[{"data":1,"prerenderedAt":109762},["ShallowReactive",2],{"post-\u002Fblog\u002F2025\u002F2025-09-27-fano-bu-deng-shi-yu-le-cam-fang-fa":3,"i-ci:tag":45580,"i-mingcute:time-line":45580},{"post":4,"nextPost":45591,"prevPost":81686},{"id":5,"title":6,"body":7,"description":13,"draft":45577,"enableCodeHighlight":45577,"enableCodeHightlight":45577,"enableComment":45578,"enableD2":45577,"enableKatex":45578,"extension":45579,"image":45543,"important":45577,"location":45580,"meta":45581,"navigation":45578,"ogImage":45580,"path":45582,"seo":45583,"stem":45584,"summary":45585,"tags":45586,"time":45589,"__hash__":45590},"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":45542},"minimark",[10,14,634,637,642,869,872,1289,1296,1303,1306,1310,1315,1540,1845,2209,2956,3052,3441,3506,3891,3895,3959,4302,4305,4490,4497,4501,4742,4981,5144,5147,5151,5155,5158,6375,6378,6582,6801,7267,7270,7934,7937,8857,8942,9059,9902,10108,10361,11099,11102,11735,12514,12518,12521,12876,12879,13316,13319,14066,14370,14409,14413,14640,15319,15660,16242,16320,16324,16356,16703,17373,17463,17467,17471,18284,18287,18703,18825,19161,19164,19538,19542,19545,21511,21514,21781,22021,22609,22725,22981,23217,23707,23739,24281,24834,25209,25212,25873,25876,26569,26775,26779,26786,27290,27384,27781,28085,28089,28093,28125,29913,30122,30608,30958,31177,31180,31688,31876,32331,32334,32338,32486,32827,33272,33275,34069,34105,34109,34186,34189,34534,34684,34752,35036,35418,35802,35805,36181,36322,36958,37064,37096,37208,37210,37733,38049,38054,38228,38235,38548,38809,39005,39307,39699,40032,40738,41068,41450,41591,41594,42547,42659,42993,43119,43415,43418,43698,43702,44748,44751,44755,44941,44944,45081,45084,45278,45281,45349,45439,45442],[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。",[5159,5160,5162,5569],"theorem",{"title":5161},"定理 3.1（Le Cam 两点方法）",[11,5163,16,5164,4741,5310,5478,5479,5508,5509,4741],{},[18,5165,5167,5197],{"className":5166,"translate":22},[21],[18,5168,5170],{"className":5169},[26],[28,5171,5172],{"xmlns":30},[32,5173,5174,5194],{},[35,5175,5176,5182,5184,5190,5192],{},[38,5177,5178,5180],{},[41,5179,77],{},[45,5181,430],{},[49,5183,52],{"separator":51},[38,5185,5186,5188],{},[41,5187,77],{},[45,5189,47],{},[49,5191,921],{},[41,5193,924],{"mathvariant":316},[86,5195,5196],{"encoding":88},"\\theta_0,\\theta_1\\in\\Theta",[18,5198,5200,5301],{"className":5199,"ariaHidden":51},[93],[18,5201,5203,5206,5246,5249,5252,5292,5295,5298],{"className":5202},[97],[18,5204],{"className":5205,"style":2890},[101],[18,5207,5209,5212],{"className":5208},[106],[18,5210,77],{"className":5211,"style":254},[106,110],[18,5213,5215],{"className":5214},[115],[18,5216,5218,5238],{"className":5217},[119,120],[18,5219,5221,5235],{"className":5220},[124],[18,5222,5224],{"className":5223,"style":129},[128],[18,5225,5226,5229],{"style":469},[18,5227],{"className":5228,"style":137},[136],[18,5230,5232],{"className":5231},[141,142,143,144],[18,5233,430],{"className":5234},[106,144],[18,5236,152],{"className":5237},[151],[18,5239,5241],{"className":5240},[124],[18,5242,5244],{"className":5243,"style":159},[128],[18,5245],{},[18,5247,52],{"className":5248},[165],[18,5250],{"className":5251,"style":170},[169],[18,5253,5255,5258],{"className":5254},[106],[18,5256,77],{"className":5257,"style":254},[106,110],[18,5259,5261],{"className":5260},[115],[18,5262,5264,5284],{"className":5263},[119,120],[18,5265,5267,5281],{"className":5266},[124],[18,5268,5270],{"className":5269,"style":129},[128],[18,5271,5272,5275],{"style":469},[18,5273],{"className":5274,"style":137},[136],[18,5276,5278],{"className":5277},[141,142,143,144],[18,5279,47],{"className":5280},[106,144],[18,5282,152],{"className":5283},[151],[18,5285,5287],{"className":5286},[124],[18,5288,5290],{"className":5289,"style":159},[128],[18,5291],{},[18,5293],{"className":5294,"style":228},[169],[18,5296,921],{"className":5297},[232],[18,5299],{"className":5300,"style":228},[169],[18,5302,5304,5307],{"className":5303},[97],[18,5305],{"className":5306,"style":600},[101],[18,5308,924],{"className":5309},[106],[18,5311,5313,5352],{"className":5312,"translate":22},[21],[18,5314,5316],{"className":5315},[26],[28,5317,5318],{"xmlns":30},[32,5319,5320,5349],{},[35,5321,5322,5324,5326,5332,5334,5340,5342,5344,5346],{},[41,5323,948],{"mathvariant":316},[49,5325,74],{"stretchy":73},[38,5327,5328,5330],{},[41,5329,77],{},[45,5331,430],{},[49,5333,52],{"separator":51},[38,5335,5336,5338],{},[41,5337,77],{},[45,5339,47],{},[49,5341,84],{"stretchy":73},[49,5343,4775],{},[45,5345,2266],{},[41,5347,5348],{},"δ",[86,5350,5351],{"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":8427},[165],[18,8429],{"className":8430,"style":170},[169],[18,8432,8434],{"className":8433},[106,681],[18,8435,8437],{"className":8436},[119],[18,8438,8440],{"className":8439},[124],[18,8441,8443,8451],{"className":8442,"style":677},[128],[18,8444,8445,8448],{"style":693},[18,8446],{"className":8447,"style":697},[136],[18,8449,77],{"className":8450,"style":254},[106,110],[18,8452,8453,8456],{"style":703},[18,8454],{"className":8455,"style":697},[136],[18,8457,8459],{"className":8458,"style":711},[710],[18,8460,664],{"className":8461},[106],[18,8463,84],{"className":8464},[267],[18,8466],{"className":8467,"style":228},[169],[18,8469,4775],{"className":8470},[232],[18,8472],{"className":8473},[169,8019],[18,8475,8477,8480,8483,8489,8492,8495,8498],{"className":8476},[97],[18,8478],{"className":8479,"style":6098},[101],[18,8481,5348],{"className":8482,"style":5477},[106,110],[18,8484,8486],{"className":8485},[250],[18,8487,945],{"className":8488},[1230,1196],[18,8490,47],{"className":8491},[106],[18,8493],{"className":8494,"style":2068},[169],[18,8496,7984],{"className":8497},[2072],[18,8499],{"className":8500,"style":2068},[169],[18,8502,8504,8507,8547,8550,8553,8556,8559],{"className":8503},[97],[18,8505],{"className":8506,"style":242},[101],[18,8508,8510,8513],{"className":8509},[106],[18,8511,939],{"className":8512,"style":1171},[106,110],[18,8514,8516],{"className":8515},[115],[18,8517,8519,8539],{"className":8518},[119,120],[18,8520,8522,8536],{"className":8521},[124],[18,8523,8525],{"className":8524,"style":129},[128],[18,8526,8527,8530],{"style":1386},[18,8528],{"className":8529,"style":137},[136],[18,8531,8533],{"className":8532},[141,142,143,144],[18,8534,430],{"className":8535},[106,144],[18,8537,152],{"className":8538},[151],[18,8540,8542],{"className":8541},[124],[18,8543,8545],{"className":8544,"style":159},[128],[18,8546],{},[18,8548,74],{"className":8549},[250],[18,8551,6395],{"className":8552,"style":3957},[106,110],[18,8554],{"className":8555,"style":228},[169],[18,8557,433],{"className":8558},[232],[18,8560],{"className":8561,"style":228},[169],[18,8563,8565,8568,8571,8574,8577,8580],{"className":8564},[97],[18,8566],{"className":8567,"style":242},[101],[18,8569,47],{"className":8570},[106],[18,8572,84],{"className":8573},[267],[18,8575],{"className":8576,"style":2068},[169],[18,8578,2253],{"className":8579},[2072],[18,8581],{"className":8582,"style":2068},[169],[18,8584,8586,8589,8629,8632,8635,8638,8641],{"className":8585},[97],[18,8587],{"className":8588,"style":242},[101],[18,8590,8592,8595],{"className":8591},[106],[18,8593,939],{"className":8594,"style":1171},[106,110],[18,8596,8598],{"className":8597},[115],[18,8599,8601,8621],{"className":8600},[119,120],[18,8602,8604,8618],{"className":8603},[124],[18,8605,8607],{"className":8606,"style":129},[128],[18,8608,8609,8612],{"style":1386},[18,8610],{"className":8611,"style":137},[136],[18,8613,8615],{"className":8614},[141,142,143,144],[18,8616,47],{"className":8617},[106,144],[18,8619,152],{"className":8620},[151],[18,8622,8624],{"className":8623},[124],[18,8625,8627],{"className":8626,"style":159},[128],[18,8628],{},[18,8630,74],{"className":8631},[250],[18,8633,6395],{"className":8634,"style":3957},[106,110],[18,8636],{"className":8637,"style":228},[169],[18,8639,433],{"className":8640},[232],[18,8642],{"className":8643,"style":228},[169],[18,8645,8647,8650,8653,8656,8662,8665],{"className":8646},[97],[18,8648],{"className":8649,"style":6098},[101],[18,8651,47],{"className":8652},[106],[18,8654,84],{"className":8655},[267],[18,8657,8659],{"className":8658},[267],[18,8660,965],{"className":8661},[1230,1196],[18,8663],{"className":8664,"style":228},[169],[18,8666,4775],{"className":8667},[232],[18,8669],{"className":8670},[169,8019],[18,8672,8674,8677,8680,8686,8689,8692,8695],{"className":8673},[97],[18,8675],{"className":8676,"style":6098},[101],[18,8678,5348],{"className":8679,"style":5477},[106,110],[18,8681,8683],{"className":8682},[250],[18,8684,74],{"className":8685},[1230,1196],[18,8687,47],{"className":8688},[106],[18,8690],{"className":8691,"style":2068},[169],[18,8693,2253],{"className":8694},[2072],[18,8696],{"className":8697,"style":2068},[169],[18,8699,8701,8704,8707,8747,8750,8753],{"className":8700},[97],[18,8702],{"className":8703,"style":242},[101],[18,8705,2248],{"className":8706},[250],[18,8708,8710,8713],{"className":8709},[106],[18,8711,939],{"className":8712,"style":1171},[106,110],[18,8714,8716],{"className":8715},[115],[18,8717,8719,8739],{"className":8718},[119,120],[18,8720,8722,8736],{"className":8721},[124],[18,8723,8725],{"className":8724,"style":129},[128],[18,8726,8727,8730],{"style":1386},[18,8728],{"className":8729,"style":137},[136],[18,8731,8733],{"className":8732},[141,142,143,144],[18,8734,430],{"className":8735},[106,144],[18,8737,152],{"className":8738},[151],[18,8740,8742],{"className":8741},[124],[18,8743,8745],{"className":8744,"style":159},[128],[18,8746],{},[18,8748],{"className":8749,"st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",[18,8861,8863,8895],{"className":8862,"translate":22},[21],[18,8864,8866],{"className":8865},[26],[28,8867,8868],{"xmlns":30},[32,8869,8870,8892],{},[35,8871,8872,8874,8876,8878,8880,8886,8888,8890],{},[41,8873,5348],{},[49,8875,74],{"stretchy":73},[45,8877,47],{},[49,8879,2253],{},[35,8881,8882,8884],{},[41,8883,5700],{"mathvariant":316},[41,8885,5703],{"mathvariant":316},[49,8887,84],{"stretchy":73},[41,8889,317],{"mathvariant":316},[45,8891,2266],{},[86,8893,8894],{"encoding":88},"\\delta(1-\\mathrm{TV})\u002F2",[18,8896,8898,8922],{"className":8897,"ariaHidden":51},[93],[18,8899,8901,8904,8907,8910,8913,8916,8919],{"className":8900},[97],[18,8902],{"className":8903,"style":242},[101],[18,8905,5348],{"className":8906,"style":5477},[106,110],[18,8908,74],{"className":8909},[250],[18,8911,47],{"className":8912},[106],[18,8914],{"className":8915,"style":2068},[169],[18,8917,2253],{"className":8918},[2072],[18,8920],{"className":8921,"style":2068},[169],[18,8923,8925,8928,8934,8937],{"className":8924},[97],[18,8926],{"className":8927,"style":242},[101],[18,8929,8931],{"className":8930},[106],[18,8932,6353],{"className":8933,"style":6352},[106,6351],[18,8935,84],{"className":8936},[267],[18,8938,8940],{"className":8939},[106],"\u002F2","。证毕。",[11,8943,8944,8945,2702,8973,9001,9002,9058],{},"如果取 ",[18,8946,8948,8961],{"className":8947,"translate":22},[21],[18,8949,8951],{"className":8950},[26],[28,8952,8953],{"xmlns":30},[32,8954,8955,8959],{},[35,8956,8957],{},[45,8958,430],{},[86,8960,430],{"encoding":88},[18,8962,8964],{"className":8963,"ariaHidden":51},[93],[18,8965,8967,8970],{"className":8966},[97],[18,8968],{"className":8969,"style":505},[101],[18,8971,430],{"className":8972},[106],[18,8974,8976,8989],{"className":8975,"translate":22},[21],[18,8977,8979],{"className":8978},[26],[28,8980,8981],{"xmlns":30},[32,8982,8983,8987],{},[35,8984,8985],{},[45,8986,47],{},[86,8988,47],{"encoding":88},[18,8990,8992],{"className":8991,"ariaHidden":51},[93],[18,8993,8995,8998],{"className":8994},[97],[18,8996],{"className":8997,"style":505},[101],[18,8999,47],{"className":9000},[106]," 损失、",[18,9003,9005,9027],{"className":9004,"translate":22},[21],[18,9006,9008],{"className":9007},[26],[28,9009,9010],{"xmlns":30},[32,9011,9012,9024],{},[35,9013,9014,9016,9018,9020,9022],{},[41,9015,5348],{},[49,9017,433],{},[45,9019,47],{},[41,9021,317],{"mathvariant":316},[45,9023,2266],{},[86,9025,9026],{"encoding":88},"\\delta=1\u002F2",[18,9028,9030,9048],{"className":9029,"ariaHidden":51},[93],[18,9031,9033,9036,9039,9042,9045],{"className":9032},[97],[18,9034],{"className":9035,"style":295},[101],[18,9037,5348],{"className":9038,"style":5477},[106,110],[18,9040],{"className":9041,"style":228},[169],[18,9043,433],{"className":9044},[232],[18,9046],{"className":9047,"style":228},[169],[18,9049,9051,9054],{"className":9050},[97],[18,9052],{"className":9053,"style":242},[101],[18,9055,9057],{"className":9056},[106],"1\u002F2","，就得到最常见的检验形式：",[18,9060,9062],{"className":9061,"translate":22},[875],[18,9063,9065,9199],{"className":9064,"translate":22},[21],[18,9066,9068],{"className":9067},[26],[28,9069,9070],{"xmlns":30,"display":884},[32,9071,9072,9196],{},[35,9073,9074,9088,9108,9110,9122,9124,9130,9132,9138,9140,9142,9148,9150,9152,9154,9156,9168,9170,9182,9192,9194],{},[890,9075,9076,9082],{},[35,9077,9078,9080],{},[41,9079,896],{},[49,9081,899],{},[658,9083,9084,9086],{"accent":51},[41,9085,77],{},[49,9087,664],{},[890,9089,9090,9096],{},[35,9091,9092,9094],{},[41,9093,5590],{},[49,9095,899],{},[35,9097,9098,9100,9102,9104,9106],{},[41,9099,5597],{},[49,9101,433],{},[45,9103,430],{},[49,9105,52],{"separator":51},[45,9107,47],{},[926,9109,928],{},[1589,9111,9112,9114,9116],{},[41,9113,939],{},[41,9115,5597],{},[35,9117,9118,9120],{},[49,9119,1599],{},[41,9121,64],{},[49,9123,74],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[658,9125,9126,9128],{"accent":51},[41,9127,77],{},[49,9129,664],{},[49,9131,2771],{"mathvariant":316},[38,9133,9134,9136],{},[41,9135,77],{},[41,9137,5597],{},[49,9139,84],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,9141,4775],{},[5651,9143,9144,9146],{},[45,9145,47],{},[45,9147,2266],{},[49,9149,74],{"fence":51,"stretchy":51,"minsize":5659,"maxsize":5659},[45,9151,47],{},[49,9153,2253],{},[49,9155,2248],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[1589,9157,9158,9160,9162],{},[41,9159,939],{},[45,9161,430],{},[35,9163,9164,9166],{},[49,9165,1599],{},[41,9167,64],{},[49,9169,2253],{},[1589,9171,9172,9174,9176],{},[41,9173,939],{},[45,9175,47],{},[35,9177,9178,9180],{},[49,9179,1599],{},[41,9181,64],{},[38,9183,9184,9186],{},[49,9185,2248],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[35,9187,9188,9190],{},[41,9189,5700],{"mathvariant":316},[41,9191,5703],{"mathvariant":316},[49,9193,84],{"fence":51,"stretchy":51,"minsize":5659,"maxsize":5659},[41,9195,968],{"mathvariant":316},[86,9197,9198],{"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).",[18,9200,9202,9492,9553,9641,9750],{"className":9201,"ariaHidden":51},[93],[18,9203,9205,9209,9286,9289,9348,9351,9354,9413,9419,9450,9453,9489],{"className":9204},[97],[18,9206],{"className":9207,"style":9208},[101],"height:1.8944em;vertical-align:-0.9365em;",[18,9210,9212],{"className":9211},[985,986],[18,9213,9215,9278],{"className":9214},[119,120],[18,9216,9218,9275],{"className":9217},[124],[18,9219,9221,9263],{"className":9220,"style":295},[128],[18,9222,9223,9226],{"style":998},[18,9224],{"className":9225,"style":697},[136],[18,9227,9229],{"className":9228},[141,142,143,144],[18,9230,9232],{"className":9231},[106,144],[18,9233,9235],{"className":9234},[106,681,144],[18,9236,9238],{"className":9237},[119],[18,9239,9241],{"className":9240},[124],[18,9242,9244,9252],{"className":9243,"style":677},[128],[18,9245,9246,9249],{"style":1022},[18,9247],{"className":9248,"style":137},[136],[18,9250,77],{"className":9251,"style":254},[106,110,144],[18,9253,9254,9257],{"style":1031},[18,9255],{"className":9256,"style":137},[136],[18,9258,9260],{"className":9259,"style":711},[710],[18,9261,664],{"className":9262},[106,144],[18,9264,9265,9268],{"style":693},[18,9266],{"className":9267,"style":697},[136],[18,9269,9270],{},[18,9271,1051,9273],{"className":9272},[985],[18,9274,1055],{"style":1054},[18,9276,152],{"className":9277},[151],[18,9279,9281],{"className":9280},[124],[18,9282,9284],{"className":9283,"style":1065},[128],[18,9285],{},[18,9287],{"className":9288,"style":170},[169],[18,9290,9292],{"className":9291},[985,986],[18,9293,9295,9340],{"className":9294},[119,120],[18,9296,9298,9337],{"className":9297},[124],[18,9299,9301,9327],{"className":9300,"style":1083},[128],[18,9302,9303,9306],{"style":5735},[18,9304],{"className":9305,"style":697},[136],[18,9307,9309],{"className":9308},[141,142,143,144],[18,9310,9312,9315,9318,9321,9324],{"className":9311},[106,144],[18,9313,5597],{"className":9314},[106,110,144],[18,9316,433],{"className":9317},[232,144],[18,9319,430],{"className":9320},[106,144],[18,9322,52],{"className":9323},[165,144],[18,9325,47],{"className":9326},[106,144],[18,9328,9329,9332],{"style":693},[18,9330],{"className":9331,"style":697},[136],[18,9333,9334],{},[18,9335,5590],{"className":9336},[985],[18,9338,152],{"className":9339},[151],[18,9341,9343],{"className":9342},[124],[18,9344,9346],{"className":9345,"style":5779},[128],[18,9347],{},[18,9349,928],{"className":9350},[169],[18,9352],{"className":9353,"style":170},[169],[18,9355,9357,9360],{"className":9356},[106],[18,9358,939],{"className":9359,"style":1171},[106,110],[18,9361,9363],{"className":9362},[115],[18,9364,9366,9404],{"className":9365},[119,120],[18,9367,9369,9401],{"className":9368},[124],[18,9370,9372,9384],{"className":9371,"style":1800},[128],[18,9373,9375,9378],{"style":9374},"top:-2.433em;margin-left:-0.1389em;margin-right:0.05em;",[18,9376],{"className":9377,"style":137},[136],[18,9379,9381],{"className":9380},[141,142,143,144],[18,9382,5597],{"className":9383},[106,110,144],[18,9385,9386,9389],{"style":1637},[18,9387],{"className":9388,"style":137},[136],[18,9390,9392],{"className":9391},[141,142,143,144],[18,9393,9395,9398],{"className":9394},[106,144],[18,9396,1599],{"className":9397},[106,144],[18,9399,64],{"className":9400},[106,110,144],[18,9402,152],{"className":9403},[151],[18,9405,9407],{"className":9406},[124],[18,9408,9411],{"className":9409,"style":9410},[128],"height:0.267em;",[18,9412],{},[18,9414,9416],{"className":9415},[250],[18,9417,74],{"className":9418},[1230,1196],[18,9420,9422],{"className":9421},[106,681],[18,9423,9425],{"className":9424},[119],[18,9426,9428],{"className":9427},[124],[18,9429,9431,9439],{"className":9430,"style":677},[128],[18,9432,9433,9436],{"style":693},[18,9434],{"className":9435,"style":697},[136],[18,9437,77],{"className":9438,"style":254},[106,110],[18,9440,9441,9444],{"style":703},[18,9442],{"className":9443,"style":697},[136],[18,9445,9447],{"className":9446,"style":711},[710],[18,9448,664],{"className":9449},[106],[18,9451],{"className":9452,"style":228},[169],[18,9454,9456,9483,9486],{"className":9455},[232],[18,9457,9459],{"className":9458},[232],[18,9460,9462],{"className":9461},[106,2878],[18,9463,9465],{"className":9464},[2882],[18,9466,9468,9471,9480],{"className":9467},[2886],[18,9469],{"className":9470,"style":2890},[101],[18,9472,9474],{"className":9473},[2894],[18,9475,9477],{"className":9476},[106],[18,9478,2901],{"className":9479},[232],[18,9481],{"className":9482},[2905],[18,9484],{"className":9485},[169,2909],[18,9487,433],{"className":9488},[232],[18,9490],{"className":9491,"style":228},[169],[18,9493,9495,9498,9538,9544,9547,9550],{"className":9494},[97],[18,9496],{"className":9497,"style":6098},[101],[18,9499,9501,9504],{"className":9500},[106],[18,9502,77],{"className":9503,"style":254},[106,110],[18,9505,9507],{"className":9506},[115],[18,9508,9510,9530],{"className":9509},[119,120],[18,9511,9513,9527],{"className":9512},[124],[18,9514,9516],{"className":9515,"style":5918},[128],[18,9517,9518,9521],{"style":469},[18,9519],{"className":9520,"style":137},[136],[18,9522,9524],{"className":9523},[141,142,143,144],[18,9525,5597],{"className":9526},[106,110,144],[18,9528,152],{"className":9529},[151],[18,9531,9533],{"className":9532},[124],[18,9534,9536],{"className":9535,"style":159},[128],[18,9537],{},[18,9539,9541],{"className":9540},[267],[18,9542,84],{"className":9543},[1230,1196],[18,9545],{"className":9546,"style":228},[169],[18,9548,4775],{"className":9549},[232],[18,9551],{"className":9552,"style":228},[169],[18,9554,9556,9560,9623,9629,9632,9635,9638],{"className":9555},[97],[18,9557],{"className":9558,"style":9559},[101],"height:2.0074em;vertical-align:-0.686em;",[18,9561,9563,9566,9620],{"className":9562},[106],[18,9564],{"className":9565},[250,6009],[18,9567,9569],{"className":9568},[5651],[18,9570,9572,9612],{"className":9571},[119,120],[18,9573,9575,9609],{"className":9574},[124],[18,9576,9579,9590,9598],{"className":9577,"style":9578},[128],"height:1.3214em;",[18,9580,9581,9584],{"style":6025},[18,9582],{"className":9583,"style":697},[136],[18,9585,9587],{"className":9586},[106],[18,9588,2266],{"className":9589},[106],[18,9591,9592,9595],{"style":6037},[18,9593],{"className":9594,"style":697},[136],[18,9596],{"className":9597,"style":6045},[6044],[18,9599,9600,9603],{"style":6048},[18,9601],{"className":9602,"style":697},[136],[18,9604,9606],{"className":9605},[106],[18,9607,47],{"className":9608},[106],[18,9610,152],{"className":9611},[151],[18,9613,9615],{"className":9614},[124],[18,9616,9618],{"className":9617,"style":6067},[128],[18,9619],{},[18,9621],{"className":9622},[267,6009],[18,9624,9626],{"className":9625},[250],[18,9627,74],{"className":9628},[1230,6079],[18,9630,47],{"className":9631},[106],[18,9633],{"className":9634,"style":2068},[169],[18,9636,2253],{"className":9637},[2072],[18,9639],{"className":9640,"style":2068},[169],[18,9642,9644,9647,9684,9741,9744,9747],{"className":9643},[97],[18,9645],{"className":9646,"style":6098},[101],[18,9648,9650],{"className":9649},[250],[18,9651,9653],{"className":9652},[1230,6105],[18,9654,9656,9676],{"className":9655},[119,120],[18,9657,9659,9673],{"className":9658},[124],[18,9660,9662],{"className":9661,"style":6115},[128],[18,9663,9664,9667],{"style":6118},[18,9665],{"className":9666,"style":6122},[136],[18,9668,9669],{"style":6125},[6127,9670,9671],{"xmlns":6129,"width":6130,"height":944,"viewBox":6131},[6133,9672],{"d":6135},[18,9674,152],{"className":9675},[151],[18,9677,9679],{"className":9678},[124],[18,9680,9682],{"className":9681,"style":6145},[128],[18,9683],{},[18,9685,9687,9690],{"className":9686},[106],[18,9688,939],{"className":9689,"style":1171},[106,110],[18,9691,9693],{"className":9692},[115],[18,9694,9696,9733],{"className":9695},[119,120],[18,9697,9699,9730],{"className":9698},[124],[18,9700,9702,9713],{"className":9701,"style":1800},[128],[18,9703,9704,9707],{"style":6168},[18,9705],{"className":9706,"style":137},[136],[18,9708,9710],{"className":9709},[141,142,143,144],[18,9711,430],{"className":9712},[106,144],[18,9714,9715,9718],{"style":1637},[18,9716],{"className":9717,"style":137},[136],[18,9719,9721],{"className":9720},[141,142,143,144],[18,9722,9724,9727],{"className":9723},[106,144],[18,9725,1599],{"className":9726},[106,144],[18,9728,64],{"className":9729},[106,110,144],[18,9731,152],{"className":9732},[151],[18,9734,9736],{"className":9735},[124],[18,9737,9739],{"className":9738,"style":6204},[128],[18,9740],{},[18,9742],{"className":9743,"style":2068},[169],[18,9745,2253],{"className":9746},[2072],[18,9748],{"className":9749,"style":2068},[169],[18,9751,9753,9756,9813,9893,9899],{"className":9752},[97],[18,9754],{"className":9755,"style":6222},[101],[18,9757,9759,9762],{"className":9758},[106],[18,9760,939],{"className":9761,"style":1171},[106,110],[18,9763,9765],{"className":9764},[115],[18,9766,9768,9805],{"className":9767},[119,120],[18,9769,9771,9802],{"className":9770},[124],[18,9772,9774,9785],{"className":9773,"style":1800},[128],[18,9775,9776,9779],{"style":6168},[18,9777],{"className":9778,"style":137},[136],[18,9780,9782],{"className":9781},[141,142,143,144],[18,9783,47],{"className":9784},[106,144],[18,9786,9787,9790],{"style":1637},[18,9788],{"className":9789,"style":137},[136],[18,9791,9793],{"className":9792},[141,142,143,144],[18,9794,9796,9799],{"className":9795},[106,144],[18,9797,1599],{"className":9798},[106,144],[18,9800,64],{"className":9801},[106,110,144],[18,9803,152],{"className":9804},[151],[18,9806,9808],{"className":9807},[124],[18,9809,9811],{"className":9810,"style":6204},[128],[18,9812],{},[18,9814,9816,9853],{"className":9815},[267],[18,9817,9819],{"className":9818},[267],[18,9820,9822],{"className":9821},[1230,6105],[18,9823,9825,9845],{"className":9824},[119,120],[18,9826,9828,9842],{"className":9827},[124],[18,9829,9831],{"className":9830,"style":6115},[128],[18,9832,9833,9836],{"style":6118},[18,9834],{"className":9835,"style":6122},[136],[18,9837,9838],{"style":6125},[6127,9839,9840],{"xmlns":6129,"width":6130,"height":944,"viewBox":6131},[6133,9841],{"d":6135},[18,9843,152],{"className":9844},[151],[18,9846,9848],{"className":9847},[124],[18,9849,9851],{"className":9850,"style":6145},[128],[18,9852],{},[18,9854,9856],{"className":9855},[115],[18,9857,9859,9885],{"className":9858},[119,120],[18,9860,9862,9882],{"className":9861},[124],[18,9863,9865],{"className":9864,"style":6332},[128],[18,9866,9867,9870],{"style":6335},[18,9868],{"className":9869,"style":137},[136],[18,9871,9873],{"className":9872},[141,142,143,144],[18,9874,9876],{"className":9875},[106,144],[18,9877,9879],{"className":9878},[106,144],[18,9880,6353],{"className":9881,"style":6352},[106,6351,144],[18,9883,152],{"className":9884},[151],[18,9886,9888],{"className":9887},[124],[18,9889,9891],{"className":9890,"style":6363},[128],[18,9892],{},[18,9894,9896],{"className":9895},[267],[18,9897,84],{"className":9898},[1230,6079],[18,9900,968],{"className":9901},[106],[11,9903,9904,9905,9940,9941,10074,10075,10107],{},"这个式子的意思是：",[717,9906,9907,9908,1990],{},"两个分布的总变差越小，任何检验的错误率就越接近 ",[18,9909,9911,9928],{"className":9910,"translate":22},[21],[18,9912,9914],{"className":9913},[26],[28,9915,9916],{"xmlns":30},[32,9917,9918,9926],{},[35,9919,9920,9922,9924],{},[45,9921,47],{},[41,9923,317],{"mathvariant":316},[45,9925,2266],{},[86,9927,9057],{"encoding":88},[18,9929,9931],{"className":9930,"ariaHidden":51},[93],[18,9932,9934,9937],{"className":9933},[97],[18,9935],{"className":9936,"style":242},[101],[18,9938,9057],{"className":9939},[106]," 当 ",[18,9942,9944,9970],{"className":9943,"translate":22},[21],[18,9945,9947],{"className":9946},[26],[28,9948,9949],{"xmlns":30},[32,9950,9951,9967],{},[35,9952,9953,9959,9961],{},[38,9954,9955,9957],{},[41,9956,939],{},[45,9958,430],{},[49,9960,433],{},[38,9962,9963,9965],{},[41,9964,939],{},[45,9966,47],{},[86,9968,9969],{"encoding":88},"P_0=P_1",[18,9971,9973,10028],{"className":9972,"ariaHidden":51},[93],[18,9974,9976,9979,10019,10022,10025],{"className":9975},[97],[18,9977],{"className":9978,"style":4563},[101],[18,9980,9982,9985],{"className":9981},[106],[18,9983,939],{"className":9984,"style":1171},[106,110],[18,9986,9988],{"className":9987},[115],[18,9989,9991,10011],{"className":9990},[119,120],[18,9992,9994,10008],{"className":9993},[124],[18,9995,9997],{"className":9996,"style":129},[128],[18,9998,9999,10002],{"style":1386},[18,10000],{"className":10001,"style":137},[136],[18,10003,10005],{"className":10004},[141,142,143,144],[18,10006,430],{"className":10007},[106,144],[18,10009,152],{"className":10010},[151],[18,10012,10014],{"className":10013},[124],[18,10015,10017],{"className":10016,"style":159},[128],[18,10018],{},[18,10020],{"className":10021,"style":228},[169],[18,10023,433],{"className":10024},[232],[18,10026],{"className":10027,"style":228},[169],[18,10029,10031,10034],{"className":10030},[97],[18,10032],{"className":10033,"style":4563},[101],[18,10035,10037,10040],{"className":10036},[106],[18,10038,939],{"className":10039,"style":1171},[106,110],[18,10041,10043],{"className":10042},[115],[18,10044,10046,10066],{"className":10045},[119,120],[18,10047,10049,10063],{"className":10048},[124],[18,10050,10052],{"className":10051,"style":129},[128],[18,10053,10054,10057],{"style":1386},[18,10055],{"className":10056,"style":137},[136],[18,10058,10060],{"className":10059},[141,142,143,144],[18,10061,47],{"className":10062},[106,144],[18,10064,152],{"className":10065},[151],[18,10067,10069],{"className":10068},[124],[18,10070,10072],{"className":10071,"style":159},[128],[18,10073],{}," 时，错误率至少是 ",[18,10076,10078,10095],{"className":10077,"translate":22},[21],[18,10079,10081],{"className":10080},[26],[28,10082,10083],{"xmlns":30},[32,10084,10085,10093],{},[35,10086,10087,10089,10091],{},[45,10088,47],{},[41,10090,317],{"mathvariant":316},[45,10092,2266],{},[86,10094,9057],{"encoding":88},[18,10096,10098],{"className":10097,"ariaHidden":51},[93],[18,10099,10101,10104],{"className":10100},[97],[18,10102],{"className":10103,"style":242},[101],[18,10105,9057],{"className":10106},[106],"，这当然——数据里根本没有信息。",[11,10109,10110,10111,10139,10140,10360],{},"定理 3.1 要求 ",[18,10112,10114,10127],{"className":10113,"translate":22},[21],[18,10115,10117],{"className":10116},[26],[28,10118,10119],{"xmlns":30},[32,10120,10121,10125],{},[35,10122,10123],{},[41,10124,948],{"mathvariant":316},[86,10126,5495],{"encoding":88},[18,10128,10130],{"className":10129,"ariaHidden":51},[93],[18,10131,10133,10136],{"className":10132},[97],[18,10134],{"className":10135,"style":295},[101],[18,10137,948],{"className":10138},[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 ",[18,10141,10143,10183],{"className":10142,"translate":22},[21],[18,10144,10146],{"className":10145},[26],[28,10147,10148],{"xmlns":30},[32,10149,10150,10180],{},[35,10151,10152,10154,10156,10158,10164,10166,10172],{},[41,10153,532],{"mathvariant":316},[49,10155,433],{},[49,10157,2248],{"stretchy":73},[38,10159,10160,10162],{},[41,10161,77],{},[45,10163,430],{},[49,10165,2253],{},[38,10167,10168,10170],{},[41,10169,77],{},[45,10171,47],{},[1589,10173,10174,10176,10178],{},[49,10175,2248],{"stretchy":73},[45,10177,2266],{},[45,10179,2266],{},[86,10181,10182],{"encoding":88},"\\Delta=\\lVert\\theta_0-\\theta_1\\rVert_2^2",[18,10184,10186,10204,10262],{"className":10185,"ariaHidden":51},[93],[18,10187,10189,10192,10195,10198,10201],{"className":10188},[97],[18,10190],{"className":10191,"style":600},[101],[18,10193,532],{"className":10194},[106],[18,10196],{"className":10197,"style":228},[169],[18,10199,433],{"className":10200},[232],[18,10202],{"className":10203,"style":228},[169],[18,10205,10207,10210,10213,10253,10256,10259],{"className":10206},[97],[18,10208],{"className":10209,"style":242},[101],[18,10211,2248],{"className":10212},[250],[18,10214,10216,10219],{"className":10215},[106],[18,10217,77],{"className":10218,"style":254},[106,110],[18,10220,10222],{"className":10221},[115],[18,10223,10225,10245],{"className":10224},[119,120],[18,10226,10228,10242],{"className":10227},[124],[18,10229,10231],{"className":10230,"style":129},[128],[18,10232,10233,10236],{"style":469},[18,10234],{"className":10235,"style":137},[136],[18,10237,10239],{"className":10238},[141,142,143,144],[18,10240,430],{"className":10241},[106,144],[18,10243,152],{"className":10244},[151],[18,10246,10248],{"className":10247},[124],[18,10249,10251],{"className":10250,"style":159},[128],[18,10252],{},[18,10254],{"className":10255,"style":2068},[169],[18,10257,2253],{"className":10258},[2072],[18,10260],{"className":10261,"style":2068},[169],[18,10263,10265,10269,10309],{"className":10264},[97],[18,10266],{"className":10267,"style":10268},[101],"height:1.0641em;vertical-align:-0.25em;",[18,10270,10272,10275],{"className":10271},[106],[18,10273,77],{"className":10274,"style":254},[106,110],[18,10276,10278],{"className":10277},[115],[18,10279,10281,10301],{"className":10280},[119,120],[18,10282,10284,10298],{"className":10283},[124],[18,10285,10287],{"className":10286,"style":129},[128],[18,10288,10289,10292],{"style":469},[18,10290],{"className":10291,"style":137},[136],[18,10293,10295],{"className":10294},[141,142,143,144],[18,10296,47],{"className":10297},[106,144],[18,10299,152],{"className":10300},[151],[18,10302,10304],{"className":10303},[124],[18,10305,10307],{"className":10306,"style":159},[128],[18,10308],{},[18,10310,10312,10315],{"className":10311},[267],[18,10313,2248],{"className":10314},[267],[18,10316,10318],{"className":10317},[115],[18,10319,10321,10352],{"className":10320},[119,120],[18,10322,10324,10349],{"className":10323},[124],[18,10325,10327,10338],{"className":10326,"style":2416},[128],[18,10328,10329,10332],{"style":2419},[18,10330],{"className":10331,"style":137},[136],[18,10333,10335],{"className":10334},[141,142,143,144],[18,10336,2266],{"className":10337},[106,144],[18,10339,10340,10343],{"style":1962},[18,10341],{"className":10342,"style":137},[136],[18,10344,10346],{"className":10345},[141,142,143,144],[18,10347,2266],{"className":10348},[106,144],[18,10350,152],{"className":10351},[151],[18,10353,10355],{"className":10354},[124],[18,10356,10358],{"className":10357,"style":2449},[128],[18,10359],{},"，则",[18,10362,10364],{"className":10363,"translate":22},[875],[18,10365,10367,10496],{"className":10366,"translate":22},[21],[18,10368,10370],{"className":10369},[26],[28,10371,10372],{"xmlns":30,"display":884},[32,10373,10374,10493],{},[35,10375,10376,10396,10412,10414,10420,10422,10428,10436,10438,10445,10447,10449,10451,10453,10465,10467,10479,10489,10491],{},[890,10377,10378,10384],{},[35,10379,10380,10382],{},[41,10381,5590],{},[49,10383,899],{},[35,10385,10386,10388,10390,10392,10394],{},[41,10387,5597],{},[49,10389,433],{},[45,10391,430],{},[49,10393,52],{"separator":51},[45,10395,47],{},[38,10397,10398,10400],{},[41,10399,934],{"mathvariant":933},[1589,10401,10402,10404,10406],{},[41,10403,939],{},[41,10405,5597],{},[35,10407,10408,10410],{},[49,10409,1599],{},[41,10411,64],{},[49,10413,2248],{"stretchy":73},[38,10415,10416,10418],{},[41,10417,77],{},[41,10419,5597],{},[49,10421,2253],{},[658,10423,10424,10426],{"accent":51},[41,10425,77],{},[49,10427,664],{},[1589,10429,10430,10432,10434],{},[49,10431,2248],{"stretchy":73},[45,10433,2266],{},[45,10435,2266],{},[49,10437,4775],{},[5651,10439,10440,10442],{},[41,10441,532],{"mathvariant":316},[45,10443,10444],{},"4",[49,10446,74],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[45,10448,47],{},[49,10450,2253],{},[49,10452,2248],{"stretchy":73},[1589,10454,10455,10457,10459],{},[41,10456,939],{},[45,10458,430],{},[35,10460,10461,10463],{},[49,10462,15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n}\\rVert_{\\mathrm{TV}}\\bigr).",[18,10497,10499,10717,10817,10905,10981],{"className":10498,"ariaHidden":51},[93],[18,10500,10502,10506,10565,10568,10665,10668,10708,10711,10714],{"className":10501},[97],[18,10503],{"className":10504,"style":10505},[101],"height:1.6138em;vertical-align:-0.8638em;",[18,10507,10509],{"className":10508},[985,986],[18,10510,10512,10557],{"className":10511},[119,120],[18,10513,10515,10554],{"className":10514},[124],[18,10516,10518,10544],{"className":10517,"style":1083},[128],[18,10519,10520,10523],{"style":5735},[18,10521],{"className":10522,"style":697},[136],[18,10524,10526],{"className":10525},[141,142,143,144],[18,10527,10529,10532,10535,10538,10541],{"className":10528},[106,144],[18,10530,5597],{"className":10531},[106,110,144],[18,10533,433],{"className":10534},[232,144],[18,10536,430],{"className":10537},[106,144],[18,10539,52],{"className":10540},[165,144],[18,10542,47],{"className":10543},[106,144],[18,10545,10546,10549],{"style":693},[18,10547],{"className":10548,"style":697},[136],[18,10550,10551],{},[18,10552,5590],{"className":10553},[985],[18,10555,152],{"className":10556},[151],[18,10558,10560],{"className":10559},[124],[18,10561,10563],{"className":10562,"style":5779},[128],[18,10564],{},[18,10566],{"className":10567,"style":170},[169],[18,10569,10571,10574],{"className":10570},[106],[18,10572,934],{"className":10573},[106,1139],[18,10575,10577],{"className":10576},[115],[18,10578,10580,10657],{"className":10579},[119,120],[18,10581,10583,10654],{"className":10582},[124],[18,10584,10586],{"className":10585,"style":1184},[128],[18,10587,10588,10591],{"style":3261},[18,10589],{"className":10590,"style":137},[136],[18,10592,10594],{"className":10593},[141,142,143,144],[18,10595,10597],{"className":10596},[106,144],[18,10598,10600,10603],{"className":10599},[106,144],[18,10601,939],{"className":10602,"style":1171},[106,110,144],[18,10604,10606],{"className":10605},[115],[18,10607,10609,10646],{"className":10608},[119,120],[18,10610,10612,10643],{"className":10611},[124],[18,10613,10615,10626],{"className":10614,"style":3289},[128],[18,10616,10617,10620],{"style":5837},[18,10618],{"className":10619,"style":1191},[136],[18,10621,10623],{"className":10622},[141,1195,1196,144],[18,10624,5597],{"className":10625},[106,110,144],[18,10627,10628,10631],{"style":3304},[18,10629],{"className":10630,"style":1191},[136],[18,10632,10634],{"className":10633},[141,1195,1196,144],[18,10635,10637,10640],{"className":10636},[106,144],[18,10638,1599],{"className":10639},[106,144],[18,10641,64],{"className":10642},[106,110,144],[18,10644,152],{"className":10645},[151],[18,10647,10649],{"className":10648},[124],[18,10650,10652],{"className":10651,"style":5873},[128],[18,10653],{},[18,10655,152],{"className":10656},[151],[18,10658,10660],{"className":10659},[124],[18,10661,10663],{"className":10662,"style":5885},[128],[18,10664],{},[18,10666,2248],{"className":10667},[250],[18,10669,10671,10674],{"className":10670},[106],[18,10672,77],{"className":10673,"style":254},[106,110],[18,10675,10677],{"className":10676},[115],[18,10678,10680,10700],{"className":10679},[119,120],[18,10681,10683,10697],{"className":10682},[124],[18,10684,10686],{"className":10685,"style":5918},[128],[18,10687,10688,10691],{"style":469},[18,10689],{"className":10690,"style":137},[136],[18,10692,10694],{"className":10693},[141,142,143,144],[18,10695,5597],{"className":10696},[106,110,144],[18,10698,152],{"className":10699},[151],[18,10701,10703],{"className":10702},[124],[18,10704,10706],{"className":10705,"style":159},[128],[18,10707],{},[18,10709],{"className":10710,"style":2068},[169],[18,10712,2253],{"className":10713},[2072],[18,10715],{"className":10716,"style":2068},[169],[18,10718,10720,10723,10754,10808,10811,10814],{"className":10719},[97],[18,10721],{"className":10722,"style":2281},[101],[18,10724,10726],{"className":10725},[106,681],[18,10727,10729],{"className":10728},[119],[18,10730,10732],{"className":10731},[124],[18,10733,10735,10743],{"className":10734,"style":677},[128],[18,10736,10737,10740],{"style":693},[18,10738],{"className":10739,"style":697},[136],[18,10741,77],{"className":10742,"style":254},[106,110],[18,10744,10745,10748],{"style":703},[18,10746],{"className":10747,"style":697},[136],[18,10749,10751],{"className":10750,"style":711},[710],[18,10752,664],{"className":10753},[106],[18,10755,10757,10760],{"className":10756},[267],[18,10758,2248],{"className":10759},[267],[18,10761,10763],{"className":10762},[115],[18,10764,10766,10799],{"className":10765},[119,120],[18,10767,10769,10796],{"className":10768},[124],[18,10770,10773,10785],{"className":10771,"style":10772},[128],"height:0.8641em;",[18,10774,10776,10779],{"style":10775},"top:-2.453em;margin-left:0em;margin-right:0.05em;",[18,10777],{"className":10778,"style":137},[136],[18,10780,10782],{"className":10781},[141,142,143,144],[18,10783,2266],{"className":10784},[106,144],[18,10786,10787,10790],{"style":1637},[18,10788],{"className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\\\\\n\\pi(X^n)=\\Pr(\\theta=\\theta_0\\mid 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",[18,11739,11741,11795],{"className":11740,"translate":22},[21],[18,11742,11744],{"className":11743},[26],[28,11745,11746],{"xmlns":30},[32,11747,11748,11792],{},[35,11749,11750,11752,11754,11756,11758,11760,11762,11768,11770,11772,11774,11776,11778,11780,11786,11788,11790],{},[41,11751,532],{"mathvariant":316},[926,11753,4024],{},[41,11755,934],{"mathvariant":933},[49,11757,945],{"stretchy":73},[41,11759,3941],{},[49,11761,74],{"stretchy":73},[1556,11763,11764,11766],{},[41,11765,43],{},[41,11767,64],{},[49,11769,84],{"stretchy":73},[49,11771,74],{"stretchy":73},[45,11773,47],{},[49,11775,2253],{},[41,11777,3941],{},[49,11779,74],{"stretchy":73},[1556,11781,11782,11784],{},[41,11783,43],{},[41,11785,64],{},[49,11787,84],{"stretchy":73},[49,11789,84],{"stretchy":73},[49,11791,965],{"stretchy":73},[86,11793,11794],{"encoding":88},"\\Delta\\,\\mathbb E[\\pi(X^n)(1-\\pi(X^n))]",[18,11796,11798,11869],{"className":11797,"ariaHidden":51},[93],[18,11799,11801,11804,11807,11810,11813,11816,11819,11822,11851,11854,11857,11860,11863,11866],{"className":11800},[97],[18,11802],{"className":11803,"style":242},[101],[18,11805,532],{"className":11806},[106],[18,11808],{"className":11809,"style":170},[169],[18,11811,934],{"className":11812},[106,1139],[18,11814,945],{"className":11815},[250],[18,11817,3941],{"className":11818,"style":3957},[106,110],[18,11820,74],{"className":11821},[250],[18,11823,11825,11828],{"className":11824},[106],[18,11826,43],{"className":11827,"style":111},[106,110],[18,11829,11831],{"className":11830},[115],[18,11832,11834],{"className":11833},[119],[18,11835,11837],{"className":11836},[124],[18,11838,11840],{"className":11839,"style":1959},[128],[18,11841,11842,11845],{"style":1962},[18,11843],{"className":11844,"style":137},[136],[18,11846,11848],{"className":11847},[141,142,143,144],[18,11849,64],{"className":11850},[106,110,144],[18,11852,84],{"className":11853},[267],[18,11855,74],{"className":11856},[250],[18,11858,47],{"className":11859},[106],[18,11861],{"className":11862,"style":2068},[169],[18,11864,2253],{"className":11865},[2072],[18,11867],{"className":11868,"style":2068},[169],[18,11870,11872,11875,11878,11881,11910],{"className":11871},[97],[18,11873],{"className":11874,"style":242},[101],[18,11876,3941],{"className":11877,"style":3957},[106,110],[18,11879,74],{"className":11880},[250],[18,11882,11884,11887],{"className":11883},[106],[18,11885,43],{"className":11886,"style":111},[106,110],[18,11888,11890],{"className":11889},[115],[18,11891,11893],{"className":11892},[119],[18,11894,11896],{"className":11895},[124],[18,11897,11899],{"className":11898,"style":1959},[128],[18,11900,11901,11904],{"style":1962},[18,11902],{"className":11903,"style":137},[136],[18,11905,11907],{"className":11906},[141,142,143,144],[18,11908,64],{"className":11909},[106,110,144],[18,11911,11913],{"className":11912},[267],"))]","。由 ",[18,11916,11918,11969],{"className":11917,"translate":22},[21],[18,11919,11921],{"className":11920},[26],[28,11922,11923],{"xmlns":30},[32,11924,11925,11966],{},[35,11926,11927,11929,11931,11933,11935,11937,11939,11941,11947,11950,11952,11954,11956,11958,11960,11962,11964],{},[41,11928,3941],{},[49,11930,74],{"stretchy":73},[45,11932,47],{},[49,11934,2253],{},[41,11936,3941],{},[49,11938,84],{"stretchy":73},[49,11940,4775],{},[5651,11942,11943,11945],{},[45,11944,47],{},[45,11946,2266],{},[41,11948,11949],{},"min",[49,11951,899],{},[49,11953,74],{"stretchy":73},[41,11955,3941],{},[49,11957,52],{"separator":51},[45,11959,47],{},[49,11961,2253],{},[41,11963,3941],{},[49,11965,84],{"stretchy":73},[86,11967,11968],{"encoding":88},"\\pi(1-\\pi)\\ge\\frac12\\min(\\pi,1-\\pi)",[18,11970,11972,11996,12017,12126],{"className":11971,"ariaHidden":51},[93],[18,11973,11975,11978,11981,11984,11987,11990,11993],{"className":11974},[97],[18,11976],{"className":11977,"style":242},[101],[18,11979,3941],{"className":11980,"style":3957},[106,110],[18,11982,74],{"className":11983},[250],[18,11985,47],{"className":11986},[106],[18,11988],{"className":11989,"style":2068},[169],[18,11991,2253],{"className":11992},[2072],[18,11994],{"className":11995,"style":2068},[169],[18,11997,11999,12002,12005,12008,12011,12014],{"className":11998},[97],[18,12000],{"className":12001,"style":242},[101],[18,12003,3941],{"className":12004,"style":3957},[106,110],[18,12006,84],{"className":12007},[267],[18,12009],{"className":12010,"style":228},[169],[18,12012,4775],{"className":12013},[232],[18,12015],{"className":12016,"style":228},[169],[18,12018,12020,12024,12096,12099,12102,12105,12108,12111,12114,12117,12120,12123],{"className":12019},[97],[18,12021],{"className":12022,"style":12023},[101],"height:1.1901em;vertical-align:-0.345em;",[18,12025,12027,12030,12093],{"className":12026},[106],[18,12028],{"className":12029},[250,6009],[18,12031,12033],{"className":12032},[5651],[18,12034,12036,12084],{"className":12035},[119,120],[18,12037,12039,12081],{"className":12038},[124],[18,12040,12043,12058,12066],{"className":12041,"style":12042},[128],"height:0.8451em;",[18,12044,12046,12049],{"style":12045},"top:-2.655em;",[18,12047],{"className":12048,"style":697},[136],[18,12050,12052],{"className":12051},[141,142,143,144],[18,12053,12055],{"className":12054},[106,144],[18,12056,2266],{"className":12057},[106,144],[18,12059,12060,12063],{"style":6037},[18,12061],{"className":12062,"style":697},[136],[18,12064],{"className":12065,"style":6045},[6044],[18,12067,12069,12072],{"style":12068},"top:-3.394em;",[18,12070],{"className":12071,"style":697},[136],[18,12073,12075],{"className":12074},[141,142,143,144],[18,12076,12078],{"className":12077},[106,144],[18,12079,47],{"className":12080},[106,144],[18,12082,152],{"className":12083},[151],[18,12085,12087],{"className":12086},[124],[18,12088,12091],{"className":12089,"style":12090},[128],"height:0.345em;",[18,12092],{},[18,12094],{"className":12095},[267,6009],[18,12097],{"className":12098,"style":170},[169],[18,12100,11949],{"className":12101},[985],[18,12103,74],{"className":12104},[250],[18,12106,3941],{"className":12107,"style":3957},[106,110],[18,12109,52],{"className":12110},[165],[18,12112],{"className":12113,"style":170},[169],[18,12115,47],{"className":12116},[106],[18,12118],{"className":12119,"style":2068},[169],[18,12121,2253],{"className":12122},[2072],[18,12124],{"className":12125,"style":2068},[169],[18,12127,12129,12132,12135],{"className":12128},[97],[18,12130],{"className":12131,"style":242},[101],[18,12133,3941],{"className":12134,"style":3957},[106,110],[18,12136,84],{"className":12137},[267]," 与均匀先验下贝叶斯检验误差的下界 ",[18,12140,12142,12200],{"className":12141,"translate":22},[21],[18,12143,12145],{"className":12144},[26],[28,12146,12147],{"xmlns":30},[32,12148,12149,12197],{},[35,12150,12151,12153,12155,12157,12159,12161,12163,12165,12167,12169,12171,12173,12175,12177,12183,12185,12187,12189,12195],{},[41,12152,934],{"mathvariant":933},[49,12154,945],{"stretchy":73},[41,12156,11949],{},[49,12158,899],{},[49,12160,74],{"stretchy":73},[41,12162,3941],{},[49,12164,52],{"separator":51},[45,12166,47],{},[49,12168,2253],{},[41,12170,3941],{},[49,12172,84],{"stretchy":73},[49,12174,965],{"stretchy":73},[49,12176,4775],{},[5651,12178,12179,12181],{},[45,12180,47],{},[45,12182,2266],{},[49,12184,74],{"stretchy":73},[45,12186,47],{},[49,12188,2253],{},[35,12190,12191,12193],{},[41,12192,5700],{"mathvariant":316},[41,12194,5703],{"mathvariant":316},[49,12196,84],{"stretchy":73},[86,12198,12199],{"encoding":88},"\\mathbb E[\\min(\\pi,1-\\pi)]\\ge\\frac12(1-\\mathrm{TV})",[18,12201,12203,12242,12264,12353],{"className":12202,"ariaHidden":51},[93],[18,12204,12206,12209,12212,12215,12218,12221,12224,12227,12230,12233,12236,12239],{"className":12205},[97],[18,12207],{"className":12208,"style":242},[101],[18,12210,934],{"className":12211},[106,1139],[18,12213,945],{"className":12214},[250],[18,12216,11949],{"className":12217},[985],[18,12219,74],{"className":12220},[250],[18,12222,3941],{"className":12223,"style":3957},[106,110],[18,12225,52],{"className":12226},[165],[18,12228],{"className":12229,"style":170},[169],[18,12231,47],{"className":12232},[106],[18,12234],{"className":12235,"style":2068},[169],[18,12237,2253],{"className":12238},[2072],[18,12240],{"className":12241,"style":2068},[169],[18,12243,12245,12248,12251,12255,12258,12261],{"className":12244},[97],[18,12246],{"className":12247,"style":242},[101],[18,12249,3941],{"className":12250,"style":3957},[106,110],[18,12252,12254],{"className":12253},[267],")]",[18,12256],{"className":12257,"style":228},[169],[18,12259,4775],{"className":12260},[232],[18,12262],{"className":12263,"style":228},[169],[18,12265,12267,12270,12338,12341,12344,12347,12350],{"className":12266},[97],[18,12268],{"className":12269,"style":12023},[101],[18,12271,12273,12276,12335],{"className":12272},[106],[18,12274],{"className":12275},[250,6009],[18,12277,12279],{"className":12278},[5651],[18,12280,12282,12327],{"className":12281},[119,120],[18,12283,12285,12324],{"className":12284},[124],[18,12286,12288,12302,12310],{"className":12287,"style":12042},[128],[18,12289,12290,12293],{"style":12045},[18,12291],{"className":12292,"style":697},[136],[18,12294,12296],{"className":12295},[141,142,143,144],[18,12297,12299],{"className":12298},[106,144],[18,12300,2266],{"className":12301},[106,144],[18,12303,12304,12307],{"style":6037},[18,12305],{"className":12306,"style":697},[136],[18,12308],{"className":12309,"style":6045},[6044],[18,12311,12312,12315],{"style":12068},[18,12313],{"className":12314,"style":697},[136],[18,12316,12318],{"className":12317},[141,142,143,144],[18,12319,12321],{"className":12320},[106,144],[18,12322,47],{"className":12323},[106,144],[18,12325,152],{"className":12326},[151],[18,12328,12330],{"className":12329},[124],[18,12331,12333],{"className":12332,"style":12090},[128],[18,12334],{},[18,12336],{"className":12337},[267,6009],[18,12339,74],{"className":12340},[250],[18,12342,47],{"className":12343},[106],[18,12345],{"className":12346,"style":2068},[169],[18,12348,2253],{"className":12349},[2072],[18,12351],{"className":12352,"style":2068},[169],[18,12354,12356,12359,12365],{"className":12355},[97],[18,12357],{"className":12358,"style":242},[101],[18,12360,12362],{"className":12361},[106],[18,12363,6353],{"className":12364,"style":6352},[106,6351],[18,12366,84],{"className":12367},[267],"，风险至少是 ",[18,12370,12372,12404],{"className":12371,"translate":22},[21],[18,12373,12375],{"className":12374},[26],[28,12376,12377],{"xmlns":30},[32,12378,12379,12401],{},[35,12380,12381,12387,12389,12391,12393,12399],{},[5651,12382,12383,12385],{},[41,12384,532],{"mathvariant":316},[45,12386,10444],{},[49,12388,74],{"stretchy":73},[45,12390,47],{},[49,12392,2253],{},[35,12394,12395,12397],{},[41,12396,5700],{"mathvariant":316},[41,12398,5703],{"mathvariant":316},[49,12400,84],{"stretchy":73},[86,12402,12403],{"encoding":88},"\\frac{\\Delta}{4}(1-\\mathrm{TV})",[18,12405,12407,12498],{"className":12406,"ariaHidden":51},[93],[18,12408,12410,12414,12483,12486,12489,12492,12495],{"className":12409},[97],[18,12411],{"className":12412,"style":12413},[101],"height:1.2173em;vertical-align:-0.345em;",[18,12415,12417,12420,12480],{"className":12416},[106],[18,12418],{"className":12419},[250,6009],[18,12421,12423],{"className":12422},[5651],[18,12424,12426,12472],{"className":12425},[119,120],[18,12427,12429,12469],{"className":12428},[124],[18,12430,12433,12447,12455],{"className":12431,"style":12432},[128],"height:0.8723em;",[18,12434,12435,12438],{"style":12045},[18,12436],{"className":12437,"style":697},[136],[18,12439,12441],{"className":12440},[141,142,143,144],[18,12442,12444],{"className":12443},[106,144],[18,12445,10444],{"className":12446},[106,144],[18,12448,12449,12452],{"style":6037},[18,12450],{"className":12451,"style":697},[136],[18,12453],{"className":12454,"style":6045},[6044],[18,12456,12457,12460],{"style":12068},[18,12458],{"className":12459,"style":697},[136],[18,12461,12463],{"className":12462},[141,142,143,144],[18,12464,12466],{"className":12465},[106,144],[18,12467,532],{"className":12468},[106,144],[18,12470,152],{"className":12471},[151],[18,12473,12475],{"className":12474},[124],[18,12476,12478],{"className":12477,"style":12090},[128],[18,12479],{},[18,12481],{"className":12482},[267,6009],[18,12484,74],{"className":12485},[250],[18,12487,47],{"className":12488},[106],[18,12490],{"className":12491,"style":2068},[169],[18,12493,2253],{"className":12494},[2072],[18,12496],{"className":12497,"style":2068},[169],[18,12499,12501,12504,12510],{"className":12500},[97],[18,12502],{"className":12503,"style":242},[101],[18,12505,12507],{"className":12506},[106],[18,12508,6353],{"className":12509,"style":6352},[106,6351],[18,12511,84],{"className":12512},[267],"。极小极大风险不小于贝叶斯风险，得证。",[1311,12515,12517],{"id":12516},"_32-从-tv-到-kl","3.2 从 TV 到 KL",[11,12519,12520],{},"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：",[18,12522,12524],{"className":12523,"translate":22},[875],[18,12525,12527,12595],{"className":12526,"translate":22},[21],[18,12528,12530],{"className":12529},[26],[28,12531,12532],{"xmlns":30,"display":884},[32,12533,12534,12592],{},[35,12535,12536,12538,12540,12542,12545,12555,12557,12590],{},[49,12537,2248],{"stretchy":73},[41,12539,939],{},[49,12541,2253],{},[41,12543,12544],{},"Q",[38,12546,12547,12549],{},[49,12548,2248],{"stretchy":73},[35,12550,12551,12553],{},[41,12552,5700],{"mathvariant":316},[41,12554,5703],{"mathvariant":316},[49,12556,4334],{},[12558,12559,12560],"msqrt",{},[35,12561,12562,12568,12580,12582,12584,12586,12588],{},[5651,12563,12564,12566],{},[45,12565,47],{},[45,12567,2266],{},[38,12569,12570,12573],{},[41,12571,12572],{},"D",[35,12574,12575,12577],{},[41,12576,5031],{"mathvariant":316},[41,12578,12579],{"mathvariant":316},"L",[49,12581,74],{"stretchy":73},[41,12583,939],{},[41,12585,2248],{"mathvariant":316},[41,12587,12544],{},[49,12589,84],{"stretchy":73},[41,12591,968],{"mathvariant":316},[86,12593,12594],{"encoding":88},"\\lVert P-Q\\rVert_{\\mathrm{TV}}\\le\\sqrt{\\frac12 D_{\\mathrm{KL}}(P\\Vert Q)}.",[18,12596,12598,12619,12683],{"className":12597,"ariaHidden":51},[93],[18,12599,12601,12604,12607,12610,12613,12616],{"className":12600},[97],[18,12602],{"className":12603,"style":242},[101],[18,12605,2248],{"className":12606},[250],[18,12608,939],{"className":12609,"style":1171},[106,110],[18,12611],{"className":12612,"style":2068},[169],[18,12614,2253],{"className":12615},[2072],[18,12617],{"className":12618,"style":2068},[169],[18,12620,12622,12625,12628,12674,12677,12680],{"className":12621},[97],[18,12623],{"className":12624,"style":242},[101],[18,12626,12544],{"className":12627},[106,110],[18,12629,12631,12634],{"className":12630},[267],[18,12632,2248],{"className":12633},[267],[18,12635,12637],{"className":12636},[115],[18,12638,12640,12666],{"className":12639},[119,120],[18,12641,12643,12663],{"className":12642},[124],[18,12644,12646],{"className":12645,"style":1152},[128],[18,12647,12648,12651],{"style":1155},[18,12649],{"className":12650,"style":137},[136],[18,12652,12654],{"className":12653},[141,142,143,144],[18,12655,12657],{"className":12656},[106,144],[18,12658,12660],{"className":12659},[106,144],[18,12661,6353],{"className":12662,"style":6352},[106,6351,144],[18,12664,152],{"className":12665},[151],[18,12667,12669],{"className":12668},[124],[18,12670,12672],{"className":12671,"style":159},[128],[18,12673],{},[18,12675],{"className":12676,"style":228},[169],[18,12678,4334],{"className":12679},[232],[18,12681],{"className":12682,"style":228},[169],[18,12684,12686,12690,12873],{"className":12685},[97],[18,12687],{"className":12688,"style":12689},[101],"height:2.44em;vertical-align:-0.7884em;",[18,12691,12694],{"className":12692},[106,12693],"sqrt",[18,12695,12697,12864],{"className":12696},[119,120],[18,12698,12700,12861],{"className":12699},[124],[18,12701,12704,12841],{"className":12702,"style":12703},[128],"height:1.6516em;",[18,12705,12709,12713],{"className":12706,"style":12708},[12707],"svg-align","top:-4.4em;",[18,12710],{"className":12711,"style":12712},[136],"height:4.4em;",[18,12714,12717,12779,12826,12829,12832,12835,12838],{"className":12715,"style":12716},[106],"padding-left:1em;",[18,12718,12720,12723,12776],{"className":12719},[106],[18,12721],{"className":12722},[250,6009],[18,12724,12726],{"className":12725},[5651],[18,12727,12729,12768],{"className":12728},[119,120],[18,12730,12732,12765],{"className":12731},[124],[18,12733,12735,12746,12754],{"className":12734,"style":9578},[128],[18,12736,12737,12740],{"style":6025},[18,12738],{"className":12739,"style":697},[136],[18,12741,12743],{"className":12742},[106],[18,12744,2266],{"className":12745},[106],[18,12747,12748,12751],{"style":6037},[18,12749],{"className":12750,"style":697},[136],[18,12752],{"className":12753,"style":6045},[6044],[18,12755,12756,12759],{"style":6048},[18,12757],{"className":12758,"style":697},[136],[18,12760,12762],{"className":12761},[106],[18,12763,47],{"className":12764},[106],[18,12766,152],{"className":12767},[151],[18,12769,12771],{"className":12770},[124],[18,12772,12774],{"className":12773,"style":6067},[128],[18,12775],{},[18,12777],{"className":12778},[267,6009],[18,12780,12782,12785],{"className":12781},[106],[18,12783,12572],{"className":12784,"style":254},[106,110],[18,12786,12788],{"className":12787},[115],[18,12789,12791,12818],{"className":12790},[119,120],[18,12792,12794,12815],{"className":12793},[124],[18,12795,12797],{"className":12796,"style":1152},[128],[18,12798,12799,12802],{"style":469},[18,12800],{"className":12801,"style":137},[136],[18,12803,12805],{"className":12804},[141,142,143,144],[18,12806,12808],{"className":12807},[106,144],[18,12809,12811],{"className":12810},[106,144],[18,12812,12814],{"className":12813},[106,6351,144],"KL",[18,12816,152],{"className":12817},[151],[18,12819,12821],{"className":12820},[124],[18,12822,12824],{"className":12823,"style":159},[128],[18,12825],{},[18,12827,74],{"className":12828},[250],[18,12830,939],{"className":12831,"style":1171},[106,110],[18,12833,2248],{"className":12834},[106],[18,12836,12544],{"className":12837},[106,110],[18,12839,84],{"className":12840},[267],[18,12842,12844,12847],{"style":12843},"top:-3.6116em;",[18,12845],{"className":12846,"style":12712},[136],[18,12848,12852],{"className":12849,"style":12851},[12850],"hide-tail","min-width:1.02em;height:2.48em;",[6127,12853,12858],{"xmlns":6129,"width":12854,"height":12855,"viewBox":12856,"preserveAspectRatio":12857},"400em","2.48em","0 0 400000 2592","xMinYMin slice",[6133,12859],{"d":12860},"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",[18,12862,152],{"className":12863},[151],[18,12865,12867],{"className":12866},[124],[18,12868,12871],{"className":12869,"style":12870},[128],"height:0.7884em;",[18,12872],{},[18,12874,968],{"className":12875},[106],[11,12877,12878],{},"而 KL 对乘积分布是可加的：",[18,12880,12882],{"className":12881,"translate":22},[875],[18,12883,12885,12973],{"className":12884,"translate":22},[21],[18,12886,12888],{"className":12887},[26],[28,12889,12890],{"xmlns":30,"display":884},[32,12891,12892,12970],{},[35,12893,12894,12904,12906,12918,12920,12932,12934,12936,12938,12940,12950,12952,12958,12960,12966,12968],{},[38,12895,12896,12898],{},[41,12897,12572],{},[35,12899,12900,12902],{},[41,12901,5031],{"mathvariant":316},[41,12903,12579],{"mathvariant":316},[49,12905,74],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[1589,12907,12908,12910,12912],{},[41,12909,939],{},[45,12911,430],{},[35,12913,12914,12916],{},[49,12915,1599],{},[41,12917,64],{},[41,12919,2248],{"mathvariant":316},[1589,12921,12922,12924,12926],{},[41,12923,939],{},[45,12925,47],{},[35,12927,12928,12930],{},[49,12929,1599],{},[41,12931,64],{},[49,12933,84],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,12935,433],{},[41,12937,64],{},[926,12939,4024],{},[38,12941,12942,12944],{},[41,12943,12572],{},[35,12945,12946,12948],{},[41,12947,5031],{"mathvariant":316},[41,12949,12579],{"mathvariant":316},[49,12951,74],{"stretchy":73},[38,12953,12954,12956],{},[41,12955,939],{},[45,12957,430],{},[41,12959,2248],{"mathvariant":316},[38,12961,12962,12964],{},[41,12963,939],{},[45,12965,47],{},[49,12967,84],{"stretchy":73},[41,12969,968],{"mathvariant":316},[86,12971,12972],{"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：",[18,13320,13322],{"className":13321,"translate":22},[875],[18,13323,13325,13451],{"className":13324,"translate":22},[21],[18,13326,13328],{"className":13327},[26],[28,13329,13330],{"xmlns":30,"display":884},[32,13331,13332,13448],{},[35,13333,13334,13354,13370,13372,13374,13380,13382,13388,13390,13392,13398,13446],{},[890,13335,13336,13342],{},[35,13337,13338,13340],{},[41,13339,5590],{},[49,13341,899],{},[35,13343,13344,13346,13348,13350,13352],{},[41,13345,5597],{},[49,13347,433],{},[45,13349,430],{},[49,13351,52],{"separator":51},[45,13353,47],{},[38,13355,13356,13358],{},[41,13357,934],{"mathvariant":933},[1589,13359,13360,13362,13364],{},[41,13361,939],{},[41,13363,5597],{},[35,13365,13366,13368],{},[49,13367,1599],{},[41,13369,64],{},[41,13371,948],{"mathvariant":316},[49,13373,74],{"stretchy":73},[38,13375,13376,13378],{},[41,13377,77],{},[41,13379,5597],{},[49,13381,52],{"separator":51},[658,13383,13384,13386],{"accent":51},[41,13385,77],{},[49,13387,664],{},[49,13389,84],{"stretchy":73},[49,13391,4775],{},[5651,13393,13394,13396],{},[41,13395,5348],{},[45,13397,2266],{},[35,13399,13400,13402,13404,13406,13444],{},[49,13401,74],{"fence":51},[45,13403,47],{},[49,13405,2253],{},[12558,13407,13408],{},[35,13409,13410,13416,13426,13428,13434,13436,13442],{},[5651,13411,13412,13414],{},[41,13413,64],{},[45,13415,2266],{},[38,13417,13418,13420],{},[41,13419,12572],{},[35,13421,13422,13424],{},[41,13423,5031],{"mathvariant":316},[41,13425,12579],{"mathvariant":316},[49,13427,74],{"stretchy":73},[38,13429,13430,13432],{},[41,13431,939],{},[45,13433,430],{},[41,13435,2248],{"mathvariant":316},[38,13437,13438,13440],{},[41,13439,939],{},[45,13441,47],{},[49,13443,84],{"stretchy":73},[49,13445,84],{"fence":51},[41,13447,968],{"mathvariant":316},[86,13449,13450],{"encoding":88},"\\max_{i=0,1}\\mathbb 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P_1)}\\right).",[18,13452,13454,13714],{"className":13453,"ariaHidden":51},[93],[18,13455,13457,13460,13519,13522,13619,13622,13625,13665,13668,13671,13702,13705,13708,13711],{"className":13456},[97],[18,13458],{"className":13459,"style":5720},[101],[18,13461,13463],{"className":13462},[985,986],[18,13464,13466,13511],{"className":13465},[119,120],[18,13467,13469,13508],{"className":13468},[124],[18,13470,13472,13498],{"className":13471,"style":1083},[128],[18,13473,13474,13477],{"style":5735},[18,13475],{"className":13476,"style":697},[136],[18,13478,13480],{"className":13479},[141,142,143,144],[18,13481,13483,13486,13489,13492,13495],{"className":13482},[106,144],[18,13484,5597],{"className":13485},[106,110,144],[18,13487,433],{"className":13488},[232,144],[18,13490,430],{"className":13491},[106,144],[18,13493,52],{"className":13494},[165,144],[18,13496,47],{"className":13497},[106,144],[18,13499,13500,13503],{"style":693},[18,13501],{"className":13502,"style":697},[136],[18,13504,13505],{},[18,13506,5590],{"className":13507},[985],[18,13509,152],{"className":13510},[151],[18,13512,13514],{"className":13513},[124],[18,13515,13517],{"className":13516,"style":5779},[128],[18,13518],{},[18,13520],{"className":13521,"style":170},[169],[18,13523,13525,13528],{"className":13524},[106],[18,13526,934],{"className":13527},[106,1139],[18,13529,13531],{"className":13530},[115],[18,13532,13534,13611],{"className":13533},[119,120],[18,13535,13537,13608],{"className":13536},[124],[18,13538,13540],{"className":13539,"style":1184},[128],[18,13541,13542,13545],{"style":3261},[18,13543],{"className":13544,"style":137},[136],[18,13546,13548],{"className":13547},[141,142,143,144],[18,13549,13551],{"className":13550},[106,144],[18,13552,13554,13557],{"className":13553},[106,144],[18,13555,939],{"className":13556,"style":1171},[106,110,144],[18,13558,13560],{"className":13559},[115],[18,13561,13563,13600],{"className":13562},[119,120],[18,13564,13566,13597],{"className":13565},[124],[18,13567,13569,13580],{"className":13568,"style":3289},[128],[18,13570,13571,13574],{"style":5837},[18,13572],{"className":13573,"style":1191},[136],[18,13575,13577],{"className":13576},[141,1195,1196,144],[18,13578,5597],{"className":13579},[106,110,144],[18,13581,13582,13585],{"style":3304},[18,13583],{"className":13584,"style":1191},[136],[18,13586,13588],{"className":13587},[141,1195,1196,144],[18,13589,13591,13594],{"className":13590},[106,144],[18,13592,1599],{"className":13593},[106,144],[18,13595,64],{"className":13596},[106,110,144],[18,13598,152],{"className":13599},[151],[18,13601,13603],{"className":13602},[124],[18,13604,13606],{"className":13605,"style":5873},[128],[18,13607],{},[18,13609,152],{"className":13610},[151],[18,13612,13614],{"className":13613},[124],[18,13615,13617],{"className":13616,"style":5885},[128],[18,13618],{},[18,13620,948],{"className":13621},[106],[18,13623,74],{"className":13624},[250],[18,13626,13628,13631],{"className":13627},[106],[18,13629,77],{"className":13630,"style":254},[106,110],[18,13632,13634],{"className":13633},[115],[18,13635,13637,13657],{"className":13636},[119,120],[18,13638,13640,13654],{"className":13639},[124],[18,13641,13643],{"className":13642,"style":5918},[128],[18,13644,13645,13648],{"style":469},[18,13646],{"className":13647,"style":137},[136],[18,13649,13651],{"className":13650},[141,142,143,144],[18,13652,5597],{"className":13653},[106,110,144],[18,13655,152],{"className":13656},[151],[18,13658,13660],{"className":13659},[124],[18,13661,13663],{"className":13662,"style":159},[128],[18,13664],{},[18,13666,52],{"className":13667},[165],[18,13669],{"className":13670,"style":170},[169],[18,13672,13674],{"className":13673},[106,681],[18,13675,13677],{"className":13676},[119],[18,13678,13680],{"className":13679},[124],[18,13681,13683,13691],{"className":13682,"style":677},[128],[18,13684,13685,13688],{"style":693},[18,13686],{"className":13687,"style":697},[136],[18,13689,77],{"className":13690,"style":254},[106,110],[18,13692,13693,13696],{"style":703},[18,13694],{"className":13695,"style":697},[136],[18,13697,13699],{"className":13698,"style":711},[710],[18,13700,664],{"className":13701},[106],[18,13703,84],{"className":13704},[267],[18,13706],{"className":13707,"style":228},[169],[18,13709,4775],{"className":13710},[232],[18,13712],{"className":13713,"style":228},[169],[18,13715,13717,13721,13783,13786,14060,14063],{"className":13716},[97],[18,13718],{"className":13719,"style":13720},[101],"height:2.4947em;vertical-align:-0.95em;",[18,13722,13724,13727,13780],{"className":13723},[106],[18,13725],{"className":13726},[250,6009],[18,13728,13730],{"className":13729},[5651],[18,13731,13733,13772],{"className":13732},[119,120],[18,13734,13736,13769],{"className":13735},[124],[18,13737,13739,13750,13758],{"className":13738,"style":6022},[128],[18,13740,13741,13744],{"style":6025},[18,13742],{"className":13743,"style":697},[136],[18,13745,13747],{"className":13746},[106],[18,13748,2266],{"className":13749},[106],[18,13751,13752,13755],{"style":6037},[18,13753],{"className":13754,"style":697},[136],[18,13756],{"className":13757,"style":6045},[6044],[18,13759,13760,13763],{"style":6048},[18,13761],{"className":13762,"style":697},[136],[18,13764,13766],{"className":13765},[106],[18,13767,5348],{"className":13768,"style":5477},[106,110],[18,13770,152],{"className":13771},[151],[18,13773,13775],{"className":13774},[124],[18,13776,13778],{"className":13777,"style":6067},[128],[18,13779],{},[18,13781],{"className":13782},[267,6009],[18,13784],{"className":13785,"style":170},[169],[18,13787,13789,13797,13800,13803,13806,13809,14054],{"className":13788},[174],[18,13790,13794],{"className":13791,"style":13793},[250,13792],"delimcenter","top:0em;",[18,13795,74],{"className":13796},[1230,143],[18,13798,47],{"className":13799},[106],[18,13801],{"className":13802,"style":2068},[169],[18,13804,2253],{"className":13805},[2072],[18,13807],{"className":13808,"style":2068},[169],[18,13810,13812],{"className":13811},[106,12693],[18,13813,13815,14045],{"className":13814},[119,120],[18,13816,13818,14042],{"className":13817},[124],[18,13819,13822,14029],{"className":13820,"style":13821},[128],"height:1.5447em;",[18,13823,13825,13828],{"className":13824,"style":12708},[12707],[18,13826],{"className":13827,"style":12712},[136],[18,13829,13831,13894,13940,13943,13983,13986,14026],{"className":13830,"style":12716},[106],[18,13832,13834,13837,13891],{"className":13833},[106],[18,13835],{"className":13836},[250,6009],[18,13838,13840],{"className":13839},[5651],[18,13841,13843,13883],{"className":13842},[119,120],[18,13844,13846,13880],{"className":13845},[124],[18,13847,13850,13861,13869],{"className":13848,"style":13849},[128],"height:1.1076em;",[18,13851,13852,13855],{"style":6025},[18,13853],{"className":13854,"style":697},[136],[18,13856,13858],{"className":13857},[106],[18,13859,2266],{"className":13860},[106],[18,13862,13863,13866],{"style":6037},[18,13864],{"className":13865,"style":697},[136],[18,13867],{"className":13868,"style":6045},[6044],[18,13870,13871,13874],{"style":6048},[18,13872],{"className":13873,"style":697},[136],[18,13875,13877],{"className":13876},[106],[18,13878,64],{"className":13879},[106,110],[18,13881,152],{"className":13882},[151],[18,13884,13886],{"className":13885},[124],[18,13887,13889],{"className":13888,"style":6067},[128],[18,13890],{},[18,13892],{"className":13893},[267,6009],[18,13895,13897,13900],{"className":13896},[106],[18,13898,12572],{"className":13899,"style":254},[106,110],[18,13901,13903],{"className":13902},[115],[18,13904,13906,13932],{"className":13905},[119,120],[18,13907,13909,13929],{"className":13908},[124],[18,13910,13912],{"className":13911,"style":1152},[128],[18,13913,13914,13917],{"style":469},[18,13915],{"className":13916,"style":137},[136],[18,13918,13920],{"className":13919},[141,142,143,144],[18,13921,13923],{"className":13922},[106,144],[18,13924,13926],{"className":13925},[106,144],[18,13927,12814],{"className":13928},[106,6351,144],[18,13930,152],{"className":13931},[151],[18,13933,13935],{"className":13934},[124],[18,13936,13938],{"className":13937,"style":159},[128],[18,13939],{},[18,13941,74],{"className":13942},[250],[18,13944,13946,13949],{"className":13945},[106],[18,13947,939],{"className":13948,"style":1171},[106,110],[18,13950,13952],{"className":13951},[115],[18,13953,13955,13975],{"className":13954},[119,120],[18,13956,13958,13972],{"className":13957},[124],[18,13959,13961],{"className":13960,"style":129},[128],[18,13962,13963,13966],{"style":1386},[18,13964],{"className":13965,"style":137},[136],[18,13967,13969],{"className":13968},[141,142,143,144],[18,13970,430],{"className":13971},[106,144],[18,13973,152],{"className":13974},[151],[18,13976,13978],{"className":13977},[124],[18,13979,13981],{"className":13980,"style":159},[128],[18,13982],{},[18,13984,2248],{"className":13985},[106],[18,13987,13989,13992],{"className":13988},[106],[18,13990,939],{"className":13991,"style":1171},[106,110],[18,13993,13995],{"className":13994},[115],[18,13996,13998,14018],{"className":13997},[119,120],[18,13999,14001,14015],{"className":14000},[124],[18,14002,14004],{"className":14003,"style":129},[128],[18,14005,14006,14009],{"style":1386},[18,14007],{"className":14008,"style":137},[136],[18,14010,14012],{"className":14011},[141,142,143,144],[18,14013,47],{"className":14014},[106,144],[18,14016,152],{"className":14017},[151],[18,14019,14021],{"className":14020},[124],[18,14022,14024],{"className":14023,"style":159},[128],[18,14025],{},[18,14027,84],{"className":14028},[267],[18,14030,14032,14035],{"style":14031},"top:-3.5047em;",[18,14033],{"className":14034,"style":12712},[136],[18,14036,14038],{"className":14037,"style":12851},[12850],[6127,14039,14040],{"xmlns":6129,"width":12854,"height":12855,"viewBox":12856,"preserveAspectRatio":12857},[6133,14041],{"d":12860},[18,14043,152],{"className":14044},[151],[18,14046,14048],{"className":14047},[124],[18,14049,14052],{"className":14050,"style":14051},[128],"height:0.8953em;",[18,14053],{},[18,14055,14057],{"className":14056,"style":13793},[267,13792],[18,14058,84],{"className":14059},[1230,143],[18,14061],{"className":14062,"style":170},[169],[18,14064,968],{"className":14065},[106],[11,14067,14068,14069,1994,14072,14101,14102,1994,14105,14190,14191,14369],{},"这个式子把下界拆成了两个因子：",[717,14070,14071],{},"损失间隔",[18,14073,14075,14089],{"className":14074,"translate":22},[21],[18,14076,14078],{"className":14077},[26],[28,14079,14080],{"xmlns":30},[32,14081,14082,14086],{},[35,14083,14084],{},[41,14085,5348],{},[86,14087,14088],{"encoding":88},"\\delta",[18,14090,14092],{"className":14091,"ariaHidden":51},[93],[18,14093,14095,14098],{"className":14094},[97],[18,14096],{"className":14097,"style":295},[101],[18,14099,5348],{"className":14100,"style":5477},[106,110],"，和",[717,14103,14104],{},"信息量",[18,14106,14108,14132],{"className":14107,"translate":22},[21],[18,14109,14111],{"className":14110},[26],[28,14112,14113],{"xmlns":30},[32,14114,14115,14129],{},[35,14116,14117,14119],{},[41,14118,64],{},[38,14120,14121,14123],{},[41,14122,12572],{},[35,14124,14125,14127],{},[41,14126,5031],{"mathvariant":316},[41,14128,12579],{"mathvariant":316},[86,14130,14131],{"encoding":88},"nD_{\\mathrm{KL}}",[18,14133,14135],{"className":14134,"ariaHidden":51},[93],[18,14136,14138,14141,14144],{"className":14137},[97],[18,14139],{"className":14140,"style":4563},[101],[18,14142,64],{"className":14143},[106,110],[18,14145,14147,14150],{"className":14146},[106],[18,14148,12572],{"className":14149,"style":254},[106,110],[18,14151,14153],{"className":14152},[115],[18,14154,14156,14182],{"className":14155},[119,120],[18,14157,14159,14179],{"className":14158},[124],[18,14160,14162],{"className":14161,"style":1152},[128],[18,14163,14164,14167],{"style":469},[18,14165],{"className":14166,"style":137},[136],[18,14168,14170],{"className":14169},[141,142,143,144],[18,14171,14173],{"className":14172},[106,144],[18,14174,14176],{"className":14175},[106,144],[18,14177,12814],{"className":14178},[106,6351,144],[18,14180,152],{"className":14181},[151],[18,14183,14185],{"className":14184},[124],[18,14186,14188],{"className":14187,"style":159},[128],[18,14189],{},"。要让下界为正，必须 ",[18,14192,14194,14226],{"className":14193,"translate":22},[21],[18,14195,14197],{"className":14196},[26],[28,14198,14199],{"xmlns":30},[32,14200,14201,14223],{},[35,14202,14203,14209,14219,14221],{},[5651,14204,14205,14207],{},[41,14206,64],{},[45,14208,2266],{},[38,14210,14211,14213],{},[41,14212,12572],{},[35,14214,14215,14217],{},[41,14216,5031],{"mathvariant":316},[41,14218,12579],{"mathvariant":316},[49,14220,6681],{},[45,14222,47],{},[86,14224,14225],{"encoding":88},"\\frac n2D_{\\mathrm{KL}}\u003C1",[18,14227,14229,14360],{"className":14228,"ariaHidden":51},[93],[18,14230,14232,14236,14305,14351,14354,14357],{"className":14231},[97],[18,14233],{"className":14234,"style":14235},[101],"height:1.0404em;vertical-align:-0.345em;",[18,14237,14239,14242,14302],{"className":14238},[106],[18,14240],{"className":14241},[250,6009],[18,14243,14245],{"className":14244},[5651],[18,14246,14248,14294],{"className":14247},[119,120],[18,14249,14251,14291],{"className":14250},[124],[18,14252,14255,14269,14277],{"className":14253,"style":14254},[128],"height:0.6954em;",[18,14256,14257,14260],{"style":12045},[18,14258],{"className":14259,"style":697},[136],[18,14261,14263],{"className":14262},[141,142,143,144],[18,14264,14266],{"className":14265},[106,144],[18,14267,2266],{"className":14268},[106,144],[18,14270,14271,14274],{"style":6037},[18,14272],{"className":14273,"style":697},[136],[18,14275],{"className":14276,"style":6045},[6044],[18,14278,14279,14282],{"style":12068},[18,14280],{"className":14281,"style":697},[136],[18,14283,14285],{"className":14284},[141,142,143,144],[18,14286,14288],{"className":14287},[106,144],[18,14289,64],{"className":14290},[106,110,144],[18,14292,152],{"className":14293},[151],[18,14295,14297],{"className":14296},[124],[18,14298,14300],{"className":14299,"style":12090},[128],[18,14301],{},[18,14303],{"className":14304},[267,6009],[18,14306,14308,14311],{"className":14307},[106],[18,14309,12572],{"className":14310,"style":254},[106,110],[18,14312,14314],{"className":14313},[115],[18,14315,14317,14343],{"className":14316},[119,120],[18,14318,14320,14340],{"className":14319},[124],[18,14321,14323],{"className":14322,"style":1152},[128],[18,14324,14325,14328],{"style":469},[18,14326],{"className":14327,"style":137},[136],[18,14329,14331],{"className":14330},[141,142,143,144],[18,14332,14334],{"className":14333},[106,144],[18,14335,14337],{"className":14336},[106,144],[18,14338,12814],{"className":14339},[106,6351,144],[18,14341,152],{"className":14342},[151],[18,14344,14346],{"className":14345},[124],[18,14347,14349],{"className":14348,"style":159},[128],[18,14350],{},[18,14352],{"className":14353,"style":228},[169],[18,14355,6681],{"className":14356},[232],[18,14358],{"className":14359,"style":228},[169],[18,14361,14363,14366],{"className":14362},[97],[18,14364],{"className":14365,"style":505},[101],[18,14367,47],{"className":14368},[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。",[11,14371,14372,14373,14408],{},"这也解释了为什么下界里的样本量总以「",[18,14374,14376,14392],{"className":14375,"translate":22},[21],[18,14377,14379],{"className":14378},[26],[28,14380,14381],{"xmlns":30},[32,14382,14383,14389],{},[35,14384,14385,14387],{},[41,14386,64],{},[49,14388,2015],{},[86,14390,14391],{"encoding":88},"n\\times",[18,14393,14395],{"className":14394,"ariaHidden":51},[93],[18,14396,14398,14402,14405],{"className":14397},[97],[18,14399],{"className":14400,"style":14401},[101],"height:0.6667em;vertical-align:-0.0833em;",[18,14403,64],{"className":14404},[106,110],[18,14406,2015],{"className":14407},[106]," 单样本信息」的形式出现：独立样本的信息是线性累加的。",[1311,14410,14412],{"id":14411},"_33-一个例子bernoulli-均值","3.3 一个例子：Bernoulli 均值",[11,14414,16,14415,4741,14534,14639],{},[18,14416,14418,14454],{"className":14417,"translate":22},[21],[18,14419,14421],{"className":14420},[26],[28,14422,14423],{"xmlns":30},[32,14424,14425,14451],{},[35,14426,14427,14433,14435,14445,14447,14449],{},[38,14428,14429,14431],{},[41,14430,43],{},[41,14432,5597],{},[49,14434,67],{},[35,14436,14437,14439,14442],{},[41,14438,3978],{"mathvariant":316},[41,14440,14441],{"mathvariant":316},"e",[41,14443,14444],{"mathvariant":316},"r",[49,14446,74],{"stretchy":73},[41,14448,11],{},[49,14450,84],{"stretchy":73},[86,14452,14453],{"encoding":88},"X_i\\sim\\mathrm{Ber}(p)",[18,14455,14457,14512],{"className":14456,"ariaHidden":51},[93],[18,14458,14460,14463,14503,14506,14509],{"className":14459},[97],[18,14461],{"className":14462,"style":4563},[101],[18,14464,14466,14469],{"className":14465},[106],[18,14467,43],{"className":14468,"style":111},[106,110],[18,14470,14472],{"className":14471},[115],[18,14473,14475,14495],{"className":14474},[119,120],[18,14476,14478,14492],{"className":14477},[124],[18,14479,14481],{"className":14480,"style":5918},[128],[18,14482,14483,14486],{"style":132},[18,14484],{"className":14485,"style":137},[136],[18,14487,14489],{"className":14488},[141,142,143,144],[18,14490,5597],{"className":14491},[106,110,144],[18,14493,152],{"className":14494},[151],[18,14496,14498],{"className":14497},[124],[18,14499,14501],{"className":14500,"style":159},[128],[18,14502],{},[18,14504],{"className":14505,"style":228},[169],[18,14507,67],{"className":14508},[232],[18,14510],{"className":14511,"style":228},[169],[18,14513,14515,14518,14525,14528,14531],{"className":14514},[97],[18,14516],{"className":14517,"style":242},[101],[18,14519,14521],{"className":14520},[106],[18,14522,14524],{"className":14523},[106,6351],"Ber",[18,14526,74],{"className":14527},[250],[18,14529,11],{"className":14530},[106,110],[18,14532,84],{"className":14533},[267],[18,14535,14537,14575],{"className":14536,"translate":22},[21],[18,14538,14540],{"className":14539},[26],[28,14541,14542],{"xmlns":30},[32,14543,14544,14572],{},[35,14545,14546,14548,14550,14552,14554,14556,14558,14560,14562,14564,14566,14568,14570],{},[41,14547,11],{},[49,14549,921],{},[49,14551,1331],{"stretchy":73},[45,14553,47],{},[41,14555,317],{"mathvariant":316},[45,14557,2266],{},[49,14559,52],{"separator":51},[45,14561,47],{},[41,14563,317],{"mathvariant":316},[45,14565,2266],{},[49,14567,7984],{},[41,14569,532],{"mathvariant":316},[49,14571,1349],{"stretchy":73},[86,14573,14574],{"encoding":88},"p\\in\\{1\u002F2,1\u002F2+\\Delta\\}",[18,14576,14578,14597,14627],{"className":14577,"ariaHidden":51},[93],[18,14579,14581,14585,14588,14591,14594],{"className":14580},[97],[18,14582],{"className":14583,"style":14584},[101],"height:0.7335em;vertical-align:-0.1944em;",[18,14586,11],{"className":14587},[106,110],[18,14589],{"className":14590,"style":228},[169],[18,14592,921],{"className":14593},[232],[18,14595],{"className":14596,"style":228},[169],[18,14598,14600,14603,14606,14609,14612,14615,14618,14621,14624],{"className":14599},[97],[18,14601],{"className":14602,"style":242},[101],[18,14604,1331],{"className":14605},[250],[18,14607,9057],{"className":14608},[106],[18,14610,52],{"className":14611},[165],[18,14613],{"className":14614,"style":170},[169],[18,14616,9057],{"className":14617},[106],[18,14619],{"className":14620,"style":2068},[169],[18,14622,7984],{"className":14623},[2072],[18,14625],{"className":14626,"style":2068},[169],[18,14628,14630,14633,14636],{"className":14629},[97],[18,14631],{"className":14632,"style":242},[101],[18,14634,532],{"className":14635},[106],[18,14637,1349],{"className":14638},[267],"，损失为平方损失。两个分布的单样本 KL 是",[18,14641,14643],{"className":14642,"translate":22},[875],[18,14644,14646,14777],{"className":14645,"translate":22},[21],[18,14647,14649],{"className":14648},[26],[28,14650,14651],{"xmlns":30,"display":884},[32,14652,14653,14774],{},[35,14654,14655,14665,14667,14675,14677,14686,14688,14690,14698,14700,14708,14710,14712,14714,14716,14718,14720,14722,14728,14731,14733,14735,14737,14739,14741,14747,14749,14752,14754,14760,14762,14764,14766,14768,14770,14772],{},[38,14656,14657,14659],{},[41,14658,12572],{},[35,14660,14661,14663],{},[41,14662,5031],{"mathvariant":316},[41,14664,12579],{"mathvariant":316},[49,14666,74],{"fence":51,"stretchy":51,"minsize":5659,"maxsize":5659},[35,14668,14669,14671,14673],{},[41,14670,3978],{"mathvariant":316},[41,14672,14441],{"mathvariant":316},[41,14674,14444],{"mathvariant":316},[49,14676,74],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[14678,14679,14680],"mstyle",{"scriptlevel":430,"displaystyle":73},[5651,14681,14682,14684],{},[45,14683,47],{},[45,14685,2266],{},[49,14687,84],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,14689,2248],{"fence":73,"stretchy":51,"minsize":5659,"maxsize":5659},[35,14691,14692,14694,14696],{},[41,14693,3978],{"mathvariant":316},[41,14695,14441],{"mathvariant":316},[41,14697,14444],{"mathvariant":316},[49,14699,74],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[14678,14701,14702],{"scriptlevel":430,"displaystyle":73},[5651,14703,14704,14706],{},[45,14705,47],{},[45,14707,2266],{},[49,14709,7984],{},[41,14711,532],{"mathvariant":316},[49,14713,84],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,14715,84],{"fence":51,"stretchy":51,"minsize":5659,"maxsize":5659},[169,14717],{"linebreak":8019},[49,14719,433],{},[49,14721,2253],{},[5651,14723,14724,14726],{},[45,14725,47],{},[45,14727,2266],{},[41,14729,14730],{},"log",[49,14732,899],{},[49,14734,74],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[45,14736,47],{},[49,14738,2253],{},[45,14740,10444],{},[1556,14742,14743,14745],{},[41,14744,532],{"mathvariant":316},[45,14746,2266],{},[49,14748,84],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,14750,14751],{},"≈",[45,14753,2266],{},[1556,14755,14756,14758],{},[41,14757,532],{"mathvariant":316},[45,14759,2266],{},[169,14761],{"width":7341},[49,14763,74],{"stretchy":73},[41,14765,532],{"mathvariant":316},[49,14767,1880],{},[45,14769,430],{},[49,14771,84],{"stretchy":73},[41,14773,968],{"mathvariant":316},[86,14775,14776],{"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).",[18,14778,14780,15057,15078,15081,15094,15193,15247,15304],{"className":14779,"ariaHidden":51},[93],[18,14781,14783,14786,14832,14838,14844,14850,14918,14924,14968,14974,14980,15048,15051,15054],{"className":14782},[97],[18,14784],{"className":14785,"style":6222},[101],[18,14787,14789,14792],{"className":14788},[106],[18,14790,12572],{"className":14791,"style":254},[106,110],[18,14793,14795],{"className":14794},[115],[18,14796,14798,14824],{"className":14797},[119,120],[18,14799,14801,14821],{"className":14800},[124],[18,14802,14804],{"className":14803,"style":1152},[128],[18,14805,14806,14809],{"style":469},[18,14807],{"className":14808,"style":137},[136],[18,14810,14812],{"className":14811},[141,142,143,144],[18,14813,14815],{"className":14814},[106,144],[18,14816,14818],{"className":14817},[106,144],[18,14819,12814],{"className":14820},[106,6351,144],[18,14822,152],{"className":14823},[151],[18,14825,14827],{"className":14826},[124],[18,14828,14830],{"className":14829,"style":159},[128],[18,14831],{},[18,14833,14835],{"className":14834},[250],[18,14836,74],{"className":14837},[1230,6079],[18,14839,14841],{"className":14840},[106],[18,14842,14524],{"className":14843},[106,6351],[18,14845,14847],{"className":14846},[250],[18,14848,74],{"className":14849},[1230,1196],[18,14851,14853,14856,14915],{"className":14852},[106],[18,14854],{"className":14855},[250,6009],[18,14857,14859],{"className":14858},[5651],[18,14860,14862,14907],{"className":14861},[119,120],[18,14863,14865,14904],{"className":14864},[124],[18,14866,14868,14882,14890],{"className":14867,"style":12042},[128],[18,14869,14870,14873],{"style":12045},[18,14871],{"className":14872,"style":697},[136],[18,14874,14876],{"className":14875},[141,142,143,144],[18,14877,14879],{"className":14878},[106,144],[18,14880,2266],{"className":14881},[106,144],[18,14883,14884,14887],{"style":6037},[18,14885],{"className":14886,"style":697},[136],[18,14888],{"className":14889,"style":6045},[6044],[18,14891,14892,14895],{"style":12068},[18,14893],{"className":14894,"style":697},[136],[18,14896,14898],{"className":14897},[141,142,143,144],[18,14899,14901],{"className":14900},[106,144],[18,14902,47],{"className":14903},[106,144],[18,14905,152],{"className":14906},[151],[18,14908,14910],{"className":14909},[124],[18,14911,14913],{"className":14912,"style":12090},[128],[18,14914],{},[18,14916],{"className":14917},[267,6009],[18,14919,14921],{"className":14920},[267],[18,14922,84],{"className":14923},[1230,1196],[18,14925,14927],{"className":14926},[106],[18,14928,14930],{"className":14929},[1230,6105],[18,14931,14933,14959],{"className":14932},[119,120],[18,14934,14936,14956],{"className":14935},[124],[18,14937,14940],{"className":14938,"style":14939},[128],"height:1.15em;",[18,14941,14943,14947],{"style":14942},"top:-3.15em;",[18,14944],{"className":14945,"style":14946},[136],"height:3.8em;",[18,14948,14950],{"style":14949},"width:0.556em;height:1.8em;",[6127,14951,14953],{"xmlns":6129,"width":6130,"height":5659,"viewBox":14952},"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}.",[18,15760,15762,15878,15971,16168],{"className":15761,"ariaHidden":51},[93],[18,15763,15765,15769,15814,15817,15820,15826,15829,15869,15872,15875],{"className":15764},[97],[18,15766],{"className":15767,"style":15768},[101],"height:1.5777em;vertical-align:-0.7277em;",[18,15770,15772],{"className":15771},[985,986],[18,15773,15775,15805],{"className":15774},[119,120],[18,15776,15778,15802],{"className":15777},[124],[18,15779,15781,15792],{"className":15780,"style":1083},[128],[18,15782,15783,15786],{"style":5735},[18,15784],{"className":15785,"style":697},[136],[18,15787,15789],{"className":15788},[141,142,143,144],[18,15790,5597],{"className":15791},[106,110,144],[18,15793,15794,15797],{"style":693},[18,15795],{"className":15796,"style":697},[136],[18,15798,15799],{},[18,15800,5590],{"className":15801},[985],[18,15803,152],{"className":15804},[151],[18,15806,15808],{"className":15807},[124],[18,15809,15812],{"className":15810,"style":15811},[128],"height:0.7277em;",[18,15813],{},[18,15815],{"className":15816,"style":170},[169],[18,15818,934],{"className":15819},[106,1139],[18,15821,15823],{"className":15822},[250],[18,15824,945],{"className":15825},[1230,1196],[18,15827,74],{"className":15828},[250],[18,15830,15832,15835],{"className":15831},[106],[18,15833,11],{"className":15834},[106,110],[18,15836,15838],{"className":15837},[115],[18,15839,15841,15861],{"className":15840},[119,120],[18,15842,15844,15858],{"className":15843},[124],[18,15845,15847],{"className":15846,"style":5918},[128],[18,15848,15849,15852],{"style":1155},[18,15850],{"className":15851,"style":137},[136],[18,15853,15855],{"className":15854},[141,142,143,144],[18,15856,5597],{"className":15857},[106,110,144],[18,15859,152],{"className":15860},[151],[18,15862,15864],{"className":15863},[124],[18,15865,15867],{"className":15866,"style":159},[128],[18,15868],{},[18,15870],{"className":15871,"style":2068},[169],[18,15873,2253],{"className":15874},[2072],[18,15876],{"className":15877,"style":2068},[169],[18,15879,15881,15884,15927,15956,15962,15965,15968],{"className":15880},[97],[18,15882],{"className":15883,"style":15199},[101],[18,15885,15887],{"className":15886},[106,681],[18,15888,15890,15918],{"className":15889},[119,120],[18,15891,15893,15915],{"className":15892},[124],[18,15894,15896,15904],{"className":15895,"style":295},[128],[18,15897,15898,15901],{"style":693},[18,15899],{"className":15900,"style":697},[136],[18,15902,11],{"className":15903},[106,110],[18,15905,15906,15909],{"style":693},[18,15907],{"className":15908,"style":697},[136],[18,15910,15912],{"className":15911,"style":711},[710],[18,15913,664],{"className":15914},[106],[18,15916,152],{"className":15917},[151],[18,15919,15921],{"className":15920},[124],[18,15922,15925],{"className":15923,"style":15924},[128],"height:0.1944em;",[18,15926],{},[18,15928,15930,15933],{"className":15929},[267],[18,15931,84],{"className":15932},[267],[18,15934,15936],{"className":15935},[115],[18,15937,15939],{"className":15938},[119],[18,15940,15942],{"className":15941},[124],[18,15943,15945],{"className":15944,"style":10772},[128],[18,15946,15947,15950],{"style":1637},[18,15948],{"className":15949,"style":137},[136],[18,15951,15953],{"className":15952},[141,142,143,144],[18,15954,2266],{"className":15955},[106,144],[18,15957,15959],{"className":15958},[267],[18,15960,965],{"className":15961},[1230,1196],[18,15963],{"className":15964,"style":228},[169],[18,15966,4775],{"className":15967},[232],[18,15969],{"className":15970,"style":228},[169],[18,15972,15974,15978,16067,16070,16159,16162,16165],{"className":15973},[97],[18,15975],{"className":15976,"style":15977},[101],"height:2.4411em;vertical-align:-0.95em;",[18,15979,15981,15984,16064],{"className":15980},[106],[18,15982],{"className":15983},[250,6009],[18,15985,15987],{"className":15986},[5651],[18,15988,15990,16056],{"className":15989},[119,120],[18,15991,15993,16053],{"className":15992},[124],[18,15994,15997,16008,16016],{"className":15995,"style":15996},[128],"height:1.4911em;",[18,15998,15999,16002],{"style":6025},[18,16000],{"className":16001,"style":697},[136],[18,16003,16005],{"className":16004},[106],[18,16006,10444],{"className":16007},[106],[18,16009,16010,16013],{"style":6037},[18,16011],{"className":16012,"style":697},[136],[18,16014],{"className":16015,"style":6045},[6044],[18,16017,16018,16021],{"style":6048},[18,16019],{"className":16020,"style":697},[136],[18,16022,16024],{"className":16023},[106],[18,16025,16027,16030],{"className":16026},[106],[18,16028,532],{"className":16029},[106],[18,16031,16033],{"className":16032},[115],[18,16034,16036],{"className":16035},[119],[18,16037,16039],{"className":16038},[124],[18,16040,16042],{"className":16041,"style":2416},[128],[18,16043,16044,16047],{"style":1962},[18,16045],{"className":16046,"style":137},[136],[18,16048,16050],{"className":16049},[141,142,143,144],[18,16051,2266],{"className":16052},[106,144],[18,16054,152],{"className":16055},[151],[18,16057,16059],{"className":16058},[124],[18,16060,16062],{"className":16061,"style":6067},[128],[18,16063],{},[18,16065],{"className":16066},[267,6009],[18,16068],{"className":16069,"style":170},[169],[18,16071,16073,16079,16082,16085,16088,16091,16153],{"className":16072},[174],[18,16074,16076],{"className":16075,"style":13793},[250,13792],[18,16077,74],{"className":16078},[1230,143],[18,16080,47],{"className":16081},[106],[18,16083],{"className":16084,"style":2068},[169],[18,16086,2253],{"className":16087},[2072],[18,16089],{"className":16090,"style":2068},[169],[18,16092,16094,16097,16150],{"className":16093},[106],[18,16095],{"className":16096},[250,6009],[18,16098,16100],{"className":16099},[5651],[18,16101,16103,16142],{"className":16102},[119,120],[18,16104,16106,16139],{"className":16105},[124],[18,16107,16109,16120,16128],{"className":16108,"style":9578},[128],[18,16110,16111,16114],{"style":6025},[18,16112],{"className":16113,"style":697},[136],[18,16115,16117],{"className":16116},[106],[18,16118,2266],{"className":16119},[106],[18,16121,16122,16125],{"style":6037},[18,16123],{"className":16124,"style":697},[136],[18,16126],{"className":16127,"style":6045},[6044],[18,16129,16130,16133],{"style":6048},[18,16131],{"className":16132,"style":697},[136],[18,16134,16136],{"className":16135},[106],[18,16137,47],{"className":16138},[106],[18,16140,152],{"className":16141},[151],[18,16143,16145],{"className":16144},[124],[18,16146,16148],{"className":16147,"style":6067},[128],[18,16149],{},[18,16151],{"className":16152},[267,6009],[18,16154,16156],{"className":16155,"style":13793},[267,13792],[18,16157,84],{"className":16158},[1230,143],[18,16160],{"className":16161,"style":228},[169],[18,16163,433],{"className":16164},[232],[18,16166],{"className":16167,"style":228},[169],[18,16169,16171,16174,16239],{"className":16170},[97],[18,16172],{"className":16173,"style":9559},[101],[18,16175,16177,16180,16236],{"className":16176},[106],[18,16178],{"className":16179},[250,6009],[18,16181,16183],{"className":16182},[5651],[18,16184,16186,16228],{"className":16185},[119,120],[18,16187,16189,16225],{"className":16188},[124],[18,16190,16192,16206,16214],{"className":16191,"style":9578},[128],[18,16193,16194,16197],{"style":6025},[18,16195],{"className":16196,"style":697},[136],[18,16198,16200,16203],{"className":16199},[106],[18,16201,15751],{"className":16202},[106],[18,16204,64],{"className":16205},[106,110],[18,16207,16208,16211],{"style":6037},[18,16209],{"className":16210,"style":697},[136],[18,16212],{"className":16213,"style":6045},[6044],[18,16215,16216,16219],{"style":6048},[18,16217],{"className":16218,"style":697},[136],[18,16220,16222],{"className":16221},[106],[18,16223,47],{"className":16224},[106],[18,16226,152],{"className":16227},[151],[18,16229,16231],{"className":16230},[124],[18,16232,16234],{"className":16233,"style":6067},[128],[18,16235],{},[18,16237],{"className":16238},[267,6009],[18,16240,968],{"className":16241},[106],[11,16243,16244,16245,16280,16281],{},"所以 Bernoulli 均值的极小极大速率是 ",[18,16246,16248,16265],{"className":16247,"translate":22},[21],[18,16249,16251],{"className":16250},[26],[28,16252,16253],{"xmlns":30},[32,16254,16255,16263],{},[35,16256,16257,16259,16261],{},[45,16258,47],{},[41,16260,317],{"mathvariant":316},[41,16262,64],{},[86,16264,322],{"encoding":88},[18,16266,16268],{"className":16267,"ariaHidden":51},[93],[18,16269,16271,16274,16277],{"className":16270},[97],[18,16272],{"className":16273,"style":242},[101],[18,16275,335],{"className":16276},[106],[18,16278,64],{"className":16279},[106,110],"。这个结论不稀奇，但推导方式值得记住：",[717,16282,16283,16284,16319],{},"我们只用了两个点，就把「",[18,16285,16287,16304],{"className":16286,"translate":22},[21],[18,16288,16290],{"className":16289},[26],[28,16291,16292],{"xmlns":30},[32,16293,16294,16302],{},[35,16295,16296,16298,16300],{},[45,16297,47],{},[41,16299,317],{"mathvariant":316},[41,16301,64],{},[86,16303,322],{"encoding":88},[18,16305,16307],{"className":16306,"ariaHidden":51},[93],[18,16308,16310,16313,16316],{"className":16309},[97],[18,16311],{"className":16312,"style":242},[101],[18,16314,335],{"className":16315},[106],[18,16317,64],{"className":16318},[106,110]," 不可能被突破」这件事证明完了。",[1311,16321,16323],{"id":16322},"_34-两点方法的局限","3.4 两点方法的局限",[11,16325,16326,16327,16355],{},"两点方法简单、常数清楚，但它只能看见一个方向。考虑 ",[18,16328,16330,16343],{"className":16329,"translate":22},[21],[18,16331,16333],{"className":16332},[26],[28,16334,16335],{"xmlns":30},[32,16336,16337,16341],{},[35,16338,16339],{},[41,16340,4031],{},[86,16342,4031],{"encoding":88},[18,16344,16346],{"className":16345,"ariaHidden":51},[93],[18,16347,16349,16352],{"className":16348},[97],[18,16350],{"className":16351,"style":295},[101],[18,16353,4031],{"className":16354},[106,110]," 维高斯均值：",[18,16357,16359],{"className":16358,"translate":22},[875],[18,16360,16362,16434],{"className":16361,"translate":22},[21],[18,16363,16365],{"className":16364},[26],[28,16366,16367],{"xmlns":30,"display":884},[32,16368,16369,16431],{},[35,16370,16371,16377,16379,16381,16383,16389,16391,16393,16395,16397,16399,16406,16413,16415,16417,16419,16421,16423,16429],{},[38,16372,16373,16375],{},[41,16374,43],{},[45,16376,47],{},[49,16378,52],{"separator":51},[49,16380,55],{},[49,16382,52],{"separator":51},[38,16384,16385,16387],{},[41,16386,43],{},[41,16388,64],{},[49,16390,67],{},[41,16392,70],{},[49,16394,74],{"stretchy":73},[41,16396,77],{},[49,16398,52],{"separator":51},[1556,16400,16401,16404],{},[41,16402,16403],{},"σ",[45,16405,2266],{},[38,16407,16408,16411],{},[41,16409,16410],{},"I",[41,16412,4031],{},[49,16414,84],{"stretchy":73},[49,16416,52],{"separator":51},[169,16418],{"width":7341},[41,16420,77],{},[49,16422,921],{},[1556,16424,16425,16427],{},[41,16426,3071],{"mathvariant":933},[41,16428,4031],{},[41,16430,968],{"mathvariant":316},[86,16432,16433],{"encoding":88},"X_1,\\dots,X_n\\sim N(\\theta,\\sigma^2 I_d),\\qquad \\theta\\in\\mathbb R^d.",[18,16435,16437,16550,16664],{"className":16436,"ariaHidden":51},[93],[18,16438,16440,16443,16483,16486,16489,16492,16495,16498,16501,16541,16544,16547],{"className":16439},[97],[18,16441],{"className":16442,"style":102},[101],[18,16444,16446,16449],{"className":16445},[106],[18,16447,43],{"className":16448,"style":111},[106,110],[18,16450,16452],{"className":16451},[115],[18,16453,16455,16475],{"className":16454},[119,120],[18,16456,16458,16472],{"className":16457},[124],[18,16459,16461],{"className":16460,"style":129},[128],[18,16462,16463,16466],{"style":132},[18,16464],{"className":16465,"style":137},[136],[18,16467,16469],{"className":16468},[141,142,143,144],[18,16470,47],{"className":16471},[106,144],[18,16473,152],{"className":16474},[151],[18,16476,16478],{"className":16477},[124],[18,16479,16481],{"className":16480,"style":159},[128],[18,16482],{},[18,16484,52],{"className":16485},[165],[18,16487],{"className":16488,"style":170},[169],[18,16490,55],{"className":16491},[174],[18,16493],{"className":16494,"style":170},[169],[18,16496,52],{"className":16497},[165],[18,16499],{"className":16500,"style":170},[169],[18,16502,16504,16507],{"className":16503},[106],[18,16505,43],{"className":16506,"style":111},[106,110],[18,16508,16510],{"className":16509},[115],[18,16511,16513,16533],{"className":16512},[119,120],[18,16514,16516,16530],{"className":16515},[124],[18,16517,16519],{"className":16518,"style":202},[128],[18,16520,16521,16524],{"style":132},[18,16522],{"className":16523,"style":137},[136],[18,16525,16527],{"className":16526},[141,142,143,144],[18,16528,64],{"className":16529},[106,110,144],[18,16531,152],{"className":16532},[151],[18,16534,16536],{"className":16535},[124],[18,16537,16539],{"className":16538,"style":159},[128],[18,16540],{},[18,16542],{"className":16543,"style":228},[169],[18,16545,67],{"className":16546},[232],[18,16548],{"className":16549,"style":228},[169],[18,16551,16553,16556,16559,16562,16565,16568,16571,16600,16640,16643,16646,16649,16652,16655,16658,16661],{"className":16552},[97],[18,16554],{"className":16555,"style":15253},[101],[18,16557,70],{"className":16558,"style":246},[106,110],[18,16560,74],{"className":16561},[250],[18,16563,77],{"className":16564,"style":254},[106,110],[18,16566,52],{"className":16567},[165],[18,16569],{"className":16570,"style":170},[169],[18,16572,16574,16577],{"className":16573},[106],[18,16575,16403],{"className":16576,"style":3957},[106,110],[18,16578,16580],{"className":16579},[115],[18,16581,16583],{"className":16582},[119],[18,16584,16586],{"className":16585},[124],[18,16587,16589],{"className":16588,"style":10772},[128],[18,16590,16591,16594],{"style":1637},[18,16592],{"className":16593,"style":137},[136],[18,16595,16597],{"className":16596},[141,142,143,144],[18,16598,2266],{"className":16599},[106,144],[18,16601,16603,16606],{"className":16602},[106],[18,16604,16410],{"className":16605,"style":111},[106,110],[18,16607,16609],{"className":16608},[115],[18,16610,16612,16632],{"className":16611},[119,120],[18,16613,16615,16629],{"className":16614},[124],[18,16616,16618],{"className":16617,"style":1383},[128],[18,16619,16620,16623],{"style":132},[18,16621],{"className":16622,"style":137},[136],[18,16624,16626],{"className":16625},[141,142,143,144],[18,16627,4031],{"className":16628},[106,110,144],[18,16630,152],{"className":16631},[151],[18,16633,16635],{"className":16634},[124],[18,16636,16638],{"className":16637,"style":159},[128],[18,16639],{},[18,16641,84],{"className":16642},[267],[18,16644,52],{"className":16645},[165],[18,16647],{"className":16648,"style":7673},[169],[18,16650],{"className":16651,"style":170},[169],[18,16653,77],{"className":16654,"style":254},[106,110],[18,16656],{"className":16657,"style":228},[169],[18,16659,921],{"className":16660},[232],[18,16662],{"className":16663,"style":228},[169],[18,16665,16667,16671,16700],{"className":16666},[97],[18,16668],{"className":16669,"style":16670},[101],"height:0.8991em;",[18,16672,16674,16677],{"className":16673},[106],[18,16675,3071],{"className":16676},[106,1139],[18,16678,16680],{"className":16679},[115],[18,16681,16683],{"className":16682},[119],[18,16684,16686],{"className":16685},[124],[18,16687,16689],{"className":16688,"style":16670},[128],[18,16690,16691,16694],{"style":1637},[18,16692],{"className":16693,"style":137},[136],[18,16695,16697],{"className":16696},[141,142,143,144],[18,16698,4031],{"className":16699},[106,110,144],[18,16701,968],{"className":16702},[106],[11,16704,16705,16706,16797,16798,16938,16939,17058,17059,17126,17127,17233,17234,17269,17270,17344,17345,1990],{},"如果只取 ",[18,16707,16709,16730],{"className":16708,"translate":22},[21],[18,16710,16712],{"className":16711},[26],[28,16713,16714],{"xmlns":30},[32,16715,16716,16728],{},[35,16717,16718,16724,16726],{},[38,16719,16720,16722],{},[41,16721,77],{},[45,16723,430],{},[49,16725,433],{},[45,16727,430],{},[86,16729,438],{"encoding":88},[18,16731,16733,16788],{"className":16732,"ariaHidden":51},[93],[18,16734,16736,16739,16779,16782,16785],{"className":16735},[97],[18,16737],{"className":16738,"style":448},[101],[18,16740,16742,16745],{"className":16741},[106],[18,16743,77],{"className":16744,"style":254},[106,110],[18,16746,16748],{"className":16747},[115],[18,16749,16751,16771],{"className":16750},[119,120],[18,16752,16754,16768],{"className":16753},[124],[18,16755,16757],{"className":16756,"style":129},[128],[18,16758,16759,16762],{"style":469},[18,16760],{"className":16761,"style":137},[136],[18,16763,16765],{"className":16764},[141,142,143,144],[18,16766,430],{"className":16767},[106,144],[18,16769,152],{"className":16770},[151],[18,16772,16774],{"className":16773},[124],[18,16775,16777],{"className":16776,"style":159},[128],[18,16778],{},[18,16780],{"className":16781,"style":228},[169],[18,16783,433],{"className":16784},[232],[18,16786],{"className":16787,"style":228},[169],[18,16789,16791,16794],{"className":16790},[97],[18,16792],{"className":16793,"style":505},[101],[18,16795,430],{"className":16796},[106],"、",[18,16799,16801,16830],{"className":16800,"translate":22},[21],[18,16802,16804],{"className":16803},[26],[28,16805,16806],{"xmlns":30},[32,16807,16808,16827],{},[35,16809,16810,16816,16818,16821],{},[38,16811,16812,16814],{},[41,16813,77],{},[45,16815,47],{},[49,16817,433],{},[41,16819,16820],{},"a",[38,16822,16823,16825],{},[41,16824,14441],{},[45,16826,47],{},[86,16828,16829],{"encoding":88},"\\theta_1=a e_1",[18,16831,16833,16888],{"className":16832,"ariaHidden":51},[93],[18,16834,16836,16839,16879,16882,16885],{"className":16835},[97],[18,16837],{"className":16838,"style":448},[101],[18,16840,16842,16845],{"className":16841},[106],[18,16843,77],{"className":16844,"style":254},[106,110],[18,16846,16848],{"className":16847},[115],[18,16849,16851,16871],{"className":16850},[119,120],[18,16852,16854,16868],{"className":16853},[124],[18,16855,16857],{"className":16856,"style":129},[128],[18,16858,16859,16862],{"style":469},[18,16860],{"className":16861,"style":137},[136],[18,16863,16865],{"className":16864},[141,142,143,144],[18,16866,47],{"className":16867},[106,144],[18,16869,152],{"className":16870},[151],[18,16872,16874],{"className":16873},[124],[18,16875,16877],{"className":16876,"style":159},[128],[18,16878],{},[18,16880],{"className":16881,"style":228},[169],[18,16883,433],{"className":16884},[232],[18,16886],{"className":16887,"style":228},[169],[18,16889,16891,16895,16898],{"className":16890},[97],[18,16892],{"className":16893,"style":16894},[101],"height:0.5806em;vertical-align:-0.15em;",[18,16896,16820],{"className":16897},[106,110],[18,16899,16901,16904],{"className":16900},[106],[18,16902,14441],{"className":16903},[106,110],[18,16905,16907],{"className":16906},[115],[18,16908,16910,16930],{"className":16909},[119,120],[18,16911,16913,16927],{"className":16912},[124],[18,16914,16916],{"className":16915,"style":129},[128],[18,16917,16918,16921],{"style":1155},[18,16919],{"className":16920,"style":137},[136],[18,16922,16924],{"className":16923},[141,142,143,144],[18,16925,47],{"className":16926},[106,144],[18,16928,152],{"className":16929},[151],[18,16931,16933],{"className":16932},[124],[18,16934,16936],{"className":16935,"style":159},[128],[18,16937],{},"，那么 KL 是 ",[18,16940,16942,16976],{"className":16941,"translate":22},[21],[18,16943,16945],{"className":16944},[26],[28,16946,16947],{"xmlns":30},[32,16948,16949,16973],{},[35,16950,16951,16953,16959,16961,16963,16965,16971],{},[41,16952,64],{},[1556,16954,16955,16957],{},[41,16956,16820],{},[45,16958,2266],{},[41,16960,317],{"mathvariant":316},[49,16962,74],{"stretchy":73},[45,16964,2266],{},[1556,16966,16967,16969],{},[41,16968,16403],{},[45,16970,2266],{},[49,16972,84],{"stretchy":73},[86,16974,16975],{"encoding":88},"n a^2\u002F(2\\sigma^2)",[18,16977,16979],{"className":16978,"ariaHidden":51},[93],[18,16980,16982,16985,16988,17017,17020,17023,17026,17055],{"className":16981},[97],[18,16983],{"className":16984,"style":10268},[101],[18,16986,64],{"className":16987},[106,110],[18,16989,16991,16994],{"className":16990},[106],[18,16992,16820],{"className":16993},[106,110],[18,16995,16997],{"className":16996},[115],[18,16998,17000],{"className":16999},[119],[18,17001,17003],{"className":17002},[124],[18,17004,17006],{"className":17005,"style":2416},[128],[18,17007,17008,17011],{"style":1962},[18,17009],{"className":17010,"style":137},[136],[18,17012,17014],{"className":17013},[141,142,143,144],[18,17015,2266],{"className":17016},[106,144],[18,17018,317],{"className":17019},[106],[18,17021,74],{"className":17022},[250],[18,17024,2266],{"className":17025},[106],[18,17027,17029,17032],{"className":17028},[106],[18,17030,16403],{"className":17031,"style":3957},[106,110],[18,17033,17035],{"className":17034},[115],[18,17036,17038],{"className":17037},[119],[18,17039,17041],{"className":17040},[124],[18,17042,17044],{"className":17043,"style":2416},[128],[18,17045,17046,17049],{"style":1962},[18,17047],{"className":17048,"style":137},[136],[18,17050,17052],{"className":17051},[141,142,143,144],[18,17053,2266],{"className":17054},[106,144],[18,17056,84],{"className":17057},[267],"，Le Cam 给出的下界是 ",[18,17060,17062,17084],{"className":17061,"translate":22},[21],[18,17063,17065],{"className":17064},[26],[28,17066,17067],{"xmlns":30},[32,17068,17069,17081],{},[35,17070,17071,17077,17079],{},[1556,17072,17073,17075],{},[41,17074,16820],{},[45,17076,2266],{},[41,17078,317],{"mathvariant":316},[45,17080,10444],{},[86,17082,17083],{"encoding":88},"a^2\u002F4",[18,17085,17087],{"className":17086,"ariaHidden":51},[93],[18,17088,17090,17093,17122],{"className":17089},[97],[18,17091],{"className":17092,"style":10268},[101],[18,17094,17096,17099],{"className":17095},[106],[18,17097,16820],{"className":17098},[106,110],[18,17100,17102],{"className":17101},[115],[18,17103,17105],{"className":17104},[119],[18,17106,17108],{"className":17107},[124],[18,17109,17111],{"className":17110,"style":2416},[128],[18,17112,17113,17116],{"style":1962},[18,17114],{"className":17115,"style":137},[136],[18,17117,17119],{"className":17118},[141,142,143,144],[18,17120,2266],{"className":17121},[106,144],[18,17123,17125],{"className":17124},[106],"\u002F4"," 的量级，取 ",[18,17128,17130,17155],{"className":17129,"translate":22},[21],[18,17131,17133],{"className":17132},[26],[28,17134,17135],{"xmlns":30},[32,17136,17137,17152],{},[35,17138,17139,17141,17144,17146,17148],{},[41,17140,16820],{},[49,17142,17143],{},"≍",[41,17145,16403],{},[41,17147,317],{"mathvariant":316},[12558,17149,17150],{},[41,17151,64],{},[86,17153,17154],{"encoding":88},"a\\asymp\\sigma\u002F\\sqrt n",[18,17156,17158,17177],{"className":17157,"ariaHidden":51},[93],[18,17159,17161,17165,17168,17171,17174],{"className":17160},[97],[18,17162],{"className":17163,"style":17164},[101],"height:0.4637em;",[18,17166,16820],{"className":17167},[106,110],[18,17169],{"className":17170,"style":228},[169],[18,17172,17143],{"className":17173},[232],[18,17175],{"className":17176,"style":228},[169],[18,17178,17180,17183,17186,17189],{"className":17179},[97],[18,17181],{"className":17182,"style":15498},[101],[18,17184,16403],{"className":17185,"style":3957},[106,110],[18,17187,317],{"className":17188},[106],[18,17190,17192],{"className":17191},[106,12693],[18,17193,17195,17225],{"className":17194},[119,120],[18,17196,17198,17222],{"className":17197},[124],[18,17199,17201,17210],{"className":17200,"style":15397},[128],[18,17202,17204,17207],{"className":17203,"style":693},[12707],[18,17205],{"className":17206,"style":697},[136],[18,17208,64],{"className":17209,"style":15407},[106,110],[18,17211,17212,17215],{"style":15410},[18,17213],{"className":17214,"style":697},[136],[18,17216,17218],{"className":17217,"style":15417},[12850],[6127,17219,17220],{"xmlns":6129,"width":12854,"height":15420,"viewBox":15421,"preserveAspectRatio":12857},[6133,17221],{"d":15424},[18,17223,152],{"className":17224},[151],[18,17226,17228],{"className":17227},[124],[18,17229,17231],{"className":17230,"style":15434},[128],[18,17232],{}," 后只有 ",[18,17235,17237,17254],{"className":17236,"translate":22},[21],[18,17238,17240],{"className":17239},[26],[28,17241,17242],{"xmlns":30},[32,17243,17244,17252],{},[35,17245,17246,17248,17250],{},[45,17247,47],{},[41,17249,317],{"mathvariant":316},[41,17251,64],{},[86,17253,322],{"encoding":88},[18,17255,17257],{"className":17256,"ariaHidden":51},[93],[18,17258,17260,17263,17266],{"className":17259},[97],[18,17261],{"className":17262,"style":242},[101],[18,17264,335],{"className":17265},[106],[18,17267,64],{"className":17268},[106,110],"。但真实的极小极大风险是 ",[18,17271,17273,17297],{"className":17272,"translate":22},[21],[18,17274,17276],{"className":17275},[26],[28,17277,17278],{"xmlns":30},[32,17279,17280,17294],{},[35,17281,17282,17288,17290,17292],{},[1556,17283,17284,17286],{},[41,17285,16403],{},[45,17287,2266],{},[41,17289,4031],{},[41,17291,317],{"mathvariant":316},[41,17293,64],{},[86,17295,17296],{"encoding":88},"\\sigma^2d\u002Fn",[18,17298,17300],{"className":17299,"ariaHidden":51},[93],[18,17301,17303,17306,17335,17338,17341],{"className":17302},[97],[18,17304],{"className":17305,"style":10268},[101],[18,17307,17309,17312],{"className":17308},[106],[18,17310,16403],{"className":17311,"style":3957},[106,110],[18,17313,17315],{"className":17314},[115],[18,17316,17318],{"className":17317},[119],[18,17319,17321],{"className":17320},[124],[18,17322,17324],{"className":17323,"style":2416},[128],[18,17325,17326,17329],{"style":1962},[18,17327],{"className":17328,"style":137},[136],[18,17330,17332],{"className":17331},[141,142,143,144],[18,17333,2266],{"className":17334},[106,144],[18,17336,4031],{"className":17337},[106,110],[18,17339,317],{"className":17340},[106],[18,17342,64],{"className":17343},[106,110],"——差了整整一个 ",[18,17346,17348,17361],{"className":17347,"translate":22},[21],[18,17349,17351],{"className":17350},[26],[28,17352,17353],{"xmlns":30},[32,17354,17355,17359],{},[35,17356,17357],{},[41,17358,4031],{},[86,17360,4031],{"encoding":88},[18,17362,17364],{"className":17363,"ariaHidden":51},[93],[18,17365,17367,17370],{"className":17366},[97],[18,17368],{"className":17369,"style":295},[101],[18,17371,4031],{"className":17372},[106,110],[11,17374,17375,17376,17404,17405,17433,17434,17462],{},"原因很直接：两点方法只探测了一个坐标方向，而 ",[18,17377,17379,17392],{"className":17378,"translate":22},[21],[18,17380,17382],{"className":17381},[26],[28,17383,17384],{"xmlns":30},[32,17385,17386,17390],{},[35,17387,17388],{},[41,17389,4031],{},[86,17391,4031],{"encoding":88},[18,17393,17395],{"className":17394,"ariaHidden":51},[93],[18,17396,17398,17401],{"className":17397},[97],[18,17399],{"className":17400,"style":295},[101],[18,17402,4031],{"className":17403},[106,110]," 维问题的困难来自 ",[18,17406,17408,17421],{"className":17407,"translate":22},[21],[18,17409,17411],{"className":17410},[26],[28,17412,17413],{"xmlns":30},[32,17414,17415,17419],{},[35,17416,17417],{},[41,17418,4031],{},[86,17420,4031],{"encoding":88},[18,17422,17424],{"className":17423,"ariaHidden":51},[93],[18,17425,17427,17430],{"className":17426},[97],[18,17428],{"className":17429,"style":295},[101],[18,17431,4031],{"className":17432},[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 ",[18,17435,17437,17450],{"className":17436,"translate":22},[21],[18,17438,17440],{"className":17439},[26],[28,17441,17442],{"xmlns":30},[32,17443,17444,17448],{},[35,17445,17446],{},[41,17447,4031],{},[86,17449,4031],{"encoding":88},[18,17451,17453],{"className":17452,"ariaHidden":51},[93],[18,17454,17456,17459],{"className":17455},[97],[18,17457],{"className":17458,"style":295},[101],[18,17460,4031],{"className":17461},[106,110]," 指数增长。这就是 Fano 不等式和 Assouad 引理的用武之地。",[638,17464,17466],{"id":17465},"四fano-不等式","四、Fano 不等式",[1311,17468,17470],{"id":17469},"_41-信息论形式","4.1 信息论形式",[5159,17472,17474,17961,18223],{"title":17473},"定理 4.1（Fano 不等式）",[11,17475,16,17476,3019,17504,17572,17573,17601,17602,4741,17630,17764,17765,10360],{},[18,17477,17479,17492],{"className":17478,"translate":22},[21],[18,17480,17482],{"className":17481},[26],[28,17483,17484],{"xmlns":30},[32,17485,17486,17490],{},[35,17487,17488],{},[41,17489,5031],{},[86,17491,5031],{"encoding":88},[18,17493,17495],{"className":17494,"ariaHidden":51},[93],[18,17496,17498,17501],{"className":17497},[97],[18,17499],{"className":17500,"style":600},[101],[18,17502,5031],{"className":17503,"style":5073},[106,110],[18,17505,17507,17533],{"className":17506,"translate":22},[21],[18,17508,17510],{"className":17509},[26],[28,17511,17512],{"xmlns":30},[32,17513,17514,17530],{},[35,17515,17516,17518,17520,17522,17524,17526,17528],{},[49,17517,1331],{"stretchy":73},[45,17519,47],{},[49,17521,52],{"separator":51},[49,17523,55],{},[49,17525,52],{"separator":51},[41,17527,3526],{},[49,17529,1349],{"stretchy":73},[86,17531,17532],{"encoding":88},"\\{1,\\dots,M\\}",[18,17534,17536],{"className":17535,"ariaHidden":51},[93],[18,17537,17539,17542,17545,17548,17551,17554,17557,17560,17563,17566,17569],{"className":17538},[97],[18,17540],{"className":17541,"style":242},[101],[18,17543,1331],{"className":17544},[250],[18,17546,47],{"className":17547},[106],[18,17549,52],{"className":17550},[165],[18,17552],{"className":17553,"style":170},[169],[18,17555,55],{"className":17556},[174],[18,17558],{"className":17559,"style":170},[169],[18,17561,52],{"className":17562},[165],[18,17564],{"className":17565,"style":170},[169],[18,17567,3526],{"className":17568,"style":246},[106,110],[18,17570,1349],{"className":17571},[267]," 上均匀分布，",[18,17574,17576,17589],{"className":17575,"translate":22},[21],[18,17577,17579],{"className":17578},[26],[28,17580,17581],{"xmlns":30},[32,17582,17583,17587],{},[35,17584,17585],{},[41,17586,43],{},[86,17588,43],{"encoding":88},[18,17590,17592],{"className":17591,"ariaHidden":51},[93],[18,17593,17595,17598],{"className":17594},[97],[18,17596],{"className":17597,"style":600},[101],[18,17599,43],{"className":17600,"style":111},[106,110]," 的分布依赖于 ",[18,17603,17605,17618],{"className":17604,"translate":22},[21],[18,17606,17608],{"className":17607},[26],[28,17609,17610],{"xmlns":30},[32,17611,17612,17616],{},[35,17613,17614],{},[41,17615,5031],{},[86,17617,5031],{"encoding":88},[18,17619,17621],{"className":17620,"ariaHidden":51},[93],[18,17622,17624,17627],{"className":17623},[97],[18,17625],{"className":17626,"style":600},[101],[18,17628,5031],{"className":17629,"style":5073},[106,110],[18,17631,17633,17665],{"className":17632,"translate":22},[21],[18,17634,17636],{"className":17635},[26],[28,17637,17638],{"xmlns":30},[32,17639,17640,17662],{},[35,17641,17642,17648,17650,17656,17658,17660],{},[658,17643,17644,17646],{"accent":51},[41,17645,5031],{},[49,17647,664],{},[49,17649,433],{},[658,17651,17652,17654],{"accent":51},[41,17653,5031],{},[49,17655,664],{},[49,17657,74],{"stretchy":73},[41,17659,43],{},[49,17661,84],{"stretchy":73},[86,17663,17664],{"encoding":88},"\\hat K=\\hat K(X)",[18,17666,17668,17717],{"className":17667,"ariaHidden":51},[93],[18,17669,17671,17675,17708,17711,17714],{"className":17670},[97],[18,17672],{"className":17673,"style":17674},[101],"height:0.9468em;",[18,17676,17678],{"className":17677},[106,681],[18,17679,17681],{"className":17680},[119],[18,17682,17684],{"className":17683},[124],[18,17685,17687,17695],{"className":17686,"style":17674},[128],[18,17688,17689,17692],{"style":693},[18,17690],{"className":17691,"style":697},[136],[18,17693,5031],{"className":17694,"style":5073},[106,110],[18,17696,17698,17701],{"style":17697},"top:-3.2523em;",[18,17699],{"className":17700,"style":697},[136],[18,17702,17705],{"className":17703,"style":17704},[710],"left:-0.1944em;",[18,17706,664],{"className":17707},[106],[18,17709],{"className":17710,"style":228},[169],[18,17712,433],{"className":17713},[232],[18,17715],{"className":17716,"style":228},[169],[18,17718,17720,17724,17755,17758,17761],{"className":17719},[97],[18,17721],{"className":17722,"style":17723},[101],"height:1.1968em;vertical-align:-0.25em;",[18,17725,17727],{"className":17726},[106,681],[18,17728,17730],{"className":17729},[119],[18,17731,17733],{"className":17732},[124],[18,17734,17736,17744],{"className":17735,"style":17674},[128],[18,17737,17738,17741],{"style":693},[18,17739],{"className":17740,"style":697},[136],[18,17742,5031],{"className":17743,"style":5073},[106,110],[18,17745,17746,17749],{"style":17697},[18,17747],{"className":17748,"style":697},[136],[18,17750,17752],{"className":17751,"style":17704},[710],[18,17753,664],{"className":17754},[106],[18,17756,74],{"className":17757},[250],[18,17759,43],{"className":17760,"style":111},[106,110],[18,17762,84],{"className":17763},[267]," 是任意估计。记 ",[18,17766,17768,17806],{"className":17767,"translate":22},[21],[18,17769,17771],{"className":17770},[26],[28,17772,17773],{"xmlns":30},[32,17774,17775,17803],{},[35,17776,17777,17783,17785,17787,17789,17791,17797,17799,17801],{},[38,17778,17779,17781],{},[41,17780,939],{},[41,17782,14441],{},[49,17784,433],{},[41,17786,11205],{},[49,17788,899],{},[49,17790,74],{"stretchy":73},[658,17792,17793,17795],{"accent":51},[41,17794,5031],{},[49,17796,664],{},[49,17798,2771],{"mathvariant":316},[41,17800,5031],{},[49,17802,84],{"stretchy":73},[86,17804,17805],{"encoding":88},"P_e=\\Pr(\\hat K\\neq K)",[18,17807,17809,17864,17949],{"className":17808,"ariaHidden":51},[93],[18,17810,17812,17815,17855,17858,17861],{"className":17811},[97],[18,17813],{"className":17814,"style":4563},[101],[18,17816,17818,17821],{"className":17817},[106],[18,17819,939],{"className":17820,"style":1171},[106,110],[18,17822,17824],{"className":17823},[115],[18,17825,17827,17847],{"className":17826},[119,120],[18,17828,17830,17844],{"className":17829},[124],[18,17831,17833],{"className":17832,"style":202},[128],[18,17834,17835,17838],{"style":1386},[18,17836],{"className":17837,"style":137},[136],[18,17839,17841],{"className":17840},[141,142,143,144],[18,17842,14441],{"className":17843},[106,110,144],[18,17845,152],{"className":17846},[151],[18,17848,17850],{"className":17849},[124],[18,17851,17853],{"className":17852,"style":159},[128],[18,17854],{},[18,17856],{"className":17857,"style":228},[169],[18,17859,433],{"className":17860},[232],[18,17862],{"className":17863,"style":228},[169],[18,17865,17867,17870,17873,17876,17907,17910,17946],{"className":17866},[97],[18,17868],{"className":17869,"style":17723},[101],[18,17871,11205],{"className":17872},[985],[18,17874,74],{"className":17875},[250],[18,17877,17879],{"className":17878},[106,681],[18,17880,17882],{"className":17881},[119],[18,17883,17885],{"className":17884},[124],[18,17886,17888,17896],{"className":17887,"style":17674},[128],[18,17889,17890,17893],{"style":693},[18,17891],{"className":17892,"style":697},[136],[18,17894,5031],{"className":17895,"style":5073},[106,110],[18,17897,17898,17901],{"style":17697},[18,17899],{"className":17900,"style":697},[136],[18,17902,17904],{"className":17903,"style":17704},[710],[18,17905,664],{"className":17906},[106],[18,17908],{"className":17909,"style":228},[169],[18,17911,17913,17940,17943],{"className":17912},[232],[18,17914,17916],{"className":17915},[232],[18,17917,17919],{"className":17918},[106,2878],[18,17920,17922],{"className":17921},[2882],[18,17923,17925,17928,17937],{"className":17924},[2886],[18,17926],{"className":17927,"style":2890},[101],[18,17929,17931],{"className":17930},[2894],[18,17932,17934],{"className":17933},[106],[18,17935,2901],{"className":17936},[232],[18,17938],{"className":17939},[2905],[18,17941],{"className":17942},[169,2909],[18,17944,433],{"className":17945},[232],[18,17947],{"className":17948,"style":228},[169],[18,17950,17952,17955,17958],{"className":17951},[97],[18,17953],{"className":17954,"style":242},[101],[18,17956,5031],{"className":17957,"style":5073},[106,110],[18,17959,84],{"className":17960},[267],[18,17962,17964],{"className":17963,"translate":22},[875],[18,17965,17967,18026],{"className":17966,"translate":22},[21],[18,17968,17970],{"className":17969},[26],[28,17971,17972],{"xmlns":30,"display":884},[32,17973,17974,18023],{},[35,17975,17976,17982,17984,17986,17988,18021],{},[38,17977,17978,17980],{},[41,17979,939],{},[41,17981,14441],{},[49,17983,4775],{},[45,17985,47],{},[49,17987,2253],{},[5651,17989,17990,18013],{},[35,17991,17992,17994,17996,17998,18001,18003,18005,18007,18009,18011],{},[41,17993,16410],{},[49,17995,74],{"stretchy":73},[41,17997,5031],{},[49,17999,18000],{"separator":51},";",[41,18002,43],{},[49,18004,84],{"stretchy":73},[49,18006,7984],{},[41,18008,14730],{},[49,18010,899],{},[45,18012,2266],{},[35,18014,18015,18017,18019],{},[41,18016,14730],{},[49,18018,899],{},[41,18020,3526],{},[49,18022,52],{"separator":51},[86,18024,18025],{"encoding":88},"P_e\\ge 1-\\frac{I(K;X)+\\log 2}{\\log M},",[18,18027,18029,18084,18103],{"className":18028,"ariaHidden":51},[93],[18,18030,18032,18035,18075,18078,18081],{"className":18031},[97],[18,18033],{"className":18034,"style":4563},[101],[18,18036,18038,18041],{"className":18037},[106],[18,18039,939],{"className":18040,"style":1171},[106,110],[18,18042,18044],{"className":18043},[115],[18,18045,18047,18067],{"className":18046},[119,120],[18,18048,18050,18064],{"className":18049},[124],[18,18051,18053],{"className":18052,"style":202},[128],[18,18054,18055,18058],{"style":1386},[18,18056],{"className":18057,"style":137},[136],[18,18059,18061],{"className":18060},[141,142,143,144],[18,18062,14441],{"className":18063},[106,110,144],[18,18065,152],{"className":18066},[151],[18,18068,18070],{"className":18069},[124],[18,18071,18073],{"className":18072,"style":159},[128],[18,18074],{},[18,18076],{"className":18077,"style":228},[169],[18,18079,4775],{"className":18080},[232],[18,18082],{"className":18083,"style":228},[169],[18,18085,18087,18091,18094,18097,18100],{"className":18086},[97],[18,18088],{"className":18089,"style":18090},[101],"height:0.7278em;vertical-align:-0.0833em;",[18,18092,47],{"className":18093},[106],[18,18095],{"className":18096,"style":2068},[169],[18,18098,2253],{"className":18099},[2072],[18,18101],{"className":18102,"style":2068},[169],[18,18104,18106,18110,18220],{"className":18105},[97],[18,18107],{"className":18108,"style":18109},[101],"height:2.3074em;vertical-align:-0.8804em;",[18,18111,18113,18116,18217],{"className":18112},[106],[18,18114],{"className":18115},[250,6009],[18,18117,18119],{"className":18118},[5651],[18,18120,18122,18208],{"className":18121},[119,120],[18,18123,18125,18205],{"className":18124},[124],[18,18126,18129,18148,18156],{"className":18127,"style":18128},[128],"height:1.427em;",[18,18130,18131,18134],{"style":6025},[18,18132],{"className":18133,"style":697},[136],[18,18135,18137,18142,18145],{"className":18136},[106],[18,18138,15171,18140],{"className":18139},[985],[18,18141,15174],{"style":6352},[18,18143],{"className":18144,"style":170},[169],[18,18146,3526],{"className":18147,"style":246},[106,110],[18,18149,18150,18153],{"style":6037},[18,18151],{"className":18152,"style":697},[136],[18,18154],{"className":18155,"style":6045},[6044],[18,18157,18158,18161],{"style":6048},[18,18159],{"className":18160,"style":697},[136],[18,18162,18164,18167,18170,18173,18176,18179,18182,18185,18188,18191,18194,18199,18202],{"className":18163},[106],[18,18165,16410],{"className":18166,"style":111},[106,110],[18,18168,74],{"className":18169},[250],[18,18171,5031],{"className":18172,"style":5073},[106,110],[18,18174,18000],{"className":18175},[165],[18,18177],{"className":18178,"style":170},[169],[18,18180,43],{"className":18181,"style":111},[106,110],[18,18183,84],{"className":18184},[267],[18,18186],{"className":18187,"style":2068},[169],[18,18189,7984],{"className":18190},[2072],[18,18192],{"className":18193,"style":2068},[169],[18,18195,15171,18197],{"className":18196},[985],[18,18198,15174],{"style":6352},[18,18200],{"className":18201,"style":170},[169],[18,18203,2266],{"className":18204},[106],[18,18206,152],{"className":18207},[151],[18,18209,18211],{"className":18210},[124],[18,18212,18215],{"className":18213,"style":18214},[128],"height:0.8804em;",[18,18216],{},[18,18218],{"className":18219},[267,6009],[18,18221,52],{"className":18222},[165],[11,18224,18225,18226,18283],{},"其中 ",[18,18227,18229,18253],{"className":18228,"translate":22},[21],[18,18230,18232],{"className":18231},[26],[28,18233,18234],{"xmlns":30},[32,18235,18236,18250],{},[35,18237,18238,18240,18242,18244,18246,18248],{},[41,18239,16410],{},[49,18241,74],{"stretchy":73},[41,18243,5031],{},[49,18245,18000],{"separator":51},[41,18247,43],{},[49,18249,84],{"stretchy":73},[86,18251,18252],{"encoding":88},"I(K;X)",[18,18254,18256],{"className":18255,"ariaHidden":51},[93],[18,18257,18259,18262,18265,18268,18271,18274,18277,18280],{"className":18258},[97],[18,18260],{"className":18261,"style":242},[101],[18,18263,16410],{"className":18264,"style":111},[106,110],[18,18266,74],{"className":18267},[250],[18,18269,5031],{"className":18270,"style":5073},[106,110],[18,18272,18000],{"className":18273},[165],[18,18275],{"className":18276,"style":170},[169],[18,18278,43],{"className":18279,"style":111},[106,110],[18,18281,84],{"className":18282},[267]," 是互信息。",[11,18285,18286],{},"证明分两步。第一步是条件熵的界：",[18,18288,18290],{"className":18289,"translate":22},[875],[18,18291,18293,18383],{"className":18292,"translate":22},[21],[18,18294,18296],{"className":18295},[26],[28,18297,18298],{"xmlns":30,"display":884},[32,18299,18300,18380],{},[35,18301,18302,18305,18307,18309,18311,18317,18319,18321,18324,18326,18332,18334,18336,18342,18344,18346,18348,18350,18352,18354,18356,18358,18360,18362,18364,18366,18372,18374,18376,18378],{},[41,18303,18304],{},"H",[49,18306,74],{"stretchy":73},[41,18308,5031],{},[49,18310,11222],{},[658,18312,18313,18315],{"accent":51},[41,18314,5031],{},[49,18316,664],{},[49,18318,84],{"stretchy":73},[49,18320,4334],{},[41,18322,18323],{},"h",[49,18325,74],{"stretchy":73},[38,18327,18328,18330],{},[41,18329,939],{},[41,18331,14441],{},[49,18333,84],{"stretchy":73},[49,18335,7984],{},[38,18337,18338,18340],{},[41,18339,939],{},[41,18341,14441],{},[41,18343,14730],{},[49,18345,899],{},[49,18347,74],{"stretchy":73},[41,18349,3526],{},[49,18351,2253],{},[45,18353,47],{},[49,18355,84],{"stretchy":73},[49,18357,4334],{},[41,18359,14730],{},[49,18361,899],{},[45,18363,2266],{},[49,18365,7984],{},[38,18367,18368,18370],{},[41,18369,939],{},[41,18371,14441],{},[41,18373,14730],{},[49,18375,899],{},[41,18377,3526],{},[49,18379,52],{"separator":51},[86,18381,18382],{"encoding":88},"H(K\\mid \\hat K)\\le h(P_e)+P_e\\log(M-1)\\le \\log 2+P_e\\log M,",[18,18384,18386,18411,18460,18524,18593,18614,18640],{"className":18385,"ariaHidden":51},[93],[18,18387,18389,18392,18396,18399,18402,18405,18408],{"className":18388},[97],[18,18390],{"className":18391,"style":242},[101],[18,18393,18304],{"className":18394,"style":18395},[106,110],"margin-right:0.0813em;",[18,18397,74],{"className":18398},[250],[18,18400,5031],{"className":18401,"style":5073},[106,110],[18,18403],{"className":18404,"style":228},[169],[18,18406,11222],{"className":18407},[232],[18,18409],{"className":18410,"style":228},[169],[18,18412,18414,18417,18448,18451,18454,18457],{"className":18413},[97],[18,18415],{"className":18416,"style":17723},[101],[18,18418,18420],{"className":18419},[106,681],[18,18421,18423],{"className":18422},[119],[18,18424,18426],{"className":18425},[124],[18,18427,18429,18437],{"className":18428,"style":17674},[128],[18,18430,18431,18434],{"style":693},[18,18432],{"className":18433,"style":697},[136],[18,18435,5031],{"className":18436,"style":5073},[106,110],[18,18438,18439,18442],{"style":17697},[18,18440],{"className":18441,"style":697},[136],[18,18443,18445],{"className":18444,"style":17704},[710],[18,18446,664],{"className":18447},[106],[18,18449,84],{"className":18450},[267],[18,18452],{"className":18453,"style":228},[169],[18,18455,4334],{"className":18456},[232],[18,18458],{"className":18459,"style":228},[169],[18,18461,18463,18466,18469,18472,18512,18515,18518,18521],{"className":18462},[97],[18,18464],{"className":18465,"style":242},[101],[18,18467,18323],{"className":18468},[106,110],[18,18470,74],{"className":18471},[250],[18,18473,18475,18478],{"className":18474},[106],[18,18476,939],{"className":18477,"style":1171},[106,110],[18,18479,18481],{"className":18480},[115],[18,18482,18484,18504],{"className":18483},[119,120],[18,18485,18487,18501],{"className":18486},[124],[18,18488,18490],{"className":18489,"style":202},[128],[18,18491,18492,18495],{"style":1386},[18,18493],{"className":18494,"style":137},[136],[18,18496,18498],{"className":18497},[141,142,143,144],[18,18499,14441],{"className":18500},[106,110,144],[18,18502,152],{"className":18503},[151],[18,18505,18507],{"className":18506},[124],[18,18508,18510],{"className":18509,"style":159},[128],[18,18511],{},[18,18513,84],{"className":18514},[267],[18,18516],{"className":18517,"style":2068},[169],[18,18519,7984],{"className":18520},[2072],[18,18522],{"className":18523,"style":2068},[169],[18,18525,18527,18530,18570,18573,18578,18581,18584,18587,18590],{"className":18526},[97],[18,18528],{"className":18529,"style":242},[101],[18,18531,18533,18536],{"className":18532},[106],[18,18534,939],{"className":18535,"style":1171},[106,110],[18,18537,18539],{"className":18538},[115],[18,18540,18542,18562],{"className":18541},[119,120],[18,18543,18545,18559],{"className":18544},[124],[18,18546,18548],{"className":18547,"style":202},[128],[18,18549,18550,18553],{"style":1386},[18,18551],{"className":18552,"style":137},[136],[18,18554,18556],{"className":18555},[141,142,143,144],[18,18557,14441],{"className":18558},[106,110,144],[18,18560,152],{"className":18561},[151],[18,18563,18565],{"className":18564},[124],[18,18566,18568],{"className":18567,"style":159},[128],[18,18569],{},[18,18571],{"className":18572,"style":170},[169],[18,18574,15171,18576],{"className":18575},[985],[18,18577,15174],{"style":6352},[18,18579,74],{"className":18580},[250],[18,18582,3526],{"className":18583,"style":246},[106,110],[18,18585],{"className":18586,"style":2068},[169],[18,18588,2253],{"className":18589},[2072],[18,18591],{"className":18592,"style":2068},[169],[18,18594,18596,18599,18602,18605,18608,18611],{"className":18595},[97],[18,18597],{"className":18598,"style":242},[101],[18,18600,47],{"className":18601},[106],[18,18603,84],{"className":18604},[267],[18,18606],{"className":18607,"style":228},[169],[18,18609,4334],{"className":18610},[232],[18,18612],{"className":18613,"style":228},[169],[18,18615,18617,18620,18625,18628,18631,18634,18637],{"className":18616},[97],[18,18618],{"className":18619,"style":2890},[101],[18,18621,15171,18623],{"className":18622},[985],[18,18624,15174],{"style":6352},[18,18626],{"className":18627,"style":170},[169],[18,18629,2266],{"className":18630},[106],[18,18632],{"className":18633,"style":2068},[169],[18,18635,7984],{"className":18636},[2072],[18,18638],{"className":18639,"style":2068},[169],[18,18641,18643,18646,18686,18689,18694,18697,18700],{"className":18642},[97],[18,18644],{"className":18645,"style":2890},[101],[18,18647,18649,18652],{"className":18648},[106],[18,18650,939],{"className":18651,"style":1171},[106,110],[18,18653,18655],{"className":18654},[115],[18,18656,18658,18678],{"className":18657},[119,120],[18,18659,18661,18675],{"className":18660},[124],[18,18662,18664],{"className":18663,"style":202},[128],[18,18665,18666,18669],{"style":1386},[18,18667],{"className":18668,"style":137},[136],[18,18670,18672],{"className":18671},[141,142,143,144],[18,18673,14441],{"className":18674},[106,110,144],[18,18676,152],{"className":18677},[151],[18,18679,18681],{"className":18680},[124],[18,18682,18684],{"className":18683,"style":159},[128],[18,18685],{},[18,18687],{"className":18688,"style":170},[169],[18,18690,15171,18692],{"className":18691},[985],[18,18693,15174],{"style":6352},[18,18695],{"className":18696,"style":170},[169],[18,18698,3526],{"className":18699,"style":246},[106,110],[18,18701,52],{"className":18702},[165],[11,18704,18225,18705,18824],{},[18,18706,18708,18740],{"className":18707,"translate":22},[21],[18,18709,18711],{"className":18710},[26],[28,18712,18713],{"xmlns":30},[32,18714,18715,18737],{},[35,18716,18717,18719,18721,18727,18729,18731,18733,18735],{},[41,18718,18323],{},[49,18720,74],{"stretchy":73},[38,18722,18723,18725],{},[41,18724,939],{},[41,18726,14441],{},[49,18728,84],{"stretchy":73},[49,18730,4334],{},[41,18732,14730],{},[49,18734,899],{},[45,18736,2266],{},[86,18738,18739],{"encoding":88},"h(P_e)\\le\\log 2",[18,18741,18743,18807],{"className":18742,"ariaHidden":51},[93],[18,18744,18746,18749,18752,18755,18795,18798,18801,18804],{"className":18745},[97],[18,18747],{"className":18748,"style":242},[101],[18,18750,18323],{"className":18751},[106,110],[18,18753,74],{"className":18754},[250],[18,18756,18758,18761],{"className":18757},[106],[18,18759,939],{"className":18760,"style":1171},[106,110],[18,18762,18764],{"className":18763},[115],[18,18765,18767,18787],{"className":18766},[119,120],[18,18768,18770,18784],{"className":18769},[124],[18,18771,18773],{"className":18772,"style":202},[128],[18,18774,18775,18778],{"style":1386},[18,18776],{"className":18777,"style":137},[136],[18,18779,18781],{"className":18780},[141,142,143,144],[18,18782,14441],{"className":18783},[106,110,144],[18,18785,152],{"className":18786},[151],[18,18788,18790],{"className":18789},[124],[18,18791,18793],{"className":18792,"style":159},[128],[18,18794],{},[18,18796,84],{"className":18797},[267],[18,18799],{"className":18800,"style":228},[169],[18,18802,4334],{"className":18803},[232],[18,18805],{"className":18806,"style":228},[169],[18,18808,18810,18813,18818,18821],{"className":18809},[97],[18,18811],{"className":18812,"style":2890},[101],[18,18814,15171,18816],{"className":18815},[985],[18,18817,15174],{"style":6352},[18,18819],{"className":18820,"style":170},[169],[18,18822,2266],{"className":18823},[106]," 是二元熵的最大值。第二步是数据处理不等式：",[18,18826,18828],{"className":18827,"translate":22},[875],[18,18829,18831,18921],{"className":18830,"translate":22},[21],[18,18832,18834],{"className":18833},[26],[28,18835,18836],{"xmlns":30,"display":884},[32,18837,18838,18918],{},[35,18839,18840,18842,18844,18846,18848,18854,18856,18858,18860,18862,18864,18866,18868,18870,18872,18874,18876,18878,18880,18882,18884,18886,18888,18890,18892,18894,18896,18898,18900,18902,18904,18906,18908,18910,18912,18914,18916],{},[41,18841,18304],{},[49,18843,74],{"stretchy":73},[41,18845,5031],{},[49,18847,11222],{},[658,18849,18850,18852],{"accent":51},[41,18851,5031],{},[49,18853,664],{},[49,18855,84],{"stretchy":73},[49,18857,4775],{},[41,18859,18304],{},[49,18861,74],{"stretchy":73},[41,18863,5031],{},[49,18865,11222],{},[41,18867,43],{},[49,18869,84],{"stretchy":73},[49,18871,433],{},[41,18873,18304],{},[49,18875,74],{"stretchy":73},[41,18877,5031],{},[49,18879,84],{"stretchy":73},[49,18881,2253],{},[41,18883,16410],{},[49,18885,74],{"stretchy":73},[41,18887,5031],{},[49,18889,18000],{"separator":51},[41,18891,43],{},[49,18893,84],{"stretchy":73},[49,18895,433],{},[41,18897,14730],{},[49,18899,899],{},[41,18901,3526],{},[49,18903,2253],{},[41,18905,16410],{},[49,18907,74],{"stretchy":73},[41,18909,5031],{},[49,18911,18000],{"separator":51},[41,18913,43],{},[49,18915,84],{"stretchy":73},[41,18917,968],{"mathvariant":316},[86,18919,18920],{"encoding":88},"H(K\\mid \\hat K)\\ge H(K\\mid X)=H(K)-I(K;X)=\\log M-I(K;X).",[18,18922,18924,18948,18997,19021,19042,19069,19105,19131],{"className":18923,"ariaHidden":51},[93],[18,18925,18927,18930,18933,18936,18939,18942,18945],{"className":18926},[97],[18,18928],{"className":18929,"style":242},[101],[18,18931,18304],{"className":18932,"style":18395},[106,110],[18,18934,74],{"className":18935},[250],[18,18937,5031],{"className":18938,"style":5073},[106,110],[18,18940],{"className":18941,"style":228},[169],[18,18943,11222],{"className":18944},[232],[18,18946],{"className":18947,"style":228},[169],[18,18949,18951,18954,18985,18988,18991,18994],{"className":18950},[97],[18,18952],{"className":18953,"style":17723},[101],[18,18955,18957],{"className":18956},[106,681],[18,18958,18960],{"className":18959},[119],[18,18961,18963],{"className":18962},[124],[18,18964,18966,18974],{"className":18965,"style":17674},[128],[18,18967,18968,18971],{"style":693},[18,18969],{"className":18970,"style":697},[136],[18,18972,5031],{"className":18973,"style":5073},[106,110],[18,18975,18976,18979],{"style":17697},[18,18977],{"className":18978,"style":697},[136],[18,18980,18982],{"className":18981,"style":17704},[710],[18,18983,664],{"className":18984},[106],[18,18986,84],{"className":18987},[267],[18,18989],{"className":18990,"style":228},[169],[18,18992,4775],{"className":18993},[232],[18,18995],{"className":18996,"style":228},[169],[18,18998,19000,19003,19006,19009,19012,19015,19018],{"className":18999},[97],[18,19001],{"className":19002,"style":242},[101],[18,19004,18304],{"className":19005,"style":18395},[106,110],[18,19007,74],{"className":19008},[250],[18,19010,5031],{"className":19011,"style":5073},[106,110],[18,19013],{"className":19014,"style":228},[169],[18,19016,11222],{"className":19017},[232],[18,19019],{"className":19020,"style":228},[169],[18,19022,19024,19027,19030,19033,19036,19039],{"className":19023},[97],[18,19025],{"className":19026,"style":242},[101],[18,19028,43],{"className":19029,"style":111},[106,110],[18,19031,84],{"className":19032},[267],[18,19034],{"className":19035,"style":228},[169],[18,19037,433],{"className":19038},[232],[18,19040],{"className":19041,"style":228},[169],[18,19043,19045,19048,19051,19054,19057,19060,19063,19066],{"className":19044},[97],[18,19046],{"className":19047,"style":242},[101],[18,19049,18304],{"className":19050,"style":18395},[106,110],[18,19052,74],{"className":19053},[250],[18,19055,5031],{"className":19056,"style":5073},[106,110],[18,19058,84],{"className":19059},[267],[18,19061],{"className":19062,"style":2068},[169],[18,19064,2253],{"className":19065},[2072],[18,19067],{"className":19068,"style":2068},[169],[18,19070,19072,19075,19078,19081,19084,19087,19090,19093,19096,19099,19102],{"className":19071},[97],[18,19073],{"className":19074,"style":242},[101],[18,19076,16410],{"className":19077,"style":111},[106,110],[18,19079,74],{"className":19080},[250],[18,19082,5031],{"className":19083,"style":5073},[106,110],[18,19085,18000],{"className":19086},[165],[18,19088],{"className":19089,"style":170},[169],[18,19091,43],{"className":19092,"style":111},[106,110],[18,19094,84],{"className":19095},[267],[18,19097],{"className":19098,"style":228},[169],[18,19100,433],{"className":19101},[232],[18,19103],{"className":19104,"style":228},[169],[18,19106,19108,19111,19116,19119,19122,19125,19128],{"className":19107},[97],[18,19109],{"className":19110,"style":2890},[101],[18,19112,15171,19114],{"className":19113},[985],[18,19115,15174],{"style":6352},[18,19117],{"className":19118,"style":170},[169],[18,19120,3526],{"className":19121,"style":246},[106,110],[18,19123],{"className":19124,"style":2068},[169],[18,19126,2253],{"className":19127},[2072],[18,19129],{"className":19130,"style":2068},[169],[18,19132,19134,19137,19140,19143,19146,19149,19152,19155,19158],{"className":19133},[97],[18,19135],{"className":19136,"style":242},[101],[18,19138,16410],{"className":19139,"style":111},[106,110],[18,19141,74],{"className":19142},[250],[18,19144,5031],{"className":19145,"style":5073},[106,110],[18,19147,18000],{"className":19148},[165],[18,19150],{"className":19151,"style":170},[169],[18,19153,43],{"className":19154,"style":111},[106,110],[18,19156,84],{"className":19157},[267],[18,19159,968],{"className":19160},[106],[11,19162,19163],{},"两式合并，整理即得。",[11,19165,19166,19167,19425,19426,19467,19468,19508,19509,19537],{},"这个不等式有一个很直观的读法：",[717,19168,19169,19170,19198,19199,19227,19228,19279,19280,19366,19367,19395,19396,19424],{},"如果 ",[18,19171,19173,19186],{"className":19172,"translate":22},[21],[18,19174,19176],{"className":19175},[26],[28,19177,19178],{"xmlns":30},[32,19179,19180,19184],{},[35,19181,19182],{},[41,19183,43],{},[86,19185,43],{"encoding":88},[18,19187,19189],{"className":19188,"ariaHidden":51},[93],[18,19190,19192,19195],{"className":19191},[97],[18,19193],{"className":19194,"style":600},[101],[18,19196,43],{"className":19197,"style":111},[106,110]," 几乎不包含关于 ",[18,19200,19202,19215],{"className":19201,"translate":22},[21],[18,19203,19205],{"className":19204},[26],[28,19206,19207],{"xmlns":30},[32,19208,19209,19213],{},[35,19210,19211],{},[41,19212,5031],{},[86,19214,5031],{"encoding":88},[18,19216,19218],{"className":19217,"ariaHidden":51},[93],[18,19219,19221,19224],{"className":19220},[97],[18,19222],{"className":19223,"style":600},[101],[18,19225,5031],{"className":19226,"style":5073},[106,110]," 的信息（",[18,19229,19231,19249],{"className":19230,"translate":22},[21],[18,19232,19234],{"className":19233},[26],[28,19235,19236],{"xmlns":30},[32,19237,19238,19246],{},[35,19239,19240,19242,19244],{},[41,19241,16410],{},[49,19243,14751],{},[45,19245,430],{},[86,19247,19248],{"encoding":88},"I\\approx0",[18,19250,19252,19270],{"className":19251,"ariaHidden":51},[93],[18,19253,19255,19258,19261,19264,19267],{"className":19254},[97],[18,19256],{"className":19257,"style":600},[101],[18,19259,16410],{"className":19260,"style":111},[106,110],[18,19262],{"className":19263,"style":228},[169],[18,19265,14751],{"className":19266},[232],[18,19268],{"className":19269,"style":228},[169],[18,19271,19273,19276],{"className":19272},[97],[18,19274],{"className":19275,"style":505},[101],[18,19277,430],{"className":19278},[106],"），那么任何猜测的错误率至少是 ",[18,19281,19283,19313],{"className":19282,"translate":22},[21],[18,19284,19286],{"className":19285},[26],[28,19287,19288],{"xmlns":30},[32,19289,19290,19310],{},[35,19291,19292,19294,19296,19298,19300,19302,19304,19306,19308],{},[45,19293,47],{},[49,19295,2253],{},[41,19297,14730],{},[49,19299,899],{},[45,19301,2266],{},[41,19303,317],{"mathvariant":316},[41,19305,14730],{},[49,19307,899],{},[41,19309,3526],{},[86,19311,19312],{"encoding":88},"1-\\log 2\u002F\\log M",[18,19314,19316,19334],{"className":19315,"ariaHidden":51},[93],[18,19317,19319,19322,19325,19328,19331],{"className":19318},[97],[18,19320],{"className":19321,"style":18090},[101],[18,19323,47],{"className":19324},[106],[18,19326],{"className":19327,"style":2068},[169],[18,19329,2253],{"className":19330},[2072],[18,19332],{"className":19333,"style":2068},[169],[18,19335,19337,19340,19345,19348,19352,19355,19360,19363],{"className":19336},[97],[18,19338],{"className":19339,"style":242},[101],[18,19341,15171,19343],{"className":19342},[985],[18,19344,15174],{"style":6352},[18,19346],{"className":19347,"style":170},[169],[18,19349,19351],{"className":19350},[106],"2\u002F",[18,19353],{"className":19354,"style":170},[169],[18,19356,15171,19358],{"className":19357},[985],[18,19359,15174],{"style":6352},[18,19361],{"className":19362,"style":170},[169],[18,19364,3526],{"className":19365,"style":246},[106,110],"，当 ",[18,19368,19370,19383],{"className":19369,"translate":22},[21],[18,19371,19373],{"className":19372},[26],[28,19374,19375],{"xmlns":30},[32,19376,19377,19381],{},[35,19378,19379],{},[41,19380,3526],{},[86,19382,3526],{"encoding":88},[18,19384,19386],{"className":19385,"ariaHidden":51},[93],[18,19387,19389,19392],{"className":19388},[97],[18,19390],{"className":19391,"style":600},[101],[18,19393,3526],{"className":19394,"style":246},[106,110]," 很大时接近 ",[18,19397,19399,19412],{"className":19398,"translate":22},[21],[18,19400,19402],{"className":19401},[26],[28,19403,19404],{"xmlns":30},[32,19405,19406,19410],{},[35,19407,19408],{},[45,19409,47],{},[86,19411,47],{"encoding":88},[18,19413,19415],{"className":19414,"ariaHidden":51},[93],[18,19416,19418,19421],{"className":19417},[97],[18,19419],{"className":19420,"style":505},[101],[18,19422,47],{"className":19423},[106],"，也就是接近「瞎猜」。"," 反过来，要让错误率小，互信息必须达到 ",[18,19427,19429,19447],{"className":19428,"translate":22},[21],[18,19430,19432],{"className":19431},[26],[28,19433,19434],{"xmlns":30},[32,19435,19436,19444],{},[35,19437,19438,19440,19442],{},[41,19439,14730],{},[49,19441,899],{},[41,19443,3526],{},[86,19445,19446],{"encoding":88},"\\log M",[18,19448,19450],{"className":19449,"ariaHidden":51},[93],[18,19451,19453,19456,19461,19464],{"className":19452},[97],[18,19454],{"className":19455,"style":2890},[101],[18,19457,15171,19459],{"className":19458},[985],[18,19460,15174],{"style":6352},[18,19462],{"className":19463,"style":170},[169],[18,19465,3526],{"className":19466,"style":246},[106,110]," 的量级——你需要至少 ",[18,19469,19471,19488],{"className":19470,"translate":22},[21],[18,19472,19474],{"className":19473},[26],[28,19475,19476],{"xmlns":30},[32,19477,19478,19486],{},[35,19479,19480,19482,19484],{},[41,19481,14730],{},[49,19483,899],{},[41,19485,3526],{},[86,19487,19446],{"encoding":88},[18,19489,19491],{"className":19490,"ariaHidden":51},[93],[18,19492,19494,19497,19502,19505],{"className":19493},[97],[18,19495],{"className":19496,"style":2890},[101],[18,19498,15171,19500],{"className":19499},[985],[18,19501,15174],{"style":6352},[18,19503],{"className":19504,"style":170},[169],[18,19506,3526],{"className":19507,"style":246},[106,110]," 奈特的信息才能区分 ",[18,19510,19512,19525],{"className":19511,"translate":22},[21],[18,19513,19515],{"className":19514},[26],[28,19516,19517],{"xmlns":30},[32,19518,19519,19523],{},[35,19520,19521],{},[41,19522,3526],{},[86,19524,3526],{"encoding":88},[18,19526,19528],{"className":19527,"ariaHidden":51},[93],[18,19529,19531,19534],{"className":19530},[97],[18,19532],{"className":19533,"style":600},[101],[18,19535,3526],{"className":19536,"style":246},[106,110]," 个等可能的选择。",[1311,19539,19541],{"id":19540},"_42-从检验到估计","4.2 从检验到估计",[11,19543,19544],{},"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。",[5159,19546,19548,19771,20024,20336,20948,21032],{"title":19547},"定理 4.2（Fano 方法，度量形式）",[11,19549,16,19550,19578,19579,19607,19608,19770],{},[18,19551,19553,19566],{"className":19552,"translate":22},[21],[18,19554,19556],{"className":19555},[26],[28,19557,19558],{"xmlns":30},[32,19559,19560,19564],{},[35,19561,19562],{},[41,19563,4031],{},[86,19565,4031],{"encoding":88},[18,19567,19569],{"className":19568,"ariaHidden":51},[93],[18,19570,19572,19575],{"className":19571},[97],[18,19573],{"className":19574,"style":295},[101],[18,19576,4031],{"className":19577},[106,110]," 是 ",[18,19580,19582,19595],{"className":19581,"translate":22},[21],[18,19583,19585],{"className":19584},[26],[28,19586,19587],{"xmlns":30},[32,19588,19589,19593],{},[35,19590,19591],{},[41,19592,924],{"mathvariant":316},[86,19594,1497],{"encoding":88},[18,19596,19598],{"className":19597,"ariaHidden":51},[93],[18,19599,19601,19604],{"className":19600},[97],[18,19602],{"className":19603,"style":600},[101],[18,19605,924],{"className":19606},[106]," 上的度量，",[18,19609,19611,19645],{"className":19610,"translate":22},[21],[18,19612,19614],{"className":19613},[26],[28,19615,19616],{"xmlns":30},[32,19617,19618,19642],{},[35,19619,19620,19626,19628,19630,19632,19638,19640],{},[38,19621,19622,19624],{},[41,19623,77],{},[45,19625,47],{},[49,19627,52],{"separator":51},[49,19629,55],{},[49,19631,52],{"separator":51},[38,19633,19634,19636],{},[41,19635,77],{},[41,19637,3526],{},[49,19639,921],{},[41,19641,924],{"mathvariant":316},[86,19643,19644],{"encoding":88},"\\theta_1,\\dots,\\theta_M\\in\\Theta",[18,19646,19648,19761],{"className":19647,"ariaHidden":51},[93],[18,19649,19651,19654,19694,19697,19700,19703,19706,19709,19712,19752,19755,19758],{"className":19650},[97],[18,19652],{"className":19653,"style":2890},[101],[18,19655,19657,19660],{"className":19656},[106],[18,19658,77],{"className":19659,"style":254},[106,110],[18,19661,19663],{"className":19662},[115],[18,19664,19666,19686],{"className":19665},[119,120],[18,19667,19669,19683],{"className":19668},[124],[18,19670,19672],{"className":19671,"style":129},[128],[18,19673,19674,19677],{"style":469},[18,19675],{"className":19676,"style":137},[136],[18,19678,19680],{"className":19679},[141,142,143,144],[18,19681,47],{"className":19682},[106,144],[18,19684,152],{"className":19685},[151],[18,19687,19689],{"className":19688},[124],[18,19690,19692],{"className":19691,"style":159},[128],[18,19693],{},[18,19695,52],{"className":19696},[165],[18,19698],{"className":19699,"style":170},[169],[18,19701,55],{"className":19702},[174],[18,19704],{"className":19705,"style":170},[169],[18,19707,52],{"className":19708},[165],[18,19710],{"className":19711,"style":170},[169],[18,19713,19715,19718],{"className":19714},[106],[18,19716,77],{"className":19717,"style":254},[106,110],[18,19719,19721],{"className":19720},[115],[18,19722,19724,19744],{"className":19723},[119,120],[18,19725,19727,19741],{"className":19726},[124],[18,19728,19730],{"className":19729,"style":1152},[128],[18,19731,19732,19735],{"style":469},[18,19733],{"className":19734,"style":137},[136],[18,19736,19738],{"className":19737},[141,142,143,144],[18,19739,3526],{"className":19740,"style":246},[106,110,144],[18,19742,152],{"className":19743},[151],[18,19745,19747],{"className":19746},[124],[18,19748,19750],{"className":19749,"style":159},[128],[18,19751],{},[18,19753],{"className":19754,"style":228},[169],[18,19756,921],{"className":19757},[232],[18,19759],{"className":19760,"style":228},[169],[18,19762,19764,19767],{"className":19763},[97],[18,19765],{"className":19766,"style":600},[101],[18,19768,924],{"className":19769},[106]," 满足",[18,19772,19774],{"className":19773,"translate":22},[875],[18,19775,19777,19830],{"className":19776,"translate":22},[21],[18,19778,19780],{"className":19779},[26],[28,19781,19782],{"xmlns":30,"display":884},[32,19783,19784,19827],{},[35,19785,19786,19788,19790,19796,19798,19805,19807,19809,19811,19813,19815,19817,19819,19821,19823,19825],{},[41,19787,4031],{},[49,19789,74],{"stretchy":73},[38,19791,19792,19794],{},[41,19793,77],{},[41,19795,5597],{},[49,19797,52],{"separator":51},[38,19799,19800,19802],{},[41,19801,77],{},[41,19803,19804],{},"j",[49,19806,84],{"stretchy":73},[49,19808,4775],{},[45,19810,2266],{},[41,19812,5348],{},[169,19814],{"width":7341},[49,19816,74],{"stretchy":73},[41,19818,5597],{},[49,19820,2771],{"mathvariant":316},[41,19822,19804],{},[49,19824,84],{"stretchy":73},[41,19826,968],{"mathvariant":316},[86,19828,19829],{"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),",[18,22142,22144,22246,22356,22457,22478,22496,22514],{"className":22143,"ariaHidden":51},[93],[18,22145,22147,22151,22154,22157,22188,22191,22194,22234,22237,22240,22243],{"className":22146},[97],[18,22148],{"className":22149,"style":22150},[101],"height:1.244em;vertical-align:-0.2861em;",[18,22152,4031],{"className":22153},[106,110],[18,22155,74],{"className":22156},[250],[18,22158,22160],{"className":22159},[106,681],[18,22161,22163],{"className":22162},[119],[18,22164,22166],{"className":22165},[124],[18,22167,22169,22177],{"className":22168,"style":677},[128],[18,22170,22171,22174],{"style":693},[18,22172],{"className":22173,"style":697},[136],[18,22175,77],{"className":22176,"style":254},[106,110],[18,22178,22179,22182],{"style":703},[18,22180],{"className":22181,"style":697},[136],[18,22183,22185],{"className":22184,"style":711},[710],[18,22186,664],{"className":22187},[106],[18,22189,52],{"className":22190},[165],[18,22192],{"className":22193,"style":170},[169],[18,22195,22197,22200],{"className":22196},[106],[18,22198,77],{"className":22199,"style":254},[106,110],[18,22201,22203],{"className":22202},[115],[18,22204,22206,22226],{"className":22205},[119,120],[18,22207,22209,22223],{"className":22208},[124],[18,22210,22212],{"className":22211,"style":5918},[128],[18,22213,22214,22217],{"style":469},[18,22215],{"className":22216,"style":137},[136],[18,22218,22220],{"className":22219},[141,142,143,144],[18,22221,19804],{"className":22222,"style":19921},[106,110,144],[18,22224,152],{"className":22225},[151],[18,22227,22229],{"className":22228},[124],[18,22230,22232],{"className":22231,"style":19931},[128],[18,22233],{},[18,22235,84],{"className":22236},[267],[18,22238],{"className":22239,"style":228},[169],[18,22241,4775],{"className":22242},[232],[18,22244],{"className":22245,"style":228},[169],[18,22247,22249,22252,22255,22258,22298,22301,22304,22344,22347,22350,22353],{"className":22248},[97],[18,22250],{"className":22251,"style":19839},[101],[18,22253,4031],{"className":22254},[106,110],[18,22256,74],{"className":22257},[250],[18,22259,22261,22264],{"className":22260},[106],[18,22262,77],{"className":22263,"style":254},[106,110],[18,22265,22267],{"className":22266},[115],[18,22268,22270,22290],{"className":22269},[119,120],[18,22271,22273,22287],{"className":22272},[124],[18,22274,22276],{"className":22275,"style":1152},[128],[18,22277,22278,22281],{"style":469},[18,22279],{"className":22280,"style":137},[136],[18,22282,22284],{"className":22283},[141,142,143,144],[18,22285,5031],{"className":22286,"style":5073},[106,110,144],[18,22288,152],{"className":22289},[151],[18,22291,22293],{"className":22292},[124],[18,22294,22296],{"className":22295,"style":159},[128],[18,22297],{},[18,22299,52],{"className":22300},[165],[18,22302],{"className":22303,"style":170},[169],[18,22305,22307,22310],{"className":22306},[106],[18,22308,77],{"className":22309,"style":254},[106,110],[18,22311,22313],{"className":22312},[115],[18,22314,22316,22336],{"className":22315},[119,120],[18,22317,22319,22333],{"className":22318},[124],[18,22320,22322],{"className":22321,"style":5918},[128],[18,22323,22324,22327],{"style":469},[18,22325],{"className":22326,"style":137},[136],[18,22328,22330],{"className":22329},[141,142,143,144],[18,22331,19804],{"className":22332,"style":19921},[106,110,144],[18,22334,152],{"className":22335},[151],[18,22337,22339],{"className":22338},[124],[18,22340,22342],{"className":22341,"style":19931},[128],[18,22343],{},[18,22345,84],{"className":22346},[267],[18,22348],{"className":22349,"style":2068},[169],[18,22351,2253],{"className":22352},[2072],[18,22354],{"className":22355,"style":2068},[169],[18,22357,22359,22362,22365,22368,22408,22411,22414,22445,22448,22451,22454],{"className":22358},[97],[18,22360],{"className":22361,"style":2281},[101],[18,22363,4031],{"className":22364},[106,110],[18,22366,74],{"className":22367},[250],[18,22369,22371,22374],{"className":22370},[106],[18,22372,77],{"className":22373,"style":254},[106,110],[18,22375,22377],{"className":22376},[115],[18,22378,22380,22400],{"className":22379},[119,120],[18,22381,22383,22397],{"className":22382},[124],[18,22384,22386],{"className":22385,"style":1152},[128],[18,22387,22388,22391],{"style":469},[18,22389],{"className":22390,"style":137},[136],[18,22392,22394],{"className":22393},[141,142,143,144],[18,22395,5031],{"className":22396,"style":5073},[106,110,144],[18,22398,152],{"className":22399},[151],[18,22401,22403],{"className":22402},[124],[18,22404,22406],{"className":22405,"style":159},[128],[18,22407],{},[18,22409,52],{"className":22410},[165],[18,22412],{"className":22413,"style":170},[169],[18,22415,22417],{"className":22416},[106,681],[18,22418,22420],{"className":22419},[119],[18,22421,22423],{"className":22422},[124],[18,22424,22426,22434],{"className":22425,"style":677},[128],[18,22427,22428,22431],{"style":693},[18,22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",[18,22613,22615,22628],{"className":22614,"translate":22},[21],[18,22616,22618],{"className":22617},[26],[28,22619,22620],{"xmlns":30},[32,22621,22622,22626],{},[35,22623,22624],{},[41,22625,5031],{},[86,22627,5031],{"encoding":88},[18,22629,22631],{"className":22630,"ariaHidden":51},[93],[18,22632,22634,22637],{"className":22633},[97],[18,22635],{"className":22636,"style":600},[101],[18,22638,5031],{"className":22639,"style":5073},[106,110]," 是唯一的最小值点，",[18,22642,22644,22666],{"className":22643,"translate":22},[21],[18,22645,22647],{"className":22646},[26],[28,22648,22649],{"xmlns":30},[32,22650,22651,22663],{},[35,22652,22653,22659,22661],{},[658,22654,22655,22657],{"accent":51},[41,22656,5031],{},[49,22658,664],{},[49,22660,433],{},[41,22662,5031],{},[86,22664,22665],{"encoding":88},"\\hat K=K",[18,22667,22669,22715],{"className":22668,"ariaHidden":51},[93],[18,22670,22672,22675,22706,22709,22712],{"className":22671},[97],[18,22673],{"className":22674,"style":17674},[101],[18,22676,22678],{"className":22677},[106,681],[18,22679,22681],{"className":22680},[119],[18,22682,22684],{"className":22683},[124],[18,22685,22687,22695],{"className":22686,"style":17674},[128],[18,22688,22689,22692],{"style":693},[18,22690],{"className":22691,"style":697},[136],[18,22693,5031],{"className":22694,"style":5073},[106,110],[18,22696,22697,22700],{"style":17697},[18,22698],{"className":22699,"style":697},[136],[18,22701,22703],{"className":22702,"style":17704},[710],[18,22704,664],{"className":22705},[106],[18,22707],{"className":22708,"style":228},[169],[18,22710,433],{"className":22711},[232],[18,22713],{"className":22714,"style":228},[169],[18,22716,22718,22721],{"className":22717},[97],[18,22719],{"className":22720,"style":600},[101],[18,22722,5031],{"className":22723,"style":5073},[106,110],"。因此",[18,22726,22728],{"className":22727,"translate":22},[875],[18,22729,22731,22789],{"className":22730,"translate":22},[21],[18,22732,22734],{"className":22733},[26],[28,22735,22736],{"xmlns":30,"display":884},[32,22737,22738,22786],{},[35,22739,22740,22742,22744,22746,22752,22754,22760,22762,22764,22766,22768,22770,22772,22778,22780,22782,22784],{},[49,22741,1331],{"stretchy":73},[41,22743,4031],{},[49,22745,74],{"stretchy":73},[38,22747,22748,22750],{},[41,22749,77],{},[41,22751,5031],{},[49,22753,52],{"separator":51},[658,22755,22756,22758],{"accent":51},[41,22757,77],{},[49,22759,664],{},[49,22761,84],{"stretchy":73},[49,22763,6681],{},[41,22765,5348],{},[49,22767,1349],{"stretchy":73},[49,22769,4548],{},[49,22771,1331],{"stretchy":73},[658,22773,22774,22776],{"accent":51},[41,22775,5031],{},[49,22777,664],{},[49,22779,433],{},[41,22781,5031],{},[49,22783,1349],{"stretchy":73},[49,22785,52],{"separator":51},[86,22787,22788],{"encoding":88},"\\{d(\\theta_K,\\hat\\theta)\u003C\\delta\\}\\subseteq\\{\\hat K=K\\},",[18,22790,22792,22896,22917,22966],{"className":22791,"ariaHidden":51},[93],[18,22793,22795,22798,22801,22804,22807,22847,22850,22853,22884,22887,22890,22893],{"className":22794},[97],[18,22796],{"className":22797,"style":2281},[101],[18,22799,1331],{"className":22800},[250],[18,22802,4031],{"className":22803},[106,110],[18,22805,74],{"className":22806},[250],[18,22808,22810,22813],{"className":22809},[106],[18,22811,77],{"className":22812,"style":254},[106,110],[18,22814,22816],{"className":22815},[115],[18,22817,22819,22839],{"className":22818},[119,120],[18,22820,22822,22836],{"className":22821},[124],[18,22823,22825],{"className":22824,"style":1152},[128],[18,22826,22827,22830],{"style":469},[18,22828],{"className":22829,"style":137},[136],[18,22831,22833],{"className":22832},[141,142,143,144],[18,22834,5031],{"className":22835,"style":5073},[106,110,144],[18,22837,152],{"className":22838},[151],[18,22840,22842],{"className":22841},[124],[18,22843,22845],{"className":22844,"style":159},[128],[18,22846],{},[18,22848,52],{"className":22849},[165],[18,22851],{"className":22852,"style":170},[169],[18,22854,22856],{"className":22855},[106,681],[18,22857,22859],{"className":22858},[119],[18,22860,22862],{"className":22861},[124],[18,22863,22865,22873],{"className":22864,"style":677},[128],[18,22866,22867,22870],{"style":693},[18,22868],{"className":22869,"style":697},[136],[18,22871,77],{"className":22872,"style":254},[106,110],[18,22874,22875,22878],{"style":703},[18,22876],{"className":22877,"style":697},[136],[18,22879,22881],{"className":22880,"style":711},[710],[18,22882,664],{"className":22883},[106],[18,22885,84],{"className":22886},[267],[18,22888],{"className":22889,"style":228},[169],[18,22891,6681],{"className":22892},[232],[18,22894],{"className":22895,"style":228},[169],[18,22897,22899,22902,22905,22908,22911,22914],{"className":22898},[97],[18,22900],{"className":22901,"style":242},[101],[18,22903,5348],{"className":22904,"style":5477},[106,110],[18,22906,1349],{"className":22907},[267],[18,22909],{"className":22910,"style":228},[169],[18,22912,4548],{"className":22913},[232],[18,22915],{"className":22916,"style":228},[169],[18,22918,22920,22923,22926,22957,22960,22963],{"className":22919},[97],[18,22921],{"className":22922,"style":17723},[101],[18,22924,1331],{"className":22925},[250],[18,22927,22929],{"className":22928},[106,681],[18,22930,22932],{"className":22931},[119],[18,22933,22935],{"className":22934},[124],[18,22936,22938,22946],{"className":22937,"style":17674},[128],[18,22939,22940,22943],{"style":693},[18,22941],{"className":22942,"style":697},[136],[18,22944,5031],{"className":22945,"style":5073},[106,110],[18,22947,22948,22951],{"style":17697},[18,22949],{"className":22950,"style":697},[136],[18,22952,22954],{"className":22953,"style":17704},[710],[18,22955,664],{"className":22956},[106],[18,22958],{"className":22959,"style":228},[169],[18,22961,433],{"className":22962},[232],[18,22964],{"className":22965,"style":228},[169],[18,22967,22969,22972,22975,22978],{"className":22968},[97],[18,22970],{"className":22971,"style":242},[101],[18,22973,5031],{"className":22974,"style":5073},[106,110],[18,22976,1349],{"className":22977},[267],[18,22979,52],{"className":22980},[165],[11,22982,22983,22984,23216],{},"从而 ",[18,22985,22987,23039],{"className":22986,"translate":22},[21],[18,22988,22990],{"className":22989},[26],[28,22991,22992],{"xmlns":30},[32,22993,22994,23036],{},[35,22995,22996,23002,23004,23006,23008,23010,23012,23014,23020,23022,23028,23030,23032,23034],{},[38,22997,22998,23000],{},[41,22999,939],{},[41,23001,14441],{},[49,23003,4334],{},[41,23005,11205],{},[49,23007,899],{},[49,23009,74],{"stretchy":73},[41,23011,4031],{},[49,23013,74],{"stretchy":73},[38,23015,23016,23018],{},[41,23017,77],{},[41,23019,5031],{},[49,23021,52],{"separator":51},[658,23023,23024,23026],{"accent":51},[41,23025,77],{},[49,23027,664],{},[49,23029,84],{"stretchy":73},[49,23031,4775],{},[41,23033,5348],{},[49,23035,84],{"stretchy":73},[86,23037,23038],{"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).",[18,23332,23334,23451,23487,23511,23572],{"className":23333,"ariaHidden":51},[93],[18,23335,23337,23341,23344,23350,23353,23356,23396,23399,23402,23433,23436,23442,23445,23448],{"className":23336},[97],[18,23338],{"className":23339,"style":23340},[101],"height:1.3079em;vertical-align:-0.35em;",[18,23342,934],{"className":23343},[106,1139],[18,23345,23347],{"className":23346},[250],[18,23348,945],{"className":23349},[1230,1196],[18,23351,4031],{"className":23352},[106,110],[18,23354,74],{"className":23355},[250],[18,23357,23359,23362],{"className":23358},[106],[18,23360,77],{"className":23361,"style":254},[106,110],[18,23363,23365],{"className":23364},[115],[18,23366,23368,23388],{"className":23367},[119,120],[18,23369,23371,23385],{"className":23370},[124],[18,23372,23374],{"className":23373,"style":1152},[128],[18,23375,23376,23379],{"style":469},[18,23377],{"className":23378,"style":137},[136],[18,23380,23382],{"className":23381},[141,142,143,144],[18,23383,5031],{"className":23384,"style":5073},[106,110,144],[18,23386,152],{"className":23387},[151],[18,23389,23391],{"className":23390},[124],[18,23392,23394],{"className":23393,"style":159},[128],[18,23395],{},[18,23397,52],{"className":23398},[165],[18,23400],{"className":23401,"style":170},[169],[18,23403,23405],{"className":23404},[106,681],[18,23406,23408],{"className":23407},[119],[18,23409,23411],{"className":23410},[124],[18,23412,23414,23422],{"className":23413,"style":677},[128],[18,23415,23416,23419],{"style":693},[18,23417],{"className":23418,"style":697},[136],[18,23420,77],{"className":23421,"style":254},[106,110],[18,23423,23424,23427],{"style":703},[18,23425],{"className":23426,"style":697},[136],[18,23428,23430],{"className":23429,"style":711},[710],[18,23431,664],{"className":23432},[106],[18,23434,84],{"className":23435},[267],[18,23437,23439],{"className":23438},[267],[18,23440,965],{"className":23441},[1230,1196],[18,23443],{"className":23444,"style":228},[169],[18,23446,4775],{"className":23447},[232],[18,23449],{"className":23450,"style":228},[169],[18,23452,23454,23457,23460,23463,23466,23469,23475,23478,23481,23484],{"className":23453},[97],[18,23455],{"className":23456,"style":6098},[101],[18,23458,5348],{"className":23459,"style":5477},[106,110],[18,23461],{"className":23462,"style":170},[169],[18,23464],{"className":23465,"style":170},[169],[18,23467,11205],{"className":23468},[985],[18,23470,23472],{"className":23471},[250],[18,23473,74],{"className":23474},[1230,1196],[18,23476,4031],{"className":23477},[106,110],[18,23479],{"className":23480,"style":228},[169],[18,23482,4775],{"className":23483},[232],[18,23485],{"className":23486,"style":228},[169],[18,23488,23490,23493,23496,23502,23505,23508],{"className":23489},[97],[18,23491],{"className":23492,"style":6098},[101],[18,23494,5348],{"className":23495,"style":5477},[106,110],[18,23497,23499],{"className":23498},[267],[18,23500,84],{"className":23501},[1230,1196],[18,23503],{"className":23504,"style":228},[169],[18,23506,4775],{"className":23507},[232],[18,23509],{"className":23510,"style":228},[169],[18,23512,23514,23517,23520,23523,23563,23566,23569],{"className":23513},[97],[18,23515],{"className":23516,"style":448},[101],[18,23518,5348],{"className":23519,"style":5477},[106,110],[18,23521],{"className":23522,"style":170},[169],[18,23524,23526,23529],{"className":23525},[106],[18,23527,939],{"className":23528,"style":1171},[106,110],[18,23530,23532],{"className":23531},[115],[18,23533,23535,23555],{"className":23534},[119,120],[18,23536,23538,23552],{"className":23537},[124],[18,23539,23541],{"className":23540,"style":202},[128],[18,23542,23543,23546],{"style":1386},[18,23544],{"className":23545,"style":137},[136],[18,23547,23549],{"className":23548},[141,142,143,144],[18,23550,14441],{"className":23551},[106,110,144],[18,23553,152],{"className":23554},[151],[18,23556,23558],{"className":23557},[124],[18,23559,23561],{"className":23560,"style":159},[128],[18,23562],{},[18,23564],{"className":23565,"style":228},[169],[18,23567,4775],{"className":23568},[232],[18,23570],{"className":23571,"style":228},[169],[18,23573,23575,23578,23581,23584,23701,23704],{"className":23574},[97],[18,23576],{"className":23577,"style":20800},[101],[18,23579,5348],{"className":23580,"style":5477},[106,110],[18,23582],{"className":23583,"style":170},[169],[18,23585,23587,23593,23596,23599,23602,23605,23695],{"className":23586},[174],[18,23588,23590],{"className":23589,"style":13793},[250,13792],[18,23591,74],{"className":23592},[1230,143],[18,23594,47],{"className":23595},[106],[18,23597],{"className":23598,"style":2068},[169],[18,23600,2253],{"className":23601},[2072],[18,23603],{"className":23604,"style":2068},[169],[18,23606,23608,23611,23692],{"className":23607},[106],[18,23609],{"className":23610},[250,6009],[18,23612,23614],{"className":23613},[5651],[18,23615,23617,23684],{"className":23616},[119,120],[18,23618,23620,23681],{"className":23619},[124],[18,23621,23623,23642,23650],{"className":23622,"style":6022},[128],[18,23624,23625,23628],{"style":6025},[18,23626],{"className":23627,"style":697},[136],[18,23629,23631,23636,23639],{"className":23630},[106],[18,23632,15171,23634],{"className":23633},[985],[18,23635,15174],{"style":6352},[18,23637],{"className":23638,"style":170},[169],[18,23640,3526],{"className":23641,"style":246},[106,110],[18,23643,23644,23647],{"style":6037},[18,23645],{"className":23646,"style":697},[136],[18,23648],{"className":23649,"style":6045},[6044],[18,23651,23652,23655],{"style":6048},[18,23653],{"className":23654,"style":697},[136],[18,23656,23658,23661,23664,23667,23670,23675,23678],{"className":23657},[106],[18,23659,16410],{"className":23660,"style":111},[106,110],[18,23662],{"className":23663,"style":2068},[169],[18,23665,7984],{"className":23666},[2072],[18,23668],{"className":23669,"style":2068},[169],[18,23671,15171,23673],{"className":23672},[985],[18,23674,15174],{"style":6352},[18,23676],{"className":23677,"style":170},[169],[18,23679,2266],{"className":23680},[106],[18,23682,152],{"className":23683},[151],[18,23685,23687],{"className":23686},[124],[18,23688,23690],{"className":23689,"style":18214},[128],[18,23691],{},[18,23693],{"className":23694},[267,6009],[18,23696,23698],{"className":23697,"style":13793},[267,13792],[18,23699,84],{"className":23700},[1230,143],[18,23702],{"className":23703,"style":170},[169],[18,23705,968],{"className":23706},[106],[11,23708,23709,23710,23738],{},"左边正是 ",[18,23711,23713,23726],{"className":23712,"translate":22},[21],[18,23714,23716],{"className":23715},[26],[28,23717,23718],{"xmlns":30},[32,23719,23720,23724],{},[35,23721,23722],{},[41,23723,3526],{},[86,23725,3526],{"encoding":88},[18,23727,23729],{"className":23728,"ariaHidden":51},[93],[18,23730,23732,23735],{"className":23731},[97],[18,23733],{"className":23734,"style":600},[101],[18,23736,3526],{"className":23737,"style":246},[106,110]," 个风险的平均，平均不超过最大，定理得证。",[11,23740,23741,23742,23823,23824,23852,23853,23932,23933,23962,23963,24174,24175,24280],{},"对平方损失 ",[18,23743,23745,23767],{"className":23744,"translate":22},[21],[18,23746,23748],{"className":23747},[26],[28,23749,23750],{"xmlns":30},[32,23751,23752,23764],{},[35,23753,23754,23756,23758],{},[41,23755,948],{"mathvariant":316},[49,23757,433],{},[1556,23759,23760,23762],{},[41,23761,4031],{},[45,23763,2266],{},[86,23765,23766],{"encoding":88},"\\ell=d^2",[18,23768,23770,23788],{"className":23769,"ariaHidden":51},[93],[18,23771,23773,23776,23779,23782,23785],{"className":23772},[97],[18,23774],{"className":23775,"style":295},[101],[18,23777,948],{"className":23778},[106],[18,23780],{"className":23781,"style":228},[169],[18,23783,433],{"className":23784},[232],[18,23786],{"className":23787,"style":228},[169],[18,23789,23791,23794],{"className":23790},[97],[18,23792],{"className":23793,"style":2416},[101],[18,23795,23797,23800],{"className":23796},[106],[18,23798,4031],{"className":23799},[106,110],[18,23801,23803],{"className":23802},[115],[18,23804,23806],{"className":23805},[119],[18,23807,23809],{"className":23808},[124],[18,23810,23812],{"className":23811,"style":2416},[128],[18,23813,23814,23817],{"style":1962},[18,23815],{"className":23816,"style":137},[136],[18,23818,23820],{"className":23819},[141,142,143,144],[18,23821,2266],{"className":23822},[106,144],"，同样的解码论证把 ",[18,23825,23827,23840],{"className":23826,"translate":22},[21],[18,23828,23830],{"className":23829},[26],[28,23831,23832],{"xmlns":30},[32,23833,23834,23838],{},[35,23835,23836],{},[41,23837,4031],{},[86,23839,4031],{"encoding":88},[18,23841,23843],{"className":23842,"ariaHidden":51},[93],[18,23844,23846,23849],{"className":23845},[97],[18,23847],{"className":23848,"style":295},[101],[18,23850,4031],{"className":23851},[106,110]," 换成 ",[18,23854,23856,23872],{"className":23855,"translate":22},[21],[18,23857,23859],{"className":23858},[26],[28,23860,23861],{"xmlns":30},[32,23862,23863,23869],{},[35,23864,23865],{},[12558,23866,23867],{},[41,23868,948],{"mathvariant":316},[86,23870,23871],{"encoding":88},"\\sqrt{\\ell}",[18,23873,23875],{"className":23874,"ariaHidden":51},[93],[18,23876,23878,23882],{"className":23877},[97],[18,23879],{"className":23880,"style":23881},[101],"height:1.04em;vertical-align:-0.1078em;",[18,23883,23885],{"className":23884},[106,12693],[18,23886,23888,23923],{"className":23887},[119,120],[18,23889,23891,23920],{"className":23890},[124],[18,23892,23895,23907],{"className":23893,"style":23894},[128],"height:0.9322em;",[18,23896,23898,23901],{"className":23897,"style":693},[12707],[18,23899],{"className":23900,"style":697},[136],[18,23902,23904],{"className":23903,"style":15407},[106],[18,23905,948],{"className":23906},[106],[18,23908,23910,23913],{"style":23909},"top:-2.8922em;",[18,23911],{"className":23912,"style":697},[136],[18,23914,23916],{"className":23915,"style":15417},[12850],[6127,23917,23918],{"xmlns":6129,"width":12854,"height":15420,"viewBox":15421,"preserveAspectRatio":12857},[6133,23919],{"d":15424},[18,23921,152],{"className":23922},[151],[18,23924,23926],{"className":23925},[124],[18,23927,23930],{"className":23928,"style":23929},[128],"height:0.1078em;",[18,23931],{},"。若平方间隔为 ",[18,23934,23936,23950],{"className":23935,"translate":22},[21],[18,23937,23939],{"className":23938},[26],[28,23940,23941],{"xmlns":30},[32,23942,23943,23947],{},[35,23944,23945],{},[41,23946,532],{"mathvariant":316},[86,23948,23949],{"encoding":88},"\\Delta",[18,23951,23953],{"className":23952,"ariaHidden":51},[93],[18,23954,23956,23959],{"className":23955},[97],[18,23957],{"className":23958,"style":600},[101],[18,23960,532],{"className":23961},[106],"，即 ",[18,23964,23966,24004],{"className":23965,"translate":22},[21],[18,23967,23969],{"className":23968},[26],[28,23970,23971],{"xmlns":30},[32,23972,23973,24001],{},[35,23974,23975,23977,23979,23985,23987,23993,23995,23997],{},[41,23976,4031],{},[49,23978,74],{"stretchy":73},[38,23980,23981,23983],{},[41,23982,77],{},[41,23984,5597],{},[49,23986,52],{"separator":51},[38,23988,23989,23991],{},[41,23990,77],{},[41,23992,19804],{},[49,23994,84],{"stretchy":73},[49,23996,4775],{},[12558,23998,23999],{},[41,24000,532],{"mathvariant":316},[86,24002,24003],{"encoding":88},"d(\\theta_i,\\theta_j)\\ge\\sqrt{\\Delta}",[18,24005,24007,24117],{"className":24006,"ariaHidden":51},[93],[18,24008,24010,24013,24016,24019,24059,24062,24065,24105,24108,24111,24114],{"className":24009},[97],[18,24011],{"className":24012,"style":19839},[101],[18,24014,4031],{"className":24015},[106,110],[18,24017,74],{"className":24018},[250],[18,24020,24022,24025],{"className":24021},[106],[18,24023,77],{"className":24024,"style":254},[106,110],[18,24026,24028],{"className":24027},[115],[18,24029,24031,24051],{"className":24030},[119,120],[18,24032,24034,24048],{"className":24033},[124],[18,24035,24037],{"className":24036,"style":5918},[128],[18,24038,24039,24042],{"style":469},[18,24040],{"className":24041,"style":137},[136],[18,24043,24045],{"className":24044},[141,142,143,144],[18,24046,5597],{"className":24047},[106,110,144],[18,24049,152],{"className":24050},[151],[18,24052,24054],{"className":24053},[124],[18,24055,24057],{"className":24056,"style":159},[128],[18,24058],{},[18,24060,52],{"className":24061},[165],[18,24063],{"className":24064,"style":170},[169],[18,24066,24068,24071],{"className":24067},[106],[18,24069,77],{"className":24070,"style":254},[106,110],[18,24072,24074],{"className":24073},[115],[18,24075,24077,24097],{"className":24076},[119,120],[18,24078,24080,24094],{"className":24079},[124],[18,24081,24083],{"className":24082,"style":5918},[128],[18,24084,24085,24088],{"style":469},[18,24086],{"className":24087,"style":137},[136],[18,24089,24091],{"className":24090},[141,142,143,144],[18,24092,19804],{"className":24093,"style":19921},[106,110,144],[18,24095,152],{"className":24096},[151],[18,24098,24100],{"className":24099},[124],[18,24101,24103],{"className":24102,"style":19931},[128],[18,24104],{},[18,24106,84],{"className":24107},[267],[18,24109],{"className":24110,"style":228},[169],[18,24112,4775],{"className":24113},[232],[18,24115],{"className":24116,"style":228},[169],[18,24118,24120,24124],{"className":24119},[97],[18,24121],{"className":24122,"style":24123},[101],"height:1.04em;vertical-align:-0.1133em;",[18,24125,24127],{"className":24126},[106,12693],[18,24128,24130,24165],{"className":24129},[119,120],[18,24131,24133,24162],{"className":24132},[124],[18,24134,24137,24149],{"className":24135,"style":24136},[128],"height:0.9267em;",[18,24138,24140,24143],{"className":24139,"style":693},[12707],[18,24141],{"className":24142,"style":697},[136],[18,24144,24146],{"className":24145,"style":15407},[106],[18,24147,532],{"className":24148},[106],[18,24150,24152,24155],{"style":24151},"top:-2.8867em;",[18,24153],{"className":24154,"style":697},[136],[18,24156,24158],{"className":24157,"style":15417},[12850],[6127,24159,24160],{"xmlns":6129,"width":12854,"height":15420,"viewBox":15421,"preserveAspectRatio":12857},[6133,24161],{"d":15424},[18,24163,152],{"className":24164},[151],[18,24166,24168],{"className":24167},[124],[18,24169,24172],{"className":24170,"style":24171},[128],"height:0.1133em;",[18,24173],{},"，则取 ",[18,24176,24178,24202],{"className":24177,"translate":22},[21],[18,24179,24181],{"className":24180},[26],[28,24182,24183],{"xmlns":30},[32,24184,24185,24199],{},[35,24186,24187,24189,24191,24195,24197],{},[41,24188,5348],{},[49,24190,433],{},[12558,24192,24193],{},[41,24194,532],{"mathvariant":316},[41,24196,317],{"mathvariant":316},[45,24198,2266],{},[86,24200,24201],{"encoding":88},"\\delta=\\sqrt{\\Delta}\u002F2",[18,24203,24205,24223],{"className":24204,"ariaHidden":51},[93],[18,24206,24208,24211,24214,24217,24220],{"className":24207},[97],[18,24209],{"className":24210,"style":295},[101],[18,24212,5348],{"className":24213,"style":5477},[106,110],[18,24215],{"className":24216,"style":228},[169],[18,24218,433],{"className":24219},[232],[18,24221],{"className":24222,"style":228},[169],[18,24224,24226,24230,24277],{"className":24225},[97],[18,24227],{"className":24228,"style":24229},[101],"height:1.1767em;vertical-align:-0.25em;",[18,24231,24233],{"className":24232},[106,12693],[18,24234,24236,24269],{"className":24235},[119,120],[18,24237,24239,24266],{"className":24238},[124],[18,24240,24242,24254],{"className":24241,"style":24136},[128],[18,24243,24245,24248],{"className":24244,"style":693},[12707],[18,24246],{"className":24247,"style":697},[136],[18,24249,24251],{"className":24250,"style":15407},[106],[18,24252,532],{"className":24253},[106],[18,24255,24256,24259],{"style":24151},[18,24257],{"className":24258,"style":697},[136],[18,24260,24262],{"className":24261,"style":15417},[12850],[6127,24263,24264],{"xmlns":6129,"width":12854,"height":15420,"viewBox":15421,"preserveAspectRatio":12857},[6133,24265],{"d":15424},[18,24267,152],{"className":24268},[151],[18,24270,24272],{"className":24271},[124],[18,24273,24275],{"className":24274,"style":24171},[128],[18,24276],{},[18,24278,8940],{"className":24279},[106],"，得到",[18,24282,24284],{"className":24283,"translate":22},[875],[18,24285,24287,24389],{"className":24286,"translate":22},[21],[18,24288,24290],{"className":24289},[26],[28,24291,24292],{"xmlns":30,"display":884},[32,24293,24294,24386],{},[35,24295,24296,24302,24308,24310,24312,24314,24316,24322,24324,24330,24332,24334,24336,24342,24384],{},[5651,24297,24298,24300],{},[45,24299,47],{},[41,24301,3526],{},[890,24303,24304,24306],{},[49,24305,20362],{},[41,24307,5597],{},[41,24309,934],{"mathvariant":933},[49,24311,945],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[41,24313,948],{"mathvariant":316},[49,24315,74],{"stretchy":73},[38,24317,24318,24320],{},[41,24319,77],{},[41,24321,5597],{},[49,24323,52],{"separator":51},[658,24325,24326,24328],{"accent":51},[41,24327,77],{},[49,24329,664],{},[49,24331,84],{"stretchy":73},[49,24333,965],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,24335,4775],{},[5651,24337,24338,24340],{},[41,24339,532],{"mathvariant":316},[45,24341,10444],{},[35,24343,24344,24346,24348,24350,24382],{},[49,24345,74],{"fence":51},[45,24347,47],{},[49,24349,2253],{},[5651,24351,24352,24374],{},[35,24353,24354,24356,24358,24360,24362,24364,24366,24368,24370,24372],{},[41,24355,16410],{},[49,24357,74],{"stretchy":73},[41,24359,5031],{},[49,24361,18000],{"separator":51},[41,24363,43],{},[49,24365,84],{"stretchy":73},[49,24367,7984],{},[41,24369,14730],{},[49,24371,899],{},[45,24373,2266],{},[35,24375,24376,24378,24380],{},[41,24377,14730],{},[49,24379,899],{},[41,24381,3526],{},[49,24383,84],{"fence":51},[41,24385,968],{"mathvariant":316},[86,24387,24388],{"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).",[18,24390,24392,24622],{"className":24391,"ariaHidden":51},[93],[18,24393,24395,24399,24461,24464,24509,24512,24515,24521,24524,24527,24567,24570,24573,24604,24607,24613,24616,24619],{"className":24394},[97],[18,24396],{"className":24397,"style":24398},[101],"height:2.5991em;vertical-align:-1.2777em;",[18,24400,24402,24405,24458],{"className":24401},[106],[18,24403],{"className":24404},[250,6009],[18,24406,24408],{"className":24407},[5651],[18,24409,24411,24450],{"className":24410},[119,120],[18,24412,24414,24447],{"className":24413},[124],[18,24415,24417,24428,24436],{"className":24416,"style":9578},[128],[18,24418,24419,24422],{"style":6025},[18,24420],{"className":24421,"style":697},[136],[18,24423,24425],{"className":24424},[106],[18,24426,3526],{"className":24427,"style":246},[106,110],[18,24429,24430,24433],{"style":6037},[18,24431],{"className":24432,"style":697},[136],[18,24434],{"className":24435,"style":6045},[6044],[18,24437,24438,24441],{"style":6048},[18,24439],{"className":24440,"style":697},[136],[18,24442,24444],{"className":24443},[106],[18,24445,47],{"className":24446},[106],[18,24448,152],{"className":24449},[151],[18,24451,24453],{"className":24452},[124],[18,24454,24456],{"className":24455,"style":6067},[128],[18,24457],{},[18,24459],{"className":24460},[267,6009],[18,24462],{"className":24463,"style":170},[169],[18,24465,24467],{"className":24466},[985,986],[18,24468,24470,24501],{"className":24469},[119,120],[18,24471,24473,24498],{"className":24472},[124],[18,24474,24477,24488],{"className":24475,"style":24476},[128],"height:1.05em;",[18,24478,24479,24482],{"style":20548},[18,24480],{"className":24481,"style":20552},[136],[18,24483,24485],{"className":24484},[141,142,143,144],[18,24486,5597],{"className":24487},[106,110,144],[18,24489,24490,24493],{"style":20570},[18,24491],{"className":24492,"style":20552},[136],[18,24494,24495],{},[18,24496,20362],{"className":24497},[985,4198,4199],[18,24499,152],{"className":24500},[151],[18,24502,24504],{"className":24503},[124],[18,24505,24507],{"className":24506,"style":20600},[128],[18,24508],{},[18,24510],{"className":24511,"style":170},[169],[18,24513,934],{"className":24514},[106,1139],[18,24516,24518],{"className":24517},[250],[18,24519,945],{"className":24520},[1230,1196],[18,24522,948],{"className":24523},[106],[18,24525,74],{"className":24526},[250],[18,24528,24530,24533],{"className":24529},[106],[18,24531,77],{"className":24532,"style":254},[106,110],[18,24534,24536],{"className":24535},[115],[18,24537,24539,24559],{"className":24538},[119,120],[18,24540,24542,24556],{"className":24541},[124],[18,24543,24545],{"className":24544,"style":5918},[128],[18,24546,24547,24550],{"style":469},[18,24548],{"className":24549,"style":137},[136],[18,24551,24553],{"className":24552},[141,142,143,144],[18,24554,5597],{"className":24555},[106,110,144],[18,24557,152],{"className":24558},[151],[18,24560,24562],{"className":24561},[124],[18,24563,24565],{"className":24564,"style":159},[128],[18,24566],{},[18,24568,52],{"className":24569},[165],[18,24571],{"className":24572,"style":170},[169],[18,24574,24576],{"className":24575},[106,681],[18,24577,24579],{"className":24578},[119],[18,24580,24582],{"className":24581},[124],[18,24583,24585,24593],{"className":24584,"style":677},[128],[18,24586,24587,24590],{"style":693},[18,24588],{"className":24589,"style":697},[136],[18,24591,77],{"className":24592,"style":254},[106,110],[18,24594,24595,24598],{"style":703},[18,24596],{"className":24597,"style":697},[136],[18,24599,24601],{"className":24600,"style":711},[710],[18,24602,664],{"className":24603},[106],[18,24605,84],{"className":24606},[267],[18,24608,24610],{"className":24609},[267],[18,24611,965],{"className":24612},[1230,1196],[18,24614],{"className":24615,"style":228},[169],[18,24617,4775],{"className":24618},[232],[18,24620],{"className":24621,"style":228},[169],[18,24623,24625,24628,24690,24693,24828,24831],{"className":24624},[97],[18,24626],{"className":24627,"style":20800},[101],[18,24629,24631,24634,24687],{"className":24630},[106],[18,24632],{"className":24633},[250,6009],[18,24635,24637],{"className":24636},[5651],[18,24638,24640,24679],{"className":24639},[119,120],[18,24641,24643,24676],{"className":24642},[124],[18,24644,24646,24657,24665],{"className":24645,"style":10842},[128],[18,24647,24648,24651],{"style":6025},[18,24649],{"className":24650,"style":697},[136],[18,24652,24654],{"className":24653},[106],[18,24655,10444],{"className":24656},[106],[18,24658,24659,24662],{"style":6037},[18,24660],{"className":24661,"style":697},[136],[18,24663],{"className":24664,"style":6045},[6044],[18,24666,24667,24670],{"style":6048},[18,24668],{"className":24669,"style":697},[136],[18,24671,24673],{"className":24672},[106],[18,24674,532],{"className":24675},[106],[18,24677,152],{"className":24678},[151],[18,24680,24682],{"className":24681},[124],[18,24683,24685],{"className":24684,"style":6067},[128],[18,24686],{},[18,24688],{"className":24689},[267,6009],[18,24691],{"className":24692,"style":170},[169],[18,24694,24696,24702,24705,24708,24711,24714,24822],{"className":24695},[174],[18,24697,24699],{"className":24698,"style":13793},[250,13792],[18,24700,74],{"className":24701},[1230,143],[18,24703,47],{"className":24704},[106],[18,24706],{"className":24707,"style":2068},[169],[18,24709,2253],{"className":24710},[2072],[18,24712],{"className":24713,"style":2068},[169],[18,24715,24717,24720,24819],{"className":24716},[106],[18,24718],{"className":24719},[250,6009],[18,24721,24723],{"className":24722},[5651],[18,24724,24726,24811],{"className":24725},[119,120],[18,24727,24729,24808],{"className":24728},[124],[18,24730,24732,24751,24759],{"className":24731,"style":18128},[128],[18,24733,24734,24737],{"style":6025},[18,24735],{"className":24736,"style":697},[136],[18,24738,24740,24745,24748],{"className":24739},[106],[18,24741,15171,24743],{"className":24742},[985],[18,24744,15174],{"style":6352},[18,24746],{"className":24747,"style":170},[169],[18,24749,3526],{"className":24750,"style":246},[106,110],[18,24752,24753,24756],{"style":6037},[18,24754],{"className":24755,"style":697},[136],[18,24757],{"className":24758,"style":6045},[6044],[18,24760,24761,24764],{"style":6048},[18,24762],{"className":24763,"style":697},[136],[18,24765,24767,24770,24773,24776,24779,24782,24785,24788,24791,24794,24797,24802,24805],{"className":24766},[106],[18,24768,16410],{"className":24769,"style":111},[106,110],[18,24771,74],{"className":24772},[250],[18,24774,5031],{"className":24775,"style":5073},[106,110],[18,24777,18000],{"className":24778},[165],[18,24780],{"className":24781,"style":170},[169],[18,24783,43],{"className":24784,"style":111},[106,110],[18,24786,84],{"className":24787},[267],[18,24789],{"className":24790,"style":2068},[169],[18,24792,7984],{"className":24793},[2072],[18,24795],{"className":24796,"style":2068},[169],[18,24798,15171,24800],{"className":24799},[985],[18,24801,15174],{"style":6352},[18,24803],{"className":24804,"style":170},[169],[18,24806,2266],{"className":24807},[106],[18,24809,152],{"className":24810},[151],[18,24812,24814],{"className":24813},[124],[18,24815,24817],{"className":24816,"style":18214},[128],[18,24818],{},[18,24820],{"className":24821},[267,6009],[18,24823,24825],{"className":24824,"style":13793},[267,13792],[18,24826,84],{"className":24827},[1230,143],[18,24829],{"className":24830,"style":170},[169],[18,24832,968],{"className":24833},[106],[11,24835,24836,24837,24898,24899,1990,25171,25208],{},"这里用了 ",[18,24838,24840,24862],{"className":24839,"translate":22},[21],[18,24841,24843],{"className":24842},[26],[28,24844,24845],{"xmlns":30},[32,24846,24847,24859],{},[35,24848,24849,24851,24853,24855,24857],{},[49,24850,1331],{"stretchy":73},[41,24852,4031],{},[49,24854,6681],{},[41,24856,5348],{},[49,24858,1349],{"stretchy":73},[86,24860,24861],{"encoding":88},"\\{d\u003C\\delta\\}",[18,24863,24865,24886],{"className":24864,"ariaHidden":51},[93],[18,24866,24868,24871,24874,24877,24880,24883],{"className":24867},[97],[18,24869],{"className":24870,"style":242},[101],[18,24872,1331],{"className":24873},[250],[18,24875,4031],{"className":24876},[106,110],[18,24878],{"className":24879,"style":228},[169],[18,24881,6681],{"className":24882},[232],[18,24884],{"className":24885,"style":228},[169],[18,24887,24889,24892,24895],{"className":24888},[97],[18,24890],{"className":24891,"style":242},[101],[18,24893,5348],{"className":24894,"style":5477},[106,110],[18,24896,1349],{"className":24897},[267]," 蕴含解码正确，因此 ",[18,24900,24902,24962],{"className":24901,"translate":22},[21],[18,24903,24905],{"className":24904},[26],[28,24906,24907],{"xmlns":30},[32,24908,24909,24959],{},[35,24910,24911,24913,24915,24921,24923,24925,24931,24933,24935,24937,24939,24941,24943,24945,24947,24953],{},[41,24912,934],{"mathvariant":933},[49,24914,945],{"stretchy":73},[1556,24916,24917,24919],{},[41,24918,4031],{},[45,24920,2266],{},[49,24922,965],{"stretchy":73},[49,24924,4775],{},[1556,24926,24927,24929],{},[41,24928,5348],{},[45,24930,2266],{},[41,24932,11205],{},[49,24934,899],{},[49,24936,74],{"stretchy":73},[41,24938,4031],{},[49,24940,4775],{},[41,24942,5348],{},[49,24944,84],{"stretchy":73},[49,24946,4775],{},[1556,24948,24949,24951],{},[41,24950,5348],{},[45,24952,2266],{},[38,24954,24955,24957],{},[41,24956,939],{},[41,24958,14441],{},[86,24960,24961],{"encoding":88},"\\mathbb E[d^2]\\ge\\delta^2\\Pr(d\\ge\\delta)\\ge\\delta^2P_e",[18,24963,24965,25018,25074,25095],{"className":24964,"ariaHidden":51},[93],[18,24966,24968,24971,24974,24977,25006,25009,25012,25015],{"className":24967},[97],[18,24969],{"className":24970,"style":10268},[101],[18,24972,934],{"className":24973},[106,1139],[18,24975,945],{"className":24976},[250],[18,24978,24980,24983],{"className":24979},[106],[18,24981,4031],{"className":24982},[106,110],[18,24984,24986],{"className":24985},[115],[18,24987,24989],{"className":24988},[119],[18,24990,24992],{"className":24991},[124],[18,24993,24995],{"className":24994,"style":2416},[128],[18,24996,24997,25000],{"style":1962},[18,24998],{"className":24999,"style":137},[136],[18,25001,25003],{"className":25002},[141,142,143,144],[18,25004,2266],{"className":25005},[106,144],[18,25007,965],{"className":25008},[267],[18,25010],{"className":25011,"style":228},[169],[18,25013,4775],{"className":25014},[232],[18,25016],{"className":25017,"style":228},[169],[18,25019,25021,25024,25053,25056,25059,25062,25065,25068,25071],{"className":25020},[97],[18,25022],{"className":25023,"style":10268},[101],[18,25025,25027,25030],{"className":25026},[106],[18,25028,5348],{"className":25029,"style":5477},[106,110],[18,25031,25033],{"className":25032},[115],[18,25034,25036],{"className":25035},[119],[18,25037,25039],{"className":25038},[124],[18,25040,25042],{"className":25041,"style":2416},[128],[18,25043,25044,25047],{"style":1962},[18,25045],{"className":25046,"style":137},[136],[18,25048,25050],{"className":25049},[141,142,143,144],[18,25051,2266],{"className":25052},[106,144],[18,25054],{"className":25055,"style":170},[169],[18,25057,11205],{"className":25058},[985],[18,25060,74],{"className":25061},[250],[18,25063,4031],{"className":25064},[106,110],[18,25066],{"className":25067,"style":228},[169],[18,25069,4775],{"className":25070},[232],[18,25072],{"className":25073,"style":228},[169],[18,25075,25077,25080,25083,25086,25089,25092],{"className":25076},[97],[18,25078],{"className":25079,"style":242},[101],[18,25081,5348],{"className":25082,"style":5477},[106,110],[18,25084,84],{"className":25085},[267],[18,25087],{"className":25088,"style":228},[169],[18,25090,4775],{"className":25091},[232],[18,25093],{"className":25094,"style":228},[169],[18,25096,25098,25102,25131],{"className":25097},[97],[18,25099],{"className":25100,"style":25101},[101],"height:0.9641em;vertical-align:-0.15em;",[18,25103,25105,25108],{"className":25104},[106],[18,25106,5348],{"className":25107,"style":5477},[106,110],[18,25109,25111],{"className":25110},[115],[18,25112,25114],{"className":25113},[119],[18,25115,25117],{"className":25116},[124],[18,25118,25120],{"className":25119,"style":2416},[128],[18,25121,25122,25125],{"style":1962},[18,25123],{"className":25124,"style":137},[136],[18,25126,25128],{"className":25127},[141,142,143,144],[18,25129,2266],{"className":25130},[106,144],[18,25132,25134,25137],{"className":25133},[106],[18,25135,939],{"className":25136,"style":1171},[106,110],[18,25138,25140],{"className":25139},[115],[18,25141,25143,25163],{"className":25142},[119,120],[18,25144,25146,25160],{"className":25145},[124],[18,25147,25149],{"className":25148,"style":202},[128],[18,25150,25151,25154],{"style":1386},[18,25152],{"className":25153,"style":137},[136],[18,25155,25157],{"className":25156},[141,142,143,144],[18,25158,14441],{"className":25159},[106,110,144],[18,25161,152],{"className":25162},[151],[18,25164,25166],{"className":25165},[124],[18,25167,25169],{"className":25168,"style":159},[128],[18,25170],{},[717,25172,25173,25174],{},"平方损失的常数是 ",[18,25175,25177,25195],{"className":25176,"translate":22},[21],[18,25178,25180],{"className":25179},[26],[28,25181,25182],{"xmlns":30},[32,25183,25184,25192],{},[35,25185,25186,25188,25190],{},[41,25187,532],{"mathvariant":316},[41,25189,317],{"mathvariant":316},[45,25191,10444],{},[86,25193,25194],{"encoding":88},"\\Delta\u002F4",[18,25196,25198],{"className":25197,"ariaHidden":51},[93],[18,25199,25201,25204],{"className":25200},[97],[18,25202],{"className":25203,"style":242},[101],[18,25205,25207],{"className":25206},[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。",[11,25210,25211],{},"剩下的任务是把互信息控制住。由定义，",[18,25213,25215],{"className":25214,"translate":22},[875],[18,25216,25218,25333],{"className":25217,"translate":22},[21],[18,25219,25221],{"className":25220},[26],[28,25222,25223],{"xmlns":30,"display":884},[32,25224,25225,25330],{},[35,25226,25227,25229,25231,25233,25235,25237,25239,25241,25247,25261,25271,25273,25279,25281,25288,25290,25292,25294,25300,25302,25308,25322,25328],{},[41,25228,16410],{},[49,25230,74],{"stretchy":73},[41,25232,5031],{},[49,25234,18000],{"separator":51},[41,25236,43],{},[49,25238,84],{"stretchy":73},[49,25240,433],{},[5651,25242,25243,25245],{},[45,25244,47],{},[41,25246,3526],{},[20358,25248,25249,25251,25259],{},[49,25250,20362],{},[35,25252,25253,25255,25257],{},[41,25254,5597],{},[49,25256,433],{},[45,25258,47],{},[41,25260,3526],{},[38,25262,25263,25265],{},[41,25264,12572],{},[35,25266,25267,25269],{},[41,25268,5031],{"mathvariant":316},[41,25270,12579],{"mathvariant":316},[49,25272,74],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[38,25274,25275,25277],{},[41,25276,939],{},[41,25278,5597],{},[41,25280,2248],{"mathvariant":316},[658,25282,25283,25285],{"accent":51},[41,25284,939],{},[49,25286,25287],{},"ˉ",[49,25289,84],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,25291,52],{"separator":51},[169,25293],{"width":7341},[658,25295,25296,25298],{"accent":51},[41,25297,939],{},[49,25299,25287],{},[49,25301,433],{},[5651,25303,25304,25306],{},[45,25305,47],{},[41,25307,3526],{},[20358,25309,25310,25312,25320],{},[49,25311,20362],{},[35,25313,25314,25316,25318],{},[41,25315,5597],{},[49,25317,433],{},[45,25319,47],{},[41,25321,3526],{},[38,25323,25324,25326],{},[41,25325,939],{},[41,25327,5597],{},[41,25329,968],{"mathvariant":316},[86,25331,25332],{"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.",[18,25334,25336,25372,25692],{"className":25335,"ariaHidden":51},[93],[18,25337,25339,25342,25345,25348,25351,25354,25357,25360,25363,25366,25369],{"className":25338},[97],[18,25340],{"className":25341,"style":242},[101],[18,25343,16410],{"className":25344,"style":111},[106,110],[18,25346,74],{"className":25347},[250],[18,25349,5031],{"className":25350,"style":5073},[106,110],[18,25352,18000],{"className":25353},[165],[18,25355],{"className":25356,"style":170},[169],[18,25358,43],{"className":25359,"style":111},[106,110],[18,25361,84],{"className":25362},[267],[18,25364],{"className":25365,"style":228},[169],[18,25367,433],{"className":25368},[232],[18,25370],{"className":25371,"style":228},[169],[18,25373,25375,25378,25440,25443,25507,25510,25556,25562,25602,25605,25637,25643,25646,25649,25652,25683,25686,25689],{"className":25374},[97],[18,25376],{"className":25377,"style":20467},[101],[18,25379,25381,25384,25437],{"className":25380},[106],[18,25382],{"className":25383},[250,6009],[18,25385,25387],{"className":25386},[5651],[18,25388,25390,25429],{"className":25389},[119,120],[18,25391,25393,25426],{"className":25392},[124],[18,25394,25396,25407,25415],{"className":25395,"style":9578},[128],[18,25397,25398,25401],{"style":6025},[18,25399],{"className":25400,"style":697},[136],[18,25402,25404],{"className":25403},[106],[18,25405,3526],{"className":25406,"style":246},[106,110],[18,25408,25409,25412],{"style":6037},[18,25410],{"className":25411,"style":697},[136],[18,25413],{"className":25414,"style":6045},[6044],[18,25416,25417,25420],{"style":6048},[18,25418],{"className":25419,"style":697},[136],[18,25421,25423],{"className":25422},[106],[18,25424,47],{"className":25425},[106],[18,25427,152],{"className":25428},[151],[18,25430,25432],{"className":25431},[124],[18,25433,25435],{"className":25434,"style":6067},[128],[18,25436],{},[18,25438],{"className":25439},[267,6009],[18,25441],{"className":25442,"style":170},[169],[18,25444,25446],{"className":25445},[985,986],[18,25447,25449,25499],{"className":25448},[119,120],[18,25450,25452,25496],{"className":25451},[124],[18,25453,25455,25475,25485],{"className":25454,"style":20545},[128],[18,25456,25457,25460],{"style":20548},[18,25458],{"className":25459,"style":20552},[136],[18,25461,25463],{"className":25462},[141,142,143,144],[18,25464,25466,25469,25472],{"className":25465},[106,144],[18,25467,5597],{"className":25468},[106,110,144],[18,25470,433],{"className":25471},[232,144],[18,25473,47],{"className":25474},[106,144],[18,25476,25477,25480],{"style":20570},[18,25478],{"className":25479,"style":20552},[136],[18,25481,25482],{},[18,25483,20362],{"className":25484},[985,4198,4199],[18,25486,25487,25490],{"style":20581},[18,25488],{"className":25489,"style":20552},[136],[18,25491,25493],{"className":25492},[141,142,143,144],[18,25494,3526],{"className":25495,"style":246},[106,110,144],[18,25497,152],{"className":25498},[151],[18,25500,25502],{"className":25501},[124],[18,25503,25505],{"className":25504,"style":20600},[128],[18,25506],{},[18,25508],{"className":25509,"style":170},[169],[18,25511,25513,25516],{"className":25512},[106],[18,25514,12572],{"className":25515,"style":254},[106,110],[18,25517,25519],{"className":25518},[115],[18,25520,25522,25548],{"className":25521},[119,120],[18,25523,25525,25545],{"className":25524},[124],[18,25526,25528],{"className":25527,"style":1152},[128],[18,25529,25530,25533],{"style":469},[18,25531],{"className":25532,"style":137},[136],[18,25534,25536],{"className":25535},[141,142,143,144],[18,25537,25539],{"className":25538},[106,144],[18,25540,25542],{"className":25541},[106,144],[18,25543,12814],{"className":25544},[106,6351,144],[18,25546,152],{"className":25547},[151],[18,25549,25551],{"className":25550},[124],[18,25552,25554],{"className":25553,"style":159},[128],[18,25555],{},[18,25557,25559],{"className":25558},[250],[18,25560,74],{"className":25561},[1230,1196],[18,25563,25565,25568],{"className":25564},[106],[18,25566,939],{"className":25567,"style":1171},[106,110],[18,25569,25571],{"className":25570},[115],[18,25572,25574,25594],{"className":25573},[119,120],[18,25575,25577,25591],{"className":25576},[124],[18,25578,25580],{"className":25579,"style":5918},[128],[18,25581,25582,25585],{"style":1386},[18,25583],{"className":25584,"style":137},[136],[18,25586,25588],{"className":25587},[141,142,143,144],[18,25589,5597],{"className":25590},[106,110,144],[18,25592,152],{"className":25593},[151],[18,25595,25597],{"className":25596},[124],[18,25598,25600],{"className":25599,"style":159},[128],[18,25601],{},[18,25603,2248],{"className":25604},[106],[18,25606,25608],{"className":25607},[106,681],[18,25609,25611],{"className":25610},[119],[18,25612,25614],{"className":25613},[124],[18,25615,25618,25626],{"className":25616,"style":25617},[128],"height:0.8201em;",[18,25619,25620,25623],{"style":693},[18,25621],{"className":25622,"style":697},[136],[18,25624,939],{"className":25625,"style":1171},[106,110],[18,25627,2562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KL 对第二个变量的凸性，",[18,25877,25879],{"className":25878,"translate":22},[875],[18,25880,25882,26012],{"className":25881,"translate":22},[21],[18,25883,25885],{"className":25884},[26],[28,25886,25887],{"xmlns":30,"display":884},[32,25888,25889,26009],{},[35,25890,25891,25893,25895,25897,25899,25901,25903,25905,25915,25933,25943,25945,25951,25953,25959,25961,25963,25979,25989,25991,25997,25999,26005,26007],{},[41,25892,16410],{},[49,25894,74],{"stretchy":73},[41,25896,5031],{},[49,25898,18000],{"separator":51},[41,25900,43],{},[49,25902,84],{"stretchy":73},[49,25904,4334],{},[5651,25906,25907,25909],{},[45,25908,47],{},[1556,25910,25911,25913],{},[41,25912,3526],{},[45,25914,2266],{},[20358,25916,25917,25919,25931],{},[49,25918,20362],{},[35,25920,25921,25923,25925,25927,25929],{},[41,25922,5597],{},[49,25924,52],{"separator":51},[41,25926,19804],{},[49,25928,433],{},[45,25930,47],{},[41,25932,3526],{},[38,25934,25935,25937],{},[41,25936,12572],{},[35,25938,25939,25941],{},[41,25940,5031],{"mathvariant":316},[41,25942,12579],{"mathvariant":316},[49,25944,74],{"stretchy":73},[38,25946,25947,25949],{},[41,25948,939],{},[41,25950,5597],{},[41,25952,2248],{"mathvariant":316},[38,25954,25955,25957],{},[41,25956,939],{},[41,25958,19804],{},[49,25960,84],{"stretchy":73},[49,25962,4334],{},[890,25964,25965,25971],{},[35,25966,25967,25969],{},[41,25968,5590],{},[49,25970,899],{},[35,25972,25973,25975,25977],{},[41,25974,5597],{},[49,25976,52],{"separator":51},[41,25978,19804],{},[38,25980,25981,25983],{},[41,25982,12572],{},[35,25984,25985,25987],{},[41,25986,5031],{"mathvariant":316},[41,25988,12579],{"mathvariant":316},[49,25990,74],{"stretchy":73},[38,25992,25993,25995],{},[41,25994,939],{},[41,25996,5597],{},[41,25998,2248],{"mathvariant":316},[38,26000,26001,26003],{},[41,26002,939],{},[41,26004,19804],{},[49,26006,84],{"stretchy":73},[41,26008,968],{"mathvariant":316},[86,26010,26011],{"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).",[18,26013,26015,26051,26369],{"className":26014,"ariaHidden":51},[93],[18,26016,26018,26021,26024,26027,26030,26033,26036,26039,26042,26045,26048],{"className":26017},[97],[18,26019],{"className":26020,"style":242},[101],[18,26022,16410],{"className":26023,"style":111},[106,110],[18,26025,74],{"className":26026},[250],[18,26028,5031],{"className":26029,"style":5073},[106,110],[18,26031,18000],{"className":26032},[165],[18,26034],{"className":26035,"style":170},[169],[18,26037,43],{"className":26038,"style":111},[106,110],[18,26040,84],{"className":26041},[267],[18,26043],{"className":26044,"style":228},[169],[18,26046,4334],{"className":26047},[232],[18,26049],{"className":26050,"style":228},[169],[18,26052,26054,26058,26148,26151,26222,26225,26271,26274,26314,26317,26357,26360,26363,26366],{"className":26053},[97],[18,26055],{"className":26056,"style":26057},[101],"height:3.2421em;vertical-align:-1.4138em;",[18,26059,26061,26064,26145],{"className":26060},[106],[18,26062],{"className":26063},[250,6009],[18,26065,26067],{"className":26066},[5651],[18,26068,26070,26137],{"className":26069},[119,120],[18,26071,26073,26134],{"className":26072},[124],[18,26074,26076,26115,26123],{"className":26075,"style":9578},[128],[18,26077,26078,26081],{"style":6025},[18,26079],{"className":26080,"style":697},[136],[18,26082,26084],{"className":26083},[106],[18,26085,26087,26090],{"className":26086},[106],[18,26088,3526],{"className":26089,"style":246},[106,110],[18,26091,26093],{"className":26092},[115],[18,26094,26096],{"className":26095},[119],[18,26097,26099],{"className":26098},[124],[18,26100,26103],{"className":26101,"style":26102},[128],"height:0.7401em;",[18,26104,26106,26109],{"style":26105},"top:-2.989em;margin-right:0.05em;",[18,26107],{"className":26108,"style":137},[136],[18,26110,26112],{"className":26111},[141,142,143,144],[18,26113,2266],{"className":26114},[106,144],[18,26116,26117,26120],{"style":6037},[18,26118],{"className":26119,"style":697},[136],[18,26121],{"className":26122,"style":6045},[6044],[18,26124,26125,26128],{"style":6048},[18,26126],{"className":26127,"style":697},[136],[18,26129,26131],{"className":26130},[106],[18,26132,47],{"className":26133},[106],[18,26135,152],{"className":26136},[151],[18,26138,26140],{"className":26139},[124],[18,26141,26143],{"className":26142,"style":6067},[128],[18,26144],{},[18,26146],{"className":26147},[267,6009],[18,26149],{"className":26150,"style":170},[169],[18,26152,26154],{"className":26153},[985,986],[18,26155,26157,26213],{"className":26156},[119,120],[18,26158,26160,26210],{"className":26159},[124],[18,26161,26163,26189,26199],{"className":26162,"style":20545},[128],[18,26164,26165,26168],{"style":20548},[18,26166],{"className":26167,"style":20552},[136],[18,26169,26171],{"className":26170},[141,142,143,144],[18,26172,26174,26177,26180,26183,26186],{"className":26173},[106,144],[18,26175,5597],{"className":26176},[106,110,144],[18,26178,52],{"className":26179},[165,144],[18,26181,19804],{"className":26182,"style":19921},[106,110,144],[18,26184,433],{"className":26185},[232,144],[18,26187,47],{"className":26188},[106,144],[18,26190,26191,26194],{"style":20570},[18,26192],{"className":26193,"style":20552},[136],[18,26195,26196],{},[18,26197,20362],{"className":26198},[985,4198,4199],[18,26200,26201,26204],{"style":20581},[18,26202],{"className":26203,"style":20552},[136],[18,26205,26207],{"className":26206},[141,142,143,144],[18,26208,3526],{"className":26209,"style":246},[106,110,144],[18,26211,152],{"className":26212},[151],[18,26214,26216],{"className":26215},[124],[18,26217,26220],{"className":26218,"style":26219},[128],"height:1.4138em;",[18,26221],{},[18,26223],{"className":26224,"style":170},[169],[18,26226,26228,26231],{"className":26227},[106],[18,26229,12572],{"className":26230,"style":254},[106,110],[18,26232,26234],{"className":26233},[115],[18,26235,26237,26263],{"className":26236},[119,120],[18,26238,26240,26260],{"className":26239},[124],[18,26241,26243],{"className":26242,"style":1152},[128],[18,26244,26245,26248],{"style":469},[18,26246],{"className":26247,"style":137},[136],[18,26249,26251],{"className":26250},[141,142,143,144],[18,26252,26254],{"className":26253},[106,144],[18,26255,26257],{"className":26256},[106,144],[18,26258,12814],{"className":26259},[106,6351,144],[18,26261,152],{"className":26262},[151],[18,26264,26266],{"className":26265},[124],[18,26267,26269],{"className":26268,"style":159},[128],[18,26270],{},[18,26272,74],{"className":26273},[250],[18,26275,26277,26280],{"className":26276},[106],[18,26278,939],{"className":26279,"style":1171},[106,110],[18,26281,26283],{"className":26282},[115],[18,26284,26286,26306],{"className":26285},[119,120],[18,26287,26289,26303],{"className":26288},[124],[18,26290,26292],{"className":26291,"style":5918},[128],[18,26293,26294,26297],{"style":1386},[18,26295],{"className":26296,"style":137},[136],[18,26298,26300],{"className":26299},[141,142,143,144],[18,26301,5597],{"className":26302},[106,110,144],[18,26304,152],{"className":26305},[151],[18,26307,26309],{"className":26308},[124],[18,26310,26312],{"className":26311,"style":159},[128],[18,26313],{},[18,26315,2248],{"className":26316},[106],[18,26318,26320,26323],{"className":26319},[106],[18,26321,939],{"className":26322,"style":1171},[106,110],[18,26324,26326],{"className":26325},[115],[18,26327,26329,26349],{"className":26328},[119,120],[18,26330,26332,26346],{"className":26331},[124],[18,26333,26335],{"className":26334,"style":5918},[128],[18,26336,26337,26340],{"style":1386},[18,26338],{"className":26339,"style":137},[136],[18,26341,26343],{"className":26342},[141,142,143,144],[18,26344,19804],{"className":26345,"style":19921},[106,110,144],[18,26347,152],{"className":26348},[151],[18,26350,26352],{"className":26351},[124],[18,26353,26355],{"className":26354,"style":19931},[128],[18,26356],{},[18,26358,84],{"className":26359},[267],[18,26361],{"className":26362,"style":228},[169],[18,26364,4334],{"className":26365},[232],[18,26367],{"className":26368,"style":228},[169],[18,26370,26372,26375,26428,26431,26477,26480,26520,26523,26563,26566],{"className":26371},[97],[18,26373],{"className":26374,"style":10505},[101],[18,26376,26378],{"className":26377},[985,986],[18,26379,26381,26420],{"className":26380},[119,120],[18,26382,26384,26417],{"className":26383},[124],[18,26385,26387,26407],{"className":26386,"style":1083},[128],[18,26388,26389,26392],{"style":5735},[18,26390],{"className":26391,"style":697},[136],[18,26393,26395],{"className":26394},[141,142,143,144],[18,26396,26398,26401,26404],{"className":26397},[106,144],[18,26399,5597],{"className":26400},[106,110,144],[18,26402,52],{"className":26403},[165,144],[18,26405,19804],{"className":26406,"style":19921},[106,110,144],[18,26408,26409,26412],{"style":693},[18,26410],{"className":26411,"style":697},[136],[18,26413,26414],{},[18,26415,5590],{"className":26416},[985],[18,26418,152],{"className":26419},[151],[18,26421,26423],{"className":26422},[124],[18,26424,26426],{"className":26425,"style":5779},[128],[18,26427],{},[18,26429],{"className":26430,"style":170},[169],[18,26432,26434,26437],{"className":26433},[106],[18,26435,12572],{"className":26436,"style":254},[106,110],[18,26438,26440],{"className":26439},[115],[18,26441,26443,26469],{"className":26442},[119,120],[18,26444,26446,26466],{"className":26445},[124],[18,26447,26449],{"className":26448,"style":1152},[128],[18,26450,26451,26454],{"style":469},[18,26452],{"className":26453,"style":137},[136],[18,26455,26457],{"className":26456},[141,142,143,144],[18,26458,26460],{"className":26459},[106,144],[18,26461,26463],{"className":26462},[106,144],[18,26464,12814],{"className":26465},[106,6351,144],[18,26467,152],{"className":26468},[151],[18,26470,26472],{"className":26471},[124],[18,26473,26475],{"className":26474,"style":159},[128],[18,26476],{},[18,26478,74],{"className":26479},[250],[18,26481,26483,26486],{"className":26482},[106],[18,26484,939],{"className":26485,"style":1171},[106,110],[18,26487,26489],{"className":26488},[115],[18,26490,26492,26512],{"className":26491},[119,120],[18,26493,26495,26509],{"className":26494},[124],[18,26496,26498],{"className":26497,"style":5918},[128],[18,26499,26500,26503],{"style":1386},[18,26501],{"className":26502,"style":137},[136],[18,26504,26506],{"className":26505},[141,142,143,144],[18,26507,5597],{"className":26508},[106,110,144],[18,26510,152],{"className":26511},[151],[18,26513,26515],{"className":26514},[124],[18,26516,26518],{"className":26517,"style":159},[128],[18,26519],{},[18,26521,2248],{"className":26522},[106],[18,26524,26526,26529],{"className":26525},[106],[18,26527,939],{"className":26528,"style":1171},[106,110],[18,26530,26532],{"className":26531},[115],[18,26533,26535,26555],{"className":26534},[119,120],[18,26536,26538,26552],{"className":26537},[124],[18,26539,26541],{"className":26540,"style":5918},[128],[18,26542,26543,26546],{"style":1386},[18,26544],{"className":26545,"style":137},[136],[18,26547,26549],{"className":26548},[141,142,143,144],[18,26550,19804],{"className":26551,"style":19921},[106,110,144],[18,26553,152],{"className":26554},[151],[18,26556,26558],{"className":26557},[124],[18,26559,26561],{"className":26560,"style":19931},[128],[18,26562],{},[18,26564,84],{"className":26565},[267],[18,26567,968],{"className":26568},[106],[11,26570,26571,26572,26642,26643,26671,26672,26700,26701,509,26741,26774],{},"如果每个 ",[18,26573,26575,26593],{"className":26574,"translate":22},[21],[18,26576,26578],{"className":26577},[26],[28,26579,26580],{"xmlns":30},[32,26581,26582,26590],{},[35,26583,26584],{},[38,26585,26586,26588],{},[41,26587,939],{},[41,26589,5597],{},[86,26591,26592],{"encoding":88},"P_i",[18,26594,26596],{"className":26595,"ariaHidden":51},[93],[18,26597,26599,26602],{"className":26598},[97],[18,26600],{"className":26601,"style":4563},[101],[18,26603,26605,26608],{"className":26604},[106],[18,26606,939],{"className":26607,"style":1171},[106,110],[18,26609,26611],{"className":26610},[115],[18,26612,26614,26634],{"className":26613},[119,120],[18,26615,26617,26631],{"className":26616},[124],[18,26618,26620],{"className":26619,"style":5918},[128],[18,26621,26622,26625],{"style":1386},[18,26623],{"className":26624,"style":137},[136],[18,26626,26628],{"className":26627},[141,142,143,144],[18,26629,5597],{"className":26630},[106,110,144],[18,26632,152],{"className":26633},[151],[18,26635,26637],{"className":26636},[124],[18,26638,26640],{"className":26639,"style":159},[128],[18,26641],{}," 都是 ",[18,26644,26646,26659],{"className":26645,"translate":22},[21],[18,26647,26649],{"className":26648},[26],[28,26650,26651],{"xmlns":30},[32,26652,26653,26657],{},[35,26654,26655],{},[41,26656,64],{},[86,26658,64],{"encoding":88},[18,26660,26662],{"className":26661,"ariaHidden":51},[93],[18,26663,26665,26668],{"className":26664},[97],[18,26666],{"className":26667,"style":1083},[101],[18,26669,64],{"className":26670},[106,110]," 个独立样本的分布，那么成对 KL 是单样本 KL 的 ",[18,26673,26675,26688],{"className":26674,"translate":22},[21],[18,26676,26678],{"className":26677},[26],[28,26679,26680],{"xmlns":30},[32,26681,26682,26686],{},[35,26683,26684],{},[41,26685,64],{},[86,26687,64],{"encoding":88},[18,26689,26691],{"className":26690,"ariaHidden":51},[93],[18,26692,26694,26697],{"className":26693},[97],[18,26695],{"className":26696,"style":1083},[101],[18,26698,64],{"className":26699},[106,110]," 倍。这样，Fano 下界就变成了「",[18,26702,26704,26721],{"className":26703,"translate":22},[21],[18,26705,26707],{"className":26706},[26],[28,26708,26709],{"xmlns":30},[32,26710,26711,26719],{},[35,26712,26713,26715,26717],{},[41,26714,14730],{},[49,26716,899],{},[41,26718,3526],{},[86,26720,19446],{"encoding":88},[18,26722,26724],{"className":26723,"ariaHidden":51},[93],[18,26725,26727,26730,26735,26738],{"className":26726},[97],[18,26728],{"className":26729,"style":2890},[101],[18,26731,15171,26733],{"className":26732},[985],[18,26734,15174],{"style":6352},[18,26736],{"className":26737,"style":170},[169],[18,26739,3526],{"className":26740,"style":246},[106,110],[18,26742,26744,26759],{"className":26743,"translate":22},[21],[18,26745,26747],{"className":26746},[26],[28,26748,26749],{"xmlns":30},[32,26750,26751,26757],{},[35,26752,26753,26755],{},[41,26754,64],{},[49,26756,2015],{},[86,26758,14391],{"encoding":88},[18,26760,26762],{"className":26761,"ariaHidden":51},[93],[18,26763,26765,26768,26771],{"className":26764},[97],[18,26766],{"className":26767,"style":14401},[101],[18,26769,64],{"className":26770},[106,110],[18,26772,2015],{"className":26773},[106],"KL 的竞争」。",[1311,26776,26778],{"id":26777},"_43-广义-fano-与局部版本","4.3 广义 Fano 与局部版本",[11,26780,26781,26782,26785],{},"定理 4.2 常被称为",[717,26783,26784],{},"广义 Fano 不等式","。它的使用可以总结成一句口诀：",[18,26787,26789],{"className":26788,"translate":22},[875],[18,26790,26792,26901],{"className":26791,"translate":22},[21],[18,26793,26795],{"className":26794},[26],[28,26796,26797],{"xmlns":30,"display":884},[32,26798,26799,26898],{},[35,26800,26801],{},[26802,26803,26805],"menclose",{"notation":26804},"box",[14678,26806,26807],{"scriptlevel":430,"displaystyle":73},[14678,26808,26809],{"scriptlevel":430,"displaystyle":73},[14678,26810,26811],{"scriptlevel":430,"displaystyle":51},[35,26812,26813,26816,26818,26820,26822,26824,26827,26829,26831,26834,26850,26860,26862,26868,26870,26876,26878,26880,26883,26889,26892,26894,26896],{},[926,26814,26815],{}," 只要 ",[41,26817,14730],{},[49,26819,899],{},[41,26821,3526],{},[926,26823,928],{},[49,26825,26826],{},"≫",[926,26828,928],{},[41,26830,64],{},[49,26832,26833],{},"⋅",[890,26835,26836,26842],{},[35,26837,26838,26840],{},[41,26839,5590],{},[49,26841,899],{},[35,26843,26844,26846,26848],{},[41,26845,5597],{},[49,26847,52],{"separator":51},[41,26849,19804],{},[38,26851,26852,26854],{},[41,26853,12572],{},[35,26855,26856,26858],{},[41,26857,5031],{"mathvariant":316},[41,26859,12579],{"mathvariant":316},[49,26861,74],{"stretchy":73},[38,26863,26864,26866],{},[41,26865,939],{},[41,26867,5597],{},[41,26869,2248],{"mathvariant":316},[38,26871,26872,26874],{},[41,26873,939],{},[41,26875,19804],{},[49,26877,84],{"stretchy":73},[49,26879,52],{"separator":51},[926,26881,26882],{}," 就有 ",[38,26884,26885,26887],{},[41,26886,3526],{"mathvariant":3525},[41,26888,64],{},[49,26890,26891],{},"≳",[41,26893,5348],{},[41,26895,968],{"mathvariant":316},[926,26897,928],{},[86,26899,26900],{"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.\\ }",[18,26902,26904],{"className":26903,"ariaHidden":51},[93],[18,26905,26907,26911],{"className":26906},[97],[18,26908],{"className":26909,"style":26910},[101],"height:2.2938em;vertical-align:-1.2038em;",[18,26912,26914],{"className":26913},[106],[18,26915,26917,27281],{"className":26916},[119,120],[18,26918,26920,27278],{"className":26919},[124],[18,26921,26924,27266],{"className":26922,"style":26923},[128],"height:1.09em;",[18,26925,26927,26931],{"style":26926},"top:-4.2938em;",[18,26928],{"className":26929,"style":26930},[136],"height:4.2938em;",[18,26932,26935],{"className":26933},[26934],"boxpad",[18,26936,26938],{"className":26937},[106],[18,26939,26941,26944,26953,26956,26959,26964,26967,26970,26973,26976,26979,26982,26985,26988,26991,26994,26997,27050,27053,27099,27102,27142,27145,27185,27188,27191,27194,27197,27204,27207,27247,27250,27254,27257,27260,27263],{"className":26940},[106],[18,26942,928],{"className":26943},[169],[18,26945,26948],{"className":26946},[106,26947],"text",[18,26949,26952],{"className":26950},[106,26951],"cjk_fallback","只要",[18,26954,928],{"className":26955},[169],[18,26957],{"className":26958,"style":170},[169],[18,26960,15171,26962],{"className":26961},[985],[18,26963,15174],{"style":6352},[18,26965],{"className":26966,"style":170},[169],[18,26968,3526],{"className":26969,"style":246},[106,110],[18,26971,928],{"className":26972},[169],[18,26974],{"className":26975,"style":228},[169],[18,26977,26826],{"className":26978},[232],[18,26980,928],{"className":26981},[169],[18,26983],{"className":26984,"style":228},[169],[18,26986,64],{"className":26987},[106,110],[18,26989],{"className":26990,"style":2068},[169],[18,26992,26833],{"className":26993},[2072],[18,26995],{"className":26996,"style":2068},[169],[18,26998,27000],{"className":26999},[985,986],[18,27001,27003,27042],{"className":27002},[119,120],[18,27004,27006,27039],{"className":27005},[124],[18,27007,27009,27029],{"className":27008,"style":1083},[128],[18,27010,27011,27014],{"style":5735},[18,27012],{"className":27013,"style":697},[136],[18,27015,27017],{"className":27016},[141,142,143,144],[18,27018,27020,27023,27026],{"className":27019},[106,144],[18,27021,5597],{"className":27022},[106,110,144],[18,27024,52],{"className":27025},[165,144],[18,27027,19804],{"className":27028,"style":19921},[106,110,144],[18,27030,27031,27034],{"style":693},[18,27032],{"className":27033,"style":697},[136],[18,27035,27036],{},[18,27037,5590],{"className":27038},[985],[18,27040,152],{"className":27041},[151],[18,27043,27045],{"className":27044},[124],[18,27046,27048],{"className":27047,"style":5779},[128],[18,27049],{},[18,27051],{"className":27052,"style":170},[169],[18,27054,27056,27059],{"className":27055},[106],[18,27057,12572],{"className":27058,"style":254},[106,110],[18,27060,27062],{"className":27061},[115],[18,27063,27065,27091],{"className":27064},[119,120],[18,27066,27068,27088],{"className":27067},[124],[18,27069,27071],{"className":27070,"style":1152},[128],[18,27072,27073,27076],{"style":469},[18,27074],{"className":27075,"style":137},[136],[18,27077,27079],{"className":27078},[141,142,143,144],[18,27080,27082],{"className":27081},[106,144],[18,27083,27085],{"className":27084},[106,144],[18,27086,12814],{"className":27087},[106,6351,144],[18,27089,152],{"className":27090},[151],[18,27092,27094],{"className":27093},[124],[18,27095,27097],{"className":27096,"style":159},[128],[18,27098],{},[18,27100,74],{"className":27101},[250],[18,27103,27105,27108],{"className":27104},[106],[18,27106,939],{"className":27107,"style":1171},[106,110],[18,27109,27111],{"className":27110},[115],[18,27112,27114,27134],{"className":27113},[119,120],[18,27115,27117,27131],{"className":27116},[124],[18,27118,27120],{"className":27119,"style":5918},[128],[18,27121,27122,27125],{"style":1386},[18,27123],{"className":27124,"style":137},[136],[18,27126,27128],{"className":27127},[141,142,143,144],[18,27129,5597],{"className":27130},[106,110,144],[18,27132,152],{"className":27133},[151],[18,27135,27137],{"className":27136},[124],[18,27138,27140],{"className":27139,"style":159},[128],[18,27141],{},[18,27143,2248],{"className":27144},[106],[18,27146,27148,27151],{"className":27147},[106],[18,27149,939],{"className":27150,"style":1171},[106,110],[18,27152,27154],{"className":27153},[115],[18,27155,27157,27177],{"className":27156},[119,120],[18,27158,27160,27174],{"className":27159},[124],[18,27161,27163],{"className":27162,"style":5918},[128],[18,27164,27165,27168],{"style":1386},[18,27166],{"className":27167,"style":137},[136],[18,27169,27171],{"className":27170},[141,142,143,144],[18,27172,19804],{"className":27173,"style":19921},[106,110,144],[18,27175,152],{"className":27176},[151],[18,27178,27180],{"className":27179},[124],[18,27181,27183],{"className":27182,"style":19931},[128],[18,27184],{},[18,27186,84],{"className":27187},[267],[18,27189,52],{"className":27190},[165],[18,27192,928],{"className":27193},[169],[18,27195],{"className":27196,"style":170},[169],[18,27198,27200],{"className":27199},[106,26947],[18,27201,27203],{"className":27202},[106,26951],"就有",[18,27205,928],{"className":27206},[169],[18,27208,27210,27213],{"className":27209},[106],[18,27211,3526],{"className":27212},[106,3607],[18,27214,27216],{"className":27215},[115],[18,27217,27219,27239],{"className":27218},[119,120],[18,27220,27222,27236],{"className":27221},[124],[18,27223,27225],{"className":27224,"style":202},[128],[18,27226,27227,27230],{"style":1155},[18,27228],{"className":27229,"style":137},[136],[18,27231,27233],{"className":27232},[141,142,143,144],[18,27234,64],{"className":27235},[106,110,144],[18,27237,152],{"className":27238},[151],[18,27240,27242],{"className":27241},[124],[18,27243,27245],{"className":27244,"style":159},[128],[18,27246],{},[18,27248],{"className":27249,"style":228},[169],[18,27251,26891],{"className":27252},[232,27253],"amsrm",[18,27255],{"className":27256,"style":228},[169],[18,27258,5348],{"className":27259,"style":5477},[106,110],[18,27261,968],{"className":27262},[106],[18,27264,928],{"className":27265},[169],[18,27267,27269,27272],{"style":27268},"top:-3.09em;",[18,27270],{"className":27271,"style":26930},[136],[18,27273],{"className":27274,"style":27277},[27275,27276],"stretchy","fbox","height:2.2938em;border-style:solid;border-width:0.04em;",[18,27279,152],{"className":27280},[151],[18,27282,27284],{"className":27283},[124],[18,27285,27288],{"className":27286,"style":27287},[128],"height:1.2038em;",[18,27289],{},[11,27291,27292,27293,27321,27322,27350,27351,27354,27355,27383],{},"这里 ",[18,27294,27296,27309],{"className":27295,"translate":22},[21],[18,27297,27299],{"className":27298},[26],[28,27300,27301],{"xmlns":30},[32,27302,27303,27307],{},[35,27304,27305],{},[41,27306,3526],{},[86,27308,3526],{"encoding":88},[18,27310,27312],{"className":27311,"ariaHidden":51},[93],[18,27313,27315,27318],{"className":27314},[97],[18,27316],{"className":27317,"style":600},[101],[18,27319,3526],{"className":27320,"style":246},[106,110]," 是「用 ",[18,27323,27325,27338],{"className":27324,"translate":22},[21],[18,27326,27328],{"className":27327},[26],[28,27329,27330],{"xmlns":30},[32,27331,27332,27336],{},[35,27333,27334],{},[41,27335,5348],{},[86,27337,14088],{"encoding":88},[18,27339,27341],{"className":27340,"ariaHidden":51},[93],[18,27342,27344,27347],{"className":27343},[97],[18,27345],{"className":27346,"style":295},[101],[18,27348,5348],{"className":27349,"style":5477},[106,110]," 分开的假设个数」，也就是损失空间里的",[717,27352,27353],{},"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 ",[18,27356,27358,27371],{"className":27357,"translate":22},[21],[18,27359,27361],{"className":27360},[26],[28,27362,27363],{"xmlns":30},[32,27364,27365,27369],{},[35,27366,27367],{},[41,27368,5348],{},[86,27370,14088],{"encoding":88},[18,27372,27374],{"className":27373,"ariaHidden":51},[93],[18,27375,27377,27380],{"className":27376},[97],[18,27378],{"className":27379,"style":295},[101],[18,27381,5348],{"className":27382,"style":5477},[106,110]," 的小球，同时又让它们在 KL 意义下足够接近？",[11,27385,27386,27387,27746,27747,27780],{},"Yang 与 Barron 的版本直接用 ",[18,27388,27390,27456],{"className":27389,"translate":22},[21],[18,27391,27393],{"className":27392},[26],[28,27394,27395],{"xmlns":30},[32,27396,27397,27453],{},[35,27398,27399,27401,27403,27405,27407,27409,27411,27413,27419,27425,27435,27437,27443,27445,27451],{},[41,27400,16410],{},[49,27402,74],{"stretchy":73},[41,27404,5031],{},[49,27406,18000],{"separator":51},[41,27408,43],{},[49,27410,84],{"stretchy":73},[49,27412,433],{},[5651,27414,27415,27417],{},[45,27416,47],{},[41,27418,3526],{},[38,27420,27421,27423],{},[49,27422,20362],{},[41,27424,5597],{},[38,27426,27427,27429],{},[41,27428,12572],{},[35,27430,27431,27433],{},[41,27432,5031],{"mathvariant":316},[41,27434,12579],{"mathvariant":316},[49,27436,74],{"stretchy":73},[38,27438,27439,27441],{},[41,27440,939],{},[41,27442,5597],{},[41,27444,2248],{"mathvariant":316},[658,27446,27447,27449],{"accent":51},[41,27448,939],{},[49,27450,25287],{},[49,27452,84],{"stretchy":73},[86,27454,27455],{"encoding":88},"I(K;X)=\\frac1M\\sum_i D_{\\mathrm{KL}}(P_i\\Vert\\bar P)",[18,27457,27459,27495],{"className":27458,"ariaHidden":51},[93],[18,27460,27462,27465,27468,27471,27474,27477,27480,27483,27486,27489,27492],{"className":27461},[97],[18,27463],{"className":27464,"style":242},[101],[18,27466,16410],{"className":27467,"style":111},[106,110],[18,27469,74],{"className":27470},[250],[18,27472,5031],{"className":27473,"style":5073},[106,110],[18,27475,18000],{"className":27476},[165],[18,27478],{"className":27479,"style":170},[169],[18,27481,43],{"className":27482,"style":111},[106,110],[18,27484,84],{"className":27485},[267],[18,27487],{"className":27488,"style":228},[169],[18,27490,433],{"className":27491},[232],[18,27493],{"className":27494,"style":228},[169],[18,27496,27498,27501,27569,27572,27617,27620,27666,27669,27709,27712,27743],{"className":27497},[97],[18,27499],{"className":27500,"style":12023},[101],[18,27502,27504,27507,27566],{"className":27503},[106],[18,27505],{"className":27506},[250,6009],[18,27508,27510],{"className":27509},[5651],[18,27511,27513,27558],{"className":27512},[119,120],[18,27514,27516,27555],{"className":27515},[124],[18,27517,27519,27533,27541],{"className":27518,"style":12042},[128],[18,27520,27521,27524],{"style":12045},[18,27522],{"className":27523,"style":697},[136],[18,27525,27527],{"className":27526},[141,142,143,144],[18,27528,27530],{"className":27529},[106,144],[18,27531,3526],{"className":27532,"style":246},[106,110,144],[18,27534,27535,27538],{"style":6037},[18,27536],{"className":27537,"style":697},[136],[18,27539],{"className":27540,"style":6045},[6044],[18,27542,27543,27546],{"style":12068},[18,27544],{"className":27545,"style":697},[136],[18,27547,27549],{"className":27548},[141,142,143,144],[18,27550,27552],{"className":27551},[106,144],[18,27553,47],{"className":27554},[106,144],[18,27556,152],{"className":27557},[151],[18,27559,27561],{"className":27560},[124],[18,27562,27564],{"className":27563,"style":12090},[128],[18,27565],{},[18,27567],{"className":27568},[267,6009],[18,27570],{"className":27571,"style":170},[169],[18,27573,27575,27580],{"className":27574},[985],[18,27576,20362],{"className":27577,"style":27579},[985,4198,27578],"small-op","position:relative;top:0em;",[18,27581,27583],{"className":27582},[115],[18,27584,27586,27608],{"className":27585},[119,120],[18,27587,27589,27605],{"className":27588},[124],[18,27590,27593],{"className":27591,"style":27592},[128],"height:0.162em;",[18,27594,27596,27599],{"style":27595},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[18,27597],{"className":27598,"style":137},[136],[18,27600,27602],{"className":27601},[141,142,143,144],[18,27603,5597],{"className":27604},[106,110,144],[18,27606,152],{"className":27607},[151],[18,27609,27611],{"className":27610},[124],[18,27612,27615],{"className":27613,"style":27614},[128],"height:0.2997em;",[18,27616],{},[18,27618],{"className":27619,"style":170},[169],[18,27621,27623,27626],{"className":27622},[106],[18,27624,12572],{"className":27625,"style":254},[106,110],[18,27627,27629],{"className":27628},[115],[18,27630,27632,27658],{"className":27631},[119,120],[18,27633,27635,27655],{"className":27634},[124],[18,27636,27638],{"className":27637,"style":1152},[128],[18,27639,27640,27643],{"style":469},[18,27641],{"className":27642,"style":137},[136],[18,27644,27646],{"className":27645},[141,142,143,144],[18,27647,27649],{"className":27648},[106,144],[18,27650,27652],{"className":27651},[106,144],[18,27653,12814],{"className":27654},[106,6351,144],[18,27656,152],{"className":27657},[151],[18,27659,27661],{"className":27660},[124],[18,27662,27664],{"className":27663,"style":159},[128],[18,27665],{},[18,27667,74],{"className":27668},[250],[18,27670,27672,27675],{"className":27671},[106],[18,27673,939],{"className":27674,"style":1171},[106,110],[18,27676,27678],{"className":27677},[115],[18,27679,27681,27701],{"className":27680},[119,120],[18,27682,27684,27698],{"className":27683},[124],[18,27685,27687],{"className":27686,"style":5918},[128],[18,27688,27689,27692],{"style":1386},[18,27690],{"className":27691,"style":137},[136],[18,27693,27695],{"className":27694},[141,142,143,144],[18,27696,5597],{"className":27697},[106,110,144],[18,27699,152],{"className":27700},[151],[18,27702,27704],{"className":27703},[124],[18,27705,27707],{"className":27706,"style":159},[128],[18,27708],{},[18,27710,2248],{"className":27711},[106],[18,27713,27715],{"className":27714},[106,681],[18,27716,27718],{"className":27717},[119],[18,27719,27721],{"className":27720},[124],[18,27722,27724,27732],{"className":27723,"style":25617},[128],[18,27725,27726,27729],{"style":693},[18,27727],{"className":27728,"style":697},[136],[18,27730,939],{"className":27731,"style":1171},[106,110],[18,27733,27734,27737],{"style":17697},[18,27735],{"className":27736,"style":697},[136],[18,27738,27740],{"className":27739,"style":711},[710],[18,27741,25287],{"className":27742},[106],[18,27744,84],{"className":27745},[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 ",[18,27748,27750,27766],{"className":27749,"translate":22},[21],[18,27751,27753],{"className":27752},[26],[28,27754,27755],{"xmlns":30},[32,27756,27757,27763],{},[35,27758,27759,27761],{},[41,27760,14730],{},[49,27762,899],{},[86,27764,27765],{"encoding":88},"\\log",[18,27767,27769],{"className":27768,"ariaHidden":51},[93],[18,27770,27772,27775],{"className":27771},[97],[18,27773],{"className":27774,"style":2890},[101],[18,27776,15171,27778],{"className":27777},[985],[18,27779,15174],{"style":6352}," 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。",[11,27782,27783,27784,27787,27788,27858,27859,27887,27888,27998,27999,28027,28028,28056,28057,1990],{},"Fano 还有一个",[717,27785,27786],{},"局部版本","：不要求 ",[18,27789,27791,27809],{"className":27790,"translate":22},[21],[18,27792,27794],{"className":27793},[26],[28,27795,27796],{"xmlns":30},[32,27797,27798,27806],{},[35,27799,27800],{},[38,27801,27802,27804],{},[41,27803,77],{},[41,27805,5597],{},[86,27807,27808],{"encoding":88},"\\theta_i",[18,27810,27812],{"className":27811,"ariaHidden":51},[93],[18,27813,27815,27818],{"className":27814},[97],[18,27816],{"className":27817,"style":448},[101],[18,27819,27821,27824],{"className":27820},[106],[18,27822,77],{"className":27823,"style":254},[106,110],[18,27825,27827],{"className":27826},[115],[18,27828,27830,27850],{"className":27829},[119,120],[18,27831,27833,27847],{"className":27832},[124],[18,27834,27836],{"className":27835,"style":5918},[128],[18,27837,27838,27841],{"style":469},[18,27839],{"className":27840,"style":137},[136],[18,27842,27844],{"className":27843},[141,142,143,144],[18,27845,5597],{"className":27846},[106,110,144],[18,27848,152],{"className":27849},[151],[18,27851,27853],{"className":27852},[124],[18,27854,27856],{"className":27855,"style":159},[128],[18,27857],{}," 覆盖整个参数空间，只要求它们落在一个半径 ",[18,27860,27862,27875],{"className":27861,"translate":22},[21],[18,27863,27865],{"className":27864},[26],[28,27866,27867],{"xmlns":30},[32,27868,27869,27873],{},[35,27870,27871],{},[41,27872,14444],{},[86,27874,14444],{"encoding":88},[18,27876,27878],{"className":27877,"ariaHidden":51},[93],[18,27879,27881,27884],{"className":27880},[97],[18,27882],{"className":27883,"style":1083},[101],[18,27885,14444],{"className":27886,"style":254},[106,110]," 的小球里。非参数速率 ",[18,27889,27891,27929],{"className":27890,"translate":22},[21],[18,27892,27894],{"className":27893},[26],[28,27895,27896],{"xmlns":30},[32,27897,27898,27926],{},[35,27899,27900],{},[1556,27901,27902,27904],{},[41,27903,64],{},[35,27905,27906,27908,27910,27912,27914,27916,27918,27920,27922,27924],{},[49,27907,2253],{},[45,27909,2266],{},[41,27911,20979],{},[41,27913,317],{"mathvariant":316},[49,27915,74],{"stretchy":73},[45,27917,2266],{},[41,27919,20979],{},[49,27921,7984],{},[41,27923,4031],{},[49,27925,84],{"stretchy":73},[86,27927,27928],{"encoding":88},"n^{-2\\beta\u002F(2\\beta+d)}",[18,27930,27932],{"className":27931,"ariaHidden":51},[93],[18,27933,27935,27939],{"className":27934},[97],[18,27936],{"className":27937,"style":27938},[101],"height:0.888em;",[18,27940,27942,27945],{"className":27941},[106],[18,27943,64],{"className":27944},[106,110],[18,27946,27948],{"className":27947},[115],[18,27949,27951],{"className":27950},[119],[18,27952,27954],{"className":27953},[124],[18,27955,27957],{"className":27956,"style":27938},[128],[18,27958,27959,27962],{"style":1962},[18,27960],{"className":27961,"style":137},[136],[18,27963,27965],{"className":27964},[141,142,143,144],[18,27966,27968,27971,27974,27977,27980,27983,27986,27989,27992,27995],{"className":27967},[106,144],[18,27969,2253],{"className":27970},[106,144],[18,27972,2266],{"className":27973},[106,144],[18,27975,20979],{"className":27976,"style":21031},[106,110,144],[18,27978,317],{"className":27979},[106,144],[18,27981,74],{"className":27982},[250,144],[18,27984,2266],{"className":27985},[106,144],[18,27987,20979],{"className":27988,"style":21031},[106,110,144],[18,27990,7984],{"className":27991},[2072,144],[18,27993,4031],{"className":27994},[106,110,144],[18,27996,84],{"className":27997},[267,144]," 正是这样来的——局部的打包数随 ",[18,28000,28002,28015],{"className":28001,"translate":22},[21],[18,28003,28005],{"className":28004},[26],[28,28006,28007],{"xmlns":30},[32,28008,28009,28013],{},[35,28010,28011],{},[41,28012,14444],{},[86,28014,14444],{"encoding":88},[18,28016,28018],{"className":28017,"ariaHidden":51},[93],[18,28019,28021,28024],{"className":28020},[97],[18,28022],{"className":28023,"style":1083},[101],[18,28025,14444],{"className":28026,"style":254},[106,110]," 增长，而 KL 随 ",[18,28029,28031,28044],{"className":28030,"translate":22},[21],[18,28032,28034],{"className":28033},[26],[28,28035,28036],{"xmlns":30},[32,28037,28038,28042],{},[35,28039,28040],{},[41,28041,14444],{},[86,28043,14444],{"encoding":88},[18,28045,28047],{"className":28046,"ariaHidden":51},[93],[18,28048,28050,28053],{"className":28049},[97],[18,28051],{"className":28052,"style":1083},[101],[18,28054,14444],{"className":28055,"style":254},[106,110]," 增长，两者平衡给出最优的 ",[18,28058,28060,28073],{"className":28059,"translate":22},[21],[18,28061,28063],{"className":28062},[26],[28,28064,28065],{"xmlns":30},[32,28066,28067,28071],{},[35,28068,28069],{},[41,28070,14444],{},[86,28072,14444],{"encoding":88},[18,28074,28076],{"className":28075,"ariaHidden":51},[93],[18,28077,28079,28082],{"className":28078},[97],[18,28080],{"className":28081,"style":1083},[101],[18,28083,14444],{"className":28084,"style":254},[106,110],[638,28086,28088],{"id":28087},"五assouad-引理","五、Assouad 引理",[1311,28090,28092],{"id":28091},"_51-超立方体","5.1 超立方体",[11,28094,28095,28096,28124],{},"Fano 处理的是「",[18,28097,28099,28112],{"className":28098,"translate":22},[21],[18,28100,28102],{"className":28101},[26],[28,28103,28104],{"xmlns":30},[32,28105,28106,28110],{},[35,28107,28108],{},[41,28109,3526],{},[86,28111,3526],{"encoding":88},[18,28113,28115],{"className":28114,"ariaHidden":51},[93],[18,28116,28118,28121],{"className":28117},[97],[18,28119],{"className":28120,"style":600},[101],[18,28122,3526],{"className":28123,"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。",[5159,28126,28128,28336,29105,29108],{"title":28127},"定理 5.1（Assouad 引理，平方损失）",[11,28129,16,28130,4741,28235,28305,28306,28335],{},[18,28131,28133,28164],{"className":28132,"translate":22},[21],[18,28134,28136],{"className":28135},[26],[28,28137,28138],{"xmlns":30},[32,28139,28140,28161],{},[35,28141,28142,28144,28146,28148,28150,28152,28154],{},[41,28143,924],{"mathvariant":316},[49,28145,433],{},[49,28147,1331],{"stretchy":73},[45,28149,430],{},[49,28151,52],{"separator":51},[45,28153,47],{},[1556,28155,28156,28158],{},[49,28157,1349],{"stretchy":73},[41,28159,28160],{},"m",[86,28162,28163],{"encoding":88},"\\Theta=\\{0,1\\}^m",[18,28165,28167,28185],{"className":28166,"ariaHidden":51},[93],[18,28168,28170,28173,28176,28179,28182],{"className":28169},[97],[18,28171],{"className":28172,"style":600},[101],[18,28174,924],{"className":28175},[106],[18,28177],{"className":28178,"style":228},[169],[18,28180,433],{"className":28181},[232],[18,28183],{"className":28184,"style":228},[169],[18,28186,28188,28191,28194,28197,28200,28203,28206],{"className":28187},[97],[18,28189],{"className":28190,"style":242},[101],[18,28192,1331],{"className":28193},[250],[18,28195,430],{"className":28196},[106],[18,28198,52],{"className":28199},[165],[18,28201],{"className":28202,"style":170},[169],[18,28204,47],{"className":28205},[106],[18,28207,28209,28212],{"className":28208},[267],[18,28210,1349],{"className":28211},[267],[18,28213,28215],{"className":28214},[115],[18,28216,28218],{"className":28217},[119],[18,28219,28221],{"className":28220},[124],[18,28222,28224],{"className":28223,"style":1959},[128],[18,28225,28226,28229],{"style":1962},[18,28227],{"className":28228,"style":137},[136],[18,28230,28232],{"className":28231},[141,142,143,144],[18,28233,28160],{"className":28234},[106,110,144],[18,28236,28238,28256],{"className":28237,"translate":22},[21],[18,28239,28241],{"className":28240},[26],[28,28242,28243],{"xmlns":30},[32,28244,28245,28253],{},[35,28246,28247],{},[38,28248,28249,28251],{},[41,28250,939],{},[41,28252,77],{},[86,28254,28255],{"encoding":88},"P_\\theta",[18,28257,28259],{"className":28258,"ariaHidden":51},[93],[18,28260,28262,28265],{"className":28261},[97],[18,28263],{"className":28264,"style":4563},[101],[18,28266,28268,28271],{"className":28267},[106],[18,28269,939],{"className":28270,"style":1171},[106,110],[18,28272,28274],{"className":28273},[115],[18,28275,28277,28297],{"className":28276},[119,120],[18,28278,28280,28294],{"className":28279},[124],[18,28281,28283],{"className":28282,"style":1383},[128],[18,28284,28285,28288],{"style":1386},[18,28286],{"className":28287,"style":137},[136],[18,28289,28291],{"className":28290},[141,142,143,144],[18,28292,77],{"className":28293,"style":254},[106,110,144],[18,28295,152],{"className":28296},[151],[18,28298,28300],{"className":28299},[124],[18,28301,28303],{"className":28302,"style":159},[128],[18,28304],{}," 是观测分布，损失为平方欧氏距离。对每个坐标 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X]",[18,29991,29993,30085,30109],{"className":29992,"ariaHidden":51},[93],[18,29994,29996,29999,30067,30070,30073,30076,30079,30082],{"className":29995},[97],[18,29997],{"className":29998,"style":2281},[101],[18,30000,30002,30033],{"className":30001},[106],[18,30003,30005],{"className":30004},[106,681],[18,30006,30008],{"className":30007},[119],[18,30009,30011],{"className":30010},[124],[18,30012,30014,30022],{"className":30013,"style":677},[128],[18,30015,30016,30019],{"style":693},[18,30017],{"className":30018,"style":697},[136],[18,30020,77],{"className":30021,"style":254},[106,110],[18,30023,30024,30027],{"style":703},[18,30025],{"className":30026,"style":697},[136],[18,30028,30030],{"className":30029,"style":711},[710],[18,30031,664],{"className":30032},[106],[18,30034,30036],{"className":30035},[115],[18,30037,30039,30059],{"className":30038},[119,120],[18,30040,30042,30056],{"className":30041},[124],[18,30043,30045],{"className":30044,"style":1152},[128],[18,30046,30047,30050],{"style":469},[18,30048],{"className":30049,"style":137},[136],[18,30051,30053],{"className":30052},[141,142,143,144],[18,30054,3978],{"className":30055,"style":11302},[106,110,144],[18,30057,152],{"className":30058},[151],[18,30060,30062],{"className":30061},[124],[18,30063,30065],{"className":30064,"style":159},[128],[18,30066],{},[18,30068,74],{"className":30069},[250],[18,30071,43],{"className":30072,"style":111},[106,110],[18,30074,84],{"className":30075},[267],[18,30077],{"className":30078,"style":228},[169],[18,30080,433],{"className":30081},[232],[18,30083],{"className":30084,"style":228},[169],[18,30086,30088,30091,30094,30097,30100,30103,30106],{"className":30087},[97],[18,30089],{"className":30090,"style":242},[101],[18,30092,934],{"className":30093},[106,1139],[18,30095,945],{"className":30096},[250],[18,30098,77],{"className":30099,"style":254},[106,110],[18,30101],{"className":30102,"style":228},[169],[18,30104,11222],{"className":30105},[232],[18,30107],{"className":30108,"style":228},[169],[18,30110,30112,30115,30118],{"className":30111},[97],[18,30113],{"className":30114,"style":242},[101],[18,30116,43],{"className":30117,"style":111},[106,110],[18,30119,965],{"className":30120},[267],"，其贝叶斯风险为",[18,30123,30125],{"className":30124,"translate":22},[875],[18,30126,30128,30234],{"className":30127,"translate":22},[21],[18,30129,30131],{"className":30130},[26],[28,30132,30133],{"xmlns":30,"display":884},[32,30134,30135,30231],{},[35,30136,30137,30151,30153,30155,30161,30163,30165,30167,30169,30171,30173,30179,30181,30183,30185,30187,30189,30191,30193,30199,30201,30203,30205,30207,30209,30211,30213,30219,30221,30223,30225,30227,30229],{},[20358,30138,30139,30141,30149],{},[49,30140,20362],{},[35,30142,30143,30145,30147],{},[41,30144,19804],{},[49,30146,433],{},[45,30148,47],{},[41,30150,28160],{},[41,30152,934],{"mathvariant":933},[49,30154,945],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[38,30156,30157,30159],{},[41,30158,3941],{},[41,30160,19804],{},[49,30162,74],{"stretchy":73},[41,30164,43],{},[49,30166,84],{"stretchy":73},[49,30168,74],{"stretchy":73},[45,30170,47],{},[49,30172,2253],{},[38,30174,30175,30177],{},[41,30176,3941],{},[41,30178,19804],{},[49,30180,74],{"stretchy":73},[41,30182,43],{},[49,30184,84],{"stretchy":73},[49,30186,84],{"stretchy":73},[49,30188,965],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,30190,52],{"separator":51},[169,30192],{"width":7341},[38,30194,30195,30197],{},[41,30196,3941],{},[41,30198,19804],{},[49,30200,74],{"stretchy":73},[41,30202,43],{},[49,30204,84],{"stretchy":73},[49,30206,433],{},[41,30208,11205],{},[49,30210,899],{},[49,30212,74],{"stretchy":73},[38,30214,30215,30217],{},[41,30216,77],{},[41,30218,19804],{},[49,30220,433],{},[45,30222,47],{},[49,30224,11222],{},[41,30226,43],{},[49,30228,84],{"stretchy":73},[41,30230,968],{"mathvariant":316},[86,30232,30233],{"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).",[18,30235,30237,30386,30514,30575,30593],{"className":30236,"ariaHidden":51},[93],[18,30238,30240,30244,30309,30312,30315,30321,30362,30365,30368,30371,30374,30377,30380,30383],{"className":30239},[97],[18,30241],{"className":30242,"style":30243},[101],"height:3.0652em;vertical-align:-1.4138em;",[18,30245,30247],{"className":30246},[985,986],[18,30248,30250,30301],{"className":30249},[119,120],[18,30251,30253,30298],{"className":30252},[124],[18,30254,30257,30277,30287],{"className":30255,"style":30256},[128],"height:1.6514em;",[18,30258,30259,30262],{"style":20548},[18,30260],{"className":30261,"style":20552},[136],[18,30263,30265],{"className":30264},[141,142,143,144],[18,30266,30268,30271,30274],{"className":30267},[106,144],[18,30269,19804],{"className":30270,"style":19921},[106,110,144],[18,30272,433],{"className":30273},[232,144],[18,30275,47],{"className":30276},[106,144],[18,30278,30279,30282],{"style":20570},[18,30280],{"className":30281,"style":20552},[136],[18,30283,30284],{},[18,30285,20362],{"className":30286},[985,4198,4199],[18,30288,30289,30292],{"style":20581},[18,30290],{"className":30291,"style":20552},[136],[18,30293,30295],{"className":30294},[141,142,143,144],[18,30296,28160],{"className":30297},[106,110,144],[18,30299,152],{"className":30300},[151],[18,30302,30304],{"className":30303},[124],[18,30305,30307],{"className":30306,"style":26219},[128],[18,30308],{},[18,30310],{"className":30311,"style":170},[169],[18,30313,934],{"className":30314},[106,1139],[18,30316,30318],{"className":30317},[250],[18,30319,945],{"className":30320},[1230,1196],[18,30322,30324,30327],{"className":30323},[106],[18,30325,3941],{"className":30326,"style":3957},[106,110],[18,30328,30330],{"className":30329},[115],[18,30331,30333,30354],{"className":30332},[119,120],[18,30334,30336,30351],{"className":30335},[124],[18,30337,30339],{"className":30338,"style":5918},[128],[18,30340,30342,30345],{"style":30341},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[18,30343],{"className":30344,"style":137},[136],[18,30346,30348],{"className":30347},[141,142,143,144],[18,30349,19804],{"className":30350,"style":19921},[106,110,144],[18,30352,152],{"className":30353},[151],[18,30355,30357],{"className":30356},[124],[18,30358,30360],{"className":30359,"style":19931},[128],[18,30361],{},[18,30363,74],{"className":30364},[250],[18,30366,43],{"className":30367,"style":111},[106,110],[18,30369,84],{"className":30370},[267],[18,30372,74],{"className":30373},[250],[18,30375,47],{"className":30376},[106],[18,30378],{"className":30379,"style":2068},[169],[18,30381,2253],{"className":30382},[2072],[18,30384],{"className":30385,"style":2068},[169],[18,30387,30389,30392,30432,30435,30438,30441,30447,30450,30453,30456,30496,30499,30502,30505,30508,30511],{"className":30388},[97],[18,30390],{"className":30391,"style":6098},[101],[18,30393,30395,30398],{"className":30394},[106],[18,30396,3941],{"className":30397,"style":3957},[106,110],[18,30399,30401],{"className":30400},[115],[18,30402,30404,30424],{"className":30403},[119,120],[18,30405,30407,30421],{"className":30406},[124],[18,30408,30410],{"className":30409,"style":5918},[128],[18,30411,30412,30415],{"style":30341},[18,30413],{"className":30414,"style":137},[136],[18,30416,30418],{"className":30417},[141,142,143,144],[18,30419,19804],{"className":30420,"style":19921},[106,110,144],[18,30422,152],{"className":30423},[151],[18,30425,30427],{"className":30426},[124],[18,30428,30430],{"className":30429,"style":19931},[128],[18,30431],{},[18,30433,74],{"className":30434},[250],[18,30436,43],{"className":30437,"style":111},[106,110],[18,30439,3431],{"className":30440},[267],[18,30442,30444],{"className":30443},[267],[18,30445,965],{"className":30446},[1230,1196],[18,30448,52],{"className":30449},[165],[18,30451],{"className":30452,"style":7673},[169],[18,30454],{"className":30455,"style":170},[169],[18,30457,30459,30462],{"className":30458},[106],[18,30460,3941],{"className":30461,"style":3957},[106,110],[18,30463,30465],{"className":30464},[115],[18,30466,30468,30488],{"className":30467},[119,120],[18,30469,30471,30485],{"className":30470},[124],[18,30472,30474],{"className":30473,"style":5918},[128],[18,30475,30476,30479],{"style":30341},[18,30477],{"className":30478,"style":137},[136],[18,30480,30482],{"className":30481},[141,142,143,144],[18,30483,19804],{"className":30484,"style":19921},[106,110,144],[18,30486,152],{"className":30487},[151],[18,30489,30491],{"className":30490},[124],[18,30492,30494],{"className":30493,"style":19931},[128],[18,30495],{},[18,30497,74],{"className":30498},[250],[18,30500,43],{"className":30501,"style":111},[106,110],[18,30503,84],{"className":30504},[267],[18,30506],{"className":30507,"style":228},[169],[18,30509,433],{"className":30510},[232],[18,30512],{"className":30513,"style":228},[169],[18,30515,30517,30520,30523,30526,30566,30569,30572],{"className":30516},[97],[18,30518],{"className":30519,"style":19839},[101],[18,30521,11205],{"className":30522},[985],[18,30524,74],{"className":30525},[250],[18,30527,30529,30532],{"className":30528},[106],[18,30530,77],{"className":30531,"style":254},[106,110],[18,30533,30535],{"className":30534},[115],[18,30536,30538,30558],{"className":30537},[119,120],[18,30539,30541,30555],{"className":30540},[124],[18,30542,30544],{"className":30543,"style":5918},[128],[18,30545,30546,30549],{"style":469},[18,30547],{"className":30548,"style":137},[136],[18,30550,30552],{"className":30551},[141,142,143,144],[18,30553,19804],{"className":30554,"style":19921},[106,110,144],[18,30556,152],{"className":30557},[151],[18,30559,30561],{"className":30560},[124],[18,30562,30564],{"className":30563,"style":19931},[128],[18,30565],{},[18,30567],{"className":30568,"style":228},[169],[18,30570,433],{"className":30571},[232],[18,30573],{"className":30574,"style":228},[169],[18,30576,30578,30581,30584,30587,30590],{"className":30577},[97],[18,30579],{"className":30580,"style":242},[101],[18,30582,47],{"className":30583},[106],[18,30585],{"className":30586,"style":228},[169],[18,30588,11222],{"className":30589},[232],[18,30591],{"className":30592,"style":228},[169],[18,30594,30596,30599,30602,30605],{"className":30595},[97],[18,30597],{"className":30598,"style":242},[101],[18,30600,43],{"className":30601,"style":111},[106,110],[18,30603,84],{"className":30604},[267],[18,30606,968],{"className":30607},[106],[11,30609,30610,30611,30639,30640,30711,30712,509,30797,30882,30883,4741],{},"固定 ",[18,30612,30614,30627],{"className":30613,"translate":22},[21],[18,30615,30617],{"className":30616},[26],[28,30618,30619],{"xmlns":30},[32,30620,30621,30625],{},[35,30622,30623],{},[41,30624,19804],{},[86,30626,19804],{"encoding":88},[18,30628,30630],{"className":30629,"ariaHidden":51},[93],[18,30631,30633,30636],{"className":30632},[97],[18,30634],{"className":30635,"style":28331},[101],[18,30637,19804],{"className":30638,"style":19921},[106,110],"。边际后验 ",[18,30641,30643,30661],{"className":30642,"translate":22},[21],[18,30644,30646],{"className":30645},[26],[28,30647,30648],{"xmlns":30},[32,30649,30650,30658],{},[35,30651,30652],{},[38,30653,30654,30656],{},[41,30655,3941],{},[41,30657,19804],{},[86,30659,30660],{"encoding":88},"\\pi_j",[18,30662,30664],{"className":30663,"ariaHidden":51},[93],[18,30665,30667,30671],{"className":30666},[97],[18,30668],{"className":30669,"style":30670},[101],"height:0.7167em;vertical-align:-0.2861em;",[18,30672,30674,30677],{"className":30673},[106],[18,30675,3941],{"className":30676,"style":3957},[106,110],[18,30678,30680],{"className":30679},[115],[18,30681,30683,30703],{"className":30682},[119,120],[18,30684,30686,30700],{"className":30685},[124],[18,30687,30689],{"className":30688,"style":5918},[128],[18,30690,30691,30694],{"style":30341},[18,30692],{"className":30693,"style":137},[136],[18,30695,30697],{"className":30696},[141,142,143,144],[18,30698,19804],{"className":30699,"style":19921},[106,110,144],[18,30701,152],{"className":30702},[151],[18,30704,30706],{"className":30705},[124],[18,30707,30709],{"className":30708,"style":19931},[128],[18,30710],{}," 正是 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在均匀先验下的后验概率。对任意 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E\\bigl[\\min(\\pi_j,1-\\pi_j)\\bigr]\n\\ge\n\\frac12\\left(1-\\lVert Q_{j,0}-Q_{j,1}\\rVert_{\\mathrm{TV}}\\right),",[18,31289,31291,31370,31434],{"className":31290,"ariaHidden":51},[93],[18,31292,31294,31297,31300,31306,31309,31312,31352,31355,31358,31361,31364,31367],{"className":31293},[97],[18,31295],{"className":31296,"style":6098},[101],[18,31298,934],{"className":31299},[106,1139],[18,31301,31303],{"className":31302},[250],[18,31304,945],{"className":31305},[1230,1196],[18,31307,11949],{"className":31308},[985],[18,31310,74],{"className":31311},[250],[18,31313,31315,31318],{"className":31314},[106],[18,31316,3941],{"className":31317,"style":3957},[106,110],[18,31319,31321],{"className":31320},[115],[18,31322,31324,31344],{"className":31323},[119,120],[18,31325,31327,31341],{"className":31326},[124],[18,31328,31330],{"className":31329,"style":5918},[128],[18,31331,31332,31335],{"style":30341},[18,31333],{"className":31334,"style":137},[136],[18,31336,31338],{"className":31337},[141,142,143,144],[18,31339,19804],{"className":31340,"style":19921},[106,110,144],[18,31342,152],{"className":31343},[151],[18,31345,31347],{"className":31346},[124],[18,31348,31350],{"className":31349,"style":19931},[128],[18,31351],{},[18,31353,52],{"className":31354},[165],[18,31356],{"className":31357,"style":170},[169],[18,31359,47],{"className":31360},[106],[18,31362],{"className":31363,"style":2068},[169],[18,31365,2253],{"className":31366},[2072],[18,31368],{"className":31369,"style":2068},[169],[18,31371,31373,31376,31416,31419,31425,31428,31431],{"className":31372},[97],[18,31374],{"className":31375,"style":6098},[101],[18,31377,31379,31382],{"className":31378},[106],[18,31380,3941],{"className":31381,"style":3957},[106,110],[18,31383,31385],{"className":31384},[115],[18,31386,31388,31408],{"className":31387},[119,120],[18,31389,31391,31405],{"className":31390},[124],[18,31392,31394],{"className":31393,"style":5918},[128],[18,31395,31396,31399],{"style":30341},[18,31397],{"className":31398,"style":137},[136],[18,31400,31402],{"className":31401},[141,142,143,144],[18,31403,19804],{"className":31404,"style":19921},[106,110,144],[18,31406,152],{"className":31407},[151],[18,31409,31411],{"className":31410},[124],[18,31412,31414],{"className":31413,"style":19931},[128],[18,31415],{},[18,31417,84],{"className":31418},[267],[18,31420,31422],{"className":31421},[267],[18,31423,965],{"className":31424},[1230,1196],[18,31426],{"className":31427,"style":228},[169],[18,31429,4775],{"className":31430},[232],[18,31432],{"className":31433,"style":228},[169],[18,31435,31437,31440,31502,31505,31682,31685],{"className":31436},[97],[18,31438],{"className":31439,"style":9559},[101],[18,31441,31443,31446,31499],{"className":31442},[106],[18,31444],{"className":31445},[250,6009],[18,31447,31449],{"className":31448},[5651],[18,31450,31452,31491],{"className":31451},[119,120],[18,31453,31455,31488],{"className":31454},[124],[18,31456,31458,31469,31477],{"className":31457,"style":9578},[128],[18,31459,31460,31463],{"style":6025},[18,31461],{"className":31462,"style":697},[136],[18,31464,31466],{"className":31465},[106],[18,31467,2266],{"className":31468},[106],[18,31470,31471,31474],{"style":6037},[18,31472],{"className":31473,"style":697},[136],[18,31475],{"className":31476,"style":6045},[6044],[18,31478,31479,31482],{"style":6048},[18,31480],{"className":31481,"style":697},[136],[18,31483,31485],{"className":31484},[106],[18,31486,47],{"className":31487},[106],[18,31489,152],{"className":31490},[151],[18,31492,31494],{"className":31493},[124],[18,31495,31497],{"className":31496,"style":6067},[128],[18,31498],{},[18,31500],{"className":31501},[267,6009],[18,31503],{"className":31504,"style":170},[169],[18,31506,31508,31511,31514,31517,31520,31523,31526,31575,31578,31581,31584,31633,31679],{"className":31507},[174],[18,31509,74],{"className":31510,"style":13793},[250,13792],[18,31512,47],{"className":31513},[106],[18,31515],{"className":31516,"style":2068},[169],[18,31518,2253],{"className":31519},[2072],[18,31521],{"className":31522,"style":2068},[169],[18,31524,2248],{"className":31525},[250],[18,31527,31529,31532],{"className":31528},[106],[18,31530,12544],{"className":31531},[106,110],[18,31533,31535],{"className":31534},[115],[18,31536,31538,31567],{"className":31537},[119,120],[18,31539,31541,31564],{"className":31540},[124],[18,31542,31544],{"className":31543,"style":5918},[128],[18,31545,31546,31549],{"style":1155},[18,31547],{"className":31548,"style":137},[136],[18,31550,31552],{"className":31551},[141,142,143,144],[18,31553,31555,31558,31561],{"className":31554},[106,144],[18,31556,19804],{"className":31557,"style":19921},[106,110,144],[18,31559,52],{"className":31560},[165,144],[18,31562,430],{"className":31563},[106,144],[18,31565,152],{"className":31566},[151],[18,31568,31570],{"className":31569},[124],[18,31571,31573],{"className":31572,"style":19931},[128],[18,31574],{},[18,31576],{"className":31577,"style":2068},[169],[18,31579,2253],{"className":31580},[2072],[18,31582],{"className":31583,"st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3.1 的检验形式。于是每个坐标贡献至少 ",[18,31692,31694,31730],{"className":31693,"translate":22},[21],[18,31695,31697],{"className":31696},[26],[28,31698,31699],{"xmlns":30},[32,31700,31701,31727],{},[35,31702,31703,31709,31711,31713,31715,31725],{},[5651,31704,31705,31707],{},[45,31706,47],{},[45,31708,10444],{},[49,31710,74],{"stretchy":73},[45,31712,47],{},[49,31714,2253],{},[38,31716,31717,31723],{},[35,31718,31719,31721],{},[41,31720,5700],{"mathvariant":316},[41,31722,5703],{"mathvariant":316},[41,31724,19804],{},[49,31726,84],{"stretchy":73},[86,31728,31729],{"encoding":88},"\\frac14(1-\\mathrm{TV}_j)",[18,31731,31733,31822],{"className":31732,"ariaHidden":51},[93],[18,31734,31736,31739,31807,31810,31813,31816,31819],{"className":31735},[97],[18,31737],{"className":31738,"style":12023},[101],[18,31740,31742,31745,31804],{"className":31741},[106],[18,31743],{"className":31744},[250,6009],[18,31746,31748],{"className":31747},[5651],[18,31749,31751,31796],{"className":31750},[119,120],[18,31752,31754,31793],{"className":31753},[124],[18,31755,31757,31771,31779],{"className":31756,"style":12042},[128],[18,31758,31759,31762],{"style":12045},[18,31760],{"className":31761,"style":697},[136],[18,31763,31765],{"className":31764},[141,142,143,144],[18,31766,31768],{"className":31767},[106,144],[18,31769,10444],{"className":31770},[106,144],[18,31772,31773,31776],{"style":6037},[18,31774],{"className":31775,"style":697},[136],[18,31777],{"className":31778,"style":6045},[6044],[18,31780,31781,31784],{"style":12068},[18,31782],{"className":31783,"style":697},[136],[18,31785,31787],{"className":31786},[141,142,143,144],[18,31788,31790],{"className":31789},[106,144],[18,31791,47],{"className":31792},[106,144],[18,31794,152],{"className":31795},[151],[18,31797,31799],{"className":31798},[124],[18,31800,31802],{"className":31801,"style":12090},[128],[18,31803],{},[18,31805],{"className":31806},[267,6009],[18,31808,74],{"className":31809},[250],[18,31811,47],{"className":31812},[106],[18,31814],{"className":31815,"style":2068},[169],[18,31817,2253],{"className":31818},[2072],[18,31820],{"className":31821,"style":2068},[169],[18,31823,31825,31828,31872],{"className":31824},[97],[18,31826],{"className":31827,"style":19839},[101],[18,31829,31831,31837],{"className":31830},[106],[18,31832,31834],{"className":31833},[106],[18,31835,6353],{"className":31836,"style":6352},[106,6351],[18,31838,31840],{"className":31839},[115],[18,31841,31843,31864],{"className":31842},[119,120],[18,31844,31846,31861],{"className":31845},[124],[18,31847,31849],{"className":31848,"style":5918},[128],[18,31850,31852,31855],{"style":31851},"top:-2.55em;margin-right:0.05em;",[18,31853],{"className":31854,"style":137},[136],[18,31856,31858],{"className":31857},[141,142,143,144],[18,31859,19804],{"className":31860,"style":19921},[106,110,144],[18,31862,152],{"className":31863},[151],[18,31865,31867],{"className":31866},[124],[18,31868,31870],{"className":31869,"style":19931},[128],[18,31871],{},[18,31873,84],{"className":31874},[267],"，求和得",[18,31877,31879],{"className":31878,"translate":22},[875],[18,31880,31882,31968],{"className":31881,"translate":22},[21],[18,31883,31885],{"className":31884},[26],[28,31886,31887],{"xmlns":30,"display":884},[32,31888,31889,31965],{},[35,31890,31891,31897,31899,31905,31963],{},[38,31892,31893,31895],{},[41,31894,3978],{"mathvariant":3525},[41,31896,64],{},[49,31898,4775],{},[5651,31900,31901,31903],{},[41,31902,28160],{},[45,31904,10444],{},[35,31906,31907,31909,31911,31913,31923,31925,31937,31939,31951,31961],{},[49,31908,74],{"fence":51},[45,31910,47],{},[49,31912,2253],{},[890,31914,31915,31921],{},[35,31916,31917,31919],{},[41,31918,5590],{},[49,31920,899],{},[41,31922,19804],{},[49,31924,2248],{"stretchy":73},[38,31926,31927,31929],{},[41,31928,12544],{},[35,31930,31931,31933,31935],{},[41,31932,19804],{},[49,31934,52],{"separator":51},[45,31936,430],{},[49,31938,2253],{},[38,31940,31941,31943],{},[41,31942,12544],{},[35,31944,31945,31947,31949],{},[41,31946,19804],{},[49,31948,52],{"separator":51},[45,31950,47],{},[38,31952,31953,31955],{},[49,31954,2248],{"stretchy":73},[35,31956,31957,31959],{},[41,31958,5700],{"mathvariant":316},[41,31960,5703],{"mathvariant":316},[49,31962,84],{"fence":51},[41,31964,968],{"mathvariant":316},[86,31966,31967],{"encoding":88},"\\mathfrak B_n\\ge\\frac m4\\left(1-\\max_j\\lVert Q_{j,0}-Q_{j,1}\\rVert_{\\mathrm{TV}}\\right).",[18,31969,31971,32027],{"className":31970,"ariaHidden":51},[93],[18,31972,31974,31978,32018,32021,32024],{"className":31973},[97],[18,31975],{"className":31976,"style":31977},[101],"height:0.8414em;vertical-align:-0.15em;",[18,31979,31981,31984],{"className":31980},[106],[18,31982,3978],{"className":31983},[106,3607],[18,31985,31987],{"className":31986},[115],[18,31988,31990,32010],{"className":31989},[119,120],[18,31991,31993,32007],{"className":31992},[124],[18,31994,31996],{"className":31995,"style":202},[128],[18,31997,31998,32001],{"style":1155},[18,31999],{"className":32000,"style":137},[136],[18,32002,32004],{"className":32003},[141,142,143,144],[18,32005,64],{"className":32006},[106,110,144],[18,32008,152],{"className":32009},[151],[18,32011,32013],{"className":32012},[124],[18,32014,32016],{"className":32015,"style":159},[128],[18,32017],{},[18,32019],{"className":32020,"style":228},[169],[18,32022,4775],{"className":32023},[232],[18,32025],{"className":32026,"style":228},[169],[18,32028,32030,32033,32095,32098,32325,32328],{"className":32029},[97],[18,32031],{"className":32032,"style":20800},[101],[18,32034,32036,32039,32092],{"className":32035},[106],[18,32037],{"className":32038},[250,6009],[18,32040,32042],{"className":32041},[5651],[18,32043,32045,32084],{"className":32044},[119,120],[18,32046,32048,32081],{"className":32047},[124],[18,32049,32051,32062,32070],{"className":32050,"style":13849},[128],[18,32052,32053,32056],{"style":6025},[18,32054],{"className":32055,"style":697},[136],[18,32057,32059],{"className":32058},[106],[18,32060,10444],{"className":32061},[106],[18,32063,32064,32067],{"style":6037},[18,32065],{"className":32066,"style":697},[136],[18,32068],{"className":32069,"style":6045},[6044],[18,32071,32072,32075],{"style":6048},[18,32073],{"className":32074,"style":697},[136],[18,32076,32078],{"className":32077},[106],[18,32079,28160],{"className":32080},[106,110],[18,32082,152],{"className":32083},[151],[18,32085,32087],{"className":32086},[124],[18,32088,32090],{"className":32089,"style":6067},[128],[18,32091],{},[18,32093],{"className":32094},[267,6009],[18,32096],{"className":32097,"style":170},[169],[18,32099,32101,32107,32110,32113,32116,32119,32163,32166,32215,32218,32221,32224,32273,32319],{"className":32100},[174],[18,32102,32104],{"className":32103,"style":13793},[250,13792],[18,32105,74],{"className":32106},[1230,143],[18,32108,47],{"className":32109},[106],[18,32111],{"className":32112,"style":2068},[169],[18,32114,2253],{"className":32115},[2072],[18,32117],{"className":32118,"style":2068},[169],[18,32120,32122],{"className":32121},[985,986],[18,32123,32125,32155],{"className":32124},[119,120],[18,32126,32128,32152],{"className":32127},[124],[18,32129,32131,32142],{"className":32130,"style":1083},[128],[18,32132,32133,32136],{"style":5735},[18,32134],{"className":32135,"style":697},[136],[18,32137,32139],{"className":32138},[141,142,143,144],[18,32140,19804],{"className":32141,"style":19921},[106,110,144],[18,32143,32144,32147],{"style":693},[18,32145],{"className":32146,"style":697},[136],[18,32148,32149],{},[18,32150,5590],{"className":32151},[985],[18,32153,152],{"className":32154},[151],[18,32156,32158],{"className":32157},[124],[18,32159,32161],{"className":32160,"style":5779},[128],[18,32162],{},[18,32164,2248],{"className":32165},[250],[18,32167,32169,32172],{"className":32168},[106],[18,32170,12544],{"className":32171},[106,110],[18,32173,32175],{"className":32174},[115],[18,32176,32178,32207],{"className":32177},[119,120],[18,32179,32181,32204],{"className":32180},[124],[18,32182,32184],{"className":32183,"style":5918},[128],[18,32185,32186,32189],{"style":1155},[18,32187],{"className":32188,"style":137},[136],[18,32190,32192],{"className":32191},[141,142,143,144],[18,32193,32195,32198,32201],{"className":32194},[106,144],[18,32196,19804],{"className":32197,"style":19921},[106,110,144],[18,32199,52],{"className":32200},[165,144],[18,32202,430],{"className":32203},[106,144],[18,32205,152],{"className":32206},[151],[18,32208,32210],{"className":32209},[124],[18,32211,32213],{"className":32212,"style":19931},[128],[18,32214],{},[18,32216],{"className":32217,"style":2068},[169],[18,32219,2253],{"className":32220},[2072],[18,32222],{"className":32223,"style":2068},[169],[18,32225,32227,32230],{"className":32226},[106],[18,32228,12544],{"className":32229},[106,110],[18,32231,32233],{"className":32232},[115],[18,32234,32236,32265],{"className":32235},[119,120],[18,32237,32239,32262],{"className":32238},[124],[18,32240,32242],{"className":32241,"style":5918},[128],[18,32243,32244,32247],{"style":1155},[18,32245],{"className":32246,"style":137},[136],[18,32248,32250],{"className":32249},[141,142,143,144],[18,32251,32253,32256,32259],{"className":32252},[106,144],[18,32254,19804],{"className":32255,"style":19921},[106,110,144],[18,32257,52],{"className":32258},[165,144],[18,32260,47],{"className":32261},[106,144],[18,322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逐坐标下界",[11,32339,32340,32341,32369,32370,32398,32399,32427,32428,32456,32457,32485],{},"Assouad 的威力在于它把 ",[18,32342,32344,32357],{"className":32343,"translate":22},[21],[18,32345,32347],{"className":32346},[26],[28,32348,32349],{"xmlns":30},[32,32350,32351,32355],{},[35,32352,32353],{},[41,32354,28160],{},[86,32356,28160],{"encoding":88},[18,32358,32360],{"className":32359,"ariaHidden":51},[93],[18,32361,32363,32366],{"className":32362},[97],[18,32364],{"className":32365,"style":1083},[101],[18,32367,28160],{"className":32368},[106,110]," 维问题拆成了 ",[18,32371,32373,32386],{"className":32372,"translate":22},[21],[18,32374,32376],{"className":32375},[26],[28,32377,32378],{"xmlns":30},[32,32379,32380,32384],{},[35,32381,32382],{},[41,32383,28160],{},[86,32385,28160],{"encoding":88},[18,32387,32389],{"className":32388,"ariaHidden":51},[93],[18,32390,32392,32395],{"className":32391},[97],[18,32393],{"className":32394,"style":1083},[101],[18,32396,28160],{"className":32397},[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，",[18,32400,32402,32415],{"className":32401,"translate":22},[21],[18,32403,32405],{"className":32404},[26],[28,32406,32407],{"xmlns":30},[32,32408,32409,32413],{},[35,32410,32411],{},[41,32412,28160],{},[86,32414,28160],{"encoding":88},[18,32416,32418],{"className":32417,"ariaHidden":51},[93],[18,32419,32421,32424],{"className":32420},[97],[18,32422],{"className":32423,"style":1083},[101],[18,32425,28160],{"className":32426},[106,110]," 个坐标就会贡献 ",[18,32429,32431,32444],{"className":32430,"translate":22},[21],[18,32432,32434],{"className":32433},[26],[28,32435,32436],{"xmlns":30},[32,32437,32438,32442],{},[35,32439,32440],{},[41,32441,28160],{},[86,32443,28160],{"encoding":88},[18,32445,32447],{"className":32446,"ariaHidden":51},[93],[18,32448,32450,32453],{"className":32449},[97],[18,32451],{"className":32452,"style":1083},[101],[18,32454,28160],{"className":32455},[106,110]," 倍的下界——这正是高维速率里 ",[18,32458,32460,32473],{"className":32459,"translate":22},[21],[18,32461,32463],{"className":32462},[26],[28,32464,32465],{"xmlns":30},[32,32466,32467,32471],{},[35,32468,32469],{},[41,32470,4031],{},[86,32472,4031],{"encoding":88},[18,32474,32476],{"className":32475,"ariaHidden":51},[93],[18,32477,32479,32482],{"className":32478},[97],[18,32480],{"className":32481,"style":295},[101],[18,32483,4031],{"className":32484},[106,110]," 的来源。",[11,32487,32488,32489,32594,32595,32687,32688,32826],{},"对 ",[18,32490,32492,32523],{"className":32491,"translate":22},[21],[18,32493,32495],{"className":32494},[26],[28,32496,32497],{"xmlns":30},[32,32498,32499,32520],{},[35,32500,32501,32503,32505,32507,32509,32511,32514],{},[41,32502,924],{"mathvariant":316},[49,32504,433],{},[49,32506,1331],{"stretchy":73},[45,32508,430],{},[49,32510,52],{"separator":51},[41,32512,32513],{},"ε",[1556,32515,32516,32518],{},[49,32517,1349],{"stretchy":73},[41,32519,28160],{},[86,32521,32522],{"encoding":88},"\\Theta=\\{0,\\varepsilon\\}^m",[18,32524,32526,32544],{"className":32525,"ariaHidden":51},[93],[18,32527,32529,32532,32535,32538,32541],{"className":32528},[97],[18,32530],{"className":32531,"style":600},[101],[18,32533,924],{"className":32534},[106],[18,32536],{"className":32537,"style":228},[169],[18,32539,433],{"className":32540},[232],[18,32542],{"className":32543,"style":228},[169],[18,32545,32547,32550,32553,32556,32559,32562,32565],{"className":32546},[97],[18,32548],{"className":32549,"style":242},[101],[18,32551,1331],{"className":32552},[250],[18,32554,430],{"className":32555},[106],[18,32557,52],{"className":32558},[165],[18,32560],{"className":32561,"style":170},[169],[18,32563,32513],{"className":32564},[106,110],[18,32566,32568,32571],{"className":32567},[267],[18,32569,1349],{"className":32570},[267],[18,32572,32574],{"className":32573},[115],[18,32575,32577],{"className":32576},[119],[18,32578,32580],{"className":32579},[124],[18,32581,32583],{"className":32582,"style":1959},[128],[18,32584,32585,32588],{"style":1962},[18,32586],{"className":32587,"style":137},[136],[18,32589,32591],{"className":32590},[141,142,143,144],[18,32592,28160],{"className":32593},[106,110,144],"，令 ",[18,32596,32598,32624],{"className":32597,"translate":22},[21],[18,32599,32601],{"className":32600},[26],[28,32602,32603],{"xmlns":30},[32,32604,32605,32621],{},[35,32606,32607,32609,32611,32613],{},[41,32608,77],{},[49,32610,433],{},[41,32612,32513],{},[1556,32614,32615,32617],{},[41,32616,77],{},[49,32618,32620],{"mathvariant":316,"lspace":32619,"rspace":32619},"0em","′",[86,32622,32623],{"encoding":88},"\\theta=\\varepsilon\\theta'",[18,32625,32627,32645],{"className":32626,"ariaHidden":51},[93],[18,32628,32630,32633,32636,32639,32642],{"className":32629},[97],[18,32631],{"className":32632,"style":295},[101],[18,32634,77],{"className":32635,"style":254},[106,110],[18,32637],{"className":32638,"style":228},[169],[18,32640,433],{"className":32641},[232],[18,32643],{"className":32644,"style":228},[169],[18,32646,32648,32652,32655],{"className":32647},[97],[18,32649],{"className":32650,"style":32651},[101],"height:0.7519em;",[18,32653,32513],{"className":32654},[106,110],[18,32656,32658,32661],{"className":32657},[106],[18,32659,77],{"className":32660,"style":254},[106,110],[18,32662,32664],{"className":32663},[115],[18,32665,32667],{"className":32666},[119],[18,32668,32670],{"className":32669},[124],[18,32671,32673],{"className":32672,"style":32651},[128],[18,32674,32675,32678],{"style":1962},[18,32676],{"className":32677,"style":137},[136],[18,32679,32681],{"className":32680},[141,142,143,144],[18,32682,32684],{"className":32683},[106,144],[18,32685,32620],{"className":32686},[106,144],"，其中 ",[18,32689,32691,32725],{"className":32690,"translate":22},[21],[18,32692,32694],{"className":32693},[26],[28,32695,32696],{"xmlns":30},[32,32697,32698,32722],{},[35,32699,32700,32706,32708,32710,32712,32714,32716],{},[1556,32701,32702,32704],{},[41,32703,77],{},[49,32705,32620],{"mathvariant":316,"lspace":32619,"rspace":32619},[49,32707,921],{},[49,32709,1331],{"stretchy":73},[45,32711,430],{},[49,32713,52],{"separator":51},[45,32715,47],{},[1556,32717,32718,32720],{},[49,32719,1349],{"stretchy":73},[41,32721,28160],{},[86,32723,32724],{"encoding":88},"\\theta'\\in\\{0,1\\}^m",[18,32726,32728,32776],{"className":32727,"ariaHidden":51},[93],[18,32729,32731,32735,32767,32770,32773],{"className":32730},[97],[18,32732],{"className":32733,"style":32734},[101],"height:0.791em;vertical-align:-0.0391em;",[18,32736,32738,32741],{"className":32737},[106],[18,32739,77],{"className":32740,"style":254},[106,110],[18,32742,32744],{"className":32743},[115],[18,32745,32747],{"className":32746},[119],[18,32748,32750],{"className":32749},[124],[18,32751,32753],{"className":32752,"style":32651},[128],[18,32754,32755,32758],{"style":1962},[18,32756],{"className":32757,"style":137},[136],[18,32759,32761],{"className":32760},[141,142,143,144],[18,32762,32764],{"className":32763},[106,144],[18,32765,32620],{"className":32766},[106,144],[18,32768],{"className":32769,"style":228},[169],[18,32771,921],{"className":32772},[232],[18,32774],{"className":32775,"style":228},[169],[18,32777,32779,32782,32785,32788,32791,32794,32797],{"className":32778},[97],[18,32780],{"className":32781,"style":242},[101],[18,32783,1331],{"className":32784},[250],[18,32786,430],{"className":32787},[106],[18,32789,52],{"className":32790},[165],[18,32792],{"className":32793,"style":170},[169],[18,32795,47],{"className":32796},[106],[18,32798,32800,32803],{"className":32799},[267],[18,32801,1349],{"className":32802},[267],[18,32804,32806],{"className":32805},[115],[18,32807,32809],{"className":32808},[119],[18,32810,32812],{"className":32811},[124],[18,32813,32815],{"className":32814,"style":1959},[128],[18,32816,32817,32820],{"style":1962},[18,32818],{"className":32819,"style":137},[136],[18,32821,32823],{"className":32822},[141,142,143,144],[18,32824,28160],{"className":32825},[106,110,144],"。平方损失满足",[18,32828,32830],{"className":32829,"translate":22},[875],[18,32831,32833,32903],{"className":32832,"translate":22},[21],[18,32834,32836],{"className":32835},[26],[28,32837,32838],{"xmlns":30,"display":884},[32,32839,32840,32900],{},[35,32841,32842,32844,32850,32852,32854,32862,32864,32870,32872,32878,32880,32882,32884,32890,32898],{},[49,32843,2248],{"stretchy":73},[658,32845,32846,32848],{"accent":51},[41,32847,77],{},[49,32849,664],{},[49,32851,2253],{},[41,32853,77],{},[1589,32855,32856,32858,32860],{},[49,32857,2248],{"stretchy":73},[45,32859,2266],{},[45,32861,2266],{},[49,32863,433],{},[1556,32865,32866,32868],{},[41,32867,32513],{},[45,32869,2266],{},[49,32871,2248],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[658,32873,32874,32876],{"accent":51},[41,32875,77],{},[49,32877,664],{},[41,32879,317],{"mathvariant":316},[41,32881,32513],{},[49,32883,2253],{},[1556,32885,32886,32888],{},[41,32887,77],{},[49,32889,32620],{"mathvariant":316,"lspace":32619,"rspace":32619},[1589,32891,32892,32894,32896],{},[49,32893,2248],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[45,32895,2266],{},[45,32897,2266],{},[49,32899,52],{"separator":51},[86,32901,32902],{"encoding":88},"\\lVert\\hat\\theta-\\theta\\rVert_2^2\n=\\varepsilon^2\\bigl\\lVert\\hat\\theta\u002F\\varepsilon-\\theta'\\bigr\\rVert_2^2,",[18,32904,32906,32955,33024,33142],{"className":32905,"ariaHidden":51},[93],[18,32907,32909,32912,32915,32946,32949,32952],{"className":32908},[97],[18,32910],{"className":32911,"style":2281},[101],[18,32913,2248],{"className":32914},[250],[18,32916,32918],{"className":32917},[106,681],[18,32919,32921],{"className":32920},[119],[18,32922,32924],{"className":32923},[124],[18,32925,32927,32935],{"className":32926,"style":677},[128],[18,32928,32929,32932],{"style":693},[18,32930],{"className":32931,"style":697},[136],[18,32933,77],{"className":32934,"style":254},[106,110],[18,32936,32937,32940],{"style":703},[18,32938],{"className":32939,"style":697},[136],[18,32941,32943],{"className":32942,"style":711},[710],[18,32944,664],{"className":32945},[106],[18,32947],{"className":32948,"style":2068},[169],[18,32950,2253],{"className":32951},[2072],[18,32953],{"className":32954,"style":2068},[169],[18,32956,32958,32961,32964,33015,33018,33021],{"className":32957},[97],[18,32959],{"className":32960,"style":15253},[101],[18,32962,77],{"className":32963,"style":254},[106,110],[18,32965,32967,32970],{"className":32966},[267],[18,32968,2248],{"className":32969},[267],[18,32971,32973],{"className":32972},[115],[18,32974,32976,33007],{"className":32975},[119,120],[18,32977,32979,33004],{"className":32978},[124],[18,32980,32982,32993],{"className":32981,"style":10772},[128],[18,32983,32984,32987],{"style":10775},[18,32985],{"className":32986,"style":137},[136],[18,32988,32990],{"className":32989},[141,142,143,144],[18,32991,2266],{"className":32992},[106,144],[18,32994,32995,32998],{"style":1637},[18,32996],{"className":32997,"style":137},[136],[18,32999,33001],{"className":33000},[141,142,143,144],[18,33002,2266],{"className":33003},[106,144],[18,33005,152],{"className":33006},[151],[18,33008,33010],{"className":33009},[124],[18,33011,33013],{"className":33012,"style":10805},[128],[18,33014],{},[18,33016],{"className":33017,"style":228},[169],[18,33019,433],{"className":33020},[232],[18,33022],{"className":33023,"style":228},[169],[18,33025,33027,33030,33059,33096,33127,33130,33133,33136,33139],{"className":33026},[97],[18,33028],{"className":33029,"style":23340},[101],[18,33031,33033,33036],{"className":33032},[106],[18,33034,32513],{"className":33035},[106,110],[18,33037,33039],{"className":33038},[115],[18,33040,33042],{"className":33041},[119],[18,33043,33045],{"className":33044},[124],[18,33046,33048],{"className":33047,"style":10772},[128],[18,33049,33050,33053],{"style":1637},[18,33051],{"className":33052,"style":137},[136],[18,33054,33056],{"className":33055},[141,142,143,144],[18,33057,2266],{"className":33058},[106,144],[18,33060,33062],{"className":33061},[250],[18,33063,33065],{"className":33064},[1230,6105],[18,33066,33068,33088],{"className":33067},[119,120],[18,33069,33071,33085],{"className":33070},[124],[18,33072,33074],{"className":33073,"style":6115},[128],[18,33075,33076,33079],{"style":6118},[18,33077],{"className":33078,"style":6122},[136],[18,33080,33081],{"style":6125},[6127,33082,33083],{"xmlns":6129,"width":6130,"height":944,"viewBox":6131},[6133,33084],{"d":6135},[18,33086,152],{"className":33087},[151],[18,33089,33091],{"className":33090},[124],[18,33092,33094],{"className":33093,"style":6145},[128],[18,33095],{},[18,33097,33099],{"className":33098},[106,681],[18,33100,33102],{"className":33101},[119],[18,33103,33105],{"className":33104},[124],[18,33106,33108,33116],{"className":33107,"style":677},[128],[18,33109,33110,33113],{"style":693},[18,33111],{"className":33112,"style":697},[136],[18,33114,77],{"className":33115,"style":254},[106,110],[18,33117,33118,33121],{"style":703},[18,33119],{"className":33120,"style":697},[136],[18,33122,33124],{"className":33123,"style":711},[710],[18,33125,664],{"className":33126},[106],[18,33128,317],{"className":33129},[106],[18,33131,32513],{"className":33132},[106,110],[18,33134],{"className":33135,"style":2068},[169],[18,33137,2253],{"className":33138},[2072],[18,33140],{"className":33141,"style":2068},[169],[18,33143,33145,33149,33182,33269],{"className":33144},[97],[18,33146],{"className":33147,"style":33148},[101],"height:1.4537em;vertical-align:-0.3997em;",[18,33150,33152,33155],{"className":33151},[106],[18,33153,77],{"className":33154,"style":254},[106,110],[18,33156,33158],{"className":33157},[115],[18,33159,33161],{"className":33160},[119],[18,33162,33164],{"className":33163},[124],[18,33165,33168],{"className":33166,"style":33167},[128],"height:0.8019em;",[18,33169,33170,33173],{"style":1637},[18,33171],{"className":33172,"style":137},[136],[18,33174,33176],{"className":33175},[141,142,143,144],[18,33177,33179],{"className":33178},[106,144],[18,33180,32620],{"className":33181},[106,144],[18,33183,33185,33222],{"className":33184},[267],[18,33186,33188],{"className":33187},[267],[18,33189,33191],{"className":33190},[1230,6105],[18,33192,33194,33214],{"className":33193},[119,120],[18,33195,33197,33211],{"className":33196},[124],[18,33198,33200],{"className":33199,"style":6115},[128],[18,33201,33202,33205],{"style":6118},[18,33203],{"className":33204,"style":6122},[136],[18,33206,33207],{"style":6125},[6127,33208,33209],{"xmlns":6129,"width":6130,"height":944,"viewBox":6131},[6133,33210],{"d":6135},[18,33212,152],{"className":33213},[151],[18,33215,33217],{"className":33216},[124],[18,33218,33220],{"className":33219,"style":6145},[128],[18,33221],{},[18,33223,33225],{"className":33224},[115],[18,33226,33228,33261],{"className":33227},[119,120],[18,33229,33231,33258],{"className":33230},[124],[18,33232,33235,33246],{"className":33233,"style":33234},[128],"height:1.054em;",[18,33236,33237,33240],{"style":6335},[18,33238],{"className":33239,"style":137},[136],[18,33241,33243],{"className":33242},[141,142,143,144],[18,33244,2266],{"className":33245},[106,144],[18,33247,33249,33252],{"style":33248},"top:-3.3029em;margin-right:0.05em;",[18,33250],{"className":33251,"style":137},[136],[18,33253,33255],{"className":33254},[141,142,143,144],[18,33256,2266],{"className":33257},[106,144],[18,33259,152],{"className":33260},[151],[18,33262,33264],{"className":33263},[124],[18,33265,33267],{"className":33266,"style":6363},[128],[18,33268],{},[18,33270,52],{"className":33271},[165],[11,33273,33274],{},"所以定理 5.1 直接给出",[18,33276,33278],{"className":33277,"translate":22},[875],[18,33279,33281,33433],{"className":33280,"translate":22},[21],[18,33282,33284],{"className":33283},[26],[28,33285,33286],{"xmlns":30,"display":884},[32,33287,33288,33430],{},[35,33289,33290,33304,33332,33334,33336,33342,33344,33346,33354,33356,33370,33428],{},[890,33291,33292,33298],{},[35,33293,33294,33296],{},[41,33295,896],{},[49,33297,899],{},[658,33299,33300,33302],{"accent":51},[41,33301,77],{},[49,33303,664],{},[890,33305,33306,33312],{},[35,33307,33308,33310],{},[41,33309,5590],{},[49,33311,899],{},[35,33313,33314,33316,33318,33320,33322,33324,33326],{},[41,33315,77],{},[49,33317,921],{},[49,33319,1331],{"stretchy":73},[45,33321,430],{},[49,33323,52],{"separator":51},[41,33325,32513],{},[1556,33327,33328,33330],{},[49,33329,1349],{"stretchy":73},[41,33331,28160],{},[41,33333,934],{"mathvariant":933},[49,33335,2248],{"stretchy":73},[658,33337,33338,33340],{"accent":51},[41,33339,77],{},[49,33341,664],{},[49,33343,2253],{},[41,33345,77],{},[1589,33347,33348,33350,33352],{},[49,33349,2248],{"stretchy":73},[45,33351,2266],{},[45,33353,2266],{},[49,33355,4775],{},[5651,33357,33358,33368],{},[35,33359,33360,33366],{},[1556,33361,33362,33364],{},[41,33363,32513],{},[45,33365,2266],{},[41,33367,28160],{},[45,33369,10444],{},[35,33371,33372,33374,33376,33378,33388,33390,33402,33404,33416,33426],{},[49,33373,74],{"fence":51},[45,33375,47],{},[49,33377,2253],{},[890,33379,33380,33386],{},[35,33381,33382,33384],{},[41,33383,5590],{},[49,33385,899],{},[41,33387,19804],{},[49,33389,2248],{"stretchy":73},[38,33391,33392,33394],{},[41,33393,12544],{},[35,33395,33396,33398,33400],{},[41,33397,19804],{},[49,33399,52],{"separator":51},[45,33401,430],{},[49,33403,2253],{},[38,33405,33406,33408],{},[41,33407,12544],{},[35,33409,33410,33412,33414],{},[41,33411,19804],{},[49,33413,52],{"separator":51},[45,33415,47],{},[38,33417,33418,33420],{},[49,33419,2248],{"stretchy":73},[35,33421,33422,33424],{},[41,33423,5700],{"mathvariant":316},[41,33425,5703],{"mathvariant":316},[49,33427,84],{"fence":51},[41,33429,968],{"mathvariant":316},[86,33431,33432],{"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：",[18,34114,34116,34139],{"className":34115,"translate":22},[21],[18,34117,34119],{"className":34118},[26],[28,34120,34121],{"xmlns":30},[32,34122,34123,34137],{},[35,34124,34125,34131,34133,34135],{},[1556,34126,34127,34129],{},[41,34128,16403],{},[45,34130,2266],{},[41,34132,4031],{},[41,34134,317],{"mathvariant":316},[41,34136,64],{},[86,34138,17296],{"encoding":88},[18,34140,34142],{"className":34141,"ariaHidden":51},[93],[18,34143,34145,34148,34177,34180,34183],{"className":34144},[97],[18,34146],{"className":34147,"style":10268},[101],[18,34149,34151,34154],{"className":34150},[106],[18,34152,16403],{"className":34153,"style":3957},[106,110],[18,34155,34157],{"className":34156},[115],[18,34158,34160],{"className":34159},[119],[18,34161,34163],{"className":34162},[124],[18,34164,34166],{"className":34165,"style":2416},[128],[18,34167,34168,34171],{"style":1962},[18,34169],{"className":34170,"style":137},[136],[18,34172,34174],{"className":34173},[141,142,143,144],[18,34175,2266],{"className":34176},[106,144],[18,34178,4031],{"className":34179},[106,110],[18,34181,317],{"className":34182},[106],[18,34184,64],{"className":34185},[106,110],[11,34187,34188],{},"模型：",[18,34190,34192],{"className":34191,"translate":22},[875],[18,34193,34195,34265],{"className":34194,"translate":22},[21],[18,34196,34198],{"className":34197},[26],[28,34199,34200],{"xmlns":30,"display":884},[32,34201,34202,34262],{},[35,34203,34204,34210,34212,34214,34216,34222,34224,34226,34228,34230,34232,34238,34244,34246,34248,34250,34252,34254,34260],{},[38,34205,34206,34208],{},[41,34207,43],{},[45,34209,47],{},[49,34211,52],{"separator":51},[49,34213,55],{},[49,34215,52],{"separator":51},[38,34217,34218,34220],{},[41,34219,43],{},[41,34221,64],{},[49,34223,67],{},[41,34225,70],{},[49,34227,74],{"stretchy":73},[41,34229,77],{},[49,34231,52],{"separator":51},[1556,34233,34234,34236],{},[41,34235,16403],{},[45,34237,2266],{},[38,34239,34240,34242],{},[41,34241,16410],{},[41,34243,4031],{},[49,34245,84],{"stretchy":73},[49,34247,52],{"separator":51},[169,34249],{"width":7341},[41,34251,77],{},[49,34253,921],{},[1556,34255,34256,34258],{},[41,34257,3071],{"mathvariant":933},[41,34259,4031],{},[49,34261,52],{"separator":51},[86,34263,34264],{"encoding":88},"X_1,\\dots,X_n\\sim N(\\theta,\\sigma^2 I_d),\\qquad \\theta\\in\\mathbb R^d,",[18,34266,34268,34381,34495],{"className":34267,"ariaHidden":51},[93],[18,34269,34271,34274,34314,34317,34320,34323,34326,34329,34332,34372,34375,34378],{"className":34270},[97],[18,34272],{"className":34273,"style":102},[101],[18,34275,34277,34280],{"className":34276},[106],[18,34278,43],{"className":34279,"style":111},[106,110],[18,34281,34283],{"className":34282},[115],[18,34284,34286,34306],{"className":34285},[119,120],[18,34287,34289,34303],{"className":34288},[124],[18,34290,34292],{"className":34291,"style":129},[128],[18,34293,34294,34297],{"style":132},[18,34295],{"className":34296,"style":137},[136],[18,34298,34300],{"className":34299},[141,142,143,144],[18,34301,47],{"className":34302},[106,144],[18,34304,152],{"className":34305},[151],[18,34307,34309],{"className":34308},[124],[18,34310,34312],{"className":34311,"style":159},[128],[18,34313],{},[18,34315,52],{"className":34316},[165],[18,34318],{"className":34319,"style":170},[169],[18,34321,55],{"className":34322},[174],[18,34324],{"className":34325,"style":170},[169],[18,34327,52],{"className":34328},[165],[18,34330],{"className":34331,"style":170},[169],[18,34333,34335,34338],{"className":34334},[106],[18,34336,43],{"className":34337,"style":111},[106,110],[18,34339,34341],{"className":34340},[115],[18,34342,34344,34364],{"className":34343},[119,120],[18,34345,34347,34361],{"className":34346},[124],[18,34348,34350],{"className":34349,"style":202},[128],[18,34351,34352,34355],{"style":132},[18,34353],{"className":34354,"style":137},[136],[18,34356,34358],{"className":34357},[141,142,143,144],[18,34359,64],{"className":34360},[106,110,144],[18,34362,152],{"className":34363},[151],[18,34365,34367],{"className":34366},[124],[18,34368,34370],{"className":34369,"style":159},[128],[18,34371],{},[18,34373],{"className":34374,"style":228},[169],[18,34376,67],{"className":34377},[232],[18,34379],{"className":34380,"style":228},[169],[18,34382,34384,34387,34390,34393,34396,34399,34402,34431,34471,34474,34477,34480,34483,34486,34489,34492],{"className":34383},[97],[18,34385],{"className":34386,"style":15253},[101],[18,34388,70],{"className":34389,"style":246},[106,110],[18,34391,74],{"className":34392},[250],[18,34394,77],{"className":34395,"style":254},[106,110],[18,34397,52],{"className":34398},[165],[18,34400],{"className":34401,"style":170},[169],[18,34403,34405,34408],{"className":34404},[106],[18,34406,16403],{"className":34407,"style":3957},[106,110],[18,34409,34411],{"className":34410},[115],[18,34412,34414],{"className":34413},[119],[18,34415,34417],{"className":34416},[124],[18,34418,34420],{"className":34419,"style":10772},[128],[18,34421,34422,34425],{"style":1637},[18,34423],{"className":34424,"style":137},[136],[18,34426,34428],{"className":34427},[141,142,143,144],[18,34429,2266],{"className":34430},[106,144],[18,34432,34434,34437],{"className":34433},[106],[18,34435,16410],{"className":34436,"style":111},[106,110],[18,34438,34440],{"className":34439},[115],[18,34441,34443,34463],{"className":34442},[119,120],[18,34444,34446,34460],{"className":34445},[124],[18,34447,34449],{"className":34448,"style":1383},[128],[18,34450,34451,34454],{"style":132},[18,34452],{"className":34453,"style":137},[136],[18,34455,34457],{"className":34456},[141,142,143,144],[18,34458,4031],{"className":34459},[106,110,144],[18,34461,152],{"className":34462},[151],[18,34464,34466],{"className":34465},[124],[18,34467,34469],{"className":34468,"style":159},[128],[18,34470],{},[18,34472,84],{"className":34473},[267],[18,34475,52],{"className":34476},[165],[18,34478],{"className":34479,"style":7673},[169],[18,34481],{"className":34482,"style":170},[169],[18,34484,77],{"className":34485,"style":254},[106,110],[18,34487],{"className":34488,"style":228},[169],[18,34490,921],{"className":34491},[232],[18,34493],{"className":34494,"style":228},[169],[18,34496,34498,34502,34531],{"className":34497},[97],[18,34499],{"className":34500,"style":34501},[101],"height:1.0935em;vertical-align:-0.1944em;",[18,34503,34505,34508],{"className":34504},[106],[18,34506,3071],{"className":34507},[106,1139],[18,34509,34511],{"className":34510},[115],[18,34512,34514],{"className":34513},[119],[18,34515,34517],{"className":34516},[124],[18,34518,34520],{"className":34519,"style":16670},[128],[18,34521,34522,34525],{"style":1637},[18,34523],{"className":34524,"style":137},[136],[18,34526,34528],{"className":34527},[141,142,143,144],[18,34529,4031],{"className":34530},[106,110,144],[18,34532,52],{"className":34533},[165],[11,34535,34536,34537,1990],{},"损失为 ",[18,34538,34540,34572],{"className":34539,"translate":22},[21],[18,34541,34543],{"className":34542},[26],[28,34544,34545],{"xmlns":30},[32,34546,34547,34569],{},[35,34548,34549,34551,34557,34559,34561],{},[49,34550,2248],{"stretchy":73},[658,34552,34553,34555],{"accent":51},[41,34554,77],{},[49,34556,664],{},[49,34558,2253],{},[41,34560,77],{},[1589,34562,34563,34565,34567],{},[49,34564,2248],{"stretchy":73},[45,34566,2266],{},[45,34568,2266],{},[86,34570,34571],{"encoding":88},"\\lVert\\hat\\theta-\\theta\\rVert_2^2",[18,34573,34575,34624],{"className":34574,"ariaHidden":51},[93],[18,34576,34578,34581,34584,34615,34618,34621],{"className":34577},[97],[18,34579],{"className":34580,"style":2281},[101],[18,34582,2248],{"className":34583},[250],[18,34585,34587],{"className":34586},[106,681],[18,34588,34590],{"className":34589},[119],[18,34591,34593],{"className":34592},[124],[18,34594,34596,34604],{"className":34595,"style":677},[128],[18,34597,34598,34601],{"style":693},[18,34599],{"className":34600,"style":697},[136],[18,34602,77],{"className":34603,"style":254},[106,110],[18,34605,34606,34609],{"style":703},[18,34607],{"className":34608,"style":697},[136],[18,34610,34612],{"className":34611,"style":711},[710],[18,34613,664],{"className":34614},[106],[18,34616],{"className":34617,"style":2068},[169],[18,34619,2253],{"className":34620},[2072],[18,34622],{"className":34623,"style":2068},[169],[18,34625,34627,34630,34633],{"className":34626},[97],[18,34628],{"className":34629,"style":10268},[101],[18,34631,77],{"className":34632,"style":254},[106,110],[18,34634,34636,34639],{"className":34635},[267],[18,34637,2248],{"className":34638},[267],[18,34640,34642],{"className":34641},[115],[18,34643,34645,34676],{"className":34644},[119,120],[18,34646,34648,34673],{"className":34647},[124],[18,34649,34651,34662],{"className":34650,"style":2416},[128],[18,34652,34653,34656],{"style":2419},[18,34654],{"className":34655,"style":137},[136],[18,34657,34659],{"className":34658},[141,142,143,144],[18,34660,2266],{"className":34661},[106,144],[18,34663,34664,34667],{"style":1962},[18,34665],{"className":34666,"style":137},[136],[18,34668,34670],{"className":34669},[141,142,143,144],[18,34671,2266],{"className":34672},[106,144],[18,34674,152],{"className":34675},[151],[18,34677,34679],{"className":34678},[124],[18,34680,34682],{"className":34681,"style":2449},[128],[18,34683],{},[11,34685,34686,34689,34690,34751],{},[717,34687,34688],{},"上界。"," 样本均值 ",[18,34691,34693,34711],{"className":34692,"translate":22},[21],[18,34694,34696],{"className":34695},[26],[28,34697,34698],{"xmlns":30},[32,34699,34700,34708],{},[35,34701,34702],{},[658,34703,34704,34706],{"accent":51},[41,34705,43],{},[49,34707,25287],{},[86,34709,34710],{"encoding":88},"\\bar X",[18,34712,34714],{"className":34713,"ariaHidden":51},[93],[18,34715,34717,34720],{"className":34716},[97],[18,34718],{"className":34719,"style":25617},[101],[18,34721,34723],{"className":34722},[106,681],[18,34724,34726],{"className":34725},[119],[18,34727,34729],{"className":34728},[124],[18,34730,34732,34740],{"className":34731,"style":25617},[128],[18,34733,34734,34737],{"style":693},[18,34735],{"className":34736,"style":697},[136],[18,34738,43],{"className":34739,"style":111},[106,110],[18,34741,34742,34745],{"style":17697},[18,34743],{"className":34744,"style":697},[136],[18,34746,34748],{"className":34747,"style":711},[710],[18,34749,25287],{"className":34750},[106]," 的风险是",[18,34753,34755],{"className":34754,"translate":22},[875],[18,34756,34758,34810],{"className":34757,"translate":22},[21],[18,34759,34761],{"className":34760},[26],[28,34762,34763],{"xmlns":30,"display":884},[32,34764,34765,34807],{},[35,34766,34767,34769,34771,34777,34779,34781,34789,34791,34805],{},[41,34768,934],{"mathvariant":933},[49,34770,2248],{"stretchy":73},[658,34772,34773,34775],{"accent":51},[41,34774,43],{},[49,34776,25287],{},[49,34778,2253],{},[41,34780,77],{},[1589,34782,34783,34785,34787],{},[49,34784,2248],{"stretchy":73},[45,34786,2266],{},[45,34788,2266],{},[49,34790,433],{},[5651,34792,34793,34803],{},[35,34794,34795,34801],{},[1556,34796,34797,34799],{},[41,34798,16403],{},[45,34800,2266],{},[41,34802,4031],{},[41,34804,64],{},[41,34806,968],{"mathvariant":316},[86,34808,34809],{"encoding":88},"\\mathbb E\\lVert\\bar X-\\theta\\rVert_2^2=\\frac{\\sigma^2d}{n}.",[18,34811,34813,34866,34935],{"className":34812,"ariaHidden":51},[93],[18,34814,34816,34820,34823,34826,34857,34860,34863],{"className":34815},[97],[18,34817],{"className":34818,"style":34819},[101],"height:1.0701em;vertical-align:-0.25em;",[18,34821,934],{"className":34822},[106,1139],[18,34824,2248],{"className":34825},[250],[18,34827,34829],{"className":34828},[106,681],[18,34830,34832],{"className":34831},[119],[18,34833,34835],{"className":34834},[124],[18,34836,34838,34846],{"className":34837,"style":25617},[128],[18,34839,34840,34843],{"style":693},[18,34841],{"className":34842,"style":697},[136],[18,34844,43],{"className":34845,"style":111},[106,110],[18,34847,34848,34851],{"style":17697},[18,34849],{"className":34850,"style":697},[136],[18,34852,34854],{"className":34853,"style":711},[710],[18,34855,25287],{"className":34856},[106],[18,34858],{"className":34859,"style":2068},[169],[18,34861,2253],{"className":34862},[2072],[18,34864],{"className":34865,"style":2068},[169],[18,34867,34869,34872,34875,34926,34929,34932],{"className":34868},[97],[18,34870],{"className":34871,"style":15253},[101],[18,34873,77],{"className":34874,"style":254},[106,110],[18,34876,34878,34881],{"className":34877},[267],[18,34879,2248],{"className":34880},[267],[18,34882,34884],{"className":34883},[115],[18,34885,34887,34918],{"className":34886},[119,120],[18,34888,34890,34915],{"className":34889},[124],[18,34891,34893,34904],{"className":34892,"style":10772},[128],[18,34894,34895,34898],{"style":10775},[18,34896],{"className":34897,"style":137},[136],[18,34899,34901],{"className":34900},[141,142,143,144],[18,34902,2266],{"className":34903},[106,144],[18,34905,34906,34909],{"style":1637},[18,34907],{"className":34908,"style":137},[136],[18,34910,34912],{"className":34911},[141,142,143,144],[18,34913,2266],{"className":34914},[106,144],[18,34916,152],{"className":34917},[151],[18,34919,34921],{"className":34920},[124],[18,34922,34924],{"className":34923,"style":10805},[128],[18,34925],{},[18,34927],{"className":34928,"style":228},[169],[18,34930,433],{"className":34931},[232],[18,34933],{"className":34934,"style":228},[169],[18,34936,34938,34942,35033],{"className":34937},[97],[18,34939],{"className":34940,"style":34941},[101],"height:2.1771em;vertical-align:-0.686em;",[18,34943,34945,34948,35030],{"className":34944},[106],[18,34946],{"className":34947},[250,6009],[18,34949,34951],{"className":34950},[5651],[18,34952,34954,35022],{"className":34953},[119,120],[18,34955,34957,35019],{"className":34956},[124],[18,34958,34960,34971,34979],{"className":34959,"style":15996},[128],[18,34961,34962,34965],{"style":6025},[18,34963],{"className":34964,"style":697},[136],[18,34966,34968],{"className":34967},[106],[18,34969,64],{"className":34970},[106,110],[18,34972,34973,34976],{"style":6037},[18,34974],{"className":34975,"style":697},[136],[18,34977],{"className":34978,"style":6045},[6044],[18,34980,34981,34984],{"style":6048},[18,34982],{"className":34983,"style":697},[136],[18,34985,34987,35016],{"className":34986},[106],[18,34988,34990,34993],{"className":34989},[106],[18,34991,16403],{"className":34992,"style":3957},[106,110],[18,34994,34996],{"className":34995},[115],[18,34997,34999],{"className":34998},[119],[18,35000,35002],{"className":35001},[124],[18,35003,35005],{"className":35004,"style":2416},[128],[18,35006,35007,35010],{"style":1962},[18,35008],{"className":35009,"style":137},[136],[18,35011,35013],{"className":35012},[141,142,143,144],[18,35014,2266],{"className":35015},[106,144],[18,35017,4031],{"className":35018},[106,110],[18,35020,152],{"className":35021},[151],[18,35023,35025],{"className":35024},[124],[18,35026,35028],{"className":35027,"style":6067},[128],[18,35029],{},[18,35031],{"className":35032},[267,6009],[18,35034,968],{"className":35035},[106],[11,35037,35038,35041,35042,35189,35190,35218,35219,509,35303,35387,35388,35417],{},[717,35039,35040],{},"下界。"," 把参数限制到超立方体 ",[18,35043,35045,35079],{"className":35044,"translate":22},[21],[18,35046,35048],{"className":35047},[26],[28,35049,35050],{"xmlns":30},[32,35051,35052,35076],{},[35,35053,35054,35060,35062,35064,35066,35068,35070],{},[38,35055,35056,35058],{},[41,35057,924],{"mathvariant":316},[45,35059,430],{},[49,35061,433],{},[49,35063,1331],{"stretchy":73},[45,35065,430],{},[49,35067,52],{"separator":51},[41,35069,32513],{},[1556,35071,35072,35074],{},[49,35073,1349],{"stretchy":73},[41,35075,4031],{},[86,35077,35078],{"encoding":88},"\\Theta_0=\\{0,\\varepsilon\\}^d",[18,35080,35082,35137],{"className":35081,"ariaHidden":51},[93],[18,35083,35085,35088,35128,35131,35134],{"className":35084},[97],[18,35086],{"className":35087,"style":4563},[101],[18,35089,35091,35094],{"className":35090},[106],[18,35092,924],{"className":35093},[106],[18,35095,35097],{"className":35096},[115],[18,35098,35100,35120],{"className":35099},[119,120],[18,35101,35103,35117],{"className":35102},[124],[18,35104,35106],{"className":35105,"style":129},[128],[18,35107,35108,35111],{"style":1155},[18,35109],{"className":35110,"style":137},[136],[18,35112,35114],{"className":35113},[141,142,143,144],[18,35115,430],{"className":35116},[106,144],[18,35118,152],{"className":35119},[151],[18,35121,35123],{"className":35122},[124],[18,35124,35126],{"className":35125,"style":159},[128],[18,35127],{},[18,35129],{"className":35130,"style":228},[169],[18,35132,433],{"className":35133},[232],[18,35135],{"className":35136,"style":228},[169],[18,35138,35140,35144,35147,35150,35153,35156,35159],{"className":35139},[97],[18,35141],{"className":35142,"style":35143},[101],"height:1.0991em;vertical-align:-0.25em;",[18,35145,1331],{"className":35146},[250],[18,35148,430],{"className":35149},[106],[18,35151,52],{"className":35152},[165],[18,35154],{"className":35155,"style":170},[169],[18,35157,32513],{"className":35158},[106,110],[18,35160,35162,35165],{"className":35161},[267],[18,35163,1349],{"className":35164},[267],[18,35166,35168],{"className":35167},[115],[18,35169,35171],{"className":35170},[119],[18,35172,35174],{"className":35173},[124],[18,35175,35178],{"className":35176,"style":35177},[128],"height:0.8491em;",[18,35179,35180,35183],{"style":1962},[18,35181],{"className":35182,"style":137},[136],[18,35184,35186],{"className":35185},[141,142,143,144],[18,35187,4031],{"className":35188},[106,110,144],"。对坐标 ",[18,35191,35193,35206],{"className":35192,"translate":22},[21],[18,35194,35196],{"className":35195},[26],[28,35197,35198],{"xmlns":30},[32,35199,35200,35204],{},[35,35201,35202],{},[41,35203,19804],{},[86,35205,19804],{"encoding":88},[18,35207,35209],{"className":35208,"ariaHidden":51},[93],[18,35210,35212,35215],{"className":35211},[97],[18,35213],{"className":35214,"style":28331},[101],[18,35216,19804],{"className":35217,"style":19921},[106,110],"，混合分布 ",[18,35220,35222,35245],{"className":35221,"translate":22},[21],[18,35223,35225],{"className":35224},[26],[28,35226,35227],{"xmlns":30},[32,35228,35229,35243],{},[35,35230,35231],{},[38,35232,35233,35235],{},[41,35234,12544],{},[35,35236,35237,35239,35241],{},[41,35238,19804],{},[49,35240,52],{"separator":51},[45,35242,430],{},[86,35244,30738],{"encoding":88},[18,35246,35248],{"className":35247,"ariaHidden":51},[93],[18,35249,35251,35254],{"className":35250},[97],[18,35252],{"className":35253,"style":28481},[101],[18,35255,35257,35260],{"className":35256},[106],[18,35258,12544],{"className":35259},[106,110],[18,35261,35263],{"className":35262},[115],[18,35264,35266,35295],{"className":35265},[119,120],[18,35267,35269,35292],{"className":35268},[124],[18,35270,35272],{"className":35271,"style":5918},[128],[18,35273,35274,35277],{"style":1155},[18,35275],{"className":35276,"style":137},[136],[18,35278,35280],{"className":35279},[141,142,143,144],[18,35281,35283,35286,35289],{"className":35282},[106,144],[18,35284,19804],{"className":35285,"style":19921},[106,110,144],[18,35287,52],{"className":35288},[165,144],[18,35290,430],{"className":35291},[106,144],[18,35293,152],{"className":35294},[151],[18,35296,35298],{"className":35297},[124],[18,35299,35301],{"className":35300,"style":19931},[128],[18,35302],{},[18,35304,35306,35329],{"className":35305,"translate":22},[21],[18,35307,35309],{"className":35308},[26],[28,35310,35311],{"xmlns":30},[32,35312,35313,35327],{},[35,35314,35315],{},[38,35316,35317,35319],{},[41,35318,12544],{},[35,35320,35321,35323,35325],{},[41,35322,19804],{},[49,35324,52],{"separator":51},[45,35326,47],{},[86,35328,30823],{"encoding":88},[18,35330,35332],{"className":35331,"ariaHidden":51},[93],[18,35333,35335,35338],{"className":35334},[97],[18,35336],{"className":35337,"style":28481},[101],[18,35339,35341,35344],{"className":35340},[106],[18,35342,12544],{"className":35343},[106,110],[18,35345,35347],{"className":35346},[115],[18,35348,35350,35379],{"className":35349},[119,120],[18,35351,35353,35376],{"className":35352},[124],[18,35354,35356],{"className":35355,"style":5918},[128],[18,35357,35358,35361],{"style":1155},[18,35359],{"className":35360,"style":137},[136],[18,35362,35364],{"className":35363},[141,142,143,144],[18,35365,35367,35370,35373],{"className":35366},[106,144],[18,35368,19804],{"className":35369,"style":19921},[106,110,144],[18,35371,52],{"className":35372},[165,144],[18,35374,47],{"className":35375},[106,144],[18,35377,152],{"className":35378},[151],[18,35380,35382],{"className":35381},[124],[18,35383,35385],{"className":35384,"style":19931},[128],[18,35386],{}," 只在这一维的均值上相差 ",[18,35389,35391,35405],{"className":35390,"translate":22},[21],[18,35392,35394],{"className":35393},[26],[28,35395,35396],{"xmlns":30},[32,35397,35398,35402],{},[35,35399,35400],{},[41,35401,32513],{},[86,35403,35404],{"encoding":88},"\\varepsilon",[18,35406,35408],{"className":35407,"ariaHidden":51},[93],[18,35409,35411,35414],{"className":35410},[97],[18,35412],{"className":35413,"style":1083},[101],[18,35415,32513],{"className":35416},[106,110],"。由 KL 的联合凸性，",[18,35419,35421],{"className":35420,"translate":22},[875],[18,35422,35424,35502],{"className":35423,"translate":22},[21],[18,35425,35427],{"className":35426},[26],[28,35428,35429],{"xmlns":30,"display":884},[32,35430,35431,35499],{},[35,35432,35433,35443,35445,35457,35459,35471,35473,35475,35497],{},[38,35434,35435,35437],{},[41,35436,12572],{},[35,35438,35439,35441],{},[41,35440,5031],{"mathvariant":316},[41,35442,12579],{"mathvariant":316},[49,35444,74],{"stretchy":73},[38,35446,35447,35449],{},[41,35448,12544],{},[35,35450,35451,35453,35455],{},[41,35452,19804],{},[49,35454,52],{"separator":51},[45,35456,430],{},[41,35458,2248],{"mathvariant":316},[38,35460,35461,35463],{},[41,35462,12544],{},[35,35464,35465,35467,35469],{},[41,35466,19804],{},[49,35468,52],{"separator":51},[45,35470,47],{},[49,35472,84],{"stretchy":73},[49,35474,4334],{},[5651,35476,35477,35487],{},[35,35478,35479,35481],{},[41,35480,64],{},[1556,35482,35483,35485],{},[41,35484,32513],{},[45,35486,2266],{},[35,35488,35489,35491],{},[45,35490,2266],{},[1556,35492,35493,35495],{},[41,35494,16403],{},[45,35496,2266],{},[41,35498,968],{"mathvariant":316},[86,35500,35501],{"encoding":88},"D_{\\mathrm{KL}}(Q_{j,0}\\Vert Q_{j,1})\n\\le \\frac{n\\varepsilon^2}{2\\sigma^2}.",[18,35503,35505,35673],{"className":35504,"ariaHidden":51},[93],[18,35506,35508,35511,35557,35560,35609,35612,35661,35664,35667,35670],{"className":35507},[97],[18,35509],{"className":35510,"style":19839},[101],[18,35512,35514,35517],{"className":35513},[106],[18,35515,12572],{"className":35516,"style":254},[106,110],[18,35518,35520],{"className":35519},[115],[18,35521,35523,35549],{"className":35522},[119,120],[18,35524,35526,35546],{"className":35525},[124],[18,35527,35529],{"className":35528,"style":1152},[128],[18,35530,35531,35534],{"style":469},[18,35532],{"className":35533,"style":137},[136],[18,35535,35537],{"className":35536},[141,142,143,144],[18,35538,35540],{"className":35539},[106,144],[18,35541,35543],{"className":35542},[106,144],[18,35544,12814],{"className":35545},[106,6351,144],[18,35547,152],{"className":35548},[151],[18,35550,35552],{"className":35551},[124],[18,35553,35555],{"className":35554,"style":159},[128],[18,35556],{},[18,35558,74],{"className":35559},[250],[18,35561,35563,35566],{"className":35562},[106],[18,35564,12544],{"className":35565},[106,110],[18,35567,35569],{"className":35568},[115],[18,35570,35572,35601],{"className":35571},[119,120],[18,35573,35575,35598],{"className":35574},[124],[18,35576,35578],{"className":35577,"style":5918},[128],[18,35579,35580,35583],{"style":1155},[18,35581],{"className":35582,"style":137},[136],[18,35584,35586],{"className":35585},[141,142,143,144],[18,35587,35589,35592,35595],{"className":35588},[106,144],[18,35590,19804],{"className":35591,"style":19921},[106,110,144],[18,35593,52],{"className":35594},[165,144],[18,35596,430],{"className":35597},[106,144],[18,35599,152],{"className":35600},[151],[18,35602,35604],{"className":35603},[124],[18,35605,35607],{"className":35606,"style":19931},[128],[18,35608],{},[18,35610,2248],{"className":35611},[106],[18,35613,35615,35618],{"className":35614},[106],[18,35616,12544],{"className":35617},[106,110],[18,35619,35621],{"className":35620},[115],[18,35622,35624,35653],{"className":35623},[119,120],[18,35625,35627,35650],{"className":35626},[124],[18,35628,35630],{"className":35629,"style":5918},[128],[18,35631,35632,35635],{"style":1155},[18,35633],{"className":35634,"style":137},[136],[18,35636,35638],{"className":35637},[141,142,143,144],[18,35639,35641,35644,35647],{"className":35640},[106,144],[18,35642,19804],{"className":35643,"style":19921},[106,110,144],[18,35645,52],{"className":35646},[165,144],[18,35648,47],{"className":35649},[106,144],[18,35651,152],{"className":35652},[151],[18,35654,35656],{"className":35655},[124],[18,35657,35659],{"className":35658,"style":19931},[128],[18,35660],{},[18,35662,84],{"className":35663},[267],[18,35665],{"className":35666,"style":228},[169],[18,35668,4334],{"className":35669},[232],[18,35671],{"className":35672,"style":228},[169],[18,35674,35676,35679,35799],{"className":35675},[97],[18,35677],{"className":35678,"style":34941},[101],[18,35680,35682,35685,35796],{"className":35681},[106],[18,35683],{"className":35684},[250,6009],[18,35686,35688],{"className":35687},[5651],[18,35689,35691,35788],{"className":35690},[119,120],[18,35692,35694,35785],{"className":35693},[124],[18,35695,35697,35737,35745],{"className":35696,"style":15996},[128],[18,35698,35699,35702],{"style":6025},[18,35700],{"className":35701,"style":697},[136],[18,35703,35705,35708],{"className":35704},[106],[18,35706,2266],{"className":35707},[106],[18,35709,35711,35714],{"className":35710},[106],[18,35712,16403],{"className":35713,"style":3957},[106,110],[18,35715,35717],{"className":35716},[115],[18,35718,35720],{"className":35719},[119],[18,35721,35723],{"className":35722},[124],[18,35724,35726],{"className":35725,"style":26102},[128],[18,35727,35728,35731],{"style":26105},[18,35729],{"className":35730,"style":137},[136],[18,35732,35734],{"className":35733},[141,142,143,144],[18,35735,2266],{"className":35736},[106,144],[18,35738,35739,35742],{"style":6037},[18,35740],{"className":35741,"style":697},[136],[18,35743],{"className":35744,"style":6045},[6044],[18,35746,35747,35750],{"style":6048},[18,35748],{"className":35749,"style":697},[136],[18,35751,35753,35756],{"className":35752},[106],[18,35754,64],{"className":35755},[106,110],[18,35757,35759,35762],{"className":35758},[106],[18,35760,32513],{"className":35761},[106,110],[18,35763,35765],{"className":35764},[115],[18,35766,35768],{"className":35767},[119],[18,35769,35771],{"className":35770},[124],[18,35772,35774],{"className":35773,"style":2416},[128],[18,35775,35776,35779],{"style":1962},[18,35777],{"className":35778,"style":137},[136],[18,35780,35782],{"className":35781},[141,142,143,144],[18,35783,2266],{"className":35784},[106,144],[18,35786,152],{"className":35787},[151],[18,35789,35791],{"className":35790},[124],[18,35792,35794],{"className":35793,"style":6067},[128],[18,35795],{},[18,35797],{"className":35798},[267,6009],[18,35800,968],{"className":35801},[106],[11,35803,35804],{},"Pinsker 给出",[18,35806,35808],{"className":35807,"translate":22},[875],[18,35809,35811,35881],{"className":35810,"translate":22},[21],[18,35812,35814],{"className":35813},[26],[28,35815,35816],{"xmlns":30,"display":884},[32,35817,35818,35878],{},[35,35819,35820,35822,35834,35836,35848,35858,35860,35876],{},[49,35821,2248],{"stretchy":73},[38,35823,35824,35826],{},[41,35825,12544],{},[35,35827,35828,35830,35832],{},[41,35829,19804],{},[49,35831,52],{"separator":51},[45,35833,430],{},[49,35835,2253],{},[38,35837,35838,35840],{},[41,35839,12544],{},[35,35841,35842,35844,35846],{},[41,35843,19804],{},[49,35845,52],{"separator":51},[45,35847,47],{},[38,35849,35850,35852],{},[49,35851,2248],{"stretchy":73},[35,35853,35854,35856],{},[41,35855,5700],{"mathvariant":316},[41,35857,5703],{"mathvariant":316},[49,35859,4334],{},[5651,35861,35862,35870],{},[35,35863,35864,35866],{},[41,35865,32513],{},[12558,35867,35868],{},[41,35869,64],{},[35,35871,35872,35874],{},[45,35873,2266],{},[41,35875,16403],{},[41,35877,968],{"mathvariant":316},[86,35879,35880],{"encoding":88},"\\lVert Q_{j,0}-Q_{j,1}\\rVert_{\\mathrm{TV}}\n\\le \\frac{\\varepsilon\\sqrt n}{2\\sigma}.",[18,35882,35884,35951,36061],{"className":35883,"ariaHidden":51},[93],[18,35885,35887,35890,35893,35942,35945,35948],{"className":35886},[97],[18,35888],{"className":35889,"style":19839},[101],[18,35891,2248],{"className":35892},[250],[18,35894,35896,35899],{"className":35895},[106],[18,35897,12544],{"className":35898},[106,110],[18,35900,35902],{"className":35901},[115],[18,35903,35905,35934],{"className":35904},[119,120],[18,35906,35908,35931],{"className":35907},[124],[18,35909,35911],{"className":35910,"style":5918},[128],[18,35912,35913,35916],{"style":1155},[18,35914],{"className":35915,"style":137},[136],[18,35917,35919],{"className":35918},[141,142,143,144],[18,35920,35922,35925,35928],{"className":35921},[106,144],[18,35923,19804],{"className":35924,"style":19921},[106,110,144],[18,35926,52],{"className":35927},[165,144],[18,35929,430],{"className":35930},[106,144],[18,35932,152],{"className":35933},[151],[18,35935,35937],{"className":35936},[124],[18,35938,35940],{"className":35939,"style":19931},[128],[18,35941],{},[18,35943],{"className":35944,"style":2068},[169],[18,35946,2253],{"className":35947},[2072],[18,35949],{"className":35950,"style":2068},[169],[18,35952,35954,35957,36006,36052,36055,36058],{"className":35953},[97],[18,35955],{"className":35956,"style":19839},[101],[18,35958,35960,35963],{"className":35959},[106],[18,35961,12544],{"className":35962},[106,110],[18,35964,35966],{"className":35965},[115],[18,35967,35969,35998],{"className":35968},[119,120],[18,35970,35972,35995],{"className":35971},[124],[18,35973,35975],{"className":35974,"style":5918},[128],[18,35976,35977,35980],{"style":1155},[18,35978],{"className":35979,"style":137},[136],[18,35981,35983],{"className":35982},[141,142,143,144],[18,35984,35986,35989,35992],{"className":35985},[106,144],[18,35987,19804],{"className":35988,"style":19921},[106,110,144],[18,35990,52],{"className":35991},[165,144],[18,35993,47],{"className":35994},[106,144],[18,35996,152],{"className":35997},[151],[18,35999,36001],{"className":36000},[124],[18,36002,36004],{"className":36003,"style":19931},[128],[18,36005],{},[18,36007,36009,36012],{"className":36008},[267],[18,36010,2248],{"className":36011},[267],[18,36013,36015],{"className":36014},[115],[18,36016,36018,36044],{"className":36017},[119,120],[18,36019,36021,36041],{"className":36020},[124],[18,36022,36024],{"className":36023,"style":1152},[128],[18,36025,36026,36029],{"style":1155},[18,36027],{"className":36028,"style":137},[136],[18,36030,36032],{"className":36031},[141,142,143,144],[18,36033,36035],{"className":36034},[106,144],[18,36036,36038],{"className":36037},[106,144],[18,36039,6353],{"className":36040,"style":6352},[106,6351,144],[18,36042,152],{"className":36043},[151],[18,36045,36047],{"className":36046},[124],[18,36048,36050],{"className":36049,"style":159},[128],[18,36051],{},[18,36053],{"className":36054,"style":228},[169],[18,36056,4334],{"className":36057},[232],[18,36059],{"className":36060,"style":228},[169],[18,36062,36064,36068,36178],{"className":36063},[97],[18,36065],{"className":36066,"style":36067},[101],"height:2.1633em;vertical-align:-0.686em;",[18,36069,36071,36074,36175],{"className":36070},[106],[18,36072],{"className":36073},[250,6009],[18,36075,36077],{"className":36076},[5651],[18,36078,36080,36167],{"className":36079},[119,120],[18,36081,36083,36164],{"className":36082},[124],[18,36084,36087,36101,36109],{"className":36085,"style":36086},[128],"height:1.4773em;",[18,36088,36089,36092],{"style":6025},[18,36090],{"className":36091,"style":697},[136],[18,36093,36095,36098],{"className":36094},[106],[18,36096,2266],{"className":36097},[106],[18,36099,16403],{"className":36100,"style":3957},[106,110],[18,36102,36103,36106],{"style":6037},[18,36104],{"className":36105,"style":697},[136],[18,36107],{"className":36108,"style":6045},[6044],[18,36110,36111,36114],{"style":6048},[18,36112],{"className":36113,"style":697},[136],[18,36115,36117,36120],{"className":36116},[106],[18,36118,32513],{"className":36119},[106,110],[18,36121,36123],{"className":36122},[106,12693],[18,36124,36126,36156],{"className":36125},[119,120],[18,36127,36129,36153],{"className":36128},[124],[18,36130,36132,36141],{"className":36131,"style":15397},[128],[18,36133,36135,36138],{"className":36134,"style":693},[12707],[18,36136],{"className":36137,"style":697},[136],[18,36139,64],{"className":36140,"style":15407},[106,110],[18,36142,36143,36146],{"style":15410},[18,36144],{"className":36145,"style":697},[136],[18,36147,36149],{"className":36148,"style":15417},[12850],[6127,36150,36151],{"xmlns":6129,"width":12854,"height":15420,"viewBox":15421,"preserveAspectRatio":12857},[6133,36152],{"d":15424},[18,36154,152],{"className":36155},[151],[18,36157,36159],{"className":36158},[124],[18,36160,36162],{"className":36161,"style":15434},[128],[18,36163],{},[18,36165,152],{"className":36166},[151],[18,36168,36170],{"className":36169},[124],[18,36171,36173],{"className":36172,"style":6067},[128],[18,36174],{},[18,36176],{"className":36177},[267,6009],[18,36179,968],{"className":36180},[106],[11,36182,36183,36184,36288,36289,36321],{},"取 ",[18,36185,36187,36211],{"className":36186,"translate":22},[21],[18,36188,36190],{"className":36189},[26],[28,36191,36192],{"xmlns":30},[32,36193,36194,36208],{},[35,36195,36196,36198,36200,36202,36204],{},[41,36197,32513],{},[49,36199,433],{},[41,36201,16403],{},[41,36203,317],{"mathvariant":316},[12558,36205,36206],{},[41,36207,64],{},[86,36209,36210],{"encoding":88},"\\varepsilon=\\sigma\u002F\\sqrt n",[18,36212,36214,36232],{"className":36213,"ariaHidden":51},[93],[18,36215,36217,36220,36223,36226,36229],{"className":36216},[97],[18,36218],{"className":36219,"style":1083},[101],[18,36221,32513],{"className":36222},[106,110],[18,36224],{"className":36225,"style":228},[169],[18,36227,433],{"className":36228},[232],[18,36230],{"className":36231,"style":228},[169],[18,36233,36235,36238,36241,36244],{"className":36234},[97],[18,36236],{"className":36237,"style":15498},[101],[18,36239,16403],{"className":36240,"style":3957},[106,110],[18,36242,317],{"className":36243},[106],[18,36245,36247],{"className":36246},[106,12693],[18,36248,36250,36280],{"className":36249},[119,120],[18,36251,36253,36277],{"className":36252},[124],[18,36254,36256,36265],{"className":36255,"style":15397},[128],[18,36257,36259,36262],{"className":36258,"style":693},[12707],[18,36260],{"className":36261,"style":697},[136],[18,36263,64],{"className":36264,"style":15407},[106,110],[18,36266,36267,36270],{"style":15410},[18,36268],{"className":36269,"style":697},[136],[18,36271,36273],{"className":36272,"style":15417},[12850],[6127,36274,36275],{"xmlns":6129,"width":12854,"height":15420,"viewBox":15421,"preserveAspectRatio":12857},[6133,36276],{"d":15424},[18,36278,152],{"className":36279},[151],[18,36281,36283],{"className":36282},[124],[18,36284,36286],{"className":36285,"style":15434},[128],[18,36287],{},"，则 TV 至多 ",[18,36290,36292,36309],{"className":36291,"translate":22},[21],[18,36293,36295],{"className":36294},[26],[28,36296,36297],{"xmlns":30},[32,36298,36299,36307],{},[35,36300,36301,36303,36305],{},[45,36302,47],{},[41,36304,317],{"mathvariant":316},[45,36306,2266],{},[86,36308,9057],{"encoding":88},[18,36310,36312],{"className":36311,"ariaHidden":51},[93],[18,36313,36315,36318],{"className":36314},[97],[18,36316],{"className":36317,"style":242},[101],[18,36319,9057],{"className":36320},[106],"。Assouad：",[18,36323,36325],{"className":36324,"translate":22},[875],[18,36326,36328,36437],{"className":36327,"translate":22},[21],[18,36329,36331],{"className":36330},[26],[28,36332,36333],{"xmlns":30,"display":884},[32,36334,36335,36434],{},[35,36336,36337,36357,36359,36361,36367,36369,36371,36379,36381,36395,36411,36413,36432],{},[890,36338,36339,36345],{},[35,36340,36341,36343],{},[41,36342,5590],{},[49,36344,899],{},[35,36346,36347,36349,36351],{},[41,36348,77],{},[49,36350,921],{},[38,36352,36353,36355],{},[41,36354,924],{"mathvariant":316},[45,36356,430],{},[41,36358,934],{"mathvariant":933},[49,36360,2248],{"stretchy":73},[658,36362,36363,36365],{"accent":51},[41,36364,77],{},[49,36366,664],{},[49,36368,2253],{},[41,36370,77],{},[1589,36372,36373,36375,36377],{},[49,36374,2248],{"stretchy":73},[45,36376,2266],{},[45,36378,2266],{},[49,36380,4775],{},[5651,36382,36383,36393],{},[35,36384,36385,36391],{},[1556,36386,36387,36389],{},[41,36388,32513],{},[45,36390,2266],{},[41,36392,4031],{},[45,36394,10444],{},[35,36396,36397,36399,36401,36403,36409],{},[49,36398,74],{"fence":51},[45,36400,47],{},[49,36402,2253],{},[5651,36404,36405,36407],{},[45,36406,47],{},[45,36408,2266],{},[49,36410,84],{"fence":51},[49,36412,433],{},[5651,36414,36415,36425],{},[35,36416,36417,36423],{},[1556,36418,36419,36421],{},[41,36420,16403],{},[45,36422,2266],{},[41,36424,4031],{},[35,36426,36427,36430],{},[45,36428,36429],{},"8",[41,36431,64],{},[41,36433,968],{"mathvariant":316},[86,36435,36436],{"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}.",[18,36438,36440,36588,36657,36855],{"className":36439,"ariaHidden":51},[93],[18,36441,36443,36447,36539,36542,36545,36548,36579,36582,36585],{"className":36442},[97],[18,36444],{"className":36445,"style":36446},[101],"height:1.8101em;vertical-align:-0.8522em;",[18,36448,36450],{"className":36449},[985,986],[18,36451,36453,36530],{"className":36452},[119,120],[18,36454,36456,36527],{"className":36455},[124],[18,36457,36459,36517],{"className":36458,"style":1083},[128],[18,36460,36461,36464],{"style":29361},[18,36462],{"className":36463,"style":697},[136],[18,36465,36467],{"className":36466},[141,142,143,144],[18,36468,36470,36473,36476],{"className":36469},[106,144],[18,36471,77],{"className":36472,"style":254},[106,110,144],[18,36474,921],{"className":36475},[232,144],[18,36477,36479,36482],{"className":36478},[106,144],[18,36480,924],{"className":36481},[106,144],[18,36483,36485],{"className":36484},[115],[18,36486,36488,36509],{"className":36487},[119,120],[18,36489,36491,36506],{"className":36490},[124],[18,36492,36494],{"className":36493,"style":7456},[128],[18,36495,36497,36500],{"style":36496},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[18,36498],{"className":36499,"style":1191},[136],[18,36501,36503],{"className":36502},[141,1195,1196,144],[18,36504,430],{"className":36505},[106,144],[18,36507,152],{"className":36508},[151],[18,36510,36512],{"className":36511},[124],[18,36513,36515],{"className":36514,"style":7478},[128],[18,36516],{},[18,36518,36519,36522],{"style":693},[18,36520],{"className":36521,"style":697},[136],[18,36523,36524],{},[18,36525,5590],{"className":36526},[985],[18,36528,152],{"className":36529},[151],[18,36531,36533],{"className":36532},[124],[18,36534,36537],{"className":36535,"style":36536},[128],"height:0.8522em;",[18,36538],{},[18,36540],{"className":36541,"style":170},[169],[18,36543,934],{"className":36544},[106,1139],[18,36546,2248],{"className":36547},[250],[18,36549,36551],{"className":36550},[106,681],[18,36552,36554],{"className":36553},[119],[18,36555,36557],{"className":36556},[124],[18,36558,36560,36568],{"className":36559,"style":677},[128],[18,36561,36562,36565],{"style":693},[18,36563],{"className":36564,"style":697},[136],[18,36566,77],{"className":36567,"style":254},[106,110],[18,36569,36570,36573],{"style":703},[18,36571],{"className":36572,"style":697},[136],[18,36574,36576],{"className":36575,"style":711},[710],[18,36577,664],{"className":36578},[106],[18,36580],{"className":36581,"style":2068},[169],[18,36583,2253],{"className":36584},[2072],[18,36586],{"className":36587,"style":2068},[169],[18,36589,36591,36594,36597,36648,36651,36654],{"className":36590},[97],[18,36592],{"className":36593,"style":15253},[101],[18,36595,77],{"className":36596,"style":254},[106,110],[18,36598,36600,36603],{"className":36599},[267],[18,36601,2248],{"className":36602},[267],[18,36604,36606],{"className":36605},[115],[18,36607,36609,36640],{"className":36608},[119,120],[18,36610,36612,36637],{"className":36611},[124],[18,36613,36615,36626],{"className":36614,"style":10772},[128],[18,36616,36617,36620],{"style":10775},[18,36618],{"className":36619,"style":137},[136],[18,36621,36623],{"className":36622},[141,142,143,144],[18,36624,2266],{"className":36625},[106,144],[18,36627,36628,36631],{"style":1637},[18,36629],{"className":36630,"style":137},[136],[18,36632,36634],{"className":36633},[141,142,143,144],[18,36635,2266],{"className":36636},[106,144],[18,36638,152],{"className":36639},[151],[18,36641,36643],{"className":36642},[124],[18,36644,36646],{"className":36645,"style":10805},[128],[18,36647],{},[18,36649],{"className":36650,"style":228},[169],[18,36652,4775],{"className":36653},[232],[18,36655],{"className":36656,"style":228},[169],[18,36658,36660,36663,36754,36757,36846,36849,36852],{"className":36659},[97],[18,36661],{"className":36662,"style":15977},[101],[18,36664,36666,36669,36751],{"className":36665},[106],[18,36667],{"className":36668},[250,6009],[18,36670,36672],{"className":36671},[5651],[18,36673,36675,36743],{"className":36674},[119,120],[18,36676,36678,36740],{"className":36677},[124],[18,36679,36681,36692,36700],{"className":36680,"style":15996},[128],[18,36682,36683,36686],{"style":6025},[18,36684],{"className":36685,"style":697},[136],[18,36687,36689],{"className":36688},[106],[18,36690,10444],{"className":36691},[106],[18,36693,36694,36697],{"style":6037},[18,36695],{"className":36696,"style":697},[136],[18,36698],{"className":36699,"style":6045},[6044],[18,36701,36702,36705],{"style":6048},[18,36703],{"className":36704,"style":697},[136],[18,36706,36708,36737],{"className":36707},[106],[18,36709,36711,36714],{"className":36710},[106],[18,36712,32513],{"className":36713},[106,110],[18,36715,36717],{"className":36716},[115],[18,36718,36720],{"className":36719},[119],[18,36721,36723],{"className":36722},[124],[18,36724,36726],{"className":36725,"style":2416},[128],[18,36727,36728,36731],{"style":1962},[18,36729],{"className":36730,"style":137},[136],[18,36732,36734],{"className":36733},[141,142,143,144],[18,36735,2266],{"className":36736},[106,144],[18,36738,4031],{"className":36739},[106,110],[18,36741,152],{"className":36742},[151],[18,36744,36746],{"className":36745},[124],[18,36747,36749],{"className":36748,"style":6067},[128],[18,36750],{},[18,36752],{"className":36753},[267,6009],[18,36755],{"className":36756,"style":170},[169],[18,36758,36760,36766,36769,36772,36775,36778,36840],{"className":36759},[174],[18,36761,36763],{"className":36762,"style":13793},[250,13792],[18,36764,74],{"className":36765},[1230,143],[18,36767,47],{"className":36768},[106],[18,36770],{"className":36771,"style":2068},[169],[18,36773,2253],{"className":36774},[2072],[18,36776],{"className":36777,"style":2068},[169],[18,36779,36781,36784,36837],{"className":36780},[106],[18,36782],{"className":36783},[250,6009],[18,36785,36787],{"className":36786},[5651],[18,36788,36790,36829],{"className":36789},[119,120],[18,36791,36793,36826],{"className":36792},[124],[18,36794,36796,36807,36815],{"className":36795,"style":9578},[128],[18,36797,36798,36801],{"style":6025},[18,36799],{"className":36800,"style":697},[136],[18,36802,36804],{"className":36803},[106],[18,36805,2266],{"className":36806},[106],[18,36808,36809,36812],{"style":6037},[18,36810],{"className":36811,"style":697},[136],[18,36813],{"className":36814,"style":6045},[6044],[18,36816,36817,36820],{"style":6048},[18,36818],{"className":36819,"style":697},[136],[18,36821,36823],{"className":36822},[106],[18,36824,47],{"className":36825},[106],[18,36827,152],{"className":36828},[151],[18,36830,36832],{"className":36831},[124],[18,36833,36835],{"className":36834,"style":6067},[128],[18,36836],{},[18,36838],{"className":36839},[267,6009],[18,36841,36843],{"className":36842,"style":13793},[267,13792],[18,36844,84],{"className":36845},[1230,143],[18,36847],{"className":36848,"style":228},[169],[18,36850,433],{"className":36851},[232],[18,36853],{"className":36854,"style":228},[169],[18,36856,36858,36861,36955],{"className":36857},[97],[18,36859],{"className":36860,"style":34941},[101],[18,36862,36864,36867,36952],{"className":36863},[106],[18,36865],{"className":36866},[250,6009],[18,36868,36870],{"className":36869},[5651],[18,36871,36873,36944],{"className":36872},[119,120],[18,36874,36876,36941],{"className":36875},[124],[18,36877,36879,36893,36901],{"className":36878,"style":15996},[128],[18,36880,36881,36884],{"style":6025},[18,36882],{"className":36883,"style":697},[136],[18,36885,36887,36890],{"className":36886},[106],[18,36888,36429],{"className":36889},[106],[18,36891,64],{"className":36892},[106,110],[18,36894,36895,36898],{"style":6037},[18,36896],{"className":36897,"style":697},[136],[18,36899],{"className":36900,"style":6045},[6044],[18,36902,36903,36906],{"style":6048},[18,36904],{"className":36905,"style":697},[136],[18,36907,36909,36938],{"className":36908},[106],[18,36910,36912,36915],{"className":36911},[106],[18,36913,16403],{"className":36914,"style":3957},[106,110],[18,36916,36918],{"className":36917},[115],[18,36919,36921],{"className":36920},[119],[18,36922,36924],{"className":36923},[124],[18,36925,36927],{"className":36926,"style":2416},[128],[18,36928,36929,36932],{"style":1962},[18,36930],{"className":36931,"style":137},[136],[18,36933,36935],{"className":36934},[141,142,143,144],[18,36936,2266],{"className":36937},[106,144],[18,36939,4031],{"className":36940},[106,110],[18,36942,152],{"className":36943},[151],[18,36945,36947],{"className":36946},[124],[18,36948,36950],{"className":36949,"style":6067},[128],[18,36951],{},[18,36953],{"className":36954},[267,6009],[18,36956,968],{"className":36957},[106],[11,36959,36960,36961,36989,36990,37063],{},"上下界只差常数 ",[18,36962,36964,36977],{"className":36963,"translate":22},[21],[18,36965,36967],{"className":36966},[26],[28,36968,36969],{"xmlns":30},[32,36970,36971,36975],{},[35,36972,36973],{},[45,36974,36429],{},[86,36976,36429],{"encoding":88},[18,36978,36980],{"className":36979,"ariaHidden":51},[93],[18,36981,36983,36986],{"className":36982},[97],[18,36984],{"className":36985,"style":505},[101],[18,36987,36429],{"className":36988},[106],"，速率 ",[18,36991,36993,37016],{"className":36992,"translate":22},[21],[18,36994,36996],{"className":36995},[26],[28,36997,36998],{"xmlns":30},[32,36999,37000,37014],{},[35,37001,37002,37008,37010,37012],{},[1556,37003,37004,37006],{},[41,37005,16403],{},[45,37007,2266],{},[41,37009,4031],{},[41,37011,317],{"mathvariant":316},[41,37013,64],{},[86,37015,17296],{"encoding":88},[18,37017,37019],{"className":37018,"ariaHidden":51},[93],[18,37020,37022,37025,37054,37057,37060],{"className":37021},[97],[18,37023],{"className":37024,"style":10268},[101],[18,37026,37028,37031],{"className":37027},[106],[18,37029,16403],{"className":37030,"style":3957},[106,110],[18,37032,37034],{"className":37033},[115],[18,37035,37037],{"className":37036},[119],[18,37038,37040],{"className":37039},[124],[18,37041,37043],{"className":37042,"style":2416},[128],[18,37044,37045,37048],{"style":1962},[18,37046],{"className":37047,"style":137},[136],[18,37049,37051],{"className":37050},[141,142,143,144],[18,37052,2266],{"className":37053},[106,144],[18,37055,4031],{"className":37056},[106,110],[18,37058,317],{"className":37059},[106],[18,37061,64],{"className":37062},[106,110]," 被夹住了。",[11,37065,37066,37067,37095],{},"这个例子也说明了 Fano 与 Assouad 的分工：Fano 需要构造一个 ",[18,37068,37070,37083],{"className":37069,"translate":22},[21],[18,37071,37073],{"className":37072},[26],[28,37074,37075],{"xmlns":30},[32,37076,37077,37081],{},[35,37078,37079],{},[41,37080,5348],{},[86,37082,14088],{"encoding":88},[18,37084,37086],{"className":37085,"ariaHidden":51},[93],[18,37087,37089,37092],{"className":37088},[97],[18,37090],{"className":37091,"style":295},[101],[18,37093,5348],{"className":37094,"style":5477},[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。",[1311,37097,37099,37100],{"id":37098},"_62-非参数回归n2β2βdn-2beta2betadn2β2βd","6.2 非参数回归：",[18,37101,37103,37140],{"className":37102,"translate":22},[21],[18,37104,37106],{"className":37105},[26],[28,37107,37108],{"xmlns":30},[32,37109,37110,37138],{},[35,37111,37112],{},[1556,37113,37114,37116],{},[41,37115,64],{},[35,37117,37118,37120,37122,37124,37126,37128,37130,37132,37134,37136],{},[49,37119,2253],{},[45,37121,2266],{},[41,37123,20979],{},[41,37125,317],{"mathvariant":316},[49,37127,74],{"stretchy":73},[45,37129,2266],{},[41,37131,20979],{},[49,37133,7984],{},[41,37135,4031],{},[49,37137,84],{"stretchy":73},[86,37139,27928],{"encoding":88},[18,37141,37143],{"className":37142,"ariaHidden":51},[93],[18,37144,37146,37149],{"className":37145},[97],[18,37147],{"className":37148,"style":27938},[101],[18,37150,37152,37155],{"className":37151},[106],[18,37153,64],{"className":37154},[106,110],[18,37156,37158],{"className":37157},[115],[18,37159,37161],{"className":37160},[119],[18,37162,37164],{"className":37163},[124],[18,37165,37167],{"className":37166,"style":27938},[128],[18,37168,37169,37172],{"style":1962},[18,37170],{"className":37171,"style":137},[136],[18,37173,37175],{"className":37174},[141,142,143,144],[18,37176,37178,37181,37184,37187,37190,37193,37196,37199,37202,37205],{"className":37177},[106,144],[18,37179,2253],{"className":37180},[106,144],[18,37182,2266],{"className":37183},[106,144],[18,37185,20979],{"className":37186,"style":21031},[106,110,144],[18,37188,317],{"className":37189},[106,144],[18,37191,74],{"className":37192},[250,144],[18,37194,2266],{"className":37195},[106,144],[18,37197,20979],{"className":37198,"style":21031},[106,110,144],[18,37200,7984],{"className":37201},[2072,144],[18,37203,4031],{"className":37204},[106,110,144],[18,37206,84],{"className":37207},[267,144],[11,37209,34188],{},[18,37211,37213],{"className":37212,"translate":22},[875],[18,37214,37216,37328],{"className":37215,"translate":22},[21],[18,37217,37219],{"className":37218},[26],[28,37220,37221],{"xmlns":30,"display":884},[32,37222,37223,37325],{},[35,37224,37225,37232,37234,37236,37238,37244,37246,37248,37255,37257,37259,37265,37267,37277,37279,37281,37283,37285,37287,37293,37295,37297,37299,37305,37307,37309,37311,37313,37315,37321,37323],{},[38,37226,37227,37230],{},[41,37228,37229],{},"Y",[41,37231,5597],{},[49,37233,433],{},[41,37235,1055],{},[49,37237,74],{"stretchy":73},[38,37239,37240,37242],{},[41,37241,43],{},[41,37243,5597],{},[49,37245,84],{"stretchy":73},[49,37247,7984],{},[38,37249,37250,37253],{},[41,37251,37252],{},"ξ",[41,37254,5597],{},[49,37256,52],{"separator":51},[169,37258],{"width":7341},[38,37260,37261,37263],{},[41,37262,43],{},[41,37264,5597],{},[49,37266,67],{},[35,37268,37269,37271,37273,37275],{},[41,37270,20047],{"mathvariant":316},[41,37272,64],{"mathvariant":316},[41,37274,5597],{"mathvariant":316},[41,37276,1055],{"mathvariant":316},[49,37278,74],{"stretchy":73},[49,37280,945],{"stretchy":73},[45,37282,430],{},[49,37284,52],{"separator":51},[45,37286,47],{},[1556,37288,37289,37291],{},[49,37290,965],{"stretchy":73},[41,37292,4031],{},[49,37294,84],{"stretchy":73},[49,37296,52],{"separator":51},[169,37298],{"width":7341},[38,37300,37301,37303],{},[41,37302,37252],{},[41,37304,5597],{},[49,37306,67],{},[41,37308,70],{},[49,37310,74],{"stretchy":73},[45,37312,430],{},[49,37314,52],{"separator":51},[1556,37316,37317,37319],{},[41,37318,16403],{},[45,37320,2266],{},[49,37322,84],{"stretchy":73},[49,37324,52],{"separator":51},[86,37326,37327],{"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),",[18,37329,37331,37387,37452,37558,37677],{"className":37330,"ariaHidden":51},[93],[18,37332,37334,37337,37378,37381,37384],{"className":37333},[97],[18,37335],{"className":37336,"style":4563},[101],[18,37338,37340,37343],{"className":37339},[106],[18,37341,37229],{"className":37342,"style":2068},[106,110],[18,37344,37346],{"className":37345},[115],[18,37347,37349,37370],{"className":37348},[119,120],[18,37350,37352,37367],{"className":37351},[124],[18,37353,37355],{"className":37354,"style":5918},[128],[18,37356,37358,37361],{"style":37357},"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;",[18,37359],{"className":37360,"style":137},[136],[18,37362,37364],{"className":37363},[141,142,143,144],[18,37365,5597],{"className":37366},[106,110,144],[18,37368,152],{"className":37369},[151],[18,37371,37373],{"className":37372},[124],[18,37374,37376],{"className":37375,"style":159},[128],[18,37377],{},[18,37379],{"className":37380,"style":228},[169],[18,37382,433],{"className":37383},[232],[18,37385],{"className":37386,"style":228},[169],[18,37388,37390,37393,37397,37400,37440,37443,37446,37449],{"className":37389},[97],[18,37391],{"className":37392,"style":242},[101],[18,37394,1055],{"className":37395,"style":37396},[106,110],"margin-right:0.1076em;",[18,37398,74],{"className":37399},[250],[18,37401,37403,37406],{"className":37402},[106],[18,37404,43],{"className":37405,"style":111},[106,110],[18,37407,37409],{"className":37408},[115],[18,37410,37412,37432],{"className":37411},[119,120],[18,37413,37415,37429],{"className":37414},[124],[18,37416,37418],{"className":37417,"style":5918},[128],[18,37419,37420,37423],{"style":132},[18,37421],{"className":37422,"style":137},[136],[18,37424,37426],{"className":37425},[141,142,143,144],[18,37427,5597],{"className":37428},[106,110,144],[18,37430,152],{"className":37431},[151],[18,37433,37435],{"className":37434},[124],[18,37436,37438],{"className":37437,"style":159},[128],[18,37439],{},[18,37441,84],{"className":37442},[267],[18,37444],{"className":37445,"style":2068},[169],[18,37447,7984],{"className":37448},[2072],[18,37450],{"className":37451,"style":2068},[169],[18,37453,37455,37458,37500,37503,37506,37509,37549,37552,37555],{"className":37454},[97],[18,37456],{"className":37457,"style":2890},[101],[18,37459,37461,37465],{"className":37460},[106],[18,37462,37252],{"className":37463,"style":37464},[106,110],"margin-right:0.046em;",[18,37466,37468],{"className":37467},[115],[18,37469,37471,37492],{"className":37470},[119,120],[18,37472,37474,37489],{"className":37473},[124],[18,37475,37477],{"className":37476,"style":5918},[128],[18,37478,37480,37483],{"style":37479},"top:-2.55em;margin-left:-0.046em;margin-right:0.05em;",[18,37481],{"className":37482,"style":137},[136],[18,37484,37486],{"className":37485},[141,142,143,144],[18,37487,5597],{"className":37488},[106,110,144],[18,37490,152],{"className":37491},[151],[18,37493,37495],{"className":37494},[124],[18,37496,37498],{"className":37497,"style":159},[128],[18,37499],{},[18,37501,52],{"className":37502},[165],[18,37504],{"className":37505,"style":7673},[169],[18,37507],{"className":37508,"style":170},[169],[18,37510,37512,37515],{"className":37511},[106],[18,37513,43],{"className":37514,"style":111},[106,110],[18,37516,37518],{"className":37517},[115],[18,37519,37521,37541],{"className":37520},[119,120],[18,37522,37524,37538],{"className":37523},[124],[18,37525,37527],{"className":37526,"style":5918},[128],[18,37528,37529,37532],{"style":132},[18,37530],{"className":37531,"style":137},[136],[18,37533,37535],{"className":37534},[141,142,143,144],[18,37536,5597],{"className":37537},[106,110,144],[18,37539,152],{"className":37540},[151],[18,37542,37544],{"className":37543},[124],[18,37545,37547],{"className":37546,"style":159},[128],[18,37548],{},[18,37550],{"className":37551,"style":228},[169],[18,37553,67],{"className":37554},[232],[18,37556],{"className":37557,"style":228},[169],[18,37559,37561,37565,37571,37575,37578,37581,37584,37587,37616,37619,37622,37625,37628,37668,37671,37674],{"className":37560},[97],[18,37562],{"className":37563,"style":37564},[101],"height:1.1491em;vertical-align:-0.25em;",[18,37566,37568],{"className":37567},[106],[18,37569,20104],{"className":37570,"style":1054},[106,6351],[18,37572,37574],{"className":37573},[250],"([",[18,37576,430],{"className":37577},[106],[18,37579,52],{"className":37580},[165],[18,37582],{"className":37583,"style":170},[169],[18,37585,47],{"className":37586},[106],[18,37588,37590,37593],{"className":37589},[267],[18,37591,965],{"className":37592},[267],[18,37594,37596],{"className":37595},[115],[18,37597,37599],{"className":37598},[119],[18,37600,37602],{"className":37601},[124],[18,37603,37605],{"className":37604,"style":16670},[128],[18,37606,37607,37610],{"style":1637},[18,37608],{"className":37609,"style":137},[136],[18,37611,37613],{"className":37612},[141,142,143,144],[18,37614,4031],{"className":37615},[106,110,144],[18,37617,84],{"classNa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属于 ",[18,37765,37767,37781],{"className":37766,"translate":22},[21],[18,37768,37770],{"className":37769},[26],[28,37771,37772],{"xmlns":30},[32,37773,37774,37778],{},[35,37775,37776],{},[41,37777,20979],{},[86,37779,37780],{"encoding":88},"\\beta",[18,37782,37784],{"className":37783,"ariaHidden":51},[93],[18,37785,37787,37790],{"className":37786},[97],[18,37788],{"className":37789,"style":2890},[101],[18,37791,20979],{"className":37792,"style":21031},[106,110],"-Hölder 球 ",[18,37795,37797,37822],{"className":37796,"translate":22},[21],[18,37798,37800],{"className":37799},[26],[28,37801,37802],{"xmlns":30},[32,37803,37804,37819],{},[35,37805,37806,37809,37811,37813,37815,37817],{},[41,37807,37808],{"mathvariant":316},"Σ",[49,37810,74],{"stretchy":73},[41,37812,20979],{},[49,37814,52],{"separator":51},[41,37816,12579],{},[49,37818,84],{"stretchy":73},[86,37820,37821],{"encoding":88},"\\Sigma(\\beta,L)",[18,37823,37825],{"className":37824,"ariaHidden":51},[93],[18,37826,37828,37831,37834,37837,37840,37843,37846,37849],{"className":37827},[97],[18,37829],{"className":37830,"style":242},[101],[18,37832,37808],{"className":37833},[106],[18,37835,74],{"className":37836},[250],[18,37838,20979],{"className":37839,"style":21031},[106,110],[18,37841,52],{"className":37842},[165],[18,37844],{"className":37845,"style":170},[169],[18,37847,12579],{"className":37848},[106,110],[18,37850,84],{"className":37851},[267],"，损失为 ",[18,37854,37856,37892],{"className":37855,"translate":22},[21],[18,37857,37859],{"className":37858},[26],[28,37860,37861],{"xmlns":30},[32,37862,37863,37889],{},[35,37864,37865,37867,37873,37875,37877],{},[49,37866,2248],{"stretchy":73},[658,37868,37869,37871],{"accent":51},[41,37870,1055],{},[49,37872,664],{},[49,37874,2253],{},[41,37876,1055],{},[1589,37878,37879,37881,37887],{},[49,37880,2248],{"stretchy":73},[1556,37882,37883,37885],{},[41,37884,12579],{},[45,37886,2266],{},[45,37888,2266],{},[86,37890,37891],{"encoding":88},"\\lVert\\hat f-f\\rVert_{L^2}^2",[18,37893,37895,37956],{"className":37894,"ariaHidden":51},[93],[18,37896,37898,37901,37904,37947,37950,37953],{"className":37897},[97],[18,37899],{"className":37900,"style":2281},[101],[18,37902,2248],{"className":37903},[250],[18,37905,37907],{"className":37906},[106,681],[18,37908,37910,37939],{"className":37909},[119,120],[18,37911,37913,37936],{"className":37912},[124],[18,37914,37916,37924],{"className":37915,"style":677},[128],[18,37917,37918,37921],{"style":693},[18,37919],{"className":37920,"style":697},[136],[18,37922,1055],{"className":37923,"style":37396},[106,110],[18,37925,37926,37929],{"style":703},[18,37927],{"className":37928,"style":697},[136],[18,37930,37933],{"className":37931,"style":37932},[710],"left:-0.0833em;",[18,37934,664],{"className":37935},[106],[18,37937,152],{"className":37938},[151],[18,37940,37942],{"className":37941},[124],[18,37943,37945],{"className":37944,"style":15924},[128],[18,37946],{},[18,37948],{"className":37949,"style":2068},[169],[18,37951,2253],{"className":37952},[2072],[18,37954],{"className":37955,"style":2068},[169],[18,37957,37959,37963,37966],{"className":37958},[97],[18,37960],{"className":37961,"style":37962},[101],"height:1.1335em;vertical-align:-0.3194em;",[18,37964,1055],{"className":37965,"style":37396},[106,110],[18,37967,37969,37972],{"className":37968},[267],[18,37970,2248],{"className":37971},[267],[18,37973,37975],{"className":37974},[115],[18,37976,37978,38040],{"className":37977},[119,120],[18,37979,37981,38037],{"className":37980},[124],[18,37982,37984,38026],{"className":37983,"style":2416},[128],[18,37985,37987,37990],{"style":37986},"top:-2.3806em;margin-left:0em;margin-right:0.05em;",[18,37988],{"className":37989,"style":137},[136],[18,37991,37993],{"className":37992},[141,142,143,144],[18,37994,37996],{"className":37995},[106,144],[18,37997,37999,38002],{"className":37998},[106,144],[18,38000,12579],{"className":38001},[106,110,144],[18,38003,38005],{"className":38004},[115],[18,38006,38008],{"className":38007},[119],[18,38009,38011],{"className":38010},[124],[18,38012,38015],{"className":38013,"style":38014},[128],"height:0.7463em;",[18,38016,38017,38020],{"style":33586},[18,38018],{"className":38019,"style":1191},[136],[18,38021,38023],{"className":38022},[141,1195,1196,144],[18,38024,2266],{"className":38025},[106,144],[18,38027,38028,38031],{"style":1962},[18,38029],{"className":38030,"style":137},[136],[18,38032,38034],{"className":38033},[141,142,143,144],[18,38035,2266],{"className":38036},[106,144],[18,38038,152],{"className":38039},[151],[18,38041,38043],{"className":38042},[124],[18,38044,38047],{"className":38045,"style":38046},[128],"height:0.3194em;",[18,38048],{},[11,38050,38051,38053],{},[717,38052,34688],{}," 核估计或局部多项式估计可以达到",[18,38055,38057],{"className":38056,"translate":22},[875],[18,38058,38060,38098],{"className":38059,"translate":22},[21],[18,38061,38063],{"className":38062},[26],[28,38064,38065],{"xmlns":30,"display":884},[32,38066,38067,38095],{},[35,38068,38069],{},[1556,38070,38071,38073],{},[41,38072,64],{},[35,38074,38075,38077],{},[49,38076,2253],{},[5651,38078,38079,38085],{},[35,38080,38081,38083],{},[45,38082,2266],{},[41,38084,20979],{},[35,38086,38087,38089,38091,38093],{},[45,38088,2266],{},[41,38090,20979],{},[49,38092,7984],{},[41,38094,4031],{},[86,38096,38097],{"encoding":88},"n^{-\\frac{2\\beta}{2\\beta+d}}",[18,38099,38101],{"className":38100,"ariaHidden":51},[93],[18,38102,38104,38108],{"className":38103},[97],[18,38105],{"className":38106,"style":38107},[101],"height:1.1167em;",[18,38109,38111,38114],{"className":38110},[106],[18,38112,64],{"className":38113},[106,110],[18,38115,38117],{"className":38116},[115],[18,38118,38120],{"className":38119},[119],[18,38121,38123],{"className":38122},[124],[18,38124,38126],{"className":38125,"style":38107},[128],[18,38127,38129,38132],{"style":38128},"top:-3.4458em;margin-right:0.05em;",[18,38130],{"className":38131,"style":697},[136],[18,38133,38135],{"className":38134},[141,142,143,144],[18,38136,38138,38141],{"className":38137},[106,144],[18,38139,2253],{"className":38140},[106,144],[18,38142,38144,38148,38225],{"className":38143},[106,144],[18,38145],{"className":38146},[250,6009,141,1195,38147],"size6",[18,38149,38151],{"className":38150},[5651],[18,38152,38154,38216],{"className":38153},[119,120],[18,38155,38157,38213],{"className":38156},[124],[18,38158,38161,38185,38195],{"className":38159,"style":38160},[128],"height:0.9584em;",[18,38162,38164,38167],{"style":38163},"top:-2.656em;",[18,38165],{"className":38166,"style":697},[136],[18,38168,38170],{"className":38169},[141,1195,1196,144],[18,38171,38173,38176,38179,38182],{"className":38172},[106,144],[18,38174,2266],{"className":38175},[106,144],[18,38177,20979],{"className":38178,"style":21031},[106,110,144],[18,38180,7984],{"className":38181},[2072,144],[18,38183,4031],{"className":38184},[106,110,144],[18,38186,38188,38191],{"style":38187},"top:-3.2255em;",[18,38189],{"className":38190,"style":697},[136],[18,38192],{"className":38193,"style":38194},[6044,144],"border-bottom-width:0.049em;",[18,38196,38198,38201],{"style":38197},"top:-3.4624em;",[18,38199],{"className":38200,"style":697},[136],[18,38202,38204],{"className":38203},[141,1195,1196,144],[18,38205,38207,38210],{"className":38206},[106,144],[18,38208,2266],{"className":38209},[106,144],[18,38211,20979],{"className":38212,"style":21031},[106,110,144],[18,38214,152],{"className":38215},[151],[18,38217,38219],{"className":38218},[124],[18,38220,38223],{"className":38221,"style":38222},[128],"height:0.4829em;",[18,38224],{},[18,38226],{"className":38227},[267,6009,141,1195,38147],[11,38229,38230,38231,38234],{},"（见 Tsybakov 第 1 章）。",[717,38232,38233],{},"下界","用 Fano。",[11,38236,36183,38237,38311,38312,38394,38395,38486,38487,38515,38516,38547],{},[18,38238,38240,38266],{"className":38239,"translate":22},[21],[18,38241,38243],{"className":38242},[26],[28,38244,38245],{"xmlns":30},[32,38246,38247,38263],{},[35,38248,38249,38251,38253,38255,38257,38259,38261],{},[41,38250,18323],{},[49,38252,921],{},[49,38254,74],{"stretchy":73},[45,38256,430],{},[49,38258,52],{"separator":51},[45,38260,47],{},[49,38262,84],{"stretchy":73},[86,38264,38265],{"encoding":88},"h\\in(0,1)",[18,38267,38269,38287],{"className":38268,"ariaHidden":51},[93],[18,38270,38272,38275,38278,38281,38284],{"className":38271},[97],[18,38273],{"className":38274,"style":1422},[101],[18,38276,18323],{"className":38277},[106,110],[18,38279],{"className":38280,"style":228},[169],[18,38282,921],{"className":38283},[232],[18,38285],{"className":38286,"style":228},[169],[18,38288,38290,38293,38296,38299,38302,38305,38308],{"className":38289},[97],[18,38291],{"className":38292,"style":242},[101],[18,38294,74],{"className":38295},[250],[18,38297,430],{"className":38298},[106],[18,38300,52],{"className":38301},[165],[18,38303],{"className":38304,"style":170},[169],[18,38306,47],{"className":38307},[106],[18,38309,84],{"className":38310},[267],"，把 ",[18,38313,38315,38341],{"className":38314,"translate":22},[21],[18,38316,38318],{"className":38317},[26],[28,38319,38320],{"xmlns":30},[32,38321,38322,38338],{},[35,38323,38324,38326,38328,38330,38332],{},[49,38325,945],{"stretchy":73},[45,38327,430],{},[49,38329,52],{"separator":51},[45,38331,47],{},[1556,38333,38334,38336],{},[49,38335,965],{"stretchy":73},[41,38337,4031],{},[86,38339,38340],{"encoding":88},"[0,1]^d",[18,38342,38344],{"className":38343,"ariaHidden":51},[93],[18,38345,38347,38350,38353,38356,38359,38362,38365],{"className":38346},[97],[18,38348],{"className":38349,"style":35143},[101],[18,38351,945],{"className":38352},[250],[18,38354,430],{"className":38355},[106],[18,38357,52],{"className":38358},[165],[18,38360],{"className":38361,"style":170},[169],[18,38363,47],{"className":38364},[106],[18,38366,38368,38371],{"className":38367},[267],[18,38369,965],{"className":38370},[267],[18,38372,38374],{"className":38373},[115],[18,38375,38377],{"className":38376},[119],[18,38378,38380],{"className":38379},[124],[18,38381,38383],{"className":38382,"style":35177},[128],[18,38384,38385,38388],{"style":1962},[18,38386],{"className":38387,"style":137},[136],[18,38389,38391],{"className":38390},[141,142,143,144],[18,38392,4031],{"className":38393},[106,110,144]," 分成 ",[18,38396,38398,38424],{"className":38397,"translate":22},[21],[18,38399,38401],{"className":38400},[26],[28,38402,38403],{"xmlns":30},[32,38404,38405,38421],{},[35,38406,38407,38409,38411],{},[41,38408,28160],{},[49,38410,433],{},[1556,38412,38413,38415],{},[41,38414,18323],{},[35,38416,38417,38419],{},[49,38418,2253],{},[41,38420,4031],{},[86,38422,38423],{"encoding":88},"m=h^{-d}",[18,38425,38427,38445],{"className":38426,"ariaHidden":51},[93],[18,38428,38430,38433,38436,38439,38442],{"className":38429},[97],[18,38431],{"className":38432,"style":1083},[101],[18,38434,28160],{"className":38435},[106,110],[18,38437],{"className":38438,"style":228},[169],[18,38440,433],{"className":38441},[232],[18,38443],{"className":38444,"style":228},[169],[18,38446,38448,38451],{"className":38447},[97],[18,38449],{"className":38450,"style":35177},[101],[18,38452,38454,38457],{"className":38453},[106],[18,38455,18323],{"className":38456},[106,110],[18,38458,38460],{"className":38459},[115],[18,38461,38463],{"className":38462},[119],[18,38464,38466],{"className":38465},[124],[18,38467,38469],{"className":38468,"style":35177},[128],[18,38470,38471,38474],{"style":1962},[18,38472],{"className":38473,"style":137},[136],[18,38475,38477],{"className":38476},[141,142,143,144],[18,38478,38480,38483],{"className":38479},[106,144],[18,38481,2253],{"className":38482},[106,144],[18,38484,4031],{"className":38485},[106,110,144]," 个边长为 ",[18,38488,38490,38503],{"className":38489,"translate":22},[21],[18,38491,38493],{"className":38492},[26],[28,38494,38495],{"xmlns":30},[32,38496,38497,38501],{},[35,38498,38499],{},[41,38500,18323],{},[86,38502,18323],{"encoding":88},[18,38504,38506],{"className":38505,"ariaHidden":51},[93],[18,38507,38509,38512],{"className":38508},[97],[18,38510],{"className":38511,"style":295},[101],[18,38513,18323],{"className":38514},[106,110]," 的小方块。取一个光滑的鼓包函数 ",[18,38517,38519,38534],{"className":38518,"translate":22},[21],[18,38520,38522],{"className":38521},[26],[28,38523,38524],{"xmlns":30},[32,38525,38526,38531],{},[35,38527,38528],{},[41,38529,38530],{},"φ",[86,38532,38533],{"encoding":88},"\\varphi",[18,38535,38537],{"className":38536,"ariaHidden":51},[93],[18,38538,38540,38544],{"className":38539},[97],[18,38541],{"className":38542,"style":38543},[101],"height:0.625em;vertical-align:-0.1944em;",[18,38545,38530],{"className":38546},[106,110],"，支撑在单位方块内，令",[18,38549,38551],{"className":38550,"translate":22},[875],[18,38552,38554,38605],{"className":38553,"translate":22},[21],[18,38555,38557],{"className":38556},[26],[28,38558,38559],{"xmlns":30,"display":884},[32,38560,38561,38602],{},[35,38562,38563,38569,38571,38574,38576,38578,38580,38582,38598,38600],{},[38,38564,38565,38567],{},[41,38566,38530],{},[41,38568,19804],{},[49,38570,74],{"stretchy":73},[41,38572,38573],{},"x",[49,38575,84],{"stretchy":73},[49,38577,433],{},[41,38579,38530],{},[49,38581,74],{"fence":51,"stretchy":51,"minsize":5659,"maxsize":5659},[5651,38583,38584,38596],{},[35,38585,38586,38588,38590],{},[41,38587,38573],{},[49,38589,2253],{},[38,38591,38592,38594],{},[41,38593,38573],{},[41,38595,19804],{},[41,38597,18323],{},[49,38599,84],{"fence":51,"stretchy":51,"minsize":5659,"maxsize":5659},[49,38601,52],{"separator":51},[86,38603,38604],{"encoding":88},"\\varphi_j(x)=\\varphi\\Bigl(\\frac{x-x_j}{h}\\Bigr),",[18,38606,38608,38672],{"className":38607,"ariaHidden":51},[93],[18,38609,38611,38614,38654,38657,38660,38663,38666,38669],{"className":38610},[97],[18,38612],{"className":38613,"style":19839},[101],[18,38615,38617,38620],{"className":38616},[106],[18,38618,38530],{"className":38619},[106,110],[18,38621,38623],{"className":38622},[115],[18,38624,38626,38646],{"className":38625},[119,120],[18,38627,38629,38643],{"className":38628},[124],[18,38630,38632],{"className":38631,"style":5918},[128],[18,38633,38634,38637],{"style":1155},[18,38635],{"className":38636,"style":137},[136],[18,38638,38640],{"className":38639},[141,142,143,144],[18,38641,19804],{"className":38642,"style":19921},[106,110,144],[18,38644,152],{"className":38645},[151],[18,38647,38649],{"className":38648},[124],[18,38650,38652],{"className":38651,"style":19931},[128],[18,38653],{},[18,38655,74],{"className":38656},[250],[18,38658,38573],{"className":38659},[106,110],[18,38661,84],{"className":38662},[267],[18,38664],{"className":38665,"style":228},[169],[18,38667,433],{"className":38668},[232],[18,38670],{"className":38671,"style":228},[169],[18,38673,38675,38679,38682,38688,38800,38806],{"className":38674},[97],[18,38676],{"className":38677,"style":38678},[101],"height:1.9463em;vertical-align:-0.686em;",[18,38680,38530],{"className":38681},[106,110],[18,38683,38685],{"className":38684},[250],[18,38686,74],{"className":38687},[1230,6079],[18,38689,38691,38694,38797],{"className":38690},[106],[18,38692],{"className":38693},[250,6009],[18,38695,38697],{"className":38696},[5651],[18,38698,38700,38789],{"className":38699},[119,120],[18,38701,38703,38786],{"className":38702},[124],[18,38704,38707,38718,38726],{"className":38705,"style":38706},[128],"height:1.2603em;",[18,38708,38709,38712],{"style":6025},[18,38710],{"className":38711,"style":697},[136],[18,38713,38715],{"className":38714},[106],[18,38716,18323],{"className":38717},[106,110],[18,38719,38720,38723],{"style":6037},[18,38721],{"className":38722,"style":697},[136],[18,38724],{"className":38725,"style":6045},[6044],[18,38727,38728,38731],{"style":6048},[18,38729],{"className":38730,"style":697},[136],[18,38732,38734,38737,38740,38743,38746],{"className":38733},[106],[18,38735,38573],{"className":38736},[106,110],[18,38738],{"className":38739,"style":2068},[169],[18,38741,2253],{"className":38742},[2072],[18,38744],{"className":38745,"style":2068},[169],[18,38747,38749,38752],{"className":38748},[106],[18,38750,38573],{"className":38751},[106,110],[18,38753,38755],{"className":38754},[115],[18,38756,38758,38778],{"className":38757},[119,120],[18,38759,38761,38775],{"className":38760},[124],[18,38762,38764],{"className":38763,"style":5918},[128],[18,38765,38766,38769],{"style":1155},[18,38767],{"className":38768,"style":137},[136],[18,38770,38772],{"className":38771},[141,142,143,144],[18,38773,19804],{"className":38774,"style":19921},[106,110,144],[18,38776,152],{"className":38777},[151],[18,38779,38781],{"className":38780},[124],[18,38782,38784],{"className":38783,"style":19931},[128],[18,38785],{},[18,38787,152],{"className":38788},[151],[18,38790,38792],{"className":38791},[124],[18,38793,38795],{"className":38794,"style":6067},[128],[18,38796],{},[18,38798],{"className":38799},[267,6009],[18,38801,38803],{"className":38802},[267],[18,38804,84],{"className":38805},[1230,6079],[18,38807,52],{"className":38808},[165],[11,38810,18225,38811,38881,38882,38910,38911,39004],{},[18,38812,38814,38832],{"className":38813,"translate":22},[21],[18,38815,38817],{"className":38816},[26],[28,38818,38819],{"xmlns":30},[32,38820,38821,38829],{},[35,38822,38823],{},[38,38824,38825,38827],{},[41,38826,38573],{},[41,38828,19804],{},[86,38830,38831],{"encoding":88},"x_j",[18,38833,38835],{"className":38834,"ariaHidden":51},[93],[18,38836,38838,38841],{"className":38837},[97],[18,38839],{"className":38840,"style":30670},[101],[18,38842,38844,38847],{"className":38843},[106],[18,38845,38573],{"className":38846},[106,110],[18,38848,38850],{"className":38849},[115],[18,38851,38853,38873],{"className":38852},[119,120],[18,38854,38856,38870],{"className":38855},[124],[18,38857,38859],{"className":38858,"style":5918},[128],[18,38860,38861,38864],{"style":1155},[18,38862],{"className":38863,"style":137},[136],[18,38865,38867],{"className":38866},[141,142,143,144],[18,38868,19804],{"className":38869,"style":19921},[106,110,144],[18,38871,152],{"className":38872},[151],[18,38874,38876],{"className":38875},[124],[18,38877,38879],{"className":38878,"style":19931},[128],[18,38880],{}," 是第 ",[18,38883,38885,38898],{"className":38884,"translate":22},[21],[18,38886,38888],{"className":38887},[26],[28,38889,38890],{"xmlns":30},[32,38891,38892,38896],{},[35,38893,38894],{},[41,38895,19804],{},[86,38897,19804],{"encoding":88},[18,38899,38901],{"className":38900,"ariaHidden":51},[93],[18,38902,38904,38907],{"className":38903},[97],[18,38905],{"className":38906,"style":28331},[101],[18,38908,19804],{"className":38909,"style":19921},[106,110]," 个方块的中心。对任意子集 ",[18,38912,38914,38945],{"className":38913,"translate":22},[21],[18,38915,38917],{"className":38916},[26],[28,38918,38919],{"xmlns":30},[32,38920,38921,38942],{},[35,38922,38923,38926,38928,38930,38932,38934,38936,38938,38940],{},[41,38924,38925],{},"S",[49,38927,4548],{},[49,38929,1331],{"stretchy":73},[45,38931,47],{},[49,38933,52],{"separator":51},[49,38935,55],{},[49,38937,52],{"separator":51},[41,38939,28160],{},[49,38941,1349],{"stretchy":73},[86,38943,38944],{"encoding":88},"S\\subseteq\\{1,\\dots,m\\}",[18,38946,38948,38968],{"className":38947,"ariaHidden":51},[93],[18,38949,38951,38955,38959,38962,38965],{"className":38950},[97],[18,38952],{"className":38953,"style":38954},[101],"height:0.8193em;vertical-align:-0.136em;",[18,38956,38925],{"className":38957,"style":38958},[106,110],"margin-right:0.0576em;",[18,38960],{"className":38961,"style":228},[169],[18,38963,4548],{"className":38964},[232],[18,38966],{"className":38967,"style":228},[169],[18,38969,38971,38974,38977,38980,38983,38986,38989,38992,38995,38998,39001],{"className":38970},[97],[18,38972],{"className":38973,"style":242},[101],[18,38975,1331],{"className":38976},[250],[18,38978,47],{"className":38979},[106],[18,38981,52],{"className":38982},[165],[18,38984],{"className":38985,"style":170},[169],[18,38987,55],{"className":38988},[174],[18,38990],{"className":38991,"style":170},[169],[18,38993,52],{"className":38994},[165],[18,38996],{"className":38997,"style":170},[169],[18,38999,28160],{"className":39000},[106,110],[18,39002,1349],{"className":39003},[267],"，定义",[18,39006,39008],{"className":39007,"translate":22},[875],[18,39009,39011,39072],{"className":39010,"translate":22},[21],[18,39012,39014],{"className":39013},[26],[28,39015,39016],{"xmlns":30,"display":884},[32,39017,39018,39069],{},[35,39019,39020,39026,39028,39030,39042,39048,39050,39052,39054,39056,39059,39061,39067],{},[38,39021,39022,39024],{},[41,39023,1055],{},[41,39025,38925],{},[49,39027,433],{},[41,39029,16820],{},[890,39031,39032,39034],{},[49,39033,20362],{},[35,39035,39036,39038,39040],{},[41,39037,19804],{},[49,39039,921],{},[41,39041,38925],{},[38,39043,39044,39046],{},[41,39045,38530],{},[41,39047,19804],{},[49,39049,52],{"separator":51},[169,39051],{"width":7341},[41,39053,16820],{},[49,39055,433],{},[41,39057,39058],{},"c",[926,39060,4024],{},[1556,39062,39063,39065],{},[41,39064,18323],{},[41,39066,20979],{},[41,39068,968],{"mathvariant":316},[86,39070,39071],{"encoding":88},"f_S=a\\sum_{j\\in S}\\varphi_j,\\qquad a=c\\,h^\\beta.",[18,39073,39075,39131,39263],{"className":39074,"ariaHidden":51},[93],[18,39076,39078,39081,39122,39125,39128],{"className":39077},[97],[18,39079],{"className":39080,"style":2890},[101],[18,39082,39084,39087],{"className":39083},[106],[18,39085,1055],{"className":39086,"style":37396},[106,110],[18,39088,39090],{"className":39089},[115],[18,39091,39093,39114],{"className":39092},[119,120],[18,39094,39096,39111],{"className":39095},[124],[18,39097,39099],{"className":39098,"style":1152},[128],[18,39100,39102,39105],{"style":39101},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[18,39103],{"className":39104,"style":137},[136],[18,39106,39108],{"className":39107},[141,142,143,144],[18,39109,38925],{"className":39110,"style":38958},[106,110,144],[18,39112,152],{"className":39113},[151],[18,39115,39117],{"className":39116},[124],[18,39118,39120],{"className":39119,"style":159},[128],[18,39121],{},[18,39123],{"className":39124,"style":228},[169],[18,39126,433],{"className":39127},[232],[18,39129],{"className":39130,"style":228},[169],[18,39132,39134,39138,39141,39144,39199,39202,39242,39245,39248,39251,39254,39257,39260],{"className":39133},[97],[18,39135],{"className":39136,"style":39137},[101],"height:2.4804em;vertical-align:-1.4304em;",[18,39139,16820],{"className":39140},[106,110],[18,39142],{"className":39143,"style":170},[169],[18,39145,39147],{"className":39146},[985,986],[18,39148,39150,39190],{"className":39149},[119,120],[18,39151,39153,39187],{"className":39152},[124],[18,39154,39156,39177],{"className":39155,"style":24476},[128],[18,39157,39159,39162],{"style":39158},"top:-1.8557em;margin-left:0em;",[18,39160],{"className":39161,"style":20552},[136],[18,39163,39165],{"className":39164},[141,142,143,144],[18,39166,39168,39171,39174],{"className":39167},[106,144],[18,39169,19804],{"className":39170,"style":19921},[106,110,144],[18,39172,921],{"className":39173},[232,144],[18,39175,38925],{"className":39176,"style":38958},[106,110,144],[18,39178,39179,39182],{"style":20570},[18,39180],{"className":39181,"style":20552},[136],[18,39183,39184],{},[18,39185,20362],{"className":39186},[985,4198,4199],[18,39188,152],{"className":39189},[151],[18,39191,39193],{"className":39192},[124],[18,39194,39197],{"className":39195,"style":39196},[128],"height:1.4304em;",[18,39198],{},[18,39200],{"className":39201,"style":170},[169],[18,39203,39205,39208],{"className":39204},[106],[18,39206,38530],{"className":39207},[106,110],[18,39209,39211],{"className":39210},[115],[18,39212,39214,39234],{"className":39213},[119,120],[18,39215,39217,39231],{"className":39216},[124],[18,39218,39220],{"className":39219,"style":5918},[128],[18,39221,39222,39225],{"style":1155},[18,39223],{"className":39224,"style":137},[136],[18,39226,39228],{"className":39227},[141,142,143,144],[18,39229,19804],{"className":39230,"style":19921},[106,110,144],[18,39232,152],{"className":39233},[151],[18,39235,39237],{"className":39236},[124],[18,39238,39240],{"className":39239,"style":19931},[128],[18,39241],{},[18,39243,52],{"className":39244},[165],[18,39246],{"className":39247,"style":7673},[169],[18,39249],{"className":39250,"style":170},[169],[18,39252,16820],{"className":39253},[106,110],[18,39255],{"className":39256,"style":228},[169],[18,39258,433],{"className":39259},[232],[18,39261],{"className":39262,"style":228},[169],[18,39264,39266,39269,39272,39275,39304],{"className":39265},[97],[18,39267],{"className":39268,"style":16670},[101],[18,39270,39058],{"className":39271},[106,110],[18,39273],{"className":39274,"style":170},[169],[18,39276,39278,39281],{"className":39277},[106],[18,39279,18323],{"className":39280},[106,110],[18,39282,39284],{"className":39283},[115],[18,39285,39287],{"className":39286},[119],[18,39288,39290],{"className":39289},[124],[18,39291,39293],{"className":39292,"style":16670},[128],[18,39294,39295,39298],{"style":1637},[18,39296],{"className":39297,"style":137},[136],[18,39299,39301],{"className":39300},[141,142,143,144],[18,39302,20979],{"className":39303,"style":21031},[106,110,144],[18,39305,968],{"className":39306},[106],[11,39308,39309,39310,39338,39339,39459,39460,39519,39520,39548,39549,39577,39578,1990],{},"当 ",[18,39311,39313,39326],{"className":39312,"translate":22},[21],[18,39314,39316],{"className":39315},[26],[28,39317,39318],{"xmlns":30},[32,39319,39320,39324],{},[35,39321,39322],{},[41,39323,39058],{},[86,39325,39058],{"encoding":88},[18,39327,39329],{"className":39328,"ariaHidden":51},[93],[18,39330,39332,39335],{"className":39331},[97],[18,39333],{"className":39334,"style":1083},[101],[18,39336,39058],{"className":39337},[106,110]," 足够小时，",[18,39340,39342,39374],{"className":39341,"translate":22},[21],[18,39343,39345],{"className":39344},[26],[28,39346,39347],{"xmlns":30},[32,39348,39349,39371],{},[35,39350,39351,39357,39359,39361,39363,39365,39367,39369],{},[38,39352,39353,39355],{},[41,39354,1055],{},[41,39356,38925],{},[49,39358,921],{},[41,39360,37808],{"mathvariant":316},[49,39362,74],{"stretchy":73},[41,39364,20979],{},[49,39366,52],{"separator":51},[41,39368,12579],{},[49,39370,84],{"stretchy":73},[86,39372,39373],{"encoding":88},"f_S\\in\\Sigma(\\beta,L)",[18,39375,39377,39432],{"className":39376,"ariaHidden":51},[93],[18,39378,39380,39383,39423,39426,39429],{"className":39379},[97],[18,39381],{"className":39382,"style":2890},[101],[18,39384,39386,39389],{"className":39385},[106],[18,39387,1055],{"className":39388,"style":37396},[106,110],[18,39390,39392],{"className":39391},[115],[18,39393,39395,39415],{"className":39394},[119,120],[18,39396,39398,39412],{"className":39397},[124],[18,39399,39401],{"className":39400,"style":1152},[128],[18,39402,39403,39406],{"style":39101},[18,39404],{"className":39405,"style":137},[136],[18,39407,39409],{"className":39408},[141,142,143,144],[18,39410,38925],{"className":39411,"style":38958},[106,110,144],[18,39413,152],{"className":39414},[151],[18,39416,39418],{"className":39417},[124],[18,39419,39421],{"className":39420,"style":159},[128],[18,39422],{},[18,39424],{"className":39425,"style":228},[169],[18,39427,921],{"className":39428},[232],[18,39430],{"className":39431,"style":228},[169],[18,39433,39435,39438,39441,39444,39447,39450,39453,39456],{"className":39434},[97],[18,39436],{"className":39437,"style":242},[101],[18,39439,37808],{"className":39440},[106],[18,39442,74],{"className":39443},[250],[18,39445,20979],{"className":39446,"style":21031},[106,110],[18,39448,52],{"className":39449},[165],[18,39451],{"className":39452,"style":170},[169],[18,39454,12579],{"className":39455},[106,110],[18,39457,84],{"className":39458},[267],"：鼓包的高度是 ",[18,39461,39463,39481],{"className":39462,"translate":22},[21],[18,39464,39466],{"className":39465},[26],[28,39467,39468],{"xmlns":30},[32,39469,39470,39478],{},[35,39471,39472],{},[1556,39473,39474,39476],{},[41,39475,18323],{},[41,39477,20979],{},[86,39479,39480],{"encoding":88},"h^\\beta",[18,39482,39484],{"className":39483,"ariaHidden":51},[93],[18,39485,39487,39490],{"className":39486},[97],[18,39488],{"className":39489,"style":35177},[101],[18,39491,39493,39496],{"className":39492},[106],[18,39494,18323],{"className":39495},[106,110],[18,39497,39499],{"className":39498},[115],[18,39500,39502],{"className":39501},[119],[18,39503,39505],{"className":39504},[124],[18,39506,39508],{"className":39507,"style":35177},[128],[18,39509,39510,39513],{"style":1962},[18,39511],{"className":39512,"style":137},[136],[18,39514,39516],{"className":39515},[141,142,143,144],[18,39517,20979],{"className":39518,"style":21031},[106,110,144],"、宽度是 ",[18,39521,39523,39536],{"className":39522,"translate":22},[21],[18,39524,39526],{"className":39525},[26],[28,39527,39528],{"xmlns":30},[32,39529,39530,39534],{},[35,39531,39532],{},[41,39533,18323],{},[86,39535,18323],{"encoding":88},[18,39537,39539],{"className":39538,"ariaHidden":51},[93],[18,39540,39542,39545],{"className":39541},[97],[18,39543],{"className":39544,"style":295},[101],[18,39546,18323],{"className":39547},[106,110],"，所以 ",[18,39550,39552,39565],{"className":39551,"translate":22},[21],[18,39553,39555],{"className":39554},[26],[28,39556,39557],{"xmlns":30},[32,39558,39559,39563],{},[35,39560,39561],{},[41,39562,20979],{},[86,39564,37780],{"encoding":88},[18,39566,39568],{"className":39567,"ariaHidden":51},[93],[18,39569,39571,39574],{"className":39570},[97],[18,39572],{"className":39573,"style":2890},[101],[18,39575,20979],{"className":39576,"style":21031},[106,110]," 阶导数的量级是 ",[18,39579,39581,39611],{"className":39580,"translate":22},[21],[18,39582,39584],{"className":39583},[26],[28,39585,39586],{"xmlns":30},[32,39587,39588,39608],{},[35,39589,39590,39596,39598,39604,39606],{},[1556,39591,39592,39594],{},[41,39593,18323],{},[41,39595,20979],{},[41,39597,317],{"mathvariant":316},[1556,39599,39600,39602],{},[41,39601,18323],{},[41,39603,20979],{},[49,39605,433],{},[45,39607,47],{},[86,39609,39610],{"encoding":88},"h^\\beta\u002Fh^\\beta=1",[18,39612,39614,39690],{"className":39613,"ariaHidden":51},[93],[18,39615,39617,39620,39649,39652,39681,39684,39687],{"className":39616},[97],[18,39618],{"className":39619,"style":35143},[101],[18,39621,39623,39626],{"className":39622},[106],[18,39624,18323],{"className":39625},[106,110],[18,39627,39629],{"className":39628},[115],[18,39630,39632],{"className":39631},[119],[18,39633,39635],{"className":39634},[124],[18,39636,39638],{"className":39637,"style":35177},[128],[18,39639,39640,39643],{"style":1962},[18,39641],{"className":39642,"style":137},[136],[18,39644,39646],{"className":39645},[141,142,143,144],[18,39647,20979],{"className":39648,"style":21031},[106,110,144],[18,39650,317],{"className":39651},[106],[18,39653,39655,39658],{"className":39654},[106],[18,39656,18323],{"className":39657},[106,110],[18,39659,39661],{"className":39660},[115],[18,39662,39664],{"className":39663},[119],[18,39665,39667],{"className":39666},[124],[18,39668,39670],{"className":39669,"style":35177},[128],[18,39671,39672,39675],{"style":1962},[18,39673],{"className":39674,"style":137},[136],[18,39676,39678],{"className":39677},[141,142,143,144],[18,39679,20979],{"className":39680,"style":21031},[106,110,144],[18,39682],{"className":39683,"style":228},[169],[18,39685,433],{"className":39686},[232],[18,39688],{"className":39689,"style":228},[169],[18,39691,39693,39696],{"className":39692},[97],[18,39694],{"className":39695,"style":505},[101],[18,39697,47],{"className":39698},[106],[11,39700,39701,39702,39863,39864,39923,39924,10360],{},"两个函数 ",[18,39703,39705,39735],{"className":39704,"translate":22},[21],[18,39706,39708],{"className":39707},[26],[28,39709,39710],{"xmlns":30},[32,39711,39712,39732],{},[35,39713,39714,39720,39722],{},[38,39715,39716,39718],{},[41,39717,1055],{},[41,39719,38925],{},[49,39721,52],{"separator":51},[38,39723,39724,39726],{},[41,39725,1055],{},[1556,39727,39728,39730],{},[41,39729,38925],{},[49,39731,32620],{"mathvariant":316,"lspace":32619,"rspace":32619},[86,39733,39734],{"encoding":88},"f_S,f_{S'}",[18,39736,39738],{"className":39737,"ariaHidden":51},[93],[18,39739,39741,39744,39784,39787,39790],{"className":39740},[97],[18,39742],{"className":39743,"style":2890},[101],[18,39745,39747,39750],{"className":39746},[106],[18,39748,1055],{"className":39749,"style":37396},[106,110],[18,39751,39753],{"className":39752},[115],[18,39754,39756,39776],{"className":39755},[119,120],[18,39757,39759,39773],{"className":39758},[124],[18,39760,39762],{"className":39761,"style":1152},[128],[18,39763,39764,39767],{"style":39101},[18,39765],{"className":39766,"style":137},[136],[18,39768,39770],{"className":39769},[141,142,143,144],[18,39771,38925],{"className":39772,"style":38958},[106,110,144],[18,39774,152],{"className":39775},[151],[18,39777,39779],{"className":39778},[124],[18,39780,39782],{"className":39781,"style":159},[128],[18,39783],{},[18,39785,52],{"className":39786},[165],[18,39788],{"className":39789,"style":170},[169],[18,39791,39793,39796],{"className":39792},[106],[18,39794,1055],{"className":39795,"style":37396},[106,110],[18,39797,39799],{"className":39798},[115],[18,39800,39802,39855],{"className":39801},[119,120],[18,39803,39805,39852],{"className":39804},[124],[18,39806,39808],{"className":39807,"style":1152},[128],[18,39809,39810,39813],{"style":39101},[18,39811],{"className":39812,"style":137},[136],[18,39814,39816],{"className":39815},[141,142,143,144],[18,39817,39819],{"className":39818},[106,144],[18,39820,39822,39825],{"className":39821},[106,144],[18,39823,38925],{"className":39824,"style":38958},[106,110,144],[18,39826,39828],{"className":39827},[115],[18,39829,39831],{"className":39830},[119],[18,39832,39834],{"className":39833},[124],[18,39835,39838],{"className":39836,"style":39837},[128],"height:0.6828em;",[18,39839,39840,39843],{"style":33586},[18,39841],{"className":39842,"style":1191},[136],[18,39844,39846],{"className":39845},[141,1195,1196,144],[18,39847,39849],{"className":39848},[106,144],[18,39850,32620],{"className":39851},[106,144],[18,39853,152],{"className":39854},[151],[18,39856,39858],{"className":39857},[124],[18,39859,39861],{"className":39860,"style":159},[128],[18,39862],{}," 的 ",[18,39865,39867,39885],{"className":39866,"translate":22},[21],[18,39868,39870],{"className":39869},[26],[28,39871,39872],{"xmlns":30},[32,39873,39874,39882],{},[35,39875,39876],{},[1556,39877,39878,39880],{},[41,39879,12579],{},[45,39881,2266],{},[86,39883,39884],{"encoding":88},"L^2",[18,39886,39888],{"className":39887,"ariaHidden":51},[93],[18,39889,39891,39894],{"className":39890},[97],[18,39892],{"className":39893,"style":2416},[101],[18,39895,39897,39900],{"className":39896},[106],[18,39898,12579],{"className":39899},[106,110],[18,39901,39903],{"className":39902},[115],[18,39904,39906],{"className":39905},[119],[18,39907,39909],{"className":39908},[124],[18,39910,39912],{"className":39911,"style":2416},[128],[18,39913,39914,39917],{"style":1962},[18,39915],{"className":39916,"style":137},[136],[18,39918,39920],{"className":39919},[141,142,143,144],[18,39921,2266],{"className":39922},[106,144]," 距离只取决于它们不同的那些鼓包。若 ",[18,39925,39927,39959],{"className":39926,"translate":22},[21],[18,39928,39930],{"className":39929},[26],[28,39931,39932],{"xmlns":30},[32,39933,39934,39956],{},[35,39935,39936,39938,39940,39943,39949,39951,39953],{},[41,39937,11222],{"mathvariant":316},[41,39939,38925],{},[41,39941,39942],{"mathvariant":316},"△",[1556,39944,39945,39947],{},[41,39946,38925],{},[49,39948,32620],{"mathvariant":316,"lspace":32619,"rspace":32619},[41,39950,11222],{"mathvariant":316},[49,39952,433],{},[41,39954,39955],{},"k",[86,39957,39958],{"encoding":88},"|S\\triangle 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个码字（",[18,40938,40940,40962],{"className":40939,"translate":22},[21],[18,40941,40943],{"className":40942},[26],[28,40944,40945],{"xmlns":30},[32,40946,40947,40959],{},[35,40948,40949,40955,40957],{},[38,40950,40951,40953],{},[41,40952,39058],{},[45,40954,430],{},[49,40956,6882],{},[45,40958,430],{},[86,40960,40961],{"encoding":88},"c_0>0",[18,40963,40965,41021],{"className":40964,"ariaHidden":51},[93],[18,40966,40968,40972,41012,41015,41018],{"className":40967},[97],[18,40969],{"className":40970,"style":40971},[101],"height:0.6891em;vertical-align:-0.15em;",[18,40973,40975,40978],{"className":40974},[106],[18,40976,39058],{"className":40977},[106,110],[18,40979,40981],{"className":40980},[115],[18,40982,40984,41004],{"className":40983},[119,120],[18,40985,40987,41001],{"className":40986},[124],[18,40988,40990],{"className":40989,"style":129},[128],[18,40991,40992,40995],{"style":1155},[18,40993],{"className":40994,"style":137},[136],[18,40996,40998],{"className":40997},[141,142,143,144],[18,40999,430],{"className":41000},[106,144],[18,41002,152],{"className":41003},[151],[18,41005,41007],{"className":41006},[124],[18,41008,41010],{"className":41009,"style":159},[128],[18,41011],{},[18,41013],{"className":41014,"style":228},[169],[18,41016,6882],{"className":41017},[232],[18,41019],{"className":41020,"style":228},[169],[18,41022,41024,41027],{"className":41023},[97],[18,41025],{"className":41026,"style":505},[101],[18,41028,430],{"className":41029},[106]," 是常数），任意两个码字至少在 ",[18,41032,41034,41052],{"className":41033,"translate":22},[21],[18,41035,41037],{"className":41036},[26],[28,41038,41039],{"xmlns":30},[32,41040,41041,41049],{},[35,41042,41043,41045,41047],{},[41,41044,28160],{},[41,41046,317],{"mathvariant":316},[45,41048,10444],{},[86,41050,41051],{"encoding":88},"m\u002F4",[18,41053,41055],{"className":41054,"ariaHidden":51},[93],[18,41056,41058,41061,41064],{"className":41057},[97],[18,41059],{"className":41060,"style":242},[101],[18,41062,28160],{"className":41063},[106,110],[18,41065,17125],{"className":41066},[106]," 个位置不同。于是",[18,41069,41071],{"className":41070,"translate":22},[875],[18,41072,41074,41138],{"className":41073,"translate":22},[21],[18,41075,41077],{"className":41076},[26],[28,41078,41079],{"xmlns":30,"display":884},[32,41080,41081,41135],{},[35,41082,41083,41085,41091,41093,41103,41115,41117,41123,41133],{},[49,41084,2248],{"stretchy":73},[38,41086,41087,41089],{},[41,41088,1055],{},[41,41090,38925],{},[49,41092,2253],{},[38,41094,41095,41097],{},[41,41096,1055],{},[1556,41098,41099,41101],{},[41,41100,38925],{},[49,41102,32620],{"mathvariant":316,"lspace":32619,"rspace":32619},[1589,41104,41105,41107,41113],{},[49,41106,2248],{"stretchy":73},[1556,41108,41109,41111],{},[41,41110,12579],{},[45,41112,2266],{},[45,41114,2266],{},[49,41116,4775],{},[38,41118,41119,41121],{},[41,41120,39058],{},[45,41122,47],{},[1556,41124,41125,41127],{},[41,41126,18323],{},[35,41128,41129,41131],{},[45,41130,2266],{},[41,41132,20979],{},[49,41134,52],{"separator":51},[86,41136,41137],{"encoding":88},"\\lVert f_S-f_{S'}\\rVert_{L^2}^2\n\\ge c_1h^{2\\beta},",[18,41139,41141,41199,41366],{"className":41140,"ariaHidden":51},[93],[18,41142,41144,41147,41150,41190,41193,41196],{"className":41143},[97],[18,41145],{"className":41146,"style":242},[101],[18,41148,2248],{"className":41149},[250],[18,41151,41153,41156],{"className":41152},[106],[18,41154,1055],{"className":41155,"style":37396},[106,110],[18,41157,41159],{"className":41158},[115],[18,41160,41162,41182],{"className":41161},[119,120],[18,41163,41165,41179],{"className":41164},[124],[18,41166,41168],{"className":41167,"style":1152},[128],[18,41169,41170,41173],{"style":39101},[18,41171],{"className":41172,"style":137},[136],[18,41174,41176],{"className":41175},[141,142,143,144],[18,41177,38925],{"className":41178,"style":38958},[106,110,144],[18,41180,152],{"className":41181},[151],[18,41183,41185],{"className":41184},[124],[18,41186,41188],{"className":41187,"style":159},[128],[18,41189],{},[18,41191],{"className":41192,"style":2068},[169],[18,41194,2253],{"className":41195},[2072],[18,41197],{"className":41198,"style":2068},[169],[18,41200,41202,41205,41277,41357,41360,41363],{"className":41201},[97],[18,41203],{"className":41204,"style":40231},[101],[18,41206,41208,41211],{"className":41207},[106],[18,41209,1055],{"className":41210,"style":37396},[106,110],[18,41212,41214],{"className":41213},[115],[18,41215,41217,41269],{"className":41216},[119,120],[18,41218,41220,41266],{"className":41219},[124],[18,41221,41223],{"className":41222,"style":1152},[128],[18,41224,41225,41228],{"style":39101},[18,41226],{"className":41227,"style":137},[136],[18,41229,41231],{"className":41230},[141,142,143,144],[18,41232,41234],{"className":41233},[106,144],[18,41235,41237,41240],{"className":41236},[106,144],[18,41238,38925],{"className":41239,"style":38958},[106,110,144],[18,41241,41243],{"className":41242},[115],[18,41244,41246],{"className":41245},[119],[18,41247,41249],{"className":41248},[124],[18,41250,41252],{"className":41251,"style":39837},[128],[18,41253,41254,41257],{"style":33586},[18,41255],{"className":41256,"style":1191},[136],[18,41258,41260],{"className":41259},[141,1195,1196,144],[18,41261,41263],{"className":41262},[106,144],[18,41264,32620],{"className":41265},[106,144],[18,41267,152],{"className":41268},[151],[18,41270,41272],{"className":41271},[124],[18,41273,41275],{"className":41274,"style":159},[128],[18,41276],{},[18,41278,41280,41283],{"className":41279},[267],[18,41281,2248],{"className":41282},[267],[18,41284,41286],{"className":41285},[115],[18,41287,41289,41349],{"className":41288},[119,120],[18,41290,41292,41346],{"className":41291},[124],[18,41293,41295,41335],{"className":41294,"style":10772},[128],[18,41296,41297,41300],{"style":40324},[18,41298],{"className":41299,"style":137},[136],[18,41301,41303],{"className":41302},[141,142,143,144],[18,41304,41306],{"className":41305},[106,144],[18,41307,41309,41312],{"className":41308},[106,144],[18,41310,12579],{"className":41311},[106,110,144],[18,41313,41315],{"className":41314},[115],[18,41316,41318],{"className":41317},[119],[18,41319,41321],{"className":41320},[124],[18,41322,41324],{"className":41323,"style":38014},[128],[18,41325,41326,41329],{"style":33586},[18,41327],{"className":41328,"style":1191},[136],[18,41330,41332],{"className":41331},[141,1195,1196,144],[18,41333,2266],{"className":41334},[106,144],[18,41336,41337,41340],{"style":1637},[18,41338],{"className":41339,"style":137},[136],[18,41341,41343],{"className":41342},[141,142,143,144],[18,41344,2266],{"className":41345},[106,144],[18,41347,152],{"className":41348},[151],[18,41350,41352],{"className":41351},[124],[18,41353,41355],{"className":41354,"style":40383},[128],[18,41356],{},[18,41358],{"className":41359,"style":228},[169],[18,41361,4775],{"className":41362},[232],[18,41364],{"className":41365,"style":228},[169],[18,41367,41369,41372,41412,41447],{"className":41368},[97],[18,41370],{"className":41371,"style":34501},[101],[18,41373,41375,41378],{"className":41374},[106],[18,41376,39058],{"className":41377},[106,110],[18,41379,41381],{"className":41380},[115],[18,41382,41384,41404],{"className":41383},[119,120],[18,41385,41387,41401],{"className":41386},[124],[18,41388,41390],{"className":41389,"style":129},[128],[18,41391,41392,41395],{"style":1155},[18,41393],{"className":41394,"style":137},[136],[18,41396,41398],{"className":41397},[141,142,143,144],[18,41399,47],{"className":41400},[106,144],[18,41402,152],{"className":41403},[151],[18,41405,41407],{"className":41406},[124],[18,41408,41410],{"className":41409,"style":159},[128],[18,41411],{},[18,41413,41415,41418],{"className":41414},[106],[18,41416,18323],{"className":41417},[106,110],[18,41419,41421],{"className":41420},[115],[18,41422,41424],{"className":41423},[119],[18,41425,41427],{"className":41426},[124],[18,41428,41430],{"className":41429,"style":16670},[128],[18,41431,41432,41435],{"style":1637},[18,41433],{"className":41434,"style":137},[136],[18,41436,41438],{"className":41437},[141,142,143,144],[18,41439,41441,41444],{"className":41440},[106,144],[18,41442,2266],{"className":41443},[106,144],[18,41445,20979],{"className":41446,"style":21031},[106,110,144],[18,41448,52],{"className":41449},[165],[11,41451,41452,41453,1990],{},"即平方间隔 ",[18,41454,41456,41488],{"className":41455,"translate":22},[21],[18,41457,41459],{"className":41458},[26],[28,41460,41461],{"xmlns":30},[32,41462,41463,41485],{},[35,41464,41465,41467,41469,41475],{},[41,41466,532],{"mathvariant":316},[49,41468,433],{},[38,41470,41471,41473],{},[41,41472,39058],{},[45,41474,47],{},[1556,41476,41477,41479],{},[41,41478,18323],{},[35,41480,41481,41483],{},[45,41482,2266],{},[41,41484,20979],{},[86,41486,41487],{"encoding":88},"\\Delta=c_1h^{2\\beta}",[18,41489,41491,41509],{"className":41490,"ariaHidden":51},[93],[18,41492,41494,41497,41500,41503,41506],{"className":41493},[97],[18,41495],{"className":41496,"style":600},[101],[18,41498,532],{"className":41499},[106],[18,41501],{"className":41502,"style":228},[169],[18,41504,433],{"className":41505},[232],[18,41507],{"className":41508,"style":228},[169],[18,41510,41512,41516,41556],{"className":41511},[97],[18,41513],{"className":41514,"style":41515},[101],"height:0.9991em;vertical-align:-0.15em;",[18,41517,41519,41522],{"className":41518},[106],[18,41520,39058],{"className":41521},[106,110],[18,41523,41525],{"className":41524},[115],[18,41526,41528,41548],{"className":41527},[119,120],[18,41529,41531,41545],{"className":41530},[124],[18,41532,41534],{"className":41533,"style":129},[128],[18,41535,41536,41539],{"style":1155},[18,41537],{"className":41538,"style":137},[136],[18,41540,41542],{"className":41541},[141,142,143,144],[18,41543,47],{"className":41544},[106,144],[18,41546,152],{"className":41547},[151],[18,41549,41551],{"className":41550},[124],[18,41552,41554],{"className":41553,"style":159},[128],[18,41555],{},[18,41557,41559,41562],{"className":41558},[106],[18,41560,18323],{"className":41561},[106,110],[18,41563,41565],{"className":41564},[115],[18,41566,41568],{"className":41567},[119],[18,41569,41571],{"className":41570},[124],[18,41572,41574],{"className":41573,"style":35177},[128],[18,41575,41576,41579],{"style":1962},[18,41577],{"className":41578,"style":137},[136],[18,41580,41582],{"className":41581},[141,142,143,144],[18,41583,41585,41588],{"className":41584},[106,144],[18,41586,2266],{"className":41587},[106,144],[18,41589,20979],{"className":41590,"style":21031},[106,110,144],[11,41592,41593],{},"另一方面，两个回归分布的 KL 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h^{2\\beta}",[18,42586,42588,42614],{"className":42587,"ariaHidden":51},[93],[18,42589,42591,42594,42599,42602,42605,42608,42611],{"className":42590},[97],[18,42592],{"className":42593,"style":2890},[101],[18,42595,15171,42597],{"className":42596},[985],[18,42598,15174],{"style":6352},[18,42600],{"className":42601,"style":170},[169],[18,42603,3526],{"className":42604,"style":246},[106,110],[18,42606],{"className":42607,"style":228},[169],[18,42609,26826],{"className":42610},[232],[18,42612],{"className":42613,"style":228},[169],[18,42615,42617,42620,42623],{"className":42616},[97],[18,42618],{"className":42619,"style":35177},[101],[18,42621,64],{"className":42622},[106,110],[18,42624,42626,42629],{"className":42625},[106],[18,42627,18323],{"className":42628},[106,110],[18,42630,42632],{"className":42631},[115],[18,42633,42635],{"className":42634},[119],[18,42636,42638],{"className":42637},[124],[18,42639,42641],{"className":42640,"style":35177},[128],[18,42642,42643,42646],{"style":1962},[18,42644],{"className":42645,"style":137},[136],[18,42647,42649],{"className":42648},[141,142,143,144],[18,42650,42652,42655],{"className":42651},[106,144],[18,42653,2266],{"className":42654},[106,144],[18,42656,20979],{"className":42657,"style":21031},[106,110,144],"，即",[18,42660,42662],{"className":42661,"translate":22},[875],[18,42663,42665,42738],{"className":42664,"translate":22},[21],[18,42666,42668],{"className":42667},[26],[28,42669,42670],{"xmlns":30,"display":884},[32,42671,42672,42735],{},[35,42673,42674,42684,42686,42688,42698,42701,42704,42706,42708,42711,42733],{},[1556,42675,42676,42678],{},[41,42677,18323],{},[35,42679,42680,42682],{},[49,42681,2253],{},[41,42683,4031],{},[49,42685,26891],{},[41,42687,64],{},[1556,42689,42690,42692],{},[41,42691,18323],{},[35,42693,42694,42696],{},[45,42695,2266],{},[41,42697,20979],{},[169,42699],{"width":42700},"1em",[49,42702,42703],{},"⟺",[169,42705],{"width":42700},[41,42707,18323],{},[49,42709,42710],{},"≲",[1556,42712,42713,42715],{},[41,42714,64],{},[35,42716,42717,42719],{},[49,42718,2253],{},[5651,42720,42721,42723],{},[45,42722,47],{},[35,42724,42725,42727,42729,42731],{},[45,42726,2266],{},[41,42728,20979],{},[49,42730,7984],{},[41,42732,4031],{},[41,42734,968],{"mathvariant":316},[86,42736,42737],{"encoding":88},"h^{-d}\\gtrsim n h^{2\\beta}\n\\quad\\Longleftrightarrow\\quad\nh\\lesssim n^{-\\frac{1}{2\\beta+d}}.",[18,42739,42741,42792,42853,42872],{"className":42740,"ariaHidden":51},[93],[18,42742,42744,42748,42783,42786,42789],{"className":42743},[97],[18,42745],{"className":42746,"style":42747},[101],"height:1.1287em;vertical-align:-0.2296em;",[18,42749,42751,42754],{"className":42750},[106],[18,42752,18323],{"className":42753},[106,110],[18,42755,42757],{"className":42756},[115],[18,42758,42760],{"className":42759},[119],[18,42761,42763],{"className":42762},[124],[18,42764,42766],{"className":42765,"style":16670},[128],[18,42767,42768,42771],{"style":1637},[18,42769],{"className":42770,"style":137},[136],[18,42772,42774],{"className":42773},[141,142,143,144],[18,42775,42777,42780],{"className":42776},[106,144],[18,42778,2253],{"className":42779},[106,144],[18,42781,4031],{"className":42782},[106,110,144],[18,42784],{"className":42785,"style":228},[169],[18,42787,26891],{"className":42788},[232,27253],[18,42790],{"className":42791,"style":228},[169],[18,42793,42795,42799,42802,42837,42841,42844,42847,42850],{"className":42794},[97],[18,42796],{"className":42797,"style":42798},[101],"height:0.9231em;vertical-align:-0.024em;",[18,42800,64],{"className":42801},[106,110],[18,42803,42805,42808],{"className":42804},[106],[18,42806,18323],{"className":42807},[106,110],[18,42809,42811],{"className":42810},[115],[18,42812,42814],{"className":42813},[119],[18,42815,42817],{"className":42816},[124],[18,42818,42820],{"className":42819,"style":16670},[128],[18,42821,42822,42825],{"style":1637},[18,42823],{"className":42824,"style":137},[136],[18,42826,42828],{"className":42827},[141,142,143,144],[18,42829,42831,42834],{"className":42830},[106,144],[18,42832,2266],{"className":42833},[106,144],[18,42835,20979],{"className":42836,"style":21031},[106,110,144],[18,42838],{"className":42839,"style":42840},[169],"margin-right:1em;",[18,42842],{"className":42843,"style":228},[169],[18,42845,42703],{"className":42846},[232],[18,42848],{"className":42849,"style":42840},[169],[18,42851],{"className":42852,"style":228},[169],[18,42854,42856,42860,42863,42866,42869],{"className":42855},[97],[18,42857],{"className":42858,"style":42859},[101],"height:0.9592em;vertical-align:-0.2296em;",[18,42861,18323],{"className":42862},[106,110],[18,42864],{"className":42865,"style":228},[169],[18,42867,42710],{"className":42868},[232,27253],[18,42870],{"className":42871,"style":228},[169],[18,42873,42875,42879,42990],{"className":42874},[97],[18,42876],{"className":42877,"style":42878},[101],"height:1.0368em;",[18,42880,42882,42885],{"className":42881},[106],[18,42883,64],{"className":42884},[106,110],[18,42886,42888],{"className":42887},[115],[18,42889,42891],{"className":42890},[119],[18,42892,42894],{"className":42893},[124],[18,42895,42897],{"className":42896,"style":42878},[128],[18,42898,42899,42902],{"style":38128},[18,42900],{"className":42901,"style":697},[136],[18,42903,42905],{"className":42904},[141,142,143,144],[18,42906,42908,42911],{"className":42907},[106,144],[18,42909,2253],{"className":42910},[106,144],[18,42912,42914,42917,42987],{"className":42913},[106,144],[18,42915],{"className":42916},[250,6009,141,1195,38147],[18,42918,42920],{"className":42919},[5651],[18,42921,42923,42979],{"className":42922},[119,120],[18,42924,42926,42976],{"className":42925},[124],[18,42927,42930,42953,42961],{"className":42928,"style":42929},[128],"height:0.8443em;",[18,42931,42932,42935],{"style":38163},[18,42933],{"className":42934,"style":697},[136],[18,42936,42938],{"className":42937},[141,1195,1196,144],[18,42939,42941,42944,42947,42950],{"className":42940},[106,144],[18,42942,2266],{"className":42943},[106,144],[18,42945,20979],{"className":42946,"style":21031},[106,110,144],[18,42948,7984],{"className":42949},[2072,144],[18,42951,4031],{"className":42952},[106,110,144],[18,42954,42955,42958],{"style":38187},[18,42956],{"className":42957,"style":697},[136],[18,42959],{"className":42960,"style":38194},[6044,144],[18,42962,42964,42967],{"style":42963},"top:-3.384em;",[18,42965],{"className":42966,"style":697},[136],[18,42968,42970],{"className":42969},[141,1195,1196,144],[18,42971,42973],{"className":42972},[106,144],[18,42974,47],{"className":42975},[106,144],[18,42977,152],{"className":42978},[151],[18,42980,42982],{"className":42981},[124],[18,42983,42985],{"className":42984,"style":38222},[128],[18,42986],{},[18,42988],{"className":42989},[267,6009,141,1195,38147],[18,42991,968],{"className":42992},[106],[11,42994,36183,42995,43118],{},[18,42996,42998,43038],{"className":42997,"translate":22},[21],[18,42999,43001],{"className":43000},[26],[28,43002,43003],{"xmlns":30},[32,43004,43005,43035],{},[35,43006,43007,43009,43011],{},[41,43008,18323],{},[49,43010,17143],{},[1556,43012,43013,43015],{},[41,43014,64],{},[35,43016,43017,43019,43021,43023,43025,43027,43029,43031,43033],{},[49,43018,2253],{},[45,43020,47],{},[41,43022,317],{"mathvariant":316},[49,43024,74],{"stretchy":73},[45,43026,2266],{},[41,43028,20979],{},[49,43030,7984],{},[41,43032,4031],{},[49,43034,84],{"stretchy":73},[86,43036,43037],{"encoding":88},"h\\asymp n^{-1\u002F(2\\beta+d)}",[18,43039,43041,43059],{"className":43040,"ariaHidden":51},[93],[18,43042,43044,43047,43050,43053,43056],{"className":43043},[97],[18,43045],{"className":43046,"style":295},[101],[18,43048,18323],{"className":43049},[106,110],[18,43051],{"className":43052,"style":228},[169],[18,43054,17143],{"className":43055},[232],[18,43057],{"className":43058,"style":228},[169],[18,43060,43062,43065],{"className":43061},[97],[18,43063],{"className":43064,"style":27938},[101],[18,43066,43068,43071],{"className":43067},[106],[18,43069,64],{"className":43070},[106,110],[18,43072,43074],{"className":43073},[115],[18,43075,43077],{"className":43076},[119],[18,43078,43080],{"className":43079},[124],[18,43081,43083],{"className":43082,"style":27938},[128],[18,43084,43085,43088],{"style":1962},[18,43086],{"className":43087,"style":137},[136],[18,43089,43091],{"className":43090},[141,142,143,144],[18,43092,43094,43097,43100,43103,43106,43109,43112,43115],{"className":43093},[106,144],[18,43095,2253],{"className":43096},[106,144],[18,43098,335],{"className":43099},[106,144],[18,43101,74],{"className":43102},[250,144],[18,43104,2266],{"className":43105},[106,144],[18,43107,20979],{"className":43108,"style":21031},[106,110,144],[18,43110,7984],{"className":43111},[2072,144],[18,43113,4031],{"className":43114},[106,110,144],[18,43116,84],{"className":43117},[267,144],"，下界就是",[18,43120,43122],{"className":43121,"translate":22},[875],[18,43123,43125,43185],{"className":43124,"translate":22},[21],[18,43126,43128],{"className":43127},[26],[28,43129,43130],{"xmlns":30,"display":884},[32,43131,43132,43182],{},[35,43133,43134,43140,43142,43152,43154,43180],{},[38,43135,43136,43138],{},[41,43137,3526],{"mathvariant":3525},[41,43139,64],{},[49,43141,26891],{},[1556,43143,43144,43146],{},[41,43145,18323],{},[35,43147,43148,43150],{},[45,43149,2266],{},[41,43151,20979],{},[49,43153,17143],{},[1556,43155,43156,43158],{},[41,43157,64],{},[35,43159,43160,43162],{},[49,43161,2253],{},[5651,43163,43164,43170],{},[35,43165,43166,43168],{},[45,43167,2266],{},[41,43169,20979],{},[35,43171,43172,43174,43176,43178],{},[45,43173,2266],{},[41,43175,20979],{},[49,43177,7984],{},[41,43179,4031],{},[49,43181,52],{"separator":51},[86,43183,43184],{"encoding":88},"\\mathfrak M_n\\gtrsim h^{2\\beta}\n\\asymp n^{-\\frac{2\\beta}{2\\beta+d}},",[18,43186,43188,43243,43293],{"className":43187,"ariaHidden":51},[93],[18,43189,43191,43194,43234,43237,43240],{"className":43190},[97],[18,43192],{"className":43193,"style":42859},[101],[18,43195,43197,43200],{"className":43196},[106],[18,43198,3526],{"className":43199},[106,3607],[18,43201,43203],{"className":43202},[115],[18,43204,43206,43226],{"className":43205},[119,120],[18,43207,43209,43223],{"className":43208},[124],[18,43210,43212],{"className":43211,"style":202},[128],[18,43213,43214,43217],{"style":1155},[18,43215],{"className":43216,"style":137},[136],[18,43218,43220],{"className":43219},[141,142,143,144],[18,43221,64],{"className":43222},[106,110,144],[18,43224,152],{"className":43225},[151],[18,43227,43229],{"className":43228},[124],[18,43230,43232],{"className":43231,"style":159},[128],[18,43233],{},[18,43235],{"className":43236,"style":228},[169],[18,43238,26891],{"className":43239},[232,27253],[18,43241],{"className":43242,"style":228},[169],[18,43244,43246,43249,43284,43287,43290],{"className":43245},[97],[18,43247],{"className":43248,"style":16670},[101],[18,43250,43252,43255],{"className":43251},[106],[18,43253,18323],{"className":43254},[106,110],[18,43256,43258],{"className":43257},[115],[18,43259,43261],{"className":43260},[119],[18,43262,43264],{"className":43263},[124],[18,43265,43267],{"className":43266,"style":16670},[128],[18,43268,43269,43272],{"style":1637},[18,43270],{"className":43271,"style":137},[136],[18,43273,43275],{"className":43274},[141,142,143,144],[18,43276,43278,43281],{"className":43277},[106,144],[18,43279,2266],{"className":43280},[106,144],[18,43282,20979],{"className":43283,"style":21031},[106,110,144],[18,43285],{"className":43286,"style":228},[169],[18,43288,17143],{"className":43289},[232],[18,43291],{"className":43292,"style":228},[169],[18,43294,43296,43300,43412],{"className":43295},[97],[18,43297],{"className":43298,"style":43299},[101],"height:1.3111em;vertical-align:-0.1944em;",[18,43301,43303,43306],{"className":43302},[106],[18,43304,64],{"className":43305},[106,110],[18,43307,43309],{"className":43308},[115],[18,43310,43312],{"className":43311},[119],[18,43313,43315],{"className":43314},[124],[18,43316,43318],{"className":43317,"style":38107},[128],[18,43319,43320,43323],{"style":38128},[18,43321],{"className":43322,"style":697},[136],[18,43324,43326],{"className":43325},[141,142,143,144],[18,43327,43329,43332],{"className":43328},[106,144],[18,43330,2253],{"className":43331},[106,144],[18,43333,43335,43338,43409],{"className":43334},[106,144],[18,43336],{"className":43337},[250,6009,141,1195,38147],[18,43339,43341],{"className":43340},[5651],[18,43342,43344,43401],{"className":43343},[119,120],[18,43345,43347,43398],{"className":43346},[124],[18,43348,43350,43373,43381],{"className":43349,"style":38160},[128],[18,43351,43352,43355],{"style":38163},[18,43353],{"className":43354,"style":697},[136],[18,43356,43358],{"className":43357},[141,1195,1196,144],[18,43359,43361,43364,43367,43370],{"className":43360},[106,144],[18,43362,2266],{"className":43363},[106,144],[18,43365,20979],{"className":43366,"style":21031},[106,110,144],[18,43368,7984],{"className":43369},[2072,144],[18,43371,4031],{"className":43372},[106,110,144],[18,43374,43375,43378],{"style":38187},[18,43376],{"className":43377,"style":697},[136],[18,43379],{"className":43380,"style":38194},[6044,144],[18,43382,43383,43386],{"style":38197},[18,43384],{"className":43385,"style":697},[136],[18,43387,43389],{"className":43388},[141,1195,1196,144],[18,43390,43392,43395],{"className":43391},[106,144],[18,43393,2266],{"className":43394},[106,144],[18,43396,20979],{"className":43397,"style":21031},[106,110,144],[18,43399,152],{"className":43400},[151],[18,43402,43404],{"className":43403},[124],[18,43405,43407],{"className":43406,"style":38222},[128],[18,43408],{},[18,43410],{"className":43411},[267,6009,141,1195,38147],[18,43413,52],{"className":43414},[165],[11,43416,43417],{},"与上界同阶。证毕。",[11,43419,43420,43421,43668,43669,43697],{},"这个构造值得反复看：",[717,43422,43423,43424,43452,43453,43522,43523,43597,43598,43667],{},"鼓包的宽度 ",[18,43425,43427,43440],{"className":43426,"translate":22},[21],[18,43428,43430],{"className":43429},[26],[28,43431,43432],{"xmlns":30},[32,43433,43434,43438],{},[35,43435,43436],{},[41,43437,18323],{},[86,43439,18323],{"encoding":88},[18,43441,43443],{"className":43442,"ariaHidden":51},[93],[18,43444,43446,43449],{"className":43445},[97],[18,43447],{"className":43448,"style":295},[101],[18,43450,18323],{"className":43451},[106,110]," 控制两件事——损失间隔（",[18,43454,43456,43478],{"className":43455,"translate":22},[21],[18,43457,43459],{"className":43458},[26],[28,43460,43461],{"xmlns":30},[32,43462,43463,43475],{},[35,43464,43465],{},[1556,43466,43467,43469],{},[41,43468,18323],{},[35,43470,43471,43473],{},[45,43472,2266],{},[41,43474,20979],{},[86,43476,43477],{"encoding":88},"h^{2\\beta}",[18,43479,43481],{"className":43480,"ariaHidden":51},[93],[18,43482,43484,43487],{"className":43483},[97],[18,43485],{"className":43486,"style":35177},[101],[18,43488,43490,43493],{"className":43489},[106],[18,43491,18323],{"className":43492},[106,110],[18,43494,43496],{"className":43495},[115],[18,43497,43499],{"className":43498},[119],[18,43500,43502],{"className":43501},[124],[18,43503,43505],{"className":43504,"style":35177},[128],[18,43506,43507,43510],{"style":1962},[18,43508],{"className":43509,"style":137},[136],[18,43511,43513],{"className":43512},[141,142,143,144],[18,43514,43516,43519],{"className":43515},[106,144],[18,43517,2266],{"className":43518},[106,144],[18,43520,20979],{"className":43521,"style":21031},[106,110,144],"）和 KL（",[18,43524,43526,43550],{"className":43525,"translate":22},[21],[18,43527,43529],{"className":43528},[26],[28,43530,43531],{"xmlns":30},[32,43532,43533,43547],{},[35,43534,43535,43537],{},[41,43536,64],{},[1556,43538,43539,43541],{},[41,43540,18323],{},[35,43542,43543,43545],{},[45,43544,2266],{},[41,43546,20979],{},[86,43548,43549],{"encoding":88},"n h^{2\\beta}",[18,43551,43553],{"className":43552,"ariaHidden":51},[93],[18,43554,43556,43559,43562],{"className":43555},[97],[18,43557],{"className":43558,"style":35177},[101],[18,43560,64],{"className":43561},[106,110],[18,43563,43565,43568],{"className":43564},[106],[18,43566,18323],{"className":43567},[106,110],[18,43569,43571],{"className":43570},[115],[18,43572,43574],{"className":43573},[119],[18,43575,43577],{"className":43576},[124],[18,43578,43580],{"className":43579,"style":35177},[128],[18,43581,43582,43585],{"style":1962},[18,43583],{"className":43584,"style":137},[136],[18,43586,43588],{"className":43587},[141,142,143,144],[18,43589,43591,43594],{"className":43590},[106,144],[18,43592,2266],{"className":43593},[106,144],[18,43595,20979],{"className":43596,"style":21031},[106,110,144],"），而鼓包的个数 ",[18,43599,43601,43623],{"className":43600,"translate":22},[21],[18,43602,43604],{"className":43603},[26],[28,43605,43606],{"xmlns":30},[32,43607,43608,43620],{},[35,43609,43610],{},[1556,43611,43612,43614],{},[41,43613,18323],{},[35,43615,43616,43618],{},[49,43617,2253],{},[41,43619,4031],{},[86,43621,43622],{"encoding":88},"h^{-d}",[18,43624,43626],{"className":43625,"ariaHidden":51},[93],[18,43627,43629,43632],{"className":43628},[97],[18,43630],{"className":43631,"style":35177},[101],[18,43633,43635,43638],{"className":43634},[106],[18,43636,18323],{"className":43637},[106,110],[18,43639,43641],{"className":43640},[115],[18,43642,43644],{"className":43643},[119],[18,43645,43647],{"className":43646},[124],[18,43648,43650],{"className":43649,"style":35177},[128],[18,43651,43652,43655],{"style":1962},[18,43653],{"className":43654,"style":137},[136],[18,43656,43658],{"className":43657},[141,142,143,144],[18,43659,43661,43664],{"className":43660},[106,144],[18,43662,2253],{"className":43663},[106,144],[18,43665,4031],{"className":43666},[106,110,144]," 控制假设的个数。"," 三者平衡，就得到非参数速率。维度 ",[18,43670,43672,43685],{"className":43671,"translate":22},[21],[18,43673,43675],{"className":43674},[26],[28,43676,43677],{"xmlns":30},[32,43678,43679,43683],{},[35,43680,43681],{},[41,43682,4031],{},[86,43684,4031],{"encoding":88},[18,43686,43688],{"className":43687,"ariaHidden":51},[93],[18,43689,43691,43694],{"className":43690},[97],[18,43692],{"className":43693,"style":295},[101],[18,43695,4031],{"className":43696},[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」。",[1311,43699,43701],{"id":43700},"_63-一些经典速率","6.3 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维高斯均值",[43727,43759,43760],{},[18,43761,43763,43789],{"className":43762,"translate":22},[21],[18,43764,43766],{"className":43765},[26],[28,43767,43768],{"xmlns":30},[32,43769,43770,43786],{},[35,43771,43772,43774,43776,43782,43784],{},[49,43773,2248],{"stretchy":73},[41,43775,77],{},[38,43777,43778,43780],{},[49,43779,2248],{"stretchy":73},[45,43781,2266],{},[49,43783,4334],{},[41,43785,3071],{},[86,43787,43788],{"encoding":88},"\\lVert\\theta\\rVert_2\\le 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d",[18,44355,44357,44418,44436],{"className":44356,"ariaHidden":51},[93],[18,44358,44360,44363,44366,44369,44409,44412,44415],{"className":44359},[97],[18,44361],{"className":44362,"style":242},[101],[18,44364,2248],{"className":44365},[250],[18,44367,77],{"className":44368,"style":254},[106,110],[18,44370,44372,44375],{"className":44371},[267],[18,44373,2248],{"className":44374},[267],[18,44376,44378],{"className":44377},[115],[18,44379,44381,44401],{"className":44380},[119,120],[18,44382,44384,44398],{"className":44383},[124],[18,44385,44387],{"className":44386,"style":129},[128],[18,44388,44389,44392],{"style":1155},[18,44390],{"className":44391,"style":137},[136],[18,44393,44395],{"className":44394},[141,142,143,144],[18,44396,430],{"className":44397},[106,144],[18,44399,152],{"className":44400},[151],[18,44402,44404],{"className":44403},[124],[18,44405,44407],{"className":44406,"style":159},[128],[18,44408],{},[18,44410],{"className":44411,"style":228},[169],[18,44413,4334],{"className":44414},[232],[18,44416],{"className":44417,"style":228},[169],[18,44419,44421,44424,44427,44430,44433],{"className":44420},[97],[18,44422],{"className":44423,"style":30921},[101],[18,44425,44302],{"className":44426},[106,110],[18,44428],{"className":44429,"style":228},[169],[18,44431,44348],{"className":44432},[232],[18,44434],{"className":44435,"style":228},[169],[18,44437,44439,44442],{"className":44438},[97],[18,44440],{"className":44441,"style":295},[101],[18,44443,4031],{"className":44444},[106,110],[43727,44446,44447],{},[18,44448,44450,44490],{"className":44449,"translate":22},[21],[18,44451,44453],{"className":44452},[26],[28,44454,44455],{"xmlns":30},[32,44456,44457,44487],{},[35,44458,44459,44475,44477,44479],{},[14678,44460,44461],{"scriptlevel":430,"displaystyle":51},[5651,44462,44463,44473],{},[35,44464,44465,44471],{},[1556,44466,44467,44469],{},[41,44468,16403],{},[45,44470,2266],{},[41,44472,44302],{},[41,44474,64],{},[41,44476,14730],{},[49,44478,899],{},[14678,44480,44481],{"scriptlevel":430,"displaystyle":51},[5651,44482,44483,44485],{},[41,44484,4031],{},[41,44486,44302],{},[86,44488,44489],{"encoding":88},"\\dfrac{\\sigma^2s}{n}\\log\\dfrac ds",[18,44491,44493],{"className":44492,"ariaHidden":51},[93],[18,44494,44496,44499,44590,44593,44598,44601],{"className":44495},[97],[18,44497],{"className":44498,"style":34941},[101],[18,44500,44502,44505,44587],{"className":44501},[106],[18,44503],{"className":44504},[250,6009],[18,44506,44508],{"className":44507},[5651],[18,44509,44511,44579],{"className":44510},[119,120],[18,44512,44514,44576],{"className":44513},[124],[18,44515,44517,44528,44536],{"className":44516,"style":15996},[128],[18,44518,44519,44522],{"style":6025},[18,44520],{"className":44521,"style":697},[136],[18,44523,44525],{"className":44524},[106],[18,44526,64],{"className":44527},[106,110],[18,44529,44530,44533],{"style":6037},[18,44531],{"className":44532,"style":697},[136],[18,44534],{"className":44535,"style":6045},[6044],[18,44537,44538,44541],{"style":6048},[18,44539],{"className":44540,"style":697},[136],[18,44542,44544,44573],{"className":44543},[106],[18,44545,44547,44550],{"className":44546},[106],[18,44548,16403],{"className":44549,"style":3957},[106,110],[18,44551,44553],{"className":44552},[115],[18,44554,44556],{"className":44555},[119],[18,44557,44559],{"className":44558},[124],[18,44560,44562],{"className":44561,"style":2416},[128],[18,44563,44564,44567],{"style":1962},[18,44565],{"className":44566,"style":137},[136],[18,44568,44570],{"className":44569},[141,142,143,144],[18,44571,2266],{"className":44572},[106,144],[18,44574,44302],{"className":44575},[106,110],[18,44577,152],{"className":44578},[151],[18,44580,44582],{"className":44581},[124],[18,44583,44585],{"className":44584,"style":6067},[128],[18,44586],{},[18,44588],{"className":44589},[267,6009],[18,44591],{"className":44592,"style":170},[169],[18,44594,15171,44596],{"className":44595},[985],[18,44597,15174],{"style":6352},[18,44599],{"className":44600,"style":170},[169],[18,44602,44604,44607,44660],{"className":44603},[106],[18,44605],{"className":44606},[250,6009],[18,44608,44610],{"className":44609},[5651],[18,44611,44613,44652],{"className":44612},[119,120],[18,44614,44616,44649],{"className":44615},[124],[18,44617,44619,44630,44638],{"className":44618,"style":6022},[128],[18,44620,44621,44624],{"style":6025},[18,44622],{"className":44623,"style":697},[136],[18,44625,44627],{"className":44626},[106],[18,44628,44302],{"className":44629},[106,110],[18,44631,44632,44635],{"style":6037},[18,44633],{"className":44634,"style":697},[136],[18,44636],{"className":44637,"style":6045},[6044],[18,44639,44640,44643],{"style":6048},[18,44641],{"className":44642,"style":697},[136],[18,44644,44646],{"className":44645},[106],[18,44647,4031],{"className":44648},[106,110],[18,44650,152],{"className":44651},[151],[18,44653,44655],{"className":44654},[124],[18,44656,44658],{"className":44657,"style":6067},[128],[18,44659],{},[18,44661],{"className":44662},[267,6009],[43709,44664,44665,44668,44671],{},[43727,44666,44667],{},"一维单调回归",[43727,44669,44670],{},"单调、有界",[43727,44672,44673],{},[18,44674,44676,44703],{"className":44675,"translate":22},[21],[18,44677,44679],{"className":44678},[26],[28,44680,44681],{"xmlns":30},[32,44682,44683,44700],{},[35,44684,44685],{},[1556,44686,44687,44689],{},[41,44688,64],{},[35,44690,44691,44693,44695,44697],{},[49,44692,2253],{},[45,44694,2266],{},[41,44696,317],{"mathvariant":316},[45,44698,44699],{},"3",[86,44701,44702],{"encoding":88},"n^{-2\u002F3}",[18,44704,44706],{"className":44705,"ariaHidden":51},[93],[18,44707,44709,44712],{"className":44708},[97],[18,44710],{"className":44711,"style":27938},[101],[18,44713,44715,44718],{"className":44714},[106],[18,44716,64],{"className":44717},[106,110],[18,44719,44721],{"className":44720},[115],[18,44722,44724],{"className":44723},[119],[18,44725,44727],{"className":44726},[124],[18,44728,44730],{"className":44729,"style":27938},[128],[18,44731,44732,44735],{"style":1962},[18,44733],{"className":44734,"style":137},[136],[18,44736,44738],{"className":44737},[141,142,143,144],[18,44739,44741,44744],{"className":44740},[106,144],[18,44742,2253],{"className":44743},[106,144],[18,44745,44747],{"className":44746},[106,144],"2\u002F3",[11,44749,44750],{},"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。",[638,44752,44754],{"id":44753},"七三种方法的比较","七、三种方法的比较",[43703,44756,44757,44776],{},[43706,44758,44759],{},[43709,44760,44761,44764,44767,44770,44773],{},[43712,44762,44763],{},"方法",[43712,44765,44766],{},"假设个数",[43712,44768,44769],{},"距离",[43712,44771,44772],{},"优势",[43712,44774,44775],{},"局限",[43722,44777,44778,44822,44866],{},[43709,44779,44780,44783,44813,44816,44819],{},[43727,44781,44782],{},"Le Cam",[43727,44784,44785],{},[18,44786,44788,44801],{"className":44787,"translate":22},[21],[18,44789,44791],{"className":44790},[26],[28,44792,44793],{"xmlns":30},[32,44794,44795,44799],{},[35,44796,44797],{},[45,44798,2266],{},[86,44800,2266],{"encoding":88},[18,44802,44804],{"className":44803,"ariaHidden":51},[93],[18,44805,44807,44810],{"className":44806},[97],[18,44808],{"className":44809,"style":505},[101],[18,44811,2266],{"className":44812},[106],[43727,44814,44815],{},"TV \u002F KL",[43727,44817,44818],{},"简单，常数清楚",[43727,44820,44821],{},"只探测一个方向，高维问题会松",[43709,44823,44824,44827,44857,44860,44863],{},[43727,44825,44826],{},"Fano",[43727,44828,44829],{},[18,44830,44832,44845],{"className":44831,"translate":22},[21],[18,44833,44835],{"className":44834},[26],[28,44836,44837],{"xmlns":30},[32,44838,44839,44843],{},[35,44840,44841],{},[41,44842,3526],{},[86,44844,3526],{"encoding":88},[18,44846,44848],{"className":44847,"ariaHidden":51},[93],[18,44849,44851,44854],{"className":44850},[97],[18,44852],{"className":44853,"style":600},[101],[18,44855,3526],{"className":44856,"style":246},[106,110],[43727,44858,44859],{},"互信息 \u002F KL",[43727,44861,44862],{},"通用，适合高维与非参数",[43727,44864,44865],{},"需要构造打包，常数较松",[43709,44867,44868,44871,44932,44935,44938],{},[43727,44869,44870],{},"Assouad",[43727,44872,44873],{},[18,44874,44876,44894],{"className":44875,"translate":22},[21],[18,44877,44879],{"className":44878},[26],[28,44880,44881],{"xmlns":30},[32,44882,44883,44891],{},[35,44884,44885],{},[1556,44886,44887,44889],{},[45,44888,2266],{},[41,44890,28160],{},[86,44892,44893],{"encoding":88},"2^m",[18,44895,44897],{"className":44896,"ariaHidden":51},[93],[18,44898,44900,44903],{"className":44899},[97],[18,44901],{"className":44902,"style":1959},[101],[18,44904,44906,44909],{"className":44905},[106],[18,44907,2266],{"className":44908},[106],[18,44910,44912],{"className":44911},[115],[18,44913,44915],{"className":44914},[119],[18,44916,44918],{"className":44917},[124],[18,44919,44921],{"className":44920,"style":1959},[128],[18,44922,44923,44926],{"style":1962},[18,44924],{"className":44925,"style":137},[136],[18,44927,44929],{"className":44928},[141,142,143,144],[18,44930,28160],{"className":44931},[106,110,144],[43727,44933,44934],{},"逐坐标 TV",[43727,44936,44937],{},"乘积结构下最锋利",[43727,44939,44940],{},"只适用于超立方体结构",[11,44942,44943],{},"三者的关系可以这样理解：",[2210,44945,44946,44952,44987],{},[2213,44947,44948,44951],{},[717,44949,44950],{},"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。",[2213,44953,44954,44957,44958,44986],{},[717,44955,44956],{},"Fano 是 Le Cam 的多假设推广。"," 它把「两个点分不清」推广为「",[18,44959,44961,44974],{"className":44960,"translate":22},[21],[18,44962,44964],{"className":44963},[26],[28,44965,44966],{"xmlns":30},[32,44967,44968,44972],{},[35,44969,44970],{},[41,44971,3526],{},[86,44973,3526],{"encoding":88},[18,44975,44977],{"className":44976,"ariaHidden":51},[93],[18,44978,44980,44983],{"className":44979},[97],[18,44981],{"className":44982,"style":600},[101],[18,44984,3526],{"className":44985,"style":246},[106,110]," 个点平均分不清」。",[2213,44988,44989,44992,44993,45051,45052,45080],{},[717,44990,44991],{},"Assouad 是 Fano 在乘积结构上的精细化。"," 它不直接对 ",[18,44994,44996,45013],{"className":44995,"translate":22},[21],[18,44997,44999],{"className":44998},[26],[28,45000,45001],{"xmlns":30},[32,45002,45003,45011],{},[35,45004,45005],{},[1556,45006,45007,45009],{},[45,45008,2266],{},[41,45010,28160],{},[86,45012,44893],{"encoding":88},[18,45014,45016],{"className":45015,"ariaHidden":51},[93],[18,45017,45019,45022],{"className":45018},[97],[18,45020],{"className":45021,"style":1959},[101],[18,45023,45025,45028],{"className":45024},[106],[18,45026,2266],{"className":45027},[106],[18,45029,45031],{"className":45030},[115],[18,45032,45034],{"className":45033},[119],[18,45035,45037],{"className":45036},[124],[18,45038,45040],{"className":45039,"style":1959},[128],[18,45041,45042,45045],{"style":1962},[18,45043],{"className":45044,"style":137},[136],[18,45046,45048],{"className":45047},[141,142,143,144],[18,45049,28160],{"className":45050},[106,110,144]," 个假设做 Fano，而是逐坐标做 Le Cam，再把 ",[18,45053,45055,45068],{"className":45054,"translate":22},[21],[18,45056,45058],{"className":45057},[26],[28,45059,45060],{"xmlns":30},[32,45061,45062,45066],{},[35,45063,45064],{},[41,45065,28160],{},[86,45067,28160],{"encoding":88},[18,45069,45071],{"className":45070,"ariaHidden":51},[93],[18,45072,45074,45077],{"className":45073},[97],[18,45075],{"className":45076,"style":1083},[101],[18,45078,28160],{"className":45079},[106,110]," 个下界加起来。",[11,45082,45083],{},"实际使用时，选择哪种方法取决于参数空间的结构：",[2210,45085,45086,45089,45133],{},[2213,45087,45088],{},"一维、局部、两样本问题：Le Cam 通常足够，而且常数最干净。",[2213,45090,45091,45092,45132],{},"高维、非参数、需要 ",[18,45093,45095,45112],{"className":45094,"translate":22},[21],[18,45096,45098],{"className":45097},[26],[28,45099,45100],{"xmlns":30},[32,45101,45102,45110],{},[35,45103,45104,45106,45108],{},[41,45105,14730],{},[49,45107,899],{},[41,45109,3526],{},[86,45111,19446],{"encoding":88},[18,45113,45115],{"className":45114,"ariaHidden":51},[93],[18,45116,45118,45121,45126,45129],{"className":45117},[97],[18,45119],{"className":45120,"style":2890},[101],[18,45122,15171,45124],{"className":45123},[985],[18,45125,15174],{"style":6352},[18,45127],{"className":45128,"style":170},[169],[18,45130,3526],{"className":45131,"style":246},[106,110]," 的地方：Fano 是标准工具。",[2213,45134,45135,45136,45206,45207,45277],{},"乘积结构、",[18,45137,45139,45157],{"className":45138,"translate":22},[21],[18,45140,45142],{"className":45141},[26],[28,45143,45144],{"xmlns":30},[32,45145,45146,45154],{},[35,45147,45148],{},[38,45149,45150,45152],{},[41,45151,948],{"mathvariant":316},[45,45153,2266],{},[86,45155,45156],{"encoding":88},"\\ell_2",[18,45158,45160],{"className":45159,"ariaHidden":51},[93],[18,45161,45163,45166],{"className":45162},[97],[18,45164],{"className":45165,"style":448},[101],[18,45167,45169,45172],{"className":45168},[106],[18,45170,948],{"className":45171},[106],[18,45173,45175],{"className":45174},[115],[18,45176,45178,45198],{"className":45177},[119,120],[18,45179,45181,45195],{"className":45180},[124],[18,45182,45184],{"className":45183,"style":129},[128],[18,45185,45186,45189],{"style":1155},[18,45187],{"className":45188,"style":137},[136],[18,45190,45192],{"className":45191},[141,142,143,144],[18,45193,2266],{"className":45194},[106,144],[18,45196,152],{"className":45197},[151],[18,45199,45201],{"className":45200},[124],[18,45202,45204],{"className":45203,"style":159},[128],[18,45205],{}," 或 ",[18,45208,45210,45228],{"className":45209,"translate":22},[21],[18,45211,45213],{"className":45212},[26],[28,45214,45215],{"xmlns":30},[32,45216,45217,45225],{},[35,45218,45219],{},[38,45220,45221,45223],{},[41,45222,948],{"mathvariant":316},[41,45224,2028],{"mathvariant":316},[86,45226,45227],{"encoding":88},"\\ell_\\infty",[18,45229,45231],{"className":45230,"ariaHidden":51},[93],[18,45232,45234,45237],{"className":45233},[97],[18,45235],{"className":45236,"style":448},[101],[18,45238,45240,45243],{"className":45239},[106],[18,45241,948],{"className":45242},[106],[18,45244,45246],{"className":45245},[115],[18,45247,45249,45269],{"className":45248},[119,120],[18,45250,45252,45266],{"className":45251},[124],[18,45253,45255],{"className":45254,"style":202},[128],[18,45256,45257,45260],{"style":1155},[18,45258],{"className":45259,"style":137},[136],[18,45261,45263],{"className":45262},[141,142,143,144],[18,45264,2028],{"className":45265},[106,144],[18,45267,152],{"className":45268},[151],[18,45270,45272],{"className":45271},[124],[18,45273,45275],{"className":45274,"style":159},[128],[18,45276],{}," 速率：Assouad 最方便。",[638,45279,45280],{"id":45280},"结语",[11,45282,45283,45284,45319,45320,45348],{},"回到开头的问题：为什么 ",[18,45285,45287,45304],{"className":45286,"translate":22},[21],[18,45288,45290],{"className":45289},[26],[28,45291,45292],{"xmlns":30},[32,45293,45294,45302],{},[35,45295,45296,45298,45300],{},[45,45297,47],{},[41,45299,317],{"mathvariant":316},[41,45301,64],{},[86,45303,322],{"encoding":88},[18,45305,45307],{"className":45306,"ariaHidden":51},[93],[18,45308,45310,45313,45316],{"className":45309},[97],[18,45311],{"className":45312,"style":242},[101],[18,45314,335],{"className":45315},[106],[18,45317,64],{"className":45318},[106,110]," 不能再小？因为两个足够接近的分布，在 ",[18,45321,45323,45336],{"className":45322,"translate":22},[21],[18,45324,45326],{"className":45325},[26],[28,45327,45328],{"xmlns":30},[32,45329,45330,45334],{},[35,45331,45332],{},[41,45333,64],{},[86,45335,64],{"encoding":88},[18,45337,45339],{"className":45338,"ariaHidden":51},[93],[18,45340,45342,45345],{"className":45341},[97],[18,45343],{"className":45344,"style":1083},[101],[18,45346,64],{"className":45347},[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。",[11,45350,45351,45352,45380,45381,45409,45410,45438],{},"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「",[18,45353,45355,45368],{"className":45354,"translate":22},[21],[18,45356,45358],{"className":45357},[26],[28,45359,45360],{"xmlns":30},[32,45361,45362,45366],{},[35,45363,45364],{},[41,45365,3526],{},[86,45367,3526],{"encoding":88},[18,45369,45371],{"className":45370,"ariaHidden":51},[93],[18,45372,45374,45377],{"className":45373},[97],[18,45375],{"className":45376,"style":600},[101],[18,45378,3526],{"className":45379,"style":246},[106,110]," 个点」，用互信息度量整体的不可区分性；Assouad 则把 ",[18,45382,45384,45397],{"className":45383,"translate":22},[21],[18,45385,45387],{"className":45386},[26],[28,45388,45389],{"xmlns":30},[32,45390,45391,45395],{},[35,45392,45393],{},[41,45394,28160],{},[86,45396,28160],{"encoding":88},[18,45398,45400],{"className":45399,"ariaHidden":51},[93],[18,45401,45403,45406],{"className":45402},[97],[18,45404],{"className":45405,"style":1083},[101],[18,45407,28160],{"className":45408},[106,110]," 维问题拆成 ",[18,45411,45413,45426],{"className":45412,"translate":22},[21],[18,45414,45416],{"className":45415},[26],[28,45417,45418],{"xmlns":30},[32,45419,45420,45424],{},[35,45421,45422],{},[41,45423,28160],{},[86,45425,28160],{"encoding":88},[18,45427,45429],{"className":45428,"ariaHidden":51},[93],[18,45430,45432,45435],{"className":45431},[97],[18,45433],{"className":45434,"style":1083},[101],[18,45436,28160],{"className":45437},[106,110]," 个一维问题，逐坐标地累加下界。",[11,45440,45441],{},"如果只记三句话：",[45443,45444,45445,45448,45539],"ol",{},[2213,45446,45447],{},"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。",[2213,45449,45450,45451,45479,45480,45538],{},"Le Cam 用两个点，Fano 用 ",[18,45452,45454,45467],{"className":45453,"translate":22},[21],[18,45455,45457],{"className":45456},[26],[28,45458,45459],{"xmlns":30},[32,45460,45461,45465],{},[35,45462,45463],{},[41,45464,3526],{},[86,45466,3526],{"encoding":88},[18,45468,45470],{"className":45469,"ariaHidden":51},[93],[18,45471,45473,45476],{"className":45472},[97],[18,45474],{"className":45475,"style":600},[101],[18,45477,3526],{"className":45478,"style":246},[106,110]," 个点，Assouad 用 ",[18,45481,45483,45500],{"className":45482,"translate":22},[21],[18,45484,45486],{"className":45485},[26],[28,45487,45488],{"xmlns":30},[32,45489,45490,45498],{},[35,45491,45492],{},[1556,45493,45494,45496],{},[45,45495,2266],{},[41,45497,28160],{},[86,45499,44893],{"encoding":88},[18,45501,45503],{"className":45502,"ariaHidden":51},[93],[18,45504,45506,45509],{"className":45505},[97],[18,45507],{"className":45508,"style":1959},[101],[18,45510,45512,45515],{"className":45511},[106],[18,45513,2266],{"className":45514},[106],[18,45516,45518],{"className":45517},[115],[18,45519,45521],{"className":45520},[119],[18,45522,45524],{"className":45523},[124],[18,45525,45527],{"className":45526,"style":1959},[128],[18,45528,45529,45532],{"style":1962},[18,45530],{"className":45531,"style":137},[136],[18,45533,45535],{"className":45534},[141,142,143,144],[18,45536,28160],{"className":45537},[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」。",[2213,45540,45541],{},"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":45543,"searchDepth":45544,"depth":45544,"links":45545},"",2,[45546,45547,45553,45559,45564,45568,45575,45576],{"id":640,"depth":45544,"text":641},{"id":1308,"depth":45544,"text":1309,"children":45548},[45549,45551,45552],{"id":1313,"depth":45550,"text":1314},3,{"id":3893,"depth":45550,"text":3894},{"id":4499,"depth":45550,"text":4500},{"id":5149,"depth":45544,"text":5150,"children":45554},[45555,45556,45557,45558],{"id":5153,"depth":45550,"text":5154},{"id":12516,"depth":45550,"text":12517},{"id":14411,"depth":45550,"text":14412},{"id":16322,"depth":45550,"text":16323},{"id":17465,"depth":45544,"text":17466,"children":45560},[45561,45562,45563],{"id":17469,"depth":45550,"text":17470},{"id":19540,"depth":45550,"text":19541},{"id":26777,"depth":45550,"text":26778},{"id":28087,"depth":45544,"text":28088,"children":45565},[45566,45567],{"id":28091,"depth":45550,"text":28092},{"id":32336,"depth":45550,"text":32337},{"id":34107,"depth":45544,"text":34108,"children":45569},[45570,45572,45574],{"id":34111,"depth":45550,"text":45571},"6.1 高斯均值：σ2d\u002Fn\\sigma^2d\u002Fnσ2d\u002Fn",{"id":37098,"depth":45550,"text":45573},"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":43700,"depth":45550,"text":43701},{"id":44753,"depth":45544,"text":44754},{"id":45280,"depth":45544,"text":45280},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。",[45587,45588],"mathematics","machine-learning","2025-09-27T16:28:39+08:00","UddcaRIuRTyzw3VagGFZOj0rS3lFxJ-QDtX0NVJjl5Q",{"id":45592,"title":45593,"body":45594,"description":45598,"draft":45577,"enableCodeHighlight":45577,"enableCodeHightlight":45577,"enableComment":45578,"enableD2":45578,"enableKatex":45578,"extension":45579,"image":45543,"important":45577,"location":45580,"meta":81678,"navigation":45578,"ogImage":45580,"path":81679,"seo":81680,"stem":81681,"summary":81682,"tags":81683,"time":81684,"__hash__":81685},"blog\u002Fblog\u002F2025\u002F2025-10-05-vapnik-chervonenkis-dimension.md","Vapnik-Chervonenkis Dimension 简论",{"type":8,"value":45595,"toc":81646},[45596,45599,45805,46065,46068,46071,46075,46079,46407,46758,46789,46829,47539,47546,47787,47791,48007,50040,50170,50202,50206,50341,50727,50763,50955,50987,50991,50995,51254,51642,51944,52147,52589,52746,52749,52753,52815,53035,53260,53797,53968,54029,54218,55209,55716,55799,56028,56384,56657,56899,57017,57131,57419,57749,58044,58427,58663,59010,59013,59079,59197,59513,59728,60037,60166,60535,60596,61074,61408,62152,62647,63151,63154,63214,63346,63349,63352,64203,64872,64876,64882,64892,64956,64963,64967,64971,64974,65325,65574,66158,66162,66560,67023,67228,67459,67823,68779,68782,69445,69448,69452,69506,69843,69907,71257,71633,71640,71644,71648,71655,71778,72547,72550,72732,73966,74250,74253,74562,74683,74687,74740,75151,75360,75539,75744,75841,75845,76048,76054,76058,76090,76318,76321,76519,76555,76558,76562,76565,76729,76791,76797,76868,77063,77070,77082,77086,77090,77282,78340,78463,78586,78590,78593,78596,78950,78990,78993,78997,79000,79005,79037,79309,79525,79557,79643,80238,80299,80422,80428,80464,80469,80558,80561,80564,80567,80572,80575,80639,80645,80648,80652,80659,80662,80725,80729,80798,80801,81631,81638,81640],[11,45597,45598],{},"先看一个只有 1 个参数的分类器：",[18,45600,45602],{"className":45601,"translate":22},[875],[18,45603,45605,45665],{"className":45604,"translate":22},[21],[18,45606,45608],{"className":45607},[26],[28,45609,45610],{"xmlns":30,"display":884},[32,45611,45612,45662],{},[35,45613,45614,45620,45622,45624,45626,45628,45631,45633,45635,45638,45640,45642,45644,45646,45648,45650,45652,45654,45656,45658,45660],{},[38,45615,45616,45618],{},[41,45617,18323],{},[41,45619,77],{},[49,45621,74],{"stretchy":73},[41,45623,38573],{},[49,45625,84],{"stretchy":73},[49,45627,433],{},[41,45629,45630],{"mathvariant":316},"sign",[49,45632,899],{},[49,45634,74],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[41,45636,45637],{},"sin",[49,45639,899],{},[49,45641,74],{"stretchy":73},[41,45643,77],{},[41,45645,38573],{},[49,45647,84],{"stretchy":73},[49,45649,84],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,45651,52],{"separator":51},[169,45653],{"width":7341},[41,45655,77],{},[49,45657,921],{},[41,45659,3071],{"mathvariant":933},[41,45661,968],{"mathvariant":316},[86,45663,45664],{"encoding":88},"h_\\theta(x)=\\operatorname{sign}\\bigl(\\sin(\\theta x)\\bigr),\\qquad \\theta\\in\\mathbb R.",[18,45666,45668,45732,45792],{"className":45667,"ariaHidden":51},[93],[18,45669,45671,45674,45714,45717,45720,45723,45726,45729],{"className":45670},[97],[18,45672],{"className":45673,"style":242},[101],[18,45675,45677,45680],{"className":45676},[106],[18,45678,18323],{"className":45679},[106,110],[18,45681,45683],{"className":45682},[115],[18,45684,45686,45706],{"className":45685},[119,120],[18,45687,45689,45703],{"className":45688},[124],[18,45690,45692],{"className":45691,"style":1383},[128],[18,45693,45694,45697],{"style":1155},[18,45695],{"className":45696,"style":137},[136],[18,45698,45700],{"className":45699},[141,142,143,144],[18,45701,77],{"className":45702,"style":254},[106,110,144],[18,45704,152],{"className":45705},[151],[18,45707,45709],{"className":45708},[124],[18,45710,45712],{"className":45711,"style":159},[128],[18,45713],{},[18,45715,74],{"className":45716},[250],[18,45718,38573],{"className":45719},[106,110],[18,45721,84],{"className":45722},[267],[18,45724],{"className":45725,"style":228},[169],[18,45727,433],{"className":45728},[232],[18,45730],{"className":45731,"style":228},[169],[18,45733,45735,45738,45744,45750,45753,45756,45759,45762,45765,45771,45774,45777,45780,45783,45786,45789],{"className":45734},[97],[18,45736],{"className":45737,"style":6098},[101],[18,45739,45741],{"className":45740},[985],[18,45742,45630],{"className":45743},[106,6351],[18,45745,45747],{"className":45746},[250],[18,45748,74],{"className":45749},[1230,1196],[18,45751,45637],{"className":45752},[985],[18,45754,74],{"className":45755},[250],[18,45757,77],{"className":45758,"style":254},[106,110],[18,45760,38573],{"className":45761},[106,110],[18,45763,84],{"className":45764},[267],[18,45766,45768],{"className":45767},[267],[18,45769,84],{"className":45770},[1230,1196],[18,45772,52],{"className":45773},[165],[18,45775],{"className":45776,"style":7673},[169],[18,45778],{"className":45779,"style":170},[169],[18,45781,77],{"className":45782,"style":254},[106,110],[18,45784],{"className":45785,"style":228},[169],[18,45787,921],{"className":45788},[232],[18,45790],{"className":45791,"style":228},[169],[18,45793,45795,45799,45802],{"className":45794},[97],[18,45796],{"className":45797,"style":45798},[101],"height:0.6889em;",[18,45800,3071],{"className":45801},[106,1139],[18,45803,968],{"className":45804},[106],[11,45806,45807,45808,46001,46002,46030,46031,46064],{},"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 ",[18,45809,45811,45861],{"className":45810,"translate":22},[21],[18,45812,45814],{"className":45813},[26],[28,45815,45816],{"xmlns":30},[32,45817,45818,45858],{},[35,45819,45820,45830,45832,45842,45844,45846,45848],{},[1556,45821,45822,45824],{},[45,45823,2266],{},[35,45825,45826,45828],{},[49,45827,2253],{},[45,45829,47],{},[49,45831,52],{"separator":51},[1556,45833,45834,45836],{},[45,45835,2266],{},[35,45837,45838,45840],{},[49,45839,2253],{},[45,45841,2266],{},[49,45843,52],{"separator":51},[49,45845,55],{},[49,45847,52],{"separator":51},[1556,45849,45850,45852],{},[45,45851,2266],{},[35,45853,45854,45856],{},[49,45855,2253],{},[41,45857,28160],{},[86,45859,45860],{"encoding":88},"2^{-1},2^{-2},\\dots,2^{-m}",[18,45862,45864],{"className":45863,"ariaHidden":51},[93],[18,45865,45867,45871,45906,45909,45912,45947,45950,45953,45956,45959,45962,45965],{"className":45866},[97],[18,45868],{"className":45869,"style":45870},[101],"height:1.0085em;vertical-align:-0.1944em;",[18,45872,45874,45877],{"className":45873},[106],[18,45875,2266],{"className":45876},[106],[18,45878,45880],{"className":45879},[115],[18,45881,45883],{"className":45882},[119],[18,45884,45886],{"className":45885},[124],[18,45887,45889],{"className":45888,"style":2416},[128],[18,45890,45891,45894],{"style":1962},[18,45892],{"className":45893,"style":137},[136],[18,45895,45897],{"className":45896},[141,142,143,144],[18,45898,45900,45903],{"className":45899},[106,144],[18,45901,2253],{"className":45902},[106,144],[18,45904,47],{"className":45905},[106,144],[18,45907,52],{"className":45908},[165],[18,45910],{"className":45911,"style":170},[169],[18,45913,45915,45918],{"className":45914},[106],[18,45916,2266],{"className":45917},[106],[18,45919,45921],{"className":45920},[115],[18,45922,45924],{"className":45923},[119],[18,45925,45927],{"className":45926},[124],[18,45928,45930],{"className":45929,"style":2416},[128],[18,45931,45932,45935],{"style":1962},[18,45933],{"className":45934,"style":137},[136],[18,45936,45938],{"className":45937},[141,142,143,144],[18,45939,45941,45944],{"className":45940},[106,144],[18,45942,2253],{"className":45943},[106,144],[18,45945,2266],{"className":45946},[106,144],[18,45948,52],{"className":45949},[165],[18,45951],{"className":45952,"style":170},[169],[18,45954,55],{"className":45955},[174],[18,45957],{"className":45958,"style":170},[169],[18,45960,52],{"className":45961},[165],[18,45963],{"className":45964,"style":170},[169],[18,45966,45968,45971],{"className":45967},[106],[18,45969,2266],{"className":45970},[106],[18,45972,45974],{"className":45973},[115],[18,45975,45977],{"className":45976},[119],[18,45978,45980],{"className":45979},[124],[18,45981,45984],{"className":45982,"style":45983},[128],"height:0.7713em;",[18,45985,45986,45989],{"style":1962},[18,45987],{"className":45988,"style":137},[136],[18,45990,45992],{"className":45991},[141,142,143,144],[18,45993,45995,45998],{"className":45994},[106,144],[18,45996,2253],{"className":45997},[106,144],[18,45999,28160],{"className":46000},[106,110,144]," 上，就总存在一个 ",[18,46003,46005,46018],{"className":46004,"translate":22},[21],[18,46006,46008],{"className":46007},[26],[28,46009,46010],{"xmlns":30},[32,46011,46012,46016],{},[35,46013,46014],{},[41,46015,77],{},[86,46017,285],{"encoding":88},[18,46019,46021],{"className":46020,"ariaHidden":51},[93],[18,46022,46024,46027],{"className":46023},[97],[18,46025],{"className":46026,"style":295},[101],[18,46028,77],{"className":46029,"style":254},[106,110],"，把任意指定的 ",[18,46032,46034,46052],{"className":46033,"translate":22},[21],[18,46035,46037],{"className":46036},[26],[28,46038,46039],{"xmlns":30},[32,46040,46041,46049],{},[35,46042,46043,46045,46047],{},[45,46044,430],{},[41,46046,317],{"mathvariant":316},[45,46048,47],{},[86,46050,46051],{"encoding":88},"0\u002F1",[18,46053,46055],{"className":46054,"ariaHidden":51},[93],[18,46056,46058,46061],{"className":46057},[97],[18,46059],{"className":46060,"style":242},[101],[18,46062,46051],{"className":46063},[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。",[11,46066,46067],{},"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。",[11,46069,46070],{},"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。",[638,46072,46074],{"id":46073},"一学习问题","一、学习问题",[1311,46076,46078],{"id":46077},"_11-设定","1.1 设定",[11,46080,46081,46082,46110,46111,46187,46188,46191,46192,46288,46289,1994,46292,46322,46323,46376,46377,46406],{},"设输入空间为 ",[18,46083,46085,46098],{"className":46084,"translate":22},[21],[18,46086,46088],{"className":46087},[26],[28,46089,46090],{"xmlns":30},[32,46091,46092,46096],{},[35,46093,46094],{},[41,46095,43],{"mathvariant":1462},[86,46097,1465],{"encoding":88},[18,46099,46101],{"className":46100,"ariaHidden":51},[93],[18,46102,46104,46107],{"className":46103},[97],[18,46105],{"className":46106,"style":600},[101],[18,46108,43],{"className":46109,"style":1479},[106,1478],"，标签空间为 ",[18,46112,46114,46140],{"className":46113,"translate":22},[21],[18,46115,46117],{"className":46116},[26],[28,46118,46119],{"xmlns":30},[32,46120,46121,46137],{},[35,46122,46123,46125,46127,46129,46131,46133,46135],{},[41,46124,37229],{"mathvariant":1462},[49,46126,433],{},[49,46128,1331],{"stretchy":73},[45,46130,430],{},[49,46132,52],{"separator":51},[45,46134,47],{},[49,46136,1349],{"stretchy":73},[86,46138,46139],{"encoding":88},"\\mathcal Y=\\{0,1\\}",[18,46141,46143,46163],{"className":46142,"ariaHidden":51},[93],[18,46144,46146,46150,46154,46157,46160],{"className":46145},[97],[18,46147],{"className":46148,"style":46149},[101],"height:0.7805em;vertical-align:-0.0972em;",[18,46151,37229],{"className":46152,"style":46153},[106,1478],"margin-right:0.0822em;",[18,46155],{"className":46156,"style":228},[169],[18,46158,433],{"className":46159},[232],[18,46161],{"className":46162,"style":228},[169],[18,46164,46166,46169,46172,46175,46178,46181,46184],{"className":46165},[97],[18,46167],{"className":46168,"style":242},[101],[18,46170,1331],{"className":46171},[250],[18,46173,430],{"className":46174},[106],[18,46176,52],{"className":46177},[165],[18,46179],{"className":46180,"style":170},[169],[18,46182,47],{"className":46183},[106],[18,46185,1349],{"className":46186},[267],"。一个",[717,46189,46190],{},"假设","是函数 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H}L_S(h).",[18,47607,47609,47664],{"className":47608,"ariaHidden":51},[93],[18,47610,47612,47615,47655,47658,47661],{"className":47611},[97],[18,47613],{"className":47614,"style":448},[101],[18,47616,47618,47621],{"className":47617},[106],[18,47619,18323],{"className":47620},[106,110],[18,47622,47624],{"className":47623},[115],[18,47625,47627,47647],{"className":47626},[119,120],[18,47628,47630,47644],{"className":47629},[124],[18,47631,47633],{"className":47632,"style":1152},[128],[18,47634,47635,47638],{"style":1155},[18,47636],{"className":47637,"style":137},[136],[18,47639,47641],{"className":47640},[141,142,143,144],[18,47642,38925],{"className":47643,"style":38958},[106,110,144],[18,47645,152],{"className":47646},[151],[18,47648,47650],{"className":47649},[124],[18,47651,47653],{"className":47652,"style":159},[128],[18,47654],{},[18,47656],{"className":47657,"style":228},[169],[18,47659,921],{"className":47660},[232],[18,47662],{"className":47663,"style":228},[169],[18,47665,47667,47671,47676,47679,47732,47735,47775,47778,47781,47784],{"className":47666},[97],[18,47668],{"className":47669,"style":47670},[101],"height:1.5295em;vertical-align:-0.7795em;",[18,47672,21635,47674],{"className":47673},[985],[18,47675,15174],{"style":6352},[18,47677],{"className":47678,"style":170},[169],[18,47680,47682],{"className":47681},[985,986],[18,47683,47685,47724],{"className":47684},[119,120],[18,47686,47688,47721],{"className":47687},[124],[18,47689,47691,47711],{"className":47690,"style":21653},[128],[18,47692,47693,47696],{"style":29361},[18,47694],{"className":47695,"style":697},[136],[18,47697,47699],{"className":47698},[141,142,143,144],[18,47700,47702,47705,47708],{"className":47701},[106,144],[18,47703,18323],{"className":47704},[106,110,144],[18,47706,921],{"className":47707},[232,144],[18,47709,18304],{"className":47710,"style":46321},[106,1478,144],[18,47712,47713,47716],{"style":693},[18,47714],{"className":47715,"style":697},[136],[18,47717,47718],{},[18,47719,11949],{"className":47720},[985],[18,47722,152],{"className":47723},[151],[18,47725,47727],{"className":47726},[124],[18,47728,47730],{"className":47729,"style":29399},[128],[18,47731],{},[18,47733],{"className":47734,"style":170},[169],[18,47736,47738,47741],{"className":47737},[106],[18,47739,12579],{"className":47740},[106,110],[18,47742,47744],{"className":47743},[115],[18,47745,47747,47767],{"className":47746},[119,120],[18,47748,47750,47764],{"className":47749},[124],[18,47751,47753],{"className":47752,"style":1152},[128],[18,47754,47755,47758],{"style":1155},[18,47756],{"className":47757,"style":137},[136],[18,47759,47761],{"className":47760},[141,142,143,144],[18,47762,38925],{"className":47763,"style":38958},[106,110,144],[18,47765,152],{"className":47766},[151],[18,47768,47770],{"className":47769},[124],[18,47771,47773],{"className":47772,"style":159},[128],[18,47774],{},[18,47776,74],{"className":47777},[250],[18,47779,18323],{"className":47780},[106,110],[18,47782,84],{"className":47783},[267],[18,47785,968],{"className":47786},[106],[1311,47788,47790],{"id":47789},"_12-误差分解","1.2 误差分解",[11,47792,47793,47794,38311,47877,48006],{},"对任意 ",[18,47795,47797,47820],{"className":47796,"translate":22},[21],[18,47798,47800],{"className":47799},[26],[28,47801,47802],{"xmlns":30},[32,47803,47804,47817],{},[35,47805,47806,47813,47815],{},[1556,47807,47808,47810],{},[41,47809,18323],{},[49,47811,47812],{},"∗",[49,47814,921],{},[41,47816,18304],{"mathvariant":1462},[86,47818,47819],{"encoding":88},"h^*\\in\\mathcal H",[18,47821,47823,47868],{"className":47822,"ariaHidden":51},[93],[18,47824,47826,47829,47859,47862,47865],{"className":47825},[97],[18,47827],{"className":47828,"style":1422},[101],[18,47830,47832,47835],{"className":47831},[106],[18,47833,18323],{"className":47834},[106,110],[18,47836,47838],{"className":47837},[115],[18,47839,47841],{"className":47840},[119],[18,47842,47844],{"className":47843},[124],[18,47845,47848],{"className":47846,"style":47847},[128],"height:0.6887em;",[18,47849,47850,47853],{"style":1962},[18,47851],{"className":47852,"style":137},[136],[18,47854,47856],{"className":47855},[141,142,143,144],[18,47857,47812],{"className":47858},[2072,144],[18,47860],{"className":47861,"style":228},[169],[18,47863,921],{"className":47864},[232],[18,47866],{"className":47867,"style":228},[169],[18,47869,47871,47874],{"className":47870},[97],[18,47872],{"className":47873,"style":600},[101],[18,47875,18304],{"className":47876,"style":46321},[106,1478],[18,47878,47880,47908],{"className":47879,"translate":22},[21],[18,47881,47883],{"className":47882},[26],[28,47884,47885],{"xmlns":30},[32,47886,47887,47905],{},[35,47888,47889,47895,47897,47903],{},[38,47890,47891,47893],{},[41,47892,12579],{},[41,47894,12572],{"mathvariant":1462},[49,47896,74],{"stretchy":73},[38,47898,47899,47901],{},[41,47900,18323],{},[41,47902,38925],{},[49,47904,84],{"stretchy":73},[86,47906,47907],{"encoding":88},"L_{\\mathcal D}(h_S)",[18,47909,47911],{"className":47910,"ariaHidden":51},[93],[18,47912,47914,47917,47960,47963,48003],{"className":47913},[97],[18,47915],{"className":47916,"style":242},[101],[18,47918,47920,47923],{"className":47919},[106],[18,47921,12579],{"className":47922},[106,110],[18,47924,47926],{"className":47925},[115],[18,47927,47929,47952],{"className":47928},[119,120],[18,47930,47932,47949],{"className":47931},[124],[18,47933,47935],{"className":47934,"style":1152},[128],[18,47936,47937,47940],{"style":1155},[18,47938],{"className":47939,"style":137},[136],[18,47941,47943],{"className":47942},[141,142,143,144],[18,47944,47946],{"className":47945},[106,144],[18,47947,12572],{"className":47948,"style":254},[106,1478,144],[18,47950,152],{"className":47951},[151],[18,47953,47955],{"className":47954},[124],[18,47956,47958],{"className":47957,"style":159},[128],[18,47959],{},[18,47961,74],{"className":47962},[250],[18,47964,47966,47969],{"className":47965},[106],[18,47967,18323],{"className":47968},[106,110],[18,47970,47972],{"className":47971},[115],[18,47973,47975,47995],{"className":47974},[119,120],[18,47976,47978,47992],{"className":47977},[124],[18,47979,47981],{"className":47980,"style":1152},[128],[18,47982,47983,47986],{"style":1155},[18,47984],{"className":47985,"style":137},[136],[18,47987,47989],{"className":47988},[141,142,143,144],[18,47990,38925],{"className":47991,"style":38958},[106,110,144],[18,47993,152],{"className":47994},[151],[18,47996,47998],{"className":47997},[124],[18,47999,48001],{"className":48000,"style":159},[128],[18,48002],{},[18,48004,84],{"className":48005},[267]," 拆开：",[18,48008,48010],{"className":48009,"translate":22},[875],[18,48011,48013,48394],{"className":48012,"translate":22},[21],[18,48014,48016],{"className":48015},[26],[28,48017,48018],{"xmlns":30,"display":884},[32,48019,48020,48391],{},[48021,48022,48025,48117,48191,48305],"mtable",{"rowspacing":48023,"columnalign":48024,"columnspacing":32619},"0.25em","right left",[48026,48027,48028,48051],"mtr",{},[48029,48030,48031],"mtd",{},[14678,48032,48033],{"scriptlevel":430,"displaystyle":51},[35,48034,48035,48041,48043,48049],{},[38,48036,48037,48039],{},[41,48038,12579],{},[41,48040,12572],{"mathvariant":1462},[49,48042,74],{"stretchy":73},[38,48044,48045,48047],{},[41,48046,18323],{},[41,48048,38925],{},[49,48050,84],{"stretchy":73},[48029,48052,48053],{},[14678,48054,48055],{"scriptlevel":430,"displaystyle":51},[35,48056,48057,48059,48061,48063,48069,48071,48077,48079,48081,48087,48089,48095,48097,48099,48101,48107,48109,48115],{},[35,48058],{},[49,48060,433],{},[49,48062,945],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[38,48064,48065,48067],{},[41,48066,12579],{},[41,48068,12572],{"mathvariant":1462},[49,48070,74],{"stretchy":73},[38,48072,48073,48075],{},[41,48074,18323],{},[41,48076,38925],{},[49,48078,84],{"stretchy":73},[49,48080,2253],{},[38,48082,48083,48085],{},[41,48084,12579],{},[41,48086,38925],{},[49,48088,74],{"stretchy":73},[38,48090,48091,48093],{},[41,48092,18323],{},[41,48094,38925],{},[49,48096,84],{"stretchy":73},[49,48098,965],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,48100,7984],{},[38,48102,48103,48105],{},[41,48104,12579],{},[41,48106,38925],{},[49,48108,74],{"stretchy":73},[38,48110,48111,48113],{},[41,48112,18323],{},[41,48114,38925],{},[49,48116,84],{"stretchy":73},[48026,48118,48119,48125],{},[48029,48120,48121],{},[14678,48122,48123],{"scriptlevel":430,"displaystyle":51},[35,48124],{},[48029,48126,48127],{},[14678,48128,48129],{"scriptlevel":430,"displaystyle":51},[35,48130,48131,48133,48135,48137,48143,48145,48151,48153,48155,48161,48163,48169,48171,48173,48175,48181,48183,48189],{},[35,48132],{},[49,48134,4334],{},[49,48136,945],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[38,48138,48139,48141],{},[41,48140,12579],{},[41,48142,12572],{"mathvariant":1462},[49,48144,74],{"stretchy":73},[38,48146,48147,48149],{},[41,48148,18323],{},[41,48150,38925],{},[49,48152,84],{"stretchy":73},[49,48154,2253],{},[38,48156,48157,48159],{},[41,48158,12579],{},[41,48160,38925],{},[49,48162,74],{"stretchy":73},[38,48164,48165,48167],{},[41,48166,18323],{},[41,48168,38925],{},[49,48170,84],{"stretchy":73},[49,48172,965],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,48174,7984],{},[38,48176,48177,48179],{},[41,48178,12579],{},[41,48180,38925],{},[49,48182,74],{"stretchy":73},[1556,48184,48185,48187],{},[41,48186,18323],{},[49,48188,47812],{},[49,48190,84],{"stretchy":73},[48026,48192,48193,48199],{},[48029,48194,48195],{},[14678,48196,48197],{"scriptlevel":430,"displaystyle":51},[35,48198],{},[48029,48200,48201],{},[14678,48202,48203],{"scriptlevel":430,"displaystyle":51},[35,48204,48205,48207,48209,48211,48217,48219,48225,48227,48229,48235,48237,48243,48245,48247,48249,48251,48257,48259,48265,48267,48269,48275,48277,48283,48285,48287,48289,48295,48297,48303],{},[35,48206],{},[49,48208,433],{},[49,48210,945],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[38,48212,48213,48215],{},[41,48214,12579],{},[41,48216,12572],{"mathvariant":1462},[49,48218,74],{"stretchy":73},[38,48220,48221,48223],{},[41,48222,18323],{},[41,48224,38925],{},[49,48226,84],{"stretchy":73},[49,48228,2253],{},[38,48230,48231,48233],{},[41,48232,12579],{},[41,48234,38925],{},[49,48236,74],{"stretchy":73},[38,48238,48239,48241],{},[41,48240,18323],{},[41,48242,38925],{},[49,48244,84],{"stretchy":73},[49,48246,965],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,48248,7984],{},[49,48250,945],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[38,48252,48253,48255],{},[41,48254,12579],{},[41,48256,38925],{},[49,48258,74],{"stretchy":73},[1556,48260,48261,48263],{},[41,48262,18323],{},[49,48264,47812],{},[49,48266,84],{"stretchy":73},[49,48268,2253],{},[38,48270,48271,48273],{},[41,48272,12579],{},[41,48274,12572],{"mathvariant":1462},[49,48276,74],{"stretchy":73},[1556,48278,48279,48281],{},[41,48280,18323],{},[49,48282,47812],{},[49,48284,84],{"stretchy":73},[49,48286,965],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,48288,7984],{},[38,48290,48291,48293],{},[41,48292,12579],{},[41,48294,12572],{"mathvariant":1462},[49,48296,74],{"stretchy":73},[1556,48298,48299,48301],{},[41,48300,18323],{},[49,48302,47812],{},[49,48304,84],{"stretchy":73},[48026,48306,48307,48313],{},[48029,48308,48309],{},[14678,48310,48311],{"scriptlevel":430,"displaystyle":51},[35,48312],{},[48029,48314,48315],{},[14678,48316,48317],{"scriptlevel":430,"displaystyle":51},[35,48318,48319,48321,48323,48329,48331,48337,48339,48341,48343,48359,48361,48367,48369,48371,48373,48375,48381,48383,48385,48387,48389],{},[35,48320],{},[49,48322,4334],{},[38,48324,48325,48327],{},[41,48326,12579],{},[41,48328,12572],{"mathvariant":1462},[49,48330,74],{"stretchy":73},[1556,48332,48333,48335],{},[41,48334,18323],{},[49,48336,47812],{},[49,48338,84],{"stretchy":73},[49,48340,7984],{},[45,48342,2266],{},[890,48344,48345,48351],{},[35,48346,48347,48349],{},[41,48348,912],{},[49,48350,899],{},[35,48352,48353,48355,48357],{},[41,48354,18323],{},[49,48356,921],{},[41,48358,18304],{"mathvariant":1462},[49,48360,11222],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[38,48362,48363,48365],{},[41,48364,12579],{},[41,48366,12572],{"mathvariant":1462},[49,48368,74],{"stretchy":73},[41,48370,18323],{},[49,48372,84],{"stretchy":73},[49,48374,2253],{},[38,48376,48377,48379],{},[41,48378,12579],{},[41,48380,38925],{},[49,48382,74],{"stretchy":73},[41,48384,18323],{},[49,48386,84],{"stretchy":73},[49,48388,11222],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[41,48390,968],{"mathvariant":316},[86,48392,48393],{"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",[717,50163,50164],{},"近似误差","：假设类里最好的那个假设有多好。第二项是",[717,50167,50168],{},"估计误差","：经验风险对真实风险的均匀偏差。",[11,50171,50172,50173,50201],{},"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 ",[18,50174,50176,50189],{"className":50175,"translate":22},[21],[18,50177,50179],{"className":50178},[26],[28,50180,50181],{"xmlns":30},[32,50182,50183,50187],{},[35,50184,50185],{},[41,50186,18304],{"mathvariant":1462},[86,50188,46308],{"encoding":88},[18,50190,50192],{"className":50191,"ariaHidden":51},[93],[18,50193,50195,50198],{"className":50194},[97],[18,50196],{"className":50197,"style":600},[101],[18,50199,18304],{"className":50200,"style":46321},[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。",[1311,50203,50205],{"id":50204},"_13-一致收敛","1.3 一致收敛",[11,50207,50208,50209,50237,50238,19366,50289,50340],{},"如果对任意分布 ",[18,50210,50212,50225],{"className":50211,"translate":22},[21],[18,50213,50215],{"className":50214},[26],[28,50216,50217],{"xmlns":30},[32,50218,50219,50223],{},[35,50220,50221],{},[41,50222,12572],{"mathvariant":1462},[86,50224,46393],{"encoding":88},[18,50226,50228],{"className":50227,"ariaHidden":51},[93],[18,50229,50231,50234],{"className":50230},[97],[18,50232],{"className":50233,"style":600},[101],[18,50235,12572],{"className":50236,"style":254},[106,1478]," 和任意 ",[18,50239,50241,50259],{"className":50240,"translate":22},[21],[18,50242,50244],{"className":50243},[26],[28,50245,50246],{"xmlns":30},[32,50247,50248,50256],{},[35,50249,50250,50252,50254],{},[41,50251,32513],{},[49,50253,6882],{},[45,50255,430],{},[86,50257,50258],{"encoding":88},"\\varepsilon>0",[18,50260,50262,50280],{"className":50261,"ariaHidden":51},[93],[18,50263,50265,50268,50271,50274,50277],{"className":50264},[97],[18,50266],{"className":50267,"style":30921},[101],[18,50269,32513],{"className":50270},[106,110],[18,50272],{"className":50273,"style":228},[169],[18,50275,6882],{"className":50276},[232],[18,50278],{"className":50279,"style":228},[169],[18,50281,50283,50286],{"className":50282},[97],[18,50284],{"className":50285,"style":505},[101],[18,50287,430],{"className":50288},[106],[18,50290,50292,50310],{"className":50291,"translate":22},[21],[18,50293,50295],{"className":50294},[26],[28,50296,50297],{"xmlns":30},[32,50298,50299,50307],{},[35,50300,50301,50303,50305],{},[41,50302,28160],{},[49,50304,1880],{},[41,50306,2028],{"mathvariant":316},[86,50308,50309],{"encoding":88},"m\\to\\infty",[18,50311,50313,50331],{"className":50312,"ariaHidden":51},[93],[18,50314,50316,50319,50322,50325,50328],{"className":50315},[97],[18,50317],{"className":50318,"style":1083},[101],[18,50320,28160],{"className":50321},[106,110],[18,50323],{"className":50324,"style":228},[169],[18,50326,1880],{"className":50327},[232],[18,50329],{"className":50330,"style":228},[169],[18,50332,50334,50337],{"className":50333},[97],[18,50335],{"className":50336,"style":1083},[101],[18,50338,2028],{"className":50339},[106]," 时",[18,50342,50344],{"className":50343,"translate":22},[875],[18,50345,50347,50425],{"className":50346,"translate":22},[21],[18,50348,50350],{"className":50349},[26],[28,50351,50352],{"xmlns":30,"display":884},[32,50353,50354,50422],{},[35,50355,50356,50358,50360,50416,50418,50420],{},[41,50357,11205],{},[49,50359,899],{},[35,50361,50362,50364,50380,50382,50388,50390,50392,50394,50396,50402,50404,50406,50408,50410,50412,50414],{},[49,50363,945],{"fence":51},[890,50365,50366,50372],{},[35,50367,50368,50370],{},[41,50369,912],{},[49,50371,899],{},[35,50373,50374,50376,50378],{},[41,50375,18323],{},[49,50377,921],{},[41,50379,18304],{"mathvariant":1462},[49,50381,11222],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[38,50383,50384,50386],{},[41,50385,12579],{},[41,50387,12572],{"mathvariant":1462},[49,50389,74],{"stretchy":73},[41,50391,18323],{},[49,50393,84],{"stretchy":73},[49,50395,2253],{},[38,50397,50398,50400],{},[41,50399,12579],{},[41,50401,38925],{},[49,50403,74],{"stretchy":73},[41,50405,18323],{},[49,50407,84],{"stretchy":73},[49,50409,11222],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,50411,6882],{},[41,50413,32513],{},[49,50415,965],{"fence":51},[49,50417,1880],{},[45,50419,430],{},[49,50421,52],{"separator":51},[86,50423,50424],{"encoding":88},"\\Pr\\left[\\sup_{h\\in\\mathcal H}\\bigl|L_{\\mathcal D}(h)-L_S(h)\\bigr|>\\varepsilon\\right]\\to 0,",[18,50426,50428,50714],{"className":50427,"ariaHidden":51},[93],[18,50429,50431,50435,50438,50441,50705,50708,50711],{"className":50430},[97],[18,50432],{"className":50433,"style":50434},[101],"height:2.4239em;vertical-align:-0.9739em;",[18,50436,11205],{"className":50437},[985],[18,50439],{"className":50440,"style":170},[169],[18,50442,50444,50450,50503,50540,50583,50586,50589,50592,50595,50598,50601,50641,50644,50647,50650,50687,50690,50693,50696,50699],{"className":50443},[174],[18,50445,50447],{"className":50446,"style":13793},[250,13792],[18,50448,945],{"className":50449},[1230,143],[18,50451,50453],{"className":50452},[985,986],[18,50454,50456,50495],{"className":50455},[119,120],[18,50457,50459,50492],{"className":50458},[124],[18,50460,50462,50482],{"className":50461,"style":1083},[128],[18,50463,50464,50467],{"style":1086},[18,50465],{"className":50466,"style":697},[136],[18,50468,50470],{"className":50469},[141,142,143,144],[18,50471,50473,50476,50479],{"className":50472},[106,144],[18,50474,18323],{"className":50475},[106,110,144],[18,50477,921],{"className":50478},[232,144],[18,50480,18304],{"className":50481,"style":46321},[106,1478,144],[18,50483,50484,50487],{"style":693},[18,50485],{"className":50486,"style":697},[136],[18,50488,50489],{},[18,50490,912],{"className":50491},[985],[18,50493,152],{"className":50494},[151],[18,50496,50498],{"className":50497},[124],[18,50499,50501],{"className":50500,"style":1124},[128],[18,50502],{},[18,50504,50506],{"className":50505},[250],[18,50507,50509],{"className":50508},[1230,6105],[18,50510,50512,50532],{"className":50511},[119,120],[18,50513,50515,50529],{"className":50514},[124],[18,50516,50518],{"className":50517,"style":6115},[128],[18,50519,50520,50523],{"style":6118},[18,50521],{"className":50522,"style":6122},[136],[18,50524,50525],{"style":49860},[6127,50526,50527],{"xmlns":6129,"width":49863,"height":944,"viewBox":49864},[6133,50528],{"d":49867},[18,50530,152],{"className":50531},[151],[18,50533,50535],{"className":50534},[124],[18,50536,50538],{"className":50537,"style":6145},[128],[18,50539],{},[18,50541,50543,50546],{"className":50542},[106],[18,50544,12579],{"className":50545},[106,110],[18,50547,50549],{"className":50548},[115],[18,50550,50552,50575],{"className":50551},[119,120],[18,50553,50555,50572],{"className":50554},[124],[18,50556,50558],{"className":50557,"style":1152},[128],[18,50559,50560,50563],{"style":1155},[18,50561],{"className":50562,"style":137},[136],[18,50564,50566],{"className":50565},[141,142,143,144],[18,50567,50569],{"className":50568},[106,144],[18,50570,12572],{"className":50571,"style":254},[106,1478,144],[18,50573,152],{"className":50574},[151],[18,50576,50578],{"className":50577},[124],[18,50579,50581],{"className":50580,"style":159},[128],[18,50582],{},[18,50584,74],{"className":50585},[250],[18,50587,18323],{"className":50588},[106,110],[18,50590,84],{"className":50591},[267],[18,50593],{"className":50594,"style":2068},[169],[18,50596,2253],{"className":50597},[2072],[18,50599],{"className":50600,"style":2068},[169],[18,50602,50604,50607],{"className":50603},[106],[18,50605,12579],{"className":50606},[106,110],[18,50608,50610],{"className":50609},[115],[18,50611,50613,50633],{"className":50612},[119,120],[18,50614,50616,50630],{"className":50615},[124],[18,50617,50619],{"className":50618,"style":1152},[128],[18,50620,50621,50624],{"style":1155},[18,50622],{"className":50623,"style":137},[136],[18,50625,50627],{"className":50626},[141,142,143,144],[18,50628,38925],{"className":50629,"style":38958},[106,110,144],[18,50631,152],{"className":50632},[151],[18,50634,50636],{"className":50635},[124],[18,50637,50639],{"className":50638,"style":159},[128],[18,50640],{},[18,50642,74],{"className":50643},[250],[18,50645,18323],{"className":50646},[106,110],[18,50648,84],{"className":50649},[267],[18,50651,50653],{"className":50652},[267],[18,50654,50656],{"className":50655},[1230,6105],[18,50657,50659,50679],{"className":50658},[119,120],[18,50660,50662,50676],{"className":50661},[124],[18,50663,50665],{"className":50664,"style":6115},[128],[18,50666,50667,50670],{"style":6118},[18,50668],{"className":50669,"style":6122},[136],[18,50671,50672],{"style":49860},[6127,50673,50674],{"xmlns":6129,"width":49863,"height":944,"viewBox":49864},[6133,50675],{"d":49867},[18,50677,152],{"className":50678},[151],[18,50680,50682],{"className":50681},[124],[18,50683,50685],{"className":50684,"style":6145},[128],[18,50686],{},[18,50688],{"className":50689,"style":228},[169],[18,50691,6882],{"className":50692},[232],[18,50694],{"className":50695,"style":228},[169],[18,50697,32513],{"className":50698},[106,110],[18,50700,50702],{"className":50701,"style":13793},[267,13792],[18,50703,965],{"className":50704},[1230,143],[18,50706],{"className":50707,"style":228},[169],[18,50709,1880],{"className":50710},[232],[18,50712],{"className":50713,"style":228},[169],[18,50715,50717,50721,50724],{"className":50716},[97],[18,50718],{"className":50719,"style":50720},[101],"height:0.8389em;vertical-align:-0.1944em;",[18,50722,430],{"className":50723},[106],[18,50725,52],{"className":50726},[165],[11,50728,50729,50730,50758,50759,50762],{},"就说 ",[18,50731,50733,50746],{"className":50732,"translate":22},[21],[18,50734,50736],{"className":50735},[26],[28,50737,50738],{"xmlns":30},[32,50739,50740,50744],{},[35,50741,50742],{},[41,50743,18304],{"mathvariant":1462},[86,50745,46308],{"encoding":88},[18,50747,50749],{"className":50748,"ariaHidden":51},[93],[18,50750,50752,50755],{"className":50751},[97],[18,50753],{"className":50754,"style":600},[101],[18,50756,18304],{"className":50757,"style":46321},[106,1478]," 具有",[717,50760,50761],{},"一致收敛性","（uniform convergence）。",[11,50764,50765,50796,50797,50825,50826,50854,50855,50925,50926,50954],{},[18,50766,50768,50784],{"className":50767,"translate":22},[21],[18,50769,50771],{"className":50770},[26],[28,50772,50773],{"xmlns":30},[32,50774,50775,50781],{},[35,50776,50777,50779],{},[41,50778,912],{},[49,50780,899],{},[86,50782,50783],{"encoding":88},"\\sup",[18,50785,50787],{"className":50786,"ariaHidden":51},[93],[18,50788,50790,50793],{"className":50789},[97],[18,50791],{"className":50792,"style":38543},[101],[18,50794,912],{"className":50795},[985]," 在概率里面，这个顺序很关键：不是「对每个 ",[18,50798,50800,50813],{"className":50799,"translate":22},[21],[18,50801,50803],{"className":50802},[26],[28,50804,50805],{"xmlns":30},[32,50806,50807,50811],{},[35,50808,50809],{},[41,50810,18323],{},[86,50812,18323],{"encoding":88},[18,50814,50816],{"className":50815,"ariaHidden":51},[93],[18,50817,50819,50822],{"className":50818},[97],[18,50820],{"className":50821,"style":295},[101],[18,50823,18323],{"className":50824},[106,110]," 分别收敛」，而是「所有 ",[18,50827,50829,50842],{"className":50828,"translate":22},[21],[18,50830,50832],{"className":50831},[26],[28,50833,50834],{"xmlns":30},[32,50835,50836,50840],{},[35,50837,50838],{},[41,50839,18323],{},[86,50841,18323],{"encoding":88},[18,50843,50845],{"className":50844,"ariaHidden":51},[93],[18,50846,50848,50851],{"className":50847},[97],[18,50849],{"className":50850,"style":295},[101],[18,50852,18323],{"className":50853},[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 ",[18,50856,50858,50876],{"className":50857,"translate":22},[21],[18,50859,50861],{"className":50860},[26],[28,50862,50863],{"xmlns":30},[32,50864,50865,50873],{},[35,50866,50867],{},[38,50868,50869,50871],{},[41,50870,18323],{},[41,50872,38925],{},[86,50874,50875],{"encoding":88},"h_S",[18,50877,50879],{"className":50878,"ariaHidden":51},[93],[18,50880,50882,50885],{"className":50881},[97],[18,50883],{"className":50884,"style":448},[101],[18,50886,50888,50891],{"className":50887},[106],[18,50889,18323],{"className":50890},[106,110],[18,50892,50894],{"className":50893},[115],[18,50895,50897,50917],{"className":50896},[119,120],[18,50898,50900,50914],{"className":50899},[124],[18,50901,50903],{"className":50902,"style":1152},[128],[18,50904,50905,50908],{"style":1155},[18,50906],{"className":50907,"style":137},[136],[18,50909,50911],{"className":50910},[141,142,143,144],[18,50912,38925],{"className":50913,"style":38958},[106,110,144],[18,50915,152],{"className":50916},[151],[18,50918,50920],{"className":50919},[124],[18,50921,50923],{"className":50922,"style":159},[128],[18,50924],{}," 本身依赖数据，你不能先固定 ",[18,50927,50929,50942],{"className":50928,"translate":22},[21],[18,50930,50932],{"className":50931},[26],[28,50933,50934],{"xmlns":30},[32,50935,50936,50940],{},[35,50937,50938],{},[41,50939,18323],{},[86,50941,18323],{"encoding":88},[18,50943,50945],{"className":50944,"ariaHidden":51},[93],[18,50946,50948,50951],{"className":50947},[97],[18,50949],{"className":50950,"style":295},[101],[18,50952,18323],{"className":50953},[106,110]," 再用大数定律。",[11,50956,50957,50958,50986],{},"于是问题变成：什么样的 ",[18,50959,50961,50974],{"className":50960,"translate":22},[21],[18,50962,50964],{"className":50963},[26],[28,50965,50966],{"xmlns":30},[32,50967,50968,50972],{},[35,50969,50970],{},[41,50971,18304],{"mathvariant":1462},[86,50973,46308],{"encoding":88},[18,50975,50977],{"className":50976,"ariaHidden":51},[93],[18,50978,50980,50983],{"className":50979},[97],[18,50981],{"className":50982,"style":600},[101],[18,50984,18304],{"className":50985,"style":46321},[106,1478]," 具有一致收敛性？答案由 VC 维给出。",[638,50988,50990],{"id":50989},"二打散与-vc-维","二、打散与 VC 维",[1311,50992,50994],{"id":50993},"_21-二分","2.1 二分",[11,50996,50997,50998,1990,51193,3019,51221,51249,51250,51253],{},"固定一个有限的点集 ",[18,50999,51001,51044],{"className":51000,"translate":22},[21],[18,51002,51004],{"className":51003},[26],[28,51005,51006],{"xmlns":30},[32,51007,51008,51041],{},[35,51009,51010,51013,51015,51017,51023,51025,51027,51029,51035,51037,51039],{},[41,51011,51012],{},"C",[49,51014,433],{},[49,51016,1331],{"stretchy":73},[38,51018,51019,51021],{},[41,51020,39058],{},[45,51022,47],{},[49,51024,52],{"separator":51},[49,51026,55],{},[49,51028,52],{"separator":51},[38,51030,51031,51033],{},[41,51032,39058],{},[41,51034,28160],{},[49,51036,1349],{"stretchy":73},[49,51038,4548],{},[41,51040,43],{"mathvariant":1462},[86,51042,51043],{"encoding":88},"C=\\{c_1,\\dots,c_m\\}\\subseteq\\mathcal X",[18,51045,51047,51065,51184],{"className":51046,"ariaHidden":51},[93],[18,51048,51050,51053,51056,51059,51062],{"className":51049},[97],[18,51051],{"className":51052,"style":600},[101],[18,51054,51012],{"className":51055,"style":5073},[106,110],[18,51057],{"className":51058,"style":228},[169],[18,51060,433],{"className":51061},[232],[18,51063],{"className":51064,"style":228},[169],[18,51066,51068,51071,51074,51114,51117,51120,51123,51126,51129,51132,51172,51175,51178,51181],{"className":51067},[97],[18,51069],{"className":51070,"style":242},[101],[18,51072,1331],{"className":51073},[250],[18,51075,51077,51080],{"className":51076},[106],[18,51078,39058],{"className":51079},[106,110],[18,51081,51083],{"className":51082},[115],[18,51084,51086,51106],{"className":51085},[119,120],[18,51087,51089,51103],{"className":51088},[124],[18,51090,51092],{"className":51091,"style":129},[128],[18,51093,51094,51097],{"style":1155},[18,51095],{"className":51096,"style":137},[136],[18,51098,51100],{"className":51099},[141,142,143,144],[18,51101,47],{"className":51102},[106,144],[18,51104,152],{"className":51105},[151],[18,51107,51109],{"className":51108},[124],[18,51110,51112],{"className":51111,"style":159},[128],[18,51113],{},[18,51115,52],{"className":51116},[165],[18,51118],{"className":51119,"style":170},[169],[18,51121,55],{"className":51122},[174],[18,51124],{"className":51125,"style":170},[169],[18,51127,52],{"className":51128},[165],[18,51130],{"className":51131,"style":170},[169],[18,51133,51135,51138],{"className":51134},[106],[18,51136,39058],{"className":51137},[106,110],[18,51139,51141],{"className":51140},[115],[18,51142,51144,51164],{"className":51143},[119,120],[18,51145,51147,51161],{"className":51146},[124],[18,51148,51150],{"className":51149,"style":202},[128],[18,51151,51152,51155],{"style":1155},[18,51153],{"className":51154,"style":137},[136],[18,51156,51158],{"className":51157},[141,142,143,144],[18,51159,28160],{"className":51160},[106,110,144],[18,51162,152],{"className":51163},[151],[18,51165,51167],{"className":51166},[124],[18,51168,51170],{"className":51169,"style":159},[128],[18,51171],{},[18,51173,1349],{"className":51174},[267],[18,51176],{"className":51177,"style":228},[169],[18,51179,4548],{"className":51180},[232],[18,51182],{"className":51183,"style":228},[169],[18,51185,51187,51190],{"className":51186},[97],[18,51188],{"className":51189,"style":600},[101],[18,51191,43],{"className":51192,"style":1479},[106,1478],[18,51194,51196,51209],{"className":51195,"translate":22},[21],[18,51197,51199],{"className":51198},[26],[28,51200,51201],{"xmlns":30},[32,51202,51203,51207],{},[35,51204,51205],{},[41,51206,18304],{"mathvariant":1462},[86,51208,46308],{"encoding":88},[18,51210,51212],{"className":51211,"ariaHidden":51},[93],[18,51213,51215,51218],{"className":51214},[97],[18,51216],{"className":51217,"style":600},[101],[18,51219,18304],{"className":51220,"style":46321},[106,1478],[18,51222,51224,51237],{"className":51223,"translate":22},[21],[18,51225,51227],{"className":51226},[26],[28,51228,51229],{"xmlns":30},[32,51230,51231,51235],{},[35,51232,51233],{},[41,51234,51012],{},[86,51236,51012],{"encoding":88},[18,51238,51240],{"className":51239,"ariaHidden":51},[93],[18,51241,51243,51246],{"className":51242},[97],[18,51244],{"className":51245,"style":600},[101],[18,51247,51012],{"className":51248,"style":5073},[106,110]," 上的",[717,51251,51252],{},"限制","（restriction）是",[18,51255,51257],{"className":51256,"translate":22},[875],[18,51258,51260,51346],{"className":51259,"translate":22},[21],[18,51261,51263],{"className":51262},[26],[28,51264,51265],{"xmlns":30,"display":884},[32,51266,51267,51343],{},[35,51268,51269,51271,51277,51279,51281,51283,51285,51287,51293,51295,51297,51299,51301,51303,51305,51311,51313,51315,51317,51319,51321,51323,51325,51327,51329,51331,51333,51335,51341],{},[41,51270,18304],{"mathvariant":1462},[38,51272,51273,51275],{},[41,51274,11222],{"mathvariant":316},[41,51276,51012],{},[49,51278,433],{},[49,51280,1331],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,51282,74],{"stretchy":73},[41,51284,18323],{},[49,51286,74],{"stretchy":73},[38,51288,51289,51291],{},[41,51290,39058],{},[45,51292,47],{},[49,51294,84],{"stretchy":73},[49,51296,52],{"separator":51},[49,51298,55],{},[49,51300,52],{"separator":51},[41,51302,18323],{},[49,51304,74],{"stretchy":73},[38,51306,51307,51309],{},[41,51308,39058],{},[41,51310,28160],{},[49,51312,84],{"stretchy":73},[49,51314,84],{"stretchy":73},[49,51316,1340],{},[41,51318,18323],{},[49,51320,921],{},[41,51322,18304],{"mathvariant":1462},[49,51324,1349],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,51326,4548],{},[49,51328,1331],{"stretchy":73},[45,51330,430],{},[49,51332,52],{"separator":51},[45,51334,47],{},[1556,51336,51337,51339],{},[49,51338,1349],{"stretchy":73},[41,51340,28160],{},[41,51342,968],{"mathvariant":316},[86,51344,51345],{"encoding":88},"\\mathcal H|_C=\\bigl\\{(h(c_1),\\dots,h(c_m)):h\\in\\mathcal H\\bigr\\}\\subseteq\\{0,1\\}^m.",[18,51347,51349,51407,51547,51565,51589],{"className":51348,"ariaHidden":51},[93],[18,51350,51352,51355,51358,51398,51401,51404],{"className":51351},[97],[18,51353],{"className":51354,"style":242},[101],[18,51356,18304],{"className":51357,"style":46321},[106,1478],[18,51359,51361,51364],{"className":51360},[106],[18,51362,11222],{"className":51363},[106],[18,51365,51367],{"className":51366},[115],[18,51368,51370,51390],{"className":51369},[119,120],[18,51371,51373,51387],{"className":51372},[124],[18,51374,51376],{"className":51375,"style":1152},[128],[18,51377,51378,51381],{"style":1155},[18,51379],{"className":51380,"style":137},[136],[18,51382,51384],{"className":51383},[141,142,143,144],[18,51385,51012],{"className":51386,"style":5073},[106,110,144],[18,51388,152],{"className":51389},[151],[18,51391,51393],{"className":51392},[124],[18,51394,51396],{"className":51395,"style":159},[128],[18,51397],{},[18,51399],{"className":51400,"style":228},[169],[18,51402,433],{"className":51403},[232],[18,51405],{"className":51406,"style":228},[169],[18,51408,51410,51413,51419,51422,51425,51428,51468,51471,51474,51477,51480,51483,51486,51489,51492,51495,51535,51538,51541,51544],{"className":51409},[97],[18,51411],{"className":51412,"style":6098},[101],[18,51414,51416],{"className":51415},[250],[18,51417,1331],{"className":51418},[1230,1196],[18,51420,74],{"className":51421},[250],[18,51423,18323],{"className":51424},[106,110],[18,51426,74],{"className":51427},[250],[18,51429,51431,51434],{"className":51430},[106],[18,51432,39058],{"className":51433},[106,110],[18,51435,51437],{"className":51436},[115],[18,51438,51440,51460],{"className":51439},[119,120],[18,51441,51443,51457],{"className":51442},[124],[18,51444,51446],{"className":51445,"style":129},[128],[18,51447,51448,51451],{"style":1155},[18,51449],{"className":51450,"style":137},[136],[18,51452,51454],{"className":51453},[141,142,143,144],[18,51455,47],{"className":51456},[106,144],[18,51458,152],{"className":51459},[151],[18,51461,51463],{"className":51462},[124],[18,51464,51466],{"className":51465,"style":159},[128],[18,51467],{},[18,51469,84],{"className":51470},[267],[18,51472,52],{"className":51473},[165],[18,51475],{"className":51476,"style":170},[169],[18,51478,55],{"className":51479},[174],[18,51481],{"className":51482,"style":170},[169],[18,51484,52],{"className":51485},[165],[18,51487],{"className":51488,"style":170},[169],[18,51490,18323],{"className":51491},[106,110],[18,51493,74],{"className":51494},[250],[18,51496,51498,51501],{"className":51497},[106],[18,51499,39058],{"className":51500},[106,110],[18,51502,51504],{"className":51503},[115],[18,51505,51507,51527],{"className":51506},[119,120],[18,51508,51510,51524],{"className":51509},[124],[18,51511,51513],{"className":51512,"style":202},[128],[18,51514,51515,51518],{"style":1155},[18,51516],{"className":51517,"style":137},[136],[18,51519,51521],{"className":51520},[141,142,143,144],[18,51522,28160],{"className":51523},[106,110,144],[18,51525,152],{"className":51526},[151],[18,51528,51530],{"className":51529},[124],[18,51531,51533],{"className":51532,"style":159},[128],[18,51534],{},[18,51536,3431],{"className":51537},[267],[18,51539],{"className":51540,"style":228},[169],[18,51542,1340],{"className":51543},[232],[18,51545],{"className":51546,"style":228},[169],[18,51548,51550,51553,51556,51559,51562],{"className":51549},[97],[18,51551],{"className":51552,"style":1422},[101],[18,51554,18323],{"className":51555},[106,110],[18,51557],{"className":51558,"style":228},[169],[18,51560,921],{"className":51561},[232],[18,51563],{"className":51564,"style":228},[169],[18,51566,51568,51571,51574,51580,51583,51586],{"className":51567},[97],[18,51569],{"className":51570,"style":6098},[101],[18,51572,18304],{"className":51573,"style":46321},[106,1478],[18,51575,51577],{"className":51576},[267],[18,51578,1349],{"className":51579},[1230,1196],[18,51581],{"className":51582,"style":228},[169],[18,51584,4548],{"className":51585},[232],[18,51587],{"className":51588,"style":228},[169],[18,51590,51592,51595,51598,51601,51604,51607,51610,51639],{"className":51591},[97],[18,51593],{"className":51594,"style":242},[101],[18,51596,1331],{"className":51597},[250],[18,51599,430],{"className":51600},[106],[18,51602,52],{"className":51603},[165],[18,51605],{"className":51606,"style":170},[169],[18,51608,47],{"className":51609},[106],[18,51611,51613,51616],{"className":51612},[267],[18,51614,1349],{"className":51615},[267],[18,51617,51619],{"className":51618},[115],[18,51620,51622],{"className":51621},[119],[18,51623,51625],{"className":51624},[124],[18,51626,51628],{"className":51627,"style":1616},[128],[18,51629,51630,51633],{"style":1637},[18,51631],{"className":51632,"style":137},[136],[18,51634,51636],{"className":51635},[141,142,143,144],[18,51637,28160],{"className":51638},[106,110,144],[18,51640,968],{"className":51641},[106],[11,51643,51644,51645,51648,51649,51734,51735,51793,51794,51852,51853,51881,51882,51914,51915,51943],{},"这个集合里的每个元素称为一个",[717,51646,51647],{},"二分","（dichotomy）。",[18,51650,51652,51676],{"className":51651,"translate":22},[21],[18,51653,51655],{"className":51654},[26],[28,51656,51657],{"xmlns":30},[32,51658,51659,51673],{},[35,51660,51661,51663,51665,51671],{},[41,51662,11222],{"mathvariant":316},[41,51664,18304],{"mathvariant":1462},[38,51666,51667,51669],{},[41,51668,11222],{"mathvariant":316},[41,51670,51012],{},[41,51672,11222],{"mathvariant":316},[86,51674,51675],{"encoding":88},"|\\mathcal H|_C|",[18,51677,51679],{"className":51678,"ariaHidden":51},[93],[18,51680,51682,51685,51688,51691,51731],{"className":51681},[97],[18,51683],{"className":51684,"style":242},[101],[18,51686,11222],{"className":51687},[106],[18,51689,18304],{"className":51690,"style":46321},[106,1478],[18,51692,51694,51697],{"className":51693},[106],[18,51695,11222],{"className":51696},[106],[18,51698,51700],{"className":51699},[115],[18,51701,51703,51723],{"className":51702},[119,120],[18,51704,51706,51720],{"className":51705},[124],[18,51707,51709],{"className":51708,"style":1152},[128],[18,51710,51711,51714],{"style":1155},[18,51712],{"className":51713,"style":137},[136],[18,51715,51717],{"className":51716},[141,142,143,144],[18,51718,51012],{"className":51719,"style":5073},[106,110,144],[18,51721,152],{"className":51722},[151],[18,51724,51726],{"className":51725},[124],[18,51727,51729],{"className":51728,"style":159},[128],[18,51730],{},[18,51732,11222],{"className":51733},[106]," 最多是 ",[18,51736,51738,51755],{"className":51737,"translate":22},[21],[18,51739,51741],{"className":51740},[26],[28,51742,51743],{"xmlns":30},[32,51744,51745,51753],{},[35,51746,51747],{},[1556,51748,51749,51751],{},[45,51750,2266],{},[41,51752,28160],{},[86,51754,44893],{"encoding":88},[18,51756,51758],{"className":51757,"ariaHidden":51},[93],[18,51759,51761,51764],{"className":51760},[97],[18,51762],{"className":51763,"style":1959},[101],[18,51765,51767,51770],{"className":51766},[106],[18,51768,2266],{"className":51769},[106],[18,51771,51773],{"className":51772},[115],[18,51774,51776],{"className":51775},[119],[18,51777,51779],{"className":51778},[124],[18,51780,51782],{"className":51781,"style":1959},[128],[18,51783,51784,51787],{"style":1962},[18,51785],{"className":51786,"style":137},[136],[18,51788,51790],{"className":51789},[141,142,143,144],[18,51791,28160],{"className":51792},[106,110,144],"。如果取到 ",[18,51795,51797,51814],{"className":51796,"translate":22},[21],[18,51798,51800],{"className":51799},[26],[28,51801,51802],{"xmlns":30},[32,51803,51804,51812],{},[35,51805,51806],{},[1556,51807,51808,51810],{},[45,51809,2266],{},[41,51811,28160],{},[86,51813,44893],{"encoding":88},[18,51815,51817],{"className":51816,"ariaHidden":51},[93],[18,51818,51820,51823],{"className":51819},[97],[18,51821],{"className":51822,"style":1959},[101],[18,51824,51826,51829],{"className":51825},[106],[18,51827,2266],{"className":51828},[106],[18,51830,51832],{"className":51831},[115],[18,51833,51835],{"className":51834},[119],[18,51836,51838],{"className":51837},[124],[18,51839,51841],{"className":51840,"style":1959},[128],[18,51842,51843,51846],{"style":1962},[18,51844],{"className":51845,"style":137},[136],[18,51847,51849],{"className":51848},[141,142,143,144],[18,51850,28160],{"className":51851},[106,110,144],"，说明对这 ",[18,51854,51856,51869],{"className":51855,"translate":22},[21],[18,51857,51859],{"className":51858},[26],[28,51860,51861],{"xmlns":30},[32,51862,51863,51867],{},[35,51864,51865],{},[41,51866,28160],{},[86,51868,28160],{"encoding":88},[18,51870,51872],{"className":51871,"ariaHidden":51},[93],[18,51873,51875,51878],{"className":51874},[97],[18,51876],{"className":51877,"style":1083},[101],[18,51879,28160],{"className":51880},[106,110]," 个点的每一种 ",[18,51883,51885,51902],{"className":51884,"translate":22},[21],[18,51886,51888],{"className":51887},[26],[28,51889,51890],{"xmlns":30},[32,51891,51892,51900],{},[35,51893,51894,51896,51898],{},[45,51895,430],{},[41,51897,317],{"mathvariant":316},[45,51899,47],{},[86,51901,46051],{"encoding":88},[18,51903,51905],{"className":51904,"ariaHidden":51},[93],[18,51906,51908,51911],{"className":51907},[97],[18,51909],{"className":51910,"style":242},[101],[18,51912,46051],{"className":51913},[106]," 标注，",[18,51916,51918,51931],{"className":51917,"translate":22},[21],[18,51919,51921],{"className":51920},[26],[28,51922,51923],{"xmlns":30},[32,51924,51925,51929],{},[35,51926,51927],{},[41,51928,18304],{"mathvariant":1462},[86,51930,46308],{"encoding":88},[18,51932,51934],{"className":51933,"ariaHidden":51},[93],[18,51935,51937,51940],{"className":51936},[97],[18,51938],{"className":51939,"style":600},[101],[18,51941,18304],{"className":51942,"style":46321},[106,1478]," 里都有一个假设恰好实现它。",[5159,51945,51947],{"title":51946},"定义 2.1（打散）",[11,51948,21783,51949,52086,52087,1994,52115,52118,52119,1990],{},[18,51950,51952,51984],{"className":51951,"translate":22},[21],[18,51953,51955],{"className":51954},[26],[28,51956,51957],{"xmlns":30},[32,51958,51959,51981],{},[35,51960,51961,51963,51965,51971,51973,51975],{},[41,51962,11222],{"mathvariant":316},[41,51964,18304],{"mathvariant":1462},[38,51966,51967,51969],{},[41,51968,11222],{"mathvariant":316},[41,51970,51012],{},[41,51972,11222],{"mathvariant":316},[49,51974,433],{},[1556,51976,51977,51979],{},[45,51978,2266],{},[41,51980,28160],{},[86,51982,51983],{"encoding":88},"|\\mathcal H|_C|=2^m",[18,51985,51987,52051],{"className":51986,"ariaHidden":51},[93],[18,51988,51990,51993,51996,51999,52039,52042,52045,52048],{"className":51989},[97],[18,51991],{"className":51992,"style":242},[101],[18,51994,11222],{"className":51995},[106],[18,51997,18304],{"className":51998,"style":46321},[106,1478],[18,52000,52002,52005],{"className":52001},[106],[18,52003,11222],{"className":52004},[106],[18,52006,52008],{"className":52007},[115],[18,52009,52011,52031],{"className":52010},[119,120],[18,52012,52014,52028],{"className":52013},[124],[18,52015,52017],{"className":52016,"style":1152},[128],[18,52018,52019,52022],{"style":1155},[18,52020],{"className":52021,"style":137},[136],[18,52023,52025],{"className":52024},[141,142,143,144],[18,52026,51012],{"className":52027,"style":5073},[106,110,144],[18,52029,152],{"className":52030},[151],[18,52032,52034],{"className":52033},[124],[18,52035,52037],{"className":52036,"style":159},[128],[18,52038],{},[18,52040,11222],{"className":52041},[106],[18,52043],{"className":52044,"style":228},[169],[18,52046,433],{"className":52047},[232],[18,52049],{"className":52050,"style":228},[169],[18,52052,52054,52057],{"className":52053},[97],[18,52055],{"className":52056,"style":1959},[101],[18,52058,52060,52063],{"className":52059},[106],[18,52061,2266],{"className":52062},[106],[18,52064,52066],{"className":52065},[115],[18,52067,52069],{"className":52068},[119],[18,52070,52072],{"className":52071},[124],[18,52073,52075],{"className":52074,"style":1959},[128],[18,52076,52077,52080],{"style":1962},[18,52078],{"className":52079,"style":137},[136],[18,52081,52083],{"className":52082},[141,142,143,144],[18,52084,28160],{"className":52085},[106,110,144],"，称 ",[18,52088,52090,52103],{"className":52089,"translate":22},[21],[18,52091,52093],{"className":52092},[26],[28,52094,52095],{"xmlns":30},[32,52096,52097,52101],{},[35,52098,52099],{},[41,52100,18304],{"mathvariant":1462},[86,52102,46308],{"encoding":88},[18,52104,52106],{"className":52105,"ariaHidden":51},[93],[18,52107,52109,52112],{"className":52108},[97],[18,52110],{"className":52111,"style":600},[101],[18,52113,18304],{"className":52114,"style":46321},[106,1478],[717,52116,52117],{},"打散","（shatter）了 ",[18,52120,52122,52135],{"className":52121,"translate":22},[21],[18,52123,52125],{"className":52124},[26],[28,52126,52127],{"xmlns":30},[32,52128,52129,52133],{},[35,52130,52131],{},[41,52132,51012],{},[86,52134,51012],{"encoding":88},[18,52136,52138],{"className":52137,"ariaHidden":51},[93],[18,52139,52141,52144],{"className":52140},[97],[18,52142],{"className":52143,"style":600},[101],[18,52145,51012],{"className":52146,"style":5073},[106,110],[5159,52148,52150,52185,52515],{"title":52149},"定义 2.2（VC 维）",[11,52151,52152,52153,39863,52181,52184],{},"假设类 ",[18,52154,52156,52169],{"className":52155,"translate":22},[21],[18,52157,52159],{"className":52158},[26],[28,52160,52161],{"xmlns":30},[32,52162,52163,52167],{},[35,52164,52165],{},[41,52166,18304],{"mathvariant":1462},[86,52168,46308],{"encoding":88},[18,52170,52172],{"className":52171,"ariaHidden":51},[93],[18,52173,52175,52178],{"className":52174},[97],[18,52176],{"className":52177,"style":600},[101],[18,52179,18304],{"className":52180,"style":46321},[106,1478],[717,52182,52183],{},"VC 维","是它能打散的最大有限集的大小：",[18,52186,52188],{"className":52187,"translate":22},[875],[18,52189,52191,52277],{"className":52190,"translate":22},[21],[18,52192,52194],{"className":52193},[26],[28,52195,52196],{"xmlns":30,"display":884},[32,52197,52198,52274],{},[35,52199,52200,52203,52205,52207,52209,52211,52213,52215,52217,52219,52221,52223,52226,52228,52230,52232,52234,52236,52238,52240,52242,52244,52246,52248,52250,52252,52254,52260,52262,52264,52270,52272],{},[41,52201,52202],{"mathvariant":316},"VCdim",[49,52204,899],{},[49,52206,74],{"stretchy":73},[41,52208,18304],{"mathvariant":1462},[49,52210,84],{"stretchy":73},[49,52212,433],{},[41,52214,912],{},[49,52216,899],{},[49,52218,1331],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[41,52220,28160],{},[49,52222,1340],{},[41,52224,52225],{"mathvariant":316},"∃",[41,52227,51012],{},[49,52229,4548],{},[41,52231,43],{"mathvariant":1462},[49,52233,52],{"separator":51},[926,52235,928],{},[41,52237,11222],{"mathvariant":316},[41,52239,51012],{},[41,52241,11222],{"mathvariant":316},[49,52243,433],{},[41,52245,28160],{},[49,52247,52],{"separator":51},[926,52249,928],{},[49,52251,11222],{"stretchy":73},[41,52253,18304],{"mathvariant":1462},[38,52255,52256,52258],{},[41,52257,11222],{"mathvariant":316},[41,52259,51012],{},[49,52261,11222],{"stretchy":73},[49,52263,433],{},[1556,52265,52266,52268],{},[45,52267,2266],{},[41,52269,28160],{},[49,52271,1349],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[41,52273,968],{"mathvariant":316},[86,52275,52276],{"encoding":88},"\\operatorname{VCdim}(\\mathcal H)=\\sup\\bigl\\{m:\\exists C\\subseteq\\mathcal X,\\ |C|=m,\\ \\lvert\\mathcal H|_C\\rvert=2^m\\bigr\\}.",[18,52278,52280,52310,52337,52359,52395,52471],{"className":52279,"ariaHidden":51},[93],[18,52281,52283,52286,52292,52295,52298,52301,52304,52307],{"className":52282},[97],[18,52284],{"className":52285,"style":242},[101],[18,52287,52289],{"className":52288},[985],[18,52290,52202],{"className":52291},[106,6351],[18,52293,74],{"className":52294},[250],[18,52296,18304],{"className":52297,"style":46321},[106,1478],[18,52299,84],{"className":52300},[267],[18,52302],{"className":52303,"style":228},[169],[18,52305,433],{"className":52306},[232],[18,52308],{"className":52309,"style":228},[169],[18,52311,52313,52316,52319,52325,52328,52331,52334],{"className":52312},[97],[18,52314],{"className":52315,"style":6098},[101],[18,52317,912],{"className":52318},[985],[18,52320,52322],{"className":52321},[250],[18,52323,1331],{"className":52324},[1230,1196],[18,52326,28160],{"className":52327},[106,110],[18,52329],{"className":52330,"style":228},[169],[18,52332,1340],{"className":52333},[232],[18,52335],{"className":52336,"style":228},[169],[18,52338,52340,52344,52347,52350,52353,52356],{"className":52339},[97],[18,52341],{"className":52342,"style":52343},[101],"height:0.8304em;vertical-align:-0.136em;",[18,52345,52225],{"className":52346},[106],[18,52348,51012],{"className":52349,"style":5073},[106,110],[18,52351],{"className":52352,"style":228},[169],[18,52354,4548],{"className":52355},[232],[18,52357],{"className":52358,"style":228},[169],[18,52360,52362,52365,52368,52371,52374,52377,52380,52383,52386,52389,52392],{"className":52361},[97],[18,52363],{"className":52364,"style":242},[101],[18,52366,43],{"className":52367,"style":1479},[106,1478],[18,52369,52],{"className":52370},[165],[18,52372,928],{"className":52373},[169],[18,52375],{"className":52376,"style":170},[169],[18,52378,11222],{"className":52379},[106],[18,52381,51012],{"className":52382,"style":5073},[106,110],[18,52384,11222],{"className":52385},[106],[18,52387],{"className":52388,"style":228},[169],[18,52390,433],{"className":52391},[232],[18,52393],{"className":52394,"style":228},[169],[18,52396,52398,52401,52404,52407,52410,52413,52416,52419,52459,52462,52465,52468],{"className":52397},[97],[18,52399],{"className":52400,"style":242},[101],[18,52402,28160],{"className":52403},[106,110],[18,52405,52],{"className":52406},[165],[18,52408,928],{"className":52409},[169],[18,52411],{"className":52412,"style":170},[169],[18,52414,11222],{"className":52415},[250],[18,52417,18304],{"className":52418,"style":46321},[106,1478],[18,52420,52422,52425],{"className":52421},[106],[18,52423,11222],{"className":52424},[106],[18,52426,52428],{"className":52427},[115],[18,52429,52431,52451],{"className":52430},[119,120],[18,52432,52434,52448],{"className":52433},[124],[18,52435,52437],{"className":52436,"style":1152},[128],[18,52438,52439,52442],{"style":1155},[18,52440],{"className":52441,"style":137},[136],[18,52443,52445],{"className":52444},[141,142,143,144],[18,52446,51012],{"className":52447,"style":5073},[106,110,144],[18,52449,152],{"className":52450},[151],[18,52452,52454],{"className":52453},[124],[18,52455,52457],{"className":52456,"style":159},[128],[18,52458],{},[18,52460,11222],{"className":52461},[267],[18,52463],{"className":52464,"style":228},[169],[18,52466,433],{"className":52467},[232],[18,52469],{"className":52470,"style":228},[169],[18,52472,52474,52477,52506,52512],{"className":52473},[97],[18,52475],{"className":52476,"style":6098},[101],[18,52478,52480,52483],{"className":52479},[106],[18,52481,2266],{"className":52482},[106],[18,52484,52486],{"className":52485},[115],[18,52487,52489],{"className":52488},[119],[18,52490,52492],{"className":52491},[124],[18,52493,52495],{"className":52494,"style":1616},[128],[18,52496,52497,52500],{"style":1637},[18,52498],{"className":52499,"style":137},[136],[18,52501,52503],{"className":52502},[141,142,143,144],[18,52504,28160],{"className":52505},[106,110,144],[18,52507,52509],{"className":52508},[267],[18,52510,1349],{"className":52511},[1230,1196],[18,52513,968],{"className":52514},[106],[11,52516,52517,52518,1990],{},"若右侧无界，则记 ",[18,52519,52521,52547],{"className":52520,"translate":22},[21],[18,52522,52524],{"className":52523},[26],[28,52525,52526],{"xmlns":30},[32,52527,52528,52544],{},[35,52529,52530,52532,52534,52536,52538,52540,52542],{},[41,52531,52202],{"mathvariant":316},[49,52533,899],{},[49,52535,74],{"stretchy":73},[41,52537,18304],{"mathvariant":1462},[49,52539,84],{"stretchy":73},[49,52541,433],{},[41,52543,2028],{"mathvariant":316},[86,52545,52546],{"encoding":88},"\\operatorname{VCdim}(\\mathcal H)=\\infty",[18,52548,52550,52580],{"className":52549,"ariaHidden":51},[93],[18,52551,52553,52556,52562,52565,52568,52571,52574,52577],{"className":52552},[97],[18,52554],{"className":52555,"style":242},[101],[18,52557,52559],{"className":52558},[985],[18,52560,52202],{"className":52561},[106,6351],[18,52563,74],{"className":52564},[250],[18,52566,18304],{"className":52567,"style":46321},[106,1478],[18,52569,84],{"className":52570},[267],[18,52572],{"className":52573,"style":228},[169],[18,52575,433],{"className":52576},[232],[18,52578],{"className":52579,"style":228},[169],[18,52581,52583,52586],{"className":52582},[97],[18,52584],{"className":52585,"style":1083},[101],[18,52587,2028],{"className":52588},[106],[11,52590,52591,52592,52595,52596,52624,52625,1994,52627,52685,52686,1994,52688,52716,52717,52745],{},"请把定义里的量词读三遍：",[717,52593,52594],{},"存在","一个点集 ",[18,52597,52599,52612],{"className":52598,"translate":22},[21],[18,52600,52602],{"className":52601},[26],[28,52603,52604],{"xmlns":30},[32,52605,52606,52610],{},[35,52607,52608],{},[41,52609,51012],{},[86,52611,51012],{"encoding":88},[18,52613,52615],{"className":52614,"ariaHidden":51},[93],[18,52616,52618,52621],{"className":52617},[97],[18,52619],{"className":52620,"style":600},[101],[18,52622,51012],{"className":52623,"style":5073},[106,110],"，使得对",[717,52626,1301],{},[18,52628,52630,52647],{"className":52629,"translate":22},[21],[18,52631,52633],{"className":52632},[26],[28,52634,52635],{"xmlns":30},[32,52636,52637,52645],{},[35,52638,52639],{},[1556,52640,52641,52643],{},[45,52642,2266],{},[41,52644,28160],{},[86,52646,44893],{"encoding":88},[18,52648,52650],{"className":52649,"ariaHidden":51},[93],[18,52651,52653,52656],{"className":52652},[97],[18,52654],{"className":52655,"style":1959},[101],[18,52657,52659,52662],{"className":52658},[106],[18,52660,2266],{"className":52661},[106],[18,52663,52665],{"className":52664},[115],[18,52666,52668],{"className":52667},[119],[18,52669,52671],{"className":52670},[124],[18,52672,52674],{"className":52673,"style":1959},[128],[18,52675,52676,52679],{"style":1962},[18,52677],{"className":52678,"style":137},[136],[18,52680,52682],{"className":52681},[141,142,143,144],[18,52683,28160],{"className":52684},[106,110,144]," 种标注，都",[717,52687,52594],{},[18,52689,52691,52704],{"className":52690,"translate":22},[21],[18,52692,52694],{"className":52693},[26],[28,52695,52696],{"xmlns":30},[32,52697,52698,52702],{},[35,52699,52700],{},[41,52701,18323],{},[86,52703,18323],{"encoding":88},[18,52705,52707],{"className":52706,"ariaHidden":51},[93],[18,52708,52710,52713],{"className":52709},[97],[18,52711],{"className":52712,"style":295},[101],[18,52714,18323],{"className":52715},[106,110],"。不是「对所有 ",[18,52718,52720,52733],{"className":52719,"translate":22},[21],[18,52721,52723],{"className":52722},[26],[28,52724,52725],{"xmlns":30},[32,52726,52727,52731],{},[35,52728,52729],{},[41,52730,51012],{},[86,52732,51012],{"encoding":88},[18,52734,52736],{"className":52735,"ariaHidden":51},[93],[18,52737,52739,52742],{"className":52738},[97],[18,52740],{"className":52741,"style":600},[101],[18,52743,51012],{"className":52744,"style":5073},[106,110],"」，也不是「对某一种标注」。",[11,52747,52748],{},"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。",[1311,52750,52752],{"id":52751},"_22-几个例子","2.2 几个例子",[11,52754,52755],{},[717,52756,52757,52758,52814],{},"例 1（半直线，",[18,52759,52761,52781],{"className":52760,"translate":22},[21],[18,52762,52764],{"className":52763},[26],[28,52765,52766],{"xmlns":30},[32,52767,52768,52778],{},[35,52769,52770,52772,52774,52776],{},[41,52771,52202],{"mathvariant":316},[49,52773,899],{},[49,52775,433],{},[45,52777,47],{},[86,52779,52780],{"encoding":88},"\\operatorname{VCdim}=1",[18,52782,52784,52805],{"className":52783,"ariaHidden":51},[93],[18,52785,52787,52790,52796,52799,52802],{"className":52786},[97],[18,52788],{"className":52789,"style":295},[101],[18,52791,52793],{"className":52792},[985],[18,52794,52202],{"className":52795},[106,6351],[18,52797],{"className":52798,"style":228},[169],[18,52800,433],{"className":52801},[232],[18,52803],{"className":52804,"style":228},[169],[18,52806,52808,52811],{"className":52807},[97],[18,52809],{"className":52810,"style":505},[101],[18,52812,47],{"className":52813},[106],"）",[11,52816,36183,52817,4741,52984,1990],{},[18,52818,52820,52866],{"className":52819,"translate":22},[21],[18,52821,52823],{"className":52822},[26],[28,52824,52825],{"xmlns":30},[32,52826,52827,52863],{},[35,52828,52829,52831,52833,52835,52837,52840,52842,52844,52846,52848,52851,52853,52855,52857,52859,52861],{},[41,52830,18304],{"mathvariant":1462},[49,52832,433],{},[49,52834,1331],{"stretchy":73},[41,52836,38573],{},[49,52838,52839],{},"↦",[45,52841,47],{"mathvariant":2764},[49,52843,945],{"stretchy":73},[41,52845,38573],{},[49,52847,4775],{},[41,52849,52850],{},"t",[49,52852,965],{"stretchy":73},[49,52854,1340],{},[41,52856,52850],{},[49,52858,921],{},[41,52860,3071],{"mathvariant":933},[49,52862,1349],{"stretchy":73},[86,52864,52865],{"encoding":88},"\\mathcal H=\\{x\\mapsto\\mathbf 1[x\\ge t]:t\\in\\mathbb R\\}",[18,52867,52869,52887,52908,52932,52953,52972],{"className":52868,"ariaHidden":51},[93],[18,52870,52872,52875,52878,52881,52884],{"className":52871},[97],[18,52873],{"className":52874,"style":600},[101],[18,52876,18304],{"className":52877,"style":46321},[106,1478],[18,52879],{"className":52880,"style":228},[169],[18,52882,433],{"className":52883},[232],[18,52885],{"className":52886,"style":228},[169],[18,52888,52890,52893,52896,52899,52902,52905],{"className":52889},[97],[18,52891],{"className":52892,"style":242},[101],[18,52894,1331],{"className":52895},[250],[18,52897,38573],{"className":52898},[106,110],[18,52900],{"className":52901,"style":228},[169],[18,52903,52839],{"className":52904},[232],[18,52906],{"className":52907,"style":228},[169],[18,52909,52911,52914,52917,52920,52923,52926,52929],{"className":52910},[97],[18,52912],{"className":52913,"style":242},[101],[18,52915,47],{"className":52916},[106,2859],[18,52918,945],{"className":52919},[250],[18,52921,38573],{"className":52922},[106,110],[18,52924],{"className":52925,"style":228},[169],[18,52927,4775],{"className":52928},[232],[18,52930],{"className":52931,"style":228},[169],[18,52933,52935,52938,52941,52944,52947,52950],{"className":52934},[97],[18,52936],{"className":52937,"style":242},[101],[18,52939,52850],{"className":52940},[106,110],[18,52942,965],{"className":52943},[267],[18,52945],{"className":52946,"style":228},[169],[18,52948,1340],{"className":52949},[232],[18,52951],{"className":52952,"style":228},[169],[18,52954,52956,52960,52963,52966,52969],{"className":52955},[97],[18,52957],{"className":52958,"style":52959},[101],"height:0.6542em;vertical-align:-0.0391em;",[18,52961,52850],{"className":52962},[106,110],[18,52964],{"className":52965,"style":228},[169],[18,52967,921],{"className":52968},[232],[18,52970],{"className":52971,"style":228},[169],[18,52973,52975,52978,52981],{"className":52974},[97],[18,52976],{"className":52977,"style":242},[101],[18,52979,3071],{"className":52980},[106,1139],[18,52982,1349],{"className":52983},[267],[18,52985,52987,53005],{"className":52986,"translate":22},[21],[18,52988,52990],{"className":52989},[26],[28,52991,52992],{"xmlns":30},[32,52993,52994,53002],{},[35,52995,52996,52998,53000],{},[41,52997,43],{"mathvariant":1462},[49,52999,433],{},[41,53001,3071],{"mathvariant":933},[86,53003,53004],{"encoding":88},"\\mathcal X=\\mathbb R",[18,53006,53008,53026],{"className":53007,"ariaHidden":51},[93],[18,53009,53011,53014,53017,53020,53023],{"className":53010},[97],[18,53012],{"className":53013,"style":600},[101],[18,53015,43],{"className":53016,"style":1479},[106,1478],[18,53018],{"className":53019,"style":228},[169],[18,53021,433],{"className":53022},[232],[18,53024],{"className":53025,"style":228},[169],[18,53027,53029,53032],{"className":53028},[97],[18,53030],{"className":53031,"style":45798},[101],[18,53033,3071],{"className":53034},[106,1139],[11,53036,53037,53038,53099,53100,53152,53153,4741,53181,53152,53232,1990],{},"一个点 ",[18,53039,53041,53063],{"className":53040,"translate":22},[21],[18,53042,53044],{"className":53043},[26],[28,53045,53046],{"xmlns":30},[32,53047,53048,53060],{},[35,53049,53050,53052,53054,53056,53058],{},[41,53051,51012],{},[49,53053,433],{},[49,53055,1331],{"stretchy":73},[45,53057,430],{},[49,53059,1349],{"stretchy":73},[86,53061,53062],{"encoding":88},"C=\\{0\\}",[18,53064,53066,53084],{"className":53065,"ariaHidden":51},[93],[18,53067,53069,53072,53075,53078,53081],{"className":53068},[97],[18,53070],{"className":53071,"style":600},[101],[18,53073,51012],{"className":53074,"style":5073},[106,110],[18,53076],{"className":53077,"style":228},[169],[18,53079,433],{"className":53080},[232],[18,53082],{"className":53083,"style":228},[169],[18,53085,53087,53090,53093,53096],{"className":53086},[97],[18,53088],{"className":53089,"style":242},[101],[18,53091,1331],{"className":53092},[250],[18,53094,430],{"className":53095},[106],[18,53097,1349],{"className":53098},[267]," 显然可以被打散：",[18,53101,53103,53121],{"className":53102,"translate":22},[21],[18,53104,53106],{"className":53105},[26],[28,53107,53108],{"xmlns":30},[32,53109,53110,53118],{},[35,53111,53112,53114,53116],{},[41,53113,52850],{},[49,53115,4334],{},[45,53117,430],{},[86,53119,53120],{"encoding":88},"t\\le 0",[18,53122,53124,53143],{"className":53123,"ariaHidden":51},[93],[18,53125,53127,53131,53134,53137,53140],{"className":53126},[97],[18,53128],{"className":53129,"style":53130},[101],"height:0.7719em;vertical-align:-0.136em;",[18,53132,52850],{"className":53133},[106,110],[18,53135],{"className":53136,"style":228},[169],[18,53138,4334],{"className":53139},[232],[18,53141],{"className":53142,"style":228},[169],[18,53144,53146,53149],{"className":53145},[97],[18,53147],{"className":53148,"style":505},[101],[18,53150,430],{"className":53151},[106]," 输出 ",[18,53154,53156,53169],{"className":53155,"translate":22},[21],[18,53157,53159],{"className":53158},[26],[28,53160,53161],{"xmlns":30},[32,53162,53163,53167],{},[35,53164,53165],{},[45,53166,47],{},[86,53168,47],{"encoding":88},[18,53170,53172],{"className":53171,"ariaHidden":51},[93],[18,53173,53175,53178],{"className":53174},[97],[18,53176],{"className":53177,"style":505},[101],[18,53179,47],{"className":53180},[106],[18,53182,53184,53202],{"className":53183,"translate":22},[21],[18,53185,53187],{"className":53186},[26],[28,53188,53189],{"xmlns":30},[32,53190,53191,53199],{},[35,53192,53193,53195,53197],{},[41,53194,52850],{},[49,53196,6882],{},[45,53198,430],{},[86,53200,53201],{"encoding":88},"t>0",[18,53203,53205,53223],{"className":53204,"ariaHidden":51},[93],[18,53206,53208,53211,53214,53217,53220],{"className":53207},[97],[18,53209],{"className":53210,"style":52959},[101],[18,53212,52850],{"className":53213},[106,110],[18,53215],{"className":53216,"style":228},[169],[18,53218,6882],{"className":53219},[232],[18,53221],{"className":53222,"style":228},[169],[18,53224,53226,53229],{"className":53225},[97],[18,53227],{"className":53228,"style":505},[101],[18,53230,430],{"className":53231},[106],[18,53233,53235,53248],{"className":53234,"translate":22},[21],[18,53236,53238],{"className":53237},[26],[28,53239,53240],{"xmlns":30},[32,53241,53242,53246],{},[35,53243,53244],{},[45,53245,430],{},[86,53247,430],{"encoding":88},[18,53249,53251],{"className":53250,"ariaHidden":51},[93],[18,53252,53254,53257],{"className":53253},[97],[18,53255],{"className":53256,"style":505},[101],[18,53258,430],{"className":53259},[106],[11,53261,53262,53263,53396,53397,53449,53450,53542,53543,53635,53636,53768,53769,1990],{},"两个点 ",[18,53264,53266,53292],{"className":53265,"translate":22},[21],[18,53267,53269],{"className":53268},[26],[28,53270,53271],{"xmlns":30},[32,53272,53273,53289],{},[35,53274,53275,53281,53283],{},[38,53276,53277,53279],{},[41,53278,38573],{},[45,53280,47],{},[49,53282,6681],{},[38,53284,53285,53287],{},[41,53286,38573],{},[45,53288,2266],{},[86,53290,53291],{"encoding":88},"x_1\u003Cx_2",[18,53293,53295,53350],{"className":53294,"ariaHidden":51},[93],[18,53296,53298,53301,53341,53344,53347],{"className":53297},[97],[18,53299],{"className":53300,"style":40971},[101],[18,53302,53304,53307],{"className":53303},[106],[18,53305,38573],{"className":53306},[106,110],[18,53308,53310],{"className":53309},[115],[18,53311,53313,53333],{"className":53312},[119,120],[18,53314,53316,53330],{"className":53315},[124],[18,53317,53319],{"className":53318,"style":129},[128],[18,53320,53321,53324],{"style":1155},[18,53322],{"className":53323,"style":137},[136],[18,53325,53327],{"className":53326},[141,142,143,144],[18,53328,47],{"className":53329},[106,144],[18,53331,152],{"className":53332},[151],[18,53334,53336],{"className":53335},[124],[18,53337,53339],{"className":53338,"style":159},[128],[18,53340],{},[18,53342],{"className":53343,"style":228},[169],[18,53345,6681],{"className":53346},[232],[18,53348],{"className":53349,"style":228},[169],[18,53351,53353,53356],{"className":53352},[97],[18,53354],{"className":53355,"style":16894},[101],[18,53357,53359,53362],{"className":53358},[106],[18,53360,38573],{"className":53361},[106,110],[18,53363,53365],{"className":53364},[115],[18,53366,53368,53388],{"className":53367},[119,120],[18,53369,53371,53385],{"className":53370},[124],[18,53372,53374],{"className":53373,"style":129},[128],[18,53375,53376,53379],{"style":1155},[18,53377],{"className":53378,"style":137},[136],[18,53380,53382],{"className":53381},[141,142,143,144],[18,53383,2266],{"className":53384},[106,144],[18,53386,152],{"className":53387},[151],[18,53389,53391],{"className":53390},[124],[18,53392,53394],{"className":53393,"style":159},[128],[18,53395],{}," 则不行：标注 ",[18,53398,53400,53422],{"className":53399,"translate":22},[21],[18,53401,53403],{"className":53402},[26],[28,53404,53405],{"xmlns":30},[32,53406,53407,53419],{},[35,53408,53409,53411,53413,53415,53417],{},[49,53410,74],{"stretchy":73},[45,53412,47],{},[49,53414,52],{"separator":51},[45,53416,430],{},[49,53418,84],{"stretchy":73},[86,53420,53421],{"encoding":88},"(1,0)",[18,53423,53425],{"className":53424,"ariaHidden":51},[93],[18,53426,53428,53431,53434,53437,53440,53443,53446],{"className":53427},[97],[18,53429],{"className":53430,"style":242},[101],[18,53432,74],{"className":53433},[250],[18,53435,47],{"className":53436},[106],[18,53438,52],{"className":53439},[165],[18,53441],{"className":53442,"style":170},[169],[18,53444,430],{"className":53445},[106],[18,53447,84],{"className":53448},[267]," 要求 ",[18,53451,53453,53475],{"className":53452,"translate":22},[21],[18,53454,53456],{"className":53455},[26],[28,53457,53458],{"xmlns":30},[32,53459,53460,53472],{},[35,53461,53462,53464,53466],{},[41,53463,52850],{},[49,53465,4334],{},[38,53467,53468,53470],{},[41,53469,38573],{},[45,53471,47],{},[86,53473,53474],{"encoding":88},"t\\le x_1",[18,53476,53478,53496],{"className":53477,"ariaHidden":51},[93],[18,53479,53481,53484,53487,53490,53493],{"className":53480},[97],[18,53482],{"className":53483,"style":53130},[101],[18,53485,52850],{"className":53486},[106,110],[18,53488],{"className":53489,"style":228},[169],[18,53491,4334],{"className":53492},[232],[18,53494],{"className":53495,"style":228},[169],[18,53497,53499,53502],{"className":53498},[97],[18,53500],{"className":53501,"style":16894},[101],[18,53503,53505,53508],{"className":53504},[106],[18,53506,38573],{"className":53507},[106,110],[18,53509,53511],{"className":53510},[115],[18,53512,53514,53534],{"className":53513},[119,120],[18,53515,53517,53531],{"className":53516},[124],[18,53518,53520],{"className":53519,"style":129},[128],[18,53521,53522,53525],{"style":1155},[18,53523],{"className":53524,"style":137},[136],[18,53526,53528],{"className":53527},[141,142,143,144],[18,53529,47],{"className":53530},[106,144],[18,53532,152],{"className":53533},[151],[18,53535,53537],{"className":53536},[124],[18,53538,53540],{"className":53539,"style":159},[128],[18,53541],{}," 且 ",[18,53544,53546,53568],{"className":53545,"translate":22},[21],[18,53547,53549],{"className":53548},[26],[28,53550,53551],{"xmlns":30},[32,53552,53553,53565],{},[35,53554,53555,53557,53559],{},[41,53556,52850],{},[49,53558,6882],{},[38,53560,53561,53563],{},[41,53562,38573],{},[45,53564,2266],{},[86,53566,53567],{"encoding":88},"t>x_2",[18,53569,53571,53589],{"className":53570,"ariaHidden":51},[93],[18,53572,53574,53577,53580,53583,53586],{"className":53573},[97],[18,53575],{"className":53576,"style":52959},[101],[18,53578,52850],{"className":53579},[106,110],[18,53581],{"className":53582,"style":228},[169],[18,53584,6882],{"className":53585},[232],[18,53587],{"className":53588,"style":228},[169],[18,53590,53592,53595],{"className":53591},[97],[18,53593],{"className":53594,"style":16894},[101],[18,53596,53598,53601],{"className":53597},[106],[18,53599,38573],{"className":53600},[106,110],[18,53602,53604],{"className":53603},[115],[18,53605,53607,53627],{"className":53606},[119,120],[18,53608,53610,53624],{"className":53609},[124],[18,53611,53613],{"className":53612,"style":129},[128],[18,53614,53615,53618],{"style":1155},[18,53616],{"className":53617,"style":137},[136],[18,53619,53621],{"className":53620},[141,142,143,144],[18,53622,2266],{"className":53623},[106,144],[18,53625,152],{"className":53626},[151],[18,53628,53630],{"className":53629},[124],[18,53631,53633],{"className":53632,"style":159},[128],[18,53634],{},"，与 ",[18,53637,53639,53664],{"className":53638,"translate":22},[21],[18,53640,53642],{"className":53641},[26],[28,53643,53644],{"xmlns":30},[32,53645,53646,53662],{},[35,53647,53648,53654,53656],{},[38,53649,53650,53652],{},[41,53651,38573],{},[45,53653,47],{},[49,53655,6681],{},[38,53657,53658,53660],{},[41,53659,38573],{},[45,53661,2266],{},[86,53663,53291],{"encoding":88},[18,53665,53667,53722],{"className":53666,"ariaHidden":51},[93],[18,53668,53670,53673,53713,53716,53719],{"className":53669},[97],[18,53671],{"className":53672,"style":40971},[101],[18,53674,53676,53679],{"className":53675},[106],[18,53677,38573],{"className":53678},[106,110],[18,53680,53682],{"className":53681},[115],[18,53683,53685,53705],{"className":53684},[119,120],[18,53686,53688,53702],{"className":53687},[124],[18,53689,53691],{"className":53690,"style":129},[128],[18,53692,53693,53696],{"style":1155},[18,53694],{"className":53695,"style":137},[136],[18,53697,53699],{"className":53698},[141,142,143,144],[18,53700,47],{"className":53701},[106,144],[18,53703,152],{"className":53704},[151],[18,53706,53708],{"className":53707},[124],[18,53709,53711],{"className":53710,"style":159},[128],[18,53712],{},[18,53714],{"className":53715,"style":228},[169],[18,53717,6681],{"className":53718},[232],[18,53720],{"className":53721,"style":228},[169],[18,53723,53725,53728],{"className":53724},[97],[18,53726],{"className":53727,"style":16894},[101],[18,53729,53731,53734],{"className":53730},[106],[18,53732,38573],{"className":53733},[106,110],[18,53735,53737],{"className":53736},[115],[18,53738,53740,53760],{"className":53739},[119,120],[18,53741,53743,53757],{"className":53742},[124],[18,53744,53746],{"className":53745,"style":129},[128],[18,53747,53748,53751],{"style":1155},[18,53749],{"className":53750,"style":137},[136],[18,53752,53754],{"className":53753},[141,142,143,144],[18,53755,2266],{"className":53756},[106,144],[18,53758,152],{"className":53759},[151],[18,53761,53763],{"className":53762},[124],[18,53764,53766],{"className":53765,"style":159},[128],[18,53767],{}," 矛盾。所以 VC 维是 ",[18,53770,53772,53785],{"className":53771,"translate":22},[21],[18,53773,53775],{"className":53774},[26],[28,53776,53777],{"xmlns":30},[32,53778,53779,53783],{},[35,53780,53781],{},[45,53782,47],{},[86,53784,47],{"encoding":88},[18,53786,53788],{"className":53787,"ariaHidden":51},[93],[18,53789,53791,53794],{"className":53790},[97],[18,53792],{"className":53793,"style":505},[101],[18,53795,47],{"className":53796},[106],[11,53798,53799,53800,53828,53829,53857,53858,53909,53910,1990],{},"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 ",[18,53801,53803,53816],{"className":53802,"translate":22},[21],[18,53804,53806],{"className":53805},[26],[28,53807,53808],{"xmlns":30},[32,53809,53810,53814],{},[35,53811,53812],{},[45,53813,430],{},[86,53815,430],{"encoding":88},[18,53817,53819],{"className":53818,"ariaHidden":51},[93],[18,53820,53822,53825],{"className":53821},[97],[18,53823],{"className":53824,"style":505},[101],[18,53826,430],{"className":53827},[106],"、后面全是 ",[18,53830,53832,53845],{"className":53831,"translate":22},[21],[18,53833,53835],{"className":53834},[26],[28,53836,53837],{"xmlns":30},[32,53838,53839,53843],{},[35,53840,53841],{},[45,53842,47],{},[86,53844,47],{"encoding":88},[18,53846,53848],{"className":53847,"ariaHidden":51},[93],[18,53849,53851,53854],{"className":53850},[97],[18,53852],{"className":53853,"style":505},[101],[18,53855,47],{"className":53856},[106],"」的形状，但它只能实现 ",[18,53859,53861,53879],{"className":53860,"translate":22},[21],[18,53862,53864],{"className":53863},[26],[28,53865,53866],{"xmlns":30},[32,53867,53868,53876],{},[35,53869,53870,53872,53874],{},[41,53871,28160],{},[49,53873,7984],{},[45,53875,47],{},[86,53877,53878],{"encoding":88},"m+1",[18,53880,53882,53900],{"className":53881,"ariaHidden":51},[93],[18,53883,53885,53888,53891,53894,53897],{"className":53884},[97],[18,53886],{"className":53887,"style":14401},[101],[18,53889,28160],{"className":53890},[106,110],[18,53892],{"className":53893,"style":2068},[169],[18,53895,7984],{"className":53896},[2072],[18,53898],{"className":53899,"style":2068},[169],[18,53901,53903,53906],{"className":53902},[97],[18,53904],{"className":53905,"style":505},[101],[18,53907,47],{"className":53908},[106]," 种二分，远少于 ",[18,53911,53913,53930],{"className":53912,"translate":22},[21],[18,53914,53916],{"className":53915},[26],[28,53917,53918],{"xmlns":30},[32,53919,53920,53928],{},[35,53921,53922],{},[1556,53923,53924,53926],{},[45,53925,2266],{},[41,53927,28160],{},[86,53929,44893],{"encoding":88},[18,53931,53933],{"className":53932,"ariaHidden":51},[93],[18,53934,53936,53939],{"className":53935},[97],[18,53937],{"className":53938,"style":1959},[101],[18,53940,53942,53945],{"className":53941},[106],[18,53943,2266],{"className":53944},[106],[18,53946,53948],{"className":53947},[115],[18,53949,53951],{"className":53950},[119],[18,53952,53954],{"className":53953},[124],[18,53955,53957],{"className":53956,"style":1959},[128],[18,53958,53959,53962],{"style":1962},[18,53960],{"className":53961,"style":137},[136],[18,53963,53965],{"className":53964},[141,142,143,144],[18,53966,28160],{"className":53967},[106,110,144],[11,53969,53970],{},[717,53971,53972,53973,52814],{},"例 2（区间，",[18,53974,53976,53996],{"className":53975,"translate":22},[21],[18,53977,53979],{"className":53978},[26],[28,53980,53981],{"xmlns":30},[32,53982,53983,53993],{},[35,53984,53985,53987,53989,53991],{},[41,53986,52202],{"mathvariant":316},[49,53988,899],{},[49,53990,433],{},[45,53992,2266],{},[86,53994,53995],{"encoding":88},"\\operatorname{VCdim}=2",[18,53997,53999,54020],{"className":53998,"ariaHidden":51},[93],[18,54000,54002,54005,54011,54014,54017],{"className":54001},[97],[18,54003],{"className":54004,"style":295},[101],[18,54006,54008],{"className":54007},[985],[18,54009,52202],{"className":54010},[106,6351],[18,54012],{"className":54013,"style":228},[169],[18,54015,433],{"className":54016},[232],[18,54018],{"className":54019,"style":228},[169],[18,54021,54023,54026],{"className":54022},[97],[18,54024],{"className":54025,"style":505},[101],[18,54027,2266],{"className":54028},[106],[11,54030,36183,54031,1990],{},[18,54032,54034,54083],{"className":54033,"translate":22},[21],[18,54035,54037],{"className":54036},[26],[28,54038,54039],{"xmlns":30},[32,54040,54041,54080],{},[35,54042,54043,54045,54047,54049,54051,54053,54055,54057,54059,54061,54063,54065,54068,54070,54072,54074,54076,54078],{},[41,54044,18304],{"mathvariant":1462},[49,54046,433],{},[49,54048,1331],{"stretchy":73},[41,54050,38573],{},[49,54052,52839],{},[45,54054,47],{"mathvariant":2764},[49,54056,945],{"stretchy":73},[41,54058,16820],{},[49,54060,4334],{},[41,54062,38573],{},[49,54064,4334],{},[41,54066,54067],{},"b",[49,54069,965],{"stretchy":73},[49,54071,1340],{},[41,54073,16820],{},[49,54075,4334],{},[41,54077,54067],{},[49,54079,1349],{"stretchy":73},[86,54081,54082],{"encoding":88},"\\mathcal H=\\{x\\mapsto\\mathbf 1[a\\le x\\le b]:a\\le b\\}",[18,54084,54086,54104,54125,54149,54167,54188,54206],{"className":54085,"ariaHidden":51},[93],[18,54087,54089,54092,54095,54098,54101],{"className":54088},[97],[18,54090],{"className":54091,"style":600},[101],[18,54093,18304],{"className":54094,"style":46321},[106,1478],[18,54096],{"className":54097,"style":228},[169],[18,54099,433],{"className":54100},[232],[18,54102],{"className":54103,"style":228},[169],[18,54105,54107,54110,54113,54116,54119,54122],{"className":54106},[97],[18,54108],{"className":54109,"style":242},[101],[18,54111,1331],{"className":54112},[250],[18,54114,38573],{"className":54115},[106,110],[18,54117],{"className":54118,"style":228},[169],[18,54120,52839],{"className":54121},[232],[18,54123],{"className":54124,"style":228},[169],[18,54126,54128,54131,54134,54137,54140,54143,54146],{"className":54127},[97],[18,54129],{"className":54130,"style":242},[101],[18,54132,47],{"className":54133},[106,2859],[18,54135,945],{"className":54136},[250],[18,54138,16820],{"className":54139},[106,110],[18,54141],{"className":54142,"style":228},[169],[18,54144,4334],{"className":54145},[232],[18,54147],{"className":54148,"style":228},[169],[18,54150,54152,54155,54158,54161,54164],{"className":54151},[97],[18,54153],{"className":54154,"style":53130},[101],[18,54156,38573],{"className":54157},[106,110],[18,54159],{"className":54160,"style":228},[169],[18,54162,4334],{"className":54163},[232],[18,54165],{"className":54166,"style":228},[169],[18,54168,54170,54173,54176,54179,54182,54185],{"className":54169},[97],[18,54171],{"className":54172,"style":242},[101],[18,54174,54067],{"className":54175},[106,110],[18,54177,965],{"className":54178},[267],[18,54180],{"className":54181,"style":228},[169],[18,54183,1340],{"className":54184},[232],[18,54186],{"className":54187,"style":228},[169],[18,54189,54191,54194,54197,54200,54203],{"className":54190},[97],[18,54192],{"className":54193,"style":53130},[101],[18,54195,16820],{"className":54196},[106,110],[18,54198],{"className":54199,"style":228},[169],[18,54201,4334],{"className":54202},[232],[18,54204],{"className":54205,"style":228},[169],[18,54207,54209,54212,54215],{"className":54208},[97],[18,54210],{"className":54211,"style":242},[101],[18,54213,54067],{"className":54214},[106,110],[18,54216,1349],{"className":54217},[267],[11,54219,53262,54220,54352,54353,54405,54406,2452,54579,54405,54630,2452,54867,54405,54919,2452,55156,55208],{},[18,54221,54223,54248],{"className":54222,"translate":22},[21],[18,54224,54226],{"className":54225},[26],[28,54227,54228],{"xmlns":30},[32,54229,54230,54246],{},[35,54231,54232,54238,54240],{},[38,54233,54234,54236],{},[41,54235,38573],{},[45,54237,47],{},[49,54239,6681],{},[38,54241,54242,54244],{},[41,54243,38573],{},[45,54245,2266],{},[86,54247,53291],{"encoding":88},[18,54249,54251,54306],{"className":54250,"ariaHidden":51},[93],[18,54252,54254,54257,54297,54300,54303],{"className":54253},[97],[18,54255],{"className":54256,"style":40971},[101],[18,54258,54260,54263],{"className":54259},[106],[18,54261,38573],{"className":54262},[106,110],[18,54264,54266],{"className":54265},[115],[18,54267,54269,54289],{"className":54268},[119,120],[18,54270,54272,54286],{"className":54271},[124],[18,54273,54275],{"className":54274,"style":129},[128],[18,54276,54277,54280],{"style":1155},[18,54278],{"className":54279,"style":137},[136],[18,54281,54283],{"className":54282},[141,142,143,144],[18,54284,47],{"className":54285},[106,144],[18,54287,152],{"className":54288},[151],[18,54290,54292],{"className":54291},[124],[18,54293,54295],{"className":54294,"style":159},[128],[18,54296],{},[18,54298],{"className":54299,"style":228},[169],[18,54301,6681],{"className":54302},[232],[18,54304],{"className":54305,"style":228},[169],[18,54307,54309,54312],{"className":54308},[97],[18,54310],{"className":54311,"style":16894},[101],[18,54313,54315,54318],{"className":54314},[106],[18,54316,38573],{"className":54317},[106,110],[18,54319,54321],{"className":54320},[115],[18,54322,54324,54344],{"className":54323},[119,120],[18,54325,54327,54341],{"className":54326},[124],[18,54328,54330],{"className":54329,"style":129},[128],[18,54331,54332,54335],{"style":1155},[18,54333],{"className":54334,"style":137},[136],[18,54336,54338],{"className":54337},[141,142,143,144],[18,54339,2266],{"className":54340},[106,144],[18,54342,152],{"className":54343},[151],[18,54345,54347],{"className":54346},[124],[18,54348,54350],{"className":54349,"style":159},[128],[18,54351],{}," 可以被打散：",[18,54354,54356,54378],{"className":54355,"translate":22},[21],[18,54357,54359],{"className":54358},[26],[28,54360,54361],{"xmlns":30},[32,54362,54363,54375],{},[35,54364,54365,54367,54369,54371,54373],{},[49,54366,74],{"stretchy":73},[45,54368,47],{},[49,54370,52],{"separator":51},[45,54372,47],{},[49,54374,84],{"stretchy":73},[86,54376,54377],{"encoding":88},"(1,1)",[18,54379,54381],{"className":54380,"ariaHidden":51},[93],[18,54382,54384,54387,54390,54393,54396,54399,54402],{"className":54383},[97],[18,54385],{"className":54386,"style":242},[101],[18,54388,74],{"className":54389},[250],[18,54391,47],{"className":54392},[106],[18,54394,52],{"className":54395},[165],[18,54397],{"className":54398,"style":170},[169],[18,54400,47],{"className":54401},[106],[18,54403,84],{"className":54404},[267]," 取 ",[18,54407,54409,54445],{"className":54408,"translate":22},[21],[18,54410,54412],{"className":54411},[26],[28,54413,54414],{"xmlns":30},[32,54415,54416,54442],{},[35,54417,54418,54420,54422,54428,54430,54432,54434,54436],{},[41,54419,16820],{},[49,54421,4334],{},[38,54423,54424,54426],{},[41,54425,38573],{},[45,54427,47],{},[49,54429,52],{"separator":51},[926,54431,928],{},[41,54433,54067],{},[49,54435,4775],{},[38,54437,54438,54440],{},[41,54439,38573],{},[45,54441,2266],{},[86,54443,54444],{"encoding":88},"a\\le x_1,\\ b\\ge x_2",[18,54446,54448,54466,54533],{"className":54447,"ariaHidden":51},[93],[18,54449,54451,54454,54457,54460,54463],{"className":54450},[97],[18,54452],{"className":54453,"style":53130},[101],[18,54455,16820],{"className":54456},[106,110],[18,54458],{"className":54459,"style":228},[169],[18,54461,4334],{"className":54462},[232],[18,54464],{"className":54465,"style":228},[169],[18,54467,54469,54472,54512,54515,54518,54521,54524,54527,54530],{"className":54468},[97],[18,54470],{"className":54471,"style":2890},[101],[18,54473,54475,54478],{"className":54474},[106],[18,54476,38573],{"className":54477},[106,110],[18,54479,54481],{"className":54480},[115],[18,54482,54484,54504],{"className":54483},[119,120],[18,54485,54487,54501],{"className":54486},[124],[18,54488,54490],{"className":54489,"style":129},[128],[18,54491,54492,54495],{"style":1155},[18,54493],{"className":54494,"style":137},[136],[18,54496,54498],{"className":54497},[141,142,143,144],[18,54499,47],{"className":54500},[106,144],[18,54502,152],{"className":54503},[151],[18,54505,54507],{"className":54506},[124],[18,54508,54510],{"className":54509,"style":159},[128],[18,54511],{},[18,54513,52],{"className":54514},[165],[18,54516,928],{"className":54517},[169],[18,54519],{"className":54520,"style":170},[169],[18,54522,54067],{"className":54523},[106,110],[18,54525],{"className":54526,"style":228},[169],[18,54528,4775],{"className":54529},[232],[18,54531],{"className":54532,"style":228},[169],[18,54534,54536,54539],{"className":54535},[97],[18,54537],{"className":54538,"style":16894},[101],[18,54540,54542,54545],{"className":54541},[106],[18,54543,38573],{"className":54544},[106,110],[18,54546,54548],{"className":54547},[115],[18,54549,54551,54571],{"className":54550},[119,120],[18,54552,54554,54568],{"className":54553},[124],[18,54555,54557],{"className":54556,"style":129},[128],[18,54558,54559,54562],{"style":1155},[18,54560],{"className":54561,"style":137},[136],[18,54563,54565],{"className":54564},[141,142,143,144],[18,54566,2266],{"className":54567},[106,144],[18,54569,152],{"className":54570},[151],[18,54572,54574],{"className":54573},[124],[18,54575,54577],{"className":54576,"style":159},[128],[18,54578],{},[18,54580,54582,54603],{"className":54581,"translate":22},[21],[18,54583,54585],{"className":54584},[26],[28,54586,54587],{"xmlns":30},[32,54588,54589,54601],{},[35,54590,54591,54593,54595,54597,54599],{},[49,54592,74],{"stretchy":73},[45,54594,47],{},[49,54596,52],{"separator":51},[45,54598,430],{},[49,54600,84],{"stretchy":73},[86,54602,53421],{"encoding":88},[18,54604,54606],{"className":54605,"ariaHidden":51},[93],[18,54607,54609,54612,54615,54618,54621,54624,54627],{"className":54608},[97],[18,54610],{"className":54611,"style":242},[101],[18,54613,74],{"className":54614},[250],[18,54616,47],{"className":54617},[106],[18,54619,52],{"className":54620},[165],[18,54622],{"className":54623,"style":170},[169],[18,54625,430],{"className":54626},[106],[18,54628,84],{"className":54629},[267],[18,54631,54633,54681],{"className":54632,"translate":22},[21],[18,54634,54636],{"className":54635},[26],[28,54637,54638],{"xmlns":30},[32,54639,54640,54678],{},[35,54641,54642,54644,54646,54652,54654,54656,54658,54660,54662,54668,54670,54676],{},[41,54643,16820],{},[49,54645,4334],{},[38,54647,54648,54650],{},[41,54649,38573],{},[45,54651,47],{},[49,54653,52],{"separator":51},[926,54655,928],{},[41,54657,54067],{},[49,54659,921],{},[49,54661,945],{"stretchy":73},[38,54663,54664,54666],{},[41,54665,38573],{},[45,54667,47],{},[49,54669,52],{"separator":51},[38,54671,54672,54674],{},[41,54673,38573],{},[45,54675,2266],{},[49,54677,84],{"stretchy":73},[86,54679,54680],{"encoding":88},"a\\le x_1,\\ b\\in[x_1,x_2)",[18,54682,54684,54702,54769],{"className":54683,"ariaHidden":51},[93],[18,54685,54687,54690,54693,54696,54699],{"className":54686},[97],[18,54688],{"className":54689,"style":53130},[101],[18,54691,16820],{"className":54692},[106,110],[18,54694],{"className":54695,"style":228},[169],[18,54697,4334],{"className":54698},[232],[18,54700],{"className":54701,"style":228},[169],[18,54703,54705,54708,54748,54751,54754,54757,54760,54763,54766],{"className":54704},[97],[18,54706],{"className":54707,"style":2890},[101],[18,54709,54711,54714],{"className":54710},[106],[18,54712,38573],{"className":54713},[106,110],[18,54715,54717],{"className":54716},[115],[18,54718,54720,54740],{"className":54719},[119,120],[18,54721,54723,54737],{"className":54722},[124],[18,54724,54726],{"className":54725,"style":129},[128],[18,54727,54728,54731],{"style":1155},[18,54729],{"className":54730,"style":137},[136],[18,54732,54734],{"className":54733},[141,142,143,144],[18,54735,47],{"className":54736},[106,144],[18,54738,152],{"className":54739},[151],[18,54741,54743],{"className":54742},[124],[18,54744,54746],{"className":54745,"style":159},[128],[18,54747],{},[18,54749,52],{"className":54750},[165],[18,54752,928],{"className":54753},[169],[18,54755],{"className":54756,"style":170},[169],[18,54758,54067],{"className":54759},[106,110],[18,54761],{"className":54762,"style":228},[169],[18,54764,921],{"className":54765},[232],[18,54767],{"className":54768,"style":228},[169],[18,54770,54772,54775,54778,54818,54821,54824,54864],{"className":54771},[97],[18,54773],{"className":54774,"style":242},[101],[18,54776,945],{"className":54777},[250],[18,54779,54781,54784],{"className":54780},[106],[18,54782,38573],{"className":54783},[106,110],[18,54785,54787],{"className":54786},[115],[18,54788,54790,54810],{"className":54789},[119,120],[18,54791,54793,54807],{"className":54792},[124],[18,54794,54796],{"className":54795,"style":129},[128],[18,54797,54798,54801],{"style":1155},[18,54799],{"className":54800,"style":137},[136],[18,54802,54804],{"className":54803},[141,142,143,144],[18,54805,47],{"className":54806},[106,144],[18,54808,152],{"className":54809},[151],[18,54811,54813],{"className":54812},[124],[18,54814,54816],{"className":54815,"style":159},[128],[18,54817],{},[18,54819,52],{"className":54820},[165],[18,54822],{"className":54823,"style":170},[169],[18,54825,54827,54830],{"className":54826},[106],[18,54828,38573],{"className":54829},[106,110],[18,54831,54833],{"className":54832},[115],[18,54834,54836,54856],{"className":54835},[119,120],[18,54837,54839,54853],{"className":54838},[124],[18,54840,54842],{"className":54841,"style":129},[128],[18,54843,54844,54847],{"style":1155},[18,54845],{"className":54846,"style":137},[136],[18,54848,54850],{"className":54849},[141,142,143,144],[18,54851,2266],{"className":54852},[106,144],[18,54854,152],{"className":54855},[151],[18,54857,54859],{"className":54858},[124],[18,54860,54862],{"className":54861,"style":159},[128],[18,54863],{},[18,54865,84],{"className":54866},[267],[18,54868,54870,54892],{"className":54869,"translate":22},[21],[18,54871,54873],{"className":54872},[26],[28,54874,54875],{"xmlns":30},[32,54876,54877,54889],{},[35,54878,54879,54881,54883,54885,54887],{},[49,54880,74],{"stretchy":73},[45,54882,430],{},[49,54884,52],{"separator":51},[45,54886,47],{},[49,54888,84],{"stretchy":73},[86,54890,54891],{"encoding":88},"(0,1)",[18,54893,54895],{"className":54894,"ariaHidden":51},[93],[18,54896,54898,54901,54904,54907,54910,54913,54916],{"className":54897},[97],[18,54899],{"className":54900,"style":242},[101],[18,54902,74],{"className":54903},[250],[18,54905,430],{"className":54906},[106],[18,54908,52],{"className":54909},[165],[18,54911],{"className":54912,"style":170},[169],[18,54914,47],{"className":54915},[106],[18,54917,84],{"className":54918},[267],[18,54920,54922,54970],{"className":54921,"translate":22},[21],[18,54923,54925],{"className":54924},[26],[28,54926,54927],{"xmlns":30},[32,54928,54929,54967],{},[35,54930,54931,54933,54935,54937,54943,54945,54951,54953,54955,54957,54959,54961],{},[41,54932,16820],{},[49,54934,921],{},[49,54936,74],{"stretchy":73},[38,54938,54939,54941],{},[41,54940,38573],{},[45,54942,47],{},[49,54944,52],{"separator":51},[38,54946,54947,54949],{},[41,54948,38573],{},[45,54950,2266],{},[49,54952,965],{"stretchy":73},[49,54954,52],{"separator":51},[926,54956,928],{},[41,54958,54067],{},[49,54960,4775],{},[38,54962,54963,54965],{},[41,54964,38573],{},[45,54966,2266],{},[86,54968,54969],{"encoding":88},"a\\in(x_1,x_2],\\ b\\ge x_2",[18,54971,54973,54991,55110],{"className":54972,"ariaHidden":51},[93],[18,54974,54976,54979,54982,54985,54988],{"className":54975},[97],[18,54977],{"className":54978,"style":30921},[101],[18,54980,16820],{"className":54981},[106,110],[18,54983],{"className":54984,"style":228},[169],[18,54986,921],{"className":54987},[232],[18,54989],{"className":54990,"style":228},[169],[18,54992,54994,54997,55000,55040,55043,55046,55086,55089,55092,55095,55098,55101,55104,55107],{"className":54993},[97],[18,54995],{"className":54996,"style":242},[101],[18,54998,74],{"className":54999},[250],[18,55001,55003,55006],{"className":55002},[106],[18,55004,38573],{"className":55005},[106,110],[18,55007,55009],{"className":55008},[115],[18,55010,55012,55032],{"className":55011},[119,120],[18,55013,55015,55029],{"className":55014},[124],[18,55016,55018],{"className":55017,"style":129},[128],[18,55019,55020,55023],{"style":1155},[18,55021],{"className":55022,"style":137},[136],[18,55024,55026],{"className":55025},[141,142,143,144],[18,55027,47],{"className":55028},[106,144],[18,55030,152],{"className":55031},[151],[18,55033,55035],{"className":55034},[124],[18,55036,55038],{"className":55037,"style":159},[128],[18,55039],{},[18,55041,52],{"className":55042},[165],[18,55044],{"className":55045,"style":170},[169],[18,55047,55049,55052],{"className":55048},[106],[18,55050,38573],{"className":55051},[106,110],[18,55053,55055],{"className":55054},[115],[18,55056,55058,55078],{"className":55057},[119,120],[18,55059,55061,55075],{"className":55060},[124],[18,55062,55064],{"className":55063,"style":129},[128],[18,55065,55066,55069],{"style":1155},[18,55067],{"className":55068,"style":137},[136],[18,55070,55072],{"className":55071},[141,142,143,144],[18,55073,2266],{"className":55074},[106,144],[18,55076,152],{"className":55077},[151],[18,55079,55081],{"className":55080},[124],[18,55082,55084],{"className":55083,"style":159},[128],[18,55085],{},[18,55087,965],{"className":55088},[267],[18,55090,52],{"className":55091},[165],[18,55093,928],{"className":55094},[169],[18,55096],{"className":55097,"style":170},[169],[18,55099,54067],{"className":55100},[106,110],[18,55102],{"className":55103,"style":228},[169],[18,55105,4775],{"className":55106},[232],[18,55108],{"className":55109,"style":228},[169],[18,55111,55113,55116],{"className":55112},[97],[18,55114],{"className":55115,"style":16894},[101],[18,55117,55119,55122],{"className":55118},[106],[18,55120,38573],{"className":55121},[106,110],[18,55123,55125],{"className":55124},[115],[18,55126,55128,55148],{"className":55127},[119,120],[18,55129,55131,55145],{"className":55130},[124],[18,55132,55134],{"className":55133,"style":129},[128],[18,55135,55136,55139],{"style":1155},[18,55137],{"className":55138,"style":137},[136],[18,55140,55142],{"className":55141},[141,142,143,144],[18,55143,2266],{"className":55144},[106,144],[18,55146,152],{"className":55147},[151],[18,55149,55151],{"className":55150},[124],[18,55152,55154],{"className":55153,"style":159},[128],[18,55155],{},[18,55157,55159,55181],{"className":55158,"translate":22},[21],[18,55160,55162],{"className":55161},[26],[28,55163,55164],{"xmlns":30},[32,55165,55166,55178],{},[35,55167,55168,55170,55172,55174,55176],{},[49,55169,74],{"stretchy":73},[45,55171,430],{},[49,55173,52],{"separator":51},[45,55175,430],{},[49,55177,84],{"stretchy":73},[86,55179,55180],{"encoding":88},"(0,0)",[18,55182,55184],{"className":55183,"ariaHidden":51},[93],[18,55185,55187,55190,55193,55196,55199,55202,55205],{"className":55186},[97],[18,55188],{"className":55189,"style":242},[101],[18,55191,74],{"className":55192},[250],[18,55194,430],{"className":55195},[106],[18,55197,52],{"className":55198},[165],[18,55200],{"className":55201,"style":170},[169],[18,55203,430],{"className":55204},[106],[18,55206,84],{"className":55207},[267]," 取一个避开两点的区间。",[11,55210,55211,55212,55408,55409,55474,55475,509,55545,55615,55616,55686,55687,55715],{},"三个点 ",[18,55213,55215,55249],{"className":55214,"translate":22},[21],[18,55216,55218],{"className":55217},[26],[28,55219,55220],{"xmlns":30},[32,55221,55222,55246],{},[35,55223,55224,55230,55232,55238,55240],{},[38,55225,55226,55228],{},[41,55227,38573],{},[45,55229,47],{},[49,55231,6681],{},[38,55233,55234,55236],{},[41,55235,38573],{},[45,55237,2266],{},[49,55239,6681],{},[38,55241,55242,55244],{},[41,55243,38573],{},[45,55245,44699],{},[86,55247,55248],{"encoding":88},"x_1\u003Cx_2\u003Cx_3",[18,55250,55252,55307,55362],{"className":55251,"ariaHidden":51},[93],[18,55253,55255,55258,55298,55301,55304],{"className":55254},[97],[18,55256],{"className":55257,"style":40971},[101],[18,55259,55261,55264],{"className":55260},[106],[18,55262,38573],{"className":55263},[106,110],[18,55265,55267],{"className":55266},[115],[18,55268,55270,55290],{"className":55269},[119,120],[18,55271,55273,55287],{"className":55272},[124],[18,55274,55276],{"className":55275,"style":129},[128],[18,55277,55278,55281],{"style":1155},[18,55279],{"className":55280,"style":137},[136],[18,55282,55284],{"className":55283},[141,142,143,144],[18,55285,47],{"className":55286},[106,144],[18,55288,152],{"className":55289},[151],[18,55291,55293],{"className":55292},[124],[18,55294,55296],{"className":55295,"style":159},[128],[18,55297],{},[18,55299],{"className":55300,"style":228},[169],[18,55302,6681],{"className":55303},[232],[18,55305],{"className":55306,"style":228},[169],[18,55308,55310,55313,55353,55356,55359],{"className":55309},[97],[18,55311],{"className":55312,"style":40971},[101],[18,55314,55316,55319],{"className":55315},[106],[18,55317,38573],{"className":55318},[106,110],[18,55320,55322],{"className":55321},[115],[18,55323,55325,55345],{"className":55324},[119,120],[18,55326,55328,55342],{"className":55327},[124],[18,55329,55331],{"className":55330,"style":129},[128],[18,55332,55333,55336],{"style":1155},[18,55334],{"className":55335,"style":137},[136],[18,55337,55339],{"className":55338},[141,142,143,144],[18,55340,2266],{"className":55341},[106,144],[18,55343,152],{"className":55344},[151],[18,55346,55348],{"className":55347},[124],[18,55349,55351],{"className":55350,"style":159},[128],[18,55352],{},[18,55354],{"className":55355,"style":228},[169],[18,55357,6681],{"className":55358},[232],[18,55360],{"className":55361,"style":228},[169],[18,55363,55365,55368],{"className":55364},[97],[18,55366],{"className":55367,"style":16894},[101],[18,55369,55371,55374],{"className":55370},[106],[18,55372,38573],{"className":55373},[106,110],[18,55375,55377],{"className":55376},[115],[18,55378,55380,55400],{"className":55379},[119,120],[18,55381,55383,55397],{"className":55382},[124],[18,55384,55386],{"className":55385,"style":129},[128],[18,55387,55388,55391],{"style":1155},[18,55389],{"className":55390,"style":137},[136],[18,55392,55394],{"className":55393},[141,142,143,144],[18,55395,44699],{"className":55396},[106,144],[18,55398,152],{"className":55399},[151],[18,55401,55403],{"className":55402},[124],[18,55404,55406],{"className":55405,"style":159},[128],[18,55407],{}," 不行：标注 ",[18,55410,55412,55438],{"className":55411,"translate":22},[21],[18,55413,55415],{"className":55414},[26],[28,55416,55417],{"xmlns":30},[32,55418,55419,55435],{},[35,55420,55421,55423,55425,55427,55429,55431,55433],{},[49,55422,74],{"stretchy":73},[45,55424,47],{},[49,55426,52],{"separator":51},[45,55428,430],{},[49,55430,52],{"separator":51},[45,55432,47],{},[49,55434,84],{"stretchy":73},[86,55436,55437],{"encoding":88},"(1,0,1)",[18,55439,55441],{"className":55440,"ariaHidden":51},[93],[18,55442,55444,55447,55450,55453,55456,55459,55462,55465,55468,55471],{"className":55443},[97],[18,55445],{"className":55446,"style":242},[101],[18,55448,74],{"className":55449},[250],[18,55451,47],{"className":55452},[106],[18,55454,52],{"className":55455},[165],[18,55457],{"className":55458,"style":170},[169],[18,55460,430],{"className":55461},[106],[18,55463,52],{"className":55464},[165],[18,55466],{"className":55467,"style":170},[169],[18,55469,47],{"className":55470},[106],[18,55472,84],{"className":55473},[267]," 要求区间同时包含 ",[18,55476,55478,55496],{"className":55477,"translate":22},[21],[18,55479,55481],{"className":55480},[26],[28,55482,55483],{"xmlns":30},[32,55484,55485,55493],{},[35,55486,55487],{},[38,55488,55489,55491],{},[41,55490,38573],{},[45,55492,47],{},[86,55494,55495],{"encoding":88},"x_1",[18,55497,55499],{"className":55498,"ariaHidden":51},[93],[18,55500,55502,55505],{"className":55501},[97],[18,55503],{"className":55504,"style":16894},[101],[18,55506,55508,55511],{"className":55507},[106],[18,55509,38573],{"className":55510},[106,110],[18,55512,55514],{"className":55513},[115],[18,55515,55517,55537],{"className":55516},[119,120],[18,55518,55520,55534],{"className":55519},[124],[18,55521,55523],{"className":55522,"style":129},[128],[18,55524,55525,55528],{"style":1155},[18,55526],{"className":55527,"style":137},[136],[18,55529,55531],{"className":55530},[141,142,143,144],[18,55532,47],{"className":55533},[106,144],[18,55535,152],{"className":55536},[151],[18,55538,55540],{"className":55539},[124],[18,55541,55543],{"className":55542,"style":159},[128],[18,55544],{},[18,55546,55548,55566],{"className":55547,"translate":22},[21],[18,55549,55551],{"className":55550},[26],[28,55552,55553],{"xmlns":30},[32,55554,55555,55563],{},[35,55556,55557],{},[38,55558,55559,55561],{},[41,55560,38573],{},[45,55562,44699],{},[86,55564,55565],{"encoding":88},"x_3",[18,55567,55569],{"className":55568,"ariaHidden":51},[93],[18,55570,55572,55575],{"className":55571},[97],[18,55573],{"className":55574,"style":16894},[101],[18,55576,55578,55581],{"className":55577},[106],[18,55579,38573],{"className":55580},[106,110],[18,55582,55584],{"className":55583},[115],[18,55585,55587,55607],{"className":55586},[119,120],[18,55588,55590,55604],{"className":55589},[124],[18,55591,55593],{"className":55592,"style":129},[128],[18,55594,55595,55598],{"style":1155},[18,55596],{"className":55597,"style":137},[136],[18,55599,55601],{"className":55600},[141,142,143,144],[18,55602,44699],{"className":55603},[106,144],[18,55605,152],{"className":55606},[151],[18,55608,55610],{"className":55609},[124],[18,55611,55613],{"className":55612,"style":159},[128],[18,55614],{},"，而区间是凸的，必然包含中间的 ",[18,55617,55619,55637],{"className":55618,"translate":22},[21],[18,55620,55622],{"className":55621},[26],[28,55623,55624],{"xmlns":30},[32,55625,55626,55634],{},[35,55627,55628],{},[38,55629,55630,55632],{},[41,55631,38573],{},[45,55633,2266],{},[86,55635,55636],{"encoding":88},"x_2",[18,55638,55640],{"className":55639,"ariaHidden":51},[93],[18,55641,55643,55646],{"className":55642},[97],[18,55644],{"className":55645,"style":16894},[101],[18,55647,55649,55652],{"className":55648},[106],[18,55650,38573],{"className":55651},[106,110],[18,55653,55655],{"className":55654},[115],[18,55656,55658,55678],{"className":55657},[119,120],[18,55659,55661,55675],{"className":55660},[124],[18,55662,55664],{"className":55663,"style":129},[128],[18,55665,55666,55669],{"style":1155},[18,55667],{"className":55668,"style":137},[136],[18,55670,55672],{"className":55671},[141,142,143,144],[18,55673,2266],{"className":55674},[106,144],[18,55676,152],{"className":55677},[151],[18,55679,55681],{"className":55680},[124],[18,55682,55684],{"className":55683,"style":159},[128],[18,55685],{},"，与中间的 ",[18,55688,55690,55703],{"className":55689,"translate":22},[21],[18,55691,55693],{"className":55692},[26],[28,55694,55695],{"xmlns":30},[32,55696,55697,55701],{},[35,55698,55699],{},[45,55700,430],{},[86,55702,430],{"encoding":88},[18,55704,55706],{"className":55705,"ariaHidden":51},[93],[18,55707,55709,55712],{"className":55708},[97],[18,55710],{"className":55711,"style":505},[101],[18,55713,430],{"className":55714},[106]," 矛盾。",[11,55717,55718],{},[717,55719,55720,55721,52814],{},"例 3（仿射超平面，",[18,55722,55724,55748],{"className":55723,"translate":22},[21],[18,55725,55727],{"className":55726},[26],[28,55728,55729],{"xmlns":30},[32,55730,55731,55745],{},[35,55732,55733,55735,55737,55739,55741,55743],{},[41,55734,52202],{"mathvariant":316},[49,55736,899],{},[49,55738,433],{},[41,55740,4031],{},[49,55742,7984],{},[45,55744,47],{},[86,55746,55747],{"encoding":88},"\\operatorname{VCdim}=d+1",[18,55749,55751,55772,55790],{"className":55750,"ariaHidden":51},[93],[18,55752,55754,55757,55763,55766,55769],{"className":55753},[97],[18,55755],{"className":55756,"style":295},[101],[18,55758,55760],{"className":55759},[985],[18,55761,52202],{"className":55762},[106,6351],[18,55764],{"className":55765,"style":228},[169],[18,55767,433],{"className":55768},[232],[18,55770],{"className":55771,"style":228},[169],[18,55773,55775,55778,55781,55784,55787],{"className":55774},[97],[18,55776],{"className":55777,"style":7221},[101],[18,55779,4031],{"className":55780},[106,110],[18,55782],{"className":55783,"style":2068},[169],[18,55785,7984],{"className":55786},[2072],[18,55788],{"className":55789,"style":2068},[169],[18,55791,55793,55796],{"className":55792},[97],[18,55794],{"className":55795,"style":505},[101],[18,55797,47],{"className":55798},[106],[11,55800,36183,55801,4741,55882,1990],{},[18,55802,55804,55826],{"className":55803,"translate":22},[21],[18,55805,55807],{"className":55806},[26],[28,55808,55809],{"xmlns":30},[32,55810,55811,55823],{},[35,55812,55813,55815,55817],{},[41,55814,43],{"mathvariant":1462},[49,55816,433],{},[1556,55818,55819,55821],{},[41,55820,3071],{"mathvariant":933},[41,55822,4031],{},[86,55824,55825],{"encoding":88},"\\mathcal X=\\mathbb R^d",[18,55827,55829,55847],{"className":55828,"ariaHidden":51},[93],[18,55830,55832,55835,55838,55841,55844],{"className":55831},[97],[18,55833],{"className":55834,"style":600},[101],[18,55836,43],{"className":55837,"style":1479},[106,1478],[18,55839],{"className":55840,"style":228},[169],[18,55842,433],{"className":55843},[232],[18,55845],{"className":55846,"style":228},[169],[18,55848,55850,55853],{"className":55849},[97],[18,55851],{"className":55852,"style":35177},[101],[18,55854,55856,55859],{"className":55855},[106],[18,55857,3071],{"className":55858},[106,1139],[18,55860,55862],{"className":55861},[115],[18,55863,55865],{"className":55864},[119],[18,55866,55868],{"className":55867},[124],[18,55869,55871],{"className":55870,"style":35177},[128],[18,55872,55873,55876],{"style":1962},[18,55874],{"className":55875,"style":137},[136],[18,55877,55879],{"className":55878},[141,142,143,144],[18,55880,4031],{"className":55881},[106,110,144],[18,55883,55885,55928],{"className":55884,"translate":22},[21],[18,55886,55888],{"className":55887},[26],[28,55889,55890],{"xmlns":30},[32,55891,55892,55925],{},[35,55893,55894,55896,55898,55900,55902,55904,55906,55908,55910,55913,55915,55917,55919,55921,55923],{},[41,55895,18304],{"mathvariant":1462},[49,55897,433],{},[49,55899,1331],{"stretchy":73},[41,55901,38573],{},[49,55903,52839],{},[41,55905,45630],{"mathvariant":316},[49,55907,899],{},[49,55909,74],{"stretchy":73},[41,55911,55912],{},"w",[49,55914,26833],{},[41,55916,38573],{},[49,55918,7984],{},[41,55920,54067],{},[49,55922,84],{"stretchy":73},[49,55924,1349],{"stretchy":73},[86,55926,55927],{"encoding":88},"\\mathcal H=\\{x\\mapsto\\operatorname{sign}(w\\cdot x+b)\\}",[18,55929,55931,55949,55970,55998,56016],{"className":55930,"ariaHidden":51},[93],[18,55932,55934,55937,55940,55943,55946],{"className":55933},[97],[18,55935],{"className":55936,"style":600},[101],[18,55938,18304],{"className":55939,"style":46321},[106,1478],[18,55941],{"className":55942,"style":228},[169],[18,55944,433],{"className":55945},[232],[18,55947],{"className":55948,"style":228},[169],[18,55950,55952,55955,55958,55961,55964,55967],{"className":55951},[97],[18,55953],{"className":55954,"style":242},[101],[18,55956,1331],{"className":55957},[250],[18,55959,38573],{"className":55960},[106,110],[18,55962],{"className":55963,"style":228},[169],[18,55965,52839],{"className":55966},[232],[18,55968],{"className":55969,"style":228},[169],[18,55971,55973,55976,55982,55985,55989,55992,55995],{"className":55972},[97],[18,55974],{"className":55975,"style":242},[101],[18,55977,55979],{"className":55978},[985],[18,55980,45630],{"className":55981},[106,6351],[18,55983,74],{"className":55984},[250],[18,55986,55912],{"className":55987,"style":55988},[106,110],"margin-right:0.0269em;",[18,55990],{"className":55991,"style":2068},[169],[18,55993,26833],{"className":55994},[2072],[18,55996],{"className":55997,"style":2068},[169],[18,55999,56001,56004,56007,56010,56013],{"className":56000},[97],[18,56002],{"className":56003,"style":14401},[101],[18,56005,38573],{"className":56006},[106,110],[18,56008],{"className":56009,"style":2068},[169],[18,56011,7984],{"className":56012},[2072],[18,56014],{"className":56015,"style":2068},[169],[18,56017,56019,56022,56025],{"className":56018},[97],[18,56020],{"className":56021,"style":242},[101],[18,56023,54067],{"className":56024},[106,110],[18,56026,46713],{"className":56027},[267],[11,56029,56030,56032,56033,56218,56219,56270,56271,56383],{},[717,56031,38233],{},"：取 ",[18,56034,56036,56078],{"className":56035,"translate":22},[21],[18,56037,56039],{"className":56038},[26],[28,56040,56041],{"xmlns":30},[32,56042,56043,56075],{},[35,56044,56045,56047,56049,56051,56053,56055,56061,56063,56065,56067,56073],{},[41,56046,51012],{},[49,56048,433],{},[49,56050,1331],{"stretchy":73},[45,56052,430],{},[49,56054,52],{"separator":51},[38,56056,56057,56059],{},[41,56058,14441],{},[45,56060,47],{},[49,56062,52],{"separator":51},[49,56064,55],{},[49,56066,52],{"separator":51},[38,56068,56069,56071],{},[41,56070,14441],{},[41,56072,4031],{},[49,56074,1349],{"stretchy":73},[86,56076,56077],{"encoding":88},"C=\\{0,e_1,\\dots,e_d\\}",[18,56079,56081,56099],{"className":56080,"ariaHidden":51},[93],[18,56082,56084,56087,56090,56093,56096],{"className":56083},[97],[18,56085],{"className":56086,"style":600},[101],[18,56088,51012],{"className":56089,"style":5073},[106,110],[18,56091],{"className":56092,"style":228},[169],[18,56094,433],{"className":56095},[232],[18,56097],{"className":56098,"style":228},[169],[18,56100,56102,56105,56108,56111,56114,56117,56157,56160,56163,56166,56169,56172,56175,56215],{"className":56101},[97],[18,56103],{"className":56104,"style":242},[101],[18,56106,1331],{"className":56107},[250],[18,56109,430],{"className":56110},[106],[18,56112,52],{"className":56113},[165],[18,56115],{"className":56116,"style":170},[169],[18,56118,56120,56123],{"className":56119},[106],[18,56121,14441],{"className":56122},[106,110],[18,56124,56126],{"className":56125},[115],[18,56127,56129,56149],{"className":56128},[119,120],[18,56130,56132,56146],{"className":56131},[124],[18,56133,56135],{"className":56134,"style":129},[128],[18,56136,56137,56140],{"style":1155},[18,56138],{"className":56139,"style":137},[136],[18,56141,56143],{"className":56142},[141,142,143,144],[18,56144,47],{"className":56145},[106,144],[18,56147,152],{"className":56148},[151],[18,56150,56152],{"className":56151},[124],[18,56153,56155],{"className":56154,"style":159},[128],[18,56156],{},[18,56158,52],{"className":56159},[165],[18,56161],{"className":56162,"style":170},[169],[18,56164,55],{"className":56165},[174],[18,56167],{"className":56168,"style":170},[169],[18,56170,52],{"className":56171},[165],[18,56173],{"className":56174,"style":170},[169],[18,56176,56178,56181],{"className":56177},[106],[18,56179,14441],{"className":56180},[106,110],[18,56182,56184],{"className":56183},[115],[18,56185,56187,56207],{"className":56186},[119,120],[18,56188,56190,56204],{"className":56189},[124],[18,56191,56193],{"className":56192,"style":1383},[128],[18,56194,56195,56198],{"style":1155},[18,56196],{"className":56197,"style":137},[136],[18,56199,56201],{"className":56200},[141,142,143,144],[18,56202,4031],{"className":56203},[106,110,144],[18,56205,152],{"className":56206},[151],[18,56208,56210],{"className":56209},[124],[18,56211,56213],{"className":56212,"style":159},[128],[18,56214],{},[18,56216,1349],{"className":56217},[267],"，这是 ",[18,56220,56222,56240],{"className":56221,"translate":22},[21],[18,56223,56225],{"className":56224},[26],[28,56226,56227],{"xmlns":30},[32,56228,56229,56237],{},[35,56230,56231,56233,56235],{},[41,56232,4031],{},[49,56234,7984],{},[45,56236,47],{},[86,56238,56239],{"encoding":88},"d+1",[18,56241,56243,56261],{"className":56242,"ariaHidden":51},[93],[18,56244,56246,56249,56252,56255,56258],{"className":56245},[97],[18,56247],{"className":56248,"style":7221},[101],[18,56250,4031],{"className":56251},[106,110],[18,56253],{"className":56254,"style":2068},[169],[18,56256,7984],{"className":56257},[2072],[18,56259],{"className":56260,"style":2068},[169],[18,56262,56264,56267],{"className":56263},[97],[18,56265],{"className":56266,"style":505},[101],[18,56268,47],{"className":56269},[106]," 个仿射无关的点。对任意标注 ",[18,56272,56274,56309],{"className":56273,"translate":22},[21],[18,56275,56277],{"className":56276},[26],[28,56278,56279],{"xmlns":30},[32,56280,56281,56306],{},[35,56282,56283,56285,56287,56289,56292,56294],{},[41,56284,46343],{},[49,56286,921],{},[49,56288,1331],{"stretchy":73},[49,56290,56291],{},"±",[45,56293,47],{},[1556,56295,56296,56298],{},[49,56297,1349],{"stretchy":73},[35,56299,56300,56302,56304],{},[41,56301,4031],{},[49,56303,7984],{},[45,56305,47],{},[86,56307,56308],{"encoding":88},"y\\in\\{\\pm1\\}^{d+1}",[18,56310,56312,56330],{"className":56311,"ariaHidden":51},[93],[18,56313,56315,56318,56321,56324,56327],{"className":56314},[97],[18,56316],{"className":56317,"style":14584},[101],[18,56319,46343],{"className":56320,"style":3957},[106,110],[18,56322],{"className":56323,"style":228},[169],[18,56325,921],{"className":56326},[232],[18,56328],{"className":56329,"style":228},[169],[18,56331,56333,56336,56339,56342,56345],{"className":56332},[97],[18,56334],{"className":56335,"style":35143},[101],[18,56337,1331],{"className":56338},[250],[18,56340,56291],{"className":56341},[106],[18,56343,47],{"className":56344},[106],[18,56346,56348,56351],{"className":56347},[267],[18,56349,1349],{"className":56350},[267],[18,56352,56354],{"className":56353},[115],[18,56355,56357],{"className":56356},[119],[18,56358,56360],{"className":56359},[124],[18,56361,56363],{"className":56362,"style":35177},[128],[18,56364,56365,56368],{"style":1962},[18,56366],{"className":56367,"style":137},[136],[18,56369,56371],{"className":56370},[141,142,143,144],[18,56372,56374,56377,56380],{"className":56373},[106,144],[18,56375,4031],{"className":56376},[106,110,144],[18,56378,7984],{"className":56379},[2072,144],[18,56381,47],{"className":56382},[106,144],"，方程组",[18,56385,56387],{"className":56386,"translate":22},[875],[18,56388,56390,56446],{"className":56389,"translate":22},[21],[18,56391,56393],{"className":56392},[26],[28,56394,56395],{"xmlns":30,"display":884},[32,56396,56397,56443],{},[35,56398,56399,56401,56403,56409,56411,56413,56415,56421,56423,56425,56427,56429,56431,56433,56435,56437,56439,56441],{},[41,56400,55912],{},[49,56402,26833],{},[38,56404,56405,56407],{},[41,56406,38573],{},[41,56408,5597],{},[49,56410,7984],{},[41,56412,54067],{},[49,56414,433],{},[38,56416,56417,56419],{},[41,56418,46343],{},[41,56420,5597],{},[49,56422,52],{"separator":51},[169,56424],{"width":7341},[41,56426,5597],{},[49,56428,433],{},[45,56430,47],{},[49,56432,52],{"separator":51},[49,56434,55],{},[49,56436,52],{"separator":51},[41,56438,4031],{},[49,56440,7984],{},[45,56442,47],{},[86,56444,56445],{"encoding":88},"w\\cdot x_i+b=y_i,\\qquad i=1,\\dots,d+1",[18,56447,56449,56468,56524,56542,56609,56648],{"className":56448,"ariaHidden":51},[93],[18,56450,56452,56456,56459,56462,56465],{"className":56451},[97],[18,56453],{"className":56454,"style":56455},[101],"height:0.4445em;",[18,56457,55912],{"className":56458,"style":55988},[106,110],[18,56460],{"className":56461,"style":2068},[169],[18,56463,26833],{"className":56464},[2072],[18,56466],{"className":56467,"style":2068},[169],[18,56469,56471,56475,56515,56518,56521],{"className":56470},[97],[18,56472],{"className":56473,"style":56474},[101],"height:0.7333em;vertical-align:-0.15em;",[18,56476,56478,56481],{"className":56477},[106],[18,56479,38573],{"className":56480},[106,110],[18,56482,56484],{"className":56483},[115],[18,56485,56487,56507],{"className":56486},[119,120],[18,56488,56490,56504],{"className":56489},[124],[18,56491,56493],{"className":56492,"style":5918},[128],[18,56494,56495,56498],{"style":1155},[18,56496],{"className":56497,"style":137},[136],[18,56499,56501],{"className":56500},[141,142,143,144],[18,56502,5597],{"className":56503},[106,110,144],[18,56505,152],{"className":56506},[151],[18,56508,56510],{"className":56509},[124],[18,56511,56513],{"className":56512,"style":159},[128],[18,56514],{},[18,56516],{"className":56517,"style":2068},[169],[18,56519,7984],{"className":56520},[2072],[18,56522],{"className":56523,"style":2068},[169],[18,56525,56527,56530,56533,56536,56539],{"className":56526},[97],[18,56528],{"className":56529,"style":295},[101],[18,56531,54067],{"className":56532},[106,110],[18,56534],{"className":56535,"style":228},[169],[18,56537,433],{"className":56538},[232],[18,56540],{"className":56541,"style":228},[169],[18,56543,56545,56548,56588,56591,56594,56597,56600,56603,56606],{"className":56544},[97],[18,56546],{"className":56547,"style":28331},[101],[18,56549,56551,56554],{"className":56550},[106],[18,56552,46343],{"className":56553,"style":3957},[106,110],[18,56555,56557],{"className":56556},[115],[18,56558,56560,56580],{"className":56559},[119,120],[18,56561,56563,56577],{"className":56562},[124],[18,56564,56566],{"className":56565,"style":5918},[128],[18,56567,56568,56571],{"style":30341},[18,56569],{"className":56570,"style":137},[136],[18,56572,56574],{"className":56573},[141,142,143,144],[18,56575,5597],{"className":56576},[106,110,144],[18,56578,152],{"className":56579},[151],[18,56581,56583],{"className":56582},[124],[18,56584,56586],{"className":56585,"style":159},[128],[18,56587],{},[18,56589,52],{"className":56590},[165],[18,56592],{"className":56593,"style":7673},[169],[18,56595],{"className":56596,"style":170},[169],[18,56598,5597],{"className":56599},[106,110],[18,56601],{"className":56602,"style":228},[169],[18,56604,433],{"className":56605},[232],[18,56607],{"className":56608,"style":228},[169],[18,56610,56612,56615,56618,56621,56624,56627,56630,56633,56636,56639,56642,56645],{"className":56611},[97],[18,56613],{"className":56614,"style":2890},[101],[18,56616,47],{"className":56617},[106],[18,56619,52],{"className":56620},[165],[18,56622],{"className":56623,"style":170},[169],[18,56625,55],{"className":56626},[174],[18,56628],{"className":56629,"style":170},[169],[18,56631,52],{"className":56632},[165],[18,56634],{"className":56635,"style":170},[169],[18,56637,4031],{"className":56638},[106,110],[18,56640],{"className":56641,"style":2068},[169],[18,56643,7984],{"className":56644},[2072],[18,56646],{"className":56647,"style":2068},[169],[18,56649,56651,56654],{"className":56650},[97],[18,56652],{"className":56653,"style":505},[101],[18,56655,47],{"className":56656},[106],[11,56658,56659,56660,56753,56754,56869,56870,56898],{},"有唯一解，因为以 ",[18,56661,56663,56689],{"className":56662,"translate":22},[21],[18,56664,56666],{"className":56665},[26],[28,56667,56668],{"xmlns":30},[32,56669,56670,56686],{},[35,56671,56672,56674,56680,56682,56684],{},[49,56673,74],{"stretchy":73},[38,56675,56676,56678],{},[41,56677,38573],{},[41,56679,5597],{},[49,56681,52],{"separator":51},[45,56683,47],{},[49,56685,84],{"stretchy":73},[86,56687,56688],{"encoding":88},"(x_i,1)",[18,56690,56692],{"className":56691,"ariaHidden":51},[93],[18,56693,56695,56698,56701,56741,56744,56747,56750],{"className":56694},[97],[18,56696],{"className":56697,"style":242},[101],[18,56699,74],{"className":56700},[250],[18,56702,56704,56707],{"className":56703},[106],[18,56705,38573],{"className":56706},[106,110],[18,56708,56710],{"className":56709},[115],[18,56711,56713,56733],{"className":56712},[119,120],[18,56714,56716,56730],{"className":56715},[124],[18,56717,56719],{"className":56718,"style":5918},[128],[18,56720,56721,56724],{"style":1155},[18,56722],{"className":56723,"style":137},[136],[18,56725,56727],{"className":56726},[141,142,143,144],[18,56728,5597],{"className":56729},[106,110,144],[18,56731,152],{"className":56732},[151],[18,56734,56736],{"className":56735},[124],[18,56737,56739],{"className":56738,"style":159},[128],[18,56740],{},[18,56742,52],{"className":56743},[165],[18,56745],{"className":56746,"style":170},[169],[18,56748,47],{"className":56749},[106],[18,56751,84],{"className":56752},[267]," 为列的 ",[18,56755,56757,56791],{"className":56756,"translate":22},[21],[18,56758,56760],{"className":56759},[26],[28,56761,56762],{"xmlns":30},[32,56763,56764,56788],{},[35,56765,56766,56768,56770,56772,56774,56776,56778,56780,56782,56784,56786],{},[49,56767,74],{"stretchy":73},[41,56769,4031],{},[49,56771,7984],{},[45,56773,47],{},[49,56775,84],{"stretchy":73},[49,56777,2015],{},[49,56779,74],{"stretchy":73},[41,56781,4031],{},[49,56783,7984],{},[45,56785,47],{},[49,56787,84],{"stretchy":73},[86,56789,56790],{"encoding":88},"(d+1)\\times(d+1)",[18,56792,56794,56815,56836,56857],{"className":56793,"ariaHidden":51},[93],[18,56795,56797,56800,56803,56806,56809,56812],{"className":56796},[97],[18,56798],{"className":56799,"style":242},[101],[18,56801,74],{"className":56802},[250],[18,56804,4031],{"className":56805},[106,110],[18,56807],{"className":56808,"style":2068},[169],[18,56810,7984],{"className":56811},[2072],[18,56813],{"className":56814,"style":2068},[169],[18,56816,56818,56821,56824,56827,56830,56833],{"className":56817},[97],[18,56819],{"className":56820,"style":242},[101],[18,56822,47],{"className":56823},[106],[18,56825,84],{"className":56826},[267],[18,56828],{"className":56829,"style":2068},[169],[18,56831,2015],{"className":56832},[2072],[18,56834],{"className":56835,"style":2068},[169],[18,56837,56839,56842,56845,56848,56851,56854],{"className":56838},[97],[18,56840],{"className":56841,"style":242},[101],[18,56843,74],{"className":56844},[250],[18,56846,4031],{"className":56847},[106,110],[18,56849],{"className":56850,"style":2068},[169],[18,56852,7984],{"className":56853},[2072],[18,56855],{"className":56856,"style":2068},[169],[18,56858,56860,56863,56866],{"className":56859},[97],[18,56861],{"className":56862,"style":242},[101],[18,56864,47],{"className":56865},[106],[18,56867,84],{"className":56868},[267]," 矩阵可逆。于是每种标注都能被某个仿射函数实现，",[18,56871,56873,56886],{"className":56872,"translate":22},[21],[18,56874,56876],{"className":56875},[26],[28,56877,56878],{"xmlns":30},[32,56879,56880,56884],{},[35,56881,56882],{},[41,56883,51012],{},[86,56885,51012],{"encoding":88},[18,56887,56889],{"className":56888,"ariaHidden":51},[93],[18,56890,56892,56895],{"className":56891},[97],[18,56893],{"className":56894,"style":600},[101],[18,56896,51012],{"className":56897,"style":5073},[106,110]," 被打散。",[11,56900,56901,56904,56905,56964,56965,57016],{},[717,56902,56903],{},"上界","：",[18,56906,56908,56926],{"className":56907,"translate":22},[21],[18,56909,56911],{"className":56910},[26],[28,56912,56913],{"xmlns":30},[32,56914,56915,56923],{},[35,56916,56917],{},[1556,56918,56919,56921],{},[41,56920,3071],{"mathvariant":933},[41,56922,4031],{},[86,56924,56925],{"encoding":88},"\\mathbb R^d",[18,56927,56929],{"className":56928,"ariaHidden":51},[93],[18,56930,56932,56935],{"className":56931},[97],[18,56933],{"className":56934,"style":35177},[101],[18,56936,56938,56941],{"className":56937},[106],[18,56939,3071],{"className":56940},[106,1139],[18,56942,56944],{"className":56943},[115],[18,56945,56947],{"className":56946},[119],[18,56948,56950],{"className":56949},[124],[18,56951,56953],{"className":56952,"style":35177},[128],[18,56954,56955,56958],{"style":1962},[18,56956],{"className":56957,"style":137},[136],[18,56959,56961],{"className":56960},[141,142,143,144],[18,56962,4031],{"className":56963},[106,110,144]," 中任意 ",[18,56966,56968,56986],{"className":56967,"translate":22},[21],[18,56969,56971],{"className":56970},[26],[28,56972,56973],{"xmlns":30},[32,56974,56975,56983],{},[35,56976,56977,56979,56981],{},[41,56978,4031],{},[49,56980,7984],{},[45,56982,2266],{},[86,56984,56985],{"encoding":88},"d+2",[18,56987,56989,57007],{"className":56988,"ariaHidden":51},[93],[18,56990,56992,56995,56998,57001,57004],{"className":56991},[97],[18,56993],{"className":56994,"style":7221},[101],[18,56996,4031],{"className":56997},[106,110],[18,56999],{"className":57000,"style":2068},[169],[18,57002,7984],{"className":57003},[2072],[18,57005],{"className":57006,"style":2068},[169],[18,57008,57010,57013],{"className":57009},[97],[18,57011],{"className":57012,"style":505},[101],[18,57014,2266],{"className":57015},[106]," 个点都仿射相关，因此可以用 Radon 定理处理。",[5159,57018,57020],{"title":57019},"定理（Radon，1921）",[11,57021,57022,56964,57080,57130],{},[18,57023,57025,57042],{"className":57024,"translate":22},[21],[18,57026,57028],{"className":57027},[26],[28,57029,57030],{"xmlns":30},[32,57031,57032,57040],{},[35,57033,57034],{},[1556,57035,57036,57038],{},[41,57037,3071],{"mathvariant":933},[41,57039,4031],{},[86,57041,56925],{"encoding":88},[18,57043,57045],{"className":57044,"ariaHidden":51},[93],[18,57046,57048,57051],{"className":57047},[97],[18,57049],{"className":57050,"style":35177},[101],[18,57052,57054,57057],{"className":57053},[106],[18,57055,3071],{"className":57056},[106,1139],[18,57058,57060],{"className":57059},[115],[18,57061,57063],{"className":57062},[119],[18,57064,57066],{"className":57065},[124],[18,57067,57069],{"className":57068,"style":35177},[128],[18,57070,57071,57074],{"style":1962},[18,57072],{"className":57073,"style":137},[136],[18,57075,57077],{"className":57076},[141,142,143,144],[18,57078,4031],{"className":57079},[106,110,144],[18,57081,57083,57100],{"className":57082,"translate":22},[21],[18,57084,57086],{"className":57085},[26],[28,57087,57088],{"xmlns":30},[32,57089,57090,57098],{},[35,57091,57092,57094,57096],{},[41,57093,4031],{},[49,57095,7984],{},[45,57097,2266],{},[86,57099,56985],{"encoding":88},[18,57101,57103,57121],{"className":57102,"ariaHidden":51},[93],[18,57104,57106,57109,57112,57115,57118],{"className":57105},[97],[18,57107],{"className":57108,"style":7221},[101],[18,57110,4031],{"className":57111},[106,110],[18,57113],{"className":57114,"style":2068},[169],[18,57116,7984],{"className":57117},[2072],[18,57119],{"className":57120,"style":2068},[169],[18,57122,57124,57127],{"className":57123},[97],[18,57125],{"className":57126,"style":505},[101],[18,57128,2266],{"className":57129},[106]," 个点都可以分成两个不相交的子集，使二者的凸包相交。",[11,57132,57133,57134,4741,57294,57344,57345,57418],{},"证明很短。把点写成齐次坐标 ",[18,57135,57137,57177],{"className":57136,"translate":22},[21],[18,57138,57140],{"className":57139},[26],[28,57141,57142],{"xmlns":30},[32,57143,57144,57174],{},[35,57145,57146,57148,57154,57156,57158,57160,57162],{},[49,57147,74],{"stretchy":73},[38,57149,57150,57152],{},[41,57151,38573],{},[41,57153,5597],{},[49,57155,52],{"separator":51},[45,57157,47],{},[49,57159,84],{"stretchy":73},[49,57161,921],{},[1556,57163,57164,57166],{},[41,57165,3071],{"mathvariant":933},[35,57167,57168,57170,57172],{},[41,57169,4031],{},[49,57171,7984],{},[45,57173,47],{},[86,57175,57176],{"encoding":88},"(x_i,1)\\in\\mathbb R^{d+1}",[18,57178,57180,57250],{"className":57179,"ariaHidden":51},[93],[18,57181,57183,57186,57189,57229,57232,57235,57238,57241,57244,57247],{"className":57182},[97],[18,57184],{"className":57185,"style":242},[101],[18,57187,74],{"className":57188},[250],[18,57190,57192,57195],{"className":57191},[106],[18,57193,38573],{"className":57194},[106,110],[18,57196,57198],{"className":57197},[115],[18,57199,57201,57221],{"className":57200},[119,120],[18,57202,57204,57218],{"className":57203},[124],[18,57205,57207],{"className":57206,"style":5918},[128],[18,57208,57209,57212],{"style":1155},[18,57210],{"className":57211,"style":137},[136],[18,57213,57215],{"className":57214},[141,142,143,144],[18,57216,5597],{"className":57217},[106,110,144],[18,57219,152],{"className":57220},[151],[18,57222,57224],{"className":57223},[124],[18,57225,57227],{"className":57226,"style":159},[128],[18,57228],{},[18,57230,52],{"className":57231},[165],[18,57233],{"className":57234,"style":170},[169],[18,57236,47],{"className":57237},[106],[18,57239,84],{"className":57240},[267],[18,57242],{"className":57243,"style":228},[169],[18,57245,921],{"className":57246},[232],[18,57248],{"className":57249,"style":228},[169],[18,57251,57253,57256],{"className":57252},[97],[18,57254],{"className":57255,"style":35177},[101],[18,57257,57259,57262],{"className":57258},[106],[18,57260,3071],{"className":57261},[106,1139],[18,57263,57265],{"className":57264},[115],[18,57266,57268],{"className":57267},[119],[18,57269,57271],{"className":57270},[124],[18,57272,57274],{"className":57273,"style":35177},[128],[18,57275,57276,57279],{"style":1962},[18,57277],{"className":57278,"style":137},[136],[18,57280,57282],{"className":57281},[141,142,143,144],[18,57283,57285,57288,57291],{"className":57284},[106,144],[18,57286,4031],{"className":57287},[106,110,144],[18,57289,7984],{"className":57290},[2072,144],[18,57292,47],{"className":57293},[106,144],[18,57295,57297,57314],{"className":57296,"translate":22},[21],[18,57298,57300],{"className":57299},[26],[28,57301,57302],{"xmlns":30},[32,57303,57304,57312],{},[35,57305,57306,57308,57310],{},[41,57307,4031],{},[49,57309,7984],{},[45,57311,2266],{},[86,57313,56985],{"encoding":88},[18,57315,57317,57335],{"className":57316,"ariaHidden":51},[93],[18,57318,57320,57323,57326,57329,57332],{"className":57319},[97],[18,57321],{"className":57322,"style":7221},[101],[18,57324,4031],{"className":57325},[106,110],[18,57327],{"className":57328,"style":2068},[169],[18,57330,7984],{"className":57331},[2072],[18,57333],{"className":57334,"style":2068},[169],[18,57336,57338,57341],{"className":57337},[97],[18,57339],{"className":57340,"style":505},[101],[18,57342,2266],{"className":57343},[106]," 个向量线性相关，故存在不全为零的系数 ",[18,57346,57348,57367],{"className":57347,"translate":22},[21],[18,57349,57351],{"className":57350},[26],[28,57352,57353],{"xmlns":30},[32,57354,57355,57364],{},[35,57356,57357],{},[38,57358,57359,57362],{},[41,57360,57361],{},"α",[41,57363,5597],{},[86,57365,57366],{"encoding":88},"\\alpha_i",[18,57368,57370],{"className":57369,"ariaHidden":51},[93],[18,57371,57373,57376],{"className":57372},[97],[18,57374],{"className":57375,"style":16894},[101],[18,57377,57379,57383],{"className":57378},[106],[18,57380,57361],{"className":57381,"style":57382},[106,110],"margin-right:0.0037em;",[18,57384,57386],{"className":57385},[115],[18,57387,57389,57410],{"className":57388},[119,120],[18,57390,57392,57407],{"className":57391},[124],[18,57393,57395],{"className":57394,"style":5918},[128],[18,57396,57398,57401],{"style":57397},"top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;",[18,57399],{"className":57400,"style":137},[136],[18,57402,57404],{"className":57403},[141,142,143,144],[18,57405,5597],{"className":57406},[106,110,144],[18,57408,152],{"className":57409},[151],[18,57411,57413],{"className":57412},[124],[18,57414,57416],{"className":57415,"style":159},[128],[18,57417],{},"，使",[18,57420,57422],{"className":57421,"translate":22},[875],[18,57423,57425,57480],{"className":57424,"translate":22},[21],[18,57426,57428],{"className":57427},[26],[28,57429,57430],{"xmlns":30,"display":884},[32,57431,57432,57477],{},[35,57433,57434,57440,57446,57452,57454,57456,57458,57460,57466,57472,57474],{},[890,57435,57436,57438],{},[49,57437,20362],{},[41,57439,5597],{},[38,57441,57442,57444],{},[41,57443,57361],{},[41,57445,5597],{},[38,57447,57448,57450],{},[41,57449,38573],{},[41,57451,5597],{},[49,57453,433],{},[45,57455,430],{},[49,57457,52],{"separator":51},[169,57459],{"width":7341},[890,57461,57462,57464],{},[49,57463,20362],{},[41,57465,5597],{},[38,57467,57468,57470],{},[41,57469,57361],{},[41,57471,5597],{},[49,57473,433],{},[45,57475,57476],{},"0.",[86,57478,57479],{"encoding":88},"\\sum_i \\alpha_i x_i=0,\\qquad \\sum_i\\alpha_i=0.",[18,57481,57483,57626,57740],{"className":57482,"ariaHidden":51},[93],[18,57484,57486,57490,57534,57537,57577,57617,57620,57623],{"className":57485},[97],[18,57487],{"className":57488,"style":57489},[101],"height:2.3277em;vertical-align:-1.2777em;",[18,57491,57493],{"className":57492},[985,986],[18,57494,57496,57526],{"className":57495},[119,120],[18,57497,57499,57523],{"className":57498},[124],[18,57500,57502,57513],{"className":57501,"style":24476},[128],[18,57503,57504,57507],{"style":20548},[18,57505],{"className":57506,"style":20552},[136],[18,57508,57510],{"className":57509},[141,142,143,144],[18,57511,5597],{"className":57512},[106,110,144],[18,57514,57515,57518],{"style":20570},[18,57516],{"className":57517,"style":20552},[136],[18,57519,57520],{},[18,57521,20362],{"className":57522},[985,4198,4199],[18,57524,152],{"className":57525},[151],[18,57527,57529],{"className":57528},[124],[18,57530,57532],{"className":57531,"style":20600},[128],[18,57533],{},[18,57535],{"className":57536,"style":170},[169],[18,57538,57540,57543],{"className":57539},[106],[18,57541,57361],{"className":57542,"style":57382},[106,110],[18,57544,57546],{"className":57545},[115],[18,57547,57549,57569],{"className":57548},[119,120],[18,57550,57552,57566],{"className":57551},[124],[18,57553,57555],{"className":57554,"style":5918},[128],[18,57556,57557,57560],{"style":57397},[18,57558],{"className":57559,"style":137},[136],[18,57561,57563],{"className":57562},[141,142,143,144],[18,57564,5597],{"className":57565},[106,110,144],[18,57567,152],{"className":57568},[151],[18,57570,57572],{"className":57571},[124],[18,57573,57575],{"className":57574,"style":159},[128],[18,57576],{},[18,57578,57580,57583],{"className":57579},[106],[18,57581,38573],{"className":57582},[106,110],[18,57584,57586],{"className":57585},[115],[18,57587,57589,57609],{"className":57588},[119,120],[18,57590,57592,57606],{"className":57591},[124],[18,57593,57595],{"className":57594,"style":5918},[128],[18,57596,57597,57600],{"style":1155},[18,57598],{"className":57599,"style":137},[136],[18,57601,57603],{"className":57602},[141,142,143,144],[18,57604,5597],{"className":57605},[106,110,144],[18,57607,152],{"className":57608},[151],[18,57610,57612],{"className":57611},[124],[18,57613,57615],{"className":57614,"style":159},[128],[18,57616],{},[18,57618],{"className":57619,"style":228},[169],[18,57621,433],{"className":57622},[232],[18,57624],{"className":57625,"style":228},[169],[18,57627,57629,57632,57635,57638,57641,57644,57688,57691,57731,57734,57737],{"className":57628},[97],[18,57630],{"className":57631,"style":57489},[101],[18,57633,430],{"className":57634},[106],[18,57636,52],{"className":57637},[165],[18,57639],{"className":57640,"style":7673},[169],[18,57642],{"className":57643,"style":170},[169],[18,57645,57647],{"className":57646},[985,986],[18,57648,57650,57680],{"className":57649},[119,120],[18,57651,57653,57677],{"className":57652},[124],[18,57654,57656,57667],{"className":57655,"style":24476},[128],[18,57657,57658,57661],{"style":20548},[18,57659],{"className":57660,"style":20552},[136],[18,57662,57664],{"className":57663},[141,142,143,144],[18,57665,5597],{"className":57666},[106,110,144],[18,57668,57669,57672],{"style":20570},[18,57670],{"className":57671,"style":20552},[136],[18,57673,57674],{},[18,57675,20362],{"className":57676},[985,4198,4199],[18,57678,152],{"className":57679},[151],[18,57681,57683],{"className":57682},[124],[18,57684,57686],{"className":57685,"style":20600},[128],[18,57687],{},[18,57689],{"className":57690,"style":170},[169],[18,57692,57694,57697],{"className":57693},[106],[18,57695,57361],{"className":57696,"style":57382},[106,110],[18,57698,57700],{"className":57699},[115],[18,57701,57703,57723],{"className":57702},[119,120],[18,57704,57706,57720],{"className":57705},[124],[18,57707,57709],{"className":57708,"style":5918},[128],[18,57710,57711,57714],{"style":57397},[18,57712],{"className":57713,"style":137},[136],[18,57715,57717],{"className":57716},[141,142,143,144],[18,57718,5597],{"className":57719},[106,110,144],[18,57721,152],{"className":57722},[151],[18,57724,57726],{"className":57725},[124],[18,57727,57729],{"className":57728,"style":159},[128],[18,57730],{},[18,57732],{"className":57733,"style":228},[169],[18,57735,433],{"className":57736},[232],[18,57738],{"className":57739,"style":228},[169],[18,57741,57743,57746],{"className":57742},[97],[18,57744],{"className":57745,"style":505},[101],[18,57747,57476],{"className":57748},[106],[11,57750,20026,57751,4741,57897,58043],{},[18,57752,57754,57788],{"className":57753,"translate":22},[21],[18,57755,57757],{"className":57756},[26],[28,57758,57759],{"xmlns":30},[32,57760,57761,57785],{},[35,57762,57763,57765,57767,57769,57771,57773,57779,57781,57783],{},[41,57764,939],{},[49,57766,433],{},[49,57768,1331],{"stretchy":73},[41,57770,5597],{},[49,57772,1340],{},[38,57774,57775,57777],{},[41,57776,57361],{},[41,57778,5597],{},[49,57780,6882],{},[45,57782,430],{},[49,57784,1349],{"stretchy":73},[86,57786,57787],{"encoding":88},"P=\\{i:\\alpha_i>0\\}",[18,57789,57791,57809,57830,57885],{"className":57790,"ariaHidden":51},[93],[18,57792,57794,57797,57800,57803,57806],{"className":57793},[97],[18,57795],{"className":57796,"style":600},[101],[18,57798,939],{"className":57799,"style":1171},[106,110],[18,57801],{"className":57802,"style":228},[169],[18,57804,433],{"className":57805},[232],[18,57807],{"className":57808,"style":228},[169],[18,57810,57812,57815,57818,57821,57824,57827],{"className":57811},[97],[18,57813],{"className":57814,"style":242},[101],[18,57816,1331],{"className":57817},[250],[18,57819,5597],{"className":57820},[106,110],[18,57822],{"className":57823,"style":228},[169],[18,57825,1340],{"className":57826},[232],[18,57828],{"className":57829,"style":228},[169],[18,57831,57833,57836,57876,57879,57882],{"className":57832},[97],[18,57834],{"className":57835,"style":40971},[101],[18,57837,57839,57842],{"className":57838},[106],[18,57840,57361],{"className":57841,"style":57382},[106,110],[18,57843,57845],{"className":57844},[115],[18,57846,57848,57868],{"className":57847},[119,120],[18,57849,57851,57865],{"className":57850},[124],[18,57852,57854],{"className":57853,"style":5918},[128],[18,57855,57856,57859],{"style":57397},[18,57857],{"className":57858,"style":137},[136],[18,57860,57862],{"className":57861},[141,142,143,144],[18,57863,5597],{"className":57864},[106,110,144],[18,57866,152],{"className":57867},[151],[18,57869,57871],{"className":57870},[124],[18,57872,57874],{"className":57873,"style":159},[128],[18,57875],{},[18,57877],{"className":57878,"style":228},[169],[18,57880,6882],{"className":57881},[232],[18,57883],{"className":57884,"style":228},[169],[18,57886,57888,57891,57894],{"className":57887},[97],[18,57889],{"className":57890,"style":242},[101],[18,57892,430],{"className":57893},[106],[18,57895,1349],{"className":57896},[267],[18,57898,57900,57934],{"className":57899,"translate":22},[21],[18,57901,57903],{"className":57902},[26],[28,57904,57905],{"xmlns":30},[32,57906,57907,57931],{},[35,57908,57909,57911,57913,57915,57917,57919,57925,57927,57929],{},[41,57910,70],{},[49,57912,433],{},[49,57914,1331],{"stretchy":73},[41,57916,5597],{},[49,57918,1340],{},[38,57920,57921,57923],{},[41,57922,57361],{},[41,57924,5597],{},[49,57926,6681],{},[45,57928,430],{},[49,57930,1349],{"stretchy":73},[86,57932,57933],{"encoding":88},"N=\\{i:\\alpha_i\u003C0\\}",[18,57935,57937,57955,57976,58031],{"className":57936,"ariaHidden":51},[93],[18,57938,57940,57943,57946,57949,57952],{"className":57939},[97],[18,57941],{"className":57942,"style":600},[101],[18,57944,70],{"className":57945,"style":246},[106,110],[18,57947],{"className":57948,"style":228},[169],[18,57950,433],{"className":57951},[232],[18,57953],{"className":57954,"style":228},[169],[18,57956,57958,57961,57964,57967,57970,57973],{"className":57957},[97],[18,57959],{"className":57960,"style":242},[101],[18,57962,1331],{"className":57963},[250],[18,57965,5597],{"className":57966},[106,110],[18,57968],{"className":57969,"style":228},[169],[18,57971,1340],{"className":57972},[232],[18,57974],{"className":57975,"style":228},[169],[18,57977,57979,57982,58022,58025,58028],{"className":57978},[97],[18,57980],{"className":57981,"style":40971},[101],[18,57983,57985,57988],{"className":57984},[106],[18,57986,57361],{"className":57987,"style":57382},[106,110],[18,57989,57991],{"className":57990},[115],[18,57992,57994,58014],{"className":57993},[119,120],[18,57995,57997,58011],{"className":57996},[124],[18,57998,58000],{"className":57999,"style":5918},[128],[18,58001,58002,58005],{"style":57397},[18,58003],{"className":58004,"style":137},[136],[18,58006,58008],{"className":58007},[141,142,143,144],[18,58009,5597],{"className":58010},[106,110,144],[18,58012,152],{"className":58013},[151],[18,58015,58017],{"className":58016},[124],[18,58018,58020],{"className":58019,"style":159},[128],[18,58021],{},[18,58023],{"className":58024,"style":228},[169],[18,58026,6681],{"className":58027},[232],[18,58029],{"className":58030,"style":228},[169],[18,58032,58034,58037,58040],{"className":58033},[97],[18,58035],{"className":58036,"style":242},[101],[18,58038,430],{"className":58039},[106],[18,58041,1349],{"className":58042},[267],"。两者都非空，且",[18,58045,58047],{"className":58046,"translate":22},[875],[18,58048,58050,58120],{"className":58049,"translate":22},[21],[18,58051,58053],{"className":58052},[26],[28,58054,58055],{"xmlns":30,"display":884},[32,58056,58057,58117],{},[35,58058,58059,58071,58077,58083,58085,58097,58099,58101,58107,58109,58115],{},[890,58060,58061,58063],{},[49,58062,20362],{},[35,58064,58065,58067,58069],{},[41,58066,5597],{},[49,58068,921],{},[41,58070,939],{},[38,58072,58073,58075],{},[41,58074,57361],{},[41,58076,5597],{},[38,58078,58079,58081],{},[41,58080,38573],{},[41,58082,5597],{},[49,58084,433],{},[890,58086,58087,58089],{},[49,58088,20362],{},[35,58090,58091,58093,58095],{},[41,58092,5597],{},[49,58094,921],{},[41,58096,70],{},[49,58098,74],{"stretchy":73},[49,58100,2253],{},[38,58102,58103,58105],{},[41,58104,57361],{},[41,58106,5597],{},[49,58108,84],{"stretchy":73},[38,58110,58111,58113],{},[41,58112,38573],{},[41,58114,5597],{},[41,58116,968],{"mathvariant":316},[86,58118,58119],{"encoding":88},"\\sum_{i\\in P}\\alpha_i x_i=\\sum_{i\\in N}(-\\alpha_i)x_i.",[18,58121,58123,58276],{"className":58122,"ariaHidden":51},[93],[18,58124,58126,58130,58184,58187,58227,58267,58270,58273],{"className":58125},[97],[18,58127],{"className":58128,"style":58129},[101],"height:2.3717em;vertical-align:-1.3217em;",[18,58131,58133],{"className":58132},[985,986],[18,58134,58136,58175],{"className":58135},[119,120],[18,58137,58139,58172],{"className":58138},[124],[18,58140,58142,58162],{"className":58141,"style":24476},[128],[18,58143,58144,58147],{"style":39158},[18,58145],{"className":58146,"style":20552},[136],[18,58148,58150],{"className":58149},[141,142,143,144],[18,58151,58153,58156,58159],{"className":58152},[106,144],[18,58154,5597],{"className":58155},[106,110,144],[18,58157,921],{"className":58158},[232,144],[18,58160,939],{"className":58161,"style":1171},[106,110,144],[18,58163,58164,58167],{"style":20570},[18,58165],{"className":58166,"style":20552},[136],[18,58168,58169],{},[18,58170,20362],{"className":58171},[985,4198,4199],[18,58173,152],{"className":58174},[151],[18,58176,58178],{"className":58177},[124],[18,58179,58182],{"className":58180,"style":58181},[128],"height:1.3217em;",[18,58183],{},[18,58185],{"className":58186,"style":170},[169],[18,58188,58190,58193],{"className":58189},[106],[18,58191,57361],{"className":58192,"style":57382},[106,110],[18,58194,58196],{"className":58195},[115],[18,58197,58199,58219],{"className":58198},[119,120],[18,58200,58202,58216],{"className":58201},[124],[18,58203,58205],{"className":58204,"style":5918},[128],[18,58206,58207,58210],{"style":57397},[18,58208],{"className":58209,"style":137},[136],[18,58211,58213],{"className":58212},[141,142,143,144],[18,58214,5597],{"className":58215},[106,110,144],[18,58217,152],{"className":58218},[151],[18,58220,58222],{"className":58221},[124],[18,58223,58225],{"className":58224,"style":159},[128],[18,58226],{},[18,58228,58230,58233],{"className":58229},[106],[18,58231,38573],{"className":58232},[106,110],[18,58234,58236],{"className":58235},[115],[18,58237,58239,58259],{"className":58238},[119,120],[18,58240,58242,58256],{"className":58241},[124],[18,58243,58245],{"className":58244,"style":5918},[128],[18,58246,58247,58250],{"style":1155},[18,58248],{"className":58249,"style":137},[136],[18,58251,58253],{"className":58252},[141,142,143,144],[18,58254,5597],{"className":58255},[106,110,144],[18,58257,152],{"className":58258},[151],[18,58260,58262],{"className":58261},[124],[18,58263,58265],{"className":58264,"style":159},[128],[18,58266],{},[18,58268],{"className":58269,"style":228},[169],[18,58271,433],{"className":58272},[232],[18,58274],{"className":58275,"style":228},[169],[18,58277,58279,58282,58335,58338,58341,58381,58384,58424],{"className":58278},[97],[18,58280],{"className":58281,"style":58129},[101],[18,58283,58285],{"className":58284},[985,986],[18,58286,58288,58327],{"className":58287},[119,120],[18,58289,58291,58324],{"className":58290},[124],[18,58292,58294,58314],{"className":58293,"style":24476},[128],[18,58295,58296,58299],{"style":39158},[18,58297],{"className":58298,"style":20552},[136],[18,58300,58302],{"className":58301},[141,142,143,144],[18,58303,58305,58308,58311],{"className":58304},[106,144],[18,58306,5597],{"className":58307},[106,110,144],[18,58309,921],{"className":58310},[232,144],[18,58312,70],{"className":58313,"style":246},[106,110,144],[18,58315,58316,58319],{"style":20570},[18,58317],{"className":58318,"style":20552},[136],[18,58320,58321],{},[18,58322,20362],{"className":58323},[985,4198,4199],[18,58325,152],{"className":58326},[151],[18,58328,58330],{"className":58329},[124],[18,58331,58333],{"className":58332,"style":58181},[128],[18,58334],{},[18,58336,74],{"className":58337},[250],[18,58339,2253],{"className":58340},[106],[18,58342,58344,58347],{"className":58343},[106],[18,58345,57361],{"className":58346,"style":57382},[106,110],[18,58348,58350],{"className":58349},[115],[18,58351,58353,58373],{"className":58352},[119,120],[18,58354,58356,58370],{"className":58355},[124],[18,58357,58359],{"className":58358,"style":5918},[128],[18,58360,58361,58364],{"style":57397},[18,58362],{"className":58363,"style":137},[136],[18,58365,58367],{"className":58366},[141,142,143,144],[18,58368,5597],{"className":58369},[106,110,144],[18,58371,152],{"className":58372},[151],[18,58374,58376],{"className":58375},[124],[18,58377,58379],{"className":58378,"style":159},[128],[18,58380],{},[18,58382,84],{"className":58383},[267],[18,58385,58387,58390],{"className":58386},[106],[18,58388,38573],{"className":58389},[106,110],[18,58391,58393],{"className":58392},[115],[18,58394,58396,58416],{"className":58395},[119,120],[18,58397,58399,58413],{"className":58398},[124],[18,58400,58402],{"className":58401,"style":5918},[128],[18,58403,58404,58407],{"style":1155},[18,58405],{"className":58406,"style":137},[136],[18,58408,58410],{"className":58409},[141,142,143,144],[18,58411,5597],{"className":58412},[106,110,144],[18,58414,152],{"className":58415},[151],[18,58417,58419],{"className":58418},[124],[18,58420,58422],{"className":58421,"style":159},[128],[18,58423],{},[18,58425,968],{"className":58426},[106],[11,58428,58429,58430,58483,58484,58512,58513,1990],{},"两边系数之和相同，记为 ",[18,58431,58433,58452],{"className":58432,"translate":22},[21],[18,58434,58436],{"className":58435},[26],[28,58437,58438],{"xmlns":30},[32,58439,58440,58449],{},[35,58441,58442,58445,58447],{},[41,58443,58444],{},"A",[49,58446,6882],{},[45,58448,430],{},[86,58450,58451],{"encoding":88},"A>0",[18,58453,58455,58474],{"className":58454,"ariaHidden":51},[93],[18,58456,58458,58462,58465,58468,58471],{"className":58457},[97],[18,58459],{"className":58460,"style":58461},[101],"height:0.7224em;vertical-align:-0.0391em;",[18,58463,58444],{"className":58464},[106,110],[18,58466],{"className":58467,"style":228},[169],[18,58469,6882],{"className":58470},[232],[18,58472],{"className":58473,"style":228},[169],[18,58475,58477,58480],{"className":58476},[97],[18,58478],{"className":58479,"style":505},[101],[18,58481,430],{"className":58482},[106],"。除以 ",[18,58485,58487,58500],{"className":58486,"translate":22},[21],[18,58488,58490],{"className":58489},[26],[28,58491,58492],{"xmlns":30},[32,58493,58494,58498],{},[35,58495,58496],{},[41,58497,58444],{},[86,58499,58444],{"encoding":88},[18,58501,58503],{"className":58502,"ariaHidden":51},[93],[18,58504,58506,58509],{"className":58505},[97],[18,58507],{"className":58508,"style":600},[101],[18,58510,58444],{"className":58511},[106,110],"，得到同一点的两个凸组合表示，于是 ",[18,58514,58516,58557],{"className":58515,"translate":22},[21],[18,58517,58519],{"className":58518},[26],[28,58520,58521],{"xmlns":30},[32,58522,58523,58554],{},[35,58524,58525,58528,58530,58532,58534,58536,58539,58541,58543,58545,58547,58549,58551],{},[41,58526,58527],{"mathvariant":316},"conv",[49,58529,899],{},[49,58531,74],{"stretchy":73},[41,58533,939],{},[49,58535,84],{"stretchy":73},[49,58537,58538],{},"∩",[41,58540,58527],{"mathvariant":316},[49,58542,899],{},[49,58544,74],{"stretchy":73},[41,58546,70],{},[49,58548,84],{"stretchy":73},[49,58550,2771],{"mathvariant":316},[41,58552,58553],{"mathvariant":316},"∅",[86,58555,58556],{"encoding":88},"\\operatorname{conv}(P)\\cap\\operatorname{conv}(N)\\neq\\varnothing",[18,58558,58560,58590,58653],{"className":58559,"ariaHidden":51},[93],[18,58561,58563,58566,58572,58575,58578,58581,58584,58587],{"className":58562},[97],[18,58564],{"className":58565,"style":242},[101],[18,58567,58569],{"className":58568},[985],[18,58570,58527],{"className":58571,"style":6352},[106,6351],[18,58573,74],{"className":58574},[250],[18,58576,939],{"className":58577,"style":1171},[106,110],[18,58579,84],{"className":58580},[267],[18,58582],{"className":58583,"style":2068},[169],[18,58585,58538],{"className":58586},[2072],[18,58588],{"className":58589,"style":2068},[169],[18,58591,58593,58596,58602,58605,58608,58611,58614,58650],{"className":58592},[97],[18,58594],{"className":58595,"style":242},[101],[18,58597,58599],{"className":58598},[985],[18,58600,58527],{"className":58601,"style":6352},[106,6351],[18,58603,74],{"className":58604},[250],[18,58606,70],{"className":58607,"style":246},[106,110],[18,58609,84],{"className":58610},[267],[18,58612],{"className":58613,"style":228},[169],[18,58615,58617,58644,58647],{"className":58616},[232],[18,58618,58620],{"className":58619},[232],[18,58621,58623],{"className":58622},[106,2878],[18,58624,58626],{"className":58625},[2882],[18,58627,58629,58632,58641],{"className":58628},[2886],[18,58630],{"className":58631,"style":2890},[101],[18,58633,58635],{"className":58634},[2894],[18,58636,58638],{"className":58637},[106],[18,58639,2901],{"className":58640},[232],[18,58642],{"className":58643},[2905],[18,58645],{"className":58646},[169,2909],[18,58648,433],{"className":58649},[232],[18,58651],{"className":58652,"style":228},[169],[18,58654,58656,58660],{"className":58655},[97],[18,58657],{"className":58658,"style":58659},[101],"height:0.6633em;vertical-align:-0.0817em;",[18,58661,58553],{"className":58662},[106,27253],[11,58664,58665,58666,58694,58695,58723,58724,58752,58753,58802,58803,58831,58832,58881,58882,58932,58933,1990],{},"现在取标注「",[18,58667,58669,58682],{"className":58668,"translate":22},[21],[18,58670,58672],{"className":58671},[26],[28,58673,58674],{"xmlns":30},[32,58675,58676,58680],{},[35,58677,58678],{},[41,58679,939],{},[86,58681,939],{"encoding":88},[18,58683,58685],{"className":58684,"ariaHidden":51},[93],[18,58686,58688,58691],{"className":58687},[97],[18,58689],{"className":58690,"style":600},[101],[18,58692,939],{"className":58693,"style":1171},[106,110]," 为正、",[18,58696,58698,58711],{"className":58697,"translate":22},[21],[18,58699,58701],{"className":58700},[26],[28,58702,58703],{"xmlns":30},[32,58704,58705,58709],{},[35,58706,58707],{},[41,58708,70],{},[86,58710,70],{"encoding":88},[18,58712,58714],{"className":58713,"ariaHidden":51},[93],[18,58715,58717,58720],{"className":58716},[97],[18,58718],{"className":58719,"style":600},[101],[18,58721,70],{"className":58722,"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 ",[18,58725,58727,58740],{"className":58726,"translate":22},[21],[18,58728,58730],{"className":58729},[26],[28,58731,58732],{"xmlns":30},[32,58733,58734,58738],{},[35,58735,58736],{},[41,58737,939],{},[86,58739,939],{"encoding":88},[18,58741,58743],{"className":58742,"ariaHidden":51},[93],[18,58744,58746,58749],{"className":58745},[97],[18,58747],{"className":58748,"style":600},[101],[18,58750,939],{"className":58751,"style":1171},[106,110]," 的半空间都包含 ",[18,58754,58756,58778],{"className":58755,"translate":22},[21],[18,58757,58759],{"className":58758},[26],[28,58760,58761],{"xmlns":30},[32,58762,58763,58775],{},[35,58764,58765,58767,58769,58771,58773],{},[41,58766,58527],{"mathvariant":316},[49,58768,899],{},[49,58770,74],{"stretchy":73},[41,58772,939],{},[49,58774,84],{"stretchy":73},[86,58776,58777],{"encoding":88},"\\operatorname{conv}(P)",[18,58779,58781],{"className":58780,"ariaHidden":51},[93],[18,58782,58784,58787,58793,58796,58799],{"className":58783},[97],[18,58785],{"className":58786,"style":242},[101],[18,58788,58790],{"className":58789},[985],[18,58791,58527],{"className":58792,"style":6352},[106,6351],[18,58794,74],{"className":58795},[250],[18,58797,939],{"className":58798,"style":1171},[106,110],[18,58800,84],{"className":58801},[267],"；任何不包含 ",[18,58804,58806,58819],{"className":58805,"translate":22},[21],[18,58807,58809],{"className":58808},[26],[28,58810,58811],{"xmlns":30},[32,58812,58813,58817],{},[35,58814,58815],{},[41,58816,70],{},[86,58818,70],{"encoding":88},[18,58820,58822],{"className":58821,"ariaHidden":51},[93],[18,58823,58825,58828],{"className":58824},[97],[18,58826],{"className":58827,"style":600},[101],[18,58829,70],{"className":58830,"style":246},[106,110]," 的半空间，其补集是凸的，因而也不包含 ",[18,58833,58835,58857],{"className":58834,"translate":22},[21],[18,58836,58838],{"className":58837},[26],[28,58839,58840],{"xmlns":30},[32,58841,58842,58854],{},[35,58843,58844,58846,58848,58850,58852],{},[41,58845,58527],{"mathvariant":316},[49,58847,899],{},[49,58849,74],{"stretchy":73},[41,58851,70],{},[49,58853,84],{"stretchy":73},[86,58855,58856],{"encoding":88},"\\operatorname{conv}(N)",[18,58858,58860],{"className":58859,"ariaHidden":51},[93],[18,58861,58863,58866,58872,58875,58878],{"className":58862},[97],[18,58864],{"className":58865,"style":242},[101],[18,58867,58869],{"className":58868},[985],[18,58870,58527],{"className":58871,"style":6352},[106,6351],[18,58873,74],{"className":58874},[250],[18,58876,70],{"className":58877,"style":246},[106,110],[18,58879,84],{"className":58880},[267],"。两者相交，矛盾。所以 ",[18,58883,58885,58902],{"className":58884,"translate":22},[21],[18,58886,58888],{"className":58887},[26],[28,58889,58890],{"xmlns":30},[32,58891,58892,58900],{},[35,58893,58894,58896,58898],{},[41,58895,4031],{},[49,58897,7984],{},[45,58899,2266],{},[86,58901,56985],{"encoding":88},[18,58903,58905,58923],{"className":58904,"ariaHidden":51},[93],[18,58906,58908,58911,58914,58917,58920],{"className":58907},[97],[18,58909],{"className":58910,"style":7221},[101],[18,58912,4031],{"className":58913},[106,110],[18,58915],{"className":58916,"style":2068},[169],[18,58918,7984],{"className":58919},[2072],[18,58921],{"className":58922,"style":2068},[169],[18,58924,58926,58929],{"className":58925},[97],[18,58927],{"className":58928,"style":505},[101],[18,58930,2266],{"className":58931},[106]," 个点无法被打散，",[18,58934,58936,58959],{"className":58935,"translate":22},[21],[18,58937,58939],{"className":58938},[26],[28,58940,58941],{"xmlns":30},[32,58942,58943,58957],{},[35,58944,58945,58947,58949,58951,58953,58955],{},[41,58946,52202],{"mathvariant":316},[49,58948,899],{},[49,58950,433],{},[41,58952,4031],{},[49,58954,7984],{},[45,58956,47],{},[86,58958,55747],{"encoding":88},[18,58960,58962,58983,59001],{"className":58961,"ariaHidden":51},[93],[18,58963,58965,58968,58974,58977,58980],{"className":58964},[97],[18,58966],{"className":58967,"style":295},[101],[18,58969,58971],{"className":58970},[985],[18,58972,52202],{"className":58973},[106,6351],[18,58975],{"className":58976,"style":228},[169],[18,58978,433],{"className":58979},[232],[18,58981],{"className":58982,"style":228},[169],[18,58984,58986,58989,58992,58995,58998],{"className":58985},[97],[18,58987],{"className":58988,"style":7221},[101],[18,58990,4031],{"className":58991},[106,110],[18,58993],{"className":58994,"style":2068},[169],[18,58996,7984],{"className":58997},[2072],[18,58999],{"className":59000,"style":2068},[169],[18,59002,59004,59007],{"className":59003},[97],[18,59005],{"className":59006,"style":505},[101],[18,59008,47],{"className":59009},[106],[11,59011,59012],{},"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。",[11,59014,59015],{},[717,59016,59017,59018,52814],{},"例 4（轴对齐矩形，",[18,59019,59021,59043],{"className":59020,"translate":22},[21],[18,59022,59024],{"className":59023},[26],[28,59025,59026],{"xmlns":30},[32,59027,59028,59040],{},[35,59029,59030,59032,59034,59036,59038],{},[41,59031,52202],{"mathvariant":316},[49,59033,899],{},[49,59035,433],{},[45,59037,2266],{},[41,59039,4031],{},[86,59041,59042],{"encoding":88},"\\operatorname{VCdim}=2d",[18,59044,59046,59067],{"className":59045,"ariaHidden":51},[93],[18,59047,59049,59052,59058,59061,59064],{"className":59048},[97],[18,59050],{"className":59051,"style":295},[101],[18,59053,59055],{"className":59054},[985],[18,59056,52202],{"className":59057},[106,6351],[18,59059],{"className":59060,"style":228},[169],[18,59062,433],{"className":59063},[232],[18,59065],{"className":59066,"style":228},[169],[18,59068,59070,59073,59076],{"className":59069},[97],[18,59071],{"className":59072,"style":295},[101],[18,59074,2266],{"className":59075},[106],[18,59077,4031],{"className":59078},[106,110],[11,59080,59081,19578,59109,59167,59168,59196],{},[18,59082,59084,59097],{"className":59083,"translate":22},[21],[18,59085,59087],{"className":59086},[26],[28,59088,59089],{"xmlns":30},[32,59090,59091,59095],{},[35,59092,59093],{},[41,59094,18304],{"mathvariant":1462},[86,59096,46308],{"encoding":88},[18,59098,59100],{"className":59099,"ariaHidden":51},[93],[18,59101,59103,59106],{"className":59102},[97],[18,59104],{"className":59105,"style":600},[101],[18,59107,18304],{"className":59108,"style":46321},[106,1478],[18,59110,59112,59129],{"className":59111,"translate":22},[21],[18,59113,59115],{"className":59114},[26],[28,59116,59117],{"xmlns":30},[32,59118,59119,59127],{},[35,59120,59121],{},[1556,59122,59123,59125],{},[41,59124,3071],{"mathvariant":933},[41,59126,4031],{},[86,59128,56925],{"encoding":88},[18,59130,59132],{"className":59131,"ariaHidden":51},[93],[18,59133,59135,59138],{"className":59134},[97],[18,59136],{"className":59137,"style":35177},[101],[18,59139,59141,59144],{"className":59140},[106],[18,59142,3071],{"className":59143},[106,1139],[18,59145,59147],{"className":59146},[115],[18,59148,59150],{"className":59149},[119],[18,59151,59153],{"className":59152},[124],[18,59154,59156],{"className":59155,"style":35177},[128],[18,59157,59158,59161],{"style":1962},[18,59159],{"className":59160,"style":137},[136],[18,59162,59164],{"className":59163},[141,142,143,144],[18,59165,4031],{"className":59166},[106,110,144]," 中所有轴对齐矩形（即 ",[18,59169,59171,59184],{"className":59170,"translate":22},[21],[18,59172,59174],{"className":59173},[26],[28,59175,59176],{"xmlns":30},[32,59177,59178,59182],{},[35,59179,59180],{},[41,59181,4031],{},[86,59183,4031],{"encoding":88},[18,59185,59187],{"className":59186,"ariaHidden":51},[93],[18,59188,59190,59193],{"className":59189},[97],[18,59191],{"className":59192,"style":295},[101],[18,59194,4031],{"className":59195},[106,110]," 个区间的乘积）的指示函数。",[11,59198,59199,59200,59256,59257,59285,59286,59314,59315,59385,59386,59414,59415,59484,59485,1990],{},"上界：任取 ",[18,59201,59203,59223],{"className":59202,"translate":22},[21],[18,59204,59206],{"className":59205},[26],[28,59207,59208],{"xmlns":30},[32,59209,59210,59220],{},[35,59211,59212,59214,59216,59218],{},[45,59213,2266],{},[41,59215,4031],{},[49,59217,7984],{},[45,59219,47],{},[86,59221,59222],{"encoding":88},"2d+1",[18,59224,59226,59247],{"className":59225,"ariaHidden":51},[93],[18,59227,59229,59232,59235,59238,59241,59244],{"className":59228},[97],[18,59230],{"className":59231,"style":7221},[101],[18,59233,2266],{"className":59234},[106],[18,59236,4031],{"className":59237},[106,110],[18,59239],{"className":59240,"style":2068},[169],[18,59242,7984],{"className":59243},[2072],[18,59245],{"className":59246,"style":2068},[169],[18,59248,59250,59253],{"className":59249},[97],[18,59251],{"className":59252,"style":505},[101],[18,59254,47],{"className":59255},[106]," 个点。若存在一点 ",[18,59258,59260,59273],{"className":59259,"translate":22},[21],[18,59261,59263],{"className":59262},[26],[28,59264,59265],{"xmlns":30},[32,59266,59267,59271],{},[35,59268,59269],{},[41,59270,11],{},[86,59272,11],{"encoding":88},[18,59274,59276],{"className":59275,"ariaHidden":51},[93],[18,59277,59279,59282],{"className":59278},[97],[18,59280],{"className":59281,"style":38543},[101],[18,59283,11],{"className":59284},[106,110],"，使它在每个坐标上都不是严格极值——即对每个坐标 ",[18,59287,59289,59302],{"className":59288,"translate":22},[21],[18,59290,59292],{"className":59291},[26],[28,59293,59294],{"xmlns":30},[32,59295,59296,59300],{},[35,59297,59298],{},[41,59299,19804],{},[86,59301,19804],{"encoding":88},[18,59303,59305],{"className":59304,"ariaHidden":51},[93],[18,59306,59308,59311],{"className":59307},[97],[18,59309],{"className":59310,"style":28331},[101],[18,59312,19804],{"className":59313,"style":19921},[106,110],"，都有别的点在 ",[18,59316,59318,59336],{"className":59317,"translate":22},[21],[18,59319,59321],{"className":59320},[26],[28,59322,59323],{"xmlns":30},[32,59324,59325,59333],{},[35,59326,59327],{},[38,59328,59329,59331],{},[41,59330,11],{},[41,59332,19804],{},[86,59334,59335],{"encoding":88},"p_j",[18,59337,59339],{"className":59338,"ariaHidden":51},[93],[18,59340,59342,59345],{"className":59341},[97],[18,59343],{"className":59344,"style":30670},[101],[18,59346,59348,59351],{"className":59347},[106],[18,59349,11],{"className":59350},[106,110],[18,59352,59354],{"className":59353},[115],[18,59355,59357,59377],{"className":59356},[119,120],[18,59358,59360,59374],{"className":59359},[124],[18,59361,59363],{"className":59362,"style":5918},[128],[18,59364,59365,59368],{"style":1155},[18,59366],{"className":59367,"style":137},[136],[18,59369,59371],{"className":59370},[141,142,143,144],[18,59372,19804],{"className":59373,"style":19921},[106,110,144],[18,59375,152],{"className":59376},[151],[18,59378,59380],{"className":59379},[124],[18,59381,59383],{"className":59382,"style":19931},[128],[18,59384],{}," 的左边和右边——那么标注「",[18,59387,59389,59402],{"className":59388,"translate":22},[21],[18,59390,59392],{"className":59391},[26],[28,59393,59394],{"xmlns":30},[32,59395,59396,59400],{},[35,59397,59398],{},[41,59399,11],{},[86,59401,11],{"encoding":88},[18,59403,59405],{"className":59404,"ariaHidden":51},[93],[18,59406,59408,59411],{"className":59407},[97],[18,59409],{"className":59410,"style":38543},[101],[18,59412,11],{"className":59413},[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 ",[18,59416,59418,59435],{"className":59417,"translate":22},[21],[18,59419,59421],{"className":59420},[26],[28,59422,59423],{"xmlns":30},[32,59424,59425,59433],{},[35,59426,59427],{},[38,59428,59429,59431],{},[41,59430,11],{},[41,59432,19804],{},[86,59434,59335],{"encoding":88},[18,59436,59438],{"className":59437,"ariaHidden":51},[93],[18,59439,59441,59444],{"className":59440},[97],[18,59442],{"className":59443,"style":30670},[101],[18,59445,59447,59450],{"className":59446},[106],[18,59448,11],{"className":59449},[106,110],[18,59451,59453],{"className":59452},[115],[18,59454,59456,59476],{"className":59455},[119,120],[18,59457,59459,59473],{"className":59458},[124],[18,59460,59462],{"className":59461,"style":5918},[128],[18,59463,59464,59467],{"style":1155},[18,59465],{"className":59466,"style":137},[136],[18,59468,59470],{"className":59469},[141,142,143,144],[18,59471,19804],{"className":59472,"style":19921},[106,110,144],[18,59474,152],{"className":59475},[151],[18,59477,59479],{"className":59478},[124],[18,59480,59482],{"className":59481,"style":19931},[128],[18,59483],{}," 两侧的值，因而必然包含 ",[18,59486,59488,59501],{"className":59487,"translate":22},[21],[18,59489,59491],{"className":59490},[26],[28,59492,59493],{"xmlns":30},[32,59494,59495,59499],{},[35,59496,59497],{},[41,59498,11],{},[86,59500,11],{"encoding":88},[18,59502,59504],{"className":59503,"ariaHidden":51},[93],[18,59505,59507,59510],{"className":59506},[97],[18,59508],{"className":59509,"style":38543},[101],[18,59511,11],{"className":59512},[106,110],[11,59514,59515,59516,59544,59545,59573,59574,59602,59603,59637,59638,59693,59694,59727],{},"这样的 ",[18,59517,59519,59532],{"className":59518,"translate":22},[21],[18,59520,59522],{"className":59521},[26],[28,59523,59524],{"xmlns":30},[32,59525,59526,59530],{},[35,59527,59528],{},[41,59529,11],{},[86,59531,11],{"encoding":88},[18,59533,59535],{"className":59534,"ariaHidden":51},[93],[18,59536,59538,59541],{"className":59537},[97],[18,59539],{"className":59540,"style":38543},[101],[18,59542,11],{"className":59543},[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 ",[18,59546,59548,59561],{"className":59547,"translate":22},[21],[18,59549,59551],{"className":59550},[26],[28,59552,59553],{"xmlns":30},[32,59554,59555,59559],{},[35,59556,59557],{},[41,59558,19804],{},[86,59560,19804],{"encoding":88},[18,59562,59564],{"className":59563,"ariaHidden":51},[93],[18,59565,59567,59570],{"className":59566},[97],[18,59568],{"className":59569,"style":28331},[101],[18,59571,19804],{"className":59572,"style":19921},[106,110]," 的最小值」和「坐标 ",[18,59575,59577,59590],{"className":59576,"translate":22},[21],[18,59578,59580],{"className":59579},[26],[28,59581,59582],{"xmlns":30},[32,59583,59584,59588],{},[35,59585,59586],{},[41,59587,19804],{},[86,59589,19804],{"encoding":88},[18,59591,59593],{"className":59592,"ariaHidden":51},[93],[18,59594,59596,59599],{"className":59595},[97],[18,59597],{"className":59598,"style":28331},[101],[18,59600,19804],{"className":59601,"style":19921},[106,110]," 的最大值」看作 ",[18,59604,59606,59622],{"className":59605,"translate":22},[21],[18,59607,59609],{"className":59608},[26],[28,59610,59611],{"xmlns":30},[32,59612,59613,59619],{},[35,59614,59615,59617],{},[45,59616,2266],{},[41,59618,4031],{},[86,59620,59621],{"encoding":88},"2d",[18,59623,59625],{"className":59624,"ariaHidden":51},[93],[18,59626,59628,59631,59634],{"className":59627},[97],[18,59629],{"className":59630,"style":295},[101],[18,59632,2266],{"className":59633},[106],[18,59635,4031],{"className":59636},[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）。",[18,59639,59641,59660],{"className":59640,"translate":22},[21],[18,59642,59644],{"className":59643},[26],[28,59645,59646],{"xmlns":30},[32,59647,59648,59658],{},[35,59649,59650,59652,59654,59656],{},[45,59651,2266],{},[41,59653,4031],{},[49,59655,7984],{},[45,59657,47],{},[86,59659,59222],{"encoding":88},[18,59661,59663,59684],{"className":59662,"ariaHidden":51},[93],[18,59664,59666,59669,59672,59675,59678,59681],{"className":59665},[97],[18,59667],{"className":59668,"style":7221},[101],[18,59670,2266],{"className":59671},[106],[18,59673,4031],{"className":59674},[106,110],[18,59676],{"className":59677,"style":2068},[169],[18,59679,7984],{"className":59680},[2072],[18,59682],{"className":59683,"style":2068},[169],[18,59685,59687,59690],{"className":59686},[97],[18,59688],{"className":59689,"style":505},[101],[18,59691,47],{"className":59692},[106]," 个点放进 ",[18,59695,59697,59712],{"className":59696,"translate":22},[21],[18,59698,59700],{"className":59699},[26],[28,59701,59702],{"xmlns":30},[32,59703,59704,59710],{},[35,59705,59706,59708],{},[45,59707,2266],{},[41,59709,4031],{},[86,59711,59621],{"encoding":88},[18,59713,59715],{"className":59714,"ariaHidden":51},[93],[18,59716,59718,59721,59724],{"className":59717},[97],[18,59719],{"className":59720,"style":295},[101],[18,59722,2266],{"className":59723},[106],[18,59725,4031],{"className":59726},[106,110]," 个位置，必有一对冲突。所以上界成立。",[11,59729,59730,59731,59764,59765,749,59816,59976,59977,1990],{},"下界是初等的但需要一点构造：确实存在 ",[18,59732,59734,59749],{"className":59733,"translate":22},[21],[18,59735,59737],{"className":59736},[26],[28,59738,59739],{"xmlns":30},[32,59740,59741,59747],{},[35,59742,59743,59745],{},[45,59744,2266],{},[41,59746,4031],{},[86,59748,59621],{"encoding":88},[18,59750,59752],{"className":59751,"ariaHidden":51},[93],[18,59753,59755,59758,59761],{"className":59754},[97],[18,59756],{"className":59757,"style":295},[101],[18,59759,2266],{"className":59760},[106],[18,59762,4031],{"className":59763},[106,110]," 个点可以被矩形打散（",[18,59766,59768,59786],{"className":59767,"translate":22},[21],[18,59769,59771],{"className":59770},[26],[28,59772,59773],{"xmlns":30},[32,59774,59775,59783],{},[35,59776,59777,59779,59781],{},[41,59778,4031],{},[49,59780,433],{},[45,59782,2266],{},[86,59784,59785],{"encoding":88},"d=2",[18,59787,59789,59807],{"className":59788,"ariaHidden":51},[93],[18,59790,59792,59795,59798,59801,59804],{"className":59791},[97],[18,59793],{"className":59794,"style":295},[101],[18,59796,4031],{"className":59797},[106,110],[18,59799],{"className":59800,"style":228},[169],[18,59802,433],{"className":59803},[232],[18,59805],{"className":59806,"style":228},[169],[18,59808,59810,59813],{"className":59809},[97],[18,59811],{"className":59812,"style":505},[101],[18,59814,2266],{"className":59815},[106],[18,59817,59819,59877],{"className":59818,"translate":22},[21],[18,59820,59822],{"className":59821},[26],[28,59823,59824],{"xmlns":30},[32,59825,59826,59874],{},[35,59827,59828,59830,59832,59834,59836,59838,59840,59842,59844,59846,59848,59850,59852,59854,59856,59858,59860,59862,59864,59866,59868,59870,59872],{},[49,59829,74],{"stretchy":73},[45,59831,47],{},[49,59833,52],{"separator":51},[45,59835,2266],{},[49,59837,84],{"stretchy":73},[49,59839,52],{"separator":51},[49,59841,74],{"stretchy":73},[45,59843,2266],{},[49,59845,52],{"separator":51},[45,59847,10444],{},[49,59849,84],{"stretchy":73},[49,59851,52],{"separator":51},[49,59853,74],{"stretchy":73},[45,59855,44699],{},[49,59857,52],{"separator":51},[45,59859,47],{},[49,59861,84],{"stretchy":73},[49,59863,52],{"separator":51},[49,59865,74],{"stretchy":73},[45,59867,10444],{},[49,59869,52],{"separator":51},[45,59871,44699],{},[49,59873,84],{"stretchy":73},[86,59875,59876],{"encoding":88},"(1,2),(2,4),(3,1),(4,3)",[18,59878,59880],{"className":59879,"ariaHidden":51},[93],[18,59881,59883,59886,59889,59892,59895,59898,59901,59904,59907,59910,59913,59916,59919,59922,59925,59928,59931,59934,59937,59940,59943,59946,59949,59952,59955,59958,59961,59964,59967,59970,59973],{"className":59882},[97],[18,59884],{"className":59885,"style":242},[101],[18,59887,74],{"className":59888},[250],[18,59890,47],{"className":59891},[106],[18,59893,52],{"className":59894},[165],[18,59896],{"className":59897,"style":170},[169],[18,59899,2266],{"className":59900},[106],[18,59902,84],{"className":59903},[267],[18,59905,52],{"className":59906},[165],[18,59908],{"className":59909,"style":170},[169],[18,59911,74],{"className":59912},[250],[18,59914,2266],{"className":59915},[106],[18,59917,52],{"className":59918},[165],[18,59920],{"className":59921,"style":170},[169],[18,59923,10444],{"className":59924},[106],[18,59926,84],{"className":59927},[267],[18,59929,52],{"className":59930},[165],[18,59932],{"className":59933,"style":170},[169],[18,59935,74],{"className":59936},[250],[18,59938,44699],{"className":59939},[106],[18,59941,52],{"className":59942},[165],[18,59944],{"className":59945,"style":170},[169],[18,59947,47],{"className":59948},[106],[18,59950,84],{"className":59951},[267],[18,59953,52],{"className":59954},[165],[18,59956],{"className":59957,"style":170},[169],[18,59959,74],{"className":59960},[250],[18,59962,10444],{"className":59963},[106],[18,59965,52],{"className":59966},[165],[18,59968],{"className":59969,"style":170},[169],[18,59971,44699],{"className":59972},[106],[18,59974,84],{"className":59975},[267]," 这四个点就可以；注意不能取正方形的四个角）。于是 ",[18,59978,59980,60001],{"className":59979,"translate":22},[21],[18,59981,59983],{"className":59982},[26],[28,59984,59985],{"xmlns":30},[32,59986,59987,59999],{},[35,59988,59989,59991,59993,59995,59997],{},[41,59990,52202],{"mathvariant":316},[49,59992,899],{},[49,59994,433],{},[45,59996,2266],{},[41,59998,4031],{},[86,60000,59042],{"encoding":88},[18,60002,60004,60025],{"className":60003,"ariaHidden":51},[93],[18,60005,60007,60010,60016,60019,60022],{"className":60006},[97],[18,60008],{"className":60009,"style":295},[101],[18,60011,60013],{"className":60012},[985],[18,60014,52202],{"className":60015},[106,6351],[18,60017],{"className":60018,"style":228},[169],[18,60020,433],{"className":60021},[232],[18,60023],{"className":60024,"style":228},[169],[18,60026,60028,60031,60034],{"className":60027},[97],[18,60029],{"className":60030,"style":295},[101],[18,60032,2266],{"className":60033},[106],[18,60035,4031],{"className":60036},[106,110],[11,60038,60039],{},[717,60040,60041,60042,52814],{},"例 5（有限类，",[18,60043,60045,60079],{"className":60044,"translate":22},[21],[18,60046,60048],{"className":60047},[26],[28,60049,60050],{"xmlns":30},[32,60051,60052,60076],{},[35,60053,60054,60056,60058,60060,60070,60072,60074],{},[41,60055,52202],{"mathvariant":316},[49,60057,899],{},[49,60059,4334],{},[38,60061,60062,60068],{},[35,60063,60064,60066],{},[41,60065,14730],{},[49,60067,899],{},[45,60069,2266],{},[41,60071,11222],{"mathvariant":316},[41,60073,18304],{"mathvariant":1462},[41,60075,11222],{"mathvariant":316},[86,60077,60078],{"encoding":88},"\\operatorname{VCdim}\\le\\log_2|\\mathcal H|",[18,60080,60082,60103],{"className":60081,"ariaHidden":51},[93],[18,60083,60085,60088,60094,60097,60100],{"className":60084},[97],[18,60086],{"className":60087,"style":52343},[101],[18,60089,60091],{"className":60090},[985],[18,60092,52202],{"className":60093},[106,6351],[18,60095],{"className":60096,"style":228},[169],[18,60098,4334],{"className":60099},[232],[18,60101],{"className":60102,"style":228},[169],[18,60104,60106,60109,60154,60157,60160,60163],{"className":60105},[97],[18,60107],{"className":60108,"style":242},[101],[18,60110,60112,60117],{"className":60111},[985],[18,60113,15171,60115],{"className":60114},[985],[18,60116,15174],{"style":6352},[18,60118,60120],{"className":60119},[115],[18,60121,60123,60145],{"className":60122},[119,120],[18,60124,60126,60142],{"className":60125},[124],[18,60127,60130],{"className":60128,"style":60129},[128],"height:0.207em;",[18,60131,60133,60136],{"style":60132},"top:-2.4559em;margin-right:0.05em;",[18,60134],{"className":60135,"style":137},[136],[18,60137,60139],{"className":60138},[141,142,143,144],[18,60140,2266],{"className":60141},[106,144],[18,60143,152],{"className":60144},[151],[18,60146,60148],{"className":60147},[124],[18,60149,60152],{"className":60150,"style":60151},[128],"height:0.2441em;",[18,60153],{},[18,60155],{"className":60156,"style":170},[169],[18,60158,11222],{"className":60159},[106],[18,60161,18304],{"className":60162,"style":46321},[106,1478],[18,60164,11222],{"className":60165},[106],[11,60167,21783,60168,60196,60197,15555,60225,39548,60409,1990],{},[18,60169,60171,60184],{"className":60170,"translate":22},[21],[18,60172,60174],{"className":60173},[26],[28,60175,60176],{"xmlns":30},[32,60177,60178,60182],{},[35,60179,60180],{},[41,60181,18304],{"mathvariant":1462},[86,60183,46308],{"encoding":88},[18,60185,60187],{"className":60186,"ariaHidden":51},[93],[18,60188,60190,60193],{"className":60189},[97],[18,60191],{"className":60192,"style":600},[101],[18,60194,18304],{"className":60195,"style":46321},[106,1478]," 有限且能打散 ",[18,60198,60200,60213],{"className":60199,"translate":22},[21],[18,60201,60203],{"className":60202},[26],[28,60204,60205],{"xmlns":30},[32,60206,60207,60211],{},[35,60208,60209],{},[41,60210,51012],{},[86,60212,51012],{"encoding":88},[18,60214,60216],{"className":60215,"ariaHidden":51},[93],[18,60217,60219,60222],{"className":60218},[97],[18,60220],{"className":60221,"style":600},[101],[18,60223,51012],{"className":60224,"style":5073},[106,110],[18,60226,60228,60274],{"className":60227,"translate":22},[21],[18,60229,60231],{"className":60230},[26],[28,60232,60233],{"xmlns":30},[32,60234,60235,60271],{},[35,60236,60237,60249,60251,60253,60255,60261,60263,60265,60267,60269],{},[1556,60238,60239,60241],{},[45,60240,2266],{},[35,60242,60243,60245,60247],{},[41,60244,11222],{"mathvariant":316},[41,60246,51012],{},[41,60248,11222],{"mathvariant":316},[49,60250,433],{},[41,60252,11222],{"mathvariant":316},[41,60254,18304],{"mathvariant":1462},[38,60256,60257,60259],{},[41,60258,11222],{"mathvariant":316},[41,60260,51012],{},[41,60262,11222],{"mathvariant":316},[49,60264,4334],{},[41,60266,11222],{"mathvariant":316},[41,60268,18304],{"mathvariant":1462},[41,60270,11222],{"mathvariant":316},[86,60272,60273],{"encoding":88},"2^{|C|}=|\\mathcal H|_C|\\le|\\mathcal H|",[18,60275,60277,60330,60394],{"className":60276,"ariaHidden":51},[93],[18,60278,60280,60283,60321,60324,60327],{"className":60279},[97],[18,60281],{"className":60282,"style":27938},[101],[18,60284,60286,60289],{"className":60285},[106],[18,60287,2266],{"className":60288},[106],[18,60290,60292],{"className":60291},[115],[18,60293,60295],{"className":60294},[119],[18,60296,60298],{"className":60297},[124],[18,60299,60301],{"className":60300,"style":27938},[128],[18,60302,60303,60306],{"style":1962},[18,60304],{"className":60305,"style":137},[136],[18,60307,60309],{"className":60308},[141,142,143,144],[18,60310,60312,60315,60318],{"className":60311},[106,144],[18,60313,11222],{"className":60314},[106,144],[18,60316,51012],{"className":60317,"style":5073},[106,110,144],[18,60319,11222],{"className":60320},[106,144],[18,60322],{"className":60323,"style":228},[169],[18,60325,433],{"className":60326},[232],[18,60328],{"className":60329,"style":228},[169],[18,60331,60333,60336,60339,60342,60382,60385,60388,60391],{"className":60332},[97],[18,60334],{"className":60335,"style":242},[101],[18,60337,11222],{"className":60338},[106],[18,60340,18304],{"className":60341,"style":46321},[106,1478],[18,60343,60345,60348],{"className":60344},[106],[18,60346,11222],{"className":60347},[106],[18,60349,60351],{"className":60350},[115],[18,60352,60354,60374],{"className":60353},[119,120],[18,60355,60357,60371],{"className":60356},[124],[18,60358,60360],{"className":60359,"style":1152},[128],[18,60361,60362,60365],{"style":1155},[18,60363],{"className":60364,"style":137},[136],[18,60366,60368],{"className":60367},[141,142,143,144],[18,60369,51012],{"className":60370,"style":5073},[106,110,144],[18,60372,152],{"className":60373},[151],[18,60375,60377],{"className":60376},[124],[18,60378,60380],{"className":60379,"style":159},[128],[18,60381],{},[18,60383,11222],{"className":60384},[106],[18,60386],{"className":60387,"style":228},[169],[18,60389,4334],{"className":60390},[232],[18,60392],{"className":60393,"style":228},[169],[18,60395,60397,60400,60403,60406],{"className":60396},[97],[18,60398],{"className":60399,"style":242},[101],[18,60401,11222],{"className":60402},[106],[18,60404,18304],{"className":60405,"style":46321},[106,1478],[18,60407,11222],{"className":60408},[106],[18,60410,60412,60448],{"className":60411,"translate":22},[21],[18,60413,60415],{"className":60414},[26],[28,60416,60417],{"xmlns":30},[32,60418,60419,60445],{},[35,60420,60421,60423,60425,60427,60429,60439,60441,60443],{},[41,60422,11222],{"mathvariant":316},[41,60424,51012],{},[41,60426,11222],{"mathvariant":316},[49,60428,4334],{},[38,60430,60431,60437],{},[35,60432,60433,60435],{},[41,60434,14730],{},[49,60436,899],{},[45,60438,2266],{},[41,60440,11222],{"mathvariant":316},[41,60442,18304],{"mathvariant":1462},[41,60444,11222],{"mathvariant":316},[86,60446,60447],{"encoding":88},"|C|\\le\\log_2|\\mathcal H|",[18,60449,60451,60475],{"className":60450,"ariaHidden":51},[93],[18,60452,60454,60457,60460,60463,60466,60469,60472],{"className":60453},[97],[18,60455],{"className":60456,"style":242},[101],[18,60458,11222],{"className":60459},[106],[18,60461,51012],{"className":60462,"style":5073},[106,110],[18,60464,11222],{"className":60465},[106],[18,60467],{"className":60468,"style":228},[169],[18,60470,4334],{"className":60471},[232],[18,60473],{"className":60474,"style":228},[169],[18,60476,60478,60481,60523,60526,60529,60532],{"className":60477},[97],[18,60479],{"className":60480,"style":242},[101],[18,60482,60484,60489],{"className":60483},[985],[18,60485,15171,60487],{"className":60486},[985],[18,60488,15174],{"style":6352},[18,60490,60492],{"className":60491},[115],[18,60493,60495,60515],{"className":60494},[119,120],[18,60496,60498,60512],{"className":60497},[124],[18,60499,60501],{"className":60500,"style":60129},[128],[18,60502,60503,60506],{"style":60132},[18,60504],{"className":60505,"style":137},[136],[18,60507,60509],{"className":60508},[141,142,143,144],[18,60510,2266],{"className":60511},[106,144],[18,60513,152],{"className":60514},[151],[18,60516,60518],{"className":60517},[124],[18,60519,60521],{"className":60520,"style":60151},[128],[18,60522],{},[18,60524],{"className":60525,"style":170},[169],[18,60527,11222],{"className":60528},[106],[18,60530,18304],{"className":60531,"style":46321},[106,1478],[18,60533,11222],{"className":60534},[106],[11,60536,60537],{},[717,60538,60539,60540,52814],{},"例 6（正弦类，",[18,60541,60543,60563],{"className":60542,"translate":22},[21],[18,60544,60546],{"className":60545},[26],[28,60547,60548],{"xmlns":30},[32,60549,60550,60560],{},[35,60551,60552,60554,60556,60558],{},[41,60553,52202],{"mathvariant":316},[49,60555,899],{},[49,60557,433],{},[41,60559,2028],{"mathvariant":316},[86,60561,60562],{"encoding":88},"\\operatorname{VCdim}=\\infty",[18,60564,60566,60587],{"className":60565,"ariaHidden":51},[93],[18,60567,60569,60572,60578,60581,60584],{"className":60568},[97],[18,60570],{"className":60571,"style":295},[101],[18,60573,60575],{"className":60574},[985],[18,60576,52202],{"className":60577},[106,6351],[18,60579],{"className":60580,"style":228},[169],[18,60582,433],{"className":60583},[232],[18,60585],{"className":60586,"style":228},[169],[18,60588,60590,60593],{"className":60589},[97],[18,60591],{"className":60592,"style":1083},[101],[18,60594,2028],{"className":60595},[106],[11,60597,36183,60598,4741,60764,60814,60815,60843,60844,60977,60978,28335],{},[18,60599,60601,60653],{"className":60600,"translate":22},[21],[18,60602,60604],{"className":60603},[26],[28,60605,60606],{"xmlns":30},[32,60607,60608,60650],{},[35,60609,60610,60612,60614,60616,60618,60620,60622,60624,60626,60628,60630,60632,60634,60636,60638,60640,60642,60644,60646,60648],{},[41,60611,18304],{"mathvariant":1462},[49,60613,433],{},[49,60615,1331],{"stretchy":73},[41,60617,38573],{},[49,60619,52839],{},[41,60621,45630],{"mathvariant":316},[49,60623,899],{},[49,60625,74],{"stretchy":73},[41,60627,45637],{},[49,60629,899],{},[49,60631,74],{"stretchy":73},[41,60633,77],{},[41,60635,38573],{},[49,60637,84],{"stretchy":73},[49,60639,84],{"stretchy":73},[49,60641,1340],{},[41,60643,77],{},[49,60645,6882],{},[45,60647,430],{},[49,60649,1349],{"stretchy":73},[86,60651,60652],{"encoding":88},"\\mathcal H=\\{x\\mapsto\\operatorname{sign}(\\sin(\\theta 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",[18,60816,60818,60831],{"className":60817,"translate":22},[21],[18,60819,60821],{"className":60820},[26],[28,60822,60823],{"xmlns":30},[32,60824,60825,60829],{},[35,60826,60827],{},[41,60828,28160],{},[86,60830,28160],{"encoding":88},[18,60832,60834],{"className":60833,"ariaHidden":51},[93],[18,60835,60837,60840],{"className":60836},[97],[18,60838],{"className":60839,"style":1083},[101],[18,60841,28160],{"className":60842},[106,110],"，取 ",[18,60845,60847,60877],{"className":60846,"translate":22},[21],[18,60848,60850],{"className":60849},[26],[28,60851,60852],{"xmlns":30},[32,60853,60854,60874],{},[35,60855,60856,60862,60864],{},[38,60857,60858,60860],{},[41,60859,38573],{},[41,60861,5597],{},[49,60863,433],{},[1556,60865,60866,60868],{},[45,60867,2266],{},[35,60869,60870,60872],{},[49,60871,2253],{},[41,60873,5597],{},[86,60875,60876],{"encoding":88},"x_i=2^{-i}",[18,60878,60880,60935],{"className":60879,"ariaHidden":51},[93],[18,60881,60883,60886,60926,60929,60932],{"className":60882},[97],[18,60884],{"className":60885,"style":16894},[101],[18,60887,60889,60892],{"className":60888},[106],[18,60890,38573],{"className":60891},[106,110],[18,60893,60895],{"className":60894},[115],[18,60896,60898,60918],{"className":60897},[119,120],[18,60899,60901,60915],{"className":60900},[124],[18,60902,60904],{"className":60903,"style":5918},[128],[18,60905,60906,60909],{"style":1155},[18,60907],{"className":60908,"style":137},[136],[18,60910,60912],{"className":60911},[141,142,143,144],[18,60913,5597],{"className":60914},[106,110,144],[18,60916,152],{"className":60917},[151],[18,60919,60921],{"className":60920},[124],[18,60922,60924],{"className":60923,"style":159},[128],[18,60925],{},[18,60927],{"className":60928,"style":228},[169],[18,60930,433],{"className":60931},[232],[18,60933],{"className":60934,"style":228},[169],[18,60936,60938,60942],{"className":60937},[97],[18,60939],{"className":60940,"style":60941},[101],"height:0.8247em;",[18,60943,60945,60948],{"className":60944},[106],[18,60946,2266],{"className":60947},[106],[18,60949,60951],{"className":60950},[115],[18,60952,60954],{"className":60953},[119],[18,60955,60957],{"className":60956},[124],[18,60958,60960],{"className":60959,"style":60941},[128],[18,60961,60962,60965],{"style":1962},[18,60963],{"className":60964,"style":137},[136],[18,60966,60968],{"className":60967},[141,142,143,144],[18,60969,60971,60974],{"className":60970},[106,144],[18,60972,2253],{"className":60973},[106,144],[18,60975,5597],{"className":60976},[106,110,144],"，给定任意标注 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取模。",[18,61552,61554,61572],{"className":61553,"translate":22},[21],[18,61555,61557],{"className":61556},[26],[28,61558,61559],{"xmlns":30},[32,61560,61561,61569],{},[35,61562,61563,61565,61567],{},[41,61564,19804],{},[49,61566,6882],{},[41,61568,5597],{},[86,61570,61571],{"encoding":88},"j>i",[18,61573,61575,61593],{"className":61574,"ariaHidden":51},[93],[18,61576,61578,61581,61584,61587,61590],{"className":61577},[97],[18,61579],{"className":61580,"style":28331},[101],[18,61582,19804],{"className":61583,"style":19921},[106,110],[18,61585],{"className":61586,"style":228},[169],[18,61588,6882],{"className":61589},[232],[18,61591],{"className":61592,"style":228},[169],[18,61594,61596,61599],{"className":61595},[97],[18,61597],{"className":61598,"style":61436},[101],[18,61600,5597],{"className":61601},[106,110]," 的项贡献 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的整数倍，可以丢掉；",[18,61638,61640,61658],{"className":61639,"translate":22},[21],[18,61641,61643],{"className":61642},[26],[28,61644,61645],{"xmlns":30},[32,61646,61647,61655],{},[35,61648,61649,61651,61653],{},[41,61650,19804],{},[49,61652,433],{},[41,61654,5597],{},[86,61656,61657],{"encoding":88},"j=i",[18,61659,61661,61679],{"className":61660,"ariaHidden":51},[93],[18,61662,61664,61667,61670,61673,61676],{"className":61663},[97],[18,61665],{"className":61666,"style":28331},[101],[18,61668,19804],{"className":61669,"style":19921},[106,110],[18,61671],{"className":61672,"style":228},[169],[18,61674,433],{"className":61675},[232],[18,61677],{"className":61678,"style":228},[169],[18,61680,61682,61685],{"className":61681},[97],[18,61683],{"className":61684,"style":61436},[101],[18,61686,5597],{"className":61687},[106,110],[18,61689,61691,61704],{"className":61690,"translate":22},[21],[18,61692,61694],{"className":61693},[26],[28,61695,61696],{"xmlns":30},[32,61697,61698,61702],{},[35,61699,61700],{},[41,61701,3941],{},[86,61703,3944],{"encoding":88},[18,61705,61707],{"className":61706,"ariaHidden":51},[93],[18,61708,61710,61713],{"className":61709},[97],[18,61711],{"className":61712,"style":1083},[101],[18,61714,3941],{"className":61715,"style":3957},[106,110],"（当 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的项合并为 ",[18,61868,61870,61898],{"className":61869,"translate":22},[21],[18,61871,61873],{"className":61872},[26],[28,61874,61875],{"xmlns":30},[32,61876,61877,61895],{},[35,61878,61879,61881,61887,61889],{},[41,61880,3941],{},[38,61882,61883,61885],{},[41,61884,39955],{},[41,61886,5597],{},[41,61888,317],{"mathvariant":316},[1556,61890,61891,61893],{},[45,61892,2266],{},[41,61894,5597],{},[86,61896,61897],{"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}.",[18,62266,62268,62326,62612],{"className":62267,"ariaHidden":51},[93],[18,62269,62271,62274,62277,62317,62320,62323],{"className":62270},[97],[18,62272],{"className":62273,"style":448},[101],[18,62275,77],{"className":62276,"style":254},[106,110],[18,62278,62280,62283],{"className":62279},[106],[18,62281,38573],{"className":62282},[106,110],[18,62284,62286],{"className":62285},[115],[18,62287,62289,62309],{"className":62288},[119,120],[18,62290,62292,62306],{"className":62291},[124],[18,62293,62295],{"className":62294,"style":5918},[128],[18,62296,62297,62300],{"style":1155},[18,62298],{"className":62299,"style":137},[136],[18,62301,62303],{"className":62302},[141,142,143,144],[18,62304,5597],{"className":62305},[106,110,144],[18,62307,152],{"className":62308},[151],[18,62310,62312],{"className":62311},[124],[18,62313,62315],{"className":62314,"style":159},[128],[18,62316],{},[18,62318],{"className":62319,"style":228},[169],[18,62321,62177],{"className":62322},[232],[18,62324],{"className":62325,"style":228},[169],[18,62327,62329,62332,62335,62338,62602,62606,62609],{"className":62328},[97],[18,62330],{"className":62331,"style":20800},[101],[18,62333,3941],{"className":62334,"style":3957},[106,110],[18,62336],{"className":62337,"style":170},[169],[18,62339,62341,62347,62388,62391,62394,62397,62400,62403,62443,62446,62449,62452,62455,62458,62461,62464,62467,62470,62596],{"className":62340},[174],[18,62342,62344],{"className":62343,"style":13793},[250,13792],[18,62345,74],{"className":62346},[1230,143],[18,62348,62350,62353],{"className":62349},[106],[18,62351,2266],{"className":62352},[106],[18,62354,62356],{"className":62355},[115],[18,62357,62359],{"className":62358},[119],[18,62360,62362],{"className":62361},[124],[18,62363,62365],{"className":62364,"style":61384},[128],[18,62366,62367,62370],{"style":1637},[18,62368],{"className":62369,"style":137},[136],[18,62371,62373],{"className":62372},[141,142,143,144],[18,62374,62376,62379,62382,62385],{"className":62375},[106,144],[18,62377,2253],{"className":62378},[106,144],[18,62380,5597],{"className":62381},[106,110,144],[18,62383,2253],{"className":62384},[2072,144],[18,62386,47],{"className":62387},[106,144],[18,62389],{"className":62390,"style":2068},[169],[18,62392,7984],{"className":62393},[2072],[18,62395],{"className":62396,"style":2068},[169],[18,62398,47],{"className":62399},[106,2859],[18,62401,945],{"className":62402},[250],[18,62404,62406,62409],{"className":62405},[106],[18,62407,46343],{"className":62408,"style":3957},[106,110],[18,62410,62412],{"className":62411},[115],[18,62413,62415,62435],{"className":62414},[119,120],[18,62416,62418,62432],{"className":62417},[124],[18,62419,62421],{"className":62420,"style":5918},[128],[18,62422,62423,62426],{"style":30341},[18,62424],{"className":62425,"style":137},[136],[18,62427,62429],{"className":62428},[141,142,143,144],[18,62430,5597],{"className":62431},[106,110,144],[18,62433,152],{"className":62434},[151],[18,62436,62438],{"className":62437},[124],[18,62439,62441],{"className":62440,"style":159},[128],[18,62442],{},[18,62444],{"className":62445,"style":228},[169],[18,62447,433],{"className":62448},[232],[18,62450],{"className":62451,"style":228},[169],[18,62453,2253],{"className":62454},[106],[18,62456,47],{"className":62457},[106],[18,62459,965],{"className":62460},[267],[18,62462],{"className":62463,"style":2068},[169],[18,62465,7984],{"className":62466},[2072],[18,62468],{"className":62469,"style":2068},[169],[18,62471,62473,62476,62593],{"className":62472},[106],[18,62474],{"className":62475},[250,6009],[18,62477,62479],{"className":62478},[5651],[18,62480,62482,62585],{"className":62481},[119,120],[18,62483,62485,62582],{"className":62484},[124],[18,62486,62488,62526,62534],{"className":62487,"style":6022},[128],[18,62489,62490,62493],{"style":6025},[18,62491],{"className":62492,"style":697},[136],[18,62494,62496],{"className":62495},[106],[18,62497,62499,62502],{"className":62498},[106],[18,62500,2266],{"className":62501},[106],[18,62503,62505],{"className":62504},[115],[18,62506,62508],{"className":62507},[119],[18,62509,62511],{"className":62510},[124],[18,62512,62515],{"className":62513,"style":62514},[128],"height:0.7507em;",[18,62516,62517,62520],{"style":26105},[18,62518],{"className":62519,"style":137},[136],[18,62521,62523],{"className":62522},[141,142,143,144],[18,62524,5597],{"className":62525},[106,110,144],[18,62527,62528,62531],{"style":6037},[18,62529],{"className":62530,"style":697},[136],[18,62532],{"className":62533,"style":6045},[6044],[18,62535,62536,62539],{"style":6048},[18,62537],{"className":62538,"style":697},[136],[18,62540,62542],{"className":62541},[106],[18,62543,62545,62548],{"className":62544},[106],[18,62546,39955],{"className":62547,"style":40031},[106,110],[18,62549,62551],{"className":62550},[115],[18,62552,62554,62574],{"className":62553},[119,120],[18,62555,62557,62571],{"className":62556},[124],[18,62558,62560],{"className":62559,"style":5918},[128],[18,62561,62562,62565],{"style":61931},[18,62563],{"className":62564,"style":137},[136],[18,62566,62568],{"className":62567},[141,142,143,144],[18,62569,5597],{"className":62570},[106,110,144],[18,62572,152],{"className":62573},[151],[18,62575,62577],{"className":62576},[124],[18,62578,62580],{"className":62579,"style":159},[128],[18,62581],{},[18,62583,152],{"className":62584},[151],[18,62586,62588],{"className":62587},[124],[18,62589,62591],{"className":62590,"style":6067},[128],[18,62592],{},[18,62594],{"className":62595},[267,6009],[18,62597,62599],{"className":62598,"style":13793},[267,13792],[18,62600,84],{"className":62601},[1230,143],[18,62603],{"className":62604},[169,62605],"allowbreak",[18,62607],{"className":62608,"style":42840},[169],[18,62610],{"className":62611,"style":170},[169],[18,62613,62615,62618,62621,62631,62635,62638,62641,62644],{"className":62614},[97],[18,62616],{"className":62617,"style":242},[101],[18,62619,74],{"className":62620},[250],[18,62622,62624],{"className":62623},[106],[18,62625,62627],{"className":62626},[106],[18,62628,62630],{"className":62629},[106,6351],"mod",[18,62632],{"className":62633,"style":62634},[169],"margin-right:0.3333em;",[18,62636,2266],{"className":62637},[106],[18,62639,3941],{"className":62640,"style":3957},[106,110],[18,62642,84],{"className":62643},[267],[18,62645,968],{"className":62646},[106],[11,62648,39309,62649,62746,62747,62798,62799,62895,62896,62948,62949,63121,63122,63150],{},[18,62650,62652,62676],{"className":62651,"translate":22},[21],[18,62653,62655],{"className":62654},[26],[28,62656,62657],{"xmlns":30},[32,62658,62659,62673],{},[35,62660,62661,62667,62669,62671],{},[38,62662,62663,62665],{},[41,62664,46343],{},[41,62666,5597],{},[49,62668,433],{},[49,62670,7984],{},[45,62672,47],{},[86,62674,62675],{"encoding":88},"y_i=+1",[18,62677,62679,62734],{"className":62678,"ariaHidden":51},[93],[18,62680,62682,62685,62725,62728,62731],{"className":62681},[97],[18,62683],{"className":62684,"style":38543},[101],[18,62686,62688,62691],{"className":62687},[106],[18,62689,46343],{"className":62690,"style":3957},[106,110],[18,62692,62694],{"className":62693},[115],[18,62695,62697,62717],{"className":62696},[119,120],[18,62698,62700,62714],{"className":62699},[124],[18,62701,62703],{"className":62702,"style":5918},[128],[18,62704,62705,62708],{"style":30341},[18,62706],{"className":62707,"style":137},[136],[18,62709,62711],{"className":62710},[141,142,143,144],[18,62712,5597],{"className":62713},[106,110,144],[18,62715,152],{"className":62716},[151],[18,62718,62720],{"className":62719},[124],[18,62721,62723],{"className":62722,"style":159},[128],[18,62724],{},[18,62726],{"className":62727,"style":228},[169],[18,62729,433],{"className":62730},[232],[18,62732],{"className":62733,"style":228},[169],[18,62735,62737,62740,62743],{"className":62736},[97],[18,62738],{"className":62739,"style":18090},[101],[18,62741,7984],{"className":62742},[106],[18,62744,47],{"className":62745},[106]," 时，括号内的值落在 ",[18,62748,62750,62771],{"className":62749,"translate":22},[21],[18,62751,62753],{"className":62752},[26],[28,62754,62755],{"xmlns":30},[32,62756,62757,62769],{},[35,62758,62759,62761,62763,62765,62767],{},[49,62760,74],{"stretchy":73},[45,62762,430],{},[49,62764,52],{"separator":51},[45,62766,47],{},[49,62768,84],{"stretchy":73},[86,62770,54891],{"encoding":88},[18,62772,62774],{"className":62773,"ariaHidden":51},[93],[18,62775,62777,62780,62783,62786,62789,62792,62795],{"className":62776},[97],[18,62778],{"className":62779,"style":242},[101],[18,62781,74],{"className":62782},[250],[18,62784,430],{"className":62785},[106],[18,62787,52],{"className":62788},[165],[18,62790],{"className":62791,"style":170},[169],[18,62793,47],{"className":62794},[106],[18,62796,84],{"className":62797},[267],"；当 ",[18,62800,62802,62825],{"className":62801,"translate":22},[21],[18,62803,62805],{"className":62804},[26],[28,62806,62807],{"xmlns":30},[32,62808,62809,62823],{},[35,62810,62811,62817,62819,62821],{},[38,62812,62813,62815],{},[41,62814,46343],{},[41,62816,5597],{},[49,62818,433],{},[49,62820,2253],{},[45,62822,47],{},[86,62824,61743],{"encoding":88},[18,62826,62828,62883],{"className":62827,"ariaHidden":51},[93],[18,62829,62831,62834,62874,62877,62880],{"className":62830},[97],[18,62832],{"className":62833,"style":38543},[101],[18,62835,62837,62840],{"className":62836},[106],[18,62838,46343],{"className":62839,"style":3957},[106,110],[18,62841,62843],{"className":62842},[115],[18,62844,62846,62866],{"className":62845},[119,120],[18,62847,62849,62863],{"className":62848},[124],[18,62850,62852],{"className":62851,"style":5918},[128],[18,62853,62854,62857],{"style":30341},[18,62855],{"className":62856,"style":137},[136],[18,62858,62860],{"className":62859},[141,142,143,144],[18,62861,5597],{"className":62862},[106,110,144],[18,62864,152],{"className":62865},[151],[18,62867,62869],{"className":62868},[124],[18,62870,62872],{"className":62871,"style":159},[128],[18,62873],{},[18,62875],{"className":62876,"style":228},[169],[18,62878,433],{"className":62879},[232],[18,62881],{"className":62882,"style":228},[169],[18,62884,62886,62889,62892],{"className":62885},[97],[18,62887],{"className":62888,"style":18090},[101],[18,62890,2253],{"className":62891},[106],[18,62893,47],{"className":62894},[106]," 时落在 ",[18,62897,62899,62921],{"className":62898,"translate":22},[21],[18,62900,62902],{"className":62901},[26],[28,62903,62904],{"xmlns":30},[32,62905,62906,62918],{},[35,62907,62908,62910,62912,62914,62916],{},[49,62909,74],{"stretchy":73},[45,62911,47],{},[49,62913,52],{"separator":51},[45,62915,2266],{},[49,62917,84],{"stretchy":73},[86,62919,62920],{"encoding":88},"(1,2)",[18,62922,62924],{"className":62923,"ariaHidden":51},[93],[18,62925,62927,62930,62933,62936,62939,62942,62945],{"className":62926},[97],[18,62928],{"className":62929,"style":242},[101],[18,62931,74],{"className":62932},[250],[18,62934,47],{"className":62935},[106],[18,62937,52],{"className":62938},[165],[18,62940],{"className":62941,"style":170},[169],[18,62943,2266],{"className":62944},[106],[18,62946,84],{"className":62947},[267],"。所以 ",[18,62950,62952,62996],{"className":62951,"translate":22},[21],[18,62953,62955],{"className":62954},[26],[28,62956,62957],{"xmlns":30},[32,62958,62959,62993],{},[35,62960,62961,62963,62965,62967,62969,62971,62973,62975,62981,62983,62985,62987],{},[41,62962,45630],{"mathvariant":316},[49,62964,899],{},[49,62966,74],{"stretchy":73},[41,62968,45637],{},[49,62970,899],{},[49,62972,74],{"stretchy":73},[41,62974,77],{},[38,62976,62977,62979],{},[41,62978,38573],{},[41,62980,5597],{},[49,62982,84],{"stretchy":73},[49,62984,84],{"stretchy":73},[49,62986,433],{},[38,62988,62989,62991],{},[41,62990,46343],{},[41,62992,5597],{},[86,62994,62995],{"encoding":88},"\\operatorname{sign}(\\sin(\\theta x_i))=y_i",[18,62997,62999,63075],{"className":62998,"ariaHidden":51},[93],[18,63000,63002,63005,63011,63014,63017,63020,63023,63063,63066,63069,63072],{"className":63001},[97],[18,63003],{"className":63004,"style":242},[101],[18,63006,63008],{"className":63007},[985],[18,63009,45630],{"className":63010},[106,6351],[18,63012,74],{"className":63013},[250],[18,63015,45637],{"className":63016},[985],[18,63018,74],{"className":63019},[250],[18,63021,77],{"className":63022,"style":254},[106,110],[18,63024,63026,63029],{"className":63025},[106],[18,63027,38573],{"className":63028},[106,110],[18,63030,63032],{"className":63031},[115],[18,63033,63035,63055],{"className":63034},[119,120],[18,63036,63038,63052],{"className":63037},[124],[18,63039,63041],{"className":63040,"style":5918},[128],[18,63042,63043,63046],{"style":1155},[18,63044],{"className":63045,"style":137},[136],[18,63047,63049],{"className":63048},[141,142,143,144],[18,63050,5597],{"className":63051},[106,110,144],[18,63053,152],{"className":63054},[151],[18,63056,63058],{"className":63057},[124],[18,63059,63061],{"className":63060,"style":159},[128],[18,63062],{},[18,63064,3431],{"className":63065},[267],[18,63067],{"className":63068,"style":228},[169],[18,63070,433],{"className":63071},[232],[18,63073],{"className":63074,"style":228},[169],[18,63076,63078,63081],{"className":63077},[97],[18,63079],{"className":63080,"style":38543},[101],[18,63082,63084,63087],{"className":63083},[106],[18,63085,46343],{"className":63086,"style":3957},[106,110],[18,63088,63090],{"className":63089},[115],[18,63091,63093,63113],{"className":63092},[119,120],[18,63094,63096,63110],{"className":63095},[124],[18,63097,63099],{"className":63098,"style":5918},[128],[18,63100,63101,63104],{"style":30341},[18,63102],{"className":63103,"style":137},[136],[18,63105,63107],{"className":63106},[141,142,143,144],[18,63108,5597],{"className":63109},[106,110,144],[18,63111,152],{"className":63112},[151],[18,63114,63116],{"className":63115},[124],[18,63117,63119],{"className":63118,"style":159},[128],[18,63120],{},"。任意 ",[18,63123,63125,63138],{"className":63124,"translate":22},[21],[18,63126,63128],{"className":63127},[26],[28,63129,63130],{"xmlns":30},[32,63131,63132,63136],{},[35,63133,63134],{},[41,63135,28160],{},[86,63137,28160],{"encoding":88},[18,63139,63141],{"className":63140,"ariaHidden":51},[93],[18,63142,63144,63147],{"className":63143},[97],[18,63145],{"className":63146,"style":1083},[101],[18,63148,28160],{"className":63149},[106,110]," 个点都能被打散，VC 维无穷。",[11,63152,63153],{},"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。",[11,63155,63156],{},[717,63157,63158,63159,52814],{},"例 7（最近邻，",[18,63160,63162,63181],{"className":63161,"translate":22},[21],[18,63163,63165],{"className":63164},[26],[28,63166,63167],{"xmlns":30},[32,63168,63169,63179],{},[35,63170,63171,63173,63175,63177],{},[41,63172,52202],{"mathvariant":316},[49,63174,899],{},[49,63176,433],{},[41,63178,2028],{"mathvariant":316},[86,63180,60562],{"encoding":88},[18,63182,63184,63205],{"className":63183,"ariaHidden":51},[93],[18,63185,63187,63190,63196,63199,63202],{"className":63186},[97],[18,63188],{"className":63189,"style":295},[101],[18,63191,63193],{"className":63192},[985],[18,63194,52202],{"className":63195},[106,6351],[18,63197],{"className":63198,"style":228},[169],[18,63200,433],{"className":63201},[232],[18,63203],{"className":63204,"style":228},[169],[18,63206,63208,63211],{"className":63207},[97],[18,63209],{"className":63210,"style":1083},[101],[18,63212,2028],{"className":63213},[106],[11,63215,63216,63217,63245,63246,63274,63275,63345],{},"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 ",[18,63218,63220,63233],{"className":63219,"translate":22},[21],[18,63221,63223],{"className":63222},[26],[28,63224,63225],{"xmlns":30},[32,63226,63227,63231],{},[35,63228,63229],{},[41,63230,51012],{},[86,63232,51012],{"encoding":88},[18,63234,63236],{"className":63235,"ariaHidden":51},[93],[18,63237,63239,63242],{"className":63238},[97],[18,63240],{"className":63241,"style":600},[101],[18,63243,51012],{"className":63244,"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 ",[18,63247,63249,63262],{"className":63248,"translate":22},[21],[18,63250,63252],{"className":63251},[26],[28,63253,63254],{"xmlns":30},[32,63255,63256,63260],{},[35,63257,63258],{},[41,63259,51012],{},[86,63261,51012],{"encoding":88},[18,63263,63265],{"className":63264,"ariaHidden":51},[93],[18,63266,63268,63271],{"className":63267},[97],[18,63269],{"className":63270,"style":600},[101],[18,63272,51012],{"className":63273,"style":5073},[106,110]," 本身（标签按标注给）；对每个 ",[18,63276,63278,63296],{"className":63277,"translate":22},[21],[18,63279,63281],{"className":63280},[26],[28,63282,63283],{"xmlns":30},[32,63284,63285,63293],{},[35,63286,63287],{},[38,63288,63289,63291],{},[41,63290,38573],{},[41,63292,5597],{},[86,63294,63295],{"encoding":88},"x_i",[18,63297,63299],{"className":63298,"ariaHidden":51},[93],[18,63300,63302,63305],{"className":63301},[97],[18,63303],{"className":63304,"style":16894},[101],[18,63306,63308,63311],{"className":63307},[106],[18,63309,38573],{"className":63310},[106,110],[18,63312,63314],{"className":63313},[115],[18,63315,63317,63337],{"className":63316},[119,120],[18,63318,63320,63334],{"className":63319},[124],[18,63321,63323],{"className":63322,"style":5918},[128],[18,63324,63325,63328],{"style":1155},[18,63326],{"className":63327,"style":137},[136],[18,63329,63331],{"className":63330},[141,142,143,144],[18,63332,5597],{"className":63333},[106,110,144],[18,63335,152],{"className":63336},[151],[18,63338,63340],{"className":63339},[124],[18,63341,63343],{"className":63342,"style":159},[128],[18,63344],{},"，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。",[11,63347,63348],{},"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。",[11,63350,63351],{},"把上面几个例子汇总：",[43703,63353,63354,63365],{},[43706,63355,63356],{},[43709,63357,63358,63360,63363],{},[43712,63359,46291],{},[43712,63361,63362],{},"输入空间",[43712,63364,52183],{},[43722,63366,63367,63433,63498,63615,63715,63832,63975,64108],{},[43709,63368,63369,63372,63403],{},[43727,63370,63371],{},"半直线",[43727,63373,63374],{},[18,63375,63377,63391],{"className":63376,"translate":22},[21],[18,63378,63380],{"className":63379},[26],[28,63381,63382],{"xmlns":30},[32,63383,63384,63388],{},[35,63385,63386],{},[41,63387,3071],{"mathvariant":933},[86,63389,63390],{"encoding":88},"\\mathbb R",[18,63392,63394],{"className":63393,"ariaHidden":51},[93],[18,63395,63397,63400],{"className":63396},[97],[18,63398],{"className":63399,"style":45798},[101],[18,63401,3071],{"className":63402},[106,1139],[43727,63404,63405],{},[18,63406,63408,63421],{"className":63407,"translate":22},[21],[18,63409,63411],{"className":63410},[26],[28,63412,63413],{"xmlns":30},[32,63414,63415,63419],{},[35,63416,63417],{},[45,63418,47],{},[86,63420,47],{"encoding":88},[18,63422,63424],{"className":63423,"ariaHidden":51},[93],[18,63425,63427,63430],{"className":63426},[97],[18,63428],{"className":63429,"style":505},[101],[18,63431,47],{"className":63432},[106],[43709,63434,63435,63438,63468],{},[43727,63436,63437],{},"区间",[43727,63439,63440],{},[18,63441,63443,63456],{"className":63442,"translate":22},[21],[18,63444,63446],{"className":63445},[26],[28,63447,63448],{"xmlns":30},[32,63449,63450,63454],{},[35,63451,63452],{},[41,63453,3071],{"mathvariant":933},[86,63455,63390],{"encoding":88},[18,63457,63459],{"className":63458,"ariaHidden":51},[93],[18,63460,63462,63465],{"className":63461},[97],[18,63463],{"className":63464,"style":45798},[101],[18,63466,3071],{"className":63467},[106,1139],[43727,63469,63470],{},[18,63471,63473,63486],{"className":63472,"translate":22},[21],[18,63474,63476],{"className":63475},[26],[28,63477,63478],{"xmlns":30},[32,63479,63480,63484],{},[35,63481,63482],{},[45,63483,2266],{},[86,63485,2266],{"encoding":88},[18,63487,63489],{"className":63488,"ariaHidden":51},[93],[18,63490,63492,63495],{"className":63491},[97],[18,63493],{"className":63494,"style":505},[101],[18,63496,2266],{"className":63497},[106],[43709,63499,63500,63503,63563],{},[43727,63501,63502],{},"仿射超平面",[43727,63504,63505],{},[18,63506,63508,63525],{"className":63507,"translate":22},[21],[18,63509,63511],{"className":63510},[26],[28,63512,63513],{"xmlns":30},[32,63514,63515,63523],{},[35,63516,63517],{},[1556,63518,63519,63521],{},[41,63520,3071],{"mathvariant":933},[41,63522,4031],{},[86,63524,56925],{"encoding":88},[18,63526,63528],{"className":63527,"ariaHidden":51},[93],[18,63529,63531,63534],{"className":63530},[97],[18,63532],{"className":63533,"style":35177},[101],[18,63535,63537,63540],{"className":63536},[106],[18,63538,3071],{"className":63539},[106,1139],[18,63541,63543],{"className":63542},[115],[18,63544,63546],{"className":63545},[119],[18,63547,63549],{"className":63548},[124],[18,63550,63552],{"className":63551,"style":35177},[128],[18,63553,63554,63557],{"style":1962},[18,63555],{"className":63556,"style":137},[136],[18,63558,63560],{"className":63559},[141,142,143,144],[18,63561,4031],{"className":63562},[106,110,144],[43727,63564,63565],{},[18,63566,63568,63585],{"className":63567,"translate":22},[21],[18,63569,63571],{"className":63570},[26],[28,63572,63573],{"xmlns":30},[32,63574,63575,63583],{},[35,63576,63577,63579,63581],{},[41,63578,4031],{},[49,63580,7984],{},[45,63582,47],{},[86,63584,56239],{"encoding":88},[18,63586,63588,63606],{"className":63587,"ariaHidden":51},[93],[18,63589,63591,63594,63597,63600,63603],{"className":63590},[97],[18,63592],{"className":63593,"style":7221},[101],[18,63595,4031],{"className":63596},[106,110],[18,63598],{"className":63599,"style":2068},[169],[18,63601,7984],{"className":63602},[2072],[18,63604],{"className":63605,"style":2068},[169],[18,63607,63609,63612],{"className":63608},[97],[18,63610],{"className":63611,"style":505},[101],[18,63613,47],{"className":63614},[106],[43709,63616,63617,63620,63680],{},[43727,63618,63619],{},"轴对齐矩形",[43727,63621,63622],{},[18,63623,63625,63642],{"className":63624,"translate":22},[21],[18,63626,63628],{"className":63627},[26],[28,63629,63630],{"xmlns":30},[32,63631,63632,63640],{},[35,63633,63634],{},[1556,63635,63636,63638],{},[41,63637,3071],{"mathvariant":933},[41,63639,4031],{},[86,63641,56925],{"encoding":88},[18,63643,63645],{"className":63644,"ariaHidden":51},[93],[18,63646,63648,63651],{"className":63647},[97],[18,63649],{"className":63650,"style":35177},[101],[18,63652,63654,63657],{"className":63653},[106],[18,63655,3071],{"className":63656},[106,1139],[18,63658,63660],{"className":63659},[115],[18,63661,63663],{"className":63662},[119],[18,63664,63666],{"className":63665},[124],[18,63667,63669],{"className":63668,"style":35177},[128],[18,63670,63671,63674],{"style":1962},[18,63672],{"className":63673,"style":137},[136],[18,63675,63677],{"className":63676},[141,142,143,144],[18,63678,4031],{"className":63679},[106,110,144],[43727,63681,63682],{},[18,63683,63685,63700],{"className":63684,"translate":22},[21],[18,63686,63688],{"className":63687},[26],[28,63689,63690],{"xmlns":30},[32,63691,63692,63698],{},[35,63693,63694,63696],{},[45,63695,2266],{},[41,63697,4031],{},[86,63699,59621],{"encoding":88},[18,63701,63703],{"className":63702,"ariaHidden":51},[93],[18,63704,63706,63709,63712],{"className":63705},[97],[18,63707],{"className":63708,"style":295},[101],[18,63710,2266],{"className":63711},[106],[18,63713,4031],{"className":63714},[106,110],[43709,63716,63717,63720,63780],{},[43727,63718,63719],{},"球",[43727,63721,63722],{},[18,63723,63725,63742],{"className":63724,"translate":22},[21],[18,63726,63728],{"className":63727},[26],[28,63729,63730],{"xmlns":30},[32,63731,63732,63740],{},[35,63733,63734],{},[1556,63735,63736,63738],{},[41,63737,3071],{"mathvariant":933},[41,63739,4031],{},[86,63741,56925],{"encoding":88},[18,63743,63745],{"className":63744,"ariaHidden":51},[93],[18,63746,63748,63751],{"className":63747},[97],[18,63749],{"className":63750,"style":35177},[101],[18,63752,63754,63757],{"className":63753},[106],[18,63755,3071],{"className":63756},[106,1139],[18,63758,63760],{"className":63759},[115],[18,63761,63763],{"className":63762},[119],[18,63764,63766],{"className":63765},[124],[18,63767,63769],{"className":63768,"style":35177},[128],[18,63770,63771,63774],{"style":1962},[18,63772],{"className":63773,"style":137},[136],[18,63775,63777],{"className":63776},[141,142,143,144],[18,63778,4031],{"className":63779},[106,110,144],[43727,63781,63782],{},[18,63783,63785,63802],{"className":63784,"translate":22},[21],[18,63786,63788],{"className":63787},[26],[28,63789,63790],{"xmlns":30},[32,63791,63792,63800],{},[35,63793,63794,63796,63798],{},[41,63795,4031],{},[49,63797,7984],{},[45,63799,47],{},[86,63801,56239],{"encoding":88},[18,63803,63805,63823],{"className":63804,"ariaHidden":51},[93],[18,63806,63808,63811,63814,63817,63820],{"className":63807},[97],[18,63809],{"className":63810,"style":7221},[101],[18,63812,4031],{"className":63813},[106,110],[18,63815],{"className":63816,"style":2068},[169],[18,63818,7984],{"className":63819},[2072],[18,63821],{"className":63822,"style":2068},[169],[18,63824,63826,63829],{"className":63825},[97],[18,63827],{"className":63828,"style":505},[101],[18,63830,47],{"className":63831},[106],[43709,63833,63834,63865,63868],{},[43727,63835,63836,63837],{},"有限类 ",[18,63838,63840,63853],{"className":63839,"translate":22},[21],[18,63841,63843],{"className":63842},[26],[28,63844,63845],{"xmlns":30},[32,63846,63847,63851],{},[35,63848,63849],{},[41,63850,18304],{"mathvariant":1462},[86,63852,46308],{"encoding":88},[18,63854,63856],{"className":63855,"ariaHidden":51},[93],[18,63857,63859,63862],{"className":63858},[97],[18,63860],{"className":63861,"style":600},[101],[18,63863,18304],{"className":63864,"style":46321},[106,1478],[43727,63866,63867],{},"任意",[43727,63869,63870],{},[18,63871,63873,63903],{"className":63872,"translate":22},[21],[18,63874,63876],{"className":63875},[26],[28,63877,63878],{"xmlns":30},[32,63879,63880,63900],{},[35,63881,63882,63884,63894,63896,63898],{},[49,63883,4334],{},[38,63885,63886,63892],{},[35,63887,63888,63890],{},[41,63889,14730],{},[49,63891,899],{},[45,63893,2266],{},[49,63895,11222],{"stretchy":73},[41,63897,18304],{"mathvariant":1462},[49,63899,11222],{"stretchy":73},[86,63901,63902],{"encoding":88},"\\le\\log_2\\lvert\\mathcal H\\rvert",[18,63904,63906,63918],{"className":63905,"ariaHidden":51},[93],[18,63907,63909,63912,63915],{"className":63908},[97],[18,63910],{"className":63911,"style":53130},[101],[18,63913,4334],{"className":63914},[232],[18,63916],{"className":63917,"style":228},[169],[18,63919,63921,63924,63966,63969,63972],{"className":63920},[97],[18,63922],{"className":63923,"style":242},[101],[18,63925,63927,63932],{"className":63926},[985],[18,63928,15171,63930],{"className":63929},[985],[18,63931,15174],{"style":6352},[18,63933,63935],{"className":63934},[115],[18,63936,63938,63958],{"className":63937},[119,120],[18,63939,63941,63955],{"className":63940},[124],[18,63942,63944],{"className":63943,"style":60129},[128],[18,63945,63946,63949],{"style":60132},[18,63947],{"className":63948,"style":137},[136],[18,63950,63952],{"className":63951},[141,142,143,144],[18,63953,2266],{"className":63954},[106,144],[18,63956,152],{"className":63957},[151],[18,63959,63961],{"className":63960},[124],[18,63962,63964],{"className":63963,"style":60151},[128],[18,63965],{},[18,63967,11222],{"className":63968},[250],[18,63970,18304],{"className":63971,"style":46321},[106,1478],[18,63973,11222],{"className":63974},[267],[43709,63976,63977,64047,64077],{},[43727,63978,63979],{},[18,63980,63982,64014],{"className":63981,"translate":22},[21],[18,63983,63985],{"className":63984},[26],[28,63986,63987],{"xmlns":30},[32,63988,63989,64011],{},[35,63990,63991,63993,63995,63997,63999,64001,64003,64005,64007,64009],{},[41,63992,45630],{"mathvariant":316},[49,63994,899],{},[49,63996,74],{"stretchy":73},[41,63998,45637],{},[49,64000,899],{},[49,64002,74],{"stretchy":73},[41,64004,77],{},[41,64006,38573],{},[49,64008,84],{"stretchy":73},[49,64010,84],{"stretchy":73},[86,64012,64013],{"encoding":88},"\\operatorname{sign}(\\sin(\\theta x))",[18,64015,64017],{"className":64016,"ariaHidden":51},[93],[18,64018,64020,64023,64029,64032,64035,64038,64041,64044],{"className":64019},[97],[18,64021],{"className":64022,"style":242},[101],[18,64024,64026],{"className":64025},[985],[18,64027,45630],{"className":64028},[106,6351],[18,64030,74],{"className":64031},[250],[18,64033,45637],{"className":64034},[985],[18,64036,74],{"className":64037},[250],[18,64039,77],{"className":64040,"style":254},[106,110],[18,64042,38573],{"className":64043},[106,110],[18,64045,3431],{"className":64046},[267],[43727,64048,64049],{},[18,64050,64052,64065],{"className":64051,"translate":22},[21],[18,64053,64055],{"className":64054},[26],[28,64056,64057],{"xmlns":30},[32,64058,64059,64063],{},[35,64060,64061],{},[41,64062,3071],{"mathvariant":933},[86,64064,63390],{"encoding":88},[18,64066,64068],{"className":64067,"ariaHidden":51},[93],[18,64069,64071,64074],{"className":64070},[97],[18,64072],{"className":64073,"style":45798},[101],[18,64075,3071],{"className":64076},[106,1139],[43727,64078,64079],{},[18,64080,64082,64096],{"className":64081,"translate":22},[21],[18,64083,64085],{"className":64084},[26],[28,64086,64087],{"xmlns":30},[32,64088,64089,64093],{},[35,64090,64091],{},[41,64092,2028],{"mathvariant":316},[86,64094,64095],{"encoding":88},"\\infty",[18,64097,64099],{"className":64098,"ariaHidden":51},[93],[18,64100,64102,64105],{"className":64101},[97],[18,64103],{"className":64104,"style":1083},[101],[18,64106,2028],{"className":64107},[106],[43709,64109,64110,64113,64173],{},[43727,64111,64112],{},"1-NN",[43727,64114,64115],{},[18,64116,64118,64135],{"className":64117,"translate":22},[21],[18,64119,64121],{"className":64120},[26],[28,64122,64123],{"xmlns":30},[32,64124,64125,64133],{},[35,64126,64127],{},[1556,64128,64129,64131],{},[41,64130,3071],{"mathvariant":933},[41,64132,4031],{},[86,64134,56925],{"encoding":88},[18,64136,64138],{"className":64137,"ariaHidden":51},[93],[18,64139,64141,64144],{"className":64140},[97],[18,64142],{"className":64143,"style":35177},[101],[18,64145,64147,64150],{"className":64146},[106],[18,64148,3071],{"className":64149},[106,1139],[18,64151,64153],{"className":64152},[115],[18,64154,64156],{"className":64155},[119],[18,64157,64159],{"className":64158},[124],[18,64160,64162],{"className":64161,"style":35177},[128],[18,64163,64164,64167],{"style":1962},[18,64165],{"className":64166,"style":137},[136],[18,64168,64170],{"className":64169},[141,142,143,144],[18,64171,4031],{"className":64172},[106,110,144],[43727,64174,64175],{},[18,64176,64178,64191],{"className":64177,"translate":22},[21],[18,64179,64181],{"className":64180},[26],[28,64182,64183],{"xmlns":30},[32,64184,64185,64189],{},[35,64186,64187],{},[41,64188,2028],{"mathvariant":316},[86,64190,64095],{"encoding":88},[18,64192,64194],{"className":64193,"ariaHidden":51},[93],[18,64195,64197,64200],{"className":64196},[97],[18,64198],{"className":64199,"style":1083},[101],[18,64201,2028],{"className":64202},[106],[5159,64204,64206],{"title":64205},"注（怎么算）",[11,64207,64208,64209,64381,64382,64478,64479,64531,64532,64724,64725,64753,64754,64813,64814,64842,64843,64871],{},"对线性分类器，固定点集 ",[18,64210,64212,64250],{"className":64211,"translate":22},[21],[18,64213,64215],{"className":64214},[26],[28,64216,64217],{"xmlns":30},[32,64218,64219,64247],{},[35,64220,64221,64223,64225,64227,64233,64235,64237,64239,64245],{},[41,64222,51012],{},[49,64224,433],{},[49,64226,1331],{"stretchy":73},[38,64228,64229,64231],{},[41,64230,38573],{},[45,64232,47],{},[49,64234,52],{"separator":51},[49,64236,55],{},[49,64238,52],{"separator":51},[38,64240,64241,64243],{},[41,64242,38573],{},[41,64244,39955],{},[49,64246,1349],{"stretchy":73},[86,64248,64249],{"encoding":88},"C=\\{x_1,\\dots,x_k\\}",[18,64251,64253,64271],{"className":64252,"ariaHidden":51},[93],[18,64254,64256,64259,64262,64265,64268],{"className":64255},[97],[18,64257],{"className":64258,"style":600},[101],[18,64260,51012],{"className":64261,"style":5073},[106,110],[18,64263],{"className":64264,"style":228},[169],[18,64266,433],{"className":64267},[232],[18,64269],{"className":64270,"style":228},[169],[18,64272,64274,64277,64280,64320,64323,64326,64329,64332,64335,64338,64378],{"className":64273},[97],[18,64275],{"className":64276,"style":242},[101],[18,64278,1331],{"className":64279},[250],[18,64281,64283,64286],{"className":64282},[106],[18,64284,38573],{"className":64285},[106,110],[18,64287,64289],{"className":64288},[115],[18,64290,64292,64312],{"className":64291},[119,120],[18,64293,64295,64309],{"className":64294},[124],[18,64296,64298],{"className":64297,"style":129},[128],[18,64299,64300,64303],{"style":1155},[18,64301],{"className":64302,"style":137},[136],[18,64304,64306],{"className":64305},[141,142,143,144],[18,64307,47],{"className":64308},[106,144],[18,64310,152],{"className":64311},[151],[18,64313,64315],{"className":64314},[124],[18,64316,64318],{"className":64317,"style":159},[128],[18,64319],{},[18,64321,52],{"className":64322},[165],[18,64324],{"className":64325,"style":170},[169],[18,64327,55],{"className":64328},[174],[18,64330],{"className":64331,"style":170},[169],[18,64333,52],{"className":64334},[165],[18,64336],{"className":64337,"style":170},[169],[18,64339,64341,64344],{"className":64340},[106],[18,64342,38573],{"className":64343},[106,110],[18,64345,64347],{"className":64346},[115],[18,64348,64350,64370],{"className":64349},[119,120],[18,64351,64353,64367],{"className":64352},[124],[18,64354,64356],{"className":64355,"style":1383},[128],[18,64357,64358,64361],{"style":1155},[18,64359],{"className":64360,"style":137},[136],[18,64362,64364],{"className":64363},[141,142,143,144],[18,64365,39955],{"className":64366,"style":40031},[106,110,144],[18,64368,152],{"className":64369},[151],[18,64371,64373],{"className":64372},[124],[18,64374,64376],{"className":64375,"style":159},[128],[18,64377],{},[18,64379,1349],{"className":64380},[267],"，对每个标注 ",[18,64383,64385,64413],{"className":64384,"translate":22},[21],[18,64386,64388],{"className":64387},[26],[28,64389,64390],{"xmlns":30},[32,64391,64392,64410],{},[35,64393,64394,64396,64398,64400,64402,64404],{},[41,64395,46343],{},[49,64397,921],{},[49,64399,1331],{"stretchy":73},[49,64401,56291],{},[45,64403,47],{},[1556,64405,64406,64408],{},[49,64407,1349],{"stretchy":73},[41,64409,39955],{},[86,64411,64412],{"encoding":88},"y\\in\\{\\pm1\\}^k",[18,64414,64416,64434],{"className":64415,"ariaHidden":51},[93],[18,64417,64419,64422,64425,64428,64431],{"className":64418},[97],[18,64420],{"className":64421,"style":14584},[101],[18,64423,46343],{"className":64424,"style":3957},[106,110],[18,64426],{"className":64427,"style":228},[169],[18,64429,921],{"className":64430},[232],[18,64432],{"className":64433,"style":228},[169],[18,64435,64437,64440,64443,64446,64449],{"className":64436},[97],[18,64438],{"className":64439,"style":35143},[101],[18,64441,1331],{"className":64442},[250],[18,64444,56291],{"className":64445},[106],[18,64447,47],{"className":64448},[106],[18,64450,64452,64455],{"className":64451},[267],[18,64453,1349],{"className":64454},[267],[18,64456,64458],{"className":64457},[115],[18,64459,64461],{"className":64460},[119],[18,64462,64464],{"className":64463},[124],[18,64465,64467],{"className":64466,"style":35177},[128],[18,64468,64469,64472],{"style":1962},[18,64470],{"className":64471,"style":137},[136],[18,64473,64475],{"className":64474},[141,142,143,144],[18,64476,39955],{"className":64477,"style":40031},[106,110,144]," 解可行性问题：是否存在 ",[18,64480,64482,64504],{"className":64481,"translate":22},[21],[18,64483,64485],{"className":64484},[26],[28,64486,64487],{"xmlns":30},[32,64488,64489,64501],{},[35,64490,64491,64493,64495,64497,64499],{},[49,64492,74],{"stretchy":73},[41,64494,55912],{},[49,64496,52],{"separator":51},[41,64498,54067],{},[49,64500,84],{"stretchy":73},[86,64502,64503],{"encoding":88},"(w,b)",[18,64505,64507],{"className":64506,"ariaHidden":51},[93],[18,64508,64510,64513,64516,64519,64522,64525,64528],{"className":64509},[97],[18,64511],{"className":64512,"style":242},[101],[18,64514,74],{"className":64515},[250],[18,64517,55912],{"className":64518,"style":55988},[106,110],[18,64520,52],{"className":64521},[165],[18,64523],{"className":64524,"style":170},[169],[18,64526,54067],{"className":64527},[106,110],[18,64529,84],{"className":64530},[267]," 使 ",[18,64533,64535,64575],{"className":64534,"translate":22},[21],[18,64536,64538],{"className":64537},[26],[28,64539,64540],{"xmlns":30},[32,64541,64542,64572],{},[35,64543,64544,64550,64552,64554,64556,64562,64564,64566,64568,64570],{},[38,64545,64546,64548],{},[41,64547,46343],{},[41,64549,5597],{},[49,64551,74],{"stretchy":73},[41,64553,55912],{},[49,64555,26833],{},[38,64557,64558,64560],{},[41,64559,38573],{},[41,64561,5597],{},[49,64563,7984],{},[41,64565,54067],{},[49,64567,84],{"stretchy":73},[49,64569,4775],{},[45,64571,47],{},[86,64573,64574],{"encoding":88},"y_i(w\\cdot x_i+b)\\ge 1",[18,64576,64578,64639,64694,64715],{"className":64577,"ariaHidden":51},[93],[18,64579,64581,64584,64624,64627,64630,64633,64636],{"className":64580},[97],[18,64582],{"className":64583,"style":242},[101],[18,64585,64587,64590],{"className":64586},[106],[18,64588,46343],{"className":64589,"style":3957},[106,110],[18,64591,64593],{"className":64592},[115],[18,64594,64596,64616],{"className":64595},[119,120],[18,64597,64599,64613],{"className":64598},[124],[18,64600,64602],{"className":64601,"style":5918},[128],[18,64603,64604,64607],{"style":30341},[18,64605],{"className":64606,"style":137},[136],[18,64608,64610],{"className":64609},[141,142,143,144],[18,64611,5597],{"className":64612},[106,110,144],[18,64614,152],{"className":64615},[151],[18,64617,64619],{"className":64618},[124],[18,64620,64622],{"className":64621,"style":159},[128],[18,64623],{},[18,64625,74],{"className":64626},[250],[18,64628,55912],{"className":64629,"style":55988},[106,110],[18,64631],{"className":64632,"style":2068},[169],[18,64634,26833],{"className":64635},[2072],[18,64637],{"className":64638,"style":2068},[169],[18,64640,64642,64645,64685,64688,64691],{"className":64641},[97],[18,64643],{"className":64644,"style":56474},[101],[18,64646,64648,64651],{"className":64647},[106],[18,64649,38573],{"className":64650},[106,110],[18,64652,64654],{"className":64653},[115],[18,64655,64657,64677],{"className":64656},[119,120],[18,64658,64660,64674],{"className":64659},[124],[18,64661,64663],{"className":64662,"style":5918},[128],[18,64664,64665,64668],{"style":1155},[18,64666],{"className":64667,"style":137},[136],[18,64669,64671],{"className":64670},[141,142,143,144],[18,64672,5597],{"className":64673},[106,110,144],[18,64675,152],{"className":64676},[151],[18,64678,64680],{"className":64679},[124],[18,64681,64683],{"className":64682,"style":159},[128],[18,64684],{},[18,64686],{"className":64687,"style":2068},[169],[18,64689,7984],{"className":64690},[2072],[18,64692],{"className":64693,"style":2068},[169],[18,64695,64697,64700,64703,64706,64709,64712],{"className":64696},[97],[18,64698],{"className":64699,"style":242},[101],[18,64701,54067],{"className":64702},[106,110],[18,64704,84],{"className":64705},[267],[18,64707],{"className":64708,"style":228},[169],[18,64710,4775],{"className":64711},[232],[18,64713],{"className":64714,"style":228},[169],[18,64716,64718,64721],{"className":64717},[97],[18,64719],{"className":64720,"style":505},[101],[18,64722,47],{"className":64723},[106]," 对所有 ",[18,64726,64728,64741],{"className":64727,"translate":22},[21],[18,64729,64731],{"className":64730},[26],[28,64732,64733],{"xmlns":30},[32,64734,64735,64739],{},[35,64736,64737],{},[41,64738,5597],{},[86,64740,5597],{"encoding":88},[18,64742,64744],{"className":64743,"ariaHidden":51},[93],[18,64745,64747,64750],{"className":64746},[97],[18,64748],{"className":64749,"style":61436},[101],[18,64751,5597],{"className":64752},[106,110]," 成立。这是一个线性规划。若 ",[18,64755,64757,64775],{"className":64756,"translate":22},[21],[18,64758,64760],{"className":64759},[26],[28,64761,64762],{"xmlns":30},[32,64763,64764,64772],{},[35,64765,64766],{},[1556,64767,64768,64770],{},[45,64769,2266],{},[41,64771,39955],{},[86,64773,64774],{"encoding":88},"2^k",[18,64776,64778],{"className":64777,"ariaHidden":51},[93],[18,64779,64781,64784],{"className":64780},[97],[18,64782],{"className":64783,"style":35177},[101],[18,64785,64787,64790],{"className":64786},[106],[18,64788,2266],{"className":64789},[106],[18,64791,64793],{"className":64792},[115],[18,64794,64796],{"className":64795},[119],[18,64797,64799],{"className":64798},[124],[18,64800,64802],{"className":64801,"style":35177},[128],[18,64803,64804,64807],{"style":1962},[18,64805],{"className":64806,"style":137},[136],[18,64808,64810],{"className":64809},[141,142,143,144],[18,64811,39955],{"className":64812,"style":40031},[106,110,144]," 个标注全部可行，",[18,64815,64817,64830],{"className":64816,"translate":22},[21],[18,64818,64820],{"className":64819},[26],[28,64821,64822],{"xmlns":30},[32,64823,64824,64828],{},[35,64825,64826],{},[41,64827,51012],{},[86,64829,51012],{"encoding":88},[18,64831,64833],{"className":64832,"ariaHidden":51},[93],[18,64834,64836,64839],{"className":64835},[97],[18,64837],{"className":64838,"style":600},[101],[18,64840,51012],{"className":64841,"style":5073},[106,110]," 就被打散；对所有 ",[18,64844,64846,64859],{"className":64845,"translate":22},[21],[18,64847,64849],{"className":64848},[26],[28,64850,64851],{"xmlns":30},[32,64852,64853,64857],{},[35,64854,64855],{},[41,64856,51012],{},[86,64858,51012],{"encoding":88},[18,64860,64862],{"className":64861,"ariaHidden":51},[93],[18,64863,64865,64868],{"className":64864},[97],[18,64866],{"className":64867,"style":600},[101],[18,64869,51012],{"className":64870,"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。",[1311,64873,64875],{"id":64874},"_23-三个误区","2.3 三个误区",[11,64877,64878,64881],{},[717,64879,64880],{},"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。",[11,64883,64884,64887,64888,64891],{},[717,64885,64886],{},"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",[717,64889,64890],{},"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。",[11,64893,64894,64897,64898,64926,64927,64955],{},[717,64895,64896],{},"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 ",[18,64899,64901,64914],{"className":64900,"translate":22},[21],[18,64902,64904],{"className":64903},[26],[28,64905,64906],{"xmlns":30},[32,64907,64908,64912],{},[35,64909,64910],{},[45,64911,47],{},[86,64913,47],{"encoding":88},[18,64915,64917],{"className":64916,"ariaHidden":51},[93],[18,64918,64920,64923],{"className":64919},[97],[18,64921],{"className":64922,"style":505},[101],[18,64924,47],{"className":64925},[106]," 的类，也可以对应 VC 维为 ",[18,64928,64930,64943],{"className":64929,"translate":22},[21],[18,64931,64933],{"className":64932},[26],[28,64934,64935],{"xmlns":30},[32,64936,64937,64941],{},[35,64938,64939],{},[41,64940,2028],{"mathvariant":316},[86,64942,64095],{"encoding":88},[18,64944,64946],{"className":64945,"ariaHidden":51},[93],[18,64947,64949,64952],{"className":64948},[97],[18,64950],{"className":64951,"style":1083},[101],[18,64953,2028],{"className":64954},[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。",[11,64957,64958,64959,64962],{},"还有一点值得提前说明：上面定义的是",[717,64960,64961],{},"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。",[638,64964,64966],{"id":64965},"三增长函数与-sauershelah-引理","三、增长函数与 Sauer–Shelah 引理",[1311,64968,64970],{"id":64969},"_31-增长函数","3.1 增长函数",[11,64972,64973],{},"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：",[18,64975,64977],{"className":64976,"translate":22},[875],[18,64978,64980,65051],{"className":64979,"translate":22},[21],[18,64981,64983],{"className":64982},[26],[28,64984,64985],{"xmlns":30,"display":884},[32,64986,64987,65048],{},[35,64988,64989,64996,64998,65000,65002,65004,65034,65036,65038,65044,65046],{},[38,64990,64991,64994],{},[41,64992,64993],{"mathvariant":316},"Π",[41,64995,18304],{"mathvariant":1462},[49,64997,74],{"stretchy":73},[41,64999,28160],{},[49,65001,84],{"stretchy":73},[49,65003,433],{},[890,65005,65006,65012],{},[35,65007,65008,65010],{},[41,65009,5590],{},[49,65011,899],{},[35,65013,65014,65016,65018,65020,65022,65024,65026,65028,65030,65032],{},[41,65015,51012],{},[49,65017,4548],{},[41,65019,43],{"mathvariant":1462},[49,65021,52],{"separator":51},[926,65023,928],{},[41,65025,11222],{"mathvariant":316},[41,65027,51012],{},[41,65029,11222],{"mathvariant":316},[49,65031,433],{},[41,65033,28160],{},[49,65035,11222],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[41,65037,18304],{"mathvariant":1462},[38,65039,65040,65042],{},[41,65041,11222],{"mathvariant":316},[41,65043,51012],{},[49,65045,11222],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[41,65047,968],{"mathvariant":316},[86,65049,65050],{"encoding":88},"\\Pi_{\\mathcal H}(m)=\\max_{C\\subseteq\\mathcal X,\\ |C|=m}\\bigl|\\mathcal H|_C\\bigr|.",[18,65052,65054,65121],{"className":65053,"ariaHidden":51},[93],[18,65055,65057,65060,65103,65106,65109,65112,65115,65118],{"className":65056},[97],[18,65058],{"className":65059,"style":242},[101],[18,65061,65063,65066],{"className":65062},[106],[18,65064,64993],{"className":65065},[106],[18,65067,65069],{"className":65068},[115],[18,65070,65072,65095],{"className":65071},[119,120],[18,65073,65075,65092],{"className":65074},[124],[18,65076,65078],{"className":65077,"style":1152},[128],[18,65079,65080,65083],{"style":1155},[18,65081],{"className":65082,"style":137},[136],[18,65084,65086],{"className":65085},[141,142,143,144],[18,65087,65089],{"className":65088},[106,144],[18,65090,18304],{"className":65091,"style":46321},[106,1478,144],[18,65093,152],{"className":65094},[151],[18,65096,65098],{"className":65097},[124],[18,65099,65101],{"className":65100,"style":159},[128],[18,65102],{},[18,65104,74],{"className":65105},[250],[18,65107,28160],{"className":65108},[106,110],[18,65110,84],{"className":65111},[267],[18,65113],{"className":65114,"style":228},[169],[18,65116,433],{"className":65117},[232],[18,65119],{"className":65120,"style":228},[169],[18,65122,65124,65128,65205,65242,65245,65285,65322],{"className":65123},[97],[18,65125],{"className":65126,"style":65127},[101],"height:1.816em;vertical-align:-0.966em;",[18,65129,65131],{"className":65130},[985,986],[18,65132,65134,65197],{"className":65133},[119,120],[18,65135,65137,65194],{"className":65136},[124],[18,65138,65140,65184],{"className":65139,"style":1083},[128],[18,65141,65142,65145],{"style":33537},[18,65143],{"className":65144,"style":697},[136],[18,65146,65148],{"className":65147},[141,142,143,144],[18,65149,65151,65154,65157,65160,65163,65169,65172,65175,65178,65181],{"className":65150},[106,144],[18,65152,51012],{"className":65153,"style":5073},[106,110,144],[18,65155,4548],{"className":65156},[232,144],[18,65158,43],{"className":65159,"style":1479},[106,1478,144],[18,65161,52],{"className":65162},[165,144],[18,65164,65166],{"className":65165},[169,144],[18,65167,928],{"className":65168},[144],[18,65170,11222],{"className":65171},[106,144],[18,65173,51012],{"className":65174,"style":5073},[106,110,144],[18,65176,11222],{"className":65177},[106,144],[18,65179,433],{"className":65180},[232,144],[18,65182,28160],{"className":65183},[106,110,144],[18,65185,65186,65189],{"style":693},[18,65187],{"className":65188,"style":697},[136],[18,65190,65191],{},[18,65192,5590],{"className":65193},[985],[18,65195,152],{"className":65196},[151],[18,65198,65200],{"className":65199},[124],[18,65201,65203],{"className":65202,"style":33615},[128],[18,65204],{},[18,65206,65208],{"className":65207},[250],[18,65209,65211],{"className":65210},[1230,6105],[18,65212,65214,65234],{"className":65213},[119,120],[18,65215,65217,65231],{"className":65216},[124],[18,65218,65220],{"className":65219,"style":6115},[128],[18,65221,65222,65225],{"style":6118},[18,65223],{"className":65224,"style":6122},[136],[18,65226,65227],{"style":49860},[6127,65228,65229],{"xmlns":6129,"width":49863,"height":944,"viewBox":49864},[6133,65230],{"d":49867},[18,65232,152],{"className":65233},[151],[18,65235,65237],{"className":65236},[124],[18,65238,65240],{"className":65239,"style":6145},[128],[18,65241],{},[18,65243,18304],{"className":65244,"style":46321},[106,1478],[18,65246,65248,65251],{"className":65247},[106],[18,65249,11222],{"className":65250},[106],[18,65252,65254],{"className":65253},[115],[18,65255,65257,65277],{"className":65256},[119,120],[18,65258,65260,65274],{"className":65259},[124],[18,65261,65263],{"className":65262,"style":1152},[128],[18,65264,65265,65268],{"style":1155},[18,65266],{"className":65267,"style":137},[136],[18,65269,65271],{"className":65270},[141,142,143,144],[18,65272,51012],{"className":65273,"style":5073},[106,110,144],[18,65275,152],{"className":65276},[151],[18,65278,65280],{"className":65279},[124],[18,65281,65283],{"className":65282,"style":159},[128],[18,65284],{},[18,65286,65288],{"className":65287},[267],[18,65289,65291],{"className":65290},[1230,6105],[18,65292,65294,65314],{"className":65293},[119,120],[18,65295,65297,65311],{"className":65296},[124],[18,65298,65300],{"className":65299,"style":6115},[128],[18,65301,65302,65305],{"style":6118},[18,65303],{"className":65304,"style":6122},[136],[18,65306,65307],{"style":49860},[6127,65308,65309],{"xmlns":6129,"width":49863,"height":944,"viewBox":49864},[6133,65310],{"d":49867},[18,65312,152],{"className":65313},[151],[18,65315,65317],{"className":65316},[124],[18,65318,65320],{"className":65319,"style":6145},[128],[18,65321],{},[18,65323,968],{"className":65324},[106],[11,65326,65327,65328,46820,65356,65359,65360,5478,65500,65573],{},"它称为 ",[18,65329,65331,65344],{"className":65330,"translate":22},[21],[18,65332,65334],{"className":65333},[26],[28,65335,65336],{"xmlns":30},[32,65337,65338,65342],{},[35,65339,65340],{},[41,65341,18304],{"mathvariant":1462},[86,65343,46308],{"encoding":88},[18,65345,65347],{"className":65346,"ariaHidden":51},[93],[18,65348,65350,65353],{"className":65349},[97],[18,65351],{"className":65352,"style":600},[101],[18,65354,18304],{"className":65355,"style":46321},[106,1478],[717,65357,65358],{},"增长函数","（growth function）。显然 ",[18,65361,65363,65395],{"className":65362,"translate":22},[21],[18,65364,65366],{"className":65365},[26],[28,65367,65368],{"xmlns":30},[32,65369,65370,65392],{},[35,65371,65372,65378,65380,65382,65384,65386],{},[38,65373,65374,65376],{},[41,65375,64993],{"mathvariant":316},[41,65377,18304],{"mathvariant":1462},[49,65379,74],{"stretchy":73},[41,65381,28160],{},[49,65383,84],{"stretchy":73},[49,65385,4334],{},[1556,65387,65388,65390],{},[45,65389,2266],{},[41,65391,28160],{},[86,65393,65394],{"encoding":88},"\\Pi_{\\mathcal H}(m)\\le 2^m",[18,65396,65398,65465],{"className":65397,"ariaHidden":51},[93],[18,65399,65401,65404,65447,65450,65453,65456,65459,65462],{"className":65400},[97],[18,65402],{"className":65403,"style":242},[101],[18,65405,65407,65410],{"className":65406},[106],[18,65408,64993],{"className":65409},[106],[18,65411,65413],{"className":65412},[115],[18,65414,65416,65439],{"className":65415},[119,120],[18,65417,65419,65436],{"className":65418},[124],[18,65420,65422],{"className":65421,"style":1152},[128],[18,65423,65424,65427],{"style":1155},[18,65425],{"className":65426,"style":137},[136],[18,65428,65430],{"className":65429},[141,142,143,144],[18,65431,65433],{"className":65432},[106,144],[18,65434,18304],{"className":65435,"style":46321},[106,1478,144],[18,65437,152],{"className":65438},[151],[18,65440,65442],{"className":65441},[124],[18,65443,65445],{"className":65444,"style":159},[128],[18,65446],{},[18,65448,74],{"className":65449},[250],[18,65451,28160],{"className":65452},[106,110],[18,65454,84],{"className":65455},[267],[18,65457],{"className":65458,"style":228},[169],[18,65460,4334],{"className":65461},[232],[18,65463],{"className":65464,"style":228},[169],[18,65466,65468,65471],{"className":65467},[97],[18,65469],{"className":65470,"style":1959},[101],[18,65472,65474,65477],{"className":65473},[106],[18,65475,2266],{"className":65476},[106],[18,65478,65480],{"className":65479},[115],[18,65481,65483],{"className":65482},[119],[18,65484,65486],{"className":65485},[124],[18,65487,65489],{"className":65488,"style":1959},[128],[18,65490,65491,65494],{"style":1962},[18,65492],{"className":65493,"style":137},[136],[18,65495,65497],{"className":65496},[141,142,143,144],[18,65498,28160],{"className":65499},[106,110,144],[18,65501,65503,65521],{"className":65502,"translate":22},[21],[18,65504,65506],{"className":65505},[26],[28,65507,65508],{"xmlns":30},[32,65509,65510,65518],{},[35,65511,65512],{},[38,65513,65514,65516],{},[41,65515,64993],{"mathvariant":316},[41,65517,18304],{"mathvariant":1462},[86,65519,65520],{"encoding":88},"\\Pi_{\\mathcal H}",[18,65522,65524],{"className":65523,"ariaHidden":51},[93],[18,65525,65527,65530],{"className":65526},[97],[18,65528],{"className":65529,"style":4563},[101],[18,65531,65533,65536],{"className":65532},[106],[18,65534,64993],{"className":65535},[106],[18,65537,65539],{"className":65538},[115],[18,65540,65542,65565],{"className":65541},[119,120],[18,65543,65545,65562],{"className":65544},[124],[18,65546,65548],{"className":65547,"style":1152},[128],[18,65549,65550,65553],{"style":1155},[18,65551],{"className":65552,"style":137},[136],[18,65554,65556],{"className":65555},[141,142,143,144],[18,65557,65559],{"className":65558},[106,144],[18,65560,18304],{"className":65561,"style":46321},[106,1478,144],[18,65563,152],{"className":65564},[151],[18,65566,65568],{"className":65567},[124],[18,65569,65571],{"className":65570,"style":159},[128],[18,65572],{}," 单调不减。",[11,65575,65576,65577,65626,65627,65767,65768,65796,65797,65825,65826,65854,65855,65883,65884,65912,65913,66009,66010,66061,66062,1990],{},"由定义，",[18,65578,65580,65602],{"className":65579,"translate":22},[21],[18,65581,65583],{"className":65582},[26],[28,65584,65585],{"xmlns":30},[32,65586,65587,65599],{},[35,65588,65589,65591,65593,65595,65597],{},[41,65590,52202],{"mathvariant":316},[49,65592,899],{},[49,65594,74],{"stretchy":73},[41,65596,18304],{"mathvariant":1462},[49,65598,84],{"stretchy":73},[86,65600,65601],{"encoding":88},"\\operatorname{VCdim}(\\mathcal H)",[18,65603,65605],{"className":65604,"ariaHidden":51},[93],[18,65606,65608,65611,65617,65620,65623],{"className":65607},[97],[18,65609],{"className":65610,"style":242},[101],[18,65612,65614],{"className":65613},[985],[18,65615,52202],{"className":65616},[106,6351],[18,65618,74],{"className":65619},[250],[18,65621,18304],{"className":65622,"style":46321},[106,1478],[18,65624,84],{"className":65625},[267]," 就是使 ",[18,65628,65630,65662],{"className":65629,"translate":22},[21],[18,65631,65633],{"className":65632},[26],[28,65634,65635],{"xmlns":30},[32,65636,65637,65659],{},[35,65638,65639,65645,65647,65649,65651,65653],{},[38,65640,65641,65643],{},[41,65642,64993],{"mathvariant":316},[41,65644,18304],{"mathvariant":1462},[49,65646,74],{"stretchy":73},[41,65648,28160],{},[49,65650,84],{"stretchy":73},[49,65652,433],{},[1556,65654,65655,65657],{},[45,65656,2266],{},[41,65658,28160],{},[86,65660,65661],{"encoding":88},"\\Pi_{\\mathcal H}(m)=2^m",[18,65663,65665,65732],{"className":65664,"ariaHidden":51},[93],[18,65666,65668,65671,65714,65717,65720,65723,65726,65729],{"className":65667},[97],[18,65669],{"className":65670,"style":242},[101],[18,65672,65674,65677],{"className":65673},[106],[18,65675,64993],{"className":65676},[106],[18,65678,65680],{"className":65679},[115],[18,65681,65683,65706],{"className":65682},[119,120],[18,65684,65686,65703],{"className":65685},[124],[18,65687,65689],{"className":65688,"style":1152},[128],[18,65690,65691,65694],{"style":1155},[18,65692],{"className":65693,"style":137},[136],[18,65695,65697],{"className":65696},[141,142,143,144],[18,65698,65700],{"className":65699},[106,144],[18,65701,18304],{"className":65702,"style":46321},[106,1478,144],[18,65704,152],{"className":65705},[151],[18,65707,65709],{"className":65708},[124],[18,65710,65712],{"className":65711,"style":159},[128],[18,65713],{},[18,65715,74],{"className":65716},[250],[18,65718,28160],{"className":65719},[106,110],[18,65721,84],{"className":65722},[267],[18,65724],{"className":65725,"style":228},[169],[18,65727,433],{"className":65728},[232],[18,65730],{"className":65731,"style":228},[169],[18,65733,65735,65738],{"className":65734},[97],[18,65736],{"className":65737,"style":1959},[101],[18,65739,65741,65744],{"className":65740},[106],[18,65742,2266],{"className":65743},[106],[18,65745,65747],{"className":65746},[115],[18,65748,65750],{"className":65749},[119],[18,65751,65753],{"className":65752},[124],[18,65754,65756],{"className":65755,"style":1959},[128],[18,65757,65758,65761],{"style":1962},[18,65759],{"className":65760,"style":137},[136],[18,65762,65764],{"className":65763},[141,142,143,144],[18,65765,28160],{"className":65766},[106,110,144]," 成立的最大的 ",[18,65769,65771,65784],{"className":65770,"translate":22},[21],[18,65772,65774],{"className":65773},[26],[28,65775,65776],{"xmlns":30},[32,65777,65778,65782],{},[35,65779,65780],{},[41,65781,28160],{},[86,65783,28160],{"encoding":88},[18,65785,65787],{"className":65786,"ariaHidden":51},[93],[18,65788,65790,65793],{"className":65789},[97],[18,65791],{"className":65792,"style":1083},[101],[18,65794,28160],{"className":65795},[106,110],"（若对所有 ",[18,65798,65800,65813],{"className":65799,"translate":22},[21],[18,65801,65803],{"className":65802},[26],[28,65804,65805],{"xmlns":30},[32,65806,65807,65811],{},[35,65808,65809],{},[41,65810,28160],{},[86,65812,28160],{"encoding":88},[18,65814,65816],{"className":65815,"ariaHidden":51},[93],[18,65817,65819,65822],{"className":65818},[97],[18,65820],{"className":65821,"style":1083},[101],[18,65823,28160],{"className":65824},[106,110]," 都成立，则为 ",[18,65827,65829,65842],{"className":65828,"translate":22},[21],[18,65830,65832],{"className":65831},[26],[28,65833,65834],{"xmlns":30},[32,65835,65836,65840],{},[35,65837,65838],{},[41,65839,2028],{"mathvariant":316},[86,65841,64095],{"encoding":88},[18,65843,65845],{"className":65844,"ariaHidden":51},[93],[18,65846,65848,65851],{"className":65847},[97],[18,65849],{"className":65850,"style":1083},[101],[18,65852,2028],{"className":65853},[106],"）。注意这个性质是「向下封闭」的：若 ",[18,65856,65858,65871],{"className":65857,"translate":22},[21],[18,65859,65861],{"className":65860},[26],[28,65862,65863],{"xmlns":30},[32,65864,65865,65869],{},[35,65866,65867],{},[41,65868,51012],{},[86,65870,51012],{"encoding":88},[18,65872,65874],{"className":65873,"ariaHidden":51},[93],[18,65875,65877,65880],{"className":65876},[97],[18,65878],{"className":65879,"style":600},[101],[18,65881,51012],{"className":65882,"style":5073},[106,110]," 被打散，",[18,65885,65887,65900],{"className":65886,"translate":22},[21],[18,65888,65890],{"className":65889},[26],[28,65891,65892],{"xmlns":30},[32,65893,65894,65898],{},[35,65895,65896],{},[41,65897,51012],{},[86,65899,51012],{"encoding":88},[18,65901,65903],{"className":65902,"ariaHidden":51},[93],[18,65904,65906,65909],{"className":65905},[97],[18,65907],{"className":65908,"style":600},[101],[18,65910,51012],{"className":65911,"style":5073},[106,110]," 的任何子集也被打散，所以一旦 ",[18,65914,65916,65944],{"className":65915,"translate":22},[21],[18,65917,65919],{"className":65918},[26],[28,65920,65921],{"xmlns":30},[32,65922,65923,65941],{},[35,65924,65925,65927,65929,65931,65933,65935],{},[41,65926,64993],{"mathvariant":316},[49,65928,74],{"stretchy":73},[41,65930,28160],{},[49,65932,84],{"stretchy":73},[49,65934,433],{},[1556,65936,65937,65939],{},[45,65938,2266],{},[41,65940,28160],{},[86,65942,65943],{"encoding":88},"\\Pi(m)=2^m",[18,65945,65947,65974],{"className":65946,"ariaHidden":51},[93],[18,65948,65950,65953,65956,65959,65962,65965,65968,65971],{"className":65949},[97],[18,65951],{"className":65952,"style":242},[101],[18,65954,64993],{"className":65955},[106],[18,65957,74],{"className":65958},[250],[18,65960,28160],{"className":65961},[106,110],[18,65963,84],{"className":65964},[267],[18,65966],{"className":65967,"style":228},[169],[18,65969,433],{"className":65970},[232],[18,65972],{"className":65973,"style":228},[169],[18,65975,65977,65980],{"className":65976},[97],[18,65978],{"className":65979,"style":1959},[101],[18,65981,65983,65986],{"className":65982},[106],[18,65984,2266],{"className":65985},[106],[18,65987,65989],{"className":65988},[115],[18,65990,65992],{"className":65991},[119],[18,65993,65995],{"className":65994},[124],[18,65996,65998],{"className":65997,"style":1959},[128],[18,65999,66000,66003],{"style":1962},[18,66001],{"className":66002,"style":137},[136],[18,66004,66006],{"className":66005},[141,142,143,144],[18,66007,28160],{"className":66008},[106,110,144],"，对所有 ",[18,66011,66013,66031],{"className":66012,"translate":22},[21],[18,66014,66016],{"className":66015},[26],[28,66017,66018],{"xmlns":30},[32,66019,66020,66028],{},[35,66021,66022,66024,66026],{},[41,66023,39955],{},[49,66025,4334],{},[41,66027,28160],{},[86,66029,66030],{"encoding":88},"k\\le m",[18,66032,66034,66052],{"className":66033,"ariaHidden":51},[93],[18,66035,66037,66040,66043,66046,66049],{"className":66036},[97],[18,66038],{"className":66039,"style":52343},[101],[18,66041,39955],{"className":66042,"style":40031},[106,110],[18,66044],{"className":66045,"style":228},[169],[18,66047,4334],{"className":66048},[232],[18,66050],{"className":66051,"style":228},[169],[18,66053,66055,66058],{"className":66054},[97],[18,66056],{"className":66057,"style":1083},[101],[18,66059,28160],{"className":66060},[106,110]," 都有 ",[18,66063,66065,66093],{"className":66064,"translate":22},[21],[18,66066,66068],{"className":66067},[26],[28,66069,66070],{"xmlns":30},[32,66071,66072,66090],{},[35,66073,66074,66076,66078,66080,66082,66084],{},[41,66075,64993],{"mathvariant":316},[49,66077,74],{"stretchy":73},[41,66079,39955],{},[49,66081,84],{"stretchy":73},[49,66083,433],{},[1556,66085,66086,66088],{},[45,66087,2266],{},[41,66089,39955],{},[86,66091,66092],{"encoding":88},"\\Pi(k)=2^k",[18,66094,66096,66123],{"className":66095,"ariaHidden":51},[93],[18,66097,66099,66102,66105,66108,66111,66114,66117,66120],{"className":66098},[97],[18,66100],{"className":66101,"style":242},[101],[18,66103,64993],{"className":66104},[106],[18,66106,74],{"className":66107},[250],[18,66109,39955],{"className":66110,"style":40031},[106,110],[18,66112,84],{"className":66113},[267],[18,66115],{"className":66116,"style":228},[169],[18,66118,433],{"className":66119},[232],[18,66121],{"className":66122,"style":228},[169],[18,66124,66126,66129],{"className":66125},[97],[18,66127],{"className":66128,"style":35177},[101],[18,66130,66132,66135],{"className":66131},[106],[18,66133,2266],{"className":66134},[106],[18,66136,66138],{"className":66137},[115],[18,66139,66141],{"className":66140},[119],[18,66142,66144],{"className":66143},[124],[18,66145,66147],{"className":66146,"style":35177},[128],[18,66148,66149,66152],{"style":1962},[18,66150],{"className":66151,"style":137},[136],[18,66153,66155],{"className":66154},[141,142,143,144],[18,66156,39955],{"className":66157,"style":40031},[106,110,144],[1311,66159,66161],{"id":66160},"_32-sauershelah-引理","3.2 Sauer–Shelah 引理",[5159,66163,66165,66288],{"title":66164},"定理 3.1（Sauer–Shelah）",[11,66166,21783,66167,21936,66260,4741],{},[18,66168,66170,66200],{"className":66169,"translate":22},[21],[18,66171,66173],{"className":66172},[26],[28,66174,66175],{"xmlns":30},[32,66176,66177,66197],{},[35,66178,66179,66181,66183,66185,66187,66189,66191,66193,66195],{},[41,66180,52202],{"mathvariant":316},[49,66182,899],{},[49,66184,74],{"stretchy":73},[41,66186,18304],{"mathvariant":1462},[49,66188,84],{"stretchy":73},[49,66190,433],{},[41,66192,4031],{},[49,66194,6681],{},[41,66196,2028],{"mathvariant":316},[86,66198,66199],{"encoding":88},"\\operatorname{VCdim}(\\mathcal H)=d\u003C\\infty",[18,66201,66203,66233,66251],{"className":66202,"ariaHidden":51},[93],[18,66204,66206,66209,66215,66218,66221,66224,66227,66230],{"className":66205},[97],[18,66207],{"className":66208,"style":242},[101],[18,66210,66212],{"className":66211},[985],[18,66213,52202],{"className":66214},[106,6351],[18,66216,74],{"className":66217},[250],[18,66219,18304],{"className":66220,"style":46321},[106,1478],[18,66222,84],{"className":66223},[267],[18,66225],{"className":66226,"style":228},[169],[18,66228,433],{"className":66229},[232],[18,66231],{"className":66232,"style":228},[169],[18,66234,66236,66239,66242,66245,66248],{"className":66235},[97],[18,66237],{"className":66238,"style":1422},[101],[18,66240,4031],{"className":66241},[106,110],[18,66243],{"className":66244,"style":228},[169],[18,66246,6681],{"className":66247},[232],[18,66249],{"className":66250,"style":228},[169],[18,66252,66254,66257],{"className":66253},[97],[18,66255],{"className":66256,"style":1083},[101],[18,66258,2028],{"className":66259},[106],[18,66261,66263,66276],{"className":66262,"translate":22},[21],[18,66264,66266],{"className":66265},[26],[28,66267,66268],{"xmlns":30},[32,66269,66270,66274],{},[35,66271,66272],{},[41,66273,28160],{},[86,66275,28160],{"encoding":88},[18,66277,66279],{"className":66278,"ariaHidden":51},[93],[18,66280,66282,66285],{"className":66281},[97],[18,66283],{"className":66284,"style":1083},[101],[18,66286,28160],{"className":66287},[106,110],[18,66289,66291],{"className":66290,"translate":22},[875],[18,66292,66294,66349],{"className":66293,"translate":22},[21],[18,66295,66297],{"className":66296},[26],[28,66298,66299],{"xmlns":30,"display":884},[32,66300,66301,66346],{},[35,66302,66303,66309,66311,66313,66315,66317,66331,66344],{},[38,66304,66305,66307],{},[41,66306,64993],{"mathvariant":316},[41,66308,18304],{"mathvariant":1462},[49,66310,74],{"stretchy":73},[41,66312,28160],{},[49,66314,84],{"stretchy":73},[49,66316,4334],{},[20358,66318,66319,66321,66329],{},[49,66320,20362],{},[35,66322,66323,66325,66327],{},[41,66324,5597],{},[49,66326,433],{},[45,66328,430],{},[41,66330,4031],{},[35,66332,66333,66335,66342],{},[49,66334,74],{"fence":51},[5651,66336,66338,66340],{"linethickness":66337},"0px",[41,66339,28160],{},[41,66341,5597],{},[49,66343,84],{"fence":51},[41,66345,968],{"mathvariant":316},[86,66347,66348],{"encoding":88},"\\Pi_{\\mathcal H}(m)\\le\\sum_{i=0}^{d}\\binom{m}{i}.",[18,66350,66352,66419],{"className":66351,"ariaHidden":51},[93],[18,66353,66355,66358,66401,66404,66407,66410,66413,66416],{"className":66354},[97],[18,66356],{"className":66357,"style":242},[101],[18,66359,66361,66364],{"className":66360},[106],[18,66362,64993],{"className":66363},[106],[18,66365,66367],{"className":66366},[115],[18,66368,66370,66393],{"className":66369},[119,120],[18,66371,66373,66390],{"className":66372},[124],[18,66374,66376],{"className":66375,"style":1152},[128],[18,66377,66378,66381],{"style":1155},[18,66379],{"className":66380,"style":137},[136],[18,66382,66384],{"className":66383},[141,142,143,144],[18,66385,66387],{"className":66386},[106,144],[18,66388,18304],{"className":66389,"style":46321},[106,1478,144],[18,66391,152],{"className":66392},[151],[18,66394,66396],{"className":66395},[124],[18,66397,66399],{"className":66398,"style":159},[128],[18,66400],{},[18,66402,74],{"className":66403},[250],[18,66405,28160],{"className":66406},[106,110],[18,66408,84],{"className":66409},[267],[18,66411],{"className":66412,"style":228},[169],[18,66414,4334],{"className":66415},[232],[18,66417],{"className":66418,"style":228},[169],[18,66420,66422,66426,66494,66497,66557],{"className":66421},[97],[18,66423],{"className":66424,"style":66425},[101],"height:3.1138em;vertical-align:-1.2777em;",[18,66427,66429],{"className":66428},[985,986],[18,66430,66432,66486],{"className":66431},[119,120],[18,66433,66435,66483],{"className":66434},[124],[18,66436,66439,66459,66469],{"className":66437,"style":66438},[128],"height:1.8361em;",[18,66440,66441,66444],{"style":20548},[18,66442],{"className":66443,"style":20552},[136],[18,66445,66447],{"className":66446},[141,142,143,144],[18,66448,66450,66453,66456],{"className":66449},[106,144],[18,66451,5597],{"className":66452},[106,110,144],[18,66454,433],{"className":66455},[232,144],[18,66457,430],{"className":66458},[106,144],[18,66460,66461,66464],{"style":20570},[18,66462],{"className":66463,"style":20552},[136],[18,66465,66466],{},[18,66467,20362],{"className":66468},[985,4198,4199],[18,66470,66471,66474],{"style":20581},[18,66472],{"className":66473,"style":20552},[136],[18,66475,66477],{"className":66476},[141,142,143,144],[18,66478,66480],{"className":66479},[106,144],[18,66481,4031],{"className":66482},[106,110,144],[18,66484,152],{"className":66485},[151],[18,66487,66489],{"className":66488},[124],[18,66490,66492],{"className":66491,"style":20600},[128],[18,66493],{},[18,66495],{"className":66496,"style":170},[169],[18,66498,66500,66506,66551],{"className":66499},[106],[18,66501,66503],{"className":66502,"style":13793},[250,13792],[18,66504,74],{"className":66505},[1230,143],[18,66507,66509],{"className":66508},[5651],[18,66510,66512,66543],{"className":66511},[119,120],[18,66513,66515,66540],{"className":66514},[124],[18,66516,66518,66529],{"className":66517,"style":13849},[128],[18,66519,66520,66523],{"style":6025},[18,66521],{"className":66522,"style":697},[136],[18,66524,66526],{"className":66525},[106],[18,66527,5597],{"className":66528},[106,110],[18,66530,66531,66534],{"style":6048},[18,66532],{"className":66533,"style":697},[136],[18,66535,66537],{"className":66536},[106],[18,66538,28160],{"className":66539},[106,110],[18,66541,152],{"className":66542},[151],[18,66544,66546],{"className":66545},[124],[18,66547,66549],{"className":66548,"style":6067},[128],[18,66550],{},[18,66552,66554],{"className":66553,"style":13793},[267,13792],[18,66555,84],{"className":66556},[1230,143],[18,66558,968],{"className":66559},[106],[11,66561,66562,66563,66614,66615,66787,66788,66946,66947,67022],{},"证明用归纳法，对 ",[18,66564,66566,66584],{"className":66565,"translate":22},[21],[18,66567,66569],{"className":66568},[26],[28,66570,66571],{"xmlns":30},[32,66572,66573,66581],{},[35,66574,66575,66577,66579],{},[41,66576,28160],{},[49,66578,7984],{},[41,66580,4031],{},[86,66582,66583],{"encoding":88},"m+d",[18,66585,66587,66605],{"className":66586,"ariaHidden":51},[93],[18,66588,66590,66593,66596,66599,66602],{"className":66589},[97],[18,66591],{"className":66592,"style":14401},[101],[18,66594,28160],{"className":66595},[106,110],[18,66597],{"className":66598,"style":2068},[169],[18,66600,7984],{"className":66601},[2072],[18,66603],{"className":66604,"style":2068},[169],[18,66606,66608,66611],{"className":66607},[97],[18,66609],{"className":66610,"style":295},[101],[18,66612,4031],{"className":66613},[106,110]," 归纳。取任意 ",[18,66616,66618,66656],{"className":66617,"translate":22},[21],[18,66619,66621],{"className":66620},[26],[28,66622,66623],{"xmlns":30},[32,66624,66625,66653],{},[35,66626,66627,66629,66631,66633,66639,66641,66643,66645,66651],{},[41,66628,51012],{},[49,66630,433],{},[49,66632,1331],{"stretchy":73},[38,66634,66635,66637],{},[41,66636,39058],{},[45,66638,47],{},[49,66640,52],{"separator":51},[49,66642,55],{},[49,66644,52],{"separator":51},[38,66646,66647,66649],{},[41,66648,39058],{},[41,66650,28160],{},[49,66652,1349],{"stretchy":73},[86,66654,66655],{"encoding":88},"C=\\{c_1,\\dots,c_m\\}",[18,66657,66659,66677],{"className":66658,"ariaHidden":51},[93],[18,66660,66662,66665,66668,66671,66674],{"className":66661},[97],[18,66663],{"className":66664,"style":600},[101],[18,66666,51012],{"className":66667,"style":5073},[106,110],[18,66669],{"className":66670,"style":228},[169],[18,66672,433],{"className":66673},[232],[18,66675],{"className":66676,"style":228},[169],[18,66678,66680,66683,66686,66726,66729,66732,66735,66738,66741,66744,66784],{"className":66679},[97],[18,66681],{"className":66682,"style":242},[101],[18,66684,1331],{"className":66685},[250],[18,66687,66689,66692],{"className":66688},[106],[18,66690,39058],{"className":66691},[106,110],[18,66693,66695],{"className":66694},[115],[18,66696,66698,66718],{"className":66697},[119,120],[18,66699,66701,66715],{"className":66700},[124],[18,66702,66704],{"className":66703,"style":129},[128],[18,66705,66706,66709],{"style":1155},[18,66707],{"className":66708,"style":137},[136],[18,66710,66712],{"className":66711},[141,142,143,144],[18,66713,47],{"className":66714},[106,144],[18,66716,152],{"className":66717},[151],[18,66719,66721],{"className":66720},[124],[18,66722,66724],{"className":66723,"style":159},[128],[18,66725],{},[18,66727,52],{"className":66728},[165],[18,66730],{"className":66731,"style":170},[169],[18,66733,55],{"className":66734},[174],[18,66736],{"className":66737,"style":170},[169],[18,66739,52],{"className":66740},[165],[18,66742],{"className":66743,"style":170},[169],[18,66745,66747,66750],{"className":66746},[106],[18,66748,39058],{"className":66749},[106,110],[18,66751,66753],{"className":66752},[115],[18,66754,66756,66776],{"className":66755},[119,120],[18,66757,66759,66773],{"className":66758},[124],[18,66760,66762],{"className":66761,"style":202},[128],[18,66763,66764,66767],{"style":1155},[18,66765],{"className":66766,"style":137},[136],[18,66768,66770],{"className":66769},[141,142,143,144],[18,66771,28160],{"className":66772},[106,110,144],[18,66774,152],{"className":66775},[151],[18,66777,66779],{"className":66778},[124],[18,66780,66782],{"className":66781,"style":159},[128],[18,66783],{},[18,66785,1349],{"className":66786},[267],"，记 ",[18,66789,66791,66826],{"className":66790,"translate":22},[21],[18,66792,66794],{"className":66793},[26],[28,66795,66796],{"xmlns":30},[32,66797,66798,66823],{},[35,66799,66800,66806,66808,66810,66813,66815,66821],{},[1556,66801,66802,66804],{},[41,66803,51012],{},[49,66805,32620],{"mathvariant":316,"lspace":32619,"rspace":32619},[49,66807,433],{},[41,66809,51012],{},[49,66811,66812],{},"∖",[49,66814,1331],{"stretchy":73},[38,66816,66817,66819],{},[41,66818,39058],{},[41,66820,28160],{},[49,66822,1349],{"stretchy":73},[86,66824,66825],{"encoding":88},"C'=C\\setminus\\{c_m\\}",[18,66827,66829,66876,66894],{"className":66828,"ariaHidden":51},[93],[18,66830,66832,66835,66867,66870,66873],{"className":66831},[97],[18,66833],{"className":66834,"style":32651},[101],[18,66836,66838,66841],{"className":66837},[106],[18,66839,51012],{"className":66840,"style":5073},[106,110],[18,66842,66844],{"className":66843},[115],[18,66845,66847],{"className":66846},[119],[18,66848,66850],{"className":66849},[124],[18,66851,66853],{"className":66852,"style":32651},[128],[18,66854,66855,66858],{"style":1962},[18,66856],{"className":66857,"style":137},[136],[18,66859,66861],{"className":66860},[141,142,143,144],[18,66862,66864],{"className":66863},[106,144],[18,66865,32620],{"className":66866},[106,144],[18,66868],{"className":66869,"style":228},[169],[18,66871,433],{"className":66872},[232],[18,66874],{"className":66875,"style":228},[169],[18,66877,66879,66882,66885,66888,66891],{"className":66878},[97],[18,66880],{"className":66881,"style":242},[101],[18,66883,51012],{"className":66884,"style":5073},[106,110],[18,66886],{"className":66887,"style":2068},[169],[18,66889,66812],{"className":66890},[2072],[18,66892],{"className":66893,"style":2068},[169],[18,66895,66897,66900,66903,66943],{"className":66896},[97],[18,66898],{"className":66899,"style":242},[101],[18,66901,1331],{"className":66902},[250],[18,66904,66906,66909],{"className":66905},[106],[18,66907,39058],{"className":66908},[106,110],[18,66910,66912],{"className":66911},[115],[18,66913,66915,66935],{"className":66914},[119,120],[18,66916,66918,66932],{"className":66917},[124],[18,66919,66921],{"className":66920,"style":202},[128],[18,66922,66923,66926],{"style":1155},[18,66924],{"className":66925,"style":137},[136],[18,66927,66929],{"className":66928},[141,142,143,144],[18,66930,28160],{"className":66931},[106,110,144],[18,66933,152],{"className":66934},[151],[18,66936,66938],{"className":66937},[124],[18,66939,66941],{"className":66940,"style":159},[128],[18,66942],{},[18,66944,1349],{"className":66945},[267],"。把 ",[18,66948,66950,66970],{"className":66949,"translate":22},[21],[18,66951,66953],{"className":66952},[26],[28,66954,66955],{"xmlns":30},[32,66956,66957,66967],{},[35,66958,66959,66961],{},[41,66960,18304],{"mathvariant":1462},[38,66962,66963,66965],{},[41,66964,11222],{"mathvariant":316},[41,66966,51012],{},[86,66968,66969],{"encoding":88},"\\mathcal H|_C",[18,66971,66973],{"className":66972,"ariaHidden":51},[93],[18,66974,66976,66979,66982],{"className":66975},[97],[18,66977],{"className":66978,"style":242},[101],[18,66980,18304],{"className":66981,"style":46321},[106,1478],[18,66983,66985,66988],{"className":66984},[106],[18,66986,11222],{"className":66987},[106],[18,66989,66991],{"className":66990},[115],[18,66992,66994,67014],{"className":66993},[119,120],[18,66995,66997,67011],{"className":66996},[124],[18,66998,67000],{"className":66999,"style":1152},[128],[18,67001,67002,67005],{"style":1155},[18,67003],{"className":67004,"style":137},[136],[18,67006,67008],{"className":67007},[141,142,143,144],[18,67009,51012],{"className":67010,"style":5073},[106,110,144],[18,67012,152],{"className":67013},[151],[18,67015,67017],{"className":67016},[124],[18,67018,67020],{"className":67019,"style":159},[128],[18,67021],{}," 按最后一个坐标分成两类：",[2210,67024,67025,67127],{},[2213,67026,67027,67097,67098,67126],{},[18,67028,67030,67048],{"className":67029,"translate":22},[21],[18,67031,67033],{"className":67032},[26],[28,67034,67035],{"xmlns":30},[32,67036,67037,67045],{},[35,67038,67039],{},[38,67040,67041,67043],{},[41,67042,58444],{},[45,67044,430],{},[86,67046,67047],{"encoding":88},"A_0",[18,67049,67051],{"className":67050,"ariaHidden":51},[93],[18,67052,67054,67057],{"className":67053},[97],[18,67055],{"className":67056,"style":4563},[101],[18,67058,67060,67063],{"className":67059},[106],[18,67061,58444],{"className":67062},[106,110],[18,67064,67066],{"className":67065},[115],[18,67067,67069,67089],{"className":67068},[119,120],[18,67070,67072,67086],{"className":67071},[124],[18,67073,67075],{"className":67074,"style":129},[128],[18,67076,67077,67080],{"style":1155},[18,67078],{"className":67079,"style":137},[136],[18,67081,67083],{"className":67082},[141,142,143,144],[18,67084,430],{"className":67085},[106,144],[18,67087,152],{"className":67088},[151],[18,67090,67092],{"className":67091},[124],[18,67093,67095],{"className":67094,"style":159},[128],[18,67096],{},"：最后一个坐标为 ",[18,67099,67101,67114],{"className":67100,"translate":22},[21],[18,67102,67104],{"className":67103},[26],[28,67105,67106],{"xmlns":30},[32,67107,67108,67112],{},[35,67109,67110],{},[45,67111,430],{},[86,67113,430],{"encoding":88},[18,67115,67117],{"className":67116,"ariaHidden":51},[93],[18,67118,67120,67123],{"className":67119},[97],[18,67121],{"className":67122,"style":505},[101],[18,67124,430],{"className":67125},[106]," 的二分；",[2213,67128,67129,67097,67199,67227],{},[18,67130,67132,67150],{"className":67131,"translate":22},[21],[18,67133,67135],{"className":67134},[26],[28,67136,67137],{"xmlns":30},[32,67138,67139,67147],{},[35,67140,67141],{},[38,67142,67143,67145],{},[41,67144,58444],{},[45,67146,47],{},[86,67148,67149],{"encoding":88},"A_1",[18,67151,67153],{"className":67152,"ariaHidden":51},[93],[18,67154,67156,67159],{"className":67155},[97],[18,67157],{"className":67158,"style":4563},[101],[18,67160,67162,67165],{"className":67161},[106],[18,67163,58444],{"className":67164},[106,110],[18,67166,67168],{"className":67167},[115],[18,67169,67171,67191],{"className":67170},[119,120],[18,67172,67174,67188],{"className":67173},[124],[18,67175,67177],{"className":67176,"style":129},[128],[18,67178,67179,67182],{"style":1155},[18,67180],{"className":67181,"style":137},[136],[18,67183,67185],{"className":67184},[141,142,143,144],[18,67186,47],{"className":67187},[106,144],[18,67189,152],{"className":67190},[151],[18,67192,67194],{"className":67193},[124],[18,67195,67197],{"className":67196,"style":159},[128],[18,67198],{},[18,67200,67202,67215],{"className":67201,"translate":22},[21],[18,67203,67205],{"className":67204},[26],[28,67206,67207],{"xmlns":30},[32,67208,67209,67213],{},[35,67210,67211],{},[45,67212,47],{},[86,67214,47],{"encoding":88},[18,67216,67218],{"className":67217,"ariaHidden":51},[93],[18,67219,67221,67224],{"className":67220},[97],[18,67222],{"className":67223,"style":505},[101],[18,67225,47],{"className":67226},[106]," 的二分。",[11,67229,16,67230,509,67299,3019,67368,67430,67431,10360],{},[18,67231,67233,67250],{"className":67232,"translate":22},[21],[18,67234,67236],{"className":67235},[26],[28,67237,67238],{"xmlns":30},[32,67239,67240,67248],{},[35,67241,67242],{},[38,67243,67244,67246],{},[41,67245,58444],{},[45,67247,430],{},[86,67249,67047],{"encoding":88},[18,67251,67253],{"className":67252,"ariaHidden":51},[93],[18,67254,67256,67259],{"className":67255},[97],[18,67257],{"className":67258,"style":4563},[101],[18,67260,67262,67265],{"className":67261},[106],[18,67263,58444],{"className":67264},[106,110],[18,67266,67268],{"className":67267},[115],[18,67269,67271,67291],{"className":67270},[119,120],[18,67272,67274,67288],{"className":67273},[124],[18,67275,67277],{"className":67276,"style":129},[128],[18,67278,67279,67282],{"style":1155},[18,67280],{"className":67281,"style":137},[136],[18,67283,67285],{"className":67284},[141,142,143,144],[18,67286,430],{"className":67287},[106,144],[18,67289,152],{"className":67290},[151],[18,67292,67294],{"className":67293},[124],[18,67295,67297],{"className":67296,"style":159},[128],[18,67298],{},[18,67300,67302,67319],{"className":67301,"translate":22},[21],[18,67303,67305],{"className":67304},[26],[28,67306,67307],{"xmlns":30},[32,67308,67309,67317],{},[35,67310,67311],{},[38,67312,67313,67315],{},[41,67314,58444],{},[45,67316,47],{},[86,67318,67149],{"encoding":88},[18,67320,67322],{"className":67321,"ariaHidden":51},[93],[18,67323,67325,67328],{"className":67324},[97],[18,67326],{"className":67327,"style":4563},[101],[18,67329,67331,67334],{"className":67330},[106],[18,67332,58444],{"className":67333},[106,110],[18,67335,67337],{"className":67336},[115],[18,67338,67340,67360],{"className":67339},[119,120],[18,67341,67343,67357],{"className":67342},[124],[18,67344,67346],{"className":67345,"style":129},[128],[18,67347,67348,67351],{"style":1155},[18,67349],{"className":67350,"style":137},[136],[18,67352,67354],{"className":67353},[141,142,143,144],[18,67355,47],{"className":67356},[106,144],[18,67358,152],{"className":67359},[151],[18,67361,67363],{"className":67362},[124],[18,67364,67366],{"className":67365,"style":159},[128],[18,67367],{},[18,67369,67371,67389],{"className":67370,"translate":22},[21],[18,67372,67374],{"className":67373},[26],[28,67375,67376],{"xmlns":30},[32,67377,67378,67386],{},[35,67379,67380],{},[1556,67381,67382,67384],{},[41,67383,51012],{},[49,67385,32620],{"mathvariant":316,"lspace":32619,"rspace":32619},[86,67387,67388],{"encoding":88},"C'",[18,67390,67392],{"className":67391,"ariaHidden":51},[93],[18,67393,67395,67398],{"className":67394},[97],[18,67396],{"className":67397,"style":32651},[101],[18,67399,67401,67404],{"className":67400},[106],[18,67402,51012],{"className":67403,"style":5073},[106,110],[18,67405,67407],{"className":67406},[115],[18,67408,67410],{"className":67409},[119],[18,67411,67413],{"className":67412},[124],[18,67414,67416],{"className":67415,"style":32651},[128],[18,67417,67418,67421],{"style":1962},[18,67419],{"className":67420,"style":137},[136],[18,67422,67424],{"className":67423},[141,142,143,144],[18,67425,67427],{"className":67426},[106,144],[18,67428,32620],{"className":67429},[106,144]," 上的投影的交集大小为 ",[18,67432,67434,67447],{"className":67433,"translate":22},[21],[18,67435,67437],{"className":67436},[26],[28,67438,67439],{"xmlns":30},[32,67440,67441,67445],{},[35,67442,67443],{},[41,67444,44302],{},[86,67446,44302],{"encoding":88},[18,67448,67450],{"className":67449,"ariaHidden":51},[93],[18,67451,67453,67456],{"className":67452},[97],[18,67454],{"className":67455,"style":1083},[101],[18,67457,44302],{"className":67458},[106,110],[18,67460,67462],{"className":67461,"translate":22},[875],[18,67463,67465,67537],{"className":67464,"translate":22},[21],[18,67466,67468],{"className":67467},[26],[28,67469,67470],{"xmlns":30,"display":884},[32,67471,67472,67534],{},[35,67473,67474,67476,67478,67484,67486,67488,67490,67496,67498,67500,67502,67508,67510,67512,67518,67520,67522,67524,67526,67528,67530,67532],{},[41,67475,11222],{"mathvariant":316},[41,67477,18304],{"mathvariant":1462},[38,67479,67480,67482],{},[41,67481,11222],{"mathvariant":316},[41,67483,51012],{},[41,67485,11222],{"mathvariant":316},[49,67487,433],{},[41,67489,11222],{"mathvariant":316},[38,67491,67492,67494],{},[41,67493,58444],{},[45,67495,430],{},[41,67497,11222],{"mathvariant":316},[49,67499,7984],{},[41,67501,11222],{"mathvariant":316},[38,67503,67504,67506],{},[41,67505,58444],{},[45,67507,47],{},[41,67509,11222],{"mathvariant":316},[49,67511,4334],{},[38,67513,67514,67516],{},[41,67515,64993],{"mathvariant":316},[41,67517,18304],{"mathvariant":1462},[49,67519,74],{"stretchy":73},[41,67521,28160],{},[49,67523,2253],{},[45,67525,47],{},[49,67527,84],{"stretchy":73},[49,67529,7984],{},[41,67531,44302],{},[41,67533,968],{"mathvariant":316},[86,67535,67536],{"encoding":88},"|\\mathcal H|_C|=|A_0|+|A_1|\\le \\Pi_{\\mathcal H}(m-1)+s.",[18,67538,67540,67604,67665,67726,67790,67811],{"className":67539,"ariaHidden":51},[93],[18,67541,67543,67546,67549,67552,67592,67595,67598,67601],{"className":67542},[97],[18,67544],{"className":67545,"style":242},[101],[18,67547,11222],{"className":67548},[106],[18,67550,18304],{"className":67551,"style":46321},[106,1478],[18,67553,67555,67558],{"className":67554},[106],[18,67556,11222],{"className":67557},[106],[18,67559,67561],{"className":67560},[115],[18,67562,67564,67584],{"className":67563},[119,120],[18,67565,67567,67581],{"className":67566},[124],[18,67568,67570],{"className":67569,"style":1152},[128],[18,67571,67572,67575],{"style":1155},[18,67573],{"className":67574,"style":137},[136],[18,67576,67578],{"className":67577},[141,142,143,144],[18,67579,51012],{"className":67580,"style":5073},[106,110,144],[18,67582,152],{"className":67583},[151],[18,67585,67587],{"className":67586},[124],[18,67588,67590],{"className":67589,"style":159},[128],[18,67591],{},[18,67593,11222],{"className":67594},[106],[18,67596],{"className":67597,"style":228},[169],[18,67599,433],{"className":67600},[232],[18,67602],{"className":67603,"style":228},[169],[18,67605,67607,67610,67613,67653,67656,67659,67662],{"className":67606},[97],[18,67608],{"className":67609,"style":242},[101],[18,67611,11222],{"className":67612},[106],[18,67614,67616,67619],{"className":67615},[106],[18,67617,58444],{"className":67618},[106,110],[18,67620,67622],{"className":67621},[115],[18,67623,67625,67645],{"className":67624},[119,120],[18,67626,67628,67642],{"className":67627},[124],[18,67629,67631],{"className":67630,"style":129},[128],[18,67632,67633,67636],{"style":1155},[18,67634],{"className":67635,"style":137},[136],[18,67637,67639],{"className":67638},[141,142,143,144],[18,67640,430],{"className":67641},[106,144],[18,67643,152],{"className":67644},[151],[18,67646,67648],{"className":67647},[124],[18,67649,67651],{"className":67650,"style":159},[128],[18,67652],{},[18,67654,11222],{"className":67655},[106],[18,67657],{"className":67658,"style":2068},[169],[18,67660,7984],{"className":67661},[2072],[18,67663],{"className":67664,"style":2068},[169],[18,67666,67668,67671,67674,67714,67717,67720,67723],{"className":67667},[97],[18,67669],{"className":67670,"style":242},[101],[18,67672,11222],{"className":67673},[106],[18,67675,67677,67680],{"className":67676},[106],[18,67678,58444],{"className":67679},[106,110],[18,67681,67683],{"className":67682},[115],[18,67684,67686,67706],{"className":67685},[119,120],[18,67687,67689,67703],{"className":67688},[124],[18,67690,67692],{"className":67691,"style":129},[128],[18,67693,67694,67697],{"style":1155},[18,67695],{"className":67696,"style":137},[136],[18,67698,67700],{"className":67699},[141,142,143,144],[18,67701,47],{"className":67702},[106,144],[18,67704,152],{"className":67705},[151],[18,67707,67709],{"className":67708},[124],[18,67710,67712],{"className":67711,"style":159},[128],[18,67713],{},[18,67715,11222],{"className":67716},[106],[18,67718],{"className":67719,"style":228},[169],[18,67721,4334],{"className":67722},[232],[18,67724],{"className":67725,"style":228},[169],[18,67727,67729,67732,67775,67778,67781,67784,67787],{"className":67728},[97],[18,67730],{"className":67731,"style":242},[101],[18,67733,67735,67738],{"className":67734},[106],[18,67736,64993],{"className":67737},[106],[18,67739,67741],{"className":67740},[115],[18,67742,67744,67767],{"className":67743},[119,120],[18,67745,67747,67764],{"className":67746},[124],[18,67748,67750],{"className":67749,"style":1152},[128],[18,67751,67752,67755],{"style":1155},[18,67753],{"className":67754,"style":137},[136],[18,67756,67758],{"className":67757},[141,142,143,144],[18,67759,67761],{"className":67760},[106,144],[18,67762,18304],{"className":67763,"style":46321},[106,1478,144],[18,67765,152],{"className":67766},[151],[18,67768,67770],{"className":67769},[124],[18,67771,67773],{"className":67772,"style":159},[128],[18,67774],{},[18,67776,74],{"className":67777},[250],[18,67779,28160],{"className":67780},[106,110],[18,67782],{"className":67783,"style":2068},[169],[18,67785,2253],{"className":67786},[2072],[18,67788],{"className":67789,"style":2068},[169],[18,67791,67793,67796,67799,67802,67805,67808],{"className":67792},[97],[18,67794],{"className":67795,"style":242},[101],[18,67797,47],{"className":67798},[106],[18,67800,84],{"className":67801},[267],[18,67803],{"className":67804,"style":2068},[169],[18,67806,7984],{"className":67807},[2072],[18,67809],{"className":67810,"style":2068},[169],[18,67812,67814,67817,67820],{"className":67813},[97],[18,67815],{"className":67816,"style":1083},[101],[18,67818,44302],{"className":67819},[106,110],[18,67821,968],{"className":67822},[106],[11,67824,67825,67826,67887,67888,67955,67956,68021,68022,68083,68084,68112,68113,68141,68142,68222,68223,68273,68274,68404,68405,1990],{},"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 ",[18,67889,67891,67913],{"className":67890,"translate":22},[21],[18,67892,67894],{"className":67893},[26],[28,67895,67896],{"xmlns":30},[32,67897,67898,67910],{},[35,67899,67900],{},[1556,67901,67902,67904],{},[41,67903,18304],{"mathvariant":1462},[35,67905,67906,67908],{},[49,67907,32620],{"mathvariant":316},[49,67909,32620],{"mathvariant":316},[86,67911,67912],{"encoding":88},"\\mathcal H''",[18,67914,67916],{"className":67915,"ariaHidden":51},[93],[18,67917,67919,67922],{"className":67918},[97],[18,67920],{"className":67921,"style":32651},[101],[18,67923,67925,67928],{"className":67924},[106],[18,67926,18304],{"className":67927,"style":46321},[106,1478],[18,67929,67931],{"className":67930},[115],[18,67932,67934],{"className":67933},[119],[18,67935,67937],{"className":67936},[124],[18,67938,67940],{"className":67939,"style":32651},[128],[18,67941,67942,67945],{"style":1962},[18,67943],{"className":67944,"style":137},[136],[18,67946,67948],{"className":67947},[141,142,143,144],[18,67949,67951],{"className":67950},[106,144],[18,67952,67954],{"className":67953},[106,144],"′′","。若 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打散了 ",[18,68023,68025,68042],{"className":68024,"translate":22},[21],[18,68026,68028],{"className":68027},[26],[28,68029,68030],{"xmlns":30},[32,68031,68032,68040],{},[35,68033,68034],{},[1556,68035,68036,68038],{},[41,68037,51012],{},[49,68039,32620],{"mathvariant":316,"lspace":32619,"rspace":32619},[86,68041,67388],{"encoding":88},[18,68043,68045],{"className":68044,"ariaHidden":51},[93],[18,68046,68048,68051],{"className":68047},[97],[18,68049],{"className":68050,"style":32651},[101],[18,68052,68054,68057],{"className":68053},[106],[18,68055,51012],{"className":68056,"style":5073},[106,110],[18,68058,68060],{"className":68059},[115],[18,68061,68063],{"className":68062},[119],[18,68064,68066],{"className":68065},[124],[18,68067,68069],{"className":68068,"style":32651},[128],[18,68070,68071,68074],{"style":1962},[18,68072],{"className":68073,"style":137},[136],[18,68075,68077],{"className":68076},[141,142,143,144],[18,68078,68080],{"className":68079},[106,144],[18,68081,32620],{"className":68082},[106,144]," 的某个 ",[18,68085,68087,68100],{"className":68086,"translate":22},[21],[18,68088,68090],{"className":68089},[26],[28,68091,68092],{"xmlns":30},[32,68093,68094,68098],{},[35,68095,68096],{},[41,68097,4031],{},[86,68099,4031],{"encoding":88},[18,68101,68103],{"className":68102,"ariaHidden":51},[93],[18,68104,68106,68109],{"className":68105},[97],[18,68107],{"className":68108,"style":295},[101],[18,68110,4031],{"className":68111},[106,110]," 元子集，那么 ",[18,68114,68116,68129],{"className":68115,"translate":22},[21],[18,68117,68119],{"className":68118},[26],[28,68120,68121],{"xmlns":30},[32,68122,68123,68127],{},[35,68124,68125],{},[41,68126,18304],{"mathvariant":1462},[86,68128,46308],{"encoding":88},[18,68130,68132],{"className":68131,"ariaHidden":51},[93],[18,68133,68135,68138],{"className":68134},[97],[18,68136],{"className":68137,"style":600},[101],[18,68139,18304],{"className":68140,"style":46321},[106,1478]," 就打散了该子集并上 ",[18,68143,68145,68167],{"className":68144,"translate":22},[21],[18,68146,68148],{"className":68147},[26],[28,68149,68150],{"xmlns":30},[32,68151,68152,68164],{},[35,68153,68154,68156,68162],{},[49,68155,1331],{"stretchy":73},[38,68157,68158,68160],{},[41,68159,39058],{},[41,68161,28160],{},[49,68163,1349],{"stretchy":73},[86,68165,68166],{"encoding":88},"\\{c_m\\}",[18,68168,68170],{"className":68169,"ariaHidden":51},[93],[18,68171,68173,68176,68179,68219],{"className":68172},[97],[18,68174],{"className":68175,"style":242},[101],[18,68177,1331],{"className":68178},[250],[18,68180,68182,68185],{"className":68181},[106],[18,68183,39058],{"className":68184},[106,110],[18,68186,68188],{"className":68187},[115],[18,68189,68191,68211],{"className":68190},[119,120],[18,68192,68194,68208],{"className":68193},[124],[18,68195,68197],{"className":68196,"style":202},[128],[18,68198,68199,68202],{"style":1155},[18,68200],{"className":68201,"style":137},[136],[18,68203,68205],{"className":68204},[141,142,143,144],[18,68206,28160],{"className":68207},[106,110,144],[18,68209,152],{"className":68210},[151],[18,68212,68214],{"className":68213},[124],[18,68215,68217],{"className":68216,"style":159},[128],[18,68218],{},[18,68220,1349],{"className":68221},[267],"，共 ",[18,68224,68226,68243],{"className":68225,"translate":22},[21],[18,68227,68229],{"className":68228},[26],[28,68230,68231],{"xmlns":30},[32,68232,68233,68241],{},[35,68234,68235,68237,68239],{},[41,68236,4031],{},[49,68238,7984],{},[45,68240,47],{},[86,68242,56239],{"encoding":88},[18,68244,68246,68264],{"className":68245,"ariaHidden":51},[93],[18,68247,68249,68252,68255,68258,68261],{"className":68248},[97],[18,68250],{"className":68251,"style":7221},[101],[18,68253,4031],{"className":68254},[106,110],[18,68256],{"className":68257,"style":2068},[169],[18,68259,7984],{"className":68260},[2072],[18,68262],{"className":68263,"style":2068},[169],[18,68265,68267,68270],{"className":68266},[97],[18,68268],{"className":68269,"style":505},[101],[18,68271,47],{"className":68272},[106]," 个点，矛盾。所以 ",[18,68275,68277,68315],{"className":68276,"translate":22},[21],[18,68278,68280],{"className":68279},[26],[28,68281,68282],{"xmlns":30},[32,68283,68284,68312],{},[35,68285,68286,68288,68290,68292,68302,68304,68306,68308,68310],{},[41,68287,52202],{"mathvariant":316},[49,68289,899],{},[49,68291,74],{"stretchy":73},[1556,68293,68294,68296],{},[41,68295,18304],{"mathvariant":1462},[35,68297,68298,68300],{},[49,68299,32620],{"mathvariant":316},[49,68301,32620],{"mathvariant":316},[49,68303,84],{"stretchy":73},[49,68305,4334],{},[41,68307,4031],{},[49,68309,2253],{},[45,68311,47],{},[86,68313,68314],{"encoding":88},"\\operatorname{VCdim}(\\mathcal H'')\\le d-1",[18,68316,68318,68377,68395],{"className":68317,"ariaHidden":51},[93],[18,68319,68321,68324,68330,68333,68365,68368,68371,68374],{"className":68320},[97],[18,68322],{"className":68323,"style":39968},[101],[18,68325,68327],{"className":68326},[985],[18,68328,52202],{"className":68329},[106,6351],[18,68331,74],{"className":68332},[250],[18,68334,68336,68339],{"className":68335},[106],[18,68337,18304],{"className":68338,"style":46321},[106,1478],[18,68340,68342],{"className":68341},[115],[18,68343,68345],{"className":68344},[119],[18,68346,68348],{"className":68347},[124],[18,68349,68351],{"className":68350,"style":32651},[128],[18,68352,68353,68356],{"style":1962},[18,68354],{"className":68355,"style":137},[136],[18,68357,68359],{"className":68358},[141,142,143,144],[18,68360,68362],{"className":68361},[106,144],[18,68363,67954],{"className":68364},[106,144],[18,68366,84],{"className":68367},[267],[18,68369],{"className":68370,"style":228},[169],[18,68372,4334],{"className":68373},[232],[18,68375],{"className":68376,"style":228},[169],[18,68378,68380,68383,68386,68389,68392],{"className":68379},[97],[18,68381],{"className":68382,"style":7221},[101],[18,68384,4031],{"className":68385},[106,110],[18,68387],{"className":68388,"style":2068},[169],[18,68390,2253],{"className":68391},[2072],[18,68393],{"className":68394,"style":2068},[169],[18,68396,68398,68401],{"className":68397},[97],[18,68399],{"className":68400,"style":505},[101],[18,68402,47],{"className":68403},[106],"，从而 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1,68780,68781],{},"代回并用 Pascal 公式：",[18,68783,68785],{"className":68784,"translate":22},[875],[18,68786,68788,68916],{"className":68787,"translate":22},[21],[18,68789,68791],{"className":68790},[26],[28,68792,68793],{"xmlns":30,"display":884},[32,68794,68795,68913],{},[35,68796,68797,68803,68805,68807,68809,68811,68825,68843,68845,68865,68883,68885,68899,68911],{},[38,68798,68799,68801],{},[41,68800,64993],{"mathvariant":316},[41,68802,18304],{"mathvariant":1462},[49,68804,74],{"stretchy":73},[41,68806,28160],{},[49,68808,84],{"stretchy":73},[49,68810,4334],{},[20358,68812,68813,68815,68823],{},[49,68814,20362],{},[35,68816,68817,68819,68821],{},[41,68818,5597],{},[49,68820,433],{},[45,68822,430],{},[41,68824,4031],{},[35,68826,68827,68829,68841],{},[49,68828,74],{"fence":51},[5651,68830,68831,68839],{"linethickness":66337},[35,68832,68833,68835,68837],{},[41,68834,28160],{},[49,68836,2253],{},[45,68838,47],{},[41,68840,5597],{},[49,68842,84],{"fence":51},[49,68844,7984],{},[20358,68846,68847,68849,68857],{},[49,68848,20362],{},[35,68850,68851,68853,68855],{},[41,68852,5597],{},[49,68854,433],{},[45,68856,430],{},[35,68858,68859,68861,68863],{},[41,68860,4031],{},[49,68862,2253],{},[45,68864,47],{},[35,68866,68867,68869,68881],{},[49,68868,74],{"fence":51},[5651,68870,68871,68879],{"linethickness":66337},[35,68872,68873,68875,68877],{},[41,68874,28160],{},[49,68876,2253],{},[45,68878,47],{},[41,68880,5597],{},[49,68882,84],{"fence":51},[49,68884,433],{},[20358,68886,68887,68889,68897],{},[49,68888,20362],{},[35,68890,68891,68893,68895],{},[41,68892,5597],{},[49,68894,433],{},[45,68896,430],{},[41,68898,4031],{},[35,68900,68901,68903,68909],{},[49,68902,74],{"fence":51},[5651,68904,68905,68907],{"linethickness":66337},[41,68906,28160],{},[41,68908,5597],{},[49,68910,84],{"fence":51},[41,68912,968],{"mathvariant":316},[86,68914,68915],{"encoding":88},"\\Pi_{\\mathcal 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维有限的假设类，在 ",[18,71261,71263,71276],{"className":71262,"translate":22},[21],[18,71264,71266],{"className":71265},[26],[28,71267,71268],{"xmlns":30},[32,71269,71270,71274],{},[35,71271,71272],{},[41,71273,28160],{},[86,71275,28160],{"encoding":88},[18,71277,71279],{"className":71278,"ariaHidden":51},[93],[18,71280,71282,71285],{"className":71281},[97],[18,71283],{"className":71284,"style":1083},[101],[18,71286,28160],{"className":71287},[106,110]," 个点上最多只有 ",[18,71290,71292,71310],{"className":71291,"translate":22},[21],[18,71293,71295],{"className":71294},[26],[28,71296,71297],{"xmlns":30},[32,71298,71299,71307],{},[35,71300,71301],{},[1556,71302,71303,71305],{},[41,71304,28160],{},[41,71306,4031],{},[86,71308,71309],{"encoding":88},"m^d",[18,71311,71313],{"className":71312,"ariaHidden":51},[93],[18,71314,71316,71319],{"className":71315},[97],[18,71317],{"className":71318,"style":35177},[101],[18,71320,71322,71325],{"className":71321},[106],[18,71323,28160],{"className":71324},[106,110],[18,71326,71328],{"className":71327},[115],[18,71329,71331],{"className":71330},[119],[18,71332,71334],{"className":71333},[124],[18,71335,71337],{"className":71336,"style":35177},[128],[18,71338,71339,71342],{"style":1962},[18,71340],{"className":71341,"style":137},[136],[18,71343,71345],{"className":71344},[141,142,143,144],[18,71346,4031],{"className":71347},[106,110,144]," 种行为，而不是 ",[18,71350,71352,71369],{"className":71351,"translate":22},[21],[18,71353,71355],{"className":71354},[26],[28,71356,71357],{"xmlns":30},[32,71358,71359,71367],{},[35,71360,71361],{},[1556,71362,71363,71365],{},[45,71364,2266],{},[41,71366,28160],{},[86,71368,44893],{"encoding":88},[18,71370,71372],{"className":71371,"ariaHidden":51},[93],[18,71373,71375,71378],{"className":71374},[97],[18,71376],{"className":71377,"style":1959},[101],[18,71379,71381,71384],{"className":71380},[106],[18,71382,2266],{"className":71383},[106],[18,71385,71387],{"className":71386},[115],[18,71388,71390],{"className":71389},[119],[18,71391,71393],{"className":71392},[124],[18,71394,71396],{"className":71395,"style":1959},[128],[18,71397,71398,71401],{"style":1962},[18,71399],{"className":71400,"style":137},[136],[18,71402,71404],{"className":71403},[141,142,143,144],[18,71405,28160],{"className":71406},[106,110,144]," 种。当 ",[18,71409,71411,71424],{"className":71410,"translate":22},[21],[18,71412,71414],{"className":71413},[26],[28,71415,71416],{"xmlns":30},[32,71417,71418,71422],{},[35,71419,71420],{},[41,71421,28160],{},[86,71423,28160],{"encoding":88},[18,71425,71427],{"className":71426,"ariaHidden":51},[93],[18,71428,71430,71433],{"className":71429},[97],[18,71431],{"className":71432,"style":1083},[101],[18,71434,28160],{"className":71435},[106,110]," 增大时，这两种增长速度的差距是压倒性的——",[18,71438,71440,71457],{"className":71439,"translate":22},[21],[18,71441,71443],{"className":71442},[26],[28,71444,71445],{"xmlns":30},[32,71446,71447,71455],{},[35,71448,71449],{},[1556,71450,71451,71453],{},[41,71452,28160],{},[41,71454,4031],{},[86,71456,71309],{"encoding":88},[18,71458,71460],{"className":71459,"ariaHidden":51},[93],[18,71461,71463,71466],{"className":71462},[97],[18,71464],{"className":71465,"style":35177},[101],[18,71467,71469,71472],{"className":71468},[106],[18,71470,28160],{"className":71471},[106,110],[18,71473,71475],{"className":71474},[115],[18,71476,71478],{"className":71477},[119],[18,71479,71481],{"className":71480},[124],[18,71482,71484],{"className":71483,"style":35177},[128],[18,71485,71486,71489],{"style":1962},[18,71487],{"className":71488,"style":137},[136],[18,71490,71492],{"className":71491},[141,142,143,144],[18,71493,4031],{"className":71494},[106,110,144]," 取对数后只有 ",[18,71497,71499,71519],{"className":71498,"translate":22},[21],[18,71500,71502],{"className":71501},[26],[28,71503,71504],{"xmlns":30},[32,71505,71506,71516],{},[35,71507,71508,71510,71512,71514],{},[41,71509,4031],{},[41,71511,14730],{},[49,71513,899],{},[41,71515,28160],{},[86,71517,71518],{"encoding":88},"d\\log m",[18,71520,71522],{"className":71521,"ariaHidden":51},[93],[18,71523,71525,71528,71531,71534,71539,71542],{"className":71524},[97],[18,71526],{"className":71527,"style":2890},[101],[18,71529,4031],{"className":71530},[106,110],[18,71532],{"className":71533,"style":170},[169],[18,71535,15171,71537],{"className":71536},[985],[18,71538,15174],{"style":6352},[18,71540],{"className":71541,"style":170},[169],[18,71543,28160],{"className":71544},[106,110],"，而 ",[18,71547,71549,71566],{"className":71548,"translate":22},[21],[18,71550,71552],{"className":71551},[26],[28,71553,71554],{"xmlns":30},[32,71555,71556,71564],{},[35,71557,71558],{},[1556,71559,71560,71562],{},[45,71561,2266],{},[41,71563,28160],{},[86,71565,44893],{"encoding":88},[18,71567,71569],{"className":71568,"ariaHidden":51},[93],[18,71570,71572,71575],{"className":71571},[97],[18,71573],{"className":71574,"style":1959},[101],[18,71576,71578,71581],{"className":71577},[106],[18,71579,2266],{"className":71580},[106],[18,71582,71584],{"className":71583},[115],[18,71585,71587],{"className":71586},[119],[18,71588,71590],{"className":71589},[124],[18,71591,71593],{"className":71592,"style":1959},[128],[18,71594,71595,71598],{"style":1962},[18,71596],{"className":71597,"style":137},[136],[18,71599,71601],{"className":71600},[141,142,143,144],[18,71602,28160],{"className":71603},[106,110,144]," 取对数后是 ",[18,71606,71608,71621],{"className":71607,"translate":22},[21],[18,71609,71611],{"className":71610},[26],[28,71612,71613],{"xmlns":30},[32,71614,71615,71619],{},[35,71616,71617],{},[41,71618,28160],{},[86,71620,28160],{"encoding":88},[18,71622,71624],{"className":71623,"ariaHidden":51},[93],[18,71625,71627,71630],{"className":71626},[97],[18,71628],{"className":71629,"style":1083},[101],[18,71631,28160],{"className":71632},[106,110],[11,71634,71635,71636,71639],{},"换句话说，",[717,71637,71638],{},"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。",[638,71641,71643],{"id":71642},"四一致收敛定理","四、一致收敛定理",[1311,71645,71647],{"id":71646},"_41-对称化","4.1 对称化",[11,71649,71650,71651,71654],{},"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",[717,71652,71653],{},"对称化","（symmetrization）。",[11,71656,16,71657,71719,71720,71748,71749,71777],{},[18,71658,71660,71678],{"className":71659,"translate":22},[21],[18,71661,71663],{"className":71662},[26],[28,71664,71665],{"xmlns":30},[32,71666,71667,71675],{},[35,71668,71669],{},[1556,71670,71671,71673],{},[41,71672,38925],{},[49,71674,32620],{"mathvariant":316,"lspace":32619,"rspace":32619},[86,71676,71677],{"encoding":88},"S'",[18,71679,71681],{"className":71680,"ariaHidden":51},[93],[18,71682,71684,71687],{"className":71683},[97],[18,71685],{"className":71686,"style":32651},[101],[18,71688,71690,71693],{"className":71689},[106],[18,71691,38925],{"className":71692,"style":38958},[106,110],[18,71694,71696],{"className":71695},[115],[18,71697,71699],{"className":71698},[119],[18,71700,71702],{"className":71701},[124],[18,71703,71705],{"className":71704,"style":32651},[128],[18,71706,71707,71710],{"style":1962},[18,71708],{"className":71709,"style":137},[136],[18,71711,71713],{"className":71712},[141,142,143,144],[18,71714,71716],{"className":71715},[106,144],[18,71717,32620],{"className":71718},[106,144]," 是与 ",[18,71721,71723,71736],{"className":71722,"translate":22},[21],[18,71724,71726],{"className":71725},[26],[28,71727,71728],{"xmlns":30},[32,71729,71730,71734],{},[35,71731,71732],{},[41,71733,38925],{},[86,71735,38925],{"encoding":88},[18,71737,71739],{"className":71738,"ariaHidden":51},[93],[18,71740,71742,71745],{"className":71741},[97],[18,71743],{"className":71744,"style":600},[101],[18,71746,38925],{"className":71747,"style":38958},[106,110]," 独立同分布的另一组 ",[18,71750,71752,71765],{"className":71751,"translate":22},[21],[18,71753,71755],{"className":71754},[26],[28,71756,71757],{"xmlns":30},[32,71758,71759,71763],{},[35,71760,71761],{},[41,71762,28160],{},[86,71764,28160],{"encoding":88},[18,71766,71768],{"className":71767,"ariaHidden":51},[93],[18,71769,71771,71774],{"className":71770},[97],[18,71772],{"className":71773,"style":1083},[101],[18,71775,28160],{"className":71776},[106,110]," 个样本。核心不等式是",[18,71779,71781],{"className":71780,"translate":22},[875],[18,71782,71784,71917],{"className":71783,"translate":22},[21],[18,71785,71787],{"className":71786},[26],[28,71788,71789],{"xmlns":30,"display":884},[32,71790,71791,71914],{},[35,71792,71793,71809,71811,71817,71819,71821,71823,71825,71831,71833,71835,71837,71839,71842,71844,71846,71862,71878,71880,71886,71888,71890,71892,71894,71904,71906,71908,71910,71912],{},[890,71794,71795,71801],{},[35,71796,71797,71799],{},[41,71798,912],{},[49,71800,899],{},[35,71802,71803,71805,71807],{},[41,71804,18323],{},[49,71806,921],{},[41,71808,18304],{"mathvariant":1462},[49,71810,11222],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[38,71812,71813,71815],{},[41,71814,12579],{},[41,71816,12572],{"mathvariant":1462},[49,71818,74],{"stretchy":73},[41,71820,18323],{},[49,71822,84],{"stretchy":73},[49,71824,2253],{},[38,71826,71827,71829],{},[41,71828,12579],{},[41,71830,38925],{},[49,71832,74],{"stretchy":73},[41,71834,18323],{},[49,71836,84],{"stretchy":73},[49,71838,11222],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[926,71840,71841],{},"  ",[49,71843,42710],{},[926,71845,71841],{},[38,71847,71848,71850],{},[41,71849,934],{"mathvariant":933},[35,71851,71852,71854,71856],{},[41,71853,38925],{},[49,71855,52],{"separator":51},[1556,71857,71858,71860],{},[41,71859,38925],{},[49,71861,32620],{"mathvariant":316,"lspace":32619,"rspace":32619},[890,71863,71864,71870],{},[35,71865,71866,71868],{},[41,71867,912],{},[49,71869,899],{},[35,71871,71872,71874,71876],{},[41,71873,18323],{},[49,71875,921],{},[41,71877,18304],{"mathvariant":1462},[49,71879,11222],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[38,71881,71882,71884],{},[41,71883,12579],{},[41,71885,38925],{},[49,71887,74],{"stretchy":73},[41,71889,18323],{},[49,71891,84],{"stretchy":73},[49,71893,2253],{},[38,71895,71896,71898],{},[41,71897,12579],{},[1556,71899,71900,71902],{},[41,71901,38925],{},[49,71903,32620],{"mathvariant":316,"lspace":32619,"rspace":32619},[49,71905,74],{"stretchy":73},[41,71907,18323],{},[49,71909,84],{"stretchy":73},[49,71911,11222],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[49,71913,52],{"separator":51},[86,71915,71916],{"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 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不等式），代价是概率意义上的常数因子。",[11,72551,72552,72553,72660,72661,72731],{},"接着引入独立同分布的 Rademacher 变量 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y]",[18,74016,74018,74082,74148],{"className":74017,"ariaHidden":51},[93],[18,74019,74021,74024,74064,74067,74070,74073,74076,74079],{"className":74020},[97],[18,74022],{"className":74023,"style":242},[101],[18,74025,74027,74030],{"className":74026},[106],[18,74028,948],{"className":74029},[106],[18,74031,74033],{"className":74032},[115],[18,74034,74036,74056],{"className":74035},[119,120],[18,74037,74039,74053],{"className":74038},[124],[18,74040,74042],{"className":74041,"style":1383},[128],[18,74043,74044,74047],{"style":1155},[18,74045],{"className":74046,"style":137},[136],[18,74048,74050],{"className":74049},[141,142,143,144],[18,74051,18323],{"className":74052},[106,110,144],[18,74054,152],{"className":74055},[151],[18,74057,74059],{"className":74058},[124],[18,74060,74062],{"className":74061,"style":159},[128],[18,74063],{},[18,74065,74],{"className":74066},[250],[18,74068,72811],{"className":74069,"style":73301},[106,110],[18,74071,84],{"className":74072},[267],[18,74074],{"className":74075,"style":228},[169],[18,74077,433],{"className":74078},[232],[18,74080],{"className":74081,"style":228},[169],[18,74083,74085,74088,74091,74094,74097,74100,74103,74106,74109,74145],{"className":74084},[97],[18,74086],{"className":74087,"style":242},[101],[18,74089,47],{"className":74090},[106,2859],[18,74092,945],{"className":74093},[250],[18,74095,18323],{"className":74096},[106,110],[18,74098,74],{"className":74099},[250],[18,74101,38573],{"className":74102},[106,110],[18,74104,84],{"className":74105},[267],[18,74107],{"className":74108,"style":228},[169],[18,74110,74112,74139,74142],{"className":74111},[232],[18,74113,74115],{"className":74114},[232],[18,74116,74118],{"className":74117},[106,2878],[18,74119,74121],{"className":74120},[2882],[18,74122,74124,74127,74136],{"className":74123},[2886],[18,74125],{"className":74126,"style":2890},[101],[18,74128,74130],{"className":74129},[2894],[18,74131,74133],{"className":74132},[106],[18,74134,2901],{"className":74135},[232],[18,74137],{"className":74138},[2905],[18,74140],{"className":74141},[169,2909],[18,74143,433],{"className":74144},[232],[18,74146],{"className":74147,"style":228},[169],[18,74149,74151,74154,74157],{"className":74150},[97],[18,74152],{"className":74153,"style":242},[101],[18,74155,46343],{"className":74156,"style":3957},[106,110],[18,74158,965],{"className":74159},[267],"。右侧就是 ",[717,74162,74163],{},"Rademacher 复杂度",[18,74165,74167,74191],{"className":74166,"translate":22},[21],[18,74168,74170],{"className":74169},[26],[28,74171,74172],{"xmlns":30},[32,74173,74174,74188],{},[35,74175,74176,74182,74184,74186],{},[38,74177,74178,74180],{},[41,74179,3071],{"mathvariant":3525},[41,74181,28160],{},[49,74183,74],{"stretchy":73},[41,74185,18304],{"mathvariant":1462},[49,74187,84],{"stretchy":73},[86,74189,74190],{"encoding":88},"\\mathfrak R_m(\\mathcal H)",[18,74192,74194],{"className":74193,"ariaHidden":51},[93],[18,74195,74197,74200,74240,74243,74246],{"className":74196},[97],[18,74198],{"className":74199,"style":242},[101],[18,74201,74203,74206],{"className":74202},[106],[18,74204,3071],{"className":74205},[106,3607],[18,74207,74209],{"className":74208},[115],[18,74210,74212,74232],{"className":74211},[119,120],[18,74213,74215,74229],{"className":74214},[124],[18,74216,74218],{"className":74217,"style":202},[128],[18,74219,74220,74223],{"style":1155},[18,74221],{"className":74222,"style":137},[136],[18,74224,74226],{"className":74225},[141,142,143,144],[18,74227,28160],{"className":74228},[106,110,144],[18,74230,152],{"className":74231},[151],[18,74233,74235],{"className":74234},[124],[18,74236,74238],{"className":74237,"style":159},[128],[18,74239],{},[18,74241,74],{"className":74242},[250],[18,74244,18304],{"className":74245,"style":46321},[106,1478],[18,74247,84],{"className":74248},[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。",[11,74251,74252],{},"对二值函数类，Rademacher 复杂度可以被增长函数控制：",[18,74254,74256],{"className":74255,"translate":22},[875],[18,74257,74259,74313],{"className":74258,"translate":22},[21],[18,74260,74262],{"className":74261},[26],[28,74263,74264],{"xmlns":30,"display":884},[32,74265,74266,74310],{},[35,74267,74268,74274,74276,74278,74280,74282,74308],{},[38,74269,74270,74272],{},[41,74271,3071],{"mathvariant":3525},[41,74273,28160],{},[49,74275,74],{"stretchy":73},[41,74277,18304],{"mathvariant":1462},[49,74279,84],{"stretchy":73},[49,74281,4334],{},[12558,74283,74284],{},[5651,74285,74286,74306],{},[35,74287,74288,74290,74292,74294,74300,74302,74304],{},[45,74289,2266],{},[41,74291,14730],{},[49,74293,899],{},[38,74295,74296,74298],{},[41,74297,64993],{"mathvariant":316},[41,74299,18304],{"mathvariant":1462},[49,74301,74],{"stretchy":73},[41,74303,28160],{},[49,74305,84],{"stretchy":73},[41,74307,28160],{},[41,74309,968],{"mathvariant":316},[86,74311,74312],{"encoding":88},"\\mathfrak R_m(\\mathcal H)\\le\\sqrt{\\frac{2\\log\\Pi_{\\mathcal H}(m)}{m}}.",[18,74314,74316,74380],{"className":74315,"ariaHidden":51},[93],[18,74317,74319,74322,74362,74365,74368,74371,74374,74377],{"className":74318},[97],[18,74320],{"className":74321,"style":242},[101],[18,74323,74325,74328],{"className":74324},[106],[18,74326,3071],{"className":74327},[106,3607],[18,74329,74331],{"className":74330},[115],[18,74332,74334,74354],{"className":74333},[119,120],[18,74335,74337,74351],{"className":74336},[124],[18,74338,74340],{"className":74339,"style":202},[128],[18,74341,74342,74345],{"style":1155},[18,74343],{"className":74344,"style":137},[136],[18,74346,74348],{"className":74347},[141,142,143,144],[18,74349,28160],{"className":74350},[106,110,144],[18,74352,152],{"className":74353},[151],[18,74355,74357],{"className":74356},[124],[18,74358,74360],{"className":74359,"style":159},[128],[18,74361],{},[18,74363,74],{"className":74364},[250],[18,74366,18304],{"className":74367,"style":46321},[106,1478],[18,74369,84],{"className":74370},[267],[18,74372],{"className":74373,"style":228},[169],[18,74375,4334],{"className":74376},[232],[18,74378],{"className":74379,"style":228},[169],[18,74381,74383,74387,74559],{"className":74382},[97],[18,74384],{"className":74385,"style":74386},[101],"height:2.44em;vertical-align:-0.7356em;",[18,74388,74390],{"className":74389},[106,12693],[18,74391,74393,74550],{"className":74392},[119,120],[18,74394,74396,74547],{"className":74395},[124],[18,74397,74400,74534],{"className":74398,"style":74399},[128],"height:1.7044em;",[18,74401,74403,74406],{"className":74402,"style":12708},[12707],[18,74404],{"className":74405,"style":12712},[136],[18,74407,74409],{"className":74408,"style":12716},[106],[18,74410,74412,74415,74531],{"className":74411},[106],[18,74413],{"className":74414},[250,6009],[18,74416,74418],{"className":74417},[5651],[18,74419,74421,74523],{"className":74420},[119,120],[18,74422,74424,74520],{"className":74423},[124],[18,74425,74427,74438,74446],{"className":74426,"style":18128},[128],[18,74428,74429,74432],{"style":6025},[18,74430],{"className":74431,"style":697},[136],[18,74433,74435],{"className":74434},[106],[18,74436,28160],{"className":74437},[106,110],[18,74439,74440,74443],{"style":6037},[18,74441],{"className":74442,"style":697},[136],[18,74444],{"className":74445,"style":6045},[6044],[18,74447,74448,74451],{"style":6048},[18,74449],{"className":74450,"style":697},[136],[18,74452,74454,74457,74460,74465,74468,74511,74514,74517],{"className":74453},[106],[18,74455,2266],{"className":74456},[106],[18,74458],{"className":74459,"style":170},[169],[18,74461,15171,74463],{"className":74462},[985],[18,74464,15174],{"style":6352},[18,74466],{"className":74467,"style":170},[169],[18,74469,74471,74474],{"className":74470},[106],[18,74472,64993],{"className":74473},[106],[18,74475,74477],{"className":74476},[115],[18,74478,74480,74503],{"className":74479},[119,120],[18,74481,74483,74500],{"className":74482},[124],[18,74484,74486],{"className":74485,"style":1152},[128],[18,74487,74488,74491],{"style":1155},[18,74489],{"className":74490,"style":137},[136],[18,74492,74494],{"className":74493},[141,142,143,144],[18,74495,74497],{"className":74496},[106,144],[18,74498,18304],{"className":74499,"style":46321},[106,1478,144],[18,74501,152],{"className":74502},[151],[18,74504,74506],{"className":74505},[124],[18,74507,74509],{"className":74508,"style":159},[128],[18,74510],{},[18,74512,74],{"className":74513},[250],[18,74515,28160],{"className":74516},[106,110],[18,74518,84],{"className":74519},[267],[18,74521,152],{"className":74522},[151],[18,74524,74526],{"className":74525},[124],[18,74527,74529],{"className":74528,"style":6067},[128],[18,74530],{},[18,74532],{"className":74533},[267,6009],[18,74535,74537,74540],{"style":74536},"top:-3.6644em;",[18,74538],{"className":74539,"style":12712},[136],[18,74541,74543],{"className":74542,"style":12851},[12850],[6127,74544,74545],{"xmlns":6129,"width":12854,"height":12855,"viewBox":12856,"preserveAspectRatio":12857},[6133,74546],{"d":12860},[18,74548,152],{"className":74549},[151],[18,74551,74553],{"className":74552},[124],[18,74554,74557],{"className":74555,"style":74556},[128],"height:0.7356em;",[18,74558],{},[18,74560,968],{"className":74561},[106],[11,74563,74564,74565,74593,74594,74682],{},"这一步用的是有限类的 Massart 引理：把 ",[18,74566,74568,74581],{"className":74567,"translate":22},[21],[18,74569,74571],{"className":74570},[26],[28,74572,74573],{"xmlns":30},[32,74574,74575,74579],{},[35,74576,74577],{},[41,74578,18304],{"mathvariant":1462},[86,74580,46308],{"encoding":88},[18,74582,74584],{"className":74583,"ariaHidden":51},[93],[18,74585,74587,74590],{"className":74586},[97],[18,74588],{"className":74589,"style":600},[101],[18,74591,18304],{"className":74592,"style":46321},[106,1478]," 限制到样本上，最多只有 ",[18,74595,74597,74621],{"className":74596,"translate":22},[21],[18,74598,74600],{"className":74599},[26],[28,74601,74602],{"xmlns":30},[32,74603,74604,74618],{},[35,74605,74606,74612,74614,74616],{},[38,74607,74608,74610],{},[41,74609,64993],{"mathvariant":316},[41,74611,18304],{"mathvariant":1462},[49,74613,74],{"stretchy":73},[41,74615,28160],{},[49,74617,84],{"stretchy":73},[86,74619,74620],{"encoding":88},"\\Pi_{\\mathcal H}(m)",[18,74622,74624],{"className":74623,"ariaHidden":51},[93],[18,74625,74627,74630,74673,74676,74679],{"className":74626},[97],[18,74628],{"className":74629,"style":242},[101],[18,74631,74633,74636],{"className":74632},[106],[18,74634,64993],{"className":74635},[106],[18,74637,74639],{"className":74638},[115],[18,74640,74642,74665],{"className":74641},[119,120],[18,74643,74645,74662],{"className":74644},[124],[18,74646,74648],{"className":74647,"style":1152},[128],[18,74649,74650,74653],{"style":1155},[18,74651],{"className":74652,"style":137},[136],[18,74654,74656],{"className":74655},[141,142,143,144],[18,74657,74659],{"className":74658},[106,144],[18,74660,18304],{"className":74661,"style":46321},[106,1478,144],[18,74663,152],{"className":74664},[151],[18,74666,74668],{"className":74667},[124],[18,74669,74671],{"className":74670,"style":159},[128],[18,74672],{},[18,74674,74],{"className":74675},[250],[18,74677,28160],{"className":74678},[106,110],[18,74680,84],{"className":74681},[267]," 种不同的二分，再用 union bound 或鞅不等式。",[1311,74684,74686],{"id":74685},"_42-泛化界","4.2 泛化界",[11,74688,74689,74690,749],{},"把上面的估计串起来，得到本文唯一需要记住的公式。当 ",[18,74691,74693,74710],{"className":74692,"translate":22},[21],[18,74694,74696],{"className":74695},[26],[28,74697,74698],{"xmlns":30},[32,74699,74700,74708],{},[35,74701,74702,74704,74706],{},[41,74703,28160],{},[49,74705,4775],{},[41,74707,4031],{},[86,74709,69474],{"encoding":88},[18,74711,74713,74731],{"className":74712,"ariaHidden":51},[93],[18,74714,74716,74719,74722,74725,74728],{"className":74715},[97],[18,74717],{"className":74718,"style":53130},[101],[18,74720,28160],{"className":74721},[106,110],[18,74723],{"className":74724,"style":228},[169],[18,74726,4775],{"className":74727},[232],[18,74729],{"className":74730,"style":228},[169],[18,74732,74734,74737],{"className":74733},[97],[18,74735],{"className":74736,"style":295},[101],[18,74738,4031],{"className":74739},[106,110],[18,74741,74743],{"className":74742,"translate":22},[875],[18,74744,74746,74838],{"className":74745,"translate":22},[21],[18,74747,74749],{"className":74748},[26],[28,74750,74751],{"xmlns":30,"display":884},[32,74752,74753,74835],{},[35,74754,74755,74761,74763,74765,74767,74769,74771,74773,74779,74781,74783,74785,74787,74789,74791,74793,74833],{},[38,74756,74757,74759],{},[41,74758,12579],{},[41,74760,12572],{"mathvariant":1462},[49,74762,74],{"stretchy":73},[41,74764,18323],{},[49,74766,84],{"stretchy":73},[926,74768,71841],{},[49,74770,4334],{},[926,74772,71841],{},[38,74774,74775,74777],{},[41,74776,12579],{},[41,74778,38925],{},[49,74780,74],{"stretchy":73},[41,74782,18323],{},[49,74784,84],{"stretchy":73},[926,74786,71841],{},[49,74788,7984],{},[926,74790,71841],{},[41,74792,51012],{},[12558,74794,74795],{},[5651,74796,74797,74831],{},[35,74798,74799,74801,74803,74805,74807,74809,74811,74813,74815,74817,74819,74821,74823,74825,74827,74829],{},[41,74800,4031],{},[41,74802,14730],{},[49,74804,899],{},[49,74806,74],{"stretchy":73},[41,74808,28160],{},[41,74810,317],{"mathvariant":316},[41,74812,4031],{},[49,74814,84],{"stretchy":73},[49,74816,7984],{},[41,74818,14730],{},[49,74820,899],{},[49,74822,74],{"stretchy":73},[45,74824,47],{},[41,74826,317],{"mathvariant":316},[41,74828,5348],{},[49,74830,84],{"stretchy":73},[41,74832,28160],{},[49,74834,52],{"separator":51},[86,74836,74837],{"encoding":88},"L_{\\mathcal D}(h)\\;\\le\\;L_S(h)\\;+\\;C\\sqrt{\\frac{d\\log(m\u002Fd)+\\log(1\u002F\\delta)}{m}},",[18,74839,74841,74914,74984],{"className":74840,"ariaHidden":51},[93],[18,74842,74844,74847,74890,74893,74896,74899,74902,74905,74908,74911],{"className":74843},[97],[18,74845],{"className":74846,"style":242},[101],[18,74848,74850,74853],{"className":74849},[106],[18,74851,12579],{"className":74852},[106,110],[18,74854,74856],{"className":74855},[115],[18,74857,74859,74882],{"className":74858},[119,120],[18,74860,74862,74879],{"className":74861},[124],[18,74863,74865],{"className":74864,"style":1152},[128],[18,74866,74867,74870],{"style":1155},[18,74868],{"className":74869,"style":137},[136],[18,74871,74873],{"className":74872},[141,142,143,144],[18,74874,74876],{"className":74875},[106,144],[18,74877,12572],{"className":74878,"style":254},[106,1478,144],[18,74880,152],{"className":74881},[151],[18,74883,74885],{"className":74884},[124],[18,74886,74888],{"className":74887,"style":159},[128],[18,74889],{},[18,74891,74],{"className":74892},[250],[18,74894,18323],{"className":74895},[106,110],[18,74897,84],{"className":74898},[267],[18,74900],{"className":74901,"style":228},[169],[18,74903],{"className":74904,"style":228},[169],[18,74906,4334],{"className":74907},[232],[18,74909],{"className":74910,"style":228},[169],[18,74912],{"className":74913,"style":228},[169],[18,74915,74917,74920,74960,74963,74966,74969,74972,74975,74978,74981],{"className":74916},[97],[18,74918],{"className":74919,"style":242},[101],[18,74921,74923,74926],{"className":74922},[106],[18,74924,12579],{"className":74925},[106,110],[18,74927,74929],{"className":74928},[115],[18,74930,74932,74952],{"className":74931},[119,120],[18,74933,74935,74949],{"className":74934},[124],[18,74936,74938],{"className":74937,"style":1152},[128],[18,74939,74940,74943],{"style":1155},[18,74941],{"className":74942,"style":137},[136],[18,74944,74946],{"className":74945},[141,142,143,144],[18,74947,38925],{"className":74948,"style":38958},[106,110,144],[18,74950,152],{"className":74951},[151],[18,74953,74955],{"className":74954},[124],[18,74956,74958],{"className":74957,"style":159},[128],[18,74959],{},[18,74961,74],{"className":74962},[250],[18,74964,18323],{"className":74965},[106,110],[18,74967,84],{"className":74968},[267],[18,74970],{"className":74971,"style":228},[169],[18,74973],{"className":74974,"style":2068},[169],[18,74976,7984],{"className":74977},[2072],[18,74979],{"className":74980,"style":228},[169],[18,74982],{"className":74983,"style":2068},[169],[18,74985,74987,74990,74993,75148],{"className":74986},[97],[18,74988],{"className":74989,"style":74386},[101],[18,74991,51012],{"className":74992,"style":5073},[106,110],[18,74994,74996],{"className":74995},[106,12693],[18,74997,74999,75140],{"className":74998},[119,120],[18,75000,75002,75137],{"className":75001},[124],[18,75003,75005,75125],{"className":75004,"style":74399},[128],[18,75006,75008,75011],{"className":75007,"style":12708},[12707],[18,75009],{"className":75010,"style":12712},[136],[18,75012,75014],{"className":75013,"style":12716},[106],[18,75015,75017,75020,75122],{"className":75016},[106],[18,75018],{"className":75019},[250,6009],[18,75021,75023],{"className":75022},[5651],[18,75024,75026,75114],{"className":75025},[119,120],[18,75027,75029,75111],{"className":75028},[124],[18,75030,75032,75043,75051],{"className":75031,"style":18128},[128],[18,75033,75034,75037],{"style":6025},[18,75035],{"className":75036,"style":697},[136],[18,75038,75040],{"className":75039},[106],[18,75041,28160],{"className":75042},[106,110],[18,75044,75045,75048],{"style":6037},[18,75046],{"className":75047,"style":697},[136],[18,75049],{"className":75050,"style":6045},[6044],[18,75052,75053,75056],{"style":6048},[18,75054],{"className":75055,"style":697},[136],[18,75057,75059,75062,75065,75070,75073,75076,75079,75082,75085,75088,75091,75094,75099,75102,75105,75108],{"className":75058},[106],[18,75060,4031],{"className":75061},[106,110],[18,75063],{"className":75064,"style":170},[169],[18,75066,15171,75068],{"className":75067},[985],[18,75069,15174],{"style":6352},[18,75071,74],{"className":75072},[250],[18,75074,28160],{"className":75075},[106,110],[18,75077,317],{"className":75078},[106],[18,75080,4031],{"className":75081},[106,110],[18,75083,84],{"className":75084},[267],[18,75086],{"className":75087,"style":2068},[169],[18,75089,7984],{"className":75090},[2072],[18,75092],{"className":75093,"style":2068},[169],[18,75095,15171,75097],{"className":75096},[985],[18,75098,15174],{"style":6352},[18,75100,74],{"className":75101},[250],[18,75103,335],{"className":75104},[106],[18,75106,5348],{"className":75107,"style":5477},[106,110],[18,75109,84],{"className":75110},[267],[18,75112,152],{"className":75113},[151],[18,75115,75117],{"className":75116},[124],[18,75118,75120],{"className":75119,"style":6067},[128],[18,75121],{},[18,75123],{"className":75124},[267,6009],[18,75126,75127,75130],{"style":74536},[18,75128],{"className":75129,"style":12712},[136],[18,75131,75133],{"className":75132,"style":12851},[12850],[6127,75134,75135],{"xmlns":6129,"width":12854,"height":12855,"viewBox":12856,"preserveAspectRatio":12857},[6133,75136],{"d":12860},[18,75138,152],{"className":75139},[151],[18,75141,75143],{"className":75142},[124],[18,75144,75146],{"className":75145,"style":74556},[128],[18,75147],{},[18,75149,52],{"className":75150},[165],[11,75152,75153,75154,75205,75206,1994,75208,75259,75260,4741,75331,75359],{},"以至少 ",[18,75155,75157,75175],{"className":75156,"translate":22},[21],[18,75158,75160],{"className":75159},[26],[28,75161,75162],{"xmlns":30},[32,75163,75164,75172],{},[35,75165,75166,75168,75170],{},[45,75167,47],{},[49,75169,2253],{},[41,75171,5348],{},[86,75173,75174],{"encoding":88},"1-\\delta",[18,75176,75178,75196],{"className":75177,"ariaHidden":51},[93],[18,75179,75181,75184,75187,75190,75193],{"className":75180},[97],[18,75182],{"className":75183,"style":18090},[101],[18,75185,47],{"className":75186},[106],[18,75188],{"className":75189,"style":2068},[169],[18,75191,2253],{"className":75192},[2072],[18,75194],{"className":75195,"style":2068},[169],[18,75197,75199,75202],{"className":75198},[97],[18,75200],{"className":75201,"style":295},[101],[18,75203,5348],{"className":75204,"style":5477},[106,110]," 的概率对",[717,75207,1301],{},[18,75209,75211,75229],{"className":75210,"translate":22},[21],[18,75212,75214],{"className":75213},[26],[28,75215,75216],{"xmlns":30},[32,75217,75218,75226],{},[35,75219,75220,75222,75224],{},[41,75221,18323],{},[49,75223,921],{},[41,75225,18304],{"mathvariant":1462},[86,75227,75228],{"encoding":88},"h\\in\\mathcal H",[18,75230,75232,75250],{"className":75231,"ariaHidden":51},[93],[18,75233,75235,75238,75241,75244,75247],{"className":75234},[97],[18,75236],{"className":75237,"style":1422},[101],[18,75239,18323],{"className":75240},[106,110],[18,75242],{"className":75243,"style":228},[169],[18,75245,921],{"className":75246},[232],[18,75248],{"className":75249,"style":228},[169],[18,75251,75253,75256],{"className":75252},[97],[18,75254],{"className":75255,"style":600},[101],[18,75257,18304],{"className":75258,"style":46321},[106,1478]," 同时成立，其中 ",[18,75261,75263,75289],{"className":75262,"translate":22},[21],[18,75264,75266],{"className":75265},[26],[28,75267,75268],{"xmlns":30},[32,75269,75270,75286],{},[35,75271,75272,75274,75276,75278,75280,75282,75284],{},[41,75273,4031],{},[49,75275,433],{},[41,75277,52202],{"mathvariant":316},[49,75279,899],{},[49,75281,74],{"stretchy":73},[41,75283,18304],{"mathvariant":1462},[49,75285,84],{"stretchy":73},[86,75287,75288],{"encoding":88},"d=\\operatorname{VCdim}(\\mathcal H)",[18,75290,75292,75310],{"className":75291,"ariaHidden":51},[93],[18,75293,75295,75298,75301,75304,75307],{"className":75294},[97],[18,75296],{"className":75297,"style":295},[101],[18,75299,4031],{"className":75300},[106,110],[18,75302],{"className":75303,"style":228},[169],[18,75305,433],{"className":75306},[232],[18,75308],{"className":75309,"style":228},[169],[18,75311,75313,75316,75322,75325,75328],{"className":75312},[97],[18,75314],{"className":75315,"style":242},[101],[18,75317,75319],{"className":75318},[985],[18,75320,52202],{"className":75321},[106,6351],[18,75323,74],{"className":75324},[250],[18,75326,18304],{"className":75327,"style":46321},[106,1478],[18,75329,84],{"className":75330},[267],[18,75332,75334,75347],{"className":75333,"translate":22},[21],[18,75335,75337],{"className":75336},[26],[28,75338,75339],{"xmlns":30},[32,75340,75341,75345],{},[35,75342,75343],{},[41,75344,51012],{},[86,75346,51012],{"encoding":88},[18,75348,75350],{"className":75349,"ariaHidden":51},[93],[18,75351,75353,75356],{"className":75352},[97],[18,75354],{"className":75355,"style":600},[101],[18,75357,51012],{"className":75358,"style":5073},[106,110]," 是某个绝对常数。",[11,75361,75362,75363,75395,75396,23852,75462,75538],{},"不同教材的常数和 ",[18,75364,75366,75381],{"className":75365,"translate":22},[21],[18,75367,75369],{"className":75368},[26],[28,75370,75371],{"xmlns":30},[32,75372,75373,75379],{},[35,75374,75375,75377],{},[41,75376,14730],{},[49,75378,899],{},[86,75380,27765],{"encoding":88},[18,75382,75384],{"className":75383,"ariaHidden":51},[93],[18,75385,75387,75390],{"className":75386},[97],[18,75388],{"className":75389,"style":2890},[101],[18,75391,15171,75393],{"className":75392},[985],[18,75394,15174],{"style":6352}," 里的具体形式略有出入（常见的一种写法是把 ",[18,75397,75399,75427],{"className":75398,"translate":22},[21],[18,75400,75402],{"className":75401},[26],[28,75403,75404],{"xmlns":30},[32,75405,75406,75424],{},[35,75407,75408,75410,75412,75414,75416,75418,75420,75422],{},[41,75409,4031],{},[41,75411,14730],{},[49,75413,899],{},[49,75415,74],{"stretchy":73},[41,75417,28160],{},[41,75419,317],{"mathvariant":316},[41,75421,4031],{},[49,75423,84],{"stretchy":73},[86,75425,75426],{"encoding":88},"d\\log(m\u002Fd)",[18,75428,75430],{"className":75429,"ariaHidden":51},[93],[18,75431,75433,75436,75439,75442,75447,75450,75453,75456,75459],{"className":75432},[97],[18,75434],{"className":75435,"style":242},[101],[18,75437,4031],{"className":75438},[106,110],[18,75440],{"className":75441,"style":170},[169],[18,75443,15171,75445],{"className":75444},[985],[18,75446,15174],{"style":6352},[18,75448,74],{"className":75449},[250],[18,75451,28160],{"className":75452},[106,110],[18,75454,317],{"className":75455},[106],[18,75457,4031],{"className":75458},[106,110],[18,75460,84],{"className":75461},[267],[18,75463,75465,75497],{"className":75464,"translate":22},[21],[18,75466,75468],{"className":75467},[26],[28,75469,75470],{"xmlns":30},[32,75471,75472,75494],{},[35,75473,75474,75476,75478,75480,75482,75484,75486,75488,75490,75492],{},[41,75475,4031],{},[41,75477,14730],{},[49,75479,899],{},[49,75481,74],{"stretchy":73},[45,75483,2266],{},[41,75485,14441],{},[41,75487,28160],{},[41,75489,317],{"mathvariant":316},[41,75491,4031],{},[49,75493,84],{"stretchy":73},[86,75495,75496],{"encoding":88},"d\\log(2em\u002Fd)",[18,75498,75500],{"className":75499,"ariaHidden":51},[93],[18,75501,75503,75506,75509,75512,75517,75520,75523,75526,75529,75532,75535],{"className":75502},[97],[18,75504],{"className":75505,"style":242},[101],[18,75507,4031],{"className":75508},[106,110],[18,75510],{"className":75511,"style":170},[169],[18,75513,15171,75515],{"className":75514},[985],[18,75516,15174],{"style":6352},[18,75518,74],{"className":75519},[250],[18,75521,2266],{"className":75522},[106],[18,75524,14441],{"className":75525},[106,110],[18,75527,28160],{"className":75528},[106,110],[18,75530,317],{"className":75531},[106],[18,75533,4031],{"className":75534},[106,110],[18,75536,84],{"className":75537},[267],"），但量级是一致的。重要的是它的结构：",[2210,75540,75541,75629,75661],{},[2213,75542,75543,75544,75572,75573,75628],{},"分子里的 ",[18,75545,75547,75560],{"className":75546,"translate":22},[21],[18,75548,75550],{"className":75549},[26],[28,75551,75552],{"xmlns":30},[32,75553,75554,75558],{},[35,75555,75556],{},[41,75557,4031],{},[86,75559,4031],{"encoding":88},[18,75561,75563],{"className":75562,"ariaHidden":51},[93],[18,75564,75566,75569],{"className":75565},[97],[18,75567],{"className":75568,"style":295},[101],[18,75570,4031],{"className":75571},[106,110]," 是容量，",[18,75574,75576,75602],{"className":75575,"translate":22},[21],[18,75577,75579],{"className":75578},[26],[28,75580,75581],{"xmlns":30},[32,75582,75583,75599],{},[35,75584,75585,75587,75589,75591,75593,75595,75597],{},[41,75586,14730],{},[49,75588,899],{},[49,75590,74],{"stretchy":73},[45,75592,47],{},[41,75594,317],{"mathvariant":316},[41,75596,5348],{},[49,75598,84],{"stretchy":73},[86,75600,75601],{"encoding":88},"\\log(1\u002F\\delta)",[18,75603,75605],{"className":75604,"ariaHidden":51},[93],[18,75606,75608,75611,75616,75619,75622,75625],{"className":75607},[97],[18,75609],{"className":75610,"style":242},[101],[18,75612,15171,75614],{"className":75613},[985],[18,75615,15174],{"style":6352},[18,75617,74],{"className":75618},[250],[18,75620,335],{"className":75621},[106],[18,75623,5348],{"className":75624,"style":5477},[106,110],[18,75626,84],{"className":75627},[267]," 是置信度；",[2213,75630,75631,75632,75660],{},"分母里的 ",[18,75633,75635,75648],{"className":75634,"translate":22},[21],[18,75636,75638],{"className":75637},[26],[28,75639,75640],{"xmlns":30},[32,75641,75642,75646],{},[35,75643,75644],{},[41,75645,28160],{},[86,75647,28160],{"encoding":88},[18,75649,75651],{"className":75650,"ariaHidden":51},[93],[18,75652,75654,75657],{"className":75653},[97],[18,75655],{"className":75656,"style":1083},[101],[18,75658,28160],{"className":75659},[106,110]," 是样本量；",[2213,75662,75663,75664,75743],{},"整个界以 ",[18,75665,75667,75687],{"className":75666,"translate":22},[21],[18,75668,75670],{"className":75669},[26],[28,75671,75672],{"xmlns":30},[32,75673,75674,75684],{},[35,75675,75676,75678,75680],{},[45,75677,47],{},[41,75679,317],{"mathvariant":316},[12558,75681,75682],{},[41,75683,28160],{},[86,75685,75686],{"encoding":88},"1\u002F\\sqrt m",[18,75688,75690],{"className":75689,"ariaHidden":51},[93],[18,75691,75693,75696,75699],{"className":75692},[97],[18,75694],{"className":75695,"style":15498},[101],[18,75697,335],{"className":75698},[106],[18,75700,75702],{"className":75701},[106,12693],[18,75703,75705,75735],{"className":75704},[119,120],[18,75706,75708,75732],{"className":75707},[124],[18,75709,75711,75720],{"className":75710,"style":15397},[128],[18,75712,75714,75717],{"className":75713,"style":693},[12707],[18,75715],{"className":75716,"style":697},[136],[18,75718,28160],{"className":75719,"style":15407},[106,110],[18,75721,75722,75725],{"style":15410},[18,75723],{"className":75724,"style":697},[136],[18,75726,75728],{"className":75727,"style":15417},[12850],[6127,75729,75730],{"xmlns":6129,"width":12854,"height":15420,"viewBox":15421,"preserveAspectRatio":12857},[6133,75731],{"d":15424},[18,75733,152],{"className":75734},[151],[18,75736,75738],{"className":75737},[124],[18,75739,75741],{"className":75740,"style":15434},[128],[18,75742],{}," 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。",[11,75745,75746,75747,75840],{},"把 ",[18,75748,75750,75776],{"className":75749,"translate":22},[21],[18,75751,75753],{"className":75752},[26],[28,75754,75755],{"xmlns":30},[32,75756,75757,75773],{},[35,75758,75759,75765,75767,75769,75771],{},[38,75760,75761,75763],{},[41,75762,64993],{"mathvariant":316},[41,75764,18304],{"mathvariant":1462},[49,75766,74],{"stretchy":73},[45,75768,2266],{},[41,75770,28160],{},[49,75772,84],{"stretchy":73},[86,75774,75775],{"encoding":88},"\\Pi_{\\mathcal H}(2m)",[18,75777,75779],{"className":75778,"ariaHidden":51},[93],[18,75780,75782,75785,75828,75831,75834,75837],{"className":75781},[97],[18,75783],{"className":75784,"style":242},[101],[18,75786,75788,75791],{"className":75787},[106],[18,75789,64993],{"className":75790},[106],[18,75792,75794],{"className":75793},[115],[18,75795,75797,75820],{"className":75796},[119,120],[18,75798,75800,75817],{"className":75799},[124],[18,75801,75803],{"className":75802,"style":1152},[128],[18,75804,75805,75808],{"style":1155},[18,75806],{"className":75807,"style":137},[136],[18,75809,75811],{"className":75810},[141,142,143,144],[18,75812,75814],{"className":75813},[106,144],[18,75815,18304],{"className":75816,"style":46321},[106,1478,144],[18,75818,152],{"className":75819},[151],[18,75821,75823],{"className":75822},[124],[18,75824,75826],{"className":75825,"style":159},[128],[18,75827],{},[18,75829,74],{"className":75830},[250],[18,75832,2266],{"className":75833},[106],[18,75835,28160],{"className":75836},[106,110],[18,75838,84],{"className":75839},[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。",[1311,75842,75844],{"id":75843},"_43-基本定理","4.3 基本定理",[5159,75846,75848,75908],{"title":75847},"定理 4.1（统计学习基本定理）",[11,75849,75850,75851,2702,75879,75907],{},"对二分类、",[18,75852,75854,75867],{"className":75853,"translate":22},[21],[18,75855,75857],{"className":75856},[26],[28,75858,75859],{"xmlns":30},[32,75860,75861,75865],{},[35,75862,75863],{},[45,75864,430],{},[86,75866,430],{"encoding":88},[18,75868,75870],{"className":75869,"ariaHidden":51},[93],[18,75871,75873,75876],{"className":75872},[97],[18,75874],{"className":75875,"style":505},[101],[18,75877,430],{"className":75878},[106],[18,75880,75882,75895],{"className":75881,"translate":22},[21],[18,75883,75885],{"className":75884},[26],[28,75886,75887],{"xmlns":30},[32,75888,75889,75893],{},[35,75890,75891],{},[45,75892,47],{},[86,75894,47],{"encoding":88},[18,75896,75898],{"className":75897,"ariaHidden":51},[93],[18,75899,75901,75904],{"className":75900},[97],[18,75902],{"className":75903,"style":505},[101],[18,75905,47],{"className":75906},[106]," 损失、分布无关（distribution-free）的设定，以下四条等价：",[45443,75909,75910,75983,76014,76017],{},[2213,75911,75912,2452],{},[18,75913,75915,75941],{"className":75914,"translate":22},[21],[18,75916,75918],{"className":75917},[26],[28,75919,75920],{"xmlns":30},[32,75921,75922,75938],{},[35,75923,75924,75926,75928,75930,75932,75934,75936],{},[41,75925,52202],{"mathvariant":316},[49,75927,899],{},[49,75929,74],{"stretchy":73},[41,75931,18304],{"mathvariant":1462},[49,75933,84],{"stretchy":73},[49,75935,6681],{},[41,75937,2028],{"mathvariant":316},[86,75939,75940],{"encoding":88},"\\operatorname{VCdim}(\\mathcal H)\u003C\\infty",[18,75942,75944,75974],{"className":75943,"ariaHidden":51},[93],[18,75945,75947,75950,75956,75959,75962,75965,75968,75971],{"className":75946},[97],[18,75948],{"className":75949,"style":242},[101],[18,75951,75953],{"className":75952},[985],[18,75954,52202],{"className":75955},[106,6351],[18,75957,74],{"className":75958},[250],[18,75960,18304],{"className":75961,"style":46321},[106,1478],[18,75963,84],{"className":75964},[267],[18,75966],{"className":75967,"style":228},[169],[18,75969,6681],{"className":75970},[232],[18,75972],{"className":75973,"style":228},[169],[18,75975,75977,75980],{"className":75976},[97],[18,75978],{"className":75979,"style":1083},[101],[18,75981,2028],{"className":75982},[106],[2213,75984,75985,76013],{},[18,75986,75988,76001],{"className":75987,"translate":22},[21],[18,75989,75991],{"className":75990},[26],[28,75992,75993],{"xmlns":30},[32,75994,75995,75999],{},[35,75996,75997],{},[41,75998,18304],{"mathvariant":1462},[86,76000,46308],{"encoding":88},[18,76002,76004],{"className":76003,"ariaHidden":51},[93],[18,76005,76007,76010],{"className":76006},[97],[18,76008],{"className":76009,"style":600},[101],[18,76011,18304],{"className":76012,"style":46321},[106,1478]," 具有一致收敛性；",[2213,76015,76016],{},"ERM 是 agnostic PAC 一致的；",[2213,76018,76019,76047],{},[18,76020,76022,76035],{"className":76021,"translate":22},[21],[18,76023,76025],{"className":76024},[26],[28,76026,76027],{"xmlns":30},[32,76028,76029,76033],{},[35,76030,76031],{},[41,76032,18304],{"mathvariant":1462},[86,76034,46308],{"encoding":88},[18,76036,76038],{"className":76037,"ariaHidden":51},[93],[18,76039,76041,76044],{"className":76040},[97],[18,76042],{"className":76043,"style":600},[101],[18,76045,18304],{"className":76046,"style":46321},[106,1478]," 是 agnostic PAC 可学习的。",[11,76049,76050,76051],{},"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：",[717,76052,76053],{},"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。",[1311,76055,76057],{"id":76056},"_44-样本复杂度","4.4 样本复杂度",[11,76059,76060,76061,76089],{},"由上面的界解出 ",[18,76062,76064,76077],{"className":76063,"translate":22},[21],[18,76065,76067],{"className":76066},[26],[28,76068,76069],{"xmlns":30},[32,76070,76071,76075],{},[35,76072,76073],{},[41,76074,28160],{},[86,76076,28160],{"encoding":88},[18,76078,76080],{"className":76079,"ariaHidden":51},[93],[18,76081,76083,76086],{"className":76082},[97],[18,76084],{"className":76085,"style":1083},[101],[18,76087,28160],{"className":76088},[106,110],"，得到 agnostic 情形下的样本复杂度：",[18,76091,76093],{"className":76092,"translate":22},[875],[18,76094,76096,76150],{"className":76095,"translate":22},[21],[18,76097,76099],{"className":76098},[26],[28,76100,76101],{"xmlns":30,"display":884},[32,76102,76103,76147],{},[35,76104,76105,76107,76109,76111,76145],{},[41,76106,28160],{},[49,76108,433],{},[41,76110,924],{"mathvariant":316},[35,76112,76113,76115,76143],{},[49,76114,74],{"fence":51},[5651,76116,76117,76137],{},[35,76118,76119,76121,76123,76125,76127,76129,76131,76133,76135],{},[41,76120,4031],{},[49,76122,7984],{},[41,76124,14730],{},[49,76126,899],{},[49,76128,74],{"stretchy":73},[45,76130,47],{},[41,76132,317],{"mathvariant":316},[41,76134,5348],{},[49,76136,84],{"stretchy":73},[1556,76138,76139,76141],{},[41,76140,32513],{},[45,76142,2266],{},[49,76144,84],{"fence":51},[49,76146,52],{"separator":51},[86,76148,76149],{"encoding":88},"m=\\Theta\\left(\\frac{d+\\log(1\u002F\\delta)}{\\varepsilon^2}\\right),",[18,76151,76153,76171],{"className":76152,"ariaHidden":51},[93],[18,76154,76156,76159,76162,76165,76168],{"className":76155},[97],[18,76157],{"className":76158,"style":1083},[101],[18,76160,28160],{"className":76161},[106,110],[18,76163],{"className":76164,"style":228},[169],[18,76166,433],{"className":76167},[232],[18,76169],{"className":76170,"style":228},[169],[18,76172,76174,76177,76180,76183,76312,76315],{"className":76173},[97],[18,76175],{"className":76176,"style":20800},[101],[18,76178,924],{"className":76179},[106],[18,76181],{"className":76182,"style":170},[169],[18,76184,76186,76192,76306],{"className":76185},[174],[18,76187,76189],{"className":76188,"style":13793},[250,13792],[18,76190,74],{"className":76191},[1230,143],[18,76193,76195,76198,76303],{"className":76194},[106],[18,76196],{"className":76197},[250,6009],[18,76199,76201],{"className":76200},[5651],[18,76202,76204,76295],{"className":76203},[119,120],[18,76205,76207,76292],{"className":76206},[124],[18,76208,76210,76247,76255],{"className":76209,"style":18128},[128],[18,76211,76212,76215],{"style":6025},[18,76213],{"className":76214,"style":697},[136],[18,76216,76218],{"className":76217},[106],[18,76219,76221,76224],{"className":76220},[106],[18,76222,32513],{"className":76223},[106,110],[18,76225,76227],{"className":76226},[115],[18,76228,76230],{"className":76229},[119],[18,76231,76233],{"className":76232},[124],[18,76234,76236],{"className":76235,"style":26102},[128],[18,76237,76238,76241],{"style":26105},[18,76239],{"className":76240,"style":137},[136],[18,76242,76244],{"className":76243},[141,142,143,144],[18,76245,2266],{"className":76246},[106,144],[18,76248,76249,76252],{"style":6037},[18,76250],{"className":76251,"style":697},[136],[18,76253],{"className":76254,"style":6045},[6044],[18,76256,76257,76260],{"style":6048},[18,76258],{"className":76259,"style":697},[136],[18,76261,76263,76266,76269,76272,76275,76280,76283,76286,76289],{"className":76262},[106],[18,76264,4031],{"className":76265},[106,110],[18,76267],{"className":76268,"style":2068},[169],[18,76270,7984],{"className":76271},[2072],[18,76273],{"className":76274,"style":2068},[169],[18,76276,15171,76278],{"className":76277},[985],[18,76279,15174],{"style":6352},[18,76281,74],{"className":76282},[250],[18,76284,335],{"className":76285},[106],[18,76287,5348],{"className":76288,"style":5477},[106,110],[18,76290,84],{"className":76291},[267],[18,76293,152],{"className":76294},[151],[18,76296,76298],{"className":76297},[124],[18,76299,76301],{"className":76300,"style":6067},[128],[18,76302],{},[18,76304],{"className":76305},[267,6009],[18,76307,76309],{"className":76308,"style":13793},[267,13792],[18,76310,84],{"className":76311},[1230,143],[18,76313],{"className":76314,"style":170},[169],[18,76316,52],{"className":76317},[165],[11,76319,76320],{},"可实现情形下则是",[18,76322,76324],{"className":76323,"translate":22},[875],[18,76325,76327,76377],{"className":76326,"translate":22},[21],[18,76328,76330],{"className":76329},[26],[28,76331,76332],{"xmlns":30,"display":884},[32,76333,76334,76374],{},[35,76335,76336,76338,76340,76342,76372],{},[41,76337,28160],{},[49,76339,433],{},[41,76341,924],{"mathvariant":316},[35,76343,76344,76346,76370],{},[49,76345,74],{"fence":51},[5651,76347,76348,76368],{},[35,76349,76350,76352,76354,76356,76358,76360,76362,76364,76366],{},[41,76351,4031],{},[49,76353,7984],{},[41,76355,14730],{},[49,76357,899],{},[49,76359,74],{"stretchy":73},[45,76361,47],{},[41,76363,317],{"mathvariant":316},[41,76365,5348],{},[49,76367,84],{"stretchy":73},[41,76369,32513],{},[49,76371,84],{"fence":51},[41,76373,968],{"mathvariant":316},[86,76375,76376],{"encoding":88},"m=\\Theta\\left(\\frac{d+\\log(1\u002F\\delta)}{\\varepsilon}\\right).",[18,76378,76380,76398],{"className":76379,"ariaHidden":51},[93],[18,76381,76383,76386,76389,76392,76395],{"className":76382},[97],[18,76384],{"className":76385,"style":1083},[101],[18,76387,28160],{"className":76388},[106,110],[18,76390],{"className":76391,"style":228},[169],[18,76393,433],{"className":76394},[232],[18,76396],{"className":76397,"style":228},[169],[18,76399,76401,76404,76407,76410,76513,76516],{"className":76400},[97],[18,76402],{"className":76403,"style":20800},[101],[18,76405,924],{"className":76406},[106],[18,76408],{"className":76409,"style":170},[169],[18,76411,76413,76419,76507],{"className":76412},[174],[18,76414,76416],{"className":76415,"style":13793},[250,13792],[18,76417,74],{"className":76418},[1230,143],[18,76420,76422,76425,76504],{"className":76421},[106],[18,76423],{"className":76424},[250,6009],[18,76426,76428],{"className":76427},[5651],[18,76429,76431,76496],{"className":76430},[119,120],[18,76432,76434,76493],{"className":76433},[124],[18,76435,76437,76448,76456],{"className":76436,"style":18128},[128],[18,76438,76439,76442],{"style":6025},[18,76440],{"className":76441,"style":697},[136],[18,76443,76445],{"className":76444},[106],[18,76446,32513],{"className":76447},[106,110],[18,76449,76450,76453],{"style":6037},[18,76451],{"className":76452,"style":697},[136],[18,76454],{"className":76455,"style":6045},[6044],[18,76457,76458,76461],{"style":6048},[18,76459],{"className":76460,"style":697},[136],[18,76462,76464,76467,76470,76473,76476,76481,76484,76487,76490],{"className":76463},[106],[18,76465,4031],{"className":76466},[106,110],[18,76468],{"className":76469,"style":2068},[169],[18,76471,7984],{"className":76472},[2072],[18,76474],{"className":76475,"style":2068},[169],[18,76477,15171,76479],{"className":76478},[985],[18,76480,15174],{"style":6352},[18,76482,74],{"className":76483},[250],[18,76485,335],{"className":76486},[106],[18,76488,5348],{"className":76489,"style":5477},[106,110],[18,76491,84],{"className":76492},[267],[18,76494,152],{"className":76495},[151],[18,76497,76499],{"className":76498},[124],[18,76500,76502],{"className":76501,"style":6067},[128],[18,76503],{},[18,76505],{"className":76506},[267,6009],[18,76508,76510],{"className":76509,"style":13793},[267,13792],[18,76511,84],{"className":76512},[1230,143],[18,76514],{"className":76515,"style":170},[169],[18,76517,968],{"className":76518},[106],[11,76520,76521,76522,76550,76551,76554],{},"这两个 ",[18,76523,76525,76538],{"className":76524,"translate":22},[21],[18,76526,76528],{"className":76527},[26],[28,76529,76530],{"xmlns":30},[32,76531,76532,76536],{},[35,76533,76534],{},[41,76535,924],{"mathvariant":316},[86,76537,1497],{"encoding":88},[18,76539,76541],{"className":76540,"ariaHidden":51},[93],[18,76542,76544,76547],{"className":76543},[97],[18,76545],{"className":76546,"style":600},[101],[18,76548,924],{"className":76549},[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",[717,76552,76553],{},"最优","的样本复杂度量级。",[11,76556,76557],{},"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。",[638,76559,76561],{"id":76560},"五vc-维的含义","五、VC 维的含义",[11,76563,76564],{},"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。",[11,76566,76567,76570,76571,76647,76648,76725,76726,1990],{},[717,76568,76569],{},"第一，它是 min-max 的。"," 定义里的 ",[18,76572,76574,76596],{"className":76573,"translate":22},[21],[18,76575,76577],{"className":76576},[26],[28,76578,76579],{"xmlns":30},[32,76580,76581,76593],{},[35,76582,76583],{},[38,76584,76585,76591],{},[35,76586,76587,76589],{},[41,76588,912],{},[49,76590,899],{},[41,76592,51012],{},[86,76594,76595],{"encoding":88},"\\sup_C",[18,76597,76599],{"className":76598,"ariaHidden":51},[93],[18,76600,76602,76606],{"className":76601},[97],[18,76603],{"className":76604,"style":76605},[101],"height:0.6747em;vertical-align:-0.2441em;",[18,76607,76609,76612],{"className":76608},[985],[18,76610,912],{"className":76611},[985],[18,76613,76615],{"className":76614},[115],[18,76616,76618,76639],{"className":76617},[119,120],[18,76619,76621,76636],{"className":76620},[124],[18,76622,76625],{"className":76623,"style":76624},[128],"height:0.2342em;",[18,76626,76627,76630],{"style":60132},[18,76628],{"className":76629,"style":137},[136],[18,76631,76633],{"className":76632},[141,142,143,144],[18,76634,51012],{"className":76635,"style":5073},[106,110,144],[18,76637,152],{"className":76638},[151],[18,76640,76642],{"className":76641},[124],[18,76643,76645],{"className":76644,"style":60151},[128],[18,76646],{}," 对应界里的 ",[18,76649,76651,76673],{"className":76650,"translate":22},[21],[18,76652,76654],{"className":76653},[26],[28,76655,76656],{"xmlns":30},[32,76657,76658,76670],{},[35,76659,76660],{},[38,76661,76662,76668],{},[35,76663,76664,76666],{},[41,76665,912],{},[49,76667,899],{},[41,76669,12572],{"mathvariant":1462},[86,76671,76672],{"encoding":88},"\\sup_{\\mathcal D}",[18,76674,76676],{"className":76675,"ariaHidden":51},[93],[18,76677,76679,76682],{"className":76678},[97],[18,76680],{"className":76681,"style":76605},[101],[18,76683,76685,76688],{"className":76684},[985],[18,76686,912],{"className":76687},[985],[18,76689,76691],{"className":76690},[115],[18,76692,76694,76717],{"className":76693},[119,120],[18,76695,76697,76714],{"className":76696},[124],[18,76698,76700],{"className":76699,"style":76624},[128],[18,76701,76702,76705],{"style":60132},[18,76703],{"className":76704,"style":137},[136],[18,76706,76708],{"className":76707},[141,142,143,144],[18,76709,76711],{"className":76710},[106,144],[18,76712,12572],{"className":76713,"style":254},[106,1478,144],[18,76715,152],{"className":76716},[151],[18,76718,76720],{"className":76719},[124],[18,76721,76723],{"className":76722,"style":60151},[128],[18,76724],{},"。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",[717,76727,76728],{},"任何情形下都不被打破的承诺",[11,76730,76731,3019,76734,2702,76762,76790],{},[717,76732,76733],{},"第二，它是充要的。",[18,76735,76737,76750],{"className":76736,"translate":22},[21],[18,76738,76740],{"className":76739},[26],[28,76741,76742],{"xmlns":30},[32,76743,76744,76748],{},[35,76745,76746],{},[45,76747,430],{},[86,76749,430],{"encoding":88},[18,76751,76753],{"className":76752,"ariaHidden":51},[93],[18,76754,76756,76759],{"className":76755},[97],[18,76757],{"className":76758,"style":505},[101],[18,76760,430],{"className":76761},[106],[18,76763,76765,76778],{"className":76764,"translate":22},[21],[18,76766,76768],{"className":76767},[26],[28,76769,76770],{"xmlns":30},[32,76771,76772,76776],{},[35,76773,76774],{},[45,76775,47],{},[86,76777,47],{"encoding":88},[18,76779,76781],{"className":76780,"ariaHidden":51},[93],[18,76782,76784,76787],{"className":76783},[97],[18,76785],{"className":76786,"style":505},[101],[18,76788,47],{"className":76789},[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。",[11,76792,76793,76796],{},[717,76794,76795],{},"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。",[11,76798,76799,76802,76803,76867],{},[717,76800,76801],{},"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 ",[18,76804,76806,76831],{"className":76805,"translate":22},[21],[18,76807,76809],{"className":76808},[26],[28,76810,76811],{"xmlns":30},[32,76812,76813,76829],{},[35,76814,76815,76817,76819,76821,76823,76825,76827],{},[49,76816,74],{"stretchy":73},[45,76818,47],{},[49,76820,52],{"separator":51},[45,76822,430],{},[49,76824,52],{"separator":51},[45,76826,47],{},[49,76828,84],{"stretchy":73},[86,76830,55437],{"encoding":88},[18,76832,76834],{"className":76833,"ariaHidden":51},[93],[18,76835,76837,76840,76843,76846,76849,76852,76855,76858,76861,76864],{"className":76836},[97],[18,76838],{"className":76839,"style":242},[101],[18,76841,74],{"className":76842},[250],[18,76844,47],{"className":76845},[106],[18,76847,52],{"className":76848},[165],[18,76850],{"className":76851,"style":170},[169],[18,76853,430],{"className":76854},[106],[18,76856,52],{"className":76857},[165],[18,76859],{"className":76860,"style":170},[169],[18,76862,47],{"className":76863},[106],[18,76865,84],{"className":76866},[267]," 这种标注）。VC 维是容量，不是质量。",[11,76869,76870,1994,76873,77033,77034,77062],{},[717,76871,76872],{},"第五，它是上界，不是预测。",[18,76874,76876,76907],{"className":76875,"translate":22},[21],[18,76877,76879],{"className":76878},[26],[28,76880,76881],{"xmlns":30},[32,76882,76883,76904],{},[35,76884,76885,76891,76893,76899,76901],{},[38,76886,76887,76889],{},[41,76888,12579],{},[41,76890,12572],{"mathvariant":1462},[49,76892,4334],{},[38,76894,76895,76897],{},[41,76896,12579],{},[41,76898,38925],{},[49,76900,7984],{},[49,76902,76903],{},"⋯",[86,76905,76906],{"encoding":88},"L_{\\mathcal D}\\le L_S+\\cdots",[18,76908,76910,76968,77023],{"className":76909,"ariaHidden":51},[93],[18,76911,76913,76916,76959,76962,76965],{"className":76912},[97],[18,76914],{"className":76915,"style":4563},[101],[18,76917,76919,76922],{"className":76918},[106],[18,76920,12579],{"className":76921},[106,110],[18,76923,76925],{"className":76924},[115],[18,76926,76928,76951],{"className":76927},[119,120],[18,76929,76931,76948],{"className":76930},[124],[18,76932,76934],{"className":76933,"style":1152},[128],[18,76935,76936,76939],{"style":1155},[18,76937],{"className":76938,"style":137},[136],[18,76940,76942],{"className":76941},[141,142,143,144],[18,76943,76945],{"className":76944},[106,144],[18,76946,12572],{"className":76947,"style":254},[106,1478,144],[18,76949,152],{"className":76950},[151],[18,76952,76954],{"className":76953},[124],[18,76955,76957],{"className":76956,"style":159},[128],[18,76958],{},[18,76960],{"className":76961,"style":228},[169],[18,76963,4334],{"className":76964},[232],[18,76966],{"className":76967,"style":228},[169],[18,76969,76971,76974,77014,77017,77020],{"className":76970},[97],[18,76972],{"className":76973,"style":4563},[101],[18,76975,76977,76980],{"className":76976},[106],[18,76978,12579],{"className":76979},[106,110],[18,76981,76983],{"className":76982},[115],[18,76984,76986,77006],{"className":76985},[119,120],[18,76987,76989,77003],{"className":76988},[124],[18,76990,76992],{"className":76991,"style":1152},[128],[18,76993,76994,76997],{"style":1155},[18,76995],{"className":76996,"style":137},[136],[18,76998,77000],{"className":76999},[141,142,143,144],[18,77001,38925],{"className":77002,"style":38958},[106,110,144],[18,77004,152],{"className":77005},[151],[18,77007,77009],{"className":77008},[124],[18,77010,77012],{"className":77011,"style":159},[128],[18,77013],{},[18,77015],{"className":77016,"style":2068},[169],[18,77018,7984],{"className":77019},[2072],[18,77021],{"className":77022,"style":2068},[169],[18,77024,77026,77030],{"className":77025},[97],[18,77027],{"className":77028,"style":77029},[101],"height:0.313em;",[18,77031,76903],{"className":77032},[174]," 是一个保证（guarantee），不是一个估计（estimate）。当界大于 ",[18,77035,77037,77050],{"className":77036,"translate":22},[21],[18,77038,77040],{"className":77039},[26],[28,77041,77042],{"xmlns":30},[32,77043,77044,77048],{},[35,77045,77046],{},[45,77047,47],{},[86,77049,47],{"encoding":88},[18,77051,77053],{"className":77052,"ariaHidden":51},[93],[18,77054,77056,77059],{"className":77055},[97],[18,77057],{"className":77058,"style":505},[101],[18,77060,47],{"className":77061},[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。",[11,77064,77065,77066,77069],{},"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",[717,77067,77068],{},"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。",[77071,77072,77073],"div",{},[77074,77075],"img",{"height":77076,"width":77077,"style":77078,"data-d2-theme":77079,"alt":77080,"src":77081},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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",[638,77083,77085],{"id":77084},"六深度学习中的-vc-维","六、深度学习中的 VC 维",[1311,77087,77089],{"id":77088},"_61-参数量估算","6.1 参数量估算",[11,77091,77092,77093,77189,77190,77218,77219,77248,77249,77281],{},"把上一节的界套到实际模型上，只需估算 ",[18,77094,77096,77118],{"className":77095,"translate":22},[21],[18,77097,77099],{"className":77098},[26],[28,77100,77101],{"xmlns":30},[32,77102,77103,77115],{},[35,77104,77105],{},[12558,77106,77107],{},[35,77108,77109,77111,77113],{},[41,77110,4031],{},[41,77112,317],{"mathvariant":316},[41,77114,28160],{},[86,77116,77117],{"encoding":88},"\\sqrt{d\u002Fm}",[18,77119,77121],{"className":77120,"ariaHidden":51},[93],[18,77122,77124,77128],{"className":77123},[97],[18,77125],{"className":77126,"style":77127},[101],"height:1.24em;vertical-align:-0.305em;",[18,77129,77131],{"className":77130},[106,12693],[18,77132,77134,77180],{"className":77133},[119,120],[18,77135,77137,77177],{"className":77136},[124],[18,77138,77141,77160],{"className":77139,"style":77140},[128],"height:0.935em;",[18,77142,77145,77148],{"className":77143,"style":77144},[12707],"top:-3.2em;",[18,77146],{"className":77147,"style":6122},[136],[18,77149,77151,77154,77157],{"className":77150,"style":12716},[106],[18,77152,4031],{"className":77153},[106,110],[18,77155,317],{"className":77156},[106],[18,77158,28160],{"className":77159},[106,110],[18,77161,77163,77166],{"style":77162},"top:-2.895em;",[18,77164],{"className":77165,"style":6122},[136],[18,77167,77170],{"className":77168,"style":77169},[12850],"min-width:1.02em;height:1.28em;",[6127,77171,77174],{"xmlns":6129,"width":12854,"height":77172,"viewBox":77173,"preserveAspectRatio":12857},"1.28em","0 0 400000 1296",[6133,77175],{"d":77176},"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",[18,77178,152],{"className":77179},[151],[18,77181,77183],{"className":77182},[124],[18,77184,77187],{"className":77185,"style":77186},[128],"height:0.305em;",[18,77188],{}," 的量级（暂时把 ",[18,77191,77193,77206],{"className":77192,"translate":22},[21],[18,77194,77196],{"className":77195},[26],[28,77197,77198],{"xmlns":30},[32,77199,77200,77204],{},[35,77201,77202],{},[41,77203,4031],{},[86,77205,4031],{"encoding":88},[18,77207,77209],{"className":77208,"ariaHidden":51},[93],[18,77210,77212,77215],{"className":77211},[97],[18,77213],{"className":77214,"style":295},[101],[18,77216,4031],{"className":77217},[106,110]," 低估成参数量 ",[18,77220,77222,77236],{"className":77221,"translate":22},[21],[18,77223,77225],{"className":77224},[26],[28,77226,77227],{"xmlns":30},[32,77228,77229,77234],{},[35,77230,77231],{},[41,77232,77233],{},"W",[86,77235,77233],{"encoding":88},[18,77237,77239],{"className":77238,"ariaHidden":51},[93],[18,77240,77242,77245],{"className":77241},[97],[18,77243],{"className":77244,"style":600},[101],[18,77246,77233],{"className":77247,"style":1171},[106,110],"，忽略 ",[18,77250,77252,77267],{"className":77251,"translate":22},[21],[18,77253,77255],{"className":77254},[26],[28,77256,77257],{"xmlns":30},[32,77258,77259,77265],{},[35,77260,77261,77263],{},[41,77262,14730],{},[49,77264,899],{},[86,77266,27765],{"encoding":88},[18,77268,77270],{"className":77269,"ariaHidden":51},[93],[18,77271,77273,77276],{"className":77272},[97],[18,77274],{"className":77275,"style":2890},[101],[18,77277,15171,77279],{"className":77278},[985],[18,77280,15174],{"style":6352}," 因子）：",[43703,77283,77284,77442],{},[43706,77285,77286],{},[43709,77287,77288,77291,77322,77353],{},[43712,77289,77290],{},"模型",[43712,77292,77293,77294],{},"参数量 ",[18,77295,77297,77310],{"className":77296,"translate":22},[21],[18,77298,77300],{"className":77299},[26],[28,77301,77302],{"xmlns":30},[32,77303,77304,77308],{},[35,77305,77306],{},[41,77307,77233],{},[86,77309,77233],{"encoding":88},[18,77311,77313],{"className":77312,"ariaHidden":51},[93],[18,77314,77316,77319],{"className":77315},[97],[18,77317],{"className":77318,"style":600},[101],[18,77320,77233],{"className":77321,"style":1171},[106,110],[43712,77323,77324,77325],{},"训练样本 ",[18,77326,77328,77341],{"className":77327,"translate":22},[21],[18,77329,77331],{"className":77330},[26],[28,77332,77333],{"xmlns":30},[32,77334,77335,77339],{},[35,77336,77337],{},[41,77338,28160],{},[86,77340,28160],{"encoding":88},[18,77342,77344],{"className":77343,"ariaHidden":51},[93],[18,77345,77347,77350],{"className":77346},[97],[18,77348],{"className":77349,"style":1083},[101],[18,77351,28160],{"className":77352},[106,110],[43712,77354,77355],{},[18,77356,77358,77380],{"className":77357,"translate":22},[21],[18,77359,77361],{"className":77360},[26],[28,77362,77363],{"xmlns":30},[32,77364,77365,77377],{},[35,77366,77367],{},[12558,77368,77369],{},[35,77370,77371,77373,77375],{},[41,77372,77233],{},[41,77374,317],{"mathvariant":316},[41,77376,28160],{},[86,77378,77379],{"encoding":88},"\\sqrt{W\u002Fm}",[18,77381,77383],{"className":77382,"ariaHidden":51},[93],[18,77384,77386,77389],{"className":77385},[97],[18,77387],{"className":77388,"style":77127},[101],[18,77390,77392],{"className":77391},[106,12693],[18,77393,77395,77434],{"className":77394},[119,120],[18,77396,77398,77431],{"className":77397},[124],[18,77399,77401,77419],{"className":77400,"style":77140},[128],[18,77402,77404,77407],{"className":77403,"style":77144},[12707],[18,77405],{"className":77406,"style":6122},[136],[18,77408,77410,77413,77416],{"className":77409,"style":12716},[106],[18,77411,77233],{"className":77412,"style":1171},[106,110],[18,77414,317],{"className":77415},[106],[18,77417,28160],{"className":77418},[106,110],[18,77420,77421,77424],{"style":77162},[18,77422],{"className":77423,"style":6122},[136],[18,77425,77427],{"className":77426,"style":77169},[12850],[6127,77428,77429],{"xmlns":6129,"width":12854,"height":77172,"viewBox":77173,"preserveAspectRatio":12857},[6133,77430],{"d":77176},[18,77432,152],{"className":77433},[151],[18,77435,77437],{"className":77436},[124],[18,77438,77440],{"className":77439,"style":77186},[128],[18,77441],{},[43722,77443,77444,77668,77894,78117],{},[43709,77445,77446,77449,77537,77622],{},[43727,77447,77448],{},"LeNet-5（MNIST）",[43727,77450,77451],{},[18,77452,77454,77478],{"className":77453,"translate":22},[21],[18,77455,77457],{"className":77456},[26],[28,77458,77459],{"xmlns":30},[32,77460,77461,77475],{},[35,77462,77463,77466,77468],{},[45,77464,77465],{},"6",[49,77467,2015],{},[1556,77469,77470,77473],{},[45,77471,77472],{},"10",[45,77474,10444],{},[86,77476,77477],{"encoding":88},"6\\times10^4",[18,77479,77481,77499],{"className":77480,"ariaHidden":51},[93],[18,77482,77484,77487,77490,77493,77496],{"className":77483},[97],[18,77485],{"className":77486,"style":18090},[101],[18,77488,77465],{"className":77489},[106],[18,77491],{"className":77492,"style":2068},[169],[18,77494,2015],{"className":77495},[2072],[18,77497],{"className":77498,"style":2068},[169],[18,77500,77502,77505,77508],{"className":77501},[97],[18,77503],{"className":77504,"style":2416},[101],[18,77506,47],{"className":77507},[106],[18,77509,77511,77514],{"className":77510},[106],[18,77512,430],{"className":77513},[106],[18,77515,77517],{"className":77516},[115],[18,77518,77520],{"className":77519},[119],[18,77521,77523],{"className":77522},[124],[18,77524,77526],{"className":77525,"style":2416},[128],[18,77527,77528,77531],{"style":1962},[18,77529],{"className":77530,"style":137},[136],[18,77532,77534],{"className":77533},[141,142,143,144],[18,77535,10444],{"className":77536},[106,144],[43727,77538,77539],{},[18,77540,77542,77563],{"className":77541,"translate":22},[21],[18,77543,77545],{"className":77544},[26],[28,77546,77547],{"xmlns":30},[32,77548,77549,77561],{},[35,77550,77551,77553,77555],{},[45,77552,77465],{},[49,77554,2015],{},[1556,77556,77557,77559],{},[45,77558,77472],{},[45,77560,10444],{},[86,77562,77477],{"encoding":88},[18,77564,77566,77584],{"className":77565,"ariaHidden":51},[93],[18,77567,77569,77572,77575,77578,77581],{"className":77568},[97],[18,77570],{"className":77571,"style":18090},[101],[18,77573,77465],{"className":77574},[106],[18,77576],{"className":77577,"style":2068},[169],[18,77579,2015],{"className":77580},[2072],[18,77582],{"className":77583,"style":2068},[169],[18,77585,77587,77590,77593],{"className":77586},[97],[18,77588],{"className":77589,"style":2416},[101],[18,77591,47],{"className":77592},[106],[18,77594,77596,77599],{"className":77595},[106],[18,77597,430],{"className":77598},[106],[18,77600,77602],{"className":77601},[115],[18,77603,77605],{"className":77604},[119],[18,77606,77608],{"className":77607},[124],[18,77609,77611],{"className":77610,"style":2416},[128],[18,77612,77613,77616],{"style":1962},[18,77614],{"className":77615,"style":137},[136],[18,77617,77619],{"className":77618},[141,142,143,144],[18,77620,10444],{"className":77621},[106,144],[43727,77623,77624],{},[18,77625,77627,77643],{"className":77626,"translate":22},[21],[18,77628,77630],{"className":77629},[26],[28,77631,77632],{"xmlns":30},[32,77633,77634,77640],{},[35,77635,77636,77638],{},[49,77637,14751],{},[45,77639,47],{},[86,77641,77642],{"encoding":88},"\\approx 1",[18,77644,77646,77659],{"className":77645,"ariaHidden":51},[93],[18,77647,77649,77653,77656],{"className":77648},[97],[18,77650],{"className":77651,"style":77652},[101],"height:0.4831em;",[18,77654,14751],{"className":77655},[232],[18,77657],{"className":77658,"style":228},[169],[18,77660,77662,77665],{"className":77661},[97],[18,77663],{"className":77664,"style":505},[101],[18,77666,47],{"className":77667},[106],[43709,77669,77670,77673,77761,77848],{},[43727,77671,77672],{},"ResNet-18（ImageNet）",[43727,77674,77675],{},[18,77676,77678,77702],{"className":77677,"translate":22},[21],[18,77679,77681],{"className":77680},[26],[28,77682,77683],{"xmlns":30},[32,77684,77685,77699],{},[35,77686,77687,77690,77692],{},[45,77688,77689],{},"1.17",[49,77691,2015],{},[1556,77693,77694,77696],{},[45,77695,77472],{},[45,77697,77698],{},"7",[86,77700,77701],{"encoding":88},"1.17\\times10^7",[18,77703,77705,77723],{"className":77704,"ariaHidden":51},[93],[18,77706,77708,77711,77714,77717,77720],{"className":77707},[97],[18,77709],{"className":77710,"style":18090},[101],[18,77712,77689],{"className":77713},[106],[18,77715],{"className":77716,"style":2068},[169],[18,77718,2015],{"className":77719},[2072],[18,77721],{"className":77722,"style":2068},[169],[18,77724,77726,77729,77732],{"className":77725},[97],[18,77727],{"className":77728,"style":2416},[101],[18,77730,47],{"className":77731},[106],[18,77733,77735,77738],{"className":77734},[106],[18,77736,430],{"className":77737},[106],[18,77739,77741],{"className":77740},[115],[18,77742,77744],{"className":77743},[119],[18,77745,77747],{"className":77746},[124],[18,77748,77750],{"className":77749,"style":2416},[128],[18,77751,77752,77755],{"style":1962},[18,77753],{"className":77754,"style":137},[136],[18,77756,77758],{"className":77757},[141,142,143,144],[18,77759,77698],{"className":77760},[106,144],[43727,77762,77763],{},[18,77764,77766,77789],{"className":77765,"translate":22},[21],[18,77767,77769],{"className":77768},[26],[28,77770,77771],{"xmlns":30},[32,77772,77773,77786],{},[35,77774,77775,77778,77780],{},[45,77776,77777],{},"1.28",[49,77779,2015],{},[1556,77781,77782,77784],{},[45,77783,77472],{},[45,77785,77465],{},[86,77787,77788],{"encoding":88},"1.28\\times10^6",[18,77790,77792,77810],{"className":77791,"ariaHidden":51},[93],[18,77793,77795,77798,77801,77804,77807],{"className":77794},[97],[18,77796],{"className":77797,"style":18090},[101],[18,77799,77777],{"className":77800},[106],[18,77802],{"className":77803,"style":2068},[169],[18,77805,2015],{"className":77806},[2072],[18,77808],{"className":77809,"style":2068},[169],[18,77811,77813,77816,77819],{"className":77812},[97],[18,77814],{"className":77815,"style":2416},[101],[18,77817,47],{"className":77818},[106],[18,77820,77822,77825],{"className":77821},[106],[18,77823,430],{"className":77824},[106],[18,77826,77828],{"className":77827},[115],[18,77829,77831],{"className":77830},[119],[18,77832,77834],{"className":77833},[124],[18,77835,77837],{"className":77836,"style":2416},[128],[18,77838,77839,77842],{"style":1962},[18,77840],{"className":77841,"style":137},[136],[18,77843,77845],{"className":77844},[141,142,143,144],[18,77846,77465],{"className":77847},[106,144],[43727,77849,77850],{},[18,77851,77853,77870],{"className":77852,"translate":22},[21],[18,77854,77856],{"className":77855},[26],[28,77857,77858],{"xmlns":30},[32,77859,77860,77867],{},[35,77861,77862,77864],{},[49,77863,14751],{},[45,77865,77866],{},"3.0",[86,77868,77869],{"encoding":88},"\\approx 3.0",[18,77871,77873,77885],{"className":77872,"ariaHidden":51},[93],[18,77874,77876,77879,77882],{"className":77875},[97],[18,77877],{"className":77878,"style":77652},[101],[18,77880,14751],{"className":77881},[232],[18,77883],{"className":77884,"style":228},[169],[18,77886,77888,77891],{"className":77887},[97],[18,77889],{"className":77890,"style":505},[101],[18,77892,77866],{"className":77893},[106],[43709,77895,77896,77899,77986,78071],{},[43727,77897,77898],{},"ResNet-50（ImageNet）",[43727,77900,77901],{},[18,77902,77904,77927],{"className":77903,"translate":22},[21],[18,77905,77907],{"className":77906},[26],[28,77908,77909],{"xmlns":30},[32,77910,77911,77924],{},[35,77912,77913,77916,77918],{},[45,77914,77915],{},"2.56",[49,77917,2015],{},[1556,77919,77920,77922],{},[45,77921,77472],{},[45,77923,77698],{},[86,77925,77926],{"encoding":88},"2.56\\times10^7",[18,77928,77930,77948],{"className":77929,"ariaHidden":51},[93],[18,77931,77933,77936,77939,77942,77945],{"className":77932},[97],[18,77934],{"className":77935,"style":18090},[101],[18,77937,77915],{"className":77938},[106],[18,77940],{"className":77941,"style":2068},[169],[18,77943,2015],{"className":77944},[2072],[18,77946],{"className":77947,"style":2068},[169],[18,77949,77951,77954,77957],{"className":77950},[97],[18,77952],{"className":77953,"style":2416},[101],[18,77955,47],{"className":77956},[106],[18,77958,77960,77963],{"className":77959},[106],[18,77961,430],{"className":77962},[106],[18,77964,77966],{"className":77965},[115],[18,77967,77969],{"className":77968},[119],[18,77970,77972],{"className":77971},[124],[18,77973,77975],{"className":77974,"style":2416},[128],[18,77976,77977,77980],{"style":1962},[18,77978],{"className":77979,"style":137},[136],[18,77981,77983],{"className":77982},[141,142,143,144],[18,77984,77698],{"className":77985},[106,144],[43727,77987,77988],{},[18,77989,77991,78012],{"className":77990,"translate":22},[21],[18,77992,77994],{"className":77993},[26],[28,77995,77996],{"xmlns":30},[32,77997,77998,78010],{},[35,77999,78000,78002,78004],{},[45,78001,77777],{},[49,78003,2015],{},[1556,78005,78006,78008],{},[45,78007,77472],{},[45,78009,77465],{},[86,78011,77788],{"encoding":88},[18,78013,78015,78033],{"className":78014,"ariaHidden":51},[93],[18,78016,78018,78021,78024,78027,78030],{"className":78017},[97],[18,78019],{"className":78020,"style":18090},[101],[18,78022,77777],{"className":78023},[106],[18,78025],{"className":78026,"style":2068},[169],[18,78028,2015],{"className":78029},[2072],[18,78031],{"className":78032,"style":2068},[169],[18,78034,78036,78039,78042],{"className":78035},[97],[18,78037],{"className":78038,"style":2416},[101],[18,78040,47],{"className":78041},[106],[18,78043,78045,78048],{"className":78044},[106],[18,78046,430],{"className":78047},[106],[18,78049,78051],{"className":78050},[115],[18,78052,78054],{"className":78053},[119],[18,78055,78057],{"className":78056},[124],[18,78058,78060],{"className":78059,"style":2416},[128],[18,78061,78062,78065],{"style":1962},[18,78063],{"className":78064,"style":137},[136],[18,78066,78068],{"className":78067},[141,142,143,144],[18,78069,77465],{"className":78070},[106,144],[43727,78072,78073],{},[18,78074,78076,78093],{"className":78075,"translate":22},[21],[18,78077,78079],{"className":78078},[26],[28,78080,78081],{"xmlns":30},[32,78082,78083,78090],{},[35,78084,78085,78087],{},[49,78086,14751],{},[45,78088,78089],{},"4.5",[86,78091,78092],{"encoding":88},"\\approx 4.5",[18,78094,78096,78108],{"className":78095,"ariaHidden":51},[93],[18,78097,78099,78102,78105],{"className":78098},[97],[18,78100],{"className":78101,"style":77652},[101],[18,78103,14751],{"className":78104},[232],[18,78106],{"className":78107,"style":228},[169],[18,78109,78111,78114],{"className":78110},[97],[18,78112],{"className":78113,"style":505},[101],[18,78115,78089],{"className":78116},[106],[43709,78118,78119,78122,78209,78294],{},[43727,78120,78121],{},"ViT-L\u002F16（ImageNet）",[43727,78123,78124],{},[18,78125,78127,78150],{"className":78126,"translate":22},[21],[18,78128,78130],{"className":78129},[26],[28,78131,78132],{"xmlns":30},[32,78133,78134,78147],{},[35,78135,78136,78139,78141],{},[45,78137,78138],{},"3.07",[49,78140,2015],{},[1556,78142,78143,78145],{},[45,78144,77472],{},[45,78146,36429],{},[86,78148,78149],{"encoding":88},"3.07\\times10^8",[18,78151,78153,78171],{"className":78152,"ariaHidden":51},[93],[18,78154,78156,78159,78162,78165,78168],{"className":78155},[97],[18,78157],{"className":78158,"style":18090},[101],[18,78160,78138],{"className":78161},[106],[18,78163],{"className":78164,"style":2068},[169],[18,78166,2015],{"className":78167},[2072],[18,78169],{"className":78170,"style":2068},[169],[18,78172,78174,78177,78180],{"className":78173},[97],[18,78175],{"className":78176,"style":2416},[101],[18,78178,47],{"className":78179},[106],[18,78181,78183,78186],{"className":78182},[106],[18,78184,430],{"className":78185},[106],[18,78187,78189],{"className":78188},[115],[18,78190,78192],{"className":78191},[119],[18,78193,78195],{"className":78194},[124],[18,78196,78198],{"className":78197,"style":2416},[128],[18,78199,78200,78203],{"style":1962},[18,78201],{"className":78202,"style":137},[136],[18,78204,78206],{"className":78205},[141,142,143,144],[18,78207,36429],{"className":78208},[106,144],[43727,78210,78211],{},[18,78212,78214,78235],{"className":78213,"translate":22},[21],[18,78215,78217],{"className":78216},[26],[28,78218,78219],{"xmlns":30},[32,78220,78221,78233],{},[35,78222,78223,78225,78227],{},[45,78224,77777],{},[49,78226,2015],{},[1556,78228,78229,78231],{},[45,78230,77472],{},[45,78232,77465],{},[86,78234,77788],{"encoding":88},[18,78236,78238,78256],{"className":78237,"ariaHidden":51},[93],[18,78239,78241,78244,78247,78250,78253],{"className":78240},[97],[18,78242],{"className":78243,"style":18090},[101],[18,78245,77777],{"className":78246},[106],[18,78248],{"className":78249,"style":2068},[169],[18,78251,2015],{"className":78252},[2072],[18,78254],{"className":78255,"style":2068},[169],[18,78257,78259,78262,78265],{"className":78258},[97],[18,78260],{"className":78261,"style":2416},[101],[18,78263,47],{"className":78264},[106],[18,78266,78268,78271],{"className":78267},[106],[18,78269,430],{"className":78270},[106],[18,78272,78274],{"className":78273},[115],[18,78275,78277],{"className":78276},[119],[18,78278,78280],{"className":78279},[124],[18,78281,78283],{"className":78282,"style":2416},[128],[18,78284,78285,78288],{"style":1962},[18,78286],{"className":78287,"style":137},[136],[18,78289,78291],{"className":78290},[141,142,143,144],[18,78292,77465],{"className":78293},[106,144],[43727,78295,78296],{},[18,78297,78299,78316],{"className":78298,"translate":22},[21],[18,78300,78302],{"className":78301},[26],[28,78303,78304],{"xmlns":30},[32,78305,78306,78313],{},[35,78307,78308,78310],{},[49,78309,14751],{},[45,78311,78312],{},"15.5",[86,78314,78315],{"encoding":88},"\\approx 15.5",[18,78317,78319,78331],{"className":78318,"ariaHidden":51},[93],[18,78320,78322,78325,78328],{"className":78321},[97],[18,78323],{"className":78324,"style":77652},[101],[18,78326,14751],{"className":78327},[232],[18,78329],{"className":78330,"style":228},[169],[18,78332,78334,78337],{"className":78333},[97],[18,78335],{"className":78336,"style":505},[101],[18,78338,78312],{"className":78339},[106],[11,78341,78342,78343,78371,78372,78400,78401,78429,78430,78462],{},"每一个都在 ",[18,78344,78346,78359],{"className":78345,"translate":22},[21],[18,78347,78349],{"className":78348},[26],[28,78350,78351],{"xmlns":30},[32,78352,78353,78357],{},[35,78354,78355],{},[45,78356,47],{},[86,78358,47],{"encoding":88},[18,78360,78362],{"className":78361,"ariaHidden":51},[93],[18,78363,78365,78368],{"className":78364},[97],[18,78366],{"className":78367,"style":505},[101],[18,78369,47],{"className":78370},[106]," 附近或远大于 ",[18,78373,78375,78388],{"className":78374,"translate":22},[21],[18,78376,78378],{"className":78377},[26],[28,78379,78380],{"xmlns":30},[32,78381,78382,78386],{},[35,78383,78384],{},[45,78385,47],{},[86,78387,47],{"encoding":88},[18,78389,78391],{"className":78390,"ariaHidden":51},[93],[18,78392,78394,78397],{"className":78393},[97],[18,78395],{"className":78396,"style":505},[101],[18,78398,47],{"className":78399},[106],"。一个「错误率上界」一旦达到或超过 ",[18,78402,78404,78417],{"className":78403,"translate":22},[21],[18,78405,78407],{"className":78406},[26],[28,78408,78409],{"xmlns":30},[32,78410,78411,78415],{},[35,78412,78413],{},[45,78414,47],{},[86,78416,47],{"encoding":88},[18,78418,78420],{"className":78419,"ariaHidden":51},[93],[18,78421,78423,78426],{"className":78422},[97],[18,78424],{"className":78425,"style":505},[101],[18,78427,47],{"className":78428},[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 ",[18,78431,78433,78448],{"className":78432,"translate":22},[21],[18,78434,78436],{"className":78435},[26],[28,78437,78438],{"xmlns":30},[32,78439,78440,78446],{},[35,78441,78442,78444],{},[41,78443,14730],{},[49,78445,899],{},[86,78447,27765],{"encoding":88},[18,78449,78451],{"className":78450,"ariaHidden":51},[93],[18,78452,78454,78457],{"className":78453},[97],[18,78455],{"className":78456,"style":2890},[101],[18,78458,15171,78460],{"className":78459},[985],[18,78461,15174],{"style":6352}," 因子。",[11,78464,78465,78466,78494,78495,78523,78524,78585],{},"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，",[18,78467,78469,78482],{"className":78468,"translate":22},[21],[18,78470,78472],{"className":78471},[26],[28,78473,78474],{"xmlns":30},[32,78475,78476,78480],{},[35,78477,78478],{},[41,78479,28160],{},[86,78481,28160],{"encoding":88},[18,78483,78485],{"className":78484,"ariaHidden":51},[93],[18,78486,78488,78491],{"className":78487},[97],[18,78489],{"className":78490,"style":1083},[101],[18,78492,28160],{"className":78493},[106,110]," 比 ",[18,78496,78498,78511],{"className":78497,"translate":22},[21],[18,78499,78501],{"className":78500},[26],[28,78502,78503],{"xmlns":30},[32,78504,78505,78509],{},[35,78506,78507],{},[41,78508,4031],{},[86,78510,4031],{"encoding":88},[18,78512,78514],{"className":78513,"ariaHidden":51},[93],[18,78515,78517,78520],{"className":78516},[97],[18,78518],{"className":78519,"style":295},[101],[18,78521,4031],{"className":78522},[106,110]," 小二十倍。",[717,78525,78526,78527,78555,78556,78584],{},"任何以 ",[18,78528,78530,78543],{"className":78529,"translate":22},[21],[18,78531,78533],{"className":78532},[26],[28,78534,78535],{"xmlns":30},[32,78536,78537,78541],{},[35,78538,78539],{},[41,78540,4031],{},[86,78542,4031],{"encoding":88},[18,78544,78546],{"className":78545,"ariaHidden":51},[93],[18,78547,78549,78552],{"className":78548},[97],[18,78550],{"className":78551,"style":295},[101],[18,78553,4031],{"className":78554},[106,110]," 为主项的界，在这个区间里都只能给出与 ",[18,78557,78559,78572],{"className":78558,"translate":22},[21],[18,78560,78562],{"className":78561},[26],[28,78563,78564],{"xmlns":30},[32,78565,78566,78570],{},[35,78567,78568],{},[45,78569,47],{},[86,78571,47],{"encoding":88},[18,78573,78575],{"className":78574,"ariaHidden":51},[93],[18,78576,78578,78581],{"className":78577},[97],[18,78579],{"className":78580,"style":505},[101],[18,78582,47],{"className":78583},[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。",[1311,78587,78589],{"id":78588},"_62-深度网络的-vc-维","6.2 深度网络的 VC 维",[11,78591,78592],{},"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。",[11,78594,78595],{},"那么它有多大？",[2210,78597,78598,78760],{},[2213,78599,78600,78603,78604,32687,78669,78697,78698,78730,78731,78759],{},[717,78601,78602],{},"阈值网络。"," Baum 与 Haussler（1989）证明，由阈值门组成的多层网络，VC 维是 ",[18,78605,78607,78634],{"className":78606,"translate":22},[21],[18,78608,78610],{"className":78609},[26],[28,78611,78612],{"xmlns":30},[32,78613,78614,78631],{},[35,78615,78616,78619,78621,78623,78625,78627,78629],{},[41,78617,78618],{},"O",[49,78620,74],{"stretchy":73},[41,78622,77233],{},[41,78624,14730],{},[49,78626,899],{},[41,78628,77233],{},[49,78630,84],{"stretchy":73},[86,78632,78633],{"encoding":88},"O(W\\log W)",[18,78635,78637],{"className":78636,"ariaHidden":51},[93],[18,78638,78640,78643,78646,78649,78652,78655,78660,78663,78666],{"className":78639},[97],[18,78641],{"className":78642,"style":242},[101],[18,78644,78618],{"className":78645,"style":254},[106,110],[18,78647,74],{"className":78648},[250],[18,78650,77233],{"className":78651,"style":1171},[106,110],[18,78653],{"className":78654,"style":170},[169],[18,78656,15171,78658],{"className":78657},[985],[18,78659,15174],{"style":6352},[18,78661],{"className":78662,"style":170},[169],[18,78664,77233],{"className":78665,"style":1171},[106,110],[18,78667,84],{"className":78668},[267],[18,78670,78672,78685],{"className":78671,"translate":22},[21],[18,78673,78675],{"className":78674},[26],[28,78676,78677],{"xmlns":30},[32,78678,78679,78683],{},[35,78680,78681],{},[41,78682,77233],{},[86,78684,77233],{"encoding":88},[18,78686,78688],{"className":78687,"ariaHidden":51},[93],[18,78689,78691,78694],{"className":78690},[97],[18,78692],{"className":78693,"style":600},[101],[18,78695,77233],{"className":78696,"style":1171},[106,110]," 是权重个数。",[18,78699,78701,78716],{"className":78700,"translate":22},[21],[18,78702,78704],{"className":78703},[26],[28,78705,78706],{"xmlns":30},[32,78707,78708,78714],{},[35,78709,78710,78712],{},[41,78711,14730],{},[49,78713,899],{},[86,78715,27765],{"encoding":88},[18,78717,78719],{"className":78718,"ariaHidden":51},[93],[18,78720,78722,78725],{"className":78721},[97],[18,78723],{"className":78724,"style":2890},[101],[18,78726,15171,78728],{"className":78727},[985],[18,78729,15174],{"style":6352}," 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 ",[18,78732,78734,78747],{"className":78733,"translate":22},[21],[18,78735,78737],{"className":78736},[26],[28,78738,78739],{"xmlns":30},[32,78740,78741,78745],{},[35,78742,78743],{},[41,78744,77233],{},[86,78746,77233],{"encoding":88},[18,78748,78750],{"className":78749,"ariaHidden":51},[93],[18,78751,78753,78756],{"className":78752},[97],[18,78754],{"className":78755,"style":600},[101],[18,78757,77233],{"className":78758,"style":1171},[106,110]," 组合增长。",[2213,78761,78762,78765,78766,78835,78836,78864,78865,78949],{},[717,78763,78764],{},"ReLU 网络。"," Bartlett、Harvey、Liaw 与 Mehrabian（2019）给出 ",[18,78767,78769,78797],{"className":78768,"translate":22},[21],[18,78770,78772],{"className":78771},[26],[28,78773,78774],{"xmlns":30},[32,78775,78776,78794],{},[35,78777,78778,78780,78782,78784,78786,78788,78790,78792],{},[41,78779,78618],{},[49,78781,74],{"stretchy":73},[41,78783,77233],{},[41,78785,12579],{},[41,78787,14730],{},[49,78789,899],{},[41,78791,77233],{},[49,78793,84],{"stretchy":73},[86,78795,78796],{"encoding":88},"O(WL\\log W)",[18,78798,78800],{"className":78799,"ariaHidden":51},[93],[18,78801,78803,78806,78809,78812,78815,78818,78821,78826,78829,78832],{"className":78802},[97],[18,78804],{"className":78805,"style":242},[101],[18,78807,78618],{"className":78808,"style":254},[106,110],[18,78810,74],{"className":78811},[250],[18,78813,77233],{"className":78814,"style":1171},[106,110],[18,78816,12579],{"className":78817},[106,110],[18,78819],{"className":78820,"style":170},[169],[18,78822,15171,78824],{"className":78823},[985],[18,78825,15174],{"style":6352},[18,78827],{"className":78828,"style":170},[169],[18,78830,77233],{"className":78831,"style":1171},[106,110],[18,78833,84],{"className":78834},[267]," 的上界（",[18,78837,78839,78852],{"className":78838,"translate":22},[21],[18,78840,78842],{"className":78841},[26],[28,78843,78844],{"xmlns":30},[32,78845,78846,78850],{},[35,78847,78848],{},[41,78849,12579],{},[86,78851,12579],{"encoding":88},[18,78853,78855],{"className":78854,"ariaHidden":51},[93],[18,78856,78858,78861],{"className":78857},[97],[18,78859],{"className":78860,"style":600},[101],[18,78862,12579],{"className":78863},[106,110]," 为层数），并给出几乎匹配的下界 ",[18,78866,78868,78905],{"className":78867,"translate":22},[21],[18,78869,78871],{"className":78870},[26],[28,78872,78873],{"xmlns":30},[32,78874,78875,78902],{},[35,78876,78877,78880,78882,78884,78886,78888,78890,78892,78894,78896,78898,78900],{},[41,78878,78879],{"mathvariant":316},"Ω",[49,78881,74],{"stretchy":73},[41,78883,77233],{},[41,78885,12579],{},[41,78887,14730],{},[49,78889,899],{},[49,78891,74],{"stretchy":73},[41,78893,77233],{},[41,78895,317],{"mathvariant":316},[41,78897,12579],{},[49,78899,84],{"stretchy":73},[49,78901,84],{"stretchy":73},[86,78903,78904],{"encoding":88},"\\Omega(WL\\log(W\u002FL))",[18,78906,78908],{"className":78907,"ariaHidden":51},[93],[18,78909,78911,78914,78917,78920,78923,78926,78929,78934,78937,78940,78943,78946],{"className":78910},[97],[18,78912],{"className":78913,"style":242},[101],[18,78915,78879],{"className":78916},[106],[18,78918,74],{"className":78919},[250],[18,78921,77233],{"className":78922,"style":1171},[106,110],[18,78924,12579],{"className":78925},[106,110],[18,78927],{"className":78928,"style":170},[169],[18,78930,15171,78932],{"className":78931},[985],[18,78933,15174],{"style":6352},[18,78935,74],{"className":78936},[250],[18,78938,77233],{"className":78939,"style":1171},[106,110],[18,78941,317],{"className":78942},[106],[18,78944,12579],{"className":78945},[106,110],[18,78947,3431],{"className":78948},[267],"；伪维（pseudo-dimension）同阶。",[11,78951,78952,78953,78989],{},"结论很清楚：",[717,78954,78955,78956,78988],{},"深度网络的 VC 维随参数线性增长（乘一个 ",[18,78957,78959,78974],{"className":78958,"translate":22},[21],[18,78960,78962],{"className":78961},[26],[28,78963,78964],{"xmlns":30},[32,78965,78966,78972],{},[35,78967,78968,78970],{},[41,78969,14730],{},[49,78971,899],{},[86,78973,27765],{"encoding":88},[18,78975,78977],{"className":78976,"ariaHidden":51},[93],[18,78978,78980,78983],{"className":78979},[97],[18,78981],{"className":78982,"style":2890},[101],[18,78984,15171,78986],{"className":78985},[985],[18,78987,15174],{"style":6352},"）。"," 「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。",[11,78991,78992],{},"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。",[1311,78994,78996],{"id":78995},"_63-三条出路","6.3 三条出路",[11,78998,78999],{},"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。",[11,79001,79002],{},[717,79003,79004],{},"（a）把容量做得更细：从假设类到算法输出",[11,79006,79007,79008,79036],{},"与其用整个 ",[18,79009,79011,79024],{"className":79010,"translate":22},[21],[18,79012,79014],{"className":79013},[26],[28,79015,79016],{"xmlns":30},[32,79017,79018,79022],{},[35,79019,79020],{},[41,79021,18304],{"mathvariant":1462},[86,79023,46308],{"encoding":88},[18,79025,79027],{"className":79026,"ariaHidden":51},[93],[18,79028,79030,79033],{"className":79029},[97],[18,79031],{"className":79032,"style":600},[101],[18,79034,18304],{"className":79035,"style":46321},[106,1478]," 的 VC 维，不如用「训练解所在的那个小子类」的容量。",[11,79038,79039,79040,16797,79101,79162,79163,79194,79195,79279,79280,79308],{},"线性情形是最好的例子。若 ",[18,79041,79043,79065],{"className":79042,"translate":22},[21],[18,79044,79046],{"className":79045},[26],[28,79047,79048],{"xmlns":30},[32,79049,79050,79062],{},[35,79051,79052,79054,79056,79058,79060],{},[49,79053,2248],{"stretchy":73},[41,79055,38573],{},[49,79057,2248],{"stretchy":73},[49,79059,4334],{},[41,79061,3071],{},[86,79063,79064],{"encoding":88},"\\lVert x\\rVert\\le R",[18,79066,79068,79092],{"className":79067,"ariaHidden":51},[93],[18,79069,79071,79074,79077,79080,79083,79086,79089],{"className":79070},[97],[18,79072],{"className":79073,"style":242},[101],[18,79075,2248],{"className":79076},[250],[18,79078,38573],{"className":79079},[106,110],[18,79081,2248],{"className":79082},[267],[18,79084],{"className":79085,"style":228},[169],[18,79087,4334],{"className":79088},[232],[18,79090],{"className":79091,"style":228},[169],[18,79093,79095,79098],{"className":79094},[97],[18,79096],{"className":79097,"style":600},[101],[18,79099,3071],{"className":79100,"style":3152},[106,110],[18,79102,79104,79126],{"className":79103,"translate":22},[21],[18,79105,79107],{"className":79106},[26],[28,79108,79109],{"xmlns":30},[32,79110,79111,79123],{},[35,79112,79113,79115,79117,79119,79121],{},[49,79114,2248],{"stretchy":73},[41,79116,55912],{},[49,79118,2248],{"stretchy":73},[49,79120,4334],{},[41,79122,3978],{},[86,79124,79125],{"encoding":88},"\\lVert w\\rVert\\le B",[18,79127,79129,79153],{"className":79128,"ariaHidden":51},[93],[18,79130,79132,79135,79138,79141,79144,79147,79150],{"className":79131},[97],[18,79133],{"className":79134,"style":242},[101],[18,79136,2248],{"className":79137},[250],[18,79139,55912],{"className":79140,"style":55988},[106,110],[18,79142,2248],{"className":79143},[267],[18,79145],{"className":79146,"style":228},[169],[18,79148,4334],{"className":79149},[232],[18,79151],{"className":79152,"style":228},[169],[18,79154,79156,79159],{"className":79155},[97],[18,79157],{"className":79158,"style":600},[101],[18,79160,3978],{"className":79161,"style":11302},[106,110],"，且分类间隔至少为 ",[18,79164,79166,79181],{"className":79165,"translate":22},[21],[18,79167,79169],{"className":79168},[26],[28,79170,79171],{"xmlns":30},[32,79172,79173,79178],{},[35,79174,79175],{},[41,79176,79177],{},"γ",[86,79179,79180],{"encoding":88},"\\gamma",[18,79182,79184],{"className":79183,"ariaHidden":51},[93],[18,79185,79187,79190],{"className":79186},[97],[18,79188],{"className":79189,"style":38543},[101],[18,79191,79177],{"className":79192,"style":79193},[106,110],"margin-right:0.0556em;","，则有效容量约为 ",[18,79196,79198,79226],{"className":79197,"translate":22},[21],[18,79199,79201],{"className":79200},[26],[28,79202,79203],{"xmlns":30},[32,79204,79205,79223],{},[35,79206,79207,79209,79211,79213,79215,79217],{},[49,79208,74],{"stretchy":73},[41,79210,3071],{},[41,79212,3978],{},[41,79214,317],{"mathvariant":316},[41,79216,79177],{},[1556,79218,79219,79221],{},[49,79220,84],{"stretchy":73},[45,79222,2266],{},[86,79224,79225],{"encoding":88},"(RB\u002F\\gamma)^2",[18,79227,79229],{"className":79228,"ariaHidden":51},[93],[18,79230,79232,79235,79238,79241,79244,79247,79250],{"className":79231},[97],[18,79233],{"className":79234,"style":10268},[101],[18,79236,74],{"className":79237},[250],[18,79239,3071],{"className":79240,"style":3152},[106,110],[18,79242,3978],{"className":79243,"style":11302},[106,110],[18,79245,317],{"className":79246},[106],[18,79248,79177],{"className":79249,"style":79193},[106,110],[18,79251,79253,79256],{"className":79252},[267],[18,79254,84],{"className":79255},[267],[18,79257,79259],{"className":79258},[115],[18,79260,79262],{"className":79261},[119],[18,79263,79265],{"className":79264},[124],[18,79266,79268],{"className":79267,"style":2416},[128],[18,79269,79270,79273],{"style":1962},[18,79271],{"className":79272,"style":137},[136],[18,79274,79276],{"className":79275},[141,142,143,144],[18,79277,2266],{"className":79278},[106,144],"，与输入维度 ",[18,79281,79283,79296],{"className":79282,"translate":22},[21],[18,79284,79286],{"className":79285},[26],[28,79287,79288],{"xmlns":30},[32,79289,79290,79294],{},[35,79291,79292],{},[41,79293,4031],{},[86,79295,4031],{"encoding":88},[18,79297,79299],{"className":79298,"ariaHidden":51},[93],[18,79300,79302,79305],{"className":79301},[97],[18,79303],{"className":79304,"style":295},[101],[18,79306,4031],{"className":79307},[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。",[11,79310,79311,79312,79351,79352,79524],{},"深度网络有类似的谱范数版本。把 ",[18,79313,79315,79333],{"className":79314,"translate":22},[21],[18,79316,79318],{"className":79317},[26],[28,79319,79320],{"xmlns":30},[32,79321,79322,79330],{},[35,79323,79324,79326,79328],{},[49,79325,2248],{"stretchy":73},[41,79327,55912],{},[49,79329,2248],{"stretchy":73},[86,79331,79332],{"encoding":88},"\\lVert w\\rVert",[18,79334,79336],{"className":79335,"ariaHidden":51},[93],[18,79337,79339,79342,79345,79348],{"className":79338},[97],[18,79340],{"className":79341,"style":242},[101],[18,79343,2248],{"className":79344},[250],[18,79346,55912],{"className":79347,"style":55988},[106,110],[18,79349,2248],{"className":79350},[267]," 换成各层谱范数的乘积 ",[18,79353,79355,79389],{"className":79354,"translate":22},[21],[18,79356,79358],{"className":79357},[26],[28,79359,79360],{"xmlns":30},[32,79361,79362,79386],{},[35,79363,79364,79372,79374,79380],{},[38,79365,79366,79369],{},[49,79367,79368],{},"∏",[41,79370,79371],{},"l",[49,79373,2248],{"stretchy":73},[38,79375,79376,79378],{},[41,79377,77233],{},[41,79379,79371],{},[38,79381,79382,79384],{},[49,79383,2248],{"stretchy":73},[41,79385,16403],{},[86,79387,79388],{"encoding":88},"\\prod_l\\lVert W_l\\rVert_\\sigma",[18,79390,79392],{"className":79391,"ariaHidden":51},[93],[18,79393,79395,79399,79441,79444,79484],{"className":79394},[97],[18,79396],{"className":79397,"style":79398},[101],"height:1.0497em;vertical-align:-0.2997em;",[18,79400,79402,79405],{"className":79401},[985],[18,79403,79368],{"className":79404,"style":27579},[985,4198,27578],[18,79406,79408],{"className":79407},[115],[18,79409,79411,79433],{"className":79410},[119,120],[18,79412,79414,79430],{"className":79413},[124],[18,79415,79418],{"className":79416,"style":79417},[128],"height:0.1864em;",[18,79419,79420,79423],{"style":27595},[18,79421],{"className":79422,"style":137},[136],[18,79424,79426],{"className":79425},[141,142,143,144],[18,79427,79371],{"className":79428,"style":79429},[106,110,144],"margin-right:0.0197em;",[18,79431,152],{"className":79432},[151],[18,79434,79436],{"className":79435},[124],[18,79437,79439],{"className":79438,"style":27614},[128],[18,79440],{},[18,79442,2248],{"className":79443},[250],[18,79445,79447,79450],{"className":79446},[106],[18,79448,77233],{"className":79449,"style":1171},[106,110],[18,79451,79453],{"className":79452},[115],[18,79454,79456,79476],{"className":79455},[119,120],[18,79457,79459,79473],{"className":79458},[124],[18,79460,79462],{"className":79461,"style":1383},[128],[18,79463,79464,79467],{"style":1386},[18,79465],{"className":79466,"style":137},[136],[18,79468,79470],{"className":79469},[141,142,143,144],[18,79471,79371],{"className":79472,"style":79429},[106,110,144],[18,79474,152],{"className":79475},[151],[18,79477,79479],{"className":79478},[124],[18,79480,79482],{"className":79481,"style":159},[128],[18,79483],{},[18,79485,79487,79490],{"className":79486},[267],[18,79488,2248],{"className":79489},[267],[18,79491,79493],{"className":79492},[115],[18,79494,79496,79516],{"className":79495},[119,120],[18,79497,79499,79513],{"className":79498},[124],[18,79500,79502],{"className":79501,"style":202},[128],[18,79503,79504,79507],{"style":1155},[18,79505],{"className":79506,"style":137},[136],[18,79508,79510],{"className":79509},[141,142,143,144],[18,79511,16403],{"className":79512,"style":3957},[106,110,144],[18,79514,152],{"className":79515},[151],[18,79517,79519],{"className":79518},[124],[18,79520,79522],{"className":79521,"style":159},[128],[18,79523],{},"，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。",[11,79526,79527,79528,79556],{},"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 ",[18,79529,79531,79544],{"className":79530,"translate":22},[21],[18,79532,79534],{"className":79533},[26],[28,79535,79536],{"xmlns":30},[32,79537,79538,79542],{},[35,79539,79540],{},[41,79541,32513],{},[86,79543,35404],{"encoding":88},[18,79545,79547],{"className":79546,"ariaHidden":51},[93],[18,79548,79550,79553],{"className":79549},[97],[18,79551],{"className":79552,"style":1083},[101],[18,79554,32513],{"className":79555},[106,110]," 的球能把假设类盖住」。二者通过 Dudley 熵积分相连。",[11,79558,79559,79562,79563,79591,79592,79642],{},[717,79560,79561],{},"PAC-Bayes"," 则把容量变成了「后验与先验的距离」。对任意后验分布 ",[18,79564,79566,79579],{"className":79565,"translate":22},[21],[18,79567,79569],{"className":79568},[26],[28,79570,79571],{"xmlns":30},[32,79572,79573,79577],{},[35,79574,79575],{},[41,79576,12544],{},[86,79578,12544],{"encoding":88},[18,79580,79582],{"className":79581,"ariaHidden":51},[93],[18,79583,79585,79588],{"className":79584},[97],[18,79586],{"className":79587,"style":102},[101],[18,79589,12544],{"className":79590},[106,110],"（通常是训练后权重的分布），以 ",[18,79593,79595,79612],{"className":79594,"translate":22},[21],[18,79596,79598],{"className":79597},[26],[28,79599,79600],{"xmlns":30},[32,79601,79602,79610],{},[35,79603,79604,79606,79608],{},[45,79605,47],{},[49,79607,2253],{},[41,79609,5348],{},[86,79611,75174],{"encoding":88},[18,79613,79615,79633],{"className":79614,"ariaHidden":51},[93],[18,79616,79618,79621,79624,79627,79630],{"className":79617},[97],[18,79619],{"className":79620,"style":18090},[101],[18,79622,47],{"className":79623},[106],[18,79625],{"className":79626,"style":2068},[169],[18,79628,2253],{"className":79629},[2072],[18,79631],{"className":79632,"style":2068},[169],[18,79634,79636,79639],{"className":79635},[97],[18,79637],{"className":79638,"style":295},[101],[18,79640,5348],{"className":79641,"style":5477},[106,110]," 的概率，",[18,79644,79646],{"className":79645,"translate":22},[875],[18,79647,79649,79782],{"className":79648,"translate":22},[21],[18,79650,79652],{"className":79651},[26],[28,79653,79654],{"xmlns":30,"display":884},[32,79655,79656,79779],{},[35,79657,79658,79670,79672,79678,79680,79682,79684,79686,79688,79690,79692,79704,79706,79712,79714,79716,79718,79720,79722,79724,79726,79728,79731,79777],{},[38,79659,79660,79662],{},[41,79661,934],{"mathvariant":933},[35,79663,79664,79666,79668],{},[41,79665,18323],{},[49,79667,67],{},[41,79669,12544],{},[49,79671,945],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[38,79673,79674,79676],{},[41,79675,12579],{},[41,79677,12572],{"mathvariant":1462},[49,79679,74],{"stretchy":73},[41,79681,18323],{},[49,79683,84],{"stretchy":73},[49,79685,965],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[926,79687,71841],{},[49,79689,4334],{},[926,79691,71841],{},[38,79693,79694,79696],{},[41,79695,934],{"mathvariant":933},[35,79697,79698,79700,79702],{},[41,79699,18323],{},[49,79701,67],{},[41,79703,12544],{},[49,79705,945],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[38,79707,79708,79710],{},[41,79709,12579],{},[41,79711,38925],{},[49,79713,74],{"stretchy":73},[41,79715,18323],{},[49,79717,84],{"stretchy":73},[49,79719,965],{"fence":51,"stretchy":51,"minsize":944,"maxsize":944},[926,79721,71841],{},[49,79723,7984],{},[926,79725,71841],{},[41,79727,78618],{},[926,79729,79730],{}," ⁣",[35,79732,79733,79735,79775],{},[49,79734,74],{"fence":51},[12558,79736,79737],{},[5651,79738,79739,79773],{},[35,79740,79741,79747,79749,79751,79753,79755,79757,79759,79761,79763,79765,79767,79769,79771],{},[35,79742,79743,79745],{},[41,79744,5031],{"mathvariant":316},[41,79746,12579],{"mathvariant":316},[49,79748,74],{"stretchy":73},[41,79750,12544],{},[41,79752,2248],{"mathvariant":316},[41,79754,939],{},[49,79756,84],{"stretchy":73},[49,79758,7984],{},[41,79760,14730],{},[49,79762,899],{},[49,79764,74],{"stretchy":73},[45,79766,47],{},[41,79768,317],{"mathvariant":316},[41,79770,5348],{},[49,79772,84],{"stretchy":73},[41,79774,28160],{},[49,79776,84],{"fence":51},[41,79778,968],{"mathvariant":316},[86,79780,79781],{"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).",[18,79783,79785,79919,80050],{"className":79784,"ariaHidden":51},[93],[18,79786,79788,79791,79840,79846,79889,79892,79895,79898,79904,79907,79910,79913,79916],{"className":79787},[97],[18,79789],{"className":79790,"style":6098},[101],[18,79792,79794,79797],{"className":79793},[106],[18,79795,934],{"className":79796},[106,1139],[18,79798,79800],{"className":79799},[115],[18,79801,79803,79832],{"className":79802},[119,120],[18,79804,79806,79829],{"className":79805},[124],[18,79807,79809],{"className":79808,"style":1383},[128],[18,79810,79811,79814],{"style":1155},[18,79812],{"className":79813,"style":137},[136],[18,79815,79817],{"className":79816},[141,142,143,144],[18,79818,79820,79823,79826],{"className":79819},[106,144],[18,79821,18323],{"className":79822},[106,110,144],[18,79824,67],{"className":79825},[232,144],[18,79827,12544],{"className":79828},[106,110,144],[18,79830,152],{"className":79831},[151],[18,79833,79835],{"className":79834},[124],[18,79836,79838],{"className":79837,"style":19931},[128],[18,79839],{},[18,79841,79843],{"className":79842},[250],[18,79844,945],{"className":79845},[1230,1196],[18,79847,79849,79852],{"className":79848},[106],[18,79850,12579],{"className":79851},[106,110],[18,79853,79855],{"className":79854},[115],[18,79856,79858,79881],{"className":79857},[119,120],[18,79859,79861,79878],{"className":79860},[124],[18,79862,79864],{"className":79863,"style":1152},[128],[18,79865,79866,79869],{"style":1155},[18,79867],{"className":79868,"style":137},[136],[18,79870,79872],{"className":79871},[141,142,143,144],[18,79873,79875],{"className":79874},[106,144],[18,79876,12572],{"className":79877,"style":254},[106,1478,144],[18,79879,152],{"className":79880},[151],[18,79882,79884],{"className":79883},[124],[18,79885,79887],{"className":79886,"style":159},[128],[18,79888],{},[18,79890,74],{"className":79891},[250],[18,79893,18323],{"className":79894},[106,110],[18,79896,84],{"className":79897},[267],[18,79899,79901],{"className":79900},[267],[18,79902,965],{"className":79903},[1230,1196],[18,79905],{"className":79906,"style":228},[169],[18,79908],{"className":79909,"style":228},[169],[18,79911,4334],{"className":79912},[232],[18,79914],{"className":79915,"style":228},[169],[18,79917],{"className":79918,"style":228},[169],[18,79920,79922,79925,79974,79980,80020,80023,80026,80029,80035,80038,80041,80044,80047],{"className":79921},[97],[18,79923],{"className":79924,"style":6098},[101],[18,79926,79928,79931],{"className":79927},[106],[18,79929,934],{"className":79930},[106,1139],[18,79932,79934],{"className":79933},[115],[18,79935,79937,79966],{"className":79936},[119,120],[18,79938,79940,79963],{"className":79939},[124],[18,79941,79943],{"className":79942,"style":1383},[128],[18,79944,79945,79948],{"style":1155},[18,79946],{"className":79947,"style":137},[136],[18,79949,79951],{"className":79950},[141,142,143,144],[18,79952,79954,79957,79960],{"className":79953},[106,144],[18,79955,18323],{"className":79956},[106,110,144],[18,79958,67],{"className":79959},[232,144],[18,79961,12544],{"className":79962},[106,110,144],[18,79964,152],{"className":79965},[151],[18,79967,79969],{"className":79968},[124],[18,79970,79972],{"className":79971,"style":19931},[128],[18,79973],{},[18,79975,79977],{"className":79976},[250],[18,79978,945],{"className":79979},[1230,1196],[18,79981,79983,79986],{"className":79982},[106],[18,79984,12579],{"className":79985},[106,110],[18,79987,79989],{"className":79988},[115],[18,79990,79992,80012],{"className":79991},[119,120],[18,79993,79995,80009],{"className":79994},[124],[18,79996,79998],{"className":79997,"style":1152},[128],[18,79999,80000,80003],{"style":1155},[18,80001],{"className":80002,"style":137},[136],[18,80004,80006],{"className":80005},[141,142,143,144],[18,80007,38925],{"className":80008,"style":38958},[106,110,144],[18,80010,152],{"className":80011},[151],[18,80013,80015],{"className":80014},[124],[18,80016,80018],{"className":80017,"style":159},[128],[18,80019],{},[18,80021,74],{"className":80022},[250],[18,80024,18323],{"className":80025},[106,110],[18,80027,84],{"className":80028},[267],[18,80030,80032],{"className":80031},[267],[18,80033,965],{"className":80034},[1230,1196],[18,80036],{"className":80037,"style":228},[169],[18,80039],{"className":80040,"style":2068},[169],[18,80042,7984],{"className":80043},[2072],[18,80045],{"className":80046,"style":228},[169],[18,80048],{"className":80049,"style":2068},[169],[18,80051,80053,80057,80060,80064,80067,80232,80235],{"className":80052},[97],[18,80054],{"className":80055,"style":80056},[101],"height:3em;vertical-align:-1.25em;",[18,80058,78618],{"className":80059,"style":254},[106,110],[18,80061],{"className":80062,"style":80063},[169],"margin-right:-0.1667em;",[18,80065],{"className":80066,"style":170},[169],[18,80068,80070,80076,80226],{"className":80069},[174],[18,80071,80073],{"className":80072,"style":13793},[250,13792],[18,80074,74],{"className":80075},[1230,61187],[18,80077,80079],{"className":80078},[106,12693],[18,80080,80082,80218],{"className":80081},[119,120],[18,80083,80085,80215],{"className":80084},[124],[18,80086,80088,80203],{"className":80087,"style":74399},[128],[18,80089,80091,80094],{"className":80090,"style":12708},[12707],[18,80092],{"className":80093,"style":12712},[136],[18,80095,80097],{"className":80096,"style":12716},[106],[18,80098,80100,80103,80200],{"className":80099},[106],[18,80101],{"className":80102},[250,6009],[18,80104,80106],{"className":80105},[5651],[18,80107,80109,80192],{"className":80108},[119,120],[18,80110,80112,80189],{"className":80111},[124],[18,80113,80115,80126,80134],{"className":80114,"style":18128},[128],[18,80116,80117,80120],{"style":6025},[18,80118],{"className":80119,"style":697},[136],[18,80121,80123],{"className":80122},[106],[18,80124,28160],{"className":80125},[106,110],[18,80127,80128,80131],{"style":6037},[18,80129],{"className":80130,"style":697},[136],[18,80132],{"className":80133,"style":6045},[6044],[18,80135,80136,80139],{"style":6048},[18,80137],{"className":80138,"style":697},[136],[18,80140,80142,80148,80151,80154,80157,80160,80163,80166,80169,80172,80177,80180,80183,80186],{"className":80141},[106],[18,80143,80145],{"className":80144},[106],[18,80146,12814],{"className":80147},[106,6351],[18,80149,74],{"className":80150},[250],[18,80152,12544],{"className":80153},[106,110],[18,80155,2248],{"className":80156},[106],[18,80158,939],{"className":80159,"style":1171},[106,110],[18,80161,84],{"className":80162},[267],[18,80164],{"className":80165,"style":2068},[169],[18,80167,7984],{"className":80168},[2072],[18,80170],{"className":80171,"style":2068},[169],[18,80173,15171,80175],{"className":80174},[985],[18,80176,15174],{"style":6352},[18,80178,74],{"className":80179},[250],[18,80181,335],{"className":80182},[106],[18,80184,5348],{"className":80185,"style":5477},[106,110],[18,80187,84],{"className":80188},[267],[18,80190,152],{"className":80191},[151],[18,80193,80195],{"className":80194},[124],[18,80196,80198],{"className":80197,"style":6067},[128],[18,80199],{},[18,80201],{"className":80202},[267,6009],[18,80204,80205,80208],{"style":74536},[18,80206],{"className":80207,"style":12712},[136],[18,80209,80211],{"className":80210,"style":12851},[12850],[6127,80212,80213],{"xmlns":6129,"width":12854,"height":12855,"viewBox":12856,"preserveAspectRatio":12857},[6133,80214],{"d":12860},[18,80216,152],{"className":80217},[151],[18,80219,80221],{"className":80220},[124],[18,80222,80224],{"className":80223,"style":74556},[128],[18,80225],{},[18,80227,80229],{"className":80228,"style":13793},[267,13792],[18,80230,84],{"className":80231},[1230,61187],[18,80233],{"className":80234,"style":170},[169],[18,80236,968],{"className":80237},[106],[11,80239,80240,80241,80269,80270,80298],{},"这里的复杂度不再是假设类的 VC 维，而是「训练后的分布 ",[18,80242,80244,80257],{"className":80243,"translate":22},[21],[18,80245,80247],{"className":80246},[26],[28,80248,80249],{"xmlns":30},[32,80250,80251,80255],{},[35,80252,80253],{},[41,80254,12544],{},[86,80256,12544],{"encoding":88},[18,80258,80260],{"className":80259,"ariaHidden":51},[93],[18,80261,80263,80266],{"className":80262},[97],[18,80264],{"className":80265,"style":102},[101],[18,80267,12544],{"className":80268},[106,110]," 离先验 ",[18,80271,80273,80286],{"className":80272,"translate":22},[21],[18,80274,80276],{"className":80275},[26],[28,80277,80278],{"xmlns":30},[32,80279,80280,80284],{},[35,80281,80282],{},[41,80283,939],{},[86,80285,939],{"encoding":88},[18,80287,80289],{"className":80288,"ariaHidden":51},[93],[18,80290,80292,80295],{"className":80291},[97],[18,80293],{"className":80294,"style":600},[101],[18,80296,939],{"className":80297,"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。",[11,80300,80301,80304,80305,80333,80334,80421],{},[717,80302,80303],{},"压缩"," 是另一条路：若训练好的网络可以被压缩成 ",[18,80306,80308,80321],{"className":80307,"translate":22},[21],[18,80309,80311],{"className":80310},[26],[28,80312,80313],{"xmlns":30},[32,80314,80315,80319],{},[35,80316,80317],{},[41,80318,39955],{},[86,80320,39955],{"encoding":88},[18,80322,80324],{"className":80323,"ariaHidden":51},[93],[18,80325,80327,80330],{"className":80326},[97],[18,80328],{"className":80329,"style":295},[101],[18,80331,39955],{"className":80332,"style":40031},[106,110]," 比特而不改变其在训练集上的预测，则泛化误差约为 ",[18,80335,80337,80359],{"className":80336,"translate":22},[21],[18,80338,80340],{"className":80339},[26],[28,80341,80342],{"xmlns":30},[32,80343,80344,80356],{},[35,80345,80346],{},[12558,80347,80348],{},[35,80349,80350,80352,80354],{},[41,80351,39955],{},[41,80353,317],{"mathvariant":316},[41,80355,28160],{},[86,80357,80358],{"encoding":88},"\\sqrt{k\u002Fm}",[18,80360,80362],{"className":80361,"ariaHidden":51},[93],[18,80363,80365,80368],{"className":80364},[97],[18,80366],{"className":80367,"style":77127},[101],[18,80369,80371],{"className":80370},[106,12693],[18,80372,80374,80413],{"className":80373},[119,120],[18,80375,80377,80410],{"className":80376},[124],[18,80378,80380,80398],{"className":80379,"style":77140},[128],[18,80381,80383,80386],{"className":80382,"style":77144},[12707],[18,80384],{"className":80385,"style":6122},[136],[18,80387,80389,80392,80395],{"className":80388,"style":12716},[106],[18,80390,39955],{"className":80391,"style":40031},[106,110],[18,80393,317],{"className":80394},[106],[18,80396,28160],{"className":80397},[106,110],[18,80399,80400,80403],{"style":77162},[18,80401],{"className":80402,"style":6122},[136],[18,80404,80406],{"className":80405,"style":77169},[12850],[6127,80407,80408],{"xmlns":6129,"width":12854,"height":77172,"viewBox":77173,"preserveAspectRatio":12857},[6133,80409],{"d":77176},[18,80411,152],{"className":80412},[151],[18,80414,80416],{"className":80415},[124],[18,80417,80419],{"className":80418,"style":77186},[128],[18,80420],{},"。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。",[11,80423,80424,80427],{},[717,80425,80426],{},"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。",[11,80429,80430,80431,80434,80435,80463],{},"这些界的共同点很重要：",[717,80432,80433],{},"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 ",[18,80436,80438,80451],{"className":80437,"translate":22},[21],[18,80439,80441],{"className":80440},[26],[28,80442,80443],{"xmlns":30},[32,80444,80445,80449],{},[35,80446,80447],{},[41,80448,18304],{"mathvariant":1462},[86,80450,46308],{"encoding":88},[18,80452,80454],{"className":80453,"ariaHidden":51},[93],[18,80455,80457,80460],{"className":80456},[97],[18,80458],{"className":80459,"style":600},[101],[18,80461,18304],{"className":80462,"style":46321},[106,1478]," 转移到了假设类、算法、数据三者共同决定的整体上。",[11,80465,80466],{},[717,80467,80468],{},"（b）算法偏置：SGD 只在一个很小的子集里搜索",[11,80470,80471,80499,80500,80528,80529,80557],{},[18,80472,80474,80487],{"className":80473,"translate":22},[21],[18,80475,80477],{"className":80476},[26],[28,80478,80479],{"xmlns":30},[32,80480,80481,80485],{},[35,80482,80483],{},[41,80484,18304],{"mathvariant":1462},[86,80486,46308],{"encoding":88},[18,80488,80490],{"className":80489,"ariaHidden":51},[93],[18,80491,80493,80496],{"className":80492},[97],[18,80494],{"className":80495,"style":600},[101],[18,80497,18304],{"className":80498,"style":46321},[106,1478]," 很大，但 SGD 不是均匀地在 ",[18,80501,80503,80516],{"className":80502,"translate":22},[21],[18,80504,80506],{"className":80505},[26],[28,80507,80508],{"xmlns":30},[32,80509,80510,80514],{},[35,80511,80512],{},[41,80513,18304],{"mathvariant":1462},[86,80515,46308],{"encoding":88},[18,80517,80519],{"className":80518,"ariaHidden":51},[93],[18,80520,80522,80525],{"className":80521},[97],[18,80523],{"className":80524,"style":600},[101],[18,80526,18304],{"className":80527,"style":46321},[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 ",[18,80530,80532,80545],{"className":80531,"translate":22},[21],[18,80533,80535],{"className":80534},[26],[28,80536,80537],{"xmlns":30},[32,80538,80539,80543],{},[35,80540,80541],{},[41,80542,18304],{"mathvariant":1462},[86,80544,46308],{"encoding":88},[18,80546,80548],{"className":80547,"ariaHidden":51},[93],[18,80549,80551,80554],{"className":80550},[97],[18,80552],{"className":80553,"style":600},[101],[18,80555,18304],{"className":80556,"style":46321},[106,1478]," 的一个很小的子集。",[11,80559,80560],{},"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。",[11,80562,80563],{},"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」。",[11,80565,80566],{},"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。",[11,80568,80569],{},[717,80570,80571],{},"（c）数据分布：最坏情况太坏",[11,80573,80574],{},"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。",[11,80576,80577,80580,80581,80609,80610,80638],{},[717,80578,80579],{},"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 ",[18,80582,80584,80597],{"className":80583,"translate":22},[21],[18,80585,80587],{"className":80586},[26],[28,80588,80589],{"xmlns":30},[32,80590,80591,80595],{},[35,80592,80593],{},[41,80594,4031],{},[86,80596,4031],{"encoding":88},[18,80598,80600],{"className":80599,"ariaHidden":51},[93],[18,80601,80603,80606],{"className":80602},[97],[18,80604],{"className":80605,"style":295},[101],[18,80607,4031],{"className":80608},[106,110]," 随 ",[18,80611,80613,80626],{"className":80612,"translate":22},[21],[18,80614,80616],{"className":80615},[26],[28,80617,80618],{"xmlns":30},[32,80619,80620,80624],{},[35,80621,80622],{},[41,80623,28160],{},[86,80625,28160],{"encoding":88},[18,80627,80629],{"className":80628,"ariaHidden":51},[93],[18,80630,80632,80635],{"className":80631},[97],[18,80633],{"className":80634,"style":1083},[101],[18,80636,28160],{"className":80637},[106,110]," 变化的区域，偏差 - 方差分解的直觉不再适用。",[11,80640,80641,80644],{},[717,80642,80643],{},"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。",[11,80646,80647],{},"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。",[1311,80649,80651],{"id":80650},"_64-一致收敛的局限","6.4 一致收敛的局限",[11,80653,80654,80655,80658],{},"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",[717,80656,80657],{},"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。",[11,80660,80661],{},"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。",[11,80663,80664,80665,2702,80693,80721,80722],{},"这里有一个容易被忽略的逻辑细节：对二分类 ",[18,80666,80668,80681],{"className":80667,"translate":22},[21],[18,80669,80671],{"className":80670},[26],[28,80672,80673],{"xmlns":30},[32,80674,80675,80679],{},[35,80676,80677],{},[45,80678,430],{},[86,80680,430],{"encoding":88},[18,80682,80684],{"className":80683,"ariaHidden":51},[93],[18,80685,80687,80690],{"className":80686},[97],[18,80688],{"className":80689,"style":505},[101],[18,80691,430],{"className":80692},[106],[18,80694,80696,80709],{"className":80695,"translate":22},[21],[18,80697,80699],{"className":80698},[26],[28,80700,80701],{"xmlns":30},[32,80702,80703,80707],{},[35,80704,80705],{},[45,80706,47],{},[86,80708,47],{"encoding":88},[18,80710,80712],{"className":80711,"ariaHidden":51},[93],[18,80713,80715,80718],{"className":80714},[97],[18,80716],{"className":80717,"style":505},[101],[18,80719,47],{"className":80720},[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。",[717,80723,80724],{},"VC 理论没有被证伪，它只是没有提供我们想要的那种解释。",[638,80726,80728],{"id":80727},"七容量度量谱系","七、容量度量谱系",[11,80730,80731,80732,80735,80736,80764,80765,80768,80769,80797],{},"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 ",[717,80733,80734],{},"pseudo-dimension","；或者要求在某个尺度 ",[18,80737,80739,80752],{"className":80738,"translate":22},[21],[18,80740,80742],{"className":80741},[26],[28,80743,80744],{"xmlns":30},[32,80745,80746,80750],{},[35,80747,80748],{},[41,80749,79177],{},[86,80751,79180],{"encoding":88},[18,80753,80755],{"className":80754,"ariaHidden":51},[93],[18,80756,80758,80761],{"className":80757},[97],[18,80759],{"className":80760,"style":38543},[101],[18,80762,79177],{"className":80763,"style":79193},[106,110]," 下仍能打散，得到 ",[717,80766,80767],{},"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 ",[18,80770,80772,80785],{"className":80771,"translate":22},[21],[18,80773,80775],{"className":80774},[26],[28,80776,80777],{"xmlns":30},[32,80778,80779,80783],{},[35,80780,80781],{},[41,80782,79177],{},[86,80784,79180],{"encoding":88},[18,80786,80788],{"className":80787,"ariaHidden":51},[93],[18,80789,80791,80794],{"className":80790},[97],[18,80792],{"className":80793,"style":38543},[101],[18,80795,79177],{"className":80796,"style":79193},[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。",[11,80799,80800],{},"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。",[43703,80802,80803,80822],{},[43706,80804,80805],{},[43709,80806,80807,80810,80813,80816,80819],{},[43712,80808,80809],{},"度量",[43712,80811,80812],{},"度量对象",[43712,80814,80815],{},"核心思想",[43712,80817,80818],{},"典型界",[43712,80820,80821],{},"主要局限",[43722,80823,80824,81032,81047,81064,81109,81155,81311,81469,81513,81614],{},[43709,80825,80826,80828,80831,80834,80972],{},[43727,80827,52183],{},[43727,80829,80830],{},"二值函数类",[43727,80832,80833],{},"最大打散集",[43727,80835,80836],{},[18,80837,80839,80881],{"className":80838,"translate":22},[21],[18,80840,80842],{"className":80841},[26],[28,80843,80844],{"xmlns":30},[32,80845,80846,80878],{},[35,80847,80848],{},[12558,80849,80850],{},[35,80851,80852,80854,80856,80858,80860,80862,80864,80866,80868,80870,80872,80874,80876],{},[49,80853,74],{"stretchy":73},[41,80855,4031],{},[49,80857,7984],{},[41,80859,14730],{},[49,80861,899],{},[49,80863,74],{"stretchy":73},[45,80865,47],{},[41,80867,317],{"mathvariant":316},[41,80869,5348],{},[49,80871,84],{"stretchy":73},[49,80873,84],{"stretchy":73},[41,80875,317],{"mathvariant":316},[41,80877,28160],{},[86,80879,80880],{"encoding":88},"\\sqrt{(d+\\log(1\u002F\\delta))\u002Fm}",[18,80882,80884],{"className":80883,"ariaHidden":51},[93],[18,80885,80887,80890],{"className":80886},[97],[18,80888],{"className":80889,"style":77127},[101],[18,80891,80893],{"className":80892},[106,12693],[18,80894,80896,80964],{"className":80895},[119,120],[18,80897,80899,80961],{"className":80898},[124],[18,80900,80902,80949],{"className":80901,"style":77140},[128],[18,80903,80905,80908],{"className":80904,"style":77144},[12707],[18,80906],{"className":80907,"style":6122},[136],[18,80909,80911,80914,80917,80920,80923,80926,80931,80934,80937,80940,80943,80946],{"className":80910,"style":12716},[106],[18,80912,74],{"className":80913},[250],[18,80915,4031],{"className":80916},[106,110],[18,80918],{"className":80919,"style":2068},[169],[18,80921,7984],{"className":80922},[2072],[18,80924],{"className":80925,"style":2068},[169],[18,80927,15171,80929],{"className":80928},[985],[18,80930,15174],{"style":6352},[18,80932,74],{"className":80933},[250],[18,80935,335],{"className":80936},[106],[18,80938,5348],{"className":80939,"style":5477},[106,110],[18,80941,3431],{"className":80942},[267],[18,80944,317],{"className":80945},[106],[18,80947,28160],{"className":80948},[106,110],[18,80950,80951,80954],{"style":77162},[18,80952],{"className":80953,"style":6122},[136],[18,80955,80957],{"className":80956,"style":77169},[12850],[6127,80958,80959],{"xmlns":6129,"width":12854,"height":77172,"viewBox":77173,"preserveAspectRatio":12857},[6133,80960],{"d":77176},[18,80962,152],{"className":80963},[151],[18,80965,80967],{"className":80966},[124],[18,80968,80970],{"className":80969,"style":77186},[128],[18,80971],{},[43727,80973,80974,80975,2702,81003,81031],{},"分布无关、最坏情况、只适用 ",[18,80976,80978,80991],{"className":80977,"translate":22},[21],[18,80979,80981],{"className":80980},[26],[28,80982,80983],{"xmlns":30},[32,80984,80985,80989],{},[35,80986,80987],{},[45,80988,430],{},[86,80990,430],{"encoding":88},[18,80992,80994],{"className":80993,"ariaHidden":51},[93],[18,80995,80997,81000],{"className":80996},[97],[18,80998],{"className":80999,"style":505},[101],[18,81001,430],{"className":81002},[106],[18,81004,81006,81019],{"className":81005,"translate":22},[21],[18,81007,81009],{"className":81008},[26],[28,81010,81011],{"xmlns":30},[32,81012,81013,81017],{},[35,81014,81015],{},[45,81016,47],{},[86,81018,47],{"encoding":88},[18,81020,81022],{"className":81021,"ariaHidden":51},[93],[18,81023,81025,81028],{"className":81024},[97],[18,81026],{"className":81027,"style":505},[101],[18,81029,47],{"className":81030},[106]," 损失",[43709,81033,81034,81036,81038,81041,81044],{},[43727,81035,65358],{},[43727,81037,80830],{},[43727,81039,81040],{},"限制类的最大规模",[43727,81042,81043],{},"Sauer 多项式界",[43727,81045,81046],{},"组合量，需要再转概率",[43709,81048,81049,81052,81055,81058,81061],{},[43727,81050,81051],{},"Pseudo-dimension",[43727,81053,81054],{},"实值函数类",[43727,81056,81057],{},"阈值化后的 VC 维",[43727,81059,81060],{},"与 VC 同阶",[43727,81062,81063],{},"仍是最坏情况",[43709,81065,81066,81069,81071,81103,81106],{},[43727,81067,81068],{},"Fat-shattering",[43727,81070,81054],{},[43727,81072,81073,81074,81102],{},"尺度 ",[18,81075,81077,81090],{"className":81076,"translate":22},[21],[18,81078,81080],{"className":81079},[26],[28,81081,81082],{"xmlns":30},[32,81083,81084,81088],{},[35,81085,81086],{},[41,81087,79177],{},[86,81089,79180],{"encoding":88},[18,81091,81093],{"className":81092,"ariaHidden":51},[93],[18,81094,81096,81099],{"className":81095},[97],[18,81097],{"className":81098,"style":38543},[101],[18,81100,79177],{"className":81101,"style":79193},[106,110]," 下的打散",[43727,81104,81105],{},"间隔界",[43727,81107,81108],{},"依赖尺度选择",[43709,81110,81111,81114,81117,81149,81152],{},[43727,81112,81113],{},"覆盖数",[43727,81115,81116],{},"度量空间",[43727,81118,81119,81120,81148],{},"用 ",[18,81121,81123,81136],{"className":81122,"translate":22},[21],[18,81124,81126],{"className":81125},[26],[28,81127,81128],{"xmlns":30},[32,81129,81130,81134],{},[35,81131,81132],{},[41,81133,32513],{},[86,81135,35404],{"encoding":88},[18,81137,81139],{"className":81138,"ariaHidden":51},[93],[18,81140,81142,81145],{"className":81141},[97],[18,81143],{"className":81144,"style":1083},[101],[18,81146,32513],{"className":81147},[106,110],"-球覆盖",[43727,81150,81151],{},"Dudley 熵积分",[43727,81153,81154],{},"计算困难",[43709,81156,81157,81159,81162,81165,81308],{},[43727,81158,74163],{},[43727,81160,81161],{},"数据相关",[43727,81163,81164],{},"随机符号下的拟合能力",[43727,81166,81167],{},[18,81168,81170,81210],{"className":81169,"translate":22},[21],[18,81171,81173],{"className":81172},[26],[28,81174,81175],{"xmlns":30},[32,81176,81177,81207],{},[35,81178,81179,81181,81183,81185],{},[45,81180,2266],{},[41,81182,3071],{"mathvariant":3525},[49,81184,7984],{},[12558,81186,81187],{},[35,81188,81189,81191,81193,81195,81197,81199,81201,81203,81205],{},[41,81190,14730],{},[49,81192,899],{},[49,81194,74],{"stretchy":73},[45,81196,47],{},[41,81198,317],{"mathvariant":316},[41,81200,5348],{},[49,81202,84],{"stretchy":73},[41,81204,317],{"mathvariant":316},[41,81206,28160],{},[86,81208,81209],{"encoding":88},"2\\mathfrak R+\\sqrt{\\log(1\u002F\\delta)\u002Fm}",[18,81211,81213,81235],{"className":81212,"ariaHidden":51},[93],[18,81214,81216,81220,81223,81226,81229,81232],{"className":81215},[97],[18,81217],{"className":81218,"style":81219},[101],"height:0.7747em;vertical-align:-0.0833em;",[18,81221,2266],{"className":81222},[106],[18,81224,3071],{"className":81225},[106,3607],[18,81227],{"className":81228,"style":2068},[169],[18,81230,7984],{"className":81231},[2072],[18,81233],{"className":81234,"style":2068},[169],[18,81236,81238,81241],{"className":81237},[97],[18,81239],{"className":81240,"style":77127},[101],[18,81242,81244],{"className":81243},[106,12693],[18,81245,81247,81300],{"className":81246},[119,120],[18,81248,81250,81297],{"className":81249},[124],[18,81251,81253,81285],{"className":81252,"style":77140},[128],[18,81254,81256,81259],{"className":81255,"style":77144},[12707],[18,81257],{"className":81258,"style":6122},[136],[18,81260,81262,81267,81270,81273,81276,81279,81282],{"className":81261,"style":12716},[106],[18,81263,15171,81265],{"className":81264},[985],[18,81266,15174],{"style":6352},[18,81268,74],{"className":81269},[250],[18,81271,335],{"className":81272},[106],[18,81274,5348],{"className":81275,"style":5477},[106,110],[18,81277,84],{"className":81278},[267],[18,81280,317],{"className":81281},[106],[18,81283,28160],{"className":81284},[106,110],[18,81286,81287,81290],{"style":77162},[18,81288],{"className":81289,"style":6122},[136],[18,81291,81293],{"className":81292,"style":77169},[12850],[6127,81294,81295],{"xmlns":6129,"width":12854,"height":77172,"viewBox":77173,"preserveAspectRatio":12857},[6133,81296],{"d":77176},[18,81298,152],{"className":81299},[151],[18,81301,81303],{"className":81302},[124],[18,81304,81306],{"className":81305,"style":77186},[128],[18,81307],{},[43727,81309,81310],{},"需要估计，通常偏松",[43709,81312,81313,81315,81318,81321,81466],{},[43727,81314,79561],{},[43727,81316,81317],{},"后验与先验",[43727,81319,81320],{},"KL 复杂度",[43727,81322,81323],{},[18,81324,81326,81372],{"className":81325,"translate":22},[21],[18,81327,81329],{"className":81328},[26],[28,81330,81331],{"xmlns":30},[32,81332,81333,81369],{},[35,81334,81335],{},[12558,81336,81337],{},[35,81338,81339,81341,81347,81349,81351,81353,81355,81357,81359,81361,81363,81365,81367],{},[49,81340,74],{"stretchy":73},[35,81342,81343,81345],{},[41,81344,5031],{"mathvariant":316},[41,81346,12579],{"mathvariant":316},[49,81348,7984],{},[41,81350,14730],{},[49,81352,899],{},[49,81354,74],{"stretchy":73},[45,81356,47],{},[41,81358,317],{"mathvariant":316},[41,81360,5348],{},[49,81362,84],{"stretchy":73},[49,81364,84],{"stretchy":73},[41,81366,317],{"mathvariant":316},[41,81368,28160],{},[86,81370,81371],{"encoding":88},"\\sqrt{(\\mathrm{KL}+\\log(1\u002F\\delta))\u002Fm}",[18,81373,81375],{"className":81374,"ariaHidden":51},[93],[18,81376,81378,81381],{"className":81377},[97],[18,81379],{"className":81380,"style":77127},[101],[18,81382,81384],{"className":81383},[106,12693],[18,81385,81387,81458],{"className":81386},[119,120],[18,81388,81390,81455],{"className":81389},[124],[18,81391,81393,81443],{"className":81392,"style":77140},[128],[18,81394,81396,81399],{"className":81395,"style":77144},[12707],[18,81397],{"className":81398,"style":6122},[136],[18,81400,81402,81405,81411,81414,81417,81420,81425,81428,81431,81434,81437,81440],{"className":81401,"style":12716},[106],[18,81403,74],{"className":81404},[250],[18,81406,81408],{"className":81407},[106],[18,81409,12814],{"className":81410},[106,6351],[18,81412],{"className":81413,"style":2068},[169],[18,81415,7984],{"className":81416},[2072],[18,81418],{"className":81419,"style":2068},[169],[18,81421,15171,81423],{"className":81422},[985],[18,81424,15174],{"style":6352},[18,81426,74],{"className":81427},[250],[18,81429,335],{"className":81430},[106],[18,81432,5348],{"className":81433,"style":5477},[106,110],[18,81435,3431],{"className":81436},[267],[18,81438,317],{"className":81439},[106],[18,81441,28160],{"className":81442},[106,110],[18,81444,81445,81448],{"style":77162},[18,81446],{"className":81447,"style":6122},[136],[18,81449,81451],{"className":81450,"style":77169},[12850],[6127,81452,81453],{"xmlns":6129,"width":12854,"height":77172,"viewBox":77173,"preserveAspectRatio":12857},[6133,81454],{"d":77176},[18,81456,152],{"className":81457},[151],[18,81459,81461],{"className":81460},[124],[18,81462,81464],{"className":81463,"style":77186},[128],[18,81465],{},[43727,81467,81468],{},"先验的选择",[43709,81470,81471,81473,81476,81479,81510],{},[43727,81472,80426],{},[43727,81474,81475],{},"算法",[43727,81477,81478],{},"替换一个样本的影响",[43727,81480,81481,81482],{},"稳定性系数 ",[18,81483,81485,81498],{"className":81484,"translate":22},[21],[18,81486,81488],{"className":81487},[26],[28,81489,81490],{"xmlns":30},[32,81491,81492,81496],{},[35,81493,81494],{},[41,81495,20979],{},[86,81497,37780],{"encoding":88},[18,81499,81501],{"className":81500,"ariaHidden":51},[93],[18,81502,81504,81507],{"className":81503},[97],[18,81505],{"className":81506,"style":2890},[101],[18,81508,20979],{"className":81509,"style":21031},[106,110],[43727,81511,81512],{},"对 SGD 难以验证",[43709,81514,81515,81517,81520,81523,81611],{},[43727,81516,80303],{},[43727,81518,81519],{},"算法输出",[43727,81521,81522],{},"描述长度",[43727,81524,81525],{},[18,81526,81528,81549],{"className":81527,"translate":22},[21],[18,81529,81531],{"className":81530},[26],[28,81532,81533],{"xmlns":30},[32,81534,81535,81547],{},[35,81536,81537],{},[12558,81538,81539],{},[35,81540,81541,81543,81545],{},[41,81542,39955],{},[41,81544,317],{"mathvariant":316},[41,81546,28160],{},[86,81548,80358],{"encoding":88},[18,81550,81552],{"className":81551,"ariaHidden":51},[93],[18,81553,81555,81558],{"className":81554},[97],[18,81556],{"className":81557,"style":77127},[101],[18,81559,81561],{"className":81560},[106,12693],[18,81562,81564,81603],{"className":81563},[119,120],[18,81565,81567,81600],{"className":81566},[124],[18,81568,81570,81588],{"className":81569,"style":77140},[128],[18,81571,81573,81576],{"className":81572,"style":77144},[12707],[18,81574],{"className":81575,"style":6122},[136],[18,81577,81579,81582,81585],{"className":81578,"style":12716},[106],[18,81580,39955],{"className":81581,"style":40031},[106,110],[18,81583,317],{"className":81584},[106],[18,81586,28160],{"className":81587},[106,110],[18,81589,81590,81593],{"style":77162},[18,81591],{"className":81592,"style":6122},[136],[18,81594,81596],{"className":81595,"style":77169},[12850],[6127,81597,81598],{"xmlns":6129,"width":12854,"height":77172,"viewBox":77173,"preserveAspectRatio":12857},[6133,81599],{"d":77176},[18,81601,152],{"className":81602},[151],[18,81604,81606],{"className":81605},[124],[18,81607,81609],{"className":81608,"style":77186},[128],[18,81610],{},[43727,81612,81613],{},"需要构造压缩方案",[43709,81615,81616,81619,81622,81625,81628],{},[43727,81617,81618],{},"NTK \u002F RKHS 范数",[43727,81620,81621],{},"核函数",[43727,81623,81624],{},"函数空间中的范数",[43727,81626,81627],{},"谱条件",[43727,81629,81630],{},"只适用于懒惰训练",[11,81632,81633,81634,81637],{},"这张表想说明一件事：",[717,81635,81636],{},"「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。",[638,81639,45280],{"id":45280},[11,81641,81642,81643],{},"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：",[717,81644,81645],{},"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":45543,"searchDepth":45544,"depth":45544,"links":81647},[81648,81653,81658,81663,81669,81670,81676,81677],{"id":46073,"depth":45544,"text":46074,"children":81649},[81650,81651,81652],{"id":46077,"depth":45550,"text":46078},{"id":47789,"depth":45550,"text":47790},{"id":50204,"depth":45550,"text":50205},{"id":50989,"depth":45544,"text":50990,"children":81654},[81655,81656,81657],{"id":50993,"depth":45550,"text":50994},{"id":52751,"depth":45550,"text":52752},{"id":64874,"depth":45550,"text":64875},{"id":64965,"depth":45544,"text":64966,"children":81659},[81660,81661,81662],{"id":64969,"depth":45550,"text":64970},{"id":66160,"depth":45550,"text":66161},{"id":69450,"depth":45550,"text":69451},{"id":71642,"depth":45544,"text":71643,"children":81664},[81665,81666,81667,81668],{"id":71646,"depth":45550,"text":71647},{"id":74685,"depth":45550,"text":74686},{"id":75843,"depth":45550,"text":75844},{"id":76056,"depth":45550,"text":76057},{"id":76560,"depth":45544,"text":76561},{"id":77084,"depth":45544,"text":77085,"children":81671},[81672,81673,81674,81675],{"id":77088,"depth":45550,"text":77089},{"id":78588,"depth":45550,"text":78589},{"id":78995,"depth":45550,"text":78996},{"id":80650,"depth":45550,"text":80651},{"id":80727,"depth":45544,"text":80728},{"id":45280,"depth":45544,"text":45280},{},"\u002Fblog\u002F2025\u002F2025-10-05-vapnik-chervonenkis-dimension",{"title":45593,"description":45598},"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。",[45587,45588],"2025-10-05T18:09:22+08:00","6HatO2q1pkHp4S2UlrkM8KhGJ3NYOLL6EHDVErNnkYM",{"id":81687,"title":81688,"body":81689,"description":45543,"draft":45577,"enableCodeHighlight":45577,"enableCodeHightlight":45577,"enableComment":45578,"enableD2":45577,"enableKatex":45578,"extension":45579,"image":45580,"important":45577,"location":45580,"meta":109754,"navigation":45578,"ogImage":45580,"path":109755,"seo":109756,"stem":109757,"summary":109758,"tags":109759,"time":109760,"__hash__":109761},"blog\u002Fblog\u002F2025\u002F2025-09-18-the-stirling-formula.md","斯特林公式",{"type":8,"value":81690,"toc":109750},[81691,81694,81697,81932,81935,82422,82552,83397,84118,84121,84700,84703,85764,85767,85854,87383,87492,88951,88954,89099,89101,89568,89957,90356,91064,91067,91607,91849,93142,93302,93305,93844,94001,94530,94631,96075,96078,96461,96464,96466,96622,96910,96913,97331,97334,98012,98015,98348,98351,99446,99448,99450,99697,99700,99962,99965,99968,100511,101023,101155,101158,101477,101550,102247,102823,102826,103339,103370,103373,103618,103621,103926,103929,104259,104438,104820,104922,105400,105402,105993,106159,106162,106507,106510,106513,106895,107093,107438,107868,107871,108302,108305,108529,108532,109083,109086,109197,109229,109506,109509],[638,81692,81693],{"id":81693},"沃利斯公式",[11,81695,81696],{},"我们先从一个定积分开始：",[18,81698,81700],{"className":81699,"translate":22},[875],[18,81701,81703,81745],{"className":81702,"translate":22},[21],[18,81704,81706],{"className":81705},[26],[28,81707,81708],{"xmlns":30,"display":884},[32,81709,81710,81742],{},[35,81711,81712,81724,81734,81736,81738,81740],{},[1589,81713,81714,81716,81718],{},[49,81715,4005],{},[45,81717,430],{},[5651,81719,81720,81722],{},[41,81721,3941],{},[45,81723,2266],{},[1556,81725,81726,81732],{},[35,81727,81728,81730],{},[41,81729,45637],{},[49,81731,899],{},[41,81733,64],{},[41,81735,38573],{},[926,81737,4024],{},[41,81739,4031],{},[41,81741,38573],{},[86,81743,81744],{"encoding":88},"\\int_{0}^{\\frac{\\pi}{2}} \\sin^n x \\, dx",[18,81746,81748],{"className":81747,"ariaHidden":51},[93],[18,81749,81751,81755,81883,81886,81917,81920,81923,81926,81929],{"className":81750},[97],[18,81752],{"className":81753,"style":81754},[101],"height:2.5089em;vertical-align:-0.9119em;",[18,81756,81758,81761],{"className":81757},[985],[18,81759,4005],{"className":81760,"style":4200},[985,4198,4199],[18,81762,81764],{"className":81763},[115],[18,81765,81767,81874],{"className":81766},[119,120],[18,81768,81770,81871],{"className":81769},[124],[18,81771,81774,81789],{"className":81772,"style":81773},[128],"height:1.597em;",[18,81775,81777,81780],{"style":81776},"top:-2.088em;margin-left:-0.4445em;margin-right:0.05em;",[18,81778],{"className":81779,"style":697},[136],[18,81781,81783],{"className":81782},[141,142,143,144],[18,81784,81786],{"className":81785},[106,144],[18,81787,430],{"className":81788},[106,144],[18,81790,81792,81795],{"style":81791},"top:-4.1129em;margin-right:0.05em;",[18,81793],{"className":81794,"style":697},[136],[18,81796,81798],{"className":81797},[141,142,143,144],[18,81799,81801],{"className":81800},[106,144],[18,81802,81804,81807,81868],{"className":81803},[106,144],[18,81805],{"className":81806},[250,6009,141,1195,38147],[18,81808,81810],{"className":81809},[5651],[18,81811,81813,81859],{"className":81812},[119,120],[18,81814,81816,81856],{"className":81815},[124],[18,81817,81820,81834,81842],{"className":81818,"style":81819},[128],"height:0.6915em;",[18,81821,81822,81825],{"style":38163},[18,81823],{"className":81824,"style":697},[136],[18,81826,81828],{"className":81827},[141,1195,1196,144],[18,81829,81831],{"className":81830},[106,144],[18,81832,2266],{"className":81833},[106,144],[18,81835,81836,81839],{"style":38187},[18,81837],{"className":81838,"style":697},[136],[18,81840],{"className":81841,"style":38194},[6044,144],[18,81843,81844,81847],{"style":42963},[18,81845],{"className":81846,"style":697},[136],[18,81848,81850],{"className":81849},[141,1195,1196,144],[18,81851,81853],{"className":81852},[106,144],[18,81854,3941],{"className":81855,"style":3957},[106,110,144],[18,81857,152],{"className":81858},[151],[18,81860,81862],{"className":81861},[124],[18,81863,81866],{"className":81864,"style":81865},[128],"height:0.344em;",[18,81867],{},[18,81869],{"className":81870},[267,6009,141,1195,38147],[18,81872,152],{"className":81873},[151],[18,81875,81877],{"className":81876},[124],[18,81878,81881],{"className":81879,"style":81880},[128],"height:0.9119em;",[18,81882],{},[18,81884],{"className":81885,"style":170},[169],[18,81887,81889,81892],{"className":81888},[985],[18,81890,45637],{"className":81891},[985],[18,81893,81895],{"className":81894},[115],[18,81896,81898],{"className":81897},[119],[18,81899,81901],{"className":81900},[124],[18,81902,81905],{"className":81903,"style":81904},[128],"height:0.7222em;",[18,81906,81908,81911],{"style":81907},"top:-3.1208em;margin-right:0.05em;",[18,81909],{"className":81910,"style":137},[136],[18,81912,81914],{"className":81913},[141,142,143,144],[18,81915,64],{"className":81916},[106,110,144],[18,81918],{"className":81919,"style":170},[169],[18,81921,38573],{"className":81922},[106,110],[18,81924],{"className":81925,"style":170},[169],[18,81927,4031],{"className":81928},[106,110],[18,81930,38573],{"className":81931},[106,110],[11,81933,81934],{},"我们可以推导出如下结果：",[18,81936,81938],{"className":81937,"translate":22},[875],[18,81939,81941,82032],{"className":81940,"translate":22},[21],[18,81942,81944],{"className":81943},[26],[28,81945,81946],{"xmlns":30,"display":884},[32,81947,81948,82029],{},[35,81949,81950,81962,81972,81974,81976,81978,81980,81982,81984,81996,82012,82014,82016,82018,82020,82023,82025,82027],{},[1589,81951,81952,81954,81956],{},[49,81953,4005],{},[45,81955,430],{},[5651,81957,81958,81960],{},[41,81959,3941],{},[45,81961,2266],{},[1556,81963,81964,81970],{},[35,81965,81966,81968],{},[41,81967,45637],{},[49,81969,899],{},[41,81971,64],{},[41,81973,38573],{},[926,81975,4024],{},[41,81977,4031],{},[41,81979,38573],{},[49,81981,433],{},[49,81983,2253],{},[1589,81985,81986,81988,81990],{},[49,81987,4005],{},[45,81989,430],{},[5651,81991,81992,81994],{},[41,81993,3941],{},[45,81995,2266],{},[1556,81997,81998,82004],{},[35,81999,82000,82002],{},[41,82001,45637],{},[49,82003,899],{},[35,82005,82006,82008,82010],{},[41,82007,64],{},[49,82009,2253],{},[45,82011,47],{},[41,82013,38573],{},[926,82015,4024],{},[41,82017,4031],{},[49,82019,74],{"stretchy":73},[41,82021,82022],{},"cos",[49,82024,899],{},[41,82026,38573],{},[49,82028,84],{"stretchy":73},[86,82030,82031],{"encoding":88},"\\int_{0}^{\\frac{\\pi}{2}} \\sin^n x \\, dx = -\\int_{0}^{\\frac{\\pi}{2}} \\sin^{n-1} x \\, d(\\cos x)",[18,82033,82035,82219],{"className":82034,"ariaHidden":51},[93],[18,82036,82038,82041,82163,82166,82195,82198,82201,82204,82207,82210,82213,82216],{"className":82037},[97],[18,82039],{"className":82040,"style":81754},[101],[18,82042,82044,82047],{"className":82043},[985],[18,82045,4005],{"className":82046,"style":4200},[985,4198,4199],[18,82048,82050],{"className":82049},[115],[18,82051,82053,82155],{"className":82052},[119,120],[18,82054,82056,82152],{"className":82055},[124],[18,82057,82059,82073],{"className":82058,"style":81773},[128],[18,82060,82061,82064],{"style":81776},[18,82062],{"className":82063,"style":697},[136],[18,82065,82067],{"className":82066},[141,142,143,144],[18,82068,82070],{"className":82069},[106,144],[18,82071,430],{"className":82072},[106,144],[18,82074,82075,82078],{"style":81791},[18,82076],{"className":82077,"style":697},[136],[18,82079,82081],{"className":82080},[141,142,143,144],[18,82082,82084],{"className":82083},[106,144],[18,82085,82087,82090,82149],{"className":82086},[106,144],[18,82088],{"className":82089},[250,6009,141,1195,38147],[18,82091,82093],{"className":82092},[5651],[18,82094,82096,82141],{"className":82095},[119,120],[18,82097,82099,82138],{"className":82098},[124],[18,82100,82102,82116,82124],{"className":82101,"style":81819},[128],[18,82103,82104,82107],{"style":38163},[18,82105],{"className":82106,"style":697},[136],[18,82108,82110],{"className":82109},[141,1195,1196,144],[18,82111,82113],{"className":82112},[106,144],[18,82114,2266],{"className":82115},[106,144],[18,82117,82118,82121],{"style":38187},[18,82119],{"className":82120,"style":697},[136],[18,82122],{"className":82123,"style":38194},[6044,144],[18,82125,82126,82129],{"style":42963},[18,82127],{"className":82128,"style":697},[136],[18,82130,82132],{"className":82131},[141,1195,1196,144],[18,82133,82135],{"className":82134},[106,144],[18,82136,3941],{"className":82137,"style":3957},[106,110,144],[18,82139,152],{"className":82140},[151],[18,82142,82144],{"className":82143},[124],[18,82145,82147],{"className":82146,"style":81865},[128],[18,82148],{},[18,82150],{"className":82151},[267,6009,141,1195,38147],[18,82153,152],{"className":82154},[151],[18,82156,82158],{"className":82157},[124],[18,82159,82161],{"className":82160,"style":81880},[128],[18,82162],{},[18,82164],{"className":82165,"style":170},[169],[18,82167,82169,82172],{"className":82168},[985],[18,82170,45637],{"className":82171},[985],[18,82173,82175],{"className":82174},[115],[18,82176,82178],{"className":82177},[119],[18,82179,82181],{"className":82180},[124],[18,82182,82184],{"className":82183,"style":81904},[128],[18,82185,82186,82189],{"style":81907},[18,82187],{"className":82188,"style":137},[136],[18,82190,82192],{"className":82191},[141,142,143,144],[18,82193,64],{"className":82194},[106,110,144],[18,82196],{"className":82197,"style":170},[169],[18,82199,38573],{"className":82200},[106,110],[18,82202],{"className":82203,"style":170},[169],[18,82205,4031],{"className":82206},[106,110],[18,82208,38573],{"className":82209},[106,110],[18,82211],{"className":82212,"style":228},[169],[18,82214,433],{"className":82215},[232],[18,82217],{"className":82218,"style":228},[169],[18,82220,82222,82225,82228,82231,82353,82356,82395,82398,82401,82404,82407,82410,82413,82416,82419],{"className":82221},[97],[18,82223],{"className":82224,"style":81754},[101],[18,82226,2253],{"className":82227},[106],[18,82229],{"className":82230,"style":170},[169],[18,82232,82234,82237],{"className":82233},[985],[18,82235,4005],{"className":82236,"style":4200},[985,4198,4199],[18,82238,82240],{"className":82239},[115],[18,82241,82243,82345],{"className":82242},[119,120],[18,82244,82246,82342],{"className":82245},[124],[18,82247,82249,82263],{"className":82248,"style":81773},[128],[18,82250,82251,82254],{"style":81776},[18,82252],{"className":82253,"style":697},[136],[18,82255,82257],{"className":82256},[141,142,143,144],[18,82258,82260],{"className":82259},[106,144],[18,82261,430],{"className":82262},[106,144],[18,82264,82265,82268],{"style":81791},[18,82266],{"className":82267,"style":697},[136],[18,82269,82271],{"className":82270},[141,142,143,144],[18,82272,82274],{"className":82273},[106,144],[18,82275,82277,82280,82339],{"className":82276},[106,144],[18,82278],{"className":82279},[250,6009,141,1195,38147],[18,82281,82283],{"className":82282},[5651],[18,82284,82286,82331],{"className":82285},[119,120],[18,82287,82289,82328],{"className":82288},[124],[18,82290,82292,82306,82314],{"className":82291,"style":81819},[128],[18,82293,82294,82297],{"style":38163},[18,82295],{"className":82296,"style":697},[136],[18,82298,82300],{"className":82299},[141,1195,1196,144],[18,82301,82303],{"className":82302},[106,144],[18,82304,2266],{"className":82305},[106,144],[18,82307,82308,82311],{"style":38187},[18,82309],{"className":82310,"style":697},[136],[18,82312],{"className":82313,"style":38194},[6044,144],[18,82315,82316,82319],{"style":42963},[18,82317],{"className":82318,"style":697},[136],[18,82320,82322],{"className":82321},[141,1195,1196,144],[18,82323,82325],{"className":82324},[106,144],[18,82326,3941],{"className":82327,"style":3957},[106,110,144],[18,82329,152],{"className":82330},[151],[18,82332,82334],{"className":82333},[124],[18,82335,82337],{"className":82336,"style":81865},[128],[18,82338],{},[18,82340],{"className":82341},[267,6009,141,1195,38147],[18,82343,152],{"className":82344},[151],[18,82346,82348],{"className":82347},[124],[18,82349,82351],{"className":82350,"style":81880},[128],[18,82352],{},[18,82354],{"className":82355,"style":170},[169],[18,82357,82359,82362],{"className":82358},[985],[18,82360,45637],{"className":82361},[985],[18,82363,82365],{"className":82364},[115],[18,82366,82368],{"className":82367},[119],[18,82369,82371],{"className":82370},[124],[18,82372,82375],{"className":82373,"style":82374},[128],"height:0.8719em;",[18,82376,82377,82380],{"style":81907},[18,82378],{"className":82379,"style":137},[136],[18,82381,82383],{"className":82382},[141,142,143,144],[18,82384,82386,82389,82392],{"className":82385},[106,144],[18,82387,64],{"className":82388},[106,110,144],[18,82390,2253],{"className":82391},[2072,144],[18,82393,47],{"className":82394},[106,144],[18,82396],{"className":82397,"style":170},[169],[18,82399,38573],{"className":82400},[106,110],[18,82402],{"className":82403,"style":170},[169],[18,82405,4031],{"className":82406},[106,110],[18,82408,74],{"className":82409},[250],[18,82411,82022],{"className":82412},[985],[18,82414],{"className":82415,"style":170},[169],[18,82417,38573],{"className":82418},[106,110],[18,82420,84],{"className":82421},[267],[11,82423,82424,82425,82551],{},"利用分部积分法，其中 ",[18,82426,82428,82470],{"className":82427,"translate":22},[21],[18,82429,82431],{"className":82430},[26],[28,82432,82433],{"xmlns":30},[32,82434,82435,82467],{},[35,82436,82437,82439,82442,82444,82446,82449,82451,82453,82455,82457,82459,82461,82463,82465],{},[49,82438,4005],{},[41,82440,82441],{},"u",[926,82443,4024],{},[41,82445,4031],{},[41,82447,82448],{},"v",[49,82450,433],{},[41,82452,82441],{},[41,82454,82448],{},[49,82456,2253],{},[49,82458,4005],{},[41,82460,82448],{},[926,82462,4024],{},[41,82464,4031],{},[41,82466,82441],{},[86,82468,82469],{"encoding":88},"\\int u \\, dv = uv - \\int v \\, du",[18,82471,82473,82508,82527],{"className":82472,"ariaHidden":51},[93],[18,82474,82476,82480,82484,82487,82490,82493,82496,82499,82502,82505],{"className":82475},[97],[18,82477],{"className":82478,"style":82479},[101],"height:1.1111em;vertical-align:-0.3061em;",[18,82481,4005],{"className":82482,"style":82483},[985,4198,27578],"margin-right:0.1945em;position:relative;top:-0.0006em;",[18,82485],{"className":82486,"style":170},[169],[18,82488,82441],{"className":82489},[106,110],[18,82491],{"className":82492,"style":170},[169],[18,82494,4031],{"className":82495},[106,110],[18,82497,82448],{"className":82498,"style":3957},[106,110],[18,82500],{"className":82501,"style":228},[169],[18,82503,433],{"className":82504},[232],[18,82506],{"className":82507,"style":228},[169],[18,82509,82511,82514,82518,82521,82524],{"className":82510},[97],[18,82512],{"className":82513,"style":14401},[101],[18,82515,82517],{"className":82516,"style":3957},[106,110],"uv",[18,82519],{"className":82520,"style":2068},[169],[18,82522,2253],{"className":82523},[2072],[18,82525],{"className":82526,"style":2068},[169],[18,82528,82530,82533,82536,82539,82542,82545,82548],{"className":82529},[97],[18,82531],{"className":82532,"style":82479},[101],[18,82534,4005],{"className":82535,"style":82483},[985,4198,27578],[18,82537],{"className":82538,"style":170},[169],[18,82540,82448],{"className":82541,"style":3957},[106,110],[18,82543],{"className":82544,"style":170},[169],[18,82546,4031],{"className":82547},[106,110],[18,82549,82441],{"className":82550},[106,110],"，可得：",[18,82553,82555],{"className":82554,"translate":22},[875],[18,82556,82558,82712],{"className":82557,"translate":22},[21],[18,82559,82561],{"className":82560},[26],[28,82562,82563],{"xmlns":30,"display":884},[32,82564,82565,82709],{},[35,82566,82567,82569,82581,82597,82599,82601,82603,82605,82607,82609,82611,82613,82615,82617,82619,82649,82661,82663,82675,82677,82679,82681,82683,82685],{},[49,82568,2253],{},[1589,82570,82571,82573,82575],{},[49,82572,4005],{},[45,82574,430],{},[5651,82576,82577,82579],{},[41,82578,3941],{},[45,82580,2266],{},[1556,82582,82583,82589],{},[35,82584,82585,82587],{},[41,82586,45637],{},[49,82588,899],{},[35,82590,82591,82593,82595],{},[41,82592,64],{},[49,82594,2253],{},[45,82596,47],{},[41,82598,38573],{},[926,82600,4024],{},[41,82602,4031],{},[49,82604,74],{"stretchy":73},[41,82606,82022],{},[49,82608,899],{},[41,82610,38573],{},[49,82612,84],{"stretchy":73},[169,82614],{"linebreak":8019},[49,82616,433],{},[49,82618,2253],{},[35,82620,82621,82623,82639,82641,82643,82645,82647],{},[49,82622,74],{"fence":51},[1556,82624,82625,82631],{},[35,82626,82627,82629],{},[41,82628,45637],{},[49,82630,899],{},[35,82632,82633,82635,82637],{},[41,82634,64],{},[49,82636,2253],{},[45,82638,47],{},[41,82640,38573],{},[41,82642,82022],{},[49,82644,899],{},[41,82646,38573],{},[49,82648,84],{"fence":51},[1589,82650,82651,82653,82655],{},[49,82652,11222],{"fence":73,"stretchy":51,"minsize":5659,"maxsize":5659},[45,82654,430],{},[5651,82656,82657,82659],{},[41,82658,3941],{},[45,82660,2266],{},[49,82662,7984],{},[1589,82664,82665,82667,82669],{},[49,82666,4005],{},[45,82668,430],{},[5651,82670,82671,82673],{},[41,82672,3941],{},[45,82674,2266],{},[41,82676,82022],{},[49,82678,899],{},[41,82680,38573],{},[926,82682,4024],{},[41,82684,4031],{},[35,82686,82687,82689,82705,82707],{},[49,82688,74],{"fence":51},[1556,82690,82691,82697],{},[35,82692,82693,82695],{},[41,82694,45637],{},[49,82696,899],{},[35,82698,82699,82701,82703],{},[41,82700,64],{},[49,82702,2253],{},[45,82704,47],{},[41,82706,38573],{},[49,82708,84],{"fence":51},[86,82710,82711],{"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)",[18,82713,82715,82917,82920,82932,83189],{"className":82714,"ariaHidden":51},[93],[18,82716,82718,82721,82724,82727,82849,82852,82890,82893,82896,82899,82902,82905,82908,82911,82914],{"className":82717},[97],[18,82719],{"className":82720,"style":81754},[101],[18,82722,2253],{"className":82723},[106],[18,82725],{"className":82726,"style":170},[169],[18,82728,82730,82733],{"className":82729},[985],[18,82731,4005],{"className":82732,"style":4200},[985,4198,4199],[18,82734,82736],{"className":82735},[115],[18,82737,82739,82841],{"className":82738},[119,120],[18,82740,82742,82838],{"className":82741},[124],[18,82743,82745,82759],{"className":82744,"style":81773},[128],[18,82746,82747,82750],{"style":81776},[18,82748],{"className":82749,"style":697},[136],[18,82751,82753],{"className":82752},[141,142,143,144],[18,82754,82756],{"className":82755},[106,144],[18,82757,430],{"className":82758},[106,144],[18,82760,82761,82764],{"style":81791},[18,82762],{"className":82763,"style":697},[136],[18,82765,82767],{"className":82766},[141,142,143,144],[18,82768,82770],{"className":82769},[106,144],[18,82771,82773,82776,82835],{"className":82772},[106,144],[18,82774],{"className":82775},[250,6009,141,1195,38147],[18,82777,82779],{"className":82778},[5651],[18,82780,82782,82827],{"className":82781},[119,120],[18,82783,82785,82824],{"className":82784},[124],[18,82786,82788,82802,82810],{"className":82787,"style":81819},[128],[18,82789,82790,82793],{"style":38163},[18,82791],{"className":82792,"style":697},[136],[18,82794,82796],{"className":82795},[141,1195,1196,144],[18,82797,82799],{"className":82798},[106,144],[18,82800,2266],{"className":82801},[106,144],[18,82803,82804,82807],{"style":38187},[18,82805],{"className":82806,"style":697},[136],[18,82808],{"className":82809,"style":38194},[6044,144],[18,82811,82812,82815],{"style":42963},[18,82813],{"className":82814,"style":697},[136],[18,82816,82818],{"className":82817},[141,1195,1196,144],[18,82819,82821],{"className":82820},[106,144],[18,82822,3941],{"className":82823,"style":3957},[106,110,144],[18,82825,152],{"className":82826},[151],[18,82828,82830],{"className":82829},[124],[18,82831,82833],{"className":82832,"style":81865},[128],[18,82834],{},[18,82836],{"className":82837},[267,6009,141,1195,38147],[18,82839,152],{"className":82840},[151],[18,82842,82844],{"className":82843},[124],[18,82845,82847],{"className":82846,"style":81880},[128],[18,82848],{},[18,82850],{"className":82851,"style":170},[169],[18,82853,82855,82858],{"className":82854},[985],[18,82856,45637],{"className":82857},[985],[18,82859,82861],{"className":82860},[115],[18,82862,82864],{"className":82863},[119],[18,82865,82867],{"className":82866},[124],[18,82868,82870],{"className":82869,"style":82374},[128],[18,82871,82872,82875],{"style":81907},[18,82873],{"className":82874,"style":137},[136],[18,82876,82878],{"className":82877},[141,142,143,144],[18,82879,82881,82884,82887],{"className":82880},[106,144],[18,82882,64],{"className":82883},[106,110,144],[18,82885,2253],{"className":82886},[2072,144],[18,82888,47],{"className":82889},[106,144],[18,82891],{"className":82892,"style":170},[169],[18,82894,38573],{"className":82895},[106,110],[18,82897],{"className":82898,"style":170},[169],[18,82900,4031],{"className":82901},[106,110],[18,82903,74],{"className":82904},[250],[18,82906,82022],{"className":82907},[985],[18,82909],{"className":82910,"style":170},[169],[18,82912,38573],{"className":82913},[106,110],[18,82915,84],{"className":82916},[267],[18,82918],{"className":82919},[169,8019],[18,82921,82923,82926,82929],{"className":82922},[97],[18,82924],{"className":82925,"style":15087},[101],[18,82927,433],{"className":82928},[232],[18,82930],{"className":82931,"style":228},[169],[18,82933,82935,82939,82942,82945,83016,83019,83180,83183,83186],{"className":82934},[97],[18,82936],{"className":82937,"style":82938},[101],"height:2.0867em;vertical-align:-0.6997em;",[18,82940,2253],{"className":82941},[106],[18,82943],{"className":82944,"style":170},[169],[18,82946,82948,82954,82992,82995,82998,83001,83004,83007,83010],{"className":82947},[174],[18,82949,82951],{"className":82950,"style":13793},[250,13792],[18,82952,74],{"className":82953},[1230,1196],[18,82955,82957,82960],{"className":82956},[985],[18,82958,45637],{"className":82959},[985],[18,82961,82963],{"className":82962},[115],[18,82964,82966],{"className":82965},[119],[18,82967,82969],{"className":82968},[124],[18,82970,82972],{"className":82971,"style":82374},[128],[18,82973,82974,82977],{"style":81907},[18,82975],{"className":82976,"style":137},[136],[18,82978,82980],{"className":82979},[141,142,143,144],[18,82981,82983,82986,82989],{"className":82982},[106,144],[18,82984,64],{"className":82985},[106,110,144],[18,82987,2253],{"className":82988},[2072,144],[18,82990,47],{"className":82991},[106,144],[18,82993],{"className":82994,"style":170},[169],[18,82996,38573],{"className":82997},[106,110],[18,82999],{"className":83000,"style":170},[169],[18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\\int_{0}^{\\frac{\\pi}{2}} \\cos^2 x \\cdot (n-1) \\sin^{n-2} x \\, dx \\\\ = (n-1) \\int_{0}^{\\frac{\\pi}{2}} \\left( \\sin^{n-2} x - \\sin^n x \\right) \\, 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\\int_{0}^{\\frac{\\pi}{2}} \\sin^n x \\, dx = (n-1) \\int_{0}^{\\frac{\\pi}{2}} \\sin^{n-2} x \\, dx \\\\ \\implies \\int_{0}^{\\frac{\\pi}{2}} \\sin^n x \\, dx = \\frac{n-1}{n} \\int_{0}^{\\frac{\\pi}{2}} \\sin^{n-2} x \\, 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为偶数时，即 ",[18,85799,85801,85821],{"className":85800,"translate":22},[21],[18,85802,85804],{"className":85803},[26],[28,85805,85806],{"xmlns":30},[32,85807,85808,85818],{},[35,85809,85810,85812,85814,85816],{},[41,85811,64],{},[49,85813,433],{},[45,85815,2266],{},[41,85817,39955],{},[86,85819,85820],{"encoding":88},"n = 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上有 ",[18,90106,90108,90180],{"className":90107,"translate":22},[21],[18,90109,90111],{"className":90110},[26],[28,90112,90113],{"xmlns":30},[32,90114,90115,90177],{},[35,90116,90117,90135,90137,90139,90153,90155,90157,90175],{},[1556,90118,90119,90125],{},[35,90120,90121,90123],{},[41,90122,45637],{},[49,90124,899],{},[35,90126,90127,90129,90131,90133],{},[45,90128,2266],{},[41,90130,39955],{},[49,90132,7984],{},[45,90134,47],{},[41,90136,38573],{},[49,90138,6681],{},[1556,90140,90141,90147],{},[35,90142,90143,90145],{},[41,90144,45637],{},[49,90146,899],{},[35,90148,90149,90151],{},[45,90150,2266],{},[41,90152,39955],{},[41,90154,38573],{},[49,90156,6681],{},[1556,90158,90159,90165],{},[35,90160,90161,90163],{},[41,90162,45637],{},[49,90164,899],{},[35,90166,90167,90169,90171,90173],{},[45,90168,2266],{},[41,90170,39955],{},[49,90172,2253],{},[45,90174,47],{},[41,90176,38573],{},[86,90178,90179],{"encoding":88},"\\sin^{2k+1} x \u003C \\sin^{2k} x \u003C \\sin^{2k-1} x",[18,90181,90183,90246,90302],{"className":90182,"ariaHidden":51},[93],[18,90184,90186,90190,90231,90234,90237,90240,90243],{"className":90185},[97],[18,90187],{"className":90188,"style":90189},[101],"height:0.946em;vertical-align:-0.0391em;",[18,90191,90193,90196],{"className":90192},[985],[18,90194,45637],{"className":90195},[985],[18,90197,90199],{"className":90198},[115],[18,90200,90202],{"className":90201},[119],[18,90203,90205],{"className":90204},[124],[18,90206,90208],{"className":90207,"style":86262},[128],[18,90209,90210,90213],{"style":81907},[18,90211],{"className":90212,"style":137},[136],[18,90214,90216],{"className":90215},[141,142,143,144],[18,90217,90219,90222,90225,90228],{"className":90218},[106,144],[18,90220,2266],{"className":90221},[106,144],[18,90223,39955],{"className":90224,"style":40031},[106,110,144],[18,90226,7984],{"className":90227},[2072,144],[18,90229,47],{"className":90230},[106,144],[18,90232],{"className":90233,"style":170},[169],[18,90235,38573],{"className":90236},[106,110],[18,90238],{"className":90239,"style":228},[169],[18,90241,6681],{"className":90242},[232],[18,90244],{"className":90245,"style":228},[169],[18,90247,90249,90252,90287,90290,90293,90296,90299],{"className":90248},[97],[18,90250],{"className":90251,"style":90189},[101],[18,90253,90255,90258],{"className":90254},[985],[18,90256,45637],{"className":90257},[985],[18,90259,90261],{"className":90260},[115],[18,90262,90264],{"className":90263},[119],[18,90265,90267],{"className":90266},[124],[18,90268,90270],{"className":90269,"style":86262},[128],[18,90271,90272,90275],{"style":81907},[18,90273],{"className":90274,"style":137},[136],[18,90276,90278],{"className":90277},[141,142,143,144],[18,90279,90281,90284],{"className":90280},[106,144],[18,90282,2266],{"className":90283},[106,144],[18,90285,39955],{"className":90286,"style":40031},[106,110,144],[18,90288],{"className":90289,"style":170},[169],[18,90291,38573],{"className":90292},[106,110],[18,90294],{"className":90295,"style":228},[169],[18,90297,6681],{"className":90298},[232],[18,90300],{"className":90301,"style":228},[169],[18,90303,90305,90308,90349,90352],{"className":90304},[97],[18,90306],{"className":90307,"style":86262},[101],[18,90309,90311,90314],{"className":90310},[985],[18,90312,45637],{"className":90313},[985],[18,90315,90317],{"className":90316},[115],[18,90318,90320],{"className":90319},[119],[18,90321,90323],{"className":90322},[124],[18,90324,90326],{"className":90325,"style":86262},[128],[18,90327,90328,90331],{"style":81907},[18,90329],{"className":90330,"style":137},[136],[18,90332,90334],{"className":90333},[141,142,143,144],[18,90335,90337,90340,90343,90346],{"className":90336},[106,144],[18,90338,2266],{"className":90339},[106,144],[18,90341,39955],{"className":90342,"style":40031},[106,110,144],[18,90344,2253],{"className":90345},[2072,144],[18,90347,47],{"className":90348},[106,144],[18,90350],{"className":90351,"style":170},[169],[18,90353,38573],{"className":90354},[106,110],"，因此有：",[18,90357,90359],{"className":90358,"translate":22},[875],[18,90360,90362,90488],{"className":90361,"translate":22},[21],[18,90363,90365],{"className":90364},[26],[28,90366,90367],{"xmlns":30,"display":884},[32,90368,90369,90485],{},[35,90370,90371,90383,90401,90403,90405,90407,90409,90411,90423,90437,90439,90441,90443,90445,90447,90459,90477,90479,90481,90483],{},[1589,90372,90373,90375,90377],{},[49,90374,4005],{},[45,90376,430],{},[5651,90378,90379,90381],{},[41,90380,3941],{},[45,90382,2266],{},[1556,90384,90385,90391],{},[35,90386,90387,90389],{},[41,90388,45637],{},[49,90390,899],{},[35,90392,90393,90395,90397,90399],{},[45,90394,2266],{},[41,90396,39955],{},[49,90398,7984],{},[45,90400,47],{},[41,90402,38573],{},[926,90404,4024],{},[41,90406,4031],{},[41,90408,38573],{},[49,90410,6681],{},[1589,90412,90413,90415,90417],{},[49,90414,4005],{},[45,90416,430],{},[5651,90418,90419,90421],{},[41,90420,3941],{},[45,90422,2266],{},[1556,90424,90425,90431],{},[35,90426,90427,90429],{},[41,90428,45637],{},[49,90430,899],{},[35,90432,90433,90435],{},[45,90434,2266],{},[41,90436,39955],{},[41,90438,38573],{},[926,90440,4024],{},[41,90442,4031],{},[41,90444,38573],{},[49,90446,6681],{},[1589,90448,90449,90451,90453],{},[49,90450,4005],{},[45,90452,430],{},[5651,90454,90455,90457],{},[41,90456,3941],{},[45,90458,2266],{},[1556,90460,90461,90467],{},[35,90462,90463,90465],{},[41,90464,45637],{},[49,90466,899],{},[35,90468,90469,90471,90473,90475],{},[45,90470,2266],{},[41,90472,39955],{},[49,90474,2253],{},[45,90476,47],{},[41,90478,38573],{},[926,90480,4024],{},[41,90482,4031],{},[41,90484,38573],{},[86,90486,90487],{"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 \\, dx",[18,90489,90491,90687,90877],{"className":90490,"ariaHidden":51},[93],[18,90492,90494,90497,90619,90622,90663,90666,90669,90672,90675,90678,90681,90684],{"className":90493},[97],[18,90495],{"className":90496,"style":81754},[101],[18,90498,90500,90503],{"className":90499},[985],[18,90501,4005],{"className":90502,"style":4200},[985,4198,4199],[18,90504,90506],{"className":90505},[115],[18,90507,90509,90611],{"className":90508},[119,120],[18,90510,90512,90608],{"className":90511},[124],[18,90513,90515,90529],{"className":90514,"style":81773},[128],[18,90516,90517,90520],{"style":81776},[18,90518],{"className":90519,"style":697},[136],[18,90521,90523],{"className":90522},[141,142,143,144],[18,90524,90526],{"className":90525},[106,144],[18,90527,430],{"className":90528},[106,144],[18,90530,90531,90534],{"style":81791},[18,90532],{"className":90533,"style":697},[136],[18,90535,90537],{"className":90536},[141,142,143,144],[18,90538,90540],{"className":90539},[106,144],[18,90541,90543,90546,90605],{"className":90542},[106,144],[18,90544],{"className":90545},[250,6009,141,1195,38147],[18,90547,90549],{"className":90548},[5651],[18,90550,90552,90597],{"className":90551},[119,120],[18,90553,90555,90594],{"className":90554},[124],[18,90556,90558,90572,90580],{"className":90557,"style":81819},[128],[18,90559,90560,90563],{"style":38163},[18,90561],{"className":90562,"style":697},[136],[18,90564,90566],{"className":90565},[141,1195,1196,144],[18,90567,90569],{"className":90568},[106,144],[18,90570,2266],{"className":90571},[106,144],[18,90573,90574,90577],{"style":38187},[18,90575],{"className":90576,"style":697},[136],[18,90578],{"className":90579,"style":38194},[6044,144],[18,90581,90582,90585],{"style":42963},[18,90583],{"className":90584,"style":697},[136],[18,90586,90588],{"className":90587},[141,1195,1196,144],[18,90589,90591],{"className":90590},[106,144],[18,90592,3941],{"className":90593,"style":3957},[106,110,144],[18,90595,152],{"className":90596},[151],[18,90598,90600],{"className":90599},[124],[18,90601,90603],{"className":90602,"style":81865},[128],[18,90604],{},[18,90606],{"className":90607},[267,6009,141,1195,38147],[18,90609,152],{"className":90610},[151],[18,90612,90614],{"className":90613},[124],[18,90615,90617],{"className":90616,"style":81880},[128],[18,90618],{},[18,90620],{"className":90621,"style":170},[169],[18,90623,90625,90628],{"className":90624},[985],[18,90626,45637],{"className":90627},[985],[18,90629,90631],{"className":90630},[115],[18,90632,90634],{"className":90633},[119],[18,90635,90637],{"className":90636},[124],[18,90638,90640],{"className":90639,"style":86262},[128],[18,90641,90642,90645],{"style":81907},[18,90643],{"className":90644,"style":137},[136],[18,90646,90648],{"className":90647},[141,142,143,144],[18,90649,90651,90654,90657,90660],{"className":90650},[106,144],[18,90652,2266],{"className":90653},[106,144],[18,90655,39955],{"className":90656,"style":40031},[106,110,144],[18,90658,7984],{"className":90659},[2072,144],[18,90661,47],{"className":90662},[106,144],[18,90664],{"className":90665,"style":170},[169],[18,90667,38573],{"className":90668},[106,110],[18,90670],{"className":90671,"style":170},[169],[18,90673,4031],{"className":90674},[106,110],[18,90676,38573],{"className":90677},[106,110],[18,90679],{"className":90680,"style":228},[169],[18,90682,6681],{"className":90683},[232],[18,90685],{"className":90686,"style":228},[169],[18,90688,90690,90693,90815,90818,90853,90856,90859,90862,90865,90868,90871,90874],{"className":90689},[97],[18,90691],{"className":90692,"style":81754},[101],[18,90694,90696,90699],{"className":90695},[985],[18,90697,4005],{"className":90698,"style":4200},[985,4198,4199],[18,90700,90702],{"className":90701},[115],[18,90703,90705,90807],{"className":90704},[119,120],[18,90706,90708,90804],{"className":90707},[124],[18,90709,90711,90725],{"className":90710,"style":81773},[128],[18,90712,90713,90716],{"style":81776},[18,90714],{"className":90715,"style":697},[136],[18,90717,90719],{"className":90718},[141,142,143,144],[18,90720,90722],{"className":90721},[106,144],[18,90723,430],{"className":90724},[106,144],[18,90726,90727,90730],{"style":81791},[18,90728],{"className":90729,"style":697},[136],[18,90731,90733],{"className":90732},[141,142,143,144],[18,90734,90736],{"className":90735},[106,144],[18,90737,90739,90742,90801],{"className":90738},[106,144],[18,90740],{"className":90741},[250,6009,141,1195,38147],[18,90743,90745],{"className":90744},[5651],[18,90746,90748,90793],{"className":90747},[119,120],[18,90749,90751,90790],{"className":90750},[124],[18,90752,90754,90768,90776],{"className":90753,"style":81819},[128],[18,90755,90756,90759],{"style":38163},[18,90757],{"className":90758,"style":697},[136],[18,90760,90762],{"className":90761},[141,1195,1196,144],[18,90763,90765],{"className":90764},[106,144],[18,90766,2266],{"className":90767},[106,144],[18,90769,90770,90773],{"style":38187},[18,90771],{"className":90772,"style":697},[136],[18,907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\u003C \\frac{(2k-1)!!}{(2k)!!} \\cdot \\frac{\\pi}{2} \u003C \\frac{(2k-2)!!}{(2k-1)!!}",[18,91204,91206,91313,91420,91497],{"className":91205,"ariaHidden":51},[93],[18,91207,91209,91212,91304,91307,91310],{"className":91208},[97],[18,91210],{"className":91211,"style":89396},[101],[18,91213,91215,91218,91301],{"className":91214},[106],[18,91216],{"className":91217},[250,6009],[18,91219,91221],{"className":91220},[5651],[18,91222,91224,91293],{"className":91223},[119,120],[18,91225,91227,91290],{"className":91226},[124],[18,91228,91230,91262,91270],{"className":91229,"style":18128},[128],[18,91231,91232,91235],{"style":6025},[18,91233],{"className":91234,"style":697},[136],[18,91236,91238,91241,91244,91247,91250,91253,91256,91259],{"className":91237},[106],[18,91239,74],{"className":91240},[250],[18,91242,2266],{"className":91243},[106],[18,91245,39955],{"className":91246,"style":40031},[106,110],[18,91248],{"className":91249,"style":2068},[169],[18,91251,7984],{"className":91252},[2072],[18,91254],{"className":91255,"style":2068},[169],[18,91257,47],{"className":91258},[106],[18,91260,89435],{"className":91261},[267],[18,91263,91264,91267],{"style":6037},[18,91265],{"className":91266,"style":697},[136],[18,91268],{"className":91269,"style":6045},[6044],[18,91271,91272,91275],{"style":6048},[18,91273],{"className":91274,"style":697},[136],[18,91276,91278,91281,91284,91287],{"className":91277},[106],[18,91279,74],{"className":91280},[250],[18,91282,2266],{"className":91283},[106],[18,91285,39955],{"className":91286,"style":40031},[106,110],[18,91288,89435],{"className":91289},[267],[18,91291,152],{"className":91292},[151],[18,91294,91296],{"className":91295},[124],[18,91297,91299],{"className":91298,"style":89485},[128],[18,91300],{},[18,91302],{"className":91303},[267,6009],[18,91305],{"className":91306,"style":228},[169],[18,91308,6681],{"className":91309},[232],[18,91311],{"className":91312,"style":228},[169],[18,91314,91316,91319,91411,91414,91417],{"className":91315},[97],[18,91317],{"className":91318,"style":89396},[101],[18,91320,91322,91325,91408],{"className":91321},[106],[18,91323],{"className":91324},[250,6009],[18,91326,91328],{"className":91327},[5651],[18,91329,91331,91400],{"className":91330},[119,120],[18,91332,91334,91397],{"className":91333},[124],[18,91335,91337,91357,91365],{"className":91336,"style":18128},[128],[18,91338,91339,91342],{"style":6025},[18,91340],{"className":91341,"style":697},[136],[18,91343,91345,91348,91351,91354],{"className":91344},[106],[18,91346,74],{"className":91347},[250],[18,91349,2266],{"className":91350},[106],[18,91352,39955],{"className":91353,"style":40031},[106,110],[18,91355,89435],{"className":91356},[267],[18,91358,91359,91362],{"style":6037},[18,91360],{"className":91361,"style":697},[136],[18,91363],{"className":91364,"style":6045},[6044],[18,91366,91367,91370],{"style":6048},[18,91368],{"className":91369,"style":697},[136],[18,91371,91373,91376,91379,91382,91385,91388,91391,91394],{"className":91372},[106],[18,91374,74],{"className":91375},[250],[18,91377,2266],{"className":91378},[106],[18,91380,39955],{"className":91381,"style":40031},[106,110],[18,91383],{"className":91384,"style":2068},[169],[18,91386,2253],{"className":91387},[2072],[18,91389],{"className":91390,"style":2068},[169],[18,91392,47],{"className":91393},[106],[18,91395,89435],{"className":91396},[267],[18,91398,152],{"className":91399},[151],[18,91401,91403],{"className":91402},[124],[18,91404,91406],{"className":91405,"style":89485},[128],[18,91407],{},[18,91409],{"className":91410},[267,6009],[18,91412],{"className":91413,"style":2068},[169],[18,91415,26833],{"className":91416},[2072],[18,91418],{"className":91419,"style":2068},[169],[18,91421,91423,91426,91488,91491,91494],{"className":91422},[97],[18,91424],{"className":91425,"style":42062},[101],[18,91427,91429,91432,91485],{"className":91428},[106],[18,91430],{"className":91431},[250,6009],[18,91433,91435],{"className":91434},[5651],[18,91436,91438,91477],{"className":91437},[119,120],[18,91439,91441,91474],{"className":91440},[124],[18,91442,91444,91455,91463],{"className":91443,"style":13849},[128],[18,91445,91446,91449],{"style":6025},[18,91447],{"className":91448,"style":697},[136],[18,91450,91452],{"className":91451},[106],[18,91453,2266],{"className":91454},[106],[18,91456,91457,91460],{"style":6037},[18,91458],{"className":91459,"style":697},[136],[18,91461],{"className":91462,"style":6045},[6044],[18,91464,91465,91468],{"style":6048},[18,91466],{"className":91467,"style":697},[136],[18,91469,91471],{"className":91470},[106],[18,91472,3941],{"className":91473,"style":3957},[106,110],[18,91475,152],{"className":91476},[151],[18,91478,91480],{"className":91479},[124],[18,91481,91483],{"className":91482,"style":6067},[128],[18,91484],{},[18,91486],{"className":91487},[267,6009],[18,91489],{"className":91490,"style":228},[169],[18,91492,6681],{"className":91493},[232],[18,91495],{"className":91496,"style":228},[169],[18,91498,91500,91503],{"className":91499},[97],[18,91501],{"className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",[18,91611,91613,91659],{"className":91612,"translate":22},[21],[18,91614,91616],{"className":91615},[26],[28,91617,91618],{"xmlns":30},[32,91619,91620,91656],{},[35,91621,91622,91634,91648,91650,91652,91654],{},[1589,91623,91624,91626,91628],{},[49,91625,4005],{},[45,91627,430],{},[5651,91629,91630,91632],{},[41,91631,3941],{},[45,91633,2266],{},[1556,91635,91636,91642],{},[35,91637,91638,91640],{},[41,91639,45637],{},[49,91641,899],{},[35,91643,91644,91646],{},[45,91645,2266],{},[41,91647,39955],{},[41,91649,38573],{},[926,91651,4024],{},[41,91653,4031],{},[41,91655,38573],{},[86,91657,91658],{"encoding":88},"\\int_{0}^{\\frac{\\pi}{2}} \\sin^{2k} x \\, 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上下界之间的差：",[18,91850,91852],{"className":91851,"translate":22},[875],[18,91853,91855,92166],{"className":91854,"translate":22},[21],[18,91856,91858],{"className":91857},[26],[28,91859,91860],{"xmlns":30,"display":884},[32,91861,91862,92163],{},[35,91863,91864,91881,91961,91963,91965,91981,92085,92087,92089,92105,92119,92121,92159,92161],{},[890,91865,91866,91873],{},[35,91867,91868,91871],{},[41,91869,91870],{},"lim",[49,91872,899],{},[35,91874,91875,91877,91879],{},[41,91876,39955],{},[49,91878,1880],{},[41,91880,2028],{"mathvariant":316},[35,91882,91883,91885,91923,91925,91959],{},[49,91884,74],{"fence":51},[5651,91886,91887,91905],{},[35,91888,91889,91891,91893,91895,91897,91899,91901,91903],{},[49,91890,74],{"stretchy":73},[45,91892,2266],{},[41,91894,39955],{},[49,91896,2253],{},[45,91898,2266],{},[49,91900,84],{"stretchy":73},[49,91902,88973],{"stretchy":73},[49,91904,88973],{"stretchy":73},[35,91906,91907,91909,91911,91913,91915,91917,91919,91921],{},[49,91908,74],{"stretchy":73},[45,91910,2266],{},[41,91912,39955],{},[49,91914,2253],{},[45,91916,47],{},[49,91918,84],{"stretchy":73},[49,91920,88973],{"stretchy":73},[49,91922,88973],{"stretchy":73},[49,91924,2253],{},[5651,91926,91927,91941],{},[35,91928,91929,91931,91933,91935,91937,91939],{},[49,91930,74],{"stretchy":73},[45,91932,2266],{},[41,91934,39955],{},[49,91936,84],{"stretchy":73},[49,91938,88973],{"stretchy":73},[49,91940,88973],{"stretchy":73},[35,91942,91943,91945,91947,91949,91951,91953,91955,91957],{},[49,91944,74],{"stretchy":73},[45,91946,2266],{},[41,91948,39955],{},[49,91950,7984],{},[45,91952,47],{},[49,91954,84],{"stretchy":73},[49,91956,88973],{"stretchy":73},[49,91958,88973],{"stretchy":73},[49,91960,84],{"fence":51},[169,91962],{"linebreak":8019},[49,91964,433],{},[890,91966,91967,91973],{},[35,91968,91969,91971],{},[41,91970,91870],{},[49,91972,899],{},[35,91974,91975,91977,91979],{},[41,91976,39955],{},[49,91978,1880],{},[41,91980,2028],{"mathvariant":316},[35,91982,91983,91985,92023,92025,92043,92045,92083],{},[49,91984,74],{"fence":51},[5651,91986,91987,92005],{},[35,91988,91989,91991,91993,91995,91997,91999,92001,92003],{},[49,91990,74],{"stretchy":73},[45,91992,2266],{},[41,91994,39955],{},[49,91996,2253],{},[45,91998,2266],{},[49,92000,84],{"stretchy":73},[49,92002,88973],{"stretchy":73},[49,92004,88973],{"stretchy":73},[35,92006,92007,92009,92011,92013,92015,92017,92019,92021],{},[49,92008,74],{"stretchy":73},[45,92010,2266],{},[41,92012,39955],{},[49,92014,2253],{},[45,92016,47],{},[49,92018,84],{"stretchy":73},[49,92020,88973],{"stretchy":73},[49,92022,88973],{"stretchy":73},[49,92024,2253],{},[5651,92026,92027,92033],{},[35,92028,92029,92031],{},[45,92030,2266],{},[41,92032,39955],{},[35,92034,92035,92037,92039,92041],{},[45,92036,2266],{},[41,92038,39955],{},[49,92040,7984],{},[45,92042,47],{},[49,92044,26833],{},[5651,92046,92047,92065],{},[35,92048,92049,92051,92053,92055,92057,92059,92061,92063],{},[49,92050,74],{"stretchy":73},[45,92052,2266],{},[41,92054,39955],{},[49,92056,2253],{},[45,92058,2266],{},[49,92060,84],{"stretchy":73},[49,92062,88973],{"stretchy":73},[49,92064,88973],{"stretchy":73},[35,92066,92067,92069,92071,92073,92075,92077,92079,92081],{},[49,92068,74],{"stretchy":73},[45,92070,2266],{},[41,92072,39955],{},[49,92074,2253],{},[45,92076,47],{},[49,92078,84],{"stretchy":73},[49,92080,88973],{"stretchy":73},[49,92082,88973],{"stretchy":73},[49,92084,84],{"fence":51},[169,92086],{"linebreak":8019},[49,92088,433],{},[890,92090,92091,92097],{},[35,92092,92093,92095],{},[41,92094,91870],{},[49,92096,899],{},[35,92098,92099,92101,92103],{},[41,92100,39955],{},[49,92102,1880],{},[41,92104,2028],{"mathvariant":316},[5651,92106,92107,92109],{},[45,92108,47],{},[35,92110,92111,92113,92115,92117],{},[45,92112,2266],{},[41,92114,39955],{},[49,92116,7984],{},[45,92118,47],{},[49,92120,26833],{},[5651,92122,92123,92141],{},[35,92124,92125,92127,92129,92131,92133,92135,92137,92139],{},[49,92126,74],{"stretchy":73},[45,92128,2266],{},[41,92130,39955],{},[49,92132,2253],{},[45,92134,2266],{},[49,92136,84],{"stretchy":73},[49,92138,88973],{"stretchy":73},[49,92140,88973],{"stretchy":73},[35,92142,92143,92145,92147,92149,92151,92153,92155,92157],{},[49,92144,74],{"stretchy":73},[45,92146,2266],{},[41,92148,39955],{},[49,92150,2253],{},[45,92152,47],{},[49,92154,84],{"stretchy":73},[49,92156,88973],{"stretchy":73},[49,92158,88973],{"stretchy":73},[49,92160,433],{},[45,92162,430],{},[86,92164,92165],{"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",[18,93187,93189,93293],{"className":93188,"ariaHidden":51},[93],[18,93190,93192,93196,93284,93287,93290],{"className":93191},[97],[18,93193],{"className":93194,"style":93195},[101],"height:1.2834em;vertical-align:-0.4033em;",[18,93197,93199,93202,93281],{"className":93198},[106],[18,93200],{"className":93201},[250,6009],[18,93203,93205],{"className":93204},[5651],[18,93206,93208,93272],{"className":93207},[119,120],[18,93209,93211,93269],{"className":93210},[124],[18,93212,93215,93238,93246],{"className":93213,"style":93214},[128],"height:0.8801em;",[18,93216,93217,93220],{"style":12045},[18,93218],{"className":93219,"style":697},[136],[18,93221,93223],{"className":93222},[141,142,143,144],[18,93224,93226,93229,93232,93235],{"className":93225},[106,144],[18,93227,2266],{"className":93228},[106,144],[18,93230,39955],{"className":93231,"style":40031},[106,110,144],[18,93233,2253],{"className":93234},[2072,144],[18,93236,47],{"className":93237},[106,144],[18,93239,93240,93243],{"style":6037},[18,93241],{"className":93242,"style":697},[136],[18,93244],{"className":93245,"style":6045},[6044],[18,93247,93248,93251],{"style":12068},[18,93249],{"className":93250,"style":697},[136],[18,93252,93254],{"className":93253},[141,142,143,144],[18,93255,93257,93260,93263,93266],{"className":93256},[106,144],[18,93258,2266],{"className":93259},[106,144],[18,93261,39955],{"className":93262,"style":40031},[106,110,144],[18,93264,2253],{"className":93265},[2072,144],[18,93267,2266],{"className":93268},[106,144],[18,93270,152],{"className":93271},[151],[18,93273,93275],{"className":93274},[124],[18,93276,93279],{"className":93277,"style":93278},[128],"height:0.4033em;",[18,93280],{},[18,93282],{"className":93283},[267,6009],[18,93285],{"className":93286,"style":228},[169],[18,93288,6681],{"className":93289},[232],[18,93291],{"className":93292,"style":228},[169],[18,93294,93296,93299],{"className":93295},[97],[18,93297],{"className":93298,"style":505},[101],[18,93300,47],{"className":93301},[106],[11,93303,93304],{},"对于不等式：",[18,93306,93308],{"className":93307,"translate":22},[875],[18,93309,93311,93440],{"className":93310,"translate":22},[21],[18,93312,93314],{"className":93313},[26],[28,93315,93316],{"xmlns":30,"display":884},[32,93317,93318,93438],{},[35,93319,93320,93354,93356,93390,93392,93398,93400],{},[5651,93321,93322,93336],{},[35,93323,93324,93326,93328,93330,93332,93334],{},[49,93325,74],{"stretchy":73},[45,93327,2266],{},[41,93329,39955],{},[49,93331,84],{"stretchy":73},[49,93333,88973],{"stretchy":73},[49,93335,88973],{"stretchy":73},[35,93337,93338,93340,93342,93344,93346,93348,93350,93352],{},[49,93339,74],{"stretchy":73},[45,93341,2266],{},[41,93343,39955],{},[49,93345,7984],{},[45,93347,47],{},[49,93349,84],{"stretchy":73},[49,93351,88973],{"stretchy":73},[49,93353,88973],{"stretchy":73},[49,93355,6681],{},[5651,93357,93358,93376],{},[35,93359,93360,93362,93364,93366,93368,93370,93372,93374],{},[49,93361,74],{"stretchy":73},[45,93363,2266],{},[41,93365,39955],{},[49,93367,2253],{},[45,93369,47],{},[49,93371,84],{"stretchy":73},[49,93373,88973],{"stretchy":73},[49,93375,88973],{"stretchy":73},[35,93377,93378,93380,93382,93384,93386,93388],{},[49,93379,74],{"stretchy":73},[45,93381,2266],{},[41,93383,39955],{},[49,93385,84],{"stretchy":73},[49,93387,88973],{"stretchy":73},[49,93389,88973],{"stretchy":73},[49,93391,26833],{},[5651,93393,93394,93396],{},[41,93395,3941],{},[45,93397,2266],{},[49,93399,6681],{},[5651,93401,93402,93420],{},[35,93403,93404,93406,93408,93410,93412,93414,93416,93418],{},[49,93405,74],{"stretchy":73},[45,93407,2266],{},[41,93409,39955],{},[49,93411,2253],{},[45,93413,2266],{},[49,93415,84],{"stretchy":73},[49,93417,88973],{"stretchy":73},[49,93419,88973],{"stretchy":73},[35,93421,93422,93424,93426,93428,93430,93432,93434,93436],{},[49,93423,74],{"stretchy":73},[45,93425,2266],{},[41,93427,39955],{},[49,93429,2253],{},[45,93431,47],{},[49,93433,84],{"stretchy":73},[49,93435,88973],{"stretchy":73},[49,93437,88973],{"stretchy":73},[86,93439,91202],{"encoding":88},[18,93441,93443,93550,93657,93734],{"className":93442,"ariaHidden":51},[93],[18,93444,93446,93449,93541,93544,93547],{"className":93445},[97],[18,93447],{"className":93448,"style":89396},[101],[18,93450,93452,93455,93538],{"className":93451},[106],[18,93453],{"className":93454},[250,6009],[18,93456,93458],{"className":93457},[5651],[18,93459,93461,93530],{"className":93460},[119,120],[18,93462,93464,93527],{"className":93463},[124],[18,93465,93467,93499,93507],{"className":93466,"style":18128},[128],[18,93468,93469,93472],{"style":6025},[18,93470],{"className":93471,"style":697},[136],[18,93473,93475,93478,93481,93484,93487,93490,93493,93496],{"className":93474},[106],[18,93476,74],{"className":93477},[250],[18,93479,2266],{"className":93480},[106],[18,93482,39955],{"className":93483,"style":40031},[106,110],[18,93485],{"className":93486,"style":2068},[169],[18,93488,7984],{"className":93489},[2072],[18,93491],{"className":93492,"style":2068},[169],[18,93494,47],{"className":93495},[106],[18,93497,89435],{"className":93498},[267],[18,93500,93501,93504],{"style":6037},[18,93502],{"className":93503,"style":697},[136],[18,93505],{"className":93506,"style":6045},[6044],[18,93508,93509,93512],{"style":6048},[18,93510],{"className":93511,"style":697},[136],[18,93513,93515,93518,93521,93524],{"className":93514},[106],[18,93516,74],{"className":93517},[250],[18,93519,2266],{"className":93520},[106],[18,93522,39955],{"className":93523,"style":40031},[106,110],[18,93525,89435],{"className":93526},[267],[18,93528,152],{"className":93529},[151],[18,93531,93533],{"className":93532},[124],[18,93534,93536],{"className":93535,"style":89485},[128],[18,93537],{},[18,93539],{"className":93540},[267,6009],[18,93542],{"className":93543,"style":228},[169],[18,93545,6681],{"className":93546},[232],[18,93548],{"className":93549,"style":228},[169],[18,93551,93553,93556,93648,93651,93654],{"className":93552},[97],[18,93554],{"className":93555,"style":89396},[101],[18,93557,93559,93562,93645],{"className":93558},[106],[18,93560],{"className":93561},[250,6009],[18,93563,93565],{"className":93564},[5651],[18,93566,93568,93637],{"className":93567},[119,120],[18,93569,93571,93634],{"className":93570},[124],[18,93572,93574,93594,93602],{"className":93573,"style":18128},[128],[18,93575,93576,93579],{"style":6025},[18,93577],{"className":93578,"style":697},[136],[18,93580,93582,93585,93588,93591],{"className":93581},[106],[18,93583,74],{"className":93584},[250],[18,93586,2266],{"className":93587},[106],[18,93589,39955],{"className":93590,"style":40031},[106,110],[18,93592,89435],{"className":93593},[267],[18,93595,93596,93599],{"style":6037},[18,93597],{"className":93598,"style":697},[136],[18,93600],{"className":93601,"style":6045},[6044],[18,93603,93604,93607],{"style":6048},[18,93605],{"className":93606,"style":697},[136],[18,93608,93610,93613,93616,93619,93622,93625,93628,93631],{"className":93609},[106],[18,93611,74],{"className":93612},[250],[18,93614,2266],{"className":93615},[106],[18,93617,39955],{"className":93618,"style":40031},[106,110],[18,93620],{"className":93621,"style":2068},[169],[18,93623,2253],{"className":93624},[2072],[18,93626],{"className":93627,"style":2068},[169],[18,93629,47],{"className":93630},[106],[18,93632,89435],{"className":93633},[267],[18,93635,152],{"className":93636},[151],[18,93638,93640],{"className":93639},[124],[18,93641,93643],{"className":93642,"style":89485},[128],[18,93644],{},[18,93646],{"className":93647},[267,6009],[18,93649],{"className":93650,"style":2068},[169],[18,93652,26833],{"className":93653},[2072],[18,93655],{"className":93656,"style":2068},[169],[18,93658,93660,93663,93725,93728,93731],{"className":93659},[97],[18,93661],{"className":93662,"style":42062},[101],[18,93664,93666,93669,93722],{"className":93665},[106],[18,93667],{"className":93668},[250,6009],[18,93670,93672],{"className":93671},[5651],[18,93673,93675,93714],{"className":93674},[119,120],[18,93676,93678,93711],{"className":93677},[124],[18,93679,93681,93692,93700],{"className":93680,"style":13849},[128],[18,93682,93683,93686],{"style":6025},[18,93684],{"className":93685,"style":697},[136],[18,93687,93689],{"className":93688},[106],[18,93690,2266],{"className":93691},[106],[18,93693,93694,93697],{"style":6037},[18,93695],{"className":93696,"style":697},[136],[18,93698],{"className":93699,"style":6045},[6044],[18,93701,93702,93705],{"style":6048},[18,93703],{"className":93704,"style":697},[136],[18,93706,93708],{"className":93707},[106],[18,93709,3941],{"className":93710,"style":3957},[106,110],[18,93712,152],{"className":93713},[151],[18,93715,93717],{"className":93716},[124],[18,93718,93720],{"className":93719,"style":6067},[128],[18,93721],{},[18,93723],{"className":93724},[267,6009],[18,93726],{"className":93727,"style":228},[169],[18,93729,6681],{"className":93730},[232],[18,93732],{"className":93733,"style":228},[169],[18,93735,93737,93740],{"className":93736},[97],[18,93738],{"className":93739,"style":89396},[101],[18,93741,93743,93746,93841],{"className":93742},[106],[18,93744],{"className":93745},[250,6009],[18,93747,93749],{"className":93748},[5651],[18,93750,93752,93833],{"className":93751},[119,120],[18,93753,93755,93830],{"className":93754},[124],[18,93756,93758,93790,93798],{"className":93757,"style":18128},[128],[18,93759,93760,93763],{"style":6025},[18,93761],{"className":93762,"style":697},[136],[18,93764,93766,93769,93772,93775,93778,93781,93784,93787],{"className":93765},[106],[18,93767,74],{"className":93768},[250],[18,93770,2266],{"className":93771},[106],[18,93773,39955],{"className":93774,"style":40031},[106,110],[18,93776],{"className":93777,"style":2068},[169],[18,93779,2253],{"className":93780},[2072],[18,93782],{"className":93783,"style":2068},[169],[18,93785,47],{"className":93786},[106],[18,93788,89435],{"className":93789},[267],[18,93791,93792,93795],{"style":6037},[18,93793],{"className":93794,"style":697},[136],[18,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\\rightarrow \\infty} \\frac{(2k)!!^2}{(2k+1)!! (2k-1)!!} = \\frac{\\pi}{2}",[18,96175,96177,96393],{"className":96176,"ariaHidden":51},[93],[18,96178,96180,96183,96236,96239,96384,96387,96390],{"className":96179},[97],[18,96181],{"className":96182,"style":94146},[101],[18,96184,96186],{"className":96185},[985,986],[18,96187,96189,96228],{"className":96188},[119,120],[18,96190,96192,96225],{"className":96191},[124],[18,96193,96195,96215],{"className":96194,"style":295},[128],[18,96196,96197,96200],{"style":29361},[18,96198],{"className":96199,"style":697},[136],[18,96201,96203],{"className":96202},[141,142,143,144],[18,96204,96206,96209,96212],{"className":96205},[106,144],[18,96207,39955],{"className":96208,"style":40031},[106,110,144],[18,96210,1880],{"className":96211},[232,144],[18,96213,2028],{"className":96214},[106,144],[18,96216,96217,96220],{"style":693},[18,96218],{"className":96219,"style":697},[136],[18,96221,96222],{},[18,96223,91870],{"className":96224},[985],[18,96226,152],{"className":96227},[151],[18,96229,96231],{"className":96230},[124],[18,96232,96234],{"className":96233,"style":92226},[128],[18,96235],{},[18,96237],{"className":96238,"style":170},[169],[18,96240,96242,96245,96381],{"className":96241},[106],[18,96243],{"className":96244},[250,6009],[18,96246,96248],{"className":96247},[5651],[18,96249,96251,96373],{"className":96250},[119,120],[18,96252,96254,96370],{"className":96253},[124],[18,96255,96257,96313,96321],{"className":96256,"style":15996},[128],[18,96258,96259,96262],{"style":6025},[18,96260],{"className":96261,"style":697},[136],[18,96263,96265,96268,96271,96274,96277,96280,96283,96286,96289,96292,96295,96298,96301,96304,96307,96310],{"className":96264},[106],[18,96266,74],{"className":96267},[250],[18,96269,2266],{"className":96270},[106],[18,96272,39955],{"className":96273,"style":40031},[106,110],[18,96275],{"className":96276,"style":2068},[169],[18,96278,7984],{"className":96279},[2072],[18,96281],{"className":96282,"style":2068},[169],[18,96284,47],{"className":96285},[106],[18,96287,89435],{"className":96288},[267],[18,96290,74],{"className":96291},[250],[18,96293,2266],{"className":96294},[106],[18,96296,39955],{"className":96297,"style":40031},[106,110],[18,96299],{"className":96300,"style":2068},[169],[18,96302,2253],{"className":96303},[2072],[18,96305],{"className":96306,"style":2068},[169],[18,96308,47],{"className":96309},[106],[18,96311,89435],{"className":96312},[267],[18,96314,96315,96318],{"style":6037},[18,96316],{"className":96317,"style":697},[136],[18,96319],{"className":96320,"style":6045},[6044],[18,96322,96323,96326],{"style":6048},[18,96324],{"className":96325,"style":697},[136],[18,96327,96329,96332,96335,96338,96341],{"className":96328},[106],[18,96330,74],{"className":96331},[250],[18,96333,2266],{"className":96334},[106],[18,96336,39955],{"className":96337,"style":40031},[106,110],[18,96339,94249],{"className":96340},[267],[18,96342,96344,96347],{"className":96343},[267],[18,96345,88973],{"className":96346},[267],[18,96348,96350],{"className":96349},[115],[18,96351,96353],{"className":96352},[119],[18,96354,96356],{"className":96355},[124],[18,96357,96359],{"className":96358,"style":2416},[128],[18,96360,96361,96364],{"style":1962},[18,96362],{"className":96363,"style":137},[136],[18,96365,96367],{"className":96366},[141,142,143,144],[18,96368,2266],{"className":96369},[106,144],[18,96371,152],{"className":96372},[151],[18,96374,96376],{"className":96375},[124],[18,96377,96379],{"className":96378,"style":89485},[128],[18,96380],{},[18,96382],{"className":96383},[267,6009],[18,96385],{"className":96386,"style":228},[169],[18,96388,433],{"className":96389},[232],[18,96391],{"className":96392,"style":228},[169],[18,96394,96396,96399],{"className":96395},[97],[18,96397],{"className":96398,"style":42062},[101],[18,96400,96402,96405,96458],{"className":96401},[106],[18,96403],{"className":96404},[250,6009],[18,96406,96408],{"className":96407},[5651],[18,96409,96411,96450],{"className":96410},[119,120],[18,96412,96414,96447],{"className":96413},[124],[18,96415,96417,96428,96436],{"className":96416,"style":13849},[128],[18,96418,96419,96422],{"style":6025},[18,96420],{"className":96421,"style":697},[136],[18,96423,96425],{"className":96424},[106],[18,96426,2266],{"className":96427},[106],[18,96429,96430,96433],{"style":6037},[18,96431],{"className":96432,"style":697},[136],[18,96434],{"className":96435,"style":6045},[6044],[18,96437,96438,96441],{"style":6048},[18,96439],{"className":96440,"style":697},[136],[18,96442,96444],{"className":96443},[106],[18,96445,3941],{"className":96446,"style":3957},[106,110],[18,96448,152],{"className":96449},[151],[18,96451,96453],{"className":96452},[124],[18,96454,96456],{"className":96455,"style":6067},[128],[18,96457],{},[18,96459],{"className":96460},[267,6009],[11,96462,96463],{},"该公式引入了沃利斯的结果，为推导斯特林公式做准备。",[638,96465,81688],{"id":81688},[11,96467,96468,96469,16797,96503,509,96562,96621],{},"斯特林公式揭示了 ",[18,96470,96472,96488],{"className":96471,"translate":22},[21],[18,96473,96475],{"className":96474},[26],[28,96476,96477],{"xmlns":30},[32,96478,96479,96485],{},[35,96480,96481,96483],{},[41,96482,64],{},[49,96484,88973],{"stretchy":73},[86,96486,96487],{"encoding":88},"n!",[18,96489,96491],{"className":96490,"ariaHidden":51},[93],[18,96492,96494,96497,96500],{"className":96493},[97],[18,96495],{"className":96496,"style":295},[101],[18,96498,64],{"className":96499},[106,110],[18,96501,88973],{"className":96502},[267],[18,96504,96506,96524],{"className":96505,"translate":22},[21],[18,96507,96509],{"className":96508},[26],[28,96510,96511],{"xmlns":30},[32,96512,96513,96521],{},[35,96514,96515],{},[1556,96516,96517,96519],{},[41,96518,64],{},[41,96520,64],{},[86,96522,96523],{"encoding":88},"n^n",[18,96525,96527],{"className":96526,"ariaHidden":51},[93],[18,96528,96530,96533],{"className":96529},[97],[18,96531],{"className":96532,"style":1959},[101],[18,96534,96536,96539],{"className":96535},[106],[18,96537,64],{"className":96538},[106,110],[18,96540,96542],{"className":96541},[115],[18,96543,96545],{"className":96544},[119],[18,96546,96548],{"className":96547},[124],[18,96549,96551],{"className":96550,"style":1959},[128],[18,96552,96553,96556],{"style":1962},[18,96554],{"className":96555,"style":137},[136],[18,96557,96559],{"className":96558},[141,142,143,144],[18,96560,64],{"className":96561},[106,110,144],[18,96563,96565,96583],{"className":96564,"translate":22},[21],[18,96566,96568],{"className":96567},[26],[28,96569,96570],{"xmlns":30},[32,96571,96572,96580],{},[35,96573,96574],{},[1556,96575,96576,96578],{},[41,96577,14441],{},[41,96579,64],{},[86,96581,96582],{"encoding":88},"e^n",[18,96584,96586],{"className":96585,"ariaHidden":51},[93],[18,96587,96589,96592],{"className":96588},[97],[18,96590],{"className":96591,"style":1959},[101],[18,96593,96595,96598],{"className":96594},[106],[18,96596,14441],{"className":96597},[106,110],[18,96599,96601],{"className":96600},[115],[18,96602,96604],{"className":96603},[119],[18,96605,96607],{"className":96606},[124],[18,96608,96610],{"className":96609,"style":1959},[128],[18,96611,96612,96615],{"style":1962},[18,96613],{"className":96614,"style":137},[136],[18,96616,96618],{"className":96617},[141,142,143,144],[18,96619,64],{"className":96620},[106,110,144]," 之间的关系。",[96623,96624,96625],"blockquote",{},[11,96626,39309,96627,96655,96656,96689,96690,1990],{},[18,96628,96630,96643],{"className":96629,"translate":22},[21],[18,96631,96633],{"className":96632},[26],[28,96634,96635],{"xmlns":30},[32,96636,96637,96641],{},[35,96638,96639],{},[41,96640,64],{},[86,96642,64],{"encoding":88},[18,96644,96646],{"className":96645,"ariaHidden":51},[93],[18,96647,96649,96652],{"className":96648},[97],[18,96650],{"className":96651,"style":1083},[101],[18,96653,64],{"className":96654},[106,110]," 趋近无穷大时，",[18,96657,96659,96674],{"className":96658,"translate":22},[21],[18,96660,96662],{"className":96661},[26],[28,96663,96664],{"xmlns":30},[32,96665,96666,96672],{},[35,96667,96668,96670],{},[41,96669,64],{},[49,96671,88973],{"stretchy":73},[86,96673,96487],{"encoding":88},[18,96675,96677],{"className":96676,"ariaHidden":51},[93],[18,96678,96680,96683,96686],{"className":96679},[97],[18,96681],{"className":96682,"style":295},[101],[18,96684,64],{"className":96685},[106,110],[18,96687,88973],{"className":96688},[267]," 约等于 ",[18,96691,96693,96731],{"className":96692,"translate":22},[21],[18,96694,96696],{"className":96695},[26],[28,96697,96698],{"xmlns":30},[32,96699,96700,96728],{},[35,96701,96702,96712],{},[12558,96703,96704],{},[35,96705,96706,96708,96710],{},[45,96707,2266],{},[41,96709,3941],{},[41,96711,64],{},[1556,96713,96714,96726],{},[35,96715,96716,96718,96724],{},[49,96717,74],{"fence":51},[5651,96719,96720,96722],{},[41,96721,64],{},[41,96723,14441],{},[49,96725,84],{"fence":51},[41,96727,64],{},[86,96729,96730],{"encoding":88},"\\sqrt{2 \\pi n} \\left(\\frac{n}{e}\\right)^n",[18,96732,96734],{"className":96733,"ariaHidden":51},[93],[18,96735,96737,96741,96797,96800],{"className":96736},[97],[18,96738],{"className":96739,"style":96740},[101],"height:1.2572em;vertical-align:-0.35em;",[18,96742,96744],{"className":96743},[106,12693],[18,96745,96747,96788],{"className":96746},[119,120],[18,96748,96750,96785],{"className":96749},[124],[18,96751,96754,96772],{"className":96752,"style":96753},[128],"height:0.9072em;",[18,96755,96757,96760],{"className":96756,"style":693},[12707],[18,96758],{"className":96759,"style":697},[136],[18,96761,96763,96766,96769],{"className":96762,"style":15407},[106],[18,96764,2266],{"className":96765},[106],[18,96767,3941],{"className":96768,"style":3957},[106,110],[18,96770,64],{"className":96771},[106,110],[18,96773,96775,96778],{"style":96774},"top:-2.8672em;",[18,96776],{"className":96777,"style":697},[136],[18,96779,96781],{"className":96780,"style":15417},[12850],[6127,96782,96783],{"xmlns":6129,"width":12854,"height":15420,"viewBox":15421,"preserveAspectRatio":12857},[6133,96784],{"d":15424},[18,96786,152],{"className":96787},[151],[18,96789,96791],{"className":96790},[124],[18,96792,96795],{"className":96793,"style":96794},[128],"height:0.1328em;",[18,96796],{},[18,96798],{"className":96799,"style":170},[169],[18,96801,96803,96886],{"className":96802},[174],[18,96804,96806,96812,96880],{"className":96805},[174],[18,96807,96809],{"className":96808,"style":13793},[250,13792],[18,96810,74],{"className":96811},[1230,1196],[18,96813,96815,96818,96877],{"className":96814},[106],[18,96816],{"className":96817},[250,6009],[18,96819,96821],{"className":96820},[5651],[18,96822,96824,96869],{"className":96823},[119,120],[18,96825,96827,96866],{"className":96826},[124],[18,96828,96830,96844,96852],{"className":96829,"style":14254},[128],[18,96831,96832,96835],{"style":12045},[18,96833],{"className":96834,"style":697},[136],[18,96836,96838],{"className":96837},[141,142,143,144],[18,96839,96841],{"className":96840},[106,144],[18,96842,14441],{"className":96843},[106,110,144],[18,96845,96846,96849],{"style":6037},[18,96847],{"className":96848,"style":697},[136],[18,96850],{"className":96851,"style":6045},[6044],[18,96853,96854,96857],{"style":12068},[18,96855],{"className":96856,"style":697},[136],[18,96858,96860],{"className":96859},[141,142,143,144],[18,96861,96863],{"className":96862},[106,144],[18,96864,64],{"className":96865},[106,110,144],[18,96867,152],{"className":96868},[151],[18,96870,96872],{"className":96871},[124],[18,96873,96875],{"className":96874,"style":12090},[128],[18,96876],{},[18,96878],{"className":96879},[267,6009],[18,96881,96883],{"className":96882,"style":13793},[267,13792],[18,96884,84],{"className":96885},[1230,1196],[18,96887,96889],{"className":96888},[115],[18,96890,96892],{"className":96891},[119],[18,96893,96895],{"className":96894},[124],[18,96896,96899],{"className":96897,"style":96898},[128],"height:0.9043em;",[18,96900,96901,96904],{"style":33248},[18,96902],{"className":96903,"style":137},[136],[18,96905,96907],{"className":96906},[141,142,143,144],[18,96908,64],{"className":96909},[106,110,144],[11,96911,96912],{},"首先引入一个极限：",[18,96914,96916],{"className":96915,"translate":22},[875],[18,96917,96919,96983],{"className":96918,"translate":22},[21],[18,96920,96922],{"className":96921},[26],[28,96923,96924],{"xmlns":30,"display":884},[32,96925,96926,96980],{},[35,96927,96928,96944,96972,96974],{},[890,96929,96930,96936],{},[35,96931,96932,96934],{},[41,96933,91870],{},[49,96935,899],{},[35,96937,96938,96940,96942],{},[41,96939,64],{},[49,96941,1880],{},[41,96943,2028],{"mathvariant":316},[1556,96945,96946,96966],{},[35,96947,96948,96950,96964],{},[49,96949,74],{"fence":51},[5651,96951,96952,96958],{},[35,96953,96954,96956],{},[41,96955,64],{},[49,96957,88973],{"stretchy":73},[1556,96959,96960,96962],{},[41,96961,64],{},[41,96963,64],{},[49,96965,84],{"fence":51},[5651,96967,96968,96970],{},[45,96969,47],{},[41,96971,64],{},[49,96973,433],{},[5651,96975,96976,96978],{},[45,96977,47],{},[41,96979,14441],{},[86,96981,96982],{"encoding":88},"\\lim_{n \\rightarrow \\infty} \\left( \\frac{n!}{n^n} \\right)^{\\frac{1}{n}} = \\frac{1}{e}",[18,96984,96986,97263],{"className":96985,"ariaHidden":51},[93],[18,96987,96989,96993,97048,97051,97254,97257,97260],{"className":96988},[97],[18,96990],{"className":96991,"style":96992},[101],"height:2.744em;vertical-align:-0.95em;",[18,96994,96996],{"className":96995},[985,986],[18,96997,96999,97039],{"className":96998},[119,120],[18,97000,97002,97036],{"className":97001},[124],[18,97003,97005,97026],{"className":97004,"style":295},[128],[18,97006,97008,97011],{"style":97007},"top:-2.4em;margin-left:0em;",[18,97009],{"className":97010,"style":697},[136],[18,97012,97014],{"className":97013},[141,142,143,144],[18,97015,97017,97020,97023],{"className":97016},[106,144],[18,97018,64],{"className":97019},[106,110,144],[18,97021,1880],{"className":97022},[232,144],[18,97024,2028],{"className":97025},[106,144],[18,97027,97028,97031],{"style":693},[18,97029],{"className":97030,"style":697},[136],[18,97032,97033],{},[18,97034,91870],{"className":97035},[985],[18,97037,152],{"className":97038},[151],[18,97040,97042],{"className":97041},[124],[18,97043,97046],{"className":97044,"style":97045},[128],"height:0.7em;",[18,97047],{},[18,97049],{"className":97050,"style":170},[169],[18,97052,97054,97161],{"className":97053},[174],[18,97055,97057,97063,97155],{"className":97056},[174],[18,97058,97060],{"className":97059,"style":13793},[250,13792],[18,97061,74],{"className":97062},[1230,143],[18,97064,97066,97069,97152],{"className":97065},[106],[18,97067],{"className":97068},[250,6009],[18,97070,97072],{"className":97071},[5651],[18,97073,97075,97144],{"className":97074},[119,120],[18,97076,97078,97141],{"className":97077},[124],[18,97079,97081,97119,97127],{"className":97080,"style":6022},[128],[18,97082,97083,97086],{"style":6025},[18,97084],{"className":97085,"style":697},[136],[18,97087,97089],{"className":97088},[106],[18,97090,97092,97095],{"className":97091},[106],[18,97093,64],{"className":97094},[106,110],[18,97096,97098],{"className":97097},[115],[18,97099,97101],{"className":97100},[119],[18,97102,97104],{"className":97103},[124],[18,97105,97108],{"className":97106,"style":97107},[128],"height:0.5904em;",[18,97109,97110,97113],{"style":26105},[18,97111],{"className":97112,"style":137},[136],[18,97114,97116],{"className":97115},[141,142,143,144],[18,97117,64],{"className":97118},[106,110,144],[18,97120,97121,97124],{"style":6037},[18,97122],{"className":97123,"style":697},[136],[18,97125],{"className":97126,"style":6045},[6044],[18,97128,97129,97132],{"style":6048},[18,97130],{"className":97131,"style":697},[136],[18,97133,97135,97138],{"className":97134},[106],[18,97136,64],{"className":97137},[106,110],[18,97139,88973],{"className":97140},[267],[18,97142,152],{"className":97143},[151],[18,97145,97147],{"className":97146},[124],[18,97148,97150],{"className":97149,"style":6067},[128],[18,97151],{},[18,97153],{"className":97154},[267,6009],[18,97156,97158],{"className":97157,"style":13793},[267,13792],[18,97159,84],{"className":97160},[1230,143],[18,97162,97164],{"className":97163},[115],[18,97165,97167],{"className":97166},[119],[18,97168,97170],{"className":97169},[124],[18,97171,97174],{"className":97172,"style":97173},[128],"height:1.7939em;",[18,97175,97177,97180],{"style":97176},"top:-4.2029em;margin-right:0.05em;",[18,97178],{"className":97179,"style":697},[136],[18,97181,97183],{"className":97182},[141,142,143,144],[18,97184,97186],{"className":97185},[106,144],[18,97187,97189,97192,97251],{"className":97188},[106,144],[18,97190],{"className":97191},[250,6009,141,1195,38147],[18,97193,97195],{"className":97194},[5651],[18,97196,97198,97243],{"className":97197},[119,120],[18,97199,97201,97240],{"className":97200},[124],[18,97202,97204,97218,97226],{"className":97203,"style":42929},[128],[18,97205,97206,97209],{"style":38163},[18,97207],{"className":97208,"style":697},[136],[18,97210,97212],{"className":97211},[141,1195,1196,144],[18,97213,97215],{"className":97214},[106,144],[18,97216,64],{"className":97217},[106,110,144],[18,97219,97220,97223],{"style":38187},[18,97221],{"className":97222,"style":697},[136],[18,97224],{"className":97225,"style":38194},[6044,144],[18,97227,97228,97231],{"style":42963},[18,97229],{"className":97230,"style":697},[136],[18,97232,97234],{"className":97233},[141,1195,1196,144],[18,97235,97237],{"className":97236},[106,144],[18,97238,47],{"className":97239},[106,144],[18,97241,152],{"className":97242},[151],[18,97244,97246],{"className":97245},[124],[18,97247,97249],{"className":97248,"style":81865},[128],[18,97250],{},[18,97252],{"className":97253},[267,6009,141,1195,38147],[18,97255],{"className":97256,"style":228},[169],[18,97258,433],{"className":97259},[232],[18,97261],{"className":97262,"style":228},[169],[18,97264,97266,97269],{"className":97265},[97],[18,97267],{"className":97268,"style":9559},[101],[18,97270,97272,97275,97328],{"className":97271},[106],[18,97273],{"className":97274},[250,6009],[18,97276,97278],{"className":97277},[5651],[18,97279,97281,97320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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)}",[18,97452,97454,97725],{"className":97453,"ariaHidden":51},[93],[18,97455,97457,97460,97513,97516,97716,97719,97722],{"className":97456},[97],[18,97458],{"className":97459,"style":96992},[101],[18,97461,97463],{"className":97462},[985,986],[18,97464,97466,97505],{"className":97465},[119,120],[18,97467,97469,97502],{"className":97468},[124],[18,97470,97472,97492],{"className":97471,"style":295},[128],[18,97473,97474,97477],{"style":97007},[18,97475],{"className":97476,"style":697},[136],[18,97478,97480],{"className":97479},[141,142,143,144],[18,97481,97483,97486,97489],{"className":97482},[106,144],[18,97484,64],{"className":97485},[106,110,144],[18,97487,1880],{"className":97488},[232,144],[18,97490,2028],{"className":97491},[106,144],[18,97493,97494,97497],{"style":693},[18,97495],{"className":97496,"style":697},[136],[18,97498,97499],{},[18,97500,91870],{"className":97501},[985],[18,97503,152],{"className":97504},[151],[18,97506,97508],{"className":97507},[124],[18,97509,97511],{"className":97510,"style":97045},[128],[18,97512],{},[18,97514],{"className":97515,"style":170},[169],[18,97517,97519,97625],{"className":97518},[174],[18,97520,97522,97528,97619],{"className":97521},[174],[18,97523,97525],{"className":97524,"style":13793},[250,13792],[18,97526,74],{"className":97527},[1230,143],[18,97529,97531,97534,97616],{"className":97530},[106],[18,97532],{"className":97533},[250,6009],[18,97535,97537],{"className":97536},[5651],[18,97538,97540,97608],{"className":97539},[119,120],[18,97541,97543,97605],{"className":97542},[124],[18,97544,97546,97583,97591],{"className":97545,"style":6022},[128],[18,97547,97548,97551],{"style":6025},[18,97549],{"className":97550,"style":697},[136],[18,97552,97554],{"className":97553},[106],[18,97555,97557,97560],{"className":97556},[106],[18,97558,64],{"className":97559},[106,110],[18,97561,97563],{"className":97562},[115],[18,97564,97566],{"className":97565},[119],[18,97567,97569],{"className":97568},[124],[18,97570,97572],{"className":97571,"style":97107},[128],[18,97573,97574,97577],{"style":26105},[18,97575],{"className":97576,"style":137},[136],[18,97578,97580],{"className":97579},[141,142,143,144],[18,97581,64],{"className":97582},[106,110,144],[18,97584,97585,97588],{"style":6037},[18,97586],{"className":97587,"style":697},[136],[18,97589],{"className":97590,"style":6045},[6044],[18,97592,97593,97596],{"style":6048},[18,97594],{"className":97595,"style":697},[136],[18,97597,97599,97602],{"className":97598},[106],[18,97600,64],{"className":97601},[106,110],[18,97603,88973],{"className":97604},[267],[18,97606,152],{"className":97607},[151],[18,97609,97611],{"className":97610},[124],[18,97612,97614],{"className":97613,"style":6067},[128],[18,97615],{},[18,97617],{"className":97618},[267,6009],[18,97620,97622],{"className":97621,"style":13793},[267,13792],[18,97623,84],{"className":97624},[1230,143],[18,97626,97628],{"className":97627},[115],[18,97629,97631],{"className":97630},[119],[18,97632,97634],{"className":97633},[124],[18,97635,97637],{"className":97636,"style":97173},[128],[18,97638,97639,97642],{"style":97176},[18,97640],{"className":97641,"style":697},[136],[18,97643,97645],{"className":97644},[141,142,143,144],[18,97646,97648],{"className":97647},[106,144],[18,97649,97651,97654,97713],{"className":97650},[106,144],[18,97652],{"className":97653},[250,6009,141,1195,38147],[18,97655,97657],{"className":97656},[5651],[18,97658,97660,97705],{"className":97659},[119,120],[18,97661,97663,97702],{"className":97662},[124],[18,97664,97666,97680,97688],{"className":97665,"style":42929},[128],[18,97667,97668,97671],{"style":38163},[18,97669],{"className":97670,"style":697},[136],[18,97672,97674],{"className":97673},[141,1195,1196,144],[18,97675,97677],{"className":97676},[106,144],[18,97678,64],{"className":97679},[106,110,144],[18,97681,97682,97685],{"style":38187},[18,97683],{"className":97684,"style":697},[136],[18,97686],{"className":97687,"style":38194},[6044,144],[18,97689,97690,97693],{"style":42963},[18,97691],{"className":97692,"style":697},[136],[18,97694,97696],{"className":97695},[141,1195,1196,144],[18,97697,97699],{"className":97698},[106,144],[18,97700,47],{"className":97701},[106,144],[18,97703,152],{"className":97704},[151],[18,97706,97708],{"className":97707},[124],[18,97709,97711],{"className":97710,"style":81865},[128],[18,97712],{},[18,97714],{"className":97715},[267,6009,141,1195,38147],[18,97717],{"className":97718,"style":228},[169],[18,97720,433],{"className":97721},[232],[18,97723],{"className":97724,"style":228},[169],[18,97726,97728,97732,97785,97788],{"className":97727},[97],[18,97729],{"className":97730,"style":97731},[101],"height:1.8105em;vertical-align:-0.7em;",[18,97733,97735],{"className":97734},[985,986],[18,97736,97738,97777],{"className":97737},[119,120],[18,97739,97741,97774],{"className":97740},[124],[18,97742,97744,97764],{"className":97743,"style":295},[128],[18,97745,97746,97749],{"style":97007},[18,97747],{"className":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41,98045,2028],{"mathvariant":316},[5651,98047,98048,98050],{},[45,98049,47],{},[41,98051,64],{},[41,98053,97425],{},[49,98055,899],{},[35,98057,98058,98060,98074],{},[49,98059,74],{"fence":51},[5651,98061,98062,98068],{},[35,98063,98064,98066],{},[41,98065,64],{},[49,98067,88973],{"stretchy":73},[1556,98069,98070,98072],{},[41,98071,64],{},[41,98073,64],{},[49,98075,84],{"fence":51},[49,98077,433],{},[49,98079,2253],{},[45,98081,47],{},[86,98083,98084],{"encoding":88},"\\lim_{n 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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",[18,99759,99761,99953],{"className":99760,"ariaHidden":51},[93],[18,99762,99764,99768,99821,99824,99915,99944,99947,99950],{"className":99763},[97],[18,99765],{"className":99766,"style":99767},[101],"height:2.0714em;vertical-align:-0.7em;",[18,99769,99771],{"className":99770},[985,986],[18,99772,99774,99813],{"className":99773},[119,120],[18,99775,99777,99810],{"className":99776},[124],[18,99778,99780,99800],{"className":99779,"style":295},[128],[18,99781,99782,99785],{"style":97007},[18,99783],{"className":99784,"style":697},[136],[18,99786,99788],{"className":99787},[141,142,143,144],[18,99789,99791,99794,99797],{"className":99790},[106,144],[18,99792,64],{"className":99793},[106,110,144],[18,99795,1880],{"className":99796},[232,144],[18,99798,2028],{"className":99799},[106,144],[18,99801,99802,99805],{"style":693},[18,99803],{"className":99804,"style":697},[136],[18,99806,99807],{},[18,99808,91870],{"className":99809},[985],[18,99811,152],{"className":99812},[151],[18,99814,99816],{"className":99815},[124],[18,99817,99819],{"className":99818,"style":97045},[128],[18,99820],{},[18,99822],{"className":99823,"style":170},[169],[18,99825,99827,99830,99912],{"className":99826},[106],[18,99828],{"className":99829},[250,6009],[18,99831,99833],{"className":99832},[5651],[18,99834,99836,99904],{"className":99835},[119,120],[18,99837,99839,99901],{"className":99838},[124],[18,99840,99842,99879,99887],{"className":99841,"style":6022},[128],[18,99843,99844,99847],{"style":6025},[18,99845],{"className":99846,"style":697},[136],[18,99848,99850],{"className":99849},[106],[18,99851,99853,99856],{"className":99852},[106],[18,99854,64],{"className":99855},[106,110],[18,99857,99859],{"className":99858},[115],[18,99860,99862],{"className":99861},[119],[18,99863,99865],{"className":99864},[124],[18,99866,99868],{"className":99867,"style":97107},[128],[18,99869,99870,99873],{"style":26105},[18,99871],{"className":99872,"style":137},[136],[18,99874,99876],{"className":99875},[141,142,143,144],[18,99877,64],{"className":99878},[106,110,144],[18,99880,99881,99884],{"style":6037},[18,99882],{"className":99883,"style":697},[136],[18,99885],{"className":99886,"style":6045},[6044],[18,99888,99889,99892],{"style":6048},[18,99890],{"className":99891,"style":697},[136],[18,99893,99895,99898],{"className":99894},[106],[18,99896,64],{"className":99897},[106,110],[18,99899,88973],{"className":99900},[267],[18,99902,152],{"className":99903},[151],[18,99905,99907],{"className":99906},[124],[18,99908,99910],{"className":99909,"style":6067},[128],[18,99911],{},[18,99913],{"className":99914},[267,6009],[18,99916,99918,99921],{"className":99917},[106],[18,99919,14441],{"className":99920},[106,110],[18,99922,99924],{"className":99923},[115],[18,99925,99927],{"className":99926},[119],[18,99928,99930],{"className":99929},[124],[18,99931,99933],{"className":99932,"style":1616},[128],[18,99934,99935,99938],{"style":1637},[18,99936],{"className":99937,"style":137},[136],[18,99939,99941],{"className":99940},[141,142,143,144],[18,99942,64],{"className":99943},[106,110,144],[18,99945],{"className":99946,"style":228},[169],[18,99948,433],{"className":99949},[232],[18,99951],{"className":99952,"style":228},[169],[18,99954,99956,99959],{"className":99955},[97],[18,99957],{"className":99958,"style":505},[101],[18,99960,47],{"className":99961},[106],[11,99963,99964],{},"我们试一下。",[11,99966,99967],{},"使用比值判别法：",[18,99969,99971],{"className":99970,"translate":22},[875],[18,99972,99974,100082],{"className":99973,"translate":22},[21],[18,99975,99977],{"className":99976},[26],[28,99978,99979],{"xmlns":30,"display":884},[32,99980,99981,100079],{},[35,99982,99983,99989,99991,100005,100011,100013,100015,100027,100029,100067],{},[38,99984,99985,99987],{},[41,99986,16820],{},[41,99988,64],{},[49,99990,433],{},[5651,99992,99993,99999],{},[35,99994,99995,99997],{},[41,99996,64],{},[49,99998,88973],{"stretchy":73},[1556,100000,100001,100003],{},[41,100002,64],{},[41,100004,64],{},[1556,100006,100007,100009],{},[41,100008,14441],{},[41,100010,64],{},[49,100012,52],{"separator":51},[169,100014],{"width":42700},[38,100016,100017,100019],{},[41,100018,16820],{},[35,100020,100021,100023,100025],{},[41,100022,64],{},[49,100024,7984],{},[45,100026,47],{},[49,100028,433],{},[5651,100030,100031,100045],{},[35,100032,100033,100035,100037,100039,100041,100043],{},[49,100034,74],{"stretchy":73},[41,100036,64],{},[49,100038,7984],{},[45,100040,47],{},[49,100042,84],{"stretchy":73},[49,100044,88973],{"stretchy":73},[35,100046,100047,100049,100051,100053,100055],{},[49,100048,74],{"stretchy":73},[41,100050,64],{},[49,100052,7984],{},[45,100054,47],{},[1556,100056,100057,100059],{},[49,100058,84],{"stretchy":73},[35,100060,100061,100063,100065],{},[41,100062,64],{},[49,100064,7984],{},[45,100066,47],{},[1556,100068,100069,100071],{},[41,100070,14441],{},[35,100072,100073,100075,100077],{},[41,100074,64],{},[49,100076,7984],{},[45,100078,47],{},[86,100080,100081],{"encoding":88},"a_n = \\frac{n!}{n^n} e^n, \\quad a_{n+1} = \\frac{(n+1)!}{(n+1)^{n+1}} e^{n+1}",[18,100083,100085,100140,100334],{"className":100084,"ariaHidden":51},[93],[18,100086,100088,100091,100131,100134,100137],{"className":100087},[97],[18,100089],{"className":100090,"style":16894},[101],[18,100092,100094,100097],{"className":100093},[106],[18,100095,16820],{"className":100096},[106,110],[18,100098,100100],{"className":100099},[115],[18,100101,100103,100123],{"className":100102},[119,120],[18,100104,100106,100120],{"className":100105},[124],[18,100107,100109],{"className":100108,"style":202},[128],[18,100110,100111,100114],{"style":1155},[18,100112],{"className":100113,"style":137},[136],[18,100115,100117],{"className":100116},[141,142,143,144],[18,100118,64],{"className":100119},[106,110,144],[18,100121,152],{"className":100122},[151],[18,100124,100126],{"className":100125},[124],[18,100127,100129],{"className":100128,"style":159},[128],[18,100130],{},[18,100132],{"className":100133,"style":228},[169],[18,100135,433],{"className":100136},[232],[18,100138],{"className":100139,"style":228},[169],[18,100141,100143,100146,100237,100266,100269,100272,100275,100325,100328,100331],{"className":100142},[97],[18,100144],{"className":100145,"style":6002},[101],[18,100147,100149,100152,100234],{"className":100148},[106],[18,100150],{"className":100151},[250,6009],[18,100153,100155],{"className":100154},[5651],[18,100156,100158,100226],{"className":100157},[119,120],[18,100159,100161,100223],{"className":100160},[124],[18,100162,100164,100201,100209],{"className":100163,"style":6022},[128],[18,100165,100166,100169],{"style":6025},[18,100167],{"className":100168,"style":697},[136],[18,100170,100172],{"className":100171},[106],[18,100173,100175,100178],{"className":100174},[106],[18,100176,64],{"className":100177},[106,110],[18,100179,100181],{"className":100180},[115],[18,100182,100184],{"className":100183},[119],[18,100185,100187],{"className":100186},[124],[18,100188,100190],{"className":100189,"style":97107},[128],[18,100191,100192,100195],{"style":26105},[18,100193],{"className":100194,"style":137},[136],[18,100196,100198],{"className":100197},[141,142,143,144],[18,100199,64],{"className":100200},[106,110,144],[18,100202,100203,100206],{"style":6037},[18,100204],{"className":100205,"style":697},[136],[18,100207],{"className":100208,"style":6045},[6044],[18,100210,100211,100214],{"style":6048},[18,100212],{"className":100213,"style":697},[136],[18,100215,100217,100220],{"className":100216},[106],[18,100218,64],{"className":100219},[106,110],[18,100221,88973],{"className":100222},[267],[18,100224,152],{"className":100225},[151],[18,100227,100229],{"className":100228},[124],[18,100230,100232],{"className":100231,"style":6067},[128],[18,100233],{},[18,100235],{"className":100236},[267,6009],[18,100238,100240,100243],{"className":100239},[106],[18,100241,14441],{"className":100242},[106,110],[18,100244,100246],{"className":100245},[115],[18,100247,100249],{"className":100248},[119],[18,100250,100252],{"className":100251},[124],[18,100253,100255],{"className":100254,"style":1616},[128],[18,100256,100257,100260],{"style":1637},[18,100258],{"className":100259,"style":137},[136],[18,100261,100263],{"className":100262},[141,142,143,144],[18,100264,64],{"className":100265},[106,110,144],[18,100267,52],{"className":100268},[165],[18,100270],{"className":100271,"style":42840},[169],[18,100273],{"className":100274,"style":170},[169],[18,100276,100278,100281],{"className":100277},[106],[18,100279,16820],{"className":100280},[106,110],[18,100282,100284],{"className":100283},[115],[18,100285,100287,100316],{"className":100286},[119,120],[18,100288,100290,100313],{"className":100289},[124],[18,100291,100293],{"className":100292,"style":129},[128],[18,100294,100295,100298],{"style":1155},[18,100296],{"className":100297,"style":137},[136],[18,100299,100301],{"className":100300},[141,142,143,144],[18,100302,100304,100307,100310],{"className":100303},[106,144],[18,100305,64],{"className":100306},[106,110,144],[18,100308,7984],{"className":100309},[2072,144],[18,100311,47],{"className":100312},[106,144],[18,100314,152],{"className":100315},[151],[18,100317,100319],{"className":100318},[124],[18,100320,100323],{"className":100321,"style":100322},[128],"height:0.2083em;",[18,100324],{},[18,100326],{"className":100327,"style":228},[169],[18,100329,433],{"className":100330},[232],[18,100332],{"className":100333,"style":228},[169],[18,100335,100337,100340,100473],{"className":100336},[97],[18,100338],{"className":100339,"style":89396},[101],[18,100341,100343,100346,100470],{"className":100342},[106],[18,100344],{"className":100345},[250,6009],[18,100347,100349],{"className":100348},[5651],[18,100350,100352,100462],{"className":100351},[119,120],[18,100353,100355,100459],{"className":100354},[124],[18,100356,100358,100422,100430],{"className":100357,"style":18128},[128],[18,100359,100360,100363],{"style":6025},[18,100361],{"className":100362,"style":697},[136],[18,100364,100366,100369,100372,100375,100378,100381,100384],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\\rightarrow \\infty} \\frac{a_{n+1}}{a_n} = \\frac{e}{\\left(1 + \\frac{1}{n}\\right)^n} > 1",[18,100596,100598,100816,101014],{"className":100597,"ariaHidden":51},[93],[18,100599,100601,100605,100658,100661,100807,100810,100813],{"className":100600},[97],[18,100602],{"className":100603,"style":100604},[101],"height:1.9436em;vertical-align:-0.836em;",[18,100606,100608],{"className":100607},[985,986],[18,100609,100611,100650],{"className":100610},[119,120],[18,100612,100614,100647],{"className":100613},[124],[18,100615,100617,100637],{"className":100616,"style":295},[128],[18,100618,100619,100622],{"style":97007},[18,100620],{"className":100621,"style":697},[136],[18,100623,100625],{"className":100624},[141,142,143,144],[18,100626,100628,100631,100634],{"className":100627},[106,144],[18,100629,64],{"className":100630},[106,110,144],[18,100632,1880],{"className":100633},[232,144],[18,100635,2028],{"className":100636},[106,144],[18,100638,100639,100642],{"style":693},[18,100640],{"className":100641,"style":697},[136],[18,100643,100644],{},[18,100645,91870],{"className":100646},[985],[18,100648,152],{"className":100649},[151],[18,100651,100653],{"className":100652},[124],[18,100654,100656],{"className":100655,"style":97045},[128],[18,100657],{},[18,100659],{"className":100660,"style":170},[169],[18,100662,100664,100667,100804],{"className":100663},[106],[18,100665],{"className":100666},[250,6009],[18,100668,100670],{"className":100669},[5651],[18,100671,100673,100795],{"className":100672},[119,120],[18,100674,100676,100792],{"className":100675},[124],[18,100677,100679,100727,100735],{"className":100678,"style":13849},[128],[18,100680,100681,100684],{"style":6025},[18,100682],{"className":100683,"style":697},[136],[18,100685,100687],{"className":100686},[106],[18,100688,100690,100693],{"className":100689},[106],[18,100691,16820],{"className":100692},[106,110],[18,100694,100696],{"className":100695},[115],[18,100697,100699,100719],{"className":100698},[119,120],[18,100700,100702,100716],{"className":100701},[124],[18,100703,100705],{"className":100704,"style":202},[128],[18,100706,100707,100710],{"style":1155},[18,100708],{"className":100709,"style":137},[136],[18,100711,100713],{"className":100712},[141,142,143,144],[18,100714,64],{"className":100715},[106,110,144],[18,100717,152],{"className":100718},[151],[18,100720,100722],{"className":100721},[124],[18,100723,100725],{"className":100724,"style":159},[128],[18,100726],{},[18,100728,100729,100732],{"style":6037},[18,100730],{"className":100731,"style":697},[136],[18,100733],{"className":100734,"style":6045},[6044],[18,100736,100737,100740],{"style":6048},[18,100738],{"className":100739,"style":697},[136],[18,100741,100743],{"className":100742},[106],[18,100744,100746,100749],{"className":100745},[106],[18,100747,16820],{"className":100748},[106,110],[18,100750,100752],{"className":100751},[115],[18,100753,100755,100784],{"className":100754},[119,120],[18,100756,100758,100781],{"className":100757},[124],[18,100759,100761],{"className":100760,"style":129},[128],[18,100762,100763,100766],{"style":1155},[18,100764],{"className":100765,"style":137},[136],[18,100767,100769],{"className":100768},[141,142,143,144],[18,100770,100772,100775,100778],{"className":100771},[106,144],[18,100773,64],{"className":100774},[106,110,144],[18,100776,7984],{"className":100777},[2072,144],[18,100779,47],{"className":100780},[106,144],[18,100782,152],{"className":100783},[151],[18,100785,100787],{"className":100786},[124],[18,100788,100790],{"className":100789,"style":100322},[128],[18,100791],{},[18,100793,152],{"className":100794},[151],[18,100796,100798],{"className":100797},[124],[18,100799,100802],{"className":100800,"style":100801},[128],"height:0.836em;",[18,100803],{},[18,100805],{"className":100806},[267,6009],[18,100808],{"className":100809,"style":228},[169],[18,100811,433],{"className":100812},[232],[18,100814],{"className":100815,"style":228},[169],[18,100817,100819,100823,101005,101008,101011],{"className":100818},[97],[18,100820],{"className":100821,"style":100822},[101],"height:2.2519em;vertical-align:-1.1443em;",[18,100824,100826,100829,101002],{"className":100825},[106],[18,100827],{"className":100828},[250,6009],[18,100830,100832],{"className":100831},[5651],[18,100833,100835,100993],{"className":100834},[119,120],[18,100836,100838,100990],{"className":100837},[124],[18,100839,100841,100971,100979],{"className":100840,"style":13849},[128],[18,100842,100844,100847],{"style":100843},"top:-2.2057em;",[18,100845],{"className":100846,"style":697},[136],[18,100848,100850],{"className":100849},[106],[18,100851,100853,100948],{"className":100852},[174],[18,100854,100856,100862,100865,100868,100871,100874,100942],{"className":100855},[174],[18,100857,100859],{"className":100858,"style":13793},[250,13792],[18,100860,74],{"className":100861},[1230,1196],[18,100863,47],{"className":100864},[106],[18,100866],{"className":100867,"style":2068},[169],[18,100869,7984],{"className":100870},[2072],[18,100872],{"className":100873,"style":2068},[169],[18,100875,100877,100880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",[18,101027,101029,101042],{"className":101028,"translate":22},[21],[18,101030,101032],{"className":101031},[26],[28,101033,101034],{"xmlns":30},[32,101035,101036,101040],{},[35,101037,101038],{},[41,101039,14441],{},[86,101041,14441],{"encoding":88},[18,101043,101045],{"className":101044,"ariaHidden":51},[93],[18,101046,101048,101051],{"className":101047},[97],[18,101049],{"className":101050,"style":1083},[101],[18,101052,14441],{"className":101053},[106,110],"，但它总是从下方趋近 ",[18,101056,101058,101071],{"className":101057,"translate":22},[21],[18,101059,101061],{"className":101060},[26],[28,101062,101063],{"xmlns":30},[32,101064,101065,101069],{},[35,101066,101067],{},[41,101068,14441],{},[86,101070,14441],{"encoding":88},[18,101072,101074],{"className":101073,"ariaHidden":51},[93],[18,101075,101077,101080],{"className":101076},[97],[18,101078],{"className":101079,"style":1083},[101],[18,101081,14441],{"className":101082},[106,110],"，因此比值仍大于 1，说明 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是发散的。",[11,101156,101157],{},"数学家们尝试了各种方法，发现：",[18,101159,101161],{"className":101160,"translate":22},[875],[18,101162,101164,101214],{"className":101163,"translate":22},[21],[18,101165,101167],{"className":101166},[26],[28,101168,101169],{"xmlns":30,"display":884},[32,101170,101171,101211],{},[35,101172,101173,101179,101181,101205],{},[38,101174,101175,101177],{},[41,101176,16820],{},[41,101178,64],{},[49,101180,433],{},[5651,101182,101183,101189],{},[35,101184,101185,101187],{},[41,101186,64],{},[49,101188,88973],{"stretchy":73},[1556,101190,101191,101193],{},[41,101192,64],{},[35,101194,101195,101197,101199],{},[41,101196,64],{},[49,101198,7984],{},[5651,101200,101201,101203],{},[45,101202,47],{},[45,101204,2266],{},[1556,101206,101207,101209],{},[41,101208,14441],{},[41,101210,64],{},[86,101212,101213],{"encoding":88},"a_n = \\frac{n!}{n^{n + \\frac{1}{2}}} e^n",[18,101215,101217,101272],{"className":101216,"ariaHidden":51},[93],[18,101218,101220,101223,101263,101266,101269],{"className":101219},[97],[18,101221],{"className":101222,"style":16894},[101],[18,101224,101226,101229],{"className":101225},[106],[18,101227,16820],{"className":101228},[106,110],[18,101230,101232],{"className":101231},[115],[18,101233,101235,101255],{"className":101234},[119,120],[18,101236,101238,101252],{"className":101237},[124],[18,101239,101241],{"className":101240,"style":202},[128],[18,101242,101243,101246],{"style":1155},[18,101244],{"className":101245,"style":137},[136],[18,101247,101249],{"className":101248},[141,142,143,144],[18,101250,64],{"className":101251},[106,110,144],[18,101253,152],{"className":101254},[151],[18,101256,101258],{"className":101257},[124],[18,101259,101261],{"className":101260,"style":159},[128],[18,101262],{},[18,101264],{"className":101265,"style":228},[169],[18,101267,433],{"className":101268},[232],[18,101270],{"className":101271,"style":228},[169],[18,101273,101275,101279,101448],{"className":101274},[97],[18,101276],{"className":101277,"style":101278},[101],"height:2.201em;vertical-align:-0.8296em;",[18,101280,101282,101285,101445],{"className":101281},[106],[18,101283],{"className":101284},[250,6009],[18,101286,101288],{"className":101287},[5651],[18,101289,101291,101436],{"className":101290},[119,120],[18,101292,101294,101433],{"className":101293},[124],[18,101295,101297,101411,101419],{"className":101296,"style":6022},[128],[18,101298,101300,101303],{"style":101299},"top:-2.1704em;",[18,101301],{"className":101302,"style":697},[136],[18,101304,101306],{"className":101305},[106],[18,101307,101309,101312],{"className":101308},[106],[18,101310,64],{"className":101311},[106,110],[18,101313,101315],{"className":101314},[115],[18,101316,101318],{"className":101317},[119],[18,101319,101321],{"className":101320},[124],[18,101322,101325],{"className":101323,"style":101324},[128],"height:0.9396em;",[18,101326,101328,101331],{"style":101327},"top:-3.3486em;margin-right:0.05em;",[18,101329],{"className":101330,"style":697},[136],[18,101332,101334],{"className":101333},[141,142,143,144],[18,101335,101337,101340,101343],{"className":101336},[106,144],[18,101338,64],{"className":101339},[106,110,144],[18,101341,7984],{"className":101342},[2072,144],[18,101344,101346,101349,101408],{"className":101345},[106,144],[18,101347],{"className":101348},[250,6009,141,1195,38147],[18,101350,101352],{"className":101351},[5651],[18,101353,101355,101400],{"className":101354},[119,120],[18,101356,101358,101397],{"className":101357},[124],[18,101359,101361,101375,101383],{"className":101360,"style":42929},[128],[18,101362,101363,101366],{"style":38163},[18,101364],{"className":101365,"style":697},[136],[18,101367,101369],{"className":101368},[141,1195,1196,144],[18,101370,101372],{"className":101371},[106,144],[18,101373,2266],{"className":101374},[106,144],[18,101376,101377,101380],{"style":38187},[18,101378],{"className":101379,"style":697},[136],[18,101381],{"className":101382,"style":38194},[6044,144],[18,101384,101385,101388],{"style":42963},[18,101386],{"className":101387,"style":697},[136],[18,101389,101391],{"className":101390},[141,1195,1196,144],[18,101392,101394],{"className":101393},[106,144],[18,101395,47],{"className":101396},[106,144],[18,101398,152],{"className":101399},[151],[18,101401,101403],{"className":101402},[124],[18,101404,101406],{"className":101405,"style":81865},[128],[18,101407],{},[18,101409],{"className":101410},[267,6009,141,1195,38147],[18,101412,101413,101416],{"style":6037},[18,101414],{"className":101415,"style":697},[136],[18,101417],{"className":101418,"style":6045},[6044],[18,101420,101421,101424],{"style":6048},[18,101422],{"className":101423,"style":697},[136],[18,101425,101427,101430],{"className":101426},[106],[18,101428,64],{"className":101429},[106,110],[18,101431,88973],{"className":101432},[267],[18,101434,152],{"className":101435},[151],[18,101437,101439],{"className":101438},[124],[18,101440,101443],{"className":101441,"style":101442},[128],"height:0.8296em;",[18,101444],{},[18,101446],{"className":101447},[267,6009],[18,101449,101451,101454],{"className":101450},[106],[18,101452,14441],{"className":101453},[106,110],[18,101455,101457],{"className":101456},[115],[18,101458,101460],{"className":101459},[119],[18,101461,101463],{"className":101462},[124],[18,101464,101466],{"className":101465,"style":1616},[128],[18,101467,101468,101471],{"style":1637},[18,101469],{"className":101470,"style":137},[136],[18,101472,101474],{"className":101473},[141,142,143,144],[18,101475,64],{"className":101476},[106,110,144],[11,101478,101479,101480,101549],{},"为了证明 ",[18,101481,101483,101500],{"className":101482,"translate":22},[21],[18,101484,101486],{"className":101485},[26],[28,101487,101488],{"xmlns":30},[32,101489,101490,101498],{},[35,101491,101492],{},[38,101493,101494,101496],{},[41,101495,16820],{},[41,101497,64],{},[86,101499,101104],{"encoding":88},[18,101501,101503],{"className":101502,"ariaHidden":51},[93],[18,101504,101506,101509],{"className":101505},[97],[18,101507],{"className":101508,"style":16894},[101],[18,101510,101512,101515],{"className":101511},[106],[18,101513,16820],{"className":101514},[106,110],[18,101516,101518],{"className":101517},[115],[18,101519,101521,101541],{"className":101520},[119,120],[18,101522,101524,101538],{"className":101523},[124],[18,101525,101527],{"className":101526,"style":202},[128],[18,101528,101529,101532],{"style":1155},[18,101530],{"className":101531,"style":137},[136],[18,101533,101535],{"className":101534},[141,142,143,144],[18,101536,64],{"className":101537},[106,110,144],[18,101539,152],{"className":101540},[151],[18,101542,101544],{"className":101543},[124],[18,101545,101547],{"className":101546,"style":159},[128],[18,101548],{}," 收敛，再次使用比值判别法：",[18,101551,101553],{"className":101552,"translate":22},[875],[18,101554,101556,101678],{"className":101555,"translate":22},[21],[18,101557,101559],{"className":101558},[26],[28,101560,101561],{"xmlns":30,"display":884},[32,101562,101563,101675],{},[35,101564,101565,101571,101573,101597,101603,101605,101607,101619,101621,101663],{},[38,101566,101567,101569],{},[41,101568,16820],{},[41,101570,64],{},[49,101572,433],{},[5651,101574,101575,101581],{},[35,101576,101577,101579],{},[41,101578,64],{},[49,101580,88973],{"stretchy":73},[1556,101582,101583,101585],{},[41,101584,64],{},[35,101586,101587,101589,101591],{},[41,101588,64],{},[49,101590,7984],{},[5651,101592,101593,101595],{},[45,101594,47],{},[45,101596,2266],{},[1556,101598,101599,101601],{},[41,101600,14441],{},[41,101602,64],{},[49,101604,52],{"separator":51},[169,101606],{"width":42700},[38,101608,101609,101611],{},[41,101610,16820],{},[35,101612,101613,101615,101617],{},[41,101614,64],{},[49,101616,7984],{},[45,101618,47],{},[49,101620,433],{},[5651,101622,101623,101637],{},[35,101624,101625,101627,101629,101631,101633,101635],{},[49,101626,74],{"stretchy":73},[41,101628,64],{},[49,101630,7984],{},[45,101632,47],{},[49,101634,84],{"stretchy":73},[49,101636,88973],{"stretchy":73},[35,101638,101639,101641,101643,101645,101647],{},[49,101640,74],{"stretchy":73},[41,101642,64],{},[49,101644,7984],{},[45,101646,47],{},[1556,101648,101649,101651],{},[49,101650,84],{"stretchy":73},[35,101652,101653,101655,101657],{},[41,101654,64],{},[49,101656,7984],{},[5651,101658,101659,101661],{},[45,101660,44699],{},[45,101662,2266],{},[1556,101664,101665,101667],{},[41,101666,14441],{},[35,101668,101669,101671,101673],{},[41,101670,64],{},[49,101672,7984],{},[45,101674,47],{},[86,101676,101677],{"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}",[18,101679,101681,101736,102003],{"className":101680,"ariaHidden":51},[93],[18,101682,101684,101687,101727,101730,101733],{"className":101683},[97],[18,101685],{"className":101686,"style":16894},[101],[18,101688,101690,101693],{"className":101689},[106],[18,101691,16820],{"className":101692},[106,110],[18,101694,101696],{"className":101695},[115],[18,101697,101699,101719],{"className":101698},[119,120],[18,101700,101702,101716],{"className":101701},[124],[18,101703,101705],{"className":101704,"style":202},[128],[18,101706,101707,101710],{"style":1155},[18,101708],{"className":101709,"style":137},[136],[18,101711,101713],{"className":101712},[141,142,143,144],[18,101714,64],{"className":101715},[106,110,144],[18,101717,152],{"className":101718},[151],[18,101720,101722],{"className":101721},[124],[18,101723,101725],{"className":101724,"style":159},[128],[18,101726],{},[18,101728],{"className":101729,"style":228},[169],[18,101731,433],{"className":101732},[232],[18,101734],{"className":101735,"style":228},[169],[18,101737,101739,101742,101907,101936,101939,101942,101945,101994,101997,102000],{"className":101738},[97],[18,101740],{"className":101741,"style":101278},[101],[18,101743,101745,101748,101904],{"className":101744},[106],[18,101746],{"className":101747},[250,6009],[18,101749,101751],{"className":101750},[5651],[18,101752,101754,101896],{"className":101753},[119,120],[18,101755,101757,101893],{"className":101756},[124],[18,101758,101760,101871,101879],{"className":101759,"style":6022},[128],[18,101761,101762,101765],{"style":101299},[18,101763],{"className":101764,"style":697},[136],[18,101766,101768],{"className":101767},[106],[18,101769,101771,101774],{"className":101770},[106],[18,101772,64],{"className":101773},[106,110],[18,101775,101777],{"className":101776},[115],[18,101778,101780],{"className":101779},[119],[18,101781,101783],{"className":101782},[124],[18,101784,101786],{"className":101785,"style":101324},[128],[18,101787,101788,101791],{"style":101327},[18,101789],{"className":101790,"style":697},[136],[18,101792,101794],{"className":101793},[141,142,143,144],[18,101795,101797,101800,101803],{"className":101796},[106,144],[18,101798,64],{"className":101799},[106,110,144],[18,101801,7984],{"className":101802},[2072,144],[18,101804,101806,101809,101868],{"className":101805},[106,144],[18,101807],{"className":101808},[250,6009,141,1195,38147],[18,101810,101812],{"className":101811},[5651],[18,101813,101815,101860],{"className":101814},[119,120],[18,101816,101818,101857],{"className":101817},[124],[18,101819,101821,101835,101843],{"className":101820,"style":42929},[128],[18,101822,101823,101826],{"style":38163},[18,101824],{"className":101825,"style":697},[136],[18,101827,101829],{"className":101828},[141,1195,1196,144],[18,101830,101832],{"className":101831},[106,144],[18,101833,2266],{"className":101834},[106,144],[18,101836,101837,101840],{"style":38187},[18,101838],{"className":101839,"style":697},[136],[18,101841],{"className":101842,"style":38194},[6044,144],[18,101844,101845,101848],{"style":42963},[18,101846],{"className":101847,"style":697},[136],[18,101849,101851],{"className":101850},[141,1195,1196,144],[18,101852,101854],{"className":101853},[106,144],[18,101855,47],{"className":101856},[106,144],[18,101858,152],{"className":101859},[151],[18,101861,101863],{"className":101862},[124],[18,101864,101866],{"className":101865,"style":81865},[128],[18,101867],{},[18,101869],{"className":101870},[267,6009,141,1195,38147],[18,101872,101873,101876],{"style":6037},[18,101874],{"className":101875,"style":697},[136],[18,101877],{"className":101878,"style":6045},[6044],[18,101880,101881,101884],{"style":6048},[18,101882],{"className":101883,"style":697},[136],[18,101885,101887,101890],{"className":101886},[106],[18,101888,64],{"className":101889},[106,110],[18,101891,88973],{"className":101892},[267],[18,101894,152],{"className":101895},[151],[18,101897,101899],{"className":101898},[124],[18,101900,101902],{"className":101901,"style":101442},[128],[18,101903],{},[18,101905],{"className":101906},[267,6009],[18,101908,101910,101913],{"className":101909},[106],[18,101911,14441],{"className":101912},[106,110],[18,101914,101916],{"className":101915},[115],[18,101917,101919],{"className":101918},[119],[18,101920,101922],{"className":101921},[124],[18,101923,101925],{"className":101924,"style":1616},[128],[18,101926,101927,101930],{"style":1637},[18,101928],{"className":101929,"style":137},[136],[18,101931,101933],{"className":101932},[141,142,143,144],[18,101934,64],{"className":101935},[106,110,144],[18,101937,52],{"className":101938},[165],[18,101940],{"className":101941,"style":42840},[169],[18,101943],{"className":101944,"style":170},[169],[18,101946,101948,101951],{"className":101947},[106],[18,101949,16820],{"className":101950},[106,110],[18,101952,101954],{"className":101953},[115],[18,101955,101957,101986],{"className":101956},[119,120],[18,101958,101960,101983],{"className":101959},[124],[18,101961,101963],{"className":101962,"style":129},[128],[18,101964,101965,101968],{"style":1155},[18,101966],{"className":101967,"style":137},[136],[18,101969,101971],{"className":101970},[141,142,143,144],[18,101972,101974,101977,101980],{"className":101973},[106,144],[18,101975,64],{"className":101976},[106,110,144],[18,101978,7984],{"className":101979},[2072,144],[18,101981,47],{"className":101982},[106,144],[18,101984,152],{"className":101985},[151],[18,101987,101989],{"className":101988},[124],[18,101990,101992],{"className":101991,"style":100322},[128],[18,101993],{},[18,101995],{"className":101996,"style":228},[169],[18,101998,433],{"className":101999},[232],[18,102001],{"className":102002,"style":228},[169],[18,102004,102006,102010,102209],{"className":102005},[97],[18,102007],{"className":102008,"style":102009},[101],"height:2.5066em;vertical-align:-1.0796em;",[18,102011,102013,102016,102206],{"className":102012},[106],[18,102014],{"className":102015},[250,6009],[18,102017,102019],{"className":102018},[5651],[18,102020,102022,102197],{"className":102021},[119,120],[18,102023,102025,102194],{"className":102024},[124],[18,102026,102028,102157,102165],{"className":102027,"style":18128},[128],[18,102029,102030,102033],{"style":101299},[18,102031],{"className":102032,"style":697},[136],[18,102034,102036,102039,102042,102045,102048,102051,102054],{"className":102035},[106],[18,102037,74],{"className":102038},[250],[18,102040,64],{"className":102041},[106,110],[18,102043],{"className":102044,"style":2068},[169],[18,102046,7984],{"className":102047},[2072],[18,102049],{"className":102050,"style":2068},[169],[18,102052,47],{"className":102053},[106],[18,102055,102057,102060],{"className":102056},[267],[18,102058,84],{"className":102059},[267],[18,102061,102063],{"className":102062},[115],[18,102064,102066],{"className":102065},[119],[18,102067,102069],{"className":102068},[124],[18,102070,102072],{"className":102071,"style":101324},[128],[18,102073,102074,102077],{"style":101327},[18,102075],{"className":102076,"style":697},[136],[18,102078,102080],{"className":102079},[141,142,143,144],[18,102081,102083,102086,102089],{"className":102082},[106,144],[18,102084,64],{"className":102085},[106,110,144],[18,102087,7984],{"className":102088},[2072,144],[18,102090,102092,102095,102154],{"className":102091},[106,144],[18,102093],{"className":102094},[250,6009,141,1195,38147],[18,102096,102098],{"className":102097},[5651],[18,102099,102101,102146],{"className":102100},[119,120],[18,102102,102104,102143],{"className":102103},[124],[18,102105,102107,102121,102129],{"className":102106,"style":42929},[128],[18,102108,102109,102112],{"style":38163},[18,102110],{"className":102111,"style":697},[136],[18,102113,102115],{"className":102114},[141,1195,1196,144],[18,102116,102118],{"className":102117},[106,144],[18,102119,2266],{"className":102120},[106,144],[18,102122,102123,102126],{"style":38187},[18,102124],{"className":102125,"style":697},[136],[18,102127],{"className":102128,"style":38194},[6044,144],[18,102130,102131,102134],{"style":42963},[18,102132],{"className":102133,"style":697},[136],[18,102135,102137],{"className":102136},[141,1195,1196,144],[18,102138,102140],{"className":102139},[106,144],[18,102141,44699],{"className":102142},[106,144],[18,102144,152],{"className":102145},[151],[18,102147,102149],{"className":102148},[124],[18,102150,102152],{"className":102151,"style":81865},[128],[18,102153],{},[18,102155],{"className":102156},[267,6009,141,1195,38147],[18,102158,102159,102162],{"style":6037},[18,102160],{"className":102161,"style":697},[136],[18,102163],{"className":102164,"style":6045},[6044],[18,102166,102167,102170],{"style":6048},[18,102168],{"className":102169,"style":697},[136],[18,102171,102173,102176,102179,102182,102185,102188,102191],{"className":102172},[106],[18,102174,74],{"className":102175},[250],[18,102177,64],{"className":102178},[106,110],[18,102180],{"className":102181,"style":2068},[169],[18,102183,7984],{"className":102184},[2072],[18,102186],{"className":102187,"style":2068},[169],[18,102189,47],{"className":102190},[106],[18,102192,94249],{"className":102193},[267],[18,102195,152],{"className":102196},[151],[18,102198,102200],{"className":102199},[124],[18,102201,102204],{"className":102202,"style":102203},[128],"height:1.0796em;",[18,102205],{},[18,102207],{"className":102208},[267,6009],[18,102210,102212,102215],{"className":102211},[106],[18,102213,14441],{"className":102214},[106,110],[18,102216,102218],{"className":102217},[115],[18,102219,102221],{"className":102220},[119],[18,102222,102224],{"className":102223},[124],[18,102225,102227],{"className":102226,"style":10772},[128],[18,102228,102229,102232],{"style":1637},[18,102230],{"className":102231,"style":137},[136],[18,102233,102235],{"className":102234},[141,142,143,144],[18,102236,102238,102241,102244],{"className":102237},[106,144],[18,102239,64],{"className":102240},[106,110,144],[18,102242,7984],{"className":102243},[2072,144],[18,102245,47],{"className":102246},[106,144],[18,102248,102250],{"className":102249,"translate":22},[875],[18,102251,102253,102337],{"className":102252,"translate":22},[21],[18,102254,102256],{"className":102255},[26],[28,102257,102258],{"xmlns":30,"display":884},[32,102259,102260,102334],{},[35,102261,102262,102278,102298,102300],{},[890,102263,102264,102270],{},[35,102265,102266,102268],{},[41,102267,91870],{},[49,102269,899],{},[35,102271,102272,102274,102276],{},[41,102273,64],{},[49,102275,1880],{},[41,102277,2028],{"mathvariant":316},[5651,102279,102280,102292],{},[38,102281,102282,102284],{},[41,102283,16820],{},[35,102285,102286,102288,102290],{},[41,102287,64],{},[49,102289,7984],{},[45,102291,47],{},[38,102293,102294,102296],{},[41,102295,16820],{},[41,102297,64],{},[49,102299,433],{},[5651,102301,102302,102304],{},[41,102303,14441],{},[1556,102305,102306,102322],{},[35,102307,102308,102310,102312,102314,102320],{},[49,102309,74],{"fence":51},[45,102311,47],{},[49,102313,7984],{},[5651,102315,102316,102318],{},[45,102317,47],{},[41,102319,64],{},[49,102321,84],{"fence":51},[35,102323,102324,102326,102328],{},[41,102325,64],{},[49,102327,7984],{},[5651,102329,102330,102332],{},[45,102331,47],{},[45,102333,2266],{},[86,102335,102336],{"encoding":88},"\\lim_{n \\rightarrow \\infty} \\frac{a_{n+1}}{a_n} = \\frac{e}{\\left(1 + \\frac{1}{n}\\right)^{n + \\frac{1}{2}}}",[18,102338,102340,102556],{"className":102339,"ariaHidden":51},[93],[18,102341,102343,102346,102399,102402,102547,102550,102553],{"className":102342},[97],[18,102344],{"className":102345,"style":100604},[101],[18,102347,102349],{"className":102348},[985,986],[18,102350,102352,102391],{"className":102351},[119,120],[18,102353,102355,102388],{"className":102354},[124],[18,102356,102358,102378],{"className":102357,"style":295},[128],[18,102359,102360,102363],{"style":97007},[18,102361],{"className":102362,"style":697},[136],[18,102364,102366],{"className":102365},[141,142,143,144],[18,102367,102369,102372,102375],{"className":102368},[106,144],[18,102370,64],{"className":102371},[106,110,144],[18,102373,1880],{"className":102374},[232,144],[18,102376,2028],{"className":102377},[106,144],[18,102379,102380,102383],{"style":693},[18,102381],{"className":102382,"style":697},[136],[18,102384,102385],{},[18,102386,91870],{"className":102387},[985],[18,102389,152],{"className":102390},[151],[18,102392,102394],{"className":102393},[124],[18,102395,102397],{"className":102396,"style":97045},[128],[18,102398],{},[18,102400],{"className":102401,"style":170},[169],[18,102403,102405,102408,102544],{"className":102404},[106],[18,102406],{"className":102407},[250,6009],[18,102409,102411],{"className":102410},[5651],[18,102412,102414,102536],{"className":102413},[119,120],[18,102415,102417,102533],{"className":102416},[124],[18,102418,102420,102468,102476],{"className":102419,"style":13849},[128],[18,102421,102422,102425],{"style":6025},[18,102423],{"className":102424,"style":697},[136],[18,102426,102428],{"className":102427},[106],[18,102429,102431,102434],{"className":102430},[106],[18,102432,16820],{"className":102433},[106,110],[18,102435,102437],{"className":102436},[115],[18,102438,102440,102460],{"className":102439},[119,120],[18,102441,102443,102457],{"className":102442},[124],[18,102444,102446],{"className":102445,"style":202},[128],[18,102447,102448,102451],{"style":1155},[18,102449],{"className":102450,"style":137},[136],[18,102452,102454],{"className":102453},[141,142,143,144],[18,102455,64],{"className":102456},[106,110,144],[18,102458,152],{"className":102459},[151],[18,102461,102463],{"className":102462},[124],[18,102464,102466],{"className":102465,"style":159},[128],[18,102467],{},[18,102469,102470,102473],{"style":6037},[18,102471],{"className":102472,"style":697},[136],[18,102474],{"className":102475,"style":6045},[6044],[18,102477,102478,102481],{"style":6048},[18,102479],{"className":102480,"style":697},[136],[18,102482,102484],{"className":102483},[106],[18,102485,102487,102490],{"className":102486},[106],[18,102488,16820],{"className":102489},[106,110],[18,102491,102493],{"className":102492},[115],[18,102494,102496,102525],{"className":102495},[119,120],[18,102497,102499,102522],{"className":102498},[124],[18,102500,102502],{"className":102501,"style":129},[128],[18,102503,102504,102507],{"style":1155},[18,102505],{"className":102506,"style":137},[136],[18,102508,102510],{"className":102509},[141,142,143,144],[18,102511,102513,102516,102519],{"className":102512},[106,144],[18,102514,64],{"className":102515},[106,110,144],[18,102517,7984],{"className":102518},[2072,144],[18,102520,47],{"className":102521},[106,144],[18,102523,152],{"className":102524},[151],[18,102526,102528],{"className":102527},[124],[18,102529,102531],{"className":102530,"style":100322},[128],[18,102532],{},[18,102534,152],{"className":102535},[151],[18,102537,102539],{"className":102538},[124],[18,102540,102542],{"className":102541,"style":100801},[128],[18,102543],{},[18,102545],{"className":102546},[267,6009],[18,102548],{"className":102549,"style":228},[169],[18,102551,433],{"className":102552},[232],[18,102554],{"className":102555,"style":228},[169],[18,102557,102559,102563],{"className":102558},[97],[18,102560],{"className":102561,"style":102562},[101],"height:2.5415em;vertical-align:-1.4339em;",[18,102564,102566,102569,102820],{"className":102565},[106],[18,102567],{"className":102568},[250,6009],[18,102570,102572],{"className":102571},[5651],[18,102573,102575,102811],{"className":102574},[119,120],[18,102576,102578,102808],{"className":102577},[124],[18,102579,102581,102787,102796],{"className":102580,"style":13849},[128],[18,102582,102584,102588],{"style":102583},"top:-2.11em;",[18,102585],{"className":102586,"style":102587},[136],"height:3.1939em;",[18,102589,102591],{"className":102590},[106],[18,102592,102594,102689],{"className":102593},[174],[18,102595,102597,102603,102606,102609,102612,102615,102683],{"className":102596},[174],[18,102598,102600],{"className":102599,"style":13793},[250,13792],[18,102601,74],{"className":102602},[1230,1196],[18,102604,47],{"className":102605},[106],[18,102607],{"className":102608,"style":2068},[169],[18,102610,7984],{"className":102611},[2072],[18,102613],{"className":102614,"style":2068},[169],[18,102616,102618,102621,102680],{"className":102617},[106],[18,102619],{"className":102620},[250,6009],[18,102622,102624],{"className":102623},[5651],[18,102625,102627,102672],{"className":102626},[119,120],[18,102628,102630,102669],{"className":102629},[124],[18,102631,102633,102647,102655],{"className":102632,"style":12042},[128],[18,102634,102635,102638],{"style":12045},[18,102636],{"className":102637,"style":697},[136],[18,102639,102641],{"className":102640},[141,142,143,144],[18,102642,102644],{"className":102643},[106,144],[18,102645,64],{"className":102646},[106,110,144],[18,102648,102649,102652],{"style":6037},[18,102650],{"className":102651,"style":697},[136],[18,102653],{"className":102654,"style":6045},[6044],[18,102656,102657,102660],{"style":12068},[18,102658],{"className":102659,"style":697},[136],[18,102661,102663],{"className":102662},[141,142,143,144],[18,102664,102666],{"className":102665},[106,144],[18,102667,47],{"className":102668},[106,144],[18,102670,152],{"className":102671},[151],[18,102673,102675],{"className":102674},[124],[18,102676,102678],{"className":102677,"style":12090},[128],[18,102679],{},[18,102681],{"className":102682},[267,6009],[18,102684,102686],{"className":102685,"style":13793},[267,13792],[18,102687,84],{"className":102688},[1230,1196],[18,102690,102692],{"className":102691},[115],[18,102693,102695],{"className":102694},[119],[18,102696,102698],{"className":102697},[124],[18,102699,102702],{"className":102700,"style":102701},[128],"height:1.1939em;",[18,102703,102704,102707],{"style":69827},[18,102705],{"className":102706,"style":697},[136],[18,102708,102710],{"className":102709},[141,142,143,144],[18,102711,102713,102716,102719],{"className":102712},[106,144],[18,102714,64],{"className":102715},[106,110,144],[18,102717,7984],{"className":102718},[2072,144],[18,102720,102722,102725,102784],{"className":102721},[106,144],[18,102723],{"className":102724},[250,6009,141,1195,38147],[18,102726,102728],{"className":102727},[5651],[18,102729,102731,102776],{"className":102730},[119,120],[18,102732,102734,102773],{"className":102733},[124],[18,102735,102737,102751,102759],{"className":102736,"style":42929},[128],[18,102738,102739,102742],{"style":38163},[18,102740],{"className":102741,"style":697},[136],[18,102743,102745],{"className":102744},[141,1195,1196,144],[18,102746,102748],{"className":102747},[106,144],[18,102749,2266],{"className":102750},[106,144],[18,102752,102753,102756],{"style":38187},[18,102754],{"className":102755,"style":697},[136],[18,102757],{"className":102758,"style":38194},[6044,144],[18,102760,102761,102764],{"style":42963},[18,102762],{"className":102763,"style":697},[136],[18,102765,102767],{"className":102766},[141,1195,1196,144],[18,102768,102770],{"className":102769},[106,144],[18,102771,47],{"className":102772},[106,144],[18,102774,152],{"className":102775},[151],[18,102777,102779],{"className":102778},[124],[18,102780,102782],{"className":102781,"style":81865},[128],[18,102783],{},[18,102785],{"className":102786},[267,6009,141,1195,38147],[18,102788,102790,102793],{"style":102789},"top:-3.4239em;",[18,102791],{"className":102792,"style":102587},[136],[18,102794],{"className":102795,"style":6045},[6044],[18,102797,102799,102802],{"style":102798},"top:-3.8709em;",[18,102800],{"className":102801,"style":102587},[136],[18,102803,102805],{"className":102804},[106],[18,102806,14441],{"className":102807},[106,110],[18,102809,152],{"className":102810},[151],[18,102812,102814],{"className":102813},[124],[18,102815,102818],{"className":102816,"style":102817},[128],"height:1.4339em;",[18,102819],{},[18,102821],{"className":102822},[267,6009],[11,102824,102825],{},"将其与下式比较：",[18,102827,102829],{"className":102828,"translate":22},[875],[18,102830,102832,102916],{"className":102831,"translate":22},[21],[18,102833,102835],{"className":102834},[26],[28,102836,102837],{"xmlns":30,"display":884},[32,102838,102839,102913],{},[35,102840,102841,102871,102873],{},[1556,102842,102843,102859],{},[35,102844,102845,102847,102849,102851,102857],{},[49,102846,74],{"fence":51},[45,102848,47],{},[49,102850,7984],{},[5651,102852,102853,102855],{},[45,102854,47],{},[41,102856,64],{},[49,102858,84],{"fence":51},[35,102860,102861,102863,102865],{},[41,102862,64],{},[49,102864,7984],{},[5651,102866,102867,102869],{},[45,102868,47],{},[45,102870,2266],{},[49,102872,433],{},[1556,102874,102875,102877],{},[41,102876,14441],{},[35,102878,102879,102881,102883,102885,102891,102893,102895,102897],{},[49,102880,74],{"stretchy":73},[41,102882,64],{},[49,102884,7984],{},[5651,102886,102887,102889],{},[45,102888,47],{},[45,102890,2266],{},[49,102892,84],{"stretchy":73},[41,102894,97425],{},[49,102896,899],{},[35,102898,102899,102901,102903,102905,102911],{},[49,102900,74],{"fence":51},[45,102902,47],{},[49,102904,7984],{},[5651,102906,102907,102909],{},[45,102908,47],{},[41,102910,64],{},[49,102912,84],{"fence":51},[86,102914,102915],{"encoding":88},"\\left(1 + \\frac{1}{n}\\right)^{n + \\frac{1}{2}} = e^{(n + \\frac{1}{2}) \\ln \\left(1 + \\frac{1}{n}\\right)}",[18,102917,102919,103123],{"className":102918,"ariaHidden":51},[93],[18,102920,102922,102925,103114,103117,103120],{"className":102921},[97],[18,102923],{"className":102924,"style":96992},[101],[18,102926,102928,103017],{"className":102927},[174],[18,102929,102931,102937,102940,102943,102946,102949,103011],{"className":102930},[174],[18,102932,102934],{"className":102933,"style":13793},[250,13792],[18,102935,74],{"className":102936},[1230,143],[18,102938,47],{"className":102939},[106],[18,102941],{"className":102942,"style":2068},[169],[18,102944,7984],{"className":102945},[2072],[18,102947],{"className":102948,"style":2068},[169],[18,102950,102952,102955,103008],{"className":102951},[106],[18,102953],{"className":102954},[250,6009],[18,102956,102958],{"className":102957},[5651],[18,102959,102961,103000],{"className":102960},[119,120],[18,102962,102964,102997],{"className":102963},[124],[18,102965,102967,102978,102986],{"className":102966,"style":9578},[128],[18,102968,102969,102972],{"style":6025},[18,102970],{"className":102971,"style":697},[136],[18,102973,102975],{"className":102974},[106],[18,102976,64],{"className":102977},[106,110],[18,102979,102980,102983],{"style":6037},[18,102981],{"className":102982,"style":697},[136],[18,102984],{"className":102985,"style":6045},[6044],[18,102987,102988,102991],{"style":6048},[18,102989],{"className":102990,"style":697},[136],[18,102992,102994],{"className":102993},[106],[18,102995,47],{"className":102996},[106],[18,102998,152],{"className":102999},[151],[18,103001,103003],{"className":103002},[124],[18,103004,103006],{"className":103005,"style":6067},[128],[18,103007],{},[18,103009],{"className":103010},[267,6009],[18,103012,103014],{"className":103013,"style":13793},[267,13792],[18,103015,84],{"className":103016},[1230,143],[18,103018,103020],{"className":103019},[115],[18,103021,103023],{"className":103022},[119],[18,103024,103026],{"className":103025},[124],[18,103027,103029],{"className":103028,"style":97173},[128],[18,103030,103031,103034],{"style":97176},[18,103032],{"className":103033,"style":697},[136],[18,103035,103037],{"className":103036},[141,142,143,144],[18,103038,103040,103043,103046],{"className":103039},[106,144],[18,103041,64],{"className":103042},[106,110,144],[18,103044,7984],{"className":103045},[2072,144],[18,103047,103049,103052,103111],{"className":103048},[106,144],[18,103050],{"className":103051},[250,6009,141,1195,38147],[18,103053,103055],{"className":103054},[5651],[18,103056,103058,103103],{"className":103057},[119,120],[18,103059,103061,103100],{"className":103060},[124],[18,103062,103064,103078,103086],{"className":103063,"style":42929},[128],[18,103065,103066,103069],{"style":38163},[18,103067],{"className":103068,"style":697},[136],[18,103070,103072],{"className":103071},[141,1195,1196,144],[18,103073,103075],{"className":103074},[106,144],[18,103076,2266],{"className":103077},[106,144],[18,103079,103080,103083],{"style":38187},[18,103081],{"className":103082,"style":697},[136],[18,103084],{"className":103085,"style":38194},[6044,144],[18,103087,103088,103091],{"style":42963},[18,103089],{"className":103090,"style":697},[136],[18,103092,103094],{"className":103093},[141,1195,1196,144],[18,103095,103097],{"className":103096},[106,144],[18,103098,47],{"className":103099},[106,144],[18,103101,152],{"className":103102},[151],[18,103104,103106],{"className":103105},[124],[18,103107,103109],{"className":103108,"style":81865},[128],[18,103110],{},[18,103112],{"className":103113},[267,6009,141,1195,38147],[18,103115],{"className":103116,"style":228},[169],[18,103118,433],{"className":103119},[232],[18,103121],{"className":103122,"style":228},[169],[18,103124,103126,103129],{"className":103125},[97],[18,103127],{"className":103128,"style":97806},[101],[18,103130,103132,103135],{"className":103131},[106],[18,103133,14441],{"className":103134},[106,110],[18,103136,103138],{"className":103137},[115],[18,103139,103141],{"className":103140},[119],[18,103142,103144],{"className":103143},[124],[18,103145,103147],{"className":103146,"style":97806},[128],[18,103148,103149,103152],{"style":97809},[18,103150],{"className":103151,"style":697},[136],[18,103153,103155],{"className":103154},[141,142,143,144],[18,103156,103158,103161,103164,103167,103235,103238,103241,103250],{"className":103157},[106,144],[18,103159,74],{"className":103160},[250,144],[18,103162,64],{"className":103163},[106,110,144],[18,103165,7984],{"className":103166},[2072,144],[18,103168,103170,103173,103232],{"className":103169},[106,144],[18,103171],{"className":103172},[250,6009,141,1195,38147],[18,103174,103176],{"className":103175},[5651],[18,103177,103179,103224],{"className":103178},[119,120],[18,103180,103182,103221],{"className":103181},[124],[18,103183,103185,103199,103207],{"className":103184,"style":42929},[128],[18,103186,103187,103190],{"style":38163},[18,103188],{"className":103189,"style":697},[136],[18,103191,103193],{"className":103192},[141,1195,1196,144],[18,103194,103196],{"className":103195},[106,144],[18,103197,2266],{"className":103198},[106,144],[18,103200,103201,103204],{"style":38187},[18,103202],{"className":103203,"style":697},[136],[18,103205],{"className":103206,"style":38194},[6044,144],[18,103208,103209,103212],{"style":42963},[18,103210],{"className":103211,"style":697},[136],[18,103213,103215],{"className":103214},[141,1195,1196,144],[18,103216,103218],{"className":103217},[106,144],[18,103219,47],{"className":103220},[106,144],[18,103222,152],{"className":103223},[151],[18,103225,103227],{"className":103226},[124],[18,103228,103230],{"className":103229,"style":81865},[128],[18,103231],{},[18,103233],{"className":103234},[267,6009,141,1195,38147],[18,103236,84],{"className":103237},[267,144],[18,103239],{"className":103240,"style":61291},[169,144],[18,103242,103244,103247],{"className":103243},[985,144],[18,103245,79371],{"className":103246},[144],[18,103248,64],{"className":103249},[144],[18,103251,103253,103259,103262,103265,103333],{"className":103252},[174,144],[18,103254,103256],{"className":103255,"style":97905},[250,141,1195,38147,144,13792],[18,103257,74],{"className":103258},[144],[18,103260,47],{"className":103261},[106,144],[18,103263,7984],{"className":103264},[2072,144],[18,103266,103268,103271,103330],{"className":103267},[106,144],[18,103269],{"className":103270},[250,6009,141,1195,38147],[18,103272,103274],{"className":103273},[5651],[18,103275,103277,103322],{"className":103276},[119,120],[18,103278,103280,103319],{"className":103279},[124],[18,103281,103283,103297,103305],{"className":103282,"style":42929},[128],[18,103284,103285,103288],{"style":38163},[18,103286],{"className":103287,"style":697},[136],[18,103289,103291],{"className":103290},[141,1195,1196,144],[18,103292,103294],{"className":103293},[106,144],[18,103295,64],{"className":103296},[106,110,144],[18,103298,103299,103302],{"style":38187},[18,103300],{"className":103301,"style":697},[136],[18,103303],{"className":103304,"style":38194},[6044,144],[18,103306,103307,103310],{"style":42963},[18,103308],{"className":103309,"style":697},[136],[18,103311,103313],{"className":103312},[141,1195,1196,144],[18,103314,103316],{"className":103315},[106,144],[18,103317,47],{"className":103318},[106,144],[18,103320,152],{"className":103321},[151],[18,103323,103325],{"className":103324},[124],[18,103326,103328],{"className":103327,"style":81865},[128],[18,103329],{},[18,103331],{"className":103332},[267,6009,141,1195,38147],[18,103334,103336],{"className":103335,"style":97905},[267,141,1195,38147,144,13792],[18,103337,84],{"className":103338},[144],[11,103340,103341,103342,1990],{},"以及 ",[18,103343,103345,103358],{"className":103344,"translate":22},[21],[18,103346,103348],{"className":103347},[26],[28,103349,103350],{"xmlns":30},[32,103351,103352,103356],{},[35,103353,103354],{},[41,103355,14441],{},[86,103357,14441],{"encoding":88},[18,103359,103361],{"className":103360,"ariaHidden":51},[93],[18,103362,103364,103367],{"className":103363},[97],[18,103365],{"className":103366,"style":1083},[101],[18,103368,14441],{"className":103369},[106,110],[11,103371,103372],{},"因此，比较：",[18,103374,103376],{"className":103375,"translate":22},[875],[18,103377,103379,103425],{"className":103378,"translate":22},[21],[18,103380,103382],{"className":103381},[26],[28,103383,103384],{"xmlns":30,"display":884},[32,103385,103386,103422],{},[35,103387,103388,103390,103392,103394,103400,103402,103404,103406],{},[49,103389,74],{"stretchy":73},[41,103391,64],{},[49,103393,7984],{},[5651,103395,103396,103398],{},[45,103397,47],{},[45,103399,2266],{},[49,103401,84],{"stretchy":73},[41,103403,97425],{},[49,103405,899],{},[35,103407,103408,103410,103412,103414,103420],{},[49,103409,74],{"fence":51},[45,103411,47],{},[49,103413,7984],{},[5651,103415,103416,103418],{},[45,103417,47],{},[41,103419,64],{},[49,103421,84],{"fence":51},[86,103423,103424],{"encoding":88},"(n + \\frac{1}{2}) \\ln \\left(1 + \\frac{1}{n}\\right)",[18,103426,103428,103449],{"className":103427,"ariaHidden":51},[93],[18,103429,103431,103434,103437,103440,103443,103446],{"className":103430},[97],[18,103432],{"className":103433,"style":242},[101],[18,103435,74],{"className":103436},[250],[18,103438,64],{"className":103439},[106,110],[18,103441],{"className":103442,"style":2068},[169],[18,103444,7984],{"className":103445},[2072],[18,103447],{"className":103448,"style":2068},[169],[18,103450,103452,103455,103517,103520,103523,103526,103529],{"className":103451},[97],[18,103453],{"className":103454,"style":20800},[101],[18,103456,103458,103461,103514],{"className":103457},[106],[18,103459],{"className":103460},[250,6009],[18,103462,103464],{"className":103463},[5651],[18,103465,103467,103506],{"className":103466},[119,120],[18,103468,103470,103503],{"className":103469},[124],[18,103471,103473,103484,103492],{"className":103472,"style":9578},[128],[18,103474,103475,103478],{"style":6025},[18,103476],{"className":103477,"style":697},[136],[18,103479,103481],{"className":103480},[106],[18,103482,2266],{"className":103483},[106],[18,103485,103486,103489],{"style":6037},[18,103487],{"className":103488,"style":697},[136],[18,103490],{"className":103491,"style":6045},[6044],[18,103493,103494,103497],{"style":6048},[18,103495],{"className":103496,"style":697},[136],[18,103498,103500],{"className":103499},[106],[18,103501,47],{"className":103502},[106],[18,103504,152],{"className":103505},[151],[18,103507,103509],{"className":103508},[124],[18,103510,103512],{"className":103511,"style":6067},[128],[18,103513],{},[18,103515],{"className":103516},[267,6009],[18,103518,84],{"className":103519},[267],[18,103521],{"className":103522,"style":170},[169],[18,103524,97425],{"className":103525},[985],[18,103527],{"className":103528,"style":170},[169],[18,103530,103532,103538,103541,103544,103547,103550,103612],{"className":103531},[174],[18,103533,103535],{"className":103534,"style":13793},[250,13792],[18,103536,74],{"className":103537},[1230,143],[18,103539,47],{"className":103540},[106],[18,103542],{"className":103543,"style":2068},[169],[18,103545,7984],{"className":103546},[2072],[18,103548],{"className":103549,"style":2068},[169],[18,103551,103553,103556,103609],{"className":103552},[106],[18,103554],{"className":103555},[250,6009],[18,103557,103559],{"className":103558},[5651],[18,103560,103562,103601],{"className":103561},[119,120],[18,103563,103565,103598],{"className":103564},[124],[18,103566,103568,103579,103587],{"className":103567,"style":9578},[128],[18,103569,103570,103573],{"style":6025},[18,103571],{"className":103572,"style":697},[136],[18,103574,103576],{"className":103575},[106],[18,103577,64],{"className":103578},[106,110],[18,103580,103581,103584],{"style":6037},[18,103582],{"className":103583,"style":697},[136],[18,103585],{"className":103586,"style":6045},[6044],[18,103588,103589,103592],{"style":6048},[18,103590],{"className":103591,"style":697},[136],[18,103593,103595],{"className":103594},[106],[18,103596,47],{"className":103597},[106],[18,103599,152],{"className":103600},[151],[18,103602,103604],{"className":103603},[124],[18,103605,103607],{"className":103606,"style":6067},[128],[18,103608],{},[18,103610],{"className":103611},[267,6009],[18,103613,103615],{"className":103614,"style":13793},[267,13792],[18,103616,84],{"className":103617},[1230,143],[11,103619,103620],{},"与 1。",[11,103622,103623,103624,509,103744,56904],{},"具体来说，比较 ",[18,103625,103627,103655],{"className":103626,"translate":22},[21],[18,103628,103630],{"className":103629},[26],[28,103631,103632],{"xmlns":30},[32,103633,103634,103652],{},[35,103635,103636,103638,103640],{},[41,103637,97425],{},[49,103639,899],{},[5651,103641,103642,103650],{},[35,103643,103644,103646,103648],{},[41,103645,64],{},[49,103647,7984],{},[45,103649,47],{},[41,103651,64],{},[86,103653,103654],{"encoding":88},"\\ln \\frac{n+1}{n}",[18,103656,103658],{"className":103657,"ariaHidden":51},[93],[18,103659,103661,103664,103667,103670],{"className":103660},[97],[18,103662],{"className":103663,"style":12023},[101],[18,103665,97425],{"className":103666},[985],[18,103668],{"className":103669,"style":170},[169],[18,103671,103673,103676,103741],{"className":103672},[106],[18,103674],{"className":103675},[250,6009],[18,103677,103679],{"className":103678},[5651],[18,103680,103682,103733],{"className":103681},[119,120],[18,103683,103685,103730],{"className":103684},[124],[18,103686,103688,103702,103710],{"className":103687,"style":12042},[128],[18,103689,103690,103693],{"style":12045},[18,103691],{"className":103692,"style":697},[136],[18,103694,103696],{"className":103695},[141,142,143,144],[18,103697,103699],{"className":103698},[106,144],[18,103700,64],{"className":103701},[106,110,144],[18,103703,103704,103707],{"style":6037},[18,103705],{"className":103706,"style":697},[136],[18,103708],{"className":103709,"style":6045},[6044],[18,103711,103712,103715],{"style":12068},[18,103713],{"className":103714,"style":697},[136],[18,103716,103718],{"className":103717},[141,142,143,144],[18,103719,103721,103724,103727],{"className":103720},[106,144],[18,103722,64],{"className":103723},[106,110,144],[18,103725,7984],{"className":103726},[2072,144],[18,103728,47],{"className":103729},[106,144],[18,103731,152],{"className":103732},[151],[18,103734,103736],{"className":103735},[124],[18,103737,103739],{"className":103738,"style":12090},[128],[18,103740],{},[18,103742],{"className":103743},[267,6009],[18,103745,103747,103775],{"className":103746,"translate":22},[21],[18,103748,103750],{"className":103749},[26],[28,103751,103752],{"xmlns":30},[32,103753,103754,103772],{},[35,103755,103756],{},[5651,103757,103758,103760],{},[45,103759,47],{},[35,103761,103762,103764,103766],{},[41,103763,64],{},[49,103765,7984],{},[5651,103767,103768,103770],{},[45,103769,47],{},[45,103771,2266],{},[86,103773,103774],{"encoding":88},"\\frac{1}{n + \\frac{1}{2}}",[18,103776,103778],{"className":103777,"ariaHidden":51},[93],[18,103779,103781,103785],{"className":103780},[97],[18,103782],{"className":103783,"style":103784},[101],"height:1.4869em;vertical-align:-0.6418em;",[18,103786,103788,103791,103923],{"className":103787},[106],[18,103789],{"className":103790},[250,6009],[18,103792,103794],{"className":103793},[5651],[18,103795,103797,103914],{"className":103796},[119,120],[18,103798,103800,103911],{"className":103799},[124],[18,103801,103803,103889,103897],{"className":103802,"style":12042},[128],[18,103804,103806,103809],{"style":103805},"top:-2.599em;",[18,103807],{"className":103808,"style":697},[136],[18,103810,103812],{"className":103811},[141,142,143,144],[18,103813,103815,103818,103821],{"className":103814},[106,144],[18,103816,64],{"className":103817},[106,110,144],[18,103819,7984],{"className":103820},[2072,144],[18,103822,103824,103827,103886],{"className":103823},[106,144],[18,103825],{"className":103826},[250,6009,141,1195,38147],[18,103828,103830],{"className":103829},[5651],[18,103831,103833,103878],{"className":103832},[119,120],[18,103834,103836,103875],{"className":103835},[124],[18,103837,103839,103853,103861],{"className":103838,"style":42929},[128],[18,103840,103841,103844],{"style":38163},[18,103842],{"className":103843,"style":697},[136],[18,103845,103847],{"className":103846},[141,1195,1196,144],[18,103848,103850],{"className":103849},[106,144],[18,103851,2266],{"className":103852},[106,144],[18,103854,103855,103858],{"style":38187},[18,103856],{"className":103857,"style":697},[136],[18,103859],{"className":103860,"style":38194},[6044,144],[18,103862,103863,103866],{"style":42963},[18,103864],{"className":103865,"style":697},[136],[18,103867,103869],{"className":103868},[141,1195,1196,144],[18,103870,103872],{"className":103871},[106,144],[18,103873,47],{"className":103874},[106,144],[18,103876,152],{"className":103877},[151],[18,103879,103881],{"className":103880},[124],[18,103882,103884],{"className":103883,"style":81865},[128],[18,103885],{},[18,103887],{"className":103888},[267,6009,141,1195,38147],[18,103890,103891,103894],{"style":6037},[18,103892],{"className":103893,"style":697},[136],[18,103895],{"className":103896,"style":6045},[6044],[18,103898,103899,103902],{"style":12068},[18,103900],{"className":103901,"style":697},[136],[18,103903,103905],{"className":103904},[141,142,143,144],[18,103906,103908],{"className":103907},[106,144],[18,103909,47],{"className":103910},[106,144],[18,103912,152],{"className":103913},[151],[18,103915,103917],{"className":103916},[124],[18,103918,103921],{"className":103919,"style":103920},[128],"height:0.6418em;",[18,103922],{},[18,103924],{"className":103925},[267,6009],[11,103927,103928],{},"由于：",[18,103930,103932],{"className":103931,"translate":22},[875],[18,103933,103935,103989],{"className":103934,"translate":22},[21],[18,103936,103938],{"className":103937},[26],[28,103939,103940],{"xmlns":30,"display":884},[32,103941,103942,103986],{},[35,103943,103944,103956,103958,103960,103962,103978,103980],{},[5651,103945,103946,103948],{},[45,103947,47],{},[35,103949,103950,103952,103954],{},[41,103951,64],{},[49,103953,7984],{},[45,103955,47],{},[49,103957,6681],{},[41,103959,97425],{},[49,103961,899],{},[35,103963,103964,103966,103968,103970,103976],{},[49,103965,74],{"fence":51},[45,103967,47],{},[49,103969,7984],{},[5651,103971,103972,103974],{},[45,103973,47],{},[41,103975,64],{},[49,103977,84],{"fence":51},[49,103979,6681],{},[5651,103981,103982,103984],{},[45,103983,47],{},[41,103985,64],{},[86,103987,103988],{"encoding":88},"\\frac{1}{n + 1} \u003C \\ln \\left(1 + \\frac{1}{n}\\right) \u003C \\frac{1}{n}",[18,103990,103992,104081,104191],{"className":103991,"ariaHidden":51},[93],[18,103993,103995,103998,104072,104075,104078],{"className":103994},[97],[18,103996],{"className":103997,"style":92871},[101],[18,103999,104001,104004,104069],{"className":104000},[106],[18,104002],{"className":104003},[250,6009],[18,104005,104007],{"className":104006},[5651],[18,104008,104010,104061],{"className":104009},[119,120],[18,104011,104013,104058],{"className":104012},[124],[18,104014,104016,104039,104047],{"className":104015,"style":9578},[128],[18,104017,104018,104021],{"style":6025},[18,104019],{"className":104020,"style":697},[136],[18,104022,104024,104027,104030,104033,104036],{"className":104023},[106],[18,104025,64],{"className":104026},[106,110],[18,104028],{"className":104029,"style":2068},[169],[18,104031,7984],{"className":104032},[2072],[18,104034],{"className":104035,"style":2068},[169],[18,104037,47],{"className":104038},[106],[18,104040,104041,104044],{"style":6037},[18,104042],{"className":104043,"style":697},[136],[18,104045],{"className":104046,"style":6045},[6044],[18,104048,104049,104052],{"style":6048},[18,104050],{"className":104051,"style":697},[136],[18,104053,104055],{"className":104054},[106],[18,104056,47],{"className":104057},[106],[18,104059,152],{"className":104060},[151],[18,104062,104064],{"className":104063},[124],[18,104065,104067],{"className":104066,"style":86777},[128],[18,104068],{},[18,104070],{"className":104071},[267,6009],[18,104073],{"className":104074,"style":228},[169],[18,104076,6681],{"className":104077},[232],[18,104079],{"className":104080,"style":228},[169],[18,104082,104084,104087,104090,104093,104182,104185,104188],{"className":104083},[97],[18,104085],{"className":104086,"style":20800},[101],[18,104088,97425],{"className":104089},[985],[18,104091],{"className":104092,"style":170},[169],[18,104094,104096,104102,104105,104108,104111,104114,104176],{"className":104095},[174],[18,104097,104099],{"className":104098,"style":13793},[250,13792],[18,104100,74],{"className":104101},[1230,143],[18,104103,47],{"className":104104},[106],[18,104106],{"className":104107,"style":2068},[169],[18,104109,7984],{"className":104110},[2072],[18,104112],{"className":104113,"style":2068},[169],[18,104115,104117,104120,104173],{"className":104116},[106],[18,104118],{"className":104119},[250,6009],[18,104121,104123],{"className":104122},[5651],[18,104124,104126,104165],{"className":104125},[119,120],[18,104127,104129,104162],{"className":104128},[124],[18,104130,104132,104143,104151],{"className":104131,"style":9578},[128],[18,104133,104134,104137],{"style":6025},[18,104135],{"className":104136,"style":697},[136],[18,104138,104140],{"className":104139},[106],[18,104141,64],{"className":104142},[106,110],[18,104144,104145,104148],{"style":6037},[18,104146],{"className":104147,"style":697},[136],[18,104149],{"className":104150,"style":6045},[6044],[18,104152,104153,104156],{"style":6048},[18,104154],{"className":104155,"style":697},[136],[18,104157,104159],{"className":104158},[106],[18,104160,47],{"className":104161},[106],[18,104163,152],{"className":104164},[151],[18,104166,104168],{"className":104167},[124],[18,104169,104171],{"className":104170,"style":6067},[128],[18,104172],{},[18,104174],{"className":104175},[267,6009],[18,104177,104179],{"className":104178,"style":13793},[267,13792],[18,104180,84],{"className":104181},[1230,143],[18,104183],{"className":104184,"style":228},[169],[18,104186,6681],{"className":104187},[232],[18,104189],{"className":104190,"style":228},[169],[18,104192,104194,104197],{"className":104193},[97],[18,104195],{"className":104196,"style":9559},[101],[18,104198,104200,104203,104256],{"className":104199},[106],[18,104201],{"className":104202},[250,6009],[18,104204,104206],{"className":104205},[5651],[18,104207,104209,104248],{"className":104208},[119,120],[18,104210,104212,104245],{"className":104211},[124],[18,104213,104215,104226,104234],{"className":104214,"style":9578},[128],[18,104216,104217,104220],{"style":6025},[18,104218],{"className":104219,"style":697},[136],[18,104221,104223],{"className":104222},[106],[18,104224,64],{"className":104225},[106,110],[18,104227,104228,104231],{"style":6037},[18,104229],{"className":104230,"style":697},[136],[18,104232],{"className":104233,"style":6045},[6044],[18,104235,104236,104239],{"style":6048},[18,104237],{"className":104238,"style":697},[136],[18,104240,104242],{"className":104241},[106],[18,104243,47],{"className":104244},[106],[18,104246,152],{"className":104247},[151],[18,104249,104251],{"className":104250},[124],[18,104252,104254],{"className":104253,"style":6067},[128],[18,104255],{},[18,104257],{"className":104258},[267,6009],[11,104260,104261,104262,104352,104353,104437],{},"注意：右侧由不等式 ",[18,104263,104265,104295],{"className":104264,"translate":22},[21],[18,104266,104268],{"className":104267},[26],[28,104269,104270],{"xmlns":30},[32,104271,104272,104292],{},[35,104273,104274,104276,104278,104280,104282,104284,104286,104288,104290],{},[41,104275,97425],{},[49,104277,899],{},[49,104279,74],{"stretchy":73},[45,104281,47],{},[49,104283,7984],{},[41,104285,38573],{},[49,104287,84],{"stretchy":73},[49,104289,6681],{},[41,104291,38573],{},[86,104293,104294],{"encoding":88},"\\ln(1 + x) \u003C x",[18,104296,104298,104322,104343],{"className":104297,"ariaHidden":51},[93],[18,104299,104301,104304,104307,104310,104313,104316,104319],{"className":104300},[97],[18,104302],{"className":104303,"style":242},[101],[18,104305,97425],{"className":104306},[985],[18,104308,74],{"className":104309},[250],[18,104311,47],{"className":104312},[106],[18,104314],{"className":104315,"style":2068},[169],[18,104317,7984],{"className":104318},[2072],[18,104320],{"className":104321,"style":2068},[169],[18,104323,104325,104328,104331,104334,104337,104340],{"className":104324},[97],[18,104326],{"className":104327,"style":242},[101],[18,104329,38573],{"className":104330},[106,110],[18,104332,84],{"className":104333},[267],[18,104335],{"className":104336,"style":228},[169],[18,104338,6681],{"className":104339},[232],[18,104341],{"className":104342,"style":228},[169],[18,104344,104346,104349],{"className":104345},[97],[18,104347],{"className":104348,"style":1083},[101],[18,104350,38573],{"className":104351},[106,110],"（其中 ",[18,104354,104356,104374],{"className":104355,"translate":22},[21],[18,104357,104359],{"className":104358},[26],[28,104360,104361],{"xmlns":30},[32,104362,104363,104371],{},[35,104364,104365,104367,104369],{},[41,104366,38573],{},[49,104368,2771],{"mathvariant":316},[45,104370,430],{},[86,104372,104373],{"encoding":88},"x \\neq 0",[18,104375,104377,104428],{"className":104376,"ariaHidden":51},[93],[18,104378,104380,104383,104386,104389,104425],{"className":104379},[97],[18,104381],{"className":104382,"style":2890},[101],[18,104384,38573],{"className":104385},[106,110],[18,104387],{"className":104388,"style":228},[169],[18,104390,104392,104419,104422],{"className":104391},[232],[18,104393,104395],{"className":104394},[232],[18,104396,104398],{"className":104397},[106,2878],[18,104399,104401],{"className":104400},[2882],[18,104402,104404,104407,104416],{"className":104403},[2886],[18,104405],{"className":104406,"style":2890},[101],[18,104408,104410],{"className":104409},[2894],[18,104411,104413],{"className":104412},[106],[18,104414,2901],{"className":104415},[232],[18,104417],{"className":104418},[2905],[18,104420],{"className":104421},[169,2909],[18,104423,433],{"className":104424},[232],[18,104426],{"className":104427,"style":228},[169],[18,104429,104431,104434],{"className":104430},[97],[18,104432],{"className":104433,"style":505},[101],[18,104435,430],{"className":104436},[106],"）得出。",[11,104439,104440,104441,104708,104709,104819],{},"左侧由 ",[18,104442,104444,104496],{"className":104443,"translate":22},[21],[18,104445,104447],{"className":104446},[26],[28,104448,104449],{"xmlns":30},[32,104450,104451,104493],{},[35,104452,104453,104455,104457,104469,104471,104473,104475],{},[45,104454,47],{},[49,104456,2253],{},[5651,104458,104459,104461],{},[41,104460,64],{},[35,104462,104463,104465,104467],{},[41,104464,64],{},[49,104466,7984],{},[45,104468,47],{},[49,104470,6681],{},[41,104472,97425],{},[49,104474,899],{},[35,104476,104477,104479,104491],{},[49,104478,74],{"fence":51},[5651,104480,104481,104489],{},[35,104482,104483,104485,104487],{},[41,104484,64],{},[49,104486,7984],{},[45,104488,47],{},[41,104490,64],{},[49,104492,84],{"fence":51},[86,104494,104495],{"encoding":88},"1 - \\frac{n}{n+1} \u003C \\ln \\left(\\frac{n+1}{n}\\right)",[18,104497,104499,104517,104607],{"className":104498,"ariaHidden":51},[93],[18,104500,104502,104505,104508,104511,104514],{"className":104501},[97],[18,104503],{"className":104504,"style":18090},[101],[18,104506,47],{"className":104507},[106],[18,104509],{"className":104510,"style":2068},[169],[18,104512,2253],{"className":104513},[2072],[18,104515],{"className":104516,"style":2068},[169],[18,104518,104520,104524,104598,104601,104604],{"className":104519},[97],[18,104521],{"className":104522,"style":104523},[101],"height:1.0987em;vertical-align:-0.4033em;",[18,104525,104527,104530,104595],{"className":104526},[106],[18,104528],{"className":104529},[250,6009],[18,104531,104533],{"className":104532},[5651],[18,104534,104536,104587],{"className":104535},[119,120],[18,104537,104539,104584],{"className":104538},[124],[18,104540,104542,104562,104570],{"className":104541,"style":14254},[128],[18,104543,104544,104547],{"style":12045},[18,104545],{"className":104546,"style":697},[136],[18,104548,104550],{"className":104549},[141,142,143,144],[18,104551,104553,104556,104559],{"className":104552},[106,144],[18,104554,64],{"className":104555},[106,110,144],[18,104557,7984],{"className":104558},[2072,144],[18,104560,47],{"className":104561},[106,144],[18,104563,104564,104567],{"style":6037},[18,104565],{"className":104566,"style":697},[136],[18,104568],{"className":104569,"style":6045},[6044],[18,104571,104572,104575],{"style":12068},[18,104573],{"className":104574,"style":697},[136],[18,104576,104578],{"className":104577},[141,142,143,144],[18,104579,104581],{"className":104580},[106,144],[18,104582,64],{"className":104583},[106,110,144],[18,104585,152],{"className":104586},[151],[18,104588,104590],{"className":104589},[124],[18,104591,104593],{"className":104592,"style":93278},[128],[18,104594],{},[18,104596],{"className":104597},[267,6009],[18,104599],{"className":104600,"style":228},[169],[18,104602,6681],{"className":104603},[232],[18,104605],{"className":104606,"style":228},[169],[18,104608,104610,104613,104616,104619],{"className":104609},[97],[18,104611],{"className":104612,"style":6098},[101],[18,104614,97425],{"className":104615},[985],[18,104617],{"className":104618,"style":170},[169],[18,104620,104622,104628,104702],{"className":104621},[174],[18,104623,104625],{"className":104624,"style":13793},[250,13792],[18,104626,74],{"className":104627},[1230,1196],[18,104629,104631,104634,104699],{"className":104630},[106],[18,104632],{"className":104633},[250,6009],[18,104635,104637],{"className":104636},[5651],[18,104638,104640,104691],{"className":104639},[119,120],[18,104641,104643,104688],{"className":104642},[124],[18,104644,104646,104660,104668],{"className":104645,"style":12042},[128],[18,104647,104648,104651],{"style":12045},[18,104649],{"className":104650,"style":697},[136],[18,104652,104654],{"className":104653},[141,142,143,144],[18,104655,104657],{"className":104656},[106,144],[18,104658,64],{"className":104659},[106,110,144],[18,104661,104662,104665],{"style":6037},[18,104663],{"className":104664,"style":697},[136],[18,104666],{"className":104667,"style":6045},[6044],[18,104669,104670,104673],{"style":12068},[18,104671],{"className":104672,"style":697},[136],[18,104674,104676],{"className":104675},[141,142,143,144],[18,104677,104679,104682,104685],{"className":104678},[106,144],[18,104680,64],{"className":104681},[106,110,144],[18,104683,7984],{"className":104684},[2072,144],[18,104686,47],{"className":104687},[106,144],[18,104689,152],{"className":104690},[151],[18,104692,104694],{"className":104693},[124],[18,104695,104697],{"className":104696,"style":12090},[128],[18,104698],{},[18,104700],{"className":104701},[267,6009],[18,104703,104705],{"className":104704,"style":13793},[267,13792],[18,104706,84],{"className":104707},[1230,1196]," 得出，其中 ",[18,104710,104712,104736],{"className":104711,"translate":22},[21],[18,104713,104715],{"className":104714},[26],[28,104716,104717],{"xmlns":30},[32,104718,104719,104733],{},[35,104720,104721],{},[5651,104722,104723,104725],{},[41,104724,64],{},[35,104726,104727,104729,104731],{},[41,104728,64],{},[49,104730,7984],{},[45,104732,47],{},[86,104734,104735],{"encoding":88},"\\frac{n}{n+1}",[18,104737,104739],{"className":104738,"ariaHidden":51},[93],[18,104740,104742,104745],{"className":104741},[97],[18,104743],{"className":104744,"style":104523},[101],[18,104746,104748,104751,104816],{"className":104747},[106],[18,104749],{"className":104750},[250,6009],[18,104752,104754],{"className":104753},[5651],[18,104755,104757,104808],{"className":104756},[119,120],[18,104758,104760,104805],{"className":104759},[124],[18,104761,104763,104783,104791],{"className":104762,"style":14254},[128],[18,104764,104765,104768],{"style":12045},[18,104766],{"className":104767,"style":697},[136],[18,104769,104771],{"className":104770},[141,142,143,144],[18,104772,104774,104777,104780],{"className":104773},[106,144],[18,104775,64],{"className":104776},[106,110,144],[18,104778,7984],{"className":104779},[2072,144],[18,104781,47],{"className":104782},[106,144],[18,104784,104785,104788],{"style":6037},[18,104786],{"className":104787,"style":697},[136],[18,104789],{"className":104790,"style":6045},[6044],[18,104792,104793,104796],{"style":12068},[18,104794],{"className":104795,"style":697},[136],[18,104797,104799],{"className":104798},[141,142,143,144],[18,104800,104802],{"className":104801},[106,144],[18,104803,64],{"className":104804},[106,110,144],[18,104806,152],{"className":104807},[151],[18,104809,104811],{"className":104810},[124],[18,104812,104814],{"className":104813,"style":93278},[128],[18,104815],{},[18,104817],{"className":104818},[267,6009]," 被视为一个整体。这是高中知识。",[11,104821,104822,104823,104921],{},"因为 ",[18,104824,104826,104844],{"className":104825,"translate":22},[21],[18,104827,104829],{"className":104828},[26],[28,104830,104831],{"xmlns":30},[32,104832,104833,104841],{},[35,104834,104835],{},[5651,104836,104837,104839],{},[45,104838,47],{},[41,104840,38573],{},[86,104842,104843],{"encoding":88},"\\frac{1}{x}",[18,104845,104847],{"className":104846,"ariaHidden":51},[93],[18,104848,104850,104853],{"className":104849},[97],[18,104851],{"className":104852,"style":12023},[101],[18,104854,104856,104859,104918],{"className":104855},[106],[18,104857],{"className":104858},[250,6009],[18,104860,104862],{"className":104861},[5651],[18,104863,104865,104910],{"className":104864},[119,120],[18,104866,104868,104907],{"className":104867},[124],[18,104869,104871,104885,104893],{"className":104870,"style":12042},[128],[18,104872,104873,104876],{"style":12045},[18,104874],{"className":104875,"style":697},[136],[18,104877,104879],{"className":104878},[141,142,143,144],[18,104880,104882],{"className":104881},[106,144],[18,104883,38573],{"className":104884},[106,110,144],[18,104886,104887,104890],{"style":6037},[18,104888],{"className":104889,"style":697},[136],[18,104891],{"className":104892,"style":6045},[6044],[18,104894,104895,104898],{"style":12068},[18,104896],{"className":104897,"style":697},[136],[18,104899,104901],{"className":104900},[141,142,143,144],[18,104902,104904],{"className":104903},[106,144],[18,104905,47],{"className":104906},[106,144],[18,104908,152],{"className":104909},[151],[18,104911,104913],{"className":104912},[124],[18,104914,104916],{"className":104915,"style":12090},[128],[18,104917],{},[18,104919],{"className":104920},[267,6009]," 是凹函数：",[18,104923,104925],{"className":104924,"translate":22},[875],[18,104926,104928,105002],{"className":104927,"translate":22},[21],[18,104929,104931],{"className":104930},[26],[28,104932,104933],{"xmlns":30,"display":884},[32,104934,104935,104999],{},[35,104936,104937,104951,104957,104959,104961,104963,104965,104967,104969,104981,104983],{},[1589,104938,104939,104941,104943],{},[49,104940,4005],{},[41,104942,64],{},[35,104944,104945,104947,104949],{},[41,104946,64],{},[49,104948,7984],{},[45,104950,47],{},[5651,104952,104953,104955],{},[45,104954,47],{},[41,104956,38573],{},[926,104958,4024],{},[41,104960,4031],{},[41,104962,38573],{},[49,104964,433],{},[41,104966,97425],{},[49,104968,899],{},[5651,104970,104971,104979],{},[35,104972,104973,104975,104977],{},[41,104974,64],{},[49,104976,7984],{},[45,104978,47],{},[41,104980,64],{},[49,104982,6882],{},[5651,104984,104985,104987],{},[45,104986,47],{},[35,104988,104989,104991,104993],{},[41,104990,64],{},[49,104992,7984],{},[5651,104994,104995,104997],{},[45,104996,47],{},[45,104998,2266],{},[86,105000,105001],{"encoding":88},"\\int_{n}^{n+1} \\frac{1}{x} \\, dx = \\ln \\frac{n+1}{n} > \\frac{1}{n + \\frac{1}{2}}",[18,105003,105005,105157,105252],{"className":105004,"ariaHidden":51},[93],[18,105006,105008,105011,105074,105077,105139,105142,105145,105148,105151,105154],{"className":105007},[97],[18,105009],{"className":105010,"style":99195},[101],[18,105012,105014,105017],{"className":105013},[985],[18,105015,4005],{"className":105016,"style":4200},[985,4198,4199],[18,105018,105020],{"className":105019},[115],[18,105021,105023,105066],{"className":105022},[119,120],[18,105024,105026,105063],{"className":105025},[124],[18,105027,105029,105043],{"className":105028,"style":99214},[128],[18,105030,105031,105034],{"style":99217},[18,105032],{"className":105033,"style":137},[136],[18,105035,105037],{"className":105036},[141,142,143,144],[18,105038,105040],{"className":105039},[106,144],[18,105041,64],{"className":105042},[106,110,144],[18,105044,105045,105048],{"style":99232},[18,105046],{"className":105047,"style":137},[136],[18,105049,105051],{"className":105050},[141,142,143,144],[18,105052,105054,105057,105060],{"className":105053},[106,144],[18,105055,64],{"className":105056},[106,110,144],[18,105058,7984],{"className":105059},[2072,144],[18,105061,47],{"className":105062},[106,144],[18,105064,152],{"className":105065},[151],[18,105067,105069],{"className":105068},[124],[18,105070,105072],{"className":105071,"style":81880},[128],[18,105073],{},[18,105075],{"className":105076,"style":170},[169],[18,105078,105080,105083,105136],{"className":105079},[106],[18,105081],{"className":105082},[250,6009],[18,105084,105086],{"className":105085},[5651],[18,105087,105089,105128],{"className":105088},[119,120],[18,105090,105092,105125],{"className":105091},[124],[18,105093,105095,105106,105114],{"className":105094,"style":9578},[128],[18,105096,105097,105100],{"style":6025},[18,105098],{"className":105099,"style":697},[136],[18,105101,105103],{"className":105102},[106],[18,105104,38573],{"className":105105},[106,110],[18,105107,105108,105111],{"style":6037},[18,105109],{"className":105110,"style":697},[136],[18,105112],{"className":105113,"style":6045},[6044],[18,105115,105116,105119],{"style":6048},[18,105117],{"className":105118,"style":697},[136],[18,105120,105122],{"className":105121},[106],[18,105123,47],{"className":105124},[106],[18,105126,152],{"className":105127},[151],[18,105129,105131],{"className":105130},[124],[18,105132,105134],{"className":105133,"style":6067},[128],[18,105135],{},[18,105137],{"className":105138},[267,6009],[18,105140],{"className":105141,"style":170},[169],[18,105143,4031],{"className":105144},[106,110],[18,105146,38573],{"className":105147},[106,110],[18,105149],{"className":105150,"style":228},[169],[18,105152,433],{"className":105153},[232],[18,105155],{"className":105156,"style":228},[169],[18,105158,105160,105163,105166,105169,105243,105246,105249],{"className":105159},[97],[18,105161],{"className":105162,"style":9559},[101],[18,105164,97425],{"className":105165},[985],[18,105167],{"className":105168,"style":170},[169],[18,105170,105172,105175,105240],{"className":105171},[106],[18,105173],{"className":105174},[250,6009],[18,105176,105178],{"className":105177},[5651],[18,105179,105181,105232],{"className":105180},[119,120],[18,105182,105184,105229],{"className":105183},[124],[18,105185,105187,105198,105206],{"className":105186,"style":9578},[128],[18,105188,105189,105192],{"style":6025},[18,105190],{"className":105191,"style":697},[136],[18,105193,105195],{"className":105194},[106],[18,105196,64],{"className":105197},[106,110],[18,105199,105200,105203],{"style":6037},[18,105201],{"className":105202,"style":697},[136],[18,105204],{"className":105205,"style":6045},[6044],[18,105207,105208,105211],{"style":6048},[18,105209],{"className":105210,"style":697},[136],[18,105212,105214,105217,105220,105223,105226],{"className":105213},[106],[18,105215,64],{"className":105216},[106,110],[18,105218],{"className":105219,"style":2068},[169],[18,105221,7984],{"className":105222},[2072],[18,105224],{"className":105225,"style":2068},[169],[18,105227,47],{"className":105228},[106],[18,105230,152],{"className":105231},[151],[18,105233,105235],{"className":105234},[124],[18,105236,105238],{"className":105237,"style":6067},[128],[18,105239],{},[18,105241],{"className":105242},[267,6009],[18,105244],{"className":105245,"style":228},[169],[18,105247,6882],{"className":105248},[232],[18,105250],{"className":105251,"style":228},[169],[18,105253,105255,105259],{"className":105254},[97],[18,105256],{"className":105257,"style":105258},[101],"height:2.4015em;vertical-align:-1.0801em;",[18,105260,105262,105265,105397],{"className":105261},[106],[18,105263],{"className":105264},[250,6009],[18,105266,105268],{"className":105267},[5651],[18,105269,105271,105388],{"className":105270},[119,120],[18,105272,105274,105385],{"className":105273},[124],[18,105275,105277,105366,105374],{"className":105276,"style":9578},[128],[18,105278,105280,105283],{"style":105279},"top:-2.2649em;",[18,105281],{"className":105282,"style":697},[136],[18,105284,105286,105289,105292,105295,105298],{"className":105285},[106],[18,105287,64],{"className":105288},[106,110],[18,105290],{"className":105291,"style":2068},[169],[18,105293,7984],{"className":105294},[2072],[18,105296],{"className":105297,"style":2068},[169],[18,105299,105301,105304,105363],{"className":105300},[106],[18,105302],{"className":105303},[250,6009],[18,105305,105307],{"className":105306},[5651],[18,105308,105310,105355],{"className":105309},[119,120],[18,105311,105313,105352],{"className":105312},[124],[18,105314,105316,105330,105338],{"className":105315,"style":12042},[128],[18,105317,105318,105321],{"style":12045},[18,105319],{"className":105320,"style":697},[136],[18,105322,105324],{"className":105323},[141,142,143,144],[18,105325,105327],{"className":105326},[106,144],[18,105328,2266],{"className":105329},[106,144],[18,105331,105332,105335],{"style":6037},[18,105333],{"className":105334,"style":697},[136],[18,105336],{"className":105337,"style":6045},[6044],[18,105339,105340,105343],{"style":12068},[18,105341],{"className":105342,"style":697},[136],[18,105344,105346],{"className":105345},[141,142,143,144],[18,105347,105349],{"className":105348},[106,144],[18,105350,47],{"className":105351},[106,144],[18,105353,152],{"className":105354},[151],[18,105356,105358],{"className":105357},[124],[18,105359,105361],{"className":105360,"style":12090},[128],[18,105362],{},[18,105364],{"className":105365},[267,6009],[18,105367,105368,105371],{"style":6037},[18,105369],{"className":105370,"style":697},[136],[18,105372],{"className":105373,"style":6045},[6044],[18,105375,105376,105379],{"style":6048},[18,105377],{"className":105378,"style":697},[136],[18,105380,105382],{"className":105381},[106],[18,105383,47],{"className":105384},[106],[18,105386,152],{"className":105387},[151],[18,105389,105391],{"className":105390},[124],[18,105392,105395],{"className":105393,"style":105394},[128],"height:1.0801em;",[18,105396],{},[18,105398],{"className":105399},[267,6009],[11,105401,84120],{},[18,105403,105405],{"className":105404,"translate":22},[875],[18,105406,105408,105496],{"className":105407,"translate":22},[21],[18,105409,105411],{"className":105410},[26],[28,105412,105413],{"xmlns":30,"display":884},[32,105414,105415,105493],{},[35,105416,105417,105433,105453,105455,105489,105491],{},[890,105418,105419,105425],{},[35,105420,105421,105423],{},[41,105422,91870],{},[49,105424,899],{},[35,105426,105427,105429,105431],{},[41,105428,64],{},[49,105430,1880],{},[41,105432,2028],{"mathvariant":316},[5651,105434,105435,105447],{},[38,105436,105437,105439],{},[41,105438,16820],{},[35,105440,105441,105443,105445],{},[41,105442,64],{},[49,105444,7984],{},[45,105446,47],{},[38,105448,105449,105451],{},[41,105450,16820],{},[41,105452,64],{},[49,105454,433],{},[5651,105456,105457,105459],{},[41,105458,14441],{},[1556,105460,105461,105477],{},[35,105462,105463,105465,105467,105469,105475],{},[49,105464,74],{"fence":51},[45,105466,47],{},[49,105468,7984],{},[5651,105470,105471,105473],{},[45,105472,47],{},[41,105474,64],{},[49,105476,84],{"fence":51},[35,105478,105479,105481,105483],{},[41,105480,64],{},[49,105482,7984],{},[5651,105484,105485,105487],{},[45,105486,47],{},[45,105488,2266],{},[49,105490,6681],{},[45,105492,47],{},[86,105494,105495],{"encoding":88},"\\lim_{n \\rightarrow \\infty} \\frac{a_{n+1}}{a_n} = \\frac{e}{\\left(1 + \\frac{1}{n}\\right)^{n + \\frac{1}{2}}} \u003C 1",[18,105497,105499,105715,105984],{"className":105498,"ariaHidden":51},[93],[18,105500,105502,105505,105558,105561,105706,105709,105712],{"className":105501},[97],[18,105503],{"className":105504,"style":100604},[101],[18,105506,105508],{"className":105507},[985,986],[18,105509,105511,105550],{"className":105510},[119,120],[18,105512,105514,105547],{"className":105513},[124],[18,105515,105517,105537],{"className":105516,"style":295},[128],[18,105518,105519,105522],{"style":97007},[18,105520],{"className":105521,"style":697},[136],[18,105523,105525],{"className":105524},[141,142,143,144],[18,105526,105528,105531,105534],{"className":105527},[106,144],[18,105529,64],{"className":105530},[106,110,144],[18,105532,1880],{"className":105533},[232,144],[18,105535,2028],{"className":105536},[106,144],[18,105538,105539,105542],{"style":693},[18,105540],{"className":105541,"style":697},[136],[18,105543,105544],{},[18,105545,91870],{"className":105546},[985],[18,105548,152],{"className":105549},[151],[18,105551,105553],{"className":105552},[124],[18,105554,105556],{"className":105555,"style":97045},[128],[18,105557],{},[18,105559],{"className":105560,"style":170},[169],[18,105562,105564,105567,105703],{"className":105563},[106],[18,105565],{"className":105566},[250,6009],[18,105568,105570],{"className":105569},[5651],[18,105571,105573,105695],{"className":105572},[119,120],[18,105574,105576,105692],{"className":105575},[124],[18,105577,105579,105627,105635],{"className":105578,"style":13849},[128],[18,105580,105581,105584],{"style":6025},[18,105582],{"className":105583,"style":697},[136],[18,105585,105587],{"className":105586},[106],[18,105588,105590,105593],{"className":105589},[106],[18,105591,16820],{"className":105592},[106,110],[18,105594,105596],{"className":105595},[115],[18,105597,105599,105619],{"className":105598},[119,120],[18,105600,105602,105616],{"className":105601},[124],[18,105603,105605],{"className":105604,"style":202},[128],[18,105606,105607,105610],{"style":1155},[18,105608],{"className":105609,"style":137},[136],[18,105611,105613],{"className":105612},[141,142,143,144],[18,105614,64],{"className":105615},[106,110,144],[18,105617,152],{"className":105618},[151],[18,105620,105622],{"className":105621},[124],[18,105623,105625],{"className":105624,"style":159},[128],[18,105626],{},[18,105628,105629,105632],{"style":6037},[18,105630],{"className":105631,"style":697},[136],[18,105633],{"className":105634,"style":6045},[6044],[18,105636,105637,105640],{"style":6048},[18,105638],{"className":105639,"style":697},[136],[18,105641,105643],{"className":105642},[106],[18,105644,105646,105649],{"className":105645},[106],[18,105647,16820],{"className":105648},[106,110],[18,105650,105652],{"className":105651},[115],[18,105653,105655,105684],{"className":105654},[119,120],[18,105656,105658,105681],{"className":105657},[124],[18,105659,105661],{"className":105660,"style":129},[128],[18,105662,105663,105666],{"style":1155},[18,105664],{"className":105665,"style":137},[136],[18,105667,105669],{"className":105668},[141,142,143,144],[18,105670,105672,105675,105678],{"className":105671},[106,144],[18,105673,64],{"className":105674},[106,110,144],[18,105676,7984],{"className":105677},[2072,144],[18,105679,47],{"className":105680},[106,144],[18,105682,152],{"className":105683},[151],[18,105685,105687],{"className":105686},[124],[18,105688,105690],{"className":105689,"style":100322},[128],[18,105691],{},[18,105693,152],{"className":105694},[151],[18,105696,105698],{"className":105697},[124],[18,105699,105701],{"className":105700,"style":100801},[128],[18,105702],{},[18,105704],{"className":105705},[267,6009],[18,105707],{"className":105708,"style":228},[169],[18,105710,433],{"className":105711},[232],[18,105713],{"className":105714,"style":228},[169],[18,105716,105718,105721,105975,105978,105981],{"className":105717},[97],[18,105719],{"className":105720,"style":102562},[101],[18,105722,105724,105727,105972],{"className":105723},[106],[18,105725],{"className":105726},[250,6009],[18,105728,105730],{"className":105729},[5651],[18,105731,105733,105964],{"className":105732},[119,120],[18,105734,105736,105961],{"className":105735},[124],[18,105737,105739,105942,105950],{"className":105738,"style":13849},[128],[18,105740,105741,105744],{"style":102583},[18,105742],{"className":105743,"style":102587},[136],[18,105745,105747],{"className":105746},[106],[18,105748,105750,105845],{"className":105749},[174],[18,105751,105753,105759,105762,105765,105768,105771,105839],{"className":105752},[174],[18,105754,105756],{"className":105755,"style":13793},[250,13792],[18,105757,74],{"className":105758},[1230,1196],[18,105760,47],{"className":105761},[106],[18,105763],{"className":105764,"style":2068},[169],[18,105766,7984],{"className":105767},[2072],[18,105769],{"className":105770,"style":2068},[169],[18,105772,105774,105777,105836],{"className":105773},[106],[18,105775],{"className":105776},[250,6009],[18,105778,105780],{"className":105779},[5651],[18,105781,105783,105828],{"className":105782},[119,120],[18,105784,105786,105825],{"className":105785},[124],[18,105787,105789,105803,105811],{"className":105788,"style":12042},[128],[18,105790,105791,105794],{"style":12045},[18,105792],{"className":105793,"style":697},[136],[18,105795,105797],{"className":105796},[141,142,143,144],[18,105798,105800],{"className":105799},[106,144],[18,105801,64],{"className":105802},[106,110,144],[18,105804,105805,105808],{"style":6037},[18,105806],{"className":105807,"style":697},[136],[18,105809],{"className":105810,"style":6045},[6044],[18,105812,105813,105816],{"style":12068},[18,105814],{"className":105815,"style":697},[136],[18,105817,105819],{"className":105818},[141,142,143,144],[18,105820,105822],{"className":105821},[106,144],[18,105823,47],{"className":105824},[106,144],[18,105826,152],{"className":105827},[151],[18,105829,105831],{"className":105830},[124],[18,105832,105834],{"className":105833,"style":12090},[128],[18,105835],{},[18,105837],{"className":105838},[267,6009],[18,105840,105842],{"className":105841,"style":13793},[267,13792],[18,105843,84],{"className":105844},[1230,1196],[18,105846,105848],{"className":105847},[115],[18,105849,105851],{"className":105850},[119],[18,105852,105854],{"className":105853},[124],[18,105855,105857],{"className":105856,"style":102701},[128],[18,105858,105859,105862],{"style":69827},[18,105860],{"className":105861,"style":697},[136],[18,105863,105865],{"className":105864},[141,142,143,144],[18,105866,105868,105871,105874],{"className":105867},[106,144],[18,105869,64],{"className":105870},[106,110,144],[18,105872,7984],{"className":105873},[2072,144],[18,105875,105877,105880,105939],{"className":105876},[106,144],[18,105878],{"className":105879},[250,6009,141,1195,38147],[18,105881,105883],{"className":105882},[5651],[18,105884,105886,105931],{"className":105885},[119,120],[18,105887,105889,105928],{"className":105888},[124],[18,105890,105892,105906,105914],{"className":105891,"style":42929},[128],[18,105893,105894,105897],{"style":38163},[18,105895],{"className":105896,"style":697},[136],[18,105898,105900],{"className":105899},[141,1195,1196,144],[18,105901,105903],{"className":105902},[106,144],[18,105904,2266],{"className":105905},[106,144],[18,105907,105908,105911],{"style":38187},[18,105909],{"className":105910,"style":697},[136],[18,105912],{"className":105913,"style":38194},[6044,144],[18,105915,105916,105919],{"style":42963},[18,105917],{"className":105918,"style":697},[136],[18,105920,105922],{"className":105921},[141,1195,1196,144],[18,105923,105925],{"className":105924},[106,144],[18,105926,47],{"className":105927},[106,144],[18,105929,152],{"className":105930},[151],[18,105932,105934],{"className":105933},[124],[18,105935,105937],{"className":105936,"style":81865},[128],[18,105938],{},[18,105940],{"className":105941},[267,6009,141,1195,38147],[18,105943,105944,105947],{"style":102789},[18,105945],{"className":105946,"style":102587},[136],[18,105948],{"className":105949,"style":6045},[6044],[18,105951,105952,105955],{"style":102798},[18,105953],{"className":105954,"style":102587},[136],[18,105956,105958],{"className":105957},[106],[18,105959,14441],{"className":105960},[106,110],[18,105962,152],{"className":105963},[151],[18,105965,105967],{"className":105966},[124],[18,105968,105970],{"className":105969,"style":102817},[128],[18,105971],{},[18,105973],{"className":105974},[267,6009],[18,105976],{"className":105977,"style":228},[169],[18,105979,6681],{"className":105980},[232],[18,105982],{"className":105983,"style":228},[169],[18,105985,105987,105990],{"className":105986},[97],[18,105988],{"className":105989,"style":505},[101],[18,105991,47],{"className":105992},[106],[11,105994,105995,105996,106065,106066,106158],{},"于是 ",[18,105997,105999,106016],{"className":105998,"translate":22},[21],[18,106000,106002],{"className":106001},[26],[28,106003,106004],{"xmlns":30},[32,106005,106006,106014],{},[35,106007,106008],{},[38,106009,106010,106012],{},[41,106011,16820],{},[41,106013,64],{},[86,106015,101104],{"encoding":88},[18,106017,106019],{"className":106018,"ariaHidden":51},[93],[18,106020,106022,106025],{"className":106021},[97],[18,106023],{"className":106024,"style":16894},[101],[18,106026,106028,106031],{"className":106027},[106],[18,106029,16820],{"className":106030},[106,110],[18,106032,106034],{"className":106033},[115],[18,106035,106037,106057],{"className":106036},[119,120],[18,106038,106040,106054],{"className":106039},[124],[18,106041,106043],{"className":106042,"style":202},[128],[18,106044,106045,106048],{"style":1155},[18,106046],{"className":106047,"style":137},[136],[18,106049,106051],{"className":106050},[141,142,143,144],[18,106052,64],{"className":106053},[106,110,144],[18,106055,152],{"className":106056},[151],[18,106058,106060],{"className":106059},[124],[18,106061,106063],{"className":106062,"style":159},[128],[18,106064],{}," 单调递减且 ",[18,106067,106069,106091],{"className":106068,"translate":22},[21],[18,106070,106072],{"className":106071},[26],[28,106073,106074],{"xmlns":30},[32,106075,106076,106088],{},[35,106077,106078,106084,106086],{},[38,106079,106080,106082],{},[41,106081,16820],{},[41,106083,64],{},[49,106085,6882],{},[45,106087,430],{},[86,106089,106090],{"encoding":88},"a_n > 0",[18,106092,106094,106149],{"className":106093,"ariaHidden":51},[93],[18,106095,106097,106100,106140,106143,106146],{"className":106096},[97],[18,106098],{"className":106099,"style":40971},[101],[18,106101,106103,106106],{"className":106102},[106],[18,106104,16820],{"className":106105},[106,110],[18,106107,106109],{"className":106108},[115],[18,106110,106112,106132],{"className":106111},[119,120],[18,106113,106115,106129],{"className":106114},[124],[18,106116,106118],{"className":106117,"style":202},[128],[18,106119,106120,106123],{"style":1155},[18,106121],{"className":106122,"style":137},[136],[18,106124,106126],{"className":106125},[141,142,143,144],[18,106127,64],{"className":106128},[106,110,144],[18,106130,152],{"className":106131},[151],[18,106133,106135],{"className":106134},[124],[18,106136,106138],{"className":106137,"style":159},[128],[18,106139],{},[18,106141],{"className":106142,"style":228},[169],[18,106144,6882],{"className":106145},[232],[18,106147],{"className":106148,"style":228},[169],[18,106150,106152,106155],{"className":106151},[97],[18,106153],{"className":106154,"style":505},[101],[18,106156,430],{"className":106157},[106],"，因此它必然收敛，即存在极限。",[11,106160,106161],{},"设：",[18,106163,106165],{"className":106164,"translate":22},[875],[18,106166,106168,106230],{"className":106167,"translate":22},[21],[18,106169,106171],{"className":106170},[26],[28,106172,106173],{"xmlns":30,"display":884},[32,106174,106175,106227],{},[35,106176,106177,106193,106217,106223,106225],{},[890,106178,106179,106185],{},[35,106180,106181,106183],{},[41,106182,91870],{},[49,106184,899],{},[35,106186,106187,106189,106191],{},[41,106188,64],{},[49,106190,1880],{},[41,106192,2028],{"mathvariant":316},[5651,106194,106195,106201],{},[35,106196,106197,106199],{},[41,106198,64],{},[49,106200,88973],{"stretchy":73},[1556,106202,106203,106205],{},[41,106204,64],{},[35,106206,106207,106209,106211],{},[41,106208,64],{},[49,106210,7984],{},[5651,106212,106213,106215],{},[45,106214,47],{},[45,106216,2266],{},[1556,106218,106219,106221],{},[41,106220,14441],{},[41,106222,64],{},[49,106224,433],{},[41,106226,16820],{},[86,106228,106229],{"encoding":88},"\\lim_{n \\rightarrow \\infty} \\frac{n!}{n^{n + \\frac{1}{2}}} e^n = a",[18,106231,106233,106498],{"className":106232,"ariaHidden":51},[93],[18,106234,106236,106239,106292,106295,106460,106489,106492,106495],{"className":106235},[97],[18,106237],{"className":106238,"style":101278},[101],[18,106240,106242],{"className":106241},[985,986],[18,106243,106245,106284],{"className":106244},[119,120],[18,106246,106248,106281],{"className":106247},[124],[18,106249,106251,106271],{"className":106250,"style":295},[128],[18,106252,106253,106256],{"style":97007},[18,106254],{"className":106255,"style":697},[136],[18,106257,106259],{"className":106258},[141,142,143,144],[18,106260,106262,106265,106268],{"className":106261},[106,144],[18,106263,64],{"className":106264},[106,110,144],[18,106266,1880],{"className":106267},[232,144],[18,106269,2028],{"className":106270},[106,144],[18,106272,106273,106276],{"style":693},[18,106274],{"className":106275,"style":697},[136],[18,106277,106278],{},[18,106279,91870],{"className":106280},[985],[18,106282,152],{"className":106283},[151],[18,106285,106287],{"className":106286},[124],[18,106288,106290],{"className":106289,"style":97045},[128],[18,106291],{},[18,106293],{"className":106294,"style":170},[169],[18,106296,106298,106301,106457],{"className":106297},[106],[18,106299],{"className":106300},[250,6009],[18,106302,106304],{"className":106303},[5651],[18,106305,106307,106449],{"className":106306},[119,120],[18,106308,106310,106446],{"className":106309},[124],[18,106311,106313,106424,106432],{"className":106312,"style":6022},[128],[18,106314,106315,106318],{"style":101299},[18,106316],{"className":106317,"style":697},[136],[18,106319,106321],{"className":106320},[106],[18,106322,106324,106327],{"className":106323},[106],[18,106325,64],{"className":106326},[106,110],[18,106328,106330],{"className":106329},[115],[18,106331,106333],{"className":106332},[119],[18,106334,106336],{"className":106335},[124],[18,106337,106339],{"className":106338,"style":101324},[128],[18,106340,106341,106344],{"style":101327},[18,106342],{"className":106343,"style":697},[136],[18,106345,106347],{"className":106346},[141,142,143,144],[18,106348,106350,106353,106356],{"className":106349},[106,144],[18,106351,64],{"className":106352},[106,110,144],[18,106354,7984],{"className":106355},[2072,144],[18,106357,106359,106362,106421],{"className":106358},[106,144],[18,106360],{"className":106361},[250,6009,141,1195,38147],[18,106363,106365],{"className":106364},[5651],[18,106366,106368,106413],{"className":106367},[119,120],[18,106369,106371,106410],{"className":106370},[124],[18,106372,106374,106388,106396],{"className":106373,"style":42929},[128],[18,106375,106376,106379],{"style":38163},[18,106377],{"className":106378,"style":697},[136],[18,106380,106382],{"className":106381},[141,1195,1196,144],[18,106383,106385],{"className":106384},[106,144],[18,106386,2266],{"className":106387},[106,144],[18,106389,106390,106393],{"style":38187},[18,106391],{"className":106392,"style":697},[136],[18,106394],{"className":106395,"style":38194},[6044,144],[18,106397,106398,106401],{"style":42963},[18,106399],{"className":106400,"style":697},[136],[18,106402,106404],{"className":106403},[141,1195,1196,144],[18,106405,106407],{"className":106406},[106,144],[18,106408,47],{"className":106409},[106,144],[18,106411,152],{"className":106412},[151],[18,106414,106416],{"className":106415},[124],[18,106417,106419],{"className":106418,"style":81865},[128],[18,106420],{},[18,106422],{"className":106423},[267,6009,141,1195,38147],[18,106425,106426,106429],{"style":6037},[18,106427],{"className":106428,"style":697},[136],[18,106430],{"className":106431,"style":6045},[6044],[18,106433,106434,106437],{"style":6048},[18,106435],{"className":106436,"style":697},[136],[18,106438,106440,106443],{"className":106439},[106],[18,106441,64],{"className":106442},[106,110],[18,106444,88973],{"className":106445},[267],[18,106447,152],{"className":106448},[151],[18,106450,106452],{"className":106451},[124],[18,106453,106455],{"className":106454,"style":101442},[128],[18,106456],{},[18,106458],{"className":106459},[267,6009],[18,106461,106463,106466],{"className":106462},[106],[18,106464,14441],{"className":106465},[106,110],[18,106467,106469],{"className":106468},[115],[18,106470,106472],{"className":106471},[119],[18,106473,106475],{"className":106474},[124],[18,106476,106478],{"className":106477,"style":1616},[128],[18,106479,106480,106483],{"style":1637},[18,106481],{"className":106482,"style":137},[136],[18,106484,106486],{"className":106485},[141,142,143,144],[18,106487,64],{"className":106488},[106,110,144],[18,106490],{"className":106491,"style":228},[169],[18,106493,433],{"className":106494},[232],[18,106496],{"className":106497,"style":228},[169],[18,106499,106501,106504],{"className":106500},[97],[18,106502],{"className":106503,"style":1083},[101],[18,106505,16820],{"className":106506},[106,110],[11,106508,106509],{},"接下来我们需要求出这个极限。",[11,106511,106512],{},"由前面的沃利斯公式：",[18,106514,106516],{"className":106515,"translate":22},[875],[18,106517,106519,106608],{"className":106518,"translate":22},[21],[18,106520,106522],{"className":106521},[26],[28,106523,106524],{"xmlns":30,"display":884},[32,106525,106526,106606],{},[35,106527,106528,106544,106598,106600],{},[890,106529,106530,106536],{},[35,106531,106532,106534],{},[41,106533,91870],{},[49,106535,899],{},[35,106537,106538,106540,106542],{},[41,106539,39955],{},[49,106541,1880],{},[41,106543,2028],{"mathvariant":316},[5651,106545,106546,106564],{},[35,106547,106548,106550,106552,106554,106556,106558],{},[49,106549,74],{"stretchy":73},[45,106551,2266],{},[41,106553,39955],{},[49,106555,84],{"stretchy":73},[49,106557,88973],{"stretchy":73},[1556,106559,106560,106562],{},[49,106561,88973],{"stretchy":73},[45,106563,2266],{},[35,106565,106566,106568,106570,106572,106574,106576,106578,106580,106582,106584,106586,106588,106590,106592,106594,106596],{},[49,106567,74],{"stretchy":73},[45,106569,2266],{},[41,106571,39955],{},[49,106573,7984],{},[45,106575,47],{},[49,106577,84],{"stretchy":73},[49,106579,88973],{"stretchy":73},[49,106581,88973],{"stretchy":73},[49,106583,74],{"stretchy":73},[45,106585,2266],{},[41,106587,39955],{},[49,106589,2253],{},[45,106591,47],{},[49,106593,84],{"stretchy":73},[49,106595,88973],{"stretchy":73},[49,106597,88973],{"stretchy":73},[49,106599,433],{},[5651,106601,106602,106604],{},[41,106603,3941],{},[45,106605,2266],{},[86,106607,96173],{"encoding":88},[18,106609,106611,106827],{"className":106610,"ariaHidden":51},[93],[18,106612,106614,106617,106670,106673,106818,106821,106824],{"className":106613},[97],[18,106615],{"className":106616,"style":94146},[101],[18,106618,106620],{"className":106619},[985,986],[18,106621,106623,106662],{"className":106622},[119,120],[18,106624,106626,106659],{"className":106625},[124],[18,106627,106629,106649],{"className":106628,"style":295},[128],[18,106630,106631,106634],{"style":29361},[18,106632],{"className":106633,"style":697},[136],[18,106635,106637],{"className":106636},[141,142,143,144],[18,106638,106640,106643,106646],{"className":106639},[106,144],[18,106641,39955],{"className":106642,"style":40031},[106,110,144],[18,106644,1880],{"className":106645},[232,144],[18,106647,2028],{"className":106648},[106,144],[18,106650,106651,106654],{"style":693},[18,106652],{"className":106653,"style":697},[136],[18,106655,106656],{},[18,106657,91870],{"className":106658},[985],[18,106660,152],{"className":106661},[151],[18,106663,106665],{"className":106664},[124],[18,106666,106668],{"className":106667,"style":92226},[128],[18,106669],{},[18,106671],{"className":106672,"style":170},[169],[18,106674,106676,106679,106815],{"className":106675},[106],[18,106677],{"className":106678},[250,6009],[18,106680,106682],{"className":106681},[5651],[18,106683,106685,106807],{"className":106684},[119,120],[18,106686,106688,106804],{"className":106687},[124],[18,106689,106691,106747,106755],{"className":106690,"style":15996},[128],[18,106692,106693,106696],{"style":6025},[18,106694],{"className":106695,"style":697},[136],[18,106697,106699,106702,106705,106708,106711,106714,106717,106720,106723,106726,106729,106732,106735,106738,106741,106744],{"className":106698},[106],[18,106700,74],{"className":106701},[250],[18,106703,2266],{"className":106704},[106],[18,106706,39955],{"className":106707,"style":40031},[106,110],[18,106709],{"className":106710,"style":2068},[169],[18,106712,7984],{"className":106713},[2072],[18,106715],{"className":106716,"style":2068},[169],[18,106718,47],{"className":106719},[106],[18,106721,89435],{"className":106722},[267],[18,106724,74],{"className":106725},[250],[18,106727,2266],{"className":106728},[106],[18,106730,39955],{"className":106731,"style":40031},[106,110],[18,106733],{"className":106734,"style":2068},[169],[18,106736,2253],{"className":106737},[2072],[18,106739],{"className":106740,"style":2068},[169],[18,106742,47],{"className":106743},[106],[18,106745,89435],{"className":106746},[267],[18,106748,106749,106752],{"style":6037},[18,106750],{"className":106751,"style":697},[136],[18,106753],{"className":106754,"style":6045},[6044],[18,106756,106757,106760],{"style":6048},[18,106758],{"className":106759,"style":697},[136],[18,106761,106763,106766,106769,106772,106775],{"className":106762},[106],[18,106764,74],{"className":106765},[250],[18,106767,2266],{"className":106768},[106],[18,106770,39955],{"className":106771,"style":40031},[106,110],[18,106773,94249],{"className":106774},[267],[18,106776,106778,106781],{"className":106777},[267],[18,106779,88973],{"className":106780},[267],[18,106782,106784],{"className":106783},[115],[18,106785,106787],{"className":106786},[119],[18,106788,106790],{"className":106789},[124],[18,106791,106793],{"className":106792,"style":2416},[128],[18,106794,106795,106798],{"style":1962},[18,106796],{"className":106797,"style":137},[136],[18,106799,106801],{"className":106800},[141,142,143,144],[18,106802,2266],{"className":106803},[106,144],[18,106805,152],{"className":106806},[151],[18,106808,106810],{"className":106809},[124],[18,106811,106813],{"className":106812,"style":89485},[128],[18,106814],{},[18,106816],{"className":106817},[267,6009],[18,106819],{"className":106820,"style":228},[169],[18,106822,433],{"className":106823},[232],[18,106825],{"className":106826,"style":228},[169],[18,106828,106830,106833],{"className":106829},[97],[18,106831],{"className":106832,"style":42062},[101],[18,106834,106836,106839,106892],{"className":106835},[106],[18,106837],{"className":106838},[250,6009],[18,106840,106842],{"className":106841},[5651],[18,106843,106845,106884],{"className":106844},[119,120],[18,106846,106848,106881],{"className":106847},[124],[18,106849,106851,106862,106870],{"className":106850,"style":13849},[128],[18,106852,106853,106856],{"style":6025},[18,106854],{"className":106855,"style":697},[136],[18,106857,106859],{"className":106858},[106],[18,106860,2266],{"className":106861},[106],[18,106863,106864,106867],{"style":6037},[18,106865],{"className":106866,"style":697},[136],[18,106868],{"className":106869,"style":6045},[6044],[18,106871,106872,106875],{"style":6048},[18,106873],{"className":106874,"style":697},[136],[18,106876,106878],{"className":106877},[106],[18,106879,3941],{"className":106880,"style":3957},[106,110],[18,106882,152],{"className":106883},[151],[18,106885,106887],{"className":106886},[124],[18,106888,106890],{"className":106889,"style":6067},[128],[18,106891],{},[18,106893],{"className":106894},[267,6009],[18,106896,106898],{"className":106897,"translate":22},[875],[18,106899,106901,106959],{"className":106900,"translate":22},[21],[18,106902,106904],{"className":106903},[26],[28,106905,106906],{"xmlns":30,"display":884},[32,106907,106908,106956],{},[35,106909,106910,106912,106914,106916,106918,106920,106922,106924,106926,106928,106930,106932,106934,106936,106938,106940,106942,106944,106946,106952,106954],{},[49,106911,74],{"stretchy":73},[45,106913,2266],{},[41,106915,64],{},[49,106917,84],{"stretchy":73},[49,106919,88973],{"stretchy":73},[49,106921,88973],{"stretchy":73},[49,106923,433],{},[45,106925,2266],{},[41,106927,64],{},[49,106929,74],{"stretchy":73},[45,106931,2266],{},[41,106933,64],{},[49,106935,2253],{},[45,106937,2266],{},[49,106939,84],{"stretchy":73},[49,106941,76903],{},[45,106943,2266],{},[49,106945,433],{},[1556,106947,106948,106950],{},[45,106949,2266],{},[41,106951,64],{},[41,106953,64],{},[49,106955,88973],{"stretchy":73},[86,106957,106958],{"encoding":88},"(2n)!! = 2n (2n-2) \\cdots 2 = 2^n n!",[18,106960,106962,106989,107019,107052],{"className":106961,"ariaHidden":51},[93],[18,106963,106965,106968,106971,106974,106977,106980,106983,106986],{"className":106964},[97],[18,106966],{"className":106967,"style":242},[101],[18,106969,74],{"className":106970},[250],[18,106972,2266],{"className":106973},[106],[18,106975,64],{"className":106976},[106,110],[18,106978,89435],{"className":106979},[267],[18,106981],{"className":106982,"style":228},[169],[18,106984,433],{"className":106985},[232],[18,106987],{"className":106988,"style":228},[169],[18,106990,106992,106995,106998,107001,107004,107007,107010,107013,107016],{"className":106991},[97],[18,106993],{"className":106994,"style":242},[101],[18,106996,2266],{"className":106997},[106],[18,106999,64],{"className":107000},[106,110],[18,107002,74],{"className":107003},[250],[18,107005,2266],{"className":107006},[106],[18,107008,64],{"className":107009},[106,110],[18,107011],{"className":107012,"style":2068},[169],[18,107014,2253],{"className":107015},[2072],[18,107017],{"className":107018,"style":2068},[169],[18,107020,107022,107025,107028,107031,107034,107037,107040,107043,107046,107049],{"className":107021},[97],[18,107023],{"className":107024,"style":242},[101],[18,107026,2266],{"className":107027},[106],[18,107029,84],{"className":107030},[267],[18,107032],{"className":107033,"style":170},[169],[18,107035,76903],{"className":107036},[174],[18,107038],{"className":107039,"style":170},[169],[18,107041,2266],{"className":107042},[106],[18,107044],{"className":107045,"style":228},[169],[18,107047,433],{"className":107048},[232],[18,107050],{"className":107051,"style":228},[169],[18,107053,107055,107058,107087,107090],{"className":107054},[97],[18,107056],{"className":107057,"style":1616},[101],[18,107059,107061,107064],{"className":107060},[106],[18,107062,2266],{"className":107063},[106],[18,107065,107067],{"className":107066},[115],[18,107068,107070],{"className":107069},[119],[18,107071,107073],{"className":107072},[124],[18,107074,107076],{"className":107075,"style":1616},[128],[18,107077,107078,107081],{"style":1637},[18,107079],{"className":107080,"style":137},[136],[18,107082,107084],{"className":107083},[141,142,143,144],[18,107085,64],{"className":107086},[106,110,144],[18,107088,64],{"className":107089},[106,110],[18,107091,88973],{"className":107092},[267],[18,107094,107096],{"className":107095,"translate":22},[875],[18,107097,107099,107185],{"className":107098,"translate":22},[21],[18,107100,107102],{"className":107101},[26],[28,107103,107104],{"xmlns":30,"display":884},[32,107105,107106,107182],{},[35,107107,107108,107110,107112,107114,107116,107118,107120,107122,107124,107126,107154,107156],{},[49,107109,74],{"stretchy":73},[45,107111,2266],{},[41,107113,64],{},[49,107115,2253],{},[45,107117,47],{},[49,107119,84],{"stretchy":73},[49,107121,88973],{"stretchy":73},[49,107123,88973],{"stretchy":73},[49,107125,433],{},[5651,107127,107128,107140],{},[35,107129,107130,107132,107134,107136,107138],{},[49,107131,74],{"stretchy":73},[45,107133,2266],{},[41,107135,64],{},[49,107137,84],{"stretchy":73},[49,107139,88973],{"stretchy":73},[35,107141,107142,107144,107146,107148,107150,107152],{},[49,107143,74],{"stretchy":73},[45,107145,2266],{},[41,107147,64],{},[49,107149,84],{"stretchy":73},[49,107151,88973],{"stretchy":73},[49,107153,88973],{"stretchy":73},[49,107155,433],{},[5651,107157,107158,107170],{},[35,107159,107160,107162,107164,107166,107168],{},[49,107161,74],{"stretchy":73},[45,107163,2266],{},[41,107165,64],{},[49,107167,84],{"stretchy":73},[49,107169,88973],{"stretchy":73},[35,107171,107172,107178,107180],{},[1556,107173,107174,107176],{},[45,107175,2266],{},[41,107177,64],{},[41,107179,64],{},[49,107181,88973],{"stretchy":73},[86,107183,107184],{"encoding":88},"(2n-1)!! = \\frac{(2n)!}{(2n)!!} = \\frac{(2n)!}{2^n n!}",[18,107186,107188,107212,107233,107328],{"className":107187,"ariaHidden":51},[93],[18,107189,107191,107194,107197,107200,107203,107206,107209],{"className":107190},[97],[18,107192],{"className":107193,"style":242},[101],[18,107195,74],{"className":107196},[250],[18,107198,2266],{"className":107199},[106],[18,107201,64],{"className":107202},[106,110],[18,107204],{"className":107205,"style":2068},[169],[18,107207,2253],{"className":107208},[2072],[18,107210],{"className":107211,"style":2068},[169],[18,107213,107215,107218,107221,107224,107227,107230],{"className":107214},[97],[18,107216],{"className":107217,"style":242},[101],[18,107219,47],{"className":107220},[106],[18,107222,89435],{"className":107223},[267],[18,107225],{"className":107226,"style":228},[169],[18,107228,433],{"className":107229},[232],[18,107231],{"className":107232,"style":228},[169],[18,107234,107236,107239,107319,107322,107325],{"className":107235},[97],[18,107237],{"className":107238,"style":89396},[101],[18,107240,107242,107245,107316],{"className":107241},[106],[18,107243],{"className":107244},[250,6009],[18,107246,107248],{"className":107247},[5651],[18,107249,107251,107308],{"className":107250},[119,120],[18,107252,107254,107305],{"className":107253},[124],[18,107255,107257,107277,107285],{"className":107256,"style":18128},[128],[18,107258,107259,107262],{"style":6025},[18,107260],{"className":107261,"style":697},[136],[18,107263,107265,107268,107271,107274],{"className":107264},[106],[18,107266,74],{"className":107267},[250],[18,107269,2266],{"className":107270},[106],[18,107272,64],{"className":107273},[106,110],[18,107275,89435],{"className":107276},[267],[18,107278,107279,107282],{"style":6037},[18,107280],{"className":107281,"style":697},[136],[18,107283],{"className":107284,"style":6045},[6044],[18,107286,107287,107290],{"style":6048},[18,107288],{"className":107289,"style":697},[136],[18,107291,107293,107296,107299,107302],{"className":107292},[106],[18,107294,74],{"className":107295},[250],[18,107297,2266],{"className":107298},[106],[18,107300,64],{"className":107301},[106,110],[18,107303,94249],{"className":107304},[267],[18,107306,152],{"className":107307},[151],[18,107309,107311],{"className":107310},[124],[18,107312,107314],{"className":107313,"style":89485},[128],[18,107315],{},[18,107317],{"className":107318},[267,6009],[18,107320],{"className":107321,"style":228},[169],[18,107323,433],{"className":107324},[232],[18,107326],{"className":107327,"style":228},[169],[18,107329,107331,107335],{"className":107330},[97],[18,107332],{"className":107333,"style":107334},[101],"height:2.113em;vertical-align:-0.686em;",[18,107336,107338,107341,107435],{"className":107337},[106],[18,107339],{"className":107340},[250,6009],[18,107342,107344],{"className":107343},[5651],[18,107345,107347,107427],{"className":107346},[119,120],[18,107348,107350,107424],{"className":107349},[124],[18,107351,107353,107396,107404],{"className":107352,"style":18128},[128],[18,107354,107355,107358],{"style":6025},[18,107356],{"className":107357,"style":697},[136],[18,107359,107361,107390,107393],{"className":107360},[106],[18,107362,107364,107367],{"className":107363},[106],[18,107365,2266],{"className":107366},[106],[18,107368,107370],{"className":107369},[115],[18,107371,107373],{"className":107372},[119],[18,107374,107376],{"className":107375},[124],[18,107377,107379],{"className":107378,"style":97107},[128],[18,107380,107381,107384],{"style":26105},[18,107382],{"className":107383,"style":137},[136],[18,107385,107387],{"className":107386},[141,142,143,144],[18,107388,64],{"className":107389},[106,110,144],[18,107391,64],{"className":107392},[106,110],[18,107394,88973],{"className":107395},[267],[18,107397,107398,107401],{"style":6037},[18,107399],{"className":107400,"style":697},[136],[18,107402],{"className":107403,"style":6045},[6044],[18,107405,107406,107409],{"style":6048},[18,107407],{"className":107408,"style":697},[136],[18,107410,107412,107415,107418,107421],{"className":107411},[106],[18,107413,74],{"className":107414},[250],[18,107416,2266],{"className":107417},[106],[18,107419,64],{"className":107420},[106,110],[18,107422,94249],{"className":107423},[267],[18,107425,152],{"className":107426},[151],[18,107428,107430],{"className":107429},[124],[18,107431,107433],{"className":107432,"style":6067},[128],[18,107434],{},[18,107436],{"className":107437},[267,6009],[18,107439,107441],{"className":107440,"translate":22},[875],[18,107442,107444,107556],{"className":107443,"translate":22},[21],[18,107445,107447],{"className":107446},[26],[28,107448,107449],{"xmlns":30,"display":884},[32,107450,107451,107553],{},[35,107452,107453,107455,107457,107459,107461,107463,107465,107467,107469,107471,107507,107509],{},[49,107454,74],{"stretchy":73},[45,107456,2266],{},[41,107458,64],{},[49,107460,7984],{},[45,107462,47],{},[49,107464,84],{"stretchy":73},[49,107466,88973],{"stretchy":73},[49,107468,88973],{"stretchy":73},[49,107470,433],{},[5651,107472,10747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= \\frac{(2n+2)!}{(2n+2)!!} = \\frac{(2n+2)!}{2^{n+1} (n+1)!}",[18,107557,107559,107583,107604,107723],{"className":107558,"ariaHidden":51},[93],[18,107560,107562,107565,107568,107571,107574,107577,107580],{"className":107561},[97],[18,107563],{"className":107564,"style":242},[101],[18,107566,74],{"className":107567},[250],[18,107569,2266],{"className":107570},[106],[18,107572,64],{"className":107573},[106,110],[18,107575],{"className":107576,"style":2068},[169],[18,107578,7984],{"className":107579},[2072],[18,107581],{"className":107582,"style":2068},[169],[18,107584,107586,107589,107592,107595,107598,107601],{"className":107585},[97],[18,107587],{"className":107588,"style":242},[101],[18,107590,47],{"className":107591},[106],[18,107593,89435],{"className":107594},[267],[18,107596],{"className":107597,"style":228},[169],[18,107599,433],{"className":107600},[232],[18,107602],{"className":107603,"style":228},[169],[18,107605,107607,107610,107714,107717,107720],{"className":107606},[97],[18,107608],{"className":107609,"style":89396},[101],[18,107611,107613,107616,107711],{"className":107612},[106],[18,107614],{"className":107615},[250,6009],[18,107617,107619],{"className":107618},[5651],[18,107620,107622,107703],{"className":107621},[119,120],[18,107623,107625,107700],{"className":107624},[124],[18,107626,107628,107660,107668],{"className":107627,"style":18128},[128],[18,107629,107630,107633],{"style":6025},[18,107631],{"className":107632,"style":697},[136],[18,107634,107636,107639,107642,107645,107648,107651,107654,107657],{"className":107635},[106],[18,107637,74],{"className":107638},[250],[18,107640,2266],{"className":107641},[106],[18,107643,64],{"className":107644},[106,110],[18,107646],{"className":107647,"style":2068},[169],[18,107649,7984],{"className":107650},[2072],[18,107652],{"className":107653,"style":2068},[169],[18,107655,2266],{"className":107656},[106],[18,107658,89435],{"className":107659},[267],[18,107661,107662,107665],{"style":6037},[18,107663],{"className":107664,"style":697},[136],[18,107666],{"className":107667,"style":6045},[6044],[18,107669,107670,107673],{"style":6048},[18,107671],{"className":107672,"style":697},[136],[18,107674,107676,107679,107682,107685,107688,107691,107694,107697],{"className":107675},[106],[18,107677,74],{"className":107678},[250],[18,107680,2266],{"className":107681},[106],[18,107683,64],{"className":107684},[106,110],[18,107686],{"className":107687,"style":2068},[169],[18,107689,7984],{"className":107690},[2072],[18,107692],{"className":107693,"style":2068},[169],[18,107695,2266],{"className":107696},[106],[18,107698,94249],{"className":107699},[267],[18,107701,152],{"className":107702},[151],[18,107704,107706],{"className":107705},[124],[18,107707,107709],{"className":107708,"style":89485},[128],[18,107710],{},[18,107712],{"className":107713},[267,6009],[18,107715],{"className":107716,"style":228},[169],[18,107718,433],{"className":107719},[232],[18,107721],{"className":107722,"style":228},[169],[18,107724,107726,107729],{"className":107725},[97],[18,107727],{"className":107728,"style":89396},[101],[18,107730,107732,107735,107865],{"className":107731},[106],[18,107733],{"className":107734},[250,6009],[18,107736,107738],{"className":107737},[5651],[18,107739,107741,107857],{"className":107740},[119,120],[18,107742,107744,107854],{"className":107743},[124],[18,107745,107747,107814,107822],{"className":107746,"style":18128},[128],[18,107748,107749,107752],{"style":6025},[18,107750],{"className":107751,"style":697},[136],[18,107753,107755,107793,107796,107799,107802,107805,107808,107811],{"className":107754},[106],[18,107756,107758,107761],{"className":107757},[106],[18,107759,2266],{"className":107760},[106],[18,107762,107764],{"className":107763},[115],[18,107765,107767],{"className":107766},[119],[18,107768,107770],{"className":107769},[124],[18,107771,107773],{"className":107772,"style":26102},[128],[18,107774,107775,107778],{"style":26105},[18,107776],{"className":107777,"style":137},[136],[18,107779,107781],{"className":107780},[141,142,143,144],[18,107782,107784,107787,107790],{"className":107783},[106,144],[18,107785,64],{"className":107786},[106,110,144],[18,107788,7984],{"className":107789},[2072,144],[18,107791,47],{"className":107792},[106,144],[18,107794,74],{"className":107795},[250],[18,107797,64],{"className":107798},[106,110],[18,107800],{"className":107801,"style":2068},[169],[18,107803,7984],{"className":107804},[2072],[18,107806],{"className":107807,"style":2068},[169],[18,107809,47],{"className":107810},[106],[18,107812,94249],{"className":107813},[267],[18,107815,107816,107819],{"style":6037},[18,107817],{"className":107818,"style":697},[136],[18,107820],{"className":107821,"style":6045},[6044],[18,107823,107824,107827],{"style":6048},[18,107825],{"className":107826,"style":697},[136],[18,107828,107830,107833,107836,107839,107842,107845,107848,107851],{"className":107829},[106],[18,107831,74],{"className":107832},[250],[18,107834,2266],{"className":107835},[106],[18,107837,64],{"className":107838},[106,110],[18,107840],{"className":107841,"style":2068},[169],[18,107843,7984],{"className":107844},[2072],[18,107846],{"className":107847,"style":2068},[169],[18,107849,2266],{"className":107850},[106],[18,107852,94249],{"className":107853},[267],[18,107855,152],{"className":107856},[151],[18,107858,107860],{"className":107859},[124],[18,107861,107863],{"className":107862,"style":89485},[128],[18,107864],{},[18,107866],{"className":107867},[267,6009],[11,107869,107870],{},"于是沃利斯公式变为：",[18,107872,107874],{"className":107873,"translate":22},[875],[18,107875,107877,107974],{"className":107876,"translate":22},[21],[18,107878,107880],{"className":107879},[26],[28,107881,107882],{"xmlns":30,"display":884},[32,107883,107884,107971],{},[35,107885,107886,107902,107963,107965],{},[890,107887,107888,107894],{},[35,107889,107890,107892],{},[41,107891,91870],{},[49,107893,899],{},[35,107895,107896,107898,107900],{},[41,107897,64],{},[49,107899,1880],{},[41,107901,2028],{"mathvariant":316},[5651,107903,107904,107937],{},[35,107905,107906,107913,107915,107917,107919,107925,107927,107929,107931,107933,107935],{},[1556,107907,107908,107911],{},[45,107909,107910],{},"16",[41,107912,64],{},[49,107914,74],{"stretchy":73},[41,107916,64],{},[49,107918,88973],{"stretchy":73},[1556,107920,107921,107923],{},[49,107922,84],{"stretchy":73},[45,107924,10444],{},[49,107926,74],{"stretchy":73},[45,107928,2266],{},[41,107930,64],{},[49,107932,7984],{},[45,107934,2266],{},[49,107936,84],{"stretchy":73},[35,107938,107939,107941,107943,107945,107947,107949,107951,107953,107955,107957,107959,107961],{},[49,107940,74],{"stretchy":73},[45,107942,2266],{},[41,107944,64],{},[49,107946,84],{"stretchy":73},[49,107948,88973],{"stretchy":73},[49,107950,74],{"stretchy":73},[45,107952,2266],{},[41,107954,64],{},[49,107956,7984],{},[45,107958,2266],{},[49,107960,84],{"stretchy":73},[49,107962,88973],{"stretchy":73},[49,107964,433],{},[5651,107966,107967,107969],{},[41,107968,3941],{},[45,107970,2266],{},[86,107972,107973],{"encoding":88},"\\lim_{n \\rightarrow \\infty} \\frac{16^n (n!)^4 (2n+2)}{(2n)!(2n+2)!} = \\frac{\\pi}{2}",[18,107975,107977,108234],{"className":107976,"ariaHidden":51},[93],[18,107978,107980,107983,108036,108039,108225,108228,108231],{"className":107979},[97],[18,107981],{"className":107982,"style":94146},[101],[18,107984,107986],{"className":107985},[985,986],[18,107987,107989,108028],{"className":107988},[119,120],[18,107990,107992,108025],{"className":107991},[124],[18,107993,107995,108015],{"className":107994,"style":295},[128],[18,107996,107997,108000],{"style":97007},[18,107998],{"className":107999,"style":697},[136],[18,108001,108003],{"className":108002},[141,142,143,144],[18,108004,108006,108009,108012],{"className":108005},[106,144],[18,108007,64],{"className":108008},[106,110,144],[18,108010,1880],{"className":108011},[232,144],[18,108013,2028],{"className":108014},[106,144],[18,108016,108017,108020],{"style":693},[18,108018],{"className":108019,"style":697},[136],[18,108021,108022],{},[18,108023,91870],{"className":108024},[985],[18,108026,152],{"className":108027},[151],[18,108029,108031],{"className":108030},[124],[18,108032,108034],{"className":108033,"style":97045},[128],[18,108035],{},[18,108037],{"className":108038,"style":170},[169],[18,108040,108042,108045,108222],{"className":108041},[106],[18,108043],{"className":108044},[250,6009],[18,108046,108048],{"className":108047},[5651],[18,108049,108051,108214],{"className":108050},[119,120],[18,108052,108054,108211],{"className":108053},[124],[18,108055,108057,108101,108109],{"className":108056,"style":15996},[128],[18,108058,108059,108062],{"style":6025},[18,108060],{"className":108061,"style":697},[136],[18,108063,108065,108068,108071,108074,108077,108080,108083,108086,108089,108092,108095,108098],{"className":108064},[106],[18,108066,74],{"className":108067},[250],[18,108069,2266],{"className":108070},[106],[18,108072,64],{"className":108073},[106,110],[18,108075,94249],{"className":108076},[267],[18,108078,74],{"className":108079},[250],[18,108081,2266],{"className":108082},[106],[18,108084,64],{"className":108085},[106,110],[18,108087],{"className":108088,"style":2068},[169],[18,108090,7984],{"className":108091},[2072],[18,108093],{"className":108094,"style":2068},[169],[18,108096,2266],{"className":108097},[106],[18,108099,94249],{"className":108100},[267],[18,108102,108103,108106],{"style":6037},[18,108104],{"className":108105,"style":697},[136],[18,108107],{"className":108108,"style":6045},[6044],[18,108110,108111,108114],{"style":6048},[18,108112],{"className":108113,"style":697},[136],[18,108115,108117,108120,108149,108152,108155,108158,108187,108190,108193,108196,108199,108202,108205,108208],{"className":108116},[106],[18,108118,47],{"className":108119},[106],[18,108121,108123,108126],{"className":108122},[106],[18,108124,77465],{"className":108125},[106],[18,108127,108129],{"className":108128},[115],[18,108130,108132],{"className":108131},[119],[18,108133,108135],{"className":108134},[124],[18,108136,108138],{"className":108137,"style":1959},[128],[18,108139,108140,108143],{"style":1962},[18,108141],{"className":108142,"style":137},[136],[18,108144,108146],{"className":108145},[141,142,143,144],[18,108147,64],{"className":108148},[106,110,144],[18,108150,74],{"className":108151},[250],[18,108153,64],{"className":108154},[106,110],[18,108156,88973],{"className":108157},[267],[18,108159,108161,108164],{"className":108160},[267],[18,108162,84],{"className":108163},[267],[18,108165,108167],{"className":108166},[115],[18,108168,108170],{"className":108169},[119],[18,108171,108173],{"className":108172},[124],[18,108174,108176],{"className":108175,"style":2416},[128],[18,108177,108178,108181],{"style":1962},[18,108179],{"className":108180,"style":137},[136],[18,108182,108184],{"className":108183},[141,142,143,144],[18,108185,10444],{"className":108186},[106,144],[18,108188,74],{"className":108189},[250],[18,108191,2266],{"className":108192},[106],[18,108194,64],{"className":108195},[106,110],[18,108197],{"className":108198,"style":2068},[169],[18,108200,7984],{"className":108201},[2072],[18,108203],{"className":108204,"style":2068},[169],[18,108206,2266],{"className":108207},[106],[18,108209,84],{"className":108210},[267],[18,108212,152],{"className":108213},[151],[18,108215,108217],{"className":108216},[124],[18,108218,108220],{"className":108219,"style":89485},[128],[18,108221],{},[18,108223],{"className":108224},[267,6009],[18,108226],{"className":108227,"style":228},[169],[18,108229,433],{"className":108230},[232],[18,108232],{"className":108233,"style":228},[169],[18,108235,108237,108240],{"className":108236},[97],[18,108238],{"className":108239,"style":42062},[101],[18,108241,108243,108246,108299],{"className":108242},[106],[18,108244],{"className":108245},[250,6009],[18,108247,108249],{"className":108248},[5651],[18,108250,108252,108291],{"className":108251},[119,120],[18,108253,108255,108288],{"className":108254},[124],[18,108256,108258,108269,108277],{"className":108257,"style":13849},[128],[18,108259,108260,108263],{"style":6025},[18,108261],{"className":108262,"style":697},[136],[18,108264,108266],{"className":108265},[106],[18,108267,2266],{"className":108268},[106],[18,108270,108271,108274],{"style":6037},[18,108272],{"className":108273,"style":697},[136],[18,108275],{"className":108276,"style":6045},[6044],[18,108278,108279,108282],{"style":6048},[18,108280],{"className":108281,"style":697},[136],[18,108283,108285],{"className":108284},[106],[18,108286,3941],{"className":108287,"style":3957},[106,110],[18,108289,152],{"className":108290},[151],[18,108292,108294],{"className":108293},[124],[18,108295,108297],{"className":108296,"style":6067},[128],[18,108298],{},[18,108300],{"className":108301},[267,6009],[11,108303,108304],{},"利用：",[18,108306,108308],{"className":108307,"translate":22},[875],[18,108309,108311,108357],{"className":108310,"translate":22},[21],[18,108312,108314],{"className":108313},[26],[28,108315,108316],{"xmlns":30,"display":884},[32,108317,108318,108354],{},[35,108319,108320,108322,108324,108326,108328,108338],{},[41,108321,64],{},[49,108323,88973],{"stretchy":73},[49,108325,67],{},[41,108327,16820],{},[1556,108329,108330,108332],{},[41,108331,14441],{},[35,108333,108334,108336],{},[49,108335,2253],{},[41,108337,64],{},[1556,108339,108340,108342],{},[41,108341,64],{},[35,108343,108344,108346,108348],{},[41,108345,64],{},[49,108347,7984],{},[5651,108349,108350,108352],{},[45,108351,47],{},[45,108353,2266],{},[86,108355,108356],{"encoding":88},"n! \\sim a e^{-n} n^{n + \\frac{1}{2}}",[18,108358,108360,108381],{"className":108359,"ariaHidden":51},[93],[18,108361,108363,108366,108369,108372,108375,108378],{"className":108362},[97],[18,108364],{"className":108365,"style":295},[101],[18,108367,64],{"className":108368},[106,110],[18,108370,88973],{"className":108371},[267],[18,108373],{"className":108374,"style":228},[169],[18,108376,67],{"className":108377},[232],[18,108379],{"className":108380,"style":228},[169],[18,108382,108384,108388,108391,108426],{"className":108383},[97],[18,108385],{"className":108386,"style":108387},[101],"height:1.004em;",[18,108389,16820],{"className":108390},[106,110],[18,108392,108394,108397],{"className":108393},[106],[18,108395,14441],{"className":108396},[106,110],[18,108398,108400],{"className":108399},[115],[18,108401,108403],{"className":108402},[119],[18,108404,108406],{"className":108405},[124],[18,108407,108409],{"className":108408,"style":1800},[128],[18,108410,108411,108414],{"style":1637},[18,108412],{"className":108413,"style":137},[136],[18,108415,108417],{"className":108416},[141,142,143,144],[18,108418,108420,108423],{"className":108419},[106,144],[18,108421,2253],{"className":108422},[106,144],[18,108424,64],{"className":108425},[106,110,144],[18,108427,108429,108432],{"className":108428},[106],[18,108430,64],{"className":108431},[106,110],[18,108433,108435],{"className":108434},[115],[18,108436,108438],{"className":108437},[119],[18,108439,108441],{"className":108440},[124],[18,108442,108444],{"className":108443,"style":108387},[128],[18,108445,108446,108449],{"style":97809},[18,108447],{"className":108448,"style":697},[136],[18,108450,108452],{"className":108451},[141,142,143,144],[18,108453,108455,108458,108461],{"className":108454},[106,144],[18,108456,64],{"className":108457},[106,110,144],[18,108459,7984],{"className":108460},[2072,144],[18,108462,108464,108467,108526],{"className":108463},[106,144],[18,108465],{"className":108466},[250,6009,141,1195,38147],[18,108468,108470],{"className":108469},[5651],[18,108471,108473,108518],{"className":108472},[119,120],[18,108474,108476,108515],{"className":108475},[124],[18,108477,108479,108493,108501],{"className":108478,"style":42929},[128],[18,108480,108481,108484],{"style":38163},[18,108482],{"className":108483,"style":697},[136],[18,108485,108487],{"className":108486},[141,1195,1196,144],[18,108488,108490],{"className":108489},[106,144],[18,108491,2266],{"className":108492},[106,144],[18,108494,108495,108498],{"style":38187},[18,108496],{"className":108497,"style":697},[136],[18,108499],{"className":108500,"style":38194},[6044,144],[18,108502,108503,108506],{"style":42963},[18,108504],{"className":108505,"style":697},[136],[18,108507,108509],{"className":108508},[141,1195,1196,144],[18,108510,108512],{"className":108511},[106,144],[18,108513,47],{"className":108514},[106,144],[18,108516,152],{"className":108517},[151],[18,108519,108521],{"className":108520},[124],[18,108522,108524],{"className":108523,"style":81865},[128],[18,108525],{},[18,108527],{"className":108528},[267,6009,141,1195,38147],[11,108530,108531],{},"代入：",[18,108533,108535],{"className":108534,"translate":22},[875],[18,108536,108538,108628],{"className":108537,"translate":22},[21],[18,108539,108541],{"className":108540},[26],[28,108542,108543],{"xmlns":30,"display":884},[32,108544,108545,108625],{},[35,108546,108547,108563,108617,108619],{},[890,108548,108549,108555],{},[35,108550,108551,108553],{},[41,108552,91870],{},[49,108554,899],{},[35,108556,108557,108559,108561],{},[41,108558,64],{},[49,108560,1880],{},[41,108562,2028],{"mathvariant":316},[5651,108564,108565,108579],{},[35,108566,108567,108573],{},[1556,108568,108569,108571],{},[41,108570,16820],{},[45,108572,2266],{},[1556,108574,108575,108577],{},[41,108576,14441],{},[45,108578,2266],{},[35,108580,108581,108583],{},[45,108582,10444],{},[1556,108584,108585,108603],{},[35,108586,108587,108589,108601],{},[49,108588,74],{"fence":51},[5651,108590,108591,108599],{},[35,108592,108593,108595,108597],{},[41,108594,64],{},[49,108596,7984],{},[45,108598,47],{},[41,108600,64],{},[49,108602,84],{"fence":51},[35,108604,108605,108607,108609,108611],{},[45,108606,2266],{},[41,108608,64],{},[49,108610,7984],{},[5651,108612,108613,108615],{},[45,108614,44699],{},[45,108616,2266],{},[49,108618,433],{},[5651,108620,108621,108623],{},[41,108622,3941],{},[45,108624,2266],{},[86,108626,108627],{"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}",[18,108629,108631,109015],{"className":108630,"ariaHidden":51},[93],[18,108632,108634,108638,108691,108694,109006,109009,109012],{"className":108633},[97],[18,108635],{"className":108636,"style":108637},[101],"height:2.925em;vertical-align:-1.4339em;",[18,108639,108641],{"className":108640},[985,986],[18,108642,108644,108683],{"className":108643},[119,120],[18,108645,108647,108680],{"className":108646},[124],[18,108648,108650,108670],{"className":108649,"style":295},[128],[18,108651,108652,108655],{"style":97007},[18,108653],{"className":108654,"style":697},[136],[18,108656,108658],{"className":108657},[141,142,143,144],[18,108659,108661,108664,108667],{"className":108660},[106,144],[18,108662,64],{"className":108663},[106,110,144],[18,108665,1880],{"className":108666},[232,144],[18,108668,2028],{"className":108669},[106,144],[18,108671,108672,108675],{"style":693},[18,108673],{"className":108674,"style":697},[136],[18,108676,108677],{},[18,108678,91870],{"className":108679},[985],[18,108681,152],{"className":108682},[151],[18,108684,108686],{"className":108685},[124],[18,108687,108689],{"className":108688,"style":97045},[128],[18,108690],{},[18,108692],{"className":108693,"style":170},[169],[18,108695,108697,108700,109003],{"className":108696},[106],[18,108698],{"className":108699},[250,6009],[18,108701,108703],{"className":108702},[5651],[18,108704,108706,108995],{"className":108705},[119,120],[18,108707,108709,108992],{"className":108708},[124],[18,108710,108712,108918,108926],{"className":108711,"style":15996},[128],[18,108713,108714,108717],{"style":102583},[18,108715],{"className":108716,"style":102587},[136],[18,108718,108720,108723,108726],{"className":108719},[106],[18,108721,10444],{"className":108722},[106],[18,108724],{"className":108725,"style":170},[169],[18,108727,108729,108818],{"className":108728},[174],[18,108730,108732,108738,108812],{"className":108731},[174],[18,108733,108735],{"className":108734,"style":13793},[250,13792],[18,108736,74],{"className":108737},[1230,1196],[18,108739,108741,108744,108809],{"className":108740},[106],[18,108742],{"className":108743},[250,6009],[18,108745,108747],{"className":108746},[5651],[18,108748,108750,108801],{"className":108749},[119,120],[18,108751,108753,108798],{"className":108752},[124],[18,108754,108756,108770,108778],{"className":108755,"style":12042},[128],[18,108757,108758,108761],{"style":12045},[18,108759],{"className":108760,"style":697},[136],[18,108762,108764],{"className":108763},[141,142,143,144],[18,108765,108767],{"className":108766},[106,144],[18,108768,64],{"className":108769},[106,110,144],[18,108771,108772,108775],{"style":6037},[18,108773],{"className":108774,"style":697},[136],[18,108776],{"className":108777,"style":6045},[6044],[18,108779,108780,108783],{"style":12068},[18,108781],{"className":108782,"style":697},[136],[18,108784,108786],{"className":108785},[141,142,143,144],[18,108787,108789,108792,108795],{"className":108788},[106,144],[18,108790,64],{"className":108791},[106,110,144],[18,108793,7984],{"className":108794},[2072,144],[18,108796,47],{"className":108797},[106,144],[18,108799,152],{"className":108800},[151],[18,108802,108804],{"className":108803},[124],[18,108805,108807],{"className":108806,"style":12090},[128],[18,108808],{},[18,108810],{"className":108811},[267,6009],[18,108813,108815],{"className":108814,"style":13793},[267,13792],[18,108816,84],{"className":108817},[1230,1196],[18,108819,108821],{"className":108820},[115],[18,108822,108824],{"className":108823},[119],[18,108825,108827],{"className":108826},[124],[18,108828,108830],{"className":108829,"style":102701},[128],[18,108831,108832,108835],{"style":69827},[18,108833],{"className":108834,"style":697},[136],[18,108836,108838],{"className":108837},[141,142,143,144],[18,108839,108841,108844,108847,108850],{"className":108840},[106,144],[18,108842,2266],{"className":108843},[106,144],[18,108845,64],{"className":108846},[106,110,144],[18,108848,7984],{"className":108849},[2072,144],[18,108851,108853,108856,108915],{"className":108852},[106,144],[18,108854],{"className":108855},[250,6009,141,1195,38147],[18,108857,108859],{"className":108858},[5651],[18,108860,108862,108907],{"className":108861},[119,120],[18,108863,108865,108904],{"className":108864},[124],[18,108866,108868,108882,108890],{"className":108867,"style":42929},[128],[18,108869,108870,108873],{"style":38163},[18,108871],{"className":108872,"style":697},[136],[18,108874,108876],{"className":108875},[141,1195,1196,144],[18,108877,108879],{"className":108878},[106,144],[18,108880,2266],{"className":108881},[106,144],[18,108883,108884,108887],{"style":38187},[18,108885],{"className":108886,"style":697},[136],[18,108888],{"className":108889,"style":38194},[6044,144],[18,108891,108892,108895],{"style":42963},[18,108893],{"className":108894,"style":697},[136],[18,108896,108898],{"className":108897},[141,1195,1196,144],[18,108899,108901],{"className":108900},[106,144],[18,108902,44699],{"className":108903},[106,144],[18,108905,152],{"className":108906},[151],[18,108908,108910],{"className":108909},[124],[18,108911,108913],{"className":108912,"style":81865},[128],[18,108914],{},[18,108916],{"className":108917},[267,6009,141,1195,38147],[18,108919,108920,108923],{"style":102789},[18,108921],{"className":108922,"style":102587},[136],[18,108924],{"className":108925,"style":6045},[6044],[18,108927,108928,108931],{"style":102798},[18,108929],{"className":108930,"style":102587},[136],[18,108932,108934,108963],{"className":108933},[106],[18,108935,108937,108940],{"className":108936},[106],[18,108938,16820],{"className":108939},[106,110],[18,108941,108943],{"className":108942},[115],[18,108944,108946],{"className":108945},[119],[18,108947,108949],{"className":108948},[124],[18,108950,108952],{"className":108951,"style":2416},[128],[18,108953,108954,108957],{"style":1962},[18,108955],{"className":108956,"style":137},[136],[18,108958,108960],{"className":108959},[141,142,143,144],[18,108961,2266],{"className":108962},[106,144],[18,108964,108966,108969],{"className":108965},[106],[18,108967,14441],{"className":108968},[106,110],[18,108970,108972],{"className":108971},[115],[18,108973,108975],{"className":108974},[119],[18,108976,108978],{"className":108977},[124],[18,108979,108981],{"className":108980,"style":2416},[128],[18,108982,108983,108986],{"style":1962},[18,108984],{"className":108985,"style":137},[136],[18,108987,108989],{"className":108988},[141,142,143,144],[18,108990,2266],{"className":108991},[106,144],[18,108993,152],{"className":108994},[151],[18,108996,108998],{"className":108997},[124],[18,108999,109001],{"className":109000,"style":102817},[128],[18,109002],{},[18,109004],{"className":109005},[267,6009],[18,109007],{"className":109008,"style":228},[169],[18,109010,433],{"className":109011},[232],[18,109013],{"className":109014,"style":228},[169],[18,109016,109018,109021],{"className":109017},[97],[18,109019],{"className":109020,"style":42062},[101],[18,109022,109024,109027,109080],{"className":109023},[106],[18,109025],{"className":109026},[250,6009],[18,109028,109030],{"className":109029},[5651],[18,109031,109033,109072],{"className":109032},[119,120],[18,109034,109036,109069],{"className":109035},[124],[18,109037,109039,109050,109058],{"className":109038,"style":13849},[128],[18,109040,109041,109044],{"style":6025},[18,109042],{"className":109043,"style":697},[136],[18,109045,109047],{"className":109046},[106],[18,109048,2266],{"className":109049},[106],[18,109051,109052,109055],{"style":6037},[18,109053],{"className":109054,"style":697},[136],[18,109056],{"className":109057,"style":6045},[6044],[18,109059,109060,109063],{"style":6048},[18,109061],{"className":109062,"style":697},[136],[18,109064,109066],{"className":109065},[106],[18,109067,3941],{"className":109068,"style":3957},[106,110],[18,109070,152],{"className":109071},[151],[18,109073,109075],{"className":109074},[124],[18,109076,109078],{"className":109077,"style":6067},[128],[18,109079],{},[18,109081],{"className":109082},[267,6009],[11,109084,109085],{},"解得：",[18,109087,109089],{"className":109088,"translate":22},[875],[18,109090,109092,109116],{"className":109091,"translate":22},[21],[18,109093,109095],{"className":109094},[26],[28,109096,109097],{"xmlns":30,"display":884},[32,109098,109099,109113],{},[35,109100,109101,109103,109105],{},[41,109102,16820],{},[49,109104,433],{},[12558,109106,109107],{},[35,109108,109109,109111],{},[45,109110,2266],{},[41,109112,3941],{},[86,109114,109115],{"encoding":88},"a = \\sqrt{2 \\pi}",[18,109117,109119,109137],{"className":109118,"ariaHidden":51},[93],[18,109120,109122,109125,109128,109131,109134],{"className":109121},[97],[18,109123],{"className":109124,"style":1083},[101],[18,109126,16820],{"className":109127},[106,110],[18,109129],{"className":109130,"style":228},[169],[18,109132,433],{"className":109133},[232],[18,109135],{"className":109136,"style":228},[169],[18,109138,109140,109144],{"className":109139},[97],[18,109141],{"className":109142,"style":109143},[101],"height:1.04em;vertical-align:-0.0839em;",[18,109145,109147],{"className":109146},[106,12693],[18,109148,109150,109188],{"className":109149},[119,120],[18,109151,109153,109185],{"className":109152},[124],[18,109154,109157,109172],{"className":109155,"style":109156},[128],"height:0.9561em;",[18,109158,109160,109163],{"className":109159,"style":693},[12707],[18,109161],{"className":109162,"style":697},[136],[18,109164,109166,109169],{"className":109165,"style":15407},[106],[18,109167,2266],{"className":109168},[106],[18,109170,3941],{"className":109171,"style":3957},[106,110],[18,109173,109175,109178],{"style":109174},"top:-2.9161em;",[18,109176],{"className":109177,"style":697},[136],[18,109179,109181],{"className":109180,"style":15417},[12850],[6127,109182,109183],{"xmlns":6129,"width":12854,"height":15420,"viewBox":15421,"preserveAspectRatio":12857},[6133,109184],{"d":15424},[18,109186,152],{"className":109187},[151],[18,109189,109191],{"className":109190},[124],[18,109192,109195],{"className":109193,"style":109194},[128],"height:0.0839em;",[18,109196],{},[11,109198,109199,109200,109228],{},"因此，当 ",[18,109201,109203,109216],{"className":109202,"translate":22},[21],[18,109204,109206],{"className":109205},[26],[28,109207,109208],{"xmlns":30},[32,109209,109210,109214],{},[35,109211,109212],{},[41,109213,64],{},[86,109215,64],{"encoding":88},[18,109217,109219],{"className":109218,"ariaHidden":51},[93],[18,109220,109222,109225],{"className":109221},[97],[18,109223],{"className":109224,"style":1083},[101],[18,109226,64],{"className":109227},[106,110]," 趋近无穷大时：",[18,109230,109232],{"className":109231,"translate":22},[875],[18,109233,109235,109287],{"className":109234,"translate":22},[21],[18,109236,109238],{"className":109237},[26],[28,109239,109240],{"xmlns":30,"display":884},[32,109241,109242,109284],{},[35,109243,109244,109246,109248,109250,109258,109268],{},[41,109245,64],{},[49,109247,88973],{"stretchy":73},[49,109249,67],{},[12558,109251,109252],{},[35,109253,109254,109256],{},[45,109255,2266],{},[41,109257,3941],{},[1556,109259,109260,109262],{},[41,109261,14441],{},[35,109263,109264,109266],{},[49,109265,2253],{},[41,109267,64],{},[1556,109269,109270,109272],{},[41,109271,64],{},[35,109273,109274,109276,109278],{},[41,109275,64],{},[49,109277,7984],{},[5651,109279,109280,109282],{},[45,109281,47],{},[45,109283,2266],{},[86,109285,109286],{"encoding":88},"n! \\sim \\sqrt{2 \\pi} e^{-n} n^{n + \\frac{1}{2}}",[18,109288,109290,109311],{"className":109289,"ariaHidden":51},[93],[18,109291,109293,109296,109299,109302,109305,109308],{"className":109292},[97],[18,109294],{"className":109295,"style":295},[101],[18,109297,64],{"className":109298},[106,110],[18,109300,88973],{"className":109301},[267],[18,109303],{"className":109304,"style":228},[169],[18,109306,67],{"className":109307},[232],[18,109309],{"className":109310,"style":228},[169],[18,109312,109314,109318,109368,109403],{"className":109313},[97],[18,109315],{"className":109316,"style":109317},[101],"height:1.0879em;vertical-align:-0.0839em;",[18,109319,109321],{"className":109320},[106,12693],[18,109322,109324,109360],{"className":109323},[119,120],[18,109325,109327,109357],{"className":109326},[124],[18,109328,109330,109345],{"className":109329,"style":109156},[128],[18,109331,109333,109336],{"className":109332,"style":693},[12707],[18,109334],{"className":109335,"style":697},[136],[18,109337,109339,109342],{"className":109338,"style":15407},[106],[18,109340,2266],{"className":109341},[106],[18,109343,3941],{"className":109344,"style":3957},[106,110],[18,109346,109347,109350],{"style":109174},[18,109348],{"className":109349,"style":697},[136],[18,109351,109353],{"className":109352,"style":15417},[12850],[6127,109354,109355],{"xmlns":6129,"width":12854,"height":15420,"viewBox":15421,"preserveAspectRatio":12857},[6133,109356],{"d":15424},[18,109358,152],{"className":109359},[151],[18,109361,109363],{"className":109362},[124],[18,109364,109366],{"className":109365,"style":109194},[128],[18,109367],{},[18,109369,109371,109374],{"className":109370},[106],[18,109372,14441],{"className":109373},[106,110],[18,109375,109377],{"className":109376},[115],[18,109378,109380],{"className":109379},[119],[18,109381,109383],{"className":109382},[124],[18,109384,109386],{"className":109385,"style":1800},[128],[18,109387,109388,109391],{"style":1637},[18,109389],{"className":109390,"style":137},[136],[18,109392,109394],{"className":109393},[141,142,143,144],[18,109395,109397,109400],{"className":109396},[106,144],[18,109398,2253],{"className":109399},[106,144],[18,109401,64],{"className":109402},[106,110,144],[18,109404,109406,109409],{"className":109405},[106],[18,109407,64],{"className":109408},[106,110],[18,109410,109412],{"className":109411},[115],[18,109413,109415],{"className":109414},[119],[18,109416,109418],{"className":109417},[124],[18,109419,109421],{"className":109420,"style":108387},[128],[18,109422,109423,109426],{"style":97809},[18,109424],{"className":109425,"style":697},[136],[18,109427,109429],{"className":109428},[141,142,143,144],[18,109430,109432,109435,109438],{"className":109431},[106,144],[18,109433,64],{"className":109434},[106,110,144],[18,109436,7984],{"className":109437},[2072,144],[18,109439,109441,109444,109503],{"className":109440},[106,144],[18,109442],{"className":109443},[250,6009,141,1195,38147],[18,109445,109447],{"className":109446},[5651],[18,109448,109450,109495],{"className":109449},[119,120],[18,109451,109453,109492],{"className":109452},[124],[18,109454,109456,109470,109478],{"className":109455,"style":42929},[128],[18,109457,109458,109461],{"style":38163},[18,109459],{"className":109460,"style":697},[136],[18,109462,109464],{"className":109463},[141,1195,1196,144],[18,109465,109467],{"className":109466},[106,144],[18,109468,2266],{"className":109469},[106,144],[18,109471,109472,109475],{"style":38187},[18,109473],{"className":109474,"style":697},[136],[18,109476],{"className":109477,"style":38194},[6044,144],[18,109479,109480,109483],{"style":42963},[18,109481],{"className":109482,"style":697},[136],[18,109484,109486],{"className":109485},[141,1195,1196,144],[18,109487,109489],{"className":109488},[106,144],[18,109490,47],{"className":109491},[106,144],[18,109493,152],{"className":109494},[151],[18,109496,109498],{"className":109497},[124],[18,109499,109501],{"className":109500,"style":81865},[128],[18,109502],{},[18,109504],{"className":109505},[267,6009,141,1195,38147],[11,109507,109508],{},"或者等价地：",[18,109510,109512],{"className":109511,"translate":22},[875],[18,109513,109515,109559],{"className":109514,"translate":22},[21],[18,109516,109518],{"className":109517},[26],[28,109519,109520],{"xmlns":30,"display":884},[32,109521,109522,109556],{},[35,109523,109524,109526,109528,109530,109540],{},[41,109525,64],{},[49,109527,88973],{"stretchy":73},[49,109529,67],{},[12558,109531,109532],{},[35,109533,109534,109536,109538],{},[45,109535,2266],{},[41,109537,3941],{},[41,109539,64],{},[1556,109541,109542,109554],{},[35,109543,109544,109546,109552],{},[49,109545,74],{"fence":51},[5651,109547,109548,109550],{},[41,109549,64],{},[41,109551,14441],{},[49,109553,84],{"fence":51},[41,109555,64],{},[86,109557,109558],{"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。",[45587],"2025-09-18T00:00:00+08:00","kCcJ1796DlLa-Cf3vQ5ANr8rCBs2ieEkP55C1qxB2oo",1794483713199]