,51021],{"className":64859,"style":5073},[106,110]," 就被打散；对所有 ",[18,64862,64864,64877],{"className":64863,"translate":22},[21],[18,64865,64867],{"className":64866},[26],[28,64868,64869],{"xmlns":30},[32,64870,64871,64875],{},[35,64872,64873],{},[41,64874,51021],{},[86,64876,51021],{"encoding":88},[18,64878,64880],{"className":64879,"ariaHidden":51},[93],[18,64881,64883,64886],{"className":64882},[97],[18,64884],{"className":64885,"style":600},[101],[18,64887,51021],{"className":64888,"style":5073},[106,110]," 取最大，就得到 VC 维。对一般的假设类，精确判定通常是组合优化问题，实践中只能靠随机采样加优化或可满足性求解来估计。",[1311,64891,64893],{"id":64892},"_23-三个误区","2.3 三个误区",[11,64895,64896,64899],{},[717,64897,64898],{},"第一，「存在」不是「所有」。"," 定义只要求存在一个点集被打散。所以 VC 维大，不意味着假设类在任意数据上都很灵活；VC 维小，也不意味着它在某些特殊点集上不灵活。",[11,64901,64902,64905,64906,64909],{},[717,64903,64904],{},"第二，打散不是「能拟合这组数据」。"," 给定一组带标签的数据，几乎任何足够灵活的类都能拟合。打散要求的是：把这组点的标签",[717,64907,64908],{},"换着花样","全部试一遍，每一种都能实现。前者是插值能力，后者是组合丰富度。",[11,64911,64912,64915,64916,64944,64945,64973],{},[717,64913,64914],{},"第三，VC 维与参数化方式无关。"," 同一个函数集合可以有一万种参数化，VC 维不变。反过来，同一个参数数目可以对应 VC 维为 ",[18,64917,64919,64932],{"className":64918,"translate":22},[21],[18,64920,64922],{"className":64921},[26],[28,64923,64924],{"xmlns":30},[32,64925,64926,64930],{},[35,64927,64928],{},[45,64929,47],{},[86,64931,47],{"encoding":88},[18,64933,64935],{"className":64934,"ariaHidden":51},[93],[18,64936,64938,64941],{"className":64937},[97],[18,64939],{"className":64940,"style":505},[101],[18,64942,47],{"className":64943},[106]," 的类，也可以对应 VC 维为 ",[18,64946,64948,64961],{"className":64947,"translate":22},[21],[18,64949,64951],{"className":64950},[26],[28,64952,64953],{"xmlns":30},[32,64954,64955,64959],{},[35,64956,64957],{},[41,64958,2028],{"mathvariant":316},[86,64960,64111],{"encoding":88},[18,64962,64964],{"className":64963,"ariaHidden":51},[93],[18,64965,64967,64970],{"className":64966},[97],[18,64968],{"className":64969,"style":1083},[101],[18,64971,2028],{"className":64972},[106]," 的类。参数只是描述函数的一种方式，容量是函数集合本身的性质。",[11,64975,64976,64977,64980],{},"还有一点值得提前说明：上面定义的是",[717,64978,64979],{},"二值","函数类的 VC 维。如果假设输出实数（回归、概率、间隔），就需要 pseudo-dimension 或 fat-shattering dimension，后文会再提。",[638,64982,64984],{"id":64983},"三增长函数与-sauershelah-引理","三、增长函数与 Sauer–Shelah 引理",[1311,64986,64988],{"id":64987},"_31-增长函数","3.1 增长函数",[11,64990,64991],{},"逐个点集去数二分太麻烦。我们只关心最坏情况下能产生多少种二分：",[18,64993,64995],{"className":64994,"translate":22},[875],[18,64996,64998,65069],{"className":64997,"translate":22},[21],[18,64999,65001],{"className":65000},[26],[28,65002,65003],{"xmlns":30,"display":884},[32,65004,65005,65066],{},[35,65006,65007,65014,65016,65018,65020,65022,65052,65054,65056,65062,65064],{},[38,65008,65009,65012],{},[41,65010,65011],{"mathvariant":316},"Π",[41,65013,18309],{"mathvariant":1462},[49,65015,74],{"stretchy":73},[41,65017,28169],{},[49,65019,84],{"stretchy":73},[49,65021,433],{},[890,65023,65024,65030],{},[35,65025,65026,65028],{},[41,65027,5593],{},[49,65029,899],{},[35,65031,65032,65034,65036,65038,65040,65042,65044,65046,65048,65050],{},[41,65033,51021],{},[49,65035,4548],{},[41,65037,43],{"mathvariant":1462},[49,65039,52],{"separator":51},[926,65041,928],{},[41,65043,11225],{"mathvariant":316},[41,65045,51021],{},[41,65047,11225],{"mathvariant":316},[49,65049,433],{},[41,65051,28169],{},[49,65053,11225],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[41,65055,18309],{"mathvariant":1462},[38,65057,65058,65060],{},[41,65059,11225],{"mathvariant":316},[41,65061,51021],{},[49,65063,11225],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[41,65065,968],{"mathvariant":316},[86,65067,65068],{"encoding":88},"\\Pi_{\\mathcal H}(m)=\\max_{C\\subseteq\\mathcal X,\\ |C|=m}\\bigl|\\mathcal H|_C\\bigr|.",[18,65070,65072,65139],{"className":65071,"ariaHidden":51},[93],[18,65073,65075,65078,65121,65124,65127,65130,65133,65136],{"className":65074},[97],[18,65076],{"className":65077,"style":242},[101],[18,65079,65081,65084],{"className":65080},[106],[18,65082,65011],{"className":65083},[106],[18,65085,65087],{"className":65086},[115],[18,65088,65090,65113],{"className":65089},[119,120],[18,65091,65093,65110],{"className":65092},[124],[18,65094,65096],{"className":65095,"style":1152},[128],[18,65097,65098,65101],{"style":1155},[18,65099],{"className":65100,"style":137},[136],[18,65102,65104],{"className":65103},[141,142,143,144],[18,65105,65107],{"className":65106},[106,144],[18,65108,18309],{"className":65109,"style":46330},[106,1478,144],[18,65111,152],{"className":65112},[151],[18,65114,65116],{"className":65115},[124],[18,65117,65119],{"className":65118,"style":159},[128],[18,65120],{},[18,65122,74],{"className":65123},[250],[18,65125,28169],{"className":65126},[106,110],[18,65128,84],{"className":65129},[267],[18,65131],{"className":65132,"style":228},[169],[18,65134,433],{"className":65135},[232],[18,65137],{"className":65138,"style":228},[169],[18,65140,65142,65146,65223,65260,65263,65303,65340],{"className":65141},[97],[18,65143],{"className":65144,"style":65145},[101],"height:1.816em;vertical-align:-0.966em;",[18,65147,65149],{"className":65148},[985,986],[18,65150,65152,65215],{"className":65151},[119,120],[18,65153,65155,65212],{"className":65154},[124],[18,65156,65158,65202],{"className":65157,"style":1083},[128],[18,65159,65160,65163],{"style":33546},[18,65161],{"className":65162,"style":697},[136],[18,65164,65166],{"className":65165},[141,142,143,144],[18,65167,65169,65172,65175,65178,65181,65187,65190,65193,65196,65199],{"className":65168},[106,144],[18,65170,51021],{"className":65171,"style":5073},[106,110,144],[18,65173,4548],{"className":65174},[232,144],[18,65176,43],{"className":65177,"style":1479},[106,1478,144],[18,65179,52],{"className":65180},[165,144],[18,65182,65184],{"className":65183},[169,144],[18,65185,928],{"className":65186},[144],[18,65188,11225],{"className":65189},[106,144],[18,65191,51021],{"className":65192,"style":5073},[106,110,144],[18,65194,11225],{"className":65195},[106,144],[18,65197,433],{"className":65198},[232,144],[18,65200,28169],{"className":65201},[106,110,144],[18,65203,65204,65207],{"style":693},[18,65205],{"className":65206,"style":697},[136],[18,65208,65209],{},[18,65210,5593],{"className":65211},[985],[18,65213,152],{"className":65214},[151],[18,65216,65218],{"className":65217},[124],[18,65219,65221],{"className":65220,"style":33624},[128],[18,65222],{},[18,65224,65226],{"className":65225},[250],[18,65227,65229],{"className":65228},[1230,6108],[18,65230,65232,65252],{"className":65231},[119,120],[18,65233,65235,65249],{"className":65234},[124],[18,65236,65238],{"className":65237,"style":6118},[128],[18,65239,65240,65243],{"style":6121},[18,65241],{"className":65242,"style":6125},[136],[18,65244,65245],{"style":49869},[6130,65246,65247],{"xmlns":6132,"width":49872,"height":944,"viewBox":49873},[6136,65248],{"d":49876},[18,65250,152],{"className":65251},[151],[18,65253,65255],{"className":65254},[124],[18,65256,65258],{"className":65257,"style":6148},[128],[18,65259],{},[18,65261,18309],{"className":65262,"style":46330},[106,1478],[18,65264,65266,65269],{"className":65265},[106],[18,65267,11225],{"className":65268},[106],[18,65270,65272],{"className":65271},[115],[18,65273,65275,65295],{"className":65274},[119,120],[18,65276,65278,65292],{"className":65277},[124],[18,65279,65281],{"className":65280,"style":1152},[128],[18,65282,65283,65286],{"style":1155},[18,65284],{"className":65285,"style":137},[136],[18,65287,65289],{"className":65288},[141,142,143,144],[18,65290,51021],{"className":65291,"style":5073},[106,110,144],[18,65293,152],{"className":65294},[151],[18,65296,65298],{"className":65297},[124],[18,65299,65301],{"className":65300,"style":159},[128],[18,65302],{},[18,65304,65306],{"className":65305},[267],[18,65307,65309],{"className":65308},[1230,6108],[18,65310,65312,65332],{"className":65311},[119,120],[18,65313,65315,65329],{"className":65314},[124],[18,65316,65318],{"className":65317,"style":6118},[128],[18,65319,65320,65323],{"style":6121},[18,65321],{"className":65322,"style":6125},[136],[18,65324,65325],{"style":49869},[6130,65326,65327],{"xmlns":6132,"width":49872,"height":944,"viewBox":49873},[6136,65328],{"d":49876},[18,65330,152],{"className":65331},[151],[18,65333,65335],{"className":65334},[124],[18,65336,65338],{"className":65337,"style":6148},[128],[18,65339],{},[18,65341,968],{"className":65342},[106],[11,65344,65345,65346,46829,65374,65377,65378,5481,65518,65591],{},"它称为 ",[18,65347,65349,65362],{"className":65348,"translate":22},[21],[18,65350,65352],{"className":65351},[26],[28,65353,65354],{"xmlns":30},[32,65355,65356,65360],{},[35,65357,65358],{},[41,65359,18309],{"mathvariant":1462},[86,65361,46317],{"encoding":88},[18,65363,65365],{"className":65364,"ariaHidden":51},[93],[18,65366,65368,65371],{"className":65367},[97],[18,65369],{"className":65370,"style":600},[101],[18,65372,18309],{"className":65373,"style":46330},[106,1478],[717,65375,65376],{},"增长函数","（growth function）。显然 ",[18,65379,65381,65413],{"className":65380,"translate":22},[21],[18,65382,65384],{"className":65383},[26],[28,65385,65386],{"xmlns":30},[32,65387,65388,65410],{},[35,65389,65390,65396,65398,65400,65402,65404],{},[38,65391,65392,65394],{},[41,65393,65011],{"mathvariant":316},[41,65395,18309],{"mathvariant":1462},[49,65397,74],{"stretchy":73},[41,65399,28169],{},[49,65401,84],{"stretchy":73},[49,65403,4334],{},[1556,65405,65406,65408],{},[45,65407,2266],{},[41,65409,28169],{},[86,65411,65412],{"encoding":88},"\\Pi_{\\mathcal H}(m)\\le 2^m",[18,65414,65416,65483],{"className":65415,"ariaHidden":51},[93],[18,65417,65419,65422,65465,65468,65471,65474,65477,65480],{"className":65418},[97],[18,65420],{"className":65421,"style":242},[101],[18,65423,65425,65428],{"className":65424},[106],[18,65426,65011],{"className":65427},[106],[18,65429,65431],{"className":65430},[115],[18,65432,65434,65457],{"className":65433},[119,120],[18,65435,65437,65454],{"className":65436},[124],[18,65438,65440],{"className":65439,"style":1152},[128],[18,65441,65442,65445],{"style":1155},[18,65443],{"className":65444,"style":137},[136],[18,65446,65448],{"className":65447},[141,142,143,144],[18,65449,65451],{"className":65450},[106,144],[18,65452,18309],{"className":65453,"style":46330},[106,1478,144],[18,65455,152],{"className":65456},[151],[18,65458,65460],{"className":65459},[124],[18,65461,65463],{"className":65462,"style":159},[128],[18,65464],{},[18,65466,74],{"className":65467},[250],[18,65469,28169],{"className":65470},[106,110],[18,65472,84],{"className":65473},[267],[18,65475],{"className":65476,"style":228},[169],[18,65478,4334],{"className":65479},[232],[18,65481],{"className":65482,"style":228},[169],[18,65484,65486,65489],{"className":65485},[97],[18,65487],{"className":65488,"style":1959},[101],[18,65490,65492,65495],{"className":65491},[106],[18,65493,2266],{"className":65494},[106],[18,65496,65498],{"className":65497},[115],[18,65499,65501],{"className":65500},[119],[18,65502,65504],{"className":65503},[124],[18,65505,65507],{"className":65506,"style":1959},[128],[18,65508,65509,65512],{"style":1962},[18,65510],{"className":65511,"style":137},[136],[18,65513,65515],{"className":65514},[141,142,143,144],[18,65516,28169],{"className":65517},[106,110,144],[18,65519,65521,65539],{"className":65520,"translate":22},[21],[18,65522,65524],{"className":65523},[26],[28,65525,65526],{"xmlns":30},[32,65527,65528,65536],{},[35,65529,65530],{},[38,65531,65532,65534],{},[41,65533,65011],{"mathvariant":316},[41,65535,18309],{"mathvariant":1462},[86,65537,65538],{"encoding":88},"\\Pi_{\\mathcal H}",[18,65540,65542],{"className":65541,"ariaHidden":51},[93],[18,65543,65545,65548],{"className":65544},[97],[18,65546],{"className":65547,"style":4563},[101],[18,65549,65551,65554],{"className":65550},[106],[18,65552,65011],{"className":65553},[106],[18,65555,65557],{"className":65556},[115],[18,65558,65560,65583],{"className":65559},[119,120],[18,65561,65563,65580],{"className":65562},[124],[18,65564,65566],{"className":65565,"style":1152},[128],[18,65567,65568,65571],{"style":1155},[18,65569],{"className":65570,"style":137},[136],[18,65572,65574],{"className":65573},[141,142,143,144],[18,65575,65577],{"className":65576},[106,144],[18,65578,18309],{"className":65579,"style":46330},[106,1478,144],[18,65581,152],{"className":65582},[151],[18,65584,65586],{"className":65585},[124],[18,65587,65589],{"className":65588,"style":159},[128],[18,65590],{}," 单调不减。",[11,65593,65594,65595,65644,65645,65785,65786,65814,65815,65843,65844,65872,65873,65901,65902,65930,65931,66027,66028,66079,66080,1990],{},"由定义，",[18,65596,65598,65620],{"className":65597,"translate":22},[21],[18,65599,65601],{"className":65600},[26],[28,65602,65603],{"xmlns":30},[32,65604,65605,65617],{},[35,65606,65607,65609,65611,65613,65615],{},[41,65608,52216],{"mathvariant":316},[49,65610,899],{},[49,65612,74],{"stretchy":73},[41,65614,18309],{"mathvariant":1462},[49,65616,84],{"stretchy":73},[86,65618,65619],{"encoding":88},"\\operatorname{VCdim}(\\mathcal H)",[18,65621,65623],{"className":65622,"ariaHidden":51},[93],[18,65624,65626,65629,65635,65638,65641],{"className":65625},[97],[18,65627],{"className":65628,"style":242},[101],[18,65630,65632],{"className":65631},[985],[18,65633,52216],{"className":65634},[106,6354],[18,65636,74],{"className":65637},[250],[18,65639,18309],{"className":65640,"style":46330},[106,1478],[18,65642,84],{"className":65643},[267]," 就是使 ",[18,65646,65648,65680],{"className":65647,"translate":22},[21],[18,65649,65651],{"className":65650},[26],[28,65652,65653],{"xmlns":30},[32,65654,65655,65677],{},[35,65656,65657,65663,65665,65667,65669,65671],{},[38,65658,65659,65661],{},[41,65660,65011],{"mathvariant":316},[41,65662,18309],{"mathvariant":1462},[49,65664,74],{"stretchy":73},[41,65666,28169],{},[49,65668,84],{"stretchy":73},[49,65670,433],{},[1556,65672,65673,65675],{},[45,65674,2266],{},[41,65676,28169],{},[86,65678,65679],{"encoding":88},"\\Pi_{\\mathcal H}(m)=2^m",[18,65681,65683,65750],{"className":65682,"ariaHidden":51},[93],[18,65684,65686,65689,65732,65735,65738,65741,65744,65747],{"className":65685},[97],[18,65687],{"className":65688,"style":242},[101],[18,65690,65692,65695],{"className":65691},[106],[18,65693,65011],{"className":65694},[106],[18,65696,65698],{"className":65697},[115],[18,65699,65701,65724],{"className":65700},[119,120],[18,65702,65704,65721],{"className":65703},[124],[18,65705,65707],{"className":65706,"style":1152},[128],[18,65708,65709,65712],{"style":1155},[18,65710],{"className":65711,"style":137},[136],[18,65713,65715],{"className":65714},[141,142,143,144],[18,65716,65718],{"className":65717},[106,144],[18,65719,18309],{"className":65720,"style":46330},[106,1478,144],[18,65722,152],{"className":65723},[151],[18,65725,65727],{"className":65726},[124],[18,65728,65730],{"className":65729,"style":159},[128],[18,65731],{},[18,65733,74],{"className":65734},[250],[18,65736,28169],{"className":65737},[106,110],[18,65739,84],{"className":65740},[267],[18,65742],{"className":65743,"style":228},[169],[18,65745,433],{"className":65746},[232],[18,65748],{"className":65749,"style":228},[169],[18,65751,65753,65756],{"className":65752},[97],[18,65754],{"className":65755,"style":1959},[101],[18,65757,65759,65762],{"className":65758},[106],[18,65760,2266],{"className":65761},[106],[18,65763,65765],{"className":65764},[115],[18,65766,65768],{"className":65767},[119],[18,65769,65771],{"className":65770},[124],[18,65772,65774],{"className":65773,"style":1959},[128],[18,65775,65776,65779],{"style":1962},[18,65777],{"className":65778,"style":137},[136],[18,65780,65782],{"className":65781},[141,142,143,144],[18,65783,28169],{"className":65784},[106,110,144]," 成立的最大的 ",[18,65787,65789,65802],{"className":65788,"translate":22},[21],[18,65790,65792],{"className":65791},[26],[28,65793,65794],{"xmlns":30},[32,65795,65796,65800],{},[35,65797,65798],{},[41,65799,28169],{},[86,65801,28169],{"encoding":88},[18,65803,65805],{"className":65804,"ariaHidden":51},[93],[18,65806,65808,65811],{"className":65807},[97],[18,65809],{"className":65810,"style":1083},[101],[18,65812,28169],{"className":65813},[106,110],"（若对所有 ",[18,65816,65818,65831],{"className":65817,"translate":22},[21],[18,65819,65821],{"className":65820},[26],[28,65822,65823],{"xmlns":30},[32,65824,65825,65829],{},[35,65826,65827],{},[41,65828,28169],{},[86,65830,28169],{"encoding":88},[18,65832,65834],{"className":65833,"ariaHidden":51},[93],[18,65835,65837,65840],{"className":65836},[97],[18,65838],{"className":65839,"style":1083},[101],[18,65841,28169],{"className":65842},[106,110]," 都成立，则为 ",[18,65845,65847,65860],{"className":65846,"translate":22},[21],[18,65848,65850],{"className":65849},[26],[28,65851,65852],{"xmlns":30},[32,65853,65854,65858],{},[35,65855,65856],{},[41,65857,2028],{"mathvariant":316},[86,65859,64111],{"encoding":88},[18,65861,65863],{"className":65862,"ariaHidden":51},[93],[18,65864,65866,65869],{"className":65865},[97],[18,65867],{"className":65868,"style":1083},[101],[18,65870,2028],{"className":65871},[106],"）。注意这个性质是「向下封闭」的：若 ",[18,65874,65876,65889],{"className":65875,"translate":22},[21],[18,65877,65879],{"className":65878},[26],[28,65880,65881],{"xmlns":30},[32,65882,65883,65887],{},[35,65884,65885],{},[41,65886,51021],{},[86,65888,51021],{"encoding":88},[18,65890,65892],{"className":65891,"ariaHidden":51},[93],[18,65893,65895,65898],{"className":65894},[97],[18,65896],{"className":65897,"style":600},[101],[18,65899,51021],{"className":65900,"style":5073},[106,110]," 被打散，",[18,65903,65905,65918],{"className":65904,"translate":22},[21],[18,65906,65908],{"className":65907},[26],[28,65909,65910],{"xmlns":30},[32,65911,65912,65916],{},[35,65913,65914],{},[41,65915,51021],{},[86,65917,51021],{"encoding":88},[18,65919,65921],{"className":65920,"ariaHidden":51},[93],[18,65922,65924,65927],{"className":65923},[97],[18,65925],{"className":65926,"style":600},[101],[18,65928,51021],{"className":65929,"style":5073},[106,110]," 的任何子集也被打散，所以一旦 ",[18,65932,65934,65962],{"className":65933,"translate":22},[21],[18,65935,65937],{"className":65936},[26],[28,65938,65939],{"xmlns":30},[32,65940,65941,65959],{},[35,65942,65943,65945,65947,65949,65951,65953],{},[41,65944,65011],{"mathvariant":316},[49,65946,74],{"stretchy":73},[41,65948,28169],{},[49,65950,84],{"stretchy":73},[49,65952,433],{},[1556,65954,65955,65957],{},[45,65956,2266],{},[41,65958,28169],{},[86,65960,65961],{"encoding":88},"\\Pi(m)=2^m",[18,65963,65965,65992],{"className":65964,"ariaHidden":51},[93],[18,65966,65968,65971,65974,65977,65980,65983,65986,65989],{"className":65967},[97],[18,65969],{"className":65970,"style":242},[101],[18,65972,65011],{"className":65973},[106],[18,65975,74],{"className":65976},[250],[18,65978,28169],{"className":65979},[106,110],[18,65981,84],{"className":65982},[267],[18,65984],{"className":65985,"style":228},[169],[18,65987,433],{"className":65988},[232],[18,65990],{"className":65991,"style":228},[169],[18,65993,65995,65998],{"className":65994},[97],[18,65996],{"className":65997,"style":1959},[101],[18,65999,66001,66004],{"className":66000},[106],[18,66002,2266],{"className":66003},[106],[18,66005,66007],{"className":66006},[115],[18,66008,66010],{"className":66009},[119],[18,66011,66013],{"className":66012},[124],[18,66014,66016],{"className":66015,"style":1959},[128],[18,66017,66018,66021],{"style":1962},[18,66019],{"className":66020,"style":137},[136],[18,66022,66024],{"className":66023},[141,142,143,144],[18,66025,28169],{"className":66026},[106,110,144],"，对所有 ",[18,66029,66031,66049],{"className":66030,"translate":22},[21],[18,66032,66034],{"className":66033},[26],[28,66035,66036],{"xmlns":30},[32,66037,66038,66046],{},[35,66039,66040,66042,66044],{},[41,66041,39964],{},[49,66043,4334],{},[41,66045,28169],{},[86,66047,66048],{"encoding":88},"k\\le m",[18,66050,66052,66070],{"className":66051,"ariaHidden":51},[93],[18,66053,66055,66058,66061,66064,66067],{"className":66054},[97],[18,66056],{"className":66057,"style":52357},[101],[18,66059,39964],{"className":66060,"style":40040},[106,110],[18,66062],{"className":66063,"style":228},[169],[18,66065,4334],{"className":66066},[232],[18,66068],{"className":66069,"style":228},[169],[18,66071,66073,66076],{"className":66072},[97],[18,66074],{"className":66075,"style":1083},[101],[18,66077,28169],{"className":66078},[106,110]," 都有 ",[18,66081,66083,66111],{"className":66082,"translate":22},[21],[18,66084,66086],{"className":66085},[26],[28,66087,66088],{"xmlns":30},[32,66089,66090,66108],{},[35,66091,66092,66094,66096,66098,66100,66102],{},[41,66093,65011],{"mathvariant":316},[49,66095,74],{"stretchy":73},[41,66097,39964],{},[49,66099,84],{"stretchy":73},[49,66101,433],{},[1556,66103,66104,66106],{},[45,66105,2266],{},[41,66107,39964],{},[86,66109,66110],{"encoding":88},"\\Pi(k)=2^k",[18,66112,66114,66141],{"className":66113,"ariaHidden":51},[93],[18,66115,66117,66120,66123,66126,66129,66132,66135,66138],{"className":66116},[97],[18,66118],{"className":66119,"style":242},[101],[18,66121,65011],{"className":66122},[106],[18,66124,74],{"className":66125},[250],[18,66127,39964],{"className":66128,"style":40040},[106,110],[18,66130,84],{"className":66131},[267],[18,66133],{"className":66134,"style":228},[169],[18,66136,433],{"className":66137},[232],[18,66139],{"className":66140,"style":228},[169],[18,66142,66144,66147],{"className":66143},[97],[18,66145],{"className":66146,"style":35186},[101],[18,66148,66150,66153],{"className":66149},[106],[18,66151,2266],{"className":66152},[106],[18,66154,66156],{"className":66155},[115],[18,66157,66159],{"className":66158},[119],[18,66160,66162],{"className":66161},[124],[18,66163,66165],{"className":66164,"style":35186},[128],[18,66166,66167,66170],{"style":1962},[18,66168],{"className":66169,"style":137},[136],[18,66171,66173],{"className":66172},[141,142,143,144],[18,66174,39964],{"className":66175,"style":40040},[106,110,144],[1311,66177,66179],{"id":66178},"_32-sauershelah-引理","3.2 Sauer–Shelah 引理",[5159,66181,66182,66308],{},[11,66183,66184,51960,66187,21943,66280,4741],{},[717,66185,66186],{},"定理 3.1（Sauer–Shelah）",[18,66188,66190,66220],{"className":66189,"translate":22},[21],[18,66191,66193],{"className":66192},[26],[28,66194,66195],{"xmlns":30},[32,66196,66197,66217],{},[35,66198,66199,66201,66203,66205,66207,66209,66211,66213,66215],{},[41,66200,52216],{"mathvariant":316},[49,66202,899],{},[49,66204,74],{"stretchy":73},[41,66206,18309],{"mathvariant":1462},[49,66208,84],{"stretchy":73},[49,66210,433],{},[41,66212,4031],{},[49,66214,6684],{},[41,66216,2028],{"mathvariant":316},[86,66218,66219],{"encoding":88},"\\operatorname{VCdim}(\\mathcal H)=d\u003C\\infty",[18,66221,66223,66253,66271],{"className":66222,"ariaHidden":51},[93],[18,66224,66226,66229,66235,66238,66241,66244,66247,66250],{"className":66225},[97],[18,66227],{"className":66228,"style":242},[101],[18,66230,66232],{"className":66231},[985],[18,66233,52216],{"className":66234},[106,6354],[18,66236,74],{"className":66237},[250],[18,66239,18309],{"className":66240,"style":46330},[106,1478],[18,66242,84],{"className":66243},[267],[18,66245],{"className":66246,"style":228},[169],[18,66248,433],{"className":66249},[232],[18,66251],{"className":66252,"style":228},[169],[18,66254,66256,66259,66262,66265,66268],{"className":66255},[97],[18,66257],{"className":66258,"style":1422},[101],[18,66260,4031],{"className":66261},[106,110],[18,66263],{"className":66264,"style":228},[169],[18,66266,6684],{"className":66267},[232],[18,66269],{"className":66270,"style":228},[169],[18,66272,66274,66277],{"className":66273},[97],[18,66275],{"className":66276,"style":1083},[101],[18,66278,2028],{"className":66279},[106],[18,66281,66283,66296],{"className":66282,"translate":22},[21],[18,66284,66286],{"className":66285},[26],[28,66287,66288],{"xmlns":30},[32,66289,66290,66294],{},[35,66291,66292],{},[41,66293,28169],{},[86,66295,28169],{"encoding":88},[18,66297,66299],{"className":66298,"ariaHidden":51},[93],[18,66300,66302,66305],{"className":66301},[97],[18,66303],{"className":66304,"style":1083},[101],[18,66306,28169],{"className":66307},[106,110],[18,66309,66311],{"className":66310,"translate":22},[875],[18,66312,66314,66369],{"className":66313,"translate":22},[21],[18,66315,66317],{"className":66316},[26],[28,66318,66319],{"xmlns":30,"display":884},[32,66320,66321,66366],{},[35,66322,66323,66329,66331,66333,66335,66337,66351,66364],{},[38,66324,66325,66327],{},[41,66326,65011],{"mathvariant":316},[41,66328,18309],{"mathvariant":1462},[49,66330,74],{"stretchy":73},[41,66332,28169],{},[49,66334,84],{"stretchy":73},[49,66336,4334],{},[20365,66338,66339,66341,66349],{},[49,66340,20369],{},[35,66342,66343,66345,66347],{},[41,66344,5600],{},[49,66346,433],{},[45,66348,430],{},[41,66350,4031],{},[35,66352,66353,66355,66362],{},[49,66354,74],{"fence":51},[5654,66356,66358,66360],{"linethickness":66357},"0px",[41,66359,28169],{},[41,66361,5600],{},[49,66363,84],{"fence":51},[41,66365,968],{"mathvariant":316},[86,66367,66368],{"encoding":88},"\\Pi_{\\mathcal H}(m)\\le\\sum_{i=0}^{d}\\binom{m}{i}.",[18,66370,66372,66439],{"className":66371,"ariaHidden":51},[93],[18,66373,66375,66378,66421,66424,66427,66430,66433,66436],{"className":66374},[97],[18,66376],{"className":66377,"style":242},[101],[18,66379,66381,66384],{"className":66380},[106],[18,66382,65011],{"className":66383},[106],[18,66385,66387],{"className":66386},[115],[18,66388,66390,66413],{"className":66389},[119,120],[18,66391,66393,66410],{"className":66392},[124],[18,66394,66396],{"className":66395,"style":1152},[128],[18,66397,66398,66401],{"style":1155},[18,66399],{"className":66400,"style":137},[136],[18,66402,66404],{"className":66403},[141,142,143,144],[18,66405,66407],{"className":66406},[106,144],[18,66408,18309],{"className":66409,"style":46330},[106,1478,144],[18,66411,152],{"className":66412},[151],[18,66414,66416],{"className":66415},[124],[18,66417,66419],{"className":66418,"style":159},[128],[18,66420],{},[18,66422,74],{"className":66423},[250],[18,66425,28169],{"className":66426},[106,110],[18,66428,84],{"className":66429},[267],[18,66431],{"className":66432,"style":228},[169],[18,66434,4334],{"className":66435},[232],[18,66437],{"className":66438,"style":228},[169],[18,66440,66442,66446,66514,66517,66577],{"className":66441},[97],[18,66443],{"className":66444,"style":66445},[101],"height:3.1138em;vertical-align:-1.2777em;",[18,66447,66449],{"className":66448},[985,986],[18,66450,66452,66506],{"className":66451},[119,120],[18,66453,66455,66503],{"className":66454},[124],[18,66456,66459,66479,66489],{"className":66457,"style":66458},[128],"height:1.8361em;",[18,66460,66461,66464],{"style":20555},[18,66462],{"className":66463,"style":20559},[136],[18,66465,66467],{"className":66466},[141,142,143,144],[18,66468,66470,66473,66476],{"className":66469},[106,144],[18,66471,5600],{"className":66472},[106,110,144],[18,66474,433],{"className":66475},[232,144],[18,66477,430],{"className":66478},[106,144],[18,66480,66481,66484],{"style":20577},[18,66482],{"className":66483,"style":20559},[136],[18,66485,66486],{},[18,66487,20369],{"className":66488},[985,4198,4199],[18,66490,66491,66494],{"style":20588},[18,66492],{"className":66493,"style":20559},[136],[18,66495,66497],{"className":66496},[141,142,143,144],[18,66498,66500],{"className":66499},[106,144],[18,66501,4031],{"className":66502},[106,110,144],[18,66504,152],{"className":66505},[151],[18,66507,66509],{"className":66508},[124],[18,66510,66512],{"className":66511,"style":20607},[128],[18,66513],{},[18,66515],{"className":66516,"style":170},[169],[18,66518,66520,66526,66571],{"className":66519},[106],[18,66521,66523],{"className":66522,"style":13796},[250,13795],[18,66524,74],{"className":66525},[1230,143],[18,66527,66529],{"className":66528},[5654],[18,66530,66532,66563],{"className":66531},[119,120],[18,66533,66535,66560],{"className":66534},[124],[18,66536,66538,66549],{"className":66537,"style":13852},[128],[18,66539,66540,66543],{"style":6028},[18,66541],{"className":66542,"style":697},[136],[18,66544,66546],{"className":66545},[106],[18,66547,5600],{"className":66548},[106,110],[18,66550,66551,66554],{"style":6051},[18,66552],{"className":66553,"style":697},[136],[18,66555,66557],{"className":66556},[106],[18,66558,28169],{"className":66559},[106,110],[18,66561,152],{"className":66562},[151],[18,66564,66566],{"className":66565},[124],[18,66567,66569],{"className":66568,"style":6070},[128],[18,66570],{},[18,66572,66574],{"className":66573,"style":13796},[267,13795],[18,66575,84],{"className":66576},[1230,143],[18,66578,968],{"className":66579},[106],[11,66581,66582,66583,66634,66635,66807,66808,66966,66967,67042],{},"证明用归纳法，对 ",[18,66584,66586,66604],{"className":66585,"translate":22},[21],[18,66587,66589],{"className":66588},[26],[28,66590,66591],{"xmlns":30},[32,66592,66593,66601],{},[35,66594,66595,66597,66599],{},[41,66596,28169],{},[49,66598,7987],{},[41,66600,4031],{},[86,66602,66603],{"encoding":88},"m+d",[18,66605,66607,66625],{"className":66606,"ariaHidden":51},[93],[18,66608,66610,66613,66616,66619,66622],{"className":66609},[97],[18,66611],{"className":66612,"style":14404},[101],[18,66614,28169],{"className":66615},[106,110],[18,66617],{"className":66618,"style":2068},[169],[18,66620,7987],{"className":66621},[2072],[18,66623],{"className":66624,"style":2068},[169],[18,66626,66628,66631],{"className":66627},[97],[18,66629],{"className":66630,"style":295},[101],[18,66632,4031],{"className":66633},[106,110]," 归纳。取任意 ",[18,66636,66638,66676],{"className":66637,"translate":22},[21],[18,66639,66641],{"className":66640},[26],[28,66642,66643],{"xmlns":30},[32,66644,66645,66673],{},[35,66646,66647,66649,66651,66653,66659,66661,66663,66665,66671],{},[41,66648,51021],{},[49,66650,433],{},[49,66652,1331],{"stretchy":73},[38,66654,66655,66657],{},[41,66656,39067],{},[45,66658,47],{},[49,66660,52],{"separator":51},[49,66662,55],{},[49,66664,52],{"separator":51},[38,66666,66667,66669],{},[41,66668,39067],{},[41,66670,28169],{},[49,66672,1349],{"stretchy":73},[86,66674,66675],{"encoding":88},"C=\\{c_1,\\dots,c_m\\}",[18,66677,66679,66697],{"className":66678,"ariaHidden":51},[93],[18,66680,66682,66685,66688,66691,66694],{"className":66681},[97],[18,66683],{"className":66684,"style":600},[101],[18,66686,51021],{"className":66687,"style":5073},[106,110],[18,66689],{"className":66690,"style":228},[169],[18,66692,433],{"className":66693},[232],[18,66695],{"className":66696,"style":228},[169],[18,66698,66700,66703,66706,66746,66749,66752,66755,66758,66761,66764,66804],{"className":66699},[97],[18,66701],{"className":66702,"style":242},[101],[18,66704,1331],{"className":66705},[250],[18,66707,66709,66712],{"className":66708},[106],[18,66710,39067],{"className":66711},[106,110],[18,66713,66715],{"className":66714},[115],[18,66716,66718,66738],{"className":66717},[119,120],[18,66719,66721,66735],{"className":66720},[124],[18,66722,66724],{"className":66723,"style":129},[128],[18,66725,66726,66729],{"style":1155},[18,66727],{"className":66728,"style":137},[136],[18,66730,66732],{"className":66731},[141,142,143,144],[18,66733,47],{"className":66734},[106,144],[18,66736,152],{"className":66737},[151],[18,66739,66741],{"className":66740},[124],[18,66742,66744],{"className":66743,"style":159},[128],[18,66745],{},[18,66747,52],{"className":66748},[165],[18,66750],{"className":66751,"style":170},[169],[18,66753,55],{"className":66754},[174],[18,66756],{"className":66757,"style":170},[169],[18,66759,52],{"className":66760},[165],[18,66762],{"className":66763,"style":170},[169],[18,66765,66767,66770],{"className":66766},[106],[18,66768,39067],{"className":66769},[106,110],[18,66771,66773],{"className":66772},[115],[18,66774,66776,66796],{"className":66775},[119,120],[18,66777,66779,66793],{"className":66778},[124],[18,66780,66782],{"className":66781,"style":202},[128],[18,66783,66784,66787],{"style":1155},[18,66785],{"className":66786,"style":137},[136],[18,66788,66790],{"className":66789},[141,142,143,144],[18,66791,28169],{"className":66792},[106,110,144],[18,66794,152],{"className":66795},[151],[18,66797,66799],{"className":66798},[124],[18,66800,66802],{"className":66801,"style":159},[128],[18,66803],{},[18,66805,1349],{"className":66806},[267],"，记 ",[18,66809,66811,66846],{"className":66810,"translate":22},[21],[18,66812,66814],{"className":66813},[26],[28,66815,66816],{"xmlns":30},[32,66817,66818,66843],{},[35,66819,66820,66826,66828,66830,66833,66835,66841],{},[1556,66821,66822,66824],{},[41,66823,51021],{},[49,66825,32629],{"mathvariant":316,"lspace":32628,"rspace":32628},[49,66827,433],{},[41,66829,51021],{},[49,66831,66832],{},"∖",[49,66834,1331],{"stretchy":73},[38,66836,66837,66839],{},[41,66838,39067],{},[41,66840,28169],{},[49,66842,1349],{"stretchy":73},[86,66844,66845],{"encoding":88},"C'=C\\setminus\\{c_m\\}",[18,66847,66849,66896,66914],{"className":66848,"ariaHidden":51},[93],[18,66850,66852,66855,66887,66890,66893],{"className":66851},[97],[18,66853],{"className":66854,"style":32660},[101],[18,66856,66858,66861],{"className":66857},[106],[18,66859,51021],{"className":66860,"style":5073},[106,110],[18,66862,66864],{"className":66863},[115],[18,66865,66867],{"className":66866},[119],[18,66868,66870],{"className":66869},[124],[18,66871,66873],{"className":66872,"style":32660},[128],[18,66874,66875,66878],{"style":1962},[18,66876],{"className":66877,"style":137},[136],[18,66879,66881],{"className":66880},[141,142,143,144],[18,66882,66884],{"className":66883},[106,144],[18,66885,32629],{"className":66886},[106,144],[18,66888],{"className":66889,"style":228},[169],[18,66891,433],{"className":66892},[232],[18,66894],{"className":66895,"style":228},[169],[18,66897,66899,66902,66905,66908,66911],{"className":66898},[97],[18,66900],{"className":66901,"style":242},[101],[18,66903,51021],{"className":66904,"style":5073},[106,110],[18,66906],{"className":66907,"style":2068},[169],[18,66909,66832],{"className":66910},[2072],[18,66912],{"className":66913,"style":2068},[169],[18,66915,66917,66920,66923,66963],{"className":66916},[97],[18,66918],{"className":66919,"style":242},[101],[18,66921,1331],{"className":66922},[250],[18,66924,66926,66929],{"className":66925},[106],[18,66927,39067],{"className":66928},[106,110],[18,66930,66932],{"className":66931},[115],[18,66933,66935,66955],{"className":66934},[119,120],[18,66936,66938,66952],{"className":66937},[124],[18,66939,66941],{"className":66940,"style":202},[128],[18,66942,66943,66946],{"style":1155},[18,66944],{"className":66945,"style":137},[136],[18,66947,66949],{"className":66948},[141,142,143,144],[18,66950,28169],{"className":66951},[106,110,144],[18,66953,152],{"className":66954},[151],[18,66956,66958],{"className":66957},[124],[18,66959,66961],{"className":66960,"style":159},[128],[18,66962],{},[18,66964,1349],{"className":66965},[267],"。把 ",[18,66968,66970,66990],{"className":66969,"translate":22},[21],[18,66971,66973],{"className":66972},[26],[28,66974,66975],{"xmlns":30},[32,66976,66977,66987],{},[35,66978,66979,66981],{},[41,66980,18309],{"mathvariant":1462},[38,66982,66983,66985],{},[41,66984,11225],{"mathvariant":316},[41,66986,51021],{},[86,66988,66989],{"encoding":88},"\\mathcal H|_C",[18,66991,66993],{"className":66992,"ariaHidden":51},[93],[18,66994,66996,66999,67002],{"className":66995},[97],[18,66997],{"className":66998,"style":242},[101],[18,67000,18309],{"className":67001,"style":46330},[106,1478],[18,67003,67005,67008],{"className":67004},[106],[18,67006,11225],{"className":67007},[106],[18,67009,67011],{"className":67010},[115],[18,67012,67014,67034],{"className":67013},[119,120],[18,67015,67017,67031],{"className":67016},[124],[18,67018,67020],{"className":67019,"style":1152},[128],[18,67021,67022,67025],{"style":1155},[18,67023],{"className":67024,"style":137},[136],[18,67026,67028],{"className":67027},[141,142,143,144],[18,67029,51021],{"className":67030,"style":5073},[106,110,144],[18,67032,152],{"className":67033},[151],[18,67035,67037],{"className":67036},[124],[18,67038,67040],{"className":67039,"style":159},[128],[18,67041],{}," 按最后一个坐标分成两类：",[2210,67044,67045,67147],{},[2213,67046,67047,67117,67118,67146],{},[18,67048,67050,67068],{"className":67049,"translate":22},[21],[18,67051,67053],{"className":67052},[26],[28,67054,67055],{"xmlns":30},[32,67056,67057,67065],{},[35,67058,67059],{},[38,67060,67061,67063],{},[41,67062,58460],{},[45,67064,430],{},[86,67066,67067],{"encoding":88},"A_0",[18,67069,67071],{"className":67070,"ariaHidden":51},[93],[18,67072,67074,67077],{"className":67073},[97],[18,67075],{"className":67076,"style":4563},[101],[18,67078,67080,67083],{"className":67079},[106],[18,67081,58460],{"className":67082},[106,110],[18,67084,67086],{"className":67085},[115],[18,67087,67089,67109],{"className":67088},[119,120],[18,67090,67092,67106],{"className":67091},[124],[18,67093,67095],{"className":67094,"style":129},[128],[18,67096,67097,67100],{"style":1155},[18,67098],{"className":67099,"style":137},[136],[18,67101,67103],{"className":67102},[141,142,143,144],[18,67104,430],{"className":67105},[106,144],[18,67107,152],{"className":67108},[151],[18,67110,67112],{"className":67111},[124],[18,67113,67115],{"className":67114,"style":159},[128],[18,67116],{},"：最后一个坐标为 ",[18,67119,67121,67134],{"className":67120,"translate":22},[21],[18,67122,67124],{"className":67123},[26],[28,67125,67126],{"xmlns":30},[32,67127,67128,67132],{},[35,67129,67130],{},[45,67131,430],{},[86,67133,430],{"encoding":88},[18,67135,67137],{"className":67136,"ariaHidden":51},[93],[18,67138,67140,67143],{"className":67139},[97],[18,67141],{"className":67142,"style":505},[101],[18,67144,430],{"className":67145},[106]," 的二分；",[2213,67148,67149,67117,67219,67247],{},[18,67150,67152,67170],{"className":67151,"translate":22},[21],[18,67153,67155],{"className":67154},[26],[28,67156,67157],{"xmlns":30},[32,67158,67159,67167],{},[35,67160,67161],{},[38,67162,67163,67165],{},[41,67164,58460],{},[45,67166,47],{},[86,67168,67169],{"encoding":88},"A_1",[18,67171,67173],{"className":67172,"ariaHidden":51},[93],[18,67174,67176,67179],{"className":67175},[97],[18,67177],{"className":67178,"style":4563},[101],[18,67180,67182,67185],{"className":67181},[106],[18,67183,58460],{"className":67184},[106,110],[18,67186,67188],{"className":67187},[115],[18,67189,67191,67211],{"className":67190},[119,120],[18,67192,67194,67208],{"className":67193},[124],[18,67195,67197],{"className":67196,"style":129},[128],[18,67198,67199,67202],{"style":1155},[18,67200],{"className":67201,"style":137},[136],[18,67203,67205],{"className":67204},[141,142,143,144],[18,67206,47],{"className":67207},[106,144],[18,67209,152],{"className":67210},[151],[18,67212,67214],{"className":67213},[124],[18,67215,67217],{"className":67216,"style":159},[128],[18,67218],{},[18,67220,67222,67235],{"className":67221,"translate":22},[21],[18,67223,67225],{"className":67224},[26],[28,67226,67227],{"xmlns":30},[32,67228,67229,67233],{},[35,67230,67231],{},[45,67232,47],{},[86,67234,47],{"encoding":88},[18,67236,67238],{"className":67237,"ariaHidden":51},[93],[18,67239,67241,67244],{"className":67240},[97],[18,67242],{"className":67243,"style":505},[101],[18,67245,47],{"className":67246},[106]," 的二分。",[11,67249,16,67250,509,67319,3019,67388,67450,67451,10363],{},[18,67251,67253,67270],{"className":67252,"translate":22},[21],[18,67254,67256],{"className":67255},[26],[28,67257,67258],{"xmlns":30},[32,67259,67260,67268],{},[35,67261,67262],{},[38,67263,67264,67266],{},[41,67265,58460],{},[45,67267,430],{},[86,67269,67067],{"encoding":88},[18,67271,67273],{"className":67272,"ariaHidden":51},[93],[18,67274,67276,67279],{"className":67275},[97],[18,67277],{"className":67278,"style":4563},[101],[18,67280,67282,67285],{"className":67281},[106],[18,67283,58460],{"className":67284},[106,110],[18,67286,67288],{"className":67287},[115],[18,67289,67291,67311],{"className":67290},[119,120],[18,67292,67294,67308],{"className":67293},[124],[18,67295,67297],{"className":67296,"style":129},[128],[18,67298,67299,67302],{"style":1155},[18,67300],{"className":67301,"style":137},[136],[18,67303,67305],{"className":67304},[141,142,143,144],[18,67306,430],{"className":67307},[106,144],[18,67309,152],{"className":67310},[151],[18,67312,67314],{"className":67313},[124],[18,67315,67317],{"className":67316,"style":159},[128],[18,67318],{},[18,67320,67322,67339],{"className":67321,"translate":22},[21],[18,67323,67325],{"className":67324},[26],[28,67326,67327],{"xmlns":30},[32,67328,67329,67337],{},[35,67330,67331],{},[38,67332,67333,67335],{},[41,67334,58460],{},[45,67336,47],{},[86,67338,67169],{"encoding":88},[18,67340,67342],{"className":67341,"ariaHidden":51},[93],[18,67343,67345,67348],{"className":67344},[97],[18,67346],{"className":67347,"style":4563},[101],[18,67349,67351,67354],{"className":67350},[106],[18,67352,58460],{"className":67353},[106,110],[18,67355,67357],{"className":67356},[115],[18,67358,67360,67380],{"className":67359},[119,120],[18,67361,67363,67377],{"className":67362},[124],[18,67364,67366],{"className":67365,"style":129},[128],[18,67367,67368,67371],{"style":1155},[18,67369],{"className":67370,"style":137},[136],[18,67372,67374],{"className":67373},[141,142,143,144],[18,67375,47],{"className":67376},[106,144],[18,67378,152],{"className":67379},[151],[18,67381,67383],{"className":67382},[124],[18,67384,67386],{"className":67385,"style":159},[128],[18,67387],{},[18,67389,67391,67409],{"className":67390,"translate":22},[21],[18,67392,67394],{"className":67393},[26],[28,67395,67396],{"xmlns":30},[32,67397,67398,67406],{},[35,67399,67400],{},[1556,67401,67402,67404],{},[41,67403,51021],{},[49,67405,32629],{"mathvariant":316,"lspace":32628,"rspace":32628},[86,67407,67408],{"encoding":88},"C'",[18,67410,67412],{"className":67411,"ariaHidden":51},[93],[18,67413,67415,67418],{"className":67414},[97],[18,67416],{"className":67417,"style":32660},[101],[18,67419,67421,67424],{"className":67420},[106],[18,67422,51021],{"className":67423,"style":5073},[106,110],[18,67425,67427],{"className":67426},[115],[18,67428,67430],{"className":67429},[119],[18,67431,67433],{"className":67432},[124],[18,67434,67436],{"className":67435,"style":32660},[128],[18,67437,67438,67441],{"style":1962},[18,67439],{"className":67440,"style":137},[136],[18,67442,67444],{"className":67443},[141,142,143,144],[18,67445,67447],{"className":67446},[106,144],[18,67448,32629],{"className":67449},[106,144]," 上的投影的交集大小为 ",[18,67452,67454,67467],{"className":67453,"translate":22},[21],[18,67455,67457],{"className":67456},[26],[28,67458,67459],{"xmlns":30},[32,67460,67461,67465],{},[35,67462,67463],{},[41,67464,44311],{},[86,67466,44311],{"encoding":88},[18,67468,67470],{"className":67469,"ariaHidden":51},[93],[18,67471,67473,67476],{"className":67472},[97],[18,67474],{"className":67475,"style":1083},[101],[18,67477,44311],{"className":67478},[106,110],[18,67480,67482],{"className":67481,"translate":22},[875],[18,67483,67485,67557],{"className":67484,"translate":22},[21],[18,67486,67488],{"className":67487},[26],[28,67489,67490],{"xmlns":30,"display":884},[32,67491,67492,67554],{},[35,67493,67494,67496,67498,67504,67506,67508,67510,67516,67518,67520,67522,67528,67530,67532,67538,67540,67542,67544,67546,67548,67550,67552],{},[41,67495,11225],{"mathvariant":316},[41,67497,18309],{"mathvariant":1462},[38,67499,67500,67502],{},[41,67501,11225],{"mathvariant":316},[41,67503,51021],{},[41,67505,11225],{"mathvariant":316},[49,67507,433],{},[41,67509,11225],{"mathvariant":316},[38,67511,67512,67514],{},[41,67513,58460],{},[45,67515,430],{},[41,67517,11225],{"mathvariant":316},[49,67519,7987],{},[41,67521,11225],{"mathvariant":316},[38,67523,67524,67526],{},[41,67525,58460],{},[45,67527,47],{},[41,67529,11225],{"mathvariant":316},[49,67531,4334],{},[38,67533,67534,67536],{},[41,67535,65011],{"mathvariant":316},[41,67537,18309],{"mathvariant":1462},[49,67539,74],{"stretchy":73},[41,67541,28169],{},[49,67543,2253],{},[45,67545,47],{},[49,67547,84],{"stretchy":73},[49,67549,7987],{},[41,67551,44311],{},[41,67553,968],{"mathvariant":316},[86,67555,67556],{"encoding":88},"|\\mathcal H|_C|=|A_0|+|A_1|\\le \\Pi_{\\mathcal H}(m-1)+s.",[18,67558,67560,67624,67685,67746,67810,67831],{"className":67559,"ariaHidden":51},[93],[18,67561,67563,67566,67569,67572,67612,67615,67618,67621],{"className":67562},[97],[18,67564],{"className":67565,"style":242},[101],[18,67567,11225],{"className":67568},[106],[18,67570,18309],{"className":67571,"style":46330},[106,1478],[18,67573,67575,67578],{"className":67574},[106],[18,67576,11225],{"className":67577},[106],[18,67579,67581],{"className":67580},[115],[18,67582,67584,67604],{"className":67583},[119,120],[18,67585,67587,67601],{"className":67586},[124],[18,67588,67590],{"className":67589,"style":1152},[128],[18,67591,67592,67595],{"style":1155},[18,67593],{"className":67594,"style":137},[136],[18,67596,67598],{"className":67597},[141,142,143,144],[18,67599,51021],{"className":67600,"style":5073},[106,110,144],[18,67602,152],{"className":67603},[151],[18,67605,67607],{"className":67606},[124],[18,67608,67610],{"className":67609,"style":159},[128],[18,67611],{},[18,67613,11225],{"className":67614},[106],[18,67616],{"className":67617,"style":228},[169],[18,67619,433],{"className":67620},[232],[18,67622],{"className":67623,"style":228},[169],[18,67625,67627,67630,67633,67673,67676,67679,67682],{"className":67626},[97],[18,67628],{"className":67629,"style":242},[101],[18,67631,11225],{"className":67632},[106],[18,67634,67636,67639],{"className":67635},[106],[18,67637,58460],{"className":67638},[106,110],[18,67640,67642],{"className":67641},[115],[18,67643,67645,67665],{"className":67644},[119,120],[18,67646,67648,67662],{"className":67647},[124],[18,67649,67651],{"className":67650,"style":129},[128],[18,67652,67653,67656],{"style":1155},[18,67654],{"className":67655,"style":137},[136],[18,67657,67659],{"className":67658},[141,142,143,144],[18,67660,430],{"className":67661},[106,144],[18,67663,152],{"className":67664},[151],[18,67666,67668],{"className":67667},[124],[18,67669,67671],{"className":67670,"style":159},[128],[18,67672],{},[18,67674,11225],{"className":67675},[106],[18,67677],{"className":67678,"style":2068},[169],[18,67680,7987],{"className":67681},[2072],[18,67683],{"className":67684,"style":2068},[169],[18,67686,67688,67691,67694,67734,67737,67740,67743],{"className":67687},[97],[18,67689],{"className":67690,"style":242},[101],[18,67692,11225],{"className":67693},[106],[18,67695,67697,67700],{"className":67696},[106],[18,67698,58460],{"className":67699},[106,110],[18,67701,67703],{"className":67702},[115],[18,67704,67706,67726],{"className":67705},[119,120],[18,67707,67709,67723],{"className":67708},[124],[18,67710,67712],{"className":67711,"style":129},[128],[18,67713,67714,67717],{"style":1155},[18,67715],{"className":67716,"style":137},[136],[18,67718,67720],{"className":67719},[141,142,143,144],[18,67721,47],{"className":67722},[106,144],[18,67724,152],{"className":67725},[151],[18,67727,67729],{"className":67728},[124],[18,67730,67732],{"className":67731,"style":159},[128],[18,67733],{},[18,67735,11225],{"className":67736},[106],[18,67738],{"className":67739,"style":228},[169],[18,67741,4334],{"className":67742},[232],[18,67744],{"className":67745,"style":228},[169],[18,67747,67749,67752,67795,67798,67801,67804,67807],{"className":67748},[97],[18,67750],{"className":67751,"style":242},[101],[18,67753,67755,67758],{"className":67754},[106],[18,67756,65011],{"className":67757},[106],[18,67759,67761],{"className":67760},[115],[18,67762,67764,67787],{"className":67763},[119,120],[18,67765,67767,67784],{"className":67766},[124],[18,67768,67770],{"className":67769,"style":1152},[128],[18,67771,67772,67775],{"style":1155},[18,67773],{"className":67774,"style":137},[136],[18,67776,67778],{"className":67777},[141,142,143,144],[18,67779,67781],{"className":67780},[106,144],[18,67782,18309],{"className":67783,"style":46330},[106,1478,144],[18,67785,152],{"className":67786},[151],[18,67788,67790],{"className":67789},[124],[18,67791,67793],{"className":67792,"style":159},[128],[18,67794],{},[18,67796,74],{"className":67797},[250],[18,67799,28169],{"className":67800},[106,110],[18,67802],{"className":67803,"style":2068},[169],[18,67805,2253],{"className":67806},[2072],[18,67808],{"className":67809,"style":2068},[169],[18,67811,67813,67816,67819,67822,67825,67828],{"className":67812},[97],[18,67814],{"className":67815,"style":242},[101],[18,67817,47],{"className":67818},[106],[18,67820,84],{"className":67821},[267],[18,67823],{"className":67824,"style":2068},[169],[18,67826,7987],{"className":67827},[2072],[18,67829],{"className":67830,"style":2068},[169],[18,67832,67834,67837,67840],{"className":67833},[97],[18,67835],{"className":67836,"style":1083},[101],[18,67838,44311],{"className":67839},[106,110],[18,67841,968],{"className":67842},[106],[11,67844,67845,67846,67907,67908,67975,67976,68041,68042,68103,68104,68132,68133,68161,68162,68242,68243,68293,68294,68424,68425,1990],{},"交集里的每个二分，都对应 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上一种「最后一位可以任意翻转」的模式。把这些模式组成的集合记为 ",[18,67909,67911,67933],{"className":67910,"translate":22},[21],[18,67912,67914],{"className":67913},[26],[28,67915,67916],{"xmlns":30},[32,67917,67918,67930],{},[35,67919,67920],{},[1556,67921,67922,67924],{},[41,67923,18309],{"mathvariant":1462},[35,67925,67926,67928],{},[49,67927,32629],{"mathvariant":316},[49,67929,32629],{"mathvariant":316},[86,67931,67932],{"encoding":88},"\\mathcal H''",[18,67934,67936],{"className":67935,"ariaHidden":51},[93],[18,67937,67939,67942],{"className":67938},[97],[18,67940],{"className":67941,"style":32660},[101],[18,67943,67945,67948],{"className":67944},[106],[18,67946,18309],{"className":67947,"style":46330},[106,1478],[18,67949,67951],{"className":67950},[115],[18,67952,67954],{"className":67953},[119],[18,67955,67957],{"className":67956},[124],[18,67958,67960],{"className":67959,"style":32660},[128],[18,67961,67962,67965],{"style":1962},[18,67963],{"className":67964,"style":137},[136],[18,67966,67968],{"className":67967},[141,142,143,144],[18,67969,67971],{"className":67970},[106,144],[18,67972,67974],{"className":67973},[106,144],"′′","。若 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打散了 ",[18,68043,68045,68062],{"className":68044,"translate":22},[21],[18,68046,68048],{"className":68047},[26],[28,68049,68050],{"xmlns":30},[32,68051,68052,68060],{},[35,68053,68054],{},[1556,68055,68056,68058],{},[41,68057,51021],{},[49,68059,32629],{"mathvariant":316,"lspace":32628,"rspace":32628},[86,68061,67408],{"encoding":88},[18,68063,68065],{"className":68064,"ariaHidden":51},[93],[18,68066,68068,68071],{"className":68067},[97],[18,68069],{"className":68070,"style":32660},[101],[18,68072,68074,68077],{"className":68073},[106],[18,68075,51021],{"className":68076,"style":5073},[106,110],[18,68078,68080],{"className":68079},[115],[18,68081,68083],{"className":68082},[119],[18,68084,68086],{"className":68085},[124],[18,68087,68089],{"className":68088,"style":32660},[128],[18,68090,68091,68094],{"style":1962},[18,68092],{"className":68093,"style":137},[136],[18,68095,68097],{"className":68096},[141,142,143,144],[18,68098,68100],{"className":68099},[106,144],[18,68101,32629],{"className":68102},[106,144]," 的某个 ",[18,68105,68107,68120],{"className":68106,"translate":22},[21],[18,68108,68110],{"className":68109},[26],[28,68111,68112],{"xmlns":30},[32,68113,68114,68118],{},[35,68115,68116],{},[41,68117,4031],{},[86,68119,4031],{"encoding":88},[18,68121,68123],{"className":68122,"ariaHidden":51},[93],[18,68124,68126,68129],{"className":68125},[97],[18,68127],{"className":68128,"style":295},[101],[18,68130,4031],{"className":68131},[106,110]," 元子集，那么 ",[18,68134,68136,68149],{"className":68135,"translate":22},[21],[18,68137,68139],{"className":68138},[26],[28,68140,68141],{"xmlns":30},[32,68142,68143,68147],{},[35,68144,68145],{},[41,68146,18309],{"mathvariant":1462},[86,68148,46317],{"encoding":88},[18,68150,68152],{"className":68151,"ariaHidden":51},[93],[18,68153,68155,68158],{"className":68154},[97],[18,68156],{"className":68157,"style":600},[101],[18,68159,18309],{"className":68160,"style":46330},[106,1478]," 就打散了该子集并上 ",[18,68163,68165,68187],{"className":68164,"translate":22},[21],[18,68166,68168],{"className":68167},[26],[28,68169,68170],{"xmlns":30},[32,68171,68172,68184],{},[35,68173,68174,68176,68182],{},[49,68175,1331],{"stretchy":73},[38,68177,68178,68180],{},[41,68179,39067],{},[41,68181,28169],{},[49,68183,1349],{"stretchy":73},[86,68185,68186],{"encoding":88},"\\{c_m\\}",[18,68188,68190],{"className":68189,"ariaHidden":51},[93],[18,68191,68193,68196,68199,68239],{"className":68192},[97],[18,68194],{"className":68195,"style":242},[101],[18,68197,1331],{"className":68198},[250],[18,68200,68202,68205],{"className":68201},[106],[18,68203,39067],{"className":68204},[106,110],[18,68206,68208],{"className":68207},[115],[18,68209,68211,68231],{"className":68210},[119,120],[18,68212,68214,68228],{"className":68213},[124],[18,68215,68217],{"className":68216,"style":202},[128],[18,68218,68219,68222],{"style":1155},[18,68220],{"className":68221,"style":137},[136],[18,68223,68225],{"className":68224},[141,142,143,144],[18,68226,28169],{"className":68227},[106,110,144],[18,68229,152],{"className":68230},[151],[18,68232,68234],{"className":68233},[124],[18,68235,68237],{"className":68236,"style":159},[128],[18,68238],{},[18,68240,1349],{"className":68241},[267],"，共 ",[18,68244,68246,68263],{"className":68245,"translate":22},[21],[18,68247,68249],{"className":68248},[26],[28,68250,68251],{"xmlns":30},[32,68252,68253,68261],{},[35,68254,68255,68257,68259],{},[41,68256,4031],{},[49,68258,7987],{},[45,68260,47],{},[86,68262,56253],{"encoding":88},[18,68264,68266,68284],{"className":68265,"ariaHidden":51},[93],[18,68267,68269,68272,68275,68278,68281],{"className":68268},[97],[18,68270],{"className":68271,"style":7224},[101],[18,68273,4031],{"className":68274},[106,110],[18,68276],{"className":68277,"style":2068},[169],[18,68279,7987],{"className":68280},[2072],[18,68282],{"className":68283,"style":2068},[169],[18,68285,68287,68290],{"className":68286},[97],[18,68288],{"className":68289,"style":505},[101],[18,68291,47],{"className":68292},[106]," 个点，矛盾。所以 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d-1",[18,68336,68338,68397,68415],{"className":68337,"ariaHidden":51},[93],[18,68339,68341,68344,68350,68353,68385,68388,68391,68394],{"className":68340},[97],[18,68342],{"className":68343,"style":39977},[101],[18,68345,68347],{"className":68346},[985],[18,68348,52216],{"className":68349},[106,6354],[18,68351,74],{"className":68352},[250],[18,68354,68356,68359],{"className":68355},[106],[18,68357,18309],{"className":68358,"style":46330},[106,1478],[18,68360,68362],{"className":68361},[115],[18,68363,68365],{"className":68364},[119],[18,68366,68368],{"className":68367},[124],[18,68369,68371],{"className":68370,"style":32660},[128],[18,68372,68373,68376],{"style":1962},[18,68374],{"className":68375,"style":137},[136],[18,68377,68379],{"className":68378},[141,142,143,144],[18,68380,68382],{"className":68381},[106,144],[18,68383,67974],{"className":68384},[106,144],[18,68386,84],{"className":68387},[267],[18,68389],{"className":68390,"style":228},[169],[18,68392,4334],{"className":68393},[232],[18,68395],{"className":68396,"style":228},[169],[18,68398,68400,68403,68406,68409,68412],{"className":68399},[97],[18,68401],{"className":68402,"style":7224},[101],[18,68404,4031],{"className":68405},[106,110],[18,68407],{"className":68408,"style":2068},[169],[18,68410,2253],{"className":68411},[2072],[18,68413],{"className":68414,"style":2068},[169],[18,68416,68418,68421],{"className":68417},[97],[18,68419],{"className":68420,"style":505},[101],[18,68422,47],{"className":68423},[106],"，从而 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1,68800,68801],{},"代回并用 Pascal 公式：",[18,68803,68805],{"className":68804,"translate":22},[875],[18,68806,68808,68936],{"className":68807,"translate":22},[21],[18,68809,68811],{"className":68810},[26],[28,68812,68813],{"xmlns":30,"display":884},[32,68814,68815,68933],{},[35,68816,68817,68823,68825,68827,68829,68831,68845,68863,68865,68885,68903,68905,68919,68931],{},[38,68818,68819,68821],{},[41,68820,65011],{"mathvariant":316},[41,68822,18309],{"mathvariant":1462},[49,68824,74],{"stretchy":73},[41,68826,28169],{},[49,68828,84],{"stretchy":73},[49,68830,4334],{},[20365,68832,68833,68835,68843],{},[49,68834,20369],{},[35,68836,68837,68839,68841],{},[41,68838,5600],{},[49,68840,433],{},[45,68842,430],{},[41,68844,4031],{},[35,68846,68847,68849,68861],{},[49,68848,74],{"fence":51},[5654,68850,68851,68859],{"linethickness":66357},[35,68852,68853,68855,68857],{},[41,68854,28169],{},[49,68856,2253],{},[45,68858,47],{},[41,68860,5600],{},[49,68862,84],{"fence":51},[49,68864,7987],{},[20365,68866,68867,68869,68877],{},[49,68868,20369],{},[35,68870,68871,68873,68875],{},[41,68872,5600],{},[49,68874,433],{},[45,68876,430],{},[35,68878,68879,68881,68883],{},[41,68880,4031],{},[49,68882,2253],{},[45,68884,47],{},[35,68886,68887,68889,68901],{},[49,68888,74],{"fence":51},[5654,68890,68891,68899],{"linethickness":66357},[35,68892,68893,68895,68897],{},[41,68894,28169],{},[49,68896,2253],{},[45,68898,47],{},[41,68900,5600],{},[49,68902,84],{"fence":51},[49,68904,433],{},[20365,68906,68907,68909,68917],{},[49,68908,20369],{},[35,68910,68911,68913,68915],{},[41,68912,5600],{},[49,68914,433],{},[45,68916,430],{},[41,68918,4031],{},[35,68920,68921,68923,68929],{},[49,68922,74],{"fence":51},[5654,68924,68925,68927],{"linethickness":66357},[41,68926,28169],{},[41,68928,5600],{},[49,68930,84],{"fence":51},[41,68932,968],{"mathvariant":316},[86,68934,68935],{"encoding":88},"\\Pi_{\\mathcal 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维有限的假设类，在 ",[18,71281,71283,71296],{"className":71282,"translate":22},[21],[18,71284,71286],{"className":71285},[26],[28,71287,71288],{"xmlns":30},[32,71289,71290,71294],{},[35,71291,71292],{},[41,71293,28169],{},[86,71295,28169],{"encoding":88},[18,71297,71299],{"className":71298,"ariaHidden":51},[93],[18,71300,71302,71305],{"className":71301},[97],[18,71303],{"className":71304,"style":1083},[101],[18,71306,28169],{"className":71307},[106,110]," 个点上最多只有 ",[18,71310,71312,71330],{"className":71311,"translate":22},[21],[18,71313,71315],{"className":71314},[26],[28,71316,71317],{"xmlns":30},[32,71318,71319,71327],{},[35,71320,71321],{},[1556,71322,71323,71325],{},[41,71324,28169],{},[41,71326,4031],{},[86,71328,71329],{"encoding":88},"m^d",[18,71331,71333],{"className":71332,"ariaHidden":51},[93],[18,71334,71336,71339],{"className":71335},[97],[18,71337],{"className":71338,"style":35186},[101],[18,71340,71342,71345],{"className":71341},[106],[18,71343,28169],{"className":71344},[106,110],[18,71346,71348],{"className":71347},[115],[18,71349,71351],{"className":71350},[119],[18,71352,71354],{"className":71353},[124],[18,71355,71357],{"className":71356,"style":35186},[128],[18,71358,71359,71362],{"style":1962},[18,71360],{"className":71361,"style":137},[136],[18,71363,71365],{"className":71364},[141,142,143,144],[18,71366,4031],{"className":71367},[106,110,144]," 种行为，而不是 ",[18,71370,71372,71389],{"className":71371,"translate":22},[21],[18,71373,71375],{"className":71374},[26],[28,71376,71377],{"xmlns":30},[32,71378,71379,71387],{},[35,71380,71381],{},[1556,71382,71383,71385],{},[45,71384,2266],{},[41,71386,28169],{},[86,71388,44902],{"encoding":88},[18,71390,71392],{"className":71391,"ariaHidden":51},[93],[18,71393,71395,71398],{"className":71394},[97],[18,71396],{"className":71397,"style":1959},[101],[18,71399,71401,71404],{"className":71400},[106],[18,71402,2266],{"className":71403},[106],[18,71405,71407],{"className":71406},[115],[18,71408,71410],{"className":71409},[119],[18,71411,71413],{"className":71412},[124],[18,71414,71416],{"className":71415,"style":1959},[128],[18,71417,71418,71421],{"style":1962},[18,71419],{"className":71420,"style":137},[136],[18,71422,71424],{"className":71423},[141,142,143,144],[18,71425,28169],{"className":71426},[106,110,144]," 种。当 ",[18,71429,71431,71444],{"className":71430,"translate":22},[21],[18,71432,71434],{"className":71433},[26],[28,71435,71436],{"xmlns":30},[32,71437,71438,71442],{},[35,71439,71440],{},[41,71441,28169],{},[86,71443,28169],{"encoding":88},[18,71445,71447],{"className":71446,"ariaHidden":51},[93],[18,71448,71450,71453],{"className":71449},[97],[18,71451],{"className":71452,"style":1083},[101],[18,71454,28169],{"className":71455},[106,110]," 增大时，这两种增长速度的差距是压倒性的——",[18,71458,71460,71477],{"className":71459,"translate":22},[21],[18,71461,71463],{"className":71462},[26],[28,71464,71465],{"xmlns":30},[32,71466,71467,71475],{},[35,71468,71469],{},[1556,71470,71471,71473],{},[41,71472,28169],{},[41,71474,4031],{},[86,71476,71329],{"encoding":88},[18,71478,71480],{"className":71479,"ariaHidden":51},[93],[18,71481,71483,71486],{"className":71482},[97],[18,71484],{"className":71485,"style":35186},[101],[18,71487,71489,71492],{"className":71488},[106],[18,71490,28169],{"className":71491},[106,110],[18,71493,71495],{"className":71494},[115],[18,71496,71498],{"className":71497},[119],[18,71499,71501],{"className":71500},[124],[18,71502,71504],{"className":71503,"style":35186},[128],[18,71505,71506,71509],{"style":1962},[18,71507],{"className":71508,"style":137},[136],[18,71510,71512],{"className":71511},[141,142,143,144],[18,71513,4031],{"className":71514},[106,110,144]," 取对数后只有 ",[18,71517,71519,71539],{"className":71518,"translate":22},[21],[18,71520,71522],{"className":71521},[26],[28,71523,71524],{"xmlns":30},[32,71525,71526,71536],{},[35,71527,71528,71530,71532,71534],{},[41,71529,4031],{},[41,71531,14733],{},[49,71533,899],{},[41,71535,28169],{},[86,71537,71538],{"encoding":88},"d\\log m",[18,71540,71542],{"className":71541,"ariaHidden":51},[93],[18,71543,71545,71548,71551,71554,71559,71562],{"className":71544},[97],[18,71546],{"className":71547,"style":2890},[101],[18,71549,4031],{"className":71550},[106,110],[18,71552],{"className":71553,"style":170},[169],[18,71555,15174,71557],{"className":71556},[985],[18,71558,15177],{"style":6355},[18,71560],{"className":71561,"style":170},[169],[18,71563,28169],{"className":71564},[106,110],"，而 ",[18,71567,71569,71586],{"className":71568,"translate":22},[21],[18,71570,71572],{"className":71571},[26],[28,71573,71574],{"xmlns":30},[32,71575,71576,71584],{},[35,71577,71578],{},[1556,71579,71580,71582],{},[45,71581,2266],{},[41,71583,28169],{},[86,71585,44902],{"encoding":88},[18,71587,71589],{"className":71588,"ariaHidden":51},[93],[18,71590,71592,71595],{"className":71591},[97],[18,71593],{"className":71594,"style":1959},[101],[18,71596,71598,71601],{"className":71597},[106],[18,71599,2266],{"className":71600},[106],[18,71602,71604],{"className":71603},[115],[18,71605,71607],{"className":71606},[119],[18,71608,71610],{"className":71609},[124],[18,71611,71613],{"className":71612,"style":1959},[128],[18,71614,71615,71618],{"style":1962},[18,71616],{"className":71617,"style":137},[136],[18,71619,71621],{"className":71620},[141,142,143,144],[18,71622,28169],{"className":71623},[106,110,144]," 取对数后是 ",[18,71626,71628,71641],{"className":71627,"translate":22},[21],[18,71629,71631],{"className":71630},[26],[28,71632,71633],{"xmlns":30},[32,71634,71635,71639],{},[35,71636,71637],{},[41,71638,28169],{},[86,71640,28169],{"encoding":88},[18,71642,71644],{"className":71643,"ariaHidden":51},[93],[18,71645,71647,71650],{"className":71646},[97],[18,71648],{"className":71649,"style":1083},[101],[18,71651,28169],{"className":71652},[106,110],[11,71654,71655,71656,71659],{},"换句话说，",[717,71657,71658],{},"有限 VC 维等价于增长函数是多项式级的","。这就是「容量」一词的数学来源：容量不是参数个数，而是假设类能产生的二分数量随样本量的增长速度。",[638,71661,71663],{"id":71662},"四一致收敛定理","四、一致收敛定理",[1311,71665,71667],{"id":71666},"_41-对称化","4.1 对称化",[11,71669,71670,71671,71674],{},"增长函数是组合量，要把它变成泛化界，需要一步概率论证，称为",[717,71672,71673],{},"对称化","（symmetrization）。",[11,71676,16,71677,71739,71740,71768,71769,71797],{},[18,71678,71680,71698],{"className":71679,"translate":22},[21],[18,71681,71683],{"className":71682},[26],[28,71684,71685],{"xmlns":30},[32,71686,71687,71695],{},[35,71688,71689],{},[1556,71690,71691,71693],{},[41,71692,38934],{},[49,71694,32629],{"mathvariant":316,"lspace":32628,"rspace":32628},[86,71696,71697],{"encoding":88},"S'",[18,71699,71701],{"className":71700,"ariaHidden":51},[93],[18,71702,71704,71707],{"className":71703},[97],[18,71705],{"className":71706,"style":32660},[101],[18,71708,71710,71713],{"className":71709},[106],[18,71711,38934],{"className":71712,"style":38967},[106,110],[18,71714,71716],{"className":71715},[115],[18,71717,71719],{"className":71718},[119],[18,71720,71722],{"className":71721},[124],[18,71723,71725],{"className":71724,"style":32660},[128],[18,71726,71727,71730],{"style":1962},[18,71728],{"className":71729,"style":137},[136],[18,71731,71733],{"className":71732},[141,142,143,144],[18,71734,71736],{"className":71735},[106,144],[18,71737,32629],{"className":71738},[106,144]," 是与 ",[18,71741,71743,71756],{"className":71742,"translate":22},[21],[18,71744,71746],{"className":71745},[26],[28,71747,71748],{"xmlns":30},[32,71749,71750,71754],{},[35,71751,71752],{},[41,71753,38934],{},[86,71755,38934],{"encoding":88},[18,71757,71759],{"className":71758,"ariaHidden":51},[93],[18,71760,71762,71765],{"className":71761},[97],[18,71763],{"className":71764,"style":600},[101],[18,71766,38934],{"className":71767,"style":38967},[106,110]," 独立同分布的另一组 ",[18,71770,71772,71785],{"className":71771,"translate":22},[21],[18,71773,71775],{"className":71774},[26],[28,71776,71777],{"xmlns":30},[32,71778,71779,71783],{},[35,71780,71781],{},[41,71782,28169],{},[86,71784,28169],{"encoding":88},[18,71786,71788],{"className":71787,"ariaHidden":51},[93],[18,71789,71791,71794],{"className":71790},[97],[18,71792],{"className":71793,"style":1083},[101],[18,71795,28169],{"className":71796},[106,110]," 个样本。核心不等式是",[18,71799,71801],{"className":71800,"translate":22},[875],[18,71802,71804,71937],{"className":71803,"translate":22},[21],[18,71805,71807],{"className":71806},[26],[28,71808,71809],{"xmlns":30,"display":884},[32,71810,71811,71934],{},[35,71812,71813,71829,71831,71837,71839,71841,71843,71845,71851,71853,71855,71857,71859,71862,71864,71866,71882,71898,71900,71906,71908,71910,71912,71914,71924,71926,71928,71930,71932],{},[890,71814,71815,71821],{},[35,71816,71817,71819],{},[41,71818,912],{},[49,71820,899],{},[35,71822,71823,71825,71827],{},[41,71824,18328],{},[49,71826,921],{},[41,71828,18309],{"mathvariant":1462},[49,71830,11225],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[38,71832,71833,71835],{},[41,71834,12582],{},[41,71836,12575],{"mathvariant":1462},[49,71838,74],{"stretchy":73},[41,71840,18328],{},[49,71842,84],{"stretchy":73},[49,71844,2253],{},[38,71846,71847,71849],{},[41,71848,12582],{},[41,71850,38934],{},[49,71852,74],{"stretchy":73},[41,71854,18328],{},[49,71856,84],{"stretchy":73},[49,71858,11225],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[926,71860,71861],{},"  ",[49,71863,42719],{},[926,71865,71861],{},[38,71867,71868,71870],{},[41,71869,934],{"mathvariant":933},[35,71871,71872,71874,71876],{},[41,71873,38934],{},[49,71875,52],{"separator":51},[1556,71877,71878,71880],{},[41,71879,38934],{},[49,71881,32629],{"mathvariant":316,"lspace":32628,"rspace":32628},[890,71883,71884,71890],{},[35,71885,71886,71888],{},[41,71887,912],{},[49,71889,899],{},[35,71891,71892,71894,71896],{},[41,71893,18328],{},[49,71895,921],{},[41,71897,18309],{"mathvariant":1462},[49,71899,11225],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[38,71901,71902,71904],{},[41,71903,12582],{},[41,71905,38934],{},[49,71907,74],{"stretchy":73},[41,71909,18328],{},[49,71911,84],{"stretchy":73},[49,71913,2253],{},[38,71915,71916,71918],{},[41,71917,12582],{},[1556,71919,71920,71922],{},[41,71921,38934],{},[49,71923,32629],{"mathvariant":316,"lspace":32628,"rspace":32628},[49,71925,74],{"stretchy":73},[41,71927,18328],{},[49,71929,84],{"stretchy":73},[49,71931,11225],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,71933,52],{"separator":51},[86,71935,71936],{"encoding":88},"\\sup_{h\\in\\mathcal H}\\bigl|L_{\\mathcal D}(h)-L_S(h)\\bigr|\n\\;\\lesssim\\;\n\\mathbb E_{S,S'}\\sup_{h\\in\\mathcal H}\\bigl|L_S(h)-L_{S'}(h)\\bigr|,",[18,71938,71940,72098,72205,72440],{"className":71939,"ariaHidden":51},[93],[18,71941,71943,71947,72000,72037,72080,72083,72086,72089,72092,72095],{"className":71942},[97],[18,71944],{"className":71945,"style":71946},[101],"height:1.8239em;vertical-align:-0.9739em;",[18,71948,71950],{"className":71949},[985,986],[18,71951,71953,71992],{"className":71952},[119,120],[18,71954,71956,71989],{"className":71955},[124],[18,71957,71959,71979],{"className":71958,"style":1083},[128],[18,71960,71961,71964],{"style":1086},[18,71962],{"className":71963,"style":697},[136],[18,71965,71967],{"className":71966},[141,142,143,144],[18,71968,71970,71973,71976],{"className":71969},[106,144],[18,71971,18328],{"className":71972},[106,110,144],[18,71974,921],{"className":71975},[232,144],[18,71977,18309],{"className":71978,"style":46330},[106,1478,144],[18,71980,71981,71984],{"style":693},[18,71982],{"className":71983,"style":697},[136],[18,71985,71986],{},[18,71987,912],{"className":71988},[985],[18,71990,152],{"className":71991},[151],[18,71993,71995],{"className":71994},[124],[18,71996,71998],{"className":71997,"style":1124},[128],[18,71999],{},[18,72001,72003],{"className":72002},[250],[18,72004,72006],{"className":72005},[1230,6108],[18,72007,72009,72029],{"className":72008},[119,120],[18,72010,72012,72026],{"className":72011},[124],[18,72013,72015],{"className":72014,"style":6118},[128],[18,72016,72017,72020],{"style":6121},[18,72018],{"className":72019,"style":6125},[136],[18,72021,72022],{"style":49869},[6130,72023,72024],{"xmlns":6132,"width":49872,"height":944,"viewBox":49873},[6136,72025],{"d":49876},[18,72027,152],{"className":72028},[151],[18,72030,72032],{"className":72031},[124],[18,72033,72035],{"className":72034,"style":6148},[128],[18,72036],{},[18,72038,72040,72043],{"className":72039},[106],[18,72041,12582],{"className":72042},[106,110],[18,72044,72046],{"className":72045},[115],[18,72047,72049,72072],{"className":72048},[119,120],[18,72050,72052,72069],{"className":72051},[124],[18,72053,72055],{"className":72054,"style":1152},[128],[18,72056,72057,72060],{"style":1155},[18,72058],{"className":72059,"style":137},[136],[18,72061,72063],{"className":72062},[141,142,143,144],[18,72064,72066],{"className":72065},[106,144],[18,72067,12575],{"className":72068,"style":254},[106,1478,144],[18,72070,152],{"className":72071},[151],[18,72073,72075],{"className":72074},[124],[18,72076,72078],{"className":72077,"style":159},[128],[18,72079],{},[18,72081,74],{"className":72082},[250],[18,72084,18328],{"className":72085},[106,110],[18,72087,84],{"className":72088},[267],[18,72090],{"className":72091,"style":2068},[169],[18,72093,2253],{"className":72094},[2072],[18,72096],{"className":72097,"style":2068},[169],[18,72099,72101,72104,72144,72147,72150,72153,72190,72193,72196,72199,72202],{"className":72100},[97],[18,72102],{"className":72103,"style":6101},[101],[18,72105,72107,72110],{"className":72106},[106],[18,72108,12582],{"className":72109},[106,110],[18,72111,72113],{"className":72112},[115],[18,72114,72116,72136],{"className":72115},[119,120],[18,72117,72119,72133],{"className":72118},[124],[18,72120,72122],{"className":72121,"style":1152},[128],[18,72123,72124,72127],{"style":1155},[18,72125],{"className":72126,"style":137},[136],[18,72128,72130],{"className":72129},[141,142,143,144],[18,72131,38934],{"className":72132,"style":38967},[106,110,144],[18,72134,152],{"className":72135},[151],[18,72137,72139],{"className":72138},[124],[18,72140,72142],{"className":72141,"style":159},[128],[18,72143],{},[18,72145,74],{"className":72146},[250],[18,72148,18328],{"className":72149},[106,110],[18,72151,84],{"className":72152},[267],[18,72154,72156],{"className":72155},[267],[18,72157,72159],{"className":72158},[1230,6108],[18,72160,72162,72182],{"className":72161},[119,120],[18,72163,72165,72179],{"className":72164},[124],[18,72166,72168],{"className":72167,"style":6118},[128],[18,72169,72170,72173],{"style":6121},[18,72171],{"className":72172,"style":6125},[136],[18,72174,72175],{"style":49869},[6130,72176,72177],{"xmlns":6132,"width":49872,"height":944,"viewBox":49873},[6136,72178],{"d":49876},[18,72180,152],{"className":72181},[151],[18,72183,72185],{"className":72184},[124],[18,72186,72188],{"className":72187,"style":6148},[128],[18,72189],{},[18,72191],{"className":72192,"style":228},[169],[18,72194],{"className":72195,"style":228},[169],[18,72197,42719],{"className":72198},[232,27260],[18,72200],{"className":72201,"style":228},[169],[18,72203],{"className":72204,"style":228},[169],[18,72206,72208,72211,72289,72292,72345,72382,72422,72425,72428,72431,72434,72437],{"className":72207},[97],[18,72209],{"className":72210,"style":71946},[101],[18,72212,72214,72217],{"className":72213},[106],[18,72215,934],{"className":72216},[106,1139],[18,72218,72220],{"className":72219},[115],[18,72221,72223,72281],{"className":72222},[119,120],[18,72224,72226,72278],{"className":72225},[124],[18,72227,72229]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不等式），代价是概率意义上的常数因子。",[11,72571,72572,72573,72680,72681,72751],{},"接着引入独立同分布的 Rademacher 变量 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y]",[18,74036,74038,74102,74168],{"className":74037,"ariaHidden":51},[93],[18,74039,74041,74044,74084,74087,74090,74093,74096,74099],{"className":74040},[97],[18,74042],{"className":74043,"style":242},[101],[18,74045,74047,74050],{"className":74046},[106],[18,74048,948],{"className":74049},[106],[18,74051,74053],{"className":74052},[115],[18,74054,74056,74076],{"className":74055},[119,120],[18,74057,74059,74073],{"className":74058},[124],[18,74060,74062],{"className":74061,"style":1383},[128],[18,74063,74064,74067],{"style":1155},[18,74065],{"className":74066,"style":137},[136],[18,74068,74070],{"className":74069},[141,142,143,144],[18,74071,18328],{"className":74072},[106,110,144],[18,74074,152],{"className":74075},[151],[18,74077,74079],{"className":74078},[124],[18,74080,74082],{"className":74081,"style":159},[128],[18,74083],{},[18,74085,74],{"className":74086},[250],[18,74088,72831],{"className":74089,"style":73321},[106,110],[18,74091,84],{"className":74092},[267],[18,74094],{"className":74095,"style":228},[169],[18,74097,433],{"className":74098},[232],[18,74100],{"className":74101,"style":228},[169],[18,74103,74105,74108,74111,74114,74117,74120,74123,74126,74129,74165],{"className":74104},[97],[18,74106],{"className":74107,"style":242},[101],[18,74109,47],{"className":74110},[106,2859],[18,74112,945],{"className":74113},[250],[18,74115,18328],{"className":74116},[106,110],[18,74118,74],{"className":74119},[250],[18,74121,38582],{"className":74122},[106,110],[18,74124,84],{"className":74125},[267],[18,74127],{"className":74128,"style":228},[169],[18,74130,74132,74159,74162],{"className":74131},[232],[18,74133,74135],{"className":74134},[232],[18,74136,74138],{"className":74137},[106,2878],[18,74139,74141],{"className":74140},[2882],[18,74142,74144,74147,74156],{"className":74143},[2886],[18,74145],{"className":74146,"style":2890},[101],[18,74148,74150],{"className":74149},[2894],[18,74151,74153],{"className":74152},[106],[18,74154,2901],{"className":74155},[232],[18,74157],{"className":74158},[2905],[18,74160],{"className":74161},[169,2909],[18,74163,433],{"className":74164},[232],[18,74166],{"className":74167,"style":228},[169],[18,74169,74171,74174,74177],{"className":74170},[97],[18,74172],{"className":74173,"style":242},[101],[18,74175,46352],{"className":74176,"style":3957},[106,110],[18,74178,965],{"className":74179},[267],"。右侧就是 ",[717,74182,74183],{},"Rademacher 复杂度",[18,74185,74187,74211],{"className":74186,"translate":22},[21],[18,74188,74190],{"className":74189},[26],[28,74191,74192],{"xmlns":30},[32,74193,74194,74208],{},[35,74195,74196,74202,74204,74206],{},[38,74197,74198,74200],{},[41,74199,3071],{"mathvariant":3525},[41,74201,28169],{},[49,74203,74],{"stretchy":73},[41,74205,18309],{"mathvariant":1462},[49,74207,84],{"stretchy":73},[86,74209,74210],{"encoding":88},"\\mathfrak R_m(\\mathcal H)",[18,74212,74214],{"className":74213,"ariaHidden":51},[93],[18,74215,74217,74220,74260,74263,74266],{"className":74216},[97],[18,74218],{"className":74219,"style":242},[101],[18,74221,74223,74226],{"className":74222},[106],[18,74224,3071],{"className":74225},[106,3607],[18,74227,74229],{"className":74228},[115],[18,74230,74232,74252],{"className":74231},[119,120],[18,74233,74235,74249],{"className":74234},[124],[18,74236,74238],{"className":74237,"style":202},[128],[18,74239,74240,74243],{"style":1155},[18,74241],{"className":74242,"style":137},[136],[18,74244,74246],{"className":74245},[141,142,143,144],[18,74247,28169],{"className":74248},[106,110,144],[18,74250,152],{"className":74251},[151],[18,74253,74255],{"className":74254},[124],[18,74256,74258],{"className":74257,"style":159},[128],[18,74259],{},[18,74261,74],{"className":74262},[250],[18,74264,18309],{"className":74265,"style":46330},[106,1478],[18,74267,84],{"className":74268},[267],"，它度量的是：给每个样本随机翻一个符号，假设类还能拟合得多好。随机符号是纯噪声，能拟合噪声就说明容量大。",[11,74271,74272],{},"对二值函数类，Rademacher 复杂度可以被增长函数控制：",[18,74274,74276],{"className":74275,"translate":22},[875],[18,74277,74279,74333],{"className":74278,"translate":22},[21],[18,74280,74282],{"className":74281},[26],[28,74283,74284],{"xmlns":30,"display":884},[32,74285,74286,74330],{},[35,74287,74288,74294,74296,74298,74300,74302,74328],{},[38,74289,74290,74292],{},[41,74291,3071],{"mathvariant":3525},[41,74293,28169],{},[49,74295,74],{"stretchy":73},[41,74297,18309],{"mathvariant":1462},[49,74299,84],{"stretchy":73},[49,74301,4334],{},[12561,74303,74304],{},[5654,74305,74306,74326],{},[35,74307,74308,74310,74312,74314,74320,74322,74324],{},[45,74309,2266],{},[41,74311,14733],{},[49,74313,899],{},[38,74315,74316,74318],{},[41,74317,65011],{"mathvariant":316},[41,74319,18309],{"mathvariant":1462},[49,74321,74],{"stretchy":73},[41,74323,28169],{},[49,74325,84],{"stretchy":73},[41,74327,28169],{},[41,74329,968],{"mathvariant":316},[86,74331,74332],{"encoding":88},"\\mathfrak R_m(\\mathcal H)\\le\\sqrt{\\frac{2\\log\\Pi_{\\mathcal H}(m)}{m}}.",[18,74334,74336,74400],{"className":74335,"ariaHidden":51},[93],[18,74337,74339,74342,74382,74385,74388,74391,74394,74397],{"className":74338},[97],[18,74340],{"className":74341,"style":242},[101],[18,74343,74345,74348],{"className":74344},[106],[18,74346,3071],{"className":74347},[106,3607],[18,74349,74351],{"className":74350},[115],[18,74352,74354,74374],{"className":74353},[119,120],[18,74355,74357,74371],{"className":74356},[124],[18,74358,74360],{"className":74359,"style":202},[128],[18,74361,74362,74365],{"style":1155},[18,74363],{"className":74364,"style":137},[136],[18,74366,74368],{"className":74367},[141,142,143,144],[18,74369,28169],{"className":74370},[106,110,144],[18,74372,152],{"className":74373},[151],[18,74375,74377],{"className":74376},[124],[18,74378,74380],{"className":74379,"style":159},[128],[18,74381],{},[18,74383,74],{"className":74384},[250],[18,74386,18309],{"className":74387,"style":46330},[106,1478],[18,74389,84],{"className":74390},[267],[18,74392],{"className":74393,"style":228},[169],[18,74395,4334],{"className":74396},[232],[18,74398],{"className":74399,"style":228},[169],[18,74401,74403,74407,74579],{"className":74402},[97],[18,74404],{"className":74405,"style":74406},[101],"height:2.44em;vertical-align:-0.7356em;",[18,74408,74410],{"className":74409},[106,12696],[18,74411,74413,74570],{"className":74412},[119,120],[18,74414,74416,74567],{"className":74415},[124],[18,74417,74420,74554],{"className":74418,"style":74419},[128],"height:1.7044em;",[18,74421,74423,74426],{"className":74422,"style":12711},[12710],[18,74424],{"className":74425,"style":12715},[136],[18,74427,74429],{"className":74428,"style":12719},[106],[18,74430,74432,74435,74551],{"className":74431},[106],[18,74433],{"className":74434},[250,6012],[18,74436,74438],{"className":74437},[5654],[18,74439,74441,74543],{"className":74440},[119,120],[18,74442,74444,74540],{"className":74443},[124],[18,74445,74447,74458,74466],{"className":74446,"style":18133},[128],[18,74448,74449,74452],{"style":6028},[18,74450],{"className":74451,"style":697},[136],[18,74453,74455],{"className":74454},[106],[18,74456,28169],{"className":74457},[106,110],[18,74459,74460,74463],{"style":6040},[18,74461],{"className":74462,"style":697},[136],[18,74464],{"className":74465,"style":6048},[6047],[18,74467,74468,74471],{"style":6051},[18,74469],{"className":74470,"style":697},[136],[18,74472,74474,74477,74480,74485,74488,74531,74534,74537],{"className":74473},[106],[18,74475,2266],{"className":74476},[106],[18,74478],{"className":74479,"style":170},[169],[18,74481,15174,74483],{"className":74482},[985],[18,74484,15177],{"style":6355},[18,74486],{"className":74487,"style":170},[169],[18,74489,74491,74494],{"className":74490},[106],[18,74492,65011],{"className":74493},[106],[18,74495,74497],{"className":74496},[115],[18,74498,74500,74523],{"className":74499},[119,120],[18,74501,74503,74520],{"className":74502},[124],[18,74504,74506],{"className":74505,"style":1152},[128],[18,74507,74508,74511],{"style":1155},[18,74509],{"className":74510,"style":137},[136],[18,74512,74514],{"className":74513},[141,142,143,144],[18,74515,74517],{"className":74516},[106,144],[18,74518,18309],{"className":74519,"style":46330},[106,1478,144],[18,74521,152],{"className":74522},[151],[18,74524,74526],{"className":74525},[124],[18,74527,74529],{"className":74528,"style":159},[128],[18,74530],{},[18,74532,74],{"className":74533},[250],[18,74535,28169],{"className":74536},[106,110],[18,74538,84],{"className":74539},[267],[18,74541,152],{"className":74542},[151],[18,74544,74546],{"className":74545},[124],[18,74547,74549],{"className":74548,"style":6070},[128],[18,74550],{},[18,74552],{"className":74553},[267,6012],[18,74555,74557,74560],{"style":74556},"top:-3.6644em;",[18,74558],{"className":74559,"style":12715},[136],[18,74561,74563],{"className":74562,"style":12854},[12853],[6130,74564,74565],{"xmlns":6132,"width":12857,"height":12858,"viewBox":12859,"preserveAspectRatio":12860},[6136,74566],{"d":12863},[18,74568,152],{"className":74569},[151],[18,74571,74573],{"className":74572},[124],[18,74574,74577],{"className":74575,"style":74576},[128],"height:0.7356em;",[18,74578],{},[18,74580,968],{"className":74581},[106],[11,74583,74584,74585,74613,74614,74702],{},"这一步用的是有限类的 Massart 引理：把 ",[18,74586,74588,74601],{"className":74587,"translate":22},[21],[18,74589,74591],{"className":74590},[26],[28,74592,74593],{"xmlns":30},[32,74594,74595,74599],{},[35,74596,74597],{},[41,74598,18309],{"mathvariant":1462},[86,74600,46317],{"encoding":88},[18,74602,74604],{"className":74603,"ariaHidden":51},[93],[18,74605,74607,74610],{"className":74606},[97],[18,74608],{"className":74609,"style":600},[101],[18,74611,18309],{"className":74612,"style":46330},[106,1478]," 限制到样本上，最多只有 ",[18,74615,74617,74641],{"className":74616,"translate":22},[21],[18,74618,74620],{"className":74619},[26],[28,74621,74622],{"xmlns":30},[32,74623,74624,74638],{},[35,74625,74626,74632,74634,74636],{},[38,74627,74628,74630],{},[41,74629,65011],{"mathvariant":316},[41,74631,18309],{"mathvariant":1462},[49,74633,74],{"stretchy":73},[41,74635,28169],{},[49,74637,84],{"stretchy":73},[86,74639,74640],{"encoding":88},"\\Pi_{\\mathcal H}(m)",[18,74642,74644],{"className":74643,"ariaHidden":51},[93],[18,74645,74647,74650,74693,74696,74699],{"className":74646},[97],[18,74648],{"className":74649,"style":242},[101],[18,74651,74653,74656],{"className":74652},[106],[18,74654,65011],{"className":74655},[106],[18,74657,74659],{"className":74658},[115],[18,74660,74662,74685],{"className":74661},[119,120],[18,74663,74665,74682],{"className":74664},[124],[18,74666,74668],{"className":74667,"style":1152},[128],[18,74669,74670,74673],{"style":1155},[18,74671],{"className":74672,"style":137},[136],[18,74674,74676],{"className":74675},[141,142,143,144],[18,74677,74679],{"className":74678},[106,144],[18,74680,18309],{"className":74681,"style":46330},[106,1478,144],[18,74683,152],{"className":74684},[151],[18,74686,74688],{"className":74687},[124],[18,74689,74691],{"className":74690,"style":159},[128],[18,74692],{},[18,74694,74],{"className":74695},[250],[18,74697,28169],{"className":74698},[106,110],[18,74700,84],{"className":74701},[267]," 种不同的二分，再用 union bound 或鞅不等式。",[1311,74704,74706],{"id":74705},"_42-泛化界","4.2 泛化界",[11,74708,74709,74710,749],{},"把上面的估计串起来，得到本文唯一需要记住的公式。当 ",[18,74711,74713,74730],{"className":74712,"translate":22},[21],[18,74714,74716],{"className":74715},[26],[28,74717,74718],{"xmlns":30},[32,74719,74720,74728],{},[35,74721,74722,74724,74726],{},[41,74723,28169],{},[49,74725,4775],{},[41,74727,4031],{},[86,74729,69494],{"encoding":88},[18,74731,74733,74751],{"className":74732,"ariaHidden":51},[93],[18,74734,74736,74739,74742,74745,74748],{"className":74735},[97],[18,74737],{"className":74738,"style":53144},[101],[18,74740,28169],{"className":74741},[106,110],[18,74743],{"className":74744,"style":228},[169],[18,74746,4775],{"className":74747},[232],[18,74749],{"className":74750,"style":228},[169],[18,74752,74754,74757],{"className":74753},[97],[18,74755],{"className":74756,"style":295},[101],[18,74758,4031],{"className":74759},[106,110],[18,74761,74763],{"className":74762,"translate":22},[875],[18,74764,74766,74858],{"className":74765,"translate":22},[21],[18,74767,74769],{"className":74768},[26],[28,74770,74771],{"xmlns":30,"display":884},[32,74772,74773,74855],{},[35,74774,74775,74781,74783,74785,74787,74789,74791,74793,74799,74801,74803,74805,74807,74809,74811,74813,74853],{},[38,74776,74777,74779],{},[41,74778,12582],{},[41,74780,12575],{"mathvariant":1462},[49,74782,74],{"stretchy":73},[41,74784,18328],{},[49,74786,84],{"stretchy":73},[926,74788,71861],{},[49,74790,4334],{},[926,74792,71861],{},[38,74794,74795,74797],{},[41,74796,12582],{},[41,74798,38934],{},[49,74800,74],{"stretchy":73},[41,74802,18328],{},[49,74804,84],{"stretchy":73},[926,74806,71861],{},[49,74808,7987],{},[926,74810,71861],{},[41,74812,51021],{},[12561,74814,74815],{},[5654,74816,74817,74851],{},[35,74818,74819,74821,74823,74825,74827,74829,74831,74833,74835,74837,74839,74841,74843,74845,74847,74849],{},[41,74820,4031],{},[41,74822,14733],{},[49,74824,899],{},[49,74826,74],{"stretchy":73},[41,74828,28169],{},[41,74830,317],{"mathvariant":316},[41,74832,4031],{},[49,74834,84],{"stretchy":73},[49,74836,7987],{},[41,74838,14733],{},[49,74840,899],{},[49,74842,74],{"stretchy":73},[45,74844,47],{},[41,74846,317],{"mathvariant":316},[41,74848,5351],{},[49,74850,84],{"stretchy":73},[41,74852,28169],{},[49,74854,52],{"separator":51},[86,74856,74857],{"encoding":88},"L_{\\mathcal D}(h)\\;\\le\\;L_S(h)\\;+\\;C\\sqrt{\\frac{d\\log(m\u002Fd)+\\log(1\u002F\\delta)}{m}},",[18,74859,74861,74934,75004],{"className":74860,"ariaHidden":51},[93],[18,74862,74864,74867,74910,74913,74916,74919,74922,74925,74928,74931],{"className":74863},[97],[18,74865],{"className":74866,"style":242},[101],[18,74868,74870,74873],{"className":74869},[106],[18,74871,12582],{"className":74872},[106,110],[18,74874,74876],{"className":74875},[115],[18,74877,74879,74902],{"className":74878},[119,120],[18,74880,74882,74899],{"className":74881},[124],[18,74883,74885],{"className":74884,"style":1152},[128],[18,74886,74887,74890],{"style":1155},[18,74888],{"className":74889,"style":137},[136],[18,74891,74893],{"className":74892},[141,142,143,144],[18,74894,74896],{"className":74895},[106,144],[18,74897,12575],{"className":74898,"style":254},[106,1478,144],[18,74900,152],{"className":74901},[151],[18,74903,74905],{"className":74904},[124],[18,74906,74908],{"className":74907,"style":159},[128],[18,74909],{},[18,74911,74],{"className":74912},[250],[18,74914,18328],{"className":74915},[106,110],[18,74917,84],{"className":74918},[267],[18,74920],{"className":74921,"style":228},[169],[18,74923],{"className":74924,"style":228},[169],[18,74926,4334],{"className":74927},[232],[18,74929],{"className":74930,"style":228},[169],[18,74932],{"className":74933,"style":228},[169],[18,74935,74937,74940,74980,74983,74986,74989,74992,74995,74998,75001],{"className":74936},[97],[18,74938],{"className":74939,"style":242},[101],[18,74941,74943,74946],{"className":74942},[106],[18,74944,12582],{"className":74945},[106,110],[18,74947,74949],{"className":74948},[115],[18,74950,74952,74972],{"className":74951},[119,120],[18,74953,74955,74969],{"className":74954},[124],[18,74956,74958],{"className":74957,"style":1152},[128],[18,74959,74960,74963],{"style":1155},[18,74961],{"className":74962,"style":137},[136],[18,74964,74966],{"className":74965},[141,142,143,144],[18,74967,38934],{"className":74968,"style":38967},[106,110,144],[18,74970,152],{"className":74971},[151],[18,74973,74975],{"className":74974},[124],[18,74976,74978],{"className":74977,"style":159},[128],[18,74979],{},[18,74981,74],{"className":74982},[250],[18,74984,18328],{"className":74985},[106,110],[18,74987,84],{"className":74988},[267],[18,74990],{"className":74991,"style":228},[169],[18,74993],{"className":74994,"style":2068},[169],[18,74996,7987],{"className":74997},[2072],[18,74999],{"className":75000,"style":228},[169],[18,75002],{"className":75003,"style":2068},[169],[18,75005,75007,75010,75013,75168],{"className":75006},[97],[18,75008],{"className":75009,"style":74406},[101],[18,75011,51021],{"className":75012,"style":5073},[106,110],[18,75014,75016],{"className":75015},[106,12696],[18,75017,75019,75160],{"className":75018},[119,120],[18,75020,75022,75157],{"className":75021},[124],[18,75023,75025,75145],{"className":75024,"style":74419},[128],[18,75026,75028,75031],{"className":75027,"style":12711},[12710],[18,75029],{"className":75030,"style":12715},[136],[18,75032,75034],{"className":75033,"style":12719},[106],[18,75035,75037,75040,75142],{"className":75036},[106],[18,75038],{"className":75039},[250,6012],[18,75041,75043],{"className":75042},[5654],[18,75044,75046,75134],{"className":75045},[119,120],[18,75047,75049,75131],{"className":75048},[124],[18,75050,75052,75063,75071],{"className":75051,"style":18133},[128],[18,75053,75054,75057],{"style":6028},[18,75055],{"className":75056,"style":697},[136],[18,75058,75060],{"className":75059},[106],[18,75061,28169],{"className":75062},[106,110],[18,75064,75065,75068],{"style":6040},[18,75066],{"className":75067,"style":697},[136],[18,75069],{"className":75070,"style":6048},[6047],[18,75072,75073,75076],{"style":6051},[18,75074],{"className":75075,"style":697},[136],[18,75077,75079,75082,75085,75090,75093,75096,75099,75102,75105,75108,75111,75114,75119,75122,75125,75128],{"className":75078},[106],[18,75080,4031],{"className":75081},[106,110],[18,75083],{"className":75084,"style":170},[169],[18,75086,15174,75088],{"className":75087},[985],[18,75089,15177],{"style":6355},[18,75091,74],{"className":75092},[250],[18,75094,28169],{"className":75095},[106,110],[18,75097,317],{"className":75098},[106],[18,75100,4031],{"className":75101},[106,110],[18,75103,84],{"className":75104},[267],[18,75106],{"className":75107,"style":2068},[169],[18,75109,7987],{"className":75110},[2072],[18,75112],{"className":75113,"style":2068},[169],[18,75115,15174,75117],{"className":75116},[985],[18,75118,15177],{"style":6355},[18,75120,74],{"className":75121},[250],[18,75123,335],{"className":75124},[106],[18,75126,5351],{"className":75127,"style":5480},[106,110],[18,75129,84],{"className":75130},[267],[18,75132,152],{"className":75133},[151],[18,75135,75137],{"className":75136},[124],[18,75138,75140],{"className":75139,"style":6070},[128],[18,75141],{},[18,75143],{"className":75144},[267,6012],[18,75146,75147,75150],{"style":74556},[18,75148],{"className":75149,"style":12715},[136],[18,75151,75153],{"className":75152,"style":12854},[12853],[6130,75154,75155],{"xmlns":6132,"width":12857,"height":12858,"viewBox":12859,"preserveAspectRatio":12860},[6136,75156],{"d":12863},[18,75158,152],{"className":75159},[151],[18,75161,75163],{"className":75162},[124],[18,75164,75166],{"className":75165,"style":74576},[128],[18,75167],{},[18,75169,52],{"className":75170},[165],[11,75172,75173,75174,75225,75226,1994,75228,75279,75280,4741,75351,75379],{},"以至少 ",[18,75175,75177,75195],{"className":75176,"translate":22},[21],[18,75178,75180],{"className":75179},[26],[28,75181,75182],{"xmlns":30},[32,75183,75184,75192],{},[35,75185,75186,75188,75190],{},[45,75187,47],{},[49,75189,2253],{},[41,75191,5351],{},[86,75193,75194],{"encoding":88},"1-\\delta",[18,75196,75198,75216],{"className":75197,"ariaHidden":51},[93],[18,75199,75201,75204,75207,75210,75213],{"className":75200},[97],[18,75202],{"className":75203,"style":18095},[101],[18,75205,47],{"className":75206},[106],[18,75208],{"className":75209,"style":2068},[169],[18,75211,2253],{"className":75212},[2072],[18,75214],{"className":75215,"style":2068},[169],[18,75217,75219,75222],{"className":75218},[97],[18,75220],{"className":75221,"style":295},[101],[18,75223,5351],{"className":75224,"style":5480},[106,110]," 的概率对",[717,75227,1301],{},[18,75229,75231,75249],{"className":75230,"translate":22},[21],[18,75232,75234],{"className":75233},[26],[28,75235,75236],{"xmlns":30},[32,75237,75238,75246],{},[35,75239,75240,75242,75244],{},[41,75241,18328],{},[49,75243,921],{},[41,75245,18309],{"mathvariant":1462},[86,75247,75248],{"encoding":88},"h\\in\\mathcal H",[18,75250,75252,75270],{"className":75251,"ariaHidden":51},[93],[18,75253,75255,75258,75261,75264,75267],{"className":75254},[97],[18,75256],{"className":75257,"style":1422},[101],[18,75259,18328],{"className":75260},[106,110],[18,75262],{"className":75263,"style":228},[169],[18,75265,921],{"className":75266},[232],[18,75268],{"className":75269,"style":228},[169],[18,75271,75273,75276],{"className":75272},[97],[18,75274],{"className":75275,"style":600},[101],[18,75277,18309],{"className":75278,"style":46330},[106,1478]," 同时成立，其中 ",[18,75281,75283,75309],{"className":75282,"translate":22},[21],[18,75284,75286],{"className":75285},[26],[28,75287,75288],{"xmlns":30},[32,75289,75290,75306],{},[35,75291,75292,75294,75296,75298,75300,75302,75304],{},[41,75293,4031],{},[49,75295,433],{},[41,75297,52216],{"mathvariant":316},[49,75299,899],{},[49,75301,74],{"stretchy":73},[41,75303,18309],{"mathvariant":1462},[49,75305,84],{"stretchy":73},[86,75307,75308],{"encoding":88},"d=\\operatorname{VCdim}(\\mathcal H)",[18,75310,75312,75330],{"className":75311,"ariaHidden":51},[93],[18,75313,75315,75318,75321,75324,75327],{"className":75314},[97],[18,75316],{"className":75317,"style":295},[101],[18,75319,4031],{"className":75320},[106,110],[18,75322],{"className":75323,"style":228},[169],[18,75325,433],{"className":75326},[232],[18,75328],{"className":75329,"style":228},[169],[18,75331,75333,75336,75342,75345,75348],{"className":75332},[97],[18,75334],{"className":75335,"style":242},[101],[18,75337,75339],{"className":75338},[985],[18,75340,52216],{"className":75341},[106,6354],[18,75343,74],{"className":75344},[250],[18,75346,18309],{"className":75347,"style":46330},[106,1478],[18,75349,84],{"className":75350},[267],[18,75352,75354,75367],{"className":75353,"translate":22},[21],[18,75355,75357],{"className":75356},[26],[28,75358,75359],{"xmlns":30},[32,75360,75361,75365],{},[35,75362,75363],{},[41,75364,51021],{},[86,75366,51021],{"encoding":88},[18,75368,75370],{"className":75369,"ariaHidden":51},[93],[18,75371,75373,75376],{"className":75372},[97],[18,75374],{"className":75375,"style":600},[101],[18,75377,51021],{"className":75378,"style":5073},[106,110]," 是某个绝对常数。",[11,75381,75382,75383,75415,75416,23859,75482,75558],{},"不同教材的常数和 ",[18,75384,75386,75401],{"className":75385,"translate":22},[21],[18,75387,75389],{"className":75388},[26],[28,75390,75391],{"xmlns":30},[32,75392,75393,75399],{},[35,75394,75395,75397],{},[41,75396,14733],{},[49,75398,899],{},[86,75400,27772],{"encoding":88},[18,75402,75404],{"className":75403,"ariaHidden":51},[93],[18,75405,75407,75410],{"className":75406},[97],[18,75408],{"className":75409,"style":2890},[101],[18,75411,15174,75413],{"className":75412},[985],[18,75414,15177],{"style":6355}," 里的具体形式略有出入（常见的一种写法是把 ",[18,75417,75419,75447],{"className":75418,"translate":22},[21],[18,75420,75422],{"className":75421},[26],[28,75423,75424],{"xmlns":30},[32,75425,75426,75444],{},[35,75427,75428,75430,75432,75434,75436,75438,75440,75442],{},[41,75429,4031],{},[41,75431,14733],{},[49,75433,899],{},[49,75435,74],{"stretchy":73},[41,75437,28169],{},[41,75439,317],{"mathvariant":316},[41,75441,4031],{},[49,75443,84],{"stretchy":73},[86,75445,75446],{"encoding":88},"d\\log(m\u002Fd)",[18,75448,75450],{"className":75449,"ariaHidden":51},[93],[18,75451,75453,75456,75459,75462,75467,75470,75473,75476,75479],{"className":75452},[97],[18,75454],{"className":75455,"style":242},[101],[18,75457,4031],{"className":75458},[106,110],[18,75460],{"className":75461,"style":170},[169],[18,75463,15174,75465],{"className":75464},[985],[18,75466,15177],{"style":6355},[18,75468,74],{"className":75469},[250],[18,75471,28169],{"className":75472},[106,110],[18,75474,317],{"className":75475},[106],[18,75477,4031],{"className":75478},[106,110],[18,75480,84],{"className":75481},[267],[18,75483,75485,75517],{"className":75484,"translate":22},[21],[18,75486,75488],{"className":75487},[26],[28,75489,75490],{"xmlns":30},[32,75491,75492,75514],{},[35,75493,75494,75496,75498,75500,75502,75504,75506,75508,75510,75512],{},[41,75495,4031],{},[41,75497,14733],{},[49,75499,899],{},[49,75501,74],{"stretchy":73},[45,75503,2266],{},[41,75505,14444],{},[41,75507,28169],{},[41,75509,317],{"mathvariant":316},[41,75511,4031],{},[49,75513,84],{"stretchy":73},[86,75515,75516],{"encoding":88},"d\\log(2em\u002Fd)",[18,75518,75520],{"className":75519,"ariaHidden":51},[93],[18,75521,75523,75526,75529,75532,75537,75540,75543,75546,75549,75552,75555],{"className":75522},[97],[18,75524],{"className":75525,"style":242},[101],[18,75527,4031],{"className":75528},[106,110],[18,75530],{"className":75531,"style":170},[169],[18,75533,15174,75535],{"className":75534},[985],[18,75536,15177],{"style":6355},[18,75538,74],{"className":75539},[250],[18,75541,2266],{"className":75542},[106],[18,75544,14444],{"className":75545},[106,110],[18,75547,28169],{"className":75548},[106,110],[18,75550,317],{"className":75551},[106],[18,75553,4031],{"className":75554},[106,110],[18,75556,84],{"className":75557},[267],"），但量级是一致的。重要的是它的结构：",[2210,75560,75561,75649,75681],{},[2213,75562,75563,75564,75592,75593,75648],{},"分子里的 ",[18,75565,75567,75580],{"className":75566,"translate":22},[21],[18,75568,75570],{"className":75569},[26],[28,75571,75572],{"xmlns":30},[32,75573,75574,75578],{},[35,75575,75576],{},[41,75577,4031],{},[86,75579,4031],{"encoding":88},[18,75581,75583],{"className":75582,"ariaHidden":51},[93],[18,75584,75586,75589],{"className":75585},[97],[18,75587],{"className":75588,"style":295},[101],[18,75590,4031],{"className":75591},[106,110]," 是容量，",[18,75594,75596,75622],{"className":75595,"translate":22},[21],[18,75597,75599],{"className":75598},[26],[28,75600,75601],{"xmlns":30},[32,75602,75603,75619],{},[35,75604,75605,75607,75609,75611,75613,75615,75617],{},[41,75606,14733],{},[49,75608,899],{},[49,75610,74],{"stretchy":73},[45,75612,47],{},[41,75614,317],{"mathvariant":316},[41,75616,5351],{},[49,75618,84],{"stretchy":73},[86,75620,75621],{"encoding":88},"\\log(1\u002F\\delta)",[18,75623,75625],{"className":75624,"ariaHidden":51},[93],[18,75626,75628,75631,75636,75639,75642,75645],{"className":75627},[97],[18,75629],{"className":75630,"style":242},[101],[18,75632,15174,75634],{"className":75633},[985],[18,75635,15177],{"style":6355},[18,75637,74],{"className":75638},[250],[18,75640,335],{"className":75641},[106],[18,75643,5351],{"className":75644,"style":5480},[106,110],[18,75646,84],{"className":75647},[267]," 是置信度；",[2213,75650,75651,75652,75680],{},"分母里的 ",[18,75653,75655,75668],{"className":75654,"translate":22},[21],[18,75656,75658],{"className":75657},[26],[28,75659,75660],{"xmlns":30},[32,75661,75662,75666],{},[35,75663,75664],{},[41,75665,28169],{},[86,75667,28169],{"encoding":88},[18,75669,75671],{"className":75670,"ariaHidden":51},[93],[18,75672,75674,75677],{"className":75673},[97],[18,75675],{"className":75676,"style":1083},[101],[18,75678,28169],{"className":75679},[106,110]," 是样本量；",[2213,75682,75683,75684,75763],{},"整个界以 ",[18,75685,75687,75707],{"className":75686,"translate":22},[21],[18,75688,75690],{"className":75689},[26],[28,75691,75692],{"xmlns":30},[32,75693,75694,75704],{},[35,75695,75696,75698,75700],{},[45,75697,47],{},[41,75699,317],{"mathvariant":316},[12561,75701,75702],{},[41,75703,28169],{},[86,75705,75706],{"encoding":88},"1\u002F\\sqrt m",[18,75708,75710],{"className":75709,"ariaHidden":51},[93],[18,75711,75713,75716,75719],{"className":75712},[97],[18,75714],{"className":75715,"style":15501},[101],[18,75717,335],{"className":75718},[106],[18,75720,75722],{"className":75721},[106,12696],[18,75723,75725,75755],{"className":75724},[119,120],[18,75726,75728,75752],{"className":75727},[124],[18,75729,75731,75740],{"className":75730,"style":15400},[128],[18,75732,75734,75737],{"className":75733,"style":693},[12710],[18,75735],{"className":75736,"style":697},[136],[18,75738,28169],{"className":75739,"style":15410},[106,110],[18,75741,75742,75745],{"style":15413},[18,75743],{"className":75744,"style":697},[136],[18,75746,75748],{"className":75747,"style":15420},[12853],[6130,75749,75750],{"xmlns":6132,"width":12857,"height":15423,"viewBox":15424,"preserveAspectRatio":12860},[6136,75751],{"d":15427},[18,75753,152],{"className":75754},[151],[18,75756,75758],{"className":75757},[124],[18,75759,75761],{"className":75760,"style":15437},[128],[18,75762],{}," 的速度下降。在 agnostic 设定下，这个速度是最优的：存在匹配的下界，样本量再少就不可能保证同样的一致收敛。",[11,75765,75766,75767,75860],{},"把 ",[18,75768,75770,75796],{"className":75769,"translate":22},[21],[18,75771,75773],{"className":75772},[26],[28,75774,75775],{"xmlns":30},[32,75776,75777,75793],{},[35,75778,75779,75785,75787,75789,75791],{},[38,75780,75781,75783],{},[41,75782,65011],{"mathvariant":316},[41,75784,18309],{"mathvariant":1462},[49,75786,74],{"stretchy":73},[45,75788,2266],{},[41,75790,28169],{},[49,75792,84],{"stretchy":73},[86,75794,75795],{"encoding":88},"\\Pi_{\\mathcal H}(2m)",[18,75797,75799],{"className":75798,"ariaHidden":51},[93],[18,75800,75802,75805,75848,75851,75854,75857],{"className":75801},[97],[18,75803],{"className":75804,"style":242},[101],[18,75806,75808,75811],{"className":75807},[106],[18,75809,65011],{"className":75810},[106],[18,75812,75814],{"className":75813},[115],[18,75815,75817,75840],{"className":75816},[119,120],[18,75818,75820,75837],{"className":75819},[124],[18,75821,75823],{"className":75822,"style":1152},[128],[18,75824,75825,75828],{"style":1155},[18,75826],{"className":75827,"style":137},[136],[18,75829,75831],{"className":75830},[141,142,143,144],[18,75832,75834],{"className":75833},[106,144],[18,75835,18309],{"className":75836,"style":46330},[106,1478,144],[18,75838,152],{"className":75839},[151],[18,75841,75843],{"className":75842},[124],[18,75844,75846],{"className":75845,"style":159},[128],[18,75847],{},[18,75849,74],{"className":75850},[250],[18,75852,2266],{"className":75853},[106],[18,75855,28169],{"className":75856},[106,110],[18,75858,84],{"className":75859},[267]," 用 Sauer 界代入，就得到只依赖 VC 维的形式。再把它代回误差分解，就得到 ERM 的泛化保证。",[1311,75862,75864],{"id":75863},"_43-基本定理","4.3 基本定理",[5159,75866,75867,75930],{},[11,75868,75869,75872,75873,2702,75901,75929],{},[717,75870,75871],{},"定理 4.1（统计学习基本定理）"," 对二分类、",[18,75874,75876,75889],{"className":75875,"translate":22},[21],[18,75877,75879],{"className":75878},[26],[28,75880,75881],{"xmlns":30},[32,75882,75883,75887],{},[35,75884,75885],{},[45,75886,430],{},[86,75888,430],{"encoding":88},[18,75890,75892],{"className":75891,"ariaHidden":51},[93],[18,75893,75895,75898],{"className":75894},[97],[18,75896],{"className":75897,"style":505},[101],[18,75899,430],{"className":75900},[106],[18,75902,75904,75917],{"className":75903,"translate":22},[21],[18,75905,75907],{"className":75906},[26],[28,75908,75909],{"xmlns":30},[32,75910,75911,75915],{},[35,75912,75913],{},[45,75914,47],{},[86,75916,47],{"encoding":88},[18,75918,75920],{"className":75919,"ariaHidden":51},[93],[18,75921,75923,75926],{"className":75922},[97],[18,75924],{"className":75925,"style":505},[101],[18,75927,47],{"className":75928},[106]," 损失、分布无关（distribution-free）的设定，以下四条等价：",[45452,75931,75932,76005,76036,76039],{},[2213,75933,75934,2452],{},[18,75935,75937,75963],{"className":75936,"translate":22},[21],[18,75938,75940],{"className":75939},[26],[28,75941,75942],{"xmlns":30},[32,75943,75944,75960],{},[35,75945,75946,75948,75950,75952,75954,75956,75958],{},[41,75947,52216],{"mathvariant":316},[49,75949,899],{},[49,75951,74],{"stretchy":73},[41,75953,18309],{"mathvariant":1462},[49,75955,84],{"stretchy":73},[49,75957,6684],{},[41,75959,2028],{"mathvariant":316},[86,75961,75962],{"encoding":88},"\\operatorname{VCdim}(\\mathcal H)\u003C\\infty",[18,75964,75966,75996],{"className":75965,"ariaHidden":51},[93],[18,75967,75969,75972,75978,75981,75984,75987,75990,75993],{"className":75968},[97],[18,75970],{"className":75971,"style":242},[101],[18,75973,75975],{"className":75974},[985],[18,75976,52216],{"className":75977},[106,6354],[18,75979,74],{"className":75980},[250],[18,75982,18309],{"className":75983,"style":46330},[106,1478],[18,75985,84],{"className":75986},[267],[18,75988],{"className":75989,"style":228},[169],[18,75991,6684],{"className":75992},[232],[18,75994],{"className":75995,"style":228},[169],[18,75997,75999,76002],{"className":75998},[97],[18,76000],{"className":76001,"style":1083},[101],[18,76003,2028],{"className":76004},[106],[2213,76006,76007,76035],{},[18,76008,76010,76023],{"className":76009,"translate":22},[21],[18,76011,76013],{"className":76012},[26],[28,76014,76015],{"xmlns":30},[32,76016,76017,76021],{},[35,76018,76019],{},[41,76020,18309],{"mathvariant":1462},[86,76022,46317],{"encoding":88},[18,76024,76026],{"className":76025,"ariaHidden":51},[93],[18,76027,76029,76032],{"className":76028},[97],[18,76030],{"className":76031,"style":600},[101],[18,76033,18309],{"className":76034,"style":46330},[106,1478]," 具有一致收敛性；",[2213,76037,76038],{},"ERM 是 agnostic PAC 一致的；",[2213,76040,76041,76069],{},[18,76042,76044,76057],{"className":76043,"translate":22},[21],[18,76045,76047],{"className":76046},[26],[28,76048,76049],{"xmlns":30},[32,76050,76051,76055],{},[35,76052,76053],{},[41,76054,18309],{"mathvariant":1462},[86,76056,46317],{"encoding":88},[18,76058,76060],{"className":76059,"ariaHidden":51},[93],[18,76061,76063,76066],{"className":76062},[97],[18,76064],{"className":76065,"style":600},[101],[18,76067,18309],{"className":76068,"style":46330},[106,1478]," 是 agnostic PAC 可学习的。",[11,76071,76072,76073],{},"在可实现（realizable）情形下，有限 VC 维同样等价于 PAC 可学习。这个定理把「组合上的有限性」和「统计上的可学习性」严格地焊在了一起：",[717,76074,76075],{},"在分布无关的框架里，有限 VC 维不只是充分条件，而是充要条件。",[1311,76077,76079],{"id":76078},"_44-样本复杂度","4.4 样本复杂度",[11,76081,76082,76083,76111],{},"由上面的界解出 ",[18,76084,76086,76099],{"className":76085,"translate":22},[21],[18,76087,76089],{"className":76088},[26],[28,76090,76091],{"xmlns":30},[32,76092,76093,76097],{},[35,76094,76095],{},[41,76096,28169],{},[86,76098,28169],{"encoding":88},[18,76100,76102],{"className":76101,"ariaHidden":51},[93],[18,76103,76105,76108],{"className":76104},[97],[18,76106],{"className":76107,"style":1083},[101],[18,76109,28169],{"className":76110},[106,110],"，得到 agnostic 情形下的样本复杂度：",[18,76113,76115],{"className":76114,"translate":22},[875],[18,76116,76118,76172],{"className":76117,"translate":22},[21],[18,76119,76121],{"className":76120},[26],[28,76122,76123],{"xmlns":30,"display":884},[32,76124,76125,76169],{},[35,76126,76127,76129,76131,76133,76167],{},[41,76128,28169],{},[49,76130,433],{},[41,76132,924],{"mathvariant":316},[35,76134,76135,76137,76165],{},[49,76136,74],{"fence":51},[5654,76138,76139,76159],{},[35,76140,76141,76143,76145,76147,76149,76151,76153,76155,76157],{},[41,76142,4031],{},[49,76144,7987],{},[41,76146,14733],{},[49,76148,899],{},[49,76150,74],{"stretchy":73},[45,76152,47],{},[41,76154,317],{"mathvariant":316},[41,76156,5351],{},[49,76158,84],{"stretchy":73},[1556,76160,76161,76163],{},[41,76162,32522],{},[45,76164,2266],{},[49,76166,84],{"fence":51},[49,76168,52],{"separator":51},[86,76170,76171],{"encoding":88},"m=\\Theta\\left(\\frac{d+\\log(1\u002F\\delta)}{\\varepsilon^2}\\right),",[18,76173,76175,76193],{"className":76174,"ariaHidden":51},[93],[18,76176,76178,76181,76184,76187,76190],{"className":76177},[97],[18,76179],{"className":76180,"style":1083},[101],[18,76182,28169],{"className":76183},[106,110],[18,76185],{"className":76186,"style":228},[169],[18,76188,433],{"className":76189},[232],[18,76191],{"className":76192,"style":228},[169],[18,76194,76196,76199,76202,76205,76334,76337],{"className":76195},[97],[18,76197],{"className":76198,"style":20807},[101],[18,76200,924],{"className":76201},[106],[18,76203],{"className":76204,"style":170},[169],[18,76206,76208,76214,76328],{"className":76207},[174],[18,76209,76211],{"className":76210,"style":13796},[250,13795],[18,76212,74],{"className":76213},[1230,143],[18,76215,76217,76220,76325],{"className":76216},[106],[18,76218],{"className":76219},[250,6012],[18,76221,76223],{"className":76222},[5654],[18,76224,76226,76317],{"className":76225},[119,120],[18,76227,76229,76314],{"className":76228},[124],[18,76230,76232,76269,76277],{"className":76231,"style":18133},[128],[18,76233,76234,76237],{"style":6028},[18,76235],{"className":76236,"style":697},[136],[18,76238,76240],{"className":76239},[106],[18,76241,76243,76246],{"className":76242},[106],[18,76244,32522],{"className":76245},[106,110],[18,76247,76249],{"className":76248},[115],[18,76250,76252],{"className":76251},[119],[18,76253,76255],{"className":76254},[124],[18,76256,76258],{"className":76257,"style":26109},[128],[18,76259,76260,76263],{"style":26112},[18,76261],{"className":76262,"style":137},[136],[18,76264,76266],{"className":76265},[141,142,143,144],[18,76267,2266],{"className":76268},[106,144],[18,76270,76271,76274],{"style":6040},[18,76272],{"className":76273,"style":697},[136],[18,76275],{"className":76276,"style":6048},[6047],[18,76278,76279,76282],{"style":6051},[18,76280],{"className":76281,"style":697},[136],[18,76283,76285,76288,76291,76294,76297,76302,76305,76308,76311],{"className":76284},[106],[18,76286,4031],{"className":76287},[106,110],[18,76289],{"className":76290,"style":2068},[169],[18,76292,7987],{"className":76293},[2072],[18,76295],{"className":76296,"style":2068},[169],[18,76298,15174,76300],{"className":76299},[985],[18,76301,15177],{"style":6355},[18,76303,74],{"className":76304},[250],[18,76306,335],{"className":76307},[106],[18,76309,5351],{"className":76310,"style":5480},[106,110],[18,76312,84],{"className":76313},[267],[18,76315,152],{"className":76316},[151],[18,76318,76320],{"className":76319},[124],[18,76321,76323],{"className":76322,"style":6070},[128],[18,76324],{},[18,76326],{"className":76327},[267,6012],[18,76329,76331],{"className":76330,"style":13796},[267,13795],[18,76332,84],{"className":76333},[1230,143],[18,76335],{"className":76336,"style":170},[169],[18,76338,52],{"className":76339},[165],[11,76341,76342],{},"可实现情形下则是",[18,76344,76346],{"className":76345,"translate":22},[875],[18,76347,76349,76399],{"className":76348,"translate":22},[21],[18,76350,76352],{"className":76351},[26],[28,76353,76354],{"xmlns":30,"display":884},[32,76355,76356,76396],{},[35,76357,76358,76360,76362,76364,76394],{},[41,76359,28169],{},[49,76361,433],{},[41,76363,924],{"mathvariant":316},[35,76365,76366,76368,76392],{},[49,76367,74],{"fence":51},[5654,76369,76370,76390],{},[35,76371,76372,76374,76376,76378,76380,76382,76384,76386,76388],{},[41,76373,4031],{},[49,76375,7987],{},[41,76377,14733],{},[49,76379,899],{},[49,76381,74],{"stretchy":73},[45,76383,47],{},[41,76385,317],{"mathvariant":316},[41,76387,5351],{},[49,76389,84],{"stretchy":73},[41,76391,32522],{},[49,76393,84],{"fence":51},[41,76395,968],{"mathvariant":316},[86,76397,76398],{"encoding":88},"m=\\Theta\\left(\\frac{d+\\log(1\u002F\\delta)}{\\varepsilon}\\right).",[18,76400,76402,76420],{"className":76401,"ariaHidden":51},[93],[18,76403,76405,76408,76411,76414,76417],{"className":76404},[97],[18,76406],{"className":76407,"style":1083},[101],[18,76409,28169],{"className":76410},[106,110],[18,76412],{"className":76413,"style":228},[169],[18,76415,433],{"className":76416},[232],[18,76418],{"className":76419,"style":228},[169],[18,76421,76423,76426,76429,76432,76535,76538],{"className":76422},[97],[18,76424],{"className":76425,"style":20807},[101],[18,76427,924],{"className":76428},[106],[18,76430],{"className":76431,"style":170},[169],[18,76433,76435,76441,76529],{"className":76434},[174],[18,76436,76438],{"className":76437,"style":13796},[250,13795],[18,76439,74],{"className":76440},[1230,143],[18,76442,76444,76447,76526],{"className":76443},[106],[18,76445],{"className":76446},[250,6012],[18,76448,76450],{"className":76449},[5654],[18,76451,76453,76518],{"className":76452},[119,120],[18,76454,76456,76515],{"className":76455},[124],[18,76457,76459,76470,76478],{"className":76458,"style":18133},[128],[18,76460,76461,76464],{"style":6028},[18,76462],{"className":76463,"style":697},[136],[18,76465,76467],{"className":76466},[106],[18,76468,32522],{"className":76469},[106,110],[18,76471,76472,76475],{"style":6040},[18,76473],{"className":76474,"style":697},[136],[18,76476],{"className":76477,"style":6048},[6047],[18,76479,76480,76483],{"style":6051},[18,76481],{"className":76482,"style":697},[136],[18,76484,76486,76489,76492,76495,76498,76503,76506,76509,76512],{"className":76485},[106],[18,76487,4031],{"className":76488},[106,110],[18,76490],{"className":76491,"style":2068},[169],[18,76493,7987],{"className":76494},[2072],[18,76496],{"className":76497,"style":2068},[169],[18,76499,15174,76501],{"className":76500},[985],[18,76502,15177],{"style":6355},[18,76504,74],{"className":76505},[250],[18,76507,335],{"className":76508},[106],[18,76510,5351],{"className":76511,"style":5480},[106,110],[18,76513,84],{"className":76514},[267],[18,76516,152],{"className":76517},[151],[18,76519,76521],{"className":76520},[124],[18,76522,76524],{"className":76523,"style":6070},[128],[18,76525],{},[18,76527],{"className":76528},[267,6012],[18,76530,76532],{"className":76531,"style":13796},[267,13795],[18,76533,84],{"className":76534},[1230,143],[18,76536],{"className":76537,"style":170},[169],[18,76539,968],{"className":76540},[106],[11,76542,76543,76544,76572,76573,76576],{},"这两个 ",[18,76545,76547,76560],{"className":76546,"translate":22},[21],[18,76548,76550],{"className":76549},[26],[28,76551,76552],{"xmlns":30},[32,76553,76554,76558],{},[35,76555,76556],{},[41,76557,924],{"mathvariant":316},[86,76559,1497],{"encoding":88},[18,76561,76563],{"className":76562,"ariaHidden":51},[93],[18,76564,76566,76569],{"className":76565},[97],[18,76567],{"className":76568,"style":600},[101],[18,76570,924],{"className":76571},[106]," 是双向的：上界由 ERM 达到，下界则是信息论下界——存在一个分布，使任何算法都需要这么多样本。所以 VC 维不仅给出了一个可用的界，还在 minimax 意义下给出了",[717,76574,76575],{},"最优","的样本复杂度量级。",[11,76578,76579],{},"这一点很重要。很多人批评 VC 界「太松」，但在分布无关、最坏情况的设定里，它并不松：你不可能用更少的样本保证同样的一致收敛。问题不在于界本身，而在于这个设定是否匹配你关心的场景。",[638,76581,76583],{"id":76582},"五vc-维的含义","五、VC 维的含义",[11,76585,76586],{},"到这里，数学已经完整了。但公式正确不等于理解。这一节把 VC 维的语义拆成五句话。",[11,76588,76589,76592,76593,76669,76670,76747,76748,1990],{},[717,76590,76591],{},"第一，它是 min-max 的。"," 定义里的 ",[18,76594,76596,76618],{"className":76595,"translate":22},[21],[18,76597,76599],{"className":76598},[26],[28,76600,76601],{"xmlns":30},[32,76602,76603,76615],{},[35,76604,76605],{},[38,76606,76607,76613],{},[35,76608,76609,76611],{},[41,76610,912],{},[49,76612,899],{},[41,76614,51021],{},[86,76616,76617],{"encoding":88},"\\sup_C",[18,76619,76621],{"className":76620,"ariaHidden":51},[93],[18,76622,76624,76628],{"className":76623},[97],[18,76625],{"className":76626,"style":76627},[101],"height:0.6747em;vertical-align:-0.2441em;",[18,76629,76631,76634],{"className":76630},[985],[18,76632,912],{"className":76633},[985],[18,76635,76637],{"className":76636},[115],[18,76638,76640,76661],{"className":76639},[119,120],[18,76641,76643,76658],{"className":76642},[124],[18,76644,76647],{"className":76645,"style":76646},[128],"height:0.2342em;",[18,76648,76649,76652],{"style":60148},[18,76650],{"className":76651,"style":137},[136],[18,76653,76655],{"className":76654},[141,142,143,144],[18,76656,51021],{"className":76657,"style":5073},[106,110,144],[18,76659,152],{"className":76660},[151],[18,76662,76664],{"className":76663},[124],[18,76665,76667],{"className":76666,"style":60167},[128],[18,76668],{}," 对应界里的 ",[18,76671,76673,76695],{"className":76672,"translate":22},[21],[18,76674,76676],{"className":76675},[26],[28,76677,76678],{"xmlns":30},[32,76679,76680,76692],{},[35,76681,76682],{},[38,76683,76684,76690],{},[35,76685,76686,76688],{},[41,76687,912],{},[49,76689,899],{},[41,76691,12575],{"mathvariant":1462},[86,76693,76694],{"encoding":88},"\\sup_{\\mathcal D}",[18,76696,76698],{"className":76697,"ariaHidden":51},[93],[18,76699,76701,76704],{"className":76700},[97],[18,76702],{"className":76703,"style":76627},[101],[18,76705,76707,76710],{"className":76706},[985],[18,76708,912],{"className":76709},[985],[18,76711,76713],{"className":76712},[115],[18,76714,76716,76739],{"className":76715},[119,120],[18,76717,76719,76736],{"className":76718},[124],[18,76720,76722],{"className":76721,"style":76646},[128],[18,76723,76724,76727],{"style":60148},[18,76725],{"className":76726,"style":137},[136],[18,76728,76730],{"className":76729},[141,142,143,144],[18,76731,76733],{"className":76732},[106,144],[18,76734,12575],{"className":76735,"style":254},[106,1478,144],[18,76737,152],{"className":76738},[151],[18,76740,76742],{"className":76741},[124],[18,76743,76745],{"className":76744,"style":60167},[128],[18,76746],{},"。VC 理论先让对手选一个最坏的分布，再让对手选一个最坏的点集，然后问：你还剩什么保证？所以它天生悲观。它给出的不是一个典型情形下的估计，而是一个",[717,76749,76750],{},"任何情形下都不被打破的承诺",[11,76752,76753,3019,76756,2702,76784,76812],{},[717,76754,76755],{},"第二，它是充要的。",[18,76757,76759,76772],{"className":76758,"translate":22},[21],[18,76760,76762],{"className":76761},[26],[28,76763,76764],{"xmlns":30},[32,76765,76766,76770],{},[35,76767,76768],{},[45,76769,430],{},[86,76771,430],{"encoding":88},[18,76773,76775],{"className":76774,"ariaHidden":51},[93],[18,76776,76778,76781],{"className":76777},[97],[18,76779],{"className":76780,"style":505},[101],[18,76782,430],{"className":76783},[106],[18,76785,76787,76800],{"className":76786,"translate":22},[21],[18,76788,76790],{"className":76789},[26],[28,76791,76792],{"xmlns":30},[32,76793,76794,76798],{},[35,76795,76796],{},[45,76797,47],{},[86,76799,47],{"encoding":88},[18,76801,76803],{"className":76802,"ariaHidden":51},[93],[18,76804,76806,76809],{"className":76805},[97],[18,76807],{"className":76808,"style":505},[101],[18,76810,47],{"className":76811},[106]," 损失、二分类、要求对所有分布一致的框架里，有限 VC 维精确刻画了可学习性。它不是「一个充分但过强的条件」；它就是那个条件。问题不在定理，而在框架。",[11,76814,76815,76818],{},[717,76816,76817],{},"第三，它是组合量，概率只出现在最后一步。"," 打散、增长函数、Sauer 引理都是纯组合的，与分布、与概率无关；对称化把组合量变成概率界。这也解释了为什么 VC 维对数据长什么样一无所知：它的输入只有假设类的集合结构。",[11,76820,76821,76824,76825,76889],{},[717,76822,76823],{},"第四，它衡量的是「能实现多少种二分」，不是「能拟合多复杂的函数」。"," 一个类可以 VC 维无穷却在某个分布上表现很好（1-NN），也可以 VC 维很小却拟合得很差（区间无法实现 ",[18,76826,76828,76853],{"className":76827,"translate":22},[21],[18,76829,76831],{"className":76830},[26],[28,76832,76833],{"xmlns":30},[32,76834,76835,76851],{},[35,76836,76837,76839,76841,76843,76845,76847,76849],{},[49,76838,74],{"stretchy":73},[45,76840,47],{},[49,76842,52],{"separator":51},[45,76844,430],{},[49,76846,52],{"separator":51},[45,76848,47],{},[49,76850,84],{"stretchy":73},[86,76852,55451],{"encoding":88},[18,76854,76856],{"className":76855,"ariaHidden":51},[93],[18,76857,76859,76862,76865,76868,76871,76874,76877,76880,76883,76886],{"className":76858},[97],[18,76860],{"className":76861,"style":242},[101],[18,76863,74],{"className":76864},[250],[18,76866,47],{"className":76867},[106],[18,76869,52],{"className":76870},[165],[18,76872],{"className":76873,"style":170},[169],[18,76875,430],{"className":76876},[106],[18,76878,52],{"className":76879},[165],[18,76881],{"className":76882,"style":170},[169],[18,76884,47],{"className":76885},[106],[18,76887,84],{"className":76888},[267]," 这种标注）。VC 维是容量，不是质量。",[11,76891,76892,1994,76895,77055,77056,77084],{},[717,76893,76894],{},"第五，它是上界，不是预测。",[18,76896,76898,76929],{"className":76897,"translate":22},[21],[18,76899,76901],{"className":76900},[26],[28,76902,76903],{"xmlns":30},[32,76904,76905,76926],{},[35,76906,76907,76913,76915,76921,76923],{},[38,76908,76909,76911],{},[41,76910,12582],{},[41,76912,12575],{"mathvariant":1462},[49,76914,4334],{},[38,76916,76917,76919],{},[41,76918,12582],{},[41,76920,38934],{},[49,76922,7987],{},[49,76924,76925],{},"⋯",[86,76927,76928],{"encoding":88},"L_{\\mathcal D}\\le L_S+\\cdots",[18,76930,76932,76990,77045],{"className":76931,"ariaHidden":51},[93],[18,76933,76935,76938,76981,76984,76987],{"className":76934},[97],[18,76936],{"className":76937,"style":4563},[101],[18,76939,76941,76944],{"className":76940},[106],[18,76942,12582],{"className":76943},[106,110],[18,76945,76947],{"className":76946},[115],[18,76948,76950,76973],{"className":76949},[119,120],[18,76951,76953,76970],{"className":76952},[124],[18,76954,76956],{"className":76955,"style":1152},[128],[18,76957,76958,76961],{"style":1155},[18,76959],{"className":76960,"style":137},[136],[18,76962,76964],{"className":76963},[141,142,143,144],[18,76965,76967],{"className":76966},[106,144],[18,76968,12575],{"className":76969,"style":254},[106,1478,144],[18,76971,152],{"className":76972},[151],[18,76974,76976],{"className":76975},[124],[18,76977,76979],{"className":76978,"style":159},[128],[18,76980],{},[18,76982],{"className":76983,"style":228},[169],[18,76985,4334],{"className":76986},[232],[18,76988],{"className":76989,"style":228},[169],[18,76991,76993,76996,77036,77039,77042],{"className":76992},[97],[18,76994],{"className":76995,"style":4563},[101],[18,76997,76999,77002],{"className":76998},[106],[18,77000,12582],{"className":77001},[106,110],[18,77003,77005],{"className":77004},[115],[18,77006,77008,77028],{"className":77007},[119,120],[18,77009,77011,77025],{"className":77010},[124],[18,77012,77014],{"className":77013,"style":1152},[128],[18,77015,77016,77019],{"style":1155},[18,77017],{"className":77018,"style":137},[136],[18,77020,77022],{"className":77021},[141,142,143,144],[18,77023,38934],{"className":77024,"style":38967},[106,110,144],[18,77026,152],{"className":77027},[151],[18,77029,77031],{"className":77030},[124],[18,77032,77034],{"className":77033,"style":159},[128],[18,77035],{},[18,77037],{"className":77038,"style":2068},[169],[18,77040,7987],{"className":77041},[2072],[18,77043],{"className":77044,"style":2068},[169],[18,77046,77048,77052],{"className":77047},[97],[18,77049],{"className":77050,"style":77051},[101],"height:0.313em;",[18,77053,76925],{"className":77054},[174]," 是一个保证（guarantee），不是一个估计（estimate）。当界大于 ",[18,77057,77059,77072],{"className":77058,"translate":22},[21],[18,77060,77062],{"className":77061},[26],[28,77063,77064],{"xmlns":30},[32,77065,77066,77070],{},[35,77067,77068],{},[45,77069,47],{},[86,77071,47],{"encoding":88},[18,77073,77075],{"className":77074,"ariaHidden":51},[93],[18,77076,77078,77081],{"className":77077},[97],[18,77079],{"className":77080,"style":505},[101],[18,77082,47],{"className":77083},[106]," 时，它没有说「误差很大」，它只是没有信息。把泛化界当成测试误差的预测器，是误用的常见来源。",[11,77086,77087,77088,77091],{},"这五句话合起来，解释了为什么 VC 理论在深度学习时代显得「失灵」：它回答的是一个比深度学习实际面对的问题",[717,77089,77090],{},"更难","的问题。深度网络不是在一个任意的分布上学习，而是在图像、文本、语音这些有强结构的分布上学习；不是用任意算法，而是用 SGD 及其变体；不是在整个假设类里搜索，而是在一个由初始化、架构和训练动力学共同决定的子集里搜索。VC 维对这三件事都不敏感。",[77093,77094,77095],"div",{},[77096,77097],"img",{"height":77098,"width":77099,"style":77100,"data-d2-theme":77101,"alt":77102,"src":77103},68,760,"max-width:100%;height:auto","default","VC 理论的逻辑链：从打散到泛化界","data:image\u002Fsvg+xml;base64,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\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",[638,77105,77107],{"id":77106},"六深度学习中的-vc-维","六、深度学习中的 VC 维",[1311,77109,77111],{"id":77110},"_61-参数量估算","6.1 参数量估算",[11,77113,77114,77115,77211,77212,77240,77241,77270,77271,77303],{},"把上一节的界套到实际模型上，只需估算 ",[18,77116,77118,77140],{"className":77117,"translate":22},[21],[18,77119,77121],{"className":77120},[26],[28,77122,77123],{"xmlns":30},[32,77124,77125,77137],{},[35,77126,77127],{},[12561,77128,77129],{},[35,77130,77131,77133,77135],{},[41,77132,4031],{},[41,77134,317],{"mathvariant":316},[41,77136,28169],{},[86,77138,77139],{"encoding":88},"\\sqrt{d\u002Fm}",[18,77141,77143],{"className":77142,"ariaHidden":51},[93],[18,77144,77146,77150],{"className":77145},[97],[18,77147],{"className":77148,"style":77149},[101],"height:1.24em;vertical-align:-0.305em;",[18,77151,77153],{"className":77152},[106,12696],[18,77154,77156,77202],{"className":77155},[119,120],[18,77157,77159,77199],{"className":77158},[124],[18,77160,77163,77182],{"className":77161,"style":77162},[128],"height:0.935em;",[18,77164,77167,77170],{"className":77165,"style":77166},[12710],"top:-3.2em;",[18,77168],{"className":77169,"style":6125},[136],[18,77171,77173,77176,77179],{"className":77172,"style":12719},[106],[18,77174,4031],{"className":77175},[106,110],[18,77177,317],{"className":77178},[106],[18,77180,28169],{"className":77181},[106,110],[18,77183,77185,77188],{"style":77184},"top:-2.895em;",[18,77186],{"className":77187,"style":6125},[136],[18,77189,77192],{"className":77190,"style":77191},[12853],"min-width:1.02em;height:1.28em;",[6130,77193,77196],{"xmlns":6132,"width":12857,"height":77194,"viewBox":77195,"preserveAspectRatio":12860},"1.28em","0 0 400000 1296",[6136,77197],{"d":77198},"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z",[18,77200,152],{"className":77201},[151],[18,77203,77205],{"className":77204},[124],[18,77206,77209],{"className":77207,"style":77208},[128],"height:0.305em;",[18,77210],{}," 的量级（暂时把 ",[18,77213,77215,77228],{"className":77214,"translate":22},[21],[18,77216,77218],{"className":77217},[26],[28,77219,77220],{"xmlns":30},[32,77221,77222,77226],{},[35,77223,77224],{},[41,77225,4031],{},[86,77227,4031],{"encoding":88},[18,77229,77231],{"className":77230,"ariaHidden":51},[93],[18,77232,77234,77237],{"className":77233},[97],[18,77235],{"className":77236,"style":295},[101],[18,77238,4031],{"className":77239},[106,110]," 低估成参数量 ",[18,77242,77244,77258],{"className":77243,"translate":22},[21],[18,77245,77247],{"className":77246},[26],[28,77248,77249],{"xmlns":30},[32,77250,77251,77256],{},[35,77252,77253],{},[41,77254,77255],{},"W",[86,77257,77255],{"encoding":88},[18,77259,77261],{"className":77260,"ariaHidden":51},[93],[18,77262,77264,77267],{"className":77263},[97],[18,77265],{"className":77266,"style":600},[101],[18,77268,77255],{"className":77269,"style":1171},[106,110],"，忽略 ",[18,77272,77274,77289],{"className":77273,"translate":22},[21],[18,77275,77277],{"className":77276},[26],[28,77278,77279],{"xmlns":30},[32,77280,77281,77287],{},[35,77282,77283,77285],{},[41,77284,14733],{},[49,77286,899],{},[86,77288,27772],{"encoding":88},[18,77290,77292],{"className":77291,"ariaHidden":51},[93],[18,77293,77295,77298],{"className":77294},[97],[18,77296],{"className":77297,"style":2890},[101],[18,77299,15174,77301],{"className":77300},[985],[18,77302,15177],{"style":6355}," 因子）：",[43712,77305,77306,77464],{},[43715,77307,77308],{},[43718,77309,77310,77313,77344,77375],{},[43721,77311,77312],{},"模型",[43721,77314,77315,77316],{},"参数量 ",[18,77317,77319,77332],{"className":77318,"translate":22},[21],[18,77320,77322],{"className":77321},[26],[28,77323,77324],{"xmlns":30},[32,77325,77326,77330],{},[35,77327,77328],{},[41,77329,77255],{},[86,77331,77255],{"encoding":88},[18,77333,77335],{"className":77334,"ariaHidden":51},[93],[18,77336,77338,77341],{"className":77337},[97],[18,77339],{"className":77340,"style":600},[101],[18,77342,77255],{"className":77343,"style":1171},[106,110],[43721,77345,77346,77347],{},"训练样本 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,[45,77661,47],{},[86,77663,77664],{"encoding":88},"\\approx 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3.0",[18,77893,77895,77907],{"className":77894,"ariaHidden":51},[93],[18,77896,77898,77901,77904],{"className":77897},[97],[18,77899],{"className":77900,"style":77674},[101],[18,77902,14754],{"className":77903},[232],[18,77905],{"className":77906,"style":228},[169],[18,77908,77910,77913],{"className":77909},[97],[18,77911],{"className":77912,"style":505},[101],[18,77914,77888],{"className":77915},[106],[43718,77917,77918,77921,78008,78093],{},[43736,77919,77920],{},"ResNet-50（ImageNet）",[43736,77922,77923],{},[18,77924,77926,77949],{"className":77925,"translate":22},[21],[18,77927,77929],{"className":77928},[26],[28,77930,77931],{"xmlns":30},[32,77932,77933,77946],{},[35,77934,77935,77938,77940],{},[45,77936,77937],{},"2.56",[49,77939,2015],{},[1556,77941,77942,77944],{},[45,77943,77494],{},[45,77945,77720],{},[86,77947,77948],{"encoding":88},"2.56\\times10^7",[18,77950,77952,77970],{"className":77951,"ariaHidden":51},[93],[18,77953,77955,77958,77961,77964,77967],{"className":77954},[97],[18,77956],{"className":77957,"style":18095},[101],[18,77959,77937],{"className":77960},[106],[18,77962],{"className":77963,"style":2068},[169],[18,77965,2015],{"className":77966},[2072],[18,77968],{"className":77969,"style":2068},[169],[18,77971,77973,77976,77979],{"className":77972},[97],[18,77974],{"className":77975,"style":2416},[101],[18,77977,47],{"className":77978},[106],[18,77980,77982,77985],{"className":77981},[106],[18,77983,430],{"className":77984},[106],[18,77986,77988],{"className":77987},[115],[18,77989,77991],{"className":77990},[119],[18,77992,77994],{"className":77993},[124],[18,77995,77997],{"className":77996,"style":2416},[128],[18,77998,77999,78002],{"style":1962},[18,78000],{"className":78001,"style":137},[136],[18,78003,78005],{"className":78004},[141,142,143,144],[18,78006,77720],{"className":78007},[106,144],[43736,78009,78010],{},[18,78011,78013,78034],{"className":78012,"translate":22},[21],[18,78014,78016],{"className":78015},[26],[28,78017,78018],{"xmlns":30},[32,78019,78020,78032],{},[35,78021,78022,78024,78026],{},[45,78023,77799],{},[49,78025,2015],{},[1556,78027,78028,78030],{},[45,78029,77494],{},[45,78031,77487],{},[86,78033,77810],{"encoding":88},[18,78035,78037,78055],{"className":78036,"ariaHidden":51},[93],[18,78038,78040,78043,78046,78049,78052],{"className":78039},[97],[18,78041],{"className":78042,"style":18095},[101],[18,78044,77799],{"className":78045},[106],[18,78047],{"className":78048,"style":2068},[169],[18,78050,2015],{"className":78051},[2072],[18,78053],{"className":78054,"style":2068},[169],[18,78056,78058,78061,78064],{"className":78057},[97],[18,78059],{"className":78060,"style":2416},[101],[18,78062,47],{"className":78063},[106],[18,78065,78067,78070],{"className":78066},[106],[18,78068,430],{"className":78069},[106],[18,78071,78073],{"className":78072},[115],[18,78074,78076],{"className":78075},[119],[18,78077,78079],{"className":78078},[124],[18,78080,78082],{"className":78081,"style":2416},[128],[18,78083,78084,78087],{"style":1962},[18,78085],{"className":78086,"style":137},[136],[18,78088,78090],{"className":78089},[141,142,143,144],[18,78091,77487],{"className":78092},[106,144],[43736,78094,78095],{},[18,78096,78098,78115],{"className":78097,"translate":22},[21],[18,78099,78101],{"className":78100},[26],[28,78102,78103],{"xmlns":30},[32,78104,78105,78112],{},[35,78106,78107,78109],{},[49,78108,14754],{},[45,78110,78111],{},"4.5",[86,78113,78114],{"encoding":88},"\\approx 4.5",[18,78116,78118,78130],{"className":78117,"ariaHidden":51},[93],[18,78119,78121,78124,78127],{"className":78120},[97],[18,78122],{"className":78123,"style":77674},[101],[18,78125,14754],{"className":78126},[232],[18,78128],{"className":78129,"style":228},[169],[18,78131,78133,78136],{"className":78132},[97],[18,78134],{"className":78135,"style":505},[101],[18,78137,78111],{"className":78138},[106],[43718,78140,78141,78144,78231,78316],{},[43736,78142,78143],{},"ViT-L\u002F16（ImageNet）",[43736,78145,78146],{},[18,78147,78149,78172],{"className":78148,"translate":22},[21],[18,78150,78152],{"className":78151},[26],[28,78153,78154],{"xmlns":30},[32,78155,78156,78169],{},[35,78157,78158,78161,78163],{},[45,78159,78160],{},"3.07",[49,78162,2015],{},[1556,78164,78165,78167],{},[45,78166,77494],{},[45,78168,36438],{},[86,78170,78171],{"encoding":88},"3.07\\times10^8",[18,78173,78175,78193],{"className":78174,"ariaHidden":51},[93],[18,78176,78178,78181,78184,78187,78190],{"className":78177},[97],[18,78179],{"className":78180,"style":18095},[101],[18,78182,78160],{"className":78183},[106],[18,78185],{"className":78186,"style":2068},[169],[18,78188,2015],{"className":78189},[2072],[18,78191],{"className":78192,"style":2068},[169],[18,78194,78196,78199,78202],{"className":78195},[97],[18,78197],{"className":78198,"style":2416},[101],[18,78200,47],{"className":78201},[106],[18,78203,78205,78208],{"className":78204},[106],[18,78206,430],{"className":78207},[106],[18,78209,78211],{"className":78210},[115],[18,78212,78214],{"className":78213},[119],[18,78215,78217],{"className":78216},[124],[18,78218,78220],{"className":78219,"style":2416},[128],[18,78221,78222,78225],{"style":1962},[18,78223],{"className":78224,"style":137},[136],[18,78226,78228],{"className":78227},[141,142,143,144],[18,78229,36438],{"className":78230},[106,144],[43736,78232,78233],{},[18,78234,78236,78257],{"className":78235,"translate":22},[21],[18,78237,78239],{"className":78238},[26],[28,78240,78241],{"xmlns":30},[32,78242,78243,78255],{},[35,78244,78245,78247,78249],{},[45,78246,77799],{},[49,78248,2015],{},[1556,78250,78251,78253],{},[45,78252,77494],{},[45,78254,77487],{},[86,78256,77810],{"encoding":88},[18,78258,78260,78278],{"className":78259,"ariaHidden":51},[93],[18,78261,78263,78266,78269,78272,78275],{"className":78262},[97],[18,78264],{"className":78265,"style":18095},[101],[18,78267,77799],{"className":78268},[106],[18,78270],{"className":78271,"style":2068},[169],[18,78273,2015],{"className":78274},[2072],[18,78276],{"className":78277,"style":2068},[169],[18,78279,78281,78284,78287],{"className":78280},[97],[18,78282],{"className":78283,"style":2416},[101],[18,78285,47],{"className":78286},[106],[18,78288,78290,78293],{"className":78289},[106],[18,78291,430],{"className":78292},[106],[18,78294,78296],{"className":78295},[115],[18,78297,78299],{"className":78298},[119],[18,78300,78302],{"className":78301},[124],[18,78303,78305],{"className":78304,"style":2416},[128],[18,78306,78307,78310],{"style":1962},[18,78308],{"className":78309,"style":137},[136],[18,78311,78313],{"className":78312},[141,142,143,144],[18,78314,77487],{"className":78315},[106,144],[43736,78317,78318],{},[18,78319,78321,78338],{"className":78320,"translate":22},[21],[18,78322,78324],{"className":78323},[26],[28,78325,78326],{"xmlns":30},[32,78327,78328,78335],{},[35,78329,78330,78332],{},[49,78331,14754],{},[45,78333,78334],{},"15.5",[86,78336,78337],{"encoding":88},"\\approx 15.5",[18,78339,78341,78353],{"className":78340,"ariaHidden":51},[93],[18,78342,78344,78347,78350],{"className":78343},[97],[18,78345],{"className":78346,"style":77674},[101],[18,78348,14754],{"className":78349},[232],[18,78351],{"className":78352,"style":228},[169],[18,78354,78356,78359],{"className":78355},[97],[18,78357],{"className":78358,"style":505},[101],[18,78360,78334],{"className":78361},[106],[11,78363,78364,78365,78393,78394,78422,78423,78451,78452,78484],{},"每一个都在 ",[18,78366,78368,78381],{"className":78367,"translate":22},[21],[18,78369,78371],{"className":78370},[26],[28,78372,78373],{"xmlns":30},[32,78374,78375,78379],{},[35,78376,78377],{},[45,78378,47],{},[86,78380,47],{"encoding":88},[18,78382,78384],{"className":78383,"ariaHidden":51},[93],[18,78385,78387,78390],{"className":78386},[97],[18,78388],{"className":78389,"style":505},[101],[18,78391,47],{"className":78392},[106]," 附近或远大于 ",[18,78395,78397,78410],{"className":78396,"translate":22},[21],[18,78398,78400],{"className":78399},[26],[28,78401,78402],{"xmlns":30},[32,78403,78404,78408],{},[35,78405,78406],{},[45,78407,47],{},[86,78409,47],{"encoding":88},[18,78411,78413],{"className":78412,"ariaHidden":51},[93],[18,78414,78416,78419],{"className":78415},[97],[18,78417],{"className":78418,"style":505},[101],[18,78420,47],{"className":78421},[106],"。一个「错误率上界」一旦达到或超过 ",[18,78424,78426,78439],{"className":78425,"translate":22},[21],[18,78427,78429],{"className":78428},[26],[28,78430,78431],{"xmlns":30},[32,78432,78433,78437],{},[35,78434,78435],{},[45,78436,47],{},[86,78438,47],{"encoding":88},[18,78440,78442],{"className":78441,"ariaHidden":51},[93],[18,78443,78445,78448],{"className":78444},[97],[18,78446],{"className":78447,"style":505},[101],[18,78449,47],{"className":78450},[106],"，就等于什么都没说——它连「模型会犯错」这种同义反复都不如，因为它没有排除任何东西。而且这还是把 VC 维低估成参数量的结果；下一小节会看到，深度网络的 VC 维还要乘上 ",[18,78453,78455,78470],{"className":78454,"translate":22},[21],[18,78456,78458],{"className":78457},[26],[28,78459,78460],{"xmlns":30},[32,78461,78462,78468],{},[35,78463,78464,78466],{},[41,78465,14733],{},[49,78467,899],{},[86,78469,27772],{"encoding":88},[18,78471,78473],{"className":78472,"ariaHidden":51},[93],[18,78474,78476,78479],{"className":78475},[97],[18,78477],{"className":78478,"style":2890},[101],[18,78480,15174,78482],{"className":78481},[985],[18,78483,15177],{"style":6355}," 因子。",[11,78486,78487,78488,78516,78517,78545,78546,78607],{},"有人会说：那就把界改紧一点。但问题不是常数，是量级。ImageNet 有 128 万张训练图，ResNet-50 有 2560 万参数，",[18,78489,78491,78504],{"className":78490,"translate":22},[21],[18,78492,78494],{"className":78493},[26],[28,78495,78496],{"xmlns":30},[32,78497,78498,78502],{},[35,78499,78500],{},[41,78501,28169],{},[86,78503,28169],{"encoding":88},[18,78505,78507],{"className":78506,"ariaHidden":51},[93],[18,78508,78510,78513],{"className":78509},[97],[18,78511],{"className":78512,"style":1083},[101],[18,78514,28169],{"className":78515},[106,110]," 比 ",[18,78518,78520,78533],{"className":78519,"translate":22},[21],[18,78521,78523],{"className":78522},[26],[28,78524,78525],{"xmlns":30},[32,78526,78527,78531],{},[35,78528,78529],{},[41,78530,4031],{},[86,78532,4031],{"encoding":88},[18,78534,78536],{"className":78535,"ariaHidden":51},[93],[18,78537,78539,78542],{"className":78538},[97],[18,78540],{"className":78541,"style":295},[101],[18,78543,4031],{"className":78544},[106,110]," 小二十倍。",[717,78547,78548,78549,78577,78578,78606],{},"任何以 ",[18,78550,78552,78565],{"className":78551,"translate":22},[21],[18,78553,78555],{"className":78554},[26],[28,78556,78557],{"xmlns":30},[32,78558,78559,78563],{},[35,78560,78561],{},[41,78562,4031],{},[86,78564,4031],{"encoding":88},[18,78566,78568],{"className":78567,"ariaHidden":51},[93],[18,78569,78571,78574],{"className":78570},[97],[18,78572],{"className":78573,"style":295},[101],[18,78575,4031],{"className":78576},[106,110]," 为主项的界，在这个区间里都只能给出与 ",[18,78579,78581,78594],{"className":78580,"translate":22},[21],[18,78582,78584],{"className":78583},[26],[28,78585,78586],{"xmlns":30},[32,78587,78588,78592],{},[35,78589,78590],{},[45,78591,47],{},[86,78593,47],{"encoding":88},[18,78595,78597],{"className":78596,"ariaHidden":51},[93],[18,78598,78600,78603],{"className":78599},[97],[18,78601],{"className":78602,"style":505},[101],[18,78604,47],{"className":78605},[106]," 同阶或更大的量级。"," 这是样本量与容量之间的量级失配，不是分析技巧的问题。",[1311,78609,78611],{"id":78610},"_62-深度网络的-vc-维","6.2 深度网络的 VC 维",[11,78613,78614],{},"先确认一件事：深度网络的 VC 维是有限的。只要参数个数有限、激活函数是分段多项式（ReLU、Leaky ReLU、MaxOut 都是），VC 维就有限。深度网络并不是一个无限容量的对象。",[11,78616,78617],{},"那么它有多大？",[2210,78619,78620,78782],{},[2213,78621,78622,78625,78626,32696,78691,78719,78720,78752,78753,78781],{},[717,78623,78624],{},"阈值网络。"," Baum 与 Haussler（1989）证明，由阈值门组成的多层网络，VC 维是 ",[18,78627,78629,78656],{"className":78628,"translate":22},[21],[18,78630,78632],{"className":78631},[26],[28,78633,78634],{"xmlns":30},[32,78635,78636,78653],{},[35,78637,78638,78641,78643,78645,78647,78649,78651],{},[41,78639,78640],{},"O",[49,78642,74],{"stretchy":73},[41,78644,77255],{},[41,78646,14733],{},[49,78648,899],{},[41,78650,77255],{},[49,78652,84],{"stretchy":73},[86,78654,78655],{"encoding":88},"O(W\\log W)",[18,78657,78659],{"className":78658,"ariaHidden":51},[93],[18,78660,78662,78665,78668,78671,78674,78677,78682,78685,78688],{"className":78661},[97],[18,78663],{"className":78664,"style":242},[101],[18,78666,78640],{"className":78667,"style":254},[106,110],[18,78669,74],{"className":78670},[250],[18,78672,77255],{"className":78673,"style":1171},[106,110],[18,78675],{"className":78676,"style":170},[169],[18,78678,15174,78680],{"className":78679},[985],[18,78681,15177],{"style":6355},[18,78683],{"className":78684,"style":170},[169],[18,78686,77255],{"className":78687,"style":1171},[106,110],[18,78689,84],{"className":78690},[267],[18,78692,78694,78707],{"className":78693,"translate":22},[21],[18,78695,78697],{"className":78696},[26],[28,78698,78699],{"xmlns":30},[32,78700,78701,78705],{},[35,78702,78703],{},[41,78704,77255],{},[86,78706,77255],{"encoding":88},[18,78708,78710],{"className":78709,"ariaHidden":51},[93],[18,78711,78713,78716],{"className":78712},[97],[18,78714],{"className":78715,"style":600},[101],[18,78717,77255],{"className":78718,"style":1171},[106,110]," 是权重个数。",[18,78721,78723,78738],{"className":78722,"translate":22},[21],[18,78724,78726],{"className":78725},[26],[28,78727,78728],{"xmlns":30},[32,78729,78730,78736],{},[35,78731,78732,78734],{},[41,78733,14733],{},[49,78735,899],{},[86,78737,27772],{"encoding":88},[18,78739,78741],{"className":78740,"ariaHidden":51},[93],[18,78742,78744,78747],{"className":78743},[97],[18,78745],{"className":78746,"style":2890},[101],[18,78748,15174,78750],{"className":78749},[985],[18,78751,15177],{"style":6355}," 因子来自网络内部的分段线性结构：不同的权重符号组合对应不同的线性区域，区域数量随 ",[18,78754,78756,78769],{"className":78755,"translate":22},[21],[18,78757,78759],{"className":78758},[26],[28,78760,78761],{"xmlns":30},[32,78762,78763,78767],{},[35,78764,78765],{},[41,78766,77255],{},[86,78768,77255],{"encoding":88},[18,78770,78772],{"className":78771,"ariaHidden":51},[93],[18,78773,78775,78778],{"className":78774},[97],[18,78776],{"className":78777,"style":600},[101],[18,78779,77255],{"className":78780,"style":1171},[106,110]," 组合增长。",[2213,78783,78784,78787,78788,78857,78858,78886,78887,78971],{},[717,78785,78786],{},"ReLU 网络。"," Bartlett、Harvey、Liaw 与 Mehrabian（2019）给出 ",[18,78789,78791,78819],{"className":78790,"translate":22},[21],[18,78792,78794],{"className":78793},[26],[28,78795,78796],{"xmlns":30},[32,78797,78798,78816],{},[35,78799,78800,78802,78804,78806,78808,78810,78812,78814],{},[41,78801,78640],{},[49,78803,74],{"stretchy":73},[41,78805,77255],{},[41,78807,12582],{},[41,78809,14733],{},[49,78811,899],{},[41,78813,77255],{},[49,78815,84],{"stretchy":73},[86,78817,78818],{"encoding":88},"O(WL\\log W)",[18,78820,78822],{"className":78821,"ariaHidden":51},[93],[18,78823,78825,78828,78831,78834,78837,78840,78843,78848,78851,78854],{"className":78824},[97],[18,78826],{"className":78827,"style":242},[101],[18,78829,78640],{"className":78830,"style":254},[106,110],[18,78832,74],{"className":78833},[250],[18,78835,77255],{"className":78836,"style":1171},[106,110],[18,78838,12582],{"className":78839},[106,110],[18,78841],{"className":78842,"style":170},[169],[18,78844,15174,78846],{"className":78845},[985],[18,78847,15177],{"style":6355},[18,78849],{"className":78850,"style":170},[169],[18,78852,77255],{"className":78853,"style":1171},[106,110],[18,78855,84],{"className":78856},[267]," 的上界（",[18,78859,78861,78874],{"className":78860,"translate":22},[21],[18,78862,78864],{"className":78863},[26],[28,78865,78866],{"xmlns":30},[32,78867,78868,78872],{},[35,78869,78870],{},[41,78871,12582],{},[86,78873,12582],{"encoding":88},[18,78875,78877],{"className":78876,"ariaHidden":51},[93],[18,78878,78880,78883],{"className":78879},[97],[18,78881],{"className":78882,"style":600},[101],[18,78884,12582],{"className":78885},[106,110]," 为层数），并给出几乎匹配的下界 ",[18,78888,78890,78927],{"className":78889,"translate":22},[21],[18,78891,78893],{"className":78892},[26],[28,78894,78895],{"xmlns":30},[32,78896,78897,78924],{},[35,78898,78899,78902,78904,78906,78908,78910,78912,78914,78916,78918,78920,78922],{},[41,78900,78901],{"mathvariant":316},"Ω",[49,78903,74],{"stretchy":73},[41,78905,77255],{},[41,78907,12582],{},[41,78909,14733],{},[49,78911,899],{},[49,78913,74],{"stretchy":73},[41,78915,77255],{},[41,78917,317],{"mathvariant":316},[41,78919,12582],{},[49,78921,84],{"stretchy":73},[49,78923,84],{"stretchy":73},[86,78925,78926],{"encoding":88},"\\Omega(WL\\log(W\u002FL))",[18,78928,78930],{"className":78929,"ariaHidden":51},[93],[18,78931,78933,78936,78939,78942,78945,78948,78951,78956,78959,78962,78965,78968],{"className":78932},[97],[18,78934],{"className":78935,"style":242},[101],[18,78937,78901],{"className":78938},[106],[18,78940,74],{"className":78941},[250],[18,78943,77255],{"className":78944,"style":1171},[106,110],[18,78946,12582],{"className":78947},[106,110],[18,78949],{"className":78950,"style":170},[169],[18,78952,15174,78954],{"className":78953},[985],[18,78955,15177],{"style":6355},[18,78957,74],{"className":78958},[250],[18,78960,77255],{"className":78961,"style":1171},[106,110],[18,78963,317],{"className":78964},[106],[18,78966,12582],{"className":78967},[106,110],[18,78969,3431],{"className":78970},[267],"；伪维（pseudo-dimension）同阶。",[11,78973,78974,78975,79011],{},"结论很清楚：",[717,78976,78977,78978,79010],{},"深度网络的 VC 维随参数线性增长（乘一个 ",[18,78979,78981,78996],{"className":78980,"translate":22},[21],[18,78982,78984],{"className":78983},[26],[28,78985,78986],{"xmlns":30},[32,78987,78988,78994],{},[35,78989,78990,78992],{},[41,78991,14733],{},[49,78993,899],{},[86,78995,27772],{"encoding":88},[18,78997,78999],{"className":78998,"ariaHidden":51},[93],[18,79000,79002,79005],{"className":79001},[97],[18,79003],{"className":79004,"style":2890},[101],[18,79006,15174,79008],{"className":79007},[985],[18,79009,15177],{"style":6355},"）。"," 「深度学习之所以泛化，是因为它的 VC 维很小」这个流行说法是错的。深度网络和线性模型一样，容量随参数增长；区别在于，深度网络的实际泛化误差可以远小于这个容量所暗示的上界。",[11,79013,79014],{},"另一个方向的反例是 1-NN：VC 维无穷，但在很多分布上表现并不差。这说明 VC 维无穷并不等于「不能用」；它只意味着不存在一个对所有分布一致的保证。",[1311,79016,79018],{"id":79017},"_63-三条出路","6.3 三条出路",[11,79020,79021],{},"既然「整个假设类的 VC 维」给不出有用的话，理论就往三个方向走。",[11,79023,79024],{},[717,79025,79026],{},"（a）把容量做得更细：从假设类到算法输出",[11,79028,79029,79030,79058],{},"与其用整个 ",[18,79031,79033,79046],{"className":79032,"translate":22},[21],[18,79034,79036],{"className":79035},[26],[28,79037,79038],{"xmlns":30},[32,79039,79040,79044],{},[35,79041,79042],{},[41,79043,18309],{"mathvariant":1462},[86,79045,46317],{"encoding":88},[18,79047,79049],{"className":79048,"ariaHidden":51},[93],[18,79050,79052,79055],{"className":79051},[97],[18,79053],{"className":79054,"style":600},[101],[18,79056,18309],{"className":79057,"style":46330},[106,1478]," 的 VC 维，不如用「训练解所在的那个小子类」的容量。",[11,79060,79061,79062,16800,79123,79184,79185,79216,79217,79301,79302,79330],{},"线性情形是最好的例子。若 ",[18,79063,79065,79087],{"className":79064,"translate":22},[21],[18,79066,79068],{"className":79067},[26],[28,79069,79070],{"xmlns":30},[32,79071,79072,79084],{},[35,79073,79074,79076,79078,79080,79082],{},[49,79075,2248],{"stretchy":73},[41,79077,38582],{},[49,79079,2248],{"stretchy":73},[49,79081,4334],{},[41,79083,3071],{},[86,79085,79086],{"encoding":88},"\\lVert x\\rVert\\le R",[18,79088,79090,79114],{"className":79089,"ariaHidden":51},[93],[18,79091,79093,79096,79099,79102,79105,79108,79111],{"className":79092},[97],[18,79094],{"className":79095,"style":242},[101],[18,79097,2248],{"className":79098},[250],[18,79100,38582],{"className":79101},[106,110],[18,79103,2248],{"className":79104},[267],[18,79106],{"className":79107,"style":228},[169],[18,79109,4334],{"className":79110},[232],[18,79112],{"className":79113,"style":228},[169],[18,79115,79117,79120],{"className":79116},[97],[18,79118],{"className":79119,"style":600},[101],[18,79121,3071],{"className":79122,"style":3152},[106,110],[18,79124,79126,79148],{"className":79125,"translate":22},[21],[18,79127,79129],{"className":79128},[26],[28,79130,79131],{"xmlns":30},[32,79132,79133,79145],{},[35,79134,79135,79137,79139,79141,79143],{},[49,79136,2248],{"stretchy":73},[41,79138,55926],{},[49,79140,2248],{"stretchy":73},[49,79142,4334],{},[41,79144,3978],{},[86,79146,79147],{"encoding":88},"\\lVert w\\rVert\\le B",[18,79149,79151,79175],{"className":79150,"ariaHidden":51},[93],[18,79152,79154,79157,79160,79163,79166,79169,79172],{"className":79153},[97],[18,79155],{"className":79156,"style":242},[101],[18,79158,2248],{"className":79159},[250],[18,79161,55926],{"className":79162,"style":56002},[106,110],[18,79164,2248],{"className":79165},[267],[18,79167],{"className":79168,"style":228},[169],[18,79170,4334],{"className":79171},[232],[18,79173],{"className":79174,"style":228},[169],[18,79176,79178,79181],{"className":79177},[97],[18,79179],{"className":79180,"style":600},[101],[18,79182,3978],{"className":79183,"style":11305},[106,110],"，且分类间隔至少为 ",[18,79186,79188,79203],{"className":79187,"translate":22},[21],[18,79189,79191],{"className":79190},[26],[28,79192,79193],{"xmlns":30},[32,79194,79195,79200],{},[35,79196,79197],{},[41,79198,79199],{},"γ",[86,79201,79202],{"encoding":88},"\\gamma",[18,79204,79206],{"className":79205,"ariaHidden":51},[93],[18,79207,79209,79212],{"className":79208},[97],[18,79210],{"className":79211,"style":38552},[101],[18,79213,79199],{"className":79214,"style":79215},[106,110],"margin-right:0.0556em;","，则有效容量约为 ",[18,79218,79220,79248],{"className":79219,"translate":22},[21],[18,79221,79223],{"className":79222},[26],[28,79224,79225],{"xmlns":30},[32,79226,79227,79245],{},[35,79228,79229,79231,79233,79235,79237,79239],{},[49,79230,74],{"stretchy":73},[41,79232,3071],{},[41,79234,3978],{},[41,79236,317],{"mathvariant":316},[41,79238,79199],{},[1556,79240,79241,79243],{},[49,79242,84],{"stretchy":73},[45,79244,2266],{},[86,79246,79247],{"encoding":88},"(RB\u002F\\gamma)^2",[18,79249,79251],{"className":79250,"ariaHidden":51},[93],[18,79252,79254,79257,79260,79263,79266,79269,79272],{"className":79253},[97],[18,79255],{"className":79256,"style":10271},[101],[18,79258,74],{"className":79259},[250],[18,79261,3071],{"className":79262,"style":3152},[106,110],[18,79264,3978],{"className":79265,"style":11305},[106,110],[18,79267,317],{"className":79268},[106],[18,79270,79199],{"className":79271,"style":79215},[106,110],[18,79273,79275,79278],{"className":79274},[267],[18,79276,84],{"className":79277},[267],[18,79279,79281],{"className":79280},[115],[18,79282,79284],{"className":79283},[119],[18,79285,79287],{"className":79286},[124],[18,79288,79290],{"className":79289,"style":2416},[128],[18,79291,79292,79295],{"style":1962},[18,79293],{"className":79294,"style":137},[136],[18,79296,79298],{"className":79297},[141,142,143,144],[18,79299,2266],{"className":79300},[106,144],"，与输入维度 ",[18,79303,79305,79318],{"className":79304,"translate":22},[21],[18,79306,79308],{"className":79307},[26],[28,79309,79310],{"xmlns":30},[32,79311,79312,79316],{},[35,79313,79314],{},[41,79315,4031],{},[86,79317,4031],{"encoding":88},[18,79319,79321],{"className":79320,"ariaHidden":51},[93],[18,79322,79324,79327],{"className":79323},[97],[18,79325],{"className":79326,"style":295},[101],[18,79328,4031],{"className":79329},[106,110]," 无关。这正是 SVM 的理论基础：在高维甚至无限维的特征空间里，只要间隔大，容量就小。",[11,79332,79333,79334,79373,79374,79546],{},"深度网络有类似的谱范数版本。把 ",[18,79335,79337,79355],{"className":79336,"translate":22},[21],[18,79338,79340],{"className":79339},[26],[28,79341,79342],{"xmlns":30},[32,79343,79344,79352],{},[35,79345,79346,79348,79350],{},[49,79347,2248],{"stretchy":73},[41,79349,55926],{},[49,79351,2248],{"stretchy":73},[86,79353,79354],{"encoding":88},"\\lVert w\\rVert",[18,79356,79358],{"className":79357,"ariaHidden":51},[93],[18,79359,79361,79364,79367,79370],{"className":79360},[97],[18,79362],{"className":79363,"style":242},[101],[18,79365,2248],{"className":79366},[250],[18,79368,55926],{"className":79369,"style":56002},[106,110],[18,79371,2248],{"className":79372},[267]," 换成各层谱范数的乘积 ",[18,79375,79377,79411],{"className":79376,"translate":22},[21],[18,79378,79380],{"className":79379},[26],[28,79381,79382],{"xmlns":30},[32,79383,79384,79408],{},[35,79385,79386,79394,79396,79402],{},[38,79387,79388,79391],{},[49,79389,79390],{},"∏",[41,79392,79393],{},"l",[49,79395,2248],{"stretchy":73},[38,79397,79398,79400],{},[41,79399,77255],{},[41,79401,79393],{},[38,79403,79404,79406],{},[49,79405,2248],{"stretchy":73},[41,79407,16406],{},[86,79409,79410],{"encoding":88},"\\prod_l\\lVert W_l\\rVert_\\sigma",[18,79412,79414],{"className":79413,"ariaHidden":51},[93],[18,79415,79417,79421,79463,79466,79506],{"className":79416},[97],[18,79418],{"className":79419,"style":79420},[101],"height:1.0497em;vertical-align:-0.2997em;",[18,79422,79424,79427],{"className":79423},[985],[18,79425,79390],{"className":79426,"style":27586},[985,4198,27585],[18,79428,79430],{"className":79429},[115],[18,79431,79433,79455],{"className":79432},[119,120],[18,79434,79436,79452],{"className":79435},[124],[18,79437,79440],{"className":79438,"style":79439},[128],"height:0.1864em;",[18,79441,79442,79445],{"style":27602},[18,79443],{"className":79444,"style":137},[136],[18,79446,79448],{"className":79447},[141,142,143,144],[18,79449,79393],{"className":79450,"style":79451},[106,110,144],"margin-right:0.0197em;",[18,79453,152],{"className":79454},[151],[18,79456,79458],{"className":79457},[124],[18,79459,79461],{"className":79460,"style":27621},[128],[18,79462],{},[18,79464,2248],{"className":79465},[250],[18,79467,79469,79472],{"className":79468},[106],[18,79470,77255],{"className":79471,"style":1171},[106,110],[18,79473,79475],{"className":79474},[115],[18,79476,79478,79498],{"className":79477},[119,120],[18,79479,79481,79495],{"className":79480},[124],[18,79482,79484],{"className":79483,"style":1383},[128],[18,79485,79486,79489],{"style":1386},[18,79487],{"className":79488,"style":137},[136],[18,79490,79492],{"className":79491},[141,142,143,144],[18,79493,79393],{"className":79494,"style":79451},[106,110,144],[18,79496,152],{"className":79497},[151],[18,79499,79501],{"className":79500},[124],[18,79502,79504],{"className":79503,"style":159},[128],[18,79505],{},[18,79507,79509,79512],{"className":79508},[267],[18,79510,2248],{"className":79511},[267],[18,79513,79515],{"className":79514},[115],[18,79516,79518,79538],{"className":79517},[119,120],[18,79519,79521,79535],{"className":79520},[124],[18,79522,79524],{"className":79523,"style":202},[128],[18,79525,79526,79529],{"style":1155},[18,79527],{"className":79528,"style":137},[136],[18,79530,79532],{"className":79531},[141,142,143,144],[18,79533,16406],{"className":79534,"style":3957},[106,110,144],[18,79536,152],{"className":79537},[151],[18,79539,79541],{"className":79540},[124],[18,79542,79544],{"className":79543,"style":159},[128],[18,79545],{},"，再乘上深度和宽度带来的多项式因子、除以间隔，就得到 Bartlett–Foster–Telgarsky（2017）与 Neyshabur 等人（2018）的界。这类界的形式大致是「分子为谱范数乘积的平方，分母为间隔与样本量」；它们在经过特殊正则的小网络上可以非空，但在标准训练的 ImageNet 网络上仍然空泛。",[11,79548,79549,79550,79578],{},"覆盖数（covering number）是同一思想的度量几何版本：VC 维回答「能打散多少点」，覆盖数回答「用多少个半径为 ",[18,79551,79553,79566],{"className":79552,"translate":22},[21],[18,79554,79556],{"className":79555},[26],[28,79557,79558],{"xmlns":30},[32,79559,79560,79564],{},[35,79561,79562],{},[41,79563,32522],{},[86,79565,35413],{"encoding":88},[18,79567,79569],{"className":79568,"ariaHidden":51},[93],[18,79570,79572,79575],{"className":79571},[97],[18,79573],{"className":79574,"style":1083},[101],[18,79576,32522],{"className":79577},[106,110]," 的球能把假设类盖住」。二者通过 Dudley 熵积分相连。",[11,79580,79581,79584,79585,79613,79614,79664],{},[717,79582,79583],{},"PAC-Bayes"," 则把容量变成了「后验与先验的距离」。对任意后验分布 ",[18,79586,79588,79601],{"className":79587,"translate":22},[21],[18,79589,79591],{"className":79590},[26],[28,79592,79593],{"xmlns":30},[32,79594,79595,79599],{},[35,79596,79597],{},[41,79598,12547],{},[86,79600,12547],{"encoding":88},[18,79602,79604],{"className":79603,"ariaHidden":51},[93],[18,79605,79607,79610],{"className":79606},[97],[18,79608],{"className":79609,"style":102},[101],[18,79611,12547],{"className":79612},[106,110],"（通常是训练后权重的分布），以 ",[18,79615,79617,79634],{"className":79616,"translate":22},[21],[18,79618,79620],{"className":79619},[26],[28,79621,79622],{"xmlns":30},[32,79623,79624,79632],{},[35,79625,79626,79628,79630],{},[45,79627,47],{},[49,79629,2253],{},[41,79631,5351],{},[86,79633,75194],{"encoding":88},[18,79635,79637,79655],{"className":79636,"ariaHidden":51},[93],[18,79638,79640,79643,79646,79649,79652],{"className":79639},[97],[18,79641],{"className":79642,"style":18095},[101],[18,79644,47],{"className":79645},[106],[18,79647],{"className":79648,"style":2068},[169],[18,79650,2253],{"className":79651},[2072],[18,79653],{"className":79654,"style":2068},[169],[18,79656,79658,79661],{"className":79657},[97],[18,79659],{"className":79660,"style":295},[101],[18,79662,5351],{"className":79663,"style":5480},[106,110]," 的概率，",[18,79666,79668],{"className":79667,"translate":22},[875],[18,79669,79671,79804],{"className":79670,"translate":22},[21],[18,79672,79674],{"className":79673},[26],[28,79675,79676],{"xmlns":30,"display":884},[32,79677,79678,79801],{},[35,79679,79680,79692,79694,79700,79702,79704,79706,79708,79710,79712,79714,79726,79728,79734,79736,79738,79740,79742,79744,79746,79748,79750,79753,79799],{},[38,79681,79682,79684],{},[41,79683,934],{"mathvariant":933},[35,79685,79686,79688,79690],{},[41,79687,18328],{},[49,79689,67],{},[41,79691,12547],{},[49,79693,945],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[38,79695,79696,79698],{},[41,79697,12582],{},[41,79699,12575],{"mathvariant":1462},[49,79701,74],{"stretchy":73},[41,79703,18328],{},[49,79705,84],{"stretchy":73},[49,79707,965],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[926,79709,71861],{},[49,79711,4334],{},[926,79713,71861],{},[38,79715,79716,79718],{},[41,79717,934],{"mathvariant":933},[35,79719,79720,79722,79724],{},[41,79721,18328],{},[49,79723,67],{},[41,79725,12547],{},[49,79727,945],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[38,79729,79730,79732],{},[41,79731,12582],{},[41,79733,38934],{},[49,79735,74],{"stretchy":73},[41,79737,18328],{},[49,79739,84],{"stretchy":73},[49,79741,965],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[926,79743,71861],{},[49,79745,7987],{},[926,79747,71861],{},[41,79749,78640],{},[926,79751,79752],{}," ⁣",[35,79754,79755,79757,79797],{},[49,79756,74],{"fence":51},[12561,79758,79759],{},[5654,79760,79761,79795],{},[35,79762,79763,79769,79771,79773,79775,79777,79779,79781,79783,79785,79787,79789,79791,79793],{},[35,79764,79765,79767],{},[41,79766,5031],{"mathvariant":316},[41,79768,12582],{"mathvariant":316},[49,79770,74],{"stretchy":73},[41,79772,12547],{},[41,79774,2248],{"mathvariant":316},[41,79776,939],{},[49,79778,84],{"stretchy":73},[49,79780,7987],{},[41,79782,14733],{},[49,79784,899],{},[49,79786,74],{"stretchy":73},[45,79788,47],{},[41,79790,317],{"mathvariant":316},[41,79792,5351],{},[49,79794,84],{"stretchy":73},[41,79796,28169],{},[49,79798,84],{"fence":51},[41,79800,968],{"mathvariant":316},[86,79802,79803],{"encoding":88},"\\mathbb E_{h\\sim Q}\\bigl[L_{\\mathcal D}(h)\\bigr]\n\\;\\le\\;\n\\mathbb E_{h\\sim Q}\\bigl[L_S(h)\\bigr]\n\\;+\\;O\\!\\left(\\sqrt{\\frac{\\mathrm{KL}(Q\\Vert P)+\\log(1\u002F\\delta)}{m}}\\right).",[18,79805,79807,79941,80072],{"className":79806,"ariaHidden":51},[93],[18,79808,79810,79813,79862,79868,79911,79914,79917,79920,79926,79929,79932,79935,79938],{"className":79809},[97],[18,79811],{"className":79812,"style":6101},[101],[18,79814,79816,79819],{"className":79815},[106],[18,79817,934],{"className":79818},[106,1139],[18,79820,79822],{"className":79821},[115],[18,79823,79825,79854],{"className":79824},[119,120],[18,79826,79828,79851],{"className":79827},[124],[18,79829,79831],{"className":79830,"style":1383},[128],[18,79832,79833,79836],{"style":1155},[18,79834],{"className":79835,"style":137},[136],[18,79837,79839],{"className":79838},[141,142,143,144],[18,79840,79842,79845,79848],{"className":79841},[106,144],[18,79843,18328],{"className":79844},[106,110,144],[18,79846,67],{"className":79847},[232,144],[18,79849,12547],{"className":79850},[106,110,144],[18,79852,152],{"className":79853},[151],[18,79855,79857],{"className":79856},[124],[18,79858,79860],{"className":79859,"style":19938},[128],[18,79861],{},[18,79863,79865],{"className":79864},[250],[18,79866,945],{"className":79867},[1230,1196],[18,79869,79871,79874],{"className":79870},[106],[18,79872,12582],{"className":79873},[106,110],[18,79875,79877],{"className":79876},[115],[18,79878,79880,79903],{"className":79879},[119,120],[18,79881,79883,79900],{"className":79882},[124],[18,79884,79886],{"className":79885,"style":1152},[128],[18,79887,79888,79891],{"style":1155},[18,79889],{"className":79890,"style":137},[136],[18,79892,79894],{"className":79893},[141,142,143,144],[18,79895,79897],{"className":79896},[106,144],[18,79898,12575],{"className":79899,"style":254},[106,1478,144],[18,79901,152],{"className":79902},[151],[18,79904,79906],{"className":79905},[124],[18,79907,79909],{"className":79908,"style":159},[128],[18,79910],{},[18,79912,74],{"className":79913},[250],[18,79915,18328],{"className":79916},[106,110],[18,79918,84],{"className":79919},[267],[18,79921,79923],{"className":79922},[267],[18,79924,965],{"className":79925},[1230,1196],[18,79927],{"className":79928,"style":228},[169],[18,79930],{"className":79931,"style":228},[169],[18,79933,4334],{"className":79934},[232],[18,79936],{"className":79937,"style":228},[169],[18,79939],{"className":79940,"style":228},[169],[18,79942,79944,79947,79996,80002,80042,80045,80048,80051,80057,80060,80063,80066,80069],{"className":79943},[97],[18,79945],{"className":79946,"style":6101},[101],[18,79948,79950,79953],{"className":79949},[106],[18,79951,934],{"className":79952},[106,1139],[18,79954,79956],{"className":79955},[115],[18,79957,79959,79988],{"className":79958},[119,120],[18,79960,79962,79985],{"className":79961},[124],[18,79963,79965],{"className":79964,"style":1383},[128],[18,79966,79967,79970],{"style":1155},[18,79968],{"className":79969,"style":137},[136],[18,79971,79973],{"className":79972},[141,142,143,144],[18,79974,79976,79979,79982],{"className":79975},[106,144],[18,79977,18328],{"className":79978},[106,110,144],[18,79980,67],{"className":79981},[232,144],[18,79983,12547],{"className":79984},[106,110,144],[18,79986,152],{"className":79987},[151],[18,79989,79991],{"className":79990},[124],[18,79992,79994],{"className":79993,"style":19938},[128],[18,79995],{},[18,79997,79999],{"className":79998},[250],[18,80000,945],{"className":80001},[1230,1196],[18,80003,80005,80008],{"className":80004},[106],[18,80006,12582],{"className":80007},[106,110],[18,80009,80011],{"className":80010},[115],[18,80012,80014,80034],{"className":80013},[119,120],[18,80015,80017,80031],{"className":80016},[124],[18,80018,80020],{"className":80019,"style":1152},[128],[18,80021,80022,80025],{"style":1155},[18,80023],{"className":80024,"style":137},[136],[18,80026,80028],{"className":80027},[141,142,143,144],[18,80029,38934],{"className":80030,"style":38967},[106,110,144],[18,80032,152],{"className":80033},[151],[18,80035,80037],{"className":80036},[124],[18,80038,80040],{"className":80039,"style":159},[128],[18,80041],{},[18,80043,74],{"className":80044},[250],[18,80046,18328],{"className":80047},[106,110],[18,80049,84],{"className":80050},[267],[18,80052,80054],{"className":80053},[267],[18,80055,965],{"className":80056},[1230,1196],[18,80058],{"className":80059,"style":228},[169],[18,80061],{"className":80062,"style":2068},[169],[18,80064,7987],{"className":80065},[2072],[18,80067],{"className":80068,"style":228},[169],[18,80070],{"className":80071,"style":2068},[169],[18,80073,80075,80079,80082,80086,80089,80254,80257],{"className":80074},[97],[18,80076],{"className":80077,"style":80078},[101],"height:3em;vertical-align:-1.25em;",[18,80080,78640],{"className":80081,"style":254},[106,110],[18,80083],{"className":80084,"style":80085},[169],"margin-right:-0.1667em;",[18,80087],{"className":80088,"style":170},[169],[18,80090,80092,80098,80248],{"className":80091},[174],[18,80093,80095],{"className":80094,"style":13796},[250,13795],[18,80096,74],{"className":80097},[1230,61203],[18,80099,80101],{"className":80100},[106,12696],[18,80102,80104,80240],{"className":80103},[119,120],[18,80105,80107,80237],{"className":80106},[124],[18,80108,80110,80225],{"className":80109,"style":74419},[128],[18,80111,80113,80116],{"className":80112,"style":12711},[12710],[18,80114],{"className":80115,"style":12715},[136],[18,80117,80119],{"className":80118,"style":12719},[106],[18,80120,80122,80125,80222],{"className":80121},[106],[18,80123],{"className":80124},[250,6012],[18,80126,80128],{"className":80127},[5654],[18,80129,80131,80214],{"className":80130},[119,120],[18,80132,80134,80211],{"className":80133},[124],[18,80135,80137,80148,80156],{"className":80136,"style":18133},[128],[18,80138,80139,80142],{"style":6028},[18,80140],{"className":80141,"style":697},[136],[18,80143,80145],{"className":80144},[106],[18,80146,28169],{"className":80147},[106,110],[18,80149,80150,80153],{"style":6040},[18,80151],{"className":80152,"style":697},[136],[18,80154],{"className":80155,"style":6048},[6047],[18,80157,80158,80161],{"style":6051},[18,80159],{"className":80160,"style":697},[136],[18,80162,80164,80170,80173,80176,80179,80182,80185,80188,80191,80194,80199,80202,80205,80208],{"className":80163},[106],[18,80165,80167],{"className":80166},[106],[18,80168,12817],{"className":80169},[106,6354],[18,80171,74],{"className":80172},[250],[18,80174,12547],{"className":80175},[106,110],[18,80177,2248],{"className":80178},[106],[18,80180,939],{"className":80181,"style":1171},[106,110],[18,80183,84],{"className":80184},[267],[18,80186],{"className":80187,"style":2068},[169],[18,80189,7987],{"className":80190},[2072],[18,80192],{"className":80193,"style":2068},[169],[18,80195,15174,80197],{"className":80196},[985],[18,80198,15177],{"style":6355},[18,80200,74],{"className":80201},[250],[18,80203,335],{"className":80204},[106],[18,80206,5351],{"className":80207,"style":5480},[106,110],[18,80209,84],{"className":80210},[267],[18,80212,152],{"className":80213},[151],[18,80215,80217],{"className":80216},[124],[18,80218,80220],{"className":80219,"style":6070},[128],[18,80221],{},[18,80223],{"className":80224},[267,6012],[18,80226,80227,80230],{"style":74556},[18,80228],{"className":80229,"style":12715},[136],[18,80231,80233],{"className":80232,"style":12854},[12853],[6130,80234,80235],{"xmlns":6132,"width":12857,"height":12858,"viewBox":12859,"preserveAspectRatio":12860},[6136,80236],{"d":12863},[18,80238,152],{"className":80239},[151],[18,80241,80243],{"className":80242},[124],[18,80244,80246],{"className":80245,"style":74576},[128],[18,80247],{},[18,80249,80251],{"className":80250,"style":13796},[267,13795],[18,80252,84],{"className":80253},[1230,61203],[18,80255],{"className":80256,"style":170},[169],[18,80258,968],{"className":80259},[106],[11,80261,80262,80263,80291,80292,80320],{},"这里的复杂度不再是假设类的 VC 维，而是「训练后的分布 ",[18,80264,80266,80279],{"className":80265,"translate":22},[21],[18,80267,80269],{"className":80268},[26],[28,80270,80271],{"xmlns":30},[32,80272,80273,80277],{},[35,80274,80275],{},[41,80276,12547],{},[86,80278,12547],{"encoding":88},[18,80280,80282],{"className":80281,"ariaHidden":51},[93],[18,80283,80285,80288],{"className":80284},[97],[18,80286],{"className":80287,"style":102},[101],[18,80289,12547],{"className":80290},[106,110]," 离先验 ",[18,80293,80295,80308],{"className":80294,"translate":22},[21],[18,80296,80298],{"className":80297},[26],[28,80299,80300],{"xmlns":30},[32,80301,80302,80306],{},[35,80303,80304],{},[41,80305,939],{},[86,80307,939],{"encoding":88},[18,80309,80311],{"className":80310,"ariaHidden":51},[93],[18,80312,80314,80317],{"className":80313},[97],[18,80315],{"className":80316,"style":600},[101],[18,80318,939],{"className":80319,"style":1171},[106,110]," 有多远」。Dziugaite 与 Roy（2017）用它在 MNIST 规模的网络上算出了非空的界，这是 PAC-Bayes 在深度网络上最早显示出实际内容的结果之一。",[11,80322,80323,80326,80327,80355,80356,80443],{},[717,80324,80325],{},"压缩"," 是另一条路：若训练好的网络可以被压缩成 ",[18,80328,80330,80343],{"className":80329,"translate":22},[21],[18,80331,80333],{"className":80332},[26],[28,80334,80335],{"xmlns":30},[32,80336,80337,80341],{},[35,80338,80339],{},[41,80340,39964],{},[86,80342,39964],{"encoding":88},[18,80344,80346],{"className":80345,"ariaHidden":51},[93],[18,80347,80349,80352],{"className":80348},[97],[18,80350],{"className":80351,"style":295},[101],[18,80353,39964],{"className":80354,"style":40040},[106,110]," 比特而不改变其在训练集上的预测，则泛化误差约为 ",[18,80357,80359,80381],{"className":80358,"translate":22},[21],[18,80360,80362],{"className":80361},[26],[28,80363,80364],{"xmlns":30},[32,80365,80366,80378],{},[35,80367,80368],{},[12561,80369,80370],{},[35,80371,80372,80374,80376],{},[41,80373,39964],{},[41,80375,317],{"mathvariant":316},[41,80377,28169],{},[86,80379,80380],{"encoding":88},"\\sqrt{k\u002Fm}",[18,80382,80384],{"className":80383,"ariaHidden":51},[93],[18,80385,80387,80390],{"className":80386},[97],[18,80388],{"className":80389,"style":77149},[101],[18,80391,80393],{"className":80392},[106,12696],[18,80394,80396,80435],{"className":80395},[119,120],[18,80397,80399,80432],{"className":80398},[124],[18,80400,80402,80420],{"className":80401,"style":77162},[128],[18,80403,80405,80408],{"className":80404,"style":77166},[12710],[18,80406],{"className":80407,"style":6125},[136],[18,80409,80411,80414,80417],{"className":80410,"style":12719},[106],[18,80412,39964],{"className":80413,"style":40040},[106,110],[18,80415,317],{"className":80416},[106],[18,80418,28169],{"className":80419},[106,110],[18,80421,80422,80425],{"style":77184},[18,80423],{"className":80424,"style":6125},[136],[18,80426,80428],{"className":80427,"style":77191},[12853],[6130,80429,80430],{"xmlns":6132,"width":12857,"height":77194,"viewBox":77195,"preserveAspectRatio":12860},[6136,80431],{"d":77198},[18,80433,152],{"className":80434},[151],[18,80436,80438],{"className":80437},[124],[18,80439,80441],{"className":80440,"style":77208},[128],[18,80442],{},"。Zhou 等人（2019）用剪枝加量化的压缩方案，在 ImageNet 规模上给出了非空（但仍然很松）的界。「非空」是里程碑，但离「解释」还很远。",[11,80445,80446,80449],{},[717,80447,80448],{},"稳定性"," 则把容量挂在算法上：若算法对替换一个样本不敏感，泛化误差就可以被稳定性系数控制（Bousquet–Elisseeff，2002）。Hardt、Recht 与 Singer（2016）把这一框架用到了 SGD 上。",[11,80451,80452,80453,80456,80457,80485],{},"这些界的共同点很重要：",[717,80454,80455],{},"它们中的大多数不再是纯粹「假设类」的属性，而是「算法输出」或「数据」的属性。"," 容量从单一的 ",[18,80458,80460,80473],{"className":80459,"translate":22},[21],[18,80461,80463],{"className":80462},[26],[28,80464,80465],{"xmlns":30},[32,80466,80467,80471],{},[35,80468,80469],{},[41,80470,18309],{"mathvariant":1462},[86,80472,46317],{"encoding":88},[18,80474,80476],{"className":80475,"ariaHidden":51},[93],[18,80477,80479,80482],{"className":80478},[97],[18,80480],{"className":80481,"style":600},[101],[18,80483,18309],{"className":80484,"style":46330},[106,1478]," 转移到了假设类、算法、数据三者共同决定的整体上。",[11,80487,80488],{},[717,80489,80490],{},"（b）算法偏置：SGD 只在一个很小的子集里搜索",[11,80492,80493,80521,80522,80550,80551,80579],{},[18,80494,80496,80509],{"className":80495,"translate":22},[21],[18,80497,80499],{"className":80498},[26],[28,80500,80501],{"xmlns":30},[32,80502,80503,80507],{},[35,80504,80505],{},[41,80506,18309],{"mathvariant":1462},[86,80508,46317],{"encoding":88},[18,80510,80512],{"className":80511,"ariaHidden":51},[93],[18,80513,80515,80518],{"className":80514},[97],[18,80516],{"className":80517,"style":600},[101],[18,80519,18309],{"className":80520,"style":46330},[106,1478]," 很大，但 SGD 不是均匀地在 ",[18,80523,80525,80538],{"className":80524,"translate":22},[21],[18,80526,80528],{"className":80527},[26],[28,80529,80530],{"xmlns":30},[32,80531,80532,80536],{},[35,80533,80534],{},[41,80535,18309],{"mathvariant":1462},[86,80537,46317],{"encoding":88},[18,80539,80541],{"className":80540,"ariaHidden":51},[93],[18,80542,80544,80547],{"className":80543},[97],[18,80545],{"className":80546,"style":600},[101],[18,80548,18309],{"className":80549,"style":46330},[106,1478]," 里搜索。从初始化出发、沿着梯度的方向、在有限步内，它只能到达 ",[18,80552,80554,80567],{"className":80553,"translate":22},[21],[18,80555,80557],{"className":80556},[26],[28,80558,80559],{"xmlns":30},[32,80560,80561,80565],{},[35,80562,80563],{},[41,80564,18309],{"mathvariant":1462},[86,80566,46317],{"encoding":88},[18,80568,80570],{"className":80569,"ariaHidden":51},[93],[18,80571,80573,80576],{"className":80572},[97],[18,80574],{"className":80575,"style":600},[101],[18,80577,18309],{"className":80578,"style":46330},[106,1478]," 的一个很小的子集。",[11,80581,80582],{},"线性可分的情形已经被严格刻画：梯度下降收敛到最大间隔解（Soudry 等，2018），即使假设类包含所有线性分类器。深度网络里也有大量关于隐式正则的工作，试图说明梯度方法偏向于某种「最小范数」或「最大间隔」的解。",[11,80584,80585],{},"神经正切核（NTK）提供了一个更清晰的图景：当网络宽度趋于无穷时，训练动力学等价于核回归，泛化由核的谱和 RKHS 范数控制。这把容量从参数空间搬到了函数空间：不是问「有多少个参数」，而是问「在核的度量下，解有多大的范数」。",[11,80587,80588],{},"但要小心：NTK 只描述「懒惰训练」（lazy regime），即特征几乎不变的情形。真实网络会学特征，这时 NTK 不再适用，隐式偏置的理论也远未完成。这是当前深度学习理论最核心的开放问题之一。",[11,80590,80591],{},[717,80592,80593],{},"（c）数据分布：最坏情况太坏",[11,80595,80596],{},"VC 界对每个分布都成立，因此必须按最坏的分布来定价。真实数据不是最坏的：图像有低维结构，类别之间有间隔，长尾分布有层次。",[11,80598,80599,80602,80603,80631,80632,80660],{},[717,80600,80601],{},"双下降","（double descent）是一个典型现象：当模型容量、训练时间或样本量越过插值阈值时，测试误差可以出现第二次下降，而不是经典 U 形曲线预言的那样一路变差（Belkin 等，2019；Nakkiran 等，2020）。需要注意，经典理论从来没有证明过 U 形曲线——它只是小模型时代的经验总结；双下降也不「推翻」VC 理论，它只是说明在 ",[18,80604,80606,80619],{"className":80605,"translate":22},[21],[18,80607,80609],{"className":80608},[26],[28,80610,80611],{"xmlns":30},[32,80612,80613,80617],{},[35,80614,80615],{},[41,80616,4031],{},[86,80618,4031],{"encoding":88},[18,80620,80622],{"className":80621,"ariaHidden":51},[93],[18,80623,80625,80628],{"className":80624},[97],[18,80626],{"className":80627,"style":295},[101],[18,80629,4031],{"className":80630},[106,110]," 随 ",[18,80633,80635,80648],{"className":80634,"translate":22},[21],[18,80636,80638],{"className":80637},[26],[28,80639,80640],{"xmlns":30},[32,80641,80642,80646],{},[35,80643,80644],{},[41,80645,28169],{},[86,80647,28169],{"encoding":88},[18,80649,80651],{"className":80650,"ariaHidden":51},[93],[18,80652,80654,80657],{"className":80653},[97],[18,80655],{"className":80656,"style":1083},[101],[18,80658,28169],{"className":80659},[106,110]," 变化的区域，偏差 - 方差分解的直觉不再适用。",[11,80662,80663,80666],{},[717,80664,80665],{},"良性过拟合","（benign overfitting）则更反直觉：在高维线性回归中，最小范数插值器可以是一致的，只要数据协方差的谱衰减得足够快（Bartlett、Long、Lugosi、Tsigler，2020）。「插值」不等于「过拟合」——这是一个已经被严格证明的现象，而它的成立强烈依赖数据分布的具体性质。",[11,80668,80669],{},"这些结果都指向同一件事：泛化不是假设类单独的性质，而是假设类、算法、分布三者共同的性质。VC 维只看到了第一个。",[1311,80671,80673],{"id":80672},"_64-一致收敛的局限","6.4 一致收敛的局限",[11,80675,80676,80677,80680],{},"还有一条更根本的批评。Nagarajan 与 Kolter（2019）构造了一类学习问题，使得真实泛化误差可以很小，但",[717,80678,80679],{},"任何","基于一致收敛的界都必然很大。换句话说，「把 VC 界改紧」这条路存在原理性的上限。",[11,80682,80683],{},"这不是说他们的构造代表了真实网络，而是说：要解释深度学习，必须放弃「对所有假设一致」的要求，转而利用算法或分布的结构。一致收敛是一个充分条件，不是必要条件。",[11,80685,80686,80687,2702,80715,80743,80744],{},"这里有一个容易被忽略的逻辑细节：对二分类 ",[18,80688,80690,80703],{"className":80689,"translate":22},[21],[18,80691,80693],{"className":80692},[26],[28,80694,80695],{"xmlns":30},[32,80696,80697,80701],{},[35,80698,80699],{},[45,80700,430],{},[86,80702,430],{"encoding":88},[18,80704,80706],{"className":80705,"ariaHidden":51},[93],[18,80707,80709,80712],{"className":80708},[97],[18,80710],{"className":80711,"style":505},[101],[18,80713,430],{"className":80714},[106],[18,80716,80718,80731],{"className":80717,"translate":22},[21],[18,80719,80721],{"className":80720},[26],[28,80722,80723],{"xmlns":30},[32,80724,80725,80729],{},[35,80726,80727],{},[45,80728,47],{},[86,80730,47],{"encoding":88},[18,80732,80734],{"className":80733,"ariaHidden":51},[93],[18,80735,80737,80740],{"className":80736},[97],[18,80738],{"className":80739,"style":505},[101],[18,80741,47],{"className":80742},[106]," 损失，有限 VC 维确实是 PAC 可学习的充要条件。所以如果深度网络在分布无关的意义下可学，它的 VC 维必须有限——而它确实有限。逻辑上没有矛盾；矛盾只在「界」与「实际」的巨大差距。",[717,80745,80746],{},"VC 理论没有被证伪，它只是没有提供我们想要的那种解释。",[638,80748,80750],{"id":80749},"七容量度量谱系","七、容量度量谱系",[11,80752,80753,80754,80757,80758,80786,80787,80790,80791,80819],{},"在前面的讨论里，VC 维一直默认假设输出是二值的。对实值函数类（回归、概率输出、间隔），它不能直接使用。标准的替代有两种：把实值函数按阈值二值化后取 VC 维，得到 ",[717,80755,80756],{},"pseudo-dimension","；或者要求在某个尺度 ",[18,80759,80761,80774],{"className":80760,"translate":22},[21],[18,80762,80764],{"className":80763},[26],[28,80765,80766],{"xmlns":30},[32,80767,80768,80772],{},[35,80769,80770],{},[41,80771,79199],{},[86,80773,79202],{"encoding":88},[18,80775,80777],{"className":80776,"ariaHidden":51},[93],[18,80778,80780,80783],{"className":80779},[97],[18,80781],{"className":80782,"style":38552},[101],[18,80784,79199],{"className":80785,"style":79215},[106,110]," 下仍能打散，得到 ",[717,80788,80789],{},"fat-shattering dimension","。后者是间隔理论的正式语言：一个类完全可以在 VC 维意义下无穷，却在每个固定的 ",[18,80792,80794,80807],{"className":80793,"translate":22},[21],[18,80795,80797],{"className":80796},[26],[28,80798,80799],{"xmlns":30},[32,80800,80801,80805],{},[35,80802,80803],{},[41,80804,79199],{},[86,80806,79202],{"encoding":88},[18,80808,80810],{"className":80809,"ariaHidden":51},[93],[18,80811,80813,80816],{"className":80812},[97],[18,80814],{"className":80815,"style":38552},[101],[18,80817,79199],{"className":80818,"style":79215},[106,110]," 下只有有限的 fat-shattering 维数——这正是「间隔大则容量小」的精确表述。",[11,80821,80822],{},"把上面提到的各种度量放在一张表里，可以看到它们各自回答的是不同的问题。",[43712,80824,80825,80844],{},[43715,80826,80827],{},[43718,80828,80829,80832,80835,80838,80841],{},[43721,80830,80831],{},"度量",[43721,80833,80834],{},"度量对象",[43721,80836,80837],{},"核心思想",[43721,80839,80840],{},"典型界",[43721,80842,80843],{},"主要局限",[43731,80845,80846,81054,81069,81086,81131,81177,81333,81491,81535,81636],{},[43718,80847,80848,80850,80853,80856,80994],{},[43736,80849,52197],{},[43736,80851,80852],{},"二值函数类",[43736,80854,80855],{},"最大打散集",[43736,80857,80858],{},[18,80859,80861,80903],{"className":80860,"translate":22},[21],[18,80862,80864],{"className":80863},[26],[28,80865,80866],{"xmlns":30},[32,80867,80868,80900],{},[35,80869,80870],{},[12561,80871,80872],{},[35,80873,80874,80876,80878,80880,80882,80884,80886,80888,80890,80892,80894,80896,80898],{},[49,80875,74],{"stretchy":73},[41,80877,4031],{},[49,80879,7987],{},[41,80881,14733],{},[49,80883,899],{},[49,80885,74],{"stretchy":73},[45,80887,47],{},[41,80889,317],{"mathvariant":316},[41,80891,5351],{},[49,80893,84],{"stretchy":73},[49,80895,84],{"stretchy":73},[41,80897,317],{"mathvariant":316},[41,80899,28169],{},[86,80901,80902],{"encoding":88},"\\sqrt{(d+\\log(1\u002F\\delta))\u002Fm}",[18,80904,80906],{"className":80905,"ariaHidden":51},[93],[18,80907,80909,80912],{"className":80908},[97],[18,80910],{"className":80911,"style":77149},[101],[18,80913,80915],{"className":80914},[106,12696],[18,80916,80918,80986],{"className":80917},[119,120],[18,80919,80921,80983],{"className":80920},[124],[18,80922,80924,80971],{"className":80923,"style":77162},[128],[18,80925,80927,80930],{"className":80926,"style":77166},[12710],[18,80928],{"className":80929,"style":6125},[136],[18,80931,80933,80936,80939,80942,80945,80948,80953,80956,80959,80962,80965,80968],{"className":80932,"style":12719},[106],[18,80934,74],{"className":80935},[250],[18,80937,4031],{"className":80938},[106,110],[18,80940],{"className":80941,"style":2068},[169],[18,80943,7987],{"className":80944},[2072],[18,80946],{"className":80947,"style":2068},[169],[18,80949,15174,80951],{"className":80950},[985],[18,80952,15177],{"style":6355},[18,80954,74],{"className":80955},[250],[18,80957,335],{"className":80958},[106],[18,80960,5351],{"className":80961,"style":5480},[106,110],[18,80963,3431],{"className":80964},[267],[18,80966,317],{"className":80967},[106],[18,80969,28169],{"className":80970},[106,110],[18,80972,80973,80976],{"style":77184},[18,80974],{"className":80975,"style":6125},[136],[18,80977,80979],{"className":80978,"style":77191},[12853],[6130,80980,80981],{"xmlns":6132,"width":12857,"height":77194,"viewBox":77195,"preserveAspectRatio":12860},[6136,80982],{"d":77198},[18,80984,152],{"className":80985},[151],[18,80987,80989],{"className":80988},[124],[18,80990,80992],{"className":80991,"style":77208},[128],[18,80993],{},[43736,80995,80996,80997,2702,81025,81053],{},"分布无关、最坏情况、只适用 ",[18,80998,81000,81013],{"className":80999,"translate":22},[21],[18,81001,81003],{"className":81002},[26],[28,81004,81005],{"xmlns":30},[32,81006,81007,81011],{},[35,81008,81009],{},[45,81010,430],{},[86,81012,430],{"encoding":88},[18,81014,81016],{"className":81015,"ariaHidden":51},[93],[18,81017,81019,81022],{"className":81018},[97],[18,81020],{"className":81021,"style":505},[101],[18,81023,430],{"className":81024},[106],[18,81026,81028,81041],{"className":81027,"translate":22},[21],[18,81029,81031],{"className":81030},[26],[28,81032,81033],{"xmlns":30},[32,81034,81035,81039],{},[35,81036,81037],{},[45,81038,47],{},[86,81040,47],{"encoding":88},[18,81042,81044],{"className":81043,"ariaHidden":51},[93],[18,81045,81047,81050],{"className":81046},[97],[18,81048],{"className":81049,"style":505},[101],[18,81051,47],{"className":81052},[106]," 损失",[43718,81055,81056,81058,81060,81063,81066],{},[43736,81057,65376],{},[43736,81059,80852],{},[43736,81061,81062],{},"限制类的最大规模",[43736,81064,81065],{},"Sauer 多项式界",[43736,81067,81068],{},"组合量，需要再转概率",[43718,81070,81071,81074,81077,81080,81083],{},[43736,81072,81073],{},"Pseudo-dimension",[43736,81075,81076],{},"实值函数类",[43736,81078,81079],{},"阈值化后的 VC 维",[43736,81081,81082],{},"与 VC 同阶",[43736,81084,81085],{},"仍是最坏情况",[43718,81087,81088,81091,81093,81125,81128],{},[43736,81089,81090],{},"Fat-shattering",[43736,81092,81076],{},[43736,81094,81095,81096,81124],{},"尺度 ",[18,81097,81099,81112],{"className":81098,"translate":22},[21],[18,81100,81102],{"className":81101},[26],[28,81103,81104],{"xmlns":30},[32,81105,81106,81110],{},[35,81107,81108],{},[41,81109,79199],{},[86,81111,79202],{"encoding":88},[18,81113,81115],{"className":81114,"ariaHidden":51},[93],[18,81116,81118,81121],{"className":81117},[97],[18,81119],{"className":81120,"style":38552},[101],[18,81122,79199],{"className":81123,"style":79215},[106,110]," 下的打散",[43736,81126,81127],{},"间隔界",[43736,81129,81130],{},"依赖尺度选择",[43718,81132,81133,81136,81139,81171,81174],{},[43736,81134,81135],{},"覆盖数",[43736,81137,81138],{},"度量空间",[43736,81140,81141,81142,81170],{},"用 ",[18,81143,81145,81158],{"className":81144,"translate":22},[21],[18,81146,81148],{"className":81147},[26],[28,81149,81150],{"xmlns":30},[32,81151,81152,81156],{},[35,81153,81154],{},[41,81155,32522],{},[86,81157,35413],{"encoding":88},[18,81159,81161],{"className":81160,"ariaHidden":51},[93],[18,81162,81164,81167],{"className":81163},[97],[18,81165],{"className":81166,"style":1083},[101],[18,81168,32522],{"className":81169},[106,110],"-球覆盖",[43736,81172,81173],{},"Dudley 熵积分",[43736,81175,81176],{},"计算困难",[43718,81178,81179,81181,81184,81187,81330],{},[43736,81180,74183],{},[43736,81182,81183],{},"数据相关",[43736,81185,81186],{},"随机符号下的拟合能力",[43736,81188,81189],{},[18,81190,81192,81232],{"className":81191,"translate":22},[21],[18,81193,81195],{"className":81194},[26],[28,81196,81197],{"xmlns":30},[32,81198,81199,81229],{},[35,81200,81201,81203,81205,81207],{},[45,81202,2266],{},[41,81204,3071],{"mathvariant":3525},[49,81206,7987],{},[12561,81208,81209],{},[35,81210,81211,81213,81215,81217,81219,81221,81223,81225,81227],{},[41,81212,14733],{},[49,81214,899],{},[49,81216,74],{"stretchy":73},[45,81218,47],{},[41,81220,317],{"mathvariant":316},[41,81222,5351],{},[49,81224,84],{"stretchy":73},[41,81226,317],{"mathvariant":316},[41,81228,28169],{},[86,81230,81231],{"encoding":88},"2\\mathfrak R+\\sqrt{\\log(1\u002F\\delta)\u002Fm}",[18,81233,81235,81257],{"className":81234,"ariaHidden":51},[93],[18,81236,81238,81242,81245,81248,81251,81254],{"className":81237},[97],[18,81239],{"className":81240,"style":81241},[101],"height:0.7747em;vertical-align:-0.0833em;",[18,81243,2266],{"className":81244},[106],[18,81246,3071],{"className":81247},[106,3607],[18,81249],{"className":81250,"style":2068},[169],[18,81252,7987],{"className":81253},[2072],[18,81255],{"className":81256,"style":2068},[169],[18,81258,81260,81263],{"className":81259},[97],[18,81261],{"className":81262,"style":77149},[101],[18,81264,81266],{"className":81265},[106,12696],[18,81267,81269,81322],{"className":81268},[119,120],[18,81270,81272,81319],{"className":81271},[124],[18,81273,81275,81307],{"className":81274,"style":77162},[128],[18,81276,81278,81281],{"className":81277,"style":77166},[12710],[18,81279],{"className":81280,"style":6125},[136],[18,81282,81284,81289,81292,81295,81298,81301,81304],{"className":81283,"style":12719},[106],[18,81285,15174,81287],{"className":81286},[985],[18,81288,15177],{"style":6355},[18,81290,74],{"className":81291},[250],[18,81293,335],{"className":81294},[106],[18,81296,5351],{"className":81297,"style":5480},[106,110],[18,81299,84],{"className":81300},[267],[18,81302,317],{"className":81303},[106],[18,81305,28169],{"className":81306},[106,110],[18,81308,81309,81312],{"style":77184},[18,81310],{"className":81311,"style":6125},[136],[18,81313,81315],{"className":81314,"style":77191},[12853],[6130,81316,81317],{"xmlns":6132,"width":12857,"height":77194,"viewBox":77195,"preserveAspectRatio":12860},[6136,81318],{"d":77198},[18,81320,152],{"className":81321},[151],[18,81323,81325],{"className":81324},[124],[18,81326,81328],{"className":81327,"style":77208},[128],[18,81329],{},[43736,81331,81332],{},"需要估计，通常偏松",[43718,81334,81335,81337,81340,81343,81488],{},[43736,81336,79583],{},[43736,81338,81339],{},"后验与先验",[43736,81341,81342],{},"KL 复杂度",[43736,81344,81345],{},[18,81346,81348,81394],{"className":81347,"translate":22},[21],[18,81349,81351],{"className":81350},[26],[28,81352,81353],{"xmlns":30},[32,81354,81355,81391],{},[35,81356,81357],{},[12561,81358,81359],{},[35,81360,81361,81363,81369,81371,81373,81375,81377,81379,81381,81383,81385,81387,81389],{},[49,81362,74],{"stretchy":73},[35,81364,81365,81367],{},[41,81366,5031],{"mathvariant":316},[41,81368,12582],{"mathvariant":316},[49,81370,7987],{},[41,81372,14733],{},[49,81374,899],{},[49,81376,74],{"stretchy":73},[45,81378,47],{},[41,81380,317],{"mathvariant":316},[41,81382,5351],{},[49,81384,84],{"stretchy":73},[49,81386,84],{"stretchy":73},[41,81388,317],{"mathvariant":316},[41,81390,28169],{},[86,81392,81393],{"encoding":88},"\\sqrt{(\\mathrm{KL}+\\log(1\u002F\\delta))\u002Fm}",[18,81395,81397],{"className":81396,"ariaHidden":51},[93],[18,81398,81400,81403],{"className":81399},[97],[18,81401],{"className":81402,"style":77149},[101],[18,81404,81406],{"className":81405},[106,12696],[18,81407,81409,81480],{"className":81408},[119,120],[18,81410,81412,81477],{"className":81411},[124],[18,81413,81415,81465],{"className":81414,"style":77162},[128],[18,81416,81418,81421],{"className":81417,"style":77166},[12710],[18,81419],{"className":81420,"style":6125},[136],[18,81422,81424,81427,81433,81436,81439,81442,81447,81450,81453,81456,81459,81462],{"className":81423,"style":12719},[106],[18,81425,74],{"className":81426},[250],[18,81428,81430],{"className":81429},[106],[18,81431,12817],{"className":81432},[106,6354],[18,81434],{"className":81435,"style":2068},[169],[18,81437,7987],{"className":81438},[2072],[18,81440],{"className":81441,"style":2068},[169],[18,81443,15174,81445],{"className":81444},[985],[18,81446,15177],{"style":6355},[18,81448,74],{"className":81449},[250],[18,81451,335],{"className":81452},[106],[18,81454,5351],{"className":81455,"style":5480},[106,110],[18,81457,3431],{"className":81458},[267],[18,81460,317],{"className":81461},[106],[18,81463,28169],{"className":81464},[106,110],[18,81466,81467,81470],{"style":77184},[18,81468],{"className":81469,"style":6125},[136],[18,81471,81473],{"className":81472,"style":77191},[12853],[6130,81474,81475],{"xmlns":6132,"width":12857,"height":77194,"viewBox":77195,"preserveAspectRatio":12860},[6136,81476],{"d":77198},[18,81478,152],{"className":81479},[151],[18,81481,81483],{"className":81482},[124],[18,81484,81486],{"className":81485,"style":77208},[128],[18,81487],{},[43736,81489,81490],{},"先验的选择",[43718,81492,81493,81495,81498,81501,81532],{},[43736,81494,80448],{},[43736,81496,81497],{},"算法",[43736,81499,81500],{},"替换一个样本的影响",[43736,81502,81503,81504],{},"稳定性系数 ",[18,81505,81507,81520],{"className":81506,"translate":22},[21],[18,81508,81510],{"className":81509},[26],[28,81511,81512],{"xmlns":30},[32,81513,81514,81518],{},[35,81515,81516],{},[41,81517,20986],{},[86,81519,37789],{"encoding":88},[18,81521,81523],{"className":81522,"ariaHidden":51},[93],[18,81524,81526,81529],{"className":81525},[97],[18,81527],{"className":81528,"style":2890},[101],[18,81530,20986],{"className":81531,"style":21038},[106,110],[43736,81533,81534],{},"对 SGD 难以验证",[43718,81536,81537,81539,81542,81545,81633],{},[43736,81538,80325],{},[43736,81540,81541],{},"算法输出",[43736,81543,81544],{},"描述长度",[43736,81546,81547],{},[18,81548,81550,81571],{"className":81549,"translate":22},[21],[18,81551,81553],{"className":81552},[26],[28,81554,81555],{"xmlns":30},[32,81556,81557,81569],{},[35,81558,81559],{},[12561,81560,81561],{},[35,81562,81563,81565,81567],{},[41,81564,39964],{},[41,81566,317],{"mathvariant":316},[41,81568,28169],{},[86,81570,80380],{"encoding":88},[18,81572,81574],{"className":81573,"ariaHidden":51},[93],[18,81575,81577,81580],{"className":81576},[97],[18,81578],{"className":81579,"style":77149},[101],[18,81581,81583],{"className":81582},[106,12696],[18,81584,81586,81625],{"className":81585},[119,120],[18,81587,81589,81622],{"className":81588},[124],[18,81590,81592,81610],{"className":81591,"style":77162},[128],[18,81593,81595,81598],{"className":81594,"style":77166},[12710],[18,81596],{"className":81597,"style":6125},[136],[18,81599,81601,81604,81607],{"className":81600,"style":12719},[106],[18,81602,39964],{"className":81603,"style":40040},[106,110],[18,81605,317],{"className":81606},[106],[18,81608,28169],{"className":81609},[106,110],[18,81611,81612,81615],{"style":77184},[18,81613],{"className":81614,"style":6125},[136],[18,81616,81618],{"className":81617,"style":77191},[12853],[6130,81619,81620],{"xmlns":6132,"width":12857,"height":77194,"viewBox":77195,"preserveAspectRatio":12860},[6136,81621],{"d":77198},[18,81623,152],{"className":81624},[151],[18,81626,81628],{"className":81627},[124],[18,81629,81631],{"className":81630,"style":77208},[128],[18,81632],{},[43736,81634,81635],{},"需要构造压缩方案",[43718,81637,81638,81641,81644,81647,81650],{},[43736,81639,81640],{},"NTK \u002F RKHS 范数",[43736,81642,81643],{},"核函数",[43736,81645,81646],{},"函数空间中的范数",[43736,81648,81649],{},"谱条件",[43736,81651,81652],{},"只适用于懒惰训练",[11,81654,81655,81656,81659],{},"这张表想说明一件事：",[717,81657,81658],{},"「容量」不是一个数，而是一个依设定而变的函数。"," 你选择哪种度量，等于选择了对「什么算简单」的回答。VC 维的回答是「组合上简单」，间隔界的回答是「几何上简单」，PAC-Bayes 的回答是「相对先验简单」，压缩的回答是「描述上简单」。它们不是互相替代的关系，而是从不同角度逼近同一个问题。",[638,81661,45289],{"id":45289},[11,81663,81664,81665],{},"回到开头那两个数字。一个只有 1 个参数的正弦分类器可以有无限 VC 维；一个 2500 万参数的 ResNet 却只能在 ImageNet 上得到一个大于 1 的界。这两件事并不矛盾，它们从两个方向说明了同一件事：",[717,81666,81667],{},"容量不是参数的函数，而是假设类结构的函数；泛化不是容量的函数，而是容量、算法与分布三者共同的函数。",{"title":45552,"searchDepth":45553,"depth":45553,"links":81669},[81670,81675,81680,81685,81691,81692,81698,81699],{"id":46082,"depth":45553,"text":46083,"children":81671},[81672,81673,81674],{"id":46086,"depth":45559,"text":46087},{"id":47798,"depth":45559,"text":47799},{"id":50213,"depth":45559,"text":50214},{"id":50998,"depth":45553,"text":50999,"children":81676},[81677,81678,81679],{"id":51002,"depth":45559,"text":51003},{"id":52765,"depth":45559,"text":52766},{"id":64892,"depth":45559,"text":64893},{"id":64983,"depth":45553,"text":64984,"children":81681},[81682,81683,81684],{"id":64987,"depth":45559,"text":64988},{"id":66178,"depth":45559,"text":66179},{"id":69470,"depth":45559,"text":69471},{"id":71662,"depth":45553,"text":71663,"children":81686},[81687,81688,81689,81690],{"id":71666,"depth":45559,"text":71667},{"id":74705,"depth":45559,"text":74706},{"id":75863,"depth":45559,"text":75864},{"id":76078,"depth":45559,"text":76079},{"id":76582,"depth":45553,"text":76583},{"id":77106,"depth":45553,"text":77107,"children":81693},[81694,81695,81696,81697],{"id":77110,"depth":45559,"text":77111},{"id":78610,"depth":45559,"text":78611},{"id":79017,"depth":45559,"text":79018},{"id":80672,"depth":45559,"text":80673},{"id":80749,"depth":45553,"text":80750},{"id":45289,"depth":45553,"text":45289},{},"\u002Fblog\u002F2025\u002F2025-10-05-vapnik-chervonenkis-dimension",{"title":45602,"description":45607},"blog\u002F2025\u002F2025-10-05-vapnik-chervonenkis-dimension","Vapnik-Chervonenkis Dimension 是统计学习理论中刻画假设类容量的核心概念。本文从打散与增长函数出发，完整推导 Sauer–Shelah 引理与一致收敛泛化界，再讨论它为何在深度网络上失效，以及间隔界、PAC-Bayes、稳定性等替代方案如何回应这一困境。",[45596,45597],"2025-10-05T18:09:22+08:00","pm0qsEbi4FVab4f8OGbmKFSlwV0EdZ0Hzz_xiuH0fJU",{"id":81709,"title":81710,"body":81711,"description":45552,"draft":45586,"enableComment":45587,"extension":45588,"image":45589,"important":45586,"location":45589,"meta":109775,"navigation":45587,"ogImage":45589,"path":109776,"seo":109777,"stem":109778,"summary":109779,"tags":109780,"time":109781,"__hash__":109782},"blog\u002Fblog\u002F2025\u002F2025-09-18-the-stirling-formula.md","斯特林公式",{"type":8,"value":81712,"toc":109771},[81713,81716,81719,81954,81957,82444,82574,83419,84140,84143,84722,84725,85786,85789,85876,87405,87514,88973,88976,89121,89123,89590,89979,90378,91086,91089,91629,91871,93164,93324,93327,93866,94023,94552,94653,96097,96100,96483,96486,96488,96644,96931,96934,97352,97355,98033,98036,98369,98372,99467,99469,99471,99718,99721,99983,99986,99989,100532,101044,101176,101179,101498,101571,102268,102844,102847,103360,103391,103394,103639,103642,103947,103950,104280,104459,104841,104943,105421,105423,106014,106180,106183,106528,106531,106534,106916,107114,107459,107889,107892,108323,108326,108550,108553,109104,109107,109218,109250,109527,109530],[638,81714,81715],{"id":81715},"沃利斯公式",[11,81717,81718],{},"我们先从一个定积分开始：",[18,81720,81722],{"className":81721,"translate":22},[875],[18,81723,81725,81767],{"className":81724,"translate":22},[21],[18,81726,81728],{"className":81727},[26],[28,81729,81730],{"xmlns":30,"display":884},[32,81731,81732,81764],{},[35,81733,81734,81746,81756,81758,81760,81762],{},[1589,81735,81736,81738,81740],{},[49,81737,4005],{},[45,81739,430],{},[5654,81741,81742,81744],{},[41,81743,3941],{},[45,81745,2266],{},[1556,81747,81748,81754],{},[35,81749,81750,81752],{},[41,81751,45646],{},[49,81753,899],{},[41,81755,64],{},[41,81757,38582],{},[926,81759,4024],{},[41,81761,4031],{},[41,81763,38582],{},[86,81765,81766],{"encoding":88},"\\int_{0}^{\\frac{\\pi}{2}} \\sin^n x \\, dx",[18,81768,81770],{"className":81769,"ariaHidden":51},[93],[18,81771,81773,81777,81905,81908,81939,81942,81945,81948,81951],{"className":81772},[97],[18,81774],{"className":81775,"style":81776},[101],"height:2.5089em;vertical-align:-0.9119em;",[18,81778,81780,81783],{"className":81779},[985],[18,81781,4005],{"className":81782,"style":4200},[985,4198,4199],[18,81784,81786],{"className":81785},[115],[18,81787,81789,81896],{"className":81788},[119,120],[18,81790,81792,81893],{"className":81791},[124],[18,81793,81796,81811],{"className":81794,"style":81795},[128],"height:1.597em;",[18,81797,81799,81802],{"style":81798},"top:-2.088em;margin-left:-0.4445em;margin-right:0.05em;",[18,81800],{"className":81801,"style":697},[136],[18,81803,81805],{"className":81804},[141,142,143,144],[18,81806,81808],{"className":81807},[106,144],[18,81809,430],{"className":81810},[106,144],[18,81812,81814,81817],{"style":81813},"top:-4.1129em;margin-right:0.05em;",[18,81815],{"className":81816,"style":697},[136],[18,81818,81820],{"className":81819},[141,142,143,144],[18,81821,81823],{"className":81822},[106,144],[18,81824,81826,81829,81890],{"className":81825},[106,144],[18,81827],{"className":81828},[250,6012,141,1195,38156],[18,81830,81832],{"className":81831},[5654],[18,81833,81835,81881],{"className":81834},[119,120],[18,81836,81838,81878],{"className":81837},[124],[18,81839,81842,81856,81864],{"className":81840,"style":81841},[128],"height:0.6915em;",[18,81843,81844,81847],{"style":38172},[18,81845],{"className":81846,"style":697},[136],[18,81848,81850],{"className":81849},[141,1195,1196,144],[18,81851,81853],{"className":81852},[106,144],[18,81854,2266],{"className":81855},[106,144],[18,81857,81858,81861],{"style":38196},[18,81859],{"className":81860,"style":697},[136],[18,81862],{"className":81863,"style":38203},[6047,144],[18,81865,81866,81869],{"style":42972},[18,81867],{"className":81868,"style":697},[136],[18,81870,81872],{"className":81871},[141,1195,1196,144],[18,81873,81875],{"className":81874},[106,144],[18,81876,3941],{"className":81877,"style":3957},[106,110,144],[18,81879,152],{"className":81880},[151],[18,81882,81884],{"className":81883},[124],[18,81885,81888],{"className":81886,"style":81887},[128],"height:0.344em;",[18,81889],{},[18,81891],{"className":81892},[267,6012,141,1195,38156],[18,81894,152],{"className":81895},[151],[18,81897,81899],{"className":81898},[124],[18,81900,81903],{"className":81901,"style":81902},[128],"height:0.9119em;",[18,81904],{},[18,81906],{"className":81907,"style":170},[169],[18,81909,81911,81914],{"className":81910},[985],[18,81912,45646],{"className":81913},[985],[18,81915,81917],{"className":81916},[115],[18,81918,81920],{"className":81919},[119],[18,81921,81923],{"className":81922},[124],[18,81924,81927],{"className":81925,"style":81926},[128],"height:0.7222em;",[18,81928,81930,81933],{"style":81929},"top:-3.1208em;margin-right:0.05em;",[18,81931],{"className":81932,"style":137},[136],[18,81934,81936],{"className":81935},[141,142,143,144],[18,81937,64],{"className":81938},[106,110,144],[18,81940],{"className":81941,"style":170},[169],[18,81943,38582],{"className":81944},[106,110],[18,81946],{"className":81947,"style":170},[169],[18,81949,4031],{"className":81950},[106,110],[18,81952,38582],{"className":81953},[106,110],[11,81955,81956],{},"我们可以推导出如下结果：",[18,81958,81960],{"className":81959,"translate":22},[875],[18,81961,81963,82054],{"className":81962,"translate":22},[21],[18,81964,81966],{"className":81965},[26],[28,81967,81968],{"xmlns":30,"display":884},[32,81969,81970,82051],{},[35,81971,81972,81984,81994,81996,81998,82000,82002,82004,82006,82018,82034,82036,82038,82040,82042,82045,82047,82049],{},[1589,81973,81974,81976,81978],{},[49,81975,4005],{},[45,81977,430],{},[5654,81979,81980,81982],{},[41,81981,3941],{},[45,81983,2266],{},[1556,81985,81986,81992],{},[35,81987,81988,81990],{},[41,81989,45646],{},[49,81991,899],{},[41,81993,64],{},[41,81995,38582],{},[926,81997,4024],{},[41,81999,4031],{},[41,82001,38582],{},[49,82003,433],{},[49,82005,2253],{},[1589,82007,82008,82010,82012],{},[49,82009,4005],{},[45,82011,430],{},[5654,82013,82014,82016],{},[41,82015,3941],{},[45,82017,2266],{},[1556,82019,82020,82026],{},[35,82021,82022,82024],{},[41,82023,45646],{},[49,82025,899],{},[35,82027,82028,82030,82032],{},[41,82029,64],{},[49,82031,2253],{},[45,82033,47],{},[41,82035,38582],{},[926,82037,4024],{},[41,82039,4031],{},[49,82041,74],{"stretchy":73},[41,82043,82044],{},"cos",[49,82046,899],{},[41,82048,38582],{},[49,82050,84],{"stretchy":73},[86,82052,82053],{"encoding":88},"\\int_{0}^{\\frac{\\pi}{2}} \\sin^n x \\, dx = -\\int_{0}^{\\frac{\\pi}{2}} \\sin^{n-1} x \\, d(\\cos x)",[18,82055,82057,82241],{"className":82056,"ariaHidden":51},[93],[18,82058,82060,82063,82185,82188,82217,82220,82223,82226,82229,82232,82235,82238],{"className":82059},[97],[18,82061],{"className":82062,"style":81776},[101],[18,82064,82066,82069],{"className":82065},[985],[18,82067,4005],{"className":82068,"style":4200},[985,4198,4199],[18,82070,82072],{"className":82071},[115],[18,82073,82075,82177],{"className":82074},[119,120],[18,82076,82078,82174],{"className":82077},[124],[18,82079,82081,82095],{"className":82080,"style":81795},[128],[18,82082,82083,82086],{"style":81798},[18,82084],{"className":82085,"style":697},[136],[18,82087,82089],{"className":82088},[141,142,143,144],[18,82090,82092],{"className":82091},[106,144],[18,82093,430],{"className":82094},[106,144],[18,82096,82097,82100],{"style":81813},[18,82098],{"className":82099,"style":697},[136],[18,82101,82103],{"className":82102},[141,142,143,144],[18,82104,82106],{"className":82105},[106,144],[18,82107,82109,82112,82171],{"className":82108},[106,144],[18,82110],{"className":82111},[250,6012,141,1195,38156],[18,82113,82115],{"className":82114},[5654],[18,82116,82118,82163],{"className":82117},[119,120],[18,82119,82121,82160],{"className":82120},[124],[18,82122,82124,82138,82146],{"className":82123,"style":81841},[128],[18,82125,82126,82129],{"style":38172},[18,82127],{"className":82128,"style":697},[136],[18,82130,82132],{"className":82131},[141,1195,1196,144],[18,82133,82135],{"className":82134},[106,144],[18,82136,2266],{"className":82137},[106,144],[18,82139,82140,82143],{"style":38196},[18,82141],{"className":82142,"style":697},[136],[18,82144],{"className":82145,"style":38203},[6047,144],[18,82147,82148,82151],{"style":42972},[18,82149],{"className":82150,"style":697},[136],[18,82152,82154],{"className":82153},[141,1195,1196,144],[18,82155,82157],{"className":82156},[106,144],[18,82158,3941],{"className":82159,"style":3957},[106,110,144],[18,82161,152],{"className":82162},[151],[18,82164,82166],{"className":82165},[124],[18,82167,82169],{"className":82168,"style":81887},[128],[18,82170],{},[18,82172],{"className":82173},[267,6012,141,1195,38156],[18,82175,152],{"className":82176},[151],[18,82178,82180],{"className":82179},[124],[18,82181,82183],{"className":82182,"style":81902},[128],[18,82184],{},[18,82186],{"className":82187,"style":170},[169],[18,82189,82191,82194],{"className":82190},[985],[18,82192,45646],{"className":82193},[985],[18,82195,82197],{"className":82196},[115],[18,82198,82200],{"className":82199},[119],[18,82201,82203],{"className":82202},[124],[18,82204,82206],{"className":82205,"style":81926},[128],[18,82207,82208,82211],{"style":81929},[18,82209],{"className":82210,"style":137},[136],[18,82212,82214],{"className":82213},[141,142,143,144],[18,82215,64],{"className":82216},[106,110,144],[18,82218],{"className":82219,"style":170},[169],[18,82221,38582],{"className":82222},[106,110],[18,82224],{"className":82225,"style":170},[169],[18,82227,4031],{"className":82228},[106,110],[18,82230,38582],{"className":82231},[106,110],[18,82233],{"className":82234,"style":228},[169],[18,82236,433],{"className":82237},[232],[18,82239],{"className":82240,"style":228},[169],[18,82242,82244,82247,82250,82253,82375,82378,82417,82420,82423,82426,82429,82432,82435,82438,82441],{"className":82243},[97],[18,82245],{"className":82246,"style":81776},[101],[18,82248,2253],{"className":82249},[106],[18,82251],{"className":82252,"style":170},[169],[18,82254,82256,82259],{"className":82255},[985],[18,82257,4005],{"className":82258,"style":4200},[985,4198,4199],[18,82260,82262],{"className":82261},[115],[18,82263,82265,82367],{"className":82264},[119,120],[18,82266,82268,82364],{"className":82267},[124],[18,82269,82271,82285],{"className":82270,"style":81795},[128],[18,82272,82273,82276],{"style":81798},[18,82274],{"className":82275,"style":697},[136],[18,82277,82279],{"className":82278},[141,142,143,144],[18,82280,82282],{"className":82281},[106,144],[18,82283,430],{"className":82284},[106,144],[18,82286,82287,82290],{"style":81813},[18,82288],{"className":82289,"style":697},[136],[18,82291,82293],{"className":82292},[141,142,143,144],[18,82294,82296],{"className":82295},[106,144],[18,82297,82299,82302,82361],{"className":82298},[106,144],[18,82300],{"className":82301},[250,6012,141,1195,38156],[18,82303,82305],{"className":82304},[5654],[18,82306,82308,82353],{"className":82307},[119,120],[18,82309,82311,82350],{"className":82310},[124],[18,82312,82314,82328,82336],{"className":82313,"style":81841},[128],[18,82315,82316,82319],{"style":38172},[18,82317],{"className":82318,"style":697},[136],[18,82320,82322],{"className":82321},[141,1195,1196,144],[18,82323,82325],{"className":82324},[106,144],[18,82326,2266],{"className":82327},[106,144],[18,82329,82330,82333],{"style":38196},[18,82331],{"className":82332,"style":697},[136],[18,82334],{"className":82335,"style":38203},[6047,144],[18,82337,82338,82341],{"style":42972},[18,82339],{"className":82340,"style":697},[136],[18,82342,82344],{"className":82343},[141,1195,1196,144],[18,82345,82347],{"className":82346},[106,144],[18,82348,3941],{"className":82349,"style":3957},[106,110,144],[18,82351,152],{"className":82352},[151],[18,82354,82356],{"className":82355},[124],[18,82357,82359],{"className":82358,"style":81887},[128],[18,82360],{},[18,82362],{"className":82363},[267,6012,141,1195,38156],[18,82365,152],{"className":82366},[151],[18,82368,82370],{"className":82369},[124],[18,82371,82373],{"className":82372,"style":81902},[128],[18,82374],{},[18,82376],{"className":82377,"style":170},[169],[18,82379,82381,82384],{"className":82380},[985],[18,82382,45646],{"className":82383},[985],[18,82385,82387],{"className":82386},[115],[18,82388,82390],{"className":82389},[119],[18,82391,82393],{"className":82392},[124],[18,82394,82397],{"className":82395,"style":82396},[128],"height:0.8719em;",[18,82398,82399,82402],{"style":81929},[18,82400],{"className":82401,"style":137},[136],[18,82403,82405],{"className":82404},[141,142,143,144],[18,82406,82408,82411,82414],{"className":82407},[106,144],[18,82409,64],{"className":82410},[106,110,144],[18,82412,2253],{"className":82413},[2072,144],[18,82415,47],{"className":82416},[106,144],[18,82418],{"className":82419,"style":170},[169],[18,82421,38582],{"className":82422},[106,110],[18,82424],{"className":82425,"style":170},[169],[18,82427,4031],{"className":82428},[106,110],[18,82430,74],{"className":82431},[250],[18,82433,82044],{"className":82434},[985],[18,82436],{"className":82437,"style":170},[169],[18,82439,38582],{"className":82440},[106,110],[18,82442,84],{"className":82443},[267],[11,82445,82446,82447,82573],{},"利用分部积分法，其中 ",[18,82448,82450,82492],{"className":82449,"translate":22},[21],[18,82451,82453],{"className":82452},[26],[28,82454,82455],{"xmlns":30},[32,82456,82457,82489],{},[35,82458,82459,82461,82464,82466,82468,82471,82473,82475,82477,82479,82481,82483,82485,82487],{},[49,82460,4005],{},[41,82462,82463],{},"u",[926,82465,4024],{},[41,82467,4031],{},[41,82469,82470],{},"v",[49,82472,433],{},[41,82474,82463],{},[41,82476,82470],{},[49,82478,2253],{},[49,82480,4005],{},[41,82482,82470],{},[926,82484,4024],{},[41,82486,4031],{},[41,82488,82463],{},[86,82490,82491],{"encoding":88},"\\int u \\, dv = uv - \\int v \\, du",[18,82493,82495,82530,82549],{"className":82494,"ariaHidden":51},[93],[18,82496,82498,82502,82506,82509,82512,82515,82518,82521,82524,82527],{"className":82497},[97],[18,82499],{"className":82500,"style":82501},[101],"height:1.1111em;vertical-align:-0.3061em;",[18,82503,4005],{"className":82504,"style":82505},[985,4198,27585],"margin-right:0.1945em;position:relative;top:-0.0006em;",[18,82507],{"className":82508,"style":170},[169],[18,82510,82463],{"className":82511},[106,110],[18,82513],{"className":82514,"style":170},[169],[18,82516,4031],{"className":82517},[106,110],[18,82519,82470],{"className":82520,"style":3957},[106,110],[18,82522],{"className":82523,"style":228},[169],[18,82525,433],{"className":82526},[232],[18,82528],{"className":82529,"style":228},[169],[18,82531,82533,82536,82540,82543,82546],{"className":82532},[97],[18,82534],{"className":82535,"style":14404},[101],[18,82537,82539],{"className":82538,"style":3957},[106,110],"uv",[18,82541],{"className":82542,"style":2068},[169],[18,82544,2253],{"className":82545},[2072],[18,82547],{"className":82548,"style":2068},[169],[18,82550,82552,82555,82558,82561,82564,82567,82570],{"className":82551},[97],[18,82553],{"className":82554,"style":82501},[101],[18,82556,4005],{"className":82557,"style":82505},[985,4198,27585],[18,82559],{"className":82560,"style":170},[169],[18,82562,82470],{"className":82563,"style":3957},[106,110],[18,82565],{"className":82566,"style":170},[169],[18,82568,4031],{"className":82569},[106,110],[18,82571,82463],{"className":82572},[106,110],"，可得：",[18,82575,82577],{"className":82576,"translate":22},[875],[18,82578,82580,82734],{"className":82579,"translate":22},[21],[18,82581,82583],{"className":82582},[26],[28,82584,82585],{"xmlns":30,"display":884},[32,82586,82587,82731],{},[35,82588,82589,82591,82603,82619,82621,82623,82625,82627,82629,82631,82633,82635,82637,82639,82641,82671,82683,82685,82697,82699,82701,82703,82705,82707],{},[49,82590,2253],{},[1589,82592,82593,82595,82597],{},[49,82594,4005],{},[45,82596,430],{},[5654,82598,82599,82601],{},[41,82600,3941],{},[45,82602,2266],{},[1556,82604,82605,82611],{},[35,82606,82607,82609],{},[41,82608,45646],{},[49,82610,899],{},[35,82612,82613,82615,82617],{},[41,82614,64],{},[49,82616,2253],{},[45,82618,47],{},[41,82620,38582],{},[926,82622,4024],{},[41,82624,4031],{},[49,82626,74],{"stretchy":73},[41,82628,82044],{},[49,82630,899],{},[41,82632,38582],{},[49,82634,84],{"stretchy":73},[169,82636],{"linebreak":8022},[49,82638,433],{},[49,82640,2253],{},[35,82642,82643,82645,82661,82663,82665,82667,82669],{},[49,82644,74],{"fence":51},[1556,82646,82647,82653],{},[35,82648,82649,82651],{},[41,82650,45646],{},[49,82652,899],{},[35,82654,82655,82657,82659],{},[41,82656,64],{},[49,82658,2253],{},[45,82660,47],{},[41,82662,38582],{},[41,82664,82044],{},[49,82666,899],{},[41,82668,38582],{},[49,82670,84],{"fence":51},[1589,82672,82673,82675,82677],{},[49,82674,11225],{"fence":73,"stretchy":51,"minsize":5662,"maxsize":5662},[45,82676,430],{},[5654,82678,82679,82681],{},[41,82680,3941],{},[45,82682,2266],{},[49,82684,7987],{},[1589,82686,82687,82689,82691],{},[49,82688,4005],{},[45,82690,430],{},[5654,82692,82693,82695],{},[41,82694,3941],{},[45,82696,2266],{},[41,82698,82044],{},[49,82700,899],{},[41,82702,38582],{},[926,82704,4024],{},[41,82706,4031],{},[35,82708,82709,82711,82727,82729],{},[49,82710,74],{"fence":51},[1556,82712,82713,82719],{},[35,82714,82715,82717],{},[41,82716,45646],{},[49,82718,899],{},[35,82720,82721,82723,82725],{},[41,82722,64],{},[49,82724,2253],{},[45,82726,47],{},[41,82728,38582],{},[49,82730,84],{"fence":51},[86,82732,82733],{"encoding":88},"-\\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 \\right)",[18,82735,82737,82939,82942,82954,83211],{"className":82736,"ariaHidden":51},[93],[18,82738,82740,82743,82746,82749,82871,82874,82912,82915,82918,82921,82924,82927,82930,82933,82936],{"className":82739},[97],[18,82741],{"className":82742,"style":81776},[101],[18,82744,2253],{"className":82745},[106],[18,82747],{"className":82748,"style":170},[169],[18,82750,82752,82755],{"className":82751},[985],[18,82753,4005],{"className":82754,"style":4200},[985,4198,4199],[18,82756,82758],{"className":82757},[115],[18,82759,82761,82863],{"className":82760},[119,120],[18,82762,82764,82860],{"className":82763},[124],[18,82765,82767,82781],{"className":82766,"style":81795},[128],[18,82768,82769,82772],{"style":81798},[18,82770],{"className":82771,"style":697},[136],[18,82773,82775],{"className":82774},[141,142,143,144],[18,82776,82778],{"className":82777},[106,144],[18,82779,430],{"className":82780},[106,144],[18,82782,82783,82786],{"style":81813},[18,82784],{"className":82785,"style":697},[136],[18,82787,82789],{"className":82788},[141,142,143,144],[18,82790,82792],{"className":82791},[106,144],[18,82793,82795,82798,82857],{"className":82794},[106,144],[18,82796],{"className":82797},[250,6012,141,1195,38156],[18,82799,82801],{"className":82800},[5654],[18,82802,82804,82849],{"className":82803},[119,120],[18,82805,82807,82846],{"className":82806},[124],[18,82808,82810,82824,82832],{"className":82809,"style":81841},[128],[18,82811,82812,82815],{"style":38172},[18,82813],{"className":82814,"style":697},[136],[18,82816,82818],{"className":82817},[141,1195,1196,144],[18,82819,82821],{"className":82820},[106,144],[18,82822,2266],{"className":82823},[106,144],[18,82825,82826,82829],{"style":38196},[18,82827],{"className":82828,"style":697},[136],[18,82830],{"className":82831,"style":38203},[6047,144],[18,82833,82834,82837],{"style":42972},[18,82835],{"className":82836,"style":697},[136],[18,82838,82840],{"className":82839},[141,1195,1196,144],[18,82841,82843],{"className":82842},[106,144],[18,82844,3941],{"className":82845,"style":3957},[106,110,144],[18,82847,152],{"className":82848},[151],[18,82850,82852],{"className":82851},[124],[18,82853,82855],{"className":82854,"style":81887},[128],[18,82856],{},[18,82858],{"className":82859},[267,6012,141,1195,38156],[18,82861,152],{"className":82862},[151],[18,82864,82866],{"className":82865},[124],[18,82867,82869],{"className":82868,"style":81902},[128],[18,82870],{},[18,82872],{"className":82873,"style":170},[169],[18,82875,82877,82880],{"className":82876},[985],[18,82878,45646],{"className":82879},[985],[18,82881,82883],{"className":82882},[115],[18,82884,82886],{"className":82885},[119],[18,82887,82889],{"className":82888},[124],[18,82890,82892],{"className":82891,"style":82396},[128],[18,82893,82894,82897],{"style":81929},[18,82895],{"className":82896,"style":137},[136],[18,82898,82900],{"className":82899},[141,142,143,144],[18,82901,82903,82906,82909],{"className":82902},[106,144],[18,82904,64],{"className":82905},[106,110,144],[18,82907,2253],{"className":82908},[2072,144],[18,82910,47],{"className":82911},[106,144],[18,82913],{"className":82914,"style":170},[169],[18,82916,38582],{"className":82917},[106,110],[18,82919],{"className":82920,"style":170},[169],[18,82922,4031],{"className":82923},[106,110],[18,82925,74],{"className":82926},[250],[18,82928,82044],{"className":82929},[985],[18,82931],{"className":82932,"style":170},[169],[18,82934,38582],{"className":82935},[106,110],[18,82937,84],{"className":82938},[267],[18,82940],{"className":82941},[169,8022],[18,82943,82945,82948,82951],{"className":82944},[97],[18,82946],{"className":82947,"style":15090},[101],[18,82949,433],{"className":82950},[232],[18,82952],{"className":82953,"style":228},[169],[18,82955,82957,82961,82964,82967,83038,83041,83202,83205,83208],{"className":82956},[97],[18,82958],{"className":82959,"style":82960},[101],"height:2.0867em;vertical-align:-0.6997em;",[18,82962,2253],{"className":82963},[106],[18,82965],{"className":82966,"style":170},[169],[18,82968,82970,82976,83014,83017,83020,83023,83026,83029,83032],{"className":82969},[174],[18,82971,82973],{"className":82972,"style":13796},[250,13795],[18,82974,74],{"className":82975},[1230,1196],[18,82977,82979,82982],{"className":82978},[985],[18,82980,45646],{"className":82981},[985],[18,82983,82985],{"className":82984},[115],[18,82986,82988],{"className":82987},[119],[18,82989,82991],{"className":82990},[124],[18,82992,82994],{"className":82993,"style":82396},[128],[18,82995,82996,82999],{"style":81929},[18,82997],{"className":82998,"style":137},[136],[18,83000,83002],{"className":83001},[141,142,143,144],[18,83003,83005,83008,83011],{"className":83004},[106,144],[18,83006,64],{"className":83007},[106,110,144],[18,83009,2253],{"className":83010},[2072,144],[18,83012,47],{"className":83013},[106,144],[18,83015],{"className":83016,"style":170},[169],[18,83018,38582],{"className":83019},[106,110],[18,83021],{"className":83022,"style":170},[169],[18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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 \\, 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为偶数时，即 ",[18,85821,85823,85843],{"className":85822,"translate":22},[21],[18,85824,85826],{"className":85825},[26],[28,85827,85828],{"xmlns":30},[32,85829,85830,85840],{},[35,85831,85832,85834,85836,85838],{},[41,85833,64],{},[49,85835,433],{},[45,85837,2266],{},[41,85839,39964],{},[86,85841,85842],{"encoding":88},"n = 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上有 ",[18,90128,90130,90202],{"className":90129,"translate":22},[21],[18,90131,90133],{"className":90132},[26],[28,90134,90135],{"xmlns":30},[32,90136,90137,90199],{},[35,90138,90139,90157,90159,90161,90175,90177,90179,90197],{},[1556,90140,90141,90147],{},[35,90142,90143,90145],{},[41,90144,45646],{},[49,90146,899],{},[35,90148,90149,90151,90153,90155],{},[45,90150,2266],{},[41,90152,39964],{},[49,90154,7987],{},[45,90156,47],{},[41,90158,38582],{},[49,90160,6684],{},[1556,90162,90163,90169],{},[35,90164,90165,90167],{},[41,90166,45646],{},[49,90168,899],{},[35,90170,90171,90173],{},[45,90172,2266],{},[41,90174,39964],{},[41,90176,38582],{},[49,90178,6684],{},[1556,90180,90181,90187],{},[35,90182,90183,90185],{},[41,90184,45646],{},[49,90186,899],{},[35,90188,90189,90191,90193,90195],{},[45,90190,2266],{},[41,90192,39964],{},[49,90194,2253],{},[45,90196,47],{},[41,90198,38582],{},[86,90200,90201],{"encoding":88},"\\sin^{2k+1} x \u003C \\sin^{2k} x \u003C \\sin^{2k-1} x",[18,90203,90205,90268,90324],{"className":90204,"ariaHidden":51},[93],[18,90206,90208,90212,90253,90256,90259,90262,90265],{"className":90207},[97],[18,90209],{"className":90210,"style":90211},[101],"height:0.946em;vertical-align:-0.0391em;",[18,90213,90215,90218],{"className":90214},[985],[18,90216,45646],{"className":90217},[985],[18,90219,90221],{"className":90220},[115],[18,90222,90224],{"className":90223},[119],[18,90225,90227],{"className":90226},[124],[18,90228,90230],{"className":90229,"style":86284},[128],[18,90231,90232,90235],{"style":81929},[18,90233],{"className":90234,"style":137},[136],[18,90236,90238],{"className":90237},[141,142,143,144],[18,90239,90241,90244,90247,90250],{"className":90240},[106,144],[18,90242,2266],{"className":90243},[106,144],[18,90245,39964],{"className":90246,"style":40040},[106,110,144],[18,90248,7987],{"className":90249},[2072,144],[18,90251,47],{"className":90252},[106,144],[18,90254],{"className":90255,"style":170},[169],[18,90257,38582],{"className":90258},[106,110],[18,90260],{"className":90261,"style":228},[169],[18,90263,6684],{"className":90264},[232],[18,90266],{"className":90267,"style":228},[169],[18,90269,90271,90274,90309,90312,90315,90318,90321],{"className":90270},[97],[18,90272],{"className":90273,"style":90211},[101],[18,90275,90277,90280],{"className":90276},[985],[18,90278,45646],{"className":90279},[985],[18,90281,90283],{"className":90282},[115],[18,90284,90286],{"className":90285},[119],[18,90287,90289],{"className":90288},[124],[18,90290,90292],{"className":90291,"style":86284},[128],[18,90293,90294,90297],{"style":81929},[18,90295],{"className":90296,"style":137},[136],[18,90298,90300],{"className":90299},[141,142,143,144],[18,90301,90303,90306],{"className":90302},[106,144],[18,90304,2266],{"className":90305},[106,144],[18,90307,39964],{"className":90308,"style":40040},[106,110,144],[18,90310],{"className":90311,"style":170},[169],[18,90313,38582],{"className":90314},[106,110],[18,90316],{"className":90317,"style":228},[169],[18,90319,6684],{"className":90320},[232],[18,90322],{"className":90323,"style":228},[169],[18,90325,90327,90330,90371,90374],{"className":90326},[97],[18,90328],{"className":90329,"style":86284},[101],[18,90331,90333,90336],{"className":90332},[985],[18,90334,45646],{"className":90335},[985],[18,90337,90339],{"className":90338},[115],[18,90340,90342],{"className":90341},[119],[18,90343,90345],{"className":90344},[124],[18,90346,90348],{"className":90347,"style":86284},[128],[18,90349,90350,90353],{"style":81929},[18,90351],{"className":90352,"style":137},[136],[18,90354,90356],{"className":90355},[141,142,143,144],[18,90357,90359,90362,90365,90368],{"className":90358},[106,144],[18,90360,2266],{"className":90361},[106,144],[18,90363,39964],{"className":90364,"style":40040},[106,110,144],[18,90366,2253],{"className":90367},[2072,144],[18,90369,47],{"className":90370},[106,144],[18,90372],{"className":90373,"style":170},[169],[18,90375,38582],{"className":90376},[106,110],"，因此有：",[18,90379,90381],{"className":90380,"translate":22},[875],[18,90382,90384,90510],{"className":90383,"translate":22},[21],[18,90385,90387],{"className":90386},[26],[28,90388,90389],{"xmlns":30,"display":884},[32,90390,90391,90507],{},[35,90392,90393,90405,90423,90425,90427,90429,90431,90433,90445,90459,90461,90463,90465,90467,90469,90481,90499,90501,90503,90505],{},[1589,90394,90395,90397,90399],{},[49,90396,4005],{},[45,90398,430],{},[5654,90400,90401,90403],{},[41,90402,3941],{},[45,90404,2266],{},[1556,90406,90407,90413],{},[35,90408,90409,90411],{},[41,90410,45646],{},[49,90412,899],{},[35,90414,90415,90417,90419,90421],{},[45,90416,2266],{},[41,90418,39964],{},[49,90420,7987],{},[45,90422,47],{},[41,90424,38582],{},[926,90426,4024],{},[41,90428,4031],{},[41,90430,38582],{},[49,90432,6684],{},[1589,90434,90435,90437,90439],{},[49,90436,4005],{},[45,90438,430],{},[5654,90440,90441,90443],{},[41,90442,3941],{},[45,90444,2266],{},[1556,90446,90447,90453],{},[35,90448,90449,90451],{},[41,90450,45646],{},[49,90452,899],{},[35,90454,90455,90457],{},[45,90456,2266],{},[41,90458,39964],{},[41,90460,38582],{},[926,90462,4024],{},[41,90464,4031],{},[41,90466,38582],{},[49,90468,6684],{},[1589,90470,90471,90473,90475],{},[49,90472,4005],{},[45,90474,430],{},[5654,90476,90477,90479],{},[41,90478,3941],{},[45,90480,2266],{},[1556,90482,90483,90489],{},[35,90484,90485,90487],{},[41,90486,45646],{},[49,90488,899],{},[35,90490,90491,90493,90495,90497],{},[45,90492,2266],{},[41,90494,39964],{},[49,90496,2253],{},[45,90498,47],{},[41,90500,38582],{},[926,90502,4024],{},[41,90504,4031],{},[41,90506,38582],{},[86,90508,90509],{"encoding":88},"\\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 \\, 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\u003C \\frac{(2k-1)!!}{(2k)!!} \\cdot \\frac{\\pi}{2} \u003C 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",[18,91633,91635,91681],{"className":91634,"translate":22},[21],[18,91636,91638],{"className":91637},[26],[28,91639,91640],{"xmlns":30},[32,91641,91642,91678],{},[35,91643,91644,91656,91670,91672,91674,91676],{},[1589,91645,91646,91648,91650],{},[49,91647,4005],{},[45,91649,430],{},[5654,91651,91652,91654],{},[41,91653,3941],{},[45,91655,2266],{},[1556,91657,91658,91664],{},[35,91659,91660,91662],{},[41,91661,45646],{},[49,91663,899],{},[35,91665,91666,91668],{},[45,91667,2266],{},[41,91669,39964],{},[41,91671,38582],{},[926,91673,4024],{},[41,91675,4031],{},[41,91677,38582],{},[86,91679,91680],{"encoding":88},"\\int_{0}^{\\frac{\\pi}{2}} \\sin^{2k} x \\, 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上下界之间的差：",[18,91872,91874],{"className":91873,"translate":22},[875],[18,91875,91877,92188],{"className":91876,"translate":22},[21],[18,91878,91880],{"className":91879},[26],[28,91881,91882],{"xmlns":30,"display":884},[32,91883,91884,92185],{},[35,91885,91886,91903,91983,91985,91987,92003,92107,92109,92111,92127,92141,92143,92181,92183],{},[890,91887,91888,91895],{},[35,91889,91890,91893],{},[41,91891,91892],{},"lim",[49,91894,899],{},[35,91896,91897,91899,91901],{},[41,91898,39964],{},[49,91900,1880],{},[41,91902,2028],{"mathvariant":316},[35,91904,91905,91907,91945,91947,91981],{},[49,91906,74],{"fence":51},[5654,91908,91909,91927],{},[35,91910,91911,91913,91915,91917,91919,91921,91923,91925],{},[49,91912,74],{"stretchy":73},[45,91914,2266],{},[41,91916,39964],{},[49,91918,2253],{},[45,91920,2266],{},[49,91922,84],{"stretchy":73},[49,91924,88995],{"stretchy":73},[49,91926,88995],{"stretchy":73},[35,91928,91929,91931,91933,91935,91937,91939,91941,91943],{},[49,91930,74],{"stretchy":73},[45,91932,2266],{},[41,91934,39964],{},[49,91936,2253],{},[45,91938,47],{},[49,91940,84],{"stretchy":73},[49,91942,88995],{"stretchy":73},[49,91944,88995],{"stretchy":73},[49,91946,2253],{},[5654,91948,91949,91963],{},[35,91950,91951,91953,91955,91957,91959,91961],{},[49,91952,74],{"stretchy":73},[45,91954,2266],{},[41,91956,39964],{},[49,91958,84],{"stretchy":73},[49,91960,88995],{"stretchy":73},[49,91962,88995],{"stretchy":73},[35,91964,91965,91967,91969,91971,91973,91975,91977,91979],{},[49,91966,74],{"stretchy":73},[45,91968,2266],{},[41,91970,39964],{},[49,91972,7987],{},[45,91974,47],{},[49,91976,84],{"stretchy":73},[49,91978,88995],{"stretchy":73},[49,91980,88995],{"stretchy":73},[49,91982,84],{"fence":51},[169,91984],{"linebreak":8022},[49,91986,433],{},[890,91988,91989,91995],{},[35,91990,91991,91993],{},[41,91992,91892],{},[49,91994,899],{},[35,91996,91997,91999,92001],{},[41,91998,39964],{},[49,92000,1880],{},[41,92002,2028],{"mathvariant":316},[35,92004,92005,92007,92045,92047,92065,92067,92105],{},[49,92006,74],{"fence":51},[5654,92008,92009,92027],{},[35,92010,92011,92013,92015,92017,92019,92021,92023,92025],{},[49,92012,74],{"stretchy":73},[45,92014,2266],{},[41,92016,39964],{},[49,92018,2253],{},[45,92020,2266],{},[49,92022,84],{"stretchy":73},[49,92024,88995],{"stretchy":73},[49,92026,88995],{"stretchy":73},[35,92028,92029,92031,92033,92035,92037,92039,92041,92043],{},[49,92030,74],{"stretchy":73},[45,92032,2266],{},[41,92034,39964],{},[49,92036,2253],{},[45,92038,47],{},[49,92040,84],{"stretchy":73},[49,92042,88995],{"stretchy":73},[49,92044,88995],{"stretchy":73},[49,92046,2253],{},[5654,92048,92049,92055],{},[35,92050,92051,92053],{},[45,92052,2266],{},[41,92054,39964],{},[35,92056,92057,92059,92061,92063],{},[45,92058,2266],{},[41,92060,39964],{},[49,92062,7987],{},[45,92064,47],{},[49,92066,26840],{},[5654,92068,92069,92087],{},[35,92070,92071,92073,92075,92077,92079,92081,92083,92085],{},[49,92072,74],{"stretchy":73},[45,92074,2266],{},[41,92076,39964],{},[49,92078,2253],{},[45,92080,2266],{},[49,92082,84],{"stretchy":73},[49,92084,88995],{"stretchy":73},[49,92086,88995],{"stretchy":73},[35,92088,92089,92091,92093,92095,92097,92099,92101,92103],{},[49,92090,74],{"stretchy":73},[45,92092,2266],{},[41,92094,39964],{},[49,92096,2253],{},[45,92098,47],{},[49,92100,84],{"stretchy":73},[49,92102,88995],{"stretchy":73},[49,92104,88995],{"stretchy":73},[49,92106,84],{"fence":51},[169,92108],{"linebreak":8022},[49,92110,433],{},[890,92112,92113,92119],{},[35,92114,92115,92117],{},[41,92116,91892],{},[49,92118,899],{},[35,92120,92121,92123,92125],{},[41,92122,39964],{},[49,92124,1880],{},[41,92126,2028],{"mathvariant":316},[5654,92128,92129,92131],{},[45,92130,47],{},[35,92132,92133,92135,92137,92139],{},[45,92134,2266],{},[41,92136,39964],{},[49,92138,7987],{},[45,92140,47],{},[49,92142,26840],{},[5654,92144,92145,92163],{},[35,92146,92147,92149,92151,92153,92155,92157,92159,92161],{},[49,92148,74],{"stretchy":73},[45,92150,2266],{},[41,92152,39964],{},[49,92154,2253],{},[45,92156,2266],{},[49,92158,84],{"stretchy":73},[49,92160,88995],{"stretchy":73},[49,92162,88995],{"stretchy":73},[35,92164,92165,92167,92169,92171,92173,92175,92177,92179],{},[49,92166,74],{"stretchy":73},[45,92168,2266],{},[41,92170,39964],{},[49,92172,2253],{},[45,92174,47],{},[49,92176,84],{"stretchy":73},[49,92178,88995],{"stretchy":73},[49,92180,88995],{"stretchy":73},[49,92182,433],{},[45,92184,430],{},[86,92186,92187],{"encoding":88},"\\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",[18,93209,93211,93315],{"className":93210,"ariaHidden":51},[93],[18,93212,93214,93218,93306,93309,93312],{"className":93213},[97],[18,93215],{"className":93216,"style":93217},[101],"height:1.2834em;vertical-align:-0.4033em;",[18,93219,93221,93224,93303],{"className":93220},[106],[18,93222],{"className":93223},[250,6012],[18,93225,93227],{"className":93226},[5654],[18,93228,93230,93294],{"className":93229},[119,120],[18,93231,93233,93291],{"className":93232},[124],[18,93234,93237,93260,93268],{"className":93235,"style":93236},[128],"height:0.8801em;",[18,93238,93239,93242],{"style":12048},[18,93240],{"className":93241,"style":697},[136],[18,93243,93245],{"className":93244},[141,142,143,144],[18,93246,93248,93251,93254,93257],{"className":93247},[106,144],[18,93249,2266],{"className":93250},[106,144],[18,93252,39964],{"className":93253,"style":40040},[106,110,144],[18,93255,2253],{"className":93256},[2072,144],[18,93258,47],{"className":93259},[106,144],[18,93261,93262,93265],{"style":6040},[18,93263],{"className":93264,"style":697},[136],[18,93266],{"className":93267,"style":6048},[6047],[18,93269,93270,93273],{"style":12071},[18,93271],{"className":93272,"style":697},[136],[18,93274,93276],{"className":93275},[141,142,143,144],[18,93277,93279,93282,93285,93288],{"className":93278},[106,144],[18,93280,2266],{"className":93281},[106,144],[18,93283,39964],{"className":93284,"style":40040},[106,110,144],[18,93286,2253],{"className":93287},[2072,144],[18,93289,2266],{"className":93290},[106,144],[18,93292,152],{"className":93293},[151],[18,93295,93297],{"className":93296},[124],[18,93298,93301],{"className":93299,"style":93300},[128],"height:0.4033em;",[18,93302],{},[18,93304],{"className":93305},[267,6012],[18,93307],{"className":93308,"style":228},[169],[18,93310,6684],{"className":93311},[232],[18,93313],{"className":93314,"style":228},[169],[18,93316,93318,93321],{"className":93317},[97],[18,93319],{"className":93320,"style":505},[101],[18,93322,47],{"className":93323},[106],[11,93325,93326],{},"对于不等式：",[18,93328,93330],{"className":93329,"translate":22},[875],[18,93331,93333,93462],{"className":93332,"translate":22},[21],[18,93334,93336],{"className":93335},[26],[28,93337,93338],{"xmlns":30,"display":884},[32,93339,93340,93460],{},[35,93341,93342,93376,93378,93412,93414,93420,93422],{},[5654,93343,93344,93358],{},[35,93345,93346,93348,93350,93352,93354,93356],{},[49,93347,74],{"stretchy":73},[45,93349,2266],{},[41,93351,39964],{},[49,93353,84],{"stretchy":73},[49,93355,88995],{"stretchy":73},[49,93357,88995],{"stretchy":73},[35,93359,93360,93362,93364,93366,93368,93370,93372,93374],{},[49,93361,74],{"stretchy":73},[45,93363,2266],{},[41,93365,39964],{},[49,93367,7987],{},[45,93369,47],{},[49,93371,84],{"stretchy":73},[49,93373,88995],{"stretchy":73},[49,93375,88995],{"stretchy":73},[49,93377,6684],{},[5654,93379,93380,93398],{},[35,93381,93382,93384,93386,93388,93390,93392,93394,93396],{},[49,93383,74],{"stretchy":73},[45,93385,2266],{},[41,93387,39964],{},[49,93389,2253],{},[45,93391,47],{},[49,93393,84],{"stretchy":73},[49,93395,88995],{"stretchy":73},[49,93397,88995],{"stretchy":73},[35,93399,93400,93402,93404,93406,93408,93410],{},[49,93401,74],{"stretchy":73},[45,93403,2266],{},[41,93405,39964],{},[49,93407,84],{"stretchy":73},[49,93409,88995],{"stretchy":73},[49,93411,88995],{"stretchy":73},[49,93413,26840],{},[5654,93415,93416,93418],{},[41,93417,3941],{},[45,93419,2266],{},[49,93421,6684],{},[5654,93423,93424,93442],{},[35,93425,93426,93428,93430,93432,93434,93436,93438,93440],{},[49,93427,74],{"stretchy":73},[45,93429,2266],{},[41,93431,39964],{},[49,93433,2253],{},[45,93435,2266],{},[49,93437,84],{"stretchy":73},[49,93439,88995],{"stretchy":73},[49,93441,88995],{"stretchy":73},[35,93443,93444,93446,93448,93450,93452,93454,93456,93458],{},[49,93445,74],{"stretchy":73},[45,93447,2266],{},[41,93449,39964],{},[49,93451,2253],{},[45,93453,47],{},[49,93455,84],{"stretchy":73},[49,93457,88995],{"stretchy":73},[49,93459,88995],{"stretchy":73},[86,93461,91224],{"encoding":88},[18,93463,93465,93572,93679,93756],{"className":93464,"ariaHidden":51},[93],[18,93466,93468,93471,93563,93566,93569],{"className":93467},[97],[18,93469],{"className":93470,"style":89418},[101],[18,93472,93474,93477,93560],{"className":93473},[106],[18,93475],{"className":93476},[250,6012],[18,93478,93480],{"className":93479},[5654],[18,93481,93483,93552],{"className":93482},[119,120],[18,93484,93486,93549],{"className":93485},[124],[18,93487,93489,93521,93529],{"className":93488,"style":18133},[128],[18,93490,93491,93494],{"style":6028},[18,93492],{"className":93493,"style":697},[136],[18,93495,93497,93500,93503,93506,93509,93512,93515,93518],{"className":93496},[106],[18,93498,74],{"className":93499},[250],[18,93501,2266],{"className":93502},[106],[18,93504,39964],{"className":93505,"style":40040},[106,110],[18,93507],{"className":93508,"style":2068},[169],[18,93510,7987],{"className":93511},[2072],[18,93513],{"className":93514,"s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\\rightarrow \\infty} \\frac{(2k)!!^2}{(2k+1)!! (2k-1)!!} = \\frac{\\pi}{2}",[18,96197,96199,96415],{"className":96198,"ariaHidden":51},[93],[18,96200,96202,96205,96258,96261,96406,96409,96412],{"className":96201},[97],[18,96203],{"className":96204,"style":94168},[101],[18,96206,96208],{"className":96207},[985,986],[18,96209,96211,96250],{"className":96210},[119,120],[18,96212,96214,96247],{"className":96213},[124],[18,96215,96217,96237],{"className":96216,"style":295},[128],[18,96218,96219,96222],{"style":29370},[18,96220],{"className":96221,"style":697},[136],[18,96223,96225],{"className":96224},[141,142,143,144],[18,96226,96228,96231,96234],{"className":96227},[106,144],[18,96229,39964],{"className":96230,"style":40040},[106,110,144],[18,96232,1880],{"className":96233},[232,144],[18,96235,2028],{"className":96236},[106,144],[18,96238,96239,96242],{"style":693},[18,96240],{"className":96241,"style":697},[136],[18,96243,96244],{},[18,96245,91892],{"className":96246},[985],[18,96248,152],{"className":96249},[151],[18,96251,96253],{"className":96252},[124],[18,96254,96256],{"className":96255,"style":92248},[128],[18,96257],{},[18,96259],{"className":96260,"style":170},[169],[18,96262,96264,96267,96403],{"className":96263},[106],[18,96265],{"className":96266},[250,6012],[18,96268,96270],{"className":96269},[5654],[18,96271,96273,96395],{"className":96272},[119,120],[18,96274,96276,96392],{"className":96275},[124],[18,96277,96279,96335,96343],{"className":96278,"style":15999},[128],[18,96280,96281,96284],{"style":6028},[18,96282],{"className":96283,"style":697},[136],[18,96285,96287,96290,96293,96296,96299,96302,96305,96308,96311,96314,96317,96320,96323,96326,96329,96332],{"className":96286},[106],[18,96288,74],{"className":96289},[250],[18,96291,2266],{"className":96292},[106],[18,96294,39964],{"className":96295,"style":40040},[106,110],[18,96297],{"className":96298,"style":2068},[169],[18,96300,7987],{"className":96301},[2072],[18,96303],{"className":96304,"style":2068},[169],[18,96306,47],{"className":96307},[106],[18,96309,89457],{"className":96310},[267],[18,96312,74],{"className":96313},[250],[18,96315,2266],{"className":96316},[106],[18,96318,39964],{"className":96319,"style":40040},[106,110],[18,96321],{"className":96322,"style":2068},[169],[18,96324,2253],{"className":96325},[2072],[18,96327],{"className":96328,"style":2068},[169],[18,96330,47],{"className":96331},[106],[18,96333,89457],{"className":96334},[267],[18,96336,96337,96340],{"style":6040},[18,96338],{"className":96339,"style":697},[136],[18,96341],{"className":96342,"style":6048},[6047],[18,96344,96345,96348],{"style":6051},[18,96346],{"className":96347,"style":697},[136],[18,96349,96351,96354,96357,96360,96363],{"className":96350},[106],[18,96352,74],{"className":96353},[250],[18,96355,2266],{"className":96356},[106],[18,96358,39964],{"className":96359,"style":40040},[106,110],[18,96361,94271],{"className":96362},[267],[18,96364,96366,96369],{"className":96365},[267],[18,96367,88995],{"className":96368},[267],[18,96370,96372],{"className":96371},[115],[18,96373,96375],{"className":96374},[119],[18,96376,96378],{"className":96377},[124],[18,96379,96381],{"className":96380,"style":2416},[128],[18,96382,96383,96386],{"style":1962},[18,96384],{"className":96385,"style":137},[136],[18,96387,96389],{"className":96388},[141,142,143,144],[18,96390,2266],{"className":96391},[106,144],[18,96393,152],{"className":96394},[151],[18,96396,96398],{"className":96397},[124],[18,96399,96401],{"className":96400,"style":89507},[128],[18,96402],{},[18,96404],{"className":96405},[267,6012],[18,96407],{"className":96408,"style":228},[169],[18,96410,433],{"className":96411},[232],[18,96413],{"className":96414,"style":228},[169],[18,96416,96418,96421],{"className":96417},[97],[18,96419],{"className":96420,"style":42071},[101],[18,96422,96424,96427,96480],{"className":96423},[106],[18,96425],{"className":96426},[250,6012],[18,96428,96430],{"className":96429},[5654],[18,96431,96433,96472],{"className":96432},[119,120],[18,96434,96436,96469],{"className":96435},[124],[18,96437,96439,96450,96458],{"className":96438,"style":13852},[128],[18,96440,96441,96444],{"style":6028},[18,96442],{"className":96443,"style":697},[136],[18,96445,96447],{"className":96446},[106],[18,96448,2266],{"className":96449},[106],[18,96451,96452,96455],{"style":6040},[18,96453],{"className":96454,"style":697},[136],[18,96456],{"className":96457,"style":6048},[6047],[18,96459,96460,96463],{"style":6051},[18,96461],{"className":96462,"style":697},[136],[18,96464,96466],{"className":96465},[106],[18,96467,3941],{"className":96468,"style":3957},[106,110],[18,96470,152],{"className":96471},[151],[18,96473,96475],{"className":96474},[124],[18,96476,96478],{"className":96477,"style":6070},[128],[18,96479],{},[18,96481],{"className":96482},[267,6012],[11,96484,96485],{},"该公式引入了沃利斯的结果，为推导斯特林公式做准备。",[638,96487,81710],{"id":81710},[11,96489,96490,96491,16800,96525,509,96584,96643],{},"斯特林公式揭示了 ",[18,96492,96494,96510],{"className":96493,"translate":22},[21],[18,96495,96497],{"className":96496},[26],[28,96498,96499],{"xmlns":30},[32,96500,96501,96507],{},[35,96502,96503,96505],{},[41,96504,64],{},[49,96506,88995],{"stretchy":73},[86,96508,96509],{"encoding":88},"n!",[18,96511,96513],{"className":96512,"ariaHidden":51},[93],[18,96514,96516,96519,96522],{"className":96515},[97],[18,96517],{"className":96518,"style":295},[101],[18,96520,64],{"className":96521},[106,110],[18,96523,88995],{"className":96524},[267],[18,96526,96528,96546],{"className":96527,"translate":22},[21],[18,96529,96531],{"className":96530},[26],[28,96532,96533],{"xmlns":30},[32,96534,96535,96543],{},[35,96536,96537],{},[1556,96538,96539,96541],{},[41,96540,64],{},[41,96542,64],{},[86,96544,96545],{"encoding":88},"n^n",[18,96547,96549],{"className":96548,"ariaHidden":51},[93],[18,96550,96552,96555],{"className":96551},[97],[18,96553],{"className":96554,"style":1959},[101],[18,96556,96558,96561],{"className":96557},[106],[18,96559,64],{"className":96560},[106,110],[18,96562,96564],{"className":96563},[115],[18,96565,96567],{"className":96566},[119],[18,96568,96570],{"className":96569},[124],[18,96571,96573],{"className":96572,"style":1959},[128],[18,96574,96575,96578],{"style":1962},[18,96576],{"className":96577,"style":137},[136],[18,96579,96581],{"className":96580},[141,142,143,144],[18,96582,64],{"className":96583},[106,110,144],[18,96585,96587,96605],{"className":96586,"translate":22},[21],[18,96588,96590],{"className":96589},[26],[28,96591,96592],{"xmlns":30},[32,96593,96594,96602],{},[35,96595,96596],{},[1556,96597,96598,96600],{},[41,96599,14444],{},[41,96601,64],{},[86,96603,96604],{"encoding":88},"e^n",[18,96606,96608],{"className":96607,"ariaHidden":51},[93],[18,96609,96611,96614],{"className":96610},[97],[18,96612],{"className":96613,"style":1959},[101],[18,96615,96617,96620],{"className":96616},[106],[18,96618,14444],{"className":96619},[106,110],[18,96621,96623],{"className":96622},[115],[18,96624,96626],{"className":96625},[119],[18,96627,96629],{"className":96628},[124],[18,96630,96632],{"className":96631,"style":1959},[128],[18,96633,96634,96637],{"style":1962},[18,96635],{"className":96636,"style":137},[136],[18,96638,96640],{"className":96639},[141,142,143,144],[18,96641,64],{"className":96642},[106,110,144]," 之间的关系。",[5159,96645,96646],{},[11,96647,39318,96648,96676,96677,96710,96711,1990],{},[18,96649,96651,96664],{"className":96650,"translate":22},[21],[18,96652,96654],{"className":96653},[26],[28,96655,96656],{"xmlns":30},[32,96657,96658,96662],{},[35,96659,96660],{},[41,96661,64],{},[86,96663,64],{"encoding":88},[18,96665,96667],{"className":96666,"ariaHidden":51},[93],[18,96668,96670,96673],{"className":96669},[97],[18,96671],{"className":96672,"style":1083},[101],[18,96674,64],{"className":96675},[106,110]," 趋近无穷大时，",[18,96678,96680,96695],{"className":96679,"translate":22},[21],[18,96681,96683],{"className":96682},[26],[28,96684,96685],{"xmlns":30},[32,96686,96687,96693],{},[35,96688,96689,96691],{},[41,96690,64],{},[49,96692,88995],{"stretchy":73},[86,96694,96509],{"encoding":88},[18,96696,96698],{"className":96697,"ariaHidden":51},[93],[18,96699,96701,96704,96707],{"className":96700},[97],[18,96702],{"className":96703,"style":295},[101],[18,96705,64],{"className":96706},[106,110],[18,96708,88995],{"className":96709},[267]," 约等于 ",[18,96712,96714,96752],{"className":96713,"translate":22},[21],[18,96715,96717],{"className":96716},[26],[28,96718,96719],{"xmlns":30},[32,96720,96721,96749],{},[35,96722,96723,96733],{},[12561,96724,96725],{},[35,96726,96727,96729,96731],{},[45,96728,2266],{},[41,96730,3941],{},[41,96732,64],{},[1556,96734,96735,96747],{},[35,96736,96737,96739,96745],{},[49,96738,74],{"fence":51},[5654,96740,96741,96743],{},[41,96742,64],{},[41,96744,14444],{},[49,96746,84],{"fence":51},[41,96748,64],{},[86,96750,96751],{"encoding":88},"\\sqrt{2 \\pi n} \\left(\\frac{n}{e}\\right)^n",[18,96753,96755],{"className":96754,"ariaHidden":51},[93],[18,96756,96758,96762,96818,96821],{"className":96757},[97],[18,96759],{"className":96760,"style":96761},[101],"height:1.2572em;vertical-align:-0.35em;",[18,96763,96765],{"className":96764},[106,12696],[18,96766,96768,96809],{"className":96767},[119,120],[18,96769,96771,96806],{"className":96770},[124],[18,96772,96775,96793],{"className":96773,"style":96774},[128],"height:0.9072em;",[18,96776,96778,96781],{"className":96777,"style":693},[12710],[18,96779],{"className":96780,"style":697},[136],[18,96782,96784,96787,96790],{"className":96783,"style":15410},[106],[18,96785,2266],{"className":96786},[106],[18,96788,3941],{"className":96789,"style":3957},[106,110],[18,96791,64],{"className":96792},[106,110],[18,96794,96796,96799],{"style":96795},"top:-2.8672em;",[18,96797],{"className":96798,"style":697},[136],[18,96800,96802],{"className":96801,"style":15420},[12853],[6130,96803,96804],{"xmlns":6132,"width":12857,"height":15423,"viewBox":15424,"preserveAspectRatio":12860},[6136,96805],{"d":15427},[18,96807,152],{"className":96808},[151],[18,96810,96812],{"className":96811},[124],[18,96813,96816],{"className":96814,"style":96815},[128],"height:0.1328em;",[18,96817],{},[18,96819],{"className":96820,"style":170},[169],[18,96822,96824,96907],{"className":96823},[174],[18,96825,96827,96833,96901],{"className":96826},[174],[18,96828,96830],{"className":96829,"style":13796},[250,13795],[18,96831,74],{"className":96832},[1230,1196],[18,96834,96836,96839,96898],{"className":96835},[106],[18,96837],{"className":96838},[250,6012],[18,96840,96842],{"className":96841},[5654],[18,96843,96845,96890],{"className":96844},[119,120],[18,96846,96848,96887],{"className":96847},[124],[18,96849,96851,96865,96873],{"className":96850,"style":14257},[128],[18,96852,96853,96856],{"style":12048},[18,96854],{"className":96855,"style":697},[136],[18,96857,96859],{"className":96858},[141,142,143,144],[18,96860,96862],{"className":96861},[106,144],[18,96863,14444],{"className":96864},[106,110,144],[18,96866,96867,96870],{"style":6040},[18,96868],{"className":96869,"style":697},[136],[18,96871],{"className":96872,"style":6048},[6047],[18,96874,96875,96878],{"style":12071},[18,96876],{"className":96877,"style":697},[136],[18,96879,96881],{"className":96880},[141,142,143,144],[18,96882,96884],{"className":96883},[106,144],[18,96885,64],{"className":96886},[106,110,144],[18,96888,152],{"className":96889},[151],[18,96891,96893],{"className":96892},[124],[18,96894,96896],{"className":96895,"style":12093},[128],[18,96897],{},[18,96899],{"className":96900},[267,6012],[18,96902,96904],{"className":96903,"style":13796},[267,13795],[18,96905,84],{"className":96906},[1230,1196],[18,96908,96910],{"className":96909},[115],[18,96911,96913],{"className":96912},[119],[18,96914,96916],{"className":96915},[124],[18,96917,96920],{"className":96918,"style":96919},[128],"height:0.9043em;",[18,96921,96922,96925],{"style":33257},[18,96923],{"className":96924,"style":137},[136],[18,96926,96928],{"className":96927},[141,142,143,144],[18,96929,64],{"className":96930},[106,110,144],[11,96932,96933],{},"首先引入一个极限：",[18,96935,96937],{"className":96936,"translate":22},[875],[18,96938,96940,97004],{"className":96939,"translate":22},[21],[18,96941,96943],{"className":96942},[26],[28,96944,96945],{"xmlns":30,"display":884},[32,96946,96947,97001],{},[35,96948,96949,96965,96993,96995],{},[890,96950,96951,96957],{},[35,96952,96953,96955],{},[41,96954,91892],{},[49,96956,899],{},[35,96958,96959,96961,96963],{},[41,96960,64],{},[49,96962,1880],{},[41,96964,2028],{"mathvariant":316},[1556,96966,96967,96987],{},[35,96968,96969,96971,96985],{},[49,96970,74],{"fence":51},[5654,96972,96973,96979],{},[35,96974,96975,96977],{},[41,96976,64],{},[49,96978,88995],{"stretchy":73},[1556,96980,96981,96983],{},[41,96982,64],{},[41,96984,64],{},[49,96986,84],{"fence":51},[5654,96988,96989,96991],{},[45,96990,47],{},[41,96992,64],{},[49,96994,433],{},[5654,96996,96997,96999],{},[45,96998,47],{},[41,97000,14444],{},[86,97002,97003],{"encoding":88},"\\lim_{n \\rightarrow \\infty} \\left( \\frac{n!}{n^n} \\right)^{\\frac{1}{n}} = \\frac{1}{e}",[18,97005,97007,97284],{"className":97006,"ariaHidden":51},[93],[18,97008,97010,97014,97069,97072,97275,97278,97281],{"className":97009},[97],[18,97011],{"className":97012,"style":97013},[101],"height:2.744em;vertical-align:-0.95em;",[18,97015,97017],{"className":97016},[985,986],[18,97018,97020,97060],{"className":97019},[119,120],[18,97021,97023,97057],{"className":97022},[124],[18,97024,97026,97047],{"className":97025,"style":295},[128],[18,97027,97029,97032],{"style":97028},"top:-2.4em;margin-left:0em;",[18,97030],{"className":97031,"style":697},[136],[18,97033,97035],{"className":97034},[141,142,143,144],[18,97036,97038,97041,97044],{"className":97037},[106,144],[18,97039,64],{"className":97040},[106,110,144],[18,97042,1880],{"className":97043},[232,144],[18,97045,2028],{"className":97046},[106,144],[18,97048,97049,97052],{"style":693},[18,97050],{"className":97051,"style":697},[136],[18,97053,97054],{},[18,97055,91892],{"className":97056},[985],[18,97058,152],{"className":97059},[151],[18,97061,97063],{"className":97062},[124],[18,97064,97067],{"className":97065,"style":97066},[128],"height:0.7em;",[18,97068],{},[18,97070],{"className":97071,"style":170},[169],[18,97073,97075,97182],{"className":97074},[174],[18,97076,97078,97084,97176],{"className":97077},[174],[18,97079,97081],{"className":97080,"style":13796},[250,13795],[18,97082,74],{"className":97083},[1230,143],[18,97085,97087,97090,97173],{"className":97086},[106],[18,97088],{"className":97089},[250,6012],[18,97091,97093],{"className":97092},[5654],[18,97094,97096,97165],{"className":97095},[119,120],[18,97097,97099,97162],{"className":97098},[124],[18,97100,97102,97140,97148],{"className":97101,"style":6025},[128],[18,97103,97104,97107],{"style":6028},[18,97105],{"className":97106,"style":697},[136],[18,97108,97110],{"className":97109},[106],[18,97111,97113,97116],{"className":97112},[106],[18,97114,64],{"className":97115},[106,110],[18,97117,97119],{"className":97118},[115],[18,97120,97122],{"className":97121},[119],[18,97123,97125],{"className":97124},[124],[18,97126,97129],{"className":97127,"style":97128},[128],"height:0.5904em;",[18,97130,97131,97134],{"style":26112},[18,97132],{"className":97133,"style":137},[136],[18,97135,97137],{"className":97136},[141,142,143,144],[18,97138,64],{"className":97139},[106,110,144],[18,97141,97142,97145],{"style":6040},[18,97143],{"className":97144,"style":697},[136],[18,97146],{"className":97147,"style":6048},[6047],[18,97149,97150,97153],{"style":6051},[18,97151],{"className":97152,"style":697},[136],[18,97154,97156,97159],{"className":97155},[106],[18,97157,64],{"className":97158},[106,110],[18,97160,88995],{"className":97161},[267],[18,97163,152],{"className":97164},[151],[18,97166,97168],{"className":97167},[124],[18,97169,97171],{"className":97170,"style":6070},[128],[18,97172],{},[18,97174],{"className":97175},[267,6012],[18,97177,97179],{"className":97178,"style":13796},[267,13795],[18,97180,84],{"className":97181},[1230,143],[18,97183,97185],{"className":97184},[115],[18,97186,97188],{"className":97187},[119],[18,97189,97191],{"className":97190},[124],[18,97192,97195],{"className":97193,"style":97194},[128],"height:1.7939em;",[18,97196,97198,97201],{"style":97197},"top:-4.2029em;margin-right:0.05em;",[18,97199],{"className":97200,"style":697},[136],[18,97202,97204],{"className":97203},[141,142,143,144],[18,97205,97207],{"className":97206},[106,144],[18,97208,97210,97213,97272],{"className":97209},[106,144],[18,97211],{"className":97212},[250,6012,141,1195,38156],[18,97214,97216],{"className":97215},[5654],[18,97217,97219,97264],{"className":97218},[119,120],[18,97220,97222,97261],{"className":97221},[124],[18,97223,97225,97239,97247],{"className":97224,"style":42938},[128],[18,97226,97227,97230],{"style":38172},[18,97228],{"className":97229,"style":697},[136],[18,97231,97233],{"className":97232},[141,1195,1196,144],[18,97234,97236],{"className":97235},[106,144],[18,97237,64],{"className":97238},[106,110,144],[18,97240,97241,97244],{"style":38196},[18,97242],{"className":97243,"style":697},[136],[18,97245],{"className":97246,"style":38203},[6047,144],[18,97248,97249,97252],{"style":42972},[18,97250],{"className":97251,"style":697},[136],[18,97253,97255],{"className":97254},[141,1195,1196,144],[18,97256,97258],{"className":97257},[106,144],[18,97259,47],{"className":97260},[106,144],[18,97262,152],{"className":97263},[151],[18,97265,97267],{"className":97266},[124],[18,97268,97270],{"className":97269,"style":81887},[128],[18,97271],{},[18,97273],{"className":97274},[267,6012,141,1195,38156],[18,97276],{"className":97277,"style":228},[169],[18,97279,433],{"className":97280},[232],[18,97282],{"className":97283,"style":228},[169],[18,97285,97287,97290],{"className":97286},[97],[18,97288],{"className":97289,"style":9562},[101],[18,97291,97293,97296,97349],{"className":97292},[106],[18,97294],{"className":97295},[250,6012],[18,97297,97299],{"className":97298},[5654],[18,97300,97302,97341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\\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)}",[18,97473,97475,97746],{"className":97474,"ariaHidden":51},[93],[18,97476,97478,97481,97534,97537,97737,97740,97743],{"className":97477},[97],[18,97479],{"className":97480,"style":97013},[101],[18,97482,97484],{"className":97483},[985,986],[18,97485,97487,97526],{"className":97486},[119,120],[18,97488,97490,97523],{"className":97489},[124],[18,97491,97493,97513],{"className":97492,"style":295},[128],[18,97494,97495,97498],{"style":97028},[18,97496],{"className":97497,"style":697},[136],[18,97499,97501],{"className":97500},[141,142,143,144],[18,97502,97504,97507,97510],{"className":97503},[106,144],[18,97505,64],{"className":97506},[106,110,144],[18,97508,1880],{"className":97509},[232,144],[18,97511,2028],{"className":97512},[106,144],[18,97514,97515,97518],{"style":693},[18,97516],{"className":97517,"style":697},[136],[18,97519,97520],{},[18,97521,91892],{"className":97522},[985],[18,97524,152],{"className":97525},[151],[18,97527,97529],{"className":97528},[124],[18,97530,97532],{"className":97531,"style":97066},[128],[18,97533],{},[18,97535],{"className":97536,"style":170},[169],[18,97538,97540,97646],{"className":97539},[174],[18,97541,97543,97549,97640],{"className":97542},[174],[18,97544,97546],{"className":97545,"style":13796},[250,13795],[18,97547,74],{"className":97548},[1230,143],[18,97550,97552,97555,97637],{"className":97551},[106],[18,97553],{"className":97554},[250,6012],[18,97556,97558],{"className":97557},[5654],[18,97559,97561,97629],{"className":97560},[119,120],[18,97562,97564,97626],{"className":97563},[124],[18,97565,97567,97604,97612],{"className":97566,"style":6025},[128],[18,97568,97569,97572],{"style":6028},[18,97570],{"className":97571,"style":697},[136],[18,97573,97575],{"className":97574},[106],[18,97576,97578,97581],{"className":97577},[106],[18,97579,64],{"className":97580},[106,110],[18,97582,97584],{"className":97583},[115],[18,97585,97587],{"className":97586},[119],[18,97588,97590],{"className":97589},[124],[18,97591,97593],{"className":97592,"style":97128},[128],[18,97594,97595,97598],{"style":26112},[18,97596],{"className":97597,"style":137},[136],[18,97599,97601],{"className":97600},[141,142,143,144],[18,97602,64],{"className":97603},[106,110,144],[18,97605,97606,97609],{"style":6040},[18,97607],{"className":97608,"style":697},[136],[18,97610],{"className":97611,"style":6048},[6047],[18,97613,97614,97617],{"style":6051},[18,97615],{"className":97616,"style":697},[136],[18,97618,97620,97623],{"className":97619},[106],[18,97621,64],{"className":97622},[106,110],[18,97624,88995],{"className":97625},[267],[18,97627,152],{"className":97628},[151],[18,97630,97632],{"className":97631},[124],[18,97633,97635],{"className":97634,"style":6070},[128],[18,97636],{},[18,97638],{"className":97639},[267,6012],[18,97641,97643],{"className":97642,"style":13796},[267,13795],[18,97644,84],{"className":97645},[1230,143],[18,97647,97649],{"className":97648},[115],[18,97650,97652],{"className":97651},[119],[18,97653,97655],{"className":97654},[124],[18,97656,97658],{"className":97657,"style":97194},[128],[18,97659,97660,97663],{"style":97197},[18,97661],{"className":97662,"style":697},[136],[18,97664,97666],{"className":97665},[141,142,143,144],[18,97667,97669],{"className":97668},[106,144],[18,97670,97672,97675,97734],{"className":97671},[106,144],[18,97673],{"className":97674},[250,6012,141,1195,38156],[18,97676,97678],{"className":97677},[5654],[18,97679,97681,97726],{"className":97680},[119,120],[18,97682,97684,97723],{"className":97683},[124],[18,97685,97687,97701,97709],{"className":97686,"style":42938},[128],[18,97688,97689,97692],{"style":38172},[18,97690],{"className":97691,"style":697},[136],[18,97693,97695],{"className":97694},[141,1195,1196,144],[18,97696,97698],{"className":97697},[106,144],[18,97699,64],{"className":97700},[106,110,144],[18,97702,97703,97706],{"style":38196},[18,97704],{"className":97705,"style":697},[136],[18,97707],{"className":97708,"style":38203},[6047,144],[18,97710,97711,97714],{"style":42972},[18,97712],{"className":97713,"style":697},[136],[18,97715,97717],{"className":97716},[141,1195,1196,144],[18,97718,97720],{"className":97719},[106,144],[18,97721,47],{"className":97722},[106,144],[18,97724,152],{"className":97725},[151],[18,97727,97729],{"className":97728},[124],[18,97730,97732],{"className":97731,"style":81887},[128],[18,97733],{},[18,97735],{"className":97736},[267,6012,141,1195,38156],[18,97738],{"className":97739,"style":228},[169],[18,97741,433],{"className":97742},[232],[18,97744],{"className":97745,"style":228},[169],[18,97747,97749,97753,97806,97809],{"className":97748},[97],[18,97750],{"className":97751,"style":97752},[101],"height:1.8105em;vertical-align:-0.7em;",[18,97754,97756],{"className":97755},[985,986],[18,97757,97759,97798],{"className":97758},[119,120],[18,97760,97762,97795],{"className":97761},[124],[18,97763,97765,97785],{"className":97764,"style":295},[128],[18,97766,97767,97770],{"style":97028},[18,97768],{"className":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41,98066,2028],{"mathvariant":316},[5654,98068,98069,98071],{},[45,98070,47],{},[41,98072,64],{},[41,98074,97446],{},[49,98076,899],{},[35,98078,98079,98081,98095],{},[49,98080,74],{"fence":51},[5654,98082,98083,98089],{},[35,98084,98085,98087],{},[41,98086,64],{},[49,98088,88995],{"stretchy":73},[1556,98090,98091,98093],{},[41,98092,64],{},[41,98094,64],{},[49,98096,84],{"fence":51},[49,98098,433],{},[49,98100,2253],{},[45,98102,47],{},[86,98104,98105],{"encoding":88},"\\lim_{n \\rightarrow \\infty} \\frac{1}{n} \\ln \\left( \\frac{n!}{n^n} \\right) = 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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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\\rightarrow \\infty} \\frac{n!}{n^n} e^n = 1",[18,99780,99782,99974],{"className":99781,"ariaHidden":51},[93],[18,99783,99785,99789,99842,99845,99936,99965,99968,99971],{"className":99784},[97],[18,99786],{"className":99787,"style":99788},[101],"height:2.0714em;vertical-align:-0.7em;",[18,99790,99792],{"className":99791},[985,986],[18,99793,99795,99834],{"className":99794},[119,120],[18,99796,99798,99831],{"className":99797},[124],[18,99799,99801,99821],{"className":99800,"style":295},[128],[18,99802,99803,99806],{"style":97028},[18,99804],{"className":99805,"style":697},[136],[18,99807,99809],{"className":99808},[141,142,143,144],[18,99810,99812,99815,99818],{"className":99811},[106,144],[18,99813,64],{"className":99814},[106,110,144],[18,99816,1880],{"className":99817},[232,144],[18,99819,2028],{"className":99820},[106,144],[18,99822,99823,99826],{"style":693},[18,99824],{"className":99825,"style":697},[136],[18,99827,99828],{},[18,99829,91892],{"className":99830},[985],[18,99832,152],{"className":99833},[151],[18,99835,99837],{"className":99836},[124],[18,99838,99840],{"className":99839,"style":97066},[128],[18,99841],{},[18,99843],{"className":99844,"style":170},[169],[18,99846,99848,99851,99933],{"className":99847},[106],[18,99849],{"className":99850},[250,6012],[18,99852,99854],{"className":99853},[5654],[18,99855,99857,99925],{"className":99856},[119,120],[18,99858,99860,99922],{"className":99859},[124],[18,99861,99863,99900,99908],{"className":99862,"style":6025},[128],[18,99864,99865,99868],{"style":6028},[18,99866],{"className":99867,"style":697},[136],[18,99869,99871],{"className":99870},[106],[18,99872,99874,99877],{"className":99873},[106],[18,99875,64],{"className":99876},[106,110],[18,99878,99880],{"className":99879},[115],[18,99881,99883],{"className":99882},[119],[18,99884,99886],{"className":99885},[124],[18,99887,99889],{"className":99888,"style":97128},[128],[18,99890,99891,99894],{"style":26112},[18,99892],{"className":99893,"style":137},[136],[18,99895,99897],{"className":99896},[141,142,143,144],[18,99898,64],{"className":99899},[106,110,144],[18,99901,99902,99905],{"style":6040},[18,99903],{"className":99904,"style":697},[136],[18,99906],{"className":99907,"style":6048},[6047],[18,99909,99910,99913],{"style":6051},[18,99911],{"className":99912,"style":697},[136],[18,99914,99916,99919],{"className":99915},[106],[18,99917,64],{"className":99918},[106,110],[18,99920,88995],{"className":99921},[267],[18,99923,152],{"className":99924},[151],[18,99926,99928],{"className":99927},[124],[18,99929,99931],{"className":99930,"style":6070},[128],[18,99932],{},[18,99934],{"className":99935},[267,6012],[18,99937,99939,99942],{"className":99938},[106],[18,99940,14444],{"className":99941},[106,110],[18,99943,99945],{"className":99944},[115],[18,99946,99948],{"className":99947},[119],[18,99949,99951],{"className":99950},[124],[18,99952,99954],{"className":99953,"style":1616},[128],[18,99955,99956,99959],{"style":1637},[18,99957],{"className":99958,"style":137},[136],[18,99960,99962],{"className":99961},[141,142,143,144],[18,99963,64],{"className":99964},[106,110,144],[18,99966],{"className":99967,"style":228},[169],[18,99969,433],{"className":99970},[232],[18,99972],{"className":99973,"style":228},[169],[18,99975,99977,99980],{"className":99976},[97],[18,99978],{"className":99979,"style":505},[101],[18,99981,47],{"className":99982},[106],[11,99984,99985],{},"我们试一下。",[11,99987,99988],{},"使用比值判别法：",[18,99990,99992],{"className":99991,"translate":22},[875],[18,99993,99995,100103],{"className":99994,"translate":22},[21],[18,99996,99998],{"className":99997},[26],[28,99999,100000],{"xmlns":30,"display":884},[32,100001,100002,100100],{},[35,100003,100004,100010,100012,100026,100032,100034,100036,100048,100050,100088],{},[38,100005,100006,100008],{},[41,100007,16823],{},[41,100009,64],{},[49,100011,433],{},[5654,100013,100014,100020],{},[35,100015,100016,100018],{},[41,100017,64],{},[49,100019,88995],{"stretchy":73},[1556,100021,100022,100024],{},[41,100023,64],{},[41,100025,64],{},[1556,100027,100028,100030],{},[41,100029,14444],{},[41,100031,64],{},[49,100033,52],{"separator":51},[169,100035],{"width":42709},[38,100037,100038,100040],{},[41,100039,16823],{},[35,100041,100042,100044,100046],{},[41,100043,64],{},[49,100045,7987],{},[45,100047,47],{},[49,100049,433],{},[5654,100051,100052,100066],{},[35,100053,100054,100056,100058,100060,100062,100064],{},[49,100055,74],{"stretchy":73},[41,100057,64],{},[49,100059,7987],{},[45,100061,47],{},[49,100063,84],{"stretchy":73},[49,100065,88995],{"stretchy":73},[35,100067,100068,100070,100072,100074,100076],{},[49,100069,74],{"stretchy":73},[41,100071,64],{},[49,100073,7987],{},[45,100075,47],{},[1556,100077,100078,100080],{},[49,100079,84],{"stretchy":73},[35,100081,100082,100084,100086],{},[41,100083,64],{},[49,100085,7987],{},[45,100087,47],{},[1556,100089,100090,100092],{},[41,100091,14444],{},[35,100093,100094,100096,100098],{},[41,100095,64],{},[49,100097,7987],{},[45,100099,47],{},[86,100101,100102],{"encoding":88},"a_n = \\frac{n!}{n^n} e^n, \\quad a_{n+1} = \\frac{(n+1)!}{(n+1)^{n+1}} e^{n+1}",[18,100104,100106,100161,100355],{"className":100105,"ariaHidden":51},[93],[18,100107,100109,100112,100152,100155,100158],{"className":100108},[97],[18,100110],{"className":100111,"style":16897},[101],[18,100113,100115,100118],{"className":100114},[106],[18,100116,16823],{"className":100117},[106,110],[18,100119,100121],{"className":100120},[115],[18,100122,100124,100144],{"className":100123},[119,120],[18,100125,100127,100141],{"className":100126},[124],[18,100128,100130],{"className":100129,"style":202},[128],[18,100131,100132,100135],{"style":1155},[18,100133],{"className":100134,"style":137},[136],[18,100136,100138],{"className":100137},[141,142,143,144],[18,100139,64],{"className":100140},[106,110,144],[18,100142,152],{"className":100143},[151],[18,100145,100147],{"className":100146},[124],[18,100148,100150],{"className":100149,"style":159},[128],[18,100151],{},[18,100153],{"className":100154,"style":228},[169],[18,100156,433],{"className":100157},[232],[18,100159],{"className":100160,"style":228},[169],[18,100162,100164,100167,100258,100287,100290,100293,100296,100346,100349,100352],{"className":100163},[97],[18,100165],{"className":100166,"style":6005},[101],[18,100168,100170,100173,100255],{"className":100169},[106],[18,100171],{"className":100172},[250,6012],[18,100174,100176],{"className":100175},[5654],[18,100177,100179,100247],{"className":100178},[119,120],[18,100180,100182,100244],{"className":100181},[124],[18,100183,100185,100222,100230],{"className":100184,"style":6025},[128],[18,100186,100187,100190],{"style":6028},[18,100188],{"className":100189,"style":697},[136],[18,100191,100193],{"className":100192},[106],[18,100194,100196,100199],{"className":100195},[106],[18,100197,64],{"className":100198},[106,110],[18,100200,100202],{"className":100201},[115],[18,100203,100205],{"className":100204},[119],[18,100206,100208],{"className":100207},[124],[18,100209,100211],{"className":100210,"style":97128},[128],[18,100212,100213,100216],{"style":26112},[18,100214],{"className":100215,"style":137},[136],[18,100217,100219],{"className":100218},[141,142,143,144],[18,100220,64],{"className":100221},[106,110,144],[18,100223,100224,100227],{"style":6040},[18,100225],{"className":100226,"style":697},[136],[18,100228],{"className":100229,"style":6048},[6047],[18,100231,100232,100235],{"style":6051},[18,100233],{"className":100234,"style":697},[136],[18,100236,100238,100241],{"className":100237},[106],[18,100239,64],{"className":100240},[106,110],[18,100242,88995],{"className":100243},[267],[18,100245,152],{"className":100246},[151],[18,100248,100250],{"className":100249},[124],[18,100251,100253],{"className":100252,"style":6070},[128],[18,100254],{},[18,100256],{"className":100257},[267,6012],[18,100259,100261,100264],{"className":100260},[106],[18,100262,14444],{"className":100263},[106,110],[18,100265,100267],{"className":100266},[115],[18,100268,100270],{"className":100269},[119],[18,100271,100273],{"className":100272},[124],[18,100274,100276],{"className":100275,"style":1616},[128],[18,100277,100278,100281],{"style":1637},[18,100279],{"className":100280,"style":137},[136],[18,100282,100284],{"className":100283},[141,142,143,144],[18,100285,64],{"className":100286},[106,110,144],[18,100288,52],{"className":100289},[165],[18,100291],{"className":100292,"style":42849},[169],[18,100294],{"className":100295,"style":170},[169],[18,100297,100299,100302],{"className":100298},[106],[18,100300,16823],{"className":100301},[106,110],[18,100303,100305],{"className":100304},[115],[18,100306,100308,100337],{"className":100307},[119,120],[18,100309,100311,100334],{"className":100310},[124],[18,100312,100314],{"className":100313,"style":129},[128],[18,100315,100316,100319],{"style":1155},[18,100317],{"className":100318,"style":137},[136],[18,100320,100322],{"className":100321},[141,142,143,144],[18,100323,100325,100328,100331],{"className":100324},[106,144],[18,100326,64],{"className":100327},[106,110,144],[18,100329,7987],{"className":100330},[2072,144],[18,100332,47],{"className":100333},[106,144],[18,100335,152],{"className":100336},[151],[18,100338,100340],{"className":100339},[124],[18,100341,100344],{"className":100342,"style":100343},[128],"height:0.2083em;",[18,100345],{},[18,100347],{"className":100348,"style":228},[169],[18,100350,433],{"className":100351},[232],[18,100353],{"className":100354,"style":228},[169],[18,100356,100358,100361,100494],{"className":100357},[97],[18,100359],{"className":100360,"style":89418},[101],[18,100362,100364,100367,100491],{"className":100363},[106],[18,100365],{"className":100366},[250,6012],[18,100368,100370],{"className":100369},[5654],[18,100371,100373,100483],{"className":100372},[119,120],[18,100374,100376,100480],{"className":100375},[124],[18,100377,100379,100443,100451],{"className":100378,"style":18133},[128],[18,100380,100381,100384],{"style":6028},[18,100382],{"className":100383,"style":697},[136],[18,100385,100387,100390,100393,100396,100399,100402,100405],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\\rightarrow \\infty} \\frac{a_{n+1}}{a_n} = \\frac{e}{\\left(1 + \\frac{1}{n}\\right)^n} > 1",[18,100617,100619,100837,101035],{"className":100618,"ariaHidden":51},[93],[18,100620,100622,100626,100679,100682,100828,100831,100834],{"className":100621},[97],[18,100623],{"className":100624,"style":100625},[101],"height:1.9436em;vertical-align:-0.836em;",[18,100627,100629],{"className":100628},[985,986],[18,100630,100632,100671],{"className":100631},[119,120],[18,100633,100635,100668],{"className":100634},[124],[18,100636,100638,100658],{"className":100637,"style":295},[128],[18,100639,100640,100643],{"style":97028},[18,100641],{"className":100642,"style":697},[136],[18,100644,100646],{"className":100645},[141,142,143,144],[18,100647,100649,100652,100655],{"className":100648},[106,144],[18,100650,64],{"className":100651},[106,110,144],[18,100653,1880],{"className":100654},[232,144],[18,100656,2028],{"className":100657},[106,144],[18,100659,100660,100663],{"style":693},[18,100661],{"className":100662,"style":697},[136],[18,100664,100665],{},[18,100666,91892],{"className":100667},[985],[18,100669,152],{"className":100670},[151],[18,100672,100674],{"className":100673},[124],[18,100675,100677],{"className":100676,"style":97066},[128],[18,100678],{},[18,100680],{"className":100681,"style":170},[169],[18,100683,100685,100688,100825],{"className":100684},[106],[18,100686],{"className":100687},[250,6012],[18,100689,100691],{"className":100690},[5654],[18,100692,100694,100816],{"className":100693},[119,120],[18,100695,100697,100813],{"className":100696},[124],[18,100698,100700,100748,100756],{"className":100699,"style":13852},[128],[18,100701,100702,100705],{"style":6028},[18,100703],{"className":100704,"style":697},[136],[18,100706,100708],{"className":100707},[106],[18,100709,100711,100714],{"className":100710},[106],[18,100712,16823],{"className":100713},[106,110],[18,100715,100717],{"className":100716},[115],[18,100718,100720,100740],{"className":100719},[119,120],[18,100721,100723,100737],{"className":100722},[124],[18,100724,100726],{"className":100725,"style":202},[128],[18,100727,100728,100731],{"style":1155},[18,100729],{"className":100730,"style":137},[136],[18,100732,100734],{"className":100733},[141,142,143,144],[18,100735,64],{"className":100736},[106,110,144],[18,100738,152],{"className":100739},[151],[18,100741,100743],{"className":100742},[124],[18,100744,100746],{"className":100745,"style":159},[128],[18,100747],{},[18,100749,100750,100753],{"style":6040},[18,100751],{"className":100752,"style":697},[136],[18,100754],{"className":100755,"style":6048},[6047],[18,100757,100758,100761],{"style":6051},[18,100759],{"className":100760,"style":697},[136],[18,100762,100764],{"className":100763},[106],[18,100765,100767,100770],{"className":100766},[106],[18,100768,16823],{"className":100769},[106,110],[18,100771,100773],{"className":100772},[115],[18,100774,100776,100805],{"className":100775},[119,120],[18,100777,100779,100802],{"className":100778},[124],[18,100780,100782],{"className":100781,"style":129},[128],[18,100783,100784,100787],{"style":1155},[18,100785],{"className":100786,"style":137},[136],[18,100788,100790],{"className":100789},[141,142,143,144],[18,100791,100793,100796,100799],{"className":100792},[106,144],[18,100794,64],{"className":100795},[106,110,144],[18,100797,7987],{"className":100798},[2072,144],[18,100800,47],{"className":100801},[106,144],[18,100803,152],{"className":100804},[151],[18,100806,100808],{"className":100807},[124],[18,100809,100811],{"className":100810,"style":100343},[128],[18,100812],{},[18,100814,152],{"className":100815},[151],[18,100817,100819],{"className":100818},[124],[18,100820,100823],{"className":100821,"style":100822},[128],"height:0.836em;",[18,100824],{},[18,100826],{"className":100827},[267,6012],[18,100829],{"className":100830,"style":228},[169],[18,100832,433],{"className":100833},[232],[18,100835],{"className":100836,"style":228},[169],[18,100838,100840,100844,101026,101029,101032],{"className":100839},[97],[18,100841],{"className":100842,"style":100843},[101],"height:2.2519em;vertical-align:-1.1443em;",[18,100845,100847,100850,101023],{"className":100846},[106],[18,100848],{"className":100849},[250,6012],[18,100851,100853],{"className":100852},[5654],[18,100854,100856,101014],{"className":100855},[119,120],[18,100857,100859,101011],{"className":100858},[124],[18,100860,100862,100992,101000],{"className":100861,"style":13852},[128],[18,100863,100865,100868],{"style":100864},"top:-2.2057em;",[18,100866],{"className":100867,"style":697},[136],[18,100869,100871],{"className":100870},[106],[18,100872,100874,100969],{"className":100873},[174],[18,100875,100877,100883,100886,100889,100892,100895,100963],{"className":100876},[174],[18,100878,100880],{"className":100879,"style":13796},[250,13795],[18,100881,74],{"className":100882},[1230,1196],[18,100884,47],{"className":100885},[106],[18,100887],{"className":100888,"style":2068},[169],[18,100890,7987],{"className":100891},[2072],[18,100893],{"className":100894,"style":2068},[169],[18,100896,100898,100901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",[18,101048,101050,101063],{"className":101049,"translate":22},[21],[18,101051,101053],{"className":101052},[26],[28,101054,101055],{"xmlns":30},[32,101056,101057,101061],{},[35,101058,101059],{},[41,101060,14444],{},[86,101062,14444],{"encoding":88},[18,101064,101066],{"className":101065,"ariaHidden":51},[93],[18,101067,101069,101072],{"className":101068},[97],[18,101070],{"className":101071,"style":1083},[101],[18,101073,14444],{"className":101074},[106,110],"，但它总是从下方趋近 ",[18,101077,101079,101092],{"className":101078,"translate":22},[21],[18,101080,101082],{"className":101081},[26],[28,101083,101084],{"xmlns":30},[32,101085,101086,101090],{},[35,101087,101088],{},[41,101089,14444],{},[86,101091,14444],{"encoding":88},[18,101093,101095],{"className":101094,"ariaHidden":51},[93],[18,101096,101098,101101],{"className":101097},[97],[18,101099],{"className":101100,"style":1083},[101],[18,101102,14444],{"className":101103},[106,110],"，因此比值仍大于 1，说明 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是发散的。",[11,101177,101178],{},"数学家们尝试了各种方法，发现：",[18,101180,101182],{"className":101181,"translate":22},[875],[18,101183,101185,101235],{"className":101184,"translate":22},[21],[18,101186,101188],{"className":101187},[26],[28,101189,101190],{"xmlns":30,"display":884},[32,101191,101192,101232],{},[35,101193,101194,101200,101202,101226],{},[38,101195,101196,101198],{},[41,101197,16823],{},[41,101199,64],{},[49,101201,433],{},[5654,101203,101204,101210],{},[35,101205,101206,101208],{},[41,101207,64],{},[49,101209,88995],{"stretchy":73},[1556,101211,101212,101214],{},[41,101213,64],{},[35,101215,101216,101218,101220],{},[41,101217,64],{},[49,101219,7987],{},[5654,101221,101222,101224],{},[45,101223,47],{},[45,101225,2266],{},[1556,101227,101228,101230],{},[41,101229,14444],{},[41,101231,64],{},[86,101233,101234],{"encoding":88},"a_n = \\frac{n!}{n^{n + \\frac{1}{2}}} e^n",[18,101236,101238,101293],{"className":101237,"ariaHidden":51},[93],[18,101239,101241,101244,101284,101287,101290],{"className":101240},[97],[18,101242],{"className":101243,"style":16897},[101],[18,101245,101247,101250],{"className":101246},[106],[18,101248,16823],{"className":101249},[106,110],[18,101251,101253],{"className":101252},[115],[18,101254,101256,101276],{"className":101255},[119,120],[18,101257,101259,101273],{"className":101258},[124],[18,101260,101262],{"className":101261,"style":202},[128],[18,101263,101264,101267],{"style":1155},[18,101265],{"className":101266,"style":137},[136],[18,101268,101270],{"className":101269},[141,142,143,144],[18,101271,64],{"className":101272},[106,110,144],[18,101274,152],{"className":101275},[151],[18,101277,101279],{"className":101278},[124],[18,101280,101282],{"className":101281,"style":159},[128],[18,101283],{},[18,101285],{"className":101286,"style":228},[169],[18,101288,433],{"className":101289},[232],[18,101291],{"className":101292,"style":228},[169],[18,101294,101296,101300,101469],{"className":101295},[97],[18,101297],{"className":101298,"style":101299},[101],"height:2.201em;vertical-align:-0.8296em;",[18,101301,101303,101306,101466],{"className":101302},[106],[18,101304],{"className":101305},[250,6012],[18,101307,101309],{"className":101308},[5654],[18,101310,101312,101457],{"className":101311},[119,120],[18,101313,101315,101454],{"className":101314},[124],[18,101316,101318,101432,101440],{"className":101317,"style":6025},[128],[18,101319,101321,101324],{"style":101320},"top:-2.1704em;",[18,101322],{"className":101323,"style":697},[136],[18,101325,101327],{"className":101326},[106],[18,101328,101330,101333],{"className":101329},[106],[18,101331,64],{"className":101332},[106,110],[18,101334,101336],{"className":101335},[115],[18,101337,101339],{"className":101338},[119],[18,101340,101342],{"className":101341},[124],[18,101343,101346],{"className":101344,"style":101345},[128],"height:0.9396em;",[18,101347,101349,101352],{"style":101348},"top:-3.3486em;margin-right:0.05em;",[18,101350],{"className":101351,"style":697},[136],[18,101353,101355],{"className":101354},[141,142,143,144],[18,101356,101358,101361,101364],{"className":101357},[106,144],[18,101359,64],{"className":101360},[106,110,144],[18,101362,7987],{"className":101363},[2072,144],[18,101365,101367,101370,101429],{"className":101366},[106,144],[18,101368],{"className":101369},[250,6012,141,1195,38156],[18,101371,101373],{"className":101372},[5654],[18,101374,101376,101421],{"className":101375},[119,120],[18,101377,101379,101418],{"className":101378},[124],[18,101380,101382,101396,101404],{"className":101381,"style":42938},[128],[18,101383,101384,101387],{"style":38172},[18,101385],{"className":101386,"style":697},[136],[18,101388,101390],{"className":101389},[141,1195,1196,144],[18,101391,101393],{"className":101392},[106,144],[18,101394,2266],{"className":101395},[106,144],[18,101397,101398,101401],{"style":38196},[18,101399],{"className":101400,"style":697},[136],[18,101402],{"className":101403,"style":38203},[6047,144],[18,101405,101406,101409],{"style":42972},[18,101407],{"className":101408,"style":697},[136],[18,101410,101412],{"className":101411},[141,1195,1196,144],[18,101413,101415],{"className":101414},[106,144],[18,101416,47],{"className":101417},[106,144],[18,101419,152],{"className":101420},[151],[18,101422,101424],{"className":101423},[124],[18,101425,101427],{"className":101426,"style":81887},[128],[18,101428],{},[18,101430],{"className":101431},[267,6012,141,1195,38156],[18,101433,101434,101437],{"style":6040},[18,101435],{"className":101436,"style":697},[136],[18,101438],{"className":101439,"style":6048},[6047],[18,101441,101442,101445],{"style":6051},[18,101443],{"className":101444,"style":697},[136],[18,101446,101448,101451],{"className":101447},[106],[18,101449,64],{"className":101450},[106,110],[18,101452,88995],{"className":101453},[267],[18,101455,152],{"className":101456},[151],[18,101458,101460],{"className":101459},[124],[18,101461,101464],{"className":101462,"style":101463},[128],"height:0.8296em;",[18,101465],{},[18,101467],{"className":101468},[267,6012],[18,101470,101472,101475],{"className":101471},[106],[18,101473,14444],{"className":101474},[106,110],[18,101476,101478],{"className":101477},[115],[18,101479,101481],{"className":101480},[119],[18,101482,101484],{"className":101483},[124],[18,101485,101487],{"className":101486,"style":1616},[128],[18,101488,101489,101492],{"style":1637},[18,101490],{"className":101491,"style":137},[136],[18,101493,101495],{"className":101494},[141,142,143,144],[18,101496,64],{"className":101497},[106,110,144],[11,101499,101500,101501,101570],{},"为了证明 ",[18,101502,101504,101521],{"className":101503,"translate":22},[21],[18,101505,101507],{"className":101506},[26],[28,101508,101509],{"xmlns":30},[32,101510,101511,101519],{},[35,101512,101513],{},[38,101514,101515,101517],{},[41,101516,16823],{},[41,101518,64],{},[86,101520,101125],{"encoding":88},[18,101522,101524],{"className":101523,"ariaHidden":51},[93],[18,101525,101527,101530],{"className":101526},[97],[18,101528],{"className":101529,"style":16897},[101],[18,101531,101533,101536],{"className":101532},[106],[18,101534,16823],{"className":101535},[106,110],[18,101537,101539],{"className":101538},[115],[18,101540,101542,101562],{"className":101541},[119,120],[18,101543,101545,101559],{"className":101544},[124],[18,101546,101548],{"className":101547,"style":202},[128],[18,101549,101550,101553],{"style":1155},[18,101551],{"className":101552,"style":137},[136],[18,101554,101556],{"className":101555},[141,142,143,144],[18,101557,64],{"className":101558},[106,110,144],[18,101560,152],{"className":101561},[151],[18,101563,101565],{"className":101564},[124],[18,101566,101568],{"className":101567,"style":159},[128],[18,101569],{}," 收敛，再次使用比值判别法：",[18,101572,101574],{"className":101573,"translate":22},[875],[18,101575,101577,101699],{"className":101576,"translate":22},[21],[18,101578,101580],{"className":101579},[26],[28,101581,101582],{"xmlns":30,"display":884},[32,101583,101584,101696],{},[35,101585,101586,101592,101594,101618,101624,101626,101628,101640,101642,101684],{},[38,101587,101588,101590],{},[41,101589,16823],{},[41,101591,64],{},[49,101593,433],{},[5654,101595,101596,101602],{},[35,101597,101598,101600],{},[41,101599,64],{},[49,101601,88995],{"stretchy":73},[1556,101603,101604,101606],{},[41,101605,64],{},[35,101607,101608,101610,101612],{},[41,101609,64],{},[49,101611,7987],{},[5654,101613,101614,101616],{},[45,101615,47],{},[45,101617,2266],{},[1556,101619,101620,101622],{},[41,101621,14444],{},[41,101623,64],{},[49,101625,52],{"separator":51},[169,101627],{"width":42709},[38,101629,101630,101632],{},[41,101631,16823],{},[35,101633,101634,101636,101638],{},[41,101635,64],{},[49,101637,7987],{},[45,101639,47],{},[49,101641,433],{},[5654,101643,101644,101658],{},[35,101645,101646,101648,101650,101652,101654,101656],{},[49,101647,74],{"stretchy":73},[41,101649,64],{},[49,101651,7987],{},[45,101653,47],{},[49,101655,84],{"stretchy":73},[49,101657,88995],{"stretchy":73},[35,101659,101660,101662,101664,101666,101668],{},[49,101661,74],{"stretchy":73},[41,101663,64],{},[49,101665,7987],{},[45,101667,47],{},[1556,101669,101670,101672],{},[49,101671,84],{"stretchy":73},[35,101673,101674,101676,101678],{},[41,101675,64],{},[49,101677,7987],{},[5654,101679,101680,101682],{},[45,101681,44708],{},[45,101683,2266],{},[1556,101685,101686,101688],{},[41,101687,14444],{},[35,101689,101690,101692,101694],{},[41,101691,64],{},[49,101693,7987],{},[45,101695,47],{},[86,101697,101698],{"encoding":88},"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}",[18,101700,101702,101757,102024],{"className":101701,"ariaHidden":51},[93],[18,101703,101705,101708,101748,101751,101754],{"className":101704},[97],[18,101706],{"className":101707,"style":16897},[101],[18,101709,101711,101714],{"className":101710},[106],[18,101712,16823],{"className":101713},[106,110],[18,101715,101717],{"className":101716},[115],[18,101718,101720,101740],{"className":101719},[119,120],[18,101721,101723,101737],{"className":101722},[124],[18,101724,101726],{"className":101725,"style":202},[128],[18,101727,101728,101731],{"style":1155},[18,101729],{"className":101730,"style":137},[136],[18,101732,101734],{"className":101733},[141,142,143,144],[18,101735,64],{"className":101736},[106,110,144],[18,101738,152],{"className":101739},[151],[18,101741,101743],{"className":101742},[124],[18,101744,101746],{"className":101745,"style":159},[128],[18,101747],{},[18,101749],{"className":101750,"style":228},[169],[18,101752,433],{"className":101753},[232],[18,101755],{"className":101756,"style":228},[169],[18,101758,101760,101763,101928,101957,101960,101963,101966,102015,102018,102021],{"className":101759},[97],[18,101761],{"className":101762,"style":101299},[101],[18,101764,101766,101769,101925],{"className":101765},[106],[18,101767],{"className":101768},[250,6012],[18,101770,101772],{"className":101771},[5654],[18,101773,101775,101917],{"className":101774},[119,120],[18,101776,101778,101914],{"className":101777},[124],[18,101779,101781,101892,101900],{"className":101780,"style":6025},[128],[18,101782,101783,101786],{"style":101320},[18,101784],{"className":101785,"style":697},[136],[18,101787,101789],{"className":101788},[106],[18,101790,101792,101795],{"className":101791},[106],[18,101793,64],{"className":101794},[106,110],[18,101796,101798],{"className":101797},[115],[18,101799,101801],{"className":101800},[119],[18,101802,101804],{"className":101803},[124],[18,101805,101807],{"className":101806,"style":101345},[128],[18,101808,101809,101812],{"style":101348},[18,101810],{"className":101811,"style":697},[136],[18,101813,101815],{"className":101814},[141,142,143,144],[18,101816,101818,101821,101824],{"className":101817},[106,144],[18,101819,64],{"className":101820},[106,110,144],[18,101822,7987],{"className":101823},[2072,144],[18,101825,101827,101830,101889],{"className":101826},[106,144],[18,101828],{"className":101829},[250,6012,141,1195,38156],[18,101831,101833],{"className":101832},[5654],[18,101834,101836,101881],{"className":101835},[119,120],[18,101837,101839,101878],{"className":101838},[124],[18,101840,101842,101856,101864],{"className":101841,"style":42938},[128],[18,101843,101844,101847],{"style":38172},[18,101845],{"className":101846,"style":697},[136],[18,101848,101850],{"className":101849},[141,1195,1196,144],[18,101851,101853],{"className":101852},[106,144],[18,101854,2266],{"className":101855},[106,144],[18,101857,101858,101861],{"style":38196},[18,101859],{"className":101860,"style":697},[136],[18,101862],{"className":101863,"style":38203},[6047,144],[18,101865,101866,101869],{"style":42972},[18,101867],{"className":101868,"style":697},[136],[18,101870,101872],{"className":101871},[141,1195,1196,144],[18,101873,101875],{"className":101874},[106,144],[18,101876,47],{"className":101877},[106,144],[18,101879,152],{"className":101880},[151],[18,101882,101884],{"className":101883},[124],[18,101885,101887],{"className":101886,"style":81887},[128],[18,101888],{},[18,101890],{"className":101891},[267,6012,141,1195,38156],[18,101893,101894,101897],{"style":6040},[18,101895],{"className":101896,"style":697},[136],[18,101898],{"className":101899,"style":6048},[6047],[18,101901,101902,101905],{"style":6051},[18,101903],{"className":101904,"style":697},[136],[18,101906,101908,101911],{"className":101907},[106],[18,101909,64],{"className":101910},[106,110],[18,101912,88995],{"className":101913},[267],[18,101915,152],{"className":101916},[151],[18,101918,101920],{"className":101919},[124],[18,101921,101923],{"className":101922,"style":101463},[128],[18,101924],{},[18,101926],{"className":101927},[267,6012],[18,101929,101931,101934],{"className":101930},[106],[18,101932,14444],{"className":101933},[106,110],[18,101935,101937],{"className":101936},[115],[18,101938,101940],{"className":101939},[119],[18,101941,101943],{"className":101942},[124],[18,101944,101946],{"className":101945,"style":1616},[128],[18,101947,101948,101951],{"style":1637},[18,101949],{"className":101950,"style":137},[136],[18,101952,101954],{"className":101953},[141,142,143,144],[18,101955,64],{"className":101956},[106,110,144],[18,101958,52],{"className":101959},[165],[18,101961],{"className":101962,"style":42849},[169],[18,101964],{"className":101965,"style":170},[169],[18,101967,101969,101972],{"className":101968},[106],[18,101970,16823],{"className":101971},[106,110],[18,101973,101975],{"className":101974},[115],[18,101976,101978,102007],{"className":101977},[119,120],[18,101979,101981,102004],{"className":101980},[124],[18,101982,101984],{"className":101983,"style":129},[128],[18,101985,101986,101989],{"style":1155},[18,101987],{"className":101988,"style":137},[136],[18,101990,101992],{"className":101991},[141,142,143,144],[18,101993,101995,101998,102001],{"className":101994},[106,144],[18,101996,64],{"className":101997},[106,110,144],[18,101999,7987],{"className":102000},[2072,144],[18,102002,47],{"className":102003},[106,144],[18,102005,152],{"className":102006},[151],[18,102008,102010],{"className":102009},[124],[18,102011,102013],{"className":102012,"style":100343},[128],[18,102014],{},[18,102016],{"className":102017,"style":228},[169],[18,102019,433],{"className":102020},[232],[18,102022],{"className":102023,"style":228},[169],[18,102025,102027,102031,102230],{"className":102026},[97],[18,102028],{"className":102029,"style":102030},[101],"height:2.5066em;vertical-align:-1.0796em;",[18,102032,102034,102037,102227],{"className":102033},[106],[18,102035],{"className":102036},[250,6012],[18,102038,102040],{"className":102039},[5654],[18,102041,102043,102218],{"className":102042},[119,120],[18,102044,102046,102215],{"className":102045},[124],[18,102047,102049,102178,102186],{"className":102048,"style":18133},[128],[18,102050,102051,102054],{"style":101320},[18,102052],{"className":102053,"style":697},[136],[18,102055,102057,102060,102063,102066,102069,102072,102075],{"className":102056},[106],[18,102058,74],{"className":102059},[250],[18,102061,64],{"className":102062},[106,110],[18,102064],{"className":102065,"style":2068},[169],[18,102067,7987],{"className":102068},[2072],[18,102070],{"className":102071,"style":2068},[169],[18,102073,47],{"className":102074},[106],[18,102076,102078,102081],{"className":102077},[267],[18,102079,84],{"className":102080},[267],[18,102082,102084],{"className":102083},[115],[18,102085,102087],{"className":102086},[119],[18,102088,102090],{"className":102089},[124],[18,102091,102093],{"className":102092,"style":101345},[128],[18,102094,102095,102098],{"style":101348},[18,102096],{"className":102097,"style":697},[136],[18,102099,102101],{"className":102100},[141,142,143,144],[18,102102,102104,102107,102110],{"className":102103},[106,144],[18,102105,64],{"className":102106},[106,110,144],[18,102108,7987],{"className":102109},[2072,144],[18,102111,102113,102116,102175],{"className":102112},[106,144],[18,102114],{"className":102115},[250,6012,141,1195,38156],[18,102117,102119],{"className":102118},[5654],[18,102120,102122,102167],{"className":102121},[119,120],[18,102123,102125,102164],{"className":102124},[124],[18,102126,102128,102142,102150],{"className":102127,"style":42938},[128],[18,102129,102130,102133],{"style":38172},[18,102131],{"className":102132,"style":697},[136],[18,102134,102136],{"className":102135},[141,1195,1196,144],[18,102137,102139],{"className":102138},[106,144],[18,102140,2266],{"className":102141},[106,144],[18,102143,102144,102147],{"style":38196},[18,102145],{"className":102146,"style":697},[136],[18,102148],{"className":102149,"style":38203},[6047,144],[18,102151,102152,102155],{"style":42972},[18,102153],{"className":102154,"style":697},[136],[18,102156,102158],{"className":102157},[141,1195,1196,144],[18,102159,102161],{"className":102160},[106,144],[18,102162,44708],{"className":102163},[106,144],[18,102165,152],{"className":102166},[151],[18,102168,102170],{"className":102169},[124],[18,102171,102173],{"className":102172,"style":81887},[128],[18,102174],{},[18,102176],{"className":102177},[267,6012,141,1195,38156],[18,102179,102180,102183],{"style":6040},[18,102181],{"className":102182,"style":697},[136],[18,102184],{"className":102185,"style":6048},[6047],[18,102187,102188,102191],{"style":6051},[18,102189],{"className":102190,"style":697},[136],[18,102192,102194,102197,102200,102203,102206,102209,102212],{"className":102193},[106],[18,102195,74],{"className":102196},[250],[18,102198,64],{"className":102199},[106,110],[18,102201],{"className":102202,"style":2068},[169],[18,102204,7987],{"className":102205},[2072],[18,102207],{"className":102208,"style":2068},[169],[18,102210,47],{"className":102211},[106],[18,102213,94271],{"className":102214},[267],[18,102216,152],{"className":102217},[151],[18,102219,102221],{"className":102220},[124],[18,102222,102225],{"className":102223,"style":102224},[128],"height:1.0796em;",[18,102226],{},[18,102228],{"className":102229},[267,6012],[18,102231,102233,102236],{"className":102232},[106],[18,102234,14444],{"className":102235},[106,110],[18,102237,102239],{"className":102238},[115],[18,102240,102242],{"className":102241},[119],[18,102243,102245],{"className":102244},[124],[18,102246,102248],{"className":102247,"style":10775},[128],[18,102249,102250,102253],{"style":1637},[18,102251],{"className":102252,"style":137},[136],[18,102254,102256],{"className":102255},[141,142,143,144],[18,102257,102259,102262,102265],{"className":102258},[106,144],[18,102260,64],{"className":102261},[106,110,144],[18,102263,7987],{"className":102264},[2072,144],[18,102266,47],{"className":102267},[106,144],[18,102269,102271],{"className":102270,"translate":22},[875],[18,102272,102274,102358],{"className":102273,"translate":22},[21],[18,102275,102277],{"className":102276},[26],[28,102278,102279],{"xmlns":30,"display":884},[32,102280,102281,102355],{},[35,102282,102283,102299,102319,102321],{},[890,102284,102285,102291],{},[35,102286,102287,102289],{},[41,102288,91892],{},[49,102290,899],{},[35,102292,102293,102295,102297],{},[41,102294,64],{},[49,102296,1880],{},[41,102298,2028],{"mathvariant":316},[5654,102300,102301,102313],{},[38,102302,102303,102305],{},[41,102304,16823],{},[35,102306,102307,102309,102311],{},[41,102308,64],{},[49,102310,7987],{},[45,102312,47],{},[38,102314,102315,102317],{},[41,102316,16823],{},[41,102318,64],{},[49,102320,433],{},[5654,102322,102323,102325],{},[41,102324,14444],{},[1556,102326,102327,102343],{},[35,102328,102329,102331,102333,102335,102341],{},[49,102330,74],{"fence":51},[45,102332,47],{},[49,102334,7987],{},[5654,102336,102337,102339],{},[45,102338,47],{},[41,102340,64],{},[49,102342,84],{"fence":51},[35,102344,102345,102347,102349],{},[41,102346,64],{},[49,102348,7987],{},[5654,102350,102351,102353],{},[45,102352,47],{},[45,102354,2266],{},[86,102356,102357],{"encoding":88},"\\lim_{n \\rightarrow \\infty} \\frac{a_{n+1}}{a_n} = \\frac{e}{\\left(1 + \\frac{1}{n}\\right)^{n + \\frac{1}{2}}}",[18,102359,102361,102577],{"className":102360,"ariaHidden":51},[93],[18,102362,102364,102367,102420,102423,102568,102571,102574],{"className":102363},[97],[18,102365],{"className":102366,"style":100625},[101],[18,102368,102370],{"className":102369},[985,986],[18,102371,102373,102412],{"className":102372},[119,120],[18,102374,102376,102409],{"className":102375},[124],[18,102377,102379,102399],{"className":102378,"style":295},[128],[18,102380,102381,102384],{"style":97028},[18,102382],{"className":102383,"style":697},[136],[18,102385,102387],{"className":102386},[141,142,143,144],[18,102388,102390,102393,102396],{"className":102389},[106,144],[18,102391,64],{"className":102392},[106,110,144],[18,102394,1880],{"className":102395},[232,144],[18,102397,2028],{"className":102398},[106,144],[18,102400,102401,102404],{"style":693},[18,102402],{"className":102403,"style":697},[136],[18,102405,102406],{},[18,102407,91892],{"className":102408},[985],[18,102410,152],{"className":102411},[151],[18,102413,102415],{"className":102414},[124],[18,102416,102418],{"className":102417,"style":97066},[128],[18,102419],{},[18,102421],{"className":102422,"style":170},[169],[18,102424,102426,102429,102565],{"className":102425},[106],[18,102427],{"className":102428},[250,6012],[18,102430,102432],{"className":102431},[5654],[18,102433,102435,102557],{"className":102434},[119,120],[18,102436,102438,102554],{"className":102437},[124],[18,102439,102441,102489,102497],{"className":102440,"style":13852},[128],[18,102442,102443,102446],{"style":6028},[18,102444],{"className":102445,"style":697},[136],[18,102447,102449],{"className":102448},[106],[18,102450,102452,102455],{"className":102451},[106],[18,102453,16823],{"className":102454},[106,110],[18,102456,102458],{"className":102457},[115],[18,102459,102461,102481],{"className":102460},[119,120],[18,102462,102464,102478],{"className":102463},[124],[18,102465,102467],{"className":102466,"style":202},[128],[18,102468,102469,102472],{"style":1155},[18,102470],{"className":102471,"style":137},[136],[18,102473,102475],{"className":102474},[141,142,143,144],[18,102476,64],{"className":102477},[106,110,144],[18,102479,152],{"className":102480},[151],[18,102482,102484],{"className":102483},[124],[18,102485,102487],{"className":102486,"style":159},[128],[18,102488],{},[18,102490,102491,102494],{"style":6040},[18,102492],{"className":102493,"style":697},[136],[18,102495],{"className":102496,"style":6048},[6047],[18,102498,102499,102502],{"style":6051},[18,102500],{"className":102501,"style":697},[136],[18,102503,102505],{"className":102504},[106],[18,102506,102508,102511],{"className":102507},[106],[18,102509,16823],{"className":102510},[106,110],[18,102512,102514],{"className":102513},[115],[18,102515,102517,102546],{"className":102516},[119,120],[18,102518,102520,102543],{"className":102519},[124],[18,102521,102523],{"className":102522,"style":129},[128],[18,102524,102525,102528],{"style":1155},[18,102526],{"className":102527,"style":137},[136],[18,102529,102531],{"className":102530},[141,142,143,144],[18,102532,102534,102537,102540],{"className":102533},[106,144],[18,102535,64],{"className":102536},[106,110,144],[18,102538,7987],{"className":102539},[2072,144],[18,102541,47],{"className":102542},[106,144],[18,102544,152],{"className":102545},[151],[18,102547,102549],{"className":102548},[124],[18,102550,102552],{"className":102551,"style":100343},[128],[18,102553],{},[18,102555,152],{"className":102556},[151],[18,102558,102560],{"className":102559},[124],[18,102561,102563],{"className":102562,"style":100822},[128],[18,102564],{},[18,102566],{"className":102567},[267,6012],[18,102569],{"className":102570,"style":228},[169],[18,102572,433],{"className":102573},[232],[18,102575],{"className":102576,"style":228},[169],[18,102578,102580,102584],{"className":102579},[97],[18,102581],{"className":102582,"style":102583},[101],"height:2.5415em;vertical-align:-1.4339em;",[18,102585,102587,102590,102841],{"className":102586},[106],[18,102588],{"className":102589},[250,6012],[18,102591,102593],{"className":102592},[5654],[18,102594,102596,102832],{"className":102595},[119,120],[18,102597,102599,102829],{"className":102598},[124],[18,102600,102602,102808,102817],{"className":102601,"style":13852},[128],[18,102603,102605,102609],{"style":102604},"top:-2.11em;",[18,102606],{"className":102607,"style":102608},[136],"height:3.1939em;",[18,102610,102612],{"className":102611},[106],[18,102613,102615,102710],{"className":102614},[174],[18,102616,102618,102624,102627,102630,102633,102636,102704],{"className":102617},[174],[18,102619,102621],{"className":102620,"style":13796},[250,13795],[18,102622,74],{"className":102623},[1230,1196],[18,102625,47],{"className":102626},[106],[18,102628],{"className":102629,"style":2068},[169],[18,102631,7987],{"className":102632},[2072],[18,102634],{"className":102635,"style":2068},[169],[18,102637,102639,102642,102701],{"className":102638},[106],[18,102640],{"className":102641},[250,6012],[18,102643,102645],{"className":102644},[5654],[18,102646,102648,102693],{"className":102647},[119,120],[18,102649,102651,102690],{"className":102650},[124],[18,102652,102654,102668,102676],{"className":102653,"style":12045},[128],[18,102655,102656,102659],{"style":12048},[18,102657],{"className":102658,"style":697},[136],[18,102660,102662],{"className":102661},[141,142,143,144],[18,102663,102665],{"className":102664},[106,144],[18,102666,64],{"className":102667},[106,110,144],[18,102669,102670,102673],{"style":6040},[18,102671],{"className":102672,"style":697},[136],[18,102674],{"className":102675,"style":6048},[6047],[18,102677,102678,102681],{"style":12071},[18,102679],{"className":102680,"style":697},[136],[18,102682,102684],{"className":102683},[141,142,143,144],[18,102685,102687],{"className":102686},[106,144],[18,102688,47],{"className":102689},[106,144],[18,102691,152],{"className":102692},[151],[18,102694,102696],{"className":102695},[124],[18,102697,102699],{"className":102698,"style":12093},[128],[18,102700],{},[18,102702],{"className":102703},[267,6012],[18,102705,102707],{"className":102706,"style":13796},[267,13795],[18,102708,84],{"className":102709},[1230,1196],[18,102711,102713],{"className":102712},[115],[18,102714,102716],{"className":102715},[119],[18,102717,102719],{"className":102718},[124],[18,102720,102723],{"className":102721,"style":102722},[128],"height:1.1939em;",[18,102724,102725,102728],{"style":69847},[18,102726],{"className":102727,"style":697},[136],[18,102729,102731],{"className":102730},[141,142,143,144],[18,102732,102734,102737,102740],{"className":102733},[106,144],[18,102735,64],{"className":102736},[106,110,144],[18,102738,7987],{"className":102739},[2072,144],[18,102741,102743,102746,102805],{"className":102742},[106,144],[18,102744],{"className":102745},[250,6012,141,1195,38156],[18,102747,102749],{"className":102748},[5654],[18,102750,102752,102797],{"className":102751},[119,120],[18,102753,102755,102794],{"className":102754},[124],[18,102756,102758,102772,102780],{"className":102757,"style":42938},[128],[18,102759,102760,102763],{"style":38172},[18,102761],{"className":102762,"style":697},[136],[18,102764,102766],{"className":102765},[141,1195,1196,144],[18,102767,102769],{"className":102768},[106,144],[18,102770,2266],{"className":102771},[106,144],[18,102773,102774,102777],{"style":38196},[18,102775],{"className":102776,"style":697},[136],[18,102778],{"className":102779,"style":38203},[6047,144],[18,102781,102782,102785],{"style":42972},[18,102783],{"className":102784,"style":697},[136],[18,102786,102788],{"className":102787},[141,1195,1196,144],[18,102789,102791],{"className":102790},[106,144],[18,102792,47],{"className":102793},[106,144],[18,102795,152],{"className":102796},[151],[18,102798,102800],{"className":102799},[124],[18,102801,102803],{"className":102802,"style":81887},[128],[18,102804],{},[18,102806],{"className":102807},[267,6012,141,1195,38156],[18,102809,102811,102814],{"style":102810},"top:-3.4239em;",[18,102812],{"className":102813,"style":102608},[136],[18,102815],{"className":102816,"style":6048},[6047],[18,102818,102820,102823],{"style":102819},"top:-3.8709em;",[18,102821],{"className":102822,"style":102608},[136],[18,102824,102826],{"className":102825},[106],[18,102827,14444],{"className":102828},[106,110],[18,102830,152],{"className":102831},[151],[18,102833,102835],{"className":102834},[124],[18,102836,102839],{"className":102837,"style":102838},[128],"height:1.4339em;",[18,102840],{},[18,102842],{"className":102843},[267,6012],[11,102845,102846],{},"将其与下式比较：",[18,102848,102850],{"className":102849,"translate":22},[875],[18,102851,102853,102937],{"className":102852,"translate":22},[21],[18,102854,102856],{"className":102855},[26],[28,102857,102858],{"xmlns":30,"display":884},[32,102859,102860,102934],{},[35,102861,102862,102892,102894],{},[1556,102863,102864,102880],{},[35,102865,102866,102868,102870,102872,102878],{},[49,102867,74],{"fence":51},[45,102869,47],{},[49,102871,7987],{},[5654,102873,102874,102876],{},[45,102875,47],{},[41,102877,64],{},[49,102879,84],{"fence":51},[35,102881,102882,102884,102886],{},[41,102883,64],{},[49,102885,7987],{},[5654,102887,102888,102890],{},[45,102889,47],{},[45,102891,2266],{},[49,102893,433],{},[1556,102895,102896,102898],{},[41,102897,14444],{},[35,102899,102900,102902,102904,102906,102912,102914,102916,102918],{},[49,102901,74],{"stretchy":73},[41,102903,64],{},[49,102905,7987],{},[5654,102907,102908,102910],{},[45,102909,47],{},[45,102911,2266],{},[49,102913,84],{"stretchy":73},[41,102915,97446],{},[49,102917,899],{},[35,102919,102920,102922,102924,102926,102932],{},[49,102921,74],{"fence":51},[45,102923,47],{},[49,102925,7987],{},[5654,102927,102928,102930],{},[45,102929,47],{},[41,102931,64],{},[49,102933,84],{"fence":51},[86,102935,102936],{"encoding":88},"\\left(1 + \\frac{1}{n}\\right)^{n + \\frac{1}{2}} = e^{(n + \\frac{1}{2}) \\ln \\left(1 + \\frac{1}{n}\\right)}",[18,102938,102940,103144],{"className":102939,"ariaHidden":51},[93],[18,102941,102943,102946,103135,103138,103141],{"className":102942},[97],[18,102944],{"className":102945,"style":97013},[101],[18,102947,102949,103038],{"className":102948},[174],[18,102950,102952,102958,102961,102964,102967,102970,103032],{"className":102951},[174],[18,102953,102955],{"className":102954,"style":13796},[250,13795],[18,102956,74],{"className":102957},[1230,143],[18,102959,47],{"className":102960},[106],[18,102962],{"className":102963,"style":2068},[169],[18,102965,7987],{"className":102966},[2072],[18,102968],{"className":102969,"style":2068},[169],[18,102971,102973,102976,103029],{"className":102972},[106],[18,102974],{"className":102975},[250,6012],[18,102977,102979],{"className":102978},[5654],[18,102980,102982,103021],{"className":102981},[119,120],[18,102983,102985,103018],{"className":102984},[124],[18,102986,102988,102999,103007],{"className":102987,"style":9581},[128],[18,102989,102990,102993],{"style":6028},[18,102991],{"className":102992,"style":697},[136],[18,102994,102996],{"className":102995},[106],[18,102997,64],{"className":102998},[106,110],[18,103000,103001,103004],{"style":6040},[18,103002],{"className":103003,"style":697},[136],[18,103005],{"className":103006,"style":6048},[6047],[18,103008,103009,103012],{"style":6051},[18,103010],{"className":103011,"style":697},[136],[18,103013,103015],{"className":103014},[106],[18,103016,47],{"className":103017},[106],[18,103019,152],{"className":103020},[151],[18,103022,103024],{"className":103023},[124],[18,103025,103027],{"className":103026,"style":6070},[128],[18,103028],{},[18,103030],{"className":103031},[267,6012],[18,103033,103035],{"className":103034,"style":13796},[267,13795],[18,103036,84],{"className":103037},[1230,143],[18,103039,103041],{"className":103040},[115],[18,103042,103044],{"className":103043},[119],[18,103045,103047],{"className":103046},[124],[18,103048,103050],{"className":103049,"style":97194},[128],[18,103051,103052,103055],{"style":97197},[18,103053],{"className":103054,"style":697},[136],[18,103056,103058],{"className":103057},[141,142,143,144],[18,103059,103061,103064,103067],{"className":103060},[106,144],[18,103062,64],{"className":103063},[106,110,144],[18,103065,7987],{"className":103066},[2072,144],[18,103068,103070,103073,103132],{"className":103069},[106,144],[18,103071],{"className":103072},[250,6012,141,1195,38156],[18,103074,103076],{"className":103075},[5654],[18,103077,103079,103124],{"className":103078},[119,120],[18,103080,103082,103121],{"className":103081},[124],[18,103083,103085,103099,103107],{"className":103084,"style":42938},[128],[18,103086,103087,103090],{"style":38172},[18,103088],{"className":103089,"style":697},[136],[18,103091,103093],{"className":103092},[141,1195,1196,144],[18,103094,103096],{"className":103095},[106,144],[18,103097,2266],{"className":103098},[106,144],[18,103100,103101,103104],{"style":38196},[18,103102],{"className":103103,"style":697},[136],[18,103105],{"className":103106,"style":38203},[6047,144],[18,103108,103109,103112],{"style":42972},[18,103110],{"className":103111,"style":697},[136],[18,103113,103115],{"className":103114},[141,1195,1196,144],[18,103116,103118],{"className":103117},[106,144],[18,103119,47],{"className":103120},[106,144],[18,103122,152],{"className":103123},[151],[18,103125,103127],{"className":103126},[124],[18,103128,103130],{"className":103129,"style":81887},[128],[18,103131],{},[18,103133],{"className":103134},[267,6012,141,1195,38156],[18,103136],{"className":103137,"style":228},[169],[18,103139,433],{"className":103140},[232],[18,103142],{"className":103143,"style":228},[169],[18,103145,103147,103150],{"className":103146},[97],[18,103148],{"className":103149,"style":97827},[101],[18,103151,103153,103156],{"className":103152},[106],[18,103154,14444],{"className":103155},[106,110],[18,103157,103159],{"className":103158},[115],[18,103160,103162],{"className":103161},[119],[18,103163,103165],{"className":103164},[124],[18,103166,103168],{"className":103167,"style":97827},[128],[18,103169,103170,103173],{"style":97830},[18,103171],{"className":103172,"style":697},[136],[18,103174,103176],{"className":103175},[141,142,143,144],[18,103177,103179,103182,103185,103188,103256,103259,103262,103271],{"className":103178},[106,144],[18,103180,74],{"className":103181},[250,144],[18,103183,64],{"className":103184},[106,110,144],[18,103186,7987],{"className":103187},[2072,144],[18,103189,103191,103194,103253],{"className":103190},[106,144],[18,103192],{"className":103193},[250,6012,141,1195,38156],[18,103195,103197],{"className":103196},[5654],[18,103198,103200,103245],{"className":103199},[119,120],[18,103201,103203,103242],{"className":103202},[124],[18,103204,103206,103220,103228],{"className":103205,"style":42938},[128],[18,103207,103208,103211],{"style":38172},[18,103209],{"className":103210,"style":697},[136],[18,103212,103214],{"className":103213},[141,1195,1196,144],[18,103215,103217],{"className":103216},[106,144],[18,103218,2266],{"className":103219},[106,144],[18,103221,103222,103225],{"style":38196},[18,103223],{"className":103224,"style":697},[136],[18,103226],{"className":103227,"style":38203},[6047,144],[18,103229,103230,103233],{"style":42972},[18,103231],{"className":103232,"style":697},[136],[18,103234,103236],{"className":103235},[141,1195,1196,144],[18,103237,103239],{"className":103238},[106,144],[18,103240,47],{"className":103241},[106,144],[18,103243,152],{"className":103244},[151],[18,103246,103248],{"className":103247},[124],[18,103249,103251],{"className":103250,"style":81887},[128],[18,103252],{},[18,103254],{"className":103255},[267,6012,141,1195,38156],[18,103257,84],{"className":103258},[267,144],[18,103260],{"className":103261,"style":61307},[169,144],[18,103263,103265,103268],{"className":103264},[985,144],[18,103266,79393],{"className":103267},[144],[18,103269,64],{"className":103270},[144],[18,103272,103274,103280,103283,103286,103354],{"className":103273},[174,144],[18,103275,103277],{"className":103276,"style":97926},[250,141,1195,38156,144,13795],[18,103278,74],{"className":103279},[144],[18,103281,47],{"className":103282},[106,144],[18,103284,7987],{"className":103285},[2072,144],[18,103287,103289,103292,103351],{"className":103288},[106,144],[18,103290],{"className":103291},[250,6012,141,1195,38156],[18,103293,103295],{"className":103294},[5654],[18,103296,103298,103343],{"className":103297},[119,120],[18,103299,103301,103340],{"className":103300},[124],[18,103302,103304,103318,103326],{"className":103303,"style":42938},[128],[18,103305,103306,103309],{"style":38172},[18,103307],{"className":103308,"style":697},[136],[18,103310,103312],{"className":103311},[141,1195,1196,144],[18,103313,103315],{"className":103314},[106,144],[18,103316,64],{"className":103317},[106,110,144],[18,103319,103320,103323],{"style":38196},[18,103321],{"className":103322,"style":697},[136],[18,103324],{"className":103325,"style":38203},[6047,144],[18,103327,103328,103331],{"style":42972},[18,103329],{"className":103330,"style":697},[136],[18,103332,103334],{"className":103333},[141,1195,1196,144],[18,103335,103337],{"className":103336},[106,144],[18,103338,47],{"className":103339},[106,144],[18,103341,152],{"className":103342},[151],[18,103344,103346],{"className":103345},[124],[18,103347,103349],{"className":103348,"style":81887},[128],[18,103350],{},[18,103352],{"className":103353},[267,6012,141,1195,38156],[18,103355,103357],{"className":103356,"style":97926},[267,141,1195,38156,144,13795],[18,103358,84],{"className":103359},[144],[11,103361,103362,103363,1990],{},"以及 ",[18,103364,103366,103379],{"className":103365,"translate":22},[21],[18,103367,103369],{"className":103368},[26],[28,103370,103371],{"xmlns":30},[32,103372,103373,103377],{},[35,103374,103375],{},[41,103376,14444],{},[86,103378,14444],{"encoding":88},[18,103380,103382],{"className":103381,"ariaHidden":51},[93],[18,103383,103385,103388],{"className":103384},[97],[18,103386],{"className":103387,"style":1083},[101],[18,103389,14444],{"className":103390},[106,110],[11,103392,103393],{},"因此，比较：",[18,103395,103397],{"className":103396,"translate":22},[875],[18,103398,103400,103446],{"className":103399,"translate":22},[21],[18,103401,103403],{"className":103402},[26],[28,103404,103405],{"xmlns":30,"display":884},[32,103406,103407,103443],{},[35,103408,103409,103411,103413,103415,103421,103423,103425,103427],{},[49,103410,74],{"stretchy":73},[41,103412,64],{},[49,103414,7987],{},[5654,103416,103417,103419],{},[45,103418,47],{},[45,103420,2266],{},[49,103422,84],{"stretchy":73},[41,103424,97446],{},[49,103426,899],{},[35,103428,103429,103431,103433,103435,103441],{},[49,103430,74],{"fence":51},[45,103432,47],{},[49,103434,7987],{},[5654,103436,103437,103439],{},[45,103438,47],{},[41,103440,64],{},[49,103442,84],{"fence":51},[86,103444,103445],{"encoding":88},"(n + \\frac{1}{2}) \\ln \\left(1 + \\frac{1}{n}\\right)",[18,103447,103449,103470],{"className":103448,"ariaHidden":51},[93],[18,103450,103452,103455,103458,103461,103464,103467],{"className":103451},[97],[18,103453],{"className":103454,"style":242},[101],[18,103456,74],{"className":103457},[250],[18,103459,64],{"className":103460},[106,110],[18,103462],{"className":103463,"style":2068},[169],[18,103465,7987],{"className":103466},[2072],[18,103468],{"className":103469,"style":2068},[169],[18,103471,103473,103476,103538,103541,103544,103547,103550],{"className":103472},[97],[18,103474],{"className":103475,"style":20807},[101],[18,103477,103479,103482,103535],{"className":103478},[106],[18,103480],{"className":103481},[250,6012],[18,103483,103485],{"className":103484},[5654],[18,103486,103488,103527],{"className":103487},[119,120],[18,103489,103491,103524],{"className":103490},[124],[18,103492,103494,103505,103513],{"className":103493,"style":9581},[128],[18,103495,103496,103499],{"style":6028},[18,103497],{"className":103498,"style":697},[136],[18,103500,103502],{"className":103501},[106],[18,103503,2266],{"className":103504},[106],[18,103506,103507,103510],{"style":6040},[18,103508],{"className":103509,"style":697},[136],[18,103511],{"className":103512,"style":6048},[6047],[18,103514,103515,103518],{"style":6051},[18,103516],{"className":103517,"style":697},[136],[18,103519,103521],{"className":103520},[106],[18,103522,47],{"className":103523},[106],[18,103525,152],{"className":103526},[151],[18,103528,103530],{"className":103529},[124],[18,103531,103533],{"className":103532,"style":6070},[128],[18,103534],{},[18,103536],{"className":103537},[267,6012],[18,103539,84],{"className":103540},[267],[18,103542],{"className":103543,"style":170},[169],[18,103545,97446],{"className":103546},[985],[18,103548],{"className":103549,"style":170},[169],[18,103551,103553,103559,103562,103565,103568,103571,103633],{"className":103552},[174],[18,103554,103556],{"className":103555,"style":13796},[250,13795],[18,103557,74],{"className":103558},[1230,143],[18,103560,47],{"className":103561},[106],[18,103563],{"className":103564,"style":2068},[169],[18,103566,7987],{"className":103567},[2072],[18,103569],{"className":103570,"style":2068},[169],[18,103572,103574,103577,103630],{"className":103573},[106],[18,103575],{"className":103576},[250,6012],[18,103578,103580],{"className":103579},[5654],[18,103581,103583,103622],{"className":103582},[119,120],[18,103584,103586,103619],{"className":103585},[124],[18,103587,103589,103600,103608],{"className":103588,"style":9581},[128],[18,103590,103591,103594],{"style":6028},[18,103592],{"className":103593,"style":697},[136],[18,103595,103597],{"className":103596},[106],[18,103598,64],{"className":103599},[106,110],[18,103601,103602,103605],{"style":6040},[18,103603],{"className":103604,"style":697},[136],[18,103606],{"className":103607,"style":6048},[6047],[18,103609,103610,103613],{"style":6051},[18,103611],{"className":103612,"style":697},[136],[18,103614,103616],{"className":103615},[106],[18,103617,47],{"className":103618},[106],[18,103620,152],{"className":103621},[151],[18,103623,103625],{"className":103624},[124],[18,103626,103628],{"className":103627,"style":6070},[128],[18,103629],{},[18,103631],{"className":103632},[267,6012],[18,103634,103636],{"className":103635,"style":13796},[267,13795],[18,103637,84],{"className":103638},[1230,143],[11,103640,103641],{},"与 1。",[11,103643,103644,103645,509,103765,56918],{},"具体来说，比较 ",[18,103646,103648,103676],{"className":103647,"translate":22},[21],[18,103649,103651],{"className":103650},[26],[28,103652,103653],{"xmlns":30},[32,103654,103655,103673],{},[35,103656,103657,103659,103661],{},[41,103658,97446],{},[49,103660,899],{},[5654,103662,103663,103671],{},[35,103664,103665,103667,103669],{},[41,103666,64],{},[49,103668,7987],{},[45,103670,47],{},[41,103672,64],{},[86,103674,103675],{"encoding":88},"\\ln \\frac{n+1}{n}",[18,103677,103679],{"className":103678,"ariaHidden":51},[93],[18,103680,103682,103685,103688,103691],{"className":103681},[97],[18,103683],{"className":103684,"style":12026},[101],[18,103686,97446],{"className":103687},[985],[18,103689],{"className":103690,"style":170},[169],[18,103692,103694,103697,103762],{"className":103693},[106],[18,103695],{"className":103696},[250,6012],[18,103698,103700],{"className":103699},[5654],[18,103701,103703,103754],{"className":103702},[119,120],[18,103704,103706,103751],{"className":103705},[124],[18,103707,103709,103723,103731],{"className":103708,"style":12045},[128],[18,103710,103711,103714],{"style":12048},[18,103712],{"className":103713,"style":697},[136],[18,103715,103717],{"className":103716},[141,142,143,144],[18,103718,103720],{"className":103719},[106,144],[18,103721,64],{"className":103722},[106,110,144],[18,103724,103725,103728],{"style":6040},[18,103726],{"className":103727,"style":697},[136],[18,103729],{"className":103730,"style":6048},[6047],[18,103732,103733,103736],{"style":12071},[18,103734],{"className":103735,"style":697},[136],[18,103737,103739],{"className":103738},[141,142,143,144],[18,103740,103742,103745,103748],{"className":103741},[106,144],[18,103743,64],{"className":103744},[106,110,144],[18,103746,7987],{"className":103747},[2072,144],[18,103749,47],{"className":103750},[106,144],[18,103752,152],{"className":103753},[151],[18,103755,103757],{"className":103756},[124],[18,103758,103760],{"className":103759,"style":12093},[128],[18,103761],{},[18,103763],{"className":103764},[267,6012],[18,103766,103768,103796],{"className":103767,"translate":22},[21],[18,103769,103771],{"className":103770},[26],[28,103772,103773],{"xmlns":30},[32,103774,103775,103793],{},[35,103776,103777],{},[5654,103778,103779,103781],{},[45,103780,47],{},[35,103782,103783,103785,103787],{},[41,103784,64],{},[49,103786,7987],{},[5654,103788,103789,103791],{},[45,103790,47],{},[45,103792,2266],{},[86,103794,103795],{"encoding":88},"\\frac{1}{n + \\frac{1}{2}}",[18,103797,103799],{"className":103798,"ariaHidden":51},[93],[18,103800,103802,103806],{"className":103801},[97],[18,103803],{"className":103804,"style":103805},[101],"height:1.4869em;vertical-align:-0.6418em;",[18,103807,103809,103812,103944],{"className":103808},[106],[18,103810],{"className":103811},[250,6012],[18,103813,103815],{"className":103814},[5654],[18,103816,103818,103935],{"className":103817},[119,120],[18,103819,103821,103932],{"className":103820},[124],[18,103822,103824,103910,103918],{"className":103823,"style":12045},[128],[18,103825,103827,103830],{"style":103826},"top:-2.599em;",[18,103828],{"className":103829,"style":697},[136],[18,103831,103833],{"className":103832},[141,142,143,144],[18,103834,103836,103839,103842],{"className":103835},[106,144],[18,103837,64],{"className":103838},[106,110,144],[18,103840,7987],{"className":103841},[2072,144],[18,103843,103845,103848,103907],{"className":103844},[106,144],[18,103846],{"className":103847},[250,6012,141,1195,38156],[18,103849,103851],{"className":103850},[5654],[18,103852,103854,103899],{"className":103853},[119,120],[18,103855,103857,103896],{"className":103856},[124],[18,103858,103860,103874,103882],{"className":103859,"style":42938},[128],[18,103861,103862,103865],{"style":38172},[18,103863],{"className":103864,"style":697},[136],[18,103866,103868],{"className":103867},[141,1195,1196,144],[18,103869,103871],{"className":103870},[106,144],[18,103872,2266],{"className":103873},[106,144],[18,103875,103876,103879],{"style":38196},[18,103877],{"className":103878,"style":697},[136],[18,103880],{"className":103881,"style":38203},[6047,144],[18,103883,103884,103887],{"style":42972},[18,103885],{"className":103886,"style":697},[136],[18,103888,103890],{"className":103889},[141,1195,1196,144],[18,103891,103893],{"className":103892},[106,144],[18,103894,47],{"className":103895},[106,144],[18,103897,152],{"className":103898},[151],[18,103900,103902],{"className":103901},[124],[18,103903,103905],{"className":103904,"style":81887},[128],[18,103906],{},[18,103908],{"className":103909},[267,6012,141,1195,38156],[18,103911,103912,103915],{"style":6040},[18,103913],{"className":103914,"style":697},[136],[18,103916],{"className":103917,"style":6048},[6047],[18,103919,103920,103923],{"style":12071},[18,103921],{"className":103922,"style":697},[136],[18,103924,103926],{"className":103925},[141,142,143,144],[18,103927,103929],{"className":103928},[106,144],[18,103930,47],{"className":103931},[106,144],[18,103933,152],{"className":103934},[151],[18,103936,103938],{"className":103937},[124],[18,103939,103942],{"className":103940,"style":103941},[128],"height:0.6418em;",[18,103943],{},[18,103945],{"className":103946},[267,6012],[11,103948,103949],{},"由于：",[18,103951,103953],{"className":103952,"translate":22},[875],[18,103954,103956,104010],{"className":103955,"translate":22},[21],[18,103957,103959],{"className":103958},[26],[28,103960,103961],{"xmlns":30,"display":884},[32,103962,103963,104007],{},[35,103964,103965,103977,103979,103981,103983,103999,104001],{},[5654,103966,103967,103969],{},[45,103968,47],{},[35,103970,103971,103973,103975],{},[41,103972,64],{},[49,103974,7987],{},[45,103976,47],{},[49,103978,6684],{},[41,103980,97446],{},[49,103982,899],{},[35,103984,103985,103987,103989,103991,103997],{},[49,103986,74],{"fence":51},[45,103988,47],{},[49,103990,7987],{},[5654,103992,103993,103995],{},[45,103994,47],{},[41,103996,64],{},[49,103998,84],{"fence":51},[49,104000,6684],{},[5654,104002,104003,104005],{},[45,104004,47],{},[41,104006,64],{},[86,104008,104009],{"encoding":88},"\\frac{1}{n + 1} \u003C \\ln \\left(1 + \\frac{1}{n}\\right) \u003C \\frac{1}{n}",[18,104011,104013,104102,104212],{"className":104012,"ariaHidden":51},[93],[18,104014,104016,104019,104093,104096,104099],{"className":104015},[97],[18,104017],{"className":104018,"style":92893},[101],[18,104020,104022,104025,104090],{"className":104021},[106],[18,104023],{"className":104024},[250,6012],[18,104026,104028],{"className":104027},[5654],[18,104029,104031,104082],{"className":104030},[119,120],[18,104032,104034,104079],{"className":104033},[124],[18,104035,104037,104060,104068],{"className":104036,"style":9581},[128],[18,104038,104039,104042],{"style":6028},[18,104040],{"className":104041,"style":697},[136],[18,104043,104045,104048,104051,104054,104057],{"className":104044},[106],[18,104046,64],{"className":104047},[106,110],[18,104049],{"className":104050,"style":2068},[169],[18,104052,7987],{"className":104053},[2072],[18,104055],{"className":104056,"style":2068},[169],[18,104058,47],{"className":104059},[106],[18,104061,104062,104065],{"style":6040},[18,104063],{"className":104064,"style":697},[136],[18,104066],{"className":104067,"style":6048},[6047],[18,104069,104070,104073],{"style":6051},[18,104071],{"className":104072,"style":697},[136],[18,104074,104076],{"className":104075},[106],[18,104077,47],{"className":104078},[106],[18,104080,152],{"className":104081},[151],[18,104083,104085],{"className":104084},[124],[18,104086,104088],{"className":104087,"style":86799},[128],[18,104089],{},[18,104091],{"className":104092},[267,6012],[18,104094],{"className":104095,"style":228},[169],[18,104097,6684],{"className":104098},[232],[18,104100],{"className":104101,"style":228},[169],[18,104103,104105,104108,104111,104114,104203,104206,104209],{"className":104104},[97],[18,104106],{"className":104107,"style":20807},[101],[18,104109,97446],{"className":104110},[985],[18,104112],{"className":104113,"style":170},[169],[18,104115,104117,104123,104126,104129,104132,104135,104197],{"className":104116},[174],[18,104118,104120],{"className":104119,"style":13796},[250,13795],[18,104121,74],{"className":104122},[1230,143],[18,104124,47],{"className":104125},[106],[18,104127],{"className":104128,"style":2068},[169],[18,104130,7987],{"className":104131},[2072],[18,104133],{"className":104134,"style":2068},[169],[18,104136,104138,104141,104194],{"className":104137},[106],[18,104139],{"className":104140},[250,6012],[18,104142,104144],{"className":104143},[5654],[18,104145,104147,104186],{"className":104146},[119,120],[18,104148,104150,104183],{"className":104149},[124],[18,104151,104153,104164,104172],{"className":104152,"style":9581},[128],[18,104154,104155,104158],{"style":6028},[18,104156],{"className":104157,"style":697},[136],[18,104159,104161],{"className":104160},[106],[18,104162,64],{"className":104163},[106,110],[18,104165,104166,104169],{"style":6040},[18,104167],{"className":104168,"style":697},[136],[18,104170],{"className":104171,"style":6048},[6047],[18,104173,104174,104177],{"style":6051},[18,104175],{"className":104176,"style":697},[136],[18,104178,104180],{"className":104179},[106],[18,104181,47],{"className":104182},[106],[18,104184,152],{"className":104185},[151],[18,104187,104189],{"className":104188},[124],[18,104190,104192],{"className":104191,"style":6070},[128],[18,104193],{},[18,104195],{"className":104196},[267,6012],[18,104198,104200],{"className":104199,"style":13796},[267,13795],[18,104201,84],{"className":104202},[1230,143],[18,104204],{"className":104205,"style":228},[169],[18,104207,6684],{"className":104208},[232],[18,104210],{"className":104211,"style":228},[169],[18,104213,104215,104218],{"className":104214},[97],[18,104216],{"className":104217,"style":9562},[101],[18,104219,104221,104224,104277],{"className":104220},[106],[18,104222],{"className":104223},[250,6012],[18,104225,104227],{"className":104226},[5654],[18,104228,104230,104269],{"className":104229},[119,120],[18,104231,104233,104266],{"className":104232},[124],[18,104234,104236,104247,104255],{"className":104235,"style":9581},[128],[18,104237,104238,104241],{"style":6028},[18,104239],{"className":104240,"style":697},[136],[18,104242,104244],{"className":104243},[106],[18,104245,64],{"className":104246},[106,110],[18,104248,104249,104252],{"style":6040},[18,104250],{"className":104251,"style":697},[136],[18,104253],{"className":104254,"style":6048},[6047],[18,104256,104257,104260],{"style":6051},[18,104258],{"className":104259,"style":697},[136],[18,104261,104263],{"className":104262},[106],[18,104264,47],{"className":104265},[106],[18,104267,152],{"className":104268},[151],[18,104270,104272],{"className":104271},[124],[18,104273,104275],{"className":104274,"style":6070},[128],[18,104276],{},[18,104278],{"className":104279},[267,6012],[11,104281,104282,104283,104373,104374,104458],{},"注意：右侧由不等式 ",[18,104284,104286,104316],{"className":104285,"translate":22},[21],[18,104287,104289],{"className":104288},[26],[28,104290,104291],{"xmlns":30},[32,104292,104293,104313],{},[35,104294,104295,104297,104299,104301,104303,104305,104307,104309,104311],{},[41,104296,97446],{},[49,104298,899],{},[49,104300,74],{"stretchy":73},[45,104302,47],{},[49,104304,7987],{},[41,104306,38582],{},[49,104308,84],{"stretchy":73},[49,104310,6684],{},[41,104312,38582],{},[86,104314,104315],{"encoding":88},"\\ln(1 + x) \u003C x",[18,104317,104319,104343,104364],{"className":104318,"ariaHidden":51},[93],[18,104320,104322,104325,104328,104331,104334,104337,104340],{"className":104321},[97],[18,104323],{"className":104324,"style":242},[101],[18,104326,97446],{"className":104327},[985],[18,104329,74],{"className":104330},[250],[18,104332,47],{"className":104333},[106],[18,104335],{"className":104336,"style":2068},[169],[18,104338,7987],{"className":104339},[2072],[18,104341],{"className":104342,"style":2068},[169],[18,104344,104346,104349,104352,104355,104358,104361],{"className":104345},[97],[18,104347],{"className":104348,"style":242},[101],[18,104350,38582],{"className":104351},[106,110],[18,104353,84],{"className":104354},[267],[18,104356],{"className":104357,"style":228},[169],[18,104359,6684],{"className":104360},[232],[18,104362],{"className":104363,"style":228},[169],[18,104365,104367,104370],{"className":104366},[97],[18,104368],{"className":104369,"style":1083},[101],[18,104371,38582],{"className":104372},[106,110],"（其中 ",[18,104375,104377,104395],{"className":104376,"translate":22},[21],[18,104378,104380],{"className":104379},[26],[28,104381,104382],{"xmlns":30},[32,104383,104384,104392],{},[35,104385,104386,104388,104390],{},[41,104387,38582],{},[49,104389,2771],{"mathvariant":316},[45,104391,430],{},[86,104393,104394],{"encoding":88},"x \\neq 0",[18,104396,104398,104449],{"className":104397,"ariaHidden":51},[93],[18,104399,104401,104404,104407,104410,104446],{"className":104400},[97],[18,104402],{"className":104403,"style":2890},[101],[18,104405,38582],{"className":104406},[106,110],[18,104408],{"className":104409,"style":228},[169],[18,104411,104413,104440,104443],{"className":104412},[232],[18,104414,104416],{"className":104415},[232],[18,104417,104419],{"className":104418},[106,2878],[18,104420,104422],{"className":104421},[2882],[18,104423,104425,104428,104437],{"className":104424},[2886],[18,104426],{"className":104427,"style":2890},[101],[18,104429,104431],{"className":104430},[2894],[18,104432,104434],{"className":104433},[106],[18,104435,2901],{"className":104436},[232],[18,104438],{"className":104439},[2905],[18,104441],{"className":104442},[169,2909],[18,104444,433],{"className":104445},[232],[18,104447],{"className":104448,"style":228},[169],[18,104450,104452,104455],{"className":104451},[97],[18,104453],{"className":104454,"style":505},[101],[18,104456,430],{"className":104457},[106],"）得出。",[11,104460,104461,104462,104729,104730,104840],{},"左侧由 ",[18,104463,104465,104517],{"className":104464,"translate":22},[21],[18,104466,104468],{"className":104467},[26],[28,104469,104470],{"xmlns":30},[32,104471,104472,104514],{},[35,104473,104474,104476,104478,104490,104492,104494,104496],{},[45,104475,47],{},[49,104477,2253],{},[5654,104479,104480,104482],{},[41,104481,64],{},[35,104483,104484,104486,104488],{},[41,104485,64],{},[49,104487,7987],{},[45,104489,47],{},[49,104491,6684],{},[41,104493,97446],{},[49,104495,899],{},[35,104497,104498,104500,104512],{},[49,104499,74],{"fence":51},[5654,104501,104502,104510],{},[35,104503,104504,104506,104508],{},[41,104505,64],{},[49,104507,7987],{},[45,104509,47],{},[41,104511,64],{},[49,104513,84],{"fence":51},[86,104515,104516],{"encoding":88},"1 - \\frac{n}{n+1} \u003C \\ln \\left(\\frac{n+1}{n}\\right)",[18,104518,104520,104538,104628],{"className":104519,"ariaHidden":51},[93],[18,104521,104523,104526,104529,104532,104535],{"className":104522},[97],[18,104524],{"className":104525,"style":18095},[101],[18,104527,47],{"className":104528},[106],[18,104530],{"className":104531,"style":2068},[169],[18,104533,2253],{"className":104534},[2072],[18,104536],{"className":104537,"style":2068},[169],[18,104539,104541,104545,104619,104622,104625],{"className":104540},[97],[18,104542],{"className":104543,"style":104544},[101],"height:1.0987em;vertical-align:-0.4033em;",[18,104546,104548,104551,104616],{"className":104547},[106],[18,104549],{"className":104550},[250,6012],[18,104552,104554],{"className":104553},[5654],[18,104555,104557,104608],{"className":104556},[119,120],[18,104558,104560,104605],{"className":104559},[124],[18,104561,104563,104583,104591],{"className":104562,"style":14257},[128],[18,104564,104565,104568],{"style":12048},[18,104566],{"className":104567,"style":697},[136],[18,104569,104571],{"className":104570},[141,142,143,144],[18,104572,104574,104577,104580],{"className":104573},[106,144],[18,104575,64],{"className":104576},[106,110,144],[18,104578,7987],{"className":104579},[2072,144],[18,104581,47],{"className":104582},[106,144],[18,104584,104585,104588],{"style":6040},[18,104586],{"className":104587,"style":697},[136],[18,104589],{"className":104590,"style":6048},[6047],[18,104592,104593,104596],{"style":12071},[18,104594],{"className":104595,"style":697},[136],[18,104597,104599],{"className":104598},[141,142,143,144],[18,104600,104602],{"className":104601},[106,144],[18,104603,64],{"className":104604},[106,110,144],[18,104606,152],{"className":104607},[151],[18,104609,104611],{"className":104610},[124],[18,104612,104614],{"className":104613,"style":93300},[128],[18,104615],{},[18,104617],{"className":104618},[267,6012],[18,104620],{"className":104621,"style":228},[169],[18,104623,6684],{"className":104624},[232],[18,104626],{"className":104627,"style":228},[169],[18,104629,104631,104634,104637,104640],{"className":104630},[97],[18,104632],{"className":104633,"style":6101},[101],[18,104635,97446],{"className":104636},[985],[18,104638],{"className":104639,"style":170},[169],[18,104641,104643,104649,104723],{"className":104642},[174],[18,104644,104646],{"className":104645,"style":13796},[250,13795],[18,104647,74],{"className":104648},[1230,1196],[18,104650,104652,104655,104720],{"className":104651},[106],[18,104653],{"className":104654},[250,6012],[18,104656,104658],{"className":104657},[5654],[18,104659,104661,104712],{"className":104660},[119,120],[18,104662,104664,104709],{"className":104663},[124],[18,104665,104667,104681,104689],{"className":104666,"style":12045},[128],[18,104668,104669,104672],{"style":12048},[18,104670],{"className":104671,"style":697},[136],[18,104673,104675],{"className":104674},[141,142,143,144],[18,104676,104678],{"className":104677},[106,144],[18,104679,64],{"className":104680},[106,110,144],[18,104682,104683,104686],{"style":6040},[18,104684],{"className":104685,"style":697},[136],[18,104687],{"className":104688,"style":6048},[6047],[18,104690,104691,104694],{"style":12071},[18,104692],{"className":104693,"style":697},[136],[18,104695,104697],{"className":104696},[141,142,143,144],[18,104698,104700,104703,104706],{"className":104699},[106,144],[18,104701,64],{"className":104702},[106,110,144],[18,104704,7987],{"className":104705},[2072,144],[18,104707,47],{"className":104708},[106,144],[18,104710,152],{"className":104711},[151],[18,104713,104715],{"className":104714},[124],[18,104716,104718],{"className":104717,"style":12093},[128],[18,104719],{},[18,104721],{"className":104722},[267,6012],[18,104724,104726],{"className":104725,"style":13796},[267,13795],[18,104727,84],{"className":104728},[1230,1196]," 得出，其中 ",[18,104731,104733,104757],{"className":104732,"translate":22},[21],[18,104734,104736],{"className":104735},[26],[28,104737,104738],{"xmlns":30},[32,104739,104740,104754],{},[35,104741,104742],{},[5654,104743,104744,104746],{},[41,104745,64],{},[35,104747,104748,104750,104752],{},[41,104749,64],{},[49,104751,7987],{},[45,104753,47],{},[86,104755,104756],{"encoding":88},"\\frac{n}{n+1}",[18,104758,104760],{"className":104759,"ariaHidden":51},[93],[18,104761,104763,104766],{"className":104762},[97],[18,104764],{"className":104765,"style":104544},[101],[18,104767,104769,104772,104837],{"className":104768},[106],[18,104770],{"className":104771},[250,6012],[18,104773,104775],{"className":104774},[5654],[18,104776,104778,104829],{"className":104777},[119,120],[18,104779,104781,104826],{"className":104780},[124],[18,104782,104784,104804,104812],{"className":104783,"style":14257},[128],[18,104785,104786,104789],{"style":12048},[18,104787],{"className":104788,"style":697},[136],[18,104790,104792],{"className":104791},[141,142,143,144],[18,104793,104795,104798,104801],{"className":104794},[106,144],[18,104796,64],{"className":104797},[106,110,144],[18,104799,7987],{"className":104800},[2072,144],[18,104802,47],{"className":104803},[106,144],[18,104805,104806,104809],{"style":6040},[18,104807],{"className":104808,"style":697},[136],[18,104810],{"className":104811,"style":6048},[6047],[18,104813,104814,104817],{"style":12071},[18,104815],{"className":104816,"style":697},[136],[18,104818,104820],{"className":104819},[141,142,143,144],[18,104821,104823],{"className":104822},[106,144],[18,104824,64],{"className":104825},[106,110,144],[18,104827,152],{"className":104828},[151],[18,104830,104832],{"className":104831},[124],[18,104833,104835],{"className":104834,"style":93300},[128],[18,104836],{},[18,104838],{"className":104839},[267,6012]," 被视为一个整体。这是高中知识。",[11,104842,104843,104844,104942],{},"因为 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是凹函数：",[18,104944,104946],{"className":104945,"translate":22},[875],[18,104947,104949,105023],{"className":104948,"translate":22},[21],[18,104950,104952],{"className":104951},[26],[28,104953,104954],{"xmlns":30,"display":884},[32,104955,104956,105020],{},[35,104957,104958,104972,104978,104980,104982,104984,104986,104988,104990,105002,105004],{},[1589,104959,104960,104962,104964],{},[49,104961,4005],{},[41,104963,64],{},[35,104965,104966,104968,104970],{},[41,104967,64],{},[49,104969,7987],{},[45,104971,47],{},[5654,104973,104974,104976],{},[45,104975,47],{},[41,104977,38582],{},[926,104979,4024],{},[41,104981,4031],{},[41,104983,38582],{},[49,104985,433],{},[41,104987,97446],{},[49,104989,899],{},[5654,104991,104992,105000],{},[35,104993,104994,104996,104998],{},[41,104995,64],{},[49,104997,7987],{},[45,104999,47],{},[41,105001,64],{},[49,105003,6885],{},[5654,105005,105006,105008],{},[45,105007,47],{},[35,105009,105010,105012,105014],{},[41,105011,64],{},[49,105013,7987],{},[5654,105015,105016,105018],{},[45,105017,47],{},[45,105019,2266],{},[86,105021,105022],{"encoding":88},"\\int_{n}^{n+1} \\frac{1}{x} \\, dx = \\ln \\frac{n+1}{n} > \\frac{1}{n + \\frac{1}{2}}",[18,105024,105026,105178,105273],{"className":105025,"ariaHidden":51},[93],[18,105027,105029,105032,105095,105098,105160,105163,105166,105169,105172,105175],{"className":105028},[97],[18,105030],{"className":105031,"style":99216},[101],[18,105033,105035,105038],{"className":105034},[985],[18,105036,4005],{"className":105037,"style":4200},[985,4198,4199],[18,105039,105041],{"className":105040},[115],[18,105042,105044,105087],{"className":105043},[119,120],[18,105045,105047,105084],{"className":105046},[124],[18,105048,105050,105064],{"className":105049,"style":99235},[128],[18,105051,105052,105055],{"style":99238},[18,105053],{"className":105054,"style":137},[136],[18,105056,105058],{"className":105057},[141,142,143,144],[18,105059,105061],{"className":105060},[106,144],[18,105062,64],{"className":105063},[106,110,144],[18,105065,105066,105069],{"style":99253},[18,105067],{"className":105068,"style":137},[136],[18,105070,105072],{"className":105071},[141,142,143,144],[18,105073,105075,105078,105081],{"className":105074},[106,144],[18,105076,64],{"className":105077},[106,110,144],[18,105079,7987],{"className":105080},[2072,144],[18,105082,47],{"className":105083},[106,144],[18,105085,152],{"className":105086},[151],[18,105088,105090],{"className":105089},[124],[18,105091,105093],{"className":105092,"style":81902},[128],[18,105094],{},[18,105096],{"className":105097,"style":170},[169],[18,105099,105101,105104,105157],{"className":105100},[106],[18,105102],{"className":105103},[250,6012],[18,105105,105107],{"className":105106},[5654],[18,105108,105110,105149],{"className":105109},[119,120],[18,105111,105113,105146],{"className":105112},[124],[18,105114,105116,105127,105135],{"className":105115,"style":9581},[128],[18,105117,105118,105121],{"style":6028},[18,105119],{"className":105120,"style":697},[136],[18,105122,105124],{"className":105123},[106],[18,105125,38582],{"className":105126},[106,110],[18,105128,105129,105132],{"style":6040},[18,105130],{"className":105131,"style":697},[136],[18,105133],{"className":105134,"style":6048},[6047],[18,105136,105137,105140],{"style":6051},[18,105138],{"className":105139,"style":697},[136],[18,105141,105143],{"className":105142},[106],[18,105144,47],{"className":105145},[106],[18,105147,152],{"className":105148},[151],[18,105150,105152],{"className":105151},[124],[18,105153,105155],{"className":105154,"style":6070},[128],[18,105156],{},[18,105158],{"className":105159},[267,6012],[18,105161],{"className":105162,"style":170},[169],[18,105164,4031],{"className":105165},[106,110],[18,105167,38582],{"className":105168},[106,110],[18,105170],{"className":105171,"style":228},[169],[18,105173,433],{"className":105174},[232],[18,105176],{"className":105177,"style":228},[169],[18,105179,105181,105184,105187,105190,105264,105267,105270],{"className":105180},[97],[18,105182],{"className":105183,"style":9562},[101],[18,105185,97446],{"className":105186},[985],[18,105188],{"className":105189,"style":170},[169],[18,105191,105193,105196,105261],{"className":105192},[106],[18,105194],{"className":105195},[250,6012],[18,105197,105199],{"className":105198},[5654],[18,105200,105202,105253],{"className":105201},[119,120],[18,105203,105205,105250],{"className":105204},[124],[18,105206,105208,105219,105227],{"className":105207,"style":9581},[128],[18,105209,105210,105213],{"style":6028},[18,105211],{"className":105212,"style":697},[136],[18,105214,105216],{"className":105215},[106],[18,105217,64],{"className":105218},[106,110],[18,105220,105221,105224],{"style":6040},[18,105222],{"className":105223,"style":697},[136],[18,105225],{"className":105226,"style":6048},[6047],[18,105228,105229,105232],{"style":6051},[18,105230],{"className":105231,"style":697},[136],[18,105233,105235,105238,105241,105244,105247],{"className":105234},[106],[18,105236,64],{"className":105237},[106,110],[18,105239],{"className":105240,"style":2068},[169],[18,105242,7987],{"className":105243},[2072],[18,105245],{"className":105246,"style":2068},[169],[18,105248,47],{"className":105249},[106],[18,105251,152],{"className":105252},[151],[18,105254,105256],{"className":105255},[124],[18,105257,105259],{"className":105258,"style":6070},[128],[18,105260],{},[18,105262],{"className":105263},[267,6012],[18,105265],{"className":105266,"style":228},[169],[18,105268,6885],{"className":105269},[232],[18,105271],{"className":105272,"style":228},[169],[18,105274,105276,105280],{"className":105275},[97],[18,105277],{"className":105278,"style":105279},[101],"height:2.4015em;vertical-align:-1.0801em;",[18,105281,105283,105286,105418],{"className":105282},[106],[18,105284],{"className":105285},[250,6012],[18,105287,105289],{"className":105288},[5654],[18,105290,105292,105409],{"className":105291},[119,120],[18,105293,105295,105406],{"className":105294},[124],[18,105296,105298,105387,105395],{"className":105297,"style":9581},[128],[18,105299,105301,105304],{"style":105300},"top:-2.2649em;",[18,105302],{"className":105303,"sty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\\rightarrow \\infty} \\frac{a_{n+1}}{a_n} = \\frac{e}{\\left(1 + \\frac{1}{n}\\right)^{n + \\frac{1}{2}}} \u003C 1",[18,105518,105520,105736,106005],{"className":105519,"ariaHidden":51},[93],[18,105521,105523,105526,105579,105582,105727,105730,105733],{"className":105522},[97],[18,105524],{"className":105525,"style":100625},[101],[18,105527,105529],{"className":105528},[985,986],[18,105530,105532,105571],{"className":105531},[119,120],[18,105533,105535,105568],{"className":105534},[124],[18,105536,105538,105558],{"className":105537,"style":295},[128],[18,105539,105540,105543],{"style":97028},[18,105541],{"className":105542,"style":697},[136],[18,105544,105546],{"className":105545},[141,142,143,144],[18,105547,105549,105552,105555],{"className":105548},[106,144],[18,105550,64],{"className":105551},[106,110,144],[18,105553,1880],{"className":105554},[232,144],[18,105556,2028],{"className":105557},[106,144],[18,105559,105560,105563],{"style":693},[18,105561],{"className":105562,"style":697},[136],[18,105564,105565],{},[18,105566,91892],{"className":105567},[985],[18,105569,152],{"className":105570},[151],[18,105572,105574],{"className":105573},[124],[18,105575,105577],{"className":105576,"style":97066},[128],[18,105578],{},[18,105580],{"className":105581,"style":170},[169],[18,105583,105585,105588,105724],{"className":105584},[106],[18,105586],{"className":105587},[250,6012],[18,105589,105591],{"className":105590},[5654],[18,105592,105594,105716],{"className":105593},[119,120],[18,105595,105597,105713],{"className":105596},[124],[18,105598,105600,105648,105656],{"className":105599,"style":13852},[128],[18,105601,105602,105605],{"style":6028},[18,105603],{"className":105604,"style":697},[136],[18,105606,105608],{"className":105607},[106],[18,105609,105611,105614],{"className":105610},[106],[18,105612,16823],{"className":105613},[106,110],[18,105615,105617],{"className":105616},[115],[18,105618,105620,105640],{"className":105619},[119,120],[18,105621,105623,105637],{"className":105622},[124],[18,105624,105626],{"className":105625,"style":202},[128],[18,105627,105628,105631],{"style":1155},[18,105629],{"className":105630,"style":137},[136],[18,105632,105634],{"className":105633},[141,142,143,144],[18,105635,64],{"className":105636},[106,110,144],[18,105638,152],{"className":105639},[151],[18,105641,105643],{"className":105642},[124],[18,105644,105646],{"className":105645,"style":159},[128],[18,105647],{},[18,105649,105650,105653],{"style":6040},[18,105651],{"className":105652,"style":697},[136],[18,105654],{"className":105655,"style":6048},[6047],[18,105657,105658,105661],{"style":6051},[18,105659],{"className":105660,"style":697},[136],[18,105662,105664],{"className":105663},[106],[18,105665,105667,105670],{"className":105666},[106],[18,105668,16823],{"className":105669},[106,110],[18,105671,105673],{"className":105672},[115],[18,105674,105676,105705],{"className":105675},[119,120],[18,105677,105679,105702],{"className":105678},[124],[18,105680,105682],{"className":105681,"style":129},[128],[18,105683,105684,105687],{"style":1155},[18,105685],{"className":105686,"style":137},[136],[18,105688,105690],{"className":105689},[141,142,143,144],[18,105691,105693,105696,105699],{"className":105692},[106,144],[18,105694,64],{"className":105695},[106,110,144],[18,105697,7987],{"className":105698},[2072,144],[18,105700,47],{"className":105701},[106,144],[18,105703,152],{"className":105704},[151],[18,105706,105708],{"className":105707},[124],[18,105709,105711],{"className":105710,"style":100343},[128],[18,105712],{},[18,105714,152],{"className":105715},[151],[18,105717,105719],{"className":105718},[124],[18,105720,105722],{"className":105721,"style":100822},[128],[18,105723],{},[18,105725],{"className":105726},[267,6012],[18,105728],{"className":105729,"style":228},[169],[18,105731,433],{"className":105732},[232],[18,105734],{"className":105735,"style":228},[169],[18,105737,105739,105742,105996,105999,106002],{"className":105738},[97],[18,105740],{"className":105741,"style":102583},[101],[18,105743,105745,105748,105993],{"className":105744},[106],[18,105746],{"className":105747},[250,6012],[18,105749,105751],{"className":105750},[5654],[18,105752,105754,105985],{"className":105753},[119,120],[18,105755,105757,105982],{"className":105756},[124],[18,105758,105760,105963,105971],{"className":105759,"style":13852},[128],[18,105761,105762,105765],{"style":102604},[18,105763],{"className":105764,"style":102608},[136],[18,105766,105768],{"className":105767},[106],[18,105769,105771,105866],{"className":105770},[174],[18,105772,105774,105780,105783,105786,105789,105792,105860],{"className":105773},[174],[18,105775,105777],{"className":105776,"style":13796},[250,13795],[18,105778,74],{"className":105779},[1230,1196],[18,105781,47],{"className":105782},[106],[18,105784],{"className":105785,"style":2068},[169],[18,105787,7987],{"className":105788},[2072],[18,105790],{"className":105791,"style":2068},[169],[18,105793,105795,105798,105857],{"className":105794},[106],[18,105796],{"className":105797},[250,6012],[18,105799,105801],{"className":105800},[5654],[18,105802,105804,105849],{"className":105803},[119,120],[18,105805,105807,105846],{"className":105806},[124],[18,105808,105810,105824,105832],{"className":105809,"style":12045},[128],[18,105811,105812,105815],{"style":12048},[18,105813],{"className":105814,"style":697},[136],[18,105816,105818],{"className":105817},[141,142,143,144],[18,105819,105821],{"className":105820},[106,144],[18,105822,64],{"className":105823},[106,110,144],[18,105825,105826,105829],{"style":6040},[18,105827],{"className":105828,"style":697},[136],[18,105830],{"className":105831,"style":6048},[6047],[18,105833,105834,105837],{"style":12071},[18,105835],{"className":105836,"style":697},[136],[18,105838,105840],{"className":105839},[141,142,143,144],[18,105841,105843],{"className":105842},[106,144],[18,105844,47],{"className":105845},[106,144],[18,105847,152],{"className":105848},[151],[18,105850,105852],{"className":105851},[124],[18,105853,105855],{"className":105854,"style":12093},[128],[18,105856],{},[18,105858],{"className":105859},[267,6012],[18,105861,105863],{"className":105862,"style":13796},[267,13795],[18,105864,84],{"className":105865},[1230,1196],[18,105867,105869],{"className":105868},[115],[18,105870,105872],{"className":105871},[119],[18,105873,105875],{"className":105874},[124],[18,105876,105878],{"className":105877,"style":102722},[128],[18,105879,105880,105883],{"style":69847},[18,105881],{"className":105882,"style":697},[136],[18,105884,105886],{"className":105885},[141,142,143,144],[18,105887,105889,105892,105895],{"className":105888},[106,144],[18,105890,64],{"className":105891},[106,110,144],[18,105893,7987],{"className":105894},[2072,144],[18,105896,105898,105901,105960],{"className":105897},[106,144],[18,105899],{"className":105900},[250,6012,141,1195,38156],[18,105902,105904],{"className":105903},[5654],[18,105905,105907,105952],{"className":105906},[119,120],[18,105908,105910,105949],{"className":105909},[124],[18,105911,105913,105927,105935],{"className":105912,"style":42938},[128],[18,105914,105915,105918],{"style":38172},[18,105916],{"className":105917,"style":697},[136],[18,105919,105921],{"className":105920},[141,1195,1196,144],[18,105922,105924],{"className":105923},[106,144],[18,105925,2266],{"className":105926},[106,144],[18,105928,105929,105932],{"style":38196},[18,105930],{"className":105931,"style":697},[136],[18,105933],{"className":105934,"style":38203},[6047,144],[18,105936,105937,105940],{"style":42972},[18,105938],{"className":105939,"style":697},[136],[18,105941,105943],{"className":105942},[141,1195,1196,144],[18,105944,105946],{"className":105945},[106,144],[18,105947,47],{"className":105948},[106,144],[18,105950,152],{"className":105951},[151],[18,105953,105955],{"className":105954},[124],[18,105956,105958],{"className":105957,"style":81887},[128],[18,105959],{},[18,105961],{"className":105962},[267,6012,141,1195,38156],[18,105964,105965,105968],{"style":102810},[18,105966],{"className":105967,"style":102608},[136],[18,105969],{"className":105970,"style":6048},[6047],[18,105972,105973,105976],{"style":102819},[18,105974],{"className":105975,"style":102608},[136],[18,105977,105979],{"className":105978},[106],[18,105980,14444],{"className":105981},[106,110],[18,105983,152],{"className":105984},[151],[18,105986,105988],{"className":105987},[124],[18,105989,105991],{"className":105990,"style":102838},[128],[18,105992],{},[18,105994],{"className":105995},[267,6012],[18,105997],{"className":105998,"style":228},[169],[18,106000,6684],{"className":106001},[232],[18,106003],{"className":106004,"style":228},[169],[18,106006,106008,106011],{"className":106007},[97],[18,106009],{"className":106010,"style":505},[101],[18,106012,47],{"className":106013},[106],[11,106015,106016,106017,106086,106087,106179],{},"于是 ",[18,106018,106020,106037],{"className":106019,"translate":22},[21],[18,106021,106023],{"className":106022},[26],[28,106024,106025],{"xmlns":30},[32,106026,106027,106035],{},[35,106028,106029],{},[38,106030,106031,106033],{},[41,106032,16823],{},[41,106034,64],{},[86,106036,101125],{"encoding":88},[18,106038,106040],{"className":106039,"ariaHidden":51},[93],[18,106041,106043,106046],{"className":106042},[97],[18,106044],{"className":106045,"style":16897},[101],[18,106047,106049,106052],{"className":106048},[106],[18,106050,16823],{"className":106051},[106,110],[18,106053,106055],{"className":106054},[115],[18,106056,106058,106078],{"className":106057},[119,120],[18,106059,106061,106075],{"className":106060},[124],[18,106062,106064],{"className":106063,"style":202},[128],[18,106065,106066,106069],{"style":1155},[18,106067],{"className":106068,"style":137},[136],[18,106070,106072],{"className":106071},[141,142,143,144],[18,106073,64],{"className":106074},[106,110,144],[18,106076,152],{"className":106077},[151],[18,106079,106081],{"className":106080},[124],[18,106082,106084],{"className":106083,"style":159},[128],[18,106085],{}," 单调递减且 ",[18,106088,106090,106112],{"className":106089,"translate":22},[21],[18,106091,106093],{"className":106092},[26],[28,106094,106095],{"xmlns":30},[32,106096,106097,106109],{},[35,106098,106099,106105,106107],{},[38,106100,106101,106103],{},[41,106102,16823],{},[41,106104,64],{},[49,106106,6885],{},[45,106108,430],{},[86,106110,106111],{"encoding":88},"a_n > 0",[18,106113,106115,106170],{"className":106114,"ariaHidden":51},[93],[18,106116,106118,106121,106161,106164,106167],{"className":106117},[97],[18,106119],{"className":106120,"style":40980},[101],[18,106122,106124,106127],{"className":106123},[106],[18,106125,16823],{"className":106126},[106,110],[18,106128,106130],{"className":106129},[115],[18,106131,106133,106153],{"className":106132},[119,120],[18,106134,106136,106150],{"className":106135},[124],[18,106137,106139],{"className":106138,"style":202},[128],[18,106140,106141,106144],{"style":1155},[18,106142],{"className":106143,"style":137},[136],[18,106145,106147],{"className":106146},[141,142,143,144],[18,106148,64],{"className":106149},[106,110,144],[18,106151,152],{"className":106152},[151],[18,106154,106156],{"className":106155},[124],[18,106157,106159],{"className":106158,"style":159},[128],[18,106160],{},[18,106162],{"className":106163,"style":228},[169],[18,106165,6885],{"className":106166},[232],[18,106168],{"className":106169,"style":228},[169],[18,106171,106173,106176],{"className":106172},[97],[18,106174],{"className":106175,"style":505},[101],[18,106177,430],{"className":106178},[106],"，因此它必然收敛，即存在极限。",[11,106181,106182],{},"设：",[18,106184,106186],{"className":106185,"translate":22},[875],[18,106187,106189,106251],{"className":106188,"translate":22},[21],[18,106190,106192],{"className":106191},[26],[28,106193,106194],{"xmlns":30,"display":884},[32,106195,106196,106248],{},[35,106197,106198,106214,106238,106244,106246],{},[890,106199,106200,106206],{},[35,106201,106202,106204],{},[41,106203,91892],{},[49,106205,899],{},[35,106207,106208,106210,106212],{},[41,106209,64],{},[49,106211,1880],{},[41,106213,2028],{"mathvariant":316},[5654,106215,106216,106222],{},[35,106217,106218,106220],{},[41,106219,64],{},[49,106221,88995],{"stretchy":73},[1556,106223,106224,106226],{},[41,106225,64],{},[35,106227,106228,106230,106232],{},[41,106229,64],{},[49,106231,7987],{},[5654,106233,106234,106236],{},[45,106235,47],{},[45,106237,2266],{},[1556,106239,106240,106242],{},[41,106241,14444],{},[41,106243,64],{},[49,106245,433],{},[41,106247,16823],{},[86,106249,106250],{"encoding":88},"\\lim_{n \\rightarrow \\infty} \\frac{n!}{n^{n + \\frac{1}{2}}} e^n = a",[18,106252,106254,106519],{"className":106253,"ariaHidden":51},[93],[18,106255,106257,106260,106313,106316,106481,106510,106513,106516],{"className":106256},[97],[18,106258],{"className":106259,"style":101299},[101],[18,106261,106263],{"className":106262},[985,986],[18,106264,106266,106305],{"className":106265},[119,120],[18,106267,106269,106302],{"className":106268},[124],[18,106270,106272,106292],{"className":106271,"style":295},[128],[18,106273,106274,106277],{"style":97028},[18,106275],{"className":106276,"style":697},[136],[18,106278,106280],{"className":106279},[141,142,143,144],[18,106281,106283,106286,106289],{"className":106282},[106,144],[18,106284,64],{"className":106285},[106,110,144],[18,106287,1880],{"className":106288},[232,144],[18,106290,2028],{"className":106291},[106,144],[18,106293,106294,106297],{"style":693},[18,106295],{"className":106296,"style":697},[136],[18,106298,106299],{},[18,106300,91892],{"className":106301},[985],[18,106303,152],{"className":106304},[151],[18,106306,106308],{"className":106307},[124],[18,106309,106311],{"className":106310,"style":97066},[128],[18,106312],{},[18,106314],{"className":106315,"style":170},[169],[18,106317,106319,106322,106478],{"className":106318},[106],[18,106320],{"className":106321},[250,6012],[18,106323,106325],{"className":106324},[5654],[18,106326,106328,106470],{"className":106327},[119,120],[18,106329,106331,106467],{"className":106330},[124],[18,106332,106334,106445,106453],{"className":106333,"style":6025},[128],[18,106335,106336,106339],{"style":101320},[18,106337],{"className":106338,"style":697},[136],[18,106340,106342],{"className":106341},[106],[18,106343,106345,106348],{"className":106344},[106],[18,106346,64],{"className":106347},[106,110],[18,106349,106351],{"className":106350},[115],[18,106352,106354],{"className":106353},[119],[18,106355,106357],{"className":106356},[124],[18,106358,106360],{"className":106359,"style":101345},[128],[18,106361,106362,106365],{"style":101348},[18,106363],{"className":106364,"style":697},[136],[18,106366,106368],{"className":106367},[141,142,143,144],[18,106369,106371,106374,106377],{"className":106370},[106,144],[18,106372,64],{"className":106373},[106,110,144],[18,106375,7987],{"className":106376},[2072,144],[18,106378,106380,106383,106442],{"className":106379},[106,144],[18,106381],{"className":106382},[250,6012,141,1195,38156],[18,106384,106386],{"className":106385},[5654],[18,106387,106389,106434],{"className":106388},[119,120],[18,106390,106392,106431],{"className":106391},[124],[18,106393,106395,106409,106417],{"className":106394,"style":42938},[128],[18,106396,106397,106400],{"style":38172},[18,106398],{"className":106399,"style":697},[136],[18,106401,106403],{"className":106402},[141,1195,1196,144],[18,106404,106406],{"className":106405},[106,144],[18,106407,2266],{"className":106408},[106,144],[18,106410,106411,106414],{"style":38196},[18,106412],{"className":106413,"style":697},[136],[18,106415],{"className":106416,"style":38203},[6047,144],[18,106418,106419,106422],{"style":42972},[18,106420],{"className":106421,"style":697},[136],[18,106423,106425],{"className":106424},[141,1195,1196,144],[18,106426,106428],{"className":106427},[106,144],[18,106429,47],{"className":106430},[106,144],[18,106432,152],{"className":106433},[151],[18,106435,106437],{"className":106436},[124],[18,106438,106440],{"className":106439,"style":81887},[128],[18,106441],{},[18,106443],{"className":106444},[267,6012,141,1195,38156],[18,106446,106447,106450],{"style":6040},[18,106448],{"className":106449,"style":697},[136],[18,106451],{"className":106452,"style":6048},[6047],[18,106454,106455,106458],{"style":6051},[18,106456],{"className":106457,"style":697},[136],[18,106459,106461,106464],{"className":106460},[106],[18,106462,64],{"className":106463},[106,110],[18,106465,88995],{"className":106466},[267],[18,106468,152],{"className":106469},[151],[18,106471,106473],{"className":106472},[124],[18,106474,106476],{"className":106475,"style":101463},[128],[18,106477],{},[18,106479],{"className":106480},[267,6012],[18,106482,106484,106487],{"className":106483},[106],[18,106485,14444],{"className":106486},[106,110],[18,106488,106490],{"className":106489},[115],[18,106491,106493],{"className":106492},[119],[18,106494,106496],{"className":106495},[124],[18,106497,106499],{"className":106498,"style":1616},[128],[18,106500,106501,106504],{"style":1637},[18,106502],{"className":106503,"style":137},[136],[18,106505,106507],{"className":106506},[141,142,143,144],[18,106508,64],{"className":106509},[106,110,144],[18,106511],{"className":106512,"style":228},[169],[18,106514,433],{"className":106515},[232],[18,106517],{"className":106518,"style":228},[169],[18,106520,106522,106525],{"className":106521},[97],[18,106523],{"className":106524,"style":1083},[101],[18,106526,16823],{"className":106527},[106,110],[11,106529,106530],{},"接下来我们需要求出这个极限。",[11,106532,106533],{},"由前面的沃利斯公式：",[18,106535,106537],{"className":106536,"translate":22},[875],[18,106538,106540,106629],{"className":106539,"translate":22},[21],[18,106541,106543],{"className":106542},[26],[28,106544,106545],{"xmlns":30,"display":884},[32,106546,106547,106627],{},[35,106548,106549,106565,106619,106621],{},[890,106550,106551,106557],{},[35,106552,106553,106555],{},[41,106554,91892],{},[49,106556,899],{},[35,106558,106559,106561,106563],{},[41,106560,39964],{},[49,106562,1880],{},[41,106564,2028],{"mathvariant":316},[5654,106566,106567,106585],{},[35,106568,106569,106571,106573,106575,106577,106579],{},[49,106570,74],{"stretchy":73},[45,106572,2266],{},[41,106574,39964],{},[49,106576,84],{"stretchy":73},[49,106578,88995],{"stretchy":73},[1556,106580,106581,106583],{},[49,106582,88995],{"stretchy":73},[45,106584,2266],{},[35,106586,106587,106589,106591,106593,106595,106597,106599,106601,106603,106605,106607,106609,106611,106613,106615,106617],{},[49,106588,74],{"stretchy":73},[45,106590,2266],{},[41,106592,39964],{},[49,106594,7987],{},[45,106596,47],{},[49,106598,84],{"stretchy":73},[49,106600,88995],{"stretchy":73},[49,106602,88995],{"stretchy":73},[49,106604,74],{"stretchy":73},[45,106606,2266],{},[41,106608,39964],{},[49,106610,2253],{},[45,106612,47],{},[49,106614,84],{"stretchy":73},[49,106616,88995],{"stretchy":73},[49,106618,88995],{"stretchy":73},[49,106620,433],{},[5654,106622,106623,106625],{},[41,106624,3941],{},[45,106626,2266],{},[86,106628,96195],{"encoding":88},[18,106630,106632,106848],{"className":106631,"ariaHidden":51},[93],[18,106633,106635,106638,106691,106694,106839,106842,106845],{"className":106634},[97],[18,106636],{"className":106637,"style":94168},[101],[18,106639,106641],{"className":106640},[985,986],[18,106642,106644,106683],{"className":106643},[119,120],[18,106645,106647,106680],{"className":106646},[124],[18,106648,106650,106670],{"className":106649,"style":295},[128],[18,106651,106652,106655],{"style":29370},[18,106653],{"className":106654,"style":697},[136],[18,106656,106658],{"className":106657},[141,142,143,144],[18,106659,106661,106664,106667],{"className":106660},[106,144],[18,106662,39964],{"className":106663,"style":40040},[106,110,144],[18,106665,1880],{"className":106666},[232,144],[18,106668,2028],{"className":106669},[106,144],[18,106671,106672,106675],{"style":693},[18,106673],{"className":106674,"style":697},[136],[18,106676,106677],{},[18,106678,91892],{"className":106679},[985],[18,106681,152],{"className":106682},[151],[18,106684,106686],{"className":106685},[124],[18,106687,106689],{"className":106688,"style":92248},[128],[18,106690],{},[18,106692],{"className":106693,"style":170},[169],[18,106695,106697,106700,106836],{"className":106696},[106],[18,106698],{"className":106699},[250,6012],[18,106701,106703],{"className":106702},[5654],[18,106704,106706,106828],{"className":106705},[119,120],[18,106707,106709,106825],{"className":106708},[124],[18,106710,106712,106768,106776],{"className":106711,"style":15999},[128],[18,106713,106714,106717],{"style":6028},[18,106715],{"className":106716,"style":697},[136],[18,106718,106720,106723,106726,106729,106732,106735,106738,106741,106744,106747,106750,106753,106756,106759,106762,106765],{"className":106719},[106],[18,106721,74],{"className":106722},[250],[18,106724,2266],{"className":106725},[106],[18,106727,39964],{"className":106728,"style":40040},[106,110],[18,106730],{"className":106731,"style":2068},[169],[18,106733,7987],{"className":106734},[2072],[18,106736],{"className":106737,"style":2068},[169],[18,106739,47],{"className":106740},[106],[18,106742,89457],{"className":106743},[267],[18,106745,74],{"className":106746},[250],[18,106748,2266],{"className":106749},[106],[18,106751,39964],{"className":106752,"style":40040},[106,110],[18,106754],{"className":106755,"style":2068},[169],[18,106757,2253],{"className":106758},[2072],[18,106760],{"className":106761,"style":2068},[169],[18,106763,47],{"className":106764},[106],[18,106766,89457],{"className":106767},[267],[18,106769,106770,106773],{"style":6040},[18,106771],{"className":106772,"style":697},[136],[18,106774],{"className":106775,"style":6048},[6047],[18,106777,106778,106781],{"style":6051},[18,106779],{"className":106780,"style":697},[136],[18,106782,106784,106787,106790,106793,106796],{"className":106783},[106],[18,106785,74],{"className":106786},[250],[18,106788,2266],{"className":106789},[106],[18,106791,39964],{"className":106792,"style":40040},[106,110],[18,106794,94271],{"className":106795},[267],[18,106797,106799,106802],{"className":106798},[267],[18,106800,88995],{"className":106801},[267],[18,106803,106805],{"className":106804},[115],[18,106806,106808],{"className":106807},[119],[18,106809,106811],{"className":106810},[124],[18,106812,106814],{"className":106813,"style":2416},[128],[18,106815,106816,106819],{"style":1962},[18,106817],{"className":106818,"style":137},[136],[18,106820,106822],{"className":106821},[141,142,143,144],[18,106823,2266],{"className":106824},[106,144],[18,106826,152],{"className":106827},[151],[18,106829,106831],{"className":106830},[124],[18,106832,106834],{"className":106833,"style":89507},[128],[18,106835],{},[18,106837],{"className":106838},[267,6012],[18,106840],{"className":106841,"style":228},[169],[18,106843,433],{"className":106844},[232],[18,106846],{"className":106847,"style":228},[169],[18,106849,106851,106854],{"className":106850},[97],[18,106852],{"className":106853,"style":42071},[101],[18,106855,106857,106860,106913],{"className":106856},[106],[18,106858],{"className":106859},[250,6012],[18,106861,106863],{"className":106862},[5654],[18,106864,106866,106905],{"className":106865},[119,120],[18,106867,106869,106902],{"className":106868},[124],[18,106870,106872,106883,106891],{"className":106871,"style":13852},[128],[18,106873,106874,106877],{"style":6028},[18,106875],{"className":106876,"style":697},[136],[18,106878,106880],{"className":106879},[106],[18,106881,2266],{"className":106882},[106],[18,106884,106885,106888],{"style":6040},[18,106886],{"className":106887,"style":697},[136],[18,106889],{"className":106890,"style":6048},[6047],[18,106892,106893,106896],{"style":6051},[18,106894],{"className":106895,"style":697},[136],[18,106897,106899],{"className":106898},[106],[18,106900,3941],{"className":106901,"style":3957},[106,110],[18,106903,152],{"className":106904},[151],[18,106906,106908],{"className":106907},[124],[18,106909,106911],{"className":106910,"style":6070},[128],[18,106912],{},[18,106914],{"className":106915},[267,6012],[18,106917,106919],{"className":106918,"translate":22},[875],[18,106920,106922,106980],{"className":106921,"translate":22},[21],[18,106923,106925],{"className":106924},[26],[28,106926,106927],{"xmlns":30,"display":884},[32,106928,106929,106977],{},[35,106930,106931,106933,106935,106937,106939,106941,106943,106945,106947,106949,106951,106953,106955,106957,106959,106961,106963,106965,106967,106973,106975],{},[49,106932,74],{"stretchy":73},[45,106934,2266],{},[41,106936,64],{},[49,106938,84],{"stretchy":73},[49,106940,88995],{"stretchy":73},[49,106942,88995],{"stretchy":73},[49,106944,433],{},[45,106946,2266],{},[41,106948,64],{},[49,106950,74],{"stretchy":73},[45,106952,2266],{},[41,106954,64],{},[49,106956,2253],{},[45,106958,2266],{},[49,106960,84],{"stretchy":73},[49,106962,76925],{},[45,106964,2266],{},[49,106966,433],{},[1556,106968,106969,106971],{},[45,106970,2266],{},[41,106972,64],{},[41,106974,64],{},[49,106976,88995],{"stretchy":73},[86,106978,106979],{"encoding":88},"(2n)!! = 2n (2n-2) \\cdots 2 = 2^n n!",[18,106981,106983,107010,107040,107073],{"className":106982,"ariaHidden":51},[93],[18,106984,106986,106989,106992,106995,106998,107001,107004,107007],{"className":106985},[97],[18,106987],{"className":106988,"style":242},[101],[18,106990,74],{"className":106991},[250],[18,106993,2266],{"className":106994},[106],[18,106996,64],{"className":106997},[106,110],[18,106999,89457],{"className":107000},[267],[18,107002],{"className":107003,"style":228},[169],[18,107005,433],{"className":107006},[232],[18,107008],{"className":107009,"style":228},[169],[18,107011,107013,107016,107019,107022,107025,107028,107031,107034,107037],{"className":107012},[97],[18,107014],{"className":107015,"style":242},[101],[18,107017,2266],{"className":107018},[106],[18,107020,64],{"className":107021},[106,110],[18,107023,74],{"className":107024},[250],[18,107026,2266],{"className":107027},[106],[18,107029,64],{"className":107030},[106,110],[18,107032],{"className":107033,"style":2068},[169],[18,107035,2253],{"className":107036},[2072],[18,107038],{"className":107039,"style":2068},[169],[18,107041,107043,107046,107049,107052,107055,107058,107061,107064,107067,107070],{"className":107042},[97],[18,107044],{"className":107045,"style":242},[101],[18,107047,2266],{"className":107048},[106],[18,107050,84],{"className":107051},[267],[18,107053],{"className":107054,"style":170},[169],[18,107056,76925],{"className":107057},[174],[18,107059],{"className":107060,"style":170},[169],[18,107062,2266],{"className":107063},[106],[18,107065],{"className":107066,"style":228},[169],[18,107068,433],{"className":107069},[232],[18,107071],{"className":107072,"style":228},[169],[18,107074,107076,107079,107108,107111],{"className":107075},[97],[18,107077],{"className":107078,"style":1616},[101],[18,107080,107082,107085],{"className":107081},[106],[18,107083,2266],{"className":107084},[106],[18,107086,107088],{"className":107087},[115],[18,107089,107091],{"className":107090},[119],[18,107092,107094],{"className":107093},[124],[18,107095,107097],{"className":107096,"style":1616},[128],[18,107098,107099,107102],{"style":1637},[18,107100],{"className":107101,"style":137},[136],[18,107103,107105],{"className":107104},[141,142,143,144],[18,107106,64],{"className":107107},[106,110,144],[18,107109,64],{"className":107110},[106,110],[18,107112,88995],{"className":107113},[267],[18,107115,107117],{"className":107116,"translate":22},[875],[18,107118,107120,107206],{"className":107119,"translate":22},[21],[18,107121,107123],{"className":107122},[26],[28,107124,107125],{"xmlns":30,"display":884},[32,107126,107127,107203],{},[35,107128,107129,107131,107133,107135,107137,107139,107141,107143,107145,107147,107175,107177],{},[49,107130,74],{"stretchy":73},[45,107132,2266],{},[41,107134,64],{},[49,107136,2253],{},[45,107138,47],{},[49,107140,84],{"stretchy":73},[49,107142,88995],{"stretchy":73},[49,107144,88995],{"stretchy":73},[49,107146,433],{},[5654,107148,107149,107161],{},[35,107150,107151,107153,107155,107157,107159],{},[49,107152,74],{"stretchy":73},[45,107154,2266],{},[41,107156,64],{},[49,107158,84],{"stretchy":73},[49,107160,88995],{"stretchy":73},[35,107162,107163,107165,107167,107169,107171,107173],{},[49,107164,74],{"stretchy":73},[45,107166,2266],{},[41,107168,64],{},[49,107170,84],{"stretchy":73},[49,107172,88995],{"stretchy":73},[49,107174,88995],{"stretchy":73},[49,107176,433],{},[5654,107178,107179,107191],{},[35,107180,107181,107183,107185,107187,107189],{},[49,107182,74],{"stretchy":73},[45,107184,2266],{},[41,107186,64],{},[49,107188,84],{"stretchy":73},[49,107190,88995],{"stretchy":73},[35,107192,107193,107199,107201],{},[1556,107194,107195,107197],{},[45,107196,2266],{},[41,107198,64],{},[41,107200,64],{},[49,107202,88995],{"stretchy":73},[86,107204,107205],{"encoding":88},"(2n-1)!! = \\frac{(2n)!}{(2n)!!} = \\frac{(2n)!}{2^n n!}",[18,107207,107209,107233,107254,107349],{"className":107208,"ariaHidden":51},[93],[18,107210,107212,107215,107218,107221,107224,107227,107230],{"className":107211},[97],[18,107213],{"className":107214,"style":242},[101],[18,107216,74],{"className":107217},[250],[18,107219,2266],{"className":107220},[106],[18,107222,64],{"className":107223},[106,110],[18,107225],{"className":107226,"style":2068},[169],[18,107228,2253],{"className":107229},[2072],[18,107231],{"className":107232,"style":2068},[169],[18,107234,107236,107239,107242,107245,107248,107251],{"className":107235},[97],[18,107237],{"className":107238,"style":242},[101],[18,107240,47],{"className":107241},[106],[18,107243,89457],{"className":107244},[267],[18,107246],{"className":107247,"style":228},[169],[18,107249,433],{"className":107250},[232],[18,107252],{"className":107253,"style":228},[169],[18,107255,107257,107260,107340,107343,107346],{"className":107256},[97],[18,107258],{"className":107259,"style":89418},[101],[18,107261,107263,107266,107337],{"className":107262},[106],[18,107264],{"className":107265},[250,6012],[18,107267,107269],{"className":107268},[5654],[18,107270,107272,107329],{"className":107271},[119,120],[18,107273,107275,107326],{"className":107274},[124],[18,107276,107278,107298,107306],{"className":107277,"style":18133},[128],[18,107279,107280,107283],{"style":6028},[18,107281],{"className":107282,"style":697},[136],[18,107284,107286,107289,107292,107295],{"className":107285},[106],[18,107287,74],{"className":107288},[250],[18,107290,2266],{"className":107291},[106],[18,107293,64],{"className":107294},[106,110],[18,107296,89457],{"className":107297},[267],[18,107299,107300,107303],{"style":6040},[18,107301],{"className":107302,"style":697},[136],[18,107304],{"className":107305,"style":6048},[6047],[18,107307,107308,107311],{"style":6051},[18,107309],{"className":107310,"style":697},[136],[18,107312,107314,107317,107320,107323],{"className":107313},[106],[18,107315,74],{"className":107316},[250],[18,107318,2266],{"className":107319},[106],[18,107321,64],{"className":107322},[106,110],[18,107324,94271],{"className":107325},[267],[18,107327,152],{"className":107328},[151],[18,107330,107332],{"className":107331},[124],[18,107333,107335],{"className":107334,"style":89507},[128],[18,107336],{},[18,107338],{"className":107339},[267,6012],[18,107341],{"className":107342,"style":228},[169],[18,107344,433],{"className":107345},[232],[18,107347],{"className":107348,"style":228},[169],[18,107350,107352,107356],{"className":107351},[97],[18,107353],{"className":107354,"style":107355},[101],"height:2.113em;vertical-align:-0.686em;",[18,107357,107359,107362,107456],{"className":107358},[106],[18,107360],{"className":107361},[250,6012],[18,107363,107365],{"className":107364},[5654],[18,107366,107368,107448],{"className":107367},[119,120],[18,107369,107371,107445],{"className":107370},[124],[18,107372,107374,107417,107425],{"className":107373,"style":18133},[128],[18,107375,107376,107379],{"style":6028},[18,107377],{"className":107378,"style":697},[136],[18,107380,107382,107411,107414],{"className":107381},[106],[18,107383,107385,107388],{"className":107384},[106],[18,107386,2266],{"className":107387},[106],[18,107389,107391],{"className":107390},[115],[18,107392,107394],{"className":107393},[119],[18,107395,107397],{"className":107396},[124],[18,107398,107400],{"className":107399,"style":97128},[128],[18,107401,107402,107405],{"style":26112},[18,107403],{"className":107404,"style":137},[136],[18,107406,107408],{"className":107407},[141,142,143,144],[18,107409,64],{"className":107410},[106,110,144],[18,107412,64],{"className":107413},[106,110],[18,107415,88995],{"className":107416},[267],[18,107418,107419,107422],{"style":6040},[18,107420],{"className":107421,"style":697},[136],[18,107423],{"className":107424,"style":6048},[6047],[18,107426,107427,107430],{"style":6051},[18,107428],{"className":107429,"style":697},[136],[18,107431,107433,107436,107439,107442],{"className":107432},[106],[18,107434,74],{"className":107435},[250],[18,107437,2266],{"className":107438},[106],[18,107440,64],{"className":107441},[106,110],[18,107443,94271],{"className":107444},[267],[18,107446,152],{"className":107447},[151],[18,107449,107451],{"className":107450},[124],[18,107452,107454],{"className":107453,"style":6070},[128],[18,107455],{},[18,107457],{"className":107458},[267,6012],[18,107460,107462],{"className":107461,"translate":22},[875],[18,107463,107465,107577],{"className":107464,"translate":22},[21],[18,107466,107468],{"className":107467},[26],[28,107469,107470],{"xmlns":30,"display":884},[32,107471,107472,107574],{},[35,107473,107474,107476,107478,107480,107482,107484,107486,107488,107490,107492,107528,107530],{},[49,107475,74],{"stretchy":73},[45,107477,2266],{},[41,107479,64],{},[49,107481,7987],{},[45,107483,47],{},[49,107485,84],{"stretchy":73},[49,107487,88995],{"stretchy":73},[49,107489,88995],{"stretchy":73},[49,107491,433],{},[5654,107493,10749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= \\frac{(2n+2)!}{(2n+2)!!} = \\frac{(2n+2)!}{2^{n+1} (n+1)!}",[18,107578,107580,107604,107625,107744],{"className":107579,"ariaHidden":51},[93],[18,107581,107583,107586,107589,107592,107595,107598,107601],{"className":107582},[97],[18,107584],{"className":107585,"style":242},[101],[18,107587,74],{"className":107588},[250],[18,107590,2266],{"className":107591},[106],[18,107593,64],{"className":107594},[106,110],[18,107596],{"className":107597,"style":2068},[169],[18,107599,7987],{"className":107600},[2072],[18,107602],{"className":107603,"style":2068},[169],[18,107605,107607,107610,107613,107616,107619,107622],{"className":107606},[97],[18,107608],{"className":107609,"style":242},[101],[18,107611,47],{"className":107612},[106],[18,107614,89457],{"className":107615},[267],[18,107617],{"className":107618,"style":228},[169],[18,107620,433],{"className":107621},[232],[18,107623],{"className":107624,"style":228},[169],[18,107626,107628,107631,107735,107738,107741],{"className":107627},[97],[18,107629],{"className":107630,"style":89418},[101],[18,107632,107634,107637,107732],{"className":107633},[106],[18,107635],{"className":107636},[250,6012],[18,107638,107640],{"className":107639},[5654],[18,107641,107643,107724],{"className":107642},[119,120],[18,107644,107646,107721],{"className":107645},[124],[18,107647,107649,107681,107689],{"className":107648,"style":18133},[128],[18,107650,107651,107654],{"style":6028},[18,107652],{"className":107653,"style":697},[136],[18,107655,107657,107660,107663,107666,107669,107672,107675,107678],{"className":107656},[106],[18,107658,74],{"className":107659},[250],[18,107661,2266],{"className":107662},[106],[18,107664,64],{"className":107665},[106,110],[18,107667],{"className":107668,"style":2068},[169],[18,107670,7987],{"className":107671},[2072],[18,107673],{"className":107674,"style":2068},[169],[18,107676,2266],{"className":107677},[106],[18,107679,89457],{"className":107680},[267],[18,107682,107683,107686],{"style":6040},[18,107684],{"className":107685,"style":697},[136],[18,107687],{"className":107688,"style":6048},[6047],[18,107690,107691,107694],{"style":6051},[18,107692],{"className":107693,"style":697},[136],[18,107695,107697,107700,107703,107706,107709,107712,107715,107718],{"className":107696},[106],[18,107698,74],{"className":107699},[250],[18,107701,2266],{"className":107702},[106],[18,107704,64],{"className":107705},[106,110],[18,107707],{"className":107708,"style":2068},[169],[18,107710,7987],{"className":107711},[2072],[18,107713],{"className":107714,"style":2068},[169],[18,107716,2266],{"className":107717},[106],[18,107719,94271],{"className":107720},[267],[18,107722,152],{"className":107723},[151],[18,107725,107727],{"className":107726},[124],[18,107728,107730],{"className":107729,"style":89507},[128],[18,107731],{},[18,107733],{"className":107734},[267,6012],[18,107736],{"className":107737,"style":228},[169],[18,107739,433],{"className":107740},[232],[18,107742],{"className":107743,"style":228},[169],[18,107745,107747,107750],{"className":107746},[97],[18,107748],{"className":107749,"style":89418},[101],[18,107751,107753,107756,107886],{"className":107752},[106],[18,107754],{"className":107755},[250,6012],[18,107757,107759],{"className":107758},[5654],[18,107760,107762,107878],{"className":107761},[119,120],[18,107763,107765,107875],{"className":107764},[124],[18,107766,107768,107835,107843],{"className":107767,"style":18133},[128],[18,107769,107770,107773],{"style":6028},[18,107771],{"className":107772,"style":697},[136],[18,107774,107776,107814,107817,107820,107823,107826,107829,107832],{"className":107775},[106],[18,107777,107779,107782],{"className":107778},[106],[18,107780,2266],{"className":107781},[106],[18,107783,107785],{"className":107784},[115],[18,107786,107788],{"className":107787},[119],[18,107789,107791],{"className":107790},[124],[18,107792,107794],{"className":107793,"style":26109},[128],[18,107795,107796,107799],{"style":26112},[18,107797],{"className":107798,"style":137},[136],[18,107800,107802],{"className":107801},[141,142,143,144],[18,107803,107805,107808,107811],{"className":107804},[106,144],[18,107806,64],{"className":107807},[106,110,144],[18,107809,7987],{"className":107810},[2072,144],[18,107812,47],{"className":107813},[106,144],[18,107815,74],{"className":107816},[250],[18,107818,64],{"className":107819},[106,110],[18,107821],{"className":107822,"style":2068},[169],[18,107824,7987],{"className":107825},[2072],[18,107827],{"className":107828,"style":2068},[169],[18,107830,47],{"className":107831},[106],[18,107833,94271],{"className":107834},[267],[18,107836,107837,107840],{"style":6040},[18,107838],{"className":107839,"style":697},[136],[18,107841],{"className":107842,"style":6048},[6047],[18,107844,107845,107848],{"style":6051},[18,107846],{"className":107847,"style":697},[136],[18,107849,107851,107854,107857,107860,107863,107866,107869,107872],{"className":107850},[106],[18,107852,74],{"className":107853},[250],[18,107855,2266],{"className":107856},[106],[18,107858,64],{"className":107859},[106,110],[18,107861],{"className":107862,"style":2068},[169],[18,107864,7987],{"className":107865},[2072],[18,107867],{"className":107868,"style":2068},[169],[18,107870,2266],{"className":107871},[106],[18,107873,94271],{"className":107874},[267],[18,107876,152],{"className":107877},[151],[18,107879,107881],{"className":107880},[124],[18,107882,107884],{"className":107883,"style":89507},[128],[18,107885],{},[18,107887],{"className":107888},[267,6012],[11,107890,107891],{},"于是沃利斯公式变为：",[18,107893,107895],{"className":107894,"translate":22},[875],[18,107896,107898,107995],{"className":107897,"translate":22},[21],[18,107899,107901],{"className":107900},[26],[28,107902,107903],{"xmlns":30,"display":884},[32,107904,107905,107992],{},[35,107906,107907,107923,107984,107986],{},[890,107908,107909,107915],{},[35,107910,107911,107913],{},[41,107912,91892],{},[49,107914,899],{},[35,107916,107917,107919,107921],{},[41,107918,64],{},[49,107920,1880],{},[41,107922,2028],{"mathvariant":316},[5654,107924,107925,107958],{},[35,107926,107927,107934,107936,107938,107940,107946,107948,107950,107952,107954,107956],{},[1556,107928,107929,107932],{},[45,107930,107931],{},"16",[41,107933,64],{},[49,107935,74],{"stretchy":73},[41,107937,64],{},[49,107939,88995],{"stretchy":73},[1556,107941,107942,107944],{},[49,107943,84],{"stretchy":73},[45,107945,10447],{},[49,107947,74],{"stretchy":73},[45,107949,2266],{},[41,107951,64],{},[49,107953,7987],{},[45,107955,2266],{},[49,107957,84],{"stretchy":73},[35,107959,107960,107962,107964,107966,107968,107970,107972,107974,107976,107978,107980,107982],{},[49,107961,74],{"stretchy":73},[45,107963,2266],{},[41,107965,64],{},[49,107967,84],{"stretchy":73},[49,107969,88995],{"stretchy":73},[49,107971,74],{"stretchy":73},[45,107973,2266],{},[41,107975,64],{},[49,107977,7987],{},[45,107979,2266],{},[49,107981,84],{"stretchy":73},[49,107983,88995],{"stretchy":73},[49,107985,433],{},[5654,107987,107988,107990],{},[41,107989,3941],{},[45,107991,2266],{},[86,107993,107994],{"encoding":88},"\\lim_{n \\rightarrow \\infty} \\frac{16^n (n!)^4 (2n+2)}{(2n)!(2n+2)!} = \\frac{\\pi}{2}",[18,107996,107998,108255],{"className":107997,"ariaHidden":51},[93],[18,107999,108001,108004,108057,108060,108246,108249,108252],{"className":108000},[97],[18,108002],{"className":108003,"style":94168},[101],[18,108005,108007],{"className":108006},[985,986],[18,108008,108010,108049],{"className":108009},[119,120],[18,108011,108013,108046],{"className":108012},[124],[18,108014,108016,108036],{"className":108015,"style":295},[128],[18,108017,108018,108021],{"style":97028},[18,108019],{"className":108020,"style":697},[136],[18,108022,108024],{"className":108023},[141,142,143,144],[18,108025,108027,108030,108033],{"className":108026},[106,144],[18,108028,64],{"className":108029},[106,110,144],[18,108031,1880],{"className":108032},[232,144],[18,108034,2028],{"className":108035},[106,144],[18,108037,108038,108041],{"style":693},[18,108039],{"className":108040,"style":697},[136],[18,108042,108043],{},[18,108044,91892],{"className":108045},[985],[18,108047,152],{"className":108048},[151],[18,108050,108052],{"className":108051},[124],[18,108053,108055],{"className":108054,"style":97066},[128],[18,108056],{},[18,108058],{"className":108059,"style":170},[169],[18,108061,108063,108066,108243],{"className":108062},[106],[18,108064],{"className":108065},[250,6012],[18,108067,108069],{"className":108068},[5654],[18,108070,108072,108235],{"className":108071},[119,120],[18,108073,108075,108232],{"className":108074},[124],[18,108076,108078,108122,108130],{"className":108077,"style":15999},[128],[18,108079,108080,108083],{"style":6028},[18,108081],{"className":108082,"style":697},[136],[18,108084,108086,108089,108092,108095,108098,108101,108104,108107,108110,108113,108116,108119],{"className":108085},[106],[18,108087,74],{"className":108088},[250],[18,108090,2266],{"className":108091},[106],[18,108093,64],{"className":108094},[106,110],[18,108096,94271],{"className":108097},[267],[18,108099,74],{"className":108100},[250],[18,108102,2266],{"className":108103},[106],[18,108105,64],{"className":108106},[106,110],[18,108108],{"className":108109,"style":2068},[169],[18,108111,7987],{"className":108112},[2072],[18,108114],{"className":108115,"style":2068},[169],[18,108117,2266],{"className":108118},[106],[18,108120,94271],{"className":108121},[267],[18,108123,108124,108127],{"style":6040},[18,108125],{"className":108126,"style":697},[136],[18,108128],{"className":108129,"style":6048},[6047],[18,108131,108132,108135],{"style":6051},[18,108133],{"className":108134,"style":697},[136],[18,108136,108138,108141,108170,108173,108176,108179,108208,108211,108214,108217,108220,108223,108226,108229],{"className":108137},[106],[18,108139,47],{"className":108140},[106],[18,108142,108144,108147],{"className":108143},[106],[18,108145,77487],{"className":108146},[106],[18,108148,108150],{"className":108149},[115],[18,108151,108153],{"className":108152},[119],[18,108154,108156],{"className":108155},[124],[18,108157,108159],{"className":108158,"style":1959},[128],[18,108160,108161,108164],{"style":1962},[18,108162],{"className":108163,"style":137},[136],[18,108165,108167],{"className":108166},[141,142,143,144],[18,108168,64],{"className":108169},[106,110,144],[18,108171,74],{"className":108172},[250],[18,108174,64],{"className":108175},[106,110],[18,108177,88995],{"className":108178},[267],[18,108180,108182,108185],{"className":108181},[267],[18,108183,84],{"className":108184},[267],[18,108186,108188],{"className":108187},[115],[18,108189,108191],{"className":108190},[119],[18,108192,108194],{"className":108193},[124],[18,108195,108197],{"className":108196,"style":2416},[128],[18,108198,108199,108202],{"style":1962},[18,108200],{"className":108201,"style":137},[136],[18,108203,108205],{"className":108204},[141,142,143,144],[18,108206,10447],{"className":108207},[106,144],[18,108209,74],{"className":108210},[250],[18,108212,2266],{"className":108213},[106],[18,108215,64],{"className":108216},[106,110],[18,108218],{"className":108219,"style":2068},[169],[18,108221,7987],{"className":108222},[2072],[18,108224],{"className":108225,"style":2068},[169],[18,108227,2266],{"className":108228},[106],[18,108230,84],{"className":108231},[267],[18,108233,152],{"className":108234},[151],[18,108236,108238],{"className":108237},[124],[18,108239,108241],{"className":108240,"style":89507},[128],[18,108242],{},[18,108244],{"className":108245},[267,6012],[18,108247],{"className":108248,"style":228},[169],[18,108250,433],{"className":108251},[232],[18,108253],{"className":108254,"style":228},[169],[18,108256,108258,108261],{"className":108257},[97],[18,108259],{"className":108260,"style":42071},[101],[18,108262,108264,108267,108320],{"className":108263},[106],[18,108265],{"className":108266},[250,6012],[18,108268,108270],{"className":108269},[5654],[18,108271,108273,108312],{"className":108272},[119,120],[18,108274,108276,108309],{"className":108275},[124],[18,108277,108279,108290,108298],{"className":108278,"style":13852},[128],[18,108280,108281,108284],{"style":6028},[18,108282],{"className":108283,"style":697},[136],[18,108285,108287],{"className":108286},[106],[18,108288,2266],{"className":108289},[106],[18,108291,108292,108295],{"style":6040},[18,108293],{"className":108294,"style":697},[136],[18,108296],{"className":108297,"style":6048},[6047],[18,108299,108300,108303],{"style":6051},[18,108301],{"className":108302,"style":697},[136],[18,108304,108306],{"className":108305},[106],[18,108307,3941],{"className":108308,"style":3957},[106,110],[18,108310,152],{"className":108311},[151],[18,108313,108315],{"className":108314},[124],[18,108316,108318],{"className":108317,"style":6070},[128],[18,108319],{},[18,108321],{"className":108322},[267,6012],[11,108324,108325],{},"利用：",[18,108327,108329],{"className":108328,"translate":22},[875],[18,108330,108332,108378],{"className":108331,"translate":22},[21],[18,108333,108335],{"className":108334},[26],[28,108336,108337],{"xmlns":30,"display":884},[32,108338,108339,108375],{},[35,108340,108341,108343,108345,108347,108349,108359],{},[41,108342,64],{},[49,108344,88995],{"stretchy":73},[49,108346,67],{},[41,108348,16823],{},[1556,108350,108351,108353],{},[41,108352,14444],{},[35,108354,108355,108357],{},[49,108356,2253],{},[41,108358,64],{},[1556,108360,108361,108363],{},[41,108362,64],{},[35,108364,108365,108367,108369],{},[41,108366,64],{},[49,108368,7987],{},[5654,108370,108371,108373],{},[45,108372,47],{},[45,108374,2266],{},[86,108376,108377],{"encoding":88},"n! \\sim a e^{-n} n^{n + \\frac{1}{2}}",[18,108379,108381,108402],{"className":108380,"ariaHidden":51},[93],[18,108382,108384,108387,108390,108393,108396,108399],{"className":108383},[97],[18,108385],{"className":108386,"style":295},[101],[18,108388,64],{"className":108389},[106,110],[18,108391,88995],{"className":108392},[267],[18,108394],{"className":108395,"style":228},[169],[18,108397,67],{"className":108398},[232],[18,108400],{"className":108401,"style":228},[169],[18,108403,108405,108409,108412,108447],{"className":108404},[97],[18,108406],{"className":108407,"style":108408},[101],"height:1.004em;",[18,108410,16823],{"className":108411},[106,110],[18,108413,108415,108418],{"className":108414},[106],[18,108416,14444],{"className":108417},[106,110],[18,108419,108421],{"className":108420},[115],[18,108422,108424],{"className":108423},[119],[18,108425,108427],{"className":108426},[124],[18,108428,108430],{"className":108429,"style":1800},[128],[18,108431,108432,108435],{"style":1637},[18,108433],{"className":108434,"style":137},[136],[18,108436,108438],{"className":108437},[141,142,143,144],[18,108439,108441,108444],{"className":108440},[106,144],[18,108442,2253],{"className":108443},[106,144],[18,108445,64],{"className":108446},[106,110,144],[18,108448,108450,108453],{"className":108449},[106],[18,108451,64],{"className":108452},[106,110],[18,108454,108456],{"className":108455},[115],[18,108457,108459],{"className":108458},[119],[18,108460,108462],{"className":108461},[124],[18,108463,108465],{"className":108464,"style":108408},[128],[18,108466,108467,108470],{"style":97830},[18,108468],{"className":108469,"style":697},[136],[18,108471,108473],{"className":108472},[141,142,143,144],[18,108474,108476,108479,108482],{"className":108475},[106,144],[18,108477,64],{"className":108478},[106,110,144],[18,108480,7987],{"className":108481},[2072,144],[18,108483,108485,108488,108547],{"className":108484},[106,144],[18,108486],{"className":108487},[250,6012,141,1195,38156],[18,108489,108491],{"className":108490},[5654],[18,108492,108494,108539],{"className":108493},[119,120],[18,108495,108497,108536],{"className":108496},[124],[18,108498,108500,108514,108522],{"className":108499,"style":42938},[128],[18,108501,108502,108505],{"style":38172},[18,108503],{"className":108504,"style":697},[136],[18,108506,108508],{"className":108507},[141,1195,1196,144],[18,108509,108511],{"className":108510},[106,144],[18,108512,2266],{"className":108513},[106,144],[18,108515,108516,108519],{"style":38196},[18,108517],{"className":108518,"style":697},[136],[18,108520],{"className":108521,"style":38203},[6047,144],[18,108523,108524,108527],{"style":42972},[18,108525],{"className":108526,"style":697},[136],[18,108528,108530],{"className":108529},[141,1195,1196,144],[18,108531,108533],{"className":108532},[106,144],[18,108534,47],{"className":108535},[106,144],[18,108537,152],{"className":108538},[151],[18,108540,108542],{"className":108541},[124],[18,108543,108545],{"className":108544,"style":81887},[128],[18,108546],{},[18,108548],{"className":108549},[267,6012,141,1195,38156],[11,108551,108552],{},"代入：",[18,108554,108556],{"className":108555,"translate":22},[875],[18,108557,108559,108649],{"className":108558,"translate":22},[21],[18,108560,108562],{"className":108561},[26],[28,108563,108564],{"xmlns":30,"display":884},[32,108565,108566,108646],{},[35,108567,108568,108584,108638,108640],{},[890,108569,108570,108576],{},[35,108571,108572,108574],{},[41,108573,91892],{},[49,108575,899],{},[35,108577,108578,108580,108582],{},[41,108579,64],{},[49,108581,1880],{},[41,108583,2028],{"mathvariant":316},[5654,108585,108586,108600],{},[35,108587,108588,108594],{},[1556,108589,108590,108592],{},[41,108591,16823],{},[45,108593,2266],{},[1556,108595,108596,108598],{},[41,108597,14444],{},[45,108599,2266],{},[35,108601,108602,108604],{},[45,108603,10447],{},[1556,108605,108606,108624],{},[35,108607,108608,108610,108622],{},[49,108609,74],{"fence":51},[5654,108611,108612,108620],{},[35,108613,108614,108616,108618],{},[41,108615,64],{},[49,108617,7987],{},[45,108619,47],{},[41,108621,64],{},[49,108623,84],{"fence":51},[35,108625,108626,108628,108630,108632],{},[45,108627,2266],{},[41,108629,64],{},[49,108631,7987],{},[5654,108633,108634,108636],{},[45,108635,44708],{},[45,108637,2266],{},[49,108639,433],{},[5654,108641,108642,108644],{},[41,108643,3941],{},[45,108645,2266],{},[86,108647,108648],{"encoding":88},"\\lim_{n \\rightarrow \\infty} \\frac{a^2 e^2}{4 \\left(\\frac{n+1}{n}\\right)^{2n + \\frac{3}{2}}} = \\frac{\\pi}{2}",[18,108650,108652,109036],{"className":108651,"ariaHidden":51},[93],[18,108653,108655,108659,108712,108715,109027,109030,109033],{"className":108654},[97],[18,108656],{"className":108657,"style":108658},[101],"height:2.925em;vertical-align:-1.4339em;",[18,108660,108662],{"className":108661},[985,986],[18,108663,108665,108704],{"className":108664},[119,120],[18,108666,108668,108701],{"className":108667},[124],[18,108669,108671,108691],{"className":108670,"style":295},[128],[18,108672,108673,108676],{"style":97028},[18,108674],{"className":108675,"style":697},[136],[18,108677,108679],{"className":108678},[141,142,143,144],[18,108680,108682,108685,108688],{"className":108681},[106,144],[18,108683,64],{"className":108684},[106,110,144],[18,108686,1880],{"className":108687},[232,144],[18,108689,2028],{"className":108690},[106,144],[18,108692,108693,108696],{"style":693},[18,108694],{"className":108695,"style":697},[136],[18,108697,108698],{},[18,108699,91892],{"className":108700},[985],[18,108702,152],{"className":108703},[151],[18,108705,108707],{"className":108706},[124],[18,108708,108710],{"className":108709,"style":97066},[128],[18,108711],{},[18,108713],{"className":108714,"style":170},[169],[18,108716,108718,108721,109024],{"className":108717},[106],[18,108719],{"className":108720},[250,6012],[18,108722,108724],{"className":108723},[5654],[18,108725,108727,109016],{"className":108726},[119,120],[18,108728,108730,109013],{"className":108729},[124],[18,108731,108733,108939,108947],{"className":108732,"style":15999},[128],[18,108734,108735,108738],{"style":102604},[18,108736],{"className":108737,"style":102608},[136],[18,108739,108741,108744,108747],{"className":108740},[106],[18,108742,10447],{"className":108743},[106],[18,108745],{"className":108746,"style":170},[169],[18,108748,108750,108839],{"className":108749},[174],[18,108751,108753,108759,108833],{"className":108752},[174],[18,108754,108756],{"className":108755,"style":13796},[250,13795],[18,108757,74],{"className":108758},[1230,1196],[18,108760,108762,108765,108830],{"className":108761},[106],[18,108763],{"className":108764},[250,6012],[18,108766,108768],{"className":108767},[5654],[18,108769,108771,108822],{"className":108770},[119,120],[18,108772,108774,108819],{"className":108773},[124],[18,108775,108777,108791,108799],{"className":108776,"style":12045},[128],[18,108778,108779,108782],{"style":12048},[18,108780],{"className":108781,"style":697},[136],[18,108783,108785],{"className":108784},[141,142,143,144],[18,108786,108788],{"className":108787},[106,144],[18,108789,64],{"className":108790},[106,110,144],[18,108792,108793,108796],{"style":6040},[18,108794],{"className":108795,"style":697},[136],[18,108797],{"className":108798,"style":6048},[6047],[18,108800,108801,108804],{"style":12071},[18,108802],{"className":108803,"style":697},[136],[18,108805,108807],{"className":108806},[141,142,143,144],[18,108808,108810,108813,108816],{"className":108809},[106,144],[18,108811,64],{"className":108812},[106,110,144],[18,108814,7987],{"className":108815},[2072,144],[18,108817,47],{"className":108818},[106,144],[18,108820,152],{"className":108821},[151],[18,108823,108825],{"className":108824},[124],[18,108826,108828],{"className":108827,"style":12093},[128],[18,108829],{},[18,108831],{"className":108832},[267,6012],[18,108834,108836],{"className":108835,"style":13796},[267,13795],[18,108837,84],{"className":108838},[1230,1196],[18,108840,108842],{"className":108841},[115],[18,108843,108845],{"className":108844},[119],[18,108846,108848],{"className":108847},[124],[18,108849,108851],{"className":108850,"style":102722},[128],[18,108852,108853,108856],{"style":69847},[18,108854],{"className":108855,"style":697},[136],[18,108857,108859],{"className":108858},[141,142,143,144],[18,108860,108862,108865,108868,108871],{"className":108861},[106,144],[18,108863,2266],{"className":108864},[106,144],[18,108866,64],{"className":108867},[106,110,144],[18,108869,7987],{"className":108870},[2072,144],[18,108872,108874,108877,108936],{"className":108873},[106,144],[18,108875],{"className":108876},[250,6012,141,1195,38156],[18,108878,108880],{"className":108879},[5654],[18,108881,108883,108928],{"className":108882},[119,120],[18,108884,108886,108925],{"className":108885},[124],[18,108887,108889,108903,108911],{"className":108888,"style":42938},[128],[18,108890,108891,108894],{"style":38172},[18,108892],{"className":108893,"style":697},[136],[18,108895,108897],{"className":108896},[141,1195,1196,144],[18,108898,108900],{"className":108899},[106,144],[18,108901,2266],{"className":108902},[106,144],[18,108904,108905,108908],{"style":38196},[18,108906],{"className":108907,"style":697},[136],[18,108909],{"className":108910,"style":38203},[6047,144],[18,108912,108913,108916],{"style":42972},[18,108914],{"className":108915,"style":697},[136],[18,108917,108919],{"className":108918},[141,1195,1196,144],[18,108920,108922],{"className":108921},[106,144],[18,108923,44708],{"className":108924},[106,144],[18,108926,152],{"className":108927},[151],[18,108929,108931],{"className":108930},[124],[18,108932,108934],{"className":108933,"style":81887},[128],[18,108935],{},[18,108937],{"className":108938},[267,6012,141,1195,38156],[18,108940,108941,108944],{"style":102810},[18,108942],{"className":108943,"style":102608},[136],[18,108945],{"className":108946,"style":6048},[6047],[18,108948,108949,108952],{"style":102819},[18,108950],{"className":108951,"style":102608},[136],[18,108953,108955,108984],{"className":108954},[106],[18,108956,108958,108961],{"className":108957},[106],[18,108959,16823],{"className":108960},[106,110],[18,108962,108964],{"className":108963},[115],[18,108965,108967],{"className":108966},[119],[18,108968,108970],{"className":108969},[124],[18,108971,108973],{"className":108972,"style":2416},[128],[18,108974,108975,108978],{"style":1962},[18,108976],{"className":108977,"style":137},[136],[18,108979,108981],{"className":108980},[141,142,143,144],[18,108982,2266],{"className":108983},[106,144],[18,108985,108987,108990],{"className":108986},[106],[18,108988,14444],{"className":108989},[106,110],[18,108991,108993],{"className":108992},[115],[18,108994,108996],{"className":108995},[119],[18,108997,108999],{"className":108998},[124],[18,109000,109002],{"className":109001,"style":2416},[128],[18,109003,109004,109007],{"style":1962},[18,109005],{"className":109006,"style":137},[136],[18,109008,109010],{"className":109009},[141,142,143,144],[18,109011,2266],{"className":109012},[106,144],[18,109014,152],{"className":109015},[151],[18,109017,109019],{"className":109018},[124],[18,109020,109022],{"className":109021,"style":102838},[128],[18,109023],{},[18,109025],{"className":109026},[267,6012],[18,109028],{"className":109029,"style":228},[169],[18,109031,433],{"className":109032},[232],[18,109034],{"className":109035,"style":228},[169],[18,109037,109039,109042],{"className":109038},[97],[18,109040],{"className":109041,"style":42071},[101],[18,109043,109045,109048,109101],{"className":109044},[106],[18,109046],{"className":109047},[250,6012],[18,109049,109051],{"className":109050},[5654],[18,109052,109054,109093],{"className":109053},[119,120],[18,109055,109057,109090],{"className":109056},[124],[18,109058,109060,109071,109079],{"className":109059,"style":13852},[128],[18,109061,109062,109065],{"style":6028},[18,109063],{"className":109064,"style":697},[136],[18,109066,109068],{"className":109067},[106],[18,109069,2266],{"className":109070},[106],[18,109072,109073,109076],{"style":6040},[18,109074],{"className":109075,"style":697},[136],[18,109077],{"className":109078,"style":6048},[6047],[18,109080,109081,109084],{"style":6051},[18,109082],{"className":109083,"style":697},[136],[18,109085,109087],{"className":109086},[106],[18,109088,3941],{"className":109089,"style":3957},[106,110],[18,109091,152],{"className":109092},[151],[18,109094,109096],{"className":109095},[124],[18,109097,109099],{"className":109098,"style":6070},[128],[18,109100],{},[18,109102],{"className":109103},[267,6012],[11,109105,109106],{},"解得：",[18,109108,109110],{"className":109109,"translate":22},[875],[18,109111,109113,109137],{"className":109112,"translate":22},[21],[18,109114,109116],{"className":109115},[26],[28,109117,109118],{"xmlns":30,"display":884},[32,109119,109120,109134],{},[35,109121,109122,109124,109126],{},[41,109123,16823],{},[49,109125,433],{},[12561,109127,109128],{},[35,109129,109130,109132],{},[45,109131,2266],{},[41,109133,3941],{},[86,109135,109136],{"encoding":88},"a = \\sqrt{2 \\pi}",[18,109138,109140,109158],{"className":109139,"ariaHidden":51},[93],[18,109141,109143,109146,109149,109152,109155],{"className":109142},[97],[18,109144],{"className":109145,"style":1083},[101],[18,109147,16823],{"className":109148},[106,110],[18,109150],{"className":109151,"style":228},[169],[18,109153,433],{"className":109154},[232],[18,109156],{"className":109157,"style":228},[169],[18,109159,109161,109165],{"className":109160},[97],[18,109162],{"className":109163,"style":109164},[101],"height:1.04em;vertical-align:-0.0839em;",[18,109166,109168],{"className":109167},[106,12696],[18,109169,109171,109209],{"className":109170},[119,120],[18,109172,109174,109206],{"className":109173},[124],[18,109175,109178,109193],{"className":109176,"style":109177},[128],"height:0.9561em;",[18,109179,109181,109184],{"className":109180,"style":693},[12710],[18,109182],{"className":109183,"style":697},[136],[18,109185,109187,109190],{"className":109186,"style":15410},[106],[18,109188,2266],{"className":109189},[106],[18,109191,3941],{"className":109192,"style":3957},[106,110],[18,109194,109196,109199],{"style":109195},"top:-2.9161em;",[18,109197],{"className":109198,"style":697},[136],[18,109200,109202],{"className":109201,"style":15420},[12853],[6130,109203,109204],{"xmlns":6132,"width":12857,"height":15423,"viewBox":15424,"preserveAspectRatio":12860},[6136,109205],{"d":15427},[18,109207,152],{"className":109208},[151],[18,109210,109212],{"className":109211},[124],[18,109213,109216],{"className":109214,"style":109215},[128],"height:0.0839em;",[18,109217],{},[11,109219,109220,109221,109249],{},"因此，当 ",[18,109222,109224,109237],{"className":109223,"translate":22},[21],[18,109225,109227],{"className":109226},[26],[28,109228,109229],{"xmlns":30},[32,109230,109231,109235],{},[35,109232,109233],{},[41,109234,64],{},[86,109236,64],{"encoding":88},[18,109238,109240],{"className":109239,"ariaHidden":51},[93],[18,109241,109243,109246],{"className":109242},[97],[18,109244],{"className":109245,"style":1083},[101],[18,109247,64],{"className":109248},[106,110]," 趋近无穷大时：",[18,109251,109253],{"className":109252,"translate":22},[875],[18,109254,109256,109308],{"className":109255,"translate":22},[21],[18,109257,109259],{"className":109258},[26],[28,109260,109261],{"xmlns":30,"display":884},[32,109262,109263,109305],{},[35,109264,109265,109267,109269,109271,109279,109289],{},[41,109266,64],{},[49,109268,88995],{"stretchy":73},[49,109270,67],{},[12561,109272,109273],{},[35,109274,109275,109277],{},[45,109276,2266],{},[41,109278,3941],{},[1556,109280,109281,109283],{},[41,109282,14444],{},[35,109284,109285,109287],{},[49,109286,2253],{},[41,109288,64],{},[1556,109290,109291,109293],{},[41,109292,64],{},[35,109294,109295,109297,109299],{},[41,109296,64],{},[49,109298,7987],{},[5654,109300,109301,109303],{},[45,109302,47],{},[45,109304,2266],{},[86,109306,109307],{"encoding":88},"n! \\sim \\sqrt{2 \\pi} e^{-n} n^{n + \\frac{1}{2}}",[18,109309,109311,109332],{"className":109310,"ariaHidden":51},[93],[18,109312,109314,109317,109320,109323,109326,109329],{"className":109313},[97],[18,109315],{"className":109316,"style":295},[101],[18,109318,64],{"className":109319},[106,110],[18,109321,88995],{"className":109322},[267],[18,109324],{"className":109325,"style":228},[169],[18,109327,67],{"className":109328},[232],[18,109330],{"className":109331,"style":228},[169],[18,109333,109335,109339,109389,109424],{"className":109334},[97],[18,109336],{"className":109337,"style":109338},[101],"height:1.0879em;vertical-align:-0.0839em;",[18,109340,109342],{"className":109341},[106,12696],[18,109343,109345,109381],{"className":109344},[119,120],[18,109346,109348,109378],{"className":109347},[124],[18,109349,109351,109366],{"className":109350,"style":109177},[128],[18,109352,109354,109357],{"className":109353,"style":693},[12710],[18,109355],{"className":109356,"style":697},[136],[18,109358,109360,109363],{"className":109359,"style":15410},[106],[18,109361,2266],{"className":109362},[106],[18,109364,3941],{"className":109365,"style":3957},[106,110],[18,109367,109368,109371],{"style":109195},[18,109369],{"className":109370,"style":697},[136],[18,109372,109374],{"className":109373,"style":15420},[12853],[6130,109375,109376],{"xmlns":6132,"width":12857,"height":15423,"viewBox":15424,"preserveAspectRatio":12860},[6136,109377],{"d":15427},[18,109379,152],{"className":109380},[151],[18,109382,109384],{"className":109383},[124],[18,109385,109387],{"className":109386,"style":109215},[128],[18,109388],{},[18,109390,109392,109395],{"className":109391},[106],[18,109393,14444],{"className":109394},[106,110],[18,109396,109398],{"className":109397},[115],[18,109399,109401],{"className":109400},[119],[18,109402,109404],{"className":109403},[124],[18,109405,109407],{"className":109406,"style":1800},[128],[18,109408,109409,109412],{"style":1637},[18,109410],{"className":109411,"style":137},[136],[18,109413,109415],{"className":109414},[141,142,143,144],[18,109416,109418,109421],{"className":109417},[106,144],[18,109419,2253],{"className":109420},[106,144],[18,109422,64],{"className":109423},[106,110,144],[18,109425,109427,109430],{"className":109426},[106],[18,109428,64],{"className":109429},[106,110],[18,109431,109433],{"className":109432},[115],[18,109434,109436],{"className":109435},[119],[18,109437,109439],{"className":109438},[124],[18,109440,109442],{"className":109441,"style":108408},[128],[18,109443,109444,109447],{"style":97830},[18,109445],{"className":109446,"style":697},[136],[18,109448,109450],{"className":109449},[141,142,143,144],[18,109451,109453,109456,109459],{"className":109452},[106,144],[18,109454,64],{"className":109455},[106,110,144],[18,109457,7987],{"className":109458},[2072,144],[18,109460,109462,109465,109524],{"className":109461},[106,144],[18,109463],{"className":109464},[250,6012,141,1195,38156],[18,109466,109468],{"className":109467},[5654],[18,109469,109471,109516],{"className":109470},[119,120],[18,109472,109474,109513],{"className":109473},[124],[18,109475,109477,109491,109499],{"className":109476,"style":42938},[128],[18,109478,109479,109482],{"style":38172},[18,109480],{"className":109481,"style":697},[136],[18,109483,109485],{"className":109484},[141,1195,1196,144],[18,109486,109488],{"className":109487},[106,144],[18,109489,2266],{"className":109490},[106,144],[18,109492,109493,109496],{"style":38196},[18,109494],{"className":109495,"style":697},[136],[18,109497],{"className":109498,"style":38203},[6047,144],[18,109500,109501,109504],{"style":42972},[18,109502],{"className":109503,"style":697},[136],[18,109505,109507],{"className":109506},[141,1195,1196,144],[18,109508,109510],{"className":109509},[106,144],[18,109511,47],{"className":109512},[106,144],[18,109514,152],{"className":109515},[151],[18,109517,109519],{"className":109518},[124],[18,109520,109522],{"className":109521,"style":81887},[128],[18,109523],{},[18,109525],{"className":109526},[267,6012,141,1195,38156],[11,109528,109529],{},"或者等价地：",[18,109531,109533],{"className":109532,"translate":22},[875],[18,109534,109536,109580],{"className":109535,"translate":22},[21],[18,109537,109539],{"className":109538},[26],[28,109540,109541],{"xmlns":30,"display":884},[32,109542,109543,109577],{},[35,109544,109545,109547,109549,109551,109561],{},[41,109546,64],{},[49,109548,88995],{"stretchy":73},[49,109550,67],{},[12561,109552,109553],{},[35,109554,109555,109557,109559],{},[45,109556,2266],{},[41,109558,3941],{},[41,109560,64],{},[1556,109562,109563,109575],{},[35,109564,109565,109567,109573],{},[49,109566,74],{"fence":51},[5654,109568,109569,109571],{},[41,109570,64],{},[41,109572,14444],{},[49,109574,84],{"fence":51},[41,109576,64],{},[86,109578,109579],{"encoding":88},"n! \\sim \\sqrt{2 \\pi n} 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n 趋向无穷大时 n!、nⁿ 与 eⁿ 之间的关系。",[45596],"2025-09-18T00:00:00+08:00","-Zl1Q4Ie-HEQw2nACoF1zCHXCl2aL6v0T6tWoExy47I",1792493793641]