[{"data":1,"prerenderedAt":66757},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F0007-dirichlet-distribution-and-evidential-deep-learning":3,"i-ci:tag":-1,"i-mingcute:time-line":-1},{"post":4,"nextPost":65034,"prevPost":65045},{"id":5,"title":6,"body":7,"description":65030,"draft":65031,"enableComment":65032,"extension":65033,"image":64988,"important":65031,"location":65034,"meta":65035,"navigation":65032,"ogImage":65034,"path":65036,"seo":65037,"stem":65038,"summary":65039,"tags":65040,"time":65043,"__hash__":65044},"blog\u002Fblog\u002F2026\u002F0007-dirichlet-distribution-and-evidential-deep-learning.md","Dirichlet 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为输入空间，",[15,66,68,109],{"className":67,"translate":19},[18],[15,69,71],{"className":70},[23],[25,72,73],{"xmlns":27},[29,74,75,106],{},[32,76,77,80,84,88,92,95,98,100,103],{},[35,78,79],{},"y",[81,82,83],"mo",{},"∈",[81,85,87],{"stretchy":86},"false","{",[89,90,91],"mn",{},"1",[81,93,94],{"separator":48},",",[81,96,97],{},"…",[81,99,94],{"separator":48},[35,101,102],{},"K",[81,104,105],{"stretchy":86},"}",[40,107,108],{"encoding":42},"y \\in \\{1, \\dots, K\\}",[15,110,112,136],{"className":111,"ariaHidden":48},[47],[15,113,115,119,124,129,133],{"className":114},[52],[15,116],{"className":117,"style":118},[56],"height:0.7335em;vertical-align:-0.1944em;",[15,120,79],{"className":121,"style":123},[61,122],"mathnormal","margin-right:0.0359em;",[15,125],{"className":126,"style":128},[127],"mspace","margin-right:0.2778em;",[15,130,83],{"className":131},[132],"mrel",[15,134],{"className":135,"style":128},[127],[15,137,139,143,147,150,154,158,162,165,168,171,175],{"className":138},[52],[15,140],{"className":141,"style":142},[56],"height:1em;vertical-align:-0.25em;",[15,144,87],{"className":145},[146],"mopen",[15,148,91],{"className":149},[61],[15,151,94],{"className":152},[153],"mpunct",[15,155],{"className":156,"style":157},[127],"margin-right:0.1667em;",[15,159,97],{"className":160},[161],"minner",[15,163],{"className":164,"style":157},[127],[15,166,94],{"className":167},[153],[15,169],{"className":170,"style":157},[127],[15,172,102],{"className":173,"style":174},[61,122],"margin-right:0.0715em;",[15,176,105],{"className":177},[178],"mclose"," 为类别标签。一个 ",[15,181,183,196],{"className":182,"translate":19},[18],[15,184,186],{"className":185},[23],[25,187,188],{"xmlns":27},[29,189,190,194],{},[32,191,192],{},[35,193,102],{},[40,195,102],{"encoding":42},[15,197,199],{"className":198,"ariaHidden":48},[47],[15,200,202,205],{"className":201},[52],[15,203],{"className":204,"style":57},[56],[15,206,102],{"className":207,"style":174},[61,122]," 分类器的输出在数学上究竟是什么？若要求它给出「属于各类别的概率」，则输出必须落在",[15,210,213],{"className":211,"translate":19},[212],"katex-display",[15,214,216,319],{"className":215,"translate":19},[18],[15,217,219],{"className":218},[23],[25,220,222],{"xmlns":27,"display":221},"block",[29,223,224,316],{},[32,225,226,242,245],{},[227,228,229,233],"msup",{},[35,230,232],{"mathvariant":231},"normal","Δ",[32,234,235,237,240],{},[35,236,102],{},[81,238,239],{},"−",[89,241,91],{},[81,243,244],{},"=",[32,246,247,249,252,254,262,265,273,276,279,281,285,314],{},[81,248,87],{"fence":48},[35,250,11],{"mathvariant":251},"bold",[81,253,83],{},[227,255,256,260],{},[35,257,259],{"mathvariant":258},"double-struck","R",[35,261,102],{},[81,263,264],{},":",[266,267,268,270],"msub",{},[35,269,11],{},[35,271,272],{},"k",[81,274,275],{},"≥",[89,277,278],{},"0",[81,280,94],{"separator":48},[282,283,284],"mtext",{}," ",[286,287,288,304,310,312],"mstyle",{"scriptlevel":278,"displaystyle":86},[289,290,291,294,302],"msubsup",{},[81,292,293],{},"∑",[32,295,296,298,300],{},[35,297,272],{},[81,299,244],{},[89,301,91],{},[35,303,102],{},[266,305,306,308],{},[35,307,11],{},[35,309,272],{},[81,311,244],{},[89,313,91],{},[81,315,105],{"fence":48},[40,317,318],{"encoding":42},"\\Delta^{K-1} = \\left\\{ \\mathbf{p} \\in \\mathbb{R}^K : p_k \\ge 0,\\ \\textstyle\\sum_{k=1}^K p_k = 1 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",[15,666,668,686],{"className":667,"translate":19},[18],[15,669,671],{"className":670},[23],[25,672,673],{"xmlns":27},[29,674,675,683],{},[32,676,677],{},[227,678,679,681],{},[35,680,259],{"mathvariant":258},[35,682,102],{},[40,684,685],{"encoding":42},"\\mathbb{R}^K",[15,687,689],{"className":688,"ariaHidden":48},[47],[15,690,692,696],{"className":691},[52],[15,693],{"className":694,"style":695},[56],"height:0.8413em;",[15,697,699,702],{"className":698},[61],[15,700,259],{"className":701},[61,427],[15,703,705],{"className":704},[338],[15,706,708],{"className":707},[342],[15,709,711],{"className":710},[346],[15,712,714],{"className":713,"style":695},[350],[15,715,717,720],{"style":716},"top:-3.063em;margin-right:0.05em;",[15,718],{"className":719,"style":358},[357],[15,721,723],{"className":722},[362,363,364,365],[15,724,102],{"className":725,"style":174},[61,122,365],"，而是一个 ",[15,728,730,754],{"className":729,"translate":19},[18],[15,731,733],{"className":732},[23],[25,734,735],{"xmlns":27},[29,736,737,751],{},[32,738,739,742,744,746,748],{},[81,740,741],{"stretchy":86},"(",[35,743,102],{},[81,745,239],{},[89,747,91],{},[81,749,750],{"stretchy":86},")",[40,752,753],{"encoding":42},"(K-1)",[15,755,757,779],{"className":756,"ariaHidden":48},[47],[15,758,760,763,766,769,773,776],{"className":759},[52],[15,761],{"className":762,"style":142},[56],[15,764,741],{"className":765},[146],[15,767,102],{"className":768,"style":174},[61,122],[15,770],{"className":771,"style":772},[127],"margin-right:0.2222em;",[15,774,239],{"className":775},[375],[15,777],{"className":778,"style":772},[127],[15,780,782,785,788],{"className":781},[52],[15,783],{"className":784,"style":142},[56],[15,786,91],{"className":787},[61],[15,789,750],{"className":790},[178]," 维的紧凸集。",[11,793,794,795,798,799,802,803,833,834,863,864,867],{},"一旦承认了这一点，一个二阶问题便随即出现：分类器对自身输出的",[661,796,797],{},"把握","有多大？若把输出建模为单纯形上的一个",[661,800,801],{},"确定点"," ",[15,804,806,820],{"className":805,"translate":19},[18],[15,807,809],{"className":808},[23],[25,810,811],{"xmlns":27},[29,812,813,817],{},[32,814,815],{},[35,816,11],{"mathvariant":251},[40,818,819],{"encoding":42},"\\mathbf{p}",[15,821,823],{"className":822,"ariaHidden":48},[47],[15,824,826,830],{"className":825},[52],[15,827],{"className":828,"style":829},[56],"height:0.6389em;vertical-align:-0.1944em;",[15,831,11],{"className":832},[61,411],"，则这个问题在表达力上是被禁止的：单纯形内部不存在可以指代「无信息」的元素，因为每个点都是一份完整的、和为 ",[15,835,837,850],{"className":836,"translate":19},[18],[15,838,840],{"className":839},[23],[25,841,842],{"xmlns":27},[29,843,844,848],{},[32,845,846],{},[89,847,91],{},[40,849,91],{"encoding":42},[15,851,853],{"className":852,"ariaHidden":48},[47],[15,854,856,860],{"className":855},[52],[15,857],{"className":858,"style":859},[56],"height:0.6444em;",[15,861,91],{"className":862},[61]," 的断言。要回答二阶问题，必须把输出从「单纯形上的点」提升为「单纯形上的分布」。承担这一角色的标准对象正是 ",[661,865,866],{},"迪利克雷分布（Dirichlet distribution）","。",[11,869,870],{},"本文的组织如下。第一部分给出迪利克雷分布的形式化定义、生成性刻画与几何图像。第二部分展开其矩结构、重参数化、共轭更新与分裂构造，并澄清「伪计数」这一常见的直观说法在何处成立、在何处失效。第三部分经由主观逻辑（subjective logic）建立迪利克雷分布与神经网络输出之间的双射，由此引入证据深度学习（Evidential Deep Learning, EDL）。第四部分推导 EDL 的损失函数族及其理论缺陷。第五部分讨论其应用形态与边界。第六部分总结。",[872,873,875],"h2",{"id":874},"一形式化","一、形式化",[877,878,880],"h3",{"id":879},"_11-概率单纯形","1.1 概率单纯形",[882,883,884],"blockquote",{},[11,885,886,889,890,943,944,996,997,1529,1530,867],{},[661,887,888],{},"定义 1.1（概率单纯形）"," 对整数 ",[15,891,893,912],{"className":892,"translate":19},[18],[15,894,896],{"className":895},[23],[25,897,898],{"xmlns":27},[29,899,900,909],{},[32,901,902,904,906],{},[35,903,102],{},[81,905,275],{},[89,907,908],{},"2",[40,910,911],{"encoding":42},"K \\ge 2",[15,913,915,934],{"className":914,"ariaHidden":48},[47],[15,916,918,922,925,928,931],{"className":917},[52],[15,919],{"className":920,"style":921},[56],"height:0.8193em;vertical-align:-0.136em;",[15,923,102],{"className":924,"style":174},[61,122],[15,926],{"className":927,"style":128},[127],[15,929,275],{"className":930},[132],[15,932],{"className":933,"style":128},[127],[15,935,937,940],{"className":936},[52],[15,938],{"className":939,"style":859},[56],[15,941,908],{"className":942},[61],"，",[15,945,947,965],{"className":946,"translate":19},[18],[15,948,950],{"className":949},[23],[25,951,952],{"xmlns":27},[29,953,954,962],{},[32,955,956,958,960],{},[35,957,102],{},[81,959,239],{},[89,961,91],{},[40,963,964],{"encoding":42},"K-1",[15,966,968,987],{"className":967,"ariaHidden":48},[47],[15,969,971,975,978,981,984],{"className":970},[52],[15,972],{"className":973,"style":974},[56],"height:0.7667em;vertical-align:-0.0833em;",[15,976,102],{"className":977,"style":174},[61,122],[15,979],{"className":980,"style":772},[127],[15,982,239],{"className":983},[375],[15,985],{"className":986,"style":772},[127],[15,988,990,993],{"className":989},[52],[15,991],{"className":992,"style":859},[56],[15,994,91],{"className":995},[61]," 维标准单纯形定义为\n",[15,998,1000,1110],{"className":999,"translate":19},[18],[15,1001,1003],{"className":1002},[23],[25,1004,1005],{"xmlns":27},[29,1006,1007,1107],{},[32,1008,1009,1021,1023],{},[227,1010,1011,1013],{},[35,1012,232],{"mathvariant":231},[32,1014,1015,1017,1019],{},[35,1016,102],{},[81,1018,239],{},[89,1020,91],{},[81,1022,244],{},[32,1024,1025,1027,1030,1032,1034,1040,1042,1044,1046,1052,1054,1056,1062,1064,1071,1073,1075,1077,1079,1105],{},[81,1026,87],{"fence":48},[35,1028,1029],{"mathvariant":251},"x",[81,1031,244],{},[81,1033,741],{"stretchy":86},[266,1035,1036,1038],{},[35,1037,1029],{},[89,1039,91],{},[81,1041,94],{"separator":48},[81,1043,97],{},[81,1045,94],{"separator":48},[266,1047,1048,1050],{},[35,1049,1029],{},[35,1051,102],{},[81,1053,750],{"stretchy":86},[81,1055,83],{},[227,1057,1058,1060],{},[35,1059,259],{"mathvariant":258},[35,1061,102],{},[81,1063,264],{},[266,1065,1066,1068],{},[35,1067,1029],{},[35,1069,1070],{},"i",[81,1072,275],{},[89,1074,278],{},[81,1076,94],{"separator":48},[282,1078,284],{},[286,1080,1081,1095,1101,1103],{"scriptlevel":278,"displaystyle":86},[289,1082,1083,1085,1093],{},[81,1084,293],{},[32,1086,1087,1089,1091],{},[35,1088,1070],{},[81,1090,244],{},[89,1092,91],{},[35,1094,102],{},[266,1096,1097,1099],{},[35,1098,1029],{},[35,1100,1070],{},[81,1102,244],{},[89,1104,91],{},[81,1106,105],{"fence":48},[40,1108,1109],{"encoding":42},"\\Delta^{K-1} = \\left\\{ \\mathbf{x} = (x_1, \\dots, x_K) \\in \\mathbb{R}^K : x_i \\ge 0,\\ \\textstyle\\sum_{i=1}^K x_i = 1 \\right\\}",[15,1111,1113,1166],{"className":1112,"ariaHidden":48},[47],[15,1114,1116,1119,1157,1160,1163],{"className":1115},[52],[15,1117],{"className":1118,"style":695},[56],[15,1120,1122,1125],{"className":1121},[61],[15,1123,232],{"className":1124},[61],[15,1126,1128],{"className":1127},[338],[15,1129,1131],{"className":1130},[342],[15,1132,1134],{"className":1133},[346],[15,1135,1137],{"className":1136,"style":695},[350],[15,1138,1139,1142],{"style":716},[15,1140],{"className":1141,"style":358},[357],[15,1143,1145],{"className":1144},[362,363,364,365],[15,1146,1148,1151,1154],{"className":1147},[61,365],[15,1149,102],{"className":1150,"style":174},[61,122,365],[15,1152,239],{"className":1153},[375,365],[15,1155,91],{"className":1156},[61,365],[15,1158],{"className":1159,"style":128},[127],[15,1161,244],{"className":1162},[132],[15,1164],{"className":1165,"style":128},[127],[15,1167,1169,1172],{"className":1168},[52],[15,1170],{"className":1171,"style":394},[56],[15,1173,1175,1181,1184,1187,1190,1193,1196,1237,1240,1243,1246,1249,1252,1255,1296,1299,1302,1305,1308,1337,1340,1343,1346,1387,1390,1393,1396,1399,1402,1405,1408,1468,1471,1511,1514,1517,1520,1523],{"className":1174},[161],[15,1176,1178],{"className":1177,"style":402},[146,401],[15,1179,87],{"className":1180},[406,407],[15,1182,1029],{"className":1183},[61,411],[15,1185],{"className":1186,"style":128},[127],[15,1188,244],{"className":1189},[132],[15,1191],{"className":1192,"style":128},[127],[15,1194,741],{"className":1195},[146],[15,1197,1199,1202],{"className":1198},[61],[15,1200,1029],{"className":1201},[61,122],[15,1203,1205],{"className":1204},[338],[15,1206,1208,1229],{"className":1207},[342,472],[15,1209,1211,1226],{"className":1210},[346],[15,1212,1215],{"className":1213,"style":1214},[350],"height:0.3011em;",[15,1216,1217,1220],{"style":482},[15,1218],{"className":1219,"style":358},[357],[15,1221,1223],{"className":1222},[362,363,364,365],[15,1224,91],{"className":1225},[61,365],[15,1227,497],{"className":1228},[496],[15,1230,1232],{"className":1231},[346],[15,1233,1235],{"className":1234,"style":504},[350],[15,1236],{},[15,1238,94],{"className":1239},[153],[15,1241],{"className":1242,"style":157},[127],[15,1244,97],{"className":1245},[161],[15,1247],{"className":1248,"style":157},[127],[15,1250,94],{"className":1251},[153],[15,1253],{"className":1254,"style":157},[127],[15,1256,1258,1261],{"className":1257},[61],[15,1259,1029],{"className":1260},[61,122],[15,1262,1264],{"className":1263},[338],[15,1265,1267,1288],{"className":1266},[342,472],[15,1268,1270,1285],{"className":1269},[346],[15,1271,1274],{"className":1272,"style":1273},[350],"height:0.3283em;",[15,1275,1276,1279],{"style":482},[15,1277],{"className":1278,"style":358},[357],[15,1280,1282],{"className":1281},[362,363,364,365],[15,1283,102],{"className":1284,"style":174},[61,122,365],[15,1286,497],{"className":1287},[496],[15,1289,1291],{"className":1290},[346],[15,1292,1294],{"className":1293,"style":504},[350],[15,1295],{},[15,1297,750],{"className":1298},[178],[15,1300],{"className":1301,"style":128},[127],[15,1303,83],{"className":1304},[132],[15,1306],{"className":1307,"style":128},[127],[15,1309,1311,1314],{"className":1310},[61],[15,1312,259],{"className":1313},[61,427],[15,1315,1317],{"className":1316},[338],[15,1318,1320],{"className":1319},[342],[15,1321,1323],{"className":1322},[346],[15,1324,1326],{"className":1325,"style":695},[350],[15,1327,1328,1331],{"style":716},[15,1329],{"className":1330,"style":358},[357],[15,1332,1334],{"className":1333},[362,363,364,365],[15,1335,102],{"className":1336,"style":174},[61,122,365],[15,1338],{"className":1339,"style":128},[127],[15,1341,264],{"className":1342},[132],[15,1344],{"className":1345,"style":128},[127],[15,1347,1349,1352],{"className":1348},[61],[15,1350,1029],{"className":1351},[61,122],[15,1353,1355],{"className":1354},[338],[15,1356,1358,1379],{"className":1357},[342,472],[15,1359,1361,1376],{"className":1360},[346],[15,1362,1365],{"className":1363,"style":1364},[350],"height:0.3117em;",[15,1366,1367,1370],{"style":482},[15,1368],{"className":1369,"style":358},[357],[15,1371,1373],{"className":1372},[362,363,364,365],[15,1374,1070],{"className":1375},[61,122,365],[15,1377,497],{"className":1378},[496],[15,1380,1382],{"className":1381},[346],[15,1383,1385],{"className":1384,"style":504},[350],[15,1386],{},[15,1388],{"className":1389,"style":128},[127],[15,1391,275],{"className":1392},[132],[15,1394],{"className":1395,"style":128},[127],[15,1397,278],{"className":1398},[61],[15,1400,94],{"className":1401},[153],[15,1403,284],{"className":1404},[127],[15,1406],{"className":1407,"style":157},[127],[15,1409,1411,1414],{"className":1410},[531],[15,1412,293],{"className":1413,"style":537},[531,535,536],[15,1415,1417],{"className":1416},[338],[15,1418,1420,1460],{"className":1419},[342,472],[15,1421,1423,1457],{"className":1422},[346],[15,1424,1426,1446],{"className":1425,"style":550},[350],[15,1427,1428,1431],{"style":553},[15,1429],{"className":1430,"style":358},[357],[15,1432,1434],{"className":1433},[362,363,364,365],[15,1435,1437,1440,1443],{"className":1436},[61,365],[15,1438,1070],{"className":1439},[61,122,365],[15,1441,244],{"className":1442},[132,365],[15,1444,91],{"className":1445},[61,365],[15,1447,1448,1451],{"style":574},[15,1449],{"className":1450,"style":358},[357],[15,1452,1454],{"className":1453},[362,363,364,365],[15,1455,102],{"className":1456,"style":174},[61,122,365],[15,1458,497],{"className":1459},[496],[15,1461,1463],{"className":1462},[346],[15,1464,1466],{"className":1465,"style":593},[350],[15,1467],{},[15,1469],{"className":1470,"style":157},[127],[15,1472,1474,1477],{"className":1473},[61],[15,1475,1029],{"className":1476},[61,122],[15,1478,1480],{"className":1479},[338],[15,1481,1483,1503],{"className":1482},[342,472],[15,1484,1486,1500],{"className":1485},[346],[15,1487,1489],{"className":1488,"style":1364},[350],[15,1490,1491,1494],{"style":482},[15,1492],{"className":1493,"style":358},[357],[15,1495,1497],{"className":1496},[362,363,364,365],[15,1498,1070],{"className":1499},[61,122,365],[15,1501,497],{"className":1502},[496],[15,1504,1506],{"className":1505},[346],[15,1507,1509],{"className":1508,"style":504},[350],[15,1510],{},[15,1512],{"className":1513,"style":128},[127],[15,1515,244],{"className":1516},[132],[15,1518],{"className":1519,"style":128},[127],[15,1521,91],{"className":1522},[61],[15,1524,1526],{"className":1525,"style":402},[178,401],[15,1527,105],{"className":1528},[406,407],"\n其内部记作 ",[15,1531,1533,1607],{"className":1532,"translate":19},[18],[15,1534,1536],{"className":1535},[23],[25,1537,1538],{"xmlns":27},[29,1539,1540,1604],{},[32,1541,1542,1545,1548,1560,1562,1564,1566,1568,1580,1582,1588,1591,1593,1595,1597,1600,1602],{},[35,1543,1544],{"mathvariant":231},"int",[81,1546,1547],{},"⁡",[227,1549,1550,1552],{},[35,1551,232],{"mathvariant":231},[32,1553,1554,1556,1558],{},[35,1555,102],{},[81,1557,239],{},[89,1559,91],{},[81,1561,244],{},[81,1563,87],{"stretchy":86},[35,1565,1029],{"mathvariant":251},[81,1567,83],{},[227,1569,1570,1572],{},[35,1571,232],{"mathvariant":231},[32,1573,1574,1576,1578],{},[35,1575,102],{},[81,1577,239],{},[89,1579,91],{},[81,1581,264],{},[266,1583,1584,1586],{},[35,1585,1029],{},[35,1587,1070],{},[81,1589,1590],{},">",[89,1592,278],{},[81,1594,94],{"separator":48},[282,1596,284],{},[35,1598,1599],{"mathvariant":231},"∀",[35,1601,1070],{},[81,1603,105],{"stretchy":86},[40,1605,1606],{"encoding":42},"\\operatorname{int}\\Delta^{K-1} = \\{\\mathbf{x} \\in \\Delta^{K-1} : x_i > 0,\\ \\forall i\\}",[15,1608,1610,1673,1694,1747,1803],{"className":1609,"ariaHidden":48},[47],[15,1611,1613,1616,1623,1626,1664,1667,1670],{"className":1612},[52],[15,1614],{"className":1615,"style":695},[56],[15,1617,1619],{"className":1618},[531],[15,1620,1544],{"className":1621},[61,1622],"mathrm",[15,1624],{"className":1625,"style":157},[127],[15,1627,1629,1632],{"className":1628},[61],[15,1630,232],{"className":1631},[61],[15,1633,1635],{"className":1634},[338],[15,1636,1638],{"className":1637},[342],[15,1639,1641],{"className":1640},[346],[15,1642,1644],{"className":1643,"style":695},[350],[15,1645,1646,1649],{"style":716},[15,1647],{"className":1648,"style":358},[357],[15,1650,1652],{"className":1651},[362,363,364,365],[15,1653,1655,1658,1661],{"className":1654},[61,365],[15,1656,102],{"className":1657,"style":174},[61,122,365],[15,1659,239],{"className":1660},[375,365],[15,1662,91],{"className":1663},[61,365],[15,1665],{"className":1666,"style":128},[127],[15,1668,244],{"className":1669},[132],[15,1671],{"className":1672,"style":128},[127],[15,1674,1676,1679,1682,1685,1688,1691],{"className":1675},[52],[15,1677],{"className":1678,"style":142},[56],[15,1680,87],{"className":1681},[146],[15,1683,1029],{"className":1684},[61,411],[15,1686],{"className":1687,"style":128},[127],[15,1689,83],{"className":1690},[132],[15,1692],{"className":1693,"style":128},[127],[15,1695,1697,1700,1738,1741,1744],{"className":1696},[52],[15,1698],{"className":1699,"style":695},[56],[15,1701,1703,1706],{"className":1702},[61],[15,1704,232],{"className":1705},[61],[15,1707,1709],{"className":1708},[338],[15,1710,1712],{"className":1711},[342],[15,1713,1715],{"className":1714},[346],[15,1716,1718],{"className":1717,"style":695},[350],[15,1719,1720,1723],{"style":716},[15,1721],{"className":1722,"style":358},[357],[15,1724,1726],{"className":1725},[362,363,364,365],[15,1727,1729,1732,1735],{"className":1728},[61,365],[15,1730,102],{"className":1731,"style":174},[61,122,365],[15,1733,239],{"className":1734},[375,365],[15,1736,91],{"className":1737},[61,365],[15,1739],{"className":1740,"style":128},[127],[15,1742,264],{"className":1743},[132],[15,1745],{"className":1746,"style":128},[127],[15,1748,1750,1754,1794,1797,1800],{"className":1749},[52],[15,1751],{"className":1752,"style":1753},[56],"height:0.6891em;vertical-align:-0.15em;",[15,1755,1757,1760],{"className":1756},[61],[15,1758,1029],{"className":1759},[61,122],[15,1761,1763],{"className":1762},[338],[15,1764,1766,1786],{"className":1765},[342,472],[15,1767,1769,1783],{"className":1768},[346],[15,1770,1772],{"className":1771,"style":1364},[350],[15,1773,1774,1777],{"style":482},[15,1775],{"className":1776,"style":358},[357],[15,1778,1780],{"className":1779},[362,363,364,365],[15,1781,1070],{"className":1782},[61,122,365],[15,1784,497],{"className":1785},[496],[15,1787,1789],{"className":1788},[346],[15,1790,1792],{"className":1791,"style":504},[350],[15,1793],{},[15,1795],{"className":1796,"style":128},[127],[15,1798,1590],{"className":1799},[132],[15,1801],{"className":1802,"style":128},[127],[15,1804,1806,1809,1812,1815,1818,1821,1824,1827],{"className":1805},[52],[15,1807],{"className":1808,"style":142},[56],[15,1810,278],{"className":1811},[61],[15,1813,94],{"className":1814},[153],[15,1816,284],{"className":1817},[127],[15,1819],{"className":1820,"style":157},[127],[15,1822,1599],{"className":1823},[61],[15,1825,1070],{"className":1826},[61,122],[15,1828,105],{"className":1829},[178],[11,1831,1832,1906,1907,1965,1966,2107,2108,2159,2160,2212,2213,2265,2266,2294],{},[15,1833,1835,1859],{"className":1834,"translate":19},[18],[15,1836,1838],{"className":1837},[23],[25,1839,1840],{"xmlns":27},[29,1841,1842,1856],{},[32,1843,1844],{},[227,1845,1846,1848],{},[35,1847,232],{"mathvariant":231},[32,1849,1850,1852,1854],{},[35,1851,102],{},[81,1853,239],{},[89,1855,91],{},[40,1857,1858],{"encoding":42},"\\Delta^{K-1}",[15,1860,1862],{"className":1861,"ariaHidden":48},[47],[15,1863,1865,1868],{"className":1864},[52],[15,1866],{"className":1867,"style":695},[56],[15,1869,1871,1874],{"className":1870},[61],[15,1872,232],{"className":1873},[61],[15,1875,1877],{"className":1876},[338],[15,1878,1880],{"className":1879},[342],[15,1881,1883],{"className":1882},[346],[15,1884,1886],{"className":1885,"style":695},[350],[15,1887,1888,1891],{"style":716},[15,1889],{"className":1890,"style":358},[357],[15,1892,1894],{"className":1893},[362,363,364,365],[15,1895,1897,1900,1903],{"className":1896},[61,365],[15,1898,102],{"className":1899,"style":174},[61,122,365],[15,1901,239],{"className":1902},[375,365],[15,1904,91],{"className":1905},[61,365]," 是 ",[15,1908,1910,1927],{"className":1909,"translate":19},[18],[15,1911,1913],{"className":1912},[23],[25,1914,1915],{"xmlns":27},[29,1916,1917,1925],{},[32,1918,1919],{},[227,1920,1921,1923],{},[35,1922,259],{"mathvariant":258},[35,1924,102],{},[40,1926,685],{"encoding":42},[15,1928,1930],{"className":1929,"ariaHidden":48},[47],[15,1931,1933,1936],{"className":1932},[52],[15,1934],{"className":1935,"style":695},[56],[15,1937,1939,1942],{"className":1938},[61],[15,1940,259],{"className":1941},[61,427],[15,1943,1945],{"className":1944},[338],[15,1946,1948],{"className":1947},[342],[15,1949,1951],{"className":1950},[346],[15,1952,1954],{"className":1953,"style":695},[350],[15,1955,1956,1959],{"style":716},[15,1957],{"className":1958,"style":358},[357],[15,1960,1962],{"className":1961},[362,363,364,365],[15,1963,102],{"className":1964,"style":174},[61,122,365]," 中的紧凸集，其顶点为坐标向量 ",[15,1967,1969,2000],{"className":1968,"translate":19},[18],[15,1970,1972],{"className":1971},[23],[25,1973,1974],{"xmlns":27},[29,1975,1976,1997],{},[32,1977,1978,1985,1987,1989,1991],{},[266,1979,1980,1983],{},[35,1981,1982],{"mathvariant":251},"e",[89,1984,91],{},[81,1986,94],{"separator":48},[81,1988,97],{},[81,1990,94],{"separator":48},[266,1992,1993,1995],{},[35,1994,1982],{"mathvariant":251},[35,1996,102],{},[40,1998,1999],{"encoding":42},"\\mathbf{e}_1, \\dots, \\mathbf{e}_K",[15,2001,2003],{"className":2002,"ariaHidden":48},[47],[15,2004,2006,2009,2049,2052,2055,2058,2061,2064,2067],{"className":2005},[52],[15,2007],{"className":2008,"style":829},[56],[15,2010,2012,2015],{"className":2011},[61],[15,2013,1982],{"className":2014},[61,411],[15,2016,2018],{"className":2017},[338],[15,2019,2021,2041],{"className":2020},[342,472],[15,2022,2024,2038],{"className":2023},[346],[15,2025,2027],{"className":2026,"style":1214},[350],[15,2028,2029,2032],{"style":482},[15,2030],{"className":2031,"style":358},[357],[15,2033,2035],{"className":2034},[362,363,364,365],[15,2036,91],{"className":2037},[61,365],[15,2039,497],{"className":2040},[496],[15,2042,2044],{"className":2043},[346],[15,2045,2047],{"className":2046,"style":504},[350],[15,2048],{},[15,2050,94],{"className":2051},[153],[15,2053],{"className":2054,"style":157},[127],[15,2056,97],{"className":2057},[161],[15,2059],{"className":2060,"style":157},[127],[15,2062,94],{"className":2063},[153],[15,2065],{"className":2066,"style":157},[127],[15,2068,2070,2073],{"className":2069},[61],[15,2071,1982],{"className":2072},[61,411],[15,2074,2076],{"className":2075},[338],[15,2077,2079,2099],{"className":2078},[342,472],[15,2080,2082,2096],{"className":2081},[346],[15,2083,2085],{"className":2084,"style":1273},[350],[15,2086,2087,2090],{"style":482},[15,2088],{"className":2089,"style":358},[357],[15,2091,2093],{"className":2092},[362,363,364,365],[15,2094,102],{"className":2095,"style":174},[61,122,365],[15,2097,497],{"className":2098},[496],[15,2100,2102],{"className":2101},[346],[15,2103,2105],{"className":2104,"style":504},[350],[15,2106],{},"。低维特例给出了全部几何直观：",[15,2109,2111,2129],{"className":2110,"translate":19},[18],[15,2112,2114],{"className":2113},[23],[25,2115,2116],{"xmlns":27},[29,2117,2118,2126],{},[32,2119,2120,2122,2124],{},[35,2121,102],{},[81,2123,244],{},[89,2125,908],{},[40,2127,2128],{"encoding":42},"K = 2",[15,2130,2132,2150],{"className":2131,"ariaHidden":48},[47],[15,2133,2135,2138,2141,2144,2147],{"className":2134},[52],[15,2136],{"className":2137,"style":57},[56],[15,2139,102],{"className":2140,"style":174},[61,122],[15,2142],{"className":2143,"style":128},[127],[15,2145,244],{"className":2146},[132],[15,2148],{"className":2149,"style":128},[127],[15,2151,2153,2156],{"className":2152},[52],[15,2154],{"className":2155,"style":859},[56],[15,2157,908],{"className":2158},[61]," 时它是一维线段，",[15,2161,2163,2182],{"className":2162,"translate":19},[18],[15,2164,2166],{"className":2165},[23],[25,2167,2168],{"xmlns":27},[29,2169,2170,2179],{},[32,2171,2172,2174,2176],{},[35,2173,102],{},[81,2175,244],{},[89,2177,2178],{},"3",[40,2180,2181],{"encoding":42},"K = 3",[15,2183,2185,2203],{"className":2184,"ariaHidden":48},[47],[15,2186,2188,2191,2194,2197,2200],{"className":2187},[52],[15,2189],{"className":2190,"style":57},[56],[15,2192,102],{"className":2193,"style":174},[61,122],[15,2195],{"className":2196,"style":128},[127],[15,2198,244],{"className":2199},[132],[15,2201],{"className":2202,"style":128},[127],[15,2204,2206,2209],{"className":2205},[52],[15,2207],{"className":2208,"style":859},[56],[15,2210,2178],{"className":2211},[61]," 时是二维三角形，",[15,2214,2216,2235],{"className":2215,"translate":19},[18],[15,2217,2219],{"className":2218},[23],[25,2220,2221],{"xmlns":27},[29,2222,2223,2232],{},[32,2224,2225,2227,2229],{},[35,2226,102],{},[81,2228,244],{},[89,2230,2231],{},"4",[40,2233,2234],{"encoding":42},"K = 4",[15,2236,2238,2256],{"className":2237,"ariaHidden":48},[47],[15,2239,2241,2244,2247,2250,2253],{"className":2240},[52],[15,2242],{"className":2243,"style":57},[56],[15,2245,102],{"className":2246,"style":174},[61,122],[15,2248],{"className":2249,"style":128},[127],[15,2251,244],{"className":2252},[132],[15,2254],{"className":2255,"style":128},[127],[15,2257,2259,2262],{"className":2258},[52],[15,2260],{"className":2261,"style":859},[56],[15,2263,2231],{"className":2264},[61]," 时是三维四面体。它与 ",[15,2267,2269,2282],{"className":2268,"translate":19},[18],[15,2270,2272],{"className":2271},[23],[25,2273,2274],{"xmlns":27},[29,2275,2276,2280],{},[32,2277,2278],{},[35,2279,102],{},[40,2281,102],{"encoding":42},[15,2283,2285],{"className":2284,"ariaHidden":48},[47],[15,2286,2288,2291],{"className":2287},[52],[15,2289],{"className":2290,"style":57},[56],[15,2292,102],{"className":2293,"style":174},[61,122]," 个取值上的离散概率分布全体一一对应，因此「分类器的输出」与「单纯形上的点」是同一件事的两种说法。",[11,2296,2297,2298,2455,2456,2681,2682,2712,2713,2773,2774,2929,2930,3003,3004,3078],{},"本文采用如下测度约定：以 ",[15,2299,2301,2337],{"className":2300,"translate":19},[18],[15,2302,2304],{"className":2303},[23],[25,2305,2306],{"xmlns":27},[29,2307,2308,2334],{},[32,2309,2310,2316,2318,2320,2322],{},[266,2311,2312,2314],{},[35,2313,1029],{},[89,2315,91],{},[81,2317,94],{"separator":48},[81,2319,97],{},[81,2321,94],{"separator":48},[266,2323,2324,2326],{},[35,2325,1029],{},[32,2327,2328,2330,2332],{},[35,2329,102],{},[81,2331,239],{},[89,2333,91],{},[40,2335,2336],{"encoding":42},"x_1, \\dots, x_{K-1}",[15,2338,2340],{"className":2339,"ariaHidden":48},[47],[15,2341,2343,2347,2387,2390,2393,2396,2399,2402,2405],{"className":2342},[52],[15,2344],{"className":2345,"style":2346},[56],"height:0.6389em;vertical-align:-0.2083em;",[15,2348,2350,2353],{"className":2349},[61],[15,2351,1029],{"className":2352},[61,122],[15,2354,2356],{"className":2355},[338],[15,2357,2359,2379],{"className":2358},[342,472],[15,2360,2362,2376],{"className":2361},[346],[15,2363,2365],{"className":2364,"style":1214},[350],[15,2366,2367,2370],{"style":482},[15,2368],{"className":2369,"style":358},[357],[15,2371,2373],{"className":2372},[362,363,364,365],[15,2374,91],{"className":2375},[61,365],[15,2377,497],{"className":2378},[496],[15,2380,2382],{"className":2381},[346],[15,2383,2385],{"className":2384,"style":504},[350],[15,2386],{},[15,2388,94],{"className":2389},[153],[15,2391],{"className":2392,"style":157},[127],[15,2394,97],{"className":2395},[161],[15,2397],{"className":2398,"style":157},[127],[15,2400,94],{"className":2401},[153],[15,2403],{"className":2404,"style":157},[127],[15,2406,2408,2411],{"className":2407},[61],[15,2409,1029],{"className":2410},[61,122],[15,2412,2414],{"className":2413},[338],[15,2415,2417,2446],{"className":2416},[342,472],[15,2418,2420,2443],{"className":2419},[346],[15,2421,2423],{"className":2422,"style":1273},[350],[15,2424,2425,2428],{"style":482},[15,2426],{"className":2427,"style":358},[357],[15,2429,2431],{"className":2430},[362,363,364,365],[15,2432,2434,2437,2440],{"className":2433},[61,365],[15,2435,102],{"className":2436,"style":174},[61,122,365],[15,2438,239],{"className":2439},[375,365],[15,2441,91],{"className":2442},[61,365],[15,2444,497],{"className":2445},[496],[15,2447,2449],{"className":2448},[346],[15,2450,2453],{"className":2451,"style":2452},[350],"height:0.2083em;",[15,2454],{}," 为自由坐标、",[15,2457,2459,2502],{"className":2458,"translate":19},[18],[15,2460,2462],{"className":2461},[23],[25,2463,2464],{"xmlns":27},[29,2465,2466,2499],{},[32,2467,2468,2474,2476,2478,2480,2493],{},[266,2469,2470,2472],{},[35,2471,1029],{},[35,2473,102],{},[81,2475,244],{},[89,2477,91],{},[81,2479,239],{},[266,2481,2482,2484],{},[81,2483,293],{},[32,2485,2486,2488,2491],{},[35,2487,1070],{},[81,2489,2490],{},"\u003C",[35,2492,102],{},[266,2494,2495,2497],{},[35,2496,1029],{},[35,2498,1070],{},[40,2500,2501],{"encoding":42},"x_K = 1 - \\sum_{i\u003CK} x_i",[15,2503,2505,2561,2580],{"className":2504,"ariaHidden":48},[47],[15,2506,2508,2512,2552,2555,2558],{"className":2507},[52],[15,2509],{"className":2510,"style":2511},[56],"height:0.5806em;vertical-align:-0.15em;",[15,2513,2515,2518],{"className":2514},[61],[15,2516,1029],{"className":2517},[61,122],[15,2519,2521],{"className":2520},[338],[15,2522,2524,2544],{"className":2523},[342,472],[15,2525,2527,2541],{"className":2526},[346],[15,2528,2530],{"className":2529,"style":1273},[350],[15,2531,2532,2535],{"style":482},[15,2533],{"className":2534,"style":358},[357],[15,2536,2538],{"className":2537},[362,363,364,365],[15,2539,102],{"className":2540,"style":174},[61,122,365],[15,2542,497],{"className":2543},[496],[15,2545,2547],{"className":2546},[346],[15,2548,2550],{"className":2549,"style":504},[350],[15,2551],{},[15,2553],{"className":2554,"style":128},[127],[15,2556,244],{"className":2557},[132],[15,2559],{"className":2560,"style":128},[127],[15,2562,2564,2568,2571,2574,2577],{"className":2563},[52],[15,2565],{"className":2566,"style":2567},[56],"height:0.7278em;vertical-align:-0.0833em;",[15,2569,91],{"className":2570},[61],[15,2572],{"className":2573,"style":772},[127],[15,2575,239],{"className":2576},[375],[15,2578],{"className":2579,"style":772},[127],[15,2581,2583,2587,2638,2641],{"className":2582},[52],[15,2584],{"className":2585,"style":2586},[56],"height:1.0771em;vertical-align:-0.3271em;",[15,2588,2590,2593],{"className":2589},[531],[15,2591,293],{"className":2592,"style":537},[531,535,536],[15,2594,2596],{"className":2595},[338],[15,2597,2599,2629],{"className":2598},[342,472],[15,2600,2602,2626],{"className":2601},[346],[15,2603,2606],{"className":2604,"style":2605},[350],"height:0.1786em;",[15,2607,2608,2611],{"style":553},[15,2609],{"className":2610,"style":358},[357],[15,2612,2614],{"className":2613},[362,363,364,365],[15,2615,2617,2620,2623],{"className":2616},[61,365],[15,2618,1070],{"className":2619},[61,122,365],[15,2621,2490],{"className":2622},[132,365],[15,2624,102],{"className":2625,"style":174},[61,122,365],[15,2627,497],{"className":2628},[496],[15,2630,2632],{"className":2631},[346],[15,2633,2636],{"className":2634,"style":2635},[350],"height:0.3271em;",[15,2637],{},[15,2639],{"className":2640,"style":157},[127],[15,2642,2644,2647],{"className":2643},[61],[15,2645,1029],{"className":2646},[61,122],[15,2648,2650],{"className":2649},[338],[15,2651,2653,2673],{"className":2652},[342,472],[15,2654,2656,2670],{"className":2655},[346],[15,2657,2659],{"className":2658,"style":1364},[350],[15,2660,2661,2664],{"style":482},[15,2662],{"className":2663,"style":358},[357],[15,2665,2667],{"className":2666},[362,363,364,365],[15,2668,1070],{"className":2669},[61,122,365],[15,2671,497],{"className":2672},[496],[15,2674,2676],{"className":2675},[346],[15,2677,2679],{"className":2678,"style":504},[350],[15,2680],{},"，并以 ",[15,2683,2685,2699],{"className":2684,"translate":19},[18],[15,2686,2688],{"className":2687},[23],[25,2689,2690],{"xmlns":27},[29,2691,2692,2696],{},[32,2693,2694],{},[35,2695,1029],{"mathvariant":251},[40,2697,2698],{"encoding":42},"\\mathbf{x}",[15,2700,2702],{"className":2701,"ariaHidden":48},[47],[15,2703,2705,2709],{"className":2704},[52],[15,2706],{"className":2707,"style":2708},[56],"height:0.4444em;",[15,2710,1029],{"className":2711},[61,411]," 上的 ",[15,2714,2716,2737],{"className":2715,"translate":19},[18],[15,2717,2719],{"className":2718},[23],[25,2720,2721],{"xmlns":27},[29,2722,2723,2735],{},[32,2724,2725,2727,2729,2731,2733],{},[81,2726,741],{"stretchy":86},[35,2728,102],{},[81,2730,239],{},[89,2732,91],{},[81,2734,750],{"stretchy":86},[40,2736,753],{"encoding":42},[15,2738,2740,2761],{"className":2739,"ariaHidden":48},[47],[15,2741,2743,2746,2749,2752,2755,2758],{"className":2742},[52],[15,2744],{"className":2745,"style":142},[56],[15,2747,741],{"className":2748},[146],[15,2750,102],{"className":2751,"style":174},[61,122],[15,2753],{"className":2754,"style":772},[127],[15,2756,239],{"className":2757},[375],[15,2759],{"className":2760,"style":772},[127],[15,2762,2764,2767,2770],{"className":2763},[52],[15,2765],{"className":2766,"style":142},[56],[15,2768,91],{"className":2769},[61],[15,2771,750],{"className":2772},[178]," 维 Lebesgue 测度 ",[15,2775,2777,2815],{"className":2776,"translate":19},[18],[15,2778,2780],{"className":2779},[23],[25,2781,2782],{"xmlns":27},[29,2783,2784,2812],{},[32,2785,2786,2789,2795,2798,2800],{},[35,2787,2788],{"mathvariant":231},"d",[266,2790,2791,2793],{},[35,2792,1029],{},[89,2794,91],{},[81,2796,2797],{},"⋯",[35,2799,2788],{"mathvariant":231},[266,2801,2802,2804],{},[35,2803,1029],{},[32,2805,2806,2808,2810],{},[35,2807,102],{},[81,2809,239],{},[89,2811,91],{},[40,2813,2814],{"encoding":42},"\\mathrm{d}x_1 \\cdots \\mathrm{d}x_{K-1}",[15,2816,2818],{"className":2817,"ariaHidden":48},[47],[15,2819,2821,2825,2828,2868,2871,2874,2877,2880],{"className":2820},[52],[15,2822],{"className":2823,"style":2824},[56],"height:0.9028em;vertical-align:-0.2083em;",[15,2826,2788],{"className":2827},[61,1622],[15,2829,2831,2834],{"className":2830},[61],[15,2832,1029],{"className":2833},[61,122],[15,2835,2837],{"className":2836},[338],[15,2838,2840,2860],{"className":2839},[342,472],[15,2841,2843,2857],{"className":2842},[346],[15,2844,2846],{"className":2845,"style":1214},[350],[15,2847,2848,2851],{"style":482},[15,2849],{"className":2850,"style":358},[357],[15,2852,2854],{"className":2853},[362,363,364,365],[15,2855,91],{"className":2856},[61,365],[15,2858,497],{"className":2859},[496],[15,2861,2863],{"className":2862},[346],[15,2864,2866],{"className":2865,"style":504},[350],[15,2867],{},[15,2869],{"className":2870,"style":157},[127],[15,2872,2797],{"className":2873},[161],[15,2875],{"className":2876,"style":157},[127],[15,2878,2788],{"className":2879},[61,1622],[15,2881,2883,2886],{"className":2882},[61],[15,2884,1029],{"className":2885},[61,122],[15,2887,2889],{"className":2888},[338],[15,2890,2892,2921],{"className":2891},[342,472],[15,2893,2895,2918],{"className":2894},[346],[15,2896,2898],{"className":2897,"style":1273},[350],[15,2899,2900,2903],{"style":482},[15,2901],{"className":2902,"style":358},[357],[15,2904,2906],{"className":2905},[362,363,364,365],[15,2907,2909,2912,2915],{"className":2908},[61,365],[15,2910,102],{"className":2911,"style":174},[61,122,365],[15,2913,239],{"className":2914},[375,365],[15,2916,91],{"className":2917},[61,365],[15,2919,497],{"className":2920},[496],[15,2922,2924],{"className":2923},[346],[15,2925,2927],{"className":2926,"style":2452},[350],[15,2928],{}," 为参考测度。在该约定下 ",[15,2931,2933,2956],{"className":2932,"translate":19},[18],[15,2934,2936],{"className":2935},[23],[25,2937,2938],{"xmlns":27},[29,2939,2940,2954],{},[32,2941,2942],{},[227,2943,2944,2946],{},[35,2945,232],{"mathvariant":231},[32,2947,2948,2950,2952],{},[35,2949,102],{},[81,2951,239],{},[89,2953,91],{},[40,2955,1858],{"encoding":42},[15,2957,2959],{"className":2958,"ariaHidden":48},[47],[15,2960,2962,2965],{"className":2961},[52],[15,2963],{"className":2964,"style":695},[56],[15,2966,2968,2971],{"className":2967},[61],[15,2969,232],{"className":2970},[61],[15,2972,2974],{"className":2973},[338],[15,2975,2977],{"className":2976},[342],[15,2978,2980],{"className":2979},[346],[15,2981,2983],{"className":2982,"style":695},[350],[15,2984,2985,2988],{"style":716},[15,2986],{"className":2987,"style":358},[357],[15,2989,2991],{"className":2990},[362,363,364,365],[15,2992,2994,2997,3000],{"className":2993},[61,365],[15,2995,102],{"className":2996,"style":174},[61,122,365],[15,2998,239],{"className":2999},[375,365],[15,3001,91],{"className":3002},[61,365]," 的体积为 ",[15,3005,3007,3037],{"className":3006,"translate":19},[18],[15,3008,3010],{"className":3009},[23],[25,3011,3012],{"xmlns":27},[29,3013,3014,3034],{},[32,3015,3016,3018,3021,3023,3025,3027,3029,3031],{},[89,3017,91],{},[35,3019,3020],{"mathvariant":231},"\u002F",[81,3022,741],{"stretchy":86},[35,3024,102],{},[81,3026,239],{},[89,3028,91],{},[81,3030,750],{"stretchy":86},[81,3032,3033],{"stretchy":86},"!",[40,3035,3036],{"encoding":42},"1\u002F(K-1)!",[15,3038,3040,3065],{"className":3039,"ariaHidden":48},[47],[15,3041,3043,3046,3050,3053,3056,3059,3062],{"className":3042},[52],[15,3044],{"className":3045,"style":142},[56],[15,3047,3049],{"className":3048},[61],"1\u002F",[15,3051,741],{"className":3052},[146],[15,3054,102],{"className":3055,"style":174},[61,122],[15,3057],{"className":3058,"style":772},[127],[15,3060,239],{"className":3061},[375],[15,3063],{"className":3064,"style":772},[127],[15,3066,3068,3071,3074],{"className":3067},[52],[15,3069],{"className":3070,"style":142},[56],[15,3072,91],{"className":3073},[61],[15,3075,3077],{"className":3076},[178],")!","，这一点在后文讨论熵时会重新出现。",[877,3080,3082],{"id":3081},"_12-形式化定义","1.2 形式化定义",[882,3084,3085],{},[11,3086,3087,3090,3091,3363,3364,3575,3576,3604,3605,3692,3693,4101,4102,4130,4131,4166,4167,4246,4247,4250],{},[661,3088,3089],{},"定义 1.2（Dirichlet 分布）"," 设参数向量 ",[15,3092,3094,3148],{"className":3093,"translate":19},[18],[15,3095,3097],{"className":3096},[23],[25,3098,3099],{"xmlns":27},[29,3100,3101,3145],{},[32,3102,3103,3107,3109,3111,3117,3119,3121,3123,3129,3131,3133],{},[35,3104,3106],{"mathvariant":3105},"bold-italic","α",[81,3108,244],{},[81,3110,741],{"stretchy":86},[266,3112,3113,3115],{},[35,3114,3106],{},[89,3116,91],{},[81,3118,94],{"separator":48},[81,3120,97],{},[81,3122,94],{"separator":48},[266,3124,3125,3127],{},[35,3126,3106],{},[35,3128,102],{},[81,3130,750],{"stretchy":86},[81,3132,83],{},[289,3134,3135,3137,3143],{},[35,3136,259],{"mathvariant":258},[32,3138,3139,3141],{},[81,3140,1590],{},[89,3142,278],{},[35,3144,102],{},[40,3146,3147],{"encoding":42},"\\boldsymbol{\\alpha} = (\\alpha_1, \\dots, \\alpha_K) \\in 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",[15,3365,3367,3407],{"className":3366,"translate":19},[18],[15,3368,3370],{"className":3369},[23],[25,3371,3372],{"xmlns":27},[29,3373,3374,3404],{},[32,3375,3376,3382,3384,3398],{},[266,3377,3378,3380],{},[35,3379,3106],{},[89,3381,278],{},[81,3383,244],{},[289,3385,3386,3388,3396],{},[81,3387,293],{},[32,3389,3390,3392,3394],{},[35,3391,1070],{},[81,3393,244],{},[89,3395,91],{},[35,3397,102],{},[266,3399,3400,3402],{},[35,3401,3106],{},[35,3403,1070],{},[40,3405,3406],{"encoding":42},"\\alpha_0 = \\sum_{i=1}^K \\alpha_i",[15,3408,3410,3465],{"className":3409,"ariaHidden":48},[47],[15,3411,3413,3416,3456,3459,3462],{"className":3412},[52],[15,3414],{"className":3415,"style":2511},[56],[15,3417,3419,3422],{"className":3418},[61],[15,3420,3106],{"className":3421,"style":3191},[61,122],[15,3423,3425],{"className":3424},[338],[15,3426,3428,3448],{"className":3427},[342,472],[15,3429,3431,3445],{"className":3430},[346],[15,3432,3434],{"className":3433,"style":1214},[350],[15,3435,3436,3439],{"style":3206},[15,3437],{"className":3438,"style":358},[357],[15,3440,3442],{"className":3441},[362,363,364,365],[15,3443,278],{"className":3444},[61,365],[15,3446,497],{"className":3447},[496],[15,3449,3451],{"className":3450},[346],[15,3452,3454],{"className":3453,"style":504},[350],[15,3455],{},[15,3457],{"className":3458,"style":128},[127],[15,3460,244],{"className":3461},[132],[15,3463],{"className":3464,"style":128},[127],[15,3466,3468,3472,3532,3535],{"className":3467},[52],[15,3469],{"className":3470,"style":3471},[56],"height:1.2809em;vertical-align:-0.2997em;",[15,3473,3475,3478],{"className":3474},[531],[15,3476,293],{"className":3477,"style":537},[531,535,536],[15,3479,3481],{"className":3480},[338],[15,3482,3484,3524],{"className":3483},[342,472],[15,3485,3487,3521],{"className":3486},[346],[15,3488,3490,3510],{"className":3489,"style":550},[350],[15,3491,3492,3495],{"style":553},[15,3493],{"className":3494,"style":358},[357],[15,3496,3498],{"className":3497},[362,363,364,365],[15,3499,3501,3504,3507],{"className":3500},[61,365],[15,3502,1070],{"className":3503},[61,122,365],[15,3505,244],{"className":3506},[132,365],[15,3508,91],{"className":3509},[61,365],[15,3511,3512,3515],{"style":574},[15,3513],{"className":3514,"style":358},[357],[15,3516,3518],{"className":3517},[362,363,364,365],[15,3519,102],{"className":3520,"style":174},[61,122,365],[15,3522,497],{"className":3523},[496],[15,3525,3527],{"className":3526},[346],[15,3528,3530],{"className":3529,"style":593},[350],[15,3531],{},[15,3533],{"className":3534,"style":157},[127],[15,3536,3538,3541],{"className":3537},[61],[15,3539,3106],{"className":3540,"style":3191},[61,122],[15,3542,3544],{"className":3543},[338],[15,3545,3547,3567],{"className":3546},[342,472],[15,3548,3550,3564],{"className":3549},[346],[15,3551,3553],{"className":3552,"style":1364},[350],[15,3554,3555,3558],{"style":3206},[15,3556],{"className":3557,"style":358},[357],[15,3559,3561],{"className":3560},[362,363,364,365],[15,3562,1070],{"className":3563},[61,122,365],[15,3565,497],{"className":3566},[496],[15,3568,3570],{"className":3569},[346],[15,3571,3573],{"className":3572,"style":504},[350],[15,3574],{},"。若随机向量 ",[15,3577,3579,3592],{"className":3578,"translate":19},[18],[15,3580,3582],{"className":3581},[23],[25,3583,3584],{"xmlns":27},[29,3585,3586,3590],{},[32,3587,3588],{},[35,3589,1029],{"mathvariant":251},[40,3591,2698],{"encoding":42},[15,3593,3595],{"className":3594,"ariaHidden":48},[47],[15,3596,3598,3601],{"className":3597},[52],[15,3599],{"className":3600,"style":2708},[56],[15,3602,1029],{"className":3603},[61,411]," 取值于 ",[15,3606,3608,3636],{"className":3607,"translate":19},[18],[15,3609,3611],{"className":3610},[23],[25,3612,3613],{"xmlns":27},[29,3614,3615,3633],{},[32,3616,3617,3619,3621],{},[35,3618,1544],{"mathvariant":231},[81,3620,1547],{},[227,3622,3623,3625],{},[35,3624,232],{"mathvariant":231},[32,3626,3627,3629,3631],{},[35,3628,102],{},[81,3630,239],{},[89,3632,91],{},[40,3634,3635],{"encoding":42},"\\operatorname{int}\\Delta^{K-1}",[15,3637,3639],{"className":3638,"ariaHidden":48},[47],[15,3640,3642,3645,3651,3654],{"className":3641},[52],[15,3643],{"className":3644,"style":695},[56],[15,3646,3648],{"className":3647},[531],[15,3649,1544],{"className":3650},[61,1622],[15,3652],{"className":3653,"style":157},[127],[15,3655,3657,3660],{"className":3656},[61],[15,3658,232],{"className":3659},[61],[15,3661,3663],{"className":3662},[338],[15,3664,3666],{"className":3665},[342],[15,3667,3669],{"className":3668},[346],[15,3670,3672],{"className":3671,"style":695},[350],[15,3673,3674,3677],{"style":716},[15,3675],{"className":3676,"style":358},[357],[15,3678,3680],{"className":3679},[362,363,364,365],[15,3681,3683,3686,3689],{"className":3682},[61,365],[15,3684,102],{"className":3685,"style":174},[61,122,365],[15,3687,239],{"className":3688},[375,365],[15,3690,91],{"className":3691},[61,365],"，且关于上述参考测度的密度为\n",[15,3694,3696,3773],{"className":3695,"translate":19},[18],[15,3697,3699],{"className":3698},[23],[25,3700,3701],{"xmlns":27},[29,3702,3703,3770],{},[32,3704,3705,3708,3710,3712,3715,3717,3719,3721,3737,3752],{},[35,3706,3707],{},"f",[81,3709,741],{"stretchy":86},[35,3711,1029],{"mathvariant":251},[81,3713,3714],{},"∣",[35,3716,3106],{"mathvariant":3105},[81,3718,750],{"stretchy":86},[81,3720,244],{},[3722,3723,3724,3726],"mfrac",{},[89,3725,91],{},[32,3727,3728,3731,3733,3735],{},[35,3729,3730],{},"B",[81,3732,741],{"stretchy":86},[35,3734,3106],{"mathvariant":3105},[81,3736,750],{"stretchy":86},[289,3738,3739,3742,3750],{},[81,3740,3741],{},"∏",[32,3743,3744,3746,3748],{},[35,3745,1070],{},[81,3747,244],{},[89,3749,91],{},[35,3751,102],{},[289,3753,3754,3756,3758],{},[35,3755,1029],{},[35,3757,1070],{},[32,3759,3760,3766,3768],{},[266,3761,3762,3764],{},[35,3763,3106],{},[35,3765,1070],{},[81,3767,239],{},[89,3769,91],{},[40,3771,3772],{"encoding":42},"f(\\mathbf{x} \\mid \\boldsymbol{\\alpha}) = \\frac{1}{B(\\boldsymbol{\\alpha})} \\prod_{i=1}^K x_i^{\\alpha_i - 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,{"className":4079,"style":4080},[350],"height:0.143em;",[15,4082],{},[15,4084,239],{"className":4085},[375,365],[15,4087,91],{"className":4088},[61,365],[15,4090,497],{"className":4091},[496],[15,4093,4095],{"className":4094},[346],[15,4096,4099],{"className":4097,"style":4098},[350],"height:0.2769em;",[15,4100],{},"\n则称 ",[15,4103,4105,4118],{"className":4104,"translate":19},[18],[15,4106,4108],{"className":4107},[23],[25,4109,4110],{"xmlns":27},[29,4111,4112,4116],{},[32,4113,4114],{},[35,4115,1029],{"mathvariant":251},[40,4117,2698],{"encoding":42},[15,4119,4121],{"className":4120,"ariaHidden":48},[47],[15,4122,4124,4127],{"className":4123},[52],[15,4125],{"className":4126,"style":2708},[56],[15,4128,1029],{"className":4129},[61,411]," 服从参数为 ",[15,4132,4134,4148],{"className":4133,"translate":19},[18],[15,4135,4137],{"className":4136},[23],[25,4138,4139],{"xmlns":27},[29,4140,4141,4145],{},[32,4142,4143],{},[35,4144,3106],{"mathvariant":3105},[40,4146,4147],{"encoding":42},"\\boldsymbol{\\alpha}",[15,4149,4151],{"className":4150,"ariaHidden":48},[47],[15,4152,4154,4157],{"className":4153},[52],[15,4155],{"className":4156,"style":2708},[56],[15,4158,4160],{"className":4159},[61],[15,4161,4163],{"className":4162},[61],[15,4164,3106],{"className":4165},[61,3166]," 的迪利克雷分布，记作 ",[15,4168,4170,4198],{"className":4169,"translate":19},[18],[15,4171,4173],{"className":4172},[23],[25,4174,4175],{"xmlns":27},[29,4176,4177,4195],{},[32,4178,4179,4181,4184,4187,4189,4191,4193],{},[35,4180,1029],{"mathvariant":251},[81,4182,4183],{},"∼",[35,4185,4186],{"mathvariant":231},"Dir",[81,4188,1547],{},[81,4190,741],{"stretchy":86},[35,4192,3106],{"mathvariant":3105},[81,4194,750],{"stretchy":86},[40,4196,4197],{"encoding":42},"\\mathbf{x} \\sim \\operatorname{Dir}(\\boldsymbol{\\alpha})",[15,4199,4201,4219],{"className":4200,"ariaHidden":48},[47],[15,4202,4204,4207,4210,4213,4216],{"className":4203},[52],[15,4205],{"className":4206,"style":2708},[56],[15,4208,1029],{"className":4209},[61,411],[15,4211],{"className":4212,"style":128},[127],[15,4214,4183],{"className":4215},[132],[15,4217],{"className":4218,"style":128},[127],[15,4220,4222,4225,4231,4234,4243],{"className":4221},[52],[15,4223],{"className":4224,"style":142},[56],[15,4226,4228],{"className":4227},[531],[15,4229,4186],{"className":4230},[61,1622],[15,4232,741],{"className":4233},[146],[15,4235,4237],{"className":4236},[61],[15,4238,4240],{"className":4239},[61],[15,4241,3106],{"className":4242},[61,3166],[15,4244,750],{"className":4245},[178],"。其中归一化常数为",[661,4248,4249],{},"多元 Beta 函数",[15,4251,4253,4322],{"className":4252,"translate":19},[18],[15,4254,4256],{"className":4255},[23],[25,4257,4258],{"xmlns":27},[29,4259,4260,4319],{},[32,4261,4262,4264,4266,4268,4270,4272],{},[35,4263,3730],{},[81,4265,741],{"stretchy":86},[35,4267,3106],{"mathvariant":3105},[81,4269,750],{"stretchy":86},[81,4271,244],{},[3722,4273,4274,4303],{},[32,4275,4276,4290,4293,4295,4301],{},[289,4277,4278,4280,4288],{},[81,4279,3741],{},[32,4281,4282,4284,4286],{},[35,4283,1070],{},[81,4285,244],{},[89,4287,91],{},[35,4289,102],{},[35,4291,4292],{"mathvariant":231},"Γ",[81,4294,741],{"stretchy":86},[266,4296,4297,4299],{},[35,4298,3106],{},[35,4300,1070],{},[81,4302,750],{"stretchy":86},[32,4304,4305,4307],{},[35,4306,4292],{"mathvariant":231},[32,4308,4309,4311,4317],{},[81,4310,741],{"fence":48},[266,4312,4313,4315],{},[35,4314,3106],{},[89,4316,278],{},[81,4318,750],{"fence":48},[40,4320,4321],{"encoding":42},"B(\\boldsymbol{\\alpha}) = \\frac{\\prod_{i=1}^K 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1.3（归一化恒等式）"," 对任意 ",[15,4617,4619,4647],{"className":4618,"translate":19},[18],[15,4620,4622],{"className":4621},[23],[25,4623,4624],{"xmlns":27},[29,4625,4626,4644],{},[32,4627,4628,4630,4632],{},[35,4629,3106],{"mathvariant":3105},[81,4631,83],{},[289,4633,4634,4636,4642],{},[35,4635,259],{"mathvariant":258},[32,4637,4638,4640],{},[81,4639,1590],{},[89,4641,278],{},[35,4643,102],{},[40,4645,4646],{"encoding":42},"\\boldsymbol{\\alpha} \\in \\mathbb{R}_{>0}^K",[15,4648,4650,4675],{"className":4649,"ariaHidden":48},[47],[15,4651,4653,4657,4666,4669,4672],{"className":4652},[52],[15,4654],{"className":4655,"style":4656},[56],"height:0.5782em;vertical-align:-0.0391em;",[15,4658,4660],{"className":4659},[61],[15,4661,4663],{"className":4662},[61],[15,4664,3106],{"className":4665},[61,3166],[15,4667],{"className":4668,"style":128},[127],[15,4670,83],{"className":4671},[132],[15,4673],{"className":4674,"style":128},[127],[15,4676,4678,4681],{"className":4677},[52],[15,4679],{"className":4680,"style":3303},[56],[15,4682,4684,4687],{"className":4683},[61],[15,4685,259],{"className":4686},[61,427],[15,4688,4690],{"className":4689},[338],[15,4691,4693,4730],{"className":4692},[342,472],[15,4694,4696,4727],{"className":4695},[346],[15,4697,4699,4716],{"className":4698,"style":695},[350],[15,4700,4701,4704],{"style":3324},[15,4702],{"className":4703,"style":358},[357],[15,4705,4707],{"className":4706},[362,363,364,365],[15,4708,4710,4713],{"className":4709},[61,365],[15,4711,1590],{"className":4712},[132,365],[15,4714,278],{"className":4715},[61,365],[15,4717,4718,4721],{"style":716},[15,4719],{"className":4720,"style":358},[357],[15,4722,4724],{"className":4723},[362,363,364,365],[15,4725,102],{"className":4726,"style":174},[61,122,365],[15,4728,497],{"className":4729},[496],[15,4731,4733],{"className":4732},[346],[15,4734,4736],{"className":4735,"style":3360},[350],[15,4737],{},"，\n",[15,4740,4742,4820],{"className":4741,"translate":19},[18],[15,4743,4745],{"className":4744},[23],[25,4746,4747],{"xmlns":27},[29,4748,4749,4817],{},[32,4750,4751,4768,4782,4800,4803,4805,4807,4809,4811,4813,4815],{},[266,4752,4753,4756],{},[81,4754,4755],{},"∫",[227,4757,4758,4760],{},[35,4759,232],{"mathvariant":231},[32,4761,4762,4764,4766],{},[35,4763,102],{},[81,4765,239],{},[89,4767,91],{},[289,4769,4770,4772,4780],{},[81,4771,3741],{},[32,4773,4774,4776,4778],{},[35,4775,1070],{},[81,4777,244],{},[89,4779,91],{},[35,4781,102],{},[289,4783,4784,4786,4788],{},[35,4785,1029],{},[35,4787,1070],{},[32,4789,4790,4796,4798],{},[266,4791,4792,4794],{},[35,4793,3106],{},[35,4795,1070],{},[81,4797,239],{},[89,4799,91],{},[282,4801,4802],{}," ",[35,4804,2788],{"mathvariant":231},[35,4806,1029],{"mathvariant":251},[81,4808,244],{},[35,4810,3730],{},[81,4812,741],{"stretchy":86},[35,4814,3106],{"mathvariant":3105},[81,4816,750],{"stretchy":86},[40,4818,4819],{"encoding":42},"\\int_{\\Delta^{K-1}} \\prod_{i=1}^K x_i^{\\alpha_i - 1} \\,\\mathrm{d}\\mathbf{x} = B(\\boldsymbol{\\alpha})",[15,4821,4823,5095],{"className":4822,"ariaHidden":48},[47],[15,4824,4826,4830,4914,4917,4977,4980,5077,5080,5083,5086,5089,5092],{"className":4825},[52],[15,4827],{"className":4828,"style":4829},[56],"height:1.3371em;vertical-align:-0.3558em;",[15,4831,4833,4837],{"className":4832},[531],[15,4834,4755],{"className":4835,"style":4836},[531,535,536],"margin-right:0.1945em;position:relative;top:-0.0006em;",[15,4838,4840],{"className":4839},[338],[15,4841,4843,4905],{"className":4842},[342,472],[15,4844,4846,4902],{"className":4845},[346],[15,4847,4850],{"className":4848,"style":4849},[350],"height:0.186em;",[15,4851,4853,4856],{"style":4852},"top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;",[15,4854],{"className":4855,"style":358},[357],[15,4857,4859],{"className":4858},[362,363,364,365],[15,4860,4862],{"className":4861},[61,365],[15,4863,4865,4868],{"className":4864},[61,365],[15,4866,232],{"className":4867},[61,365],[15,4869,4871],{"className":4870},[338],[15,4872,4874],{"className":4873},[342],[15,4875,4877],{"className":4876},[346],[15,4878,4881],{"className":4879,"style":4880},[350],"height:0.7741em;",[15,4882,4884,4887],{"style":4883},"top:-2.786em;margin-right:0.0714em;",[15,4885],{"className":4886,"style":4062},[357],[15,4888,4890],{"className":4889},[362,4066,4067,365],[15,4891,4893,4896,4899],{"className":4892},[61,365],[15,4894,102],{"className":4895,"style":174},[61,122,365],[15,4897,239],{"className":4898},[375,365],[15,4900,91],{"className":4901},[61,365],[15,4903,497],{"className":4904},[496],[15,4906,4908],{"className":4907},[346],[15,4909,4912],{"className":4910,"style":4911},[350],"height:0.3558em;",[15,4913],{},[15,4915],{"className":4916,"style":157},[127],[15,4918,4920,4923],{"className":4919},[531],[15,4921,3741],{"className":4922,"style":537},[531,535,536],[15,4924,4926],{"className":4925},[338],[15,4927,4929,4969],{"className":4928},[342,472],[15,4930,4932,4966],{"className":4931},[346],[15,4933,4935,4955],{"className":4934,"style":550},[350],[15,4936,4937,4940],{"style":553},[15,4938],{"className":4939,"style":358},[357],[15,4941,4943],{"className":4942},[362,363,364,365],[15,4944,4946,4949,4952],{"className":4945},[61,365],[15,4947,1070],{"className":4948},[61,122,365],[15,4950,244],{"className":4951},[132,365],[15,4953,91],{"className":4954},[61,365],[15,4956,4957,4960],{"style":574},[15,4958],{"className":4959,"style":358},[357],[15,4961,4963],{"className":4962},[362,363,364,365],[15,4964,102],{"className":4965,"style":174},[61,122,365],[15,4967,497],{"className":4968},[496],[15,4970,4972],{"className":4971},[346],[15,4973,4975],{"className":4974,"style":593},[350],[15,4976],{},[15,4978],{"className":4979,"style":157},[127],[15,4981,4983,4986],{"className":4982},[61],[15,4984,1029],{"className":4985},[61,122],[15,4987,4989],{"className":4988},[338],[15,4990,4992,5069],{"className":4991},[342,472],[15,4993,4995,5066],{"className":4994},[346],[15,4996,4998,5009],{"className":4997,"style":4012},[350],[15,4999,5000,5003],{"style":4015},[15,5001],{"className":5002,"style":358},[357],[15,5004,5006],{"className":5005},[362,363,364,365],[15,5007,1070],{"className":5008},[61,122,365],[15,5010,5011,5014],{"style":4027},[15,5012],{"className":5013,"style":358},[357],[15,5015,5017],{"className":5016},[362,363,364,365],[15,5018,5020,5060,5063],{"className":5019},[61,365],[15,5021,5023,5026],{"className":5022},[61,365],[15,5024,3106],{"className":5025,"style":3191},[61,122,365],[15,5027,5029],{"className":5028},[338],[15,5030,5032,5052],{"className":5031},[342,472],[15,5033,5035,5049],{"className":5034},[346],[15,5036,5038],{"className":5037,"style":4055},[350],[15,5039,5040,5043],{"style":4058},[15,5041],{"className":5042,"style":4062},[357],[15,5044,5046],{"className":5045},[362,4066,4067,365],[15,5047,1070],{"className":5048},[61,122,365],[15,5050,497],{"className":5051},[496],[15,5053,5055],{"className":5054},[346],[15,5056,5058],{"className":5057,"style":4080},[350],[15,5059],{},[15,5061,239],{"className":5062},[375,365],[15,5064,91],{"className":5065},[61,365],[15,5067,497],{"className":5068},[496],[15,5070,5072],{"className":5071},[346],[15,5073,5075],{"className":5074,"style":4098},[350],[15,5076],{},[15,5078],{"className":5079,"style":157},[127],[15,5081,2788],{"className":5082},[61,1622],[15,5084,1029],{"className":5085},[61,411],[15,5087],{"className":5088,"style":128},[127],[15,5090,244],{"className":5091},[132],[15,5093],{"className":5094,"style":128},[127],[15,5096,5098,5101,5104,5107,5116],{"className":5097},[52],[15,5099],{"className":5100,"style":142},[56],[15,5102,3730],{"className":5103,"style":3871},[61,122],[15,5105,741],{"className":5106},[146],[15,5108,5110],{"className":5109},[61],[15,5111,5113],{"className":5112},[61],[15,5114,3106],{"className":5115},[61,3166],[15,5117,750],{"className":5118},[178],[11,5120,5121,5124,5125,5218],{},[661,5122,5123],{},"证明"," 考虑 ",[15,5126,5128,5152],{"className":5127,"translate":19},[18],[15,5129,5131],{"className":5130},[23],[25,5132,5133],{"xmlns":27},[29,5134,5135,5149],{},[32,5136,5137],{},[289,5138,5139,5141,5147],{},[35,5140,259],{"mathvariant":258},[32,5142,5143,5145],{},[81,5144,1590],{},[89,5146,278],{},[35,5148,102],{},[40,5150,5151],{"encoding":42},"\\mathbb{R}_{>0}^K",[15,5153,5155],{"className":5154,"ariaHidden":48},[47],[15,5156,5158,5161],{"className":5157},[52],[15,5159],{"className":5160,"style":3303},[56],[15,5162,5164,5167],{"className":5163},[61],[15,5165,259],{"className":5166},[61,427],[15,5168,5170],{"className":5169},[338],[15,5171,5173,5210],{"className":5172},[342,472],[15,5174,5176,5207],{"className":5175},[346],[15,5177,5179,5196],{"className":5178,"style":695},[350],[15,5180,5181,5184],{"style":3324},[15,5182],{"className":5183,"style":358},[357],[15,5185,5187],{"className":5186},[362,363,364,365],[15,5188,5190,5193],{"className":5189},[61,365],[15,5191,1590],{"className":5192},[132,365],[15,5194,278],{"className":5195},[61,365],[15,5197,5198,5201],{"style":716},[15,5199],{"className":5200,"style":358},[357],[15,5202,5204],{"className":5203},[362,363,364,365],[15,5205,102],{"className":5206,"style":174},[61,122,365],[15,5208,497],{"className":5209},[496],[15,5211,5213],{"className":5212},[346],[15,5214,5216],{"className":5215,"style":3360},[350],[15,5217],{}," 上的 Gamma 积分之积：",[15,5220,5222],{"className":5221,"translate":19},[212],[15,5223,5225,5335],{"className":5224,"translate":19},[18],[15,5226,5228],{"className":5227},[23],[25,5229,5230],{"xmlns":27,"display":221},[29,5231,5232,5332],{},[32,5233,5234,5249,5251,5253,5259,5261,5263,5279,5293,5312,5326,5328,5330],{},[5235,5236,5237,5239,5247],"munderover",{},[81,5238,3741],{},[32,5240,5241,5243,5245],{},[35,5242,1070],{},[81,5244,244],{},[89,5246,91],{},[35,5248,102],{},[35,5250,4292],{"mathvariant":231},[81,5252,741],{"stretchy":86},[266,5254,5255,5257],{},[35,5256,3106],{},[35,5258,1070],{},[81,5260,750],{"stretchy":86},[81,5262,244],{},[266,5264,5265,5267],{},[81,5266,4755],{},[289,5268,5269,5271,5277],{},[35,5270,259],{"mathvariant":258},[32,5272,5273,5275],{},[81,5274,1590],{},[89,5276,278],{},[35,5278,102],{},[5235,5280,5281,5283,5291],{},[81,5282,3741],{},[32,5284,5285,5287,5289],{},[35,5286,1070],{},[81,5288,244],{},[89,5290,91],{},[35,5292,102],{},[289,5294,5295,5298,5300],{},[35,5296,5297],{},"t",[35,5299,1070],{},[32,5301,5302,5308,5310],{},[266,5303,5304,5306],{},[35,5305,3106],{},[35,5307,1070],{},[81,5309,239],{},[89,5311,91],{},[227,5313,5314,5316],{},[35,5315,1982],{},[32,5317,5318,5320],{},[81,5319,239],{},[266,5321,5322,5324],{},[35,5323,5297],{},[35,5325,1070],{},[282,5327,4802],{},[35,5329,2788],{"mathvariant":231},[35,5331,5297],{"mathvariant":251},[40,5333,5334],{"encoding":42},"\\prod_{i=1}^K \\Gamma(\\alpha_i) = \\int_{\\mathbb{R}_{>0}^K} \\prod_{i=1}^K t_i^{\\alpha_i - 1} e^{-t_i} 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解耦：",[15,8083,8085],{"className":8084,"translate":19},[212],[15,8086,8088,8305],{"className":8087,"translate":19},[18],[15,8089,8091],{"className":8090},[23],[25,8092,8093],{"xmlns":27,"display":221},[29,8094,8095,8302],{},[32,8096,8097,8111,8113,8115,8121,8123,8125,8171,8231,8233,8236,8238,8240,8246,8248,8264,8278,8296,8298,8300],{},[5235,8098,8099,8101,8109],{},[81,8100,3741],{},[32,8102,8103,8105,8107],{},[35,8104,1070],{},[81,8106,244],{},[89,8108,91],{},[35,8110,102],{},[35,8112,4292],{"mathvariant":231},[81,8114,741],{"stretchy":86},[266,8116,8117,8119],{},[35,8118,3106],{},[35,8120,1070],{},[81,8122,750],{"stretchy":86},[81,8124,244],{},[32,8126,8127,8129,8137,8153,8163,8165,8167,8169],{},[81,8128,741],{"fence":48},[289,8130,8131,8133,8135],{},[81,8132,4755],{},[89,8134,278],{},[35,8136,6796],{"mathvariant":231},[227,8138,8139,8141],{},[35,8140,5864],{},[32,8142,8143,8149,8151],{},[266,8144,8145,8147],{},[35,8146,3106],{},[89,8148,278],{},[81,8150,239],{},[89,8152,91],{},[227,8154,8155,8157],{},[35,8156,1982],{},[32,8158,8159,8161],{},[81,8160,239],{},[35,8162,5864],{},[282,8164,4802],{},[35,8166,2788],{"mathvariant":231},[35,8168,5864],{},[81,8170,750],{"fence":48},[32,8172,8173,8175,8191,8205,8223,8225,8227,8229],{},[81,8174,741],{"fence":48},[266,8176,8177,8179],{},[81,8178,4755],{},[227,8180,8181,8183],{},[35,8182,232],{"mathvariant":231},[32,8184,8185,8187,8189],{},[35,8186,102],{},[81,8188,239],{},[89,8190,91],{},[5235,8192,8193,8195,8203],{},[81,8194,3741],{},[32,8196,8197,8199,8201],{},[35,8198,1070],{},[81,8200,244],{},[89,8202,91],{},[35,8204,102],{},[289,8206,8207,8209,8211],{},[35,8208,1029],{},[35,8210,1070],{},[32,8212,8213,8219,8221],{},[266,8214,8215,8217],{},[35,8216,3106],{},[35,8218,1070],{},[81,8220,239],{},[89,8222,91],{},[282,8224,4802],{},[35,8226,2788],{"mathvariant":231},[35,8228,1029],{"mathvariant":251},[81,8230,750],{"fence":48},[81,8232,244],{},[127,8234],{"linebreak":8235},"newline",[35,8237,4292],{"mathvariant":231},[81,8239,741],{"stretchy":86},[266,8241,8242,8244],{},[35,8243,3106],{},[89,8245,278],{},[81,8247,750],{"stretchy":86},[266,8249,8250,8252],{},[81,8251,4755],{},[227,8253,8254,8256],{},[35,8255,232],{"mathvariant":231},[32,8257,8258,8260,8262],{},[35,8259,102],{},[81,8261,239],{},[89,8263,91],{},[5235,8265,8266,8268,8276],{},[81,8267,3741],{},[32,8269,8270,8272,8274],{},[35,8271,1070],{},[81,8273,244],{},[89,8275,91],{},[35,8277,102],{},[289,8279,8280,8282,8284],{},[35,8281,1029],{},[35,8283,1070],{},[32,8285,8286,8292,8294],{},[266,8287,8288,8290],{},[35,8289,3106],{},[35,8291,1070],{},[81,8293,239],{},[89,8295,91],{},[282,8297,4802],{},[35,8299,2788],{"mathvariant":231},[35,8301,1029],{"mathvariant":251},[40,8303,8304],{"encoding":42},"\\prod_{i=1}^K \\Gamma(\\alpha_i) = \\left(\\int_0^\\infty s^{\\alpha_0 - 1} e^{-s} \\,\\mathrm{d}s\\right) \\left(\\int_{\\Delta^{K-1}} \\prod_{i=1}^K x_i^{\\alpha_i - 1} \\,\\mathrm{d}\\mathbf{x}\\right) = \\\\ \\Gamma(\\alpha_0) \\int_{\\Delta^{K-1}} \\prod_{i=1}^K x_i^{\\alpha_i - 1} 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即得结论。",[11,9318,9319,9320,9348,9349,9377,9378,9381],{},"命题 1.3 的证明过程本身包含了比结论更多的信息：",[15,9321,9323,9336],{"className":9322,"translate":19},[18],[15,9324,9326],{"className":9325},[23],[25,9327,9328],{"xmlns":27},[29,9329,9330,9334],{},[32,9331,9332],{},[35,9333,5864],{},[40,9335,5864],{"encoding":42},[15,9337,9339],{"className":9338,"ariaHidden":48},[47],[15,9340,9342,9345],{"className":9341},[52],[15,9343],{"className":9344,"style":6032},[56],[15,9346,5864],{"className":9347},[61,122]," 与 ",[15,9350,9352,9365],{"className":9351,"translate":19},[18],[15,9353,9355],{"className":9354},[23],[25,9356,9357],{"xmlns":27},[29,9358,9359,9363],{},[32,9360,9361],{},[35,9362,1029],{"mathvariant":251},[40,9364,2698],{"encoding":42},[15,9366,9368],{"className":9367,"ariaHidden":48},[47],[15,9369,9371,9374],{"className":9370},[52],[15,9372],{"className":9373,"style":2708},[56],[15,9375,1029],{"className":9376},[61,411]," 的",[661,9379,9380],{},"完全可分离性","。下一小节的生成性刻画正是对这一可分离性的直接读解。",[877,9383,9385],{"id":9384},"_13-生成性刻画gamma-商表示","1.3 生成性刻画：Gamma 商表示",[882,9387,9388],{},[11,9389,9390,9393,9394,9538,9539,9706,9707,9777,9778,9806,9807,10335,10336,10412,10413,5987,10492,867],{},[661,9391,9392],{},"命题 1.4（Gamma 商表示）"," 设 ",[15,9395,9397,9428],{"className":9396,"translate":19},[18],[15,9398,9400],{"className":9399},[23],[25,9401,9402],{"xmlns":27},[29,9403,9404,9425],{},[32,9405,9406,9413,9415,9417,9419],{},[266,9407,9408,9411],{},[35,9409,9410],{},"γ",[89,9412,91],{},[81,9414,94],{"separator":48},[81,9416,97],{},[81,9418,94],{"separator":48},[266,9420,9421,9423],{},[35,9422,9410],{},[35,9424,102],{},[40,9426,9427],{"encoding":42},"\\gamma_1, \\dots, \\gamma_K",[15,9429,9431],{"className":9430,"ariaHidden":48},[47],[15,9432,9434,9438,9480,9483,9486,9489,9492,9495,9498],{"className":9433},[52],[15,9435],{"className":9436,"style":9437},[56],"height:0.625em;vertical-align:-0.1944em;",[15,9439,9441,9445],{"className":9440},[61],[15,9442,9410],{"className":9443,"style":9444},[61,122],"margin-right:0.0556em;",[15,9446,9448],{"className":9447},[338],[15,9449,9451,9472],{"className":9450},[342,472],[15,9452,9454,9469],{"className":9453},[346],[15,9455,9457],{"className":9456,"style":1214},[350],[15,9458,9460,9463],{"style":9459},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[15,9461],{"className":9462,"style":358},[357],[15,9464,9466],{"className":9465},[362,363,364,365],[15,9467,91],{"className":9468},[61,365],[15,9470,497],{"className":9471},[496],[15,9473,9475],{"className":9474},[346],[15,9476,9478],{"className":9477,"style":504},[350],[15,9479],{},[15,9481,94],{"className":9482},[153],[15,9484],{"className":9485,"style":157},[127],[15,9487,97],{"className":9488},[161],[15,9490],{"className":9491,"style":157},[127],[15,9493,94],{"className":9494},[153],[15,9496],{"className":9497,"style":157},[127],[15,9499,9501,9504],{"className":9500},[61],[15,9502,9410],{"className":9503,"style":9444},[61,122],[15,9505,9507],{"className":9506},[338],[15,9508,9510,9530],{"className":9509},[342,472],[15,9511,9513,9527],{"className":9512},[346],[15,9514,9516],{"className":9515,"style":1273},[350],[15,9517,9518,9521],{"style":9459},[15,9519],{"className":9520,"style":358},[357],[15,9522,9524],{"className":9523},[362,363,364,365],[15,9525,102],{"className":9526,"style":174},[61,122,365],[15,9528,497],{"className":9529},[496],[15,9531,9533],{"className":9532},[346],[15,9534,9536],{"className":9535,"style":504},[350],[15,9537],{}," 相互独立且 ",[15,9540,9542,9581],{"className":9541,"translate":19},[18],[15,9543,9545],{"className":9544},[23],[25,9546,9547],{"xmlns":27},[29,9548,9549,9578],{},[32,9550,9551,9557,9559,9562,9564,9566,9572,9574,9576],{},[266,9552,9553,9555],{},[35,9554,9410],{},[35,9556,1070],{},[81,9558,4183],{},[35,9560,9561],{"mathvariant":231},"Gamma",[81,9563,1547],{},[81,9565,741],{"stretchy":86},[266,9567,9568,9570],{},[35,9569,3106],{},[35,9571,1070],{},[81,9573,94],{"separator":48},[89,9575,91],{},[81,9577,750],{"stretchy":86},[40,9579,9580],{"encoding":42},"\\gamma_i \\sim \\operatorname{Gamma}(\\alpha_i, 1)",[15,9582,9584,9639],{"className":9583,"ariaHidden":48},[47],[15,9585,9587,9590,9630,9633,9636],{"className":9586},[52],[15,9588],{"className":9589,"style":9437},[56],[15,9591,9593,9596],{"className":9592},[61],[15,9594,9410],{"className":9595,"style":9444},[61,122],[15,9597,9599],{"className":9598},[338],[15,9600,9602,9622],{"className":9601},[342,472],[15,9603,9605,9619],{"className":9604},[346],[15,9606,9608],{"className":9607,"style":1364},[350],[15,9609,9610,9613],{"style":9459},[15,9611],{"className":9612,"style":358},[357],[15,9614,9616],{"className":9615},[362,363,364,365],[15,9617,1070],{"className":9618},[61,122,365],[15,9620,497],{"className":9621},[496],[15,9623,9625],{"className":9624},[346],[15,9626,9628],{"className":9627,"style":504},[350],[15,9629],{},[15,9631],{"className":9632,"style":128},[127],[15,9634,4183],{"className":9635},[132],[15,9637],{"className":9638,"style":128},[127],[15,9640,9642,9645,9651,9654,9694,9697,9700,9703],{"className":9641},[52],[15,9643],{"className":9644,"style":142},[56],[15,9646,9648],{"className":9647},[531],[15,9649,9561],{"className":9650},[61,1622],[15,9652,741],{"className":9653},[146],[15,9655,9657,9660],{"className":9656},[61],[15,9658,3106],{"className":9659,"style":3191},[61,122],[15,9661,9663],{"className":9662},[338],[15,9664,9666,9686],{"className":9665},[342,472],[15,9667,9669,9683],{"className":9668},[346],[15,9670,9672],{"className":9671,"style":1364},[350],[15,9673,9674,9677],{"style":3206},[15,9675],{"className":9676,"style":358},[357],[15,9678,9680],{"className":9679},[362,363,364,365],[15,9681,1070],{"className":9682},[61,122,365],[15,9684,497],{"className":9685},[496],[15,9687,9689],{"className":9688},[346],[15,9690,9692],{"className":9691,"style":504},[350],[15,9693],{},[15,9695,94],{"className":9696},[153],[15,9698],{"className":9699,"style":157},[127],[15,9701,91],{"className":9702},[61],[15,9704,750],{"className":9705},[178],"（形状 ",[15,9708,9710,9728],{"className":9709,"translate":19},[18],[15,9711,9713],{"className":9712},[23],[25,9714,9715],{"xmlns":27},[29,9716,9717,9725],{},[32,9718,9719],{},[266,9720,9721,9723],{},[35,9722,3106],{},[35,9724,1070],{},[40,9726,9727],{"encoding":42},"\\alpha_i",[15,9729,9731],{"className":9730,"ariaHidden":48},[47],[15,9732,9734,9737],{"className":9733},[52],[15,9735],{"className":9736,"style":2511},[56],[15,9738,9740,9743],{"className":9739},[61],[15,9741,3106],{"className":9742,"style":3191},[61,122],[15,9744,9746],{"className":9745},[338],[15,9747,9749,9769],{"className":9748},[342,472],[15,9750,9752,9766],{"className":9751},[346],[15,9753,9755],{"className":9754,"style":1364},[350],[15,9756,9757,9760],{"style":3206},[15,9758],{"className":9759,"style":358},[357],[15,9761,9763],{"className":9762},[362,363,364,365],[15,9764,1070],{"className":9765},[61,122,365],[15,9767,497],{"className":9768},[496],[15,9770,9772],{"className":9771},[346],[15,9773,9775],{"className":9774,"style":504},[350],[15,9776],{},"、率 ",[15,9779,9781,9794],{"className":9780,"translate":19},[18],[15,9782,9784],{"className":9783},[23],[25,9785,9786],{"xmlns":27},[29,9787,9788,9792],{},[32,9789,9790],{},[89,9791,91],{},[40,9793,91],{"encoding":42},[15,9795,9797],{"className":9796,"ariaHidden":48},[47],[15,9798,9800,9803],{"className":9799},[52],[15,9801],{"className":9802,"style":859},[56],[15,9804,91],{"className":9805},[61],"）。令\n",[15,9808,9810,9899],{"className":9809,"translate":19},[18],[15,9811,9813],{"className":9812},[23],[25,9814,9815],{"xmlns":27},[29,9816,9817,9896],{},[32,9818,9819,9821,9823,9837,9843,9845,9848,9854,9856,9866,9868,9870,9872,9874,9876,9882,9884,9886,9888,9894],{},[35,9820,5864],{},[81,9822,244],{},[289,9824,9825,9827,9835],{},[81,9826,293],{},[32,9828,9829,9831,9833],{},[35,9830,1070],{},[81,9832,244],{},[89,9834,91],{},[35,9836,102],{},[266,9838,9839,9841],{},[35,9840,9410],{},[35,9842,1070],{},[81,9844,94],{"separator":48},[127,9846],{"width":9847},"2em",[266,9849,9850,9852],{},[35,9851,1029],{},[35,9853,1070],{},[81,9855,244],{},[3722,9857,9858,9864],{},[266,9859,9860,9862],{},[35,9861,9410],{},[35,9863,1070],{},[35,9865,5864],{},[81,9867,94],{"separator":48},[127,9869],{"width":9847},[35,9871,1029],{"mathvariant":251},[81,9873,244],{},[81,9875,741],{"stretchy":86},[266,9877,9878,9880],{},[35,9879,1029],{},[89,9881,91],{},[81,9883,94],{"separator":48},[81,9885,97],{},[81,9887,94],{"separator":48},[266,9889,9890,9892],{},[35,9891,1029],{},[35,9893,102],{},[81,9895,750],{"stretchy":86},[40,9897,9898],{"encoding":42},"s = \\sum_{i=1}^K \\gamma_i, \\qquad x_i = \\frac{\\gamma_i}{s}, \\qquad \\mathbf{x} = (x_1, \\dots, x_K)",[15,9900,9902,9920,10088,10225],{"className":9901,"ariaHidden":48},[47],[15,9903,9905,9908,9911,9914,9917],{"className":9904},[52],[15,9906],{"className":9907,"style":6032},[56],[15,9909,5864],{"className":9910},[61,122],[15,9912],{"className":9913,"style":128},[127],[15,9915,244],{"className":9916},[132],[15,9918],{"className":9919,"style":128},[127],[15,9921,9923,9926,9986,9989,10029,10032,10036,10039,10079,10082,10085],{"className":9922},[52],[15,9924],{"className":9925,"style":3471},[56],[15,9927,9929,9932],{"className":9928},[531],[15,9930,293],{"className":9931,"style":537},[531,535,536],[15,9933,9935],{"className":9934},[338],[15,9936,9938,9978],{"className":9937},[342,472],[15,9939,9941,9975],{"className":9940},[346],[15,9942,9944,9964],{"className":9943,"style":550},[350],[15,9945,9946,9949],{"style":553},[15,9947],{"className":9948,"style":358},[357],[15,9950,9952],{"className":9951},[362,363,364,365],[15,9953,9955,9958,9961],{"className":9954},[61,365],[15,9956,1070],{"className":9957},[61,122,365],[15,9959,244],{"className":9960},[132,365],[15,9962,91],{"className":9963},[61,365],[15,9965,9966,9969],{"style":574},[15,9967],{"className":9968,"style":358},[357],[15,9970,9972],{"className":9971},[362,363,364,365],[15,9973,102],{"className":9974,"style":174},[61,122,365],[15,9976,497],{"className":9977},[496],[15,9979,9981],{"className":9980},[346],[15,9982,9984],{"className":9983,"style":593},[350],[15,9985],{},[15,9987],{"className":9988,"style":157},[127],[15,9990,9992,9995],{"className":9991},[61],[15,9993,9410],{"className":9994,"style":9444},[61,122],[15,9996,9998],{"className":9997},[338],[15,9999,10001,10021],{"className":10000},[342,472],[15,10002,10004,10018],{"className":10003},[346],[15,10005,10007],{"className":10006,"style":1364},[350],[15,10008,10009,10012],{"style":9459},[15,10010],{"className":10011,"style":358},[357],[15,10013,10015],{"className":10014},[362,363,364,365],[15,10016,1070],{"className":10017},[61,122,365],[15,10019,497],{"className":10020},[496],[15,10022,10024],{"className":10023},[346],[15,10025,10027],{"className":10026,"style":504},[350],[15,10028],{},[15,10030,94],{"className":10031},[153],[15,10033],{"className":10034,"style":10035},[127],"margin-right:2em;",[15,10037],{"className":10038,"style":157},[127],[15,10040,10042,10045],{"className":10041},[61],[15,10043,1029],{"className":10044},[61,122],[15,10046,10048],{"className":10047},[338],[15,10049,10051,10071],{"className":10050},[342,472],[15,10052,10054,10068],{"className":10053},[346],[15,10055,10057],{"className":10056,"style":1364},[350],[15,10058,10059,10062],{"style":482},[15,10060],{"className":10061,"style":358},[357],[15,10063,10065],{"className":10064},[362,363,364,365],[15,10066,1070],{"className":10067},[61,122,365],[15,10069,497],{"className":10070},[496],[15,10072,10074],{"className":10073},[346],[15,10075,10077],{"className":10076,"style":504},[350],[15,10078],{},[15,10080],{"className":10081,"style":128},[127],[15,10083,244],{"className":10084},[132],[15,10086],{"className":10087,"style":128},[127],[15,10089,10091,10095,10204,10207,10210,10213,10216,10219,10222],{"className":10090},[52],[15,10092],{"className":10093,"style":10094},[56],"height:1.0925em;vertical-align:-0.345em;",[15,10096,10098,10101,10201],{"className":10097},[61],[15,10099],{"className":10100},[146,3841],[15,10102,10104],{"className":10103},[3722],[15,10105,10107,10192],{"className":10106},[342,472],[15,10108,10110,10189],{"className":10109},[346],[15,10111,10114,10128,10136],{"className":10112,"style":10113},[350],"height:0.7475em;",[15,10115,10116,10119],{"style":3857},[15,10117],{"className":10118,"style":3861},[357],[15,10120,10122],{"className":10121},[362,363,364,365],[15,10123,10125],{"className":10124},[61,365],[15,10126,5864],{"className":10127},[61,122,365],[15,10129,10130,10133],{"style":3889},[15,10131],{"className":10132,"style":3861},[357],[15,10134],{"className":10135,"style":3897},[3896],[15,10137,10139,10142],{"style":10138},"top:-3.4461em;",[15,10140],{"className":10141,"style":3861},[357],[15,10143,10145],{"className":10144},[362,363,364,365],[15,10146,10148],{"className":10147},[61,365],[15,10149,10151,10154],{"className":10150},[61,365],[15,10152,9410],{"className":10153,"style":9444},[61,122,365],[15,10155,10157],{"className":10156},[338],[15,10158,10160,10181],{"className":10159},[342,472],[15,10161,10163,10178],{"className":10162},[346],[15,10164,10166],{"className":10165,"style":4055},[350],[15,10167,10169,10172],{"style":10168},"top:-2.357em;margin-left:-0.0556em;margin-right:0.0714em;",[15,10170],{"className":10171,"style":4062},[357],[15,10173,10175],{"className":10174},[362,4066,4067,365],[15,10176,1070],{"className":10177},[61,122,365],[15,10179,497],{"className":10180},[496],[15,10182,10184],{"className":10183},[346],[15,10185,10187],{"className":10186,"style":4080},[350],[15,10188],{},[15,10190,497],{"className":10191},[496],[15,10193,10195],{"className":10194},[346],[15,10196,10199],{"className":10197,"style":10198},[350],"height:0.345em;",[15,10200],{},[15,10202],{"className":10203},[178,3841],[15,10205,94],{"className":10206},[153],[15,10208],{"className":10209,"style":10035},[127],[15,10211],{"className":10212,"style":157},[127],[15,10214,1029],{"className":10215},[61,411],[15,10217],{"className":10218,"style":128},[127],[15,10220,244],{"className":10221},[132],[15,10223],{"className":10224,"style":128},[127],[15,10226,10228,10231,10234,10274,10277,10280,10283,10286,10289,10292,10332],{"className":10227},[52],[15,10229],{"className":10230,"style":142},[56],[15,10232,741],{"className":10233},[146],[15,10235,10237,10240],{"className":10236},[61],[15,10238,1029],{"className":10239},[61,122],[15,10241,10243],{"className":10242},[338],[15,10244,10246,10266],{"className":10245},[342,472],[15,10247,10249,10263],{"className":10248},[346],[15,10250,10252],{"className":10251,"style":1214},[350],[15,10253,10254,10257],{"style":482},[15,10255],{"className":10256,"style":358},[357],[15,10258,10260],{"className":10259},[362,363,364,365],[15,10261,91],{"className":10262},[61,365],[15,10264,497],{"className":10265},[496],[15,10267,10269],{"className":10268},[346],[15,10270,10272],{"className":10271,"style":504},[350],[15,10273],{},[15,10275,94],{"className":10276},[153],[15,10278],{"className":10279,"style":157},[127],[15,10281,97],{"className":10282},[161],[15,10284],{"className":10285,"style":157},[127],[15,10287,94],{"className":10288},[153],[15,10290],{"className":10291,"style":157},[127],[15,10293,10295,10298],{"className":10294},[61],[15,10296,1029],{"className":10297},[61,122],[15,10299,10301],{"className":10300},[338],[15,10302,10304,10324],{"className":10303},[342,472],[15,10305,10307,10321],{"className":10306},[346],[15,10308,10310],{"className":10309,"style":1273},[350],[15,10311,10312,10315],{"style":482},[15,10313],{"className":10314,"style":358},[357],[15,10316,10318],{"className":10317},[362,363,364,365],[15,10319,102],{"className":10320,"style":174},[61,122,365],[15,10322,497],{"className":10323},[496],[15,10325,10327],{"className":10326},[346],[15,10328,10330],{"className":10329,"style":504},[350],[15,10331],{},[15,10333,750],{"className":10334},[178],"\n则 ",[15,10337,10339,10364],{"className":10338,"translate":19},[18],[15,10340,10342],{"className":10341},[23],[25,10343,10344],{"xmlns":27},[29,10345,10346,10362],{},[32,10347,10348,10350,10352,10354,10356,10358,10360],{},[35,10349,1029],{"mathvariant":251},[81,10351,4183],{},[35,10353,4186],{"mathvariant":231},[81,10355,1547],{},[81,10357,741],{"stretchy":86},[35,10359,3106],{"mathvariant":3105},[81,10361,750],{"stretchy":86},[40,10363,4197],{"encoding":42},[15,10365,10367,10385],{"className":10366,"ariaHidden":48},[47],[15,10368,10370,10373,10376,10379,10382],{"className":10369},[52],[15,10371],{"className":10372,"style":2708},[56],[15,10374,1029],{"className":10375},[61,411],[15,10377],{"className":10378,"style":128},[127],[15,10380,4183],{"className":10381},[132],[15,10383],{"className":10384,"style":128},[127],[15,10386,10388,10391,10397,10400,10409],{"className":10387},[52],[15,10389],{"className":10390,"style":142},[56],[15,10392,10394],{"className":10393},[531],[15,10395,4186],{"className":10396},[61,1622],[15,10398,741],{"className":10399},[146],[15,10401,10403],{"className":10402},[61],[15,10404,10406],{"className":10405},[61],[15,10407,3106],{"className":10408},[61,3166],[15,10410,750],{"className":10411},[178],"，且 ",[15,10414,10416,10443],{"className":10415,"translate":19},[18],[15,10417,10419],{"className":10418},[23],[25,10420,10421],{"xmlns":27},[29,10422,10423,10440],{},[32,10424,10425,10427,10430,10432,10434,10436,10438],{},[35,10426,1029],{"mathvariant":251},[81,10428,10429],{},"⊥",[282,10431,6799],{},[282,10433,6799],{},[282,10435,6799],{},[81,10437,10429],{},[35,10439,5864],{},[40,10441,10442],{"encoding":42},"\\mathbf{x} \\perp\\!\\!\\!\\perp s",[15,10444,10446,10471,10483],{"className":10445,"ariaHidden":48},[47],[15,10447,10449,10453,10456,10459,10462,10465,10468],{"className":10448},[52],[15,10450],{"className":10451,"style":10452},[56],"height:0.6944em;",[15,10454,1029],{"className":10455},[61,411],[15,10457],{"className":10458,"style":128},[127],[15,10460,10429],{"className":10461},[132],[15,10463],{"className":10464,"style":7105},[127],[15,10466],{"className":10467,"style":7105},[127],[15,10469],{"className":10470,"style":7105},[127],[15,10472,10474,10477,10480],{"className":10473},[52],[15,10475],{"className":10476,"style":10452},[56],[15,10478,10429],{"className":10479},[132],[15,10481],{"className":10482,"style":128},[127],[15,10484,10486,10489],{"className":10485},[52],[15,10487],{"className":10488,"style":6032},[56],[15,10490,5864],{"className":10491},[61,122],[15,10493,10495,10529],{"className":10494,"translate":19},[18],[15,10496,10498],{"className":10497},[23],[25,10499,10500],{"xmlns":27},[29,10501,10502,10526],{},[32,10503,10504,10506,10508,10510,10512,10514,10520,10522,10524],{},[35,10505,5864],{},[81,10507,4183],{},[35,10509,9561],{"mathvariant":231},[81,10511,1547],{},[81,10513,741],{"stretchy":86},[266,10515,10516,10518],{},[35,10517,3106],{},[89,10519,278],{},[81,10521,94],{"separator":48},[89,10523,91],{},[81,10525,750],{"stretchy":86},[40,10527,10528],{"encoding":42},"s \\sim \\operatorname{Gamma}(\\alpha_0, 1)",[15,10530,10532,10550],{"className":10531,"ariaHidden":48},[47],[15,10533,10535,10538,10541,10544,10547],{"className":10534},[52],[15,10536],{"className":10537,"style":6032},[56],[15,10539,5864],{"className":10540},[61,122],[15,10542],{"className":10543,"style":128},[127],[15,10545,4183],{"className":10546},[132],[15,10548],{"className":10549,"style":128},[127],[15,10551,10553,10556,10562,10565,10605,10608,10611,10614],{"className":10552},[52],[15,10554],{"className":10555,"style":142},[56],[15,10557,10559],{"className":10558},[531],[15,10560,9561],{"className":10561},[61,1622],[15,10563,741],{"className":10564},[146],[15,10566,10568,10571],{"className":10567},[61],[15,10569,3106],{"className":10570,"style":3191},[61,122],[15,10572,10574],{"className":10573},[338],[15,10575,10577,10597],{"className":10576},[342,472],[15,10578,10580,10594],{"className":10579},[346],[15,10581,10583],{"className":10582,"style":1214},[350],[15,10584,10585,10588],{"style":3206},[15,10586],{"className":10587,"style":358},[357],[15,10589,10591],{"className":10590},[362,363,364,365],[15,10592,278],{"className":10593},[61,365],[15,10595,497],{"className":10596},[496],[15,10598,10600],{"className":10599},[346],[15,10601,10603],{"className":10602,"style":504},[350],[15,10604],{},[15,10606,94],{"className":10607},[153],[15,10609],{"className":10610,"style":157},[127],[15,10612,91],{"className":10613},[61],[15,10615,750],{"className":10616},[178],[11,10618,10619,10621,10622,10658,10659,10999,11000,11138],{},[661,10620,5123],{}," 由独立性，",[15,10623,10625,10639],{"className":10624,"translate":19},[18],[15,10626,10628],{"className":10627},[23],[25,10629,10630],{"xmlns":27},[29,10631,10632,10636],{},[32,10633,10634],{},[35,10635,9410],{"mathvariant":3105},[40,10637,10638],{"encoding":42},"\\boldsymbol{\\gamma}",[15,10640,10642],{"className":10641,"ariaHidden":48},[47],[15,10643,10645,10648],{"className":10644},[52],[15,10646],{"className":10647,"style":829},[56],[15,10649,10651],{"className":10650},[61],[15,10652,10654],{"className":10653},[61],[15,10655,9410],{"className":10656,"style":10657},[61,3166],"margin-right:0.0639em;"," 的联合密度为 ",[15,10660,10662,10726],{"className":10661,"translate":19},[18],[15,10663,10665],{"className":10664},[23],[25,10666,10667],{"xmlns":27},[29,10668,10669,10723],{},[32,10670,10671,10677,10695,10709,10711,10713,10715,10721],{},[266,10672,10673,10675],{},[81,10674,3741],{},[35,10676,1070],{},[289,10678,10679,10681,10683],{},[35,10680,9410],{},[35,10682,1070],{},[32,10684,10685,10691,10693],{},[266,10686,10687,10689],{},[35,10688,3106],{},[35,10690,1070],{},[81,10692,239],{},[89,10694,91],{},[227,10696,10697,10699],{},[35,10698,1982],{},[32,10700,10701,10703],{},[81,10702,239],{},[266,10704,10705,10707],{},[35,10706,9410],{},[35,10708,1070],{},[35,10710,3020],{"mathvariant":231},[35,10712,4292],{"mathvariant":231},[81,10714,741],{"stretchy":86},[266,10716,10717,10719],{},[35,10718,3106],{},[35,10720,1070],{},[81,10722,750],{"stretchy":86},[40,10724,10725],{"encoding":42},"\\prod_i \\gamma_i^{\\alpha_i - 1} e^{-\\gamma_i} \u002F \\Gamma(\\alpha_i)",[15,10727,10729],{"className":10728,"ariaHidden":48},[47],[15,10730,10732,10736,10776,10779,10877,10949,10953,10956,10996],{"className":10731},[52],[15,10733],{"className":10734,"style":10735},[56],"height:1.1957em;vertical-align:-0.2997em;",[15,10737,10739,10742],{"className":10738},[531],[15,10740,3741],{"className":10741,"style":537},[531,535,536],[15,10743,10745],{"className":10744},[338],[15,10746,10748,10768],{"className":10747},[342,472],[15,10749,10751,10765],{"className":10750},[346],[15,10752,10754],{"className":10753,"style":6070},[350],[15,10755,10756,10759],{"style":553},[15,10757],{"className":10758,"style":358},[357],[15,10760,10762],{"className":10761},[362,363,364,365],[15,10763,1070],{"className":10764},[61,122,365],[15,10766,497],{"className":10767},[496],[15,10769,10771],{"className":10770},[346],[15,10772,10774],{"className":10773,"style":593},[350],[15,10775],{},[15,10777],{"className":10778,"style":157},[127],[15,10780,10782,10785],{"className":10781},[61],[15,10783,9410],{"className":10784,"style":9444},[61,122],[15,10786,10788],{"className":10787},[338],[15,10789,10791,10869],{"className":10790},[342,472],[15,10792,10794,10866],{"className":10793},[346],[15,10795,10797,10809],{"className":10796,"style":4012},[350],[15,10798,10800,10803],{"style":10799},"top:-2.4231em;margin-left:-0.0556em;margin-right:0.05em;",[15,10801],{"className":10802,"style":358},[357],[15,10804,10806],{"className":10805},[362,363,364,365],[15,10807,1070],{"className":10808},[61,122,365],[15,10810,10811,10814],{"style":4027},[15,10812],{"className":10813,"style":358},[357],[15,10815,10817],{"className":10816},[362,363,364,365],[15,10818,10820,10860,10863],{"className":10819},[61,365],[15,10821,10823,10826],{"className":10822},[61,365],[15,10824,3106],{"className":10825,"style":3191},[61,122,365],[15,10827,10829],{"className":10828},[338],[15,10830,10832,10852],{"className":10831},[342,472],[15,10833,10835,10849],{"className":10834},[346],[15,10836,10838],{"className":10837,"style":4055},[350],[15,10839,10840,10843],{"style":4058},[15,10841],{"className":10842,"style":4062},[357],[15,10844,10846],{"className":10845},[362,4066,4067,365],[15,10847,1070],{"className":10848},[61,122,365],[15,10850,497],{"className":10851},[496],[15,10853,10855],{"className":10854},[346],[15,10856,10858],{"className":10857,"style":4080},[350],[15,10859],{},[15,10861,239],{"className":10862},[375,365],[15,10864,91],{"className":10865},[61,365],[15,10867,497],{"className":10868},[496],[15,10870,10872],{"className":10871},[346],[15,10873,10875],{"className":10874,"style":4098},[350],[15,10876],{},[15,10878,10880,10883],{"className":10879},[61],[15,10881,1982],{"className":10882},[61,122],[15,10884,10886],{"className":10885},[338],[15,10887,10889],{"className":10888},[342],[15,10890,10892],{"className":10891},[346],[15,10893,10895],{"className":10894,"style":8016},[350],[15,10896,10897,10900],{"style":716},[15,10898],{"className":10899,"style":358},[357],[15,10901,10903],{"className":10902},[362,363,364,365],[15,10904,10906,10909],{"className":10905},[61,365],[15,10907,239],{"className":10908},[61,365],[15,10910,10912,10915],{"className":10911},[61,365],[15,10913,9410],{"className":10914,"style":9444},[61,122,365],[15,10916,10918],{"className":10917},[338],[15,10919,10921,10941],{"className":10920},[342,472],[15,10922,10924,10938],{"className":10923},[346],[15,10925,10927],{"className":10926,"style":4055},[350],[15,10928,10929,10932],{"style":10168},[15,10930],{"className":10931,"style":4062},[357],[15,10933,10935],{"className":10934},[362,4066,4067,365],[15,10936,1070],{"className":10937},[61,122,365],[15,10939,497],{"className":10940},[496],[15,10942,10944],{"className":10943},[346],[15,10945,10947],{"className":10946,"style":4080},[350],[15,10948],{},[15,10950,10952],{"className":10951},[61],"\u002FΓ",[15,10954,741],{"className":10955},[146],[15,10957,10959,10962],{"className":10958},[61],[15,10960,3106],{"className":10961,"style":3191},[61,122],[15,10963,10965],{"className":10964},[338],[15,10966,10968,10988],{"className":10967},[342,472],[15,10969,10971,10985],{"className":10970},[346],[15,10972,10974],{"className":10973,"style":1364},[350],[15,10975,10976,10979],{"style":3206},[15,10977],{"className":10978,"style":358},[357],[15,10980,10982],{"className":10981},[362,363,364,365],[15,10983,1070],{"className":10984},[61,122,365],[15,10986,497],{"className":10987},[496],[15,10989,10991],{"className":10990},[346],[15,10992,10994],{"className":10993,"style":504},[350],[15,10995],{},[15,10997,750],{"className":10998},[178],"。施行与命题 1.3 相同的替换 ",[15,11001,11003,11031],{"className":11002,"translate":19},[18],[15,11004,11006],{"className":11005},[23],[25,11007,11008],{"xmlns":27},[29,11009,11010,11028],{},[32,11011,11012,11018,11020,11022],{},[266,11013,11014,11016],{},[35,11015,9410],{},[35,11017,1070],{},[81,11019,244],{},[35,11021,5864],{},[266,11023,11024,11026],{},[35,11025,1029],{},[35,11027,1070],{},[40,11029,11030],{"encoding":42},"\\gamma_i = s x_i",[15,11032,11034,11089],{"className":11033,"ariaHidden":48},[47],[15,11035,11037,11040,11080,11083,11086],{"className":11036},[52],[15,11038],{"className":11039,"style":9437},[56],[15,11041,11043,11046],{"className":11042},[61],[15,11044,9410],{"className":11045,"style":9444},[61,122],[15,11047,11049],{"className":11048},[338],[15,11050,11052,11072],{"className":11051},[342,472],[15,11053,11055,11069],{"className":11054},[346],[15,11056,11058],{"className":11057,"style":1364},[350],[15,11059,11060,11063],{"style":9459},[15,11061],{"className":11062,"style":358},[357],[15,11064,11066],{"className":11065},[362,363,364,365],[15,11067,1070],{"className":11068},[61,122,365],[15,11070,497],{"className":11071},[496],[15,11073,11075],{"className":11074},[346],[15,11076,11078],{"className":11077,"style":504},[350],[15,11079],{},[15,11081],{"className":11082,"style":128},[127],[15,11084,244],{"className":11085},[132],[15,11087],{"className":11088,"style":128},[127],[15,11090,11092,11095,11098],{"className":11091},[52],[15,11093],{"className":11094,"style":2511},[56],[15,11096,5864],{"className":11097},[61,122],[15,11099,11101,11104],{"className":11100},[61],[15,11102,1029],{"className":11103},[61,122],[15,11105,11107],{"className":11106},[338],[15,11108,11110,11130],{"className":11109},[342,472],[15,11111,11113,11127],{"className":11112},[346],[15,11114,11116],{"className":11115,"style":1364},[350],[15,11117,11118,11121],{"style":482},[15,11119],{"className":11120,"style":358},[357],[15,11122,11124],{"className":11123},[362,363,364,365],[15,11125,1070],{"className":11126},[61,122,365],[15,11128,497],{"className":11129},[496],[15,11131,11133],{"className":11132},[346],[15,11134,11136],{"className":11135,"style":504},[350],[15,11137],{},"，联合密度变换为",[15,11140,11142],{"className":11141,"translate":19},[212],[15,11143,11145,11335],{"className":11144,"translate":19},[18],[15,11146,11148],{"className":11147},[23],[25,11149,11150],{"xmlns":27,"display":221},[29,11151,11152,11332],{},[32,11153,11154,11156,11158,11160,11162,11164,11166,11168,11212,11215,11291,11293,11295,11297,11299,11301,11304,11310,11312,11314,11316,11318,11320,11322,11324,11326,11328,11330],{},[35,11155,11],{},[81,11157,741],{"stretchy":86},[35,11159,5864],{},[81,11161,94],{"separator":48},[35,11163,1029],{"mathvariant":251},[81,11165,750],{"stretchy":86},[81,11167,244],{},[3722,11169,11170,11198],{},[32,11171,11172,11188],{},[227,11173,11174,11176],{},[35,11175,5864],{},[32,11177,11178,11184,11186],{},[266,11179,11180,11182],{},[35,11181,3106],{},[89,11183,278],{},[81,11185,239],{},[89,11187,91],{},[227,11189,11190,11192],{},[35,11191,1982],{},[32,11193,11194,11196],{},[81,11195,239],{},[35,11197,5864],{},[32,11199,11200,11202,11204,11210],{},[35,11201,4292],{"mathvariant":231},[81,11203,741],{"stretchy":86},[266,11205,11206,11208],{},[35,11207,3106],{},[89,11209,278],{},[81,11211,750],{"stretchy":86},[81,11213,11214],{},"⋅",[3722,11216,11217,11263],{},[32,11218,11219,11221,11223,11229,11231,11245],{},[35,11220,4292],{"mathvariant":231},[81,11222,741],{"stretchy":86},[266,11224,11225,11227],{},[35,11226,3106],{},[89,11228,278],{},[81,11230,750],{"stretchy":86},[5235,11232,11233,11235,11243],{},[81,11234,3741],{},[32,11236,11237,11239,11241],{},[35,11238,1070],{},[81,11240,244],{},[89,11242,91],{},[35,11244,102],{},[289,11246,11247,11249,11251],{},[35,11248,1029],{},[35,11250,1070],{},[32,11252,11253,11259,11261],{},[266,11254,11255,11257],{},[35,11256,3106],{},[35,11258,1070],{},[81,11260,239],{},[89,11262,91],{},[32,11264,11265,11279,11281,11283,11289],{},[5235,11266,11267,11269,11277],{},[81,11268,3741],{},[32,11270,11271,11273,11275],{},[35,11272,1070],{},[81,11274,244],{},[89,11276,91],{},[35,11278,102],{},[35,11280,4292],{"mathvariant":231},[81,11282,741],{"stretchy":86},[266,11284,11285,11287],{},[35,11286,3106],{},[35,11288,1070],{},[81,11290,750],{"stretchy":86},[81,11292,244],{},[35,11294,9561],{"mathvariant":231},[81,11296,1547],{},[81,11298,741],{"stretchy":86},[35,11300,5864],{},[81,11302,11303],{"separator":48},";",[266,11305,11306,11308],{},[35,11307,3106],{},[89,11309,278],{},[81,11311,94],{"separator":48},[89,11313,91],{},[81,11315,750],{"stretchy":86},[81,11317,11214],{},[35,11319,4186],{"mathvariant":231},[81,11321,1547],{},[81,11323,741],{"stretchy":86},[35,11325,1029],{"mathvariant":251},[81,11327,11303],{"separator":48},[35,11329,3106],{"mathvariant":3105},[81,11331,750],{"stretchy":86},[40,11333,11334],{"encoding":42},"p(s, \\mathbf{x}) = \\frac{s^{\\alpha_0 - 1} e^{-s}}{\\Gamma(\\alpha_0)} \\cdot \\frac{\\Gamma(\\alpha_0) \\prod_{i=1}^K x_i^{\\alpha_i - 1}}{\\prod_{i=1}^K \\Gamma(\\alpha_i)}\n= \\operatorname{Gamma}(s; \\alpha_0, 1) \\cdot \\operatorname{Dir}(\\mathbf{x}; 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,[15,11932,3106],{"className":11933,"style":3191},[61,122,365],[15,11935,11937],{"className":11936},[338],[15,11938,11940,11960],{"className":11939},[342,472],[15,11941,11943,11957],{"className":11942},[346],[15,11944,11946],{"className":11945,"style":4055},[350],[15,11947,11948,11951],{"style":4058},[15,11949],{"className":11950,"style":4062},[357],[15,11952,11954],{"className":11953},[362,4066,4067,365],[15,11955,1070],{"className":11956},[61,122,365],[15,11958,497],{"className":11959},[496],[15,11961,11963],{"className":11962},[346],[15,11964,11966],{"className":11965,"style":4080},[350],[15,11967],{},[15,11969,239],{"className":11970},[375,365],[15,11972,91],{"className":11973},[61,365],[15,11975,497],{"className":11976},[496],[15,11978,11980],{"className":11979},[346],[15,11981,11983],{"className":11982,"style":4098},[350],[15,11984],{},[15,11986,497],{"className":11987},[496],[15,11989,11991],{"className":11990},[346],[15,11992,11995],{"className":11993,"style":11994},[350],"height:1.1709em;",[15,11996],{},[15,11998],{"className":11999},[178,3841],[15,12001],{"className":12002,"style":128},[127],[15,12004,244],{"className":12005},[132],[15,12007],{"className":12008,"style":128},[127],[15,12010,12012,12015,12021,12024,12027,12030,12033,12073,12076,12079,12082,12085,12088,12091],{"className":12011},[52],[15,12013],{"className":12014,"style":142},[56],[15,12016,12018],{"className":12017},[531],[15,12019,9561],{"className":12020},[61,1622],[15,12022,741],{"className":12023},[146],[15,12025,5864],{"className":12026},[61,122],[15,12028,11303],{"className":12029},[153],[15,12031],{"className":12032,"style":157},[127],[15,12034,12036,12039],{"className":12035},[61],[15,12037,3106],{"className":12038,"style":3191},[61,122],[15,12040,12042],{"className":12041},[338],[15,12043,12045,12065],{"className":12044},[342,472],[15,12046,12048,12062],{"className":12047},[346],[15,12049,12051],{"className":12050,"style":1214},[350],[15,12052,12053,12056],{"style":3206},[15,12054],{"className":12055,"style":358},[357],[15,12057,12059],{"className":12058},[362,363,364,365],[15,12060,278],{"className":12061},[61,365],[15,12063,497],{"className":12064},[496],[15,12066,12068],{"className":12067},[346],[15,12069,12071],{"className":12070,"style":504},[350],[15,12072],{},[15,12074,94],{"className":12075},[153],[15,12077],{"className":12078,"style":157},[127],[15,12080,91],{"className":12081},[61],[15,12083,750],{"className":12084},[178],[15,12086],{"className":12087,"style":772},[127],[15,12089,11214],{"className":12090},[375],[15,12092],{"className":12093,"style":772},[127],[15,12095,12097,12100,12106,12109,12112,12115,12118,12127],{"className":12096},[52],[15,12098],{"className":12099,"style":142},[56],[15,12101,12103],{"className":12102},[531],[15,12104,4186],{"className":12105},[61,1622],[15,12107,741],{"className":12108},[146],[15,12110,1029],{"className":12111},[61,411],[15,12113,11303],{"className":12114},[153],[15,12116],{"className":12117,"style":157},[127],[15,12119,12121],{"className":12120},[61],[15,12122,12124],{"className":12123},[61],[15,12125,3106],{"className":12126},[61,3166],[15,12128,750],{"className":12129},[178],[11,12131,12132,12133,12161,12162,12190,12191,9348,12219,12247],{},"右侧分解为仅含 ",[15,12134,12136,12149],{"className":12135,"translate":19},[18],[15,12137,12139],{"className":12138},[23],[25,12140,12141],{"xmlns":27},[29,12142,12143,12147],{},[32,12144,12145],{},[35,12146,5864],{},[40,12148,5864],{"encoding":42},[15,12150,12152],{"className":12151,"ariaHidden":48},[47],[15,12153,12155,12158],{"className":12154},[52],[15,12156],{"className":12157,"style":6032},[56],[15,12159,5864],{"className":12160},[61,122]," 与仅含 ",[15,12163,12165,12178],{"className":12164,"translate":19},[18],[15,12166,12168],{"className":12167},[23],[25,12169,12170],{"xmlns":27},[29,12171,12172,12176],{},[32,12173,12174],{},[35,12175,1029],{"mathvariant":251},[40,12177,2698],{"encoding":42},[15,12179,12181],{"className":12180,"ariaHidden":48},[47],[15,12182,12184,12187],{"className":12183},[52],[15,12185],{"className":12186,"style":2708},[56],[15,12188,1029],{"className":12189},[61,411]," 的两个因子的乘积，故 ",[15,12192,12194,12207],{"className":12193,"translate":19},[18],[15,12195,12197],{"className":12196},[23],[25,12198,12199],{"xmlns":27},[29,12200,12201,12205],{},[32,12202,12203],{},[35,12204,5864],{},[40,12206,5864],{"encoding":42},[15,12208,12210],{"className":12209,"ariaHidden":48},[47],[15,12211,12213,12216],{"className":12212},[52],[15,12214],{"className":12215,"style":6032},[56],[15,12217,5864],{"className":12218},[61,122],[15,12220,12222,12235],{"className":12221,"translate":19},[18],[15,12223,12225],{"className":12224},[23],[25,12226,12227],{"xmlns":27},[29,12228,12229,12233],{},[32,12230,12231],{},[35,12232,1029],{"mathvariant":251},[40,12234,2698],{"encoding":42},[15,12236,12238],{"className":12237,"ariaHidden":48},[47],[15,12239,12241,12244],{"className":12240},[52],[15,12242],{"className":12243,"style":2708},[56],[15,12245,1029],{"className":12246},[61,411]," 独立，且各自的边缘分布如命题所述。",[11,12249,12250,12251,12297,12298,12326,12327,12509,12510,12513],{},"命题 1.4 给出了三条即时的推论。其一，它提供了一个 ",[15,12252,12254,12275],{"className":12253,"translate":19},[18],[15,12255,12257],{"className":12256},[23],[25,12258,12259],{"xmlns":27},[29,12260,12261,12272],{},[32,12262,12263,12266,12268,12270],{},[35,12264,12265],{},"O",[81,12267,741],{"stretchy":86},[35,12269,102],{},[81,12271,750],{"stretchy":86},[40,12273,12274],{"encoding":42},"O(K)",[15,12276,12278],{"className":12277,"ariaHidden":48},[47],[15,12279,12281,12284,12288,12291,12294],{"className":12280},[52],[15,12282],{"className":12283,"style":142},[56],[15,12285,12265],{"className":12286,"style":12287},[61,122],"margin-right:0.0278em;",[15,12289,741],{"className":12290},[146],[15,12292,102],{"className":12293,"style":174},[61,122],[15,12295,750],{"className":12296},[178]," 的精确采样算法：独立抽取 ",[15,12299,12301,12314],{"className":12300,"translate":19},[18],[15,12302,12304],{"className":12303},[23],[25,12305,12306],{"xmlns":27},[29,12307,12308,12312],{},[32,12309,12310],{},[35,12311,102],{},[40,12313,102],{"encoding":42},[15,12315,12317],{"className":12316,"ariaHidden":48},[47],[15,12318,12320,12323],{"className":12319},[52],[15,12321],{"className":12322,"style":57},[56],[15,12324,102],{"className":12325,"style":174},[61,122]," 个 Gamma 变量再归一化，无需拒绝采样或逆变换。其二，它说明参数 ",[15,12328,12330,12362],{"className":12329,"translate":19},[18],[15,12331,12333],{"className":12332},[23],[25,12334,12335],{"xmlns":27},[29,12336,12337,12359],{},[32,12338,12339,12345,12347,12353],{},[266,12340,12341,12343],{},[35,12342,3106],{},[89,12344,278],{},[81,12346,244],{},[266,12348,12349,12351],{},[81,12350,293],{},[35,12352,1070],{},[266,12354,12355,12357],{},[35,12356,3106],{},[35,12358,1070],{},[40,12360,12361],{"encoding":42},"\\alpha_0 = \\sum_i \\alpha_i",[15,12363,12365,12420],{"className":12364,"ariaHidden":48},[47],[15,12366,12368,12371,12411,12414,12417],{"className":12367},[52],[15,12369],{"className":12370,"style":2511},[56],[15,12372,12374,12377],{"className":12373},[61],[15,12375,3106],{"className":12376,"style":3191},[61,122],[15,12378,12380],{"className":12379},[338],[15,12381,12383,12403],{"className":12382},[342,472],[15,12384,12386,12400],{"className":12385},[346],[15,12387,12389],{"className":12388,"style":1214},[350],[15,12390,12391,12394],{"style":3206},[15,12392],{"className":12393,"style":358},[357],[15,12395,12397],{"className":12396},[362,363,364,365],[15,12398,278],{"className":12399},[61,365],[15,12401,497],{"className":12402},[496],[15,12404,12406],{"className":12405},[346],[15,12407,12409],{"className":12408,"style":504},[350],[15,12410],{},[15,12412],{"className":12413,"style":128},[127],[15,12415,244],{"className":12416},[132],[15,12418],{"className":12419,"style":128},[127],[15,12421,12423,12426,12466,12469],{"className":12422},[52],[15,12424],{"className":12425,"style":6051},[56],[15,12427,12429,12432],{"className":12428},[531],[15,12430,293],{"className":12431,"style":537},[531,535,536],[15,12433,12435],{"className":12434},[338],[15,12436,12438,12458],{"className":12437},[342,472],[15,12439,12441,12455],{"className":12440},[346],[15,12442,12444],{"className":12443,"style":6070},[350],[15,12445,12446,12449],{"style":553},[15,12447],{"className":12448,"style":358},[357],[15,12450,12452],{"className":12451},[362,363,364,365],[15,12453,1070],{"className":12454},[61,122,365],[15,12456,497],{"className":12457},[496],[15,12459,12461],{"className":12460},[346],[15,12462,12464],{"className":12463,"style":593},[350],[15,12465],{},[15,12467],{"className":12468,"style":157},[127],[15,12470,12472,12475],{"className":12471},[61],[15,12473,3106],{"className":12474,"style":3191},[61,122],[15,12476,12478],{"className":12477},[338],[15,12479,12481,12501],{"className":12480},[342,472],[15,12482,12484,12498],{"className":12483},[346],[15,12485,12487],{"className":12486,"style":1364},[350],[15,12488,12489,12492],{"style":3206},[15,12490],{"className":12491,"style":358},[357],[15,12493,12495],{"className":12494},[362,363,364,365],[15,12496,1070],{"className":12497},[61,122,365],[15,12499,497],{"className":12500},[496],[15,12502,12504],{"className":12503},[346],[15,12505,12507],{"className":12506,"style":504},[350],[15,12508],{}," 在概率意义上是",[661,12511,12512],{},"先于归一化的总质量","，这一角色将在 2.2 节被重新参数化为一个精度参数。其三，它把迪利克雷分布与 Gamma 分布、进而与 Poisson 过程联系起来，这正是 2.5 节分裂构造的来源。",[877,12515,12517],{"id":12516},"_14-与-beta-分布多项分布的关系","1.4 与 Beta 分布、多项分布的关系",[11,12519,12520,12523,12524,12574,12575,12731,12732,12802],{},[661,12521,12522],{},"退化到 Beta 分布。"," 当 ",[15,12525,12527,12544],{"className":12526,"translate":19},[18],[15,12528,12530],{"className":12529},[23],[25,12531,12532],{"xmlns":27},[29,12533,12534,12542],{},[32,12535,12536,12538,12540],{},[35,12537,102],{},[81,12539,244],{},[89,12541,908],{},[40,12543,2128],{"encoding":42},[15,12545,12547,12565],{"className":12546,"ariaHidden":48},[47],[15,12548,12550,12553,12556,12559,12562],{"className":12549},[52],[15,12551],{"className":12552,"style":57},[56],[15,12554,102],{"className":12555,"style":174},[61,122],[15,12557],{"className":12558,"style":128},[127],[15,12560,244],{"className":12561},[132],[15,12563],{"className":12564,"style":128},[127],[15,12566,12568,12571],{"className":12567},[52],[15,12569],{"className":12570,"style":859},[56],[15,12572,908],{"className":12573},[61]," 时，约束 ",[15,12576,12578,12608],{"className":12577,"translate":19},[18],[15,12579,12581],{"className":12580},[23],[25,12582,12583],{"xmlns":27},[29,12584,12585,12605],{},[32,12586,12587,12593,12595,12601,12603],{},[266,12588,12589,12591],{},[35,12590,1029],{},[89,12592,91],{},[81,12594,7762],{},[266,12596,12597,12599],{},[35,12598,1029],{},[89,12600,908],{},[81,12602,244],{},[89,12604,91],{},[40,12606,12607],{"encoding":42},"x_1 + x_2 = 1",[15,12609,12611,12667,12722],{"className":12610,"ariaHidden":48},[47],[15,12612,12614,12618,12658,12661,12664],{"className":12613},[52],[15,12615],{"className":12616,"style":12617},[56],"height:0.7333em;vertical-align:-0.15em;",[15,12619,12621,12624],{"className":12620},[61],[15,12622,1029],{"className":12623},[61,122],[15,12625,12627],{"className":12626},[338],[15,12628,12630,12650],{"className":12629},[342,472],[15,12631,12633,12647],{"className":12632},[346],[15,12634,12636],{"className":12635,"style":1214},[350],[15,12637,12638,12641],{"style":482},[15,12639],{"className":12640,"style":358},[357],[15,12642,12644],{"className":12643},[362,363,364,365],[15,12645,91],{"className":12646},[61,365],[15,12648,497],{"className":12649},[496],[15,12651,12653],{"className":12652},[346],[15,12654,12656],{"className":12655,"style":504},[350],[15,12657],{},[15,12659],{"className":12660,"style":772},[127],[15,12662,7762],{"className":12663},[375],[15,12665],{"className":12666,"style":772},[127],[15,12668,12670,12673,12713,12716,12719],{"className":12669},[52],[15,12671],{"className":12672,"style":2511},[56],[15,12674,12676,12679],{"className":12675},[61],[15,12677,1029],{"className":12678},[61,122],[15,12680,12682],{"className":12681},[338],[15,12683,12685,12705],{"className":12684},[342,472],[15,12686,12688,12702],{"className":12687},[346],[15,12689,12691],{"className":12690,"style":1214},[350],[15,12692,12693,12696],{"style":482},[15,12694],{"className":12695,"style":358},[357],[15,12697,12699],{"className":12698},[362,363,364,365],[15,12700,908],{"className":12701},[61,365],[15,12703,497],{"className":12704},[496],[15,12706,12708],{"className":12707},[346],[15,12709,12711],{"className":12710,"style":504},[350],[15,12712],{},[15,12714],{"className":12715,"style":128},[127],[15,12717,244],{"className":12718},[132],[15,12720],{"className":12721,"style":128},[127],[15,12723,12725,12728],{"className":12724},[52],[15,12726],{"className":12727,"style":859},[56],[15,12729,91],{"className":12730},[61]," 使分布由单一变量 ",[15,12733,12735,12753],{"className":12734,"translate":19},[18],[15,12736,12738],{"className":12737},[23],[25,12739,12740],{"xmlns":27},[29,12741,12742,12750],{},[32,12743,12744],{},[266,12745,12746,12748],{},[35,12747,1029],{},[89,12749,91],{},[40,12751,12752],{"encoding":42},"x_1",[15,12754,12756],{"className":12755,"ariaHidden":48},[47],[15,12757,12759,12762],{"className":12758},[52],[15,12760],{"className":12761,"style":2511},[56],[15,12763,12765,12768],{"className":12764},[61],[15,12766,1029],{"className":12767},[61,122],[15,12769,12771],{"className":12770},[338],[15,12772,12774,12794],{"className":12773},[342,472],[15,12775,12777,12791],{"className":12776},[346],[15,12778,12780],{"className":12779,"style":1214},[350],[15,12781,12782,12785],{"style":482},[15,12783],{"className":12784,"style":358},[357],[15,12786,12788],{"className":12787},[362,363,364,365],[15,12789,91],{"className":12790},[61,365],[15,12792,497],{"className":12793},[496],[15,12795,12797],{"className":12796},[346],[15,12798,12800],{"className":12799,"style":504},[350],[15,12801],{}," 完全决定，密度退化为",[15,12804,12806],{"className":12805,"translate":19},[212],[15,12807,12809,12980],{"className":12808,"translate":19},[18],[15,12810,12812],{"className":12811},[23],[25,12813,12814],{"xmlns":27,"display":221},[29,12815,12816,12977],{},[32,12817,12818,12820,12822,12828,12830,12836,12838,12844,12846,12848,12920,12922,12924,12927,12929,12931,12933,12935,12941,12943,12949,12951,12954,12957,12959,12961,12967,12969,12975],{},[35,12819,3707],{},[81,12821,741],{"stretchy":86},[266,12823,12824,12826],{},[35,12825,1029],{},[89,12827,91],{},[81,12829,3714],{},[266,12831,12832,12834],{},[35,12833,3106],{},[89,12835,91],{},[81,12837,94],{"separator":48},[266,12839,12840,12842],{},[35,12841,3106],{},[89,12843,908],{},[81,12845,750],{"stretchy":86},[81,12847,244],{},[3722,12849,12850,12898],{},[32,12851,12852,12870,12872,12874,12876,12882],{},[289,12853,12854,12856,12858],{},[35,12855,1029],{},[89,12857,91],{},[32,12859,12860,12866,12868],{},[266,12861,12862,12864],{},[35,12863,3106],{},[89,12865,91],{},[81,12867,239],{},[89,12869,91],{},[81,12871,741],{"stretchy":86},[89,12873,91],{},[81,12875,239],{},[266,12877,12878,12880],{},[35,12879,1029],{},[89,12881,91],{},[227,12883,12884,12886],{},[81,12885,750],{"stretchy":86},[32,12887,12888,12894,12896],{},[266,12889,12890,12892],{},[35,12891,3106],{},[89,12893,908],{},[81,12895,239],{},[89,12897,91],{},[32,12899,12900,12902,12904,12910,12912,12918],{},[35,12901,3730],{},[81,12903,741],{"stretchy":86},[266,12905,12906,12908],{},[35,12907,3106],{},[89,12909,91],{},[81,12911,94],{"separator":48},[266,12913,12914,12916],{},[35,12915,3106],{},[89,12917,908],{},[81,12919,750],{"stretchy":86},[81,12921,94],{"separator":48},[127,12923],{"width":9847},[282,12925,12926],{},"即",[127,12928],{"width":9847},[35,12930,4186],{"mathvariant":231},[81,12932,1547],{},[81,12934,741],{"stretchy":86},[266,12936,12937,12939],{},[35,12938,3106],{},[89,12940,91],{},[81,12942,94],{"separator":48},[266,12944,12945,12947],{},[35,12946,3106],{},[89,12948,908],{},[81,12950,750],{"stretchy":86},[81,12952,12953],{},"≡",[35,12955,12956],{"mathvariant":231},"Beta",[81,12958,1547],{},[81,12960,741],{"stretchy":86},[266,12962,12963,12965],{},[35,12964,3106],{},[89,12966,91],{},[81,12968,94],{"separator":48},[266,12970,12971,12973],{},[35,12972,3106],{},[89,12974,908],{},[81,12976,750],{"stretchy":86},[40,12978,12979],{"encoding":42},"f(x_1 \\mid \\alpha_1, \\alpha_2) = \\frac{x_1^{\\alpha_1 - 1}(1 - x_1)^{\\alpha_2 - 1}}{B(\\alpha_1, \\alpha_2)},\n\\qquad \\text{即} \\qquad \\operatorname{Dir}(\\alpha_1, \\alpha_2) \\equiv \\operatorname{Beta}(\\alpha_1, \\alpha_2)",[15,12981,12983,13044,13148,13666],{"className":12982,"ariaHidden":48},[47],[15,12984,12986,12989,12992,12995,13035,13038,13041],{"className":12985},[52],[15,12987],{"className":12988,"style":142},[56],[15,12990,3707],{"className":12991,"style":3785},[61,122],[15,12993,741],{"className":12994},[146],[15,12996,12998,13001],{"className":12997},[61],[15,12999,1029],{"className":13000},[61,122],[15,13002,13004],{"className":13003},[338],[15,13005,13007,13027],{"className":13006},[342,472],[15,13008,13010,13024],{"className":13009},[346],[15,13011,13013],{"className":13012,"style":1214},[350],[15,13014,13015,13018],{"style":482},[15,13016],{"className":13017,"style":358},[357],[15,13019,13021],{"className":13020},[362,363,364,365],[15,13022,91],{"className":13023},[61,365],[15,13025,497],{"className":13026},[496],[15,13028,13030],{"className":13029},[346],[15,13031,13033],{"className":13032,"style":504},[350],[15,13034],{},[15,13036],{"className":13037,"style":128},[127],[15,13039,3714],{"className":13040},[132],[15,13042],{"className":13043,"style":128},[127],[15,13045,13047,13050,13090,13093,13096,13136,13139,13142,13145],{"className":13046},[52],[15,13048],{"className":13049,"style":142},[56],[15,13051,13053,13056],{"className":13052},[61],[15,13054,3106],{"className":13055,"style":3191},[61,122],[15,13057,13059],{"className":13058},[338],[15,13060,13062,13082],{"className":13061},[342,472],[15,13063,13065,13079],{"className":13064},[346],[15,13066,13068],{"className":13067,"style":1214},[350],[15,13069,13070,13073],{"style":3206},[15,13071],{"className":13072,"style":358},[357],[15,13074,13076],{"className":13075},[362,363,364,365],[15,13077,91],{"className":13078},[61,365],[15,13080,497],{"className":13081},[496],[15,13083,13085],{"className":13084},[346],[15,13086,13088],{"className":13087,"style":504},[350],[15,13089],{},[15,13091,94],{"className":13092},[153],[15,13094],{"className":13095,"style":157},[127],[15,13097,13099,13102],{"className":13098},[61],[15,13100,3106],{"className":13101,"style":3191},[61,122],[15,13103,13105],{"className":13104},[338],[15,13106,13108,13128],{"className":13107},[342,472],[15,13109,13111,13125],{"className":13110},[346],[15,13112,13114],{"className":13113,"style":1214},[350],[15,13115,13116,13119],{"style":3206},[15,13117],{"className":13118,"style":358},[357],[15,13120,13122],{"className":13121},[362,363,364,365],[15,13123,908],{"className":13124},[61,365],[15,13126,497],{"className":13127},[496],[15,13129,13131],{"className":13130},[346],[15,13132,13134],{"className":13133,"style":504},[350],[15,13135],{},[15,13137,750],{"className":13138},[178],[15,13140],{"className":13141,"style":128},[127],[15,13143,244],{"className":13144},[132],[15,13146],{"className":13147,"style":128},[127],[15,13149,13151,13155,13536,13539,13542,13545,13553,13556,13559,13565,13568,13608,13611,13614,13654,13657,13660,13663],{"className":13150},[52],[15,13152],{"className":13153,"style":13154},[56],"height:2.509em;vertical-align:-0.936em;",[15,13156,13158,13161,13533],{"className":13157},[61],[15,13159],{"className":13160},[146,3841],[15,13162,13164],{"className":13163},[3722],[15,13165,13167,13525],{"className":13166},[342,472],[15,13168,13170,13522],{"className":13169},[346],[15,13171,13174,13277,13285],{"className":13172,"style":13173},[350],"height:1.573em;",[15,13175,13176,13179],{"style":11402},[15,13177],{"className":13178,"style":3861},[357],[15,13180,13182,13185,13188,13228,13231,13234,13274],{"className":13181},[61],[15,13183,3730],{"className":13184,"style":3871},[61,122],[15,13186,741],{"className":13187},[146],[15,13189,13191,13194],{"className":13190},[61],[15,13192,3106],{"className":13193,"style":3191},[61,122],[15,13195,13197],{"className":13196},[338],[15,13198,13200,13220],{"className":13199},[342,472],[15,13201,13203,13217],{"className":13202},[346],[15,13204,13206],{"className":13205,"style":1214},[350],[15,13207,13208,13211],{"style":3206},[15,13209],{"className":13210,"style":358},[357],[15,13212,13214],{"className":13213},[362,363,364,365],[15,13215,91],{"className":13216},[61,365],[15,13218,497],{"className":13219},[496],[15,13221,13223],{"className":13222},[346],[15,13224,13226],{"className":13225,"style":504},[350],[15,13227],{},[15,13229,94],{"className":13230},[153],[15,13232],{"className":13233,"style":157},[127],[15,13235,13237,13240],{"className":13236},[61],[15,13238,3106],{"className":13239,"style":3191},[61,122],[15,13241,13243],{"className":13242},[338],[15,13244,13246,13266],{"className":13245},[342,472],[15,13247,13249,13263],{"className":13248},[346],[15,13250,13252],{"className":13251,"style":1214},[350],[15,13253,13254,13257],{"style":3206},[15,13255],{"className":13256,"style":358},[357],[15,13258,13260],{"className":13259},[362,363,364,365],[15,13261,908],{"className":13262},[61,365],[15,13264,497],{"className":13265},[496],[15,13267,13269],{"className":13268},[346],[15,13270,13272],{"className":13271,"style":504},[350],[15,13273],{},[15,13275,750],{"className":13276},[178],[15,13278,13279,13282],{"style":3889},[15,13280],{"className":13281,"style":3861},[357],[15,13283],{"className":13284,"style":3897},[3896],[15,13286,13287,13290],{"style":11468},[15,13288],{"className":13289,"style":3861},[357],[15,13291,13293,13392,13395,13398,13401,13404,13407,13447],{"className":13292},[61],[15,13294,13296,13299],{"className":13295},[61],[15,13297,1029],{"className":13298},[61,122],[15,13300,13302],{"className":13301},[338],[15,13303,13305,13383],{"className":13304},[342,472],[15,13306,13308,13380],{"className":13307},[346],[15,13309,13311,13323],{"className":13310,"style":4012},[350],[15,13312,13314,13317],{"style":13313},"top:-2.4337em;margin-left:0em;margin-right:0.05em;",[15,13315],{"className":13316,"style":358},[357],[15,13318,13320],{"className":13319},[362,363,364,365],[15,13321,91],{"className":13322},[61,365],[15,13324,13325,13328],{"style":4027},[15,13326],{"className":13327,"style":358},[357],[15,13329,13331],{"className":13330},[362,363,364,365],[15,13332,13334,13374,13377],{"className":13333},[61,365],[15,13335,13337,13340],{"className":13336},[61,365],[15,13338,3106],{"className":13339,"style":3191},[61,122,365],[15,13341,13343],{"className":13342},[338],[15,13344,13346,13366],{"className":13345},[342,472],[15,13347,13349,13363],{"className":13348},[346],[15,13350,13352],{"className":13351,"style":4425},[350],[15,13353,13354,13357],{"style":4058},[15,13355],{"className":13356,"style":4062},[357],[15,13358,13360],{"className":13359},[362,4066,4067,365],[15,13361,91],{"className":13362},[61,365],[15,13364,497],{"className":13365},[496],[15,13367,13369],{"className":13368},[346],[15,13370,13372],{"className":13371,"style":4080},[350],[15,13373],{},[15,13375,239],{"className":13376},[375,365],[15,13378,91],{"className":13379},[61,365],[15,13381,497],{"className":13382},[496],[15,13384,13386],{"className":13385},[346],[15,13387,13390],{"className":13388,"style":13389},[350],"height:0.2663em;",[15,13391],{},[15,13393,741],{"className":13394},[146],[15,13396,91],{"className":13397},[61],[15,13399],{"className":13400,"style":772},[127],[15,13402,239],{"className":13403},[375],[15,13405],{"className":13406,"style":772},[127],[15,13408,13410,13413],{"className":13409},[61],[15,13411,1029],{"className":13412},[61,122],[15,13414,13416],{"className":13415},[338],[15,13417,13419,13439],{"className":13418},[342,472],[15,13420,13422,13436],{"className":13421},[346],[15,13423,13425],{"className":13424,"style":1214},[350],[15,13426,13427,13430],{"style":482},[15,13428],{"className":13429,"style":358},[357],[15,13431,13433],{"className":13432},[362,363,364,365],[15,13434,91],{"className":13435},[61,365],[15,13437,497],{"className":13438},[496],[15,13440,13442],{"className":13441},[346],[15,13443,13445],{"className":13444,"style":504},[350],[15,13446],{},[15,13448,13450,13453],{"className":13449},[178],[15,13451,750],{"className":13452},[178],[15,13454,13456],{"className":13455},[338],[15,13457,13459],{"className":13458},[342],[15,13460,13462],{"className":13461},[346],[15,13463,13465],{"className":13464,"style":7905},[350],[15,13466,13467,13470],{"style":716},[15,13468],{"className":13469,"style":358},[357],[15,13471,13473],{"className":13472},[362,363,364,365],[15,13474,13476,13516,13519],{"className":13475},[61,365],[15,13477,13479,13482],{"className":13478},[61,365],[15,13480,3106],{"className":13481,"style":3191},[61,122,365],[15,13483,13485],{"className":13484},[338],[15,13486,13488,13508],{"className":13487},[342,472],[15,13489,13491,13505],{"className":13490},[346],[15,13492,13494],{"className":13493,"style":4425},[350],[15,13495,13496,13499],{"style":4058},[15,13497],{"className":13498,"style":4062},[357],[15,13500,13502],{"className":13501},[362,4066,4067,365],[15,13503,908],{"className":13504},[61,365],[15,13506,497],{"className":13507},[496],[15,13509,13511],{"className":13510},[346],[15,13512,13514],{"className":13513,"style":4080},[350],[15,13515],{},[15,13517,239],{"className":13518},[375,365],[15,13520,91],{"className":13521},[61,365],[15,13523,497],{"className":13524},[496],[15,13526,13528],{"className":13527},[346],[15,13529,13531],{"className":13530,"style":11594},[350],[15,13532],{},[15,13534],{"className":13535},[178,3841],[15,13537,94],{"className":13538},[153],[15,13540],{"className":13541,"style":10035},[127],[15,13543],{"className":13544,"style":157},[127],[15,13546,13549],{"className":13547},[61,13548],"text",[15,13550,12926],{"className":13551},[61,13552],"cjk_fallback",[15,13554],{"className":13555,"style":10035},[127],[15,13557],{"className":13558,"style":157},[127],[15,13560,13562],{"className":13561},[531],[15,13563,4186],{"className":13564},[61,1622],[15,13566,741],{"className":13567},[146],[15,13569,13571,13574],{"className":13570},[61],[15,13572,3106],{"className":13573,"style":3191},[61,122],[15,13575,13577],{"className":13576},[338],[15,13578,13580,13600],{"className":13579},[342,472],[15,13581,13583,13597],{"className":13582},[346],[15,13584,13586],{"className":13585,"style":1214},[350],[15,13587,13588,13591],{"style":3206},[15,13589],{"className":13590,"style":358},[357],[15,13592,13594],{"className":13593},[362,363,364,365],[15,13595,91],{"className":13596},[61,365],[15,13598,497],{"className":13599},[496],[15,13601,13603],{"className":13602},[346],[15,13604,13606],{"className":13605,"style":504},[350],[15,13607],{},[15,13609,94],{"className":13610},[153],[15,13612],{"className":13613,"style":157},[127],[15,13615,13617,13620],{"className":13616},[61],[15,13618,3106],{"className":13619,"style":3191},[61,122],[15,13621,13623],{"className":13622},[338],[15,13624,13626,13646],{"className":13625},[342,472],[15,13627,13629,13643],{"className":13628},[346],[15,13630,13632],{"className":13631,"style":1214},[350],[15,13633,13634,13637],{"style":3206},[15,13635],{"className":13636,"style":358},[357],[15,13638,13640],{"className":13639},[362,363,364,365],[15,13641,908],{"className":13642},[61,365],[15,13644,497],{"className":13645},[496],[15,13647,13649],{"className":13648},[346],[15,13650,13652],{"className":13651,"style":504},[350],[15,13653],{},[15,13655,750],{"className":13656},[178],[15,13658],{"className":13659,"style":128},[127],[15,13661,12953],{"className":13662},[132],[15,13664],{"className":13665,"style":128},[127],[15,13667,13669,13672,13678,13681,13721,13724,13727,13767],{"className":13668},[52],[15,13670],{"className":13671,"style":142},[56],[15,13673,13675],{"className":13674},[531],[15,13676,12956],{"className":13677},[61,1622],[15,13679,741],{"className":13680},[146],[15,13682,13684,13687],{"className":13683},[61],[15,13685,3106],{"className":13686,"style":3191},[61,122],[15,13688,13690],{"className":13689},[338],[15,13691,13693,13713],{"className":13692},[342,472],[15,13694,13696,13710],{"className":13695},[346],[15,13697,13699],{"className":13698,"style":1214},[350],[15,13700,13701,13704],{"style":3206},[15,13702],{"className":13703,"style":358},[357],[15,13705,13707],{"className":13706},[362,363,364,365],[15,13708,91],{"className":13709},[61,365],[15,13711,497],{"className":13712},[496],[15,13714,13716],{"className":13715},[346],[15,13717,13719],{"className":13718,"style":504},[350],[15,13720],{},[15,13722,94],{"className":13723},[153],[15,13725],{"className":13726,"style":157},[127],[15,13728,13730,13733],{"className":13729},[61],[15,13731,3106],{"className":13732,"style":3191},[61,122],[15,13734,13736],{"className":13735},[338],[15,13737,13739,13759],{"className":13738},[342,472],[15,13740,13742,13756],{"className":13741},[346],[15,13743,13745],{"className":13744,"style":1214},[350],[15,13746,13747,13750],{"style":3206},[15,13748],{"className":13749,"style":358},[357],[15,13751,13753],{"className":13752},[362,363,364,365],[15,13754,908],{"className":13755},[61,365],[15,13757,497],{"className":13758},[496],[15,13760,13762],{"className":13761},[346],[15,13763,13765],{"className":13764,"style":504},[350],[15,13766],{},[15,13768,750],{"className":13769},[178],[11,13771,13772,13773,13824],{},"故迪利克雷分布是 Beta 分布在维度上的推广。下文所有关于迪利克雷分布的陈述，在 ",[15,13774,13776,13794],{"className":13775,"translate":19},[18],[15,13777,13779],{"className":13778},[23],[25,13780,13781],{"xmlns":27},[29,13782,13783,13791],{},[32,13784,13785,13787,13789],{},[35,13786,102],{},[81,13788,244],{},[89,13790,908],{},[40,13792,13793],{"encoding":42},"K=2",[15,13795,13797,13815],{"className":13796,"ariaHidden":48},[47],[15,13798,13800,13803,13806,13809,13812],{"className":13799},[52],[15,13801],{"className":13802,"style":57},[56],[15,13804,102],{"className":13805,"style":174},[61,122],[15,13807],{"className":13808,"style":128},[127],[15,13810,244],{"className":13811},[132],[15,13813],{"className":13814,"style":128},[127],[15,13816,13818,13821],{"className":13817},[52],[15,13819],{"className":13820,"style":859},[56],[15,13822,908],{"className":13823},[61]," 时都应退化为 Beta 分布的相应陈述，这构成了检验公式正确性的一个廉价判据。",[11,13826,13827,13830,13831,13901,13902,14109,14110,14263,14264,14430,14431,14500],{},[661,13828,13829],{},"边缘分布仍为 Beta。"," 更一般地，任意单个分量 ",[15,13832,13834,13852],{"className":13833,"translate":19},[18],[15,13835,13837],{"className":13836},[23],[25,13838,13839],{"xmlns":27},[29,13840,13841,13849],{},[32,13842,13843],{},[266,13844,13845,13847],{},[35,13846,1029],{},[35,13848,1070],{},[40,13850,13851],{"encoding":42},"x_i",[15,13853,13855],{"className":13854,"ariaHidden":48},[47],[15,13856,13858,13861],{"className":13857},[52],[15,13859],{"className":13860,"style":2511},[56],[15,13862,13864,13867],{"className":13863},[61],[15,13865,1029],{"className":13866},[61,122],[15,13868,13870],{"className":13869},[338],[15,13871,13873,13893],{"className":13872},[342,472],[15,13874,13876,13890],{"className":13875},[346],[15,13877,13879],{"className":13878,"style":1364},[350],[15,13880,13881,13884],{"style":482},[15,13882],{"className":13883,"style":358},[357],[15,13885,13887],{"className":13886},[362,363,364,365],[15,13888,1070],{"className":13889},[61,122,365],[15,13891,497],{"className":13892},[496],[15,13894,13896],{"className":13895},[346],[15,13897,13899],{"className":13898,"style":504},[350],[15,13900],{}," 的边缘分布是 ",[15,13903,13905,13947],{"className":13904,"translate":19},[18],[15,13906,13908],{"className":13907},[23],[25,13909,13910],{"xmlns":27},[29,13911,13912,13944],{},[32,13913,13914,13916,13918,13920,13926,13928,13934,13936,13942],{},[35,13915,12956],{"mathvariant":231},[81,13917,1547],{},[81,13919,741],{"stretchy":86},[266,13921,13922,13924],{},[35,13923,3106],{},[35,13925,1070],{},[81,13927,94],{"separator":48},[266,13929,13930,13932],{},[35,13931,3106],{},[89,13933,278],{},[81,13935,239],{},[266,13937,13938,13940],{},[35,13939,3106],{},[35,13941,1070],{},[81,13943,750],{"stretchy":86},[40,13945,13946],{"encoding":42},"\\operatorname{Beta}(\\alpha_i, \\alpha_0 - \\alpha_i)",[15,13948,13950,14060],{"className":13949,"ariaHidden":48},[47],[15,13951,13953,13956,13962,13965,14005,14008,14011,14051,14054,14057],{"className":13952},[52],[15,13954],{"className":13955,"style":142},[56],[15,13957,13959],{"className":13958},[531],[15,13960,12956],{"className":13961},[61,1622],[15,13963,741],{"className":13964},[146],[15,13966,13968,13971],{"className":13967},[61],[15,13969,3106],{"className":13970,"style":3191},[61,122],[15,13972,13974],{"className":13973},[338],[15,13975,13977,13997],{"className":13976},[342,472],[15,13978,13980,13994],{"className":13979},[346],[15,13981,13983],{"className":13982,"style":1364},[350],[15,13984,13985,13988],{"style":3206},[15,13986],{"className":13987,"style":358},[357],[15,13989,13991],{"className":13990},[362,363,364,365],[15,13992,1070],{"className":13993},[61,122,365],[15,13995,497],{"className":13996},[496],[15,13998,14000],{"className":13999},[346],[15,14001,14003],{"className":14002,"style":504},[350],[15,14004],{},[15,14006,94],{"className":14007},[153],[15,14009],{"className":14010,"style":157},[127],[15,14012,14014,14017],{"className":14013},[61],[15,14015,3106],{"className":14016,"style":3191},[61,122],[15,14018,14020],{"className":14019},[338],[15,14021,14023,14043],{"className":14022},[342,472],[15,14024,14026,14040],{"className":14025},[346],[15,14027,14029],{"className":14028,"style":1214},[350],[15,14030,14031,14034],{"style":3206},[15,14032],{"className":14033,"style":358},[357],[15,14035,14037],{"className":14036},[362,363,364,365],[15,14038,278],{"className":14039},[61,365],[15,14041,497],{"className":14042},[496],[15,14044,14046],{"className":14045},[346],[15,14047,14049],{"className":14048,"style":504},[350],[15,14050],{},[15,14052],{"className":14053,"style":772},[127],[15,14055,239],{"className":14056},[375],[15,14058],{"className":14059,"style":772},[127],[15,14061,14063,14066,14106],{"className":14062},[52],[15,14064],{"className":14065,"style":142},[56],[15,14067,14069,14072],{"className":14068},[61],[15,14070,3106],{"className":14071,"style":3191},[61,122],[15,14073,14075],{"className":14074},[338],[15,14076,14078,14098],{"className":14077},[342,472],[15,14079,14081,14095],{"className":14080},[346],[15,14082,14084],{"className":14083,"style":1364},[350],[15,14085,14086,14089],{"style":3206},[15,14087],{"className":14088,"style":358},[357],[15,14090,14092],{"className":14091},[362,363,364,365],[15,14093,1070],{"className":14094},[61,122,365],[15,14096,497],{"className":14097},[496],[15,14099,14101],{"className":14100},[346],[15,14102,14104],{"className":14103,"style":504},[350],[15,14105],{},[15,14107,750],{"className":14108},[178],"。这一点可由命题 1.4 直接看出：",[15,14111,14113,14147],{"className":14112,"translate":19},[18],[15,14114,14116],{"className":14115},[23],[25,14117,14118],{"xmlns":27},[29,14119,14120,14144],{},[32,14121,14122,14128,14130,14136,14138,14140,14142],{},[266,14123,14124,14126],{},[35,14125,1029],{},[35,14127,1070],{},[81,14129,244],{},[266,14131,14132,14134],{},[35,14133,9410],{},[35,14135,1070],{},[35,14137,3020],{"mathvariant":231},[81,14139,741],{"stretchy":86},[35,14141,5864],{},[81,14143,750],{"stretchy":86},[40,14145,14146],{"encoding":42},"x_i = \\gamma_i \u002F (s)",[15,14148,14150,14205],{"className":14149,"ariaHidden":48},[47],[15,14151,14153,14156,14196,14199,14202],{"className":14152},[52],[15,14154],{"className":14155,"style":2511},[56],[15,14157,14159,14162],{"className":14158},[61],[15,14160,1029],{"className":14161},[61,122],[15,14163,14165],{"className":14164},[338],[15,14166,14168,14188],{"className":14167},[342,472],[15,14169,14171,14185],{"className":14170},[346],[15,14172,14174],{"className":14173,"style":1364},[350],[15,14175,14176,14179],{"style":482},[15,14177],{"className":14178,"style":358},[357],[15,14180,14182],{"className":14181},[362,363,364,365],[15,14183,1070],{"className":14184},[61,122,365],[15,14186,497],{"className":14187},[496],[15,14189,14191],{"className":14190},[346],[15,14192,14194],{"className":14193,"style":504},[350],[15,14195],{},[15,14197],{"className":14198,"style":128},[127],[15,14200,244],{"className":14201},[132],[15,14203],{"className":14204,"style":128},[127],[15,14206,14208,14211,14251,14254,14257,14260],{"className":14207},[52],[15,14209],{"className":14210,"style":142},[56],[15,14212,14214,14217],{"className":14213},[61],[15,14215,9410],{"className":14216,"style":9444},[61,122],[15,14218,14220],{"className":14219},[338],[15,14221,14223,14243],{"className":14222},[342,472],[15,14224,14226,14240],{"className":14225},[346],[15,14227,14229],{"className":14228,"style":1364},[350],[15,14230,14231,14234],{"style":9459},[15,14232],{"className":14233,"style":358},[357],[15,14235,14237],{"className":14236},[362,363,364,365],[15,14238,1070],{"className":14239},[61,122,365],[15,14241,497],{"className":14242},[496],[15,14244,14246],{"className":14245},[346],[15,14247,14249],{"className":14248,"style":504},[350],[15,14250],{},[15,14252,3020],{"className":14253},[61],[15,14255,741],{"className":14256},[146],[15,14258,5864],{"className":14259},[61,122],[15,14261,750],{"className":14262},[178],"，而 ",[15,14265,14267,14301],{"className":14266,"translate":19},[18],[15,14268,14270],{"className":14269},[23],[25,14271,14272],{"xmlns":27},[29,14273,14274,14298],{},[32,14275,14276,14282,14284,14286,14288,14290,14296],{},[266,14277,14278,14280],{},[35,14279,9410],{},[35,14281,1070],{},[81,14283,10429],{},[81,14285,741],{"stretchy":86},[35,14287,5864],{},[81,14289,239],{},[266,14291,14292,14294],{},[35,14293,9410],{},[35,14295,1070],{},[81,14297,750],{"stretchy":86},[40,14299,14300],{"encoding":42},"\\gamma_i \\perp (s - \\gamma_i)",[15,14302,14304,14360,14381],{"className":14303,"ariaHidden":48},[47],[15,14305,14307,14311,14351,14354,14357],{"className":14306},[52],[15,14308],{"className":14309,"style":14310},[56],"height:0.8889em;vertical-align:-0.1944em;",[15,14312,14314,14317],{"className":14313},[61],[15,14315,9410],{"className":14316,"style":9444},[61,122],[15,14318,14320],{"className":14319},[338],[15,14321,14323,14343],{"className":14322},[342,472],[15,14324,14326,14340],{"className":14325},[346],[15,14327,14329],{"className":14328,"style":1364},[350],[15,14330,14331,14334],{"style":9459},[15,14332],{"className":14333,"style":358},[357],[15,14335,14337],{"className":14336},[362,363,364,365],[15,14338,1070],{"className":14339},[61,122,365],[15,14341,497],{"className":14342},[496],[15,14344,14346],{"className":14345},[346],[15,14347,14349],{"className":14348,"style":504},[350],[15,14350],{},[15,14352],{"className":14353,"style":128},[127],[15,14355,10429],{"className":14356},[132],[15,14358],{"className":14359,"style":128},[127],[15,14361,14363,14366,14369,14372,14375,14378],{"className":14362},[52],[15,14364],{"className":14365,"style":142},[56],[15,14367,741],{"className":14368},[146],[15,14370,5864],{"className":14371},[61,122],[15,14373],{"className":14374,"style":772},[127],[15,14376,239],{"className":14377},[375],[15,14379],{"className":14380,"style":772},[127],[15,14382,14384,14387,14427],{"className":14383},[52],[15,14385],{"className":14386,"style":142},[56],[15,14388,14390,14393],{"className":14389},[61],[15,14391,9410],{"className":14392,"style":9444},[61,122],[15,14394,14396],{"className":14395},[338],[15,14397,14399,14419],{"className":14398},[342,472],[15,14400,14402,14416],{"className":14401},[346],[15,14403,14405],{"className":14404,"style":1364},[350],[15,14406,14407,14410],{"style":9459},[15,14408],{"className":14409,"style":358},[357],[15,14411,14413],{"className":14412},[362,363,364,365],[15,14414,1070],{"className":14415},[61,122,365],[15,14417,497],{"className":14418},[496],[15,14420,14422],{"className":14421},[346],[15,14423,14425],{"className":14424,"style":504},[350],[15,14426],{},[15,14428,750],{"className":14429},[178],"，故 ",[15,14432,14434,14451],{"className":14433,"translate":19},[18],[15,14435,14437],{"className":14436},[23],[25,14438,14439],{"xmlns":27},[29,14440,14441,14449],{},[32,14442,14443],{},[266,14444,14445,14447],{},[35,14446,1029],{},[35,14448,1070],{},[40,14450,13851],{"encoding":42},[15,14452,14454],{"className":14453,"ariaHidden":48},[47],[15,14455,14457,14460],{"className":14456},[52],[15,14458],{"className":14459,"style":2511},[56],[15,14461,14463,14466],{"className":14462},[61],[15,14464,1029],{"className":14465},[61,122],[15,14467,14469],{"className":14468},[338],[15,14470,14472,14492],{"className":14471},[342,472],[15,14473,14475,14489],{"className":14474},[346],[15,14476,14478],{"className":14477,"style":1364},[350],[15,14479,14480,14483],{"style":482},[15,14481],{"className":14482,"style":358},[357],[15,14484,14486],{"className":14485},[362,363,364,365],[15,14487,1070],{"className":14488},[61,122,365],[15,14490,497],{"className":14491},[496],[15,14493,14495],{"className":14494},[346],[15,14496,14498],{"className":14497,"style":504},[350],[15,14499],{}," 是相互独立的两个 Gamma 变量之比，其分布为 Beta。",[11,14502,14503,14506,14507,14510,14511,14545,14546,14576,14577,867,14664],{},[661,14504,14505],{},"与多项分布的共轭关系。"," 迪利克雷分布之所以在贝叶斯统计中占据中心位置，在于它是多项分布（等价地，分类分布）的",[661,14508,14509],{},"共轭先验","。其确切含义与算术后果见 2.4 节。此处只指出其直觉：若先验以「伪计数」",[15,14512,14514,14527],{"className":14513,"translate":19},[18],[15,14515,14517],{"className":14516},[23],[25,14518,14519],{"xmlns":27},[29,14520,14521,14525],{},[32,14522,14523],{},[35,14524,3106],{"mathvariant":3105},[40,14526,4147],{"encoding":42},[15,14528,14530],{"className":14529,"ariaHidden":48},[47],[15,14531,14533,14536],{"className":14532},[52],[15,14534],{"className":14535,"style":2708},[56],[15,14537,14539],{"className":14538},[61],[15,14540,14542],{"className":14541},[61],[15,14543,3106],{"className":14544},[61,3166]," 描述对各类别出现次数的初始信念，则观测到计数 ",[15,14547,14549,14564],{"className":14548,"translate":19},[18],[15,14550,14552],{"className":14551},[23],[25,14553,14554],{"xmlns":27},[29,14555,14556,14561],{},[32,14557,14558],{},[35,14559,14560],{"mathvariant":251},"n",[40,14562,14563],{"encoding":42},"\\mathbf{n}",[15,14565,14567],{"className":14566,"ariaHidden":48},[47],[15,14568,14570,14573],{"className":14569},[52],[15,14571],{"className":14572,"style":2708},[56],[15,14574,14560],{"className":14575},[61,411]," 后，信念的更新只是 ",[15,14578,14580,14603],{"className":14579,"translate":19},[18],[15,14581,14583],{"className":14582},[23],[25,14584,14585],{"xmlns":27},[29,14586,14587,14600],{},[32,14588,14589,14591,14594,14596,14598],{},[35,14590,3106],{"mathvariant":3105},[81,14592,14593],{},"↦",[35,14595,3106],{"mathvariant":3105},[81,14597,7762],{},[35,14599,14560],{"mathvariant":251},[40,14601,14602],{"encoding":42},"\\boldsymbol{\\alpha} \\mapsto \\boldsymbol{\\alpha} + \\mathbf{n}",[15,14604,14606,14631,14655],{"className":14605,"ariaHidden":48},[47],[15,14607,14609,14613,14622,14625,14628],{"className":14608},[52],[15,14610],{"className":14611,"style":14612},[56],"height:0.522em;vertical-align:-0.011em;",[15,14614,14616],{"className":14615},[61],[15,14617,14619],{"className":14618},[61],[15,14620,3106],{"className":14621},[61,3166],[15,14623],{"className":14624,"style":128},[127],[15,14626,14593],{"className":14627},[132],[15,14629],{"className":14630,"style":128},[127],[15,14632,14634,14637,14646,14649,14652],{"className":14633},[52],[15,14635],{"className":14636,"style":6737},[56],[15,14638,14640],{"className":14639},[61],[15,14641,14643],{"className":14642},[61],[15,14644,3106],{"className":14645},[61,3166],[15,14647],{"className":14648,"style":772},[127],[15,14650,7762],{"className":14651},[375],[15,14653],{"className":14654,"style":772},[127],[15,14656,14658,14661],{"className":14657},[52],[15,14659],{"className":14660,"style":2708},[56],[15,14662,14560],{"className":14663},[61,411],[661,14665,14666],{},"贝叶斯更新的全部计算复杂度，在这个共轭对中退化为一次向量加法。",[872,14668,14670],{"id":14669},"二解析性质","二、解析性质",[877,14672,14674],{"id":14673},"_21-矩结构","2.1 矩结构",[882,14676,14677],{},[11,14678,14679,9393,14682,14758,14759,14804],{},[661,14680,14681],{},"命题 2.1（期望、方差与协方差）",[15,14683,14685,14710],{"className":14684,"translate":19},[18],[15,14686,14688],{"className":14687},[23],[25,14689,14690],{"xmlns":27},[29,14691,14692,14708],{},[32,14693,14694,14696,14698,14700,14702,14704,14706],{},[35,14695,1029],{"mathvariant":251},[81,14697,4183],{},[35,14699,4186],{"mathvariant":231},[81,14701,1547],{},[81,14703,741],{"stretchy":86},[35,14705,3106],{"mathvariant":3105},[81,14707,750],{"stretchy":86},[40,14709,4197],{"encoding":42},[15,14711,14713,14731],{"className":14712,"ariaHidden":48},[47],[15,14714,14716,14719,14722,14725,14728],{"className":14715},[52],[15,14717],{"className":14718,"style":2708},[56],[15,14720,1029],{"className":14721},[61,411],[15,14723],{"className":14724,"style":128},[127],[15,14726,4183],{"className":14727},[132],[15,14729],{"className":14730,"style":128},[127],[15,14732,14734,14737,14743,14746,14755],{"className":14733},[52],[15,14735],{"className":14736,"style":142},[56],[15,14738,14740],{"className":14739},[531],[15,14741,4186],{"className":14742},[61,1622],[15,14744,741],{"className":14745},[146],[15,14747,14749],{"className":14748},[61],[15,14750,14752],{"className":14751},[61],[15,14753,3106],{"className":14754},[61,3166],[15,14756,750],{"className":14757},[178],"。则对任意 ",[15,14760,14762,14781],{"className":14761,"translate":19},[18],[15,14763,14765],{"className":14764},[23],[25,14766,14767],{"xmlns":27},[29,14768,14769,14778],{},[32,14770,14771,14773,14775],{},[35,14772,1070],{},[81,14774,94],{"separator":48},[35,14776,14777],{},"j",[40,14779,14780],{"encoding":42},"i, 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= \\frac{\\alpha_i}{\\alpha_0}, \\qquad \\operatorname{Var}(x_i) = \\frac{\\alpha_i(\\alpha_0 - \\alpha_i)}{\\alpha_0^2(\\alpha_0 + 1)}, \\qquad \\operatorname{Cov}(x_i, x_j) = \\frac{-\\alpha_i \\alpha_j}{\\alpha_0^2(\\alpha_0 + 1)} \\quad (i \\ne 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s",[15,16090,16092,16147],{"className":16091,"ariaHidden":48},[47],[15,16093,16095,16098,16138,16141,16144],{"className":16094},[52],[15,16096],{"className":16097,"style":2511},[56],[15,16099,16101,16104],{"className":16100},[61],[15,16102,1029],{"className":16103},[61,122],[15,16105,16107],{"className":16106},[338],[15,16108,16110,16130],{"className":16109},[342,472],[15,16111,16113,16127],{"className":16112},[346],[15,16114,16116],{"className":16115,"style":1364},[350],[15,16117,16118,16121],{"style":482},[15,16119],{"className":16120,"style":358},[357],[15,16122,16124],{"className":16123},[362,363,364,365],[15,16125,1070],{"className":16126},[61,122,365],[15,16128,497],{"className":16129},[496],[15,16131,16133],{"className":16132},[346],[15,16134,16136],{"className":16135,"style":504},[350],[15,16137],{},[15,16139],{"className":16140,"style":128},[127],[15,16142,244],{"className":16143},[132],[15,16145],{"className":16146,"style":128},[127],[15,16148,16150,16153,16193,16196],{"className":16149},[52],[15,16151],{"className":16152,"style":142},[56],[15,16154,16156,16159],{"className":16155},[61],[15,16157,9410],{"className":16158,"style":9444},[61,122],[15,16160,16162],{"className":16161},[338],[15,16163,16165,16185],{"className":16164},[342,472],[15,16166,16168,16182],{"className":16167},[346],[15,16169,16171],{"className":16170,"style":1364},[350],[15,16172,16173,16176],{"style":9459},[15,16174],{"className":16175,"style":358},[357],[15,16177,16179],{"className":16178},[362,363,364,365],[15,16180,1070],{"className":16181},[61,122,365],[15,16183,497],{"className":16184},[496],[15,16186,16188],{"className":16187},[346],[15,16189,16191],{"className":16190,"style":504},[350],[15,16192],{},[15,16194,3020],{"className":16195},[61],[15,16197,5864],{"className":16198},[61,122]," 且 ",[15,16201,16203,16225],{"className":16202,"translate":19},[18],[15,16204,16206],{"className":16205},[23],[25,16207,16208],{"xmlns":27},[29,16209,16210,16222],{},[32,16211,16212,16218,16220],{},[266,16213,16214,16216],{},[35,16215,9410],{},[35,16217,1070],{},[81,16219,10429],{},[35,16221,5864],{},[40,16223,16224],{"encoding":42},"\\gamma_i \\perp s",[15,16226,16228,16283],{"className":16227,"ariaHidden":48},[47],[15,16229,16231,16234,16274,16277,16280],{"className":16230},[52],[15,16232],{"className":16233,"style":14310},[56],[15,16235,16237,16240],{"className":16236},[61],[15,16238,9410],{"className":16239,"style":9444},[61,122],[15,16241,16243],{"className":16242},[338],[15,16244,16246,16266],{"className":16245},[342,472],[15,16247,16249,16263],{"className":16248},[346],[15,16250,16252],{"className":16251,"style":1364},[350],[15,16253,16254,16257],{"style":9459},[15,16255],{"className":16256,"style":358},[357],[15,16258,16260],{"className":16259},[362,363,364,365],[15,16261,1070],{"className":16262},[61,122,365],[15,16264,497],{"className":16265},[496],[15,16267,16269],{"className":16268},[346],[15,16270,16272],{"className":16271,"style":504},[350],[15,16273],{},[15,16275],{"className":16276,"style":128},[127],[15,16278,10429],{"className":16279},[132],[15,16281],{"className":16282,"style":128},[127],[15,16284,16286,16289],{"className":16285},[52],[15,16287],{"className":16288,"style":6032},[56],[15,16290,5864],{"className":16291},[61,122],[15,16293,16295,16367],{"className":16294,"translate":19},[18],[15,16296,16298],{"className":16297},[23],[25,16299,16300],{"xmlns":27},[29,16301,16302,16364],{},[32,16303,16304,16306,16308,16314,16316,16318,16320,16322,16328,16330,16332,16334,16336,16338,16340,16342,16344,16346,16352,16354],{},[35,16305,14818],{"mathvariant":258},[81,16307,14821],{"stretchy":86},[266,16309,16310,16312],{},[35,16311,1029],{},[35,16313,1070],{},[81,16315,14830],{"stretchy":86},[81,16317,244],{},[35,16319,14818],{"mathvariant":258},[81,16321,14821],{"stretchy":86},[266,16323,16324,16326],{},[35,16325,9410],{},[35,16327,1070],{},[81,16329,14830],{"stretchy":86},[282,16331,4802],{},[35,16333,14818],{"mathvariant":258},[81,16335,14821],{"stretchy":86},[89,16337,91],{},[35,16339,3020],{"mathvariant":231},[35,16341,5864],{},[81,16343,14830],{"stretchy":86},[81,16345,244],{},[266,16347,16348,16350],{},[35,16349,3106],{},[35,16351,1070],{},[81,16353,11214],{},[3722,16355,16356,16358],{},[89,16357,91],{},[266,16359,16360,16362],{},[35,16361,3106],{},[89,16363,278],{},[40,16365,16366],{"encoding":42},"\\mathbb{E}[x_i] = \\mathbb{E}[\\gamma_i]\\,\\mathbb{E}[1\u002Fs] = \\alpha_i \\cdot \\frac{1}{\\alpha_0}",[15,16368,16370,16434,16516,16572],{"className":16369,"ariaHidden":48},[47],[15,16371,16373,16376,16379,16382,16422,16425,16428,16431],{"className":16372},[52],[15,16374],{"className":16375,"style":142},[56],[15,16377,14818],{"className":16378},[61,427],[15,16380,14821],{"className":16381},[146],[15,16383,16385,16388],{"className":16384},[61],[15,16386,1029],{"className":16387},[61,122],[15,16389,16391],{"className":16390},[338],[15,16392,16394,16414],{"className":16393},[342,472],[15,16395,16397,16411],{"className":16396},[346],[15,16398,16400],{"className":16399,"style":1364},[350],[15,16401,16402,16405],{"style":482},[15,16403],{"className":16404,"style":358},[357],[15,16406,16408],{"className":16407},[362,363,364,365],[15,16409,1070],{"className":16410},[61,122,365],[15,16412,497],{"className":16413},[496],[15,16415,16417],{"className":16416},[346],[15,16418,16420],{"className":16419,"style":504},[350],[15,16421],{},[15,16423,14830],{"className":16424},[178],[15,16426],{"className":16427,"style":128},[127],[15,16429,244],{"className":16430},[132],[15,16432],{"className":16433,"style":128},[127],[15,16435,16437,16440,16443,16446,16486,16489,16492,16495,16498,16501,16504,16507,16510,16513],{"className":16436},[52],[15,16438],{"className":16439,"style":142},[56],[15,16441,14818],{"className":16442},[61,427],[15,16444,14821],{"className":16445},[146],[15,16447,16449,16452],{"className":16448},[61],[15,16450,9410],{"className":16451,"style":9444},[61,122],[15,16453,16455],{"className":16454},[338],[15,16456,16458,16478],{"className":16457},[342,472],[15,16459,16461,16475],{"className":16460},[346],[15,16462,16464],{"className":16463,"style":1364},[350],[15,16465,16466,16469],{"style":9459},[15,16467],{"className":16468,"style":358},[357],[15,16470,16472],{"className":16471},[362,363,364,365],[15,16473,1070],{"className":16474},[61,122,365],[15,16476,497],{"className":16477},[496],[15,16479,16481],{"className":16480},[346],[15,16482,16484],{"className":16483,"style":504},[350],[15,16485],{},[15,16487,14830],{"className":16488},[178],[15,16490],{"className":16491,"style":157},[127],[15,16493,14818],{"className":16494},[61,427],[15,16496,14821],{"className":16497},[146],[15,16499,3049],{"className":16500},[61],[15,16502,5864],{"className":16503},[61,122],[15,16505,14830],{"className":16506},[178],[15,16508],{"className":16509,"style":128},[127],[15,16511,244],{"className":16512},[132],[15,16514],{"className":16515,"style":128},[127],[15,16517,16519,16523,16563,16566,16569],{"className":16518},[52],[15,16520],{"className":16521,"style":16522},[56],"height:0.5945em;vertical-align:-0.15em;",[15,16524,16526,16529],{"className":16525},[61],[15,16527,3106],{"className":16528,"style":3191},[61,122],[15,16530,16532],{"className":16531},[338],[15,16533,16535,16555],{"className":16534},[342,472],[15,16536,16538,16552],{"className":16537},[346],[15,16539,16541],{"className":16540,"style":1364},[350],[15,16542,16543,16546],{"style":3206},[15,16544],{"className":16545,"style":358},[357],[15,16547,16549],{"className":16548},[362,363,364,365],[15,16550,1070],{"className":16551},[61,122,365],[15,16553,497],{"className":16554},[496],[15,16556,16558],{"className":16557},[346],[15,16559,16561],{"className":16560,"style":504},[350],[15,16562],{},[15,16564],{"className":16565,"style":772},[127],[15,16567,11214],{"className":16568},[375],[15,16570],{"className":16571,"style":772},[127],[15,16573,16575,16579],{"className":16574},[52],[15,16576],{"className":16577,"style":16578},[56],"height:1.2902em;vertical-align:-0.4451em;",[15,16580,16582,16585,16681],{"className":16581},[61],[15,16583],{"className":16584},[146,3841],[15,16586,16588],{"className":16587},[3722],[15,16589,16591,16673],{"className":16590},[342,472],[15,16592,16594,16670],{"className":16593},[346],[15,16595,16597,16648,16656],{"className":16596,"style":3854},[350],[15,16598,16599,16602],{"style":3857},[15,16600],{"className":16601,"style":3861},[357],[15,16603,16605],{"className":16604},[362,363,364,365],[15,16606,16608],{"className":16607},[61,365],[15,16609,16611,16614],{"className":16610},[61,365],[15,16612,3106],{"className":16613,"style":3191},[61,122,365],[15,16615,16617],{"className":16616},[338],[15,16618,16620,16640],{"className":16619},[342,472],[15,16621,16623,16637],{"className":16622},[346],[15,16624,16626],{"className":16625,"style":4425},[350],[15,16627,16628,16631],{"style":4058},[15,16629],{"className":16630,"style":4062},[357],[15,16632,16634],{"className":16633},[362,4066,4067,365],[15,16635,278],{"className":16636},[61,365],[15,16638,497],{"className":16639},[496],[15,16641,16643],{"className":16642},[346],[15,16644,16646],{"className":16645,"style":4080},[350],[15,16647],{},[15,16649,16650,16653],{"style":3889},[15,16651],{"className":16652,"style":3861},[357],[15,16654],{"className":16655,"style":3897},[3896],[15,16657,16658,16661],{"style":3900},[15,16659],{"className":16660,"style":3861},[357],[15,16662,16664],{"className"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",[15,16686,16688,16718],{"className":16687,"translate":19},[18],[15,16689,16691],{"className":16690},[23],[25,16692,16693],{"xmlns":27},[29,16694,16695,16715],{},[32,16696,16697,16699,16701,16703,16709,16711,16713],{},[35,16698,9561],{"mathvariant":231},[81,16700,1547],{},[81,16702,741],{"stretchy":86},[266,16704,16705,16707],{},[35,16706,3106],{},[89,16708,278],{},[81,16710,94],{"separator":48},[89,16712,91],{},[81,16714,750],{"stretchy":86},[40,16716,16717],{"encoding":42},"\\operatorname{Gamma}(\\alpha_0,1)",[15,16719,16721],{"className":16720,"ariaHidden":48},[47],[15,16722,16724,16727,16733,16736,16776,16779,16782,16785],{"className":16723},[52],[15,16725],{"className":16726,"style":142},[56],[15,16728,16730],{"className":16729},[531],[15,16731,9561],{"className":16732},[61,1622],[15,16734,741],{"className":16735},[146],[15,16737,16739,16742],{"className":16738},[61],[15,16740,3106],{"className":16741,"style":3191},[61,122],[15,16743,16745],{"className":16744},[338],[15,16746,16748,16768],{"className":16747},[342,472],[15,16749,16751,16765],{"className":16750},[346],[15,16752,16754],{"className":16753,"style":1214},[350],[15,16755,16756,16759],{"style":3206},[15,16757],{"className":16758,"style":358},[357],[15,16760,16762],{"className":16761},[362,363,364,365],[15,16763,278],{"className":16764},[61,365],[15,16766,497],{"className":16767},[496],[15,16769,16771],{"className":16770},[346],[15,16772,16774],{"className":16773,"style":504},[350],[15,16775],{},[15,16777,94],{"className":16778},[153],[15,16780],{"className":16781,"style":157},[127],[15,16783,91],{"className":16784},[61],[15,16786,750],{"className":16787},[178]," 的倒数的期望 ",[15,16790,16792,16836],{"className":16791,"translate":19},[18],[15,16793,16795],{"className":16794},[23],[25,16796,16797],{"xmlns":27},[29,16798,16799,16833],{},[32,16800,16801,16803,16805,16807,16809,16811,16813,16815,16817,16819,16821,16827,16829,16831],{},[35,16802,14818],{"mathvariant":258},[81,16804,14821],{"stretchy":86},[89,16806,91],{},[35,16808,3020],{"mathvariant":231},[35,16810,5864],{},[81,16812,14830],{"stretchy":86},[81,16814,244],{},[89,16816,91],{},[35,16818,3020],{"mathvariant":231},[81,16820,741],{"stretchy":86},[266,16822,16823,16825],{},[35,16824,3106],{},[89,16826,278],{},[81,16828,239],{},[89,16830,91],{},[81,16832,750],{"stretchy":86},[40,16834,16835],{"encoding":42},"\\mathbb{E}[1\u002Fs] = 1\u002F(\\alpha_0 - 1)",[15,16837,16839,16869,16930],{"className":16838,"ariaHidden":48},[47],[15,16840,16842,16845,16848,16851,16854,16857,16860,16863,16866],{"className":16841},[52],[15,16843],{"className":16844,"style":142},[56],[15,16846,14818],{"className":16847},[61,427],[15,16849,14821],{"className":16850},[146],[15,16852,3049],{"className":16853},[61],[15,16855,5864],{"className":16856},[61,122],[15,16858,14830],{"className":16859},[178],[15,16861],{"className":16862,"style":128},[127],[15,16864,244],{"className":16865},[132],[15,16867],{"className":16868,"style":128},[127],[15,16870,16872,16875,16878,16881,16921,16924,16927],{"className":16871},[52],[15,16873],{"className":16874,"style":142},[56],[15,16876,3049],{"className":16877},[61],[15,16879,741],{"className":16880},[146],[15,16882,16884,16887],{"className":16883},[61],[15,16885,3106],{"className":16886,"style":3191},[61,122],[15,16888,16890],{"className":16889},[338],[15,16891,16893,16913],{"className":16892},[342,472],[15,16894,16896,16910],{"className":16895},[346],[15,16897,16899],{"className":16898,"style":1214},[350],[15,16900,16901,16904],{"style":3206},[15,16902],{"className":16903,"style":358},[357],[15,16905,16907],{"className":16906},[362,363,364,365],[15,16908,278],{"className":16909},[61,365],[15,16911,497],{"className":16912},[496],[15,16914,16916],{"className":16915},[346],[15,16917,16919],{"className":16918,"style":504},[350],[15,16920],{},[15,16922],{"className":16923,"style":772},[127],[15,16925,239],{"className":16926},[375],[15,16928],{"className":16929,"style":772},[127],[15,16931,16933,16936,16939],{"className":16932},[52],[15,16934],{"className":16935,"style":142},[56],[15,16937,91],{"className":16938},[61],[15,16940,750],{"className":16941},[178],[15,16943,16945,16975],{"className":16944,"translate":19},[18],[15,16946,16948],{"className":16947},[23],[25,16949,16950],{"xmlns":27},[29,16951,16952,16972],{},[32,16953,16954,16960,16962,16968,16970],{},[266,16955,16956,16958],{},[35,16957,3106],{},[89,16959,278],{},[81,16961,14593],{},[266,16963,16964,16966],{},[35,16965,3106],{},[89,16967,278],{},[81,16969,239],{},[89,16971,91],{},[40,16973,16974],{"encoding":42},"\\alpha_0 \\mapsto \\alpha_0 - 1",[15,16976,16978,17034,17089],{"className":16977,"ariaHidden":48},[47],[15,16979,16981,16985,17025,17028,17031],{"className":16980},[52],[15,16982],{"className":16983,"style":16984},[56],"height:0.661em;vertical-align:-0.15em;",[15,16986,16988,16991],{"className":16987},[61],[15,16989,3106],{"className":16990,"style":3191},[61,122],[15,16992,16994],{"className":16993},[338],[15,16995,16997,17017],{"className":16996},[342,472],[15,16998,17000,17014],{"className":16999},[346],[15,17001,17003],{"className":17002,"style":1214},[350],[15,17004,17005,17008],{"style":3206},[15,17006],{"className":17007,"style":358},[357],[15,17009,17011],{"className":17010},[362,363,364,365],[15,17012,278],{"className":17013},[61,365],[15,17015,497],{"className":17016},[496],[15,17018,17020],{"className":17019},[346],[15,17021,17023],{"className":17022,"style":504},[350],[15,17024],{},[15,17026],{"className":17027,"style":128},[127],[15,17029,14593],{"className":17030},[132],[15,17032],{"className":17033,"style":128},[127],[15,17035,17037,17040,17080,17083,17086],{"className":17036},[52],[15,17038],{"className":17039,"style":12617},[56],[15,17041,17043,17046],{"className":17042},[61],[15,17044,3106],{"className":17045,"style":3191},[61,122],[15,17047,17049],{"className":17048},[338],[15,17050,17052,17072],{"className":17051},[342,472],[15,17053,17055,17069],{"className":17054},[346],[15,17056,17058],{"className":17057,"style":1214},[350],[15,17059,17060,17063],{"style":3206},[15,17061],{"className":17062,"style":358},[357],[15,17064,17066],{"className":17065},[362,363,364,365],[15,17067,278],{"className":17068},[61,365],[15,17070,497],{"className":17071},[496],[15,17073,17075],{"className":17074},[346],[15,17076,17078],{"className":17077,"style":504},[350],[15,17079],{},[15,17081],{"className":17082,"style":772},[127],[15,17084,239],{"className":17085},[375],[15,17087],{"className":17088,"style":772},[127],[15,17090,17092,17095],{"className":17091},[52],[15,17093],{"className":17094,"style":859},[56],[15,17096,91],{"className":17097},[61]," 的平移技巧，或直接由命题 1.3 的矩恒等式 ",[15,17100,17102,17163],{"className":17101,"translate":19},[18],[15,17103,17105],{"className":17104},[23],[25,17106,17107],{"xmlns":27},[29,17108,17109,17160],{},[32,17110,17111,17113,17115,17121,17134,17136,17138,17140,17142,17144,17146,17148,17150,17152,17154,17156,17158],{},[35,17112,14818],{"mathvariant":258},[81,17114,14821],{"stretchy":86},[266,17116,17117,17119],{},[81,17118,3741],{},[35,17120,1070],{},[289,17122,17123,17125,17127],{},[35,17124,1029],{},[35,17126,1070],{},[266,17128,17129,17132],{},[35,17130,17131],{},"β",[35,17133,1070],{},[81,17135,14830],{"stretchy":86},[81,17137,244],{},[35,17139,3730],{},[81,17141,741],{"stretchy":86},[35,17143,3106],{"mathvariant":3105},[81,17145,7762],{},[35,17147,17131],{"mathvariant":3105},[81,17149,750],{"stretchy":86},[35,17151,3020],{"mathvariant":231},[35,17153,3730],{},[81,17155,741],{"stretchy":86},[35,17157,3106],{"mathvariant":3105},[81,17159,750],{"stretchy":86},[40,17161,17162],{"encoding":42},"\\mathbb{E}[\\prod_i x_i^{\\beta_i}] = B(\\boldsymbol{\\alpha} + \\boldsymbol{\\beta})\u002FB(\\boldsymbol{\\alpha})",[15,17164,17166,17329,17359],{"className":17165,"ariaHidden":48},[47],[15,17167,17169,17173,17176,17179,17219,17222,17317,17320,17323,17326],{"className":17168},[52],[15,17170],{"className":17171,"style":17172},[56],"height:1.2667em;vertical-align:-0.2997em;",[15,17174,14818],{"className":17175},[61,427],[15,17177,14821],{"className":17178},[146],[15,17180,17182,17185],{"className":17181},[531],[15,17183,3741],{"className":17184,"style":537},[531,535,536],[15,17186,17188],{"className":17187},[338],[15,17189,17191,17211],{"className":17190},[342,472],[15,17192,17194,17208],{"className":17193},[346],[15,17195,17197],{"className":17196,"style":6070},[350],[15,17198,17199,17202],{"style":553},[15,17200],{"className":17201,"style":358},[357],[15,17203,17205],{"className":17204},[362,363,364,365],[15,17206,1070],{"className":17207},[61,122,365],[15,17209,497],{"className":17210},[496],[15,17212,17214],{"className":17213},[346],[15,17215,17217],{"className":17216,"style":593},[350],[15,17218],{},[15,17220],{"className":17221,"style":157},[127],[15,17223,17225,17228],{"className":17224},[61],[15,17226,1029],{"className":17227},[61,122],[15,17229,17231],{"className":17230},[338],[15,17232,17234,17309],{"className":17233},[342,472],[15,17235,17237,17306],{"className":17236},[346],[15,17238,17241,17252],{"className":17239,"style":17240},[350],"height:0.967em;",[15,17242,17243,17246],{"style":4015},[15,17244],{"className":17245,"style":358},[357],[15,17247,17249],{"className":17248},[362,363,364,365],[15,17250,1070],{"className":17251},[61,122,365],[15,17253,17255,17258],{"style":17254},"top:-3.1809em;margin-right:0.05em;",[15,17256],{"className":17257,"style":358},[357],[15,17259,17261],{"className":17260},[362,363,364,365],[15,17262,17264],{"className":17263},[61,365],[15,17265,17267,17271],{"className":17266},[61,365],[15,17268,17131],{"className":17269,"style":17270},[61,122,365],"margin-right:0.0528em;",[15,17272,17274],{"className":17273},[338],[15,17275,17277,17298],{"className":17276},[342,472],[15,17278,17280,17295],{"className":17279},[346],[15,17281,17283],{"className":17282,"style":4055},[350],[15,17284,17286,17289],{"style":17285},"top:-2.357em;margin-left:-0.0528em;margin-right:0.0714em;",[15,17287],{"className":17288,"style":4062},[357],[15,17290,17292],{"className":17291},[362,4066,4067,365],[15,17293,1070],{"className":17294},[61,122,365],[15,17296,497],{"className":17297},[496],[15,17299,17301],{"className":17300},[346],[15,17302,17304],{"className":17303,"style":4080},[350],[15,17305],{},[15,17307,497],{"className":17308},[496],[15,17310,17312],{"className":17311},[346],[15,17313,17315],{"className":17314,"style":4098},[350],[15,17316],{},[15,17318,14830],{"className":17319},[178],[15,17321],{"className":17322,"style":128},[127],[15,17324,244],{"className":17325},[132],[15,17327],{"className":17328,"style":128},[127],[15,17330,17332,17335,17338,17341,17350,17353,17356],{"className":17331},[52],[15,17333],{"className":17334,"style":142},[56],[15,17336,3730],{"className":17337,"style":3871},[61,122],[15,17339,741],{"className":17340},[146],[15,17342,17344],{"className":17343},[61],[15,17345,17347],{"className":17346},[61],[15,17348,3106],{"className":17349},[61,3166],[15,17351],{"className":17352,"style":772},[127],[15,17354,7762],{"className":17355},[375],[15,17357],{"className":17358,"style":772},[127],[15,17360,17362,17365,17375,17378,17381,17384,17387,17396],{"className":17361},[52],[15,17363],{"className":17364,"style":142},[56],[15,17366,17368],{"className":17367},[61],[15,17369,17371],{"className":17370},[61],[15,17372,17131],{"className":17373,"style":17374},[61,3166],"margin-right:0.034em;",[15,17376,750],{"className":17377},[178],[15,17379,3020],{"className":17380},[61],[15,17382,3730],{"className":17383,"style":3871},[61,122],[15,17385,741],{"className":17386},[146],[15,17388,17390],{"className":17389},[61],[15,17391,17393],{"className":17392},[61],[15,17394,3106],{"className":17395},[61,3166],[15,17397,750],{"className":17398},[178]," 取 ",[15,17401,17403,17425],{"className":17402,"translate":19},[18],[15,17404,17406],{"className":17405},[23],[25,17407,17408],{"xmlns":27},[29,17409,17410,17422],{},[32,17411,17412,17418,17420],{},[266,17413,17414,17416],{},[35,17415,17131],{},[35,17417,1070],{},[81,17419,244],{},[89,17421,91],{},[40,17423,17424],{"encoding":42},"\\beta_i = 1",[15,17426,17428,17484],{"className":17427,"ariaHidden":48},[47],[15,17429,17431,17434,17475,17478,17481],{"className":17430},[52],[15,17432],{"className":17433,"style":14310},[56],[15,17435,17437,17440],{"className":17436},[61],[15,17438,17131],{"className":17439,"style":17270},[61,122],[15,17441,17443],{"className":17442},[338],[15,17444,17446,17467],{"className":17445},[342,472],[15,17447,17449,17464],{"className":17448},[346],[15,17450,17452],{"className":17451,"style":1364},[350],[15,17453,17455,17458],{"style":17454},"top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;",[15,17456],{"className":17457,"style":358},[357],[15,17459,17461],{"className":17460},[362,363,364,365],[15,17462,1070],{"className":17463},[61,122,365],[15,17465,497],{"className":17466},[496],[15,17468,17470],{"className":17469},[346],[15,17471,17473],{"className":17472,"style":504},[350],[15,17474],{},[15,17476],{"className":17477,"style":128},[127],[15,17479,244],{"className":17480},[132],[15,17482],{"className":17483,"style":128},[127],[15,17485,17487,17490],{"className":17486},[52],[15,17488],{"className":17489,"style":859},[56],[15,17491,91],{"className":17492},[61]," 得到。二阶矩由 ",[15,17495,17497,17571],{"className":17496,"translate":19},[18],[15,17498,17500],{"className":17499},[23],[25,17501,17502],{"xmlns":27},[29,17503,17504,17568],{},[32,17505,17506,17508,17510,17518,17520,17522],{},[35,17507,14818],{"mathvariant":258},[81,17509,14821],{"stretchy":86},[289,17511,17512,17514,17516],{},[35,17513,1029],{},[35,17515,1070],{},[89,17517,908],{},[81,17519,14830],{"stretchy":86},[81,17521,244],{},[3722,17523,17524,17546],{},[32,17525,17526,17532,17534,17540,17542,17544],{},[266,17527,17528,17530],{},[35,17529,3106],{},[35,17531,1070],{},[81,17533,741],{"stretchy":86},[266,17535,17536,17538],{},[35,17537,3106],{},[35,17539,1070],{},[81,17541,7762],{},[89,17543,91],{},[81,17545,750],{"stretchy":86},[32,17547,17548,17554,17556,17562,17564,17566],{},[266,17549,17550,17552],{},[35,17551,3106],{},[89,17553,278],{},[81,17555,741],{"stretchy":86},[266,17557,17558,17560],{},[35,17559,3106],{},[89,17561,278],{},[81,17563,7762],{},[89,17565,91],{},[81,17567,750],{"stretchy":86},[40,17569,17570],{"encoding":42},"\\mathbb{E}[x_i^2] = \\frac{\\alpha_i(\\alpha_i+1)}{\\alpha_0(\\alpha_0+1)}",[15,17572,17574,17652],{"className":17573,"ariaHidden":48},[47],[15,17575,17577,17581,17584,17587,17640,17643,17646,17649],{"className":17576},[52],[15,17578],{"className":17579,"style":17580},[56],"height:1.0728em;vertical-align:-0.2587em;",[15,17582,14818],{"className":17583},[61,427],[15,17585,14821],{"className":17586},[146],[15,17588,17590,17593],{"className":17589},[61],[15,17591,1029],{"className":17592},[61,122],[15,17594,17596],{"className":17595},[338],[15,17597,17599,17631],{"className":17598},[342,472],[15,17600,17602,17628],{"className":17601},[346],[15,17603,17605,17617],{"className":17604,"style":7905},[350],[15,17606,17608,17611],{"style":17607},"top:-2.4413em;margin-left:0em;margin-right:0.05em;",[15,17609],{"className":17610,"style":358},[357],[15,17612,17614],{"className":17613},[362,363,364,365],[15,17615,1070],{"className":17616},[61,122,365],[15,17618,17619,17622],{"style":716},[15,17620],{"className":17621,"style":358},[357],[15,17623,17625],{"className":17624},[362,363,364,365],[15,17626,908],{"className":17627},[61,365],[15,17629,497],{"className":17630},[496],[15,17632,17634],{"className":17633},[346],[15,17635,17638],{"className":17636,"style":17637},[350],"height:0.2587em;",[15,17639],{},[15,17641,14830],{"className":17642},[178],[15,17644],{"className":17645,"style":128},[127],[15,17647,244],{"className":17648},[132],[15,17650],{"className":17651,"style":128},[127],[15,17653,17655,17659],{"className":17654},[52],[15,17656],{"className":17657,"style":17658},[56],"height:1.53em;vertical-align:-0.52em;",[15,17660,17662,17665,17902],{"className":17661},[61],[15,17663],{"className":17664},[146,3841],[15,17666,17668],{"className":17667},[3722],[15,17669,17671,17894],{"className":17670},[342,472],[15,17672,17674,17891],{"className":17673},[346],[15,17675,17677,17780,17788],{"className":17676,"style":15321},[350],[15,17678,17679,17682],{"style":3857},[15,17680],{"className":17681,"style":3861},[357],[15,17683,17685],{"className":17684},[362,363,364,365],[15,17686,17688,17728,17731,17771,17774,17777],{"className":17687},[61,365],[15,17689,17691,17694],{"className":17690},[61,365],[15,17692,3106],{"className":17693,"style":3191},[61,122,365],[15,17695,17697],{"className":17696},[338],[15,17698,17700,17720],{"className":17699},[342,472],[15,17701,17703,17717],{"className":17702},[346],[15,17704,17706],{"className":17705,"style":4425},[350],[15,17707,17708,17711],{"style":4058},[15,17709],{"className":17710,"style":4062},[357],[15,17712,17714],{"className":17713},[362,4066,4067,365],[15,17715,278],{"className":17716},[61,365],[15,17718,497],{"className":17719},[496],[15,17721,17723],{"className":17722},[346],[15,17724,17726],{"className":17725,"style":4080},[350],[15,17727],{},[15,17729,741],{"className":17730},[146,365],[15,17732,17734,17737],{"className":17733},[61,365],[15,17735,3106],{"className":17736,"style":3191},[61,122,365],[15,17738,17740],{"className":17739},[338],[15,17741,17743,17763],{"className":17742},[342,472],[15,17744,17746,17760],{"className":17745},[346],[15,17747,17749],{"className":17748,"style":4425},[350],[15,17750,17751,17754],{"style":4058},[15,17752],{"className":17753,"style":4062},[357],[15,17755,17757],{"className":17756},[362,4066,4067,365],[15,17758,278],{"className":17759},[61,365],[15,17761,497],{"className":17762},[496],[15,17764,17766],{"className":17765},[346],[15,17767,17769],{"className":17768,"style":4080},[350],[15,17770],{},[15,17772,7762],{"className":17773},[375,365],[15,17775,91],{"className":17776},[61,365],[15,17778,750],{"className":17779},[178,365],[15,17781,17782,17785],{"style":3889},[15,17783],{"className":17784,"style":3861},[357],[15,17786],{"className":17787,"style":3897},[3896],[15,17789,17790,17793],{"style":15450},[15,17791],{"className":17792,"style":3861},[357],[15,17794,17796],{"className":17795},[362,363,364,365],[15,17797,17799,17839,17842,17882,17885,17888],{"className":17798},[61,365],[15,17800,17802,17805],{"className":17801},[61,365],[15,17803,3106],{"className":17804,"style":3191},[61,122,365],[15,17806,17808],{"className":17807},[338],[15,17809,17811,17831],{"className":17810},[342,472],[15,17812,17814,17828],{"className":17813},[346],[15,17815,17817],{"className":17816,"style":4055},[350],[15,17818,17819,17822],{"style":4058},[15,17820],{"className":17821,"style":4062},[357],[15,17823,17825],{"className":17824},[362,4066,4067,365],[15,17826,1070],{"className":17827},[61,122,365],[15,17829,497],{"className":17830},[496],[15,17832,17834],{"className":17833},[346],[15,17835,17837],{"className":17836,"style":4080},[350],[15,17838],{},[15,17840,741],{"className":17841},[146,365],[15,17843,17845,17848],{"className":17844},[61,365],[15,17846,3106],{"className":17847,"style":3191},[61,122,365],[15,17849,17851],{"className":17850},[338],[15,17852,17854,17874],{"className":17853},[342,472],[15,17855,17857,17871],{"className":17856},[346],[15,17858,17860],{"className":17859,"s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x_j] = \\frac{\\alpha_i \\alpha_j}{\\alpha_0(\\alpha_0+1)}",[15,17979,17981,18086],{"className":17980,"ariaHidden":48},[47],[15,17982,17984,17988,17991,17994,18034,18074,18077,18080,18083],{"className":17983},[52],[15,17985],{"className":17986,"style":17987},[56],"height:1.0361em;vertical-align:-0.2861em;",[15,17989,14818],{"className":17990},[61,427],[15,17992,14821],{"className":17993},[146],[15,17995,17997,18000],{"className":17996},[61],[15,17998,1029],{"className":17999},[61,122],[15,18001,18003],{"className":18002},[338],[15,18004,18006,18026],{"className":18005},[342,472],[15,18007,18009,18023],{"className":18008},[346],[15,18010,18012],{"className":18011,"style":1364},[350],[15,18013,18014,18017],{"style":482},[15,18015],{"className":18016,"style":358},[357],[15,18018,18020],{"className":18019},[362,363,364,365],[15,18021,1070],{"className":18022},[61,122,365],[15,18024,497],{"className":18025},[496],[15,18027,18029],{"className":18028},[346],[15,18030,18032],{"className":18031,"style":504},[350],[15,18033],{},[15,18035,18037,18040],{"className":18036},[61],[15,18038,1029],{"className":18039},[61,122],[15,18041,18043],{"className":18042},[338],[15,18044,18046,18066],{"className":18045},[342,472],[15,18047,18049,18063],{"className":18048},[346],[15,18050,18052],{"className":18051,"style":1364},[350],[15,18053,18054,18057],{"style":482},[15,18055],{"className":18056,"style":358},[357],[15,18058,18060],{"className":18059},[362,363,364,365],[15,18061,14777],{"className":18062,"style":14803},[61,122,365],[15,18064,497],{"className":18065},[496],[15,18067,18069],{"className":18068},[346],[15,18070,18072],{"className":18071,"style":15707},[350],[15,18073],{},[15,18075,14830],{"className":18076},[178],[15,18078],{"className":18079,"style":128},[127],[15,18081,244],{"className":18082},[132],[15,18084],{"className":18085,"style":128},[127],[15,18087,18089,18093],{"className":18088},[52],[15,18090],{"className":18091,"style":18092},[56],"height:1.3287em;vertical-align:-0.52em;",[15,18094,18096,18099,18325],{"className":18095},[61],[15,18097],{"className":18098},[146,3841],[15,18100,18102],{"className":18101},[3722],[15,18103,18105,18317],{"className":18104},[342,472],[15,18106,18108,18314],{"className":18107},[346],[15,18109,18112,18215,18223],{"className":18110,"style":18111},[350],"height:0.8087em;",[15,18113,18114,18117],{"style":3857},[15,18115],{"className":18116,"style":3861},[357],[15,18118,18120],{"className":18119},[362,363,364,365],[15,18121,18123,18163,18166,18206,18209,18212],{"className":18122},[61,365],[15,18124,18126,18129],{"className":18125},[61,365],[15,18127,3106],{"className":18128,"style":3191},[61,122,365],[15,18130,18132],{"className":18131},[338],[15,18133,18135,18155],{"className":18134},[342,472],[15,18136,18138,18152],{"className":18137},[346],[15,18139,18141],{"className":18140,"style":4425},[350],[15,18142,18143,18146],{"style":4058},[15,18144],{"className":18145,"style":4062},[357],[15,18147,18149],{"className":18148},[362,4066,4067,365],[15,18150,278],{"className":18151},[61,365],[15,18153,497],{"className":18154},[496],[15,18156,18158],{"className":18157},[346],[15,18159,18161],{"className":18160,"style":4080},[350],[15,18162],{},[15,18164,741],{"className":18165},[146,365],[15,18167,18169,18172],{"className":18168},[61,365],[15,18170,3106],{"className":18171,"style":3191},[61,122,365],[15,18173,18175],{"className":18174},[338],[15,18176,18178,18198],{"className":18177},[342,472],[15,18179,18181,18195],{"className":18180},[346],[15,18182,18184],{"className":18183,"style":4425},[350],[15,18185,18186,18189],{"style":4058},[15,18187],{"className":18188,"style":4062},[357],[15,18190,18192],{"className":18191},[362,4066,4067,365],[15,18193,278],{"className":18194},[61,365],[15,18196,497],{"className":18197},[496],[15,18199,18201],{"className":18200},[346],[15,18202,18204],{"className":18203,"style":4080},[350],[15,18205],{},[15,18207,7762],{"className":18208},[375,365],[15,18210,91],{"className":18211},[61,365],[15,18213,750],{"className":18214},[178,365],[15,18216,18217,18220],{"style":3889},[15,18218],{"className":18219,"style":3861},[357],[15,18221],{"className":18222,"style":3897},[3896],[15,18224,18225,18228],{"style":15872},[15,18226],{"className":18227,"style":3861},[357],[15,18229,18231],{"className":18230},[362,363,364,365],[15,18232,18234,18274],{"className":18233},[61,365],[15,18235,18237,18240],{"className":18236},[61,365],[15,18238,3106],{"className":18239,"style":3191},[61,122,365],[15,18241,18243],{"className":18242},[338],[15,18244,18246,18266],{"className":18245},[342,472],[15,18247,18249,18263],{"className":18248},[346],[15,18250,18252],{"className":18251,"style":4055},[350],[15,18253,18254,18257],{"style":4058},[15,18255],{"className":18256,"style":4062},[357],[15,18258,18260],{"className":18259},[362,4066,4067,365],[15,18261,1070],{"className":18262},[61,122,365],[15,18264,497],{"className":18265},[496],[15,18267,18269],{"className":18268},[346],[15,18270,18272],{"className":18271,"style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相减即得。",[11,18330,18331,18332,18527],{},"上述结果可写成更紧凑的矩阵形式。记 ",[15,18333,18335,18380],{"className":18334,"translate":19},[18],[15,18336,18338],{"className":18337},[23],[25,18339,18340],{"xmlns":27},[29,18341,18342,18377],{},[32,18343,18344,18347,18349,18351,18353,18359,18361,18363,18365],{},[35,18345,18346],{"mathvariant":3105},"μ",[81,18348,244],{},[35,18350,3106],{"mathvariant":3105},[35,18352,3020],{"mathvariant":231},[266,18354,18355,18357],{},[35,18356,3106],{},[89,18358,278],{},[81,18360,83],{},[35,18362,1544],{"mathvariant":231},[81,18364,1547],{},[227,18366,18367,18369],{},[35,18368,232],{"mathvariant":231},[32,18370,18371,18373,18375],{},[35,18372,102],{},[81,18374,239],{},[89,18376,91],{},[40,18378,18379],{"encoding":42},"\\boldsymbol{\\mu} = \\boldsymbol{\\alpha}\u002F\\alpha_0 \\in \\operatorname{int}\\Delta^{K-1}",[15,18381,18383,18407,18474],{"className":18382,"ariaHidden":48},[47],[15,18384,18386,18389,18398,18401,18404],{"className":18385},[52],[15,18387],{"className":18388,"style":829},[56],[15,18390,18392],{"className":18391},[61],[15,18393,18395],{"className":18394},[61],[15,18396,18346],{"className":18397},[61,3166],[15,18399],{"className":18400,"style":128},[127],[15,18402,244],{"className":18403},[132],[15,18405],{"className":18406,"style":128},[127],[15,18408,18410,18413,18422,18425,18465,18468,18471],{"className":18409},[52],[15,18411],{"className":18412,"style":142},[56],[15,18414,18416],{"className":18415},[61],[15,18417,18419],{"className":18418},[61],[15,18420,3106],{"className":18421},[61,3166],[15,18423,3020],{"className":18424},[61],[15,18426,18428,18431],{"className":18427},[61],[15,18429,3106],{"className":18430,"style":3191},[61,122],[15,18432,18434],{"className":18433},[338],[15,18435,18437,18457],{"className":18436},[342,472],[15,18438,18440,18454],{"className":18439},[346],[15,18441,18443],{"className":18442,"style":1214},[350],[15,18444,18445,18448],{"style":3206},[15,18446],{"className":18447,"style":358},[357],[15,18449,18451],{"className":18450},[362,363,364,365],[15,18452,278],{"className":18453},[61,365],[15,18455,497],{"className":18456},[496],[15,18458,18460],{"className":18459},[346],[15,18461,18463],{"className":18462,"style":504},[350],[15,18464],{},[15,18466],{"className":18467,"style":128},[127],[15,18469,83],{"className":18470},[132],[15,18472],{"className":18473,"style":128},[127],[15,18475,18477,18480,18486,18489],{"className":18476},[52],[15,18478],{"className":18479,"style":695},[56],[15,18481,18483],{"className":18482},[531],[15,18484,1544],{"className":18485},[61,1622],[15,18487],{"className":18488,"style":157},[127],[15,18490,18492,18495],{"className":18491},[61],[15,18493,232],{"className":18494},[61],[15,18496,18498],{"className":18497},[338],[15,18499,18501],{"className":18500},[342],[15,18502,18504],{"className":18503},[346],[15,18505,18507],{"className":18506,"style":695},[350],[15,18508,18509,18512],{"style":716},[15,18510],{"className":18511,"style":358},[357],[15,18513,18515],{"className":18514},[362,363,364,365],[15,18516,18518,18521,18524],{"className":18517},[61,365],[15,18519,102],{"className":18520,"style":174},[61,122,365],[15,18522,239],{"className":18523},[375,365],[15,18525,91],{"className":18526},[61,365]," 为均值向量，则",[15,18529,18531],{"className":18530,"translate":19},[212],[15,18532,18534,18607],{"className":18533,"translate":19},[18],[15,18535,18537],{"className":18536},[23],[25,18538,18539],{"xmlns":27,"display":221},[29,18540,18541,18604],{},[32,18542,18543,18546,18548,18550,18552,18554,18556,18558,18560,18576],{},[35,18544,18545],{"mathvariant":251},"Σ",[81,18547,244],{},[35,18549,14926],{"mathvariant":231},[81,18551,1547],{},[81,18553,741],{"stretchy":86},[35,18555,1029],{"mathvariant":251},[81,18557,750],{"stretchy":86},[81,18559,244],{},[3722,18561,18562,18564],{},[89,18563,91],{},[32,18565,18566,18572,18574],{},[266,18567,18568,18570],{},[35,18569,3106],{},[89,18571,278],{},[81,18573,7762],{},[89,18575,91],{},[32,18577,18578,18580,18583,18585,18587,18589,18591,18593,18595,18602],{},[81,18579,741],{"fence":48},[35,18581,18582],{"mathvariant":231},"diag",[81,18584,1547],{},[81,18586,741],{"stretchy":86},[35,18588,18346],{"mathvariant":3105},[81,18590,750],{"stretchy":86},[81,18592,239],{},[35,18594,18346],{"mathvariant":3105},[227,18596,18597,18599],{},[35,18598,18346],{"mathvariant":3105},[35,18600,18601],{"mathvariant":231},"⊤",[81,18603,750],{"fence":48},[40,18605,18606],{"encoding":42},"\\boldsymbol{\\Sigma} = \\operatorname{Cov}(\\mathbf{x}) = \\frac{1}{\\alpha_0 + 1}\\left(\\operatorname{diag}(\\boldsymbol{\\mu}) - \\boldsymbol{\\mu}\\boldsymbol{\\mu}^\\top\\right)",[15,18608,18610,18635,18665],{"className":18609,"ariaHidden":48},[47],[15,18611,18613,18617,18626,18629,18632],{"className":18612},[52],[15,18614],{"className":18615,"style":18616},[56],"height:0.6861em;",[15,18618,18620],{"className":18619},[61],[15,18621,18623],{"className":18622},[61],[15,18624,18545],{"className":18625},[61,411],[15,18627],{"className":18628,"style":128},[127],[15,18630,244],{"className":18631},[132],[15,18633],{"className":18634,"style":128},[127],[15,18636,18638,18641,18647,18650,18653,18656,18659,18662],{"className":18637},[52],[15,18639],{"className":18640,"style":142},[56],[15,18642,18644],{"className":18643},[531],[15,18645,14926],{"className":18646,"style":15619},[61,1622],[15,18648,741],{"className":18649},[146],[15,18651,1029],{"className":18652},[61,411],[15,18654,750],{"className":18655},[178],[15,18657],{"className":18658,"style":128},[127],[15,18660,244],{"className":18661},[132],[15,18663],{"className":18664,"style":128},[127],[15,18666,18668,18672,18785,18788],{"className":18667},[52],[15,18669],{"className":18670,"style":18671},[56],"height:2.1574em;vertical-align:-0.836em;",[15,18673,18675,18678,18782],{"className":18674},[61],[15,18676],{"className":18677},[146,3841],[15,18679,18681],{"className":18680},[3722],[15,18682,18684,18773],{"className":18683},[342,472],[15,18685,18687,18770],{"className":18686},[346],[15,18688,18691,18751,18759],{"className":18689,"style":18690},[350],"height:1.3214em;",[15,18692,18693,18696],{"style":11402},[15,18694],{"className":18695,"style":3861},[357],[15,18697,18699,18739,18742,18745,18748],{"className":18698},[61],[15,18700,18702,18705],{"className":18701},[61],[15,18703,3106],{"className":18704,"style":3191},[61,122],[15,18706,18708],{"className":18707},[338],[15,18709,18711,18731],{"className":18710},[342,472],[15,18712,18714,18728],{"className":18713},[346],[15,18715,18717],{"className":18716,"style":1214},[350],[15,18718,18719,18722],{"style":3206},[15,18720],{"className":18721,"style":358},[357],[15,18723,18725],{"className":18724},[362,363,364,365],[15,18726,278],{"className":18727},[61,365],[15,18729,497],{"className":18730},[496],[15,18732,18734],{"className":18733},[346],[15,18735,18737],{"className":18736,"style":504},[350],[15,18738],{},[15,18740],{"className":18741,"style":772},[127],[15,18743,7762],{"className":18744},[375],[15,18746],{"className":18747,"style":772},[127],[15,18749,91],{"className":18750},[61],[15,18752,18753,18756],{"style":3889},[15,18754],{"className":18755,"style":3861},[357],[15,18757],{"className":18758,"style":3897},[3896],[15,18760,18761,18764],{"style":11468},[15,18762],{"className":18763,"style":3861},[357],[15,18765,18767],{"className":18766},[61],[15,18768,91],{"className":18769},[61],[15,18771,497],{"className":18772},[496],[15,18774,18776],{"className":18775},[346],[15,18777,18780],{"className":18778,"style":18779},[350],"height:0.836em;",[15,18781],{},[15,18783],{"className":18784},[178,3841],[15,18786],{"className":18787,"style":157},[127],[15,18789,18791,18797,18803,18806,18815,18818,18821,18824,18827,18836,18872],{"className":18790},[161],[15,18792,18794],{"className":18793,"style":402},[146,401],[15,18795,741],{"className":18796},[406,4067],[15,18798,18800],{"className":18799},[531],[15,18801,18582],{"className":18802,"style":15619},[61,1622],[15,18804,741],{"className":18805},[146],[15,18807,18809],{"className":18808},[61],[15,18810,18812],{"className":18811},[61],[15,18813,18346],{"className":18814},[61,3166],[15,18816,750],{"className":18817},[178],[15,18819],{"className":18820,"style":772},[127],[15,18822,239],{"className":18823},[375],[15,18825],{"className":18826,"style":772},[127],[15,18828,18830],{"className":18829},[61],[15,18831,18833],{"className":18832},[61],[15,18834,18346],{"className":18835},[61,3166],[15,18837,18839,18848],{"className":18838},[61],[15,18840,18842],{"className":18841},[61],[15,18843,18845],{"className":18844},[61],[15,18846,18346],{"className":18847},[61,3166],[15,18849,18851],{"className":18850},[338],[15,18852,18854],{"className":18853},[342],[15,18855,18857],{"className":18856},[346],[15,18858,18861],{"className":18859,"style":18860},[350],"height:0.8991em;",[15,18862,18863,18866],{"style":353},[15,18864],{"className":18865,"style":358},[357],[15,18867,18869],{"className":18868},[362,363,364,365],[15,18870,18601],{"className":18871},[61,365],[15,18873,18875],{"className":18874,"style":402},[178,401],[15,18876,750],{"className":18877},[406,4067],[11,18879,18880,18881,18931,18932,943,19030,19065,19066,19095,19096,19156,19157,19398,19399,19427],{},"这一形式揭示了两个结构性事实。第一，协方差矩阵的秩为 ",[15,18882,18884,18901],{"className":18883,"translate":19},[18],[15,18885,18887],{"className":18886},[23],[25,18888,18889],{"xmlns":27},[29,18890,18891,18899],{},[32,18892,18893,18895,18897],{},[35,18894,102],{},[81,18896,239],{},[89,18898,91],{},[40,18900,964],{"encoding":42},[15,18902,18904,18922],{"className":18903,"ariaHidden":48},[47],[15,18905,18907,18910,18913,18916,18919],{"className":18906},[52],[15,18908],{"className":18909,"style":974},[56],[15,18911,102],{"className":18912,"style":174},[61,122],[15,18914],{"className":18915,"style":772},[127],[15,18917,239],{"className":18918},[375],[15,18920],{"className":18921,"style":772},[127],[15,18923,18925,18928],{"className":18924},[52],[15,18926],{"className":18927,"style":859},[56],[15,18929,91],{"className":18930},[61],"：由于 ",[15,18933,18935,18961],{"className":18934,"translate":19},[18],[15,18936,18938],{"className":18937},[23],[25,18939,18940],{"xmlns":27},[29,18941,18942,18958],{},[32,18943,18944,18950,18952,18954,18956],{},[227,18945,18946,18948],{},[89,18947,91],{"mathvariant":251},[35,18949,18601],{"mathvariant":231},[35,18951,18545],{"mathvariant":251},[89,18953,91],{"mathvariant":251},[81,18955,244],{},[89,18957,278],{},[40,18959,18960],{"encoding":42},"\\mathbf{1}^\\top \\boldsymbol{\\Sigma} \\mathbf{1} = 0",[15,18962,18964,19021],{"className":18963,"ariaHidden":48},[47],[15,18965,18967,18971,19000,19009,19012,19015,19018],{"className":18966},[52],[15,18968],{"className":18969,"style":18970},[56],"height:0.8491em;",[15,18972,18974,18977],{"className":18973},[61],[15,18975,91],{"className":18976},[61,411],[15,18978,18980],{"className":18979},[338],[15,18981,18983],{"className":18982},[342],[15,18984,18986],{"className":18985},[346],[15,18987,18989],{"className":18988,"style":18970},[350],[15,18990,18991,18994],{"style":716},[15,18992],{"className":18993,"style":358},[357],[15,18995,18997],{"className":18996},[362,363,364,365],[15,18998,18601],{"className":18999},[61,365],[15,19001,19003],{"className":19002},[61],[15,19004,19006],{"className":19005},[61],[15,19007,18545],{"className":19008},[61,411],[15,19010,91],{"className":19011},[61,411],[15,19013],{"className":19014,"style":128},[127],[15,19016,244],{"className":19017},[132],[15,19019],{"className":19020,"style":128},[127],[15,19022,19024,19027],{"className":19023},[52],[15,19025],{"className":19026,"style":859},[56],[15,19028,278],{"className":19029},[61],[15,19031,19033,19047],{"className":19032,"translate":19},[18],[15,19034,19036],{"className":19035},[23],[25,19037,19038],{"xmlns":27},[29,19039,19040,19044],{},[32,19041,19042],{},[35,19043,18545],{"mathvariant":251},[40,19045,19046],{"encoding":42},"\\boldsymbol{\\Sigma}",[15,19048,19050],{"className":19049,"ariaHidden":48},[47],[15,19051,19053,19056],{"className":19052},[52],[15,19054],{"className":19055,"style":18616},[56],[15,19057,19059],{"className":19058},[61],[15,19060,19062],{"className":19061},[61],[15,19063,18545],{"className":19064},[61,411]," 必然奇异，其零空间由 ",[15,19067,19069,19083],{"className":19068,"translate":19},[18],[15,19070,19072],{"className":19071},[23],[25,19073,19074],{"xmlns":27},[29,19075,19076,19080],{},[32,19077,19078],{},[89,19079,91],{"mathvariant":251},[40,19081,19082],{"encoding":42},"\\mathbf{1}",[15,19084,19086],{"className":19085,"ariaHidden":48},[47],[15,19087,19089,19092],{"className":19088},[52],[15,19090],{"className":19091,"style":859},[56],[15,19093,91],{"className":19094},[61,411]," 张成——这正是「分布在 ",[15,19097,19099,19120],{"className":19098,"translate":19},[18],[15,19100,19102],{"className":19101},[23],[25,19103,19104],{"xmlns":27},[29,19105,19106,19118],{},[32,19107,19108,19110,19112,19114,19116],{},[81,19109,741],{"stretchy":86},[35,19111,102],{},[81,19113,239],{},[89,19115,91],{},[81,19117,750],{"stretchy":86},[40,19119,753],{"encoding":42},[15,19121,19123,19144],{"className":19122,"ariaHidden":48},[47],[15,19124,19126,19129,19132,19135,19138,19141],{"className":19125},[52],[15,19127],{"className":19128,"style":142},[56],[15,19130,741],{"className":19131},[146],[15,19133,102],{"className":19134,"style":174},[61,122],[15,19136],{"className":19137,"style":772},[127],[15,19139,239],{"className":19140},[375],[15,19142],{"className":19143,"style":772},[127],[15,19145,19147,19150,19153],{"className":19146},[52],[15,19148],{"className":19149,"style":142},[56],[15,19151,91],{"className":19152},[61],[15,19154,750],{"className":19155},[178]," 维对象上」的代数痕迹。第二，全部协方差为负：",[15,19158,19160,19210],{"className":19159,"translate":19},[18],[15,19161,19163],{"className":19162},[23],[25,19164,19165],{"xmlns":27},[29,19166,19167,19207],{},[32,19168,19169,19171,19173,19175,19181,19183,19189,19191,19193,19195,19197,19199,19201,19203,19205],{},[35,19170,14926],{"mathvariant":231},[81,19172,1547],{},[81,19174,741],{"stretchy":86},[266,19176,19177,19179],{},[35,19178,1029],{},[35,19180,1070],{},[81,19182,94],{"separator":48},[266,19184,19185,19187],{},[35,19186,1029],{},[35,19188,14777],{},[81,19190,750],{"stretchy":86},[81,19192,2490],{},[89,19194,278],{},[282,19196,284],{},[81,19198,741],{"stretchy":86},[35,19200,1070],{},[81,19202,14999],{"mathvariant":231},[35,19204,14777],{},[81,19206,750],{"stretchy":86},[40,19208,19209],{"encoding":42},"\\operatorname{Cov}(x_i, x_j) \u003C 0\\ (i \\ne j)",[15,19211,19213,19326,19386],{"className":19212,"ariaHidden":48},[47],[15,19214,19216,19219,19225,19228,19268,19271,19274,19314,19317,19320,19323],{"className":19215},[52],[15,19217],{"className":19218,"style":17987},[56],[15,19220,19222],{"className":19221},[531],[15,19223,14926],{"className":19224,"style":15619},[61,1622],[15,19226,741],{"className":19227},[146],[15,19229,19231,19234],{"className":19230},[61],[15,19232,1029],{"className":19233},[61,122],[15,19235,19237],{"className":19236},[338],[15,19238,19240,19260],{"className":19239},[342,472],[15,19241,19243,19257],{"className":19242},[346],[15,19244,19246],{"className":19245,"style":1364},[350],[15,19247,19248,19251],{"style":482},[15,19249],{"className":19250,"style":358},[357],[15,19252,19254],{"className":19253},[362,363,364,365],[15,19255,1070],{"className":19256},[61,122,365],[15,19258,497],{"className":19259},[496],[15,19261,19263],{"className":19262},[346],[15,19264,19266],{"className":19265,"style":504},[350],[15,19267],{},[15,19269,94],{"className":19270},[153],[15,19272],{"className":19273,"style":157},[127],[15,19275,19277,19280],{"className":19276},[61],[15,19278,1029],{"className":19279},[61,122],[15,19281,19283],{"className":19282},[338],[15,19284,19286,19306],{"className":19285},[342,472],[15,19287,19289,19303],{"className":19288},[346],[15,19290,19292],{"className":19291,"style":1364},[350],[15,19293,19294,19297],{"style":482},[15,19295],{"className":19296,"style":358},[357],[15,19298,19300],{"className":19299},[362,363,364,365],[15,19301,14777],{"className":19302,"style":14803},[61,122,365],[15,19304,497],{"className":19305},[496],[15,19307,19309],{"className":19308},[346],[15,19310,19312],{"className":19311,"style":15707},[350],[15,19313],{},[15,19315,750],{"className":19316},[178],[15,19318],{"className":19319,"style":128},[127],[15,19321,2490],{"className":19322},[132],[15,19324],{"className":19325,"style":128},[127],[15,19327,19329,19332,19335,19338,19341,19344,19347,19383],{"className":19328},[52],[15,19330],{"className":19331,"style":142},[56],[15,19333,278],{"className":19334},[61],[15,19336,284],{"className":19337},[127],[15,19339,741],{"className":19340},[146],[15,19342,1070],{"className":19343},[61,122],[15,19345],{"className":19346,"style":128},[127],[15,19348,19350,19377,19380],{"className":19349},[132],[15,19351,19353],{"className":19352},[132],[15,19354,19356],{"className":19355},[61,16002],[15,19357,19359],{"className":19358},[16006],[15,19360,19362,19365,19374],{"className":19361},[16010],[15,19363],{"className":19364,"style":14310},[56],[15,19366,19368],{"className":19367},[16017],[15,19369,19371],{"className":19370},[61],[15,19372,16024],{"className":19373},[132],[15,19375],{"className":19376},[16028],[15,19378],{"className":19379},[127,16032],[15,19381,244],{"className":19382},[132],[15,19384],{"className":19385,"style":128},[127],[15,19387,19389,19392,19395],{"className":19388},[52],[15,19390],{"className":19391,"style":142},[56],[15,19393,14777],{"className":19394,"style":14803},[61,122],[15,19396,750],{"className":19397},[178],"，因为分量之间必须通过和为 ",[15,19400,19402,19415],{"className":19401,"translate":19},[18],[15,19403,19405],{"className":19404},[23],[25,19406,19407],{"xmlns":27},[29,19408,19409,19413],{},[32,19410,19411],{},[89,19412,91],{},[40,19414,91],{"encoding":42},[15,19416,19418],{"className":19417,"ariaHidden":48},[47],[15,19419,19421,19424],{"className":19420},[52],[15,19422],{"className":19423,"style":859},[56],[15,19425,91],{"className":19426},[61]," 的约束相互竞争，任何一个分量的增加都以其他分量的期望减少为代价。",[877,19429,19431],{"id":19430},"_22-均值精度重参数化","2.2 均值—精度重参数化",[11,19433,19434,19435,9348,19470,19540],{},"命题 2.1 中的 ",[15,19436,19438,19452],{"className":19437,"translate":19},[18],[15,19439,19441],{"className":19440},[23],[25,19442,19443],{"xmlns":27},[29,19444,19445,19449],{},[32,19446,19447],{},[35,19448,18346],{"mathvariant":3105},[40,19450,19451],{"encoding":42},"\\boldsymbol{\\mu}",[15,19453,19455],{"className":19454,"ariaHidden":48},[47],[15,19456,19458,19461],{"className":19457},[52],[15,19459],{"className":19460,"style":829},[56],[15,19462,19464],{"className":19463},[61],[15,19465,19467],{"className":19466},[61],[15,19468,18346],{"className":19469},[61,3166],[15,19471,19473,19491],{"className":19472,"translate":19},[18],[15,19474,19476],{"className":19475},[23],[25,19477,19478],{"xmlns":27},[29,19479,19480,19488],{},[32,19481,19482],{},[266,19483,19484,19486],{},[35,19485,3106],{},[89,19487,278],{},[40,19489,19490],{"encoding":42},"\\alpha_0",[15,19492,19494],{"className":19493,"ariaHidden":48},[47],[15,19495,19497,19500],{"className":19496},[52],[15,19498],{"className":19499,"style":2511},[56],[15,19501,19503,19506],{"className":19502},[61],[15,19504,3106],{"className":19505,"style":3191},[61,122],[15,19507,19509],{"className":19508},[338],[15,19510,19512,19532],{"className":19511},[342,472],[15,19513,19515,19529],{"className":19514},[346],[15,19516,19518],{"className":19517,"style":1214},[350],[15,19519,19520,19523],{"style":3206},[15,19521],{"className":19522,"style":358},[357],[15,19524,19526],{"className":19525},[362,363,364,365],[15,19527,278],{"className":19528},[61,365],[15,19530,497],{"className":19531},[496],[15,19533,19535],{"className":19534},[346],[15,19536,19538],{"className":19537,"style":504},[350],[15,19539],{}," 是两个正交的自由度：前者决定「概率质量落在何处」，后者决定「分布在该处聚集得多紧」。据此引入重参数化",[15,19542,19544],{"className":19543,"translate":19},[212],[15,19545,19547,19609],{"className":19546,"translate":19},[18],[15,19548,19550],{"className":19549},[23],[25,19551,19552],{"xmlns":27,"display":221},[29,19553,19554,19606],{},[32,19555,19556,19558,19560,19566,19568,19570,19572,19578,19580,19582,19584,19586,19588,19590,19592,19594],{},[35,19557,3106],{"mathvariant":3105},[81,19559,244],{},[266,19561,19562,19564],{},[35,19563,3106],{},[89,19565,278],{},[35,19567,18346],{"mathvariant":3105},[81,19569,94],{"separator":48},[127,19571],{"width":9847},[266,19573,19574,19576],{},[35,19575,3106],{},[89,19577,278],{},[81,19579,1590],{},[89,19581,278],{},[81,19583,94],{"separator":48},[282,19585,284],{},[35,19587,18346],{"mathvariant":3105},[81,19589,83],{},[35,19591,1544],{"mathvariant":231},[81,19593,1547],{},[227,19595,19596,19598],{},[35,19597,232],{"mathvariant":231},[32,19599,19600,19602,19604],{},[35,19601,102],{},[81,19603,239],{},[89,19605,91],{},[40,19607,19608],{"encoding":42},"\\boldsymbol{\\alpha} = \\alpha_0 \\boldsymbol{\\mu}, \\qquad \\alpha_0 > 0,\\ \\boldsymbol{\\mu} \\in \\operatorname{int}\\Delta^{K-1}",[15,19610,19612,19636,19749,19786],{"className":19611,"ariaHidden":48},[47],[15,19613,19615,19618,19627,19630,19633],{"className":19614},[52],[15,19616],{"className":19617,"style":2708},[56],[15,19619,19621],{"className":19620},[61],[15,19622,19624],{"className":19623},[61],[15,19625,3106],{"className":19626},[61,3166],[15,19628],{"className":19629,"style":128},[127],[15,19631,244],{"className":19632},[132],[15,19634],{"className":19635,"style":128},[127],[15,19637,19639,19642,19682,19691,19694,19697,19700,19740,19743,19746],{"className":19638},[52],[15,19640],{"className":19641,"style":118},[56],[15,19643,19645,19648],{"className":19644},[61],[15,19646,3106],{"className":19647,"style":3191},[61,122],[15,19649,19651],{"className":19650},[338],[15,19652,19654,19674],{"className":19653},[342,472],[15,19655,19657,19671],{"className":19656},[346],[15,19658,19660],{"className":19659,"style":1214},[350],[15,19661,19662,19665],{"style":3206},[15,19663],{"className":19664,"style":358},[357],[15,19666,19668],{"className":19667},[362,363,364,365],[15,19669,278],{"className":19670},[61,365],[15,19672,497],{"className":19673},[496],[15,19675,19677],{"className":19676},[346],[15,19678,19680],{"className":19679,"style":504},[350],[15,19681],{},[15,19683,19685],{"className":19684},[61],[15,19686,19688],{"className":19687},[61],[15,19689,18346],{"className":19690},[61,3166],[15,19692,94],{"className":19693},[153],[15,19695],{"className":19696,"style":10035},[127],[15,19698],{"className":19699,"style":157},[127],[15,19701,19703,19706],{"className":19702},[61],[15,19704,3106],{"className":19705,"style":3191},[61,122],[15,19707,19709],{"className":19708},[338],[15,19710,19712,19732],{"className":19711},[342,472],[15,19713,19715,19729],{"className":19714},[346],[15,19716,19718],{"className":19717,"style":1214},[350],[15,19719,19720,19723],{"style":3206},[15,19721],{"className":19722,"style":358},[357],[15,19724,19726],{"className":19725},[362,363,364,365],[15,19727,278],{"className":19728},[61,365],[15,19730,497],{"className":19731},[496],[15,19733,19735],{"className":19734},[346],[15,19736,19738],{"className":19737,"style":504},[350],[15,19739],{},[15,19741],{"className":19742,"style":128},[127],[15,19744,1590],{"className":19745},[132],[15,19747],{"className":19748,"style":128},[127],[15,19750,19752,19756,19759,19762,19765,19768,19777,19780,19783],{"className":19751},[52],[15,19753],{"className":19754,"style":19755},[56],"height:0.8389em;vertical-align:-0.1944em;",[15,19757,278],{"className":19758},[61],[15,19760,94],{"className":19761},[153],[15,19763,284],{"className":19764},[127],[15,19766],{"className":19767,"style":157},[127],[15,19769,19771],{"className":19770},[61],[15,19772,19774],{"className":19773},[61],[15,19775,18346],{"className":19776},[61,3166],[15,19778],{"className":19779,"style":128},[127],[15,19781,83],{"className":19782},[132],[15,19784],{"className":19785,"style":128},[127],[15,19787,19789,19792,19798,19801],{"className":19788},[52],[15,19790],{"className":19791,"style":328},[56],[15,19793,19795],{"className":19794},[531],[15,19796,1544],{"className":19797},[61,1622],[15,19799],{"className":19800,"style":157},[127],[15,19802,19804,19807],{"className":19803},[61],[15,19805,232],{"className":19806},[61],[15,19808,19810],{"className":19809},[338],[15,19811,19813],{"className":19812},[342],[15,19814,19816],{"className":19815},[346],[15,19817,19819],{"className":19818,"style":328},[350],[15,19820,19821,19824],{"style":353},[15,19822],{"className":19823,"style":358},[357],[15,19825,19827],{"className":19826},[362,363,364,365],[15,19828,19830,19833,19836],{"className":19829},[61,365],[15,19831,102],{"className":19832,"style":174},[61,122,365],[15,19834,239],{"className":19835},[375,365],[15,19837,91],{"className":19838},[61,365],[11,19840,19841,19842,19845,19846,19896,19897,19900,19901,943,20184,20292,20293,20362],{},"它把参数空间分解为一个",[661,19843,19844],{},"方向","（",[15,19847,19849,19866],{"className":19848,"translate":19},[18],[15,19850,19852],{"className":19851},[23],[25,19853,19854],{"xmlns":27},[29,19855,19856,19864],{},[32,19857,19858,19860,19862],{},[35,19859,102],{},[81,19861,239],{},[89,19863,91],{},[40,19865,964],{"encoding":42},[15,19867,19869,19887],{"className":19868,"ariaHidden":48},[47],[15,19870,19872,19875,19878,19881,19884],{"className":19871},[52],[15,19873],{"className":19874,"style":974},[56],[15,19876,102],{"className":19877,"style":174},[61,122],[15,19879],{"className":19880,"style":772},[127],[15,19882,239],{"className":19883},[375],[15,19885],{"className":19886,"style":772},[127],[15,19888,19890,19893],{"className":19889},[52],[15,19891],{"className":19892,"style":859},[56],[15,19894,91],{"className":19895},[61]," 维单纯形）与一个",[661,19898,19899],{},"尺度","（正实数）。在此参数化下，协方差为 ",[15,19902,19904,19956],{"className":19903,"translate":19},[18],[15,19905,19907],{"className":19906},[23],[25,19908,19909],{"xmlns":27},[29,19910,19911,19953],{},[32,19912,19913,19915,19917,19933,19935,19937,19939,19941,19943,19945,19951],{},[35,19914,18545],{"mathvariant":251},[81,19916,244],{},[3722,19918,19919,19921],{},[89,19920,91],{},[32,19922,19923,19929,19931],{},[266,19924,19925,19927],{},[35,19926,3106],{},[89,19928,278],{},[81,19930,7762],{},[89,19932,91],{},[81,19934,741],{"stretchy":86},[35,19936,18582],{"mathvariant":231},[81,19938,1547],{},[35,19940,18346],{"mathvariant":3105},[81,19942,239],{},[35,19944,18346],{"mathvariant":3105},[227,19946,19947,19949],{},[35,19948,18346],{"mathvariant":3105},[35,19950,18601],{"mathvariant":231},[81,19952,750],{"stretchy":86},[40,19954,19955],{"encoding":42},"\\boldsymbol{\\Sigma} = \\frac{1}{\\alpha_0+1}(\\operatorname{diag}\\boldsymbol{\\mu} - \\boldsymbol{\\mu}\\boldsymbol{\\mu}^\\top)",[15,19957,19959,19983,20130],{"className":19958,"ariaHidden":48},[47],[15,19960,19962,19965,19974,19977,19980],{"className":19961},[52],[15,19963],{"className":19964,"style":18616},[56],[15,19966,19968],{"className":19967},[61],[15,19969,19971],{"className":19970},[61],[15,19972,18545],{"className":19973},[61,411],[15,19975],{"className":19976,"style":128},[127],[15,19978,244],{"className":19979},[132],[15,19981],{"className":19982,"style":128},[127],[15,19984,19986,19989,20100,20103,20109,20112,20121,20124,20127],{"className":19985},[52],[15,19987],{"className":19988,"style":16578},[56],[15,19990,19992,19995,20097],{"className":19991},[61],[15,19993],{"className":19994},[146,3841],[15,19996,19998],{"className":19997},[3722],[15,19999,20001,20089],{"className":20000},[342,472],[15,20002,20004,20086],{"className":20003},[346],[15,20005,20007,20064,20072],{"className":20006,"style":3854},[350],[15,20008,20009,20012],{"style":3857},[15,20010],{"className":20011,"style":3861},[357],[15,20013,20015],{"className":20014},[362,363,364,365],[15,20016,20018,20058,20061],{"className":20017},[61,365],[15,20019,20021,20024],{"className":20020},[61,365],[15,20022,3106],{"className":20023,"style":3191},[61,122,365],[15,20025,20027],{"className":20026},[338],[15,20028,20030,20050],{"className":20029},[342,472],[15,20031,20033,20047],{"className":20032},[346],[15,20034,20036],{"className":20035,"style":4425},[350],[15,20037,20038,20041],{"style":4058},[15,20039],{"className":20040,"style":4062},[357],[15,20042,20044],{"className":20043},[362,4066,4067,365],[15,20045,278],{"className":20046},[61,365],[15,20048,497],{"className":20049},[496],[15,20051,20053],{"className":20052},[346],[15,20054,20056],{"className":20055,"style":4080},[350],[15,20057],{},[15,20059,7762],{"className":20060},[375,365],[15,20062,91],{"className":20063},[61,365],[15,20065,20066,20069],{"style":3889},[15,20067],{"className":20068,"style":3861},[357],[15,20070],{"className":20071,"style":3897},[3896],[15,20073,20074,20077],{"style":3900},[15,20075],{"className":20076,"style":3861},[357],[15,20078,20080],{"className":20079},[362,363,364,365],[15,20081,20083],{"className":20082},[61,365],[15,20084,91],{"className":20085},[61,365],[15,20087,497],{"className":20088},[496],[15,20090,20092],{"className":20091},[346],[15,20093,20095],{"className":20094,"style":15220},[350],[15,20096],{},[15,20098],{"className":20099},[178,3841],[15,20101,741],{"className":20102},[146],[15,20104,20106],{"className":20105},[531],[15,20107,18582],{"className":20108,"style":15619},[61,1622],[15,20110],{"className":20111,"style":157},[127],[15,20113,20115],{"className":20114},[61],[15,20116,20118],{"className":20117},[61],[15,20119,18346],{"className":20120},[61,3166],[15,20122],{"className":20123,"style":772},[127],[15,20125,239],{"className":20126},[375],[15,20128],{"className":20129,"style":772},[127],[15,20131,20133,20137,20146,20181],{"className":20132},[52],[15,20134],{"className":20135,"style":20136},[56],"height:1.0991em;vertical-align:-0.25em;",[15,20138,20140],{"className":20139},[61],[15,20141,20143],{"className":20142},[61],[15,20144,18346],{"className":20145},[61,3166],[15,20147,20149,20158],{"className":20148},[61],[15,20150,20152],{"className":20151},[61],[15,20153,20155],{"className":20154},[61],[15,20156,18346],{"className":20157},[61,3166],[15,20159,20161],{"className":20160},[338],[15,20162,20164],{"className":20163},[342],[15,20165,20167],{"className":20166},[346],[15,20168,20170],{"className":20169,"style":18970},[350],[15,20171,20172,20175],{"style":716},[15,20173],{"className":20174,"style":358},[357],[15,20176,20178],{"className":20177},[362,363,364,365],[15,20179,18601],{"className":20180},[61,365],[15,20182,750],{"className":20183},[178],[661,20185,20186,20187,20221,20222,20291],{},"与 ",[15,20188,20190,20203],{"className":20189,"translate":19},[18],[15,20191,20193],{"className":20192},[23],[25,20194,20195],{"xmlns":27},[29,20196,20197,20201],{},[32,20198,20199],{},[35,20200,18346],{"mathvariant":3105},[40,20202,19451],{"encoding":42},[15,20204,20206],{"className":20205,"ariaHidden":48},[47],[15,20207,20209,20212],{"className":20208},[52],[15,20210],{"className":20211,"style":829},[56],[15,20213,20215],{"className":20214},[61],[15,20216,20218],{"className":20217},[61],[15,20219,18346],{"className":20220},[61,3166]," 的取值无关、只随 ",[15,20223,20225,20242],{"className":20224,"translate":19},[18],[15,20226,20228],{"className":20227},[23],[25,20229,20230],{"xmlns":27},[29,20231,20232,20240],{},[32,20233,20234],{},[266,20235,20236,20238],{},[35,20237,3106],{},[89,20239,278],{},[40,20241,19490],{"encoding":42},[15,20243,20245],{"className":20244,"ariaHidden":48},[47],[15,20246,20248,20251],{"className":20247},[52],[15,20249],{"className":20250,"style":2511},[56],[15,20252,20254,20257],{"className":20253},[61],[15,20255,3106],{"className":20256,"style":3191},[61,122],[15,20258,20260],{"className":20259},[338],[15,20261,20263,20283],{"className":20262},[342,472],[15,20264,20266,20280],{"className":20265},[346],[15,20267,20269],{"className":20268,"style":1214},[350],[15,20270,20271,20274],{"style":3206},[15,20272],{"className":20273,"style":358},[357],[15,20275,20277],{"className":20276},[362,363,364,365],[15,20278,278],{"className":20279},[61,365],[15,20281,497],{"className":20282},[496],[15,20284,20286],{"className":20285},[346],[15,20287,20289],{"className":20288,"style":504},[350],[15,20290],{}," 单调收缩","。因此 ",[15,20294,20296,20313],{"className":20295,"translate":19},[18],[15,20297,20299],{"className":20298},[23],[25,20300,20301],{"xmlns":27},[29,20302,20303,20311],{},[32,20304,20305],{},[266,20306,20307,20309],{},[35,20308,3106],{},[89,20310,278],{},[40,20312,19490],{"encoding":42},[15,20314,20316],{"className":20315,"ariaHidden":48},[47],[15,20317,20319,20322],{"className":20318},[52],[15,20320],{"className":20321,"style":2511},[56],[15,20323,20325,20328],{"className":20324},[61],[15,20326,3106],{"className":20327,"style":3191},[61,122],[15,20329,20331],{"className":20330},[338],[15,20332,20334,20354],{"className":20333},[342,472],[15,20335,20337,20351],{"className":20336},[346],[15,20338,20340],{"className":20339,"style":1214},[350],[15,20341,20342,20345],{"style":3206},[15,20343],{"className":20344,"style":358},[357],[15,20346,20348],{"className":20347},[362,363,364,365],[15,20349,278],{"className":20350},[61,365],[15,20352,497],{"className":20353},[496],[15,20355,20357],{"className":20356},[346],[15,20358,20360],{"className":20359,"style":504},[350],[15,20361],{}," 是一个纯粹的集中度（或称精度、强度）参数：",[882,20364,20365],{},[11,20366,20367,20370,20371,20486,20487,20683,20684,20777,20778,20879,20880,20914,20915,867],{},[661,20368,20369],{},"命题 2.2（集中性极限）"," 固定 ",[15,20372,20374,20406],{"className":20373,"translate":19},[18],[15,20375,20377],{"className":20376},[23],[25,20378,20379],{"xmlns":27},[29,20380,20381,20403],{},[32,20382,20383,20385,20387,20389,20391],{},[35,20384,18346],{"mathvariant":3105},[81,20386,83],{},[35,20388,1544],{"mathvariant":231},[81,20390,1547],{},[227,20392,20393,20395],{},[35,20394,232],{"mathvariant":231},[32,20396,20397,20399,20401],{},[35,20398,102],{},[81,20400,239],{},[89,20402,91],{},[40,20404,20405],{"encoding":42},"\\boldsymbol{\\mu} \\in \\operatorname{int}\\Delta^{K-1}",[15,20407,20409,20433],{"className":20408,"ariaHidden":48},[47],[15,20410,20412,20415,20424,20427,20430],{"className":20411},[52],[15,20413],{"className":20414,"style":118},[56],[15,20416,20418],{"className":20417},[61],[15,20419,20421],{"className":20420},[61],[15,20422,18346],{"className":20423},[61,3166],[15,20425],{"className":20426,"style":128},[127],[15,20428,83],{"className":20429},[132],[15,20431],{"className":20432,"style":128},[127],[15,20434,20436,20439,20445,20448],{"className":20435},[52],[15,20437],{"className":20438,"style":695},[56],[15,20440,20442],{"className":20441},[531],[15,20443,1544],{"className":20444},[61,1622],[15,20446],{"className":20447,"style":157},[127],[15,20449,20451,20454],{"className":20450},[61],[15,20452,232],{"className":20453},[61],[15,20455,20457],{"className":20456},[338],[15,20458,20460],{"className":20459},[342],[15,20461,20463],{"className":20462},[346],[15,20464,20466],{"className":20465,"style":695},[350],[15,20467,20468,20471],{"style":716},[15,20469],{"className":20470,"style":358},[357],[15,20472,20474],{"className":20473},[362,363,364,365],[15,20475,20477,20480,20483],{"className":20476},[61,365],[15,20478,102],{"className":20479,"style":174},[61,122,365],[15,20481,239],{"className":20482},[375,365],[15,20484,91],{"className":20485},[61,365],"，令 ",[15,20488,20490,20528],{"className":20489,"translate":19},[18],[15,20491,20493],{"className":20492},[23],[25,20494,20495],{"xmlns":27},[29,20496,20497,20525],{},[32,20498,20499,20515,20517,20523],{},[227,20500,20501,20503],{},[35,20502,3106],{"mathvariant":3105},[32,20504,20505,20507,20513],{},[81,20506,741],{"stretchy":86},[266,20508,20509,20511],{},[35,20510,3106],{},[89,20512,278],{},[81,20514,750],{"stretchy":86},[81,20516,244],{},[266,20518,20519,20521],{},[35,20520,3106],{},[89,20522,278],{},[35,20524,18346],{"mathvariant":3105},[40,20526,20527],{"encoding":42},"\\boldsymbol{\\alpha}^{(\\alpha_0)} = \\alpha_0 \\boldsymbol{\\mu}",[15,20529,20531,20628],{"className":20530,"ariaHidden":48},[47],[15,20532,20534,20538,20619,20622,20625],{"className":20533},[52],[15,20535],{"className":20536,"style":20537},[56],"height:0.888em;",[15,20539,20541,20550],{"className":20540},[61],[15,20542,20544],{"className":20543},[61],[15,20545,20547],{"className":20546},[61],[15,20548,3106],{"className":20549},[61,3166],[15,20551,20553],{"className":20552},[338],[15,20554,20556],{"className":20555},[342],[15,20557,20559],{"className":20558},[346],[15,20560,20562],{"className":20561,"style":20537},[350],[15,20563,20564,20567],{"style":716},[15,20565],{"className":20566,"style":358},[357],[15,20568,20570],{"className":20569},[362,363,364,365],[15,20571,20573,20576,20616],{"className":20572},[61,365],[15,20574,741],{"className":20575},[146,365],[15,20577,20579,20582],{"className":20578},[61,365],[15,20580,3106],{"className":20581,"style":3191},[61,122,365],[15,20583,20585],{"className":20584},[338],[15,20586,20588,20608],{"className":20587},[342,472],[15,20589,20591,20605],{"className":20590},[346],[15,20592,20594],{"className":20593,"style":4425},[350],[15,20595,20596,20599],{"style":4058},[15,20597],{"className":20598,"style":4062},[357],[15,20600,20602],{"className":20601},[362,4066,4067,365],[15,20603,278],{"className":20604},[61,365],[15,20606,497],{"className":20607},[496],[15,20609,20611],{"className":20610},[346],[15,20612,20614],{"className":20613,"style":4080},[350],[15,20615],{},[15,20617,750],{"className":20618},[178,365],[15,20620],{"className":20621,"style":128},[127],[15,20623,244],{"className":20624},[132],[15,20626],{"className":20627,"style":128},[127],[15,20629,20631,20634,20674],{"className":20630},[52],[15,20632],{"className":20633,"style":829},[56],[15,20635,20637,20640],{"className":20636},[61],[15,20638,3106],{"className":20639,"style":3191},[61,122],[15,20641,20643],{"className":20642},[338],[15,20644,20646,20666],{"className":20645},[342,472],[15,20647,20649,20663],{"className":20648},[346],[15,20650,20652],{"className":20651,"style":1214},[350],[15,20653,20654,20657],{"style":3206},[15,20655],{"className":20656,"style":358},[357],[15,20658,20660],{"className":20659},[362,363,364,365],[15,20661,278],{"className":20662},[61,365],[15,20664,497],{"className":20665},[496],[15,20667,20669],{"className":20668},[346],[15,20670,20672],{"className":20671,"style":504},[350],[15,20673],{},[15,20675,20677],{"className":20676},[61],[15,20678,20680],{"className":20679},[61],[15,20681,18346],{"className":20682},[61,3166],"。则当 ",[15,20685,20687,20710],{"className":20686,"translate":19},[18],[15,20688,20690],{"className":20689},[23],[25,20691,20692],{"xmlns":27},[29,20693,20694,20707],{},[32,20695,20696,20702,20705],{},[266,20697,20698,20700],{},[35,20699,3106],{},[89,20701,278],{},[81,20703,20704],{},"→",[35,20706,6796],{"mathvariant":231},[40,20708,20709],{"encoding":42},"\\alpha_0 \\to \\infty",[15,20711,20713,20768],{"className":20712,"ariaHidden":48},[47],[15,20714,20716,20719,20759,20762,20765],{"className":20715},[52],[15,20717],{"className":20718,"style":2511},[56],[15,20720,20722,20725],{"className":20721},[61],[15,20723,3106],{"className":20724,"style":3191},[61,122],[15,20726,20728],{"className":20727},[338],[15,20729,20731,20751],{"className":20730},[342,472],[15,20732,20734,20748],{"className":20733},[346],[15,20735,20737],{"className":20736,"style":1214},[350],[15,20738,20739,20742],{"style":3206},[15,20740],{"className":20741,"style":358},[357],[15,20743,20745],{"className":20744},[362,363,364,365],[15,20746,278],{"className":20747},[61,365],[15,20749,497],{"className":20750},[496],[15,20752,20754],{"className":20753},[346],[15,20755,20757],{"className":20756,"style":504},[350],[15,20758],{},[15,20760],{"className":20761,"style":128},[127],[15,20763,20704],{"className":20764},[132],[15,20766],{"className":20767,"style":128},[127],[15,20769,20771,20774],{"className":20770},[52],[15,20772],{"className":20773,"style":6032},[56],[15,20775,6796],{"className":20776},[61]," 时，",[15,20779,20781,20809],{"className":20780,"translate":19},[18],[15,20782,20784],{"className":20783},[23],[25,20785,20786],{"xmlns":27},[29,20787,20788,20806],{},[32,20789,20790,20792,20794,20796,20802,20804],{},[35,20791,4186],{"mathvariant":231},[81,20793,1547],{},[81,20795,741],{"stretchy":86},[266,20797,20798,20800],{},[35,20799,3106],{},[89,20801,278],{},[35,20803,18346],{"mathvariant":3105},[81,20805,750],{"stretchy":86},[40,20807,20808],{"encoding":42},"\\operatorname{Dir}(\\alpha_0 \\boldsymbol{\\mu})",[15,20810,20812],{"className":20811,"ariaHidden":48},[47],[15,20813,20815,20818,20824,20827,20867,20876],{"className":20814},[52],[15,20816],{"className":20817,"style":142},[56],[15,20819,20821],{"className":20820},[531],[15,20822,4186],{"className":20823},[61,1622],[15,20825,741],{"className":20826},[146],[15,20828,20830,20833],{"className":20829},[61],[15,20831,3106],{"className":20832,"style":3191},[61,122],[15,20834,20836],{"className":20835},[338],[15,20837,20839,20859],{"className":20838},[342,472],[15,20840,20842,20856],{"className":20841},[346],[15,20843,20845],{"className":20844,"style":1214},[350],[15,20846,20847,20850],{"style":3206},[15,20848],{"className":20849,"style":358},[357],[15,20851,20853],{"className":20852},[362,363,364,365],[15,20854,278],{"className":20855},[61,365],[15,20857,497],{"className":20858},[496],[15,20860,20862],{"className":20861},[346],[15,20863,20865],{"className":20864,"style":504},[350],[15,20866],{},[15,20868,20870],{"className":20869},[61],[15,20871,20873],{"className":20872},[61],[15,20874,18346],{"className":20875},[61,3166],[15,20877,750],{"className":20878},[178]," 弱收敛于位于 ",[15,20881,20883,20896],{"className":20882,"translate":19},[18],[15,20884,20886],{"className":20885},[23],[25,20887,20888],{"xmlns":27},[29,20889,20890,20894],{},[32,20891,20892],{},[35,20893,18346],{"mathvariant":3105},[40,20895,19451],{"encoding":42},[15,20897,20899],{"className":20898,"ariaHidden":48},[47],[15,20900,20902,20905],{"className":20901},[52],[15,20903],{"className":20904,"style":829},[56],[15,20906,20908],{"className":20907},[61],[15,20909,20911],{"className":20910},[61],[15,20912,18346],{"className":20913},[61,3166]," 的退化测度 ",[15,20916,20918,20937],{"className":20917,"translate":19},[18],[15,20919,20921],{"className":20920},[23],[25,20922,20923],{"xmlns":27},[29,20924,20925,20934],{},[32,20926,20927],{},[266,20928,20929,20932],{},[35,20930,20931],{},"δ",[35,20933,18346],{"mathvariant":3105},[40,20935,20936],{"encoding":42},"\\delta_{\\boldsymbol{\\mu}}",[15,20938,20940],{"className":20939,"ariaHidden":48},[47],[15,20941,20943,20947],{"className":20942},[52],[15,20944],{"className":20945,"style":20946},[56],"height:0.9805em;vertical-align:-0.2861em;",[15,20948,20950,20954],{"className":20949},[61],[15,20951,20931],{"className":20952,"style":20953},[61,122],"margin-right:0.0379em;",[15,20955,20957],{"className":20956},[338],[15,20958,20960,20991],{"className":20959},[342,472],[15,20961,20963,20988],{"className":20962},[346],[15,20964,20967],{"className":20965,"style":20966},[350],"height:0.1611em;",[15,20968,20970,20973],{"style":20969},"top:-2.55em;margin-left:-0.0379em;margin-right:0.05em;",[15,20971],{"className":20972,"style":358},[357],[15,20974,20976],{"className":20975},[362,363,364,365],[15,20977,20979],{"className":20978},[61,365],[15,20980,20982],{"className":20981},[61,365],[15,20983,20985],{"className":20984},[61,365],[15,20986,18346],{"className":20987},[61,3166,365],[15,20989,497],{"className":20990},[496],[15,20992,20994],{"className":20993},[346],[15,20995,20997],{"className":20996,"style":15707},[350],[15,20998],{},[11,21000,21001,21003,21004,16199,21076,21218,21219,21271,21272,21484,21485,867],{},[661,21002,5123],{}," 由命题 2.1，",[15,21005,21007,21031],{"className":21006,"translate":19},[18],[15,21008,21010],{"className":21009},[23],[25,21011,21012],{"xmlns":27},[29,21013,21014,21028],{},[32,21015,21016,21018,21020,21022,21024,21026],{},[35,21017,14818],{"mathvariant":258},[81,21019,14821],{"stretchy":86},[35,21021,1029],{"mathvariant":251},[81,21023,14830],{"stretchy":86},[81,21025,244],{},[35,21027,18346],{"mathvariant":3105},[40,21029,21030],{"encoding":42},"\\mathbb{E}[\\mathbf{x}] = \\boldsymbol{\\mu}",[15,21032,21034,21061],{"className":21033,"ariaHidden":48},[47],[15,21035,21037,21040,21043,21046,21049,21052,21055,21058],{"className":21036},[52],[15,21038],{"className":21039,"style":142},[56],[15,21041,14818],{"className":21042},[61,427],[15,21044,14821],{"className":21045},[146],[15,21047,1029],{"className":21048},[61,411],[15,21050,14830],{"className":21051},[178],[15,21053],{"className":21054,"style":128},[127],[15,21056,244],{"className":21057},[132],[15,21059],{"className":21060,"style":128},[127],[15,21062,21064,21067],{"className":21063},[52],[15,21065],{"className":21066,"style":829},[56],[15,21068,21070],{"className":21069},[61],[15,21071,21073],{"className":21072},[61],[15,21074,18346],{"className":21075},[61,3166],[15,21077,21079,21115],{"className":21078,"translate":19},[18],[15,21080,21082],{"className":21081},[23],[25,21083,21084],{"xmlns":27},[29,21085,21086,21112],{},[32,21087,21088,21090,21092,21094,21096,21098,21100,21106,21108,21110],{},[35,21089,18545],{"mathvariant":251},[81,21091,244],{},[35,21093,12265],{},[81,21095,741],{"stretchy":86},[89,21097,91],{},[35,21099,3020],{"mathvariant":231},[266,21101,21102,21104],{},[35,21103,3106],{},[89,21105,278],{},[81,21107,750],{"stretchy":86},[81,21109,20704],{},[89,21111,278],{"mathvariant":251},[40,21113,21114],{"encoding":42},"\\boldsymbol{\\Sigma} = O(1\u002F\\alpha_0) \\to \\mathbf{0}",[15,21116,21118,21142,21209],{"className":21117,"ariaHidden":48},[47],[15,21119,21121,21124,21133,21136,21139],{"className":21120},[52],[15,21122],{"className":21123,"style":18616},[56],[15,21125,21127],{"className":21126},[61],[15,21128,21130],{"className":21129},[61],[15,21131,18545],{"className":21132},[61,411],[15,21134],{"className":21135,"style":128},[127],[15,21137,244],{"className":21138},[132],[15,21140],{"className":21141,"style":128},[127],[15,21143,21145,21148,21151,21154,21157,21197,21200,21203,21206],{"className":21144},[52],[15,21146],{"className":21147,"style":142},[56],[15,21149,12265],{"className":21150,"style":12287},[61,122],[15,21152,741],{"className":21153},[146],[15,21155,3049],{"className":21156},[61],[15,21158,21160,21163],{"className":21159},[61],[15,21161,3106],{"className":21162,"style":3191},[61,122],[15,21164,21166],{"className":21165},[338],[15,21167,21169,21189],{"className":21168},[342,472],[15,21170,21172,21186],{"className":21171},[346],[15,21173,21175],{"className":21174,"style":1214},[350],[15,21176,21177,21180],{"style":3206},[15,21178],{"className":21179,"style":358},[357],[15,21181,21183],{"className":21182},[362,363,364,365],[15,21184,278],{"className":21185},[61,365],[15,21187,497],{"className":21188},[496],[15,21190,21192],{"className":21191},[346],[15,21193,21195],{"className":21194,"style":504},[350],[15,21196],{},[15,21198,750],{"className":21199},[178],[15,21201],{"className":21202,"style":128},[127],[15,21204,20704],{"className":21205},[132],[15,21207],{"className":21208,"style":128},[127],[15,21210,21212,21215],{"className":21211},[52],[15,21213],{"className":21214,"style":859},[56],[15,21216,278],{"className":21217},[61,411],"。对任意 ",[15,21220,21222,21241],{"className":21221,"translate":19},[18],[15,21223,21225],{"className":21224},[23],[25,21226,21227],{"xmlns":27},[29,21228,21229,21238],{},[32,21230,21231,21234,21236],{},[35,21232,21233],{},"ε",[81,21235,1590],{},[89,21237,278],{},[40,21239,21240],{"encoding":42},"\\varepsilon > 0",[15,21242,21244,21262],{"className":21243,"ariaHidden":48},[47],[15,21245,21247,21250,21253,21256,21259],{"className":21246},[52],[15,21248],{"className":21249,"style":4656},[56],[15,21251,21233],{"className":21252},[61,122],[15,21254],{"className":21255,"style":128},[127],[15,21257,1590],{"className":21258},[132],[15,21260],{"className":21261,"style":128},[127],[15,21263,21265,21268],{"className":21264},[52],[15,21266],{"className":21267,"style":859},[56],[15,21269,278],{"className":21270},[61],"，由 Chebyshev 不等式 ",[15,21273,21275,21331],{"className":21274,"translate":19},[18],[15,21276,21278],{"className":21277},[23],[25,21279,21280],{"xmlns":27},[29,21281,21282,21328],{},[32,21283,21284,21287,21289,21292,21294,21296,21298,21300,21302,21304,21306,21309,21312,21314,21316,21318,21324,21326],{},[35,21285,21286],{"mathvariant":258},"P",[81,21288,741],{"stretchy":86},[35,21290,21291],{"mathvariant":231},"∥",[35,21293,1029],{"mathvariant":251},[81,21295,239],{},[35,21297,18346],{"mathvariant":3105},[35,21299,21291],{"mathvariant":231},[81,21301,1590],{},[35,21303,21233],{},[81,21305,750],{"stretchy":86},[81,21307,21308],{},"≤",[35,21310,21311],{"mathvariant":231},"tr",[81,21313,1547],{},[35,21315,18545],{"mathvariant":251},[35,21317,3020],{"mathvariant":231},[227,21319,21320,21322],{},[35,21321,21233],{},[89,21323,908],{},[81,21325,20704],{},[89,21327,278],{},[40,21329,21330],{"encoding":42},"\\mathbb{P}(\\|\\mathbf{x} - \\boldsymbol{\\mu}\\| > \\varepsilon) \\le \\operatorname{tr}\\boldsymbol{\\Sigma}\u002F\\varepsilon^2 \\to 0",[15,21332,21334,21361,21388,21409,21475],{"className":21333,"ariaHidden":48},[47],[15,21335,21337,21340,21343,21346,21349,21352,21355,21358],{"className":21336},[52],[15,21338],{"className":21339,"style":142},[56],[15,21341,21286],{"className":21342},[61,427],[15,21344,741],{"className":21345},[146],[15,21347,21291],{"className":21348},[61],[15,21350,1029],{"className":21351},[61,411],[15,21353],{"className":21354,"style":772},[127],[15,21356,239],{"className":21357},[375],[15,21359],{"className":21360,"style":772},[127],[15,21362,21364,21367,21376,21379,21382,21385],{"className":21363},[52],[15,21365],{"className":21366,"style":142},[56],[15,21368,21370],{"className":21369},[61],[15,21371,21373],{"className":21372},[61],[15,21374,18346],{"className":21375},[61,3166],[15,21377,21291],{"className":21378},[61],[15,21380],{"className":21381,"style":128},[127],[15,21383,1590],{"className":21384},[132],[15,21386],{"className":21387,"style":128},[127],[15,21389,21391,21394,21397,21400,21403,21406],{"className":21390},[52],[15,21392],{"className":21393,"style":142},[56],[15,21395,21233],{"className":21396},[61,122],[15,21398,750],{"className":21399},[178],[15,21401],{"className":21402,"style":128},[127],[15,21404,21308],{"className":21405},[132],[15,21407],{"className":21408,"style":128},[127],[15,21410,21412,21416,21422,21425,21434,21437,21466,21469,21472],{"className":21411},[52],[15,21413],{"className":21414,"style":21415},[56],"height:1.0641em;vertical-align:-0.25em;",[15,21417,21419],{"className":21418},[531],[15,21420,21311],{"className":21421},[61,1622],[15,21423],{"className":21424,"style":157},[127],[15,21426,21428],{"className":21427},[61],[15,21429,21431],{"className":21430},[61],[15,21432,18545],{"className":21433},[61,411],[15,21435,3020],{"className":21436},[61],[15,21438,21440,21443],{"className":21439},[61],[15,21441,21233],{"className":21442},[61,122],[15,21444,21446],{"className":21445},[338],[15,21447,21449],{"className":21448},[342],[15,21450,21452],{"className":21451},[346],[15,21453,21455],{"className":21454,"style":7905},[350],[15,21456,21457,21460],{"style":716},[15,21458],{"className":21459,"style":358},[357],[15,21461,21463],{"className":21462},[362,363,364,365],[15,21464,908],{"className":21465},[61,365],[15,21467],{"className":21468,"style":128},[127],[15,21470,20704],{"className":21471},[132],[15,21473],{"className":21474,"style":128},[127],[15,21476,21478,21481],{"className":21477},[52],[15,21479],{"className":21480,"style":859},[56],[15,21482,278],{"className":21483},[61],"，故依概率收敛，进而弱收敛于 ",[15,21486,21488,21506],{"className":21487,"translate":19},[18],[15,21489,21491],{"className":21490},[23],[25,21492,21493],{"xmlns":27},[29,21494,21495,21503],{},[32,21496,21497],{},[266,21498,21499,21501],{},[35,21500,20931],{},[35,21502,18346],{"mathvariant":3105},[40,21504,21505],{"encoding":42},"\\delta_{\\boldsymbol\\mu}",[15,21507,21509],{"className":21508,"ariaHidden":48},[47],[15,21510,21512,21515],{"className":21511},[52],[15,21513],{"className":21514,"style":20946},[56],[15,21516,21518,21521],{"className":21517},[61],[15,21519,20931],{"className":21520,"style":20953},[61,122],[15,21522,21524],{"className":21523},[338],[15,21525,21527,21556],{"className":21526},[342,472],[15,21528,21530,21553],{"className":21529},[346],[15,21531,21533],{"className":21532,"style":20966},[350],[15,21534,21535,21538],{"style":20969},[15,21536],{"className":21537,"style":358},[357],[15,21539,21541],{"className":21540},[362,363,364,365],[15,21542,21544],{"className":21543},[61,365],[15,21545,21547],{"className":21546},[61,365],[15,21548,21550],{"className":21549},[61,365],[15,21551,18346],{"className":21552},[61,3166,365],[15,21554,497],{"className":21555},[496],[15,21557,21559],{"className":21558},[346],[15,21560,21562],{"className":21561,"style":15707},[350],[15,21563],{},[11,21565,21566,21567,21683],{},"这一命题给出了本文后续所有「不确定性」语汇的几何基础：",[661,21568,21569,21570,21647,21648,21682],{},"迪利克雷分布的「模糊程度」等于 ",[15,21571,21573,21595],{"className":21572,"translate":19},[18],[15,21574,21576],{"className":21575},[23],[25,21577,21578],{"xmlns":27},[29,21579,21580,21592],{},[32,21581,21582,21584,21586],{},[89,21583,91],{},[35,21585,3020],{"mathvariant":231},[266,21587,21588,21590],{},[35,21589,3106],{},[89,21591,278],{},[40,21593,21594],{"encoding":42},"1\u002F\\alpha_0",[15,21596,21598],{"className":21597,"ariaHidden":48},[47],[15,21599,21601,21604,21607],{"className":21600},[52],[15,21602],{"className":21603,"style":142},[56],[15,21605,3049],{"className":21606},[61],[15,21608,21610,21613],{"className":21609},[61],[15,21611,3106],{"className":21612,"style":3191},[61,122],[15,21614,21616],{"className":21615},[338],[15,21617,21619,21639],{"className":21618},[342,472],[15,21620,21622,21636],{"className":21621},[346],[15,21623,21625],{"className":21624,"style":1214},[350],[15,21626,21627,21630],{"style":3206},[15,21628],{"className":21629,"style":358},[357],[15,21631,21633],{"className":21632},[362,363,364,365],[15,21634,278],{"className":21635},[61,365],[15,21637,497],{"className":21638},[496],[15,21640,21642],{"className":21641},[346],[15,21643,21645],{"className":21644,"style":504},[350],[15,21646],{}," 的量级，而它的「中心」由 ",[15,21649,21651,21664],{"className":21650,"translate":19},[18],[15,21652,21654],{"className":21653},[23],[25,21655,21656],{"xmlns":27},[29,21657,21658,21662],{},[32,21659,21660],{},[35,21661,18346],{"mathvariant":3105},[40,21663,19451],{"encoding":42},[15,21665,21667],{"className":21666,"ariaHidden":48},[47],[15,21668,21670,21673],{"className":21669},[52],[15,21671],{"className":21672,"style":829},[56],[15,21674,21676],{"className":21675},[61],[15,21677,21679],{"className":21678},[61],[15,21680,18346],{"className":21681},[61,3166]," 给出。"," 两者在同一族分布中被参数化为彼此独立的量——这正是证据深度学习能够把「预测什么」与「有多确定」分离开来的数学根据。",[877,21685,21687],{"id":21686},"_23-众数与熵","2.3 众数与熵",[882,21689,21690],{},[11,21691,21692,21695,21696,21788,21789,21818,21819,21891,21892,21965,21966],{},[661,21693,21694],{},"命题 2.3（众数）"," 若 ",[15,21697,21699,21721],{"className":21698,"translate":19},[18],[15,21700,21702],{"className":21701},[23],[25,21703,21704],{"xmlns":27},[29,21705,21706,21718],{},[32,21707,21708,21714,21716],{},[266,21709,21710,21712],{},[35,21711,3106],{},[35,21713,1070],{},[81,21715,1590],{},[89,21717,91],{},[40,21719,21720],{"encoding":42},"\\alpha_i > 1",[15,21722,21724,21779],{"className":21723,"ariaHidden":48},[47],[15,21725,21727,21730,21770,21773,21776],{"className":21726},[52],[15,21728],{"className":21729,"style":1753},[56],[15,21731,21733,21736],{"className":21732},[61],[15,21734,3106],{"className":21735,"style":3191},[61,122],[15,21737,21739],{"className":21738},[338],[15,21740,21742,21762],{"className":21741},[342,472],[15,21743,21745,21759],{"className":21744},[346],[15,21746,21748],{"className":21747,"style":1364},[350],[15,21749,21750,21753],{"style":3206},[15,21751],{"className":21752,"style":358},[357],[15,21754,21756],{"className":21755},[362,363,364,365],[15,21757,1070],{"className":21758},[61,122,365],[15,21760,497],{"className":21761},[496],[15,21763,21765],{"className":21764},[346],[15,21766,21768],{"className":21767,"style":504},[350],[15,21769],{},[15,21771],{"className":21772,"style":128},[127],[15,21774,1590],{"className":21775},[132],[15,21777],{"className":21778,"style":128},[127],[15,21780,21782,21785],{"className":21781},[52],[15,21783],{"className":21784,"style":859},[56],[15,21786,91],{"className":21787},[61]," 对一切 ",[15,21790,21792,21805],{"className":21791,"translate":19},[18],[15,21793,21795],{"className":21794},[23],[25,21796,21797],{"xmlns":27},[29,21798,21799,21803],{},[32,21800,21801],{},[35,21802,1070],{},[40,21804,1070],{"encoding":42},[15,21806,21808],{"className":21807,"ariaHidden":48},[47],[15,21809,21811,21815],{"className":21810},[52],[15,21812],{"className":21813,"style":21814},[56],"height:0.6595em;",[15,21816,1070],{"className":21817},[61,122]," 成立，则 ",[15,21820,21822,21846],{"className":21821,"translate":19},[18],[15,21823,21825],{"className":21824},[23],[25,21826,21827],{"xmlns":27},[29,21828,21829,21843],{},[32,21830,21831,21833,21835,21837,21839,21841],{},[35,21832,3707],{},[81,21834,741],{"stretchy":86},[35,21836,1029],{"mathvariant":251},[81,21838,3714],{},[35,21840,3106],{"mathvariant":3105},[81,21842,750],{"stretchy":86},[40,21844,21845],{"encoding":42},"f(\\mathbf{x} \\mid 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",[15,21893,21895,21918],{"className":21894,"translate":19},[18],[15,21896,21898],{"className":21897},[23],[25,21899,21900],{"xmlns":27},[29,21901,21902,21916],{},[32,21903,21904],{},[227,21905,21906,21908],{},[35,21907,232],{"mathvariant":231},[32,21909,21910,21912,21914],{},[35,21911,102],{},[81,21913,239],{},[89,21915,91],{},[40,21917,1858],{"encoding":42},[15,21919,21921],{"className":21920,"ariaHidden":48},[47],[15,21922,21924,21927],{"className":21923},[52],[15,21925],{"className":21926,"style":695},[56],[15,21928,21930,21933],{"className":21929},[61],[15,21931,232],{"className":21932},[61],[15,21934,21936],{"className":21935},[338],[15,21937,21939],{"className":21938},[342],[15,21940,21942],{"className":21941},[346],[15,21943,21945],{"className":21944,"style":695},[350],[15,21946,21947,21950],{"style":716},[15,21948],{"className":21949,"style":358},[357],[15,21951,21953],{"className":21952},[362,363,364,365],[15,21954,21956,21959,21962],{"className":21955},[61,365],[15,21957,102],{"className":21958,"style":174},[61,122,365],[15,21960,239],{"className":21961},[375,365],[15,21963,91],{"className":21964},[61,365]," 的内部有唯一众数\n",[15,21967,21969,22054],{"className":21968,"translate":19},[18],[15,21970,21972],{"className":21971},[23],[25,21973,21974],{"xmlns":27},[29,21975,21976,22051],{},[32,21977,21978,21985,21987],{},[227,21979,21980,21982],{},[35,21981,1029],{"mathvariant":251},[81,21983,21984],{},"⋆",[81,21986,244],{},[32,21988,21989,21991,22017,22019,22021,22023,22049],{},[81,21990,741],{"fence":48},[3722,21992,21993,22005],{},[32,21994,21995,22001,22003],{},[266,21996,21997,21999],{},[35,21998,3106],{},[89,22000,91],{},[81,22002,239],{},[89,22004,91],{},[32,22006,22007,22013,22015],{},[266,22008,22009,22011],{},[35,22010,3106],{},[89,22012,278],{},[81,22014,239],{},[35,22016,102],{},[81,22018,94],{"separator":48},[81,22020,97],{},[81,22022,94],{"separator":48},[3722,22024,22025,22037],{},[32,22026,22027,22033,22035],{},[266,22028,22029,22031],{},[35,22030,3106],{},[35,22032,102],{},[81,22034,239],{},[89,22036,91],{},[32,22038,22039,22045,22047],{},[266,22040,22041,22043],{},[35,22042,3106],{},[89,22044,278],{},[81,22046,239],{},[35,22048,102],{},[81,22050,750],{"fence":48},[40,22052,22053],{"encoding":42},"\\mathbf{x}^\\star = \\left( \\frac{\\alpha_1 - 1}{\\alpha_0 - K}, \\dots, \\frac{\\alpha_K - 1}{\\alpha_0 - K} \\right)",[15,22055,22057,22102],{"className":22056,"ariaHidden":48},[47],[15,22058,22060,22064,22093,22096,22099],{"className":22059},[52],[15,22061],{"className":22062,"style":22063},[56],"height:0.6887em;",[15,22065,22067,22070],{"className":22066},[61],[15,22068,1029],{"className":22069},[61,411],[15,22071,22073],{"className":22072},[338],[15,22074,22076],{"className":22075},[342],[15,22077,22079],{"className":22078},[346],[15,22080,22082],{"className":22081,"style":22063},[350],[15,22083,22084,22087],{"style":716},[15,22085],{"className":22086,"style":358},[357],[15,22088,22090],{"className":22089},[362,363,364,365],[15,22091,21984],{"className":22092},[375,365],[15,22094],{"className":22095,"style":128},[127],[15,22097,244],{"className":22098},[132],[15,22100],{"className":22101,"style":128},[127],[15,22103,22105,22108],{"className":22104},[52],[15,22106],{"className":22107,"style":394},[56],[15,22109,22111,22117,22272,22275,22278,22281,22284,22287,22290,22449],{"className":22110},[161],[15,22112,22114],{"className":22113,"style":402},[146,401],[15,22115,741],{"className":22116},[406,407],[15,22118,22120,22123,22269],{"className":22119},[61],[15,22121],{"className":22122},[146,3841],[15,22124,22126],{"className":22125},[3722],[15,22127,22129,22261],{"className":22128},[342,472],[15,22130,22132,22258],{"className":22131},[346],[15,22133,22136,22193,22201],{"className":22134,"style":22135},[350],"height:0.8612em;",[15,22137,22138,22141],{"style":3857},[15,22139],{"className":22140,"style":3861},[357],[15,22142,22144],{"className":22143},[362,363,364,365],[15,22145,22147,22187,22190],{"className":22146},[61,365],[15,22148,22150,22153],{"className":22149},[61,365],[15,22151,3106],{"className":22152,"style":3191},[61,122,365],[15,22154,22156],{"className":22155},[338],[15,22157,22159,22179],{"className":22158},[342,472],[15,22160,22162,22176],{"className":22161},[346],[15,22163,22165],{"className":22164,"style":4425},[350],[15,22166,22167,22170],{"style":4058},[15,22168],{"className":22169,"style":4062},[357],[15,22171,22173],{"className":22172},[362,4066,4067,365],[15,22174,278],{"className":22175},[61,365],[15,22177,497],{"className":22178},[496],[15,22180,22182],{"className":22181},[346],[15,22183,22185],{"className":22184,"style":4080},[350],[15,22186],{},[15,22188,239],{"className":22189},[375,365],[15,22191,102],{"className":22192,"style":174},[61,122,365],[15,22194,22195,22198],{"style":3889},[15,22196],{"className":22197,"style":3861},[357],[15,22199],{"className":22200,"style":3897},[3896],[15,22202,22203,22206],{"style":15161},[15,22204],{"className":22205,"style":3861},[357],[15,22207,22209],{"className":22208},[362,363,364,365],[15,22210,22212,22252,22255],{"className":22211},[61,365],[15,22213,22215,22218],{"className":22214},[61,365],[15,22216,3106],{"className":22217,"style":3191},[61,122,365],[15,22219,22221],{"className":22220},[338],[15,22222,22224,22244],{"className":22223},[342,472],[15,22225,22227,22241],{"className":22226},[346],[15,22228,22230],{"className":22229,"style":4425},[350],[15,22231,22232,22235],{"style":4058},[15,22233],{"className":22234,"style":4062},[357],[15,22236,22238],{"className":22237},[362,4066,4067,365],[15,22239,91],{"className":22240},[61,365],[15,22242,497],{"className":22243},[496],[15,22245,22247],{"className":22246},[346],[15,22248,22250],{"className":22249,"style":4080},[350],[15,22251],{},[15,22253,239],{"className":22254},[375,365],[15,22256,91],{"className":22257},[61,365],[15,22259,497],{"className":22260},[496],[15,22262,22264],{"className":22263},[346],[15,22265,22267],{"className":22266,"style":15220},[350],[15,22268],{},[15,22270],{"className":22271},[178,3841],[15,22273,94],{"className":22274},[153],[15,22276],{"className":22277,"style":157},[127],[15,22279,97],{"className":22280},[161],[15,22282],{"className":22283,"style":157},[127],[15,22285,94],{"className":22286},[153],[15,22288],{"className":22289,"style":157},[127],[15,22291,22293,22296,22446],{"className":22292},[61],[15,22294],{"className":22295},[146,3841],[15,22297,22299],{"className":22298},[3722],[15,22300,22302,22438],{"className":22301},[342,472],[15,22303,22305,22435],{"className":22304},[346],[15,22306,22309,22366,22374],{"className":22307,"style":22308},[350],"height:0.8614em;",[15,22310,22311,22314],{"style":3857},[15,22312],{"className":22313,"style":3861},[357],[15,22315,22317],{"className":22316},[362,363,364,365],[15,22318,22320,22360,22363],{"className":22319},[61,365],[15,22321,22323,22326],{"className":22322},[61,365],[15,22324,3106],{"className":22325,"style":3191},[61,122,365],[15,22327,22329],{"className":22328},[338],[15,22330,22332,22352],{"className":22331},[342,472],[15,22333,22335,22349],{"className":22334},[346],[15,22336,22338],{"className":22337,"style":4425},[350],[15,22339,22340,22343],{"style":4058},[15,22341],{"className":22342,"style":4062},[357],[15,22344,22346],{"className":22345},[362,4066,4067,365],[15,22347,278],{"className":22348},[61,365],[15,22350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",[15,22459,22461,22483],{"className":22460,"translate":19},[18],[15,22462,22464],{"className":22463},[23],[25,22465,22466],{"xmlns":27},[29,22467,22468,22480],{},[32,22469,22470,22476,22478],{},[266,22471,22472,22474],{},[35,22473,3106],{},[35,22475,1070],{},[81,22477,21308],{},[89,22479,91],{},[40,22481,22482],{"encoding":42},"\\alpha_i \\le 1",[15,22484,22486,22542],{"className":22485,"ariaHidden":48},[47],[15,22487,22489,22493,22533,22536,22539],{"className":22488},[52],[15,22490],{"className":22491,"style":22492},[56],"height:0.786em;vertical-align:-0.15em;",[15,22494,22496,22499],{"className":22495},[61],[15,22497,3106],{"className":22498,"style":3191},[61,122],[15,22500,22502],{"className":22501},[338],[15,22503,22505,22525],{"className":22504},[342,472],[15,22506,22508,22522],{"className":22507},[346],[15,22509,22511],{"className":22510,"style":1364},[350],[15,22512,22513,22516],{"style":3206},[15,22514],{"className":22515,"style":358},[357],[15,22517,22519],{"className":22518},[362,363,364,365],[15,22520,1070],{"className":22521},[61,122,365],[15,22523,497],{"className":22524},[496],[15,22526,22528],{"className":22527},[346],[15,22529,22531],{"className":22530,"style":504},[350],[15,22532],{},[15,22534],{"className":22535,"style":128},[127],[15,22537,21308],{"className":22538},[132],[15,22540],{"className":22541,"style":128},[127],[15,22543,22545,22548],{"className":22544},[52],[15,22546],{"className":22547,"style":859},[56],[15,22549,91],{"className":22550},[61]," 时密度在对应边界面上无界，众数退化到边界，此时分布呈现 2.6 节所述的稀疏形态。",[882,22553,22554],{},[11,22555,22556,22559,22560,22636,22637,23138,23139,23234],{},[661,22557,22558],{},"命题 2.4（微分熵）"," 在 1.1 节的测度约定下，",[15,22561,22563,22588],{"className":22562,"translate":19},[18],[15,22564,22566],{"className":22565},[23],[25,22567,22568],{"xmlns":27},[29,22569,22570,22586],{},[32,22571,22572,22574,22576,22578,22580,22582,22584],{},[35,22573,1029],{"mathvariant":251},[81,22575,4183],{},[35,22577,4186],{"mathvariant":231},[81,22579,1547],{},[81,22581,741],{"stretchy":86},[35,22583,3106],{"mathvariant":3105},[81,22585,750],{"stretchy":86},[40,22587,4197],{"encoding":42},[15,22589,22591,22609],{"className":22590,"ariaHidden":48},[47],[15,22592,22594,22597,22600,22603,22606],{"className":22593},[52],[15,22595],{"className":22596,"style":2708},[56],[15,22598,1029],{"className":22599},[61,411],[15,22601],{"className":22602,"style":128},[127],[15,22604,4183],{"className":22605},[132],[15,22607],{"className":22608,"style":128},[127],[15,22610,22612,22615,22621,22624,22633],{"className":22611},[52],[15,22613],{"className":22614,"style":142},[56],[15,22616,22618],{"className":22617},[531],[15,22619,4186],{"className":22620},[61,1622],[15,22622,741],{"className":22623},[146],[15,22625,22627],{"className":22626},[61],[15,22628,22630],{"className":22629},[61],[15,22631,3106],{"className":22632},[61,3166],[15,22634,750],{"className":22635},[178]," 的微分熵为\n",[15,22638,22640,22751],{"className":22639,"translate":19},[18],[15,22641,22643],{"className":22642},[23],[25,22644,22645],{"xmlns":27},[29,22646,22647,22748],{},[32,22648,22649,22652,22654,22656,22658,22660,22663,22665,22667,22669,22671,22673,22675,22677,22683,22685,22687,22689,22691,22694,22696,22702,22704,22706,22720,22722,22728,22730,22732,22734,22736,22738,22740,22746],{},[35,22650,22651],{},"H",[81,22653,741],{"stretchy":86},[35,22655,1029],{"mathvariant":251},[81,22657,750],{"stretchy":86},[81,22659,244],{},[35,22661,22662],{},"log",[81,22664,1547],{},[35,22666,3730],{},[81,22668,741],{"stretchy":86},[35,22670,3106],{"mathvariant":3105},[81,22672,750],{"stretchy":86},[81,22674,7762],{},[81,22676,741],{"stretchy":86},[266,22678,22679,22681],{},[35,22680,3106],{},[89,22682,278],{},[81,22684,239],{},[35,22686,102],{},[81,22688,750],{"stretchy":86},[282,22690,4802],{},[35,22692,22693],{},"ψ",[81,22695,741],{"stretchy":86},[266,22697,22698,22700],{},[35,22699,3106],{},[89,22701,278],{},[81,22703,750],{"stretchy":86},[81,22705,239],{},[289,22707,22708,22710,22718],{},[81,22709,293],{},[32,22711,22712,22714,22716],{},[35,22713,1070],{},[81,22715,244],{},[89,22717,91],{},[35,22719,102],{},[81,22721,741],{"stretchy":86},[266,22723,22724,22726],{},[35,22725,3106],{},[35,22727,1070],{},[81,22729,239],{},[89,22731,91],{},[81,22733,750],{"stretchy":86},[282,22735,4802],{},[35,22737,22693],{},[81,22739,741],{"stretchy":86},[266,22741,22742,22744],{},[35,22743,3106],{},[35,22745,1070],{},[81,22747,750],{"stretchy":86},[40,22749,22750],{"encoding":42},"H(\\mathbf{x}) = \\log B(\\boldsymbol{\\alpha}) + (\\alpha_0 - K)\\,\\psi(\\alpha_0) - \\sum_{i=1}^K (\\alpha_i - 1)\\,\\psi(\\alpha_i)",[15,22752,22754,22782,22825,22883,22956,23074],{"className":22753,"ariaHidden":48},[47],[15,22755,22757,22760,22764,22767,22770,22773,22776,22779],{"className":22756},[52],[15,22758],{"className":22759,"style":142},[56],[15,22761,22651],{"className":22762,"style":22763},[61,122],"margin-right:0.0813em;",[15,22765,741],{"className":22766},[146],[15,22768,1029],{"className":22769},[61,411],[15,22771,750],{"className":22772},[178],[15,22774],{"className":22775,"style":128},[127],[15,22777,244],{"className":22778},[132],[15,22780],{"className":22781,"style":128},[127],[15,22783,22785,22788,22795,22798,22801,22804,22813,22816,22819,22822],{"className":22784},[52],[15,22786],{"className":22787,"style":142},[56],[15,22789,22791,22792],{"className":22790},[531],"lo",[15,22793,22794],{"style":15619},"g",[15,22796],{"className":22797,"style":157},[127],[15,22799,3730],{"className":22800,"style":3871},[61,122],[15,22802,741],{"className":22803},[146],[15,22805,22807],{"className":22806},[61],[15,22808,22810],{"className":22809},[61],[15,22811,3106],{"className":22812},[61,3166],[15,22814,750],{"className":22815},[178],[15,22817],{"className":22818,"style":772},[127],[15,22820,7762],{"className":22821},[375],[15,22823],{"className":22824,"style":772},[127],[15,22826,22828,22831,22834,22874,22877,22880],{"className":22827},[52],[15,22829],{"className":22830,"style":142},[56],[15,22832,741],{"className":22833},[146],[15,22835,22837,22840],{"className":22836},[61],[15,22838,3106],{"className":22839,"style":3191},[61,122],[15,22841,22843],{"className":22842},[338],[15,22844,22846,22866],{"className":22845},[342,472],[15,22847,22849,22863],{"className":22848},[346],[15,22850,22852],{"className":22851,"style":1214},[350],[15,22853,22854,22857],{"style":3206},[15,22855],{"className":22856,"style":358},[357],[15,22858,22860],{"className":22859},[362,363,364,365],[15,22861,278],{"className":22862},[61,365],[15,22864,497],{"className":22865},[496],[15,22867,22869],{"className":22868},[346],[15,22870,22872],{"className":22871,"style":504},[350],[15,22873],{},[15,22875],{"className":22876,"style":772},[127],[15,22878,239],{"className":22879},[375],[15,22881],{"className":22882,"style":772},[127],[15,22884,22886,22889,22892,22895,22898,22901,22904,22944,22947,22950,22953],{"className":22885},[52],[15,22887],{"className":22888,"style":142},[56],[15,22890,102],{"className":22891,"style":174},[61,122],[15,22893,750],{"className":22894},[178],[15,22896],{"className":22897,"style":157},[127],[15,22899,22693],{"className":22900,"style":123},[61,122],[15,22902,741],{"className":22903},[146],[15,22905,22907,22910],{"className":22906},[61],[15,22908,3106],{"className":22909,"style":3191},[61,122],[15,22911,22913],{"className":22912},[338],[15,22914,22916,22936],{"className":22915},[342,472],[15,22917,22919,22933],{"className":22918},[346],[15,22920,22922],{"className":22921,"style":1214},[350],[15,22923,22924,22927],{"style":3206},[15,22925],{"className":22926,"style":358},[357],[15,22928,22930],{"className":22929},[362,363,364,365],[15,22931,278],{"className":22932},[61,365],[15,22934,497],{"className":22935},[496],[15,22937,22939],{"className":22938},[346],[15,22940,22942],{"className":22941,"style":504},[350],[15,22943],{},[15,22945,750],{"className":22946},[178],[15,22948],{"className":22949,"style":772},[127],[15,22951,239],{"className":22952},[375],[15,22954],{"className":22955,"style":772},[127],[15,22957,22959,22962,23022,23025,23065,23068,23071],{"className":22958},[52],[15,22960],{"className":22961,"style":3471},[56],[15,22963,22965,22968],{"className":22964},[531],[15,22966,293],{"className":22967,"style":537},[531,535,536],[15,22969,22971],{"className":22970},[338],[15,22972,22974,23014],{"className":22973},[342,472],[15,22975,22977,23011],{"className":22976},[346],[15,22978,22980,23000],{"className":22979,"style":550},[350],[15,22981,22982,22985],{"style":553},[15,22983],{"className":22984,"style":358},[357],[15,22986,22988],{"className":22987},[362,363,364,365],[15,22989,22991,22994,22997],{"className":22990},[61,365],[15,22992,1070],{"className":22993},[61,122,365],[15,22995,244],{"className":22996},[132,365],[15,22998,91],{"className":22999},[61,365],[15,23001,23002,23005],{"style":574},[15,23003],{"className":23004,"style":358},[357],[15,23006,23008],{"className":23007},[362,363,364,365],[15,23009,102],{"className":23010,"style":174},[61,122,365],[15,23012,497],{"className":23013},[496],[15,23015,23017],{"className":23016},[346],[15,23018,23020],{"className":23019,"style":593},[350],[15,23021],{},[15,23023,741],{"className":23024},[146],[15,23026,23028,23031],{"className":23027},[61],[15,23029,3106],{"className":23030,"style":3191},[61,122],[15,23032,23034],{"className":23033},[338],[15,23035,23037,23057],{"className":23036},[342,472],[15,23038,23040,23054],{"className":23039},[346],[15,23041,23043],{"className":23042,"style":1364},[350],[15,23044,23045,23048],{"style":3206},[15,23046],{"className":23047,"style":358},[357],[15,23049,23051],{"className":23050},[362,363,364,365],[15,23052,1070],{"className":23053},[61,122,365],[15,23055,497],{"className":23056},[496],[15,23058,23060],{"className":23059},[346],[15,23061,23063],{"className":23062,"style":504},[350],[15,23064],{},[15,23066],{"className":23067,"style":772},[127],[15,23069,239],{"className":23070},[375],[15,23072],{"className":23073,"style":772},[127],[15,23075,23077,23080,23083,23086,23089,23092,23095,23135],{"className":23076},[52],[15,23078],{"className":23079,"style":142},[56],[15,23081,91],{"className":23082},[61],[15,23084,750],{"className":23085},[178],[15,23087],{"className":23088,"style":157},[127],[15,23090,22693],{"className":23091,"style":123},[61,122],[15,23093,741],{"className":23094},[146],[15,23096,23098,23101],{"className":23097},[61],[15,23099,3106],{"className":23100,"style":3191},[61,122],[15,23102,23104],{"className":23103},[338],[15,23105,23107,23127],{"className":23106},[342,472],[15,23108,23110,23124],{"className":23109},[346],[15,23111,23113],{"className":23112,"style":1364},[350],[15,23114,23115,23118],{"style":3206},[15,23116],{"className":23117,"style":358},[357],[15,23119,23121],{"className":23120},[362,363,364,365],[15,23122,1070],{"className":23123},[61,122,365],[15,23125,497],{"className":23126},[496],[15,23128,23130],{"className":23129},[346],[15,23131,23133],{"className":23132,"style":504},[350],[15,23134],{},[15,23136,750],{"className":23137},[178],"\n其中 ",[15,23140,23142,23170],{"className":23141,"translate":19},[18],[15,23143,23145],{"className":23144},[23],[25,23146,23147],{"xmlns":27},[29,23148,23149,23167],{},[32,23150,23151,23153,23155,23163,23165],{},[35,23152,22693],{},[81,23154,244],{},[227,23156,23157,23159],{},[35,23158,4292],{"mathvariant":231},[81,23160,23162],{"mathvariant":231,"lspace":23161,"rspace":23161},"0em","′",[35,23164,3020],{"mathvariant":231},[35,23166,4292],{"mathvariant":231},[40,23168,23169],{"encoding":42},"\\psi = \\Gamma'\u002F\\Gamma",[15,23171,23173,23191],{"className":23172,"ariaHidden":48},[47],[15,23174,23176,23179,23182,23185,23188],{"className":23175},[52],[15,23177],{"className":23178,"style":14310},[56],[15,23180,22693],{"className":23181,"style":123},[61,122],[15,23183],{"className":23184,"style":128},[127],[15,23186,244],{"className":23187},[132],[15,23189],{"className":23190,"style":128},[127],[15,23192,23194,23198,23231],{"className":23193},[52],[15,23195],{"className":23196,"style":23197},[56],"height:1.0019em;vertical-align:-0.25em;",[15,23199,23201,23204],{"className":23200},[61],[15,23202,4292],{"className":23203},[61],[15,23205,23207],{"className":23206},[338],[15,23208,23210],{"className":23209},[342],[15,23211,23213],{"className":23212},[346],[15,23214,23217],{"className":23215,"style":23216},[350],"height:0.7519em;",[15,23218,23219,23222],{"style":716},[15,23220],{"className":23221,"style":358},[357],[15,23223,23225],{"className":23224},[362,363,364,365],[15,23226,23228],{"className":23227},[61,365],[15,23229,23162],{"className":23230},[61,365],[15,23232,10952],{"className":23233},[61]," 为 digamma 函数。",[11,23236,23237,23239,23240,23328,23329,9348,23635,23888,23889,23959],{},[661,23238,5123],{}," 直接计算 ",[15,23241,23243,23271],{"className":23242,"translate":19},[18],[15,23244,23246],{"className":23245},[23],[25,23247,23248],{"xmlns":27},[29,23249,23250,23268],{},[32,23251,23252,23254,23256,23258,23260,23262,23264,23266],{},[35,23253,22651],{},[81,23255,244],{},[81,23257,239],{},[81,23259,4755],{},[35,23261,3707],{},[35,23263,22662],{},[81,23265,1547],{},[35,23267,3707],{},[40,23269,23270],{"encoding":42},"H = -\\int f \\log f",[15,23272,23274,23292],{"className":23273,"ariaHidden":48},[47],[15,23275,23277,23280,23283,23286,23289],{"className":23276},[52],[15,23278],{"className":23279,"style":57},[56],[15,23281,22651],{"className":23282,"style":22763},[61,122],[15,23284],{"className":23285,"style":128},[127],[15,23287,244],{"className":23288},[132],[15,23290],{"className":23291,"style":128},[127],[15,23293,23295,23299,23302,23305,23308,23311,23314,23317,23322,23325],{"className":23294},[52],[15,23296],{"className":23297,"style":23298},[56],"height:1.1111em;vertical-align:-0.3061em;",[15,23300,239],{"className":23301},[61],[15,23303],{"className":23304,"style":157},[127],[15,23306,4755],{"className":23307,"style":4836},[531,535,536],[15,23309],{"className":23310,"style":157},[127],[15,23312,3707],{"className":23313,"style":3785},[61,122],[15,23315],{"className":23316,"style":157},[127],[15,23318,22791,23320],{"className":23319},[531],[15,23321,22794],{"style":15619},[15,23323],{"className":23324,"style":157},[127],[15,23326,3707],{"className":23327,"style":3785},[61,122],"，代入 ",[15,23330,23332,23398],{"className":23331,"translate":19},[18],[15,23333,23335],{"className":23334},[23],[25,23336,23337],{"xmlns":27},[29,23338,23339,23395],{},[32,23340,23341,23343,23345,23347,23349,23351,23353,23355,23357,23359,23361,23363,23365,23371,23373,23379,23381,23383,23385,23387,23389],{},[81,23342,239],{},[35,23344,22662],{},[81,23346,1547],{},[35,23348,3707],{},[81,23350,244],{},[35,23352,22662],{},[81,23354,1547],{},[35,23356,3730],{},[81,23358,741],{"stretchy":86},[35,23360,3106],{"mathvariant":3105},[81,23362,750],{"stretchy":86},[81,23364,239],{},[266,23366,23367,23369],{},[81,23368,293],{},[35,23370,1070],{},[81,23372,741],{"stretchy":86},[266,23374,23375,23377],{},[35,23376,3106],{},[35,23378,1070],{},[81,23380,239],{},[89,23382,91],{},[81,23384,750],{"stretchy":86},[35,23386,22662],{},[81,23388,1547],{},[266,23390,23391,23393],{},[35,23392,1029],{},[35,23394,1070],{},[40,23396,23397],{"encoding":42},"-\\log f = \\log B(\\boldsymbol{\\alpha}) - \\sum_i (\\alpha_i - 1)\\log x_i",[15,23399,23401,23433,23474,23572],{"className":23400,"ariaHidden":48},[47],[15,23402,23404,23407,23410,23413,23418,23421,23424,23427,23430],{"className":23403},[52],[15,23405],{"className":23406,"style":14310},[56],[15,23408,239],{"className":23409},[61],[15,23411],{"className":23412,"style":157},[127],[15,23414,22791,23416],{"className":23415},[531],[15,23417,22794],{"style":15619},[15,23419],{"className":23420,"style":157},[127],[15,23422,3707],{"className":23423,"style":3785},[61,122],[15,23425],{"className":23426,"style":128},[127],[15,23428,244],{"className":23429},[132],[15,23431],{"className":23432,"style":128},[127],[15,23434,23436,23439,23444,23447,23450,23453,23462,23465,23468,23471],{"className":23435},[52],[15,23437],{"className":23438,"style":142},[56],[15,23440,22791,23442],{"className":23441},[531],[15,23443,22794],{"style":15619},[15,23445],{"className":23446,"style":157},[127],[15,23448,3730],{"className":23449,"style":3871},[61,122],[15,23451,741],{"className":23452},[146],[15,23454,23456],{"className":23455},[61],[15,23457,23459],{"className":23458},[61],[15,23460,3106],{"className":23461},[61,3166],[15,23463,750],{"className":23464},[178],[15,23466],{"className":23467,"style":772},[127],[15,23469,239],{"className":23470},[375],[15,23472],{"className":23473,"style":772},[127],[15,23475,23477,23480,23520,23523,23563,23566,23569],{"className":23476},[52],[15,23478],{"className":23479,"style":6051},[56],[15,23481,23483,23486],{"className":23482},[531],[15,23484,293],{"className":23485,"style":537},[531,535,536],[15,23487,23489],{"className":23488},[338],[15,23490,23492,23512],{"className":23491},[342,472],[15,23493,23495,23509],{"className":23494},[346],[15,23496,23498],{"className":23497,"style":6070},[350],[15,23499,23500,23503],{"style":553},[15,23501],{"className":23502,"style":358},[357],[15,23504,23506],{"className":23505},[362,363,364,365],[15,23507,1070],{"className":23508},[61,122,365],[15,23510,497],{"className":23511},[496],[15,23513,23515],{"className":23514},[346],[15,23516,23518],{"className":23517,"style":593},[350],[15,23519],{},[15,23521,741],{"className":23522},[146],[15,23524,23526,23529],{"className":23525},[61],[15,23527,3106],{"className":23528,"style":3191},[61,122],[15,23530,23532],{"className":23531},[338],[15,23533,23535,23555],{"className":23534},[342,472],[15,23536,23538,23552],{"className":23537},[346],[15,23539,23541],{"className":23540,"style":1364},[350],[15,23542,23543,23546],{"style":3206},[15,23544],{"className":23545,"style":358},[357],[15,23547,23549],{"className":23548},[362,363,364,365],[15,23550,1070],{"className":23551},[61,122,365],[15,23553,497],{"className":23554},[496],[15,23556,23558],{"className":23557},[346],[15,23559,23561],{"className":23560,"style":504},[350],[15,23562],{},[15,23564],{"className":23565,"style":772},[127],[15,23567,239],{"className":23568},[375],[15,23570],{"className":23571,"style":772},[127],[15,23573,23575,23578,23581,23584,23587,23592,23595],{"className":23574},[52],[15,23576],{"className":23577,"style":142},[56],[15,23579,91],{"className":23580},[61],[15,23582,750],{"className":23583},[178],[15,23585],{"className":23586,"style":157},[127],[15,23588,22791,23590],{"className":23589},[531],[15,23591,22794],{"style":15619},[15,23593],{"className":23594,"style":157},[127],[15,23596,23598,23601],{"className":23597},[61],[15,23599,1029],{"className":23600},[61,122],[15,23602,23604],{"className":23603},[338],[15,23605,23607,23627],{"className":23606},[342,472],[15,23608,23610,23624],{"className":23609},[346],[15,23611,23613],{"className":23612,"style":1364},[350],[15,23614,23615,23618],{"style":482},[15,23616],{"className":23617,"style":358},[357],[15,23619,23621],{"className":23620},[362,363,364,365],[15,23622,1070],{"className":23623},[61,122,365],[15,23625,497],{"className":23626},[496],[15,23628,23630],{"className":23629},[346],[15,23631,23633],{"className":23632,"style":504},[350],[15,23634],{},[15,23636,23638,23694],{"className":23637,"translate":19},[18],[15,23639,23641],{"className":23640},[23],[25,23642,23643],{"xmlns":27},[29,23644,23645,23691],{},[32,23646,23647,23649,23651,23653,23655,23661,23663,23665,23667,23669,23675,23677,23679,23681,23683,23689],{},[35,23648,14818],{"mathvariant":258},[81,23650,14821],{"stretchy":86},[35,23652,22662],{},[81,23654,1547],{},[266,23656,23657,23659],{},[35,23658,1029],{},[35,23660,1070],{},[81,23662,14830],{"stretchy":86},[81,23664,244],{},[35,23666,22693],{},[81,23668,741],{"stretchy":86},[266,23670,23671,23673],{},[35,23672,3106],{},[35,23674,1070],{},[81,23676,750],{"stretchy":86},[81,23678,239],{},[35,23680,22693],{},[81,23682,741],{"stretchy":86},[266,23684,23685,23687],{},[35,23686,3106],{},[89,23688,278],{},[81,23690,750],{"stretchy":86},[40,23692,23693],{"encoding":42},"\\mathbb{E}[\\log x_i] = \\psi(\\alpha_i) - \\psi(\\alpha_0)",[15,23695,23697,23769,23833],{"className":23696,"ariaHidden":48},[47],[15,23698,23700,23703,23706,23709,23714,23717,23757,23760,23763,23766],{"className":23699},[52],[15,23701],{"className":23702,"style":142},[56],[15,23704,14818],{"className":23705},[61,427],[15,23707,14821],{"className":23708},[146],[15,23710,22791,23712],{"className":23711},[531],[15,23713,22794],{"style":15619},[15,23715],{"className":23716,"style":157},[127],[15,23718,23720,23723],{"className":23719},[61],[15,23721,1029],{"className":23722},[61,122],[15,23724,23726],{"className":23725},[338],[15,23727,23729,23749],{"className":23728},[342,472],[15,23730,23732,23746],{"className":23731},[346],[15,23733,23735],{"className":23734,"style":1364},[350],[15,23736,23737,23740],{"style":482},[15,23738],{"className":23739,"style":358},[357],[15,23741,23743],{"className":23742},[362,363,364,365],[15,23744,1070],{"className":23745},[61,122,365],[15,23747,497],{"className":23748},[496],[15,23750,23752],{"className":23751},[346],[15,23753,23755],{"className":23754,"style":504},[350],[15,23756],{},[15,23758,14830],{"className":23759},[178],[15,23761],{"className":23762,"style":128},[127],[15,23764,244],{"className":23765},[132],[15,23767],{"className":23768,"style":128},[127],[15,23770,23772,23775,23778,23781,23821,23824,23827,23830],{"className":23771},[52],[15,23773],{"className":23774,"style":142},[56],[15,23776,22693],{"className":23777,"style":123},[61,122],[15,23779,741],{"className":23780},[146],[15,23782,23784,23787],{"className":23783},[61],[15,23785,3106],{"className":23786,"style":3191},[61,122],[15,23788,23790],{"className":23789},[338],[15,23791,23793,23813],{"className":23792},[342,472],[15,23794,23796,23810],{"className":23795},[346],[15,23797,23799],{"className":23798,"style":1364},[350],[15,23800,23801,23804],{"style":3206},[15,23802],{"className":23803,"style":358},[357],[15,23805,23807],{"className":23806},[362,363,364,365],[15,23808,1070],{"className":23809},[61,122,365],[15,23811,497],{"className":23812},[496],[15,23814,23816],{"className":23815},[346],[15,23817,23819],{"className":23818,"style":504},[350],[15,23820],{},[15,23822,750],{"className":23823},[178],[15,23825],{"className":23826,"style":772},[127],[15,23828,239],{"className":23829},[375],[15,23831],{"className":23832,"style":772},[127],[15,23834,23836,23839,23842,23845,23885],{"className":23835},[52],[15,23837],{"className":23838,"style":142},[56],[15,23840,22693],{"className":23841,"style":123},[61,122],[15,23843,741],{"className":23844},[146],[15,23846,23848,23851],{"className":23847},[61],[15,23849,3106],{"className":23850,"style":3191},[61,122],[15,23852,23854],{"className":23853},[338],[15,23855,23857,23877],{"className":23856},[342,472],[15,23858,23860,23874],{"className":23859},[346],[15,23861,23863],{"className":23862,"style":1214},[350],[15,23864,23865,23868],{"style":3206},[15,23866],{"className":23867,"style":358},[357],[15,23869,23871],{"className":23870},[362,363,364,365],[15,23872,278],{"className":23873},[61,365],[15,23875,497],{"className":23876},[496],[15,23878,23880],{"className":23879},[346],[15,23881,23883],{"className":23882,"style":504},[350],[15,23884],{},[15,23886,750],{"className":23887},[178],"（由命题 1.3 的矩恒等式对 ",[15,23890,23892,23910],{"className":23891,"translate":19},[18],[15,23893,23895],{"className":23894},[23],[25,23896,23897],{"xmlns":27},[29,23898,23899,23907],{},[32,23900,23901],{},[266,23902,23903,23905],{},[35,23904,17131],{},[35,23906,1070],{},[40,23908,23909],{"encoding":42},"\\beta_i",[15,23911,23913],{"className":23912,"ariaHidden":48},[47],[15,23914,23916,23919],{"className":23915},[52],[15,23917],{"className":23918,"style":14310},[56],[15,23920,23922,23925],{"className":23921},[61],[15,23923,17131],{"className":23924,"style":17270},[61,122],[15,23926,23928],{"className":23927},[338],[15,23929,23931,23951],{"className":23930},[342,472],[15,23932,23934,23948],{"className":23933},[346],[15,23935,23937],{"className":23936,"style":1364},[350],[15,23938,23939,23942],{"style":17454},[15,23940],{"className":23941,"style":358},[357],[15,23943,23945],{"className":23944},[362,363,364,365],[15,23946,1070],{"className":23947},[61,122,365],[15,23949,497],{"className":23950},[496],[15,23952,23954],{"className":23953},[346],[15,23955,23957],{"className":23956,"style":504},[350],[15,23958],{}," 求导得到），整理即得。",[11,23961,23962,23963,24025,24026,21891,24054,24105,24106,24243,24244,24336,24337,24387,24388,24391,24392,24465],{},"在对称情形 ",[15,23964,23966,23986],{"className":23965,"translate":19},[18],[15,23967,23969],{"className":23968},[23],[25,23970,23971],{"xmlns":27},[29,23972,23973,23983],{},[32,23974,23975,23977,23979,23981],{},[35,23976,3106],{"mathvariant":3105},[81,23978,244],{},[35,23980,3106],{},[89,23982,91],{"mathvariant":251},[40,23984,23985],{"encoding":42},"\\boldsymbol{\\alpha} = \\alpha \\mathbf{1}",[15,23987,23989,24013],{"className":23988,"ariaHidden":48},[47],[15,23990,23992,23995,24004,24007,24010],{"className":23991},[52],[15,23993],{"className":23994,"style":2708},[56],[15,23996,23998],{"className":23997},[61],[15,23999,24001],{"className":24000},[61],[15,24002,3106],{"className":24003},[61,3166],[15,24005],{"className":24006,"style":128},[127],[15,24008,244],{"className":24009},[132],[15,24011],{"className":24012,"style":128},[127],[15,24014,24016,24019,24022],{"className":24015},[52],[15,24017],{"className":24018,"style":859},[56],[15,24020,3106],{"className":24021,"style":3191},[61,122],[15,24023,91],{"className":24024},[61,411]," 下，",[15,24027,24029,24042],{"className":24028,"translate":19},[18],[15,24030,24032],{"className":24031},[23],[25,24033,24034],{"xmlns":27},[29,24035,24036,24040],{},[32,24037,24038],{},[35,24039,22651],{},[40,24041,22651],{"encoding":42},[15,24043,24045],{"className":24044,"ariaHidden":48},[47],[15,24046,24048,24051],{"className":24047},[52],[15,24049],{"className":24050,"style":57},[56],[15,24052,22651],{"className":24053,"style":22763},[61,122],[15,24055,24057,24075],{"className":24056,"translate":19},[18],[15,24058,24060],{"className":24059},[23],[25,24061,24062],{"xmlns":27},[29,24063,24064,24072],{},[32,24065,24066,24068,24070],{},[35,24067,3106],{},[81,24069,244],{},[89,24071,91],{},[40,24073,24074],{"encoding":42},"\\alpha = 1",[15,24076,24078,24096],{"className":24077,"ariaHidden":48},[47],[15,24079,24081,24084,24087,24090,24093],{"className":24080},[52],[15,24082],{"className":24083,"style":6032},[56],[15,24085,3106],{"className":24086,"style":3191},[61,122],[15,24088],{"className":24089,"style":128},[127],[15,24091,244],{"className":24092},[132],[15,24094],{"className":24095,"style":128},[127],[15,24097,24099,24102],{"className":24098},[52],[15,24100],{"className":24101,"style":859},[56],[15,24103,91],{"className":24104},[61]," 处取最大值 ",[15,24107,24109,24155],{"className":24108,"translate":19},[18],[15,24110,24112],{"className":24111},[23],[25,24113,24114],{"xmlns":27},[29,24115,24116,24152],{},[32,24117,24118,24120,24122,24124,24126,24128,24130,24132,24134,24136,24138,24140,24142,24144,24146,24148,24150],{},[81,24119,239],{},[35,24121,22662],{},[81,24123,1547],{},[35,24125,4292],{"mathvariant":231},[81,24127,741],{"stretchy":86},[35,24129,102],{},[81,24131,750],{"stretchy":86},[81,24133,244],{},[81,24135,239],{},[35,24137,22662],{},[81,24139,1547],{},[81,24141,741],{"stretchy":86},[35,24143,102],{},[81,24145,239],{},[89,24147,91],{},[81,24149,750],{"stretchy":86},[81,24151,3033],{"stretchy":86},[40,24153,24154],{"encoding":42},"-\\log \\Gamma(K) = -\\log (K-1)!",[15,24156,24158,24199,24231],{"className":24157,"ariaHidden":48},[47],[15,24159,24161,24164,24167,24170,24175,24178,24181,24184,24187,24190,24193,24196],{"className":24160},[52],[15,24162],{"className":24163,"style":142},[56],[15,24165,239],{"className":24166},[61],[15,24168],{"className":24169,"style":157},[127],[15,24171,22791,24173],{"className":24172},[531],[15,24174,22794],{"style":15619},[15,24176],{"className":24177,"style":157},[127],[15,24179,4292],{"className":24180},[61],[15,24182,741],{"className":24183},[146],[15,24185,102],{"className":24186,"style":174},[61,122],[15,24188,750],{"className":24189},[178],[15,24191],{"className":24192,"style":128},[127],[15,24194,244],{"className":24195},[132],[15,24197],{"className":24198,"style":128},[127],[15,24200,24202,24205,24208,24211,24216,24219,24222,24225,24228],{"className":24201},[52],[15,24203],{"className":24204,"style":142},[56],[15,24206,239],{"className":24207},[61],[15,24209],{"className":24210,"style":157},[127],[15,24212,22791,24214],{"className":24213},[531],[15,24215,22794],{"style":15619},[15,24217,741],{"className":24218},[146],[15,24220,102],{"className":24221,"style":174},[61,122],[15,24223],{"className":24224,"style":772},[127],[15,24226,239],{"className":24227},[375],[15,24229],{"className":24230,"style":772},[127],[15,24232,24234,24237,24240],{"className":24233},[52],[15,24235],{"className":24236,"style":142},[56],[15,24238,91],{"className":24239},[61],[15,24241,3077],{"className":24242},[178],"。该值为负，并不矛盾：它是相对于体积为 ",[15,24245,24247,24279],{"className":24246,"translate":19},[18],[15,24248,24250],{"className":24249},[23],[25,24251,24252],{"xmlns":27},[29,24253,24254,24276],{},[32,24255,24256,24258,24260,24262,24264,24266,24268,24270,24272,24274],{},[89,24257,91],{},[35,24259,3020],{"mathvariant":231},[81,24261,741],{"stretchy":86},[35,24263,102],{},[81,24265,239],{},[89,24267,91],{},[81,24269,750],{"stretchy":86},[81,24271,3033],{"stretchy":86},[81,24273,2490],{},[89,24275,91],{},[40,24277,24278],{"encoding":42},"1\u002F(K-1)! \u003C 1",[15,24280,24282,24306,24327],{"className":24281,"ariaHidden":48},[47],[15,24283,24285,24288,24291,24294,24297,24300,24303],{"className":24284},[52],[15,24286],{"className":24287,"style":142},[56],[15,24289,3049],{"className":24290},[61],[15,24292,741],{"className":24293},[146],[15,24295,102],{"className":24296,"style":174},[61,122],[15,24298],{"className":24299,"style":772},[127],[15,24301,239],{"className":24302},[375],[15,24304],{"className":24305,"style":772},[127],[15,24307,24309,24312,24315,24318,24321,24324],{"className":24308},[52],[15,24310],{"className":24311,"style":142},[56],[15,24313,91],{"className":24314},[61],[15,24316,3077],{"className":24317},[178],[15,24319],{"className":24320,"style":128},[127],[15,24322,2490],{"className":24323},[132],[15,24325],{"className":24326,"style":128},[127],[15,24328,24330,24333],{"className":24329},[52],[15,24331],{"className":24332,"style":859},[56],[15,24334,91],{"className":24335},[61]," 的支撑集而言的。",[15,24338,24340,24357],{"className":24339,"translate":19},[18],[15,24341,24343],{"className":24342},[23],[25,24344,24345],{"xmlns":27},[29,24346,24347,24355],{},[32,24348,24349,24351,24353],{},[35,24350,3106],{},[81,24352,244],{},[89,24354,91],{},[40,24356,24074],{"encoding":42},[15,24358,24360,24378],{"className":24359,"ariaHidden":48},[47],[15,24361,24363,24366,24369,24372,24375],{"className":24362},[52],[15,24364],{"className":24365,"style":6032},[56],[15,24367,3106],{"className":24368,"style":3191},[61,122],[15,24370],{"className":24371,"style":128},[127],[15,24373,244],{"className":24374},[132],[15,24376],{"className":24377,"style":128},[127],[15,24379,24381,24384],{"className":24380},[52],[15,24382],{"className":24383,"style":859},[56],[15,24385,91],{"className":24386},[61]," 对应单纯形上的",[661,24389,24390],{},"均匀分布","——一个在其支撑集上平坦、却因支撑集体积小于一而具有负微分熵的密度。一般地，由于 ",[15,24393,24395,24418],{"className":24394,"translate":19},[18],[15,24396,24398],{"className":24397},[23],[25,24399,24400],{"xmlns":27},[29,24401,24402,24416],{},[32,24403,24404],{},[227,24405,24406,24408],{},[35,24407,232],{"mathvariant":231},[32,24409,24410,24412,24414],{},[35,24411,102],{},[81,24413,239],{},[89,24415,91],{},[40,24417,1858],{"encoding":42},[15,24419,24421],{"className":24420,"ariaHidden":48},[47],[15,24422,24424,24427],{"className":24423},[52],[15,24425],{"className":24426,"style":695},[56],[15,24428,24430,24433],{"className":24429},[61],[15,24431,232],{"className":24432},[61],[15,24434,24436],{"className":24435},[338],[15,24437,24439],{"className":24438},[342],[15,24440,24442],{"className":24441},[346],[15,24443,24445],{"className":24444,"style":695},[350],[15,24446,24447,24450],{"style":716},[15,24448],{"className":24449,"style":358},[357],[15,24451,24453],{"className":24452},[362,363,364,365],[15,24454,24456,24459,24462],{"className":24455},[61,365],[15,24457,102],{"className":24458,"style":174},[61,122,365],[15,24460,239],{"className":24461},[375,365],[15,24463,91],{"className":24464},[61,365]," 紧且测度有限，均匀分布是最大熵分布，故",[15,24467,24469],{"className":24468,"translate":19},[212],[15,24470,24472,24529],{"className":24471,"translate":19},[18],[15,24473,24475],{"className":24474},[23],[25,24476,24477],{"xmlns":27,"display":221},[29,24478,24479,24526],{},[32,24480,24481,24483,24485,24487,24489,24491,24493,24495,24497,24499,24501,24503,24505,24507,24509,24512,24515,24518,24520,24522,24524],{},[35,24482,22651],{},[81,24484,741],{"stretchy":86},[35,24486,1029],{"mathvariant":251},[81,24488,750],{"stretchy":86},[81,24490,21308],{},[81,24492,239],{},[35,24494,22662],{},[81,24496,1547],{},[35,24498,4292],{"mathvariant":231},[81,24500,741],{"stretchy":86},[35,24502,102],{},[81,24504,750],{"stretchy":86},[81,24506,94],{"separator":48},[127,24508],{"width":9847},[282,24510,24511],{},"等号成立",[282,24513,24514],{},"  ",[81,24516,24517],{},"⟺",[282,24519,24514],{},[35,24521,3106],{"mathvariant":3105},[81,24523,244],{},[89,24525,91],{"mathvariant":251},[40,24527,24528],{"encoding":42},"H(\\mathbf{x}) \\le -\\log \\Gamma(K), \\qquad \\text{等号成立} \\iff \\boldsymbol{\\alpha} = \\mathbf{1}",[15,24530,24532,24559,24621,24645],{"className":24531,"ariaHidden":48},[47],[15,24533,24535,24538,24541,24544,24547,24550,24553,24556],{"className":24534},[52],[15,24536],{"className":24537,"style":142},[56],[15,24539,22651],{"className":24540,"style":22763},[61,122],[15,24542,741],{"className":24543},[146],[15,24545,1029],{"className":24546},[61,411],[15,24548,750],{"className":24549},[178],[15,24551],{"className":24552,"style":128},[127],[15,24554,21308],{"className":24555},[132],[15,24557],{"className":24558,"style":128},[127],[15,24560,24562,24565,24568,24571,24576,24579,24582,24585,24588,24591,24594,24597,24600,24606,24609,24612,24615,24618],{"className":24561},[52],[15,24563],{"className":24564,"style":142},[56],[15,24566,239],{"className":24567},[61],[15,24569],{"className":24570,"style":157},[127],[15,24572,22791,24574],{"className":24573},[531],[15,24575,22794],{"style":15619},[15,24577],{"className":24578,"style":157},[127],[15,24580,4292],{"className":24581},[61],[15,24583,741],{"className":24584},[146],[15,24586,102],{"className":24587,"style":174},[61,122],[15,24589,750],{"className":24590},[178],[15,24592,94],{"className":24593},[153],[15,24595],{"className":24596,"style":10035},[127],[15,24598],{"className":24599,"style":157},[127],[15,24601,24603],{"className":24602},[61,13548],[15,24604,24511],{"className":24605},[61,13552],[15,24607],{"className":24608,"style":128},[127],[15,24610],{"className":24611,"style":128},[127],[15,24613,24517],{"className":24614},[132],[15,24616],{"className":24617,"style":128},[127],[15,24619],{"className":24620,"style":128},[127],[15,24622,24624,24627,24636,24639,24642],{"className":24623},[52],[15,24625],{"className":24626,"style":2708},[56],[15,24628,24630],{"className":24629},[61],[15,24631,24633],{"className":24632},[61],[15,24634,3106],{"className":24635},[61,3166],[15,24637],{"className":24638,"style":128},[127],[15,24640,244],{"className":24641},[132],[15,24643],{"className":24644,"style":128},[127],[15,24646,24648,24651],{"className":24647},[52],[15,24649],{"className":24650,"style":859},[56],[15,24652,91],{"className":24653},[61,411],[877,24655,24657],{"id":24656},"_24-共轭性与贝叶斯更新","2.4 共轭性与贝叶斯更新",[882,24659,24660],{},[11,24661,24662,24665,24666,24751,24752,24787,24788,19845,24960,943,25054,25196,25197],{},[661,24663,24664],{},"命题 2.5（Dirichlet–Multinomial 共轭）"," 设先验 ",[15,24667,24669,24696],{"className":24668,"translate":19},[18],[15,24670,24672],{"className":24671},[23],[25,24673,24674],{"xmlns":27},[29,24675,24676,24693],{},[32,24677,24678,24681,24683,24685,24687,24689,24691],{},[35,24679,24680],{"mathvariant":3105},"π",[81,24682,4183],{},[35,24684,4186],{"mathvariant":231},[81,24686,1547],{},[81,24688,741],{"stretchy":86},[35,24690,3106],{"mathvariant":3105},[81,24692,750],{"stretchy":86},[40,24694,24695],{"encoding":42},"\\boldsymbol{\\pi} \\sim \\operatorname{Dir}(\\boldsymbol{\\alpha})",[15,24697,24699,24724],{"className":24698,"ariaHidden":48},[47],[15,24700,24702,24705,24715,24718,24721],{"className":24701},[52],[15,24703],{"className":24704,"style":2708},[56],[15,24706,24708],{"className":24707},[61],[15,24709,24711],{"className":24710},[61],[15,24712,24680],{"className":24713,"style":24714},[61,3166],"margin-right:0.037em;",[15,24716],{"className":24717,"style":128},[127],[15,24719,4183],{"className":24720},[132],[15,24722],{"className":24723,"style":128},[127],[15,24725,24727,24730,24736,24739,24748],{"className":24726},[52],[15,24728],{"className":24729,"style":142},[56],[15,24731,24733],{"className":24732},[531],[15,24734,4186],{"className":24735},[61,1622],[15,24737,741],{"className":24738},[146],[15,24740,24742],{"className":24741},[61],[15,24743,24745],{"className":24744},[61],[15,24746,3106],{"className":24747},[61,3166],[15,24749,750],{"className":24750},[178],"，给定 ",[15,24753,24755,24769],{"className":24754,"translate":19},[18],[15,24756,24758],{"className":24757},[23],[25,24759,24760],{"xmlns":27},[29,24761,24762,24766],{},[32,24763,24764],{},[35,24765,24680],{"mathvariant":3105},[40,24767,24768],{"encoding":42},"\\boldsymbol{\\pi}",[15,24770,24772],{"className":24771,"ariaHidden":48},[47],[15,24773,24775,24778],{"className":24774},[52],[15,24776],{"className":24777,"style":2708},[56],[15,24779,24781],{"className":24780},[61],[15,24782,24784],{"className":24783},[61],[15,24785,24680],{"className":24786,"style":24714},[61,3166]," 观测到计数向量 ",[15,24789,24791,24829],{"className":24790,"translate":19},[18],[15,24792,24794],{"className":24793},[23],[25,24795,24796],{"xmlns":27},[29,24797,24798,24826],{},[32,24799,24800,24802,24804,24806,24812,24814,24816,24818,24824],{},[35,24801,14560],{"mathvariant":251},[81,24803,244],{},[81,24805,741],{"stretchy":86},[266,24807,24808,24810],{},[35,24809,14560],{},[89,24811,91],{},[81,24813,94],{"separator":48},[81,24815,97],{},[81,24817,94],{"separator":48},[266,24819,24820,24822],{},[35,24821,14560],{},[35,24823,102],{},[81,24825,750],{"stretchy":86},[40,24827,24828],{"encoding":42},"\\mathbf{n} = (n_1, \\dots, n_K)",[15,24830,24832,24850],{"className":24831,"ariaHidden":48},[47],[15,24833,24835,24838,24841,24844,24847],{"className":24834},[52],[15,24836],{"className":24837,"style":2708},[56],[15,24839,14560],{"className":24840},[61,411],[15,24842],{"className":24843,"style":128},[127],[15,24845,244],{"className":24846},[132],[15,24848],{"className":24849,"style":128},[127],[15,24851,24853,24856,24859,24899,24902,24905,24908,24911,24914,24917,24957],{"className":24852},[52],[15,24854],{"className":24855,"style":142},[56],[15,24857,741],{"className":24858},[146],[15,24860,24862,24865],{"className":24861},[61],[15,24863,14560],{"className":24864},[61,122],[15,24866,24868],{"className":24867},[338],[15,24869,24871,24891],{"className":24870},[342,472],[15,24872,24874,24888],{"className":24873},[346],[15,24875,24877],{"className":24876,"style":1214},[350],[15,24878,24879,24882],{"style":482},[15,24880],{"className":24881,"style":358},[357],[15,24883,24885],{"className":24884},[362,363,364,365],[15,24886,91],{"className":24887},[61,365],[15,24889,497],{"className":24890},[496],[15,24892,24894],{"className":24893},[346],[15,24895,24897],{"className":24896,"style":504},[350],[15,24898],{},[15,24900,94],{"className":24901},[153],[15,24903],{"className":24904,"style":157},[127],[15,24906,97],{"className":24907},[161],[15,24909],{"className":24910,"style":157},[127],[15,24912,94],{"className":24913},[153],[15,24915],{"className":24916,"style":157},[127],[15,24918,24920,24923],{"className":24919},[61],[15,24921,14560],{"className":24922},[61,122],[15,24924,24926],{"className":24925},[338],[15,24927,24929,24949],{"className":24928},[342,472],[15,24930,24932,24946],{"className":24931},[346],[15,24933,24935],{"className":24934,"style":1273},[350],[15,24936,24937,24940],{"style":482},[15,24938],{"className":24939,"style":358},[357],[15,24941,24943],{"className":24942},[362,363,364,365],[15,24944,102],{"className":24945,"style":174},[61,122,365],[15,24947,497],{"className":24948},[496],[15,24950,24952],{"className":24951},[346],[15,24953,24955],{"className":24954,"style":504},[350],[15,24956],{},[15,24958,750],{"className":24959},[178],[15,24961,24963,24986],{"className":24962,"translate":19},[18],[15,24964,24966],{"className":24965},[23],[25,24967,24968],{"xmlns":27},[29,24969,24970,24983],{},[32,24971,24972,24978,24980],{},[266,24973,24974,24976],{},[35,24975,14560],{},[35,24977,1070],{},[81,24979,83],{},[35,24981,24982],{"mathvariant":258},"N",[40,24984,24985],{"encoding":42},"n_i \\in \\mathbb{N}",[15,24987,24989,25044],{"className":24988,"ariaHidden":48},[47],[15,24990,24992,24995,25035,25038,25041],{"className":24991},[52],[15,24993],{"className":24994,"style":1753},[56],[15,24996,24998,25001],{"className":24997},[61],[15,24999,14560],{"className":25000},[61,122],[15,25002,25004],{"className":25003},[338],[15,25005,25007,25027],{"className":25006},[342,472],[15,25008,25010,25024],{"className":25009},[346],[15,25011,25013],{"className":25012,"style":1364},[350],[15,25014,25015,25018],{"style":482},[15,25016],{"className":25017,"style":358},[357],[15,25019,25021],{"className":25020},[362,363,364,365],[15,25022,1070],{"className":25023},[61,122,365],[15,25025,497],{"className":25026},[496],[15,25028,25030],{"className":25029},[346],[15,25031,25033],{"className":25032,"style":504},[350],[15,25034],{},[15,25036],{"className":25037,"style":128},[127],[15,25039,83],{"className":25040},[132],[15,25042],{"className":25043,"style":128},[127],[15,25045,25047,25051],{"className":25046},[52],[15,25048],{"className":25049,"style":25050},[56],"height:0.6889em;",[15,25052,24982],{"className":25053},[61,427],[15,25055,25057,25085],{"className":25056,"translate":19},[18],[15,25058,25060],{"className":25059},[23],[25,25061,25062],{"xmlns":27},[29,25063,25064,25082],{},[32,25065,25066,25068,25070,25076],{},[35,25067,24982],{},[81,25069,244],{},[266,25071,25072,25074],{},[81,25073,293],{},[35,25075,1070],{},[266,25077,25078,25080],{},[35,25079,14560],{},[35,25081,1070],{},[40,25083,25084],{"encoding":42},"N = \\sum_i n_i",[15,25086,25088,25107],{"className":25087,"ariaHidden":48},[47],[15,25089,25091,25094,25098,25101,25104],{"className":25090},[52],[15,25092],{"className":25093,"style":57},[56],[15,25095,24982],{"className":25096,"style":25097},[61,122],"margin-right:0.109em;",[15,25099],{"className":25100,"style":128},[127],[15,25102,244],{"className":25103},[132],[15,25105],{"className":25106,"style":128},[127],[15,25108,25110,25113,25153,25156],{"className":25109},[52],[15,25111],{"className":25112,"style":6051},[56],[15,25114,25116,25119],{"className":25115},[531],[15,25117,293],{"className":25118,"style":537},[531,535,536],[15,25120,25122],{"className":25121},[338],[15,25123,25125,25145],{"className":25124},[342,472],[15,25126,25128,25142],{"className":25127},[346],[15,25129,25131],{"className":25130,"style":6070},[350],[15,25132,25133,25136],{"style":553},[15,25134],{"className":25135,"style":358},[357],[15,25137,25139],{"className":25138},[362,363,364,365],[15,25140,1070],{"className":25141},[61,122,365],[15,25143,497],{"className":25144},[496],[15,25146,25148],{"className":25147},[346],[15,25149,25151],{"className":25150,"style":593},[350],[15,25152],{},[15,25154],{"className":25155,"style":157},[127],[15,25157,25159,25162],{"className":25158},[61],[15,25160,14560],{"className":25161},[61,122],[15,25163,25165],{"className":25164},[338],[15,25166,25168,25188],{"className":25167},[342,472],[15,25169,25171,25185],{"className":25170},[346],[15,25172,25174],{"className":25173,"style":1364},[350],[15,25175,25176,25179],{"style":482},[15,25177],{"className":25178,"style":358},[357],[15,25180,25182],{"className":25181},[362,363,364,365],[15,25183,1070],{"className":25184},[61,122,365],[15,25186,497],{"className":25187},[496],[15,25189,25191],{"className":25190},[346],[15,25192,25194],{"className":25193,"style":504},[350],[15,25195],{},"），则后验为\n",[15,25198,25200,25234],{"className":25199,"translate":19},[18],[15,25201,25203],{"className":25202},[23],[25,25204,25205],{"xmlns":27},[29,25206,25207,25231],{},[32,25208,25209,25211,25213,25215,25217,25219,25221,25223,25225,25227,25229],{},[35,25210,24680],{"mathvariant":3105},[81,25212,3714],{},[35,25214,14560],{"mathvariant":251},[81,25216,4183],{},[35,25218,4186],{"mathvariant":231},[81,25220,1547],{},[81,25222,741],{"stretchy":86},[35,25224,3106],{"mathvariant":3105},[81,25226,7762],{},[35,25228,14560],{"mathvariant":251},[81,25230,750],{"stretchy":86},[40,25232,25233],{"encoding":42},"\\boldsymbol{\\pi} \\mid \\mathbf{n} \\sim \\operatorname{Dir}(\\boldsymbol{\\alpha} + \\mathbf{n})",[15,25235,25237,25261,25279,25312],{"className":25236,"ariaHidden":48},[47],[15,25238,25240,25243,25252,25255,25258],{"className":25239},[52],[15,25241],{"className":25242,"style":142},[56],[15,25244,25246],{"className":25245},[61],[15,25247,25249],{"className":25248},[61],[15,25250,24680],{"className":25251,"style":24714},[61,3166],[15,25253],{"className":25254,"style":128},[127],[15,25256,3714],{"className":25257},[132],[15,25259],{"className":25260,"style":128},[127],[15,25262,25264,25267,25270,25273,25276],{"className":25263},[52],[15,25265],{"className":25266,"style":2708},[56],[15,25268,14560],{"className":25269},[61,411],[15,25271],{"className":25272,"style":128},[127],[15,25274,4183],{"className":25275},[132],[15,25277],{"className":25278,"style":128},[127],[15,25280,25282,25285,25291,25294,25303,25306,25309],{"className":25281},[52],[15,25283],{"className":25284,"style":142},[56],[15,25286,25288],{"className":25287},[531],[15,25289,4186],{"className":25290},[61,1622],[15,25292,741],{"className":25293},[146],[15,25295,25297],{"className":25296},[61],[15,25298,25300],{"className":25299},[61],[15,25301,3106],{"className":25302},[61,3166],[15,25304],{"className":25305,"style":772},[127],[15,25307,7762],{"className":25308},[375],[15,25310],{"className":25311,"style":772},[127],[15,25313,25315,25318,25321],{"className":25314},[52],[15,25316],{"className":25317,"style":142},[56],[15,25319,14560],{"className":25320},[61,411],[15,25322,750],{"className":25323},[178],[11,25325,25326,25328,25329,25507],{},[661,25327,5123],{}," 多项似然正比于 ",[15,25330,25332,25362],{"className":25331,"translate":19},[18],[15,25333,25335],{"className":25334},[23],[25,25336,25337],{"xmlns":27},[29,25338,25339,25359],{},[32,25340,25341,25347],{},[266,25342,25343,25345],{},[81,25344,3741],{},[35,25346,1070],{},[289,25348,25349,25351,25353],{},[35,25350,24680],{},[35,25352,1070],{},[266,25354,25355,25357],{},[35,25356,14560],{},[35,25358,1070],{},[40,25360,25361],{"encoding":42},"\\prod_i \\pi_i^{n_i}",[15,25363,25365],{"className":25364,"ariaHidden":48},[47],[15,25366,25368,25371,25411,25414],{"className":25367},[52],[15,25369],{"className":25370,"style":6051},[56],[15,25372,25374,25377],{"className":25373},[531],[15,25375,3741],{"className":25376,"style":537},[531,535,536],[15,25378,25380],{"className":25379},[338],[15,25381,25383,25403],{"className":25382},[342,472],[15,25384,25386,25400],{"className":25385},[346],[15,25387,25389],{"className":25388,"style":6070},[350],[15,25390,25391,25394],{"style":553},[15,25392],{"className":25393,"style":358},[357],[15,25395,25397],{"className":25396},[362,363,364,365],[15,25398,1070],{"className":25399},[61,122,365],[15,25401,497],{"className":25402},[496],[15,25404,25406],{"className":25405},[346],[15,25407,25409],{"className":25408,"style":593},[350],[15,25410],{},[15,25412],{"className":25413,"style":157},[127],[15,25415,25417,25420],{"className":25416},[61],[15,25418,24680],{"className":25419,"style":123},[61,122],[15,25421,25423],{"className":25422},[338],[15,25424,25426,25499],{"className":25425},[342,472],[15,25427,25429,25496],{"className":25428},[346],[15,25430,25433,25445],{"className":25431,"style":25432},[350],"height:0.7463em;",[15,25434,25436,25439],{"style":25435},"top:-2.4231em;margin-left:-0.0359em;margin-right:0.05em;",[15,25437],{"className":25438,"style":358},[357],[15,25440,25442],{"className":25441},[362,363,364,365],[15,25443,1070],{"className":25444},[61,122,365],[15,25446,25447,25450],{"style":4027},[15,25448],{"className":25449,"style":358},[357],[15,25451,25453],{"className":25452},[362,363,364,365],[15,25454,25456],{"className":25455},[61,365],[15,25457,25459,25462],{"className":25458},[61,365],[15,25460,14560],{"className":25461},[61,122,365],[15,25463,25465],{"className":25464},[338],[15,25466,25468,25488],{"className":25467},[342,472],[15,25469,25471,25485],{"className":25470},[346],[15,25472,25474],{"className":25473,"style":4055},[350],[15,25475,25476,25479],{"style":5809},[15,25477],{"className":25478,"style":4062},[357],[15,25480,25482],{"className":25481},[362,4066,4067,365],[15,25483,1070],{"className":25484},[61,122,365],[15,25486,497],{"className":25487},[496],[15,25489,25491],{"className":25490},[346],[15,25492,25494],{"className":25493,"style":4080},[350],[15,25495],{},[15,25497,497],{"className":25498},[496],[15,25500,25502],{"className":25501},[346],[15,25503,25505],{"className":25504,"style":4098},[350],[15,25506],{},"，与先验密度相乘得",[15,25509,25511],{"className":25510,"translate":19},[212],[15,25512,25514,25581],{"className":25513,"translate":19},[18],[15,25515,25517],{"className":25516},[23],[25,25518,25519],{"xmlns":27,"display":221},[29,25520,25521,25578],{},[32,25522,25523,25525,25527,25529,25531,25533,25535,25538,25552],{},[35,25524,11],{},[81,25526,741],{"stretchy":86},[35,25528,24680],{"mathvariant":3105},[81,25530,3714],{},[35,25532,14560],{"mathvariant":251},[81,25534,750],{"stretchy":86},[81,25536,25537],{},"∝",[5235,25539,25540,25542,25550],{},[81,25541,3741],{},[32,25543,25544,25546,25548],{},[35,25545,1070],{},[81,25547,244],{},[89,25549,91],{},[35,25551,102],{},[289,25553,25554,25556,25558],{},[35,25555,24680],{},[35,25557,1070],{},[32,25559,25560,25566,25568,25574,25576],{},[266,25561,25562,25564],{},[35,25563,3106],{},[35,25565,1070],{},[81,25567,7762],{},[266,25569,25570,25572],{},[35,25571,14560],{},[35,25573,1070],{},[81,25575,239],{},[89,25577,91],{},[40,25579,25580],{"encoding":42},"p(\\boldsymbol{\\pi} \\mid \\mathbf{n}) \\propto \\prod_{i=1}^K \\pi_i^{\\alpha_i + n_i - 1}",[15,25582,25584,25614,25635],{"className":25583,"ariaHidden":48},[47],[15,25585,25587,25590,25593,25596,25605,25608,25611],{"className":25586},[52],[15,25588],{"className":25589,"style":142},[56],[15,25591,11],{"className":25592},[61,122],[15,25594,741],{"className":25595},[146],[15,25597,25599],{"className":25598},[61],[15,25600,25602],{"className":25601},[61],[15,25603,24680],{"className":25604,"style":24714},[61,3166],[15,25606],{"className":25607,"style":128},[127],[15,25609,3714],{"className":25610},[132],[15,25612],{"className":25613,"style":128},[127],[15,25615,25617,25620,25623,25626,25629,25632],{"className":25616},[52],[15,25618],{"className":25619,"style":142},[56],[15,25621,14560],{"className":25622},[61,411],[15,25624,750],{"className":25625},[178],[15,25627],{"className":25628,"style":128},[127],[15,25630,25537],{"className":25631},[132],[15,25633],{"className":25634,"style":128},[127],[15,25636,25638,25641,25705,25708],{"className":25637},[52],[15,25639],{"className":25640,"style":5344},[56],[15,25642,25644],{"className":25643},[531,5348],[15,25645,25647,25697],{"className":25646},[342,472],[15,25648,25650,25694],{"className":25649},[346],[15,25651,25653,25673,25683],{"className":25652,"style":5358},[350],[15,25654,25655,25658],{"style":5361},[15,25656],{"className":25657,"style":5365},[357],[15,25659,25661],{"className":25660},[362,363,364,365],[15,25662,25664,25667,25670],{"className":25663},[61,365],[15,25665,1070],{"className":25666},[61,122,365],[15,25668,244],{"className":25669},[132,365],[15,25671,91],{"className":25672},[61,365],[15,25674,25675,25678],{"style":5383},[15,25676],{"className":25677,"style":5365},[357],[15,25679,25680],{},[15,25681,3741],{"className":25682},[531,535,5392],[15,25684,25685,25688],{"style":5395},[15,25686],{"className":25687,"style":5365},[357],[15,25689,25691],{"className":25690},[362,363,364,365],[15,25692,102],{"className":25693,"style":174},[61,122,365],[15,25695,497],{"className":25696},[496],[15,25698,25700],{"className":25699},[346],[15,25701,25703],{"className":25702,"style":5414},[350],[15,25704],{},[15,25706],{"className":25707,"style":157},[127],[15,25709,25711,25714],{"className":25710},[61],[15,25712,24680],{"className":25713,"style":123},[61,122],[15,25715,25717],{"className":25716},[338],[15,25718,25720,25840],{"className":25719},[342,472],[15,25721,25723,25837],{"className":25722},[346],[15,25724,25726,25737],{"className":25725,"style":4012},[350],[15,25727,25728,25731],{"style":25435},[15,25729],{"className":25730,"style":358},[357],[15,25732,25734],{"className":25733},[362,363,364,365],[15,25735,1070],{"className":25736},[61,122,365],[15,25738,25739,25742],{"style":4027},[15,25740],{"className":25741,"style":358},[357],[15,25743,25745],{"className":25744},[362,363,364,365],[15,25746,25748,25788,25791,25831,25834],{"className":25747},[61,365],[15,25749,25751,25754],{"className":25750},[61,365],[15,25752,3106],{"className":25753,"style":3191},[61,122,365],[15,25755,25757],{"className":25756},[338],[15,25758,25760,25780],{"className":25759},[342,472],[15,25761,25763,25777],{"className":25762},[346],[15,25764,25766],{"className":25765,"style":4055},[350],[15,25767,25768,25771],{"style":4058},[15,25769],{"className":25770,"style":4062},[357],[15,25772,25774],{"className":25773},[362,4066,4067,365],[15,25775,1070],{"className":25776},[61,122,365],[15,25778,497],{"className":25779},[496],[15,25781,25783],{"className":25782},[346],[15,25784,25786],{"className":25785,"style":4080},[350],[15,25787],{},[15,25789,7762],{"className":25790},[375,365],[15,25792,25794,25797],{"className":25793},[61,365],[15,25795,14560],{"className":25796},[61,122,365],[15,25798,25800],{"className":25799},[338],[15,25801,25803,25823],{"className":25802},[342,472],[15,25804,25806,25820],{"className":25805},[346],[15,25807,25809],{"className":25808,"style":4055},[350],[15,25810,25811,25814],{"style":5809},[15,25812],{"className":25813,"style":4062},[357],[15,25815,25817],{"className":25816},[362,4066,4067,365],[15,25818,1070],{"className":25819},[61,122,365],[15,25821,497],{"className":25822},[496],[15,25824,25826],{"className":25825},[346],[15,25827,25829],{"className":25828,"style":4080},[350],[15,25830],{},[15,25832,239],{"className":25833},[375,365],[15,25835,91],{"className":25836},[61,365],[15,25838,497],{"className":25839},[496],[15,25841,25843],{"className":25842},[346],[15,25844,25846],{"className":25845,"style":4098},[350],[15,25847],{},[11,25849,25850,25851,25928],{},"该核与 ",[15,25852,25854,25880],{"className":25853,"translate":19},[18],[15,25855,25857],{"className":25856},[23],[25,25858,25859],{"xmlns":27},[29,25860,25861,25877],{},[32,25862,25863,25865,25867,25869,25871,25873,25875],{},[35,25864,4186],{"mathvariant":231},[81,25866,1547],{},[81,25868,741],{"stretchy":86},[35,25870,3106],{"mathvariant":3105},[81,25872,7762],{},[35,25874,14560],{"mathvariant":251},[81,25876,750],{"stretchy":86},[40,25878,25879],{"encoding":42},"\\operatorname{Dir}(\\boldsymbol{\\alpha} + \\mathbf{n})",[15,25881,25883,25916],{"className":25882,"ariaHidden":48},[47],[15,25884,25886,25889,25895,25898,25907,25910,25913],{"className":25885},[52],[15,25887],{"className":25888,"style":142},[56],[15,25890,25892],{"className":25891},[531],[15,25893,4186],{"className":25894},[61,1622],[15,25896,741],{"className":25897},[146],[15,25899,25901],{"className":25900},[61],[15,25902,25904],{"className":25903},[61],[15,25905,3106],{"className":25906},[61,3166],[15,25908],{"className":25909,"style":772},[127],[15,25911,7762],{"className":25912},[375],[15,25914],{"className":25915,"style":772},[127],[15,25917,25919,25922,25925],{"className":25918},[52],[15,25920],{"className":25921,"style":142},[56],[15,25923,14560],{"className":25924},[61,411],[15,25926,750],{"className":25927},[178]," 的核成比例，归一化即得。",[11,25930,25931],{},"由共轭性可立即导出两个推论，它们在应用中的地位不低于命题本身。",[882,25933,25934],{},[11,25935,25936,25939,25940,25968,25969],{},[661,25937,25938],{},"推论 2.6（Dirichlet–Multinomial 分布）"," 计数向量 ",[15,25941,25943,25956],{"className":25942,"translate":19},[18],[15,25944,25946],{"className":25945},[23],[25,25947,25948],{"xmlns":27},[29,25949,25950,25954],{},[32,25951,25952],{},[35,25953,14560],{"mathvariant":251},[40,25955,14563],{"encoding":42},[15,25957,25959],{"className":25958,"ariaHidden":48},[47],[15,25960,25962,25965],{"className":25961},[52],[15,25963],{"className":25964,"style":2708},[56],[15,25966,14560],{"className":25967},[61,411]," 的边缘似然为\n",[15,25970,25972,26138],{"className":25971,"translate":19},[18],[15,25973,25975],{"className":25974},[23],[25,25976,25977],{"xmlns":27},[29,25978,25979,26135],{},[32,25980,25981,25983,25985,25987,25989,25991,25993,25995,26019,26021,26047,26049,26083,26097],{},[35,25982,11],{},[81,25984,741],{"stretchy":86},[35,25986,14560],{"mathvariant":251},[81,25988,3714],{},[35,25990,3106],{"mathvariant":3105},[81,25992,750],{"stretchy":86},[81,25994,244],{},[3722,25996,25997,26003],{},[32,25998,25999,26001],{},[35,26000,24982],{},[81,26002,3033],{"stretchy":86},[32,26004,26005,26011,26017],{},[266,26006,26007,26009],{},[81,26008,3741],{},[35,26010,1070],{},[266,26012,26013,26015],{},[35,26014,14560],{},[35,26016,1070],{},[81,26018,3033],{"stretchy":86},[81,26020,11214],{},[3722,26022,26023,26037],{},[32,26024,26025,26027,26029,26031,26033,26035],{},[35,26026,3730],{},[81,26028,741],{"stretchy":86},[35,26030,3106],{"mathvariant":3105},[81,26032,7762],{},[35,26034,14560],{"mathvariant":251},[81,26036,750],{"stretchy":86},[32,26038,26039,26041,26043,26045],{},[35,26040,3730],{},[81,26042,741],{"stretchy":86},[35,26044,3106],{"mathvariant":3105},[81,26046,750],{"stretchy":86},[81,26048,244],{},[3722,26050,26051,26065],{},[32,26052,26053,26055,26057,26063],{},[35,26054,4292],{"mathvariant":231},[81,26056,741],{"stretchy":86},[266,26058,26059,26061],{},[35,26060,3106],{},[89,26062,278],{},[81,26064,750],{"stretchy":86},[32,26066,26067,26069,26071,26077,26079,26081],{},[35,26068,4292],{"mathvariant":231},[81,26070,741],{"stretchy":86},[266,26072,26073,26075],{},[35,26074,3106],{},[89,26076,278],{},[81,26078,7762],{},[35,26080,24982],{},[81,26082,750],{"stretchy":86},[289,26084,26085,26087,26095],{},[81,26086,3741],{},[32,26088,26089,26091,26093],{},[35,26090,1070],{},[81,26092,244],{},[89,26094,91],{},[35,26096,102],{},[3722,26098,26099,26121],{},[32,26100,26101,26103,26105,26111,26113,26119],{},[35,26102,4292],{"mathvariant":231},[81,26104,741],{"stretchy":86},[266,26106,26107,26109],{},[35,26108,3106],{},[35,26110,1070],{},[81,26112,7762],{},[266,26114,26115,26117],{},[35,26116,14560],{},[35,26118,1070],{},[81,26120,750],{"stretchy":86},[32,26122,26123,26125,26127,26133],{},[35,26124,4292],{"mathvariant":231},[81,26126,741],{"stretchy":86},[266,26128,26129,26131],{},[35,26130,3106],{},[35,26132,1070],{},[81,26134,750],{"stretchy":86},[40,26136,26137],{"encoding":42},"p(\\mathbf{n} \\mid \\boldsymbol{\\alpha}) = \\frac{N!}{\\prod_i n_i!} \\cdot \\frac{B(\\boldsymbol{\\alpha} + \\mathbf{n})}{B(\\boldsymbol{\\alpha})} = \\frac{\\Gamma(\\alpha_0)}{\\Gamma(\\alpha_0 + N)} \\prod_{i=1}^K \\frac{\\Gamma(\\alpha_i + n_i)}{\\Gamma(\\alpha_i)}",[15,26139,26141,26165,26192,26364,26483],{"className":26140,"ariaHidden":48},[47],[15,26142,26144,26147,26150,26153,26156,26159,26162],{"className":26143},[52],[15,26145],{"className":26146,"style":142},[56],[15,26148,11],{"className":26149},[61,122],[15,26151,741],{"className":26152},[146],[15,26154,14560],{"className":26155},[61,411],[15,26157],{"className":26158,"style":128},[127],[15,26160,3714],{"className":26161},[132],[15,26163],{"className":26164,"style":128},[127],[15,26166,26168,26171,26180,26183,26186,26189],{"className":26167},[52],[15,26169],{"className":26170,"style":142},[56],[15,26172,26174],{"className":26173},[61],[15,26175,26177],{"className":26176},[61],[15,26178,3106],{"className":26179},[61,3166],[15,26181,750],{"className":26182},[178],[15,26184],{"className":26185,"style":128},[127],[15,26187,244],{"className":26188},[132],[15,26190],{"className":26191,"style":128},[127],[15,26193,26195,26199,26355,26358,26361],{"className":26194},[52],[15,26196],{"className":26197,"style":26198},[56],"height:1.4501em;vertical-align:-0.57em;",[15,26200,26202,26205,26352],{"className":26201},[61],[15,26203],{"className":26204},[146,3841],[15,26206,26208],{"className":26207},[3722],[15,26209,26211,26343],{"className":26210},[342,472],[15,26212,26214,26340],{"className":26213},[346],[15,26215,26218,26315,26323],{"className":26216,"style":26217},[350],"height:0.8801em;",[15,26219,26220,26223],{"style":3857},[15,26221],{"className":26222,"style":3861},[357],[15,26224,26226],{"className":26225},[362,363,364,365],[15,26227,26229,26269,26272,26312],{"className":26228},[61,365],[15,26230,26232,26235],{"className":26231},[531,365],[15,26233,3741],{"className":26234,"style":537},[531,535,536,365],[15,26236,26238],{"className":26237},[338],[15,26239,26241,26261],{"className":26240},[342,472],[15,26242,26244,26258],{"className":26243},[346],[15,26245,26247],{"className":26246,"style":7478},[350],[15,26248,26249,26252],{"style":4495},[15,26250],{"className":26251,"style":4062},[357],[15,26253,26255],{"className":26254},[362,4066,4067,365],[15,26256,1070],{"className":26257},[61,122,365],[15,26259,497],{"className":26260},[496],[15,26262,26264],{"className":26263},[346],[15,26265,26267],{"className":26266,"style":4535},[350],[15,26268],{},[15,26270],{"className":26271,"style":4541},[127,365],[15,26273,26275,26278],{"className":26274},[61,365],[15,26276,14560],{"className":26277},[61,122,365],[15,26279,26281],{"className":26280},[338],[15,26282,26284,26304],{"className":26283},[342,472],[15,26285,26287,26301],{"className":26286},[346],[15,26288,26290],{"className":26289,"style":4055},[350],[15,26291,26292,26295],{"style":5809},[15,26293],{"className":26294,"style":4062},[357],[15,26296,26298],{"className":26297},[362,4066,4067,365],[15,26299,1070],{"className":26300},[61,122,365],[15,26302,497],{"className":26303},[496],[15,26305,26307],{"className":26306},[346],[15,26308,26310],{"className":26309,"style":4080},[350],[15,26311],{},[15,26313,3033],{"className":26314},[178,365],[15,26316,26317,26320],{"style":3889},[15,26318],{"className":26319,"style":3861},[357],[15,26321],{"className":26322,"style":3897},[3896],[15,26324,26325,26328],{"style":3900},[15,26326],{"className":26327,"style":3861},[357],[15,26329,26331],{"className":26330},[362,363,364,365],[15,26332,26334,26337],{"className":26333},[61,365],[15,26335,24982],{"className":26336,"style":25097},[61,122,365],[15,26338,3033],{"className":26339},[178,365],[15,26341,497],{"className":26342},[496],[15,26344,26346],{"className":26345},[346],[15,26347,26350],{"className":26348,"style":26349},[350],"height:0.57em;",[15,26351],{},[15,26353],{"className":26354},[178,3841],[15,26356],{"className":26357,"style":772},[127],[15,26359,11214],{"className":26360},[375],[15,26362],{"className":26363,"style":772},[127],[15,26365,26367,26370,26474,26477,26480],{"className":26366},[52],[15,26368],{"className":26369,"style":17658},[56],[15,26371,26373,26376,26471],{"className":26372},[61],[15,26374],{"className":26375},[146,3841],[15,26377,26379],{"className":26378},[3722],[15,26380,26382,26463],{"className":26381},[342,472],[15,26383,26385,26460],{"className":26384},[346],[15,26386,26388,26417,26425],{"className":26387,"style":15321},[350],[15,26389,26390,26393],{"style":3857},[15,26391],{"className":26392,"style":3861},[357],[15,26394,26396],{"className":26395},[362,363,364,365],[15,26397,26399,26402,26405,26414],{"className":26398},[61,365],[15,26400,3730],{"className":26401,"style":3871},[61,122,365],[15,26403,741],{"className":26404},[146,365],[15,26406,26408],{"className":26407},[61,365],[15,26409,26411],{"className":26410},[61,365],[15,26412,3106],{"className":26413},[61,3166,365],[15,26415,750],{"className":26416},[178,365],[15,26418,26419,26422],{"style":3889},[15,26420],{"className":26421,"style":3861},[357],[15,26423],{"className":26424,"style":3897},[3896],[15,26426,26427,26430],{"style":15450},[15,26428],{"className":26429,"style":3861},[3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lassName":28039,"style":22404},[350],[15,28041,28042,28045],{"style":27234},[15,28043],{"className":28044,"style":4062},[357],[15,28046,28048],{"className":28047},[362,4066,4067,365],[15,28049,272],{"className":28050,"style":492},[61,122,365],[15,28052,497],{"className":28053},[496],[15,28055,28057],{"className":28056},[346],[15,28058,28060],{"className":28059,"style":27253},[350],[15,28061],{},[15,28063,497],{"className":28064},[496],[15,28066,28068],{"className":28067},[346],[15,28069,28071],{"className":28070,"style":15220},[350],[15,28072],{},[15,28074],{"className":28075},[178,3841],[15,28077],{"className":28078,"style":772},[127],[15,28080,7762],{"className":28081},[375],[15,28083],{"className":28084,"style":772},[127],[15,28086,28088,28092,28314,28317,28320],{"className":28087},[52],[15,28089],{"className":28090,"style":28091},[56],"height:3.5227em;vertical-align:-2.1623em;",[15,28093,28095],{"className":28094},[161,6881],[15,28096,28098,28306],{"className":28097},[342,472],[15,28099,28101,28303],{"className":28100},[346],[15,28102,28105,28124],{"className":28103,"style":28104},[350],"height:1.3603em;",[15,28106,28108,28112],{"style":28107},"top:-1.198em;",[15,28109],{"className":28110,"style":28111},[357],"height:3.3603em;",[15,28113,28115],{"className":28114},[362,363,364,365],[15,28116,28118],{"className":28117},[61,365],[15,28119,28121],{"className":28120},[61,13548,365],[15,28122,27529],{"className":28123},[61,13552,365],[15,28125,28127,28130],{"style":28126},"top:-3.3603em;",[15,28128],{"className":28129,"style":28111},[357],[15,28131,28133],{"className":28132},[161,6881],[15,28134,28136,28295],{"className":28135},[342,472],[15,28137,28139,28292],{"className":28138},[346],[15,28140,28142,28173],{"className":28141,"style":28104},[350],[15,28143,28146,28149],{"className":28144,"style":28145},[27690],"top:-1.8763em;",[15,28147],{"className":28148,"style":28111},[357],[15,28150,28152,28159,28166],{"className":28151,"style":27699},[27698],[15,28153,28155],{"className":28154,"style":27704},[27703],[27706,28156,28157],{"xmlns":27708,"width":27709,"height":27710,"viewBox":27711,"preserveAspectRatio":27712},[27714,28158],{"d":27716},[15,28160,28162],{"className":28161,"style":27704},[27720],[27706,28163,28164],{"xmlns":27708,"width":27709,"height":27710,"viewBox":27711,"preserveAspectRatio":27723},[27714,28165],{"d":27726},[15,28167,28169],{"className":28168,"style":27704},[27730],[27706,28170,28171],{"xmlns":27708,"width":27709,"height":27710,"viewBox":27711,"preserveAspectRatio":27733},[27714,28172],{"d":27736},[15,28174,28175,28178],{"style":28126},[15,28176],{"className":28177,"style":28111},[357],[15,28179,28181],{"className":28180},[61],[15,28182,28184,28187,28289],{"className":28183},[61],[15,28185],{"className":28186},[146,3841],[15,28188,28190],{"className":28189},[3722],[15,28191,28193,28281],{"className":28192},[342,472],[15,28194,28196,28278],{"className":28195},[346],[15,28197,28199,28259,28267],{"className":28198,"style":28104},[350],[15,28200,28201,28204],{"style":11402},[15,28202],{"className":28203,"style":3861},[357],[15,28205,28207,28247,28250,28253,28256],{"className":28206},[61],[15,28208,28210,28213],{"className":28209},[61],[15,28211,3106],{"className":28212,"style":3191},[61,122],[15,28214,28216],{"className":28215},[338],[15,28217,28219,28239],{"className":28218},[342,472],[15,28220,28222,28236],{"className":28221},[346],[15,28223,28225],{"className":28224,"style":1214},[350],[15,28226,28227,28230],{"style":3206},[15,28228],{"className":28229,"style":358},[357],[15,28231,28233],{"className":28232},[362,363,364,365],[15,28234,278],{"className":28235},[61,365],[15,28237,497],{"className":28238},[496],[15,28240,28242],{"className":28241},[346],[15,28243,28245],{"className":28244,"style":504},[350],[15,28246],{},[15,28248],{"className":28249,"style":772},[127],[15,28251,7762],{"className":28252},[375],[15,28254],{"className":28255,"style":772},[127],[15,28257,24982],{"className":28258,"style":25097},[61,122],[15,28260,28261,28264],{"style":3889},[15,28262],{"className":28263,"style":3861},[357],[15,28265],{"className":28266,"style":3897},[3896],[15,28268,28269,28272],{"style":11468},[15,28270],{"className":28271,"style":3861},[357],[15,28273,28275],{"className":28274},[61],[15,28276,24982],{"className":28277,"style":25097},[61,122],[15,28279,497],{"className":28280},[496],[15,28282,28284],{"className":28283},[346],[15,28285,28287],{"className":28286,"style":18779},[350],[15,28288],{},[15,28290],{"className":28291},[178,3841],[15,28293,497],{"className":28294},[496],[15,28296,28298],{"className":28297},[346],[15,28299,28301],{"className":28300,"style":27902},[350],[15,28302],{},[15,28304,497],{"className":28305},[496],[15,28307,28309],{"className":28308},[346],[15,28310,28312],{"className":28311,"style":27914},[350],[15,28313],{},[15,28315],{"className":28316,"style":772},[127],[15,28318,11214],{"className":28319},[375],[15,28321],{"className":28322,"style":772},[127],[15,28324,28326,28330],{"className":28325},[52],[15,28327],{"className":28328,"style":28329},[56],"height:1.0623em;vertical-align:-0.345em;",[15,28331,28333,28336,28432],{"className":28332},[61],[15,28334],{"className":28335},[146,3841],[15,28337,28339],{"className":28338},[3722],[15,28340,28342,28424],{"className":28341},[342,472],[15,28343,28345,28421],{"className":28344},[346],[15,28346,28348,28362,28370],{"className":28347,"style":27951},[350],[15,28349,28350,28353],{"style":3857},[15,28351],{"className":28352,"style":3861},[357],[15,28354,28356],{"className":28355},[362,363,364,365],[15,28357,28359],{"className":28358},[61,365],[15,28360,24982],{"className":28361,"style":25097},[61,122,365],[15,28363,28364,28367],{"style":3889},[15,28365],{"className":28366,"style":3861},[357],[15,28368],{"className":28369,"style":3897},[3896],[15,28371,28372,28375],{"style":27204},[15,28373],{"className":28374,"style":3861},[357],[15,28376,28378],{"className":28377},[362,363,364,365],[15,28379,28381],{"className":28380},[61,365],[15,28382,28384,28387],{"className":28383},[61,365],[15,28385,14560],{"className":28386},[61,122,365],[15,28388,28390],{"className":28389},[338],[15,28391,28393,28413],{"className":28392},[342,472],[15,28394,28396,28410],{"className":28395},[346],[15,28397,28399],{"className":28398,"style":22404},[350],[15,28400,28401,28404],{"style":27279},[15,28402],{"className":28403,"style":4062},[357],[15,28405,28407],{"className":28406},[362,4066,4067,365],[15,28408,272],{"className":28409,"style":492},[61,122,365],[15,28411,497],{"className":28412},[496],[15,28414,28416],{"className":28415},[346],[15,28417,28419],{"className":28418,"style":27253},[350],[15,28420],{},[15,28422,497],{"className":28423},[496],[15,28425,28427],{"className":28426},[346],[15,28428,28430],{"className":28429,"style":10198},[350],[15,28431],{},[15,28433],{"className":28434},[178,3841],[11,28436,28437,28438,17399,28540,28632,28633,28724,28725,28817],{},"推论 2.7 用一句话概括了加法平滑的全部含义：",[661,28439,28440,28441,28510,28511,28539],{},"先验强度的倒数 ",[15,28442,28444,28461],{"className":28443,"translate":19},[18],[15,28445,28447],{"className":28446},[23],[25,28448,28449],{"xmlns":27},[29,28450,28451,28459],{},[32,28452,28453],{},[266,28454,28455,28457],{},[35,28456,3106],{},[89,28458,278],{},[40,28460,19490],{"encoding":42},[15,28462,28464],{"className":28463,"ariaHidden":48},[47],[15,28465,28467,28470],{"className":28466},[52],[15,28468],{"className":28469,"style":2511},[56],[15,28471,28473,28476],{"className":28472},[61],[15,28474,3106],{"className":28475,"style":3191},[61,122],[15,28477,28479],{"className":28478},[338],[15,28480,28482,28502],{"className":28481},[342,472],[15,28483,28485,28499],{"className":28484},[346],[15,28486,28488],{"className":28487,"style":1214},[350],[15,28489,28490,28493],{"style":3206},[15,28491],{"className":28492,"style":358},[357],[15,28494,28496],{"className":28495},[362,363,364,365],[15,28497,278],{"className":28498},[61,365],[15,28500,497],{"className":28501},[496],[15,28503,28505],{"className":28504},[346],[15,28506,28508],{"className":28507,"style":504},[350],[15,28509],{}," 相对于样本量 ",[15,28512,28514,28527],{"className":28513,"translate":19},[18],[15,28515,28517],{"className":28516},[23],[25,28518,28519],{"xmlns":27},[29,28520,28521,28525],{},[32,28522,28523],{},[35,28524,24982],{},[40,28526,24982],{"encoding":42},[15,28528,28530],{"className":28529,"ariaHidden":48},[47],[15,28531,28533,28536],{"className":28532},[52],[15,28534],{"className":28535,"style":57},[56],[15,28537,24982],{"className":28538,"style":25097},[61,122]," 的大小，就是数据与先验的信任分配比例。",[15,28541,28543,28565],{"className":28542,"translate":19},[18],[15,28544,28546],{"className":28545},[23],[25,28547,28548],{"xmlns":27},[29,28549,28550,28562],{},[32,28551,28552,28558,28560],{},[266,28553,28554,28556],{},[35,28555,3106],{},[35,28557,1070],{},[81,28559,244],{},[89,28561,91],{},[40,28563,28564],{"encoding":42},"\\alpha_i = 1",[15,28566,28568,28623],{"className":28567,"ariaHidden":48},[47],[15,28569,28571,28574,28614,28617,28620],{"className":28570},[52],[15,28572],{"className":28573,"style":2511},[56],[15,28575,28577,28580],{"className":28576},[61],[15,28578,3106],{"className":28579,"style":3191},[61,122],[15,28581,28583],{"className":28582},[338],[15,28584,28586,28606],{"className":28585},[342,472],[15,28587,28589,28603],{"className":28588},[346],[15,28590,28592],{"className":28591,"style":1364},[350],[15,28593,28594,28597],{"style":3206},[15,28595],{"className":28596,"style":358},[357],[15,28598,28600],{"className":28599},[362,363,364,365],[15,28601,1070],{"className":28602},[61,122,365],[15,28604,497],{"className":28605},[496],[15,28607,28609],{"className":28608},[346],[15,28610,28612],{"className":28611,"style":504},[350],[15,28613],{},[15,28615],{"className":28616,"style":128},[127],[15,28618,244],{"className":28619},[132],[15,28621],{"className":28622,"style":128},[127],[15,28624,28626,28629],{"className":28625},[52],[15,28627],{"className":28628,"style":859},[56],[15,28630,91],{"className":28631},[61]," 得 Laplace 平滑，取 ",[15,28634,28636,28657],{"className":28635,"translate":19},[18],[15,28637,28639],{"className":28638},[23],[25,28640,28641],{"xmlns":27},[29,28642,28643,28655],{},[32,28644,28645,28651,28653],{},[266,28646,28647,28649],{},[35,28648,3106],{},[35,28650,1070],{},[81,28652,1590],{},[89,28654,91],{},[40,28656,21720],{"encoding":42},[15,28658,28660,28715],{"className":28659,"ariaHidden":48},[47],[15,28661,28663,28666,28706,28709,28712],{"className":28662},[52],[15,28664],{"className":28665,"style":1753},[56],[15,28667,28669,28672],{"className":28668},[61],[15,28670,3106],{"className":28671,"style":3191},[61,122],[15,28673,28675],{"className":28674},[338],[15,28676,28678,28698],{"className":28677},[342,472],[15,28679,28681,28695],{"className":28680},[346],[15,28682,28684],{"className":28683,"style":1364},[350],[15,28685,28686,28689],{"style":3206},[15,28687],{"className":28688,"style":358},[357],[15,28690,28692],{"className":28691},[362,363,364,365],[15,28693,1070],{"className":28694},[61,122,365],[15,28696,497],{"className":28697},[496],[15,28699,28701],{"className":28700},[346],[15,28702,28704],{"className":28703,"style":504},[350],[15,28705],{},[15,28707],{"className":28708,"style":128},[127],[15,28710,1590],{"className":28711},[132],[15,28713],{"className":28714,"style":128},[127],[15,28716,28718,28721],{"className":28717},[52],[15,28719],{"className":28720,"style":859},[56],[15,28722,91],{"className":28723},[61]," 得 Lidstone 平滑，取 ",[15,28726,28728,28750],{"className":28727,"translate":19},[18],[15,28729,28731],{"className":28730},[23],[25,28732,28733],{"xmlns":27},[29,28734,28735,28747],{},[32,28736,28737,28743,28745],{},[266,28738,28739,28741],{},[35,28740,3106],{},[35,28742,1070],{},[81,28744,20704],{},[89,28746,278],{},[40,28748,28749],{"encoding":42},"\\alpha_i \\to 0",[15,28751,28753,28808],{"className":28752,"ariaHidden":48},[47],[15,28754,28756,28759,28799,28802,28805],{"className":28755},[52],[15,28757],{"className":28758,"style":2511},[56],[15,28760,28762,28765],{"className":28761},[61],[15,28763,3106],{"className":28764,"style":3191},[61,122],[15,28766,28768],{"className":28767},[338],[15,28769,28771,28791],{"className":28770},[342,472],[15,28772,28774,28788],{"className":28773},[346],[15,28775,28777],{"className":28776,"style":1364},[350],[15,28778,28779,28782],{"style":3206},[15,28780],{"className":28781,"style":358},[357],[15,28783,28785],{"className":28784},[362,363,364,365],[15,28786,1070],{"className":28787},[61,122,365],[15,28789,497],{"className":28790},[496],[15,28792,28794],{"className":28793},[346],[15,28795,28797],{"className":28796,"style":504},[350],[15,28798],{},[15,28800],{"className":28801,"style":128},[127],[15,28803,20704],{"className":28804},[132],[15,28806],{"className":28807,"style":128},[127],[15,28809,28811,28814],{"className":28810},[52],[15,28812],{"className":28813,"style":859},[56],[15,28815,278],{"className":28816},[61]," 得最大似然估计——这三个统计实践中彼此孤立出现的技巧，在此被识别为同一族参数的不同取值。",[877,28819,28821],{"id":28820},"_25-聚合性中立性与分裂构造","2.5 聚合性、中立性与分裂构造",[882,28823,28824],{},[11,28825,28826,9393,28829,943,28905,1906,29059,29127,29128,29371,29372,5987,29450,867],{},[661,28827,28828],{},"命题 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与其余分量的归一化版本 ",[15,29802,29804,29854],{"className":29803,"translate":19},[18],[15,29805,29807],{"className":29806},[23],[25,29808,29809],{"xmlns":27},[29,29810,29811,29851],{},[32,29812,29813],{},[266,29814,29815,29843],{},[32,29816,29817,29819,29825,29827,29829,29831,29833,29839,29841],{},[81,29818,741],{"fence":48},[266,29820,29821,29823],{},[35,29822,1029],{},[35,29824,1070],{},[35,29826,3020],{"mathvariant":231},[81,29828,741],{"stretchy":86},[89,29830,91],{},[81,29832,239],{},[266,29834,29835,29837],{},[35,29836,1029],{},[35,29838,272],{},[81,29840,750],{"stretchy":86},[81,29842,750],{"fence":48},[32,29844,29845,29847,29849],{},[35,29846,1070],{},[81,29848,14999],{"mathvariant":231},[35,29850,272],{},[40,29852,29853],{"encoding":42},"\\left(x_i\u002F(1-x_k)\\right)_{i \\ne k}",[15,29855,29857],{"className":29856,"ariaHidden":48},[47],[15,29858,29860,29864],{"className":29859},[52],[15,29861],{"className":29862,"style":29863},[56],"height:1.1858em;vertical-align:-0.4358em;",[15,29865,29867,29977],{"className":29866},[161],[15,29868,29870,29873,29913,29916,29919,29922,29925,29928,29931,29971,29974],{"className":29869},[161],[15,29871,741],{"className":29872,"style":402},[146,401],[15,29874,29876,29879],{"className":29875},[61],[15,29877,1029],{"className":29878},[61,122],[15,29880,29882],{"className":29881},[338],[15,29883,29885,29905],{"className":29884},[342,472],[15,29886,29888,29902],{"className":29887},[346],[15,29889,29891],{"className":29890,"style":1364},[350],[15,29892,29893,29896],{"style":482},[15,29894],{"className":29895,"style":358},[357],[15,29897,29899],{"className":29898},[362,363,364,365],[15,29900,1070],{"className":29901},[61,122,365],[15,29903,497],{"className":29904},[496],[15,29906,29908],{"className":29907},[346],[15,29909,29911],{"className":29910,"style":504},[350],[15,29912],{},[15,29914,3020],{"className":29915},[61],[15,29917,741],{"className":29918},[146],[15,29920,91],{"className":29921},[61],[15,29923],{"className":29924,"style":772},[127],[15,29926,239],{"className":29927},[375],[15,29929],{"className":29930,"style":772},[127],[15,29932,29934,29937],{"className":29933},[61],[15,29935,1029],{"className":29936},[61,122],[15,29938,29940],{"className":29939},[338],[15,29941,29943,29963],{"className":29942},[342,472],[15,29944,29946,29960],{"className":29945},[346],[15,29947,29949],{"className":29948,"style":479},[350],[15,29950,29951,29954],{"style":482},[15,29952],{"className":29953,"style":358},[357],[15,29955,29957],{"className":29956},[362,363,364,365],[15,29958,272],{"className":29959,"style":492},[61,122,365],[15,29961,497],{"className":29962},[496],[15,29964,29966],{"className":29965},[346],[15,29967,29969],{"className":29968,"style":504},[350],[15,29970],{},[15,29972,750],{"className":29973},[178],[15,29975,750],{"className":29976,"style":402},[178,401],[15,29978,29980],{"className":29979},[338],[15,29981,29983,30047],{"className":29982},[342,472],[15,29984,29986,30044],{"className":29985},[346],[15,29987,29990],{"className":29988,"style":29989},[350],"height:0.1864em;",[15,29991,29993,29996],{"style":29992},"top:-2.4003em;margin-right:0.05em;",[15,29994],{"className":29995,"style":358},[357],[15,29997,29999],{"className":29998},[362,363,364,365],[15,30000,30002,30005,30041],{"className":30001},[61,365],[15,30003,1070],{"className":30004},[61,122,365],[15,30006,30008,30035,30038],{"className":30007},[132,365],[15,30009,30011],{"className":30010},[132,365],[15,30012,30014],{"className":30013},[61,16002,365],[15,30015,30017],{"className":30016},[16006,365],[15,30018,30020,30023,30032],{"className":30019},[16010,365],[15,30021],{"className":30022,"style":14310},[56],[15,30024,30026],{"className":30025},[16017],[15,30027,30029],{"className":30028},[61,365],[15,30030,16024],{"className":30031},[132,365],[15,30033],{"className":30034},[16028],[15,30036],{"className":30037},[127,16032,365],[15,30039,244],{"className":30040},[132,365],[15,30042,272],{"className":30043,"style":492},[61,122,365],[15,30045,497],{"className":30046},[496],[15,30048,30050],{"className":30049},[346],[15,30051,30054],{"className":30052,"style":30053},[350],"height:0.4358em;",[15,30055],{}," 相互独立，且后者服从 ",[15,30058,30060,30090],{"className":30059,"translate":19},[18],[15,30061,30063],{"className":30062},[23],[25,30064,30065],{"xmlns":27},[29,30066,30067,30087],{},[32,30068,30069,30071,30073,30075,30085],{},[35,30070,4186],{"mathvariant":231},[81,30072,1547],{},[81,30074,741],{"stretchy":86},[266,30076,30077,30079],{},[35,30078,3106],{"mathvariant":3105},[32,30080,30081,30083],{},[81,30082,239],{},[35,30084,272],{},[81,30086,750],{"stretchy":86},[40,30088,30089],{"encoding":42},"\\operatorname{Dir}(\\boldsymbol{\\alpha}_{-k})",[15,30091,30093],{"className":30092,"ariaHidden":48},[47],[15,30094,30096,30099,30105,30108,30161],{"className":30095},[52],[15,30097],{"className":30098,"style":142},[56],[15,30100,30102],{"className":30101},[531],[15,30103,4186],{"className":30104},[61,1622],[15,30106,741],{"className":30107},[146],[15,30109,30111,30120],{"className":30110},[61],[15,30112,30114],{"className":30113},[61],[15,30115,30117],{"className":30116},[61],[15,30118,3106],{"className":30119},[61,3166],[15,30121,30123],{"className":30122},[338],[15,30124,30126,30153],{"className":30125},[342,472],[15,30127,30129,30150],{"className":30128},[346],[15,30130,30132],{"className":30131,"style":479},[350],[15,30133,30135,30138],{"style":30134},"top:-2.55em;margin-right:0.05em;",[15,30136],{"className":30137,"style":358},[357],[15,30139,30141],{"className":30140},[362,363,364,365],[15,30142,30144,30147],{"className":30143},[61,365],[15,30145,239],{"className":30146},[61,365],[15,30148,272],{"className":30149,"style":492},[61,122,365],[15,30151,497],{"className":30152},[496],[15,30154,30156],{"className":30155},[346],[15,30157,30159],{"className":30158,"style":2452},[350],[15,30160],{},[15,30162,750],{"className":30163},[178]," 而 ",[15,30166,30168,30218],{"className":30167,"translate":19},[18],[15,30169,30171],{"className":30170},[23],[25,30172,30173],{"xmlns":27},[29,30174,30175,30215],{},[32,30176,30177,30183,30185,30187,30189,30191,30197,30199,30205,30207,30213],{},[266,30178,30179,30181],{},[35,30180,1029],{},[35,30182,272],{},[81,30184,4183],{},[35,30186,12956],{"mathvariant":231},[81,30188,1547],{},[81,30190,741],{"stretchy":86},[266,30192,30193,30195],{},[35,30194,3106],{},[35,30196,272],{},[81,30198,94],{"separator":48},[266,30200,30201,30203],{},[35,30202,3106],{},[89,30204,278],{},[81,30206,239],{},[266,30208,30209,30211],{},[35,30210,3106],{},[35,30212,272],{},[81,30214,750],{"stretchy":86},[40,30216,30217],{"encoding":42},"x_k \\sim \\operatorname{Beta}(\\alpha_k, \\alpha_0 - \\alpha_k)",[15,30219,30221,30276,30386],{"className":30220,"ariaHidden":48},[47],[15,30222,30224,30227,30267,30270,30273],{"className":30223},[52],[15,30225],{"className":30226,"style":2511},[56],[15,30228,30230,30233],{"className":30229},[61],[15,30231,1029],{"className":30232},[61,122],[15,30234,30236],{"className":30235},[338],[15,30237,30239,30259],{"className":30238},[342,472],[15,30240,30242,30256],{"className":30241},[346],[15,30243,30245],{"className":30244,"style":479},[350],[15,30246,30247,30250],{"style":482},[15,30248],{"className":30249,"style":358},[357],[15,30251,30253],{"className":30252},[362,363,364,365],[15,30254,272],{"className":30255,"style":492},[61,122,365],[15,30257,497],{"className":30258},[496],[15,30260,30262],{"className":30261},[346],[15,30263,30265],{"className":30264,"style":504},[350],[15,30266],{},[15,30268],{"className":30269,"style":128},[127],[15,30271,4183],{"className":30272},[132],[15,30274],{"className":30275,"style":128},[127],[15,30277,30279,30282,30288,30291,30331,30334,30337,30377,30380,30383],{"className":30278},[52],[15,30280],{"className":30281,"style":142},[56],[15,30283,30285],{"className":30284},[531],[15,30286,12956],{"className":30287},[61,1622],[15,30289,741],{"className":30290},[146],[15,30292,30294,30297],{"className":30293},[61],[15,30295,3106],{"className":30296,"style":3191},[61,122],[15,30298,30300],{"className":30299},[338],[15,30301,30303,30323],{"className":30302},[342,472],[15,30304,30306,30320],{"className":30305},[346],[15,30307,30309],{"className":30308,"style":479},[350],[15,30310,30311,30314],{"style":3206},[15,30312],{"className":30313,"style":358},[357],[15,30315,30317],{"className":30316},[362,363,364,365],[15,30318,272],{"className":30319,"style":492},[61,122,365],[15,30321,497],{"className":30322},[496],[15,30324,30326],{"className":30325},[346],[15,30327,30329],{"className":30328,"style":504},[350],[15,30330],{},[15,30332,94],{"className":30333},[153],[15,30335],{"className":30336,"style":157},[127],[15,30338,30340,30343],{"className":30339},[61],[15,30341,3106],{"className":30342,"style":3191},[61,122],[15,30344,30346],{"className":30345},[338],[15,30347,30349,30369],{"className":30348},[342,472],[15,30350,30352,30366],{"className":30351},[346],[15,30353,30355],{"className":30354,"style":1214},[350],[15,30356,30357,30360],{"style":3206},[15,30358],{"className":30359,"style":358},[357],[15,30361,30363],{"className":30362},[362,363,364,365],[15,30364,278],{"className":30365},[61,365],[15,30367,497],{"className":30368},[496],[15,30370,30372],{"className":30371},[346],[15,30373,30375],{"className":30374,"style":504},[350],[15,30376],{},[15,30378],{"className":30379,"style":772},[127],[15,30381,239],{"className":30382},[375],[15,30384],{"className":30385,"style":772},[127],[15,30387,30389,30392,30432],{"className":30388},[52],[15,30390],{"className":30391,"style":142},[56],[15,30393,30395,30398],{"className":30394},[61],[15,30396,3106],{"className":30397,"style":3191},[61,122],[15,30399,30401],{"className":30400},[338],[15,30402,30404,30424],{"className":30403},[342,472],[15,30405,30407,30421],{"className":30406},[346],[15,30408,30410],{"className":30409,"style":479},[350],[15,30411,30412,30415],{"style":3206},[15,30413],{"className":30414,"style":358},[357],[15,30416,30418],{"className":30417},[362,363,364,365],[15,30419,272],{"className":30420,"style":492},[61,122,365],[15,30422,497],{"className":30423},[496],[15,30425,30427],{"className":30426},[346],[15,30428,30430],{"className":30429,"style":504},[350],[15,30431],{},[15,30433,750],{"className":30434},[178],"。反复施用中立性，得到如下构造性表示：",[882,30437,30438],{},[11,30439,30440,9393,30443,30734,30735,10335,31203,867],{},[661,30441,30442],{},"命题 2.9（stick-breaking 表示）",[15,30444,30446,30505],{"className":30445,"translate":19},[18],[15,30447,30449],{"className":30448},[23],[25,30450,30451],{"xmlns":27},[29,30452,30453,30502],{},[32,30454,30455,30462,30464,30466,30468,30470],{},[266,30456,30457,30460],{},[35,30458,30459],{},"v",[35,30461,272],{},[81,30463,4183],{},[35,30465,12956],{"mathvariant":231},[81,30467,1547],{},[282,30469,6799],{},[32,30471,30472,30474,30480,30482,30494,30500],{},[81,30473,741],{"fence":48},[266,30475,30476,30478],{},[35,30477,3106],{},[35,30479,272],{},[81,30481,94],{"separator":48},[266,30483,30484,30486],{},[81,30485,293],{},[32,30487,30488,30490,30492],{},[35,30489,14777],{},[81,30491,1590],{},[35,30493,272],{},[266,30495,30496,30498],{},[35,30497,3106],{},[35,30499,14777],{},[81,30501,750],{"fence":48},[40,30503,30504],{"encoding":42},"v_k \\sim \\operatorname{Beta}\\!\\left(\\alpha_k, \\sum_{j>k} \\alpha_j\\right)",[15,30506,30508,30563],{"className":30507,"ariaHidden":48},[47],[15,30509,30511,30514,30554,30557,30560],{"className":30510},[52],[15,30512],{"className":30513,"style":2511},[56],[15,30515,30517,30520],{"className":30516},[61],[15,30518,30459],{"className":30519,"style":123},[61,122],[15,30521,30523],{"className":30522},[338],[15,30524,30526,30546],{"className":30525},[342,472],[15,30527,30529,30543],{"className":30528},[346],[15,30530,30532],{"className":30531,"style":479},[350],[15,30533,30534,30537],{"style":27036},[15,30535],{"className":30536,"style":358},[357],[15,30538,30540],{"className":30539},[362,363,364,365],[15,30541,272],{"className":30542,"style":492},[61,122,365],[15,30544,497],{"className":30545},[496],[15,30547,30549],{"className":30548},[346],[15,30550,30552],{"className":30551,"style":504},[350],[15,30553],{},[15,30555],{"className":30556,"style":128},[127],[15,30558,4183],{"className":30559},[132],[15,30561],{"className":30562,"style":128},[127],[15,30564,30566,30569,30575,30578,30581],{"className":30565},[52],[15,30567],{"className":30568,"style":394},[56],[15,30570,30572],{"className":30571},[531],[15,30573,12956],{"className":30574},[61,1622],[15,30576],{"className":30577,"style":7105},[127],[15,30579],{"className":30580,"style":157},[127],[15,30582,30584,30590,30630,30633,30636,30685,30688,30728],{"className":30583},[161],[15,30585,30587],{"className":30586,"style":402},[146,401],[15,30588,741],{"className":30589},[406,407],[15,30591,30593,30596],{"className":30592},[61],[15,30594,3106],{"className":30595,"style":3191},[61,122],[15,30597,30599],{"className":30598},[338],[15,30600,30602,30622],{"className":30601},[342,472],[15,30603,30605,30619],{"className":30604},[346],[15,30606,30608],{"className":30607,"style":479},[350],[15,30609,30610,30613],{"style":3206},[15,30611],{"className":30612,"style":358},[357],[15,30614,30616],{"className":30615},[362,363,364,365],[15,30617,272],{"className":30618,"style":492},[61,122,365],[15,30620,497],{"className":30621},[496],[15,30623,30625],{"className":30624},[346],[15,30626,30628],{"className":30627,"style":504},[350],[15,30629],{},[15,30631,94],{"className":30632},[153],[15,30634],{"className":30635,"style":157},[127],[15,30637,30639,30642],{"className":30638},[531],[15,30640,293],{"className":30641,"style":537},[531,535,536],[15,30643,30645],{"className":30644},[338],[15,30646,30648,30677],{"className":30647},[342,472],[15,30649,30651,30674],{"className":30650},[346],[15,30652,30654],{"className":30653,"style":29989},[350],[15,30655,30656,30659],{"style":553},[15,30657],{"className":30658,"style":358},[357],[15,30660,30662],{"className":30661},[362,363,364,365],[15,30663,30665,30668,30671],{"className":30664},[61,365],[15,30666,14777],{"className":30667,"style":14803},[61,122,365],[15,30669,1590],{"className":30670},[132,365],[15,30672,272],{"className":30673,"style":492},[61,122,365],[15,30675,497],{"className":30676},[496],[15,30678,30680],{"className":30679},[346],[15,30681,30683],{"className":30682,"style":30053},[350],[15,30684],{},[15,30686],{"className":30687,"style":157},[127],[15,30689,30691,30694],{"className":30690},[61],[15,30692,3106],{"className":30693,"style":3191},[61,122],[15,30695,30697],{"className":30696},[338],[15,30698,30700,30720],{"className":30699},[342,472],[15,30701,30703,30717],{"className":30702},[346],[15,30704,30706],{"className":30705,"style":1364},[350],[15,30707,30708,30711],{"style":3206},[15,30709],{"className":30710,"style":358},[357],[15,30712,30714],{"className":30713},[362,363,364,365],[15,30715,14777],{"className":30716,"style":14803},[61,122,365],[15,30718,497],{"className":30719},[496],[15,30721,30723],{"className":30722},[346],[15,30724,30726],{"className":30725,"style":15707},[350],[15,30727],{},[15,30729,30731],{"className":30730,"style":402},[178,401],[15,30732,750],{"className":30733},[406,407]," 相互独立。令\n",[15,30736,30738,30828],{"className":30737,"translate":19},[18],[15,30739,30741],{"className":30740},[23],[25,30742,30743],{"xmlns":27},[29,30744,30745,30825],{},[32,30746,30747,30753,30755,30761,30763,30765,30771,30773,30779,30791,30793,30795,30797,30803,30805,30807,30809,30811,30813,30815,30817,30819,30821,30823],{},[266,30748,30749,30751],{},[35,30750,1029],{},[89,30752,91],{},[81,30754,244],{},[266,30756,30757,30759],{},[35,30758,30459],{},[89,30760,91],{},[81,30762,94],{"separator":48},[127,30764],{"width":9847},[266,30766,30767,30769],{},[35,30768,1029],{},[35,30770,272],{},[81,30772,244],{},[266,30774,30775,30777],{},[35,30776,30459],{},[35,30778,272],{},[266,30780,30781,30783],{},[81,30782,3741],{},[32,30784,30785,30787,30789],{},[35,30786,14777],{},[81,30788,2490],{},[35,30790,272],{},[81,30792,741],{"stretchy":86},[89,30794,91],{},[81,30796,239],{},[266,30798,30799,30801],{},[35,30800,30459],{},[35,30802,14777],{},[81,30804,750],{"stretchy":86},[127,30806],{"width":6451},[81,30808,741],{"stretchy":86},[35,30810,272],{},[81,30812,244],{},[89,30814,908],{},[81,30816,94],{"separator":48},[81,30818,97],{},[81,30820,94],{"separator":48},[35,30822,102],{},[81,30824,750],{"stretchy":86},[40,30826,30827],{"encoding":42},"x_1 = v_1, \\qquad x_k = v_k \\prod_{j \u003C k}(1 - v_j) \\quad (k = 2, \\dots, K)",[15,30829,30831,30886,30990,31103,31170],{"className":30830,"ariaHidden":48},[47],[15,30832,30834,30837,30877,30880,30883],{"className":30833},[52],[15,30835],{"className":30836,"style":2511},[56],[15,30838,30840,30843],{"className":30839},[61],[15,30841,1029],{"className":30842},[61,122],[15,30844,30846],{"className":30845},[338],[15,30847,30849,30869],{"className":30848},[342,472],[15,30850,30852,30866],{"className":30851},[346],[15,30853,30855],{"className":30854,"style":1214},[350],[15,30856,30857,30860],{"style":482},[15,30858],{"className":30859,"style":358},[357],[15,30861,30863],{"className":30862},[362,363,364,365],[15,30864,91],{"className":30865},[61,365],[15,30867,497],{"className":30868},[496],[15,30870,30872],{"className":30871},[346],[15,30873,30875],{"className":30874,"style":504},[350],[15,30876],{},[15,30878],{"className":30879,"style":128},[127],[15,30881,244],{"className":30882},[132],[15,30884],{"className":30885,"style":128},[127],[15,30887,30889,30892,30932,30935,30938,30941,30981,30984,30987],{"className":30888},[52],[15,30890],{"className":30891,"style":9437},[56],[15,30893,30895,30898],{"className":30894},[61],[15,30896,30459],{"className":30897,"style":123},[61,122],[15,30899,30901],{"className":30900},[338],[15,30902,30904,30924],{"className":30903},[342,472],[15,30905,30907,30921],{"className":30906},[346],[15,30908,30910],{"className":30909,"style":1214},[350],[15,30911,30912,30915],{"style":27036},[15,30913],{"className":30914,"style":358},[357],[15,30916,30918],{"className":30917},[362,363,364,365],[15,30919,91],{"className":30920},[61,365],[15,30922,497],{"className":30923},[496],[15,30925,30927],{"className":30926},[346],[15,30928,30930],{"className":30929,"style":504},[350],[15,30931],{},[15,30933,94],{"className":30934},[153],[15,30936],{"className":30937,"style":10035},[127],[15,30939],{"className":30940,"style":157},[127],[15,30942,30944,30947],{"className":30943},[61],[15,30945,1029],{"className":30946},[61,122],[15,30948,30950],{"className":30949},[338],[15,30951,30953,30973],{"className":30952},[342,472],[15,30954,30956,30970],{"className":30955},[346],[15,30957,30959],{"className":30958,"style":479},[350],[15,30960,30961,30964],{"style":482},[15,30962],{"className":30963,"style":358},[357],[15,30965,30967],{"className":30966},[362,363,364,365],[15,30968,272],{"className":30969,"style":492},[61,122,365],[15,30971,497],{"className":30972},[496],[15,30974,30976],{"className":30975},[346],[15,30977,30979],{"className":30978,"style":504},[350],[15,30980],{},[15,30982],{"className":30983,"style":128},[127],[15,30985,244],{"className":30986},[132],[15,30988],{"className":30989,"style":128},[127],[15,30991,30993,30996,31036,31039,31088,31091,31094,31097,31100],{"className":30992},[52],[15,30994],{"className":30995,"style":29863},[56],[15,30997,30999,31002],{"className":30998},[61],[15,31000,30459],{"className":31001,"style":123},[61,122],[15,31003,31005],{"className":31004},[338],[15,31006,31008,31028],{"className":31007},[342,472],[15,31009,31011,31025],{"className":31010},[346],[15,31012,31014],{"className":31013,"style":479},[350],[15,31015,31016,31019],{"style":27036},[15,31017],{"className":31018,"style":358},[357],[15,31020,31022],{"className":31021},[362,363,364,365],[15,31023,272],{"className":31024,"style":492},[61,122,365],[15,31026,497],{"className":31027},[496],[15,31029,31031],{"className":31030},[346],[15,31032,31034],{"className":31033,"style":504},[350],[15,31035],{},[15,31037],{"className":31038,"style":157},[127],[15,31040,31042,31045],{"className":31041},[531],[15,31043,3741],{"className":31044,"style":537},[531,535,536],[15,31046,31048],{"className":31047},[338],[15,31049,31051,31080],{"className":31050},[342,472],[15,31052,31054,31077],{"className":31053},[346],[15,31055,31057],{"className":31056,"style":29989},[350],[15,31058,31059,31062],{"style":553},[15,31060],{"className":31061,"style":358},[357],[15,31063,31065],{"className":31064},[362,363,364,365],[15,31066,31068,31071,31074],{"className":31067},[61,365],[15,31069,14777],{"className":31070,"style":14803},[61,122,365],[15,31072,2490],{"className":31073},[132,365],[15,31075,272],{"className":31076,"style":492},[61,122,365],[15,31078,497],{"className":31079},[496],[15,31081,31083],{"className":31082},[346],[15,31084,31086],{"className":31085,"style":30053},[350],[15,31087],{},[15,31089,741],{"className":31090},[146],[15,31092,91],{"className":31093},[61],[15,31095],{"className":31096,"style":772},[127],[15,31098,239],{"className":31099},[375],[15,31101],{"className":31102,"style":772},[127],[15,31104,31106,31109,31149,31152,31155,31158,31161,31164,31167],{"className":31105},[52],[15,31107],{"className":31108,"style":17987},[56],[15,31110,31112,31115],{"className":31111},[61],[15,31113,30459],{"className":31114,"style":123},[61,122],[15,31116,31118],{"className":31117},[338],[15,31119,31121,31141],{"className":31120},[342,472],[15,31122,31124,31138],{"className":31123},[346],[15,31125,31127],{"c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\\dots, x_K) \\sim \\operatorname{Dir}(\\boldsymbol{\\alpha})",[15,31253,31255,31374],{"className":31254,"ariaHidden":48},[47],[15,31256,31258,31261,31264,31304,31307,31310,31313,31316,31319,31322,31362,31365,31368,31371],{"className":31257},[52],[15,31259],{"className":31260,"style":142},[56],[15,31262,741],{"className":31263},[146],[15,31265,31267,31270],{"className":31266},[61],[15,31268,1029],{"className":31269},[61,122],[15,31271,31273],{"className":31272},[338],[15,31274,31276,31296],{"className":31275},[342,472],[15,31277,31279,31293],{"className":31278},[346],[15,31280,31282],{"className":31281,"style":1214},[350],[15,31283,31284,31287],{"style":482},[15,31285],{"className":31286,"style":358},[357],[15,31288,31290],{"className":31289},[362,363,364,365],[15,31291,91],{"className":31292},[61,365],[15,31294,497],{"className":31295},[496],[15,31297,31299],{"className":31298},[346],[15,31300,31302],{"className":31301,"style":504},[350],[15,31303],{},[15,31305,94],{"className":31306},[153],[15,31308],{"className":31309,"style":157},[127],[15,31311,97],{"className":31312},[161],[15,31314],{"className":31315,"style":157},[127],[15,31317,94],{"className":31318},[153],[15,31320],{"className":31321,"style":157},[127],[15,31323,31325,31328],{"className":31324},[61],[15,31326,1029],{"className":31327},[61,122],[15,31329,31331],{"className":31330},[338],[15,31332,31334,31354],{"className":31333},[342,472],[15,31335,31337,31351],{"className":31336},[346],[15,31338,31340],{"className":31339,"style":1273},[350],[15,31341,31342,31345],{"style":482},[15,31343],{"className":31344,"style":358},[357],[15,31346,31348],{"className":31347},[362,363,364,365],[15,31349,102],{"className":31350,"style":174},[61,122,365],[15,31352,497],{"className":31353},[496],[15,31355,31357],{"className":31356},[346],[15,31358,31360],{"className":31359,"style":504},[350],[15,31361],{},[15,31363,750],{"className":31364},[178],[15,31366],{"className":31367,"style":128},[127],[15,31369,4183],{"className":31370},[132],[15,31372],{"className":31373,"style":128},[127],[15,31375,31377,31380,31386,31389,31398],{"className":31376},[52],[15,31378],{"className":31379,"style":142},[56],[15,31381,31383],{"className":31382},[531],[15,31384,4186],{"className":31385},[61,1622],[15,31387,741],{"className":31388},[146],[15,31390,31392],{"className":31391},[61],[15,31393,31395],{"className":31394},[61],[15,31396,3106],{"className":31397},[61,3166],[15,31399,750],{"className":31400},[178],[11,31402,31403,31404,31432,31433,31503],{},"命题 2.9 是 Sethuraman 式的「折断木棍」构造在有限维的情形：从长度为 ",[15,31405,31407,31420],{"className":31406,"translate":19},[18],[15,31408,31410],{"className":31409},[23],[25,31411,31412],{"xmlns":27},[29,31413,31414,31418],{},[32,31415,31416],{},[89,31417,91],{},[40,31419,91],{"encoding":42},[15,31421,31423],{"className":31422,"ariaHidden":48},[47],[15,31424,31426,31429],{"className":31425},[52],[15,31427],{"className":31428,"style":859},[56],[15,31430,91],{"className":31431},[61]," 的木棍中依次折断出比例 ",[15,31434,31436,31454],{"className":31435,"translate":19},[18],[15,31437,31439],{"className":31438},[23],[25,31440,31441],{"xmlns":27},[29,31442,31443,31451],{},[32,31444,31445],{},[266,31446,31447,31449],{},[35,31448,30459],{},[35,31450,272],{},[40,31452,31453],{"encoding":42},"v_k",[15,31455,31457],{"className":31456,"ariaHidden":48},[47],[15,31458,31460,31463],{"className":31459},[52],[15,31461],{"className":31462,"style":2511},[56],[15,31464,31466,31469],{"className":31465},[61],[15,31467,30459],{"className":31468,"style":123},[61,122],[15,31470,31472],{"className":31471},[338],[15,31473,31475,31495],{"className":31474},[342,472],[15,31476,31478,31492],{"className":31477},[346],[15,31479,31481],{"className":31480,"style":479},[350],[15,31482,31483,31486],{"style":27036},[15,31484],{"className":31485,"style":358},[357],[15,31487,31489],{"className":31488},[362,363,364,365],[15,31490,272],{"className":31491,"style":492},[61,122,365],[15,31493,497],{"className":31494},[496],[15,31496,31498],{"className":31497},[346],[15,31499,31501],{"className":31500,"style":504},[350],[15,31502],{}," 的一段。它在无穷维极限下给出 Dirichlet 过程，因而把本文的有限维话题与贝叶斯非参数方法连接起来。",[877,31505,31507],{"id":31506},"_26-参数的直观伪计数及其边界","2.6 参数的直观：伪计数及其边界",[11,31509,31510,31511,31545],{},"综合 2.1 至 2.5 节，可以把形状参数 ",[15,31512,31514,31527],{"className":31513,"translate":19},[18],[15,31515,31517],{"className":31516},[23],[25,31518,31519],{"xmlns":27},[29,31520,31521,31525],{},[32,31522,31523],{},[35,31524,3106],{"mathvariant":3105},[40,31526,4147],{"encoding":42},[15,31528,31530],{"className":31529,"ariaHidden":48},[47],[15,31531,31533,31536],{"className":31532},[52],[15,31534],{"className":31535,"style":2708},[56],[15,31537,31539],{"className":31538},[61],[15,31540,31542],{"className":31541},[61],[15,31543,3106],{"className":31544},[61,3166]," 的语义概括为三点：",[31547,31548,31549,31661,31837],"ol",{},[31550,31551,31552,31590,31591,31660],"li",{},[661,31553,31554,31555,31589],{},"在共轭更新中，",[15,31556,31558,31571],{"className":31557,"translate":19},[18],[15,31559,31561],{"className":31560},[23],[25,31562,31563],{"xmlns":27},[29,31564,31565,31569],{},[32,31566,31567],{},[35,31568,3106],{"mathvariant":3105},[40,31570,4147],{"encoding":42},[15,31572,31574],{"className":31573,"ariaHidden":48},[47],[15,31575,31577,31580],{"className":31576},[52],[15,31578],{"className":31579,"style":2708},[56],[15,31581,31583],{"className":31582},[61],[15,31584,31586],{"className":31585},[61],[15,31587,3106],{"className":31588},[61,3166]," 起先验样本计数的角色","（命题 2.5），其总强度 ",[15,31592,31594,31611],{"className":31593,"translate":19},[18],[15,31595,31597],{"className":31596},[23],[25,31598,31599],{"xmlns":27},[29,31600,31601,31609],{},[32,31602,31603],{},[266,31604,31605,31607],{},[35,31606,3106],{},[89,31608,278],{},[40,31610,19490],{"encoding":42},[15,31612,31614],{"className":31613,"ariaHidden":48},[47],[15,31615,31617,31620],{"className":31616},[52],[15,31618],{"className":31619,"style":2511},[56],[15,31621,31623,31626],{"className":31622},[61],[15,31624,3106],{"className":31625,"style":3191},[61,122],[15,31627,31629],{"className":31628},[338],[15,31630,31632,31652],{"className":31631},[342,472],[15,31633,31635,31649],{"className":31634},[346],[15,31636,31638],{"className":31637,"style":1214},[350],[15,31639,31640,31643],{"style":3206},[15,31641],{"className":31642,"style":358},[357],[15,31644,31646],{"className":31645},[362,363,364,365],[15,31647,278],{"className":31648},[61,365],[15,31650,497],{"className":31651},[496],[15,31653,31655],{"className":31654},[346],[15,31656,31658],{"className":31657,"style":504},[350],[15,31659],{}," 是「先验等价样本量」（推论 2.7）。",[31550,31662,31663,31801,31802,31836],{},[661,31664,31665,31666,31695,31696,31730,31731,31800],{},"在几何上，",[15,31667,31669,31683],{"className":31668,"translate":19},[18],[15,31670,31672],{"className":31671},[23],[25,31673,31674],{"xmlns":27},[29,31675,31676,31680],{},[32,31677,31678],{},[35,31679,3106],{},[40,31681,31682],{"encoding":42},"\\alpha",[15,31684,31686],{"className":31685,"ariaHidden":48},[47],[15,31687,31689,31692],{"className":31688},[52],[15,31690],{"className":31691,"style":6032},[56],[15,31693,3106],{"className":31694,"style":3191},[61,122]," 由方向 ",[15,31697,31699,31712],{"className":31698,"translate":19},[18],[15,31700,31702],{"className":31701},[23],[25,31703,31704],{"xmlns":27},[29,31705,31706,31710],{},[32,31707,31708],{},[35,31709,18346],{"mathvariant":3105},[40,31711,19451],{"encoding":42},[15,31713,31715],{"className":31714,"ariaHidden":48},[47],[15,31716,31718,31721],{"className":31717},[52],[15,31719],{"className":31720,"style":829},[56],[15,31722,31724],{"className":31723},[61],[15,31725,31727],{"className":31726},[61],[15,31728,18346],{"className":31729},[61,3166]," 与精度 ",[15,31732,31734,31751],{"className":31733,"translate":19},[18],[15,31735,31737],{"className":31736},[23],[25,31738,31739],{"xmlns":27},[29,31740,31741,31749],{},[32,31742,31743],{},[266,31744,31745,31747],{},[35,31746,3106],{},[89,31748,278],{},[40,31750,19490],{"encoding":42},[15,31752,31754],{"className":31753,"ariaHidden":48},[47],[15,31755,31757,31760],{"className":31756},[52],[15,31758],{"className":31759,"style":2511},[56],[15,31761,31763,31766],{"className":31762},[61],[15,31764,3106],{"className":31765,"style":3191},[61,122],[15,31767,31769],{"className":31768},[338],[15,31770,31772,31792],{"className":31771},[342,472],[15,31773,31775,31789],{"className":31774},[346],[15,31776,31778],{"className":31777,"style":1214},[350],[15,31779,31780,31783],{"style":3206},[15,31781],{"className":31782,"style":358},[357],[15,31784,31786],{"className":31785},[362,363,364,365],[15,31787,278],{"className":31788},[61,365],[15,31790,497],{"className":31791},[496],[15,31793,31795],{"className":31794},[346],[15,31796,31798],{"className":31797,"style":504},[350],[15,31799],{}," 两个正交分量构成","（2.2 节），精度决定围绕 ",[15,31803,31805,31818],{"className":31804,"translate":19},[18],[15,31806,31808],{"className":31807},[23],[25,31809,31810],{"xmlns":27},[29,31811,31812,31816],{},[32,31813,31814],{},[35,31815,18346],{"mathvariant":3105},[40,31817,19451],{"encoding":42},[15,31819,31821],{"className":31820,"ariaHidden":48},[47],[15,31822,31824,31827],{"className":31823},[52],[15,31825],{"className":31826,"style":829},[56],[15,31828,31830],{"className":31829},[61],[15,31831,31833],{"className":31832},[61],[15,31834,18346],{"className":31835},[61,3166]," 的收缩程度。",[31550,31838,31839,14430,32005,32097,32098,32190],{},[661,31840,31841,31842,31934,31935,32004],{},"在似然意义上，",[15,31843,31845,31867],{"className":31844,"translate":19},[18],[15,31846,31848],{"className":31847},[23],[25,31849,31850],{"xmlns":27},[29,31851,31852,31864],{},[32,31853,31854,31860,31862],{},[266,31855,31856,31858],{},[35,31857,3106],{},[35,31859,1070],{},[81,31861,239],{},[89,31863,91],{},[40,31865,31866],{"encoding":42},"\\alpha_i - 1",[15,31868,31870,31925],{"className":31869,"ariaHidden":48},[47],[15,31871,31873,31876,31916,31919,31922],{"className":31872},[52],[15,31874],{"className":31875,"style":12617},[56],[15,31877,31879,31882],{"className":31878},[61],[15,31880,3106],{"className":31881,"style":3191},[61,122],[15,31883,31885],{"className":31884},[338],[15,31886,31888,31908],{"className":31887},[342,472],[15,31889,31891,31905],{"className":31890},[346],[15,31892,31894],{"className":31893,"style":1364},[350],[15,31895,31896,31899],{"style":3206},[15,31897],{"className":31898,"style":358},[357],[15,31900,31902],{"className":31901},[362,363,364,365],[15,31903,1070],{"className":31904},[61,122,365],[15,31906,497],{"className":31907},[496],[15,31909,31911],{"className":31910},[346],[15,31912,31914],{"className":31913,"style":504},[350],[15,31915],{},[15,31917],{"className":31918,"style":772},[127],[15,31920,239],{"className":31921},[375],[15,31923],{"className":31924,"style":772},[127],[15,31926,31928,31931],{"className":31927},[52],[15,31929],{"className":31930,"style":859},[56],[15,31932,91],{"className":31933},[61]," 是对数密度中 ",[15,31936,31938,31955],{"className":31937,"translate":19},[18],[15,31939,31941],{"className":31940},[23],[25,31942,31943],{"xmlns":27},[29,31944,31945,31953],{},[32,31946,31947],{},[266,31948,31949,31951],{},[35,31950,1029],{},[35,31952,1070],{},[40,31954,13851],{"encoding":42},[15,31956,31958],{"className":31957,"ariaHidden":48},[47],[15,31959,31961,31964],{"className":31960},[52],[15,31962],{"className":31963,"style":2511},[56],[15,31965,31967,31970],{"className":31966},[61],[15,31968,1029],{"className":31969},[61,122],[15,31971,31973],{"className":31972},[338],[15,31974,31976,31996],{"className":31975},[342,472],[15,31977,31979,31993],{"className":31978},[346],[15,31980,31982],{"className":31981,"style":1364},[350],[15,31983,31984,31987],{"style":482},[15,31985],{"className":31986,"style":358},[357],[15,31988,31990],{"className":31989},[362,363,364,365],[15,31991,1070],{"className":31992},[61,122,365],[15,31994,497],{"className":31995},[496],[15,31997,31999],{"className":31998},[346],[15,32000,32002],{"className":32001,"style":504},[350],[15,32003],{}," 的指数",[15,32006,32008,32030],{"className":32007,"translate":19},[18],[15,32009,32011],{"className":32010},[23],[25,32012,32013],{"xmlns":27},[29,32014,32015,32027],{},[32,32016,32017,32023,32025],{},[266,32018,32019,32021],{},[35,32020,3106],{},[35,32022,1070],{},[81,32024,2490],{},[89,32026,91],{},[40,32028,32029],{"encoding":42},"\\alpha_i \u003C 1",[15,32031,32033,32088],{"className":32032,"ariaHidden":48},[47],[15,32034,32036,32039,32079,32082,32085],{"className":32035},[52],[15,32037],{"className":32038,"style":1753},[56],[15,32040,32042,32045],{"className":32041},[61],[15,32043,3106],{"className":32044,"style":3191},[61,122],[15,32046,32048],{"className":32047},[338],[15,32049,32051,32071],{"className":32050},[342,472],[15,32052,32054,32068],{"className":32053},[346],[15,32055,32057],{"className":32056,"style":1364},[350],[15,32058,32059,32062],{"style":3206},[15,32060],{"className":32061,"style":358},[357],[15,32063,32065],{"className":32064},[362,363,364,365],[15,32066,1070],{"className":32067},[61,122,365],[15,32069,497],{"className":32070},[496],[15,32072,32074],{"className":32073},[346],[15,32075,32077],{"className":32076,"style":504},[350],[15,32078],{},[15,32080],{"className":32081,"style":128},[127],[15,32083,2490],{"className":32084},[132],[15,32086],{"className":32087,"style":128},[127],[15,32089,32091,32094],{"className":32090},[52],[15,32092],{"className":32093,"style":859},[56],[15,32095,91],{"className":32096},[61]," 意味着密度在 ",[15,32099,32101,32123],{"className":32100,"translate":19},[18],[15,32102,32104],{"className":32103},[23],[25,32105,32106],{"xmlns":27},[29,32107,32108,32120],{},[32,32109,32110,32116,32118],{},[266,32111,32112,32114],{},[35,32113,1029],{},[35,32115,1070],{},[81,32117,20704],{},[89,32119,278],{},[40,32121,32122],{"encoding":42},"x_i \\to 0",[15,32124,32126,32181],{"className":32125,"ariaHidden":48},[47],[15,32127,32129,32132,32172,32175,32178],{"className":32128},[52],[15,32130],{"className":32131,"style":2511},[56],[15,32133,32135,32138],{"className":32134},[61],[15,32136,1029],{"className":32137},[61,122],[15,32139,32141],{"className":32140},[338],[15,32142,32144,32164],{"className":32143},[342,472],[15,32145,32147,32161],{"className":32146},[346],[15,32148,32150],{"className":32149,"style":1364},[350],[15,32151,32152,32155],{"style":482},[15,32153],{"className":32154,"style":358},[357],[15,32156,32158],{"className":32157},[362,363,364,365],[15,32159,1070],{"className":32160},[61,122,365],[15,32162,497],{"className":32163},[496],[15,32165,32167],{"className":32166},[346],[15,32168,32170],{"className":32169,"style":504},[350],[15,32171],{},[15,32173],{"className":32174,"style":128},[127],[15,32176,20704],{"className":32177},[132],[15,32179],{"className":32180,"style":128},[127],[15,32182,32184,32187],{"className":32183},[52],[15,32185],{"className":32186,"style":859},[56],[15,32188,278],{"className":32189},[61]," 处发散。",[11,32192,32193,32194,32197,32198,32259],{},"对于",[661,32195,32196],{},"对称","情形 ",[15,32199,32201,32220],{"className":32200,"translate":19},[18],[15,32202,32204],{"className":32203},[23],[25,32205,32206],{"xmlns":27},[29,32207,32208,32218],{},[32,32209,32210,32212,32214,32216],{},[35,32211,3106],{"mathvariant":3105},[81,32213,244],{},[35,32215,3106],{},[89,32217,91],{"mathvariant":251},[40,32219,23985],{"encoding":42},[15,32221,32223,32247],{"className":32222,"ariaHidden":48},[47],[15,32224,32226,32229,32238,32241,32244],{"className":32225},[52],[15,32227],{"className":32228,"style":2708},[56],[15,32230,32232],{"className":32231},[61],[15,32233,32235],{"className":32234},[61],[15,32236,3106],{"className":32237},[61,3166],[15,32239],{"className":32240,"style":128},[127],[15,32242,244],{"className":32243},[132],[15,32245],{"className":32246,"style":128},[127],[15,32248,32250,32253,32256],{"className":32249},[52],[15,32251],{"className":32252,"style":859},[56],[15,32254,3106],{"className":32255,"style":3191},[61,122],[15,32257,91],{"className":32258},[61,411],"，这三重语义给出一条完整的形态谱：",[32261,32262,32263,32282],"table",{},[32264,32265,32266],"thead",{},[21311,32267,32268,32273,32276,32279],{},[32269,32270,32272],"th",{"align":32271},"left","参数区间",[32269,32274,32275],{"align":32271},"分布的形态",[32269,32277,32278],{"align":32271},"采样的典型特征",[32269,32280,32281],{"align":32271},"典型用途",[32283,32284,32285,32459,32522],"tbody",{},[21311,32286,32287,32341,32425,32456],{},[32288,32289,32290],"td",{"align":32271},[15,32291,32293,32311],{"className":32292,"translate":19},[18],[15,32294,32296],{"className":32295},[23],[25,32297,32298],{"xmlns":27},[29,32299,32300,32308],{},[32,32301,32302,32304,32306],{},[35,32303,3106],{},[81,32305,20704],{},[89,32307,278],{},[40,32309,32310],{"encoding":42},"\\alpha \\to 0",[15,32312,32314,32332],{"className":32313,"ariaHidden":48},[47],[15,32315,32317,32320,32323,32326,32329],{"className":32316},[52],[15,32318],{"className":32319,"style":6032},[56],[15,32321,3106],{"className":32322,"style":3191},[61,122],[15,32324],{"className":32325,"style":128},[127],[15,32327,20704],{"className":32328},[132],[15,32330],{"className":32331,"style":128},[127],[15,32333,32335,32338],{"className":32334},[52],[15,32336],{"className":32337,"style":859},[56],[15,32339,278],{"className":32340},[61],[32288,32342,32343,32344,32424],{"align":32271},"质量集中于顶点 ",[15,32345,32347,32369],{"className":32346,"translate":19},[18],[15,32348,32350],{"className":32349},[23],[25,32351,32352],{"xmlns":27},[29,32353,32354,32366],{},[32,32355,32356,32358,32364],{},[81,32357,87],{"stretchy":86},[266,32359,32360,32362],{},[35,32361,1982],{"mathvariant":251},[35,32363,1070],{},[81,32365,105],{"stretchy":86},[40,32367,32368],{"encoding":42},"\\{\\mathbf{e}_i\\}",[15,32370,32372],{"className":32371,"ariaHidden":48},[47],[15,32373,32375,32378,32381,32421],{"className":32374},[52],[15,32376],{"className":32377,"style":142},[56],[15,32379,87],{"className":32380},[146],[15,32382,32384,32387],{"className":32383},[61],[15,32385,1982],{"className":32386},[61,411],[15,32388,32390],{"className":32389},[338],[15,32391,32393,32413],{"className":32392},[342,472],[15,32394,32396,32410],{"className":32395},[346],[15,32397,32399],{"className":32398,"style":1364},[350],[15,32400,32401,32404],{"style":482},[15,32402],{"className":32403,"style":358},[357],[15,32405,32407],{"className":32406},[362,363,364,365],[15,32408,1070],{"className":32409},[61,122,365],[15,32411,497],{"className":32412},[496],[15,32414,32416],{"className":32415},[346],[15,32417,32419],{"className":32418,"style":504},[350],[15,32420],{},[15,32422,105],{"className":32423},[178]," 邻域",[32288,32426,32427,32428],{"align":32271},"极端稀疏，单个分量接近 ",[15,32429,32431,32444],{"className":32430,"translate":19},[18],[15,32432,32434],{"className":32433},[23],[25,32435,32436],{"xmlns":27},[29,32437,32438,32442],{},[32,32439,32440],{},[89,32441,91],{},[40,32443,91],{"encoding":42},[15,32445,32447],{"className":32446,"ariaHidden":48},[47],[15,32448,32450,32453],{"className":32449},[52],[15,32451],{"className":32452,"style":859},[56],[15,32454,91],{"className":32455},[61],[32288,32457,32458],{"align":32271},"LDA 文档—主题混合（文档只涉少数主题）",[21311,32460,32461,32513,32516,32519],{},[32288,32462,32463],{"align":32271},[15,32464,32466,32483],{"className":32465,"translate":19},[18],[15,32467,32469],{"className":32468},[23],[25,32470,32471],{"xmlns":27},[29,32472,32473,32481],{},[32,32474,32475,32477,32479],{},[35,32476,3106],{},[81,32478,244],{},[89,32480,91],{},[40,32482,24074],{"encoding":42},[15,32484,32486,32504],{"className":32485,"ariaHidden":48},[47],[15,32487,32489,32492,32495,32498,32501],{"className":32488},[52],[15,32490],{"className":32491,"style":6032},[56],[15,32493,3106],{"className":32494,"style":3191},[61,122],[15,32496],{"className":32497,"style":128},[127],[15,32499,244],{"className":32500},[132],[15,32502],{"className":32503,"style":128},[127],[15,32505,32507,32510],{"className":32506},[52],[15,32508],{"className":32509,"style":859},[56],[15,32511,91],{"className":32512},[61],[32288,32514,32515],{"align":32271},"单纯形上的均匀分布",[32288,32517,32518],{"align":32271},"各分量量级相当，无偏好",[32288,32520,32521],{"align":32271},"无信息先验",[21311,32523,32524,32577,32649,32689],{},[32288,32525,32526],{"align":32271},[15,32527,32529,32547],{"className":32528,"translate":19},[18],[15,32530,32532],{"className":32531},[23],[25,32533,32534],{"xmlns":27},[29,32535,32536,32544],{},[32,32537,32538,32540,32542],{},[35,32539,3106],{},[81,32541,1590],{},[89,32543,91],{},[40,32545,32546],{"encoding":42},"\\alpha > 1",[15,32548,32550,32568],{"className":32549,"ariaHidden":48},[47],[15,32551,32553,32556,32559,32562,32565],{"className":32552},[52],[15,32554],{"className":32555,"style":4656},[56],[15,32557,3106],{"className":32558,"style":3191},[61,122],[15,32560],{"className":32561,"style":128},[127],[15,32563,1590],{"className":32564},[132],[15,32566],{"className":32567,"style":128},[127],[15,32569,32571,32574],{"className":32570},[52],[15,32572],{"className":32573,"style":859},[56],[15,32575,91],{"className":32576},[61],[32288,32578,32579,32580,32619,32620,32648],{"align":32271},"集中于中心 ",[15,32581,32583,32601],{"className":32582,"translate":19},[18],[15,32584,32586],{"className":32585},[23],[25,32587,32588],{"xmlns":27},[29,32589,32590,32598],{},[32,32591,32592,32594,32596],{},[89,32593,91],{"mathvariant":251},[35,32595,3020],{"mathvariant":231},[35,32597,102],{},[40,32599,32600],{"encoding":42},"\\mathbf{1}\u002FK",[15,32602,32604],{"className":32603,"ariaHidden":48},[47],[15,32605,32607,32610,32613,32616],{"className":32606},[52],[15,32608],{"className":32609,"style":142},[56],[15,32611,91],{"className":32612},[61,411],[15,32614,3020],{"className":32615},[61],[15,32617,102],{"className":32618,"style":174},[61,122],"，随 ",[15,32621,32623,32636],{"className":32622,"translate":19},[18],[15,32624,32626],{"className":32625},[23],[25,32627,32628],{"xmlns":27},[29,32629,32630,32634],{},[32,32631,32632],{},[35,32633,3106],{},[40,32635,31682],{"encoding":42},[15,32637,32639],{"className":32638,"ariaHidden":48},[47],[15,32640,32642,32645],{"className":32641},[52],[15,32643],{"className":32644,"style":6032},[56],[15,32646,3106],{"className":32647,"style":3191},[61,122]," 增大而收缩",[32288,32650,32651,32652,32688],{"align":32271},"各分量接近 ",[15,32653,32655,32673],{"className":32654,"translate":19},[18],[15,32656,32658],{"className":32657},[23],[25,32659,32660],{"xmlns":27},[29,32661,32662,32670],{},[32,32663,32664,32666,32668],{},[89,32665,91],{},[35,32667,3020],{"mathvariant":231},[35,32669,102],{},[40,32671,32672],{"encoding":42},"1\u002FK",[15,32674,32676],{"className":32675,"ariaHidden":48},[47],[15,32677,32679,32682,32685],{"className":32678},[52],[15,32680],{"className":32681,"style":142},[56],[15,32683,3049],{"className":32684},[61],[15,32686,102],{"className":32687,"style":174},[61,122]," 的平滑向量",[32288,32690,32691],{"align":32271},"强正则化、收缩估计",[11,32693,32694,32695,32764,32765,32857,32858,20777,32949,33064,33065,33286],{},"此处必须澄清一个常见但过强的说法：把 ",[15,32696,32698,32715],{"className":32697,"translate":19},[18],[15,32699,32701],{"className":32700},[23],[25,32702,32703],{"xmlns":27},[29,32704,32705,32713],{},[32,32706,32707],{},[266,32708,32709,32711],{},[35,32710,3106],{},[35,32712,1070],{},[40,32714,9727],{"encoding":42},[15,32716,32718],{"className":32717,"ariaHidden":48},[47],[15,32719,32721,32724],{"className":32720},[52],[15,32722],{"className":32723,"style":2511},[56],[15,32725,32727,32730],{"className":32726},[61],[15,32728,3106],{"className":32729,"style":3191},[61,122],[15,32731,32733],{"className":32732},[338],[15,32734,32736,32756],{"className":32735},[342,472],[15,32737,32739,32753],{"className":32738},[346],[15,32740,32742],{"className":32741,"style":1364},[350],[15,32743,32744,32747],{"style":3206},[15,32745],{"className":32746,"style":358},[357],[15,32748,32750],{"className":32749},[362,363,364,365],[15,32751,1070],{"className":32752},[61,122,365],[15,32754,497],{"className":32755},[496],[15,32757,32759],{"className":32758},[346],[15,32760,32762],{"className":32761,"style":504},[350],[15,32763],{}," 直接称为「观测到的次数」只在 ",[15,32766,32768,32790],{"className":32767,"translate":19},[18],[15,32769,32771],{"className":32770},[23],[25,32772,32773],{"xmlns":27},[29,32774,32775,32787],{},[32,32776,32777,32783,32785],{},[266,32778,32779,32781],{},[35,32780,3106],{},[35,32782,1070],{},[81,32784,275],{},[89,32786,91],{},[40,32788,32789],{"encoding":42},"\\alpha_i \\ge 1",[15,32791,32793,32848],{"className":32792,"ariaHidden":48},[47],[15,32794,32796,32799,32839,32842,32845],{"className":32795},[52],[15,32797],{"className":32798,"style":22492},[56],[15,32800,32802,32805],{"className":32801},[61],[15,32803,3106],{"className":32804,"style":3191},[61,122],[15,32806,32808],{"className":32807},[338],[15,32809,32811,32831],{"className":32810},[342,472],[15,32812,32814,32828],{"className":32813},[346],[15,32815,32817],{"className":32816,"style":1364},[350],[15,32818,32819,32822],{"style":3206},[15,32820],{"className":32821,"style":358},[357],[15,32823,32825],{"className":32824},[362,363,364,365],[15,32826,1070],{"className":32827},[61,122,365],[15,32829,497],{"className":32830},[496],[15,32832,32834],{"className":32833},[346],[15,32835,32837],{"className":32836,"style":504},[350],[15,32838],{},[15,32840],{"className":32841,"style":128},[127],[15,32843,275],{"className":32844},[132],[15,32846],{"className":32847,"style":128},[127],[15,32849,32851,32854],{"className":32850},[52],[15,32852],{"className":32853,"style":859},[56],[15,32855,91],{"className":32856},[61]," 时在字面上成立。当 ",[15,32859,32861,32882],{"className":32860,"translate":19},[18],[15,32862,32864],{"className":32863},[23],[25,32865,32866],{"xmlns":27},[29,32867,32868,32880],{},[32,32869,32870,32876,32878],{},[266,32871,32872,32874],{},[35,32873,3106],{},[35,32875,1070],{},[81,32877,2490],{},[89,32879,91],{},[40,32881,32029],{"encoding":42},[15,32883,32885,32940],{"className":32884,"ariaHidden":48},[47],[15,32886,32888,32891,32931,32934,32937],{"className":32887},[52],[15,32889],{"className":32890,"style":1753},[56],[15,32892,32894,32897],{"className":32893},[61],[15,32895,3106],{"className":32896,"style":3191},[61,122],[15,32898,32900],{"className":32899},[338],[15,32901,32903,32923],{"className":32902},[342,472],[15,32904,32906,32920],{"className":32905},[346],[15,32907,32909],{"className":32908,"style":1364},[350],[15,32910,32911,32914],{"style":3206},[15,32912],{"className":32913,"style":358},[357],[15,32915,32917],{"className":32916},[362,363,364,365],[15,32918,1070],{"className":32919},[61,122,365],[15,32921,497],{"className":32922},[496],[15,32924,32926],{"className":32925},[346],[15,32927,32929],{"className":32928,"style":504},[350],[15,32930],{},[15,32932],{"className":32933,"style":128},[127],[15,32935,2490],{"className":32936},[132],[15,32938],{"className":32939,"style":128},[127],[15,32941,32943,32946],{"className":32942},[52],[15,32944],{"className":32945,"style":859},[56],[15,32947,91],{"className":32948},[61],[15,32950,32952,32978],{"className":32951,"translate":19},[18],[15,32953,32955],{"className":32954},[23],[25,32956,32957],{"xmlns":27},[29,32958,32959,32975],{},[32,32960,32961,32967,32969,32971,32973],{},[266,32962,32963,32965],{},[35,32964,3106],{},[35,32966,1070],{},[81,32968,239],{},[89,32970,91],{},[81,32972,2490],{},[89,32974,278],{},[40,32976,32977],{"encoding":42},"\\alpha_i - 1 \u003C 0",[15,32979,32981,33036,33055],{"className":32980,"ariaHidden":48},[47],[15,32982,32984,32987,33027,33030,33033],{"className":32983},[52],[15,32985],{"className":32986,"style":12617},[56],[15,32988,32990,32993],{"className":32989},[61],[15,32991,3106],{"className":32992,"style":3191},[61,122],[15,32994,32996],{"className":32995},[338],[15,32997,32999,33019],{"className":32998},[342,472],[15,33000,33002,33016],{"className":33001},[346],[15,33003,33005],{"className":33004,"style":1364},[350],[15,33006,33007,33010],{"style":3206},[15,33008],{"className":33009,"style":358},[357],[15,33011,33013],{"className":33012},[362,363,364,365],[15,33014,1070],{"className":33015},[61,122,365],[15,33017,497],{"className":33018},[496],[15,33020,33022],{"className":33021},[346],[15,33023,33025],{"className":33024,"style":504},[350],[15,33026],{},[15,33028],{"className":33029,"style":772},[127],[15,33031,239],{"className":33032},[375],[15,33034],{"className":33035,"style":772},[127],[15,33037,33039,33043,33046,33049,33052],{"className":33038},[52],[15,33040],{"className":33041,"style":33042},[56],"height:0.6835em;vertical-align:-0.0391em;",[15,33044,91],{"className":33045},[61],[15,33047],{"className":33048,"style":128},[127],[15,33050,2490],{"className":33051},[132],[15,33053],{"className":33054,"style":128},[127],[15,33056,33058,33061],{"className":33057},[52],[15,33059],{"className":33060,"style":859},[56],[15,33062,278],{"className":33063},[61],"，隐含的「先验计数」为负，这在计数语义下不可解释，却对应于密度在边界处的发散——一个在稀疏建模中必不可少的形态。准确的说法是：",[661,33066,33067,33101,33102,33193,33194,33285],{},[15,33068,33070,33083],{"className":33069,"translate":19},[18],[15,33071,33073],{"className":33072},[23],[25,33074,33075],{"xmlns":27},[29,33076,33077,33081],{},[32,33078,33079],{},[35,33080,3106],{"mathvariant":3105},[40,33082,4147],{"encoding":42},[15,33084,33086],{"className":33085,"ariaHidden":48},[47],[15,33087,33089,33092],{"className":33088},[52],[15,33090],{"className":33091,"style":2708},[56],[15,33093,33095],{"className":33094},[61],[15,33096,33098],{"className":33097},[61],[15,33099,3106],{"className":33100},[61,3166]," 是伪计数概念的连续化，其在 ",[15,33103,33105,33126],{"className":33104,"translate":19},[18],[15,33106,33108],{"className":33107},[23],[25,33109,33110],{"xmlns":27},[29,33111,33112,33124],{},[32,33113,33114,33120,33122],{},[266,33115,33116,33118],{},[35,33117,3106],{},[35,33119,1070],{},[81,33121,275],{},[89,33123,91],{},[40,33125,32789],{"encoding":42},[15,33127,33129,33184],{"className":33128,"ariaHidden":48},[47],[15,33130,33132,33135,33175,33178,33181],{"className":33131},[52],[15,33133],{"className":33134,"style":22492},[56],[15,33136,33138,33141],{"className":33137},[61],[15,33139,3106],{"className":33140,"style":3191},[61,122],[15,33142,33144],{"className":33143},[338],[15,33145,33147,33167],{"className":33146},[342,472],[15,33148,33150,33164],{"className":33149},[346],[15,33151,33153],{"className":33152,"style":1364},[350],[15,33154,33155,33158],{"style":3206},[15,33156],{"className":33157,"style":358},[357],[15,33159,33161],{"className":33160},[362,363,364,365],[15,33162,1070],{"className":33163},[61,122,365],[15,33165,497],{"className":33166},[496],[15,33168,33170],{"className":33169},[346],[15,33171,33173],{"className":33172,"style":504},[350],[15,33174],{},[15,33176],{"className":33177,"style":128},[127],[15,33179,275],{"className":33180},[132],[15,33182],{"className":33183,"style":128},[127],[15,33185,33187,33190],{"className":33186},[52],[15,33188],{"className":33189,"style":859},[56],[15,33191,91],{"className":33192},[61]," 区域退化回字面意义上的计数解释，而在 ",[15,33195,33197,33218],{"className":33196,"translate":19},[18],[15,33198,33200],{"className":33199},[23],[25,33201,33202],{"xmlns":27},[29,33203,33204,33216],{},[32,33205,33206,33212,33214],{},[266,33207,33208,33210],{},[35,33209,3106],{},[35,33211,1070],{},[81,33213,2490],{},[89,33215,91],{},[40,33217,32029],{"encoding":42},[15,33219,33221,33276],{"className":33220,"ariaHidden":48},[47],[15,33222,33224,33227,33267,33270,33273],{"className":33223},[52],[15,33225],{"className":33226,"style":1753},[56],[15,33228,33230,33233],{"className":33229},[61],[15,33231,3106],{"className":33232,"style":3191},[61,122],[15,33234,33236],{"className":33235},[338],[15,33237,33239,33259],{"className":33238},[342,472],[15,33240,33242,33256],{"className":33241},[346],[15,33243,33245],{"className":33244,"style":1364},[350],[15,33246,33247,33250],{"style":3206},[15,33248],{"className":33249,"style":358},[357],[15,33251,33253],{"className":33252},[362,363,364,365],[15,33254,1070],{"className":33255},[61,122,365],[15,33257,497],{"className":33258},[496],[15,33260,33262],{"className":33261},[346],[15,33263,33265],{"className":33264,"style":504},[350],[15,33266],{},[15,33268],{"className":33269,"style":128},[127],[15,33271,2490],{"className":33272},[132],[15,33274],{"className":33275,"style":128},[127],[15,33277,33279,33282],{"className":33278},[52],[15,33280],{"className":33281,"style":859},[56],[15,33283,91],{"className":33284},[61]," 区域承担的是稀疏性先验的角色。"," 这一区分在 3.4 节将直接决定 EDL 中「证据」一词的适用边界。",[872,33288,33290],{"id":33289},"三主观逻辑到证据深度学习","三、主观逻辑到证据深度学习",[877,33292,33294],{"id":33293},"_31-主观逻辑的-opinion-表示","3.1 主观逻辑的 opinion 表示",[11,33296,33297,33298,33301],{},"证据深度学习的直接理论前身是 Josang 的",[661,33299,33300],{},"主观逻辑（subjective logic）","，它为「对一个命题的相信程度」建立了一套带不确定性维度的代数。",[882,33303,33304,33496,34083],{},[11,33305,33306,33309,33310,33403,33404,33495],{},[661,33307,33308],{},"定义 3.1（opinion）"," 在有限域 ",[15,33311,33313,33344],{"className":33312,"translate":19},[18],[15,33314,33316],{"className":33315},[23],[25,33317,33318],{"xmlns":27},[29,33319,33320,33341],{},[32,33321,33322,33325,33327,33329,33331,33333,33335,33337,33339],{},[35,33323,33324],{"mathvariant":37},"Y",[81,33326,244],{},[81,33328,87],{"stretchy":86},[89,33330,91],{},[81,33332,94],{"separator":48},[81,33334,97],{},[81,33336,94],{"separator":48},[35,33338,102],{},[81,33340,105],{"stretchy":86},[40,33342,33343],{"encoding":42},"\\mathcal{Y} = \\{1, \\dots, K\\}",[15,33345,33347,33367],{"className":33346,"ariaHidden":48},[47],[15,33348,33350,33354,33358,33361,33364],{"className":33349},[52],[15,33351],{"className":33352,"style":33353},[56],"height:0.7805em;vertical-align:-0.0972em;",[15,33355,33324],{"className":33356,"style":33357},[61,62],"margin-right:0.0822em;",[15,33359],{"className":33360,"style":128},[127],[15,33362,244],{"className":33363},[132],[15,33365],{"className":33366,"style":128},[127],[15,33368,33370,33373,33376,33379,33382,33385,33388,33391,33394,33397,33400],{"className":33369},[52],[15,33371],{"className":33372,"style":142},[56],[15,33374,87],{"className":33375},[146],[15,33377,91],{"className":33378},[61],[15,33380,94],{"className":33381},[153],[15,33383],{"className":33384,"style":157},[127],[15,33386,97],{"className":33387},[161],[15,33389],{"className":33390,"style":157},[127],[15,33392,94],{"className":33393},[153],[15,33395],{"className":33396,"style":157},[127],[15,33398,102],{"className":33399,"style":174},[61,122],[15,33401,105],{"className":33402},[178]," 上的一个 opinion 是三元组 ",[15,33405,33407,33441],{"className":33406,"translate":19},[18],[15,33408,33410],{"className":33409},[23],[25,33411,33412],{"xmlns":27},[29,33413,33414,33438],{},[32,33415,33416,33419,33421,33423,33426,33428,33431,33433,33436],{},[35,33417,33418],{},"ω",[81,33420,244],{},[81,33422,741],{"stretchy":86},[35,33424,33425],{"mathvariant":251},"b",[81,33427,94],{"separator":48},[35,33429,33430],{},"u",[81,33432,94],{"separator":48},[35,33434,33435],{"mathvariant":251},"a",[81,33437,750],{"stretchy":86},[40,33439,33440],{"encoding":42},"\\omega = (\\mathbf{b}, u, \\mathbf{a})",[15,33442,33444,33462],{"className":33443,"ariaHidden":48},[47],[15,33445,33447,33450,33453,33456,33459],{"className":33446},[52],[15,33448],{"className":33449,"style":6032},[56],[15,33451,33418],{"className":33452,"style":123},[61,122],[15,33454],{"className":33455,"style":128},[127],[15,33457,244],{"className":33458},[132],[15,33460],{"className":33461,"style":128},[127],[15,33463,33465,33468,33471,33474,33477,33480,33483,33486,33489,33492],{"className":33464},[52],[15,33466],{"className":33467,"style":142},[56],[15,33469,741],{"className":33470},[146],[15,33472,33425],{"className":33473},[61,411],[15,33475,94],{"className":33476},[153],[15,33478],{"className":33479,"style":157},[127],[15,33481,33430],{"className":33482},[61,122],[15,33484,94],{"className":33485},[153],[15,33487],{"className":33488,"style":157},[127],[15,33490,33435],{"className":33491},[61,411],[15,33493,750],{"className":33494},[178],"，其中",[33497,33498,33499,33753,33832],"ul",{},[31550,33500,33501,33749,33750,867],{},[15,33502,33504,33558],{"className":33503,"translate":19},[18],[15,33505,33507],{"className":33506},[23],[25,33508,33509],{"xmlns":27},[29,33510,33511,33555],{},[32,33512,33513,33515,33517,33519,33525,33527,33529,33531,33537,33539,33541,33543,33545,33547,33549],{},[35,33514,33425],{"mathvariant":251},[81,33516,244],{},[81,33518,741],{"stretchy":86},[266,33520,33521,33523],{},[35,33522,33425],{},[89,33524,91],{},[81,33526,94],{"separator":48},[81,33528,97],{},[81,33530,94],{"separator":48},[266,33532,33533,33535],{},[35,33534,33425],{},[35,33536,102],{},[81,33538,750],{"stretchy":86},[81,33540,83],{},[81,33542,14821],{"stretchy":86},[89,33544,278],{},[81,33546,94],{"separator":48},[89,33548,91],{},[227,33550,33551,33553],{},[81,33552,14830],{"stretchy":86},[35,33554,102],{},[40,33556,33557],{"encoding":42},"\\mathbf{b} = (b_1, \\dots, b_K) \\in [0,1]^K",[15,33559,33561,33579,33698],{"className":33560,"ariaHidden":48},[47],[15,33562,33564,33567,33570,33573,33576],{"className":33563},[52],[15,33565],{"className":33566,"style":10452},[56],[15,33568,33425],{"className":33569},[61,411],[15,33571],{"className":33572,"style":128},[127],[15,33574,244],{"className":33575},[132],[15,33577],{"className":33578,"style":128},[127],[15,33580,33582,33585,33588,33628,33631,33634,33637,33640,33643,33646,33686,33689,33692,33695],{"className":33581},[52],[15,33583],{"className":33584,"style":142},[56],[15,33586,741],{"className":33587},[146],[15,33589,33591,33594],{"className":33590},[61],[15,33592,33425],{"className":33593},[61,122],[15,33595,33597],{"className":33596},[338],[15,33598,33600,33620],{"className":33599},[342,472],[15,33601,33603,33617],{"className":33602},[346],[15,33604,33606],{"className":33605,"style":1214},[350],[15,33607,33608,33611],{"style":482},[15,33609],{"className":33610,"style":358},[357],[15,33612,33614],{"className":33613},[362,363,364,365],[15,33615,91],{"className":33616},[61,365],[15,33618,497],{"className":33619},[496],[15,33621,33623],{"className":33622},[346],[15,33624,33626],{"className":33625,"style":504},[350],[15,33627],{},[15,33629,94],{"className":33630},[153],[15,33632],{"className":33633,"style":157},[127],[15,33635,97],{"className":33636},[161],[15,33638],{"className":33639,"style":157},[127],[15,33641,94],{"className":33642},[153],[15,33644],{"className":33645,"style":157},[127],[15,33647,33649,33652],{"className":33648},[61],[15,33650,33425],{"className":33651},[61,122],[15,33653,33655],{"className":33654},[338],[15,33656,33658,33678],{"className":33657},[342,472],[15,33659,33661,33675],{"className":33660},[346],[15,33662,33664],{"className":33663,"style":1273},[350],[15,33665,33666,33669],{"style":482},[15,33667],{"className":33668,"style":358},[357],[15,33670,33672],{"className":33671},[362,363,364,365],[15,33673,102],{"className":33674,"style":174},[61,122,365],[15,33676,497],{"className":33677},[496],[15,33679,33681],{"className":33680},[346],[15,33682,33684],{"className":33683,"style":504},[350],[15,33685],{},[15,33687,750],{"className":33688},[178],[15,33690],{"className":33691,"style":128},[127],[15,33693,83],{"className":33694},[132],[15,33696],{"className":33697,"style":128},[127],[15,33699,33701,33705,33708,33711,33714,33717,33720],{"className":33700},[52],[15,33702],{"className":33703,"style":33704},[56],"height:1.0913em;vertical-align:-0.25em;",[15,33706,14821],{"className":33707},[146],[15,33709,278],{"className":33710},[61],[15,33712,94],{"className":33713},[153],[15,33715],{"className":33716,"style":157},[127],[15,33718,91],{"className":33719},[61],[15,33721,33723,33726],{"className":33722},[178],[15,33724,14830],{"className":33725},[178],[15,33727,33729],{"className":33728},[338],[15,33730,33732],{"className":33731},[342],[15,33733,33735],{"className":33734},[346],[15,33736,33738],{"className":33737,"style":695},[350],[15,33739,33740,33743],{"style":716},[15,33741],{"className":33742,"style":358},[357],[15,33744,33746],{"className":33745},[362,363,364,365],[15,33747,102],{"className":33748,"style":174},[61,122,365]," 为",[661,33751,33752],{},"信念质量（belief mass）",[31550,33754,33755,33749,33829,867],{},[15,33756,33758,33784],{"className":33757,"translate":19},[18],[15,33759,33761],{"className":33760},[23],[25,33762,33763],{"xmlns":27},[29,33764,33765,33781],{},[32,33766,33767,33769,33771,33773,33775,33777,33779],{},[35,33768,33430],{},[81,33770,83],{},[81,33772,14821],{"stretchy":86},[89,33774,278],{},[81,33776,94],{"separator":48},[89,33778,91],{},[81,33780,14830],{"stretchy":86},[40,33782,33783],{"encoding":42},"u \\in [0,1]",[15,33785,33787,33805],{"className":33786,"ariaHidden":48},[47],[15,33788,33790,33793,33796,33799,33802],{"className":33789},[52],[15,33791],{"className":33792,"style":4656},[56],[15,33794,33430],{"className":33795},[61,122],[15,33797],{"className":33798,"style":128},[127],[15,33800,83],{"className":33801},[132],[15,33803],{"className":33804,"style":128},[127],[15,33806,33808,33811,33814,33817,33820,33823,33826],{"className":33807},[52],[15,33809],{"className":33810,"style":142},[56],[15,33812,14821],{"className":33813},[146],[15,33815,278],{"className":33816},[61],[15,33818,94],{"className":33819},[153],[15,33821],{"className":33822,"style":157},[127],[15,33824,91],{"className":33825},[61],[15,33827,14830],{"className":33828},[178],[661,33830,33831],{},"不确定性质量（uncertainty mass）",[31550,33833,33834,943,33938,34079,34080,867],{},[15,33835,33837,33867],{"className":33836,"translate":19},[18],[15,33838,33840],{"className":33839},[23],[25,33841,33842],{"xmlns":27},[29,33843,33844,33864],{},[32,33845,33846,33848,33850,33852,33854,33856,33858],{},[35,33847,33435],{"mathvariant":251},[81,33849,83],{},[81,33851,14821],{"stretchy":86},[89,33853,278],{},[81,33855,94],{"separator":48},[89,33857,91],{},[227,33859,33860,33862],{},[81,33861,14830],{"stretchy":86},[35,33863,102],{},[40,33865,33866],{"encoding":42},"\\mathbf{a} \\in [0,1]^K",[15,33868,33870,33888],{"className":33869,"ariaHidden":48},[47],[15,33871,33873,33876,33879,33882,33885],{"className":33872},[52],[15,33874],{"className":33875,"style":4656},[56],[15,33877,33435],{"className":33878},[61,411],[15,33880],{"className":33881,"style":128},[127],[15,33883,83],{"className":33884},[132],[15,33886],{"className":33887,"style":128},[127],[15,33889,33891,33894,33897,33900,33903,33906,33909],{"className":33890},[52],[15,33892],{"className":33893,"style":33704},[56],[15,33895,14821],{"className":33896},[146],[15,33898,278],{"className":33899},[61],[15,33901,94],{"className":33902},[153],[15,33904],{"className":33905,"style":157},[127],[15,33907,91],{"className":33908},[61],[15,33910,33912,33915],{"className":33911},[178],[15,33913,14830],{"className":33914},[178],[15,33916,33918],{"className":33917},[338],[15,33919,33921],{"className":33920},[342],[15,33922,33924],{"className":33923},[346],[15,33925,33927],{"className":33926,"style":695},[350],[15,33928,33929,33932],{"style":716},[15,33930],{"className":33931,"style":358},[357],[15,33933,33935],{"className":33934},[362,363,364,365],[15,33936,102],{"className":33937,"style":174},[61,122,365],[15,33939,33941,33969],{"className":33940,"translate":19},[18],[15,33942,33944],{"className":33943},[23],[25,33945,33946],{"xmlns":27},[29,33947,33948,33966],{},[32,33949,33950,33956,33962,33964],{},[266,33951,33952,33954],{},[81,33953,293],{},[35,33955,272],{},[266,33957,33958,33960],{},[35,33959,33435],{},[35,33961,272],{},[81,33963,244],{},[89,33965,91],{},[40,33967,33968],{"encoding":42},"\\sum_k a_k = 1",[15,33970,33972,34070],{"className":33971,"ariaHidden":48},[47],[15,33973,33975,33978,34018,34021,34061,34064,34067],{"className":33974},[52],[15,33976],{"className":33977,"style":6051},[56],[15,33979,33981,33984],{"className":33980},[531],[15,33982,293],{"className":33983,"style":537},[531,535,536],[15,33985,33987],{"className":33986},[338],[15,33988,33990,34010],{"className":33989},[342,472],[15,33991,33993,34007],{"className":33992},[346],[15,33994,33996],{"className":33995,"style":29989},[350],[15,33997,33998,34001],{"style":553},[15,33999],{"className":34000,"style":358},[357],[15,34002,34004],{"className":34003},[362,363,364,365],[15,34005,272],{"className":34006,"style":492},[61,122,365],[15,34008,497],{"className":34009},[496],[15,34011,34013],{"className":34012},[346],[15,34014,34016],{"className":34015,"style":593},[350],[15,34017],{},[15,34019],{"className":34020,"style":157},[127],[15,34022,34024,34027],{"className":34023},[61],[15,34025,33435],{"className":34026},[61,122],[15,34028,34030],{"className":34029},[338],[15,34031,34033,34053],{"className":34032},[342,472],[15,34034,34036,34050],{"className":34035},[346],[15,34037,34039],{"className":34038,"style":479},[350],[15,34040,34041,34044],{"style":482},[15,34042],{"className":34043,"style":358},[357],[15,34045,34047],{"className":34046},[362,363,364,365],[15,34048,272],{"className":34049,"style":492},[61,122,365],[15,34051,497],{"className":34052},[496],[15,34054,34056],{"className":34055},[346],[15,34057,34059],{"className":34058,"style":504},[350],[15,34060],{},[15,34062],{"className":34063,"style":128},[127],[15,34065,244],{"className":34066},[132],[15,34068],{"className":34069,"style":128},[127],[15,34071,34073,34076],{"className":34072},[52],[15,34074],{"className":34075,"style":859},[56],[15,34077,91],{"className":34078},[61],"，为",[661,34081,34082],{},"基准率（base rate）",[11,34084,34085,34086,867],{},"并满足单纯形约束 ",[15,34087,34089,34129],{"className":34088,"translate":19},[18],[15,34090,34092],{"className":34091},[23],[25,34093,34094],{"xmlns":27},[29,34095,34096,34126],{},[32,34097,34098,34112,34118,34120,34122,34124],{},[289,34099,34100,34102,34110],{},[81,34101,293],{},[32,34103,34104,34106,34108],{},[35,34105,272],{},[81,34107,244],{},[89,34109,91],{},[35,34111,102],{},[266,34113,34114,34116],{},[35,34115,33425],{},[35,34117,272],{},[81,34119,7762],{},[35,34121,33430],{},[81,34123,244],{},[89,34125,91],{},[40,34127,34128],{"encoding":42},"\\sum_{k=1}^K b_k + u = 1",[15,34130,34132,34250,34268],{"className":34131,"ariaHidden":48},[47],[15,34133,34135,34138,34198,34201,34241,34244,34247],{"className":34134},[52],[15,34136],{"className":34137,"style":3471},[56],[15,34139,34141,34144],{"className":34140},[531],[15,34142,293],{"className":34143,"style":537},[531,535,536],[15,34145,34147],{"className":34146},[338],[15,34148,34150,34190],{"className":34149},[342,472],[15,34151,34153,34187],{"className":34152},[346],[15,34154,34156,34176],{"className":34155,"style":550},[350],[15,34157,34158,34161],{"style":553},[15,34159],{"className":34160,"style":358},[357],[15,34162,34164],{"className":34163},[362,363,364,365],[15,34165,34167,34170,34173],{"className":34166},[61,365],[15,34168,272],{"className":34169,"style":492},[61,122,365],[15,34171,244],{"className":34172},[132,365],[15,34174,91],{"className":34175},[61,365],[15,34177,34178,34181],{"style":574},[15,34179],{"className":34180,"style":358},[357],[15,34182,34184],{"className":34183},[362,363,364,365],[15,34185,102],{"className":34186,"style":174},[61,122,365],[15,34188,497],{"className":34189},[496],[15,34191,34193],{"className":34192},[346],[15,34194,34196],{"className":34195,"style":593},[350],[15,34197],{},[15,34199],{"className":34200,"style":157},[127],[15,34202,34204,34207],{"className":34203},[61],[15,34205,33425],{"className":34206},[61,122],[15,34208,34210],{"className":34209},[338],[15,34211,34213,34233],{"className":34212},[342,472],[15,34214,34216,34230],{"className":34215},[346],[15,34217,34219],{"className":34218,"style":479},[350],[15,34220,34221,34224],{"style":482},[15,34222],{"className":34223,"style":358},[357],[15,34225,34227],{"className":34226},[362,363,364,365],[15,34228,272],{"className":34229,"style":492},[61,122,365],[15,34231,497],{"className":34232},[496],[15,34234,34236],{"className":34235},[346],[15,34237,34239],{"className":34238,"style":504},[350],[15,34240],{},[15,34242],{"className":34243,"style":772},[127],[15,34245,7762],{"className":34246},[375],[15,34248],{"className":34249,"style":772},[127],[15,34251,34253,34256,34259,34262,34265],{"className":34252},[52],[15,34254],{"className":34255,"style":6032},[56],[15,34257,33430],{"className":34258},[61,122],[15,34260],{"className":34261,"style":128},[127],[15,34263,244],{"className":34264},[132],[15,34266],{"className":34267,"style":128},[127],[15,34269,34271,34274],{"className":34270},[52],[15,34272],{"className":34273,"style":859},[56],[15,34275,91],{"className":34276},[61],[11,34278,34279,9377,34308,34311,34312,34517,34518,20777,34617,867],{},[15,34280,34282,34296],{"className":34281,"translate":19},[18],[15,34283,34285],{"className":34284},[23],[25,34286,34287],{"xmlns":27},[29,34288,34289,34293],{},[32,34290,34291],{},[35,34292,33418],{},[40,34294,34295],{"encoding":42},"\\omega",[15,34297,34299],{"className":34298,"ariaHidden":48},[47],[15,34300,34302,34305],{"className":34301},[52],[15,34303],{"className":34304,"style":6032},[56],[15,34306,33418],{"className":34307,"style":123},[61,122],[661,34309,34310],{},"概率投影（probability projection）"," 定义为 ",[15,34313,34315,34351],{"className":34314,"translate":19},[18],[15,34316,34318],{"className":34317},[23],[25,34319,34320],{"xmlns":27},[29,34321,34322,34348],{},[32,34323,34324,34330,34332,34338,34340,34346],{},[266,34325,34326,34328],{},[35,34327,21286],{},[35,34329,272],{},[81,34331,244],{},[266,34333,34334,34336],{},[35,34335,33425],{},[35,34337,272],{},[81,34339,7762],{},[266,34341,34342,34344],{},[35,34343,33435],{},[35,34345,272],{},[35,34347,33430],{},[40,34349,34350],{"encoding":42},"P_k = b_k + a_k u",[15,34352,34354,34412,34468],{"className":34353,"ariaHidden":48},[47],[15,34355,34357,34361,34403,34406,34409],{"className":34356},[52],[15,34358],{"className":34359,"style":34360},[56],"height:0.8333em;vertical-align:-0.15em;",[15,34362,34364,34368],{"className":34363},[61],[15,34365,21286],{"className":34366,"style":34367},[61,122],"margin-right:0.1389em;",[15,34369,34371],{"className":34370},[338],[15,34372,34374,34395],{"className":34373},[342,472],[15,34375,34377,34392],{"className":34376},[346],[15,34378,34380],{"className":34379,"style":479},[350],[15,34381,34383,34386],{"style":34382},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[15,34384],{"className":34385,"style":358},[357],[15,34387,34389],{"className":34388},[362,363,364,365],[15,34390,272],{"className":34391,"style":492},[61,122,365],[15,34393,497],{"className":34394},[496],[15,34396,34398],{"className":34397},[346],[15,34399,34401],{"className":34400,"style":504},[350],[15,34402],{},[15,34404],{"className":34405,"style":128},[127],[15,34407,244],{"className":34408},[132],[15,34410],{"className":34411,"style":128},[127],[15,34413,34415,34419,34459,34462,34465],{"className":34414},[52],[15,34416],{"className":34417,"style":34418},[56],"height:0.8444em;vertical-align:-0.15em;",[15,34420,34422,34425],{"className":34421},[61],[15,34423,33425],{"className":34424},[61,122],[15,34426,34428],{"className":34427},[338],[15,34429,34431,34451],{"className":34430},[342,472],[15,34432,34434,34448],{"className":34433},[346],[15,34435,34437],{"className":34436,"style":479},[350],[15,34438,34439,34442],{"style":482},[15,34440],{"className":34441,"style":358},[357],[15,34443,34445],{"className":34444},[362,363,364,365],[15,34446,272],{"className":34447,"style":492},[61,122,365],[15,34449,497],{"className":34450},[496],[15,34452,34454],{"className":34453},[346],[15,34455,34457],{"className":34456,"style":504},[350],[15,34458],{},[15,34460],{"className":34461,"style":772},[127],[15,34463,7762],{"className":34464},[375],[15,34466],{"className":34467,"style":772},[127],[15,34469,34471,34474,34514],{"className":34470},[52],[15,34472],{"className":34473,"style":2511},[56],[15,34475,34477,34480],{"className":34476},[61],[15,34478,33435],{"className":34479},[61,122],[15,34481,34483],{"className":34482},[338],[15,34484,34486,34506],{"className":34485},[342,472],[15,34487,34489,34503],{"className":34488},[346],[15,34490,34492],{"className":34491,"style":479},[350],[15,34493,34494,34497],{"style":482},[15,34495],{"className":34496,"style":358},[357],[15,34498,34500],{"className":34499},[362,363,364,365],[15,34501,272],{"className":34502,"style":492},[61,122,365],[15,34504,497],{"className":34505},[496],[15,34507,34509],{"className":34508},[346],[15,34510,34512],{"className":34511,"style":504},[350],[15,34513],{},[15,34515,33430],{"className":34516},[61,122],"：在不了解具体类别时退回基准率，在有所了解时按信念质量分配。当基准率取均匀值 ",[15,34519,34521,34547],{"className":34520,"translate":19},[18],[15,34522,34524],{"className":34523},[23],[25,34525,34526],{"xmlns":27},[29,34527,34528,34544],{},[32,34529,34530,34536,34538,34540,34542],{},[266,34531,34532,34534],{},[35,34533,33435],{},[35,34535,272],{},[81,34537,244],{},[89,34539,91],{},[35,34541,3020],{"mathvariant":231},[35,34543,102],{},[40,34545,34546],{"encoding":42},"a_k = 1\u002FK",[15,34548,34550,34605],{"className":34549,"ariaHidden":48},[47],[15,34551,34553,34556,34596,34599,34602],{"className":34552},[52],[15,34554],{"className":34555,"style":2511},[56],[15,34557,34559,34562],{"className":34558},[61],[15,34560,33435],{"className":34561},[61,122],[15,34563,34565],{"className":34564},[338],[15,34566,34568,34588],{"className":34567},[342,472],[15,34569,34571,34585],{"className":34570},[346],[15,34572,34574],{"className":34573,"style":479},[350],[15,34575,34576,34579],{"style":482},[15,34577],{"className":34578,"style":358},[357],[15,34580,34582],{"className":34581},[362,363,364,365],[15,34583,272],{"className":34584,"style":492},[61,122,365],[15,34586,497],{"className":34587},[496],[15,34589,34591],{"className":34590},[346],[15,34592,34594],{"className":34593,"style":504},[350],[15,34595],{},[15,34597],{"className":34598,"style":128},[127],[15,34600,244],{"className":34601},[132],[15,34603],{"className":34604,"style":128},[127],[15,34606,34608,34611,34614],{"className":34607},[52],[15,34609],{"className":34610,"style":142},[56],[15,34612,3049],{"className":34613},[61],[15,34615,102],{"className":34616,"style":174},[61,122],[15,34618,34620,34654],{"className":34619,"translate":19},[18],[15,34621,34623],{"className":34622},[23],[25,34624,34625],{"xmlns":27},[29,34626,34627,34651],{},[32,34628,34629,34635,34637,34643,34645,34647,34649],{},[266,34630,34631,34633],{},[35,34632,21286],{},[35,34634,272],{},[81,34636,244],{},[266,34638,34639,34641],{},[35,34640,33425],{},[35,34642,272],{},[81,34644,7762],{},[35,34646,33430],{},[35,34648,3020],{"mathvariant":231},[35,34650,102],{},[40,34652,34653],{"encoding":42},"P_k = b_k + u\u002FK",[15,34655,34657,34712,34767],{"className":34656,"ariaHidden":48},[47],[15,34658,34660,34663,34703,34706,34709],{"className":34659},[52],[15,34661],{"className":34662,"style":34360},[56],[15,34664,34666,34669],{"className":34665},[61],[15,34667,21286],{"className":34668,"style":34367},[61,122],[15,34670,34672],{"className":34671},[338],[15,34673,34675,34695],{"className":34674},[342,472],[15,34676,34678,34692],{"className":34677},[346],[15,34679,34681],{"className":34680,"style":479},[350],[15,34682,34683,34686],{"style":34382},[15,34684],{"className":34685,"style":358},[357],[15,34687,34689],{"className":34688},[362,363,364,365],[15,34690,272],{"className":34691,"style":492},[61,122,365],[15,34693,497],{"className":34694},[496],[15,34696,34698],{"className":34697},[346],[15,34699,34701],{"className":34700,"style":504},[350],[15,34702],{},[15,34704],{"className":34705,"style":128},[127],[15,34707,244],{"className":34708},[132],[15,34710],{"className":34711,"style":128},[127],[15,34713,34715,34718,34758,34761,34764],{"className":34714},[52],[15,34716],{"className":34717,"style":34418},[56],[15,34719,34721,34724],{"className":34720},[61],[15,34722,33425],{"className":34723},[61,122],[15,34725,34727],{"className":34726},[338],[15,34728,34730,34750],{"className":34729},[342,472],[15,34731,34733,34747],{"className":34732},[346],[15,34734,34736],{"className":34735,"style":479},[350],[15,34737,34738,34741],{"style":482},[15,34739],{"className":34740,"style":358},[357],[15,34742,34744],{"className":34743},[362,363,364,365],[15,34745,272],{"className":34746,"style":492},[61,122,365],[15,34748,497],{"className":34749},[496],[15,34751,34753],{"className":34752},[346],[15,34754,34756],{"className":34755,"style":504},[350],[15,34757],{},[15,34759],{"className":34760,"style":772},[127],[15,34762,7762],{"className":34763},[375],[15,34765],{"className":34766,"style":772},[127],[15,34768,34770,34773,34776,34779],{"className":34769},[52],[15,34771],{"className":34772,"style":142},[56],[15,34774,33430],{"className":34775},[61,122],[15,34777,3020],{"className":34778},[61],[15,34780,102],{"className":34781,"style":174},[61,122],[11,34783,34784,34785,34847,34848,34876,34877,34950],{},"定义 3.1 的关键在于：opinion 所栖居的空间是 ",[15,34786,34788,34806],{"className":34787,"translate":19},[18],[15,34789,34791],{"className":34790},[23],[25,34792,34793],{"xmlns":27},[29,34794,34795,34803],{},[32,34796,34797],{},[227,34798,34799,34801],{},[35,34800,232],{"mathvariant":231},[35,34802,102],{},[40,34804,34805],{"encoding":42},"\\Delta^{K}",[15,34807,34809],{"className":34808,"ariaHidden":48},[47],[15,34810,34812,34815],{"className":34811},[52],[15,34813],{"className":34814,"style":695},[56],[15,34816,34818,34821],{"className":34817},[61],[15,34819,232],{"className":34820},[61],[15,34822,34824],{"className":34823},[338],[15,34825,34827],{"className":34826},[342],[15,34828,34830],{"className":34829},[346],[15,34831,34833],{"className":34832,"style":695},[350],[15,34834,34835,34838],{"style":716},[15,34836],{"className":34837,"style":358},[357],[15,34839,34841],{"className":34840},[362,363,364,365],[15,34842,34844],{"className":34843},[61,365],[15,34845,102],{"className":34846,"style":174},[61,122,365],"——一个 ",[15,34849,34851,34864],{"className":34850,"translate":19},[18],[15,34852,34854],{"className":34853},[23],[25,34855,34856],{"xmlns":27},[29,34857,34858,34862],{},[32,34859,34860],{},[35,34861,102],{},[40,34863,102],{"encoding":42},[15,34865,34867],{"className":34866,"ariaHidden":48},[47],[15,34868,34870,34873],{"className":34869},[52],[15,34871],{"className":34872,"style":57},[56],[15,34874,102],{"className":34875,"style":174},[61,122]," 维单纯形，比 ",[15,34878,34880,34903],{"className":34879,"translate":19},[18],[15,34881,34883],{"className":34882},[23],[25,34884,34885],{"xmlns":27},[29,34886,34887,34901],{},[32,34888,34889],{},[227,34890,34891,34893],{},[35,34892,232],{"mathvariant":231},[32,34894,34895,34897,34899],{},[35,34896,102],{},[81,34898,239],{},[89,34900,91],{},[40,34902,1858],{"encoding":42},[15,34904,34906],{"className":34905,"ariaHidden":48},[47],[15,34907,34909,34912],{"className":34908},[52],[15,34910],{"className":34911,"style":695},[56],[15,34913,34915,34918],{"className":34914},[61],[15,34916,232],{"className":34917},[61],[15,34919,34921],{"className":34920},[338],[15,34922,34924],{"className":34923},[342],[15,34925,34927],{"className":34926},[346],[15,34928,34930],{"className":34929,"style":695},[350],[15,34931,34932,34935],{"style":716},[15,34933],{"className":34934,"style":358},[357],[15,34936,34938],{"className":34937},[362,363,364,365],[15,34939,34941,34944,34947],{"className":34940},[61,365],[15,34942,102],{"className":34943,"style":174},[61,122,365],[15,34945,239],{"className":34946},[375,365],[15,34948,91],{"className":34949},[61,365]," 多出一个维度。多出来的这一维正是「不确定性」的自由度，而它恰恰是 softmax 输出所没有的。",[877,34952,34954],{"id":34953},"_32-dirichlet-与-opinion-的双射","3.2 Dirichlet 与 opinion 的双射",[11,34956,34957,34958,867],{},"主观逻辑与迪利克雷分布之间不存在类比关系，而是存在严格的",[661,34959,34960],{},"双射",[882,34962,34963],{},[11,34964,34965,34968,34969,35021,35022,35120,35121],{},[661,34966,34967],{},"命题 3.2（双射）"," 设先验权重 ",[15,34970,34972,34991],{"className":34971,"translate":19},[18],[15,34973,34975],{"className":34974},[23],[25,34976,34977],{"xmlns":27},[29,34978,34979,34988],{},[32,34980,34981,34984,34986],{},[35,34982,34983],{},"W",[81,34985,244],{},[35,34987,102],{},[40,34989,34990],{"encoding":42},"W = K",[15,34992,34994,35012],{"className":34993,"ariaHidden":48},[47],[15,34995,34997,35000,35003,35006,35009],{"className":34996},[52],[15,34998],{"className":34999,"style":57},[56],[15,35001,34983],{"className":35002,"style":34367},[61,122],[15,35004],{"className":35005,"style":128},[127],[15,35007,244],{"className":35008},[132],[15,35010],{"className":35011,"style":128},[127],[15,35013,35015,35018],{"className":35014},[52],[15,35016],{"className":35017,"style":57},[56],[15,35019,102],{"className":35020,"style":174},[61,122],"、基准率 ",[15,35023,35025,35050],{"className":35024,"translate":19},[18],[15,35026,35028],{"className":35027},[23],[25,35029,35030],{"xmlns":27},[29,35031,35032,35048],{},[32,35033,35034,35040,35042,35044,35046],{},[266,35035,35036,35038],{},[35,35037,33435],{},[35,35039,272],{},[81,35041,244],{},[89,35043,91],{},[35,35045,3020],{"mathvariant":231},[35,35047,102],{},[40,35049,34546],{"encoding":42},[15,35051,35053,35108],{"className":35052,"ariaHidden":48},[47],[15,35054,35056,35059,35099,35102,35105],{"className":35055},[52],[15,35057],{"className":35058,"style":2511},[56],[15,35060,35062,35065],{"className":35061},[61],[15,35063,33435],{"className":35064},[61,122],[15,35066,35068],{"className":35067},[338],[15,35069,35071,35091],{"className":35070},[342,472],[15,35072,35074,35088],{"className":35073},[346],[15,35075,35077],{"className":35076,"style":479},[350],[15,35078,35079,35082],{"style":482},[15,35080],{"className":35081,"style":358},[357],[15,35083,35085],{"className":35084},[362,363,364,365],[15,35086,272],{"className":35087,"style":492},[61,122,365],[15,35089,497],{"className":35090},[496],[15,35092,35094],{"className":35093},[346],[15,35095,35097],{"className":35096,"style":504},[350],[15,35098],{},[15,35100],{"className":35101,"style":128},[127],[15,35103,244],{"className":35104},[132],[15,35106],{"className":35107,"style":128},[127],[15,35109,35111,35114,35117],{"className":35110},[52],[15,35112],{"className":35113,"style":142},[56],[15,35115,3049],{"className":35116},[61],[15,35118,102],{"className":35119,"style":174},[61,122],"。则如下两个映射互逆：\n",[15,35122,35124,35226],{"className":35123,"translate":19},[18],[15,35125,35127],{"className":35126},[23],[25,35128,35129],{"xmlns":27},[29,35130,35131,35223],{},[32,35132,35133,35135,35137,35153,35155,35157,35159,35161,35171,35173,35175,35177,35183,35185,35201,35203,35205,35207,35209,35215,35217],{},[35,35134,33425],{"mathvariant":251},[81,35136,244],{},[3722,35138,35139,35147],{},[32,35140,35141,35143,35145],{},[35,35142,3106],{"mathvariant":3105},[81,35144,239],{},[89,35146,91],{"mathvariant":251},[266,35148,35149,35151],{},[35,35150,3106],{},[89,35152,278],{},[81,35154,94],{"separator":48},[127,35156],{"width":9847},[35,35158,33430],{},[81,35160,244],{},[3722,35162,35163,35165],{},[35,35164,102],{},[266,35166,35167,35169],{},[35,35168,3106],{},[89,35170,278],{},[127,35172],{"width":9847},[81,35174,24517],{},[127,35176],{"width":9847},[266,35178,35179,35181],{},[35,35180,3106],{},[35,35182,272],{},[81,35184,244],{},[3722,35186,35187,35199],{},[32,35188,35189,35191,35193],{},[35,35190,102],{},[282,35192,4802],{},[266,35194,35195,35197],{},[35,35196,33425],{},[35,35198,272],{},[35,35200,33430],{},[81,35202,7762],{},[89,35204,91],{},[81,35206,94],{"separator":48},[127,35208],{"width":9847},[266,35210,35211,35213],{},[35,35212,3106],{},[89,35214,278],{},[81,35216,244],{},[3722,35218,35219,35221],{},[35,35220,102],{},[35,35222,33430],{},[40,35224,35225],{"encoding":42},"\\mathbf{b} = \\frac{\\boldsymbol{\\alpha} - \\mathbf{1}}{\\alpha_0}, \\qquad u = \\frac{K}{\\alpha_0} \\qquad \\Longleftrightarrow \\qquad \\alpha_k = \\frac{K\\, b_k}{u} + 1, \\qquad \\alpha_0 = \\frac{K}{u}",[15,35227,35229,35247,35391,35519,35574,35702,35769],{"className":35228,"ariaHidden":48},[47],[15,35230,35232,35235,35238,35241,35244],{"className":35231},[52],[15,35233],{"className":35234,"style":10452},[56],[15,35236,33425],{"className":35237},[61,411],[15,35239],{"className":35240,"style":128},[127],[15,35242,244],{"className":35243},[132],[15,35245],{"className":35246,"style":128},[127],[15,35248,35250,35253,35370,35373,35376,35379,35382,35385,35388],{"className":35249},[52],[15,35251],{"className":35252,"style":16578},[56],[15,35254,35256,35259,35367],{"className":35255},[61],[15,35257],{"className":35258},[146,3841],[15,35260,35262],{"className":35261},[3722],[15,35263,35265,35359],{"className":35264},[342,472],[15,35266,35268,35356],{"className":35267},[346],[15,35269,35271,35322,35330],{"className":35270,"style":3854},[350],[15,35272,35273,35276],{"style":3857},[15,35274],{"className":35275,"style":3861},[357],[15,35277,35279],{"className":35278},[362,363,364,365],[15,35280,35282],{"className":35281},[61,365],[15,35283,35285,35288],{"className":35284},[61,365],[15,35286,3106],{"className":35287,"style":3191},[61,122,365],[15,35289,35291],{"className":35290},[338],[15,35292,35294,35314],{"className":35293},[342,472],[15,35295,35297,35311],{"className":35296},[346],[15,35298,35300],{"className":35299,"style":4425},[350],[15,35301,35302,35305],{"style":4058},[15,35303],{"className":35304,"style":4062},[357],[15,35306,35308],{"className":35307},[362,4066,4067,365],[15,35309,278],{"className":35310},[61,365],[15,35312,497],{"className":35313},[496],[15,35315,35317],{"className":35316},[346],[15,35318,35320],{"className":35319,"style":4080},[350],[15,35321],{},[15,35323,35324,35327],{"style":3889},[15,35325],{"className":35326,"style":3861},[357],[15,35328],{"className":35329,"style":3897},[3896],[15,35331,35332,35335],{"style":3900},[15,35333],{"className":35334,"style":3861},[357],[15,35336,35338],{"className":35337},[362,363,364,365],[15,35339,35341,35350,35353],{"className":35340},[61,365],[15,35342,35344],{"className":35343},[61,365],[15,35345,35347],{"className":35346},[61,365],[15,35348,3106],{"className":35349},[61,3166,365],[15,35351,239],{"className":35352},[375,365],[15,35354,91],{"className":35355},[61,411,365],[15,35357,497],{"className":35358},[496],[15,35360,35362],{"className":35361},[346],[15,35363,35365],{"className":35364,"style":15220},[350],[15,35366],{},[15,35368],{"className":35369},[178,3841],[15,35371,94],{"className":35372},[153],[15,35374],{"className":35375,"style":10035},[127],[15,35377],{"className":35378,"style":157},[127],[15,35380,33430],{"className":35381},[61,122],[15,35383],{"className":35384,"style":128},[127],[15,35386,244],{"className":35387},[132],[15,35389],{"className":35390,"style":128},[127],[15,35392,35394,35398,35504,35507,35510,35513,35516],{"className":35393},[52],[15,35395],{"className":35396,"style":35397},[56],"height:1.3174em;vertical-align:-0.4451em;",[15,35399,35401,35404,35501],{"className":35400},[61],[15,35402],{"className":35403},[146,3841],[15,35405,35407],{"className":35406},[3722],[15,35408,35410,35493],{"className":35409},[342,472],[15,35411,35413,35490],{"className":35412},[346],[15,35414,35417,35468,35476],{"className":35415,"style":35416},[350],"height:0.8723em;",[15,35418,35419,35422],{"style":3857},[15,35420],{"className":35421,"style":3861},[357],[15,35423,35425],{"className":35424},[362,363,364,365],[15,35426,35428],{"className":35427},[61,365],[15,35429,35431,35434],{"className":35430},[61,365],[15,35432,3106],{"className":35433,"style":3191},[61,122,365],[15,35435,35437],{"className":35436},[338],[15,35438,35440,35460],{"className":35439},[342,472],[15,35441,35443,35457],{"className":35442},[346],[15,35444,35446],{"className":35445,"style":4425},[350],[15,35447,35448,35451],{"style":4058},[15,35449],{"className":35450,"style":4062},[357],[15,35452,35454],{"className":35453},[362,4066,4067,365],[15,35455,278],{"className":35456},[61,365],[15,35458,497],{"className":35459},[496],[15,35461,35463],{"className":35462},[346],[15,35464,35466],{"className":35465,"style":4080},[350],[15,35467],{},[15,35469,35470,35473],{"style":3889},[15,35471],{"className":35472,"style":3861},[357],[15,35474],{"className":35475,"style":3897},[3896],[15,35477,35478,35481],{"style":3900},[15,35479],{"className":35480,"style":3861},[357],[15,35482,35484],{"className":35483},[362,363,364,365],[15,35485,35487],{"className":35486},[61,365],[15,35488,102],{"className":35489,"style":174},[61,122,365],[15,35491,497],{"className":35492},[496],[15,35494,35496],{"className":35495},[346],[15,35497,35499],{"className":35498,"style":15220},[350],[15,35500],{},[15,35502],{"className":35503},[178,3841],[15,35505],{"className":35506,"style":10035},[127],[15,35508],{"className":35509,"style":128},[127],[15,35511,24517],{"className":35512},[132],[15,35514],{"className":35515,"style":10035},[127],[15,35517],{"className":35518,"style":128},[127],[15,35520,35522,35525,35565,35568,35571],{"className":35521},[52],[15,35523],{"className":35524,"style":2511},[56],[15,35526,35528,35531],{"className":35527},[61],[15,35529,3106],{"className":35530,"style":3191},[61,122],[15,35532,35534],{"className":35533},[338],[15,35535,35537,35557],{"className":35536},[342,472],[15,35538,35540,35554],{"className":35539},[346],[15,35541,35543],{"className":35542,"style":479},[350],[15,35544,35545,35548],{"style":3206},[15,35546],{"className":35547,"style":358},[357],[15,35549,35551],{"className":35550},[362,363,364,365],[15,35552,272],{"className":35553,"style":492},[61,122,365],[15,35555,497],{"className":35556},[496],[15,35558,35560],{"className":35559},[346],[15,35561,35563],{"className":35562,"style":504},[350],[15,35564],{},[15,35566],{"className":35567,"style":128},[127],[15,35569,244],{"className":35570},[132],[15,35572],{"className":35573,"style":128},[127],[15,35575,35577,35581,35693,35696,35699],{"className":35576},[52],[15,35578],{"className":35579,"style":35580},[56],"height:1.247em;vertical-align:-0.345em;",[15,35582,35584,35587,35690],{"className":35583},[61],[15,35585],{"className":35586},[146,3841],[15,35588,35590],{"className":35589},[3722],[15,35591,35593,35682],{"className":35592},[342,472],[15,35594,35596,35679],{"className":35595},[346],[15,35597,35600,35614,35622],{"className":35598,"style":35599},[350],"height:0.902em;",[15,35601,35602,35605],{"style":3857},[15,35603],{"className":35604,"style":3861},[357],[15,35606,35608],{"className":35607},[362,363,364,365],[15,35609,35611],{"className":35610},[61,365],[15,35612,33430],{"className":35613},[61,122,365],[15,35615,35616,35619],{"style":3889},[15,35617],{"className":35618,"style":3861},[357],[15,35620],{"className":35621,"style":3897},[3896],[15,35623,35624,35627],{"style":27204},[15,35625],{"className":35626,"style":3861},[357],[15,35628,35630],{"className":35629},[362,363,364,365],[15,35631,35633,35636,35639],{"className":35632},[61,365],[15,35634,102],{"className":35635,"style":174},[61,122,365],[15,35637],{"className":35638,"style":4541},[127,365],[15,35640,35642,35645],{"className":35641},[61,365],[15,35643,33425],{"className":35644},[61,122,365],[15,35646,35648],{"className":35647},[338],[15,35649,35651,35671],{"className":35650},[342,472],[15,35652,35654,35668],{"className":35653},[346],[15,35655,35657],{"className":35656,"style":22404},[350],[15,35658,35659,35662],{"style":27279},[15,35660],{"className":35661,"style":4062},[357],[15,35663,35665],{"className":35664},[362,4066,4067,365],[15,35666,272],{"className":35667,"style":492},[61,122,365],[15,35669,497],{"className":35670},[496],[15,35672,35674],{"className":35673},[346],[15,35675,35677],{"className":35676,"style":27253},[350],[15,35678],{},[15,35680,497],{"className":35681},[496],[15,35683,35685],{"className":35684},[346],[15,35686,35688],{"className":35687,"style":10198},[350],[15,35689],{},[15,35691],{"className":35692},[178,3841],[15,35694],{"className":35695,"style":772},[127],[15,35697,7762],{"className":35698},[375],[15,35700],{"className":35701,"style":772},[127],[15,35703,35705,35708,35711,35714,35717,35720,35760,35763,35766],{"className":35704},[52],[15,35706],{"className":35707,"style":19755},[56],[15,35709,91],{"className":35710},[61],[15,35712,94],{"className":35713},[153],[15,35715],{"className":35716,"style":10035},[127],[15,35718],{"className":35719,"style":157},[127],[15,35721,35723,35726],{"className":35722},[61],[15,35724,3106],{"className":35725,"style":3191},[61,122],[15,35727,35729],{"className":35728},[338],[15,35730,35732,35752],{"className":35731},[342,472],[15,35733,35735,35749],{"className":35734},[346],[15,35736,35738],{"className":35737,"style":1214},[350],[15,35739,35740,35743],{"style":3206},[15,35741],{"className":35742,"style":358},[357],[15,35744,35746],{"className":35745},[362,363,364,365],[15,35747,278],{"className":35748},[61,365],[15,35750,497],{"className":35751},[496],[15,35753,35755],{"className":35754},[346],[15,35756,35758],{"className":35757,"style":504},[350],[15,35759],{},[15,35761],{"className":35762,"style":128},[127],[15,35764,244],{"className":35765},[132],[15,35767],{"className":35768,"style":128},[127],[15,35770,35772,35776],{"className":35771},[52],[15,35773],{"className":35774,"style":35775},[56],"height:1.2173em;vertical-align:-0.345em;",[15,35777,35779,35782,35841],{"className":35778},[61],[15,35780],{"className":35781},[146,3841],[15,35783,35785],{"className":35784},[3722],[15,35786,35788,35833],{"className":35787},[342,472],[15,35789,35791,35830],{"className":35790},[346],[15,35792,35794,35808,35816],{"className":35793,"style":35416},[350],[15,35795,35796,35799],{"style":3857},[15,35797],{"className":35798,"style":3861},[357],[15,35800,35802],{"className":35801},[362,363,364,365],[15,35803,35805],{"className":35804},[61,365],[15,35806,33430],{"className":35807},[61,122,365],[15,35809,35810,35813],{"style":3889},[15,35811],{"className":35812,"style":3861},[357],[15,35814],{"className":35815,"style":3897},[3896],[15,35817,35818,35821],{"style":3900},[15,35819],{"className":35820,"style":3861},[357],[15,35822,35824],{"className":35823},[362,363,364,365],[15,35825,35827],{"className":35826},[61,365],[15,35828,102],{"className":35829,"style":174},[61,122,365],[15,35831,497],{"className":35832},[496],[15,35834,35836],{"className":35835},[346],[15,35837,35839],{"className":35838,"style":10198},[350],[15,35840],{},[15,35842],{"className":35843},[178,3841],[11,35845,35846,35849,35850,36332],{},[661,35847,35848],{},"验证"," 正向：",[15,35851,35853,35919],{"className":35852,"translate":19},[18],[15,35854,35856],{"className":35855},[23],[25,35857,35858],{"xmlns":27},[29,35859,35860,35916],{},[32,35861,35862,35868,35874,35876,35878,35880,35900,35902,35912,35914],{},[266,35863,35864,35866],{},[81,35865,293],{},[35,35867,272],{},[266,35869,35870,35872],{},[35,35871,33425],{},[35,35873,272],{},[81,35875,7762],{},[35,35877,33430],{},[81,35879,244],{},[3722,35881,35882,35894],{},[32,35883,35884,35890,35892],{},[266,35885,35886,35888],{},[35,35887,3106],{},[89,35889,278],{},[81,35891,239],{},[35,35893,102],{},[266,35895,35896,35898],{},[35,35897,3106],{},[89,35899,278],{},[81,35901,7762],{},[3722,35903,35904,35906],{},[35,35905,102],{},[266,35907,35908,35910],{},[35,35909,3106],{},[89,35911,278],{},[81,35913,244],{},[89,35915,91],{},[40,35917,35918],{"encoding":42},"\\sum_k b_k + u = \\frac{\\alpha_0 - K}{\\alpha_0} + \\frac{K}{\\alpha_0} = 1",[15,35920,35922,36020,36038,36203,36323],{"className":35921,"ariaHidden":48},[47],[15,35923,35925,35928,35968,35971,36011,36014,36017],{"className":35924},[52],[15,35926],{"className":35927,"style":6051},[56],[15,35929,35931,35934],{"className":35930},[531],[15,35932,293],{"className":35933,"style":537},[531,535,536],[15,35935,35937],{"className":35936},[338],[15,35938,35940,35960],{"className":35939},[342,472],[15,35941,35943,35957],{"className":35942},[346],[15,35944,35946],{"className":35945,"style":29989},[350],[15,35947,35948,35951],{"style":553},[15,35949],{"className":35950,"style":358},[357],[15,35952,35954],{"className":35953},[362,363,364,365],[15,35955,272],{"className":35956,"style":492},[61,122,365],[15,35958,497],{"className":35959},[496],[15,35961,35963],{"className":35962},[346],[15,35964,35966],{"className":35965,"style":593},[350],[15,35967],{},[15,35969],{"className":35970,"style":157},[127],[15,35972,35974,35977],{"className":35973},[61],[15,35975,33425],{"className":35976},[61,122],[15,35978,35980],{"className":35979},[338],[15,35981,35983,36003],{"className":35982},[342,472],[15,35984,35986,36000],{"className":35985},[346],[15,35987,35989],{"className":35988,"style":479},[350],[15,35990,35991,35994],{"style":482},[15,35992],{"className":35993,"style":358},[357],[15,35995,35997],{"className":35996},[362,363,364,365],[15,35998,272],{"className":35999,"style":492},[61,122,365],[15,36001,497],{"className":36002},[496],[15,36004,36006],{"className":36005},[346],[15,36007,36009],{"className":36008,"style":504},[350],[15,36010],{},[15,36012],{"className":36013,"style":772},[127],[15,36015,7762],{"className":36016},[375],[15,36018],{"className":36019,"style":772},[127],[15,36021,36023,36026,36029,36032,36035],{"className":36022},[52],[15,36024],{"className":36025,"style":6032},[56],[15,36027,33430],{"className":36028},[61,122],[15,36030],{"className":36031,"style":128},[127],[15,36033,244],{"className":36034},[132],[15,36036],{"className":36037,"style":128},[127],[15,36039,36041,36045,36194,36197,36200],{"className":36040},[52],[15,36042],{"className":36043,"style":36044},[56],"height:1.3335em;vertical-align:-0.4451em;",[15,36046,36048,36051,36191],{"className":36047},[61],[15,36049],{"className":36050},[146,3841],[15,36052,36054],{"className":36053},[3722],[15,36055,36057,36183],{"className":36056},[342,472],[15,36058,36060,36180],{"className":36059},[346],[15,36061,36064,36115,36123],{"className":36062,"style":36063},[350],"height:0.8884em;",[15,36065,36066,36069],{"style":3857},[15,36067],{"className":36068,"style":3861},[357],[15,36070,36072],{"className":36071},[362,363,364,365],[15,36073,36075],{"className":36074},[61,365],[15,36076,36078,36081],{"className":36077},[61,365],[15,36079,3106],{"className":36080,"style":3191},[61,122,365],[15,36082,36084],{"className":36083},[338],[15,36085,36087,36107],{"className":36086},[342,472],[15,36088,36090,36104],{"className":36089},[346],[15,36091,36093],{"className":36092,"style":4425},[350],[15,36094,36095,36098],{"style":4058},[15,36096],{"className":36097,"style":4062},[357],[15,36099,36101],{"className":36100},[362,4066,4067,365],[15,36102,278],{"className":36103},[61,365],[15,36105,497],{"className":36106},[496],[15,36108,36110],{"className":36109},[346],[15,36111,36113],{"className":36112,"style":4080},[350],[15,36114],{},[15,36116,36117,36120],{"style":3889},[15,36118],{"className":36119,"style":3861},[357],[15,36121],{"className":36122,"style":3897},[3896],[15,36124,36125,36128],{"style":15161},[15,36126],{"className":36127,"style":3861},[357],[15,36129,36131],{"className":36130},[362,363,364,365],[15,36132,36134,36174,36177],{"className":36133},[61,365],[15,36135,36137,36140],{"className":36136},[61,365],[15,36138,3106],{"className":36139,"style":3191},[61,122,365],[15,36141,36143],{"className":36142},[338],[15,36144,36146,36166],{"className":36145},[342,472],[15,36147,36149,36163],{"className":36148},[346],[15,36150,36152],{"className":36151,"style":4425},[350],[15,36153,36154,36157],{"style":4058},[15,36155],{"className":36156,"style":4062},[357],[15,36158,36160],{"className":36159},[362,4066,4067,365],[15,36161,278],{"className":36162},[61,365],[15,36164,497],{"className":36165},[496],[15,36167,36169],{"className":36168},[346],[15,36170,36172],{"className":36171,"style":4080},[350],[15,36173],{},[15,36175,239],{"className":36176},[375,365],[15,36178,102],{"className":36179,"style":174},[61,122,365],[15,36181,497],{"className":36182},[496],[15,36184,36186],{"className":36185},[346],[15,36187,36189],{"className":36188,"style":15220},[350],[15,36190],{},[15,36192],{"className":36193},[178,3841],[15,36195],{"className":36196,"style":772},[127],[15,36198,7762],{"className":36199},[375],[15,36201],{"className":36202,"style":772},[127],[15,36204,36206,36209,36314,36317,36320],{"className":36205},[52],[15,36207],{"className":36208,"style":35397},[56],[15,36210,36212,36215,36311],{"className":36211},[61],[15,36213],{"className":36214},[146,3841],[15,36216,36218],{"className":36217},[3722],[15,36219,36221,36303],{"className":36220},[342,472],[15,36222,36224,36300],{"className":36223},[346],[15,36225,36227,36278,36286],{"className":36226,"style":35416},[350],[15,36228,36229,36232],{"style":3857},[15,36230],{"className":36231,"style":3861},[357],[15,36233,36235],{"className":36234},[362,363,364,365],[15,36236,36238],{"className":36237},[61,365],[15,36239,36241,36244],{"className":36240},[61,365],[15,36242,3106],{"className":36243,"style":3191},[61,122,365],[15,36245,36247],{"className":36246},[338],[15,36248,36250,36270],{"className":36249},[342,472],[15,36251,36253,36267],{"className":36252},[346],[15,36254,36256],{"className":36255,"style":4425},[350],[15,36257,36258,36261],{"style":4058},[15,36259],{"className":36260,"style":4062},[357],[15,36262,36264],{"className":36263},[362,4066,4067,365],[15,36265,278],{"className":36266},[61,365],[15,36268,497],{"className":36269},[496],[15,36271,36273],{"className":36272},[346],[15,36274,36276],{"className":36275,"style":4080},[350],[15,36277],{},[15,36279,36280,36283],{"style":3889},[15,36281],{"className":36282,"style":3861},[357],[15,36284],{"className":36285,"style":3897},[3896],[15,36287,36288,36291],{"style":3900},[15,36289],{"className":36290,"style":3861},[357],[15,36292,36294],{"className":36293},[362,363,364,365],[15,36295,36297],{"className":36296},[61,365],[15,36298,102],{"className":36299,"style":174},[61,122,365],[15,36301,497],{"className":36302},[496],[15,36304,36306],{"className":36305},[346],[15,36307,36309],{"className":36308,"style":15220},[350],[15,36310],{},[15,36312],{"className":36313},[178,3841],[15,36315],{"className":36316,"style":128},[127],[15,36318,244],{"className":36319},[132],[15,36321],{"className":36322,"style":128},[127],[15,36324,36326,36329],{"className":36325},[52],[15,36327],{"className":36328,"style":859},[56],[15,36330,91],{"className":36331},[61],"，满足定义 3.1 的约束。反向：代入即得恒等。此外，概率投影与迪利克雷期望重合：",[15,36334,36336],{"className":36335,"translate":19},[212],[15,36337,36339,36437],{"className":36338,"translate":19},[18],[15,36340,36342],{"className":36341},[23],[25,36343,36344],{"xmlns":27,"display":221},[29,36345,36346,36434],{},[32,36347,36348,36354,36356,36362,36364,36370,36372,36392,36394,36404,36406,36420,36422,36424,36426,36432],{},[266,36349,36350,36352],{},[35,36351,21286],{},[35,36353,272],{},[81,36355,244],{},[266,36357,36358,36360],{},[35,36359,33425],{},[35,36361,272],{},[81,36363,7762],{},[3722,36365,36366,36368],{},[35,36367,33430],{},[35,36369,102],{},[81,36371,244],{},[3722,36373,36374,36386],{},[32,36375,36376,36382,36384],{},[266,36377,36378,36380],{},[35,36379,3106],{},[35,36381,272],{},[81,36383,239],{},[89,36385,91],{},[266,36387,36388,36390],{},[35,36389,3106],{},[89,36391,278],{},[81,36393,7762],{},[3722,36395,36396,36398],{},[89,36397,91],{},[266,36399,36400,36402],{},[35,36401,3106],{},[89,36403,278],{},[81,36405,244],{},[3722,36407,36408,36414],{},[266,36409,36410,36412],{},[35,36411,3106],{},[35,36413,272],{},[266,36415,36416,36418],{},[35,36417,3106],{},[89,36419,278],{},[81,36421,244],{},[35,36423,14818],{"mathvariant":258},[81,36425,14821],{"stretchy":86},[266,36427,36428,36430],{},[35,36429,1029],{},[35,36431,272],{},[81,36433,14830],{"stretchy":86},[40,36435,36436],{"encoding":42},"P_k = b_k + \\frac{u}{K} = \\frac{\\alpha_k - 1}{\\alpha_0} + \\frac{1}{\\alpha_0} = \\frac{\\alpha_k}{\\alpha_0} = \\mathbb{E}[x_k]",[15,36438,36440,36495,36550,36629,36792,36906,37058],{"className":36439,"ariaHidden":48},[47],[15,36441,36443,36446,36486,36489,36492],{"className":36442},[52],[15,36444],{"className":36445,"style":34360},[56],[15,36447,36449,36452],{"className":36448},[61],[15,36450,21286],{"className":36451,"style":34367},[61,122],[15,36453,36455],{"className":36454},[338],[15,36456,36458,36478],{"className":36457},[342,472],[15,36459,36461,36475],{"className":36460},[346],[15,36462,36464],{"className":36463,"style":479},[350],[15,36465,36466,36469],{"style":34382},[15,36467],{"className":36468,"style":358},[357],[15,36470,36472],{"className":36471},[362,363,364,365],[15,36473,272],{"className":36474,"style":492},[61,122,365],[15,36476,497],{"className":36477},[496],[15,36479,36481],{"className":36480},[346],[15,36482,36484],{"className":36483,"style":504},[350],[15,36485],{},[15,36487],{"className":36488,"style":128},[127],[15,36490,244],{"className":36491},[132],[15,36493],{"className":36494,"style":128},[127],[15,36496,36498,36501,36541,36544,36547],{"className":36497},[52],[15,36499],{"className":36500,"style":34418},[56],[15,36502,36504,36507],{"className":36503},[61],[15,36505,33425],{"className":36506},[61,122],[15,36508,36510],{"className":36509},[338],[15,36511,36513,36533],{"className":36512},[342,472],[15,36514,36516,36530],{"className":36515},[346],[15,36517,36519],{"className":36518,"style":479},[350],[15,36520,36521,36524],{"style":482},[15,36522],{"className":36523,"style":358},[357],[15,36525,36527],{"className":36526},[362,363,364,365],[15,36528,272],{"className":36529,"style":492},[61,122,365],[15,36531,497],{"className":36532},[496],[15,36534,36536],{"className":36535},[346],[15,36537,36539],{"className":36538,"style":504},[350],[15,36540],{},[15,36542],{"className":36543,"style":772},[127],[15,36545,7762],{"className":36546},[375],[15,36548],{"className":36549,"style":772},[127],[15,36551,36553,36557,36620,36623,36626],{"className":36552},[52],[15,36554],{"className":36555,"style":36556},[56],"height:1.7936em;vertical-align:-0.686em;",[15,36558,36560,36563,36617],{"className":36559},[61],[15,36561],{"className":36562},[146,3841],[15,36564,36566],{"className":36565},[3722],[15,36567,36569,36608],{"className":36568},[342,472],[15,36570,36572,36605],{"className":36571},[346],[15,36573,36575,36586,36594],{"className":36574,"style":27649},[350],[15,36576,36577,36580],{"style":11402},[15,36578],{"className":36579,"style":3861},[357],[15,36581,36583],{"className":36582},[61],[15,36584,102],{"className":36585,"style":174},[61,122],[15,36587,36588,36591],{"style":3889},[15,36589],{"className":36590,"style":3861},[357],[15,36592],{"className":36593,"style":3897},[3896],[15,36595,36596,36599],{"style":11468},[15,36597],{"className":36598,"style":3861},[357],[15,36600,36602],{"className":36601},[61],[15,36603,33430],{"className":36604},[61,122],[15,36606,497],{"className":36607},[496],[15,36609,36611],{"className":36610},[346],[15,36612,36615],{"className":36613,"style":36614},[350],"height:0.686em;",[15,36616],{},[15,36618],{"className":36619},[178,3841],[15,36621],{"className":36622,"style":128},[127],[15,36624,244],{"className":36625},[132],[15,36627],{"className":36628,"style":128},[127],[15,36630,36632,36635,36783,36786,36789],{"className":36631},[52],[15,36633],{"className":36634,"style":18671},[56],[15,36636,36638,36641,36780],{"className":36637},[61],[15,36639],{"className":36640},[146,3841],[15,36642,36644],{"className":36643},[3722],[15,36645,36647,36772],{"className":36646},[342,472],[15,36648,36650,36769],{"className":36649},[346],[15,36651,36653,36701,36709],{"className":36652,"style":18690},[350],[15,36654,36655,36658],{"style":11402},[15,36656],{"className":36657,"style":3861},[357],[15,36659,36661],{"className":36660},[61],[15,36662,36664,36667],{"className":36663},[61],[15,36665,3106],{"className":36666,"style":3191},[61,122],[15,36668,36670],{"className":36669},[338],[15,36671,36673,36693],{"className":36672},[342,472],[15,36674,36676,36690],{"className":36675},[346],[15,36677,36679],{"className":36678,"style":1214},[350],[15,36680,36681,36684],{"style":3206},[15,36682],{"className":36683,"style":358},[357],[15,36685,36687],{"className":36686},[362,363,364,365],[15,36688,278],{"className":36689},[61,365],[15,36691,497],{"className":36692},[496],[15,36694,36696],{"className":36695},[346],[15,36697,36699],{"className":36698,"style":504},[350],[15,36700],{},[15,36702,36703,36706],{"style":3889},[15,36704],{"className":36705,"style":3861},[357],[15,36707],{"className":36708,"style":3897},[3896],[15,36710,36711,36714],{"style":11468},[15,36712],{"className":36713,"style":3861},[357],[15,36715,36717,36757,36760,36763,36766],{"className":36716},[61],[15,36718,36720,36723],{"className":36719},[61],[15,36721,3106],{"className":36722,"style":3191},[61,122],[15,36724,36726],{"className":36725},[338],[15,36727,36729,36749],{"className":36728},[342,472],[15,36730,36732,36746],{"className":36731},[346],[15,36733,36735],{"className":36734,"style":479},[350],[15,36736,36737,36740],{"style":3206},[15,36738],{"className":36739,"style":358},[357],[15,36741,36743],{"className":36742},[362,363,364,365],[15,36744,272],{"className":36745,"style":492},[61,122,365],[15,36747,497],{"className":36748},[496],[15,36750,36752],{"className":36751},[346],[15,36753,36755],{"className":36754,"style":504},[350],[15,36756],{},[15,36758],{"className":36759,"style":772},[127],[15,36761,239],{"className":36762},[375],[15,36764],{"className":36765,"style":772},[127],[15,36767,91],{"className":36768},[61],[15,36770,497],{"className":36771},[496],[15,36773,36775],{"className":36774},[346],[15,36776,36778],{"className":36777,"style":18779},[350],[15,36779],{},[15,36781],{"className":36782},[178,3841],[15,36784],{"className":36785,"style":772},[127],[15,36787,7762],{"className":36788},[375],[15,36790],{"className":36791,"style":772},[127],[15,36793,36795,36798,36897,36900,36903],{"className":36794},[52],[15,36796],{"className":36797,"style":18671},[56],[15,36799,36801,36804,36894],{"className":36800},[61],[15,36802],{"className":36803},[146,3841],[15,36805,36807],{"className":36806},[3722],[15,36808,36810,36886],{"className":36809},[342,472],[15,36811,36813,36883],{"className":36812},[346],[15,36814,36816,36864,36872],{"className":36815,"style":18690},[350],[15,36817,36818,36821],{"style":11402},[15,36819],{"className":36820,"style":3861},[357],[15,36822,36824],{"className":36823},[61],[15,36825,36827,36830],{"className":36826},[61],[15,36828,3106],{"className":36829,"style":3191},[61,122],[15,36831,36833],{"className":36832},[338],[15,36834,36836,36856],{"className":36835},[342,472],[15,36837,36839,36853],{"className":36838},[346],[15,36840,36842],{"className":36841,"style":1214},[350],[15,36843,36844,36847],{"style":3206},[15,36845],{"className":36846,"style":358},[357],[15,36848,36850],{"className":36849},[362,363,364,365],[15,36851,278],{"className":36852},[61,365],[15,36854,497],{"className":36855},[496],[15,36857,36859],{"className":36858},[346],[15,36860,36862],{"className":36861,"style":504},[350],[15,36863],{},[15,36865,36866,36869],{"style":3889},[15,36867],{"className":36868,"style":3861},[357],[15,36870],{"className":36871,"style":3897},[3896],[15,36873,36874,36877],{"style":11468},[15,36875],{"className":36876,"style":3861},[357],[15,36878,36880],{"className":36879},[61],[15,36881,91],{"className":36882},[61],[15,36884,497],{"className":36885},[496],[15,36887,36889],{"className":36888},[346],[15,36890,36892],{"className":36891,"style":18779},[350],[15,36893],{},[15,36895],{"className":36896},[178,3841],[15,36898],{"className":36899,"style":128},[127],[15,36901,244],{"className":36902},[132],[15,36904],{"className":36905,"style":128},[127],[15,36907,36909,36913,37049,37052,37055],{"className":36908},[52],[15,36910],{"className":36911,"style":36912},[56],"height:1.9436em;vertical-align:-0.836em;",[15,36914,36916,36919,37046],{"className":36915},[61],[15,36917],{"className":36918},[146,3841],[15,36920,36922],{"className":36921},[3722],[15,36923,36925,37038],{"className":36924},[342,472],[15,36926,36928,37035],{"className":36927},[346],[15,36929,36931,36979,36987],{"className":36930,"style":27649},[350],[15,36932,36933,36936],{"style":11402},[15,36934],{"className":36935,"style":3861},[357],[15,36937,36939],{"className":36938},[61],[15,36940,36942,36945],{"className":36941},[61],[15,36943,3106],{"className":36944,"style":3191},[61,122],[15,36946,36948],{"className":36947},[338],[15,36949,36951,36971],{"className":36950},[342,472],[15,36952,36954,36968],{"className":36953},[346],[15,36955,36957],{"className":36956,"style":1214},[350],[15,36958,36959,36962],{"style":3206},[15,36960],{"className":36961,"style":358},[357],[15,36963,36965],{"className":36964},[362,363,364,365],[15,36966,278],{"className":36967},[61,365],[15,36969,497],{"className":36970},[496],[15,36972,36974],{"className":36973},[346],[15,36975,36977],{"className":36976,"style":504},[350],[15,36978],{},[15,36980,36981,36984],{"style":3889},[15,36982],{"className":36983,"style":3861},[357],[15,36985],{"className":36986,"style":3897},[3896],[15,36988,36989,36992],{"style":11468},[15,36990],{"className":36991,"style":3861},[357],[15,36993,36995],{"className":36994},[61],[15,36996,36998,37001],{"className":36997},[61],[15,36999,3106],{"className":37000,"style":3191},[61,122],[15,37002,37004],{"className":37003},[338],[15,37005,37007,37027],{"className":37006},[342,472],[15,37008,37010,37024],{"className":37009},[346],[15,37011,37013],{"className":37012,"style":479},[350],[15,37014,37015,37018],{"style":3206},[15,37016],{"className":37017,"style":358},[357],[15,37019,37021],{"className":37020},[362,363,364,365],[15,37022,272],{"className":37023,"style":492},[61,122,365],[15,37025,497],{"className":37026},[496],[15,37028,37030],{"className":37029},[346],[15,37031,37033],{"className":37032,"style":504},[350],[15,37034],{},[15,37036,497],{"className":37037},[496],[15,37039,37041],{"className":37040},[346],[15,37042,37044],{"className":37043,"style":18779},[350],[15,37045],{},[15,37047],{"className":37048},[178,3841],[15,37050],{"className":37051,"style":128},[127],[15,37053,244],{"className":37054},[132],[15,37056],{"className":37057,"style":128},[127],[15,37059,37061,37064,37067,37070,37110],{"className":37060},[52],[15,37062],{"className":37063,"style":142},[56],[15,37065,14818],{"className":37066},[61,427],[15,37068,14821],{"className":37069},[146],[15,37071,37073,37076],{"className":37072},[61],[15,37074,1029],{"className":37075},[61,122],[15,37077,37079],{"className":37078},[338],[15,37080,37082,37102],{"className":37081},[342,472],[15,37083,37085,37099],{"className":37084},[346],[15,37086,37088],{"className":37087,"style":479},[350],[15,37089,37090,37093],{"style":482},[15,37091],{"className":37092,"style":358},[357],[15,37094,37096],{"className":37095},[362,363,364,365],[15,37097,272],{"className":37098,"style":492},[61,122,365],[15,37100,497],{"className":37101},[496],[15,37103,37105],{"className":37104},[346],[15,37106,37108],{"className":37107,"style":504},[350],[15,37109],{},[15,37111,14830],{"className":37112},[178],[11,37114,37115],{},"其中最后一个等号由命题 2.1 给出。",[11,37117,37118,37119,37122],{},"命题 3.2 是整篇文章的枢纽：它说明",[661,37120,37121],{},"在单纯形上输出一个概率分布，与在 opinion 空间中输出一个带不确定性度量的判断，是同一件事。"," 于是，「给网络加一个不确定性维度」这一看似需要新架构的任务，被归结为「让网络的输出参数化一个迪利克雷分布」。证据深度学习正是沿着这条路径展开的。",[877,37124,37126],{"id":37125},"_33-softmax-的翻译不变性与过度自信","3.3 Softmax 的翻译不变性与过度自信",[11,37128,37129,37130,37286,37287,37404,37405,37477,37478,37784],{},"设神经网络 ",[15,37131,37133,37164],{"className":37132,"translate":19},[18],[15,37134,37136],{"className":37135},[23],[25,37137,37138],{"xmlns":27},[29,37139,37140,37161],{},[32,37141,37142,37149,37151,37153,37155],{},[266,37143,37144,37146],{},[35,37145,3707],{},[35,37147,37148],{"mathvariant":3105},"θ",[81,37150,264],{},[35,37152,38],{"mathvariant":37},[81,37154,20704],{},[227,37156,37157,37159],{},[35,37158,259],{"mathvariant":258},[35,37160,102],{},[40,37162,37163],{"encoding":42},"f_{\\boldsymbol{\\theta}}: \\mathcal{X} \\to \\mathbb{R}^K",[15,37165,37167,37233,37251],{"className":37166,"ariaHidden":48},[47],[15,37168,37170,37173,37224,37227,37230],{"className":37169},[52],[15,37171],{"className":37172,"style":14310},[56],[15,37174,37176,37179],{"className":37175},[61],[15,37177,3707],{"className":37178,"style":3785},[61,122],[15,37180,37182],{"className":37181},[338],[15,37183,37185,37216],{"className":37184},[342,472],[15,37186,37188,37213],{"className":37187},[346],[15,37189,37191],{"className":37190,"style":479},[350],[15,37192,37194,37197],{"style":37193},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[15,37195],{"className":37196,"style":358},[357],[15,37198,37200],{"className":37199},[362,363,364,365],[15,37201,37203],{"className":37202},[61,365],[15,37204,37206],{"className":37205},[61,365],[15,37207,37209],{"className":37208},[61,365],[15,37210,37148],{"className":37211,"style":37212},[61,3166,365],"margin-right:0.0319em;",[15,37214,497],{"className":37215},[496],[15,37217,37219],{"className":37218},[346],[15,37220,37222],{"className":37221,"style":504},[350],[15,37223],{},[15,37225],{"className":37226,"style":128},[127],[15,37228,264],{"className":37229},[132],[15,37231],{"className":37232,"style":128},[127],[15,37234,37236,37239,37242,37245,37248],{"className":37235},[52],[15,37237],{"className":37238,"style":57},[56],[15,37240,38],{"className":37241,"style":63},[61,62],[15,37243],{"className":37244,"style":128},[127],[15,37246,20704],{"className":37247},[132],[15,37249],{"className":37250,"style":128},[127],[15,37252,37254,37257],{"className":37253},[52],[15,37255],{"className":37256,"style":695},[56],[15,37258,37260,37263],{"className":37259},[61],[15,37261,259],{"className":37262},[61,427],[15,37264,37266],{"className":37265},[338],[15,37267,37269],{"className":37268},[342],[15,37270,37272],{"className":37271},[346],[15,37273,37275],{"className":37274,"style":695},[350],[15,37276,37277,37280],{"style":716},[15,37278],{"className":37279,"style":358},[357],[15,37281,37283],{"className":37282},[362,363,364,365],[15,37284,102],{"className":37285,"style":174},[61,122,365]," 输出 logits ",[15,37288,37290,37319],{"className":37289,"translate":19},[18],[15,37291,37293],{"className":37292},[23],[25,37294,37295],{"xmlns":27},[29,37296,37297,37316],{},[32,37298,37299,37302,37304,37310,37312,37314],{},[35,37300,37301],{"mathvariant":251},"z",[81,37303,244],{},[266,37305,37306,37308],{},[35,37307,3707],{},[35,37309,37148],{"mathvariant":3105},[81,37311,741],{"stretchy":86},[35,37313,1029],{"mathvariant":251},[81,37315,750],{"stretchy":86},[40,37317,37318],{"encoding":42},"\\mathbf{z} = f_{\\boldsymbol{\\theta}}(\\mathbf{x})",[15,37320,37322,37340],{"className":37321,"ariaHidden":48},[47],[15,37323,37325,37328,37331,37334,37337],{"className":37324},[52],[15,37326],{"className":37327,"style":2708},[56],[15,37329,37301],{"className":37330},[61,411],[15,37332],{"className":37333,"style":128},[127],[15,37335,244],{"className":37336},[132],[15,37338],{"className":37339,"style":128},[127],[15,37341,37343,37346,37395,37398,37401],{"className":37342},[52],[15,37344],{"className":37345,"style":142},[56],[15,37347,37349,37352],{"className":37348},[61],[15,37350,3707],{"className":37351,"style":3785},[61,122],[15,37353,37355],{"className":37354},[338],[15,37356,37358,37387],{"className":37357},[342,472],[15,37359,37361,37384],{"className":37360},[346],[15,37362,37364],{"className":37363,"style":479},[350],[15,37365,37366,37369],{"style":37193},[15,37367],{"className":37368,"style":358},[357],[15,37370,37372],{"className":37371},[362,363,364,365],[15,37373,37375],{"className":37374},[61,365],[15,37376,37378],{"className":37377},[61,365],[15,37379,37381],{"className":37380},[61,365],[15,37382,37148],{"className":37383,"style":37212},[61,3166,365],[15,37385,497],{"className":37386},[496],[15,37388,37390],{"className":37389},[346],[15,37391,37393],{"className":37392,"style":504},[350],[15,37394],{},[15,37396,741],{"className":37397},[146],[15,37399,1029],{"className":37400},[61,411],[15,37402,750],{"className":37403},[178],"，softmax 给出 ",[15,37406,37408,37435],{"className":37407,"translate":19},[18],[15,37409,37411],{"className":37410},[23],[25,37412,37413],{"xmlns":27},[29,37414,37415,37432],{},[32,37416,37417,37419,37421,37424,37426,37428,37430],{},[35,37418,11],{"mathvariant":251},[81,37420,244],{},[35,37422,37423],{"mathvariant":231},"softmax",[81,37425,1547],{},[81,37427,741],{"stretchy":86},[35,37429,37301],{"mathvariant":251},[81,37431,750],{"stretchy":86},[40,37433,37434],{"encoding":42},"\\mathbf{p} = \\operatorname{softmax}(\\mathbf{z})",[15,37436,37438,37456],{"className":37437,"ariaHidden":48},[47],[15,37439,37441,37444,37447,37450,37453],{"className":37440},[52],[15,37442],{"className":37443,"style":829},[56],[15,37445,11],{"className":37446},[61,411],[15,37448],{"className":37449,"style":128},[127],[15,37451,244],{"className":37452},[132],[15,37454],{"className":37455,"style":128},[127],[15,37457,37459,37462,37468,37471,37474],{"className":37458},[52],[15,37460],{"className":37461,"style":142},[56],[15,37463,37465],{"className":37464},[531],[15,37466,37423],{"className":37467},[61,1622],[15,37469,741],{"className":37470},[146],[15,37472,37301],{"className":37473},[61,411],[15,37475,750],{"className":37476},[178],"，即 ",[15,37479,37481,37529],{"className":37480,"translate":19},[18],[15,37482,37484],{"className":37483},[23],[25,37485,37486],{"xmlns":27},[29,37487,37488,37526],{},[32,37489,37490,37496,37498,37508,37510,37516],{},[266,37491,37492,37494],{},[35,37493,11],{},[35,37495,272],{},[81,37497,244],{},[227,37499,37500,37502],{},[35,37501,1982],{},[266,37503,37504,37506],{},[35,37505,37301],{},[35,37507,272],{},[35,37509,3020],{"mathvariant":231},[266,37511,37512,37514],{},[81,37513,293],{},[35,37515,14777],{},[227,37517,37518,37520],{},[35,37519,1982],{},[266,37521,37522,37524],{},[35,37523,37301],{},[35,37525,14777],{},[40,37527,37528],{"encoding":42},"p_k = e^{z_k}\u002F\\sum_j e^{z_j}",[15,37530,37532,37587],{"className":37531,"ariaHidden":48},[47],[15,37533,37535,37538,37578,37581,37584],{"className":37534},[52],[15,37536],{"className":37537,"style":9437},[56],[15,37539,37541,37544],{"className":37540},[61],[15,37542,11],{"className":37543},[61,122],[15,37545,37547],{"className":37546},[338],[15,37548,37550,37570],{"className":37549},[342,472],[15,37551,37553,37567],{"className":37552},[346],[15,37554,37556],{"className":37555,"style":479},[350],[15,37557,37558,37561],{"style":482},[15,37559],{"className":37560,"style":358},[357],[15,37562,37564],{"className":37563},[362,363,364,365],[15,37565,272],{"className":37566,"style":492},[61,122,365],[15,37568,497],{"className":37569},[496],[15,37571,37573],{"className":37572},[346],[15,37574,37576],{"className":37575,"style":504},[350],[15,37577],{},[15,37579],{"className":37580,"style":128},[127],[15,37582,244],{"className":37583},[132],[15,37585],{"className":37586,"style":128},[127],[15,37588,37590,37593,37665,37668,37671,37711,37714],{"className":37589},[52],[15,37591],{"className":37592,"style":29863},[56],[15,37594,37596,37599],{"className":37595},[61],[15,37597,1982],{"className":37598},[61,122],[15,37600,37602],{"className":37601},[338],[15,37603,37605],{"className":37604},[342],[15,37606,37608],{"className":37607},[346],[15,37609,37612],{"className":37610,"style":37611},[350],"height:0.6644em;",[15,37613,37614,37617],{"style":716},[15,37615],{"className":37616,"style":358},[357],[15,37618,37620],{"className":37619},[362,363,364,365],[15,37621,37623],{"className":37622},[61,365],[15,37624,37626,37630],{"className":37625},[61,365],[15,37627,37301],{"className":37628,"style":37629},[61,122,365],"margin-right:0.044em;",[15,37631,37633],{"className":37632},[338],[15,37634,37636,37657],{"className":37635},[342,472],[15,37637,37639,37654],{"className":37638},[346],[15,37640,37642],{"className":37641,"style":22404},[350],[15,37643,37645,37648],{"style":37644},"top:-2.3488em;margin-left:-0.044em;margin-right:0.0714em;",[15,37646],{"className":37647,"style":4062},[357],[15,37649,37651],{"className":37650},[362,4066,4067,365],[15,37652,272],{"className":37653,"style":492},[61,122,365],[15,37655,497],{"className":37656},[496],[15,37658,37660],{"className":37659},[346],[15,37661,37663],{"className":37662,"style":27253},[350],[15,37664],{},[15,37666,3020],{"className":37667},[61],[15,37669],{"className":37670,"style":157},[127],[15,37672,37674,37677],{"className":37673},[531],[15,37675,293],{"className":37676,"style":537},[531,535,536],[15,37678,37680],{"className":37679},[338],[15,37681,37683,37703],{"className":37682},[342,472],[15,37684,37686,37700],{"className":37685},[346],[15,37687,37689],{"className":37688,"style":6070},[350],[15,37690,37691,37694],{"style":553},[15,37692],{"className":37693,"style":358},[357],[15,37695,37697],{"className":37696},[362,363,364,365],[15,37698,14777],{"className":37699,"style":14803},[61,122,365],[15,37701,497],{"className":37702},[496],[15,37704,37706],{"className":37705},[346],[15,37707,37709],{"className":37708,"style":30053},[350],[15,37710],{},[15,37712],{"className":37713,"style":157},[127],[15,37715,37717,37720],{"className":37716},[61],[15,37718,1982],{"className":37719},[61,122],[15,37721,37723],{"className":37722},[338],[15,37724,37726],{"className":37725},[342],[15,37727,37729],{"className":37728},[346],[15,37730,37732],{"className":37731,"style":37611},[350],[15,37733,37734,37737],{"style":716},[15,37735],{"className":37736,"style":358},[357],[15,37738,37740],{"className":37739},[362,363,364,365],[15,37741,37743],{"className":37742},[61,365],[15,37744,37746,37749],{"className":37745},[61,365],[15,37747,37301],{"className":37748,"style":37629},[61,122,365],[15,37750,37752],{"className":37751},[338],[15,37753,37755,37776],{"className":37754},[342,472],[15,37756,37758,37773],{"className":37757},[346],[15,37759,37761],{"className":37760,"style":4055},[350],[15,37762,37764,37767],{"style":37763},"top:-2.357em;margin-left:-0.044em;margin-right:0.0714em;",[15,37765],{"className":37766,"style":4062},[357],[15,37768,37770],{"className":37769},[362,4066,4067,365],[15,37771,14777],{"className":37772,"style":14803},[61,122,365],[15,37774,497],{"className":37775},[496],[15,37777,37779],{"className":37778},[346],[15,37780,37782],{"className":37781,"style":15963},[350],[15,37783],{},"。这一映射存在两个结构性缺陷。",[11,37786,37787,37790,37791,943,37820,37938,37939,9377,38020,38023,38024,38075,38076,38150,38151,38189],{},[661,37788,37789],{},"其一，平移不变性抹去了「无信息」这一状态。"," 对任意常数 ",[15,37792,37794,37808],{"className":37793,"translate":19},[18],[15,37795,37797],{"className":37796},[23],[25,37798,37799],{"xmlns":27},[29,37800,37801,37806],{},[32,37802,37803],{},[35,37804,37805],{},"c",[40,37807,37805],{"encoding":42},[15,37809,37811],{"className":37810,"ariaHidden":48},[47],[15,37812,37814,37817],{"className":37813},[52],[15,37815],{"className":37816,"style":6032},[56],[15,37818,37805],{"className":37819},[61,122],[15,37821,37823,37863],{"className":37822,"translate":19},[18],[15,37824,37826],{"className":37825},[23],[25,37827,37828],{"xmlns":27},[29,37829,37830,37860],{},[32,37831,37832,37834,37836,37838,37840,37842,37844,37846,37848,37850,37852,37854,37856,37858],{},[35,37833,37423],{"mathvariant":231},[81,37835,1547],{},[81,37837,741],{"stretchy":86},[35,37839,37301],{"mathvariant":251},[81,37841,7762],{},[35,37843,37805],{},[89,37845,91],{"mathvariant":251},[81,37847,750],{"stretchy":86},[81,37849,244],{},[35,37851,37423],{"mathvariant":231},[81,37853,1547],{},[81,37855,741],{"stretchy":86},[35,37857,37301],{"mathvariant":251},[81,37859,750],{"stretchy":86},[40,37861,37862],{"encoding":42},"\\operatorname{softmax}(\\mathbf{z} + c\\mathbf{1}) = \\operatorname{softmax}(\\mathbf{z})",[15,37864,37866,37893,37917],{"className":37865,"ariaHidden":48},[47],[15,37867,37869,37872,37878,37881,37884,37887,37890],{"className":37868},[52],[15,37870],{"className":37871,"style":142},[56],[15,37873,37875],{"className":37874},[531],[15,37876,37423],{"className":37877},[61,1622],[15,37879,741],{"className":37880},[146],[15,37882,37301],{"className":37883},[61,411],[15,37885],{"className":37886,"style":772},[127],[15,37888,7762],{"className":37889},[375],[15,37891],{"className":37892,"style":772},[127],[15,37894,37896,37899,37902,37905,37908,37911,37914],{"className":37895},[52],[15,37897],{"className":37898,"style":142},[56],[15,37900,37805],{"className":37901},[61,122],[15,37903,91],{"className":37904},[61,411],[15,37906,750],{"className":37907},[178],[15,37909],{"className":37910,"style":128},[127],[15,37912,244],{"className":37913},[132],[15,37915],{"className":37916,"style":128},[127],[15,37918,37920,37923,37929,37932,37935],{"className":37919},[52],[15,37921],{"className":37922,"style":142},[56],[15,37924,37926],{"className":37925},[531],[15,37927,37423],{"className":37928},[61,1622],[15,37930,741],{"className":37931},[146],[15,37933,37301],{"className":37934},[61,411],[15,37936,750],{"className":37937},[178],"。因此 logits 的绝对水平不含任何信息，网络的输出只依赖于 ",[15,37940,37942,37964],{"className":37941,"translate":19},[18],[15,37943,37945],{"className":37944},[23],[25,37946,37947],{"xmlns":27},[29,37948,37949,37961],{},[32,37950,37951,37953,37959],{},[81,37952,87],{"stretchy":86},[266,37954,37955,37957],{},[35,37956,37301],{},[35,37958,272],{},[81,37960,105],{"stretchy":86},[40,37962,37963],{"encoding":42},"\\{z_k\\}",[15,37965,37967],{"className":37966,"ariaHidden":48},[47],[15,37968,37970,37973,37976,38017],{"className":37969},[52],[15,37971],{"className":37972,"style":142},[56],[15,37974,87],{"className":37975},[146],[15,37977,37979,37982],{"className":37978},[61],[15,37980,37301],{"className":37981,"style":37629},[61,122],[15,37983,37985],{"className":37984},[338],[15,37986,37988,38009],{"className":37987},[342,472],[15,37989,37991,38006],{"className":37990},[346],[15,37992,37994],{"className":37993,"style":479},[350],[15,37995,37997,38000],{"style":37996},"top:-2.55em;margin-left:-0.044em;margin-right:0.05em;",[15,37998],{"className":37999,"style":358},[357],[15,38001,38003],{"className":38002},[362,363,364,365],[15,38004,272],{"className":38005,"style":492},[61,122,365],[15,38007,497],{"className":38008},[496],[15,38010,38012],{"className":38011},[346],[15,38013,38015],{"className":38014,"style":504},[350],[15,38016],{},[15,38018,105],{"className":38019},[178],[661,38021,38022],{},"相对差","。后果是：",[15,38025,38027,38045],{"className":38026,"translate":19},[18],[15,38028,38030],{"className":38029},[23],[25,38031,38032],{"xmlns":27},[29,38033,38034,38042],{},[32,38035,38036,38038,38040],{},[35,38037,37301],{"mathvariant":251},[81,38039,244],{},[89,38041,278],{"mathvariant":251},[40,38043,38044],{"encoding":42},"\\mathbf{z} = \\mathbf{0}",[15,38046,38048,38066],{"className":38047,"ariaHidden":48},[47],[15,38049,38051,38054,38057,38060,38063],{"className":38050},[52],[15,38052],{"className":38053,"style":2708},[56],[15,38055,37301],{"className":38056},[61,411],[15,38058],{"className":38059,"style":128},[127],[15,38061,244],{"className":38062},[132],[15,38064],{"className":38065,"style":128},[127],[15,38067,38069,38072],{"className":38068},[52],[15,38070],{"className":38071,"style":859},[56],[15,38073,278],{"className":38074},[61,411],"（「全零 logits」）与 ",[15,38077,38079,38102],{"className":38078,"translate":19},[18],[15,38080,38082],{"className":38081},[23],[25,38083,38084],{"xmlns":27},[29,38085,38086,38099],{},[32,38087,38088,38090,38092,38095,38097],{},[35,38089,37301],{"mathvariant":251},[81,38091,244],{},[89,38093,38094],{},"100",[81,38096,11214],{},[89,38098,91],{"mathvariant":251},[40,38100,38101],{"encoding":42},"\\mathbf{z} = 100 \\cdot \\mathbf{1}",[15,38103,38105,38123,38141],{"className":38104,"ariaHidden":48},[47],[15,38106,38108,38111,38114,38117,38120],{"className":38107},[52],[15,38109],{"className":38110,"style":2708},[56],[15,38112,37301],{"className":38113},[61,411],[15,38115],{"className":38116,"style":128},[127],[15,38118,244],{"className":38119},[132],[15,38121],{"className":38122,"style":128},[127],[15,38124,38126,38129,38132,38135,38138],{"className":38125},[52],[15,38127],{"className":38128,"style":859},[56],[15,38130,38094],{"className":38131},[61],[15,38133],{"className":38134,"style":772},[127],[15,38136,11214],{"className":38137},[375],[15,38139],{"className":38140,"style":772},[127],[15,38142,38144,38147],{"className":38143},[52],[15,38145],{"className":38146,"style":859},[56],[15,38148,91],{"className":38149},[61,411]," 不可区分，二者都对应均匀分布 ",[15,38152,38154,38171],{"className":38153,"translate":19},[18],[15,38155,38157],{"className":38156},[23],[25,38158,38159],{"xmlns":27},[29,38160,38161,38169],{},[32,38162,38163,38165,38167],{},[89,38164,91],{"mathvariant":251},[35,38166,3020],{"mathvariant":231},[35,38168,102],{},[40,38170,32600],{"encoding":42},[15,38172,38174],{"className":38173,"ariaHidden":48},[47],[15,38175,38177,38180,38183,38186],{"className":38176},[52],[15,38178],{"className":38179,"style":142},[56],[15,38181,91],{"className":38182},[61,411],[15,38184,3020],{"className":38185},[61],[15,38187,102],{"className":38188,"style":174},[61,122],"。既然均匀分布既可由「无任何信号」也可由「极其强烈的信号恰好相互抵消」产生，输出端就不可能把前者作为一个可辨识的状态标记出来。",[11,38191,38192,38195,38196,38328,38329,38332,38333,867],{},[661,38193,38194],{},"其二，交叉熵的最优解在可分数据上把 logits 推向发散。"," 在训练集线性可分时，以交叉熵为目标的极大似然估计不存在有限最优解：沿方向 ",[15,38197,38199,38235],{"className":38198,"translate":19},[18],[15,38200,38202],{"className":38201},[23],[25,38203,38204],{"xmlns":27},[29,38205,38206,38232],{},[32,38207,38208,38210,38212,38214,38216,38222,38224,38226,38228,38230],{},[35,38209,37301],{"mathvariant":251},[81,38211,20704],{},[35,38213,37805],{},[282,38215,4802],{},[227,38217,38218,38220],{},[35,38219,37301],{"mathvariant":251},[81,38221,21984],{},[81,38223,94],{"separator":48},[282,38225,284],{},[35,38227,37805],{},[81,38229,20704],{},[35,38231,6796],{"mathvariant":231},[40,38233,38234],{"encoding":42},"\\mathbf{z} \\to c\\,\\mathbf{z}^\\star,\\ c \\to \\infty",[15,38236,38238,38256,38319],{"className":38237,"ariaHidden":48},[47],[15,38239,38241,38244,38247,38250,38253],{"className":38240},[52],[15,38242],{"className":38243,"style":2708},[56],[15,38245,37301],{"className":38246},[61,411],[15,38248],{"className":38249,"style":128},[127],[15,38251,20704],{"className":38252},[132],[15,38254],{"className":38255,"style":128},[127],[15,38257,38259,38263,38266,38269,38298,38301,38304,38307,38310,38313,38316],{"className":38258},[52],[15,38260],{"className":38261,"style":38262},[56],"height:0.8831em;vertical-align:-0.1944em;",[15,38264,37805],{"className":38265},[61,122],[15,38267],{"className":38268,"style":157},[127],[15,38270,38272,38275],{"className":38271},[61],[15,38273,37301],{"className":38274},[61,411],[15,38276,38278],{"className":38277},[338],[15,38279,38281],{"className":38280},[342],[15,38282,38284],{"className":38283},[346],[15,38285,38287],{"className":38286,"style":22063},[350],[15,38288,38289,38292],{"style":716},[15,38290],{"className":38291,"style":358},[357],[15,38293,38295],{"className":38294},[362,363,364,365],[15,38296,21984],{"className":38297},[375,365],[15,38299,94],{"className":38300},[153],[15,38302,284],{"className":38303},[127],[15,38305],{"className":38306,"style":157},[127],[15,38308,37805],{"className":38309},[61,122],[15,38311],{"className":38312,"style":128},[127],[15,38314,20704],{"className":38315},[132],[15,38317],{"className":38318,"style":128},[127],[15,38320,38322,38325],{"className":38321},[52],[15,38323],{"className":38324,"style":6032},[56],[15,38326,6796],{"className":38327},[61]," 可将损失单调压向零。换言之，",[661,38330,38331],{},"过度自信不是训练不足的症状，而是 MLE 在可分数据上的退化解","——模型越是被「训练充分」，其 softmax 输出越接近 one-hot，而这一行为",[661,38334,38335],{},"完全不受输入是否为分布内样本的约束",[11,38337,38338],{},"两个缺陷合起来解释了 5.1 节的经验事实：以最大 softmax 概率（MSP）作为分布外检测分数存在系统性偏差，因为 softmax 对任何使 logits 相对间距增大的输入都会返回高置信度的断言，而无从表达「这个输入对我而言是陌生的」。",[877,38340,38342],{"id":38341},"_34-证据参数化","3.4 证据参数化",[11,38344,38345,38346],{},"证据深度学习对上述缺陷的回应是：",[661,38347,38348],{},"不再用 softmax 把 logits 压缩为概率，而是用非负激活函数把 logits 提升为证据。",[882,38350,38351],{},[11,38352,38353,9393,38356,38545,38546,38549,38550,38809,38810,39047,39048,867],{},[661,38354,38355],{},"定义 3.3（证据网络）",[15,38357,38359,38395],{"className":38358,"translate":19},[18],[15,38360,38362],{"className":38361},[23],[25,38363,38364],{"xmlns":27},[29,38365,38366,38392],{},[32,38367,38368,38374,38376,38378,38380],{},[266,38369,38370,38372],{},[35,38371,22794],{},[35,38373,37148],{"mathvariant":3105},[81,38375,264],{},[35,38377,38],{"mathvariant":37},[81,38379,20704],{},[289,38381,38382,38384,38390],{},[35,38383,259],{"mathvariant":258},[32,38385,38386,38388],{},[81,38387,275],{},[89,38389,278],{},[35,38391,102],{},[40,38393,38394],{"encoding":42},"g_{\\boldsymbol{\\theta}}: \\mathcal{X} \\to \\mathbb{R}_{\\ge 0}^K",[15,38396,38398,38462,38480],{"className":38397,"ariaHidden":48},[47],[15,38399,38401,38404,38453,38456,38459],{"className":38400},[52],[15,38402],{"className":38403,"style":9437},[56],[15,38405,38407,38410],{"className":38406},[61],[15,38408,22794],{"className":38409,"style":123},[61,122],[15,38411,38413],{"className":38412},[338],[15,38414,38416,38445],{"className":38415},[342,472],[15,38417,38419,38442],{"className":38418},[346],[15,38420,38422],{"className":38421,"style":479},[350],[15,38423,38424,38427],{"style":27036},[15,38425],{"className":38426,"style":358},[357],[15,38428,38430],{"className":38429},[362,363,364,365],[15,38431,38433],{"className":38432},[61,365],[15,38434,38436],{"className":38435},[61,365],[15,38437,38439],{"className":38438},[61,365],[15,38440,37148],{"className":38441,"style":37212},[61,3166,365],[15,38443,497],{"className":38444},[496],[15,38446,38448],{"className":38447},[346],[15,38449,38451],{"className":38450,"style":504},[350],[15,38452],{},[15,38454],{"className":38455,"style":128},[127],[15,38457,264],{"className":38458},[132],[15,38460],{"className":38461,"style":128},[127],[15,38463,38465,38468,38471,38474,38477],{"className":38464},[52],[15,38466],{"className":38467,"style":57},[56],[15,38469,38],{"className":38470,"style":63},[61,62],[15,38472],{"className":38473,"style":128},[127],[15,38475,20704],{"className":38476},[132],[15,38478],{"className":38479,"style":128},[127],[15,38481,38483,38487],{"className":38482},[52],[15,38484],{"className":38485,"style":38486},[56],"height:1.1846em;vertical-align:-0.3433em;",[15,38488,38490,38493],{"className":38489},[61],[15,38491,259],{"className":38492},[61,427],[15,38494,38496],{"className":38495},[338],[15,38497,38499,38536],{"className":38498},[342,472],[15,38500,38502,38533],{"className":38501},[346],[15,38503,38505,38522],{"className":38504,"style":695},[350],[15,38506,38507,38510],{"style":3324},[15,38508],{"className":38509,"style":358},[357],[15,38511,38513],{"className":38512},[362,363,364,365],[15,38514,38516,38519],{"className":38515},[61,365],[15,38517,275],{"className":38518},[132,365],[15,38520,278],{"className":38521},[61,365],[15,38523,38524,38527],{"style":716},[15,38525],{"className":38526,"style":358},[357],[15,38528,38530],{"className":38529},[362,363,364,365],[15,38531,102],{"className":38532,"style":174},[61,122,365],[15,38534,497],{"className":38535},[496],[15,38537,38539],{"className":38538},[346],[15,38540,38543],{"className":38541,"style":38542},[350],"height:0.3433em;",[15,38544],{}," 是由主干网络与",[661,38547,38548],{},"非负激活函数","复合而成的证据函数，输出 ",[15,38551,38553,38605],{"className":38552,"translate":19},[18],[15,38554,38556],{"className":38555},[23],[25,38557,38558],{"xmlns":27},[29,38559,38560,38602],{},[32,38561,38562,38564,38566,38572,38574,38576,38578,38580,38582,38588,38590,38592,38594,38600],{},[35,38563,1982],{"mathvariant":251},[81,38565,244],{},[266,38567,38568,38570],{},[35,38569,22794],{},[35,38571,37148],{"mathvariant":3105},[81,38573,741],{"stretchy":86},[35,38575,1029],{"mathvariant":251},[81,38577,750],{"stretchy":86},[81,38579,244],{},[81,38581,741],{"stretchy":86},[266,38583,38584,38586],{},[35,38585,1982],{},[89,38587,91],{},[81,38589,94],{"separator":48},[81,38591,97],{},[81,38593,94],{"separator":48},[266,38595,38596,38598],{},[35,38597,1982],{},[35,38599,102],{},[81,38601,750],{"stretchy":86},[40,38603,38604],{"encoding":42},"\\mathbf{e} = g_{\\boldsymbol{\\theta}}(\\mathbf{x}) = (e_1, \\dots, e_K)",[15,38606,38608,38626,38699],{"className":38607,"ariaHidden":48},[47],[15,38609,38611,38614,38617,38620,38623],{"className":38610},[52],[15,38612],{"className":38613,"style":2708},[56],[15,38615,1982],{"className":38616},[61,411],[15,38618],{"className":38619,"style":128},[127],[15,38621,244],{"className":38622},[132],[15,38624],{"className":38625,"style":128},[127],[15,38627,38629,38632,38681,38684,38687,38690,38693,38696],{"className":38628},[52],[15,38630],{"className":38631,"style":142},[56],[15,38633,38635,38638],{"className":38634},[61],[15,38636,22794],{"className":38637,"style":123},[61,122],[15,38639,38641],{"className":38640},[338],[15,38642,38644,38673],{"className":38643},[342,472],[15,38645,38647,38670],{"className":38646},[346],[15,38648,38650],{"className":38649,"style":479},[350],[15,38651,38652,38655],{"style":27036},[15,38653],{"className":38654,"style":358},[357],[15,38656,38658],{"className":38657},[362,363,364,365],[15,38659,38661],{"className":38660},[61,365],[15,38662,38664],{"className":38663},[61,365],[15,38665,38667],{"className":38666},[61,365],[15,38668,37148],{"className":38669,"style":37212},[61,3166,365],[15,38671,497],{"className":38672},[496],[15,38674,38676],{"className":38675},[346],[15,38677,38679],{"className":38678,"style":504},[350],[15,38680],{},[15,38682,741],{"className":38683},[146],[15,38685,1029],{"className":38686},[61,411],[15,38688,750],{"className":38689},[178],[15,38691],{"className":38692,"style":128},[127],[15,38694,244],{"className":38695},[132],[15,38697],{"className":38698,"style":128},[127],[15,38700,38702,38705,38708,38748,38751,38754,38757,38760,38763,38766,38806],{"className":38701},[52],[15,38703],{"className":38704,"style":142},[56],[15,38706,741],{"className":38707},[146],[15,38709,38711,38714],{"className":38710},[61],[15,38712,1982],{"className":38713},[61,122],[15,38715,38717],{"className":38716},[338],[15,38718,38720,38740],{"className":38719},[342,472],[15,38721,38723,38737],{"className":38722},[346],[15,38724,38726],{"className":38725,"style":1214},[350],[15,38727,38728,38731],{"style":482},[15,38729],{"className":38730,"style":358},[357],[15,38732,38734],{"className":38733},[362,363,364,365],[15,38735,91],{"className":38736},[61,365],[15,38738,497],{"className":38739},[496],[15,38741,38743],{"className":38742},[346],[15,38744,38746],{"className":38745,"style":504},[350],[15,38747],{},[15,38749,94],{"className":38750},[153],[15,38752],{"className":38753,"style":157},[127],[15,38755,97],{"className":38756},[161],[15,38758],{"className":38759,"style":157},[127],[15,38761,94],{"className":38762},[153],[15,38764],{"className":38765,"style":157},[127],[15,38767,38769,38772],{"className":38768},[61],[15,38770,1982],{"className":38771},[61,122],[15,38773,38775],{"className":38774},[338],[15,38776,38778,38798],{"className":38777},[342,472],[15,38779,38781,38795],{"className":38780},[346],[15,38782,38784],{"className":38783,"style":1273},[350],[15,38785,38786,38789],{"style":482},[15,38787],{"className":38788,"style":358},[357],[15,38790,38792],{"className":38791},[362,363,364,365],[15,38793,102],{"className":38794,"style":174},[61,122,365],[15,38796,497],{"className":38797},[496],[15,38799,38801],{"className":38800},[346],[15,38802,38804],{"className":38803,"style":504},[350],[15,38805],{},[15,38807,750],{"className":38808},[178],"。迪利克雷参数取为\n",[15,38811,38813,38861],{"className":38812,"translate":19},[18],[15,38814,38816],{"className":38815},[23],[25,38817,38818],{"xmlns":27},[29,38819,38820,38858],{},[32,38821,38822,38828,38830,38836,38838,38840,38842,38844,38846,38848,38850,38852,38854,38856],{},[266,38823,38824,38826],{},[35,38825,3106],{},[35,38827,272],{},[81,38829,244],{},[266,38831,38832,38834],{},[35,38833,1982],{},[35,38835,272],{},[81,38837,7762],{},[89,38839,91],{},[81,38841,94],{"separator":48},[127,38843],{"width":9847},[282,38845,12926],{},[127,38847],{"width":9847},[35,38849,3106],{"mathvariant":3105},[81,38851,244],{},[35,38853,1982],{"mathvariant":251},[81,38855,7762],{},[89,38857,91],{"mathvariant":251},[40,38859,38860],{"encoding":42},"\\alpha_k = e_k + 1, \\qquad \\text{即} \\qquad \\boldsymbol{\\alpha} = \\mathbf{e} + \\mathbf{1}",[15,38862,38864,38919,38974,39020,39038],{"className":38863,"ariaHidden":48},[47],[15,38865,38867,38870,38910,38913,38916],{"className":38866},[52],[15,38868],{"className":38869,"style":2511},[56],[15,38871,38873,38876],{"className":38872},[61],[15,38874,3106],{"className":38875,"style":3191},[61,122],[15,38877,38879],{"className":38878},[338],[15,38880,38882,38902],{"className":38881},[342,472],[15,38883,38885,38899],{"className":38884},[346],[15,38886,38888],{"className":38887,"style":479},[350],[15,38889,38890,38893],{"style":3206},[15,38891],{"className":38892,"style":358},[357],[15,38894,38896],{"className":38895},[362,363,364,365],[15,38897,272],{"className":38898,"style":492},[61,122,365],[15,38900,497],{"className":38901},[496],[15,38903,38905],{"className":38904},[346],[15,38906,38908],{"className":38907,"style":504},[350],[15,38909],{},[15,38911],{"className":38912,"style":128},[127],[15,38914,244],{"className":38915},[132],[15,38917],{"className":38918,"style":128},[127],[15,38920,38922,38925,38965,38968,38971],{"className":38921},[52],[15,38923],{"className":38924,"style":12617},[56],[15,38926,38928,38931],{"className":38927},[61],[15,38929,1982],{"className":38930},[61,122],[15,38932,38934],{"className":38933},[338],[15,38935,38937,38957],{"className":38936},[342,472],[15,38938,38940,38954],{"className":38939},[346],[15,38941,38943],{"className":38942,"style":479},[350],[15,38944,38945,38948],{"style":482},[15,38946],{"className":38947,"style":358},[357],[15,38949,38951],{"className":38950},[362,363,364,365],[15,38952,272],{"className":38953,"style":492},[61,122,365],[15,38955,497],{"className":38956},[496],[15,38958,38960],{"className":38959},[346],[15,38961,38963],{"className":38962,"style":504},[350],[15,38964],{},[15,38966],{"className":38967,"style":772},[127],[15,38969,7762],{"className":38970},[375],[15,38972],{"className":38973,"style":772},[127],[15,38975,38977,38981,38984,38987,38990,38993,38999,39002,39011,39014,39017],{"className":38976},[52],[15,38978],{"className":38979,"style":38980},[56],"height:0.8778em;vertical-align:-0.1944em;",[15,38982,91],{"className":38983},[61],[15,38985,94],{"className":38986},[153],[15,38988],{"className":38989,"style":10035},[127],[15,38991],{"className":38992,"style":157},[127],[15,38994,38996],{"className":38995},[61,13548],[15,38997,12926],{"className":38998},[61,13552],[15,39000],{"className":39001,"style":10035},[127],[15,39003,39005],{"className":39004},[61],[15,39006,39008],{"className":39007},[61],[15,39009,3106],{"className":39010},[61,3166],[15,39012],{"className":39013,"style":128},[127],[15,39015,244],{"className":39016},[132],[15,39018],{"className":39019,"style":128},[127],[15,39021,39023,39026,39029,39032,39035],{"className":39022},[52],[15,39024],{"className":39025,"style":6737},[56],[15,39027,1982],{"className":39028},[61,411],[15,39030],{"className":39031,"style":772},[127],[15,39033,7762],{"className":39034},[375],[15,39036],{"className":39037,"style":772},[127],[15,39039,39041,39044],{"className":39040},[52],[15,39042],{"className":39043,"style":859},[56],[15,39045,91],{"className":39046},[61,411],"\n网络对输入的预测由此被定义为 ",[15,39049,39051,39073],{"className":39050,"translate":19},[18],[15,39052,39054],{"className":39053},[23],[25,39055,39056],{"xmlns":27},[29,39057,39058,39070],{},[32,39059,39060,39062,39064,39066,39068],{},[35,39061,4186],{"mathvariant":231},[81,39063,1547],{},[81,39065,741],{"stretchy":86},[35,39067,3106],{"mathvariant":3105},[81,39069,750],{"stretchy":86},[40,39071,39072],{"encoding":42},"\\operatorname{Dir}(\\boldsymbol{\\alpha})",[15,39074,39076],{"className":39075,"ariaHidden":48},[47],[15,39077,39079,39082,39088,39091,39100],{"className":39078},[52],[15,39080],{"className":39081,"style":142},[56],[15,39083,39085],{"className":39084},[531],[15,39086,4186],{"className":39087},[61,1622],[15,39089,741],{"className":39090},[146],[15,39092,39094],{"className":39093},[61],[15,39095,39097],{"className":39096},[61],[15,39098,3106],{"className":39099},[61,3166],[15,39101,750],{"className":39102},[178],[11,39104,39105,39106,39185,39186,39287,39288,39358,39359,39387],{},"参数化的语义可直接由命题 3.2 读出：",[15,39107,39109,39131],{"className":39108,"translate":19},[18],[15,39110,39112],{"className":39111},[23],[25,39113,39114],{"xmlns":27},[29,39115,39116,39128],{},[32,39117,39118,39120,39122,39124,39126],{},[35,39119,1982],{"mathvariant":251},[81,39121,244],{},[35,39123,3106],{"mathvariant":3105},[81,39125,239],{},[89,39127,91],{"mathvariant":251},[40,39129,39130],{"encoding":42},"\\mathbf{e} = \\boldsymbol{\\alpha} - \\mathbf{1}",[15,39132,39134,39152,39176],{"className":39133,"ariaHidden":48},[47],[15,39135,39137,39140,39143,39146,39149],{"className":39136},[52],[15,39138],{"className":39139,"style":2708},[56],[15,39141,1982],{"className":39142},[61,411],[15,39144],{"className":39145,"style":128},[127],[15,39147,244],{"className":39148},[132],[15,39150],{"className":39151,"style":128},[127],[15,39153,39155,39158,39167,39170,39173],{"className":39154},[52],[15,39156],{"className":39157,"style":6737},[56],[15,39159,39161],{"className":39160},[61],[15,39162,39164],{"className":39163},[61],[15,39165,3106],{"className":39166},[61,3166],[15,39168],{"className":39169,"style":772},[127],[15,39171,239],{"className":39172},[375],[15,39174],{"className":39175,"style":772},[127],[15,39177,39179,39182],{"className":39178},[52],[15,39180],{"className":39181,"style":859},[56],[15,39183,91],{"className":39184},[61,411]," 正是",[661,39187,39188,39189,39286],{},"超出均匀先验 ",[15,39190,39192,39216],{"className":39191,"translate":19},[18],[15,39193,39195],{"className":39194},[23],[25,39196,39197],{"xmlns":27},[29,39198,39199,39213],{},[32,39200,39201,39207,39209,39211],{},[266,39202,39203,39205],{},[35,39204,33435],{},[35,39206,272],{},[35,39208,34983],{},[81,39210,244],{},[89,39212,91],{},[40,39214,39215],{"encoding":42},"a_k W = 1",[15,39217,39219,39277],{"className":39218,"ariaHidden":48},[47],[15,39220,39222,39225,39265,39268,39271,39274],{"className":39221},[52],[15,39223],{"className":39224,"style":34360},[56],[15,39226,39228,39231],{"className":39227},[61],[15,39229,33435],{"className":39230},[61,122],[15,39232,39234],{"className":39233},[338],[15,39235,39237,39257],{"className":39236},[342,472],[15,39238,39240,39254],{"className":39239},[346],[15,39241,39243],{"className":39242,"style":479},[350],[15,39244,39245,39248],{"style":482},[15,39246],{"className":39247,"style":358},[357],[15,39249,39251],{"className":39250},[362,363,364,365],[15,39252,272],{"className":39253,"style":492},[61,122,365],[15,39255,497],{"className":39256},[496],[15,39258,39260],{"className":39259},[346],[15,39261,39263],{"className":39262,"style":504},[350],[15,39264],{},[15,39266,34983],{"className":39267,"style":34367},[61,122],[15,39269],{"className":39270,"style":128},[127],[15,39272,244],{"className":39273},[132],[15,39275],{"className":39276,"style":128},[127],[15,39278,39280,39283],{"className":39279},[52],[15,39281],{"className":39282,"style":859},[56],[15,39284,91],{"className":39285},[61]," 的那部分伪计数","，即 2.6 节意义上「可解释为计数」的区域。这就是「证据」一词的来源：",[15,39289,39291,39309],{"className":39290,"translate":19},[18],[15,39292,39294],{"className":39293},[23],[25,39295,39296],{"xmlns":27},[29,39297,39298,39306],{},[32,39299,39300],{},[266,39301,39302,39304],{},[35,39303,1982],{},[35,39305,272],{},[40,39307,39308],{"encoding":42},"e_k",[15,39310,39312],{"className":39311,"ariaHidden":48},[47],[15,39313,39315,39318],{"className":39314},[52],[15,39316],{"className":39317,"style":2511},[56],[15,39319,39321,39324],{"className":39320},[61],[15,39322,1982],{"className":39323},[61,122],[15,39325,39327],{"className":39326},[338],[15,39328,39330,39350],{"className":39329},[342,472],[15,39331,39333,39347],{"className":39332},[346],[15,39334,39336],{"className":39335,"style":479},[350],[15,39337,39338,39341],{"style":482},[15,39339],{"className":39340,"style":358},[357],[15,39342,39344],{"className":39343},[362,363,364,365],[15,39345,272],{"className":39346,"style":492},[61,122,365],[15,39348,497],{"className":39349},[496],[15,39351,39353],{"className":39352},[346],[15,39354,39356],{"className":39355,"style":504},[350],[15,39357],{}," 是与类别 ",[15,39360,39362,39375],{"className":39361,"translate":19},[18],[15,39363,39365],{"className":39364},[23],[25,39366,39367],{"xmlns":27},[29,39368,39369,39373],{},[32,39370,39371],{},[35,39372,272],{},[40,39374,272],{"encoding":42},[15,39376,39378],{"className":39377,"ariaHidden":48},[47],[15,39379,39381,39384],{"className":39380},[52],[15,39382],{"className":39383,"style":10452},[56],[15,39385,272],{"className":39386,"style":492},[61,122]," 相关的、由网络从输入中提取出的支持性观测。",[11,39389,39390],{},"非负激活函数的选择影响显著，值得逐一说明：",[33497,39392,39393,39713,39833,40135],{},[31550,39394,39395,39538,39539,39595,39596,39655,39656,39712],{},[15,39396,39398,39441],{"className":39397,"translate":19},[18],[15,39399,39401],{"className":39400},[23],[25,39402,39403],{"xmlns":27},[29,39404,39405,39438],{},[32,39406,39407,39410,39412,39414,39416,39418,39420,39422,39424,39426,39428,39430,39436],{},[35,39408,39409],{"mathvariant":231},"softplus",[81,39411,1547],{},[81,39413,741],{"stretchy":86},[35,39415,37301],{},[81,39417,750],{"stretchy":86},[81,39419,244],{},[35,39421,22662],{},[81,39423,1547],{},[81,39425,741],{"stretchy":86},[89,39427,91],{},[81,39429,7762],{},[227,39431,39432,39434],{},[35,39433,1982],{},[35,39435,37301],{},[81,39437,750],{"stretchy":86},[40,39439,39440],{"encoding":42},"\\operatorname{softplus}(z) = \\log(1 + e^z)",[15,39442,39444,39474,39500],{"className":39443,"ariaHidden":48},[47],[15,39445,39447,39450,39456,39459,39462,39465,39468,39471],{"className":39446},[52],[15,39448],{"className":39449,"style":142},[56],[15,39451,39453],{"className":39452},[531],[15,39454,39409],{"className":39455},[61,1622],[15,39457,741],{"className":39458},[146],[15,39460,37301],{"className":39461,"style":37629},[61,122],[15,39463,750],{"className":39464},[178],[15,39466],{"className":39467,"style":128},[127],[15,39469,244],{"className":39470},[132],[15,39472],{"className":39473,"style":128},[127],[15,39475,39477,39480,39485,39488,39491,39494,39497],{"className":39476},[52],[15,39478],{"className":39479,"style":142},[56],[15,39481,22791,39483],{"className":39482},[531],[15,39484,22794],{"style":15619},[15,39486,741],{"className":39487},[146],[15,39489,91],{"className":39490},[61],[15,39492],{"className":39493,"style":772},[127],[15,39495,7762],{"className":39496},[375],[15,39498],{"className":39499,"style":772},[127],[15,39501,39503,39506,39535],{"className":39502},[52],[15,39504],{"className":39505,"style":142},[56],[15,39507,39509,39512],{"className":39508},[61],[15,39510,1982],{"className":39511},[61,122],[15,39513,39515],{"className":39514},[338],[15,39516,39518],{"className":39517},[342],[15,39519,39521],{"className":39520},[346],[15,39522,39524],{"className":39523,"style":37611},[350],[15,39525,39526,39529],{"style":716},[15,39527],{"className":39528,"style":358},[357],[15,39530,39532],{"className":39531},[362,363,364,365],[15,39533,37301],{"className":39534,"style":37629},[61,122,365],[15,39536,750],{"className":39537},[178],"：恒正、处处可微、在 ",[15,39540,39542,39562],{"className":39541,"translate":19},[18],[15,39543,39545],{"className":39544},[23],[25,39546,39547],{"xmlns":27},[29,39548,39549,39559],{},[32,39550,39551,39553,39555,39557],{},[35,39552,37301],{},[81,39554,20704],{},[81,39556,239],{},[35,39558,6796],{"mathvariant":231},[40,39560,39561],{"encoding":42},"z \\to -\\infty",[15,39563,39565,39583],{"className":39564,"ariaHidden":48},[47],[15,39566,39568,39571,39574,39577,39580],{"className":39567},[52],[15,39569],{"className":39570,"style":6032},[56],[15,39572,37301],{"className":39573,"style":37629},[61,122],[15,39575],{"className":39576,"style":128},[127],[15,39578,20704],{"className":39579},[132],[15,39581],{"className":39582,"style":128},[127],[15,39584,39586,39589,39592],{"className":39585},[52],[15,39587],{"className":39588,"style":6737},[56],[15,39590,239],{"className":39591},[61],[15,39593,6796],{"className":39594},[61]," 时以 ",[15,39597,39599,39617],{"className":39598,"translate":19},[18],[15,39600,39602],{"className":39601},[23],[25,39603,39604],{"xmlns":27},[29,39605,39606,39614],{},[32,39607,39608],{},[227,39609,39610,39612],{},[35,39611,1982],{},[35,39613,37301],{},[40,39615,39616],{"encoding":42},"e^z",[15,39618,39620],{"className":39619,"ariaHidden":48},[47],[15,39621,39623,39626],{"className":39622},[52],[15,39624],{"className":39625,"style":37611},[56],[15,39627,39629,39632],{"className":39628},[61],[15,39630,1982],{"className":39631},[61,122],[15,39633,39635],{"className":39634},[338],[15,39636,39638],{"className":39637},[342],[15,39639,39641],{"className":39640},[346],[15,39642,39644],{"className":39643,"style":37611},[350],[15,39645,39646,39649],{"style":716},[15,39647],{"className":39648,"style":358},[357],[15,39650,39652],{"className":39651},[362,363,364,365],[15,39653,37301],{"className":39654,"style":37629},[61,122,365]," 的速度趋零、在 ",[15,39657,39659,39679],{"className":39658,"translate":19},[18],[15,39660,39662],{"className":39661},[23],[25,39663,39664],{"xmlns":27},[29,39665,39666,39676],{},[32,39667,39668,39670,39672,39674],{},[35,39669,37301],{},[81,39671,20704],{},[81,39673,7762],{},[35,39675,6796],{"mathvariant":231},[40,39677,39678],{"encoding":42},"z \\to +\\infty",[15,39680,39682,39700],{"className":39681,"ariaHidden":48},[47],[15,39683,39685,39688,39691,39694,39697],{"className":39684},[52],[15,39686],{"className":39687,"style":6032},[56],[15,39689,37301],{"className":39690,"style":37629},[61,122],[15,39692],{"className":39693,"style":128},[127],[15,39695,20704],{"className":39696},[132],[15,39698],{"className":39699,"style":128},[127],[15,39701,39703,39706,39709],{"className":39702},[52],[15,39704],{"className":39705,"style":6737},[56],[15,39707,7762],{"className":39708},[61],[15,39710,6796],{"className":39711},[61]," 时线性增长。它既提供了精确的零证据状态，又避免了指数爆炸，是实践中默认的选择。",[31550,39714,39715,39762,39763,39832],{},[15,39716,39718,39741],{"className":39717,"translate":19},[18],[15,39719,39721],{"className":39720},[23],[25,39722,39723],{"xmlns":27},[29,39724,39725,39738],{},[32,39726,39727,39730,39732,39734,39736],{},[35,39728,39729],{},"exp",[81,39731,1547],{},[81,39733,741],{"stretchy":86},[35,39735,37301],{},[81,39737,750],{"stretchy":86},[40,39739,39740],{"encoding":42},"\\exp(z)",[15,39742,39744],{"className":39743,"ariaHidden":48},[47],[15,39745,39747,39750,39753,39756,39759],{"className":39746},[52],[15,39748],{"className":39749,"style":142},[56],[15,39751,39729],{"className":39752},[531],[15,39754,741],{"className":39755},[146],[15,39757,37301],{"className":39758,"style":37629},[61,122],[15,39760,750],{"className":39761},[178],"：严格正、光滑，但无上界，使 ",[15,39764,39766,39783],{"className":39765,"translate":19},[18],[15,39767,39769],{"className":39768},[23],[25,39770,39771],{"xmlns":27},[29,39772,39773,39781],{},[32,39774,39775],{},[266,39776,39777,39779],{},[35,39778,3106],{},[89,39780,278],{},[40,39782,19490],{"encoding":42},[15,39784,39786],{"className":39785,"ariaHidden":48},[47],[15,39787,39789,39792],{"className":39788},[52],[15,39790],{"className":39791,"style":2511},[56],[15,39793,39795,39798],{"className":39794},[61],[15,39796,3106],{"className":39797,"style":3191},[61,122],[15,39799,39801],{"className":39800},[338],[15,39802,39804,39824],{"className":39803},[342,472],[15,39805,39807,39821],{"className":39806},[346],[15,39808,39810],{"className":39809,"style":1214},[350],[15,39811,39812,39815],{"style":3206},[15,39813],{"className":39814,"style":358},[357],[15,39816,39818],{"className":39817},[362,363,364,365],[15,39819,278],{"className":39820},[61,365],[15,39822,497],{"className":39823},[496],[15,39825,39827],{"className":39826},[346],[15,39828,39830],{"className":39829,"style":504},[350],[15,39831],{}," 可任意大，易造成数值溢出与损失对证据的不稳定响应。",[31550,39834,39835,39885,39886,39937,39938,40030,40031,40034,40035,40105,40106,40134],{},[15,39836,39838,39861],{"className":39837,"translate":19},[18],[15,39839,39841],{"className":39840},[23],[25,39842,39843],{"xmlns":27},[29,39844,39845,39858],{},[32,39846,39847,39850,39852,39854,39856],{},[35,39848,39849],{"mathvariant":231},"ReLU",[81,39851,1547],{},[81,39853,741],{"stretchy":86},[35,39855,37301],{},[81,39857,750],{"stretchy":86},[40,39859,39860],{"encoding":42},"\\operatorname{ReLU}(z)",[15,39862,39864],{"className":39863,"ariaHidden":48},[47],[15,39865,39867,39870,39876,39879,39882],{"className":39866},[52],[15,39868],{"className":39869,"style":142},[56],[15,39871,39873],{"className":39872},[531],[15,39874,39849],{"className":39875},[61,1622],[15,39877,741],{"className":39878},[146],[15,39880,37301],{"className":39881,"style":37629},[61,122],[15,39883,750],{"className":39884},[178],"：计算廉价，但在 ",[15,39887,39889,39907],{"className":39888,"translate":19},[18],[15,39890,39892],{"className":39891},[23],[25,39893,39894],{"xmlns":27},[29,39895,39896,39904],{},[32,39897,39898,39900,39902],{},[35,39899,37301],{},[81,39901,2490],{},[89,39903,278],{},[40,39905,39906],{"encoding":42},"z \u003C 0",[15,39908,39910,39928],{"className":39909,"ariaHidden":48},[47],[15,39911,39913,39916,39919,39922,39925],{"className":39912},[52],[15,39914],{"className":39915,"style":4656},[56],[15,39917,37301],{"className":39918,"style":37629},[61,122],[15,39920],{"className":39921,"style":128},[127],[15,39923,2490],{"className":39924},[132],[15,39926],{"className":39927,"style":128},[127],[15,39929,39931,39934],{"className":39930},[52],[15,39932],{"className":39933,"style":859},[56],[15,39935,278],{"className":39936},[61]," 处梯度恒为零。处于该区域的单元输出证据 ",[15,39939,39941,39963],{"className":39940,"translate":19},[18],[15,39942,39944],{"className":39943},[23],[25,39945,39946],{"xmlns":27},[29,39947,39948,39960],{},[32,39949,39950,39956,39958],{},[266,39951,39952,39954],{},[35,39953,1982],{},[35,39955,272],{},[81,39957,244],{},[89,39959,278],{},[40,39961,39962],{"encoding":42},"e_k = 0",[15,39964,39966,40021],{"className":39965,"ariaHidden":48},[47],[15,39967,39969,39972,40012,40015,40018],{"className":39968},[52],[15,39970],{"className":39971,"style":2511},[56],[15,39973,39975,39978],{"className":39974},[61],[15,39976,1982],{"className":39977},[61,122],[15,39979,39981],{"className":39980},[338],[15,39982,39984,40004],{"className":39983},[342,472],[15,39985,39987,40001],{"className":39986},[346],[15,39988,39990],{"className":39989,"style":479},[350],[15,39991,39992,39995],{"style":482},[15,39993],{"className":39994,"style":358},[357],[15,39996,39998],{"className":39997},[362,363,364,365],[15,39999,272],{"className":40000,"style":492},[61,122,365],[15,40002,497],{"className":40003},[496],[15,40005,40007],{"className":40006},[346],[15,40008,40010],{"className":40009,"style":504},[350],[15,40011],{},[15,40013],{"className":40014,"style":128},[127],[15,40016,244],{"className":40017},[132],[15,40019],{"className":40020,"style":128},[127],[15,40022,40024,40027],{"className":40023},[52],[15,40025],{"className":40026,"style":859},[56],[15,40028,278],{"className":40029},[61]," 且",[661,40032,40033],{},"永远无法通过梯度逃离","——",[15,40036,40038,40056],{"className":40037,"translate":19},[18],[15,40039,40041],{"className":40040},[23],[25,40042,40043],{"xmlns":27},[29,40044,40045,40053],{},[32,40046,40047],{},[266,40048,40049,40051],{},[35,40050,3106],{},[35,40052,272],{},[40,40054,40055],{"encoding":42},"\\alpha_k",[15,40057,40059],{"className":40058,"ariaHidden":48},[47],[15,40060,40062,40065],{"className":40061},[52],[15,40063],{"className":40064,"style":2511},[56],[15,40066,40068,40071],{"className":40067},[61],[15,40069,3106],{"className":40070,"style":3191},[61,122],[15,40072,40074],{"className":40073},[338],[15,40075,40077,40097],{"className":40076},[342,472],[15,40078,40080,40094],{"className":40079},[346],[15,40081,40083],{"className":40082,"style":479},[350],[15,40084,40085,40088],{"style":3206},[15,40086],{"className":40087,"style":358},[357],[15,40089,40091],{"className":40090},[362,363,364,365],[15,40092,272],{"className":40093,"style":492},[61,122,365],[15,40095,497],{"className":40096},[496],[15,40098,40100],{"className":40099},[346],[15,40101,40103],{"className":40102,"style":504},[350],[15,40104],{}," 被锁在 ",[15,40107,40109,40122],{"className":40108,"translate":19},[18],[15,40110,40112],{"className":40111},[23],[25,40113,40114],{"xmlns":27},[29,40115,40116,40120],{},[32,40117,40118],{},[89,40119,91],{},[40,40121,91],{"encoding":42},[15,40123,40125],{"className":40124,"ariaHidden":48},[47],[15,40126,40128,40131],{"className":40127},[52],[15,40129],{"className":40130,"style":859},[56],[15,40132,91],{"className":40133},[61],"，形成死区。这使 ReLU 在证据函数中通常劣于 softplus。",[31550,40136,40137,40209,40210,40238],{},[15,40138,40140,40167],{"className":40139,"translate":19},[18],[15,40141,40143],{"className":40142},[23],[25,40144,40145],{"xmlns":27},[29,40146,40147,40164],{},[32,40148,40149,40152,40154,40156,40158,40160,40162],{},[35,40150,40151],{"mathvariant":231},"ELU",[81,40153,1547],{},[81,40155,741],{"stretchy":86},[35,40157,37301],{},[81,40159,750],{"stretchy":86},[81,40161,7762],{},[89,40163,91],{},[40,40165,40166],{"encoding":42},"\\operatorname{ELU}(z) + 1",[15,40168,40170,40200],{"className":40169,"ariaHidden":48},[47],[15,40171,40173,40176,40182,40185,40188,40191,40194,40197],{"className":40172},[52],[15,40174],{"className":40175,"style":142},[56],[15,40177,40179],{"className":40178},[531],[15,40180,40151],{"className":40181},[61,1622],[15,40183,741],{"className":40184},[146],[15,40186,37301],{"className":40187,"style":37629},[61,122],[15,40189,750],{"className":40190},[178],[15,40192],{"className":40193,"style":772},[127],[15,40195,7762],{"className":40196},[375],[15,40198],{"className":40199,"style":772},[127],[15,40201,40203,40206],{"className":40202},[52],[15,40204],{"className":40205,"style":859},[56],[15,40207,91],{"className":40208},[61],"：具有负饱和下界 ",[15,40211,40213,40226],{"className":40212,"translate":19},[18],[15,40214,40216],{"className":40215},[23],[25,40217,40218],{"xmlns":27},[29,40219,40220,40224],{},[32,40221,40222],{},[89,40223,278],{},[40,40225,278],{"encoding":42},[15,40227,40229],{"className":40228,"ariaHidden":48},[47],[15,40230,40232,40235],{"className":40231},[52],[15,40233],{"className":40234,"style":859},[56],[15,40236,278],{"className":40237},[61],"，梯度在负半轴非零，是介于二者之间的折中。",[877,40240,40242],{"id":40241},"_35-预测概率与不确定性的解析形式","3.5 预测概率与不确定性的解析形式",[882,40244,40245],{},[11,40246,40247,40250,40251,40647,40648],{},[661,40248,40249],{},"命题 3.4（EDL 的预测与不确定性）"," 在定义 3.3 的参数化下，记总强度 ",[15,40252,40254,40325],{"className":40253,"translate":19},[18],[15,40255,40257],{"className":40256},[23],[25,40258,40259],{"xmlns":27},[29,40260,40261,40322],{},[32,40262,40263,40266,40268,40274,40276,40290,40296,40298,40300,40302,40316],{},[35,40264,40265],{},"S",[81,40267,244],{},[266,40269,40270,40272],{},[35,40271,3106],{},[89,40273,278],{},[81,40275,244],{},[289,40277,40278,40280,40288],{},[81,40279,293],{},[32,40281,40282,40284,40286],{},[35,40283,272],{},[81,40285,244],{},[89,40287,91],{},[35,40289,102],{},[266,40291,40292,40294],{},[35,40293,3106],{},[35,40295,272],{},[81,40297,244],{},[35,40299,102],{},[81,40301,7762],{},[289,40303,40304,40306,40314],{},[81,40305,293],{},[32,40307,40308,40310,40312],{},[35,40309,272],{},[81,40311,244],{},[89,40313,91],{},[35,40315,102],{},[266,40317,40318,40320],{},[35,40319,1982],{},[35,40321,272],{},[40,40323,40324],{"encoding":42},"S = \\alpha_0 = \\sum_{k=1}^K \\alpha_k = K + \\sum_{k=1}^K e_k",[15,40326,40328,40347,40402,40520,40538],{"className":40327,"ariaHidden":48},[47],[15,40329,40331,40334,40338,40341,40344],{"className":40330},[52],[15,40332],{"className":40333,"style":57},[56],[15,40335,40265],{"className":40336,"style":40337},[61,122],"margin-right:0.0576em;",[15,40339],{"className":40340,"style":128},[127],[15,40342,244],{"className":40343},[132],[15,40345],{"className":40346,"style":128},[127],[15,40348,40350,40353,40393,40396,40399],{"className":40349},[52],[15,40351],{"className":40352,"style":2511},[56],[15,40354,40356,40359],{"className":40355},[61],[15,40357,3106],{"className":40358,"style":3191},[61,122],[15,40360,40362],{"className":40361},[338],[15,40363,40365,40385],{"className":40364},[342,472],[15,40366,40368,40382],{"className":40367},[346],[15,40369,40371],{"className":40370,"style":1214},[350],[15,40372,40373,40376],{"style":3206},[15,40374],{"className":40375,"style":358},[357],[15,40377,40379],{"className":40378},[362,363,364,365],[15,40380,278],{"className":40381},[61,365],[15,40383,497],{"className":40384},[496],[15,40386,40388],{"className":40387},[346],[15,40389,40391],{"className":40390,"style":504},[350],[15,40392],{},[15,40394],{"className":40395,"style":128},[127],[15,40397,244],{"className":40398},[132],[15,40400],{"className":40401,"style":128},[127],[15,40403,40405,40408,40468,40471,40511,40514,40517],{"className":40404},[52],[15,40406],{"className":40407,"style":3471},[56],[15,40409,40411,40414],{"className":40410},[531],[15,40412,293],{"className":40413,"style":537},[531,535,536],[15,40415,40417],{"className":40416},[338],[15,40418,40420,40460],{"className":40419},[342,472],[15,40421,40423,40457],{"className":40422},[346],[15,40424,40426,40446],{"className":40425,"style":550},[350],[15,40427,40428,40431],{"style":553},[15,40429],{"className":40430,"style":358},[357],[15,40432,40434],{"className":40433},[362,363,364,365],[15,40435,40437,40440,40443],{"className":40436},[61,365],[15,40438,272],{"className":40439,"style":492},[61,122,365],[15,40441,244],{"className":40442},[132,365],[15,40444,91],{"className":40445},[61,365],[15,40447,40448,40451],{"style":574},[15,40449],{"className":40450,"style":358},[357],[15,40452,40454],{"className":40453},[362,363,364,365],[15,40455,102],{"className":40456,"style":174},[61,122,365],[15,40458,497],{"className":40459},[496],[15,40461,40463],{"className":40462},[346],[15,40464,40466],{"className":40465,"style":593},[350],[15,40467],{},[15,40469],{"className":40470,"style":157},[127],[15,40472,40474,40477],{"className":40473},[61],[15,40475,3106],{"className":40476,"style":3191},[61,122],[15,40478,40480],{"className":40479},[338],[15,40481,40483,40503],{"className":40482},[342,472],[15,40484,40486,40500],{"className":40485},[346],[15,40487,40489],{"className":40488,"style":479},[350],[15,40490,40491,40494],{"style":3206},[15,40492],{"className":40493,"style":358},[357],[15,40495,40497],{"className":40496},[362,363,364,365],[15,40498,272],{"className":40499,"style":492},[61,122,365],[15,40501,497],{"className":40502},[496],[15,40504,40506],{"className":40505},[346],[15,40507,40509],{"className":40508,"style":504},[350],[15,40510],{},[15,40512],{"className":40513,"style":128},[127],[15,40515,244],{"className":40516},[132],[15,40518],{"className":40519,"style":128},[127],[15,40521,40523,40526,40529,40532,40535],{"className":40522},[52],[15,40524],{"className":40525,"style":974},[56],[15,40527,102],{"className":40528,"style":174},[61,122],[15,40530],{"className":40531,"style":772},[127],[15,40533,7762],{"className":40534},[375],[15,40536],{"className":40537,"style":772},[127],[15,40539,40541,40544,40604,40607],{"className":40540},[52],[15,40542],{"className":40543,"style":3471},[56],[15,40545,40547,40550],{"className":40546},[531],[15,40548,293],{"className":40549,"style":537},[531,535,536],[15,40551,40553],{"className":40552},[338],[15,40554,40556,40596],{"className":40555},[342,472],[15,40557,40559,40593],{"className":40558},[346],[15,40560,40562,40582],{"className":40561,"style":550},[350],[15,40563,40564,40567],{"style":553},[15,40565],{"className":40566,"style":358},[357],[15,40568,40570],{"className":40569},[362,363,364,365],[15,40571,40573,40576,40579],{"className":40572},[61,365],[15,40574,272],{"className":40575,"style":492},[61,122,365],[15,40577,244],{"className":40578},[132,365],[15,40580,91],{"className":40581},[61,365],[15,40583,40584,40587],{"style":574},[15,40585],{"className":40586,"style":358},[357],[15,40588,40590],{"className":40589},[362,363,364,365],[15,40591,102],{"className":40592,"style":174},[61,122,365],[15,40594,497],{"className":40595},[496],[15,40597,40599],{"className":40598},[346],[15,40600,40602],{"className":40601,"style":593},[350],[15,40603],{},[15,40605],{"className":40606,"style":157},[127],[15,40608,40610,40613],{"className":40609},[61],[15,40611,1982],{"className":40612},[61,122],[15,40614,40616],{"className":40615},[338],[15,40617,40619,40639],{"className":40618},[342,472],[15,40620,40622,40636],{"className":40621},[346],[15,40623,40625],{"className":40624,"style":479},[350],[15,40626,40627,40630],{"style":482},[15,40628],{"className":40629,"style":358},[357],[15,40631,40633],{"className":40632},[362,363,364,365],[15,40634,272],{"className":40635,"style":492},[61,122,365],[15,40637,497],{"className":40638},[496],[15,40640,40642],{"className":40641},[346],[15,40643,40645],{"className":40644,"style":504},[350],[15,40646],{},"。则\n",[15,40649,40651,40709],{"className":40650,"translate":19},[18],[15,40652,40654],{"className":40653},[23],[25,40655,40656],{"xmlns":27},[29,40657,40658,40706],{},[32,40659,40660,40666,40668,40670,40672,40678,40680,40682,40692,40694,40696,40698,40700],{},[266,40661,40662,40664],{},[35,40663,11],{},[35,40665,272],{},[81,40667,244],{},[35,40669,14818],{"mathvariant":258},[81,40671,14821],{"stretchy":86},[266,40673,40674,40676],{},[35,40675,1029],{},[35,40677,272],{},[81,40679,14830],{"stretchy":86},[81,40681,244],{},[3722,40683,40684,40690],{},[266,40685,40686,40688],{},[35,40687,3106],{},[35,40689,272],{},[35,40691,40265],{},[81,40693,94],{"separator":48},[127,40695],{"width":9847},[35,40697,33430],{},[81,40699,244],{},[3722,40701,40702,40704],{},[35,40703,102],{},[35,40705,40265],{},[40,40707,40708],{"encoding":42},"p_k = \\mathbb{E}[x_k] = \\frac{\\alpha_k}{S}, \\qquad u = \\frac{K}{S}",[15,40710,40712,40767,40831,40963],{"className":40711,"ariaHidden":48},[47],[15,40713,40715,40718,40758,40761,40764],{"className":40714},[52],[15,40716],{"className":40717,"style":9437},[56],[15,40719,40721,40724],{"className":40720},[61],[15,40722,11],{"className":40723},[61,122],[15,40725,40727],{"className":40726},[338],[15,40728,40730,40750],{"className":40729},[342,472],[15,40731,40733,40747],{"className":40732},[346],[15,40734,40736],{"className":40735,"style":479},[350],[15,40737,40738,40741],{"style":482},[15,40739],{"className":40740,"style":358},[357],[15,40742,40744],{"className":40743},[362,363,364,365],[15,40745,272],{"className":40746,"style":492},[61,122,365],[15,40748,497],{"className":40749},[496],[15,40751,40753],{"className":40752},[346],[15,40754,40756],{"className":40755,"style":504},[350],[15,40757],{},[15,40759],{"className":40760,"style":128},[127],[15,40762,244],{"className":40763},[132],[15,40765],{"className":40766,"style":128},[127],[15,40768,40770,40773,40776,40779,40819,40822,40825,40828],{"className":40769},[52],[15,40771],{"className":40772,"style":142},[56],[15,40774,14818],{"className":40775},[61,427],[15,40777,14821],{"className":40778},[146],[15,40780,40782,40785],{"className":40781},[61],[15,40783,1029],{"className":40784},[61,122],[15,40786,40788],{"className":40787},[338],[15,40789,40791,40811],{"className":40790},[342,472],[15,40792,40794,40808],{"className":40793},[346],[15,40795,40797],{"className":40796,"style":479},[350],[15,40798,40799,40802],{"style":482},[15,40800],{"className":40801,"style":358},[357],[15,40803,40805],{"className":40804},[362,363,364,365],[15,40806,272],{"className":40807,"style":492},[61,122,365],[15,40809,497],{"className":40810},[496],[15,40812,40814],{"className":40813},[346],[15,40815,40817],{"className":40816,"style":504},[350],[15,40818],{},[15,40820,14830],{"className":40821},[178],[15,40823],{"className":40824,"style":128},[127],[15,40826,244],{"className":40827},[132],[15,40829],{"className":40830,"style":128},[127],[15,40832,40834,40837,40942,40945,40948,40951,40954,40957,40960],{"className":40833},[52],[15,40835],{"className":40836,"style":28329},[56],[15,40838,40840,40843,40939],{"className":40839},[61],[15,40841],{"className":40842},[146,3841],[15,40844,40846],{"className":40845},[3722],[15,40847,40849,40931],{"className":40848},[342,472],[15,40850,40852,40928],{"className":40851},[346],[15,40853,40855,40869,40877],{"className":40854,"style":27951},[350],[15,40856,40857,40860],{"style":3857},[15,40858],{"className":40859,"style":3861},[357],[15,40861,40863],{"className":40862},[362,363,364,365],[15,40864,40866],{"className":40865},[61,365],[15,40867,40265],{"className":40868,"style":40337},[61,122,365],[15,40870,40871,40874],{"style":3889},[15,40872],{"className":40873,"style":3861},[357],[15,40875],{"className":40876,"style":3897},[3896],[15,40878,40879,40882],{"style":27204},[15,40880],{"className":40881,"style":3861},[357],[15,40883,40885],{"className":40884},[362,363,364,365],[15,40886,40888],{"className":40887},[61,365],[15,40889,40891,40894],{"className":40890},[61,365],[15,40892,3106],{"className":40893,"style":3191},[61,122,365],[15,40895,40897],{"className":40896},[338],[15,40898,40900,40920],{"className":40899},[342,472],[15,40901,40903,40917],{"className":40902},[346],[15,40904,40906],{"className":40905,"style":22404},[350],[15,40907,40908,40911],{"style":27234},[15,40909],{"className":40910,"style":4062},[357],[15,40912,40914],{"className":40913},[362,4066,4067,365],[15,40915,272],{"className":40916,"style":492},[61,122,365],[15,40918,497],{"className":40919},[496],[15,40921,40923],{"className":40922},[346],[15,40924,40926],{"className":40925,"style":27253},[350],[15,40927],{},[15,40929,497],{"className":40930},[496],[15,40932,40934],{"className":40933},[346],[15,40935,40937],{"className":40936,"style":10198},[350],[15,40938],{},[15,40940],{"className":40941},[178,3841],[15,40943,94],{"className":40944},[153],[15,40946],{"className":40947,"style":10035},[127],[15,40949],{"className":40950,"style":157},[127],[15,40952,33430],{"className":40953},[61,122],[15,40955],{"className":40956,"style":128},[127],[15,40958,244],{"className":40959},[132],[15,40961],{"className":40962,"style":128},[127],[15,40964,40966,40969],{"className":40965},[52],[15,40967],{"className":40968,"style":35775},[56],[15,40970,40972,40975,41034],{"className":40971},[61],[15,40973],{"className":40974},[146,3841],[15,40976,40978],{"className":40977},[3722],[15,40979,40981,41026],{"className":40980},[342,472],[15,40982,40984,41023],{"className":40983},[346],[15,40985,40987,41001,41009],{"className":40986,"style":35416},[350],[15,40988,40989,40992],{"style":3857},[15,40990],{"className":40991,"style":3861},[357],[15,40993,40995],{"className":40994},[362,363,364,365],[15,40996,40998],{"className":40997},[61,365],[15,40999,40265],{"className":41000,"style":40337},[61,122,365],[15,41002,41003,41006],{"style":3889},[15,41004],{"className":41005,"style":3861},[357],[15,41007],{"className":41008,"style":3897},[3896],[15,41010,41011,41014],{"style":3900},[15,41012],{"className":41013,"style":3861},[357],[15,41015,41017],{"className":41016},[362,363,364,365],[15,41018,41020],{"className":41019},[61,365],[15,41021,102],{"className":41022,"style":174},[61,122,365],[15,41024,497],{"className":41025},[496],[15,41027,41029],{"className":41028},[346],[15,41030,41032],{"className":41031,"style":10198},[350],[15,41033],{},[15,41035],{"className":41036},[178,3841],[11,41038,41039,41041,41042,41070,41071,41140,41141,867],{},[661,41040,5123],{}," 第一式即命题 2.1。第二式即命题 3.2。对 ",[15,41043,41045,41058],{"className":41044,"translate":19},[18],[15,41046,41048],{"className":41047},[23],[25,41049,41050],{"xmlns":27},[29,41051,41052,41056],{},[32,41053,41054],{},[35,41055,33430],{},[40,41057,33430],{"encoding":42},[15,41059,41061],{"className":41060,"ariaHidden":48},[47],[15,41062,41064,41067],{"className":41063},[52],[15,41065],{"className":41066,"style":6032},[56],[15,41068,33430],{"className":41069},[61,122]," 的推导亦可独立地由聚合性（命题 2.8）得到：若把全部类别合并为一个事件，其迪利克雷参数为 ",[15,41072,41074,41091],{"className":41073,"translate":19},[18],[15,41075,41077],{"className":41076},[23],[25,41078,41079],{"xmlns":27},[29,41080,41081,41089],{},[32,41082,41083],{},[266,41084,41085,41087],{},[35,41086,3106],{},[89,41088,278],{},[40,41090,19490],{"encoding":42},[15,41092,41094],{"className":41093,"ariaHidden":48},[47],[15,41095,41097,41100],{"className":41096},[52],[15,41098],{"className":41099,"style":2511},[56],[15,41101,41103,41106],{"className":41102},[61],[15,41104,3106],{"className":41105,"style":3191},[61,122],[15,41107,41109],{"className":41108},[338],[15,41110,41112,41132],{"className":41111},[342,472],[15,41113,41115,41129],{"className":41114},[346],[15,41116,41118],{"className":41117,"style":1214},[350],[15,41119,41120,41123],{"style":3206},[15,41121],{"className":41122,"style":358},[357],[15,41124,41126],{"className":41125},[362,363,364,365],[15,41127,278],{"className":41128},[61,365],[15,41130,497],{"className":41131},[496],[15,41133,41135],{"className":41134},[346],[15,41136,41138],{"className":41137,"style":504},[350],[15,41139],{},"，不确定性质量即为 ",[15,41142,41144,41166],{"className":41143,"translate":19},[18],[15,41145,41147],{"className":41146},[23],[25,41148,41149],{"xmlns":27},[29,41150,41151,41163],{},[32,41152,41153,41155,41157],{},[35,41154,102],{},[35,41156,3020],{"mathvariant":231},[266,41158,41159,41161],{},[35,41160,3106],{},[89,41162,278],{},[40,41164,41165],{"encoding":42},"K\u002F\\alpha_0",[15,41167,41169],{"className":41168,"ariaHidden":48},[47],[15,41170,41172,41175,41178,41181],{"className":41171},[52],[15,41173],{"className":41174,"style":142},[56],[15,41176,102],{"className":41177,"style":174},[61,122],[15,41179,3020],{"className":41180},[61],[15,41182,41184,41187],{"className":41183},[61],[15,41185,3106],{"className":41186,"style":3191},[61,122],[15,41188,41190],{"className":41189},[338],[15,41191,41193,41213],{"className":41192},[342,472],[15,41194,41196,41210],{"className":41195},[346],[15,41197,41199],{"className":41198,"style":1214},[350],[15,41200,41201,41204],{"style":3206},[15,41202],{"className":41203,"style":358},[357],[15,41205,41207],{"className":41206},[362,363,364,365],[15,41208,278],{"className":41209},[61,365],[15,41211,497],{"className":41212},[496],[15,41214,41216],{"className":41215},[346],[15,41217,41219],{"className":41218,"style":504},[350],[15,41220],{},[11,41222,41223,41224,41252],{},"在 ",[15,41225,41227,41240],{"className":41226,"translate":19},[18],[15,41228,41230],{"className":41229},[23],[25,41231,41232],{"xmlns":27},[29,41233,41234,41238],{},[32,41235,41236],{},[35,41237,33430],{},[40,41239,33430],{"encoding":42},[15,41241,41243],{"className":41242,"ariaHidden":48},[47],[15,41244,41246,41249],{"className":41245},[52],[15,41247],{"className":41248,"style":6032},[56],[15,41250,33430],{"className":41251},[61,122]," 的定义下，预测概率具有一个在解释上极有用的分解：",[15,41254,41256],{"className":41255,"translate":19},[212],[15,41257,41259,41373],{"className":41258,"translate":19},[18],[15,41260,41262],{"className":41261},[23],[25,41263,41264],{"xmlns":27,"display":221},[29,41265,41266,41370],{},[32,41267,41268,41274,41276,41301,41303,41318,41320,41322,41324,41326,41328,41330,41332,41358,41360,41362,41364],{},[266,41269,41270,41272],{},[35,41271,11],{},[35,41273,272],{},[81,41275,244],{},[6881,41277,41278,41298],{},[6881,41279,41280,41296],{},[3722,41281,41282,41294],{},[32,41283,41284,41290,41292],{},[266,41285,41286,41288],{},[35,41287,3106],{},[35,41289,272],{},[81,41291,239],{},[89,41293,91],{},[35,41295,40265],{},[81,41297,27483],{"stretchy":48},[282,41299,41300],{},"证据部分",[81,41302,7762],{},[6881,41304,41305,41315],{},[6881,41306,41307,41313],{},[3722,41308,41309,41311],{},[89,41310,91],{},[35,41312,40265],{},[81,41314,27483],{"stretchy":48},[282,41316,41317],{},"先验部分",[81,41319,244],{},[81,41321,741],{"stretchy":86},[89,41323,91],{},[81,41325,239],{},[35,41327,33430],{},[81,41329,750],{"stretchy":86},[282,41331,4802],{},[3722,41333,41334,41346],{},[32,41335,41336,41342,41344],{},[266,41337,41338,41340],{},[35,41339,3106],{},[35,41341,272],{},[81,41343,239],{},[89,41345,91],{},[32,41347,41348,41354,41356],{},[266,41349,41350,41352],{},[35,41351,3106],{},[89,41353,278],{},[81,41355,239],{},[35,41357,102],{},[81,41359,7762],{},[35,41361,33430],{},[81,41363,11214],{},[3722,41365,41366,41368],{},[89,41367,91],{},[35,41369,102],{},[40,41371,41372],{"encoding":42},"p_k = \\underbrace{\\frac{\\alpha_k - 1}{S}}_{\\text{证据部分}} + \\underbrace{\\frac{1}{S}}_{\\text{先验部分}} = (1 - u)\\,\\frac{\\alpha_k - 1}{\\alpha_0 - K} + u \\cdot \\frac{1}{K}",[15,41374,41376,41431,41670,41853,41874,42058,42077],{"className":41375,"ariaHidden":48},[47],[15,41377,41379,41382,41422,41425,41428],{"className":41378},[52],[15,41380],{"className":41381,"style":9437},[56],[15,41383,41385,41388],{"className":41384},[61],[15,41386,11],{"className":41387},[61,122],[15,41389,41391],{"className":41390},[338],[15,41392,41394,41414],{"className":41393},[342,472],[15,41395,41397,41411],{"className":41396},[346],[15,41398,41400],{"className":41399,"style":479},[350],[15,41401,41402,41405],{"style":482},[15,41403],{"className":41404,"style":358},[357],[15,41406,41408],{"className":41407},[362,363,364,365],[15,41409,272],{"className":41410,"style":492},[61,122,365],[15,41412,497],{"className":41413},[496],[15,41415,41417],{"className":41416},[346],[15,41418,41420],{"className":41419,"style":504},[350],[15,41421],{},[15,41423],{"className":41424,"style":128},[127],[15,41426,244],{"className":41427},[132],[15,41429],{"className":41430,"style":128},[127],[15,41432,41434,41438,41661,41664,41667],{"className":41433},[52],[15,41435],{"className":41436,"style":41437},[56],"height:3.3338em;vertical-align:-2.0123em;",[15,41439,41441],{"className":41440},[161,6881],[15,41442,41444,41652],{"className":41443},[342,472],[15,41445,41447,41649],{"className":41446},[346],[15,41448,41450,41469],{"className":41449,"style":18690},[350],[15,41451,41453,41457],{"style":41452},"top:-1.3091em;",[15,41454],{"className":41455,"style":41456},[357],"height:3.3214em;",[15,41458,41460],{"className":41459},[362,363,364,365],[15,41461,41463],{"className":41462},[61,365],[15,41464,41466],{"className":41465},[61,13548,365],[15,41467,41300],{"className":41468},[61,13552,365],[15,41470,41472,41475],{"style":41471},"top:-3.3214em;",[15,41473],{"className":41474,"style":41456},[357],[15,41476,41478],{"className":41477},[161,6881],[15,41479,41481,41640],{"className":41480},[342,472],[15,41482,41484,41637],{"className":41483},[346],[15,41485,41487,41518],{"className":41486,"style":18690},[350],[15,41488,41491,41494],{"className":41489,"style":41490},[27690],"top:-1.9874em;",[15,41492],{"className":41493,"style":41456},[357],[15,41495,41497,41504,41511],{"className":41496,"style":27699},[27698],[15,41498,41500],{"className":41499,"style":27704},[27703],[27706,41501,41502],{"xmlns":27708,"width":27709,"height":27710,"viewBox":27711,"preserveAspectRatio":27712},[27714,41503],{"d":27716},[15,41505,41507],{"className":41506,"style":27704},[27720],[27706,41508,41509],{"xmlns":27708,"width":27709,"height":27710,"viewBox":27711,"preserveAspectRatio":27723},[27714,41510],{"d":27726},[15,41512,41514],{"className":41513,"style":27704},[27730],[27706,41515,41516],{"xmlns":27708,"width":27709,"height":27710,"viewBox":27711,"preserveAspectRatio":27733},[27714,41517],{"d":27736},[15,41519,41520,41523],{"style":41471},[15,41521],{"className":41522,"style":41456},[357],[15,41524,41526],{"className":41525},[61],[15,41527,41529,41532,41634],{"className":41528},[61],[15,41530],{"className":41531},[146,3841],[15,41533,41535],{"className":41534},[3722],[15,41536,41538,41626],{"className":41537},[342,472],[15,41539,41541,41623],{"className":41540},[346],[15,41542,41544,41555,41563],{"className":41543,"style":18690},[350],[15,41545,41546,41549],{"style":11402},[15,41547],{"className":41548,"style":3861},[357],[15,41550,41552],{"className":41551},[61],[15,41553,40265],{"className":41554,"style":40337},[61,122],[15,41556,41557,41560],{"style":3889},[15,41558],{"className":41559,"style":3861},[357],[15,41561],{"className":41562,"style":3897},[3896],[15,41564,41565,41568],{"style":11468},[15,41566],{"className":41567,"style":3861},[357],[15,41569,41571,41611,41614,41617,41620],{"className":41570},[61],[15,41572,41574,41577],{"className":41573},[61],[15,41575,3106],{"className":41576,"style":3191},[61,122],[15,41578,41580],{"className":41579},[338],[15,41581,41583,41603],{"className":41582},[342,472],[15,41584,41586,41600],{"className":41585},[346],[15,41587,41589],{"className":41588,"style":479},[350],[15,41590,41591,41594],{"style":3206},[15,41592],{"className":41593,"style":358},[357],[15,41595,41597],{"className":41596},[362,363,364,365],[15,41598,272],{"className":41599,"style":492},[61,122,365],[15,41601,497],{"className":41602},[496],[15,41604,41606],{"className":41605},[346],[15,41607,41609],{"className":41608,"style":504},[350],[15,41610],{},[15,41612],{"className":41613,"style":772},[127],[15,41615,239],{"className":41616},[375],[15,41618],{"className":41619,"style":772},[127],[15,41621,91],{"className":41622},[61],[15,41624,497],{"className":41625},[496],[15,41627,41629],{"className":41628},[346],[15,41630,41632],{"className":41631,"style":36614},[350],[15,41633],{},[15,41635],{"className":41636},[178,3841],[15,41638,497],{"className":41639},[496],[15,41641,41643],{"className":41642},[346],[15,41644,41647],{"className":41645,"style":41646},[350],"height:1.334em;",[15,41648],{},[15,41650,497],{"className":41651},[496],[15,41653,41655],{"className":41654},[346],[15,41656,41659],{"className":41657,"style":41658},[350],"height:2.0123em;",[15,41660],{},[15,41662],{"className":41663,"style":772},[127],[15,41665,7762],{"className":41666},[375],[15,41668],{"className":41669,"style":772},[127],[15,41671,41673,41676,41844,41847,41850],{"className":41672},[52],[15,41674],{"className":41675,"style":41437},[56],[15,41677,41679],{"className":41678},[161,6881],[15,41680,41682,41836],{"className":41681},[342,472],[15,41683,41685,41833],{"className":41684},[346],[15,41686,41688,41705],{"className":41687,"style":18690},[350],[15,41689,41690,41693],{"style":41452},[15,41691],{"className":41692,"style":41456},[357],[15,41694,41696],{"className":41695},[362,363,364,365],[15,41697,41699],{"className":41698},[61,365],[15,41700,41702],{"className":41701},[61,13548,365],[15,41703,41317],{"className":41704},[61,13552,365],[15,41706,41707,41710],{"style":41471},[15,41708],{"className":41709,"style":41456},[357],[15,41711,41713],{"className":41712},[161,6881],[15,41714,41716,41825],{"className":41715},[342,472],[15,41717,41719,41822],{"className":41718},[346],[15,41720,41722,41752],{"className":41721,"style":18690},[350],[15,41723,41725,41728],{"className":41724,"style":41490},[27690],[15,41726],{"className":41727,"style":41456},[357],[15,41729,41731,41738,41745],{"className":41730,"style":27699},[27698],[15,41732,41734],{"className":41733,"style":27704},[27703],[27706,41735,41736],{"xmlns":27708,"width":27709,"height":27710,"viewBox":27711,"preserveAspectRatio":27712},[27714,41737],{"d":27716},[15,41739,41741],{"className":41740,"style":27704},[27720],[27706,41742,41743],{"xmlns":27708,"width":27709,"height":27710,"viewBox":27711,"preserveAspectRatio":27723},[27714,41744],{"d":27726},[15,41746,41748],{"className":41747,"style":27704},[27730],[27706,41749,41750],{"xmlns":27708,"width":27709,"height":27710,"viewBox":27711,"preserveAspectRatio":27733},[27714,41751],{"d":27736},[15,41753,41754,41757],{"style":41471},[15,41755],{"className":41756,"style":41456},[357],[15,41758,41760],{"className":41759},[61],[15,41761,41763,41766,41819],{"className":41762},[61],[15,41764],{"className":41765},[146,3841],[15,41767,41769],{"className":41768},[3722],[15,41770,41772,41811],{"className":41771},[342,472],[15,41773,41775,41808],{"className":41774},[346],[15,41776,41778,41789,41797],{"className":41777,"style":18690},[350],[15,41779,41780,41783],{"style":11402},[15,41781],{"className":41782,"style":3861},[357],[15,41784,41786],{"className":41785},[61],[15,41787,40265],{"className":41788,"style":40337},[61,122],[15,41790,41791,41794],{"style":3889},[15,41792],{"className":41793,"style":3861},[357],[15,41795],{"className":41796,"style":3897},[3896],[15,41798,41799,41802],{"style":11468},[15,41800],{"className":41801,"style":3861},[357],[15,41803,41805],{"className":41804},[61],[15,41806,91],{"className":41807},[61],[15,41809,497],{"className":41810},[496],[15,41812,41814],{"className":41813},[346],[15,41815,41817],{"className":41816,"style":36614},[350],[15,41818],{},[15,41820],{"className":41821},[178,3841],[15,41823,497],{"className":41824},[496],[15,41826,41828],{"className":41827},[346],[15,41829,41831],{"className":41830,"style":41646},[350],[15,41832],{},[15,41834,497],{"className":41835},[496],[15,41837,41839],{"className":41838},[346],[15,41840,41842],{"className":41841,"style":41658},[350],[15,41843],{},[15,41845],{"className":41846,"style":128},[127],[15,41848,244],{"className":41849},[132],[15,41851],{"className":41852,"style":128},[127],[15,41854,41856,41859,41862,41865,41868,41871],{"className":41855},[52],[15,41857],{"className":41858,"style":142},[56],[15,41860,741],{"className":41861},[146],[15,41863,91],{"className":41864},[61],[15,41866],{"className":41867,"style":772},[127],[15,41869,239],{"className":41870},[375],[15,41872],{"className":41873,"style":772},[127],[15,41875,41877,41880,41883,41886,41889,42049,42052,42055],{"className":41876},[52],[15,41878],{"className":41879,"style":18671},[56],[15,41881,33430],{"className":41882},[61,122],[15,41884,750],{"className":41885},[178],[15,41887],{"className":41888,"style":157},[127],[15,41890,41892,41895,42046],{"className":41891},[61],[15,41893],{"className":41894},[146,3841],[15,41896,41898],{"className":41897},[3722],[15,41899,41901,42038],{"className":41900},[342,472],[15,41902,41904,42035],{"className":41903},[346],[15,41905,41907,41967,41975],{"className":41906,"style":18690},[350],[15,41908,41909,41912],{"style":11402},[15,41910],{"className":41911,"style":3861},[357],[15,41913,41915,41955,41958,41961,41964],{"className":41914},[61],[15,41916,41918,41921],{"className":41917},[61],[15,41919,3106],{"className":41920,"style":3191},[61,122],[15,41922,41924],{"className":41923},[338],[15,41925,41927,41947],{"className":41926},[342,472],[15,41928,41930,41944],{"className":41929},[346],[15,41931,41933],{"className":41932,"style":1214},[350],[15,41934,41935,41938],{"style":3206},[15,41936],{"className":41937,"style":358},[357],[15,41939,41941],{"className":41940},[362,363,364,365],[15,41942,278],{"className":41943},[61,365],[15,41945,497],{"className":41946},[496],[15,41948,41950],{"className":41949},[346],[15,41951,41953],{"className":41952,"style":504},[350],[15,41954],{},[15,41956],{"className":41957,"style":772},[127],[15,41959,239],{"className":41960},[375],[15,41962],{"className":41963,"style":772},[127],[15,41965,102],{"className":41966,"style":174},[61,122],[15,41968,41969,41972],{"style":3889},[15,41970],{"className":41971,"style":3861},[357],[15,41973],{"className":41974,"style":3897},[3896],[15,41976,41977,41980],{"style":11468},[15,41978],{"className":41979,"style":3861},[357],[15,41981,41983,42023,42026,42029,42032],{"className":41982},[61],[15,41984,41986,41989],{"className":41985},[61],[15,41987,3106],{"className":41988,"style":3191},[61,122],[15,41990,41992],{"className":41991},[338],[15,41993,41995,42015],{"className":41994},[342,472],[15,41996,41998,42012],{"className":41997},[346],[15,41999,42001],{"className":42000,"style":479},[350],[15,42002,42003,42006],{"style":3206},[15,42004],{"className":42005,"style":358},[357],[15,42007,42009],{"className":42008},[362,363,364,365],[15,42010,272],{"className":42011,"style":492},[61,122,365],[15,42013,497],{"className":42014},[496],[15,42016,42018],{"className":42017},[346],[15,42019,42021],{"className":42020,"style":504},[350],[15,42022],{},[15,42024],{"className":42025,"style":772},[127],[15,42027,239],{"className":42028},[375],[15,42030],{"className":42031,"style":772},[127],[15,42033,91],{"className":42034},[61],[15,42036,497],{"className":42037},[496],[15,42039,42041],{"className":42040},[346],[15,42042,42044],{"className":42043,"style":18779},[350],[15,42045],{},[15,42047],{"className":42048},[178,3841],[15,42050],{"className":42051,"style":772},[127],[15,42053,7762],{"className":42054},[375],[15,42056],{"className":42057,"style":772},[127],[15,42059,42061,42065,42068,42071,42074],{"className":42060},[52],[15,42062],{"className":42063,"style":42064},[56],"height:0.4445em;",[15,42066,33430],{"className":42067},[61,122],[15,42069],{"className":42070,"style":772},[127],[15,42072,11214],{"className":42073},[375],[15,42075],{"className":42076,"style":772},[127],[15,42078,42080,42084],{"className":42079},[52],[15,42081],{"className":42082,"style":42083},[56],"height:2.0074em;vertical-align:-0.686em;",[15,42085,42087,42090,42143],{"className":42086},[61],[15,42088],{"className":42089},[146,3841],[15,42091,42093],{"className":42092},[3722],[15,42094,42096,42135],{"className":42095},[342,472],[15,42097,42099,42132],{"className":42098},[346],[15,42100,42102,42113,42121],{"className":42101,"style":18690},[350],[15,42103,42104,42107],{"style":11402},[15,42105],{"className":42106,"style":3861},[357],[15,42108,42110],{"className":42109},[61],[15,42111,102],{"className":42112,"style":174},[61,122],[15,42114,42115,42118],{"style":3889},[15,42116],{"className":42117,"style":3861},[357],[15,42119],{"className":42120,"style":3897},[3896],[15,42122,42123,42126],{"style":11468},[15,42124],{"className":42125,"style":3861},[357],[15,42127,42129],{"className":42128},[61],[15,42130,91],{"className":42131},[61],[15,42133,497],{"className":42134},[496],[15,42136,42138],{"className":42137},[346],[15,42139,42141],{"className":42140,"style":36614},[350],[15,42142],{},[15,42144],{"className":42145},[178,3841],[11,42147,12926,42148,42226,42227,42278,42279,42330],{},[661,42149,42150,42151,42225],{},"预测概率是「证据归一化的类比例」与「均匀分布」按 ",[15,42152,42154,42180],{"className":42153,"translate":19},[18],[15,42155,42157],{"className":42156},[23],[25,42158,42159],{"xmlns":27},[29,42160,42161,42177],{},[32,42162,42163,42165,42167,42169,42171,42173,42175],{},[81,42164,741],{"stretchy":86},[89,42166,91],{},[81,42168,239],{},[35,42170,33430],{},[81,42172,94],{"separator":48},[35,42174,33430],{},[81,42176,750],{"stretchy":86},[40,42178,42179],{"encoding":42},"(1-u, u)",[15,42181,42183,42204],{"className":42182,"ariaHidden":48},[47],[15,42184,42186,42189,42192,42195,42198,42201],{"className":42185},[52],[15,42187],{"className":42188,"style":142},[56],[15,42190,741],{"className":42191},[146],[15,42193,91],{"className":42194},[61],[15,42196],{"className":42197,"style":772},[127],[15,42199,239],{"className":42200},[375],[15,42202],{"className":42203,"style":772},[127],[15,42205,42207,42210,42213,42216,42219,42222],{"className":42206},[52],[15,42208],{"className":42209,"style":142},[56],[15,42211,33430],{"className":42212},[61,122],[15,42214,94],{"className":42215},[153],[15,42217],{"className":42218,"style":157},[127],[15,42220,33430],{"className":42221},[61,122],[15,42223,750],{"className":42224},[178]," 的凸组合","。当证据充分时（",[15,42228,42230,42248],{"className":42229,"translate":19},[18],[15,42231,42233],{"className":42232},[23],[25,42234,42235],{"xmlns":27},[29,42236,42237,42245],{},[32,42238,42239,42241,42243],{},[35,42240,33430],{},[81,42242,20704],{},[89,42244,278],{},[40,42246,42247],{"encoding":42},"u \\to 0",[15,42249,42251,42269],{"className":42250,"ariaHidden":48},[47],[15,42252,42254,42257,42260,42263,42266],{"className":42253},[52],[15,42255],{"className":42256,"style":6032},[56],[15,42258,33430],{"className":42259},[61,122],[15,42261],{"className":42262,"style":128},[127],[15,42264,20704],{"className":42265},[132],[15,42267],{"className":42268,"style":128},[127],[15,42270,42272,42275],{"className":42271},[52],[15,42273],{"className":42274,"style":859},[56],[15,42276,278],{"className":42277},[61],"），预测趋于证据的方向。当证据消失时（",[15,42280,42282,42300],{"className":42281,"translate":19},[18],[15,42283,42285],{"className":42284},[23],[25,42286,42287],{"xmlns":27},[29,42288,42289,42297],{},[32,42290,42291,42293,42295],{},[35,42292,33430],{},[81,42294,20704],{},[89,42296,91],{},[40,42298,42299],{"encoding":42},"u \\to 1",[15,42301,42303,42321],{"className":42302,"ariaHidden":48},[47],[15,42304,42306,42309,42312,42315,42318],{"className":42305},[52],[15,42307],{"className":42308,"style":6032},[56],[15,42310,33430],{"className":42311},[61,122],[15,42313],{"className":42314,"style":128},[127],[15,42316,20704],{"className":42317},[132],[15,42319],{"className":42320,"style":128},[127],[15,42322,42324,42327],{"className":42323},[52],[15,42325],{"className":42326,"style":859},[56],[15,42328,91],{"className":42329},[61],"），预测退回均匀分布。极限行为因此完全符合直觉：",[15,42332,42334],{"className":42333,"translate":19},[212],[15,42335,42337,42442],{"className":42336,"translate":19},[18],[15,42338,42340],{"className":42339},[23],[25,42341,42342],{"xmlns":27,"display":221},[29,42343,42344,42439],{},[32,42345,42346,42348,42350,42352,42354,42356,42358,42360,42362,42364,42366,42368,42370,42372,42374,42376,42379,42381,42383,42391,42393,42395,42397,42399,42401,42403,42405,42407,42409,42411,42413,42415,42417,42419,42421],{},[35,42347,40265],{},[81,42349,20704],{},[35,42351,102],{},[282,42353,284],{},[81,42355,741],{"stretchy":86},[35,42357,1982],{"mathvariant":251},[81,42359,20704],{},[89,42361,278],{"mathvariant":251},[81,42363,750],{"stretchy":86},[81,42365,264],{},[127,42367],{"width":6451},[35,42369,33430],{},[81,42371,20704],{},[89,42373,91],{},[81,42375,94],{"separator":48},[282,42377,42378],{},"  ",[35,42380,11],{"mathvariant":251},[81,42382,20704],{},[286,42384,42385],{"scriptlevel":278,"displaystyle":86},[3722,42386,42387,42389],{},[89,42388,91],{},[35,42390,102],{},[89,42392,91],{"mathvariant":251},[127,42394],{"width":9847},[127,42396],{"width":9847},[35,42398,40265],{},[81,42400,20704],{},[35,42402,6796],{"mathvariant":231},[81,42404,264],{},[127,42406],{"width":6451},[35,42408,33430],{},[81,42410,20704],{},[89,42412,278],{},[81,42414,94],{"separator":48},[282,42416,42378],{},[35,42418,11],{"mathvariant":251},[81,42420,20704],{},[3722,42422,42423,42425],{},[35,42424,1982],{"mathvariant":251},[32,42426,42427,42433],{},[6881,42428,42429,42431],{},[81,42430,293],{},[35,42432,14777],{},[266,42434,42435,42437],{},[35,42436,1982],{},[35,42438,14777],{},[40,42440,42441],{"encoding":42},"S \\to K \\ (\\mathbf{e} \\to \\mathbf{0}): \\quad u \\to 1,\\ \\ \\mathbf{p} \\to \\tfrac{1}{K}\\mathbf{1}\n\\qquad\\qquad\nS \\to \\infty: \\quad u \\to 0,\\ \\ \\mathbf{p} \\to \\frac{\\mathbf{e}}{\\sum_j e_j}",[15,42443,42445,42463,42490,42514,42532,42565,42661,42682,42700,42733],{"className":42444,"ariaHidden":48},[47],[15,42446,42448,42451,42454,42457,42460],{"className":42447},[52],[15,42449],{"className":42450,"style":57},[56],[15,42452,40265],{"className":42453,"style":40337},[61,122],[15,42455],{"className":42456,"style":128},[127],[15,42458,20704],{"className":42459},[132],[15,42461],{"className":42462,"style":128},[127],[15,42464,42466,42469,42472,42475,42478,42481,42484,42487],{"className":42465},[52],[15,42467],{"className":42468,"style":142},[56],[15,42470,102],{"className":42471,"style":174},[61,122],[15,42473,284],{"className":42474},[127],[15,42476,741],{"className":42477},[146],[15,42479,1982],{"className":42480},[61,411],[15,42482],{"className":42483,"style":128},[127],[15,42485,20704],{"className":42486},[132],[15,42488],{"className":42489,"style":128},[127],[15,42491,42493,42496,42499,42502,42505,42508,42511],{"className":42492},[52],[15,42494],{"className":42495,"style":142},[56],[15,42497,278],{"className":42498},[61,411],[15,42500,750],{"className":42501},[178],[15,42503],{"className":42504,"style":128},[127],[15,42506,264],{"className":42507},[132],[15,42509],{"className":42510,"style":15983},[127],[15,42512],{"className":42513,"style":128},[127],[15,42515,42517,42520,42523,42526,42529],{"className":42516},[52],[15,42518],{"className":42519,"style":6032},[56],[15,42521,33430],{"className":42522},[61,122],[15,42524],{"className":42525,"style":128},[127],[15,42527,20704],{"className":42528},[132],[15,42530],{"className":42531,"style":128},[127],[15,42533,42535,42538,42541,42544,42547,42550,42553,42556,42559,42562],{"className":42534},[52],[15,42536],{"className":42537,"style":19755},[56],[15,42539,91],{"className":42540},[61],[15,42542,94],{"className":42543},[153],[15,42545,284],{"className":42546},[127],[15,42548,284],{"className":42549},[127],[15,42551],{"className":42552,"style":157},[127],[15,42554,11],{"className":42555},[61,411],[15,42557],{"className":42558,"style":128},[127],[15,42560,20704],{"className":42561},[132],[15,42563],{"className":42564,"style":128},[127],[15,42566,42568,42572,42640,42643,42646,42649,42652,42655,42658],{"className":42567},[52],[15,42569],{"className":42570,"style":42571},[56],"height:1.1901em;vertical-align:-0.345em;",[15,42573,42575,42578,42637],{"className":42574},[61],[15,42576],{"className":42577},[146,3841],[15,42579,42581],{"className":42580},[3722],[15,42582,42584,42629],{"className":42583},[342,472],[15,42585,42587,42626],{"className":42586},[346],[15,42588,42590,42604,42612],{"className":42589,"style":3854},[350],[15,42591,42592,42595],{"style":3857},[15,42593],{"className":42594,"style":3861},[357],[15,42596,42598],{"className":42597},[362,363,364,365],[15,42599,42601],{"className":42600},[61,365],[15,42602,102],{"className":42603,"style":174},[61,122,365],[15,42605,42606,42609],{"style":3889},[15,42607],{"className":42608,"style":3861},[357],[15,42610],{"className":42611,"style":3897},[3896],[15,42613,42614,42617],{"style":3900},[15,42615],{"className":42616,"style":3861},[357],[15,42618,42620],{"className":42619},[362,363,364,365],[15,42621,42623],{"className":42622},[61,365],[15,42624,91],{"className":42625},[61,365],[15,42627,497],{"className":42628},[496],[15,42630,42632],{"className":42631},[346],[15,42633,42635],{"className":42634,"style":10198},[350],[15,42636],{},[15,42638],{"className":42639},[178,3841],[15,42641,91],{"className":42642},[61,411],[15,42644],{"className":42645,"style":10035},[127],[15,42647],{"className":42648,"style":10035},[127],[15,42650,40265],{"className":42651,"style":40337},[61,122],[15,42653],{"className":42654,"style":128},[127],[15,42656,20704],{"className":42657},[132],[15,42659],{"className":42660,"style":128},[127],[15,42662,42664,42667,42670,42673,42676,42679],{"className":42663},[52],[15,42665],{"className":42666,"style":6032},[56],[15,42668,6796],{"className":42669},[61],[15,42671],{"className":42672,"style":128},[127],[15,42674,264],{"className":42675},[132],[15,42677],{"className":42678,"style":15983},[127],[15,42680],{"className":42681,"style":128},[127],[15,42683,42685,42688,42691,42694,42697],{"className":42684},[52],[15,42686],{"className":42687,"style":6032},[56],[15,42689,33430],{"className":42690},[61,122],[15,42692],{"className":42693,"style":128},[127],[15,42695,20704],{"className":42696},[132],[15,42698],{"className":42699,"style":128},[127],[15,42701,42703,42706,42709,42712,42715,42718,42721,42724,42727,42730],{"className":42702},[52],[15,42704],{"className":42705,"style":19755},[56],[15,42707,278],{"className":42708},[61],[15,42710,94],{"className":42711},[153],[15,42713,284],{"className":42714},[127],[15,42716,284],{"className":42717},[127],[15,42719],{"className":42720,"style":157},[127],[15,42722,11],{"className":42723},[61,411],[15,42725],{"className":42726,"style":128},[127],[15,42728,20704],{"className":42729},[132],[15,42731],{"className":42732,"style":128},[127],[15,42734,42736,42740],{"className":42735},[52],[15,42737],{"className":42738,"style":42739},[56],"height:2.2433em;vertical-align:-1.1218em;",[15,42741,42743,42746,42881],{"className":42742},[61],[15,42744],{"className":42745},[146,3841],[15,42747,42749],{"className":42748},[3722],[15,42750,42752,42872],{"className":42751},[342,472],[15,42753,42755,42869],{"className":42754},[346],[15,42756,42759,42850,42858],{"className":42757,"style":42758},[350],"height:1.1214em;",[15,42760,42761,42764],{"style":11402},[15,42762],{"className":42763,"style":3861},[357],[15,42765,42767,42807,42810],{"className":42766},[61],[15,42768,42770,42773],{"className":42769},[531],[15,42771,293],{"className":42772,"style":537},[531,535,536],[15,42774,42776],{"className":42775},[338],[15,42777,42779,42799],{"className":42778},[342,472],[15,42780,42782,42796],{"className":42781},[346],[15,42783,42785],{"className":42784,"style":6070},[350],[15,42786,42787,42790],{"style":553},[15,42788],{"className":42789,"style":358},[357],[15,42791,42793],{"className":42792},[362,363,364,365],[15,42794,14777],{"className":42795,"style":14803},[61,122,365],[15,42797,497],{"className":42798},[496],[15,42800,42802],{"className":42801},[346],[15,42803,42805],{"className":42804,"style":30053},[350],[15,42806],{},[15,42808],{"className":42809,"style":157},[127],[15,42811,42813,42816],{"className":42812},[61],[15,42814,1982],{"className":42815},[61,122],[15,42817,42819],{"className":42818},[338],[15,42820,42822,42842],{"className":42821},[342,472],[15,42823,42825,42839],{"className":42824},[346],[15,42826,42828],{"className":42827,"style":1364},[350],[15,42829,42830,42833],{"style":482},[15,42831],{"className":42832,"style":358},[357],[15,42834,42836],{"className":42835},[362,363,364,365],[15,42837,14777],{"className":42838,"style":14803},[61,122,365],[15,42840,497],{"className":42841},[496],[15,42843,42845],{"className":42844},[346],[15,42846,42848],{"className":42847,"style":15707},[350],[15,42849],{},[15,42851,42852,42855],{"style":3889},[15,42853],{"className":42854,"style":3861},[357],[15,42856],{"className":42857,"style":3897},[3896],[15,42859,42860,42863],{"style":11468},[15,42861],{"className":42862,"style":3861},[357],[15,42864,42866],{"className":42865},[61],[15,42867,1982],{"className":42868},[61,411],[15,42870,497],{"className":42871},[496],[15,42873,42875],{"className":42874},[346],[15,42876,42879],{"className":42877,"style":42878},[350],"height:1.1218em;",[15,42880],{},[15,42882],{"className":42883},[178,3841],[11,42885,42886,42887,42890,42891,42919,42920,42948,42949,42952],{},"命题 3.4 同时也暴露了该参数化的一个",[661,42888,42889],{},"结构性局限","：",[15,42892,42894,42907],{"className":42893,"translate":19},[18],[15,42895,42897],{"className":42896},[23],[25,42898,42899],{"xmlns":27},[29,42900,42901,42905],{},[32,42902,42903],{},[35,42904,33430],{},[40,42906,33430],{"encoding":42},[15,42908,42910],{"className":42909,"ariaHidden":48},[47],[15,42911,42913,42916],{"className":42912},[52],[15,42914],{"className":42915,"style":6032},[56],[15,42917,33430],{"className":42918},[61,122]," 只是总强度 ",[15,42921,42923,42936],{"className":42922,"translate":19},[18],[15,42924,42926],{"className":42925},[23],[25,42927,42928],{"xmlns":27},[29,42929,42930,42934],{},[32,42931,42932],{},[35,42933,40265],{},[40,42935,40265],{"encoding":42},[15,42937,42939],{"className":42938,"ariaHidden":48},[47],[15,42940,42942,42945],{"className":42941},[52],[15,42943],{"className":42944,"style":57},[56],[15,42946,40265],{"className":42947,"style":40337},[61,122]," 的函数，与证据在各类别之间的",[661,42950,42951],{},"分布","无关。因此 EDL 的不确定性分数无法区分「少量证据集中于一类」与「大量证据均匀分散」，也无法区分「证据落在正确类别上」与「证据落在错误类别上」。这一局限将在 4.5 节与 5.6 节被重新讨论。",[877,42954,42956],{"id":42955},"_36-不确定性的分解偶然与认知","3.6 不确定性的分解：偶然与认知",[11,42958,42959,42960,43021,43022,43025],{},"命题 3.4 给出的 ",[15,42961,42963,42985],{"className":42962,"translate":19},[18],[15,42964,42966],{"className":42965},[23],[25,42967,42968],{"xmlns":27},[29,42969,42970,42982],{},[32,42971,42972,42974,42976,42978,42980],{},[35,42973,33430],{},[81,42975,244],{},[35,42977,102],{},[35,42979,3020],{"mathvariant":231},[35,42981,40265],{},[40,42983,42984],{"encoding":42},"u = K\u002FS",[15,42986,42988,43006],{"className":42987,"ariaHidden":48},[47],[15,42989,42991,42994,42997,43000,43003],{"className":42990},[52],[15,42992],{"className":42993,"style":6032},[56],[15,42995,33430],{"className":42996},[61,122],[15,42998],{"className":42999,"style":128},[127],[15,43001,244],{"className":43002},[132],[15,43004],{"className":43005,"style":128},[127],[15,43007,43009,43012,43015,43018],{"className":43008},[52],[15,43010],{"className":43011,"style":142},[56],[15,43013,102],{"className":43014,"style":174},[61,122],[15,43016,3020],{"className":43017},[61],[15,43019,40265],{"className":43020,"style":40337},[61,122]," 是一个",[661,43023,43024],{},"启发式","的不确定性标量。与之相对，Malinin 与 Gales 在 Prior Networks 的工作中给出了一个有信息论根据的分解，它在迪利克雷族内具有闭式解。",[11,43027,13,43028,16199,43110,43216],{},[15,43029,43031,43056],{"className":43030,"translate":19},[18],[15,43032,43034],{"className":43033},[23],[25,43035,43036],{"xmlns":27},[29,43037,43038,43054],{},[32,43039,43040,43042,43044,43046,43048,43050,43052],{},[35,43041,24680],{"mathvariant":3105},[81,43043,4183],{},[35,43045,4186],{"mathvariant":231},[81,43047,1547],{},[81,43049,741],{"stretchy":86},[35,43051,3106],{"mathvariant":3105},[81,43053,750],{"stretchy":86},[40,43055,24695],{"encoding":42},[15,43057,43059,43083],{"className":43058,"ariaHidden":48},[47],[15,43060,43062,43065,43074,43077,43080],{"className":43061},[52],[15,43063],{"className":43064,"style":2708},[56],[15,43066,43068],{"className":43067},[61],[15,43069,43071],{"className":43070},[61],[15,43072,24680],{"className":43073,"style":24714},[61,3166],[15,43075],{"className":43076,"style":128},[127],[15,43078,4183],{"className":43079},[132],[15,43081],{"className":43082,"style":128},[127],[15,43084,43086,43089,43095,43098,43107],{"className":43085},[52],[15,43087],{"className":43088,"style":142},[56],[15,43090,43092],{"className":43091},[531],[15,43093,4186],{"className":43094},[61,1622],[15,43096,741],{"className":43097},[146],[15,43099,43101],{"className":43100},[61],[15,43102,43104],{"className":43103},[61],[15,43105,3106],{"className":43106},[61,3166],[15,43108,750],{"className":43109},[178],[15,43111,43113,43144],{"className":43112,"translate":19},[18],[15,43114,43116],{"className":43115},[23],[25,43117,43118],{"xmlns":27},[29,43119,43120,43141],{},[32,43121,43122,43124,43126,43128,43130,43133,43135,43137,43139],{},[35,43123,79],{},[81,43125,3714],{},[35,43127,24680],{"mathvariant":3105},[81,43129,4183],{},[35,43131,43132],{"mathvariant":231},"Cat",[81,43134,1547],{},[81,43136,741],{"stretchy":86},[35,43138,24680],{"mathvariant":3105},[81,43140,750],{"stretchy":86},[40,43142,43143],{"encoding":42},"y \\mid \\boldsymbol{\\pi} \\sim \\operatorname{Cat}(\\boldsymbol{\\pi})",[15,43145,43147,43165,43189],{"className":43146,"ariaHidden":48},[47],[15,43148,43150,43153,43156,43159,43162],{"className":43149},[52],[15,43151],{"className":43152,"style":142},[56],[15,43154,79],{"className":43155,"style":123},[61,122],[15,43157],{"className":43158,"style":128},[127],[15,43160,3714],{"className":43161},[132],[15,43163],{"className":43164,"style":128},[127],[15,43166,43168,43171,43180,43183,43186],{"className":43167},[52],[15,43169],{"className":43170,"style":2708},[56],[15,43172,43174],{"className":43173},[61],[15,43175,43177],{"className":43176},[61],[15,43178,24680],{"className":43179,"style":24714},[61,3166],[15,43181],{"className":43182,"style":128},[127],[15,43184,4183],{"className":43185},[132],[15,43187],{"className":43188,"style":128},[127],[15,43190,43192,43195,43201,43204,43213],{"className":43191},[52],[15,43193],{"className":43194,"style":142},[56],[15,43196,43198],{"className":43197},[531],[15,43199,43132],{"className":43200},[61,1622],[15,43202,741],{"className":43203},[146],[15,43205,43207],{"className":43206},[61],[15,43208,43210],{"className":43209},[61],[15,43211,24680],{"className":43212,"style":24714},[61,3166],[15,43214,750],{"className":43215},[178],"。考虑三个量：",[31547,43218,43219,43569,43722],{},[31550,43220,43221,43224,43225,5987,43304,867],{},[661,43222,43223],{},"总不确定性（total uncertainty）","：预测分布的熵 ",[15,43226,43228,43258],{"className":43227,"translate":19},[18],[15,43229,43231],{"className":43230},[23],[25,43232,43233],{"xmlns":27},[29,43234,43235,43255],{},[32,43236,43237,43239,43241,43243,43245,43247,43249,43251,43253],{},[35,43238,22651],{},[81,43240,14821],{"stretchy":86},[35,43242,21286],{"mathvariant":258},[81,43244,741],{"stretchy":86},[35,43246,79],{},[81,43248,3714],{},[35,43250,1029],{"mathvariant":251},[81,43252,750],{"stretchy":86},[81,43254,14830],{"stretchy":86},[40,43256,43257],{"encoding":42},"H[\\mathbb{P}(y \\mid \\mathbf{x})]",[15,43259,43261,43291],{"className":43260,"ariaHidden":48},[47],[15,43262,43264,43267,43270,43273,43276,43279,43282,43285,43288],{"className":43263},[52],[15,43265],{"className":43266,"style":142},[56],[15,43268,22651],{"className":43269,"style":22763},[61,122],[15,43271,14821],{"className":43272},[146],[15,43274,21286],{"className":43275},[61,427],[15,43277,741],{"className":43278},[146],[15,43280,79],{"className":43281,"style":123},[61,122],[15,43283],{"className":43284,"style":128},[127],[15,43286,3714],{"className":43287},[132],[15,43289],{"className":43290,"style":128},[127],[15,43292,43294,43297,43300],{"className":43293},[52],[15,43295],{"className":43296,"style":142},[56],[15,43298,1029],{"className":43299},[61,411],[15,43301,43303],{"className":43302},[178],")]",[15,43305,43307,43359],{"className":43306,"translate":19},[18],[15,43308,43310],{"className":43309},[23],[25,43311,43312],{"xmlns":27},[29,43313,43314,43356],{},[32,43315,43316,43318,43320,43322,43324,43326,43328,43330,43332,43334,43340,43342,43348,43350],{},[35,43317,21286],{"mathvariant":258},[81,43319,741],{"stretchy":86},[35,43321,79],{},[81,43323,244],{},[35,43325,272],{},[81,43327,3714],{},[35,43329,1029],{"mathvariant":251},[81,43331,750],{"stretchy":86},[81,43333,244],{},[266,43335,43336,43338],{},[35,43337,11],{},[35,43339,272],{},[81,43341,244],{},[266,43343,43344,43346],{},[35,43345,3106],{},[35,43347,272],{},[35,43349,3020],{"mathvariant":231},[266,43351,43352,43354],{},[35,43353,3106],{},[89,43355,278],{},[40,43357,43358],{"encoding":42},"\\mathbb{P}(y = k \\mid \\mathbf{x}) = p_k = \\alpha_k\u002F\\alpha_0",[15,43360,43362,43386,43404,43425,43480],{"className":43361,"ariaHidden":48},[47],[15,43363,43365,43368,43371,43374,43377,43380,43383],{"className":43364},[52],[15,43366],{"className":43367,"style":142},[56],[15,43369,21286],{"className":43370},[61,427],[15,43372,741],{"className":43373},[146],[15,43375,79],{"className":43376,"style":123},[61,122],[15,43378],{"className":43379,"style":128},[127],[15,43381,244],{"className":43382},[132],[15,43384],{"className":43385,"style":128},[127],[15,43387,43389,43392,43395,43398,43401],{"className":43388},[52],[15,43390],{"className":43391,"style":142},[56],[15,43393,272],{"className":43394,"style":492},[61,122],[15,43396],{"className":43397,"style":128},[127],[15,43399,3714],{"className":43400},[132],[15,43402],{"className":43403,"style":128},[127],[15,43405,43407,43410,43413,43416,43419,43422],{"className":43406},[52],[15,43408],{"className":43409,"style":142},[56],[15,43411,1029],{"className":43412},[61,411],[15,43414,750],{"className":43415},[178],[15,43417],{"className":43418,"style":128},[127],[15,43420,244],{"className":43421},[132],[15,43423],{"className":43424,"style":128},[127],[15,43426,43428,43431,43471,43474,43477],{"className":43427},[52],[15,43429],{"className":43430,"style":9437},[56],[15,43432,43434,43437],{"className":43433},[61],[15,43435,11],{"className":43436},[61,122],[15,43438,43440],{"className":43439},[338],[15,43441,43443,43463],{"className":43442},[342,472],[15,43444,43446,43460],{"className":43445},[346],[15,43447,43449],{"className":43448,"style":479},[350],[15,43450,43451,43454],{"style":482},[15,43452],{"className":43453,"style":358},[357],[15,43455,43457],{"className":43456},[362,363,364,365],[15,43458,272],{"className":43459,"style":492},[61,122,365],[15,43461,497],{"className":43462},[496],[15,43464,43466],{"className":43465},[346],[15,43467,43469],{"className":43468,"style":504},[350],[15,43470],{},[15,43472],{"className":43473,"style":128},[127],[15,43475,244],{"className":43476},[132],[15,43478],{"className":43479,"style":128},[127],[15,43481,43483,43486,43526,43529],{"className":43482},[52],[15,43484],{"className":43485,"style":142},[56],[15,43487,43489,43492],{"className":43488},[61],[15,43490,3106],{"className":43491,"style":3191},[61,122],[15,43493,43495],{"className":43494},[338],[15,43496,43498,43518],{"className":43497},[342,472],[15,43499,43501,43515],{"className":43500},[346],[15,43502,43504],{"className":43503,"style":479},[350],[15,43505,43506,43509],{"style":3206},[15,43507],{"className":43508,"style":358},[357],[15,43510,43512],{"className":43511},[362,363,364,365],[15,43513,272],{"className":43514,"style":492},[61,122,365],[15,43516,497],{"className":43517},[496],[15,43519,43521],{"className":43520},[346],[15,43522,43524],{"className":43523,"style":504},[350],[15,43525],{},[15,43527,3020],{"className":43528},[61],[15,43530,43532,43535],{"className":43531},[61],[15,43533,3106],{"className":43534,"style":3191},[61,122],[15,43536,43538],{"className":43537},[338],[15,43539,43541,43561],{"className":43540},[342,472],[15,43542,43544,43558],{"className":43543},[346],[15,43545,43547],{"className":43546,"style":1214},[350],[15,43548,43549,43552],{"style":3206},[15,43550],{"className":43551,"style":358},[357],[15,43553,43555],{"className":43554},[362,363,364,365],[15,43556,278],{"className":43557},[61,365],[15,43559,497],{"className":43560},[496],[15,43562,43564],{"className":43563},[346],[15,43565,43567],{"className":43566,"style":504},[350],[15,43568],{},[31550,43570,43571,43574,43575,43609,43610,867],{},[661,43572,43573],{},"偶然不确定性（aleatoric uncertainty）","：给定 ",[15,43576,43578,43591],{"className":43577,"translate":19},[18],[15,43579,43581],{"className":43580},[23],[25,43582,43583],{"xmlns":27},[29,43584,43585,43589],{},[32,43586,43587],{},[35,43588,24680],{"mathvariant":3105},[40,43590,24768],{"encoding":42},[15,43592,43594],{"className":43593,"ariaHidden":48},[47],[15,43595,43597,43600],{"className":43596},[52],[15,43598],{"className":43599,"style":2708},[56],[15,43601,43603],{"className":43602},[61],[15,43604,43606],{"className":43605},[61],[15,43607,24680],{"className":43608,"style":24714},[61,3166]," 后仍存在的熵的期望 ",[15,43611,43613,43643],{"className":43612,"translate":19},[18],[15,43614,43616],{"className":43615},[23],[25,43617,43618],{"xmlns":27},[29,43619,43620,43640],{},[32,43621,43622,43628,43630,43632,43634,43636,43638],{},[266,43623,43624,43626],{},[35,43625,14818],{"mathvariant":258},[35,43627,24680],{"mathvariant":3105},[81,43629,14821],{"stretchy":86},[35,43631,22651],{},[81,43633,741],{"stretchy":86},[35,43635,24680],{"mathvariant":3105},[81,43637,750],{"stretchy":86},[81,43639,14830],{"stretchy":86},[40,43641,43642],{"encoding":42},"\\mathbb{E}_{\\boldsymbol{\\pi}}[H(\\boldsymbol{\\pi})]",[15,43644,43646],{"className":43645,"ariaHidden":48},[47],[15,43647,43649,43652,43701,43704,43707,43710,43719],{"className":43648},[52],[15,43650],{"className":43651,"style":142},[56],[15,43653,43655,43658],{"className":43654},[61],[15,43656,14818],{"className":43657},[61,427],[15,43659,43661],{"className":43660},[338],[15,43662,43664,43693],{"className":43663},[342,472],[15,43665,43667,43690],{"className":43666},[346],[15,43668,43670],{"className":43669,"style":20966},[350],[15,43671,43672,43675],{"style":482},[15,43673],{"className":43674,"style":358},[357],[15,43676,43678],{"className":43677},[362,363,364,365],[15,43679,43681],{"className":43680},[61,365],[15,43682,43684],{"className":43683},[61,365],[15,43685,43687],{"className":43686},[61,365],[15,43688,24680],{"className":43689,"style":24714},[61,3166,365],[15,43691,497],{"className":43692},[496],[15,43694,43696],{"className":43695},[346],[15,43697,43699],{"className":43698,"style":504},[350],[15,43700],{},[15,43702,14821],{"className":43703},[146],[15,43705,22651],{"className":43706,"style":22763},[61,122],[15,43708,741],{"className":43709},[146],[15,43711,43713],{"className":43712},[61],[15,43714,43716],{"className":43715},[61],[15,43717,24680],{"className":43718,"style":24714},[61,3166],[15,43720,43303],{"className":43721},[178],[31550,43723,43724,43727,43728,9348,43756,43790],{},[661,43725,43726],{},"认知不确定性（epistemic uncertainty）","：二者的差，即 ",[15,43729,43731,43744],{"className":43730,"translate":19},[18],[15,43732,43734],{"className":43733},[23],[25,43735,43736],{"xmlns":27},[29,43737,43738,43742],{},[32,43739,43740],{},[35,43741,79],{},[40,43743,79],{"encoding":42},[15,43745,43747],{"className":43746,"ariaHidden":48},[47],[15,43748,43750,43753],{"className":43749},[52],[15,43751],{"className":43752,"style":9437},[56],[15,43754,79],{"className":43755,"style":123},[61,122],[15,43757,43759,43772],{"className":43758,"translate":19},[18],[15,43760,43762],{"className":43761},[23],[25,43763,43764],{"xmlns":27},[29,43765,43766,43770],{},[32,43767,43768],{},[35,43769,24680],{"mathvariant":3105},[40,43771,24768],{"encoding":42},[15,43773,43775],{"className":43774,"ariaHidden":48},[47],[15,43776,43778,43781],{"className":43777},[52],[15,43779],{"className":43780,"style":2708},[56],[15,43782,43784],{"className":43783},[61],[15,43785,43787],{"className":43786},[61],[15,43788,24680],{"className":43789,"style":24714},[61,3166]," 之间的互信息。",[882,43792,43793],{},[11,43794,43795,43798,43799,943,43983,40647,44075,44362,44761],{},[661,43796,43797],{},"命题 3.5（Dirichlet 族内的不确定性分解）"," 记 ",[15,43800,43802,43836],{"className":43801,"translate":19},[18],[15,43803,43805],{"className":43804},[23],[25,43806,43807],{"xmlns":27},[29,43808,43809,43833],{},[32,43810,43811,43817,43819,43825,43827],{},[266,43812,43813,43815],{},[35,43814,11],{},[35,43816,272],{},[81,43818,244],{},[266,43820,43821,43823],{},[35,43822,3106],{},[35,43824,272],{},[35,43826,3020],{"mathvariant":231},[266,43828,43829,43831],{},[35,43830,3106],{},[89,43832,278],{},[40,43834,43835],{"encoding":42},"p_k = \\alpha_k\u002F\\alpha_0",[15,43837,43839,43894],{"className":43838,"ariaHidden":48},[47],[15,43840,43842,43845,43885,43888,43891],{"className":43841},[52],[15,43843],{"className":43844,"style":9437},[56],[15,43846,43848,43851],{"className":43847},[61],[15,43849,11],{"className":43850},[61,122],[15,43852,43854],{"className":43853},[338],[15,43855,43857,43877],{"className":43856},[342,472],[15,43858,43860,43874],{"className":43859},[346],[15,43861,43863],{"className":43862,"style":479},[350],[15,43864,43865,43868],{"style":482},[15,43866],{"className":43867,"style":358},[357],[15,43869,43871],{"className":43870},[362,363,364,365],[15,43872,272],{"className":43873,"style":492},[61,122,365],[15,43875,497],{"className":43876},[496],[15,43878,43880],{"className":43879},[346],[15,43881,43883],{"className":43882,"style":504},[350],[15,43884],{},[15,43886],{"className":43887,"style":128},[127],[15,43889,244],{"className":43890},[132],[15,43892],{"className":43893,"style":128},[127],[15,43895,43897,43900,43940,43943],{"className":43896},[52],[15,43898],{"className":43899,"style":142},[56],[15,43901,43903,43906],{"className":43902},[61],[15,43904,3106],{"className":43905,"style":3191},[61,122],[15,43907,43909],{"className":43908},[338],[15,43910,43912,43932],{"className":43911},[342,472],[15,43913,43915,43929],{"className":43914},[346],[15,43916,43918],{"className":43917,"style":479},[350],[15,43919,43920,43923],{"style":3206},[15,43921],{"className":43922,"style":358},[357],[15,43924,43926],{"className":43925},[362,363,364,365],[15,43927,272],{"className":43928,"style":492},[61,122,365],[15,43930,497],{"className":43931},[496],[15,43933,43935],{"className":43934},[346],[15,43936,43938],{"className":43937,"style":504},[350],[15,43939],{},[15,43941,3020],{"className":43942},[61],[15,43944,43946,43949],{"className":43945},[61],[15,43947,3106],{"className":43948,"style":3191},[61,122],[15,43950,43952],{"className":43951},[338],[15,43953,43955,43975],{"className":43954},[342,472],[15,43956,43958,43972],{"className":43957},[346],[15,43959,43961],{"className":43960,"style":1214},[350],[15,43962,43963,43966],{"style":3206},[15,43964],{"className":43965,"style":358},[357],[15,43967,43969],{"className":43968},[362,363,364,365],[15,43970,278],{"className":43971},[61,365],[15,43973,497],{"className":43974},[496],[15,43976,43978],{"className":43977},[346],[15,43979,43981],{"className":43980,"style":504},[350],[15,43982],{},[15,43984,43986,44008],{"className":43985,"translate":19},[18],[15,43987,43989],{"className":43988},[23],[25,43990,43991],{"xmlns":27},[29,43992,43993,44005],{},[32,43994,43995,43997,43999],{},[35,43996,40265],{},[81,43998,244],{},[266,44000,44001,44003],{},[35,44002,3106],{},[89,44004,278],{},[40,44006,44007],{"encoding":42},"S = \\alpha_0",[15,44009,44011,44029],{"className":44010,"ariaHidden":48},[47],[15,44012,44014,44017,44020,44023,44026],{"className":44013},[52],[15,44015],{"className":44016,"style":57},[56],[15,44018,40265],{"className":44019,"style":40337},[61,122],[15,44021],{"className":44022,"style":128},[127],[15,44024,244],{"className":44025},[132],[15,44027],{"className":44028,"style":128},[127],[15,44030,44032,44035],{"className":44031},[52],[15,44033],{"className":44034,"style":2511},[56],[15,44036,44038,44041],{"className":44037},[61],[15,44039,3106],{"className":44040,"style":3191},[61,122],[15,44042,44044],{"className":44043},[338],[15,44045,44047,44067],{"className":44046},[342,472],[15,44048,44050,44064],{"className":44049},[346],[15,44051,44053],{"className":44052,"style":1214},[350],[15,44054,44055,44058],{"style":3206},[15,44056],{"className":44057,"style":358},[357],[15,44059,44061],{"className":44060},[362,363,364,365],[15,44062,278],{"className":44063},[61,365],[15,44065,497],{"className":44066},[496],[15,44068,44070],{"className":44069},[346],[15,44071,44073],{"className":44072,"style":504},[350],[15,44074],{},[15,44076,44078,44142],{"className":44077,"translate":19},[18],[15,44079,44081],{"className":44080},[23],[25,44082,44083],{"xmlns":27},[29,44084,44085,44139],{},[32,44086,44087,44089,44091,44093,44095,44097,44099,44101,44103,44105,44107,44109,44123,44129,44131,44133],{},[35,44088,22651],{},[81,44090,14821],{"stretchy":86},[35,44092,21286],{"mathvariant":258},[81,44094,741],{"stretchy":86},[35,44096,79],{},[81,44098,3714],{},[35,44100,1029],{"mathvariant":251},[81,44102,750],{"stretchy":86},[81,44104,14830],{"stretchy":86},[81,44106,244],{},[81,44108,239],{},[289,44110,44111,44113,44121],{},[81,44112,293],{},[32,44114,44115,44117,44119],{},[35,44116,272],{},[81,44118,244],{},[89,44120,91],{},[35,44122,102],{},[266,44124,44125,44127],{},[35,44126,11],{},[35,44128,272],{},[35,44130,22662],{},[81,44132,1547],{},[266,44134,44135,44137],{},[35,44136,11],{},[35,44138,272],{},[40,44140,44141],{"encoding":42},"H[\\mathbb{P}(y \\mid \\mathbf{x})] = -\\sum_{k=1}^K p_k \\log p_k",[15,44143,44145,44175,44196],{"className":44144,"ariaHidden":48},[47],[15,44146,44148,44151,44154,44157,44160,44163,44166,44169,44172],{"className":44147},[52],[15,44149],{"className":44150,"style":142},[56],[15,44152,22651],{"className":44153,"style":22763},[61,122],[15,44155,14821],{"className":44156},[146],[15,44158,21286],{"className":44159},[61,427],[15,44161,741],{"className":44162},[146],[15,44164,79],{"className":44165,"style":123},[61,122],[15,44167],{"className":44168,"style":128},[127],[15,44170,3714],{"className":44171},[132],[15,44173],{"className":44174,"style":128},[127],[15,44176,44178,44181,44184,44187,44190,44193],{"className":44177},[52],[15,44179],{"className":44180,"style":142},[56],[15,44182,1029],{"className":44183},[61,411],[15,44185,43303],{"className":44186},[178],[15,44188],{"className":44189,"style":128},[127],[15,44191,244],{"className":44192},[132],[15,44194],{"className":44195,"style":128},[127],[15,44197,44199,44202,44205,44208,44268,44271,44311,44314,44319,44322],{"className":44198},[52],[15,44200],{"className":44201,"style":3471},[56],[15,44203,239],{"className":44204},[61],[15,44206],{"className":44207,"style":157},[127],[15,44209,44211,44214],{"className":44210},[531],[15,44212,293],{"className":44213,"style":537},[531,535,536],[15,44215,44217],{"className":44216},[338],[15,44218,44220,44260],{"className":44219},[342,472],[15,44221,44223,44257],{"className":44222},[346],[15,44224,44226,44246],{"className":44225,"style":550},[350],[15,44227,44228,44231],{"style":553},[15,44229],{"className":44230,"style":358},[357],[15,44232,44234],{"className":44233},[362,363,364,365],[15,44235,44237,44240,44243],{"className":44236},[61,365],[15,44238,272],{"className":44239,"style":492},[61,122,365],[15,44241,244],{"className":44242},[132,365],[15,44244,91],{"className":44245},[61,365],[15,44247,44248,44251],{"style":574},[15,44249],{"className":44250,"style":358},[357],[15,44252,44254],{"className":44253},[362,363,364,365],[15,44255,102],{"className":44256,"style":174},[61,122,365],[15,44258,497],{"className":44259},[496],[15,44261,44263],{"className":44262},[346],[15,44264,44266],{"className":44265,"style":593},[350],[15,44267],{},[15,44269],{"className":44270,"style":157},[127],[15,44272,44274,44277],{"className":44273},[61],[15,44275,11],{"className":44276},[61,122],[15,44278,44280],{"className":44279},[338],[15,44281,44283,44303],{"className":44282},[342,472],[15,44284,44286,44300],{"className":44285},[346],[15,44287,44289],{"className":44288,"style":479},[350],[15,44290,44291,44294],{"style":482},[15,44292],{"className":44293,"style":358},[357],[15,44295,44297],{"className":44296},[362,363,364,365],[15,44298,272],{"className":44299,"style":492},[61,122,365],[15,44301,497],{"className":44302},[496],[15,44304,44306],{"className":44305},[346],[15,44307,44309],{"className":44308,"style":504},[350],[15,44310],{},[15,44312],{"className":44313,"style":157},[127],[15,44315,22791,44317],{"className":44316},[531],[15,44318,22794],{"style":15619},[15,44320],{"className":44321,"style":157},[127],[15,44323,44325,44328],{"className":44324},[61],[15,44326,11],{"className":44327},[61,122],[15,44329,44331],{"className":44330},[338],[15,44332,44334,44354],{"className":44333},[342,472],[15,44335,44337,44351],{"className":44336},[346],[15,44338,44340],{"className":44339,"style":479},[350],[15,44341,44342,44345],{"style":482},[15,44343],{"className":44344,"style":358},[357],[15,44346,44348],{"className":44347},[362,363,364,365],[15,44349,272],{"className":44350,"style":492},[61,122,365],[15,44352,497],{"className":44353},[496],[15,44355,44357],{"className":44356},[346],[15,44358,44360],{"className":44359,"style":504},[350],[15,44361],{},[15,44363,44365,44449],{"className":44364,"translate":19},[18],[15,44366,44368],{"className":44367},[23],[25,44369,44370],{"xmlns":27},[29,44371,44372,44446],{},[32,44373,44374,44380,44382,44384,44386,44388,44390,44392,44394,44396,44398,44400,44402,44404,44406,44408,44422,44428,44430,44432,44434,44440,44442,44444],{},[266,44375,44376,44378],{},[35,44377,14818],{"mathvariant":258},[35,44379,24680],{"mathvariant":3105},[81,44381,14821],{"stretchy":86},[35,44383,22651],{},[81,44385,741],{"stretchy":86},[35,44387,24680],{"mathvariant":3105},[81,44389,750],{"stretchy":86},[81,44391,14830],{"stretchy":86},[81,44393,244],{},[35,44395,22693],{},[81,44397,741],{"stretchy":86},[35,44399,40265],{},[81,44401,7762],{},[89,44403,91],{},[81,44405,750],{"stretchy":86},[81,44407,239],{},[289,44409,44410,44412,44420],{},[81,44411,293],{},[32,44413,44414,44416,44418],{},[35,44415,272],{},[81,44417,244],{},[89,44419,91],{},[35,44421,102],{},[266,44423,44424,44426],{},[35,44425,11],{},[35,44427,272],{},[282,44429,4802],{},[35,44431,22693],{},[81,44433,741],{"stretchy":86},[266,44435,44436,44438],{},[35,44437,3106],{},[35,44439,272],{},[81,44441,7762],{},[89,44443,91],{},[81,44445,750],{"stretchy":86},[40,44447,44448],{"encoding":42},"\\mathbb{E}_{\\boldsymbol{\\pi}}[H(\\boldsymbol{\\pi})] = \\psi(S + 1) - \\sum_{k=1}^K p_k \\,\\psi(\\alpha_k + 1)",[15,44450,44452,44537,44561,44582,44749],{"className":44451,"ariaHidden":48},[47],[15,44453,44455,44458,44507,44510,44513,44516,44525,44528,44531,44534],{"className":44454},[52],[15,44456],{"className":44457,"style":142},[56],[15,44459,44461,44464],{"className":44460},[61],[15,44462,14818],{"className":44463},[61,427],[15,44465,44467],{"className":44466},[338],[15,44468,44470,44499],{"className":44469},[342,472],[15,44471,44473,44496],{"className":44472},[346],[15,44474,44476],{"className":44475,"style":20966},[350],[15,44477,44478,44481],{"style":482},[15,44479],{"className":44480,"style":358},[357],[15,44482,44484],{"className":44483},[362,363,364,365],[15,44485,44487],{"className":44486},[61,365],[15,44488,44490],{"className":44489},[61,365],[15,44491,44493],{"className":44492},[61,365],[15,44494,24680],{"className":44495,"style":24714},[61,3166,365],[15,44497,497],{"className":44498},[496],[15,44500,44502],{"className":44501},[346],[15,44503,44505],{"className":44504,"style":504},[350],[15,44506],{},[15,44508,14821],{"className":44509},[146],[15,44511,22651],{"className":44512,"style":22763},[61,122],[15,44514,741],{"className":44515},[146],[15,44517,44519],{"className":44518},[61],[15,44520,44522],{"className":44521},[61],[15,44523,24680],{"className":44524,"style":24714},[61,3166],[15,44526,43303],{"className":44527},[178],[15,44529],{"className":44530,"style":128},[127],[15,44532,244],{"className":44533},[132],[15,44535],{"className":44536,"style":128},[127],[15,44538,44540,44543,44546,44549,44552,44555,44558],{"className":44539},[52],[15,44541],{"className":44542,"style":142},[56],[15,44544,22693],{"className":44545,"style":123},[61,122],[15,44547,741],{"className":44548},[146],[15,44550,40265],{"className":44551,"style":40337},[61,122],[15,44553],{"className":44554,"style":772},[127],[15,44556,7762],{"className":44557},[375],[15,44559],{"className":44560,"style":772},[127],[15,44562,44564,44567,44570,44573,44576,44579],{"className":44563},[52],[15,44565],{"className":44566,"style":142},[56],[15,44568,91],{"className":44569},[61],[15,44571,750],{"className":44572},[178],[15,44574],{"className":44575,"style":772},[127],[15,44577,239],{"className":44578},[375],[15,44580],{"className":44581,"style":772},[127],[15,44583,44585,44588,44648,44651,44691,44694,44697,44700,44740,44743,44746],{"className":44584},[52],[15,44586],{"className":44587,"style":3471},[56],[15,44589,44591,44594],{"className":44590},[531],[15,44592,293],{"className":44593,"style":537},[531,535,536],[15,44595,44597],{"className":44596},[338],[15,44598,44600,44640],{"className":44599},[342,472],[15,44601,44603,44637],{"className":44602},[346],[15,44604,44606,44626],{"className":44605,"style":550},[350],[15,44607,44608,44611],{"style":553},[15,44609],{"className":44610,"style":358},[357],[15,44612,44614],{"className":44613},[362,363,364,365],[15,44615,44617,44620,44623],{"className":44616},[61,365],[15,44618,272],{"className":44619,"style":492},[61,122,365],[15,44621,244],{"className":44622},[132,365],[15,44624,91],{"className":44625},[61,365],[15,44627,44628,44631],{"style":574},[15,44629],{"className":44630,"style":358},[357],[15,44632,44634],{"className":44633},[362,363,364,365],[15,44635,102],{"className":44636,"style":174},[61,122,365],[15,44638,497],{"className":44639},[496],[15,44641,44643],{"className":44642},[346],[15,44644,44646],{"className":44645,"style":593},[350],[15,44647],{},[15,44649],{"className":44650,"style":157},[127],[15,44652,44654,44657],{"className":44653},[61],[15,44655,11],{"className":44656},[61,122],[15,44658,44660],{"className":44659},[338],[15,44661,44663,44683],{"className":44662},[342,472],[15,44664,44666,44680],{"className":44665},[346],[15,44667,44669],{"className":44668,"style":479},[350],[15,44670,44671,44674],{"style":482},[15,44672],{"className":44673,"style":358},[357],[15,44675,44677],{"className":44676},[362,363,364,365],[15,44678,272],{"className":44679,"style":492},[61,122,365],[15,44681,497],{"className":44682},[496],[15,44684,44686],{"className":44685},[346],[15,44687,44689],{"className":44688,"style":504},[350],[15,44690],{},[15,44692],{"className":44693,"style":157},[127],[15,44695,22693],{"className":44696,"style":123},[61,122],[15,44698,741],{"className":44699},[146],[15,44701,44703,44706],{"className":44702},[61],[15,44704,3106],{"className":44705,"style":3191},[61,122],[15,44707,44709],{"className":44708},[338],[15,44710,44712,44732],{"className":44711},[342,472],[15,44713,44715,44729],{"className":44714},[346],[15,44716,44718],{"className":44717,"style":479},[350],[15,44719,44720,44723],{"style":3206},[15,44721],{"className":44722,"style":358},[357],[15,44724,44726],{"className":44725},[362,363,364,365],[15,44727,272],{"className":44728,"style":492},[61,122,365],[15,44730,497],{"className":44731},[496],[15,44733,44735],{"className":44734},[346],[15,44736,44738],{"className":44737,"style":504},[350],[15,44739],{},[15,44741],{"className":44742,"style":772},[127],[15,44744,7762],{"className":44745},[375],[15,44747],{"className":44748,"style":772},[127],[15,44750,44752,44755,44758],{"className":44751},[52],[15,44753],{"className":44754,"style":142},[56],[15,44756,91],{"className":44757},[61],[15,44759,750],{"className":44760},[178],[15,44762,44764,44897],{"className":44763,"translate":19},[18],[15,44765,44767],{"className":44766},[23],[25,44768,44769],{"xmlns":27},[29,44770,44771,44894],{},[32,44772,44773,44775,44777,44779,44781,44783,44785,44787,44789,44791,44793,44795,44797,44799,44801,44803,44805,44807,44813,44815,44817,44819,44821,44823,44825,44827,44841,44847,44850,44852,44854,44860,44862,44864,44866,44868,44870,44872,44878,44880,44882,44884,44886,44888,44890,44892],{},[35,44774,28923],{},[81,44776,741],{"stretchy":86},[35,44778,79],{},[81,44780,11303],{"separator":48},[35,44782,24680],{"mathvariant":3105},[81,44784,750],{"stretchy":86},[81,44786,244],{},[35,44788,22651],{},[81,44790,14821],{"stretchy":86},[35,44792,21286],{"mathvariant":258},[81,44794,741],{"stretchy":86},[35,44796,79],{},[81,44798,3714],{},[35,44800,1029],{"mathvariant":251},[81,44802,750],{"stretchy":86},[81,44804,14830],{"stretchy":86},[81,44806,239],{},[266,44808,44809,44811],{},[35,44810,14818],{"mathvariant":258},[35,44812,24680],{"mathvariant":3105},[81,44814,14821],{"stretchy":86},[35,44816,22651],{},[81,44818,741],{"stretchy":86},[35,44820,24680],{"mathvariant":3105},[81,44822,750],{"stretchy":86},[81,44824,14830],{"stretchy":86},[81,44826,244],{},[289,44828,44829,44831,44839],{},[81,44830,293],{},[32,44832,44833,44835,44837],{},[35,44834,272],{},[81,44836,244],{},[89,44838,91],{},[35,44840,102],{},[266,44842,44843,44845],{},[35,44844,11],{},[35,44846,272],{},[81,44848,741],{"fence":86,"stretchy":48,"minsize":44849,"maxsize":44849},"1.2em",[35,44851,22693],{},[81,44853,741],{"stretchy":86},[266,44855,44856,44858],{},[35,44857,3106],{},[35,44859,272],{},[81,44861,7762],{},[89,44863,91],{},[81,44865,750],{"stretchy":86},[81,44867,239],{},[35,44869,22662],{},[81,44871,1547],{},[266,44873,44874,44876],{},[35,44875,11],{},[35,44877,272],{},[81,44879,750],{"fence":86,"stretchy":48,"minsize":44849,"maxsize":44849},[81,44881,239],{},[35,44883,22693],{},[81,44885,741],{"stretchy":86},[35,44887,40265],{},[81,44889,7762],{},[89,44891,91],{},[81,44893,750],{"stretchy":86},[40,44895,44896],{"encoding":42},"I(y; \\boldsymbol{\\pi}) = H[\\mathbb{P}(y \\mid \\mathbf{x})] - \\mathbb{E}_{\\boldsymbol{\\pi}}[H(\\boldsymbol{\\pi})] = \\sum_{k=1}^K p_k\\big(\\psi(\\alpha_k + 1) - \\log p_k\\big) - \\psi(S + 1)",[15,44898,44900,44942,44972,44993,45078,45249,45270,45340,45364],{"className":44899,"ariaHidden":48},[47],[15,44901,44903,44906,44909,44912,44915,44918,44921,44930,44933,44936,44939],{"className":44902},[52],[15,44904],{"className":44905,"style":142},[56],[15,44907,28923],{"className":44908,"style":28962},[61,122],[15,44910,741],{"className":44911},[146],[15,44913,79],{"className":44914,"style":123},[61,122],[15,44916,11303],{"className":44917},[153],[15,44919],{"className":44920,"style":157},[127],[15,44922,44924],{"className":44923},[61],[15,44925,44927],{"className":44926},[61],[15,44928,24680],{"className":44929,"style":24714},[61,3166],[15,44931,750],{"className":44932},[178],[15,44934],{"className":44935,"style":128},[127],[15,44937,244],{"className":44938},[132],[15,44940],{"className":44941,"style":128},[127],[15,44943,44945,44948,44951,44954,44957,44960,44963,44966,44969],{"className":44944},[52],[15,44946],{"className":44947,"style":142},[56],[15,44949,22651],{"className":44950,"style":22763},[61,122],[15,44952,14821],{"className":44953},[146],[15,44955,21286],{"className":44956},[61,427],[15,44958,741],{"className":44959},[146],[15,44961,79],{"className":44962,"style":123},[61,122],[15,44964],{"className":44965,"style":128},[127],[15,44967,3714],{"className":44968},[132],[15,44970],{"className":44971,"style":128},[127],[15,44973,44975,44978,44981,44984,44987,44990],{"className":44974},[52],[15,44976],{"className":44977,"style":142},[56],[15,44979,1029],{"className":44980},[61,411],[15,44982,43303],{"className":44983},[178],[15,44985],{"className":44986,"style":772},[127],[15,44988,239],{"className":44989},[375],[15,44991],{"className":44992,"style":772},[127],[15,44994,44996,44999,45048,45051,45054,45057,45066,45069,45072,45075],{"className":44995},[52],[15,44997],{"className":44998,"style":142},[56],[15,45000,45002,45005],{"className":45001},[61],[15,45003,14818],{"className":45004},[61,427],[15,45006,45008],{"className":45007},[338],[15,45009,45011,45040],{"className":45010},[342,472],[15,45012,45014,45037],{"className":45013},[346],[15,45015,45017],{"className":45016,"style":20966},[350],[15,45018,45019,45022],{"style":482},[15,45020],{"className":45021,"style":358},[357],[15,45023,45025],{"className":45024},[362,363,364,365],[15,45026,45028],{"className":45027},[61,365],[15,45029,45031],{"className":45030},[61,365],[15,45032,45034],{"className":45033},[61,365],[15,45035,24680],{"className":45036,"style":24714},[61,3166,365],[15,45038,497],{"className":45039},[496],[15,45041,45043],{"className":45042},[346],[15,45044,45046],{"className":45045,"style":504},[350],[15,45047],{},[15,45049,14821],{"className":45050},[146],[15,45052,22651],{"className":45053,"style":22763},[61,122],[15,45055,741],{"className":45056},[146],[15,45058,45060],{"className":45059},[61],[15,45061,45063],{"className":45062},[61],[15,45064,24680],{"className":45065,"style":24714},[61,3166],[15,45067,43303],{"className":45068},[178],[15,45070],{"className":45071,"style":128},[127],[15,45073,244],{"className":45074},[132],[15,45076],{"className":45077,"style":128},[127],[15,45079,45081,45085,45145,45148,45188,45194,45197,45200,45240,45243,45246],{"className":45080},[52],[15,45082],{"className":45083,"style":45084},[56],"height:1.3312em;vertical-align:-0.35em;",[15,45086,45088,45091],{"className":45087},[531],[15,45089,293],{"className":45090,"style":537},[531,535,536],[15,45092,45094],{"className":45093},[338],[15,45095,45097,45137],{"className":45096},[342,472],[15,45098,45100,45134],{"className":45099},[346],[15,45101,45103,45123],{"className":45102,"style":550},[350],[15,45104,45105,45108],{"style":553},[15,45106],{"className":45107,"style":358},[357],[15,45109,45111],{"className":45110},[362,363,364,365],[15,45112,45114,45117,45120],{"className":45113},[61,365],[15,45115,272],{"className":45116,"style":492},[61,122,365],[15,45118,244],{"className":45119},[132,365],[15,45121,91],{"className":45122},[61,365],[15,45124,45125,45128],{"style":574},[15,45126],{"className":45127,"style":358},[357],[15,45129,45131],{"className":45130},[362,363,364,365],[15,45132,102],{"className":45133,"style":174},[61,122,365],[15,45135,497],{"className":45136},[496],[15,45138,45140],{"className":45139},[346],[15,45141,45143],{"className":45142,"style":593},[350],[15,45144],{},[15,45146],{"className":45147,"style":157},[127],[15,45149,45151,45154],{"className":45150},[61],[15,45152,11],{"className":45153},[61,122],[15,45155,45157],{"className":45156},[338],[15,45158,45160,45180],{"className":45159},[342,472],[15,45161,45163,45177],{"className":45162},[346],[15,45164,45166],{"className":45165,"style":479},[350],[15,45167,45168,45171],{"style":482},[15,45169],{"className":45170,"style":358},[357],[15,45172,45174],{"className":45173},[362,363,364,365],[15,45175,272],{"className":45176,"style":492},[61,122,365],[15,45178,497],{"className":45179},[496],[15,45181,45183],{"className":45182},[346],[15,45184,45186],{"className":45185,"style":504},[350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关键在于 ",[15,45382,45384,45430],{"className":45383,"translate":19},[18],[15,45385,45387],{"className":45386},[23],[25,45388,45389],{"xmlns":27},[29,45390,45391,45427],{},[32,45392,45393,45395,45397,45399,45401,45403,45405,45411,45417,45419,45421],{},[35,45394,22651],{},[81,45396,741],{"stretchy":86},[35,45398,24680],{"mathvariant":3105},[81,45400,750],{"stretchy":86},[81,45402,244],{},[81,45404,239],{},[266,45406,45407,45409],{},[81,45408,293],{},[35,45410,272],{},[266,45412,45413,45415],{},[35,45414,24680],{},[35,45416,272],{},[35,45418,22662],{},[81,45420,1547],{},[266,45422,45423,45425],{},[35,45424,24680],{},[35,45426,272],{},[40,45428,45429],{"encoding":42},"H(\\boldsymbol{\\pi}) = -\\sum_k \\pi_k \\log 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= -\\sum_k \\mathbb{E}[\\pi_k \\log \\pi_k]",[15,45674,45676,45715],{"className":45675,"ariaHidden":48},[47],[15,45677,45679,45682,45685,45688,45691,45694,45703,45706,45709,45712],{"className":45678},[52],[15,45680],{"className":45681,"style":142},[56],[15,45683,14818],{"className":45684},[61,427],[15,45686,14821],{"className":45687},[146],[15,45689,22651],{"className":45690,"style":22763},[61,122],[15,45692,741],{"className":45693},[146],[15,45695,45697],{"className":45696},[61],[15,45698,45700],{"className":45699},[61],[15,45701,24680],{"className":45702,"style":24714},[61,3166],[15,45704,43303],{"className":45705},[178],[15,45707],{"className":45708,"style":128},[127],[15,45710,244],{"className":45711},[132],[15,45713],{"className":45714,"style":128},[127],[15,45716,45718,45721,45724,45727,45767,45770,45773,45776,45816,45819,45824,45827,45867],{"className":45717},[52],[15,45719],{"className":45720,"style":6051},[56],[15,45722,239],{"className":45723},[61],[15,45725],{"className":45726,"style":157},[127],[15,45728,45730,45733],{"className":45729},[531],[15,45731,293],{"className":45732,"style":537},[531,535,536],[15,45734,45736],{"className":45735},[338],[15,45737,45739,45759],{"className":45738},[342,472],[15,45740,45742,45756],{"className":45741},[346],[15,45743,45745],{"className":45744,"style":29989},[350],[15,45746,45747,45750],{"style":553},[15,45748],{"className":45749,"style":358},[357],[15,45751,45753],{"className":45752},[362,363,364,365],[15,45754,272],{"className":45755,"style":492},[61,122,365],[15,45757,497],{"className":45758},[496],[15,45760,45762],{"className":45761},[346],[15,45763,45765],{"className":45764,"style":593},[350],[15,45766],{},[15,45768],{"className":45769,"style":157},[127],[15,45771,14818],{"className":45772},[61,427],[15,45774,14821],{"className":45775},[146],[15,45777,45779,45782],{"className":45778},[61],[15,45780,24680],{"className":45781,"style":123},[61,122],[15,45783,45785],{"className":45784},[338],[15,45786,45788,45808],{"className":45787},[342,472],[15,45789,45791,45805],{"className":45790},[346],[15,45792,45794],{"className":45793,"style":479},[350],[15,45795,45796,45799],{"style":27036},[15,45797],{"className":45798,"style":358},[357],[15,45800,45802],{"className":45801},[362,363,364,365],[15,45803,272],{"className":45804,"style":492},[61,122,365],[15,45806,497],{"className":45807},[496],[15,45809,45811],{"className":45810},[346],[15,45812,45814],{"className":45813,"style":504},[350],[15,45815],{},[15,45817],{"className":45818,"style":157},[127],[15,45820,22791,45822],{"className":45821},[531],[15,45823,22794],{"style":15619},[15,45825],{"className":45826,"style":157},[127],[15,45828,45830,45833],{"className":45829},[61],[15,45831,24680],{"className":45832,"style":123},[61,122],[15,45834,45836],{"className":45835},[338],[15,45837,45839,45859],{"className":45838},[342,472],[15,45840,45842,45856],{"className":45841},[346],[15,45843,45845],{"className":45844,"style":479},[350],[15,45846,45847,45850],{"style":27036},[15,45848],{"className":45849,"style":358},[357],[15,45851,45853],{"className":45852},[362,363,364,365],[15,45854,272],{"className":45855,"style":492},[61,122,365],[15,45857,497],{"className":45858},[496],[15,45860,45862],{"className":45861},[346],[15,45863,45865],{"className":45864,"style":504},[350],[15,45866],{},[15,45868,14830],{"className":45869},[178],"。由 1.4 节的边缘 Beta 性，",[15,45872,45874,45920],{"className":45873,"translate":19},[18],[15,45875,45877],{"className":45876},[23],[25,45878,45879],{"xmlns":27},[29,45880,45881,45917],{},[32,45882,45883,45889,45891,45893,45895,45897,45903,45905,45907,45909,45915],{},[266,45884,45885,45887],{},[35,45886,24680],{},[35,45888,272],{},[81,45890,4183],{},[35,45892,12956],{"mathvariant":231},[81,45894,1547],{},[81,45896,741],{"stretchy":86},[266,45898,45899,45901],{},[35,45900,3106],{},[35,45902,272],{},[81,45904,94],{"separator":48},[35,45906,40265],{},[81,45908,239],{},[266,45910,45911,45913],{},[35,45912,3106],{},[35,45914,272],{},[81,45916,750],{"stretchy":86},[40,45918,45919],{"encoding":42},"\\pi_k \\sim \\operatorname{Beta}(\\alpha_k, S - \\alpha_k)",[15,45921,45923,45978,46051],{"className":45922,"ariaHidden":48},[47],[15,45924,45926,45929,45969,45972,45975],{"className":45925},[52],[15,45927],{"className":45928,"style":2511},[56],[15,45930,45932,45935],{"className":45931},[61],[15,45933,24680],{"className":45934,"style":123},[61,122],[15,45936,45938],{"className":45937},[338],[15,45939,45941,45961],{"className":45940},[342,472],[15,45942,45944,45958],{"className":45943},[346],[15,45945,45947],{"className":45946,"style":479},[350],[15,45948,45949,45952],{"style":27036},[15,45950],{"className":45951,"style":358},[357],[15,45953,45955],{"className":45954},[362,363,364,365],[15,45956,272],{"className":45957,"style":492},[61,122,365],[15,45959,497],{"className":45960},[496],[15,45962,45964],{"className":45963},[346],[15,45965,45967],{"className":45966,"style":504},[350],[15,45968],{},[15,45970],{"className":45971,"style":128},[127],[15,45973,4183],{"className":45974},[132],[15,45976],{"className":45977,"style":128},[127],[15,45979,45981,45984,45990,45993,46033,46036,46039,46042,46045,46048],{"className":45980},[52],[15,45982],{"className":45983,"style":142},[56],[15,45985,45987],{"className":45986},[531],[15,45988,12956],{"className":45989},[61,1622],[15,45991,741],{"className":45992},[146],[15,45994,45996,45999],{"className":45995},[61],[15,45997,3106],{"className":45998,"style":3191},[61,122],[15,46000,46002],{"className":46001},[338],[15,46003,46005,46025],{"className":46004},[342,472],[15,46006,46008,46022],{"className":46007},[346],[15,46009,46011],{"className":46010,"style":479},[350],[15,46012,46013,46016],{"style":3206},[15,46014],{"className":46015,"style":358},[357],[15,46017,46019],{"className":46018},[362,363,364,365],[15,46020,272],{"className":46021,"style":492},[61,122,365],[15,46023,497],{"className":46024},[496],[15,46026,46028],{"className":46027},[346],[15,46029,46031],{"className":46030,"style":504},[350],[15,46032],{},[15,46034,94],{"className":46035},[153],[15,46037],{"className":46038,"style":157},[127],[15,46040,40265],{"className":46041,"style":40337},[61,122],[15,46043],{"className":46044,"style":772},[127],[15,46046,239],{"className":46047},[375],[15,46049],{"className":46050,"style":772},[127],[15,46052,46054,46057,46097],{"className":46053},[52],[15,46055],{"className":46056,"style":142},[56],[15,46058,46060,46063],{"className":46059},[61],[15,46061,3106],{"className":46062,"style":3191},[61,122],[15,46064,46066],{"className":46065},[338],[15,46067,46069,46089],{"className":46068},[342,472],[15,46070,46072,46086],{"className":46071},[346],[15,46073,46075],{"className":46074,"style":479},[350],[15,46076,46077,46080],{"style":3206},[15,46078],{"className":46079,"style":358},[357],[15,46081,46083],{"className":46082},[362,363,364,365],[15,46084,272],{"className":46085,"style":492},[61,122,365],[15,46087,497],{"className":46088},[496],[15,46090,46092],{"className":46091},[346],[15,46093,46095],{"className":46094,"style":504},[350],[15,46096],{},[15,46098,750],{"className":46099},[178],"，而对该 Beta 分布有恒等式 ",[15,46102,46104,46178],{"className":46103,"translate":19},[18],[15,46105,46107],{"className":46106},[23],[25,46108,46109],{"xmlns":27},[29,46110,46111,46175],{},[32,46112,46113,46115,46117,46119,46121,46123,46125,46127,46129,46141,46143,46145,46147,46149,46151,46153,46155,46157,46159,46161,46163,46165,46167,46169,46171,46173],{},[35,46114,14818],{"mathvariant":258},[81,46116,14821],{"stretchy":86},[35,46118,38],{},[35,46120,22662],{},[81,46122,1547],{},[35,46124,38],{},[81,46126,14830],{"stretchy":86},[81,46128,244],{},[3722,46130,46131,46133],{},[35,46132,33435],{},[32,46134,46135,46137,46139],{},[35,46136,33435],{},[81,46138,7762],{},[35,46140,33425],{},[81,46142,741],{"fence":86,"stretchy":48,"minsize":44849,"maxsize":44849},[35,46144,22693],{},[81,46146,741],{"stretchy":86},[35,46148,33435],{},[81,46150,7762],{},[89,46152,91],{},[81,46154,750],{"stretchy":86},[81,46156,239],{},[35,46158,22693],{},[81,46160,741],{"stretchy":86},[35,46162,33435],{},[81,46164,7762],{},[35,46166,33425],{},[81,46168,7762],{},[89,46170,91],{},[81,46172,750],{"stretchy":86},[81,46174,750],{"fence":86,"stretchy":48,"minsize":44849,"maxsize":44849},[40,46176,46177],{"encoding":42},"\\mathbb{E}[X \\log X] = \\frac{a}{a+b}\\big(\\psi(a+1) - \\psi(a+b+1)\\big)",[15,46179,46181,46222,46329,46350,46374,46393],{"className":46180,"ariaHidden":48},[47],[15,46182,46184,46187,46190,46193,46196,46199,46204,46207,46210,46213,46216,46219],{"className":46183},[52],[15,46185],{"className":46186,"style":142},[56],[15,46188,14818],{"className":46189},[61,427],[15,46191,14821],{"className":46192},[146],[15,46194,38],{"className":46195,"style":28962},[61,122],[15,46197],{"className":46198,"style":157},[127],[15,46200,22791,46202],{"className":46201},[531],[15,46203,22794],{"style":15619},[15,46205],{"className":46206,"style":157},[127],[15,46208,38],{"className":46209,"style":28962},[61,122],[15,46211,14830],{"className":46212},[178],[15,46214],{"className":46215,"style":128},[127],[15,46217,244],{"className":46218},[132],[15,46220],{"className":46221,"style":128},[127],[15,46223,46225,46229,46305,46311,46314,46317,46320,46323,46326],{"className":46224},[52],[15,46226],{"className":46227,"style":46228},[56],"height:1.2533em;vertical-align:-0.4033em;",[15,46230,46232,46235,46302],{"className":46231},[61],[15,46233],{"className":46234},[146,3841],[15,46236,46238],{"className":46237},[3722],[15,46239,46241,46293],{"className":46240},[342,472],[15,46242,46244,46290],{"className":46243},[346],[15,46245,46248,46268,46276],{"className":46246,"style":46247},[350],"height:0.6954em;",[15,46249,46250,46253],{"style":3857},[15,46251],{"className":46252,"style":3861},[357],[15,46254,46256],{"className":46255},[362,363,364,365],[15,46257,46259,46262,46265],{"className":46258},[61,365],[15,46260,33435],{"className":46261},[61,122,365],[15,46263,7762],{"className":46264},[375,365],[15,46266,33425],{"className":46267},[61,122,365],[15,46269,46270,46273],{"style":3889},[15,46271],{"className":46272,"style":3861},[357],[15,46274],{"className":46275,"style":3897},[3896],[15,46277,46278,46281],{"style":3900},[15,46279],{"className":46280,"style":3861},[357],[15,46282,46284],{"className":46283},[362,363,364,365],[15,46285,46287],{"className":46286},[61,365],[15,46288,33435],{"className":46289},[61,122,365],[15,46291,497],{"className":46292},[496],[15,46294,46296],{"className":46295},[346],[15,46297,46300],{"className":46298,"style":46299},[350],"height:0.4033em;",[15,46301],{},[15,46303],{"className":46304},[178,3841],[15,46306,46308],{"className":46307},[61],[15,46309,741],{"className":46310},[406,4067],[15,46312,22693],{"className":46313,"style":123},[61,122],[15,46315,741],{"className":46316},[146],[15,46318,33435],{"className":46319},[61,122],[15,46321],{"className":46322,"style":772},[127],[15,46324,7762],{"className":46325},[375],[15,46327],{"className":46328,"style":772},[127],[15,46330,46332,46335,46338,46341,46344,46347],{"className":46331},[52],[15,46333],{"className":46334,"style":142},[56],[15,46336,91],{"className":46337},[61],[15,46339,750],{"className":46340},[178],[15,46342],{"className":46343,"style":772},[127],[15,46345,239],{"className":46346},[375],[15,46348],{"className":46349,"style":772},[127],[15,46351,46353,46356,46359,46362,46365,46368,46371],{"className":46352},[52],[15,46354],{"className":46355,"style":142},[56],[15,46357,22693],{"className":46358,"style":123},[61,122],[15,46360,741],{"className":46361},[146],[15,46363,33435],{"className":46364},[61,122],[15,46366],{"className":46367,"style":772},[127],[15,46369,7762],{"className":46370},[375],[15,46372],{"className":46373,"style":772},[127],[15,46375,46377,46381,46384,46387,46390],{"className":46376},[52],[15,46378],{"className":46379,"style":46380},[56],"height:0.7778em;vertical-align:-0.0833em;",[15,46382,33425],{"className":46383},[61,122],[15,46385],{"className":46386,"style":772},[127],[15,46388,7762],{"className":46389},[375],[15,46391],{"className":46392,"style":772},[127],[15,46394,46396,46399,46402,46405],{"className":46395},[52],[15,46397],{"className":46398,"style":45276},[56],[15,46400,91],{"className":46401},[61],[15,46403,750],{"className":46404},[178],[15,46406,46408],{"className":46407},[61],[15,46409,750],{"className":46410},[406,4067],"（此处 ",[15,46413,46415,46437],{"className":46414,"translate":19},[18],[15,46416,46418],{"className":46417},[23],[25,46419,46420],{"xmlns":27},[29,46421,46422,46434],{},[32,46423,46424,46426,46428],{},[35,46425,33435],{},[81,46427,244],{},[266,46429,46430,46432],{},[35,46431,3106],{},[35,46433,272],{},[40,46435,46436],{"encoding":42},"a = \\alpha_k",[15,46438,46440,46458],{"className":46439,"ariaHidden":48},[47],[15,46441,46443,46446,46449,46452,46455],{"className":46442},[52],[15,46444],{"className":46445,"style":6032},[56],[15,46447,33435],{"className":46448},[61,122],[15,46450],{"className":46451,"style":128},[127],[15,46453,244],{"className":46454},[132],[15,46456],{"className":46457,"style":128},[127],[15,46459,46461,46464],{"className":46460},[52],[15,46462],{"className":46463,"style":2511},[56],[15,46465,46467,46470],{"className":46466},[61],[15,46468,3106],{"className":46469,"style":3191},[61,122],[15,46471,46473],{"className":46472},[338],[15,46474,46476,46496],{"className":46475},[342,472],[15,46477,46479,46493],{"className":46478},[346],[15,46480,46482],{"className":46481,"style":479},[350],[15,46483,46484,46487],{"style":3206},[15,46485],{"className":46486,"style":358},[357],[15,46488,46490],{"className":46489},[362,363,364,365],[15,46491,272],{"className":46492,"style":492},[61,122,365],[15,46494,497],{"className":46495},[496],[15,46497,46499],{"className":46498},[346],[15,46500,46502],{"className":46501,"style":504},[350],[15,46503],{},[15,46505,46507,46529],{"className":46506,"translate":19},[18],[15,46508,46510],{"className":46509},[23],[25,46511,46512],{"xmlns":27},[29,46513,46514,46526],{},[32,46515,46516,46518,46520,46522,46524],{},[35,46517,33435],{},[81,46519,7762],{},[35,46521,33425],{},[81,46523,244],{},[35,46525,40265],{},[40,46527,46528],{"encoding":42},"a + b = S",[15,46530,46532,46550,46568],{"className":46531,"ariaHidden":48},[47],[15,46533,46535,46538,46541,46544,46547],{"className":46534},[52],[15,46536],{"className":46537,"style":6737},[56],[15,46539,33435],{"className":46540},[61,122],[15,46542],{"className":46543,"style":772},[127],[15,46545,7762],{"className":46546},[375],[15,46548],{"className":46549,"style":772},[127],[15,46551,46553,46556,46559,46562,46565],{"className":46552},[52],[15,46554],{"className":46555,"style":10452},[56],[15,46557,33425],{"className":46558},[61,122],[15,46560],{"className":46561,"style":128},[127],[15,46563,244],{"className":46564},[132],[15,46566],{"className":46567,"style":128},[127],[15,46569,46571,46574],{"className":46570},[52],[15,46572],{"className":46573,"style":57},[56],[15,46575,40265],{"className":46576,"style":40337},[61,122],"）。代入 ",[15,46579,46581,46615],{"className":46580,"translate":19},[18],[15,46582,46584],{"className":46583},[23],[25,46585,46586],{"xmlns":27},[29,46587,46588,46612],{},[32,46589,46590,46592,46594,46596,46598,46600,46602,46604,46606],{},[35,46591,33435],{},[35,46593,3020],{"mathvariant":231},[81,46595,741],{"stretchy":86},[35,46597,33435],{},[81,46599,7762],{},[35,46601,33425],{},[81,46603,750],{"stretchy":86},[81,46605,244],{},[266,46607,46608,46610],{},[35,46609,11],{},[35,46611,272],{},[40,46613,46614],{"encoding":42},"a\u002F(a+b) = p_k",[15,46616,46618,46645,46666],{"className":46617,"ariaHidden":48},[47],[15,46619,46621,46624,46627,46630,46633,46636,46639,46642],{"className":46620},[52],[15,46622],{"className":46623,"style":142},[56],[15,46625,33435],{"className":46626},[61,122],[15,46628,3020],{"className":46629},[61],[15,46631,741],{"className":46632},[146],[15,46634,33435],{"className":46635},[61,122],[15,46637],{"className":46638,"style":772},[127],[15,46640,7762],{"className":46641},[375],[15,46643],{"className":46644,"style":772},[127],[15,46646,46648,46651,46654,46657,46660,46663],{"className":46647},[52],[15,46649],{"className":46650,"style":142},[56],[15,46652,33425],{"className":46653},[61,122],[15,46655,750],{"className":46656},[178],[15,46658],{"className":46659,"style":128},[127],[15,46661,244],{"className":46662},[132],[15,46664],{"className":46665,"style":128},[127],[15,46667,46669,46672],{"className":46668},[52],[15,46670],{"className":46671,"style":9437},[56],[15,46673,46675,46678],{"className":46674},[61],[15,46676,11],{"className":46677},[61,122],[15,46679,46681],{"className":46680},[338],[15,46682,46684,46704],{"className":46683},[342,472],[15,46685,46687,46701],{"className":46686},[346],[15,46688,46690],{"className":46689,"style":479},[350],[15,46691,46692,46695],{"style":482},[15,46693],{"className":46694,"style":358},[357],[15,46696,46698],{"className":46697},[362,363,364,365],[15,46699,272],{"className":46700,"style":492},[61,122,365],[15,46702,497],{"className":46703},[496],[15,46705,46707],{"className":46706},[346],[15,46708,46710],{"className":46709,"style":504},[350],[15,46711],{}," 即得第二式。互信息由 ",[15,46714,46716,46760],{"className":46715,"translate":19},[18],[15,46717,46719],{"className":46718},[23],[25,46720,46721],{"xmlns":27},[29,46722,46723,46757],{},[32,46724,46725,46727,46729,46731,46733,46735,46737,46739,46741,46743,46745,46747,46749,46751,46753,46755],{},[35,46726,28923],{},[81,46728,244],{},[35,46730,22651],{},[81,46732,14821],{"stretchy":86},[35,46734,14818],{"mathvariant":258},[282,46736,4802],{},[35,46738,24680],{"mathvariant":3105},[81,46740,14830],{"stretchy":86},[81,46742,239],{},[35,46744,14818],{"mathvariant":258},[81,46746,14821],{"stretchy":86},[35,46748,22651],{},[81,46750,741],{"stretchy":86},[35,46752,24680],{"mathvariant":3105},[81,46754,750],{"stretchy":86},[81,46756,14830],{"stretchy":86},[40,46758,46759],{"encoding":42},"I = H[\\mathbb{E}\\,\\boldsymbol{\\pi}] - \\mathbb{E}[H(\\boldsymbol{\\pi})]",[15,46761,46763,46781,46820],{"className":46762,"ariaHidden":48},[47],[15,46764,46766,46769,46772,46775,46778],{"className":46765},[52],[15,46767],{"className":46768,"style":57},[56],[15,46770,28923],{"className":46771,"style":28962},[61,122],[15,46773],{"className":46774,"style":128},[127],[15,46776,244],{"className":46777},[132],[15,46779],{"className":46780,"style":128},[127],[15,46782,46784,46787,46790,46793,46796,46799,46808,46811,46814,46817],{"className":46783},[52],[15,46785],{"className":46786,"style":142},[56],[15,46788,22651],{"className":46789,"style":22763},[61,122],[15,46791,14821],{"className":46792},[146],[15,46794,14818],{"className":46795},[61,427],[15,46797],{"className":46798,"style":157},[127],[15,46800,46802],{"className":46801},[61],[15,46803,46805],{"className":46804},[61],[15,46806,24680],{"className":46807,"style":24714},[61,3166],[15,46809,14830],{"className":46810},[178],[15,46812],{"className":46813,"style":772},[127],[15,46815,239],{"className":46816},[375],[15,46818],{"className":46819,"style":772},[127],[15,46821,46823,46826,46829,46832,46835,46838,46847],{"className":46822},[52],[15,46824],{"className":46825,"style":142},[56],[15,46827,14818],{"className":46828},[61,427],[15,46830,14821],{"className":46831},[146],[15,46833,22651],{"className":46834,"style":22763},[61,122],[15,46836,741],{"className":46837},[146],[15,46839,46841],{"className":46840},[61],[15,46842,46844],{"className":46843},[61],[15,46845,24680],{"className":46846,"style":24714},[61,3166],[15,46848,43303],{"className":46849},[178]," 得到。",[11,46852,46853,46854,46857,46858,21891,46921,46972,46973,47033,47034,47063,47064,47092,47093,47096,47097,47125],{},"这一分解的意义在于：",[661,46855,46856],{},"它把「数据本身的随机性」与「关于类别分布的认知不足」在数学上分离成了两个可分别计算的量。"," 认知项 ",[15,46859,46861,46885],{"className":46860,"translate":19},[18],[15,46862,46864],{"className":46863},[23],[25,46865,46866],{"xmlns":27},[29,46867,46868,46882],{},[32,46869,46870,46872,46874,46876,46878,46880],{},[35,46871,28923],{},[81,46873,741],{"stretchy":86},[35,46875,79],{},[81,46877,11303],{"separator":48},[35,46879,24680],{"mathvariant":3105},[81,46881,750],{"stretchy":86},[40,46883,46884],{"encoding":42},"I(y;\\boldsymbol{\\pi})",[15,46886,46888],{"className":46887,"ariaHidden":48},[47],[15,46889,46891,46894,46897,46900,46903,46906,46909,46918],{"className":46890},[52],[15,46892],{"className":46893,"style":142},[56],[15,46895,28923],{"className":46896,"style":28962},[61,122],[15,46898,741],{"className":46899},[146],[15,46901,79],{"className":46902,"style":123},[61,122],[15,46904,11303],{"className":46905},[153],[15,46907],{"className":46908,"style":157},[127],[15,46910,46912],{"className":46911},[61],[15,46913,46915],{"className":46914},[61],[15,46916,24680],{"className":46917,"style":24714},[61,3166],[15,46919,750],{"className":46920},[178],[15,46922,46924,46942],{"className":46923,"translate":19},[18],[15,46925,46927],{"className":46926},[23],[25,46928,46929],{"xmlns":27},[29,46930,46931,46939],{},[32,46932,46933,46935,46937],{},[35,46934,40265],{},[81,46936,20704],{},[35,46938,6796],{"mathvariant":231},[40,46940,46941],{"encoding":42},"S \\to \\infty",[15,46943,46945,46963],{"className":46944,"ariaHidden":48},[47],[15,46946,46948,46951,46954,46957,46960],{"className":46947},[52],[15,46949],{"className":46950,"style":57},[56],[15,46952,40265],{"className":46953,"style":40337},[61,122],[15,46955],{"className":46956,"style":128},[127],[15,46958,20704],{"className":46959},[132],[15,46961],{"className":46962,"style":128},[127],[15,46964,46966,46969],{"className":46965},[52],[15,46967],{"className":46968,"style":6032},[56],[15,46970,6796],{"className":46971},[61],"（证据充分）时趋于零，在证据缺失时增大，与 ",[15,46974,46976,46997],{"className":46975,"translate":19},[18],[15,46977,46979],{"className":46978},[23],[25,46980,46981],{"xmlns":27},[29,46982,46983,46995],{},[32,46984,46985,46987,46989,46991,46993],{},[35,46986,33430],{},[81,46988,244],{},[35,46990,102],{},[35,46992,3020],{"mathvariant":231},[35,46994,40265],{},[40,46996,42984],{"encoding":42},[15,46998,47000,47018],{"className":46999,"ariaHidden":48},[47],[15,47001,47003,47006,47009,47012,47015],{"className":47002},[52],[15,47004],{"className":47005,"style":6032},[56],[15,47007,33430],{"className":47008},[61,122],[15,47010],{"className":47011,"style":128},[127],[15,47013,244],{"className":47014},[132],[15,47016],{"className":47017,"style":128},[127],[15,47019,47021,47024,47027,47030],{"className":47020},[52],[15,47022],{"className":47023,"style":142},[56],[15,47025,102],{"className":47026,"style":174},[61,122],[15,47028,3020],{"className":47029},[61],[15,47031,40265],{"className":47032,"style":40337},[61,122]," 的定性行为一致。但它是通过 ",[15,47035,47037,47051],{"className":47036,"translate":19},[18],[15,47038,47040],{"className":47039},[23],[25,47041,47042],{"xmlns":27},[29,47043,47044,47048],{},[32,47045,47046],{},[35,47047,22693],{},[40,47049,47050],{"encoding":42},"\\psi",[15,47052,47054],{"className":47053,"ariaHidden":48},[47],[15,47055,47057,47060],{"className":47056},[52],[15,47058],{"className":47059,"style":14310},[56],[15,47061,22693],{"className":47062,"style":123},[61,122]," 函数的差构造的，捕捉了 ",[15,47065,47067,47080],{"className":47066,"translate":19},[18],[15,47068,47070],{"className":47069},[23],[25,47071,47072],{"xmlns":27},[29,47073,47074,47078],{},[32,47075,47076],{},[35,47077,33430],{},[40,47079,33430],{"encoding":42},[15,47081,47083],{"className":47082,"ariaHidden":48},[47],[15,47084,47086,47089],{"className":47085},[52],[15,47087],{"className":47088,"style":6032},[56],[15,47090,33430],{"className":47091},[61,122]," 所忽略的",[661,47094,47095],{},"证据分布","信息，因而是比 ",[15,47098,47100,47113],{"className":47099,"translate":19},[18],[15,47101,47103],{"className":47102},[23],[25,47104,47105],{"xmlns":27},[29,47106,47107,47111],{},[32,47108,47109],{},[35,47110,33430],{},[40,47112,33430],{"encoding":42},[15,47114,47116],{"className":47115,"ariaHidden":48},[47],[15,47117,47119,47122],{"className":47118},[52],[15,47120],{"className":47121,"style":6032},[56],[15,47123,33430],{"className":47124},[61,122]," 更精细的分数。",[11,47127,47128,47129,47163,47164,47198,47199,47293,47294,20292,47297,47359,47360,47364,47365,47368],{},"不过，这一分解的「认知」标签必须谨慎对待。它在形式上是「关于 ",[15,47130,47132,47145],{"className":47131,"translate":19},[18],[15,47133,47135],{"className":47134},[23],[25,47136,47137],{"xmlns":27},[29,47138,47139,47143],{},[32,47140,47141],{},[35,47142,24680],{"mathvariant":3105},[40,47144,24768],{"encoding":42},[15,47146,47148],{"className":47147,"ariaHidden":48},[47],[15,47149,47151,47154],{"className":47150},[52],[15,47152],{"className":47153,"style":2708},[56],[15,47155,47157],{"className":47156},[61],[15,47158,47160],{"className":47159},[61],[15,47161,24680],{"className":47162,"style":24714},[61,3166]," 的不确定性」，而 ",[15,47165,47167,47180],{"className":47166,"translate":19},[18],[15,47168,47170],{"className":47169},[23],[25,47171,47172],{"xmlns":27},[29,47173,47174,47178],{},[32,47175,47176],{},[35,47177,24680],{"mathvariant":3105},[40,47179,24768],{"encoding":42},[15,47181,47183],{"className":47182,"ariaHidden":48},[47],[15,47184,47186,47189],{"className":47185},[52],[15,47187],{"className":47188,"style":2708},[56],[15,47190,47192],{"className":47191},[61],[15,47193,47195],{"className":47194},[61],[15,47196,24680],{"className":47197,"style":24714},[61,3166]," 在 EDL 中是 ",[15,47200,47202,47226],{"className":47201,"translate":19},[18],[15,47203,47205],{"className":47204},[23],[25,47206,47207],{"xmlns":27},[29,47208,47209,47223],{},[32,47210,47211,47217,47219,47221],{},[266,47212,47213,47215],{},[35,47214,22794],{},[35,47216,37148],{"mathvariant":3105},[81,47218,741],{"stretchy":86},[35,47220,1029],{"mathvariant":251},[81,47222,750],{"stretchy":86},[40,47224,47225],{"encoding":42},"g_{\\boldsymbol{\\theta}}(\\mathbf{x})",[15,47227,47229],{"className":47228,"ariaHidden":48},[47],[15,47230,47232,47235,47284,47287,47290],{"className":47231},[52],[15,47233],{"className":47234,"style":142},[56],[15,47236,47238,47241],{"className":47237},[61],[15,47239,22794],{"className":47240,"style":123},[61,122],[15,47242,47244],{"className":47243},[338],[15,47245,47247,47276],{"className":47246},[342,472],[15,47248,47250,47273],{"className":47249},[346],[15,47251,47253],{"className":47252,"style":479},[350],[15,47254,47255,47258],{"style":27036},[15,47256],{"className":47257,"style":358},[357],[15,47259,47261],{"className":47260},[362,363,364,365],[15,47262,47264],{"className":47263},[61,365],[15,47265,47267],{"className":47266},[61,365],[15,47268,47270],{"className":47269},[61,365],[15,47271,37148],{"className":47272,"style":37212},[61,3166,365],[15,47274,497],{"className":47275},[496],[15,47277,47279],{"className":47278},[346],[15,47280,47282],{"className":47281,"style":504},[350],[15,47283],{},[15,47285,741],{"className":47286},[146],[15,47288,1029],{"className":47289},[61,411],[15,47291,750],{"className":47292},[178]," 的输出，是",[661,47295,47296],{},"网络参数的确定性函数",[15,47298,47300,47323],{"className":47299,"translate":19},[18],[15,47301,47303],{"className":47302},[23],[25,47304,47305],{"xmlns":27},[29,47306,47307,47321],{},[32,47308,47309,47311,47313,47315,47317,47319],{},[35,47310,28923],{},[81,47312,741],{"stretchy":86},[35,47314,79],{},[81,47316,11303],{"separator":48},[35,47318,24680],{"mathvariant":3105},[81,47320,750],{"stretchy":86},[40,47322,46884],{"encoding":42},[15,47324,47326],{"className":47325,"ariaHidden":48},[47],[15,47327,47329,47332,47335,47338,47341,47344,47347,47356],{"className":47328},[52],[15,47330],{"className":47331,"style":142},[56],[15,47333,28923],{"className":47334,"style":28962},[61,122],[15,47336,741],{"className":47337},[146],[15,47339,79],{"className":47340,"style":123},[61,122],[15,47342,11303],{"className":47343},[153],[15,47345],{"className":47346,"style":157},[127],[15,47348,47350],{"className":47349},[61],[15,47351,47353],{"className":47352},[61],[15,47354,24680],{"className":47355,"style":24714},[61,3166],[15,47357,750],{"className":47358},[178]," 度量的是",[47361,47362,47363],"em",{},"类概率向量","的不确定，而不是",[47361,47366,47367],{},"模型参数","的不确定——后者才是贝叶斯认识论意义上的 epistemic uncertainty。二者的距离，正是 4.5 节全部批评的技术根源。",[872,47370,47372],{"id":47371},"四证据深度学习的损失函数","四、证据深度学习的损失函数",[11,47374,47375,47376,47404,47405,47459],{},"EDL 的训练目标不再是「使 ",[15,47377,47379,47392],{"className":47378,"translate":19},[18],[15,47380,47382],{"className":47381},[23],[25,47383,47384],{"xmlns":27},[29,47385,47386,47390],{},[32,47387,47388],{},[35,47389,11],{"mathvariant":251},[40,47391,819],{"encoding":42},[15,47393,47395],{"className":47394,"ariaHidden":48},[47],[15,47396,47398,47401],{"className":47397},[52],[15,47399],{"className":47400,"style":829},[56],[15,47402,11],{"className":47403},[61,411]," 逼近 one-hot 标签」，而是「使整族 ",[15,47406,47408,47429],{"className":47407,"translate":19},[18],[15,47409,47411],{"className":47410},[23],[25,47412,47413],{"xmlns":27},[29,47414,47415,47427],{},[32,47416,47417,47419,47421,47423,47425],{},[35,47418,4186],{"mathvariant":231},[81,47420,1547],{},[81,47422,741],{"stretchy":86},[35,47424,3106],{"mathvariant":3105},[81,47426,750],{"stretchy":86},[40,47428,39072],{"encoding":42},[15,47430,47432],{"className":47431,"ariaHidden":48},[47],[15,47433,47435,47438,47444,47447,47456],{"className":47434},[52],[15,47436],{"className":47437,"style":142},[56],[15,47439,47441],{"className":47440},[531],[15,47442,4186],{"className":47443},[61,1622],[15,47445,741],{"className":47446},[146],[15,47448,47450],{"className":47449},[61],[15,47451,47453],{"className":47452},[61],[15,47454,3106],{"className":47455},[61,3166],[15,47457,750],{"className":47458},[178]," 逼近一个理想的分布」。损失函数因此被组织为两项：一项拟合分类目标，一项约束不确定性的行为。",[877,47461,47463],{"id":47462},"_41-type-ii-极大似然与对数风险","4.1 Type-II 极大似然与对数风险",[11,47465,47466,47467,47470],{},"最自然的构造是极小化对数损失在迪利克雷先验下的",[661,47468,47469],{},"贝叶斯风险","（亦称 Type-II 极大似然）：",[15,47472,47474],{"className":47473,"translate":19},[212],[15,47475,47477,47535],{"className":47476,"translate":19},[18],[15,47478,47480],{"className":47479},[23],[25,47481,47482],{"xmlns":27,"display":221},[29,47483,47484,47532],{},[32,47485,47486,47494,47496,47516,47518,47520,47522,47524,47530],{},[227,47487,47488,47491],{},[35,47489,47490],{"mathvariant":37},"L",[282,47492,47493],{},"NLL",[81,47495,244],{},[266,47497,47498,47500],{},[35,47499,14818],{"mathvariant":258},[32,47501,47502,47504,47506,47508,47510,47512,47514],{},[35,47503,24680],{"mathvariant":3105},[81,47505,4183],{},[35,47507,4186],{"mathvariant":231},[81,47509,1547],{},[81,47511,741],{"stretchy":86},[35,47513,3106],{"mathvariant":3105},[81,47515,750],{"stretchy":86},[81,47517,14821],{"fence":86,"stretchy":48,"minsize":44849,"maxsize":44849},[81,47519,239],{},[35,47521,22662],{},[81,47523,1547],{},[266,47525,47526,47528],{},[35,47527,24680],{},[35,47529,79],{},[81,47531,14830],{"fence":86,"stretchy":48,"minsize":44849,"maxsize":44849},[40,47533,47534],{"encoding":42},"\\mathcal{L}^{\\text{NLL}} = \\mathbb{E}_{\\boldsymbol{\\pi} \\sim \\operatorname{Dir}(\\boldsymbol{\\alpha})}\\big[-\\log \\pi_y\\big]",[15,47536,47538,47588,47685],{"className":47537,"ariaHidden":48},[47],[15,47539,47541,47544,47579,47582,47585],{"className":47540},[52],[15,47542],{"className":47543,"style":328},[56],[15,47545,47547,47550],{"className":47546},[61],[15,47548,47490],{"className":47549},[61,62],[15,47551,47553],{"className":47552},[338],[15,47554,47556],{"className":47555},[342],[15,47557,47559],{"className":47558},[346],[15,47560,47562],{"className":47561,"style":328},[350],[15,47563,47564,47567],{"style":353},[15,47565],{"className":47566,"style":358},[357],[15,47568,47570],{"className":47569},[362,363,364,365],[15,47571,47573],{"className":47572},[61,365],[15,47574,47576],{"className":47575},[61,13548,365],[15,47577,47493],{"className":47578},[61,365],[15,47580],{"className":47581,"style":128},[127],[15,47583,244],{"className":47584},[132],[15,47586],{"className":47587,"style":128},[127],[15,47589,47591,47595,47670,47676,47679,47682],{"className":47590},[52],[15,47592],{"className":47593,"style":47594},[56],"height:1.2052em;vertical-align:-0.3552em;",[15,47596,47598,47601],{"className":47597},[61],[15,47599,14818],{"className":47600},[61,427],[15,47602,47604],{"className":47603},[338],[15,47605,47607,47661],{"className":47606},[342,472],[15,47608,47610,47658],{"className":47609},[346],[15,47611,47613],{"className":47612,"style":22404},[350],[15,47614,47616,47619],{"style":47615},"top:-2.5198em;margin-left:0em;margin-right:0.05em;",[15,47617],{"className":47618,"style":358},[357],[15,47620,47622],{"className":47621},[362,363,364,365],[15,47623,47625,47634,47637,47643,47646,47655],{"className":47624},[61,365],[15,47626,47628],{"className":47627},[61,365],[15,47629,47631],{"className":47630},[61,365],[15,47632,24680],{"className":47633,"style":24714},[61,3166,365],[15,47635,4183],{"className":47636},[132,365],[15,47638,47640],{"className":47639},[531,365],[15,47641,4186],{"className":47642},[61,1622,365],[15,47644,741],{"className":47645},[146,365],[15,47647,47649],{"className":47648},[61,365],[15,47650,47652],{"className":47651},[61,365],[15,47653,3106],{"className":47654},[61,3166,365],[15,47656,750],{"className":47657},[178,365],[15,47659,497],{"className":47660},[496],[15,47662,47664],{"className":47663},[346],[15,47665,47668],{"className":47666,"style":47667},[350],"height:0.3552em;",[15,47669],{},[15,47671,47673],{"className":47672},[61],[15,47674,14821],{"className":47675},[406,4067],[15,47677],{"className":47678,"style":772},[127],[15,47680,239],{"className":47681},[375],[15,47683],{"className":47684,"style":772},[127],[15,47686,47688,47691,47696,47699,47739],{"className":47687},[52],[15,47689],{"className":47690,"style":45276},[56],[15,47692,22791,47694],{"className":47693},[531],[15,47695,22794],{"style":15619},[15,47697],{"className":47698,"style":157},[127],[15,47700,47702,47705],{"className":47701},[61],[15,47703,24680],{"className":47704,"style":123},[61,122],[15,47706,47708],{"className":47707},[338],[15,47709,47711,47731],{"className":47710},[342,472],[15,47712,47714,47728],{"className":47713},[346],[15,47715,47717],{"className":47716,"style":27033},[350],[15,47718,47719,47722],{"style":27036},[15,47720],{"className":47721,"style":358},[357],[15,47723,47725],{"className":47724},[362,363,364,365],[15,47726,79],{"className":47727,"style":123},[61,122,365],[15,47729,497],{"className":47730},[496],[15,47732,47734],{"className":47733},[346],[15,47735,47737],{"className":47736,"style":15707},[350],[15,47738],{},[15,47740,47742],{"className":47741},[61],[15,47743,14830],{"className":47744},[406,4067],[882,47746,47747],{},[11,47748,47749,47752,47753,47781,47782],{},[661,47750,47751],{},"命题 4.1（对数风险的闭式）"," 设真实标签为 ",[15,47754,47756,47769],{"className":47755,"translate":19},[18],[15,47757,47759],{"className":47758},[23],[25,47760,47761],{"xmlns":27},[29,47762,47763,47767],{},[32,47764,47765],{},[35,47766,79],{},[40,47768,79],{"encoding":42},[15,47770,47772],{"className":47771,"ariaHidden":48},[47],[15,47773,47775,47778],{"className":47774},[52],[15,47776],{"className":47777,"style":9437},[56],[15,47779,79],{"className":47780,"style":123},[61,122],"，则\n",[15,47783,47785,47827],{"className":47784,"translate":19},[18],[15,47786,47788],{"className":47787},[23],[25,47789,47790],{"xmlns":27},[29,47791,47792,47824],{},[32,47793,47794,47800,47802,47804,47806,47808,47810,47812,47814,47816,47822],{},[227,47795,47796,47798],{},[35,47797,47490],{"mathvariant":37},[282,47799,47493],{},[81,47801,244],{},[35,47803,22693],{},[81,47805,741],{"stretchy":86},[35,47807,40265],{},[81,47809,750],{"stretchy":86},[81,47811,239],{},[35,47813,22693],{},[81,47815,741],{"stretchy":86},[266,47817,47818,47820],{},[35,47819,3106],{},[35,47821,79],{},[81,47823,750],{"stretchy":86},[40,47825,47826],{"encoding":42},"\\mathcal{L}^{\\text{NLL}} = \\psi(S) - \\psi(\\alpha_y)",[15,47828,47830,47880,47907],{"className":47829,"ariaHidden":48},[47],[15,47831,47833,47836,47871,47874,47877],{"className":47832},[52],[15,47834],{"className":47835,"style":695},[56],[15,47837,47839,47842],{"className":47838},[61],[15,47840,47490],{"className":47841},[61,62],[15,47843,47845],{"className":47844},[338],[15,47846,47848],{"className":47847},[342],[15,47849,47851],{"className":47850},[346],[15,47852,47854],{"className":47853,"style":695},[350],[15,47855,47856,47859],{"style":716},[15,47857],{"className":47858,"style":358},[357],[15,47860,47862],{"className":47861},[362,363,364,365],[15,47863,47865],{"className":47864},[61,365],[15,47866,47868],{"className":47867},[61,13548,365],[15,47869,47493],{"className":47870},[61,365],[15,47872],{"className":47873,"style":128},[127],[15,47875,244],{"className":47876},[132],[15,47878],{"className":47879,"style":128},[127],[15,47881,47883,47886,47889,47892,47895,47898,47901,47904],{"className":47882},[52],[15,47884],{"className":47885,"style":142},[56],[15,47887,22693],{"className":47888,"style":123},[61,122],[15,47890,741],{"className":47891},[146],[15,47893,40265],{"className":47894,"style":40337},[61,122],[15,47896,750],{"className":47897},[178],[15,47899],{"className":47900,"style":772},[127],[15,47902,239],{"className":47903},[375],[15,47905],{"className":47906,"style":772},[127],[15,47908,47910,47913,47916,47919,47959],{"className":47909},[52],[15,47911],{"className":47912,"style":17987},[56],[15,47914,22693],{"className":47915,"style":123},[61,122],[15,47917,741],{"className":47918},[146],[15,47920,47922,47925],{"className":47921},[61],[15,47923,3106],{"className":47924,"style":3191},[61,122],[15,47926,47928],{"className":47927},[338],[15,47929,47931,47951],{"className":47930},[342,472],[15,47932,47934,47948],{"className":47933},[346],[15,47935,47937],{"className":47936,"style":27033},[350],[15,47938,47939,47942],{"style":3206},[15,47940],{"className":47941,"style":358},[357],[15,47943,47945],{"className":47944},[362,363,364,365],[15,47946,79],{"className":47947,"style":123},[61,122,365],[15,47949,497],{"className":47950},[496],[15,47952,47954],{"className":47953},[346],[15,47955,47957],{"className":47956,"style":15707},[350],[15,47958],{},[15,47960,750],{"className":47961},[178],[11,47963,47964,47966,47967,48220,48221,48272,48273,48307],{},[661,47965,5123],{}," 由 ",[15,47968,47970,48026],{"className":47969,"translate":19},[18],[15,47971,47973],{"className":47972},[23],[25,47974,47975],{"xmlns":27},[29,47976,47977,48023],{},[32,47978,47979,47981,47983,47985,47987,47993,47995,47997,47999,48001,48007,48009,48011,48013,48015,48021],{},[35,47980,14818],{"mathvariant":258},[81,47982,14821],{"stretchy":86},[35,47984,22662],{},[81,47986,1547],{},[266,47988,47989,47991],{},[35,47990,1029],{},[35,47992,272],{},[81,47994,14830],{"stretchy":86},[81,47996,244],{},[35,47998,22693],{},[81,48000,741],{"stretchy":86},[266,48002,48003,48005],{},[35,48004,3106],{},[35,48006,272],{},[81,48008,750],{"stretchy":86},[81,48010,239],{},[35,48012,22693],{},[81,48014,741],{"stretchy":86},[266,48016,48017,48019],{},[35,48018,3106],{},[89,48020,278],{},[81,48022,750],{"stretchy":86},[40,48024,48025],{"encoding":42},"\\mathbb{E}[\\log x_k] = \\psi(\\alpha_k) - \\psi(\\alpha_0)",[15,48027,48029,48101,48165],{"className":48028,"ariaHidden":48},[47],[15,48030,48032,48035,48038,48041,48046,48049,48089,48092,48095,48098],{"className":48031},[52],[15,48033],{"className":48034,"style":142},[56],[15,48036,14818],{"className":48037},[61,427],[15,48039,14821],{"className":48040},[146],[15,48042,22791,48044],{"className":48043},[531],[15,48045,22794],{"style":15619},[15,48047],{"className":48048,"style":157},[127],[15,48050,48052,48055],{"className":48051},[61],[15,48053,1029],{"className":48054},[61,122],[15,48056,48058],{"className":48057},[338],[15,48059,48061,48081],{"className":48060},[342,472],[15,48062,48064,48078],{"className":48063},[346],[15,48065,48067],{"className":48066,"style":479},[350],[15,48068,48069,48072],{"style":482},[15,48070],{"className":48071,"style":358},[357],[15,48073,48075],{"className":48074},[362,363,364,365],[15,48076,272],{"className":48077,"style":492},[61,122,365],[15,48079,497],{"className":48080},[496],[15,48082,48084],{"className":48083},[346],[15,48085,48087],{"className":48086,"style":504},[350],[15,48088],{},[15,48090,14830],{"className":48091},[178],[15,48093],{"className":48094,"style":128},[127],[15,48096,244],{"className":48097},[132],[15,48099],{"className":48100,"style":128},[127],[15,48102,48104,48107,48110,48113,48153,48156,48159,48162],{"className":48103},[52],[15,48105],{"className":48106,"style":142},[56],[15,48108,22693],{"className":48109,"style":123},[61,122],[15,48111,741],{"className":48112},[146],[15,48114,48116,48119],{"className":48115},[61],[15,48117,3106],{"className":48118,"style":3191},[61,122],[15,48120,48122],{"className":48121},[338],[15,48123,48125,48145],{"className":48124},[342,472],[15,48126,48128,48142],{"className":48127},[346],[15,48129,48131],{"className":48130,"style":479},[350],[15,48132,48133,48136],{"style":3206},[15,48134],{"className":48135,"style":358},[357],[15,48137,48139],{"className":48138},[362,363,364,365],[15,48140,272],{"className":48141,"style":492},[61,122,365],[15,48143,497],{"className":48144},[496],[15,48146,48148],{"className":48147},[346],[15,48149,48151],{"className":48150,"style":504},[350],[15,48152],{},[15,48154,750],{"className":48155},[178],[15,48157],{"className":48158,"style":772},[127],[15,48160,239],{"className":48161},[375],[15,48163],{"className":48164,"style":772},[127],[15,48166,48168,48171,48174,48177,48217],{"className":48167},[52],[15,48169],{"className":48170,"style":142},[56],[15,48172,22693],{"className":48173,"style":123},[61,122],[15,48175,741],{"className":48176},[146],[15,48178,48180,48183],{"className":48179},[61],[15,48181,3106],{"className":48182,"style":3191},[61,122],[15,48184,48186],{"className":48185},[338],[15,48187,48189,48209],{"className":48188},[342,472],[15,48190,48192,48206],{"className":48191},[346],[15,48193,48195],{"className":48194,"style":1214},[350],[15,48196,48197,48200],{"style":3206},[15,48198],{"className":48199,"style":358},[357],[15,48201,48203],{"className":48202},[362,363,364,365],[15,48204,278],{"className":48205},[61,365],[15,48207,497],{"className":48208},[496],[15,48210,48212],{"className":48211},[346],[15,48213,48215],{"className":48214,"style":504},[350],[15,48216],{},[15,48218,750],{"className":48219},[178],"（命题 2.4 的证明中已给出），取 ",[15,48222,48224,48242],{"className":48223,"translate":19},[18],[15,48225,48227],{"className":48226},[23],[25,48228,48229],{"xmlns":27},[29,48230,48231,48239],{},[32,48232,48233,48235,48237],{},[35,48234,272],{},[81,48236,244],{},[35,48238,79],{},[40,48240,48241],{"encoding":42},"k = y",[15,48243,48245,48263],{"className":48244,"ariaHidden":48},[47],[15,48246,48248,48251,48254,48257,48260],{"className":48247},[52],[15,48249],{"className":48250,"style":10452},[56],[15,48252,272],{"className":48253,"style":492},[61,122],[15,48255],{"className":48256,"style":128},[127],[15,48258,244],{"className":48259},[132],[15,48261],{"className":48262,"style":128},[127],[15,48264,48266,48269],{"className":48265},[52],[15,48267],{"className":48268,"style":9437},[56],[15,48270,79],{"className":48271,"style":123},[61,122]," 并乘以 ",[15,48274,48276,48292],{"className":48275,"translate":19},[18],[15,48277,48279],{"className":48278},[23],[25,48280,48281],{"xmlns":27},[29,48282,48283,48289],{},[32,48284,48285,48287],{},[81,48286,239],{},[89,48288,91],{},[40,48290,48291],{"encoding":42},"-1",[15,48293,48295],{"className":48294,"ariaHidden":48},[47],[15,48296,48298,48301,48304],{"className":48297},[52],[15,48299],{"className":48300,"style":2567},[56],[15,48302,239],{"className":48303},[61],[15,48305,91],{"className":48306},[61]," 即得。",[11,48309,48310],{},"该损失的梯度结构值得注意：",[15,48312,48314],{"className":48313,"translate":19},[212],[15,48315,48317,48444],{"className":48316,"translate":19},[18],[15,48318,48320],{"className":48319},[23],[25,48321,48322],{"xmlns":27,"display":221},[29,48323,48324,48441],{},[32,48325,48326,48349,48351,48357,48359,48361,48363,48365,48371,48373,48379,48381,48383,48385,48387,48389,48411,48413,48419,48421,48423,48425,48427,48429,48431,48433,48435,48437,48439],{},[3722,48327,48328,48339],{},[32,48329,48330,48333],{},[35,48331,48332],{"mathvariant":231},"∂",[227,48334,48335,48337],{},[35,48336,47490],{"mathvariant":37},[282,48338,47493],{},[32,48340,48341,48343],{},[35,48342,48332],{"mathvariant":231},[266,48344,48345,48347],{},[35,48346,3106],{},[35,48348,79],{},[81,48350,244],{},[227,48352,48353,48355],{},[35,48354,22693],{},[81,48356,23162],{"mathvariant":231,"lspace":23161,"rspace":23161},[81,48358,741],{"stretchy":86},[35,48360,40265],{},[81,48362,750],{"stretchy":86},[81,48364,239],{},[227,48366,48367,48369],{},[35,48368,22693],{},[81,48370,23162],{"mathvariant":231,"lspace":23161,"rspace":23161},[81,48372,741],{"stretchy":86},[266,48374,48375,48377],{},[35,48376,3106],{},[35,48378,79],{},[81,48380,750],{"stretchy":86},[81,48382,21308],{},[89,48384,278],{},[81,48386,94],{"separator":48},[127,48388],{"width":9847},[3722,48390,48391,48401],{},[32,48392,48393,48395],{},[35,48394,48332],{"mathvariant":231},[227,48396,48397,48399],{},[35,48398,47490],{"mathvariant":37},[282,48400,47493],{},[32,48402,48403,48405],{},[35,48404,48332],{"mathvariant":231},[266,48406,48407,48409],{},[35,48408,3106],{},[35,48410,14777],{},[81,48412,244],{},[227,48414,48415,48417],{},[35,48416,22693],{},[81,48418,23162],{"mathvariant":231,"lspace":23161,"rspace":23161},[81,48420,741],{"stretchy":86},[35,48422,40265],{},[81,48424,750],{"stretchy":86},[81,48426,1590],{},[89,48428,278],{},[127,48430],{"width":6451},[81,48432,741],{"stretchy":86},[35,48434,14777],{},[81,48436,14999],{"mathvariant":231},[35,48438,79],{},[81,48440,750],{"stretchy":86},[40,48442,48443],{"encoding":42},"\\frac{\\partial \\mathcal{L}^{\\text{NLL}}}{\\partial \\alpha_y} = \\psi'(S) - \\psi'(\\alpha_y) \\le 0,\n\\qquad\n\\frac{\\partial \\mathcal{L}^{\\text{NLL}}}{\\partial \\alpha_j} = \\psi'(S) > 0 \\quad (j \\ne y)",[15,48445,48447,48602,48660,48754,48918,48974,49034],{"className":48446,"ariaHidden":48},[47],[15,48448,48450,48454,48593,48596,48599],{"className":48449},[52],[15,48451],{"className":48452,"style":48453},[56],"height:2.4904em;vertical-align:-0.9721em;",[15,48455,48457,48460,48590],{"className":48456},[61],[15,48458],{"className":48459},[146,3841],[15,48461,48463],{"className":48462},[3722],[15,48464,48466,48581],{"className":48465},[342,472],[15,48467,48469,48578],{"className":48468},[346],[15,48470,48473,48524,48532],{"className":48471,"style":48472},[350],"height:1.5183em;",[15,48474,48475,48478],{"style":11402},[15,48476],{"className":48477,"style":3861},[357],[15,48479,48481,48484],{"className":48480},[61],[15,48482,48332],{"className":48483,"style":9444},[61],[15,48485,48487,48490],{"className":48486},[61],[15,48488,3106],{"className":48489,"style":3191},[61,122],[15,48491,48493],{"className":48492},[338],[15,48494,48496,48516],{"className":48495},[342,472],[15,48497,48499,48513],{"className":48498},[346],[15,48500,48502],{"className":48501,"style":27033},[350],[15,48503,48504,48507],{"style":3206},[15,48505],{"className":48506,"style":358},[357],[15,48508,48510],{"className":48509},[362,363,364,365],[15,48511,79],{"className":48512,"style":123},[61,122,365],[15,48514,497],{"className":48515},[496],[15,48517,48519],{"className":48518},[346],[15,48520,48522],{"className":48521,"style":15707},[350],[15,48523],{},[15,48525,48526,48529],{"style":3889},[15,48527],{"className":48528,"style":3861},[357],[15,48530],{"className":48531,"style":3897},[3896],[15,48533,48534,48537],{"style":11468},[15,48535],{"className":48536,"style":3861},[357],[15,48538,48540,48543],{"className":48539},[61],[15,48541,48332],{"className":48542,"style":9444},[61],[15,48544,48546,48549],{"className":48545},[61],[15,48547,47490],{"className":48548},[61,62],[15,48550,48552],{"className":48551},[338],[15,48553,48555],{"className":48554},[342],[15,48556,48558],{"className":48557},[346],[15,48559,48561],{"className":48560,"style":695},[350],[15,48562,48563,48566],{"style":716},[15,48564],{"className":48565,"style":358},[357],[15,48567,48569],{"className":48568},[362,363,364,365],[15,48570,48572],{"className":48571},[61,365],[15,48573,48575],{"className":48574},[61,13548,365],[15,48576,47493],{"className":48577},[61,365],[15,48579,497],{"className":48580},[496],[15,48582,48584],{"className":48583},[346],[15,48585,48588],{"className":48586,"style":48587},[350],"height:0.9721em;",[15,48589],{},[15,48591],{"className":48592},[178,3841],[15,48594],{"className":48595,"style":128},[127],[15,48597,244],{"className":48598},[132],[15,48600],{"className":48601,"style":128},[127],[15,48603,48605,48609,48642,48645,48648,48651,48654,48657],{"className":48604},[52],[15,48606],{"className":48607,"style":48608},[56],"height:1.0519em;vertical-align:-0.25em;",[15,48610,48612,48615],{"className":48611},[61],[15,48613,22693],{"className":48614,"style":123},[61,122],[15,48616,48618],{"className":48617},[338],[15,48619,48621],{"className":48620},[342],[15,48622,48624],{"className":48623},[346],[15,48625,48628],{"className":48626,"style":48627},[350],"height:0.8019em;",[15,48629,48630,48633],{"style":353},[15,48631],{"className":48632,"style":358},[357],[15,48634,48636],{"className":48635},[362,363,364,365],[15,48637,48639],{"className":48638},[61,365],[15,48640,23162],{"className":48641},[61,365],[15,48643,741],{"className":48644},[146],[15,48646,40265],{"className":48647,"style":40337},[61,122],[15,48649,750],{"className":48650},[178],[15,48652],{"className":48653,"style":772},[127],[15,48655,239],{"className":48656},[375],[15,48658],{"className":48659,"style":772},[127],[15,48661,48663,48667,48699,48702,48742,48745,48748,48751],{"className":48662},[52],[15,48664],{"className":48665,"style":48666},[56],"height:1.088em;vertical-align:-0.2861em;",[15,48668,48670,48673],{"className":48669},[61],[15,48671,22693],{"className":48672,"style":123},[61,122],[15,48674,48676],{"className":48675},[338],[15,48677,48679],{"className":48678},[342],[15,48680,48682],{"className":48681},[346],[15,48683,48685],{"className":48684,"style":48627},[350],[15,48686,48687,48690],{"style":353},[15,48688],{"className":48689,"style":358},[357],[15,48691,48693],{"className":48692},[362,363,364,365],[15,48694,48696],{"className":48695},[61,365],[15,48697,23162],{"className":48698},[61,365],[15,48700,741],{"className":48701},[146],[15,48703,48705,48708],{"className":48704},[61],[15,48706,3106],{"className":48707,"style":3191},[61,122],[15,48709,48711],{"className":48710},[338],[15,48712,48714,48734],{"className":48713},[342,472],[15,48715,48717,48731],{"className":48716},[346],[15,48718,48720],{"className":48719,"style":27033},[350],[15,48721,48722,48725],{"style":3206},[15,48723],{"className":48724,"style":358},[357],[15,48726,48728],{"className":48727},[362,363,364,365],[15,48729,79],{"className":48730,"style":123},[61,122,365],[15,48732,497],{"className":48733},[496],[15,48735,48737],{"className":48736},[346],[15,48738,48740],{"className":48739,"style":15707},[350],[15,48741],{},[15,48743,750],{"className":48744},[178],[15,48746],{"className":48747,"style":128},[127],[15,48749,21308],{"className":48750},[132],[15,48752],{"className":48753,"style":128},[127],[15,48755,48757,48760,48763,48766,48769,48772,48909,48912,48915],{"className":48756},[52],[15,48758],{"className":48759,"style":48453},[56],[15,48761,278],{"className":48762},[61],[15,48764,94],{"className":48765},[153],[15,48767],{"className":48768,"style":10035},[127],[15,48770],{"className":48771,"style":157},[127],[15,48773,48775,48778,48906],{"className":48774},[61],[15,48776],{"className":48777},[146,3841],[15,48779,48781],{"className":48780},[3722],[15,48782,48784,48898],{"className":48783},[342,472],[15,48785,48787,48895],{"className":48786},[346],[15,48788,48790,48841,48849],{"className":48789,"style":48472},[350],[15,48791,48792,48795],{"style":11402},[15,48793],{"className":48794,"style":3861},[357],[15,48796,48798,48801],{"className":48797},[61],[15,48799,48332],{"className":48800,"style":9444},[61],[15,48802,48804,48807],{"className":48803},[61],[15,48805,3106],{"className":48806,"style":3191},[61,122],[15,48808,48810],{"className":48809},[338],[15,48811,48813,48833],{"className":48812},[342,472],[15,48814,48816,48830],{"className":48815},[346],[15,48817,48819],{"className":48818,"style":1364},[350],[15,48820,48821,48824],{"style":3206},[15,48822],{"className":48823,"style":358},[357],[15,48825,48827],{"className":48826},[362,363,364,365],[15,48828,14777],{"className":48829,"style":14803},[61,122,365],[15,48831,497],{"className":48832},[496],[15,48834,48836],{"className":48835},[346],[15,48837,48839],{"className":48838,"style":15707},[350],[15,48840],{},[15,48842,48843,48846],{"style":3889},[15,48844],{"className":48845,"style":3861},[357],[15,48847],{"className":48848,"style":3897},[3896],[15,48850,48851,48854],{"style":11468},[15,48852],{"className":48853,"style":3861},[357],[15,48855,48857,48860],{"className":48856},[61],[15,48858,48332],{"className":48859,"style":9444},[61],[15,48861,48863,48866],{"className":48862},[61],[15,48864,47490],{"className":48865},[61,62],[15,48867,48869],{"className":48868},[338],[15,48870,48872],{"className":48871},[342],[15,48873,48875],{"className":48874},[346],[15,48876,48878],{"className":48877,"style":695},[350],[15,48879,48880,48883],{"style":716},[15,48881],{"className":48882,"style":358},[357],[15,48884,48886],{"className":48885},[362,363,364,365],[15,48887,48889],{"className":48888},[61,365],[15,48890,48892],{"className":48891},[61,13548,365],[15,48893,47493],{"className":48894},[61,365],[15,48896,497],{"className":48897},[496],[15,48899,48901],{"className":48900},[346],[15,48902,48904],{"className":48903,"style":48587},[350],[15,48905],{},[15,48907],{"className":48908},[178,3841],[15,48910],{"className":48911,"style":128},[127],[15,48913,244],{"className":48914},[132],[15,48916],{"className":48917,"style":128},[127],[15,48919,48921,48924,48956,48959,48962,48965,48968,48971],{"className":48920},[52],[15,48922],{"className":48923,"style":48608},[56],[15,48925,48927,48930],{"className":48926},[61],[15,48928,22693],{"className":48929,"style":123},[61,122],[15,48931,48933],{"className":48932},[338],[15,48934,48936],{"className":48935},[342],[15,48937,48939],{"className":48938},[346],[15,48940,48942],{"className":48941,"style":48627},[350],[15,48943,48944,48947],{"style":353},[15,48945],{"className":48946,"style":358},[357],[15,48948,48950],{"className":48949},[362,363,364,365],[15,48951,48953],{"className":48952},[61,365],[15,48954,23162],{"className":48955},[61,365],[15,48957,741],{"className":48958},[146],[15,48960,40265],{"className":48961,"style":40337},[61,122],[15,48963,750],{"className":48964},[178],[15,48966],{"className":48967,"style":128},[127],[15,48969,1590],{"className":48970},[132],[15,48972],{"className":48973,"style":128},[127],[15,48975,48977,48980,48983,48986,48989,48992,48995,49031],{"className":48976},[52],[15,48978],{"className":48979,"style":142},[56],[15,48981,278],{"className":48982},[61],[15,48984],{"className":48985,"style":15983},[127],[15,48987,741],{"className":48988},[146],[15,48990,14777],{"className":48991,"style":14803},[61,122],[15,48993],{"className":48994,"style":128},[127],[15,48996,48998,49025,49028],{"className":48997},[132],[15,48999,49001],{"className":49000},[132],[15,49002,49004],{"className":49003},[61,16002],[15,49005,49007],{"className":49006},[16006],[15,49008,49010,49013,49022],{"className":49009},[16010],[15,49011],{"className":49012,"style":14310},[56],[15,49014,49016],{"className":49015},[16017],[15,49017,49019],{"className":49018},[61],[15,49020,16024],{"className":49021},[132],[15,49023],{"className":49024},[16028],[15,49026],{"className":49027},[127,16032],[15,49029,244],{"className":49030},[132],[15,49032],{"className":49033,"style":128},[127],[15,49035,49037,49040,49043],{"className":49036},[52],[15,49038],{"className":49039,"style":142},[56],[15,49041,79],{"className":49042,"style":123},[61,122],[15,49044,750],{"className":49045},[178],[11,49047,49048,49049,49112,49113,19845,49116,49187,49188,19845,49191,49261,49262,49331,49332,49360,49361,49389],{},"其中 ",[15,49050,49052,49070],{"className":49051,"translate":19},[18],[15,49053,49055],{"className":49054},[23],[25,49056,49057],{"xmlns":27},[29,49058,49059,49067],{},[32,49060,49061],{},[227,49062,49063,49065],{},[35,49064,22693],{},[81,49066,23162],{"mathvariant":231,"lspace":23161,"rspace":23161},[40,49068,49069],{"encoding":42},"\\psi'",[15,49071,49073],{"className":49072,"ariaHidden":48},[47],[15,49074,49076,49080],{"className":49075},[52],[15,49077],{"className":49078,"style":49079},[56],"height:0.9463em;vertical-align:-0.1944em;",[15,49081,49083,49086],{"className":49082},[61],[15,49084,22693],{"className":49085,"style":123},[61,122],[15,49087,49089],{"className":49088},[338],[15,49090,49092],{"className":49091},[342],[15,49093,49095],{"className":49094},[346],[15,49096,49098],{"className":49097,"style":23216},[350],[15,49099,49100,49103],{"style":716},[15,49101],{"className":49102,"style":358},[357],[15,49104,49106],{"className":49105},[362,363,364,365],[15,49107,49109],{"className":49108},[61,365],[15,49110,23162],{"className":49111},[61,365]," 为 trigamma 函数（恒正、单调递减）。因此梯度沿两个方向作用：",[661,49114,49115],{},"提升真实类别的证据",[15,49117,49119,49137],{"className":49118,"translate":19},[18],[15,49120,49122],{"className":49121},[23],[25,49123,49124],{"xmlns":27},[29,49125,49126,49134],{},[32,49127,49128],{},[266,49129,49130,49132],{},[35,49131,3106],{},[35,49133,79],{},[40,49135,49136],{"encoding":42},"\\alpha_y",[15,49138,49140],{"className":49139,"ariaHidden":48},[47],[15,49141,49143,49147],{"className":49142},[52],[15,49144],{"className":49145,"style":49146},[56],"height:0.7167em;vertical-align:-0.2861em;",[15,49148,49150,49153],{"className":49149},[61],[15,49151,3106],{"className":49152,"style":3191},[61,122],[15,49154,49156],{"className":49155},[338],[15,49157,49159,49179],{"className":49158},[342,472],[15,49160,49162,49176],{"className":49161},[346],[15,49163,49165],{"className":49164,"style":27033},[350],[15,49166,49167,49170],{"style":3206},[15,49168],{"className":49169,"style":358},[357],[15,49171,49173],{"className":49172},[362,363,364,365],[15,49174,79],{"className":49175,"style":123},[61,122,365],[15,49177,497],{"className":49178},[496],[15,49180,49182],{"className":49181},[346],[15,49183,49185],{"className":49184,"style":15707},[350],[15,49186],{}," 增大），",[661,49189,49190],{},"压低其余类别的证据",[15,49192,49194,49212],{"className":49193,"translate":19},[18],[15,49195,49197],{"className":49196},[23],[25,49198,49199],{"xmlns":27},[29,49200,49201,49209],{},[32,49202,49203],{},[266,49204,49205,49207],{},[35,49206,3106],{},[35,49208,14777],{},[40,49210,49211],{"encoding":42},"\\alpha_j",[15,49213,49215],{"className":49214,"ariaHidden":48},[47],[15,49216,49218,49221],{"className":49217},[52],[15,49219],{"className":49220,"style":49146},[56],[15,49222,49224,49227],{"className":49223},[61],[15,49225,3106],{"className":49226,"style":3191},[61,122],[15,49228,49230],{"className":49229},[338],[15,49231,49233,49253],{"className":49232},[342,472],[15,49234,49236,49250],{"className":49235},[346],[15,49237,49239],{"className":49238,"style":1364},[350],[15,49240,49241,49244],{"style":3206},[15,49242],{"className":49243,"style":358},[357],[15,49245,49247],{"className":49246},[362,363,364,365],[15,49248,14777],{"className":49249,"style":14803},[61,122,365],[15,49251,497],{"className":49252},[496],[15,49254,49256],{"className":49255},[346],[15,49257,49259],{"className":49258,"style":15707},[350],[15,49260],{}," 减小）。第二条作用有一个退化倾向：若标签之外的证据可被任意压低而不受惩罚，网络便有动机把 ",[15,49263,49265,49282],{"className":49264,"translate":19},[18],[15,49266,49268],{"className":49267},[23],[25,49269,49270],{"xmlns":27},[29,49271,49272,49280],{},[32,49273,49274],{},[266,49275,49276,49278],{},[35,49277,3106],{},[35,49279,14777],{},[40,49281,49211],{"encoding":42},[15,49283,49285],{"className":49284,"ariaHidden":48},[47],[15,49286,49288,49291],{"className":49287},[52],[15,49289],{"className":49290,"style":49146},[56],[15,49292,49294,49297],{"className":49293},[61],[15,49295,3106],{"className":49296,"style":3191},[61,122],[15,49298,49300],{"className":49299},[338],[15,49301,49303,49323],{"className":49302},[342,472],[15,49304,49306,49320],{"className":49305},[346],[15,49307,49309],{"className":49308,"style":1364},[350],[15,49310,49311,49314],{"style":3206},[15,49312],{"className":49313,"style":358},[357],[15,49315,49317],{"className":49316},[362,363,364,365],[15,49318,14777],{"className":49319,"style":14803},[61,122,365],[15,49321,497],{"className":49322},[496],[15,49324,49326],{"className":49325},[346],[15,49327,49329],{"className":49328,"style":15707},[350],[15,49330],{}," 全部驱向 ",[15,49333,49335,49348],{"className":49334,"translate":19},[18],[15,49336,49338],{"className":49337},[23],[25,49339,49340],{"xmlns":27},[29,49341,49342,49346],{},[32,49343,49344],{},[89,49345,91],{},[40,49347,91],{"encoding":42},[15,49349,49351],{"className":49350,"ariaHidden":48},[47],[15,49352,49354,49357],{"className":49353},[52],[15,49355],{"className":49356,"style":859},[56],[15,49358,91],{"className":49359},[61],"，从而收缩总证据 ",[15,49362,49364,49377],{"className":49363,"translate":19},[18],[15,49365,49367],{"className":49366},[23],[25,49368,49369],{"xmlns":27},[29,49370,49371,49375],{},[32,49372,49373],{},[35,49374,40265],{},[40,49376,40265],{"encoding":42},[15,49378,49380],{"className":49379,"ariaHidden":48},[47],[15,49381,49383,49386],{"className":49382},[52],[15,49384],{"className":49385,"style":57},[56],[15,49387,40265],{"className":49388,"style":40337},[61,122],"——这正是 4.3 节引入正则项的直接原因。",[882,49391,49392],{},[11,49393,49394,802,49397,49462,49463],{},[661,49395,49396],{},"推论 4.2（与插值交叉熵的关系）",[15,49398,49400,49418],{"className":49399,"translate":19},[18],[15,49401,49403],{"className":49402},[23],[25,49404,49405],{"xmlns":27},[29,49406,49407,49415],{},[32,49408,49409],{},[227,49410,49411,49413],{},[35,49412,47490],{"mathvariant":37},[282,49414,47493],{},[40,49416,49417],{"encoding":42},"\\mathcal{L}^{\\text{NLL}}",[15,49419,49421],{"className":49420,"ariaHidden":48},[47],[15,49422,49424,49427],{"className":49423},[52],[15,49425],{"className":49426,"style":695},[56],[15,49428,49430,49433],{"className":49429},[61],[15,49431,47490],{"className":49432},[61,62],[15,49434,49436],{"className":49435},[338],[15,49437,49439],{"className":49438},[342],[15,49440,49442],{"className":49441},[346],[15,49443,49445],{"className":49444,"style":695},[350],[15,49446,49447,49450],{"style":716},[15,49448],{"className":49449,"style":358},[357],[15,49451,49453],{"className":49452},[362,363,364,365],[15,49454,49456],{"className":49455},[61,365],[15,49457,49459],{"className":49458},[61,13548,365],[15,49460,47493],{"className":49461},[61,365]," 是 one-hot 标签下插值交叉熵的 Jensen 上界：\n",[15,49464,49466,49570],{"className":49465,"translate":19},[18],[15,49467,49469],{"className":49468},[23],[25,49470,49471],{"xmlns":27},[29,49472,49473,49567],{},[32,49474,49475,49481,49483,49485,49487,49489,49495,49497,49503,49527,49529,49531,49533,49549,49551,49553,49555,49557,49559,49561],{},[227,49476,49477,49479],{},[35,49478,47490],{"mathvariant":37},[282,49480,47493],{},[81,49482,244],{},[81,49484,239],{},[35,49486,22662],{},[81,49488,1547],{},[266,49490,49491,49493],{},[35,49492,11],{},[35,49494,79],{},[81,49496,7762],{},[3722,49498,49499,49501],{},[89,49500,91],{},[89,49502,908],{},[32,49504,49505,49507,49517,49519,49525],{},[81,49506,741],{"fence":48},[3722,49508,49509,49511],{},[89,49510,91],{},[266,49512,49513,49515],{},[35,49514,3106],{},[35,49516,79],{},[81,49518,239],{},[3722,49520,49521,49523],{},[89,49522,91],{},[35,49524,40265],{},[81,49526,750],{"fence":48},[81,49528,7762],{},[35,49530,12265],{},[282,49532,6799],{},[32,49534,49535,49537,49547],{},[81,49536,741],{"fence":48},[227,49538,49539,49541],{},[35,49540,40265],{},[32,49542,49543,49545],{},[81,49544,239],{},[89,49546,908],{},[81,49548,750],{"fence":48},[282,49550,284],{},[81,49552,275],{},[282,49554,284],{},[81,49556,239],{},[35,49558,22662],{},[81,49560,1547],{},[266,49562,49563,49565],{},[35,49564,11],{},[35,49566,79],{},[40,49568,49569],{"encoding":42},"\\mathcal{L}^{\\text{NLL}} = -\\log p_y + \\frac{1}{2}\\left(\\frac{1}{\\alpha_y} - \\frac{1}{S}\\right) + O\\!\\left(S^{-2}\\right) \\ \\ge \\ -\\log p_y",[15,49571,49573,49623,49692,49976,50056],{"className":49572,"ariaHidden":48},[47],[15,49574,49576,49579,49614,49617,49620],{"className":49575},[52],[15,49577],{"className":49578,"style":695},[56],[15,49580,49582,49585],{"className":49581},[61],[15,49583,47490],{"className":49584},[61,62],[15,49586,49588],{"className":49587},[338],[15,49589,49591],{"className":49590},[342],[15,49592,49594],{"className":49593},[346],[15,49595,49597],{"className":49596,"style":695},[350],[15,49598,49599,49602],{"style":716},[15,49600],{"className":49601,"style":358},[357],[15,49603,49605],{"className":49604},[362,363,364,365],[15,49606,49608],{"className":49607},[61,365],[15,49609,49611],{"className":49610},[61,13548,365],[15,49612,47493],{"className":49613},[61,365],[15,49615],{"className":49616,"style":128},[127],[15,49618,244],{"className":49619},[132],[15,49621],{"className":49622,"style":128},[127],[15,49624,49626,49629,49632,49635,49640,49643,49683,49686,49689],{"className":49625},[52],[15,49627],{"className":49628,"style":20946},[56],[15,49630,239],{"className":49631},[61],[15,49633],{"className":49634,"style":157},[127],[15,49636,22791,49638],{"className":49637},[531],[15,49639,22794],{"style":15619},[15,49641],{"className":49642,"style":157},[127],[15,49644,49646,49649],{"className":49645},[61],[15,49647,11],{"className":49648},[61,122],[15,49650,49652],{"className":49651},[338],[15,49653,49655,49675],{"className":49654},[342,472],[15,49656,49658,49672],{"className":49657},[346],[15,49659,49661],{"className":49660,"style":27033},[350],[15,49662,49663,49666],{"style":482},[15,49664],{"className":49665,"style":358},[357],[15,49667,49669],{"className":49668},[362,363,364,365],[15,49670,79],{"className":49671,"style":123},[61,122,365],[15,49673,497],{"className":49674},[496],[15,49676,49678],{"className":49677},[346],[15,49679,49681],{"className":49680,"style":15707},[350],[15,49682],{},[15,49684],{"className":49685,"style":772},[127],[15,49687,7762],{"className":49688},[375],[15,49690],{"className":49691,"style":772},[127],[15,49693,49695,49698,49766,49769,49967,49970,49973],{"className":49694},[52],[15,49696],{"className":49697,"style":394},[56],[15,49699,49701,49704,49763],{"className":49700},[61],[15,49702],{"className":49703},[146,3841],[15,49705,49707],{"className":49706},[3722],[15,49708,49710,49755],{"className":49709},[342,472],[15,49711,49713,49752],{"className":49712},[346],[15,49714,49716,49730,49738],{"className":49715,"style":3854},[350],[15,49717,49718,49721],{"style":3857},[15,49719],{"className":49720,"style":3861},[357],[15,49722,49724],{"className":49723},[362,363,364,365],[15,49725,49727],{"className":49726},[61,365],[15,49728,908],{"className":49729},[61,365],[15,49731,49732,49735],{"style":3889},[15,49733],{"className":49734,"style":3861},[357],[15,49736],{"className":49737,"style":3897},[3896],[15,49739,49740,49743],{"style":3900},[15,49741],{"className":49742,"style":3861},[357],[15,49744,49746],{"className":49745},[362,363,364,365],[15,49747,49749],{"className":49748},[61,365],[15,49750,91],{"className":49751},[61,365],[15,49753,497],{"className":49754},[496],[15,49756,49758],{"className":49757},[346],[15,49759,49761],{"className":49760,"style":10198},[350],[15,49762],{},[15,49764],{"className":49765},[178,3841],[15,49767],{"className":49768,"style":157},[127],[15,49770,49772,49778,49884,49887,49890,49893,49961],{"className":49771},[161],[15,49773,49775],{"className":49774,"style":402},[146,401],[15,49776,741],{"className":49777},[406,407],[15,49779,49781,49784,49881],{"className":49780},[61],[15,49782],{"className":49783},[146,3841],[15,49785,49787],{"className":49786},[3722],[15,49788,49790,49872],{"className":49789},[342,472],[15,49791,49793,49869],{"className":49792},[346],[15,49794,49796,49847,49855],{"className":49795,"style":3854},[350],[15,49797,49798,49801],{"style":3857},[15,49799],{"className":49800,"style":3861},[357],[15,49802,49804],{"className":49803},[362,363,364,365],[15,49805,49807],{"className":49806},[61,365],[15,49808,49810,49813],{"className":49809},[61,365],[15,49811,3106],{"className":49812,"style":3191},[61,122,365],[15,49814,49816],{"className":49815},[338],[15,49817,49819,49839],{"className":49818},[342,472],[15,49820,49822,49836],{"className":49821},[346],[15,49823,49825],{"className":49824,"style":29292},[350],[15,49826,49827,49830],{"style":4058},[15,49828],{"className":49829,"style":4062},[357],[15,49831,49833],{"className":49832},[362,4066,4067,365],[15,49834,79],{"className":49835,"style":123},[61,122,365],[15,49837,497],{"className":49838},[496],[15,49840,49842],{"className":49841},[346],[15,49843,49845],{"className":49844,"style":15963},[350],[15,49846],{},[15,49848,49849,49852],{"style":3889},[15,49850],{"className":49851,"style":3861},[357],[15,49853],{"className":49854,"style":3897},[3896],[15,49856,49857,49860],{"style":3900},[15,49858],{"className":49859,"style":3861},[357],[15,49861,49863],{"className":49862},[362,363,364,365],[15,49864,49866],{"className":49865},[61,365],[15,49867,91],{"className":49868},[61,365],[15,49870,497],{"className":49871},[496],[15,49873,49875],{"className":49874},[346],[15,49876,49879],{"className":49877,"style":49878},[350],"height:0.5423em;",[15,49880],{},[15,49882],{"className":49883},[178,3841],[15,49885],{"className":49886,"style":772},[127],[15,49888,239],{"className":49889},[375],[15,49891],{"className":49892,"style":772},[127],[15,49894,49896,49899,49958],{"className":49895},[61],[15,49897],{"className":49898},[146,3841],[15,49900,49902],{"className":49901},[3722],[15,49903,49905,49950],{"className":49904},[342,472],[15,49906,49908,49947],{"className":49907},[346],[15,49909,49911,49925,49933],{"className":49910,"style":3854},[350],[15,49912,49913,49916],{"style":3857},[15,49914],{"className":49915,"style":3861},[357],[15,49917,49919],{"className":49918},[362,363,364,365],[15,49920,49922],{"className":49921},[61,365],[15,49923,40265],{"className":49924,"style":40337},[61,122,365],[15,49926,49927,49930],{"style":3889},[15,49928],{"className":49929,"style":3861},[357],[15,49931],{"className":49932,"style":3897},[3896],[15,49934,49935,49938],{"style":3900},[15,49936],{"className":49937,"style":3861},[357],[15,49939,49941],{"className":49940},[362,363,364,365],[15,49942,49944],{"className":49943},[61,365],[15,49945,91],{"className":49946},[61,365],[15,49948,497],{"className":49949},[496],[15,49951,49953],{"className":49952},[346],[15,49954,49956],{"className":49955,"style":10198},[350],[15,49957],{},[15,49959],{"className":49960},[178,3841],[15,49962,49964],{"className":49963,"style":402},[178,401],[15,49965,750],{"className":49966},[406,407],[15,49968],{"className":49969,"style":772},[127],[15,49971,7762],{"className":49972},[375],[15,49974],{"className":49975,"style":772},[127],[15,49977,49979,49982,49985,49988,49991,50041,50044,50047,50050,50053],{"className":49978},[52],[15,49980],{"className":49981,"style":45276},[56],[15,49983,12265],{"className":49984,"style":12287},[61,122],[15,49986],{"className":49987,"style":7105},[127],[15,49989],{"className":49990,"style":157},[127],[15,49992,49994,50000,50035],{"className":49993},[161],[15,49995,49997],{"className":49996,"style":402},[146,401],[15,49998,741],{"className":49999},[406,4067],[15,50001,50003,50006],{"className":50002},[61],[15,50004,40265],{"className":50005,"style":40337},[61,122],[15,50007,50009],{"className":50008},[338],[15,50010,50012],{"className":50011},[342],[15,50013,50015],{"className":50014},[346],[15,50016,50018],{"className":50017,"style":7905},[350],[15,50019,50020,50023],{"style":716},[15,50021],{"className":50022,"style":358},[357],[15,50024,50026],{"className":50025},[362,363,364,365],[15,50027,50029,50032],{"className":50028},[61,365],[15,50030,239],{"className":50031},[61,365],[15,50033,908],{"className":50034},[61,365],[15,50036,50038],{"className":50037,"style":402},[178,401],[15,50039,750],{"className":50040},[406,4067],[15,50042,284],{"className":50043},[127],[15,50045],{"className":50046,"style":128},[127],[15,50048,275],{"className":50049},[132],[15,50051,284],{"className":50052},[127],[15,50054],{"className":50055,"style":128},[127],[15,50057,50059,50062,50065,50068,50073,50076],{"className":50058},[52],[15,50060],{"className":50061,"style":20946},[56],[15,50063,239],{"className":50064},[61],[15,50066],{"className":50067,"style":157},[127],[15,50069,22791,50071],{"className":50070},[531],[15,50072,22794],{"style":15619},[15,50074],{"className":50075,"style":157},[127],[15,50077,50079,50082],{"className":50078},[61],[15,50080,11],{"className":50081},[61,122],[15,50083,50085],{"className":50084},[338],[15,50086,50088,50108],{"className":50087},[342,472],[15,50089,50091,50105],{"className":50090},[346],[15,50092,50094],{"className":50093,"style":27033},[350],[15,50095,50096,50099],{"style":482},[15,50097],{"className":50098,"style":358},[357],[15,50100,50102],{"className":50101},[362,363,364,365],[15,50103,79],{"className":50104,"style":123},[61,122,365],[15,50106,497],{"className":50107},[496],[15,50109,50111],{"className":50110},[346],[15,50112,50114],{"className":50113,"style":15707},[350],[15,50115],{},[11,50117,50118,50120,50121,14430,50383,50532,50533,50786,50787,50880],{},[661,50119,5123],{}," 由 Jensen 不等式 ",[15,50122,50124,50182],{"className":50123,"translate":19},[18],[15,50125,50127],{"className":50126},[23],[25,50128,50129],{"xmlns":27},[29,50130,50131,50179],{},[32,50132,50133,50135,50137,50139,50141,50147,50149,50151,50153,50155,50157,50159,50165,50167,50169,50171,50173],{},[35,50134,14818],{"mathvariant":258},[81,50136,14821],{"stretchy":86},[35,50138,22662],{},[81,50140,1547],{},[266,50142,50143,50145],{},[35,50144,24680],{},[35,50146,79],{},[81,50148,14830],{"stretchy":86},[81,50150,21308],{},[35,50152,22662],{},[81,50154,1547],{},[35,50156,14818],{"mathvariant":258},[81,50158,14821],{"stretchy":86},[266,50160,50161,50163],{},[35,50162,24680],{},[35,50164,79],{},[81,50166,14830],{"stretchy":86},[81,50168,244],{},[35,50170,22662],{},[81,50172,1547],{},[266,50174,50175,50177],{},[35,50176,11],{},[35,50178,79],{},[40,50180,50181],{"encoding":42},"\\mathbb{E}[\\log \\pi_y] \\le \\log \\mathbb{E}[\\pi_y] = \\log p_y",[15,50183,50185,50257,50329],{"className":50184,"ariaHidden":48},[47],[15,50186,50188,50191,50194,50197,50202,50205,50245,50248,50251,50254],{"className":50187},[52],[15,50189],{"className":50190,"style":17987},[56],[15,50192,14818],{"className":50193},[61,427],[15,50195,14821],{"className":50196},[146],[15,50198,22791,50200],{"className":50199},[531],[15,50201,22794],{"style":15619},[15,50203],{"className":50204,"style":157},[127],[15,50206,50208,50211],{"className":50207},[61],[15,50209,24680],{"className":50210,"style":123},[61,122],[15,50212,50214],{"className":50213},[338],[15,50215,50217,50237],{"className":50216},[342,472],[15,50218,50220,50234],{"className":50219},[346],[15,50221,50223],{"className":50222,"style":27033},[350],[15,50224,50225,50228],{"style":27036},[15,50226],{"className":50227,"style":358},[357],[15,50229,50231],{"className":50230},[362,363,364,365],[15,50232,79],{"className":50233,"style":123},[61,122,365],[15,50235,497],{"className":50236},[496],[15,50238,50240],{"className":50239},[346],[15,50241,50243],{"className":50242,"style":15707},[350],[15,50244],{},[15,50246,14830],{"className":50247},[178],[15,50249],{"className":50250,"style":128},[127],[15,50252,21308],{"className":50253},[132],[15,50255],{"className":50256,"style":128},[127],[15,50258,50260,50263,50268,50271,50274,50277,50317,50320,50323,50326],{"className":50259},[52],[15,50261],{"className":50262,"style":17987},[56],[15,50264,22791,50266],{"className":50265},[531],[15,50267,22794],{"style":15619},[15,50269],{"className":50270,"style":157},[127],[15,50272,14818],{"className":50273},[61,427],[15,50275,14821],{"className":50276},[146],[15,50278,50280,50283],{"className":50279},[61],[15,50281,24680],{"className":50282,"style":123},[61,122],[15,50284,50286],{"className":50285},[338],[15,50287,50289,50309],{"className":50288},[342,472],[15,50290,50292,50306],{"className":50291},[346],[15,50293,50295],{"className":50294,"style":27033},[350],[15,50296,50297,50300],{"style":27036},[15,50298],{"className":50299,"style":358},[357],[15,50301,50303],{"className":50302},[362,363,364,365],[15,50304,79],{"className":50305,"style":123},[61,122,365],[15,50307,497],{"className":50308},[496],[15,50310,50312],{"className":50311},[346],[15,50313,50315],{"className":50314,"style":15707},[350],[15,50316],{},[15,50318,14830],{"className":50319},[178],[15,50321],{"className":50322,"style":128},[127],[15,50324,244],{"className":50325},[132],[15,50327],{"className":50328,"style":128},[127],[15,50330,50332,50335,50340,50343],{"className":50331},[52],[15,50333],{"className":50334,"style":20946},[56],[15,50336,22791,50338],{"className":50337},[531],[15,50339,22794],{"style":15619},[15,50341],{"className":50342,"style":157},[127],[15,50344,50346,50349],{"className":50345},[61],[15,50347,11],{"className":50348},[61,122],[15,50350,50352],{"className":50351},[338],[15,50353,50355,50375],{"className":50354},[342,472],[15,50356,50358,50372],{"className":50357},[346],[15,50359,50361],{"className":50360,"style":27033},[350],[15,50362,50363,50366],{"style":482},[15,50364],{"className":50365,"style":358},[357],[15,50367,50369],{"className":50368},[362,363,364,365],[15,50370,79],{"className":50371,"style":123},[61,122,365],[15,50373,497],{"className":50374},[496],[15,50376,50378],{"className":50377},[346],[15,50379,50381],{"className":50380,"style":15707},[350],[15,50382],{},[15,50384,50386,50418],{"className":50385,"translate":19},[18],[15,50387,50389],{"className":50388},[23],[25,50390,50391],{"xmlns":27},[29,50392,50393,50415],{},[32,50394,50395,50401,50403,50405,50407,50409],{},[227,50396,50397,50399],{},[35,50398,47490],{"mathvariant":37},[282,50400,47493],{},[81,50402,275],{},[81,50404,239],{},[35,50406,22662],{},[81,50408,1547],{},[266,50410,50411,50413],{},[35,50412,11],{},[35,50414,79],{},[40,50416,50417],{"encoding":42},"\\mathcal{L}^{\\text{NLL}} \\ge -\\log p_y",[15,50419,50421,50472],{"className":50420,"ariaHidden":48},[47],[15,50422,50424,50428,50463,50466,50469],{"className":50423},[52],[15,50425],{"className":50426,"style":50427},[56],"height:0.9773em;vertical-align:-0.136em;",[15,50429,50431,50434],{"className":50430},[61],[15,50432,47490],{"className":50433},[61,62],[15,50435,50437],{"className":50436},[338],[15,50438,50440],{"className":50439},[342],[15,50441,50443],{"className":50442},[346],[15,50444,50446],{"className":50445,"style":695},[350],[15,50447,50448,50451],{"style":716},[15,50449],{"className":50450,"style":358},[357],[15,50452,50454],{"className":50453},[362,363,364,365],[15,50455,50457],{"className":50456},[61,365],[15,50458,50460],{"className":50459},[61,13548,365],[15,50461,47493],{"className":50462},[61,365],[15,50464],{"className":50465,"style":128},[127],[15,50467,275],{"className":50468},[132],[15,50470],{"className":50471,"style":128},[127],[15,50473,50475,50478,50481,50484,50489,50492],{"className":50474},[52],[15,50476],{"className":50477,"style":20946},[56],[15,50479,239],{"className":50480},[61],[15,50482],{"className":50483,"style":157},[127],[15,50485,22791,50487],{"className":50486},[531],[15,50488,22794],{"style":15619},[15,50490],{"className":50491,"style":157},[127],[15,50493,50495,50498],{"className":50494},[61],[15,50496,11],{"className":50497},[61,122],[15,50499,50501],{"className":50500},[338],[15,50502,50504,50524],{"className":50503},[342,472],[15,50505,50507,50521],{"className":50506},[346],[15,50508,50510],{"className":50509,"style":27033},[350],[15,50511,50512,50515],{"style":482},[15,50513],{"className":50514,"style":358},[357],[15,50516,50518],{"className":50517},[362,363,364,365],[15,50519,79],{"className":50520,"style":123},[61,122,365],[15,50522,497],{"className":50523},[496],[15,50525,50527],{"className":50526},[346],[15,50528,50530],{"className":50529,"style":15707},[350],[15,50531],{},"。渐近展开由 ",[15,50534,50536,50594],{"className":50535,"translate":19},[18],[15,50537,50539],{"className":50538},[23],[25,50540,50541],{"xmlns":27},[29,50542,50543,50591],{},[32,50544,50545,50547,50549,50551,50553,50555,50557,50559,50561,50563,50573,50575,50577,50579,50589],{},[35,50546,22693],{},[81,50548,741],{"stretchy":86},[35,50550,33435],{},[81,50552,750],{"stretchy":86},[81,50554,244],{},[35,50556,22662],{},[81,50558,1547],{},[35,50560,33435],{},[81,50562,239],{},[3722,50564,50565,50567],{},[89,50566,91],{},[32,50568,50569,50571],{},[89,50570,908],{},[35,50572,33435],{},[81,50574,7762],{},[35,50576,12265],{},[81,50578,741],{"stretchy":86},[227,50580,50581,50583],{},[35,50582,33435],{},[32,50584,50585,50587],{},[81,50586,239],{},[89,50588,908],{},[81,50590,750],{"stretchy":86},[40,50592,50593],{"encoding":42},"\\psi(a) = \\log a - \\frac{1}{2a} + O(a^{-2})",[15,50595,50597,50624,50650,50736],{"className":50596,"ariaHidden":48},[47],[15,50598,50600,50603,50606,50609,50612,50615,50618,50621],{"className":50599},[52],[15,50601],{"className":50602,"style":142},[56],[15,50604,22693],{"className":50605,"style":123},[61,122],[15,50607,741],{"className":50608},[146],[15,50610,33435],{"className":50611},[61,122],[15,50613,750],{"className":50614},[178],[15,50616],{"className":50617,"style":128},[127],[15,50619,244],{"className":50620},[132],[15,50622],{"className":50623,"style":128},[127],[15,50625,50627,50630,50635,50638,50641,50644,50647],{"className":50626},[52],[15,50628],{"className":50629,"style":14310},[56],[15,50631,22791,50633],{"className":50632},[531],[15,50634,22794],{"style":15619},[15,50636],{"className":50637,"style":157},[127],[15,50639,33435],{"className":50640},[61,122],[15,50642],{"className":50643,"style":772},[127],[15,50645,239],{"className":50646},[375],[15,50648],{"className":50649,"style":772},[127],[15,50651,50653,50656,50727,50730,50733],{"className":50652},[52],[15,50654],{"className":50655,"style":42571},[56],[15,50657,50659,50662,50724],{"className":50658},[61],[15,50660],{"className":50661},[146,3841],[15,50663,50665],{"className":50664},[3722],[15,50666,50668,50716],{"className":50667},[342,472],[15,50669,50671,50713],{"className":50670},[346],[15,50672,50674,50691,50699],{"className":50673,"style":3854},[350],[15,50675,50676,50679],{"style":3857},[15,50677],{"className":50678,"style":3861},[357],[15,50680,50682],{"className":50681},[362,363,364,365],[15,50683,50685,50688],{"className":50684},[61,365],[15,50686,908],{"className":50687},[61,365],[15,50689,33435],{"className":50690},[61,122,365],[15,50692,50693,50696],{"style":3889},[15,50694],{"className":50695,"style":3861},[357],[15,50697],{"className":50698,"style":3897},[3896],[15,50700,50701,50704],{"style":3900},[15,50702],{"className":50703,"style":3861},[357],[15,50705,50707],{"className":50706},[362,363,364,365],[15,50708,50710],{"className":50709},[61,365],[15,50711,91],{"className":50712},[61,365],[15,50714,497],{"className":50715},[496],[15,50717,50719],{"className":50718},[346],[15,50720,50722],{"className":50721,"style":10198},[350],[15,50723],{},[15,50725],{"className":50726},[178,3841],[15,50728],{"className":50729,"style":772},[127],[15,50731,7762],{"className":50732},[375],[15,50734],{"className":50735,"style":772},[127],[15,50737,50739,50742,50745,50748,50783],{"className":50738},[52],[15,50740],{"className":50741,"style":21415},[56],[15,50743,12265],{"className":50744,"style":12287},[61,122],[15,50746,741],{"className":50747},[146],[15,50749,50751,50754],{"className":50750},[61],[15,50752,33435],{"className":50753},[61,122],[15,50755,50757],{"className":50756},[338],[15,50758,50760],{"className":50759},[342],[15,50761,50763],{"className":50762},[346],[15,50764,50766],{"className":50765,"style":7905},[350],[15,50767,50768,50771],{"style":716},[15,50769],{"className":50770,"style":358},[357],[15,50772,50774],{"className":50773},[362,363,364,365],[15,50775,50777,50780],{"className":50776},[61,365],[15,50778,239],{"className":50779},[61,365],[15,50781,908],{"className":50782},[61,365],[15,50784,750],{"className":50785},[178]," 给出，而 ",[15,50788,50790,50812],{"className":50789,"translate":19},[18],[15,50791,50793],{"className":50792},[23],[25,50794,50795],{"xmlns":27},[29,50796,50797,50809],{},[32,50798,50799,50805,50807],{},[266,50800,50801,50803],{},[35,50802,3106],{},[35,50804,79],{},[81,50806,21308],{},[35,50808,40265],{},[40,50810,50811],{"encoding":42},"\\alpha_y \\le S",[15,50813,50815,50871],{"className":50814,"ariaHidden":48},[47],[15,50816,50818,50822,50862,50865,50868],{"className":50817},[52],[15,50819],{"className":50820,"style":50821},[56],"height:0.9221em;vertical-align:-0.2861em;",[15,50823,50825,50828],{"className":50824},[61],[15,50826,3106],{"className":50827,"style":3191},[61,122],[15,50829,50831],{"className":50830},[338],[15,50832,50834,50854],{"className":50833},[342,472],[15,50835,50837,50851],{"className":50836},[346],[15,50838,50840],{"className":50839,"style":27033},[350],[15,50841,50842,50845],{"style":3206},[15,50843],{"className":50844,"style":358},[357],[15,50846,50848],{"className":50847},[362,363,364,365],[15,50849,79],{"className":50850,"style":123},[61,122,365],[15,50852,497],{"className":50853},[496],[15,50855,50857],{"className":50856},[346],[15,50858,50860],{"className":50859,"style":15707},[350],[15,50861],{},[15,50863],{"className":50864,"style":128},[127],[15,50866,21308],{"className":50867},[132],[15,50869],{"className":50870,"style":128},[127],[15,50872,50874,50877],{"className":50873},[52],[15,50875],{"className":50876,"style":57},[56],[15,50878,40265],{"className":50879,"style":40337},[61,122]," 保证修正项非负。",[11,50882,50883,50884,50887,50888,50916],{},"推论 4.2 揭示了一个在文献中常被忽略的事实：",[661,50885,50886],{},"EDL 的对数损失并非多项—迪利克雷边缘似然的负对数，而是它加上一个显式的、随证据稀缺而增大的修正项。"," 该项在 ",[15,50889,50891,50904],{"className":50890,"translate":19},[18],[15,50892,50894],{"className":50893},[23],[25,50895,50896],{"xmlns":27},[29,50897,50898,50902],{},[32,50899,50900],{},[35,50901,40265],{},[40,50903,40265],{"encoding":42},[15,50905,50907],{"className":50906,"ariaHidden":48},[47],[15,50908,50910,50913],{"className":50909},[52],[15,50911],{"className":50912,"style":57},[56],[15,50914,40265],{"className":50915,"style":40337},[61,122]," 大时消失，在证据不足时提供额外压力，其作用与下一小节方差惩罚项在平方损失中的地位完全平行。",[877,50918,50920],{"id":50919},"_42-平方损失的贝叶斯风险","4.2 平方损失的贝叶斯风险",[11,50922,50923],{},"Sensoy 等人的原始工作采用了平方损失而非对数损失，其闭式更为直观。",[882,50925,50926],{},[11,50927,50928,9393,50931,50960,50961],{},[661,50929,50930],{},"命题 4.3（平方风险的闭式）",[15,50932,50934,50948],{"className":50933,"translate":19},[18],[15,50935,50937],{"className":50936},[23],[25,50938,50939],{"xmlns":27},[29,50940,50941,50945],{},[32,50942,50943],{},[35,50944,79],{"mathvariant":251},[40,50946,50947],{"encoding":42},"\\mathbf{y}",[15,50949,50951],{"className":50950,"ariaHidden":48},[47],[15,50952,50954,50957],{"className":50953},[52],[15,50955],{"className":50956,"style":829},[56],[15,50958,79],{"className":50959,"style":29413},[61,411]," 为 one-hot 标签，则\n",[15,50962,50964,51089],{"className":50963,"translate":19},[18],[15,50965,50967],{"className":50966},[23],[25,50968,50969],{"xmlns":27},[29,50970,50971,51086],{},[32,50972,50973,50980,50982,50988,51008,51010,51024],{},[227,50974,50975,50977],{},[35,50976,47490],{"mathvariant":37},[282,50978,50979],{},"MSE",[81,50981,244],{},[266,50983,50984,50986],{},[35,50985,14818],{"mathvariant":258},[35,50987,24680],{"mathvariant":3105},[32,50989,50990,50992,50994,50996,50998,51000,51006],{},[81,50991,14821],{"fence":48},[35,50993,21291],{"mathvariant":231},[35,50995,79],{"mathvariant":251},[81,50997,239],{},[35,50999,24680],{"mathvariant":3105},[227,51001,51002,51004],{},[35,51003,21291],{"mathvariant":231},[89,51005,908],{},[81,51007,14830],{"fence":48},[81,51009,244],{},[289,51011,51012,51014,51022],{},[81,51013,293],{},[32,51015,51016,51018,51020],{},[35,51017,272],{},[81,51019,244],{},[89,51021,91],{},[35,51023,102],{},[32,51025,51026,51028,51030,51036,51038,51044,51050,51052,51084],{},[81,51027,14821],{"fence":48},[81,51029,741],{"stretchy":86},[266,51031,51032,51034],{},[35,51033,79],{},[35,51035,272],{},[81,51037,239],{},[266,51039,51040,51042],{},[35,51041,11],{},[35,51043,272],{},[227,51045,51046,51048],{},[81,51047,750],{"stretchy":86},[89,51049,908],{},[81,51051,7762],{},[3722,51053,51054,51076],{},[32,51055,51056,51062,51064,51066,51068,51074],{},[266,51057,51058,51060],{},[35,51059,11],{},[35,51061,272],{},[81,51063,741],{"stretchy":86},[89,51065,91],{},[81,51067,239],{},[266,51069,51070,51072],{},[35,51071,11],{},[35,51073,272],{},[81,51075,750],{"stretchy":86},[32,51077,51078,51080,51082],{},[35,51079,40265],{},[81,51081,7762],{},[89,51083,91],{},[81,51085,14830],{"fence":48},[40,51087,51088],{"encoding":42},"\\mathcal{L}^{\\text{MSE}} = \\mathbb{E}_{\\boldsymbol{\\pi}}\\left[\\|\\mathbf{y} - \\boldsymbol{\\pi}\\|^2\\right] = \\sum_{k=1}^K \\left[(y_k - p_k)^2 + \\frac{p_k(1 - p_k)}{S + 1}\\right]",[15,51090,51092,51142,51277],{"className":51091,"ariaHidden":48},[47],[15,51093,51095,51098,51133,51136,51139],{"className":51094},[52],[15,51096],{"className":51097,"style":695},[56],[15,51099,51101,51104],{"className":51100},[61],[15,51102,47490],{"className":51103},[61,62],[15,51105,51107],{"className":51106},[338],[15,51108,51110],{"className":51109},[342],[15,51111,51113],{"className":51112},[346],[15,51114,51116],{"className":51115,"style":695},[350],[15,51117,51118,51121],{"style":716},[15,51119],{"className":51120,"style":358},[357],[15,51122,51124],{"className":51123},[362,363,364,365],[15,51125,51127],{"className":51126},[61,365],[15,51128,51130],{"className":51129},[61,13548,365],[15,51131,50979],{"className":51132},[61,365],[15,51134],{"className":51135,"style":128},[127],[15,51137,244],{"className":51138},[132],[15,51140],{"className":51141,"style":128},[127],[15,51143,51145,51148,51197,51200,51268,51271,51274],{"className":51144},[52],[15,51146],{"className":51147,"style":45276},[56],[15,51149,51151,51154],{"className":51150},[61],[15,51152,14818],{"className":51153},[61,427],[15,51155,51157],{"className":51156},[338],[15,51158,51160,51189],{"className":51159},[342,472],[15,51161,51163,51186],{"className":51162},[346],[15,51164,51166],{"className":51165,"style":20966},[350],[15,51167,51168,51171],{"style":482},[15,51169],{"className":51170,"style":358},[357],[15,51172,51174],{"className":51173},[362,363,364,365],[15,51175,51177],{"className":51176},[61,365],[15,51178,51180],{"className":51179},[61,365],[15,51181,51183],{"className":51182},[61,365],[15,51184,24680],{"className":51185,"style":24714},[61,3166,365],[15,51187,497],{"className":51188},[496],[15,51190,51192],{"className":51191},[346],[15,51193,51195],{"className":51194,"style":504},[350],[15,51196],{},[15,51198],{"className":51199,"style":157},[127],[15,51201,51203,51209,51212,51215,51218,51221,51224,51233,51262],{"className":51202},[161],[15,51204,51206],{"className":51205,"style":402},[146,401],[15,51207,14821],{"className":51208},[406,4067],[15,51210,21291],{"className":51211},[61],[15,51213,79],{"className":51214,"style":29413},[61,411],[15,51216],{"className":51217,"style":772},[127],[15,51219,239],{"className":51220},[375],[15,51222],{"className":51223,"style":772},[127],[15,51225,51227],{"className":51226},[61],[15,51228,51230],{"className":51229},[61],[15,51231,24680],{"className":51232,"style":24714},[61,3166],[15,51234,51236,51239],{"className":51235},[61],[15,51237,21291],{"className":51238},[61],[15,51240,51242],{"className":51241},[338],[15,51243,51245],{"className":51244},[342],[15,51246,51248],{"className":51247},[346],[15,51249,51251],{"className":51250,"style":7905},[350],[15,51252,51253,51256],{"style":716},[15,51254],{"className":51255,"style":358},[357],[15,51257,51259],{"className":51258},[362,363,364,365],[15,51260,908],{"className":51261},[61,365],[15,51263,51265],{"className":51264,"style":402},[178,401],[15,51266,14830],{"className":51267},[406,4067],[15,51269],{"className":51270,"style":128},[127],[15,51272,244],{"className":51273},[132],[15,51275],{"className":51276,"style":128},[127],[15,51278,51280,51283,51343,51346],{"className":51279},[52],[15,51281],{"className":51282,"style":394},[56],[15,51284,51286,51289],{"className":51285},[531],[15,51287,293],{"className":51288,"style":537},[531,535,536],[15,51290,51292],{"className":51291},[338],[15,51293,51295,51335],{"className":51294},[342,472],[15,51296,51298,51332],{"className":51297},[346],[15,51299,51301,51321],{"className":51300,"style":550},[350],[15,51302,51303,51306],{"style":553},[15,51304],{"className":51305,"style":358},[357],[15,51307,51309],{"className":51308},[362,363,364,365],[15,51310,51312,51315,51318],{"className":51311},[61,365],[15,51313,272],{"className":51314,"style":492},[61,122,365],[15,51316,244],{"className":51317},[132,365],[15,51319,91],{"className":51320},[61,365],[15,51322,51323,51326],{"style":574},[15,51324],{"className":51325,"style":358},[357],[15,51327,51329],{"className":51328},[362,363,364,365],[15,51330,102],{"className":51331,"style":174},[61,122,365],[15,51333,497],{"className":51334},[496],[15,51336,51338],{"className":51337},[346],[15,51339,51341],{"className":51340,"style":593},[350],[15,51342],{},[15,51344],{"className":51345,"style":157},[127],[15,51347,51349,51355,51358,51398,51401,51404,51407,51447,51476,51479,51482,51485,51648],{"className":51348},[161],[15,51350,51352],{"className":51351,"style":402},[146,401],[15,51353,14821],{"className":51354},[406,407],[15,51356,741],{"className":51357},[146],[15,51359,51361,51364],{"className":51360},[61],[15,51362,79],{"className":51363,"style":123},[61,122],[15,51365,51367],{"className":51366},[338],[15,51368,51370,51390],{"className":51369},[342,472],[15,51371,51373,51387],{"className":51372},[346],[15,51374,51376],{"className":51375,"style":479},[350],[15,51377,51378,51381],{"style":27036},[15,51379],{"className":51380,"style":358},[357],[15,51382,51384],{"className":51383},[362,363,364,365],[15,51385,272],{"className":51386,"style":492},[61,122,365],[15,51388,497],{"className":51389},[496],[15,51391,51393],{"className":51392},[346],[15,51394,51396],{"className":51395,"style":504},[350],[15,51397],{},[15,51399],{"className":51400,"style":772},[127],[15,51402,239],{"className":51403},[375],[15,51405],{"className":51406,"style":772},[127],[15,51408,51410,51413],{"className":51409},[61],[15,51411,11],{"className":51412},[61,122],[15,51414,51416],{"className":51415},[338],[15,51417,51419,51439],{"className":51418},[342,472],[15,51420,51422,51436],{"className":51421},[346],[15,51423,51425],{"className":51424,"style":479},[350],[15,51426,51427,51430],{"style":482},[15,51428],{"className":51429,"style":358},[357],[15,51431,51433],{"className":51432},[362,363,364,365],[15,51434,272],{"className":51435,"style":492},[61,122,365],[15,51437,497],{"className":51438},[496],[15,51440,51442],{"className":51441},[346],[15,51443,51445],{"className":51444,"style":504},[350],[15,51446],{},[15,51448,51450,51453],{"className":51449},[178],[15,51451,750],{"className":51452},[178],[15,51454,51456],{"className":51455},[338],[15,51457,51459],{"className":51458},[342],[15,51460,51462],{"className":51461},[346],[15,51463,51465],{"className":51464,"style":7905},[350],[15,51466,51467,51470],{"style":716},[15,51468],{"className":51469,"style":358},[357],[15,51471,51473],{"className":51472},[362,363,364,365],[15,51474,908],{"className":51475},[61,365],[15,51477],{"className":51478,"style":772},[127],[15,51480,7762],{"className":51481},[375],[15,51483],{"className":51484,"style":772},[127],[15,51486,51488,51491,51645],{"className":51487},[61],[15,51489],{"className":51490},[146,3841],[15,51492,51494],{"className":51493},[3722],[15,51495,51497,51637],{"className":51496},[342,472],[15,51498,51500,51634],{"className":51499},[346],[15,51501,51503,51523,51531],{"className":51502,"style":15321},[350],[15,51504,51505,51508],{"style":3857},[15,51506],{"className":51507,"style":3861},[357],[15,51509,51511],{"className":51510},[362,363,364,365],[15,51512,51514,51517,51520],{"className":51513},[61,365],[15,51515,40265],{"className":51516,"style":40337},[61,122,365],[15,51518,7762],{"className":51519},[375,365],[15,51521,91],{"className":51522},[61,365],[15,51524,51525,51528],{"style":3889},[15,51526],{"className":51527,"style":3861},[357],[15,51529],{"className":51530,"style":3897},[3896],[15,51532,51533,51536],{"style":15450},[15,51534],{"className":51535,"style":3861},[357],[15,51537,51539],{"className":51538},[362,363,364,365],[15,51540,51542,51582,51585,51588,51591,51631],{"className":51541},[61,365],[15,51543,51545,51548],{"className":51544},[61,365],[15,51546,11],{"className":51547},[61,122,365],[15,51549,51551],{"className":51550},[338],[15,51552,51554,51574],{"className":51553},[342,472],[15,51555,51557,51571],{"className":51556},[346],[15,51558,51560],{"className":51559,"style":22404},[350],[15,51561,51562,51565],{"style":27279},[15,51563],{"className":51564,"style":4062},[357],[15,51566,51568],{"className":51567},[362,4066,4067,365],[15,51569,272],{"className":51570,"style":492},[61,122,365],[15,51572,497],{"className":51573},[496],[15,51575,51577],{"className":51576},[346],[15,51578,51580],{"className":51579,"style":27253},[350],[15,51581],{},[15,51583,741],{"className":51584},[146,365],[15,51586,91],{"className":51587},[61,365],[15,51589,239],{"className":51590},[375,365],[15,51592,51594,51597],{"className":51593},[61,365],[15,51595,11],{"className":51596},[61,122,365],[15,51598,51600],{"className":51599},[338],[15,51601,51603,51623],{"className":51602},[342,472],[15,51604,51606,51620],{"className":51605},[346],[15,51607,51609],{"className":51608,"style":22404},[350],[15,51610,51611,51614],{"style":27279},[15,51612],{"className":51613,"style":4062},[357],[15,51615,51617],{"className":51616},[362,4066,4067,365],[15,51618,272],{"className":51619,"style":492},[61,122,365],[15,51621,497],{"className":51622},[496],[15,51624,51626],{"className":51625},[346],[15,51627,51629],{"className":51628,"style":27253},[350],[15,51630],{},[15,51632,750],{"className":51633},[178,365],[15,51635,497],{"className":51636},[496],[15,51638,51640],{"className":51639},[346],[15,51641,51643],{"className":51642,"style":46299},[350],[15,51644],{},[15,51646],{"className":51647},[178,3841],[15,51649,51651],{"className":51650,"style":402},[178,401],[15,51652,14830],{"className":51653},[406,407],[11,51655,51656,51658,51659,52094,52095,52655],{},[661,51657,5123],{}," 逐项展开：",[15,51660,51662,51742],{"className":51661,"translate":19},[18],[15,51663,51665],{"className":51664},[23],[25,51666,51667],{"xmlns":27},[29,51668,51669,51739],{},[32,51670,51671,51673,51675,51677,51683,51685,51691,51697,51699,51701,51703,51709,51711,51717,51723,51725,51727,51729,51731,51737],{},[35,51672,14818],{"mathvariant":258},[81,51674,14821],{"stretchy":86},[81,51676,741],{"stretchy":86},[266,51678,51679,51681],{},[35,51680,79],{},[35,51682,272],{},[81,51684,239],{},[266,51686,51687,51689],{},[35,51688,24680],{},[35,51690,272],{},[227,51692,51693,51695],{},[81,51694,750],{"stretchy":86},[89,51696,908],{},[81,51698,14830],{"stretchy":86},[81,51700,244],{},[81,51702,741],{"stretchy":86},[266,51704,51705,51707],{},[35,51706,79],{},[35,51708,272],{},[81,51710,239],{},[266,51712,51713,51715],{},[35,51714,11],{},[35,51716,272],{},[227,51718,51719,51721],{},[81,51720,750],{"stretchy":86},[89,51722,908],{},[81,51724,7762],{},[35,51726,14853],{"mathvariant":231},[81,51728,1547],{},[81,51730,741],{"stretchy":86},[266,51732,51733,51735],{},[35,51734,24680],{},[35,51736,272],{},[81,51738,750],{"stretchy":86},[40,51740,51741],{"encoding":42},"\\mathbb{E}[(y_k - \\pi_k)^2] = (y_k - p_k)^2 + \\operatorname{Var}(\\pi_k)",[15,51743,51745,51807,51894,51952,52036],{"className":51744,"ariaHidden":48},[47],[15,51746,51748,51751,51754,51758,51798,51801,51804],{"className":51747},[52],[15,51749],{"className":51750,"style":142},[56],[15,51752,14818],{"className":51753},[61,427],[15,51755,51757],{"className":51756},[146],"[(",[15,51759,51761,51764],{"className":51760},[61],[15,51762,79],{"className":51763,"style":123},[61,122],[15,51765,51767],{"className":51766},[338],[15,51768,51770,51790],{"className":51769},[342,472],[15,51771,51773,51787],{"className":51772},[346],[15,51774,51776],{"className":51775,"style":479},[350],[15,51777,51778,51781],{"style":27036},[15,51779],{"className":51780,"style":358},[357],[15,51782,51784],{"className":51783},[362,363,364,365],[15,51785,272],{"className":51786,"style":492},[61,122,365],[15,51788,497],{"className":51789},[496],[15,51791,51793],{"className":51792},[346],[15,51794,51796],{"className":51795,"style":504},[350],[15,51797],{},[15,51799],{"className":51800,"style":772},[127],[15,51802,239],{"className":51803},[375],[15,51805],{"className":51806,"style":772},[127],[15,51808,51810,51813,51853,51882,51885,51888,51891],{"className":51809},[52],[15,51811],{"className":51812,"style":21415},[56],[15,51814,51816,51819],{"className":51815},[61],[15,51817,24680],{"className":51818,"style":123},[61,122],[15,51820,51822],{"className":51821},[338],[15,51823,51825,51845],{"className":51824},[342,472],[15,51826,51828,51842],{"className":51827},[346],[15,51829,51831],{"className":51830,"style":479},[350],[15,51832,51833,51836],{"style":27036},[15,51834],{"className":51835,"style":358},[357],[15,51837,51839],{"className":51838},[362,363,364,365],[15,51840,272],{"className":51841,"style":492},[61,122,365],[15,51843,497],{"className":51844},[496],[15,51846,51848],{"className":51847},[346],[15,51849,51851],{"className":51850,"style":504},[350],[15,51852],{},[15,51854,51856,51859],{"className":51855},[178],[15,51857,750],{"className":51858},[178],[15,51860,51862],{"className":51861},[338],[15,51863,51865],{"className":51864},[342],[15,51866,51868],{"className":51867},[346],[15,51869,51871],{"className":51870,"style":7905},[350],[15,51872,51873,51876],{"style":716},[15,51874],{"className":51875,"style":358},[357],[15,51877,51879],{"className":51878},[362,363,364,365],[15,51880,908],{"className":51881},[61,365],[15,51883,14830],{"className":51884},[178],[15,51886],{"className":51887,"style":128},[127],[15,51889,244],{"className":51890},[132],[15,51892],{"className":51893,"style":128},[127],[15,51895,51897,51900,51903,51943,51946,51949],{"className":51896},[52],[15,51898],{"className":51899,"style":142},[56],[15,51901,741],{"className":51902},[146],[15,51904,51906,51909],{"className":51905},[61],[15,51907,79],{"className":51908,"style":123},[61,122],[15,51910,51912],{"className":51911},[338],[15,51913,51915,51935],{"className":51914},[342,472],[15,51916,51918,51932],{"className":51917},[346],[15,51919,51921],{"className":51920,"style":479},[350],[15,51922,51923,51926],{"style":27036},[15,51924],{"className":51925,"style":358},[357],[15,51927,51929],{"className":51928},[362,363,364,365],[15,51930,272],{"className":51931,"style":492},[61,122,365],[15,51933,497],{"className":51934},[496],[15,51936,51938],{"className":51937},[346],[15,51939,51941],{"className":51940,"style":504},[350],[15,51942],{},[15,51944],{"className":51945,"style":772},[127],[15,51947,239],{"className":51948},[375],[15,51950],{"className":51951,"style":772},[127],[15,51953,51955,51958,51998,52027,52030,52033],{"className":51954},[52],[15,51956],{"className":51957,"style":21415},[56],[15,51959,51961,51964],{"className":51960},[61],[15,51962,11],{"className":51963},[61,122],[15,51965,51967],{"className":51966},[338],[15,51968,51970,51990],{"className":51969},[342,472],[15,51971,51973,51987],{"className":51972},[346],[15,51974,51976],{"className":51975,"style":479},[350],[15,51977,51978,51981],{"style":482},[15,51979],{"className":51980,"style":358},[357],[15,51982,51984],{"className":51983},[362,363,364,365],[15,51985,272],{"className":51986,"style":492},[61,122,365],[15,51988,497],{"className":51989},[496],[15,51991,51993],{"className":51992},[346],[15,51994,51996],{"className":51995,"style":504},[350],[15,51997],{},[15,51999,52001,52004],{"className":52000},[178],[15,52002,750],{"className":52003},[178],[15,52005,52007],{"className":52006},[338],[15,52008,52010],{"className":52009},[342],[15,52011,52013],{"className":52012},[346],[15,52014,52016],{"className":52015,"style":7905},[350],[15,52017,52018,52021],{"style":716},[15,52019],{"className":52020,"style":358},[357],[15,52022,52024],{"className":52023},[362,363,364,365],[15,52025,908],{"className":52026},[61,365],[15,52028],{"className":52029,"style":772},[127],[15,52031,7762],{"className":52032},[375],[15,52034],{"className":52035,"style":772},[127],[15,52037,52039,52042,52048,52051,52091],{"className":52038},[52],[15,52040],{"className":52041,"style":142},[56],[15,52043,52045],{"className":52044},[531],[15,52046,14853],{"className":52047},[61,1622],[15,52049,741],{"className":52050},[146],[15,52052,52054,52057],{"className":52053},[61],[15,52055,24680],{"className":52056,"style":123},[61,122],[15,52058,52060],{"className":52059},[338],[15,52061,52063,52083],{"className":52062},[342,472],[15,52064,52066,52080],{"className":52065},[346],[15,52067,52069],{"className":52068,"style":479},[350],[15,52070,52071,52074],{"style":27036},[15,52072],{"className":52073,"style":358},[357],[15,52075,52077],{"className":52076},[362,363,364,365],[15,52078,272],{"className":52079,"style":492},[61,122,365],[15,52081,497],{"className":52082},[496],[15,52084,52086],{"className":52085},[346],[15,52087,52089],{"className":52088,"style":504},[350],[15,52090],{},[15,52092,750],{"className":52093},[178],"。由命题 2.1，",[15,52096,52098,52202],{"className":52097,"translate":19},[18],[15,52099,52101],{"className":52100},[23],[25,52102,52103],{"xmlns":27},[29,52104,52105,52199],{},[32,52106,52107,52109,52111,52113,52119,52121,52123,52165,52167],{},[35,52108,14853],{"mathvariant":231},[81,52110,1547],{},[81,52112,741],{"stretchy":86},[266,52114,52115,52117],{},[35,52116,24680],{},[35,52118,272],{},[81,52120,750],{"stretchy":86},[81,52122,244],{},[3722,52124,52125,52147],{},[32,52126,52127,52133,52135,52137,52139,52145],{},[266,52128,52129,52131],{},[35,52130,3106],{},[35,52132,272],{},[81,52134,741],{"stretchy":86},[35,52136,40265],{},[81,52138,239],{},[266,52140,52141,52143],{},[35,52142,3106],{},[35,52144,272],{},[81,52146,750],{"stretchy":86},[32,52148,52149,52155,52157,52159,52161,52163],{},[227,52150,52151,52153],{},[35,52152,40265],{},[89,52154,908],{},[81,52156,741],{"stretchy":86},[35,52158,40265],{},[81,52160,7762],{},[89,52162,91],{},[81,52164,750],{"stretchy":86},[81,52166,244],{},[3722,52168,52169,52191],{},[32,52170,52171,52177,52179,52181,52183,52189],{},[266,52172,52173,52175],{},[35,52174,11],{},[35,52176,272],{},[81,52178,741],{"stretchy":86},[89,52180,91],{},[81,52182,239],{},[266,52184,52185,52187],{},[35,52186,11],{},[35,52188,272],{},[81,52190,750],{"stretchy":86},[32,52192,52193,52195,52197],{},[35,52194,40265],{},[81,52196,7762],{},[89,52198,91],{},[40,52200,52201],{"encoding":42},"\\operatorname{Var}(\\pi_k) = \\frac{\\alpha_k(S - \\alpha_k)}{S^2(S+1)} = \\frac{p_k(1-p_k)}{S+1}",[15,52203,52205,52272,52485],{"className":52204,"ariaHidden":48},[47],[15,52206,52208,52211,52217,52220,52260,52263,52266,52269],{"className":52207},[52],[15,52209],{"className":52210,"style":142},[56],[15,52212,52214],{"className":52213},[531],[15,52215,14853],{"className":52216},[61,1622],[15,52218,741],{"className":52219},[146],[15,52221,52223,52226],{"className":52222},[61],[15,52224,24680],{"className":52225,"style":123},[61,122],[15,52227,52229],{"className":52228},[338],[15,52230,52232,52252],{"className":52231},[342,472],[15,52233,52235,52249],{"className":52234},[346],[15,52236,52238],{"className":52237,"style":479},[350],[15,52239,52240,52243],{"style":27036},[15,52241],{"className":52242,"style":358},[357],[15,52244,52246],{"className":52245},[362,363,364,365],[15,52247,272],{"className":52248,"style":492},[61,122,365],[15,52250,497],{"className":52251},[496],[15,52253,52255],{"className":52254},[346],[15,52256,52258],{"className":52257,"style":504},[350],[15,52259],{},[15,52261,750],{"className":52262},[178],[15,52264],{"className":52265,"style":128},[127],[15,52267,244],{"className":52268},[132],[15,52270],{"className":52271,"style":128},[127],[15,52273,52275,52278,52476,52479,52482],{"className":52274},[52],[15,52276],{"className":52277,"style":17658},[56],[15,52279,52281,52284,52473],{"className":52280},[61],[15,52282],{"className":52283},[146,3841],[15,52285,52287],{"className":52286},[3722],[15,52288,52290,52465],{"className":52289},[342,472],[15,52291,52293,52462],{"className":52292},[346],[15,52294,52296,52351,52359],{"className":52295,"style":15321},[350],[15,52297,52298,52301],{"style":3857},[15,52299],{"className":52300,"style":3861},[357],[15,52302,52304],{"className":52303},[362,363,364,365],[15,52305,52307,52336,52339,52342,52345,52348],{"className":52306},[61,365],[15,52308,52310,52313],{"className":52309},[61,365],[15,52311,40265],{"className":52312,"style":40337},[61,122,365],[15,52314,52316],{"className":52315},[338],[15,52317,52319],{"className":52318},[342],[15,52320,52322],{"className":52321},[346],[15,52323,52325],{"className":52324,"style":25432},[350],[15,52326,52327,52330],{"style":4883},[15,52328],{"className":52329,"style":4062},[357],[15,52331,52333],{"className":52332},[362,4066,4067,365],[15,52334,908],{"className":52335},[61,365],[15,52337,741],{"className":52338},[146,365],[15,52340,40265],{"className":52341,"style":40337},[61,122,365],[15,52343,7762],{"className":52344},[375,365],[15,52346,91],{"className":52347},[61,365],[15,52349,750],{"className":52350},[178,365],[15,52352,52353,52356],{"style":3889},[15,52354],{"className":52355,"style":3861},[357],[15,52357],{"className":52358,"style":3897},[3896],[15,52360,52361,52364],{"style":15450},[15,52362],{"className":52363,"style":3861},[357],[15,52365,52367],{"className":52366},[362,363,364,365],[15,52368,52370,52410,52413,52416,52419,52459],{"className":52369},[61,365],[15,52371,52373,52376],{"className":52372},[61,365],[15,52374,3106],{"className":52375,"style":3191},[61,122,365],[15,52377,52379],{"className":52378},[338],[15,52380,52382,52402],{"className":52381},[342,472],[15,52383,52385,52399],{"className":52384},[346],[15,52386,52388],{"className":52387,"style":22404},[350],[15,52389,52390,52393],{"style":27234},[15,52391],{"className":52392,"style":4062},[357],[15,52394,52396],{"className":52395},[362,4066,4067,365],[15,52397,272],{"className":52398,"style":492},[61,122,365],[15,52400,497],{"className":52401},[496],[15,52403,52405],{"className":52404},[346],[15,52406,52408],{"className":52407,"style":27253},[350],[15,52409],{},[15,52411,741],{"className":52412},[146,365],[15,52414,40265],{"className":52415,"style":40337},[61,122,365],[15,52417,239],{"className":52418},[375,365],[15,52420,52422,52425],{"className":52421},[61,365],[15,52423,3106],{"className":52424,"style":3191},[61,122,365],[15,52426,52428],{"className":52427},[338],[15,52429,52431,52451],{"className":52430},[342,472],[15,52432,52434,52448],{"className":52433},[346],[15,52435,52437],{"className":52436,"style":22404},[350],[15,52438,52439,52442],{"style":27234},[15,52440],{"className":52441,"style":4062},[357],[15,52443,52445],{"className":52444},[362,4066,4067,365],[15,52446,272],{"className":52447,"style":492},[61,122,365],[15,52449,497],{"className":52450},[496],[15,52452,52454],{"className":52453},[346],[15,52455,52457],{"className":52456,"style":27253},[350],[15,52458],{},[15,52460,750],{"className":52461},[178,365],[15,52463,497],{"className":52464},[496],[15,52466,52468],{"className":52467},[346],[15,52469,52471],{"className":52470,"style":3922},[350],[15,52472],{},[15,52474],{"className":52475},[178,3841],[15,52477],{"className":52478,"style":128},[127],[15,52480,244],{"className":52481},[132],[15,52483],{"className":52484,"style":128},[127],[15,52486,52488,52492],{"className":52487},[52],[15,52489],{"className":52490,"style":52491},[56],"height:1.4133em;vertical-align:-0.4033em;",[15,52493,52495,52498,52652],{"className":52494},[61],[15,52496],{"className":52497},[146,3841],[15,52499,52501],{"className":52500},[3722],[15,52502,52504,52644],{"className":52503},[342,472],[15,52505,52507,52641],{"className":52506},[346],[15,52508,52510,52530,52538],{"className":52509,"style":15321},[350],[15,52511,52512,52515],{"style":3857},[15,52513],{"className":52514,"style":3861},[357],[15,52516,52518],{"className":52517},[362,363,364,365],[15,52519,52521,52524,52527],{"className":52520},[61,365],[15,52522,40265],{"className":52523,"style":40337},[61,122,365],[15,52525,7762],{"className":52526},[375,365],[15,52528,91],{"className":52529},[61,365],[15,52531,52532,52535],{"style":3889},[15,52533],{"className":52534,"style":3861},[357],[15,52536],{"className":52537,"style":3897},[3896],[15,52539,52540,52543],{"style":15450},[15,52541],{"className":52542,"style":3861},[357],[15,52544,52546],{"className":52545},[362,363,364,365],[15,52547,52549,52589,52592,52595,52598,52638],{"className":52548},[61,365],[15,52550,52552,52555],{"className":52551},[61,365],[15,52553,11],{"className":52554},[61,122,365],[15,52556,52558],{"className":52557},[338],[15,52559,52561,52581],{"className":52560},[342,472],[15,52562,52564,52578],{"className":52563},[346],[15,52565,52567],{"className":52566,"style":22404},[350],[15,52568,52569,52572],{"style":27279},[15,52570],{"className":52571,"style":4062},[357],[15,52573,52575],{"className":52574},[362,4066,4067,365],[15,52576,272],{"className":52577,"style":492},[61,122,365],[15,52579,497],{"className":52580},[496],[15,52582,52584],{"className":52583},[346],[15,52585,52587],{"className":52586,"style":27253},[350],[15,52588],{},[15,52590,741],{"className":52591},[146,365],[15,52593,91],{"className":52594},[61,365],[15,52596,239],{"className":52597},[375,365],[15,52599,52601,52604],{"className":52600},[61,365],[15,52602,11],{"className":52603},[61,122,365],[15,52605,52607],{"className":52606},[338],[15,52608,52610,52630],{"className":52609},[342,472],[15,52611,52613,52627],{"className":52612},[346],[15,52614,52616],{"className":52615,"style":22404},[350],[15,52617,52618,52621],{"style":27279},[15,52619],{"className":52620,"style":4062},[357],[15,52622,52624],{"className":52623},[362,4066,4067,365],[15,52625,272],{"className":52626,"style":492},[61,122,365],[15,52628,497],{"className":52629},[496],[15,52631,52633],{"className":52632},[346],[15,52634,52636],{"className":52635,"style":27253},[350],[15,52637],{},[15,52639,750],{"className":52640},[178,365],[15,52642,497],{"className":52643},[496],[15,52645,52647],{"className":52646},[346],[15,52648,52650],{"className":52649,"style":46299},[350],[15,52651],{},[15,52653],{"className":52654},[178,3841],"。求和即得。",[11,52657,52658,52659,52662,52663,52733,52734,52737,52738,52886,52887,52938],{},"该损失的两项具有不同的角色。",[661,52660,52661],{},"第一项是拟合项","：在 ",[15,52664,52666,52684],{"className":52665,"translate":19},[18],[15,52667,52669],{"className":52668},[23],[25,52670,52671],{"xmlns":27},[29,52672,52673,52681],{},[32,52674,52675],{},[266,52676,52677,52679],{},[35,52678,11],{},[35,52680,272],{},[40,52682,52683],{"encoding":42},"p_k",[15,52685,52687],{"className":52686,"ariaHidden":48},[47],[15,52688,52690,52693],{"className":52689},[52],[15,52691],{"className":52692,"style":9437},[56],[15,52694,52696,52699],{"className":52695},[61],[15,52697,11],{"className":52698},[61,122],[15,52700,52702],{"className":52701},[338],[15,52703,52705,52725],{"className":52704},[342,472],[15,52706,52708,52722],{"className":52707},[346],[15,52709,52711],{"className":52710,"style":479},[350],[15,52712,52713,52716],{"style":482},[15,52714],{"className":52715,"style":358},[357],[15,52717,52719],{"className":52718},[362,363,364,365],[15,52720,272],{"className":52721,"style":492},[61,122,365],[15,52723,497],{"className":52724},[496],[15,52726,52728],{"className":52727},[346],[15,52729,52731],{"className":52730,"style":504},[350],[15,52732],{}," 退化为确定性预测时，它退化为 one-hot 编码下的均方误差。",[661,52735,52736],{},"第二项是方差惩罚项","：它正比于每个分量的伯努利方差 ",[15,52739,52741,52773],{"className":52740,"translate":19},[18],[15,52742,52744],{"className":52743},[23],[25,52745,52746],{"xmlns":27},[29,52747,52748,52770],{},[32,52749,52750,52756,52758,52760,52762,52768],{},[266,52751,52752,52754],{},[35,52753,11],{},[35,52755,272],{},[81,52757,741],{"stretchy":86},[89,52759,91],{},[81,52761,239],{},[266,52763,52764,52766],{},[35,52765,11],{},[35,52767,272],{},[81,52769,750],{"stretchy":86},[40,52771,52772],{"encoding":42},"p_k(1-p_k)",[15,52774,52776,52837],{"className":52775,"ariaHidden":48},[47],[15,52777,52779,52782,52822,52825,52828,52831,52834],{"className":52778},[52],[15,52780],{"className":52781,"style":142},[56],[15,52783,52785,52788],{"className":52784},[61],[15,52786,11],{"className":52787},[61,122],[15,52789,52791],{"className":52790},[338],[15,52792,52794,52814],{"className":52793},[342,472],[15,52795,52797,52811],{"className":52796},[346],[15,52798,52800],{"className":52799,"style":479},[350],[15,52801,52802,52805],{"style":482},[15,52803],{"className":52804,"style":358},[357],[15,52806,52808],{"className":52807},[362,363,364,365],[15,52809,272],{"className":52810,"style":492},[61,122,365],[15,52812,497],{"className":52813},[496],[15,52815,52817],{"className":52816},[346],[15,52818,52820],{"className":52819,"style":504},[350],[15,52821],{},[15,52823,741],{"className":52824},[146],[15,52826,91],{"className":52827},[61],[15,52829],{"className":52830,"style":772},[127],[15,52832,239],{"className":52833},[375],[15,52835],{"className":52836,"style":772},[127],[15,52838,52840,52843,52883],{"className":52839},[52],[15,52841],{"className":52842,"style":142},[56],[15,52844,52846,52849],{"className":52845},[61],[15,52847,11],{"className":52848},[61,122],[15,52850,52852],{"className":52851},[338],[15,52853,52855,52875],{"className":52854},[342,472],[15,52856,52858,52872],{"className":52857},[346],[15,52859,52861],{"className":52860,"style":479},[350],[15,52862,52863,52866],{"style":482},[15,52864],{"className":52865,"style":358},[357],[15,52867,52869],{"className":52868},[362,363,364,365],[15,52870,272],{"className":52871,"style":492},[61,122,365],[15,52873,497],{"className":52874},[496],[15,52876,52878],{"className":52877},[346],[15,52879,52881],{"className":52880,"style":504},[350],[15,52882],{},[15,52884,750],{"className":52885},[178],"，反比于总证据 ",[15,52888,52890,52908],{"className":52889,"translate":19},[18],[15,52891,52893],{"className":52892},[23],[25,52894,52895],{"xmlns":27},[29,52896,52897,52905],{},[32,52898,52899,52901,52903],{},[35,52900,40265],{},[81,52902,7762],{},[89,52904,91],{},[40,52906,52907],{"encoding":42},"S + 1",[15,52909,52911,52929],{"className":52910,"ariaHidden":48},[47],[15,52912,52914,52917,52920,52923,52926],{"className":52913},[52],[15,52915],{"className":52916,"style":974},[56],[15,52918,40265],{"className":52919,"style":40337},[61,122],[15,52921],{"className":52922,"style":772},[127],[15,52924,7762],{"className":52925},[375],[15,52927],{"className":52928,"style":772},[127],[15,52930,52932,52935],{"className":52931},[52],[15,52933],{"className":52934,"style":859},[56],[15,52936,91],{"className":52937},[61],"。其作用是双重的：",[31547,52940,52941,52977],{},[31550,52942,52943,52944,52972,52973,52976],{},"当证据稀缺（",[15,52945,52947,52960],{"className":52946,"translate":19},[18],[15,52948,52950],{"className":52949},[23],[25,52951,52952],{"xmlns":27},[29,52953,52954,52958],{},[32,52955,52956],{},[35,52957,40265],{},[40,52959,40265],{"encoding":42},[15,52961,52963],{"className":52962,"ariaHidden":48},[47],[15,52964,52966,52969],{"className":52965},[52],[15,52967],{"className":52968,"style":57},[56],[15,52970,40265],{"className":52971,"style":40337},[61,122]," 小）时，方差项较大，迫使模型",[661,52974,52975],{},"不得","以低方差的方式给出尖锐预测——这是一种自适应的软惩罚，其强度由数据本身决定，而非人为设定的超参数。",[31550,52978,52979,52980,9348,53008,53077,53078,53106],{},"当证据变化时，",[15,52981,52983,52996],{"className":52982,"translate":19},[18],[15,52984,52986],{"className":52985},[23],[25,52987,52988],{"xmlns":27},[29,52989,52990,52994],{},[32,52991,52992],{},[35,52993,40265],{},[40,52995,40265],{"encoding":42},[15,52997,52999],{"className":52998,"ariaHidden":48},[47],[15,53000,53002,53005],{"className":53001},[52],[15,53003],{"className":53004,"style":57},[56],[15,53006,40265],{"className":53007,"style":40337},[61,122],[15,53009,53011,53028],{"className":53010,"translate":19},[18],[15,53012,53014],{"className":53013},[23],[25,53015,53016],{"xmlns":27},[29,53017,53018,53026],{},[32,53019,53020],{},[266,53021,53022,53024],{},[35,53023,11],{},[35,53025,272],{},[40,53027,52683],{"encoding":42},[15,53029,53031],{"className":53030,"ariaHidden":48},[47],[15,53032,53034,53037],{"className":53033},[52],[15,53035],{"className":53036,"style":9437},[56],[15,53038,53040,53043],{"className":53039},[61],[15,53041,11],{"className":53042},[61,122],[15,53044,53046],{"className":53045},[338],[15,53047,53049,53069],{"className":53048},[342,472],[15,53050,53052,53066],{"className":53051},[346],[15,53053,53055],{"className":53054,"style":479},[350],[15,53056,53057,53060],{"style":482},[15,53058],{"className":53059,"style":358},[357],[15,53061,53063],{"className":53062},[362,363,364,365],[15,53064,272],{"className":53065,"style":492},[61,122,365],[15,53067,497],{"className":53068},[496],[15,53070,53072],{"className":53071},[346],[15,53073,53075],{"className":53074,"style":504},[350],[15,53076],{}," 同时进入两项，形成张力：增大 ",[15,53079,53081,53094],{"className":53080,"translate":19},[18],[15,53082,53084],{"className":53083},[23],[25,53085,53086],{"xmlns":27},[29,53087,53088,53092],{},[32,53089,53090],{},[35,53091,40265],{},[40,53093,40265],{"encoding":42},[15,53095,53097],{"className":53096,"ariaHidden":48},[47],[15,53098,53100,53103],{"className":53099},[52],[15,53101],{"className":53102,"style":57},[56],[15,53104,40265],{"className":53105,"style":40337},[61,122]," 可以压低方差项，却会强化拟合项。因此最优解不会退化为「证据全为零」的平凡解。",[877,53108,53110],{"id":53109},"_43-证据正则化与退火-kl","4.3 证据正则化与退火 KL",[11,53112,53113,53114,53117,53118,53195,53196,53199],{},"仅有拟合损失仍不足以约束错误预测上的证据。原因在于 4.1 节的梯度分析：拟合项只对",[661,53115,53116],{},"真实类别","的证据给出直接的提升压力，而对错误类别上的证据，它只通过 ",[15,53119,53121,53145],{"className":53120,"translate":19},[18],[15,53122,53124],{"className":53123},[23],[25,53125,53126],{"xmlns":27},[29,53127,53128,53142],{},[32,53129,53130,53136,53138,53140],{},[227,53131,53132,53134],{},[35,53133,22693],{},[81,53135,23162],{"mathvariant":231,"lspace":23161,"rspace":23161},[81,53137,741],{"stretchy":86},[35,53139,40265],{},[81,53141,750],{"stretchy":86},[40,53143,53144],{"encoding":42},"\\psi'(S)",[15,53146,53148],{"className":53147,"ariaHidden":48},[47],[15,53149,53151,53154,53186,53189,53192],{"className":53150},[52],[15,53152],{"className":53153,"style":23197},[56],[15,53155,53157,53160],{"className":53156},[61],[15,53158,22693],{"className":53159,"style":123},[61,122],[15,53161,53163],{"className":53162},[338],[15,53164,53166],{"className":53165},[342],[15,53167,53169],{"className":53168},[346],[15,53170,53172],{"className":53171,"style":23216},[350],[15,53173,53174,53177],{"style":716},[15,53175],{"className":53176,"style":358},[357],[15,53178,53180],{"className":53179},[362,363,364,365],[15,53181,53183],{"className":53182},[61,365],[15,53184,23162],{"className":53185},[61,365],[15,53187,741],{"className":53188},[146],[15,53190,40265],{"className":53191,"style":40337},[61,122],[15,53193,750],{"className":53194},[178]," 给出一个整体的压制信号，这一信号与证据在各类别间的分配无关。后果是：",[661,53197,53198],{},"当一个样本被错误分类时，网络可以通过在错误类别上堆积证据来降低损失","，而这恰恰是我们希望惩罚的行为。",[11,53201,53202],{},"正则项的构造基于一个「反事实」的迪利克雷参数：抹去真实类别的证据，保留其余。",[882,53204,53205],{},[11,53206,53207,53210,53211,53392,53393,53421,53422],{},[661,53208,53209],{},"定义 4.4（反事实参数与证据正则）"," 令 ",[15,53212,53214,53255],{"className":53213,"translate":19},[18],[15,53215,53217],{"className":53216},[23],[25,53218,53219],{"xmlns":27},[29,53220,53221,53252],{},[32,53222,53223,53231,53233,53235,53237,53239,53241,53243,53245,53247,53250],{},[53224,53225,53226,53228],"mover",{"accent":48},[35,53227,3106],{"mathvariant":3105},[81,53229,53230],{},"~",[81,53232,244],{},[35,53234,79],{"mathvariant":251},[81,53236,7762],{},[81,53238,741],{"stretchy":86},[89,53240,91],{"mathvariant":251},[81,53242,239],{},[35,53244,79],{"mathvariant":251},[81,53246,750],{"stretchy":86},[81,53248,53249],{},"⊙",[35,53251,3106],{"mathvariant":3105},[40,53253,53254],{"encoding":42},"\\tilde{\\boldsymbol{\\alpha}} = \\mathbf{y} + (\\mathbf{1} - \\mathbf{y}) \\odot \\boldsymbol{\\alpha}",[15,53256,53258,53316,53335,53356,53377],{"className":53257,"ariaHidden":48},[47],[15,53259,53261,53265,53307,53310,53313],{"className":53260},[52],[15,53262],{"className":53263,"style":53264},[56],"height:0.6813em;",[15,53266,53269],{"className":53267},[61,53268],"accent",[15,53270,53272],{"className":53271},[342],[15,53273,53275],{"className":53274},[346],[15,53276,53278,53293],{"className":53277,"style":53264},[350],[15,53279,53281,53284],{"style":53280},"top:-3em;",[15,53282],{"className":53283,"style":3861},[357],[15,53285,53287],{"className":53286},[61],[15,53288,53290],{"className":53289},[61],[15,53291,3106],{"className":53292},[61,3166],[15,53294,53296,53299],{"style":53295},"top:-3.3634em;",[15,53297],{"className":53298,"style":3861},[357],[15,53300,53304],{"className":53301,"style":53303},[53302],"accent-body","left:-0.25em;",[15,53305,53230],{"className":53306},[61],[15,53308],{"className":53309,"style":128},[127],[15,53311,244],{"className":53312},[132],[15,53314],{"className":53315,"style":128},[127],[15,53317,53319,53323,53326,53329,53332],{"className":53318},[52],[15,53320],{"className":53321,"style":53322},[56],"height:0.7778em;vertical-align:-0.1944em;",[15,53324,79],{"className":53325,"style":29413},[61,411],[15,53327],{"className":53328,"style":772},[127],[15,53330,7762],{"className":53331},[375],[15,53333],{"className":53334,"style":772},[127],[15,53336,53338,53341,53344,53347,53350,53353],{"className":53337},[52],[15,53339],{"className":53340,"style":142},[56],[15,53342,741],{"className":53343},[146],[15,53345,91],{"className":53346},[61,411],[15,53348],{"className":53349,"style":772},[127],[15,53351,239],{"className":53352},[375],[15,53354],{"className":53355,"style":772},[127],[15,53357,53359,53362,53365,53368,53371,53374],{"className":53358},[52],[15,53360],{"className":53361,"style":142},[56],[15,53363,79],{"className":53364,"style":29413},[61,411],[15,53366,750],{"className":53367},[178],[15,53369],{"className":53370,"style":772},[127],[15,53372,53249],{"className":53373},[375],[15,53375],{"className":53376,"style":772},[127],[15,53378,53380,53383],{"className":53379},[52],[15,53381],{"className":53382,"style":2708},[56],[15,53384,53386],{"className":53385},[61],[15,53387,53389],{"className":53388},[61],[15,53390,3106],{"className":53391},[61,3166],"，即把真实类别处的参数置为 ",[15,53394,53396,53409],{"className":53395,"translate":19},[18],[15,53397,53399],{"className":53398},[23],[25,53400,53401],{"xmlns":27},[29,53402,53403,53407],{},[32,53404,53405],{},[89,53406,91],{},[40,53408,91],{"encoding":42},[15,53410,53412],{"className":53411,"ariaHidden":48},[47],[15,53413,53415,53418],{"className":53414},[52],[15,53416],{"className":53417,"style":859},[56],[15,53419,91],{"className":53420},[61],"、其余保持不变。证据正则项定义为\n",[15,53423,53425,53487],{"className":53424,"translate":19},[18],[15,53426,53428],{"className":53427},[23],[25,53429,53430],{"xmlns":27},[29,53431,53432,53484],{},[32,53433,53434,53441,53443,53446,53448],{},[227,53435,53436,53438],{},[35,53437,47490],{"mathvariant":37},[282,53439,53440],{},"reg",[81,53442,244],{},[35,53444,53445],{"mathvariant":231},"KL",[81,53447,1547],{},[32,53449,53450,53452,53454,53456,53458,53464,53466,53468,53470,53472,53474,53476,53478,53480,53482],{},[81,53451,14821],{"fence":48},[35,53453,4186],{"mathvariant":231},[81,53455,1547],{},[81,53457,741],{"stretchy":86},[53224,53459,53460,53462],{"accent":48},[35,53461,3106],{"mathvariant":3105},[81,53463,53230],{},[81,53465,750],{"stretchy":86},[282,53467,4802],{},[81,53469,21291],{"fence":86,"stretchy":48,"minsize":44849,"maxsize":44849},[282,53471,4802],{},[35,53473,4186],{"mathvariant":231},[81,53475,1547],{},[81,53477,741],{"stretchy":86},[89,53479,91],{"mathvariant":251},[81,53481,750],{"stretchy":86},[81,53483,14830],{"fence":48},[40,53485,53486],{"encoding":42},"\\mathcal{L}^{\\text{reg}} = \\operatorname{KL}\\left[\\operatorname{Dir}(\\tilde{\\boldsymbol{\\alpha}}) \\,\\big\\|\\, \\operatorname{Dir}(\\mathbf{1})\\right]",[15,53488,53490,53540],{"className":53489,"ariaHidden":48},[47],[15,53491,53493,53496,53531,53534,53537],{"className":53492},[52],[15,53494],{"className":53495,"style":57},[56],[15,53497,53499,53502],{"className":53498},[61],[15,53500,47490],{"className":53501},[61,62],[15,53503,53505],{"className":53504},[338],[15,53506,53508],{"className":53507},[342],[15,53509,53511],{"className":53510},[346],[15,53512,53514],{"className":53513,"style":37611},[350],[15,53515,53516,53519],{"style":716},[15,53517],{"className":53518,"style":358},[357],[15,53520,53522],{"className":53521},[362,363,364,365],[15,53523,53525],{"className":53524},[61,365],[15,53526,53528],{"className":53527},[61,13548,365],[15,53529,53440],{"className":53530},[61,365],[15,53532],{"className":53533,"style":128},[127],[15,53535,244],{"className":53536},[132],[15,53538],{"className":53539,"style":128},[127],[15,53541,53543,53546,53552,53555],{"className":53542},[52],[15,53544],{"className":53545,"style":45276},[56],[15,53547,53549],{"className":53548},[531],[15,53550,53445],{"className":53551},[61,1622],[15,53553],{"className":53554,"style":157},[127],[15,53556,53558,53564,53570,53573,53610,53613,53616,53662,53665,53668,53674,53677,53680,53683],{"className":53557},[161],[15,53559,53561],{"className":53560,"style":402},[146,401],[15,53562,14821],{"className":53563},[406,4067],[15,53565,53567],{"className":53566},[531],[15,53568,4186],{"className":53569},[61,1622],[15,53571,741],{"className":53572},[146],[15,53574,53576],{"className":53575},[61,53268],[15,53577,53579],{"className":53578},[342],[15,53580,53582],{"className":53581},[346],[15,53583,53585,53599],{"className":53584,"style":53264},[350],[15,53586,53587,53590],{"style":53280},[15,53588],{"className":53589,"style":3861},[357],[15,53591,53593],{"className":53592},[61],[15,53594,53596],{"className":53595},[61],[15,53597,3106],{"className":53598},[61,3166],[15,53600,53601,53604],{"style":53295},[15,53602],{"className":53603,"style":3861},[357],[15,53605,53607],{"className":53606,"style":53303},[53302],[15,53608,53230],{"className":53609},[61],[15,53611,750],{"className":53612},[178],[15,53614],{"className":53615,"style":157},[127],[15,53617,53619],{"className":53618},[61],[15,53620,53623],{"className":53621},[406,53622],"mult",[15,53624,53626,53653],{"className":53625},[342,472],[15,53627,53629,53650],{"className":53628},[346],[15,53630,53633],{"className":53631,"style":53632},[350],"height:0.85em;",[15,53634,53636,53640],{"style":53635},"top:-2.85em;",[15,53637],{"className":53638,"style":53639},[357],"height:3.2em;",[15,53641,53643],{"style":53642},"width:0.556em;height:1.2em;",[27706,53644,53647],{"xmlns":27708,"width":53645,"height":44849,"viewBox":53646},"0.556em","0 0 556 1200",[27714,53648],{"d":53649},"M145 15 v585 v0 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z\nM367 15 v585 v0 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M410 15 H367 v585 v0 v585 h43z",[15,53651,497],{"className":53652},[496],[15,53654,53656],{"className":53655},[346],[15,53657,53660],{"className":53658,"style":53659},[350],"height:0.35em;",[15,53661],{},[15,53663],{"className":53664,"style":157},[127],[15,53666],{"className":53667,"style":157},[127],[15,53669,53671],{"className":53670},[531],[15,53672,4186],{"className":53673},[61,1622],[15,53675,741],{"className":53676},[146],[15,53678,91],{"className":53679},[61,411],[15,53681,750],{"className":53682},[178],[15,53684,53686],{"className":53685,"style":402},[178,401],[15,53687,14830],{"className":53688},[406,4067],[11,53690,53691,53692,53741,53742,53829],{},"注意 ",[15,53693,53695,53717],{"className":53694,"translate":19},[18],[15,53696,53698],{"className":53697},[23],[25,53699,53700],{"xmlns":27},[29,53701,53702,53714],{},[32,53703,53704,53706,53708,53710,53712],{},[35,53705,4186],{"mathvariant":231},[81,53707,1547],{},[81,53709,741],{"stretchy":86},[89,53711,91],{"mathvariant":251},[81,53713,750],{"stretchy":86},[40,53715,53716],{"encoding":42},"\\operatorname{Dir}(\\mathbf{1})",[15,53718,53720],{"className":53719,"ariaHidden":48},[47],[15,53721,53723,53726,53732,53735,53738],{"className":53722},[52],[15,53724],{"className":53725,"style":142},[56],[15,53727,53729],{"className":53728},[531],[15,53730,4186],{"className":53731},[61,1622],[15,53733,741],{"className":53734},[146],[15,53736,91],{"className":53737},[61,411],[15,53739,750],{"className":53740},[178]," 正是无信息均匀先验。该 KL 散度只依赖于非目标类别上的证据：若网络在错误类别上给出大量证据，",[15,53743,53745,53771],{"className":53744,"translate":19},[18],[15,53746,53748],{"className":53747},[23],[25,53749,53750],{"xmlns":27},[29,53751,53752,53768],{},[32,53753,53754,53756,53758,53760,53766],{},[35,53755,4186],{"mathvariant":231},[81,53757,1547],{},[81,53759,741],{"stretchy":86},[53224,53761,53762,53764],{"accent":48},[35,53763,3106],{"mathvariant":3105},[81,53765,53230],{},[81,53767,750],{"stretchy":86},[40,53769,53770],{"encoding":42},"\\operatorname{Dir}(\\tilde{\\boldsymbol{\\alpha}})",[15,53772,53774],{"className":53773,"ariaHidden":48},[47],[15,53775,53777,53780,53786,53789,53826],{"className":53776},[52],[15,53778],{"className":53779,"style":142},[56],[15,53781,53783],{"className":53782},[531],[15,53784,4186],{"className":53785},[61,1622],[15,53787,741],{"className":53788},[146],[15,53790,53792],{"className":53791},[61,53268],[15,53793,53795],{"className":53794},[342],[15,53796,53798],{"className":53797},[346],[15,53799,53801,53815],{"className":53800,"style":53264},[350],[15,53802,53803,53806],{"style":53280},[15,53804],{"className":53805,"style":3861},[357],[15,53807,53809],{"className":53808},[61],[15,53810,53812],{"className":53811},[61],[15,53813,3106],{"className":53814},[61,3166],[15,53816,53817,53820],{"style":53295},[15,53818],{"className":53819,"style":3861},[357],[15,53821,53823],{"className":53822,"style":53303},[53302],[15,53824,53230],{"className":53825},[61],[15,53827,750],{"className":53828},[178]," 将显著偏离均匀分布，KL 随之增大。",[882,53831,53832],{},[11,53833,53834,53837,53838,4738,53978,54664,54665,54722,54723,54809,54810],{},[661,53835,53836],{},"命题 4.5（Dirichlet 的 KL 散度）"," 对 ",[15,53839,53841,53873],{"className":53840,"translate":19},[18],[15,53842,53844],{"className":53843},[23],[25,53845,53846],{"xmlns":27},[29,53847,53848,53870],{},[32,53849,53850,53852,53854,53856,53858],{},[35,53851,3106],{"mathvariant":3105},[81,53853,94],{"separator":48},[35,53855,17131],{"mathvariant":3105},[81,53857,83],{},[289,53859,53860,53862,53868],{},[35,53861,259],{"mathvariant":258},[32,53863,53864,53866],{},[81,53865,1590],{},[89,53867,278],{},[35,53869,102],{},[40,53871,53872],{"encoding":42},"\\boldsymbol{\\alpha}, \\boldsymbol{\\beta} \\in \\mathbb{R}_{>0}^K",[15,53874,53876,53915],{"className":53875,"ariaHidden":48},[47],[15,53877,53879,53882,53891,53894,53897,53906,53909,53912],{"className":53878},[52],[15,53880],{"className":53881,"style":14310},[56],[15,53883,53885],{"className":53884},[61],[15,53886,53888],{"className":53887},[61],[15,53889,3106],{"className":53890},[61,3166],[15,53892,94],{"className":53893},[153],[15,53895],{"className":53896,"style":157},[127],[15,53898,53900],{"className":53899},[61],[15,53901,53903],{"className":53902},[61],[15,53904,17131],{"className":53905,"style":17374},[61,3166],[15,53907],{"className":53908,"style":128},[127],[15,53910,83],{"className":53911},[132],[15,53913],{"className":53914,"style":128},[127],[15,53916,53918,53921],{"className":53917},[52],[15,53919],{"className":53920,"style":3303},[56],[15,53922,53924,53927],{"className":53923},[61],[15,53925,259],{"className":53926},[61,427],[15,53928,53930],{"className":53929},[338],[15,53931,53933,53970],{"className":53932},[342,472],[15,53934,53936,53967],{"className":53935},[346],[15,53937,53939,53956],{"className":53938,"style":695},[350],[15,53940,53941,53944],{"style":3324},[15,53942],{"className":53943,"style":358},[357],[15,53945,53947],{"className":53946},[362,363,364,365],[15,53948,53950,53953],{"className":53949},[61,365],[15,53951,1590],{"className":53952},[132,365],[15,53954,278],{"className":53955},[61,365],[15,53957,53958,53961],{"style":716},[15,53959],{"className":53960,"style":358},[357],[15,53962,53964],{"className":53963},[362,363,364,365],[15,53965,102],{"className":53966,"style":174},[61,122,365],[15,53968,497],{"className":53969},[496],[15,53971,53973],{"className":53972},[346],[15,53974,53976],{"className":53975,"style":3360},[350],[15,53977],{},[15,53979,53981,54121],{"className":53980,"translate":19},[18],[15,53982,53984],{"className":53983},[23],[25,53985,53986],{"xmlns":27},[29,53987,53988,54118],{},[32,53989,53990,53992,53994,54026,54028,54030,54032,54054,54056,54070,54072,54078,54080,54086,54088,54090,54092,54094,54100,54102,54104,54106,54108,54114,54116],{},[35,53991,53445],{"mathvariant":231},[81,53993,1547],{},[32,53995,53996,53998,54000,54002,54004,54006,54008,54010,54012,54014,54016,54018,54020,54022,54024],{},[81,53997,14821],{"fence":48},[35,53999,4186],{"mathvariant":231},[81,54001,1547],{},[81,54003,741],{"stretchy":86},[35,54005,3106],{"mathvariant":3105},[81,54007,750],{"stretchy":86},[282,54009,4802],{},[81,54011,21291],{"fence":86,"stretchy":48,"minsize":44849,"maxsize":44849},[282,54013,4802],{},[35,54015,4186],{"mathvariant":231},[81,54017,1547],{},[81,54019,741],{"stretchy":86},[35,54021,17131],{"mathvariant":3105},[81,54023,750],{"stretchy":86},[81,54025,14830],{"fence":48},[81,54027,244],{},[35,54029,22662],{},[81,54031,1547],{},[3722,54033,54034,54044],{},[32,54035,54036,54038,54040,54042],{},[35,54037,3730],{},[81,54039,741],{"stretchy":86},[35,54041,17131],{"mathvariant":3105},[81,54043,750],{"stretchy":86},[32,54045,54046,54048,54050,54052],{},[35,54047,3730],{},[81,54049,741],{"stretchy":86},[35,54051,3106],{"mathvariant":3105},[81,54053,750],{"stretchy":86},[81,54055,7762],{},[289,54057,54058,54060,54068],{},[81,54059,293],{},[32,54061,54062,54064,54066],{},[35,54063,272],{},[81,54065,244],{},[89,54067,91],{},[35,54069,102],{},[81,54071,741],{"stretchy":86},[266,54073,54074,54076],{},[35,54075,3106],{},[35,54077,272],{},[81,54079,239],{},[266,54081,54082,54084],{},[35,54083,17131],{},[35,54085,272],{},[81,54087,750],{"stretchy":86},[81,54089,741],{"fence":86,"stretchy":48,"minsize":44849,"maxsize":44849},[35,54091,22693],{},[81,54093,741],{"stretchy":86},[266,54095,54096,54098],{},[35,54097,3106],{},[35,54099,272],{},[81,54101,750],{"stretchy":86},[81,54103,239],{},[35,54105,22693],{},[81,54107,741],{"stretchy":86},[266,54109,54110,54112],{},[35,54111,3106],{},[89,54113,278],{},[81,54115,750],{"stretchy":86},[81,54117,750],{"fence":86,"stretchy":48,"minsize":44849,"maxsize":44849},[40,54119,54120],{"encoding":42},"\\operatorname{KL}\\left[\\operatorname{Dir}(\\boldsymbol{\\alpha}) \\,\\big\\|\\, \\operatorname{Dir}(\\boldsymbol{\\beta})\\right] = \\log\\frac{B(\\boldsymbol{\\beta})}{B(\\boldsymbol{\\alpha})} + \\sum_{k=1}^K (\\alpha_k - \\beta_k)\\big(\\psi(\\alpha_k) - \\psi(\\alpha_0)\\big)",[15,54122,54124,54251,54372,54490,54603],{"className":54123,"ariaHidden":48},[47],[15,54125,54127,54130,54136,54139,54242,54245,54248],{"className":54126},[52],[15,54128],{"className":54129,"style":45276},[56],[15,54131,54133],{"className":54132},[531],[15,54134,53445],{"className":54135},[61,1622],[15,54137],{"className":54138,"style":157},[127],[15,54140,54142,54148,54154,54157,54166,54169,54172,54209,54212,54215,54221,54224,54233,54236],{"className":54141},[161],[15,54143,54145],{"className":54144,"style":402},[146,401],[15,54146,14821],{"className":54147},[406,4067],[15,54149,54151],{"className":54150},[531],[15,54152,4186],{"className":54153},[61,1622],[15,54155,741],{"className":54156},[146],[15,54158,54160],{"className":54159},[61],[15,54161,54163],{"className":54162},[61],[15,54164,3106],{"className":54165},[61,3166],[15,54167,750],{"className":54168},[178],[15,54170],{"className":54171,"style":157},[127],[15,54173,54175],{"className":54174},[61],[15,54176,54178],{"className":54177},[406,53622],[15,54179,54181,54201],{"className":54180},[342,472],[15,54182,54184,54198],{"className":54183},[346],[15,54185,54187],{"className":54186,"style":53632},[350],[15,54188,54189,54192],{"style":53635},[15,54190],{"className":54191,"style":53639},[357],[15,54193,54194],{"style":53642},[27706,54195,54196],{"xmlns":27708,"width":53645,"height":44849,"viewBox":53646},[27714,54197],{"d":53649},[15,54199,497],{"className":54200},[496],[15,54202,54204],{"className":54203},[346],[15,54205,54207],{"className":54206,"style":53659},[350],[15,54208],{},[15,54210],{"className":54211,"style":157},[127],[15,54213],{"className":54214,"style":157},[127],[15,54216,54218],{"className":54217},[531],[15,54219,4186],{"className":54220},[61,1622],[15,54222,741],{"className":54223},[146],[15,54225,54227],{"className":54226},[61],[15,54228,54230],{"className":54229},[61],[15,54231,17131],{"className":54232,"style":17374},[61,3166],[15,54234,750],{"className":54235},[178],[15,54237,54239],{"className":54238,"style":402},[178,401],[15,54240,14830],{"className":54241},[406,4067],[15,54243],{"className":54244,"style":128},[127],[15,54246,244],{"className":54247},[132],[15,54249],{"className":54250,"style":128},[127],[15,54252,54254,54257,54262,54265,54363,54366,54369],{"className":54253},[52],[15,54255],{"className":54256,"style":17658},[56],[15,54258,22791,54260],{"className":54259},[531],[15,54261,22794],{"style":15619},[15,54263],{"className":54264,"style":157},[127],[15,54266,54268,54271,54360],{"className":54267},[61],[15,54269],{"className":54270},[146,3841],[15,54272,54274],{"className":54273},[3722],[15,54275,54277,54352],{"className":54276},[342,472],[15,54278,54280,54349],{"className":54279},[346],[15,54281,54283,54312,54320],{"className":54282,"style":15321},[350],[15,54284,54285,54288],{"style":3857},[15,54286],{"className":54287,"style":3861},[357],[15,54289,54291],{"className":54290},[362,363,364,365],[15,54292,54294,54297,54300,54309],{"className":54293},[61,365],[15,54295,3730],{"className":54296,"style":3871},[61,122,365],[15,54298,741],{"className":54299},[146,365],[15,54301,54303],{"className":54302},[61,365],[15,54304,54306],{"className":54305},[61,365],[15,54307,3106],{"className":54308},[61,3166,365],[15,54310,750],{"className":54311},[178,365],[15,54313,54314,54317],{"style":3889},[15,54315],{"className":54316,"style":3861},[357],[15,54318],{"className":54319,"style":3897},[3896],[15,54321,54322,54325],{"style":15450},[15,54323],{"className":54324,"style":3861},[357],[15,54326,54328],{"className":54327},[362,363,364,365],[15,54329,54331,54334,54337,54346],{"className":54330},[61,365],[15,54332,3730],{"className":54333,"style":3871},[61,122,365],[15,54335,741],{"className":54336},[146,365],[15,54338,54340],{"className":54339},[61,365],[15,54341,54343],{"className":54342},[61,365],[15,54344,17131],{"className":54345,"style":17374},[61,3166,365],[15,54347,750],{"className":54348},[178,365],[15,54350,497],{"className":54351},[496],[15,54353,54355],{"className":54354},[346],[15,54356,54358],{"className":54357,"style":3922},[350],[15,54359],{},[15,54361],{"className":54362},[178,3841],[15,54364],{"className":54365,"style":772},[127],[15,54367,7762],{"className":54368},[375],[15,54370],{"className":54371,"style":772},[127],[15,54373,54375,54378,54438,54441,54481,54484,54487],{"className":54374},[52],[15,54376],{"className":54377,"style":3471},[56],[15,54379,54381,54384],{"className":54380},[531],[15,54382,293],{"className":54383,"style":537},[531,535,536],[15,54385,54387],{"className":54386},[338],[15,54388,54390,54430],{"className":54389},[342,472],[15,54391,54393,54427],{"className":54392},[346],[15,54394,54396,54416],{"className":54395,"style":550},[350],[15,54397,54398,54401],{"style":553},[15,54399],{"className":54400,"style":358},[357],[15,54402,54404],{"className":54403},[362,363,364,365],[15,54405,54407,54410,54413],{"className":54406},[61,365],[15,54408,272],{"className":54409,"style":492},[61,122,365],[15,54411,244],{"className":54412},[132,365],[15,54414,91],{"className":54415},[61,365],[15,54417,54418,54421],{"style":574},[15,54419],{"className":54420,"style":358},[357],[15,54422,54424],{"className":54423},[362,363,364,365],[15,54425,102],{"className":54426,"style":174},[61,122,365],[15,54428,497],{"className":54429},[496],[15,54431,54433],{"className":54432},[346],[15,54434,54436],{"className":54435,"style":593},[350],[15,54437],{},[15,54439,741],{"className":54440},[146],[15,54442,54444,54447],{"className":54443},[61],[15,54445,3106],{"className":54446,"style":3191},[61,122],[15,54448,54450],{"className":54449},[338],[15,54451,54453,54473],{"className":54452},[342,472],[15,54454,54456,54470],{"className":54455},[346],[15,54457,54459],{"className":54458,"style":479},[350],[15,54460,54461,54464],{"style":3206},[15,54462],{"className":54463,"style":358},[357],[15,54465,54467],{"className":54466},[362,363,364,365],[15,54468,272],{"className":54469,"style":492},[61,122,365],[15,54471,497],{"className":54472},[496],[15,54474,54476],{"className":54475},[346],[15,54477,54479],{"className":54478,"style":504},[350],[15,54480],{},[15,54482],{"className":54483,"style":772},[127],[15,54485,239],{"className":54486},[375],[15,54488],{"className":54489,"style":772},[127],[15,54491,54493,54496,54536,54539,54545,54548,54551,54591,54594,54597,54600],{"className":54492},[52],[15,54494],{"className":54495,"style":45276},[56],[15,54497,54499,54502],{"className":54498},[61],[15,54500,17131],{"className":54501,"style":17270},[61,122],[15,54503,54505],{"className":54504},[338],[15,54506,54508,54528],{"className":54507},[342,472],[15,54509,54511,54525],{"className":54510},[346],[15,54512,54514],{"className":54513,"style":479},[350],[15,54515,54516,54519],{"style":17454},[15,54517],{"className":54518,"style":358},[357],[15,54520,54522],{"className":54521},[362,363,364,365],[15,54523,272],{"className":54524,"style":492},[61,122,365],[15,54526,497],{"className":54527},[496],[15,54529,54531],{"className":54530},[346],[15,54532,54534],{"className":54533,"style":504},[350],[15,54535],{},[15,54537,750],{"className":54538},[178],[15,54540,54542],{"className":54541},[61],[15,54543,741],{"className":54544},[406,4067],[15,54546,22693],{"className":54547,"style":123},[61,122],[15,54549,741],{"className":54550},[146],[15,54552,54554,54557],{"className":54553},[61],[15,54555,3106],{"className":54556,"style":3191},[61,122],[15,54558,54560],{"className":54559},[338],[15,54561,54563,54583],{"className":54562},[342,472],[15,54564,54566,54580],{"className":54565},[346],[15,54567,54569],{"className":54568,"style":479},[350],[15,54570,54571,54574],{"style":3206},[15,54572],{"className":54573,"style":358},[357],[15,54575,54577],{"className":54576},[362,363,364,365],[15,54578,272],{"className":54579,"style":492},[61,122,365],[15,54581,497],{"className":54582},[496],[15,54584,54586],{"className":54585},[346],[15,54587,54589],{"className":54588,"style":504},[350],[15,54590],{},[15,54592,750],{"className":54593},[178],[15,54595],{"className":54596,"style":772},[127],[15,54598,239],{"className":54599},[375],[15,54601],{"className":54602,"style":772},[127],[15,54604,54606,54609,54612,54615,54655,54658],{"className":54605},[52],[15,54607],{"className":54608,"style":45276},[56],[15,54610,22693],{"className":54611,"style":123},[61,122],[15,54613,741],{"className":54614},[146],[15,54616,54618,54621],{"className":54617},[61],[15,54619,3106],{"className":54620,"style":3191},[61,122],[15,54622,54624],{"className":54623},[338],[15,54625,54627,54647],{"className":54626},[342,472],[15,54628,54630,54644],{"className":54629},[346],[15,54631,54633],{"className":54632,"style":1214},[350],[15,54634,54635,54638],{"style":3206},[15,54636],{"className":54637,"style":358},[357],[15,54639,54641],{"className":54640},[362,363,364,365],[15,54642,278],{"className":54643},[61,365],[15,54645,497],{"className":54646},[496],[15,54648,54650],{"className":54649},[346],[15,54651,54653],{"className":54652,"style":504},[350],[15,54654],{},[15,54656,750],{"className":54657},[178],[15,54659,54661],{"className":54660},[61],[15,54662,750],{"className":54663},[406,4067],"\n特别地，取 ",[15,54666,54668,54686],{"className":54667,"translate":19},[18],[15,54669,54671],{"className":54670},[23],[25,54672,54673],{"xmlns":27},[29,54674,54675,54683],{},[32,54676,54677,54679,54681],{},[35,54678,17131],{"mathvariant":3105},[81,54680,244],{},[89,54682,91],{"mathvariant":251},[40,54684,54685],{"encoding":42},"\\boldsymbol{\\beta} = \\mathbf{1}",[15,54687,54689,54713],{"className":54688,"ariaHidden":48},[47],[15,54690,54692,54695,54704,54707,54710],{"className":54691},[52],[15,54693],{"className":54694,"style":14310},[56],[15,54696,54698],{"className":54697},[61],[15,54699,54701],{"className":54700},[61],[15,54702,17131],{"className":54703,"style":17374},[61,3166],[15,54705],{"className":54706,"style":128},[127],[15,54708,244],{"className":54709},[132],[15,54711],{"className":54712,"style":128},[127],[15,54714,54716,54719],{"className":54715},[52],[15,54717],{"className":54718,"style":859},[56],[15,54720,91],{"className":54721},[61,411],"（此时 ",[15,54724,54726,54760],{"className":54725,"translate":19},[18],[15,54727,54729],{"className":54728},[23],[25,54730,54731],{"xmlns":27},[29,54732,54733,54757],{},[32,54734,54735,54737,54739,54741,54743,54745,54747,54749,54751,54753,54755],{},[35,54736,3730],{},[81,54738,741],{"stretchy":86},[89,54740,91],{"mathvariant":251},[81,54742,750],{"stretchy":86},[81,54744,244],{},[89,54746,91],{},[35,54748,3020],{"mathvariant":231},[35,54750,4292],{"mathvariant":231},[81,54752,741],{"stretchy":86},[35,54754,102],{},[81,54756,750],{"stretchy":86},[40,54758,54759],{"encoding":42},"B(\\mathbf{1}) = 1\u002F\\Gamma(K)",[15,54761,54763,54790],{"className":54762,"ariaHidden":48},[47],[15,54764,54766,54769,54772,54775,54778,54781,54784,54787],{"className":54765},[52],[15,54767],{"className":54768,"style":142},[56],[15,54770,3730],{"className":54771,"style":3871},[61,122],[15,54773,741],{"className":54774},[146],[15,54776,91],{"className":54777},[61,411],[15,54779,750],{"className":54780},[178],[15,54782],{"className":54783,"style":128},[127],[15,54785,244],{"className":54786},[132],[15,54788],{"className":54789,"style":128},[127],[15,54791,54793,54796,54800,54803,54806],{"className":54792},[52],[15,54794],{"className":54795,"style":142},[56],[15,54797,54799],{"className":54798},[61],"1\u002FΓ",[15,54801,741],{"className":54802},[146],[15,54804,102],{"className":54805,"style":174},[61,122],[15,54807,750],{"className":54808},[178],"）得\n",[15,54811,54813,54975],{"className":54812,"translate":19},[18],[15,54814,54816],{"className":54815},[23],[25,54817,54818],{"xmlns":27},[29,54819,54820,54972],{},[32,54821,54822,54828,54830,54832,54834,54836,54838,54848,54850,54852,54866,54868,54870,54872,54874,54884,54886,54888,54890,54892,54894,54896,54898,54900,54902,54916,54918,54928,54930,54932,54934,54936,54938,54940,54950,54952,54954,54956,54958,54968,54970],{},[227,54823,54824,54826],{},[35,54825,47490],{"mathvariant":37},[282,54827,53440],{},[81,54829,244],{},[35,54831,22662],{},[81,54833,1547],{},[35,54835,4292],{"mathvariant":231},[81,54837,741],{"stretchy":86},[266,54839,54840,54846],{},[53224,54841,54842,54844],{"accent":48},[35,54843,3106],{},[81,54845,53230],{},[89,54847,278],{},[81,54849,750],{"stretchy":86},[81,54851,239],{},[289,54853,54854,54856,54864],{},[81,54855,293],{},[32,54857,54858,54860,54862],{},[35,54859,272],{},[81,54861,244],{},[89,54863,91],{},[35,54865,102],{},[35,54867,22662],{},[81,54869,1547],{},[35,54871,4292],{"mathvariant":231},[81,54873,741],{"stretchy":86},[266,54875,54876,54882],{},[53224,54877,54878,54880],{"accent":48},[35,54879,3106],{},[81,54881,53230],{},[35,54883,272],{},[81,54885,750],{"stretchy":86},[81,54887,239],{},[35,54889,22662],{},[81,54891,1547],{},[35,54893,4292],{"mathvariant":231},[81,54895,741],{"stretchy":86},[35,54897,102],{},[81,54899,750],{"stretchy":86},[81,54901,7762],{},[289,54903,54904,54906,54914],{},[81,54905,293],{},[32,54907,54908,54910,54912],{},[35,54909,272],{},[81,54911,244],{},[89,54913,91],{},[35,54915,102],{},[81,54917,741],{"stretchy":86},[266,54919,54920,54926],{},[53224,54921,54922,54924],{"accent":48},[35,54923,3106],{},[81,54925,53230],{},[35,54927,272],{},[81,54929,239],{},[89,54931,91],{},[81,54933,750],{"stretchy":86},[81,54935,741],{"fence":86,"stretchy":48,"minsize":44849,"maxsize":44849},[35,54937,22693],{},[81,54939,741],{"stretchy":86},[266,54941,54942,54948],{},[53224,54943,54944,54946],{"accent":48},[35,54945,3106],{},[81,54947,53230],{},[35,54949,272],{},[81,54951,750],{"stretchy":86},[81,54953,239],{},[35,54955,22693],{},[81,54957,741],{"stretchy":86},[266,54959,54960,54966],{},[53224,54961,54962,54964],{"accent":48},[35,54963,3106],{},[81,54965,53230],{},[89,54967,278],{},[81,54969,750],{"stretchy":86},[81,54971,750],{"fence":86,"stretchy":48,"minsize":44849,"maxsize":44849},[40,54973,54974],{"encoding":42},"\\mathcal{L}^{\\text{reg}} = \\log\\Gamma(\\tilde{\\alpha}_0) - \\sum_{k=1}^K \\log\\Gamma(\\tilde{\\alpha}_k) - \\log\\Gamma(K) + \\sum_{k=1}^K (\\tilde{\\alpha}_k - 1)\\big(\\psi(\\tilde{\\alpha}_k) - \\psi(\\tilde{\\alpha}_0)\\big)",[15,54976,54978,55028,55131,55294,55329,55475,55579],{"className":54977,"ariaHidden":48},[47],[15,54979,54981,54984,55019,55022,55025],{"className":54980},[52],[15,54982],{"className":54983,"style":57},[56],[15,54985,54987,54990],{"className":54986},[61],[15,54988,47490],{"className":54989},[61,62],[15,54991,54993],{"className":54992},[338],[15,54994,54996],{"className":54995},[342],[15,54997,54999],{"className":54998},[346],[15,55000,55002],{"className":55001,"style":37611},[350],[15,55003,55004,55007],{"style":716},[15,55005],{"className":55006,"style":358},[357],[15,55008,55010],{"className":55009},[362,363,364,365],[15,55011,55013],{"className":55012},[61,365],[15,55014,55016],{"className":55015},[61,13548,365],[15,55017,53440],{"className":55018},[61,365],[15,55020],{"className":55021,"style":128},[127],[15,55023,244],{"className":55024},[132],[15,55026],{"className":55027,"style":128},[127],[15,55029,55031,55034,55039,55042,55045,55048,55119,55122,55125,55128],{"className":55030},[52],[15,55032],{"className":55033,"style":142},[56],[15,55035,22791,55037],{"className":55036},[531],[15,55038,22794],{"style":15619},[15,55040],{"className":55041,"style":157},[127],[15,55043,4292],{"className":55044},[61],[15,55046,741],{"className":55047},[146],[15,55049,55051,55085],{"className":55050},[61],[15,55052,55054],{"className":55053},[61,53268],[15,55055,55057],{"className":55056},[342],[15,55058,55060],{"className":55059},[346],[15,55061,55064,55072],{"className":55062,"style":55063},[350],"height:0.6679em;",[15,55065,55066,55069],{"style":53280},[15,55067],{"className":55068,"style":3861},[357],[15,55070,3106],{"className":55071,"style":3191},[61,122],[15,55073,55075,55078],{"style":55074},"top:-3.35em;",[15,55076],{"className":55077,"style":3861},[357],[15,55079,55082],{"className":55080,"style":55081},[53302],"left:-0.2222em;",[15,55083,53230],{"className":55084},[61],[15,55086,55088],{"className":55087},[338],[15,55089,55091,55111],{"className":55090},[342,472],[15,55092,55094,55108],{"className":55093},[346],[15,55095,55097],{"className":55096,"style":1214},[350],[15,55098,55099,55102],{"style":3206},[15,55100],{"className":55101,"style":358},[357],[15,55103,55105],{"className":55104},[362,363,364,365],[15,55106,278],{"className":55107},[61,365],[15,55109,497],{"className":55110},[496],[15,55112,55114],{"className":55113},[346],[15,55115,55117],{"className":55116,"style":504},[350],[15,55118],{},[15,55120,750],{"className":55121},[178],[15,55123],{"className":55124,"style":772},[127],[15,55126,239],{"className":55127},[375],[15,55129],{"className":55130,"style":772},[127],[15,55132,55134,55137,55197,55200,55205,55208,55211,55214,55282,55285,55288,55291],{"className":55133},[52],[15,55135],{"className":55136,"style":3471},[56],[15,55138,55140,55143],{"className":55139},[531],[15,55141,293],{"className":55142,"style":537},[531,535,536],[15,55144,55146],{"className":55145},[338],[15,55147,55149,55189],{"className":55148},[342,472],[15,55150,55152,55186],{"className":55151},[346],[15,55153,55155,55175],{"className":55154,"style":550},[350],[15,55156,55157,55160],{"style":553},[15,55158],{"className":55159,"style":358},[357],[15,55161,55163],{"className":55162},[362,363,364,365],[15,55164,55166,55169,55172],{"className":55165},[61,365],[15,55167,272],{"className":55168,"style":492},[61,122,365],[15,55170,244],{"className":55171},[132,365],[15,55173,91],{"className":55174},[61,365],[15,55176,55177,55180],{"style":574},[15,55178],{"className":55179,"style":358},[357],[15,55181,55183],{"className":55182},[362,363,364,365],[15,55184,102],{"className":55185,"style":174},[61,122,365],[15,55187,497],{"className":55188},[496],[15,55190,55192],{"className":55191},[346],[15,55193,55195],{"className":55194,"style":593},[350],[15,55196],{},[15,55198],{"className":55199,"style":157},[127],[15,55201,22791,55203],{"className":55202},[531],[15,55204,22794],{"style":15619},[15,55206],{"className":55207,"style":157},[127],[15,55209,4292],{"className":55210},[61],[15,55212,741],{"className":55213},[146],[15,55215,55217,55248],{"className":55216},[61],[15,55218,55220],{"className":55219},[61,53268],[15,55221,55223],{"className":55222},[342],[15,55224,55226],{"className":55225},[346],[15,55227,55229,55237],{"className":55228,"style":55063},[350],[15,55230,55231,55234],{"style":53280},[15,55232],{"className":55233,"style":3861},[357],[15,55235,3106],{"className":55236,"style":3191},[61,122],[15,55238,55239,55242],{"style":55074},[15,55240],{"className":55241,"style":3861},[357],[15,55243,55245],{"className":55244,"style":55081},[53302],[15,55246,53230],{"className":55247},[61],[15,55249,55251],{"className":55250},[338],[15,55252,55254,55274],{"className":55253},[342,472],[15,55255,55257,55271],{"className":55256},[346],[15,55258,55260],{"className":55259,"style":479},[350],[15,55261,55262,55265],{"style":3206},[15,55263],{"className":55264,"style":358},[357],[15,55266,55268],{"className":55267},[362,363,364,365],[15,55269,272],{"className":55270,"style":492},[61,122,365],[15,55272,497],{"className":55273},[496],[15,55275,55277],{"className":55276},[346],[15,55278,55280],{"className":55279,"style":504},[350],[15,55281],{},[15,55283,750],{"className":55284},[178],[15,55286],{"className":55287,"style":772},[127],[15,55289,239],{"className":55290},[375],[15,55292],{"className":55293,"style":772},[127],[15,55295,55297,55300,55305,55308,55311,55314,55317,55320,55323,55326],{"className":55296},[52],[15,55298],{"className":55299,"style":142},[56],[15,55301,22791,55303],{"className":55302},[531],[15,55304,22794],{"style":15619},[15,55306],{"className":55307,"style":157},[127],[15,55309,4292],{"className":55310},[61],[15,55312,741],{"className":55313},[146],[15,55315,102],{"className":55316,"style":174},[61,122],[15,55318,750],{"className":55319},[178],[15,55321],{"className":55322,"style":772},[127],[15,55324,7762],{"className":55325},[375],[15,55327],{"className":55328,"style":772},[127],[15,55330,55332,55335,55395,55398,55466,55469,55472],{"className":55331},[52],[15,55333],{"className":55334,"style":3471},[56],[15,55336,55338,55341],{"className":55337},[531],[15,55339,293],{"className":55340,"style":537},[531,535,536],[15,55342,55344],{"className":55343},[338],[15,55345,55347,55387],{"className":55346},[342,472],[15,55348,55350,55384],{"className":55349},[346],[15,55351,55353,55373],{"className":55352,"style":550},[350],[15,55354,55355,55358],{"style":553},[15,55356],{"className":55357,"style":358},[357],[15,55359,55361],{"className":55360},[362,363,364,365],[15,55362,55364,55367,55370],{"className":55363},[61,365],[15,55365,272],{"className":55366,"style":492},[61,122,365],[15,55368,244],{"className":55369},[132,365],[15,55371,91],{"className":55372},[61,365],[15,55374,55375,55378],{"style":574},[15,55376],{"className":55377,"style":358},[357],[15,55379,55381],{"className":55380},[362,363,364,365],[15,55382,102],{"className":55383,"style":174},[61,122,365],[15,55385,497],{"className":55386},[496],[15,55388,55390],{"className":55389},[346],[15,55391,55393],{"className":55392,"style":593},[350],[15,55394],{},[15,55396,741],{"className":55397},[146],[15,55399,55401,55432],{"className":55400},[61],[15,55402,55404],{"className":55403},[61,53268],[15,55405,55407],{"className":55406},[342],[15,55408,55410],{"className":55409},[346],[15,55411,55413,55421],{"className":55412,"style":55063},[350],[15,55414,55415,55418],{"style":53280},[15,55416],{"className":55417,"style":3861},[357],[15,55419,3106],{"className":55420,"style":3191},[61,122],[15,55422,55423,55426],{"style":55074},[15,55424],{"className":55425,"style":3861},[357],[15,55427,55429],{"className":55428,"style":55081},[53302],[15,55430,53230],{"className":55431},[61],[15,55433,55435],{"className":55434},[338],[15,55436,55438,55458],{"className":55437},[342,472],[15,55439,55441,55455],{"className":55440},[346],[15,55442,55444],{"className":55443,"style":479},[350],[15,55445,55446,55449],{"style":3206},[15,55447],{"className":55448,"style":358},[357],[15,55450,55452],{"className":55451},[362,363,364,365],[15,55453,272],{"className":55454,"style":492},[61,122,365],[15,55456,497],{"className":55457},[496],[15,55459,55461],{"className":55460},[346],[15,55462,55464],{"className":55463,"style":504},[350],[15,55465],{},[15,55467],{"className":55468,"style":772},[127],[15,55470,239],{"className":55471},[375],[15,55473],{"className":55474,"style":772},[127],[15,55476,55478,55481,55484,55487,55493,55496,55499,55567,55570,55573,55576],{"className":55477},[52],[15,55479],{"className":55480,"style":45276},[56],[15,55482,91],{"className":55483},[61],[15,55485,750],{"className":55486},[178],[15,55488,55490],{"className":55489},[61],[15,55491,741],{"className":55492},[406,4067],[15,55494,22693],{"className":55495,"style":123},[61,122],[15,55497,741],{"className":55498},[146],[15,55500,55502,55533],{"className":55501},[61],[15,55503,55505],{"className":55504},[61,53268],[15,55506,55508],{"className":55507},[342],[15,55509,55511],{"className":55510},[346],[15,55512,55514,55522],{"className":55513,"style":55063},[350],[15,55515,55516,55519],{"style":53280},[15,55517],{"className":55518,"style":3861},[357],[15,55520,3106],{"className":55521,"style":3191},[61,122],[15,55523,55524,55527],{"style":55074},[15,55525],{"className":55526,"style":3861},[357],[15,55528,55530],{"className":55529,"style":55081},[53302],[15,55531,53230],{"className":55532},[61],[15,55534,55536],{"className":55535},[338],[15,55537,55539,55559],{"className":55538},[342,472],[15,55540,55542,55556],{"className":55541},[346],[15,55543,55545],{"className":55544,"style":479},[350],[15,55546,55547,55550],{"style":3206},[15,55548],{"className":55549,"style":358},[357],[15,55551,55553],{"className":55552},[362,363,364,365],[15,55554,272],{"className":55555,"style":492},[61,122,365],[15,55557,497],{"className":55558},[496],[15,55560,55562],{"className":55561},[346],[15,55563,55565],{"className":55564,"style":504},[350],[15,55566],{},[15,55568,750],{"className":55569},[178],[15,55571],{"className":55572,"style":772},[127],[15,55574,239],{"className":55575},[375],[15,55577],{"className":55578,"style":772},[127],[15,55580,55582,55585,55588,55591,55659,55662],{"className":55581},[52],[15,55583],{"className":55584,"style":45276},[56],[15,55586,22693],{"className":55587,"style":123},[61,122],[15,55589,741],{"className":55590},[146],[15,55592,55594,55625],{"className":55593},[61],[15,55595,55597],{"className":55596},[61,53268],[15,55598,55600],{"className":55599},[342],[15,55601,55603],{"className":55602},[346],[15,55604,55606,55614],{"className":55605,"style":55063},[350],[15,55607,55608,55611],{"style":53280},[15,55609],{"className":55610,"style":3861},[357],[15,55612,3106],{"className":55613,"style":3191},[61,122],[15,55615,55616,55619],{"style":55074},[15,55617],{"className":55618,"style":3861},[357],[15,55620,55622],{"className":55621,"style":55081},[53302],[15,55623,53230],{"className":55624},[61],[15,55626,55628],{"className":55627},[338],[15,55629,55631,55651],{"className":55630},[342,472],[15,55632,55634,55648],{"className":55633},[346],[15,55635,55637],{"className":55636,"style":1214},[350],[15,55638,55639,55642],{"style":3206},[15,55640],{"className":55641,"style":358},[357],[15,55643,55645],{"className":55644},[362,363,364,365],[15,55646,278],{"className":55647},[61,365],[15,55649,497],{"className":55650},[496],[15,55652,55654],{"className":55653},[346],[15,55655,55657],{"className":55656,"style":504},[350],[15,55658],{},[15,55660,750],{"className":55661},[178],[15,55663,55665],{"className":55664},[61],[15,55666,750],{"className":55667},[406,4067],[11,55669,55670,55672,55673,55940,55941,9348,56244,56671],{},[661,55671,5123],{}," 由指数族的一般 KL 公式：",[15,55674,55676,55726],{"className":55675,"translate":19},[18],[15,55677,55679],{"className":55678},[23],[25,55680,55681],{"xmlns":27},[29,55682,55683,55723],{},[32,55684,55685,55687,55689,55691,55697,55699,55701,55703,55709,55711,55713,55715,55721],{},[35,55686,53445],{"mathvariant":231},[81,55688,1547],{},[81,55690,244],{},[266,55692,55693,55695],{},[35,55694,14818],{"mathvariant":258},[35,55696,3106],{"mathvariant":3105},[81,55698,14821],{"stretchy":86},[35,55700,22662],{},[81,55702,1547],{},[266,55704,55705,55707],{},[35,55706,11],{},[35,55708,3106],{"mathvariant":3105},[81,55710,239],{},[35,55712,22662],{},[81,55714,1547],{},[266,55716,55717,55719],{},[35,55718,11],{},[35,55720,17131],{"mathvariant":3105},[81,55722,14830],{"stretchy":86},[40,55724,55725],{"encoding":42},"\\operatorname{KL} = \\mathbb{E}_{\\boldsymbol{\\alpha}}[\\log p_{\\boldsymbol{\\alpha}} - \\log p_{\\boldsymbol{\\beta}}]",[15,55727,55729,55750,55874],{"className":55728,"ariaHidden":48},[47],[15,55730,55732,55735,55741,55744,55747],{"className":55731},[52],[15,55733],{"className":55734,"style":57},[56],[15,55736,55738],{"className":55737},[531],[15,55739,53445],{"className":55740},[61,1622],[15,55742],{"className":55743,"style":128},[127],[15,55745,244],{"className":55746},[132],[15,55748],{"className":55749,"style":128},[127],[15,55751,55753,55756,55805,55808,55813,55816,55865,55868,55871],{"className":55752},[52],[15,55754],{"className":55755,"style":142},[56],[15,55757,55759,55762],{"className":55758},[61],[15,55760,14818],{"className":55761},[61,427],[15,55763,55765],{"className":55764},[338],[15,55766,55768,55797],{"className":55767},[342,472],[15,55769,55771,55794],{"className":55770},[346],[15,55772,55774],{"className":55773,"style":20966},[350],[15,55775,55776,55779],{"style":482},[15,55777],{"className":55778,"style":358},[357],[15,55780,55782],{"className":55781},[362,363,364,365],[15,55783,55785],{"className":55784},[61,365],[15,55786,55788],{"className":55787},[61,365],[15,55789,55791],{"className":55790},[61,365],[15,55792,3106],{"className":55793},[61,3166,365],[15,55795,497],{"className":55796},[496],[15,55798,55800],{"className":55799},[346],[15,55801,55803],{"className":55802,"style":504},[350],[15,55804],{},[15,55806,14821],{"className":55807},[146],[15,55809,22791,55811],{"className":55810},[531],[15,55812,22794],{"style":15619},[15,55814],{"className":55815,"style":157},[127],[15,55817,55819,55822],{"className":55818},[61],[15,55820,11],{"className":55821},[61,122],[15,55823,55825],{"className":55824},[338],[15,55826,55828,55857],{"className":55827},[342,472],[15,55829,55831,55854],{"className":55830},[346],[15,55832,55834],{"className":55833,"style":20966},[350],[15,55835,55836,55839],{"style":482},[15,55837],{"className":55838,"style":358},[357],[15,55840,55842],{"className":55841},[362,363,364,365],[15,55843,55845],{"className":55844},[61,365],[15,55846,55848],{"className":55847},[61,365],[15,55849,55851],{"className":55850},[61,365],[15,55852,3106],{"className":55853},[61,3166,365],[15,55855,497],{"className":55856},[496],[15,55858,55860],{"className":55859},[346],[15,55861,55863],{"className":55862,"style":504},[350],[15,55864],{},[15,55866],{"className":55867,"style":772},[127],[15,55869,239],{"className":55870},[375],[15,55872],{"className":55873,"style":772},[127],[15,55875,55877,55880,55885,55888,55937],{"className":55876},[52],[15,55878],{"className":55879,"style":17987},[56],[15,55881,22791,55883],{"className":55882},[531],[15,55884,22794],{"style":15619},[15,55886],{"className":55887,"style":157},[127],[15,55889,55891,55894],{"className":55890},[61],[15,55892,11],{"className":55893},[61,122],[15,55895,55897],{"className":55896},[338],[15,55898,55900,55929],{"className":55899},[342,472],[15,55901,55903,55926],{"className":55902},[346],[15,55904,55906],{"className":55905,"style":479},[350],[15,55907,55908,55911],{"style":482},[15,55909],{"className":55910,"style":358},[357],[15,55912,55914],{"className":55913},[362,363,364,365],[15,55915,55917],{"className":55916},[61,365],[15,55918,55920],{"className":55919},[61,365],[15,55921,55923],{"className":55922},[61,365],[15,55924,17131],{"className":55925,"style":17374},[61,3166,365],[15,55927,497],{"className":55928},[496],[15,55930,55932],{"className":55931},[346],[15,55933,55935],{"className":55934,"style":15707},[350],[15,55936],{},[15,55938,14830],{"className":55939},[178],"。代入 1.2 节的密度并利用 ",[15,55942,55944,56004],{"className":55943,"translate":19},[18],[15,55945,55947],{"className":55946},[23],[25,55948,55949],{"xmlns":27},[29,55950,55951,56001],{},[32,55952,55953,55959,55961,55963,55965,55971,55973,55975,55977,55979,55985,55987,55989,55991,55993,55999],{},[266,55954,55955,55957],{},[35,55956,14818],{"mathvariant":258},[35,55958,3106],{"mathvariant":3105},[81,55960,14821],{"stretchy":86},[35,55962,22662],{},[81,55964,1547],{},[266,55966,55967,55969],{},[35,55968,1029],{},[35,55970,272],{},[81,55972,14830],{"stretchy":86},[81,55974,244],{},[35,55976,22693],{},[81,55978,741],{"stretchy":86},[266,55980,55981,55983],{},[35,55982,3106],{},[35,55984,272],{},[81,55986,750],{"stretchy":86},[81,55988,239],{},[35,55990,22693],{},[81,55992,741],{"stretchy":86},[266,55994,55995,55997],{},[35,55996,3106],{},[89,55998,278],{},[81,56000,750],{"stretchy":86},[40,56002,56003],{"encoding":42},"\\mathbb{E}_{\\boldsymbol{\\alpha}}[\\log x_k] = \\psi(\\alpha_k) - \\psi(\\alpha_0)",[15,56005,56007,56125,56189],{"className":56006,"ariaHidden":48},[47],[15,56008,56010,56013,56062,56065,56070,56073,56113,56116,56119,56122],{"className":56009},[52],[15,56011],{"className":56012,"style":142},[56],[15,56014,56016,56019],{"className":56015},[61],[15,56017,14818],{"className":56018},[61,427],[15,56020,56022],{"className":56021},[338],[15,56023,56025,56054],{"className":56024},[342,472],[15,56026,56028,56051],{"className":56027},[346],[15,56029,56031],{"className":56030,"style":20966},[350],[15,56032,56033,56036],{"style":482},[15,56034],{"className":56035,"style":358},[357],[15,56037,56039],{"className":56038},[362,363,364,365],[15,56040,56042],{"className":56041},[61,365],[15,56043,56045],{"className":56044},[61,365],[15,56046,56048],{"className":56047},[61,365],[15,56049,3106],{"className":56050},[61,3166,365],[15,56052,497],{"className":56053},[496],[15,56055,56057],{"className":56056},[346],[15,56058,56060],{"className":56059,"style":504},[350],[15,56061],{},[15,56063,14821],{"className":56064},[146],[15,56066,22791,56068],{"className":56067},[531],[15,56069,22794],{"style":15619},[15,56071],{"className":56072,"style":157},[127],[15,56074,56076,56079],{"className":56075},[61],[15,56077,1029],{"className":56078},[61,122],[15,56080,56082],{"className":56081},[338],[15,56083,56085,56105],{"className":56084},[342,472],[15,56086,56088,56102],{"className":56087},[346],[15,56089,56091],{"className":56090,"style":479},[350],[15,56092,56093,56096],{"style":482},[15,56094],{"className":56095,"style":358},[357],[15,56097,56099],{"className":56098},[362,363,364,365],[15,56100,272],{"className":56101,"style":492},[61,122,365],[15,56103,497],{"className":56104},[496],[15,56106,56108],{"className":56107},[346],[15,56109,56111],{"className":56110,"style":504},[350],[15,56112],{},[15,56114,14830],{"className":56115},[178],[15,56117],{"className":56118,"style":128},[127],[15,56120,244],{"className":56121},[132],[15,56123],{"className":56124,"style":128},[127],[15,56126,56128,56131,56134,56137,56177,56180,56183,56186],{"className":56127},[52],[15,56129],{"className":56130,"style":142},[56],[15,56132,22693],{"className":56133,"style":123},[61,122],[15,56135,741],{"className":56136},[146],[15,56138,56140,56143],{"className":56139},[61],[15,56141,3106],{"className":56142,"style":3191},[61,122],[15,56144,56146],{"className":56145},[338],[15,56147,56149,56169],{"className":56148},[342,472],[15,56150,56152,56166],{"className":56151},[346],[15,56153,56155],{"className":56154,"style":479},[350],[15,56156,56157,56160],{"style":3206},[15,56158],{"className":56159,"style":358},[357],[15,56161,56163],{"className":56162},[362,363,364,365],[15,56164,272],{"className":56165,"style":492},[61,122,365],[15,56167,497],{"className":56168},[496],[15,56170,56172],{"className":56171},[346],[15,56173,56175],{"className":56174,"style":504},[350],[15,56176],{},[15,56178,750],{"className":56179},[178],[15,56181],{"className":56182,"style":772},[127],[15,56184,239],{"className":56185},[375],[15,56187],{"className":56188,"style":772},[127],[15,56190,56192,56195,56198,56201,56241],{"className":56191},[52],[15,56193],{"className":56194,"style":142},[56],[15,56196,22693],{"className":56197,"style":123},[61,122],[15,56199,741],{"className":56200},[146],[15,56202,56204,56207],{"className":56203},[61],[15,56205,3106],{"className":56206,"style":3191},[61,122],[15,56208,56210],{"className":56209},[338],[15,56211,56213,56233],{"className":56212},[342,472],[15,56214,56216,56230],{"className":56215},[346],[15,56217,56219],{"className":56218,"style":1214},[350],[15,56220,56221,56224],{"style":3206},[15,56222],{"className":56223,"style":358},[357],[15,56225,56227],{"className":56226},[362,363,364,365],[15,56228,278],{"className":56229},[61,365],[15,56231,497],{"className":56232},[496],[15,56234,56236],{"className":56235},[346],[15,56237,56239],{"className":56238,"style":504},[350],[15,56240],{},[15,56242,750],{"className":56243},[178],[15,56245,56247,56337],{"className":56246,"translate":19},[18],[15,56248,56250],{"className":56249},[23],[25,56251,56252],{"xmlns":27},[29,56253,56254,56334],{},[32,56255,56256,56262,56264,56266,56268,56270,56276,56278,56280,56282,56284,56290,56292,56294,56296,56298,56300,56306,56308,56310,56312,56314,56316,56318,56320,56322,56324,56326,56332],{},[266,56257,56258,56260],{},[81,56259,293],{},[35,56261,272],{},[35,56263,14818],{"mathvariant":258},[81,56265,14821],{"stretchy":86},[35,56267,22662],{},[81,56269,1547],{},[266,56271,56272,56274],{},[35,56273,1029],{},[35,56275,272],{},[81,56277,14830],{"stretchy":86},[81,56279,244],{},[35,56281,22693],{},[81,56283,741],{"stretchy":86},[266,56285,56286,56288],{},[35,56287,3106],{},[89,56289,278],{},[81,56291,750],{"stretchy":86},[81,56293,239],{},[35,56295,102],{},[35,56297,22693],{},[81,56299,741],{"stretchy":86},[266,56301,56302,56304],{},[35,56303,3106],{},[89,56305,278],{},[81,56307,750],{"stretchy":86},[81,56309,244],{},[81,56311,239],{},[81,56313,741],{"stretchy":86},[35,56315,102],{},[81,56317,239],{},[89,56319,91],{},[81,56321,750],{"stretchy":86},[35,56323,22693],{},[81,56325,741],{"stretchy":86},[266,56327,56328,56330],{},[35,56329,3106],{},[89,56331,278],{},[81,56333,750],{"stretchy":86},[40,56335,56336],{"encoding":42},"\\sum_k \\mathbb{E}[\\log x_k] = \\psi(\\alpha_0) - K\\psi(\\alpha_0) = -(K-1)\\psi(\\alpha_0)",[15,56338,56340,56455,56519,56586,56610],{"className":56339,"ariaHidden":48},[47],[15,56341,56343,56346,56386,56389,56392,56395,56400,56403,56443,56446,56449,56452],{"className":56342},[52],[15,56344],{"className":56345,"style":6051},[56],[15,56347,56349,56352],{"className":56348},[531],[15,56350,293],{"className":56351,"style":537},[531,535,536],[15,56353,56355],{"className":56354},[338],[15,56356,56358,56378],{"className":56357},[342,472],[15,56359,56361,56375],{"className":56360},[346],[15,56362,56364],{"className":56363,"style":29989},[350],[15,56365,56366,56369],{"style":553},[15,56367],{"className":56368,"style":358},[357],[15,56370,56372],{"className":56371},[362,363,364,365],[15,56373,272],{"className":56374,"style":492},[61,122,365],[15,56376,497],{"className":56377},[496],[15,56379,56381],{"className":56380},[346],[15,56382,56384],{"className":56383,"style":593},[350],[15,56385],{},[15,56387],{"className":56388,"style":157},[127],[15,56390,14818],{"className":56391},[61,427],[15,56393,14821],{"className":56394},[146],[15,56396,22791,56398],{"className":56397},[531],[15,56399,22794],{"style":15619},[15,56401],{"className":56402,"style":157},[127],[15,56404,56406,56409],{"className":56405},[61],[15,56407,1029],{"className":56408},[61,122],[15,56410,56412],{"className":56411},[338],[15,56413,56415,56435],{"className":56414},[342,472],[15,56416,56418,56432],{"className":56417},[346],[15,56419,56421],{"className":56420,"style":479},[350],[15,56422,56423,56426],{"style":482},[15,56424],{"className":56425,"style":358},[357],[15,56427,56429],{"className":56428},[362,363,364,365],[15,56430,272],{"className":56431,"style":492},[61,122,365],[15,56433,497],{"className":56434},[496],[15,56436,56438],{"className":56437},[346],[15,56439,56441],{"className":56440,"style":504},[350],[15,56442],{},[15,56444,14830],{"className":56445},[178],[15,56447],{"className":56448,"style":128},[127],[15,56450,244],{"className":56451},[132],[15,56453],{"className":56454,"style":128},[127],[15,56456,56458,56461,56464,56467,56507,56510,56513,56516],{"className":56457},[52],[15,56459],{"className":56460,"style":142},[56],[15,56462,22693],{"className":56463,"style":123},[61,122],[15,56465,741],{"className":56466},[146],[15,56468,56470,56473],{"className":56469},[61],[15,56471,3106],{"className":56472,"style":3191},[61,122],[15,56474,56476],{"className":56475},[338],[15,56477,56479,56499],{"className":56478},[342,472],[15,56480,56482,56496],{"className":56481},[346],[15,56483,56485],{"className":56484,"style":1214},[350],[15,56486,56487,56490],{"style":3206},[15,56488],{"className":56489,"style":358},[357],[15,56491,56493],{"className":56492},[362,363,364,365],[15,56494,278],{"className":56495},[61,365],[15,56497,497],{"className":56498},[496],[15,56500,56502],{"className":56501},[346],[15,56503,56505],{"className":56504,"style":504},[350],[15,56506],{},[15,56508,750],{"className":56509},[178],[15,56511],{"className":56512,"style":772},[127],[15,56514,239],{"className":56515},[375],[15,56517],{"className":56518,"style":772},[127],[15,56520,56522,56525,56528,56531,56534,56574,56577,56580,56583],{"className":56521},[52],[15,56523],{"className":56524,"style":142},[56],[15,56526,102],{"className":56527,"style":174},[61,122],[15,56529,22693],{"className":56530,"style":123},[61,122],[15,56532,741],{"className":56533},[146],[15,56535,56537,56540],{"className":56536},[61],[15,56538,3106],{"className":56539,"style":3191},[61,122],[15,56541,56543],{"className":56542},[338],[15,56544,56546,56566],{"className":56545},[342,472],[15,56547,56549,56563],{"className":56548},[346],[15,56550,56552],{"className":56551,"style":1214},[350],[15,56553,56554,56557],{"style":3206},[15,56555],{"className":56556,"style":358},[357],[15,56558,56560],{"className":56559},[362,363,364,365],[15,56561,278],{"className":56562},[61,365],[15,56564,497],{"className":56565},[496],[15,56567,56569],{"className":56568},[346],[15,56570,56572],{"className":56571,"style":504},[350],[15,56573],{},[15,56575,750],{"className":56576},[178],[15,56578],{"className":56579,"style":128},[127],[15,56581,244],{"className":56582},[132],[15,56584],{"className":56585,"style":128},[127],[15,56587,56589,56592,56595,56598,56601,56604,56607],{"className":56588},[52],[15,56590],{"className":56591,"style":142},[56],[15,56593,239],{"className":56594},[61],[15,56596,741],{"className":56597},[146],[15,56599,102],{"className":56600,"style":174},[61,122],[15,56602],{"className":56603,"style":772},[127],[15,56605,239],{"className":56606},[375],[15,56608],{"className":56609,"style":772},[127],[15,56611,56613,56616,56619,56622,56625,56628,56668],{"className":56612},[52],[15,56614],{"className":56615,"style":142},[56],[15,56617,91],{"className":56618},[61],[15,56620,750],{"className":56621},[178],[15,56623,22693],{"className":56624,"style":123},[61,122],[15,56626,741],{"className":56627},[146],[15,56629,56631,56634],{"className":56630},[61],[15,56632,3106],{"className":56633,"style":3191},[61,122],[15,56635,56637],{"className":56636},[338],[15,56638,56640,56660],{"className":56639},[342,472],[15,56641,56643,56657],{"className":56642},[346],[15,56644,56646],{"className":56645,"style":1214},[350],[15,56647,56648,56651],{"style":3206},[15,56649],{"className":56650,"style":358},[357],[15,56652,56654],{"className":56653},[362,363,364,365],[15,56655,278],{"className":56656},[61,365],[15,56658,497],{"className":56659},[496],[15,56661,56663],{"className":56662},[346],[15,56664,56666],{"className":56665,"style":504},[350],[15,56667],{},[15,56669,750],{"className":56670},[178]," 即可化简。",[11,56673,56674,56675,42890],{},"总损失为两项的加权和，其中权重按训练步",[661,56676,56677],{},"退火",[15,56679,56681],{"className":56680,"translate":19},[212],[15,56682,56684,56765],{"className":56683,"translate":19},[18],[15,56685,56687],{"className":56686},[23],[25,56688,56689],{"xmlns":27,"display":221},[29,56690,56691,56762],{},[32,56692,56693,56695,56697,56704,56706,56713,56715,56721,56723,56725,56731,56733,56736,56738,56740],{},[35,56694,47490],{"mathvariant":37},[81,56696,244],{},[227,56698,56699,56701],{},[35,56700,47490],{"mathvariant":37},[282,56702,56703],{},"fit",[81,56705,7762],{},[266,56707,56708,56711],{},[35,56709,56710],{},"λ",[35,56712,5297],{},[282,56714,4802],{},[227,56716,56717,56719],{},[35,56718,47490],{"mathvariant":37},[282,56720,53440],{},[81,56722,94],{"separator":48},[127,56724],{"width":9847},[266,56726,56727,56729],{},[35,56728,56710],{},[35,56730,5297],{},[81,56732,244],{},[35,56734,56735],{},"min",[81,56737,1547],{},[282,56739,6799],{},[32,56741,56742,56744,56746,56748,56760],{},[81,56743,741],{"fence":48},[89,56745,91],{},[81,56747,94],{"separator":48},[3722,56749,56750,56752],{},[35,56751,5297],{},[266,56753,56754,56757],{},[35,56755,56756],{},"T",[282,56758,56759],{},"anneal",[81,56761,750],{"fence":48},[40,56763,56764],{"encoding":42},"\\mathcal{L} = \\mathcal{L}^{\\text{fit}} + \\lambda_t \\,\\mathcal{L}^{\\text{reg}},\n\\qquad\n\\lambda_t = \\min\\!\\left(1, \\frac{t}{T_{\\text{anneal}}}\\right)",[15,56766,56768,56786,56837,56982],{"className":56767,"ariaHidden":48},[47],[15,56769,56771,56774,56777,56780,56783],{"className":56770},[52],[15,56772],{"className":56773,"style":57},[56],[15,56775,47490],{"className":56776},[61,62],[15,56778],{"className":56779,"style":128},[127],[15,56781,244],{"className":56782},[132],[15,56784],{"className":56785,"style":128},[127],[15,56787,56789,56793,56828,56831,56834],{"className":56788},[52],[15,56790],{"className":56791,"style":56792},[56],"height:0.9824em;vertical-align:-0.0833em;",[15,56794,56796,56799],{"className":56795},[61],[15,56797,47490],{"className":56798},[61,62],[15,56800,56802],{"className":56801},[338],[15,56803,56805],{"className":56804},[342],[15,56806,56808],{"className":56807},[346],[15,56809,56811],{"className":56810,"style":18860},[350],[15,56812,56813,56816],{"style":353},[15,56814],{"className":56815,"style":358},[357],[15,56817,56819],{"className":56818},[362,363,364,365],[15,56820,56822],{"className":56821},[61,365],[15,56823,56825],{"className":56824},[61,13548,365],[15,56826,56703],{"className":56827},[61,365],[15,56829],{"className":56830,"style":772},[127],[15,56832,7762],{"className":56833},[375],[15,56835],{"className":56836,"style":772},[127],[15,56838,56840,56844,56885,56888,56924,56927,56930,56933,56973,56976,56979],{"className":56839},[52],[15,56841],{"className":56842,"style":56843},[56],"height:0.9088em;vertical-align:-0.1944em;",[15,56845,56847,56850],{"className":56846},[61],[15,56848,56710],{"className":56849},[61,122],[15,56851,56853],{"className":56852},[338],[15,56854,56856,56877],{"className":56855},[342,472],[15,56857,56859,56874],{"className":56858},[346],[15,56860,56863],{"className":56861,"style":56862},[350],"height:0.2806em;",[15,56864,56865,56868],{"style":482},[15,56866],{"className":56867,"style":358},[357],[15,56869,56871],{"className":56870},[362,363,364,365],[15,56872,5297],{"className":56873},[61,122,365],[15,56875,497],{"className":56876},[496],[15,56878,56880],{"className":56879},[346],[15,56881,56883],{"className":56882,"style":504},[350],[15,56884],{},[15,56886],{"className":56887,"style":157},[127],[15,56889,56891,56894],{"className":56890},[61],[15,56892,47490],{"className":56893},[61,62],[15,56895,56897],{"className":56896},[338],[15,56898,56900],{"className":56899},[342],[15,56901,56903],{"className":56902},[346],[15,56904,56907],{"className":56905,"style":56906},[350],"height:0.7144em;",[15,56908,56909,56912],{"style":353},[15,56910],{"className":56911,"style":358},[357],[15,56913,56915],{"className":56914},[362,363,364,365],[15,56916,56918],{"className":56917},[61,365],[15,56919,56921],{"className":56920},[61,13548,365],[15,56922,53440],{"className":56923},[61,365],[15,56925,94],{"className":56926},[153],[15,56928],{"className":56929,"style":10035},[127],[15,56931],{"className":56932,"style":157},[127],[15,56934,56936,56939],{"className":56935},[61],[15,56937,56710],{"className":56938},[61,122],[15,56940,56942],{"className":56941},[338],[15,56943,56945,56965],{"className":56944},[342,472],[15,56946,56948,56962],{"className":56947},[346],[15,56949,56951],{"className":56950,"style":56862},[350],[15,56952,56953,56956],{"style":482},[15,56954],{"className":56955,"style":358},[357],[15,56957,56959],{"className":56958},[362,363,364,365],[15,56960,5297],{"className":56961},[61,122,365],[15,56963,497],{"className":56964},[496],[15,56966,56968],{"className":56967},[346],[15,56969,56971],{"className":56970,"style":504},[350],[15,56972],{},[15,56974],{"className":56975,"style":128},[127],[15,56977,244],{"className":56978},[132],[15,56980],{"className":56981,"style":128},[127],[15,56983,56985,56988,56991,56994,56997],{"className":56984},[52],[15,56986],{"className":56987,"style":6514},[56],[15,56989,56735],{"className":56990},[531],[15,56992],{"className":56993,"style":7105},[127],[15,56995],{"className":56996,"style":157},[127],[15,56998,57000,57006,57009,57012,57015,57121],{"className":56999},[161],[15,57001,57003],{"className":57002,"style":402},[146,401],[15,57004,741],{"className":57005},[406,364],[15,57007,91],{"className":57008},[61],[15,57010,94],{"className":57011},[153],[15,57013],{"className":57014,"style":157},[127],[15,57016,57018,57021,57118],{"className":57017},[61],[15,57019],{"className":57020},[146,3841],[15,57022,57024],{"className":57023},[3722],[15,57025,57027,57110],{"className":57026},[342,472],[15,57028,57030,57107],{"className":57029},[346],[15,57031,57034,57088,57096],{"className":57032,"style":57033},[350],"height:1.2921em;",[15,57035,57036,57039],{"style":11402},[15,57037],{"className":57038,"style":3861},[357],[15,57040,57042],{"className":57041},[61],[15,57043,57045,57048],{"className":57044},[61],[15,57046,56756],{"className":57047,"style":34367},[61,122],[15,57049,57051],{"className":57050},[338],[15,57052,57054,57080],{"className":57053},[342,472],[15,57055,57057,57077],{"className":57056},[346],[15,57058,57060],{"className":57059,"style":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",[15,57131,57133,57156],{"className":57132,"translate":19},[18],[15,57134,57136],{"className":57135},[23],[25,57137,57138],{"xmlns":27},[29,57139,57140,57153],{},[32,57141,57142,57148,57151],{},[266,57143,57144,57146],{},[35,57145,56710],{},[35,57147,5297],{},[81,57149,57150],{},"≈",[89,57152,278],{},[40,57154,57155],{"encoding":42},"\\lambda_t \\approx 0",[15,57157,57159,57214],{"className":57158,"ariaHidden":48},[47],[15,57160,57162,57165,57205,57208,57211],{"className":57161},[52],[15,57163],{"className":57164,"style":34418},[56],[15,57166,57168,57171],{"className":57167},[61],[15,57169,56710],{"className":57170},[61,122],[15,57172,57174],{"className":57173},[338],[15,57175,57177,57197],{"className":57176},[342,472],[15,57178,57180,57194],{"className":57179},[346],[15,57181,57183],{"className":57182,"style":56862},[350],[15,57184,57185,57188],{"style":482},[15,57186],{"className":57187,"style":358},[357],[15,57189,57191],{"className":57190},[362,363,364,365],[15,57192,5297],{"className":57193},[61,122,365],[15,57195,497],{"className":57196},[496],[15,57198,57200],{"className":57199},[346],[15,57201,57203],{"className":57202,"style":504},[350],[15,57204],{},[15,57206],{"className":57207,"style":128},[127],[15,57209,57150],{"className":57210},[132],[15,57212],{"className":57213,"style":128},[127],[15,57215,57217,57220],{"className":57216},[52],[15,57218],{"className":57219,"style":859},[56],[15,57221,278],{"className":57222},[61],"，网络可以自由地按拟合损失调整证据的量级，避免在证据尚未成形时被正则项过早地压向零。训练后期 ",[15,57225,57227,57249],{"className":57226,"translate":19},[18],[15,57228,57230],{"className":57229},[23],[25,57231,57232],{"xmlns":27},[29,57233,57234,57246],{},[32,57235,57236,57242,57244],{},[266,57237,57238,57240],{},[35,57239,56710],{},[35,57241,5297],{},[81,57243,20704],{},[89,57245,91],{},[40,57247,57248],{"encoding":42},"\\lambda_t \\to 1",[15,57250,57252,57307],{"className":57251,"ariaHidden":48},[47],[15,57253,57255,57258,57298,57301,57304],{"className":57254},[52],[15,57256],{"className":57257,"style":34418},[56],[15,57259,57261,57264],{"className":57260},[61],[15,57262,56710],{"className":57263},[61,122],[15,57265,57267],{"className":57266},[338],[15,57268,57270,57290],{"className":57269},[342,472],[15,57271,57273,57287],{"className":57272},[346],[15,57274,57276],{"className":57275,"style":56862},[350],[15,57277,57278,57281],{"style":482},[15,57279],{"className":57280,"style":358},[357],[15,57282,57284],{"className":57283},[362,363,364,365],[15,57285,5297],{"className":57286},[61,122,365],[15,57288,497],{"className":57289},[496],[15,57291,57293],{"className":57292},[346],[15,57294,57296],{"className":57295,"style":504},[350],[15,57297],{},[15,57299],{"className":57300,"style":128},[127],[15,57302,20704],{"className":57303},[132],[15,57305],{"className":57306,"style":128},[127],[15,57308,57310,57313],{"className":57309},[52],[15,57311],{"className":57312,"style":859},[56],[15,57314,91],{"className":57315},[61],"，正则项接管，迫使模型对错误预测给出高不确定性。消融实验一致表明，缺少退火（即使用常数 ",[15,57318,57320,57334],{"className":57319,"translate":19},[18],[15,57321,57323],{"className":57322},[23],[25,57324,57325],{"xmlns":27},[29,57326,57327,57331],{},[32,57328,57329],{},[35,57330,56710],{},[40,57332,57333],{"encoding":42},"\\lambda",[15,57335,57337],{"className":57336,"ariaHidden":48},[47],[15,57338,57340,57343],{"className":57339},[52],[15,57341],{"className":57342,"style":10452},[56],[15,57344,56710],{"className":57345},[61,122],"）会显著损害性能——这与该设计的解释相符，而非仅仅是超参数调优的产物。",[877,57348,57350],{"id":57349},"_44-统一视角与变体","4.4 统一视角与变体",[11,57352,57353],{},"上述两项可以统一在一个更一般的框架下。所有基于 EDL 的拟合损失都具有形式",[15,57355,57357],{"className":57356,"translate":19},[212],[15,57358,57360,57419],{"className":57359,"translate":19},[18],[15,57361,57363],{"className":57362},[23],[25,57364,57365],{"xmlns":27,"display":221},[29,57366,57367,57416],{},[32,57368,57369,57375,57377,57397],{},[227,57370,57371,57373],{},[35,57372,47490],{"mathvariant":37},[282,57374,56703],{},[81,57376,244],{},[266,57378,57379,57381],{},[35,57380,14818],{"mathvariant":258},[32,57382,57383,57385,57387,57389,57391,57393,57395],{},[35,57384,24680],{"mathvariant":3105},[81,57386,4183],{},[35,57388,4186],{"mathvariant":231},[81,57390,1547],{},[81,57392,741],{"stretchy":86},[35,57394,3106],{"mathvariant":3105},[81,57396,750],{"stretchy":86},[32,57398,57399,57401,57404,57406,57408,57410,57412,57414],{},[81,57400,14821],{"fence":48},[35,57402,57403],{"mathvariant":231},"ℓ",[81,57405,741],{"stretchy":86},[35,57407,79],{"mathvariant":251},[81,57409,94],{"separator":48},[35,57411,24680],{"mathvariant":3105},[81,57413,750],{"stretchy":86},[81,57415,14830],{"fence":48},[40,57417,57418],{"encoding":42},"\\mathcal{L}^{\\text{fit}} = \\mathbb{E}_{\\boldsymbol{\\pi} \\sim \\operatorname{Dir}(\\boldsymbol{\\alpha})}\\left[\\ell(\\mathbf{y}, \\boldsymbol{\\pi})\\right]",[15,57420,57422,57472],{"className":57421,"ariaHidden":48},[47],[15,57423,57425,57428,57463,57466,57469],{"className":57424},[52],[15,57426],{"className":57427,"style":18860},[56],[15,57429,57431,57434],{"className":57430},[61],[15,57432,47490],{"className":57433},[61,62],[15,57435,57437],{"className":57436},[338],[15,57438,57440],{"className":57439},[342],[15,57441,57443],{"className":57442},[346],[15,57444,57446],{"className":57445,"style":18860},[350],[15,57447,57448,57451],{"style":353},[15,57449],{"className":57450,"style":358},[357],[15,57452,57454],{"className":57453},[362,363,364,365],[15,57455,57457],{"className":57456},[61,365],[15,57458,57460],{"className":57459},[61,13548,365],[15,57461,56703],{"className":57462},[61,365],[15,57464],{"className":57465,"style":128},[127],[15,57467,244],{"className":57468},[132],[15,57470],{"className":57471,"style":128},[127],[15,57473,57475,57479,57552,57555],{"className":57474},[52],[15,57476],{"className":57477,"style":57478},[56],"height:1.1052em;vertical-align:-0.3552em;",[15,57480,57482,57485],{"className":57481},[61],[15,57483,14818],{"className":57484},[61,427],[15,57486,57488],{"className":57487},[338],[15,57489,57491,57544],{"className":57490},[342,472],[15,57492,57494,57541],{"className":57493},[346],[15,57495,57497],{"className":57496,"style":22404},[350],[15,57498,57499,57502],{"style":47615},[15,57500],{"className":57501,"style":358},[357],[15,57503,57505],{"className":57504},[362,363,364,365],[15,57506,57508,57517,57520,57526,57529,57538],{"className":57507},[61,365],[15,57509,57511],{"className":57510},[61,365],[15,57512,57514],{"className":57513},[61,365],[15,57515,24680],{"className":57516,"style":24714},[61,3166,365],[15,57518,4183],{"className":57519},[132,365],[15,57521,57523],{"className":57522},[531,365],[15,57524,4186],{"className":57525},[61,1622,365],[15,57527,741],{"className":57528},[146,365],[15,57530,57532],{"className":57531},[61,365],[15,57533,57535],{"className":57534},[61,365],[15,57536,3106],{"className":57537},[61,3166,365],[15,57539,750],{"className":57540},[178,365],[15,57542,497],{"className":57543},[496],[15,57545,57547],{"className":57546},[346],[15,57548,57550],{"className":57549,"style":47667},[350],[15,57551],{},[15,57553],{"className":57554,"style":157},[127],[15,57556,57558,57561,57564,57567,57570,57573,57576,57585,57588],{"className":57557},[161],[15,57559,14821],{"className":57560,"style":402},[146,401],[15,57562,57403],{"className":57563},[61],[15,57565,741],{"className":57566},[146],[15,57568,79],{"className":57569,"style":29413},[61,411],[15,57571,94],{"className":57572},[153],[15,57574],{"className":57575,"style":157},[127],[15,57577,57579],{"className":57578},[61],[15,57580,57582],{"className":57581},[61],[15,57583,24680],{"className":57584,"style":24714},[61,3166],[15,57586,750],{"className":57587},[178],[15,57589,14830],{"className":57590,"style":402},[178,401],[11,57592,12926,57593,57626,57627,57739,57740,57859],{},[661,57594,57595,57596,57625],{},"对某个一阶损失 ",[15,57597,57599,57613],{"className":57598,"translate":19},[18],[15,57600,57602],{"className":57601},[23],[25,57603,57604],{"xmlns":27},[29,57605,57606,57610],{},[32,57607,57608],{},[35,57609,57403],{"mathvariant":231},[40,57611,57612],{"encoding":42},"\\ell",[15,57614,57616],{"className":57615,"ariaHidden":48},[47],[15,57617,57619,57622],{"className":57618},[52],[15,57620],{"className":57621,"style":10452},[56],[15,57623,57403],{"className":57624},[61]," 的先验期望","。取 ",[15,57628,57630,57658],{"className":57629,"translate":19},[18],[15,57631,57633],{"className":57632},[23],[25,57634,57635],{"xmlns":27},[29,57636,57637,57655],{},[32,57638,57639,57641,57643,57645,57647,57649],{},[35,57640,57403],{"mathvariant":231},[81,57642,244],{},[81,57644,239],{},[35,57646,22662],{},[81,57648,1547],{},[266,57650,57651,57653],{},[35,57652,24680],{},[35,57654,79],{},[40,57656,57657],{"encoding":42},"\\ell = -\\log \\pi_y",[15,57659,57661,57679],{"className":57660,"ariaHidden":48},[47],[15,57662,57664,57667,57670,57673,57676],{"className":57663},[52],[15,57665],{"className":57666,"style":10452},[56],[15,57668,57403],{"className":57669},[61],[15,57671],{"className":57672,"style":128},[127],[15,57674,244],{"className":57675},[132],[15,57677],{"className":57678,"style":128},[127],[15,57680,57682,57685,57688,57691,57696,57699],{"className":57681},[52],[15,57683],{"className":57684,"style":20946},[56],[15,57686,239],{"className":57687},[61],[15,57689],{"className":57690,"style":157},[127],[15,57692,22791,57694],{"className":57693},[531],[15,57695,22794],{"style":15619},[15,57697],{"className":57698,"style":157},[127],[15,57700,57702,57705],{"className":57701},[61],[15,57703,24680],{"className":57704,"style":123},[61,122],[15,57706,57708],{"className":57707},[338],[15,57709,57711,57731],{"className":57710},[342,472],[15,57712,57714,57728],{"className":57713},[346],[15,57715,57717],{"className":57716,"style":27033},[350],[15,57718,57719,57722],{"style":27036},[15,57720],{"className":57721,"style":358},[357],[15,57723,57725],{"className":57724},[362,363,364,365],[15,57726,79],{"className":57727,"style":123},[61,122,365],[15,57729,497],{"className":57730},[496],[15,57732,57734],{"className":57733},[346],[15,57735,57737],{"className":57736,"style":15707},[350],[15,57738],{}," 得命题 4.1 的 digamma 形式。取 ",[15,57741,57743,57773],{"className":57742,"translate":19},[18],[15,57744,57746],{"className":57745},[23],[25,57747,57748],{"xmlns":27},[29,57749,57750,57770],{},[32,57751,57752,57754,57756,57758,57760,57762,57764],{},[35,57753,57403],{"mathvariant":231},[81,57755,244],{},[35,57757,21291],{"mathvariant":231},[35,57759,79],{"mathvariant":251},[81,57761,239],{},[35,57763,24680],{"mathvariant":3105},[227,57765,57766,57768],{},[35,57767,21291],{"mathvariant":231},[89,57769,908],{},[40,57771,57772],{"encoding":42},"\\ell = \\|\\mathbf{y} - \\boldsymbol{\\pi}\\|^2",[15,57774,57776,57794,57815],{"className":57775,"ariaHidden":48},[47],[15,57777,57779,57782,57785,57788,57791],{"className":57778},[52],[15,57780],{"className":57781,"style":10452},[56],[15,57783,57403],{"className":57784},[61],[15,57786],{"className":57787,"style":128},[127],[15,57789,244],{"className":57790},[132],[15,57792],{"className":57793,"style":128},[127],[15,57795,57797,57800,57803,57806,57809,57812],{"className":57796},[52],[15,57798],{"className":57799,"style":142},[56],[15,57801,21291],{"className":57802},[61],[15,57804,79],{"className":57805,"style":29413},[61,411],[15,57807],{"className":57808,"style":772},[127],[15,57810,239],{"className":57811},[375],[15,57813],{"className":57814,"style":772},[127],[15,57816,57818,57821,57830],{"className":57817},[52],[15,57819],{"className":57820,"style":21415},[56],[15,57822,57824],{"className":57823},[61],[15,57825,57827],{"className":57826},[61],[15,57828,24680],{"className":57829,"style":24714},[61,3166],[15,57831,57833,57836],{"className":57832},[61],[15,57834,21291],{"className":57835},[61],[15,57837,57839],{"className":57838},[338],[15,57840,57842],{"className":57841},[342],[15,57843,57845],{"className":57844},[346],[15,57846,57848],{"className":57847,"style":7905},[350],[15,57849,57850,57853],{"style":716},[15,57851],{"className":57852,"style":358},[357],[15,57854,57856],{"className":57855},[362,363,364,365],[15,57857,908],{"className":57858},[61,365]," 得命题 4.3 的平方形式。这一视角把「EDL 的损失设计」转化为「选择哪个一阶损失并在迪利克雷族上求其贝叶斯风险」的标准问题，从而使新损失的设计有章可循。",[11,57861,57862],{},"在此基础上，文献中出现的主要变体可归入三类：",[31547,57864,57865,57971,57981],{},[31550,57866,57867,57870,57871,57941,57942,57970],{},[661,57868,57869],{},"正则方式不同","。除 4.3 节的 KL 正则外，也有工作直接以「认知不确定性应随证据增加而衰减」作为约束，或对证据施加 ",[15,57872,57874,57892],{"className":57873,"translate":19},[18],[15,57875,57877],{"className":57876},[23],[25,57878,57879],{"xmlns":27},[29,57880,57881,57889],{},[32,57882,57883],{},[266,57884,57885,57887],{},[35,57886,47490],{},[89,57888,908],{},[40,57890,57891],{"encoding":42},"L_2",[15,57893,57895],{"className":57894,"ariaHidden":48},[47],[15,57896,57898,57901],{"className":57897},[52],[15,57899],{"className":57900,"style":34360},[56],[15,57902,57904,57907],{"className":57903},[61],[15,57905,47490],{"className":57906},[61,122],[15,57908,57910],{"className":57909},[338],[15,57911,57913,57933],{"className":57912},[342,472],[15,57914,57916,57930],{"className":57915},[346],[15,57917,57919],{"className":57918,"style":1214},[350],[15,57920,57921,57924],{"style":482},[15,57922],{"className":57923,"style":358},[357],[15,57925,57927],{"className":57926},[362,363,364,365],[15,57928,908],{"className":57929},[61,365],[15,57931,497],{"className":57932},[496],[15,57934,57936],{"className":57935},[346],[15,57937,57939],{"className":57938,"style":504},[350],[15,57940],{}," 惩罚。其共同目标都是阻止 ",[15,57943,57945,57958],{"className":57944,"translate":19},[18],[15,57946,57948],{"className":57947},[23],[25,57949,57950],{"xmlns":27},[29,57951,57952,57956],{},[32,57953,57954],{},[35,57955,40265],{},[40,57957,40265],{"encoding":42},[15,57959,57961],{"className":57960,"ariaHidden":48},[47],[15,57962,57964,57967],{"className":57963},[52],[15,57965],{"className":57966,"style":57},[56],[15,57968,40265],{"className":57969,"style":40337},[61,122]," 的退化。",[31550,57972,57973,57976,57977,57980],{},[661,57974,57975],{},"监督方式不同","。Prior Networks 不在期望风险上训练，而是显式地构造一个",[661,57978,57979],{},"目标迪利克雷分布","（例如：分布内样本用尖锐的 Dirichlet 匹配、分布外样本用平坦的 Dirichlet 匹配），并在分布层面直接计算 KL。这需要分布外样本来定义目标，因此换取的是更强的不确定性控制与更弱的即插即用性。",[31550,57982,57983,57986],{},[661,57984,57985],{},"回归情形的推广","。把分类的迪利克雷替换为 Normal-Inverse-Gamma 先验，即得 Deep Evidential Regression（5.5 节）。",[11,57988,57989,57990,57993,57994,57997],{},"统一框架也澄清了一个理论定位问题：EDL 与贝叶斯神经网络（BNN）、深度集成（deep ensembles）、MC Dropout 属于",[661,57991,57992],{},"不同的","不确定性来源。后三者通过对网络参数（或其近似后验）的分布求积分得到预测分布。EDL 则在固定的参数点估计上，对",[661,57995,57996],{},"输出层的二阶分布","求积分。二者的区分不是工程实现上的，而是概率模型结构上的，这正是下一小节批评的出发点。",[877,57999,58001],{"id":58000},"_45-使用边界","4.5 使用边界",[11,58003,58004],{},"EDL 在实践中表现良好，但其理论地位常被过度解读。把边界集中列出，与前文的正式推导互为对照：",[31547,58006,58007,58918,59048,59314,59375],{},[31550,58008,58009,58012,58013,58323,58324,58394,58395,867,58610,58613,58614,58797,58798,9348,58826,58888,58889,58917],{},[661,58010,58011],{},"它不是贝叶斯不确定性。"," 严格的后验预测分布应由对网络参数的边缘化给出：\n",[15,58014,58016,58102],{"className":58015,"translate":19},[18],[15,58017,58019],{"className":58018},[23],[25,58020,58021],{"xmlns":27},[29,58022,58023,58099],{},[32,58024,58025,58027,58029,58031,58033,58035,58037,58040,58042,58044,58047,58049,58051,58053,58055,58057,58059,58061,58063,58065,58067,58069,58071,58073,58075,58077,58079,58081,58083,58085,58087,58089,58091,58093,58095,58097],{},[35,58026,11],{},[81,58028,741],{"stretchy":86},[35,58030,79],{},[81,58032,3714],{},[35,58034,1029],{"mathvariant":251},[81,58036,94],{"separator":48},[35,58038,58039],{"mathvariant":37},"D",[81,58041,750],{"stretchy":86},[81,58043,244],{},[81,58045,58046],{},"∬",[35,58048,11],{},[81,58050,741],{"stretchy":86},[35,58052,79],{},[81,58054,3714],{},[35,58056,24680],{"mathvariant":3105},[81,58058,750],{"stretchy":86},[282,58060,4802],{},[35,58062,11],{},[81,58064,741],{"stretchy":86},[35,58066,24680],{"mathvariant":3105},[81,58068,3714],{},[35,58070,37148],{"mathvariant":3105},[81,58072,750],{"stretchy":86},[282,58074,4802],{},[35,58076,11],{},[81,58078,741],{"stretchy":86},[35,58080,37148],{"mathvariant":3105},[81,58082,3714],{},[35,58084,58039],{"mathvariant":37},[81,58086,750],{"stretchy":86},[282,58088,4802],{},[35,58090,2788],{"mathvariant":231},[35,58092,24680],{"mathvariant":3105},[282,58094,4802],{},[35,58096,2788],{"mathvariant":231},[35,58098,37148],{"mathvariant":3105},[40,58100,58101],{"encoding":42},"p(y \\mid \\mathbf{x}, \\mathcal{D}) = \\iint p(y \\mid \\boldsymbol{\\pi})\\,p(\\boldsymbol{\\pi} \\mid \\boldsymbol{\\theta})\\,p(\\boldsymbol{\\theta} \\mid \\mathcal{D})\\,\\mathrm{d}\\boldsymbol{\\pi}\\,\\mathrm{d}\\boldsymbol{\\theta}",[15,58103,58105,58129,58159,58191,58236,58281],{"className":58104,"ariaHidden":48},[47],[15,58106,58108,58111,58114,58117,58120,58123,58126],{"className":58107},[52],[15,58109],{"className":58110,"style":142},[56],[15,58112,11],{"className":58113},[61,122],[15,58115,741],{"className":58116},[146],[15,58118,79],{"className":58119,"style":123},[61,122],[15,58121],{"className":58122,"style":128},[127],[15,58124,3714],{"className":58125},[132],[15,58127],{"className":58128,"style":128},[127],[15,58130,58132,58135,58138,58141,58144,58147,58150,58153,58156],{"className":58131},[52],[15,58133],{"className":58134,"style":142},[56],[15,58136,1029],{"className":58137},[61,411],[15,58139,94],{"className":58140},[153],[15,58142],{"className":58143,"style":157},[127],[15,58145,58039],{"className":58146,"style":12287},[61,62],[15,58148,750],{"className":58149},[178],[15,58151],{"className":58152,"style":128},[127],[15,58154,244],{"className":58155},[132],[15,58157],{"className":58158,"style":128},[127],[15,58160,58162,58166,58170,58173,58176,58179,58182,58185,58188],{"className":58161},[52],[15,58163],{"className":58164,"style":58165},[56],"height:1.111em;vertical-align:-0.306em;",[15,58167,58046],{"className":58168,"style":58169},[531,535,536],"margin-right:0.1945em;position:relative;top:-0.0005em;",[15,58171],{"className":58172,"style":157},[127],[15,58174,11],{"className":58175},[61,122],[15,58177,741],{"className":58178},[146],[15,58180,79],{"className":58181,"style":123},[61,122],[15,58183],{"className":58184,"style":128},[127],[15,58186,3714],{"className":58187},[132],[15,58189],{"className":58190,"style":128},[127],[15,58192,58194,58197,58206,58209,58212,58215,58218,58227,58230,58233],{"className":58193},[52],[15,58195],{"className":58196,"style":142},[56],[15,58198,58200],{"className":58199},[61],[15,58201,58203],{"className":58202},[61],[15,58204,24680],{"className":58205,"style":24714},[61,3166],[15,58207,750],{"className":58208},[178],[15,58210],{"className":58211,"style":157},[127],[15,58213,11],{"className":58214},[61,122],[15,58216,741],{"className":58217},[146],[15,58219,58221],{"className":58220},[61],[15,58222,58224],{"className":58223},[61],[15,58225,24680],{"className":58226,"style":24714},[61,3166],[15,58228],{"className":58229,"style":128},[127],[15,58231,3714],{"className":58232},[132],[15,58234],{"className":58235,"style":128},[127],[15,58237,58239,58242,58251,58254,58257,58260,58263,58272,58275,58278],{"className":58238},[52],[15,58240],{"className":58241,"style":142},[56],[15,58243,58245],{"className":58244},[61],[15,58246,58248],{"className":58247},[61],[15,58249,37148],{"className":58250,"style":37212},[61,3166],[15,58252,750],{"className":58253},[178],[15,58255],{"className":58256,"style":157},[127],[15,58258,11],{"className":58259},[61,122],[15,58261,741],{"className":58262},[146],[15,58264,58266],{"className":58265},[61],[15,58267,58269],{"className":58268},[61],[15,58270,37148],{"className":58271,"style":37212},[61,3166],[15,58273],{"className":58274,"style":128},[127],[15,58276,3714],{"className":58277},[132],[15,58279],{"className":58280,"style":128},[127],[15,58282,58284,58287,58290,58293,58296,58299,58308,58311,58314],{"className":58283},[52],[15,58285],{"className":58286,"style":142},[56],[15,58288,58039],{"className":58289,"style":12287},[61,62],[15,58291,750],{"className":58292},[178],[15,58294],{"className":58295,"style":157},[127],[15,58297,2788],{"className":58298},[61,1622],[15,58300,58302],{"className":58301},[61],[15,58303,58305],{"className":58304},[61],[15,58306,24680],{"className":58307,"style":24714},[61,3166],[15,58309],{"className":58310,"style":157},[127],[15,58312,2788],{"className":58313},[61,1622],[15,58315,58317],{"className":58316},[61],[15,58318,58320],{"className":58319},[61],[15,58321,37148],{"className":58322,"style":37212},[61,3166],"\n而 EDL 以点估计 ",[15,58325,58327,58346],{"className":58326,"translate":19},[18],[15,58328,58330],{"className":58329},[23],[25,58331,58332],{"xmlns":27},[29,58333,58334,58343],{},[32,58335,58336],{},[53224,58337,58338,58340],{"accent":48},[35,58339,37148],{"mathvariant":3105},[81,58341,58342],{},"^",[40,58344,58345],{"encoding":42},"\\hat{\\boldsymbol{\\theta}}",[15,58347,58349],{"className":58348,"ariaHidden":48},[47],[15,58350,58352,58356],{"className":58351},[52],[15,58353],{"className":58354,"style":58355},[56],"height:0.9579em;",[15,58357,58359],{"className":58358},[61,53268],[15,58360,58362],{"className":58361},[342],[15,58363,58365],{"className":58364},[346],[15,58366,58368,58382],{"className":58367,"style":58355},[350],[15,58369,58370,58373],{"style":53280},[15,58371],{"className":58372,"style":3861},[357],[15,58374,58376],{"className":58375},[61],[15,58377,58379],{"className":58378},[61],[15,58380,37148],{"className":58381,"style":37212},[61,3166],[15,58383,58385,58388],{"style":58384},"top:-3.2634em;",[15,58386],{"className":58387,"style":3861},[357],[15,58389,58391],{"className":58390,"style":53303},[53302],[15,58392,58342],{"className":58393},[61]," 替换外层积分，取 ",[15,58396,58398,58452],{"className":58397,"translate":19},[18],[15,58399,58401],{"className":58400},[23],[25,58402,58403],{"xmlns":27},[29,58404,58405,58449],{},[32,58406,58407,58409,58411,58413,58415,58417,58419,58421,58423,58425,58427,58429,58431,58433,58435,58437,58439,58445,58447],{},[35,58408,11],{},[81,58410,741],{"stretchy":86},[35,58412,24680],{"mathvariant":3105},[81,58414,3714],{},[35,58416,1029],{"mathvariant":251},[81,58418,750],{"stretchy":86},[81,58420,244],{},[35,58422,4186],{"mathvariant":231},[81,58424,1547],{},[81,58426,741],{"fence":86,"stretchy":48,"minsize":44849,"maxsize":44849},[35,58428,24680],{"mathvariant":3105},[81,58430,11303],{"separator":48},[35,58432,3106],{"mathvariant":3105},[81,58434,741],{"stretchy":86},[35,58436,1029],{"mathvariant":251},[81,58438,11303],{"separator":48},[53224,58440,58441,58443],{"accent":48},[35,58442,37148],{"mathvariant":3105},[81,58444,58342],{},[81,58446,750],{"stretchy":86},[81,58448,750],{"fence":86,"stretchy":48,"minsize":44849,"maxsize":44849},[40,58450,58451],{"encoding":42},"p(\\boldsymbol{\\pi} \\mid \\mathbf{x}) = \\operatorname{Dir}\\big(\\boldsymbol{\\pi}; \\boldsymbol{\\alpha}(\\mathbf{x}; \\hat{\\boldsymbol{\\theta}})\\big)",[15,58453,58455,58485,58506],{"className":58454,"ariaHidden":48},[47],[15,58456,58458,58461,58464,58467,58476,58479,58482],{"className":58457},[52],[15,58459],{"className":58460,"style":142},[56],[15,58462,11],{"className":58463},[61,122],[15,58465,741],{"className":58466},[146],[15,58468,58470],{"className":58469},[61],[15,58471,58473],{"className":58472},[61],[15,58474,24680],{"className":58475,"style":24714},[61,3166],[15,58477],{"className":58478,"style":128},[127],[15,58480,3714],{"className":58481},[132],[15,58483],{"className":58484,"style":128},[127],[15,58486,58488,58491,58494,58497,58500,58503],{"className":58487},[52],[15,58489],{"className":58490,"style":142},[56],[15,58492,1029],{"className":58493},[61,411],[15,58495,750],{"className":58496},[178],[15,58498],{"className":58499,"style":128},[127],[15,58501,244],{"className":58502},[132],[15,58504],{"className":58505,"style":128},[127],[15,58507,58509,58513,58519,58522,58528,58537,58540,58543,58552,58555,58558,58561,58564,58601,58604],{"className":58508},[52],[15,58510],{"className":58511,"style":58512},[56],"height:1.3079em;vertical-align:-0.35em;",[15,58514,58516],{"className":58515},[531],[15,58517,4186],{"className":58518},[61,1622],[15,58520],{"className":58521,"style":157},[127],[15,58523,58525],{"className":58524},[61],[15,58526,741],{"className":58527},[406,4067],[15,58529,58531],{"className":58530},[61],[15,58532,58534],{"className":58533},[61],[15,58535,24680],{"className":58536,"style":24714},[61,3166],[15,58538,11303],{"className":58539},[153],[15,58541],{"className":58542,"style":157},[127],[15,58544,58546],{"className":58545},[61],[15,58547,58549],{"className":58548},[61],[15,58550,3106],{"className":58551},[61,3166],[15,58553,741],{"className":58554},[146],[15,58556,1029],{"className":58557},[61,411],[15,58559,11303],{"className":58560},[153],[15,58562],{"className":58563,"style":157},[127],[15,58565,58567],{"className":58566},[61,53268],[15,58568,58570],{"className":58569},[342],[15,58571,58573],{"className":58572},[346],[15,58574,58576,58590],{"className":58575,"style":58355},[350],[15,58577,58578,58581],{"style":53280},[15,58579],{"className":58580,"style":3861},[357],[15,58582,58584],{"className":58583},[61],[15,58585,58587],{"className":58586},[61],[15,58588,37148],{"className":58589,"style":37212},[61,3166],[15,58591,58592,58595],{"style":58384},[15,58593],{"className":58594,"style":3861},[357],[15,58596,58598],{"className":58597,"style":53303},[53302],[15,58599,58342],{"className":58600},[61],[15,58602,750],{"className":58603},[178],[15,58605,58607],{"className":58606},[61],[15,58608,750],{"className":58609},[406,4067],[661,58611,58612],{},"结果是一个插值（plug-in）估计","，其内层积分给出 ",[15,58615,58617,58650],{"className":58616,"translate":19},[18],[15,58618,58620],{"className":58619},[23],[25,58621,58622],{"xmlns":27},[29,58623,58624,58648],{},[32,58625,58626,58632,58634,58640,58642],{},[266,58627,58628,58630],{},[35,58629,11],{},[35,58631,272],{},[81,58633,244],{},[266,58635,58636,58638],{},[35,58637,3106],{},[35,58639,272],{},[35,58641,3020],{"mathvariant":231},[266,58643,58644,58646],{},[35,58645,3106],{},[89,58647,278],{},[40,58649,43835],{"encoding":42},[15,58651,58653,58708],{"className":58652,"ariaHidden":48},[47],[15,58654,58656,58659,58699,58702,58705],{"className":58655},[52],[15,58657],{"className":58658,"style":9437},[56],[15,58660,58662,58665],{"className":58661},[61],[15,58663,11],{"className":58664},[61,122],[15,58666,58668],{"className":58667},[338],[15,58669,58671,58691],{"className":58670},[342,472],[15,58672,58674,58688],{"className":58673},[346],[15,58675,58677],{"className":58676,"style":479},[350],[15,58678,58679,58682],{"style":482},[15,58680],{"className":58681,"style":358},[357],[15,58683,58685],{"className":58684},[362,363,364,365],[15,58686,272],{"className":58687,"style":492},[61,122,365],[15,58689,497],{"className":58690},[496],[15,58692,58694],{"className":58693},[346],[15,58695,58697],{"className":58696,"style":504},[350],[15,58698],{},[15,58700],{"className":58701,"style":128},[127],[15,58703,244],{"className":58704},[132],[15,58706],{"className":58707,"style":128},[127],[15,58709,58711,58714,58754,58757],{"className":58710},[52],[15,58712],{"className":58713,"style":142},[56],[15,58715,58717,58720],{"className":58716},[61],[15,58718,3106],{"className":58719,"style":3191},[61,122],[15,58721,58723],{"className":58722},[338],[15,58724,58726,58746],{"className":58725},[342,472],[15,58727,58729,58743],{"className":58728},[346],[15,58730,58732],{"className":58731,"style":479},[350],[15,58733,58734,58737],{"style":3206},[15,58735],{"className":58736,"style":358},[357],[15,58738,58740],{"className":58739},[362,363,364,365],[15,58741,272],{"className":58742,"style":492},[61,122,365],[15,58744,497],{"className":58745},[496],[15,58747,58749],{"className":58748},[346],[15,58750,58752],{"className":58751,"style":504},[350],[15,58753],{},[15,58755,3020],{"className":58756},[61],[15,58758,58760,58763],{"className":58759},[61],[15,58761,3106],{"className":58762,"style":3191},[61,122],[15,58764,58766],{"className":58765},[338],[15,58767,58769,58789],{"className":58768},[342,472],[15,58770,58772,58786],{"className":58771},[346],[15,58773,58775],{"className":58774,"style":1214},[350],[15,58776,58777,58780],{"style":3206},[15,58778],{"className":58779,"style":358},[357],[15,58781,58783],{"className":58782},[362,363,364,365],[15,58784,278],{"className":58785},[61,365],[15,58787,497],{"className":58788},[496],[15,58790,58792],{"className":58791},[346],[15,58793,58795],{"className":58794,"style":504},[350],[15,58796],{},"（由聚合性），确实正确。但外层积分被抛弃，",[15,58799,58801,58814],{"className":58800,"translate":19},[18],[15,58802,58804],{"className":58803},[23],[25,58805,58806],{"xmlns":27},[29,58807,58808,58812],{},[32,58809,58810],{},[35,58811,33430],{},[40,58813,33430],{"encoding":42},[15,58815,58817],{"className":58816,"ariaHidden":48},[47],[15,58818,58820,58823],{"className":58819},[52],[15,58821],{"className":58822,"style":6032},[56],[15,58824,33430],{"className":58825},[61,122],[15,58827,58829,58852],{"className":58828,"translate":19},[18],[15,58830,58832],{"className":58831},[23],[25,58833,58834],{"xmlns":27},[29,58835,58836,58850],{},[32,58837,58838,58840,58842,58844,58846,58848],{},[35,58839,28923],{},[81,58841,741],{"stretchy":86},[35,58843,79],{},[81,58845,11303],{"separator":48},[35,58847,24680],{"mathvariant":3105},[81,58849,750],{"stretchy":86},[40,58851,46884],{"encoding":42},[15,58853,58855],{"className":58854,"ariaHidden":48},[47],[15,58856,58858,58861,58864,58867,58870,58873,58876,58885],{"className":58857},[52],[15,58859],{"className":58860,"style":142},[56],[15,58862,28923],{"className":58863,"style":28962},[61,122],[15,58865,741],{"className":58866},[146],[15,58868,79],{"className":58869,"style":123},[61,122],[15,58871,11303],{"className":58872},[153],[15,58874],{"className":58875,"style":157},[127],[15,58877,58879],{"className":58878},[61],[15,58880,58882],{"className":58881},[61],[15,58883,24680],{"className":58884,"style":24714},[61,3166],[15,58886,750],{"className":58887},[178]," 所度量的是「对输出分布的把握」，而非「对模型的把握」。二者在分布外输入上可以严重分歧：一个被错误训练的网络可以以极低的 ",[15,58890,58892,58905],{"className":58891,"translate":19},[18],[15,58893,58895],{"className":58894},[23],[25,58896,58897],{"xmlns":27},[29,58898,58899,58903],{},[32,58900,58901],{},[35,58902,33430],{},[40,58904,33430],{"encoding":42},[15,58906,58908],{"className":58907,"ariaHidden":48},[47],[15,58909,58911,58914],{"className":58910},[52],[15,58912],{"className":58913,"style":6032},[56],[15,58915,33430],{"className":58916},[61,122]," 输出一个错误的类别。",[31550,58919,58920,58923,58924,58984,58985,59047],{},[661,58921,58922],{},"不确定性分数丢弃了证据的分布信息。"," 如 3.5 节所述，",[15,58925,58927,58948],{"className":58926,"translate":19},[18],[15,58928,58930],{"className":58929},[23],[25,58931,58932],{"xmlns":27},[29,58933,58934,58946],{},[32,58935,58936,58938,58940,58942,58944],{},[35,58937,33430],{},[81,58939,244],{},[35,58941,102],{},[35,58943,3020],{"mathvariant":231},[35,58945,40265],{},[40,58947,42984],{"encoding":42},[15,58949,58951,58969],{"className":58950,"ariaHidden":48},[47],[15,58952,58954,58957,58960,58963,58966],{"className":58953},[52],[15,58955],{"className":58956,"style":6032},[56],[15,58958,33430],{"className":58959},[61,122],[15,58961],{"className":58962,"style":128},[127],[15,58964,244],{"className":58965},[132],[15,58967],{"className":58968,"style":128},[127],[15,58970,58972,58975,58978,58981],{"className":58971},[52],[15,58973],{"className":58974,"style":142},[56],[15,58976,102],{"className":58977,"style":174},[61,122],[15,58979,3020],{"className":58980},[61],[15,58982,40265],{"className":58983,"style":40337},[61,122]," 仅是总证据的函数。它无法表达「证据分散但充分」与「证据集中但贫乏」的区别，也无法区分证据落在何类。",[15,58986,58988,59011],{"className":58987,"translate":19},[18],[15,58989,58991],{"className":58990},[23],[25,58992,58993],{"xmlns":27},[29,58994,58995,59009],{},[32,58996,58997,58999,59001,59003,59005,59007],{},[35,58998,28923],{},[81,59000,741],{"stretchy":86},[35,59002,79],{},[81,59004,11303],{"separator":48},[35,59006,24680],{"mathvariant":3105},[81,59008,750],{"stretchy":86},[40,59010,46884],{"encoding":42},[15,59012,59014],{"className":59013,"ariaHidden":48},[47],[15,59015,59017,59020,59023,59026,59029,59032,59035,59044],{"className":59016},[52],[15,59018],{"className":59019,"style":142},[56],[15,59021,28923],{"className":59022,"style":28962},[61,122],[15,59024,741],{"className":59025},[146],[15,59027,79],{"className":59028,"style":123},[61,122],[15,59030,11303],{"className":59031},[153],[15,59033],{"className":59034,"style":157},[127],[15,59036,59038],{"className":59037},[61],[15,59039,59041],{"className":59040},[61],[15,59042,24680],{"className":59043,"style":24714},[61,3166],[15,59045,750],{"className":59046},[178]," 缓解了这一点，但仍不引入任何关于模型参数的信息。",[31550,59049,59050,59053,59054,59309,59310,59313],{},[661,59051,59052],{},"拟合损失带有一个与强度耦合的偏置。"," 推论 4.2 表明，对数形式等价于交叉熵加上 ",[15,59055,59057,59099],{"className":59056,"translate":19},[18],[15,59058,59060],{"className":59059},[23],[25,59061,59062],{"xmlns":27},[29,59063,59064,59096],{},[32,59065,59066,59068,59082,59084,59094],{},[81,59067,741],{"stretchy":86},[3722,59069,59070,59072],{},[89,59071,91],{},[32,59073,59074,59076],{},[89,59075,908],{},[266,59077,59078,59080],{},[35,59079,3106],{},[35,59081,79],{},[81,59083,239],{},[3722,59085,59086,59088],{},[89,59087,91],{},[32,59089,59090,59092],{},[89,59091,908],{},[35,59093,40265],{},[81,59095,750],{"stretchy":86},[40,59097,59098],{"encoding":42},"(\\frac{1}{2\\alpha_y} - \\frac{1}{2S})",[15,59100,59102,59229],{"className":59101,"ariaHidden":48},[47],[15,59103,59105,59109,59112,59220,59223,59226],{"className":59104},[52],[15,59106],{"className":59107,"style":59108},[56],"height:1.3874em;vertical-align:-0.5423em;",[15,59110,741],{"className":59111},[146],[15,59113,59115,59118,59217],{"className":59114},[61],[15,59116],{"className":59117},[146,3841],[15,59119,59121],{"className":59120},[3722],[15,59122,59124,59209],{"className":59123},[342,472],[15,59125,59127,59206],{"className":59126},[346],[15,59128,59130,59184,59192],{"className":59129,"style":3854},[350],[15,59131,59132,59135],{"style":3857},[15,59133],{"className":59134,"style":3861},[357],[15,59136,59138],{"className":59137},[362,363,364,365],[15,59139,59141,59144],{"className":59140},[61,365],[15,59142,908],{"className":59143},[61,365],[15,59145,59147,59150],{"className":59146},[61,365],[15,59148,3106],{"className":59149,"style":3191},[61,122,365],[15,59151,59153],{"className":59152},[338],[15,59154,59156,59176],{"className":59155},[342,472],[15,59157,59159,59173],{"className":59158},[346],[15,59160,59162],{"className":59161,"style":29292},[350],[15,59163,59164,59167],{"style":4058},[15,59165],{"className":59166,"style":4062},[357],[15,59168,59170],{"className":59169},[362,4066,4067,365],[15,59171,79],{"className":59172,"style":123},[61,122,365],[15,59174,497],{"className":59175},[496],[15,59177,59179],{"className":59178},[346],[15,59180,59182],{"className":59181,"style":15963},[350],[15,59183],{},[15,59185,59186,59189],{"style":3889},[15,59187],{"className":59188,"style":3861},[357],[15,59190],{"className":59191,"style":3897},[3896],[15,59193,59194,59197],{"style":3900},[15,59195],{"className":59196,"style":3861},[357],[15,59198,59200],{"className":59199},[362,363,364,365],[15,59201,59203],{"className":59202},[61,365],[15,59204,91],{"className":59205},[61,365],[15,59207,497],{"className":59208},[496],[15,59210,59212],{"className":59211},[346],[15,59213,59215],{"className":59214,"style":49878},[350],[15,59216],{},[15,59218],{"className":59219},[178,3841],[15,59221],{"className":59222,"style":772},[127],[15,59224,239],{"className":59225},[375],[15,59227],{"className":59228,"style":772},[127],[15,59230,59232,59235,59306],{"className":59231},[52],[15,59233],{"className":59234,"style":42571},[56],[15,59236,59238,59241,59303],{"className":59237},[61],[15,59239],{"className":59240},[146,3841],[15,59242,59244],{"className":59243},[3722],[15,59245,59247,59295],{"className":59246},[342,472],[15,59248,59250,59292],{"className":59249},[346],[15,59251,59253,59270,59278],{"className":59252,"style":3854},[350],[15,59254,59255,59258],{"style":3857},[15,59256],{"className":59257,"style":3861},[357],[15,59259,59261],{"className":59260},[362,363,364,365],[15,59262,59264,59267],{"className":59263},[61,365],[15,59265,908],{"className":59266},[61,365],[15,59268,40265],{"className":59269,"style":40337},[61,122,365],[15,59271,59272,59275],{"style":3889},[15,59273],{"className":59274,"style":3861},[357],[15,59276],{"className":59277,"style":3897},[3896],[15,59279,59280,59283],{"style":3900},[15,59281],{"className":59282,"style":3861},[357],[15,59284,59286],{"className":59285},[362,363,364,365],[15,59287,59289],{"className":59288},[61,365],[15,59290,91],{"className":59291},[61,365],[15,59293,497],{"className":59294},[496],[15,59296,59298],{"className":59297},[346],[15,59299,59301],{"className":59300,"style":10198},[350],[15,59302],{},[15,59304],{"className":59305},[178,3841],[15,59307,750],{"className":59308},[178]," 的修正。因此损失的最小值并不位于「插值交叉熵的最优解」，而是位于一个把证据量级也纳入权衡的点上。这不是缺陷本身，但它意味着",[661,59311,59312],{},"EDL 的优化目标与「拟合条件分布」并非同一个目标","，将其损失解释为似然的做法在术语上应作区分。",[31550,59315,59316,59319,59320,59374],{},[661,59317,59318],{},"「以损失最小化实现不确定性量化」这一范式受到根本性质疑。"," 有研究指出，当认知不确定性被定义为某个二阶损失（即关于 ",[15,59321,59323,59344],{"className":59322,"translate":19},[18],[15,59324,59326],{"className":59325},[23],[25,59327,59328],{"xmlns":27},[29,59329,59330,59342],{},[32,59331,59332,59334,59336,59338,59340],{},[35,59333,4186],{"mathvariant":231},[81,59335,1547],{},[81,59337,741],{"stretchy":86},[35,59339,3106],{"mathvariant":3105},[81,59341,750],{"stretchy":86},[40,59343,39072],{"encoding":42},[15,59345,59347],{"className":59346,"ariaHidden":48},[47],[15,59348,59350,59353,59359,59362,59371],{"className":59349},[52],[15,59351],{"className":59352,"style":142},[56],[15,59354,59356],{"className":59355},[531],[15,59357,4186],{"className":59358},[61,1622],[15,59360,741],{"className":59361},[146],[15,59363,59365],{"className":59364},[61],[15,59366,59368],{"className":59367},[61],[15,59369,3106],{"className":59370},[61,3166],[15,59372,750],{"className":59373},[178]," 的泛函）的极小化者时，损失可以在与真实认知状态无关的第二阶分布上取得同一最小值，从而该极小化者一般并不落在具有正确认知解释的分布上。换言之，「损失小」并不蕴含「认知不确定性估计正确」。这一批评与本条前三点在方向上是独立的：即便接受 EDL 的目标函数，其最优解也未必具有所宣称的语义。",[31550,59376,59377,59380,59381,59497,59498,59526,59527,867],{},[661,59378,59379],{},"它对主干特征的可分性有隐含依赖。"," 证据由 ",[15,59382,59384,59412],{"className":59383,"translate":19},[18],[15,59385,59387],{"className":59386},[23],[25,59388,59389],{"xmlns":27},[29,59390,59391,59409],{},[32,59392,59393,59395,59397,59403,59405,59407],{},[35,59394,1982],{"mathvariant":251},[81,59396,244],{},[266,59398,59399,59401],{},[35,59400,22794],{},[35,59402,37148],{"mathvariant":3105},[81,59404,741],{"stretchy":86},[35,59406,1029],{"mathvariant":251},[81,59408,750],{"stretchy":86},[40,59410,59411],{"encoding":42},"\\mathbf{e} = g_{\\boldsymbol{\\theta}}(\\mathbf{x})",[15,59413,59415,59433],{"className":59414,"ariaHidden":48},[47],[15,59416,59418,59421,59424,59427,59430],{"className":59417},[52],[15,59419],{"className":59420,"style":2708},[56],[15,59422,1982],{"className":59423},[61,411],[15,59425],{"className":59426,"style":128},[127],[15,59428,244],{"className":59429},[132],[15,59431],{"className":59432,"style":128},[127],[15,59434,59436,59439,59488,59491,59494],{"className":59435},[52],[15,59437],{"className":59438,"style":142},[56],[15,59440,59442,59445],{"className":59441},[61],[15,59443,22794],{"className":59444,"style":123},[61,122],[15,59446,59448],{"className":59447},[338],[15,59449,59451,59480],{"className":59450},[342,472],[15,59452,59454,59477],{"className":59453},[346],[15,59455,59457],{"className":59456,"style":479},[350],[15,59458,59459,59462],{"style":27036},[15,59460],{"className":59461,"style":358},[357],[15,59463,59465],{"className":59464},[362,363,364,365],[15,59466,59468],{"className":59467},[61,365],[15,59469,59471],{"className":59470},[61,365],[15,59472,59474],{"className":59473},[61,365],[15,59475,37148],{"className":59476,"style":37212},[61,3166,365],[15,59478,497],{"className":59479},[496],[15,59481,59483],{"className":59482},[346],[15,59484,59486],{"className":59485,"style":504},[350],[15,59487],{},[15,59489,741],{"className":59490},[146],[15,59492,1029],{"className":59493},[61,411],[15,59495,750],{"className":59496},[178]," 给出，因此 ",[15,59499,59501,59514],{"className":59500,"translate":19},[18],[15,59502,59504],{"className":59503},[23],[25,59505,59506],{"xmlns":27},[29,59507,59508,59512],{},[32,59509,59510],{},[35,59511,1029],{"mathvariant":251},[40,59513,2698],{"encoding":42},[15,59515,59517],{"className":59516,"ariaHidden":48},[47],[15,59518,59520,59523],{"className":59519},[52],[15,59521],{"className":59522,"style":2708},[56],[15,59524,1029],{"className":59525},[61,411]," 落在分布外时能否给出低证据，完全取决于特征提取器是否把该区域映射到低响应区。若主干网络对分布外输入仍产生高幅 logits，EDL 的不确定性同样失效——",[661,59528,59529],{},"它把 OOD 问题从输出层转移到了特征层，而非解决它",[11,59531,59532,59533,59536],{},"综合以上点，恰当的使用立场是：",[661,59534,59535],{},"把 EDL 视为一种参数高效、结构可解释的不确定性启发式，而非贝叶斯推理的替代品。"," 它在「单次前向代价下同时输出预测与不确定性」这一约束下的性价比是突出的。但凡是论断依赖于「该不确定性具有贝叶斯后验语义」的场合，都应改回到对参数做边缘化的方法，或至少给出与后者的对照实验。",[872,59538,59540],{"id":59539},"五应用","五、应用",[877,59542,59544],{"id":59543},"_51-分布外检测与开放集识别","5.1 分布外检测与开放集识别",[11,59546,59547,59548,43021,59608,59611,59612,59640,59641,59672],{},"由命题 3.4，",[15,59549,59551,59572],{"className":59550,"translate":19},[18],[15,59552,59554],{"className":59553},[23],[25,59555,59556],{"xmlns":27},[29,59557,59558,59570],{},[32,59559,59560,59562,59564,59566,59568],{},[35,59561,33430],{},[81,59563,244],{},[35,59565,102],{},[35,59567,3020],{"mathvariant":231},[35,59569,40265],{},[40,59571,42984],{"encoding":42},[15,59573,59575,59593],{"className":59574,"ariaHidden":48},[47],[15,59576,59578,59581,59584,59587,59590],{"className":59577},[52],[15,59579],{"className":59580,"style":6032},[56],[15,59582,33430],{"className":59583},[61,122],[15,59585],{"className":59586,"style":128},[127],[15,59588,244],{"className":59589},[132],[15,59591],{"className":59592,"style":128},[127],[15,59594,59596,59599,59602,59605],{"className":59595},[52],[15,59597],{"className":59598,"style":142},[56],[15,59600,102],{"className":59601,"style":174},[61,122],[15,59603,3020],{"className":59604},[61],[15,59606,40265],{"className":59607,"style":40337},[61,122],[661,59609,59610],{},"单调反比于总证据","的量，因此可以直接充当分布外（OOD）分数：",[15,59613,59615,59628],{"className":59614,"translate":19},[18],[15,59616,59618],{"className":59617},[23],[25,59619,59620],{"xmlns":27},[29,59621,59622,59626],{},[32,59623,59624],{},[35,59625,33430],{},[40,59627,33430],{"encoding":42},[15,59629,59631],{"className":59630,"ariaHidden":48},[47],[15,59632,59634,59637],{"className":59633},[52],[15,59635],{"className":59636,"style":6032},[56],[15,59638,33430],{"className":59639},[61,122]," 高意味着输入未能激活任何类别的证据。以阈值 ",[15,59642,59644,59659],{"className":59643,"translate":19},[18],[15,59645,59647],{"className":59646},[23],[25,59648,59649],{"xmlns":27},[29,59650,59651,59656],{},[32,59652,59653],{},[35,59654,59655],{},"τ",[40,59657,59658],{"encoding":42},"\\tau",[15,59660,59662],{"className":59661,"ariaHidden":48},[47],[15,59663,59665,59668],{"className":59664},[52],[15,59666],{"className":59667,"style":6032},[56],[15,59669,59655],{"className":59670,"style":59671},[61,122],"margin-right:0.1132em;"," 作拒识判决",[15,59674,59676],{"className":59675,"translate":19},[212],[15,59677,59679,59794],{"className":59678,"translate":19},[18],[15,59680,59682],{"className":59681},[23],[25,59683,59684],{"xmlns":27,"display":221},[29,59685,59686,59791],{},[32,59687,59688,59694,59696,59698,59700,59702],{},[53224,59689,59690,59692],{"accent":48},[35,59691,79],{},[81,59693,58342],{},[81,59695,741],{"stretchy":86},[35,59697,1029],{"mathvariant":251},[81,59699,750],{"stretchy":86},[81,59701,244],{},[32,59703,59704,59706],{},[81,59705,87],{"fence":48},[6447,59707,59710,59760],{"rowspacing":59708,"columnalign":59709,"columnspacing":6451},"0.36em","left left",[6453,59711,59712,59742],{},[6456,59713,59714],{},[286,59715,59716],{"scriptlevel":278,"displaystyle":86},[32,59717,59718,59721,59723,59734,59740],{},[35,59719,59720],{},"arg",[81,59722,1547],{},[266,59724,59725,59732],{},[32,59726,59727,59730],{},[35,59728,59729],{},"max",[81,59731,1547],{},[35,59733,272],{},[266,59735,59736,59738],{},[35,59737,11],{},[35,59739,272],{},[81,59741,94],{"separator":48},[6456,59743,59744],{},[286,59745,59746],{"scriptlevel":278,"displaystyle":86},[32,59747,59748,59750,59752,59754,59756,59758],{},[35,59749,33430],{},[81,59751,741],{"stretchy":86},[35,59753,1029],{"mathvariant":251},[81,59755,750],{"stretchy":86},[81,59757,21308],{},[35,59759,59655],{},[6453,59761,59762,59773],{},[6456,59763,59764],{},[286,59765,59766],{"scriptlevel":278,"displaystyle":86},[32,59767,59768,59771],{},[282,59769,59770],{},"reject",[81,59772,94],{"separator":48},[6456,59774,59775],{},[286,59776,59777],{"scriptlevel":278,"displaystyle":86},[32,59778,59779,59781,59783,59785,59787,59789],{},[35,59780,33430],{},[81,59782,741],{"stretchy":86},[35,59784,1029],{"mathvariant":251},[81,59786,750],{"stretchy":86},[81,59788,1590],{},[35,59790,59655],{},[40,59792,59793],{"encoding":42},"\\hat{y}(\\mathbf{x}) = \\begin{cases} \\arg\\max_k p_k, & u(\\mathbf{x}) \\le \\tau \\\\ \\text{reject}, & u(\\mathbf{x}) > \\tau \\end{cases}",[15,59795,59797,59865],{"className":59796,"ariaHidden":48},[47],[15,59798,59800,59803,59847,59850,59853,59856,59859,59862],{"className":59799},[52],[15,59801],{"className":59802,"style":142},[56],[15,59804,59806],{"className":59805},[61,53268],[15,59807,59809,59838],{"className":59808},[342,472],[15,59810,59812,59835],{"className":59811},[346],[15,59813,59815,59823],{"className":59814,"style":10452},[350],[15,59816,59817,59820],{"style":53280},[15,59818],{"className":59819,"style":3861},[357],[15,59821,79],{"className":59822,"style":123},[61,122],[15,59824,59825,59828],{"style":53280},[15,59826],{"className":59827,"style":3861},[357],[15,59829,59832],{"className":59830,"style":59831},[53302],"left:-0.1944em;",[15,59833,58342],{"className":59834},[61],[15,59836,497],{"className":59837},[496],[15,59839,59841],{"className":59840},[346],[15,59842,59845],{"className":59843,"style":59844},[350],"height:0.1944em;",[15,59846],{},[15,59848,741],{"className":59849},[146],[15,59851,1029],{"className":59852},[61,411],[15,59854,750],{"className":59855},[178],[15,59857],{"className":59858,"style":128},[127],[15,59860,244],{"className":59861},[132],[15,59863],{"className":59864,"style":128},[127],[15,59866,59868,59872],{"className":59867},[52],[15,59869],{"className":59870,"style":59871},[56],"height:3em;vertical-align:-1.25em;",[15,59873,59875,59881,60127],{"className":59874},[161],[15,59876,59878],{"className":59877,"style":402},[146,401],[15,59879,87],{"className":59880},[406,8646],[15,59882,59884],{"className":59883},[61],[15,59885,59887,60036,60040],{"className":59886},[6447],[15,59888,59891],{"className":59889},[59890],"col-align-l",[15,59892,59894,60027],{"className":59893},[342,472],[15,59895,59897,60024],{"className":59896},[346],[15,59898,59901,60006],{"className":59899,"style":59900},[350],"height:1.69em;",[15,59902,59904,59908],{"style":59903},"top:-3.69em;",[15,59905],{"className":59906,"style":59907},[357],"height:3.008em;",[15,59909,59911,59917,59920,59960,59963,60003],{"className":59910},[61],[15,59912,59914,59915],{"className":59913},[531],"ar",[15,59916,22794],{"style":15619},[15,59918],{"className":59919,"style":157},[127],[15,59921,59923,59926],{"className":59922},[531],[15,59924,59729],{"className":59925},[531],[15,59927,59929],{"className":59928},[338],[15,59930,59932,59952],{"className":59931},[342,472],[15,59933,59935,59949],{"className":59934},[346],[15,59936,59938],{"className":59937,"style":479},[350],[15,59939,59940,59943],{"style":30134},[15,59941],{"className":59942,"style":358},[357],[15,59944,59946],{"className":59945},[362,363,364,365],[15,59947,272],{"className":59948,"style":492},[61,122,365],[15,59950,497],{"className":59951},[496],[15,59953,59955],{"className":59954},[346],[15,59956,59958],{"className":59957,"style":504},[350],[15,59959],{},[15,59961],{"className":59962,"style":157},[127],[15,59964,59966,59969],{"className":59965},[61],[15,59967,11],{"className":59968},[61,122],[15,59970,59972],{"className":59971},[338],[15,59973,59975,59995],{"className":59974},[342,472],[15,59976,59978,59992],{"className":59977},[346],[15,59979,59981],{"className":59980,"style":479},[350],[15,59982,59983,59986],{"style":482},[15,59984],{"className":59985,"style":358},[357],[15,59987,59989],{"className":59988},[362,363,364,365],[15,59990,272],{"className":59991,"style":492},[61,122,365],[15,59993,497],{"className":59994},[496],[15,59996,59998],{"className":59997},[346],[15,59999,60001],{"className":60000,"style":504},[350],[15,60002],{},[15,60004,94],{"className":60005},[153],[15,60007,60009,60012],{"style":60008},"top:-2.25em;",[15,60010],{"className":60011,"style":59907},[357],[15,60013,60015,60021],{"className":60014},[61],[15,60016,60018],{"className":60017},[61,13548],[15,60019,59770],{"className":60020},[61],[15,60022,94],{"className":60023},[153],[15,60025,497],{"className":60026},[496],[15,60028,60030],{"className":60029},[346],[15,60031,60034],{"className":60032,"style":60033},[350],"height:1.19em;",[15,60035],{},[15,60037],{"className":60038,"style":60039},[6663],"width:1em;",[15,60041,60043],{"className":60042},[59890],[15,60044,60046,60119],{"className":60045},[342,472],[15,60047,60049,60116],{"className":60048},[346],[15,60050,60052,60084],{"className":60051,"style":59900},[350],[15,60053,60054,60057],{"style":59903},[15,60055],{"className":60056,"style":59907},[357],[15,60058,60060,60063,60066,60069,60072,60075,60078,60081],{"className":60059},[61],[15,60061,33430],{"className":60062},[61,122],[15,60064,741],{"className":60065},[146],[15,60067,1029],{"className":60068},[61,411],[15,60070,750],{"className":60071},[178],[15,60073],{"className":60074,"style":128},[127],[15,60076,21308],{"className":60077},[132],[15,60079],{"className":60080,"style":128},[127],[15,60082,59655],{"className":60083,"style":59671},[61,122],[15,60085,60086,60089],{"style":60008},[15,60087],{"className":60088,"style":59907},[357],[15,60090,60092,60095,60098,60101,60104,60107,60110,60113],{"className":60091},[61],[15,60093,33430],{"className":60094},[61,122],[15,60096,741],{"className":60097},[146],[15,60099,1029],{"className":60100},[61,411],[15,60102,750],{"className":60103},[178],[15,60105],{"className":60106,"style":128},[127],[15,60108,1590],{"className":60109},[132],[15,60111],{"className":60112,"style":128},[127],[15,60114,59655],{"className":60115,"style":59671},[61,122],[15,60117,497],{"className":60118},[496],[15,60120,60122],{"className":60121},[346],[15,60123,60125],{"className":60124,"style":60033},[350],[15,60126],{},[15,60128],{"className":60129},[178,3841],[11,60131,60132,60133,60340],{},"即得开放集分类器。与之竞争的分数包括最大 softmax 概率（MSP）、预测熵、ODIN 的温度与输入扰动、Mahalanobis 距离，以及能量分数 ",[15,60134,60136,60181],{"className":60135,"translate":19},[18],[15,60137,60139],{"className":60138},[23],[25,60140,60141],{"xmlns":27},[29,60142,60143,60178],{},[32,60144,60145,60148,60150,60152,60154,60156,60158,60160,60162,60168],{},[35,60146,60147],{"mathvariant":231},"LSE",[81,60149,1547],{},[81,60151,741],{"stretchy":86},[35,60153,37301],{"mathvariant":251},[81,60155,750],{"stretchy":86},[81,60157,244],{},[35,60159,22662],{},[81,60161,1547],{},[266,60163,60164,60166],{},[81,60165,293],{},[35,60167,272],{},[227,60169,60170,60172],{},[35,60171,1982],{},[266,60173,60174,60176],{},[35,60175,37301],{},[35,60177,272],{},[40,60179,60180],{"encoding":42},"\\operatorname{LSE}(\\mathbf{z}) = \\log\\sum_k e^{z_k}",[15,60182,60184,60214],{"className":60183,"ariaHidden":48},[47],[15,60185,60187,60190,60196,60199,60202,60205,60208,60211],{"className":60186},[52],[15,60188],{"className":60189,"style":142},[56],[15,60191,60193],{"className":60192},[531],[15,60194,60147],{"className":60195},[61,1622],[15,60197,741],{"className":60198},[146],[15,60200,37301],{"className":60201},[61,411],[15,60203,750],{"className":60204},[178],[15,60206],{"className":60207,"style":128},[127],[15,60209,244],{"className":60210},[132],[15,60212],{"className":60213,"style":128},[127],[15,60215,60217,60220,60225,60228,60268,60271],{"className":60216},[52],[15,60218],{"className":60219,"style":6051},[56],[15,60221,22791,60223],{"className":60222},[531],[15,60224,22794],{"style":15619},[15,60226],{"className":60227,"style":157},[127],[15,60229,60231,60234],{"className":60230},[531],[15,60232,293],{"className":60233,"style":537},[531,535,536],[15,60235,60237],{"className":60236},[338],[15,60238,60240,60260],{"className":60239},[342,472],[15,60241,60243,60257],{"className":60242},[346],[15,60244,60246],{"className":60245,"style":29989},[350],[15,60247,60248,60251],{"style":553},[15,60249],{"className":60250,"style":358},[357],[15,60252,60254],{"className":60253},[362,363,364,365],[15,60255,272],{"className":60256,"style":492},[61,122,365],[15,60258,497],{"className":60259},[496],[15,60261,60263],{"className":60262},[346],[15,60264,60266],{"className":60265,"style":593},[350],[15,60267],{},[15,60269],{"className":60270,"style":157},[127],[15,60272,60274,60277],{"className":60273},[61],[15,60275,1982],{"className":60276},[61,122],[15,60278,60280],{"className":60279},[338],[15,60281,60283],{"className":60282},[342],[15,60284,60286],{"className":60285},[346],[15,60287,60289],{"className":60288,"style":37611},[350],[15,60290,60291,60294],{"style":716},[15,60292],{"className":60293,"style":358},[357],[15,60295,60297],{"className":60296},[362,363,364,365],[15,60298,60300],{"className":60299},[61,365],[15,60301,60303,60306],{"className":60302},[61,365],[15,60304,37301],{"className":60305,"style":37629},[61,122,365],[15,60307,60309],{"className":60308},[338],[15,60310,60312,60332],{"className":60311},[342,472],[15,60313,60315,60329],{"className":60314},[346],[15,60316,60318],{"className":60317,"style":22404},[350],[15,60319,60320,60323],{"style":37644},[15,60321],{"className":60322,"style":4062},[357],[15,60324,60326],{"className":60325},[362,4066,4067,365],[15,60327,272],{"className":60328,"style":492},[61,122,365],[15,60330,497],{"className":60331},[496],[15,60333,60335],{"className":60334},[346],[15,60336,60338],{"className":60337,"style":27253},[350],[15,60339],{},"。将它们在性质上加以对照：",[32261,60342,60343,60359],{},[32264,60344,60345],{},[21311,60346,60347,60350,60353,60356],{},[32269,60348,60349],{"align":32271},"分数",[32269,60351,60352],{"align":32271},"依赖的结构",[32269,60354,60355],{"align":32271},"是否需要 OOD 数据",[32269,60357,60358],{"align":32271},"不确定性语汇",[32283,60360,60361,60375,60388,60401],{},[21311,60362,60363,60366,60369,60372],{},[32288,60364,60365],{"align":32271},"MSP \u002F 熵",[32288,60367,60368],{"align":32271},"softmax 输出",[32288,60370,60371],{"align":32271},"否",[32288,60373,60374],{"align":32271},"无（受 3.3 节缺陷支配）",[21311,60376,60377,60380,60383,60385],{},[32288,60378,60379],{"align":32271},"Mahalanobis",[32288,60381,60382],{"align":32271},"各特征层的类条件高斯",[32288,60384,60371],{"align":32271},[32288,60386,60387],{"align":32271},"距离，几何语义",[21311,60389,60390,60393,60396,60398],{},[32288,60391,60392],{"align":32271},"能量分数",[32288,60394,60395],{"align":32271},"logits 的对数配分",[32288,60397,60371],{"align":32271},[32288,60399,60400],{"align":32271},"无显式不确定性",[21311,60402,60403,60436,60467,60471],{},[32288,60404,60405],{"align":32271},[661,60406,60407,60408],{},"EDL 的 ",[15,60409,60411,60424],{"className":60410,"translate":19},[18],[15,60412,60414],{"className":60413},[23],[25,60415,60416],{"xmlns":27},[29,60417,60418,60422],{},[32,60419,60420],{},[35,60421,33430],{},[40,60423,33430],{"encoding":42},[15,60425,60427],{"className":60426,"ariaHidden":48},[47],[15,60428,60430,60433],{"className":60429},[52],[15,60431],{"className":60432,"style":6032},[56],[15,60434,33430],{"className":60435},[61,122],[32288,60437,60438,60439],{"align":32271},"迪利克雷总强度 ",[15,60440,60442,60455],{"className":60441,"translate":19},[18],[15,60443,60445],{"className":60444},[23],[25,60446,60447],{"xmlns":27},[29,60448,60449,60453],{},[32,60450,60451],{},[35,60452,40265],{},[40,60454,40265],{"encoding":42},[15,60456,60458],{"className":60457,"ariaHidden":48},[47],[15,60459,60461,60464],{"className":60460},[52],[15,60462],{"className":60463,"style":57},[56],[15,60465,40265],{"className":60466,"style":40337},[61,122],[32288,60468,60469],{"align":32271},[661,60470,60371],{},[32288,60472,60473],{"align":32271},"证据缺失量，可解释",[11,60475,60476,60477,60480],{},"EDL 在此项比较中的结构性优势在于两点。第一，它的不确定性是",[661,60478,60479],{},"训练目标的一部分","（由 4.3 节的正则项直接优化），而非对训练好的网络做事后加工的产物。第二，它给出的是类概率向量的分布，因此可以在不接触任何 OOD 样本的前提下产生 OOD 分数——这是先验网络之外少有的性质（先验网络需要 OOD 样本以构造目标分布）。",[11,60482,60483],{},"限度同样明确：如 4.5 节第 5 点所述，其成败取决于特征空间的可分性。评价应使用 AUROC、AUPR（In\u002FOut 与 Out\u002FIn 分别报告）、FPR@95TPR 以及开放集风险—覆盖曲线，而非仅报单一的准确率。",[877,60485,60487],{"id":60486},"_52-校准与选择性预测","5.2 校准与选择性预测",[11,60489,60490,60493,60494,60714],{},[661,60491,60492],{},"校准","关心的是 ",[15,60495,60497,60542],{"className":60496,"translate":19},[18],[15,60498,60500],{"className":60499},[23],[25,60501,60502],{"xmlns":27},[29,60503,60504,60539],{},[32,60505,60506,60508,60510,60516,60518,60520,60522,60528,60530,60533,60535,60537],{},[35,60507,21286],{"mathvariant":258},[81,60509,741],{"stretchy":86},[53224,60511,60512,60514],{"accent":48},[35,60513,79],{},[81,60515,58342],{},[81,60517,244],{},[35,60519,79],{},[81,60521,3714],{},[53224,60523,60524,60526],{"accent":48},[35,60525,11],{},[81,60527,58342],{},[81,60529,244],{},[35,60531,60532],{},"q",[81,60534,750],{"stretchy":86},[81,60536,244],{},[35,60538,60532],{},[40,60540,60541],{"encoding":42},"\\mathbb{P}(\\hat{y} = y \\mid \\hat{p} = q) = q",[15,60543,60545,60608,60626,60684,60705],{"className":60544,"ariaHidden":48},[47],[15,60546,60548,60551,60554,60557,60599,60602,60605],{"className":60547},[52],[15,60549],{"className":60550,"style":142},[56],[15,60552,21286],{"className":60553},[61,427],[15,60555,741],{"className":60556},[146],[15,60558,60560],{"className":60559},[61,53268],[15,60561,60563,60591],{"className":60562},[342,472],[15,60564,60566,60588],{"className":60565},[346],[15,60567,60569,60577],{"className":60568,"style":10452},[350],[15,60570,60571,60574],{"style":53280},[15,60572],{"className":60573,"style":3861},[357],[15,60575,79],{"className":60576,"style":123},[61,122],[15,60578,60579,60582],{"style":53280},[15,60580],{"className":60581,"style":3861},[357],[15,60583,60585],{"className":60584,"style":59831},[53302],[15,60586,58342],{"className":60587},[61],[15,60589,497],{"className":60590},[496],[15,60592,60594],{"className":60593},[346],[15,60595,60597],{"className":60596,"style":59844},[350],[15,60598],{},[15,60600],{"className":60601,"style":128},[127],[15,60603,244],{"className":60604},[132],[15,60606],{"className":60607,"style":128},[127],[15,60609,60611,60614,60617,60620,60623],{"className":60610},[52],[15,60612],{"className":60613,"style":142},[56],[15,60615,79],{"className":60616,"style":123},[61,122],[15,60618],{"className":60619,"style":128},[127],[15,60621,3714],{"className":60622},[132],[15,60624],{"className":60625,"style":128},[127],[15,60627,60629,60632,60675,60678,60681],{"className":60628},[52],[15,60630],{"className":60631,"style":14310},[56],[15,60633,60635],{"className":60634},[61,53268],[15,60636,60638,60667],{"className":60637},[342,472],[15,60639,60641,60664],{"className":60640},[346],[15,60642,60644,60652],{"className":60643,"style":10452},[350],[15,60645,60646,60649],{"style":53280},[15,60647],{"className":60648,"style":3861},[357],[15,60650,11],{"className":60651},[61,122],[15,60653,60654,60657],{"style":53280},[15,60655],{"className":60656,"style":3861},[357],[15,60658,60661],{"className":60659,"style":60660},[53302],"left:-0.1667em;",[15,60662,58342],{"className":60663},[61],[15,60665,497],{"className":60666},[496],[15,60668,60670],{"className":60669},[346],[15,60671,60673],{"className":60672,"style":59844},[350],[15,60674],{},[15,60676],{"className":60677,"style":128},[127],[15,60679,244],{"className":60680},[132],[15,60682],{"className":60683,"style":128},[127],[15,60685,60687,60690,60693,60696,60699,60702],{"className":60686},[52],[15,60688],{"className":60689,"style":142},[56],[15,60691,60532],{"className":60692,"style":123},[61,122],[15,60694,750],{"className":60695},[178],[15,60697],{"className":60698,"style":128},[127],[15,60700,244],{"className":60701},[132],[15,60703],{"className":60704,"style":128},[127],[15,60706,60708,60711],{"className":60707},[52],[15,60709],{"className":60710,"style":9437},[56],[15,60712,60532],{"className":60713,"style":123},[61,122]," 是否成立，常用期望校准误差",[15,60716,60718],{"className":60717,"translate":19},[212],[15,60719,60721,60807],{"className":60720,"translate":19},[18],[15,60722,60724],{"className":60723},[23],[25,60725,60726],{"xmlns":27,"display":221},[29,60727,60728,60804],{},[32,60729,60730,60733,60735,60749,60765,60767,60770,60773,60775,60777,60783,60785,60787,60790,60792,60794,60800,60802],{},[282,60731,60732],{},"ECE",[81,60734,244],{},[5235,60736,60737,60739,60747],{},[81,60738,293],{},[32,60740,60741,60743,60745],{},[35,60742,29146],{},[81,60744,244],{},[89,60746,91],{},[35,60748,28938],{},[3722,60750,60751,60763],{},[32,60752,60753,60755,60761],{},[35,60754,3714],{"mathvariant":231},[266,60756,60757,60759],{},[35,60758,3730],{"mathvariant":37},[35,60760,29146],{},[35,60762,3714],{"mathvariant":231},[35,60764,14560],{},[282,60766,4802],{},[81,60768,3714],{"fence":86,"stretchy":48,"minsize":60769,"maxsize":60769},"1.8em",[35,60771,60772],{"mathvariant":231},"acc",[81,60774,1547],{},[81,60776,741],{"stretchy":86},[266,60778,60779,60781],{},[35,60780,3730],{"mathvariant":37},[35,60782,29146],{},[81,60784,750],{"stretchy":86},[81,60786,239],{},[35,60788,60789],{"mathvariant":231},"conf",[81,60791,1547],{},[81,60793,741],{"stretchy":86},[266,60795,60796,60798],{},[35,60797,3730],{"mathvariant":37},[35,60799,29146],{},[81,60801,750],{"stretchy":86},[81,60803,3714],{"fence":86,"stretchy":48,"minsize":60769,"maxsize":60769},[40,60805,60806],{"encoding":42},"\\text{ECE} = \\sum_{m=1}^{M} \\frac{|\\mathcal{B}_m|}{n}\\,\\Big|\\operatorname{acc}(\\mathcal{B}_m) - \\operatorname{conf}(\\mathcal{B}_m)\\Big|",[15,60808,60810,60831,61130],{"className":60809,"ariaHidden":48},[47],[15,60811,60813,60816,60822,60825,60828],{"className":60812},[52],[15,60814],{"className":60815,"style":57},[56],[15,60817,60819],{"className":60818},[61,13548],[15,60820,60732],{"className":60821},[61],[15,60823],{"className":60824,"style":128},[127],[15,60826,244],{"className":60827},[132],[15,60829],{"className":60830,"style":128},[127],[15,60832,60834,60838,60907,60910,61018,61021,61066,61069,61075,61078,61118,61121,61124,61127],{"className":60833},[52],[15,60835],{"className":60836,"style":60837},[56],"height:3.0954em;vertical-align:-1.2671em;",[15,60839,60841],{"className":60840},[531,5348],[15,60842,60844,60898],{"className":60843},[342,472],[15,60845,60847,60895],{"className":60846},[346],[15,60848,60850,60871,60881],{"className":60849,"style":5358},[350],[15,60851,60853,60856],{"style":60852},"top:-1.8829em;margin-left:0em;",[15,60854],{"className":60855,"style":5365},[357],[15,60857,60859],{"className":60858},[362,363,364,365],[15,60860,60862,60865,60868],{"className":60861},[61,365],[15,60863,29146],{"className":60864},[61,122,365],[15,60866,244],{"className":60867},[132,365],[15,60869,91],{"className":60870},[61,365],[15,60872,60873,60876],{"style":5383},[15,60874],{"className":60875,"style":5365},[357],[15,60877,60878],{},[15,60879,293],{"className":60880},[531,535,5392],[15,60882,60883,60886],{"style":5395},[15,60884],{"className":60885,"style":5365},[357],[15,60887,60889],{"className":60888},[362,363,364,365],[15,60890,60892],{"className":60891},[61,365],[15,60893,28938],{"className":60894,"style":25097},[61,122,365],[15,60896,497],{"className":60897},[496],[15,60899,60901],{"className":60900},[346],[15,60902,60905],{"className":60903,"style":60904},[350],"height:1.2671em;",[15,60906],{},[15,60908],{"className":60909,"style":157},[127],[15,60911,60913,60916,61015],{"className":60912},[61],[15,60914],{"className":60915},[146,3841],[15,60917,60919],{"className":60918},[3722],[15,60920,60922,61007],{"className":60921},[342,472],[15,60923,60925,61004],{"className":60924},[346],[15,60926,60929,60940,60948],{"className":60927,"style":60928},[350],"height:1.427em;",[15,60930,60931,60934],{"style":11402},[15,60932],{"className":60933,"style":3861},[357],[15,60935,60937],{"className":60936},[61],[15,60938,14560],{"className":60939},[61,122],[15,60941,60942,60945],{"style":3889},[15,60943],{"className":60944,"style":3861},[357],[15,60946],{"className":60947,"style":3897},[3896],[15,60949,60950,60953],{"style":11468},[15,60951],{"className":60952,"style":3861},[357],[15,60954,60956,60959,61001],{"className":60955},[61],[15,60957,3714],{"className":60958},[61],[15,60960,60962,60966],{"className":60961},[61],[15,60963,3730],{"className":60964,"style":60965},[61,62],"margin-right:0.0304em;",[15,60967,60969],{"className":60968},[338],[15,60970,60972,60993],{"className":60971},[342,472],[15,60973,60975,60990],{"className":60974},[346],[15,60976,60978],{"className":60977,"style":27033},[350],[15,60979,60981,60984],{"style":60980},"top:-2.55em;margin-left:-0.0304em;margin-right:0.05em;",[15,60982],{"className":60983,"style":358},[357],[15,60985,60987],{"className":60986},[362,363,364,365],[15,60988,29146],{"className":60989},[61,122,365],[15,60991,497],{"className":60992},[496],[15,60994,60996],{"className":60995},[346],[15,60997,60999],{"className":60998,"style":504},[350],[15,61000],{},[15,61002,3714],{"className":61003},[61],[15,61005,497],{"className":61006},[496],[15,61008,61010],{"className":61009},[346],[15,61011,61013],{"className":61012,"style":36614},[350],[15,61014],{},[15,61016],{"className":61017},[178,3841],[15,61019],{"className":61020,"style":157},[127],[15,61022,61024],{"className":61023},[61],[15,61025,61027],{"className":61026},[406,53622],[15,61028,61030,61057],{"className":61029},[342,472],[15,61031,61033,61054],{"className":61032},[346],[15,61034,61037],{"className":61035,"style":61036},[350],"height:1.15em;",[15,61038,61040,61044],{"style":61039},"top:-3.15em;",[15,61041],{"className":61042,"style":61043},[357],"height:3.8em;",[15,61045,61047],{"style":61046},"width:0.333em;height:1.8em;",[27706,61048,61051],{"xmlns":27708,"width":61049,"height":60769,"viewBox":61050},"0.333em","0 0 333 1800",[27714,61052],{"d":61053},"M145 15 v585 v600 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v-600 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v600 v585 h43z",[15,61055,497],{"className":61056},[496],[15,61058,61060],{"className":61059},[346],[15,61061,61064],{"className":61062,"style":61063},[350],"height:0.65em;",[15,61065],{},[15,61067],{"className":61068,"style":157},[127],[15,61070,61072],{"className":61071},[531],[15,61073,60772],{"className":61074},[61,1622],[15,61076,741],{"className":61077},[146],[15,61079,61081,61084],{"className":61080},[61],[15,61082,3730],{"className":61083,"style":60965},[61,62],[15,61085,61087],{"className":61086},[338],[15,61088,61090,61110],{"className":61089},[342,472],[15,61091,61093,61107],{"className":61092},[346],[15,61094,61096],{"className":61095,"style":27033},[350],[15,61097,61098,61101],{"style":60980},[15,61099],{"className":61100,"style":358},[357],[15,61102,61104],{"className":61103},[362,363,364,365],[15,61105,29146],{"className":61106},[61,122,365],[15,61108,497],{"className":61109},[496],[15,61111,61113],{"className":61112},[346],[15,61114,61116],{"className":61115,"style":504},[350],[15,61117],{},[15,61119,750],{"className":61120},[178],[15,61122],{"className":61123,"style":772},[127],[15,61125,239],{"className":61126},[375],[15,61128],{"className":61129,"style":772},[127],[15,61131,61133,61136,61143,61146,61186,61189],{"className":61132},[52],[15,61134],{"className":61135,"style":394},[56],[15,61137,61139],{"className":61138},[531],[15,61140,60789],{"className":61141,"style":61142},[61,1622],"margin-right:0.0778em;",[15,61144,741],{"className":61145},[146],[15,61147,61149,61152],{"className":61148},[61],[15,61150,3730],{"className":61151,"style":60965},[61,62],[15,61153,61155],{"className":61154},[338],[15,61156,61158,61178],{"className":61157},[342,472],[15,61159,61161,61175],{"className":61160},[346],[15,61162,61164],{"className":61163,"style":27033},[350],[15,61165,61166,61169],{"style":60980},[15,61167],{"className":61168,"style":358},[357],[15,61170,61172],{"className":61171},[362,363,364,365],[15,61173,29146],{"className":61174},[61,122,365],[15,61176,497],{"className":61177},[496],[15,61179,61181],{"className":61180},[346],[15,61182,61184],{"className":61183,"style":504},[350],[15,61185],{},[15,61187,750],{"className":61188},[178],[15,61190,61192],{"className":61191},[61],[15,61193,61195],{"className":61194},[406,53622],[15,61196,61198,61218],{"className":61197},[342,472],[15,61199,61201,61215],{"className":61200},[346],[15,61202,61204],{"className":61203,"style":61036},[350],[15,61205,61206,61209],{"style":61039},[15,61207],{"className":61208,"style":61043},[357],[15,61210,61211],{"style":61046},[27706,61212,61213],{"xmlns":27708,"width":61049,"height":60769,"viewBox":61050},[27714,61214],{"d":61053},[15,61216,497],{"className":61217},[496],[15,61219,61221],{"className":61220},[346],[15,61222,61224],{"className":61223,"style":61063},[350],[15,61225],{},[11,61227,61228,61229,61299],{},"度量，其中 ",[15,61230,61232,61250],{"className":61231,"translate":19},[18],[15,61233,61235],{"className":61234},[23],[25,61236,61237],{"xmlns":27},[29,61238,61239,61247],{},[32,61240,61241],{},[266,61242,61243,61245],{},[35,61244,3730],{"mathvariant":37},[35,61246,29146],{},[40,61248,61249],{"encoding":42},"\\mathcal{B}_m",[15,61251,61253],{"className":61252,"ariaHidden":48},[47],[15,61254,61256,61259],{"className":61255},[52],[15,61257],{"className":61258,"style":34360},[56],[15,61260,61262,61265],{"className":61261},[61],[15,61263,3730],{"className":61264,"style":60965},[61,62],[15,61266,61268],{"className":61267},[338],[15,61269,61271,61291],{"className":61270},[342,472],[15,61272,61274,61288],{"className":61273},[346],[15,61275,61277],{"className":61276,"style":27033},[350],[15,61278,61279,61282],{"style":60980},[15,61280],{"className":61281,"style":358},[357],[15,61283,61285],{"className":61284},[362,363,364,365],[15,61286,29146],{"className":61287},[61,122,365],[15,61289,497],{"className":61290},[496],[15,61292,61294],{"className":61293},[346],[15,61295,61297],{"className":61296,"style":504},[350],[15,61298],{}," 是置信度分箱。EDL 的训练目标隐含地抑制了 3.3 节所述的 logits 发散退化，因此在若干基准上比 MSP 更校准。但这不是免费的：如 4.5 节第 7 点，EDL 的一阶校准同样需要验证，不应默认成立。",[11,61301,61302,1906,61305,61333,61334,61400,61401,61433,61434,61462,61463,61525],{},[661,61303,61304],{},"选择性预测（selective prediction）",[15,61306,61308,61321],{"className":61307,"translate":19},[18],[15,61309,61311],{"className":61310},[23],[25,61312,61313],{"xmlns":27},[29,61314,61315,61319],{},[32,61316,61317],{},[35,61318,33430],{},[40,61320,33430],{"encoding":42},[15,61322,61324],{"className":61323,"ariaHidden":48},[47],[15,61325,61327,61330],{"className":61326},[52],[15,61328],{"className":61329,"style":6032},[56],[15,61331,33430],{"className":61332},[61,122]," 的更自然的落点。经典 Chow 规则指出，在已知真实后验 ",[15,61335,61337,61361],{"className":61336,"translate":19},[18],[15,61338,61340],{"className":61339},[23],[25,61341,61342],{"xmlns":27},[29,61343,61344,61358],{},[32,61345,61346,61348,61350,61352,61354,61356],{},[35,61347,11],{},[81,61349,741],{"stretchy":86},[35,61351,79],{},[81,61353,3714],{},[35,61355,1029],{"mathvariant":251},[81,61357,750],{"stretchy":86},[40,61359,61360],{"encoding":42},"p(y \\mid \\mathbf{x})",[15,61362,61364,61388],{"className":61363,"ariaHidden":48},[47],[15,61365,61367,61370,61373,61376,61379,61382,61385],{"className":61366},[52],[15,61368],{"className":61369,"style":142},[56],[15,61371,11],{"className":61372},[61,122],[15,61374,741],{"className":61375},[146],[15,61377,79],{"className":61378,"style":123},[61,122],[15,61380],{"className":61381,"style":128},[127],[15,61383,3714],{"className":61384},[132],[15,61386],{"className":61387,"style":128},[127],[15,61389,61391,61394,61397],{"className":61390},[52],[15,61392],{"className":61393,"style":142},[56],[15,61395,1029],{"className":61396},[61,411],[15,61398,750],{"className":61399},[178]," 时，最优拒识策略是「当最大后验概率低于某阈值时拒识」，其最优性",[661,61402,61403,61404,61432],{},"依赖于 ",[15,61405,61407,61420],{"className":61406,"translate":19},[18],[15,61408,61410],{"className":61409},[23],[25,61411,61412],{"xmlns":27},[29,61413,61414,61418],{},[32,61415,61416],{},[35,61417,11],{},[40,61419,11],{"encoding":42},[15,61421,61423],{"className":61422,"ariaHidden":48},[47],[15,61424,61426,61429],{"className":61425},[52],[15,61427],{"className":61428,"style":9437},[56],[15,61430,11],{"className":61431},[61,122]," 本身的正确性","。EDL 提供了一条与之互补的路径：直接使用二阶量 ",[15,61435,61437,61450],{"className":61436,"translate":19},[18],[15,61438,61440],{"className":61439},[23],[25,61441,61442],{"xmlns":27},[29,61443,61444,61448],{},[32,61445,61446],{},[35,61447,33430],{},[40,61449,33430],{"encoding":42},[15,61451,61453],{"className":61452,"ariaHidden":48},[47],[15,61454,61456,61459],{"className":61455},[52],[15,61457],{"className":61458,"style":6032},[56],[15,61460,33430],{"className":61461},[61,122]," 或 ",[15,61464,61466,61489],{"className":61465,"translate":19},[18],[15,61467,61469],{"className":61468},[23],[25,61470,61471],{"xmlns":27},[29,61472,61473,61487],{},[32,61474,61475,61477,61479,61481,61483,61485],{},[35,61476,28923],{},[81,61478,741],{"stretchy":86},[35,61480,79],{},[81,61482,11303],{"separator":48},[35,61484,24680],{"mathvariant":3105},[81,61486,750],{"stretchy":86},[40,61488,46884],{"encoding":42},[15,61490,61492],{"className":61491,"ariaHidden":48},[47],[15,61493,61495,61498,61501,61504,61507,61510,61513,61522],{"className":61494},[52],[15,61496],{"className":61497,"style":142},[56],[15,61499,28923],{"className":61500,"style":28962},[61,122],[15,61502,741],{"className":61503},[146],[15,61505,79],{"className":61506,"style":123},[61,122],[15,61508,11303],{"className":61509},[153],[15,61511],{"className":61512,"style":157},[127],[15,61514,61516],{"className":61515},[61],[15,61517,61519],{"className":61518},[61],[15,61520,24680],{"className":61521,"style":24714},[61,3166],[15,61523,750],{"className":61524},[178]," 作为拒识分数，从而绕开对一阶校准的依赖。其代价是引入了一个新的、校准状况未知的量。实践中应同时考察两条路径并报告风险—覆盖曲线下的面积（AURC）。",[877,61527,61529],{"id":61528},"_53-主动学习与半监督学习","5.3 主动学习与半监督学习",[11,61531,61532,61535,61536,61462,61580,61658,61659,61687,61688,61716,61717,61720,61721,61749],{},[661,61533,61534],{},"主动学习。"," 采集函数可以直接取 ",[15,61537,61539,61559],{"className":61538,"translate":19},[18],[15,61540,61542],{"className":61541},[23],[25,61543,61544],{"xmlns":27},[29,61545,61546,61556],{},[32,61547,61548,61550,61552,61554],{},[35,61549,33430],{},[81,61551,741],{"stretchy":86},[35,61553,1029],{"mathvariant":251},[81,61555,750],{"stretchy":86},[40,61557,61558],{"encoding":42},"u(\\mathbf{x})",[15,61560,61562],{"className":61561,"ariaHidden":48},[47],[15,61563,61565,61568,61571,61574,61577],{"className":61564},[52],[15,61566],{"className":61567,"style":142},[56],[15,61569,33430],{"className":61570},[61,122],[15,61572,741],{"className":61573},[146],[15,61575,1029],{"className":61576},[61,411],[15,61578,750],{"className":61579},[178],[15,61581,61583,61613],{"className":61582,"translate":19},[18],[15,61584,61586],{"className":61585},[23],[25,61587,61588],{"xmlns":27},[29,61589,61590,61610],{},[32,61591,61592,61594,61596,61598,61600,61602,61604,61606,61608],{},[35,61593,28923],{},[81,61595,741],{"stretchy":86},[35,61597,79],{},[81,61599,11303],{"separator":48},[35,61601,24680],{"mathvariant":3105},[81,61603,750],{"stretchy":86},[81,61605,741],{"stretchy":86},[35,61607,1029],{"mathvariant":251},[81,61609,750],{"stretchy":86},[40,61611,61612],{"encoding":42},"I(y;\\boldsymbol{\\pi})(\\mathbf{x})",[15,61614,61616],{"className":61615,"ariaHidden":48},[47],[15,61617,61619,61622,61625,61628,61631,61634,61637,61646,61649,61652,61655],{"className":61618},[52],[15,61620],{"className":61621,"style":142},[56],[15,61623,28923],{"className":61624,"style":28962},[61,122],[15,61626,741],{"className":61627},[146],[15,61629,79],{"className":61630,"style":123},[61,122],[15,61632,11303],{"className":61633},[153],[15,61635],{"className":61636,"style":157},[127],[15,61638,61640],{"className":61639},[61],[15,61641,61643],{"className":61642},[61],[15,61644,24680],{"className":61645,"style":24714},[61,3166],[15,61647,750],{"className":61648},[178],[15,61650,741],{"className":61651},[146],[15,61653,1029],{"className":61654},[61,411],[15,61656,750],{"className":61657},[178],"：前者是「总证据最少」，后者是「关于类概率的知识增益最大」。选择 ",[15,61660,61662,61675],{"className":61661,"translate":19},[18],[15,61663,61665],{"className":61664},[23],[25,61666,61667],{"xmlns":27},[29,61668,61669,61673],{},[32,61670,61671],{},[35,61672,28923],{},[40,61674,28923],{"encoding":42},[15,61676,61678],{"className":61677,"ariaHidden":48},[47],[15,61679,61681,61684],{"className":61680},[52],[15,61682],{"className":61683,"style":57},[56],[15,61685,28923],{"className":61686,"style":28962},[61,122]," 而非 ",[15,61689,61691,61704],{"className":61690,"translate":19},[18],[15,61692,61694],{"className":61693},[23],[25,61695,61696],{"xmlns":27},[29,61697,61698,61702],{},[32,61699,61700],{},[35,61701,33430],{},[40,61703,33430],{"encoding":42},[15,61705,61707],{"className":61706,"ariaHidden":48},[47],[15,61708,61710,61713],{"className":61709},[52],[15,61711],{"className":61712,"style":6032},[56],[15,61714,33430],{"className":61715},[61,122]," 的好处在于它同时考虑证据的量与分布，等价于 BALD 在迪利克雷族内的闭式对应物。EDL 在此场景的最大优势是",[661,61718,61719],{},"代价","：一次前向即可得到采集分数，无需 MC 采样、无需多次前向、无需集成子模型。其风险同样直接来自 4.5 节第 1 点：若 ",[15,61722,61724,61737],{"className":61723,"translate":19},[18],[15,61725,61727],{"className":61726},[23],[25,61728,61729],{"xmlns":27},[29,61730,61731,61735],{},[32,61732,61733],{},[35,61734,33430],{},[40,61736,33430],{"encoding":42},[15,61738,61740],{"className":61739,"ariaHidden":48},[47],[15,61741,61743,61746],{"className":61742},[52],[15,61744],{"className":61745,"style":6032},[56],[15,61747,33430],{"className":61748},[61,122]," 因特征层的原因而系统性高估某些区域，主动学习会持续采样这些区域，形成采样偏差。缓解办法是在采集函数中叠加批内多样性项。",[11,61751,61752,61755,61756,61784,61785,61836,61837,61865],{},[661,61753,61754],{},"半监督与伪标签。"," 伪标签流程中，",[15,61757,61759,61772],{"className":61758,"translate":19},[18],[15,61760,61762],{"className":61761},[23],[25,61763,61764],{"xmlns":27},[29,61765,61766,61770],{},[32,61767,61768],{},[35,61769,33430],{},[40,61771,33430],{"encoding":42},[15,61773,61775],{"className":61774,"ariaHidden":48},[47],[15,61776,61778,61781],{"className":61777},[52],[15,61779],{"className":61780,"style":6032},[56],[15,61782,33430],{"className":61783},[61,122]," 提供了比「最大概率」更细的过滤准则：取 ",[15,61786,61788,61806],{"className":61787,"translate":19},[18],[15,61789,61791],{"className":61790},[23],[25,61792,61793],{"xmlns":27},[29,61794,61795,61803],{},[32,61796,61797,61799,61801],{},[35,61798,33430],{},[81,61800,2490],{},[35,61802,59655],{},[40,61804,61805],{"encoding":42},"u \u003C \\tau",[15,61807,61809,61827],{"className":61808,"ariaHidden":48},[47],[15,61810,61812,61815,61818,61821,61824],{"className":61811},[52],[15,61813],{"className":61814,"style":4656},[56],[15,61816,33430],{"className":61817},[61,122],[15,61819],{"className":61820,"style":128},[127],[15,61822,2490],{"className":61823},[132],[15,61825],{"className":61826,"style":128},[127],[15,61828,61830,61833],{"className":61829},[52],[15,61831],{"className":61832,"style":6032},[56],[15,61834,59655],{"className":61835,"style":59671},[61,122]," 的样本赋予伪标签，可以同时排除「概率高但实为分布外」的样本——这正是 MSP 过滤所无法处理的失败模式。反之，当 ",[15,61838,61840,61853],{"className":61839,"translate":19},[18],[15,61841,61843],{"className":61842},[23],[25,61844,61845],{"xmlns":27},[29,61846,61847,61851],{},[32,61848,61849],{},[35,61850,59655],{},[40,61852,59658],{"encoding":42},[15,61854,61856],{"className":61855,"ariaHidden":48},[47],[15,61857,61859,61862],{"className":61858},[52],[15,61860],{"className":61861,"style":6032},[56],[15,61863,59655],{"className":61864,"style":59671},[61,122]," 过紧时，模型会倾向于只在自己已经熟悉的区域采样伪标签，导致训练集分布收缩、对稀有模式进一步失明。该权衡与 5.6 节讨论的长尾问题同源。",[877,61867,61869],{"id":61868},"_54-安全关键场景","5.4 安全关键场景",[11,61871,61872,61873,42890],{},"在医学影像、自动驾驶与工业质检等场景中，「放弃预测并移交人类」是一个具有明确成本结构的决策，而非纯粹的统计操作。EDL 在这里的价值不是给出更准的标签，而是提供一个",[661,61874,61875],{},"可嵌入决策规则的二阶量",[33497,61877,61878,61968,62026,62061],{},[31550,61879,61880,61883,61884,61938,61939,61967],{},[661,61881,61882],{},"分割与体素级不确定性","：对每个体素输出 ",[15,61885,61887,61908],{"className":61886,"translate":19},[18],[15,61888,61890],{"className":61889},[23],[25,61891,61892],{"xmlns":27},[29,61893,61894,61906],{},[32,61895,61896,61898,61900,61902,61904],{},[35,61897,4186],{"mathvariant":231},[81,61899,1547],{},[81,61901,741],{"stretchy":86},[35,61903,3106],{"mathvariant":3105},[81,61905,750],{"stretchy":86},[40,61907,39072],{"encoding":42},[15,61909,61911],{"className":61910,"ariaHidden":48},[47],[15,61912,61914,61917,61923,61926,61935],{"className":61913},[52],[15,61915],{"className":61916,"style":142},[56],[15,61918,61920],{"className":61919},[531],[15,61921,4186],{"className":61922},[61,1622],[15,61924,741],{"className":61925},[146],[15,61927,61929],{"className":61928},[61],[15,61930,61932],{"className":61931},[61],[15,61933,3106],{"className":61934},[61,3166],[15,61936,750],{"className":61937},[178],"，得到逐像素的 ",[15,61940,61942,61955],{"className":61941,"translate":19},[18],[15,61943,61945],{"className":61944},[23],[25,61946,61947],{"xmlns":27},[29,61948,61949,61953],{},[32,61950,61951],{},[35,61952,33430],{},[40,61954,33430],{"encoding":42},[15,61956,61958],{"className":61957,"ariaHidden":48},[47],[15,61959,61961,61964],{"className":61960},[52],[15,61962],{"className":61963,"style":6032},[56],[15,61965,33430],{"className":61966},[61,122],"，可据此生成「需人工复核区域」的掩码。",[31550,61969,61970,61973,61974,62025],{},[661,61971,61972],{},"转诊（referral）机制","：将 ",[15,61975,61977,61995],{"className":61976,"translate":19},[18],[15,61978,61980],{"className":61979},[23],[25,61981,61982],{"xmlns":27},[29,61983,61984,61992],{},[32,61985,61986,61988,61990],{},[35,61987,33430],{},[81,61989,1590],{},[35,61991,59655],{},[40,61993,61994],{"encoding":42},"u > \\tau",[15,61996,61998,62016],{"className":61997,"ariaHidden":48},[47],[15,61999,62001,62004,62007,62010,62013],{"className":62000},[52],[15,62002],{"className":62003,"style":4656},[56],[15,62005,33430],{"className":62006},[61,122],[15,62008],{"className":62009,"style":128},[127],[15,62011,1590],{"className":62012},[132],[15,62014],{"className":62015,"style":128},[127],[15,62017,62019,62022],{"className":62018},[52],[15,62020],{"className":62021,"style":6032},[56],[15,62023,59655],{"className":62024,"style":59671},[61,122]," 的样本路由至专家，使自动化系统在其能力边界处显式地退出。",[31550,62027,62028,62031,62032,62060],{},[661,62029,62030],{},"开放集动作识别","：视频中存在未见动作时，",[15,62033,62035,62048],{"className":62034,"translate":19},[18],[15,62036,62038],{"className":62037},[23],[25,62039,62040],{"xmlns":27},[29,62041,62042,62046],{},[32,62043,62044],{},[35,62045,33430],{},[40,62047,33430],{"encoding":42},[15,62049,62051],{"className":62050,"ariaHidden":48},[47],[15,62052,62054,62057],{"className":62053},[52],[15,62055],{"className":62056,"style":6032},[56],[15,62058,33430],{"className":62059},[61,122]," 提供拒绝信号。",[31550,62062,62063,62066],{},[661,62064,62065],{},"分子性质预测与虚拟筛选","：在化学空间中标识训练分布之外的候选物，避免对高不确定性的预测投入昂贵的实验验证。",[11,62068,62069,62070,62073,62074,62102],{},"这些应用的共同点是：",[661,62071,62072],{},"错误的代价不对称，且「不预测」是允许的动作。"," 这也划定了 EDL 不适用的场合——当系统必须对所有输入给出一个标签、且拒识没有对应的动作时，",[15,62075,62077,62090],{"className":62076,"translate":19},[18],[15,62078,62080],{"className":62079},[23],[25,62081,62082],{"xmlns":27},[29,62083,62084,62088],{},[32,62085,62086],{},[35,62087,33430],{},[40,62089,33430],{"encoding":42},[15,62091,62093],{"className":62092,"ariaHidden":48},[47],[15,62094,62096,62099],{"className":62095},[52],[15,62097],{"className":62098,"style":6032},[56],[15,62100,33430],{"className":62101},[61,122]," 至多是一个监控指标。",[877,62104,62106],{"id":62105},"_55-回归情形的推广","5.5 回归情形的推广",[11,62108,62109,62110,42890],{},"分类的目标是单纯形上的向量，回归的目标是实数。相应地把迪利克雷替换为",[661,62111,62112],{},"正态—逆 Gamma（Normal-Inverse-Gamma, NIG）先验",[15,62114,62116],{"className":62115,"translate":19},[212],[15,62117,62119,62193],{"className":62118,"translate":19},[18],[15,62120,62122],{"className":62121},[23],[25,62123,62124],{"xmlns":27,"display":221},[29,62125,62126,62190],{},[32,62127,62128,62130,62132,62134,62136,62158,62160,62162,62168,62170,62180,62182,62184,62186,62188],{},[35,62129,18346],{},[81,62131,4183],{},[35,62133,24982],{"mathvariant":37},[282,62135,6799],{},[32,62137,62138,62140,62142,62144,62156],{},[81,62139,741],{"fence":48},[35,62141,9410],{},[81,62143,94],{"separator":48},[3722,62145,62146,62153],{},[227,62147,62148,62151],{},[35,62149,62150],{},"σ",[89,62152,908],{},[35,62154,62155],{},"υ",[81,62157,750],{"fence":48},[81,62159,94],{"separator":48},[127,62161],{"width":9847},[227,62163,62164,62166],{},[35,62165,62150],{},[89,62167,908],{},[81,62169,4183],{},[227,62171,62172,62174],{},[35,62173,4292],{"mathvariant":231},[32,62175,62176,62178],{},[81,62177,239],{},[89,62179,91],{},[81,62181,741],{"stretchy":86},[35,62183,3106],{},[81,62185,94],{"separator":48},[35,62187,17131],{},[81,62189,750],{"stretchy":86},[40,62191,62192],{"encoding":42},"\\mu \\sim \\mathcal{N}\\!\\left(\\gamma, \\frac{\\sigma^2}{\\upsilon}\\right),\n\\qquad\n\\sigma^2 \\sim \\Gamma^{-1}(\\alpha, \\beta)",[15,62194,62196,62214,62393],{"className":62195,"ariaHidden":48},[47],[15,62197,62199,62202,62205,62208,62211],{"className":62198},[52],[15,62200],{"className":62201,"style":9437},[56],[15,62203,18346],{"className":62204},[61,122],[15,62206],{"className":62207,"style":128},[127],[15,62209,4183],{"className":62210},[132],[15,62212],{"className":62213,"style":128},[127],[15,62215,62217,62221,62225,62228,62231,62343,62346,62349,62352,62355,62384,62387,62390],{"className":62216},[52],[15,62218],{"className":62219,"style":62220},[56],"height:2.4411em;vertical-align:-0.95em;",[15,62222,24982],{"className":62223,"style":62224},[61,62],"margin-right:0.1474em;",[15,62226],{"className":62227,"style":7105},[127],[15,62229],{"className":62230,"style":157},[127],[15,62232,62234,62240,62243,62246,62249,62337],{"className":62233},[161],[15,62235,62237],{"className":62236,"style":402},[146,401],[15,62238,741],{"className":62239},[406,364],[15,62241,9410],{"className":62242,"style":9444},[61,122],[15,62244,94],{"className":62245},[153],[15,62247],{"className":62248,"style":157},[127],[15,62250,62252,62255,62334],{"className":62251},[61],[15,62253],{"className":62254},[146,3841],[15,62256,62258],{"className":62257},[3722],[15,62259,62261,62326],{"className":62260},[342,472],[15,62262,62264,62323],{"className":62263},[346],[15,62265,62267,62278,62286],{"className":62266,"style":11399},[350],[15,62268,62269,62272],{"style":11402},[15,62270],{"className":62271,"style":3861},[357],[15,62273,62275],{"className":62274},[61],[15,62276,62155],{"className":62277,"style":123},[61,122],[15,62279,62280,62283],{"style":3889},[15,62281],{"className":62282,"style":3861},[357],[15,62284],{"className":62285,"style":3897},[3896],[15,62287,62288,62291],{"style":11468},[15,62289],{"className":62290,"style":3861},[357],[15,62292,62294],{"className":62293},[61],[15,62295,62297,62300],{"className":62296},[61],[15,62298,62150],{"className":62299,"style":123},[61,122],[15,62301,62303],{"className":62302},[338],[15,62304,62306],{"className":62305},[342],[15,62307,62309],{"className":62308},[346],[15,62310,62312],{"className":62311,"style":7905},[350],[15,62313,62314,62317],{"style":716},[15,62315],{"className":62316,"style":358},[357],[15,62318,62320],{"className":62319},[362,363,364,365],[15,62321,908],{"className":62322},[61,365],[15,62324,497],{"className":62325},[496],[15,62327,62329],{"className":62328},[346],[15,62330,62332],{"className":62331,"style":36614},[350],[15,62333],{},[15,62335],{"className":62336},[178,3841],[15,62338,62340],{"className":62339,"style":402},[178,401],[15,62341,750],{"className":62342},[406,364],[15,62344],{"className":62345,"style":157},[127],[15,62347,94],{"className":62348},[153],[15,62350],{"className":62351,"style":10035},[127],[15,62353],{"className":62354,"style":157},[127],[15,62356,62358,62361],{"className":62357},[61],[15,62359,62150],{"className":62360,"style":123},[61,122],[15,62362,62364],{"className":62363},[338],[15,62365,62367],{"className":62366},[342],[15,62368,62370],{"className":62369},[346],[15,62371,62373],{"className":62372,"style":7363},[350],[15,62374,62375,62378],{"style":353},[15,62376],{"className":62377,"style":358},[357],[15,62379,62381],{"className":62380},[362,363,364,365],[15,62382,908],{"className":62383},[61,365],[15,62385],{"className":62386,"style":128},[127],[15,62388,4183],{"className":62389},[132],[15,62391],{"className":62392,"style":128},[127],[15,62394,62396,62400,62435,62438,62441,62444,62447,62450],{"className":62395},[52],[15,62397],{"className":62398,"style":62399},[56],"height:1.1141em;vertical-align:-0.25em;",[15,62401,62403,62406],{"className":62402},[61],[15,62404,4292],{"className":62405},[61],[15,62407,62409],{"className":62408},[338],[15,62410,62412],{"className":62411},[342],[15,62413,62415],{"className":62414},[346],[15,62416,62418],{"className":62417,"style":7363},[350],[15,62419,62420,62423],{"style":353},[15,62421],{"className":62422,"style":358},[357],[15,62424,62426],{"className":62425},[362,363,364,365],[15,62427,62429,62432],{"className":62428},[61,365],[15,62430,239],{"className":62431},[61,365],[15,62433,91],{"className":62434},[61,365],[15,62436,741],{"className":62437},[146],[15,62439,3106],{"className":62440,"style":3191},[61,122],[15,62442,94],{"className":62443},[153],[15,62445],{"className":62446,"style":157},[127],[15,62448,17131],{"className":62449,"style":17270},[61,122],[15,62451,750],{"className":62452},[178],[11,62454,62455,62456,62534,62535,62569,62570,62599,62600,62671,62672,62755,62756,62876,62877,62569,62910,62938],{},"网络对每个输入输出四元组 ",[15,62457,62459,62489],{"className":62458,"translate":19},[18],[15,62460,62462],{"className":62461},[23],[25,62463,62464],{"xmlns":27},[29,62465,62466,62486],{},[32,62467,62468,62470,62472,62474,62476,62478,62480,62482,62484],{},[81,62469,741],{"stretchy":86},[35,62471,9410],{},[81,62473,94],{"separator":48},[35,62475,62155],{},[81,62477,94],{"separator":48},[35,62479,3106],{},[81,62481,94],{"separator":48},[35,62483,17131],{},[81,62485,750],{"stretchy":86},[40,62487,62488],{"encoding":42},"(\\gamma, \\upsilon, \\alpha, \\beta)",[15,62490,62492],{"className":62491,"ariaHidden":48},[47],[15,62493,62495,62498,62501,62504,62507,62510,62513,62516,62519,62522,62525,62528,62531],{"className":62494},[52],[15,62496],{"className":62497,"style":142},[56],[15,62499,741],{"className":62500},[146],[15,62502,9410],{"className":62503,"style":9444},[61,122],[15,62505,94],{"className":62506},[153],[15,62508],{"className":62509,"style":157},[127],[15,62511,62155],{"className":62512,"style":123},[61,122],[15,62514,94],{"className":62515},[153],[15,62517],{"className":62518,"style":157},[127],[15,62520,3106],{"className":62521,"style":3191},[61,122],[15,62523,94],{"className":62524},[153],[15,62526],{"className":62527,"style":157},[127],[15,62529,17131],{"className":62530,"style":17270},[61,122],[15,62532,750],{"className":62533},[178],"，边缘预测为自由度 ",[15,62536,62538,62554],{"className":62537,"translate":19},[18],[15,62539,62541],{"className":62540},[23],[25,62542,62543],{"xmlns":27},[29,62544,62545,62551],{},[32,62546,62547,62549],{},[89,62548,908],{},[35,62550,3106],{},[40,62552,62553],{"encoding":42},"2\\alpha",[15,62555,62557],{"className":62556,"ariaHidden":48},[47],[15,62558,62560,62563,62566],{"className":62559},[52],[15,62561],{"className":62562,"style":859},[56],[15,62564,908],{"className":62565},[61],[15,62567,3106],{"className":62568,"style":3191},[61,122]," 的 Student-",[15,62571,62573,62586],{"className":62572,"translate":19},[18],[15,62574,62576],{"className":62575},[23],[25,62577,62578],{"xmlns":27},[29,62579,62580,62584],{},[32,62581,62582],{},[35,62583,5297],{},[40,62585,5297],{"encoding":42},[15,62587,62589],{"className":62588,"ariaHidden":48},[47],[15,62590,62592,62596],{"className":62591},[52],[15,62593],{"className":62594,"style":62595},[56],"height:0.6151em;",[15,62597,5297],{"className":62598},[61,122]," 分布。其预测方差可精确分解为偶然部分 ",[15,62601,62603,62629],{"className":62602,"translate":19},[18],[15,62604,62606],{"className":62605},[23],[25,62607,62608],{"xmlns":27},[29,62609,62610,62626],{},[32,62611,62612,62614,62616,62618,62620,62622,62624],{},[35,62613,17131],{},[35,62615,3020],{"mathvariant":231},[81,62617,741],{"stretchy":86},[35,62619,3106],{},[81,62621,239],{},[89,62623,91],{},[81,62625,750],{"stretchy":86},[40,62627,62628],{"encoding":42},"\\beta\u002F(\\alpha - 1)",[15,62630,62632,62659],{"className":62631,"ariaHidden":48},[47],[15,62633,62635,62638,62641,62644,62647,62650,62653,62656],{"className":62634},[52],[15,62636],{"className":62637,"style":142},[56],[15,62639,17131],{"className":62640,"style":17270},[61,122],[15,62642,3020],{"className":62643},[61],[15,62645,741],{"className":62646},[146],[15,62648,3106],{"className":62649,"style":3191},[61,122],[15,62651],{"className":62652,"style":772},[127],[15,62654,239],{"className":62655},[375],[15,62657],{"className":62658,"style":772},[127],[15,62660,62662,62665,62668],{"className":62661},[52],[15,62663],{"className":62664,"style":142},[56],[15,62666,91],{"className":62667},[61],[15,62669,750],{"className":62670},[178]," 与认知部分 ",[15,62673,62675,62707],{"className":62674,"translate":19},[18],[15,62676,62678],{"className":62677},[23],[25,62679,62680],{"xmlns":27},[29,62681,62682,62704],{},[32,62683,62684,62686,62688,62690,62692,62694,62696,62698,62700,62702],{},[35,62685,17131],{},[35,62687,3020],{"mathvariant":231},[81,62689,14821],{"stretchy":86},[35,62691,62155],{},[81,62693,741],{"stretchy":86},[35,62695,3106],{},[81,62697,239],{},[89,62699,91],{},[81,62701,750],{"stretchy":86},[81,62703,14830],{"stretchy":86},[40,62705,62706],{"encoding":42},"\\beta\u002F[\\upsilon(\\alpha - 1)]",[15,62708,62710,62743],{"className":62709,"ariaHidden":48},[47],[15,62711,62713,62716,62719,62722,62725,62728,62731,62734,62737,62740],{"className":62712},[52],[15,62714],{"className":62715,"style":142},[56],[15,62717,17131],{"className":62718,"style":17270},[61,122],[15,62720,3020],{"className":62721},[61],[15,62723,14821],{"className":62724},[146],[15,62726,62155],{"className":62727,"style":123},[61,122],[15,62729,741],{"className":62730},[146],[15,62732,3106],{"className":62733,"style":3191},[61,122],[15,62735],{"className":62736,"style":772},[127],[15,62738,239],{"className":62739},[375],[15,62741],{"className":62742,"style":772},[127],[15,62744,62746,62749,62752],{"className":62745},[52],[15,62747],{"className":62748,"style":142},[56],[15,62750,91],{"className":62751},[61],[15,62753,43303],{"className":62754},[178],"，二者之和 ",[15,62757,62759,62801],{"className":62758,"translate":19},[18],[15,62760,62762],{"className":62761},[23],[25,62763,62764],{"xmlns":27},[29,62765,62766,62798],{},[32,62767,62768,62770,62772,62774,62776,62778,62780,62782,62784,62786,62788,62790,62792,62794,62796],{},[35,62769,17131],{},[81,62771,741],{"stretchy":86},[35,62773,62155],{},[81,62775,7762],{},[89,62777,91],{},[81,62779,750],{"stretchy":86},[35,62781,3020],{"mathvariant":231},[81,62783,14821],{"stretchy":86},[35,62785,62155],{},[81,62787,741],{"stretchy":86},[35,62789,3106],{},[81,62791,239],{},[89,62793,91],{},[81,62795,750],{"stretchy":86},[81,62797,14830],{"stretchy":86},[40,62799,62800],{"encoding":42},"\\beta(\\upsilon + 1)\u002F[\\upsilon(\\alpha - 1)]",[15,62802,62804,62828,62864],{"className":62803,"ariaHidden":48},[47],[15,62805,62807,62810,62813,62816,62819,62822,62825],{"className":62806},[52],[15,62808],{"className":62809,"style":142},[56],[15,62811,17131],{"className":62812,"style":17270},[61,122],[15,62814,741],{"className":62815},[146],[15,62817,62155],{"className":62818,"style":123},[61,122],[15,62820],{"className":62821,"style":772},[127],[15,62823,7762],{"className":62824},[375],[15,62826],{"className":62827,"style":772},[127],[15,62829,62831,62834,62837,62840,62843,62846,62849,62852,62855,62858,62861],{"className":62830},[52],[15,62832],{"className":62833,"style":142},[56],[15,62835,91],{"className":62836},[61],[15,62838,750],{"className":62839},[178],[15,62841,3020],{"className":62842},[61],[15,62844,14821],{"className":62845},[146],[15,62847,62155],{"className":62848,"style":123},[61,122],[15,62850,741],{"className":62851},[146],[15,62853,3106],{"className":62854,"style":3191},[61,122],[15,62856],{"className":62857,"style":772},[127],[15,62859,239],{"className":62860},[375],[15,62862],{"className":62863,"style":772},[127],[15,62865,62867,62870,62873],{"className":62866},[52],[15,62868],{"className":62869,"style":142},[56],[15,62871,91],{"className":62872},[61],[15,62874,43303],{"className":62875},[178]," 即自由度 ",[15,62878,62880,62895],{"className":62879,"translate":19},[18],[15,62881,62883],{"className":62882},[23],[25,62884,62885],{"xmlns":27},[29,62886,62887,62893],{},[32,62888,62889,62891],{},[89,62890,908],{},[35,62892,3106],{},[40,62894,62553],{"encoding":42},[15,62896,62898],{"className":62897,"ariaHidden":48},[47],[15,62899,62901,62904,62907],{"className":62900},[52],[15,62902],{"className":62903,"style":859},[56],[15,62905,908],{"className":62906},[61],[15,62908,3106],{"className":62909,"style":3191},[61,122],[15,62911,62913,62926],{"className":62912,"translate":19},[18],[15,62914,62916],{"className":62915},[23],[25,62917,62918],{"xmlns":27},[29,62919,62920,62924],{},[32,62921,62922],{},[35,62923,5297],{},[40,62925,5297],{"encoding":42},[15,62927,62929],{"className":62928,"ariaHidden":48},[47],[15,62930,62932,62935],{"className":62931},[52],[15,62933],{"className":62934,"style":62595},[56],[15,62936,5297],{"className":62937},[61,122]," 分布的方差，形式上与命题 3.5 的分解对应。该推广在分子性质预测、三维重建与不确定性感知的姿态估计中均有应用。",[11,62940,62941],{},"需要指出的是，这一推广的理论状况比分类情形更受争议：有后续研究指出，此类模型所报告的「认知」分量并不随训练数据的增加而按预期衰减，其与偶然分量的分离缺乏理论保证。因此 4.5 节的谨慎立场在回归情形下应进一步加强。",[877,62943,62945],{"id":62944},"_56-与原型学习超球面几何的融合","5.6 与原型学习、超球面几何的融合",[11,62947,62948,62949,63111,63112,63261,63262,63420,63421,63561],{},"最后讨论一个尚未定型、但在方法上自然的方向：把 EDL 与基于原型的超球面表示学习结合。设主干网络输出归一化特征 ",[15,62950,62952,62990],{"className":62951,"translate":19},[18],[15,62953,62955],{"className":62954},[23],[25,62956,62957],{"xmlns":27},[29,62958,62959,62987],{},[32,62960,62961,62967,62969,62971,62973,62975],{},[266,62962,62963,62965],{},[35,62964,3707],{"mathvariant":251},[35,62966,37148],{"mathvariant":3105},[81,62968,741],{"stretchy":86},[35,62970,1029],{"mathvariant":251},[81,62972,750],{"stretchy":86},[81,62974,83],{},[227,62976,62977,62979],{},[35,62978,40265],{"mathvariant":258},[32,62980,62981,62983,62985],{},[35,62982,2788],{},[81,62984,239],{},[89,62986,91],{},[40,62988,62989],{"encoding":42},"\\mathbf{f}_{\\boldsymbol{\\theta}}(\\mathbf{x}) \\in \\mathbb{S}^{d-1}",[15,62991,62993,63067],{"className":62992,"ariaHidden":48},[47],[15,62994,62996,62999,63049,63052,63055,63058,63061,63064],{"className":62995},[52],[15,62997],{"className":62998,"style":142},[56],[15,63000,63002,63005],{"className":63001},[61],[15,63003,3707],{"className":63004,"style":25097},[61,411],[15,63006,63008],{"className":63007},[338],[15,63009,63011,63041],{"className":63010},[342,472],[15,63012,63014,63038],{"className":63013},[346],[15,63015,63017],{"className":63016,"style":479},[350],[15,63018,63020,63023],{"style":63019},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[15,63021],{"className":63022,"style":358},[357],[15,63024,63026],{"className":63025},[362,363,364,365],[15,63027,63029],{"className":63028},[61,365],[15,63030,63032],{"className":63031},[61,365],[15,63033,63035],{"className":63034},[61,365],[15,63036,37148],{"className":63037,"style":37212},[61,3166,365],[15,63039,497],{"className":63040},[496],[15,63042,63044],{"className":63043},[346],[15,63045,63047],{"className":63046,"style":504},[350],[15,63048],{},[15,63050,741],{"className":63051},[146],[15,63053,1029],{"className":63054},[61,411],[15,63056,750],{"className":63057},[178],[15,63059],{"className":63060,"style":128},[127],[15,63062,83],{"className":63063},[132],[15,63065],{"className":63066,"style":128},[127],[15,63068,63070,63073],{"className":63069},[52],[15,63071],{"className":63072,"style":18970},[56],[15,63074,63076,63079],{"className":63075},[61],[15,63077,40265],{"className":63078},[61,427],[15,63080,63082],{"className":63081},[338],[15,63083,63085],{"className":63084},[342],[15,63086,63088],{"className":63087},[346],[15,63089,63091],{"className":63090,"style":18970},[350],[15,63092,63093,63096],{"style":716},[15,63094],{"className":63095,"style":358},[357],[15,63097,63099],{"className":63098},[362,363,364,365],[15,63100,63102,63105,63108],{"className":63101},[61,365],[15,63103,2788],{"className":63104},[61,122,365],[15,63106,239],{"className":63107},[375,365],[15,63109,91],{"className":63110},[61,365],"，各类原型为 ",[15,63113,63115,63152],{"className":63114,"translate":19},[18],[15,63116,63118],{"className":63117},[23],[25,63119,63120],{"xmlns":27},[29,63121,63122,63149],{},[32,63123,63124,63126,63132,63134,63137],{},[81,63125,87],{"stretchy":86},[266,63127,63128,63130],{},[35,63129,30459],{"mathvariant":251},[35,63131,272],{},[81,63133,105],{"stretchy":86},[81,63135,63136],{},"⊂",[227,63138,63139,63141],{},[35,63140,40265],{"mathvariant":258},[32,63142,63143,63145,63147],{},[35,63144,2788],{},[81,63146,239],{},[89,63148,91],{},[40,63150,63151],{"encoding":42},"\\{\\mathbf{v}_k\\} \\subset \\mathbb{S}^{d-1}",[15,63153,63155,63217],{"className":63154,"ariaHidden":48},[47],[15,63156,63158,63161,63164,63205,63208,63211,63214],{"className":63157},[52],[15,63159],{"className":63160,"style":142},[56],[15,63162,87],{"className":63163},[146],[15,63165,63167,63170],{"className":63166},[61],[15,63168,30459],{"className":63169,"style":29413},[61,411],[15,63171,63173],{"className":63172},[338],[15,63174,63176,63197],{"className":63175},[342,472],[15,63177,63179,63194],{"className":63178},[346],[15,63180,63182],{"className":63181,"style":479},[350],[15,63183,63185,63188],{"style":63184},"top:-2.55em;margin-left:-0.016em;margin-right:0.05em;",[15,63186],{"className":63187,"style":358},[357],[15,63189,63191],{"className":63190},[362,363,364,365],[15,63192,272],{"className":63193,"style":492},[61,122,365],[15,63195,497],{"className":63196},[496],[15,63198,63200],{"className":63199},[346],[15,63201,63203],{"className":63202,"style":504},[350],[15,63204],{},[15,63206,105],{"className":63207},[178],[15,63209],{"className":63210,"style":128},[127],[15,63212,63136],{"className":63213},[132],[15,63215],{"className":63216,"style":128},[127],[15,63218,63220,63223],{"className":63219},[52],[15,63221],{"className":63222,"style":18970},[56],[15,63224,63226,63229],{"className":63225},[61],[15,63227,40265],{"className":63228},[61,427],[15,63230,63232],{"className":63231},[338],[15,63233,63235],{"className":63234},[342],[15,63236,63238],{"className":63237},[346],[15,63239,63241],{"className":63240,"style":18970},[350],[15,63242,63243,63246],{"style":716},[15,63244],{"className":63245,"style":358},[357],[15,63247,63249],{"className":63248},[362,363,364,365],[15,63250,63252,63255,63258],{"className":63251},[61,365],[15,63253,2788],{"className":63254},[61,122,365],[15,63256,239],{"className":63257},[375,365],[15,63259,91],{"className":63260},[61,365],"，相似度 ",[15,63263,63265,63301],{"className":63264,"translate":19},[18],[15,63266,63268],{"className":63267},[23],[25,63269,63270],{"xmlns":27},[29,63271,63272,63298],{},[32,63273,63274,63280,63282,63285,63287,63289,63295],{},[266,63275,63276,63278],{},[35,63277,5864],{},[35,63279,272],{},[81,63281,244],{},[81,63283,63284],{"stretchy":86},"⟨",[35,63286,3707],{"mathvariant":251},[81,63288,94],{"separator":48},[266,63290,63291,63293],{},[35,63292,30459],{"mathvariant":251},[35,63294,272],{},[81,63296,63297],{"stretchy":86},"⟩",[40,63299,63300],{"encoding":42},"s_k = \\langle \\mathbf{f}, \\mathbf{v}_k \\rangle",[15,63302,63304,63359],{"className":63303,"ariaHidden":48},[47],[15,63305,63307,63310,63350,63353,63356],{"className":63306},[52],[15,63308],{"className":63309,"style":2511},[56],[15,63311,63313,63316],{"className":63312},[61],[15,63314,5864],{"className":63315},[61,122],[15,63317,63319],{"className":63318},[338],[15,63320,63322,63342],{"className":63321},[342,472],[15,63323,63325,63339],{"className":63324},[346],[15,63326,63328],{"className":63327,"style":479},[350],[15,63329,63330,63333],{"style":482},[15,63331],{"className":63332,"style":358},[357],[15,63334,63336],{"className":63335},[362,363,364,365],[15,63337,272],{"className":63338,"style":492},[61,122,365],[15,63340,497],{"className":63341},[496],[15,63343,63345],{"className":63344},[346],[15,63346,63348],{"className":63347,"style":504},[350],[15,63349],{},[15,63351],{"className":63352,"style":128},[127],[15,63354,244],{"className":63355},[132],[15,63357],{"className":63358,"style":128},[127],[15,63360,63362,63365,63368,63371,63374,63377,63417],{"className":63361},[52],[15,63363],{"className":63364,"style":142},[56],[15,63366,63284],{"className":63367},[146],[15,63369,3707],{"className":63370,"style":25097},[61,411],[15,63372,94],{"className":63373},[153],[15,63375],{"className":63376,"style":157},[127],[15,63378,63380,63383],{"className":63379},[61],[15,63381,30459],{"className":63382,"style":29413},[61,411],[15,63384,63386],{"className":63385},[338],[15,63387,63389,63409],{"className":63388},[342,472],[15,63390,63392,63406],{"className":63391},[346],[15,63393,63395],{"className":63394,"style":479},[350],[15,63396,63397,63400],{"style":63184},[15,63398],{"className":63399,"style":358},[357],[15,63401,63403],{"className":63402},[362,363,364,365],[15,63404,272],{"className":63405,"style":492},[61,122,365],[15,63407,497],{"className":63408},[496],[15,63410,63412],{"className":63411},[346],[15,63413,63415],{"className":63414,"style":504},[350],[15,63416],{},[15,63418,63297],{"className":63419},[178],"。传统原型方法以 ",[15,63422,63424,63453],{"className":63423,"translate":19},[18],[15,63425,63427],{"className":63426},[23],[25,63428,63429],{"xmlns":27},[29,63430,63431,63450],{},[32,63432,63433,63439,63441,63444],{},[266,63434,63435,63437],{},[35,63436,37301],{"mathvariant":251},[35,63438,272],{},[81,63440,244],{},[35,63442,63443],{},"κ",[266,63445,63446,63448],{},[35,63447,5864],{},[35,63449,272],{},[40,63451,63452],{"encoding":42},"\\mathbf{z}_k = \\kappa s_k",[15,63454,63456,63512],{"className":63455,"ariaHidden":48},[47],[15,63457,63459,63463,63503,63506,63509],{"className":63458},[52],[15,63460],{"className":63461,"style":63462},[56],"height:0.5944em;vertical-align:-0.15em;",[15,63464,63466,63469],{"className":63465},[61],[15,63467,37301],{"className":63468},[61,411],[15,63470,63472],{"className":63471},[338],[15,63473,63475,63495],{"className":63474},[342,472],[15,63476,63478,63492],{"className":63477},[346],[15,63479,63481],{"className":63480,"style":479},[350],[15,63482,63483,63486],{"style":482},[15,63484],{"className":63485,"style":358},[357],[15,63487,63489],{"className":63488},[362,363,364,365],[15,63490,272],{"className":63491,"style":492},[61,122,365],[15,63493,497],{"className":63494},[496],[15,63496,63498],{"className":63497},[346],[15,63499,63501],{"className":63500,"style":504},[350],[15,63502],{},[15,63504],{"className":63505,"style":128},[127],[15,63507,244],{"className":63508},[132],[15,63510],{"className":63511,"style":128},[127],[15,63513,63515,63518,63521],{"className":63514},[52],[15,63516],{"className":63517,"style":2511},[56],[15,63519,63443],{"className":63520},[61,122],[15,63522,63524,63527],{"className":63523},[61],[15,63525,5864],{"className":63526},[61,122],[15,63528,63530],{"className":63529},[338],[15,63531,63533,63553],{"className":63532},[342,472],[15,63534,63536,63550],{"className":63535},[346],[15,63537,63539],{"className":63538,"style":479},[350],[15,63540,63541,63544],{"style":482},[15,63542],{"className":63543,"style":358},[357],[15,63545,63547],{"className":63546},[362,363,364,365],[15,63548,272],{"className":63549,"style":492},[61,122,365],[15,63551,497],{"className":63552},[496],[15,63554,63556],{"className":63555},[346],[15,63557,63559],{"className":63558,"style":504},[350],[15,63560],{}," 作为 logits 并配 softmax 交叉熵，由 3.3 节，这仍受平移不变性与 logits 发散两项缺陷的支配。",[11,63563,63564],{},"一个直接的融合方案是把相似度送入证据函数：",[15,63566,63568],{"className":63567,"translate":19},[212],[15,63569,63571,63681],{"className":63570,"translate":19},[18],[15,63572,63574],{"className":63573},[23],[25,63575,63576],{"xmlns":27,"display":221},[29,63577,63578,63678],{},[32,63579,63580,63586,63588,63590,63592,63594,63596,63602,63604,63606,63608,63614,63616,63618,63620,63622,63624,63626,63628,63634,63636,63638,63640,63642,63644,63646,63648,63662,63664,63666,63668,63670,63676],{},[266,63581,63582,63584],{},[35,63583,1982],{},[35,63585,272],{},[81,63587,244],{},[35,63589,22794],{},[81,63591,741],{"stretchy":86},[35,63593,63443],{},[282,63595,4802],{},[266,63597,63598,63600],{},[35,63599,5864],{},[35,63601,272],{},[81,63603,750],{"stretchy":86},[81,63605,94],{"separator":48},[127,63607],{"width":9847},[266,63609,63610,63612],{},[35,63611,3106],{},[35,63613,272],{},[81,63615,244],{},[89,63617,91],{},[81,63619,7762],{},[35,63621,22794],{},[81,63623,741],{"stretchy":86},[35,63625,63443],{},[282,63627,4802],{},[266,63629,63630,63632],{},[35,63631,5864],{},[35,63633,272],{},[81,63635,750],{"stretchy":86},[81,63637,94],{"separator":48},[127,63639],{"width":9847},[35,63641,40265],{},[81,63643,244],{},[35,63645,102],{},[81,63647,7762],{},[5235,63649,63650,63652,63660],{},[81,63651,293],{},[32,63653,63654,63656,63658],{},[35,63655,272],{},[81,63657,244],{},[89,63659,91],{},[35,63661,102],{},[35,63663,22794],{},[81,63665,741],{"stretchy":86},[35,63667,63443],{},[282,63669,4802],{},[266,63671,63672,63674],{},[35,63673,5864],{},[35,63675,272],{},[81,63677,750],{"stretchy":86},[40,63679,63680],{"encoding":42},"e_k = g(\\kappa\\, s_k), \\qquad \\alpha_k = 1 + g(\\kappa\\, s_k), \\qquad S = K + \\sum_{k=1}^K g(\\kappa\\, s_k)",[15,63682,63684,63739,63858,63876,63958,63976],{"className":63683,"ariaHidden":48},[47],[15,63685,63687,63690,63730,63733,63736],{"className":63686},[52],[15,63688],{"className":63689,"style":2511},[56],[15,63691,63693,63696],{"className":63692},[61],[15,63694,1982],{"className":63695},[61,122],[15,63697,63699],{"className":63698},[338],[15,63700,63702,63722],{"className":63701},[342,472],[15,63703,63705,63719],{"className":63704},[346],[15,63706,63708],{"className":63707,"style":479},[350],[15,63709,63710,63713],{"style":482},[15,63711],{"className":63712,"style":358},[357],[15,63714,63716],{"className":63715},[362,363,364,365],[15,63717,272],{"className":63718,"style":492},[61,122,365],[15,63720,497],{"className":63721},[496],[15,63723,63725],{"className":63724},[346],[15,63726,63728],{"className":63727,"style":504},[350],[15,63729],{},[15,63731],{"className":63732,"style":128},[127],[15,63734,244],{"className":63735},[132],[15,63737],{"className":63738,"style":128},[127],[15,63740,63742,63745,63748,63751,63754,63757,63797,63800,63803,63806,63809,63849,63852,63855],{"className":63741},[52],[15,63743],{"className":63744,"style":142},[56],[15,63746,22794],{"className":63747,"style":123},[61,122],[15,63749,741],{"className":63750},[146],[15,63752,63443],{"className":63753},[61,122],[15,63755],{"className":63756,"style":157},[127],[15,63758,63760,63763],{"className":63759},[61],[15,63761,5864],{"className":63762},[61,122],[15,63764,63766],{"className":63765},[338],[15,63767,63769,63789],{"className":63768},[342,472],[15,63770,63772,63786],{"className":63771},[346],[15,63773,63775],{"className":63774,"style":479},[350],[15,63776,63777,63780],{"style":482},[15,63778],{"className":63779,"style":358},[357],[15,63781,63783],{"className":63782},[362,363,364,365],[15,63784,272],{"className":63785,"style":492},[61,122,365],[15,63787,497],{"className":63788},[496],[15,63790,63792],{"className":63791},[346],[15,63793,63795],{"className":63794,"style":504},[350],[15,63796],{},[15,63798,750],{"className":63799},[178],[15,63801,94],{"className":63802},[153],[15,63804],{"className":63805,"style":10035},[127],[15,63807],{"className":63808,"style":157},[127],[15,63810,63812,63815],{"className":63811},[61],[15,63813,3106],{"className":63814,"style":3191},[61,122],[15,63816,63818],{"className":63817},[338],[15,63819,63821,63841],{"className":63820},[342,472],[15,63822,63824,63838],{"className":63823},[346],[15,63825,63827],{"className":63826,"style":479},[350],[15,63828,63829,63832],{"style":3206},[15,63830],{"className":63831,"style":358},[357],[15,63833,63835],{"className":63834},[362,363,364,365],[15,63836,272],{"className":63837,"style":492},[61,122,365],[15,63839,497],{"className":63840},[496],[15,63842,63844],{"className":63843},[346],[15,63845,63847],{"className":63846,"style":504},[350],[15,63848],{},[15,63850],{"className":63851,"style":128},[127],[15,63853,244],{"className":63854},[132],[15,63856],{"className":63857,"style":128},[127],[15,63859,63861,63864,63867,63870,63873],{"className":63860},[52],[15,63862],{"className":63863,"style":2567},[56],[15,63865,91],{"className":63866},[61],[15,63868],{"className":63869,"style":772},[127],[15,63871,7762],{"className":63872},[375],[15,63874],{"className":63875,"style":772},[127],[15,63877,63879,63882,63885,63888,63891,63894,63934,63937,63940,63943,63946,63949,63952,63955],{"className":63878},[52],[15,63880],{"className":63881,"style":142},[56],[15,63883,22794],{"className":63884,"style":123},[61,122],[15,63886,741],{"className":63887},[146],[15,63889,63443],{"className":63890},[61,122],[15,63892],{"className":63893,"style":157},[127],[15,63895,63897,63900],{"className":63896},[61],[15,63898,5864],{"className":63899},[61,122],[15,63901,63903],{"className":63902},[338],[15,63904,63906,63926],{"className":63905},[342,472],[15,63907,63909,63923],{"className":63908},[346],[15,63910,63912],{"className":63911,"style":479},[350],[15,63913,63914,63917],{"style":482},[15,63915],{"className":63916,"style":358},[357],[15,63918,63920],{"className":63919},[362,363,364,365],[15,63921,272],{"className":63922,"style":492},[61,122,365],[15,63924,497],{"className":63925},[496],[15,63927,63929],{"className":63928},[346],[15,63930,63932],{"className":63931,"style":504},[350],[15,63933],{},[15,63935,750],{"className":63936},[178],[15,63938,94],{"className":63939},[153],[15,63941],{"className":63942,"style":10035},[127],[15,63944],{"className":63945,"style":157},[127],[15,63947,40265],{"className":63948,"style":40337},[61,122],[15,63950],{"className":63951,"style":128},[127],[15,63953,244],{"className":63954},[132],[15,63956],{"className":63957,"style":128},[127],[15,63959,63961,63964,63967,63970,63973],{"className":63960},[52],[15,63962],{"className":63963,"style":974},[56],[15,63965,102],{"className":63966,"style":174},[61,122],[15,63968],{"className":63969,"style":772},[127],[15,63971,7762],{"className":63972},[375],[15,63974],{"className":63975,"style":772},[127],[15,63977,63979,63983,64049,64052,64055,64058,64061,64064,64104],{"className":63978},[52],[15,63980],{"className":63981,"style":63982},[56],"height:3.1304em;vertical-align:-1.3021em;",[15,63984,63986],{"className":63985},[531,5348],[15,63987,63989,64040],{"className":63988},[342,472],[15,63990,63992,64037],{"className":63991},[346],[15,63993,63995,64016,64026],{"className":63994,"style":5358},[350],[15,63996,63998,64001],{"style":63997},"top:-1.8479em;margin-left:0em;",[15,63999],{"className":64000,"style":5365},[357],[15,64002,64004],{"className":64003},[362,363,364,365],[15,64005,64007,64010,64013],{"className":64006},[61,365],[15,64008,272],{"className":64009,"style":492},[61,122,365],[15,64011,244],{"className":64012},[132,365],[15,64014,91],{"className":64015},[61,365],[15,64017,64018,64021],{"style":5383},[15,64019],{"className":64020,"style":5365},[357],[15,64022,64023],{},[15,64024,293],{"className":64025},[531,535,5392],[15,64027,64028,64031],{"style":5395},[15,64029],{"className":64030,"style":5365},[357],[15,64032,64034],{"className":64033},[362,363,364,365],[15,64035,102],{"className":64036,"style":174},[61,122,365],[15,64038,497],{"className":64039},[496],[15,64041,64043],{"className":64042},[346],[15,64044,64047],{"className":64045,"style":64046},[350],"height:1.3021em;",[15,64048],{},[15,64050],{"className":64051,"style":157},[127],[15,64053,22794],{"className":64054,"style":123},[61,122],[15,64056,741],{"className":64057},[146],[15,64059,63443],{"className":64060},[61,122],[15,64062],{"className":64063,"style":157},[127],[15,64065,64067,64070],{"className":64066},[61],[15,64068,5864],{"className":64069},[61,122],[15,64071,64073],{"className":64072},[338],[15,64074,64076,64096],{"className":64075},[342,472],[15,64077,64079,64093],{"className":64078},[346],[15,64080,64082],{"className":64081,"style":479},[350],[15,64083,64084,64087],{"style":482},[15,64085],{"className":64086,"style":358},[357],[15,64088,64090],{"className":64089},[362,363,364,365],[15,64091,272],{"className":64092,"style":492},[61,122,365],[15,64094,497],{"className":64095},[496],[15,64097,64099],{"className":64098},[346],[15,64100,64102],{"className":64101,"style":504},[350],[15,64103],{},[15,64105,750],{"className":64106},[178],[11,64108,49048,64109,64137,64138,64167,64168,64238,64239,64267,64268,64296,64297,64366,64367,867,64417],{},[15,64110,64112,64125],{"className":64111,"translate":19},[18],[15,64113,64115],{"className":64114},[23],[25,64116,64117],{"xmlns":27},[29,64118,64119,64123],{},[32,64120,64121],{},[35,64122,22794],{},[40,64124,22794],{"encoding":42},[15,64126,64128],{"className":64127,"ariaHidden":48},[47],[15,64129,64131,64134],{"className":64130},[52],[15,64132],{"className":64133,"style":9437},[56],[15,64135,22794],{"className":64136,"style":123},[61,122]," 取 3.4 节的非负激活，",[15,64139,64141,64155],{"className":64140,"translate":19},[18],[15,64142,64144],{"className":64143},[23],[25,64145,64146],{"xmlns":27},[29,64147,64148,64152],{},[32,64149,64150],{},[35,64151,63443],{},[40,64153,64154],{"encoding":42},"\\kappa",[15,64156,64158],{"className":64157,"ariaHidden":48},[47],[15,64159,64161,64164],{"className":64160},[52],[15,64162],{"className":64163,"style":6032},[56],[15,64165,63443],{"className":64166},[61,122]," 沿用 vMF 分布的集中参数语义。该构造把几何与不确定性直接挂钩：特征落在某原型附近时，",[15,64169,64171,64189],{"className":64170,"translate":19},[18],[15,64172,64174],{"className":64173},[23],[25,64175,64176],{"xmlns":27},[29,64177,64178,64186],{},[32,64179,64180],{},[266,64181,64182,64184],{},[35,64183,5864],{},[35,64185,272],{},[40,64187,64188],{"encoding":42},"s_k",[15,64190,64192],{"className":64191,"ariaHidden":48},[47],[15,64193,64195,64198],{"className":64194},[52],[15,64196],{"className":64197,"style":2511},[56],[15,64199,64201,64204],{"className":64200},[61],[15,64202,5864],{"className":64203},[61,122],[15,64205,64207],{"className":64206},[338],[15,64208,64210,64230],{"className":64209},[342,472],[15,64211,64213,64227],{"className":64212},[346],[15,64214,64216],{"className":64215,"style":479},[350],[15,64217,64218,64221],{"style":482},[15,64219],{"className":64220,"style":358},[357],[15,64222,64224],{"className":64223},[362,363,364,365],[15,64225,272],{"className":64226,"style":492},[61,122,365],[15,64228,497],{"className":64229},[496],[15,64231,64233],{"className":64232},[346],[15,64234,64236],{"className":64235,"style":504},[350],[15,64237],{}," 接近 ",[15,64240,64242,64255],{"className":64241,"translate":19},[18],[15,64243,64245],{"className":64244},[23],[25,64246,64247],{"xmlns":27},[29,64248,64249,64253],{},[32,64250,64251],{},[89,64252,91],{},[40,64254,91],{"encoding":42},[15,64256,64258],{"className":64257,"ariaHidden":48},[47],[15,64259,64261,64264],{"className":64260},[52],[15,64262],{"className":64263,"style":859},[56],[15,64265,91],{"className":64266},[61],"、证据充分、",[15,64269,64271,64284],{"className":64270,"translate":19},[18],[15,64272,64274],{"className":64273},[23],[25,64275,64276],{"xmlns":27},[29,64277,64278,64282],{},[32,64279,64280],{},[35,64281,33430],{},[40,64283,33430],{"encoding":42},[15,64285,64287],{"className":64286,"ariaHidden":48},[47],[15,64288,64290,64293],{"className":64289},[52],[15,64291],{"className":64292,"style":6032},[56],[15,64294,33430],{"className":64295},[61,122]," 小。特征落在所有原型之外（其与全部原型的相似度均低或为负）时，各 ",[15,64298,64300,64317],{"className":64299,"translate":19},[18],[15,64301,64303],{"className":64302},[23],[25,64304,64305],{"xmlns":27},[29,64306,64307,64315],{},[32,64308,64309],{},[266,64310,64311,64313],{},[35,64312,1982],{},[35,64314,272],{},[40,64316,39308],{"encoding":42},[15,64318,64320],{"className":64319,"ariaHidden":48},[47],[15,64321,64323,64326],{"className":64322},[52],[15,64324],{"className":64325,"style":2511},[56],[15,64327,64329,64332],{"className":64328},[61],[15,64330,1982],{"className":64331},[61,122],[15,64333,64335],{"className":64334},[338],[15,64336,64338,64358],{"className":64337},[342,472],[15,64339,64341,64355],{"className":64340},[346],[15,64342,64344],{"className":64343,"style":479},[350],[15,64345,64346,64349],{"style":482},[15,64347],{"className":64348,"style":358},[357],[15,64350,64352],{"className":64351},[362,363,364,365],[15,64353,272],{"className":64354,"style":492},[61,122,365],[15,64356,497],{"className":64357},[496],[15,64359,64361],{"className":64360},[346],[15,64362,64364],{"className":64363,"style":504},[350],[15,64365],{}," 同时趋零、",[15,64368,64370,64387],{"className":64369,"translate":19},[18],[15,64371,64373],{"className":64372},[23],[25,64374,64375],{"xmlns":27},[29,64376,64377,64385],{},[32,64378,64379,64381,64383],{},[35,64380,33430],{},[81,64382,20704],{},[89,64384,91],{},[40,64386,42299],{"encoding":42},[15,64388,64390,64408],{"className":64389,"ariaHidden":48},[47],[15,64391,64393,64396,64399,64402,64405],{"className":64392},[52],[15,64394],{"className":64395,"style":6032},[56],[15,64397,33430],{"className":64398},[61,122],[15,64400],{"className":64401,"style":128},[127],[15,64403,20704],{"className":64404},[132],[15,64406],{"className":64407,"style":128},[127],[15,64409,64411,64414],{"className":64410},[52],[15,64412],{"className":64413,"style":859},[56],[15,64415,91],{"className":64416},[61],[661,64418,64419],{},"「离原型远」与「证据不足」在此被同一个机制表达。",[11,64421,64422,64423,64426,64427,64430,64431,64503,64504,64532,64533,64536,64537,64571],{},"但这一构造的成立范围必须严格界定。迪利克雷分布定义在",[661,64424,64425],{},"类概率单纯形","上，原型方法定义在",[661,64428,64429],{},"球面特征空间","上，二者的「不确定性」不是同一个对象：前者是关于类概率向量的二阶分布，后者是关于特征位置的一阶几何量。",[15,64432,64434,64458],{"className":64433,"translate":19},[18],[15,64435,64437],{"className":64436},[23],[25,64438,64439],{"xmlns":27},[29,64440,64441,64455],{},[32,64442,64443,64445,64447,64449,64451,64453],{},[35,64444,1029],{"mathvariant":251},[81,64446,14593],{},[35,64448,3106],{"mathvariant":3105},[81,64450,741],{"stretchy":86},[35,64452,1029],{"mathvariant":251},[81,64454,750],{"stretchy":86},[40,64456,64457],{"encoding":42},"\\mathbf{x} \\mapsto \\boldsymbol{\\alpha}(\\mathbf{x})",[15,64459,64461,64479],{"className":64460,"ariaHidden":48},[47],[15,64462,64464,64467,64470,64473,64476],{"className":64463},[52],[15,64465],{"className":64466,"style":14612},[56],[15,64468,1029],{"className":64469},[61,411],[15,64471],{"className":64472,"style":128},[127],[15,64474,14593],{"className":64475},[132],[15,64477],{"className":64478,"style":128},[127],[15,64480,64482,64485,64494,64497,64500],{"className":64481},[52],[15,64483],{"className":64484,"style":142},[56],[15,64486,64488],{"className":64487},[61],[15,64489,64491],{"className":64490},[61],[15,64492,3106],{"className":64493},[61,3166],[15,64495,741],{"className":64496},[146],[15,64498,1029],{"className":64499},[61,411],[15,64501,750],{"className":64502},[178]," 仍然是一个点估计的映射，因此 4.5 节第 1 点的批评在此完全适用——",[15,64505,64507,64520],{"className":64506,"translate":19},[18],[15,64508,64510],{"className":64509},[23],[25,64511,64512],{"xmlns":27},[29,64513,64514,64518],{},[32,64515,64516],{},[35,64517,33430],{},[40,64519,33430],{"encoding":42},[15,64521,64523],{"className":64522,"ariaHidden":48},[47],[15,64524,64526,64529],{"className":64525},[52],[15,64527],{"className":64528,"style":6032},[56],[15,64530,33430],{"className":64531},[61,122]," 度量的是「特征到原型的距离」，而不是「模型对该映射的把握」。要真正结合二者，需要构造（特征空间，类概率空间）的",[661,64534,64535],{},"联合","模型，例如令 ",[15,64538,64540,64553],{"className":64539,"translate":19},[18],[15,64541,64543],{"className":64542},[23],[25,64544,64545],{"xmlns":27},[29,64546,64547,64551],{},[32,64548,64549],{},[35,64550,3106],{"mathvariant":3105},[40,64552,4147],{"encoding":42},[15,64554,64556],{"className":64555,"ariaHidden":48},[47],[15,64557,64559,64562],{"className":64558},[52],[15,64560],{"className":64561,"style":2708},[56],[15,64563,64565],{"className":64564},[61],[15,64566,64568],{"className":64567},[61],[15,64569,3106],{"className":64570},[61,3166]," 自身携带分布（层级模型），或以 vMF 构造特征的生成式模型、以迪利克雷构造其上的分类头，使两个不确定性来源可以在同一层级结构中分别标识。",[11,64573,64574],{},"即便在当前的简化形式下，该方向仍有两个值得推进的具体问题：",[31547,64576,64577,64641],{},[31550,64578,64579,64582,64583,64611,64612,64640],{},[661,64580,64581],{},"长尾与稀有类。"," 稀有类的原型天然缺乏证据支撑，",[15,64584,64586,64599],{"className":64585,"translate":19},[18],[15,64587,64589],{"className":64588},[23],[25,64590,64591],{"xmlns":27},[29,64592,64593,64597],{},[32,64594,64595],{},[35,64596,33430],{},[40,64598,33430],{"encoding":42},[15,64600,64602],{"className":64601,"ariaHidden":48},[47],[15,64603,64605,64608],{"className":64604},[52],[15,64606],{"className":64607,"style":6032},[56],[15,64609,33430],{"className":64610},[61,122]," 更高。这既可以被用作保护机制（在损失中对高 ",[15,64613,64615,64628],{"className":64614,"translate":19},[18],[15,64616,64618],{"className":64617},[23],[25,64619,64620],{"xmlns":27},[29,64621,64622,64626],{},[32,64623,64624],{},[35,64625,33430],{},[40,64627,33430],{"encoding":42},[15,64629,64631],{"className":64630,"ariaHidden":48},[47],[15,64632,64634,64637],{"className":64633},[52],[15,64635],{"className":64636,"style":6032},[56],[15,64638,33430],{"className":64639},[61,122]," 样本降低梯度权重），也可能加剧对稀有类的欠拟合（模型学会用「不确定」来逃避稀有类）。二者孰占主导是一个可以实验判定的问题。",[31550,64642,64643,64646,64647,61462,64675,64737,64738],{},[661,64644,64645],{},"与开集\u002F分布外任务的重叠。"," 在代码漏洞检测这类任务中，异常样本（漏洞）与正常样本在特征空间中高度重叠，其本质是开集问题而非单纯的类别不平衡。",[15,64648,64650,64663],{"className":64649,"translate":19},[18],[15,64651,64653],{"className":64652},[23],[25,64654,64655],{"xmlns":27},[29,64656,64657,64661],{},[32,64658,64659],{},[35,64660,33430],{},[40,64662,33430],{"encoding":42},[15,64664,64666],{"className":64665,"ariaHidden":48},[47],[15,64667,64669,64672],{"className":64668},[52],[15,64670],{"className":64671,"style":6032},[56],[15,64673,33430],{"className":64674},[61,122],[15,64676,64678,64701],{"className":64677,"translate":19},[18],[15,64679,64681],{"className":64680},[23],[25,64682,64683],{"xmlns":27},[29,64684,64685,64699],{},[32,64686,64687,64689,64691,64693,64695,64697],{},[35,64688,28923],{},[81,64690,741],{"stretchy":86},[35,64692,79],{},[81,64694,11303],{"separator":48},[35,64696,24680],{"mathvariant":3105},[81,64698,750],{"stretchy":86},[40,64700,46884],{"encoding":42},[15,64702,64704],{"className":64703,"ariaHidden":48},[47],[15,64705,64707,64710,64713,64716,64719,64722,64725,64734],{"className":64706},[52],[15,64708],{"className":64709,"style":142},[56],[15,64711,28923],{"className":64712,"style":28962},[61,122],[15,64714,741],{"className":64715},[146],[15,64717,79],{"className":64718,"style":123},[61,122],[15,64720,11303],{"className":64721},[153],[15,64723],{"className":64724,"style":157},[127],[15,64726,64728],{"className":64727},[61],[15,64729,64731],{"className":64730},[61],[15,64732,24680],{"className":64733,"style":24714},[61,3166],[15,64735,750],{"className":64736},[178]," 可用作告警与拒识的分数，但必须与 4.5 节第 5 点的前提一并验证：",[661,64739,64740],{},"只有当特征提取器确实把漏洞样本映射到低证据区时，这一机制才成立，而这正是需要独立验证的假设。",[872,64742,64743],{"id":64743},"结语",[11,64745,64746],{},"回顾全文，迪利克雷分布在本文中依次扮演了三个层次的角色：",[33497,64748,64749,64859,64865],{},[31550,64750,64751,64754,64755,31730,64789,64858],{},[661,64752,64753],{},"作为单纯形上的分布","，它是「关于概率的概率」这一二阶对象的标准载体，其参数由方向 ",[15,64756,64758,64771],{"className":64757,"translate":19},[18],[15,64759,64761],{"className":64760},[23],[25,64762,64763],{"xmlns":27},[29,64764,64765,64769],{},[32,64766,64767],{},[35,64768,18346],{"mathvariant":3105},[40,64770,19451],{"encoding":42},[15,64772,64774],{"className":64773,"ariaHidden":48},[47],[15,64775,64777,64780],{"className":64776},[52],[15,64778],{"className":64779,"style":829},[56],[15,64781,64783],{"className":64782},[61],[15,64784,64786],{"className":64785},[61],[15,64787,18346],{"className":64788},[61,3166],[15,64790,64792,64809],{"className":64791,"translate":19},[18],[15,64793,64795],{"className":64794},[23],[25,64796,64797],{"xmlns":27},[29,64798,64799,64807],{},[32,64800,64801],{},[266,64802,64803,64805],{},[35,64804,3106],{},[89,64806,278],{},[40,64808,19490],{"encoding":42},[15,64810,64812],{"className":64811,"ariaHidden":48},[47],[15,64813,64815,64818],{"className":64814},[52],[15,64816],{"className":64817,"style":2511},[56],[15,64819,64821,64824],{"className":64820},[61],[15,64822,3106],{"className":64823,"style":3191},[61,122],[15,64825,64827],{"className":64826},[338],[15,64828,64830,64850],{"className":64829},[342,472],[15,64831,64833,64847],{"className":64832},[346],[15,64834,64836],{"className":64835,"style":1214},[350],[15,64837,64838,64841],{"style":3206},[15,64839],{"className":64840,"style":358},[357],[15,64842,64844],{"className":64843},[362,363,364,365],[15,64845,278],{"className":64846},[61,365],[15,64848,497],{"className":64849},[496],[15,64851,64853],{"className":64852},[346],[15,64854,64856],{"className":64855,"style":504},[350],[15,64857],{}," 两个正交分量构成（命题 2.2）。",[31550,64860,64861,64864],{},[661,64862,64863],{},"作为多项分布的共轭先验","，它把贝叶斯更新化简为一次向量加法（命题 2.5），并在聚合性、中立性与 stick-breaking 构造下形成一个结构完整、边界清晰的分布族（命题 2.8、2.9）。",[31550,64866,64867,64870],{},[661,64868,64869],{},"作为神经网络的输出层","，经由与主观逻辑的严格双射（命题 3.2），它把分类器的输出从「单纯形上的点」提升为「单纯形上的分布」，从而首次让「我不知道」成为一个可表示、可优化的对象。",[11,64872,64873,64874,64877,64878,64938],{},"证据深度学习的核心洞见可以压缩为一句话：",[661,64875,64876],{},"softmax 的平移不变性丢弃了 logits 的绝对水平，因而无法表示无信息状态。证据函数不是平移不变的，它把这一被丢弃的自由度重新赋值为「证据的量级」。"," 命题 3.4 中 ",[15,64879,64881,64902],{"className":64880,"translate":19},[18],[15,64882,64884],{"className":64883},[23],[25,64885,64886],{"xmlns":27},[29,64887,64888,64900],{},[32,64889,64890,64892,64894,64896,64898],{},[35,64891,33430],{},[81,64893,244],{},[35,64895,102],{},[35,64897,3020],{"mathvariant":231},[35,64899,40265],{},[40,64901,42984],{"encoding":42},[15,64903,64905,64923],{"className":64904,"ariaHidden":48},[47],[15,64906,64908,64911,64914,64917,64920],{"className":64907},[52],[15,64909],{"className":64910,"style":6032},[56],[15,64912,33430],{"className":64913},[61,122],[15,64915],{"className":64916,"style":128},[127],[15,64918,244],{"className":64919},[132],[15,64921],{"className":64922,"style":128},[127],[15,64924,64926,64929,64932,64935],{"className":64925},[52],[15,64927],{"className":64928,"style":142},[56],[15,64930,102],{"className":64931,"style":174},[61,122],[15,64933,3020],{"className":64934},[61],[15,64936,40265],{"className":64937,"style":40337},[61,122]," 的全部行为——证据充分时趋于确定、证据消失时退回均匀——都是这一重新赋值的直接后果。",[11,64940,64941],{},"配合 4.3 节的退火 KL 正则，模型不仅被要求「预测得对」，还被要求「在预测错时承认自己不确定」。",[11,64943,64944,64945,64947,64948,64976,64977],{},"但本文同样花了相当篇幅界定这一范式的边界。EDL 的不确定性是",[661,64946,43024],{},"的，不是贝叶斯后验不确定性：它以点估计替换了对网络参数的边缘化，因而度量的是对输出分布的把握而非对模型的把握（4.5 节第 1 点）。",[15,64949,64951,64964],{"className":64950,"translate":19},[18],[15,64952,64954],{"className":64953},[23],[25,64955,64956],{"xmlns":27},[29,64957,64958,64962],{},[32,64959,64960],{},[35,64961,33430],{},[40,64963,33430],{"encoding":42},[15,64965,64967],{"className":64966,"ariaHidden":48},[47],[15,64968,64970,64973],{"className":64969},[52],[15,64971],{"className":64972,"style":6032},[56],[15,64974,33430],{"className":64975},[61,122]," 丢弃了证据在类别间的分布信息（第 2 点）。其拟合损失带有与证据量级耦合的偏置，不等同于边缘似然（第 3 点）。并且「以损失最小化实现不确定性量化」这一范式本身受到根本性质疑（第 4 点）。这些限制并不取消该方法的实用价值——在单次前向的代价约束下同时给出预测与不确定性，其性价比是突出的——但它们决定了正确的使用姿态：",[661,64978,64979],{},"在需要二阶量的场合使用它，在需要后验语义的场合验证它，并在两者都不确定时，回到对参数做边缘化的方法上去。",[11,64981,64982,64983,64986],{},"最后应当记住的是，不确定性量化的可靠程度，与因果推断一样，",[661,64984,64985],{},"等于你愿意且能够承担的假设的强度","。迪利克雷分布给出的是一个精确而优美的数学结构。把一个精确的结构接到一个近似的映射（神经网络）上，得到的精确性只属于前者。识别出哪一部分是定理、哪一部分是启发式，是使用这套工具时最有价值的一步。",{"title":64988,"searchDepth":64989,"depth":64989,"links":64990},"",2,[64991,64998,65006,65014,65021,65029],{"id":874,"depth":64989,"text":875,"children":64992},[64993,64995,64996,64997],{"id":879,"depth":64994,"text":880},3,{"id":3081,"depth":64994,"text":3082},{"id":9384,"depth":64994,"text":9385},{"id":12516,"depth":64994,"text":12517},{"id":14669,"depth":64989,"text":14670,"children":64999},[65000,65001,65002,65003,65004,65005],{"id":14673,"depth":64994,"text":14674},{"id":19430,"depth":64994,"text":19431},{"id":21686,"depth":64994,"text":21687},{"id":24656,"depth":64994,"text":24657},{"id":28820,"depth":64994,"text":28821},{"id":31506,"depth":64994,"text":31507},{"id":33289,"depth":64989,"text":33290,"children":65007},[65008,65009,65010,65011,65012,65013],{"id":33293,"depth":64994,"text":33294},{"id":34953,"depth":64994,"text":34954},{"id":37125,"depth":64994,"text":37126},{"id":38341,"depth":64994,"text":38342},{"id":40241,"depth":64994,"text":40242},{"id":42955,"depth":64994,"text":42956},{"id":47371,"depth":64989,"text":47372,"children":65015},[65016,65017,65018,65019,65020],{"id":47462,"depth":64994,"text":47463},{"id":50919,"depth":64994,"text":50920},{"id":53109,"depth":64994,"text":53110},{"id":57349,"depth":64994,"text":57350},{"id":58000,"depth":64994,"text":58001},{"id":59539,"depth":64989,"text":59540,"children":65022},[65023,65024,65025,65026,65027,65028],{"id":59543,"depth":64994,"text":59544},{"id":60486,"depth":64994,"text":60487},{"id":61528,"depth":64994,"text":61529},{"id":61868,"depth":64994,"text":61869},{"id":62105,"depth":64994,"text":62106},{"id":62944,"depth":64994,"text":62945},{"id":64743,"depth":64989,"text":64743},"设 X\\mathcal{X}X 为输入空间，y∈{1,…,K}y \\in \\{1, \\dots, K\\}y∈{1,…,K} 为类别标签。一个 KKK 分类器的输出在数学上究竟是什么？若要求它给出「属于各类别的概率」，则输出必须落在",false,true,"md",null,{},"\u002Fblog\u002F2026\u002F0007-dirichlet-distribution-and-evidential-deep-learning",{"title":6,"description":65030},"blog\u002F2026\u002F0007-dirichlet-distribution-and-evidential-deep-learning","Dirichlet 是单纯形上的分布、多项分布的共轭先验，也是「关于概率的概率」这一二阶对象的自然载体。本文先给出它在定义、矩结构、共轭更新与分裂构造上的完整形式化，神经网络的输出重解释为证据并参数化迪利克雷分布，推导证据深度学习的对数损失等应用。",[65041,65042],"statistics","machine-learning","2026-10-01T18:52:06+08:00","Mepc-PK7MsRVGk2QQbFNDWa8P860uqFfJNNZDD7bKTo",{"id":65046,"title":65047,"body":65048,"description":65052,"draft":65031,"enableComment":65032,"extension":65033,"image":64988,"important":65031,"location":66747,"meta":66748,"navigation":65032,"ogImage":65034,"path":66749,"seo":66750,"stem":66751,"summary":66752,"tags":66753,"time":66755,"__hash__":66756},"blog\u002Fblog\u002F2026\u002F0006-markdown-ke-xue-xie-zuo-tong-dian-wen-ti.md","Markdown 并不适合科学写作",{"type":8,"value":65049,"toc":66735},[65050,65053,65056,65064,65067,65070,65075,65081,65084,65087,65090,65096,65099,65104,65107,65110,65113,65116,65119,65122,65125,65128,65131,65134,65140,65143,65146,65149,65152,65190,65193,65196,65203,65206,65209,65212,65215,65218,65221,65224,65227,65230,65233,65239,65242,65361,65683,65694,65697,65700,65703,65706,65709,65712,65715,65721,65832,65839,65846,65849,65852,65855,65862,65869,65872,65875,65878,65896,65899,66014,66021,66024,66027,66137,66161,66167,66173,66182,66185,66191,66194,66201,66315,66422,66426,66429,66432,66439,66556,66559,66563,66566,66572,66575,66585,66588,66595,66597,66600,66603,66606,66609,66612,66619,66726,66731],[11,65051,65052],{},"写博客了很长一段时间了，但是个人觉得，Markdown 并不适合科学写作。读研究生期间，搞搞科研，有时候会阶段性地总结一下自己的研究 idea，或者梳理自己的知识，就需要系统性地写作总结。这部分我经常写在自己的博客里，就比如之前所发表的博客内容。",[11,65054,65055],{},"我这个博客网站是基于 Markdown + Unified.js + Nuxt Content 工具链实现的，每篇博客源格式都是 markdown 写成，再编译成 HTML 发表在网页上。",[11,65057,65058,65059,65063],{},"但是，在写作过程中，我经常会有种非常别扭且痛苦的感觉。前几天写了一篇文章，",[33435,65060,65062],{"href":65061},"\u002Fblog\u002F2026\u002F0005-yin-guo-jie-gou-mo-xing-jian-lun","因果结构模型简论","，这是一篇偏学术和展开论述的文章。",[11,65065,65066],{},"但是把它们写成 Markdown 博客，却花了我几乎两周的时间去校对和验证。最后精力不支，只能交给 LLM 后仓促发表。类似的，再撰写这类科学写作的文章，Markdown 让我深深感受到了这种文档的不足和局限。",[11,65068,65069],{},"写作对工具不挑剔，但工具的选择会悄悄影响你写作的组织方式。用记事本写，你倾向于纯线性；用 Notion 写，你可能会不自觉地建数据库、打标签；用 Markdown 写，你会习惯用标题层级来切分内容。",[11,65071,65072],{},[661,65073,65074],{},"一般来说，写作分为几种类型：日常写作、科学写作、技术写作。",[11,65076,65077,65078],{},"三者虽然都叫「写作」，但是背后所使用的行文组织、写作思路等完全不同。",[661,65079,65080],{},"使用的认知模型完全不同。",[11,65082,65083],{},"它们对「段落」「证据」「结构」「读者」的预设都不一样。Markdown 的流行，很大程度上是因为它恰好适配了技术写作和一部分日常写作，但科学写作的核心需求它确实覆盖得很差。",[11,65085,65086],{},"Markdown 与其说「不适合写科学内容」，不如说它的抽象层级主要是「轻量文档标记」，而科学写作本质上更接近「结构化知识建模」。",[872,65088,65089],{"id":65089},"日常写作",[11,65091,65092,65095],{},[661,65093,65094],{},"首先是日常写作。"," 日常写作在互联网上随处可见，比如我们写的沟通、记录、表达。比如邮件、笔记、社交媒体、日记等等，它的核心是「交流感受、讲述经验、维持关系」。读者一般是自己或熟人，共享大量语境。",[11,65097,65098],{},"日常写作是「线性的」。比如：",[882,65100,65101],{},[11,65102,65103],{},"今天去了一趟实验室，发现服务器又挂了。折腾了半天，最后发现是 CUDA 环境的问题。",[11,65105,65106],{},"这样的好处，就是随意、口语化、结构松散，容错率高，歧义靠语境补全。",[11,65108,65109],{},"它的基本结构就是：想法 → 句子 → 段落 → 文章。",[11,65111,65112],{},"Markdown 对这种东西非常舒服。标题、粗体、列表、引用，已经足够了。而且这个链条是单向、可增量的。写一段是一段，不必先有全局骨架。这些已经覆盖了日常写作 90% 的标记需求。日常写作本来也不需要那么精确的视觉层级。",[11,65114,65115],{},"更重要的是，Markdown 的纯文本性让它在任何设备、任何编辑器里都能打开和修改。手机备忘录、VS Code、Obsidian、Notion、Typora，甚至 GitHub 的 issue 框，都能写。这种低门槛 + 跨平台，正好匹配日常写作「随时随地、随手就写」的场景。",[11,65117,65118],{},"它允许模糊、跳跃、情绪化，结构常常是情绪曲线或联想链条。博客、散文、社交媒体都属于这一类。Markdown 在这里够用，因为它只需要轻量地标记标题、列表、强调、链接。",[11,65120,65121],{},"作者不需要预先设计结构，想到哪写到哪。读者也不指望从中提取精确知识，能跟上情绪和大致信息就行。逻辑跳了、指代不清、重复啰嗦，都不致命，语境会补全。改一句话不影响全局，不需要回头调整编号、引文、交叉引用。",[11,65123,65124],{},"所以，对于日常写作，工具要求实际上并不高，任意文本编辑器、笔记软件如 MS Word，Evernote，Notion 都可以轻松应对，甚至手机自带的笔记本 App，Windows 文本编辑器都能应对。",[11,65126,65127],{},"日常写作还有一个特点：结构可以后补。你先随手写一堆碎片，之后再整理成「标题 + 段落 + 列表」，完全来得及。因为日常写作不依赖精确的交叉引用和编号，结构调整不会引发连锁修改。",[11,65129,65130],{},"所以，日常写作是三种写作里对工具最宽容、对结构最随意、对修改最不敏感的一类。Markdown 在这里如鱼得水，不是因为它多强大，而是因为日常写作本来就不需要多强大。",[872,65132,65133],{"id":65133},"技术写作",[11,65135,65136,65139],{},[661,65137,65138],{},"其次是技术写作。"," 技术写作的核心是「让读者完成操作或理解系统」。回答「怎么做」。比如 API 文档、用户手册，核心是让读者准确、高效地完成操作，追求可用性和任务完成率。",[11,65141,65142],{},"它追求可检索、可复用、可版本控制、可多端发布。文档、API 说明、教程、README 是典型。大多数开源项目的文档、技术博客等等，都属于这种类型。",[11,65144,65145],{},"技术写作是「操作性的」。技术文档的写作内容，更像是目标、环境、安装、配置、运行、错误处理、结果一连串起来。",[11,65147,65148],{},"文章的每个部分都是任务导向，模块化。比如，读者可跳读。且每个部分都可以在自己的内部领域展开——这就是典型的树形文章组织。",[11,65150,65151],{},"因此 Markdown 的标题树非常自然，以一个开源项目的 Github Readme 文档的组织结构为例：",[65153,65154,65158],"pre",{"className":65155,"code":65156,"language":65157,"meta":64988,"style":64988},"language-markdown shiki shiki-themes github-dark","# Installation\n## Requirements\n## Install\n# Usage\n# Troubleshooting\n","markdown",[65159,65160,65161,65168,65173,65178,65184],"code",{"__ignoreMap":64988},[15,65162,65165],{"class":65163,"line":65164},"line",1,[15,65166,65167],{},"# Installation\n",[15,65169,65170],{"class":65163,"line":64989},[15,65171,65172],{},"## Requirements\n",[15,65174,65175],{"class":65163,"line":64994},[15,65176,65177],{},"## Install\n",[15,65179,65181],{"class":65163,"line":65180},4,[15,65182,65183],{},"# Usage\n",[15,65185,65187],{"class":65163,"line":65186},5,[15,65188,65189],{},"# Troubleshooting\n",[11,65191,65192],{},"例如，如果读者已经 install 了这个软件，就可以直接跳过 installation 和 requirements 这两章节，直接跳转到 Usage 章节。",[11,65194,65195],{},"Markdown 的核心模型是树状文档——标题下挂段落和列表，段落和列表之下再无更深的语义结构。就像这样：",[11,65197,65198],{},[65199,65200],"img",{"alt":65201,"src":65202},"Markdown 层次写作模型","https:\u002F\u002Fimage-assets.dreams.plus\u002F202609222216870.png",[11,65204,65205],{},"Markdown 几乎是为这类写作量身定做的：纯文本、易 diff、易转 HTML、易嵌入代码块。技术写作的「正确性」往往由代码运行结果或操作是否成功来验证，不需要复杂的引注和排版。Markdown 的设计目标不是「表达知识结构」，而是「让纯文本容易读、容易写、容易转 HTML」。它的语法元素——标题、列表、强调、链接、代码块、引用——本质上都是视觉标记，不是语义对象。",[11,65207,65208],{},"所以 Markdown 甚至非常适合技术写作。例如 GitHub README、API 文档、开发文档，本质上就是让一个人沿着信息路径完成一个任务。",[872,65210,65211],{"id":65211},"科学写作",[11,65213,65214],{},"但科学写作不是信息排列，而是在构造一个可以被检验的论证系统。核心是提出假设、展示证据、论证结论。",[11,65216,65217],{},"这一点在技术写作和科学写作里就完全不同了。技术写作的结构往往是预先设计的，科学写作的结构更是由论证逻辑和引用关系约束的。一旦结构变了，引文、图表编号、交叉引用都要跟着变。",[11,65219,65220],{},"以写一篇论文为例，真正的逻辑并不是「Introduction → Method → Experiment → Conclusion」这样的排版顺序，而更接近：先有一个现象，再追问为什么现有解释不足，进而提出新的问题表述，建立数学模型，推导出模型应当具有的可验证性质，设计方法去利用这个性质，用实验去验证这些性质是否真的存在，检验方法是否真正解决了问题，最后排除其他可能的解释。",[11,65222,65223],{},"这条链条可以进一步抽象为 Claim → Assumption → Model → Derivation → Prediction → Evidence → Conclusion，这已经不是普通意义上的「文档结构」，而更像一张证明图、因果图或知识图。",[11,65225,65226],{},"由此可以看到，科学写作其实存在两套结构。读者看到的是表层结构，也就是论文的 IMRaD 结构（引言 - 方法 - 结果 - 讨论），这一线性过程：Introduction、Related Work、Method、Experiments、Conclusion。",[11,65228,65229],{},"但真正决定论文质量的是深层结构——从 Research Question 出发得到 Claim，Claim 一边连接 Assumption，一边连接 Evidence，Assumption 导向 Model，Evidence 来自 Experiment 并与 Model 互相印证，Model 产生 Prediction，最终汇聚为 Conclusion。",[11,65231,65232],{},"一个好的论文段落内部也遵循类似的链条：Claim → Reason → Evidence → Interpretation → Connection to next claim。这种写作模型如下图所示",[11,65234,65235],{},[65199,65236],{"alt":65237,"src":65238},"科学写作的写作模型","https:\u002F\u002Fimage-assets.dreams.plus\u002F202609222126754.png",[11,65240,65241],{},"可以说，科学论文本质上非常接近一种有向图结构，这就产生了一个有意思的矛盾：Markdown 擅长表达「文档树」，而科学写作真正需要表达的是「论证图」。这条链条在认知科学和论证理论里也有对应。比如 Toulmin 的论证模型：Claim、Ground、Warrant、Backing、Qualifier、Rebuttal。",[11,65243,65244,65245,65360],{},"这也是为什么 ",[15,65246,65248,65263],{"className":65247,"translate":19},[18],[15,65249,65251],{"className":65250},[23],[25,65252,65253],{"xmlns":27},[29,65254,65255,65260],{},[32,65256,65257],{},[282,65258,65259],{},"LaTeX",[40,65261,65262],{"encoding":42},"\\LaTeX",[15,65264,65266],{"className":65265,"ariaHidden":48},[47],[15,65267,65269,65273],{"className":65268},[52],[15,65270],{"className":65271,"style":65272},[56],"height:0.8988em;vertical-align:-0.2155em;",[15,65274,65276,65280,65284,65306,65310],{"className":65275},[61,13548],[15,65277,47490],{"className":65278},[61,65279],"textrm",[15,65281],{"className":65282,"style":65283},[127],"margin-right:-0.36em;",[15,65285,65287],{"className":65286},[342],[15,65288,65290],{"className":65289},[346],[15,65291,65293],{"className":65292,"style":57},[350],[15,65294,65296,65299],{"style":65295},"top:-2.905em;",[15,65297],{"className":65298,"style":358},[357],[15,65300,65302],{"className":65301},[61],[15,65303,65305],{"className":65304},[61,65279,365,362,363,364],"A",[15,65307],{"className":65308,"style":65309},[127],"margin-right:-0.15em;",[15,65311,65313,65316,65319,65353,65357],{"className":65312},[61,13548],[15,65314,56756],{"className":65315},[61,65279],[15,65317],{"className":65318,"style":7105},[127],[15,65320,65322,65344],{"className":65321},[342,472],[15,65323,65325,65341],{"className":65324},[346],[15,65326,65329],{"className":65327,"style":65328},[350],"height:0.4678em;",[15,65330,65332,65335],{"style":65331},"top:-2.7845em;",[15,65333],{"className":65334,"style":3861},[357],[15,65336,65338],{"className":65337},[61],[15,65339,14818],{"className":65340},[61,65279],[15,65342,497],{"className":65343},[496],[15,65345,65347],{"className":65346},[346],[15,65348,65351],{"className":65349,"style":65350},[350],"height:0.2155em;",[15,65352],{},[15,65354],{"className":65355,"style":65356},[127],"margin-right:-0.125em;",[15,65358,38],{"className":65359},[61,65279]," 在科学写作中显得格外强大，且在 互联网时代 Markdown 盛行的情况下仍然未被淘汰的原因。",[11,65362,65363,65466,65467,65470,65471,65574,65575,65678,65679,65682],{},[15,65364,65366,65379],{"className":65365,"translate":19},[18],[15,65367,65369],{"className":65368},[23],[25,65370,65371],{"xmlns":27},[29,65372,65373,65377],{},[32,65374,65375],{},[282,65376,65259],{},[40,65378,65262],{"encoding":42},[15,65380,65382],{"className":65381,"ariaHidden":48},[47],[15,65383,65385,65388],{"className":65384},[52],[15,65386],{"className":65387,"style":65272},[56],[15,65389,65391,65394,65397,65417,65420],{"className":65390},[61,13548],[15,65392,47490],{"className":65393},[61,65279],[15,65395],{"className":65396,"style":65283},[127],[15,65398,65400],{"className":65399},[342],[15,65401,65403],{"className":65402},[346],[15,65404,65406],{"className":65405,"style":57},[350],[15,65407,65408,65411],{"style":65295},[15,65409],{"className":65410,"style":358},[357],[15,65412,65414],{"className":65413},[61],[15,65415,65305],{"className":65416},[61,65279,365,362,363,364],[15,65418],{"className":65419,"style":65309},[127],[15,65421,65423,65426,65429,65460,65463],{"className":65422},[61,13548],[15,65424,56756],{"className":65425},[61,65279],[15,65427],{"className":65428,"style":7105},[127],[15,65430,65432,65452],{"className":65431},[342,472],[15,65433,65435,65449],{"className":65434},[346],[15,65436,65438],{"className":65437,"style":65328},[350],[15,65439,65440,65443],{"style":65331},[15,65441],{"className":65442,"style":3861},[357],[15,65444,65446],{"className":65445},[61],[15,65447,14818],{"className":65448},[61,65279],[15,65450,497],{"className":65451},[496],[15,65453,65455],{"className":65454},[346],[15,65456,65458],{"className":65457,"style":65350},[350],[15,65459],{},[15,65461],{"className":65462,"style":65356},[127],[15,65464,38],{"className":65465},[61,65279]," 表面上看起来比 Markdown 古老得多，但它解决的其实不只是排版问题，它隐含的是 ",[661,65468,65469],{},"科学对象是一等公民"," 这样一种思想。",[15,65472,65474,65487],{"className":65473,"translate":19},[18],[15,65475,65477],{"className":65476},[23],[25,65478,65479],{"xmlns":27},[29,65480,65481,65485],{},[32,65482,65483],{},[282,65484,65259],{},[40,65486,65262],{"encoding":42},[15,65488,65490],{"className":65489,"ariaHidden":48},[47],[15,65491,65493,65496],{"className":65492},[52],[15,65494],{"className":65495,"style":65272},[56],[15,65497,65499,65502,65505,65525,65528],{"className":65498},[61,13548],[15,65500,47490],{"className":65501},[61,65279],[15,65503],{"className":65504,"style":65283},[127],[15,65506,65508],{"className":65507},[342],[15,65509,65511],{"className":65510},[346],[15,65512,65514],{"className":65513,"style":57},[350],[15,65515,65516,65519],{"style":65295},[15,65517],{"className":65518,"style":358},[357],[15,65520,65522],{"className":65521},[61],[15,65523,65305],{"className":65524},[61,65279,365,362,363,364],[15,65526],{"className":65527,"style":65309},[127],[15,65529,65531,65534,65537,65568,65571],{"className":65530},[61,13548],[15,65532,56756],{"className":65533},[61,65279],[15,65535],{"className":65536,"style":7105},[127],[15,65538,65540,65560],{"className":65539},[342,472],[15,65541,65543,65557],{"className":65542},[346],[15,65544,65546],{"className":65545,"style":65328},[350],[15,65547,65548,65551],{"style":65331},[15,65549],{"className":65550,"style":3861},[357],[15,65552,65554],{"className":65553},[61],[15,65555,14818],{"className":65556},[61,65279],[15,65558,497],{"className":65559},[496],[15,65561,65563],{"className":65562},[346],[15,65564,65566],{"className":65565,"style":65350},[350],[15,65567],{},[15,65569],{"className":65570,"style":65356},[127],[15,65572,38],{"className":65573},[61,65279]," 内置的 equation、theorem、lemma、proof、definition、figure、table、citation、cross-reference、appendix 等等组件，这些东西在 ",[15,65576,65578,65591],{"className":65577,"translate":19},[18],[15,65579,65581],{"className":65580},[23],[25,65582,65583],{"xmlns":27},[29,65584,65585,65589],{},[32,65586,65587],{},[282,65588,65259],{},[40,65590,65262],{"encoding":42},[15,65592,65594],{"className":65593,"ariaHidden":48},[47],[15,65595,65597,65600],{"className":65596},[52],[15,65598],{"className":65599,"style":65272},[56],[15,65601,65603,65606,65609,65629,65632],{"className":65602},[61,13548],[15,65604,47490],{"className":65605},[61,65279],[15,65607],{"className":65608,"style":65283},[127],[15,65610,65612],{"className":65611},[342],[15,65613,65615],{"className":65614},[346],[15,65616,65618],{"className":65617,"style":57},[350],[15,65619,65620,65623],{"style":65295},[15,65621],{"className":65622,"style":358},[357],[15,65624,65626],{"className":65625},[61],[15,65627,65305],{"className":65628},[61,65279,365,362,363,364],[15,65630],{"className":65631,"style":65309},[127],[15,65633,65635,65638,65641,65672,65675],{"className":65634},[61,13548],[15,65636,56756],{"className":65637},[61,65279],[15,65639],{"className":65640,"style":7105},[127],[15,65642,65644,65664],{"className":65643},[342,472],[15,65645,65647,65661],{"className":65646},[346],[15,65648,65650],{"className":65649,"style":65328},[350],[15,65651,65652,65655],{"style":65331},[15,65653],{"className":65654,"style":3861},[357],[15,65656,65658],{"className":65657},[61],[15,65659,14818],{"className":65660},[61,65279],[15,65662,497],{"className":65663},[496],[15,65665,65667],{"className":65666},[346],[15,65668,65670],{"className":65669,"style":65350},[350],[15,65671],{},[15,65673],{"className":65674,"style":65356},[127],[15,65676,38],{"className":65677},[61,65279]," 里不是普通文本，而是具有实际意义的科学对象（Object）。在科学写作中，讲论证图组织起来的方式，就是 ",[65159,65680,65681],{},"\\ref"," 各个科学对象。",[11,65684,65685,65686,65689,65690,65693],{},"写一个公式并加上 ",[65159,65687,65688],{},"\\label{eq:loss}","，再用 ",[65159,65691,65692],{},"\\ref{eq:loss}"," 去引用它，本质上不是在说「这里有一段数学」，而是在声明这是一个具有编号、引用关系和语义身份的 Equation。此外，在科学写作中，定理环境（theorem）也是很重要的，有时候我会在展开论述的时候，给出一些定义、推论、引理等等。在其他文章部分，会经常 refer 到这些。这样一来，文中就建立起了知识对象之间的引用关系，这已经比 Markdown 里单纯加粗一个 Loss function 高了一个抽象层次。",[11,65695,65696],{},"而仅 Markdown 则完全做不到这些。在进行科学写作的时候，只能手动去标注哪些是「定理」，用到了哪些部分，然后再手动写出引用位置，实在无法做到随心所欲去 refer 每一个对象。这些对象和关系，在 LaTeX 编译时会被自动编号、自动解析、自动维护。你增删一个公式，所有引用它的地方都会自动更新。这就是「图结构」在工具层面的体现。Markdown 没有这些语义对象。你只能手动写「如公式 (3) 所示」，然后手动维护编号。一旦增删，就是灾难。",[11,65698,65699],{},"更麻烦的是，科学写作并不是一日之功，很多时候要经过好几天甚至数月时间才能写成文章，在此期间，校对、增删文章时都会重构整个对象引用关系，如果论述过程简单还尚可，对系统性长文，手动维护各个对象之间的关系，简直是灾难级别的事故。",[11,65701,65702],{},"我个人也尝试去寻找相关工具链，很遗憾，我没有找到任何能适用于科学写作处理的 Markdown 文档工具链。",[11,65704,65705],{},"虽然社区有一些尝试，比如 Pandoc 的 Markdown 扩展、Quarto、MyST、Jupyter Book，它们确实支持公式、引用、交叉引用。但它们的抽象层级仍然是「文档 + 一些科学标记」，而不是「科学知识图」。它们能让你在 Markdown 里写公式和引用，但无法让你像 LaTeX 那样自然地声明和引用定理、引理、证明、假设、证据。",[11,65707,65708],{},"更重要的是，它们没有解决上面提到的核心问题：科学写作的深层结构是图，而不是树。工具如果只提供树形结构，写作者就只能手动在脑子里维护图，然后把它压扁成树。这个「压扁」过程，就是让我觉得别扭和痛苦的根源。",[872,65710,65711],{"id":65711},"压扁的代价",[11,65713,65714],{},"上面说的「压扁」不是一个比喻，它是一件有具体成本的事。",[11,65716,65717,65718],{},"写作并不是「先在脑子里想好一张完整的图，再把它抄成文字」。真实的过程往往相反：图和文本互相塑造。你在写第三段的时候，才想清楚第一段那个 Claim 到底需要什么假设；你在写实验的时候，才发现原来的模型少了一个条件。",[661,65719,65720],{},"图是在写作过程中长出来的，而 Markdown 要求你在图还没有定形的时候，就把它线性地固定下来。",[11,65722,65723,65724,65827,65828,65831],{},"举一个具体的例子。假设我在写一篇关于因果推断的文章，第三节给出了「后门准则」的定义，第五节有一个命题依赖这个定义，第七节的分析又依赖那个命题。现在我想在第三节中间再插入一个识别性假设。在 ",[15,65725,65727,65740],{"className":65726,"translate":19},[18],[15,65728,65730],{"className":65729},[23],[25,65731,65732],{"xmlns":27},[29,65733,65734,65738],{},[32,65735,65736],{},[282,65737,65259],{},[40,65739,65262],{"encoding":42},[15,65741,65743],{"className":65742,"ariaHidden":48},[47],[15,65744,65746,65749],{"className":65745},[52],[15,65747],{"className":65748,"style":65272},[56],[15,65750,65752,65755,65758,65778,65781],{"className":65751},[61,13548],[15,65753,47490],{"className":65754},[61,65279],[15,65756],{"className":65757,"style":65283},[127],[15,65759,65761],{"className":65760},[342],[15,65762,65764],{"className":65763},[346],[15,65765,65767],{"className":65766,"style":57},[350],[15,65768,65769,65772],{"style":65295},[15,65770],{"className":65771,"style":358},[357],[15,65773,65775],{"className":65774},[61],[15,65776,65305],{"className":65777},[61,65279,365,362,363,364],[15,65779],{"className":65780,"style":65309},[127],[15,65782,65784,65787,65790,65821,65824],{"className":65783},[61,13548],[15,65785,56756],{"className":65786},[61,65279],[15,65788],{"className":65789,"style":7105},[127],[15,65791,65793,65813],{"className":65792},[342,472],[15,65794,65796,65810],{"className":65795},[346],[15,65797,65799],{"className":65798,"style":65328},[350],[15,65800,65801,65804],{"style":65331},[15,65802],{"className":65803,"style":3861},[357],[15,65805,65807],{"className":65806},[61],[15,65808,14818],{"className":65809},[61,65279],[15,65811,497],{"className":65812},[496],[15,65814,65816],{"className":65815},[346],[15,65817,65819],{"className":65818,"style":65350},[350],[15,65820],{},[15,65822],{"className":65823,"style":65356},[127],[15,65825,38],{"className":65826},[61,65279]," 里这是一个局部操作：加一个 ",[65159,65829,65830],{},"\\label","，被影响的地方编译时会提醒我。在 Markdown 里这是一次全局手术：我得记住哪些地方引用了「上面那个定义」，哪些公式的编号要变，哪些「如公式 (3) 所示」要改成 (4)。而 Markdown 不会告诉我它们在哪里。",[11,65833,65834,65835,65838],{},"不过编号错乱其实还不是最贵的部分，因为编辑器至少能搜到。真正贵的是",[661,65836,65837],{},"关系的丢失","。在 Markdown 里，「这一段是为了支持那个 Claim」这个事实不存在于文件里，它只存在于我的脑子里。今天写的时候当然记得，三个月后回来改的时候就不记得了。",[11,65840,65841,65842,65845],{},"这带来一个更隐蔽的后果：",[661,65843,65844],{},"别人无法验证我的论证","。合作者和审稿人拿到的是一段线性文本，他们只能重新读一遍全文，然后在自己脑子里把这张图重建出来，再检查它成不成立。每个人都要重建一遍。引用是显式的，论证是隐式的。",[11,65847,65848],{},"所以那种「别扭」的感觉可以更精确地描述：写作时，我的大脑同时在维护两个表征——一张还在变的论证图，和一条正在往前走的文本流——并且要保证它们在每一步都一致。Markdown 为前者提供的支撑是零。它不是不好用，是它根本没打算管这件事。",[872,65850,65851],{"id":65851},"读者也在走这张图",[11,65853,65854],{},"线性化的成本不只由写作者承担，读者也在付。",[11,65856,65857,65858,65861],{},"读一篇论文的真实动作从来不是从头读到尾。读到中间的 Claim，我要翻回前面看那个定义到底是什么；看到「实验表明」，我要跳到结果章节确认；想知道这个结论有多强，我要去附录看证明的边界条件。",[661,65859,65860],{},"阅读本身就是在图上做遍历","，而且是带随机跳转的遍历。",[11,65863,65864,65865,65868],{},"IMRaD 与其说是一种逻辑结构，不如说是",[661,65866,65867],{},"一种为印刷媒介做的妥协","。纸是顺序的，装订是顺序的，翻页是顺序的，所以我们把一张图压成一条线，然后指望读者自己再把它拆开、重新组装。",[11,65870,65871],{},"这里有一个有点讽刺的地方：网页其实是天然支持非线性阅读的。折叠、跳转、悬浮预览、双向链接，技术上全都存在。但我们依然在用线性的写作工具，生产线性的制品，再把它放到一个非线性的媒介上。最后得到的，是一篇被伪装成网页的 PDF。",[11,65873,65874],{},"所以「压扁」这件事，写的人付一次，读的人付一次，每个合作者再各付一次。它是一个被反复征收的成本。",[11,65876,65877],{},"有人会说，图结构的写作工具不是早就有了吗？Obsidian、Logseq、Roam Research 的双向链接，不就是图吗？",[11,65879,65880,65881,65884,65885,867,65888,65891,65892,65895],{},"是图，但粒度不对。这些工具的节点是",[661,65882,65883],{},"笔记","，不是",[661,65886,65887],{},"命题",[65159,65889,65890],{},"[[因果推断]]"," 这样一条链接只说了「这两篇相关」，没有说「这篇支持那篇」还是「这篇反驳那篇」。而当边没有类型的时候，一张论证图就退化成了一个共现网络。图上只剩下相关性，可科学写作里最要紧的东西恰恰是",[661,65893,65894],{},"边的类型","：支持、反驳、假设、推导、检验。",[11,65897,65898],{},"Zettelkasten 也是同样的情况。它的原意是让想法自由连接、自然涌现，服务于「我该想什么」，而不是「我这个论证成不成立」。卡片之间可以链接，但链接不带论证语义。",[11,65900,65901,65902,65905,65906,66013],{},"另一个方向是 Pandoc、Quarto、MyST、Jupyter Book 这一类。它们的贡献是实打实的：让你能在 Markdown 里写公式、写 ",[65159,65903,65904],{},"@fig:xxx"," 这样的交叉引用。但它们做的是",[661,65907,65908,65909,66012],{},"把 ",[15,65910,65912,65925],{"className":65911,"translate":19},[18],[15,65913,65915],{"className":65914},[23],[25,65916,65917],{"xmlns":27},[29,65918,65919,65923],{},[32,65920,65921],{},[282,65922,65259],{},[40,65924,65262],{"encoding":42},[15,65926,65928],{"className":65927,"ariaHidden":48},[47],[15,65929,65931,65934],{"className":65930},[52],[15,65932],{"className":65933,"style":65272},[56],[15,65935,65937,65940,65943,65963,65966],{"className":65936},[61,13548],[15,65938,47490],{"className":65939},[61,65279],[15,65941],{"className":65942,"style":65283},[127],[15,65944,65946],{"className":65945},[342],[15,65947,65949],{"className":65948},[346],[15,65950,65952],{"className":65951,"style":57},[350],[15,65953,65954,65957],{"style":65295},[15,65955],{"className":65956,"style":358},[357],[15,65958,65960],{"className":65959},[61],[15,65961,65305],{"className":65962},[61,65279,365,362,363,364],[15,65964],{"className":65965,"style":65309},[127],[15,65967,65969,65972,65975,66006,66009],{"className":65968},[61,13548],[15,65970,56756],{"className":65971},[61,65279],[15,65973],{"className":65974,"style":7105},[127],[15,65976,65978,65998],{"className":65977},[342,472],[15,65979,65981,65995],{"className":65980},[346],[15,65982,65984],{"className":65983,"style":65328},[350],[15,65985,65986,65989],{"style":65331},[15,65987],{"className":65988,"style":3861},[357],[15,65990,65992],{"className":65991},[61],[15,65993,14818],{"className":65994},[61,65279],[15,65996,497],{"className":65997},[496],[15,65999,66001],{"className":66000},[346],[15,66002,66004],{"className":66003,"style":65350},[350],[15,66005],{},[15,66007],{"className":66008,"style":65356},[127],[15,66010,38],{"className":66011},[61,65279]," 的对象搬进 Markdown 的语法","，并没有动 Markdown 的骨架。文档的骨架依然是一个标题树，对象只是挂在树上的装饰。至于「这个公式是从哪个假设推出来的」这类关系，它们依然没有地方可以放。",[11,66015,66016,66017,66020],{},"两个方向的工具缺的是同一样东西：",[661,66018,66019],{},"它们都没有把「对象」和「关系」变成编辑器里可以直接操作的对象，也没有让文本变成这张图的一个投影。"," 一个是把边的类型丢了，一个是把对象降格成了树的装饰。而科学写作要求这两件事同时成立。",[872,66022,66023],{"id":66023},"科学写作需要什么",[11,66025,66026],{},"如果要认真地解决这件事，工具大概需要具备以下几个性质。这是一张草图，不是一份产品方案。",[11,66028,66029,66032,66033,66136],{},[661,66030,66031],{},"第一，科学对象是一等公民。"," 这一点其实 ",[15,66034,66036,66049],{"className":66035,"translate":19},[18],[15,66037,66039],{"className":66038},[23],[25,66040,66041],{"xmlns":27},[29,66042,66043,66047],{},[32,66044,66045],{},[282,66046,65259],{},[40,66048,65262],{"encoding":42},[15,66050,66052],{"className":66051,"ariaHidden":48},[47],[15,66053,66055,66058],{"className":66054},[52],[15,66056],{"className":66057,"style":65272},[56],[15,66059,66061,66064,66067,66087,66090],{"className":66060},[61,13548],[15,66062,47490],{"className":66063},[61,65279],[15,66065],{"className":66066,"style":65283},[127],[15,66068,66070],{"className":66069},[342],[15,66071,66073],{"className":66072},[346],[15,66074,66076],{"className":66075,"style":57},[350],[15,66077,66078,66081],{"style":65295},[15,66079],{"className":66080,"style":358},[357],[15,66082,66084],{"className":66083},[61],[15,66085,65305],{"className":66086},[61,65279,365,362,363,364],[15,66088],{"className":66089,"style":65309},[127],[15,66091,66093,66096,66099,66130,66133],{"className":66092},[61,13548],[15,66094,56756],{"className":66095},[61,65279],[15,66097],{"className":66098,"style":7105},[127],[15,66100,66102,66122],{"className":66101},[342,472],[15,66103,66105,66119],{"className":66104},[346],[15,66106,66108],{"className":66107,"style":65328},[350],[15,66109,66110,66113],{"style":65331},[15,66111],{"className":66112,"style":3861},[357],[15,66114,66116],{"className":66115},[61],[15,66117,14818],{"className":66118},[61,65279],[15,66120,497],{"className":66121},[496],[15,66123,66125],{"className":66124},[346],[15,66126,66128],{"className":66127,"style":65350},[350],[15,66129],{},[15,66131],{"className":66132,"style":65356},[127],[15,66134,38],{"className":66135},[61,65279]," 已经做到了，但是其他的写作工具仍然做得不彻底。一个 Definition、一个 Assumption、一个 Claim、一条 Evidence、一个 Prediction，应该是编辑器里真实存在的对象：有稳定 ID，有类型，有内容，而不是一段被加粗的文字。类型本身携带信息——修改一个 Assumption 和修改一个 Definition，后果完全不同。",[11,66138,66139,802,66142,6154,66145,6154,66148,6154,66151,6154,66154,6154,66157,66160],{},[661,66140,66141],{},"第二，边是有类型的。",[65159,66143,66144],{},"supports",[65159,66146,66147],{},"contradicts",[65159,66149,66150],{},"assumes",[65159,66152,66153],{},"derives",[65159,66155,66156],{},"tests",[65159,66158,66159],{},"qualifies","。这是全文最核心的一条。只有节点类型和边的类型同时存在，才配叫论证图；只有节点，那是笔记库；只有无类型的边，那是共现网络。",[11,66162,66163,66166],{},[661,66164,66165],{},"第三，文本是图的一个投影。"," 意思是：论文不是唯一制品，而是某一张图在某一个视图下的渲染结果。同一张图应该能渲染成论文、组会 slides、审稿回复、以及一篇博客。现在的情况是反过来的——文字是本体的全部，图和 slides 都是它的下脚料。",[11,66168,66169,66172],{},[661,66170,66171],{},"第四，视图要可切换。"," 写作时用线性视图，检查覆盖面用大纲视图，审自己的逻辑用图视图。这三个视图必须操作同一份数据，否则又会退回到「手动同步」的老问题里。",[11,66174,66175,66178,66179,66181],{},[661,66176,66177],{},"第五，一致性检查应该上升到语义层。"," 编译器检查 ",[65159,66180,65681],{}," 指向的 label 是否存在，这是最浅的一层。再往上一层，工具应该能问：这个 Claim 有 Evidence 支撑吗？它的 Assumption 声明了吗？现有的 Evidence 支撑的是不是只是这个 Claim 的弱化版本？结论有没有超出实验能支持的范围？这些问题，本质上就是读者读论文时会问的那几个问题——工具应该先替他们问一遍。",[872,66183,66184],{"id":66184},"为什么这件事一直没做成",[11,66186,66187,66188,867],{},"既然需求这么明确，为什么没有这样的工具？因为",[661,66189,66190],{},"读一张图很便宜，写一张图很贵",[11,66192,66193],{},"一个理想的知识编辑系统，要求作者在想法还没成形的时候就声明「这个支持那个」「这个是从那个推出来的」。但正如前面说的，图是在写作过程中长出来的。要求作者提前声明关系，等于要求他在思考完成之前就交出思考的结论。这个交互本身就是反人性的。",[11,66195,66196,66197,66200],{},"这不是新问题。Bush 在上世纪四十年代设想的 Memex，Nelson 在六十年代提出的 Xanadu，都是「让人在非线性的知识网络里写作」的方案。它们没有普及，原因通常被归为工程难度和商业失败，但我更倾向于另一个解释：",[661,66198,66199],{},"人的思维展开是序列式的，图是事后归纳出来的。"," 我们不是先有图再写文章，而是先写文章才发现图长什么样。任何强迫作者「先建图再写作」的工具，都会在第一步被放弃。",[11,66202,66203,66204,66307,66308,66310,66311,66314],{},"这反过来解释了一件我一直觉得有意思的事：",[15,66205,66207,66220],{"className":66206,"translate":19},[18],[15,66208,66210],{"className":66209},[23],[25,66211,66212],{"xmlns":27},[29,66213,66214,66218],{},[32,66215,66216],{},[282,66217,65259],{},[40,66219,65262],{"encoding":42},[15,66221,66223],{"className":66222,"ariaHidden":48},[47],[15,66224,66226,66229],{"className":66225},[52],[15,66227],{"className":66228,"style":65272},[56],[15,66230,66232,66235,66238,66258,66261],{"className":66231},[61,13548],[15,66233,47490],{"className":66234},[61,65279],[15,66236],{"className":66237,"style":65283},[127],[15,66239,66241],{"className":66240},[342],[15,66242,66244],{"className":66243},[346],[15,66245,66247],{"className":66246,"style":57},[350],[15,66248,66249,66252],{"style":65295},[15,66250],{"className":66251,"style":358},[357],[15,66253,66255],{"className":66254},[61],[15,66256,65305],{"className":66257},[61,65279,365,362,363,364],[15,66259],{"className":66260,"style":65309},[127],[15,66262,66264,66267,66270,66301,66304],{"className":66263},[61,13548],[15,66265,56756],{"className":66266},[61,65279],[15,66268],{"className":66269,"style":7105},[127],[15,66271,66273,66293],{"className":66272},[342,472],[15,66274,66276,66290],{"className":66275},[346],[15,66277,66279],{"className":66278,"style":65328},[350],[15,66280,66281,66284],{"style":65331},[15,66282],{"className":66283,"style":3861},[357],[15,66285,66287],{"className":66286},[61],[15,66288,14818],{"className":66289},[61,65279],[15,66291,497],{"className":66292},[496],[15,66294,66296],{"className":66295},[346],[15,66297,66299],{"className":66298,"style":65350},[350],[15,66300],{},[15,66302],{"className":66303,"style":65356},[127],[15,66305,38],{"className":66306},[61,65279]," 为什么活到了今天。它并不要求你放弃线性写作，它只要求你在必要的地方声明对象。你可以照常从头写到尾，只在需要的时候加一个 ",[65159,66309,65830],{},"。它是一个",[661,66312,66313],{},"树骨架 + 少量图边","的混合体。这个折中在工程上极其成功——它把「线性写作」和「图结构」之间的冲突压到了大多数人都能接受的水平。",[11,66316,66317,66318,66421],{},"所以 ",[15,66319,66321,66334],{"className":66320,"translate":19},[18],[15,66322,66324],{"className":66323},[23],[25,66325,66326],{"xmlns":27},[29,66327,66328,66332],{},[32,66329,66330],{},[282,66331,65259],{},[40,66333,65262],{"encoding":42},[15,66335,66337],{"className":66336,"ariaHidden":48},[47],[15,66338,66340,66343],{"className":66339},[52],[15,66341],{"className":66342,"style":65272},[56],[15,66344,66346,66349,66352,66372,66375],{"className":66345},[61,13548],[15,66347,47490],{"className":66348},[61,65279],[15,66350],{"className":66351,"style":65283},[127],[15,66353,66355],{"className":66354},[342],[15,66356,66358],{"className":66357},[346],[15,66359,66361],{"className":66360,"style":57},[350],[15,66362,66363,66366],{"style":65295},[15,66364],{"className":66365,"style":358},[357],[15,66367,66369],{"className":66368},[61],[15,66370,65305],{"className":66371},[61,65279,365,362,363,364],[15,66373],{"className":66374,"style":65309},[127],[15,66376,66378,66381,66384,66415,66418],{"className":66377},[61,13548],[15,66379,56756],{"className":66380},[61,65279],[15,66382],{"className":66383,"style":7105},[127],[15,66385,66387,66407],{"className":66386},[342,472],[15,66388,66390,66404],{"className":66389},[346],[15,66391,66393],{"className":66392,"style":65328},[350],[15,66394,66395,66398],{"style":65331},[15,66396],{"className":66397,"style":3861},[357],[15,66399,66401],{"className":66400},[61],[15,66402,14818],{"className":66403},[61,65279],[15,66405,497],{"className":66406},[496],[15,66408,66410],{"className":66409},[346],[15,66411,66413],{"className":66412,"style":65350},[350],[15,66414],{},[15,66416],{"className":66417,"style":65356},[127],[15,66419,38],{"className":66420},[61,65279]," 的地位并不只是历史惯性。它在科学写作里没有被淘汰，是因为它是目前唯一一个既把对象和引用做成了语言原语、又不强迫作者放弃线性写作的系统。它的不足也在这里：图边只有引用这一种类型，而且维护成本随文章长度急剧上升——这正是文章开头我花两周校对的原因。",[872,66423,66425],{"id":66424},"在工具出现之前我现在怎么工作","在工具出现之前，我现在怎么工作",[11,66427,66428],{},"没有现成工具，只能用流程凑。我目前的办法是把「图」和「文」分成两层，中间只做一次机械转换。",[11,66430,66431],{},"草图层不管顺序，只管对象和关系。给每个对象一个短 ID：",[65153,66433,66437],{"className":66434,"code":66436,"language":13548,"meta":64988},[66435],"language-text","C1   后门准则可以替代 do-演算完成识别\nA1   因果充分性假设（不存在未观测混杂）\nA2   可交换性\nM1   SCM 中图结构与分布的对应关系\nD1   由 M1 + A1 推出 C1 的充分性方向\nE1   洒水器模型上的模拟实验\nE2   真实数据集上的估计\n\nC1 \u003C- supported by D1, E1\nC1 \u003C- assumes A1, A2\nC1 \u003C- contradicts 朴素相关分析\n",[65159,66438,66436],{"__ignoreMap":64988},[11,66440,66441,66442,66545,66546,66548,66549,66551,66552,66555],{},"然后在成文层，把这些 ID 落成 ",[15,66443,66445,66458],{"className":66444,"translate":19},[18],[15,66446,66448],{"className":66447},[23],[25,66449,66450],{"xmlns":27},[29,66451,66452,66456],{},[32,66453,66454],{},[282,66455,65259],{},[40,66457,65262],{"encoding":42},[15,66459,66461],{"className":66460,"ariaHidden":48},[47],[15,66462,66464,66467],{"className":66463},[52],[15,66465],{"className":66466,"style":65272},[56],[15,66468,66470,66473,66476,66496,66499],{"className":66469},[61,13548],[15,66471,47490],{"className":66472},[61,65279],[15,66474],{"className":66475,"style":65283},[127],[15,66477,66479],{"className":66478},[342],[15,66480,66482],{"className":66481},[346],[15,66483,66485],{"className":66484,"style":57},[350],[15,66486,66487,66490],{"style":65295},[15,66488],{"className":66489,"style":358},[357],[15,66491,66493],{"className":66492},[61],[15,66494,65305],{"className":66495},[61,65279,365,362,363,364],[15,66497],{"className":66498,"style":65309},[127],[15,66500,66502,66505,66508,66539,66542],{"className":66501},[61,13548],[15,66503,56756],{"className":66504},[61,65279],[15,66506],{"className":66507,"style":7105},[127],[15,66509,66511,66531],{"className":66510},[342,472],[15,66512,66514,66528],{"className":66513},[346],[15,66515,66517],{"className":66516,"style":65328},[350],[15,66518,66519,66522],{"style":65331},[15,66520],{"className":66521,"style":3861},[357],[15,66523,66525],{"className":66524},[61],[15,66526,14818],{"className":66527},[61,65279],[15,66529,497],{"className":66530},[496],[15,66532,66534],{"className":66533},[346],[15,66535,66537],{"className":66536,"style":65350},[350],[15,66538],{},[15,66540],{"className":66541,"style":65356},[127],[15,66543,38],{"className":66544},[61,65279]," 的 ",[65159,66547,65830],{}," 和 ",[65159,66550,65681],{},"。这样做的好处是：",[661,66553,66554],{},"压扁只发生在最后一步，而且是可逆的","——草图层始终在，任何时候都能回到图上检查。改动时先动草图层，再重新生成文本层的引用，而不是反过来。",[11,66557,66558],{},"诚实地说，这只是把痛苦从脑子里挪到了纸上，并没有消除手动维护。但它至少让「图」变成了一个可以被检查的实体，而不是一个只存在于我记忆里的东西。而且它带来一个意外的收益：这些 ID 本身，恰好可以当作和 LLM 交流的接口。",[872,66560,66562],{"id":66561},"当-llm-进入这张图","当 LLM 进入这张图",[11,66564,66565],{},"回到文章开头那个略显狼狈的结尾：那篇文章我写了两周，最后精力不支，交给了 LLM 仓促发表。",[11,66567,66568,66569],{},"当时我把它当成一次失败。但现在回头看，它其实指出了一个更重要的变化：",[661,66570,66571],{},"LLM 是历史上第一个有可能承担「图维护」这件事的东西。",[11,66573,66574],{},"前面说「写图很贵」，贵在哪里？贵在三件事：声明关系、保持一致、以及结构变动后重新检查下游。这三件事恰好都是 LLM 擅长的。它能读完一段文本后抽出隐含的 Claim 和 Evidence，能在插入新假设之后重新检查所有依赖它的结论，能在你改动一个定义时把下游的引用全部找出来。",[11,66576,66577,66578,66581,66582,867],{},"但这里有一个前提，而且是致命的前提：",[661,66579,66580],{},"LLM 在显式的图上工作，远好于在隐式的图上工作。"," 如果关系从来不曾落在文件里，LLM 也只能像我一样去猜，而且会猜得同样不可靠。所以「把图显式化」这件事的性质变了——它不再纯粹是作者的额外负担，而是让机器能够介入的前提。也许真正可行的分工不是「作者建图、作者维护图」，而是",[661,66583,66584],{},"作者建图、机器维护图",[11,66586,66587],{},"不过有一个反向的危险，值得单独说。LLM 天然擅长生成流畅的线性文本，它会不由自主地把你论证里的裂缝糊上。一段话读起来通顺，和这个论证成不成立，是两件完全不同的事，而 LLM 恰好最擅长制造前者。我用 LLM 校对那篇两周的文章时就吃过这个亏：它把很多衔接不畅的地方改顺了，但那些地方原本的不畅，其实是论证真的没接上。",[11,66589,66590,66591,66594],{},"所以 LLM 在这件事上的正确用法，是当图的审查者，而不是文本的润色者。",[661,66592,66593],{},"让它问我「这个 Claim 的证据在哪里」，而不是问它「这段话怎么写得更好」。"," 前者会暴露问题，后者只会掩盖问题。",[872,66596,64743],{"id":64743},[11,66598,66599],{},"如果把三种写作对应到三种数据结构，会看得更清楚：日常写作对应的是序列（Sequence），句子接句子，核心是语言流。技术写作对应的是树（Tree），文档下分节，节下分概念与流程，核心是信息层级，Markdown 对此非常合适。",[11,66601,66602],{},"而科学写作对应的则是图（Graph），一个 Claim 由 Evidence 支持、由 Assumption 建立基础、由 Model 推导而来、由 Experiment 加以检验，核心是命题之间的逻辑关系。到了这一层，Markdown 就开始显得太扁平了。",[11,66604,66605],{},"无论是读者还是写作者，实际上一直在追问：这个 Claim 从哪里来的？这个 Assumption 合理吗？这个公式是否真的从前面的定义推出？这个实验是否真的能够验证这个 Claim？这个结果还有没有其他解释？结论有没有超出实验能够支持的范围？所以科学写作，本质上是在维护一张 Claim、Evidence、Reasoning 三者互相印证的关系网络。",[11,66607,66608],{},"这也解释了为什么科学写作写久之后，我会逐渐不满足于 Markdown。",[11,66610,66611],{},"不是因为它不够漂亮，因为真正需要的已经不再是一个好用的文本编辑器，而是一个能够表达定义、命题、假设、推导、证据、引用和论证关系的科学知识编辑系统。",[11,66613,66614,66615,66618],{},"这三种数据结构也解释了前面所有具体的痛苦。序列容错，所以日常写作拿记事本就能对付；树可以局部修改，所以技术文档在 Markdown 里如鱼得水；而",[661,66616,66617],{},"图一旦被压成线，每一次改动都要付出全局的代价","——那两周的校对时间，大部分就是花在这里的。",[11,66620,66621,66622,66725],{},"所以在那个系统出现之前，我大概还会继续用 Markdown 写博客、用 ",[15,66623,66625,66638],{"className":66624,"translate":19},[18],[15,66626,66628],{"className":66627},[23],[25,66629,66630],{"xmlns":27},[29,66631,66632,66636],{},[32,66633,66634],{},[282,66635,65259],{},[40,66637,65262],{"encoding":42},[15,66639,66641],{"className":66640,"ariaHidden":48},[47],[15,66642,66644,66647],{"className":66643},[52],[15,66645],{"className":66646,"style":65272},[56],[15,66648,66650,66653,66656,66676,66679],{"className":66649},[61,13548],[15,66651,47490],{"className":66652},[61,65279],[15,66654],{"className":66655,"style":65283},[127],[15,66657,66659],{"className":66658},[342],[15,66660,66662],{"className":66661},[346],[15,66663,66665],{"className":66664,"style":57},[350],[15,66666,66667,66670],{"style":65295},[15,66668],{"className":66669,"style":358},[357],[15,66671,66673],{"className":66672},[61],[15,66674,65305],{"className":66675},[61,65279,365,362,363,364],[15,66677],{"className":66678,"style":65309},[127],[15,66680,66682,66685,66688,66719,66722],{"className":66681},[61,13548],[15,66683,56756],{"className":66684},[61,65279],[15,66686],{"className":66687,"style":7105},[127],[15,66689,66691,66711],{"className":66690},[342,472],[15,66692,66694,66708],{"className":66693},[346],[15,66695,66697],{"className":66696,"style":65328},[350],[15,66698,66699,66702],{"style":65331},[15,66700],{"className":66701,"style":3861},[357],[15,66703,66705],{"className":66704},[61],[15,66706,14818],{"className":66707},[61,65279],[15,66709,497],{"className":66710},[496],[15,66712,66714],{"className":66713},[346],[15,66715,66717],{"className":66716,"style":65350},[350],[15,66718],{},[15,66720],{"className":66721,"style":65356},[127],[15,66723,38],{"className":66724},[61,65279]," 写论文，并且在两者之间手动维护一张只有我自己看得见的图。这篇文章本身也是用 Markdown 写的——这大概是它最后一个恰当的讽刺。",[882,66727,66728],{},[11,66729,66730],{},"Markdown 的问题不是它表达不了科学内容，而是它把科学内容中最重要的那部分——对象之间的论证关系——降格成了文本。",[66732,66733,66734],"style",{},"html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}",{"title":64988,"searchDepth":64989,"depth":64989,"links":66736},[66737,66738,66739,66740,66741,66742,66743,66744,66745,66746],{"id":65089,"depth":64989,"text":65089},{"id":65133,"depth":64989,"text":65133},{"id":65211,"depth":64989,"text":65211},{"id":65711,"depth":64989,"text":65711},{"id":65851,"depth":64989,"text":65851},{"id":66023,"depth":64989,"text":66023},{"id":66184,"depth":64989,"text":66184},{"id":66424,"depth":64989,"text":66425},{"id":66561,"depth":64989,"text":66562},{"id":64743,"depth":64989,"text":64743},"甘肃兰州",{},"\u002Fblog\u002F2026\u002F0006-markdown-ke-xue-xie-zuo-tong-dian-wen-ti",{"title":65047,"description":65052},"blog\u002F2026\u002F0006-markdown-ke-xue-xie-zuo-tong-dian-wen-ti","写作的格式工具会悄悄影响到你的行文组织方式。Markdown 与其说「不适合写科学内容」，不如说它的抽象层级主要是「轻量文档标记」，而科学写作本质上更接近「结构化知识建模」。",[66754],"thoughts","2026-09-22T22:53:33+08:00","5_F5AFfH5x0FqoxmQpb4yz2mbUriKEidgISRIkmGWX8",1790855355309]